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\begin_body

\begin_layout Chapter
Soft Symmetry Improvement
\begin_inset CommandInset label
LatexCommand label
name "chap:chap5"

\end_inset


\end_layout

\begin_layout Section
Synopsis
\begin_inset CommandInset label
LatexCommand label
name "sec:ch5-Synopsis"

\end_inset


\end_layout

\begin_layout Standard
Chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:chap4"

\end_inset

 investigated the symmetry improvement formalism for 
\begin_inset Formula $n$
\end_inset

PIEA, an attempt to improve the symmetry properties of solutions of non-perturba
tive equations of motion by directly imposing the appropriate Ward identities
 as constraints.
 As discussed there are some successes attributable to this method.
 However, there are also pathologies.
 In particular: truncations of SI-
\begin_inset Formula $n$
\end_inset

PIEA may not admit solutions, too low order truncations of SI-
\begin_inset Formula $n$
\end_inset

PIEA (
\begin_inset Formula $\mathcal{O}\left(\hbar^{l}\right)$
\end_inset

 with 
\begin_inset Formula $l<n$
\end_inset

) are not fully self-consistent due to over-imposed constraints, and, as
 will be shown in Chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:chap6"

\end_inset

, solutions of SI-
\begin_inset Formula $n$
\end_inset

PI equations of motion likely do not exist beyond equilibrium (certainly
 not in the linear response approximation) because violations of 
\begin_inset Formula $>n$
\end_inset

th order WIs feed back into the solutions for the 
\begin_inset Formula $\leq n$
\end_inset

th order correlation functions.
 All of this suggests that symmetry improvement 
\emph on
over-imposes
\emph default
 the WIs.
 This motivates the investigation of methods which only enforce WIs approximatel
y rather than exactly.
 The idea is that the extra freedom may allow solutions to exist while violation
s of the WIs may be acceptably small phenomenologically depending on the
 application one has in mind.
 This chapter investigates a novel method called 
\emph on
soft symmetry improvement
\emph default
 (SSI) with this aim.
\end_layout

\begin_layout Standard
The idea of soft symmetry improvement is to relax the WI constraint of the
 symmetry improvement method.
 Instead, the WI is enforced 
\emph on
softly
\emph default
 in the sense of least squares: a solution is found which minimises at the
 same time violations of the WI and the violation of the unimproved equations
 of motion.
 The relative cost of WI violations is controlled by a new 
\emph on
stiffness
\emph default
 parameter 
\begin_inset Formula $\xi$
\end_inset

, such that 
\begin_inset Formula $\xi\to0\left(\infty\right)$
\end_inset

 reduces to the SI(unimproved)-2PIEA respectively.
 The motivating hope behind the investigation is that for some range of
 finite values of 
\begin_inset Formula $\xi$
\end_inset

 the extra freedom to allow small violations of the WIs leads to a formalism
 which is phenomenologically useful in the sense that physical quantities
 can be computed to some desired accuracy fixed by the particular application
 in mind.
 It is shown in this chapter that this program cannot be fully realised,
 at least in the infinite volume / low temperature limit, due to the infrared
 sensitivity of the SSI-2PIEA: the limit is a subtle one and the only consistent
 limiting procedures reduce to either the unimproved or SI-2PIEA, or to
 a novel limit which has pathological properties of its own.
\end_layout

\begin_layout Standard
This chapter formulates the SSI-2PIEA, then renormalises and solves the
 resulting equations of motion in the Hartree-Fock approximation in thermal
 equilibrium.
 The infinite volume / low temperature limit (
\begin_inset Formula $\mathrm{V}\beta\to\infty$
\end_inset

) is examined carefully and three consistent limiting procedures are found.
 Two reduce to previous work (the unimproved and SI-2PIEAs respectively)
 but the third is new.
 The properties of this new limit are examined in equilibrium at finite
 and zero temperature.
 All of this work is new and forms the basis of a paper by the author 
\begin_inset CommandInset citation
LatexCommand citep
key "Brown2016"

\end_inset

.
\end_layout

\begin_layout Section
Soft Symmetry Improvement of 2PIEA
\begin_inset CommandInset label
LatexCommand label
name "sec:ch5-SSI-2PIEA"

\end_inset


\end_layout

\begin_layout Standard
The simplest way to arrive at the definition of the SSI-2PIEA is to start
 with the standard 2PIEA 
\begin_inset Formula $\Gamma\left[\varphi,\Delta\right]$
\end_inset

 (suppressing indices and spacetime arguments where these just clutter)
 and the trivial identity
\begin_inset Formula 
\begin{equation}
\exp\left(\frac{i}{\hbar}\Gamma\left[\varphi,\Delta\right]\right)=\int\mathcal{D}\phi\delta\left(\phi-\varphi\right)\exp\left(\frac{i}{\hbar}\Gamma\left[\phi,\Delta\right]\right).
\end{equation}

\end_inset

The usual symmetry improved action 
\begin_inset Formula $\Gamma^{\mathrm{SI}}\left[\varphi,\Delta\right]$
\end_inset

 is then defined by inserting a delta function
\begin_inset Formula 
\begin{equation}
\exp\left(\frac{i}{\hbar}\Gamma^{\mathrm{SI}}\left[\varphi,\Delta\right]\right)=N\int\mathcal{D}\phi\delta\left(\phi-\varphi\right)\exp\left(\frac{i}{\hbar}\Gamma\left[\phi,\Delta\right]\right)\delta\left(\mathcal{W}\left[\phi,\Delta\right]\right)
\end{equation}

\end_inset

where 
\begin_inset Formula $\mathcal{W}\left[\phi,\Delta\right]=0$
\end_inset

 is the Ward identity and the normalisation factor 
\begin_inset Formula $N$
\end_inset

 is chosen so that 
\begin_inset Formula $\Gamma^{\mathrm{SI}}\left[\varphi,\Delta\right]$
\end_inset

 numerically equals 
\begin_inset Formula $\Gamma\left[\varphi,\Delta\right]$
\end_inset

 when the arguments satisfy the Ward identity
\begin_inset Foot
status collapsed

\begin_layout Plain Layout
Formally 
\begin_inset Formula $N=\left[\delta\left(0\right)\right]^{-1}$
\end_inset

 though it is not necessary to worry about rigorously defining this here.
\end_layout

\end_inset


\begin_inset Foot
status collapsed

\begin_layout Plain Layout
Note that if one wants invariance under redefinitions of 
\begin_inset Formula $\mathcal{W}$
\end_inset

 then one also needs to insert a factor of 
\begin_inset Formula $\mathrm{Det}\left(\frac{\delta\mathcal{W}}{\delta\phi}\right)$
\end_inset

.
 This can be handled straightforwardly in the functional formalism through
 the introduction of Faddeev-Popov ghost fields
\begin_inset CommandInset citation
LatexCommand citep
key "Zinn-Justin1990,Peskin1995,Srednicki2007"

\end_inset

.
 However, this is not necessary here because there is no strong reason to
 consider transformations of 
\begin_inset Formula $\mathcal{W}$
\end_inset

.
\end_layout

\end_inset

.
 
\begin_inset Formula $\Gamma^{\mathrm{SI}}\left[\varphi,\Delta\right]$
\end_inset

 is defined only for field configurations satisfying the Ward identity,
 and equals the usual effective action on those configurations.
 Thus 
\begin_inset Formula $\Gamma^{\mathrm{SI}}\left[\varphi,\Delta\right]$
\end_inset

 is nothing but the SI-2PIEA as discussed in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:chap4"

\end_inset

 arrived at in a new way.
\end_layout

\begin_layout Standard
It seems reasonable that the problems encountered with PT symmetry improvement
 are due to the singular nature of the constraint surface, as embodied by
 the delta function above.
 Due to these problems a 
\emph on
soft symmetry improved
\emph default
 (SSI) effective action 
\begin_inset Formula $\Gamma_{\xi}^{\mathrm{SSI}}\left[\varphi,\Delta\right]$
\end_inset

 can be introduced where the Ward identity is no longer strictly enforced.
 Small violations 
\begin_inset Formula $\mathcal{W}\neq0$
\end_inset

 are allowed but punished in the functional integral.
 A new free parameter controls how strictly the constraint is enforced.
 The hope is that the added freedom allows consistent solutions with non-trivial
 dynamics (e.g.
 linear response to external sources), while the stiffness can be tuned
 to make violations of the Ward identity acceptably small in practice.
 To achieve this replace the delta function by a smoothed version 
\begin_inset Formula $\delta\left(\mathcal{W}\right)\to\delta_{\xi}\left(\mathcal{W}\right)$
\end_inset

 defined as follows
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\exp\left(\frac{i}{\hbar}\Gamma_{\xi}^{\mathrm{SSI}}\left[\varphi,\Delta\right]\right) & =N_{0}\int\mathcal{D}\phi\delta\left(\phi-\varphi\right)\exp\left(\frac{i}{\hbar}\Gamma\left[\phi,\Delta\right]\right)\delta_{\xi}\left(\mathcal{W}\left[\phi,\Delta\right]\right)\nonumber \\
 & =N_{1}\int\mathcal{D}\left[\phi,\lambda_{\phi},\lambda_{W}\right]\exp\left(\frac{i}{\hbar}\left[\lambda_{\phi}\left(\phi-\varphi\right)+\Gamma\left[\phi,\Delta\right]+\lambda_{W}\mathcal{W}-\frac{1}{2}\xi\lambda_{W}^{2}\right]\right)\nonumber \\
 & =N_{2}\int\mathcal{D}\left[\phi,\lambda_{\phi}\right]\exp\left(\frac{i}{\hbar}\left[\lambda_{\phi}\left(\phi-\varphi\right)+\Gamma\left[\phi,\Delta\right]+\frac{1}{2\xi}\mathcal{W}^{2}\right]\right)\nonumber \\
 & =\exp\left(\frac{i}{\hbar}\left[\Gamma\left[\varphi,\Delta\right]+\frac{1}{2\xi}\mathcal{W}^{2}\left[\varphi,\Delta\right]\right]\right).
\end{align}

\end_inset

The first line is a formal expression that is defined by the next line.
 The Fourier representation of the delta functions are used to replace 
\begin_inset Formula $\delta\left(\phi-\varphi\right)\to\int\mathcal{D}\lambda_{\phi}\exp\frac{i}{\hbar}\lambda_{\phi}\left(\phi-\varphi\right)$
\end_inset

 etc.
 The 
\begin_inset Formula $\frac{1}{2}\xi\lambda_{W}^{2}$
\end_inset

 term is the one responsible for smoothing the delta function, with the
 limit 
\begin_inset Formula $\xi\to0$
\end_inset

 corresponding to a stiffening of the constraint.
 In the third line the integral over 
\begin_inset Formula $\lambda_{W}$
\end_inset

, which is Gaussian, is performed.
 Finally, the integral over 
\begin_inset Formula $\lambda_{\phi}$
\end_inset

 yields a delta function which kills the 
\begin_inset Formula $\phi$
\end_inset

 integral, resulting in
\begin_inset Formula 
\begin{equation}
\Gamma_{\xi}^{\mathrm{SSI}}\left[\varphi,\Delta\right]=\Gamma\left[\varphi,\Delta\right]+\frac{1}{2\xi}\mathcal{W}^{2}\left[\varphi,\Delta\right].
\end{equation}

\end_inset

In a slight generalisation of this derivation one can use the smoothing
 term 
\begin_inset Formula $-\frac{1}{2}\xi\lambda_{W}R^{-1}\lambda_{W}$
\end_inset

 where 
\begin_inset Formula $R^{-1}$
\end_inset

 is an arbitrary positive definite symmetric kernel which may depend on
 
\begin_inset Formula $\varphi$
\end_inset

 and 
\begin_inset Formula $\Delta$
\end_inset

, which gives
\begin_inset Formula 
\begin{equation}
\Gamma_{\xi R}^{\mathrm{SSI}}\left[\varphi,\Delta\right]=\Gamma\left[\varphi,\Delta\right]+\frac{1}{2\xi}\mathcal{W}R\mathcal{W}-\frac{i\hbar}{2}\mathrm{Tr}\ln R.
\end{equation}

\end_inset

This is the most general form of the definition of the SSI-2PIEA.
 The simpler form 
\begin_inset Formula $\Gamma_{\xi}^{\mathrm{SSI}}\left[\varphi,\Delta\right]$
\end_inset

 corresponds to a trivial kernel (now with indices explicit) 
\begin_inset Formula 
\begin{equation}
R_{ab}^{AB}\left(x,y\right)=\delta^{AB}\delta_{ab}\delta\left(x-y\right),
\end{equation}

\end_inset

which is used exclusively in the following, though the freedom to choose
 a non-trivial 
\begin_inset Formula $R$
\end_inset

 in the definition of the SSI effective action may be useful in certain
 circumstances.
 The end result is simply that 
\begin_inset Formula $\mathcal{W}=0$
\end_inset

 is enforced in the sense of (possibly weighted if 
\begin_inset Formula $R$
\end_inset

 is non-trivial) least-squared error, rather than as a strict constraint.
\end_layout

\begin_layout Standard
Taking as the SSI equations of motion 
\begin_inset Formula $\delta\Gamma_{\xi}^{\mathrm{SSI}}=0$
\end_inset

, it is straightforward to derive the equations of motion, now including
 indices:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\frac{\delta\Gamma\left[\varphi,\Delta\right]}{\delta\varphi_{a}} & =-\frac{1}{\xi}\mathcal{W}_{c}^{A}\left[\varphi,\Delta\right]\frac{\delta}{\delta\varphi_{a}}\mathcal{W}_{c}^{A}\left[\varphi,\Delta\right]\nonumber \\
 & =-\frac{1}{\xi}\left(\Delta_{cf}^{-1}T_{fg}^{A}\varphi_{g}\right)\Delta_{cd}^{-1}T_{da}^{A},\\
\frac{\delta\Gamma\left[\varphi,\Delta\right]}{\delta\Delta_{ab}} & =-\frac{1}{\xi}\mathcal{W}_{c}^{A}\left[\varphi,\Delta\right]\frac{\delta}{\delta\Delta_{ab}}\mathcal{W}_{c}^{A}\left[\varphi,\Delta\right]\nonumber \\
 & =\frac{1}{\xi}\left(\Delta_{cf}^{-1}T_{fg}^{A}\varphi_{g}\right)\Delta_{ca}^{-1}\left(\Delta_{bd}^{-1}T_{de}^{A}\varphi_{e}\right).
\end{align}

\end_inset

Now the spontaneous symmetry breaking ansatz 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:SSB-ansatz-Delta"

\end_inset

, reproduced here
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\varphi_{a} & =v\delta_{aN},\\
\Delta_{ab}^{-1} & =\begin{cases}
\Delta_{G}^{-1} & a=b\neq N,\\
\Delta_{H}^{-1} & a=b=N,
\end{cases}
\end{align}

\end_inset

can be used, yielding
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\frac{\delta\Gamma\left[\varphi,\Delta\right]}{\delta\varphi_{g}\left(x\right)} & =0,\ \left(g\neq N\right),\\
\frac{\delta\Gamma\left[\varphi,\Delta\right]}{\delta\varphi_{N}\left(x\right)} & =\frac{1}{\xi}2\left(N-1\right)v\int_{yz}\Delta_{G}^{-1}\left(y,z\right)\Delta_{G}^{-1}\left(y,x\right)\nonumber \\
 & =\frac{1}{\xi}2\left(N-1\right)vm_{G}^{4},\\
\frac{\delta\Gamma\left[\varphi,\Delta\right]}{\delta\Delta_{G}\left(x,y\right)} & =-\frac{1}{\xi}2v^{2}\int_{wrz}\Delta_{G}^{-1}\left(w,r\right)\Delta_{G}^{-1}\left(w,x\right)\Delta_{G}^{-1}\left(y,z\right)\nonumber \\
 & =\frac{1}{\xi}2v^{2}m_{G}^{6},\\
\frac{\delta\Gamma\left[\varphi,\Delta\right]}{\delta\Delta_{H}} & =0.
\end{align}

