Advanced usage of FlavorKit to calculate new Wilson coefficients

The user can also implement new operators and obtain analytical expressions for their Wilson coefficients. In this case, he will need to use PreSARAH which, with the help of FeynArts and FormCalc, provides generic expressions for the coefficients, later to be adapted to specific models with SARAH.

Introduction

New operators can be implemented by extending the content of the folder

 /FlavorKit/[$Type]/Operators/

In the current version of FlavorKit, 3- and 4-point operators are supported. Each operator is defined by a .m -file. These files contain information about the external particles, the kind of considered diagrams (tree-level, self-energies, penguins, boxes) as well as generic expressions for the coefficients. These expressions, derived from the generic Feynman diagrams contributing to the coefficients, are written in the form of a Mathematica code, which can be used to generate Fortrancode.

For the automatization of the underlying calculations we have created an additional Mathematica package called PreSARAH, which can be used to create the files for all 4-fermion as well as 2-fermion-1-boson operators. This package creates not only the infrastructure to include the operators in the SPheno output of SARAH but makes also use of FeynArts and FormCalc to calculate the amplitudes and to extract the coefficient of the demanded operators. It takes into account all topologies with 2 or 4 external fermions and all possible, renormalisable couplings.

Input for PreSARAH

In order to derive the results for the Wilson coefficients, PreSARAH needs an input file with the following information:

ConsideredProcess : A string which defines the generic type for the process
- 4Fermion
- 2Fermion1Scalar
- 2Fermion1Vector
NameProcess : A string to uniquely define the process
ExternalFields : The external fields. Possible names are ChargedLepton, Neutrino, DownQuark, UpQuark, ScalarHiggs, PseudoScalar, Zboson, Wboson The particles.m file is used to define for each model which particle corresponds to SM states using the Description statement together with Leptons, Neutrinos, Down-Quarks, Up-Quarks, Higgs, Pseudo-Scalar Higgs, Z-Boson, W-Boson. If there is a mixture between the SM particles and other states (like in R-parity violating SUSY or in models with additional vector quarks/leptons) the combined state has to be labeled according to the description for the SM state. Notice that in the SM Pseudo-Scalar Higgs is just the neutral Goldstone boson. If an external state is not present in a given model or has not been defined as such in the particles.m file the corresponding Wilson coefficients are not calculated by SPheno.
FermionOrderExternal : the fermion order to apply the Fierz transformation (see the FormCalc manual for more details)
NeglectMasses : which external masses can be neglected (a list of integers counting the external fields)
ColorFlow : defines the color flow in the case of four quark operators. To contract the colors of external fields, ColorDelta is used, i.e ColorFlow = ColorDelta[1,2]*ColorDelta[3,4] assigns (\bar{q}^\alpha \Gamma q_\alpha)(\bar{q}^\beta \Gamma' q_\beta).
AllOperators : a list with the definition of the operators. This is a two dimensional list, where the first entry defines the name of the operator and the second one the Lorentz structure. The operators are expressed in the chiral basis and the syntax for Dirac chains in FormCalc is used:
- 6 forP_L = \frac{1}{2}(1-\gamma_5), 7 forP_R = \frac{1}{2}(1-\gamma_5)
- Lor[1], Lor[2] for \gamma_\mu, \gamma_\nu.
- ec[3] for the helicity of an external gauge boson.
- k[N] for the momentum of the external particle N (N is an integer).
- Pair[A,B] is used to contract Lorentz indices. For instance, Pair[k[1],ec[3]] stands for k_μ¹ϵ^μ, *
- A Dirac chain starting with a negative first entry is taken to be anti-symmetrized.
See the FormCalc manual for more details. To make the definitions more readable, not the full DiracChain object of FeynArts /FormCalc has to be defined: PreSARAH puts everything with the head Op into a Dirac chain using the defined fermion order. For 4-fermion operators the combination of both operators is written as dot product. For instance Op[6].Op[6] is internally translated into
```
 DiracChain[Spinor[k[1],MassEx1,-1],6,Spinor[k[2],MassEx2,1]]*
             DiracChain[Spinor[k[3],MassEx3,-1],6,Spinor[k[4],MassEx4,1]]
```
while Op[6] Pair[ec[3],k[1] becomes
```
 DiracChain[Spinor[k[1],MassEx1,-1],6,Spinor[k[2],MassEx2,1]] Pair[ec[3],k[1]]
```
CombinationGenerations : the combination of external generations for which the operators are calculated by SPheno
Filters : a list of filters to drop specific diagrams. Possible entries are NoBoxes, NoPenguins, NoTree, NoCrossedDiagrams.
- Filters = {NoBoxes, NoPenguins} can be used for processes which are already triggered at tree-level
- Filters = {NoPenguins} might be useful for processes which at the 1-loop level are only induced by box diagrams
- Filters = {NoCrossedDiagrams} is used to drop diagrams which only differ by a permutation of the external fields.

