The BFKL equation at next-to-leading level and beyond1

22 April 1999

Physics Letters B 452 Ž1999. 372–378

The BFKL equation at next-to-leading level and beyond M. Ciafaloni 2 , D. Colferai

1

3

Dipartimento di Fisica, UniÕersita` di Firenze, and INFN, Sezione di Firenze Largo E. Fermi, 2 - 50125 Firenze, Italy Received 28 December 1998 Editor: R. Gatto

Abstract On the basis of a renormalization group analysis of the kernel and of the solutions of the BFKL equation with subleading corrections, we propose and calculate a novel expansion of a properly defined effective eigenvalue function. We argue that in this formulation the collinear properties of the kernel are taken into account to all orders, and that the ensuing next-to-leading truncation provides a much more stable estimate of hard Pomeron and of resummed anomalous dimensions. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 12.38.Cy

1. Introduction The recent calculation w1,2x of the next-to-leading log s ŽNL. corrections to the kernel of the BFKL equation has solved a long-standing problem w3x, but has also raised some puzzling questions. In fact, such corrections – which decrease both the gluon anomalous dimension and the hard pomeron intercept – turn out to be so large, that they cannot be taken literally for reasonable values of a s G 0.1. The question then arises: what is the origin of such large corrections? and, what use can we make of them? We do not suggest here to perform exact higher order calculations, not only because they are awkward, but also because they mix with unitarity effects w4x and thus cannot be described within the

BFKL equation alone. Our purpose is rather to illustrate the physics of subleading corrections and to perform partial resummations on the basis of a renormalization group analysis which improves the perturbative formulation of the small-x problem. Our argument follows two steps. First, we propose an ansatz for the BFKL solutions which is automatically consistent with the R.G. for a general dependence of the kernel on the running coupling. Secondly, we identify the subleading corrections to be resummed, mainly the scale-dependent ones, as recently suggested w5,6x, and the collinear singular ones w7x. The improved NL expansion follows naturally.

2. Renormalization group analysis 1

Work supported in part by the E.U. QCDNET contract FMRX-CT98-0194 and by MURST ŽItaly.. 2 E-mail: [email protected] 3 E-mail: [email protected]

In small-x physics Ž s 4 Q 2 4 L2 ., Regge theory and the renormalization group come to a sort of

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 2 8 1 - 6

M. Ciafaloni, D. Colferair Physics Letters B 452 (1999) 372–378

clash, which provides nontrivial consistency requirements. It is known w8,9x that the BFKL equation satisfies R.G. factorization in an asymptotic way. Introducing, by k-factorization w10x, the NL gluon Green’s function Gv Ž t ,t 0 . s v y a s Ž t . Ž K 0 q a s Ž m 2 . K 1 .

ž

as s

Nc a s

p

/

y1

,

,

Ž 2.1 .

its asymptotic form for t ' log k 2rL 2 4 t 0 is given by Gv Ž t ,t 0 . , C Ž v , a s Ž t . . exp

t q

Ht g

v

Ž a s Ž t . . dt

0

= C0 Ž v , a s Ž t 0 . . ,

Ž 2.2 .

where gvq is the larger eigenvalue of the singlet anomalous dimension matrix, defined by the equation

v s a s Ž t . Ž x 0 Ž gvq . q a s Ž m 2 . x 1 Ž gvq . . .

Ž 2.3 .

Solutions to this equation are found so long as v G v P Ž a s ., the ‘‘hard pomeron’’ intercept, which in this treatment is estimated from Eq. Ž2.3. and from the NL calculations w2x to be

v P Ž a s . s a s Ž x 0 Ž 12 . q a s x 1 Ž 12 . . , 2.77a s Ž 1 y 6.47a s . ,

Ž 2.4 .

leading to the anomalously large NL corrections mentioned before. While the t-dependent coefficient C Ž v , a s Ž t .. in Eq. Ž2.2. is perturbatively calculable, the t 0-dependent one C0 is not, because it contains the leading v-singularity, that we call simply the ‘‘pomeron’’. The latter turns out to be dependent on how a s Ž t . is smoothed out or cut-off around t s 0 Ž k 2 s L 2 ., in order to avoid the Landau pole. However if t 0 4 1, it is suppressed by a power of L2rk 02 . Perturbative calculations are thus hampered by two v-singularities. The first one, the hard pomeron v P Ž a s Ž t .. is a singularity of the anomalous dimension expansion, not necessarily of the full amplitude, and is our main concern here. It dominates an inter-

