PHYSICAL REVIEW D VOLUME 47, NUMBER 1 1 JANUARY 1993 Relating the @CD Pomemn to an effective gluon mass F. Halzen Deparment of Physics, Uniuersity of Wisconsin, Madison, Wisconsin M706 G. Krein and A. A. Natale Instituto de Fssica Tedrica, Unieersi dade Estadua/ Paulista, Rua Pamplona 1/5, 011I05, Sao Paulo, Sao Paulo, Brazil (Received 29 May 1992) We construct the Pomeron as an exchange of two nonperturbative gluons, where the nonpertur- bative gluon propagator is described by an approximate solution of the Schwinger-Dyson equation which contains a dynamically generated gluon mass. We compute the total and elastic differential (der/dt) cross sections for pp scattering, obtaining agreement with the experimental data for a gluon mass m = 370 MeV for AgcD ——300 MeV, In particular, the Pomeron effectively behaves like a photon-exchange diagram with a coupling determined by the gluon mass. PACS number(s): 12.38.Lg, 12.38.@k, 12.40.Gg, 13.85.Dz It is known that Pomeron exchange gives a good de- scription of the experimental data for total cross sections in hadron-hadron collisions, small-~t~ elastic scattering, and difFraction dissociation [1]. The phenomenological Pomeron exchange can be understood within QCD as the exchange of two gluons [2]; however, the perturbative cal- culation of the elastic quark-quark scattering amplitude through two-gluon exchange shows a singularity at t = 0. Although this singularity can be canceled when incorpo- rating quarks in the proton wave function, this procedure is not able to reproduce the t dependence of the differen- tial cross section observed in experiment [3]. The origin of this singularity is the pole in the gluon propagator at k = 0, and it is clear that the small-~t~ behavior of the cross section cannot be explained in perturbative QCD. It was noticed by Landshoff and Nachtmann [4] that, if the gluon propagator is finite at k = 0, the singularity in the two-gluon calculation of hadron-hadron scattering is eliminated. A behavior softer than a pole of the gluon propagator at small k is expected from nonperturbative effects of the QCD vacuum which leads to confinement. In this model the Pomeron is described by the exchange of two nonperturbative gluons, where by nonperturbative we mean a gluon whose propagator does not show a pole at k2 = 0. Several phenomenological consequences of such an approach have already been discussed in the lit- erature, assuming a simple form for the propagator [5]. Recently, Cudell and Ross [6] have solved an approximate Schwinger-Dyson (SD) equation for the gluon propagator in the axial gauge, and found a solution whose infrared behavior is less singular than a pole at A: = 0. This new solution for the gluon propagator, when incorporated in the LandshofF-Nachtmann model, gives reasonable agree- ment with the experimental data. In the following we will show how a gluon propaga- tor regularized by a dynamically generated gluon mass, derived by Cornwall some years ago [7], successfully describes nucleon-nucleon scattering in the two-gluon- exchange model. The propagator represents an approx- imate solution of the Schwinger-Dyson equation and, 2= 1 P =36. d2k g D(k2) (2) The origin is familiar from QED: multiple-photon ex- changes only modify the single-exchange amplitude by a Coulomb phase. The advantage of using an infrared- finite propagator is that the integral in Eq. (2) converges. Particularly, this solves the problems with factorization discussed in detail by Landshoff and Nachtmann [4]. This is not the case for most of the Schwinger-Dyson solu- tions for the gluon propagator found in the literature [6, 11]. We will show that in our model the phenomeno- logical value Po of 2 GeV [1], determined from the total cross section, is accommodated by a gluon mass m 2 (1.2)AqcD for AqcD = 150 (300) MeV. We will show that these parameters also describe the experimen- tal features of forward proton-proton scattering. The possibility that the gluon has a dynamically gen- erated mass has been advocated by Cornwall in a se- ries of papers [7, 8]. Lattice calculations also indicate that the gluon behaves like a massive particle [9]. A bare gluon mass has attractive