- Email: [email protected]
An investigation of a possible hard contribution to φ photoproduction
16 December 1999
Physics Letters B 470 Ž1999. 200–208
An investigation of a possible hard contribution to f photoproduction Erasmo Ferreira a
a,1
, Uri Maor
a,b,2
Instituto de Fisica, UniÕersidade Federal do Rio de Janeiro, Rio de Janeiro RJ21945-970, Brazil b School of Physics and Astronomy, Tel AÕiÕ UniÕersity, Ramat AÕiÕ 69978, Israel Received 29 April 1999; received in revised form 3 September 1999; accepted 29 October 1999 Editor: R. Gatto
Abstract We investigate the possibility that the process of f photoproduction may have a significant hard perturbative QCD component. This suggestion is based on a study of the energy dependence of the forward f photoproduction cross section followed by a calculation where we show that a coherent sum of the pQCD and conventional soft Pomeron contributions provides an excellent reproduction of the experimental data. Our results suggest that the transition from the predominantly soft photoproduction of light r and v vector mesons to the predominantly hard photoproduction of heavy JrC and F is smooth and gradual, similar to the transition observed in deep inelastic scattering studies of the proton structure function in the small x limit. Our predictions for higher HERA energies are presented. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 12.38.-t; 12.38.Bx; 12.38.Lg; 11.55.Jy; 13.60.-r; 13.60.Le
1. Introduction Over the past few years we have seen a vigorous phenomenological investigation of the Pomeron through the study of hadronic total, elastic and diffractive cross sections, as well as the study of the proton deep inelastic scattering ŽDIS. structure function. In particular, Donnachie and Landshoff ŽDL. have promoted w1x an appealing and very simple
Regge parameterization of the total hadronic cross sections in which
stot s X
s
D
s
ž / ž / s0
qY
s0
yh
.
Ž 1.
The two key ingredients of this approach are the Regge trajectories
a R Ž t . s a R Ž 0 . q a RX t ,
Ž 2.
where a R Ž0. s 1 y h and the Pomeron trajectory, which dominates at high energies, 1 2
E-mail: [email protected] E-mail: [email protected]
a P Ž t . s a P Ž 0 . q a PX t ,
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 1 2 8 7 - 3
Ž 3.
E. Ferreira, U. Maor r Physics Letters B 470 (1999) 200–208
where a P Ž0. s 1 q D. The DL study establishes universal values D s 0.0808 and h s 0.4525. This study is supplemented by the analysis of Block, Kang and White w2x who determine the slope a PX of the Pomeron trajectory to be a PX , 0.2 y 0.3 GeVy2 . The same approach is also applicable to the analyses of real photoproduction and of the proton DIS structure function w3x. While the energy dependence of the photoproduction total cross section follows the DL pattern, it has been observed w4x that F2 Ž x,Q 2 . behaves, for small enough x, like xyl , where l is slowly growing with Q 2 . This growth is given by
ls
E ln Ž F2 Ž x ,Q 2 . . E ln Ž 1rx .
.
Ž 4.
It has been recognized for quite some time that the transition from the predominantly soft real photoproduction Ž Q 2 s 0. to the predominantly hard DIS processes, with high enough Q 2 , is smooth and gradual w3,5–7x. This observation, regardless of its theoretical interpretation, is evident once we examine the energy dependence of F2 Ž x,Q 2 . in the small x limit with Q 2 ranging from zero to a few GeV 2 . It is also well known that real photoproduction of light vector mesons, r and v , is dominated by a soft Pomeron exchange w8,9x, whereas photoproduction of heavy vectors, JrC and F , is well reproduced by a perturbative QCD ŽpQCD. calculation w10–14x, where M V2r4 replaces Q 2 as a measure of the process hardness. Although M V2r4 is a discrete variable, while Q 2 is a continuous DIS variable, it is interesting to check if the transition from soft to hard photoproduction of vector mesons follows the behavior pattern observed in DIS. To this end, the study of f photoproduction is particularly instructive, as Mf2r4 s 0.26 GeV 2 , and we know that the energy dependence of F2 , with small x and Q 2 as low as 0.2–0.3 GeV 2 , is somewhat steeper than the energy dependence of stot Žg p .. Our investigation is susceptible to both experimental and theoretical uncertainties. Experimentally, a systematic study of the integrated f photoproduction cross section and the forward differential cross section slope w15–18x are not very reliable, as different experimental groups have utilized different, not
201
always mutually consistent, methods to extract and relate these quantities. To overcome this difficulty, we have analyzed the measured differential cross sections rather than integrated quantities. Even so, the two higher energy data points w17,18x are averaged over wide energy bins. This, combined with the overall poor quality of the reported data, may make a detailed analysis non conclusive at this stage. Theoretically, our calculation is done at the boundary of pQCD applicability and as such should be considered as a rough estimate. Technically, a pQCD calculation of w d s Žg p f p .rdt x ts0 depends on our knowledge of the gluon structure function at Q 2 s 0.26 GeV 2 . Such information requires an extrapolation of a given parton distribution below its initial evolution threshold Q 02 . For this purpose we adopt a linear extrapolation which was successfully utilized in previous calculations w6,19x. As we shall see, there is a significant difference between the MRST w20x and GRV98 w21x input gluon distributions. We have chosen to use the GRV98 distribution and shall explain our motivations for doing so. The purpose of this Letter is to examine these issues in some detail from different points of view. We present an analysis of the existing f photoproduction forward differential cross section data which suggests an energy dependence which is steeper than the typical energy dependence associated with the soft Pomeron w1x. We then present a pQCD calculation from which we deduce that the hard component is responsible for about a half of the f photoproduction amplitude in the forward direction at presently available energies. We then proceed to show that a coherent sum of the calculated pQCD amplitude and a conventional soft Pomeron exchange contribution provides an excellent reproduction of the available data w15–18x.
