A NOTE ON THE EXPONENTIAL FACTOR IN QED RADIATIVE CORRECTIONS
J.P. LEROY and J. MICHELI Laboratoire de Physique Thdorique et Hautes Energies*, Universit~ de Paris-Sud, 91405 Orsay, France
Received 23 July 1979 (Final version received 29 November 1979)
The dependence of the exponential factor on the fou r -momen tum carried by soft photons is studied. The dominant singularities with respect to q are taken into account to all orders in ~ and factored out. A method is proposed to evaluate the non-singular factor perturbatively.
1. Introduction
The infrared-divergence problem has been extensively studied in the literature [1-5]. The main result is the exponentiation of the infrared corrections established by Yennie, Frautschi and Suura, which states that the infrared-divergent contribu- tion can be summed up to all orders in a while the remaining parts are treated perturbatively. Let us recall briefly what happens. Consider the differential cross section do./dE for some process, E being the energy taken away by the undetected photons. The naive perturbation expansion would give
do- - - ~ h o b ( e ) + o~h l ( e ) , de
the ho term corresponding to no photon being emitted. As a result of the exponentiation procedure the cross section actually reads:
d~_ hou,(E) + a~ql(E) de
where q~ represents the effect of the emission of an arbitrary number of soft pho- tons and ah~ is obtained from the convolution of ~, with the cross section for emitting "hard" photons. In this case an explicit form of ~ can easily be computed [1]. An important feature of this result is the replacement of the ~ distribution by a
* Laboratoire associ6 au Centre National de la Recherche Scientifique.
141
142 J.-P. Leroy, J. Micheli / O E D radiative corrections
smoother function. A similar phenomenon takes place when one looks at the differential cross section with respect to the four-momentum q. However it is impossible in this case to obtain the corresponding ~(q) in closed form. In the papers quoted above this difficulty is overcome at the cost of introducing a fictitious dependence on a parameter, the value of which is to be chosen in a convenient way for each experimental situation. Our aim in this paper is to try to avoid this by writing qb as the product of two terms one of which is known exactly (to all orders in c~), while the second can safely be expanded. In sect. 2 we exhibit the singulari- ties of qb and its form in their neighbourhood. In particular we show that qb can be computed exactly in the neighbourhood of q = qo. We then show that, once the singularities are factored out, the remaining function F satisfies a set of four integral equations. In sect. 3 we show how these equations can be solved pertur- batively.
The q-dependence of the function obtained here in a general way is the same as the one reached by Chahine [6] by using a specific approximation to the infrared emission factor.
2. Light-cone behaviour of the exponential factor
2.1. Recalling the standard treatment of the infrared problem
We first briefly recall, for the sake of definiteness, the general features of the well-known treatment of the infrared problem by Yennie, Frautschi and Suura (YFS) [2]. We denote by pi(p'~)the momenta of the incoming (outgoing) charged particles; by Ki(KI) the momenta of the eventual observed incoming (outgoing) neutrals, and by ki the momenta of the unobserved produced photons. Then the differential cross section for the process is given by
dE- l im H d3pl I-I d3K~ ~ ~ n.ll f [ fi d3ki ]
, , ( , z ) xo'.({pz},{pi},{K.},{Kk},{ki})6 ~pj+2K'~ +2k~-2K.- p , . (2.1) - j k l n
As shown by YFS, the infrared-divergent part of the corrections due to virtual photons can be factorized to give
.3 , d3K , 1' n O P i r l i
dye- lm lI ~gT~,, II ~-/7772-,~,~ exp [2,~B({p,.}, {p~}, A)] x.~o , E(pg) i E(Kj)
d3kl ~ r r a
x,~ ~'~ p'i 2 g j + E k ~ - 2 p m - g k . (2.2) i I m
Z-P. Leroy, Z Micheli / OED radiative corrections 143
The real photons are dealt with by rewriting (cfi ref. [2]):
t3n [{p,}, {p;}, {K,}, {K'} , {kk}]
= 2 2 dd{pm},{p}},{K,},{K'},k~.. .kp]d(G+l)X.. .xd(k,), p - 0 partitions
(2.3)
where the fie remain finite when the ki go to zero and the divergence is contained in the S factors. Defining the function qb by
° ° I f d3kj ° 4 r ) X r~O ~ .= • j=l fi ~ / ~ S ( k , ) ( ~ . (j~lkj-q= , (2.4)
we can rewrite (2.2) as
d 3-' "~3K' ~ 1 t" Pi u i Z d3kt . . . . , , d £ - [ I ~-7-77,,[I .-v-s;77..,, Y 2SJ ll-7-flpL{pm},{pi},{gk},{g,},{kt}]_lK, i l:ztpi) j l z~ l~ j )p=op . =
[ , ,+ xr~ {p~},{pj} ,~p~-~p~ y~KI-YK i kl . m i i J
Our aim in this note is the study of the function qb. The quantities involved in (2.4) are defined as
The behaviour with respect to A of the two exponential factors is well-known:
2aB[{p}, {p'}, A ] - aA Log (A) + b ({p}, {p'}),
I d3k t ~ / . x -iky , / ~ inK) e - - a A Log (a)+ b({p}, {p'}, y).
