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Asymptotic coherent states and colour screening
Volume 168B, number 3
PHYSICS LETTERS
6 March 1986
ASYMPTOTIC C O H E R E N T STATES A N D COLOUR SCREENING
S. CATANI, M. C I A F A L O N I Dipartimento di Fisica, Universitb di Firenze, 1-50125 Florence, Italy and liVEN, Sezione di Firenze, 1-50125 Florence, Italy
and G. M A R C H E S I N I Dipartimento di Fisica, Universiti~ di Parma, I-43100 Parma, Italy and INFN, Sezione di Milano, 1-20133 Milan, Italy Received 13 December 1985
The recent construction of QCD coherent states provides constraints on the momenta and colour charges of possible partonic asymptotic states. By using the structure of the leading and first subleading infrared singularities for the case of incoming q~l pairs we argue that asymptotic colour separation is not allowed.
The Faddeev-Kulish method [1] of asymptotic dynamics has recently been generalised to QCD [2,3] and used for the construction [4,5] of infrared (IR) coherent states up to the first subleading singularity level. In this note we investigate possible partonic asymptotic states in this context and we argue, on the basis of a fourth order perturbative result and of the structure of the coherent states, that only low mass asymptotic colour singlets are self-consistent, i.e., that eolour separation is not allowed. In the framework of asymptotic dynamics the partonic Fock space is decomposed in a hard and in a soft part ( ~ = ~ H ® ~ S ) according to a scale E, and a similar decomposition is made for the hamiltonian. IR singularities are then factorised in operator form, i.e., S = ~_EI"SH~2E ,
(1)
where S H is IR finite [2,6] and the operator ~2E embodies soft gluon interactions up to energy E and leaves unchanged the momenta of the hard partons (i.e., those withE(Pi)>> E). Candidates for singularities free incoming (outgoing) asymptotic states are then 284
constructed by means of the M611er operators ~2+E?(I2E_t), so as to lead to a f'mite S-matrix, i.e., [hard, -+) = ~2~ E t Ihard),
(2)
where Ihard) E ~ H is a superposition of partonic Fock states lPi, ai) with E(Pi) >>E. However, for the states (2) to be acceptable in a partonic framework, it is plausible to require that gauge invariant [7] hard partonic operators O H be measurable * 1, i.e. that the expectation values (hard,-+ IOH Ihard, +) = (hard 112tOH~2~ [hard),
(3)
by IR finite. We shall see in the following that the perturbative evaluation of (3) is similar to that of a Drell-Yan type process [8] and that in fact analogous IR divergences [9] arise for the finwanted configurations of initial state partons. Similar singularities also occur [10] in the unitary relation for the transformation matrix
,1 We always refer here to globally colour singlet Ihard) states, for which the gauge invariance o f (2) has been checked up to relative order gsa [7] (el. eq. (6)).
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PHYSICS LETTERS
(Pi, {3i1~- £2t+IPi, ai) = M~,a_(Pl .... , Pn) ,
(4)
which relates the in and out states (2). Our results agree with those obtained in refs. [9,10] in a different context. It is convenient to work out the matrix elements of ~_+ between hard states by defining H (01Y~ IO)H [U+-E(pl.... , Pn)]a,a_
=- (Pi, ~il~2E lPi , ai> ,
(5)
where UE is the "coherent state" operator, i.e. a matrix in the colours (ai) of the hard partons, and an operator in the soft gluon Hilbert space ~ s . Its explicit expression in terms of gluon annihilation (creation) operators Aua(q) (A L ( q ) ) in the interaction representation has been worked out in ref. [5] and found to be
6 March 1986
~ - O2
= exp +ia s k
J [ia s E dw ~ t . w . t ~ O A i / ) , , ~ X P t ° e x p x + - 2 - . x w i¢/ % =
1 - 1,
oi: --
(8)
[1 - p 2 p : / ( p i . p / ) 2 ] * / : ,
where the first factor is the massless quark contribution, which depends on Q, the.total hard parton colour charge and is therefore trivial for Q = 0. The second factor only depends on A//= vii 1 - 1 and contains radiative corrections of typeIl(q) due to the wevolution of the t-matrices t.o
¢o
tea = [U eik(Pi)]ab tib .
