Martin GabelmannAlready defined Operators in FlavorKit
Lagrangian
In this section we present our notation and conventions for the operators (and their corresponding Wilson coefficients) implemented in PreSARAH. Although a more complete list of flavor violating operators can be built, we will concentrate on those implemented in PreSARAH. If necessary, the user can extend it by adding his/her own operators.
The interaction Lagrangian relevant for flavor violating processes can be written as
{\mathcal L}_{\text{FV}} = {\mathcal L}_{\text{LFV}} + {\mathcal L}_{\text{QFV}} \, .
The first piece contains the operators that can trigger lepton flavor violation whereas the second piece contains the operators responsible for quark flavor violation.
The general Lagrangian relevant for lepton flavor violation can be written as
{\mathcal L}_{\text{LFV}} = {\mathcal L}_{\ell \ell \gamma} + {\mathcal L}_{\ell \ell Z} + {\mathcal L}_{\ell \ell h} + {\mathcal L}_{4 \ell} + {\mathcal L}_{2 \ell 2q} \, .
The first term contains theℓ − ℓ − γ interaction, given by
{\mathcal L}_{\ell \ell \gamma} = e \, \bar \ell_\beta \left[ \gamma^\mu \left(K_1^L P_L + K_1^R P_R \right) + i m_{\ell_\alpha} \sigma^{\mu \nu} q_\nu \left(K_2^L P_L + K_2^R P_R \right) \right] \ell_\alpha A_\mu + h.c.
Heree is the electric charge,q the photon momentum,$P_{L,R} = \frac{1}{2} (1 \mp \gamma_5)$ are the usual chirality projectors andℓα, β denote the lepton flavors. For practical reasons, we will always consider the photonic contributions independently, and we will not include them in other vector operators. On the contrary, theZ- and Higgs boson contributions will be included whenever possible. Therefore, theℓ − ℓ − Z andℓ − ℓ − h interaction Lagrangians will only be used for observables involving realZ- and Higgs bosons. These two Lagrangians can be written as
{\mathcal L}_{\ell \ell Z} = \bar \ell_\beta \left[ \gamma^\mu \left(R_1^L P_L + R_1^R P_R \right) + p^\mu \left(R_2^L P_L + R_2^R P_R \right) \right] \ell_\alpha Z_\mu \, ,
wherep is theℓβ 4-momentum, and
{\mathcal L}_{\ell \ell h} = \bar \ell_\beta \left(S_L P_L + S_R P_R \right) \ell_\alpha h \, .
The general4ℓ 4-fermion interaction Lagrangian can be written as
{\mathcal L}_{4 \ell} = \sum_{\substack{I=S,V,T\\X,Y=L,R}} A_{XY}^I \bar \ell_\beta \Gamma_I P_X \ell_\alpha \bar \ell_\delta \Gamma_I P_Y \ell_\gamma + h.c. \, ,
whereℓα, β, γ, δ denote the lepton flavors andΓS = 1,ΓV = γμ andΓT = σμ**ν. We omit flavor indices in the Wilson coefficients for the sake of clarity. This Lagrangian contains the most general form compatible with Lorentz invariance. The Wilson coefficientsAL**RS andAR**LS were included in , but absent in . As previously stated, the coefficients in Eq. do not include photonic contributions, but they include Z-boson and scalar ones. Finally, the general2ℓ2q four fermion interaction Lagrangian at the quark level is given by
{\mathcal L}_{2 \ell 2q} = {\mathcal L}_{2 \ell 2d} + {\mathcal L}_{2 \ell 2u}
where
{\mathcal L}_{2 \ell 2d} = \sum_{\substack{I=S,V,T\\X,Y=L,R}} B_{XY}^I \bar \ell_\beta \Gamma_I P_X \ell_\alpha \bar d_\gamma \Gamma_I P_Y d_\gamma + h.c. \\ {\mathcal L}_{2 \ell 2u} = \left. {\mathcal L}_{2 \ell 2d} \right|_{d \to u, \, B \to C} \, .
Heredγ denotes the d-quark flavor.
Let us now consider the Lagrangian relevant for quark flavor violation. This can be written as
{\mathcal L}_{\text{QFV}} = {\mathcal L}_{q q \gamma} + {\mathcal L}_{q q g} + {\mathcal L}_{4 d} + {\mathcal L}_{2d2l} + {\mathcal L}_{2d2\nu} + {\mathcal L}_{du\ell\nu} + {\mathcal L}_{d d H} \, .
The first two terms correspond to operators that couple quark bilinears to massless gauge bosons. These are
{\mathcal L}_{q q \gamma} = e \left[ \bar d_\beta \sigma_{\mu \nu} \left( m_{d_\beta} Q_1^L P_L + m_{d_\alpha} Q_1^R P_R \right) d_\alpha \right] F^{\mu \nu} \\ {\mathcal L}_{q q g} = g_s \left[ \bar d_\beta \sigma_{\mu \nu} \left( m_{d_\beta} Q_2^L P_L + m_{d_\alpha} Q_2^R P_R \right) T^a d_\alpha \right] G_a^{\mu \nu} \, .
