- Email: [email protected]
Pomeron intercept and BFKL gluon dynamics in the pT(D∗) spectra at HERA
8 July 1999
Physics Letters B 458 Ž1999. 389–392
Pomeron intercept and BFKL gluon dynamics in the p T žD ) / spectra at HERA S.P. Baranov b
a,1
, N.P. Zotov
b,2
a P.N.LebedeÕ Institute of Physics, Leninsky prosp. 53, 117924 Moscow, Russia D.V.Skobeltsyn Institute of Nuclear Physics, M.V.LomonosoÕ Moscow State UniÕersity, 119899 Moscow, Russia
Received 3 February 1999; received in revised form 22 March 1999 Editor: P.V. Landshoff
Abstract In the framework of semihard QCD approach, we consider the differential cross sections of inclusive D ) " meson production at HERA. We find that the slopes of the calculated pT spectra are sensitive to the Pomeron intercept parameter D. We present a comparison of the theoretical results with available ZEUS data and derive an estimation for the value of D. q 1999 Elsevier Science B.V. All rights reserved. Keywords: QCD; Semihard approach; BFKL equation; Charm electroproduction; Pomeron
1. Introduction Recently H1 and ZEUS collaborations have reported w1,2x experimental data on the differential cross section d 2srdh dpT of inclusive D ) " electroproduction at low Q 2 . A comparison of these results with NLO pQCD calculations in the ‘‘massive’’ w3x and ‘‘massless’’ w4x charm schemes shows w5x that both approaches underestimate the cross section in the intermediate p T Ž D ) . and forward h Ž D ) . regions. The massive scheme meets similar difficulties also at Tevatron conditions. To reproduce the heavy quark p T spectra one usually attributes some primordial transverse momentum k T to the incoming partons. The size of this k T cannot be predicted within the model itself and is required to be of about 1 or 2 1 2
Electronic address: [email protected] Electronic address: [email protected]
GeV to fit the data. The massless scheme is only valid in the asymptotic limit p T 4 m and does not suit to intermediate p T Ž D ) . values w6x. Besides, the calculations use special assumptions on the c ™ D ) fragmentation and imply an unusually low value of the Peterson parameter e , 0.02 w7x. In view of the above problems, it would be certainly reasonable to try a different way. In the present note, we focus on the so called semihard approach w8x ŽSHA., which we had applied earlier w9x to open charm and JrC photoproduction. The inherent BFKL gluon evolution automatically generates k T distributions, and thus allows to keep this theoretical input under control. At the HERA energies and beyond, the interaction dynamics is governed by the properties of parton distributions in the small x region. This domain is characterized by the double inequality s 4 m2 , sˆ 4 L2 , which shows that the typical parton interaction scale m is much higher than the QCD parameter
0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 5 2 9 - 8
S.P. BaranoÕ, N.P. ZotoÕr Physics Letters B 458 (1999) 389–392
390
L, but is much lower than the total c.m.s. energy 's . The situation is therefore classified as ‘‘semihard’’. The resummation w8,10–12x of the terms wlnŽ m 2 rL 2 . a s x n , wlnŽ m 2 rL 2 . lnŽ1rx . a s x n and wlnŽ1rx . a s x n in SHA results in the unintegrated parton distributions w i Ž x,k T2 , m ., which determine the probability to find a parton of type i carrying the longitudinal momentum fraction x and transverse momentum k T at the probing scale m2 . They obey the BFKL w13x equation and reduce to the conventional parton densities Fi Ž x, m2 . once the k T dependence is integrated out: 2
H0m w Ž x ,k i
2 2 T ,m
. dk T2 s x Fi Ž x , m2 . .
Ž 1.
