FeynCalc manual (development version)

FeynRule

FeynRule[lag, {fields}] derives the Feynman rule corresponding to the field configuration fields of the Lagrangian lag.

FeynRule does not calculate propagator Feynman rules.

The option ZeroMomentumInsertion can be used for twist-2 and higher twist operators.

FeynRule is not very versatile and was primarily developed for QCD calculations. It is often more useful when dealing with bosonic fields than with fermions. If you need a more powerful and universal solution for deriving Feynman rules, have a look at the standalone Mathematica Package FeynRules (not related to FeynCalc).

See also

Overview

Examples

?Lagrangian

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ϕ4\phi ^4 Feynman rule

- \[Lambda]/4! QuantumField[\[Phi]]^4 
 
FeynRule[%, {QuantumField[\[Phi]][p1], QuantumField[\[Phi]][p2], 
   QuantumField[\[Phi]][p3], QuantumField[\[Phi]][p4]}]

λϕ424-\frac{\lambda \phi ^4}{24}

iλ-i \lambda

Quark-gluon vertex Feynman rule

I QuantumField[AntiQuarkField] . GA[\[Mu]] . CovariantD[\[Mu]] . QuantumField[QuarkField] 
 
FeynRule[%, {QuantumField[GaugeField, {\[Mu]}, {a}][p1], 
   QuantumField[QuarkField][p2], QuantumField[AntiQuarkField][p3]}]

iψˉ.γˉμ.Dμ.ψi \bar{\psi }.\bar{\gamma }^{\mu }.D_{\mu }.\psi

iTagsγˉμi T^a g_s \bar{\gamma }^{\mu }

4-gluon vertex Feynman rule

-(1/4) FieldStrength[\[Alpha], \[Beta], i] . FieldStrength[\[Alpha], \[Beta], i] 
 
FeynRule[%, {QuantumField[GaugeField, {\[Mu]}, {a}][p1], QuantumField[GaugeField, {\[Nu]}, {b}][p2], 
    QuantumField[GaugeField, {\[Rho]}, {c}][p3], QuantumField[GaugeField, {\[Sigma]}, {d}][p4]}] 
 
GluonVertex[{p, \[Mu], a}, {q, \[Nu], b}, {r, \[Rho], c}, {s, \[Sigma], d}, Dimension -> 4, Explicit -> True] 
 
FCCanonicalizeDummyIndices[% - %%] // Factor

14Fαβi.Fαβi-\frac{1}{4} F_{\alpha \beta }^i.F_{\alpha \beta }^i

igs2fadFCGV(si49)fbcFCGV(si49)(gˉμρgˉνσgˉμνgˉρσ)+igs2facFCGV(si49)fbdFCGV(si49)(gˉμσgˉνρgˉμνgˉρσ)+igs2fabFCGV(si49)fcdFCGV(si49)(gˉμσgˉνρgˉμρgˉνσ)i g_s^2 f^{ad\text{FCGV}(\text{si49})} f^{bc\text{FCGV}(\text{si49})} \left(\bar{g}^{\mu \rho } \bar{g}^{\nu \sigma }-\bar{g}^{\mu \nu } \bar{g}^{\rho \sigma }\right)+i g_s^2 f^{ac\text{FCGV}(\text{si49})} f^{bd\text{FCGV}(\text{si49})} \left(\bar{g}^{\mu \sigma } \bar{g}^{\nu \rho }-\bar{g}^{\mu \nu } \bar{g}^{\rho \sigma }\right)+i g_s^2 f^{ab\text{FCGV}(\text{si49})} f^{cd\text{FCGV}(\text{si49})} \left(\bar{g}^{\mu \sigma } \bar{g}^{\nu \rho }-\bar{g}^{\mu \rho } \bar{g}^{\nu \sigma }\right)

