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\begin_body

\begin_layout Chapter
Symmetry Improvement of 
\begin_inset Formula $n$
\end_inset

PIEA through Lagrange Multipliers
\begin_inset CommandInset label
LatexCommand label
name "chap:chap4"

\end_inset


\end_layout

\begin_layout Section
Synopsis
\end_layout

\begin_layout Standard
Chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:chap2"

\end_inset

 introduced 
\begin_inset Formula $n$
\end_inset

PIEAs as powerful tools for computations in QFT, particularly used for their
 ability to handle non-equilibrium situations through time contour methods.
 The previous chapter discussed global symmetries and Ward identities in
 
\begin_inset Formula $n$
\end_inset

PIEA, showing that the solutions of truncated 
\begin_inset Formula $n$
\end_inset

PI equations of motion do not obey the 1PI Ward identities.
 Several studies have attempted to find a remedy for this problem (see,
 e.g., 
\begin_inset CommandInset citation
LatexCommand citep
key "Pilaftsis2013"

\end_inset

 and references therein).
 The last chapter discussed the external propagator method, which has been
 frequently used in the literature and does yield massless Goldstone bosons.
 However, the external propagator is not the propagator used in loop graphs,
 so the loop corrections still contain massive Goldstone bosons leading
 to incorrect thresholds, decay rates and violations of unitarity.
 In order to avoid these problems a manifestly self-consistent scheme must
 be used.
 One such scheme is the large 
\begin_inset Formula $N$
\end_inset

 approximation, which was also briefly reviewed.
 The Goldstone theorem and second order phase transition do hold to leading
 order in the large 
\begin_inset Formula $N$
\end_inset

 approximation, but these attractive features are lost at higher orders.
 Another approach which is not discussed in detail here is to abandon the
 
\begin_inset Formula $n$
\end_inset

PIEA formalism entirely, using instead the Schwinger-Dyson equations.
 The Schwinger-Dyson equations require a closure ansatz which can be chosen
 to respect appropriate Ward identities.
 However, this choice still involves some degree of arbitrariness and self-consi
stency is not guaranteed.
\end_layout

\begin_layout Standard
This chapter discusses the symmetry improvement method introduced by Pilaftsis
 and Teresi to circumvent these difficulties 
\begin_inset CommandInset citation
LatexCommand citep
key "Pilaftsis2013"

\end_inset

 for the 2PIEA.
 The idea is simply to impose the desired Ward identities directly on the
 free correlation functions.
 This is consistently implemented by using Lagrange multipliers.
 The remarkable point is that the resulting equations of motion can be put
 into a form that completely eliminates the Lagrange multiplier field.
 They achieve this by taking a limit in which the Lagrange multiplier vanishes
 from all but one of the equations of motion, and this remaining equation
 of motion is replaced with the constraint to obtain a closed system.
 Here the symmetry improved 2PIEA (SI-2PIEA) is reviewed and extended to
 general 
\begin_inset Formula $\mathrm{O}\left(N\right)$
\end_inset

 theories.
 An ambiguity of the constraint scheme is pointed out which was not recognised
 in the original literature.
 This ambiguity has no influence on the equilibrium results of this chapter,
 but is relevant to the discussion of Chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:chap6"

\end_inset

.
 Following that the method is generalised to the 3PIEA (reporting on work
 published by the author 
\begin_inset CommandInset citation
LatexCommand citep
key "Brown2015a"

\end_inset

).
 The generalisation is non-trivial, requiring a careful consideration of
 the variational procedure leading to an infinity of possible schemes.
 A new principle is introduced to choose between the schemes and is called
 the 
\emph on
d'Alembert formalism
\emph default
 by analogy to the constrained variational problem in mechanics.
\end_layout

\begin_layout Standard
The motivation for extending symmetry improvement to the 3PIEA is threefold.
 First, the 3PIEA is known to be the required starting point to obtain a
 self-consistent non-equilibrium kinetic theory of gauge theories.
 The accurate calculation of transport coefficients and thermalisation times
 in gauge theories requires the use of 
\begin_inset Formula $n$
\end_inset

PIEA with 
\begin_inset Formula $n\geq3$
\end_inset

 (see, e.g.
 
\begin_inset CommandInset citation
LatexCommand citep
key "Carrington2009,Smolic2012,Berges2004"

\end_inset

 and references therein for discussion).
 The fundamental reason for this is that the 3PIEA includes medium induced
 effects on the three-point vertex at leading order.
 The 2PIEA in gauge theory contains a dressed propagator but not a dressed
 vertex, leading not only to an inconsistency of the resulting kinetic equation
 but also to a spurious gauge dependence analogous to the failure of global
 symmetries in truncations as discussed in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:chap3"

\end_inset

.
 Thus this work is a stepping stone towards a fully self-consistent, non-perturb
ative and manifestly gauge invariant treatment of out of equilibrium gauge
 theories.
\end_layout

\begin_layout Standard
Second, 
\begin_inset Formula $n$
\end_inset

PIEAs allow one to accurately describe the initial value problem with 1-
 to 
\begin_inset Formula $n$
\end_inset

-point connected correlation functions in the initial state.
 For example, the 2PIEA allows one to solve the initial value problem for
 initial states with a Gaussian density matrix.
 However, the physical applications one has in mind typically start from
 a near thermal equilibrium state which is not well approximated by a Gaussian
 density matrix.
 This leads to problems with renormalisation, unphysical transient responses
 and thermalisation to the wrong temperature 
\begin_inset CommandInset citation
LatexCommand citep
key "Garny2009"

\end_inset

.
 This is addressed in 
\begin_inset CommandInset citation
LatexCommand citep
key "Garny2009,VanLeeuwen2012,Stefanucci2013"

\end_inset

 by the addition of an infinite set of non-local vertices which only have
 support at the initial time.
 Going to 
\begin_inset Formula $n>2$
\end_inset

 allows one to better describe the initial state, thereby reducing the need
 for additional non-local vertices.
\end_layout

\begin_layout Standard
Lastly, the infinite hierarchy of 
\begin_inset Formula $n$
\end_inset

PIEA is the natural home for the 2PIEA and provides the clearest route for
 systematic improvements over existing treatments.
 Thus investigating symmetry improvement of 3PIEA is a well motived step
 in the development of non-perturbative QFT.
\end_layout

\begin_layout Section
Symmetry Improvement of the 2PIEA
\begin_inset ERT
status open

\begin_layout Plain Layout

}{
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "sec:SI-2PIEA"

\end_inset


\end_layout

\begin_layout Standard
The essence of symmetry improvement is to impose the WIs derived for the
 1PI correlation functions on the 
\begin_inset Formula $n$
\end_inset

PI correlation functions.
 Effectively, one changes 
\begin_inset Formula 
\begin{equation}
\mathcal{W}_{a_{1}\cdots a_{j}}^{A}\left(\Delta_{\text{1PI}},V_{\text{1PI}},\cdots,V_{\text{1PI}}^{\left(n\right)}\right)\to\mathcal{W}_{a_{1}\cdots a_{j}}^{A}\left(\Delta_{n\text{PI}},V_{n\text{PI}},\cdots,V_{n\text{PI}}^{\left(n\right)}\right),
\end{equation}

\end_inset

where 
\begin_inset Formula $\mathcal{W}_{a_{1}\cdots a_{j}}^{A}$
\end_inset

 for 
\begin_inset Formula $j=1,\cdots,n-1$
\end_inset

 are the WIs and the arguments change but not the functional form.
 Note that higher order WIs cannot be enforced on an 
\begin_inset Formula $n$
\end_inset

PIEA since only the 1- through 
\begin_inset Formula $n$
\end_inset

-point functions are free.
 Thus, in the SI-2PIEA a single constraint is enforced, namely 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:1PI-Delta-WI"

\end_inset

, which recall is
\begin_inset Formula 
\begin{equation}
0=\mathcal{W}_{c}^{A}=\Delta_{ca}^{-1}T_{ab}^{A}\varphi_{b}-J_{a}T_{ac}^{A},
\end{equation}

\end_inset

where 
\begin_inset Formula $\Delta$
\end_inset

 is thought of as the 2PI function.
 Only the case 
\begin_inset Formula $J_{a}=0$
\end_inset

 will be considered here.
 
\begin_inset CommandInset citation
LatexCommand citet
key "Pilaftsis2013"

\end_inset

 considered the case of a translationally invariant 
\begin_inset Formula $J_{a}\neq0$
\end_inset

 to define a symmetry improved effective potential.
 The situation with a general 
\begin_inset Formula $J_{a}\neq0$
\end_inset

 is discussed in Chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:chap6"

\end_inset

.
\end_layout

\begin_layout Standard
Symmetry improvement imposes 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:1PI-Delta-WI"

\end_inset

 as a constraint on the allowable values of 
\begin_inset Formula $\varphi$
\end_inset

 and 
\begin_inset Formula $\Delta$
\end_inset

 in the 2PIEA.
 This proceeds through the introduction of Lagrange multiplier fields 
\begin_inset Formula $\ell_{A}^{d}\left(x\right)$
\end_inset

 and a shift of the action 
\begin_inset Formula $\Gamma\to\Gamma-\mathcal{C}$
\end_inset

 where 
\begin_inset Formula $\mathcal{C}\sim\ell\mathcal{W}$
\end_inset

 is the constraint.
 Note however that Pilaftsis and Teresi introduced symmetry improvement
 non-covariantly, leading to two possible covariant symmetry improvement
 schemes that reduce to theirs in equilibrium.
 The first is to take
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\mathcal{C}=\frac{i}{2}\ell_{A}^{c}\mathcal{W}_{c}^{A}.\label{eq:simple-constraint}
\end{equation}

\end_inset

The second follows the author's previous paper 
\begin_inset CommandInset citation
LatexCommand citep
key "Brown2015a"

\end_inset

 by including a transverse projector 
\begin_inset Formula $P_{ab}^{\perp}\left(x\right)=\delta_{ab}-\varphi_{a}\left(x\right)\varphi_{b}\left(x\right)/\varphi^{2}\left(x\right)$
\end_inset

: 
\begin_inset Formula 
\begin{equation}
\mathcal{C}'=\frac{i}{2}\ell_{A}^{c}P_{cd}^{\perp}\mathcal{W}_{d}^{A},\label{eq:projected-constraint}
\end{equation}

\end_inset

to ensure that only Goldstone modes are involved in the constraint.
 The choice between 
\begin_inset Formula $\mathcal{C}$
\end_inset

 and 
\begin_inset Formula $\mathcal{C}'$
\end_inset

 turns out to make no difference in equilibrium.
 However, the two constraints lead to different schemes beyond equilibrium,
 both of which are pathological as discussed in the author's paper 
\begin_inset CommandInset citation
LatexCommand citep
key "Brown_2016"

\end_inset

 and in Chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:chap6"

\end_inset

.
 For now the simple constraint 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:simple-constraint"

\end_inset

 is taken.
\end_layout

\begin_layout Standard
The equations of motion following from the symmetry improved effective action
 are 
\begin_inset Formula 
\begin{align}
\mathcal{W}_{c}^{A} & =0,\\
\frac{\delta\Gamma}{\delta\varphi_{d}\left(z\right)} & =\frac{i}{2}\int_{x}\ell_{A}^{c}\left(x\right)\Delta_{ca}^{-1}\left(x,z\right)T_{ad}^{A}-J_{d}\left(z\right)-\int_{w}K_{de}\left(z,w\right)\varphi_{e}\left(w\right),\label{eq:si-eom-vev}\\
\frac{\delta\Gamma}{\delta\Delta_{de}\left(z,w\right)} & =\frac{i}{2}\int_{x}\ell_{A}^{c}\left(x\right)\int_{y}\frac{\delta\Delta_{ca}^{-1}\left(x,y\right)}{\delta\Delta_{de}\left(z,w\right)}T_{ab}^{A}\varphi_{b}\left(y\right)-\frac{1}{2}i\hbar K_{de}\left(z,w\right),
\end{align}

\end_inset

where the last equation simplifies to 
\begin_inset Formula 
\begin{equation}
\frac{\delta\Gamma}{\delta\Delta_{de}\left(z,w\right)}=-\frac{i}{2}\int_{x}\ell_{A}^{c}\left(x\right)\Delta_{cd}^{-1}\left(x,z\right)\int_{y}\Delta_{ea}^{-1}\left(w,y\right)T_{ab}^{A}\varphi_{b}\left(y\right)-\frac{1}{2}i\hbar K_{de}\left(z,w\right),\label{eq:si-eom-delta}
\end{equation}

\end_inset

on using the identity 
\begin_inset Formula $\delta\Delta_{ca}^{-1}/\delta\Delta_{de}=-\Delta_{cd}^{-1}\Delta_{ea}^{-1}$
\end_inset

.
 In the present discussion only the equilibrium case with 
\begin_inset Formula $J=K=0$
\end_inset

 is relevant (this is reviewing 
\begin_inset CommandInset citation
LatexCommand citep
key "Pilaftsis2013,Brown2015a"

\end_inset

, the more general case is discussed in 
\begin_inset CommandInset citation
LatexCommand citep
key "Brown_2016"

\end_inset

 and Chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:chap6"

\end_inset

).
 The only non-trivial WIs are 
\begin_inset Formula $\mathcal{W}_{c}^{gN}=-ivm_{G}^{2}P_{cg}^{\perp}$
\end_inset

, so that the constraint enforces 
\begin_inset Formula $vm_{G}^{2}=0$
\end_inset

, i.e.
 the Goldstone mass vanishes if 
\begin_inset Formula $v\neq0$
\end_inset

 as expected.
 Using homogeneity 
\begin_inset Formula $\ell_{A}^{c}\left(x\right)=\ell_{A}^{c}$
\end_inset

, and the SSB ansatz 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:SSB-ansatz-Delta"

\end_inset

 the other equations of motion become 
\begin_inset Formula 
\begin{align}
\frac{\partial\Gamma/\mathrm{V}T}{\partial v} & =\ell_{cN}^{c}m_{G}^{2},\\
\frac{\delta\Gamma}{\delta\Delta_{G}\left(z,w\right)} & =\ell_{cN}^{c}vm_{G}^{4},\\
\frac{\delta\Gamma}{\delta\Delta_{H}\left(z,w\right)} & =0,\label{eq:SI-2PIEA-H-eom-general}
\end{align}

\end_inset

where 
\begin_inset Formula $\mathrm{V}T$
\end_inset

 is the volume of spacetime
\begin_inset Foot
status collapsed

\begin_layout Plain Layout
Recall that 
\begin_inset Formula $\left(\delta/\delta\varphi\left(x\right)\right)\int_{y}\varphi\left(y\right)J\left(y\right)\equiv J\left(x\right)$
\end_inset

, however, for a translation invariant 
\begin_inset Formula $\varphi\left(x\right)=v$
\end_inset

 and 
\begin_inset Formula $J\left(x\right)=J$
\end_inset

, 
\begin_inset Formula $\partial_{v}\int_{y}\varphi\left(x\right)J\left(x\right)=\partial_{v}\int_{x}vJ=\mathrm{V}TJ$
\end_inset

 so that 
\begin_inset Formula $\delta/\delta\varphi=\left(\mathrm{V}T\right)^{-1}\partial_{v}$
\end_inset

 for constant fields.
\end_layout

\end_inset

.
 The symmetry improvement constraint is a singular one: one must require
 
\begin_inset Formula $\ell\to\infty$
\end_inset

 as 
\begin_inset Formula $v\int\Delta_{G}^{-1}\to0$
\end_inset

.
 The reason for this is that 
\begin_inset Formula $m_{G}^{2}\to0$
\end_inset

 so the right hand sides vanish unless 
\begin_inset Formula $\ell_{cN}^{c}\to\infty$
\end_inset

.
 Applying the constraint with 
\begin_inset Formula $v\neq0$
\end_inset

 directly in the equations of motion would give zero right hand sides, reducing
 to the standard 2PI formalism.
 This is valid in the full theory because the Ward identity is satisfied.
 However, this is impossible in the case where the 2PI effective action
 is truncated at finite loop order because the actual Ward identity obeyed
 by the 2PIEA is 
\emph on
not
\emph default
 
\begin_inset Formula $\mathcal{W}_{c}^{A}$
\end_inset

 as discussed in Chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:chap3"

\end_inset

.
\end_layout

\begin_layout Standard
The divergence is regulated by setting 
\begin_inset Formula $vm_{G}^{2}=\eta m^{3}$
\end_inset

 and taking the limit 
\begin_inset Formula $\eta\to0$
\end_inset

 such that 
\begin_inset Formula $\eta\ell_{cN}^{c}/v=\ell_{0}$
\end_inset

 is a constant.
 This gives
\begin_inset Formula 
\begin{align}
\frac{\partial\Gamma/\mathrm{V}T}{\partial v} & =\ell_{0}m^{3},\label{eq:SI-2PIEA-v-eom-general}\\
\frac{\delta\Gamma}{\delta\Delta_{G}\left(z,w\right)} & =0.\label{eq:SI-2PIEA-G-eom-general}
\end{align}

\end_inset

(Note that one can consistently set 
\begin_inset Formula 
\begin{equation}
\ell_{bN}^{a}=-\ell_{Nb}^{a}=P_{ab}^{\perp}\left(\frac{1}{N-1}\ell_{cN}^{c}\right),\label{eq:lagrange-multipliers}
\end{equation}

\end_inset

and all other components of 
\begin_inset Formula $\ell_{A}^{c}$
\end_inset

 to zero.) Thus the propagator equations of motion are unmodified and the
 vev equation is modified by the presence of a homogeneous force that acts
 to push 
\begin_inset Formula $v$
\end_inset

 away from the minimum of the effective potential to the point where 
\begin_inset Formula $m_{G}^{2}=0$
\end_inset

.
 In practice, in the symmetry broken phase, one simply discards the vev
 equation of motion and solves the propagator ones in conjunction with the
 Ward identity, which suffices to give a closed system.
 In the symmetric phase 
\begin_inset Formula $v=0$
\end_inset

 and the Ward identity is trivial, but 
\begin_inset Formula $\Gamma$
\end_inset

 also does not depend linearly on 
\begin_inset Formula $v$
\end_inset

, hence one can take the previous equations of motion with 
\begin_inset Formula $\ell_{0}=0$
\end_inset

.
 Note that one can keep a non-zero 
\begin_inset Formula $m_{G}^{2}$
\end_inset

 in the intermediate stages of the computation to serve as an infrared regulator.
\end_layout

\begin_layout Standard
To recap the procedure: first define a symmetry improved effective action
 using Lagrange multipliers and compute the equations of motion.
 Second, note that the equations of motion are singular when the constraints
 are applied.
 Third, regulate the singularity by slightly violating the constraint.
 Fourth, pass to a suitable limit where violation of the constraint tends
 to zero while requiring the limiting procedure to be universal in the sense
 that no additional data (arbitrary forms of the Lagrange multiplier fields)
 need be introduced into the theory.
\end_layout

\begin_layout Section
Renormalisation and solution of the SI-2PI Hartree-Fock gap equations
\begin_inset ERT
status open

\begin_layout Plain Layout

}{
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "sec:Renormalisation-and-solution-SI-2PIEA-HF"

\end_inset


\end_layout

\begin_layout Standard
The renormalisation of the SI-2PIEA in the Hartree-Fock approximation follows
 the same procedure as in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Renormalisation-and-solution-2PI-HF"

\end_inset

 with minor alterations.
 The gap equations 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:ren-2PI-HF-G-eom"

\end_inset

-
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:fin-2PI-HF-H-eom"

\end_inset

 are the same, as are the finite versions 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:fin-2PI-HF-G-eom"

\end_inset

-
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:fin-2PI-HF-H-eom"

\end_inset

, reproduced here:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
m_{G}^{2} & =m^{2}+\frac{\lambda}{6}v^{2}+\frac{\hbar\lambda}{6}\left(N+1\right)\mathcal{T}_{G}^{\mathrm{fin}}+\frac{\hbar\lambda}{6}\mathcal{T}_{H}^{\mathrm{fin}}+\mathcal{O}\left(\lambda^{2}\right),\label{eq:fin-SI-2PI-HF-G-eom}\\
m_{H}^{2} & =m^{2}+\frac{\lambda}{2}v^{2}+\frac{\hbar\lambda}{6}\left(N-1\right)\mathcal{T}_{G}^{\mathrm{fin}}+\frac{\hbar\lambda}{2}\mathcal{T}_{H}^{\mathrm{fin}}+\mathcal{O}\left(\lambda^{2}\right),\label{eq:fin-SI-2PI-HF-H-eom}
\end{align}

\end_inset

however the vev equation 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:fin-2PI-HF-vev-eom"

\end_inset

 is replaced by Goldstone's theorem
\begin_inset Formula 
\begin{equation}
0=vm_{G}^{2},\label{eq:fin-SI-2PI-HF-Goldstone-thm}
\end{equation}

\end_inset

which does not require renormalisation (more precisely, the WI is multiplicative
ly renormalised, however since 
\begin_inset Formula $\mathcal{W}_{c}^{A}=0$
\end_inset

 an overall factor makes no difference).
 The counter-terms 
\begin_inset Formula $\delta m_{0}^{2}$
\end_inset

 and 
\begin_inset Formula $\delta\lambda_{0}$
\end_inset

 do not enter into the renormalisation of the equations of motion and 
\begin_inset Formula $Z_{\Delta}$
\end_inset

 is again redundant.
 Altogether there are six remaining constants 
\begin_inset Formula $Z$
\end_inset

, 
\begin_inset Formula $\delta m_{1}^{2}$
\end_inset

, and 
\begin_inset Formula $\delta\lambda_{1,2}^{A,B}$
\end_inset

 which are constrained by eight independent equations.
 That there is a solution is a non-trivial check on the calculations and
 one find (for details refer to the 
\noun on
Mathematica 
\begin_inset CommandInset citation
LatexCommand citep
key "mathematica"

\end_inset


\noun default
 notebook in Appendix 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Appendix-Renormalisation-Details"

\end_inset

)
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
Z=Z_{\Delta} & =1,\\
\delta m_{1}^{2} & =-\frac{\hbar\lambda}{6}\left(N+2\right)\left(c_{0}\Lambda^{2}+c_{1}m^{2}\ln\left(\frac{\Lambda^{2}}{\mu^{2}}\right)\right)\frac{\delta\lambda_{1}^{A}+\lambda}{\delta\lambda_{1}^{B}+\lambda},\\
\delta\lambda_{1}^{A}=\delta\lambda_{2}^{A} & =\left(1+\frac{3\left(N+2\right)}{6+c_{1}\left(N+2\right)\lambda\hbar\ln\left(\frac{\Lambda^{2}}{\mu^{2}}\right)}\right)\delta\lambda_{1}^{B},\\
\delta\lambda_{1}^{B}=\delta\lambda_{2}^{B} & =\left(-1+\frac{3}{3+c_{1}\lambda\hbar\ln\left(\frac{\Lambda^{2}}{\mu^{2}}\right)}\right)\lambda,
\end{align}