phenomenological impli- cations [10], although it leads to known problems with unitarity. This problem is avoided in the case of a dy- namical mass. Approximate solutions of the Schwinger- Dyson equation, obtained in a particular gauge, do not unlike the propagator obtained by Cudell and Ross, it asymptotically obeys the renormalization group equation and is actually finite at A:2 = 0 as a result of the effective gluon mass. The model is described by a single parame- ter which is taken to be the ratio of the gluon mass and A = AgCD. In the Landshoff-Nachtmann model Pomeron exchange between quarks behaves like a photon-exchange diagram with an amplitude iPo (u7„u) (uy" u), where Po is the strength of the Pomeron coupling to quarks which is given by 295 296 F. HALZEN, G. KREIN, AND A. A. NATALE 47 q' + 4m'(q'& D '(qz) = q + m (qz) bg ln (3) result in a dynamical gluon mass generation [6, 11]. Corn- wall has, however, identified a gauge-invariant set of di- agrams for the Schwinger-Dyson equation which allows for a finite (and different from zero) result at kz = 0. Most importantly, he also found a massive trial propaga- tor that gives an excellent fit to the numerical solution. This solution incorporates the correct ultraviolet behav- ior of the gluon propagator according to the renormal- ization group. Because of ambiguities in numerical solu- tions of the Schwinger-Dyson equation it is much more important to have an approximate solution which obeys the renormalization group than a single numerical result where such behavior is not directly observed [7]. In the Feynman gauge the propagator is given by D„= ig„D—(q ), where (in Euclidean space) [7] two-gluon exchange can be written as [14] A(s, t) = is8n, [Ti —Tz], with d2kD + I D I Q q O (6) d kD —+k D ——k G„,k —— X qG2(q, o) —G, (q, q ——), (q) where n, = gz/4vr is the strong coupling constant, D(k) the gluon propagator given by Eq. (3), and G„(q, k) a convolution of proton wave functions: with the momentum-dependent dynamical mass given by G~(q, k) = d p dn g' (n, p) @(n, p —k —nq) . (8) ~2( 2) ~2 ( ) ln 4m~ A~ (4) In the above equations m is the gluon mass, and 6 = (33 —2ny)/48mz is the leading-order coefficient of the P function of the renormalization group equation. The coupling g is frozen and g D(q ) is formally independent of g. The solution is valid only for m ) A/2 [7]. Re- lating the gluon mass to the gluon condensate, Cornwall estimated m = 500 + 200 MeV for AqcD = 300 MeV. The description of the Pomeron by the exchange of two nonperturbative gluons is necessarily crude, e.g. , the calculation does not include multiple-gluon effects de- scribed by the I.ipatov equation [12]. The model does not incorporate an intercept larger than 1 and there- fore cross sections are independent of energy. In QCD- inspired models [13] of the total cross section our model of the Pomeron would describe the energy-independent part of the cross section associated with the interac- tions of the valence quarks. The rise of the cross sec- tion with energy is associated with the increasing num- ber of gluons in very-high-energy hadrons. Using a Regge parametrization the total pp cross section is fit- ted by oT (pp) 22.7(s/m„)Dos, where m~ is the pro- ton mass [1], and the "bare" Pomeron thus accounts for 22.7 mb energy-independent cross section. The model should also reproduce the differential cross section for pp elastic scattering up to t, 0.5 GeV, fo—r larger ~t~ val- ues the cross section will not be purely imaginary and will receive important contributions from multiple Pomeron exchange [1]. Notice that in @CD-inspired models the energy in- dependent part of the total cross section is roughly 30 mb [13]. The variation from model to model for the value of the constant part of the total cross section is just a refIection of the different fits to the energy dependence exhibited by the data. We will use the previously quoted value of 22.7 mb as representative. The amplitude for elastic proton-proton scattering via The wave function g(n, p) describes the amplitude for a quark of transverse momentum p and fractional longi- tudinal momentum o;. T~ represents contributions from diagrams where both gluons are attached to the same quark within the proton, whereas Tz comes from di- agrams in which the gluons are attached to difFerent quarks. G„(q, 0) is given by the Dirac form factor of the proton 4m' —2.79t 1 4m2 —t (1 —t/0. 