To calculate the cross section of a physical process, the unintegrated functions w i have to be convoluted with off mass shell matrix elements corresponding to the relevant partonic subprocesses. The specific properties of semihard theory may manifest in several ways. With respect to inclusive production properties, one points out an additional contribution to the cross sections due to the integration over the k T2 region above m2 and the broadening of the p T spectra due to extra transverse momentum of the interacting gluons w8–10x. It is important that the gluons are not on mass shell but are characterized by virtual masses proportional to their transverse momentum: m2 s yk T2rŽ1 y x .. This also assumes a modification of the polarization density matrix. A striking consequence of this fact on the Jrc spin alignement has been demonstrated in w14x. Now, consider the SHA predictions for D ) photoproduction at HERA.
)"
2. D electroproduction in the semihard QCD approach The differential cross section of the process eŽ p . q g Ž k . ™ eX Ž pX . q cŽ p 1 . q cŽ p 2 . is calculated according to the formula e. d s ; LŽma H mn Ha b LŽnbg .wG Ž x ,k T2 . 2 2 =dp1T dp 2T dpXT2 dy1 dy 2 .
Ž 2.
When evaluating the spin average of the matrix element squared, we substitute the full lepton tensor for the photon polarization matrix Žincluding also the
photon propagator factor and photon-lepton coupling.: e 1me 1) n ; LŽmne. 2
s 4pa 8 p m p n y 4 Ž pk 1 . g mn r Ž k 12 . , Ž 3 . where k 1 s pX y p. The virtual gluon polarization tensor is taken w8x: LŽmng . s e 2me 2) n s k Tm k Tnr< k T < 2 . Ž 4. This prescription has a clear analogy with the photon polarization matrix. Neglecting the second term in the right hand side in Ž3. in the small x limit, p 4 k 1 , one immediately arrives at the spin structure e me ) n ; p m p n . The latter may be rewritten in the form Ž4. if to use the parametrization for the 4-vectors k s xp q k T and to apply a gauge shift e m ™ e m y k mrx. With respect to another essential ingredient, the unintegrated gluon distribution wG Ž x,k T2 , m2 ., we follow the prescriptions of paper w15x. The proposed method lies upon a straightforward perturbative solution of the BFKL equation where the collinear gluon density x GŽ x, m2 . is used as the boundary condition in the integral form Ž1.. Technically, the unintegrated gluon density is calculated as a convolution of collinear gluon density with universal weight factors w15x: 2 wG Ž x ,k H , m2 . x x 2 1 2 s G Ž h ,k H , m2 . G , m dh , Ž 5. h h x
ž
H
GŽ
2 h ,k H
s
,m
as 2 xk H
2
/
.
ž(
2 J0 2 a s ln Ž 1rh . ln Ž m2rk H . ,
/
2 kH - m2 , 2 G Ž h ,k H , m2 . as 2 s 2 I0 2 a s ln Ž 1rh . ln Ž k H rm2 . , xk H
ž(
Ž 6.
/
2 kH ) m2 , Ž 7. where J0 and I0 stand for Bessel functions Žof real and imaginary arguments, respectively., and a s s 3 a srp . The latter parameter is connected with the Pomeron trajectory intercept: D s a s4ln2 in the LO and D s a s4ln2 y N a s2 in the NLO approximations, respectively, where N is a number w16x. The described method provides a common smooth parametrization for the gluon distributions in the
S.P. BaranoÕ, N.P. ZotoÕr Physics Letters B 458 (1999) 389–392
whole k T range. For the input collinear densities we use the standard GRV set w17x that fixes the absolute normalization of wG . The use of a different set of collinear structure functions does not introduce extra uncertainties because all the modern parametrizations are numerically close to each other since they are fitted to the same experimental data. The fragmentation c ™ D ) is described by Peterson fragmentation function w18x with the usual parameter e s 0.06. This value reproduces the shape of the fragmentation function obtained within LO QCD approximation. The theoretical results are also sensitive to the quark mass value m c and the probing scale m2 . However, the dependence of SHA predictions to the choice of these parameters is weaker than in the case of fixed order calculations because of the resummation procedure. In calculations we have tried several values for the charm mass m c s 1.3 or 1.5 GeV, the pQCD scale m2 s sr4 ˆ or mT2 , and the parameter D varying from 0.166 to 0.53.
