S Munier

R Peschanski

DAPNIA/SPP Ch Royon

Hard di raction at HERA in the dipole model of BFKL dynamics

Using the QCD dipole picture of the hard BFKL pomeron, we derive the general expressions of the elastic and inelastic components of the proton di ractive structure functions. We obtain a good 7 parameter t (including a secondary reggeon contribution) to data taken at HERA by the H1 and ZEUS collaborations. The main characteristic features of the model in reproducing the data are discussed, namely the e ective pomeron intercept, the scaling violations and the beta dependence. A di erence obtained in the separate H1 and ZEUS ts leads us to analyse directly the di erences between both measurements. Predictions on R, the ratio of longitudinal to transverse photon cross sections are performed and lead to very large values at high beta and large virtuality Q which may lead to a discrimination between models.

Introduction

The HERA data for deep-inelastic scattering at high energy and high Q 2 contain a sizeable fraction of events in which one observes a large rapidity gap in the forward region 1, 2]. These events result from a colour-singlet exchange between the di ractively dissociated virtual photon and the proton. Since this phenomenon is present even for high virtuality of the photon, it is called hard di raction, at variance with soft di ractive phenomena. Several theoretical formulations have been proposed 3]. Among the most popular models, the one based on a pointlike structure of the pomeron 4] has been studied quantitatively using a non-perturbative input supplemented by a DGLAP evolution 1]. In this formulation, it is assumed that the exchanged object, the pomeron, is a colour-singlet quasi-particle whose structure is probed in deep-inelastic scattering.

There exists a di erent approach 5, 6] in which the cross sections are determined by the interaction between colour dipole states 7] describing the photon and the proton. Indeed, it is well-known that the photon can be analyzed in terms of q q con gurations 8] while it has been shown 9] that the small-x structure function of the proton can be described by a collection of primordial dipoles with subsequent perturbative QCD evolution. More speci cally 6], the combination of the dipole description of perturbative QCD at high energy and the Good-Walker mechanism 10] leads to a uni ed description of the proton total and di ractive structure functions 11]. However, a quantitative description of the data following 6] was still lacking. The aim of the present paper is to provide a simple and quantitative formulation tting the experimental data.

In the dipole approach, two components are shown to contribute to the di ractive structure function (see g.1). First, an elastic component ( g.1-a) corresponds to the elastic interaction of two dipole con gurations. It is expected to be dominant in the nite region, i.e. for small relative masses of the di ractive system. Second, there is an inelastic component ( g.1-b) where the initial photon dipole con guration is di ractively dissociated in multi-dipole states by the target. This process is expected to be important at small (large masses). Note that both components are obtained in ref . 6] through perturbative QCD resummation corresponding to the BFKL approach 12]. In these respects, it is based on a \hard pomeron" approach at variance with e.g. a non perturbative pomeron input evolved using DGLAP evolution 1].

We present here ts of the published di ractive data 1, 2] based on a simple with 7 free parameters and analytic parametrization for the di ractive structure functions in the QCD dipole model. Both ts are successful showing that this approach is a good candidate for an understanding of hard di raction. The main input of the calculation is a formulation of the elastic QCD amplitude T (r; ; b; x P ) for the di usion of a dipole of size r on a dipole of size with rapidity gap log 1=x P at xed impact parameter b. This b-dependent input allows one to factorize the di erent integrals involved in the perturbative calculation of the di ractive components. The dipole model for the proton 9] is used to provide predictions for the HERA data. We also include a phenomenological secondary Regge trajectory which is known to play a rôle in the limited domain of large mass and small rapidity gap 1].

The outline of the paper is as follows: in section 2, we introduce the dipole formalism and in particular the input elastic dipole-dipole amplitude T (r; ; b; x P ) following the requirements of the BFKL dynamics. Using this amplitude, we derive the general expressions for the elastic (2.9, 2.10), and inelastic (2.15) di ractive components for both transverse and longitudinal polarizations of the photon. In section 3 we obtain an analytical form of the proton di ractive amplitudes used to t the HERA data. In section 4, the resulting t is displayed. The simple form of the amplitudes makes transparent the discussion of the model properties e.g. the relative contribution of the two components, the e ective exponent for the rapidity gap dependence, the origin of the scaling violations, and the dependence. Conclusions and outlook are given in the nal section 5.

The QCD dipole formalism

As mentionned in the introduction, the main input of our approach to the structure functions is the interaction amplitude T (r; ; b; x P ) for a dipole of size with a dipole of size r, at impact parameter b where log 1=x P is the rapidity gap. The knowledge of the amplitude as a function of b is essential for the calculation of the two components of hard di raction 6]. It is thus compulsory to start with a correct BFKL amplitude in the whole impact parameter space which goes beyond the commonly used expression for the forward elastic amplitude T (forward) = R d 2 b T (r; ; b; x P ). For this sake, one uses the conformal invariance of the BFKL kernel in transverse coordinates 13]. We shall propose such a b-dependent amplitude by asking for the two following requirements:

(i) The integral over impact parameter space boils down to the known exact expression for the forward amplitude T (forward) 12, 13].

(ii) It has the correct high impact parameter approximation 7] including the modi cation by a scale factor which has been deduced from conformal invariance 13].

Using the inverse-Mellin transform which appears in the solution of the BFKL equations 12], the amplitude reads:

T (r; ; b; x P ) = 1 2 2 r 2 b 4 Z d 2i (1 ? 2 )d( ) 16b 2 r ! 2
x ? P ( ) P ;

(2.1)

where the factor 16 is the scale factor determined by conformal invariance.

