DoPolarizationSums[exp, k, ...] acts on an expression exp that must contain a polarization vector \varepsilon(k) and its complex conjugate (e.g. exp can be a matrix element squared).
Depending on the arguments of the function, it will perform a sum over the polarization of \varepsilon(k) and its c.c.
DoPolarizationSums[exp, k] sums over the three physical polarizations of an external massive vector boson with the 4-momentum k and the mass k^2.DoPolarizationSums[exp, k, 0] replaces the polarization sum of an external massless vector boson with the momentum k by -g^{\mu \nu}. This corresponds to the summation over all 4 polarizations, including the unphysical ones.DoPolarizationSums[exp, k, n] sums over physical (transverse) polarizations of an external massless vector boson with the momentum k, where n is an auxiliary 4-vector from the gauge-dependent polarization sum formula.Cf. PolarizationSum for more examples and explanations on different polarizations.
DoPolarizationSums also work with D-dimensional amplitudes.
Overview, Polarization, PolarizationSum, NumberOfPolarizations, VirtualBoson, Uncontract.
The standard formula for massless vector bosons is valid for all types of the corresponding particles, including gluons.
FCClearScalarProducts[]
SP[p] = 0;
Pair[LorentzIndex[\[Mu]], Momentum[Polarization[p, -I]]] Pair[LorentzIndex[\[Nu]],
Momentum[Polarization[p, I]]]\bar{\varepsilon }^{*\mu }(p) \bar{\varepsilon }^{\nu }(p)
DoPolarizationSums[%, p, n]-\frac{\overline{n}^2 \overline{p}^{\mu } \overline{p}^{\nu }}{(\overline{n}\cdot \overline{p})^2}-\bar{g}^{\mu \nu }+\frac{\overline{n}^{\nu } \overline{p}^{\mu }}{\overline{n}\cdot \overline{p}}+\frac{\overline{n}^{\mu } \overline{p}^{\nu }}{\overline{n}\cdot \overline{p}}
In QED the gauge invariance ensures the cancellation of the unphysical polarizations so that for photons one can also employ the simpler replacement with the metric tensor.
FCClearScalarProducts[]
SP[p] = 0;
Pair[LorentzIndex[\[Mu]], Momentum[Polarization[p, -I]]] Pair[LorentzIndex[\[Nu]],
Momentum[Polarization[p, I]]]\bar{\varepsilon }^{*\mu }(p) \bar{\varepsilon }^{\nu }(p)
DoPolarizationSums[%, p, 0]-\bar{g}^{\mu \nu }
You can also use this trick in QCD, provided that the unphysical degrees of freedom are subtracted using ghosts at a later stage.
Notice that in this case you should not make the polarization vectors transverse using the Transversality option.
Furthermore, the averaging over the polarizations of the initial gluons must be done on the physical amplitude squared, i.e. after the ghost contributions have been subtracted.
Massive vector bosons (e.g. W or Z) have 3 degrees of freedom and require no auxiliary vector.
FCClearScalarProducts[]
SP[p] = m^2;
Pair[LorentzIndex[\[Mu]], Momentum[Polarization[p, -I]]] Pair[LorentzIndex[\[Nu]],
Momentum[Polarization[p, I]]]\bar{\varepsilon }^{*\mu }(p) \bar{\varepsilon }^{\nu }(p)
DoPolarizationSums[%, p]\frac{\overline{p}^{\mu } \overline{p}^{\nu }}{m^2}-\bar{g}^{\mu \nu }
A more realistic example of summing over the polarizations of the photons in e^+e^ \to \gamma \gamma
ClearAll[s, t, u];
FCClearScalarProducts[];
SP[k1] = 0;
SP[k2] = 0;
amp = (-((Spinor[Momentum[p1], 0, 1] . GS[Polarization[k1, I,
Transversality -> True]] . GS[k2 - p2] . GS[Polarization[k2, I,
Transversality -> True]] . Spinor[-Momentum[p2], 0, 1]*SMP["e"]^2)/t) -
