DiracSimplify
DiracSimplify[exp]
simplifies products of Dirac matrices in exp
and expands noncommutative products. The simplifications are done by applying Contract
, DiracEquation
, DiracTrick
, SpinorChainTrick
and SirlinSimplify
. All γ5, γ6 and γ7 are moved to the right. The order of the other Dirac matrices is not changed, unless the option DiracOrder is set to True
.
See also
Overview, Contract, DiracEquation, DiracSigmaExplicit, DiracSubstitute5, DiracSubstitute67, DiracGamma, DiracGammaExpand, DiracOrder, DiracTrace, DiracTraceEvaluate, DiracTrick, FCDiracIsolate, SirlinSimplify, SpinorChainTrick, SpinorChainEvaluate, ToDiracGamma67.
Examples
Simplify a 4-dimensional Dirac matrix chain with a dummy Lorentz index
GA[\[Mu], \[Nu], \[Mu]]
DiracSimplify[%]
γˉμ.γˉν.γˉμ
−2γˉν
Another common simplification concerns Dirac matrices contracted to the same 4-vector
GS[p] . GS[p]
DiracSimplify[%]
(γˉ⋅p).(γˉ⋅p)
p2
Unlike DiracTrick
, DiracSimplify
also carries out noncommutative expansions
GS[a + b] . GS[p] . GS[c + d] . GS[p]
DiracSimplify[%]
(γˉ⋅(a+b)).(γˉ⋅p).(γˉ⋅(c+d)).(γˉ⋅p)
2(c⋅p)(γˉ⋅a).(γˉ⋅p)−p2(γˉ⋅a).(γˉ⋅c)+2(d⋅p)(γˉ⋅a).(γˉ⋅p)−p2(γˉ⋅a).(γˉ⋅d)+2(c⋅p)(γˉ⋅b).(γˉ⋅p)−p2(γˉ⋅b).(γˉ⋅c)+2(d⋅p)(γˉ⋅b).(γˉ⋅p)−p2(γˉ⋅b).(γˉ⋅d)
DiracTrick[GS[a + b] . GS[p] . GS[c + d] . GS[p]]
2(γˉ⋅(a+b)).(γˉ⋅p)((c+d)⋅p)−p2(γˉ⋅(a+b)).(γˉ⋅(c+d))
Some of those expansions can be inhibited via the option Expanding
.
DiracSimplify[GS[a + b] . GS[p] . GS[c + d] . GS[p], Expanding -> False]
−p2(γˉ⋅a+γˉ⋅b).(γˉ⋅c+γˉ⋅d)+2(c⋅p)(γˉ⋅a+γˉ⋅b).(γˉ⋅p)+2(d⋅p)(γˉ⋅a+γˉ⋅b).(γˉ⋅p)
The matrix chain may also live in D dimensions
GAD[\[Mu], \[Nu], \[Mu]]
DiracSimplify[%]
γμ.γν.γμ
2γν−Dγν
GSD[p] . GAD[\[Alpha], \[Beta]] . GSD[p]
DiracSimplify[%]
(γ⋅p).γα.γβ.(γ⋅p)
p2γα.γβ+2pαγβ.(γ⋅p)−2pβγα.(γ⋅p)
GAD @@ Join[{\[Mu]}, Table[Subscript[\[Nu], i], {i, 6}], {\[Mu]}]
DiracSimplify[%]
γμ.γν1.γν2.γν3.γν4.γν5.γν6.γμ
−12γν1.γν2.γν3.γν4.γν5.γν6+Dγν1.γν2.γν3.γν4.γν5.γν6+4γν3.γν4.γν5.γν6gν1ν2−4γν2.γν4.γν5.γν6gν1ν3+4γν2.γν3.γν5.γν6gν1ν4−4γν2.γν3.γν4.γν6gν1ν5+4γν2.γν3.γν4.γν5gν1ν6+4γν1.γν4.γν5.γν6gν2ν3−4γν1.γν3.γν5.γν6gν2ν4+4γν1.γν3.γν4.γν6gν2ν5−4γν1.γν3.γν4.γν5gν2ν6+4γν1.γν2.γν5.γν6gν3ν4−4γν1.γν2.γν4.γν6gν3ν5+4γν1.γν2.γν4.γν5gν3ν6+4γν1.γν2.γν3.γν6gν4ν5−4γν1.γν2.γν3.γν5gν4ν6+4γν1.γν2.γν3.γν4gν5ν6