\end_inset


\end_layout

\begin_layout Standard
Note that if one takes 
\begin_inset Formula $\xi\to0$
\end_inset

 proportionally to 
\begin_inset Formula $vm_{G}^{4}$
\end_inset

 one obtains for the non-trivial right hand sides above 
\begin_inset Formula $2\left(N-1\right)vm_{G}^{4}/\xi\to\text{constant}$
\end_inset

 and 
\begin_inset Formula $2v^{2}m_{G}^{6}/\xi\to\left(\text{const.}\right)\times vm_{G}^{2}\to0$
\end_inset

 and one recovers the usual SI-2PIEA scheme in the limit (c.f.
 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:SI-2PIEA-H-eom-general"

\end_inset

-
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:SI-2PIEA-G-eom-general"

\end_inset

).
 In fact it is shown through a careful treatment of the infinite volume
 limit in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Broken-Phase-with-massless-Goldstones"

\end_inset

 that this limit is not quite correct, but there does exist a scaling limit
 with 
\begin_inset Formula $\xi\to0$
\end_inset

 which reduces to the SI-2PIEA.
 This confirms the intuition that 
\begin_inset Formula $\xi\to0$
\end_inset

 approaches hard symmetry improvement and that 
\begin_inset Formula $\Gamma_{\xi}^{\mathrm{SSI}}\left[\varphi,\Delta\right]\to\Gamma^{\mathrm{SI}}\left[\varphi,\Delta\right]$
\end_inset

 which really is just the standard symmetry improved effective action.
 In the next sections these equations of motion are renormalised and solved
 in the Hartree-Fock approximation.
\end_layout

\begin_layout Section
Renormalisation of the Hartree-Fock truncation
\begin_inset CommandInset label
LatexCommand label
name "sec:ch5-Renormalisation-of-the-HF-truncation"

\end_inset


\end_layout

\begin_layout Standard
Since it turns out that the SSI method is sensitive to the 
\begin_inset Formula $\mathrm{V}\beta\to\infty$
\end_inset

 limit the theory will be formulated in Euclidean spacetime (i.e.
 the Matsubara formalism) in a box of volume 
\begin_inset Formula $\mathrm{V}=L^{3}$
\end_inset

 with periodic boundary conditions of period 
\begin_inset Formula $L$
\end_inset

 in the space directions and 
\begin_inset Formula $\beta$
\end_inset

 in the time 
\begin_inset Formula $\tau=it$
\end_inset

 direction.
 This mirrors the treatment of finite temperature scalar fields in section
 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Time-contours-Green-functions-NEQFT"

\end_inset

.
 The Euclidean continuation leads to 
\begin_inset Formula $\partial_{\mu}\partial^{\mu}\to-\nabla^{2}$
\end_inset

, 
\begin_inset Formula $\int_{x}\to-i\int_{x_{E}}$
\end_inset

 and the conventions
\begin_inset Formula 
\begin{align}
f\left(x_{E}\right) & =\frac{1}{\mathrm{V}\beta}\sum_{n,\boldsymbol{k}}\mathrm{e}^{i\left(\omega_{n}\tau+\boldsymbol{k}\cdot\boldsymbol{x}\right)}f\left(n,\boldsymbol{k}\right),\\
f\left(n,\boldsymbol{q}\right) & =\int_{x_{E}}\mathrm{e}^{-i\left(\omega_{m}\tau+\boldsymbol{q}\cdot\boldsymbol{x}\right)}f\left(x_{E}\right),
\end{align}

\end_inset

for Fourier transforms.
 This leads to the Fourier transforms of the propagators
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\Delta_{G/H}\left(x,y\right) & =\frac{1}{\mathrm{V}\beta}\sum_{n,\boldsymbol{k}}\mathrm{e}^{i\left(\omega_{n}\left(\tau_{x}-\tau_{y}\right)+\boldsymbol{k}\cdot\left(\boldsymbol{x}-\boldsymbol{y}\right)\right)}\Delta_{G/H}\left(n,\boldsymbol{k}\right),\\
\Delta_{G/H}^{-1}\left(x,y\right) & =\frac{1}{\mathrm{V}\beta}\sum_{n,\boldsymbol{k}}\mathrm{e}^{i\left(\omega_{n}\left(\tau_{x}-\tau_{y}\right)+\boldsymbol{k}\cdot\left(\boldsymbol{x}-\boldsymbol{y}\right)\right)}\Delta_{G/H}^{-1}\left(n,\boldsymbol{k}\right),
\end{align}

\end_inset

and
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\Delta_{G/H}\left(n,\boldsymbol{k}\right)=\int_{x-y}\mathrm{e}^{-i\left(\omega_{n}\left(\tau_{x}-\tau_{y}\right)+\boldsymbol{k}\cdot\left(\boldsymbol{x}-\boldsymbol{y}\right)\right)}\Delta_{G/H}\left(x,y\right),
\end{equation}

\end_inset

etc., where the Matsubara frequencies are 
\begin_inset Formula $\omega_{n}=2\pi n/\beta$
\end_inset

 and the wave vectors 
\begin_inset Formula $\boldsymbol{k}$
\end_inset

 are discretised on a lattice of spacing 
\begin_inset Formula $2\pi/L$
\end_inset

.
 The four dimensional Euclidean shorthand 
\begin_inset Formula $k_{E}=\left(\omega_{n},\boldsymbol{k}\right)$
\end_inset

 is often useful.
 
\end_layout

\begin_layout Standard
It is straightforward to modify the real time 2PIEA of chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:chap2"

\end_inset

 to the Euclidean formalism.
 The results are collected here for easy reference.
 The defining functional integral of the field theory is the partition function
\begin_inset Formula 
\begin{equation}
Z\left[J,K\right]=\int\mathcal{D}\left[\phi\right]\exp\left(-S_{E}\left[\phi\right]-J_{a}\phi_{a}-\frac{1}{2}\phi_{a}K_{ab}\phi_{b}\right),
\end{equation}

\end_inset

where
\begin_inset Formula 
\begin{equation}
S_{E}\left[\phi\right]=\int_{x}\frac{1}{2}\left(\nabla\phi_{a}\right)^{2}+\frac{1}{2}m^{2}\phi_{a}\phi_{a}+\frac{1}{4!}\lambda\left(\phi_{a}\phi_{a}\right)^{2},
\end{equation}

\end_inset

is the Euclidean action.
 Then 
\begin_inset Formula $W\left[J,K\right]=-\ln Z\left[J,K\right]$
\end_inset

 is the connected generating functional and
\begin_inset Formula 
\begin{equation}
\Gamma\left[\varphi,\Delta\right]=W-J\frac{\delta W}{\delta J}-K\frac{\delta W}{\delta K},
\end{equation}

\end_inset

is the standard 2PIEA once 
\begin_inset Formula $J$
\end_inset

 and 
\begin_inset Formula $K$
\end_inset

 are eliminated in terms of 
\begin_inset Formula $\varphi$
\end_inset

 and 
\begin_inset Formula $\Delta$
\end_inset

 using
\begin_inset Formula 
\begin{align}
\frac{\delta W}{\delta J_{a}} & =\left\langle \phi_{a}\right\rangle =\varphi_{a},\\
\frac{\delta W}{\delta K_{ab}} & =\frac{1}{2}\left\langle \phi_{a}\phi_{b}\right\rangle =\frac{1}{2}\left(\Delta_{ab}+\varphi_{a}\varphi_{b}\right).
\end{align}

\end_inset

As before, the Legendre transform can be evaluated by the saddle point method,
 which results in
\begin_inset Formula 
\begin{equation}
\Gamma=S_{E}\left[\varphi\right]+\frac{1}{2}\mathrm{Tr}\ln\left(\Delta^{-1}\right)+\frac{1}{2}\mathrm{Tr}\left(\Delta_{0}^{-1}\Delta-1\right)+\Gamma_{2},
\end{equation}

\end_inset

where 
\begin_inset Formula $\Gamma_{2}$
\end_inset

 is the set of two particle irreducible graphs and 
\begin_inset Formula $\Delta_{0}^{-1}=\delta^{2}S_{E}/\delta\phi\delta\phi$
\end_inset

 is the unperturbed propagator
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\Delta_{0ab}^{-1}=\left(-\nabla^{2}+m^{2}+\frac{1}{6}\lambda\varphi^{2}\right)\delta_{ab}+\frac{1}{3}\lambda\varphi_{a}\varphi_{b}.
\end{equation}

\end_inset

To two loop order
\begin_inset Formula 
\begin{align}
\Gamma_{2} & =\frac{1}{4!}\lambda\Delta_{aa}\Delta_{bb}+\frac{1}{12}\lambda\Delta_{ab}\Delta_{ab}-\frac{1}{36}\lambda^{2}\varphi_{b}\Delta_{ac}\Delta_{ac}\Delta_{bd}\varphi_{d}-\frac{1}{18}\lambda^{2}\varphi_{b}\Delta_{ac}\Delta_{ad}\Delta_{bc}\varphi_{d}+\cdots.
\end{align}

\end_inset


\end_layout

\begin_layout Standard
To form 
\begin_inset Formula $\Gamma_{\xi}^{\mathrm{SSI}}$
\end_inset

 one adds the soft-symmetry improvement term 
\begin_inset Formula $-\frac{1}{2\xi}\mathcal{W}^{2}$
\end_inset

 where the Ward identity 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:1PI-Delta-WI"

\end_inset

 is
\begin_inset Formula 
\begin{equation}
\mathcal{W}_{a}^{A}=\Delta_{ab}^{-1}T_{bc}^{A}\varphi_{c}.
\end{equation}

\end_inset

(Note that 
\begin_inset Formula $\mathcal{W}$
\end_inset

 is pure imaginary due to the 
\begin_inset Formula $i$
\end_inset

 in 
\begin_inset Formula $T^{A}$
\end_inset

 so 
\begin_inset Formula $-\mathcal{W}^{2}$
\end_inset

 is positive definite.) Using the spontaneous symmetry breaking ansatz and
 inserting the ten renormalisation constants 
\begin_inset Formula $Z$
\end_inset

, 
\begin_inset Formula $Z_{\Delta}$
\end_inset

, 
\begin_inset Formula $\delta m_{0,1}^{2}$
\end_inset

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

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

, 
\begin_inset Formula $\delta\lambda$
\end_inset

 (c.f.
 section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Renormalisation-and-solution-2PI-HF"

\end_inset

) and a new one 
\begin_inset Formula $Z_{\xi}$
\end_inset

 for 
\begin_inset Formula $\xi$
\end_inset

 gives the renormalised SSI effective action:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\Gamma_{\xi}^{\mathrm{SSI}}\left[\varphi,\Delta\right] & =\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)\nonumber \\
 & +\frac{1}{2}\left(N-1\right)\mathrm{Tr}\ln\left(Z^{-1}Z_{\Delta}^{-1}\Delta_{G}^{-1}\right)+\frac{1}{2}\mathrm{Tr}\ln\left(Z^{-1}Z_{\Delta}^{-1}\Delta_{H}^{-1}\right)\nonumber \\
 & +\frac{1}{2}\left(N-1\right)\mathrm{Tr}\left[\left(-ZZ_{\Delta}\nabla^{2}+m^{2}+\delta m_{1}^{2}+Z_{\Delta}\frac{\lambda+\delta\lambda_{1}^{A}}{6}v^{2}\right)\Delta_{G}\right]\nonumber \\
 & +\frac{1}{2}\mathrm{Tr}\left[\left(-ZZ_{\Delta}\nabla^{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}\right]\nonumber \\
 & +\Gamma_{2}+Z_{\xi}^{-1}\frac{1}{\xi}\left(N-1\right)v^{2}Z^{-1}Z_{\Delta}^{-2}\int_{xyz}\Delta_{G}^{-1}\left(x,y\right)\Delta_{G}^{-1}\left(x,z\right)
\end{align}

\end_inset

with
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\Gamma_{2} & =\frac{1}{4!}Z_{\Delta}^{2}\left(\lambda+\delta\lambda_{2}^{A}\right)\Delta_{aa}\Delta_{bb}+\frac{1}{12}Z_{\Delta}^{2}\left(\lambda+\delta\lambda_{2}^{B}\right)\Delta_{ab}\Delta_{ab}\nonumber \\
 & -\frac{1}{36}Z_{\Delta}^{3}\left(\lambda+\delta\lambda\right)^{2}\varphi_{b}\Delta_{ac}\Delta_{ac}\Delta_{bd}\varphi_{d}-\frac{1}{18}Z_{\Delta}^{3}\left(\lambda+\delta\lambda\right)^{2}\varphi_{b}\Delta_{ac}\Delta_{ad}\Delta_{bc}\varphi_{d}+\cdots\\
 & =\frac{1}{4!}Z_{\Delta}^{2}\left(N-1\right)\left[\left(N+1\right)\lambda+\left(N-1\right)\delta\lambda_{2}^{A}+2\delta\lambda_{2}^{B}\right]\Delta_{G}\Delta_{G}\nonumber \\
 & +\frac{1}{4!}Z_{\Delta}^{2}\left(\lambda+\delta\lambda_{2}^{A}\right)2\left(N-1\right)\Delta_{G}\Delta_{H}+\frac{1}{4!}Z_{\Delta}^{2}\left[3\lambda+\delta\lambda_{2}^{A}+2\delta\lambda_{2}^{B}\right]\Delta_{H}\Delta_{H}\nonumber \\
 & -\frac{1}{36}\left(N-1\right)Z_{\Delta}^{3}\left(\lambda+\delta\lambda\right)^{2}v^{2}\Delta_{G}\Delta_{G}\Delta_{H}-\frac{1}{12}Z_{\Delta}^{3}\left(\lambda+\delta\lambda\right)^{2}v^{2}\Delta_{H}\Delta_{H}\Delta_{H}+\cdots.
\end{align}

\end_inset

This can be simplified using the mode expansions for 
\begin_inset Formula $\Delta_{G/H}$
\end_inset

 and doing the integrals, giving
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\Gamma_{\xi}^{\mathrm{SSI}}\left[\varphi,\Delta\right] & =\mathrm{V}\beta\left(Z_{\Delta}^{-1}\frac{m^{2}+\delta m_{0}^{2}}{2}v^{2}+\frac{\lambda+\delta\lambda_{0}}{4!}v^{4}\right)+\frac{1}{2}N\mathrm{V}\beta\ln\left(Z^{-1}Z_{\Delta}^{-1}\right)\nonumber \\
 & +\frac{1}{2}\left(N-1\right)\sum_{n,\boldsymbol{k}}\ln\frac{1}{\Delta_{G}\left(n,\boldsymbol{k}\right)}+\frac{1}{2}\sum_{n,\boldsymbol{k}}\ln\frac{1}{\Delta_{H}\left(n,\boldsymbol{k}\right)}\nonumber \\
 & +\frac{1}{2}\left(N-1\right)\sum_{n,\boldsymbol{k}}\left(ZZ_{\Delta}k_{E}^{2}+m^{2}+\delta m_{1}^{2}+Z_{\Delta}\frac{\lambda+\delta\lambda_{1}^{A}}{6}v^{2}\right)\Delta_{G}\left(n,\boldsymbol{k}\right)\nonumber \\
 & +\frac{1}{2}\sum_{n,\boldsymbol{k}}\left(ZZ_{\Delta}k_{E}^{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(n,\boldsymbol{k}\right)\nonumber \\
 & +\Gamma_{2}+Z_{\xi}^{-1}\frac{1}{\xi}\left(N-1\right)v^{2}Z^{-1}Z_{\Delta}^{-2}\mathrm{V}\beta\left[\Delta_{G}^{-1}\left(0,\boldsymbol{0}\right)\right]^{2},
\end{align}