373

mediate-x regime, characterized by quite large anomalous dimensions. The second singularity – the pomeron v P – is the leading v-singularity of the amplitude and dominates the very small-x regime, but is non-perturbative and is not directly related to scaling violations. The treatment of the hard pomeron just summarized has, however, the limitation that a s Ž m 2 . in Eq. Ž2.1. is kept frozen in front of K 1 , while the overall factor a s Ž t . is allowed to run. Even if consistent at NL level, this approach should be improved in order to perform partial resummations to all orders in a s Ž t ., as we envisage here. How can this be achieved? Our main proposal is to use v as the expansion parameter of the BFKL solutions, instead of a s Ž t .. More precisely, we look for the Žregular. solution w9x of the homogeneous BFKL equation

v Fv Ž t . s w K v Fv x Ž t . ,

Ž 2.5 .

where the kernel K v has a general a s Ž t .-dependence of the form Žsee Section 3. 2

K v Ž k , kX . s a s Ž t . K 0v Ž k , kX . q a s Ž t . K 1v Ž k , kX . q PPP , t ' log

k2

L2

,

Ž 2.6 .

and the K nv :n s 0,1, PPP are scale invariant kernels which may be v-dependent. We then assume the ansatz

Fv Ž t . s

1

1 k

dg

qi`

2

H12 yi`

2p i

1

e g ty bv X Ž g , v . ,

2

b'

p 11 Nc y 2 Nf Nc

12p

,

Ž 2.7 .

where X Žg , v . is to be found by solving Eq. Ž2.5. , as a power series in v . The relevance of such solution is that it is shown to represent w11x the asymptotic form of the Green’s function Ž2.1. for a general form of the kernel K, as follows: Gv Ž t ,t 0 . , Fv Ž t . F˜v Ž t 0 . ,

Ž t 4 t0 R 1.

Ž 2.8 .

M. Ciafaloni, D. Colferair Physics Letters B 452 (1999) 372–378

374

where, for a given regularization of the Landau pole, Fv Ž t . has the form Ž2.7., while F˜v Ž t 0 . contains the Žregularization-dependent. pomeron singularity. While the detailed properties of X Žg , v . are dependent on how a s Ž t . in Eq. Ž2.6. is smoothed out or cut-off around t s 0, the large-t behaviour of Eq. Ž2.7. is instead universal, provided a saddle point gv Ž t . of the exponent Ev Žg ,t . of Eq. Ž2.7. exists and is stable. We thus assume, in the asymptotic regime bt R v1 4 1, the expansion Ev Ž g ,t . s Ev Ž gv ,t . y 1 y

6 bv

1

x X Ž gv , v . Ž g y gv .

2 bv

3

Ž 2.9 . with the saddle point conditions bv t ' x Ž gv Ž t . , v . s X X Ž gv Ž t . , v . ,

E

X

Ž a. '

Eg

Ž a. ,

Ž 2.10 .

which need to be checked ‘‘a posteriori’’. The crucial property of the representation Ž2.7. around the saddle point of Eq. Ž2.9. and Eq. Ž2.10. is that it takes a form consistent with the R.G., namely k 2Fv Ž t . ;

1 X

(2p Žyx Ž g Ž t . , v . . v

= exp  Ev Ž gv Ž t . ,t . 4 = Ž 1 q O Ž v . . , Ž 2.11 . where, for any v-dependence, Ev Ž gv Ž t . ,t . s gv Ž t . t y s

t

1 bv

Žg . qv

qv

qb

ž

2

x 1v Ž g . x 0v Ž g .

1

x 0v Ž g . x 1v Ž g . x 0v Ž g .

ž

x 2v Ž g . x 0v Ž g .

y

ž

x 1v Ž g . x 0v Ž g .

2

/

X

//

qO Ž v3. ,

Ž 2.13 .

which has been pushed up to NNL level in v . Some of the terms of the v-expansion Ž2.13. follow from the trivial replacement 1rt s bvrx 0v Žg . q PPP , X while the term b Ž x 1v Ž g . rx 0v Ž g . . needs a careful treatment of fluctuations. This result is confirmed, and extended to higher orders, by the replacement t ™ Eg in the BFKL equation, which leads to a nonlinear differential equation for x w11x. The result Ž2.13. is the basic formula that we present here, in order to resum some large subleading corrections, and to obtain a smoother NL truncation for the remaining ones. Basically, the coefficients of the v-expansion Ž2.13. turn out to be less singular than the ones in the a s-expansion in the collinear limits g ™ 0 Žg ™ 1., related to t 4 t 0 Ž t 0 4 t .. This is due to the powers of x 0 in the denominators which damp the collinear behaviour and to cancellations to be explained below. This feature will yield a smoother expansion in the region g , 1r2 also, which is the relevant one for the hard pomeron estimate.

