Non-cancelling infrared divergences in QCD coherent states

Nuclear Physics B264 (1986) 588-620 © North-Holland Publishing Company

NON-CANCELLING INFRARED DIVERGENCES IN QCD COHERENT STATES S. CATANI and M. CIAFALONI Dipartimento di Fisica, Universith di Firenze and INFN, Sezione di Firenze, Italy

G. MARCHESINI Dipartimento di Fisica, Universith di Parma and INFN, Sezione di Milano, Italy

Received 8 July 1985

The Faddeev-Kulish approach of asymptotic dynamics is used to construct the QCD coherent states at the level of leading and first subleading infrared singularities. Gluon correlations are properly taken into account and an arbitrary number of initial and final partons are considered. The violation of the Bloch-Nordsieck theorem and the cancellation of the Coulomb phase in many-parton initiated processes are confirmed. In particular, the non-cancelling terms are given in operator form and are shown to be in general of higher twist type. The physical interpretation and the gauge dependence of the results are also discussed.

1. Introduction This paper deals with a perturbative hamiltonian a p p r o a c h [1] to coherent states in q u a n t u m c h r o m o d y n a m i c s (QCD), which is able in principle to treat subleading infrared (IR) singularities [2] and is pushed here up to the next subleading ones. T h e motivation for such an effort is twofold. On the one hand, a large a m o u n t of work on h a d r o n i c jet physics [3] has suggested that for a variety of physical processes higher-order Q C D corrections are important, due to coherent infrared effects. A n explicit form of the coherent state operator has been given [2, 4] and f o u n d to reproduce all interesting physical predictions first obtained by F e y n m a n d i a g r a m techniques. This result improves previous work on the subject [5-7] by including gluon correlations previously neglected [5] and by making the hamiltonian a p p r o a c h [6] more explicit to all leading orders [2]. It seems therefore worthwhile to try to include subleading singularities in a systematic way and to investigate their physical effects. O n the other hand, it has been pointed out by various authors that: (i) the Bloch-Nordsieck cancellation theorem is not valid, [8-10] i.e. subleading infrared singularities do not cancel in m a n y - p a r t o n initiated processes, even after summing 588

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over final soft gluons and colour averaging over the initial state and (ii) the factorisation of collinear singularities may not be valid [11-14] in Drell-Yan type processes, when the initial state interactions of spectator partons are also taken into account. At the present stage, both problems (i) and (ii) seem to be partly overcome, since various proofs are available to show that both effects are higher twist [9,13] and that, moreover, the leading twist factorisation of collinear singularities in colour-singlet initiated processes is recovered [14]. However, the diagrammatic techniques used so far in this context are rather cumbersome, and it is difficult to get a direct understanding of the results and a systematic way of approaching them. The hamihonian method presented here seems to us better defined and more systematic and simplifies some aspects of the problem, if not all of them. The starting point [1, 6] is a separation of the QCD interaction hamiltonian in a hard and soft part. The hard part contains only gluonic frequencies larger than some scale E, while the soft part is the remaining one, i.e. in the interaction representation. H i ( t ) = HE(t) + H / ( t ) ,

(1.1)

where the precise cut-off definition will be given in sect. 2. Correspondingly, infrared effects can be factored out in the form

OE*CeOE

(1.2)

where 82e_+= UE(0, -Y-o¢) are MSller operators for the soft hamiltonian and S(~ is now supposed to be free of infrared singularities [7]. Of course, this formal argument relies on the assumption that, after the factorisation (1.2), the operator S(~ is well defined, which probably is not true outside the gauge-invariant sector of the Hilbert space. Nevertheless, we shall use eq. (1.2) in perturbation theory with an infrared (or finite-time) cut-off )% with the final goal of defining gauge-invariant and infrared finite processes. According to eq. (1.2) finite matrix elements should be obtained on the coherent state basis, defined by in IPl, al .... Pn, I/,1, oq .... Pn, or,; Out> = ,~E+,az_+.

(1.3)

where Pi, ai denote hard parton momenta (E, >> E) and colour indices, respectively. We shall actually be concerned in the following with the construction of the transformation matrix HH=HHU~g,i,~,},

(1.4)

where, in the Hilbert space % - - ~ H ~ ~S' [0>H has no hard partons but may

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contain soft ones and therefore

U E

is an operator in the soft-gluon Hilbert space

~(~s. The trouble with the Bloch-Nordsieck theorem comes from the fact that processes of the Drell-Yan type involve two or more partons in the initial state. This can be seen by the following argument [15]. The general cross section initiated by a partonic state l i) is given by

(a> = ~ ~]I (fl~2-~*TH~2+ E li>l2 ,

(1.5)

f,E c

where Y~f,E denotes a sum over final soft gluons up to energy E and Y'.,. denotes a sum over final colours and averaging over the initial ones. In the present formulation it is clear that final state infrared effects cancel out in (1.5) just because the ~2E operator is unitary (actually when restricted to the K s space, see sect. 5), leading (roughly) to the formula Tr s(0[

e,

H*

H

i qi E10>s,

(1.6)

where T(fi~ refers to a definite set (i), (f) of hard states, U(i~ is the restriction of ~2~(i) to K s given in (1.4) and colour indices are understood. On the other hand, initial state interactions do not generally cancel, because in QCD tt'(fi) T H t T"(fi), H E ~ 0, except U(i)] when there is only one parton in the initial state, as in deep inelastic scattering. It is known that infrared leading singularities do cancel in eq. (1.6), the first non-trivial effect [8] being O(as21og(E/?Q), where ?~ denotes the IR cutoff. In the present formulation this fact is recovered in a transparent way: the leading-order coherent state operator is not only unitary, but turns out [4] to commute at different colours, i.e., u E , in

(# .... ),

u E , in

]

(#;,~:)J = 0

(1.7)

in the Hilbert space K s. This allows use of the colour trace in (1.6) as a completeness sum to yield the expected cancellation (cf. sect. 3). When subleading singularities are introduced (sect. 5), the commutativity property (1.7) no longer holds, leading to a non-cancelling term that we are able to give in operator form in sect. 5. This brief introduction to the hamiltonian method (that will be completed in sect. 2) allows us to illustrate some more advantages of this approach. We have already mentioned the automatic cancellation of final state singularities, due to the explicit time ordering. We can see two other main points: Firstly, since we always deal with on-shell gluons, the eikonal approximation for vertices and propagators of the hard partons is fully justified, even for Coulomb-like interactions. The same can be obtained by Feynman diagram techniques only by contour distortion for the case of virtual gluon contributions. Secondly, due again to time ordering (which is sys-

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tematically replaced by energy ordering in sect. 2) the factorisation of the non-abelian Coulomb phase is particularly transparent at least to all leading orders. In the massless quark case, it turns out that, due to coherent gluonic contributions, this phase is factorised in the final state and disappears, to this order, from processes inclusive over soft gluons. Of course, a disadvantage of our approach is that we introduce from the beginning the energy-like scales E and ~, and time-ordering, so that our results are not explicitly Lorentz covariant. It turns out, however, that we shall be able to show the Lorentz covariance of the physical quantities we shall obtain without too much effort (appendix B). A related problem is the one of gauge independence. Here we have full control of the leading orders (sect. 3) where we can exhibit the gauge dependence of the coherent state operators and the gauge invariance of the physical quantities obtained. As for the non-cancelling operator, we have checked that the Feynman gauge and the Coulomb gauge give the same integrated quantities, but we have no direct proof of the general gauge equivalence. Also, we shall use the Feynman gauge in most of the paper for obvious Lorentz covariance reasons, and also because the Coulomb gauge hamiltonian in QCD is defined only as a power series in the gluon field. The contents of the paper are as follows. In sect. 2 we specify better the hamiltonian approach, and we give a method for switching from time ordering to energy ordering in a systematic way. This allows definition of the hierarchy of IR singularities in operator form. The leading-order coherent state operator is discussed in detail in sect. 3, where also the relation to previous work is given. In sect. 4 we discuss the non-abelian Coulomb phase to all leading orders and we exhibit its leading twist cancellation from Drell-Yan type processes. The violation of the Bloch-Nordsieck theorem is discussed in sect. 5 where the non-cancelling operator is constructed. We also discuss here the inclusion of next subleading singularities in the coherent state operator and their physical consequences. All calculational details of this section are given in appendices A and B, while we discuss our results in the concluding sect. 6.