\end_inset

noting that the integrals in the SSI term give
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\int_{xyz}\Delta_{G}^{-1}\left(x,y\right)\Delta_{G}^{-1}\left(x,z\right)=\mathrm{V}\beta\left[\Delta_{G}^{-1}\left(0,\boldsymbol{0}\right)\right]^{2}.
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Now one can make a basic consistency check by examining the tree level equations
 of motion, which are (setting renormalisation constants to their trivial
 values)
\begin_inset Formula 
\begin{align}
0 & =\mathrm{V}\beta v\left\{ \left(m^{2}+\frac{\lambda}{6}v^{2}\right)+\frac{2\left(N-1\right)}{\xi}\left[\Delta_{G}^{-1}\left(0,\boldsymbol{0}\right)\right]^{2}\right\} ,\\
\frac{1}{\Delta_{G}\left(n,\boldsymbol{k}\right)} & =k_{E}^{2}+m^{2}+\frac{\lambda}{6}v^{2},\ n,\boldsymbol{k}\neq0,\\
\Delta_{G}^{-1}\left(0,\boldsymbol{0}\right) & =m^{2}+\frac{\lambda}{6}v^{2}-\frac{4\mathrm{V}\beta}{\xi}v^{2}\left[\Delta_{G}^{-1}\left(0,\boldsymbol{0}\right)\right]^{3},\\
\frac{1}{\Delta_{H}\left(n,\boldsymbol{k}\right)} & =k_{E}^{2}+m^{2}+\frac{\lambda}{2}v^{2}.
\end{align}

\end_inset

Indeed these have 
\begin_inset Formula $v^{2}=-6m^{2}/\lambda$
\end_inset

, 
\begin_inset Formula $\Delta_{G}^{-1}\left(n,\boldsymbol{k}\right)=k_{E}^{2}$
\end_inset

 and 
\begin_inset Formula $\Delta_{H}^{-1}\left(n,\boldsymbol{k}\right)=k_{E}^{2}+m_{H}^{2}=k_{E}^{2}+\frac{\lambda}{3}v^{2}$
\end_inset

 as a consistent solution as expected.
 Since these equations are self-consistent there is a valid concern about
 possible spurious solutions.
 This can be investigated by solving the 
\begin_inset Formula $v$
\end_inset

 and 
\begin_inset Formula $\Delta_{G}^{-1}\left(0,\boldsymbol{0}\right)$
\end_inset

 equations together on the assumption that 
\begin_inset Formula $v^{2}\neq0,-6m^{2}/\lambda$
\end_inset

.
 Using the 
\begin_inset Formula $v$
\end_inset

 equation to reduce the degree of the 
\begin_inset Formula $\Delta_{G}^{-1}\left(0,\boldsymbol{0}\right)$
\end_inset

 to first order gives the potential spurious solution
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
-\frac{\xi}{2\left(N-1\right)} & =\left(m^{2}+\frac{\lambda}{6}v^{2}\right)/\left[1-\frac{2\mathrm{V}\beta}{N-1}v^{2}\left(m^{2}+\frac{\lambda}{6}v^{2}\right)\right]^{2},\\
\Delta_{G}^{-1}\left(0,\boldsymbol{0}\right) & =\left(m^{2}+\frac{\lambda}{6}v^{2}\right)/\left[1-\frac{2\mathrm{V}\beta}{N-1}v^{2}\left(m^{2}+\frac{\lambda}{6}v^{2}\right)\right].
\end{align}

\end_inset

However, the condition that there are no tachyons (i.e.
 excitations with 
\begin_inset Formula $m^{2}<0$
\end_inset

 indicating the instability of the vacuum) requires 
\begin_inset Formula $\Delta_{G}^{-1}\left(0,\boldsymbol{0}\right)\geq0$
\end_inset

 which implies
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
0\leq m^{2}+\frac{\lambda}{6}v^{2}<\frac{N-1}{2\mathrm{V}\beta v^{2}},
\end{equation}

\end_inset

which then implies that the right hand side of the first equation is positive,
 but then the left hand side 
\begin_inset Formula $\propto-\xi$
\end_inset

 is negative leading to a contradiction.
 Thus the only spurious solutions are tachyonic and so easily dismissable.
\end_layout

\begin_layout Standard
At the Hartree-Fock level
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\Gamma_{2} & =\frac{1}{4!}Z_{\Delta}^{2}\left(N-1\right)\left[\left(N+1\right)\lambda+\left(N-1\right)\delta\lambda_{2}^{A}+2\delta\lambda_{2}^{B}\right]\frac{1}{\mathrm{V}\beta}\sum_{n,\boldsymbol{k}}\Delta_{G}\left(n,\boldsymbol{k}\right)\sum_{m,\boldsymbol{q}}\Delta_{G}\left(m,\boldsymbol{q}\right)\nonumber \\
 & +\frac{1}{4!}Z_{\Delta}^{2}\left(\lambda+\delta\lambda_{2}^{A}\right)2\left(N-1\right)\frac{1}{\mathrm{V}\beta}\sum_{n,\boldsymbol{k}}\Delta_{G}\left(n,\boldsymbol{k}\right)\sum_{m,\boldsymbol{q}}\Delta_{H}\left(m,\boldsymbol{q}\right)\nonumber \\
 & +\frac{1}{4!}Z_{\Delta}^{2}\left[3\lambda+\delta\lambda_{2}^{A}+2\delta\lambda_{2}^{B}\right]\frac{1}{\mathrm{V}\beta}\sum_{n,\boldsymbol{k}}\Delta_{H}\left(n,\boldsymbol{k}\right)\sum_{m,\boldsymbol{q}}\Delta_{H}\left(m,\boldsymbol{q}\right).
\end{align}

\end_inset

Further, since 
\begin_inset Formula $\xi$
\end_inset

 is so far arbitrary, there is no loss of generality in setting 
\begin_inset Formula $Z_{\xi}=Z_{\Delta}^{-1}$
\end_inset

.
 The resulting equations of motion are for the vev:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
0 & =\mathrm{V}\beta\left(Z_{\Delta}^{-1}\frac{m^{2}+\delta m_{0}^{2}}{2}2v+\frac{\lambda+\delta\lambda_{0}}{4!}4v^{3}\right)+\frac{1}{2}\left(N-1\right)\left(Z_{\Delta}\frac{\lambda+\delta\lambda_{1}^{A}}{6}2v\right)\sum_{n,\boldsymbol{k}}\Delta_{G}\left(n,\boldsymbol{k}\right)\nonumber \\
 & +\frac{1}{2}\left(Z_{\Delta}\frac{3\lambda+\delta\lambda_{1}^{A}+2\delta\lambda_{1}^{B}}{6}2v\right)\sum_{n,\boldsymbol{k}}\Delta_{H}\left(n,\boldsymbol{k}\right)+\frac{1}{\xi}\left(N-1\right)2vZ^{-1}Z_{\Delta}^{-1}\mathrm{V}\beta\left[\Delta_{G}^{-1}\left(0,\boldsymbol{0}\right)\right]^{2},
\end{align}

\end_inset

for the Goldstone propagator:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\frac{1}{\Delta_{G}\left(n,\boldsymbol{k}\right)} & =ZZ_{\Delta}k_{E}^{2}+m^{2}+\delta m_{1}^{2}+Z_{\Delta}\frac{\lambda+\delta\lambda_{1}^{A}}{6}v^{2}\nonumber \\
 & +2\frac{1}{4!}Z_{\Delta}^{2}\left[\left(N+1\right)\lambda+\left(N-1\right)\delta\lambda_{2}^{A}+2\delta\lambda_{2}^{B}\right]\frac{1}{\mathrm{V}\beta}2\sum_{m,\boldsymbol{q}}\Delta_{G}\left(m,\boldsymbol{q}\right)\nonumber \\
 & +2\frac{1}{4!}Z_{\Delta}^{2}\left(\lambda+\delta\lambda_{2}^{A}\right)2\frac{1}{\mathrm{V}\beta}\sum_{m,\boldsymbol{q}}\Delta_{H}\left(m,\boldsymbol{q}\right)\nonumber \\
 & +2\delta_{n0}\delta_{\boldsymbol{k}\boldsymbol{0}}\frac{1}{\xi}v^{2}Z^{-1}Z_{\Delta}^{-1}\mathrm{V}\beta\left(-2\right)\left[\Delta_{G}^{-1}\left(0,\boldsymbol{0}\right)\right]^{3},
\end{align}

\end_inset

and
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\frac{1}{\Delta_{H}\left(n,\boldsymbol{k}\right)} & =ZZ_{\Delta}k_{E}^{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 \\
 & +2\frac{1}{4!}Z_{\Delta}^{2}\left(\lambda+\delta\lambda_{2}^{A}\right)2\left(N-1\right)\frac{1}{\mathrm{V}\beta}\sum_{m,\boldsymbol{q}}\Delta_{G}\left(m,\boldsymbol{q}\right)\nonumber \\
 & +2\frac{1}{4!}Z_{\Delta}^{2}\left[3\lambda+\delta\lambda_{2}^{A}+2\delta\lambda_{2}^{B}\right]\frac{1}{\mathrm{V}\beta}2\sum_{m,\boldsymbol{q}}\Delta_{H}\left(m,\boldsymbol{q}\right)
\end{align}

\end_inset

for the Higgs propagator.
\end_layout

\begin_layout Standard
Since the self-energies are momentum independent (except for the 
\begin_inset Formula $\delta_{n0}\delta_{\boldsymbol{k}\boldsymbol{0}}$
\end_inset

 term in 
\begin_inset Formula $\Delta_{G}$
\end_inset

) the propagators can be written as
\begin_inset Formula 
\begin{eqnarray}
\Delta_{G}\left(n,\boldsymbol{k}\right) & = & \begin{cases}
\Delta_{G}\left(0,\boldsymbol{0}\right) & n=\boldsymbol{k}=0\\
\frac{1}{k_{E}^{2}+m_{G}^{2}} & n,\boldsymbol{k}\neq0
\end{cases},\\
\Delta_{H}\left(n,\boldsymbol{k}\right) & = & \frac{1}{k_{E}^{2}+m_{H}^{2}}.
\end{eqnarray}

\end_inset

Substituting these in the propagator equations of motion gives 
\begin_inset Formula $ZZ_{\Delta}=1$
\end_inset

 and
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
m_{G}^{2} & =m^{2}+\delta m_{1}^{2}+Z_{\Delta}\frac{\lambda+\delta\lambda_{1}^{A}}{6}v^{2}\nonumber \\
 & +2\frac{1}{4!}Z_{\Delta}^{2}\left[\left(N+1\right)\lambda+\left(N-1\right)\delta\lambda_{2}^{A}+2\delta\lambda_{2}^{B}\right]\frac{1}{\mathrm{V}\beta}2\sum_{m,\boldsymbol{q}}\Delta_{G}\left(m,\boldsymbol{q}\right)\nonumber \\
 & +2\frac{1}{4!}Z_{\Delta}^{2}\left(\lambda+\delta\lambda_{2}^{A}\right)2\frac{1}{\mathrm{V}\beta}\sum_{m,\boldsymbol{q}}\Delta_{H}\left(m,\boldsymbol{q}\right),
\end{align}

\end_inset

from the non-zero Goldstone modes,
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\frac{1}{\Delta_{G}\left(0,\boldsymbol{0}\right)} & =m^{2}+\delta m_{1}^{2}+Z_{\Delta}\frac{\lambda+\delta\lambda_{1}^{A}}{6}v^{2}\nonumber \\
 & +2\frac{1}{4!}Z_{\Delta}^{2}\left[\left(N+1\right)\lambda+\left(N-1\right)\delta\lambda_{2}^{A}+2\delta\lambda_{2}^{B}\right]\frac{1}{\mathrm{V}\beta}2\sum_{m,\boldsymbol{q}}\Delta_{G}\left(m,\boldsymbol{q}\right)\nonumber \\
 & +2\frac{1}{4!}Z_{\Delta}^{2}\left(\lambda+\delta\lambda_{2}^{A}\right)2\frac{1}{\mathrm{V}\beta}\sum_{m,\boldsymbol{q}}\Delta_{H}\left(m,\boldsymbol{q}\right)\nonumber \\
 & +2\frac{1}{\xi}v^{2}\mathrm{V}\beta\left(-2\right)\left[\Delta_{G}^{-1}\left(0,\boldsymbol{0}\right)\right]^{3}\nonumber \\
 & =m_{G}^{2}-4\frac{1}{\xi}v^{2}\mathrm{V}\beta\left[\Delta_{G}^{-1}\left(0,\boldsymbol{0}\right)\right]^{3},
\end{align}

\end_inset

for the zero mode and
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
m_{H}^{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 \\
 & +2\frac{1}{4!}Z_{\Delta}^{2}\left(\lambda+\delta\lambda_{2}^{A}\right)2\left(N-1\right)\frac{1}{\mathrm{V}\beta}\sum_{m,\boldsymbol{q}}\Delta_{G}\left(m,\boldsymbol{q}\right)\nonumber \\
 & +2\frac{1}{4!}Z_{\Delta}^{2}\left[3\lambda+\delta\lambda_{2}^{A}+2\delta\lambda_{2}^{B}\right]\frac{1}{\mathrm{V}\beta}2\sum_{m,\boldsymbol{q}}\Delta_{H}\left(m,\boldsymbol{q}\right)
\end{align}

\end_inset

for the Higgs mass.
\end_layout

\begin_layout Standard
Now there are two cases which must be distinguished.
 In the first, 
\begin_inset Formula $m_{G}^{2}=0$
\end_inset

 and the zero mode equation has the solutions
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\Delta_{G}^{-1}\left(0,\boldsymbol{0}\right) & =\begin{cases}
0,\\
\pm i\sqrt{\frac{\xi}{4v^{2}\mathrm{V}\beta}},
\end{cases}
\end{align}

\end_inset

the latter two of which are clearly unphysical.
 However the first solution is just what one would have if 
\begin_inset Formula $\Delta_{G}\left(n,\boldsymbol{k}\right)=k_{E}^{-2}$
\end_inset

 as usual for a massless particle (i.e.
 the zero mode need no longer be treated separately).
 Then 
\begin_inset Formula $\sum_{m,\boldsymbol{q}}\Delta_{G}\left(m,\boldsymbol{q}\right)$
\end_inset

 and 
\begin_inset Formula $\sum_{m,\boldsymbol{q}}\Delta_{H}\left(m,\boldsymbol{q}\right)$
\end_inset

 are just the familiar Hartree-Fock tadpole sums, which in the infinite
 volume limit are
\begin_inset Formula 
\begin{eqnarray}
\sum_{n,\boldsymbol{k}}\Delta_{G}\left(n,\boldsymbol{k}\right) & = & \beta\mathrm{V}\left(\mathcal{T}_{G}^{\infty}+\mathcal{T}_{G}^{\mathrm{fin}}+\mathcal{T}_{G}^{\mathrm{th}}\right),\\
\sum_{n,\boldsymbol{k}}\Delta_{H}\left(n,\boldsymbol{k}\right) & = & \beta\mathrm{V}\left(\mathcal{T}_{H}^{\infty}+\mathcal{T}_{H}^{\mathrm{fin}}+\mathcal{T}_{H}^{\mathrm{th}}\right),
\end{eqnarray}

\end_inset

where
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray}
\mathcal{T}_{G/H}^{\infty} & = & c_{0}\Lambda^{2}+c_{1}m_{G/H}^{2}\ln\frac{\Lambda^{2}}{\mu^{2}},\\
\mathcal{T}_{G/H}^{\mathrm{fin}} & = & \frac{m_{G/H}^{2}}{16\pi^{2}}\ln\frac{m_{G/H}^{2}}{\mu^{2}},
\end{eqnarray}

\end_inset

are the vacuum contribution and 
\begin_inset Formula $\mathcal{T}_{G/H}^{\mathrm{th}}$
\end_inset

 are the Bose-Einstein integrals 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Bose-Einstein-thermal-tadpole"

\end_inset


\begin_inset Formula 
\begin{equation}
\mathcal{T}_{G/H}^{\mathrm{th}}=\int_{\boldsymbol{k}}\frac{1}{\omega_{\boldsymbol{k}}}\frac{1}{\mathrm{e}^{\beta\omega_{\boldsymbol{k}}}-1},\ \omega_{\boldsymbol{k}}=\sqrt{\boldsymbol{k}^{2}+m_{G/H}^{2}}.
\end{equation}