igs2(fadFCGV(u56)fbcFCGV(u56)(gˉμνgˉρσgˉμρgˉνσ)+facFCGV(u56)fbdFCGV(u56)(gˉμνgˉρσgˉμσgˉνρ)+fabFCGV(u56)fcdFCGV(u56)(gˉμρgˉνσgˉμσgˉνρ))-i g_s^2 \left(f^{ad\text{FCGV}(\text{u56})} f^{bc\text{FCGV}(\text{u56})} \left(\bar{g}^{\mu \nu } \bar{g}^{\rho \sigma }-\bar{g}^{\mu \rho } \bar{g}^{\nu \sigma }\right)+f^{ac\text{FCGV}(\text{u56})} f^{bd\text{FCGV}(\text{u56})} \left(\bar{g}^{\mu \nu } \bar{g}^{\rho \sigma }-\bar{g}^{\mu \sigma } \bar{g}^{\nu \rho }\right)+f^{ab\text{FCGV}(\text{u56})} f^{cd\text{FCGV}(\text{u56})} \left(\bar{g}^{\mu \rho } \bar{g}^{\nu \sigma }-\bar{g}^{\mu \sigma } \bar{g}^{\nu \rho }\right)\right)

00

3-gluon vertex Feynman rule

-(1/4) FieldStrength[\[Alpha], \[Beta], i] . FieldStrength[\[Alpha], \[Beta], i] 
 
FeynRule[%, {QuantumField[GaugeField, {\[Mu]}, {a}][p], QuantumField[GaugeField, {\[Nu]}, {b}][q], 
    QuantumField[GaugeField, {\[Rho]}, {c}][r]}] 
 
GluonVertex[{p, \[Mu], a}, {q, \[Nu], b}, {r, \[Rho], c}, Dimension -> 4, Explicit -> True] 
 
ExpandScalarProduct[% - %%] // Factor

14Fαβi.Fαβi-\frac{1}{4} F_{\alpha \beta }^i.F_{\alpha \beta }^i

gsfabc(gˉμν(pρqρ)gˉμρ(pνrν)+gˉνρ(qμrμ))g_s f^{abc} \left(\bar{g}^{\mu \nu } \left(\overline{p}^{\rho }-\overline{q}^{\rho }\right)-\bar{g}^{\mu \rho } \left(\overline{p}^{\nu }-\overline{r}^{\nu }\right)+\bar{g}^{\nu \rho } \left(\overline{q}^{\mu }-\overline{r}^{\mu }\right)\right)

gsfabc(gˉμν(pq)ρ+gˉμρ(rp)ν+gˉνρ(qr)μ)g_s f^{abc} \left(\bar{g}^{\mu \nu } \left(\overline{p}-\overline{q}\right)^{\rho }+\bar{g}^{\mu \rho } \left(\overline{r}-\overline{p}\right)^{\nu }+\bar{g}^{\nu \rho } \left(\overline{q}-\overline{r}\right)^{\mu }\right)

00

Higgs EFT interaction vertex

heftInt = -(1/4) CH FieldStrength[mu, nu, a] . FieldStrength[mu, nu, a] . QuantumField[H]

14  CHFmu  nua.Fmu  nua.H-\frac{1}{4} \;\text{CH} F_{\text{mu}\;\text{nu}}^a.F_{\text{mu}\;\text{nu}}^a.H

HggHgg vertex Feynman rules

FeynRule[heftInt, {QuantumField[GaugeField, {i}, {a}][p1], QuantumField[GaugeField, 
     {j}, {b}][p2], QuantumField[H][p3]}]

i  CHδab(p2ip1jgˉij(p1p2))-i \;\text{CH} \delta ^{ab} \left(\overline{\text{p2}}^i \overline{\text{p1}}^j-\bar{g}^{ij} \left(\overline{\text{p1}}\cdot \overline{\text{p2}}\right)\right)

HgggHggg vertex Feynman rules

FeynRule[heftInt, {QuantumField[GaugeField, {i}, {a}][p1], QuantumField[GaugeField, 
      {j}, {b}][p2], QuantumField[GaugeField, {k}, {c}][p3], QuantumField[H][p4]}] // Simplify

CHgsfabc(gˉij(p1kp2k)gˉik(p1jp3j)+gˉjk(p2ip3i))\text{CH} g_s f^{abc} \left(\bar{g}^{ij} \left(\overline{\text{p1}}^k-\overline{\text{p2}}^k\right)-\bar{g}^{ik} \left(\overline{\text{p1}}^j-\overline{\text{p3}}^j\right)+\bar{g}^{jk} \left(\overline{\text{p2}}^i-\overline{\text{p3}}^i\right)\right)

