- Email: [email protected]
Threshold infrared behaviour in perturbative QCD
Nuclear Physics B194 (1982) 328-348 O North-Holland Publishing Company
THRESHOLD INFRARED B E H A V I O U R IN PERTURBATIVE QCD J. FRENKEL and M.L. FRENKEL Instituto de Ffsica, USP, Sdo Paulo, Brasil and DAMTP, University of Cambridge, Cambridge, England J.C. TAYLOR DAMTP, University of Cambridge, Cambridge, England Received 11 May 1981 We study a class of Bloch-Nordsieck uncancelled infrared divergences which are relevant near threshold in inelastic reactions between coloured particles. We show that the dominant threshold contributions factorize, in the cross section, from the leading non-cancelling infrared singularities.
1. Introduction It has b e e n shown [ 1 - 7 ] that the B l o c h - N o r d s i e c k [8] m e c h a n i s m for the cancellation of I R divergences does not work in general for the scattering of coloured massive quarks or antiquarks in perturbative Q C D . In particular, in refs. [6, 7] we s u m m e d all I R divergences of leading non-cancelling o r d e r as(as In A) ~, where as = g2/47r is the strong coupling constant and h d e n o t e s an I R cut-off. T h e aim of the present p a p e r is to study the behaviour of non-cancelling I R divergences b e y o n d the leading order. W e have not been able to do this in general. Instead we take advantage of simplifications which occur near threshold, that is to say for B << 1, w h e r e / 3 is the speed of one q u a r k in the rest frame of the other. W e seek the non-leading divergences of o r d e r /3 0a s2 (as In h)". In fact, we go further and sum contributions of o r d e r /3as(as/3-1)m (as In h )~.
(1)
As a guide 'to our search for terms of this order, we note that each transverse gluon gives a factor /32, whereas each C o u l o m b exchange can give a factor /3-1 [6, 7]. Consequently, we expect the right sort of b e h a v i o u r to c o m e f r o m graphs containing one transverse gluon and m a n y C o u l o m b exchanges. H o w e v e r , we find (see appendices A and B) that such graphs give zero contributions, provided all the C o u l o m b m o m e n t a are soft, so that the eikonal approximation m a y be made. But, one must r e m e m b e r that the hard parts of the graph (with m o m e n t a exceeding s o m e arbitrary small a m o u n t / t ) , in which the eikonal a p p r o x i m a t i o n is 328
329
J. Frenkel et al. / Infrared behaviour
C::I
q"
C:l
q
Ca)
(b)
Fig. 1. The small circles represent the virtual photon y* which attaches to the quark (q) and antiquark (q). The wavy line denotes a soft gluon and the dashed-dotted line stands for a Coulomb line. The right-hand side of graphs represent the complex-conjugate part of the corresponding diagram.
inadmissible, can also give inverse powers of/3. We therefore obtain contributions of the form (1) and we find that they factorize (see below). For definiteness we study the inelastic reaction involving two massive particles of mass M : q+Cl~ 3'* +soft gluons,
(2)
which is the simplest process exhibiting Bloch-Nordsieck non-cancellation. We use the Coulomb gauge in the rest frame of the antiquark Ct, and /3 is the speed of the quark q. As shown in [6], graphs like fig. l a have at least one more power of /3-1 than graphs with three-gluon vertices like fig. lb; so in the remainder of the paper we ignore skeleton graphs containing three-gluon vertices [but, of course, we do consider renormalization graphs involving these vertices]. Consider a more general class of graphs exemplified in fig. 2a, containing n Coulomb exchanges. In sect. 2 we show that they factorize into the product of the two classes of graphs shown respectively in figs. 2b, 2c. The first class contains only hard Coulomb lines and yields precisely the expression for the non-relativistic inelastic Coulomb cross section [9] (see also appendix C):
[
l
O-i"=O'B°rn 1--exp(--I
i] '
(3a)
where l CF/3-12"n'CZsand the Casimir invariant CF for the representation associated with the quarks is defined by CFt~ab = Tr tatb. The second class contains only two soft gluons, one being a transverse gluon and the other a Coulomb exchange, as shown in fig. 2c. This contribution can be calculated in the eikonal approximation [1-6] and gives, in the non-relativistic limit, the result =
A ~CFCyM/3O/s
= ~CFCyM~O/s
In
.
Here GyM is the Casimir invariant of the adjoint representation defined by
[,,bcfabd, where fabc are the structure constants.
(3b) CTMScd =
330
J. Frenkel et al. / Infrared behaviour
(a)
Cb)
C¢)
Fig. 2. (a) A typical diagram which gives contributions of order (1). Dotted-dashed lines denote soft or hard Coulomb exchanges. (b) and (c) represent the factors resulting from (a). Dashed (dotted) lines stand respectively for hard (soft) Coulomb exchanges.
In sect. 3 we study the effects resulting f r o m the inclusion of the r e n o r m a l i z a t i o n graphs, which lead to the a p p e a r a n c e of the running coupling constant. In the leading logarithm a p p r o x i m a t i o n , which is consistent with the terms considered in (1), as(k/l*) is given by [10]
as
- 1 + (11/6rr)CyMas(ix) In ( k / ~ ) '
(4)
w h e r e ~ is an arbitrary mass introduced in the renormalization process. W e find that, in this a p p r o x i m a t i o n , the factorization previously m e n t i o n e d continues to hold, resulting in graphs like the one shown in fig. 3. T h e inclusion of the running coupling constant in the hard C o u l o m b diagrams (see fig. 3a) leads to an expression similar to that of eq. (3a), except that as is replaced by the effective coupling constant a s ( t i M / # ) : ~rin = O'Born[ 1 -- ex~ ( - i ) l
(5a)
where l = CFfl-127ro~(BM//x).
(a)
(b)
Fig. 3. The blobs represent schematically the inclusion of vertex and self-energy corrections into the skeleton graphs.
J. Frenkel et al. / Infrared behaviour
331
Turning now to the contributions resulting from soft graphs like fig. 3b, one obtains [7] that the effect of the running coupling constant results in the following expression, as a -* 0 [compare with (3b)]:
a dko2(k) where we have used a principal value prescription. In the non-relativistic limit, when ~-las(M~/l~) >> 1, we can neglect the exponential function in (5a). Multiplying then the expressions (5a) and (5b) we obtain: / - 8 7 r 2 ' ~ C 2 a { M[3"~a { A'~
OrBorn~k--H--)
F s~---)
s~7 ) ,
(5C)
which represents a finite non-zero result. Finally, we discuss at the end of sect. 3 the possibility of extending these results to the case of the inelastic scattering of two quarks. One of the motivations for our work has been to see whether the sum of an infinite n u m b e r of non-cancelling infrared divergences could give a finite result. This expectation is based on the unitarity of the theory. Indeed, if one could sum exactly all the classes of non-cancelled infrared singularities the complete answer must be finite since the S-matrix elements satisfy E I&nl 2 = a .
n
However, we have been able to sum only the uncancelled infrared divergences corresponding to a particular class relevant in the non-relativistic limit. For this class, it was consistent to use the leading logarithm approximation for the effective coupling constant as(k). Nevertheless, one must r e m e m b e r that in this case as(k) may become negative for small enough values of k, behaviour which is unsatisfactory from a physical point of view.
2. Factorization of skeleton diagrams We begin this section by briefly describing a convenient method for the calculation of uncancelled infrared divergences in process (2). (For more details see refs. [6, 7].) This method is based on the observation that in the reaction q + ft + soft gluons-~ y*
(6)
the infrared divergences cancel out, essentially as a consequence of unitarity. But note that in this case the soft gluons are incoming, as opposed to reaction (2) where they are outgoing. Let us denote the graphs containing the outgoing real gluons contributing to (2) by R °ut, and the graphs containing the ingoing real gluons contributing to (6) by R i". Furthermore, we denote by V the purely virtual graphs
332
J. Frenkel et al. / Infrared behaviour
k
(a)
(b)
(c)
(d)
Fig. 4. (a) and (b) represent graphs which give a~ In A uncancelled infrared divergences. (c) and (d) denote lowest order non-trivial diagrams contributing aJ3 -1 terms to the inelastic Coulomb cross section.
contributing to q + q ~
y*. T h e n we can write for reaction (2): R °u' + V = D ,
(7)
where D d e n o t e s the uncancelled infrared divergences, while for process (6) we have instead R i" + V = 0 .
(8)
T h u s we see that we can obtain D as follows: D
---- R ° U t - R
in .