\end_inset


\begin_inset Formula $\mu$
\end_inset

, 
\begin_inset Formula $\Lambda$
\end_inset

 and the dimensionless constants 
\begin_inset Formula $c_{0}$
\end_inset

 and 
\begin_inset Formula $c_{1}$
\end_inset

 are regularisation scheme dependent.
 In 
\begin_inset Formula $\overline{\mathrm{MS}}$
\end_inset

 in 
\begin_inset Formula $4-2\eta$
\end_inset

 dimensions, 
\begin_inset Formula $c_{0}=0$
\end_inset

, 
\begin_inset Formula $c_{1}=-1/16\pi^{2}$
\end_inset

 and 
\begin_inset Formula $\Lambda^{2}=4\pi\mu^{2}\exp\left(\frac{1}{\eta}-\gamma+1\right)$
\end_inset

.
\end_layout

\begin_layout Standard
If, on the other hand, 
\begin_inset Formula $m_{G}^{2}\neq0$
\end_inset

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

 no longer has the usual form and the Goldstone tadpole must be handled
 differently.
 In this case it can be rewritten as
\begin_inset Formula 
\begin{eqnarray}
\sum_{m,\boldsymbol{q}}\Delta_{G}\left(m,\boldsymbol{q}\right) & = & \sum_{m,\boldsymbol{q}\neq0}\Delta_{G}\left(m,\boldsymbol{q}\right)+\Delta_{G}\left(0,\boldsymbol{0}\right)\nonumber \\
 & = & \sum_{m,\boldsymbol{q}\neq0}\Delta_{G}\left(m,\boldsymbol{q}\right)+\frac{1}{m_{G}^{2}}+\Delta_{G}\left(0,\boldsymbol{0}\right)-\frac{1}{m_{G}^{2}}\nonumber \\
 & = & \sum_{m,\boldsymbol{q}}\tilde{\Delta}_{G}\left(m,\boldsymbol{q}\right)+\Delta_{G}\left(0,\boldsymbol{0}\right)-\frac{1}{m_{G}^{2}}
\end{eqnarray}

\end_inset

where 
\begin_inset Formula $\tilde{\Delta}_{G}$
\end_inset

 is an auxiliary propagator defined to have the usual form
\begin_inset Formula 
\begin{equation}
\tilde{\Delta}_{G}\left(n,\boldsymbol{k}\right)=\frac{1}{k_{E}^{2}+m_{G}^{2}}.
\end{equation}

\end_inset

Then 
\begin_inset Formula $\sum_{m,\boldsymbol{q}}\tilde{\Delta}_{G}\left(m,\boldsymbol{q}\right)$
\end_inset

 is just the familiar Hartree-Fock tadpole sum for a particle of mass 
\begin_inset Formula $m_{G}$
\end_inset

.
 The terms 
\begin_inset Formula $\Delta_{G}\left(0,\boldsymbol{0}\right)-\frac{1}{m_{G}^{2}}$
\end_inset

 in the Goldstone tadpole account for the shift in the zero mode propagator
 from its usual value.
 Focusing on the zero mode equation, let
\begin_inset Formula 
\begin{equation}
\Delta_{G}\left(0,\boldsymbol{0}\right)=\frac{1}{m_{G}^{2}\epsilon},
\end{equation}

\end_inset

which defines 
\begin_inset Formula $\epsilon$
\end_inset

.
 Then,
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\epsilon & =1-\frac{4v^{2}\mathrm{V}\beta m_{G}^{4}}{\xi}\epsilon^{3}=1-\frac{4\epsilon^{3}}{27\hat{\xi}},
\end{align}

\end_inset

where
\begin_inset Formula 
\begin{equation}
\hat{\xi}=\frac{\xi}{27v^{2}\mathrm{V}\beta m_{G}^{4}}
\end{equation}

\end_inset

and the numeric factor is chosen for later convenience.
 The real solution of this cubic equation is
\begin_inset Formula 
\begin{equation}
\epsilon=\frac{3}{2}\sqrt[3]{\hat{\xi}\left(\sqrt{\hat{\xi}+1}-1\right)}\left(\sqrt[3]{\frac{\hat{\xi}}{\left(\sqrt{\hat{\xi}+1}-1\right)^{2}}}-1\right),\label{eq:epsilon-solution}
\end{equation}

\end_inset

which is monotonically increasing from 
\begin_inset Formula $0$
\end_inset

 to 
\begin_inset Formula $1$
\end_inset

 as 
\begin_inset Formula $\hat{\xi}$
\end_inset

 goes from 
\begin_inset Formula $0$
\end_inset

 to 
\begin_inset Formula $+\infty$
\end_inset

 and behaves asymptotically as
\begin_inset Formula 
\begin{equation}
\epsilon\sim\begin{cases}
\frac{3\hat{\xi}^{1/3}}{2^{2/3}}+\mathcal{O}\left(\hat{\xi}^{2/3}\right), & \hat{\xi}\to0,\\
1-\frac{4}{27\hat{\xi}}+\mathcal{O}\left(\hat{\xi}^{-2}\right), & \hat{\xi}\to\infty.
\end{cases}\label{eq:epsilon-asymptotic-behaviour}
\end{equation}

\end_inset

This is shown in Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:goldstone-zero-mode-epsilon"

\end_inset

.
\end_layout

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

\begin_layout Plain Layout
\begin_inset Graphics
	filename images/ch5/zero_mode_epsilon.eps
	width 100col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:goldstone-zero-mode-epsilon"

\end_inset

Plot of 
\begin_inset Formula $\epsilon$
\end_inset

 (blue) versus 
\begin_inset Formula $\hat{\xi}$
\end_inset

 solving the Goldstone boson zero mode equation 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:epsilon-solution"

\end_inset

.
 The line at 
\begin_inset Formula $\epsilon=1$
\end_inset

 is to guide the eye.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
The remaining equations are renormalised by demanding that kinematically
 distinct divergences vanish.
 Details of this laborious and unenlightening calculation are included in
 the supplemental 
\noun on
Mathematica 
\begin_inset CommandInset citation
LatexCommand citep
key "mathematica"

\end_inset


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

\end_inset

).
 The resulting renormalisation constants are
\begin_inset Formula 
\begin{eqnarray}
\delta m_{0}^{2} & = & \delta m_{1}^{2}=-\frac{\hbar\lambda}{3}\left(c_{0}\Lambda^{2}+c_{1}m^{2}\ln\frac{\Lambda^{2}}{\mu^{2}}\right)\left(\frac{\delta\lambda_{1}^{A}}{\delta\lambda_{1}^{B}}-1\right),\\
\delta\lambda_{0} & = & \delta\lambda_{1}^{A}+2\delta\lambda_{1}^{B},\\
\delta\lambda_{1}^{A} & = & \delta\lambda_{2}^{A}=\left(1+\frac{3\left(N+2\right)}{6+\left(N+2\right)\hbar\lambda c_{1}\ln\frac{\Lambda^{2}}{\mu^{2}}}\right)\delta\lambda_{1}^{B},\\
\delta\lambda_{1}^{B} & = & \delta\lambda_{2}^{B}=\lambda\left(-1+\frac{3}{3+\hbar\lambda c_{1}\ln\frac{\Lambda^{2}}{\mu^{2}}}\right),\\
Z & = & Z_{\Delta}=Z_{\xi}=1,
\end{eqnarray}

\end_inset

and the resulting finite equations of motion are
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
v & =0,\ \text{or}\\
0 & =m^{2}+\frac{\lambda}{6}v^{2}+\frac{1}{2}\left(N-1\right)\frac{\lambda}{3}\left(\frac{m_{G}^{2}}{16\pi^{2}}\ln\frac{m_{G}^{2}}{\mu^{2}}+\mathcal{T}_{G}^{\mathrm{th}}+\frac{1}{\mathrm{V}\beta}\frac{1}{m_{G}^{2}}\left(\frac{1}{\epsilon}-1\right)\right)\nonumber \\
 & +\frac{1}{2}\lambda\left(\frac{m_{H}^{2}}{16\pi^{2}}\ln\frac{m_{H}^{2}}{\mu^{2}}+\mathcal{T}_{H}^{\mathrm{th}}\right)+\frac{1}{\xi}\left(N-1\right)2\left[m_{G}^{2}\epsilon\right]^{2},\label{eq:SSI-2PI-HF-vev-eom}
\end{align}

\end_inset

for the vev, and
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
m_{G}^{2} & =m^{2}+\frac{\lambda}{6}v^{2}+\frac{1}{6}\left(N+1\right)\lambda\left(\frac{m_{G}^{2}}{16\pi^{2}}\ln\frac{m_{G}^{2}}{\mu^{2}}+\mathcal{T}_{G}^{\mathrm{th}}+\frac{1}{\mathrm{V}\beta}\frac{1}{m_{G}^{2}}\left(\frac{1}{\epsilon}-1\right)\right)\nonumber \\
 & +\frac{1}{6}\lambda\left(\frac{m_{H}^{2}}{16\pi^{2}}\ln\frac{m_{H}^{2}}{\mu^{2}}+\mathcal{T}_{H}^{\mathrm{th}}\right),\label{eq:SSI-2PI-HF-G-eom}
\end{align}

\end_inset

and
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
m_{H}^{2} & =m^{2}+\frac{\lambda}{2}v^{2}+\frac{1}{6}\lambda\left(N-1\right)\left(\frac{m_{G}^{2}}{16\pi^{2}}\ln\frac{m_{G}^{2}}{\mu^{2}}+\mathcal{T}_{G}^{\mathrm{th}}+\frac{1}{\mathrm{V}\beta}\frac{1}{m_{G}^{2}}\left(\frac{1}{\epsilon}-1\right)\right)\nonumber \\
 & +\frac{1}{2}\lambda\left(\frac{m_{H}^{2}}{16\pi^{2}}\ln\frac{m_{H}^{2}}{\mu^{2}}+\mathcal{T}_{H}^{\mathrm{th}}\right),\label{eq:SSI-2PI-HF-H-eom}
\end{align}

\end_inset

for the Goldstone and Higgs masses, respectively.
\end_layout

\begin_layout Section
Solution in the infinite volume / low temperature limit
\begin_inset CommandInset label
LatexCommand label
name "sec:ch5-Solution-in-the-limit"

\end_inset


\end_layout

\begin_layout Subsection
Symmetric phase
\begin_inset CommandInset label
LatexCommand label
name "subsec:Symmetric-Phase"

\end_inset


\end_layout

\begin_layout Standard
At high temperatures there should be a symmetric phase solution to the equations
 of motion.
 Therefore examine the 
\begin_inset Formula $v\to0$
\end_inset

 limit of the equations of motion.
 As 
\begin_inset Formula $v\to0$
\end_inset

 at fixed 
\begin_inset Formula $\xi$
\end_inset

, 
\begin_inset Formula $\epsilon\to1-\frac{4v^{2}\mathrm{V}\beta m_{G}^{4}}{\xi}$
\end_inset

 so long as 
\begin_inset Formula $m_{G}$
\end_inset

 does not go to infinity faster than 
\begin_inset Formula $1/\sqrt{v}$
\end_inset

.
 Then
\begin_inset Formula 
\begin{equation}
\frac{1}{\mathrm{V}\beta}\frac{1}{m_{G}^{2}}\left(\frac{1}{\epsilon}-1\right)\to\frac{1}{\mathrm{V}\beta}\frac{1}{m_{G}^{2}}\left(\frac{1}{1-\frac{4v^{2}\mathrm{V}\beta m_{G}^{4}}{\xi}}-1\right)\to\frac{4v^{2}m_{G}^{2}}{\xi}\to0,
\end{equation}

\end_inset

and the equations of motion 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:SSI-2PI-HF-vev-eom"

\end_inset

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

\end_inset

 reduce to
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
m_{G}^{2}=m_{H}^{2}=m^{2}+\frac{1}{6}\left(N+2\right)\lambda\left(\frac{m_{H}^{2}}{16\pi^{2}}\ln\frac{m_{H}^{2}}{\mu^{2}}+\mathcal{T}_{H}^{\mathrm{th}}\right),
\end{equation}

\end_inset

which is a symmetric (i.e.
 equal mass) phase as expected.
 Indeed, the gap equation is unmodified by the presence of soft symmetry
 improvement in the symmetric phase.
 This phase terminates at the critical point 
\begin_inset Formula $m_{G}^{2}=m_{H}^{2}=0$
\end_inset

 which gives the critical temperature
\begin_inset Formula 
\begin{equation}
T_{\star}=\sqrt{\frac{12\bar{v}^{2}}{N+2}},
\end{equation}

\end_inset

where 
\begin_inset Formula $m^{2}=-\lambda\bar{v}^{2}/6$
\end_inset

.
 That 
\begin_inset Formula $T_{\star}$
\end_inset

 is independent of 
\begin_inset Formula $\xi$
\end_inset

 follows because 
\begin_inset Formula $\mathcal{W}=0$
\end_inset

 trivially in the symmetric phase, and is consistent with the previously
 noted result that the same critical point is found in all schemes (c.f.
 chapter 
\begin_inset CommandInset ref
LatexCommand eqref
reference "chap:chap4"

\end_inset

).
 There is no subtlety involved in the 
\begin_inset Formula $\mathrm{V}\beta\to\infty$
\end_inset

 limit in this case.
\end_layout

\begin_layout Subsection
Broken phase with 
\begin_inset Formula $m_{G}^{2}\neq0$
\end_inset


\begin_inset CommandInset label
LatexCommand label
name "subsec:Broken-Phase-with-massive-Goldstones"

\end_inset


\end_layout

\begin_layout Standard
Attempting to describe the broken phase with the SSI equations of motion
 is rather more complicated than the symmetric phase.
 Decreasing temperature at 
\emph on
fixed
\emph default
 
\begin_inset Formula $\xi$
\end_inset

 gives 
\begin_inset Formula $\hat{\xi}\to0$
\end_inset

 and
\begin_inset Formula 
\begin{equation}
\frac{1}{\mathrm{V}\beta}\frac{1}{m_{G}^{2}}\left(\frac{1}{\epsilon}-1\right)\to\frac{1}{\mathrm{V}\beta}\frac{1}{m_{G}^{2}}\left(\frac{2^{2/3}}{3\left(\frac{\xi}{27v^{2}\mathrm{V}\beta m_{G}^{4}}\right)^{1/3}}-1\right)\to\left(\frac{4v^{2}}{\xi m_{G}^{2}\left(\mathrm{V}\beta\right)^{2}}\right)^{1/3},
\end{equation}

\end_inset

so the equations of motion 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:SSI-2PI-HF-vev-eom"

\end_inset

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

\end_inset

 become
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
0 & =m^{2}+\frac{\lambda}{6}v^{2}+\frac{1}{2}\left(N-1\right)\frac{\lambda}{3}\left(\frac{m_{G}^{2}}{16\pi^{2}}\ln\frac{m_{G}^{2}}{\mu^{2}}+\mathcal{T}_{G}^{\mathrm{th}}+\left(\frac{4v^{2}}{\xi m_{G}^{2}\left(\mathrm{V}\beta\right)^{2}}\right)^{1/3}\right)\nonumber \\
 & +\frac{1}{2}\lambda\left(\frac{m_{H}^{2}}{16\pi^{2}}\ln\frac{m_{H}^{2}}{\mu^{2}}+\mathcal{T}_{H}^{\mathrm{th}}\right)+\frac{1}{\xi}\left(N-1\right)\left(\frac{\xi m_{G}^{2}}{\sqrt{2}v^{2}\mathrm{V}\beta}\right)^{2/3},
\end{align}

\end_inset

for the vev, and
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
m_{G}^{2} & =m^{2}+\frac{\lambda}{6}v^{2}+\frac{1}{6}\left(N+1\right)\lambda\left(\frac{m_{G}^{2}}{16\pi^{2}}\ln\frac{m_{G}^{2}}{\mu^{2}}+\mathcal{T}_{G}^{\mathrm{th}}+\left(\frac{4v^{2}}{\xi m_{G}^{2}\left(\mathrm{V}\beta\right)^{2}}\right)^{1/3}\right)\nonumber \\
 & +\frac{1}{6}\lambda\left(\frac{m_{H}^{2}}{16\pi^{2}}\ln\frac{m_{H}^{2}}{\mu^{2}}+\mathcal{T}_{H}^{\mathrm{th}}\right),
\end{align}

