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
ϕ 4 \phi ^4 ϕ 4 Feynman rule
- \ [ Lambda] / 4 ! QuantumField[ \ [ Phi]] ^ 4
FeynRule[ % , { QuantumField[ \ [ Phi]][ p1], QuantumField[ \ [ Phi]][ p2],
QuantumField[ \ [ Phi]][ p3], QuantumField[ \ [ Phi]][ p4]}]
− λ ϕ 4 24 -\frac{\lambda \phi ^4}{24} − 24 λ ϕ 4
− i λ -i \lambda − iλ
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 ψ ˉ . γ ˉ μ . D μ . ψ i \bar{\psi }.\bar{\gamma }^{\mu }.D_{\mu }.\psi i ψ ˉ . γ ˉ μ . D μ . ψ
i T a g s γ ˉ μ i T^a g_s \bar{\gamma }^{\mu } i T a g s γ ˉ μ
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
− 1 4 F α β i . F α β i -\frac{1}{4} F_{\alpha \beta }^i.F_{\alpha \beta }^i − 4 1 F α β i . F α β i
i g s 2 f a d FCGV ( si49 ) f b c FCGV ( si49 ) ( g ˉ μ ρ g ˉ ν σ − g ˉ μ ν g ˉ ρ σ ) + i g s 2 f a c FCGV ( si49 ) f b d FCGV ( si49 ) ( g ˉ μ σ g ˉ ν ρ − g ˉ μ ν g ˉ ρ σ ) + i g s 2 f a b FCGV ( si49 ) f c d FCGV ( si49 ) ( g ˉ μ σ g ˉ ν ρ − g ˉ μ ρ g ˉ ν σ ) 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) i g s 2 f a d FCGV ( si49 ) f b c FCGV ( si49 ) ( g ˉ μ ρ g ˉ ν σ − g ˉ μν g ˉ ρ σ ) + i g s 2 f a c FCGV ( si49 ) f b d FCGV ( si49 ) ( g ˉ μ σ g ˉ ν ρ − g ˉ μν g ˉ ρ σ ) + i g s 2 f ab FCGV ( si49 ) f c d FCGV ( si49 ) ( g ˉ μ σ g ˉ ν ρ − g ˉ μ ρ g ˉ ν σ )
− i g s 2 ( f a d FCGV ( u56 ) f b c FCGV ( u56 ) ( g ˉ μ ν g ˉ ρ σ − g ˉ μ ρ g ˉ ν σ ) + f a c FCGV ( u56 ) f b d FCGV ( u56 ) ( g ˉ μ ν g ˉ ρ σ − g ˉ μ σ g ˉ ν ρ ) + f a b FCGV ( u56 ) f c d FCGV ( u56 ) ( g ˉ μ ρ g ˉ ν σ − g ˉ μ σ g ˉ ν ρ ) ) -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) − i g s 2 ( f a d FCGV ( u56 ) f b c FCGV ( u56 ) ( g ˉ μν g ˉ ρ σ − g ˉ μ ρ g ˉ ν σ ) + f a c FCGV ( u56 ) f b d FCGV ( u56 ) ( g ˉ μν g ˉ ρ σ − g ˉ μ σ g ˉ ν ρ ) + f ab FCGV ( u56 ) f c d FCGV ( u56 ) ( g ˉ μ ρ g ˉ ν σ − g ˉ μ σ g ˉ ν ρ ) )
0 0 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 ]}]
GluonVertex[{ p , \ [ Mu], a }, { q , \ [ Nu], b }, { r , \ [ Rho], c }, Dimension -> 4 , Explicit -> True ]
ExpandScalarProduct[ % - %% ] // Factor
− 1 4 F α β i . F α β i -\frac{1}{4} F_{\alpha \beta }^i.F_{\alpha \beta }^i − 4 1 F α β i . F α β i
g s f a b c ( g ˉ μ ν ( p ‾ ρ − q ‾ ρ ) − g ˉ μ ρ ( p ‾ ν − r ‾ ν ) + g ˉ ν ρ ( q ‾ μ − r ‾ μ ) ) 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) g s f ab c ( g ˉ μν ( p ρ − q ρ ) − g ˉ μ ρ ( p ν − r ν ) + g ˉ ν ρ ( q μ − r μ ) )