P ( ) = s N c ( ); ( ) = 2 (1) ? ( ) ? (1 ? ) (2.2)
is the -dependent QCD pomeron intercept 12], and

d( ) = 2 s 16 2 (1 ? ) 2 (2.3)
is the two-gluon exchange elementary dipole-dipole cross section. As already mentioned, the amplitude meets the abovementionned requirements, since:

(i) the integration over b leads to the known expression for the forward amplitude (for r > ):

Z +1 r=4 Z +1 r=4 2 b db T (r; ; b; x P ) = 8 2 Z d 2i d( ) r ! 2
x ? P ( ) P T (forward) :

(2.4) Note that the lower integration bound r=4 on b is required to recover the correct expression for the forward amplitude. In the absence of an exact conformal invariant solution for all impact parameter 14], this cuto will play an important role in the determination of the hard di raction components.

(ii) A saddle point approximation of the integral over in formula (2.1) gives the approximate expressions in the large b approximation used in ref. 6,[START_REF] Bialas | Inelastic component[END_REF], up to the scale factor 16. This factor is strictly speaking not present in the large b approximation, but appears when one takes into account the global conformal invariance of the BFKL kernel 13]. This scale factor leads to a much more central impact parameter distribution than the approximate expressions used in ref. 6,[START_REF] Bialas | Inelastic component[END_REF], and plays a major rôle in di raction, as we shall see in the following.

Using formula (2.1) as an input leads to an analytic formulation of the di ractive amplitudes. In this section, we shall rst consider hard di raction on a dipole of given size r. It will be shown in the next section how, starting from this case, it is possible to derive phenomenological expressions for hard di raction on a proton. One has the following identities for the di ractive structure functions 6]: and f T (z) z 2 (1 ? z) 2 (z 2 + (1 ? z) 2 ) f L (z) 4z 3 (1 ? z) 3 :

F D T;L (Q 2 ; x P ; M 2 ) = Q 4 4 2 em 1 x P d T;L dM 2 = Q 2 4 2
(2.8)

The notation \ X " is de ned by X =(z(1 ? z)) 1=2 X.

Following ref. 6], the formulae (2.6{2.8) describe the interaction of a q q con guration of transverse size of the virtual photon with a dipole of size r. The input elastic amplitude is taken from (2.1). The photon wave function in terms of q q con gurations is projected on its transverse (T) and longitudinal (L) polarization states which di er only by the factors f T;L (z) and by the order of the Bessel functions 15] involved: K 1 , J 1 are associated to the transverse component, and K 0 , J 0 to the longitudinal one. z (resp. (1 ?z)) is the longitudinal momentum fraction carried by the antiquark (resp. quark). Note that interference e ects are present 6] which explain the product of Bessel functions of di erent kinds (K and J). The cuto in impact parameter space appears twice in formulae (2.6) and in (2.7), rst by the upper bound 4b on and second by a lower bound r=4 on the impact parameter b. In practice, we will release the rst bound, which will greatly simplify the calculations and allow to nd an analytical form for the di ractive amplitude. This approximation is expected to change only the normalisations.

Inserting expression (2.1) for the dipole-dipole amplitude in eq. (2.6, 2.7), one obtains the results for F D T;L (2.5) involving integrals over the impact parameter b, the momentum fraction z and the q q-pair size . Interestingly enough, these integrations can be factorized and exactly computed using in particular the change of variable ! ^ = (z(1?z)) 1=2 . After a tedious but straightforward calculation using well-known identities 15], the successive integrations give

F D(el) T = 16 3 2x P N c e 2 2 2 (1 ? ) Q 0 Q ! 2 Z Y j=1;2 2 4 d j 2i Q 2Q 0 p ! 2 j x ? P ( j ) P 3 5 Y j=1;2
f(1?2 j )d( j )?(2? j )?(3? j ) 2 F 1 (2? j ; ?1+ j ; 2; 1? )g 1 + 2

( 1 + 2 ?1)(3?2( 1 + 2 )) B( 1 + 2 ; 1=2) ; (2.9)

F D(el) L = 16 3 x P N c e 2 2 3 Q 0 Q ! 2 Z Y j=1;2 2 4 d j 2i Q 2Q 0 p ! 2 j x ? P ( j ) P 3 5 Y j=1;2 n (1?2 j )d( j )? 2 (2? j ) 2 F 1 (2? j ; ?1+ j ; 1; 1? ) o 1 3?2( 1 + 2 ) B( 1 + 2 ; 1=2) ; (2.10)
where we have introduced the scale Q 0 2=r. Note that these expressions involve two coupled inverse-Mellin transforms in 1 and 2 , each corresponding to one elastic dipole interaction, (see g.1-a).

b) Inelastic component

As shown in g. 1.b, the inelastic component stems from the following process: the initial q q state of the virtual photon develops a set of colour dipoles through cascading 7] and the di ractive component is due to the interactions of two of the produced dipoles with a target dipole, each of these being described by the amplitude (2.1). The cross section can be expressed as 6]:

d (in) T;L d = 16 3 em e 2 5 s N 2 c 1 Q 2 Z 1 r=4 d 2 b b 2 D % 2 E 2 Z d 2i b Q 2 ! 2
(1 ? ) 3 H T;L ( ) ? P ( ) ;

