Emission of two hard photons in large-angle Bhabha scattering

El ELSEVIER

Nuclear Physics B 483 (1997) 83-94

Emission of two hard photons in large-angle Bhabha scattering A.B. Arbuzov a.,, V.A. Astakhov b, E.A. Kuraev a, N.E Merenkov c, L. Trentadue d, E.V. Zemlyanaya e a Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, 141980, Russia b Budker Institute for Nuclear Physics, Prospect Nauki 11, Novosibirsk, 630090, Russia c Institute of Physics and Technology, Kharkov, 310108, Ukraine J Dipartimento di Fisica, Universitd di Parma, 43100 Parma, Italy and INFN Sezione di Milano, Milan, Italy e Laboratory of Computing Techniques and Automation, JINR, Dubna, 141980, Russia Received 29 November 1995; revised 2 October 1996; accepted 4 October 1996

Abstract

A closed expression for the differential cross section of the large-angle Bhabha e+e - scattering which explicitly takes into account the leading and next-to-leading contributions due to the emission of two hard photons is presented. Both collinear and semi-collinear kinematical regions are considered. The results are illustrated by numerical calculations. PACS: 12.20.-m; 12.20.Ds Keywords: Bhabha scattering; Double bremsstrahlung; Large angles; High energy

1. I n t r o d u c t i o n

The large-angle Bhabha process is well suited for the determination of the luminosity /~ at e+e - colliders of the intermediate energy range x/~ = 2e ~ 1 GeV [ 1,2]. Small scattering angle kinematics of Bhabha scattering is used for high-energy colliders such as LEP I [3]. As far as 0.1% accuracy is desirable in the determination of /2, the corresponding requirement * Corresponding author. E-mail: [email protected] 0550-3213/97/$17.00 Copyright (~) 1997 Elsevier Science B.V. All rights reserved PII S 0 5 5 0 - 3 2 13 ( 9 6 ) 0 0 5 5 6 - 1

A.B. Arbuzovet aL/Nuclear PhysicsB 483 (1997) 83-94

84 --~ ~< 10 -3

(1)

on the Bhabha cross section theoretical description appears. The quantity Ao- is an unknown uncertainty in the cross section due to higher order radiative corrections. Much attention was paid to this process during the last decades (see review [4] and references therein). The Born cross section with weak interactions taken into account and the first order QED radiative corrections to it were studied in detail [5]. Both contributions, the one enhanced by the large logarithmic multiplier L -- In ( s / m 2) (where s = (p÷ + p_)2 = 482 is the total center-of-mass (CM) energy squared, m is the electron mass), and the one without L are to be kept in the limits (1): aL/¢r, o
The total two-loop ( ~ (a/Tr) 2) correction could be constructed from: ( 1) the two-loop corrections arising from the emission of two virtual photons; (2) the one-loop corrections to a single real (soft and hard) photon emission; (3) the ones arising from the emission of two real photons; (4) the virtual and real e+e - pair production [6]. As for the corrections in the third order of perturbation theory, only the leading ones proportional to (olL/Tr) 3 are to be taken into account. In this paper we consider the emission of two real hard photons:

e÷(p+) + e - ( p _ ) ---* e+(q+) + e - ( q _ ) + y ( k l ) + y(k2).

(3)

The relevant contribution to the experimental cross section has the following form: O'exp

=/

(4)

do- 0 + 0 _ ,

where O+ and O_ are the experimental restrictions providing the simultaneous detection of both the scattered electron and positron. First, this means that their energy fractions should be larger than a certain (small) quantity eth/e, eth is the energy threshold of the detectors. The second condition restricts their angles with respect to the beam axes. They should be larger than a certain finite value ~P0 (00 '~ 35 ° in the experimental conditions accepted in Ref. [ 1] ) :

~--~0 > 0-, 0÷ > 00,

0± = q ~ _ ,

(5)

where 0± are the polar angles of the scattered leptons with respect to the beam axes ( p _ ) . We accept the condition on the energy threshold of the charged-particle registration qO > eth. Both photons are assumed to be hard. Their minimal energy O)min = ZI$:,