\end_inset

and
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
m_{H}^{2} & =m^{2}+\frac{\lambda}{2}v^{2}+\frac{1}{6}\lambda\left(N-1\right)\left(\frac{m_{G}^{2}}{16\pi^{2}}\ln\frac{m_{G}^{2}}{\mu^{2}}+\mathcal{T}_{G}^{\mathrm{th}}+\left(\frac{4v^{2}}{\xi m_{G}^{2}\left(\mathrm{V}\beta\right)^{2}}\right)^{1/3}\right)\nonumber \\
 & +\frac{1}{2}\lambda\left(\frac{m_{H}^{2}}{16\pi^{2}}\ln\frac{m_{H}^{2}}{\mu^{2}}+\mathcal{T}_{H}^{\mathrm{th}}\right).
\end{align}

\end_inset

Note that all of the soft symmetry improvement terms vanish in the limit
 
\begin_inset Formula $\mathrm{V}\beta\to\infty$
\end_inset

.
 Thus the SSI-2PIEA reduces to the unimproved 2PIEA if 
\begin_inset Formula $\mathrm{V}\beta\to\infty$
\end_inset

 at fixed 
\begin_inset Formula $\xi$
\end_inset

.
 It is necessary to allow 
\begin_inset Formula $\xi$
\end_inset

 to vary as the 
\begin_inset Formula $\mathrm{V}\beta\to\infty$
\end_inset

 limit is taken to obtain a non-trivial correction to the unimproved 2PIEA.
 This is the first sign that the limit is non-trivial.
\end_layout

\begin_layout Standard
This section examines the simplest scheme to find a non-trivial limit, letting
 
\begin_inset Formula $\xi$
\end_inset

 vary with 
\begin_inset Formula $\mathrm{V}\beta$
\end_inset

 as 
\begin_inset Formula $\xi=\left(\mathrm{V}\beta\right)^{\alpha}\zeta$
\end_inset

 where 
\begin_inset Formula $\zeta$
\end_inset

 is a constant (with mass dimension 
\begin_inset Formula $\left[\zeta\right]=2+4\alpha$
\end_inset

).
 If 
\begin_inset Formula $\alpha\geq1$
\end_inset

 the SSI terms vanish in the limit.
 If 
\begin_inset Formula $\alpha<1$
\end_inset


\begin_inset Formula 
\begin{equation}
\epsilon\to\left(\frac{\left(\mathrm{V}\beta\right)^{\alpha}\zeta}{4v^{2}\mathrm{V}\beta m_{G}^{4}}\right)^{1/3},
\end{equation}

\end_inset

and the symmetry improvement terms are
\begin_inset Formula 
\begin{eqnarray}
\frac{1}{\mathrm{V}\beta}\frac{1}{m_{G}^{2}}\left(\frac{1}{\epsilon}-1\right) & \to & \frac{1}{m_{G}^{2}}\left(\left(\frac{4v^{2}m_{G}^{4}}{\zeta}\right)^{1/3}\left(\mathrm{V}\beta\right)^{\left(-\alpha-2\right)/3}-\frac{1}{\mathrm{V}\beta}\right),\\
\frac{1}{\xi}\left(N-1\right)2\left[m_{G}^{2}\epsilon\right]^{2} & \to & \frac{1}{\zeta}\left(N-1\right)\left(\frac{\zeta m_{G}^{2}}{\sqrt{2}v^{2}}\right)^{2/3}\left(\mathrm{V}\beta\right)^{\left(-2-\alpha\right)/3}.
\end{eqnarray}

\end_inset

The only non-trivial possibility is 
\begin_inset Formula $\alpha=-2$
\end_inset

, for which
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray}
\epsilon & \to & \frac{1}{\mathrm{V}\beta}\left(\frac{\zeta}{4v^{2}m_{G}^{4}}\right)^{1/3}\to0,\\
\frac{1}{\mathrm{V}\beta}\frac{1}{m_{G}^{2}}\left(\frac{1}{\epsilon}-1\right) & \to & \left(\frac{4v^{2}}{\zeta m_{G}^{2}}\right)^{1/3},\\
\frac{1}{\xi}\left(N-1\right)2\left[m_{G}^{2}\epsilon\right]^{2} & \to & \frac{1}{\zeta}\left(N-1\right)\left(\frac{\zeta m_{G}^{2}}{\sqrt{2}v^{2}}\right)^{2/3}.
\end{eqnarray}

\end_inset


\end_layout

\begin_layout Standard
The equations of motion are then
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
0 & =m^{2}+\frac{\lambda}{6}v^{2}+\frac{1}{2}\left(N-1\right)\frac{\lambda}{3}\left(\frac{m_{G}^{2}}{16\pi^{2}}\ln\frac{m_{G}^{2}}{\mu^{2}}+\mathcal{T}_{G}^{\mathrm{th}}+\left(\frac{4v^{2}}{\zeta m_{G}^{2}}\right)^{1/3}\right)\nonumber \\
 & +\frac{1}{2}\lambda\left(\frac{m_{H}^{2}}{16\pi^{2}}\ln\frac{m_{H}^{2}}{\mu^{2}}+\mathcal{T}_{H}^{\mathrm{th}}\right)+\frac{1}{\zeta}\left(N-1\right)\left(\frac{\zeta m_{G}^{2}}{\sqrt{2}v^{2}}\right)^{2/3},
\end{align}

\end_inset

for the vev, and
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
m_{G}^{2} & =m^{2}+\frac{\lambda}{6}v^{2}+\frac{1}{6}\left(N+1\right)\lambda\left(\frac{m_{G}^{2}}{16\pi^{2}}\ln\frac{m_{G}^{2}}{\mu^{2}}+\mathcal{T}_{G}^{\mathrm{th}}+\left(\frac{4v^{2}}{\zeta m_{G}^{2}}\right)^{1/3}\right)\nonumber \\
 & +\frac{1}{6}\lambda\left(\frac{m_{H}^{2}}{16\pi^{2}}\ln\frac{m_{H}^{2}}{\mu^{2}}+\mathcal{T}_{H}^{\mathrm{th}}\right),
\end{align}

\end_inset

and
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
m_{H}^{2} & =m^{2}+\frac{\lambda}{2}v^{2}+\frac{1}{6}\lambda\left(N-1\right)\left(\frac{m_{G}^{2}}{16\pi^{2}}\ln\frac{m_{G}^{2}}{\mu^{2}}+\mathcal{T}_{G}^{\mathrm{th}}+\left(\frac{4v^{2}}{\zeta m_{G}^{2}}\right)^{1/3}\right)\nonumber \\
 & +\frac{1}{2}\lambda\left(\frac{m_{H}^{2}}{16\pi^{2}}\ln\frac{m_{H}^{2}}{\mu^{2}}+\mathcal{T}_{H}^{\mathrm{th}}\right),
\end{align}

\end_inset

for the Goldstone and Higgs masses, respectively.
 Importantly, note that the mass appearing in Goldstone's theorem is 
\begin_inset Formula $\epsilon m_{G}^{2}$
\end_inset

 (from the definition of 
\begin_inset Formula $\epsilon$
\end_inset

: 
\begin_inset Formula $\Delta_{G}^{-1}\left(0,\boldsymbol{0}\right)=\epsilon m_{G}^{2}$
\end_inset

), which obeys 
\begin_inset Formula $\epsilon m_{G}^{2}\to0$
\end_inset

 as 
\begin_inset Formula $\mathrm{V}\beta\to\infty$
\end_inset

 thanks to the scaling chosen for 
\begin_inset Formula $\xi$
\end_inset

.
 Thus this scheme satisfies Goldstone's theorem even if 
\begin_inset Formula $m_{G}^{2}\neq0$
\end_inset

.
 What 
\begin_inset Formula $m_{G}^{2}\neq0$
\end_inset

 indicates here is not actually a violation of Goldstone's theorem, but
 a 
\emph on
non
\emph default
-communication of the masslessness of the Goldstone zero mode to the other
 modes.
\end_layout

\begin_layout Standard
Defining the dimensionless variables
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{flalign}
x & =v^{2}/\mu^{2}, & y & =m_{G}^{2}/\mu^{2}, & z & =m_{H}^{2}/\mu^{2},\\
\bar{x} & =\bar{v}^{2}/\mu^{2}, & \bar{X} & =-6m^{2}/\lambda\mu^{2}, & \hat{\zeta} & =\zeta\mu^{6},\\
T_{G/H} & =\mu^{-2}\mathcal{T}_{G/H}^{\mathrm{th}},
\end{flalign}

\end_inset

(note the distinction between the Lagrangian parameter 
\begin_inset Formula $\bar{X}$
\end_inset

 and the zero temperature value of the mean field 
\begin_inset Formula $\bar{x}$
\end_inset

, which happen to be equal at tree level and in the usual renormalisation
 scheme for the Hartree-Fock approximation) this system becomes
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
0 & =\frac{\lambda}{6}\left(x-\bar{X}\right)+\frac{1}{6}\left(N-1\right)\lambda\left(\frac{1}{16\pi^{2}}y\ln y+T_{G}+\left(\frac{4x}{\hat{\zeta}y}\right)^{1/3}\right)+\frac{1}{2}\lambda\left(\frac{1}{16\pi^{2}}z\ln z+T_{H}\right)\nonumber \\
 & +\frac{1}{\hat{\zeta}}\left(N-1\right)\left(\frac{\hat{\zeta}y}{\sqrt{2}x}\right)^{2/3},\label{eq:ssi-hf-veom-nondim}\\
y & =\frac{\lambda}{6}\left(x-\bar{X}\right)+\frac{1}{6}\left(N+1\right)\lambda\left(\frac{1}{16\pi^{2}}y\ln y+T_{G}+\left(\frac{4x}{\hat{\zeta}y}\right)^{1/3}\right)+\frac{1}{6}\lambda\left(\frac{1}{16\pi^{2}}z\ln z+T_{H}\right),\label{eq:ssi-hf-geom-nondim}\\
z & =\frac{\lambda}{3}x-\frac{1}{\hat{\zeta}}\left(N-1\right)\left(\frac{\hat{\zeta}y}{\sqrt{2}x}\right)^{2/3}.\label{eq:ssi-hf-heom-nondim}
\end{align}

\end_inset


\end_layout

\begin_layout Standard
Looking for a zero temperature solution gives the system
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
0 & =\frac{\lambda}{6}\left(\bar{x}-\bar{X}\right)+\frac{1}{6}\left(N-1\right)\lambda\left(\frac{1}{16\pi^{2}}\bar{y}\ln\bar{y}+\left(\frac{4\bar{x}}{\hat{\zeta}\bar{y}}\right)^{1/3}\right)+\frac{1}{2}\lambda\frac{1}{16\pi^{2}}\bar{z}\ln\bar{z}+\left(N-1\right)\left(\frac{\bar{y}^{2}}{2\hat{\zeta}\bar{x}^{2}}\right)^{1/3},\label{eq:ssi-hf-veom-nondim-zerot}\\
\bar{y} & =\frac{\lambda}{6}\left(\bar{x}-\bar{X}\right)+\frac{1}{6}\left(N+1\right)\lambda\left(\frac{1}{16\pi^{2}}\bar{y}\ln\bar{y}+\left(\frac{4\bar{x}}{\hat{\zeta}\bar{y}}\right)^{1/3}\right)+\frac{1}{6}\lambda\frac{1}{16\pi^{2}}\bar{z}\ln\bar{z},\label{eq:ssi-hf-geom-nondim-zerot}\\
\bar{z} & =\frac{\lambda\bar{x}}{3}-\left(N-1\right)\left(\frac{\bar{y}^{2}}{2\hat{\zeta}\bar{x}^{2}}\right)^{1/3}.\label{eq:ssi-hf-heom-nondim-zerot}
\end{align}

\end_inset

First, ignoring the SSI terms one finds the usual unimproved 2PI solution
 
\begin_inset Formula $\bar{x}=\bar{X}$
\end_inset

, 
\begin_inset Formula $\bar{y}=0$
\end_inset

, 
\begin_inset Formula $\bar{z}=\lambda\bar{x}/3=1$
\end_inset

.
 Now examine the large 
\begin_inset Formula $N$
\end_inset

 limit of these equations, taking as the scaling limit 
\begin_inset Formula $g=\lambda N=\text{constant}$
\end_inset

 and 
\begin_inset Formula $\bar{x},\bar{X}\sim N^{1}$
\end_inset

, 
\begin_inset Formula $\bar{y},\bar{z}\sim N^{0}$
\end_inset

 and 
\begin_inset Formula $\hat{\zeta}\sim N^{a}$
\end_inset

 with 
\begin_inset Formula $a$
\end_inset

 to be determined.
 To leading order
\begin_inset Formula 
\begin{align}
0 & =\frac{g}{6N}\left(\bar{x}-\bar{X}\right)+\frac{g}{6}\left(\frac{1}{16\pi^{2}}\bar{y}\ln\bar{y}+N^{\left(1-a\right)/3}\left(\frac{4\bar{x}/N}{\left(\hat{\zeta}/N^{a}\right)\bar{y}}\right)^{1/3}\right)\nonumber \\
 & +N^{\left(1-a\right)/3}\left(\frac{\bar{y}^{2}}{2\left(\hat{\zeta}/N^{a}\right)\left(\bar{x}/N\right)^{2}}\right)^{1/3},\\
\bar{y} & =\frac{g}{6N}\left(\bar{x}-\bar{X}\right)+\frac{g}{6}\left(\frac{1}{16\pi^{2}}\bar{y}\ln\bar{y}+N^{\left(1-a\right)/3}\left(\frac{4\bar{x}/N}{\left(\hat{\zeta}/N^{a}\right)\bar{y}}\right)^{1/3}\right),\\
\bar{z} & =\frac{g\bar{x}}{3N}-N^{\left(1-a\right)/3}\left(\frac{\bar{y}^{2}}{2\left(\hat{\zeta}/N^{a}\right)\left(\bar{x}/N\right)^{2}}\right)^{1/3}.
\end{align}

\end_inset

Scaling limits exist if 
\begin_inset Formula $a\geq1$
\end_inset

.
 Note that the SSI term in 
\begin_inset Formula $\Gamma_{\xi}^{\mathrm{SSI}}$
\end_inset

 goes as 
\begin_inset Formula $\xi^{-1}Nv^{2}\sim N^{3-a}$
\end_inset

 so that one needs 
\begin_inset Formula $a\geq2$
\end_inset

 for a scaling limit for 
\begin_inset Formula $\Gamma_{\xi}^{\mathrm{SSI}}$
\end_inset

 to exist.
 
\begin_inset Formula $a=1$
\end_inset

 can also be ruled out by considering the equations of motion, for in this
 case the leading approximation is
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
0 & =\frac{g}{6N}\left(\bar{x}-\bar{X}\right)+\frac{g}{6}\left(\frac{1}{16\pi^{2}}\bar{y}\ln\bar{y}+\left(\frac{4\bar{x}/N}{\left(\hat{\zeta}/N\right)\bar{y}}\right)^{1/3}\right)+\left(\frac{\bar{y}^{2}}{2\left(\hat{\zeta}/N\right)\left(\bar{x}/N\right)^{2}}\right)^{1/3},\\
\bar{y} & =\frac{g}{6N}\left(\bar{x}-\bar{X}\right)+\frac{g}{6}\left(\frac{1}{16\pi^{2}}\bar{y}\ln\bar{y}+\left(\frac{4\bar{x}/N}{\left(\hat{\zeta}/N\right)\bar{y}}\right)^{1/3}\right),\\
\bar{z} & =\frac{g\bar{x}}{3N}-\left(\frac{\bar{y}^{2}}{2\left(\hat{\zeta}/N\right)\left(\bar{x}/N\right)^{2}}\right)^{1/3}.
\end{align}

\end_inset

Using the first equation to simplify the second,
\begin_inset Formula 
\begin{equation}
\bar{y}=-\left(\frac{\bar{y}^{2}}{2\left(\hat{\zeta}/N\right)\left(\bar{x}/N\right)^{2}}\right)^{1/3},
\end{equation}

\end_inset

which has the solution
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\bar{y}=-\frac{1}{2\left(\hat{\zeta}/N\right)\left(\bar{x}/N\right)^{2}}<0,
\end{equation}

\end_inset

with an unphysical tachyonic Goldstone 
\begin_inset Formula $m_{G}^{2}<0$
\end_inset

.
 This is not 
\emph on
necessarily
\emph default
 a problem because the zero mode 
\begin_inset Formula $\Delta_{G}\left(0,\boldsymbol{0}\right)$
\end_inset

 is always positive and, in finite volume with 
\begin_inset Formula $\beta$
\end_inset

 and 
\begin_inset Formula $L$
\end_inset

 sufficiently small (i.e.
 
