Leading logarithmic evolution of the off-forward distributions

5 March 1998

Physics Letters B 421 Ž1998. 312–318

Leading logarithmic evolution of the off-forward distributions A.V. Belitsky

a,1,b

c , B. Geyer c , D. Muller , A. Schafer ¨ ¨

a

a

b c

Institut fur ¨ Theoretische Physik, UniÕersitat ¨ Regensburg, D-93040 Regensburg, Germany BogoliuboÕ Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia Institute for Theoretical Physics, Center of Theoretical Science, Leipzig UniÕersity, 04109 Leipzig, Germany Received 6 November 1997; revised 15 December 1997 Editor: P.V. Landshoff

Abstract We have found the analytical solution of the LO-evolution equation for off-forward distributions which arise in the processes of deeply virtual Compton scattering or exclusive production of mesons. We present the predictions for their evolution with an input distribution taken from recent bag model calculations. q 1998 Elsevier Science B.V. PACS: 11.10.Hi; 13.60.Fz Keywords: Deeply virtual Compton scattering; Evolution equation; Conformal operators; Bag model

1. Introduction Recently, Ji proposed to explore deeply virtual Compton scattering ŽDVCS. Žsee Fig. 1. to get deeper insight into the spin structure of the polarized nucleon and, namely, to determine the total quark angular momentum Žspin and orbital angular momentum. in the nucleon w1,2x. It turned out that this process is very interesting in its own right, since it allows to obtain information about some non-perturbative off-forward parton distributions ŽOFPDs., which are inaccessible in ordinary inclusive measurements such as deep inelastic scattering ŽDIS.. Planning of experiments to measure DVCS is under

1

way, but it is hampered by the lack of reliable theoretical predictions in the interesting Ž v , h , Q 2 .range. A crucial step towards this goal is to understand the Q 2-evolution of OFPD. In the present short note we, therefore, present the Q 2-evolution of the nonpolarized quark distribution function in the leading order ŽLO. approximation starting form MIT bag model predictions at a low energy scale w3x.

Alexander von Humboldt Fellow.

0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 0 0 7 - 0

Fig. 1. Generic form of the DVCS amplitude.

A.V. Belitsky et al.r Physics Letters B 421 (1998) 312–318

In the configuration space a Compton process w1,4–11x is given by the following amplitude: T mn Ž v ,h ,Q 2 . s i d 4 z e i z Q ² p 2 < T  J m Ž yzr2. J n Ž zr2 . 4 < p 1 : ,

H

Ž 1. and is dominated by singularities of the T-product of electromagnetic currents Jm on the light-cone. In momentum space this dominance corresponds to the generalized Bjorken limit: yQ 2 ™ `,

v , D 2 -fixed,

PQ ™ `,

where the variables are defined as follows Q s 12 Ž q1 q q2 ., P s Ž p 1 q p 2 ., D s p 1 y p 2 s q2 y q1. DVCS is characterized by two scaling variables v s DQ w x y PQ Q 2 and h s PQ 4 , the former is analogous to the Bjorken one in ordinary DIS while the latter gives the magnitude of the skewedness of the process in question and parametrizes different high-energy two-photon reactions. To the LO of perturbation theory only transverse helicity amplitudes contribute and we can write, e.g. for unpolarized scattering which will be the focus of our present study: T Ž lX , l . s e 2m Ž lX . Tmn e 1n Ž l . f 12 e 2n Ž lX . e 1n Ž l . dl1r2 dlX 1r2 Tmm ,

Ž 2.

and for Tmm we have 2 : LO Tmm s

1

2 i

i

Hy1dt Ý Q q Ž t ,h , D

2

.

i

=

½v

v

v

5

. Ž 3. v t q 1 y i0 Here the off-forward distribution is defined in terms of the light-cone Fourier transformation of the renormalized twist-2 light-ray operator: 2

q Ž t ,h , D , m

2

q

t y 1 q i0

dk

. sH

2p

i

e

2

k t Pq

=² p 2 < c Ž ykr2 n . gq c Ž kr2 n .

m2 <

p 1 :D q s h Pq , Ž 4.