g s f a b c ( g ˉ μ ν ( p ‾ − q ‾ ) ρ + g ˉ μ ρ ( r ‾ − p ‾ ) ν + g ˉ ν ρ ( q ‾ − r ‾ ) 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) g s f ab c ( g ˉ μν ( p − q ) ρ + g ˉ μ ρ ( r − p ) ν + g ˉ ν ρ ( q − r ) μ )
0 0 0
Higgs EFT interaction vertex
heftInt = - (1 / 4 ) CH FieldStrength[ mu, nu, a ] . FieldStrength[ mu, nu, a ] . QuantumField[ H ]
− 1 4 CH F mu nu a . F mu nu a . H -\frac{1}{4} \;\text{CH} F_{\text{mu}\;\text{nu}}^a.F_{\text{mu}\;\text{nu}}^a.H − 4 1 CH F mu nu a . F mu nu a . H
H g g Hgg H gg vertex Feynman rules
FeynRule[ heftInt, { QuantumField[ GaugeField, { i }, { a }][ p1], QuantumField[ GaugeField,
{ j }, { b }][ p2], QuantumField[ H ][ p3]}]
− i CH δ a b ( p2 ‾ i p1 ‾ j − g ˉ i j ( p1 ‾ ⋅ p2 ‾ ) ) -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) − i CH δ ab ( p2 i p1 j − g ˉ ij ( p1 ⋅ p2 ) )
H g g g Hggg H ggg vertex Feynman rules
FeynRule[ heftInt, { QuantumField[ GaugeField, { i }, { a }][ p1], QuantumField[ GaugeField,
{ j }, { b }][ p2], QuantumField[ GaugeField, { k }, { c }][ p3], QuantumField[ H ][ p4]}] // Simplify
CH g s f a b c ( g ˉ i j ( p1 ‾ k − p2 ‾ k ) − g ˉ i k ( p1 ‾ j − p3 ‾ j ) + g ˉ j k ( p2 ‾ i − p3 ‾ i ) ) \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) CH g s f ab c ( g ˉ ij ( p1 k − p2 k ) − g ˉ ik ( p1 j − p3 j ) + g ˉ jk ( p2 i − p3 i ) )
H g g g g Hgggg H gggg 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 CH g s 2 ( f a d e f b c e ( g ˉ i k g ˉ j l − g ˉ i j g ˉ k l ) + f a c e f b d e ( g ˉ i l g ˉ j k − g ˉ i j g ˉ k l ) + f a b e f c d e ( g ˉ i l g ˉ j k − g ˉ i k g ˉ j l ) ) 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) i CH g s 2 ( f a d e f b ce ( g ˉ ik g ˉ j l − g ˉ ij g ˉ k l ) + f a ce f b d e ( g ˉ i l g ˉ jk − g ˉ ij g ˉ k l ) + f ab e f c d e ( g ˉ i l g ˉ jk − g ˉ ik g ˉ j l ) )
Some OPE-related examples:
2-quark Feynman rules (unpolarized).
Lagrangian[ "oqu" ]
FeynRule[ % , { QuantumField[ QuarkField][ p ], QuantumField[ AntiQuarkField][ q ]},
ZeroMomentumInsertion -> False ] // Factor
i m ψ ˉ . ( γ ˉ ⋅ Δ ) . D Δ m − 1 . ψ i^m \bar{\psi }.\left(\bar{\gamma }\cdot \Delta \right).D_{\Delta }{}^{m-1}.\psi i m ψ ˉ . ( γ ˉ ⋅ Δ ) . D Δ m − 1 . ψ
− i i m ( γ ˉ ⋅ Δ ) . ( ∂ ⃗ Δ ) m − 1 -i i^m \left(\bar{\gamma }\cdot \Delta \right).\left(\vec{\partial }_{\Delta }\right){}^{m-1} − i i m ( γ ˉ ⋅ Δ ) . ( ∂ Δ ) m − 1