-1/2 GA[5] . (GAD[\[Mu]] . GSD[v] - FVD[v, \[Mu]]) FVD[v, \[Mu]]
DiracSimplify[%]
−21vμγˉ5.(γμ.(γ⋅v)−vμ)
0
γ5 and the chirality projectors are always moved to the right
GA[5, \[Mu], \[Nu]]
DiracSimplify[%]
γˉ5.γˉμ.γˉν
γˉμ.γˉν.γˉ5
GA[6] . GS[p + q]
DiracSimplify[%]
γˉ6.(γˉ⋅(p+q))
(γˉ⋅p).γˉ7+(γˉ⋅q).γˉ7
The properties of the chirality projectors are taken into account without substituting explicit expressions for γ6 and γ7.
GA[\[Mu]] . (c1 GA[6] + c2 GA[7]) . (GA[p] + m) . (c3 GA[6] + c4 GA[7]) . GA[\[Mu]]
DiracSimplify[%]
γˉμ.(c1γˉ6+c2γˉ7).(γˉp+m).(c3γˉ6+c4γˉ7).γˉμ
4c1c3mγˉ7−2c1c4γˉp.γˉ6−2c2c3γˉp.γˉ7+4c2c4mγˉ6
Moreover, 21(1±γ5) is automatically replaced by γ6/7.
(1/2 - GA[5]/2) . (-((a + GS[p + q])/b)) . (1/2 + GA[5]/2)
DiracSimplify[%]
(21−2γˉ5).(−bγˉ⋅(p+q)+a).(2γˉ5+21)
−b(γˉ⋅p).γˉ6−b(γˉ⋅q).γˉ6
Suitable combinations of γ5 will not be rewritten in terms of chirality projectors, if the option ToDiracGamma67
is set to False
.
DiracSimplify[(1/2 - GA[5]/2) . (-((a + GS[p + q])/b)) . (1/2 + GA[5]/2),
ToDiracGamma67 -> False]
−2bγˉ⋅p−2b(γˉ⋅p).γˉ5−2bγˉ⋅q−2b(γˉ⋅q).γˉ5
However, it the final result must contain only γ5 but not γ6 or γ7, it is better to invoke the option DiracSubstitute67
. This way DiracSimplify can perform more intermediate simplifications before presenting the final result.
DiracSimplify[(1/2 - GA[5]/2) . (-((a + GS[p + q])/b)) . (1/2 + GA[5]/2),
DiracSubstitute67 -> True]
−2bγˉ⋅p−2b(γˉ⋅p).γˉ5−2bγˉ⋅q−2b(γˉ⋅q).γˉ5
It is also possible to eliminate γ5 by rewriting it in terms of the chirality projectors
DiracSimplify[GA[5, \[Mu], \[Nu]], DiracSubstitute5 -> True]
γˉμ.γˉν.γˉ6−γˉμ.γˉν.γˉ7
The Dirac equation is routinely used to simplify closed spinor chains.
(SpinorVBar[Subscript[p, 2], Subscript[m, 2]] . (GS[Subscript[p, 1]] +
Subscript[m, 1]) . SpinorU[Subscript[p, 1], Subscript[m, 1]])
DiracSimplify[%]
vˉ(p2,m2).(γˉ⋅p1+m1).u(p1,m1)
2m1(φ(−p2,m2)).(φ(p1,m1))
SpinorVBar[p] . GS[p] . SpinorU[q]
DiracSimplify[%]
vˉ(p).(γˉ⋅p).u(q)
0
Use the option DiracEquation
to deactivate this type of simplifications.