2. QCD asymptotic dynamics: energy transfer evolution Here we shall concentrate on the study of the soft interaction hamiltonian of eq. (1.1), which is essentially a small frequency form [6] of the complete one. The final goal is the explicit construction of the coherent state operators (1.4) in terms of soft-gluons fields, by exploiting the hierarchy of the infrared singularities in perturbation theory. The main idea [2] in this construction is to replace time ordering with frequency ordering in a systematic way, since the latter are more directly related to infrared logarithms.

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We shall use in the following a standard on-mass-shell hamiltonian approach in the interaction representation. The time evolution frequencies u are given in this approach by the energy transfers at the interaction vertices, i.e.,

v = [Eoi~i(qi)l,

Eoiqi = O,

i

(2.1)

i

where ~oi(qi ) denote the energies and o i = _+1 the energy signs of the incoming partons at the vertex. The hard and soft interaction hamiltonians of eq. (l.1) are then defined according to whether v >
HsE( t ) = H ( ( t ) + HgE( t ) , where the fermionic part is defined by

fdIp]d[q]pf(p)~A]°(q)e-'"~'q'O(E-~.q),

H f e = g s ~_,

(2.3)

o=+

and we use the notation d[q] = d3q/(2~r)32~Oq,

13~= p ~ / E p ,

o f ( P ) = E bti~(p)(tia)~bi~(P),

tq~= t~,

t~a = _ tT,

i=q,Tz1

[ A]°(q ), A~:°'(q')] = -g""'oSo. _o,Saa,(21r)32~0q83(q - q'),

(2.4)

where A ~a + (q), A~-(q) are gluon destruction (creation) operators and I: = E p "¢- ¢UOq - - E p + q

~- tO q -

l).q

= &q~

(2.5)

is the energy transfer in the eikonal form, valid for lq[ << [P[ (or in the collinear region).

S. Catani et al. / Infrared diuergences

593

In the gluonic part of the hamiltonian we are not generally entitled to use the eikonal approximation for the vertex or the energy transfer (except for the leading singularities). We then write 1

n =-igsFL,o o ' E

f d[qi]d[q2]d[q3](2~r)383(~io,q,)

0i= ±

v a ° ,a l t t l .a°2 . a 2 ~ 2 . .so, a 3 1 , 3 - -r~,~2~,,.~ \Vl"/1, a2q2, 03q3)e ....

- i ~ o,~ , : ) , ' O I E _ [ ~i oi~i I , (2.6)

where /'~1~2~3(ql, q2, q3) = g,,~2(q' - q2),, + perm.

(2.7)

is the 3-gluon vertex. The first step in the construction of the coherent state operators is to give a manageable expression for the soft Mbller operators ~2_+=e _ UE(0 ' -T-m). It is convenient to single out the dependence on the energy transfer v by defining positive and negative frequency hamiltonian densities h - ( 9 ) such that H ~ ( t ) = o=~+ foEdVh°(v)exp(-iovt)

(2.8)

.

The general term in the perturbative expansion of ~2e+ then takes the form

rEh°"(P,,)exp[--i(o,,v,, + . . . Oll,1)t ] £a{+")= ~2¢Jx

h°l(pl) -

o - ~ . + : :; + a l ~ + ~

"'"

-

Ol/.' 1

+ie

dv 1 ...

dv n . (2.9)

Infrared singularities in (2.9) come from vanishing energy denominators (so that an infrared cut-off ~ is needed). The leading logarithms come from the region where the v's are strongly ordered, i.e., << P l << p2 <<

(2.1o)

" " " << E ,

so that we can use in (2.9) the approximation (Onl*n + "'"

0"191 + i e ) - 1 ' ' "

(O'll' 1 +

i e ) - 1 = H (OiVi)

1

{9(Vn ' ' "

Vl),

i=1

{2.11)

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which yields the expression E_ dv s2+P,exp/"E E --h°(v), JX

(2.12)

e OV

where O(v n .-- vl) and P~ refer to the v-ordering. Subleading singularities come from integration regions where two or more energy transfers are of the same order. In QED such terms essentially cancel in (2.9) upon symmetrisation over the vi's, except for the Coulomb phase, which is due to the c-number commutator of two h °'s (see below). In QCD, subleading singularities do not cancel, but the result (2.12) suggests using the v-ordered exponentials in a systematic way. This is done as follows. Let us define the E-evolution hamiltonian A( E ) by d

d E UE( t' -T- oo ) = a + ( E )UE( t, -T- oo ) .

(2.13)

From the perturbative expansion (2.9), or the corresponding time-evolution equation it is easy to show that, e.g.,

dt'UE(t, t')121E(t')UtE(t, t'),

A+(E) = -if' --00

fiE(t ) = y" hO( E ) e x p ( - ioEt )

(2.14)

O

and therefore (by setting t = 0 and dropping the + index) that

a~d(E)=a(E)=-if

° dtoI:lE(to)-f ° --00

d t a f tx dto[HE(tl),I2Ie(to)] O0

--00

+(-i)3f_°oodt2f~2 ~ dtxf~' dto[ He(t2), [He(q),/:/u(to)] ] + ....

(2.15)

The first term in the expansion (2.15) is just the exponent in (2.12). The others roughly correspond to the clustering (or wrong ordering) of 2, or 3, or more v,'s, the remaining frequencies still being in the natural order (2.10). Therefore, (2.15) defines the hierarchy of infrared approximations we were looking for.

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This conclusion is better seen by switching to/1-integrals, in terms of which (2.15) is rewritten as

(E)

= a(,(e)

+ zx(:)(E) + a O ) ( E )

(2.16a)

+ ...

h°(E) E [h°~(/11),h°°(E)] =~"~ o----E-+ O0,Y"Ol fx d/110°looE(ooE+o,vl+ie) E

(2.16b)

[h02(/12) , [hal(/.,1) , hO°(E)]]

+ E f

d/11d/120o1Oo2 . . . . . . . . (~oE(ooE + 01/11 + iE)(ooE + 01/21 oi"X

--[- 02//2 --}- i 8 ) --k - - .

,

(2.16c) where O~j= O(/1~-/1j),(i, j = 0 , 1 , 2 ) , denote the /1-ordering (instead of the time ordering) which turns out to be opposite to the natural one. The formal solution of our problem becomes therefore

E_ P~expfEd/1(A(X)(/1) I2+-aX

+ A(2)(/1) + A(3)(v) + ..- ) .

(2.17)

Furthermore, the analogue expression for $2e- is simply obtained by reversing the ie prescriptions in (2.16), due to the different integration path (0 < t < + oo). A simple example of this approach is provided by QED, where only AO) and A(2) are non-vanishing. More precisely, we have

zVr o= ef

a [ p ] d[ q]p(p)(~q)-l~.

(A(q) - At(q)) 6(v -/3q), ^

(2.18a)

^

ZI~)ED= _i~re2 f d[pl]d[ P2] d[q]PlPz--~TqS((Pl P,'P2 - P 2 ) " q)8(v - b ~ q ) , (2.18b) where we have used the fact that [h °o, h °,] -OO~oo,axpop1 commutes with expressions (2.18) yield immediately the QED coherent state

h °. The

q ]pP" (p .- A t ) e x p ( - i ~ c ) ,

(2.19)

where the /1-ordering has disappeared because the operators ill(q)= commute, and the Coulomb phase ~c is given, from eq. (2.18b), by

A(q)- A*(q)

I2+e'QED = exp[e f d[ p ] d [

~c=½af d[px]d[p2] [(pl.p2)2_p~p~]l/2 log

.