2 Recently, all NLO coefficient functions w12,10x as well as a s-corrections to the eigenfunctions of the two-loop QQ- and QG-evolution kernels became available.

313

and we have omitted the path ordered link factor which ensures gauge invariance. The integration range of the variable t in Eq. Ž2. results from the spectral properties of the off-forward distribution amplitude that can be derived in a straightforward manner with the help of Jaffe’s approach w13x for studying the support properties of parton densities. In our conventions the variable t has no direct interpretation as longitudinal momentum fraction, but it has the advantage that its region is process independent: y1 F t F 1. The same feature is appropriate to the quark Fza Ž x . Ž0 F x F 1. and anti-quark Fz a Ž x . Žy1 F x F 0. non-forward distributions introduced by Radyushkin w5x. The z-dependent limits appear only when one attempts to combine them into one function F˜za Ž x . s u Ž x .u Ž x . Fz a Ž x . y u Ž z y x .u Ž x q 1 y z . Fza Ž z y x . which is exactly the OFPD introduced above in Eq. Ž4. in different notations. The non-diagonal distribution 3 f g Ž x 1 , x 2 . of Collins, Frankfurt and Strikman w6x is parametrized in terms of the momentum fraction of the considered partons and consequently their limits depend on the skewedness of the processes. Thus, the momentum fractions of the incoming and outgoing partons are h ty h x s 1t q q h , and x y z s 1 q h , respectively, with z 2h s 1 q h . For t ) h both momentum fractions are positive and, therefore, a quark field with longitudinal momentum xp 1 arise from the incoming hadron and enters with momentum Ž x y z . p 1 into the outgoing hadron. For yh ) t both momentum fractions are negative and the analogous picture for two antiquarks holds true. In these regions, the corresponding off-forward quark and anti-quark distributions merge in the forward case to the well-known quark and anti-quark distributions with support 0 F t F 1 and y1 F t F 0, respectively. For yh - t - h , x is positive while x y z is negative, so that both partons come from the same hadron. Here the OFPD looks like a distribution amplitude. Indeed, inserting the vacuum and one-meson state in the definition Ž4. instead of incoming and outgoing nucleon state, we recover the definition of the meson distribution amplitude depending on the momentum fraction 0 F x

3

In the Radyushkin’s conventions f g Ž x, x y z . s F˜z g Ž x .r x Ž x y z . w5x.

A.V. Belitsky et al.r Physics Letters B 421 (1998) 312–318

314

s Ž1 q t .r2 F 1, where h s 1 w14x. Thus, OFPDs are hybrids and their probability interpretation depends upon the interplay between the respective values for t and h.

For h ™ 0 the evolution Eq. Ž5. turns into the DGLAP one with a splitting function given by the limit: 1 z 1 P Ž z . s lim V , . Ž 8. h h q h™0 < h <

ž /

2. Evolution equation 3. Solution of the evolution equation The evolution of OFPDs arises technically from the renormalization procedure for the light-ray operators w15x, which satisfy in the non-singlet channel the well-known renormalization group equation w16–19x. Employing the definition Ž4., it is straightforward to derive from the latter the evolution equation for the OFPD: Q2

d dQ 2

q NS Ž t ,h , D 2 ,Q 2 .

t tX s V , 2p y1
as

1

H

dtX

ž /

The evolution equation analogous to Eq. Ž5., but for gluons, was attempted to be integrated numerically w8x. However, it remains an important issue to find an analytical solution. The way to do that is based on the use of the conformal invariance of the free field theory which ensures that conformal 2-particle operators do not mix under renormalization at LO w14,21–23,10x. Such conformal operators are given in terms of Gegenbauer polynomials Q Ojl s l c Ž i Eq . lgq C j DqrEq c and their off-forward matrix elements can be obtained from the moments: 3 2

X

q NS Ž t ,h , D 2 ,Q 2 . .

ž

/

q

3

Ž 5.

2

q j Ž h , D ,Q

2

. sh H

0F x , yF1 .

Ž 6.