\end_inset

which is actually identical to the expressions for the counter-terms in
 the unimproved 2PIEA case.
 This should accord with intuition: symmetry improvement is an infrared
 modification of the theory which does not alter the ultraviolet divergence
 structure at all.
 However, such a result is not automatically assured in a self-consistent
 approximation scheme such as 
\begin_inset Formula $n$
\end_inset

PIEA, since the non-linearity of the equations of motion results in a coupling
 of scales.
 Furthermore, section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:SI-3PIEA"

\end_inset

 shows that the SI-3PIEA truncated to two loops is 
\emph on
not
\emph default
 equivalent to the SI-2PIEA.
 Instead the Higgs propagator equation of motion is modified as are the
 counter-terms.
 Full self-consistency of the theory is then lost with various physical
 effects that will be discussed.
\end_layout

\begin_layout Standard
Remarkably there is an exact analytical solution for the SI-2PI HF approximation.
 Subtracting 
\begin_inset Formula $3m_{G}^{2}$
\end_inset

 from the 
\begin_inset Formula $m_{H}^{2}$
\end_inset

 equation to cancel the 
\begin_inset Formula $\lambda v^{2}/2$
\end_inset

 and 
\begin_inset Formula $\hbar\lambda\mathcal{T}_{H}^{\mathrm{fin}}/2$
\end_inset

 terms, and using that 
\begin_inset Formula $m_{G}^{2}=0$
\end_inset

 in the symmetry breaking regime, one finds
\begin_inset Formula 
\begin{equation}
m_{H}^{2}=\bar{m}_{H}^{2}-\frac{\hbar\lambda}{3}\left(N+2\right)\mathcal{T}_{G}^{\mathrm{fin}}+\mathcal{O}\left(\lambda^{2}\right),
\end{equation}

\end_inset

where 
\begin_inset Formula $\mathcal{T}_{G}^{\mathrm{fin}}=T^{2}/12$
\end_inset

 since 
\begin_inset Formula $m_{G}^{2}=0$
\end_inset

.
 The critical temperature occurs when 
\begin_inset Formula $m_{H}^{2}=0$
\end_inset

, giving
\begin_inset Formula 
\begin{equation}
T_{\star}=\sqrt{\frac{12\bar{v}^{2}}{\hbar\left(N+2\right)}}+\mathcal{O}\left(\lambda^{2}\right),
\end{equation}

\end_inset

which is the same as in the unimproved case 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:critical-Temp"

\end_inset

.
 However, unlike the unimproved case, the SI-2PI solution has a second order
 phase transition.
 Indeed, using the previous expression for 
\begin_inset Formula $T_{\star}$
\end_inset

 in 
\begin_inset Formula $m_{H}$
\end_inset

 gives
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
m_{H}^{2}=\bar{m}_{H}^{2}\left(1-\frac{T^{2}}{T_{\star}^{2}}\right)+\mathcal{O}\left(\lambda^{2}\right),
\end{equation}

\end_inset

which is the expression obtained from the classic Landau theory of second
 order phase transitions 
\begin_inset CommandInset citation
LatexCommand citep
key "Landau2008"

\end_inset

.
 The value of 
\begin_inset Formula $v$
\end_inset

 is found by using this result in the 
\begin_inset Formula $m_{G}^{2}$
\end_inset

 equation, obtaining
\begin_inset Formula 
\begin{equation}
v^{2}=\bar{v}^{2}-\hbar\left(\left(N+1\right)\frac{T^{2}}{12}+\mathcal{T}_{H}^{\mathrm{th}}\right)+\mathcal{O}\left(\lambda\right),
\end{equation}

\end_inset

where 
\begin_inset Formula $m_{H}$
\end_inset

 is the mass appearing in 
\begin_inset Formula $\mathcal{T}_{H}^{\mathrm{th}}$
\end_inset

.
 At the critical temperature
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
v^{2}=0+\mathcal{O}\left(\lambda\right),
\end{equation}

\end_inset

as expected.
 
\begin_inset Formula $v^{2}$
\end_inset

 smoothly interpolates between 
\begin_inset Formula $\bar{v}^{2}$
\end_inset

 at 
\begin_inset Formula $T=0$
\end_inset

 and zero at 
\begin_inset Formula $T=T_{\star}$
\end_inset

, then remains at zero at all higher temperatures.
 There is no unphysical metastable phase with broken symmetry at high temperatur
e.
 This agrees with the behaviour of the 
\begin_inset Formula $\mathrm{O}\left(N\right)$
\end_inset

 model in lattice simulations (see, e.g.
 
\begin_inset CommandInset citation
LatexCommand citep
key "Strouthos2003,Campostrini2002"

\end_inset

).
\end_layout

\begin_layout Standard
The solution of the SI-2PI gap equations in the Hartree-Fock approximation
 is shown in Figures 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:si2pi-hf-vev-comparison"

\end_inset

, 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:si-2pi-hf-m_H-comparison"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:si-2pi-hf-m_G-comparison"

\end_inset

 using the parameters 
\begin_inset Formula $N=4$
\end_inset

, 
\begin_inset Formula $\bar{v}=93\ \mathrm{MeV}$
\end_inset

 and 
\begin_inset Formula $\bar{m}_{H}=500\ \mathrm{MeV}$
\end_inset

 to enable direct comparison to the results of Chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:chap3"

\end_inset

 (which have been reproduced here for ease of comparison).
 One sees that in the SI-2PI HF approximation the unphysical metastable
 phase is removed and replaced by a continuous (2nd order) phase transition.
 Furthermore the Goldstone theorem is satisfied, as expected, in the broken
 phase, in contrast to the unimproved 2PI solution.
 This behaviour has been noted before in the literature (
\begin_inset CommandInset citation
LatexCommand citep
key "Pilaftsis2013"

\end_inset

 in the case 
\begin_inset Formula $N=2$
\end_inset

, 
\begin_inset CommandInset citation
LatexCommand citep
key "Mao2014"

\end_inset

 in the case 
\begin_inset Formula $N=4$
\end_inset

, and 
\begin_inset CommandInset citation
LatexCommand citep
key "Brown2015a"

\end_inset

 and here for general 
\begin_inset Formula $N$
\end_inset

).
\end_layout

\begin_layout Standard
These results can be contrasted with the large 
\begin_inset Formula $N$
\end_inset

 approximation discussed in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Contrast-to-the-large-N-method"

\end_inset

 and 
\begin_inset CommandInset citation
LatexCommand citep
key "Mao2014"

\end_inset

.
 A second order phase transition is found and Goldstone's theorem is satisfied
 in both schemes.
 However, the large 
\begin_inset Formula $N$
\end_inset

 estimate for the critical temperature is larger than the symmetry improved
 value by a factor 
\begin_inset Formula $\sqrt{\left(N+2\right)/N}$
\end_inset

 which is 
\begin_inset Formula $\sqrt{3/2}\approx1.22$
\end_inset

 for 
\begin_inset Formula $N=4$
\end_inset

, accurate to the expected 
\begin_inset Formula $\mathcal{O}\left(1/N\right)=25\%$
\end_inset

.
 This makes the critical temperature 
\begin_inset Formula $\approx162\ \mathrm{MeV}$
\end_inset

 for the numerical values used previously.
 This difference is the due to the neglect of the Higgs boson thermal contributi
on, which is indeed swamped by Goldstone modes as 
\begin_inset Formula $N\to\infty$
\end_inset

.
 Thus the leading large 
\begin_inset Formula $N$
\end_inset

 approximation has several qualitative similarities to the symmetry improved
 2PIEA, and is quantitatively consistent with it to the expected 
\begin_inset Formula $\mathcal{O}\left(1/N\right)$
\end_inset

 accuracy.
\end_layout

\begin_layout Standard
There are some important qualitative differences however.
 First, the Goldstone theorem is lost at higher orders in 
\begin_inset Formula $1/N$
\end_inset

, while the SI-2PIEA explicitly enforces it at all orders.
 Second, the contribution of the Higgs to the phase transition thermodynamics
 is totally ignored in the large 
\begin_inset Formula $N$
\end_inset

 approximation, while it is included fully self-consistently in the SI-2PIEA.
 Third, the large 
\begin_inset Formula $N$
\end_inset

 approximation only resums 
\begin_inset Formula $\left\langle \phi_{a}\phi_{a}\right\rangle $
\end_inset

 and is not fully self-consistent in the sense that all possible Feynman
 diagram topologies with skeletons of a given order are fully resummed,
 but the SI-2PIEA computes all of the 
\begin_inset Formula $\Delta_{ab}\propto\left\langle \phi_{a}\phi_{b}\right\rangle $
\end_inset

 fully self-consistently and hence is a more accurate method when applicable.
 Finally, the large 
\begin_inset Formula $N$
\end_inset

 approximation is more natural in the sense that it does not rely upon any
 
\emph on
ad hoc
\emph default
 modification of the effective action, unlike the artificially imposed Ward
 identities of the SI-2PIEA.
 This means that when a large 
\begin_inset Formula $N$
\end_inset

 limit exists in a given field theory, it is fairly well behaved.
 However, the fact that the SI-2PIEA is constrained leads to difficulties
 in certain truncations (see section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:The-Two-Loop-Solution"

\end_inset

) and out of equilibrium (see Chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:chap6"

\end_inset

).
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset space \hfill{}
\end_inset


\begin_inset Graphics
	filename images/ch4/si2pi-vev-comparison.pdf
	width 100col%

\end_inset


\begin_inset space \hfill{}
\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:si2pi-hf-vev-comparison"

\end_inset

Field expectation value 
\begin_inset Formula $v$
\end_inset

 from the unimproved 2PI (solid) and SI-2PI (dash-dot) in the Hartree-Fock
 approximation with parameters 
\begin_inset Formula $N=4$
\end_inset

, 
\begin_inset Formula $\bar{v}=93\ \mathrm{MeV}$
\end_inset

 and 
\begin_inset Formula $\bar{m}_{H}=500\ \mathrm{MeV}$
\end_inset

.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset space \hfill{}
\end_inset


\begin_inset Graphics
	filename images/ch4/si2pi-higgs-mass-comparison.pdf
	width 100col%

\end_inset


\begin_inset space \hfill{}
\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:si-2pi-hf-m_H-comparison"

\end_inset

Higgs mass 
\begin_inset Formula $m_{H}^{2}$
\end_inset

 from the unimproved 2PI (solid) and SI-2PI (dash-dot) in the Hartree-Fock
 approximation with parameters 
\begin_inset Formula $N=4$
\end_inset

, 
\begin_inset Formula $\bar{v}=93\ \mathrm{MeV}$
\end_inset

 and 
\begin_inset Formula $\bar{m}_{H}=500\ \mathrm{MeV}$
\end_inset

.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset space \hfill{}
\end_inset


\begin_inset Graphics
	filename images/ch4/si2pi-goldstone-mass-comparison.pdf
	width 100col%

\end_inset


\begin_inset space \hfill{}
\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:si-2pi-hf-m_G-comparison"

\end_inset

Goldstone mass 
\begin_inset Formula $m_{G}^{2}$
\end_inset

 from the unimproved 2PI (solid) and SI-2PI (dash-dot) in the Hartree-Fock
 approximation with parameters 
\begin_inset Formula $N=4$
\end_inset

, 
\begin_inset Formula $\bar{v}=93\ \mathrm{MeV}$
\end_inset

 and 
\begin_inset Formula $\bar{m}_{H}=500\ \mathrm{MeV}$
\end_inset

.
 Note that the Goldstone theorem is satisfied, i.e.
 
\begin_inset Formula $m_{G}^{2}=0$
\end_inset

, in the broken phase for the SI-2PI solution but not the unimproved 2PI
 solution.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Section
The two loop solution (or lack thereof)
\begin_inset ERT
status open

\begin_layout Plain Layout

}{
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "sec:The-Two-Loop-Solution"

\end_inset


\end_layout

\begin_layout Standard
In order to investigate higher orders in the SI-2PIEA it is necessary to
 discuss more generally the problem of renormalising 
\begin_inset Formula $n$
\end_inset

PIEA.
 Self-consistent equations of motion are in general far harder to renormalise
 than perturbation theory and several methods of renormalisation have been
 introduced in the literature, though the general problem is still open.
 The 2PIEA is known to be renormalisable, either by an implicit construction
 involving Bethe-Salpeter integral equations or an explicit algebraic Bogoliubov
-Parasiuk-Hepp-Zimmerman (BPHZ) style construction which has non-trivial
 consistency requirements (but which has been shown to be equivalent to
 the Bethe-Salpeter method) 
\begin_inset CommandInset citation
LatexCommand citep
key "Berges2005,Fejos2008,Patkos2009,Marko2012"

\end_inset

.
 A recent discussion of the general theory can be found in Chapter 4 of
 
\begin_inset CommandInset citation
LatexCommand citep
key "Jakov_c_2016"

\end_inset

, which however is light on specific examples.
 The BPHZ style procedure of 
\begin_inset CommandInset citation
LatexCommand citep
key "Fejos2008,Patkos2008,Patkos2009"

\end_inset

 was adapted to symmetry improved 2PIEA by 
\begin_inset CommandInset citation
LatexCommand citep
key "Pilaftsis2013"

\end_inset

 and to 3PIEA for three dimensional pure glue QCD (without symmetry improvement)
 by 
\begin_inset CommandInset citation
LatexCommand citep
key "York2012"

\end_inset

 and is the one used here.
\end_layout

\begin_layout Standard
The essence of the procedure is quite simple.
 Consider for example the quadratically divergent integral 
\begin_inset Formula $\int_{q}i\Delta_{G/H}\left(q\right)$
\end_inset

.
 Since 
\begin_inset Formula $\Delta_{G/H}\left(q\right)$
\end_inset

 is determined self-consistently this is a complicated integral which must
 be evaluated numerically.
 However, the UV behaviour of the propagator should approach 
\begin_inset Formula $q^{-2}$
\end_inset

 as 
\begin_inset Formula $q\to\infty$
\end_inset

.
 Indeed, Weinberg's theorem 
\begin_inset CommandInset citation
LatexCommand citep
key "Weinberg1960"

\end_inset

 implies that the self-consistent propagators have this form up to powers
 of logarithms 
\begin_inset CommandInset citation
LatexCommand citep
key "Patkos2008"

\end_inset


\begin_inset Foot
status collapsed

\begin_layout Plain Layout
Though note that renormalisation group theory shows the true large-momentum
 behaviour of the propagators is a power law with an anomalous dimension.
 This implies that a truncated 
\begin_inset Formula $n$
\end_inset

PIEA does not effect a resummation of large logarithms, and that a 2PI (or
 indeed 
\begin_inset Formula $n$
\end_inset

PI) expansion is generally only valid within a finite energy range about
 the renormalisation point.
\end_layout

\end_inset

.
 Now one can add and subtract an integral with the same UV asymptotic behaviour
 as the divergent integral: 
\begin_inset Formula 
\begin{equation}
\int_{q}i\Delta_{G/H}\left(q\right)=\int_{q}\left[i\Delta_{G/H}\left(q\right)-\frac{i}{q^{2}-\mu^{2}+i\epsilon}\right]+\int_{q}\frac{i}{q^{2}-\mu^{2}+i\epsilon},
\end{equation}

\end_inset

where 
\begin_inset Formula $\mu$
\end_inset

 is an arbitrary mass subtraction scale (not a cutoff scale).
 The first term is now only logarithmically divergent and the second term
 can be evaluated analytically in a chosen regularisation scheme such as
 dimensional regularisation.
 A further subtraction of this kind can render the first term finite.
\end_layout

\begin_layout Standard
The renormalised propagators can be written as
\begin_inset Formula 
\begin{align}
\Delta_{G/H}^{-1} & =p^{2}-m_{G/H}^{2}-\Sigma_{G/H}\left(p\right),\\
\Sigma_{G/H}\left(p\right) & =\Sigma_{G/H}^{a}\left(p\right)+\Sigma_{G/H}^{0}\left(p\right)+\Sigma_{G/H}^{r}\left(p\right),
\end{align}

\end_inset

where 
\begin_inset Formula $m_{G/H}$
\end_inset

 is the physical mass and the (renormalised) self-energies have been separated
 into pieces according to their asymptotic behaviour: 
\begin_inset Formula $\Sigma_{G/H}^{a}\left(p\right)\sim p^{2}\left(\ln p\right)^{d_{1}}$
\end_inset

, 
\begin_inset Formula $\Sigma_{G/H}^{0}\sim\left(\ln p\right)^{d_{2}}$
\end_inset

 and 
\begin_inset Formula $\Sigma_{G/H}^{r}\sim p^{-2}$
\end_inset

 as 
\begin_inset Formula $p\to\infty$
\end_inset

 respectively.
 The pole condition requires 
\begin_inset Formula 
\begin{equation}
\Sigma_{G/H}\left(p^{2}=m_{G/H}^{2}\right)=0.
\end{equation}

\end_inset

It is useful to introduce the auxiliary propagator 
\begin_inset Formula $\Delta_{G/H}^{\mu}=\left(p^{2}-\mu^{2}-\Sigma_{G/H}^{a}\left(p\right)\right)^{-1}$
\end_inset

.
 Then 
\begin_inset Formula $\Delta_{G/H}$
\end_inset

 can be expanded in 
\begin_inset Formula $\Delta_{G/H}^{\mu}$
\end_inset

 as
\begin_inset Formula 
\begin{align}
\Delta_{G/H}\left(p\right) & =\Delta_{G/H}^{\mu}\left(p\right)+\left[\Delta_{G/H}^{\mu}\left(p\right)\right]^{2}\left(m_{G/H}^{2}-\mu^{2}+\Sigma_{G/H}^{0}+\Sigma_{G/H}^{r}\right)\nonumber \\
 & +\mathcal{O}\left(\left[\Delta_{G/H}^{\mu}\left(p\right)\right]^{3}\left[\Sigma_{G/H}^{0}\left(p\right)\right]^{2}\right).
\end{align}

\end_inset

Proceeding systematically in this manner allows one to extract the leading
 order asymptotics of diagrams as 
\begin_inset Formula $p\to\infty$
\end_inset

.
 In the two loop truncation it turns out that 
\begin_inset Formula $\Sigma_{G/H}^{a}\left(p\right)=0$
\end_inset

 so one can simply use a common 
\begin_inset Formula $\Delta^{\mu}$
\end_inset

.
\end_layout

\begin_layout Standard
Substituting the expressions for bare fields and parameters in terms of
 the renormalised fields and parameters according to
\begin_inset Formula 
\begin{align}
\left(\phi,\varphi,v\right) & \to Z^{1/2}\left(\phi,\varphi,v\right),\\
m^{2} & \to Z^{-1}Z_{\Delta}^{-1}\left(m^{2}+\delta m^{2}\right),\\
\lambda & \to Z^{-2}\left(\lambda+\delta\lambda\right),\\
\Delta & \to ZZ_{\Delta}\Delta,
\end{align}

\end_inset

with separate counter-terms 
\begin_inset Formula $\delta m_{0,1}^{2},\delta\lambda_{0},\delta\lambda_{1,2}^{A,B}$
\end_inset

 as done before for the Hartree-Fock truncation in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Renormalisation-and-solution-2PI-HF"

\end_inset

, with an additional counter-term 
\begin_inset Formula $\delta\lambda$
\end_inset

 for the sunset diagram
\begin_inset Foot
status open

\begin_layout Plain Layout
Note that the majority of the quantum field theory literature does not call
 this the sunset diagram, instead using that to refer to the self-energy
 term with two four point vertices.
 This thesis follows the convention that has appeared in the symmetry improvemen
t literature, e.g.
 