Using these forms in (2.8) leads immediately to
• [{p'}, {p}, tq] = t'~A-4*[{p}, {p'}, q].
We note incidentally that one proves in the same way that ~ is homogeneous of degree - a A in the variables p. This gives an overall energy dimension of (-4) in agreement with naive counting. We can now rewrite ~ in the form
The integration over yo can be performed explicitly and gives
dp[{p'}, {p}, q] __- 2[F(4- ceA )/ (277")4]qo A-4 e -~AC e °~
O(3
1 xRe{ei '~A I t2 dt f df~yil_tx~. ~_ie)4 ~A
0
exp[ Ido lo l ogll i, (2.10)
C is Euler's constant, ~ is a constant (with respect to q) which can easily be obtained from B, but the form of which is not needed at present. We remark that qb is Lorentz invariant, although Y3 and ~ df~k S'(Dk)Log (1 + ie - tffc) separately are not. Let us look at the singularities of the integral in (2.10). The behaviour of the exponential for t going to infinity is easily found to be t -~A. Thus the infinite range of integration in (2.10) leads to no divergence as long as x is non-zero. However x = 0 is not a Lorentz-invariant point so that we can conclude that no singularity will arise from the behaviour at infinity of the integrand. The factor S'(flk) in the exponential is regular. So the exponential can be singular only when the singularity of the logarithm at )7/~ = (1 + iE)/t becomes an end-point of the integral, i.e., for t = ±(1 + ie) (of course, t = - 1 is outside the integration range and will give no trouble). The first factor in the integral in (2.10) is singular when ~f = ( 1 - ie)/tx. Upon integration over fly this will give rise to singularities in t = ± ( 1 - iE)/x. Thus, we see that the only singularity of the integral will be at x = 1, when the two singularities t = 1 + iE and t = (1 - iE)/x pinch the integration contour in t.
To study the behaviour of the integral near x = 1 we rewrite it as
O(3
f t=dtfdf~y[1-i'-txq'~] ~A-4 0
xexp [ - f dflkS~'(f~k)Log (1-ie- t l~ " ~)+ 2irrO(tl~" ; - 1 ) ] , (2.11)
146 J.-P. Leroy, J. Micheli / OED radiative corrections
The first term is no longer singular since the two coincident singularities now lie on the same side of the integration path. The range of the second integral is actu- ally [1, co] because of the 0 function, so that the singularity in x = 1 now appears as an end-point singularity. Taking advantage of the fact that it stems from the contribution of the neighbourhood of t = 1, c~ = 1 , /~ = 1, we can obtain the lead- ing term in (2.12) as
c o
exp [ - y dakS'(f lk)Log ( 1 - i e - O ' / ~ ) ] ff t 2 d t f df~y(1-ie-txO" f ) ~ a 4
0
x [exp (-4i~r2d'(flq)(t- 1)0(t - 1)) - 1].
Performing the fly integration, developing the exponential and integrating finally over t, we then arrive at the following expression for the leading singularity
Finally inserting this in (2.10) we obtain for the behaviour of • near x = 1: I I-- o ~ A - 1
~[{p'}, {p}, q]= F(1-aA) sin (1raA)qsa-4( 1 1 r -J-~o) S'(f~q)
xexp { N - a A C - I dnkg'(nk) Log (1-i,-4. i)}. (2. 14)
An explicit expression for the exponential factors can be found in appendix A.