™
2. Data analysis Our data analysis investigates whether the f photoproduction cross section follows a power dependence on the c.m. energy W, and whether this power is larger than the value determined from the energy dependence of the g p total cross section. Following
202
E. Ferreira, U. Maor r Physics Letters B 470 (1999) 200–208
DL w1x and Block et al. w2x, we expect the f photoproduction cross section to behave like W 4 D and the forward slope to behave like 4a PX lnW. The analysis of f photoproduction data is seemingly easy, as this process proceeds exclusively through Pomeron exchange, since the various Regge exchanges cancel each other. The problem is that the published analyses w1,18x depend on a comparison between integrated cross section data taken by different groups who have utilized different, and not always mutually consistent, procedures. In addition, because the f forward slope is shrinking, the interpretation of the integrated cross section behaving as a fixed power of W is somewhat ambiguous. In order to bypass these difficulties, we have limited ourselves to the analysis of the individual differential cross sections d srdt as reported by the experimental groups w15–18x. We have used data with W ) 6 GeV corresponding to x - 0.025.
Fig. 1 shows that Ž d srdt . 0 in the available energy range is, indeed, well fitted by an effective power of W,
ž
ds Ž g p
™f p.
dt
/
sA ts0
2
W
ž / W0
4l
,
Ž 5.
where W0 s 1 GeV. Our best fit, for 5 data points with 6.7 F W F 70 GeV, has a x 2rn.d. f.s 0.22, corresponding to 3 degrees of freedom, where the fitted parameters are A2 s 0.76 " 0.18 m brGeV 2 and l s 0.135 " 0.024. For comparison we show also a fit Žpresented by a dashed line. with 4 degrees of freedom where we fix the power to its DL value 4 D s 0.3232 and obtain A2DL s 1.21 " 0.12 m br GeV 2 with x 2rn.d. f.s 0.92, which cannot be discredited. Our inability to determine the power unambiguously results from the big error coupled to the ZEUS high energy data point w18x. It is clear that an
Fig. 1. Our best fit of Eq. Ž5. Ždenoted FM. compared with DL prediction for w d s Žg p
™ f p.rdt x
ts 0 .
All are given in m brGeV 2 .
E. Ferreira, U. Maor r Physics Letters B 470 (1999) 200–208
improvement of the HERA data point at ²W : s 70 GeV and additional relevant data at higher energies will enable us to conclusively distinguish between a DL type interpretation and ours. In order to further examine the suggestion that the dependence of f photoproduction on W is steeper than the behavior implied by a soft Pomeron exchange, we have studied the ratio Rs
Ž d srdt . 0 , sP2 Ž g p .
Ž 6.
where sP Žg p . is the soft Pomeron exchange contribution to stot Žg p .. The ratio is shown in Fig. 2. Using the DL parameterization w1x we have
sP Ž g p . s 67.7
W
ž / W0
203
shown in Fig. 2 indicates the onsetting of a hard component. The best fit that we have obtained implies that R behaves as W 0.215" 0.040 . A constant ratio provides an acceptable fit which implies that at this stage this option cannot be definitely excluded. To summarize: The available experimental data on Ž d s Žg p f p .rdt . 0 is consistently well described as having an energy dependence steeper than the one implied by a soft Pomeron exchange. Nevertheless, the assumption of a pure soft production mechanism cannot be unambiguously eliminated. Note, that our results suggest that f photoproduction is somewhat harder than F2 at Q 2 s 0.26 GeV 2 .
™
0.1616
mb .
Ž 7.
predominantly soft production mechanism for g p f p means that the ratio presented in Fig. 2 ™Ashould be a constant. The suggested increase with W
3. A pQCD calculation Our pQCD calculation of the forward f photoproduction follows earlier pQCD calculations of the forward photoproduction cross section of heavy vec-
Fig. 2. R data and our best fit. The ratio is given in w m b GeV 2 xy1 .
E. Ferreira, U. Maor r Physics Letters B 470 (1999) 200–208
204
tor mesons w10–14x. These calculations are considerably simplified once we assume a non-relativistic wave function for those vector meson states. This assumption, which is also valid for f , enables us to write a leading-order expression ds Ž g p
™f p.
dt s
ts0
a S2 Gefe 3 a E M Mf5
16p
3
ž
xG x ,
Mf2 4
2
/
,
Ž 8.
™
where x s Ž MfrW . 2 and Gefe is the partial decay width of f eqey. The above LO pQCD calculation is done at a very small scale of 0.26 GeV 2 . Even though we shall include an error estimate due to higher order contributions, our calculation is only approximate. Nevertheless, in our opinion, an estimate of a possible hard pQCD contribution to f photoproduction is of interest and may become necessary once the ZEUS data w18x are improved. In the following we follow Refs. w12–14x and, after calculating the cross section resulting from the imaginary forward amplitude in leading order, we correct for the real part of the amplitude. The energy dependence of Ž d s Ž g p f p . rdt . 0 is determined by the x dependence of xGŽ x,Q 2 .. Assuming that xGŽ x,Q 2 . behaves, in the small x limit, as xyl , we have
™
ls
E ln Ž xG Ž x ,Q 2 . . E ln Ž 1rx .
.
Ž 9.
Our suggested calculation depends on the knowledge of the gluon structure function xGŽ x,Q 2 . at Q 2 s Mf2r4 s 0.26 GeV 2 . This Q 2 value is well below Q02 , the minimum value of Q 2 above which we can use the updated parton distribution parameterizations w20,21x. We recall the general property of the gluon structure function which is linear in Q 2 in the limit of very small virtuality and use, accordingly, a linear extrapolation xG Ž x ,Q 2 . s
Q2 Q 02
xG Ž x ,Q02 . , Q 2 - Q 02 .
Ž 10 .
This approximation has been successfully used in previous theoretical DIS investigations w6,19x in which the knowledge of xG in the small Q 2 region was required.
