CovariantD
CovariantD[mu] is a generic covariant derivative with
Lorentz index μ \mu μ
CovariantD[x, mu] is a generic covariant derivative with
respect to x μ x^{\mu } x μ
CovariantD[mu, a, b] is a covariant derivative for a
bosonic field that acts on QuantumField[f, {}, {a, b}],
where f is some field name and a and
b are two S U ( N ) SU(N) S U ( N )
CovariantD[OPEDelta, a, b] is a short form for
CovariantD[mu, a, b] FV[OPEDelta, mu].
CovariantD[{OPEDelta, a, b}, {n}] yields the product of
n operators, where n is an integer.
CovariantD[OPEDelta, a, b, {m, n}] gives the expanded
form of CovariantD[OPEDelta, a, b]^m up to order g n g^n g n n n n g g g CouplingConstant.
CovariantD[OPEDelta, {m, n}] gives the expanded form of
CovariantD[OPEDelta]^m up to order g n g^n g n Explicit must be set to True.
See also
Overview
Examples
D μ D_{\mu } D μ
D μ a b D_{\mu }^{ab} D μ ab
[ \ [ Mu], Explicit -> True ] ∂ ⃗ μ − i g s T c19 . A μ c19 \vec{\partial }_{\mu }-i g_s
T^{\text{c19}}.A_{\mu }^{\text{c19}} ∂ μ − i g s T c19 . A μ c19
The first argument of CovariantD is interpreted as type
LorentzIndex, except for OPEDelta, which is
type Momentum.
D Δ D_{\Delta } D Δ
[ OPEDelta, a , b ] D Δ a b D_{\Delta }^{ab} D Δ ab
[ OPEDelta, a , b , Explicit -> True ] δ a b ∂ ⃗ Δ − g s A Δ c20 f a b c20 \delta ^{ab} \vec{\partial }_{\Delta }-g_s
A_{\Delta }^{\text{c20}} f^{ab\text{c20}} δ ab ∂ Δ − g s A Δ c20 f ab c20
[ OPEDelta, Explicit -> True ] ∂ ⃗ Δ − i g s T c21 . A Δ c21 \vec{\partial }_{\Delta }-i g_s
T^{\text{c21}}.A_{\Delta }^{\text{c21}} ∂ Δ − i g s T c21 . A Δ c21
[ OPEDelta, a , b , { 2 }] ( δ a c22 ∂ ⃗ Δ − g s A Δ e23 f a c22 e23 ) . ( δ b c22 ∂ ⃗ Δ − g s A Δ e24 f c22 b e24 ) \left(\delta ^{a\text{c22}} \vec{\partial
}_{\Delta }-g_s A_{\Delta }^{\text{e23}}
f^{a\text{c22}\;\text{e23}}\right).\left(\delta ^{b\text{c22}}
\vec{\partial }_{\Delta }-g_s A_{\Delta }^{\text{e24}}
f^{\text{c22}b\text{e24}}\right) ( δ a c22 ∂ Δ − g s A Δ e23 f a c22 e23 ) . ( δ b c22 ∂ Δ − g s A Δ e24 f c22 b e24 )
This gives m ∗ ∂ ⃗ Δ m *
\vec{\partial}_{\Delta} m ∗ ∂ Δ ∂ ⃗ μ \vec{\partial}_{\mu } ∂ μ Δ μ \Delta ^{\mu } Δ μ
[ OPEDelta, a , b , { OPEm, 0 }] δ a b ( ∂ ⃗ Δ ) m \delta ^{ab} \left(\vec{\partial }_{\Delta
}\right){}^m δ ab ( ∂ Δ ) m
The expansion up to first order in the coupling constant g s g_s g s FeynCalcOPESum)
[ OPEDelta, a , b , { OPEm, 1 }] δ a b ( ∂ ⃗ Δ ) m − g s ( ∑ i = 0 − 1 + m ( ∂ ⃗ Δ ) i . A Δ c34 1 . ( ∂ ⃗ Δ ) − 1 − i + m f a b c34 1 ) \delta ^{ab} \left(\vec{\partial }_{\Delta
}\right){}^m-g_s \left(\sum _{i=0}^{-1+m} \left(\vec{\partial }_{\Delta
}\right){}^i.A_{\Delta }^{\text{c34}_1}.\left(\vec{\partial }_{\Delta
