Non-asymptotic effects in the photon-pomeron coupling
L. L. JENKOVSZKY(1)(*), E. S. MARTYNOV(1)(**) and F. PACCANONI(2)(***)
(1) Bogoliubov Institute for Theoretical Physics, Academy of Sciences of the Ukrain 252143 Kiev, Ukrain
(2) Dipartimento di Fisica, Università di Padova and INFN, Sezione di Padova via F. Marzolo 8, I-35131 Padova, Italy
(ricevuto il 14 Luglio 1997; approvato il 4 Agosto 1997)
Summary. — Pomeron-photon coupling is studied in a model for the pomeron based
on the analyticity properties of the gluon propagator. Non-asymptotic effects are considered and their relevance in explaining the rise of the proton structure function at HERA is estimated.
PACS 13.60.Hb – Total and inclusive cross-sections (including deep-inelastic processes). PACS 12.38 – Quantum chromodynamics.
PACS 11.55.Jy – Regge formalism.
1. – Introduction
The description of diffractive hadronic physics in terms of a pomeron exchange has proved remarkably successful in predicting the high-energy behaviour of hadron-hadron scattering. The real photoproduction cross-section has a similar energy dependence well reproduced by fits of Regge type [1-4].
The rapid rise with decreasing Bjorken x of the proton structure function [5, 6], corresponding to the strong increase with energy of the virtual photon-proton cross-section, seems to be an exception. It supports, for large Q2, a perturbative calculation of the pomeron [7, 8] that differs from the «classical» pomeron both in its s and t dependence. GLAP evolution equation [9] completes this theoretical description determining the Q2 behaviour of the amplitude. At small x, gluon density dominates over quark densities and its growth can be obtained from the evaluation of a gluonic ladder graph with radiative corrections. The result of the GLAP equation is recovered by integrating over intermediate gluons with ordered longitudinal momenta and strongly ordered transverse momenta.
(*) E-mail address: jenkHgluk.apc.org (**) E-mail address: martynovHgluk.apc.org (***) E-mail address: paccanoniHpadova.infn.it
In any case one considers perturbative and non-perturbative contributions to the same object [10], the Regge trajectory with vacuum quantum numbers. The attributes of hard and soft, referred to the pomeron, will be used in the following just in this sense and having in mind its uniqueness. An estimate of the relative relevance of the hard and soft component in DIS represents a difficult problem, but in order to reach a meaningful comparison between different theoretical models one must confront with it. The presence of pre-asymptotic contributions like, for example, the f trajectory, further complicates this program. The neglect of secondary Regge trajectories where dynamics is not yet really asymptotic can influence the interpretation of the rapid rise of the virtual photon-proton cross-section with energy [3]. In search of other non-asymptotic effects, a quantity of interest is the g *-pomeron-g * vertex. At first sight, this vertex and its Q2-dependence represents the main difference with an on-shell hadron-hadron process. In order to clarify this point we need to investigate non-leading effects related to the quark loop that couples the photon to the pomeron, renouncing in part the strong simplification coming from the ordering of longitudinal and transverse momenta. This is the purpose of this paper.
In the following we will consider only the soft pomeron, the coupling of the BFKL pomeron [7, 8] to quarks has been investigated in ref. [11] and a beautiful review of perturbative small-x physics is given in ref. [12]. We first discuss a non-perturbative model developed earlier in order to evaluate nucleon-nucleon cross-sections at high energies [13, 10]. Then the result of the calculation for the absorptive part of the virtual forward g * -p amplitude is compared with experimental data [5, 6] for the proton structure function at small x. The relevance of non-leading terms in the evaluation of the g *-pomeron-g * vertex appears clearly in this comparison.
2. – The model for the pomeron
As in similar approaches, developed by many authors [14, 15], the «Born term» for the pomeron is the two-gluons exchange. The rise of the cross-section with s can be realized by increasing the number of gluonic rungs in the ladder. At the order g6 for nucleon-nucleon scattering, or e2Q g6 for photon-proton, the cross-section will increase as ln (s).
Before starting the calculation we briefly summarize the main properties of the model. The final result will not depend on technical details, that can be found in [13], but only on the good convergence of all integrals in consequence of the chosen model for the propagator of a non-perturbative gluon. The analyticity properties of the gauge field propagator [16] lead to a two-term expression for the gluon structure function
D(2k2). The first term, of the form c/(k2
2 m2)2, enhances low-frequency modes and provides for screening as can be seen from the evaluation of the non-relativistic potential [17]. In the following this term will be referred to as the non-perturbative gluon propagator. The second term, 21 /(k2
2 M2), M2
cm2, represents the perturbative gluon propagator since c , m2 and M2 should run and vanish with the coupling. These parameters, together with as, will be considered as constants, for simplicity; their value can be determined from heavy-quark spectroscopy [17] and proton-proton scattering [13].
