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.

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

\phi ^4 Feynman rule

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

-\frac{\lambda \phi ^4}{24}

-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]}]

i \bar{\psi }.\bar{\gamma }^{\mu }.D_{\mu }.\psi

i 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

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

i g_s^2 \bar{g}^{\mu \rho } \bar{g}^{\nu \sigma } f^{ad\text{FCGV}(\text{sun621})} f^{bc\text{FCGV}(\text{sun621})}-i g_s^2 \bar{g}^{\mu \nu } \bar{g}^{\rho \sigma } f^{ad\text{FCGV}(\text{sun621})} f^{bc\text{FCGV}(\text{sun621})}+i g_s^2 \bar{g}^{\mu \sigma } \bar{g}^{\nu \rho } f^{ac\text{FCGV}(\text{sun621})} f^{bd\text{FCGV}(\text{sun621})}-i g_s^2 \bar{g}^{\mu \nu } \bar{g}^{\rho \sigma } f^{ac\text{FCGV}(\text{sun621})} f^{bd\text{FCGV}(\text{sun621})}+i g_s^2 \bar{g}^{\mu \sigma } \bar{g}^{\nu \rho } f^{ab\text{FCGV}(\text{sun621})} f^{cd\text{FCGV}(\text{sun621})}-i g_s^2 \bar{g}^{\mu \rho } \bar{g}^{\nu \sigma } f^{ab\text{FCGV}(\text{sun621})} f^{cd\text{FCGV}(\text{sun621})}

-i g_s^2 \left(f^{ad\text{FCGV}(\text{u66})} f^{bc\text{FCGV}(\text{u66})} \left(\bar{g}^{\mu \nu } \bar{g}^{\rho \sigma }-\bar{g}^{\mu \rho } \bar{g}^{\nu \sigma }\right)+f^{ac\text{FCGV}(\text{u66})} f^{bd\text{FCGV}(\text{u66})} \left(\bar{g}^{\mu \nu } \bar{g}^{\rho \sigma }-\bar{g}^{\mu \sigma } \bar{g}^{\nu \rho }\right)+f^{ab\text{FCGV}(\text{u66})} f^{cd\text{FCGV}(\text{u66})} \left(\bar{g}^{\mu \rho } \bar{g}^{\nu \sigma }-\bar{g}^{\mu \sigma } \bar{g}^{\nu \rho }\right)\right)

0

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]}, Collecting -> {SUNF}] 
 
GluonVertex[{p, \[Mu], a}, {q, \[Nu], b}, {r, \[Rho], c}, Dimension -> 4, Explicit -> True] 
 
ExpandScalarProduct[% - %%] // Factor

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

g_s f^{abc} \left(-\overline{p}^{\nu } \bar{g}^{\mu \rho }+\overline{p}^{\rho } \bar{g}^{\mu \nu }+\overline{q}^{\mu } \bar{g}^{\nu \rho }-\overline{q}^{\rho } \bar{g}^{\mu \nu }-\overline{r}^{\mu } \bar{g}^{\nu \rho }+\overline{r}^{\nu } \bar{g}^{\mu \rho }\right)

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)

0

Higgs EFT interaction vertex

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

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

Hgg vertex Feynman rules

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

-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)

Hggg vertex Feynman rules

FeynRule[heftInt, {QuantumField[GaugeField, {i}, {a}][p1], QuantumField[GaugeField, 
     {j}, {b}][p2], QuantumField[GaugeField, {k}, {c}][p3], QuantumField[H][p4]}, Collecting -> {SUNF}]

\text{CH} g_s f^{abc} \left(-\overline{\text{p1}}^j \bar{g}^{ik}+\overline{\text{p1}}^k \bar{g}^{ij}+\overline{\text{p2}}^i \bar{g}^{jk}-\overline{\text{p2}}^k \bar{g}^{ij}-\overline{\text{p3}}^i \bar{g}^{jk}+\overline{\text{p3}}^j \bar{g}^{ik}\right)

Hgggg 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]}, 
  SUNIndexNames -> {e}, Collecting -> {SUNF}, FCFactorOut -> I CH SMP["g_s"]^2, FCVerbose -> 0]

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)

FeynRule[QuantumField[FCPartialD[LorentzIndex[b]], GaugeField, 
    LorentzIndex[a], SUNIndex[aa]] . QuantumField[
    FCPartialD[LorentzIndex[b]], GaugeField, LorentzIndex[a], 
    SUNIndex[aa]], {QuantumField[GaugeField, {mu1}, {i1}][p1], 
    QuantumField[GaugeField, {mu2}, {i2}][p2]}]

-2 i \delta ^{\text{i1}\;\text{i2}} \bar{g}^{\text{mu1}\;\text{mu2}} \left(\overline{\text{p1}}\cdot \overline{\text{p2}}\right)