\begin_inset Formula $\beta,L<2\pi/\left|m_{G}\right|$
\end_inset

), each mode 
\begin_inset Formula $\Delta_{G}\left(n,\boldsymbol{k}\right)=1/\left(\omega_{n}^{2}+\boldsymbol{k}^{2}+m_{G}^{2}\right)$
\end_inset

 with 
\begin_inset Formula $n,\boldsymbol{k}\neq\boldsymbol{0}$
\end_inset

 is still positive.
 Physically, confinement energy is stabilising the tachyon.
 However, a second condition is that the imaginary part of the first equation
 of motion vanishes, yielding
\begin_inset Formula 
\begin{equation}
0=-\frac{1}{16\pi^{2}}\left|\bar{y}\right|\left(2k+1\right)\pi-\sin\left(\frac{\left(2k+1\right)\pi}{3}\right)\left(\frac{4\bar{x}/N}{\left(\hat{\zeta}/N\right)\left|\bar{y}\right|}\right)^{1/3},
\end{equation}

\end_inset

where the branch chosen is 
\begin_inset Formula $\bar{y}=\left|\bar{y}\right|\exp\left(i\pi\left(2k+1\right)\right)$
\end_inset

 where 
\begin_inset Formula $k$
\end_inset

 is an integer.
 Using the solution for 
\begin_inset Formula $\bar{y}$
\end_inset

 this becomes
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
k=-\frac{1}{2}-32\pi\left(\hat{\zeta}/N\right)\left(\bar{x}/N\right)^{3}\sin\left(\frac{\left(2k+1\right)\pi}{3}\right),
\end{equation}

\end_inset

which only has solutions of the form 
\begin_inset Formula $k=3j$
\end_inset

 if
\begin_inset Formula 
\begin{equation}
j=-\frac{1}{6}-\frac{16\pi}{\sqrt{3}}\left(\hat{\zeta}/N\right)\left(\bar{x}/N\right)^{3},
\end{equation}

\end_inset

is an integer.
 The existence of solutions only for certain discrete values of 
\begin_inset Formula $\hat{\zeta}\bar{x}^{3}$
\end_inset

 is troubling and highly counter-intuitive (note especially that the relationshi
p between 
\begin_inset Formula $\hat{\zeta}$
\end_inset

 and 
\begin_inset Formula $\bar{x}$
\end_inset

 for a given 
\begin_inset Formula $j$
\end_inset

 is independent of 
\begin_inset Formula $g$
\end_inset

, so that no matter how 
\begin_inset Formula $g$
\end_inset

 is varied at fixed 
\begin_inset Formula $m^{2}$
\end_inset

 and 
\begin_inset Formula $\hat{\zeta}$
\end_inset

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

 is fixed even though one expects 
\begin_inset Formula $\bar{x}\sim N/g$
\end_inset

).
\end_layout

\begin_layout Standard
If 
\begin_inset Formula $a>1$
\end_inset

 the SSI terms in the equation of motion are of higher order and the leading
 large 
\begin_inset Formula $N$
\end_inset

 approximation is just the standard one, i.e.
\begin_inset Formula 
\begin{align}
0 & =\frac{g}{6N}\left(\bar{x}-\bar{X}\right)+\frac{g}{6}\frac{1}{16\pi^{2}}\bar{y}\ln\bar{y},\\
\bar{y} & =\frac{g}{6N}\left(\bar{x}-\bar{X}\right)+\frac{g}{6}\frac{1}{16\pi^{2}}\bar{y}\ln\bar{y},\\
\bar{z} & =\frac{g\bar{x}}{3N},
\end{align}

\end_inset

which has the solution 
\begin_inset Formula $\bar{x}=\bar{X}$
\end_inset

, 
\begin_inset Formula $\bar{y}=0$
\end_inset

 and 
\begin_inset Formula $\bar{z}=\lambda\bar{x}/3$
\end_inset

 as expected.
 Now note that if 
\begin_inset Formula $1<a<4$
\end_inset

 the SSI terms goes as a fractional power of 
\begin_inset Formula $N$
\end_inset

 between 
\begin_inset Formula $N^{0}$
\end_inset

 and 
\begin_inset Formula $N^{-1}$
\end_inset

 which cannot balance any of the terms coming from diagrams, which all go
 as integer powers of 
\begin_inset Formula $N^{-1}$
\end_inset

.
 This implies that if 
\begin_inset Formula $a>1$
\end_inset

 it must be of the form 
\begin_inset Formula $a=4+3k$
\end_inset

 where 
\begin_inset Formula $k=0,1,2,\cdots$
\end_inset

.
 The SSI terms then scale as 
\begin_inset Formula $N^{-\left(1+k\right)}$
\end_inset

 in the equation of motion and 
\begin_inset Formula $N^{-1-3k}$
\end_inset

 in 
\begin_inset Formula $\Gamma_{\xi}^{\mathrm{SSI}}$
\end_inset

.
 Thus the SSI equations of motion possess a satisfactory leading large 
\begin_inset Formula $N$
\end_inset

 limit, but only if the scaling is such that the SSI terms are of higher
 order.
 This is the first sign that the SSI terms are problematic.
\end_layout

\begin_layout Standard
Now consider the case where symmetry improvement is only weakly imposed,
 i.e.
 the SSI terms are a small perturbation.
 Intuitively this can be achieved by taking 
\begin_inset Formula $\hat{\zeta}$
\end_inset

 sufficiently large.
 It is thus natural to solve the equations of motion 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:ssi-hf-veom-nondim-zerot"

\end_inset

-
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:ssi-hf-heom-nondim-zerot"

\end_inset

 perturbatively in powers of 
\begin_inset Formula $\hat{\zeta}^{-1/3}$
\end_inset

 as 
\begin_inset Formula $\hat{\zeta}\to\infty$
\end_inset

.
 Writing 
\begin_inset Formula $\bar{x}=\bar{x}_{0}+\hat{\zeta}^{-1/3}\bar{x}_{1}+\hat{\zeta}^{-2/3}\bar{x}_{2}+\cdots$
\end_inset

 and so on, the leading equations of motion are just the unimproved 2PI
 ones
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
0 & =\frac{\lambda}{6}\left(\bar{x}_{0}-\bar{X}\right)+\frac{1}{6}\left(N-1\right)\lambda\frac{1}{16\pi^{2}}\bar{y}_{0}\ln\bar{y}_{0}+\frac{1}{2}\lambda\frac{1}{16\pi^{2}}\bar{z}_{0}\ln\bar{z}_{0},\\
\bar{y}_{0} & =\frac{\lambda}{6}\left(\bar{x}_{0}-\bar{X}\right)+\frac{1}{6}\left(N+1\right)\lambda\frac{1}{16\pi^{2}}\bar{y}_{0}\ln\bar{y}_{0}+\frac{1}{6}\lambda\frac{1}{16\pi^{2}}\bar{z}_{0}\ln\bar{z}_{0},\\
\bar{z}_{0} & =\frac{\lambda\bar{x}_{0}}{3}.
\end{align}

\end_inset

The first order perturbation obeys a system of equations which can be arranged
 as the matrix equation
\begin_inset Formula 
\begin{equation}
\left(\begin{array}{ccc}
\frac{\lambda}{6} & \frac{\left(N-1\right)\lambda}{96\pi^{2}}\left(1+\ln\bar{y}_{0}\right) & \frac{\lambda}{32\pi^{2}}\left(1+\ln\bar{z}_{0}\right)\\
\frac{\lambda}{6} & \frac{\left(N+1\right)\lambda}{96\pi^{2}}\left(1+\ln\bar{y}_{0}\right)-1 & \frac{\lambda}{96\pi^{2}}\left(1+\ln\bar{z}_{0}\right)\\
\frac{\lambda}{3} & 0 & -1
\end{array}\right)\left(\begin{array}{c}
\bar{x}_{1}\\
\bar{y}_{1}\\
\bar{z}_{1}
\end{array}\right)=\left(N-1\right)\left(\frac{\bar{y}_{0}^{2}}{2\bar{x}_{0}^{2}}\right)^{1/3}\left(\begin{array}{c}
-\frac{\lambda\bar{x}_{0}}{3\bar{y}_{0}}-1\\
-\frac{N+1}{N-1}\frac{\lambda\bar{x}_{0}}{3\bar{y}_{0}}\\
1
\end{array}\right).
\end{equation}

\end_inset

Note that this equation is singular in the limit 
\begin_inset Formula $\bar{y}_{0}\to0$
\end_inset

.
 The solution for 
\begin_inset Formula $\bar{y}_{1}$
\end_inset

 in this limit is
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\bar{y}_{1}\to-\frac{32\pi^{2}}{\ln\bar{y}_{0}}\left(\frac{\bar{x}_{0}}{2\bar{y}_{0}}\right)^{1/3}\to\infty.
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
There is no sense in which the SSI terms are a small perturbation, no matter
 the value of 
\begin_inset Formula $\hat{\zeta}$
\end_inset

.
 This can also be seen from a direct examination of the full equations of
 motion.
 In the limit 
\begin_inset Formula $\bar{y}\to0$
\end_inset

 the 
\begin_inset Formula $\left(\frac{4\bar{x}}{\hat{\zeta}\bar{y}}\right)^{1/3}$
\end_inset

 terms always dominate for any finite value of 
\begin_inset Formula $\hat{\zeta}$
\end_inset

.
 The result is that the SSI solution must always have 
\begin_inset Formula $\bar{y}\neq0$
\end_inset

, even at zero temperature.
 For the same reason a perturbation analysis near the critical temperature
 also fails and, in fact, real valued solutions do not exist in a (
\begin_inset Formula $\hat{\zeta}$
\end_inset

 dependent) range of temperatures beneath the critical temperature.
 Further, 
\begin_inset Formula $m_{G}^{2}$
\end_inset

 appears to increase as the SSI terms are more strongly imposed.
 Physically, the unimproved 2PI equations of motion 
\begin_inset Quotes eld
\end_inset

would like to have
\begin_inset Quotes erd
\end_inset

 a non-zero Goldstone mass.
 When the mass of the zero mode is forced to vanish the SSI-2PIEA adjusts
 by 
\emph on
increasing
\emph default
 the mass of the other modes.
 This can be verified by examining numerical solutions.
\end_layout

\begin_layout Standard
Numerical solutions of the 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:ssi-hf-veom-nondim"

\end_inset

-
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:ssi-hf-heom-nondim"

\end_inset

 are shown in Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:ssi-hf-solns"

\end_inset

 for 
\begin_inset Formula $\lambda=10$
\end_inset

, 
\begin_inset Formula $N=4$
\end_inset

, 
\begin_inset Formula $\bar{X}=0.3$
\end_inset

 and several values of 
\begin_inset Formula $\hat{\zeta}$
\end_inset

 from 
\begin_inset Formula $10^{4}$
\end_inset

 to 
\begin_inset Formula $\infty$
\end_inset

.
 The critical temperature for these values is 
\begin_inset Formula $T_{\star}/\mu\approx0.775$
\end_inset

.
 For very large 
\begin_inset Formula $\hat{\zeta}$
\end_inset

 the solution is near the unimproved solution.
 However, as 
\begin_inset Formula $\hat{\zeta}$
\end_inset

 is decreased, 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $z$
\end_inset

 decrease and 
\begin_inset Formula $y$
\end_inset

 
\emph on
increases
\emph default
 (this is consistent with the perturbation 
\begin_inset Formula $y_{1}$
\end_inset

 being positive).
 Broken phase solutions cease to exist above the upper spinodal temperature
 
\begin_inset Formula $T_{\mathrm{us}}\left(\hat{\zeta}\right)$
\end_inset

 which depends on 
\begin_inset Formula $\hat{\zeta}$
\end_inset

.
 Note that 
\begin_inset Formula $T_{\mathrm{us}}\left(\hat{\zeta}\right)$
\end_inset

 drops below 
\begin_inset Formula $T_{\star}$
\end_inset

 for all 
\begin_inset Formula $\hat{\zeta}<\hat{\zeta}_{c}$
\end_inset

 where 
\begin_inset Formula $\hat{\zeta}_{c}$
\end_inset

 is somewhere between 
\begin_inset Formula $10^{6}$
\end_inset

 and 
\begin_inset Formula $10^{7}$
\end_inset

.
 This means that, for 
\begin_inset Formula $\hat{\zeta}<\hat{\zeta}_{c}$
\end_inset

 there is 
\emph on
no solution
\emph default
 between 
\begin_inset Formula $T_{\mathrm{us}}\left(\hat{\zeta}\right)$
\end_inset

 and 
\begin_inset Formula $T_{\star}$
\end_inset

.
 Further, as 
\begin_inset Formula $\hat{\zeta}\to0$
\end_inset

, 
\begin_inset Formula $T_{\mathrm{us}}\left(\hat{\zeta}\right)\to0$
\end_inset

.
 This behaviour can be seen in Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:ssi-hf-spinodal-temps"

\end_inset

.
 At a critical value 
\begin_inset Formula $\hat{\zeta}=\hat{\zeta}_{\star}$
\end_inset

 one has 
\begin_inset Formula $T_{\mathrm{us}}\left(\hat{\zeta}_{\star}\right)=0$
\end_inset

 and real solutions cease to exist for all 
\begin_inset Formula $\hat{\zeta}\leq\hat{\zeta}_{\star}$
\end_inset

.
\end_layout

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

\begin_layout Plain Layout
\begin_inset Graphics
	filename images/ch5/ssi-solns.eps
	width 100col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:ssi-hf-solns"

\end_inset

Solutions of 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:ssi-hf-veom-nondim"

\end_inset

-
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:ssi-hf-heom-nondim"

\end_inset

, the SSI equations of motion in the Hartree-Fock approximation.
 Shown are broken phase 
\begin_inset Formula $x$
\end_inset

 (solid), 
\begin_inset Formula $y$
\end_inset

 (dashed) and 
\begin_inset Formula $z$
\end_inset

 (dot-dashed) and symmetric phase 
\begin_inset Formula $y=z$
\end_inset

 (solid black) versus temperature for 
\begin_inset Formula $\lambda=10$
\end_inset

, 
\begin_inset Formula $N=4$
\end_inset

, 
\begin_inset Formula $\bar{X}=0.3$
\end_inset

 and several values of 
\begin_inset Formula $\hat{\zeta}$
\end_inset

 from 
\begin_inset Formula $10^{5}$
\end_inset

 to 
\begin_inset Formula $\infty$
\end_inset

 (unimproved).
 The critical temperature is 
\begin_inset Formula $T_{\star}/\mu\approx0.775$
\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 Graphics
	filename images/ch5/ssi-spinodal-temps.eps
	width 100col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:ssi-hf-spinodal-temps"

\end_inset

The upper spinodal temperature 
\begin_inset Formula $T_{\mathrm{us}}$
\end_inset

 versus 
\begin_inset Formula $\hat{\zeta}$
\end_inset

 for broken phase solutions of the SSI equations of motion in the Hartree-Fock
 approximation for 
\begin_inset Formula $\lambda=10$
\end_inset

, 
\begin_inset Formula $N=4$
\end_inset

, and 
\begin_inset Formula $\bar{X}=0.3$
\end_inset

.
 The critical temperature is 
\begin_inset Formula $T_{\star}/\mu\approx0.775$
\end_inset

