On an exact calculation of the lowest-order electromagnetic correction to the point particle elastic scattering

Nuclear Physics 13127 (1977) 242 258 © North-Ilolland Publishing Company

ON AN EXACT CALCULATION OF THE LOWEST-ORDER ELECTROMAGNETIC CORRECTION TO THE POINT PARTICLE ELASTIC SCATTERING D.Yu. BARDIN and N.M. SHUMEIKO * Joint Institute for Nuclear Research, Laboratory o f Theoretical Physics, Dubna, USSR Received 10 November 1976 (Revised 14 June 1977)

A covariant procedure has been developed for separation of the infrared-divergent part of the lowest-order bremsstrahlung cross section. The contribution of this part to the electromagnetic correction to the point particle elastic scattering is calculated exactly for the experiments which measure the energy or the energy and angle of one of the final particles and which do not distinguish between the elastic reaction and bremsstrahlung.

1. Introduction The increase in energy and in measurement accuracy of modern experiments aimed at studying elementary particle reactions requires the calculation of radiative corrections to their cross sections. The radiative correction of order n is the ratio of the cross section of order n in the interaction constant to the cross section in the lowest order (Born approximation). The latter is sometimes called the non-radiative cross section. The radiative corrections in the electromagnetic interaction are often called the electromagnetic corrections (EC). To calculate them, it is necessary to remove the infrared divergences (in addition to the ultraviolet ones). For instance, to find the lowest order EC to the lepton current L

I

L

implies calculating the expression

* Byelorussian State University, Minsk. 242

D. Yu. llardin, N.M. Shumeiko / Hectromagnetic correction

243

Tile first sunnnand of (1) diverges for tile virtual photon nlomentunl k ~ 0, the second at small momenta of the real photon, but tile whole expression (1) is infrared free. The contributions of the first and second terms of eq. (I) are to be calculated separately as characterising physically different processes. Tills requires a uniform regularization for both the terms. For this purpose it is accepted either to introduce an infinitesimally small photon mass X [11 or to use dimensional regularizatiou [21 where the divergences appear as simple poles at n = 4, where n is the dimension of space-time. Natural{y, tile final result contains no regularization parameters. Now we discuss in detail the calculation of tile contribution to the EC to a nonradiative cross section dNo (iV is the number of essential variables) from diagrmns with virtual photon exchange (V contribution) and from diagrams with reM photon emission (R contribution). We shall have in mind the distinction between tile analytic calculation of the contribution of a diagram to the EC by quantum electrodynamic roles and the calculation of the saJne contribution under concrete experimental conditions. Since the kinematics of the V contribution is the same as for the non-radiative process, the experimental conditions are taken into account in their calculations in like manner. The mlalytic calculations precede this consideration and are therefore independent of the latter. As for the R contribution, the differential cross section dN+ 3a corresponds to it. It is integrated up to the wanted dNo over that phase-space part where the bremsstrahlung is experimentally indistinguishable from the non-radiative process. Depending on concrete experimental conditions, the boundary of that part may include either all the three additional dimensions or just some of them. For this reason, it is often impossible to calculate the R contribution analytically and it should be integrated numerically [3]. Further, due to finite resolutions the phase-space boundary can be diffused, and the phase space itself can be distorted by experimental conditions that requires one to use a Monte-Carlo method [4]. However, there exist such types of experiment where the R contribution can be analytically integrated either partly or completely. For instance, one may think of such experimental conditions under which the bremsstrahlung is totally indistinguishable from the non-radiative process. Then the R contribution should be calculated through integrating over the whole phase space of photons. In principle, this can be performed analytically without any approximations. Such calculations are free of many uncertainties, connected with the experimental conditions which are inevitably present in the first type of calculations. In what follows, we shall distinguish between two types of experiment. In the experiment of type I, the background process of bremsstrahlung is to some extent distinguishable from the non-radiative process and is'not considered in the data processing, while in the experiment of type II such a discrimination is either impossible or is simply neglected. Since in the first type of experiments the calculation of the R contribution essentially depends on the experimental conditions, the problem arises of how to cancel