J.-P. Leroy, J. Micheli / OED radiative corrections 147
To conclude this subsection we shall derive another integral representation for which has the advantage of exhibiting its support properties and of involving only integrals over a finite domain.
Returning to eq. (2.10) we notice that it is convergent provided we integrate first over df~y because this integration leads to a behaviour in t -4. However, in order to be able to manipulate the integral we introduce a factor A2/(A 2 - t2+ ie) and let A go to infinity at the end of the calculation. Using the fact that the integrand is invariant under the transformation t + - t , y ~ - y we can rewrite the left-hand side (up to numerical factors) as
+co c~.9>0 • A 2
I=Re{e '~A f t2dtA2_t2+i E f d~y ( l - t xq ' f -i~)~a 4 o o
(2.15)
Introducing the new variable z -- t4 • f, it is easily ascertained from the explicit formulas given in appendix A that the function
is, in fact, a function of z and t 2 with a branch point at t 2 = 1 + iE as its only singularity. The dfly integration gives rise to an extra branch point at t = 1Ix- iE. Thus the integrand (for the t-integration) will have two poles at t = +(A + ie) and three branch points 1Ix-iE and +(1 +&). One can then close the integration contour in the upper half plane and rewrite (2.15) as
,~.9>o
I = Re {e i~mA f t 2 d/A2~)2+te I d ~ y ( 1 - t x 4 , f- iE) °~A-4
148 J.-P. Leroy, J. Micheli / O E D radiative corrections
Making the contribution of the pole term explicit, one gets A
I=Re{ei"a[-iTrAZ f dz(1-xz- ie) "a 4g(A,z)
0
0(3 t
I'd'A 2: I ]} - - - - l e ) DISC g(t, z) . (2.17) +iE dz(1-xz . o~A 4 •
1 0
In the second term of eq. (2.17) we rewrite
t ( l + r l ) / x t
f d z = I dz+ I dz, 0 0 ( l + v l ) / x
with 77 arbitrary but positive. Let us first consider the integral (1 + ' o ) / x
f dz(1 -xz - &),~A 4 Disc g(t, z). 0
Since the integration limits no longer depend on t the z integration gives rise to no additional singularity, so that it is equal to
(l+n)/x
Disc[ f (1-xz-ie)~A-4g(t,z)dz] . 0
As noticed above, g is in fact a function of t 2, thus the contribution of this term to (2.17) just amounts to a dispersion integral with respect to t 2 for the function
( l + n ) / x
I dz(1-xz • o ~ A 4 - re) g(t , z ) ,
0
leading to the result
( l+n)/x
i'n'A2 I dz(1 • x a A 4 i . -xz -re) gtz~, z). 0
When combined with the first term in (2.17) it gives (l+n)/x
iTrA 2 f dz(1 -xz - ie) ~a 4g(A, z) . (2.18)
A
J.-P. Leroy, J. Micheli / OED radiative corrections 149
To deal with the second integral we split the t-domain into two parts: o~ (l +~)/x oc
fat= I at+ i at, 1 1 (1 +~5)/x
with, again, ( positive but arbitrarily small. Consider first the contribution t
A 2 f tdtA2_t2+i E f dz(1-xz-&)~A-4Discg(t,z) •
( l + ~ ) / x ( l + r t ) / x
The quantity xz is always greater than 1 in the integration domain so that the factor (1-xz-&) ~A 4 will have the phase e -i '~A. Recalling that the real part of e i='~A times this quantity will have to be taken and that Disc g is pure imaginary, one sees that only the 6 ( t 2 - A 2) part of A2/(A2-t2+ie) will produce a non-zero contribution. The net contribution of this term to I will then be:
A
dz(zx - 1) ~A 4 Im (g(A, z)). ~ ~ ~ 2
(l+n)/x
Using the same phase considerations the contribution stemming from (2.18) will exactly cancel this one. We are thus finally left with
(l+~)/x 1
I=Re{e i~r~a f t2d t f d(~)Disc(g(t,z)) 1 ( l+~) / t x
× (1 - tx~f~ - ie)~a-4}. (2.19)
For x greater than 1 the integration ranges over values of t smaller than 1 for which Disc g = 0, which shows that the domain of qb is indeed 0 ~< x ~< 1.