HereTa areS**U(3) matrices. The Wilson coefficientsQ1, 2L, R can be easily related to the usualC7, 8(′) coefficients, sometimes normalized with an additional\frac{1}{16 \pi^2} factor. The4d four fermion interaction Lagrangian can be written as
{\mathcal L}_{4 d} = \sum_{\substack{I=S,V,T\\X,Y=L,R}} D_{XY}^I \bar d_\beta \Gamma_I P_X d_\alpha \bar d_\delta \Gamma_I P_Y d_\gamma + h.c. \, ,
wheredα, β, γ, δ denote the lepton flavors. Again, we omit flavor indices in the Wilson coefficients for the sake of clarity. The2d2ℓ four fermion interaction Lagrangian is given by
{\mathcal L}_{2d 2 \ell} = \sum_{\substack{I=S,V,T\\X,Y=L,R}} E_{XY}^I \bar d_\beta \Gamma_I P_X d_\alpha \bar \ell_\gamma \, \Gamma_I P_Y \ell_\gamma + hc \, .
Hereℓγ denotes the lepton flavor.{\mathcal L}_{2d 2 \ell} should not be confused with{\mathcal L}_{2 \ell 2d}. In the former case one has QFV operators, whereas in the latter one has LFV operators. This distinction has been made for practical reasons. The2d2ν andd**uℓν terms of the QFV Lagrangian are
{\mathcal L}_{2d 2 \nu} = \sum_{X,Y=L,R} F_{XY}^V \bar d_\beta \gamma_\mu P_X d_\alpha \bar \nu_\gamma \gamma^\mu P_Y \nu_\gamma + hc \\ {\mathcal L}_{du\ell\nu} = \sum_{\substack{I=S,V\\X,Y=L,R}} G_{XY}^I \bar d_\beta \Gamma_I P_X u_\alpha \bar \ell_\gamma \, \Gamma_I P_Y \nu_\gamma + hc \, .
Note that we have not introduced scalar or tensor2d2ν operators, nor tensord**uℓν ones, and that lepton flavor (denoted by the indexγ) is conserved in these operators. Finally, we have also included a term in the Lagrangian accounting for operators of the type(d̄Γd)S and(d̄Γd)P, whereS (P) is a virtual [1] scalar (pseudoscalar) state. This piece can be written as
{\mathcal L}_{d d H} = \bar d_\beta \left(H_L^S P_L + H_R^S P_R \right) d_\alpha S + \bar d_\beta \left(H_L^P P_L + H_R^P P_R \right) d_\alpha P \, .
Operators available by default in the SPheno output of SARAH
The operators presented abvoe have been implemented by using the results of PreSARAH in SARAH. Those are exported to SPheno. In the following a list of all internal names for these operators, which can be used in the calculation of new flavor observables is given.
2-Fermion-1-Boson operators
These operators are arrays with either two or three elements. While operators involving vector bosons have always dimension3 × 3, those with scalars have dimension3 × 3 × ng.ng is the number of generations of the considered scalar and forng = 1 the last index is dropped.
(d̄βσμ**νΓ**dα)Fμ**ν and(d̄βσμ**νΓ**dα)Gμ**ν
| Variable | Operator | Name |
|---|---|---|
| CC7 | e**mdβ(d̄βσμ**νPLdα)Fμ**ν | Q1L |
| CC7p | e**mdα(d̄βσμ**νPRdα)Fμ**ν | Q1R |
| CC8 | gsmdβ(d̄βσμ**νPLdα)Gμ**ν | Q2L |
| CC8p | gsmdα(d̄βσμ**νPRdα)Gμ**ν | Q2R |
These operators are derived by PreSARAH with the following input files
NameProcess="Gamma2Q";
ConsideredProcess = "2Fermion1Vector";
FermionOrderExternal={1,2};
NeglectMasses={3};
ExternalFields= {bar[BottomQuark], BottomQuark,Photon};
CombinationGenerations = {{3,2}};
AllOperators={
{OA2qSL,Op[7] Pair[ec[3],k[1]]},
{OA2qSR,Op[6] Pair[ec[3],k[1]]},
{OA2qVL,Op[7,ec[3]]},
{OA2qVR,Op[6,ec[3]]}
};
OutputFile = "Gamma2Q.m";
Filters = {};
NameProcess="Gluon2Q";
ConsideredProcess = "2Fermion1Vector";
FermionOrderExternal={1,2};