1). Yu. Bardin, N.M. Shumeiko /t:'h'ctlzmta~m'th' correction

244

the infrared divergences i~1 the V and R contributions, calculated in a different way. This problem is usually solved by splitting the R contribution into two parts: the part corresponding to soft photons, the S contribution (k o < ~max), and the one corresponding to bard photons (ko ~ ~-'.)max), the H contribution [5]. Naturally, the infrared divergence is present in the S contribution only. The quantity COmax is determined by experimental conditions. It is just a boundary of the region of isotropically distributed unobserved photons inside which the experimental conditions limiting the phase space of bremsstraldung are not exhibited. As follows from the definition of ~o,m~x, the method of calculation o f the V and S contributions under concrete experimental conditions is the same as that which provides the cancellation of infrared divergences at the stage of analytic calculation prior to consideration of experimental conditions. Thus, the conditions of the type I experiment influence mainly the calculation of the infrared free contribution of hard photons. Under the conditions of the type II experiment, the calculation procedure is the stone for the V and for the whole R contributions. In some simple cases, the R contribution can be integrated straightforwardly, the infrared divergence being cancelled in the sum V + R [6]. However, in more complicated cases, the computational difficulties require the splitting R = S ( ~ ) + H(co)

(2)

to be performed v& introducing an infinitesimal parameter D independent of the experimental conditions. The R contribution, of course, does not depend on it, but S and H do depend [7] on it. This paper reports the calculation procedure for the R contribution to the EC to the point charged particle elastic scattering under the conditions of the type I1 experiment. For definiteness, we consider the reaction 1+1

~ ~ 1 + ~1 ,

(3)

e.g., pe scattering. We have used the splitting (2), the dependence of S on the "softness" parameter w and o11 the photon mass X and the dependence of H on co being factored in temas of the simplest fornr. Our calculation is perfonned without any approximations and in covariant fore1. The paper is organized as follows. In sect. 2 we perform the simple separation of the infrared divergent part from the bremsstr',flllung cross section. In sect. 3 we calculate the soft photon contribution and a part of the hard photon contribution. The other contributions to the EC can be found in ref. [8]. In conclusion we discuss the changes necessary for applying our procedure to type i experiments. In this case our method of calculation resembles the one used in ref. [3] for the exact calculation of the lowest order EC to the process e+e - -->p + p though it differs from it in details.

D. Yu. Bardin. N.M. Shumeiko / Electromagnetic correction

245

2. Separation of the infrared divergence from the bremsstrahlung cross section Tile process 1(/91 ) + 2(k I ) -+ l(P2) + 2(k2) + 7(k)

(4)

in the lowest order in a is described by four diagrams ill which 3' quanta are emitted from external lines. Consider their contribution to the measured cross section of (3) when one of the final particles is detected under the type 11 experimental conditions. For definiteness, we calculate doR(S, X ) / d X and dog(S, Y ) / d Y where S=

2pl " kl ,

X =

2pl " k2 ,

Y= (k 2 - k l ) 2 .

(5)

In the rest frame of particle 2(kl ) (lab system) the invariant Y is determined by the energy, and X by the energy and the scattering angle of particle 2(k2). The phase space is related to the above invariants as follows fdl'- =jrdak2d3kd3~2~(k2+k+p2-kl2k~2o 2ko 2P2o -Pl)

= 4 ~ x s f d X d Y ~ o 8 [ ( Q - k)2 + M2]O(Qo - ko) .

(6)

Here ?'s = $2 - 4m2M2, Q = P l + k l - k2, M(m) is the mass of particle 1(2). When M > m, the invariants in the process (4) in the sequence Y, X or X, Y change in the following limits

O <~ y <~ y1 = k s ~ ( S + m 2 + M 2 ) , 1

Xmi n - 2m 2 [ ( Y + 2 m 2 ) S - V ~ m ~ s ]

<~X<.S-

Y,

~,m = y2 + 41712y .

(7)

2raM ~< X ~< S - Yl ,

Ymin ~< Y ~< Yrnax ,

1

Ymax,min - 2M 2 [SX - 4m2M 2 + x/Xs~x] ,

S-

Yl <<,X<~S,

XX = X 2

4mZM 2 •

Ymin~Y~S-X.

