Quark contribution to the gluon-gluon-reggeon vertex in QCD

Physics Letters B 294 (1992) 286-292 North-Holland

PHYSICS LETTERS B

Quark contribution to the gluon-gluon-reggeon vertex in QCD V. F a d i n i a n d R. F i o r e 2 Dipartimento di Fisica, Universit?t della Calabria, and Istituto Nazionale di Fisica Nucleate, Gruppo collegato di Cosenza, Arcavacata di Rende, 1-87036 Cosenza, Italy

Received 17 August 1992

The quark loop contribution to the gluon-gluon-reggeonvertex is calculated in QCD, where the reggeonis the reggeizedgluon. This contribution exhibits helicity non-conservation as well as a gluon loop contribution. We find an exact cancellationbetween these two contributions in the case of three masslessquark flavours.

1. I n t r o d u c t i o n

It is known [ 1 ] that in the leading logarithmic approximation (LLA) gauge particles in non-abelian gauge theories are reggeized with the trajectory j(t)=

(1)

l +to(t) ,

where for the gauge group SU (N) to(t)=

= d2kL

g2t

(2n)3 N f ( k 2 _ m Z ) [ ( q _ k ) 2 _ m a ] .

(2)

Here the integration is performed over the two-dimensional m o m e n t u m orthogonal to the initial particle mom e n t u m plane, g is the coupling constant of the gauge theory, q is the m o m e n t u m transfer, t = q2 and m is the mass that the gauge particle acquires in the case of a spontaneously broken gauge symmetry. For Q C D one has to put N = 3 a n d m = 0. Let us consider the multi-gluon production in the multi-Regge kinematics: s = (PA +PB)2 >> sl, s2, ..., s,+l >> Iq 21 ~ Iq 2 1. . . . . P,=q,+l-q,,

Po=--PA ' ,

P,+I=---PB ' ,

I q2+l I , Si =

(Pi-1 +pi)2,

SI"S2"...'S,+I=s'p21±'P~±'...'P2±,

(3)

where Pa, PB and PA,, Pn, are the m o m e n t a of the colliding and scattered particles respectively, Pl, ..., Pn are the m o m e n t a of the produced gluons a n d Pl ± ..... p,± are their transverse components (P].t = _p2± ). Due to the gluon reggeization in the LLA, the amplitude of the multi-gluon production in the above kinematics has a simple multi-Regge form [ 1 ]:

~A'BVI...VnB'

__o/"~il

--o~ ~ ,

StO(tl ) °1 ~IVI [~ tl aVili2[tll,

s ~ ( t2)

q2)

__ t2

~co( tn+ l )

y~23(q2' q3)... °n+_........L~ 1 in+t tn+l Fsn, ,

(4)

* Work supported in part by the Ministero dell'Universit/t e della Ricerca Scientificae Tecnologica. 1 Address after 9 July 1992: Budker Institute for Nuclear Physics, Novosibirsk, Russian Federation. 2 E-mailaddress: 40330::FIORE (Decnet), [email protected] (Bitnet). 286

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where 7V(q, q, ) and F ~ , are the reggeon-reggeon-gluon (RRG) and the particle-particle-reggeon (PPR) vertices respectively. The former can be written as yV(q, q')

-~v-*aTijge~l,a • ~u(q, q') •

(5)

Here (Ta);j= - i f ~ o represent the colour group generators in the adjoint representation withf~bc the group structure constants, ~, and eft are the colour wave functions and the polarization vectors of the produced gluons respectively. In the lowest order of the perturbation theory the vector ~u(q, q' ) is given by

~u(q,q')=q+q'+pA (q2pvpA --------+2 PvPn) --Pn ( q , Z + 2 PvPA), PAPB} \PvPB PaPB}

(6)

with Pv= q' --q. In this order PPR vertices conserve the helicity of each of the colliding particles. They take the form

F~, = - v / 2 g(A' l Ti[A)~aaz,, ,

(7)

where 2A is the helicity of the particle A and (A' I T"IA) represents the matrix element of the group generator in the corresponding representation (i.e. adj pint for gluons and fundamental, T~= t~= ½2~, for quarks). Here it is assumed that the polarization vectors of the scattered particles are obtained from those of the initial particles by rotation around the axis orthogonal to the scattering plane. The imaginary part of the elastic scattering amplitude in the LLA is obtained by using the unitarity condition in terms of products of the inelastic amplitudes (4). For colourless object scattering the infrared singularities in the gluon trajectory ( 1 ) are cancelled with those in the real gluon emission. The integral equation for t-channel partial waves with vacuum quantum numbers [ 1 ] is free of singularities. Its kernel is determined by the gluon Regge trajectory ( I ) and RRG vertices (5). The calculation of the radiative corrections to this kernel was started by Lipatov and one of the authors (V.F.) in ref. [2], where the calculation program was presented. As a necessary step in this program one needs to calculate one-loop corrections to the PPR vertices. The calculation of the gluon loop contribution to the gluongluon-reggeon vertex was performed by Lipatov and one of the authors (V.F.) [ 3 ]. One of the interesting results of this calculation is the helicity non-conservation for each of the two colliding particles. As it was pointed out in ref. [ 3 ], it would also be interesting to calculate the contribution coming from the quark loop in order to check the possibility of cancelling this effect. In this paper we are presenting the results of the calculation of the quark loop contribution and discussing the cancellation.