NeglectMasses={3};
ExternalFields= {bar[BottomQuark], BottomQuark,Gluon};
CombinationGenerations = {{3,2}};
AllOperators={
{OG2qSL,Op[7] Pair[ec[3],k[1]]},
{OG2qSR,Op[6] Pair[ec[3],k[1]]}
};
OutputFile = "Gluon2Q.m";
Filters = {};The normalization is changed to match the standard definitions by
ProcessWrapper = True;
NameProcess = "Gamma2Q"
ExternalFields = {bar[BottomQuark], BottomQuark, Photon};
SumContributionsOperators["Gamma2Q"] = {
{CC7, OA2qSL},
{CC7p, OA2qSR}
};
NormalizationOperators["Gamma2Q"] ={
"CC7(2,:) = 0.25_dp*CC7(2,:)/sqrt(Alpha_160*4*Pi)/MFd(2)",
"CC7(3,:) = 0.25_dp*CC7(3,:)/sqrt(Alpha_160*4*Pi)/MFd(3)",
"CC7p(2,:) = 0.25_dp*CC7p(2,:)/sqrt(Alpha_160*4*Pi)/MFd(2)",
"CC7p(3,:) = 0.25_dp*CC7p(3,:)/sqrt(Alpha_160*4*Pi)/MFd(3)",
"CC7SM(2,:) = 0.25_dp*CC7SM(2,:)/sqrt(Alpha_160*4*Pi)/MFd(2)",
"CC7SM(3,:) = 0.25_dp*CC7SM(3,:)/sqrt(Alpha_160*4*Pi)/MFd(3)",
"CC7pSM(2,:) = 0.25_dp*CC7pSM(2,:)/sqrt(Alpha_160*4*Pi)/MFd(2)",
"CC7pSM(3,:) = 0.25_dp*CC7pSM(3,:)/sqrt(Alpha_160*4*Pi)/MFd(3)"
};
ProcessWrapper = True;
NameProcess = "Gluon2Q"
ExternalFields = {bar[BottomQuark], BottomQuark, Gluon};
SumContributionsOperators["Gluon2Q"] = {
{CC8, OG2qSL},
{CC8p, OG2qSR}};
NormalizationOperators["Gluon2Q"] ={
"CC8(2,:) = 0.25_dp*CC8(2,:)/sqrt(AlphaS_160*4*Pi)/MFd(2)",
"CC8(3,:) = 0.25_dp*CC8(3,:)/sqrt(AlphaS_160*4*Pi)/MFd(3)",
"CC8p(2,:) = 0.25_dp*CC8p(2,:)/sqrt(AlphaS_160*4*Pi)/MFd(2)",
"CC8p(3,:) = 0.25_dp*CC8p(3,:)/sqrt(AlphaS_160*4*Pi)/MFd(3)",
"CC8SM(2,:) = 0.25_dp*CC8SM(2,:)/sqrt(AlphaS_160*4*Pi)/MFd(2)",
"CC8SM(3,:) = 0.25_dp*CC8SM(3,:)/sqrt(AlphaS_160*4*Pi)/MFd(3)",
"CC8pSM(2,:) = 0.25_dp*CC8pSM(2,:)/sqrt(AlphaS_160*4*Pi)/MFd(2)",
"CC8pSM(3,:) = 0.25_dp*CC8pSM(3,:)/sqrt(AlphaS_160*4*Pi)/MFd(3)"
};\bar \ell_\beta \left( q^2 \gamma^\mu + i m_{\ell_\alpha} \sigma^{\mu \nu} q_\nu \right) \ell_\alpha A_\mu
| Variable | Operator | Name |
|---|---|---|
| K2L | e m_{\ell_\alpha} (\bar \ell_\beta \sigma_{\mu\nu} P_L \ell_\alpha) q^{\nu} A^\mu |
K2L |
| K1L | q^2 (\bar \ell_\beta \gamma_\mu P_L \ell_\alpha) A^\mu |
K1L |
| K2R | e m_{\ell_\alpha} (\bar \ell_\beta \sigma_{\mu\nu} P_R \ell_\alpha) q^{\nu} A^\mu |
K2L |
| K1R | q^2 (\bar \ell_\beta \gamma_\nu P_R \ell_\alpha) A^\mu |
K1R |
These operators are derived by PreSARAH with the following input files
NameProcess="Gamma2l";
ConsideredProcess = "2Fermion1Vector";
FermionOrderExternal={1,2};
NeglectMasses={3};
ExternalFields= {bar[ChargedLepton], ChargedLepton,Photon};
CombinationGenerations = {{2,1},{3,1},{3,2}};
AllOperators={
{OA2lSL,Op[6] Pair[ec[3],k[1]]},
{OA2lSR,Op[7] Pair[ec[3],k[1]]},
{OA1L,Op[6,ec[3]] Pair[k[3],k[3]]},
{OA1R,Op[7,ec[3]] Pair[k[3],k[3]]}
};
OutputFile = "Gamma2l.m";
Filters = {};The normalization is changed to match the standard definitions by
ProcessWrapper = True;
NameProcess = "Gamma2l"
ExternalFields = {bar[ChargedLepton], ChargedLepton, Photon};
SumContributionsOperators["Gamma2l"] = {
{K1L, OA1L},
{K1R, OA1R},
{K2L, OA2lSL},
{K2R, OA2lSR}};
NormalizationOperators["Gamma2l"] ={
"K1L = K1L/sqrt(Alpha_MZ*4*Pi)",
"K1R = K1R/sqrt(Alpha_MZ*4*Pi)",
"K2L(2,:) = -0.5_dp*K2L(2,:)/sqrt(Alpha_MZ*4*Pi)/MFe(2)",
"K2L(3,:) = -0.5_dp*K2L(3,:)/sqrt(Alpha_MZ*4*Pi)/MFe(3)",
"K2R(2,:) = -0.5_dp*K2R(2,:)/sqrt(Alpha_MZ*4*Pi)/MFe(2)",