(2.11)

where D % 2 E = Z 4b 0 d% % % 2 T (r; %; b; x P ) 4 2 s % 2 (2.12)
and H T ( ) = V ( ) 16 2 (1? ) 4 ?(3? )? 3 (2? )?(2+ )?(1+ ) ?(4?2 )?(2+2 )

H L ( ) = V ( ) 8 (1? ) 3 (1+ )(2? )
?(3? )? 3 (2? )?(2+ )?(1+ ) ?(4?2 )?(2+2 ) (2.13) with V ( ) = 3 F 2 (1? ; 1? ; 1; 3=2; 1):

(2.14)

The dipole-dipole interaction is represented by formula (2.12) which can be interpreted as the average squared transverse sizes of the dipoles seen at the impact parameter b. Indeed, the quantity T (r; %; b; x P )=4 2 s % 2 is nothing else than the density of dipoles of size % emitted from a dipole of size r at impact parameter b 7, 13]. The inverse Mellin-transform and its integrand factor V ( ) (2.14) comes from the calculation of the 1 ! 2 dipole branching at the virtual photon vertex 6]. The di erent prefactors appearing in H T and H L in formula (2.13) come from the transverse and longitudinal wave-functions of the virtual photon, respectively.

Inserting the expression (2.12) into eq. (2.11) and integrating over the impact parameter b, one gets:

F D(in) T;L = 1 x P 16 5 3 e 2 s N 2 c Z Y j=1;2 ( d j 2i 1?2 j 2?2 j d( j )x ? P ( j ) P ) Z d 2i Q 4Q 0 ! 2 ? P ( ) (1? ) 3 H T;L ( ) 1 2? ? 1 ? 2 : (2.15)
The formulae (2.9, 2.10, 2.15) condense in an analytical form the perturbative QCD predictions for the di ractive structure functions of a dipole of size 2=Q 0 .

3 Proton di ractive structure functions in the dipole model

In the previous section, we have obtained the expressions for the di ractive structure functions, when the target is a single dipole of size r = 2=Q 0 . Our aim in this section is to use these formulae to obtain the di ractive structure functions of the proton. We have thus to implement a model for the proton which could describe it as a collection of primordial dipoles.

a) Elastic component

Applying the steepest-descent method, the saddle-point is determined by the usual BFKL-type integrand (between the square brackets in (2.9, 2.10)). It reads:

sp = 1 2 1 ? a(x P ) log Q 2Q 0 p ! : (3.16)
where a(x) = =(7 (3) s N c log 1=x) is the k ? -di usion coe cient coming from the BFKL evolution 12,[START_REF] Prudnikov | Integrals and Series[END_REF]. We have to use a special treatment for the pole at 1 + 2 = 1 in formula (2.9), since it contributes for the asymptotic saddle points at 1 = 2 = 1=2 when x P ! 0. All the other singularities are not relevant, since they are far from the saddle points.

Noting that in the kinematical domain of interest, 1 ? 2 sp = a(x P ) log Q=2Q 0 p 1, and working at lowest order, the results are:

F D(el) T = 12 N c e 2 4 s x ?2 P +1 P a 3 (x P ) log 3 Q 2Q 0 p e ?a(x P ) log 2 Q 2Q 0 p
(1? ) 2 F 1 ? 1 2 ; 3 2 ; 2; 1 ? 2 ; (3.17)

F D(el) L = 16
N c e 2 4 s x ?2 P +1 P a 3 (x P ) log 2 Q 2Q 0 p e ?a(x P ) log 2 Q 2Q 0 p 2 2 F 1 ? 1 2 ; 3 2 ; 1; 1 ? 2 ; (3.18) with

P 1 + P 1 2 = 1 + s N c 4 log 2 : (3.19) The pole at 1 + 2 =1 in F D(el)
T results in an extra log Q=2Q 0 p which may be large.

The formulae (3.17, 3.18) give an explicit expression for the elastic components of di raction in the QCD dipole model. The x P -dependent prefactor x ?2 P +1 P a 3 (x P ) plays the role of the (hard) pomeron ux factor already discussed in references 6,11]. Revealing new features of the model, the present explicit calculation shows that the relevant scale factor responsible for the scaling violations is 2Q 0 p where Q 0 is the scale of the proton-dipole amplitude. The -dependence is found to be non polynomial.

b) Inelastic component

Using again the steepest-descent method for evaluating the three integrals in , 1 and 2 of formula (2.15), we nd saddle-points at sp = 1 2 (1 ? a( ) log Q 4Q 0 ) ; j;sp = 1 2 (1 ? a(x P )) :

(3.20)

Note that there is no coupling of the integrals at the saddle points; however, one should take into account the prefactor zeros at j = 1=2, see eq. (2.15). All in all, the result reads:

F D(in) T;L = 2 9 s 2 H T;L 1 2
N c e 2 5 s 4 x ?2 P +1 P a 3 (x P ) Q Q 0 e ? a( ) 2 log 2 Q 4Q 0 a 1 2 ( ) ? P : (3.21) Comparing our expression (3.21) with the original result (formula (11) in the second paper of ref. 6]), we nd identical results but for the scale of the transverse momentum in the BFKL evolution: 4Q 0 instead of Q 0 . This change comes from the scale factor determined by conformal invariance (see 2.1) and was not considered in the previous approximation 6]. This modi ed scale factor has an important phenomenological e ect.