2. Contribution of the quark loop to the gluon-gluon scattering amplitude The simplest way to find the quark loop contribution to the gluon-gluon-reggeon vertex is to calculate the ~4B contribution of the quark loop to the gluon-gluon scattering amplitude d~j~ '. As well as in ref. [ 3 ], we use the dispersive method based on analyticity and t-channel unitarity. Here we are interested in the quark-antiquark intermediate state in the t-channel, therefore we need the quark-gluon scattering amplitudes ~¢]jv' and alger,. Since our reggeon has the quantum numbers of the gluon, we consider only the octet contribution in the tchannel. This contribution has the following form:

~ f f ' = ( A , lTtlA)(F, lttlF)( -~g 1 2 )cA, *or" eAU(PF,) ot(

,,A

-

-

, , A 11

4

)

X ya',F __ mfya,--ya,.r,+__,_mfTa+7(gaa,~A+qa, Ta--qaya') U(PF), where

(8)

m:is the J:flavour quark mass and 287

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q=PA'--PA =PF--PF, t=q 2.

12 November 1992 (9)

By evident substitution in eqs. (8), (9) we get the corresponding expression of d~FF,. We consider on mass-shell amplitudes, therefore, we are free to choose a gauge for each of the gluon polarization vectors separately. It is convenient to use the light-cone gauge with gauge fixing vectorspn (Pn,) for eA (cA,) and vice versa. That means

eapA=eAPB=O, eBp~=eBpA=O,

eA'PA'=eA'PB'=O, en, PB,=en, pA,=O.

(10)

We shall use polarization states of definite helicity. Let us define the polarization vector for particle A with helicity 2 by the relation i2 1 { u e~(PA)=--7~ _~l~q±--

2i2 ¢u~,aaqupAppBa), S

(11)

where ~0123 ~--"1

and

_qU - p~ ~qPB q~_-p~--.

qPa

(12)

pA PB

Polarization vectors with the same helicity ;t for particles A', B and B' can be obtained from eqs. ( 11 ), (12) with the following substitutions respectively:

e~(PA'): A--.A',B~B' ; e~(pn):

A~--,B;

e~(pn,): A ~ B ' , B ~ A ' . Polarization vectors so defined satisfy conditions (10), and polarization vectors of the scattered particles (A', B' ) are obtained from those ones of the initial particles (A, B) by rotation around the axis transverse to the scattering plane. In the Regge region the following relations:

e~(PA)e~'~(PA')gu~=--SZ~' , e~(Pa)e~r(PA,) (guv--2 qutqv) =--Sa.-~,,

(13,14)

for particles A, A', and analogous relations for particles B, B' hold. In order to calculate the contribution of the quark-ant±quark intermediate state in the t-channel to the giuongluon scattering amplitude ~¢jjn,, we need to calculate the t-channel imaginary part of this amplitude, in order to restore the whole amplitude and to perform analytical continuation in the Regge region. The result we obtain can be written as follows:

dOpednpF ' t~(n)(pe--q--pv, ) MA,F, .cIB, F m~+ie) ~'AF ~'nV''

~ y ' ( q u a r k - l o o p ) = i v,e,Z (2n)O(p2_m~+ie)(p2

(15)

The sum here stands for polarization, colour and flavour states; D is the space-time dimension which is not equal to four to regularize the ultraviolet divergences. In order to have a dimensionless coupling constant, we have to substitute in the above formulas gZ by g21~4-°. In the expression (2) for the gluon trajectory we also substitute dZk±/ (2 r~)3 by d n - 2k ± / (2•) n - ~. Let us note that one can put into correspondence with the right-hand side ofeq. ( 15 ) a definite set of Feynman diagrams. This leads to a better understanding of the connection between the usual calculations in terms of 288

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Feynman diagrams and calculations that make use of the t-channel unitarity condition. We prefer following this last method of calculation. The reason is that it gives us at once a set of only Feynman diagrams which may be relevant to our problem. Then we obtain considerable simplifications by decomposing the scattering amplitudes into the sum of two terms: ~¢AF~"= ~¢j~e, (as) + dAj~F"( n a ) .