"K2R(3,:) = -0.5_dp*K2R(3,:)/sqrt(Alpha_MZ*4*Pi)/MFe(3)"
};(\bar \ell \Gamma \ell) Z
| Variable | Operator | Name |
|---|---|---|
| OZ2lVL | (\bar{\ell} \, \gamma^\mu P_L \ell) Z_\mu |
R1L |
| OZ2lSL | (\bar{\ell} p^\mu P_L \ell) Z_\mu |
R2L |
| OZ2lVR | (\bar{\ell} \, \gamma^\mu P_R \ell) Z_\mu |
R1R |
| OZ2lSR | (\bar{\ell} p^\mu P_R \ell) Z_\mu |
R2R |
In the following we omit flavor indices for the sake of simplicity. These operators are derived by PreSARAH with the following input files
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] Pair[ec[3],k[1]]}, {OZ2lSR,Op[6] Pair[ec[3],k[1]]},
{OZ2lVL,Op[7,ec[3]]}, {OZ2lVR,Op[6,ec[3]]}
};
OutputFile = "Z2l.m";
Filters = {};(\bar{\ell} \Gamma \ell) h
| Variable | Operator | Name |
|---|---|---|
| OH2lSL | \bar{\ell} P_L \ell \, h |
SL |
| OH2lSR | \bar{\ell} P_R \ell \, h |
SR |
These operators are derived by PreSARAH with the following input files
NameProcess="H2l";
ConsideredProcess = "2Fermion1Scalar";
FermionOrderExternal={1,2};
NeglectMasses={1,2};
ExternalFields= {ChargedLepton,bar[ChargedLepton],HiggsBoson};
CombinationGenerations = {{1,2,ALL},{1,3,ALL},{2,3,ALL}};
AllOperators={{OH2lSL,Op[7]},
{OH2lSR,Op[6]}
};
OutputFile = "H2l.m";
Filters = {};(d̄Γd)S and(d̄Γd)P
| Variable | Operator | Name |
|---|---|---|
| OH2qSL | d̄**PLd S | H_L^S |
| OH2qSR | d̄**PRd S | HRS |
| OAh2qSL | d̄**PLd P | H_L^P |
| OAh2qSR | d̄**PRd P | HRP |
These auxiliary [2] operators are derived by PreSARAH with the following input files
NameProcess="H2q";
(* operators needed for double penguins with internal scalars *)
(* we neglect therefore the mass of the scalar in the loop functions *)
(* and treat it as massless *)
ConsideredProcess = "2Fermion1Scalar";
FermionOrderExternal={2,1};
NeglectMasses={3};
ExternalFields= {DownQuark,bar[DownQuark],HiggsBoson};
CombinationGenerations = {{2,1,ALL},{3,1,ALL},{3,2,ALL}};
AllOperators={{OH2qSL,Op[7]},
{OH2qSR,Op[6]}
};
OutputFile = "H2q.m";
Filters = {};
NameProcess="A2q";
(* operators needed for double penguins with internal scalars *)
(* we neglect therefore the mass of the scalar in the loop functions *)
(* and treat it as massless *)
ConsideredProcess = "2Fermion1Scalar";
FermionOrderExternal={2,1};
NeglectMasses={3};
ExternalFields= {DownQuark,bar[DownQuark],PseudoScalar};
CombinationGenerations = {{2,1,ALL},{3,1,ALL},{3,2,ALL}};
AllOperators={{OAh2qSL,Op[7]},
{OAh2qSR,Op[6]}
};
OutputFile = "A2q.m";
Filters = {};4-Fermion operators
All operators listed below carry four indices and have dimension3 × 3 × 3 × 3. In addition, the user can access the different contributions of all operators from tree-level diagrams, as well as penguin and box diagrams. The name conventions are as follows: for each operator op the additional parameter exist
-
TSop: tree-level contributions with scalar propagator -
TVop: tree-level contributions with scalar propagator -
PSop: sum of penguin and self-energy contributions with scalar propagator -
PVop: sum of penguin and self-energy contributions with scalar propagator -
Bop: box contributions.
We will denote the 4-fermion operators involving two leptons and two down-type quarks depending on whether they lead to LFV or to QFV processes ℓℓd**d for LFV andd**dℓℓ for QFV.