(Spinor[Momentum[p1], 0, 1] . GS[Polarization[k2, I,
Transversality -> True]] . GS[k1 - p2] . GS[Polarization[k1, I,
Transversality -> True]] . Spinor[-Momentum[p2], 0, 1]*SMP["e"]^2)/u)*
(-((Spinor[-Momentum[p2], 0, 1] . GS[Polarization[k1, -I,
Transversality -> True]] . GS[k1 - p2] . GS[Polarization[k2, -I,
Transversality -> True]] . Spinor[Momentum[p1], 0, 1]*
SMP["e"]^2)/u) - (Spinor[-Momentum[p2], 0, 1] . GS[Polarization[k2, -I,
Transversality -> True]] . GS[k2 - p2] . GS[Polarization[k1, -I,
Transversality -> True]] . Spinor[Momentum[p1], 0, 1]*SMP["e"]^2)/t)\left(-\frac{\text{e}^2 \left(\varphi (\overline{\text{p1}})\right).\left(\bar{\gamma }\cdot \bar{\varepsilon }(\text{k1})\right).\left(\bar{\gamma }\cdot \left(\overline{\text{k2}}-\overline{\text{p2}}\right)\right).\left(\bar{\gamma }\cdot \bar{\varepsilon }(\text{k2})\right).\left(\varphi (-\overline{\text{p2}})\right)}{t}-\frac{\text{e}^2 \left(\varphi (\overline{\text{p1}})\right).\left(\bar{\gamma }\cdot \bar{\varepsilon }(\text{k2})\right).\left(\bar{\gamma }\cdot \left(\overline{\text{k1}}-\overline{\text{p2}}\right)\right).\left(\bar{\gamma }\cdot \bar{\varepsilon }(\text{k1})\right).\left(\varphi (-\overline{\text{p2}})\right)}{u}\right) \left(-\frac{\text{e}^2 \left(\varphi (-\overline{\text{p2}})\right).\left(\bar{\gamma }\cdot \bar{\varepsilon }^*(\text{k2})\right).\left(\bar{\gamma }\cdot \left(\overline{\text{k2}}-\overline{\text{p2}}\right)\right).\left(\bar{\gamma }\cdot \bar{\varepsilon }^*(\text{k1})\right).\left(\varphi (\overline{\text{p1}})\right)}{t}-\frac{\text{e}^2 \left(\varphi (-\overline{\text{p2}})\right).\left(\bar{\gamma }\cdot \bar{\varepsilon }^*(\text{k1})\right).\left(\bar{\gamma }\cdot \left(\overline{\text{k1}}-\overline{\text{p2}}\right)\right).\left(\bar{\gamma }\cdot \bar{\varepsilon }^*(\text{k2})\right).\left(\varphi (\overline{\text{p1}})\right)}{u}\right)
amp // DoPolarizationSums[#, k1, 0] & // DoPolarizationSums[#, k2, 0] &\frac{\text{e}^4 \left(\varphi (-\overline{\text{p2}})\right).\bar{\gamma }^{\text{\$MU}(\text{\$27})}.\left(\bar{\gamma }\cdot \left(\overline{\text{k1}}-\overline{\text{p2}}\right)\right).\bar{\gamma }^{\text{\$MU}(\text{\$29})}.\left(\varphi (\overline{\text{p1}})\right) \left(\varphi (\overline{\text{p1}})\right).\bar{\gamma }^{\text{\$MU}(\text{\$27})}.\left(\bar{\gamma }\cdot \left(\overline{\text{k2}}-\overline{\text{p2}}\right)\right).\bar{\gamma }^{\text{\$MU}(\text{\$29})}.\left(\varphi (-\overline{\text{p2}})\right)}{t u}+\frac{\text{e}^4 \left(\varphi (\overline{\text{p1}})\right).\bar{\gamma }^{\text{\$MU}(\text{\$29})}.\left(\bar{\gamma }\cdot \left(\overline{\text{k1}}-\overline{\text{p2}}\right)\right).\bar{\gamma }^{\text{\$MU}(\text{\$27})}.\left(\varphi (-\overline{\text{p2}})\right) \left(\varphi (-\overline{\text{p2}})\right).\bar{\gamma }^{\text{\$MU}(\text{\$29})}.\left(\bar{\gamma }\cdot \left(\overline{\text{k2}}-\overline{\text{p2}}\right)\right).\bar{\gamma }^{\text{\$MU}(\text{\$27})}.\left(\varphi (\overline{\text{p1}})\right)}{t u}+\frac{\text{e}^4 \left(\varphi (\overline{\text{p1}})\right).\bar{\gamma }^{\text{\$MU}(\text{\$29})}.\left(\bar{\gamma }\cdot \left(\overline{\text{k1}}-\overline{\text{p2}}\right)\right).\bar{\gamma }^{\text{\$MU}(\text{\$27})}.