CovariantD[mu] is a generic covariant derivative with
Lorentz index \mu.
CovariantD[x, mu] is a generic covariant derivative with
respect to x^{\mu }.
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 SU(N) indices in
the adjoint representation.
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 for the gluon, where n is an integer and g the coupling constant indicated by the
setting of the option CouplingConstant.
CovariantD[OPEDelta, {m, n}] gives the expanded form of
CovariantD[OPEDelta]^m up to order g^n of the fermionic field. To obtain the
explicit expression for a particular covariant derivative, the option
Explicit must be set to True.
CovariantD[\[Mu]]D_{\mu }
CovariantD[\[Mu], a, b]D_{\mu }^{ab}
CovariantD[\[Mu], Explicit -> True]\vec{\partial }_{\mu }-i g_s T^{\text{c19}}.A_{\mu }^{\text{c19}}
The first argument of CovariantD is interpreted as type
LorentzIndex, except for OPEDelta, which is
type Momentum.
CovariantD[OPEDelta]D_{\Delta }
CovariantD[OPEDelta, a, b]D_{\Delta }^{ab}
CovariantD[OPEDelta, a, b, Explicit -> True]\delta ^{ab} \vec{\partial }_{\Delta }-g_s A_{\Delta }^{\text{c20}} f^{ab\text{c20}}
CovariantD[OPEDelta, Explicit -> True]\vec{\partial }_{\Delta }-i g_s T^{\text{c21}}.A_{\Delta }^{\text{c21}}
CovariantD[OPEDelta, a, b, {2}]\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)
This gives m * \vec{\partial}_{\Delta}, the partial derivative \vec{\partial}_{\mu } contracted with \Delta ^{\mu }
CovariantD[OPEDelta, a, b, {OPEm, 0}]\delta ^{ab} \left(\vec{\partial }_{\Delta }\right){}^m
The expansion up to first order in the coupling constant g_s (the sum is the
FeynCalcOPESum)
CovariantD[OPEDelta, a, b, {OPEm, 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)
The expansion up to second order in the g_s
CovariantD[OPEDelta, a, b, {OPEm, 2}]-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
CovariantD[OPEDelta, a, b]^OPEm\left(D_{\Delta }^{ab}\right){}^m
CovariantD[OPEDelta, {OPEm, 2}]-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
CovariantD[OPEDelta, Explicit -> True] // StandardForm
(*RightPartialD[Momentum[OPEDelta]] - I SUNT[SUNIndex[c62]] . QuantumField[GaugeField, Momentum[OPEDelta], SUNIndex[c62]] SMP["g_s"]*)CovariantD[\[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]*)