arXiv:1110.1750v1 [hep-ph] 8 Oct 2011
Quadratic electroweak corrections for polarized Møller scattering
Memorial University, Corner Brook, Canada
Acadia University, Wolfville, Canada
University of California, Berkeley, USA
Joint Institute for Nuclear Research, Dubna, Russia
Belarussian State University of Transport, Gomel, Belarus
The paper discusses the two-loop (NNLO) electroweak radiative corrections to the parity vio-
lating e−e−→ e−e−(γ)(γγ) scattering asymmetry induced by squaring one-loop diagrams. The
calculations are relevant for the ultra-precise 11 GeV MOLLER experiment planned at Jefferson
Laboratory and experiments at future high-energy colliders. The imaginary parts of the amplitudes
are taken into consideration consistently in both the infrared-finite and divergent terms. The size
of the obtained partial correction is significant, which indicates a need for a complete study of the
two-loop electroweak radiative corrections in order to meet the precision goals of future experiments.
12.15.Lk, 13.88.+e, 25.30.Bf
Polarized Møller scattering has been a well-studied low-energy reaction for close to eight
decades now , but has attracted especially active interest from both experimental and theo-
retical communities due to the recent rapid progress in measuring spin-dependent observables.
Since the nineties the interaction has allowed the high-precision determination of the electron-
beam polarization at SLC , SLAC  , JLab  and MIT-Bates . A Møller polarimeter
may also be useful in future experiments planned at the ILC . In addition, polarized Møller
scattering can be an excellent tool for measuring parity-violating (PV) weak interaction asym-
The first observation of Parity Violation in Møller scattering was made by the E-158 ex-
periment at SLAC , which studied Møller scattering of 45- to 48-GeV polarized electrons
on the unpolarized electrons in a hydrogen target.
APV = (1.31 ± 0.14 (stat.) ± 0.10 (syst.)) × 10−7 allowed one of the most important
parameters in the Standard Model – the sine of the Weinberg angle – to be determined with an
Its result at low Q2= 0.026 GeV2,
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accuracy of 0.5% (sin2θW= 0.2403 ± 0.0013 in the MS scheme). A very promising experiment
measuring the e-p scattering asymmetry currently running at Jefferson Lab, Qweak , aims to
determine sin2θWwith relative precision of 0.3%. The next-generation experiment to study e-e
scattering – MOLLER, planned at JLab following the 11 GeV upgrade – will offer a new level
of sensitivity and measure the parity-violating asymmetry in the scattering of longitudinally
polarized electrons off an unpolarized target to a precision of 0.73 ppb. That would allow a de-
termination of the weak mixing angle with an uncertainty of ±0.00026 (stat.)±0.00013 (syst.)
, or about 0.1%, an improvement of a factor of five in fractional precision when compared
with the E-158 measurement.
Since Møller scattering is a very clean process with a well-known initial energy and low back-
grounds, any inconsistency with the Standard Model will signal new physics. Møller scattering
experiments can provide indirect access to physics at multi-TeV scales and play an important
complementary role to the LHC research program .
Obviously, before we can extract reliable information from the experimental data, it is nec-
essary to take into account higher order effects of electroweak theory, i.e. electroweak radiative
corrections (EWC). The inclusion of EWC is an indispensable part of any modern experiment,
but will be of the paramount importance for the ultra-precise measurement of the weak mixing
angle via 11 GeV Møller scattering planned at JLab. In general, from the theory point of view,
the interpretability of e-e scattering is exceptionally good. However, to match the precision of
MOLLER experiment, theoretical predictions for the PV e-e scattering asymmetry must in-
clude not only full treatment of one-loop radiative corrections (NLO) but also leading two-loop
A significant theoretical effort has been dedicated to one-loop radiative corrections already.
A short review of the literature to date on that topic is done in . In , we also calculated
a full gauge-invariant set of the one-loop EWC both numerically with no simplifications using
FeynArts , FormCalc , LoopTools  and Form  as the base languages and by
hand in a compact form analytically free from nonphysical parameters. The total correction
was found to be close to −70%, and we found no significant theoretical uncertainty coming
from the largest possible source, the hadronic contributions to the vacuum polarization. The
dependence on other uncertain input parameters, like the mass of the Higgs boson, was below
It is possible that a much larger theoretical uncertainty in the prediction for the asymmetry
may come from two-loop corrections. Paper  argued that the higher order corrections are
suppressed by a factor of either about 0.1% or 5% (depending on a type of corrections) relative
to the one-loop result. However, since the one-loop weak corrections for Møller scattering are so
large and since the 11 GeV MOLLER experiment is striving for such unprecedented precision,
we believe it is now worth looking into evaluating two-loop weak corrections.