(8) (9)

When m ~>M the limits are defined by formulae (7) and (9). 111 the elastic scattering (3) the invariants which obey the relation So-Xo-

Yo = 0

(10)

change within the limits 0 ~< Yo ~ Ymo,

So

Ylo ~
D. Yu. Bardin, N.M. Shumeiko / Electromagnetic correction

246

Y,o:)~/(So+m 2 +M2),

X,~=S g

4mZM 2.

(11)

Obviously, at S = S O tile limits of Y and Yo variation are the same for any ratio of masses of particles, while for X and X o tire condition Z < X o (region (8)) is 'allowed when M > m. By using the identity transformation, we represent the differential cross section do k of the process (4) in the form dOR--=dOR- da m + d o l RR = d c r ~ + d o ~ R ,

(12)

where do m is the infrared divergent part and do~ is finite as k --* O. The separation o f d o ~R is not unique, in general, but for the type II experiments it allows a unique procedure of its construction which we will present here. For the photon momentum k -* O, d(r R can be written in the form 4~3

dOR Ik--*0 - //.2%/~S

Tx2dF,

(13)

where T = ~ Sp(/~2 + im)

X =t\kl.k

~/a(~l + im) ~[~S p ~ 2 + iM) "Ya~l

k-2Sk

, xel'k

P2"k

+/M) ")'t~ ,

'

and f = +'1, if particles 1 and 2 are of the same sign, and f = - 1 , otherwise. In terms of the invariants (5) and t = (Pl - P2) 2, T and X2 are "T=½(S 2 + X 2 - 2 m 2 t - 2 M

m2zz m2z2 +~Y+2m2]zzl / + SY(X+X+u+zvzl

X2=4[~( f2 [

2Y),

M2

t

M2 +

.v

/j'

S-V-Zvz (14)

where z =

2k2"k,

2kl"k=z+t-Y, u=-2pl"k=Sx t, Y, Sx = S - X. to the leading terms ~l/k 2 in the limit k -+ 0 which just

Zl=

v =-2p2 • k =Sx-

As is seen, in addition produce the infrared divergence, the product TX2 also contains subsequent terms of the expansion in k. A further separation of the infrared divergence will be carried out under the condition that only one particle (2(k2)) is detected,the momentum of which enters into the invariants Xand Y. Therefore we require that T, the numerators of all terms o f x 2 and the factors t -1 and t -2 contain only the invariants S, X and Y. This can be achieved

D. Yu. Bardin, N.M. Shunwiko / Eh,ctromagnetic correction

247

by omitting the invariants z, z I, u and v linear in k from those quantities. A part of IR file cross section (13) resulting from this separation will be used as ool~ for the identity transfommtion (12). Thtls •

do IR -

4a3 TF m d X d Y d3k 5[(Q - k) 2 + M 2] 0(O 0 - ko) ¢rXsy2 ko ,

(15)

where T-][S 2 + X 2 _

2Y(m 2 + M / ) I ,

I

~4R_

m2

m2

z2

d

+ f2

Y+2m2 +

ZZ 1

M2

M2

//2

02 +

(X +f

X + OZ 1

S OZ

S) ~Z

y + 2M 2) UO

/

3. Calculation of the soft and hard photon contributions

Now we separate the infrared part in the single differential cross section dolRa/dy. Since do IR diverges as k -~ 0, it should be regularized at small k integration. For this purpose, we use the dimensional regularization of infrared divergences [2] (DR). From (15) we have d a l R _ 4a3

S- Y

f XsY 2

d X TI(Y, X ) ,

(16)

1 rd3k I(Y, X ) = ~rJ l-~-o S[(Q _ k) 2 + MZl O(Qo _ ko) F m .

(17)

dY

Xmin

whe re

Take in (16) the variable v = S X -- Y which varies wifllin the limits 0 <~ v ~< Vmax - 2m1 2 [%/~SXm-

Y(S+2mZ)l

(18)

and split the integration range into two parts: (i) 0 ~ v ~ g

(ii) U ~ v ~ Vmax

where the parameter gis chosen as follows: g > 0 and g<
Y, m Z , M 2 .