(9a)
(3) The residual factor ~3r V = I + i A R = 1+ i i<]$sJabctiat]b
U E± k rV'*, . '--,Pn) n
= Ii=1 I U~ik(Pi)U+Ek C(Pl , "", Pn) vEXCal ..... Pn)
(6a)
h
= ~r~_+~Ex I-I 4fl~(pi), (6b) i where ~ (~, < rn < E) is an IR cut-off (to be understood in the following) which should disappear from all physical quantities. The notation in eq. (6) is the following: (1) The factorised term in front of(6a) is the leading order (eikonal) coherent state, a functional of the field II(q) = i(At - A) only, given by the energy ordered exponential E U~k(P) = Pro exp -( i g s f
d[q]
P • II~°q
t" ) '
d[q] - d3q/(2rr)32Wq , ~
(7a)
× fd[q]GRt*(q)(A#c
+ A;c),
contains partonic correlations due to mixed Coulombradiation terms of type (A + At), where the function Gty Rv is given in detail in ref. [5]. Finally, the operators V, 0 c are unitarily related to Uc , V by 0 C = U~eikUC Ueik, and similarly for V. Therefore 0 c differs from U C by the replacement
ti°~ -* ~iat° =- [UEkt (i) U~k(i)]abtib ,
UeE'$ = H UE~c(Pi) •
IIua (q)
This implies that hard parton colour lines can be treated, to some extend, independently. (iii) Leading order commutativity:
_
- [l~eik(q)]abII~b(q ), II~°L = IIL .
(7b)
(2) ~ is the non-abelian "Coulomb-phase" matrix* 2 .2 The Coulomb-phase operator is here classified as subleading because of the lack of collinear logarithms. It contains,
however, leading IR logs and radiative corrections of type rRq) (eq. (9)).
(9b)
and V can be set equal to V within the present accuracy. For further use, let us emphasize the following properties of(6) proved in ref. [5]: (i) Unitarity: U_+ Et UE = 1. This implies the automatic cancellation of final state interactions. (ii) Leading order factorisation:
where t a is the colour matrix of parton p and the transverse and longitudinal parts of II w are given by to.[
(10)
!
[(UEk)t~, (UEk)t3,~_,] = 0 ,
(11)
(12)
due to uE~ being a functional of II(q) only. This implies the cancellation of IR leading initial state inter285
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PHYSICS LETTERS
actions in colour averaged cross sections, and is also valid for the Coulomb-phase operator. For our discussion, the most important feature of eq. (6) is the violation of factorisation [11,12] and/or o f the Bloch-Nordsieck theorem [8] in many-parton initiated processes. This fact is due to long-ran~ge twobody correlations, either of Coulomb-type (U~) or with accompanied radiation (V), and will be recalled in the following. Let us consider in eq. (2), for simplicity of language, an operator O H of Drell-Yan type, which measures the number of colour singlet qFt pairs with widely separated momenta. According to (3) and (5), this process is described by (OH) = (hardl ~2+EOH~2+ Et Ihard)
(13a)
= S(0I [ U f ( P l , - . . , Pn)OH u+Et(ff 1 , ..., pn)],--,a_10)S, (13b) where we have set Ihard) = I P l a l , ...,Pnan ) and OH in (13b) is a colour matrix given by ,a (On)t@',/~ = ~31 3~5/31/~Z i~l~l,2 ~3~3i "
(14)
6 March 1986
4
Fig. 1. Colour indices for the U~+and OH operators in the text. Crosses denote emission and absorption of soft gluorm.