}\right){}^{-1-i+m} f^{ab\text{c34}_1}\right) δ ab ( ∂ Δ ) m − g s ( i = 0 ∑ − 1 + m ( ∂ Δ ) i . A Δ c34 1 . ( ∂ Δ ) − 1 − i + m f ab c34 1 )
The expansion up to second order in the g s g_s g s
[ OPEDelta, a , b , { OPEm, 2 }] − g s ( ∑ i = 0 − 1 + m ( ∂ ⃗ Δ ) i . A Δ c42 1 . ( ∂ ⃗ Δ ) − 1 − i + m f a b c42 1 ) − g s 2 ( ∑ j = 0 − 2 + m ( ∑ i = 0 j ( ∂ ⃗ Δ ) i . A Δ c46 1 . ( ∂ ⃗ Δ ) − i + j . A Δ c46 2 . ( ∂ ⃗ Δ ) − 2 − j + m f a c46 1 e45 1 f b c46 2 e45 1 ) ) + δ a b ( ∂ ⃗ Δ ) m -g_s \left(\sum _{i=0}^{-1+m}
\left(\vec{\partial }_{\Delta }\right){}^i.A_{\Delta
}^{\text{c42}_1}.\left(\vec{\partial }_{\Delta }\right){}^{-1-i+m}
f^{ab\text{c42}_1}\right)-g_s^2 \left(\sum _{j=0}^{-2+m} \left(\sum
_{i=0}^j \left(\vec{\partial }_{\Delta }\right){}^i.A_{\Delta
}^{\text{c46}_1}.\left(\vec{\partial }_{\Delta
}\right){}^{-i+j}.A_{\Delta }^{\text{c46}_2}.\left(\vec{\partial
}_{\Delta }\right){}^{-2-j+m} f^{a\text{c46}_1\text{e45}_1}
f^{b\text{c46}_2\text{e45}_1}\right)\right)+\delta ^{ab}
\left(\vec{\partial }_{\Delta }\right){}^m − g s ( i = 0 ∑ − 1 + m ( ∂ Δ ) i . A Δ c42 1 . ( ∂ Δ ) − 1 − i + m f ab c42 1 ) − g s 2 ( j = 0 ∑ − 2 + m ( i = 0 ∑ j ( ∂ Δ ) i . A Δ c46 1 . ( ∂ Δ ) − i + j . A Δ c46 2 . ( ∂ Δ ) − 2 − j + m f a c46 1 e45 1 f b c46 2 e45 1 ) ) + δ ab ( ∂ Δ ) m
[ OPEDelta, a , b ] ^ OPEm( D Δ a b ) m \left(D_{\Delta
}^{ab}\right){}^m ( D Δ ab ) m
[ OPEDelta, { OPEm, 2 }] − i g s ( ∑ i = 0 − 1 + m ( ∂ ⃗ Δ ) i . A Δ c55 1 . ( ∂ ⃗ Δ ) − 1 − i + m T c55 1 ) − g s 2 ( ∑ j = 0 − 2 + m ( ∑ i = 0 j T c59 1 . T c59 2 ( ∂ ⃗ Δ ) i . A Δ c59 1 . ( ∂ ⃗ Δ ) − i + j . A Δ c59 2 . ( ∂ ⃗ Δ ) − 2 − j + m ) ) + ( ∂ ⃗ Δ ) m -i g_s \left(\sum _{i=0}^{-1+m}
\left(\vec{\partial }_{\Delta }\right){}^i.A_{\Delta
}^{\text{c55}_1}.\left(\vec{\partial }_{\Delta }\right){}^{-1-i+m}
T^{\text{c55}_1}\right)-g_s^2 \left(\sum _{j=0}^{-2+m} \left(\sum
_{i=0}^j T^{\text{c59}_1}.T^{\text{c59}_2} \left(\vec{\partial }_{\Delta
}\right){}^i.A_{\Delta }^{\text{c59}_1}.\left(\vec{\partial }_{\Delta
}\right){}^{-i+j}.A_{\Delta }^{\text{c59}_2}.\left(\vec{\partial
}_{\Delta }\right){}^{-2-j+m}\right)\right)+\left(\vec{\partial
}_{\Delta }\right){}^m − i g s ( i = 0 ∑ − 1 + m ( ∂ Δ ) i . A Δ c55 1 . ( ∂ Δ ) − 1 − i + m T c55 1 ) − g s 2 ( j = 0 ∑ − 2 + m ( i = 0 ∑ j T c59 1 . T c59 2 ( ∂ Δ ) i . A Δ c59 1 . ( ∂ Δ ) − i + j . A Δ c59 2 . ( ∂ Δ ) − 2 − j + m ) ) + ( ∂ Δ ) m
[ OPEDelta, Explicit -> True ] // StandardForm (*RightPartialD[Momentum[OPEDelta]] - I SUNT[SUNIndex[c62]] . QuantumField[GaugeField, Momentum[OPEDelta], SUNIndex[c62]] SMP["g_s"]*) [ \ [ Mu], a , b , Explicit -> True ] // StandardForm (*RightPartialD[LorentzIndex[\[Mu]]] SUNDelta[a, b] - QuantumField[GaugeField, LorentzIndex[\[Mu]], SUNIndex[c63]] SMP["g_s"] SUNF[a, b, c63]*)