Application of QCD structure function method to calculate of NLO corrections to Bhabha scattering (I) Soft and Virtual photons

Abstract

Soft and virtual loop photonic contributions to the second order next-to-leading QED radiative corrections to Bhabha scattering are calculated with help of the renormalization group approach. The results are in agreement with earlier calculations, where other methods were used.

Full text

Page 1

Application of QCD structure function method to
calculate of NLO corrections to Bhabha scattering
(I) Soft and Virtual photons
A. B. Arbuzova∗, E. S. Scherbakovaa†
aBogoliubov Laboratory of Theoretical Physics JINR, Dubna, 141980 Russia
Abstract
Soft and virtual loop photonic contributions to the second order next-to-leading QED
radiative corrections to Bhabha scattering are calculated with help of the renormalization
group approach. The results are in agreement with earlier calculations, where other methods
were used.
1Introduction
Bhabha scattering (e+e−−→ e+e−) is one of the fundamental processes in particle physics.
Precision theoretical predictions for the differential cross section of Bhabha scattering are of
ultimate importance for all experiments at electron-positron colliders. They are required for
normalization purposes including luminosity determination, for several searches of new physics,
and as a background contribution to many other processes studied at e+e−colliders. To provide
the required accuracy of the predictions, one should take into account radiative corrections in
the first and higher order of perturbative QED.
In this paper we present the calculation of the second order next-to-leading virtual and soft
photonic radiative corrections to the cross section of Bhabha scattering. To get the corrections
we use the renormalization group techniques borrowed from QCD. The result is found to be in
agreement with earlier calculations [1, 2]. The advantage of our approach is its universality: in
the same way one can get all the other remaining contributions to the radiative corrections in
O?α2L?. For the case of small angle Bhabha scattering the complete result for the O?α2L?
systematized, while there is a number of results for particular contributions scattered in the
literature.
is known [3], but for the general case of large angle scattering terms of this order are still not
2Initiative
The differential Bhabha cross section as a series in α is:
dσ = dσBorn+ dσ(1)+ dσ(2)+ O?α3?,
where dσBornrepresents the Born level cross section, and dσ(1,2)are the pure QED contributions
of the first and second order corrections.
The first order contribution is usually decomposed into three parts: dσ(1)= dσV+ dσS+
dσH, where superscripts ”V”, ”S”, and ”H” are used to denote the virtual, soft, and hard
(1)
∗e-mail: arbuzov@theor.jinr.ru
†e-mail: scherbak@theor.jinr.ru
1

Page 2

photonic corrections, respectively. The small parameter ∆ (∆ ? 1) subdivides the kinematical
domain of real photon emission into soft and hard parts with photon energy below and above
∆·Ebeam, where Ebeamis the beam energy. The one-loop contributions are well known, see i.e.
Refs. [4, 1, 5].
In the second order we construct a similar decomposition:
dσ(2)
=dσVV+ dσSV+ dσSS+ dσVH+ dσSH+ dσHH,(2)
where the superscripts have the same meaning as stated above, so that for instance “VH”
denotes the contribution due to emission of one hard photon accompanied by the effect of a
single virtual loop.
In this stady we don’t take into account the so-called pair contributions, related to emission
of real or virtual pairs (e+e−, π+π−etc.). Their numerical contribution to the observed cross
section is typically small compared to the photonic correction, see Ref. [6, 7].
It is natural to expand the QED part of radiative corrections to Bhabha scattering into
a series in the fine structure constant α and in powers of the so-called large logarithm L =
ln(M2/m2
e), where M is a large energy scale related to the beam energy, M ? me.
The second order contribution dσ(2)is decomposed into series in the powers of the large
logarithm. There in particular contributions we meet terms of the orders O?α2L4,3,2,1,0?. The
photon contributions.
The Soft-Soft (SS) and Soft-Virtual (SV) contributions we can found with using the factori-
sation properties of soft photon radiation [8]:
terms with the fourth and the third powers of L will cancel out in the sum of virtual and soft
dσSS
=
1
2!(δS)2dσBorn,dσSV= δSδVdσBorn,(3)
where δS,V= dσS,V/dσBorn. Note that for the contribution of double soft photon emission, dσSS,
in the formula above we apply an upper limit on the energy of each of the photons independently.
The Virtual-Virtual (VV) contribution can’t be received in such a simple manner. Below we
will show how to reconstruct the logarithmically enhanced part of it using the renormalization
group techniques or in other words the electron structure function approach.
3Structure Function Approach
The structure function approach, widely used in QCD can be applied to QED problems [9, 10,
11, 13]. With help of it we can analytically find the most important contributions reinforced
by the large logarithm L, since they can be treated as electron mass singularities.
We are going to drop the pair contributions, so we need here the pure photonic part of the
non-singlet structure (fragmentation) functions for the initial (final) state corrections. These
functions describe the probability to find a massless (massive) electron with energy fraction z
in the given massive (massless) electron. In our case with the next-to-leading accuracy we have
?
?α
+
Dstr,frg
ee
(z) = δ(1 − z) +α
?2?1
O?α2L0,α3?,
2π
d1(z,µ0,me) + LP(0)
ee(z)
?
+
2π2L2P(0)
ee⊗ P(0)
ee(z) + LP(0)
ee⊗ d1(z,µ0,me) + LP(1,γ)str,frg
ee
(z)
?
(4)
where the superscripts “str” and “frg” are used to mark the structure and fragmentation func-
tions, respectively. The difference between the functions appear only due to the difference in
the next-to-order splitting functions P(1,γ). For our calculation used the modified minimal sub-
traction scheme MS with the factorizations scale equal to M, and the renormalization scale µ0
2