Moreover, the continuation to the whole t,tX-plane is unique w4x, so that the extended evolution kernel V Ž t,tX . can be easily recovered from the corresponding ER-BL analogues. From the support restrictions and charge conjugation symmetry a representation was derived in w20x, which is valid in leading order: 0 V Ž t ,tX . s Q 11 Ž t y tX ,t y 1 . V Ž t ,tX .

q Ž t ™ yt ,tX ™ ytX . ,

Ž 7.

w h e re Q 101 Ž x 1 , x 2 . s Ž u Ž x 1 . u Ž y x 2 . y u Ž x 2 .u Žyx 1 ..rŽ x 1 y x 2 ., and V Ž t,tX . is an analytic function of t and tX . Thus, the extended kernel can be obtained from the exclusive ER-BL one by replacing the ordinary u-functions in front of V Ž t,tX . by the generalized ones, namely, u Ž t y tX .rŽ1 y tX . 0Ž ™ Q 11 t y tX ,t y 1.. For concrete examples the reader is referred to Ref. w10x. Beyond LO, further contributions appear in Eq. Ž7., but they are not of relevance for our present consideration.

dt C j2

y1

In the restricted region < t,tX < F 1 the kernel V Ž t,tX . coincides with the Efremov-Radyushkin-BrodskyLepage ŽER-BL. one w14x: VER - BL Ž x , y . sV Ž 2 x y 1,2 y y 1 .

1

j

2 s

Pqjq1

¦p

2

t

ž /Ž h

q t ,h , D 2 ,Q 2 .

< Q Oj j Ž 0 .

m 2 sQ 2

< p1 : . Ž 9.

In the following, we restrict ourselves to the flavour non-singlet channel since up to now no estimation of the off-forward gluon distribution has been performed yet as long as available strong interaction models do not contain the gluon fields at all. The generalization of our results to the singlet channel is straightforward. The only complication which is due to quark-gluon mixing can be treated in the same way as in DIS. For h s 1 it is well-known that the Gegenbauer polynomials diagonalize the ER-BLkernel. Moreover, the support of the extended evolution kernel ensures this property for arbitrary h : 3

1

Hy1

dt C j2

t

t tX , h h

ž / ž / h

V

3

s g j C j2 q

tX

ž / h

.

Ž 10 .

The remaining problem is to find the inverse Mellin transformation of the moments Ž9.. The support property 4 of q NS Ž t ,h ,Q 2 . allows to expand it with 4

In the subsequent discussion we omit, for brevity, the dependence of the distributions on the momentum transferred squared.

A.V. Belitsky et al.r Physics Letters B 421 (1998) 312–318

315

respect to an appropriate complete set of polynomials Ckn Ž t . which are orthogonal in the domain y1 F t F 1 with the weight function w Ž t < n . s Ž1 y t 2 . ny : 1 2

w Ž t < 32 .

`

q

NS

2

Ž t ,h ,Q . s Ý

Nj Ž 32 .

js0

3

Cj2 Ž t . a j Ž h ,Q 2 < 32 . ,

Ž 11 . 2

1 2 Ž .

where Nj Ž n . s 2y2 nq1 GG Ž2 n.ŽGn Ž n. j! . is a normalization factor. It is straightforward to calculate the expansion coefficients in terms of the conformal moments Ž9.: 2 qj

qj

j

a j Ž h ,Q 2 < 32 . s

Ý a jk Ž h < 32 . qkNS Ž h ,Q02 . ks0

½

=exp g k

HQQ

2

2 0

ds as Ž s .

s

2p

5

,

Ž 12 .

where g j s CF wy2 c Ž j q 2. q 2 c Ž1. q 1rŽ j q 1. y 1rŽ j q 2. q 3r2x are the well-known forward nonsinglet anomalous dimensions, and a jk Žh < 32 . can be written in a very compact manner by employing the definition of hypergeometric functions 5 :

1

wŽ t
Hy1dt h N Ž n . C k

n k

Ž t . C jn Ž h t . jyk

Ž y1. s 12 u jk 1 q Ž y1 .