(\bar{d} \Gamma d) (\bar{\ell} \Gamma^\prime \ell) and(d̄Γd)(ν̄**Γ′ν)
| Variable | Operator | Name |
|---|---|---|
| OddllSLL | (\bar{d} P_L d) (\bar{\ell} P_L \ell) |
EL**LS |
| OddllSRR | (\bar{d} P_R d) (\bar{\ell} P_R \ell) |
ER**RS |
| OddllSLR | (\bar{d} P_L d) (\bar{\ell} P_R \ell) |
EL**RS |
| OddllSRL | (\bar{d} P_R d) (\bar{\ell} P_L \ell) |
ER**LS |
| OddllVLL | (\bar{d} \gamma_\mu P_L d) (\bar{\ell} \gamma^\mu P_L \ell) |
EL**LV |
| OddvvVLL | (d̄**γμPLd)(ν̄**γμPRν) | FL**LV |
| OddllVRR | (\bar{d} \gamma_\mu P_R d) (\bar{\ell} \gamma^\mu P_R \ell) |
ER**RV |
| OddvvVRR | (d̄**γμPRd)(ν̄**γμPRν) | FR**RV |
| OddllVLR | (\bar{d} \gamma_\mu P_L d) (\bar{\ell} \gamma^\mu P_R \ell) |
EL**RV |
| OddvvVLR | (d̄**γμPLd)(ν̄**γμPRν) | FL**RV |
| OddllVRL | (\bar{d} \gamma_\mu P_R d) (\bar{\ell} \gamma^\mu P_L \ell) |
ER**LV |
| OddvvVRL | (d̄**γμPRd)(ν̄**γμPLν) | FR**LV |
| OddllTLL | (\bar{d} \sigma_{\mu\nu} P_L d) (\bar{\ell} \sigma^{\mu\nu} P_L \ell) |
EL**LT |
| OddllTRR | (\bar{d} \sigma_{\mu\nu} P_R d) (\bar{\ell} \sigma^{\mu\nu} P_R \ell) |
ER**RT |
| OddllTLR | (\bar{d} \sigma_{\mu\nu} P_L d) (\bar{\ell} \sigma^{\mu\nu} P_R \ell) |
EL**RT |
| OddllTRL | (\bar{d} \sigma_{\mu\nu} P_R d) (\bar{\ell} \sigma^{\mu\nu} P_L \ell) |
ER**LT |
These operators are derived by PreSARAH with the following input files
NameProcess="2d2L";
ConsideredProcess = "4Fermion";
FermionOrderExternal={2,1,4,3};
NeglectMasses={1,2,3,4};
ExternalFields= {DownQuark,bar[DownQuark],ChargedLepton,bar[ChargedLepton]};
CombinationGenerations = {{3,1,1,1}, {3,1,2,2}, {3,1,3,3},
{3,2,1,1}, {3,2,2,2}, {3,2,3,3}};
AllOperators={{OddllSLL,Op[7].Op[7]},
{OddllSRR,Op[6].Op[6]},
{OddllSRL,Op[6].Op[7]},
{OddllSLR,Op[7].Op[6]},
{OddllVRR,Op[7,Lor[1]].Op[7,Lor[1]]},
{OddllVLL,Op[6,Lor[1]].Op[6,Lor[1]]},
{OddllVRL,Op[7,Lor[1]].Op[6,Lor[1]]},
{OddllVLR,Op[6,Lor[1]].Op[7,Lor[1]]},
{OddllTLL,Op[-7,Lor[1],Lor[2]].Op[-7,Lor[1],Lor[2]]},
{OddllTLR,Op[-7,Lor[1],Lor[2]].Op[-6,Lor[1],Lor[2]]},
{OddllTRL,Op[-6,Lor[1],Lor[2]].Op[-7,Lor[1],Lor[2]]},
{OddllTRR,Op[-6,Lor[1],Lor[2]].Op[-6,Lor[1],Lor[2]]}
};
NameProcess="2d2nu";
ConsideredProcess = "4Fermion";
FermionOrderExternal={2,1,4,3};
NeglectMasses={1,2,3,4};
ExternalFields= {DownQuark,bar[DownQuark],Neutrino,bar[Neutrino]};
CombinationGenerations = Flatten[Table[{{2,1, neutrino1, neutrino2},
{3,1, neutrino1, neutrino2},{3,2, neutrino1, neutrino2}},
{neutrino1,1,3},{neutrino2,1,3}],2];
AllOperators={{OddvvVRR,Op[7,Lor[1]].Op[7,Lor[1]]},
{OddvvVLL,Op[6,Lor[1]].Op[6,Lor[1]]},
{OddvvVRL,Op[7,Lor[1]].Op[6,Lor[1]]},
{OddvvVLR,Op[6,Lor[1]].Op[7,Lor[1]]}
};(\bar{\ell} \Gamma \ell) (\bar{d} \Gamma^\prime d) and(\bar{\ell} \Gamma \ell) (\bar{u} \Gamma^\prime u)
| Variable | Operator | Name |
|---|---|---|