The results of our gluon structure function extrapolation for MRST w20x and GRV98 w21x, at the relevant Q 2 s 0.26 GeV 2 value, are shown in Fig. 3. Note that Q02 s 1.2 GeV 2 for MRST and 0.8 GeV 2 for GRV98. Fig. 3 also shows the Pomeron term of ALLM97 w3x. In our calculations we have used the GRV98 extrapolated distribution. Following is our motivation: Ž1. Both MRST and GRV98 start their evolution with non realistic input gluon distributions which have valence like x dependences. MRST starts its evolution from m2 s 1.0 GeV 2 while GRV98 starts from m2 s 0.4 GeV 2 . The non valence small x structure of xGŽ x,Q 2 ., typical at higher Q 2 values, is driven by the Q 2 evolution. However, we note that this change is faster in the GRV98 parametrization. As a result we consider GRV98 to provide a more realistic parameterization of xG in the Ž x,Q 2 . limit of interest. Indeed, both the normalization and x dependence of GRV98 provide an estimate of the hard component which is compatible with the data analysis described before. On the other hand, the xG obtained from MRST is too small and too soft to be considered as a candidate for the hard component of interest. Note that the above difference between MRST and GRV98 slowly diminishes as Q 2 is increased. Ž2. We note that in the kinematic domain relevant to this investigation, the GRV98 x dependence of xG in the small x limit, given by Eq. Ž9., is somewhat steeper than the corresponding x dependence of F2 at Q 2 s 0.26 GeV 2 , given by Eq. Ž4.. The difference is accounted for by the softer contributions to F2 which do not contribute to the phi channel. The above is compatible with the conclusions of Section 2. Ž3. From a different point of view, we note that MRST is very close to ALLM97 in the x region of interest. This re-enforces our interpretation that the MRST input at small Q 2 is predominantly soft, such as is ALLM97, and thus, less suitable for our analysis where we attempt a calculation of the hard component on its own. On the other hand, GRV98 provides an xG distribution which is harder than ALLM and as such seems suitable to describe the hard distribution with a low risk of a soft contamination leading to double counting. We follow Ref. w12,14x and consider the following corrections to the leading order cross section
E. Ferreira, U. Maor r Physics Letters B 470 (1999) 200–208
205
Fig. 3. ALLM97 and the extrapolated MRST and GRV98 parameterizations for the gluon structure function xGŽ x,Q 2 . at Q 2 s 0.26 GeV 2 .
written in Eq. Ž8.: Ž1. Eq. Ž8. corresponds to the imaginary part of the forward amplitude and should be corrected by Ž1 q r 2 ., where r s Re FrIm F, and F is the amplitude under consideration. We recall
that r , plr2, where, in our case, l is given by Eq. Ž9.. We note that l s 0.145 for W s 70 GeV and decreases very slowly as the energy decreases. Ž2. The next to leading order corrections are estimated
Table 1 Ž d srdt . ts 0 data and calculations. FM refers to the hybrid model calculation. W is given in GeV and the differential cross section in m brGeV 2 exp
FM
soft
pQCD
W
Ž ddts . 0
Ž ddts . 0
Ž ddts . 0
Ž ddts . 0
6.7 7.3 8.0 13.7 30 50 70 100 150 200 250
2.01 " 0.19 2.48 " 0.51 2.39 " 0.22 3.11 " 0.33
2.16 2.19 2.31 3.03 4.30 5.56 6.04 6.88 7.89 8.67 9.24
0.55 0.57 0.58 0.70 0.90 1.06 1.18 1.32 1.51 1.66 1.78
0.52 0.53 0.57 0.82 1.27 1.77 1.88 2.17 2.50 2.75 2.91
7.20 " 2.10
206
E. Ferreira, U. Maor r Physics Letters B 470 (1999) 200–208
Fig. 4. Data and our hybrid calculation Ždenoted FM. for Ž d srdt . ts 0 given in m brGeV 2 . Also shown are our separate calculations of the soft and hard cross sections.
by w1 q 0.5 a S Ž Mf2r4.x. For a S , small enough, this is a reasonably realistic estimate. In our case, where a S is relatively large, the NLO corrections amount to 35–40%. However, it is probable that this big correction will be reduced by even higher orders which cannot be neglected. We, thus, consider the NLO estimate as a measure of our possible error in the above LO calculation which is, obviously, a rough estimate due to the high value of a S . Ž3. Relativistic corrections and the effects of intermediate off diagonal partons in f photoproduction are rather small and have been neglected. The results of our calculated hard cross section are given in Table 1 and displayed in Fig. 4.
4. A hybrid pQCD and soft Pomeron model Although the pQCD term calculated in the previous section provides a significant contribution to f
photoproduction, there is no doubt that, in the energy range under consideration, the production mechanism initiated by soft Pomeron exchange is of major importance. Accordingly, we attempt to fit the data with a simple hybrid two- component model with the following prescriptions: Ž1. The first component is a soft DL Pomeron with an a P Ž t . intercept of 1.0808, namely, ds
ž / dt
soft
s A2S ts0
W
ž / W0
0.3232
.
Ž 11 . pQCD
Ž2. A hard pQCD component Ž ddts . ts0 , as calculated in the previous section. Ž3. A coherent sum of the above two components specified as follows: ds
ž / dt
2
2
s Ž ReF . q Ž Im F . , 0
Ž 12 .
E. Ferreira, U. Maor r Physics Letters B 470 (1999) 200–208
where F s Fsoft q FpQCD . Note, that for both the soft and the pQCD components we have Re Fi s r i Im Fi i and ŽIm Fi . 2 s Ž d srdt . 0rrŽ1 q r i2 ., where i s soft, pQCD. We fit the 5 data points of Ž d srdt . 0 with one parameter, the normalization A S of the DL Pomeron. We obtain a best fit Ž4 degrees of freedom. with A2S s 0.30 " 0.08 m b, corresponding to x 2rn.d. f.s 0.34. Our fitted cross sections Žcalled FM. are presented in Table 1, which contains also our predictions for higher HERA energies, and are displayed in Fig. 4. Our fit should be compared with the power fit presented earlier, which has a x 2rn.d. f.s 0.22, and the conventional DL fit which has x 2rn.d. f.s 0.92 Žsee Fig. 1.. Clearly, the presently available data is not sufficient to rule out any of these options. Once again we note that this ambiguity results from the big error associated to the ²W : s 70 GeV point. Improvement of the quality of this point and additional HERA data will enable a more discriminative analysis. The hybrid model that we have just suggested can be further examined by considering the differential cross sections. Such data is available w17,18x at ²W : s 13.7 and 70 GeV. For the purpose of our calculations we need to know the t-dependence of both the pQCD and soft Pomeron amplitudes. To this end we assume each of these dependences to be approximated by an exponential, Bi
t
Fi s Fi Ž 0 . e 2 ,
i s soft, pQCD .