(2.20)

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596

3. Leading-order coherent state operator Before exploiting the full expansion (2.16) it is convenient to discuss the properties of the leading-order operator (2.12) [4,2]. Even if the u frequencies are simply ordered in (2.12), we know that the leading singularities come from the strongly ordered region (2.10), where also gluon energies are ordered. This automatically introduces a hierarchy of softness at each 3-gluon vertex, so that one emitted (or absorbed) gluon is much softer than the other two. Therefore, we are entitled to use the eikonal form of the 3-gluon vertex

o(Iql] \lpl ]

-p + q,-q)=

= 2 p ~ g ~ 2 + longitudinal terms,

(3.1)

where actually the eikonal term is the first one. We shall now argue that the extra terms in (3.1) (which insure the fulfilment of the W a r d identities [16,17]) actually provide the decoupling of the longitudinal gluons. In fact, by introducing (3.1) into (2.6) we find that soft gluons are coupled to the interaction current ((Ta)hc = ifb~,.),

jr:

_A~(p)TAX(p)p~+½[(p.A+(p))TA~(p)+h.c.].

(3.2)

By introducing a small gluon mass ~ and physical polarizations e~>(p)(a = 1,2) with e<~>.p = 0 we can replace the expansion

A•(p)

= Eef(p)A<~'"(p)

+

p~

--~AL(P)

(3.3)

O~

into (3.2). We then find that the longitudinal field that*

AL

cancels out (for ?~ ~ 0) so

(3.4)

and the interaction hamiltonian takes the form HeiEk(t) =

gsfd[ p] d[q]Pa(P)P~A~u°(q)e

i(pq)°t~)( g - ~ q ) O ( Ep - ~q),

(3.5)

* This rough argument, based on the introduction of a small gluon mass, is actually not needed for gauge-invariant quantities, because the longitudinal terms in (3.1) vanish by Ward identities [16,17]. It is useful, however, in order to discuss the gauge dependence of the leading-ordercoherent state (see below).

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597

where the subscript "elk" recalls that we are using the eikonal form (3.1) of the 3-gluon vertex, and we have defined the total colour charge density P a ( P ) = Pfa(P) + Pag(P) = Pfa(P) + V A t(p) ( a ) T a A(a) (p),

(3.6)

C~

which only contains the physical gluonic annihilation (and creation) operators A ( ~ = -e(~)-A. Note that the factor O(Ep- ~q) in (3.5) defines q~ as the softest gluons at each 3-gluon vertex. Note also that the longitudinal ghions in (3.5), which only occur in A~(q), have no self-interactions, while transverse gluons do. In other words, p~ does not simply act as a source of softer gluons, but gives rise to nontrivial self-interactions that we are going to explicitate. Let us now recall [2] that, to leading order, the coherent state operator (1.4), including soft gluon interactions, can be defined on single partonic operators by the expression

bf(p)= $2eikb,,(p)~i E+ ek=

Uff(H)bo(p),

(3.7)

where, by (3.5) and (2.12),

= Ue'~(O,-~)=PFxp[igsfxE d[q] f%d[p]Pa(P)

p'Ha(q) p.q

(3.8a)

= P~exp fxE d 1,A(~(p),

iH~a(q) = A~(q ) - A~*(q)

(3.8b)

and be denotes the annihilation operator of a hard fermion (Ip[ >> E). A similar formula holds for hard gluons with the colour index a (in the fundamental representation) replaced by a (in the adjoint one). Note that to this order the _+ie prescription is not relevant, so that 12+ and ~2 coincide. Let us emphasize that the result (3.7) is not obvious, because U(II) is a functional of the soft field (3.8b) only. A first important consequence of (3.7) is that the transformation matrix (1.4) factorises in the colour space of the hard partons, i.e.,

H(Pl, /31 "'" p,/3,1~2~,lpl, al "'" P,, a , ) H --

{~,,,~,}=

UE

U~f(H),

(3.9)

where U p,E is given by (3.7) as operator on the space ~ s . In order to prove (3.7), [2], we use the equation of motion for the hard operator bf which is only sensitive to the charge density p ( p ) in the hamiltonian. We

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598

uqE~,,K

UpE

A a ,

Fig. 1. Tree-like picture of the leading order E-evolution equation (3.10) for the quark incoming coherent state operator. The prime denotes differentiation with respect to E, and the crosses stand for insertions of quark annihilation operators or of gluon fields H,(q).

find (fig. 1):

d be(P) = £~eikE+[ba(p) ' ~2 -h ~-( E ) ] ~ a E ,,

dE

=igsf d[q]8(E-b'q)(G)~'

~. Hff(q) ~.q b~(p).

(3.10)

Furthermore, the same equation holds for the operator HE(q) itself, due to the fact that [H~(q), Hb(q')] = 0. Therefore we get the final solutions

upE( H) = P~exp[igsfxd[ q ] P" l-~Pa'q(q)p.q GO(E -/3q)], H~(q) = uq~,(H)17b(q).

(3.11a) (3.11b)

In other words Upe exponentiates a "moving" field HPaq(q)(including soft gluons up to energy v =~bq). The v-evolution of the latter (given by U~h in (3.11b)) is found by writing eq. (3.11a) in the adjoint representation (t a ~ Ta). If the U operator in (3.11b) were omitted by replacing H with a free field, eqs. (3.11a) and (3.9) would be simply interpreted as giving gluon bremsstrahlung off the non-abelian current

L=£

i Piq

t,,

(3.12at

where the sum runs over the initial state hard partons. The effect of the evolution (3.11b) is to take into account further radiation of softer gluons of energy v.

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More precisely, the non-linear set of eqs. (3.11) can be solved iteratively to yield the expansion Ups(/-/) = Po,exp{ igsLEd[ qt] Jox" II(ql) z

x 2 / ' E > 6°1 > t°2

z

x 3 / ' E > t~l > 6°2 > 603

-~igs) Jx

d[ql]d[q2][J°l"Hx'J12"H2]

+Jigs) Jx

d[ql]d[q2]d[q3]

×([[ Jol. n , J,3. n3], s 2. n2] +[Jol. Hl,[J12.H2, J:3.II311)+ ...),

(3.13)

where

J;t =

q~tk qk" qt

- -

(3.12b)

are the eikonal currents, including gluonic contributions, and we have replaced the v-ordering with energy ordering, which turns out to be the same to leading order. Note that the "dressed" operator II"(q) in the exponent of (3.11) contains gluon correlation terms of any order, which correspond, by (3.13), to a sum of three diagrams with energy ordering. Note also that gluon emission amplitudes also contain loop contributions, described in ref. [4], due to the fact that (3.13) is not normal ordered (fig. 2). The leading-order result (3.11) leads to a trivial cancellation of IR singularities in inclusive quantities, as remarked in the introduction. In fact, the coherent state operator U E in (3.11) and (3.9) is unitary:

U{~, #}U{B,v} El = "(~,#}"{#, fret ire v} = ~(,*,v}

(3.14)

and this automatically implies the cancellation of final state soft interactions.

a

X

j

i

pE ©

Fig. 2. Schematic branching structure for matrix elements of a quark leading order outgoing coherent state. The amplitudes S H are generalisationsof the Sudakovform factor, due to virtual corrections,which are defined and studied in ref. [4].

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S. Catani et al. / Infrared divergences a:

h~

h2

h1

h2

¥ U E'in aT a 1

) a 2 a~

(a)

(b)

Fig. 3. Colour indices saturation scheme in Drell-Yan type processes for (a) Colour-averaged cross sections and (b) Colour-singlet incoming states. The crosses denote emission and absorption of soft gluons, and ~ 2 is the hard overlap function defined in the test. The incoming coherent state U E(in) factorises in colour only to leading order.

Furthermore, since

[ v~f(u), v:,',,~(m] = o

(3.15)

due to [Ha(q), Hb(q')] = 0, this also implies the cancellation of initial state interactions in colour-averaged cross sections. Typically we have, as in (1.6), (fig. 3a),

= ~

~ (Pf, afI~EtTH/2~+Ip,,,~i>

{al,af } f,E

=

~

7rEt ~ 1rE (0[,~(~,~}j,~(~,B}~(~,~)[0),

(3.16)

a,fl,y

where we have denoted by 6"~(y,fl } = (TfiHt TfiH ){T,B)'

(3.17)

the hard scattering matrix overlap function in the space of initial state colours, denoted collectively by ( 7 }, ( fl }. By then using the commutativity property (3.15) and again unitarity, ( o ) in (3.16) simply reduces to Tr 4 , which verifies the Bloch-Nordsieck theorem to this order.