which contains uncancelled double logs. For large N C we find (OH)q~ "" exp [-- (Nceq/n)F(Ol2) log (E/k)]
= S1A2(E, ;k),
(17)
Note first that in (13b) the Ueik(Pi) factors of the "spectator" partons (i :/: 1,2) cancel out by unitarity, so as to give
where SA12 is the Sudakov form factor in the adjoint representation (cf., e.g. ref. [13]) and
(OH)-a"~-
F(v) = (1/2 o) log [(1 + v)/(1 - v)] - 1,
(15)
= S(01(~+ VUeik(1,2) OH Uteik(l,2)V~ 0_C)~_,,a_I0)S. The question now arises of whether the IR singularities of the remaining U and V factors cancel out in eq. (15). If we did not know about quark confinement, we could think of measuring O H on a large mass q~l singlet of "active" (unconfined) quarks by setting o~1 = o~2, ! t OtI = O~2 (fig. 1). This measure is inconsistent already at the leading IR singularity level. By setting U c = V = 1 in (15) we get in fact the expression (OH)q~ = (1/Nc)s(OlTr[Uteik(1) Ueik(2)] X Tr [Uefik(2) Ueik(1)] t0) S ,
(16)
o12 = [1 -- p2.2/r~ 1/-'2 w l " P 2 ) 2] 1/2 ,
(18)
is the bremsstrahlung function. Therefore this divergence disappears only for o12 ~ 0 (Pl = P2), i.e. for a singlet of small mass. The lack of cancellation of real emission and virtual corrections in e q. (16) (somewhat similar to preconFmement [14] in the final state) will occur for any non trivial colour correlation of the active partons (i = 1,2). However, if we perform an incoherent I r average over colours, by setting tx1 = or1, ct2 = or2, the leading IR singularities disappear. In fact, if Uc and V are neglected, by the computativity property (iii) we obtain Tr [Ueik(1,2) O u _U~eik(1,2)]
,3 The exact gauge invariance of the short-distance operator OH so defined requires in general gluonic string contributions, which however are power suppressed in the infrared region. 286
= Tr [OnUetik(1,2) Ueik(1,2)] = T r O n , which proves the cancellation stated before.
(19)
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It is common understanding that colour averaging is realised because the active partons belong to two different q~ colour singlets ((1,3) and (2,4) in fig. 1). However, in such case correlations (among all quarks) due to UC and V play an important role, and we shall show that subleading IR singularities will not cancel, unless pairs (1,3) and (2, 4) have small mass. In order to see that, let us perform a complete fourth order evaluation of eq. (15) for two incoming q~ s ~ e t s . To be consistent, the Coulomb-phase matrix U~ will be expanded, by use of eq. (9), in the form = e x p ( + i ~ c ) ( l --+iAC + ...), ¢)C = ~ asl°g E~k~i:~k Aik ti" tk '
(20a)
AC=4-~Jabc i
(20b)
where AI.k --- u~ 1 - 1. By then replacing eqs. (10) and (20) into (15) we obtain
6 March 1986
8R = -2a21°g(E/X)(I12 - 114 - 132 +I34),
(23)
where, by the definition o f A R in eq. (10) [5], Ii/ =
fd [q] Gi~(q ) (Plu/Pi" q - P2u/P2" q)
× (47r)2/log(E/X)]-1
= ~ [F(oi2 ) + F(Ol/) - F(oi2 ) - F(Uli)] Ai/.
(24)
The result (23) parallels those for the DrelI-Yan cross section [9] in the corresponding kinematics and is similarly of higher twist type, because Ai/= vii 1 -- 1 = 0 in the massless quark case. However, the infrared divergence is still there for/)2 ~ 0, and the O Hmeasure is therefore inconsistent, unless the singlets (1,3) and (2,4) have small mass with respect to the large momentum transferred by O H . In order to see that, care should be taken of regulating the IR divergence for small mass colour singlets. In the eikonal approximation for the hard incoming partons (cf. refs. [11,13]) one can introduce the impact parameters b i of spectator partons and the finite colour singlet radius R by
( ~ , xl i) = (hi, (Pi/Ei)t) ,
-i[O~,AR
+ A c ] ) l a l = a 3 , a 2 = a 4 ),
(21)
e ~ et where 0~1 = Ugi k O H Uei k and we have used the fact that [On , ~ c ] has vanishing colour factor. The action o f A R and AC can then be evaluated, up to order gs4, by the replacement
[O~H,A ua(q)] ~ gs [OH, J~a(q)],
(22 a)
Jun(q) = ~ (Pku/Pk " q)tka • k
(22b)
bl=bl=0
(b2)=R 2
iv2 f E d2 q, : ' s JI/T ~
lIT
d2q
- - 2 A12 7rq