(19)

248

D. Yu. Bardin, N.M. Shumeiko / Electromagnetic correction

Then we get df° Y ~ -d Xs 4a3 y2

4°~3 ff~max

do TI n (g, v) + ~S- ~

dos

d°H

(20)

do TI(Y, v) =- --dg + .d. .Y. "

0

The first sununand in (20), doS/dy, infrared-divergent, (as v ~" 0) is a covariant analog of tile soft-photon contribution. The function In(y, u) is the integral (17) regularized by the DR method. The second sumnlmld doH/dy, free of the divergence, (since there v ~> b > 0) is a part of the hard-photon contribution. First, we shall calculate just this integral. Consider the identity /~max 40~3 d°H - 4~3 d d V [ T j ( Y , v ) - - ToJ(Y,O)] + ~ s y 2

dY

Xs y2 F

V

ToJ(Y,O)f i;

Vmax

dv --V '

(21)

where

J(Y, v) = vI(Y, v), TO = ]'1 [S 2 + X 2

J(Y, O) = J(Y, v)lv= o , 2 Y ( m 2 + M 2 )]

(22)

and the index " 0 " means that invariants are taken at the point v = 0, i.e. obey the kinematics of the elastic scattering process (3) (relation (10)). Obviously, S o -= S and also Yo = Y if experimentally just the cross section d o / d Y is measured. The structure of the identity transformation (21) is such that the in tegrand of the first separated integral is finite as v ~ 0. Therefore, in this integral [;may be put equal to zero because of the inequality (19). The final form of (21) can be written as follows do H

d H

- d ° ° ¢~ (6 + 61H) + - ° z

dY

dY rr

.

(23)

dY

Here the first tema with the factorized cross section of the elastic process (3): d o o _ 47r0¢2

dY

(24)

Xsy2 To

contains spiMndependent corrections

= J(Y, O) In Vmax F

'

(25)

Omax

dVv(Y,v) J(Y, 0)I, 0

0

(26)

D. Yu. Bardin, N.M. Shumeiko / Electromagnetic correction

249

and the second term is doll _ 4a3 j~max d_U(T- To)J{Y,v) " dY XsY2 o v

(27)

The ful'lction J(Y, u) is J(g, v) = J[F ll~] = - m Z J l z - 2 ] - m 2 j [ z - ( 2 ] + ( Y + 2m2)j[(zzl ) -1 ]

+f{ XJ[(lIZ)-l] +XJ[211] -SJ[Z-I]U +f2 _M2J[u-2] - 7

J[1] + - - - - v

SJ[(IgZl)-I]} J[u-ll

'

(28)

wh e r e

J[A] -

v

d3k

7r

ko

(29)

8 [ ( Q - k) 2 +M 2] O(Qo - k o ) A .

Integrals (29) are defined by the formulae J[z - 2 ] = J [ Z l 2 ] = 1/m 2

,

(30)

9 in X/'Xm + y J[(ZZl)-l]= 2 L m - x / ~ m ~m-g Jl(uz)-l] : L x = ~

1

(31)

X +~ffXx In ~ ,

v S - Y + V'-~x J [ z l l ] =~"~X l n s - Y - V ~ x , '

(32)

A x = (S - y)2 _ 4m2~.,

7=M2+o, o

j[z-ll=~slnS

(33) v+v/-~s S - v'-x/-As '

A s=(S-v)

2-4m2r,

(34)

1 in S + x/~-s, J[(uz l ) - l ] = Ls = ~ s s

(35)

J[t1-2 ] = 1/M 2 ,

(36)

U2

(37)

Jill

-

j[tt_ l ]

~

,

o

o+ r+2j142 +'~y

),,y = (v + y)2 + 4M 2 y .

(38)

250

D. Yu. Bard#t, N.M. Shtm,eiko / Kh'ctromagnetic correction

J(Y, 0) is J(Y,O)=2{(Y+2m2)Lm- I +f(XoL°

The function

SLs)+fZ[(Y+~2)LM-I]), (39)

where 1 , V'~6y + r

_

z.~, -~/~,,, ~-

(40)

r.