\begin_inset CommandInset citation
LatexCommand citep
key "Pilaftsis2013"

\end_inset

.
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{fmffile}{sunset-diagram}
\end_layout

\begin_layout Plain Layout


\backslash
begin{equation}
\end_layout

\begin_layout Plain Layout


\backslash
Phi_2 = 
\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,40)
\end_layout

\begin_layout Plain Layout


\backslash
fmfleft{v1}
\end_layout

\begin_layout Plain Layout


\backslash
fmfright{v2}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain,left=1}{v1,v2}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain,right=1}{v1,v2}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain}{v1,v2}
\end_layout

\begin_layout Plain Layout


\backslash
fmfdot{v1}
\backslash
fmfdot{v2}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmfgraph}},
\end_layout

\begin_layout Plain Layout


\backslash
end{equation}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmffile}
\end_layout

\end_inset

 gives the renormalised effective action in momentum space 
\begin_inset Formula 
\begin{align}
\Gamma & =\int_{x}\left(-Z_{\Delta}^{-1}\frac{m^{2}+\delta m_{0}^{2}}{2}v^{2}-\frac{\lambda+\delta\lambda_{0}}{4!}v^{4}\right)+\frac{i\hbar}{2}\left(N-1\right)\mathrm{Tr}\ln\left(\Delta_{G}^{-1}\right)+\frac{i\hbar}{2}\mathrm{Tr}\ln\left(\Delta_{H}^{-1}\right)\nonumber \\
 & -\frac{i\hbar}{2}\left(N-1\right)\int_{k}\left(-ZZ_{\Delta}k^{2}+m^{2}+\delta m_{1}^{2}+Z_{\Delta}\frac{\lambda+\delta\lambda_{1}^{A}}{6}v^{2}\right)\Delta_{G}\left(k\right)\nonumber \\
 & -\frac{i\hbar}{2}\int_{k}\left(-ZZ_{\Delta}k^{2}+m^{2}+\delta m_{1}^{2}+Z_{\Delta}\frac{3\lambda+\delta\lambda_{1}^{A}+2\delta\lambda_{1}^{B}}{6}v^{2}\right)\Delta_{H}\left(k\right)\nonumber \\
 & +\frac{\hbar^{2}}{24}\left[\left(N+1\right)\lambda+\left(N-1\right)\delta\lambda_{2}^{A}+2\delta\lambda_{2}^{B}\right]Z_{\Delta}^{2}\left(N-1\right)\left(\int_{k}\Delta_{G}\left(k\right)\right)^{2}\nonumber \\
 & +\frac{\hbar^{2}}{24}\left[3\lambda+\delta\lambda_{2}^{A}+2\delta\lambda_{2}^{B}\right]Z_{\Delta}^{2}\left(\int_{k}\Delta_{H}\left(k\right)\right)^{2}+\frac{\hbar^{2}}{24}2\left(\lambda+\delta\lambda_{2}^{A}\right)Z_{\Delta}^{2}\left(N-1\right)\int_{k}\Delta_{G}\left(k\right)\int_{q}\Delta_{H}\left(q\right)\nonumber \\
 & +\frac{\hbar^{2}}{4}\left[\frac{\left(\lambda+\delta\lambda\right)v}{3}\right]^{2}Z_{\Delta}^{3}\left(N-1\right)\int_{kl}\Delta_{H}\left(k\right)\Delta_{G}\left(l\right)\Delta_{G}\left(k+l\right)\nonumber \\
 & +\frac{\hbar^{2}}{12}\left[\left(\lambda+\delta\lambda\right)v\right]^{2}Z_{\Delta}^{3}\int_{kl}\Delta_{H}\left(k\right)\Delta_{H}\left(l\right)\Delta_{H}\left(k+l\right).
\end{align}

\end_inset


\end_layout

\begin_layout Standard
The equations of motion following from this are
\begin_inset Formula 
\begin{align}
\Delta_{G}^{-1}\left(k\right) & =ZZ_{\Delta}k^{2}-m^{2}-\delta m_{1}^{2}-Z_{\Delta}\frac{\lambda+\delta\lambda_{1}^{A}}{6}v^{2}\nonumber \\
 & -\frac{\hbar}{6}\left[\left(N+1\right)\lambda+\left(N-1\right)\delta\lambda_{2}^{A}+2\delta\lambda_{2}^{B}\right]Z_{\Delta}^{2}\mathcal{T}_{G}\nonumber \\
 & -\frac{\hbar}{6}\left(\lambda+\delta\lambda_{2}^{A}\right)Z_{\Delta}^{2}\mathcal{T}_{H}+i\hbar\left[\frac{\left(\lambda+\delta\lambda\right)v}{3}\right]^{2}Z_{\Delta}^{3}\mathcal{I}_{HG}\left(k\right),
\end{align}

\end_inset

and
\begin_inset Formula 
\begin{align}
\Delta_{H}^{-1} & =ZZ_{\Delta}k^{2}-m^{2}-\delta m_{1}^{2}-Z_{\Delta}\frac{3\lambda+\delta\lambda_{1}^{A}+2\delta\lambda_{1}^{B}}{6}v^{2}\nonumber \\
 & -\frac{\hbar}{6}\left[3\lambda+\delta\lambda_{2}^{A}+2\delta\lambda_{2}^{B}\right]Z_{\Delta}^{2}\mathcal{T}_{H}-\frac{\hbar}{6}\left(\lambda+\delta\lambda_{2}^{A}\right)Z_{\Delta}^{2}\left(N-1\right)\mathcal{T}_{G}\nonumber \\
 & +\frac{i\hbar}{2}\left[\frac{\left(\lambda+\delta\lambda\right)v}{3}\right]^{2}Z_{\Delta}^{3}\left(N-1\right)\mathcal{I}_{GG}\left(k\right)+\frac{i\hbar}{2}\left[\left(\lambda+\delta\lambda\right)v\right]^{2}Z_{\Delta}^{3}\mathcal{I}_{HH}\left(k\right),
\end{align}

\end_inset

where the (divergent) integrals are
\begin_inset Formula 
\begin{align}
\mathcal{T}_{G/H} & =i\int_{q}\Delta_{G/H}\left(q\right),\\
\mathcal{I}_{AB}\left(k\right) & =\int_{q}\Delta_{A}\left(q\right)\Delta_{B}\left(k-q\right),
\end{align}

\end_inset

where 
\begin_inset Formula $A,B=H,G$
\end_inset

 in the last equation.
\end_layout

\begin_layout Standard
\begin_inset Formula $\mathcal{I}_{AB}\left(p\right)$
\end_inset

 can be rendered finite by a single subtraction 
\begin_inset Formula 
\begin{equation}
\mathcal{I}_{AB}\left(p\right)=\mathcal{I}^{\mu}+\mathcal{I}_{AB}^{\text{fin}}\left(p\right),\label{eq:Ifinng}
\end{equation}

\end_inset

where 
\begin_inset Formula $\mathcal{I}^{\mu}=\int_{q}\left[i\Delta^{\mu}\left(q\right)\right]^{2}$
\end_inset

.
 Since the propagators are written with the physical masses explicit, it
 is crucial to also subtract a portion of the finite piece 
\begin_inset Formula $\mathcal{I}_{HG}^{\text{fin}}\left(m_{G/H}\right)$
\end_inset

 so that the pole of the propagator is fixed at the physical mass of the
 Goldstone/Higgs propagator respectively.
 Making this subtraction separately allows a universal 
\begin_inset Formula $\mathcal{I}^{\mu}$
\end_inset

.
\end_layout

\begin_layout Standard
The tadpole integrals 
\begin_inset Formula $\mathcal{T}_{G/H}$
\end_inset

 require two subtractions each since 
\begin_inset Formula $\int_{q}i\left[\Delta^{\mu}\left(q\right)\right]^{2}\Sigma_{G/H}^{0}\left(q\right)$
\end_inset

 is logarithmically divergent.
 To that end introduce 
\begin_inset Formula 
\begin{equation}
\Sigma^{\mu}\left(q\right)=-i\hbar\left[\frac{\left(\lambda+\delta\lambda\right)v}{3}\right]^{2}Z_{\Delta}^{3}\left[\int_{\ell}i\Delta^{\mu}\left(\ell\right)i\Delta^{\mu}\left(q+\ell\right)-\mathcal{I}^{\mu}\right],
\end{equation}

\end_inset

 which is asymptotically the same as 
\begin_inset Formula $\Sigma_{G/H}^{0}\left(q\right)$
\end_inset

, so that 
\begin_inset Formula $\int_{q}i\left[\Delta^{\mu}\left(q\right)\right]^{2}\left[\Sigma_{G/H}^{0}\left(q\right)-\Sigma^{\mu}\left(q\right)\right]$
\end_inset

 is finite.
 For later convenience write
\begin_inset Formula 
\begin{equation}
\int_{q}i\left[\Delta^{\mu}\left(q\right)\right]^{2}\Sigma^{\mu}\left(q\right)=\hbar\left[\frac{\left(\lambda+\delta\lambda\right)v}{3}\right]^{2}Z_{\Delta}^{3}c^{\mu}.
\end{equation}

\end_inset

Then 
\begin_inset Formula 
\begin{equation}
\mathcal{T}_{G/H}=\mathcal{T}^{\mu}-i\left(m_{G/H}^{2}-\mu^{2}\right)\mathcal{I}^{\mu}+\hbar\left[\frac{\left(\lambda+\delta\lambda\right)v}{3}\right]^{2}Z_{\Delta}^{3}c^{\mu}+\mathcal{T}_{G/H}^{\text{fin}},\label{eq:Tfing}
\end{equation}

\end_inset

where 
\begin_inset Formula $\mathcal{T}^{\mu}=\int_{q}i\Delta^{\mu}\left(q\right)$
\end_inset

.
 Note that 
\begin_inset Formula $\mathcal{T}^{\mu}$
\end_inset

 and 
\begin_inset Formula $c^{\mu}$
\end_inset

 are real and 
\begin_inset Formula $\mathcal{I}^{\mu}$
\end_inset

 is imaginary, so that all of the subtractions can be absorbed into real
 counter-terms.
 By dimensional analysis these can be written in terms of regularisation
 scheme dependent constants as
\begin_inset Formula 
\begin{align}
\mathcal{T}^{\mu} & =c_{0}\Lambda^{2}+c_{1}\mu^{2}\ln\frac{\Lambda^{2}}{\mu^{2}},\\
\mathcal{I}^{\mu} & =c_{2}\ln\frac{\Lambda^{2}}{\mu^{2}},\\
c^{\mu} & =a_{0}\left(\ln\frac{\Lambda^{2}}{\mu^{2}}\right)^{2}+a_{1}\ln\frac{\Lambda^{2}}{\mu^{2}}.
\end{align}

\end_inset


\end_layout

\begin_layout Standard
The counter-terms are found by eliminating 
\begin_inset Formula $m_{G/H}^{2}$
\end_inset

 and demanding that the divergences proportional to different powers of
 
\begin_inset Formula $v^{2}$
\end_inset

 and 
\begin_inset Formula $\mathcal{T}_{G/H}^{\text{fin}}$
\end_inset

 separately vanish.
 Further renormalisation conditions are the pole conditions 
\begin_inset Formula $\Delta_{G}^{-1}\left(m_{G}\right)=0$
\end_inset

 and 
\begin_inset Formula $\Delta_{H}^{-1}\left(m_{H}\right)=0$
\end_inset

 and that the counter-terms are momentum independent.
 This gives eight equations for the seven constants 
\begin_inset Formula $Z,Z_{\Delta},\delta m_{1}^{2},\delta\lambda_{1}^{A},\delta\lambda_{2}^{A},\delta\lambda_{2}^{B}$
\end_inset

 and 
\begin_inset Formula $\delta\lambda$
\end_inset

, however one of them is redundant and a solution exists.
 Details of this calculation are included in a 
\noun on
Mathematica
\noun default
 
\begin_inset CommandInset citation
LatexCommand citep
key "mathematica"

\end_inset

 notebook in Appendix 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Appendix-Renormalisation-Details"

\end_inset

.
 After all subtractions the finite equations of motion are
\begin_inset Formula 
\begin{align}
\Delta_{G}^{-1}\left(k\right) & =k^{2}-m^{2}-\frac{\lambda}{6}v^{2}-\frac{\hbar}{6}\left(N+1\right)\lambda\mathcal{T}_{G}^{\mathrm{fin}}-\frac{\hbar}{6}\lambda\mathcal{T}_{H}^{\mathrm{fin}}+i\hbar\left(\frac{\lambda v}{3}\right)^{2}\left[\mathcal{I}_{HG}^{\mathrm{fin}}\left(k\right)-\mathcal{I}_{HG}^{\mathrm{fin}}\left(m_{G}\right)\right],
\end{align}

\end_inset

and
\begin_inset Formula 
\begin{align}
\Delta_{H}^{-1} & =k^{2}-m^{2}-\frac{\lambda}{2}v^{2}-\frac{\hbar}{6}\lambda\left(N-1\right)\mathcal{T}_{G}^{\mathrm{fin}}-\frac{\hbar}{2}\lambda\mathcal{T}_{H}^{\mathrm{fin}}\nonumber \\
 & +\frac{i\hbar}{2}\left(\frac{\lambda v}{3}\right)^{2}\left(N-1\right)\left[\mathcal{I}_{GG}^{\mathrm{fin}}\left(k\right)-\mathcal{I}_{GG}^{\mathrm{fin}}\left(m_{H}\right)\right]+\frac{i\hbar}{2}\left(\lambda v\right)^{2}\left[\mathcal{I}_{HH}^{\mathrm{fin}}\left(k\right)-\mathcal{I}_{HH}^{\mathrm{fin}}\left(m_{H}\right)\right],
\end{align}

\end_inset

and, of course, 
\begin_inset Formula $vm_{G}^{2}=0$
\end_inset

.
\end_layout

\begin_layout Standard
Note that due to the momentum dependence of the self-energies these equations
 are significantly more complicated than their Hartree-Fock counter-parts.
 No analytical solutions of these equations are possible (to the author's
 knowledge).
 Numerical solutions have been investigated in 
\begin_inset CommandInset citation
LatexCommand citep
key "Pilaftsis2013,Marko2016"

\end_inset

, which have found dramatically differing results using somewhat different
 methods.
 These results will be discussed but not reproduced here.
 
\begin_inset CommandInset citation
LatexCommand citet
key "Pilaftsis2013"

\end_inset

 renormalised using the 
\begin_inset Formula $\overline{\mathrm{MS}}$
\end_inset

 scheme at zero temperature in four dimensional Euclidean space, then solved
 the gap equations by an iterative relaxation method in momentum space using
 a lattice in 
\begin_inset Formula $\left|k_{E}\right|$
\end_inset

, using rotational invariance to simplify the numerics, then performing
 a numerical analytic continuation back to Minkowski space using Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximants.
 They used parameter values relevant to Higgs boson physics, with 
\begin_inset Formula $m_{H}=125\ \mathrm{GeV}$
\end_inset

 and 
\begin_inset Formula $v=246\ \mathrm{GeV}$
\end_inset

 (though with an unphysical value of 
\begin_inset Formula $N=2$
\end_inset

!).
 The solutions they found were well behaved at all momentum scales included
 in the calculation and had physically reasonable self-energies, including
 imaginary parts representing quantitatively accurate decay rates and reaction
 thresholds for processes such as 
\begin_inset Formula $H\to GG$
\end_inset

 and decays of off-shell states such as 
\begin_inset Formula $G^{\star}\to HG$
\end_inset

 and 
\begin_inset Formula $H^{\star}\to HH^{\star}\to HGG$
\end_inset

.
 In short, all of the results found in 
\begin_inset CommandInset citation
LatexCommand citep
key "Pilaftsis2013"

\end_inset

 indicate that the two loop SI-2PIEA approximation is physically well behaved.
\end_layout

\begin_layout Standard
In contrast, 
\begin_inset CommandInset citation
LatexCommand citet
key "Marko2016"

\end_inset

 used several different varieties of cutoff regularisation with three different
 parameter sets (chosen for illustrative, not physical, purposes, and all
 with 
\begin_inset Formula $N=2$
\end_inset

) at zero and finite temperature in 
\begin_inset Formula $d=4$
\end_inset

 Euclidean space to argue that, contrary to physical intuition and renormalisati
on group based experience, the IR behaviour of the SI-2PI propagators is
 highly sensitive to the UV structure of the regulator used to render the
 theory finite.
 The general conclusion of their investigation, not rigorously proven though
 supported by semi-analytical and numerical arguments, is that for any 
\begin_inset Quotes eld
\end_inset

sufficiently smooth
\begin_inset Quotes erd
\end_inset

 UV regulator
\begin_inset Foot
status collapsed

\begin_layout Plain Layout
Which they define to be any regulator that does not induce an anomalous
 dimension of the Goldstone propagator in the IR, or in other words any
 UV regulator which does not directly alter the IR physics.
\end_layout

\end_inset

, the SI-2PI equations of motion lack a solution in the physical infinite
 volume limit.
 Their general argument proceeds in two steps.
 First: since 
\begin_inset Formula $m_{G}^{2}=0$
\end_inset

 in the broken phase by construction, 
\begin_inset Formula $\Delta_{G}\left(k\right)$
\end_inset

 must have an anomalous behaviour 
\begin_inset Formula $\sim k^{-2\alpha}$
\end_inset

 with 
\begin_inset Formula $\alpha<1$
\end_inset

 as 
\begin_inset Formula $k^{2}\to0$
\end_inset

, otherwise the term 
\begin_inset Formula $\mathcal{I}_{GG}^{\mathrm{fin}}\left(k\right)$
\end_inset

 in the Higgs equation of motion diverges in the IR and the only solution
 is 
\begin_inset Formula $v=0$
\end_inset

.
 Second: if 
\begin_inset Formula $m_{H}^{2}\neq0$
\end_inset

 and the UV regulator does not produce an anomalous dimension in 
\begin_inset Formula $\Delta_{G}$
\end_inset

, an anomalous dimension in 
\begin_inset Formula $\Delta_{G}$
\end_inset

 is incompatible with the Goldstone equation of motion which can be rewritten
 as
\begin_inset Formula 
\begin{equation}
\Delta_{G}^{-1}\left(k\right)=k^{2}+i\hbar\left(\frac{\lambda v}{3}\right)^{2}\left[\mathcal{I}_{HG}^{\mathrm{fin}}\left(k\right)-\mathcal{I}_{HG}^{\mathrm{fin}}\left(0\right)\right],
\end{equation}

\end_inset

using 
\begin_inset Formula $\Delta_{G}^{-1}\left(k=0\right)=m_{G}^{2}=0$
\end_inset

 to eliminate the constant and 
\begin_inset Formula $\mathcal{T}_{G/H}^{\mathrm{fin}}$
\end_inset

 terms.
 This equation is incompatible with an anomalous dimension because 
\begin_inset Formula $\mathcal{I}_{HG}^{\mathrm{fin}}\left(k\right)$
\end_inset

 has a regular (i.e.
 analytic) dependence on 
\begin_inset Formula $k$
\end_inset

 in neighbourhood of zero
\begin_inset Foot
status collapsed

\begin_layout Plain Layout
This follows because the momentum 
\begin_inset Formula $k$
\end_inset

 can be routed through the Higgs propagator which is regular in the IR,
 regardless of the form of 
\begin_inset Formula $\Delta_{G}$
\end_inset

 there.
 This is also the reason for the assumption 
\begin_inset Formula $m_{H}^{2}=0$
\end_inset

 in the argument.
\end_layout

\end_inset

.
 Hence, in the two loop truncation of the SI-2PIEA 
\emph on
either
\emph default
 a UV regulator is chosen which produces an anomalous scaling behaviour
 of 
\begin_inset Formula $\Delta_{G}$
\end_inset

 at large distances, in which case the large distance behaviour of the theory
 is sensitive to the specific UV regulator chosen, 
\emph on
or
\emph default
 there is no physically reasonable solution for a broken symmetry phase
 in the infinite volume limit.
 While 
\begin_inset CommandInset citation
LatexCommand citep
key "Marko2016"

\end_inset

 does not directly address the issue, the likely reason these effects are
 not seen in the analysis of 
\begin_inset CommandInset citation
LatexCommand citep
key "Pilaftsis2013"

\end_inset

 is the latter's use of dimensional regularisation, which 
\emph on
does
\emph default
 directly alter the IR properties of the theory
\begin_inset Foot
status collapsed

\begin_layout Plain Layout
Dimensional regularisation requires careful treatment of the infinite volume
 limit since IR divergences automatically vanish in the standard version
 of the scheme, see e.g.
 
\begin_inset CommandInset citation
LatexCommand citep
key "Zinn-Justin1990"

\end_inset

.
\end_layout

\end_inset

.
 One should also note that there was no conclusive (numerical) observation
 in 
\begin_inset CommandInset citation
LatexCommand citep
key "Marko2016"

\end_inset

 of loss of solution for sufficiently small 
\begin_inset Formula $\lambda$
\end_inset

 with a smooth UV regulator.
 Also, the behaviour of the IR regulator at higher loop orders remains an
 open problem.
\end_layout

\begin_layout Standard
This situation is unfortunate, although not necessarily disastrous, since
 there are situations in which the UV cutoff is physical (e.g.
 in condensed matter physics) and/or the infinite volume limit is merely
 an idealisation which may not be strictly required.
 For example, the universe could be placed in a box with no observable consequen
ces so long as the box is sufficiently large compared to the Hubble volume.
 This provides a natural IR regulator which, depending on the parameter
 values, can allow a physically reasonable solution to exist.
 The essential message is that the IR and UV sensitivity of solutions of
 the SI-2PI gap equations should be checked on a case by case basis to establish
 the plausibility of the results obtained.
\end_layout

\begin_layout Section
Symmetry Improvement of the 3PIEA
\begin_inset ERT
status open

\begin_layout Plain Layout

}{
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "sec:SI-3PIEA"

\end_inset


\end_layout

\begin_layout Standard
This section examines the extension of symmetry improvement to the 3PIEA,
 closely following the work published by the author in 
\begin_inset CommandInset citation
LatexCommand citep
key "Brown2015a"

\end_inset

.
 In this section the equations of motion are derived but not solved.
 Solutions for the (over simplified) Hartree-Fock truncation are presented
 in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Solution-of-the-SI-3PI-HF-Approx"

\end_inset

.
 Unfortunately, due to the added numerical difficulty, solutions of the
 more interesting (and physically informative) two and three loop truncations
 are left for future work.
 For ease of reference the explicit forms of the 3PIEA and its equations
 of motion reproduced from 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:3piea"

\end_inset

 and following are
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\Gamma^{\left(3\right)}\left[\varphi,\Delta,V\right]=S\left[\varphi\right]+\frac{i\hbar}{2}\mathrm{Tr}\ln\Delta^{-1}+\frac{i\hbar}{2}\mathrm{Tr}\left(\Delta_{0}^{-1}\Delta-1\right)+\Gamma_{3}\left[\varphi,\Delta,V\right],
\end{equation}

\end_inset

where
\begin_inset Formula 
\begin{equation}
\Gamma_{3}=\Phi_{1}-\Phi_{2}+\frac{\hbar^{2}}{3!}V_{0abc}\Delta_{ad}\Delta_{be}\Delta_{cf}V_{def}+\Phi_{3}+\Phi_{4}+\Phi_{5}+\mathcal{O}\left(\hbar^{4}\right),
\end{equation}

\end_inset

where the 
\begin_inset Formula $\Phi$
\end_inset

s are
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray}
\Phi_{1} & = & -\frac{\hbar^{2}}{8}W_{abcd}\Delta_{ab}\Delta_{cd},\\
\Phi_{2} & = & \frac{\hbar^{2}}{12}V_{abc}V_{def}\Delta_{ad}\Delta_{be}\Delta_{cf},\\
\Phi_{3} & = & \frac{i\hbar^{3}}{4!}V_{abc}V_{def}V_{ghi}V_{jkl}\Delta_{ad}\Delta_{bg}\Delta_{cj}\Delta_{eh}\Delta_{fk}\Delta_{il},\\
\Phi_{4} & = & -\frac{i\hbar^{3}}{8}W_{abcd}V_{efg}V_{hij}\Delta_{ae}\Delta_{bf}\Delta_{ch}\Delta_{di}\Delta_{gj},\\
\Phi_{5} & = & \frac{i\hbar^{3}}{48}W_{abcd}W_{efgh}\Delta_{ae}\Delta_{bf}\Delta_{cg}\Delta_{dh}.
\end{eqnarray}