One way to find some indication of the size of higher-order contributions is to compare results
that are expressed in terms of quantities related to different renormalization schemes. In ,
we provided a tuned comparison between the result obtained with different renormalization
conditions, first within one scheme then between two schemes. Our calculations in the on-shell
and CDR schemes show a difference of about 11%, which is comparable with the difference
of 10% between MS  and the on-shell scheme . It is also worth noting that although
two-loop corrections to the cross section may seem to be small, it is much harder to estimate
their scale and behaviour for such a complicated observable as the parity-violating asymmetry
to be measured by MOLLER experiment.
FIG. 1: Diagrams describing nonradiative Møller scattering in the (1) t- and (2) u-channels.
The two-loop EWC to the Born (∼ M0M+
Q-part induced by quadratic one-loop amplitudes (∼ M1M+
to the interference of the Born and two-loop diagrams (∼ 2ReM0M+
paper is to calculate the Q-part exactly. We show that the Q-part is much higher than the
planned experimental uncertainty of MOLLER, which means that the two-loop EWC may be
larger that previously thought. The large size of the Q-part demands a detailed and consistent
consideration of the T-part, and that will be the next task of our group.
0) cross section can be divided into two classes: the
1), and the T-part corresponding
2−loop). The goal of this
II.GENERAL NOTATIONS AND MATRIX ELEMENTS
Let us start by writing the cross section of polarized Møller scattering with the Born kine-
matics shown in Fig. 1,
e−(k1) + e−(p1) → e−(k2) + e−(p2),(1)
such that, with the appropriate accuracy for the present paper, we find:
Here, σ ≡ dσ/dcosθ, θ is the scattering angle of the detected electron with 4-momentum k2
in the center-of-mass system of the initial electrons. The 4-momenta of initial (k1and p1) and
final (k2and p2) electrons generate a standard set of Mandelstam variables:
s = (k1+ p1)2, t = (k1− k2)2, u = (k2− p1)2.(3)
It should also be noted that the electron mass m is disregarded wherever possible, in particular
if m2≪ s,−t,−u.
M0and M1are the Born (O(α)) and one-loop (O(α2)) amplitudes (matrix elements), respec-
tively. Let us describe the structure of M0:
M0= M0,t− M0,u, M0,u= M0,t(k2↔ p2), M0,t=
where the t-channel upper and lower electron leg currents are
µ= ¯ u(k2)γµ(vj− ajγ5)u(k1), Jj
The squared Born amplitude M0forms the Born cross section:
µ= ¯ u(p2)γµ(vj− ajγ5)u(p1).(5)
−(u2DitDjt+ t2DiuDju) + λi,j
+s2(Dit+ Diu)(Djt+ Dju)]. (6)
A handy structure to use in the present study is
r − m2
(i = γ,Z; r = t,u),(7)
which depends on the Z-boson mass mZor on the photon mass mγ≡ λ. The photon mass is
set to zero everywhere with the exception of specially-indicated cases where the photon mass is
taken to be an infinitesimal parameter that regularizes the infrared divergence (IRD). Another
set of useful functions is
These are combinations of coupling constants and pB(T), where pB(T)are the degrees of polar-
ization of electrons with 4-momentum k1(p1). More specifically,
V= vivj+ aiaj, λi,j
A= viaj+ aivj,(9)
vγ= 1, aγ= 0, vZ= (I3
W)/(2sWcW), aZ= I3
The subscripts L and R on the cross sections correspond to pB(T)= −1 and pB(T)= +1, where
the first subscript indicates the degree of polarization for the 4-momentum k1and the second
one indicates the degree of polarization for the 4-momentum p1. Let us recall that I3
and sW(cW) is the sine (cosine) of the Weinberg angle expressed in terms of the Z- and W-boson
masses according to the rules of the Standard Model:
cW= mW/mZ, sW=
1 − c2
We can present the one-loop amplitude M1as a sum of boson self-energy (BSE), vertex (Ver)
and box diagrams:
M1= M1,t− M1,u, M1,u= M1,t(k2↔ p2),
M1,t= MBSE,t+ MVer,t+ MBox,t.(12)
We use the on-shell renormalization scheme from [21, 22], so there are no contributions from
the electron self-energies. The question of the dependence of EWC on renormalization schemes
and renormalization conditions (within the same scheme) was addressed in our earlier paper
FIG. 2: One-loop t-channel diagrams for the Møller process. The circles represent the contributions of self-energies and
vertex functions. The u-channel diagrams are obtained via the interchange k2 ↔ p2.