| OllddSLL | (\bar{\ell} P_L \ell) (\bar{d} P_L d) |
BL**LS |
| OlluuSLL | (\bar{\ell} P_L \ell) (\bar{u} P_L u) |
CL**LS |
| OllddSRR | (\bar{\ell} P_R \ell) (\bar{d} P_R d) |
BR**RS |
| OlluuSRR | (\bar{\ell} P_R \ell) (\bar{u} P_R u) |
CR**RS |
| OllddSRL | (\bar{\ell} P_R \ell) (\bar{d} P_L d) |
BR**LS |
| OlluuSRL | (\bar{\ell} P_R \ell) (\bar{u} P_L u) |
CR**LS |
| OllddSLR | (\bar{\ell} P_L \ell) (\bar{d} P_R d) |
BL**RS |
| OlluuSLR | (\bar{\ell} P_L \ell) (\bar{u} P_R u) |
CL**RS |
| OllddVLL | (\bar{\ell} \gamma_\mu P_L \ell) (\bar{d} \gamma^\mu P_L d) |
BL**LV |
| OlluuVLL | (\bar{\ell} \gamma_\mu P_L \ell) (\bar{u} \gamma^\mu P_L u) |
CL**LV |
| OllddVRR | (\bar{\ell} \gamma_\mu P_R \ell) (\bar{d} \gamma^\mu P_R d) |
BR**RV |
| OlluuVRR | (\bar{\ell} \gamma_\mu P_R \ell) (\bar{u} \gamma^\mu P_R u) |
CR**RV |
| OllddVLR | (\bar{\ell} \gamma_\mu P_L \ell) (\bar{d} \gamma^\mu P_R d) |
BL**RV |
| OlluuVLR | (\bar{\ell} \gamma_\mu P_L \ell) (\bar{u} \gamma^\mu P_R u) |
CL**RV |
| OllddVRL | (\bar{\ell} \gamma_\mu P_R \ell) (\bar{d} \gamma^\mu P_L d) |
BR**LV |
| OlluuVRL | (\bar{\ell} \gamma_\mu P_R \ell) (\bar{u} \gamma^\mu P_L u) |
CR**LV |
| OllddTLL | (\bar{\ell} \sigma_{\mu\nu} P_L \ell) (\bar{d} \sigma^{\mu\nu} P_L d) |
BL**LT |
| OlluuTLL | (\bar{\ell} \sigma_{\mu\nu} P_L \ell) (\bar{u} \sigma^{\mu\nu} P_L u) |
CL**LT |
| OllddTRR | (\bar{\ell} \sigma_{\mu\nu} P_R \ell) (\bar{d} \sigma^{\mu\nu} P_R d) |
BR**RT |
| OlluuTRR | (\bar{\ell} \sigma_{\mu\nu} P_R \ell) (\bar{u} \sigma^{\mu\nu} P_R u) |
CR**RT |
| OllddTLR | (\bar{\ell} \sigma_{\mu\nu} P_L \ell) (\bar{d} \sigma^{\mu\nu} P_R d) |
BL**RT |
| OlluuTLR | (\bar{\ell} \sigma_{\mu\nu} P_L \ell) (\bar{u} \sigma^{\mu\nu} P_R u) |
CL**RT |
| OllddTRL | (\bar{\ell} \sigma_{\mu\nu} P_R \ell) (\bar{d} \sigma^{\mu\nu} P_L d) |
BR**LT |
| OlluuTRL | (\bar{\ell} \sigma_{\mu\nu} P_R \ell) (\bar{u} \sigma^{\mu\nu} P_L u) |
CR**LT |
NameProcess="2L2d";
ConsideredProcess = "4Fermion";
FermionOrderExternal={2,1,4,3};
NeglectMasses={1,2,3,4};
ExternalFields= {ChargedLepton,bar[ChargedLepton],DownQuark,bar[DownQuark]};
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}};
AllOperators={{OllddSLL,Op[7].Op[7]},
{OllddSRR,Op[6].Op[6]},
{OllddSRL,Op[6].Op[7]},
{OllddSLR,Op[7].Op[6]},
{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]]},
{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]]}
};
NameProcess="2L2u";
ConsideredProcess = "4Fermion";
FermionOrderExternal={2,1,4,3};
NeglectMasses={1,2,3,4};
ExternalFields= {ChargedLepton,bar[ChargedLepton],UpQuark,bar[UpQuark]};
CombinationGenerations = {{2,1,1,1},{3,1,1,1},{3,2,1,1}};
AllOperators={{OlluuSLL,Op[7].Op[7]},
{OlluuSRR,Op[6].Op[6]},
{OlluuSRL,Op[6].Op[7]},
{OlluuSLR,Op[7].Op[6]},