Ž 13 .
This simple approximation is sufficient here, considering the quality of the available data on f photoproduction. We then proceed as follows: Ž1. We take for the pQCD amplitude an energy independent exponential slope. We derive its value from the high energy differential cross section on JrC photoproduction combined with the observation w22x that this slope is energy independent, corresponding to a flat hard Pomeron. In our analysis we have taken BpQCD s 4.6 GeVy2 , which corresponds to the H1 measurement w23x. A good fit is obtained, also, with the ZEUS value w24x of BpQCD s 4.0 GeVy2 . We have also treated BpQCD as a free parameter without a significant change. Ž2. For the soft Pomeron ampli-
207
Fig. 5. Data and our calculations for Ž d s r dt ., given in m brGeV 2 , at ²W : s13.7 and 70 GeV.
tude we assume a conventional Regge type exponential slope depending on two parameters, Bsoft s B0S q 2 a PX ln
W
ž / W0
2
.
Ž 14 .
B0S is a free parameter and we fix a PX s 0.25 based on high energy hadron-hadron phenomenology w2x. This hybrid model was fitted to reproduce 13 Ž d srdt . data points measured at ²W : s 13.7 and 70 GeV with one parameter, for which we obtain B0S s 6.00 " 0.92 GeVy2 with x 2rn.d. f.s 0.65 for 12 degrees of freedom. The data and our fit are shown in Fig. 5. Although our Ž d srdt . fit corroborates our proposition that high energy f real photoproduction has a significant hard component, we do not consider our success to be decisive. Indeed, we ran a fit in which only the soft DL component contributes to Ž d srdt .. In this fit A2DL s 1.21 is taken from Section 2 so that the only fitted parameter is B0S. We obtain B0S s 2.98 " 0.14 GeVy2 with x 2rn.d. f.s 1.29 for 12 degrees of freedom. This fit is of a lesser quality than the one obtained for the hybrid model, but we caution against reaching too strong a conclusion based on limited data. In general, we note that our hybrid model provides a natural explanation for the higher t curvature suggested by the data w17x. Clearly, and not only for the purpose of our analysis, additional knowledge on higher t behavior will help to clarify the picture. 5. Discussion Following are the main conclusions of our study and some general remarks: Ž1. The data analysis of
E. Ferreira, U. Maor r Physics Letters B 470 (1999) 200–208
208
forward real f photoproduction suggests the possible existence of a significant hard pQCD contribution in the energy range of 6 F W F 70 GeV. Ž2. The above suggestion is corroborated by a LO pQCD f p .rdt x 0 using the calculation of w d s Ž g p GRV98 gluon distribution extrapolated to Q 2 s 0.26 GeV 2 . Ž3. A hybrid model, in which we coherently add the pQCD hard component and a DL type soft component, provides an excellent overall reproduction of the data. We note that the ratio of hard to soft contribution in our model is compatible with the ratio obtained in a detailed study w6x of DIS in comparable small Q 2 values. Ž4. Our hybrid model is significantly different from the two Pomeron model suggested in Ref. w7x. In our model the t s 0 intercept of the hard effective Pomeron is deduced from Eq. Ž9. and, as such, it is a moving pole. In the model of Ref. w7x, the hard Pomeron is a fixed pole with a comparatively high l. Ž5. Theoretically, the validity of our calculation rests on: Ži. The validity of a pQCD calculation done at small Q 2 . Žii. The legitimacy of our Q 2 extrapolation of xGŽ x,Q 2 . below Q 02 . Žiii. Our choice of GRV98 for the input gluon structure function. Ž6. Our overall analysis suggests the existence of a significant hard component contributing to g p f p. However, a decisive quantitative conclusion depends on improving and extending the HERA data on f photoproduction.
™
™
Acknowledgements We thank Eran Naftali for his help in the numerical computations. U.M. wishes to thank UFRJ and FAPERJ ŽBrazil. for their support. This research was supported in part by the Israel Academy of Science and Humanities.
