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arXiv:1110.1750v1 [hep-ph] 8 Oct 2011

Quadratic electroweak corrections for polarized Møller scattering

Aleksandrs Aleksejevs∗

Memorial University, Corner Brook, Canada

Svetlana Barkanova†

Acadia University, Wolfville, Canada

Yury Kolomensky‡

University of California, Berkeley, USA

Eduard Kuraev§

Joint Institute for Nuclear Research, Dubna, Russia

Vladimir Zykunov¶

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.

PACS numbers:

Keywords:

12.15.Lk, 13.88.+e, 25.30.Bf

I.INTRODUCTION

Polarized Møller scattering has been a well-studied low-energy reaction for close to eight

decades now [1], 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 [2], SLAC [3] [4], JLab [5] and MIT-Bates [6]. A Møller polarimeter

may also be useful in future experiments planned at the ILC [7]. In addition, polarized Møller

scattering can be an excellent tool for measuring parity-violating (PV) weak interaction asym-

metries [8].

The first observation of Parity Violation in Møller scattering was made by the E-158 ex-

periment at SLAC [9], 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[10] 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,

∗Electronic address: aaleksejevs@swgc.mun.ca

†Electronic address: svetlana.barkanova@acadiau.ca

‡Electronic address: yury@physics.berkeley.edu

§Electronic address: kuraev@theor.jinr.ru

¶Electronic address: vladimir.zykunov@cern.ch

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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 [11], 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.)

[12], 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 [13].

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

corrections (NNLO).

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 [14]. In [14], we also calculated

a full gauge-invariant set of the one-loop EWC both numerically with no simplifications using

FeynArts [15], FormCalc [16], LoopTools [16] and Form [17] 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

0.1%.

It is possible that a much larger theoretical uncertainty in the prediction for the asymmetry

may come from two-loop corrections. Paper [18] 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 [19],

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 [20] and the on-shell scheme [18]. 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.

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γ, Z

e−

e−

e−

e−

k1

k2

p1

p2

γ, Z

e−

e−

e−

e−

k1

k2

p1

p2

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:

σ =π3

2s|M0+ M1|2=π3

2s(M0M+

0+ 2ReM1M+

0+ M1M+

1).(2)

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=

?

j=γ,Z

Mj

t, Mj

t= iα

πIj

µDjtJµ,j,(4)

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where the t-channel upper and lower electron leg currents are

Ij

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)

σ0=π3

2sM0M+

0=πα2

s

?

i,j=γ,Z

[λi,j

−(u2DitDjt+ t2DiuDju) + λi,j

+s2(Dit+ Diu)(Djt+ Dju)]. (6)

A handy structure to use in the present study is

Dir=

1

r − m2

i

(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

λ±i,k= λ1

i,k

Bλ1

i,k

T± λ2

i,k

Bλ2

i,k

T,(8)

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,

λ1

i,j

B(T)= λi,j

V− pB(T)λi,j

A, λ2

i,j

B(T)= λi,j

A− pB(T)λi,j

V,

λi,j

V= vivj+ aiaj, λi,j

A= viaj+ aivj,(9)

where

vγ= 1, aγ= 0, vZ= (I3

e+ 2s2

W)/(2sWcW), aZ= I3

e/(2sWcW).(10)

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:

e= −1/2

cW= mW/mZ, sW=

?

1 − c2

W.(11)

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

[19].

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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:

MBSE,t= iα

π

?

i,j=γ,Z

Ii

µDijt

SJµ,j,(13)

with

Dijr

S= −DirˆΣij

T(r)Djr,(14)

whereˆΣij

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 [21], replacing the coupling constants vj, ajwith

, aγ(Z)→ δFγ(Z)e

?

VA

. Then,

MVer,t=

j=γ,Z

Mj/B,t+ Mj/H,t

?