\end_inset

The equations of motion are
\begin_inset Formula 
\begin{eqnarray}
\frac{\delta\Gamma^{\left(3\right)}}{\delta\varphi_{a}} & = & -J_{a}-K_{ab}^{\left(2\right)}\varphi_{b}-\frac{1}{2}K_{abc}^{\left(3\right)}\left(i\hbar\Delta_{bc}+\varphi_{b}\varphi_{c}\right),\\
\frac{\delta\Gamma^{\left(3\right)}}{\delta\Delta_{ab}} & = & -\frac{i\hbar}{2}K_{ab}^{\left(2\right)}-\frac{i\hbar}{2}K_{abc}^{\left(3\right)}\varphi_{c}\nonumber \\
 &  & -\frac{\hbar^{2}}{3!2}\left(K_{acd}^{\left(3\right)}\Delta_{ce}\Delta_{df}V_{bef}+K_{bcd}^{\left(3\right)}\Delta_{ce}\Delta_{df}V_{aef}\right)\\
\frac{\delta\Gamma^{\left(3\right)}}{\delta V_{abc}} & = & -\frac{1}{3!}\hbar^{2}\Delta_{ad}\Delta_{be}\Delta_{cf}K_{def}^{\left(3\right)},\label{eq:3pi-vertex-eom-rhs}
\end{eqnarray}

\end_inset

where the left hand sides are
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\frac{\delta\Gamma^{\left(3\right)}}{\delta\varphi_{a}} & =\frac{\delta S}{\delta\varphi_{a}}+\frac{i\hbar}{2}V_{0abc}\Delta_{cb}+\frac{\hbar^{2}}{3!}W_{abcd}\Delta_{be}\Delta_{cf}\Delta_{dg}V_{efg}+\mathcal{O}\left(\hbar^{3}\right),\\
\frac{\delta\Gamma^{\left(3\right)}}{\delta\Delta_{ab}} & =-\frac{i\hbar}{2}\left(\Delta_{ab}^{-1}+\mathrm{Tr}\Delta_{0ab}^{-1}-\Sigma_{ab}\right)+\mathcal{O}\left(\hbar^{4}\right),\\
\frac{\delta\Gamma^{\left(3\right)}}{\delta V_{abc}} & =\frac{\hbar^{2}}{3!}\left(V_{0def}-V_{def}\right)\Delta_{ad}\Delta_{be}\Delta_{cf}+\frac{i\hbar^{3}}{3!}V_{def}V_{ghi}V_{jkl}\Delta_{ad}\Delta_{bg}\Delta_{cj}\Delta_{eh}\Delta_{fk}\Delta_{il}\nonumber \\
 & -\frac{i\hbar^{3}}{4}W_{hijd}V_{efg}\left(\frac{1}{3}\Delta_{ah}\Delta_{bi}\Delta_{ej}\Delta_{df}\Delta_{cg}+\text{cyclic permutations }\left(a\to b\to c\to a\right)\right)\nonumber \\
 & +\mathcal{O}\left(\hbar^{4}\right),\label{eq:3pi-vertex-eom-lhs}
\end{align}

\end_inset


\end_layout

\begin_layout Standard
It is now necessary to impose a WI on the three point vertex function.
 To that end introduce the symmetry improved 3PIEA
\begin_inset Formula 
\begin{equation}
\tilde{\Gamma}^{\left(3\right)}=\Gamma^{\left(3\right)}-\frac{i}{2}\int_{x}\ell_{A}^{d}\left(x\right)\mathcal{W}_{c}^{A}\left(x\right)\left[P^{\perp}\left(\varphi,x\right)\right]_{cd}-Bf\left[\mathcal{W}_{cd}^{A}\right],
\end{equation}

\end_inset

where the second term is the same as the 2PI symmetry improvement term (note
 that the projected version of the constraint is used to match 
\begin_inset CommandInset citation
LatexCommand citep
key "Brown2015a"

\end_inset

) and the third term contains the extended symmetry improvement where, recall
 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:1PI-V-WI"

\end_inset

, the WI is
\begin_inset Formula 
\begin{equation}
\mathcal{W}_{cd}^{A}=V_{dca}T_{ab}^{A}\varphi_{b}+\Delta_{ca}^{-1}T_{ad}^{A}+\Delta_{da}^{-1}T_{ac}^{A},
\end{equation}

\end_inset


\begin_inset Formula $B$
\end_inset

 is the new Lagrange multiplier and 
\begin_inset Formula $f\left[\mathcal{W}_{cd}^{A\left(1\right)}\right]$
\end_inset

 is an arbitrary functional which vanishes if and only if its argument vanishes.
 Substituting the SSB ansatz 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:SSB-ansatz-Delta"

\end_inset

 gives
\begin_inset Formula 
\begin{equation}
\tilde{\Gamma}^{\left(3\right)}=\Gamma^{\left(3\right)}-\ell\mathcal{W}_{1}-Bf\left[\mathcal{W}_{2}\right],
\end{equation}

\end_inset

where
\begin_inset Formula 
\begin{align}
\mathcal{W}_{1} & =\int_{z}\Delta_{G}^{-1}\left(x,z\right)v,\\
\mathcal{W}_{2} & =\int_{z}\bar{V}\left(x,y,z\right)v+\Delta_{G}^{-1}\left(x,y\right)-\Delta_{H}^{-1}\left(x,y\right),
\end{align}

\end_inset

are the WIs in terms of the ansatz variables.
\end_layout

\begin_layout Standard
The equations of motion are
\begin_inset Formula 
\begin{align}
\frac{\partial\Gamma^{\left(3\right)}/VT}{\partial v} & =\ell_{0}m^{3}+B\int_{xz}\frac{\delta f}{\delta\mathcal{W}_{2}\left(x,y\right)}\bar{V}\left(x,y,z\right),\label{eq:si3pi-v-eom}\\
\frac{\delta\Gamma^{\left(3\right)}}{\delta\Delta_{G}\left(r,s\right)} & =-B\int_{xy}\frac{\delta f}{\delta\mathcal{W}_{2}\left(x,y\right)}\Delta_{G}^{-1}\left(x,r\right)\Delta_{G}^{-1}\left(s,y\right),\label{eq:si3pi-deltaG-eom}\\
\frac{\delta\Gamma^{\left(3\right)}}{\delta\Delta_{H}\left(r,s\right)} & =B\int_{xy}\frac{\delta f}{\delta\mathcal{W}_{2}\left(x,y\right)}\Delta_{H}^{-1}\left(x,r\right)\Delta_{H}^{-1}\left(s,y\right),\label{eq:si3pi-deltaN-eom}\\
\frac{\delta\Gamma^{\left(3\right)}}{\delta\bar{V}\left(r,s,t\right)} & =vB\int_{xyz}\frac{\delta f}{\delta\mathcal{W}_{2}\left(x,y\right)}\delta^{\left(4\right)}\left(x-r\right)\delta^{\left(4\right)}\left(y-s\right)\delta^{\left(4\right)}\left(z-t\right),\label{eq:si3pi-barV-eom}
\end{align}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\frac{\delta\Gamma^{\left(3\right)}}{\delta V_{N}\left(r,s,t\right)} & =0,\label{eq:si3pi-VN-eom}\\
\mathcal{W}_{1} & =0,\label{eq:si3pi-deltaG-WI}\\
\mathcal{W}_{2} & =0,\label{eq:si3pi-barV-WI}
\end{align}

\end_inset

where the limiting procedure to eliminate 
\begin_inset Formula $\ell$
\end_inset

 and 
\begin_inset Formula $\mathcal{W}_{1}$
\end_inset

 is already taken as in the 2PI case.
 A factor of 
\begin_inset Formula $1=\int_{z}\delta^{\left(4\right)}\left(z-t\right)$
\end_inset

 has been inserted in 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:si3pi-barV-eom"

\end_inset

 for later convenience.
 Now a limiting procedure must be found such that the right hand sides of
 two of 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:si3pi-deltaG-eom"

\end_inset

, 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:si3pi-deltaN-eom"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:si3pi-barV-eom"

\end_inset

 vanish.
 The remaining equation must be chosen so that it can be replaced by the
 constraint 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:si3pi-barV-WI"

\end_inset

 and still give a closed system.
 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:si3pi-barV-eom"

\end_inset

 cannot be eliminated because there is not enough information to reconstruct
 
\begin_inset Formula $\bar{V}$
\end_inset

 from 
\begin_inset Formula $\Delta_{G/H}$
\end_inset

 using 
\begin_inset Formula $\mathcal{W}_{2}$
\end_inset

 (the integral in 
\begin_inset Formula $\mathcal{W}_{2}$
\end_inset

 irreversibly loses information about 
\begin_inset Formula $\bar{V}$
\end_inset

).
 Thus one must eliminate either 
\begin_inset Formula $\Delta_{G}$
\end_inset

 or 
\begin_inset Formula $\Delta_{H}$
\end_inset

, or else artificially restrict the form of 
\begin_inset Formula $\bar{V}$
\end_inset

.
\end_layout

\begin_layout Standard
The desired simplification of the equations of motion can be achieved without
 restricting 
\begin_inset Formula $\bar{V}$
\end_inset

 under the assumption that 
\begin_inset Formula $\delta f/\delta\mathcal{W}_{2}\left(x,y\right)$
\end_inset

 is a spacetime independent constant.
 Note that this is not required by Poincar
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 invariance (only the weaker condition 
\begin_inset Formula $\delta f/\delta\mathcal{W}_{2}\left(x,y\right)=g\left(\left|x-y\right|^{2}\right)$
\end_inset

 is mandated).
 This assumption will be justified in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Justifying-the-d'Alembert-Formalism"

\end_inset

 with the introduction of the so-called 
\emph on
d'Alembert formalism
\emph default
.
 Proceed now by adopting it for the sake of argument.
\end_layout

\begin_layout Standard
Computing the left hand side of 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:si3pi-barV-eom"

\end_inset

 using the explicit form of the 3PIEA 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:3piea"

\end_inset

 and displaying only the two loop terms explicitly gives 
\begin_inset Formula 
\begin{align}
\frac{\delta\Gamma^{\left(3\right)}}{\delta\bar{V}\left(r,s,t\right)} & =-\frac{\hbar^{2}}{2}\left(N-1\right)\int_{xyz}\left(\bar{V}\left(x,y,z\right)+\frac{\lambda v}{3}\delta\left(x-y\right)\delta\left(x-z\right)\right)\nonumber \\
 & \times\Delta_{H}\left(x,r\right)\Delta_{G}\left(y,s\right)\Delta_{G}\left(z,t\right)+\mathcal{O}\left(\hbar^{3}\right).
\end{align}

\end_inset

Without symmetry improvement one sets this quantity to zero, giving an equation
 equivalent to the usual 3PI vertex equation of motion.
 This equation is now modified by the symmetry improvement to
\begin_inset Formula 
\begin{multline}
-\frac{\hbar^{2}}{2}\left(N-1\right)\int_{xyz}\left(\bar{V}\left(x,y,z\right)+\frac{\lambda v}{3}\delta\left(x-y\right)\delta\left(x-z\right)\right)\\
\times\Delta_{H}\left(x,r\right)\Delta_{G}\left(y,s\right)\Delta_{G}\left(z,t\right)+\mathcal{O}\left(\hbar^{3}\right)\\
=vB\int_{xyz}\frac{\delta f}{\delta\mathcal{W}_{2}\left(x,y\right)}\delta^{\left(4\right)}\left(x-r\right)\delta^{\left(4\right)}\left(y-s\right)\delta^{\left(4\right)}\left(z-t\right).
\end{multline}

\end_inset

Convolving with the inverse propagators 
\begin_inset Formula $\Delta_{H}^{-1}\Delta_{G}^{-1}\Delta_{G}^{-1}$
\end_inset

 gives 
\begin_inset Formula 
\begin{multline}
-\frac{\hbar^{2}}{2}\left(N-1\right)\left(\bar{V}\left(a,b,c\right)+\frac{\lambda v}{3}\delta\left(a-b\right)\delta\left(a-c\right)\right)+\mathcal{O}\left(\hbar^{3}\right)\\
=vB\int_{xyz}\frac{\delta f}{\delta\mathcal{W}_{2}\left(x,y\right)}\Delta_{H}^{-1}\left(x,a\right)\Delta_{G}^{-1}\left(y,b\right)\Delta_{G}^{-1}\left(z,c\right),
\end{multline}

\end_inset

which can be recognised as
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\left[B\int_{xy}\frac{\delta f}{\delta\mathcal{W}_{2}\left(x,y\right)}\Delta_{H}^{-1}\left(x,a\right)\Delta_{G}^{-1}\left(y,b\right)\right]\mathcal{W}_{1}.
\end{equation}

\end_inset

The right hand side now vanishes due to 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:si3pi-deltaG-WI"

\end_inset

.
 With the regulator 
\begin_inset Formula $vm_{G}^{2}=\eta m^{3}$
\end_inset

 in place the symmetry improvement term in the 
\begin_inset Formula $\bar{V}$
\end_inset

 equation of motion vanishes faster than the naive 
\begin_inset Formula $B\delta f/\delta\mathcal{W}_{2}$
\end_inset

 scaling manifest in 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:si3pi-barV-eom"

\end_inset

.
 Schematically, the right hand side scales as 
\begin_inset Formula $B\left(\delta f/\delta\mathcal{W}_{2}\right)m_{H}^{2}m_{G}^{4}v\sim B\left(\delta f/\delta\mathcal{W}_{2}\right)m_{H}^{8}\eta^{2}/v$
\end_inset

.
 So long as 
\begin_inset Formula $B\delta f/\delta\mathcal{W}_{2}$
\end_inset

 does not blow up as fast as 
\begin_inset Formula $\eta^{-2}$
\end_inset

 as 
\begin_inset Formula $\eta\to0$
\end_inset

 the symmetry improvement has no effect on 
\begin_inset Formula $\bar{V}$
\end_inset

.
\end_layout

\begin_layout Standard
Now consider the Goldstone propagator.
 Substituting 
\begin_inset Formula $\Gamma^{\left(3\right)}$
\end_inset

 into 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:si3pi-deltaG-eom"

\end_inset

 the symmetry improved equation of motion for 
\begin_inset Formula $\Delta_{G}$
\end_inset

 is found to be
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\Delta_{G}^{-1}\left(r,s\right)=\Delta_{0G}^{-1}\left(r,s\right)-\tilde{\Sigma}_{G}\left(r,s\right),
\end{equation}

\end_inset

where the 3PI symmetry improved self-energy 
\begin_inset Formula $\tilde{\Sigma}_{G}$
\end_inset

 is 
\begin_inset Formula 
\begin{align}
\tilde{\Sigma}_{G}\left(r,s\right) & \equiv\frac{2i}{\hbar\left(N-1\right)}\left[\frac{\delta\Gamma_{3}}{\delta\Delta_{G}\left(r,s\right)}+B\int_{xy}\frac{\delta f}{\delta\mathcal{W}_{2}\left(x,y\right)}\Delta_{G}^{-1}\left(x,r\right)\Delta_{G}^{-1}\left(s,y\right)\right].
\end{align}

\end_inset

Substituting in 
\begin_inset Formula $\Gamma_{3}^{\left(3\right)}$
\end_inset

 to two loop order, 
\begin_inset Formula 
\begin{align}
\tilde{\Sigma}_{G}\left(r,s\right) & =\frac{i\hbar\lambda}{6}\mathrm{Tr}\left[\left(N+1\right)\Delta_{G}+\Delta_{H}\right]\delta^{\left(4\right)}\left(r-s\right)\nonumber \\
 & -i\hbar\int_{abcd}\left(\bar{V}\left(a,b,r\right)+\frac{2\lambda v}{3}\delta^{\left(4\right)}\left(a-r\right)\delta^{\left(4\right)}\left(b-r\right)\right)\bar{V}\left(c,d,s\right)\Delta_{H}\left(a,c\right)\Delta_{G}\left(b,d\right)\nonumber \\
 & +\frac{2i}{\hbar\left(N-1\right)}B\int_{xy}\frac{\delta f}{\delta\mathcal{W}_{2}\left(x,y\right)}\Delta_{G}^{-1}\left(x,r\right)\Delta_{G}^{-1}\left(s,y\right)+\mathcal{O}\left(\hbar^{2}\right).
\end{align}

\end_inset

The first term corresponds to the Hartree-Fock diagram, the second term
 corresponds to the sunset diagrams and the third term is the symmetry improveme
nt term.
 The equation of motion for 
\begin_inset Formula $\Delta_{H}$
\end_inset

 can be written in the same form with a suitable definition for a symmetry
 improved self energy 
\begin_inset Formula $\tilde{\Sigma}_{H}\left(r,s\right)$
\end_inset

, where the symmetry improvement term now has the form 
\begin_inset Formula $\sim B\int\left(\delta f/\delta\mathcal{W}_{2}\right)\Delta_{H}^{-1}\Delta_{H}^{-1}$
\end_inset

.
\end_layout

\begin_layout Standard
If 
\begin_inset Formula $\delta f/\delta\mathcal{W}_{2}$
\end_inset

 is constant, then 
\begin_inset Formula $\tilde{\Sigma}_{G}$
\end_inset

 and 
\begin_inset Formula $\tilde{\Sigma}_{H}$
\end_inset

 scale as 
\begin_inset Formula 
\begin{equation}
\left(B\delta f/\delta\mathcal{W}_{2}\right)m_{G}^{4}\sim\left(B\delta f/\delta\mathcal{W}_{2}\right)m_{H}^{6}\eta^{2}/v^{2},
\end{equation}

\end_inset

and 
\begin_inset Formula 
\begin{equation}
\left(B\delta f/\delta\mathcal{W}_{2}\right)m_{H}^{4}\sim\left(B\delta f/\delta\mathcal{W}_{2}\right)m_{H}^{4}\eta^{0},
\end{equation}

\end_inset

respectively.
 Thus, by choosing a regulator such that 
\begin_inset Formula $B\delta f/\delta\mathcal{W}_{2}$
\end_inset

 goes to a finite limit, the equations of motion for 
\begin_inset Formula $\bar{V}$
\end_inset

 and 
\begin_inset Formula $\Delta_{G}$
\end_inset

 are unmodified and the equation of motion for 
\begin_inset Formula $\Delta_{H}$
\end_inset

 is modified by a finite term.
 This is the desired limiting procedure.
 Adopting it gives the final set of SI-3PI equations of motion: 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\frac{\delta\Gamma^{\left(3\right)}}{\delta\Delta_{G}\left(r,s\right)} & =0,\\
\frac{\delta\Gamma^{\left(3\right)}}{\delta\bar{V}\left(r,s,t\right)} & =0,\\
\frac{\delta\Gamma^{\left(3\right)}}{\delta V_{N}\left(r,s,t\right)} & =0,\\
\mathcal{W}_{1} & =0,\\
\mathcal{W}_{2} & =0.
\end{align}

\end_inset

That is the vev and Higgs propagator equations of motion are replaced by
 WIs, but the Goldstone propagator and vertex equations of motion are unmodified.
\end_layout

\begin_layout Section
Justifying the d'Alembert formalism
\begin_inset ERT
status open

\begin_layout Plain Layout

}{
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "sec:Justifying-the-d'Alembert-Formalism"

\end_inset


\end_layout

\begin_layout Standard
The assumption that 
\begin_inset Formula $\delta f/\delta\mathcal{W}_{2}$
\end_inset

 can be consistently taken to be constant requires explanation.
 Constrained Lagrangian problems are generally under-specified unless some
 principle like d'Alembert's principle (that the constraint forces are 
\begin_inset Quotes eld
\end_inset

ideal
\begin_inset Quotes erd
\end_inset

, i.e.
 they do no work on the system) is invoked to specify the constraint forces.
 Note that while it is usually stated that enforcing constraints through
 Lagrange multipliers is equivalent to applying d'Alembert's principle,
 this is no longer automatically the case if the constraints involve a singular
 limit as happens in the field theory case.
 This leads to a real ambiguity in the procedure which requires the analyst
 to input physical information to resolve it.
 In the case of mechanical systems the analyst is expected to be able to
 furnish the correct form of the constraints by inspection of the system.
 However, the interpretation of 
\begin_inset Quotes eld
\end_inset

work
\begin_inset Quotes erd
\end_inset

 and 
\begin_inset Quotes eld
\end_inset

constraint force
\begin_inset Quotes erd
\end_inset

 in the field theory case is subtle and the appropriate generalisation is
 not obvious.
 Here it is argued, by way of a simple mechanical analogy, that the procedure
 which leads to the maximum simplification of the equations of motion is
 the correct field theory analogue of d'Alembert's principle in mechanics.
\end_layout

\begin_layout Standard
d'Alembert's principle is empirically verifiable for a given mechanical
 system, but here it forms part of the 
\emph on
definition
\emph default
 of the SI-3PIEA approximation scheme, which can be referred to as the 
\begin_inset Quotes eld
\end_inset

d'Alembert formalism.
\begin_inset Quotes erd
\end_inset

 The result of Section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:SI-3PIEA"

\end_inset

 was a set of unambiguous 
\begin_inset Formula $f$
\end_inset

-independent equations of motion and constraint at some fixed order of the
 loop expansion, say 
\begin_inset Formula $l$
\end_inset

-loops.
 The use of any other limiting procedure requires the analyst to specify
 a spacetime function's worth of data ahead of time, representing the 
\begin_inset Quotes eld
\end_inset

work
\begin_inset Quotes erd
\end_inset

 that the constraint forces do.
 The resulting equations of motion represent a different formulation of
 the system and will have a different solution depending on the choice of
 
\begin_inset Quotes eld
\end_inset

work
\begin_inset Quotes erd
\end_inset

 function.
\end_layout

\begin_layout Standard
Imagine that one is competing against another analyst to find the most accurate
 solution for a particular system.
 It is possible that the competing smart analyst could choose a work function
 that results in a more accurate solution, also working at 
\begin_inset Formula $l$
\end_inset

-loops.
 However, one could still beat the other analyst by working in the d'Alembert
 formalism but at higher loop order.
 A reasonable conjecture is that the optimum choice of work function (in
 the sense of guaranteeing the optimum accuracy of the resulting solution
 of the 
\begin_inset Formula $l$
\end_inset

-loop equations) is merely a clever repackaging of information contained
 in 
\begin_inset Formula $>l$
\end_inset

-loop corrections.
 (This conjecture is given without proof.
 Indeed it is hard to see if any alternative to the d'Alembert formalism
 is practicable.) Additionally the d'Alembert formalism has the virtue of
 being a definite procedure requiring little cleverness from the analyst,
 at the cost of potentially having a sub-optimal accuracy for a given loop
 order.
\end_layout

\begin_layout Standard
To illustrate the connection with a mechanics problem consider a classical
 particle in 2D constrained to 
\begin_inset Formula $x^{2}+y^{2}=r^{2}$
\end_inset

.
 The motion is uniform circular:
\begin_inset Formula 
\begin{align}
\left(\begin{array}{c}
x\\
y
\end{array}\right) & =r\left(\begin{array}{c}
\cos\left(\omega t+\phi\right)\\
\sin\left(\omega t+\phi\right)
\end{array}\right).
\end{align}