The integral (26) is calculated straightforwardly and the result reads

Eftv~-~:'~-4

~," :.r{~-~

L~, ~/~

Xo 1 -

,i~__~. ~v~=x.~ ~ ~/~+Xo

]

qbi~X _JrV/~max'~ ,tN/~OX + %/~max~ cl)i 2V~X

__(i)[

2N/~X ~1

_ _1[

o o t ~kxLx

~g/~x hlX° + ~

Xo]n°max+m2~ ~(li/2 Xmin +4£mZ M2 ] + Xmin ~ n l i n

+H¢(Xo, X°, Xo,-1)

Xo +k/~x~ 11!2Xo ---~N/r~]

H¢(S,Xs,-S-2m2,-1)} (41)

Here

4m2Vmax ,

~max = ~0 O(x) =

~,min = Xmi 2 n - 4m2M2 .

/dtlnll--tl o

t

is tire Spence flmction, and the function

He is defined

as:

+ aVmax + X/~()t + 2aVmax + V2max)l/2

2X 4

(42)

251

D. Yu. Bardin, N.M. Shumeiko / Electromagnetic correction

At b 2 = 1, t] have the form tl = - b ,

t2 =

s j = ( 1 , 1,

b(s + X/X) - 2a s -X/X

1,-1),

'

t3

- b ( s - X/X) + 2a =

s+x~X

t4 = b

'

"

(43)

t m = [(X+2aVmax+V2x) l / 2 - x / ~ ] / v m a x .

(44)

It is not difficult to also calculate tile integral (27). However, in practical calculations, it is convenient to add the cross section do~/d Y to the other hard-photon contribution d o ~ / d Y , given as a single integral in ref. [8] and then to integrate both numerically. Consider now the soft photon contribution doS/d Y. Due to the condition (19), T = To in the first integral (20) a n d / n ( y , v) is calculated neglecting v as compared to other invariants. Then we have doS _ d°o ~ ~soft dY dY 7r '

(45)

6 s°ft = f d v / n ( y , V) o

(46)

where

a n d / n ( y , v)follows from (17)by the replacement [2] dak

d n - 1K

2rcn/2-1

(2.)Sko-~f (2~-ZTKL-(2n)._,m(½n_1)flKm-3 dlKI f

n

sinn-SOdO.

o

(47)

First, using the 5-function, we integrate/n(y, v) over Ko = IKI in the system Q = 0 and write explicitly the angular dependence of FIR: 2 In(r'v)=(2X/~)n-OP(½n

l(V]n-4 - 1)u\~!

where F(a, ~)= [ [ K I 2 F m ] o ~l=v/2M -

Y + 2rn2 + ~:~o(~ -

+

["

m~ K]o( 1 _ ~1~)2 Xo

+f t p}o(~Z~x~)~ ~3~) 2

Xo Mklo(1 - 31~)

t +1 f da f (1 - ~ 2 ) ( n - 4 ) / 2 F ( c ~ , ~ ) , o -~ (48)

S p}o(1

S M k 2 0 ( 1 - ~2~)] + f 2 [ _

MPlo(1 - 13p~)d "

m2 K~o( 1 _ 32,~)2

~s~) 2

M2 p20(1 - 3p~) 2

1

(49)

D. Yu. Bardin, N.M. S h u m c i k o / A'h'ctronmgnetic correction

252

1,1 the equality (48) ~ = cos 0 and the integral over ~emeans the Feynmml parametrization of propagators when required. In formula (49) [Ji = Ikil/kio,

k3=kl~+k2(l

- 00,

pX=plct+k2(1

t5x = Ipx I/Pxo,

Ps = P Ict + k I ( 1 - ~ ) , {Jp =

~),

{3S = Ips I/Pso ,

(50)

Ip I I/plo .

Next we integrate (46) over v and expand the result in a Laurent series around n = 4: ~soft = 6~ + 8 S ,

(5 I )

where 1

+1

6 ) : ( 'n - ~l n 2 V ~ + ½ C + ~ 1 4 n-~-I ) lf/ d ~T

27

6s = 1 4

do:

0

-,

d~F({~,~),

(52)

d~ ln(1 - ~2) F(ot, ~).