3. Improved next-to-leading expansion X Ž gv Ž t . , v .

H g Ž t . dt q const. , v

x Žg ,v .

s x 0v

2

x XX Ž gv , v . Ž g y gv . q PPP ,

x X Ž gv Ž t . , v . - 0 ,

K 0v , K 1v , PPP up to the relevant order in g y g , we find the solution

Ž 2.12 .

so that Eq. Ž2.8. takes the form anticipated in Eq. Ž2.2. for the gluon Green’s function. Furthermore, the form of x Žg , v . can be found, as an expansion in v , from the original Eq. Ž2.5. , expanded around the saddle point. For instance, if we let the coupling run up to NL level, by expanding the eigenvalue functions x 0v Žg ., x 1v Žg ., PPP of

Before proceeding further, we need to understand the physics of subleading corrections, in order to single out the ones that need to be resummed. We limit ourselves, for simplicity, to the gluonic contributions, because the qq ones turn out to be small w1x. The explicit eigenvalues of the gluonic part of the kernels K 0 and K 1 are w2x

x 0 Ž g . s 2 c Ž 1. y c Ž g . y c Ž 1 y g . 1 , g™0

g

y 2 c Ž 1 . g 2 q PPP

Ž 3.1 .

M. Ciafaloni, D. Colferair Physics Letters B 452 (1999) 372–378

and

ž

x 1Ž g . s y

11 24

Ž x 02 Ž g . q x 0X Ž g . . 1

x 0XX Ž g .

q y 4 1 q 4

y 9

= 11 q 3 q 2

q

x0 Ž g .

cospg 3 Ž 1 y 2g .

Ž 1 q 2 g . Ž 3 y 2g .

/5

p3 z Ž 3. q

n Ý Ž y.

ns0

/

/

g Ž1yg .

`

F Žg . '

3 2

sinpg

ž

p2

67

p

ž

y

½ž

/

4sinpg

yF Žg . ,

Ž 3.2 .

c Ž n q 1 q g . y c Ž 1.

Ž nqg .

c Ž n q 2 y g . y c Ž 1.

Ž nq1yg .

2

2

.

Here we quote the form of x 1 referring to the choice s0 s kk 0 of the energy scale in the NL k-factorization formula w10x: ds

AB

dv s

H 2p i

and kX . Their explicit form can be shown w6x to shift the effective scale to the symmetrical one q 2 s Ž k y kX . 2 , which is the one occurring in the phase space n R q 2 , where n is the longitudinal part of the two-gluon subenergy in the s-channel. In g-space, the running coupling terms show a double pole at g s 1 only. The collinear and scale-dependent terms have multiple poles at both g s 0 and g s 1 due to the behaviour of the kernel for k 4 kX and kX 4 k. All these poles are actually of collinear origin, and the cubic ones are dependent on the choice of the scale of the energy in Eq. Ž3.3. , whether it is s0 s kk 0 Žthe symmetrical choice., or instead s1 s k 2 Ž s2 s k 02 ., which is the choice leading to the correct scaling variable k 2rs Ž k 02rs . for k 4 k 0 Ž k 0 4 k .. It has been already remarked w12x that the choice s0 s kk 0 is not collinear safe and leads to the nasty singularities of type 1rg 3 Ž1rŽ1 y g . 3 . of x 1 in Eq. Ž3.2. , which correspond to the double logarithmic behaviour ; log 2 krkX of the kernel when k 4 kX Ž kX 4 k .. On the other hand, complete information about the collinear behaviour of the kernel in such regions comes from the renormalization group equations. In general the kernel K v Ž t,tX . in Eq. Ž2.6. is collinear finite and symmetrical in its arguments for s0 s kk 0 , so that it must have the form K v Ž a s Ž m2 . , m2 , k , kX . s

d2 k d2 k 0 h A Ž k . h B Ž k 0 . Gv Ž k , k 0 .

s

ž / kk 0

v

,

375

s

asŽ t . k2 a s Ž tX . kX 2

Kˆv Ž a s Ž t . ;t ,tX . Kˆv Ž a s Ž tX . ;tX ,t . ,

Ž 3.3 .