HggggHgggg vertex Feynman rules

FeynRule[heftInt, {QuantumField[GaugeField, {i}, {a}][p1], QuantumField[GaugeField, {j}, 
       {b}][p2], QuantumField[GaugeField, {k}, {c}][p3], 
     QuantumField[GaugeField, {l}, {d}][p4], QuantumField[H][p5]}] // 
   FCCanonicalizeDummyIndices[#, SUNIndexNames -> {e}] & // Collect2[#, SUNF, 
    FCFactorOut -> I CH SMP["g_s"]^2] &

i  CHgs2(fadefbce(gˉikgˉjlgˉijgˉkl)+facefbde(gˉilgˉjkgˉijgˉkl)+fabefcde(gˉilgˉjkgˉikgˉjl))i \;\text{CH} g_s^2 \left(f^{ade} f^{bce} \left(\bar{g}^{ik} \bar{g}^{jl}-\bar{g}^{ij} \bar{g}^{kl}\right)+f^{ace} f^{bde} \left(\bar{g}^{il} \bar{g}^{jk}-\bar{g}^{ij} \bar{g}^{kl}\right)+f^{abe} f^{cde} \left(\bar{g}^{il} \bar{g}^{jk}-\bar{g}^{ik} \bar{g}^{jl}\right)\right)

Some OPE-related examples:

2-quark Feynman rules (unpolarized).

Lagrangian["oqu"]
FeynRule[%, {QuantumField[QuarkField][p], QuantumField[AntiQuarkField][q]}, 
   ZeroMomentumInsertion -> False] // Factor

imψˉ.(γˉΔ).DΔm1.ψi^m \bar{\psi }.\left(\bar{\gamma }\cdot \Delta \right).D_{\Delta }{}^{m-1}.\psi

iim(γˉΔ).(Δ)m1-i i^m \left(\bar{\gamma }\cdot \Delta \right).\left(\vec{\partial }_{\Delta }\right){}^{m-1}

?ZeroMomentumInsertion

1cklte43ogg8m

ExpandPartialD[Lagrangian["oqu"] /. OPEm -> 3]

igs2Tc152.Tc153.(γˉΔ).ψˉ.AΔc152.AΔc153.ψgsTc152.(γˉΔ).ψˉ.AΔc152.(Δψ)gsTc153.(γˉΔ).ψˉ.AΔc153.(Δψ)gsTc153.(γˉΔ).ψˉ.(ΔAΔc153).ψi(γˉΔ).ψˉ.(ΔΔψ)i g_s^2 T^{\text{c152}}.T^{\text{c153}}.\left(\bar{\gamma }\cdot \Delta \right).\bar{\psi }.A_{\Delta }^{\text{c152}}.A_{\Delta }^{\text{c153}}.\psi -g_s T^{\text{c152}}.\left(\bar{\gamma }\cdot \Delta \right).\bar{\psi }.A_{\Delta }^{\text{c152}}.\left(\partial _{\Delta }\psi \right)-g_s T^{\text{c153}}.\left(\bar{\gamma }\cdot \Delta \right).\bar{\psi }.A_{\Delta }^{\text{c153}}.\left(\partial _{\Delta }\psi \right)-g_s T^{\text{c153}}.\left(\bar{\gamma }\cdot \Delta \right).\bar{\psi }.\left(\partial _{\Delta }A_{\Delta }^{\text{c153}}\right).\psi -i \left(\bar{\gamma }\cdot \Delta \right).\bar{\psi }.\left(\partial _{\Delta }\partial _{\Delta }\psi \right)

Lagrangian["oqu"]
FeynRule[% /. OPEm -> 3, {QuantumField[QuarkField][p], QuantumField[AntiQuarkField][q], 
   QuantumField[GaugeField, {\[Mu]}, {a}][r]}, ZeroMomentumInsertion -> True]
FCReplaceMomenta[%, {r -> -p - q}] // ExpandScalarProduct // Expand

imψˉ.(γˉΔ).DΔm1.ψi^m \bar{\psi }.\left(\bar{\gamma }\cdot \Delta \right).D_{\Delta }{}^{m-1}.\psi