S. Catani et al. / Infrared divergences

601

Note that this cancellation of initial state soft interactions is also valid if we refer to colour-singlet initial states as in hadron-hadron scattering instead of colour-averaged cross sections. This result follows from the factorisation property (3.9) (which allows translation of the colour singlet condition into a colour trace, fig. 3b) and again from commutativity, eq. (3.15). Therefore no violation of factorisation arises either. Finally let us briefly discuss the gauge dependence of the expression (3.8). We shall introduce a gauge transformation which to leading order is simply

A;(q) ~ A;(q) + q,A°(q),

(3.18)

where A ° commutes with the physical fields A (")(q) and has c-number commutators with the longitudinal ones. Then the A contribution to the exponent in (3.8) is simply

fJ),E d [ q ]O( wq)( A +(q ) - A - ( q ) ) ,

(3.19)

Q(w) = f d [ p ] p ( p ) O ( E v - w)

(3.20)

where

denotes the total colour charge of partons with energy larger than w. Since (3.19) commutes with A(~(w) for ~0> Wq, the corresponding contribution to (3.8) factorises to the left, i.e., ~o~A

= P,g x p f xE d[q]Q(wq)(A + ( q ) - A - ( q ) ) n [ ~ k

= G(A)I2eeik •

(3.21)

From (3.21), the following gauge transformation properties follow: (i) The S-matrix operator is invariant. In fact,

S a = ~2*_GtASHGa$2+=I2t_Sn~2+

(3.22)

because S H is purely hard, and colour conserving ([SH, GA] = 0). (ii) The transition matrix elements between coloured states are gauge dependent, due to the gauge transformation property

IP, a) ~ e-iq'alp, a) -- (eliO"A ) B,lp, B), a a = ffd[q]i(a+~(q) - a a ( q ) ) ,

(3.23)

where Q(w) has been replaced by the total colour charge QH on hard partonic states. (iii) Correspondingly, the soft coherent state operator only changes because of the total colour charge of the hard partons. In fact, by (3.23), the b E operators are

602

S. Catani et al. / Infrared divergences

invariant, i.e.,

b f ' a = b f = U~ba

(3.24)

and therefore the soft coherent state operator (which is the transformation matrix to the b representation) transforms by multiplication to the right:

UE ~ UEA= UeeiO ".A, U~E(II)--* UPe(H)e " ' a .

(3.25)

The results (i)-(iii) are as expected, with the warning that we have not touched on the issue of the gauge transformation at infinite times. In particular, the " i n " states of eq. (1.4) are not changed by our transformation, as confirmed by eq. (3.24). In general, one expects that evolution operators transform as

U(O, T ) --* Go(A )U(O, T)GtT(A )

(3.26)

and the existence (or less) of the T--, _+ oo limit in (3.26) is related to a proper definition of the asymptotic states of the theory that we expect to be neutral to start with. The attitude we have taken here is the perturbative one, according to which coloured coherent states as (3.11) are defined with an IR cutoff, by then looking for cutoff-independent physical quantities: the investigation of many-parton initiated processes goes precisely in this direction. 4. The non-abelian Coulomb phase: a cancellation theorem

The eikonal hamiltonian (3.5) contributes to all orders to the expression (2.16) of A(E). However, if we stick to leading infrared logarithms (terms of type ( a s l o g ( E / X ) ) n) only A(x) and A(2) may contribute. While A(1) has both leading IR and collinear logarithms (double logs), A(2) has only leading IR logs and does not yield soft gluon radiation, because it is of pp type*. Its expression results from the commutator of two soft operators in h~-~, so that only a term - +8(~0 v + OlV) in (2.16b) contributes. We get A(~)(v) = - i ~ r g ~ f d [ q ] f ~ q d [ p l ] d [ p 2 ] p a t p x ) p a t p 2 ) - ~ l " q

X ~((/~1 - - P 2 ) " q ) 6 ( v - - ~ , q )

-

ias f d [ p l ] d[p2] P~ "P2P~P2~ ~9(E~ - v ) 6 ) ( E 2 - v) 2v ~ W )/I'PIP ' 2~-Pip222 2 ' (4.1)

* A(2) also contains pA ÷A terms coming from the commutation of two hard operators, which however are infrared subleading.

S. Catani et al. / Infrared divergences

603

which differs from the abelian Coulomb phase (2.20) by its non-abelian indices and by the lower cutoff v. Note that (4.1) changes sign under reversal of the ie prescriptions. It is interesting to see how the MSller operators 52e_+evolve under the action of (A(eX~ + A(~), which is the complete IR leading evolution operator. This is particularly simple in the limit (pip2) 2 >>p2p2 which can be seen as the limit of large S = (p~ +p2) 2, or massless quarks. In such case one can set v~j = 1, where

vij= ( p,. pj)- l[ ( p, pj)2 - p2p2] 1/2,

(4.2)

so that one has simply

A(~) _

ias

2~,0.~, O.(v)=fdtp]p(p)O(Ep-p),

(4.3)

where Q(v) is the total colour charge of partons with energy larger than v of eq. (3.20). Therefore, to this order in the expansion (2.16), 52E_ ± - P~expf E(A(~ _+ A~)) dv = Vc-* + 52ea e, ,

v?= (vc)*= (Vc)-1,

=

(4.4)

factorises to the left into a Coulomb interaction operator Vc and the eikonal evolution operator discussed before. In fact, A(~)(v) in (4.3) commutes with the operator A(~(v') for v' > v which occur to its left because of the v-ordering. The simple result (4.4) is due to a coherent effect of various Coulomb interactions which only depend on the total colour charge in the massless limit, according to (4.3). A trivial, but important consequence of (4.4) is the cancellation of the non-abelian Coulomb phase in this case of massless (on-shell) partons, if the customary sum over final soft gluons is performed. In fact, the general S-matrix of sect. 1, with the soft interaction described by (4.4) reads E+

E

_

Et

2

E

S -~- 52elk VcSHVc52eik - 52eik VdSH~'~eik,

(4.5)

where we have used the fact that [S n, Q,] = 0 (v ~
604

S. Catani et al. / Infrared divergences

cannot always set v~j = 1. In fact partons inside an incoming hadron may interact with a small relative velocity, so that 1 -v~j is not necessarily higher twist in this case. In order to evaluate the left-over contributions, let us consider the expansion E+= e ~2 Pvexp fx du( A(ct~_+A~)) + E = V~2eitP~ex p _+f x E d v A , ~)(u) ,

(4.6)

where the evolution operator ~'~') (r) = ~ a ' ~ (~) ~a~,

a'c(~')i%fd[pl]d[p2]( 1 . . . . . 2u v12

)

1 Pl"P2

(4.7)

is defined by subtraction of (4.1) and (4.3). To be more definite, let us consider a 4-parton initial state corresponding to the colour singlets h I = (11) and h 2 = ( 2 2) and the Drell-Yan cross section, which is of the form DY = (hlh 21~2+E,T{qTH~ + t E ihlh 2>"

(4.8)

By introducing the factorisation (4.6) in (4.8) and by evaluating the hard parton matrix elements we obtain (fig. 3b).

(o)Dv= (hlh2lP~exp(- f dl'A'(c~)(r)) x UE*°-Ogi2UEP~exp(f dvA'~c~)(v ) )lhxh2> ,

(4.9)

where 6 ~ 1 2 = TfiHtTfy is the overlap function in the active partons colour space (1, 2) and by A'(c~)(r) we now mean the restriction of (4.7a) to the soft space 9Cs, i.e., .0/S P i
= - i as E Aij( U:*Uf )..ti~tjb,

(4.10a)

/J i < j

t~ ) = U:g't,b = Ui;btib,

(4.10b)

where the colour matrices t~ are evolved by the eikonal coherent state of the

S. Catani et al. / Infrared divergences

605

corresponding momentum Pi, and Ai/= v/~1 -- 1 is

(4.11)

O(m4/S 2) whenever S = (pi "P/) >> P~.