{OlluuVRR,Op[7,Lor[1]].Op[7,Lor[1]]},
{OlluuVLL,Op[6,Lor[1]].Op[6,Lor[1]]},
{OlluuVRL,Op[7,Lor[1]].Op[6,Lor[1]]},
{OlluuVLR,Op[6,Lor[1]].Op[7,Lor[1]]},
{OlluuTLL,Op[-7,Lor[1],Lor[2]].Op[-7,Lor[1],Lor[2]]},
{OlluuTLR,Op[-7,Lor[1],Lor[2]].Op[-6,Lor[1],Lor[2]]},
{OlluuTRL,Op[-6,Lor[1],Lor[2]].Op[-7,Lor[1],Lor[2]]},
{OlluuTRR,Op[-6,Lor[1],Lor[2]].Op[-6,Lor[1],Lor[2]]}
};(d̄Γd)(d̄**Γ′d) and(\bar{\ell} \Gamma \ell) (\bar{\ell} \Gamma^\prime \ell)
| Variable | Operator | Name |
|---|---|---|
| O4dSLL | (d̄**PLd)(d̄**PLd) | DL**LS |
| O4lSLL | (\bar{\ell} P_L \ell) (\bar{\ell} P_L \ell) |
AL**LS |
| O4dSRR | (d̄**PRd)(d̄**PRd) | DR**RS |
| O4lSRR | (\bar{\ell} P_R \ell) (\bar{\ell} P_R \ell) |
AR**RS |
| O4dSLR | (d̄**PLd)(d̄**PRd) | DL**RS |
| O4lSLR | (\bar{\ell} P_L \ell) (\bar{\ell} P_R \ell) |
AL**RS |
| O4dSRL | (d̄**PRd)(d̄**PLd) | DR**LS |
| O4lSRL | (\bar{\ell} P_R \ell) (\bar{\ell} P_L \ell) |
AR**LS |
| O4dVLL | (d̄**γμPLd)(d̄**γμPLd) | DL**LV |
| O4lVLL | (\bar{\ell} \gamma_\mu P_L \ell) (\bar{\ell} \gamma^\mu P_L \ell) |
AL**LV |
| O4dVRR | (d̄**γμPRd)(d̄**γμPRd) | DR**RV |
| O4lVRR | (\bar{\ell} \gamma_\mu P_R \ell) (\bar{\ell} \gamma^\mu P_R \ell) |
AR**RV |
| O4dVLR | (d̄**γμPLd)(d̄**γμPRd) | DL**RV |
| O4lVLR | (\bar{\ell} \gamma_\mu P_L \ell) (\bar{\ell} \gamma^\mu P_R \ell) |
AL**RV |
| O4dVRL | (d̄**γμPRd)(d̄**γμPLd) | DR**LV |
| O4lVRL | (\bar{\ell} \gamma_\mu P_R \ell) (\bar{\ell} \gamma^\mu P_L \ell) |
AR**LV |
| O4dTLL | (d̄**σμ**νPLd)(d̄**σμ**νPLd) | DL**LT |
| O4lTLL | (\bar{\ell} \sigma_{\mu\nu} P_L \ell) (\bar{\ell} \sigma^{\mu\nu} P_L \ell) |
AL**LT |
| O4dTRR | (d̄**σμ**νPRd)(d̄**σμ**νPRd) | DR**RT |
| O4lTRR | (\bar{\ell} \sigma_{\mu\nu} P_R \ell) (\bar{\ell} \sigma^{\mu\nu} P_R \ell) |
AR**RT |
| O4dTLR | (d̄**σμ**νPLd)(d̄**σμ**νPRd) | DL**RT |
| O4lTLR | (\bar{\ell} \sigma_{\mu\nu} P_L \ell) (\bar{\ell} \sigma^{\mu\nu} P_R \ell) |
AL**RT |
| O4dTRL | (d̄**σμ**νPRd)(d̄**σμ**νPLd) | DR**LT |
| O4lTRL | (\bar{\ell} \sigma_{\mu\nu} P_R \ell) (\bar{\ell} \sigma^{\mu\nu} P_L \ell) |
AR**LT |
NameProcess="4d";
ConsideredProcess = "4Fermion";
FermionOrderExternal={2,1,4,3};
NeglectMasses={1,2,3,4};
ExternalFields= {DownQuark,bar[DownQuark],DownQuark,bar[DownQuark]};
ColorFlow = ColorDelta[1,2] ColorDelta[3,4];
CombinationGenerations = {{3,1,3,1},{3,2,3,2},{2,1,2,1}};
AllOperators={{O4dSLL,Op[7].Op[7]},
{O4dSRR,Op[6].Op[6]},
{O4dSRL,Op[6].Op[7]},
{O4dSLR,Op[7].Op[6]},
{O4dVRR,Op[7,Lor[1]].Op[7,Lor[1]]},
{O4dVLL,Op[6,Lor[1]].Op[6,Lor[1]]},
{O4dVRL,Op[7,Lor[1]].Op[6,Lor[1]]},
{O4dVLR,Op[6,Lor[1]].Op[7,Lor[1]]},
{O4dTLL,Op[-7,Lor[1],Lor[2]].Op[-7,Lor[1],Lor[2]]},
{O4dTLR,Op[-7,Lor[1],Lor[2]].Op[-6,Lor[1],Lor[2]]},
{O4dTRL,Op[-6,Lor[1],Lor[2]].Op[-7,Lor[1],Lor[2]]},
{O4dTRR,Op[-6,Lor[1],Lor[2]].Op[-6,Lor[1],Lor[2]]}