The infrared-finite BSE term can easily be expressed as:
The longitudinal parts of the boson self-energy make contributions that are proportional to
m2/r; therefore they are very small and are not considered here.
In order to get the electron vertex amplitude (2nd and 3rd diagrams in Fig. 2), we use the
form factors δFje
form factors vγ(Z)→ δFγ(Z)e
where the electron currents with vertices look like
T(r) is the transverse part of the renormalized photon, Z-boson and γZ self-energies.
V,Ain the manner of paper , replacing the coupling constants vj, ajwith
, aγ(Z)→ δFγ(Z)e
, Mj/B,t= iα
µDjtJµ,j, Mj/H,t= iα
V, aj→ δFje
A), Hµ,j= Jµ,j(vj→ δFje
V, aj→ δFje
The infrared singularity is regularized by giving photon a small mass λ and in the t-channel
vertex amplitude can be extracted in the form:
where e is the base of the natural logarithm. The rest (infrared-finite) part of t-channel vertex
amplitude has the simple form The remaining (infrared-finite) part of the t-channel vertex
amplitude has a simple form convenient for analysis and coding:
Ver,t= MVer,t− Mλ
Ver,t= MVer,t(λ2→ s).(18)
The box term can be presented as a sum of all two-boson contributions:
MBox,t= Mγγ,t+ MγZ,t+ MZγ,t+ MZZ,t+ MWW,t.(19)
We need to account for both direct and crossed γγ, γZ and ZZ-boxes:
ij,t(i,j = γ,Z),(20)
ij,tgiven by exact expressions in 4-dimensional integral form (4-point functions)
×¯ u(k2)γµ(vj− ajγ5)(ˆk1−ˆk + m)γν(vi− aiγ5)u(k1) ×
×¯ u(p2)γµ(vj− ajγ5)(ˆ p1+ˆk + m)γν(vi− aiγ5u(p1),
(k2− 2k1k)(k2+ 2p1k)((q − k)2− m2
×¯ u(k2)γµ(vj− ajγ5)(ˆk1−ˆk + m)γν(vi− aiγ5)u(k1) ×
×¯ u(p2)γν(vi− aiγ5)(ˆ p2−ˆk + m)γµ(vj− ajγ5)u(p1).
(k2− 2k1k)(k2− 2p2k)((q − k)2− m2
Obviously, for WW-boxes we only need the crossed expression (22).
The infrared parts of the γγ- and γZ-boxes in the t-channel are similarly given by
2+ iπ logs
Using asymptotic methods, we can significantly simplify the box amplitudes containing at least
one heavy boson (see, for example, , where simplifications were done on the cross-section
¯ u(k2)γµ(vZ− aZγ5)(−γα)γνu(k1) · ¯ u(p2)γµ(vZ− aZγ5)γαγνu(p1) +
¯ u(k2)γµ(vZ− aZγ5)γαγνu(k1) · ¯ u(p2)γνγαγµ(vZ− aZγ5)u(p1)
¯ u(k2)γµ(vB− aBγ5)(−γα)γνu(k1) · ¯ u(p2)γµ(vB− aBγ5)γαγνu(p1) +
+¯ u(k2)γµ(vB− aBγ5)γαγνu(k1) · ¯ u(p2)γνγαγµ(vB− aBγ5)u(p1)
¯ u(k2)γµ(vC− aCγ5)γαγνu(k1) · ¯ u(p2)γνγαγµ(vC− aCγ5)u(p1)
with the coupling-constants combinations for ZZ- and WW-boxes
2, aB= 2vZaZ, vC= aC= 1/(4s2
Now we are ready to present the one-loop complex amplitude as the sum of IR and IR-finite
1= MBSE+ Mf
1= 4B log
and the complex value B can be presented in form (see, for example, )
B = log
m2s− 1 + iπ.(30)
The amplitudes from the non-factorized part of the boxes are given by
u+ π2)M0,t− (L2
where Lr= log(−s/r).