½

a k Ž 1,Q 2 < 32 . s qkNS Ž 1,Q 02 . exp g k

ž ž

G nq

jyk

G Žnqk. G 1q

=2 F1



2

,n q

jqk

nqkq1

2

h

2

0

2 0

2

ds as Ž s .

s

2p

5

,

Ž 14 .

jqk 2 jyk 2

/ /

a j Ž 0,Q 2 < 32 . jyk

Ž y1.

j

s

Ý ks0, kyj even

kyj

HQQ

3 2

k

2

Let us turn to the consideration of the limiting cases. Since 2 F1Žyn,a q n;a q 1;1. s dn0 , we obtain for h s 1:

and, therefore, from Eq. Ž11. we recover the wellknown LO result for the evolution of the meson distribution amplitude w14x. In the limit h ™ 0, the solution of the DGLAP equation expanded with respect to Gegenbauer polynomials Ck , has the following coefficients:

a jk Ž h < n . s

Fig. 2. The toy input OFPD Žsolid line. given at Q 2 s 0.5 GeV 2 is evolved up to Q 2 s 5 GeV 2 , where LQC D s 220 MeV. The dashed and dotted lines show the evolution for h s 0.5 and h s 0, respectively, while for h s1 the input distribution does not evolve since it has already the asymptotic shape.

.

Ž 13 .

G

ž

2

G

3q2k

/ ž G

2

½

=qkNS Ž 0,Q02 . exp g k

ž

3qjqk

HQQ

2

2 0

2

ds as Ž s .

s

/

2 2qjyk

4p

5

/ .

Ž 15 .

In this limit conformal moments coincide up to an overall normalization with the usual ones in DIS: 5

To be general we have not specified the index n : for the quark operators n s 3r2, for gluons n s 5r2. Moreover, all equations derived in main text can be applied in a straightforward way to the polarized quark distributions.

qkNS Ž 0,Q02 . s

2 kG Ž n q k . k! G Ž n . k

1

H0 dxx

k

q NS Ž x ,0,Q02 .

q Ž y1 . q NS Ž yx ,0,Q 02 . .

Ž 16 .

A.V. Belitsky et al.r Physics Letters B 421 (1998) 312–318

316

For asymptotically large Q 2 all conformal moments with k ) 0 will be suppressed due to non-zero anomalous dimensions and, consequently, only jeven expansion coefficients survive in this limit: aasj Ž h < 32 . s lim a j Ž h ,Q < 23 . Q™`

j 2

s

G Ž 32 . G

3qj

ž / ž ž /

Ž y1. G

2 2qj

=q0 Ž h ,Q02 .

2 F1

yjr2, Ž 3 q j . r2 2 h 5r2

/

2

for j even.

Ž 17 .

It is not hard to check that they belong to the following asymptotic off-forward distribution which

was originally found in Ref. w5x: q as Ž t ,h . A

1
ž

u 1y

t2

h2

/ž

1y

t2

h2

/

.

Ž 18 .

It turns into d Ž t . when h ™ 0. From the asymptotic form found so far, we can argue that, in general, OFPDs will be enhanced in the region 0 F < t < - h and suppressed for h - < t < F 1 when evolved upwards in Q 2 . To demonstrate these effects in a clear manner, we choose a h-independent toy-model input distribution 3r4Ž1 y t 2 . Žwhich is in fact unphysical for arbitrary h .. For h s 1 this distribution possesses already the asymptotic form, while for smaller h ’s the evolution provides the mentioned behaviour as illustrated in Fig. 2. The evolution has been done with the help of Eqs. Ž11. – Ž13., while the conformal

Fig. 3. The evolution of the bag model motivated OFPD Ž20. for LQC D s 220 MeV, where the scales are Q02 s m2bag Ža., Q 2 s 2 GeV 2 Žb., Q 2 s 200 GeV 2 Žc., and asymptotically large Q 2 Žd..