\end_inset

 The action is 
\begin_inset Formula 
\begin{align}
S & =\int L\mathrm{d}t-\lambda f\left[w\right],\\
L & =\frac{1}{2}m\left(\dot{x}^{2}+\dot{y}^{2}\right),
\end{align}

\end_inset

 where the constraint is 
\begin_inset Formula $w\left(t\right)=x\left(t\right)^{2}+y\left(t\right)^{2}-r^{2}=0$
\end_inset

 and 
\begin_inset Formula $f\left[w\right]=0$
\end_inset

 if 
\begin_inset Formula $w\left(t\right)=0$
\end_inset

.
 The equations of motion are 
\begin_inset Formula 
\begin{align}
m\ddot{x}\left(t\right) & =-2\lambda\frac{\delta f}{\delta w\left(t\right)}x\left(t\right),\\
m\ddot{y}\left(t\right) & =-2\lambda\frac{\delta f}{\delta w\left(t\right)}y\left(t\right).
\end{align}

\end_inset


\end_layout

\begin_layout Standard
In this mechanics problem one could set 
\begin_inset Formula $f\left[w\right]=\int w\left(t\right)\mathrm{d}t$
\end_inset

 and carry through the problem in the standard way without any complications.
 But to mimic the field theory case, where a limiting procedure is required,
 take 
\begin_inset Formula $f\left[w\right]=\int w\left(t\right)^{2}\mathrm{d}t$
\end_inset

.
 In this case 
\begin_inset Formula 
\begin{equation}
\frac{\delta f}{\delta w\left(t\right)}=2w\left(t\right)\to0\text{\ as\ }w\to0.
\end{equation}

\end_inset

This requires 
\begin_inset Formula $\lambda\to\infty$
\end_inset

 such that 
\begin_inset Formula $\lambda\delta f/\delta w$
\end_inset

 approaches a finite limit.
 Importantly, it must approach a 
\begin_inset Formula $t$
\end_inset

 independent limit, otherwise an unspecified function of time enters the
 equations of motion: 
\begin_inset Formula $m\ddot{x}\left(t\right)=-k\left(t\right)x\left(t\right),$
\end_inset

 etc.
 This limit can be achieved by restricting the class of variations considered.
 Let 
\begin_inset Formula $x\left(t\right)=r\left(t\right)\cos\theta\left(t\right)$
\end_inset

 and 
\begin_inset Formula $y\left(t\right)=r\left(t\right)\sin\theta\left(t\right)$
\end_inset

, where 
\begin_inset Formula $\delta r\left(t\right)=r\left(t\right)-r$
\end_inset

 parametrises deviations from the constraint surface.
 Then 
\begin_inset Formula $w\left(t\right)=2r\delta r\left(t\right)+\mathcal{O}\left(\delta r^{2}\right)$
\end_inset

.
 One needs 
\begin_inset Formula $\dot{w}\left(t\right)=0$
\end_inset

 which is obviously satisfied by 
\begin_inset Formula $\delta r\left(t\right)=\delta r$
\end_inset

.
\end_layout

\begin_layout Standard
The present section is arguing that one only consider variations of this
 restricted form.
 The variations along the constraint surface (i.e.
 variations of 
\begin_inset Formula $\theta\left(t\right)$
\end_inset

) are unrestricted as they should be.
 Only variations orthogonal to the constraint surface are restricted.
 This is equivalent to the classical d'Alembert principle.
 To see this compute the second derivative of 
\begin_inset Formula $r^{2}$
\end_inset

 to obtain 
\begin_inset Formula $\dot{\theta}^{2}-k=\frac{\ddot{r}}{r}.$
\end_inset

 When the constraint is enforced 
\begin_inset Formula $\ddot{r}=0,$
\end_inset

 hence 
\begin_inset Formula $\dot{k}\neq0$
\end_inset

 implies 
\begin_inset Formula $\ddot{\theta}\neq0$
\end_inset

: the constraint forces are causing angular accelerations, doing work on
 the particle.
 At constant radius, the centripetal force only changes if the angular velocity
 changes.
\end_layout

\begin_layout Standard
In the field theory case the constraint is 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:si3pi-barV-WI"

\end_inset

.
 For any given value of 
\begin_inset Formula $\bar{V}$
\end_inset

 and 
\begin_inset Formula $\Delta_{G}$
\end_inset

 only one value of 
\begin_inset Formula $\Delta_{H}$
\end_inset

 satisfies the constraint, given by
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\Delta_{H}^{-1\star}\left(x,y\right)=\int_{z}\bar{V}\left(x,y,z\right)v+\Delta_{G}^{-1}\left(x,y\right),
\end{equation}

\end_inset

where the 
\begin_inset Formula $\star$
\end_inset

 denotes the constraint solution.
 This is a holonomic constraint: in principle one could substitute this
 into the effective action directly and not worry about Lagrange multipliers
 at all (this is very messy analytically, though it may be numerically feasible).
 In the d'Alembert formalism, then, one restricts variations of 
\begin_inset Formula $\Delta_{H}^{-1}$
\end_inset

 to be of the form 
\begin_inset Formula $\Delta_{H}^{-1\star}\left(x,y\right)+\delta k$
\end_inset

, where 
\begin_inset Formula $\delta k$
\end_inset

 is a spacetime independent constant.
 This guarantees 
\begin_inset Formula 
\begin{equation}
\frac{\delta f}{\delta\mathcal{W}_{2}\left(x,y\right)}=2\mathcal{W}_{2}\left(x,y\right)=-2\delta k=\text{const},
\end{equation}

\end_inset

and all the desired simplifications go through.
 Variations of the other variables are unrestricted.
 Because the constraint force 
\begin_inset Formula $B\delta f/\delta\mathcal{W}_{2}$
\end_inset

 disappears from the 
\begin_inset Formula $\Delta_{G}$
\end_inset

, 
\begin_inset Formula $\bar{V}$
\end_inset

 and 
\begin_inset Formula $V_{N}$
\end_inset

 equations of motion the constraint 
\begin_inset Quotes eld
\end_inset

does no work
\begin_inset Quotes erd
\end_inset

 on these variables, and the other variables (
\begin_inset Formula $v$
\end_inset

 and 
\begin_inset Formula $\Delta_{H}$
\end_inset

) are determined solely by the constraint equations.
 This seems a fitting field theory analogy for the classical d'Alembert
 principle.
\end_layout

\begin_layout Section
Renormalisation of the SI-3PIEA
\begin_inset ERT
status open

\begin_layout Plain Layout

}{
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "sec:Renormalisation-of-the-SI-3PIEA"

\end_inset


\end_layout

\begin_layout Subsection
General comments
\begin_inset CommandInset label
LatexCommand label
name "subsec:SI-3PIEA-Renormalisation-General-comments"

\end_inset


\end_layout

\begin_layout Standard
Before examining the renormalisation problem for the SI-3PIEA in detail
 a few general comments are in order.
 Detailed considerations follow in Sections 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:SI-3PIEA-Two-loop-truncation"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:SI-3PIEA-Three-loop-truncation"

\end_inset

 for the two and three loop truncations at zero temperature respectively.
 The two loop renormalisation of the theory in Section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:SI-3PIEA-Two-loop-truncation"

\end_inset

 is non-trivial already even though the vertex equation of motion can be
 solved trivially.
 This is because the symmetry improvement breaks the 
\begin_inset Formula $n$
\end_inset

PIEA equivalence hierarchy by modifying the Higgs equation of motion.
 The renormalisation of the Hartree-Fock approximation is deferred to Section
 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Solution-of-the-SI-3PI-HF-Approx"

\end_inset

 where it is discussed along with its numerical solution at finite temperature.
\end_layout

\begin_layout Standard
Generically, modifications of the equations of motion following from the
 2PIEA will lead to an inconsistency of the renormalisation procedure since
 the 2PIEA is self-consistently complete at two loop order (in the action,
 i.e.
 one loop order in the equations of motion).
 However, the wavefunction and propagator renormalisation constants (normally
 trivial in 
\begin_inset Formula $\phi^{4}$
\end_inset

 at one loop) provide the extra freedom required to obtain consistency in
 the present case.
 The theory will be renormalised at three loops in Section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:SI-3PIEA-Three-loop-truncation"

\end_inset

.
 Non-perturbative counter-term calculations are generally much more difficult
 than the analogous perturbative calculations, hence many of the manipulations
 are left in a supplemental 
\noun on
Mathematica
\noun default
 
\begin_inset CommandInset citation
LatexCommand citep
key "mathematica"

\end_inset

 notebook (Appendix 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Appendix-Renormalisation-Details"

\end_inset

).
 The results of this section are finite equations of motion for renormalised
 quantities which must be solved numerically.
 Except in the case of the Hartree-Fock approximation, the numerical implementat
ion of the equations of motion requires a much greater numerical effort
 than anything else that has been attempted in this thesis and so is left
 to future work.
\end_layout

\begin_layout Standard
To demonstrate the renormalisability of the SI-3PI equations of motion first
 consider the symmetric phase since, on physical grounds SSB, an infrared
 phenomenon, is irrelevant to renormalisability.
 In the symmetric phase 
\begin_inset Formula $v=0$
\end_inset

 and the Ward identity 
\begin_inset Formula $\mathcal{W}_{1}$
\end_inset

 is trivially satisfied, while 
\begin_inset Formula $\mathcal{W}_{2}$
\end_inset

 requires 
\begin_inset Formula $\Delta_{G}=\Delta_{H}$
\end_inset

 as expected.
 Further, iteration shows that vertex equations have the solution 
\begin_inset Formula $\bar{V}=V_{N}=0$
\end_inset

 as expected on general grounds: there is no three point vertex in the symmetric
 phase.
 As a result, the symmetry improved 3PIEA in the symmetric phase is equivalent
 to the ordinary 2PIEA
\begin_inset Formula 
\begin{equation}
\tilde{\Gamma}^{\left(3\right)}\left[\varphi=0,\Delta,V=0\right]=\Gamma^{\left(2\right)}\left[\varphi=0,\Delta\right],
\end{equation}

\end_inset

which is known to be renormalisable, as discussed in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:The-Two-Loop-Solution"

\end_inset

.
 Thus, only divergences arising from non-zero 
\begin_inset Formula $V$
\end_inset

 pose any new conceptual problems.
\end_layout

\begin_layout Standard
Now an analysis similar to the one done for the 2PIEA in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:The-Two-Loop-Solution"

\end_inset

 is done to isolate the leading asymptotics for 
\begin_inset Formula $V$
\end_inset

 at large momentum.
 This thesis follows this approach because it maps easily to that used for
 the SI-2PIEA above and in the literature.
 However, it is not known whether this procedure works in four dimensions.
 So far, counter-term renormalisation has not been applied to 3PIEA in four
 dimensions and it is unknown whether subdivergences, which are ignored
 here, are fully controlled in this method.
 In three dimensions the renormalisation theory of the 3PIEA is much simpler
 and this method goes through.
 However, the argument given here falls short of a proof of renormalisability
 of the SI-3PIEA in four dimensions and the general problem remains open.
\end_layout

\begin_layout Standard
Suppressing 
\begin_inset Formula $\mathrm{O}\left(N\right)$
\end_inset

 indices one can write 
\begin_inset Formula $V\left(p_{1},p_{2},p_{3}\right)=\lambda vf\left(\frac{p_{1}}{v},\frac{p_{2}}{v},\frac{p_{3}}{v}\right)$
\end_inset

 where 
\begin_inset Formula $p_{1}+p_{2}+p_{3}=0$
\end_inset

.
 Now 
\begin_inset Formula $V\to0$
\end_inset

 as 
\begin_inset Formula $v\to0$
\end_inset

 implies that 
\begin_inset Formula $f\left(\chi_{1},\chi_{2},\chi_{3}\right)\sim\chi^{\alpha}\left(\ln\chi\right)^{d_{3}}$
\end_inset

, where 
\begin_inset Formula $\alpha<1$
\end_inset

 and 
\begin_inset Formula $\chi$
\end_inset

 is representative of the largest scale among 
\begin_inset Formula $\chi_{1}$
\end_inset

, 
\begin_inset Formula $\chi_{2}$
\end_inset

 and 
\begin_inset Formula $\chi_{3}$
\end_inset

.
 Now consider the vertex equation of motion 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:3pi-vertex-eom-rhs"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:3pi-vertex-eom-lhs"

\end_inset

, or
\begin_inset ERT
status collapsed

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\backslash
fmfsurroundn{i}{3} 
\backslash
fmf{plain}{i1,v1} 
\backslash
fmf{plain}{i2,v2} 
\backslash
fmf{plain}{i3,v3} 
\backslash
fmf{plain,tension=0.25}{v1,v2,v3,v1} 
\backslash
fmfdot{v1,v2,v3} 
\backslash
end{fmfgraph}}
\end_layout

\begin_layout Plain Layout

+
\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,40) 
\backslash
fmfsurroundn{i}{3} 
\backslash
fmf{plain}{i1,v} 
\backslash
fmf{plain}{i2,w} 
\backslash
fmf{plain}{i3,w} 
\backslash
fmf{plain,left=1,tension=0.25}{v,w,v} 
\backslash
fmfdot{v,w} 
\backslash
end{fmfgraph}}.
\backslash
label{eq:3pi-vertex-eom}
\end_layout

\begin_layout Plain Layout


\backslash
end{equation}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmffile}
\end_layout

\end_inset

where the open circle vertex denotes the bare function 
\begin_inset Formula $V_{0abc}$
\end_inset

 and a sum over cyclic permutations of the legs of the last diagram is understoo
d.
 The triangle graph goes like
\begin_inset Foot
status collapsed

\begin_layout Plain Layout
Recall that the asymptotic form of the self-energies as 
\begin_inset Formula $p\to\infty$
\end_inset

 is given by terms going like 
\begin_inset Formula $\Sigma_{G/H}^{a}\left(p\right)\sim p^{2}\left(\ln p\right)^{d_{1}}$
\end_inset

 and 
\begin_inset Formula $\Sigma_{G/H}^{0}\sim\left(\ln p\right)^{d_{2}}$
\end_inset

 .
\end_layout

\end_inset

 
\begin_inset Formula $\int_{\ell}\ell^{3\alpha-6}\left(\ln\ell\right)^{3d_{3}-3d_{1}}\sim\chi^{3\alpha-2}\left(\ln\chi\right)^{3d_{3}-3d_{1}}$
\end_inset

 if 
\begin_inset Formula $\alpha\neq2/3$
\end_inset

 or 
\begin_inset Formula $\left(\ln\chi\right)^{1+3d_{3}-3d_{1}}$
\end_inset

 if 
\begin_inset Formula $\alpha=2/3$
\end_inset

, which is dominated by the bubble graph which goes like 
\begin_inset Formula $\int_{\ell}\ell^{\alpha-4}\left(\ln\ell\right)^{d_{3}-2d_{1}}\sim\chi^{\alpha}\left(\ln\chi\right)^{d_{3}-2d_{1}}$
\end_inset

 if 
\begin_inset Formula $\alpha\neq0$
\end_inset

 or 
\begin_inset Formula $\left(\ln\chi\right)^{1+d_{3}-2d_{1}}$
\end_inset

 if 
\begin_inset Formula $\alpha=0$
\end_inset

.
 Thus to a leading approximation the large momentum behaviour is obtained
 by dropping the triangle graph from the equation of motion.
 This can also be seen by taking 
\begin_inset Formula $v\to0$
\end_inset

 at fixed 
\begin_inset Formula $p_{i}$
\end_inset

 which suppresses the triangle graph relative to the bubble graph.
\end_layout

\begin_layout Standard
Now define auxiliary vertex functions 
\begin_inset Formula $\bar{V}^{\mu}$
\end_inset

 and 
\begin_inset Formula $V_{N}^{\mu}$
\end_inset

 which have the same asymptotic behaviour as 
\begin_inset Formula $\bar{V}$
\end_inset

 and 
\begin_inset Formula $V_{N}$
\end_inset

 respectively, though depend only on the auxiliary propagators.
 
\begin_inset Formula $\bar{V}^{\mu}$
\end_inset

 and 
\begin_inset Formula $V_{N}^{\mu}$
\end_inset

 are defined by taking the equations of motion for 
\begin_inset Formula $\bar{V}$
\end_inset

 and 
\begin_inset Formula $V_{N}$
\end_inset

, dropping the triangle graphs, and making the replacements 
\begin_inset Formula $\bar{V}\to\bar{V}^{\mu}$
\end_inset

, 
\begin_inset Formula $V_{N}\to V_{N}^{\mu}$
\end_inset

 (represented diagrammatically by a shaded blob), and 
\begin_inset Formula $\Delta_{G/H}\to\Delta_{G/H}^{\mu}$
\end_inset

 (represented by dashed lines), resulting in
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{fmffile}{aux-v-eom}
\end_layout

\begin_layout Plain Layout


\backslash
begin{equation}
\end_layout

\begin_layout Plain Layout


\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,40) 
\backslash
fmfsurroundn{i}{3} 
\backslash
fmfrpolyn{shaded,tension=0.5}{v}{3} 
\backslash
fmf{plain}{i1,v3} 
\backslash
fmf{plain}{i2,v2} 
\backslash
fmf{plain}{i3,v1} 
\backslash
end{fmfgraph}}=
\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,40) 
\backslash
fmfsurroundn{i}{3} 
\backslash
fmf{plain}{i1,v} 
\backslash
fmf{plain}{i2,v} 
\backslash
fmf{plain}{i3,v} 
\backslash
fmfv{decor.shape=circle,decor.filled=empty,decor.size=2thick}{v} 
\backslash
end{fmfgraph}}
\end_layout

\begin_layout Plain Layout

+
\backslash
parbox{22mm}{
\backslash
begin{fmfgraph}(60,40) 
\backslash
fmfsurroundn{i}{3} 
\backslash
fmfrpolyn{shaded,tension=0.2}{v}{3} 
\backslash
fmf{plain}{i1,v3} 
\backslash
fmf{plain}{i2,w} 
\backslash
fmf{plain}{i3,w} 
\backslash
fmf{dashes,left=.75,tension=0.33}{v1,w,v2} 
\backslash
fmfdot{w} 
\backslash
end{fmfgraph}}.
\backslash
label{eq:3pi-aux-vertex-eom}
\end_layout

\begin_layout Plain Layout


\backslash
end{equation}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmffile}
\end_layout

\end_inset

 This gives a pair of coupled linear integral equations, analogous to the
 Bethe-Salpeter equations, for 
\begin_inset Formula $\bar{V}^{\mu}$
\end_inset

 and 
\begin_inset Formula $V_{N}^{\mu}$
\end_inset

 which can be solved explicitly by iteration.
 (Details of this calculation are presented in Section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:SI-3PIEA-Three-loop-truncation"

\end_inset

).
 Unfortunately the problem is only tractable in fewer than 1+3 dimensions,
 so for the present any analytical results depending on the explicit forms
 of 
\begin_inset Formula $\bar{V}^{\mu}$
\end_inset

 and 
\begin_inset Formula $V_{N}^{\mu}$
\end_inset

 are confined to this case.
 In 
\begin_inset Formula $1+2$
\end_inset

 or 
\begin_inset Formula $3$
\end_inset

 dimensions the problem simplifies dramatically as there are no divergent
 vertex corrections.
 In the physically most interesting case of 1+3 dimensions, 
\begin_inset Formula $\bar{V}^{\mu}$
\end_inset

 and 
\begin_inset Formula $V_{N}^{\mu}$
\end_inset

 must be numerically determined at the same time as 
\begin_inset Formula $\bar{V}$
\end_inset

 and 
\begin_inset Formula $V_{N}$
\end_inset

 and it is unkown whether this renormalisation procedure actually works.
\end_layout

\begin_layout Subsection
Two loop truncation
\begin_inset ERT
status open

\begin_layout Plain Layout

}{
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "subsec:SI-3PIEA-Two-loop-truncation"

\end_inset


\end_layout

\begin_layout Standard
The theory simplifies dramatically at two loop order.
 It follows from 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:3pi-vertex-eom-lhs"

\end_inset

 that at this order 
\begin_inset Formula $V\to V_{0}$
\end_inset

 up to a renormalisation (this is just the equivalence hierarchy at work).
 Substituting this into the action gives, apart from the symmetry improvement
 terms, the standard 2PIEA.
 Another simplification is that the logarithmic enhancement of the propagators
 in the UV due to 
\begin_inset Formula $\Sigma_{G/H}^{a}$
\end_inset

 vanishes at this level (
\begin_inset Formula $\Sigma_{G/H}^{a}$
\end_inset

 is generated by the diagram 
\begin_inset Formula $\Phi_{5}$
\end_inset

 appearing at three loop order).
 In this case 
\begin_inset Formula $\Delta_{G}^{\mu}=\Delta_{H}^{\mu}\equiv\Delta^{\mu}=\left(p^{2}-\mu^{2}\right)^{-1}$
\end_inset

.
 However, the reduction is not trivial because now the Higgs equation of
 motion has been replaced by a Ward identity.
 The equations of motion reduce to
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\Delta_{G}^{-1}\left(x,y\right) & =-\left(\partial_{\mu}\partial^{\mu}+m^{2}+\frac{\lambda}{6}v^{2}\right)\delta^{\left(4\right)}\left(x-y\right)\nonumber \\
 & -\frac{i\hbar}{6}\left(N+1\right)\lambda\Delta_{G}\left(x,x\right)\delta^{\left(4\right)}\left(x-y\right)\nonumber \\
 & -\frac{i\hbar}{6}\lambda\Delta_{H}\left(x,x\right)\delta^{\left(4\right)}\left(x-y\right)\nonumber \\
 & -\frac{i\hbar}{9}\lambda^{2}v^{2}\Delta_{H}\left(x,y\right)\Delta_{G}\left(x,y\right),\\
\Delta_{H}^{-1}\left(x,y\right) & =-\frac{\lambda v^{2}}{3}\delta^{\left(4\right)}\left(x-y\right)+\Delta_{G}^{-1}\left(x,y\right),\\
vm_{G}^{2} & =0.
\end{align}