III.EXTRACTION OF INFRARED DIVERGENCES
Now we should make sure that the infrared divergences are cancelled. In a similar way as
it was done for amplitudes, we present the complex interference term ˆ σ1and differential cross
section σQas sums of λ-dependent (IRD-terms) and λ-independent (infrared-finite) parts:
The one-loop cross section which we denote σ1= Reˆ σ1was carefully evaluated with full control
of the uncertainties in paper . The term σQ(see (2)) is called the Q-part of the two-loop
EWC and is the main subject of the present paper.
If we substitute the amplitudes derived in Section II to the left-hand-side of (2), and compare
the result with the right-hand side of this equation, we will get the same expression for σ1as
given in . The simplest form for σλ
1(see formula (42) of ) is then:
The infrared-finite part σf
1can be conveniently to presented via the relative dimensionless
After some transformations, the value σλ
Qis given by
Finally, the infrared-finite part σf
Qexpressed via the relative dimensionless corrections has form
IV.BREMSSTRAHLUNG AND CANCELLATION OF INFRARED DIVERGENCES
To evaluate the cross section induced by the emission of one soft photon with energy less
then ω, we follow the methods of  (see also ). Then this cross section can be expressed
1,2have the similar factorized structure based on the factorization of soft-photon
1+ R1)∗ˆ σ1
R1= −4B log
em2+ 1 −π2
The first part of the soft-photon cross section, σγ
while the second part, σγ
cancellation of IRD in the Q-part and the other half going to treat IRD in the T-part:
1, cancels the IRD at the one-loop order,
2, cancels the IRD a the two-loop order, with half of σγ
2going to the
To obtain the term −δλ
the phase space of one real soft photon. It can be done according to  in c.m.s:
1+ R1in Eq. (38), we must calculate the 3-dimensional integral over
1+ R1= L(λ,ω) = −1
The difference between the estimation relying on the soft part only and the result obtained by
separation into the soft and hard parts at lowest order is rather small (see ), so we believe
that the soft cross section will provide the sufficient accuracy at second order as well.
At last, the cross section induced by the emission of two soft photons with a total energy less
then ω can be written as:
3π2|B|2. The detailed calculations of σγγare shown in Appendix A.
Just like σγ, the cross section σγγis divided into equal halves, with a half going to cancel the
IRD in the Q-part and a half going to the T-part:
2is a statistical factor caused by the indistinguishability of two final photons and R2=
Combining all the terms together, we get the infrared-finite result at both the first and second
orders. The first(second) order is given by the first(second) term on the LHS of the equation
V. NUMERICAL RESULTS
For the numerical calculations we use α = 1/137.035999, mW = 80.398 GeV, and mZ =
91.1876 GeV as input parameters in accordance with . The electron, muon, and τ-lepton
masses are taken to be me= 0.510998910 MeV, mµ= 0.105658367 GeV, mτ= 1.77684 GeV,
while the quark masses for vector boson self-energy loop contributions are taken to be mu=
0.06983 GeV, mc= 1.2 GeV, mt= 174 GeV, md= 0.06984 GeV, ms= 0.15 GeV, and mb=
4.6 GeV. The values of the light quark masses were extracted using the fact that they provide
shifts in the fine structure constant due to hadronic vacuum polarization ∆α(5)
and Qfis the electric charge of fermion f in proton charge units q (q =√4πα).
On the other hand, the contribution of hadronic vacuum polarization to the fine structure
constant also can be evaluated using the dispersion relation:
had(s) = −
s′− s − i0,(47)
where P means that the principle value of the integral should be considered and σh(s) is the
cross section of hadron production in e+e−annihilation. In the case of small energies this cross
section can be approximated by the cross section of the pion production channel e+e−→ π+π−:
thus giving the following contribution to ∆α(5)
?1 + βπ
1 − βπ
Using Eq. (46) and Eq. (49) we can incorporate the use of the light quark masses as parameters
regulated by the hadronic vacuum polarization in our calculations.
FIG. 3: Virtual and bremstrahlung contributions to the relative correction to unpolarized cross section vs. the photon
mass λ at θ = 90oand Elab = 11 GeV.
Finally, for the mass of the Higgs boson, we take mH = 115 GeV. Although this mass is
still to be determined experimentally, the dependence of EWC on mHis rather weak. For the
maximum soft photon energy we use ω = 0.05√s, according to  and .