\left(\varphi (-\overline{\text{p2}})\right) \left(\varphi (-\overline{\text{p2}})\right).\bar{\gamma }^{\text{\$MU}(\text{\$27})}.\left(\bar{\gamma }\cdot \left(\overline{\text{k1}}-\overline{\text{p2}}\right)\right).\bar{\gamma }^{\text{\$MU}(\text{\$29})}.\left(\varphi (\overline{\text{p1}})\right)}{u^2}+\frac{\text{e}^4 \left(\varphi (\overline{\text{p1}})\right).\bar{\gamma }^{\text{\$MU}(\text{\$27})}.\left(\bar{\gamma }\cdot \left(\overline{\text{k2}}-\overline{\text{p2}}\right)\right).\bar{\gamma }^{\text{\$MU}(\text{\$29})}.\left(\varphi (-\overline{\text{p2}})\right) \left(\varphi (-\overline{\text{p2}})\right).\bar{\gamma }^{\text{\$MU}(\text{\$29})}.\left(\bar{\gamma }\cdot \left(\overline{\text{k2}}-\overline{\text{p2}}\right)\right).\bar{\gamma }^{\text{\$MU}(\text{\$27})}.\left(\varphi (\overline{\text{p1}})\right)}{t^2}
This is a small piece of the matrix element squared for g g to Q \bar{Q}.
The proper summation over the polarizations of the gluons requires a choice of two auxiliary vectors (unless we subtract the unphysical contributions using ghosts).
It is customary to take the 4-momentum of another gluon as the auxiliary vector in the summation formula.
The option ExtraFactor is used to average over the polarizations of the initial gluons.
ClearAll[s, t, u];
FCClearScalarProducts[];
SP[p1] = 0;
SP[p2] = 0;amp = 1/(s^2 SUNN (1 - SUNN^2) u^2) 2 SMP["g_s"]^4 SP[k1,
Polarization[p2, -I, Transversality -> True]] SP[k1, Polarization[p2,
I, Transversality -> True]] (2 s^2 SP[k1, Polarization[p1, I,
Transversality -> True]] SP[k2, Polarization[p1, -I,
Transversality -> True]] + 2 s SUNN^2 t SP[k1, Polarization[p1, I,
Transversality -> True]] SP[k2, Polarization[p1, -I,
Transversality -> True]] + s SUNN^2 u SP[k1, Polarization[p1, I,
Transversality -> True]] SP[k2, Polarization[p1, -I,
Transversality -> True]] + 2 s^2 SP[k1, Polarization[p1, -I,
Transversality -> True]] SP[k2, Polarization[p1, I,
Transversality -> True]] + 2 s SUNN^2 t SP[k1, Polarization[p1, -I,
Transversality -> True]] SP[k2, Polarization[p1, I,
Transversality -> True]] + s SUNN^2 u SP[k1, Polarization[p1, -I,
Transversality -> True]] SP[k2, Polarization[p1, I,
Transversality -> True]] + 2 SUNN^2 u^2 SP[k2,
Polarization[p1, -I, Transversality -> True]] SP[k2,
Polarization[p1, I, Transversality -> True]])\frac{1}{N \left(1-N^2\right) s^2 u^2}2 g_s^4 \left(\overline{\text{k1}}\cdot \bar{\varepsilon }^*(\text{p2})\right) \left(\overline{\text{k1}}\cdot \bar{\varepsilon }(\text{p2})\right) \left(2 N^2 s t \left(\overline{\text{k1}}\cdot \bar{\varepsilon }(\text{p1})\right) \left(\overline{\text{k2}}\cdot \bar{\varepsilon }^*(\text{p1})\right)+2 N^2 s t \left(\overline{\text{k1}}\cdot \bar{\varepsilon }^*(\text{p1})\right) \left(\overline{\text{k2}}\cdot \bar{\varepsilon }(\text{p1})\right)+N^2 s u \left(\overline{\text{k1}}\cdot \bar{\varepsilon }(\text{p1})\right) \left(\overline{\text{k2}}\cdot \bar{\varepsilon }^*(\text{p1})\right)+N^2 s u \left(\overline{\text{k1}}\cdot \bar{\varepsilon }^*(\text{p1})\right) \left(\overline{\text{k2}}\cdot \bar{\varepsilon }(\text{p1})\right)+2 s^2 \left(\overline{\text{k1}}\cdot \bar{\varepsilon }(\text{p1})\right) \left(\overline{\text{k2}}\cdot \bar{\varepsilon }^*(\text{p1})\right)+2 s^2 \left(\overline{\text{k1}}\cdot \bar{\varepsilon }^*(\text{p1})\right) \left(\overline{\text{k2}}\cdot \bar{\varepsilon }(\text{p1})\right)+2 N^2 u^2 \left(\overline{\text{k2}}\cdot \bar{\varepsilon }^*(\text{p1})\right) \left(\overline{\text{k2}}\cdot \bar{\varepsilon }(\text{p1})\right)\right)