(9)
This f o r m u l a is convenient because using this m e t h o d we do not have to calculate the purely virtual graphs, which are usually m o r e difficult to c o m p u t e . F u r t h e r m o r e , when all gluons are soft, R in c a n be obtained directly f r o m R ° u t j u s t by changing the signs of all ie in the fermion propagators. Since, for the would be leading infrared divergences of the f o r m (as In A)n, we k n o w that we can disregard these ie, eq. (9) implies that these divergences will always cancel, to all orders. Thus, we are left with the divergences of the form a~+"(lnA)" which are uncancelled in general, since in this case the difference in sign of the ie in R °ut and R in is crucial. F o r instance, consider the graphs shown in figs. 4a, b, where both gluons k' and k are soft. W e k n o w that the sum of these two yield in R ° U t - R in a single infrared divergence of the form a 2 ~ u -1 du, which arises from the region where both k and k' are simultaneously scaled to zero by a c o m m o n factor u. That is, each one of the m o m e n t a k or k' is controlling. In order to understand the factorization p r o p e r t y described in sect. 1 consider next the diagrams shown in fig. 5, where h d e n o t e s a hard gluon. Using the notation H = q o l q • h, K = q o l q • k , etc., after p e r f o r m i n g the k~, ho integrations, we obtain that the hard parts of these graphs are respectively p r o p o r -
333
J. Frenkelet al. / Infrared behaviour k
k
<.
(a')
<<-2
(b)
(b')
Fig. 5. Sixth order diagrams which exemplify the factorization property. tional to
[qolh2+H+K'+K+k+ie] [qolhZ+H+K+k+ie]
-1,
-1 ,
[qolh2+H+K+k-ie]
-1 ,
(10a)
[qolh2+H+K'+K+k-ie]
-1 .
(10b)
Since k' is controlling in diagrams (a) and (b'), we may replace these expressions by
[qolh2+H+K+k+ie]
-1,
[qolh2+H+K+k-ie]
-1 ,
(lla)
[qolh2+H+K+k+ie]
-1,
[qolh2+H+K+k-ie]
-1 .
(llb)
Thus each of the graphs of figs. 4a, b is multiplied by
[qolh 2 + H + K + k +ie]-l + [ q o l h 2 + H + K + k - i e ] -~ .
(12a)
If we neglect here the soft m o m e n t a k compared with the hard one h, we obtain
[qoah 2 + H + ie] -1 + [qolh 2 + H - ie] -1 .
(12b)
which corresponds to the contribution arising from diagrams 4c and 4d. This would lead to the factorization property mentioned in the introduction. (Note that for the abelian graphs it is actually a property valid for all values of/3.) The error made in using the above approximation is
{[qolh2+H+K+k+ie]
a+[qolhZ+H+K+k_ie]
a
- [ q o l h 2 + h +iv] 1 - [ q o l h Z + H - i e ] - l } [ ( 4 a ) + ( 4 b ) ] .
(13)
But, since either k or k' is controlling in [(4a) + (4b)], this is an infrared convergent expression, and hence not relevant for our purposes. Consider now a higher order graph, shown in fig. 6, where k, kl, k2 are soft m o m e n t a , and h denotes a hard gluon. The hard parts of these graphs are respectively proportional to
[qolh2+H+Kl+K2+K+k+ie]-l,
[qolhZ+H+K+k_ie]
1,
(14a)
334
J. Frenkel et al. / Infrared behaviour k
(a)
:
(a')
,
i
(b)
(b')
(c)
(c')
i
Fig. 6. Example of an eighth order factorizable diagram.
[qoah2+H+K2+K+k+ie]-1, [qoXh2+H+Kx+K+k-ie] -1, [qoahz+H+K+k+ie]-1, [qolhZ+H+Kl+Kz+K+k-ie] -1.
(14b) (14c)
We observe that, since kl is controlling in fig.'6a, we can leave out Ka in first terms of eq. (14a). Similarly since in fig. 6a' ka is controlling, we can insert K~ in the second denominator in (14a). Proceeding in a similar way, we insert K2 in the first denominator in (14c), and neglect Kz in the second denominator. Then we obtain the factor
[qolh2+H+Kz+K+k+ie]-X+[qo~h2+H+Kl+K+k-ie]-l,
(15)
which multiplies the integrands corresponding to the graphs shown in fig. 7. In the sum of graphs any one of the m o m e n t a k, kl or k2 is controlling, so we can replace
(a)
(b)
(c) Fig. 7. Sixth order soft diagrams resulting from the factorization of graphs corresponding to fig. 6.
J. Frenkel et al. / Infrared behaviour
335
~J
Fig. 8. Example of a general graph whose contributions cancel in R ° u t - R ~n.
(15) by eq. (12b), which corresponds to the contributions resulting from fig. 4c + 4d. Thus we have shown the factorization of diagrams in fig. 6 into classes containing only soft or hard gluons, respectively. Actually, as we will now show, the contributions resulting from the graphs of fig. 7 cancel in R ° U t - R in, so that only graphs with a single soft Coulomb line with the structure shown in fig. 2c will contribute to the dominant infrared divergences. The above property results when considering a general graph like the one of fig. 8, which contains m + n soft Coulomb lines. Graphs obtained from fig. 8 by moving the transverse line k in all possible positions give essentially the same contribution [i.e. apart from a combinatorial constant and colour factors (see appendix B)]. When considering together all such graphs, we find that all contributions except for the one proportional to (CyM) m÷" cancel out. The contribution proportional to (CvM) m-n, which results precisely from the diagram shown in fig. 8, is calculated in appendix A. We show there that, when summing over all values of m and n, we obtain for R °ut a contribution which has only a single infrared divergence of the form I f dk[ 27rA/3 -1 -k-[exp ~ 1 ) _
1] =In ( ~ ) [ 1
1 2rrA/3 -1 ~ ~. + " "]
(16)
1 where A = aCyMas. In this series which defines the Bernoulli numbers (see appendix A) all terms with odd index vanish, except for the lowest order one. For R i", which can be obtained from R °ut by changing the sign of ie in the fermion denominators, we obtain a similar series, except that A is replaced by - A . Hence, in R out_ R in, only the term proportional to as, which corresponds to graphs with a single Coulomb soft line, does contribute. More complicated graphs can be treated in a similar way in order to verify the factorization property discussed in the preceding section. This property depends essentially on the fact that when all gluons are soft, general graphs like the one shown in fig. 12 (see appendix B) do not contribute to R ° U t - R in. Consider now the diagrams in figs. 4c, d, which contribute the lowest order corrections to the inelastic Coulomb cross section. They yield a contribution given by oe 3 as f d h 1 (17a)
2O'Bo,nCFM~-~2 Ja -~ h2+q . h "
J. Frenkelet aL / Infraredbehaviour
336
(a)
(b)
(c) Fig. 9. Diagrams of order a~ which contribute to the inelastic Coulomb cross section.
Choosing the z axis along q and doing the angular integrations we obtain, using Iq[ = M 3 , 1
~
h+Mfl
O'aoroCF2;S-~fa ? l n l h _ M
fl
=O'B°rnCF2OLs ~77" ~--~-O ~'~ where we have chosen A <
[ Cvas~ 2~r2 CrBom~---ff--) y .
(18)
Therefore, to this order, adding the Born diagram contribution with the one resulting from (17b) and (18), we obtain
r
1 Cvas2~r 1
O'"°rn[1-+2
--/3
1/CFas2"rr]2
+-62!~-- -fl
"] +" " "J] "
(19a)
The general pattern is clear: for each Coulomb line we get a factor Cva~2rr/[3 with a coefficient which is characteristic of the Bernoulli numbers. This suggests that the general result for the inelastic Coulomb cross section is
CFas2 7r/ 3 ~ra°r"l -- exp (-- Cvas2zr/3)"
(19b)
Indeed, this is in accordance with the result obtained in non-relativistic quantum mechanics for the inelastic scattering of slow particles in a Coulomb potential [9]. In that case, the reaction cross section is obtained from ]~bc(O)[2, where ~c(X) is the
J. Frenkel et al. / Infrared behaviour
337
exact wave function of a particle in a Coulomb field. This then leads precisely to eq. (19). Consider now the leading infrared contributions arising from graphs like 2c, having one soft Coulomb line and one soft transverse gluon. As shown in ref. [6] the infrared-divergent contributions resulting from these fourth-order soft diagrams are 2 Of s
2 C F C y M ~ - F ( f l ) In A ~-.
(20a)
Here F(/3) denotes the bremsstrahlung probability function: ~] F(/3) = 2-~ In \(1~ -+/3~
-
1 =
1~2 + . . . .
(20b)
Eq. (20) was derived using the eikonal approximation. In this approximation one neglects k 2 compared with k ' q in the fermion denominators. For consistency, remembering that ksoft < A, we must therefore satisfy in the non-relativistic limit the condition A <
3. Discussion
of running
coupling
effects
In this section we wish to study the effects of the inclusion of self-energies and vertex UV-divergent graphs, which account for the appearance of the running coupling constant. To this end consider the diagrams in fig. 10, which represent corrections to the Coulomb line. As is well known, in virtue of the Ward identities in the Coulomb gauge, the contributions resulting from diagrams ( b ) . . . (e) cancel out, and the self-energy graph in fig. 10a gives the complete coupling constant renormalization [11]. For the transverse lines [7] both vertex and self-energy corrections are necessary in order to obtain the running coupling constant % ( k / t z ) which is given in the leading
~} i
,
(a)
i
"
I
,
(b)
(d)
i I
(c)
(e)
Fig. 10. Example of self-energy and vertex corrections associated with the Coulomb line. Graph (a) includes all second-order self-energy parts constructed from Coulomb and transverse gluons.