+ (A42 - A41) exp(iq" b4) ] (exp(iq -b3) - 1) X (exp(iq • b4) - 1)) ~ rv20(1),
(26a)
E 8 R =--2t~2112 f
- - ( I / N c ) Tr(OH t 1 "t2)(8C + 8R),
X
(25)
X ([(A31 -- A32) exp(iq" b3)
(O H) - (I/N2c) Tr O H
= -
(i4:1,2),
which also implies a finite parton lifetime T i ~- EiR2. As a consequence, the eikonal current (22b) is replaced by Jui(q)exp(iq • hi), while (because of the small mass) the relative velocities of fast quarks become degenerate: vii ~ - o12. We thus obtain the regulated expressions for the uncancelled terms: 8C = -
We thus f'md that the contributions of A and A t cancel in II = -i(A - At), so that the A C contribution vanishes also. Finally, by performing the colour algebra of the remaining terms, we obtain the result
,
('a
3 +
x24 -
(AI2 -- A14 -- A23 + A34) ,
'h4
- zx23)
d2q
7rq2
X ((exp(iq • b3) - 1)(exp(iq • b4) - 1)~ -~ - 2ct2112 log E R ,
(26b) 287
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PHYSICS LETTERS
where an average has been taken over initial state wave functions. We can see that in eq. (26) the gluon energies are cut-off at co > 1JR so that the result is ),-independent. This is due to the fact that, when active and spectator partons act coherently (R < ¢o-1 < 7) the interaction is cancelled because o f the colour singlet condition. For the remaining time (E -1 < 6o-1 < R) only the A - A interaction survives in the radiative term, leading to the logarithmic, higher twise, value of eq. (26b). O n t h e othe~ hand, the Coulomb contribution (26a) is washed out completely because (i) the 60ordering implies 60' > 6 0 >~ 1/R but (ii) the second gluon 60' has no A - , 4 interactions and therefore 60' <~ 1/R. The fact that in the time interval E -1 < 60-1 < R the A - A interaction is ineffective is due to the commutativity property in eq. (12), and is valid for higher orders also, the Coulomb-phase operator depending only o n n ( q ) . We thus conclude that asymptotic coherent states are inconsistent in perturbation theory, unless the q~ pairs arrange in colour singlets of small mass. This conclusion relies on the operator factorisation o f IR singularities (1) with the form (6) o f the coherent state, and on the requirement that hard operators of type O H be measurable. An additional point is that in and out states are in general not unitarily related, in agreement with ref. [ 10]. In fact, the transformation matrix o f eq. (4) is, because o f the expression (6), given by
M(Pl ..... Pn) = S(01(6"C) 210)S,
(27)
where the factors Ueik, V (which are ie-independent) drop out by unitarity. Therefore one has (in the colour space o f { P l , ... , Pn ))
288
6 March 1986
1 - MtM = n~O IS(01(crC-)2 In>s 12 > g 6 is(0tAC i 1)S 12 ,
(28)
and the RHS of (28) is, because of (20b), generally divergent, unless (again) the state (Pl, "", Pn} is arranged in colour singlets o f small mass. The divergences o f eq. (28) are of different type from those of eq. (23) (since they involve H-fields), but seem to point to the same conclusion.
References [1] P.P. Kulish and L.D. Faddeev, Teor. Mat. Fiz. 4 (1970) 153,745. [2] C.A. Nelson, Nuel. Phys. B181 (1981) 141. [3] M. Ciafaloni, Phys. Lett. 150B (1985) 379. [4] S. Catani and M. Ciafalorti, Nucl. Phys. B249 (1985) 301. [5] S. Catani, M. Ciagaloni and G. Marchesini, Nucl. Phys., to be published. [6] J. Frenkel, J.G.M. Gatheral and J.C. Taylor, Nucl. Phys. B194 (1982) 172. [7 ] S. Catani and M. Ciafaloni, unpublished. [ 8 ] A. Andrasi, M. Day, R. Doda, J. Ftenkel and J.C. Taylor, Nucl. Phys. B182 (1981) 104. [9] P.H. Sbrensen and J.C. Taylor, Nuel. Phys. B238 (1984) 284; J. Frenkel, P. S6rensen and J.C. Taylor, DAMPT report 85-4 (1985). [10] A.Yu. Kamensehik and N.A. Sveshnikov, Phys. Lett. 123B (1983) 255. [11] C.T. Bodwin, S.I. Brodsky and G.P. Lepage, Phys. Rev. Lett. 47 (1981) 1799; SLAC reports SLAC PUB 2927 (1982). [12] W.W. Lindsay, R.A. Ross and C.T. Sachrajda, Nucl. Phys. B214 (1983) 61. [13] A. Bassetto, M. Ciafaloni and G. Marchesini, Phys. Rep. i00 (1983) 201. [14] D. Amati and G. Veneziano, Phys. Lett. 83B (1979) 87.