DiracSimplify[SpinorVBar[p] . GS[p] . SpinorU[q], DiracEquation -> False]
(φ(−p)).(γˉ⋅p).(φ(q))
Suitable products of 4-dimensional spinor chains are simplified using Sirlin’s identities
(SpinorUBar[Subscript[p, 3], Subscript[m, 3]] . GA[\[Mu], \[Rho], \[Nu], 6] . SpinorU[Subscript[p, 1],
Subscript[m, 1]] SpinorUBar[Subscript[p, 4],
Subscript[m, 4]] . GA[\[Mu], \[Tau], \[Nu], 6] . SpinorU[Subscript[p, 2], Subscript[m, 2]])
DiracSimplify[%]
uˉ(p3,m3).γˉμ.γˉρ.γˉν.γˉ6.u(p1,m1)uˉ(p4,m4).γˉμ.γˉτ.γˉν.γˉ6.u(p2,m2)
(φ(p3,m3)).γˉμ.γˉρ.γˉν.γˉ6.(φ(p1,m1))(φ(p4,m4)).γˉμ.γˉτ.γˉν.γˉ6.(φ(p2,m2))
The applications of Sirlin’s identities can be disabled by setting the option SirlinSimplify
to False
.
DiracSimplify[SpinorUBar[Subscript[p, 3], Subscript[m, 3]] . GA[\[Mu], \[Rho], \[Nu],
6] . SpinorU[Subscript[p, 1], Subscript[m, 1]]*
SpinorUBar[Subscript[p, 4], Subscript[m, 4]] . GA[\[Mu], \[Tau], \[Nu],
6] . SpinorU[Subscript[p, 2], Subscript[m, 2]], SirlinSimplify -> False]
(φ(p3,m3)).γˉμ.γˉρ.γˉν.γˉ6.(φ(p1,m1))(φ(p4,m4)).γˉμ.γˉτ.γˉν.γˉ6.(φ(p2,m2))
Even when the usage of Sirlin’s identities is disabled, DiracSimplify will still try to perform some simplifications on the spinor chains, e.g. by canonicalizing the dummy indices.
(c1 SpinorUBar[Subscript[p, 3], Subscript[m, 3]] . GA[\[Mu], \[Rho], \[Nu], 6] . SpinorU[Subscript[p,
1], Subscript[m, 1]] SpinorUBar[Subscript[p, 4], Subscript[m,
4]] . GA[\[Mu], \[Tau], \[Nu], 6] . SpinorU[Subscript[p, 2], Subscript[m, 2]] +
c2 SpinorUBar[Subscript[p, 3], Subscript[m, 3]] . GA[\[Alpha], \[Rho],
\[Nu], 6] . SpinorU[Subscript[p, 1], Subscript[m, 1]] SpinorUBar[Subscript[p,
4], Subscript[m, 4]] . GA[\[Alpha], \[Tau], \[Nu], 6] . SpinorU[Subscript[p, 2], Subscript[m, 2]])
DiracSimplify[%, SirlinSimplify -> False] // Factor
c1uˉ(p3,m3).γˉμ.γˉρ.γˉν.γˉ6.u(p1,m1)uˉ(p4,m4).γˉμ.γˉτ.γˉν.γˉ6.u(p2,m2)+c2uˉ(p3,m3).γˉα.γˉρ.γˉν.γˉ6.u(p1,m1)uˉ(p4,m4).γˉα.γˉτ.γˉν.γˉ6.u(p2,m2)
c1(φ(p3,m3)).γˉμ.γˉρ.γˉν.γˉ6.(φ(p1,m1))(φ(p4,m4)).γˉμ.γˉτ.γˉν.γˉ6.(φ(p2,m2))+c2(φ(p3,m3)).γˉα.γˉρ.γˉν.γˉ6.(φ(p1,m1))(φ(p4,m4)).γˉα.γˉτ.γˉν.γˉ6.(φ(p2,m2))
Setting SpinorChainTrick
to `False disables this behavior.