,4 <<

1,

(6)

A.B. Arbuzov et al./Nuclear Physics B 483 (1997) 83-94

85

could be considered as the threshold of the photon registration. The main (,,, (aL/~r) 2) contribution to the total cross section (5) comes from the collinear region: when both the emitted photons move within narrow cones along the charged-particle momenta (they may go along the same particle). So we will distinguish 16 kinematical regions: akl and m --<<00<<1, e

ak2 < 0o, a --b b,

akl

and

bk2 < 00,

a, b = p _ , p + , q _ , q + .

(7)

The matrix element module square summed over spin states in the regions (7) is of the form of the Born matrix element multiplied by the so-called collinear factors. The contribution to the cross section of each region has also the form of 2 --+ 2 Bhabha cross sections in the Born approximation multiplied by factors of the form

do'7°ll: diToi

/ 8202 X [ i~202 X -I ai(xS,y,)ln=t:-~2° ) +b,(x,,y,,ln(,:-2#)],

(8)

where xj = tojle, Yl = q ° le, Y2 = q° le are the energy fractions of the photons and of the scattered electron and positron, respectively. The dependence on the auxiliary parameter 00 will be cancelled in the sum of the contributions of the collinear and semicollinear regions. The last region corresponds to the kinematics, when only one photon is emitted inside the narrow cone 01 < 00 along one of the charged-particle momenta. And the second photon is emitted outside any cone of that sort along charged particles (02 > 00): do':C=

°Zlnf4e2"~do'roi(k2),

(9)

where do-~i has the known form of the single hard bremsstrahlung cross section in the Born approximation [7]. Below we show explicitly that the result of the integration over the single hard photon emission in Eq. (9) in the kinematical region 01 > 00 (01 is the emission angle of the second hard photon with respect to the direction of one of the four charged particles) has the following form: i

02 do-~i(k2)=-21n(4)ai(x,y)

do'~ + d&i.

(10)

The collinear factors in the double bremsstrahlung process were first considered in papers of the CALCUL collaboration [ 8]. Unfortunately they have a rather complicated form which is less convenient for further analytical integration in comparison with the expressions given below. The method of calculation of the collinear factors may be considered as a generalization of the quasi-real electron method [9] to the case of multiple bremsstrahlung. Another generalization is required for the calculations of the cross section of process e+e - --->2e+2e - [6].

86

A.B. Arbuzov et aL/Nuclear Physics B 483 (1997) 83-94

It is interesting that the collinear factors for the kinematical region of the two hard photon emission along the projectile and the scattered electron are found to be the same as for the electron-proton scattering process considered by one of us in Ref. [ 10]. There are forty Feynman diagrams of the tree type which describe the double bremsstrahlung process in e+e - collisions. The differential cross section in terms of helicity amplitudes was computed about ten years ago [ 8,11 ]. It has a very complicated form. We note that the contribution from the kinematical region in which the angles (in the CM system) between any two final particles are large compared with m/e is of the order

a2r~m2 ,~ 10 -36 cm 2

(11)

qT"2t32

(ro is the classical electron radius). So, statistics at colliders with luminosity £ ~ cases of double bremsstrahlung imitating e + e - y , which corresponds to the emission momenta.

the corresponding events will possess poor 1031-1032 cm -2 s -1. More probable are the the processes e+e - ~ e+e - or e+e of one or two photons along charged-particle

2. Kinematics in the collinear region It is convenient to introduce, in the collinear region, new variables and transform the phase volume of the final state in the following way (from now on we will work in the CM system) :

/dF=fd3q-d3q+d3kld3k26 (4) 16qOqOwlw2(27r)8 1

m4-rr2

-

1

2rr

4(2~r)6/dxl/dX2XlX2f

f dFq=f

A

d3q-d3q+

A

4qOqO (2rr) 2

zl,2 =

/12q+)