In the QCD dipole model, there is a tight connection between the total and di ractive structure functions. In refs. 9,11], the QCD dipole model has been successfully applied to the proton total structure function in the small-x kinematical domain. A model for the primordial dipole con gurations in the proton has been introduced through a distribution with average size and the non perturbative features of the proton target manifest themselves only through this scale and the global normalization constant related to the density of primordial dipoles (for a more complete discussion, see ref. 9]). We thus use this representation of the nucleon target in formulae (3.17, 3.18, 3.21) to derive the model for the proton di ractive structure functions. In this context, the scale parameter is reinterpreted as Q 0 = 2=<r> and we introduce the arbitrary normalizations N (el) T , N (el) L , N (in) for F D(el) T;L and F D(in) T;L multiplying formulae (3.17, 3.18, 3.21) respectively. These unknown normalizations re ect the non-perturbative primordial dipole con gurations contributing to the various proton structure functions.

With this model, we are ready to write a full parametrization adequate for the description of the data. The free parameters of the dipole model are P , which is related to the xed coupling constant s in the BFKL scheme at leading order (see eq. (3.19)), Q 0 , corresponding to a non-perturbative scale for the proton, and the three normalizations N (el) T , N (el) L , N (in) . As is now well-known, a secondary trajectory based on reggeon exchange is added in order to take into account the large-mass and small rapidity gap domain. Reggeon exchange can here be simply parametrized in the following way:

F D(R) 2 (x P ; ; Q 2 ) = f R (x P )F R 2 ( ; Q 2 ) ; (3.22)
where the reggeon ux f R (x P ) is assumed to follow a Regge behaviour with a linear trajectory R (t) = R (0) + 0 R t : f R (x P ) = Z t min tcut dt e B R t x 2 R (t)?1 P ;

(3.23)

where jt min j is the minimum kinematically allowed value of jtj and t cut = ?1 GeV 2 is the limit of the measurement. The values of B R and 0 R are xed with data from hadron-hadron collisions 1]. The reggeon structure function is assumed to be the pion structure function 17]. The free parameters for this component are the reggeon normalisation N R and exponent R . The nal parametrization used for the t is the sum of the three contributions detailed above (3.17, 3.18, 3.21, 3.22), namely:

F D(3) 2 = 1 N C e 2 N in (F D(in) T + F D(in) L ) + N el T F D(el) T + N el L F D(el) L + N R F R 2 :
(3.24)

Only one parameter (N in ) is used for the normalisation of the inelastic component since

F D(in) L =F D(in) T = 2=9.
4 Phenomenology 4.1 Fits to the H1 and ZEUS data A t to the recently published H1 1] and ZEUS 2] di ractive structure function data is performed separately. The result of both ts is shown in Figure 2 for H1 and Figure 3 for ZEUS, and their parameters are given in Table 1. Let us comment the interesting features of the ts. The t to the H1 data leads to a very good 2 (1.17 per degree of freedom with statistical errors only). The value of R is consistent with the usual values found for secondary reggeon contributions if interference e ects are taken into account 1]. The value of P is found to be consistent with the expected intercept for a hard BFKL pomeron 12]. This intercept is higher than the value obtained from the t to the structure function F 2 9]. Q 0 is a typical non perturbative scale for the proton and very close to the value obtained in the proton structure function t. It should be noted that the scale Q 0 appears in a quite non trivial way as the virtuality in the inelastic component (Q=4Q 0 ), and in the elastic one (Q=2 p Q 0 ). The validity of these results can be checked by starting with two di erent values of Q 0 for each component and the result of the t leads to same values within errors, with the same 2 . Furthermore, imposing a constant scale for the elastic component, i.e. Q=2 p Q 0 ! Q=Q 0 ; leads to a very bad quality t. The other features of this description will be discussed further on. The t to the ZEUS data leads to a worse 2 ( 2 =dof = 1:95 with statistical errors only). The result is shown in Figure 3 in continuous line, and by comparison, the dashed curve corresponds to the parameters obtained in the H1 t. It can be noticed that a di erence is clearly seen both in the low Q 2 ; low and the high Q 2 ; high bins, in opposite way, the H1 t going from below to above the Zeus t.

In order to investigate the origin of these di erences, a direct comparison between ZEUS and H1 data has been performed 18]. The H1 data have been interpolated to the ZEUS closest bins in and Q 2 using the dipole model t. This interpolation is weakly sensitive to the model used as the interpolation in the kinematical variables is very small. It was checked that the use of the model by Bartels et al. 19] gives a similar result. The result of this comparison is displayed in Figure 4. The striking feature is that the main di erence between both ts noticed in Figure 3 comes from the region where the data are most di erent. The di erences occur both at small and high (i.e. small and high masses) and the solution of this puzzle cannot be due only to the di erence in selecting di ractive events -either the rapidity gap selection 1] or the M X substraction method 2] -in both experiments. In fact, only high-mass events could be a ected by these di erent selections. A global shift of ZEUS data of about 15% would be possible as they do not correct for proton dissociation but the di erence between H1 and ZEUS data is not only due to normalisation e ects. Indeed, it is clear in Figure 4 that, depending on the values of Q 2 and , H1 points lie either higher or lower than ZEUS data. The analysis of new accumulated data in this kinematical region would help solving the experimental puzzle and would be of great interest.