Let us define the relative corrections to the Born cross section due to a specific type of
contributions (labeled by C) as
δC= (σC− σ0)/σ0, C = 1-loop,Q,T,...
The parity-violating asymmetry is defined in a traditional way,
ALR=σLL+ σLR− σRL− σRR
σLL+ σLR+ σRL+ σRR
σLL+ 2σLR+ σRR,(50)
and the relative correction to the Born asymmetry due to C-contribution is defined as
Fig. 3, plotted for θ = 90oand Elab= 11 GeV, clearly demonstrates that the relative correction
to the unpolarized cross section is independent of the photon mass λ. We can also see a
quadratic dependence on the log scale of λ for both the virtual and bremstrahlung contributions.
The left frame of Fig. 4 depicts the relative corrections to the asymmetry at Elab= 11 GeV
versus the scattering angle θ in c.m.s. The lower line shows the corrections to the asymmetry
with only one-loop EWC taken into account, and the upper line shows the combined one-loop
and Q-part corrections. As expected, both of them are symmetric along the line θ = π/2, have
a minimum at θ = 90o, and depend on the scattering angle quite weakly.
The difference of these two effects is an absolute correction defined by
FIG. 4: The relative corrections to the asymmetry (left) and the absolute correction ∆A (right) vs scattering angle θ.
and depicted in the right frame of Fig. 4. Here we can see that the Q-part gives quite a
significant contribution, with ∆A reaching a maximum of 0.0419 at θ = 90o. Taking into
account that MOLLER’s planned experimental error to the PV asymmetry is ∼ 2% or less, we
see that it is necessary to continue to work on the two-loop EWC, staring from the T-part.
Fig. 5 shows the relative (labeled as 1-loop and 1-loop+Q) corrections and absolute ∆A
corrections (labeled by Q) versus√s at θ = 90o. In the high-energy region (√s ≥ 50 GeV) our
one-loop result (see ) is in excellent agreement with the result from  if we use the same set
of Standard Model parameters. As one can see from Fig. 5, the scale of the Q-part contribution
in the low-energy region is approximately constant, but grows sharply at√s ≥ mZ, where the
weak contribution becomes comparable to the electromagnetic. This increasing importance of
the two-loop contribution at higher energies may have a significant effect on the asymmetry
measured at future e−e−-colliders.
Experimental investigation of Møller scattering is not only one of the oldest tools of modern
physics, but also a powerful probe of new physics effects. The new ultra-precise measurement
of the weak mixing angle via 11 GeV Møller scattering planned at JLab (MOLLER) – as
well as experiments proposed at future high-energy electron colliders – will require that the
higher-order effects to be taken into account with the highest precision possible.
In this work, we build on the study of the one-loop electroweak radiative corrections to the
cross-section asymmetry of the polarized Møller scattering at 11 GeV initiated by our group in
, and address some of the two-loop effects. At this stage, we perform a detailed calculation
for the part of the two-loop electroweak radiative correction induced by squaring one-loop
FIG. 5: Relative (labeled by 1-loop+Q and 1-loop) and absolute (labeled by Q) corrections to PV-asymmetry vs√s. The
filled circle corresponds to our predictions for the MOLLER experiment.
The two-loop EWC to the Born (∼ M0M+
the interference of Born and two-loop diagrams (∼ 2ReM0M+
quadratic one-loop amplitudes (∼ M1M+
both numerical and analytical form, with the infrared divergence explicitly cancelled. Also, we
clearly demonstrate the important role of the imaginary part of amplitude, which is consistently
taken into consideration both in the infrared-finite and divergent terms.
As one can see from our numerical data, at the MOLLER kinematic conditions, the part of
the NNLO EWC we considered in this work can increase the asymmetry by up to ∼ 4%. The
corrections depend quite significantly on the energy and scattering angles; at the high-energy
region of√s ∼ 1000 GeV achievable in the planned experimental program of the ILC, the
estimated contribution of the quadratic EWC can reach +14%; for 3 TeV at CLIC, it would be
+42%. We see that the large size of the Q-part demands detailed and consistent consideration
of the T-part, which will be the next task of our group. It is impossible to say at this time
if the Q-part will be partially enhanced or cancelled by other two-loop radiative corrections,
although it seems probable that the two-loop EWC may be larger than previously thought.
