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Electroweak radiative corrections to the parity­violating
asymmetry for SLAC experiment E158
Vladimir A. Zykunov
Gomel State Technical University
Gomel, Belarus
Abstract
Electroweak radiative corrections to observable quantities of MÜller scattering of polarized par­
ticles are calculated. We emphasize the contribution induced by infrared divergent parts of cross
section. The covariant method is used to remove infrared divergences, so that our results do not
involve any unphysical parameters. When applied to the kinematics of SLAC E158 experiment, these
corrections reduce the parity violating asymmetry by about ­6.5% at E = 48 GeV and y=0.5, and
kinematically weighted ''hard'' bremsstrahlung e#ect for SLAC E158 is # 1%.
1 Introduction
The SLAC experiment E158 [1] is aimed to measure the parity violating left­right polarization asymmetry
APV in MÜller scattering with a precision not reached before: error of measurements #APV /APV # ±8%.
E158 will determine sin 2 (# W ) at momentum transfer Q 2
# 0.02 GeV 2 with uncertainty # sin 2 (# W ) =
±0.0008 making it the most accurate determination of sin 2 (# W ) at low energies. To this aim the 45­48
GeV polarized electron beam scattering o# unpolarized electrons in a hydrogen target is used.
To extract the reliable data with high precision, it is necessary to consider higher order electroweak
radiative corrections (EWRC). The EWRC to E158 experiment were estimated by Czarnecki and Mar­
ciano [2], Denner, Pozzorini [3] and Petriello [4]. We see at least two reasons for new calculation of the
EWRC: 1) the problem of radiative corrections is of crucial importance and it is necessary to have various
independent calculations, 2) in papers cited above a scheme of infrared singularity removal is used, so
the result contains unphysical parameters.
In this paper a calculation of the lowest order EWRC is carried out using the on­shell renormalization
scheme, Feynman gauge and the covariant approach in order to cancel explicitly the infrared divergences.
We carry out the calculation at E158 energies (and for the situation when only one electron is detected),
emphasizing on a contribution induced by infrared divergent parts of cross section and discuss the nu­
merical estimation of corrections.
2 Born cross section of e - e - # e - e -
The Born cross section of MÜller scattering can be written as:
d# 0
dy
= 2## 2
s #
i,j=#,Z
[# ij
- (u 2 D it D jt + t 2 D iu D ju ) + # ij
+ s 2 (D it +D iu )(D jt +D ju )], (1)
where the four­momenta of the initial and final electrons k 1 , p 1 and k 2 , p 2 (see Fig.1) can be combined
to form the Mandelstam invariants
s = (k 1 + p 1 ) 2 , t = (k 1 - k 2 ) 2 , u = (k 2 - p 1 ) 2 . (2)
The kinematic variable y is defined as
y = - t
s #
1 - cos #
2
E #
E , (3)
where # is the center of mass scattering angle of the detected electron with momentum k 2 . E(E # ) is the
energy of the initial (detected) electron, respectively. Whenever possible, we ignore the electron mass m
(this cannot be done in the collinear singularity regions discussed below).
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The matrix elements in the Born cross section (1) are expressed through the photon and Z 0 propa­
gators
D ik = 1
k -m 2
i
(i = #, Z). (4)
When squaring matrix elements, we used the combinations of coupling constants and the polarizations
of the beam and target electrons:
# ij
± = # 1
ij
B # 1
ij
T ± # 2
ij
B # 2
ij
T , (5)
# 1
ij
B(T ) = # ij
V - p B(T ) # ij
A , # 2
ij
B(T ) = # ij
A - p B(T ) # ij
V , (6)
# ij
V
= v i v j + a i a j , # ij
A
= v i a j + a i v j , (7)
where
v # = 1, a # = 0, v Z = (I 3
e + 2s 2
W )/(2s W c W ), a Z = I 3
e /(2s W c W ), (8)
I 3
e = -1/2 and s W (c W ) are sine (cosine) of the Weinberg angle.
3 One­loop electroweak radiative corrections
We apply the on­shell renormalization scheme of electroweak standard model to our calculation of the
one­loop electroweak radiative corrections. The building blocks needed for explicit calculations according
to this scheme have been worked out in the paper of B˜ohm et al. [5]. We use the results for gauge boson
self­energies and vertex functions taken from [5].