E ective pomeron intercept

Let us now study the dependence on x P which is directly related to the rapidity gap dependence. One can de ne an e ective pomeron intercept in the following way: where the t dependence is integrated out (the data are mainly at t 0). This e ective exponent can also be determinated for the inelastic and elastic components separately as a function of x P (Figure 5) or Q 2 (Figure 6). In Figure 5, The e ective intercepts of both components and their sums have been compared to the soft pomeron intercept 20] (dashed line) and to the bare hard BFKL pomeron obtained in the t ( P = 1:395; cf. Table 1). The range of obtained values sits essentially between these two limits except in the large x P region (x P 10 ?2 ). It is clearly not consistent with the soft pomeron value (1.08). It is as well much lower than the bare pomeron intercept 6,11]. This can be explained by the large logarithmic corrections induced by the a 3 (x P ) term, proportional to log 3 (1=x P ), present in both di ractive components (see formulae 3.17, 3.18, 3.21). The e ect of this logarithmic term induces also an x P dependence of the intercept. Moreover, in Figure 5, it can be seen that the x P dependence of the intercept is di erent between the elastic and the inelastic components. This induces a breaking of factorisation directly for the di ractive components of this model, which comes in addition to the known factorisation breaking due to secondary trajectories. In addition, the elastic component itself does not factorise, as it can be seen in Figure 5 for di erent Q 2 values. This comes from the mixing of the x P and Q 2 dependence in formula 3.17, 3.18.

As seen in Figure 6, the Q 2 dependence of the pomeron intercept (taken for instance at x P =0.001 where the reggeon contribution is negligible) is weak. It is in good agreement with the H1 determination ( P = 1:204 :02) indicated in Figure 6. It is thus intermediate between the soft and hard pomeron lines but inconsistent with both of them. In our model, this softening of the bare pomeron comes from the large logarithmic corrections which cannot be neglected in perturbative calculations as was mentioned in the last paragraph.

Scaling violations

One striking feature of the di ractive proton structure functions was the Q 2 dependance at xed x P as a function of as was pointed out experimentally by the H1 collaboration 1] (see Figure 7) and con rmed at lower Q 2 21]. The structure functions are increasing with Q 2 even at very high (see Figure 7) at variance with the behaviour of the total proton structure function as a function of x . In the QCD dipole model, this experimental feature is described by a non trivial interplay between the two di ractive components. In Figure 7, the dipole t is compared with the H1 result showing the contribution of each component: at small , the inelastic component dominated and vary quasi linearly in log Q 2 , and at high , this component is depressed similarly to the total structure function, but is progressively substituted by the elastic component. Note that the enhancement present in the data at high and low Q 2 which is probably due to vector meson production is partially reproduced by the model. It is expected that including the speci c vector meson contributions in addition to the elastic q q component will even improve the result of the model in this kinematical range. It is striking that it is possible to describe the observed scaling violations in a very di erent framework as the one given by the DGLAP evolution as was performed by the H1 collaboration 1].

?dependence

In Figure 8 is displayed the dependence of the proton di ractive structure function for a xed value of x P = 0:003 for di erent values of Q 2 . The dependence is quite weak in all Q 2 bins and correctly reproduced by the interplay of the two components of the QCD dipole model. While at low Q 2 , this e ect is essentially due to the inelastic component, at high Q 2 the interplay between both components is required to describe the observed ?dependence.

The interplay of the di erent components can be analysed in more details in Figure 9 where the elastic, inelastic, reggeon components and their sum are displayed for three di erent values of x P and four di erent values of Q 2 as a function of . The reggeon component is only important at high value of x P as expected and dominates at low . It disappears at smaller x P . At low Q 2 , the inelastic component dominates in almost the full range, while at higher Q 2 , it is only important at low and replaced at low masses by the inelastic component.

Longitudinal contribution to the di ractive structure function

In Figure 10 are displayed as a function of in di erent Q 2 , x P bins the longitudinal and transverse components of the proton di ractive structure functions. Note that the x P values are chosen a bit smaller than in the previous gure to depress the reggeon contribution. Note also that we did not separate the longitudinal and the transverse contributions for the inelastic component as they are directly proportional by a factor H L (1=2)=H T (1=2) = 2=9 (see formula 3.21). In addition to the already mentioned dominance of the inelastic component at small Q 2 , the longitudinal elastic component is found to be high at high and crosses over the transverse component near 0:8. It is thus expected to obtain high values of the ratio R of the longitudinal to the transverse components at high values of . The result of our t prediction is displayed in Figure 11. We note that the R ratio remains small ( 0:2) in almost the full kinematical plane except notably at high where it may reach high values such as 2. Note that this value is in the range of the measured R ratio with vector meson production 3]. A measurement of R in di raction would thus be of great interest and would be a good test of the model. The question arises whether such high values of R would not modify the F D 2 measurement itself as it was assumed to be 0 1]. We have checked that the parameters and the quality of the t are not much in uenced if only data points with y < 0:3, where the in uence of R is small, are taken.

It is instructive to notice that another model of di raction based on selecting q q and q qg components of the photon 19] also leads to a large contribution of the longitudinal q q contribution at high . However, a distinct feature of the dipole model is the above mentioned di erence between the high and low Q 2 behaviour where at low Q 2 , the inelastic component dominates and induces a small value of R. We thus expect thet a measurement of R in di raction at low Q 2 would be a way to distinguish both models.

Conclusion and outlook

Let us summarize the results we have obtained. First of all, the colour dipole model has provided us with a interesting theoretical and phenomenological framework to study hard di raction. On thetheoretical ground, it enables to interpret hard di raction in terms of BFKL dynamics of (resummed) perturbative QCD, which allows one to make theoretical estimates. On the other hand, the success of the phenomenological t using analytical amplitudes allow a discussion of the various interesting and intringuing aspects of the data.

