Small-Q2 extension of DGLAP-constrained Regge residues 4 0 0 G. Soyez∗ 2 n February 1, 2008 a J 3 2 Abstract 1 v In a previous paper, we have shown that it was possible to use the 7 DGLAP evolution equation to constrain the high-Q2 (Q2 10 GeV2) 7 behaviouroftheresiduesofahigh-energyReggemodel,and≥weapplied 1 thedevelopedmethodtothetriple-polepomeronmodel. Weshowhere 1 that one can obtain a description of the low-Q2 γ(∗)p data matching 0 4 the high-Q2 results at Q2 =10 GeV2. 0 / We know that one can use Regge theory [1] to describe high-energy h p hadronic interactions. Particularly, using a triple-pole pomeron model [2, - 3, 4], one can reproduce the hadronic total cross-sections, the γp and γγ p p e cross-sections, and also the proton and photon structure functions F and 2 h γ F . In the latter case, one must point out that Regge theory is applied at : 2 v all values of Q2. Xi On the other hand, it is well known that the high-Q2 behaviour of the r proton structure function can be reproduced using the DGLAP evolution a equation [5]. Therefore, we would like to find a model compatible both with Regge theory and with DGLAP evolution at high Q2. We have shown [6] that it is possible to extract the behaviour of the triple-pole pomeron residues at high Q2 from DGLAP evolution. In such an analysis, we need information not only on F but also on parton distributions. One easily 2 shows that the minimal number of quark distributions needed to reproduce p F is 2: one flavour-non-singlet distribution 2 T(x,Q2) = x (u++c+ +t+) (d+ +s++b+) , − ∗e-mail: [email protected] (cid:2) (cid:3) 1 2 with q+ = q + q¯, evolving alone with xP as splitting function, and one qq flavour-singlet distribution Σ(x,Q2) = x (u++c+ +t+)+(d+ +s++b+) , (cid:2) (cid:3) coupled with the gluon distribution xg(x,Q2) and evolving with the full splitting matrix. Before going into the main subject of this paper, we shall summarisethetechniques developedinthispreviourpaper[6]andshowhow we can extend the results down to Q2 = 0. First of all, given that F can be parametrised at small x by a log2(1/x) 2 term, we have parametrised the quark content of the proton in the most natural way i.e. using a triple-pole pomeron term and an f/a reggeon 2 terms. After a few manipulations, we end up with the following functions T(x,Q2) = d∗xη(1 x)b2, 0 T − Σ(x,Q2) = a log2(1/x)+b log(1/x)+c∗(1 x)b1 0 Σ Σ Σ − + d xη(1 x)b2 Σ − xg(x,Q2) = a log2(1/x)+b log(1/x)+c∗(1 x)b1. (1) 0 G G G − SinceReggetheorydoesnotextenduptox= 1,weusedtheGRVparametri- sation for x x = 0.15 and imposed that our distributions match Regge ≥ GRV’s at x = x . This requirement constrains the parameters marked Regge ∗ with a superscript in eq. (1). Thus, the 7 parameters a , b , d , a , b , Σ Σ Σ G G b and b need to be extracted from DGLAP evolution. 1 2 SinceDGLAPevolutiongeneratesanessentialsingularityinthecomplex- j plane at j = 1, the only place where we can use the Regge model is in the initial distributions at Q2 = Q2. In such a case, we shall not worry about 0 the presence of an essential singularity for Q2 = Q2 and consider the result 6 0 ofDGLAPevolution asanumericalapproximationtoatriple-polepomeron. One can therefore extract the residues of the Regge model at high Q2 using the following method: 1. choose an initial scale Q2, 0 2. choose a value for the parameters in the initial distribution, 3. compute the parton distributions for Q2 Q2 Q2 using forward 0 ≤ ≤ max DGLAP evolution and for Q2 Q2 Q2 using backward DGLAP min ≤ ≤ 