DiracSimplify[c1 SpinorUBar[Subscript[p, 3], Subscript[m, 3]] . GA[\[Mu], \[Rho],
\[Nu], 6] . SpinorU[Subscript[p, 1], Subscript[m, 1]] SpinorUBar[Subscript[p,
4], Subscript[m, 4]] . GA[\[Mu], \[Tau], \[Nu], 6] . SpinorU[Subscript[p, 2],
Subscript[m, 2]] + c2 SpinorUBar[Subscript[p, 3], Subscript[m,
3]] . GA[\[Alpha], \[Rho], \[Nu], 6] . SpinorU[Subscript[p, 1], Subscript[m,
1]] SpinorUBar[Subscript[p, 4], Subscript[m, 4]] . GA[\[Alpha], \[Tau],
\[Nu], 6] . SpinorU[Subscript[p, 2], Subscript[m, 2]],
SirlinSimplify -> False, SpinorChainTrick -> False]
c1(φ(p3,m3)).γˉμ.γˉρ.γˉν.γˉ6.(φ(p1,m1))(φ(p4,m4)).γˉμ.γˉτ.γˉν.γˉ6.(φ(p2,m2))+c2(φ(p3,m3)).γˉα.γˉρ.γˉν.γˉ6.(φ(p1,m1))(φ(p4,m4)).γˉα.γˉτ.γˉν.γˉ6.(φ(p2,m2))
DiracSimplify
will not reorder Dirac matrices lexicographically, but can be forced to do so via the option DiracOrder
.
DiracSimplify[GA[\[Nu], \[Mu]]]
DiracSimplify[GA[\[Nu], \[Mu]], DiracOrder -> True]
γˉν.γˉμ
2gˉμν−γˉμ.γˉν
Setting InsideDiracTrace
to True
$ makes the function assume that it is acting inside a Dirac trace. For instance, chains with an odd number of Dirac matrices will be set to zero.
GA[\[Mu], \[Nu], \[Rho]]
DiracSimplify[%, InsideDiracTrace -> True]
γˉμ.γˉν.γˉρ
0
Yet, it will not explicitly calculate the trace
GA[\[Mu], \[Nu], \[Rho], \[Sigma]]
DiracSimplify[%, InsideDiracTrace -> True]
γˉμ.γˉν.γˉρ.γˉσ
γˉμ.γˉν.γˉρ.γˉσ
Since FeynCalc 9.3, DiracSimplify
will automatically evaluate Dirac traces in the input expression
DiracTrace[GA[\[Mu], \[Nu], \[Rho], \[Sigma]]]
DiracSimplify[%]
tr(γˉμ.γˉν.γˉρ.γˉσ)
4gˉμσgˉνρ−4gˉμρgˉνσ+4gˉμνgˉρσ
DiracTrace[(-GSD[q] + SMP["m_e"]) . GAD[\[Nu]] . (GSD[p - q] + SMP["m_e"]) . GAD[\[Mu]]]
DiracSimplify[%]
tr((me−γ⋅q).γν.(me+γ⋅(p−q)).γμ)
4me2gμν+4gμν(p⋅q)−4q2gμν−4pνqμ−4pμqν+8qμqν
This will not happen if the option DiracTraceEvaluate
is set to False
. However, DiracSimplify
will still perform some simplifications inside the trace, without evaluating it explicitly.
DiracSimplify[DiracTrace[(-GSD[q] + SMP["m_e"]) . GAD[\[Nu]] . (GSD[p - q] +
SMP["m_e"]) . GAD[\[Mu]]] , DiracTraceEvaluate -> False]
tr(me2γν.γμ+meγν.(γ⋅p).γμ−meγν.(γ⋅q).γμ−me(γ⋅q).γν.γμ−(γ⋅q).γν.(γ⋅p).γμ−q2γν.γμ+2qν(γ⋅q).γμ)
Set DiracTrace
to False
if you want DiracSimplify
not to touch the Dirac traces.