PHYSICS LETTERS
6 March 1986
ASYMPTOTIC C O H E R E N T STATES A N D COLOUR SCREENING
S. CATANI, M. C I A F A L O N I Dipartimento di Fisica, Universitb di Firenze, 1-50125 Florence, Italy and liVEN, Sezione di Firenze, 1-50125 Florence, Italy
and G. M A R C H E S I N I Dipartimento di Fisica, Universiti~ di Parma, I-43100 Parma, Italy and INFN, Sezione di Milano, 1-20133 Milan, Italy Received 13 December 1985
The recent construction of QCD coherent states provides constraints on the momenta and colour charges of possible partonic asymptotic states. By using the structure of the leading and first subleading infrared singularities for the case of incoming q~l pairs we argue that asymptotic colour separation is not allowed.
The Faddeev-Kulish method [1] of asymptotic dynamics has recently been generalised to QCD [2,3] and used for the construction [4,5] of infrared (IR) coherent states up to the first subleading singularity level. In this note we investigate possible partonic asymptotic states in this context and we argue, on the basis of a fourth order perturbative result and of the structure of the coherent states, that only low mass asymptotic colour singlets are self-consistent, i.e., that eolour separation is not allowed. In the framework of asymptotic dynamics the partonic Fock space is decomposed in a hard and in a soft part ( ~ = ~ H ® ~ S ) according to a scale E, and a similar decomposition is made for the hamiltonian. IR singularities are then factorised in operator form, i.e., S = ~_EI"SH~2E ,
(1)
where S H is IR finite [2,6] and the operator ~2E embodies soft gluon interactions up to energy E and leaves unchanged the momenta of the hard partons (i.e., those withE(Pi)>> E). Candidates for singularities free incoming (outgoing) asymptotic states are then 284
constructed by means of the M611er operators ~2+E?(I2E_t), so as to lead to a f'mite S-matrix, i.e., [hard, -+) = ~2~ E t Ihard),
(2)
where Ihard) E ~ H is a superposition of partonic Fock states lPi, ai) with E(Pi) >>E. However, for the states (2) to be acceptable in a partonic framework, it is plausible to require that gauge invariant [7] hard partonic operators O H be measurable * 1, i.e. that the expectation values (hard,-+ IOH Ihard, +) = (hard 112tOH~2~ [hard),
(3)
by IR finite. We shall see in the following that the perturbative evaluation of (3) is similar to that of a Drell-Yan type process [8] and that in fact analogous IR divergences [9] arise for the finwanted configurations of initial state partons. Similar singularities also occur [10] in the unitary relation for the transformation matrix
,1 We always refer here to globally colour singlet Ihard) states, for which the gauge invariance o f (2) has been checked up to relative order gsa [7] (el. eq. (6)).
0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 168B, number 3
PHYSICS LETTERS
(Pi, {3i1~- £2t+IPi, ai) = M~,a_(Pl .... , Pn) ,
(4)
which relates the in and out states (2). Our results agree with those obtained in refs. [9,10] in a different context. It is convenient to work out the matrix elements of ~_+ between hard states by defining H (01Y~ IO)H [U+-E(pl.... , Pn)]a,a_
=- (Pi, ~il~2E lPi , ai> ,
(5)
where UE is the "coherent state" operator, i.e. a matrix in the colours (ai) of the hard partons, and an operator in the soft gluon Hilbert space ~ s . Its explicit expression in terms of gluon annihilation (creation) operators Aua(q) (A L ( q ) ) in the interaction representation has been worked out in ref. [5] and found to be
6 March 1986
~ - O2
= exp +ia s k
J [ia s E dw ~ t . w . t ~ O A i / ) , , ~ X P t ° e x p x + - 2 - . x w i¢/ % =
1 - 1,
oi: --
(8)
[1 - p 2 p : / ( p i . p / ) 2 ] * / : ,
where the first factor is the massless quark contribution, which depends on Q, the.total hard parton colour charge and is therefore trivial for Q = 0. The second factor only depends on A//= vii 1 - 1 and contains radiative corrections of typeIl(q) due to the wevolution of the t-matrices t.o
¢o
tea = [U eik(Pi)]ab tib .