( f l i p - + r/2p+ -- , ~ l q - --

0

¢~(4)(7/1 p _ +

~b=kl±k2±,

d~b

z0

z0

fdzlfdz2fdrq, 0

r/2p+ O.1i xi=--,

(12)

0 Alq-

-

z0=

A2q+),

>>1,

A-

¢Omin

,

where Oi (i = 1,2) is the polar angle of the /-photon emission with respect to the momentum of the charged particle that emits the photon; 7/+ and A+ depend on the specific emission kinematics, they are given in Table 1. The columns of Table 1 correspond to a certain choice of the kinematics in the following way: p_p_ means the emission of both the photons along the projectile electron, p+q_ means that the first of the photons goes along the projectile positron; the second, along the scattered electron, and so on. The contributions from 6 remaining kinematical regions (when the photons in the last 6 columns are interchanged) could

A.B. Arbuzov et al./Nuclear Physics B 483 (1997) 83-94

87

Table 1 r/i and ,~i for different collinear kinematics p-p-

q-q-

p+p+

q+q+

P-P+

q-q+

p-q-

p+q+

p-q+

P+q-

1 1 1

1 -- xl t l+XZ

1 1 - xl 1

I - xl 1 1

1 1 - x~ 1 + x2

l

1

1+

1+

rh 1"]2 A~

3' 1 1

1 1 -~

1 y 1

1 1 1

I -- Xl 1 -- x2 1

A2

1

1

1

1

|

•

I --X I

Yl

I--X,~

x--z Y2

x-2v,

Yl

1

be found by the simple substitution Xl ~ x2. We will use the momentum conservation law '01P-+72P+

=

(13)

Alq- + A 2 q + ,

and the following relations coming from it:

711 q'- 72 =,AlYl + ,A2Y2,

,'~IYl s i n 0 _

q°,2 Yl,2 = - -

= ,A2Y2 s i n 0 + ,

7,~ + 722 + (,722 - n?)c (14)

~-2Y2 = 71 -~- 72 + ('02 -- '01) c "

Each of the 16 contributions to the cross section of process (3) can be expressed in terms of the corresponding Born-like cross section multiplied by its collinear factor

(15)

do'coil = 2[ \ 2 ¢ r J - - ~ Z K(7, A) dO'0(7, A) dxl dx2, (ma) d~o(7, a)

dli(7,,~)

2a2 B(7, a) dI(7, a), =s =/

d3-~-~q+

B(7, A) = (~2 + 72 + g~-~2 _ ~ J'

S(4)(71P - + 72P+ - ,~lq--

q-q+ 4~7172 dc a ~ a ~ [ c ( T z - m ) + 7~ + 72] 2, z0

z0

K(7,a)=m4fdzl/ o

o

2~r

P

dz2

ddp

J ~-~E (7, ,~), o

?= (71P- - Alq-) 2 = - s ~=(71P_

A2q+)

7 1 ( 1 -- C)

71 + 72 + ( 7 2 - 71)c '

+ 72P+)2=4e27172=s7172

,

~+T+I~

=0.

The sum over (7, A) means the sum over 16 collinear kinematical regions. The corresponding (7, A) can be found in Table 1. The quantities gSi(7, A) are as follows:

88

A.B. Arbuzov et al./Nuclear Physics B 483 (1997) 83-94

1 C ( p _ p _ ) = 2 ¢ 4 ( A i , A2, A , Xl, x2, y ) , Y 1C(q_q_)=2y,4(Bl

Bz,B,-xj

-x2 Y

Y

1 Y

IC(p+p+ ) = 2.A(C1, C2, C, Xl, x2, y),

Y

IC(q+q+) = 2 y . A ( D t , D 2 , D , - x ~

- x 2 l_) Y Y

Y •A ( A I , A 2 , A , x l , x 2 )

yAl + l + y 2 AZA2 xlx2AiA2

yA2 AZA1

-

-

71-

r~ + yr2 AAlxlx2 -

-

-~ yr__________ r 3 -4- L + 2m2(y 2 -4- rl2) + 2m2(y 2 -4- r~) AA2xlx2

AA2x2

AA2xl

'