The main point is now that (4.10a) is always a higher twist contribution, either because (i) A ij is small for large interaction energies (1 2,1 2,1 2,1 2), or because (ii) the operator factor in front yields a cancelling phase (1 1, 2 2). In fact, for nearly parallel pi and p~ one has

(Uit~U/")ab=8~b+O(asm2JS),

(/j = 1 1 , 2 2 ) ,

(4.12)

and the 8 factor (corresponding to no colour fluctuations) yields the c-number exponents ta • ti, t 2 • t~ which cancel out in (4.9). Furthermore, since only the octet part of fife12 contributes to (4.9), the O(as) correction in (4.12) involving a single/7 field has vanishing colour factor. Therefore, the higher twist terms of type (i) still give the leading non-cancelling contributions to (o) DY -- ( 0 )(0), of order m 4//S 2. We have so far discussed the Coulomb phase for colour singlet incoming states. It should be remarked, however, that an incoherent average over initial colours leads again to a complete cancellation. In fact, the A'c dependent factor in (4.9), due to eq. (4.10), is a functional of the soft gluon field /7 only. Therefore, initial state effects cancel by unitarity and commutativity, just as for the leading order operator, eq. (3.15). In order to get non-trivial effects after colour average we need soft operators which do not commute with H, and these are provided by the mixed Coulomb-radiation terms which will be discussed in the next section. Finally, let us note that the A(4) term in the expansion (2.16) also contains a Coulomb-like contribution, which however comes simply from an IR subleading renormalisation of the Coulomb potential, and therefore follows the same pattern as the leading term discussed here.

5. Subleading terms and non-cancelling operator When higher-order terms in A(E) of eq. (2.14) are included, we have to use the full expression (2.6) of the gluonic hamiltonian. Moreover, it is not generally possible to define the coherent state operators in the space of soft gluons by starting from a single hard parton operator as in (3.7). We shall then use, as in (1.4), the matrix elements in the hard parton Fock space

idpaB,,,

•"

p.B.IS2~lpx,xl," " ,

p.a.>H=H<01~U+ Ok E'in /H U {B,,a,),

(5.1)

where the first factor is the purely soft part of ~2E+(which is generally not trivial), and we have defined

S. Catani et al. / Infrared divergences

606 u{E, in t B. . . . }I, P l

* • "" P n ) = H ( O [ f i E biB biailO)H , i=1

bg~ = oEtt, oE+ , ~ + ~ia.~

(5.2)

and we shall drop the " i n " superscript from now on. A formal expression for U in terms of soft operators is obtained by writing I-I~( t ) = I-1:~ > ~( t ) + ~ ~

(5.3)

< E( t ) ,

where now H~p < E only contains soft gluon fields (with energy less than E). By the definition (5.1) we then have the expressions (5.4a)

H(O',QE+ 'O)H = T e x p [ - i f ? d t H g E p < E ( t ) ] ,

{~,,~,} = Texp

.A,a (q)e o~

d[q] ,

(5.4b)

i

where A°~ t) is in the interaction representation defined by (5.4a). This result displays the unitarity property of U E in the soft gluon Fock space ~s- A trivial, but important consequence of unitarity is that final state infrared effects cancel in physical cross sections (which are inclusive over soft final gluons) for subleading singularities as well. In order to analyze the effect of subleading terms for initial state interactions it is convenient to factor out the "eikonal" term of sect. 3 by writing the full evolution operators as

/-/~(t) =/L~,(t) +/-/;(t), heOik ( P )

(1) ae~(~)= Z --, o=+_

OP

(5.5)

where now Heik and A(~ include soft operators as well and give rise to the leading order coherent state of eqs. (3.8) and (3.11). We then write (5.6) where, by (3.7), b aelk ( p ) = ~ b a ( p ) l " 2 e i Ek = W f f ( H ) b f l ( p )

(5.7)

takes into account the eikonal evolution,

lie = .Uei o ~ tko.~~+- -- P . e x p f e d v A , ~ ) ( v ) , aX

(5.8)

S. Catani et al. / Infrared divergences

607

evolves in (,) with the "interaction representation" operator

x(')(,)

=

"* A (r) ~2.ik Oe~k '

(5.9)

"

and Uf~ = V~ Uf~(/7)VE is a unitary redefinition of the eikonal coherent state and therefore still commutes at different colours: [ U::, U~,~] = 0 .

(5.10)

Due to (5.10), the evolution which may give rise to non-cancelling singularities is the one coming from A'(~) in the interaction representation, i.e.,

+ fxE[[bfl,/1'('°)(%)],/1'('l)(,1)] Ool dr0drl + - - =bfl + fxEdro [ bfl, 5'2~f(Vo) + 5'3),ff("0)]" + " " ,

(5.11a) (5.11b)

where in (5.11b) we have neglected terms O(gs4) or higher which will turn out not to contribute to next subleading order*. The calculation of the effective evolution hamiltonian in (5.11) requires some algebra (based on eqs. (2.16), (5.5) and (5.9)) which, due to a mixing of perturbative orders, is performed in appendix A. We find the relatively simple result A(2?f( ' o ) =

+ h~"l ( , 1 ) , h~.~( PO)] Ea, fxEd'1Ih (,1)Ool aoVo(°oro + a l r l + is)

'

/l(3)¢ff~troj ~ = ~ f E d ' l dr2 a~

X

[[h:~k(,1)O01 + hg,.1 ( , 1 ) , h ~ ( V o ) ] , h e ~a( , 2 ) O o 2 + hg'02( '2)] 00,00"2,2(001P 0 "Or01,1 "~ 02/12 +

ie) (5.12)

The form of (5.12) is somewhat reminiscent of (2.16), but it has got selected v-orderings and A(~f)f has also a different energy denominator, typical of /1(3)+ [A(2),/1(1)]. * The g~ Coulomb term factorises and cancels out as in sect. 4, and the remaining O(g4) terms contain radiation fields, contributing to order O(a3slog (E/X)) or higher.

608

S. Catani et al. / Infrared divergences

Let us first discuss qualitatively what the effect of (5.12) on (5.11) may be, by distinguishing hard operators of type O, which has a non-vanishing commutator with b, and soft operators S which commute with b. Since heik - P" S, while h' - S', it is easily realized that A(2~fcontains pp- and p. S-type terms, while A(3f)f is of ooS or pS type. Therefore, the r.h.s, of (5.11) is of the form

V;ba(p)Ve= ba(p) + S(~°)(p)t~a,ba, + f d[p']S(,1)(p, p')t~a,{ p~(p'), ba,} ,

(5.13) where S <°) and S m are soft operators of the type S <°) = gZ("A2 " + gs"A 3 " + • • • ),

(5.14a)

S (1) ~_. g 2 ( " 1 "

(5.14b)

+ gs"A " + . . . )

and "A"" denotes the power of gluon fields occurring in the various soft operators. By replacing (5.13) into (5.2) we can see that the modified coherent state operator, up to order gs3, takes the form

VE(,,,l"''t,.)=

(l+S:>(p,)t;)+

0":

L'ab \ Y i ~ i¢k

(5.15) where colour indices (fli, ai } are understood, and U f f is now unitarily related to the eikonal coherent state Uf~ by the purely soft operator ia(O[ Ve[O) H. We can see from (5.15) that the only non-factorized term comes from S <1> in (5.13), which also contains the Coulomb phase contribution discussed in sect. 4 (the gs t term in (5.14b)). It turns out that the non-cancelling effects in the initial state also come from the S <1) term, while the S <°) terms do not contribute. In fact, after summing over soft final gluons the colour averaged cross section takes the general form ( o ) = Trs(01UetaOlLUe10)s,

°)lL{a.... }= (TrimTfiH){a.... }.