References w1x A. Donnachie, P.V. Landshoff, Phys. Lett. B 296 Ž1992. 227. w2x M.M. Block, K. Kang, A.R. White, Int. J. Mod. Phys. 7 Ž1992. 4449. w3x H. Abramowicz, E.M. Levin, A. Levy, U. Maor, Phys. Lett. B 269 Ž1991. 465; H. Abramowicz, A. Levy, DESY 97-251; hep-phr9712415. w4x H1 Collaboration, S. Aid et al., Nucl. Phys. B 470 Ž1996. 3; H1 Collaboration, C. Adloff et al., Nucl. Phys. B 497 Ž1997. 3; ZEUS Collaboration, J. Breitweg et al., Phys. Lett. B 407 Ž1997. 432. w5x A. Capella, A. Kaidalov, C. Merino, J. Tran Than Van, Phys. Lett. B 337 Ž1994. 358. w6x E. Gotsman, E.M. Levin, U. Maor, Eur. Phys. J. C 5 Ž1998. 303; E. Gotsman, E.M. Levin, U. Maor, E. Naftali, Eur. Phys. J. C 10 Ž1999. 689. w7x A. Donnachie, P.V. Landshoff, Phys. Lett. B 437 Ž1998. 408. w8x A. Donnachie, P. Landshoff, Phys. Lett. B 348 Ž1995. 213. w9x E. Gotsman, E.M. Levin, U. Maor, Phys. Lett. B 347 Ž1995. 424. w10x M.G. Ryskin, Z. Phys. C 57 Ž1993. 89. w11x S.J. Brodsky et al., Phys. Rev. D 50 Ž1994. 3134. w12x M.G. Ryskin, R.G. Roberts, A.D. Martin, E.M. Levin, Z. Phys. C 76 Ž1997. 231. w13x L. Frankfurt, M. McDermott, M. Strikman, hep-phr9812316. w14x A.D. Martin, M.G. Ryskin, T. Teubner, Phys. Lett. B 454 Ž1999. 339. w15x D. Aston et al., Nucl. Phys. B 172 Ž1980. 1. w16x Omega Photon Collaboration, M. Atkinson et al., Z. Phys. C 27 Ž1985. 233. w17x J. Busenitz et al., Phys. Rev. D 40 Ž1989. 1. w18x ZEUS Collaboration, M. Derrick et al., Phys. Lett. B 377 Ž1996. 259. w19x E. Gotsman, E.M. Levin, U. Maor, Phys. Lett. B 425 Ž1998. 369; E. Gotsman, E.M. Levin, U. Maor, E. Naftali, Nucl. Phys. B 539 Ž1999. 535. w20x A.D. Martin, R.G. Roberts, W.J. Stirling, R.S. Thorne, Eur. Phys. J. C 4 Ž1998. 463. w21x M. Gluck, E. Reya, A. Vogt, Eur. Phys. J. C 5 Ž1998. 461. w22x A. Levy, Phys. Lett. B 424 Ž1998. 191. w23x H1 Collaboration, S. Aid et al., Nucl. Phys. B 472 Ž1996. 3. w24x ZEUS Collaboration, M. Derrick et al., Z. Phys. C 75 Ž1997. 215.
Physics Letters B 470 Ž1999. 200–208
An investigation of a possible hard contribution to f photoproduction Erasmo Ferreira a
a,1
, Uri Maor
a,b,2
Instituto de Fisica, UniÕersidade Federal do Rio de Janeiro, Rio de Janeiro RJ21945-970, Brazil b School of Physics and Astronomy, Tel AÕiÕ UniÕersity, Ramat AÕiÕ 69978, Israel Received 29 April 1999; received in revised form 3 September 1999; accepted 29 October 1999 Editor: R. Gatto
Abstract We investigate the possibility that the process of f photoproduction may have a significant hard perturbative QCD component. This suggestion is based on a study of the energy dependence of the forward f photoproduction cross section followed by a calculation where we show that a coherent sum of the pQCD and conventional soft Pomeron contributions provides an excellent reproduction of the experimental data. Our results suggest that the transition from the predominantly soft photoproduction of light r and v vector mesons to the predominantly hard photoproduction of heavy JrC and F is smooth and gradual, similar to the transition observed in deep inelastic scattering studies of the proton structure function in the small x limit. Our predictions for higher HERA energies are presented. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 12.38.-t; 12.38.Bx; 12.38.Lg; 11.55.Jy; 13.60.-r; 13.60.Le
1. Introduction Over the past few years we have seen a vigorous phenomenological investigation of the Pomeron through the study of hadronic total, elastic and diffractive cross sections, as well as the study of the proton deep inelastic scattering ŽDIS. structure function. In particular, Donnachie and Landshoff ŽDL. have promoted w1x an appealing and very simple
Regge parameterization of the total hadronic cross sections in which
stot s X
s
D
s
ž / ž / s0
qY
s0
yh
.
Ž 1.
The two key ingredients of this approach are the Regge trajectories
a R Ž t . s a R Ž 0 . q a RX t ,
Ž 2.
where a R Ž0. s 1 y h and the Pomeron trajectory, which dominates at high energies, 1 2
E-mail: [email protected] E-mail: [email protected]
a P Ž t . s a P Ž 0 . q a PX t ,
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 1 2 8 7 - 3
Ž 3.
E. Ferreira, U. Maor r Physics Letters B 470 (1999) 200–208
where a P Ž0. s 1 q D. The DL study establishes universal values D s 0.0808 and h s 0.4525. This study is supplemented by the analysis of Block, Kang and White w2x who determine the slope a PX of the Pomeron trajectory to be a PX , 0.2 y 0.3 GeVy2 . The same approach is also applicable to the analyses of real photoproduction and of the proton DIS structure function w3x. While the energy dependence of the photoproduction total cross section follows the DL pattern, it has been observed w4x that F2 Ž x,Q 2 . behaves, for small enough x, like xyl , where l is slowly growing with Q 2 . This growth is given by
ls
E ln Ž F2 Ž x ,Q 2 . . E ln Ž 1rx .
.
Ž 4.
It has been recognized for quite some time that the transition from the predominantly soft real photoproduction Ž Q 2 s 0. to the predominantly hard DIS processes, with high enough Q 2 , is smooth and gradual w3,5–7x. This observation, regardless of its theoretical interpretation, is evident once we examine the energy dependence of F2 Ž x,Q 2 . in the small x limit with Q 2 ranging from zero to a few GeV 2 . It is also well known that real photoproduction of light vector mesons, r and v , is dominated by a soft Pomeron exchange w8,9x, whereas photoproduction of heavy vectors, JrC and F , is well reproduced by a perturbative QCD ŽpQCD. calculation w10–14x, where M V2r4 replaces Q 2 as a measure of the process hardness. Although M V2r4 is a discrete variable, while Q 2 is a continuous DIS variable, it is interesting to check if the transition from soft to hard photoproduction of vector mesons follows the behavior pattern observed in DIS. To this end, the study of f photoproduction is particularly instructive, as Mf2r4 s 0.26 GeV 2 , and we know that the energy dependence of F2 , with small x and Q 2 as low as 0.2–0.3 GeV 2 , is somewhat steeper than the energy dependence of stot Žg p .. Our investigation is susceptible to both experimental and theoretical uncertainties. Experimentally, a systematic study of the integrated f photoproduction cross section and the forward differential cross section slope w15–18x are not very reliable, as different experimental groups have utilized different, not
201
always mutually consistent, methods to extract and relate these quantities. To overcome this difficulty, we have analyzed the measured differential cross sections rather than integrated quantities. Even so, the two higher energy data points w17,18x are averaged over wide energy bins. This, combined with the overall poor quality of the reported data, may make a detailed analysis non conclusive at this stage. Theoretically, our calculation is done at the boundary of pQCD applicability and as such should be considered as a rough estimate. Technically, a pQCD calculation of w d s Žg p f p .rdt x ts0 depends on our knowledge of the gluon structure function at Q 2 s 0.26 GeV 2 . Such information requires an extrapolation of a given parton distribution below its initial evolution threshold Q 02 . For this purpose we adopt a linear extrapolation which was successfully utilized in previous calculations w6,19x. As we shall see, there is a significant difference between the MRST w20x and GRV98 w21x input gluon distributions. We have chosen to use the GRV98 distribution and shall explain our motivations for doing so. The purpose of this Letter is to examine these issues in some detail from different points of view. We present an analysis of the existing f photoproduction forward differential cross section data which suggests an energy dependence which is steeper than the typical energy dependence associated with the soft Pomeron w1x. We then present a pQCD calculation from which we deduce that the hard component is responsible for about a half of the f photoproduction amplitude in the forward direction at presently available energies. We then proceed to show that a coherent sum of the calculated pQCD amplitude and a conventional soft Pomeron exchange contribution provides an excellent reproduction of the available data w15–18x.