(53)

--1

As is clear from (52), the infrared divergence is factored in 8) as a pole at n = 4. The transition to the standard regularization scheme [9] is achieved by the change

( n - 4 ) -1 + l C - l n 2 x / ~

(54)

lnX,

i.e. the two schemes are equivalent [2]. By using (54), we calculate in (52), (53) the other integrals. In the covariant form we have 8~

= J(Y,

8S

= J ( Y , O)In 2 + ½(SL S + Xo L ° ) + S . ( Y

O) ln 2 ~ '

(55)

+ f ( S o , ( X o , )t ° , S, Y, X ° )

+ 2m 2, ~-m, S, X 0

-- 2 M 2 , X0 )

S . ( S , Xs, Xo, Y,.X° )

t

X0 +~

-

_[

2V~ °

'~

,/_

2V'X°

~']

[~'!,-v'~:Srol

WX} + xdJ

~ll J +/= Itl

+ (Y + ~ : ) LM

{I}[- 2V"Xs

~~s+slJ

+ ~Y+2M 2

2x/-xo I ~ ( ~ o _

2x/f °

?ZM2) -'~'

S - 2,/2s

(~,~ 2x/~°

F*(-2X/-Xs "~

L ~,~;--sl

)]}

o+g+2M 5

,

(56)

D. Yu. Bardin, N.M. Shumeiko / Electromagnetic correction

253

where

ZvAISI{In S2-Nfr~2.t_VA2 S a , ( s l , Xl, s2, s3, X3) = ~ _ ~ - - - In $2

(t- tl)(t-

t3)

( t - t 2 ) ( t - t4)

4 + ~ ( i,j = 1 -

dp

t -

1)i+'sj(~ijlln2(t

ti

t=

- ti)+(1-~)ij)Ihl(ti--ti)hl(t-ti)

tu

(57)

Parmneters for the function (57) are given by the formulae *

tl,2 = 2M 2 S1 S 2 -----~1 _~ 2

S3 - 2 M 2

~2

,

13,4 = 2M 2 -s1 +-N~I $2 + V ~ 2 +$3 +2M2 -V/~-2, 1 tt = --X/X2 + ~

[s2s3 + 2M2(sE

Sl)]

tu = X / ~ s - x / ~ z ,

(58)

)t 2 = s 2 - 4 m 2 M 2 .

Thus, the infrared divergence and tile dependence on the "softness" parameter-0 have been factored in S and H contributions in terms (55) and (25) of the simplest form. Naturally, the total R contribution does not contain g which is merely a regularization parameter. Indeed, summing ~- and 6~ we find their contribution to the cross section daXR _ do o a 6 ~ , dY dY n

(59)

where ~

= ~ + 6as = J ( Y , 0)In Vmax

2MX

(60)

That is, the infrared divergent part of the R contribution (59) corresponds to the contribution of soft and hard photons which in the second-type experiments are physically indistinguishable. Now let us show how the photon mass X vanishes when (59) is summed with the contribution of the virtual photon exchange diagrams d o v / d Y . The latter can be

* Here we present the functions SO in a form somewhat different from that given in ref. [9 ], as more suitable for obtaining approximate expressions.

D. Yu. Bardin, N.M. Shumeiko / Electromagnetic correction

254

represented as a sum of tire linite do~)/d Y and infrared divergent dolJt/d Y parts. The cross section do~zR/dy is usually written in tile form 1101 douR _ doo o~ 8 ip, , dY dY rr

(61)

wh e re ~IgR = K(kl, k l )

K(kl,k2)+2f[K(Pl,kl)-K(pl,k2)

+fZ[K(pt,Pl)

] (62)

K(Pl, P2)] •

The function K(PA,PB ) is defined by the integral [10]

K(PA,PB) = PA ' PB

/dc~ -p~ _-2TIn X-W" 0

(63)

Pa

where p~ = 0~pA + (1 -- a) PB. Obviously, K(PA , PB) = K(PB, PA)- Since PA and PB are time-like vectors, p~ is also a time-like vector, i.e. p~ < 0, and the integral (63) has no poles. If we represent (63) as a sum of two terms

K(PA,PB) = L ( P A , P B ) + P A ' P B l n - ~ /J(PA, PB) , A-

(64)

where da -p~ ~ In

L(PA, PB) = PA" PB 0

P~

/-/(PA, PB) =

~ '

/

d0~ 2T,

0

Pa

(65)

then 8bR also splits into two terms 8 L and 8~, containing L and/~ functions, respectively: 81¢R = 8 L + 8~;. For 8 L we have