( 16 )

The first term on the RHS of eq. (16) contains the asymptotic contribution for the Regge kinematics

PAPF'~PAPF >>PAPA': SC~FF(as)= (A' IT~IA ) ( F' I:IF) (--2g2lz4-°) ( ~ )

~(pF, ),AU(pF) .

(17)

The non-asymptotic part ~¢~j:' (na) contains the remaining terms of the amplitude (8). We decompose the amplitude ~¢s~v in the same way. After substituting in eq. ( 15 ), we neglect the product of the two non-asymptotic parts, because they cannot yield s order terms. Indeed, to get s order contributions, we need to have

SA'SB~S,

(18)

with

SA=(pA+pF) 2, SB=(pn+pF,) z.

(19)

Hence we are left with three contributions. The first one comes from the product of the asymptotic parts of the amplitudes in eq. (15 ) and turns out to be

sl~'nn,(as.as)=(A, ITClA)(B, ITelB > 2g

(e?~,ea)(e~,en)p~p~ ~ ( q ) ,

(20)

where ~)(q)=½

i fJ

d~'P trD,~(tk+m:)~,"(lk-(t+m:)] (27t) ° (pZ-m~ +i~) [ (p-q)2-m~+iE] "

(21)

The calculation of the integral in eq. (21 ) is quite easy and yields the well-known result 1

4 F ( 2 - ½O) ! d x x ( 1- x ) ~ ~)(q)-(47~)D/2 (--guvq2+ququ) [m~_q2x(l_x)_i¢]2-o/2"

(22)

The second contribution comes from the product of the asymptotic part of the amplitude ~tAFF' and the nonasymptotic part of the amplitude ~¢nnF F. Not all the terms in this product are of the desired order s. Taking into account only those terms which can lead to such a result, we obtain (g2~4--D) 2

~Asn'(as'na)= (A'lTClA) (B'ITClB)

(e~,eA)e~'egp.~ ~. ~u##' (f) (Pn, q ) , t

(23)

f

where

~r'u~,(pB, q)=i ~ d°p

tr [ (~+ mf)yp( -~n+l~+ms)Yp,(l~-O+mf)yu] (2n) ° (p2--m} +iE) [ (p--q)2--m~ +iE] [ (pB--p)2--m} +i¢] •

(24)

The s order contribution in eq. (23) comes only from those terms of tensor Us##, if) (Ps, q) which containp~ (we stress that the tensor ~u##' tr) is expressed through Pn and q). Then, since

eBpB=eB' PB' =eB'(PB--q)=O , 289

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we can disregard the terms which contain (ps)a and replace (PB)p, by qp,. As a consequence of that we are left with only the terms having in their structure (PB)/,gaa' and (PB)~,qpqp,.The result we arrive at is 1

1

4(PB)# ! ! dxI dx20(1 -Xl -x2) /~##' tPB, q) = (4n)o/z (m} -xlx2q z - i e ) z-°/2

f(f)-

( 41-'(3-½D)xlxz( 1 - x l - x 2 ) . ) × - g # a . ( 3 - D ) F ( 2 - J D ) ( 2 - x l -x2)-qaqp' m}_xlx2q2_i~ •

(25)

The third contribution, coming from the product ~j)r, (na)d~j~e, (as), of course, can be obtained by the evident substitution A*-,B. Summing up the three contributions we get 2

4--D

~j~s, (quark_loop) = (A, IT¢IA >(B, IT¢IB ) (2g2/t4_D) _s,,.,a, ~a~.j,~8 2g~/z4-L ~ .~ ~c),#a,' t~a ~A~a, ~n (4rr)D/2 :

(26)

where the quantity -~~/d,p#,, that takes count of the fermion loop, is given by

~2,##, =gaa,gpp,F(2-½D) (

X -2

1

fldx,dx,(3-D)O(l-x,-x2)(2-x,-x.).) 1 1

dxx(l-x)

!