For instance, the PreSARAH input to calculate the coefficient of the(\bar{\ell}\Gamma \ell)(\bar{d} \Gamma' d) operator reads

NameProcess="2L2d";
ConsideredProcess = "4Fermion";
ExternalFields={{ChargedLepton,bar[ChargedLepton],
                   DownQuark,bar[DownQuark]}};

FermionOrderExternal={2,1,4,3};
NeglectMasses={1,2,3,4};


AllOperators={
   (* scalar operators*)
   {OllddSLL,Op[7].Op[7]},
   {OllddSRR,Op[6].Op[6]},
   {OllddSRL,Op[6].Op[7]},
   {OllddSLR,Op[7].Op[6]},

   (* vector operators*)
   {OllddVRR,Op[7,Lor[1]].Op[7,Lor[1]]},
   {OllddVLL,Op[6,Lor[1]].Op[6,Lor[1]]},
   {OllddVRL,Op[7,Lor[1]].Op[6,Lor[1]]},
   {OllddVLR,Op[6,Lor[1]].Op[7,Lor[1]]},

   (* tensor operators*)
   {OllddTLL,Op[-7,Lor[1],Lor[2]].Op[-7,Lor[1],Lor[2]]},
   {OllddTLR,Op[-7,Lor[1],Lor[2]].Op[-6,Lor[1],Lor[2]]},
   {OllddTRL,Op[-6,Lor[1],Lor[2]].Op[-7,Lor[1],Lor[2]]},
   {OllddTRR,Op[-6,Lor[1],Lor[2]].Op[-6,Lor[1],Lor[2]]}
};

CombinationGenerations = {{2,1,1,1}, {3,1,1,1}, {3,2,1,1},
                          {2,1,2,2}, {3,1,2,2}, {3,2,2,2}};

Filters = {};

Here, we neglect all external masses in the operators (NeglectMasses={1,2,3,4} ), and the different coefficients of the scalar operators (\bar{\ell}P_X \ell)(\bar{d} P_Y d) are called OllddSXY, the ones for the vector operators (\bar{\ell} P_X \gamma_\mu \ell)(\bar{d} P_Y \gamma^\mu d) are called OllddVYX and the ones for the tensor operators(\bar{\ell} P_X \sigma_{\mu \nu} \ell)(\bar{d} \sigma^{\mu \nu} P_Y d) OllddTYX, with X,Y=L,R. Notice that FormCalc returns the results in form ofP_Xγ_μ while in the literature the orderγ_μP_X is often used. Finally, SPheno will not calculate all possible combinations of external states, but only some specific cases \mu e d d, \tau e d d, \tau \mu d d, \mu e s s, \tau e s s, \tau \mu s s[1]. Here we usedd for the first generation of down-type quarks while in the rest of this manual it is used to summarize all three families. [2]

The input file to calculate the coefficients of the ℓ − ℓ − Z operators (\bar{\ell} \gamma_\mu P_{L,R} \ell) Z^\muand(\bar{\ell} p_\mu P_{L,R} \gamma_\mu \ell) Z^\mu is

NameProcess="Z2l";

ConsideredProcess = "2Fermion1Vector";
FermionOrderExternal={1,2};
NeglectMasses={1,2};


ExternalFields= {ChargedLepton,bar[ChargedLepton],Zboson};
CombinationGenerations = {{1,2},{1,3},{2,3}};


AllOperators={
   {OZ2lSL,Op[7]}, {OZ2lSR,Op[6]},
   {OZ2lVL,Op[7,ec[3]]}, {OZ2lVR,Op[6,ec[3]]}
};

OutputFile = "Z2l.m";

Filters = {};

Note that ExternalFields must contain first the involved fermions and the boson at the end. Furthermore, in the case of processes involving scalars one can define

ExternalFields= {ChargedLepton,bar[ChargedLepton],ScalarHiggs};
CombinationGenerations = {{1,2,ALL}, {1,3,ALL}, {2,3,ALL}};

In this case the operators for all Higgs states present in the considered model will be computed.

Operators with massless gauge bosons

We have to add a few more remarks concerning 2-fermion-1-boson operators with massless gauge bosons since those are treated in a special way. It is common for these operators to include terms in the amplitude which are proportional to the external masses. Therefore, if one proceeds in the usual way and neglects the external momenta, some inconsistencies would be obtained. For this reason, a special treatment is in order. In PreSARAH, when one uses

ConsideredProcess = "2Fermion1Vector";
FermionOrderExternal={1,2};
NeglectMasses={3};

the dependence on the two fermion masses is neglected in the resulting Passarino-Veltman integrals but terms proportional tom_f₁ andm_f₂ are kept. This solves the aforementioned potential inconsistencies.

