Meta code

Table of Contents

1 Re-usable meta code packages
2 References

1 Re-usable meta code packages

FlexibleSUSY contains several packages with general algorithms needed in Quantum Field Theory calculations and implementations of analytic expressions from the literature.

1.1 Passarino-Veltman loop functions

meta/LoopFunctions.m contains an implementation of Passarino-Veltman 1-loop functions A_0, B_0, B_1, B_{00}, B_{11}, etc.

1.2 Passarino-Veltman loop functions for zero external momenta

meta/LoopFunctionsZeroMomentum.m contains an implementation of the Passarino-Veltman 1-loop functions A_0, B_0, \frac{\partial B_0}{\partial p^2}, C_0, D_0, E_0 and F_0 for zero external momenta.

Example:

Get["meta/LoopFunctionsZeroMomentum.m"];
F0[x,x,x,x,x,x,q] //. loopFunctionsZeroMomentum

yields:

1/(20*x^4)

1.3 Renormalization group equation integrator

meta/RGIntegrator.m contains a routine, which perturbatively integrates a system of coupled renormalization group equations:

Get["meta/RGIntegrator.m"];
?RGIntegrate

Example 1: Integrating a single RGE up to 2-loop order:

Get["meta/RGIntegrator.m"];
betas = { {g, h a g^2} };
RGIntegrate[betas, Q1, Q2, loopOrder -> 2]

yields:

{g[Q1] -> g[Q2] + a h g[Q2]^2 Log[Q1/Q2] + a^2 h^2 g[Q2]^3 Log[Q1/Q2]^2}

Example 2: Interation of the following coupled system:

Get["meta/RGIntegrator.m"];
betas = { {g, h a g^2, h^2 b g^2 l^2},
          {l, h (c l^2 + d g^2), h^2 (e l^4 + f g^2 l^2)} };
RGIntegrate[betas, Q1, Q2]

yields:

{g[Q1] -> g[Q2] + a*h*g[Q2]^2*Log[Q1/Q2] +
    h^2*(b*g[Q2]^2*l[Q2]^2*Log[Q1/Q2] + a^2*g[Q2]^3*Log[Q1/Q2]^2),
  l[Q1] -> l[Q2] + h*(d*g[Q2]^2*Log[Q1/Q2] + c*l[Q2]^2*Log[Q1/Q2]) +
    h^2*(f*g[Q2]^2*l[Q2]^2*Log[Q1/Q2] + e*l[Q2]^4*Log[Q1/Q2] +
      a*d*g[Q2]^3*Log[Q1/Q2]^2 + c*d*g[Q2]^2*l[Q2]*Log[Q1/Q2]^2 +
      c^2*l[Q2]^3*Log[Q1/Q2]^2)}

1.4 Beta functions

1.4.1 Standard Model

meta/ThreeLoopSM.m contains a routine which returns the beta functions of the Standard Model up to the 5-loop level. The beta functions are stored in the files meta/SM/beta_*.m, which have been obtained at 3-loop level from [1303.4364], [1504.05200] with some additional 4- and 5-loop corrections from [1508.00912], [1508.02680], [1604.00853], [1606.08659].

Note

The loop factors 1/(4\pi)^2 have been omitted.

Example: Extracting the 5-loop QCD beta function for g_3:

b = Get["meta/SM/beta_g3.m"];

bg3 = { k^1 b[[1]],
        k^2 b[[2]],
        k^3 b[[3]],
        k^4 b[[4]],
        k^5 b[[5]] };

bg3 /. { g1 -> 0, g2 -> 0, gb -> 0, gt -> 0 } // Chop

Output:

{ -7*g3^3*k,
  -26*g3^5*k^2,
  (65*g3^7*k^3)/2,
  -2472.2837425797156*g3^9*k^4,
  271.4283824198132*g3^11*k^5 }

1.4.2 MSSM

meta/ThreeLoopMSSM.m contains a routine which returns the beta functions of the MSSM up to the 3-loop level. The beta functions are stored in the files meta/MSSM/beta_*.m, which have been obtained from http://www.liv.ac.uk/~dij/betas/allgennb.log [hep-ph:0308231].