2.4. Integral equations for cb
Going back to the definition, (2.4), of qb we transform it using the following trick due to Hearn, Kuo and Yennie [3]: thanks to the 6 4 o n e can write it as
r ~j=l k~/q ~', # = 0, 1, 2, 3. One then inserts the standard integral representation for 6 4 and uses the fact that limA~o exp [2aB({p}, {p'}, ,~)] = 0 to get:
d 3 K . ~ ~ f ei(q -K)'x q"~P({p~},{p~},q)=f--~--K S(K) 1 ' lim -- d4x ~o (2zr) J
[ [ " d3k , ~ , . , ik.x'] x exp [2aB({pi}, {pl}, a)] exp [ j ~ / ~ o t K ) e j . (2.20)
150 J.-P. Leroy, J. MicheIi / O E D radiative corrections
Comparing eqs. (2.15) and (2.8) and making the support properties of do explicit, one derives four integral equations for do:
P d 3 k q~do({p,}, {pl}, q) = J -~-k~,S(k)do({p}, {p'}, q - k)O(qo - ko-Iq - kl). (2.21)
Note that the support property of do is consistently implied by these equations since O(qo-ko - lq -k I) is equal to zero if qo<0 or qo<q. It is a simple matter to verify from (2.16), by forming the combination
Note, however, that making the support property of do explicit is essential in getting this result. We shall now concentrate on the equation fo r / , ---0 and using (2.9) we shall convert it into an equation for 0p. With x = Iql/qo we get
1 t 2 dt
+({p},{p'},x, tlq)--(1-x2) ~a ' f dtlk y x/(1-tx4"k ) 2 - ( 1 - / 2 ) ( 1 - x 2) 0
X g ' ( a ~ ) [ 1 - t x 4 ' /~ +~/(1- tx 4 "l~)2-(1-t2)(1-x2)] '-'~A
1 - t 2 × ~({p}, {p'}, t, fit,), (2.22)
xtq" / ~ - / 2 + x / ( 1 - t x q ' / ~ ) 2 - ( 1 - t 2 ) ( 1 - X 2)
where we have set
q l~= k 1- tx 4 • l~-~/(1-tx4 .Ic)2-(1-t2)(1-x2)tl ~ 4 = I q l ' [ k l ' v = x 4 - 1 - t 2 '
The integrand in (2.22) is non-negative so that the integral cannot vanish. Thus eq. (2.22) confirms what we have found in subsect. 2.3 concerning the behaviour of dO near x = 1. Defining the function F by
J.-P. Leroy, J. Micheli / QED radiative corrections 151
We know from subsect. 2.3 that the singularities of F in the physical region can only be at x = 1. Even if such a singularity does exist we also know that F must be finite in x = 1. It can be seen by direct inspection that eq. (2.24) is compatible with the absence of any singularity in F at this point. We shall simply suppose that the eventual singularity of F is independent of c~. More precisely, we shall make the hypothesis that F can be expanded in a. This will be justified in sect. 4. This is in contrast with the situation for ~P or +, where the effective expansion parameter would have been a log (qo) or a log (1 -x ) , making the expansion useless in the neighbourhood of qo = 0 or x = 1. Our next task will be to try to obtain such a development.
However, before doing that, we shall verify that the dependence exhibited in eq. (2.14) leads to an integrable cross section.
To this end let us consider a function f(q) which we suppose, for simplicity, to be derivable with respect to q at q = 0. Neglecting "gentle" factors we have to consider the integral
J = s i n (TTolA) f d4q S'(~q)q~ A 4(1--q00" ]ql](~A 4f(q)'
D
where the domain D is bounded by qo = E, Iql ~<qo. In terms of the variables qo, x = [q[/qo, Oq and ~ = q/Iq], it becomes
E 1
J =sin ( ¢ r a A ) f d q o q ~ A - l f t 2 d , ( 1 - , ) ° ~ ' f dOqS'(Oq)f(qo, tqo4). 0 0
Integrating by parts with
0
1
0
respect to qo gives
1
t 2 d t ( 1 - t ) ~A-1 f , dfIq S (flq)f(0, 0) d
t 2 dt(1 - t) ~A-' f dO. S'(Oq)f(E, tE~)
E 1 1 d
f I } . 0 0
Repeating the same procedure for the x integration and remembering that S'(Oq) is proportional to a and satisfies
" f ~ " [ x f d f ~ k ~ f ( E , EO)- f dqo(q~A-1) f df~k S(~k) ddqo
0
1
sinOmA) f dt[(1--t)~a--l_a(1--t)~A+l--1 (1-- t)aA+2-- 1] + c ~ [_ c~A aA+l q -.a-A+2
0
E
x sin oA aA f dqo(q~ A - 1)
o
X f d t [ ( 1 - t ) " A - 1 2 (1 - t )~a+ l - -1 (1--t)'~A+2--1]
f(qo, qoc~)}
0
d 2 x f dflk S'([lk) d-~qof(qo, tqoq).