Z Frenkel et al. / Infrared behaviour
338
approximation by as
- 1 + (11CvM/6"tr)O~s(kt ) In =a~(tt)[l
(k/#)
l16-C~vMad/.t)ln(k)+...],
(21)
where/z is the arbitrary mass unit introduced in the renormalization process. It is not difficult to see that in this approximation all the previous arguments for the factorization of graphs are still valid. For instance consider the effect of (21) on eq. (16). In order to obtain the leading uncancelled infrared divergences we rescale all momenta in fig. 8 by a common factor u and keep only the dominant contributions as u ~ 0: In ki =- In
/'//)i ~
In u.
(22a)
In this case the term proportional to the nth Bernoulli number B , in (16) will now be multiplied by a term proportional to [cadu)] p p =o
-(lnu) fxa-du b/
p~
p = ~ [ca~(#)] p /7=0
[(ln A)P+I-(lnA)P+I].
(22b)
Clearly the same modification occurs also in _Ri', so that in R ° U t - R i" only contributions proportional to B1 remain. These correspond precisely to corrections arising from self-energy insertions in one soft Coulomb line, as exemplified in fig. 3b. Furthermore, the leading logarithm expression (21) for the running coupling constant is consistent with the order of the terms considered in (1) for the calculation of non-leading uncancelled infrared divergences. We will now consider the effect of this coupling constant in the graphs shown in fig. 11, which contribute to the inelastic Coulomb cross section. Diagrams (a) and
(a)
(b)
Fig. 11. Example of hard Coulomb diagrams representing the effect of the running coupling constant given by eq. (4).
J. Frenkel et al. / Infrared behaviour
339
(b) give the contribution [compare with eq. (17)]
M~+h ~rB°r~CF~ ~ Ja
In M----~_ h ,
l+Cln(h/iz
(23a)
where 11
C = 7 - - CyMO/s(/2 ) . o7)"
Defining a new variable x via h = M/3x and neglecting terms of order A/M/3, we obtain 2CF f ~ d x [ ces(/z) °'n°rn 7r/3 J0 x - l + C l n ( M / 3 / l x ) + C l n x
]In
x+l -x--~-l"
(23b)
In order to evaluate this integral we expand the first bracket in a series of powers of In x and integrate term by term. We find
2Cv "7/'2i-.+1 . . . . 2 + 1 . . . . 4 + O'Bor, ~.---~as~---~-] ~ - [ l ~Trt. ) 3 ~ - ~ ) "" '],
(24)
where 1
In the non-relativistic limit, as /3 ~ 0 we have C'<< 1. In this case the dominant contributions to (24) arise just from the first term in the bracket which yields OrBorn7"/'CF~ a s ( - - ~ ) .
(25)
Note that (25) can be obtained directly from (17b) by replacing in that equation as by the effective coupling constant ces(M/3/tz). Similarly, graphs (c) and (d) will lead to a contribution like (18), except that in this case as will be replaced to as(M/3/lx). Clearly, given the above approximations, this generalizes to higher orders and we obtain for the (hard) inelastic cross section the result [compare with (19)]
Cv/3 las(M/3/iz)2~r O-Bor, {1 - exp [-CF/3-1as(M/3/tz)27r]} "
(26a)
If the relation/3-1a~(M/3/tz) >>1 is satisfied in the non-relativistic limit, then eq. (26a) reduces to O'Bo~nCe/3-1as ( ~ )
27r.
(26b)
340
J. Frenkel et al. / Infrared behaviour
The inclusion of running coupling (4) (see fig. 3b) has also the effect of modifying (20) into an IR finite expression given in eq. (5b). The product of this with (26b) gives
--877-2 OrBorn~ - - C2as(~) as(~ -~)
(27)
which represents a finite, non-zero result. [See the remarks following eq. (5c).] These results can presumably be extended to the case of inelastic scattering of two quarks. We must then replace as by - a s since the charge of the antiquark is opposite to that of the quark. Furthermore, in the above equations, CF will be replaced by the matrix ta ® Ta, where ta and Ta are the matrix generators of the representations to which the two quarks belong. Then we obtain for Orqq i.
(-ta ® Ta)~-las(M[3/lz )27r
O'B°r"{1--exp [t~ ® Tafl-las(M[3/tz)2Tr]} "
(28)
When the condition fl-las(Mfl/ix)>> 1 is satisfied, the behaviour of this relation depends on the colour state and representation of the quark-quark system. If the system is in an eigenstate of ta ® Ta with negative eigenvalue, (28) reduces to
O-Born(-ta ® Ta)~-las(-M~--)2~
(29)
[compare with eq. (26b)]. This is similar to the quark-antiquark system. On the other hand, if the quark-quark system is in an eigenstate of ta ® Ta with positive eigenvalue, (28) reduces to
O-Born(-ta ® ra)~-las(-h~-)27r exp [ (-ta @ Ta)~-lce~(-~-)27r] .
(30)
In this case the cross section tends to zero. The exponential factor by which (29) [or (26b)] and (30) differ can be interpreted, in analogy with the quantum mechanical case [9], as the probability of passage through the effective Coulomb potential barrier. We are grateful to R. Taylor for his proof presented in appendix B as well as for numerically checking (C.9). J.F. is grateful to Conselho National de Pesquisas (CNPq) and to Fundaqio de Amparo ~ Pesquisa do Estado de Silo Paulo (FAPESP) for financial support. He would like to thank Professor G.K. Batchelor for the hospitality extended to him at D A M T P in Cambridge.
Appendix A In this appendix we present a derivation of eq. (16). To this end, we first perform the k ° . . . . . /~0 integrations using the Cauchy theorem and find, in the non-rela-
J. Frenkelet al. / Infraredbehaviour tivistic
limit,
a
341
contribution proportional to
lf~dk
" A " m+" 1
k
fd3kl
×
yd3/~l
x
1
fd3k,~
k 2 Kx+ie"" 1
~2 K l - i e
1
1
k 2 Km+ieKl+" fd3/~n
• "'
1
"K~+k+ie 1
---'2
(A.1)
k. I ( ~ - i e R l + ' " I ( ~ + k - i e '
1 where A = aCv~aas and Ki = ki • q/qo. Choosing the z-axis along q, we can further perform the k~ integrations, writing k~ = (k ~)2 + (k~)=. Performing the angular integration and defining
t~k ? = y,k, we
t~k ? = z , k ,
can write for I m'" the following expression:
I""
(Aft ,)m+.fadkfOO d y l " ' ' d y ~ I ? dz,'''dz. re!n! Ja k J0 (iY0 (iY~) (-izl) (-iz.) x I+-°~l +d iYY f+-o~ °~ 1 Z ~ z g ( y - y l
.
.
.
.
ym)~(Z-Zl-"
"zn).
(A.2)
Introducing the representation for the g-function, .+oo
a(x)=~ and summing over m and
n, we
l ~ dc e icx,
obtain
' = l n ah ~+°°da f_T db exp {iA[3-' f ~ d~yy[eXp (-iby)-exp (-iay)]}
1 f+°"dYeiaVf+°°dZeibZ × ( ~ ) 2 o~ 1 + - ~ J-~ 1 - i Z With the help of the relations +ood Y
e laY
oo l ~ l Y - 217"e-aO(a) , ~o~ dZ e i b z o~ 1 - i Z = 2rr ebO(--b),
(A.3)
342
J.
Frenkelet al. / Infraredbehaviour
we o b t a i n from (A.3) the expression I = In
da e a
db
eb
co
XexpIiA~-l f;dY[cosby--cosay--isinby~-isinay]l-
,
Y
(A°4)
P e r f o r m i n g the y i n t e g r a t i o n s [12], we t h e n find, c h a n g i n g b ~ - b ,
I=lnde-=At3--A 110
dae-aaiAt3 110
dbe
bb-iAO-1
.
(A.5)
U s i n g the r e p r e s e n t a t i o n for the F f u n c t i o n / o oo
F(z)
= Jo dt e - ' t z-1 ,
(A.5) reduces to the following expression: I = In A e ~ra/3 ' F ( 1 + A
iA[3-1)F(1
- iA[3 -1)
d 2rrA/3 -1 = In A exp (2~rAB -1) - 1 "
(A.6)
This is precisely the g e n e r a t i n g f u n c t i o n for the B e r n o u l l i n u m b e r s [12] in e I - 1 - ~ =o 1
where Bo = 1. In this series all terms with odd index vanish except for B~ - - 2 .
Appendix B In this a p p e n d i x we discuss a m o r e g e n e r a l c o n f i g u r a t i o n of soft gluons, which is exemplified in the d i a g r a m s h o w n in fig. 12.
Fig. 12. Example of the soft skeleton graph, containing one transverse real gluon k and a total of N Coulomb lines kl . . . . . kN.