3. Results and discussions The theoretical predictions for p T Ž D ) . and Ž h D ) . distributions made with different parameter
Fig. 1. D ) transverse momentum distribution, d s r dpT . Theoretical calculations: Ž1. solid curve: m c s1.3 GeV, m2 s sr4, Ds ˆ 0.35; Ž2. boldface dashed curve: m c s1.5 GeV, m2 s mT2 , D s 0.35; Ž3. dashed curve: m c s1.5 GeV, m2 s mT2 , D s 0.53; Ž4. D s 0.166; Ž5. boldface dotted curve: m c s1.3 GeV, m2 s sr4, ˆ dotted curve: m c s1.5 GeV, m2 s sr4, D s 0.35. v: ZEUS data ˆ w5x.
391
Fig. 2. D ) pseudorapidity distribution, d s r dh , for pT ) 3 GeV. Theoretical curves are as in Fig. 1. v: ZEUS data w5x.
sets are presented in Figs. 1 and 2, respectively, together with the experimental data w5x obtained by ZEUS collaboration at HERA. The theoretical curves 1 and 5 illustrate the dependence on m c . The value m c s 1.3 GeV leads to an exceeding cross section only at small p T . The curves 2 and 5 show the dependence on the scale m2 , which is found to be rather weak. We find that, among the model parameters, it is the value of the Pomeron intercept, which has the most pronounced influence on the shape of the p T spectrum. In contrast, the parameters m c and m2 do mainly change the overall normalization of the cross section, with only a negligeable effect on the p T slope. The curves 1, 2 and 5, which correspond to D s 0.35 almost coincide with each other, irrespectively to the values of m c and m. All these curves demonstrate rather good agreement with the experimental data. We would like to note that the p T spectra obtained within SHA are typically broader than those in fixed order calculations. The latter number is apparently higher than the one extracted from the rapidity gap events in photoproduction and DIS at HERA w19x. However, the estimated value of D represents only the first approximation based on the LO kernel in the BFKL equation, whereas the NLO corrections to the BFKL kernel are known to be rather large w20x. In the same time, our estimations appear to be close to the theoretical results of Ref. w16x, where the resummation of double transverse logarithms has been performed in a manner consistent with the full NLO BFKL kernel. Concerning the pseudorapidity distributions ŽFig. . 2 , semihard approach is also in a satisfactory agree-
392
S.P. BaranoÕ, N.P. ZotoÕr Physics Letters B 458 (1999) 389–392
ment with data Žexcept for region of positive h .. The agreement looks even better than in the pQCD massless scheme, although the latter has spent special efforts, such as the modification of the charm fragmentation function. On the other hand, the charm production cross section at large positive h may probably be related to other contributing mechanisms, such as, for example, the resolved photon.
4. Conclusions In the framework of semihard QCD approach, we consider the differential cross sections of inclusive D ) " meson production at HERA. We see that semihard approach reproduces the experimental p T distribution. As is expected, it is broader than that predicted in the pQCD massive and massless schemes. The shape of the gluon k T distribution is under control by BFKL equation, which contrasts with the conventional pQCD schemes where the gluon k T value is taken an arbitrary parameter. The slope of the final state p T spectrum mainly depends on the Pomeron intercept. We have obtained for it the value D s 0.35, which reasonably fits the experimental data even at different values for charm quark mass and QCD scale parameter m.
Acknowledgements We thank L. Gladilin for providing us with ZEUS experimental data.