(5.16)

The S <°) dependent factor contributes terms which are linear in t/ and cancel because of the colour trace. More in detail Tr(0l

I-I ( OP'et~?)iLJ*"E)~ S(~°)(p/)tTlO) i

= Tr(01 cArtE J

(5.17a)

j

ObP'~etbS(~O)(pg)]0)

= 0,

(5.17b)

S. Catani et al. / Infrared divergences

609

Fig. 4. Typical diagrams contributing to Ate3~f(q) in eq. (5.19). Note however that the denominators in (5.19) are written in old-fashioned perturbation theory: the uncancelled part roughly corresponds to commutatorfunctionsinstead of Feynmanpropagators. where we have used the commutativity property (5.10) and the fact that

uPEtauPEt = UPft b ,

(5.18)

i.e., the t-matrices evolve in the adjoint representation. Since the Coulomb contribution cancels, as discussed in sect. 4, we shall concentrate, in the following, on the g3A term of S ~1), which defines the non-cancelling insertion operator. This one comes from the ppS term of A(3f)f,which is calculated in appendix A, and turns out to be (fig. 4)

f d[qlAO)fr(q)]ops= - ½(2~gs)3f d[plld[psld[qx]d[qEld[q3] X Eifa,asa3 oi

(pa~(pl),pa2(p2) } el" q,/32 "q2

[/31

"/3:/32" A a~ (q2)2,0~ ~(q~ -- ql )

X

° 2 ° 1 ( / 3 : - P l ) "ql +/32"q2

~'~ ~ F/~1/~2/~3 \( O"l trl/ l ~ ' ' ' ' 1 P'l~lP'2P'2

+-2

03603 + °1vl

+ ie°2

o

03q3)A,3a3(q3)~ 3

(;o,,)

"ql + °sV:" q: + ie (5.19a)

=-

½if,,,o2o3fd[Pl] d[Ps] d[q] { P~1, O~2} ~-,A~a3(q)Go~(Px, P2, q)o

(5.19b) By then replacing (5.19) into (5.1!) we get for the Oab~'(1)/n\Yl'Ps)=ifabc

d[q])-~.G~(Pl, Ps,q)A~ ( q ) , o

where G~ is defined by (5.19b).

g3A term of S (1) (5.20)

610

S. Catani et al. / Infrared divergences

We are now in a position to discuss the non-cancellation of subleading singularities in the initial state. By replacing (5.20) into (5.15) and (5.16), we obtain

( o ) - Tr aA]L=Tr(Ol[~j UP,etG)]LUP~e, ~., c(1)(pi, pk)] t iat kb10), ~ab

(5.21)

i4=k

where S (~) still contains the soft field A as in (5.20). The effect of the latter on U, up to order g4 included, is simply to replace A with the eikonal current, i.e.,

A tac( q ) ~ -gs~_,t[ Pit i Pi'q

c gsJ~(q).

(5.22)

We thus obtain, after performing the colour algebra,

( o ) - Tr°S~L=gsN~ Y" f d[q]jki(q).Go(pi, pk,q)Tr(CAlLti.tk), iq=k ~

(5.23) where

j~(q)

P~ = -Pk'q

PC

(5.24)

Pi'q

and we have used the identity

ifabcTr([ 9]L, tf]t~t~,) = Nc(Sjk-

8ik)Tr ( 63]Lti. t k ).

(5.25)

The integrals occurring in (5.23) are performed in appendix B, where we explicitly show that the final result is Lorentz invariant. We finally obtain ( o ) - Tr 9 L = -~sN¢log ~-

E T r ( g L t i . tk)(v/~ 1 -

1)F(vik),

(5.26)

i~k

where 1 F(O12)

va2 -

1 -~- /312

--2o12l°g 1 ~

v12

) [~

, "2

2 211/2

-PIP2]

(5 27)

is the bremsstrahlung function at relative velocity v12. Therefore, (5.26) is a highertwist effect. The result (5.26), when specialised to 2-parton annihilation, checks, to order a~log (E/2~), the original findings of J. C. Taylor et al. [8] for the Drell-Yan process.

S. Catani et al. / Infrared divergences

611

Since final state interactions have been shown to cancel to all orders, the result (5.25) applies as well to multiple jet production in parton-parton or multi-parton scattering, as seen e.g., in collider physics. The extension of (5.26) to all next subleading orders et~(log ( E/~ ))( a slog ( E/~ ))" is rather direct, since it is obtained by including the complete u-evolution in the definition (5.11) of Aar(v ). This means that A~f in (5.19a) should be replaced by*

(5.28)

UUeik~'aelf ~,t/7 UUeik ,

where ~2~'~a,is the eikonal coherent state operator. The effect of (5.28) on (5.21) is to replace A (q) and the colour matrices t~ with evolved ones as in (5.18), up to energy O~q.This cancels, by unitarity, part of the 9L-evolution, by leaving the one related to the energy interval ~q < ~0 < E. In other words, (5.21) should be replaced by

(o)

-

Tr tAIL= if,~b~ Y'. i4,k,o

f d[q]G#(p,p , ,

q)

×Tr(Ol[G31L, J)f%)(q)]t~f°',)t~'~)lO),

(5.29)

where t[ae'~) evolves as in (5.18), but with the operator

U p~e'°) = UpEu p'~*.

(5.30)

By then performing the colour algebra, by using (5.18) and by introducing the definition of the Sudakov form factor [4]

(01U '¢E )Ua"g¢E )IO) =

8bcS12( e , ,o),

(5.31)

we obtain ced~0

.

( ° ) - - TrgL= --a~Nc E Jx --d- S ' k ( E ' ~ ) i-'#k

×Tr(6)Et,'tk)(V~ 1 - 1) F(vik) •

(5.32)

This result differs from the lowest-order formula (5.26) by the introduction [9] of the * Strictly speaking, we should use in (5.28) the v variable instead of % . This difference is inessential, because we are looking for leading O((aslog(E/A)) n) corrections to the a21og(E/h) result: any scale of order % will do the same.

612

S. Catani et al. / Infrared divergences

Sudakov form factor in the adjoint representation, which is, to leading order [3]

Sa2(E, oa)=exp(- N~--~F( v12)log(E ) ) .

(5.33)

The introduction of the factor (5.33) into (5.26) can be interpreted [12] as a lack of compensation of virtual corrections by gluon radiation in the octet part of ~L projected out by (5.26). Due to the damping for ~0 << E, it translates [9] the infrared divergence in (5.26) into a finite violation of the factorisation of collinear singularities. Here we would remark that, besides checking previous results [8,9] by the coherent state formalism, eqs. (5.19) and (5.29) provide the structure of the general non-cancelling operator to next subleading order. As such, it could be applied to colour singlet initial states as well, thus giving an expression for the uncancelled Coulomb phase to this order, for both inclusive and exclusive processes. This analysis is deferred to further investigations.

6. Discussion

We have developed here a perturbative hamiltonian approach to IR singularities in QCD, which is based on the energy transfer evolution equations of sect. 2 and allows the explicit construction of QCD coherent states as operators in the soft-gluon Hilbert space. After summarising the properties of the leading-order coherent state operator previously obtained, [2, 4] our main concern here has been to include next subleading singularities in the coherent state, and to analyse in this context possible non-cancellations in many-parton initiated processes. Besides confirming previous results [9,12] we basically gain here a more systematic understanding of the matter. The main new results on this issue are the following: (i) The non-abelian Coulomb phase (due to low-momentum transfer interactions without radiation) factorises, in the massless quark limit, in the (sect. 4), contrary to previous expectations [11,18]. This fact can be applied, with some extra care, to Drell-Yan type processes to prove the leading twist cancellation of the Coulomb interaction. An important tool in this discussion is the coherent emission of Coulomb gluons. (ii) The uncancelled subleading terms in colour-averaged cross sections of DrellYan type are given in operator form (sect. 5), up to order a~log ( E / ~ ) ( a s l o g (E/~))n. Therefore the calculation applies to processes with more general initial and final states, like those of collider type, or those with more than two initial partons. Furthermore, the same formalism applies as well to hadronic wave functions, or to colour-averaged processes, the basic tool being the subleading coherent state operator of eq. (5.15).

finalstate

S. Catani et al. / Infrared divergences

613

An important remark on (5.15) is that it lacks the factorisation in the colour space of the hard partons - found to leading order, (eq. (3.9)) - in an essential way: in fact the correlation operator S~ ) in (5.20) contains radiation fields. This means that, if we were to construct coherent initial states (eq. 1.3), providing automatically IR finite matrix elements, the latter would be a property of the whole set of partons and not of each single coloured one. It is not possible to construct a single parton coherent state at IR subleading level, even in a perturbative approach: this is perhaps the main lesson learned from uncancelled IR divergences. This conclusion does not say, by itself, that the only physical asymptotic states are overall colour singlets. However, it may be possible to investigate this issue in the coherent state approach because the latter automatically contains [4] the Sudakov form factors typical of colour separation at infinite times. We known [3] that, for leading singularities, Sudakov factors disappear after summing over final soft gluons, even for coloured asymptotic states: but this is presumably what may not happen for IR subleading ones, due to the coherent phenomenon mentioned before. A related deep question is whether the whole approach of separating IR singularities as in (1.1) and (1.2) is valid in QCD outside the gauge-invariant sector of the Hilbert space. No counterexamples are known [7] so far, but it may well be that this is not the case. We have examined here the gauge transformation properties of the coherent states at time t = 0 (sect. 3), but what one should really do is to study the case of large times, which is related to possible left-over divergences. An analysis of these problems is left to future investigations.