The announced result is immediately seen from this formula. Moreover we see that as o~ goes to zero J is proportional to ~rf(0, 0). This means that when a goes to zero ~b behaves upon integration over d4q as a 84(q) distribution, which, of course, was to be expected in view of the definition (2.4). On the other hand, if we first let E tend to zero, J behaves as E ~A, a well-known result [5] which reflects the fact that no scattering is possible without an energy loss. Thus the singularities exhibited in qb ensure that ~ has the desired properties that (a) it is integrable and behaves as a 6 (4) in the limit a = 0; and (b) its integral vanishes in the limit of a null energy loss.
Finally we can go a step further in the evaluation of J. One then sees that the next term involves the integral over d3q of f, taken at qo =q; i.e., the next term behaves upon integration as a 6(qo-q): by the same mechanism as for the 34(q), the 3(qo-q) distribution corresponding to the emission of a single photon has been smoothed by the exponentiation procedure, leading to the ( l - x 2 ) "A 1 factor.
3. Perturbative treatment of the integral equations
3.1. Preliminary remarks
In sect. 2 we have exhibited some singularities of the function ~b(q) in the phy- sical region Iq[-< qo; singularities which insure that ~ ( q ) ~ 64(q) when a ~ 0. The
J.-P. Leroy, J. Micheli / Q E D radiative corrections 153
nature of these singularities is obtained taking into account all orders in a. We also know that the function F defined in (2.18) is finite in the whole physical region, so we can hope that we do not have to isolate any more terms which should be cal- culated to all orders in a, and that F can be expanded in a power series in a. Let us show now that this property is actually satisfied.
Let us consider first the Fourier transforms of the four integral equations. Denoting by ~(x) the Fourier transform of qb(q),
~(x) = f d4q e iq.x c~(q) ,
we obtain
d3k , i O~O(x) = O(x) f --£-k,,S(k) e ,k.x
Ox~
The solution of this set of four equations is unique:
{ I" d3k,z . . . . ik.x J ~ - 3 t x ) t e - g(k))} O(x) exp
up to the arbitrary function g(k) which must go to 1 when k ~ 0 in order to avoid the infrared divergence at k = 0. It will give rise to an arbitrary multiplicative factor. We of course obtain, up to a factor, the Fourier transform of the expression of q~ given in eq. (2.7).
In subsect. 3.2 we shall exhibit a solution of the integral equation for/~ = 0 which can be expanded in a. Because of the unicity we shall be sure that this solution is the good one defined in sect. 2 if it also obeys the three other equations. But, as shown in appendix B, a solution of the equation for # = 0 also fulfills the three others. This seems in contradiction with what has just been said above. The Fourier transform of the equation for U = 0 has the following general solution:
where h(x) is an arbitrary function of x; so that 01 is not a solution of the full system unless h happens to be a constant. However, we must realize that the equation for ~bl is not equivalent to the equation for ~: it has forgotten the support conditions which • must obey so that the general solution written above will not fulfill these support conditions.
In conclusion all we have to do now is to find a perturbative solution of eq. (2.24) for the function F.
3.2. Perturbative expansion ofF
Let us remark first that the kernel G of eq. (2.24) cannot be directly expanded in a power series of o~ because the singularity at t = 1 becomes non-integrable when
154 J.-P. Leroy, J. Micheli / OED radiative corrections
a is set equal to zero. Nevertheless we can do it in the following way: consider an integral of the type: I = Jo ~ a A ( 1 - t)~a-aR(t)dt. By integration by parts it becomes I = R(1) +~o 1 [(1 - t) '~a - 1](d/dt)R(t) dt. The distribution aA(1 - t) '~a-1 on the interval [0, 1] is then equal to 8 ( t - 1 ) + [ ( 1 - t) ~ a - 1] d/dt. So denoting by P the operator which projects any function on the subspace of functions of ~q only through PH(x, l)q) = H(1, (~q), we can decompose G into a sum of two terms: G = OP + K with
2 . . . . 1-x4" I~ -x4"/~)4-(1--X2)/~) OP= ~-A ~tt- l ) l + x2-2x4 " /~,(2x(1 l + xZ-2x4 " I(: (3.1)
2 1-x4" I~ k d , ( 2 x ( 1 - x 4 • / ~ ) 4 - ( 1 - x 2 ) / ~ K=~-A6(t-1)l+xT~-2x4. * i+~----2ff~7; ]
x [ ( 1 - x 4 ' /~)-'~A- 1 ] + ~ A [ ( 1 - t ) '~A--1]dt2(1 + t) '~A-1
where d/dt must be applied to all functions that follow it, including the function of (t, f~q) on which G acts. The formulas above must be understood as integrals over k.