The model predicts two contributions: an elastic one for which the nal state consists in the hadronization of a quark-antiquark pair, and an inelastic one, including soft gluon radiation reproduced in terms of a cascading process involving colour dipoles. Both contribute in quite di erent kinematical domains since they are characterized by di erent mass distributions (di erent ) and for di erent photon polarization. In this paper, we have obtained a suitable parametrization and a satisfactory t to the H1 data along these lines. The t to Zeus data is less satisfactory and with di erent parameter values. By a model-independent analysis of both data, we could identify the reason of the discrepancy in a genuine di erence between data sets when interpolating in the same bins. This point deserves a detailed experimental study in the near future, since it may be of importance to doiscriminate between models. in particular the con rmation of the trend of present H1 data would be in favor of the dipole model.

On the theoretical ground, we have computed amplitudes obeying some of the requirements of BFKL dynamics for di ractive amplitudes (on a dipole). However, exact calculations of these rather complicated amplitudes are still under way 22] and we hope in the near future obtain de nite predictions with all requirements of BFKL dynamics. the evaluation of next-leading BFKL e ects would also be welcome since they already play a rôle for the total structure functions.

Finally, as emphasized in ref . 11] the colour dipole model formalism calls for a uni ed description of the di ractive and total deep-inelastic scattering events, e.g. including events with no rapidity gap. We showed that within the precision of the current data, there are quite a few indications (similar scale Q 0 ; softening of the gard Pomeron by logarithmic facors in di raction,etc...) of such a common theoretical ground. However, further tests of the model are deserving. The rst one would be a confrontation of the predicted R ratio with the data if available: indeed, the various models should predict quite di erent contributions from the two polarization states of the photon. Other useful tests concern the nal states. For instance one can compute the predictions for di ractive vector meson production and confront them to the recent data. Such tests might help distinguish between the few di erent models for hard di raction which are able to describe the data. de nitely, the QCD dipole model of BFKL dynamics is one of them.

Acknowledgements

A fruitful collaboration with Andrzej Bialas and Henri Navelet which has much inspired the present paper is acknowledged. We also thank them for very useful physical and technical remarks and suggestions during the elaboration of the paper. Figure 4 Direct comparison between H1 (squares) and ZEUS (triangles) data. The H1 data have been interpolated to the ZEUS bins by using the dipole parametrisation. Both data are found compatible within error bars. Note however that sensitive di erences are seen in particular in three di erent and Q 2 bins (Q 2 =8 GeV 2 for =0.2, Q 2 =60 GeV 2 , for =0.7 and 0.9) (see text).

Figure 5 E ective pomeron intercept as a function of x P . The dependence of the e ective pomeron intercept P on x P is shown for the inelastic component (independent of Q 2 ; continuous line) and for the inelastic one for 3 di erent values of Q 2 . The value of is xed at 0:3. The straight lines at P =1.395 and P =1.08 correspond respectively to the bare BFKL intercept as found by the t (see table 1), and the soft Donnachie Landsho prediction 20].

Figure 6 E ective pomeron intercept as a function of Q 2 . The e ective pomeron intercept of the model is displayed: elastic component (dashed curve), inelastic component (dotted curve), total (full curve). The value of and x P are xed for reference at ( = 0:3 and x P = 10 ?3 ). The bare and soft pomeron intercepts are given by dotted-dashed straight lines. The H1 determination 1] lies in the region between the two dashed straight lines. x IP Figure 11

Figure 3

 3 Figure 3 Result of the F D 2 t to the ZEUS data 2]. The data are displayed by triangles with statistical and systematic errors added in quadrature as a function of x P in and Q 2 bins. The t has been performed with statistical errors only and is displayed in full line (see text). For comparison, he result of the t of the H1 F D 2 data is displayed in dashed line.

Figure 7

 7 Figure 7 Scaling violations. The dependence of x P F D 2 on Q 2 for di erent values of at xed x P (3:10 ?3 ) is shown together with the dipole model t. Dotted line: elastic component, dashed line: inelastic component, full line: total.

Figure 8

 8 Figure 8 Beta dependence. The dependence of x P F D 2 on for di erent values of Q 2 at xed x P (3:10 ?3 ) is shown together with the dipole model t. Dotted line: elastic component, dashed line: inelastic component, full line: total.

Figure 9

 9 Figure 9 Di erent components of the H1 F D 2 t. Dotted line: elastic component, dashed line: inelastic component, dashed-dotted line: secondary reggeon component, full line: sum of all components.

Figure 10

 10 Figure 10 Di erent components of the H1 F D 2 t. Longitudinal versus transverse. Dotted line: tranverse elastic component, dotted-dashed line: longitudinal elastic component, dashed line: total (F T + F L with F L =F T = 2=9, see text) inelastic component, full line: result of the t.

Figure 11

 11 Figure 11 Prediction for R, the ratio of the longitudinal to the transverse cross section. Note the di erent scale for the high bins.

Figure 1 F

 1 Figure1

TABLE CAPTION Table I

 CAPTIONI Parameters obtained for the F D 2 t.The t has been performed with statistical errors only. The rst error quoted is the statistical one and the second the systematic one.