Example: Extracting the 3-loop QCD beta function for g_3 in the MSSM:

trace[args__] := Tr[Dot[args]];
Adj = ConjugateTranspose;
Yt = Yb = Ye = Array[0&, {3,3}];

b = Get["meta/MSSM/beta_g3.m"];

bg3 = { k^1 b[[1]],
        k^2 b[[2]],
        k^3 b[[3]] };

bg3 /. { g1 -> 0, g2 -> 0 }

Output:

{-3*g3^3*k, 14*g3^5*k^2, (347*g3^7*k^3)/3}

1.5 Loop corrections to masses

1.5.1 Standard Model

meta/SM/Mh2_effpot.m contains the QCD contributions to the 4-loop effective Higgs potential in the Standard Model from [1508.00912]

Example:

Get["meta/SM/Mh2_effpot.m"];

k = 1/(4 Pi)^2;
yt = 0.9;
g3 = 1.166;
v = 247.5;
\[Lambda] = 0.25;
mu2 = -8.55 10^3;
mt = yt v / Sqrt[2];
Q = 173.34;

Sqrt[DMh2]

Output:

124.926

meta/ThreeLoopQCD.m contains a routine, which returns the ratio of the \overline{\text{MS}} top mass over the top pole mass in the SM up to 3-loop level in QCD from [hep-ph:9912391], Eq. (10). The expression contains the full renormalization scale dependence, which has been taken from [hep-ph:9911434].

Example:

Get["meta/ThreeLoopQCD.m"];
Start["SM"];
FlexibleSUSY`M[Fu] = mt;

h = k (4 Pi)^2;

MfOvermf = GetMTopPoleOverMTopMSbar[{1,h,h^2,h^3}] /. {
    Log[Q^2/mt^2] -> -Lbar[t],
    Log[mt^2/Q^2] -> Lbar[t]
};

Mt = mt N[Collect[MfOvermf, {k, g3, Lbar[__]}, Simplify]]

Output:

mt*(1 +
    g3^2*k*(5.333333333333333 - 4*Lbar[t]) +
    g3^4*k^2*(131.78498721717762 - 80.66666666666667*Lbar[t] + 22*Lbar[t]^2) +
    g3^6*k^3*(4712.740192659316 - 2031.1382275647934*Lbar[t] + 710*Lbar[t]^2 - 132*Lbar[t]^3))

meta/TwoLoopQCD.m contains routines, which return the ratio of the top pole mass over the running top mass up to the 2-loop level in the \overline{\text{MS}} and \overline{\text{DR}} schemes [hep-ph:0210258], [hep-ph:9803493].

1.5.2 MSSM

meta/TwoLoopMSSM.m contains routines, which return the analytic 2-loop corrections to the Higgs masses in the CP-conserving MSSM [hep-ph:0105096].

1.6 Threshold corrections

1.6.1 Standard Model

meta/SM/mf_3loop_qcd.m contains the 3-loop relation between a quark pole mass and the corresponding running \overline{\text{MS}} mass from [hep-ph:9912391], [hep-ph:9911434].

Example:

Get["meta/SM/mf_3loop_qcd.m"];

L = Lbar[t];
NL = 5; (* number of light quark masses *)
NH = 1; (* number of heavy quark masses *)

Mt = mt N[Collect[MfOvermf, {k, g3, Lbar[__]}, Simplify]]

Output:

mt*(1 +
    g3^2*k*(5.333333333333333 - 4*Lbar[t]) +
    g3^4*k^2*(131.78498721717762 - 80.66666666666667*Lbar[t] + 22*Lbar[t]^2) +
    g3^6*k^3*(4712.740192659316 - 2031.1382275647934*Lbar[t] + 710*Lbar[t]^2 - 132*Lbar[t]^3))

meta/SM/mt_4loop_qcd.m contains the 4-loop QCD relation between the top quark pole mass and the corresponding running \overline{\text{MS}} mass from [1604.01134], [1502.01030], [1606.06754].

Example:

Get["meta/SM/mt_4loop_qcd.m"];

L = Lbar[t];

Mt = mt N[Collect[MtOvermt, {k, g3, Lbar[__]}, Simplify]]

Output:

mt*(1 +
    g3^2*k*(5.333333333333333 - 4*Lbar[t]) +
    g3^4*k^2*(131.78498721717762 - 80.66666666666667*Lbar[t] + 22*Lbar[t]^2) +
    g3^6*k^3*(4712.740192659316 - 2031.1382275647934*Lbar[t] + 710*Lbar[t]^2 - 132*Lbar[t]^3) +
    g3^8*k^4*(211681.74421123447 - 104673.38261571848*Lbar[t] + 22162.91142653778*Lbar[t]^2 - 5638*Lbar[t]^3 + 825*Lbar[t]^4))

meta/SM/mt_2loop_gaugeless.m contains the 2-loop relation between the top quark pole mass and the corresponding running \overline{\text{MS}} mass from [1604.01134] in the gaugeless limit. Contributions of O(\alpha_s^2, \alpha_s\alpha_t, \alpha_t^2, \alpha_t\lambda^n) are included. The relation is gauge-independent.

meta/SM/as_4loop_qcd.m contains the 4-loop QCD relation between the strong coupling \alpha_s^{n_f} with n_f flavours and \alpha_s^{n_l} with n_l = n_f - 1 flavours in the \overline{\text{MS}} scheme [hep-ph:0512060].