J. Frenkel et al. / Infrared behaviour
343
We denote by n the total n u m b e r of Coulomb lines which join the fermion lines outside the region delimited by the transverse gluon and the small circles. Then the colour factor associated with this graph is C V\IC F)~N-"IC ~" " \ F - - 2 1C YMJ
(B.1)
Consider now, apart from the colour factor, the infrared-divergent contribution resulting from this diagram. Summing over all configurations which correspond to a given value of n, and symmetrizing over the n outside lines as well as over the remaining N - n Coulomb lines, we obtain 1
1
D
rl! ( N - - t l ) !
1 N[ }~D (/3, N,n,a) (/3, N,n,o~)=-~.~n
•
(B.2)
H e r e D(/3, N, n, a) denotes the infrared-divergent contribution depending apart from /3, N, n, also on a set of parameters like the infrared cut-off, the coupling constant, etc, which we have designated by a. In order to understand how the function D might depend on its parameters, let us consider the situation in Q E D . As is well known [13], when summing over all possible configurations, the contribution arising from the soft real boson factorizes. The other part, coming from the soft C o u l o m b lines, gives a vanishing contribution to the cross section. Hence, using (B.2), the function D(/3, N, n, a ) must satisfy
n=0
for all values of/3, o~, N. A simple way of satisfying (B.3) is if
D(/3, N, n, a ) = (-1)riD(/3, N, a) .
(B.4)
At the end of this appendix, we shall prove that the simple relation (B.4) is in fact correct. Using (B.1), (B.2) and (B.4), we obtain that the infrared-divergent contribution corresponding to a given value of n is
In Q C D , as opposed to the case in Q E D , when the sum over all values of n we no longer get zero. Instead, because of the colour factors we find
CF(½CyM)NN-1.D (/3, N, or).
(B.6)
We thus see that we are left with a contribution proportional to (CyM) N, which corresponds to a graph having all Coulomb lines outside the real gluon line. This is precisely the case exemplified in fig. 8 which determines, as shown in appendix A, D(/3, N, a ) in terms of the Bernoulli numbers BN.
344
J. F r e n k e l e t al. / I n f r a r e d b e h a v i o u r
We now turn to the proof of our conjecture• To this end consider the configuration depicted in fig. 12, where l denotes the number of Coulomb lines outside the real line k on the right-hand side of the diagram• Consequently there will be n - l outside Coulomb lines on the left-hand side of the graph. Furthermore, if we denote by m the number of Coulomb lines inside the real line on the right-hand side of the diagram there will be N - ( n + m ) Coulomb lines inside the real line on the left-hand side of the diagram. Finally, let r be any integer between N - ( n + m) and N - ( l + m). Here N - ( l + m) denotes the total number of Coulomb lines on the left-hand side and we have r + j = N - ( n + m ) . We want to show that the contributions resulting from such a configuration are, apart from a factor ( - 1 ) " (N), independent of n. More precisely we will show that the integrands corresponding to these configurations are the same, except for the above factor, provided we symmetrize over the contributions arising from the Coulomb lines• In what follows this symmetrization is to be understood. Consider then the integrand resulting from such a configuration. Using a similar procedure to that in appendix A, and defining the variables xi via ¢ t k ~ =- x i k , we obtain N-r
I
~.. i=0
l
(--1) N
r-i+lD-l(x,
A =
where the summation over I is between max (0, N D(x,
h)
l)
(B.7a)
,
r) and min ( N - r, n) and
= X l ( X 1 n L x 2 ) • " " (X 1-1-" " " " ] - X n _ l ) ( , ~
- } - X l "nt-" " " - } - X n - l )
• " • ( A "~-X 1 nL . " • " t - X r ) " " " ( A - I - X 1 nt-" " " ~ - X r + j ) X N • • • (xN
+'.
• +XN-~+I)(XN
+"
• "+Xl,r
t+l--,~)
• • "(xN+" • " + x , + j + I - A ) .
(B.7b)
In our case A = i, but since the identity we want to prove is independent of this particular value, we consider A as a general parameter. We will show that these integrands depend on n only through a factor (-1)"(,u). To this end consider the pole at A = - x l . . . . Xr in (B.7). We will show that the residue at this pole is given by A ( - 1 ) ' [ ( x l ) • • • ( X N ) ( X 1 -}-" • • - I - X N ) ] 1 , (B.8) where A is a combinatorial factor which depends only on N and n. Using (B.7) we obtain that the residue is
E N -Z, - I (--1)N-r+"+1(--1)/+~ ( n r ) t j=0
l
D"-1 (x, a)
(B.9a)
'
where a - x l + • • • + Xr and l ~ ( x , a ) = ( x O " " " ( x , ) ( X , + l ) . . . (Xr+I + " " " + X r + k ) ( X u ) • " " ( X N -1-" " " J r - X N - I + I ) ( X N
• • • (xN +" • "Xr+j+l + a ) .
-}-" " " J C X N - - I + I
+a)
(B.9b)
J. Frenkel et al. / Infrared behaviour
345
Here the symmetrization over x1, . . . , x, has been performed explicitly and (B.9) new variables via is understood to be symmetrized over xrtl, . . . , xN. Introducing yl = XN, . . . , YN-r = X,+ 1, and comparing (B.8) and (B.9) we need to show that c N$-’ (-I)‘-‘( 1
n ;
j=O
I)yi
. * ’ y&,fiP’(y,
a) = A(-l)N+n+l[a
+ y1 +.
. * + yN_,]-’
,
(B.lOa) where d(y, &Y,
a) is given by a)=(yd..
’ (y1+.
* *+y,)(a
* ’ * (a+yl+*
*
*
+
+y1+.
* *+yr)
YN-r_j)(yN-r)
*
*
*
(YN-~
+
*
’
’
+
YN_,-j+l)
.
(B.lOb)
TO this end let us compare the poles on both sides of this equation at a = -(y1 f. . . + y,), where p is any integer between I and N-r-j. Symmetrizing the left-hand side in the variables yi, for i
THRESHOLD INFRARED B E H A V I O U R IN PERTURBATIVE QCD J. FRENKEL and M.L. FRENKEL Instituto de Ffsica, USP, Sdo Paulo, Brasil and DAMTP, University of Cambridge, Cambridge, England J.C. TAYLOR DAMTP, University of Cambridge, Cambridge, England Received 11 May 1981 We study a class of Bloch-Nordsieck uncancelled infrared divergences which are relevant near threshold in inelastic reactions between coloured particles. We show that the dominant threshold contributions factorize, in the cross section, from the leading non-cancelling infrared singularities.
1. Introduction It has b e e n shown [ 1 - 7 ] that the B l o c h - N o r d s i e c k [8] m e c h a n i s m for the cancellation of I R divergences does not work in general for the scattering of coloured massive quarks or antiquarks in perturbative Q C D . In particular, in refs. [6, 7] we s u m m e d all I R divergences of leading non-cancelling o r d e r as(as In A) ~, where as = g2/47r is the strong coupling constant and h d e n o t e s an I R cut-off. T h e aim of the present p a p e r is to study the behaviour of non-cancelling I R divergences b e y o n d the leading order. W e have not been able to do this in general. Instead we take advantage of simplifications which occur near threshold, that is to say for B << 1, w h e r e / 3 is the speed of one q u a r k in the rest frame of the other. W e seek the non-leading divergences of o r d e r /3 0a s2 (as In h)". In fact, we go further and sum contributions of o r d e r /3as(as/3-1)m (as In h )~.
(1)
As a guide 'to our search for terms of this order, we note that each transverse gluon gives a factor /32, whereas each C o u l o m b exchange can give a factor /3-1 [6, 7]. Consequently, we expect the right sort of b e h a v i o u r to c o m e f r o m graphs containing one transverse gluon and m a n y C o u l o m b exchanges. H o w e v e r , we find (see appendices A and B) that such graphs give zero contributions, provided all the C o u l o m b m o m e n t a are soft, so that the eikonal approximation m a y be made. But, one must r e m e m b e r that the hard parts of the graph (with m o m e n t a exceeding s o m e arbitrary small a m o u n t / t ) , in which the eikonal a p p r o x i m a t i o n is 328
329
J. Frenkel et al. / Infrared behaviour
C::I
q"
C:l
q
Ca)
(b)
Fig. 1. The small circles represent the virtual photon y* which attaches to the quark (q) and antiquark (q). The wavy line denotes a soft gluon and the dashed-dotted line stands for a Coulomb line. The right-hand side of graphs represent the complex-conjugate part of the corresponding diagram.
inadmissible, can also give inverse powers of/3. We therefore obtain contributions of the form (1) and we find that they factorize (see below). For definiteness we study the inelastic reaction involving two massive particles of mass M : q+Cl~ 3'* +soft gluons,
(2)
which is the simplest process exhibiting Bloch-Nordsieck non-cancellation. We use the Coulomb gauge in the rest frame of the antiquark Ct, and /3 is the speed of the quark q. As shown in [6], graphs like fig. l a have at least one more power of /3-1 than graphs with three-gluon vertices like fig. lb; so in the remainder of the paper we ignore skeleton graphs containing three-gluon vertices [but, of course, we do consider renormalization graphs involving these vertices]. Consider a more general class of graphs exemplified in fig. 2a, containing n Coulomb exchanges. In sect. 2 we show that they factorize into the product of the two classes of graphs shown respectively in figs. 2b, 2c. The first class contains only hard Coulomb lines and yields precisely the expression for the non-relativistic inelastic Coulomb cross section [9] (see also appendix C):
[
l
O-i"=O'B°rn 1--exp(--I
i] '
(3a)
where l CF/3-12"n'CZsand the Casimir invariant CF for the representation associated with the quarks is defined by CFt~ab = Tr tatb. The second class contains only two soft gluons, one being a transverse gluon and the other a Coulomb exchange, as shown in fig. 2c. This contribution can be calculated in the eikonal approximation [1-6] and gives, in the non-relativistic limit, the result =
A ~CFCyM/3O/s
= ~CFCyM~O/s
In
.