.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
The mathematical origin of this loss of solution can be understood by considerin
g the zero temperature equations of motion, reproduced below.
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
0 & =\frac{\lambda}{6}\left(\bar{x}-\bar{X}\right)+\frac{1}{6}\left(N-1\right)\lambda\left(\frac{1}{16\pi^{2}}\bar{y}\ln\bar{y}+\left(\frac{4\bar{x}}{\hat{\zeta}\bar{y}}\right)^{1/3}\right)+\frac{1}{2}\lambda\frac{1}{16\pi^{2}}\bar{z}\ln\bar{z}+\left(N-1\right)\left(\frac{\bar{y}^{2}}{2\hat{\zeta}\bar{x}^{2}}\right)^{1/3},\label{eq:xb-eom}\\
\bar{y} & =\frac{\lambda}{6}\left(\bar{x}-\bar{X}\right)+\frac{1}{6}\left(N+1\right)\lambda\left(\frac{1}{16\pi^{2}}\bar{y}\ln\bar{y}+\left(\frac{4\bar{x}}{\hat{\zeta}\bar{y}}\right)^{1/3}\right)+\frac{1}{6}\lambda\frac{1}{16\pi^{2}}\bar{z}\ln\bar{z},\label{eq:yb-eom}\\
\bar{z} & =\frac{\lambda\bar{x}}{3}-\left(N-1\right)\left(\frac{\bar{y}^{2}}{2\hat{\zeta}\bar{x}^{2}}\right)^{1/3}.\label{eq:zb-eom}
\end{align}

\end_inset

Use 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:zb-eom"

\end_inset

 to eliminate 
\begin_inset Formula $\bar{z}$
\end_inset

 in the first two equations and consider the real and imaginary parts of
 the right hand sides of these equations as functions of 
\begin_inset Formula $\bar{x}$
\end_inset

 and 
\begin_inset Formula $\bar{y}$
\end_inset

.
 The relevant regions of the 
\begin_inset Formula $\bar{x}-\bar{y}$
\end_inset

 plane are shown in Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:loss-of-solution-plots"

\end_inset

.
 The blue regions satisfy 
\begin_inset Formula $\Re\left(\eqref{eq:xb-eom}\right)>0$
\end_inset

, yellow regions satisfy 
\begin_inset Formula $\Im\left(\eqref{eq:xb-eom}\right)\neq0$
\end_inset

 and the green regions satisfy 
\begin_inset Formula $\Re\left(\eqref{eq:yb-eom}\right)>\bar{y}$
\end_inset

.
 Note that, for 
\begin_inset Formula $\bar{x},\bar{y}>0$
\end_inset

, 
\begin_inset Formula $\Im\left(\eqref{eq:xb-eom}\right)\neq0$
\end_inset

 is equivalent to 
\begin_inset Formula $\bar{z}>0$
\end_inset

, as is 
\begin_inset Formula $\Im\left(\eqref{eq:yb-eom}\right)\neq0$
\end_inset

 which does not give anything new.
\end_layout

\begin_layout Standard
Valid solutions of the equations of motion are on the boundary of the blue
 and green regions simultaneously 
\emph on
and
\emph default
 outside of the yellow region.
 As 
\begin_inset Formula $\hat{\zeta}$
\end_inset

 is decreased it can be seen that the blue region 
\begin_inset Quotes eld
\end_inset

closes in
\begin_inset Quotes erd
\end_inset

 towards the origin, the green region grows upwards, and the yellow region
 grows to the right.
 Solutions cease to exist for 
\begin_inset Formula $\hat{\zeta}=\hat{\zeta}_{\star}\approx12200$
\end_inset

 where all three regions intersect at a common point.
 For all 
\begin_inset Formula $\hat{\zeta}<\hat{\zeta}_{\star}$
\end_inset

 there are no solutions (intersection points between the blue and green
 curves) which are also real (outside the yellow region).
 If the temperature is non-zero, the thermal contributions increase the
 real parts of the right hand sides which, in comparison with Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:loss-of-solution-plots"

\end_inset

, hastens the onset of loss of solutions, which is therefore achieved at
 a greater value 
\begin_inset Formula $\hat{\zeta}$
\end_inset

.
 Conversely, the temperature at which solutions are lost for a given 
\begin_inset Formula $\hat{\zeta}$
\end_inset

 increases as a function of 
\begin_inset Formula $\hat{\zeta}$
\end_inset

, which, of course, matches the behaviour seen in Figures 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:ssi-hf-solns"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:ssi-hf-spinodal-temps"

\end_inset

.
\end_layout

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

\begin_layout Plain Layout
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset Graphics
	filename images/ch5/ssi-hf-soln-existence-32768.eps
	width 45col%

\end_inset


\begin_inset Caption Standard

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\end_inset


\begin_inset space \hfill{}
\end_inset


\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset Graphics
	filename images/ch5/ssi-hf-soln-existence-16384.eps
	width 45col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset Graphics
	filename images/ch5/ssi-hf-soln-existence-12200.eps
	width 45col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\end_inset


\begin_inset space \hfill{}
\end_inset


\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset Graphics
	filename images/ch5/ssi-hf-soln-existence-10000.eps
	width 45col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:loss-of-solution-plots"

\end_inset

Real and imaginary parts of the right hand sides of 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:xb-eom"

\end_inset

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

\end_inset

 as functions of 
\begin_inset Formula $\bar{x}$
\end_inset

 and 
\begin_inset Formula $\bar{y}$
\end_inset

 for 
\begin_inset Formula $\lambda=10$
\end_inset

, 
\begin_inset Formula $N=4$
\end_inset

, 
\begin_inset Formula $\bar{X}=0.3$
\end_inset

 and 
\begin_inset Formula $\hat{\zeta}=2^{15}$
\end_inset

 (upper left), 
\begin_inset Formula $2^{14}$
\end_inset

 (upper right), 
\begin_inset Formula $12200$
\end_inset

 (lower left) and 
\begin_inset Formula $10^{4}$
\end_inset

 (lower right).
 The blue regions satisfy 
\begin_inset Formula $\Re\left(\eqref{eq:xb-eom}\right)>0$
\end_inset

, the yellow regions satisfy 
\begin_inset Formula $\Im\left(\eqref{eq:xb-eom}\right)\neq0$
\end_inset

 and the green regions satisfy 
\begin_inset Formula $\Re\left(\eqref{eq:yb-eom}\right)>\bar{y}$
\end_inset

.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Broken phase with 
\begin_inset Formula $m_{G}^{2}\to0$
\end_inset


\begin_inset CommandInset label
LatexCommand label
name "subsec:Broken-Phase-with-massless-Goldstones"

\end_inset


\end_layout

\begin_layout Standard
In order to find a broken phase solution without the pathological properties
 of the previous section one can try to find solutions with 
\begin_inset Formula $m_{G}^{2}\to0$
\end_inset

 in the 
\begin_inset Formula $\mathrm{V}\beta\to\infty$
\end_inset

 limit.
 To achieve this, take the scalings
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\xi & =\left(\mathrm{V}\beta\right)^{\alpha}\zeta=\left(\mathrm{V}\beta\right)^{\alpha}\mu^{2+4\alpha}\hat{\zeta},\\
m_{G}^{2} & =\left(\mathrm{V}\beta\right)^{-\gamma}\mu^{2-4\gamma}y,
\end{align}

\end_inset

where 
\begin_inset Formula $\gamma>0$
\end_inset

.
 The definitions of the other dimensionless variables (
\begin_inset Formula $x$
\end_inset

, 
\begin_inset Formula $z$
\end_inset

, etc.) are as before.
 Then
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\epsilon\sim\begin{cases}
\left(\frac{\left(\mathrm{V}\beta\right)^{\alpha+2\gamma-1}}{\left(\mu^{-4}\right)^{\alpha+2\gamma-1}}\right)^{1/3}\left(\frac{\hat{\zeta}}{4xy^{2}}\right)^{1/3}, & \alpha+2\gamma-1<0,\\
1-\frac{\left(\mu^{-4}\right)^{\alpha+2\gamma-1}}{\left(\mathrm{V}\beta\right)^{\alpha+2\gamma-1}}\frac{4xy^{2}}{\hat{\zeta}}, & \alpha+2\gamma-1>0.
\end{cases}
\end{equation}

\end_inset

One can take the equations of motion 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:SSI-2PI-HF-vev-eom"

\end_inset

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

\end_inset

 with the prescription 
\begin_inset Formula $\frac{1}{\mathrm{V}\beta}\frac{1}{m_{G}^{2}}\left(\frac{1}{\epsilon}-1\right)\to0$
\end_inset

 because, as discussed in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:ch5-Renormalisation-of-the-HF-truncation"

\end_inset

, the Goldstone tadpole reduces to the unmodified form in the massless case.
 The result is the equation of motion
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
0=m^{2}+\frac{\lambda}{6}v^{2}+\frac{1}{2}\left(N-1\right)\frac{\lambda}{3}\frac{T^{2}}{12}+\frac{1}{2}\lambda\left(\frac{m_{H}^{2}}{16\pi^{2}}\ln\frac{m_{H}^{2}}{\mu^{2}}+\mathcal{T}_{H}^{\mathrm{th}}\right)+\frac{1}{\xi}\left(N-1\right)2\left[m_{G}^{2}\epsilon\right]^{2},
\end{equation}

\end_inset

for the vev, and
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
0=m^{2}+\frac{\lambda}{6}v^{2}+\frac{1}{6}\left(N+1\right)\lambda\frac{T^{2}}{12}+\frac{1}{6}\lambda\left(\frac{m_{H}^{2}}{16\pi^{2}}\ln\frac{m_{H}^{2}}{\mu^{2}}+\mathcal{T}_{H}^{\mathrm{th}}\right),
\end{equation}

\end_inset

and
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
m_{H}^{2}=m^{2}+\frac{\lambda}{2}v^{2}+\frac{1}{6}\lambda\left(N-1\right)\frac{T^{2}}{12}+\frac{1}{2}\lambda\left(\frac{m_{H}^{2}}{16\pi^{2}}\ln\frac{m_{H}^{2}}{\mu^{2}}+\mathcal{T}_{H}^{\mathrm{th}}\right).
\end{equation}

\end_inset

Note that, remarkably, the last two equations are nothing but the SI-2PIEA
 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-H-eom"

\end_inset

, having used that 
\begin_inset Formula $\mathcal{T}_{G}^{\mathrm{fin}}=T^{2}/12$
\end_inset

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

.
 The only thing new is the modification of the vev equation by the term
 
\begin_inset Formula $\frac{1}{\xi}\left(N-1\right)2\left[m_{G}^{2}\epsilon\right]^{2}$
\end_inset

.
 To examine this further one must consider the three cases 
\begin_inset Formula $\alpha+2\gamma-1\gtreqless0$
\end_inset

 which govern the possible scaling behaviours of this term.
\end_layout

\begin_layout Standard
In the 
\begin_inset Formula $\alpha+2\gamma-1>0$
\end_inset

 case, 
\begin_inset Formula $\epsilon\to1$
\end_inset

 and
\begin_inset Formula 
\begin{equation}
\frac{1}{\xi}\left(N-1\right)2\left[m_{G}^{2}\epsilon\right]^{2}\to\mu^{2}\frac{\left(\mu^{-4}\right)^{\alpha+2\gamma}}{\left(\mathrm{V}\beta\right)^{\alpha+2\gamma}}\frac{\left(N-1\right)2y^{2}}{\hat{\zeta}}\to0.
\end{equation}

\end_inset

If on the other hand 
\begin_inset Formula $\alpha+2\gamma-1=0$
\end_inset

, 
\begin_inset Formula $\epsilon$
\end_inset

 is a constant as 
\begin_inset Formula $\mathrm{V}\beta\to\infty$
\end_inset

 and
\begin_inset Formula 
\begin{equation}
\frac{1}{\xi}\left(N-1\right)2\left[m_{G}^{2}\epsilon\right]^{2}\to\mu^{2}\frac{\left(\mu^{-4}\right)^{\alpha+2\gamma}}{\left(\mathrm{V}\beta\right)^{\alpha+2\gamma}}\frac{1}{\hat{\zeta}}\left(N-1\right)2y^{2}\epsilon^{2}=\mu^{2}\frac{\left(\mu^{-4}\right)}{\left(\mathrm{V}\beta\right)}\frac{1}{\hat{\zeta}}\left(N-1\right)2y^{2}\epsilon^{2}\to0.
\end{equation}

\end_inset

In both of these cases the vev equation is unmodified by SSI and cannot
 hold at the same time as the other two equations of motion.
 To see this solve the SI-2PI equations (c.f.
 section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Renormalisation-and-solution-SI-2PIEA-HF"

\end_inset

) to get
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
m_{H}^{2}=-2m^{2}-\frac{1}{3}\lambda\left(N+2\right)\frac{T^{2}}{12},
\end{equation}

\end_inset

and
\begin_inset Formula 
\begin{equation}
\frac{\lambda}{6}v^{2}=-m^{2}-\frac{1}{6}\left(N+1\right)\lambda\frac{T^{2}}{12}-\frac{1}{6}\lambda\left(\frac{m_{H}^{2}}{16\pi^{2}}\ln\frac{m_{H}^{2}}{\mu^{2}}+\mathcal{T}_{H}^{\mathrm{th}}\right).
\end{equation}

\end_inset

Now use these in the vev equation to get
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\frac{T^{2}}{12}=\frac{m_{H}^{2}}{16\pi^{2}}\ln\frac{m_{H}^{2}}{\mu^{2}}+\mathcal{T}_{H}^{\mathrm{th}},
\end{equation}

\end_inset

which only holds at 
\begin_inset Formula $T=0$
\end_inset

 (for 
\begin_inset Formula $\mu=\bar{m}_{H}$
\end_inset

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

.
 There is no solution at any other temperature.
\end_layout

\begin_layout Standard
The remaining case is 
\begin_inset Formula $\alpha+2\gamma-1<0$
\end_inset

.
 For this case the SSI term becomes
\begin_inset Formula 
\begin{equation}
\frac{1}{\xi}\left(N-1\right)2\left[m_{G}^{2}\epsilon\right]^{2}\to\mu^{2}\left(\frac{\left(\mu^{-4}\right)^{\alpha+2\gamma+2}}{\left(\mathrm{V}\beta\right)^{\alpha+2\gamma+2}}\right)^{1/3}\left(N-1\right)\left(\frac{y^{2}}{2\hat{\zeta}x^{2}}\right)^{1/3}.
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
The only consistent solution possible is 
\begin_inset Formula $\alpha+2\gamma+2=0$
\end_inset

 (which automatically satisfies the condition 
\begin_inset Formula $\alpha+2\gamma-1<0$
\end_inset

).
 In this case the vev equation of motion reduces to (in terms of dimensionless
 variables now)
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
0=-\frac{\lambda}{6}\bar{X}+\frac{\lambda}{6}x+\frac{1}{6}\left(N-1\right)\lambda\frac{T^{2}/\mu^{2}}{12}+\frac{1}{2}\lambda\left(\frac{z}{16\pi^{2}}\ln z+T_{H}\right)+\left(N-1\right)\left(\frac{y^{2}}{2\hat{\zeta}x^{2}}\right)^{1/3}.
\end{equation}

\end_inset

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

 equation from the vev equation gives
\begin_inset Formula 
\begin{equation}
\left(N-1\right)\left(\frac{y^{2}}{2\hat{\zeta}x^{2}}\right)^{1/3}=\frac{\lambda}{3}x-z,
\end{equation}

\end_inset

which can be easily solved for 
\begin_inset Formula $y$
\end_inset

, giving
\begin_inset Formula 
\begin{equation}
y^{2}=2\hat{\zeta}x^{2}\left(\frac{\frac{\lambda}{3}x-z}{N-1}\right)^{3}.
\end{equation}

\end_inset

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

 regardless of the value of 
\begin_inset Formula $y$
\end_inset

.
 The only constraint is that 
\begin_inset Formula $0\leq y<\infty$
\end_inset

 which requires 
\begin_inset Formula $\lambda x/3\geq z$
\end_inset

.
 This can be verified using the solution of the SI-2PI equations of motion
\begin_inset Formula 
\begin{align}
z & =1-\frac{T^{2}}{T_{\star}^{2}},\\
x & =\bar{X}-\left(\left(N+1\right)\frac{T^{2}/\mu^{2}}{12}+T_{H}\right),
\end{align}