TaΔμgs((γˉΔ))(2(Δq)+Δr)T^a \Delta ^{\mu } g_s \left(-\left(\bar{\gamma }\cdot \Delta \right)\right) \left(2 \left(\Delta \cdot \overline{q}\right)+\Delta \cdot \overline{r}\right)

TaΔμgsγˉΔ(Δp)TaΔμgsγˉΔ(Δq)T^a \Delta ^{\mu } g_s \bar{\gamma }\cdot \Delta \left(\Delta \cdot \overline{p}\right)-T^a \Delta ^{\mu } g_s \bar{\gamma }\cdot \Delta \left(\Delta \cdot \overline{q}\right)

Lagrangian["oqu"]
FeynRule[% /. OPEm -> 4, {QuantumField[QuarkField][p], QuantumField[AntiQuarkField][q], 
   QuantumField[GaugeField, {\[Mu]}, {a}][r]}, ZeroMomentumInsertion -> True]
FCReplaceMomenta[%, {r -> -p - q}] // ExpandScalarProduct // Expand

imψˉ.(γˉΔ).DΔm1.ψi^m \bar{\psi }.\left(\bar{\gamma }\cdot \Delta \right).D_{\Delta }{}^{m-1}.\psi

TaΔμgsγˉΔ(3(Δq)2+(Δr)2+3(Δq)(Δr))T^a \Delta ^{\mu } g_s \bar{\gamma }\cdot \Delta \left(3 (\Delta \cdot \overline{q})^2+(\Delta \cdot \overline{r})^2+3 \left(\Delta \cdot \overline{q}\right) \left(\Delta \cdot \overline{r}\right)\right)

TaΔμgsγˉΔ(Δp)2+TaΔμgsγˉΔ(Δq)2TaΔμgsγˉΔ(Δp)(Δq)T^a \Delta ^{\mu } g_s \bar{\gamma }\cdot \Delta (\Delta \cdot \overline{p})^2+T^a \Delta ^{\mu } g_s \bar{\gamma }\cdot \Delta (\Delta \cdot \overline{q})^2-T^a \Delta ^{\mu } g_s \bar{\gamma }\cdot \Delta \left(\Delta \cdot \overline{p}\right) \left(\Delta \cdot \overline{q}\right)

2-gluon Feynman rules (unpolarized).

Lagrangian["ogu"]
FeynRule[%, {QuantumField[GaugeField, {\[Mu]}, {a}][p], QuantumField[GaugeField, {\[Nu]}, 
      {b}][q]}, ZeroMomentumInsertion -> False] // Factor

12im1FFCGV(α)ΔFCGV(a).(DΔFCGV(a)FCGV(b))m2.FFCGV(α)ΔFCGV(b)\frac{1}{2} i^{m-1} F_{\text{FCGV}(\alpha )\Delta }^{\text{FCGV}(\text{a})}.\left(D_{\Delta }^{\text{FCGV}(\text{a})\text{FCGV}(\text{b})}\right){}^{m-2}.F_{\text{FCGV}(\alpha )\Delta }^{\text{FCGV}(\text{b})}

imδab(Δ)m2(gˉμν(Δp)(Δq)+Δνqμ(Δp)+Δμpν(Δq)ΔμΔν(pq))-i^m \delta ^{ab} \left(\vec{\partial }_{\Delta }\right){}^{m-2} \left(-\bar{g}^{\mu \nu } \left(\Delta \cdot \overline{p}\right) \left(\Delta \cdot \overline{q}\right)+\Delta ^{\nu } \overline{q}^{\mu } \left(\Delta \cdot \overline{p}\right)+\Delta ^{\mu } \overline{p}^{\nu } \left(\Delta \cdot \overline{q}\right)-\Delta ^{\mu } \Delta ^{\nu } \left(\overline{p}\cdot \overline{q}\right)\right)

2-gluon Feynman rules (polarized).