3.1 Virtual corrections (V ­contribution)
The virtual contributions to MÜller scattering can be classified into three categories (see Fig.2): boson self­
energies, vertex functions and boxes. In renormalization ''on­mass shell'' scheme there is no contribution
from the self­energy of electrons. The total virtual cross section is the following sum:
d# V
dy
= d# S
dy
+ d# V er
dy
+ d# B
dy
. (9)
The self­energies of # and Z­boson (including the photon vacuum polarization associated with light
quarks) have been studied extensively (see [2, 3, 4] and references therein). Calculating the lepton
vertices corrections we used the form factors #F je
V,A
from [5] taken at k 2 = t, u. Substituting the coupling
constants for the vertex form­factors (e.g. v #
# #F #e
V
) in the expressions for the functions #± , we get
the vertex part of the cross section
d# V er
dy
= 4## 2
s
Re #
i,j=#,Z
[(# F i jij
- + # ijF i j
- )(u 2 D it D jt + t 2 D iu D ju ) +
+(# F i jij
+ + # ijF i j
+ )s 2 (D it +D iu )(D jt +D ju )]. (10)
The box diagrams with at least one photon (e.g. forth and fifth diagrams in Fig.2) also contain infrared
divergences. The diagrams with two Z or two W bosons are infrared­convergent. The IR­finite part of
cross section looks like
d# B
fin
dy
= -
2# 3
s
4
#
(ij)=1
#
k=#,Z
B k
(ij) + # t # u # , (11)
where double subscript (ij) runs (ij) = {1, 2, 3, 4} = {##, #Z, ZZ,WW}.
The terms B have the form
B k
(##) = D kt # #k
- # 1
(##) + (D kt +D ku )# #k
+ # 2
(##) , B k
(#Z) = D kt # Zk
- # 1
(#Z) + (D kt +D ku )# Zk
+ # 2
(#Z) ,
B k
(ZZ)
= D kt # Bk
- # 1
(ZZ)
+ (D kt +D ku )# Bk
+ # 2
(ZZ) ,
B k
(WW )
= D kt # Ck
- # 1
(WW )
+ (D kt +D ku )# Ck
+ # 2
(WW ) . (12)
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The coupling constants for two heavy bosons look like
v B = (v Z ) 2
+ (a Z ) 2
, a B = 2v Z a Z , v C = a C = 1/(4s 2
W ). (13)
The expressions # 1,2
(ij) have the form (here we used the low energy approximation: s, |t|, |u| # m 2
Z ):
# 1
(##) = L 2
s (s 2 + u 2 )/(2t) - L s u - (L 2
x + # 2 )u 2 /t,
# 2
(##) = L 2
s s 2 /t + L x s - (L 2
x + # 2 )(s 2 + u 2 )/(2t),
# 1
(#Z) = 8u 2 (4I #Z - “
I #Z ), # 2
(#Z) = 8s 2 (I #Z - 4 “
I #Z ),
# 1
(ZZ) = 3u 2 /(2m 2
Z ), # 2
(ZZ) = -3s 2 /(2m 2
Z ),
# 1
(WW ) = 2u 2 /m 2
W , # 2
(WW ) = s 2 /(2m 2
W ); (14)
logarithms from pure electromagnetic boxes are L s = ln(s/|t|), L x = ln(u/t). The scalar integrals in
#Z­part are
I #Z = 1
2 # -u
# 1
0
zdz # 1
0
dx
1
# #
ln |
xz # -u - # #
xz # -u + # # |, “
I #Z = I #Z | u#-s ,
# = -ux 2 z 2 + 4(1 - z)(tz(x - 1) +m 2
Z ).
3.2 Extraction of infrared singularity from the one­loop virtual cross section
Thus, let us present the total virtual one­loop cross section as the sum of infrared (IR) divergent and
IR­finite parts
d# V
dy
= d# V
IR
dy
+ d# V
dy
(# 2
# s), (15)
where # is infinitesimal photon mass.