(16)

lC ( p _ p + ) = 2K1K2,

1C(p_q+) = - 2 K I K 3 ,

1C(p+q_ ) = - 2 K 4 K s ,

=2K6KT,

IC(p_q_ ) = -2KtKs,

1C(p+q+) = -2K4K3,

E(q-q+)

K, =

gl AlXlr I

K3 =

g4 D2x2t2

K5 = KT=

+ 2m----~2

K4 =

2m 2 B2

g2

2m 2

D2x2

D2 ,

C2 ,

g----'--L--I + 2m----~2

D2 '

g3

tl = yl + xZ,

g------L--2 + 2m2 C2xzr2

2m 2

B2x2tl

gl = l + r21,

/(2 =

A2 '

Clxl rl

gl K6 - BlXt --

'

rl = 1 - x l ,

g2 = l + r~, t2 = yz + x2,

C2 '

2rn 2 B2 , r2 = I - x 2 ,

g3 = y~ + t21,

y=l-Xl-X2.

g4 = y2 + t~,

(17)

Yl, Y2 are the energy fractions of the scattered electron and positron defined in Eq. (14). Expressions (17) agree with the results of Ref. [8] except for a simpler form of l C ( q _ q + ) . As for Eq. (16) it has an evident advantage in comparison to the corresponding formulae given in Ref. [8]. Let us note that the remaining factors/C(p, q) could be obtained from the ones given in Eq. (17) using relations of the following type: I C ( p _ q _ ) (Xl, x2, AI, B 2 ) = 1C(q_p_ ) (x2, Xl, A2, B1 ).

(18)

Note also that terms of the kind m 4 / ( B2C 2) do not give logarithmically enhanced contributions, and we will omit them below. The denominators of the propagators entering into Eqs. (16), (17) are Ai = ( p - - ki) 2 - m 2,

A = (p_ - kl - k2) 2 - m 2,

Bi = ( q - + ki) 2 - m 2,

B = (q_ + kl

Ci = (ki _ p + ) 2 _ m 2,

C = (kl -4- k2 - p+)2 _ m 2,

Di = (q+ + ki) 2 - m 2,

D=(q++kl+kz)

+ k2) 2 -

m 2,

2 - m 2.

(19)

A.B. Arbuzovet al./Nuclear PhysicsB 483 (1997)83-94

89

For further integration it is useful to rewrite the denominators in terms of the photon energy fractions Xl,2 and their emission angles. In the case of the emission of both the photons along p_ we would have A m--~ = - x l ( 1 + zl ) - x2( 1 + z2) + xlx2(zl + z2) "F-2XlX2ZV/-~COS q~,

Ai

V

(20)

= - - X i ( 1 -]- Z i ) ,

where zi = (eOi/m) 2, qb is the azimuthal angle between the planes containing the space vector pairs (p_, kl ) and (p_, k2). In the same way one can obtain in the case

kl ,kzl[q-: B = Xl(1 + y~z,) + x'-~2(1 + yZlz2) + x,x2(zl + z2) + 2XlX2z~/~cos ~b, m 2 Yl Yl B--L/=x/(1 -k- y2zi). m 2 yj

(21)

Then we perform the elementary azimuthal angle integration and the integration over zl, z2 within the logarithmical accuracy using the procedure suggested in Ref. [ 10] : ZO

ZO

2~r

a=ma f dzl f dz2 f ~-~-~a. 0

0

(22)

0

The list of relevant integrals is given in Appendix A. In this way one obtains the differential cross section in the collinear region as do-col1