Due to the factor a in front of S, OP is independent of a while K is of order a and can be directly expanded in a power series in c~.
We shall show, now, that this integral equation has a solution which can be computed perturbatively provided certain conditions are verified. Then we shall show that this is actually the case.
To begin with, let us rewrite POP in a bra notation: POP= Is)(ul where Is) represents the function of f~q only S(Oq)/aA and (ul = 6(x- 1); [(ulF)=
df~q F(1, f~q)]. Now, expanding F: F = ~ , c~nF,, the equation for the lowest-order term is Fo = OPFo, the solution of which is obviously Fo = OP]S).
The equation for the nth-order term is an inhomogeneous integro-differential equation:
IF.)--OeIF.)+ E KplF.-p>. p=l
J.-P. Leroy, J. Micheli / Q E D radiative corrections 155
Projecting it on (u I we obtain a necessary condition for this equation to have a solution:
(ulKp]F. ~)=0. (3.3) p = l
It is straightforward to see that this condition is sufficient and that the solution is then
IF. ) = a, IFo) + (1 + OP) ~ K v IF,, p), (3.4) p = l
where a , is an unfixed coefficient. Thus F, is obtained as a function of the Fv's of smaller order. The complete
solution is easily seen to be given (formally) by
IF) = c[1 - (1 + OP)K]-'IFo).
We shall now turn to verifying that the conditions (3.3) are actually satisfied. To this end we first establish the following property of the kernel G:
f dx df~qX2(1 -x2)~A-1G(x, = t2(1 -t2) ~A 1 nq; t, f~k) (3.5) I
Returning to the kernel G of the integral equation for ~, let Y(q) be the result of applying G to an arbitrary function X(q):
Eq. (3.6) leads to the corresponding relation between Z and T:
I dx df~x2(1-x2)"A-1T(x,a . ) = f dtdf~KtZ(1-t2)"a-lZ(t,~)K),
whence, using (3.7),
= f dt dflKZ(t, f~K)t2(1 --t2) ~a 1.
Since this equation must hold for any regular function Z, eq. (3.5) follows immedi- ately. Expanding both sides of (3.5) with respect to a we get for the nth-order terms the equality
n-1 "-P J f LogO(l-x) d 2x 2 Log ~-p °(I+X)Kp(x ' <u[K,+ 2 2 A"-P dxdOq ~o;t,~)k)
p=oq=l q! dx l + x ( n - p - q ) !
~, LogP(1-t) d 2t 2 Log ~ P(l+t) =A"
p=lZ'~ p! dt l + t (n-p)! '
where we have set Ko = OP on the left-hand side. Letting both sides of these (operatorial) equations act on the functions Fs(s = O, n -1) defined above and summing, we arrive at
n - 1 n - 1 . - s y, (ulK ~ ,iFo)= ~ ~ A ~ S jd td f lKLOg P ( l - t ) d 2t 2
s=o s=o p=l p! dt l + t
Log" P-s(1 + t)Fs(t ' Y~K) , - 1 , ~ - , , s-p f × - E E E j dxdl'~qA" ~ "
( n - p - s ) ! s = o 19=0 q = l
Log"(1-x) d 2x 2 Log ~ ~-P-°(l+x) l" x j dt df~KKp(x, ~o t, f~K)Fs(t, ftK)
q! dx l + x ( n - s - p - q ) ! (3.8)
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The last term is rewritten as
n - - l ~ r f LogO(l-x) d Log n-r q ( l + x ) 2 x 2 -r=o ~ q=l dxdflqAn r q! dx (n-r -q) ! l + x
r I xp~] °= dtdflKKp(x,f~q;t,f~K)F~_p(t,~K).