Here GyM is the Casimir invariant of the adjoint representation defined by
[,,bcfabd, where fabc are the structure constants.
(3b) CTMScd =
330
J. Frenkel et al. / Infrared behaviour
(a)
Cb)
C¢)
Fig. 2. (a) A typical diagram which gives contributions of order (1). Dotted-dashed lines denote soft or hard Coulomb exchanges. (b) and (c) represent the factors resulting from (a). Dashed (dotted) lines stand respectively for hard (soft) Coulomb exchanges.
In sect. 3 we study the effects resulting f r o m the inclusion of the r e n o r m a l i z a t i o n graphs, which lead to the a p p e a r a n c e of the running coupling constant. In the leading logarithm a p p r o x i m a t i o n , which is consistent with the terms considered in (1), as(k/l*) is given by [10]
as
- 1 + (11/6rr)CyMas(ix) In ( k / ~ ) '
(4)
w h e r e ~ is an arbitrary mass introduced in the renormalization process. W e find that, in this a p p r o x i m a t i o n , the factorization previously m e n t i o n e d continues to hold, resulting in graphs like the one shown in fig. 3. T h e inclusion of the running coupling constant in the hard C o u l o m b diagrams (see fig. 3a) leads to an expression similar to that of eq. (3a), except that as is replaced by the effective coupling constant a s ( t i M / # ) : ~rin = O'Born[ 1 -- ex~ ( - i ) l
(5a)
where l = CFfl-127ro~(BM//x).
(a)
(b)
Fig. 3. The blobs represent schematically the inclusion of vertex and self-energy corrections into the skeleton graphs.
J. Frenkel et al. / Infrared behaviour
331
Turning now to the contributions resulting from soft graphs like fig. 3b, one obtains [7] that the effect of the running coupling constant results in the following expression, as a -* 0 [compare with (3b)]:
a dko2(k) where we have used a principal value prescription. In the non-relativistic limit, when ~-las(M~/l~) >> 1, we can neglect the exponential function in (5a). Multiplying then the expressions (5a) and (5b) we obtain: / - 8 7 r 2 ' ~ C 2 a { M[3"~a { A'~
OrBorn~k--H--)
F s~---)
s~7 ) ,
(5C)
which represents a finite non-zero result. Finally, we discuss at the end of sect. 3 the possibility of extending these results to the case of the inelastic scattering of two quarks. One of the motivations for our work has been to see whether the sum of an infinite n u m b e r of non-cancelling infrared divergences could give a finite result. This expectation is based on the unitarity of the theory. Indeed, if one could sum exactly all the classes of non-cancelled infrared singularities the complete answer must be finite since the S-matrix elements satisfy E I&nl 2 = a .
n
However, we have been able to sum only the uncancelled infrared divergences corresponding to a particular class relevant in the non-relativistic limit. For this class, it was consistent to use the leading logarithm approximation for the effective coupling constant as(k). Nevertheless, one must r e m e m b e r that in this case as(k) may become negative for small enough values of k, behaviour which is unsatisfactory from a physical point of view.
2. Factorization of skeleton diagrams We begin this section by briefly describing a convenient method for the calculation of uncancelled infrared divergences in process (2). (For more details see refs. [6, 7].) This method is based on the observation that in the reaction q + ft + soft gluons-~ y*
(6)
the infrared divergences cancel out, essentially as a consequence of unitarity. But note that in this case the soft gluons are incoming, as opposed to reaction (2) where they are outgoing. Let us denote the graphs containing the outgoing real gluons contributing to (2) by R °ut, and the graphs containing the ingoing real gluons contributing to (6) by R i". Furthermore, we denote by V the purely virtual graphs
332
J. Frenkel et al. / Infrared behaviour
k
(a)
(b)
(c)
(d)
Fig. 4. (a) and (b) represent graphs which give a~ In A uncancelled infrared divergences. (c) and (d) denote lowest order non-trivial diagrams contributing aJ3 -1 terms to the inelastic Coulomb cross section.
contributing to q + q ~
y*. T h e n we can write for reaction (2): R °u' + V = D ,
(7)
where D d e n o t e s the uncancelled infrared divergences, while for process (6) we have instead R i" + V = 0 .
(8)
T h u s we see that we can obtain D as follows: D
---- R ° U t - R
in .
(9)
This f o r m u l a is convenient because using this m e t h o d we do not have to calculate the purely virtual graphs, which are usually m o r e difficult to c o m p u t e . F u r t h e r m o r e , when all gluons are soft, R in c a n be obtained directly f r o m R ° u t j u s t by changing the signs of all ie in the fermion propagators. Since, for the would be leading infrared divergences of the f o r m (as In A)n, we k n o w that we can disregard these ie, eq. (9) implies that these divergences will always cancel, to all orders. Thus, we are left with the divergences of the form a~+"(lnA)" which are uncancelled in general, since in this case the difference in sign of the ie in R °ut and R in is crucial. F o r instance, consider the graphs shown in figs. 4a, b, where both gluons k' and k are soft. W e k n o w that the sum of these two yield in R ° U t - R in a single infrared divergence of the form a 2 ~ u -1 du, which arises from the region where both k and k' are simultaneously scaled to zero by a c o m m o n factor u. That is, each one of the m o m e n t a k or k' is controlling. In order to understand the factorization p r o p e r t y described in sect. 1 consider next the diagrams shown in fig. 5, where h d e n o t e s a hard gluon. Using the notation H = q o l q • h, K = q o l q • k , etc., after p e r f o r m i n g the k~, ho integrations, we obtain that the hard parts of these graphs are respectively p r o p o r -
333
J. Frenkelet al. / Infrared behaviour k
k
<.
(a')
<<-2
(b)
(b')
Fig. 5. Sixth order diagrams which exemplify the factorization property. tional to
[qolh2+H+K'+K+k+ie] [qolhZ+H+K+k+ie]
-1,
-1 ,
[qolh2+H+K+k-ie]
-1 ,
(10a)
[qolh2+H+K'+K+k-ie]
-1 .
(10b)
Since k' is controlling in diagrams (a) and (b'), we may replace these expressions by
[qolh2+H+K+k+ie]
-1,
[qolh2+H+K+k-ie]
-1 ,
(lla)
[qolh2+H+K+k+ie]
-1,
[qolh2+H+K+k-ie]
-1 .
(llb)
Thus each of the graphs of figs. 4a, b is multiplied by
[qolh 2 + H + K + k +ie]-l + [ q o l h 2 + H + K + k - i e ] -~ .
(12a)
If we neglect here the soft m o m e n t a k compared with the hard one h, we obtain
[qoah 2 + H + ie] -1 + [qolh 2 + H - ie] -1 .
(12b)
which corresponds to the contribution arising from diagrams 4c and 4d. This would lead to the factorization property mentioned in the introduction. (Note that for the abelian graphs it is actually a property valid for all values of/3.) The error made in using the above approximation is
{[qolh2+H+K+k+ie]
a+[qolhZ+H+K+k_ie]
a
- [ q o l h 2 + h +iv] 1 - [ q o l h Z + H - i e ] - l } [ ( 4 a ) + ( 4 b ) ] .
(13)
But, since either k or k' is controlling in [(4a) + (4b)], this is an infrared convergent expression, and hence not relevant for our purposes. Consider now a higher order graph, shown in fig. 6, where k, kl, k2 are soft m o m e n t a , and h denotes a hard gluon. The hard parts of these graphs are respectively proportional to
[qolh2+H+Kl+K2+K+k+ie]-l,
[qolhZ+H+K+k_ie]
1,
(14a)
334
J. Frenkel et al. / Infrared behaviour k
(a)
:
(a')
,
i
(b)
(b')
(c)
(c')
i
Fig. 6. Example of an eighth order factorizable diagram.
[qoah2+H+K2+K+k+ie]-1, [qoXh2+H+Kx+K+k-ie] -1, [qoahz+H+K+k+ie]-1, [qolhZ+H+Kl+Kz+K+k-ie] -1.
(14b) (14c)
We observe that, since kl is controlling in fig.'6a, we can leave out Ka in first terms of eq. (14a). Similarly since in fig. 6a' ka is controlling, we can insert K~ in the second denominator in (14a). Proceeding in a similar way, we insert K2 in the first denominator in (14c), and neglect Kz in the second denominator. Then we obtain the factor
[qolh2+H+Kz+K+k+ie]-X+[qo~h2+H+Kl+K+k-ie]-l,
(15)
which multiplies the integrands corresponding to the graphs shown in fig. 7. In the sum of graphs any one of the m o m e n t a k, kl or k2 is controlling, so we can replace
(a)
(b)
(c) Fig. 7. Sixth order soft diagrams resulting from the factorization of graphs corresponding to fig. 6.