References w1x H1 Collaboration, S. Aid, Nucl. Phys. B 472 Ž1996. 32. w2x ZEUS Collaboration, J. Breitweg, Phys. Lett. B 401 Ž1997. 192. w3x S. Frixione, M.L. Mangano, P. Nason, G. Ridolfi, Phys. Lett. B 348 Ž1995. 633; Nucl. Phys. B 454 Ž1995. 3. w4x B.A. Kniehl, G. Kramer, M. Spira, Z. Phys. C 76 Ž1997. 689; J. Binnewies, B.A. Kniehl, G. Kramer, preprint DESY 97241. w5x ZEUS Collaboration, J. Breitweg et al., preprint DESY 98085. w6x B.A. Kniehl, M. Kramer, G. Kramer, M. Spira, Phys. Lett. B ¨ 356 Ž1995. 539. w7x M. Cacciari, M. Greco, Z. Phys. C 69 Ž1996. 459; preprint DESY 97-029. w8x L.V. Gribov, E.M. Levin, M.G. Ryskin, Phys. Rep. 100 Ž1983. 1. w9x V.A. Saleev, N.P. Zotov, Mod. Phys. Lett. A 9 Ž1994. 151; A 11 Ž1996. 25. w10x E.M. Levin, M.G. Ryskin, Yu.M. Shabelski, A.G. Shuvaev, Sov. J. Nucl. Phys. 53 Ž1991. 1059; 54 Ž1991. 1420. w11x S. Catani, M. Ciafoloni, F. Hautmann, Nucl. Phys. B 366 Ž1991. 135. w12x J.C. Collins, R.K. Ellis, Nucl. Phys. B 360 Ž1991. 3. w13x E.A. Kuraev, L.N. Lipatov, V.S. Fadin, Sov. Phys. JETP 45 Ž1977. 199; Ya. Balitsky, L.N. Lipatov, Sov. J. Nucl. Phys. 28 Ž1978. 822. w14x S.P. Baranov, Phys. Lett. B 428 Ž1998. 377. w15x J. Blumlein, preprint DESY 95-121. w16x G.P. Salam, hep-phr9806482. w17x M. Gluck, ¨ E. Reya, A. Vogt, Z. Phys. C 67 Ž1995. 433. w18x C. Peterson, D. Schlatter, I. Schmitt, P. Zerwas, Phys. Rev. D 27 Ž1983. 105. w19x ZEUS Collaboration, J. Breitweg et al., preprint DESY 97061; H1 Collaboration, C. Adloff et al., preprint DESY 97-158. w20x V.S. Fadin, L.N. Lipatov, hep-phr9802290; M. Ciafaloni, G. Camici, hep-phr9803389.
Physics Letters B 458 Ž1999. 389–392
Pomeron intercept and BFKL gluon dynamics in the p T žD ) / spectra at HERA S.P. Baranov b
a,1
, N.P. Zotov
b,2
a P.N.LebedeÕ Institute of Physics, Leninsky prosp. 53, 117924 Moscow, Russia D.V.Skobeltsyn Institute of Nuclear Physics, M.V.LomonosoÕ Moscow State UniÕersity, 119899 Moscow, Russia
Received 3 February 1999; received in revised form 22 March 1999 Editor: P.V. Landshoff
Abstract In the framework of semihard QCD approach, we consider the differential cross sections of inclusive D ) " meson production at HERA. We find that the slopes of the calculated pT spectra are sensitive to the Pomeron intercept parameter D. We present a comparison of the theoretical results with available ZEUS data and derive an estimation for the value of D. q 1999 Elsevier Science B.V. All rights reserved. Keywords: QCD; Semihard approach; BFKL equation; Charm electroproduction; Pomeron
1. Introduction Recently H1 and ZEUS collaborations have reported w1,2x experimental data on the differential cross section d 2srdh dpT of inclusive D ) " electroproduction at low Q 2 . A comparison of these results with NLO pQCD calculations in the ‘‘massive’’ w3x and ‘‘massless’’ w4x charm schemes shows w5x that both approaches underestimate the cross section in the intermediate p T Ž D ) . and forward h Ž D ) . regions. The massive scheme meets similar difficulties also at Tevatron conditions. To reproduce the heavy quark p T spectra one usually attributes some primordial transverse momentum k T to the incoming partons. The size of this k T cannot be predicted within the model itself and is required to be of about 1 or 2 1 2
Electronic address: [email protected] Electronic address: [email protected]
GeV to fit the data. The massless scheme is only valid in the asymptotic limit p T 4 m and does not suit to intermediate p T Ž D ) . values w6x. Besides, the calculations use special assumptions on the c ™ D ) fragmentation and imply an unusually low value of the Peterson parameter e , 0.02 w7x. In view of the above problems, it would be certainly reasonable to try a different way. In the present note, we focus on the so called semihard approach w8x ŽSHA., which we had applied earlier w9x to open charm and JrC photoproduction. The inherent BFKL gluon evolution automatically generates k T distributions, and thus allows to keep this theoretical input under control. At the HERA energies and beyond, the interaction dynamics is governed by the properties of parton distributions in the small x region. This domain is characterized by the double inequality s 4 m2 , sˆ 4 L2 , which shows that the typical parton interaction scale m is much higher than the QCD parameter