(9a)
(3) The residual factor ~3r V = I + i A R = 1+ i i<]$sJabctiat]b
U E± k rV'*, . '--,Pn) n
= Ii=1 I U~ik(Pi)U+Ek C(Pl , "", Pn) vEXCal ..... Pn)
(6a)
h
= ~r~_+~Ex I-I 4fl~(pi), (6b) i where ~ (~, < rn < E) is an IR cut-off (to be understood in the following) which should disappear from all physical quantities. The notation in eq. (6) is the following: (1) The factorised term in front of(6a) is the leading order (eikonal) coherent state, a functional of the field II(q) = i(At - A) only, given by the energy ordered exponential E U~k(P) = Pro exp -( i g s f
d[q]
P • II~°q
t" ) '
d[q] - d3q/(2rr)32Wq , ~
(7a)
× fd[q]GRt*(q)(A#c
+ A;c),
contains partonic correlations due to mixed Coulombradiation terms of type (A + At), where the function Gty Rv is given in detail in ref. [5]. Finally, the operators V, 0 c are unitarily related to Uc , V by 0 C = U~eikUC Ueik, and similarly for V. Therefore 0 c differs from U C by the replacement
ti°~ -* ~iat° =- [UEkt (i) U~k(i)]abtib ,
UeE'$ = H UE~c(Pi) •
IIua (q)
This implies that hard parton colour lines can be treated, to some extend, independently. (iii) Leading order commutativity:
_
- [l~eik(q)]abII~b(q ), II~°L = IIL .
(7b)
(2) ~ is the non-abelian "Coulomb-phase" matrix* 2 .2 The Coulomb-phase operator is here classified as subleading because of the lack of collinear logarithms. It contains,
however, leading IR logs and radiative corrections of type rRq) (eq. (9)).
(9b)
and V can be set equal to V within the present accuracy. For further use, let us emphasize the following properties of(6) proved in ref. [5]: (i) Unitarity: U_+ Et UE = 1. This implies the automatic cancellation of final state interactions. (ii) Leading order factorisation:
where t a is the colour matrix of parton p and the transverse and longitudinal parts of II w are given by to.[
(10)
!
[(UEk)t~, (UEk)t3,~_,] = 0 ,
(11)
(12)
due to uE~ being a functional of II(q) only. This implies the cancellation of IR leading initial state inter285
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actions in colour averaged cross sections, and is also valid for the Coulomb-phase operator. For our discussion, the most important feature of eq. (6) is the violation of factorisation [11,12] and/or o f the Bloch-Nordsieck theorem [8] in many-parton initiated processes. This fact is due to long-ran~ge twobody correlations, either of Coulomb-type (U~) or with accompanied radiation (V), and will be recalled in the following. Let us consider in eq. (2), for simplicity of language, an operator O H of Drell-Yan type, which measures the number of colour singlet qFt pairs with widely separated momenta. According to (3) and (5), this process is described by (OH) = (hardl ~2+EOH~2+ Et Ihard)
(13a)
= S(0I [ U f ( P l , - . . , Pn)OH u+Et(ff 1 , ..., pn)],--,a_10)S, (13b) where we have set Ihard) = I P l a l , ...,Pnan ) and OH in (13b) is a colour matrix given by ,a (On)t@',/~ = ~31 3~5/31/~Z i~l~l,2 ~3~3i "
(14)
6 March 1986
4
Fig. 1. Colour indices for the U~+and OH operators in the text. Crosses denote emission and absorption of soft gluorm.