, Mj/B,t= iα

πBj

µDjtJµ,j, Mj/H,t= iα

πIj

µDjtHµ,j,(15)

Bj

µ= Ij

µ(vj→ δFje

V, aj→ δFje

A), Hµ,j= Jµ,j(vj→ δFje

V, aj→ δFje

A).(16)

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:

Mλ

Ver,t= −α

π

?

log−t

m2− 1

?

logs

λ2M0,t,(17)

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

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amplitude has a simple form convenient for analysis and coding:

Mf

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:

Mij,t= MD

ij,t+ MC

ij,t(i,j = γ,Z),(20)

with MD

by

ij,tand MC

ij,tgiven by exact expressions in 4-dimensional integral form (4-point functions)

MD

ij,t= −i

?α

π

?2

·

i

(2π)2

?

×¯ 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),

d4k

(k2− 2k1k)(k2+ 2p1k)((q − k)2− m2

j)(k2− m2

i)×

(21)

MC

ij,t= −i

?α

π

?2

·

i

(2π)2

?

×¯ 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).

d4k

(k2− 2k1k)(k2− 2p2k)((q − k)2− m2

j)(k2− m2

i)×

(22)

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

Mλ

γγ(γZ+Zγ),t=−α

π

?1

2log−u

s

log−us

λ4+π2

2+ iπ logs

λ2

?

Mγ(Z)

t

.(23)

Using asymptotic methods, we can significantly simplify the box amplitudes containing at least

one heavy boson (see, for example, [14], where simplifications were done on the cross-section

level). Then

Mf

γZ,t+Mf

Zγ,t=

?

MγZ,t+ MZγ,t

?

−

?

Mλ

γZ,t+ Mλ

Zγ,t

?

= −2i

?α

π

?2

×

×

??

?

3

2+ logm2

Z

s

?

?

¯ u(k2)γµ(vZ− aZγ5)(−γα)γνu(k1) · ¯ u(p2)γµ(vZ− aZγ5)γαγνu(p1) +

+

3

2+ logm2

Z

−u

¯ u(k2)γµ(vZ− aZγ5)γαγνu(k1) · ¯ u(p2)γνγαγµ(vZ− aZγ5)u(p1)

?

,(24)

MZZ,t= −i

?α

π

?2

1

16m2

Z

?

¯ 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)

?

, (25)

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MWW,t= −i

?α

π

?2

1

16m2

W

?

¯ u(k2)γµ(vC− aCγ5)γαγνu(k1) · ¯ u(p2)γνγαγµ(vC− aCγ5)u(p1)

?

, (26)

with the coupling-constants combinations for ZZ- and WW-boxes

vB= (vZ)

2+ (aZ)

2, aB= 2vZaZ, vC= aC= 1/(4s2

W). (27)

Now we are ready to present the one-loop complex amplitude as the sum of IR and IR-finite

parts:

M1= Mλ

1+ Mf

1, Mλ

1=α

π

1

2δλ

1M0, Mf

1= MBSE+ Mf

Ver+ Mf

Box+ Ma,(28)

where

δλ

1= 4B log

λ

√s,

(29)

and the complex value B can be presented in form (see, for example, [23])

B = log

tu

m2s− 1 + iπ.(30)

The amplitudes from the non-factorized part of the boxes are given by

ReMa= −α

2π

?

(L2

u+ π2)M0,t− (L2

t+ π2)M0,u

?

.(31)

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:

ˆ σ1=π3

sM1M+

0= σλ

1+ σf

1, σQ=π3

2sM1M+

1= σλ

Q+ σf

Q.(32)

The one-loop cross section which we denote σ1= Reˆ σ1was carefully evaluated with full control

of the uncertainties in paper [14]. 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 [14]. The simplest form for σλ

1(see formula (42) of [14]) is then:

σλ

1=α

πδλ

1σ0.(33)

The infrared-finite part σf

correction:

1can be conveniently to presented via the relative dimensionless

σf

1=α

πδf

1σ0.(34)

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After some transformations, the value σλ

Qis given by

σλ

Q=π3

2sMλ

1

+?Mλ

1+ 2Mf

1

?=1

4

?α

π

?2Re

?

δλ

1

∗(δλ

1+ 2δf

1)

?