0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 5 2 9 - 8
S.P. BaranoÕ, N.P. ZotoÕr Physics Letters B 458 (1999) 389–392
390
L, but is much lower than the total c.m.s. energy 's . The situation is therefore classified as ‘‘semihard’’. The resummation w8,10–12x of the terms wlnŽ m 2 rL 2 . a s x n , wlnŽ m 2 rL 2 . lnŽ1rx . a s x n and wlnŽ1rx . a s x n in SHA results in the unintegrated parton distributions w i Ž x,k T2 , m ., which determine the probability to find a parton of type i carrying the longitudinal momentum fraction x and transverse momentum k T at the probing scale m2 . They obey the BFKL w13x equation and reduce to the conventional parton densities Fi Ž x, m2 . once the k T dependence is integrated out: 2
H0m w Ž x ,k i
2 2 T ,m
. dk T2 s x Fi Ž x , m2 . .
Ž 1.
To calculate the cross section of a physical process, the unintegrated functions w i have to be convoluted with off mass shell matrix elements corresponding to the relevant partonic subprocesses. The specific properties of semihard theory may manifest in several ways. With respect to inclusive production properties, one points out an additional contribution to the cross sections due to the integration over the k T2 region above m2 and the broadening of the p T spectra due to extra transverse momentum of the interacting gluons w8–10x. It is important that the gluons are not on mass shell but are characterized by virtual masses proportional to their transverse momentum: m2 s yk T2rŽ1 y x .. This also assumes a modification of the polarization density matrix. A striking consequence of this fact on the Jrc spin alignement has been demonstrated in w14x. Now, consider the SHA predictions for D ) photoproduction at HERA.
)"
2. D electroproduction in the semihard QCD approach The differential cross section of the process eŽ p . q g Ž k . ™ eX Ž pX . q cŽ p 1 . q cŽ p 2 . is calculated according to the formula e. d s ; LŽma H mn Ha b LŽnbg .wG Ž x ,k T2 . 2 2 =dp1T dp 2T dpXT2 dy1 dy 2 .
Ž 2.
When evaluating the spin average of the matrix element squared, we substitute the full lepton tensor for the photon polarization matrix Žincluding also the
photon propagator factor and photon-lepton coupling.: e 1me 1) n ; LŽmne. 2
s 4pa 8 p m p n y 4 Ž pk 1 . g mn r Ž k 12 . , Ž 3 . where k 1 s pX y p. The virtual gluon polarization tensor is taken w8x: LŽmng . s e 2me 2) n s k Tm k Tnr< k T < 2 . Ž 4. This prescription has a clear analogy with the photon polarization matrix. Neglecting the second term in the right hand side in Ž3. in the small x limit, p 4 k 1 , one immediately arrives at the spin structure e me ) n ; p m p n . The latter may be rewritten in the form Ž4. if to use the parametrization for the 4-vectors k s xp q k T and to apply a gauge shift e m ™ e m y k mrx. With respect to another essential ingredient, the unintegrated gluon distribution wG Ž x,k T2 , m2 ., we follow the prescriptions of paper w15x. The proposed method lies upon a straightforward perturbative solution of the BFKL equation where the collinear gluon density x GŽ x, m2 . is used as the boundary condition in the integral form Ž1.. Technically, the unintegrated gluon density is calculated as a convolution of collinear gluon density with universal weight factors w15x: 2 wG Ž x ,k H , m2 . x x 2 1 2 s G Ž h ,k H , m2 . G , m dh , Ž 5. h h x
ž
H
GŽ
2 h ,k H
s
,m
as 2 xk H
2
/
.