DiracSimplify[DiracTrace[(-GSD[q] + SMP["m_e"]) . GAD[\[Nu]] . (GSD[p - q] +
SMP["m_e"]) . GAD[\[Mu]]] , DiracTraceEvaluate -> False, DiracTrace -> False]
tr((me−γ⋅q).γν.(me+γ⋅(p−q)).γμ)
When doing calculations at one loop and above, you may encounter expressions that contain D- and 4-dimensional objects.
Although DiracSimplify
can handle such terms effortlessly, it will not do so unless you certify that you fully understand what you are doing: being sloppy with the dimensions easily leads to inconsistencies and wrong results!
GAD[\[Mu]] . (GA[p] + m) . GAD[\[Mu]]
DiracSimplify[%]
γμ.(γˉp+m).γμ
![161ti5temvheu](img/161ti5temvheu.svg)
$Aborted
By default, FeynCalc uses the naive dimensional regularization (NDR) scheme, where all Dirac matrices are taken to be D-dimensional. Therefore, in NDR you may not have mixtures of Dirac matrices living in D and 4 dimensions. However, such expressions are possible in the t’Hooft-Veltman-Breitenlohner-Maison (BMHV) scheme.
FCSetDiracGammaScheme["BMHV"];
DiracSimplify[GAD[\[Mu]] . (GA[p] + m) . GAD[\[Mu]]]
−Dγˉp+2γˉp+Dm
FCSetDiracGammaScheme["NDR"];
The BMHV scheme is a special prescription for handling γ5 in dimensional regularization. Do not activate this scheme mindlessly just to get rid of errors from DiracSimplify! If you are doing a calculation in NDR or a calculation that does not involve γ5, better make sure that your input expressions are correctly written to be D-dimensional objects.
Traces that contain an odd number of γ5 or chirality projectors cannot be calculated unambiguously in NDR. To avoid inconsistencies, DiracTrace will refuse to evaluate such traces in NDR.
DiracTrace[GAD[\[Mu], \[Nu], \[Rho], \[Sigma], \[Alpha], \[Beta]] . GA[5]]
DiracSimplify[%]
tr(γμ.γν.γρ.γσ.γα.γβ.γˉ5)
tr(γμ.γν.γρ.γσ.γα.γβ.γˉ5)
Such traces can be consistently calculated in the BMHV scheme. Our scheme choice as of course also possible, but those are not implemented in FeynCalc.
FCSetDiracGammaScheme["BMHV"];
DiracSimplify[DiracTrace[GAD[\[Mu], \[Nu], \[Rho], \[Sigma], \[Alpha], \[Beta]] . GA[5]]]
−4igαβϵˉμνρσ−4igαμϵˉβνρσ+4igανϵˉβμρσ−4igαρϵˉβμνσ+4igασϵˉβμνρ+4igβμϵˉανρσ−4igβνϵˉαμρσ+4igβρϵˉαμνσ−4igβσϵˉαμνρ−4igμνϵˉαβρσ+4igμρϵˉαβνσ−4igμσϵˉαβνρ−4igνρϵˉαβμσ+4igνσϵˉαβμρ−4igρσϵˉαβμν
FCSetDiracGammaScheme["NDR"];
Keep in mind that the BMHV scheme violates axial Ward identities and requires special model-dependent counter-terms to restore those. Therefore, just setting FCSetDiracGammaScheme["BMHV"]
does not magically resolve all your troubles with γ5 in D-dimensions. The proper treatment of γ5 in dimensional regularization is an intricate issue that cannot be boiled down to a simple and universal recipe. FeynCalc merely carries out the algebraic operations that you request, but it is still your task to ensure that what you do makes sense.
Since FeynCalc 9.3 it is also possible to simplify Dirac matrices with Cartesian or temporal indices. However, the support of nonrelativistic calculations is a very new feature, so that things may not work as smooth as they do for manifestly Lorentz covariant expressions.