J. Frenkel et al. / Infrared behaviour
335
~J
Fig. 8. Example of a general graph whose contributions cancel in R ° u t - R ~n.
(15) by eq. (12b), which corresponds to the contributions resulting from fig. 4c + 4d. Thus we have shown the factorization of diagrams in fig. 6 into classes containing only soft or hard gluons, respectively. Actually, as we will now show, the contributions resulting from the graphs of fig. 7 cancel in R ° U t - R in, so that only graphs with a single soft Coulomb line with the structure shown in fig. 2c will contribute to the dominant infrared divergences. The above property results when considering a general graph like the one of fig. 8, which contains m + n soft Coulomb lines. Graphs obtained from fig. 8 by moving the transverse line k in all possible positions give essentially the same contribution [i.e. apart from a combinatorial constant and colour factors (see appendix B)]. When considering together all such graphs, we find that all contributions except for the one proportional to (CyM) m÷" cancel out. The contribution proportional to (CvM) m-n, which results precisely from the diagram shown in fig. 8, is calculated in appendix A. We show there that, when summing over all values of m and n, we obtain for R °ut a contribution which has only a single infrared divergence of the form I f dk[ 27rA/3 -1 -k-[exp ~ 1 ) _
1] =In ( ~ ) [ 1
1 2rrA/3 -1 ~ ~. + " "]
(16)
1 where A = aCyMas. In this series which defines the Bernoulli numbers (see appendix A) all terms with odd index vanish, except for the lowest order one. For R i", which can be obtained from R °ut by changing the sign of ie in the fermion denominators, we obtain a similar series, except that A is replaced by - A . Hence, in R out_ R in, only the term proportional to as, which corresponds to graphs with a single Coulomb soft line, does contribute. More complicated graphs can be treated in a similar way in order to verify the factorization property discussed in the preceding section. This property depends essentially on the fact that when all gluons are soft, general graphs like the one shown in fig. 12 (see appendix B) do not contribute to R ° U t - R in. Consider now the diagrams in figs. 4c, d, which contribute the lowest order corrections to the inelastic Coulomb cross section. They yield a contribution given by oe 3 as f d h 1 (17a)
2O'Bo,nCFM~-~2 Ja -~ h2+q . h "
J. Frenkelet aL / Infraredbehaviour
336
(a)
(b)
(c) Fig. 9. Diagrams of order a~ which contribute to the inelastic Coulomb cross section.
Choosing the z axis along q and doing the angular integrations we obtain, using Iq[ = M 3 , 1
~
h+Mfl
O'aoroCF2;S-~fa ? l n l h _ M
fl
=O'B°rnCF2OLs ~77" ~--~-O ~'~ where we have chosen A <
[ Cvas~ 2~r2 CrBom~---ff--) y .
(18)
Therefore, to this order, adding the Born diagram contribution with the one resulting from (17b) and (18), we obtain
r
1 Cvas2~r 1
O'"°rn[1-+2
--/3
1/CFas2"rr]2
+-62!~-- -fl
"] +" " "J] "
(19a)
The general pattern is clear: for each Coulomb line we get a factor Cva~2rr/[3 with a coefficient which is characteristic of the Bernoulli numbers. This suggests that the general result for the inelastic Coulomb cross section is
CFas2 7r/ 3 ~ra°r"l -- exp (-- Cvas2zr/3)"
(19b)
Indeed, this is in accordance with the result obtained in non-relativistic quantum mechanics for the inelastic scattering of slow particles in a Coulomb potential [9]. In that case, the reaction cross section is obtained from ]~bc(O)[2, where ~c(X) is the
J. Frenkel et al. / Infrared behaviour
337
exact wave function of a particle in a Coulomb field. This then leads precisely to eq. (19). Consider now the leading infrared contributions arising from graphs like 2c, having one soft Coulomb line and one soft transverse gluon. As shown in ref. [6] the infrared-divergent contributions resulting from these fourth-order soft diagrams are 2 Of s
2 C F C y M ~ - F ( f l ) In A ~-.
(20a)
Here F(/3) denotes the bremsstrahlung probability function: ~] F(/3) = 2-~ In \(1~ -+/3~
-
1 =
1~2 + . . . .
(20b)
Eq. (20) was derived using the eikonal approximation. In this approximation one neglects k 2 compared with k ' q in the fermion denominators. For consistency, remembering that ksoft < A, we must therefore satisfy in the non-relativistic limit the condition A <
3. Discussion
of running
coupling
effects
In this section we wish to study the effects of the inclusion of self-energies and vertex UV-divergent graphs, which account for the appearance of the running coupling constant. To this end consider the diagrams in fig. 10, which represent corrections to the Coulomb line. As is well known, in virtue of the Ward identities in the Coulomb gauge, the contributions resulting from diagrams ( b ) . . . (e) cancel out, and the self-energy graph in fig. 10a gives the complete coupling constant renormalization [11]. For the transverse lines [7] both vertex and self-energy corrections are necessary in order to obtain the running coupling constant % ( k / t z ) which is given in the leading
~} i
,
(a)
i
"
I
,
(b)
(d)
i I
(c)
(e)
Fig. 10. Example of self-energy and vertex corrections associated with the Coulomb line. Graph (a) includes all second-order self-energy parts constructed from Coulomb and transverse gluons.
Z Frenkel et al. / Infrared behaviour
338
approximation by as
- 1 + (11CvM/6"tr)O~s(kt ) In =a~(tt)[l
(k/#)
l16-C~vMad/.t)ln(k)+...],
(21)
where/z is the arbitrary mass unit introduced in the renormalization process. It is not difficult to see that in this approximation all the previous arguments for the factorization of graphs are still valid. For instance consider the effect of (21) on eq. (16). In order to obtain the leading uncancelled infrared divergences we rescale all momenta in fig. 8 by a common factor u and keep only the dominant contributions as u ~ 0: In ki =- In
/'//)i ~
In u.
(22a)
In this case the term proportional to the nth Bernoulli number B , in (16) will now be multiplied by a term proportional to [cadu)] p p =o
-(lnu) fxa-du b/
p~
p = ~ [ca~(#)] p /7=0
[(ln A)P+I-(lnA)P+I].
(22b)
Clearly the same modification occurs also in _Ri', so that in R ° U t - R i" only contributions proportional to B1 remain. These correspond precisely to corrections arising from self-energy insertions in one soft Coulomb line, as exemplified in fig. 3b. Furthermore, the leading logarithm expression (21) for the running coupling constant is consistent with the order of the terms considered in (1) for the calculation of non-leading uncancelled infrared divergences. We will now consider the effect of this coupling constant in the graphs shown in fig. 11, which contribute to the inelastic Coulomb cross section. Diagrams (a) and
(a)
(b)
Fig. 11. Example of hard Coulomb diagrams representing the effect of the running coupling constant given by eq. (4).
J. Frenkel et al. / Infrared behaviour
339
(b) give the contribution [compare with eq. (17)]
M~+h ~rB°r~CF~ ~ Ja
In M----~_ h ,
l+Cln(h/iz
(23a)
where 11
C = 7 - - CyMO/s(/2 ) . o7)"
Defining a new variable x via h = M/3x and neglecting terms of order A/M/3, we obtain 2CF f ~ d x [ ces(/z) °'n°rn 7r/3 J0 x - l + C l n ( M / 3 / l x ) + C l n x
]In
x+l -x--~-l"
(23b)
In order to evaluate this integral we expand the first bracket in a series of powers of In x and integrate term by term. We find
2Cv "7/'2i-.+1 . . . . 2 + 1 . . . . 4 + O'Bor, ~.---~as~---~-] ~ - [ l ~Trt. ) 3 ~ - ~ ) "" '],
(24)
where 1
In the non-relativistic limit, as /3 ~ 0 we have C'<< 1. In this case the dominant contributions to (24) arise just from the first term in the bracket which yields OrBorn7"/'CF~ a s ( - - ~ ) .
(25)
Note that (25) can be obtained directly from (17b) by replacing in that equation as by the effective coupling constant ces(M/3/tz). Similarly, graphs (c) and (d) will lead to a contribution like (18), except that in this case as will be replaced to as(M/3/lx). Clearly, given the above approximations, this generalizes to higher orders and we obtain for the (hard) inelastic cross section the result [compare with (19)]
Cv/3 las(M/3/iz)2~r O-Bor, {1 - exp [-CF/3-1as(M/3/tz)27r]} "
(26a)
If the relation/3-1a~(M/3/tz) >>1 is satisfied in the non-relativistic limit, then eq. (26a) reduces to O'Bo~nCe/3-1as ( ~ )
27r.