ž(
2 J0 2 a s ln Ž 1rh . ln Ž m2rk H . ,
/
2 kH - m2 , 2 G Ž h ,k H , m2 . as 2 s 2 I0 2 a s ln Ž 1rh . ln Ž k H rm2 . , xk H
ž(
Ž 6.
/
2 kH ) m2 , Ž 7. where J0 and I0 stand for Bessel functions Žof real and imaginary arguments, respectively., and a s s 3 a srp . The latter parameter is connected with the Pomeron trajectory intercept: D s a s4ln2 in the LO and D s a s4ln2 y N a s2 in the NLO approximations, respectively, where N is a number w16x. The described method provides a common smooth parametrization for the gluon distributions in the
S.P. BaranoÕ, N.P. ZotoÕr Physics Letters B 458 (1999) 389–392
whole k T range. For the input collinear densities we use the standard GRV set w17x that fixes the absolute normalization of wG . The use of a different set of collinear structure functions does not introduce extra uncertainties because all the modern parametrizations are numerically close to each other since they are fitted to the same experimental data. The fragmentation c ™ D ) is described by Peterson fragmentation function w18x with the usual parameter e s 0.06. This value reproduces the shape of the fragmentation function obtained within LO QCD approximation. The theoretical results are also sensitive to the quark mass value m c and the probing scale m2 . However, the dependence of SHA predictions to the choice of these parameters is weaker than in the case of fixed order calculations because of the resummation procedure. In calculations we have tried several values for the charm mass m c s 1.3 or 1.5 GeV, the pQCD scale m2 s sr4 ˆ or mT2 , and the parameter D varying from 0.166 to 0.53.
3. Results and discussions The theoretical predictions for p T Ž D ) . and Ž h D ) . distributions made with different parameter
Fig. 1. D ) transverse momentum distribution, d s r dpT . Theoretical calculations: Ž1. solid curve: m c s1.3 GeV, m2 s sr4, Ds ˆ 0.35; Ž2. boldface dashed curve: m c s1.5 GeV, m2 s mT2 , D s 0.35; Ž3. dashed curve: m c s1.5 GeV, m2 s mT2 , D s 0.53; Ž4. D s 0.166; Ž5. boldface dotted curve: m c s1.3 GeV, m2 s sr4, ˆ dotted curve: m c s1.5 GeV, m2 s sr4, D s 0.35. v: ZEUS data ˆ w5x.
391
Fig. 2. D ) pseudorapidity distribution, d s r dh , for pT ) 3 GeV. Theoretical curves are as in Fig. 1. v: ZEUS data w5x.
sets are presented in Figs. 1 and 2, respectively, together with the experimental data w5x obtained by ZEUS collaboration at HERA. The theoretical curves 1 and 5 illustrate the dependence on m c . The value m c s 1.3 GeV leads to an exceeding cross section only at small p T . The curves 2 and 5 show the dependence on the scale m2 , which is found to be rather weak. We find that, among the model parameters, it is the value of the Pomeron intercept, which has the most pronounced influence on the shape of the p T spectrum. In contrast, the parameters m c and m2 do mainly change the overall normalization of the cross section, with only a negligeable effect on the p T slope. The curves 1, 2 and 5, which correspond to D s 0.35 almost coincide with each other, irrespectively to the values of m c and m. All these curves demonstrate rather good agreement with the experimental data. We would like to note that the p T spectra obtained within SHA are typically broader than those in fixed order calculations. The latter number is apparently higher than the one extracted from the rapidity gap events in photoproduction and DIS at HERA w19x. However, the estimated value of D represents only the first approximation based on the LO kernel in the BFKL equation, whereas the NLO corrections to the BFKL kernel are known to be rather large w20x. In the same time, our estimations appear to be close to the theoretical results of Ref. w16x, where the resummation of double transverse logarithms has been performed in a manner consistent with the full NLO BFKL kernel. Concerning the pseudorapidity distributions ŽFig. . 2 , semihard approach is also in a satisfactory agree-
392
S.P. BaranoÕ, N.P. ZotoÕr Physics Letters B 458 (1999) 389–392
ment with data Žexcept for region of positive h .. The agreement looks even better than in the pQCD massless scheme, although the latter has spent special efforts, such as the modification of the charm fragmentation function. On the other hand, the charm production cross section at large positive h may probably be related to other contributing mechanisms, such as, for example, the resolved photon.