\end_inset

The first line is the tree level term, the second and third lines are the
 Hartree-Fock self-energies, the fourth line is the sunset self-energy,
 and the last two lines are the Ward identities 
\begin_inset Formula $\mathcal{W}_{2}$
\end_inset

 and 
\begin_inset Formula $\mathcal{W}_{1}$
\end_inset

 respectively.
\end_layout

\begin_layout Standard
To renormalise the theory one follows the by now standard procedure by introduci
ng the renormalised parameters 
\begin_inset Formula 
\begin{align}
\left(\phi,\varphi,v\right) & \to Z^{1/2}\left(\phi,\varphi,v\right),\\
m^{2} & \to Z^{-1}Z_{\Delta}^{-1}\left(m^{2}+\delta m^{2}\right),\\
\lambda & \to Z^{-2}\left(\lambda+\delta\lambda\right),\\
\Delta & \to ZZ_{\Delta}\Delta,\\
V & \to Z^{-3/2}Z_{V}V,
\end{align}

\end_inset

and demanding that all kinematically distinct divergences vanish.
 Details of this computation and explicit expressions for the counter-terms
 are given in Appendix 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Appendix-Renormalisation-Details"

\end_inset

.
 The finite equations of motion are found to be
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\Delta_{G}^{-1}\left(p\right) & =p^{2}-m^{2}-\frac{\lambda}{6}v^{2}-\frac{\hbar}{6}\left(N+1\right)\lambda\mathcal{T}_{G}^{\text{fin}}-\frac{\hbar}{6}\lambda\mathcal{T}_{H}^{\text{fin}}\nonumber \\
 & +i\hbar\left(\frac{\lambda v}{3}\right)^{2}\left[\mathcal{I}_{HG}^{\text{fin}}\left(p\right)-\mathcal{I}_{HG}^{\text{fin}}\left(m_{G}\right)\right],\\
\Delta_{H}^{-1}\left(p\right) & =p^{2}-m^{2}-\frac{\lambda v^{2}}{3}-\frac{\lambda}{6}v^{2}-\frac{\hbar}{6}\left(N+1\right)\lambda\mathcal{T}_{G}^{\text{fin}}\nonumber \\
 & -\frac{\hbar}{6}\lambda\mathcal{T}_{H}^{\text{fin}}+i\hbar\left(\frac{\lambda v}{3}\right)^{2}\left[\mathcal{I}_{HG}^{\text{fin}}\left(p\right)-\mathcal{I}_{HG}^{\text{fin}}\left(m_{H}\right)\right].
\end{align}

\end_inset

The finite parts 
\begin_inset Formula $\mathcal{T}_{G/H}^{\text{fin}}$
\end_inset

 and 
\begin_inset Formula $\mathcal{I}_{HG}^{\text{fin}}\left(p\right)$
\end_inset

 are
\begin_inset Formula 
\begin{align}
\mathcal{I}_{HG}^{\text{fin}}\left(p\right) & =\mathcal{I}_{HG}\left(p\right)-\mathcal{I}^{\mu},\\
\mathcal{T}_{G/H}^{\text{fin}} & =\mathcal{T}_{G/H}-\mathcal{T}^{\mu}+i\left(m_{G/H}^{2}-\mu^{2}\right)\mathcal{I}^{\mu}-\int_{q}i\left[\Delta^{\mu}\left(q\right)\right]^{2}\Sigma^{\mu}\left(q\right),
\end{align}

\end_inset

where the auxiliary quantities are 
\begin_inset Formula 
\begin{align}
\mathcal{T}^{\mu} & =\int_{q}i\Delta^{\mu}\left(q\right),\\
\mathcal{I}^{\mu} & =\int_{q}\left[i\Delta^{\mu}\left(q\right)\right]^{2},\\
\Sigma^{\mu}\left(q\right) & =-i\hbar\left(\frac{\lambda v}{3}\right)^{2}\left[\int_{\ell}i\Delta^{\mu}\left(\ell\right)i\Delta^{\mu}\left(q+\ell\right)-\mathcal{I}^{\mu}\right].
\end{align}

\end_inset

These equations are the main result of this section.
 They could presumably be solved numerically using techniques similar to
 
\begin_inset CommandInset citation
LatexCommand citep
key "Marko2012"

\end_inset

, though this is deferred as the analytical results given in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:SI-3PIEA-Optical-theorem-and-dispersion-relations"

\end_inset

 indicate this is an unsatisfactory truncation.
\end_layout

\begin_layout Subsection
Three loop truncation
\begin_inset ERT
status open

\begin_layout Plain Layout

}{
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "subsec:SI-3PIEA-Three-loop-truncation"

\end_inset


\end_layout

\begin_layout Standard
Now consider the three loop truncation of the SI-3PIEA.
 The vertex equation of motion 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:3pi-vertex-eom"

\end_inset


\begin_inset Note Note
status open

\begin_layout Plain Layout
not broken: refers to an fmffile above
\end_layout

\end_inset

 now includes one loop corrections and, as shown above, the leading asymptotics
 at large momentum are captured by the auxiliary vertex defined by its equation
 of motion 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:3pi-aux-vertex-eom"

\end_inset


\begin_inset Note Note
status open

\begin_layout Plain Layout
not broken: refers to an fmffile above
\end_layout

\end_inset

.
 Subtracting these two equations shows that the right hand side is finite
 (indeed the auxiliary vertices were constructed to guarantee this).
 Thus the problem of renormalising the vertex equation of motion reduces
 to the problem of renormalising the auxiliary vertex equation of motion.
\end_layout

\begin_layout Standard
It is temporarily more convenient to go back to the 
\begin_inset Formula $\mathrm{O}\left(N\right)$
\end_inset

 covariant form of the equations of motion before introducing the SSB ansatz.
 Introduce the covariant auxiliary vertex 
\begin_inset Formula $V_{abc}^{\mu}$
\end_inset

 which is related to 
\begin_inset Formula $\bar{V}^{\mu}$
\end_inset

 and 
\begin_inset Formula $V_{N}^{\mu}$
\end_inset

 by the analogue of 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:SSB-ansatz-Vertex"

\end_inset

.
 Iterating the auxiliary vertex equation of motion gives the solution 
\begin_inset Formula 
\begin{equation}
V_{abc}^{\mu}=\mathcal{K}_{abcdef}V_{0def},\label{eq:aux-vertex-soln}
\end{equation}

\end_inset

where the six point kernel 
\begin_inset Formula $\mathcal{K}_{abcdef}$
\end_inset

 obeys the Bethe-Salpeter like equation 
\begin_inset Formula 
\begin{equation}
\mathcal{K}_{abcdef}=\delta_{ad}\delta_{be}\delta_{cf}+\frac{1}{3!}\sum_{\pi}\left(-\frac{3i\hbar}{2}\right)\delta_{\pi\left(a\right)h}W_{\pi\left(b\right)\pi\left(c\right)kg}\Delta_{ki}^{\mu}\Delta_{gj}^{\mu}\mathcal{K}_{hijdef},\label{eq:K-6pt-kernel-eom}
\end{equation}

\end_inset

where 
\begin_inset Formula $\sum_{\pi}$
\end_inset

 is a sum over permutations of the incoming legs.
 Graphically this is
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{fmffile}{aux-v-soln} 
\backslash
begin{equation} 
\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,40) 
\backslash
fmfstraight 
\backslash
fmfleftn{i}{3} 
\backslash
fmfpolyn{shaded}{v}{3} 
\backslash
fmfright{p}
\backslash
fmf{phantom}{p,v1}
\backslash
fmf{phantom}{p,v3} 
\backslash
fmf{plain,tension=2}{i2,v2}
\backslash
fmffreeze 
\backslash
fmf{plain,right=.33}{i1,v3} 
\backslash
fmf{plain,left=.33}{i3,v1} 
\backslash
end{fmfgraph}}=
\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,40) 
\backslash
fmfstraight 
\backslash
fmfleftn{i}{3} 
\backslash
fmfpolyn{shaded,tension=2}{K}{6} 
\backslash
fmfright{v} 
\backslash
fmf{plain,tension=3}{K1,v} 
\backslash
fmf{plain,left=.5,tension=.1}{K2,v} 
\backslash
fmf{plain,right=.5,tension=.1}{K6,v} 
\backslash
fmf{plain,tension=1}{i1,K5} 
\backslash
fmf{plain,tension=1}{i2,K4} 
\backslash
fmf{plain,tension=1}{i3,K3} 
\backslash
fmfv{decor.shape=circle,decor.filled=empty,decor.size=2thick}{v} 
\backslash
end{fmfgraph}},  
\backslash
end{equation} 
\backslash
end{fmffile}
\end_layout

\end_inset

where the shaded hexagonal blob is 
\begin_inset Formula $\mathcal{K}$
\end_inset

 which obeys
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{fmffile}{BS-kernel-K-eom}
\end_layout

\begin_layout Plain Layout


\backslash
begin{equation}
\end_layout

\begin_layout Plain Layout


\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,40) 
\backslash
fmfstraight 
\backslash
fmfleftn{i}{3} 
\backslash
fmfrightn{o}{3} 
\backslash
fmfpolyn{shaded}{K}{6} 
\backslash
fmf{plain}{i1,K5} 
\backslash
fmf{plain}{i2,K4} 
\backslash
fmf{plain}{i3,K3} 
\backslash
fmf{plain}{o1,K6} 
\backslash
fmf{plain}{o2,K1} 
\backslash
fmf{plain}{o3,K2} 
\backslash
end{fmfgraph}} =
\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,40) 
\backslash
fmfstraight 
\backslash
fmfleftn{i}{3} 
\backslash
fmfrightn{o}{3} 
\backslash
fmf{plain}{i1,o1} 
\backslash
fmf{plain}{i2,o2} 
\backslash
fmf{plain}{i3,o3} 
\backslash
end{fmfgraph}} +
\backslash
parbox{27mm}{
\backslash
begin{fmfgraph}(75,40) 
\backslash
fmfstraight 
\backslash
fmfleftn{i}{3} 
\backslash
fmfrightn{o}{3} 
\backslash
fmfpolyn{full}{Pi}{6} 
\backslash
fmf{plain}{i1,Pi5} 
\backslash
fmf{plain}{i2,Pi4} 
\backslash
fmf{plain}{i3,Pi3} 
\backslash
fmfpolyn{shaded}{K}{6} 
\backslash
fmf{plain,tension=0.5}{K5,Pi6} 
\backslash
fmf{phantom,tension=0.5}{Pi1,K4} 
\backslash
fmf{phantom,tension=0.5}{Pi2,K3} 
\backslash
fmf{plain}{o1,K6} 
\backslash
fmf{plain}{o2,K1} 
\backslash
fmf{plain}{o3,K2} 
\backslash
fmffreeze 
\backslash
fmf{plain}{v,Pi1} 
\backslash
fmf{plain}{v,Pi2} 
\backslash
fmf{plain}{v,K3}
\backslash
fmf{plain}{v,K4} 
\backslash
fmfdot{v} 
\backslash
end{fmfgraph}},
\end_layout

\begin_layout Plain Layout


\backslash
end{equation}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmffile}
\end_layout

\end_inset

where the solid hexagonal blob is the sum over all permutations of the lines
 coming in from the left and connecting to those on the right.
 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:K-6pt-kernel-eom"

\end_inset

 can be written in a form that makes explicit all divergences (see Appendix
 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Auxiliary-vertex-and-renorm"

\end_inset

) and replaces the bare vertex 
\begin_inset Formula $W$
\end_inset

 by a four point kernel 
\begin_inset Formula $\mathcal{K}_{abcd}^{\left(4\right)}\sim\lambda/\left(1+\lambda\mathcal{I}^{\mu}\right)$
\end_inset

.
\end_layout

\begin_layout Standard
In fewer than four dimensions 
\begin_inset Formula $\mathcal{K}_{abcd}^{\left(4\right)}$
\end_inset

 is finite and every correction to the tree level value is asymptotically
 sub-dominant.
 Thus the leading term at large momentum is the tree level term and, instead
 of the full auxiliary vertex as defined above, one can simply take 
\begin_inset Formula $V_{abc}^{\mu}=V_{0abc}$
\end_inset

, dramatically simplifying the renormalisation theory.
 A similar simplification happens to the auxiliary propagator due to the
 logarithmic (rather than quadratic) divergence of the 
\begin_inset Formula $\Phi_{5}$
\end_inset

-generated self-energy in 
\begin_inset Formula $<1+3$
\end_inset

 dimensions.
 This confirms statements made in the literature (supported by numerical
 evidence though without proof, to the author's knowledge) to the effect
 that the asymptotic behaviour of Green functions is free (e.g.
 
\begin_inset CommandInset citation
LatexCommand citep
key "York2012"

\end_inset

).
\end_layout

\begin_layout Standard
Unfortunately, the situation is much more difficult in four dimensions and
 the renormalisation of the 
\begin_inset Formula $n$
\end_inset

PIEA for 
\begin_inset Formula $n\geq3$
\end_inset

 in 
\begin_inset Formula $d>3$
\end_inset

 remains an open problem, both in general and in the present case.
 The problem can be seen from the behaviour of the auxiliary vertex which
 is discussed further in Appendix 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Auxiliary-vertex-and-renorm"

\end_inset

.
 For the sake of obtaining analytical results the rest of this section is
 restricted to 
\begin_inset Formula $<1+3$
\end_inset

 dimensions.
 Because of the extra complication of the 
\begin_inset Formula $d>3$
\end_inset

 case a full numerical implementation is required before one can 
\begin_inset Quotes eld
\end_inset

get off the ground,
\begin_inset Quotes erd
\end_inset

 so to speak, so the renormalisation of the 
\begin_inset Formula $1+3$
\end_inset

 dimensional case is left to future work.
\end_layout

\begin_layout Standard
The counter-terms for 
\begin_inset Formula $1+2$
\end_inset

 dimensions are derived following essentially the same procedure used several
 times now already.
 Details are given in Appendix 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Deriving-Counter-terms-for-3-Loop-Truncations"

\end_inset

.
 The are only two interesting comments about this derivation: the first
 is that an additional (non-universal) counter-term is required for the
 sunset graph linear in 
\begin_inset Formula $V$
\end_inset

; the second is that only 
\begin_inset Formula $\delta m_{1}^{2}$
\end_inset

 is required to UV-renormalise the theory.
 This is consistent with the known fact that 
\begin_inset Formula $\phi^{4}$
\end_inset

 theory is 
\emph on
super-renormalisable
\emph default
 in 2+1 dimensions, that is, only a finite number of primitive divergent
 graphs exist and these graphs only contribute to the mass shift.
 Every other counter-term is finite and exists solely to maintain the pole
 condition for the Higgs propagator despite the vertex Ward identity.
 The resulting finite equations of motion are
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\Delta_{G}^{-1}=-\left(\partial_{\mu}\partial^{\mu}+m^{2}+\frac{\lambda}{6}v^{2}\right)-\left[\Sigma_{G}^{0}\left(p\right)-\Sigma_{G}^{0}\left(m_{G}\right)\right],\label{eq:three-loop-geom-finite}
\end{equation}

\end_inset

for the Goldstone propagator,
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\bar{V} & =-\frac{\lambda v}{3}+i\hbar\left[V_{N}\left(\bar{V}\right)^{2}\left(\Delta_{H}\right)^{2}\Delta_{G}+\left(\bar{V}\right)^{3}\Delta_{H}\left(\Delta_{G}\right)^{2}\right]\nonumber \\
 & +\frac{i\hbar\lambda}{6}\left[V_{N}\left(\Delta_{H}\right)^{2}+\left(N+1\right)\bar{V}\left(\Delta_{G}\right)^{2}+4\bar{V}\Delta_{G}\Delta_{H}\right],\label{eq:three-loop-Vbar-eom-finite}
\end{align}

\end_inset

for the Higgs-Goldstone-Goldstone vertex, and
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
V_{N} & =-\lambda v+i\hbar\left[\left(N-1\right)\left(\bar{V}\right)^{3}\left(\Delta_{G}\right)^{3}+\left(V_{N}\right)^{3}\left(\Delta_{H}\right)^{3}\right]\nonumber \\
 & +\frac{i\hbar\lambda}{2}\left[\left(N-1\right)\bar{V}\Delta_{G}\Delta_{G}+3V_{N}\left(\Delta_{H}\right)^{2}\right],\label{eq:three-loop-V_N-eom-finite}
\end{align}

\end_inset

for the triple Higgs vertex.
\end_layout

\begin_layout Standard
The finite Goldstone self-energy is
\begin_inset Formula 
\begin{align}
-\Sigma_{G}^{0}\left(p\right) & =-\frac{\hbar}{6}\left(N+1\right)\lambda\left(\mathcal{T}_{G}-\mathcal{T}^{\mu}\right)-\frac{\hbar}{6}\lambda\left(\mathcal{T}_{H}-\mathcal{T}^{\mu}\right)\nonumber \\
 & -i\hbar\left[-2\frac{\lambda v}{3}-\bar{V}\right]\Delta_{H}\Delta_{G}\bar{V}+\hbar^{2}\left[V_{N}\left(\bar{V}\right)^{3}\left(\Delta_{H}\right)^{3}\left(\Delta_{G}\right)^{2}+\left(\bar{V}\right)^{4}\Delta_{H}\Delta_{H}\left(\Delta_{G}\right)^{3}\right]\nonumber \\
 & +\frac{\hbar^{2}\lambda}{3}\left[\bar{V}V_{N}\left(\Delta_{H}\right)^{3}\Delta_{G}+\left(N+1\right)\bar{V}\bar{V}\Delta_{H}\left(\Delta_{G}\right)^{3}+3\bar{V}\bar{V}\left(\Delta_{G}\right)^{2}\Delta_{H}\Delta_{H}\right]\nonumber \\
 & +\frac{\hbar^{2}\lambda^{2}}{18}\left[\left(N+1\right)\left(\Delta_{G}\right)^{3}+\Delta_{H}\Delta_{H}\Delta_{G}-\left(N+2\right)\mathcal{B}^{\mu}\right],\label{eq:three-loop-goldstone-self-energy-finite}
\end{align}

\end_inset

where the BBALL integral is 
\begin_inset Formula $\mathcal{B}^{\mu}=\int_{qp}\Delta^{\mu}\left(q\right)\Delta^{\mu}\left(p\right)\Delta^{\mu}\left(p+q\right)$
\end_inset

.
 The graph topologies appearing in this equation are
\begin_inset ERT
status collapsed

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end{align*}
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end{fmffile}
\end_layout

\end_inset

Finally, the Higgs equation of motion is
\begin_inset Formula 
\begin{equation}
\Delta_{H}^{-1}\left(p\right)=\left(m_{G}^{2}+\Sigma_{G}^{0}\left(m_{H}\right)-m_{H}^{2}\right)\frac{\bar{V}\left(p,-p,0\right)}{\bar{V}\left(m_{H},-m_{H},0\right)}+\Delta_{G}^{-1}\left(p\right).\label{eq:three-loop-Higgs-eom-finite}
\end{equation}

\end_inset

 The unusual form of this equation is a result of the pole condition 
\begin_inset Formula $\Delta_{H}^{-1}\left(m_{H}\right)=0$
\end_inset

.
 As discussed previously, numerical implementation of these equations is
 deferred to future work.
\end_layout

\begin_layout Section
Solution of the SI-3PI Hartree-Fock approximation
\begin_inset ERT
status open

\begin_layout Plain Layout

}{
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "sec:Solution-of-the-SI-3PI-HF-Approx"

\end_inset


\end_layout

\begin_layout Standard
In the Hartree-Fock approximation one drops the 
\begin_inset Formula $\mathcal{I}_{HG}\left(p\right)$
\end_inset

 term in the two loop equations of motion, or equivalently drops the sunset
 diagram.
 In this case the problem simplifies dramatically because the self-energy
 is momentum independent.
 The machinery of the auxiliary propagators introduced previously is now
 unnecessary and 
\begin_inset Formula $\mathcal{T}_{G/H}=\mathcal{T}_{G/H}^{\infty}+\mathcal{T}_{G/H}^{\text{fin}}$
\end_inset

 can be written as the sum of divergent and finite parts which can be evaluated
 in closed form.
 Renormalisation follows through almost exactly as in the unimproved and
 SI-2PI HF cases, with the resulting renormalisation constants (details
 in Appendix 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Appendix-Renormalisation-Details"

\end_inset

)
\begin_inset Formula 
\begin{align}
Z=Z_{\Delta} & =1,\\
\delta m_{1}^{2} & =-\frac{\left(N+2\right)\hbar\lambda\left(c_{0}\Lambda^{2}+c_{1}m^{2}\ln\left(\frac{\Lambda^{2}}{\mu^{2}}\right)\right)}{6+c_{1}\left(N+2\right)\hbar\lambda\ln\left(\frac{\Lambda^{2}}{\mu^{2}}\right)},\\
\delta\lambda_{1}^{A} & =-\frac{c_{1}\left(N+4\right)\hbar\lambda^{2}\ln\left(\frac{\Lambda^{2}}{\mu^{2}}\right)}{6+c_{1}\left(N+2\right)\hbar\lambda\ln\left(\frac{\Lambda^{2}}{\mu^{2}}\right)},\\
\delta\lambda_{2}^{A} & =\delta\lambda_{2}^{B}=\frac{N+2}{N+4}\delta\lambda_{1}^{A}.
\end{align}

\end_inset

Note that the undetermined constant 
\begin_inset Formula $\delta\lambda$
\end_inset

 can be consistently set to zero at this order.
 The finite equations of motion are 
\begin_inset Formula 
\begin{align}
m_{G}^{2} & =m^{2}+\frac{\lambda}{6}v^{2}+\frac{\hbar\lambda}{6}\left(N+1\right)\mathcal{T}_{G}^{\text{fin}}+\frac{\hbar\lambda}{6}\mathcal{T}_{H}^{\text{fin}},\label{eq:mg2-hartree-gap-eq}\\
m_{H}^{2} & =\frac{\lambda v^{2}}{3}+m_{G}^{2},\label{eq:mn2-hartree-gap-eq}\\
vm_{G}^{2} & =0.\label{eq:goldstone-theorem-hartree-gap-eq}
\end{align}

\end_inset

Finally, as before, requiring the zero temperature tree level relation 
\begin_inset Formula $v^{2}\left(T=0\right)\equiv\bar{v}^{2}=-6m^{2}/\lambda$
\end_inset

 sets the renormalisation point 
\begin_inset Formula $\mu^{2}=\bar{m}_{H}^{2}\equiv m_{H}^{2}\left(T=0\right)=\lambda\bar{v}^{2}/3$
\end_inset

.
 These are to be contrasted with the SI-2PI equations of motion 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:fin-SI-2PI-HF-G-eom"

\end_inset

-
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:fin-SI-2PI-HF-Goldstone-thm"