Page 3

will be taken equal to me. More details on application of the approach to calculation of second
order next-to-leading QED corrections can be found in Refs. [10, 13]. Also the description of
functions d1, P(0), P(1)can be found in Ref. [12].
The master formula describing the radiatively corrected Bhabha cross section in the struc-
ture function approach reads [5]:
?1
?1
dσ=
¯ z1
dz1
?1
¯ z2
dz2Dstr
?1
ee(z1)Dstr
ee(z2)
?
Y2),
dσBorn(z1,z2) + d¯ σ(1)(z1,z2) + O?α2L0??
ee(y2
×
¯ y1
dy1
Y1
¯ y2
dy2
Y2
Dfrg
ee(y1
Y1)Dfrg
(5)
where d¯ σ(1)is the O (α) correction to the massless Bhabha scattering, calculated using the
MS scheme to subtract the lepton mass singularities. Energy fractions of incoming partons are
z1,2, and Y1,2are the energy fractions of the outcoming electron and positron.
Here we are interested in the contributions due to virtual and soft photons, so all the four
integrals will have the same lower limit being equal to 1 − ∆. In this way ( also ∆ ? 1) this
function gives the probability to find such a situation where one looses in total due to photon
emission ∆Ebeamfrom the total energy of the process under consideration.
Let us fix now the factorization scale M =√s and define Ls≡ ln(s/m2
Later on we will consider another choice of the scale.
Convolution of the function found above with the Born part of the kernel cross section gives
us the corresponding part to the cross section (with the upper limit on the energy lost):
e), s = 4E2
beam.
?1
?α
1−∆
D⊗4(z)dσBorn(z)dz = dσBorn
?2?
?
1 +α
2π
?
4Ls
?
2ln∆ +3
2
?
+ O?L0
s
??
,
+
2π
8L2
s
?
P(0)?⊗2
∆
+ 16Ls(P0⊗ d1)∆+ 4LsP(1)
∆
?
+ O?α2L0
s,α3??
(6)
where we used subscript ∆ to specify the so-called ∆-part of the corresponding function (see
i.e. Refs. [11]).
Convolution with the d¯ σ(1)is more complicated since the latter is a non-trivial function of
z. But we restricted ourselves to consider only the terms reinforced by the large logarithm and
need to compute only the following part:
?1
Using the techniques of dealing with the singular functions regularized by introduction of the
∆ and Θ parts [11, 14] we cast it into the following form:
?1
where
?1
δ(¯ σ(1)) =
1−∆
4α
2πLs
1−∆
dz[P(0)⊗ d¯ σ(1)](z). (7)
4α
2π
1−∆
dy
?1
0
dz
z
LsP(0)?y
z
?
d¯ σ(1)(z) = 4α
2πd¯ σ(1)
∆LsP(0)
∆+ 4α
2πLsδ(¯ σ(1)),(8)
d¯ σ(1)
∆=
1−∆
?1
d¯ σ(1)(z)dz = dσV+ dσS− dσBornα
?
2π4LsP(0)
∆,
d¯ σ(1)(z) 2 ln1 −1 − z
∆
?
dz.(9)
The virtual loop contribution to the function d¯ σ(1)doesn’t contribute to the result of the
integral. And we can put the upper limit of the integral over z to be equal to 1 − ∆1, ∆1?
3