8L = L(kl ' kl ) _ L(kl ' k2 ) + 2f[L(p 1, kl ) _ L(Pl ' k2)] + f 2 [L(p I , P l ) -- LI201 , P 2 ) ]

,

(66)

where

L(kl,kl)=O,

M2 L(Pl,Pl)=lnm2 , Y + 2m2V [ 2V~m

_ dp[ 2X/~m 1] ,

1

2X/~y

_[

2X/X~

t,/g

D. Yu. Bardin, N.M. Shumeiko / LTectromagnetic correction S L(Pi, kl)==Ls z

In

2MXs m IN/~8(M 2 - m 2 ) + S(M 2 + m 2 ) - 4m2M 2 [

F [2M 2 - S - V ~ S

S

255

1

[2m 2

-S+~,,Ixs~]

s

S-

stJ"

(L(Pl, k2) can be obtained from L(Pl , kl ) by the substitution S ~ Xo). Using the definitions (64), (65) and calculating the/a functions

u(kl, kl) = -1/m 2

/-t(pl,Pl) = -1/M 2

la(kl k2) = - 2 L m

U(Pi , P 2 ) = --2LM ,

la(Pl, k l ) = - L s ,

U(Pl, k2) = - L °

,

(67)

for 6 x we iliad

fix = j ( y , 0 ) I n ---~.

(68)

m

As is seen, the sum o f f x and 6XR does not contain X and equals

6X = 6X + 6)t =J(Y, 0)Ill Omax 2raM

(69)

Finally, we trace the main steps o f the calculation of the cross section d o I g / d X in analogy with dolR/dy. From (15) we have do IR = 4o~_~_3 (~rnax

dX

Xs Jo

dv

(Sx - v) 2

TI(X,

U)

(70) '

where Omax = S x - Ymin ,

(71)

and T a n d I(X, v) are given by formulae (15), (17) with Y= S x - v. Separating in (70), as in (20), the contributions of soft and hard photons, we write do H _ doo ~.(~- + a lH) + _d _o n dX dX 7r dX '

(72)

where * doo _ 4rr~ 2 dX XsYg T ° ' To = ~1[$2 +X2 - 2Yo( m2 +MZ)] ,

* When the cross section do/dX is measured, X 0 = X.

Yo = YIv=o = S x ,

(73)

256

D. Yu. Bardin, N.M. Shumeiko / l:'lectromagnctic correction = J(X, 0) In Vmax -v ' ~H =

fmax

dv

V

0

(74)

[J(x, v) J(X, o)1,

do~. 4cd ~ma× dv[( dX

XS o

v

T

(75)

~x]j(x,v )

Sx

(76)

v) z

The quantities J(X, v) and J(X, 0) result from J(Y, v) and J(Y, 0) if we put Y = S x - v in J(Y, v), and Y = S x in J(Y, 0). For 8~ we obtain goH=H*(Sx+2m2,?t°rn,

Sx

2 V~m + S x In ~ + f [ H m ( X ,

2m 2,

Xx, X

X/Xm - Sx

1 ) + l n 2 v,,mln+Ymin. ~/V7-. N/Xmin Ymin 2m 2, 1 ) - H ¢ , ( S , X s , - S

1-%

2m z - 1 ) ]

,

m

+

-

~,x/xk - SM/

+ ®t,/

TsMl

'

where Xmax = )t° - 4M z Vmax, ~,min = Y2min + 4m2 Ymin, SM = SX + ~ 2 . The soft photon contribution is calculated and the photon mass X is cancelled in the same way, as for doS/dy. In the fommlae obtained for d o S / d y above, only the substitutions d o / d Y -+ do/dX, In(y, v) ~ In(X, v), J( Y, O) ~ J(X, 0), Y ~ S X should be made and Vmax should be detemlined by eq. (71). In this way, we find the cross section do~a/dX in the whole kinematic region of X (9) for m ~> M. In order to obtain dolR/dX for M > m in the type II experiment, it is necessary to consider also the contribution of the region (8) which has the form do~ _ 4a 3 V/max dX

dv

ks Vdmin ( S x - 0)2

TI(X, v)

(78)