[mJ-x(1-x)q2-ie]2-DI2-- 0 0

(mJ-xlx2q2-ie)2-°12

1 l

- 2 (g.~, qpqa' +gpp'q . q . ) r ( 3 - ½D) -J "_jdx~ dx20( 1 - x l -x2)xlx2(1 -Xl -x2) ( m} - xl x2 qZ- i¢ ) 3-D/2

(27)

0 0

One integration in the two-dimensional integrals of the RHS of eq. (27) can be performed without difficulties. However, the resulting expression for ~c),pp, becomes more complicated, therefore we do not write it for the general case. It takes a considerably simplified form when rn1= 0: 1 .t@~']J#' Itll,l f = O - -

F Z ( J D - 1)

F(D)

(__qZ)Z--O/2

×[½(DZ-2D-4)F(2-½D)g.a, ga.,+2F(3-½D)~g.a,-~-+g.#,~)]. [

qaq#'

(28)

3. Quark contribution to the gluon-gluon-reggeon vertex

The usual Regge expression for the elastic scattering amplitude with the gluon quantum numbers in the tchannel takes the following form: S I"/

,,co(t)

~'Jj~"=F~, ~ L ( - ~ t )

/

.x c o ( t ) - i

+ (-~t)

J/"Jl.,,

(29)

where w(t), in the one-loop order, is given by eq. (2) and the PPR-vertices, in the lowest approximation, are presented in eq. (7). Comparing eqs. (26) and (29), and using expression (7) and relations (13), (14), we obtain, in the helicity basis, 290

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]

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I

I

(4-D) ( V+(f) _). r~,(quark-loop)=(-x/~glt2-D/2(A'lTilA)) 292/1 (4/t)D/2 ~,~,!,4,~ +~a,4-aA,Z v(f) f

(30)

f

In this equation, where the factor between small parentheses accounts for the Born approximation, the term describing the helicity conserving part is given by

[! 1

V+~) = - r ( 2 -

½D)

x,l_x)

[ m } - t x ( 1 - x ) - i e ] 2-D/2

I 1 .......

x,x.l_x,_x.,

(m~_tXlX2_iE)2_D/2k~,O--l-))(Z--Xl--X2)+(2--½D)

(--~f__t--~lX;~l~))] ,

(31)

while the term describing the helicity non-conservation is given by 1 1

vg) =F(3-ID)t ~ f dx~ dx2 0(1--X 1--X2)XlX2(1--X 1 - - X 2 ) 0 0

( m} - t x i x 2 - - i ~ ) 3 - D / 2

(32)

For massless quarks eqs. (31 ) and (32) reduce to

v+~) I,.:=o

=

1 2(½D) ' (_t)2_o/2 r ( 2 - ½D) / "F(D)

V~-) [mf=O- ( _-~:-o/2 -~ F( 3 - 1]l.J) " " F2(½D1)

(33,34)

4. D i s c u s s i o n and conclusions

It was shown in ref. [ 3 ] that the gluon loop contribution to the gluon-gluon-reggeon vertex in the helicity basis has the form F~t~,(gluon-loop) = ( - x / ~ g # 2 - D / 2 < A ' I T i I A ) )

X

[(

(D-3)~u(3-½D)-Zq/(½D-2)+¥(1)

Ng2ltt4-D) F( 2 - ½D) F2 ( ½D- 1) (4n)D/2(-t) 2-°/2 F ( D - 2 )

4,01,1

~.a.,

°_4

2(D_I)~A,_~A.

],

(35)

where ~u(z) is the logarithmic derivative of the gamma function:

r'(z)

~,(z) = r(z---(" It can be seen from eqs. (30) and (35) that the total contribution to the helicity non-conserving part of the gluon-gluon-reggeon vertex is proportional to 2 ~ /'2(½D- 1) y.: Vv_)+N2 ( - itO) -2-n/2F(3-'D) F(D)

(36)

Taking into account eq. (34), we conclude that the helicity non-conserving contributions cancel in the theory with massless quarks, if the number of quark flavours is given by

nf=N.½(D-2).

(37)

For QCD in the four-dimensional space-time this number equals 3. For the actual case of three light and three heavy flavours it could mean that there is an approximate conser291

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vation o f the helicity, for each o f the two colliding gluons, in the regime o f large energy a n d small, c o m p a r e d to the mass o f the c-quark, m o m e n t u m transfer. This conclusion seems rather intriguing a n d could have interesting phenomenological consequences.

Acknowledgement One o f us (V.F.) thanks the Physics D e p a r t m e n t o f the University o f Calabria for its w a r m hospitality.

References [ 1] V.S. Fadin, E.A. Kuraev and L.N. Lipatov, Phys. Lett. B 60 ( 1975 ) 50; E.A. Kuraev, L.N. Lipatov and V.S. Fadin, Soy. Phys. JETP 44 (1976) 443; 45 (1977 ) 199. [2 ] L.N. Lipatov and V.S. Fadin, Zh. Eksp. Teor. Fiz. Pis'ma 46 (1989) 311; Yad. Fiz. 50 (1989) I 141. [ 3 ] V.S. Fadin and L.N. Lipatov, High order corrections to QCD scattering amplitudes in a multi-Regge kinematics, in: Proc. Zeuthen Workshop on Elementary particles theory: deep inelastic scattering (Teupitz, April 1992), to appear.

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