Ž 3.4 .

where k and k 0 are the transverse momenta of the outgoing jets in the fragmentation regions of the incoming partons A and B respectively. In the expansion Ž3.2. we have singled out some contributions which have a natural physical interpretation, namely the running coupling terms Žin round brackets., the scale-dependent terms Žin square brackets. and the collinear terms Žin curly brackets.. The running coupling terms are proportional to the one-loop beta function coefficient b and refer to the choice of Eq. Ž2.6. of factorizing the running coupling a s Ž t . at the upper scale k 2 of the kernel K v Ž k, kX ., instead of a symmetrical combination of k

where Kˆ can be expanded in a s Ž t . with scale-invariant coefficients, as assumed in Eq. Ž2.6. . Furthermore, calling K vŽ1. Ž K vŽ2. . the kernel at energy scale k 2 Ž k 02 ., it acquires for kXrk ™ 0 Ž krkX ™ 0. the collinear singularities due to the non singular part of the gluon anomalous dimension in the Q 0-scheme w13x which, neglecting the qq part, is as g˜ Ž v . s gg g Ž v . y v s a s A1 Ž v . q a s2 A 2 Ž v . q PPP , Ž 3.5 . 11 A1 Ž v . s y q O Ž v . , A2 Ž v . s 0 q O Ž v . , 12

M. Ciafaloni, D. Colferair Physics Letters B 452 (1999) 372–378

376

the singular part being taken into account by the BFKL iteration itself. It follows that, for k 4 kX , K vŽ1. Ž a s Ž t . ;t ,tX . ,

asŽ t . k

,

2

asŽ t . k

2

t

Ht g˜ Ž v , a Ž t . . dt

exp

ž

s

X

1 y ba s Ž t . log

y

k2 k

A 1Ž v . b

X2

/

,

Ž 3.6 .

with a similar behaviour, with t and tX interchanged, for K vŽ2. in the opposite limit kX 4 k. In order to derive from Eq. Ž3.6. the g-dependent singularities of K v care should be taken of the relationships w12x X

Kv Ž k , k . s

kX

ž / k

v

K vŽ1.

X

Ž k,k . s

k

v

ž / kX

K vŽ2.

Ž k,k .

which shift the g-singularities of K vŽ1. Ž K vŽ2. . by yvr2 Žqvr2.. As a consequence, by Eq. Ž3.6. , the g-singularities of the eigenvalue functions xnv of the kernels K nv in Eq. Ž2.6. are 1 P A1 Ž A1 q b . PPP Ž A1 q Ž n y 1 . b .

ž ,

gq

1 2

nq1

v

,

/

Ž g < 1. 1 P Ž A1 y b . Ž A1 y 2 b . PPP Ž A1 y nb .

ž

1ygq

1 2

nq 1

v

,

/ Ž 3.8 .

x 0v

In particular, has symmetrical singularities and x 1v slightly asymmetrical ones, as follows: 1 1 x 0v Ž g . , q , Ž 3.9 . 1 1 gq v 1ygq v 2 2 A1 A1 y b x 1v Ž g . , q . 2 2 1 1 gq v 1ygq v 2 2 Ž 3.10 .

/ ž

y c Ž 1 y g q 12 v . ,

Ž 3.11 .

which corresponds to the kernel K 0v

X

X

Ž k,k . sK0Ž k,k .

v

k-

ž /

,

k)

k - ' min  k ,kX 4 ,

k ) ' max  k ,kX 4 . Ž 3.12 . v Here the v-dependent factor Ž k - rk ) . can be directly interpreted as an s-channel threshold factor w15x in properly defined Toller variables w16x. The NL terms in Eq. Ž2.13. are now simply identified from Eq. Ž3.2. after subtraction of the singular part of the scale-dependent terms already taken into account in Eq. Ž3.11.. This subtraction eliminates the cubic singularities in x 1 , but leaves double and single poles which need to be shifted by the procedure of Ref. w5x. This amounts to the replacement of x 1 by

x 1v Ž g . ' x 1 Ž g . q

1 2

p2 x0 Ž g .

sin2 Ž pg .

p2

Ž 1 y g < 1. .

ž

x 0v Ž g . s c Ž 1 . y c Ž g q 12 v . q c Ž 1 .

X

Ž 3.7 .

xnv Ž g . ,

the correct scale-dependent terms Ž; 1rg 3 , 1rŽ1 y g . 3 . by the v-expansion of Eq. Ž3.9.. On the other hand, Eq. Ž3.10. provides the correct collinear behaviour of K vŽ1. Ž K vŽ2. . close to g s 0 Žg s 1. at NL order, and Eq. Ž3.8. generalizes it to all orders. The simplest realization of Eq. Ž3.9. is to assume, following Ref. w5x, that x 0v in Eq. Ž2.6. and Eq. Ž2.13. is provided by the eigenvalue function of the Lund model w14x

/

The shift by "vr2 in Eqs. Ž3.8. – Ž3.10. already represents a resummation of the scale-dependent singularities, as noticed by Salam w5x, and it provides