=ce4La~2sd3q+ d3q-qoqO dXl l +X2 dx2 x (l

791'2)(~r~I~(L+21)glg5

+(y2 + r 4) In xzr2 + x,x2(y - xlx2) - 2rigs] [Bp_p_t3p_p_ + Bp+p+6p+p,]

xly

J

+.--.-~_2 1 [~ (L + 21+41ny)glgs+(Y 2 +r4) lnxlr~ +XlX2(y_xtx2)_2rlgll yr I x2y ×[Bq_q ~q_q_ d- Bq+q+~q+q+]Jr-Bp_p+~p_p+[(L -q-2l)glg2rlr2 - 2gl-r, 2g!]r2J +Bq_q+6q_q+ ](L + 2l + 2 ln(r, r2)) gig2 - 2 gl - 2g2[ / rlr2 rl r2J +[Bp q Sp q +Bp~q Sp+q_]l(L+2l+21nyl) gig3 .... _ rlyltl

2g__~1_ 2 g___2_3[

rl

yltl J

q-[Bp~q~6p+q+ +Bp_q+Sp_q+][(L+21+21ny2) g, g4 2 gl - 2 g4 ] } rlY2t--"-~-- r-'T ~ J " (23)

A.B. Arbuzov et al./Nuclear Physics B 483 (1997) 83-94

90

We used the symbol 791,2 for the interchange operator (791,2f(xl, x2) = f ( x 2 , xl )). We used the notation (see also Eq. (17) ) l = In

(°4)

,

g5 = y2 + r~,

(24)

where 00 is the collinear parameter. The delta function 8p,q corresponds to the specific conservation law of the kinematical situation defined by the pair p , q (see Table 1): ~p,q = ~(4)(r/2p+ + r/lp- - ,~lq- - A2q+). Besides, we imply that the first photon is emitted along the momentum p; and the second along the momentum q (p, q = p_, p+, q_, q+). These 8 functions could be taken into account in the integration as is done in the expression for dI(~7, ,~) (see Eq. (15)). Finally, we define

(•2s

alt

Bp,q = ~,,~lt -k- --rl2s + 1

)2

t = (p_ - q_)2.

,

(25)

3. Contribution of the semi-collinear region We will suggest for definiteness that the photon with momentum k2 moves inside a narrow cone along the momentum direction of one of the charged particles, while the other photon moves in any direction outside that cone along any charged particle. This choice allows us to omit the statistical factor 1/2!. The quasi-real electron method [9] may be used to obtain the cross section do "so-

d3k2( )UP- 6, R ,

d3q-d3q+d3kl

°(4

32s-~

q-q+

1

)Uq_

(26)

+ Ep~ ~p~Rp+ + Rq + )Uq+ ~q+Rq+~. p+k2 q-k2 qq+k2 J -

We omitted the terms of the kind m 2 / ( p _ k 2 ) 2 in Eq. (26) because their contribution does not contain the large logarithm L. The quantities entering Eq. (26) are given by V-

s kip+ • k i p -

-~

st

+

blt

kip+ • k l q -

klq+ • k l q -

+

tt

t

kip+ • klq+

kip- • klq-

U

(27)

klq+ • k i p - "

V is the known accompanying radiation factor; the )Ui are the single photon emission collinear factors: )up_ = )Up+ -

g2

x2r2 '

)Uq-

=

y2 + (Yl + x2) 2 x2(Yl "}- X2) '

)Uq+=

Y22+ (Y2 + x2) 2 x2(y2 -3WX2)

The quantities Ri read Rp_ =R[sr2, t r 2 , u r 2 , s ' , t t , u ' ] ,

Rp÷ = R[sr2, t,u, st,ttr2,utr2],

(28)

A.B. Arbuzov et al./Nuclear Physics B 483 (1997) 83-94 eq_ =R[s, tt-L,u, s t t l , f f , u ' q ] , Yl Yl Yl

gq+ =R[s,t, u t 2 , s ' t 2 , t ' t 2 , u ' ] , Y2 Y2 Y2

91 (29)

where the function R has the form [ 12]

R[s, t, u, s', if, u'] - ~ [sst(s 2 + s '2) + tfl(t 2 + if2) + uut(u 2 + u,2)] - ss'tF s=(p+ + p _ ) 2, s'=(q+ + q _ ) 2, t = ( p _ - q _ ) 2, ff=(p+_q+)2, u = ( p _ - q + ) 2 u~=(p+_q_)2.