σ0.(35)

Finally, the infrared-finite part σf

Qexpressed via the relative dimensionless corrections has form

σf

Q=π3

2sMf

1Mf

1

+=

?α

π

?2δf

Qσ0.(36)

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 [24] (see also [25]). Then this cross section can be expressed

as:

σγ= σγ

1+ σγ

2,(37)

where σγ

bremsstrahlung:

1,2have the similar factorized structure based on the factorization of soft-photon

σγ

1=α

πRe?−δλ

1+ R1

?σ0,σγ

2=α

πRe?(−δλ

1+ R1)∗ˆ σ1

?,(38)

where

R1= −4B log

√s

2ω− log2

s

em2+ 1 −π2

3+ log2u

t.

(39)

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

σγ

Q= σγ

T=1

2σγ

2.(40)

To obtain the term −δλ

the phase space of one real soft photon. It can be done according to [24] in c.m.s:

1+ R1in Eq. (38), we must calculate the 3-dimensional integral over

− δλ

1+ R1= L(λ,ω) = −1

4π

?

k0<ω

d3k

k0

Tβ(k)Tβ(k),(41)

where

Tα(k) =kα

1

k1k−kα

2

k2k+pα

1

p1k−pα

2

p2k.

(42)

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 [14]), 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:

σγγ=1

2

?α

π

?2?????−δλ

1+ R1

????

2

− R2

?

σ0,(43)

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where1

8

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=

σγγ

Q= σγγ

T=1

2σγγ.(44)

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

below:

Re[σ1+σγ

1]+(σQ+σγ

Q+σγγ

Q) =α

πRe[R1+δf

1]σ0+

?α

π

?2Re[1

2R∗

1δf

1+δf

Q+1

4R∗

1R1−1

4R2]σ0. (45)

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 [26]. 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)

[27], where

had(m2

Z)=0.02757

∆α(5)

had(s) =

α

3π

?

f=u,d,s

Q2

f

?

log

s

m2

f

−5

3

?

,(46)

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:

∆α(5)

had(s) = −

s

4π2αP

∞

?

2M2

π

ds′

σh(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−:

σh(s) =πα2

3sβ3

π,βπ=

?

1 −4M2

π

s

,(48)

thus giving the following contribution to ∆α(5)

had(s):

∆α(5)

had(s) =α

π

?1

12log

?1 + βπ

1 − βπ

?

−2

3− 2β2

π

?

.(49)

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.

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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 [14] and [28].

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− σRR

σLL+ 2σLR+ σRR,(50)

and the relative correction to the Born asymmetry due to C-contribution is defined as

δC

A= (AC

LR− A0

LR)/A0

LR.

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

∆A= (A1−loop+Q

LR

− A0

LR)/A0

LR− (A1−loop

LR

− A0

LR)/A0

LR= (A1−loop+Q

LR

− A1−loop

LR

)/A0

LR

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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 [14]) is in excellent agreement with the result from [28] 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.

VI. CONCLUSIONS

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

[14], 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

diagrams.

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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 [18], for example), we are reluctant to conclude that

they can be dismissed, especially in the light of 2% uncertainty to the asymmetry promised by

MOLLER.

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

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13

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 ([14], [19]),

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.

VII.ACKNOWLEDGMENTS

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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Anthony et al., Phys. Rev. Lett. 92, 181602 (2004) [hep-ex/0312035].

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[12] J. Benesch et al., www.jlab.org/~armd/moller_proposal.pdf (2008)

[13] C. A. Heusch, Int. J. Mod. Phys A 15, 2347 (2000); J. L. Feng, Int. J. Mod. Phys. A 15, 2355 (2000).

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[17] J. Vermaseren, (2000) [math-ph/0010025].

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[19] A. Aleksejevs et al., arXiv:1010.4185v3 [hep-ph].

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14

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).