Furthermore, for the dipole operators, defined by

   {DipoleL,Op[6] Pair[ec[3],k[1]]},
   {DipoleR,Op[7] Pair[ec[3],k[1]]},

we are using the results obtained by FeynArts and FormCalc and have implemented all special cases for the involved loop integrals (C_0, C_{00}, C_1, C_2, C_{11}, C_{12}, C_{22}) with identical or vanishing internal masses in SPheno. This guarantees the numerical stability of the results [2].

The monopole operators of the form q²(f̄**γ_μf)V^μ are only non-zero for off-shell external gauge bosons, while PreSARAH always treats all fields as on-shell. Because of this, and to stabilize the numerical evaluation later on, these operators are treated differently to all other operators: the coefficients are not calculated by FeynArts and FormCalc but instead we have included the generic expressions in PreSARAH using a special set of loop functions in SPheno. In these loop functions the resulting Passarino-Veltman integrals are already combined, leading to well-known expressions in the literature, see . They have been cross-checked with the package Peng4BSM@LO. To get the coefficients for the monopole operators, these have to be given always in the form

   {MonopoleL,Op[6,ec[3]] Pair[k[3],k[3]]},
   {MonopoleR,Op[7,ec[3]] Pair[k[3],k[3]]}

in the input of PreSARAH.

Combination and normalization of operators

The user can define new operators as combination of existing operators. For this purpose wrapper files containing the definition of the operators can be included in the FlavorKit directories. These files have to begin with ProcessWrapper = True;. This function is also used by PreSARAH in the case of 4-fermion operators: for these operators the contributions stemming from tree-level, box- and penguin- diagrams are saved separately and summed up at the end. Thus, the wrapper code for the 4-lepton operators written by PreSARAH reads

ProcessWrapper = True;
NameProcess = "4L"
ExternalFields = {ChargedLepton, bar[ChargedLepton], ChargedLepton, bar[ChargedLepton]};
SumContributionsOperators["4L"] = {
{O4lSLL, BO4lSLL + PSO4lSLL + PVO4lSLL + TSO4lSLL + TVO4lSLL},
{O4lSRR, BO4lSRR + PSO4lSRR + PVO4lSRR + TSO4lSRR + TVO4lSRR},
...
};

It is also possible to use these wrapper files to change the normalization of the operators. We have made use of this functionality for the operators with external photons and gluons to match the standard definition used in literature: it is common to write these operators ase m_f(f̄**σ_μ**νf)F^μ**ν, i.e. with the electric coupling (or strong coupling for gluon operators) and fermion mass factored out. This is done with the files Photon_wrapper.m and Gluon_wrapper.m, included in the FlavorKit directory of SARAH :

ProcessWrapper = True;
NameProcess = "Gamma2Q"
ExternalFields = {bar[BottomQuark], BottomQuark, Photon};

SumContributionsOperators["Gamma2Q"] = {
{CC7, OA2qSL},
{CC7p, OA2qSR}
};

NormalizationOperators["Gamma2Q"] ={
"CC7(3,:) = 0.25_dp*CC7(3,:)/sqrt(Alpha_160*4*Pi)/mf_d_160(3)",
"CC7p(3,:) = 0.25_dp*CC7p(3,:)/sqrt(Alpha_160*4*Pi)/mf_d_160(3)",

"CC7SM(3,:) = 0.25_dp*CC7SM(3,:)/sqrt(Alpha_160*4*Pi)/mf_d_160(3)",
"CC7pSM(3,:) = 0.25_dp*CC7pSM(3,:)/sqrt(Alpha_160*4*Pi)/mf_d_160(3)"
};

First, the coefficients OA2qSL and OA2qSR derived with PreSARAH are matched to the new coefficients CC7 and CC7p. The same matching is automatically applied also to the SM coefficients OA2qSLSM and OA2qSRSM. In a second step, these operators are re-normalized to the standard definition of the Wilson coefficientsC₇ andC′₇. The initial coefficients OA2qSR, OA2qSL are not discarded, but co-exist besides CC7, CC7p. Thus, the user can choose in the implementation of the observables which operators are more suitable for his purposes.

Notes

[1] Here we usedd for the first generation of down-type quarks while in the rest of this manual it is used to summarize all three families.

[2] We note that the coefficients for the operators defined above (\bar{f} \gamma_\mu f \, V^\mu) are by a factor of 2 (4) larger than the coefficients of the standard definition for the dipole operators *f̄**σ**μ**ν**P**L**f**q**ν**V**μ* (\bar{f} \sigma_{\mu\nu}P_L f F^{\mu\nu}).

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