Page 4

∆ ? 1. Therefore we need to consider only the soft photon part of the function:
d¯ σ(1,S)(z)
dz4π2
dz
|?k|2+ λ2
α
2π4
=dσBorn
?
−αd|?k|
|?k|2
?
??2π
0
dφ
?1
−1
dc
?pe+
pe+k−pe
pek+
p?e
p?ek−
p?
p?
e+
e+k
?2?
−
?
d1(z) + LsP(0)(z)
??
,(10)
where 1 − z = |?k|/Ebeam; λ is a fictitious photon mass, λ ? me; pe (pe+) is momentum of
incoming electron (positron), p?e, p?
the standard techniques of calculations of soft photon contributions we get
?1−∆1
=
2π
e+, and k are the momenta of the outgoing particles. Using
δ(¯ σ(1))
dσBorn=
4α
1
dσBorn
1−∆
dzd¯ σ(1,S)(z)
?
dz
2ln
?
1 −1 − z
??
∆
?
?
−4ζ(2)ln1 − c
1 + c+ 8ζ(3) − ζ(2)ln∆, (11)
where c is the cosine of the electron scattering angle, c = cos?
next-to-leading second order contributions to quasi-elastic Bhabha cross section, where the total
energy loss is limited by ∆Ebeam. It is useful to describe also the case when the energies of the
soft photons (if they are two) are limited independently. The transition between the two cases
was derived in Ref. [8]. Applying it we get the final result:
?
?
+
?
function f(x) can be found in Ref. [1]. In the formula above besides the logarithmically enhanced
terms derived here, we included also the known contribution without the large logs, δ(2)
is given by Eq. (3) from Ref. [2].
? pe? p?e.
Summing up the contributions in Eq. (8) and then Eqs. (3,6,8) we receive the leading and
dσVV+ dσSV+ dσSS
=
?α
2π
?2dσBorn
2(Li2(1−x)−Li2(x)) + 3ln
24Li2(1 − x) − 24Li2(x) + 12f(x) + 24ζ(3) −93
?m2
L2
s
?
32ln2∆+48ln∆+18
?
+f(x) − 7−2π2
+Ls
?
64
?
ln
?
x
1 − x
?
?
−1
?
ln2∆
+ 16
?
x
1 − x
?
3
ln∆
2
−10π2
+ 4δ(2)
0
+ O
e
s
??
,x ≡1 − c
2
, (12)
0, which
4Numerical Results
Let us compare the numerical values of the leading, next-to-leading, and next-to-next-to-leading
corrections for two choices of the factorization scale: M =√s, which has been used in Refs. [1,
2], and M =√−t, which has been advocated in Ref. [3]. Since we have the complete answer (12),
we can easily choose any other the factorization scale by changing the argument of the large
logarithm, while the total sum is kept unchanged.
t = −xs, Ls= Lt− lnx, Lt≡ ln−t
In Figures 1–4 we show the values of the second order soft and virtual photonic radiative
corrections in different approximations with respect to the power of the large logarithm. Values
of the corrections are given in terms of 10−3· dσBorn. In particular, r(2)
O?α2L2?, the next-to-leading O?α2L1?, and the next-to-next-to-leading O?α2L0?relative con-
disappear in the sum of the virtual and soft corrections with the remaining three contributions
For this purpose we use the relations:
m2
e.
2,1,0represent the leading
tributions to the cross section, respectively. Since the dependence on the parameter ∆ should
4

Page 5

(see Eq. (2)), we put ∆ = 1. In this way we receive only an estimate of the magnitude and the
relative size of the corrections in different approximations. Nevertheless this evaluation helps
us to get an idea about the size of the unknown second order contributions and to estimate the
theoretical uncertainties.
-20
-15
-10
-5
0
5
10
15
0123456
θ

103 · r0
103 · r1
103 · r2
(2)
(2)
(2)
-1
0
1
2
3
4
5
6
7
8
9
0123456
θ

103 · r0
103 · r1
103 · r2
(2)
(2)
(2)
-15
-10
-5
0
5
10
15
0 2040 6080 100120 140160 180
θ

103 · r0
103 · r1
103 · r2
(2)
(2)
(2)
-15
-10
-5
0
5
10
15
0 20406080 100120140 160180
θ

103 · r0
103 · r1
103 · r2
(2)
(2)
(2)
Figure 1: Soft and virtual second order photonic radiative corrections versus the scattering
angle in degrees for ∆ = 1,
side.
√s=100 GeV; M =√s on the left side and M =√−t on the right
Fig. 1 gives us results for the small and large angle Bhabha scattering at LEP/SLC, re-
spectively. We checked that for√s=200 GeV the plots are very close the the ones shown for
√s=100 GeV.
Figures 2 give as and idea, how do the corrections behave at higher energies, which can be
reached at a future linear collider.
Looking at the plots representing the contributions of different powers in L, we conclude
that with the proper choice of the factorization scale M =√−t, the magnitude of the non-
logarithmic corrections is below 1·10−4%, everywhere except the region of very large scattering
angles (θ >∼160◦). That region requires a special treatment, and it doesn’t seem to be of
interest for the experiments. Note that the estimate of the size of the non-logarithmic second
order corrections agree with the one made earlier in Ref. [3]. At the same moment it is clear
that to reach the 1·10−4% level in the precision of theoretical description of Bhabha scattering
we should take into account the complete O?α2L0?calculations including the effects of virtual
and real corrections due to pairs and photons.
5 Conclusions
In this way we received the photonic part of the second order next-to-leading logarithmic contri-
bution to Bhabha cross section. The result agrees with earlier calculations by means of different
5