Other Green’s functions, necessary to complete the evaluation of the relevant diagrams, are determined from the solution of the Schwinger-Dyson equation for the ghost propagator in Landau gauge. Ghosts are not important in our calculation, they do
not even appear at the order we consider, but the correction factor for the ghost propagator can be used for identifying renormalization constants [18]. It turns out [13] that the correction factor for the ghost propagator, at the scale m2, is of the order m2/M2, that is a very small quantity.
We get further constraints on the model from the Slavnov-Taylor identity for the three-gluon vertex. In the limit of any one of the incoming momenta going to zero, the emission of a non-perturbative gluon from a perturbative one is strongly suppressed because of the presence of the correction factor for the ghost propagator. If all incoming momenta go to zero, we get the perturbative answer multiplied by a factor that changes one of the non-perturbative propagators to a perturbative one. Another important consequence of the model regards the emission of two non-perturbative gluons from a quark. In this case, each quark-gluon vertex is multiplied by the factor
m2/M2[18] and, approximately, one of the two gluons becomes perturbative.
Since in the triple vertex function at least one gluon must be perturbative, we can say that the non-perturbative gluon is «Abelian-like». In the limit c K0, we recover perturbative QCD if we add diagrams with scalar fields that give a mass M to the gluon through the Higgs mechanism. It turns out, however, that the contribution of diagrams where only perturbative gluons are exchanged is negligible because the mass m is very small, approximately 50 MeV from spectroscopy [17].
In fig. 1a), b) we show two diagrams (with weight two) that contribute to the amplitude at the order e2Q g6. The gluonic part, where the spiral lines represent non-perturbative gluons, exemplify the two possible gluon configurations. It has been shown in [13] that, for t 40, cancellations occur between radiative corrections and the diagrams with three-gluon exchange in the t-channel. Hence, only four diagrams, with weight two, must be evaluated. Two of them are purely perturbative and will be neglected. Another simplification regards the choice of the gauge. While the model has been established in Landau gauge, Feynman gauge can be used throughout the
Fig. 1. – Diagrams evaluated in the text for the g *-quark scattering. Spirals denote non-perturbative gluons.
calculation. The resulting structure of the pomeron, one soft and one hard gluon in the
t-channel, is characteristic of other models [19].
The evaluation of the diagram in fig. 1b) will follow the method of momentum space technique [20] for the gluon ladder and use Sudakov variables for the quark loop: k 4
ap 1bq1kA, kA2G 0. The transition from quark to nucleon scattering for the quark p,
p2
4 0, relies on the non-relativistic quark model [21]. In a coordinate system where the large components of the momenta of the external particles are along the z-axis, let
p 4 (v, 0, 2v) . Then (q2 G 0 ) q 4 1 2
g
q2v p Q q 1 p Q q v , 0 , p Q q v 2 q2v p Q qh
and the expression for k is easily obtained from the Sudakov parametrization. Now we can evaluate the amplitude for the forward scattering of the virtual photon on the proton target and, from its imaginary part, the DIS structure functions
Wmn4 2W1
g
gmn2 qmqn q2h
1 W2 Mp2g
pm2 q Q p q2 qmhg
pn2 q Q p q2 qnh
. (1)The proton structure function can be obtained, for example, from the coefficient of pmqn
in (1) 2q Q p q2 Q W2 Mp2 4 2 F2 Mp2q2 ,
where Mpis the proton mass.
For fig. 1b), we must evaluate the imaginary part of
216 ig6eF2
d4k ( 2 p)4 1 (k2 2 m21 ie)[ (k 2 q)22 m21 ie] 3 (2) 3 d 4 k1 ( 2 p)4 d4k2 ( 2 p)4 N(b) D(b) F(k2 2) ,where the factor 2 16 comprises the color factor, the diagram with the reversed quark arrows, the different disposition of non-perturbative gluons and a sum over the final proton spin. eFis the charge of the quark of color F and, in the calculation, the proton
wave function F(k22) will become [21]
F 4 k2 » 2 w21 k2 »2 , (3) where w2 4 1 arp2b B 0.113 GeV2.
