QCD fits in diffractive DIS revisited
aa r X i v : . [ h e p - ph ] O c t QCD fits in diffractive DIS revisited
F. A. Ceccopieri , L. Favart IFPA, Universit´e de Li`ege, All´ee du 6 aoˆut, Bˆat B5a, 4000 Li`ege, Belgium. Universit´e Libre de Bruxelles, Boulevard du Triomphe, 1050 Bruxelles, Belgium.
Abstract.
A new method of extracting diffractive parton distributions is presented whichavoids the use of Regge theory ansatz and is in much closer relation with the factorizationtheorem for diffarctive hard processes.
1. Introduction
Diffractive parton distributions functions (DPDF) are essential ingredients in the understandingand description of hard diffractive processes. The factorization theorem for diffractive processin Deep Inelastic Scattering (DIS) [1, 2] enables one to factorize the diffractive DIS cross-sections into the long-distance contribution parametrised by DPDF’s, from the short-distance,pertubatively calculable, one. Although DPDF’s encode non-perturbative effects of QCDdynamics, their variation with respect to the factorization scale is predicted by pQCD [3, 4].Moreover the short distance cross-sections is the same as inclusive DIS [1] so that higher ordercorrections can be systematically evaluated. Due to the factorization theorem, DPDF’s arepredicted to be universal distributions in the context of diffractive DIS. Next-to-leading orderpredictions based on DPDF’s have been found to well describe DIS diffractive dijet cross-sections [5, 6], thus confirming factorization. On the contrary, the issue of factorization is stilldebated in diffractive photoproduction of dijets since the H1 [7] and ZEUS [8] collaborations doreport conflicting results.DPDF’s give the joint probability for a parton in a proton of four-momentum P to initiate a hardscattering keeping the proton intact with a four-momentum P ′ in the final state. Diffractiveparton distributions are a special case of extended fracture functions [4] in the kinematic range x IP ≤ − and | t | ≤ , where x IP is the fractional energy loss of the final state protonand t the invariant momentum transfer, t = ( P − P ′ ) . The presence of large and positive scaleviolations up to the largest parton fractional momentum, β , accessible to experiments revealsthat DPDF’s are gluon dominated distributions. Diffractive gluon distributions are howeverhardly determined by the diffractive structure functions alone and this fact has stimulatedexperimental collaborations to measure quantity that are directly sensitive to it [5, 6, 9] and toinclude such data in global fits.The general method used to extract DPDF’s from available data heavily relies on a number ofassumptions motivated by Regge phenomenology. The purpose of this short note is to presenta new method which does not require any Regge assumptions and is much closer in spirit to thefactorization theorem. . Data set and observable For this analysis we will use H1 [10] data in which DIS diffractive events, ep → eXY , are selectedby requiring a large rapidity gap between the hadronic system X and the low mass dissociativesystem Y (which may consist in the scattered p only). The kinematics of diffractive events isspecified by the following variables: β = Q Q + M X ; x IP = x B β ; y = Q sx B , (1)where Q is the photon virtuality, M X is the invariant mass of the hadronic final state X , x B is the usual x -Bjorken DIS variable and √ s = 318 GeV is the HERA center of mass energy.The measurament is integrated over the hadronic final state Y mass region M Y < . | t | < , which then define the extracted DPDF’s [11].The kinematical coverage is wide ranging from 3 . ≤ Q ≤ , 3 · − ≤ x IP ≤ · − and 10 − ≤ β ≤ .