For IR­part we find the expression which is proportional to Born cross section
d# V
IR
dy
= -
2#
#
log s
# 2
(log tu
m 2 s - 1) d# 0
dy
. (16)
3.3 The photon bremsstrahlung e - e - # e - e - #
To complete the lowest order radiative corrections (and to get an infrared finite result) one needs to
include the real bremsstrahlung diagrams (see Fig.3) (R­contribution).
The di#erential cross section for the process with the emission of one real photon reads
d# R
dy
= -
# 3
4s#
v max
# 0
dv # d 3 k
k 0
#[(k 1 + p 1 - k 2 - k) 2
-m 2 ]
4
#
j,i=1
M R
ij (-1) i+j . (17)
For the calculation of squared matrix elements M R
ij , where i, j = (1, 2, 3, 4) = (#t, #u, Zt, Zu), we used
the standard Feynman rules.
As the kinematic variables of the bremsstrahlung process we use in this case
z = 2kk 2 , z 1 = 2kk 1 = z - t 1 + t, t 1 = (p 2 - p 1 ) 2 ,
v 1 = 2kp 1 = s + u + t 1 - 4m 2 , v = 2kp 2 = s + u + t - 4m 2 , (18)
where k is a 4­momentum of the radiated photon.
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The integration region for variable v is given by the Chew--Low diagram [6] (see Fig.4). The upper
bound of v at s # 0.05 GeV 2 is denoted by solid line. We can observe the asymptotic behavior of the
function in the regions around v = s + t and t = 0. As the upper border v = v max corresponds to point
u = 0 (collinear singularity), we must cut the region of integration up to the value, which corresponds to
experimental set­up of E158 -- the energy of detected particle in the lab. system E L
# 13 GeV. In this
case u max = 2m(m -E L ), and v max = s + t + u max
- 4m 2 (dotted line).
According to the covariant method of Bardin and Shumeiko [7] we can present the cross section of
bremsstrahlung by splitting it into a soft infrared­divergent part and IR--finite contribution
d# R
dy
= d# R
IR
dy
+ d# R
F
dy
. (19)
The infrared divergent part (the first term) of the expression (19) after integrations over k and v and
#­parametrization reads
d# R
IR
dy
= 2#
#
(ln (v max ) 2
4m 2 # 2
(ln tu
m 2 s - 1) + # S + # H
1 ) d# 0
dy
. (20)
The corrections # S and # H
1 can be found in [7]:
# S
1 = ln s(s + t)
m 4 -
1
2 l m ln s 2 (s + t) 2
-tm 6 -
1
2 l 2
r - 2l r l m + l m - l 2
m - # 3
3 + 1, (21)
# H
1 =
v max
# 0
dv # 2
v # ln(1 - v
t
) - ln(1 - v
s
) + ln(1 - v
s + t
) -
1
2 ln(1 + v
m 2
) # +
+ 2
s + t - v
ln s + t - v
m 2 -
1
s - v
ln (s - v) 2
m 2 # -
1
v - t
ln (v - t) 2
m 2 # -
1
# # ,
l m = ln -t
m 2 , l r = ln s + t
s
, # = v +m 2 . (22)
The second term of (19) is the IR­finite part of bremsstrahlung. To calculate this part it is necessary
to integrate analytically over k­4­momenta of photon and then numerically over v. The integral over
whole phase space of the radiated photon can be presented in the form
I [A] = 1
#
# d 3 k
k 0
#[(k 1 + p 1 - k 2 - k) 2
-m 2 ][A] = 1
#
t max
1
#
t min
1
dt 1
z max
#
z min
dz
# R z
[A], (23)
where R z is proportional to the Gram determinant. The limits of the double integration z min/max and
t min/max
1 are the roots of equations R z = 0 and z min = z max , respectively.
During the calculation of ''hard'' part of squared matrix elements M R
ij we calculated more then 100
scalar integrals. Exact results for this part are rather cumbersome and we do not present them here. The
total list of expression M R
ij as a set of output REDUCE files and scalar integrals as subroutine­functions
can be found in text of FORTRAN code RCORR2A1 (''Radiative CORRections TO asymmetry A1'').