Although an argument can be made that the two-loop corrections are suppressed by a factor of
απ relative to the one-loop corrections (see , for example), we are reluctant to conclude that
they can be dismissed, especially in the light of 2% uncertainty to the asymmetry promised by
Since the problem of EWC for the Møller scattering asymmetry is rather involved, a tuned
step-by-step comparison between different calculation approaches is essential. One of the im-
0) cross section is divided into the T-part, which is
2−loop), and the Q-part, induced by
1), which we evaluate here. The results are presented in
portant results of this work is the correctness of our calculations, which was controlled by a
comparison of the results obtained from the equations derived by hand with the numerical data
obtained by a semi-automatic approach based on FeynArts, FormCalc, LoopTools and Form.
These base languages have already been successfully employed in similar projects (, ),
so we are highly confident in their reliability.
In the future, we plan to address the remaining two-loop electroweak corrections which may be
required by the promised experimental precision of the MOLLER experiment and experiments
planned at ILC.
We are grateful to Y. Bystritskiy and T. Hahn for stimulating discussions. A. A. and S. B.
thank the Theory Center at Jefferson Lab, and V. Z. thanks Acadia University for hospitality
in 2011. This work was supported by the Natural Sciences and Engineering Research Council
of Canada and Belarus scientific program ”Convergence”.
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FIG. 6: Double-bremsstrahlung diagrams for Møller scattering in t-channel corresponding to Mi
are obtained by interchange k2 ↔ p2.
11. The u-channel diagrams
Appendix A: Detailes of calculations for the case of emission of two real soft photons
First, let us calculate the amplitudes corresponding to the emission of two real soft photons
(see Fig. 6),
e−(k1,ξ) + e−(p1,η) → e−(k2) + e−(p2) + γ(k) + γ(p)
in t- and u-channels with i-boson exchange (i = γ, Z).
γ(p)) emitted photon: 1 – emitted from electron e−(k1), 2 – from electron e−(k2), 3 – from
electron e−(p1) and 4 – from electron e−(p2). The exact expression for Mi
The amplitudes are labeled as
13,..., where the first (second) subscript denotes the origin of the first γ(k) (second
11is the following:
ˆk3− mγβeβ(k)u(k1) ·
11=i(2πe)4Nk1Nk2Np1Np2NpNk· ¯ u(k2)γµ(vi− aiγ5)
¯ v(p2)γµ(vi− aiγ5)v(p1) ·
· δ(k1+ p1− k2− p2− k − p),(A2)
√2k0. Using the Dirac equation and taking k → 0, we can simplify
ˆk1−ˆk + m
(k1− k)2− m2γβu(k1) ≈
Analogously, at k,p → 0,
ˆk4− mγαu(k1) =
Finally, the amplitude Mi
tion from the Born amplitude:
1+ γβ[−ˆk1+ m])u(k1) = −kβ
−k1(k + p) + kpu(k1).(A4)
11at k,p → 0 has the following form, with the convenient factoriza-
11|k,p→0=e2NpNk· eα(p)eβ(k) ·
(−k1k)(−k1(k + p) + kp)· Mi
In the same manner, we get
22|k,p→0=e2NpNk· eα(p)eβ(k) ·
(k2p)(k2(k + p) + kp)· Mi
(−p1k)(−p1(k + p) + kp)· Mi
(p2p)(p2(k + p) + kp)· Mi
12|k,p→0=e2NpNk· eα(p)eβ(k) ·
33|k,p→0=e2NpNk· eα(p)eβ(k) ·
44|k,p→0=e2NpNk· eα(p)eβ(k) ·
34|k,p→0=e2NpNk· eα(p)eβ(k) ·
13|k,p→0=e2NpNk· eα(p)eβ(k) ·
14|k,p→0=e2NpNk· eα(p)eβ(k) ·
23|k,p→0=e2NpNk· eα(p)eβ(k) ·
24|k,p→0=e2NpNk· eα(p)eβ(k) ·
Now we need to sum the terms generated by the substitution k ↔ p. For the 11–, 22–, 33–,
44–cases it works as the following:
11|k,p→0+ (k ↔ p) =e2NpNkMi
(−k1k)(−k1(k + p) + kp)+
(−k1p)(−k1(k + p) + kp)
(−k1(k + p) + kp)
As a result, the total t(u)-channel amplitude is given by
t(u)|k,p→0= e2NpNk· eα(p)eβ(k) · Tα(p)Tβ(k) · Mi
Then, the cross section of (A1) has the form
d3kd3p · (eNk)2(eNp)2Tα(p)Tα(p)Tβ(k)Tβ(k)