By hypothesis the Fr's satisfy IFr)= ~ = o KplFr p) so that the second and third terms on the right-hand side of (3.8) cancel, which achieves the proof.
So starting from one of the four integral equations satisfied by 6p and isolating the singular part of ap, a perturbation expansion of the remaining term F(x, l)q) is possible. We know from subsect. 3.1 that the function ~ so obtained is identical up to a normalization constant to the function defined in sect. 2. There is no problem in determining the normalization constant since we know explicitly the exact value of F(1, ~a).
Now a question arises: are the singular terms of the form (qo) '~A-4 and ( l - x2) '~A 1 the only ones that must be exponentiated? In fact direct inspection of formula (2.14) shows that for x = 1 and at high energy the term exp (-~ d~k Log (1 -q/()S'(flK)) which appears in F(1, ~K) becomes important and must be exponentiated. This comes from the fact that, at high energies, the terms 1/pi]~ which appear in S'(flK) become singular outside but very near the physical region.
In order to take into account the presence of these singularities we propose a slightly different method to solve the integral equation for F:
(1 - K ) I F ) = OPIF).
From sect. 1 and appendix A we know exactly the right-hand side which depends only on F(1, l)o). So now that we have shown that the integral equation has a unique solution which can be expanded in a, we can calculate F from the formula IF) = (1 -K) 10PIF) where we expand (1 - K ) 1 with respect to ~ but keep the exact value of OPIF).
To summarize, we have obtained two expressions for the exponential factor which exhibit its singular dependence on q in the form
"h -qo }
[eqs. (2.19) and (2.23)]. Either of these expressions seem to be suitable for practical calculations. Examples of such applications will appear in a forthcoming paper.
158 J.-P. Leroy, J. Micheli / QED radiative corrections
Appendix A
We give in this appendix the explicit form of the expression
- I dflk Log (1 -4" I =
which appears in eq. (2.14). For a given pair of external momenta pi, P/we define
6 , = [ ( p i ' ,2 2 211/2 EJ-P i '4 PJ) - m i m j to = (Pi -pi) 2 , xir = ' E i - P i "
air = pi • Pr - 4~ir, bit = p~ • pj + ¢b,.
The formulas below are written in the case mi > mr (although they are, in fact, symmetrical under the exchange pi ~pj) :
[ xqti, ] fm2-bq z t +Li2/ -- -Li2 (1- xi/)(bij-m/ )-I t tii xijti] _1
F m 2-bi/ ai i-m 2] - Li2| xij - -
L tij tij 3
If tq<O:
_ f bq -m2 ] [(m2-ai j )xq-(bq-m2)] g(xi,)-Log[(-~2i_----~i,) 1 i Log tq
+ Log \|/xii------TJaiim2-bi'\_ m i , Log [(m2-bii)xi ' -(aq-m2!]-½Log(xii , \ a q,
a . . - 2 2
-L~2( " t-Sj m'.- 1 - xq/+ Li2[ ( l x q " --xij),l~mi--bij]71
-Li2[(1 m2 - aij'] r bq - m 2 1 - Xij] -x,)--77--. J +g,2[ ~ x, a'
x Li2 is the standard dilogarithm: Li2(x) = -Io (dt/t) Log (1 - t ) . Note that for P~Pi >> mimj, g(x) is given to a very good accuracy by the approximate expression
g(x) = -2 ! Log (x) Log (m~/m2ix).
The physical range for xq extends from 2 2 (ai i-mi)/(mi -bii) to (bii 2 2 - mj )/(mj - aii). Finally the following properties of g may be useful:
g(1) = g(m2/m 2) = O, g(x) = g(m2/m2x) ;
g has an extremum at the point x = mjmi. As for the expression
I dO, k S'(D.k) Log (1 - t~ . /~ + ie)
which appears in formula (2.10), it appears as a sum of various terms of the form
Log (1 - t a + ie)+ V o - t ~ " V [(Vo- tl 3. V)2-(1-t2)(V2 o- V2)] 1/2
where (Vo, V) is either one of the external momenta or a combination of two of them of the form [xp~ +(1 -x)pj], O<~x <~1.