A.V. Belitsky et al.r Physics Letters B 421 (1998) 312–318

moments of our toy distribution at the input scale Q 2 s 0.5 GeV 2 was computed from Eq. Ž9.. Taking the first 80 terms in Eq. Ž11. the OFPD was evolved to Q 2 s 5 GeV 2 for h s  5r10,04 and t s 0," 1r20," 2r20, . . . ,1. To avoid numerical problems arising from the oscillation of the Gegenbauer polynomials the calculation were done exactly; and, finally, the result was interpolated to a smooth function.

317

ŽQQ mn is a quark part of the energy-momentum tensor., we neglect the h-dependence altogether, which is not far from the bag model results. Since the main goal of the present study is to acquire some intuition about the evolution properties of OFPD, we do not pursue the aim for construction of more realistic models; rather motivated by the given results, we take a very simple parametrization of the OFPD in the non-singlet channel, namely: 3

q NS Ž t ,h ,Q 02 s m2bag . s 60 u Ž t . t 2 Ž 1 y t . ,

Ž 20 .

2

4. Q -evolution of the bag model motivated OFPD A first non-perturbative estimation of the off-forward valence quark distribution function has been done using the MIT bag model 6 w3x. In this model calculation it turned out that at a scale m 2bag , 0.2 GeV 2 the h-dependence of the off-forward u and d quark distributions is extremely weak 7 and these functions vanish for negative t as well as for t ™ 1. ŽThe nonzero result of the calculated t-dependence in the vicinity of unity is a model artifact, which reflects the fact that the incoming and outgoing protons are not good momentum eigenstates.. For positive t the distributions are positive with a maximum at t f 0.4. To be able to satisfy the sum rules which are obeyed by the off-forward distributions, namely, the first two moments should be independent of the skewedness parameter h : Pq Jq Ž 0 . 2 1 2 ² < < p1 : , dt q Ž t ,h ,Q . s 2 p 2 Q qq t Pq Q Ž 0. y1 1

H

½5

½

5

Ž 19 .

6 Quite recently there appeared a paper w24x with evaluation of the OFPDs in the instanton inspired chiral quark-soliton model of the nucleon. However, it is not possible to use their results here as only flavour singlet quark density for particular values of the skewedness was considered in the paper. 7 The singet electric-type OFPD derived in Ref. w24x exhibits rather strong h-dependence. However, if only the contribution from the discrete Dirac levels is taken into account in their model, that roughly corresponds to the valence quark distribution, it gives similar smooth function in Ž x,h .-space as discussed in the main text. Thus, the same qualitative behaviour due to evolution is expected for the chiral quark-soliton model result as will be obtained below for the bag-model ansatz Ž20..

with a first moment normalized to unity. In Fig. 3, we evolve this input as described previously Žwe took also discrete values for h s 0,1r10, . . . ,1 and interpolated the result with respect to t and h . up to the scales Q 2 s 2 GeV 2 , Q 2 s 200 GeV 2 , and asymptotically large Q 2 ; for other parameters, we set Nf s 3, LQCD s 220 MeV in a s Ž Q 2 .. As expected from our previous discussion, the distribution spreads in t for larger value of h and shrinks for smaller ones. These graphs clearly show that during the evolution the large-h part spreads over the whole range of the momentum fraction while the small-h distribution is pushed towards t ™ 0 and concentrates in the vicinity of zero. Since we have omitted in the present analysis the gluon sector, we can expect that taking it into account we will get, probably, an even more enhanced function in the small-t Žand h ™ 0. region for moderate values of Q 2 . As is easily seen from the above figures with growing scale the OFPD approaches the asymptotic form. Namely, for h ™ 1 it takes the Ž1 y t 2 .shape of the asymptotic distribution amplitude, while for small h it becomes d Ž t .. For intermediate h it smoothly interpolates between these limits. This is a quite general feature that should be obeyed by any reasonable Žread physical. off-forward parton density.

Acknowledgements We would like to thank A.V. Radyushkin for careful reading of the manuscript and useful comments and O.V. Teryaev for discussions. A.B. was supported by the Russian Foundation for Fundamental Research, grant N 96-02-17631, Deutsche

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A.V. Belitsky et al.r Physics Letters B 421 (1998) 312–318

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