	H1 1.395 0.005 0.003 0.682 0.046 0.049 0.428 0.011 0.001 N in 0.00244 0.00028 0.00033 0.00374 0.00015 0.00076 ZEUS P 1.327 0.001 0.016 R | Q 0 0.241 0.002 0.014 N el T 40.0 1.6 3.6 126.9 0.5 37.6 N el L 12.8 1.2 1.1 24.43 0.51 8.12 N R 7.44 1.84 3.50 0 58 0 2 255.4 for 226 pts 89.6 for 53 pts 1.17/d.o.f. 1.95/d.o.f.

Table I FIGURE

 I CAPTIONS Figure 1 Schematic description of the QCD dipole model of di raction. 1-a: elastic component, 1-b: inelastic component Figure 2 Result of the F D 2 t to the H1 data 1]. The data are displayed by triangles (with statistical and systematic errors added in quadrature) as a function of x P in and Q 2 bins. The t has been performed with statistical errors only and is displayed in full line (see text).


Dennis Diefenbach
email: dennis.diefenbach@univ-st-etienne.fr

Pierre-René Lhérisson
email: pr.lherisson@univ-st-etienne.fr

Fabrice Muhlenbach
email: fabrice.muhlenbach@univ-st-etienne.fr

Pierre Maret
email: pierre.maret@univ-st-etienne.fr

Computing the Semantic Relatedness of Music Genres using Semantic Web Data

Keywords: CCS Concepts, Information systems → Similarity measures, Language models, Vector Representations, Semantic Relatedness, Music Genres, Recommendation Systems

Computing the semantic relatedness between two entities has many applications domains. In this paper, we show a new way to compute the semantic relatedness between two resources using semantic web data. Moreover, we show how this measure can be used to compute the semantic relatedness between music genres which can be used for music recommendation systems.

We first describe how to build a vector representations for resources in an ontology. Subsequently we show how these vector representations can be used to compute the semantic relatedness of two resources. Finally, as an application, we show that our measure can be used to compute the semantic relatedness of music genres.

INTRODUCTION

The work presented in this paper is the result of a research work on music recommendation systems conducted in collaboration with a french startup company called 1D Lab 1 . The aim is to create a music recommendation system over a database of independent artists, i.e. artists who are not under contract with the major labels. The independent, emerging artists, suffer from a lack of visibility on the web. To make reccomandations we therefore have to relay on the few information we have about their tracks, like their music genre. Moreover we face the so-called cold-start problem, i.e. we cannot use user profiles to generate the recommendations.

While it is nearly impossible to find information about independent artists in the Semantic Web it is easy to find information about the music genre. These include songs of a particular genre, bands that play the genre, characteristic instruments for the genre, influences, regional distributions and many more.

The problem we address here is how to exploit semantic web data to answer question like "How similar is Ska and Reggae?" or "How similar is Ska and Classical Music?". The answer to these questions can then be used to recommend to a person that is listening Ska a song of a different but close music genre.

Answering the above questions is equivalent to find out what is the semantic relatedness between two music genres. We propose first a way to construct for resource of an ontology a vector representation of it. Using these vector representations we propose a new way to compute the semantic relatedness of two resources in an ontology. Finally, we show the results we get if we apply it to music genres.

The publication is organized as follows. In section 2 we discuss the related work. In section 3 we describe how to compute the semantic relatedness between two resources in an ontology. In section 4 we show our results when applied to music genres and finally in section 5 we discuss future work.

RELATED WORK

Recommender Systems are designed to guide users through large volumes of data. One of the most popular methods is collaborative-filtering where the recommendation is based on users' preferences and behaviors [START_REF] Resnick | Recommender systems[END_REF]. Another approach is content-based recommendation which uses information associated to items like unstructured text for web pages, blogs, keywords, attributes and properties for music, and tries to match them to the user's profile [START_REF] Lops | Content-based recommender systems: State of the art and trends[END_REF]. For example Pandora 2 , a radio streaming platform, uses as features tags set by experts to recommend music. Since human tagging is a time-consuming task there are methods to retrieve information automatically. [START_REF] Turnbull | Semantic annotation and retrieval of music and sound effects. Audio, Speech, and Language Processing[END_REF] uses signal processing on music and inferred high-level semantic descriptors (timber, rhythm, tempo). Other meta-data provided with a track like the music genre can be used. Linked Open Data and Encyclopedic Knowledge Source can also help learning more accurate items features. On the artist's level, [START_REF] Passant | dbrec music recommendations using dbpedia[END_REF] propose dbrec, a recommender system based on DBpedia. DBpedia is also used in the movie recommendation field [START_REF] Mirizzi | Movie recommendation with dbpedia[END_REF]. They use the genres, but also information like the actors, directors, writers, to create a vector for movies and compute similarities with a Semantic version of the classical Vector Space Model (sVSM) that they created. These work require detailed information on each artist or movie that can be recommended. Since we are working with a dataset of independent artists we don't find machine readable data for most of them in the Linked Data Cloud. We therefore choose to use the music genre that is provided with our data to make recommendations. We have seen in [START_REF] Bogdanov | Content-based music recommendation based on user preference examples[END_REF] that there is no big difference between a content-based semantic distance and a simple genre-based baseline.

We therefore concentrated on the problem of using semantic data to find the semantic relatedness of music genres. There are different works that propose semantic relatedness measures over ontologies [START_REF] Hulpuş | Path-based semantic relatedness on linked data and its use to word and entity disambiguation[END_REF][START_REF] Passant | Measuring semantic distance on linking data and using it for resources recommendations[END_REF]. The first proposes a semantic relatedness based on the top-k path connecting two resources. First the top-k paths are found. Then the relatedness is computed based on the properties appearing in these paths. The second is based on the number of different types of connection that exits between two resources x and y. It considers direct connections between x and y and connections where a third resource z is connected to x and y with the same relation.