3.4 The result of infrared singularity cancellation
Adding the IR­parts of V ­ and R­ contributions (formulas (16) and (20))
d# C
dy
= d# V
IR
dy
+ d# R
IR
dy
= #
#
(4 ln v max
m # s
(ln tu
m 2 s - 1) + # S
1 + # H
1 ) d# 0
dy
, (24)
we obtain the final finite expression which is free of infrared divergences and unphysical parameters.
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4 Numerical estimates
Numerical calculations of the electroweak radiative corrections to the asymmetry APV in MÜller scattering
at the energy of longitudinally polarized electron beam of SLAC experiment E158 were performed using
the FORTRAN code RCORR2A1 [8]. The structure of code allows us to estimate the corrections for
arbitrary experimental conditions and successfully apply them to the E158 Monte Carlo simulation [9].
The asymmetry corresponds to the usual expression
APV = #LL + #LR - #RL - #RR
#LL + #LR + #RL + #RR
, # # d#/dy. (25)
So we are interested in the following basic contributions: 1) infrared­finite parts: self­energy of gauge
bosons (for this part authors of [2, 3, 4] have found the correction to the asymmetry of # -50%),
heavy vertices (give rather small contribution to asymmetry), heavy boxes (ZZ and WW) (give +3%
to asymmetry); 2) infrared­divergent parts (IR parts): the rest part of virtual 1­loop contributions,
which consist of electron vertices with photon , #­#-- and #­Z--boxes, infrared singularity cancellation
(this log­term is proportional to Born cross section and does not change the asymmetry), and ''hard''
bremsstrahlung.
The correction to the asymmetry is defined as follows
#APV = A RC
PV -A 0
PV
A 0
PV
, (26)
where A 0
PV is the Born asymmetry, and A RC
PV is the asymmetry taking into consideration the electroweak
radiative corrections.
The influence of the electroweak radiative corrections to the asymmetry is shown in Fig.5. One can
observe that for E = 48 GeV at y = 0.5 the correction #APV amounts to # -6.5%, it is minimal at
moderate y, and this minimum shifts to small y with the increasing energy. At last
-6.5% = +1% (''hard'' brems.) + (-7.5%) (the rest part).
Kinematically weighted ''hard'' initial and final state radiation e#ect for observable APV under condition
of E158 is
F b = 1.01 ± 0.01
(notation of [9]), it is our contribution to radiative correction procedure for SLAC E158.
References
[1] K.S. Kumar et al., Mod. Phys. Lett. A10 (1995) 2979; R. Carr et al., SLAC­PROPOSAL­E­158
[2] A. Czarnecki and W.J. Marciano, Phys. Rev. D53 (1996) 1066
[3] A. Denner and S. Pozzorini, Eur. Phys. J. C7 (1999) 185
[4] F.J. Petriello, SLAC­PUB­9532, October 2002 (hep­ph/0210259)
[5] M. B˜ohm et al., Forschr. Phys. 34 (1986) 687
[6] G.F. Chew and F.E. Low, Phys. Rev. 113 (1959) 1640
[7] D.Yu. Bardin and N.M. Shumeiko, Nucl. Phys. B127 (1977) 242
[8] V.A. Zykunov, Yad. Fiz., 67 (2004) 1366
[9] P.L. Anthony et al. (SLAC E158 Collaboration), hep­ex/0312035
145

Figure 1: Neutral current t­channel (1) and u­channel (2) amplitudes leading to the asymmetry APV at
tree level.
Figure 2: The virtual t­channel one­loop diagrams for e - e - # e - e - process. The contributions to the
self­energies and vertex corrections are symbolized by the empty loops.
Figure 3: Bremsstrahlung t­channel diagrams for e - e - # e - e - # process.
146

0
0.01
0.02
0.03
0.04
0.05
0.06
­0.06 ­0.05 ­0.04 ­0.03 ­0.02 ­0.01 0
Figure 4: Chew­Low diagram at s # 0.05 GeV 2 . Mass of electron is real (solid line) and had been raised
in 20 times for illustration (dashed line). Dotted line corresponds to v max for experimental conditions of
E158.
­0.1
­0.09
­0.08
­0.07
­0.06
­0.05
­0.04
­0.03
­0.02
­0.01
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7
Figure 5: Corrections to polarization asymmetry APV as a functions of y at di#erent energies which are
denoted by numbers on curves.
147