(26b)
340
J. Frenkel et al. / Infrared behaviour
The inclusion of running coupling (4) (see fig. 3b) has also the effect of modifying (20) into an IR finite expression given in eq. (5b). The product of this with (26b) gives
--877-2 OrBorn~ - - C2as(~) as(~ -~)
(27)
which represents a finite, non-zero result. [See the remarks following eq. (5c).] These results can presumably be extended to the case of inelastic scattering of two quarks. We must then replace as by - a s since the charge of the antiquark is opposite to that of the quark. Furthermore, in the above equations, CF will be replaced by the matrix ta ® Ta, where ta and Ta are the matrix generators of the representations to which the two quarks belong. Then we obtain for Orqq i.
(-ta ® Ta)~-las(M[3/lz )27r
O'B°r"{1--exp [t~ ® Tafl-las(M[3/tz)2Tr]} "
(28)
When the condition fl-las(Mfl/ix)>> 1 is satisfied, the behaviour of this relation depends on the colour state and representation of the quark-quark system. If the system is in an eigenstate of ta ® Ta with negative eigenvalue, (28) reduces to
O-Born(-ta ® Ta)~-las(-M~--)2~
(29)
[compare with eq. (26b)]. This is similar to the quark-antiquark system. On the other hand, if the quark-quark system is in an eigenstate of ta ® Ta with positive eigenvalue, (28) reduces to
O-Born(-ta ® ra)~-las(-h~-)27r exp [ (-ta @ Ta)~-lce~(-~-)27r] .
(30)
In this case the cross section tends to zero. The exponential factor by which (29) [or (26b)] and (30) differ can be interpreted, in analogy with the quantum mechanical case [9], as the probability of passage through the effective Coulomb potential barrier. We are grateful to R. Taylor for his proof presented in appendix B as well as for numerically checking (C.9). J.F. is grateful to Conselho National de Pesquisas (CNPq) and to Fundaqio de Amparo ~ Pesquisa do Estado de Silo Paulo (FAPESP) for financial support. He would like to thank Professor G.K. Batchelor for the hospitality extended to him at D A M T P in Cambridge.
Appendix A In this appendix we present a derivation of eq. (16). To this end, we first perform the k ° . . . . . /~0 integrations using the Cauchy theorem and find, in the non-rela-
J. Frenkelet al. / Infraredbehaviour tivistic
limit,
a
341
contribution proportional to
lf~dk
" A " m+" 1
k
fd3kl
×
yd3/~l
x
1
fd3k,~
k 2 Kx+ie"" 1
~2 K l - i e
1
1
k 2 Km+ieKl+" fd3/~n
• "'
1
"K~+k+ie 1
---'2
(A.1)
k. I ( ~ - i e R l + ' " I ( ~ + k - i e '
1 where A = aCv~aas and Ki = ki • q/qo. Choosing the z-axis along q, we can further perform the k~ integrations, writing k~ = (k ~)2 + (k~)=. Performing the angular integration and defining
t~k ? = y,k, we
t~k ? = z , k ,
can write for I m'" the following expression:
I""
(Aft ,)m+.fadkfOO d y l " ' ' d y ~ I ? dz,'''dz. re!n! Ja k J0 (iY0 (iY~) (-izl) (-iz.) x I+-°~l +d iYY f+-o~ °~ 1 Z ~ z g ( y - y l
.
.
.
.
ym)~(Z-Zl-"
"zn).
(A.2)
Introducing the representation for the g-function, .+oo
a(x)=~ and summing over m and
n, we
l ~ dc e icx,
obtain
' = l n ah ~+°°da f_T db exp {iA[3-' f ~ d~yy[eXp (-iby)-exp (-iay)]}
1 f+°"dYeiaVf+°°dZeibZ × ( ~ ) 2 o~ 1 + - ~ J-~ 1 - i Z With the help of the relations +ood Y
e laY
oo l ~ l Y - 217"e-aO(a) , ~o~ dZ e i b z o~ 1 - i Z = 2rr ebO(--b),
(A.3)
342
J.
Frenkelet al. / Infraredbehaviour
we o b t a i n from (A.3) the expression I = In
da e a
db
eb
co
XexpIiA~-l f;dY[cosby--cosay--isinby~-isinay]l-
,
Y
(A°4)
P e r f o r m i n g the y i n t e g r a t i o n s [12], we t h e n find, c h a n g i n g b ~ - b ,
I=lnde-=At3--A 110
dae-aaiAt3 110
dbe
bb-iAO-1
.
(A.5)
U s i n g the r e p r e s e n t a t i o n for the F f u n c t i o n / o oo
F(z)
= Jo dt e - ' t z-1 ,
(A.5) reduces to the following expression: I = In A e ~ra/3 ' F ( 1 + A
iA[3-1)F(1
- iA[3 -1)
d 2rrA/3 -1 = In A exp (2~rAB -1) - 1 "
(A.6)
This is precisely the g e n e r a t i n g f u n c t i o n for the B e r n o u l l i n u m b e r s [12] in e I - 1 - ~ =o 1
where Bo = 1. In this series all terms with odd index vanish except for B~ - - 2 .
Appendix B In this a p p e n d i x we discuss a m o r e g e n e r a l c o n f i g u r a t i o n of soft gluons, which is exemplified in the d i a g r a m s h o w n in fig. 12.
Fig. 12. Example of the soft skeleton graph, containing one transverse real gluon k and a total of N Coulomb lines kl . . . . . kN.
J. Frenkel et al. / Infrared behaviour
343
We denote by n the total n u m b e r of Coulomb lines which join the fermion lines outside the region delimited by the transverse gluon and the small circles. Then the colour factor associated with this graph is C V\IC F)~N-"IC ~" " \ F - - 2 1C YMJ
(B.1)
Consider now, apart from the colour factor, the infrared-divergent contribution resulting from this diagram. Summing over all configurations which correspond to a given value of n, and symmetrizing over the n outside lines as well as over the remaining N - n Coulomb lines, we obtain 1
1
D
rl! ( N - - t l ) !
1 N[ }~D (/3, N,n,a) (/3, N,n,o~)=-~.~n
•
(B.2)
H e r e D(/3, N, n, a) denotes the infrared-divergent contribution depending apart from /3, N, n, also on a set of parameters like the infrared cut-off, the coupling constant, etc, which we have designated by a. In order to understand how the function D might depend on its parameters, let us consider the situation in Q E D . As is well known [13], when summing over all possible configurations, the contribution arising from the soft real boson factorizes. The other part, coming from the soft C o u l o m b lines, gives a vanishing contribution to the cross section. Hence, using (B.2), the function D(/3, N, n, a ) must satisfy
n=0
for all values of/3, o~, N. A simple way of satisfying (B.3) is if
D(/3, N, n, a ) = (-1)riD(/3, N, a) .
(B.4)
At the end of this appendix, we shall prove that the simple relation (B.4) is in fact correct. Using (B.1), (B.2) and (B.4), we obtain that the infrared-divergent contribution corresponding to a given value of n is
In Q C D , as opposed to the case in Q E D , when the sum over all values of n we no longer get zero. Instead, because of the colour factors we find
CF(½CyM)NN-1.D (/3, N, or).
(B.6)
We thus see that we are left with a contribution proportional to (CyM) N, which corresponds to a graph having all Coulomb lines outside the real gluon line. This is precisely the case exemplified in fig. 8 which determines, as shown in appendix A, D(/3, N, a ) in terms of the Bernoulli numbers BN.
344
J. F r e n k e l e t al. / I n f r a r e d b e h a v i o u r
We now turn to the proof of our conjecture• To this end consider the configuration depicted in fig. 12, where l denotes the number of Coulomb lines outside the real line k on the right-hand side of the diagram• Consequently there will be n - l outside Coulomb lines on the left-hand side of the graph. Furthermore, if we denote by m the number of Coulomb lines inside the real line on the right-hand side of the diagram there will be N - ( n + m ) Coulomb lines inside the real line on the left-hand side of the diagram. Finally, let r be any integer between N - ( n + m) and N - ( l + m). Here N - ( l + m) denotes the total number of Coulomb lines on the left-hand side and we have r + j = N - ( n + m ) . We want to show that the contributions resulting from such a configuration are, apart from a factor ( - 1 ) " (N), independent of n. More precisely we will show that the integrands corresponding to these configurations are the same, except for the above factor, provided we symmetrize over the contributions arising from the Coulomb lines• In what follows this symmetrization is to be understood. Consider then the integrand resulting from such a configuration. Using a similar procedure to that in appendix A, and defining the variables xi via ¢ t k ~ =- x i k , we obtain N-r
I
~.. i=0
l
(--1) N
r-i+lD-l(x,
A =
where the summation over I is between max (0, N D(x,
h)
l)
(B.7a)
,
r) and min ( N - r, n) and
= X l ( X 1 n L x 2 ) • " " (X 1-1-" " " " ] - X n _ l ) ( , ~
- } - X l "nt-" " " - } - X n - l )
• " • ( A "~-X 1 nL . " • " t - X r ) " " " ( A - I - X 1 nt-" " " ~ - X r + j ) X N • • • (xN
+'.
• +XN-~+I)(XN
+"
• "+Xl,r
t+l--,~)
• • "(xN+" • " + x , + j + I - A ) .