160 J.-P. Leroy, J. Micheli / OED radiative corrections
Appendix B
We want to show that if qb satisfies eq. (2.21) for ~ = O, it also satisfies the three other integral equations, which we can rewrite as
q'dP(qo, q)O(qo-]q l) qo
= 2 I dKo I d3K(qi- Ki)•[(q - g)?]o(g°-IKI)S(q - K)*(Ko, K). 0
We shall show in fact that the moments with respect to the variable O of the two members of these equations are equal. So, defining
f " " i i yi ...... i~'i(qo) = d3qq"q ~2... q ~q dP(qo, q)O(qo-lql),
we must show that qo
Yi~i"'i(q°)=2 I d K ° f d3Kd3k(kil+Ki')'"(ki"+Ki~)ki
0
x 6[(No-go) 2- k2]S(k),(go, K)O(Ko-IKI/. (B.1)
Introducing the quantities
D i~i''i = ; dflk/~i,.../~i,/~i~,(flk) '
eq. (B.1) can be written qO
Yi'i~'i(qo)= ~ 2 Pi ' i" ' i I dKo(qo-Ko)Pyi"+'i"(qo), (B.2) p = O il...i~
~(h...iJi+~...i.) o
where ~ ( ) means a sum over all partitions of ix . . . . . i. in two groups of p and n - p indices. Now, since • is an homogeneous function of degree a A - 4 in the variable q, we know that
Y il"''i~'i(qO) ~il . . . i ,i otA+n = C ~ q o •
So eq. (B.2) reduces to
C i l ' ' ' i n ' i : ~ ~ Di~G'ici~+ri"B(aA+n-p,p+l) (B.3) p-o ~,(j~ ...~i~;'+~...,,)
where B is the Euler function B(x, y)= F(x)F(y)/F(x + y). The equality must hold provided eq. (2.21) for/~ = 0 is satisfied, i.e.,
J.-P. Leroy, J. Micheli / QED radiative corrections 161
Writing the equality between the moments of the two members of this equation we obtain
whence
n + l c i l " i n ' i = ~ 2
p=O it...i,,,i ~(j~...g/g+~...
J..~)
D i ' ¢ C i ~ + ' i " + ' B ( a A +.n + 1 - p , p + 1),
cil...iw i a A + n + l "+1 . . . . ~ D ' I " " ~ C ¥ + ' " , , + ' B ( A + n + l - p , p + l ) t 1 . . . tn t Fl + 1 p= 1 ~(it,...il/ip+l..4n+l)
One can show by recurrence that the solution of these equations is, up to a multi- plicative constant,
1 C 6 ¢ ' ~ = ~ Y~' H ( r i - 1 ) ! D (r')
a A ( a A + l ) . . . ( c e A + n ) ~ l ~( i l . . . i~ , i ) i ~.r i -n+l
where D ('> is a coefficient with rg indices and Y~,( ~ means a sum over all partitions of il . . . ini in group of indices rl, r2 . . . . . each term appearing only once if some of the r~'s are equal. Replacing in the right-hand side of eq. (B.3) the coefficients C ¢ + ' i " by their value leads to the expression
n
1 ~= Y~ P! Y~ E' H(rg-1)rO ('>. a A ( a A + l ) ( a A + n ) p o~ . . . . . . . . . . 1 ~(j ...... i.) i
• " " ( /1 ""ig*ip+l'"in) ~r i=n_ p
Each term of the form D~">. . . D %> appears with a coefficient ( S 1 - - 1 ) ! . . . (sq- 1)!. So this term is equal to C i ' ¢ ' i and eq. (B.3) is satisfied.
Note added
After this work was completed, we received a paper by D. Zwanziger (preprint NYU/TR2/79) which deals with the same subject and contains similar results• We thank D. Zwanziger for drawing our attention to this work.
References
[1] F. Bloch and A• Nordsieck, Phys. Rev. 52 (1937) 54. [2] D.R. Yennie, S.C. Frautschi and H. Suura, Ann. of Phys. 13 (1961) 379. [3] A.C. Hearn, P.K. Kuo and D.R. Yennie, Phys. Rev. 187 (1969) 1950. [4] G. Grammer Jr., and D.R. Yennie, Phys. Rev. D8 (1973) 4332. [5] D. Zwanziger, Phys. Rev. D7 (1973) 1082; D l l (1975) 3481, 3504. [6] C. Chahine, Brandeis University preprint LPC 7802 (1978).