Example:

nl = 5;

L = Log[Q^2/mf[Q]^2];

alphaS = Get["meta/SM/as_4loop_qcd.m"];

1.6.2 2HDM

meta/THDM/Thresholds_1L_full.m contains the implementation of the complete analytic 1-loop threshold corrections of the THDM and the THDM + Higgsinos + gauginos to the MSSM [0901.2065].

Example:

Get["meta/THDM/Thresholds_1L_full.m"];

tc = (4 Pi)^2 GetTHDMThresholds1L[];

$Assumptions = { Element[ht, Reals], Element[Mu, Reals], Element[At, Reals] };

Yu[i_, k_] := DiagonalMatrix[{0,0,ht}][[i,k]];
Tu[i_, k_] := DiagonalMatrix[{0,0,ht At}][[i,k]];
Yd[__] := 0;
Ye[__] := 0;
Td[__] := 0;
Te[__] := 0;
g2 = gY = 0;

{l1, l2, l3, l4, l5, l6, l7} = Collect[tc, ht, Simplify]

Output:

{ -3*ht^4*Mu^4*D0[msq[3], msq[3], msu[3], msu[3]],
  -3*ht^4*(B0[msq[3], msq[3], Q] + B0[msu[3], msu[3], Q] + At^2*(2*C0[msq[3], msq[3], msu[3]] + 2*C0[msq[3], msu[3], msu[3]] + At^2*D0[msq[3], msq[3], msu[3], msu[3]])),
  -3*ht^4*Mu^2*(C0[msq[3], msu[3], msu[3]] + At^2*D0[msq[3], msq[3], msu[3], msu[3]]),
  -3*ht^4*Mu^2*(C0[msq[3], msq[3], msu[3]] + At^2*D0[msq[3], msq[3], msu[3], msu[3]]),
  -3*At^2*ht^4*Mu^2*D0[msq[3], msq[3], msu[3], msu[3]],
   3*At*ht^4*Mu^3*D0[msq[3], msq[3], msq[3], msq[3]],
   3*At*ht^4*Mu*(C0[msq[3], msq[3], msu[3]] + C0[msq[3], msu[3], msu[3]] + At^2*D0[msq[3], msq[3], msq[3], msq[3]]) }

1.6.3 MSSM

meta/MSSM/tquark_2loop_strong.m contains the analytic expression for the 2-loop relation O(\alpha_s^2) between the top quark pole mass and the \overline{\text{DR}} top mass in the MSSM [hep-ph:0210258], [hep-ph:0507139].

meta/MSSM/bquark_2loop_sqcd_decoupling.m contains the analytic expression for the 2-loop relation O(\alpha_s^2) between the \overline{\text{MS}} bottom quark mass in the Standard Model (without the top quark) and the \overline{\text{DR}} bottom mass in the MSSM [0707.0650].

meta/MSSM/dmtauas2.m contains the analytic expression for the 2-loop relation between the tau lepton pole mass and the \overline{\text{DR}} tau mass in the MSSM.

meta/MSSM/das2.m contains the analytic expression for the 2-loop relation between the \overline{\text{MS}} \alpha_s in the Standard Model (without the top quark) and the \overline{\text{DR}} value in the MSSM [hep-ph:0509048], [0810.5101], [1009.5455].

1.7 C++ source code formatting

meta/TextFormatting.m contains routines for text formatting of long expressions in C/C++ form, see WrapText[] and IndentText[].

Example: Formatting long expression:

Get["meta/TextFormatting.m"];

(* long expression *)
dmt = Get["meta/MSSM/tquark_2loop_strong.m"];

maxWidth = 70;
indent = 3;

"dmt = " <> WrapText[ToString[dmt, CForm], maxWidth, indent]

Output:

dmt = (Power(GS,4)*((-11*colorCA*colorCF*MGl*mmst1*s2t)/(-mmgl + mmst1) + (6
   *Power(colorCF,2)*MGl*mmst1*s2t)/(-mmgl + mmst1) - (6*Power(colorCF
   ,2)*MGl*mmst2*s2t)/(-mmgl + mmst1) + (6*Power(colorCF,2)*MGl*mmst1*
   mmst2*s2t)/((-mmgl + mmst1)*(mmst1 - mmst2)) - (6*Power(colorCF,2)*
   MGl*Power(mmst2,2)*s2t)/((-mmgl + mmst1)*(mmst1 - mmst2)) + (11*
   colorCA*colorCF*MGl*mmst2*s2t)/(-mmgl + mmst2) - (6*Power(colorCF,2
   )*MGl*mmst2*s2t)/(-mmgl + mmst2) - (Power(colorCF,2)*MGl*mmst1*
   Power(s2t,3))/(-mmgl + mmst1) + (7*Power(colorCF,2)*MGl*mmst2*Power
   (s2t,3))/(-mmgl + mmst1) - (6*Power(colorCF,2)*MGl*mmst1*mmst2*[...]