0 evolution, 4. repeat 2 and 3 until the value of the parameters reproducing the F 2 data for Q2 > Q2 and x x is found. min ≤ Regge 3 5. This gives the residues at the scale Q2 and steps 1 to 4 are repeated 0 in order to obtain the residues at all Q2 values. We have applied this method to the parametrisation (1) within the do- main 10 Q2 1000GeV2, cos≤(θt) =≤2√xmQ2p ≥ 492Gme2pV2, (2) ensuringthat both Regge theory and DGLAP evolution can beapplied, and required1 x < 0.15. Using the residues of the triple-pole pomeron obtained in this way, we have a description of Fp for Q2 10 GeV2 with a χ2/nop 2 ≥ of 1.02 for 560 experimental points. Since the method explained here gives us the Regge residues at large scales, one may ask if it is possible to extend the results down to Q2 = 0. The main problem here is that, instead of using x and Q2, we must use ν and Q2 if we want to obtain a relevant expression for the total cross section. p Of course, we shall only extend the F predictions instead of the parton 2 distributions T and Σ. As a starting point, we shall not consider the powers of (1 x) since, − at low Q2, there are no point inside the Regge domain beyond x = 0.003, which means that it is just a correction of a few percents. At low Q2, we require that F has the same form as used in [2] 2 Q2 F (ν,Q2)= A(Q2) log(2ν) B(Q2) 2+C(Q2)+D(Q2)(2ν)−η . 2 4π2α − e n o (cid:2) (cid:3) (3) The total γp cross-section is then σ = A(0)[log(s) B(0)]2+C(0)+D(0)s−η. (4) γp − At Q2 = Q2, the form factors A, B, C and D are related to the parametri- 0 sation (1) by the relations 4π2α A(Q2) = ea , 0 Q2 0 0 b B(Q2) = log(Q2) 0 , 0 0 − 2a 0 (5) 4π2α b2 C(Q2) = e c 0 , 0 Q2 (cid:18) 0− 4a (cid:19) 0 0 4π2α D(Q2) = ed (Q2)η. 0 Q2 0 0 0 1This limit is only effective at large Q2. 4 where the subscript to refer to the form factors obtained at Q2 = Q2 from 0 0 DGLAP evolution. AtsmallQ2,theunknownfunctionsA,B,C andD(Q2)areparametrised in the same way as in [2] Q2 εa A(Q2) = A a , a(cid:18)Q2+Q2(cid:19) a Q2 εb B(Q2) = A +A′, b(cid:18)Q2+Q2(cid:19) b b (6) Q2 εc C(Q2) = A c , c(cid:18)Q2+Q2(cid:19) c Q2 εd D(Q2) = A d . d(cid:18)Q2+Q2(cid:19) d ′ If we use the relations (5) to fix the parameters A , A , A and A in a b c d (6), we find the final form of the small-Q2 form factors: 4π2α Q2+Q2 εa A(Q2) = ea 0 , Q2 0(cid:18)Q2 +Q2(cid:19) 0 a b Q2 εb Q2 εb B(Q2) = log(Q2) 0 +A 0 , 0 − 2a b(cid:20)(cid:18)Q2+Q2(cid:19) −(cid:18)Q2+Q2(cid:19) (cid:21) 0 b b 0 (7) 4π2α b2 Q2+Q2 εc C(Q2) = e c 0 0 , Q2 (cid:18) 0− 4a (cid:19)(cid:18)Q2+Q2(cid:19) 0 0 c 4π2α Q2+Q2 εd D(Q2) = ed (Q2)η 0 . Q2 0 0 (cid:18)Q2+Q2(cid:19) 0 d If we now want to reinsert the large-x corrections, we need to multiply c and d by some power of (1 x). This gives − 4π2α eF (x,Q2) = A(Q2)log(1/x) log(1/x)+2 log(Q2) B(Q2) Q2 2 − (cid:8) (cid:2) (cid:3)(cid:9) + A(Q2) log(Q2) B(Q2) 2+C(Q2) (1 x)b1 − − n o (cid:2)Q2 −η (cid:3) + D(Q2) (1 x)b2. (cid:18) x (cid:19) − These large-x corrections do not modify the expression of the total cross section since, when Q2 0 → 2ν 1 x = 1 1. − − Q2 → 5 Parameter value error A 69.151 0.055 b Q2 25.099 0.088 a Q2 4.943 0.086 b Q2 0.002468 0.000042 c Q2 0.01292 0.00074 d ε 1.5745 0.0046 a ε 0.08370 0.00052 b ε 0.92266 0.00019 c ε 0.3336 0.0029 d Table 1: Values of the parameters for the low-Q2 fit (0 Q2 Q2). ≤ ≤ 0 Experiment n χ2 χ2/n E665 69 59.811 0.867 H1 99 104.924 1.060 NMC 37 28.392 0.767 ZEUS 216 201.790 0.934 p F 421 394.916 0.938 2 σ 30 17.171 0.572 γp Total 451 412.086 0.914 Table 2: χ2 resulting from the small-Q2 Regge fit. The results are given for p all F experiments and for the total cross-section. 