Page 6

-12
-10
-8
-6
-4
-2
0
2
4
6
0123456
θ

103 · r0
103 · r1
103 · r2
(2)
(2)
(2)
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0123456
θ

103 · r0
103 · r1
103 · r2
(2)
(2)
(2)
-10
-8
-6
-4
-2
0
2
4
6
0 204060 80 100120 140 160180
θ

103 · r0
103 · r1
103 · r2
(2)
(2)
(2)
-10
-8
-6
-4
-2
0
2
4
6
0 20 406080 100120140160 180
θ

103 · r0
103 · r1
103 · r2
(2)
(2)
(2)
Figure 2: Soft and virtual second order photonic radiative corrections versus the scattering
angle in degrees for ∆ = 1,
side.
√s=500 GeV; M =√s on the left side and M =√−t on the right
methods. Our approach allows to get all the next-to-leading contributions systematically. It
can be applied to any kind of a process, where one has to look for the radiative corrections
enhanced by large logarithms. In particular, we applied the same approach to the description
of the contribution of real photon emission to Bhabha scattering [to be described elsewhere].
Taking the result of the present study and the O?α2L?results of papers [6], where the pair
we arrive at the complete result for the second order next-to-leading radiative corrections to
Bhabha scattering. The results are valid both for the small and large angle scattering. To apply
the results to data analysis of modern and future experiments at electron-positron colliders, we
are going to implement them into the Monte Carlo event generators LABSMC [16] and SAMBHA [17]
for large and small angle scattering, respectively.
corrections were evaluated, and of Refs. [15], where real photon radiation was taken into account,
We are grateful to L. Trentadue and A. Penin for discussions. This work was supported by
RFBR grant 04-02-17192. One of us (A.A.) thanks also the grant of the President RF (Scientific
Schools 2027.2003.2).
References
[1] E.W.N. Glover, J.B. Tausk and J.J. Van der Bij, Phys. Lett. B 516 33 (2001).
[2] A.A. Penin, Phys. Rev. Lett. 95 010408 (2005).
6

Page 7

[3] A.B. Arbuzov, V.S. Fadin, E.A. Kuraev, L.N. Lipatov, N.P. Merenkov, L. Trentadue,
Nucl. Phys. B 485 457 (1997).
[4] W. Beenakker, F.A. Berends and S.C. van der Marck, Nucl. Phys. B 349 323 (1991).
[5] A.B. Arbuzov, G.V. Fedotovich, E.A. Kuraev, N.P. Merenkov, V.D. Rushai, L.
Trentadue, JHEP 10 001 (1997).
[6] A.B. Arbuzov, E.A. Kuraev, N.P. Merenkov, L. Trentadue, Nucl. Phys. B 474 271 (1996);
A.B. Arbuzov, E.A. Kuraev, N.P. Merenkov, L. Trentadue, Phys. Atom. Nucl. 60 591
(1997).
[7] A.B. Arbuzov, JHEP 0107 (2001) 043.
[8] A.B. Arbuzov, E.A. Kuraev, B.G. Shaikhatdenov, Mod. Phys. Lett. A 13 2305 (1998).
[9] E.A. Kuraev and V.S. Fadin, Sov. J. Nucl. Phys. 41 466 (1985).
[10] F.A. Berends, W.L. van Neerven and G.J.H. Burgers, Nucl. Phys. B 297 429 (1988)
[Erratum-ibid. B 304 921 (1988)].
[11] M. Skrzypek, Acta Phys. Polon. B 23 135 (1992); A.B. Arbuzov, Phys. Lett. B 470 252
(1999).
[12] A. Arbuzov, E. Scherbakova, JETP. Lett. 83, 427 (2006)
[13] A. Arbuzov, K. Melnikov, Phys. Rev. D 66 093003 (2002).
[14] A.B. Arbuzov, Tables of convolution integrals, hep-ph/0304063.
[15] A.B. Arbuzov, V.A. Astakhov, E.A. Kuraev, N.P. Merenkov, L. Trentadue, E.V.
Zemlyanaya, Nucl. Phys. B 483 83 (1997); A.B. Arbuzov, E.A. Kuraev, B.G. Shaikhat-
denov, JETP 88 213 (1999), ibid Erratum, 97 858 (2003).
[16] A.B. Arbuzov, LABSMC: Monte Carlo event generator for large-angle Bhabha scattering,
hep-ph/9907298.
[17] A.B. Arbuzov, D. Haidt, C. Matteuzzi, M. Paganoni and L. Trentadue, Eur. Phys. J. C
34 267 (2004).
7