Here we propose a new semantic relatedness measure. As we have shown in [START_REF] Diefenbach | A scalable approach for computing semantic relatedness using semantic web data[END_REF] our method has the advantage of considering all relations and classes of the ontology. Also, notice that [START_REF] Passant | Measuring semantic distance on linking data and using it for resources recommendations[END_REF] considers distance between artist in DBpedia, while we are considering music genres, because as previously said, lots of independent artists are not described on the Web nor in the Semantic Web.

VECTOR REPRESENTATION AND SE-MANTIC RELATEDNESS

In this section we describe how we compute the semantic relatedness for two resources of an ontology. We first associate to each resource r a vector representation that captures the semantic of r. Then, we use this representation to compute the semantic relatedness of two resources.

We interpret the ontology as a graph G. We transverse the graph starting from r up to a depth of k in an directed or undirected way. Let Rr,1 be the set of resources whose distance from r is 1, Rr,2 the set of resources whose distance from r is 2 and so on. We construct a vector representation of r which we denote by vr as follows. vr lives in a vector-space that has as many dimensions as the number of 2 pandora.com resources of the ontology. The i-th component is defined as:

(vr)i =          0 if i / ∈ Rr,1 ∪ ... ∪ R r,k 1 if i ∈ Rr,1 . . . 1 k if i ∈ R r,k
i.e. it contains zero-entries for all resources, except a 1 for the resources in Rr,1, a 1/2 for the resource Rr,2 and so on.

To compute the semantic relatedness between two resources r and s we compute the cosine similarity between the two corresponding vectors vr and vs:

< vr, vs >= i (vr)i • (vs)i ||vr|| • ||vs||
This will return a number between 0 and 1. There are two possible interpretations for this similarity measure. The first is that the similarity of the resources is based on the angle between the corresponding vector representations. The second is to interpret it as the overlap of some sub-graphs of G. The vector of r contains non-zero entries for all resources that are at a distance of maximum k from r. Let Gr be the sub-graph containing these resources. Analogously for the vector of s. This means that the sum in the cosine similarity is non-zero for all resources that are at a distance of k from both r and s, i.e. we compute how big is the overlap between the sub-graphs Gr and Gs.

SEMANTIC RELATEDNESS OF MUSIC GENRES

As an application of our similarity measure we have computed the similarity between music genres. As an ontology we considered the English DBpedia 3 ontology. We created a list of music genre. Many of them could be identified using the dbo:MusicGenre class. In a next step we computed the vector representation of all music genre. We constructed the vectors as described above by setting k = 2 and by considering the ontology graph as an undirected graph. Note that for a music genre r the corresponding vector-representation contains non-zero entries for example for: instruments that are characteristic for the genre, the region where it comes from and it's stylistic origin. Moreover since we consider the graph as undirected it contains also bands that play this genre, albums and tracks that are of this genre and many more. We do not restrict the search to a particular properties but we consider them all.

All the computations where done using the work presented in [START_REF] Diefenbach | A scalable approach for computing semantic relatedness using semantic web data[END_REF]. It shows how to transverse efficiently an ontology using sparse matrix multiplications and it was used to construct the vector representations presented above.

We used the vector representation to compute the pairwise semantic relatedness of the music genre. The result is a square dense matrix with one row and one column for each music genre. The i-th row j-th column contains the semantic similarity between the music genre i and j, i.e. a double between 0 and 1. Since it is difficult to judge the quality of the result in this format we decided to visualize them. We do not consider all connections, but using a threshold, only the connections with a high degree of similarity. Otherwise it would be difficult to visualize all the data, since we have 132 The data is displayed using the Force Atlas 2 [START_REF] Jacomy | Forceatlas2, a continuous graph layout algorithm for handy network visualization designed for the gephi software[END_REF] algorithm provided by Gephi [START_REF] Bastian | Gephi: an open source software for exploring and manipulating networks[END_REF]. The main idea behind the visualizing algorithm is that all nodes repulse each other while two connected nodes attract each other.

The entire graph is showed in figure 1 while a subgraph including electronic music genres is showed in figure 2. Both show that we were able to capture the semantic relatedness between the music genres.

FUTURE WORKS

We are working on a music recommendation system for independent artists using the numbers that we presented here. Moreover we would like to compare our similarity measure to existing ones and see how it performs. Another interesting direction is to make the semantic relatedness more flexible. The semantic relatedness between music genre is not absolute. A user could be more interested in music genre that come from the same geographical area, music genre that arose in the same period of time or that are characterised by the same instruments. By weighting accordingly different relations it would be possible to reflect these interests and give accordingly recommendations to the users.

1

  http://en.1d-lab.eu c 2016 Copyright held by the author/owner(s). SEMANTICS 2016: Posters and Demos Track September 13-14, 2016, Leipzig, Germany

Figure 1 :

 1 Figure1: Graph of the music genre constructed using the corresponding semantic relatedness and visualized using the Force Atlas 2 algorithm of Gephi

Figure 2 :

 2 Figure 2: Graph of electronic music genre constructed using the corresponding semantic relatedness and visualized using the Force Atlas 2 algorithm of Gephi

ACKNOWLEDGMENTS

This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 642795.