\end_inset

 reproduced here
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
m_{G}^{2} & =m^{2}+\frac{\lambda}{6}v^{2}+\frac{\hbar\lambda}{6}\left(N+1\right)\mathcal{T}_{G}^{\mathrm{fin}}+\frac{\hbar\lambda}{6}\mathcal{T}_{H}^{\mathrm{fin}},\label{eq:si2pi-goldstone-gap-eq}\\
m_{H}^{2} & =m^{2}+\frac{\lambda}{2}v^{2}+\frac{\hbar\lambda}{6}\left(N-1\right)\mathcal{T}_{G}^{\mathrm{fin}}+\frac{\hbar\lambda}{2}\mathcal{T}_{H}^{\mathrm{fin}},\label{eq:si2pi-higgs-gap-eq}\\
vm_{G}^{2} & =0.\label{eq:si2pi-goldstone-theorem}
\end{align}

\end_inset

Note that only the Higgs equation of motion differs from the SI-2PI case
 as expected.
 These equations of motion possess a phase transition and a critical point
 where 
\begin_inset Formula $m_{H}^{2}=m_{G}^{2}=v^{2}=0$
\end_inset

 with the same value of the critical temperature 
\begin_inset Formula 
\begin{equation}
T_{\star}=\sqrt{\frac{12\bar{v}^{2}}{\hbar\left(N+2\right)}},
\end{equation}

\end_inset

independent of the formalism used (apart from large 
\begin_inset Formula $N$
\end_inset

).
 However, the order of the phase transition differs in the three cases.
\end_layout

\begin_layout Standard
Numerical solutions of the SI-3PIEA in the Hartree-Fock approximation are
 presented in Figures 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:hartree-fock-vev-comparison"

\end_inset

, 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:hartree-fock-mn-comparison"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:hartree-fock-mg-comparison"

\end_inset

 with the same parameters as before, namely 
\begin_inset Formula $N=4$
\end_inset

, 
\begin_inset Formula $v=93\ \mathrm{MeV}$
\end_inset

 and 
\begin_inset Formula $\bar{m}_{H}=500\ \mathrm{MeV}$
\end_inset

.
 The solution is implemented in 
\noun on
Python
\noun default
 as an iterative root finder based on 
\emph on
scipy.optimize.root
\emph default
 
\begin_inset CommandInset citation
LatexCommand citep
key "Jones"

\end_inset

 with an estimated Jacobian or, if that fails to converge, a direct iteration
 of the gap equations.
 The Bose-Einstein integrals are precomputed to save time.
\end_layout

\begin_layout Standard
Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:hartree-fock-vev-comparison"

\end_inset

 shows 
\begin_inset Formula $v\left(T\right)$
\end_inset

, the order parameter of the phase transition.
 Below the critical temperature there is a broken phase with 
\begin_inset Formula $v\neq0$
\end_inset

, but the symmetry is restored when 
\begin_inset Formula $v=0$
\end_inset

 above the critical temperature.
 Note, however, that the unimproved 2PIEA and symmetry improved 3PIEA have
 unphysical metastable broken phases at 
\begin_inset Formula $T>T_{\star}$
\end_inset

, signalling a first order phase transition.
 The symmetry improved 2PIEA correctly predicts the second order nature
 of the phase transition.
 Though unphysical, the symmetry improved 3PIEA behaviour is much more reasonabl
e than the unimproved 2PIEA in that the strength of the first order phase
 transition is greatly reduced and the metastable phase ceases to exist
 at a temperature much closer to the critical temperature than for the unimprove
d 2PIEA.
 Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:hartree-fock-mn-comparison"

\end_inset

 shows the Higgs mass 
\begin_inset Formula $m_{H}\left(T\right)$
\end_inset

.
 The same phase transition behaviour is seen again, and again all three
 methods agree in the symmetric phase, giving the usual thermal mass effect.
 Finally, Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:hartree-fock-mg-comparison"

\end_inset

 shows the Goldstone boson mass.
 The unimproved 2PIEA strongly violates the Goldstone theorem, but both
 symmetry improvement methods satisfy it as expected.
 Note that the Goldstone theorem is even satisfied in the unphysical metastable
 phase predicted by the symmetry improved 3PIEA.
 All three methods correctly predict 
\begin_inset Formula $m_{G}=m_{H}$
\end_inset

 in the symmetric phase.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset space \hfill{}
\end_inset


\begin_inset Graphics
	filename images/ch4/vev-comparison.eps
	width 100col%

\end_inset


\begin_inset space \hfill{}
\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:hartree-fock-vev-comparison"

\end_inset

Expectation value of the scalar field 
\begin_inset Formula $v=\left\langle \phi\right\rangle $
\end_inset

 as a function of temperature 
\begin_inset Formula $T$
\end_inset

 computed in the Hartree-Fock approximation using the unimproved 2PIEA (solid
 black), the Pilaftsis and Teresi symmetry improved 2PIEA (dash dotted blue)
 and the symmetry improved 3PIEA (solid green).
 In the symmetric phase (dashed black) all methods agree.
 The vertical grey line at 
\begin_inset Formula $T\approx131.5\ \mathrm{MeV}$
\end_inset

 corresponds to the critical temperature which is the same in all methods.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset space \hfill{}
\end_inset


\begin_inset Graphics
	filename images/ch4/higgs-mass-comparison.eps
	width 100col%

\end_inset


\begin_inset space \hfill{}
\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:hartree-fock-mn-comparison"

\end_inset

The Higgs mass 
\begin_inset Formula $m_{H}$
\end_inset

 as a function of temperature 
\begin_inset Formula $T$
\end_inset

 computed in the Hartree-Fock approximation using the unimproved 2PIEA (solid
 black), the Pilaftsis and Teresi symmetry improved 2PIEA (dash dotted blue)
 and the symmetry improved 3PIEA (solid green).
 In the symmetric phase (dashed black) all methods agree.
 The vertical grey line at 
\begin_inset Formula $T\approx131.5\ \mathrm{MeV}$
\end_inset

 corresponds to the critical temperature which is the same in all methods.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset space \hfill{}
\end_inset


\begin_inset Graphics
	filename images/ch4/goldstone-mass-comparison.eps
	width 100col%

\end_inset


\begin_inset space \hfill{}
\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:hartree-fock-mg-comparison"

\end_inset

The Goldstone mass 
\begin_inset Formula $m_{G}$
\end_inset

 as a function of temperature 
\begin_inset Formula $T$
\end_inset

 computed in the Hartree-Fock approximation using the unimproved 2PIEA (solid
 black), the Pilaftsis and Teresi symmetry improved 2PIEA (dash dotted blue)
 and the symmetry improved 3PIEA (solid green).
 In the symmetric phase (dashed black) all methods agree.
 The vertical grey line at 
\begin_inset Formula $T\approx131.5\ \mathrm{MeV}$
\end_inset

 corresponds to the critical temperature which is the same in all methods.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Section
Two dimensions and the Coleman-Mermin-Wagner theorem
\begin_inset ERT
status open

\begin_layout Plain Layout

}{
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "sec:SI-3PIEA-Two-dimensions-and-CMW-Thm"

\end_inset


\end_layout

\begin_layout Standard
The Coleman-Mermin-Wagner theorem 
\begin_inset CommandInset citation
LatexCommand citep
key "Coleman1973a"

\end_inset

, which has been interpreted as a breakdown of the Goldstone theorem (e.g.
 
\begin_inset CommandInset citation
LatexCommand citep
key "Shifman2012"

\end_inset

), is a general result stating that the spontaneous breaking of a continuous
 symmetry is impossible in 
\begin_inset Formula $d=2$
\end_inset

 or 
\begin_inset Formula $d=1+1$
\end_inset

 dimensions.
 This occurs due to the infrared divergence of the massless scalar propagator
 in two dimensions.
 The symmetry improved gap equations satisfy this theorem despite the direct
 imposition of Goldstone's theorem.
 Thus symmetry improvement passes a test that any robust quantum field theoretic
al method must satisfy.
 (Note that symmetry improvement is not 
\emph on
required
\emph default
 to obtain consistency of 
\begin_inset Formula $n$
\end_inset

PIEA with the Coleman-Mermin-Wagner theorem, but neither does it ruin it.)
\end_layout

\begin_layout Standard
The general statement of the result is that 
\begin_inset Formula $\int_{x}\Sigma\left(x,0\right)$
\end_inset

 diverges in the IR whenever massless particles appear in loops in 
\begin_inset Formula $d=2$
\end_inset

.
 Since by the Dyson equation Goldstone's theorem can be written 
\begin_inset Formula $0=\left(m_{G}^{2}+\int_{x}\Sigma_{G}\left(x,0\right)\right)v,$
\end_inset

 this implies 
\begin_inset Formula $v=0$
\end_inset

 and a mass gap is generated.
 This can be shown explicitly using the Hartree-Fock gap equations where,
 in two dimensions, 
\begin_inset Formula 
\begin{equation}
\mathcal{T}_{a}^{\text{vac}}\left(\overline{\text{MS}}\right)=-\frac{1}{4\pi}\ln\left(\frac{m_{a}^{2}}{\mu^{2}}\right).
\end{equation}

\end_inset

(Note that the renormalisation can be carried through without difficulty
 in two dimensions.
 Only the 
\begin_inset Formula $\delta m_{1}^{2}$
\end_inset

 counter-term is needed.) One must show that the gap equations possess no
 solution for 
\begin_inset Formula $m_{G}^{2}=0$
\end_inset

.
 It is clear that if 
\begin_inset Formula $v\neq0$
\end_inset

, 
\begin_inset Formula $\mathcal{T}_{G}^{\text{vac}}$
\end_inset

 diverges as 
\begin_inset Formula $m_{G}^{2}\to0$
\end_inset

 if 
\begin_inset Formula $\mu$
\end_inset

 is taken as a constant, and 
\begin_inset Formula $\mathcal{T}_{H}^{\text{vac}}$
\end_inset

 diverges if one takes 
\begin_inset Formula $\mu^{2}\propto m_{G}^{2}$
\end_inset

 as 
\begin_inset Formula $m_{G}^{2}\to0$
\end_inset

.
 Either way there is no solution.
 At finite temperature the Bose-Einstein integral 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Bose-Einstein-thermal-tadpole"

\end_inset

 also has an infrared divergence as 
\begin_inset Formula $m_{a}\to0$
\end_inset

 which does not cancel against the singularity of the vacuum term.
 It can be shown that the singularity is due to the Matsubara zero mode.
\end_layout

\begin_layout Standard
For 
\begin_inset Formula $v=0$
\end_inset

 on the other hand, the gap equations reduce to 
\begin_inset Formula 
\begin{equation}
m_{H}^{2}=m_{G}^{2}=m^{2}-\frac{1}{4\pi}\frac{\hbar}{6}\left(N+2\right)\lambda\ln\left(\frac{m_{G}^{2}}{\mu^{2}}\right),
\end{equation}

\end_inset

which always has a positive solution.
 If 
\begin_inset Formula $m^{2}>0$
\end_inset

 then one can choose the renormalisation point 
\begin_inset Formula $\mu^{2}=m_{G}^{2}$
\end_inset

 so that the tree level relationship 
\begin_inset Formula $m_{G}^{2}=m^{2}$
\end_inset

 holds.
 If 
\begin_inset Formula $m^{2}<0$
\end_inset

 a positive mass is dynamically generated and one requires a renormalisation
 point 
\begin_inset Formula $\mu^{2}>m_{G}^{2}\exp\left(\frac{24\pi\left|m^{2}\right|}{\hbar\left(N+2\right)\lambda}\right)$
\end_inset

 non-perturbatively large in the ratio 
\begin_inset Formula $\lambda/\left|m^{2}\right|$
\end_inset

, reflecting the fact that perturbation theory is bound to fail in this
 case.
\end_layout

\begin_layout Section
Optical theorem and dispersion relations
\begin_inset ERT
status open

\begin_layout Plain Layout

}{
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "sec:SI-3PIEA-Optical-theorem-and-dispersion-relations"

\end_inset


\end_layout

\begin_layout Standard
In this section the analytic structure of propagators and self-energies
 is studied in the symmetry improved 3PI formalism.
 A physical quantity of particular interest is the decay width 
\begin_inset Formula $\Gamma_{H}$
\end_inset

 of the Higgs, which is dominated by decays to two Goldstone bosons.
 
\begin_inset Formula $\Gamma_{H}$
\end_inset

 is given by the optical theorem in terms of the imaginary part of the self-ener
gy evaluated on-shell (see, e.g.
 
\begin_inset CommandInset citation
LatexCommand citep
key "Peskin1995"

\end_inset

 Chapter 7): 
\begin_inset Formula 
\begin{equation}
-m_{H}\Gamma_{H}=\mathrm{Im}\Sigma_{H}\left(m_{H}\right).
\end{equation}

\end_inset

(This is valid so long as 
\begin_inset Formula $\Gamma_{H}\ll m_{H}$
\end_inset

, otherwise the full energy dependence of 
\begin_inset Formula $\Sigma_{H}\left(p\right)$
\end_inset

 must be taken into account.) The standard one loop perturbative result gives
 (see, e.g., the 
\begin_inset Quotes eld
\end_inset

Kinematics
\begin_inset Quotes erd
\end_inset

 review in 
\begin_inset CommandInset citation
LatexCommand citep
key "Olive2014"

\end_inset

)
\begin_inset Formula 
\begin{equation}
\Gamma_{H}=\frac{N-1}{2}\frac{\hbar}{16\pi m_{H}}\left(\frac{\lambda v}{3}\right)^{2},
\end{equation}

\end_inset

which comes entirely from the Goldstone loop sunset graph.
 Each part of this expression has a simple interpretation in relation to
 the tree level decay graph
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{fmffile}{Higgs-decay-tree-graph}
\end_layout

\begin_layout Plain Layout


\backslash
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\end_layout

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\backslash
parbox{20mm}{
\backslash
begin{fmfgraph*}(40,20) 
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\backslash
fmflabel{$H$}{i} 
\backslash
fmflabel{$G$}{o1} 
\backslash
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\backslash
end{fmfgraph*}}.
\end_layout

\begin_layout Plain Layout


\backslash
end{equation*}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmffile}
\end_layout

\end_inset

The 
\begin_inset Formula $N-1$
\end_inset

 is due to the sum over final state Goldstone flavours, the factor of 
\begin_inset Formula $1/2$
\end_inset

 is due to the Bose statistics of the two particles in the final state,
 the 
\begin_inset Formula $\hbar/16\pi m_{H}$
\end_inset

 is due to the final state phase space integration and the 
\begin_inset Formula $\left(\lambda v/3\right)^{2}$
\end_inset

 is the absolute square of the invariant decay amplitude.
\end_layout

\begin_layout Standard
The Hartree-Fock approximation fails to reproduce this result regardless
 of the use or not of symmetry improvement.
 This is because there is no self-energy apart from a mass correction.
 Thus the Hartree-Fock approximation always predicts that the Higgs is stable.
 Attempts to repair the Hartree-Fock approximation through the use of an
 external propagator lead to a non-zero but still incorrect result.
 This is because an unphysical value of 
\begin_inset Formula $m_{G}$
\end_inset

 still appears in loops.
 Satisfactory results are obtained within the symmetry improved 2PI formalism
 for both on- and off-shell Higgs 
\begin_inset CommandInset citation
LatexCommand citep
key "Pilaftsis2013"

\end_inset

 (so long as the two loop solutions exist).
 Here it is shown that the symmetry improved 3PIEA can not yield a satisfactory
 value for 
\begin_inset Formula $\Gamma_{H}$
\end_inset

 at the two loop level.
\end_layout

\begin_layout Standard
From the Higgs equation of motion 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:three-loop-Higgs-eom-finite"

\end_inset

 the two loop truncation gives 
\begin_inset Formula 
\begin{align}
\mathrm{Im}\Sigma_{H}\left(p\right) & =\frac{\hbar}{6}\left(N+1\right)\lambda\mathrm{Im}\mathcal{T}_{G}^{\text{fin}}+\frac{\hbar}{6}\lambda\mathrm{Im}\mathcal{T}_{H}^{\text{fin}}-\hbar\left(\frac{\lambda v}{3}\right)^{2}\mathrm{Im}\left[i\mathcal{I}_{HG}^{\text{fin}}\left(p\right)\right]\nonumber \\
 & =\frac{\hbar}{6}\left(N+1\right)\lambda\mathrm{Im}\mathcal{T}_{G}+\frac{\hbar}{6}\lambda\mathrm{Im}\mathcal{T}_{H}-\hbar\left(\frac{\lambda v}{3}\right)^{2}\mathrm{Im}\left[i\mathcal{I}_{HG}\left(p\right)\right],
\end{align}

\end_inset

which can be written in terms of the un-subtracted 
\begin_inset Formula $\mathcal{T}_{G/H}$
\end_inset

 and 
\begin_inset Formula $\mathcal{I}_{HG}$
\end_inset

 because all of the subtractions are manifestly real.
 Now it is shown that 
\begin_inset Formula $\mathrm{Im}\mathcal{T}_{G/H}=0$
\end_inset

.
 To do this introduce the K
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
"a
\end_layout

\end_inset

ll
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

n-Lehmann spectral representation of the propagators 
\begin_inset CommandInset citation
LatexCommand citep
key "Weinberg2005"

\end_inset

 
\begin_inset Formula 
\begin{equation}
\Delta_{G/H}\left(q\right)=\int_{0}^{\infty}\mathrm{d}s\frac{\rho_{G/H}\left(s\right)}{q^{2}-s+i\epsilon},
\end{equation}

\end_inset

where the spectral densities 
\begin_inset Formula $\rho_{G/H}\left(s\right)$
\end_inset

 are real and positive for 
\begin_inset Formula $s\geq0$
\end_inset

 and obey the sum rule
\begin_inset Formula 
\begin{equation}
\int_{0}^{\infty}\mathrm{d}s\rho_{G/H}\left(s\right)=\left(ZZ_{\Delta}\right)^{-1}=1,
\end{equation}

\end_inset

where the last equality holds at two loop order (the standard formula has
 been adapted for the current renormalisation scheme)
\begin_inset Foot
status collapsed

\begin_layout Plain Layout
Note that this standard theory actually conflicts with the asymptotic 
\begin_inset Formula $p^{2}\left(\ln p^{2}\right)^{d_{1}}$
\end_inset

 form assumed for the self-energy when the 
\begin_inset Formula $\Phi_{5}$
\end_inset

 graph is included, so that this argument must be refined at the three loop
 level.
 The essential problem is that the self-consistent 
\begin_inset Formula $n$
\end_inset

PI propagator is not resumming large logarithms.
 However, it seems unlikely that a refinement of the argument to account
 for this fact will change the qualitative conclusions of this section since,
 as will be shown shortly, the predicted 
\begin_inset Formula $\Gamma_{H}$
\end_inset

 is wrong by group theory factors in addition to the 
\begin_inset Formula $\mathcal{O}\left(1\right)$
\end_inset

 factors which could be compensated by a modification of 
\begin_inset Formula $\rho_{G/H}$
\end_inset

.
\end_layout

\end_inset

.
\end_layout

\begin_layout Standard
Then 
\begin_inset Formula 
\begin{equation}
\mathrm{Im}\int_{q}i\int_{0}^{\infty}\mathrm{d}\mu^{2}\frac{\rho_{G/H}\left(\mu^{2}\right)}{q^{2}-\mu^{2}+i\epsilon}=\mathrm{Im}\int_{0}^{\infty}\mathrm{d}\mu^{2}\rho_{G/H}\left(\mu^{2}\right)\mathcal{T}^{\mu}=0.
\end{equation}

\end_inset

This allows one to obtain a dispersion relation relating the real and imaginary
 parts of the self-energies 
\begin_inset Formula 
\begin{align}
0 & =\mathrm{Im}\int_{q}i\frac{1}{q^{2}-m_{G/H}^{2}-\Sigma_{G/H}\left(q\right)}\nonumber \\
 & =\int_{q}\frac{q^{2}-m_{G/H}^{2}-\mathrm{Re}\Sigma_{G/H}\left(q\right)}{\left[q^{2}-m_{G/H}^{2}-\mathrm{Re}\Sigma_{G/H}\left(q\right)\right]^{2}+\left[\mathrm{Im}\Sigma_{G/H}\left(q\right)\right]^{2}}.
\end{align}

\end_inset


\end_layout

\begin_layout Standard
Finally one must compute 
\begin_inset Formula $\mathrm{Im}\left[i\mathcal{I}_{HG}\left(p\right)\right]$
\end_inset

, which can be written
\begin_inset Formula 
\begin{align}
\mathrm{Im}\left[i\mathcal{I}_{HG}\left(p\right)\right] & =\mathrm{Im}i\int_{q}\int_{0}^{\infty}\mathrm{d}s_{1}\int_{0}^{\infty}\mathrm{d}s_{2}\frac{i\rho_{H}\left(s_{1}\right)}{q^{2}-s_{1}+i\epsilon}\frac{i\rho_{G}\left(s_{2}\right)}{\left(p-q\right)^{2}-s_{2}+i\epsilon}\nonumber \\
 & =\mathrm{Im}i\int_{0}^{\infty}\mathrm{d}s_{1}\int_{0}^{\infty}\mathrm{d}s_{2}\rho_{H}\left(s_{1}\right)\rho_{G}\left(s_{2}\right)\int_{q}\frac{i}{q^{2}-s_{1}+i\epsilon}\frac{i}{\left(p-q\right)^{2}-s_{2}+i\epsilon}\nonumber \\
 & =\frac{1}{16\pi^{2}}\int_{0}^{\infty}\mathrm{d}s_{1}\int_{0}^{\infty}\mathrm{d}s_{2}\rho_{H}\left(s_{1}\right)\rho_{G}\left(s_{2}\right)\nonumber \\
 & \times\mathrm{Im}\int_{0}^{1}\mathrm{d}x\ln\left(\frac{\mu^{2}}{-x\left(1-x\right)p^{2}+xs_{1}+\left(1-x\right)s_{2}-i\epsilon}\right).
\end{align}

\end_inset

The imaginary part of the 
\begin_inset Formula $x$
\end_inset

 integral is only non-zero for 
\begin_inset Formula $\sqrt{s_{1}}+\sqrt{s_{2}}<\sqrt{p^{2}}$
\end_inset