};
Filters = {NoPenguins};
NameProcess="4L";
ConsideredProcess = "4Fermion";
FermionOrderExternal={2,1,4,3};
NeglectMasses={1,2,3,4};
ExternalFields= {ChargedLepton,bar[ChargedLepton],ChargedLepton,bar[ChargedLepton]};
CombinationGenerations = {{2,1,1,1},{3,1,1,1},{3,2,2,2}};
AllOperators={{O4lSLL,Op[7].Op[7]},
{O4lSRR,Op[6].Op[6]},
{O4lSRL,Op[6].Op[7]},
{O4lSLR,Op[7].Op[6]},
{O4lVRR,Op[7,Lor[1]].Op[7,Lor[1]]},
{O4lVLL,Op[6,Lor[1]].Op[6,Lor[1]]},
{O4lVRL,Op[7,Lor[1]].Op[6,Lor[1]]},
{O4lVLR,Op[6,Lor[1]].Op[7,Lor[1]]},
{O4lTLL,Op[-7,Lor[1],Lor[2]].Op[-7,Lor[1],Lor[2]]},
{O4lTLR,Op[-7,Lor[1],Lor[2]].Op[-6,Lor[1],Lor[2]]},
{O4lTRL,Op[-6,Lor[1],Lor[2]].Op[-7,Lor[1],Lor[2]]},
{O4lTRR,Op[-6,Lor[1],Lor[2]].Op[-6,Lor[1],Lor[2]]}
};
Filters = {NoCrossedDiagrams};(\bar{d} \Gamma u) (\bar{\ell} \Gamma^\prime \nu)
| Variable | Operator | Name |
|---|---|---|
| OdulvVLL | (\bar{d} \gamma_\mu P_L u) (\bar{\ell} \gamma^\mu P_L \nu) |
GL**LV |
| OdulvSLL | (\bar{d} P_L u) (\bar{\ell} P_L \nu) |
GL**LS |
| OdulvVRR | (\bar{d} \gamma_\mu P_R u) (\bar{\ell} \gamma^\mu P_R \nu) |
GR**RV |
| OdulvSRR | (\bar{d} P_R u) (\bar{\ell} P_R \nu) |
GR**RS |
| OdulvVLR | (\bar{d} \gamma_\mu P_L u) (\bar{\ell} \gamma^\mu P_R \nu) |
GL**RV |
| OdulvSLR | (\bar{d} P_L u) (\bar{\ell} P_R \nu) |
GL**RS |
| OdulvVRL | (\bar{d} \gamma_\mu P_R u) (\bar{\ell} \gamma^\mu P_L \nu) |
GR**LV |
| OdulvSRL | (\bar{d} P_R u) (\bar{\ell} P_L \nu) |
GR**LS |
NameProcess="dulv";
ConsideredProcess = "4Fermion";
FermionOrderExternal={2,1,3,4};
NeglectMasses={1,2,3,4};
ExternalFields= {DownQuark,bar[UpQuark], Neutrino, bar[ChargedLepton]};
CombinationGenerations =
Flatten[Table[{{3,1,i,j},{3,2,i,j},{2,2,i,j},{2,1,i,j}},{i,1,3},{j,1,3}],2];
Clear[i,j];
AllOperators={{OdulvSLL,Op[7].Op[7]},
{OdulvSRR,Op[6].Op[6]},
{OdulvSRL,Op[6].Op[7]},
{OdulvSLR,Op[7].Op[6]},
{OdulvVRR,Op[7,Lor[1]].Op[7,Lor[1]]},
{OdulvVLL,Op[6,Lor[1]].Op[6,Lor[1]]},
{OdulvVRL,Op[7,Lor[1]].Op[6,Lor[1]]},
{OdulvVLR,Op[6,Lor[1]].Op[7,Lor[1]]}
};
Filters = {NoBoxes, NoPenguins};See also
[1] We would like to emphasize that our implementation of these operators is only valid for virtual scalars and pseudoscalars. They have been introduced in order to provide the 1-loop vertices necessary for the computation of the double penguin contributions toΔ**MBq. Therefore, they are not valid for observables in which the scalar or pseudoscalar states are real particles.
[2] The(d̄Γd)S and(d̄Γd)P operators have been introduced to compute double penguin corrections toΔ**MBq, whereS andP appear as intermediate (virtual) particles. They should not be used in processes where the scalar or pseudoscalar states are real particles because the loop functions are calculated with vanishing external momenta.