With the help of an algebraic program [22] it is not difficult to determine the coefficient of pmqnin N(b) 4 Tr [gs(k×2 k×12 q× 1m) gn(k×2 k×11m) gl(k×1m) gm(k×2 q× 1m) ]3 3 (Alps1 Aspl1 Bgls1 Cplps) , where Al4 6[p Q (k22 k1) k1 l2 p Q k2k2 l] , B 42[pQ (2k12 k2) ]2, C 42(k11 k2)2
in Feynman gauge. Only the coefficient of C will give terms increasing like ln (p Q q) in the final result and will be retained in the following.
D(b)in this model is
D(b)4 [ (k 2 k1)22 m21 ie] [ (k 2 k12 q)22 m21 ie] [ (p 1 k2)21 ie] G , where the gluonic part G is
G21 4 c2
»
i 41 2 [ (ki22 M21 ie)21(ki22 m21 ie)22] 3 3 ][ (k12 k2)22 M21 ie]212 c[ (k12 k2)22 m21 ie]22( . Mass parameters can be taken as in [17] but the determination of g2 and c is more subtle since spectroscopy determines only their product and their values depend on the scale of the gluon momenta. This is true also for m and M, but a small variation of these last quantities does not change the result sensibly.3. – Calculation and results
The involved structure of expression (2) makes a fit of g2 and c problematic and suggests the following procedure. After an integration over the light-cone variables
k1 6 and k2 6 following the technique of [20] for obtaining the leading part of the
s-channel discontinuity, we perform another integration over a closing on the pole a 4 ( 1 2b) 2 q22 (k» 2 1 m2) 1ie 2( 1 2b) pQq .
We firstly evaluate analytically the integrals over k1 », k2 »and the relative angle and verify that, for q2
4 0 and one value of (p Q q), the result is reasonable by comparing it with g-p cross-section data [4]. The value chosen for (p Q q) corresponds to a center-of-mass energy ks 4195 GeV. We included in the computation the contribution of the diagram in fig. 1a) that can be obtained from eq. (2) with a change of sign, setting
k1 »4 0 in N(b) and modifying properly D(b). Whereas for p-p scattering, at the same energy, the value of as4 g2/( 4 p) comes out to be near 0.4 [13], in this case we find a smaller coupling: asC 0.33.
The above result is gratifying since we succeeded in estimating the photon-proton cross-section with parameters obtained from spectroscopy and a sensible value for the strong coupling. Integration in (2), however, leads to cumbersome expressions already for q2
4 0 and is practically impossible for q2c0 if we want, at the same time, impose a lower bound on the pomeron subenergy. As we will see later this condition has both physical and technical grounds.
The integrals over k1and k2in (2), after integrating over the Sudakov variable a and taking into account both diagrams in fig. 1, give a contribution proportional to
0 Q 0 Q du dv u( 2(u 1v)1M 2 2 c) (u 1M2 )(u 1m2)2 (v 1M2 )(v 1m2)2F(v) g(u) (4)and F(v) is as in (3). With the definitions
a 422b2 ( 1 2b)2q2 1 ( 1 2 2 b 1 2 b2)(k» 2 1 m2) and b 42b(12b) q2 1 k» 2 1 m2 we get g(u) 4 1 u
u
1 2 2 a b 1 2 a 2b2uk
(b 1u)22 4 k» 2 uv
B 2[k» 2 ( 2 a 2b)2a(u1b) ] b[ (b 1u)2 2 2 k» 2u] .Due to the good convergence of the integrals in (4) and the smallness of the mass m the main contribution to (4) will come from the region of small u. The drastic simplification of substituting the integral (4) with a constant multiplying g( 0 ) can be justified, in the model we considered, also with numerical tests. Results, for large Nq2N, will be underestimated but the structure of non-leading terms appears as in the exact computation. Apart from a multiplicative factor and constant terms, that cannot be determined within the method adopted, we get (x 4k»
2) F2 Q2 4 const ln ( 2 p Q q)
0 xmax dx bmin bmax db ab 2x(2a2b) b4 , (5) with Q24 2q2 and a , b as before. The integration limits will be defined below and all the parameters of the starting model disappeared. The latter was however necessary to justify the approximation leading to (5).
A constraint on the integration region is necessary for the validity of the momentum space technique applied to the gluon ladder. The integration limits in (5) will be determined from the physical condition that the pomeron exchange dominates only if the energy is sufficiently high. This condition requires that
(k 1p)2 D s0
and, above s0, the pomeron contribution is important. This bound gives ( 0 EbE1) b 1
2
u
1 1s02 m26
k
( 2 p Q q 1q21 s02 m2)22 4( 2 p Q q 1 q2)(s01 x)2 p Q q 1q2
v
,whereas xmax is determined from the vanishing of the square root in (6). The true parameter is now the mass of the quark in the loop, m, since the dependence of the result from s0 is rather weak. The unknown factor «const» includes all other parameters in the crude approximation explained above and will be determined by comparison with phenomenological parametrisations for the soft pomeron, that predict this high-energy behavior.