4. Conclusions In the framework of semihard QCD approach, we consider the differential cross sections of inclusive D ) " meson production at HERA. We see that semihard approach reproduces the experimental p T distribution. As is expected, it is broader than that predicted in the pQCD massive and massless schemes. The shape of the gluon k T distribution is under control by BFKL equation, which contrasts with the conventional pQCD schemes where the gluon k T value is taken an arbitrary parameter. The slope of the final state p T spectrum mainly depends on the Pomeron intercept. We have obtained for it the value D s 0.35, which reasonably fits the experimental data even at different values for charm quark mass and QCD scale parameter m.
Acknowledgements We thank L. Gladilin for providing us with ZEUS experimental data.
References w1x H1 Collaboration, S. Aid, Nucl. Phys. B 472 Ž1996. 32. w2x ZEUS Collaboration, J. Breitweg, Phys. Lett. B 401 Ž1997. 192. w3x S. Frixione, M.L. Mangano, P. Nason, G. Ridolfi, Phys. Lett. B 348 Ž1995. 633; Nucl. Phys. B 454 Ž1995. 3. w4x B.A. Kniehl, G. Kramer, M. Spira, Z. Phys. C 76 Ž1997. 689; J. Binnewies, B.A. Kniehl, G. Kramer, preprint DESY 97241. w5x ZEUS Collaboration, J. Breitweg et al., preprint DESY 98085. w6x B.A. Kniehl, M. Kramer, G. Kramer, M. Spira, Phys. Lett. B ¨ 356 Ž1995. 539. w7x M. Cacciari, M. Greco, Z. Phys. C 69 Ž1996. 459; preprint DESY 97-029. w8x L.V. Gribov, E.M. Levin, M.G. Ryskin, Phys. Rep. 100 Ž1983. 1. w9x V.A. Saleev, N.P. Zotov, Mod. Phys. Lett. A 9 Ž1994. 151; A 11 Ž1996. 25. w10x E.M. Levin, M.G. Ryskin, Yu.M. Shabelski, A.G. Shuvaev, Sov. J. Nucl. Phys. 53 Ž1991. 1059; 54 Ž1991. 1420. w11x S. Catani, M. Ciafoloni, F. Hautmann, Nucl. Phys. B 366 Ž1991. 135. w12x J.C. Collins, R.K. Ellis, Nucl. Phys. B 360 Ž1991. 3. w13x E.A. Kuraev, L.N. Lipatov, V.S. Fadin, Sov. Phys. JETP 45 Ž1977. 199; Ya. Balitsky, L.N. Lipatov, Sov. J. Nucl. Phys. 28 Ž1978. 822. w14x S.P. Baranov, Phys. Lett. B 428 Ž1998. 377. w15x J. Blumlein, preprint DESY 95-121. w16x G.P. Salam, hep-phr9806482. w17x M. Gluck, ¨ E. Reya, A. Vogt, Z. Phys. C 67 Ž1995. 433. w18x C. Peterson, D. Schlatter, I. Schmitt, P. Zerwas, Phys. Rev. D 27 Ž1983. 105. w19x ZEUS Collaboration, J. Breitweg et al., preprint DESY 97061; H1 Collaboration, C. Adloff et al., preprint DESY 97-158. w20x V.S. Fadin, L.N. Lipatov, hep-phr9802290; M. Ciafaloni, G. Camici, hep-phr9803389.