™
2. Data analysis Our data analysis investigates whether the f photoproduction cross section follows a power dependence on the c.m. energy W, and whether this power is larger than the value determined from the energy dependence of the g p total cross section. Following
202
E. Ferreira, U. Maor r Physics Letters B 470 (1999) 200–208
DL w1x and Block et al. w2x, we expect the f photoproduction cross section to behave like W 4 D and the forward slope to behave like 4a PX lnW. The analysis of f photoproduction data is seemingly easy, as this process proceeds exclusively through Pomeron exchange, since the various Regge exchanges cancel each other. The problem is that the published analyses w1,18x depend on a comparison between integrated cross section data taken by different groups who have utilized different, and not always mutually consistent, procedures. In addition, because the f forward slope is shrinking, the interpretation of the integrated cross section behaving as a fixed power of W is somewhat ambiguous. In order to bypass these difficulties, we have limited ourselves to the analysis of the individual differential cross sections d srdt as reported by the experimental groups w15–18x. We have used data with W ) 6 GeV corresponding to x - 0.025.
Fig. 1 shows that Ž d srdt . 0 in the available energy range is, indeed, well fitted by an effective power of W,
ž
ds Ž g p
™f p.
dt
/
sA ts0
2
W
ž / W0
4l
,
Ž 5.
where W0 s 1 GeV. Our best fit, for 5 data points with 6.7 F W F 70 GeV, has a x 2rn.d. f.s 0.22, corresponding to 3 degrees of freedom, where the fitted parameters are A2 s 0.76 " 0.18 m brGeV 2 and l s 0.135 " 0.024. For comparison we show also a fit Žpresented by a dashed line. with 4 degrees of freedom where we fix the power to its DL value 4 D s 0.3232 and obtain A2DL s 1.21 " 0.12 m br GeV 2 with x 2rn.d. f.s 0.92, which cannot be discredited. Our inability to determine the power unambiguously results from the big error coupled to the ZEUS high energy data point w18x. It is clear that an
Fig. 1. Our best fit of Eq. Ž5. Ždenoted FM. compared with DL prediction for w d s Žg p
™ f p.rdt x
ts 0 .
All are given in m brGeV 2 .
E. Ferreira, U. Maor r Physics Letters B 470 (1999) 200–208
improvement of the HERA data point at ²W : s 70 GeV and additional relevant data at higher energies will enable us to conclusively distinguish between a DL type interpretation and ours. In order to further examine the suggestion that the dependence of f photoproduction on W is steeper than the behavior implied by a soft Pomeron exchange, we have studied the ratio Rs
Ž d srdt . 0 , sP2 Ž g p .
Ž 6.
where sP Žg p . is the soft Pomeron exchange contribution to stot Žg p .. The ratio is shown in Fig. 2. Using the DL parameterization w1x we have
sP Ž g p . s 67.7
W
ž / W0
203
shown in Fig. 2 indicates the onsetting of a hard component. The best fit that we have obtained implies that R behaves as W 0.215" 0.040 . A constant ratio provides an acceptable fit which implies that at this stage this option cannot be definitely excluded. To summarize: The available experimental data on Ž d s Žg p f p .rdt . 0 is consistently well described as having an energy dependence steeper than the one implied by a soft Pomeron exchange. Nevertheless, the assumption of a pure soft production mechanism cannot be unambiguously eliminated. Note, that our results suggest that f photoproduction is somewhat harder than F2 at Q 2 s 0.26 GeV 2 .
™
0.1616
mb .
Ž 7.
predominantly soft production mechanism for g p f p means that the ratio presented in Fig. 2 ™Ashould be a constant. The suggested increase with W
3. A pQCD calculation Our pQCD calculation of the forward f photoproduction follows earlier pQCD calculations of the forward photoproduction cross section of heavy vec-
Fig. 2. R data and our best fit. The ratio is given in w m b GeV 2 xy1 .
E. Ferreira, U. Maor r Physics Letters B 470 (1999) 200–208
204
tor mesons w10–14x. These calculations are considerably simplified once we assume a non-relativistic wave function for those vector meson states. This assumption, which is also valid for f , enables us to write a leading-order expression ds Ž g p
™f p.
dt s
ts0
a S2 Gefe 3 a E M Mf5
16p
3
ž
xG x ,
Mf2 4
2
/
,
Ž 8.