ExpandPartialD[ Lagrangian[ "oqu" ] /. OPEm -> 3 ]
i g s 2 T c152 . T c153 . ( γ ˉ ⋅ Δ ) . ψ ˉ . A Δ c152 . A Δ c153 . ψ − g s T c152 . ( γ ˉ ⋅ Δ ) . ψ ˉ . A Δ c152 . ( ∂ Δ ψ ) − g s T c153 . ( γ ˉ ⋅ Δ ) . ψ ˉ . A Δ c153 . ( ∂ Δ ψ ) − g s T c153 . ( γ ˉ ⋅ Δ ) . ψ ˉ . ( ∂ Δ A Δ c153 ) . ψ − i ( γ ˉ ⋅ Δ ) . ψ ˉ . ( ∂ Δ ∂ Δ ψ ) 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) i g s 2 T c152 . T c153 . ( γ ˉ ⋅ Δ ) . ψ ˉ . A Δ c152 . A Δ c153 . ψ − g s T c152 . ( γ ˉ ⋅ Δ ) . ψ ˉ . A Δ c152 . ( ∂ Δ ψ ) − g s T c153 . ( γ ˉ ⋅ Δ ) . ψ ˉ . A Δ c153 . ( ∂ Δ ψ ) − g s T c153 . ( γ ˉ ⋅ Δ ) . ψ ˉ . ( ∂ Δ A Δ c153 ) . ψ − i ( γ ˉ ⋅ Δ ) . ψ ˉ . ( ∂ Δ ∂ Δ ψ )
Lagrangian[ "oqu" ]
FeynRule[ % /. OPEm -> 3 , { QuantumField[ QuarkField][ p ], QuantumField[ AntiQuarkField][ q ],
QuantumField[ GaugeField, { \ [ Mu]}, { a }][ r ]}, ZeroMomentumInsertion -> True ]
FCReplaceMomenta[ % , { r -> - p - q }] // ExpandScalarProduct // Expand
i m ψ ˉ . ( γ ˉ ⋅ Δ ) . D Δ m − 1 . ψ i^m \bar{\psi }.\left(\bar{\gamma }\cdot \Delta \right).D_{\Delta }{}^{m-1}.\psi i m ψ ˉ . ( γ ˉ ⋅ Δ ) . D Δ m − 1 . ψ
T a Δ μ g s ( − ( γ ˉ ⋅ Δ ) ) ( 2 ( Δ ⋅ q ‾ ) + Δ ⋅ r ‾ ) 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) T a Δ μ g s ( − ( γ ˉ ⋅ Δ ) ) ( 2 ( Δ ⋅ q ) + Δ ⋅ r )
T a Δ μ g s γ ˉ ⋅ Δ ( Δ ⋅ p ‾ ) − T a Δ μ g s γ ˉ ⋅ Δ ( Δ ⋅ q ‾ ) 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) T a Δ μ g s γ ˉ ⋅ Δ ( Δ ⋅ p ) − T a Δ μ g s γ ˉ ⋅ Δ ( Δ ⋅ q )
Lagrangian[ "oqu" ]
FeynRule[ % /. OPEm -> 4 , { QuantumField[ QuarkField][ p ], QuantumField[ AntiQuarkField][ q ],
QuantumField[ GaugeField, { \ [ Mu]}, { a }][ r ]}, ZeroMomentumInsertion -> True ]
FCReplaceMomenta[ % , { r -> - p - q }] // ExpandScalarProduct // Expand
i m ψ ˉ . ( γ ˉ ⋅ Δ ) . D Δ m − 1 . ψ i^m \bar{\psi }.\left(\bar{\gamma }\cdot \Delta \right).D_{\Delta }{}^{m-1}.\psi i m ψ ˉ . ( γ ˉ ⋅ Δ ) . D Δ m − 1 . ψ
T a Δ μ g s γ ˉ ⋅ Δ ( 3 ( Δ ⋅ q ‾ ) 2 + ( Δ ⋅ r ‾ ) 2 + 3 ( Δ ⋅ q ‾ ) ( Δ ⋅ r ‾ ) ) 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) T a Δ μ g s γ ˉ ⋅ Δ ( 3 ( Δ ⋅ q ) 2 + ( Δ ⋅ r ) 2 + 3 ( Δ ⋅ q ) ( Δ ⋅ r ) )
T a Δ μ g s γ ˉ ⋅ Δ ( Δ ⋅ p ‾ ) 2 + T a Δ μ g s γ ˉ ⋅ Δ ( Δ ⋅ q ‾ ) 2 − T a Δ μ g s γ ˉ ⋅ Δ ( Δ ⋅ p ‾ ) ( Δ ⋅ q ‾ ) 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) T a Δ μ g s γ ˉ ⋅ Δ ( Δ ⋅ p ) 2 + T a Δ μ g s γ ˉ ⋅ Δ ( Δ ⋅ q ) 2 − T a Δ μ g s γ ˉ ⋅ Δ ( Δ ⋅ p ) ( Δ ⋅ q )
2-gluon Feynman rules (unpolarized).