71)2 (9) To estimate G~(q, k —q/2) we assume a proton wave func- tion peaked at n = 1/3 and use [6] G„(q, k —q/2) = Fi(q2 + 9~ kz —q2/4~ ). From the amplitude of Eq. (5) we obtain both the total cross section oT = A(s, 0)/is and the differential cross section der/dt = ]A(s, t)~z/(16vrsz). For very small ~t~- values the elastic differential cross section can be fitted as do/dt = Ae i, where B is the logarithmic slope of the forward amplitude. The validity of the model requires that a reasonable value of the gluon mass m yields o.o —— 22.7 mb. B and do/dt are subsequently predicted. We will make this comparison for a c.m. energy of ~s = 53 GeV, i.e. , an energy where the rising component of the total cross section does not significantly contribute. At this energy B = 11.06+0.11 GeV and (do'/dt)i 94.0 + 1.0 mb/GeV~ [15]. In Fig. 1 we show the total cross section for pp scat- tering as a function of the gluon mass. The different curves are for different values of A. Although the model is basically parametrized in terms of their ratio, there is some residual dependence on the gluon mass and A. Notice that for increasing A agreement with o.T is only obtained by decreasing the ratio m/A. Increasing A only yields the proper value of oz- for smaller m, and here one must remember that Eq. (3) only solves the Schwinger- Dyson equation for m ) A/2 [7]. Only a small region in (mni A) parameter space remains when fitting the full set of data (crT or Po and B). We obtain excellent agreement 47 RELATING THE QCD POMERON TO AN EFFECTIVE GLUON MASS 297 210 —I t I I f I t I I I 60— 101 40— 100 20— b 1Q 1 0 I I 0.2 0.3 M (Gev) 0.4 0.5 1p 2 0.3 (Gev ) 0. 1 0.2 Q 4 Q 5 FIG. 1. Total cross section (oT ) as a function of the gluon mass m for different values of A: A = 300 MeV (solid curve), A = 200 MeV (dashed curve), and A = 150 MeV (dotted curve . FIG. 3. Differential pp elastic cross section obtained from double nonperturbative gluon exchange, with m = 370 MeV and A = 300 MeV (solid curve). Data from Ref. [15] at v s = 53 GeV. between theory and experiment for m = 370 MeV and A = 300 MeV. These parameters yield oc = 21.6 mb, B = 11.5 GeV 2, and Pc = 2.3 GeV In Fig. 2 we show the slope B for small [t[ as a function of A. The curves represent difFerent values of the ratio m/A. For A in the interval of 150 to 300 GeV, we verify that smaller A values require larger masses. That this simple model nicely accommodates do. /dt can be seen in Fig. 3. Our results have to be adjusted by a normalization factor s ~ which accounts for the energy- dependent part of do/dt [1]. The excellent fit implies that the energy dependent part cannot have a drastically different structure in impact parameter. This is true in most models, see, e.g. , Ref. [13]. For larger [t[ values, double-Pomeron exchange and three-gluon exchange are known to be important; we therefore do not expect to describe the data near and above t 0.5 Ge—V [1, 16]. T T i & 1 t &» I & & i & I » & & t 7 20 The above fits assumed a nominal value of 22.7 mb for the energy-independent part of the total cross section. Different models for the energy dependence of the total cross section lead to a range of values between 20 and 30 mb. This range corresponds to a spread in the gluon mass of m 2 (1.2)AqgD for AqeD = 150 (300) MeV. In Eq. (5) the amplitude Ti where the gluons couple to the same quark recovers the factorizable, additive quark model, T2 does not. For the Btted values of the gluon mass Tq is slightly larger than Tq with their ratio in- creasing for larger gluon masses. The fact that Tq is larger than T2 implies that the additive quark model is valid in an approximate sense. In our model this result is not critically dependent on the form of hadronic wave functions. For [t] of the order of 3 GeV or larger, triple-gluon exchange is expected to dominate the pp scattering cross section, and its contribution has the form [16] Ct [n, (t)] 16 14 12 10 I I I I I I I I I I I I I I I I + J I 0. 