.
 Denote the region of 
\begin_inset Formula $s_{1,2}$
\end_inset

 integration by 
\begin_inset Formula $\Omega$
\end_inset

.
 Then the imaginary part of the 
\begin_inset Formula $x$
\end_inset

 integral can be evaluated straightforwardly, giving
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\mathrm{Im}\left[i\mathcal{I}_{HG}\left(p\right)\right] & =\frac{1}{16\pi p^{2}}\int_{\Omega}\mathrm{d}s_{1}\mathrm{d}s_{2}\rho_{H}\left(s_{1}\right)\rho_{G}\left(s_{2}\right)\nonumber \\
 & \times\sqrt{p^{2}-\left(\sqrt{s_{1}}+\sqrt{s_{2}}\right)^{2}}\sqrt{p^{2}-\left(\sqrt{s_{1}}-\sqrt{s_{2}}\right)^{2}}.
\end{align}

\end_inset


\end_layout

\begin_layout Standard
Now, since the each term of the integrand is positive and the square root
 is 
\begin_inset Formula $\leq p^{2}$
\end_inset

, 
\begin_inset Formula 
\begin{equation}
\mathrm{Im}\left[i\mathcal{I}_{HG}\left(p\right)\right]\leq\frac{1}{16\pi}\int_{\Omega}\mathrm{d}s_{1}\mathrm{d}s_{2}\rho_{H}\left(s_{1}\right)\rho_{G}\left(s_{2}\right)\leq\frac{1}{16\pi},
\end{equation}

\end_inset

using the sum rule for 
\begin_inset Formula $\rho_{H/G}\left(s\right)$
\end_inset

.
 Finally, 
\begin_inset Formula 
\begin{equation}
\Gamma_{H}\leq\frac{\hbar}{16\pi m_{H}}\left(\frac{\lambda v}{3}\right)^{2},
\end{equation}

\end_inset

which is smaller than the expected value for all 
\begin_inset Formula $N>3$
\end_inset

.
 For the 
\begin_inset Formula $N=2$
\end_inset

 and 
\begin_inset Formula $3$
\end_inset

 cases one could possibly obtain an (accidentally) reasonable result, depending
 on the precise form of the spectral functions, but it is clear that one
 should not generically expect a correct prediction of 
\begin_inset Formula $\Gamma_{H}$
\end_inset

 from the symmetry improved 3PIEA at two loops.
 The source of the problem is the derivation of the two loop truncation
 where the vertex correction term is dropped, resulting in a truncation
 of the true Ward identity that keeps the one loop graphs in 
\begin_inset Formula $\Sigma_{G}$
\end_inset

 but not in 
\begin_inset Formula $\bar{V}$
\end_inset

.
 The diagram contributing to 
\begin_inset Formula $\Gamma_{H}$
\end_inset

 above is thus the sunset Goldstone self-energy
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{fmffile}{sunset-g-sigma}
\end_layout

\begin_layout Plain Layout


\backslash
begin{equation*}
\end_layout

\begin_layout Plain Layout


\backslash
parbox{20mm}{
\backslash
begin{fmfgraph*}(40,40) 
\backslash
fmfleft{i}
\backslash
fmfright{o} 
\backslash
fmf{phantom}{i,v1,v2,o}
\backslash
fmffreeze 
\backslash
fmf{plain,left=1,label=$G$}{v1,v2} 
\backslash
fmf{plain,right=1,label=$H$}{v1,v2} 
\backslash
fmf{plain}{i,v1}
\backslash
fmf{plain}{o,v2} 
\backslash
fmfdot{v1,v2} 
\backslash
fmflabel{$G$}{i} 
\backslash
fmflabel{$G$}{o} 
\backslash
end{fmfgraph*}},
\end_layout

\begin_layout Plain Layout


\backslash
end{equation*}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmffile}
\end_layout

\end_inset

which has the incorrect kinematics and lacks the required group theory 
\begin_inset Formula $\left(N-1\right)$
\end_inset

 and Bose symmetry 
\begin_inset Formula $\left(1/2\right)$
\end_inset

 factors as well.
 In fact, a perturbative evaluation of this diagram gives 
\begin_inset Formula $\Gamma_{H}=0$
\end_inset

 due to the threshold at 
\begin_inset Formula $p^{2}=m_{H}^{2}$
\end_inset

! What has been shown here is that no matter the form of the exact spectral
 functions, there cannot be a non-perturbative enhancement of this graph
 large enough to give the correct 
\begin_inset Formula $\Gamma_{H}$
\end_inset

 for 
\begin_inset Formula $N>3$
\end_inset

.
 The neglected vertex corrections give a leading 
\begin_inset Formula $\mathcal{O}\left(\hbar\right)$
\end_inset

 contribution to 
\begin_inset Formula $\Gamma_{H}$
\end_inset

 which must be included.
\end_layout

\begin_layout Standard
Now consider the three loop truncation of the symmetry improved 3PIEA.
 Since one loop vertex corrections appear at this order one expects that
 
\begin_inset Formula $\Gamma_{H}$
\end_inset

 should be correct at least to 
\begin_inset Formula $\mathcal{O}\left(\hbar\right)$
\end_inset

.
 Since the previous result was incorrect by group theory factors already
 at 
\begin_inset Formula $\mathcal{O}\left(\hbar\right)$
\end_inset

 one is motivated in seeking only the 
\begin_inset Formula $\mathcal{O}\left(\hbar\right)$
\end_inset

 decay width at first, delaying the requirement for numerical computation
 as much as possible.
 Thus one can make use of only the one loop terms in the Higgs equation
 of motion, which are
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{fmffile}{higg-one-loop-self-energy}
\end_layout

\begin_layout Plain Layout


\backslash
begin{align*}
\end_layout

\begin_layout Plain Layout


\backslash
parbox{15mm}{
\backslash
begin{fmfgraph*}(40,40) 
\backslash
fmfleft{i}
\backslash
fmfright{o} 
\backslash
fmf{plain}{i,v,v,o} 
\backslash
fmfdot{v} 
\backslash
fmflabel{$G$}{i} 
\backslash
fmflabel{$G$}{o} 
\backslash
end{fmfgraph*}}&&
\end_layout

\begin_layout Plain Layout


\backslash
parbox{15mm}{
\backslash
begin{fmfgraph*}(40,40) 
\backslash
fmfleft{i}
\backslash
fmfright{o} 
\backslash
fmf{phantom}{i,v1,v2,o}
\backslash
fmffreeze 
\backslash
fmf{plain,left=1}{v1,v2} 
\backslash
fmf{plain,right=1}{v1,v2} 
\backslash
fmf{plain}{i,v1}
\backslash
fmf{plain}{o,v2} 
\backslash
fmfdot{v1,v2} 
\backslash
fmflabel{$G$}{i} 
\backslash
fmflabel{$G$}{o} 
\backslash
end{fmfgraph*}}&&
\end_layout

\begin_layout Plain Layout


\backslash
parbox{20mm}{
\backslash
begin{fmfgraph*}(40,40) 
\backslash
fmfleft{i}
\backslash
fmfright{o}
\backslash
fmfbottom{b} 
\backslash
fmf{phantom}{i,v1,v2,o}
\backslash
fmffreeze 
\backslash
fmf{plain}{i,v1,v2,o} 
\backslash
fmf{plain}{i,v1}
\backslash
fmf{plain}{o,v2} 
\backslash
fmfdot{v1,v2} 
\backslash
fmfv{decor.shape=cross,decor.filled=full,decor.size=2thick}{b} 
\backslash
fmf{plain,tension=2}{b,v3} 
\backslash
fmf{plain}{v1,v3,v2} 
\backslash
fmfdot{v3} 
\backslash
fmflabel{$H$}{i} 
\backslash
fmflabel{$G$}{o} 
\backslash
end{fmfgraph*}}
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
parbox{15mm}{
\backslash
begin{fmfgraph*}(40,40) 
\backslash
fmfleft{i}
\backslash
fmfright{o}
\backslash
fmfbottom{b} 
\backslash
fmf{phantom}{i,v1,o}
\backslash
fmffreeze 
\backslash
fmf{plain}{i,v1,o} 
\backslash
fmfv{decor.shape=cross,decor.filled=full,decor.size=2thick}{b} 
\backslash
fmf{plain,tension=2}{b,v3} 
\backslash
fmf{plain,left=1}{v1,v3,v1} 
\backslash
fmfdot{v1,v3} 
\backslash
fmflabel{$H$}{i} 
\backslash
fmflabel{$G$}{o} 
\backslash
end{fmfgraph*}}&&
\end_layout

\begin_layout Plain Layout


\backslash
parbox{15mm}{
\backslash
begin{fmfgraph*}(40,40) 
\backslash
fmfleft{i}
\backslash
fmfright{o} 
\backslash
fmf{phantom}{i,v1,v2,o}
\backslash
fmffreeze 
\backslash
fmf{plain,left=1}{v1,v2} 
\backslash
fmf{plain,right=1}{v1,v2} 
\backslash
fmf{plain}{i,v1}
\backslash
fmf{plain}{o,v2} 
\backslash
fmfdot{v1,v2} 
\backslash
fmflabel{$H$}{i} 
\backslash
fmflabel{$G$}{o} 
\backslash
fmfbottom{b} 
\backslash
fmf{plain}{b,v2} 
\backslash
fmfv{decor.shape=cross,decor.filled=full,decor.size=2thick}{b} 
\backslash
end{fmfgraph*}}&&
\end_layout

\begin_layout Plain Layout


\backslash
parbox{20mm}{
\backslash
begin{fmfgraph*}(40,40) 
\backslash
fmfleft{i}
\backslash
fmfright{o} 
\backslash
fmf{phantom}{i,v1,v2,o}
\backslash
fmffreeze 
\backslash
fmf{plain,left=1}{v1,v2} 
\backslash
fmf{plain,right=1}{v1,v2} 
\backslash
fmf{plain}{i,v1}
\backslash
fmf{plain}{o,v2} 
\backslash
fmfdot{v1,v2} 
\backslash
fmflabel{$H$}{i} 
\backslash
fmflabel{$G$}{o} 
\backslash
fmfbottom{b} 
\backslash
fmf{plain}{b,v1} 
\backslash
fmfv{decor.shape=cross,decor.filled=full,decor.size=2thick}{b} 
\backslash
end{fmfgraph*}}.
\end_layout

\begin_layout Plain Layout


\backslash
end{align*}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmffile}
\end_layout

\end_inset

The crossed vertices represent the injection of a zero momentum Goldstone
 boson with amplitude 
\begin_inset Formula $v$
\end_inset

.
 Furthermore, by iterating the equations of motion one may replace all propagato
rs and vertices by their perturbative values to 
\begin_inset Formula $\mathcal{O}\left(\hbar\right)$
\end_inset

.
 This will leave out contributions of higher order decay processes such
 as 
\begin_inset Formula $H\to GGGG$
\end_inset

.
 Evaluating the decay width predicted by the SI-3PIEA more accurately requires
 a numerical solution of the equations of motion and so is left to future
 work.
\end_layout

\begin_layout Standard
The contributions of the various terms to the imaginary part of 
\begin_inset Formula $\Sigma_{H}$
\end_inset

 can be determined using Cutkosky cutting rules 
\begin_inset CommandInset citation
LatexCommand citep
key "Peskin1995"

\end_inset

.
 In particular, the Hartree-Fock diagram and the first bubble vertex correction
 diagram (left diagram, bottom row above) have no cuts such that all cut
 lines can be put on shell.
 Also, cuts through intermediate states with both Goldstone and Higgs lines
 contribute to the process 
\begin_inset Formula $H\to HG$
\end_inset

, which is prohibited due to the unbroken 
\begin_inset Formula $\mathrm{O}\left(N-1\right)$
\end_inset

 symmetry and anyway vanishes due to the zero phase space at threshold.
 This means one can drop the sunset diagram and the last bubble vertex correctio
n (right diagram, bottom row).
 Similarly cuts through two intermediate Higgs lines can be dropped since
 
\begin_inset Formula $H\to HH$
\end_inset

 is impossible on shell.
 This means one can drop the contributions to the triangle and remaining
 bubble diagram where the leftmost vertex is 
\begin_inset Formula $V_{N}$
\end_inset

 rather than 
\begin_inset Formula $\bar{V}$
\end_inset

.
 The relevant contributions can now be displayed explicitly:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
-\Sigma_{H} & \supset\bar{V}v\nonumber \\
 & \supset v\left[i\hbar\left(-\frac{\lambda v}{3}\right)^{3}\int_{\ell}\frac{1}{\left(\ell-p\right)^{2}-m_{G}^{2}+i\epsilon}\frac{1}{\ell^{2}-m_{G}^{2}+i\epsilon}\frac{1}{\ell^{2}-m_{H}^{2}+i\epsilon}\right.\nonumber \\
 & +\left.\frac{i\hbar\lambda}{6}\left(N+1\right)\left(-\frac{\lambda v}{3}\right)\int_{\ell}\frac{1}{\left(\ell-p\right)^{2}-m_{G}^{2}+i\epsilon}\frac{1}{\ell^{2}-m_{G}^{2}+i\epsilon}\right],
\end{align}

\end_inset

where the first and second term are the triangle and bubble graphs respectively.
 Now cut the Goldstone lines by replacing each cut propagator 
\begin_inset Formula 
\begin{equation}
\left(p^{2}-m_{G}^{2}+i\epsilon\right)^{-1}\to-2\pi i\delta\left(p^{2}-m_{G}^{2}\right),
\end{equation}

\end_inset

to give 
\begin_inset Formula $-2i\mathrm{Im}\Sigma_{H}$
\end_inset

 (because the cutting rules give the 
\emph on
discontinuity
\emph default
 of the diagram, which is 
\begin_inset Formula $2i$
\end_inset

 times the imaginary part), yielding
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
-2i\mathrm{Im}\Sigma_{H} & \supset-i\hbar v\left[\left(-\frac{\lambda v}{3}\right)^{3}\frac{1}{-m_{H}^{2}}+\frac{\lambda}{6}\left(N+1\right)\left(-\frac{\lambda v}{3}\right)\right]\int_{\ell}2\pi\delta\left(\left(\ell-p\right)^{2}\right)2\pi\delta\left(\ell^{2}\right)\nonumber \\
 & =i\hbar\frac{\lambda^{2}v^{2}}{3^{2}2}\left(N-1\right)\int_{\ell}2\pi\delta\left(\left(\ell-p\right)^{2}\right)2\pi\delta\left(\ell^{2}\right),
\end{align}

\end_inset


\end_layout

\begin_layout Standard
The integral can be evaluated by elementary techniques, giving
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\int_{\ell}2\pi\delta\left(\left(\ell-p\right)^{2}\right)2\pi\delta\left(\ell^{2}\right) & =\frac{1}{4\pi^{2}}\int\mathrm{d}^{4}\ell\delta\left(\ell^{2}-2\ell\cdot p+p^{2}\right)\delta\left(\ell^{2}\right)\nonumber \\
 & =\frac{1}{4\pi^{2}}\int\mathrm{d}\ell_{0}\mathrm{d}l4\pi l^{2}\delta\left(-2\ell_{0}m_{H}+m_{H}^{2}\right)\delta\left(\ell_{0}^{2}-l^{2}\right)\nonumber \\
 & =\frac{1}{\pi}\int\mathrm{d}ll^{2}\frac{1}{2m_{H}}\frac{\delta\left(\frac{m_{H}}{2}-l\right)}{2\frac{m_{H}}{2}}\nonumber \\
 & =\frac{1}{8\pi},
\end{align}

\end_inset

and finally
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
-\mathrm{Im}\Sigma_{H}\left(m_{H}\right) & =\frac{N-1}{2}\frac{\hbar}{16\pi}\left(\frac{\lambda v}{3}\right)^{2}+\mathcal{O}\left(\hbar^{2}\right).
\end{align}

\end_inset

This exactly matches the expected 
\begin_inset Formula $\Gamma_{H}$
\end_inset

, including group theory and Bose symmetry factors.
 The full non-perturbative solution will give corrections to this accounting
 for loop corrections as well as cascade decay processes 
\begin_inset Formula $H\to GG\to\left(GG\right)^{2}\to\cdots$
\end_inset

.
 While the full evaluation of 
\begin_inset Formula $\Gamma_{H}$
\end_inset

 is left to future work as discussed, it has been shown that the one loop
 vertex corrections are required to get the correct 
\begin_inset Formula $\Gamma_{H}$
\end_inset

 at leading order.
\end_layout

\begin_layout Section
Summary
\begin_inset ERT
status open

\begin_layout Plain Layout

}{
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "sec:ch4-Summary"

\end_inset


\end_layout

\begin_layout Standard
The symmetry improvement formalism of Pilaftsis and Teresi is able to enforce
 the preservation of global symmetries in two particle irreducible effective
 actions, allowing among other things the accurate description of phase
 transitions in strongly coupled theories with spontaneously broken global
 symmetries.
 It restores Goldstone's theorem and produces physically reasonable absorptive
 parts in propagators 
\begin_inset CommandInset citation
LatexCommand citep
key "Pilaftsis2013"

\end_inset

 and has been shown to restore the second order phase transition of the
 
\begin_inset Formula $\mathrm{O}\left(4\right)$
\end_inset

 linear sigma model in the Hartree-Fock approximation 
\begin_inset CommandInset citation
LatexCommand citep
key "Mao2014"

\end_inset

.
 It has been used to study pion strings evolving in the thermal bath of
 a heavy ion collision 
\begin_inset CommandInset citation
LatexCommand citep
key "Lu2015"

\end_inset

 and has been demonstrated to improve the evaluation of the effective potential
 of the standard model by taming the infrared divergences of the Higgs sector,
 treated as an 
\begin_inset Formula $\mathrm{O}\left(4\right)$
\end_inset

 scalar field theory with gauge interactions turned off 
\begin_inset CommandInset citation
LatexCommand citep
key "Pilaftsis2015,Pilaftsis2015a"

\end_inset

.
\end_layout

\begin_layout Standard
However, the development of a first principles non-perturbative kinetic
 theory for the gauge theories of real physical interest requires the use
 of 
\begin_inset Formula $n$
\end_inset

-particle irreducible effective actions with 
\begin_inset Formula $n\geq3$
\end_inset

 
\begin_inset CommandInset citation
LatexCommand citep
key "Carrington2009,Smolic2012,Berges2004"

\end_inset

.
 A step in this direction has been achieved by extending the symmetry improvemen
t formalism to the 3PIEA for a scalar field theory with spontaneous breaking
 of a global 
\begin_inset Formula $\mathrm{O}\left(N\right)$
\end_inset

 symmetry.
 In the SI-3PIEA formalism an extra Ward identity involving the vertex function
 must be imposed.
 Since the constraints are singular this requires a careful consideration
 of the variational procedure, namely one must be careful to impose constraints
 in a way that satisfies d'Alembert's principle.
 Once this is done the theory can be renormalised in a more or less standard
 way, though the counter-terms differ in value from the unimproved case.
 The finite equations of motion and counter-terms for the Hartree-Fock truncatio
n, two loop truncation, and three loop truncation of the effective action
 were found.
\end_layout

\begin_layout Standard
Several important qualitative results have been found already in this investigat
ion.
 First, symmetry improvement breaks the equivalence hierarchy of 
\begin_inset Formula $n$
\end_inset

PIEA.
 Second, the numerical solution of the Hartree-Fock truncation gave mixed
 results: Goldstone's theorem was satisfied, but the order of the phase
 transition was incorrectly predicted to be weakly first order (though there
 was still a large quantitative improvement over the unimproved 2PIEA case).
 Third, the two loop truncation incorrectly predicts the Higgs decay width
 as a consequence of the optical theorem, though the three loop truncation
 gives the correct value, at least to 
\begin_inset Formula $\mathcal{O}\left(\hbar\right)$
\end_inset

.
 These results could be considered strong circumstantial evidence that one
 should not apply symmetry improvement to 
\begin_inset Formula $n$
\end_inset

PIEA at a truncation to less than 
\begin_inset Formula $n$
\end_inset

 loops.
 One could test this conjecture further by, for example, computing the symmetry
 improved 4PIEA.
 The prediction is that unsatisfactory results of some kind will be found
 for any truncation of this to 
\begin_inset Formula $<4$
\end_inset

 loops.
\end_layout

\begin_layout Standard
The results presented here are limited in several ways due to the lack of
 numerical solutions for the full SI-3PI equations of motion.
 The results presented are for the Hartree-Fock truncation, which completely
 misses vertex corrections and therefore most of the physical content of
 the 3PI scheme.
 In other words, the results presented are essentially those for the SI-2PI
 effective action with the Higgs propagator equation of motion replaced
 by a truncated version of the vertex Ward identity.
 There is therefore no basis, given the results presented here, to conclude
 that the SI-3PI is problematic or not once the vertex corrections are taken
 into account.
 Further limitations consider the renormalisation procedure.
 The renormalisation of the theory at two and three loops was performed
 in vacuum.
 The only finite temperature computation performed here was for the Hartree-Fock
 approximation.
 The extension of the two or three loop truncations to finite temperature,
 or an extension to non-equilibrium situations, will require a much heavier
 numerical effort than has yet been attempted and so falls beyond the scope
 of the present thesis.
 Likewise, it would be interesting to compare the self-consistent Higgs
 decay rate in the symmetry improved 2PI and 3PI formalisms.
 Similarly, analytical results for the renormalisation of the three loop
 truncation were only presented in 
\begin_inset Formula $1+2$
\end_inset

 dimensions, since the renormalisation was not analytically tractable in
 
\begin_inset Formula $1+3$
\end_inset

 dimensions.
 Because these investigations involve a numerical effort substantially greater
 than anything else in this thesis they are left to future work.
\end_layout

\begin_layout Standard
The general renormalisation theory presented here, based on counter-terms,
 is difficult to use in practice.
 It would be interesting to see if symmetry improvement could work along
 with the counter-term-free functional renormalisation group approach 
\begin_inset CommandInset citation
LatexCommand citep
key "Carrington2015"

\end_inset

.
 Such an approach may not be easier to set up in the first instance, but
 once developed would likely be easier to extend to higher loop order and
 
\begin_inset Formula $n$
\end_inset

 than the current method.
 Of course it will be interesting to see if this work can be extended to
 gauge symmetries and, eventually, the standard model of particle physics.
 If successful, such an effort could serve to open a new window to the non-pertu
rbative physics of these theories in high temperature, high density and
 strong coupling regimes.
 In an attempt to address some of problems with the SI-
\begin_inset Formula $n$
\end_inset

PIEA, a novel method of 
\begin_inset Quotes eld
\end_inset

soft
\begin_inset Quotes erd
\end_inset

 symmetry improvement is studied in the next chapter.
 Following that, as a stepping stone towards non-equilibrium applications,
 the theory of the linear response of a system to external perturbations
 is studied in the SI-2PIEA formalism in Chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:chap6"

\end_inset

.
\end_layout

\end_body
\end_document