Appendix A EVALUATION OF THE EFFECTIVE u-EVOLUTION OPERATOR

In this appendix we give the details about the evaluation of the effective evolution hamiltonian in eqs. (5.11b), (5.12), and (5.19). The calculation leading to (5.11b), (5.12) is straightforward and involves only some algebraic manipulations. Let us start from eq. (5.9) by making the eikonal u-evolution explicit, i.e.,

~'(~°)(Uo)

~ t

,

,

(:'(uo)+ f dUl [At(1)(b'o),A(el)(ul)]O01)

-

+ {:)(uo)+f UP1[A(2)(Uo),A(I)(P1)]OOl +

fdvldu2OolOo2[[A'(1)(Uo),

5(1)¢ik~'vlJl ~l A(1)eik~[U2:1 ~l

= Am)(Vo) + A'(:)(%) +/V(3)(Uo) + . . . ,

(A.1)

where A,O)(p) = A(1)(u) -- A(~(v), 0/j = 0(u~ -- uj) and we have neglected terms O(g~)

614

S. Catani et al. / Infrared divergences

or higher. Since the only O(gs) term in A'(")(1') is A'O)(11) ( a purely soft operator) the commutators in (5.11a) start at order gs2. By inserting (A.1) into eq. (5.11a) we have:

V;aeV~=bB+f d1)o[bB,,a'(2)(11o)]

-.t-Sd1)o[bfT,mt(3)(110)-t-Sd1)1~)o1[m'(2)(1)o) At(1)(1)l)]]-t-... =bB+

fxEd1"o[b~, ~(2'efftc11O/x + ~,~(110) +

'

]

(A.2)

where the last term comes from [[b~, A'(2)(1"o)], A'O)(1)0] by using [b~, A'(1)(1)1)] = 0 and the Jacobi identity. The two commutators in (A.2) define the effective evolution hamiltonian A(~ + A(~f. To recast this in the simple form of eq. (5.12) we only have to rewrite the A(~) operators in terms of h"(1') in eq. (2.8). By writing h = h eik+ h' as in (5.5), from eqs. (2.16), (A.1) and (A.2) we obtain: ( [hl'ho] [ho'h~ik] ) f'~1)--O-eft\aa)tllo)=fd1)od1)xOol 1'o(11o+1"1)q" 1)01'1 ' (A.3)

f fd1)°A(:f)f(1)°) .,

[h2,[hl, ho]]

=-d1)°d111d112 1)0(110-1-1)1)(1)0-t-1'1+1"2) 001002 + [ [hl'hOl'he2ik]

1)0(1'0q'- 1'1)1'2

001002 +

[[ht, hol, h'2]

[[h~,h~ ik] h~ik] ' 001012 1)01111'2 [[h~),po1)11' h~]'2 hS] [~OlOO2}' (A.4)

where h i = h(vi) and the energy signs o i are understood. By exchanging the uintegration variables and by omitting purely soft operators (i.e. terms with only h' contributions), eqs. (A.3) and (A.4) give, after some algebra,

fdvoA¢~(1)o)=fdvodvlIh~ikOol+ h;' h;ik] Vo(Uo+ vl)

duo dvl

,

(a.5)

d1) 2

f d"°a':~'(11°)=f1"o1)2(~o+111+1)2) × { [[h~, hg~<], n~] Oo1002 +

[[h~<,h~<],h'~]Ool

× [[h~, h~,'], h~] Oo~+ [[h~, h~,~], h~]}, (A.6) that is, eq. (5.12).

615

S. Catani et aL / Infrared divergences

The effective evolution hamiltonian A(3~f in eq. (5.12) contains the ppS type non-cancelling operator. However the direct evaluation of this operator by starting from (5.12) is involved because of the ;,-ordering. Therefore we perform first of all the colour algebra of the hard operator p. This allows elimination of the v-ordering. Let us start by decomposing the v-hamiltonian h i in hard and soft operators. We write h ~ = p,Si + S,',

(A.7)

where Pi is the hard charge density, Si is a soft operator which is linear in A°'(q;) and S/ is cubic in the soft gluon field. By replacing (A.7) in (5.12) we see that Sg and h~ do not contribute to the ppS type terms due to the fact that the pp type operator in [h~a'Ool + h~, h~"~] does not contain soft operators. Therefore we get: dv ° dv 1 dr2

f dv°A°~'(v°)~s =f VoV:(Vo+Vl+V:) X

([[plS1, PoSo],P2S2]001002-I- [[hi, PoSo], P2S2] 002 + [[s;, poSo], p~s~]Oo#o~+ [[pA, s~], p~s~]Oo#o~).

(A.8) Let us consider the first commutator on the r.h.s of eq. (A.8). For the ppS type operators we get:

[Pl, Po]Po[Sl, So]S2 + [Po, P2]Pl[S1, So]S2 + [P,, Po]P:([So, S:]S~ + [S~, S:]So) •

(A.9)

Then by replacing (A.9) in (A.8) and by exchanging the v-integration variables, we obtain: dvodvldv2

_ f duo dvx dr2

~

( OmO02 001012 010@12 020021)

[p~,p~]po[S~,So]S~((Vo+V~+~)--v~XOxoO~). 1

(A.10) The last term in this equation vanishes. In fact if we reintroduce the energy signs o;,

S. Catani et al. / Infrareddivergences

616

we get IS 1, S o l ~ r o y I - [ A ~ , A~o]/OoVoOlV 1 = 0. Therefore we have:

term=/VoV2(v---~O+Vl--V2)[[plSl'S°]'p2S2]p°" dv o d v I dv 2

oS

(A.11)

Let us now consider the h~ term in (A.8). Due to the fact that [So, S2] is a c-number, by using the Jacobi identity we can make the replacement: h~ term ~ ½{ [[h~, p0S0], p2S2] + (0 o 2)}.

(A.12)

Then by exchanging the (Vo, v2) integration variables in (A.8), we get dv o dv 1 d v 2

h'~ term= ½f vov2(Vo + Vx + p2) [[ h'~, OoSo], 02S2] .

(A.13)

After similar manipulations we have for the S/ terms in (A.8) (S; + S d) terms dv o dv x dv 2 "~-1"~2)[[S['p°S°]'p2S2]"

=½fvov2(Vo+Vl

(A.14)

Putting together eqs. (A.13) and (A.14), we reconstruct the 3g soft hamiltonian h g = h' + S' and so, by replacing (A.11), (A.13), (A.14) in (A.8), the non-cancelling insertion operator of a (3) err can be recast in the following form without v-ordering,

-if

f d~oa'f?r(~o)oos -

2

dv°dvldV2 PlP2(l~o -'t-" 1J1 -'1- v2)

i g [[OoSo+2ho, S 1 ] , o 2 s 2 l o l - h . c . (A.15)

The next step is to perform the Fock algebra in eq. (A.15). From eqs. (2.3) and (2.6) we have

(A.16a)

os ° = g s f d[p]d[q]oo(p)p.Ao (q)~(v - k q ) 1

XF~'lS2p'3(°:l,,i"~i,'--p.,]'d°la'(ql)mp.2 (q2)A~,,

(q3) 6 ov-~_,aso~

,

J

(A.16b)

617

S. Catani et al. / Infrared divergences

from which we get [[poSo, S1],P2S2]Pl=ig3f~l~:a~fd[p,]d[pz]d[ql]d[q2]d[q3]P,P2 >( El31"/32 P2" A 20.O8oo,-o,(2~')383(q3 - ql) oi X 8(v 0 -- ~2q1)8(v, --/~}lql) 8(v2 -- ~b2q2),

[[ ho~, s~], o~s~] ~ =

(A.17a)

,g~fo,o~o~fd[ p,]d[ p2]d[ q,]d[ q2]d[ q3 ] PiP2 3 3 o~

X 0"10"28(0"01P0-E0"iO)i)8(/)1 -- P l q l ) 8 ( P 2 - P2q2)'

(A.17b) By replacing eqs. (A.17) in (A.15) we obtain (5.19).