D, Yu. Bardin.N.M. Shumeiko / l:'h'ctroma.~,m'ticcorrection

257

whe re

SX

Omax,min =

Ymin,max -

(79)

The remaining part of R, dol~/dX, is again given by formulae (21), (22), (25) and (28) of ref. [8] by changing the integration over X to Y in the limits (9) for m ~> M and in tile limits (8), (q) for m < M. In the cross section (78) representing the part of the hard-photoll contribution, we separate off (as in do H) the spin-independent correction ((53H):then do H _ d o o a 6 ~ + -do4" dX dX n dX '

(80)

where Omax

,. = f Vmin

dvj(x, v),

(81)

V

d°~ - 4a3 VmaxdoI( T .~Xl J(X, v) . dX Xs f v Sx v) 2 -

(82)

Vrnin

Taking the integral (81), we have Omax

6~ = J(X, 0)In Vmax + f Omin

d~ [J(X, v) - J(X, 0)l

Umin

=J(X, 0)ln Vmax + 6 ~ - 6-~1 .

(83)

Omin

Here 6~ is given by (75), (77) and ~5-~1follows from 6~ by changing Omax to Omin and Ymin to Ymax" It is convenient to add tile cross sections (82) and (76) to do~/dX for the subsequent numerical integration. Note again, that the contribution (78) and the one from the region (8) to do v differ from zero only for M > m.

4. Conclusion

We have calculated the contribution to the lowest order EC to the process (3) from soft photons (45)-(58), the spin-independent part of the contribution from hard photons (45)-(58), the spin-independent part of the contribution from ruing the X dependent terms in the V and R contributions. The calculation has been performed in the covariant form without any approximations. The latter has caused a rather cumbersome result of the calculations. However,

258

D. Yu. Bardin, N.M. Shumeiko /Electromagnetic correction

the exact formulae are certainly important, as from these any approximate results call be obtained. Covariance is "also useful for straightforward application of the c',dculated result to other cases, e.g., for the EC contribution to the deep inelastic ~2N scattering in the parton model [11]. The final expressions do not depend on ~, the covariant analog of co;however, this independence o f the parameter, that separates the contributions of soft and hard photons, takes place only in the type II experiments in which the processes (3) and (4) are indistinguishable throughout the whole phase space. In the type I experiments, the whole second term do H in (2) corresponding to the hard photon contribution should be calculated with d o ~ . Also, it should be represented in the form o f the entirely differential cross section in order that the procedure o f calculation adequate to the experimental conditions can be used. The contribution of soft photons dcrs (the first term in (20)) is calculated by using the same procedure as for the V contribution, that provides the cancellation of ~. Note, that do s is of the same form as under type II experimental conditions. In conclusion we stress that the considered performed allows a simple generalization to the bremsstrahlung by particles with other spins. For instance, for the process 1 + 7, it is sufficient to change T = T (1/2) to ½T(°) = S X - Y M 2 in the 0 + ½ --*0 + ~formulae we have obtained. We are grateful to G.V. Micelmacher for useful discussions.

References [1] R.P. Feynman, Phys. Rev. 74 (1953) 1430. [2] W.J. Marciano and A. Sirlin, Nucl. Phys. B88 (1975) 86; W.J. Marciano, Phys. Rev. DI2 (1975) 3861~ [3] F.A. Berends, K.J.F. Gaemers and R. Gastmans, Nucl. Phys. B57 (1973) 381; B63 (1973) 381. [4] D.Yu. Bardin, G.V. Micelmacher and N.M. Shumeiko, JINR communication E2-7235 (1972). [5] D. Yennie, S. Frautschi and H. Suura, Ann. of Phys. 13 (1961) 379. [6] K.O. Mikaelian and J. Smith, Phys. Rev. D5 (1972) 1763. [7] J. Kiskis, Phys. Rev. D8 (1973) 2129. [8] D.Yu. Bardin, O.M. Fedorenko and N.M. Shumeiko, JINR communication P2-10114 (1976). [9] D.Yu. Bardin and N.M. Shumeiko, JINR preprint P2-10113 (1976). [10] J. Kahane, Phys. Rev. 135 (1964) B975. [11] D.Yu. Bardin and N.M. Shumeiko, JINR preprint P2-9940 (1976).