1 0.2 0.4 0.5 I 0.:3 A (reV) FIG. 2. Slope of the differential py scattering cross section as a function of A. The different curves represent different values of the ratio m/A: m/A = 0.9 (dashed curve), m/A = 1 (solid curve), m/A = 1.1 (dotted curve), and m/A = 1.3 (dot- dashed curve). The experimental data can be fitted by a t s behavior with a coeKcient C which is approximately independent of energy. However, if n, in Eq. (10) is allowed to run agreement between the theory and experiment is ruined. Notice however, that if the exchange of the three gluons is described by the propagator of Eq. (3) which is of the O(1/n, ), the coupling constants in Eq. (10) are exactly canceled. In conclusion, we verified that a model where the Pomeron is described by the exchange of two nonper- turbative gluons with massive propagators provides an excellent description of small-angle pp scattering. Even for moderate values of ]t], where triple gluon exchange dominates the behavior of the differential cross section, the propagator given by Eq. (3) may give a hint of how an apparent contradiction between theory and experi- ment [16] can be resolved. Finally, the determination of a dynamical gluon mass of order 1.2 2AqpD is consis- F. HALZEN, G. KREIN, AND A. A. NATALE 47 tent with totally unrelated estimates [7]. If we assume an energy-independent part of the total cross section of 22.7 mb, then a best Gt to the experimental data is ob- tained for m = 370 MeV when A@~0 ——300 MeV. Since a massive gluon propagator has many other phenomeno- logical implications [5, 10], a study of these together with the total cross section calculation, may provide a tighter relation between m and AqcD. We thank Jean-Rene Cudell and Peter Landshoff for many constructive comments on an early version of this paper. This research was supported in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation, by the U.S. Department of Energy under Contract No. DE-AC02-76ER00881, by the Texas Na- tional Research Laboratory Commission under Grant No. RGFY9173, and by the Conselho Nacional de De- senvolvimento Cientifico e Tecnologico (CNPq). [1] P. V. LandshofF, in Proceedings of the Joint Interna tional Lepton-Photon Symposium and Europhysics Con- ference on High Energy Physics, Geneva, Switzerland, 1991, edited by S. Hegarty, K. Potter, and E. Quercigh (World Scientific, Singapore, 1992); Report No. CERN- TH-6277/91 (unpublished) . [2] F. E. Low, Phys. Rev. D 12, 163 (1975); S. Nussinov, Phys. Rev. Lett. 34, 1268 (1975). [3] D. G. Richards, Nucl. Phys. B258, 267 (1985). [4] P. V. Landshoff and O. Nachtmann, Z. Phys. C 35, 405 (1987). [5] A. Donnachie and P. V. Landshoff, Nucl. Phys. B311, 509 (1988/89); J. R. Cudell, A. Dorm achie, and P. V. Landshoff, ibid. B322, 55 (1989); J. R. Cudell, ibid B336,. 1 (1990). [6] J. R. Cudell and D. A. Ross, Nucl. Phys. B359, 247 (1991). [7] J. M. Cornwall, Phys. Rev. D 26, 1453 (1982). [8] J. M. Cornwall, in Deeper Pathioays in High Energy- Physics, edited by B. Kursunoglu, A. Perlmutter, and L. Scott (Plenum, New York, 1977), p. 683; Nucl. Phys. B157, 392 (1979). [9] C. Bernard, Phys. Lett. 108B,431 (1982); P. A. Amund- sen and J. Greensite, Phys. Lett. B 173, 179 (1986); J. E. Mandula and M. Ogilvie, ibid. 185, 127 (1987). [10] G. Parisi and R. Petronzio, Phys. Lett. 94B, 51 (1980). [11] M. Baker, J. S. Ball, and F. Zachariasen, Nucl. Phys. B186, 531 (1981); B186, 560 (1981); N. Brown and M. R. Pennington, Phys. Rev. D 38, 2266 (1988); 39, 2723 (1989). [12] L. N. Lipatov, and G. V. Frolov, Yad. Fiz. 13, 588 (1971.) [Sov. J. Nucl. Phys. 13, 333 (1971)];E. Kuraev, L. Lipa- tov, and V. Fadin, Zh. Eksp. Teor. Fiz. 72, 377 (1977) [Sov. Phys. JETP 45, 199 (1977)];Y. Balitsky and L. Li- patov, Yad. Fiz. 28, 1597 (1978) [Sov. J. Nucl. Phys. 28, 822 (1978)]. [13] M. M. Block, F. Halzen, and B. Margolis, Phys. Rev. D 45, 839 (1992), and references therein. [14] J. F. Gunion and D. Soper, Phys. Rev. D 15, 2617 (1977); E. M. Levin and M. G. Ryskin, Yad. Fiz. 41, 1622 (1985) [Sov. J. Nucl. Phys. 41, 1027 (1985)]. [15] A. Breakstone et a/. , Nucl. Phys. B248, 253 (1984). [16] A. Donnachie and P. V. Landshoif, Nucl. Phys. B231, 189 (1984).