(30)

Finally, we define

t3p_ =~(4)(p_r 2 + p+ -- q+ -- q_ -- kl), ~p4 = ~(4)(p_ + p + r 2 - q+ - q - - kl), ~q_=6(4)(p_ + p + _ q + _ q

Yl+X~

kl),

Yl ~q+ =~(4)(p_ _{_p+ _q+y2+x.._________2 q- - k l ) . Ye

(31)

Performing the integration over angular variables of the collinear photon we obtain do-sc = °~4L d3q_ d3q+ d3kl { 16sTr3 qO_qOk0 dx2V_]~,p_[Rp t~p_ + Rp+~p+]

+~2](~q+Rq+t~q+q-~--7]~q-eq-~q-} •

(32)

To see that the sum of the cross sections, (23) and (32) do"sT = do-c°ll +

dO1 \ dOl J '

(33)

does not depend on the auxiliary parameter 00. We verify that the terms L . l from Eq. (23) cancel out with the terms

Lk°q° f 6ol klqi

- L . l,

(34)

which arise from 16 regions in the semi-collinear kinematics.

4. Numerical results and discussion We separated the contribution of the collinear and semi-collinear regions using the auxiliary parameter 00. By direct numerical integration according to the presented formulae we had convinced ourselves that the total result is independent on the choice of 00. It is convenient to compare the cross section of double hard photon emission with the Born cross section

92

A.B. Arbuzov et al./Nuclear Physics B 483 (1997) 83-94 O"B°rn = o~2'77" cos~P0

f

2

\T-L-Sc/

dc.

(35)

-- COS~#0

For illustrations we integrated over some typical experimental angular acceptance and chose the following values of the parameters: @0 = zr/4,

v/s = 0.9 GeV,

L = 15.0,

A1 =

0.4,

A = 0.05,

00 = 0.05,

l = -7.38,

(36)

where A1 defines the energy threshold for the registration of the final electron and positron: qO > eth = e a t . Note that the restrictions (7) and (12) on O0 (z0 = exp{L + l} >> 1) are fulfilled. For the chosen parameters we get o-coil O "B°rn =

1.2 mkb,

o-SC o.Bor-------~ X 1 0 0 ° ~ = 0 . 8 1 °~o,

- -

o-Born

× 100% = - 0 . 2 5 %, o-SC @ O"c°ll

~O"t°t --

trBorn

X 100% = 0.56 %.

(37)

The phenomenon of a negative contribution to the cross section from the collinear kinematics is an artifact of our approach. Namely, we systematically omitted positive terms without large logarithms, among them we dropped terms proportional to l 2. The cancellation of l 2 terms can be seen only after adding the contribution of the non-collinear kinematics (when both photons are emitted outside narrow cones along charged-particle momenta). The non-collinear kinematics does not provide any large logarithm L. Both quantities tr c°ll and o-sc depend on the auxiliary parameter 00. We eliminated by hand from Eq. (23) the terms proportional to l and obtained the following quantity: bare

°'c°l----~l× 100% = 1.43 %.

o-Born

(38)

This quantity corresponds to an approximation for the correction under consideration in which one considers only the collinear regions and takes into account only terms proportional to L 2 and L (all terms dependent on 00 are to be omitted). Having in mind the cancellation of 00 dependence in the sum of the collinear and semi-collinear contributions, we may subtract from the value of the semi-collinear contribution the part which is associated with l: bare bare x O'sc = O-se -+ (O'coll -- O-coil ), .bare

°so

o-Born

x 100%=-0.87%.

Looking at bare quantities one can get an idea of the relative impact of the two regions considered. We see that at the precision level of 0.1% the next-to-leading contributions of semi-collinear regions are important.

A.B. Arbuzov et al./Nuclear Physics B 483 (1997) 83-94

93

20

1E5 ¸

½

.................

A~=(.,.%

. . . . . .

A1=(,,.3

o"~',._ | 0,

A1=(,.1 ,\ \ , "%

\ *,. '%

B-

0 0.001

0.01

0.