which contains uncancelled double logs. For large N C we find (OH)q~ "" exp [-- (Nceq/n)F(Ol2) log (E/k)]
= S1A2(E, ;k),
(17)
Note first that in (13b) the Ueik(Pi) factors of the "spectator" partons (i :/: 1,2) cancel out by unitarity, so as to give
where SA12 is the Sudakov form factor in the adjoint representation (cf., e.g. ref. [13]) and
(OH)-a"~-
F(v) = (1/2 o) log [(1 + v)/(1 - v)] - 1,
(15)
= S(01(~+ VUeik(1,2) OH Uteik(l,2)V~ 0_C)~_,,a_I0)S. The question now arises of whether the IR singularities of the remaining U and V factors cancel out in eq. (15). If we did not know about quark confinement, we could think of measuring O H on a large mass q~l singlet of "active" (unconfined) quarks by setting o~1 = o~2, ! t OtI = O~2 (fig. 1). This measure is inconsistent already at the leading IR singularity level. By setting U c = V = 1 in (15) we get in fact the expression (OH)q~ = (1/Nc)s(OlTr[Uteik(1) Ueik(2)] X Tr [Uefik(2) Ueik(1)] t0) S ,
(16)
o12 = [1 -- p2.2/r~ 1/-'2 w l " P 2 ) 2] 1/2 ,
(18)
is the bremsstrahlung function. Therefore this divergence disappears only for o12 ~ 0 (Pl = P2), i.e. for a singlet of small mass. The lack of cancellation of real emission and virtual corrections in e q. (16) (somewhat similar to preconFmement [14] in the final state) will occur for any non trivial colour correlation of the active partons (i = 1,2). However, if we perform an incoherent I r average over colours, by setting tx1 = or1, ct2 = or2, the leading IR singularities disappear. In fact, if Uc and V are neglected, by the computativity property (iii) we obtain Tr [Ueik(1,2) O u _U~eik(1,2)]
,3 The exact gauge invariance of the short-distance operator OH so defined requires in general gluonic string contributions, which however are power suppressed in the infrared region. 286
= Tr [OnUetik(1,2) Ueik(1,2)] = T r O n , which proves the cancellation stated before.
(19)
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It is common understanding that colour averaging is realised because the active partons belong to two different q~ colour singlets ((1,3) and (2,4) in fig. 1). However, in such case correlations (among all quarks) due to UC and V play an important role, and we shall show that subleading IR singularities will not cancel, unless pairs (1,3) and (2, 4) have small mass. In order to see that, let us perform a complete fourth order evaluation of eq. (15) for two incoming q~ s ~ e t s . To be consistent, the Coulomb-phase matrix U~ will be expanded, by use of eq. (9), in the form = e x p ( + i ~ c ) ( l --+iAC + ...), ¢)C = ~ asl°g E~k~i:~k Aik ti" tk '
(20a)
AC=4-~Jabc i
(20b)
where AI.k --- u~ 1 - 1. By then replacing eqs. (10) and (20) into (15) we obtain
6 March 1986
8R = -2a21°g(E/X)(I12 - 114 - 132 +I34),
(23)
where, by the definition o f A R in eq. (10) [5], Ii/ =
fd [q] Gi~(q ) (Plu/Pi" q - P2u/P2" q)
× (47r)2/log(E/X)]-1
= ~ [F(oi2 ) + F(Ol/) - F(oi2 ) - F(Uli)] Ai/.
(24)
The result (23) parallels those for the DrelI-Yan cross section [9] in the corresponding kinematics and is similarly of higher twist type, because Ai/= vii 1 -- 1 = 0 in the massless quark case. However, the infrared divergence is still there for/)2 ~ 0, and the O Hmeasure is therefore inconsistent, unless the singlets (1,3) and (2,4) have small mass with respect to the large momentum transferred by O H . In order to see that, care should be taken of regulating the IR divergence for small mass colour singlets. In the eikonal approximation for the hard incoming partons (cf. refs. [11,13]) one can introduce the impact parameters b i of spectator partons and the finite colour singlet radius R by
( ~ , xl i) = (hi, (Pi/Ei)t) ,
-i[O~,AR
+ A c ] ) l a l = a 3 , a 2 = a 4 ),
(21)
e ~ et where 0~1 = Ugi k O H Uei k and we have used the fact that [On , ~ c ] has vanishing colour factor. The action o f A R and AC can then be evaluated, up to order gs4, by the replacement
[O~H,A ua(q)] ~ gs [OH, J~a(q)],
(22 a)
Jun(q) = ~ (Pku/Pk " q)tka • k
(22b)
bl=bl=0
(b2)=R 2
iv2 f E d2 q, : ' s JI/T ~
lIT
d2q
- - 2 A12 7rq