™
where x s Ž MfrW . 2 and Gefe is the partial decay width of f eqey. The above LO pQCD calculation is done at a very small scale of 0.26 GeV 2 . Even though we shall include an error estimate due to higher order contributions, our calculation is only approximate. Nevertheless, in our opinion, an estimate of a possible hard pQCD contribution to f photoproduction is of interest and may become necessary once the ZEUS data w18x are improved. In the following we follow Refs. w12–14x and, after calculating the cross section resulting from the imaginary forward amplitude in leading order, we correct for the real part of the amplitude. The energy dependence of Ž d s Ž g p f p . rdt . 0 is determined by the x dependence of xGŽ x,Q 2 .. Assuming that xGŽ x,Q 2 . behaves, in the small x limit, as xyl , we have
™
ls
E ln Ž xG Ž x ,Q 2 . . E ln Ž 1rx .
.
Ž 9.
Our suggested calculation depends on the knowledge of the gluon structure function xGŽ x,Q 2 . at Q 2 s Mf2r4 s 0.26 GeV 2 . This Q 2 value is well below Q02 , the minimum value of Q 2 above which we can use the updated parton distribution parameterizations w20,21x. We recall the general property of the gluon structure function which is linear in Q 2 in the limit of very small virtuality and use, accordingly, a linear extrapolation xG Ž x ,Q 2 . s
Q2 Q 02
xG Ž x ,Q02 . , Q 2 - Q 02 .
Ž 10 .
This approximation has been successfully used in previous theoretical DIS investigations w6,19x in which the knowledge of xG in the small Q 2 region was required.
The results of our gluon structure function extrapolation for MRST w20x and GRV98 w21x, at the relevant Q 2 s 0.26 GeV 2 value, are shown in Fig. 3. Note that Q02 s 1.2 GeV 2 for MRST and 0.8 GeV 2 for GRV98. Fig. 3 also shows the Pomeron term of ALLM97 w3x. In our calculations we have used the GRV98 extrapolated distribution. Following is our motivation: Ž1. Both MRST and GRV98 start their evolution with non realistic input gluon distributions which have valence like x dependences. MRST starts its evolution from m2 s 1.0 GeV 2 while GRV98 starts from m2 s 0.4 GeV 2 . The non valence small x structure of xGŽ x,Q 2 ., typical at higher Q 2 values, is driven by the Q 2 evolution. However, we note that this change is faster in the GRV98 parametrization. As a result we consider GRV98 to provide a more realistic parameterization of xG in the Ž x,Q 2 . limit of interest. Indeed, both the normalization and x dependence of GRV98 provide an estimate of the hard component which is compatible with the data analysis described before. On the other hand, the xG obtained from MRST is too small and too soft to be considered as a candidate for the hard component of interest. Note that the above difference between MRST and GRV98 slowly diminishes as Q 2 is increased. Ž2. We note that in the kinematic domain relevant to this investigation, the GRV98 x dependence of xG in the small x limit, given by Eq. Ž9., is somewhat steeper than the corresponding x dependence of F2 at Q 2 s 0.26 GeV 2 , given by Eq. Ž4.. The difference is accounted for by the softer contributions to F2 which do not contribute to the phi channel. The above is compatible with the conclusions of Section 2. Ž3. From a different point of view, we note that MRST is very close to ALLM97 in the x region of interest. This re-enforces our interpretation that the MRST input at small Q 2 is predominantly soft, such as is ALLM97, and thus, less suitable for our analysis where we attempt a calculation of the hard component on its own. On the other hand, GRV98 provides an xG distribution which is harder than ALLM and as such seems suitable to describe the hard distribution with a low risk of a soft contamination leading to double counting. We follow Ref. w12,14x and consider the following corrections to the leading order cross section
E. Ferreira, U. Maor r Physics Letters B 470 (1999) 200–208
205
Fig. 3. ALLM97 and the extrapolated MRST and GRV98 parameterizations for the gluon structure function xGŽ x,Q 2 . at Q 2 s 0.26 GeV 2 .
written in Eq. Ž8.: Ž1. Eq. Ž8. corresponds to the imaginary part of the forward amplitude and should be corrected by Ž1 q r 2 ., where r s Re FrIm F, and F is the amplitude under consideration. We recall
that r , plr2, where, in our case, l is given by Eq. Ž9.. We note that l s 0.145 for W s 70 GeV and decreases very slowly as the energy decreases. Ž2. The next to leading order corrections are estimated
Table 1 Ž d srdt . ts 0 data and calculations. FM refers to the hybrid model calculation. W is given in GeV and the differential cross section in m brGeV 2 exp
FM
soft
pQCD
W
Ž ddts . 0
Ž ddts . 0
Ž ddts . 0
Ž ddts . 0
6.7 7.3 8.0 13.7 30 50 70 100 150 200 250
2.01 " 0.19 2.48 " 0.51 2.39 " 0.22 3.11 " 0.33
2.16 2.19 2.31 3.03 4.30 5.56 6.04 6.88 7.89 8.67 9.24
0.55 0.57 0.58 0.70 0.90 1.06 1.18 1.32 1.51 1.66 1.78
0.52 0.53 0.57 0.82 1.27 1.77 1.88 2.17 2.50 2.75 2.91
7.20 " 2.10
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E. Ferreira, U. Maor r Physics Letters B 470 (1999) 200–208
Fig. 4. Data and our hybrid calculation Ždenoted FM. for Ž d srdt . ts 0 given in m brGeV 2 . Also shown are our separate calculations of the soft and hard cross sections.
by w1 q 0.5 a S Ž Mf2r4.x. For a S , small enough, this is a reasonably realistic estimate. In our case, where a S is relatively large, the NLO corrections amount to 35–40%. However, it is probable that this big correction will be reduced by even higher orders which cannot be neglected. We, thus, consider the NLO estimate as a measure of our possible error in the above LO calculation which is, obviously, a rough estimate due to the high value of a S . Ž3. Relativistic corrections and the effects of intermediate off diagonal partons in f photoproduction are rather small and have been neglected. The results of our calculated hard cross section are given in Table 1 and displayed in Fig. 4.