8. We notice however that the { β, Q } coverage changes with x IP due tokinematic constraints and this fact will reflect on DPDF’s extraction. Data are presented as athree-fold reduced e + p cross section which depends on the diffractive structure functions F D (3)2 and F D (3) L . In the one-photon exchange approximation, it reads: σ D (3) r ( β, Q , x IP ) = F D (3)2 ( β, Q , x IP ) − y − y ) F D (3) L ( β, Q , x IP ) . (2)
3. The new method
The widely used approach [5, 10, 12, 13] to extract DPDF’s is to assume proton vertexfactorization, i.e. that DPDF’s can be factorized into a flux factor depending only on x IP and t and a term depending only on β and Q : F Di ( β, Q , x IP , t ) = f IP/P ( x IP , t ) F IPi ( β, Q ) + f IR/P ( x IP , t ) F IRi ( β, Q ) + ... (3)Each term in the expansion, according to Regge theory, is supposed to give a dominantcontribution in a given range of x IP , the pomeron ( IP ) at low x IP and the reggeon ( IR ) athigher value of x IP . The flux factor f IP/P ( f IR/P ) can be interpreted as the probability that apomeron (reggeon) with a given value of x IP and t couples to the proton. The fluxes x IP and t dependences used in the fits are motivated by Regge theory. This approach requires that partondistributions of the pomeron, F IPi , and of the reggeon, F IRi , must be simultaneously extractedfrom data. This procedure introduces a large number of parameters in the fit and it is potentiallybiased by the choices of the flux factors. A common choice used in phenomenological applicationis to fix F IRi to be equal to pion parton distribution functions. Although such an approach hasbeen proven to be supported by phenomenological analyses within the precision obtained withHERA-I data, it is not routed in perturbative QCD and may show up to be not satisfactorywith the expected precision increase of HERA-II data and H1+ZEUS complete combination.The alternative method we propose is instead inspired by the factorization theorem [1] fordiffractive DIS itself. The latter states that factorization holds at fixed values of x IP and t sothat the parton content described by F Di is uniquely fixed by the kinematics of the outgoingproton and it is in principle different for different values of x IP . In practice this idea is realizedperforming a series of separate pQCD fits at fixed values of x IP with a common initial conditioncontrolled by a set of parameters { p i } . This procedure guide us to infere the approximatedependence of parameters { p i } on x IP allowing the construction of initial condition in the { β, x IP } space to be used in a global fit, without any further model dependent assumption. able 1. Fit results at fixed xpom. Fits are performed taking into account only statisticalerrors only. Only data for which M X ≥ and Q ≥ . are included in the fits. x IP χ / d.o.f0.001 0.9210.003 0.8750.01 0.8820.03 0.472
4. Fit procedure and results
In each x IP -bin for which data are presented by the experimental collaboration we perform aseparate pQCD fits. For the singlet and gluon distributions at the arbitrary scale Q we choose: β Σ( β, Q ) = A q β B q (1 − β ) C q e − . − z ,β g ( β, Q ) = A g e − . − z , (4)so that there are four free parameters. We make the common assumption that all lights quarkdistributions are equal to each other and the exponential dumping exponential factor allowsmore freedom in the variation of the parameters C q at large β . The functional form in eq. (4)is identical to the one used by H1 collaboration in Ref. [10]. Since diffractive DIS data hardlydiscriminate [10] between different behaviour of gluon distributions at large β we choose thesimpler one in which the gluon is a costant at Q . Such distributions are then evolved withthe QCDNUM17 [14] program within a fixed flavour number scheme to next-to-leading orderaccuracy. Heavy flavours contributions are taken into account in the general massive scheme.The convolution engine of
QCDNUM17 is used to obtain F D (3)2 and F D (3) L structure functions atnext-to-leading order which are then minimized versus data. In order to avoid the resonanceregion, a cut on the invariant mass of the hadronic system X is applied, M X ≥ . Single x IP -fit results are sensitive to the choice of the mininum value Q of data to be included inthe fits. The inclusion of data with Q < . in general worsen the χ and induce largefluctuation in the gluon distribution. This instability has been already noticed in Ref. [10] andavoided by including in the fit only data for which Q ≥ . . We will adopt here the samestrategy.An essential condition for the procedure to work is that good quality fits are all obtained withthe common initial condition, eq. (4). From fits results presented in Table 1 we conclude thatthe initial condition provided by eq. (4) is general enough to describe data in all x IP -bins. Thedependence of the parameters on x IP is reported in Figures from 1 to 4. Red dots are the resultsfrom pQCD fits at fixed x IP . The singlet normalization A q behaves as an inverse power of x IP .In order to improve the description at higher x IP , however, an additional term is also included: A q ( x IP ) = A q, ( x IP ) A q, (1 − x IP ) A q, . (5)The gluon normalization is compatible with a single inverse power behaviour of the type: A g ( x IP ) = A g, ( x IP ) A g, . (6) IP A q Figure 1. A q as a function of x IP . x IP A g Figure 2. A g as a function of x IP . x IP B q Figure 3. B q as a function of x IP . x IP C q Figure 4. C q as a function of x IP .The coefficients B q and C q which control the β -shape of the singlet distribution are well describedby: B q ( x IP ) = B q, + B q, x IP , (7) C q ( x IP ) = C q, + C q, x IP . (8)In order to facilitate the comparison, curves resulting from fits with functional form specifiedin eqs. (6-9) are superimposed to points in Figures 1-4. Since the approximate behaviour ofthe parameters against x IP is now known, we can perform a x IP -bin combined fit, using thegeneralized initial condition β Σ( β, Q , x IP ) = A q ( x IP ) β B q ( x IP ) (1 − β ) C q ( x IP ) e − . − z , (9) β g ( β, Q , x IP ) = A g ( x IP ) e − . − z , (10)with the parameters depending on x IP as specified in eqs. (6-9). The combined fit has nine freeparameters. Following the procedure described in Ref. [15], to each systematic errors quotedin the experimental paper is assigned a free systematic parameters which is then minimized inthe fit along with parameters. As for the single- x IP fits, only data for which M X ≥ and Q ≥ . are included in the fit. The latter has a appreciable sensitivity on the scale Q due to the relative stiffness of the initial condition in eq. (4). The choice of Q is then optimizedperforming a scan which gives the best χ value for Q = 2 . . The partial results for the χ function in the various x IP bin are reported in Table 2. The best fit returns a χ = 166 for 182degrees of freedom which is of comparable quality as the one presented in Ref. [10]. The initialcondition, eq. (9), allows the singlet and gluon normalization, A q and A g respectively, to have adifferent power behaviour. It is therefore interesting to notice that if the condition A q, = A g, is imposed, this result in a global χ = 171 for 183 degree of freedom. If one further neglects able 2. Combined x IP fit result. x IP χ fitted totalpoints points0.0003 3.1 3 120.0010 26.2 19 400.0030 38.9 39 590.0100 58.6 61 800.0300 36.4 68 85the x IP -dependence of B q and C q by setting B q, = C q, = 0 the χ increases to 188 units for185 degree of freedom. This is an a posteriori confirmation that not only diffractive partondistributions change their magnitude versus x IP but also that a modulation in their β -shape (forthe singlet, in this case) is necessary to better fit the data.
5. Conclusions
We have outlined a new method to extract diffractive parton distributions inspired by thefactorization theorem for diffractive DIS. From a series of pQCD fits at fixed x IP we infere thedependence of parameters on such a variable and this allows us to construct a generalized initialcondition without assuming neither proton vertex factorization nor the existence of a series ofRegge trajectories. Although the quality of the resulting fit gives a χ / d.o.f close to unity, asthe Regge based pQCD fit [10], the new procedure treats the non-perturbative x IP -dependence ofthe cross-sections in a controlled and less model dependent way. This feature will allow furtherfactorization test to be performed with the improved precision of published [16] and forthcomingdata [17, 18]. References [1] J. C. Collins,
Phys. Rev.
D57 (1998) 3051.[2] M. Grazzini, L. Trentadue, G. Veneziano,
Nucl. Phys.
B519 (1998) 394.[3] L. Trentadue, G. Veneziano,
Phys. Lett.
B323 (1994) 201.[4] G. Camici, M. Grazzini, L. Trentadue,
Phys. Lett.
B439 (1998) 382.[5] H1 Collaboration (A. Aktas et al.)
JHEP (2007) 0710:042.[6] ZEUS Collaboration (S. Chekanov et al.)
Eur. Phys. J.
C52 (2007)813.[7] H1 Collaboration (F.D. Aaron et al.)
Eur. Phys. J.
C70 (2010) 15.[8] ZEUS Collaboration (S. Chekanov et al.)
Eur. Phys. J.
C55 (2008) 177.[9] H1 Collaboration (F.D. Aaron et al.) e-Print: arXiv :1107.3420.[10] H1 Collaboration (A. Aktas et al.)
Eur. Phys. J.
C48 (2006) 715.[11] F. A. Ceccopieri, L. Trentadue,
Phys. Lett.
B655 (2007) 15.[12] ZEUS Collaboration (S. Chekanov et al.)
Nucl. Phys.
B831 (2010) 1.[13] S. Taheri Monfared, Ali N. Khorramian, S. Atashbar Tehrani, e-Print: arXiv :1109.0912.[14] M. Botje,
Comput. Phys. Commun. (2011) 490.[15] C. Pascaud and F. Zomer, LAL-95-05.[16] H1 Collaboration (F.D. Aaron et al.)
Eur. Phys. J.
C71 (2011) 1578.[17] H1 Coll., [H1prelim-10-011].[18] H1 Coll., [H1prelim-10-014], T. Hreus,