(B.7b)
In our case A = i, but since the identity we want to prove is independent of this particular value, we consider A as a general parameter. We will show that these integrands depend on n only through a factor (-1)"(,u). To this end consider the pole at A = - x l . . . . Xr in (B.7). We will show that the residue at this pole is given by A ( - 1 ) ' [ ( x l ) • • • ( X N ) ( X 1 -}-" • • - I - X N ) ] 1 , (B.8) where A is a combinatorial factor which depends only on N and n. Using (B.7) we obtain that the residue is
E N -Z, - I (--1)N-r+"+1(--1)/+~ ( n r ) t j=0
l
D"-1 (x, a)
(B.9a)
'
where a - x l + • • • + Xr and l ~ ( x , a ) = ( x O " " " ( x , ) ( X , + l ) . . . (Xr+I + " " " + X r + k ) ( X u ) • " " ( X N -1-" " " J r - X N - I + I ) ( X N
• • • (xN +" • "Xr+j+l + a ) .
-}-" " " J C X N - - I + I
+a)
(B.9b)
J. Frenkel et al. / Infrared behaviour
345
Here the symmetrization over x1, . . . , x, has been performed explicitly and (B.9) new variables via is understood to be symmetrized over xrtl, . . . , xN. Introducing yl = XN, . . . , YN-r = X,+ 1, and comparing (B.8) and (B.9) we need to show that c N$-’ (-I)‘-‘( 1
n ;
j=O
I)yi
. * ’ y&,fiP’(y,
a) = A(-l)N+n+l[a
+ y1 +.
. * + yN_,]-’
,
(B.lOa) where d(y, &Y,
a) is given by a)=(yd..
’ (y1+.
* *+y,)(a
* ’ * (a+yl+*
*
*
+
+y1+.
* *+yr)
YN-r_j)(yN-r)
*
*
*
(YN-~
+
*
’
’
+
YN_,-j+l)
.
(B.lOb)
TO this end let us compare the poles on both sides of this equation at a = -(y1 f. . . + y,), where p is any integer between I and N-r-j. Symmetrizing the left-hand side in the variables yi, for i
p, we obtain for its residue
while the residue
of the right-hand
side gives
N+n+l
C-1)
A,
forp=N-r, (B.llb) forp
0, We note that the sum over j in (B.lla) case, (B.lla) reduces to min(kNpr) f=max
(n-r,
is zero unless
r
( 0)
we have p = N - r. In this
,)(N;r),
(B.12a)
n-
Using mathematical induction, together with the convention that (G)= 0 when not usually defined, we can show that the above sum is actually independent of r. With m = n - 1, we obtain:
,=m:$+;:,O, Comparing
with (B.llb)
factor independent
kkN--;>
= (:)
we see that this relation
determines
(B.12b)
’
A as a combinatorial
of r
A~(-l)~+“+l
(> ;
.
Together with (B.S), this result shows that the infrared-divergent corresponding to distinct values of n differ only by a factor (-l)“(E), with (B.2) and (B.4).
(B.13) contributions in accordance
J. Frenkel et al. / Infrared behaviour
346
Appendix In this appendix
we calculate
C
the contributions
resulting
from the diagram
shown
in fig. 9. Since, as we will show explicitly, the infrared divergences associated with the Coulomb phase cancel in the cross section, we have found it more convenient to extend the integrations of h1 and hZ over the whole gives, after performing the h: and hi integrations,
range.
Then
graph
(9a)
1
X
(C.1)
(h~+h2)[(hl+h2)*+(I.(hl+h2)-i&]’
It will be useful to perform a shift of variables hi+ hi -$I. Then we perform angular integrations d& choosing the z axis along (the new) hi. Next, doing angular dRi integrations by taking the z axis along q, we find
dh2A
X
With the change
h2+h
of variables dx x2-l-_-n
In
(hl+h2)2-~$q2-i~ (hl-h2)2-+q2-iE
I ’
hl = iqx; h2 = :qy and h* = 4h2/M2P2, ln
(x+1)2+‘2 (x-l)2+/\2
the the
II
mdY 2Y_21n 0 Y +A
we get
(y+x)2-1-i7j (Y-x)2-i-in
I
(C.2) We now half-plane as well as calculation
extend the y integration from ---COto +CO and perform it in the upper with the help of Cauchy theorem. Here we encounter a pole at y = ih cuts coming from the logarithmic function. After a straightforward we find 2
cc
cF%
(7)
I0 X&i
[
In
1
1
dx
X2-l+/ (iA+x)2-1-i?j
I
(ih-x)2-1-i?J
+ln I
(X+J1+i71)2+h2 I (-X+J1+iTj)*+X2
II .
(C.3)
We recall that we are actually interested in the real part of this expression, as the relevant contribution to the cross section must be real. The part that results from 8(x2 - 1) in the first denominator gives (C.4)
347
J. Frenkel et al. / Infrared behaviour
We turn now to the real part resulting from the principal value of (x 2 - 1) -1, which is more difficult to calculate. Only the first logarithm in the bracket of (C.3) will in this case give a relevant contribution which is
(--C~) 2
(-~')
IO~
dx
1
((Xq-1)2q-A2~
2 In x -1 ( x - 1 ) 2 + A 2]
X [F0 ( l q- ~2 -- X2),/T q._arctan
{
)I
2Xx
\x z _ ~2 _
1}J "
(C.5)
After some calculation we obtain, for the part coming from the O-function, the result
(CFas~ \ 7rfl ]
2 2 _~._[3(1n X2)2_3 In 2
In X2+3(ln 2)2- 1A2"n'2].
(C.6)
Z
In order to calculate the part arising from the arctangent function in (C.5) we first note that since it is strongly peaked along x ~ 1, we can effectively restrict the range of integration to 1 - a < x < 1 + a, where a is a convenient p a r a m e t e r satisfying a >>~. Furthermore, defining a new variable z = (x - 1 ) / ] we can write this part, up to corrections of order (A/a) as follows: (CFOLs~2--7T " + ~ d z ~r31 2 J_~ ~ - l n The contribution coming from relation [12]:
IOa
[X2(Z4Q_
1)] arctan ( ~ ) .
In (4/A 2)
(C.7)
can be calculated with the help of the
1
2 +q2
dXx(l_p2xa) arctan qx =¼7rlnP P2
so that we find for it
(C
q2- a(lnM2 2" 2
~-3 ]
4 \
--7-)
"
(c.8)
We must still consider the second contribution in (C.7). Since it is regular as A -->O, making the change of variable x 2 = (1 +z2) -1 we get for this part
( Cl~O~s~2
\ ~'3 / (2rr)
Io1 __dxarcsin x In x - - 21 x
x -1
•
We have only been able to show analytically that this integral is larger than ~.2. A numerical evaluation gave 1.28 which is consistent with the value ~4~-3. Assuming this value we get
( CFas~27r 4 ~'/3 / 1 2 '
(C.9)
Collecting the result coming from equations (C.4), (C.6), (C.8) and (C.9) and
J. Frenkel et al. / Infrared behaviour
348
multiplying
by 2 since diagram
9b gives same real contribution,
we find ((2.10)
On the other hand, since the one-loop
graph in fig. 9c gives ,
its contribution
to the cross section
will be (C.11)
Adding (C.10) and (C.ll) we see that the infrared-divergent expected. The finite part gives c ( p3 in accordance
with the result stated
part cancels
out as
=lr*
FaS
)
in eq. (18).
References [l] [2] [3] [4]
R. Doria, J. Frenkel and J.C. Taylor, Nucl. Phys. B168 (1980) 93 A. Andrasi, M. Day, R. Doria, J. Frenkel and J.C. Taylor, Nucl. Phys. B182 (1981) 104 C. Di’Lieto, S. Gendron, LG. Halliday and C.T. Sachrajda, Nucl. Phys. B183 (1981) 223 C.A. Nelson, Nucl. Phys. B186 (1981) 187 PI N. Yoshida, University of Tokyo preprints UT-353 and 354 (1981) f61 A. Andrasi, C. Carneiro, M. Day, R. Doria, J. Frenkel, J.C. Taylor and M.T. Thomaz, Proc. 20th Int. Conf. on High-energy physics, Madison (1980), ed. L. Durand and L. Pondrom [71 C. Carneiro, M. Day, J. Frenkel, J.C. Taylor and M. Thomaz, Nucl. Phys. B183 (1981) 445 @I F. Bloch and A. Nordsieck, Phys. Rev. 52 (1939) 54 r91 L. Landau and E.M. Lifshitz, Quantum mechanics, Third edition (Pergamon Press, 1977) r101 W. Marciano and H. Pagels, Phys. Reports 36 (1978) 137 [Ill LB. Khriplovich, Sov. J. Nucl. Phys. 10 (1970) 235; J. Frenkel and J.C. Taylor, Nucl. Phys. B109 (1976) 439 [I21 IS. Gradshteyn and I.M. Ryzhik, Table of integrals, series and products, Fourth edition (Academic Press, 1965) r131 G. Grammer and D.R. Yennie, Phys. Rev. D8 (1973) 4332