Lagrangian["ogp"]
FeynRule[%, {QuantumField[GaugeField, {\[Mu]}, {a}][p], QuantumField[GaugeField, {\[Nu]}, 
       {b}][q]}, ZeroMomentumInsertion -> False] // Factor 
 
Factor2[Calc[% /. p -> -q, Assumptions -> Automatic]]

12imϵˉFCGV(α)FCGV(β)FCGV(γ)Δ.FFCGV(β)FCGV(γ)FCGV(a).(DΔFCGV(a)FCGV(b))m2.FFCGV(α)ΔFCGV(b)\frac{1}{2} i^m \bar{\epsilon }^{\text{FCGV}(\alpha )\text{FCGV}(\beta )\text{FCGV}(\gamma )\Delta }.F_{\text{FCGV}(\beta )\text{FCGV}(\gamma )}^{\text{FCGV}(\text{a})}.\left(D_{\Delta }^{\text{FCGV}(\text{a})\text{FCGV}(\text{b})}\right){}^{m-2}.F_{\text{FCGV}(\alpha )\Delta }^{\text{FCGV}(\text{b})}

im+1δab(Δ)m2(ΔμϵˉνΔpqΔνϵˉμΔpq(Δp)ϵˉμνΔq+(Δq)ϵˉμνΔp)i^{m+1} \delta ^{ab} \left(\vec{\partial }_{\Delta }\right){}^{m-2} \left(\Delta ^{\mu } \bar{\epsilon }^{\nu \Delta \overline{p}\overline{q}}-\Delta ^{\nu } \bar{\epsilon }^{\mu \Delta \overline{p}\overline{q}}-\left(\Delta \cdot \overline{p}\right) \bar{\epsilon }^{\mu \nu \Delta \overline{q}}+\left(\Delta \cdot \overline{q}\right) \bar{\epsilon }^{\mu \nu \Delta \overline{p}}\right)

00

Compare with the Feynman rule tabulated in Twist2GluonOperator.

Twist2GluonOperator[q, {\[Mu], a}, {\[Nu], b}, Polarization -> 1, Explicit -> True]

i(1(1)m)δab(Δq)m1ϵμνΔqi \left(1-(-1)^m\right) \delta ^{ab} (\Delta \cdot q)^{m-1} \overset{\text{}}{\epsilon }^{\mu \nu \Delta q}

quark-quark -gluon-gluon Feynman rule (unpolarized).

Lagrangian["oqu"] 
 
frule = FeynRule[%, {QuantumField[QuarkField][p], QuantumField[AntiQuarkField][q], 
     QuantumField[GaugeField, {\[Mu]}, {a}][r], QuantumField[GaugeField, {\[Nu]}, {b}][s]}, 
    ZeroMomentumInsertion -> True, InitialFunction -> Identity];

imψˉ.(γˉΔ).DΔm1.ψi^m \bar{\psi }.\left(\bar{\gamma }\cdot \Delta \right).D_{\Delta }{}^{m-1}.\psi

1utkt5lswqq9j

$Aborted\text{\$Aborted}

LeafCount[frule]

11

Twist2QuarkOperator[{p}, {q}, {r, \[Mu], a}, {s, \[Nu], b}, Polarization -> 0]

(1)mΔμΔνgs2(γˉΔ).(Ta.Tb(i=03+m    (i+1)(Δq)j((Δp))i+m3(Δq+Δr)ij)+Tb.Ta(i=03+m    (i+1)(Δq)j((Δp))i+m3(Δq+Δs)ij))-(-1)^m \Delta ^{\mu } \Delta ^{\nu } g_s^2 \left(\bar{\gamma }\cdot \Delta \right).\left(T^a.T^b \left(\sum _{i=0}^{-3+m} \;\text{}\;\text{} (i+1)(\Delta \cdot q)^j (-(\Delta \cdot p))^{-i+m-3} (\Delta \cdot q+\Delta \cdot r)^{i-j}\right)+T^b.T^a \left(\sum _{i=0}^{-3+m} \;\text{}\;\text{} (i+1)(\Delta \cdot q)^j (-(\Delta \cdot p))^{-i+m-3} (\Delta \cdot q+\Delta \cdot s)^{i-j}\right)\right)

(*Twist2QuarkOperator[{p},{q},{r,\[Mu],a},{s,\[Nu],b},Polarization->0]
Calc[frule-%%/.OPEm->5/.s->-p-q-r/.D->4]*)