amp // DoPolarizationSums[#, p1, p2, ExtraFactor -> 1/2] & //
DoPolarizationSums[#, p2, p1, ExtraFactor -> 1/2] & // Simplify-\frac{1}{N \left(N^2-1\right) s^2 u^2 (\overline{\text{p1}}\cdot \overline{\text{p2}})^2}g_s^4 \left(2 \left(\overline{\text{k1}}\cdot \overline{\text{p1}}\right) \left(\overline{\text{k1}}\cdot \overline{\text{p2}}\right)-\overline{\text{k1}}^2 \left(\overline{\text{p1}}\cdot \overline{\text{p2}}\right)\right) \left(s \left(\overline{\text{k1}}\cdot \overline{\text{p2}}\right) \left(\overline{\text{k2}}\cdot \overline{\text{p1}}\right) \left(N^2 (2 t+u)+2 s\right)+s \left(\overline{\text{k1}}\cdot \overline{\text{p1}}\right) \left(\overline{\text{k2}}\cdot \overline{\text{p2}}\right) \left(N^2 (2 t+u)+2 s\right)-2 N^2 s t \left(\overline{\text{k1}}\cdot \overline{\text{k2}}\right) \left(\overline{\text{p1}}\cdot \overline{\text{p2}}\right)-N^2 s u \left(\overline{\text{k1}}\cdot \overline{\text{k2}}\right) \left(\overline{\text{p1}}\cdot \overline{\text{p2}}\right)-2 s^2 \left(\overline{\text{k1}}\cdot \overline{\text{k2}}\right) \left(\overline{\text{p1}}\cdot \overline{\text{p2}}\right)+2 N^2 u^2 \left(\overline{\text{k2}}\cdot \overline{\text{p1}}\right) \left(\overline{\text{k2}}\cdot \overline{\text{p2}}\right)-N^2 u^2 \overline{\text{k2}}^2 \left(\overline{\text{p1}}\cdot \overline{\text{p2}}\right)\right)
We can also do the same calculation in D-dimensions
ClearAll[s, t, u];
FCClearScalarProducts[];
SPD[p1] = 0;
SPD[p2] = 0;
ChangeDimension[amp, D] // DoPolarizationSums[#, p1, p2, ExtraFactor -> 1/2] & //
DoPolarizationSums[#, p2, p1, ExtraFactor -> 1/2] & // Simplify-\frac{1}{N \left(N^2-1\right) s^2 u^2 (\text{p1}\cdot \;\text{p2})^2}g_s^4 \left(2 (\text{k1}\cdot \;\text{p1}) (\text{k1}\cdot \;\text{p2})-\text{k1}^2 (\text{p1}\cdot \;\text{p2})\right) \left(s (\text{k1}\cdot \;\text{p2}) (\text{k2}\cdot \;\text{p1}) \left(N^2 (2 t+u)+2 s\right)+s (\text{k1}\cdot \;\text{p1}) (\text{k2}\cdot \;\text{p2}) \left(N^2 (2 t+u)+2 s\right)-2 N^2 s t (\text{k1}\cdot \;\text{k2}) (\text{p1}\cdot \;\text{p2})-N^2 s u (\text{k1}\cdot \;\text{k2}) (\text{p1}\cdot \;\text{p2})-2 s^2 (\text{k1}\cdot \;\text{k2}) (\text{p1}\cdot \;\text{p2})+2 N^2 u^2 (\text{k2}\cdot \;\text{p1}) (\text{k2}\cdot \;\text{p2})-\text{k2}^2 N^2 u^2 (\text{p1}\cdot \;\text{p2})\right)
DoPolarizationSums will complain if you try to sum over the polarizations of a massless vector boson that is not on-shell
PolarizationVector[p, mu] ComplexConjugate[PolarizationVector[p, mu]]
DoPolarizationSums[%, p, 0]\bar{\varepsilon }^{*\text{mu}}(p) \bar{\varepsilon }^{\text{mu}}(p)
-4
The obvious solution to remove this warning is to put the boson on-shell
FCClearScalarProducts[]
ScalarProduct[p, p] = 0
PolarizationVector[p, mu] ComplexConjugate[PolarizationVector[p, mu]]
DoPolarizationSums[%, p, 0]0
\bar{\varepsilon }^{*\text{mu}}(p) \bar{\varepsilon }^{\text{mu}}(p)
-4
However, if you have a massless virtual boson in the final state that by definition cannot be on-shell, (e.g. in the process q \bar{q} \to g \gamma^\ast), you can tell this to the function by setting the option VirtualBoson to True.