A Fig. 1. The ratio O'collbare//°'B°m in percentas functionsof d for ~'0 = ~r/4 and differentvalues of Al. In Fig. 1 we illustrated the dependence on the parameter A of the bare collinear contribution for different fixed values of A1. A large increase in the region of small d corresponds to an infrared singularity, which will be cancelled after adding contributions of virtual and soft photon emission.

Acknowledgements

We are grateful to S. Eidelman, G. Fedotovich, E Franzini and G. Pancheri for fruitful discussions. For the help in numerical calculations we thank I.V. Amirkhanov, T.A. Strizh and T.E Puzynina. We are also grateful to INFN, Parma university for hospitality. The work was partially supported by INTAS grant 93-1867 and RFBR grant 96-02-17512. One of us (A.B.A.) is thankful to the INTAS foundation for financial support via an ICFPM grant.

Appendix A

We present here the list of integrals (see Eqs. ( 1 9 ) - ( 2 2 ) ): A2 AEAI

Lo [1 x2 r2 XlX2] XlX2r2 ~Lo + In x i y - 1 + Y ,

1 _ LO [1 x2r21] AA1 XlXErl -~Lo + In x l y J ,

m2 -A-A 2 =

Lo x2x2rl ,

A.B. Arbuzov et aL/Nuclear Physics B 483 (1997) 83-94

94

1

L~

AIA2

XlX2

Lo=lnzo-L+l,

--

1

AIB2

-

Lo --(Lo+21nyl),

ylXlX2

/=In

( ~ 02)

,

(4e2 '~

L=In\m2j.

(A.1)

The remainingintegrals could be obtainedby simple substitutionsdefined in Eqs. ( 19)(22). References [ I] S.I. Dolinsky et al., Phys. Rep. 202 (1991) 99. [2] A. Aloisio et al., preprint LNF-92/019 (IR); also in The DAqSNE Physics Handbook, Vol. 2, 1993; E. Drago and G. Venanzoni, KLOE MEMO 59/96; P. Franzini, in The Second DA(bNE Physics Handbook, ed. L. Maiani, G. Pancheri and N. Paver, Vol. 2, p. 471, 1995. [31 A.B. Arbuzov, V.S. Fadin, E.A. Kuraev, LN. Lipatov, N.P. Merenkov and L. Trentadue, preprint CERNTH-95-313, hep-ph/9512344, to appear in Nucl. Phys. B; S. Jadach and O. Nicrosini (conv.), Event generators for Bhabha scattering, CERN Yellow Report 96-01, Vol. 2, 1996. [411 W. Beenakker, E.A. Berends and S.C. van der Mark, Nucl. Phys. B 349 (1991) 323. [5J R. Budny, Phys. Lett. B 55 (1975) 227; M.L.C. Redhead, Proc. Roy. Soc. 220 (1953) 219; R.V. Polovin, ZhETF 31 (1956) 449; F.A. Berends et al., Nucl. Phys. B 68 (1974) 541; F.A. Berends et al., Nucl. Phys. B 206 (1982) 61. [61 A.B. Arbnzov, E.A. Kuraev, N.P. Merenkov and L. Trentadue, Nucl. Phys. B 474 (1996) 271; A.B. Arbuzov et al., JINR preprint E2-95-421, Dubna, 1995, to appear in Yad. Fiz. {7] EA. Berends and R. Kleiss, Nucl. Phys. B 228 (1983) 537. [8] F.A. Berends et al., Nucl. Phys. B 264 (1986) 243. [9] V.N. Baler, V.S. Fadin and V.A. Khoze, Nucl. Phys. B 65 (1973) 381. [10] N.P. Merenkov, Soy. J. Nucl. Phys. 48 (1988) 1073. [11] E.A. Kuraev and A.N. Peryshkin, Yad. Fiz. 42 (1985) 1195. [12] F.A. Berends et al., Nucl. Phys. B 206 (1982) 59; Phys. Lett. B 103 (1981) 124.