4. A hybrid pQCD and soft Pomeron model Although the pQCD term calculated in the previous section provides a significant contribution to f
photoproduction, there is no doubt that, in the energy range under consideration, the production mechanism initiated by soft Pomeron exchange is of major importance. Accordingly, we attempt to fit the data with a simple hybrid two- component model with the following prescriptions: Ž1. The first component is a soft DL Pomeron with an a P Ž t . intercept of 1.0808, namely, ds
ž / dt
soft
s A2S ts0
W
ž / W0
0.3232
.
Ž 11 . pQCD
Ž2. A hard pQCD component Ž ddts . ts0 , as calculated in the previous section. Ž3. A coherent sum of the above two components specified as follows: ds
ž / dt
2
2
s Ž ReF . q Ž Im F . , 0
Ž 12 .
E. Ferreira, U. Maor r Physics Letters B 470 (1999) 200–208
where F s Fsoft q FpQCD . Note, that for both the soft and the pQCD components we have Re Fi s r i Im Fi i and ŽIm Fi . 2 s Ž d srdt . 0rrŽ1 q r i2 ., where i s soft, pQCD. We fit the 5 data points of Ž d srdt . 0 with one parameter, the normalization A S of the DL Pomeron. We obtain a best fit Ž4 degrees of freedom. with A2S s 0.30 " 0.08 m b, corresponding to x 2rn.d. f.s 0.34. Our fitted cross sections Žcalled FM. are presented in Table 1, which contains also our predictions for higher HERA energies, and are displayed in Fig. 4. Our fit should be compared with the power fit presented earlier, which has a x 2rn.d. f.s 0.22, and the conventional DL fit which has x 2rn.d. f.s 0.92 Žsee Fig. 1.. Clearly, the presently available data is not sufficient to rule out any of these options. Once again we note that this ambiguity results from the big error associated to the ²W : s 70 GeV point. Improvement of the quality of this point and additional HERA data will enable a more discriminative analysis. The hybrid model that we have just suggested can be further examined by considering the differential cross sections. Such data is available w17,18x at ²W : s 13.7 and 70 GeV. For the purpose of our calculations we need to know the t-dependence of both the pQCD and soft Pomeron amplitudes. To this end we assume each of these dependences to be approximated by an exponential, Bi
t
Fi s Fi Ž 0 . e 2 ,
i s soft, pQCD .
Ž 13 .
This simple approximation is sufficient here, considering the quality of the available data on f photoproduction. We then proceed as follows: Ž1. We take for the pQCD amplitude an energy independent exponential slope. We derive its value from the high energy differential cross section on JrC photoproduction combined with the observation w22x that this slope is energy independent, corresponding to a flat hard Pomeron. In our analysis we have taken BpQCD s 4.6 GeVy2 , which corresponds to the H1 measurement w23x. A good fit is obtained, also, with the ZEUS value w24x of BpQCD s 4.0 GeVy2 . We have also treated BpQCD as a free parameter without a significant change. Ž2. For the soft Pomeron ampli-
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Fig. 5. Data and our calculations for Ž d s r dt ., given in m brGeV 2 , at ²W : s13.7 and 70 GeV.
tude we assume a conventional Regge type exponential slope depending on two parameters, Bsoft s B0S q 2 a PX ln
W
ž / W0
2
.
Ž 14 .
B0S is a free parameter and we fix a PX s 0.25 based on high energy hadron-hadron phenomenology w2x. This hybrid model was fitted to reproduce 13 Ž d srdt . data points measured at ²W : s 13.7 and 70 GeV with one parameter, for which we obtain B0S s 6.00 " 0.92 GeVy2 with x 2rn.d. f.s 0.65 for 12 degrees of freedom. The data and our fit are shown in Fig. 5. Although our Ž d srdt . fit corroborates our proposition that high energy f real photoproduction has a significant hard component, we do not consider our success to be decisive. Indeed, we ran a fit in which only the soft DL component contributes to Ž d srdt .. In this fit A2DL s 1.21 is taken from Section 2 so that the only fitted parameter is B0S. We obtain B0S s 2.98 " 0.14 GeVy2 with x 2rn.d. f.s 1.29 for 12 degrees of freedom. This fit is of a lesser quality than the one obtained for the hybrid model, but we caution against reaching too strong a conclusion based on limited data. In general, we note that our hybrid model provides a natural explanation for the higher t curvature suggested by the data w17x. Clearly, and not only for the purpose of our analysis, additional knowledge on higher t behavior will help to clarify the picture. 5. Discussion Following are the main conclusions of our study and some general remarks: Ž1. The data analysis of
E. Ferreira, U. Maor r Physics Letters B 470 (1999) 200–208
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forward real f photoproduction suggests the possible existence of a significant hard pQCD contribution in the energy range of 6 F W F 70 GeV. Ž2. The above suggestion is corroborated by a LO pQCD f p .rdt x 0 using the calculation of w d s Ž g p GRV98 gluon distribution extrapolated to Q 2 s 0.26 GeV 2 . Ž3. A hybrid model, in which we coherently add the pQCD hard component and a DL type soft component, provides an excellent overall reproduction of the data. We note that the ratio of hard to soft contribution in our model is compatible with the ratio obtained in a detailed study w6x of DIS in comparable small Q 2 values. Ž4. Our hybrid model is significantly different from the two Pomeron model suggested in Ref. w7x. In our model the t s 0 intercept of the hard effective Pomeron is deduced from Eq. Ž9. and, as such, it is a moving pole. In the model of Ref. w7x, the hard Pomeron is a fixed pole with a comparatively high l. Ž5. Theoretically, the validity of our calculation rests on: Ži. The validity of a pQCD calculation done at small Q 2 . Žii. The legitimacy of our Q 2 extrapolation of xGŽ x,Q 2 . below Q 02 . Žiii. Our choice of GRV98 for the input gluon structure function. Ž6. Our overall analysis suggests the existence of a significant hard component contributing to g p f p. However, a decisive quantitative conclusion depends on improving and extending the HERA data on f photoproduction.
™
™
Acknowledgements We thank Eran Naftali for his help in the numerical computations. U.M. wishes to thank UFRJ and FAPERJ ŽBrazil. for their support. This research was supported in part by the Israel Academy of Science and Humanities.
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