FCClearScalarProducts[]
PolarizationVector[p, mu] ComplexConjugate[PolarizationVector[p, mu]]
DoPolarizationSums[%, p, 0, VirtualBoson -> True]\bar{\varepsilon }^{*\text{mu}}(p) \bar{\varepsilon }^{\text{mu}}(p)
-4
It may happen that your expression is not directly proportional to a pair of polarization vectors. In this case terms that are free of polarization vectors will be multiplied by the suitable number of polarizations. This behavior is controlled by the option NumberOfPolarizations. The default value Automatic means that the function will automatically figure out the correct number of polarizations.
Here we have 2 physical polarizations (massless vector boson)
FCClearScalarProducts[];
ScalarProduct[p, p] = 0;
PolarizationVector[p, mu] ComplexConjugate[PolarizationVector[p, mu]] + xyz
DoPolarizationSums[%, p, n]\bar{\varepsilon }^{*\text{mu}}(p) \bar{\varepsilon }^{\text{mu}}(p)+\text{xyz}
\text{DoPolarizationSums: The input expression contains terms free of polarization vectors. Those will be multiplied with the number of polarizations given by }2.
2 \;\text{xyz}-2
In D dimensions the number of polarizations becomes D-2
FCClearScalarProducts[];
ScalarProduct[p, p] = 0;
ChangeDimension[PolarizationVector[p, mu]*
ComplexConjugate[PolarizationVector[p, mu]] + xyz, D]
DoPolarizationSums[%, p, n]\varepsilon ^{*\text{mu}}(p) \varepsilon ^{\text{mu}}(p)+\text{xyz}
\text{DoPolarizationSums: The input expression contains terms free of polarization vectors. Those will be multiplied with the number of polarizations given by }D-2.
(D-2) \;\text{xyz}-D+2
A massive vector boson has 3 physical polarizations in 4 dimensions
FCClearScalarProducts[];
ScalarProduct[p, p] = M^2;
PolarizationVector[p, mu] ComplexConjugate[PolarizationVector[p, mu]] + xyz
DoPolarizationSums[%, p]\bar{\varepsilon }^{*\text{mu}}(p) \bar{\varepsilon }^{\text{mu}}(p)+\text{xyz}
\text{DoPolarizationSums: The input expression contains terms free of polarization vectors. Those will be multiplied with the number of polarizations given by }3.
3 \;\text{xyz}-3
or D-1 physical polarizations in D dimensions
FCClearScalarProducts[];
ScalarProduct[p, p] = M^2;
ChangeDimension[PolarizationVector[p, mu]*
ComplexConjugate[PolarizationVector[p, mu]] + xyz, D]
DoPolarizationSums[%, p]\varepsilon ^{*\text{mu}}(p) \varepsilon ^{\text{mu}}(p)+\text{xyz}
\text{DoPolarizationSums: The input expression contains terms free of polarization vectors. Those will be multiplied with the number of polarizations given by }D-1.
(D-1) \;\text{xyz}-D+1
In the case of a standalone expression that contains no polarization vectors whatsoever, the function has no way to determine the correct number of polarizations.
DoPolarizationSums[xyz, p, n]
\text{\$Aborted}
Here additional user input is needed
DoPolarizationSums[xyz, p, NumberOfPolarizations -> 2]\text{DoPolarizationSums: The input expression contains terms free of polarization vectors. Those will be multiplied with the number of polarizations given by }2.
2 \;\text{xyz}