Lagrangian[ "ogu" ]
FeynRule[ % , { QuantumField[ GaugeField, { \ [ Mu]}, { a }][ p ], QuantumField[ GaugeField, { \ [ Nu]},
{ b }][ q ]}, ZeroMomentumInsertion -> False ] // Factor
1 2 i m − 1 F FCGV ( α ) Δ FCGV ( a ) . ( D Δ FCGV ( a ) FCGV ( b ) ) m − 2 . F FCGV ( α ) Δ FCGV ( 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})} 2 1 i m − 1 F FCGV ( α ) Δ FCGV ( a ) . ( D Δ FCGV ( a ) FCGV ( b ) ) m − 2 . F FCGV ( α ) Δ FCGV ( b )
− i m δ a b ( ∂ ⃗ Δ ) m − 2 ( − g ˉ μ ν ( Δ ⋅ p ‾ ) ( Δ ⋅ q ‾ ) + Δ ν q ‾ μ ( Δ ⋅ p ‾ ) + Δ μ p ‾ ν ( Δ ⋅ q ‾ ) − Δ μ Δ ν ( p ‾ ⋅ q ‾ ) ) -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) − i m δ ab ( ∂ Δ ) m − 2 ( − g ˉ μν ( Δ ⋅ p ) ( Δ ⋅ q ) + Δ ν q μ ( Δ ⋅ p ) + Δ μ p ν ( Δ ⋅ q ) − Δ μ Δ ν ( p ⋅ q ) )
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 ]]
1 2 i m ϵ ˉ FCGV ( α ) FCGV ( β ) FCGV ( γ ) Δ . F FCGV ( β ) FCGV ( γ ) FCGV ( a ) . ( D Δ FCGV ( a ) FCGV ( b ) ) m − 2 . F FCGV ( α ) Δ FCGV ( 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})} 2 1 i m ϵ ˉ FCGV ( α ) FCGV ( β ) FCGV ( γ ) Δ . F FCGV ( β ) FCGV ( γ ) FCGV ( a ) . ( D Δ FCGV ( a ) FCGV ( b ) ) m − 2 . F FCGV ( α ) Δ FCGV ( b )
i m + 1 δ a b ( ∂ ⃗ Δ ) m − 2 ( Δ μ ϵ ˉ ν Δ p ‾ q ‾ − Δ ν ϵ ˉ μ Δ p ‾ q ‾ − ( Δ ⋅ p ‾ ) ϵ ˉ μ ν Δ q ‾ + ( Δ ⋅ q ‾ ) ϵ ˉ μ ν Δ p ‾ ) 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) i m + 1 δ ab ( ∂ Δ ) m − 2 ( Δ μ ϵ ˉ ν Δ p q − Δ ν ϵ ˉ μ Δ p q − ( Δ ⋅ p ) ϵ ˉ μν Δ q + ( Δ ⋅ q ) ϵ ˉ μν Δ p )
0 0 0
Compare with the Feynman rule tabulated in Twist2GluonOperator.
Twist2GluonOperator[ q , { \ [ Mu], a }, { \ [ Nu], b }, Polarization -> 1 , Explicit -> True ]
i ( 1 − ( − 1 ) m ) δ a b ( Δ ⋅ q ) m − 1 ϵ μ ν Δ q i \left(1-(-1)^m\right) \delta ^{ab} (\Delta \cdot q)^{m-1} \overset{\text{}}{\epsilon }^{\mu \nu \Delta q} i ( 1 − ( − 1 ) m ) δ ab ( Δ ⋅ q ) m − 1 ϵ μν Δ 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 ] ;
i m ψ ˉ . ( γ ˉ ⋅ Δ ) . D Δ m − 1 . ψ i^m \bar{\psi }.\left(\bar{\gamma }\cdot \Delta \right).D_{\Delta }{}^{m-1}.\psi i m ψ ˉ . ( γ ˉ ⋅ Δ ) . D Δ m − 1 . ψ
$Aborted \text{\$Aborted} $Aborted
1 1 1
Twist2QuarkOperator[{ p }, { q }, { r , \ [ Mu], a }, { s , \ [ Nu], b }, Polarization -> 0 ]
− ( − 1 ) m Δ μ Δ ν g s 2 ( γ ˉ ⋅ Δ ) . ( T a . T b ( ∑ i = 0 − 3 + m ( i + 1 ) ( Δ ⋅ q ) j ( − ( Δ ⋅ p ) ) − i + m − 3 ( Δ ⋅ q + Δ ⋅ r ) i − j ) + T b . T a ( ∑ i = 0 − 3 + m ( i + 1 ) ( Δ ⋅ q ) j ( − ( Δ ⋅ p ) ) − i + m − 3 ( Δ ⋅ q + Δ ⋅ s ) i − j ) ) -(-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) − ( − 1 ) m Δ μ Δ ν g s 2 ( γ ˉ ⋅ Δ ) . ( T a . T b ( i = 0 ∑ − 3 + m ( i + 1 ) ( Δ ⋅ q ) j ( − ( Δ ⋅ p ) ) − i + m − 3 ( Δ ⋅ q + Δ ⋅ r ) i − j ) + T b . T a ( i = 0 ∑ − 3 + m ( i + 1 ) ( Δ ⋅ q ) j ( − ( Δ ⋅ p ) ) − i + m − 3 ( Δ ⋅ q + Δ ⋅ s ) i − j ) )
(*Twist2QuarkOperator[{p},{q},{r,\[Mu],a},{s,\[Nu],b},Polarization->0]
Calc[frule-%%/.OPEm->5/.s->-p-q-r/.D->4]*)