Appendix B MOMENTUM INTEGRALS AND LORENTZ COVARIANCE

Here we want to prove the expression (5.26) of the non-cancelled term on the basis of eq. (5.23), and check its Lorentz invariance. By decomposing the function G~(plp2; q) of eq. (5.19b) in its fermionic and gluonic parts we can write the momentum integral in (5.23) as 4 ( -~ ~ ) ( I f ( p l , P 2 ) + l g ( p l , pz)), g s ~ f d[q]J12(q).Go=gslog o

(B.1)

where we have factored out the logarithmic divergence and we have defined If ~

[ Pl "P2P2"J12(q2) +1~ f Pld_[q qlP2 1] Ad q 2]] E 0201(/02--~/~ 1----)-~q7 +--p2q-----2+ i e0"2 "q2 ,'1o2

218(q~-1)

"

(B.2) Ig = f d[ q3!d[ q2____]d[._q3] (2rr)383(Eeiqi)8(q 2 - 1) Px" qlP2" q X

p 1111p2~2r~l~2~3(O'qi) J12~3(q3) 03093 + Ol!'~1"ql + O2t~2" q2 "}- ie

(B.3)

S. Cataniet al. / Infrareddivergences

618

Since the logarithmically divergent integration has been performed, (B.2) and (B.3) only contain the ratio of energy variables, whose scale has been fixed by setting Iqll = 1 with a &function. In order to prove the Lorentz-invariance of (B.1) let us remark that the scale fixing &function is somewhat arbitrary. In fact, the integrals (B.2) and (B.3) are invariant under the replacement

3(q? -

1) ~ 3( F( ql, q2) - 1),

(B.4)

where F is homogeneous of degree 2 in the q,~'s but is otherwise arbitrary. This can be checked by the qg dependent scale transformation q~ ---, q~. ~(ql, q2): the 8-function is replaced by 3(q 2 - 1 / ~ 2) and, due to scale invariance, we get the jacobian factor 5det

g~'dJii-t ~. Oqj~ =-~ l +qi" Oqi~

'

which is unity if (1/~ 2 - 1) is quadratic in the qj, s. We may therefore set F = -ql~q~ or F = -q2,q~ or any other Lorentz invariant combination, if we wish to. Consider first the fermionic integral (B.2). By symmetrising with respect to ql ~ - ql, Ol ~ - Ol and by defining qO = Vl. ql, (B.2) takes the explicitly Lorentz invariant form

1

if= Eo,~ f

d4qld4q23+(q___ 22)3(P_______ix~ql) (Pl"Pz)P2"J,z(q2) 3(q? + 1 ) + ( 1 *-* 2), (2~r)6(_qZ)o2(pzq2) p2(olqt + ozq2) + ie (B.6)

where now ( - q ~ ) - l = ( q 2 - ( v l . q l ) 2 ) - I replaces the previous on-shell factor (0:a - vx" ql) -1, and we have set the scale by q2 = _ 1. It is clear from (B.6) that the eikonal representation (B.2) is obtained by going to one fermionic and one gluonic pole. Furthermore, the 0:2 integration in (B.6) is easily performed in the complex plane and, summing over oi = + 1, yields a factor (2¢r)23(p2 • ql)- We end up with the expression* d4qa

I r = - r (3- ~3 )( p l q l ) 3 ( p 2 q l ) 3 ( q 2 + 1 ) p l ' p 2 J ~

(

f d~22 --P2"P2tt2 1

1 g(u12), (4,tr) 2 U12 * When exchanging 1 ~ 2 we set the scale by q2 = cumbersome integrals.

--p1.P102 )

2

{1+u12 ~

F(/)12 ) = 2-~121og~ _----~12 1 } - 1, (B.7) 1. This trick (here and in the following)avoids

619

S. Catani et al. / Infrared divergences

where F(v12 ) is the bremsstrahlung function of the relative speed v~2 defined by (5.27). Let us then consider the gluonic integral (B.3), which requires a more elaborate procedure, due to the fact that 0)3 occurs in two denominators. By choosing, e.g., the Pl contribution to J12 that we call lg(1), and by writing, e.g.,

(0)3 "1- Vl" ql + 192"q2) -1(0)3 -- Vl" q3) -1

= (v2"q2 + 131 "(qx + q3))-l[(0)3-vl"q3) -1 -(0)3 + 191"ql

-[- 192 " q2)-1]

,

(B.8) we can repeat the symmetrisation for ( q ~ , o i ) o ( - q ~ , - o i ) on all denominators coming from the first term in the r.h.s, of (B.8). This yields the off-shell representation

d4qx d4q2~ (Plql) ~(P2q2)

I,=Eo f (2~r) qlq2[(ql-q2)

+tea]

PlbtlP2~2 /'t~2~,(ql,

q2, --ql -- q2) Pf' Px(qx + q2) 3(F(ql, q2) - 1),

×

(B.9)

while the second term in (B.8) gives a vanishing contribution, because the corresponding numerator is antisymmetric under qx ~ q3. We can see from (B.9) that the eikonal representation (B.3) is this time obtained from the two fermionic poles. By adding to (B.9) the contribution 1(2) from P2 and working out the numerators, we obtain _

:d4qld4q28(Plq1)8(P2q2)3(q?+1)( ,, 2Px'q2'~ (2~r)rq?q~[(q~+q--~2)~+ie--~ Pl"P2--zP2 p--f--.~, qt } + (1 <--)2),

Ig = r t e j

(B.10) where we have set the scale by q~ = - l ( q ~ = - 1 ) in the first (second) term. Since this integral is manifestly covariant, we can evaluate it in the frame v2 = 0 by performing the q2 integration h la Feynman, and yielding

Ig=-J~-~ (q2 -- (ql • Vl) 2 - l/ ef0 . X

d3q2(1 - 2x) f J [q~ + x ( 1 - xq21)-ie]

2

+(1

(2~.)3

2).

(B.11)

620

S. Cutani

Finally,

by noting

that

angular

integration

becomes

I,&= -(4a)-2F(vr2),

et ul. /

the principal

Infrared

value

divergences

integral

is simply

iq;l(l

-

the

q;‘),

trivial, and yields u12=

(P1.P2)-1[(P1.P2)2-P:P5]1’2r

03.w

where we have replaced u1(u2) by ur2. By putting (B.7) and (B.12) together, we get the final result (5.26). Note that the trick (B.4) has considerably simplified the integrals, compared to previous calculations [S, 91.

References [1] [2] [3] [4] [5] [6] [7] [8]

[9] [lo]

[11] [12]

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[13] W. W. Lindsay, D. A. Ross and C. T. Sachrajda, Nucl. Phys. B214 (1983) 61: W. W. Lindsay, Nucl. Phys. B229 (1983) 231 [14] G. T. Bodwin, Phys. Rev. D31 (1985) 2616; J. C. Collins, D. E. Soper and G. Sterman, Oregon Report Ol-TS-287 (1985) [15] A. H. Mueller, Phys. Reports 73 (1981) 237 [16] S. Catani and M. Ciafaloni, Nucl. Phys. B236 (1984) 61 1171 J. M. F. Labastida and G. Sterman, Nucl. Phys. B254 (1985) 425 [18] J. P. Ralston and B. Pire, Phys. Rev. Lett. 49 (1982) 1605; Phys. Lett. 117B (1982) 233