Mi

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

(A1)

The amplitudes are labeled as

11, Mi

12, Mi

13,..., where the first (second) subscript denotes the origin of the first γ(k) (second

11is the following:

1

ˆk3− mγβeβ(k)u(k1) ·

Mi

11=i(2πe)4Nk1Nk2Np1Np2NpNk· ¯ u(k2)γµ(vi− aiγ5)

1

ˆk4− mγαeα(p)

¯ v(p2)γµ(vi− aiγ5)v(p1) ·

1

q2− m2

i

· δ(k1+ p1− k2− p2− k − p),(A2)

where Nk=

1

(2π)3/2

1

√2k0. Using the Dirac equation and taking k → 0, we can simplify

ˆk1−ˆk + m

(k1− k)2− m2γβu(k1) ≈

1

−2k1k(2kβ

Analogously, at k,p → 0,

1

ˆk4− mγαu(k1) =

Finally, the amplitude Mi

tion from the Born amplitude:

1

ˆk3− mγβu(k1)=

ˆk1+ m

−2k1kγβu(k1) =

=

1+ γβ[−ˆk1+ m])u(k1) = −kβ

1

k1ku(k1).(A3)

kα

1

−k1(k + p) + kpu(k1).(A4)

11at k,p → 0 has the following form, with the convenient factoriza-

Mi

11|k,p→0=e2NpNk· eα(p)eβ(k) ·

kα

1kβ

1

(−k1k)(−k1(k + p) + kp)· Mi

0.(A5)

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15

In the same manner, we get

Mi

22|k,p→0=e2NpNk· eα(p)eβ(k) ·

kα

2kβ

2

(k2p)(k2(k + p) + kp)· Mi

kα

1

(k2p)(−k1k)· Mi

pα

(−p1k)(−p1(k + p) + kp)· Mi

pα

2

(p2p)(p2(k + p) + kp)· Mi

pα

1

(p2p)(−p1k)· Mi

kα

1

(−k1p)(−p1k)· Mi

kα

2

(−k1p)(p2k)· Mi

kα

1

(k2p)(−p1k)· Mi

kα

2

(k2p)(p2k)· Mi

0,

Mi

12|k,p→0=e2NpNk· eα(p)eβ(k) ·

2kβ

0,

Mi

33|k,p→0=e2NpNk· eα(p)eβ(k) ·

1pβ

1

0,

Mi

44|k,p→0=e2NpNk· eα(p)eβ(k) ·

2pβ

0,

Mi

34|k,p→0=e2NpNk· eα(p)eβ(k) ·

2pβ

0,

Mi

13|k,p→0=e2NpNk· eα(p)eβ(k) ·

1pβ

0,

Mi

14|k,p→0=e2NpNk· eα(p)eβ(k) ·

1pβ

0,

Mi

23|k,p→0=e2NpNk· eα(p)eβ(k) ·

2pβ

0,

Mi

24|k,p→0=e2NpNk· eα(p)eβ(k) ·

2pβ

0.(A6)

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:

Mi

11|k,p→0+ (k ↔ p) =e2NpNkMi

0

?

eα(p)eβ(k)kα

(−k1k)(−k1(k + p) + kp)+

= e2NpNkMi

0

−k1k+

≈ e2NpNkMi

(k1k)(k1p)

1kβ

1

eα(k)eβ(p)kα

(−k1p)(−k1(k + p) + kp)

?eα(p)eβ(k)kα

.

1kβ

1

?

?

eα(p)eβ(k)kα

1

1

−k1p

1kβ

1kβ

1

(−k1(k + p) + kp)

0

1

(A7)

As a result, the total t(u)-channel amplitude is given by

Mi

t(u)|k,p→0= e2NpNk· eα(p)eβ(k) · Tα(p)Tβ(k) · Mi

Then, the cross section of (A1) has the form

0,t(u).(A8)

σγγ=σ01

2

?

k0+p0<ω

?α

d3kd3p · (eNk)2(eNp)2Tα(p)Tα(p)Tβ(k)Tβ(k)

= σ0

π

?21

2

?1

4π

?2

?

k0+p0<ω

d3k

k0

d3p

p0

· Tα(p)Tα(p)Tβ(k)Tβ(k).(A9)