A typical example of a soft pomeron, with an intercept near one, is the one adopted in refs. [1, 2] and [23], where it has been proposed that non-perturbative solutions could help in explaining HERA data. While agreeing on the latter point, we will be more conservative in the choice of the soft phenomenological pomeron. We consider a dipole pomeron with unit intercept that predicts logarithmically rising total cross-sections [24]. One can interpret it as the coalition of two Regge poles with unit intercept or as the derivative, with respect to a(t), of a Regge pole.
Consequently we choose the parametrization for the real photon-proton cross-section, that will determine the unknown factor in (5), in the form [3]:
sgptot4 g1ln (s) 1g21 gfsaf( 0 ) 21,
(7)
where af( 0 ) varies between 0.7 and 0.8 [2, 3] depending on the model for the pomeron
adopted. The choice af( 0 ) 40.74 seems preferred in this calculation since the constant
Fig. 2. – Dipole pomeron fit of stotgp(dashed line) and contribution of the logarithmic term (dotted and full lines). Data from the CS database of the Particle Physics Data System [4].
Fig. 3. – Prediction of the model (see text) for the proton structure function F2(x , Q2), at different Q2values, vs. x (full lines). Data from [5, 6].
term, g2, becomes very small: g2A 20.0012 mb. With the values g14 0.0142 mb and gf4
0.167 mb we obtain the fit shown in fig. 2 with a dashed line. In the same figure the dotted line represents the contribution from the logarithm in (7). Since
sgptot4 4 p2a F2 Q2
N
Q2 4 0 and, from (5), F2 Q2N
Q2 4 0 C const 2 m2 ln (s) 1Oy
ln( s ) sz
we get
const 4 m 2 3.958
with m expressed in GeV (s04 100 GeV2). The full expression for the increasing part of sgptot has been plotted in fig. 2 (continuous line) and it is practically independent of m2.
On the contrary, F2 in (5) does depend strongly on m. The value m C412 MeV, about three pions, leads to the continuous curves in fig. 3 where ZEUS data [5] (squares) and H1 data [6] (circles) are also drawn. We have not considered higher values of Q2 because of the approximations inherent in the momentum space technique.
4. – Discussion
Equation (5) is the main result of this paper. It takes into account non-asymptotic effects related to the quark loop in fig. 1 and it contains a lower bound for the pomeron subenergy. The latter condition turns out to be important in explaining the predicted trend of the proton structure function F2(x , Q2) in fig. 3. The result depends only weakly on the precise value of the pomeron threshold s0.
The conclusion we can draw from this calculation is that preasymptotic effects can explain the main features of present experimental data for F2 in a limited, but interesting, Q2region. Limitations are due to approximations, in the calculation of the two-loop subgraph with gluons, that could change, through integration, the Q2 dependence.
It is well possible that the BFKL pomeron will become essential in explaining future data at smaller x. In such a case unitarity limit will be satisfied with the onset of shadowing corrections [25]. In ref. [26] present HERA data were shown to be compatible with a logarithmic parametrisation of the structure function. In that case, as in [3] and in the present approach, the Froissart bound is satisfied for all x and therefore the solution does not need unitarity corrections.
Asymptotically the structure function F2in (5) increases like ln ( 1 /x). This behavior cannot match with the result of perturbative QCD as far as the x-dependence, for small
x, is concerned. On the contrary, the Q2-dependence is near to the prediction of the GLAP equations, that is weaker than any power of ln (Q2). Deviations from perturbative QCD can be estimated numerically in eq. (5) and are O( ln ln Q2), for large Q2and fixed x. The validity of eq. (5), however, is limited to values of Q2
G 15 GeV2and, in order to proceed outside this region, we can only use our result as an initial condition to perturbative evolution with, eventually, shadowing corrections taken into account.
Many problems are left open because of technical difficulties. Some features of the result, however, are promising. The curved shape of F2(x), particularly evident for Q2
4 1.5 GeV2, simulates the screening of the gluon density predicted from the non-linear GLR equation [25]. Moreover, non-asymptotic contributions coming from the photon-pomeron coupling seem to persist till rather low x-values, 1023
2 1024. For
x D531023 the contribution from other Regge trajectories becomes important. The inclusion of secondary trajectories, like the f, requires however a different approach. This problem, as well as the correction of some shortcomings present in the model, will be considered elsewhere.
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