+ (A42 - A41) exp(iq" b4) ] (exp(iq -b3) - 1) X (exp(iq • b4) - 1)) ~ rv20(1),
(26a)
E 8 R =--2t~2112 f
- - ( I / N c ) Tr(OH t 1 "t2)(8C + 8R),
X
(25)
X ([(A31 -- A32) exp(iq" b3)
(O H) - (I/N2c) Tr O H
= -
(i4:1,2),
which also implies a finite parton lifetime T i ~- EiR2. As a consequence, the eikonal current (22b) is replaced by Jui(q)exp(iq • hi), while (because of the small mass) the relative velocities of fast quarks become degenerate: vii ~ - o12. We thus obtain the regulated expressions for the uncancelled terms: 8C = -
We thus f'md that the contributions of A and A t cancel in II = -i(A - At), so that the A C contribution vanishes also. Finally, by performing the colour algebra of the remaining terms, we obtain the result
,
('a
3 +
x24 -
(AI2 -- A14 -- A23 + A34) ,
'h4
- zx23)
d2q
7rq2
X ((exp(iq • b3) - 1)(exp(iq • b4) - 1)~ -~ - 2ct2112 log E R ,
(26b) 287
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where an average has been taken over initial state wave functions. We can see that in eq. (26) the gluon energies are cut-off at co > 1JR so that the result is ),-independent. This is due to the fact that, when active and spectator partons act coherently (R < ¢o-1 < 7) the interaction is cancelled because o f the colour singlet condition. For the remaining time (E -1 < 6o-1 < R) only the A - A interaction survives in the radiative term, leading to the logarithmic, higher twise, value of eq. (26b). O n t h e othe~ hand, the Coulomb contribution (26a) is washed out completely because (i) the 60ordering implies 60' > 6 0 >~ 1/R but (ii) the second gluon 60' has no A - , 4 interactions and therefore 60' <~ 1/R. The fact that in the time interval E -1 < 60-1 < R the A - A interaction is ineffective is due to the commutativity property in eq. (12), and is valid for higher orders also, the Coulomb-phase operator depending only o n n ( q ) . We thus conclude that asymptotic coherent states are inconsistent in perturbation theory, unless the q~ pairs arrange in colour singlets of small mass. This conclusion relies on the operator factorisation o f IR singularities (1) with the form (6) o f the coherent state, and on the requirement that hard operators of type O H be measurable. An additional point is that in and out states are in general not unitarily related, in agreement with ref. [ 10]. In fact, the transformation matrix o f eq. (4) is, because o f the expression (6), given by
M(Pl ..... Pn) = S(01(6"C) 210)S,
(27)
where the factors Ueik, V (which are ie-independent) drop out by unitarity. Therefore one has (in the colour space o f { P l , ... , Pn ))
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6 March 1986
1 - MtM = n~O IS(01(crC-)2 In>s 12 > g 6 is(0tAC i 1)S 12 ,
(28)
and the RHS of (28) is, because of (20b), generally divergent, unless (again) the state (Pl, "", Pn} is arranged in colour singlets o f small mass. The divergences o f eq. (28) are of different type from those of eq. (23) (since they involve H-fields), but seem to point to the same conclusion.
References [1] P.P. Kulish and L.D. Faddeev, Teor. Mat. Fiz. 4 (1970) 153,745. [2] C.A. Nelson, Nuel. Phys. B181 (1981) 141. [3] M. Ciafaloni, Phys. Lett. 150B (1985) 379. [4] S. Catani and M. Ciafalorti, Nucl. Phys. B249 (1985) 301. [5] S. Catani, M. Ciagaloni and G. Marchesini, Nucl. Phys., to be published. [6] J. Frenkel, J.G.M. Gatheral and J.C. Taylor, Nucl. Phys. B194 (1982) 172. [7 ] S. Catani and M. Ciafaloni, unpublished. [ 8 ] A. Andrasi, M. Day, R. Doda, J. Ftenkel and J.C. Taylor, Nucl. Phys. B182 (1981) 104. [9] P.H. Sbrensen and J.C. Taylor, Nuel. Phys. B238 (1984) 284; J. Frenkel, P. S6rensen and J.C. Taylor, DAMPT report 85-4 (1985). [10] A.Yu. Kamensehik and N.A. Sveshnikov, Phys. Lett. 123B (1983) 255. [11] C.T. Bodwin, S.I. Brodsky and G.P. Lepage, Phys. Rev. Lett. 47 (1981) 1799; SLAC reports SLAC PUB 2927 (1982). [12] W.W. Lindsay, R.A. Ross and C.T. Sachrajda, Nucl. Phys. B214 (1983) 61. [13] A. Bassetto, M. Ciafaloni and G. Marchesini, Phys. Rep. i00 (1983) 201. [14] D. Amati and G. Veneziano, Phys. Lett. 83B (1979) 87.