CGA[i] . CGA[i]
DiracSimplify[%]
γi.γi
−3
CGA[i] . CGS[p] . CGA[j] . CGS[p + q]
DiracSimplify[%]
γi.(γ⋅p).γj.(γ⋅(p+q))
p2γi.γj−2pjγi.(γ⋅p)+γi.(γ⋅p).γj.(γ⋅q)
CGA[i] . CGS[p] . CGA[j] . CGS[p + q] KD[i, j]
DiracSimplify[%]
δˉijγi.(γ⋅p).γj.(γ⋅(p+q))
(γ⋅p).(γ⋅q)−p2
TGA[] . CGA[i] . TGA[]
DiracSimplify[%]
γˉ0.γi.γˉ0
−γi
DiracTrace[CGAD[i, j, k, l]]
DiracSimplify[%]
tr(γi.γj.γk.γl)
4δilδjk−4δikδjl+4δijδkl
For performance reasons DiracSimplify
will not canonically order Dirac matrices and canonicalize Lorentz/Cartesian indices by default. However, amplitudes involving 4-fermion operators may require such additional simplifications. In this case they should explicitly activated by the user. Of course, this will somewhat slow down the evaluation.
ex = (Spinor[-Momentum[p1, D], mb, 1] . GAD[\[Mu]] . GA[7] . GAD[\[Nu]] . GAD[\[Alpha]] .
GAD[\[Beta]] . GAD[\[Delta]] . GA[7] . Spinor[-Momentum[p4, D], 0, 1] Spinor[Momentum[p3, D], 0,
1] . GAD[\[Alpha]] . GAD[\[Beta]] . GAD[\[Delta]] . GA[7] . GAD[\[Nu]] . GAD[\[Mu]] .
GA[7] . Spinor[Momentum[p2, D], 0, 1])
DiracSimplify[ex]
(φ(p3)).γα.γβ.γδ.γˉ7.γν.γμ.γˉ7.(φ(p2))(φ(−p1,mb)).γμ.γˉ7.γν.γα.γβ.γδ.γˉ7.(φ(−p4))
(φ(p3)).γα.γβ.γδ.γν.γμ.γˉ7.(φ(p2))(φ(−p1,mb)).γμ.γν.γα.γβ.γδ.γˉ7.(φ(−p4))
DiracSimplify[ex, DiracOrder -> True, LorentzIndexNames -> {i1, i2, i3, i4, i5}]
−24D2(φ(p3)).γi1.γˉ7.(φ(p2))(φ(−p1,mb)).γi1.γˉ7.(φ(−p4))+14D(φ(p3)).γi1.γi2.γi3.γˉ7.(φ(p2))(φ(−p1,mb)).γi1.γi2.γi3.γˉ7.(φ(−p4))+112D(φ(p3)).γi1.γˉ7.(φ(p2))(φ(−p1,mb)).γi1.γˉ7.(φ(−p4))−(φ(p3)).γi1.γi2.γi3.γi4.γi5.γˉ7.(φ(p2))(φ(−p1,mb)).γi1.γi2.γi3.γi4.γi5.γˉ7.(φ(−p4))−36(φ(p3)).γi1.γi2.γi3.γˉ7.(φ(p2))(φ(−p1,mb)).γi1.γi2.γi3.γˉ7.(φ(−p4))−64(φ(p3)).γi1.γˉ7.(φ(p2))(φ(−p1,mb)).γi1.γˉ7.(φ(−p4))
DiracSimplify
automatically evaluates suitable spinor products with equal momenta, e.g.
ex = SpinorUBar[p, m] . SpinorU[p, m]
DiracSimplify[ex]
uˉ(p,m).u(p,m)
2m
This behavior can be turned off by setting the option SpinorChainEvaluate
to False
DiracSimplify[ex, SpinorChainEvaluate -> False]
(φ(p,m)).(φ(p,m))