2 Moreover, since the large-x corrections are only a few percents effects, we shall keep the exponents b and b constant and equal to their value at 1 2 Q2 = Q2. 0 p Now, we may adjust the parameters in the form factors by fitting F in 2 the Regge domain ν 49GeV2, ≥ cos(θt) = 2√xmQ2p ≥ 492Gme2pV2, (8) Q2 10GeV2,  ≤   together with the total cross-section for √s 7 GeV. The resulting param- ≥ eters are presented in Table 1 and the form factor are plotted in Figure 1. 6 0.03 0.1 0.08 0.025 0.06 0.02 0.04 a b 0.02 0.015 0 0.01 -0.02 -0.04 0.005 -0.06 0 -0.08 0.01 0.1 1 10 0.01 0.1 1 10 Q2 Q2 0.4 2 0.3 1.8 1.6 0.2 1.4 0.1 1.2 d 0 1 -0.1 0.8 0.6 -0.2 0.4 -0.3 0.2 -0.4 0 0.01 0.1 1 10 0.01 0.1 1 10 Q2 Q2 Figure 1: Regge theory predictions for the form factors at small values of Q2. The lines show the analytical curve for 0 Q2 10 GeV2 and the ≤ ≤ points are the results obtained in [6] from DGLAP evolution. As we can see from Table 2 and from Figures 2 and 3, this gives a very good extension in the soft region (see Table 2). To conclude, we have seen that, using a triple-pole-pomeron model, one can obtain a description of the γ(∗)p interactions at all values of Q2 com- patible with the DGLAP equation at large Q2. It should be interesting, in the future, to test this method with other Regge models and to see if the results are compatible with the t-channel unitarity relations obtained in [7] and if they can give useful information on how to link perturbative and non-perturbative QCD. Acknowledgments REFERENCES 7 0.2 0.19 0.18 0.17 p 0.16 (cid:13) (cid:27) 0.15 0.14 0.13 0.12 0.11 10 100 p s Figure 2: Fit for the total γp cross-section. I would like to thanks J.-R. Cudell for useful discussions. This work is supported by the National Fund for Scientific Research (FNRS), Belgium. References [1] The reader who wants a modern overview of Regge theory and diffrac- tion can read the books by S. Donnachie, G. Dosch, P. Landshoff and O. Nachtmann, Pomeron Phy sics and QCD (Cambridge University Press, Cambridge, 2002), and by V. Barone and E. Predazzi , High- Energy Particle Diffraction (Springer, Berlin Heidelberg, 2002). [2] J. R. Cudell and G. Soyez, Phys. Lett. B 516 (2001) 77 [arXiv:hep-ph/0106307]. [3] P. Desgrolard and E. Martynov, Eur. Phys. J. C 22 (2001) 479 [arXiv:hep-ph/0105277]. [4] J. R. Cudell et al. [COMPETE Collaboration], Phys. Rev. Lett. 89 (2002) 201801 [arXiv:hep-ph/0206172]. REFERENCES 8 [5] V.N. Gribov and L.N. Lipatov, Sov. J. Nucl. Phys. 15 (1972) 438. G. Altarelli andG.Parisi, Nucl. Phys.B126(1977) 298.Yu.L.Dokshitzer, Sov. Phys. JETP 46 (1977) 641. [6] G. Soyez, arXiv:hep-ph/0306113. [7] J. R. Cudell, E. Martynov and G. Soyez, arXiv:hep-ph/0207196. REFERENCES 9 10-6 10-4 10-2 10-6 10-4 10-2 10-6 10-4 10-2 0.045 GeV 2 0.065 GeV 2 0.085 GeV 2 0.2 H1 ZEUS E665 0.1 NMC fit 0.0 0.4 0.11 GeV 2 0.15 GeV 2 0.2 GeV 2 0.25 GeV 2 0.4 0.2 0.2 0.0 0.0 0.3 GeV 2 0.4 GeV 2 0.5 GeV 2 0.65 GeV 2 0.6 0.6 0.4 0.4 fi 2 F 0.2 0.2 0.0 0.0 0.8 GeV 2 1.2 GeV 2 1.496 GeV 2 1.5 GeV 2 1.0 1.0 0.5 0.5 0.0 0.0 2.0 GeV 2 2.5 GeV 2 3.5 GeV 2 4.5 GeV 2 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0 5.0 GeV 2 6.5 GeV 2 8.5 GeV 2 10.0 GeV 2 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0 10-6 10-4 10-2 10-6 10-4 10-2 10-6 10-4 10-2 10-6 10-4 10-2 x fi Figure 3: Fit for the Fp at low Q2. Only the most populated Q2 bins are 2 shown.