Ž x 0v Ž g . y x 0 Ž g . . 6 q A1 Ž v . c X Ž g q 12 v . y A1 Ž 0 . c X Ž g . q

q Ž A1 Ž v . y b . c X Ž 1 y g q 12 v . y Ž A1 Ž 0 . y b . c X Ž 1 y g . ' A1 Ž v . c X Ž g q 12 v . q Ž A1 Ž v . y b . c X Ž 1 y g q 12 v .

p2

x v Ž g . q x˜ 1 Ž g . , Ž 3.13 . 6 0 where now x˜ 1Žg . has no g s 0 nor g s 1 singularities at all. What about higher subleading orders in Eq. Ž2.13.? Here the crucial observation is that, after substitution q

M. Ciafaloni, D. Colferair Physics Letters B 452 (1999) 372–378

of the behaviour Ž3.8. in the NNL term of Eq. Ž2.13., the leading g-singularities cancel out close to both g s 0 and g s 1. Therefore, the coefficient of v 2 in Eq. Ž2.13. has no g-singularities at all, and no further resummation is needed. This peculiar feature, confirmed at higher orders w11x, is due to the fact that running coupling effects are already taken into account by the representation Ž2.7. with the solution Ž2.13.. Our truncated NL effective eigenvalue function Ž. then reads

x Ž g , v . s x 0v Ž g . q v

x 1v Ž g . x 0v Ž g .

qO Ž v2 . ,

Ž 3.14 . and contains only the shifted single poles in g , while the neglected terms have no leading twist g-singularities. This expression performs the resummation of the scale-dependent singularities proposed in Ref. w5x, but it also takes into account the collinear singularities of Eq. Ž3.8. to all orders. This resummation effect is perhaps more easily seen by using Eq. Ž3.14. in order to recast Eq. Ž2.10. in the form bv t s x 0v

1y

1 bt

ž

x 1v x 0v

y1

qO Ž v .

/

,

Ž 3.15 .

which defines an effective x-function as a power series in a s . By specializing Eq. Ž3.15. to the scale k 2 Žshift g q vr2 ™ g ., it is apparent that all the small g poles Ž A1Ž v .rbt . nrg nq1 are resummed as

377

a geometric series, to provide the full one-loop anomalous dimension of Eq. Ž3.5.. The expression Ž3.14. for x Žg , v . is not fully symmetrical under the replacement g l Ž1 y g ., but keeps the asymmetrical ybvx 0X r2 x 0 contribution of Eq. Ž3.2.. The breakdown of the g l Ž1 y g . symmetry in Eq. Ž2.7. follows from that of scale invariance by running coupling effects. It is easy to recognize that the contribution ybvx 0X r2 x 0 to x Žg , v . yields a factor x 0v Ž g . in the representation Ž2.7., and this factor is required w11x by the continuum normalization of the eigenfunctions of K v provided by the Jacobian X X Žg , v . s x 0v Žg . q O Ž v .. Therefore, even if the kernel K v and the Green’s function Gv are symmetrical operators, the effective eigenvalue function x Žg , v . must have some asymmetry, due to running coupling effects. It appears from Fig. 1 that x Žg , v ., with the b-dependent asymmetry pointed out before, is a decreasing function of v because of the negative sign of NL corrections, and contains a stable minimum for all the relevant v values. This means that in the present approach the hard pomeron can be estimated up to sizeable values of a s s 0.2 % 0.3, and it turns out to be v P s 0.26 % 0.32 in this range, which is a reasonable value for the HERA data w17x at intermediate values or Q 2 . Since in the expansion of Eq. Ž3.14. the neglected terms – of order v 2 or higher – have no g-singularities, they only affect the evaluation of Eq. Ž2.10. by a roughly g-independent uncertainty which corre-

(

Fig. 1. Resummed eigenvalue function x Žg , v . for various values of v .

378

M. Ciafaloni, D. Colferair Physics Letters B 452 (1999) 372–378

sponds to a change of scale DŽ bt . s O Ž v ., or Da s s O Ž v . a s2 . This means that the error affecting the NL truncation is uniformly of NNL order for all values of g – whether O Ž a s ., O Ž a srv . or O Ž1. – and therefore it cannot change too much the v P estimate given above. A detailed analysis of the anomalous dimensions and of the hard pomeron properties in the present approach is left to a parallel investigation w11x.

w4x

w5x w6x w7x w8x w9x w10x

Acknowledgements We wish to thank Gavin Salam for interesting discussions and suggestions.

References

w11x w12x w13x w14x w15x

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w16x

w17x

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