\end_inset

(recalling 
\begin_inset Formula $\bar{z}=1$
\end_inset

 and 
\begin_inset Formula $\bar{X}=3/\lambda$
\end_inset

 are the zero temperature solutions and 
\begin_inset Formula $T_{\star}^{2}=12\bar{X}\mu^{2}/\left(N+2\right)$
\end_inset

) so that
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\frac{\lambda}{3}x-z & =\frac{\lambda}{3}\left[\bar{X}-\left(\left(N+1\right)\frac{T^{2}/\mu^{2}}{12}+T_{H}\right)\right]-\left(1-\frac{T^{2}}{T_{\star}^{2}}\right)\nonumber \\
 & =\left(N+2\right)\frac{T^{2}/\mu^{2}}{12\bar{X}}-\left(N+1\right)\frac{T^{2}/\mu^{2}}{12\bar{X}}-\frac{\lambda}{3}T_{H}=\frac{1}{\bar{X}}\left(\frac{T^{2}/\mu^{2}}{12}-T_{H}\right)\geq0,
\end{align}

\end_inset

since the thermal integral 
\begin_inset Formula $T_{H}$
\end_inset

 is maximised for massless particles.
 Thus 
\begin_inset Formula $0\leq y^{2}<\infty$
\end_inset

 and 
\begin_inset Formula $y$
\end_inset

 can always be chosen in 
\begin_inset Formula $0\leq y<\infty$
\end_inset

.
 Thus 
\begin_inset Formula $\xi\sim\left(\mathrm{V}\beta\right)^{-2\gamma-2}$
\end_inset

 and 
\begin_inset Formula $m_{G}^{2}\sim\left(\mathrm{V}\beta\right)^{-\gamma}$
\end_inset

 with 
\begin_inset Formula $\gamma>0$
\end_inset

 is identified as the unique limiting procedure that gives back the old
 SI-2PIEA from the SSI-2PIEA.
\end_layout

\begin_layout Standard
One can see that this procedure is the unique way of connecting the SI and
 SSI methods by directly matching the SSI term in 
\begin_inset Formula $\Gamma_{\xi}^{\mathrm{SSI}}$
\end_inset

 with the Lagrange multiplier term in 
\begin_inset Formula $\Gamma^{\mathrm{SI}}$
\end_inset

.
 Recall that the constraint term in the SI-2PIEA is (c.f.
 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:simple-constraint"

\end_inset

)
\begin_inset Formula 
\begin{equation}
\mathcal{C}=\frac{i}{2}\ell_{A}^{a}\mathcal{W}_{a}^{A}.
\end{equation}

\end_inset

Recall also that the constraint is singular.
 One must proceed by violating the constraint by an amount 
\begin_inset Formula $\sim\eta$
\end_inset

 then taking a limit 
\begin_inset Formula $\eta\to0$
\end_inset

 such that 
\begin_inset Formula $\ell\eta$
\end_inset

 is a constant.
 Previously this procedure was carried out at the level of the equations
 of motion.
 Now it is convenient to implement this at the level of the action by shifting
 the constraint term to
\begin_inset Formula 
\begin{equation}
\frac{i}{2}\ell_{A}^{a}\left(\mathcal{W}_{a}^{A}-i\mathcal{F}_{a}^{A}\right),
\end{equation}

\end_inset

where 
\begin_inset Formula $\mathcal{F}_{a}^{A}\sim\eta$
\end_inset

 is the regulator written in 
\begin_inset Formula $\mathrm{O}\left(N\right)$
\end_inset

-covariant form.
 Setting the SI constraint term equal to the SSI term gives
\begin_inset Formula 
\begin{equation}
\frac{i}{2}\ell_{A}^{a}\left(\mathcal{W}_{a}^{A}-i\mathcal{F}_{a}^{A}\right)=-\frac{1}{2\xi}\mathcal{W}_{a}^{A}\mathcal{W}_{a}^{A}.
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Substituting 
\begin_inset Formula $\mathcal{W}_{a}^{A}$
\end_inset

 gives
\begin_inset Formula 
\begin{equation}
2\frac{i}{2}\mathrm{V}\beta\ell_{cN}^{a}\left(iP_{ca}^{\perp}\left[\Delta_{G}^{-1}\left(0,\boldsymbol{0}\right)\right]v-i\mathcal{F}_{a}^{cN}\right)=-\frac{1}{2\xi}\left(-2\left(N-1\right)v^{2}\mathrm{V}\beta\left[\Delta_{G}^{-1}\left(0,\boldsymbol{0}\right)\right]^{2}\right),
\end{equation}

\end_inset

having used 
\begin_inset Formula $\int_{y}\Delta_{G}^{-1}\left(x,y\right)=\Delta_{G}^{-1}\left(0,\boldsymbol{0}\right)$
\end_inset

.
 Without loss of generality one can set
\begin_inset Formula 
\begin{align}
\ell_{cN}^{a} & =P_{ac}^{\perp}\left(\frac{1}{N-1}\ell_{dN}^{d}\right),\\
\mathcal{F}_{a}^{cN} & =P_{ac}^{\perp}\mathcal{F},
\end{align}

\end_inset

and find
\begin_inset Formula 
\begin{equation}
-\ell_{cN}^{c}\left(\Delta_{G}^{-1}\left(0,\boldsymbol{0}\right)v-\mathcal{F}\right)=\frac{1}{\xi}\left(N-1\right)v^{2}\left[\Delta_{G}^{-1}\left(0,\boldsymbol{0}\right)\right]^{2}.
\end{equation}

\end_inset

Now recall that the usual form of the SI regulator is 
\begin_inset Formula $\Delta_{G}^{-1}\left(0,\boldsymbol{0}\right)v=m_{G}^{2}v=\eta m^{3}$
\end_inset

 where 
\begin_inset Formula $m$
\end_inset

 is some arbitrary mass scale (it is convenient to take 
\begin_inset Formula $m=\mu$
\end_inset

).
 This identifies 
\begin_inset Formula $\mathcal{F}=\eta m^{3}$
\end_inset

.
 The 
\begin_inset Formula $\eta\to0$
\end_inset

 limit is taken so that 
\begin_inset Formula $\eta\ell_{cN}^{c}=\ell_{0}v$
\end_inset

 is a constant.
 Using this and 
\begin_inset Formula $\Delta_{G}^{-1}\left(0,\boldsymbol{0}\right)=\epsilon m_{G}^{2}$
\end_inset

 gives
\begin_inset Formula 
\begin{equation}
-\frac{\ell_{0}v}{\eta}\left(\epsilon m_{G}^{2}v-\eta\mu^{3}\right)=\frac{1}{\xi}\left(N-1\right)v^{2}\left[\epsilon m_{G}^{2}\right]^{2}.
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
It is now convenient to take 
\begin_inset Formula $\eta=\left(\mathrm{V}\beta\right)^{-\delta}\mu^{-4\delta}$
\end_inset

 with 
\begin_inset Formula $\delta>0$
\end_inset

.
 Taking also the usual scalings for 
\begin_inset Formula $\xi$
\end_inset

 and 
\begin_inset Formula $m_{G}^{2}$
\end_inset

 one finds
\begin_inset Formula 
\begin{equation}
-\left(\mathrm{V}\beta\right)^{\delta-\gamma}\mu^{4\left(\delta-\gamma\right)}\epsilon\ell_{0}xy+\ell_{0}\sqrt{x}=\frac{\mu^{-4\alpha-8\gamma}}{\left(\mathrm{V}\beta\right)^{\alpha+2\gamma}}\epsilon^{2}\left(N-1\right)\frac{xy^{2}}{\hat{\zeta}}.
\end{equation}

\end_inset

If 
\begin_inset Formula $\alpha+2\gamma-1>0$
\end_inset

 asymptotic balance is impossible (dominant terms can be matched, but not
 subdominant terms).
 Likewise, balance cannot be achieved for 
\begin_inset Formula $\alpha+2\gamma-1=0$
\end_inset

.
 If 
\begin_inset Formula $\alpha+2\gamma-1<0$
\end_inset

, however,
\begin_inset Formula 
\begin{multline}
-\left(\mathrm{V}\beta\right)^{\delta-\gamma}\mu^{4\left(\delta-\gamma\right)}\left(\frac{\left(\mathrm{V}\beta\right)^{\alpha+2\gamma-1}}{\left(\mu^{-4}\right)^{\alpha+2\gamma-1}}\right)^{1/3}\left(\frac{\hat{\zeta}}{4xy^{2}}\right)^{1/3}\ell_{0}xy+\ell_{0}\sqrt{x}\\
=\frac{\mu^{-4\alpha-8\gamma}}{\left(\mathrm{V}\beta\right)^{\alpha+2\gamma}}\left(\frac{\left(\mathrm{V}\beta\right)^{\alpha+2\gamma-1}}{\left(\mu^{-4}\right)^{\alpha+2\gamma-1}}\right)^{2/3}\left(\frac{\hat{\zeta}}{4xy^{2}}\right)^{2/3}\left(N-1\right)\frac{xy^{2}}{\hat{\zeta}}.
\end{multline}

\end_inset

Matching powers of 
\begin_inset Formula $\left(\mathrm{V}\beta\right)$
\end_inset

 on both sides gives
\begin_inset Formula 
\begin{align}
0 & =3\delta-1+\alpha-\gamma,\\
0 & =\alpha+2\gamma+2,
\end{align}

\end_inset

which has the solution
\begin_inset Formula 
\begin{align}
\alpha & =-2\delta,\\
\gamma & =\delta-1.
\end{align}

\end_inset


\begin_inset Formula $\gamma>0$
\end_inset

 requires 
\begin_inset Formula $\delta>1$
\end_inset

.
 Substituting this into the SI=SSI equation gives
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
-\left(\frac{\hat{\zeta}}{4xy^{2}}\right)^{1/3}\ell_{0}xy+\ell_{0}\sqrt{x}=\left(\frac{\hat{\zeta}}{4xy^{2}}\right)^{2/3}\left(N-1\right)\frac{xy^{2}}{\hat{\zeta}},
\end{equation}

\end_inset

which can be solved for 
\begin_inset Formula $\hat{\zeta}$
\end_inset

, giving
\begin_inset Formula 
\begin{equation}
\hat{\zeta}^{1/3}=\left(\frac{1}{2\sqrt{x}y}\right)^{1/3}\left(1\pm\sqrt{1-\left(N-1\right)\frac{y}{\ell_{0}}}\right).
\end{equation}

\end_inset

This is the desired connection between the SSI stiffness parameter 
\begin_inset Formula $\hat{\zeta}$
\end_inset

 and the SI Lagrange multiplier 
\begin_inset Formula $\ell_{0}$
\end_inset

.
\end_layout

\begin_layout Section
Summary
\begin_inset CommandInset label
LatexCommand label
name "sec:ch5-Summary"

\end_inset


\end_layout

\begin_layout Standard
This chapter introduced, for the first time to the author's knowledge, a
 new method of symmetry improvement called 
\emph on
soft symmetry improvement
\emph default
 or SSI.
 As opposed to the 
\emph on
hard
\emph default
 symmetry improvement (SI) of Chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:chap4"

\end_inset

, Ward identities are imposed weakly in the sense of least squared error.
 The relative strength of the soft constraint is determined by a new stiffness
 parameter 
\begin_inset Formula $\xi$
\end_inset

 which can be tuned from zero (equivalent to hard SI) to 
\begin_inset Formula $\infty$
\end_inset

 (equivalent to no constraint), with the hope that the added freedom allows
 the SSI method to circumvent some of the problems of the SI method for
 some range of finite 
\begin_inset Formula $\xi$
\end_inset

.
 Here the SSI method was studied for the 2PIEA in the Hartree-Fock approximation
, though the generalisation to the 
\begin_inset Formula $n$
\end_inset

PIEA and higher order approximations is conceptually straightforward (leading
 only to greater technical complication).
\end_layout

\begin_layout Standard
Unfortunately the goals of this program cannot be fully realised in the
 most interesting case of the infinite volume limit 
\begin_inset Formula $\beta\mathrm{V}\to\infty$
\end_inset

.
 Due to an infrared sensitivity of the SSI method, 
\begin_inset Formula $\xi$
\end_inset

 must be scaled with 
\begin_inset Formula $\beta\mathrm{V}$
\end_inset

 in the limit in order to obtain non-trivial behaviour.
 Three consistent limiting behaviours were found.
 Two of these are equivalent to the unimproved 2PIEA and SI-2PIEA, respectively,
 both of which have been studied previously.
 The remaining case is a novel one and was studied in detail in section
 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Broken-Phase-with-massive-Goldstones"

\end_inset

.
\end_layout

\begin_layout Standard
In the novel limit the Goldstone self-energy is momentum dependent even
 in the Hartree-Fock approximation.
 The zero momentum Goldstone mode is massless as required by Goldstone's
 theorem, but the remaining modes are massive with a constant 
\begin_inset Formula $m_{G}^{2}\neq0$
\end_inset

.
 Perhaps counter-intuitively, 
\begin_inset Formula $m_{G}^{2}$
\end_inset

 
\emph on
increases
\emph default
 as the symmetry improvement constraint is more strongly imposed.
 A strong first order phase transition is found for all values of the (scaled)
 stiffness parameter 
\begin_inset Formula $\hat{\zeta}$
\end_inset

.
 Further, there is a critical value 
\begin_inset Formula $\hat{\zeta}_{c}$
\end_inset

 such that for 
\begin_inset Formula $\hat{\zeta}<\hat{\zeta}_{c}$
\end_inset

 solutions cease to exist for a range of temperatures below the critical
 temperature.
 As 
\begin_inset Formula $\hat{\zeta}\to0$
\end_inset

 this range increases and at some finite value 
\begin_inset Formula $\hat{\zeta}_{\star}$
\end_inset

 the solution ceases to exist even at zero temperature.
 This loss of solution was confirmed both analytically and numerically.
\end_layout

\begin_layout Standard
This behaviour highlights the difficulty faced by the symmetry improvement
 program.
 Ward identities are by nature infrared properties of a field theory and
 imposing them amounts to modifying a theory at large distances, while trying
 not to modify anything at short distance scales.
 However, due to the use of self-consistent propagators and vertices, 
\begin_inset Formula $n$
\end_inset

PIEAs inherently couple IR and UV behaviour.
 This can be seen in the loss of solutions of the SI-2PIEA in the two loop
 truncation as discussed in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:The-Two-Loop-Solution"

\end_inset

: it cannot be guaranteed for the SI-2PIEA in a given truncation that all
 UV regulators lead to the same IR physics.
 In the SSI method in the new limit, the imposition of the Ward identity
 (even weakly) is incompatible with the existence of massless non-zero modes
 of the Goldstone propagator in the 
\begin_inset Formula $\beta\mathrm{V}\to0$
\end_inset

 limit.
 This again leads to loss of solutions.
 Further, due to the unavoidable scaling of an IR quantity (the SSI term)
 in the 
\begin_inset Formula $\beta\mathrm{V}\to\infty$
\end_inset

 limit, the unimproved, SI and new limits become decoupled.
 There is no version of the theory which smoothly interpolates between the
 unimproved and symmetry improved cases in the 
\begin_inset Formula $\beta\mathrm{V}\to\infty$
\end_inset

 limit.
 To put it colloquially, infinite volume is too big a space to 
\begin_inset Quotes eld
\end_inset

softly
\begin_inset Quotes erd
\end_inset

 impose anything on.
\end_layout

\begin_layout Standard
It would be interesting to examine whether at finite 
\begin_inset Formula $\beta\mathrm{V}$
\end_inset

 any physically reasonable results could be obtained.
 This could be studied by, for example, putting the universe in a box the
 size of the Hubble scale.
 However, this would required redoing the renormalisation procedure in finite
 volume and re-evaluating the thermal sums which are no longer trivial.
 Thus, this investigation is deferred to future work.
\end_layout

\end_body
\end_document