Central exclusive production of axial-vector $f_{1}$ mesons in proton-proton collisions within the tensor-pomeron approach

We discuss the central exclusive production of $f_{1}$ mesons in proton-proton collisions. The diffractive pomeron-pomeron fusion process within the tensor-pomeron approach is considered. Two ways to construct the pomeron-pomeron-$f_{1}$ coupling are discussed. The theoretical calculation of coupling constants is a challenging problem of nonperturbative QCD. We adjust the parameters of the model to the WA102 experimental data. The total cross section and differential distributions are presented. Predictions for LHC experiments are given. Detailed analysis of the distributions in $\phi_{pp}$ the azimuthal angle between the transverse momenta of the outgoing protons can help to check different models and to study real pattern of the absorption effects.


Introduction
In this contribution we discuss central exclusive production (CEP) of axial-vector f 1 (J P C = 1 ++ ) mesons in proton-proton collisions where p a,b , p 1,2 and λ a,b , λ 1,2 = ±1/2 denote the four-momenta and helicities of the protons, and k and λ = 0, ±1 denote the four-momentum and helicity of the f 1 meson, respectively. Here f 1 stands for one of the axial-vector mesons with J P C = 1 ++ , i.e. f 1 (1285) or f 1 (1420). This presentation summarises some of the key results of [1] to which we refer the reader for further details. CEP of f 1 (1285) and f 1 (1420) mesons was measured by WA102 Collaboration [2][3][4].
Their internal structure (qq, tetraquark, KK molecule) remains to be established. At high energies the double-pomeron exchange mechanism ( Figure 1) is expected to be dominant. The pomeron ( ) is essential object for understanding diffractive phenomena. Within QCD is a color singlet, predominantly gluonic object, thus the CEP of mesons has long been regarded as a potential source of glueballs. For soft reactions, calculations of the pomeron from first principle are currently not possible, and one has to retreat to Regge models to describe soft high-energy diffractive scattering. Until recently, the spin structure of the pomeron has not received much attention. It is well known that the pomeron carries vacuum quantum numbers with regard to charge, color, isospin and charge conjugation. But what about spin? It has been shown some time ago that the charge-conjugation C = +1 pomeron can be regarded as a coherent sum of elementary spin 2 + 4 + 6 + . . . exchanges [5]. The tensor-pomeron model introduced in [6] assumes this property. We treat the reaction (1) in this model, in which the pomeron exchange is described as effective rank 2 symmetric tensor exchange. This approach has a good basis from nonperturbative QCD using functional integral techniques [5]. A tensor character of the pomeron is also preferred in holographic QCD models [7][8][9][10].
The tensor-pomeron model was applied to two-body hadronic reactions [6,11,12], to photoproduction of π + π − pairs [13], to low-x deep inelastic lepton-nucleon scattering and photoproduction [14], and especially to CEP reactions see e.g. [15][16][17][18][19][20][21]. In this model the C = −1 odderon [22] is described as effective vector exchange. Exclusive reactions suitable for studies of the odderon exchange at high energies were discussed in [13,19,20]. Conceptually, vector-type couplings of the pomeron turn out to be rather questionable. For example, a vector pomeron implies that the total cross sections for pp and pp scattering at high energy have opposite sign [11]. But, of course, quantum field theory forbids negative cross sections. A further argument against a vector pomeron was shown in [14], mainly it does not give any contribution to photoproduction data. One may also ask about the possibility of a scalar coupling of the pomeron to external particles. While possible from the point of view of QFT, such a coupling is experimentally disfavoured. In [11] it was shown that STAR data [23] on polarised elastic pp scattering are compatible with the tensor-pomeron ansatz but clearly rule out a scalar character of the soft pomeron what its coupling concerns. Also some historical remarks on different views of the pomeron were made in [11]. In the light of our discussion here we cannot support the conclusions of [24,25] that the pomeron behaves (couples) like vector current.
The theoretical calculation of f 1 coupling is a challenging problem of nonperturbative QCD. We argue that the pomeron couplings play an important role, and that they should be treated as tensor couplings. Using our model we perform a fit to the available WA102 data [2,4] and we analyse whether our study could shed light on the f 1 couplings. In the future the model parameters ( f 1 coupling constants, cutoff parameters in form factors) could be adjusted by comparison with precise experimental data from both RHIC and the LHC. The π + π − π + π − channel seems well suited to measure the f 1 (1285) CEP at high energies. For a preliminary data of the reaction pp → pp2π + 2π − measured at LHC@13TeV by the ATLAS Collaboration see [26].
The relevant kinematic quantities are is the polarisation vector of the f 1 meson, ∆ ( ) and Γ ( pp) denote the effective propagator and proton vertex function, respectively, for the tensor-pomeron exchange [6]. The new quantity, to be studied here, is the f 1 coupling (vertex function). In our analysis we should also include absorption effects to the Born amplitude. Then the full amplitude is The amplitude including the pp-rescattering corrections can be written as (within the one-channeleikonal approach) where p t ,1 and p t ,2 are the transverse components of the momenta of the outgoing protons and k t is the transverse momentum carried around the pomeron loop. M Born pp→pp f 1 is the Born amplitude given by (3) withp t ,1 = p t ,1 − k t andp t ,2 = p t ,2 + k t . M pp→pp is the elastic pp scattering amplitude for large s and with the momentum transfer t = −k 2 t . In practice we work with the amplitudes in the high-energy approximation, i.e. assuming s-channel helicity conservation in the pomeron-proton vertex.

The pomeron-pomeronf 1 coupling
We follow two strategies for constructing the f 1 coupling and the vertex function. (1) Phenomenological approach. First we consider a fictitious process: the fusion of two "real spin-2 pomerons" (or tensor glueballs) of mass m giving an f 1 meson of J P C = 1 ++ . We make an angular momentum analysis of this reaction in its c.m. system, the rest system of the f 1 meson: . The spin 2 of these "pomerons" can be combined to a total spin S (0 S 4) and this must be combined with the orbital angular momentum l to give the J P C = 1 ++ values of the f 1 . There are two possibilities, (l, S) = (2, 2) and (4, 4) (see Appendix A of [15]), and corresponding coupling Lagrangians f 1 are: where M 0 ≡ 1 GeV (introduced for dimensional reasons), g ′ f 1 and g ′′ f 1 are dimensionless coupling constants, κλ is the effective field, U α is the f 1 field, and Γ (8) , Γ (10) are known tensor functions [1].
(2) Our second approach uses holographic QCD, in particular the Sakai-Sugimoto model [27][28][29] where the f 1 coupling is determined by the mixed axial-gravitational anomaly of QCD. In this approach (see Appendix B of [1]) with ′ a dimensionless constant and ′′ a constant of dimension GeV −2 . For the CEP reaction, we use the f 1 vertex derived from (9) supplemented by suitable form factor (11). For our fictitious reaction ( + → f 1 ) there is strict equivalence L CS = L (2,2) + L (4,4) if the couplings satisfy the relations For our CEP reaction (1) we are dealing with pomerons of mass squared t 1 , t 2 < 0 and, in general, t 1 = t 2 . Then, the equivalence relation for small values |t 1 | and |t 2 | will still be approximately true and we confirm this by explicit numerical studies (see Fig. 11 of [1]).
For realistic applications we should multiply the "bare" vertex Γ ( f 1 ) (q 1 , q 2 ) as derived from a corresponding coupling Lagrangian by a form factorF ( f 1 ) (t 1 , t 2 , k 2 ) which we take in the factorised ansatzF where the cutoff constant Λ E should be adjusted to experimental data. As discussed in Appendix B of [1], the prediction for ′′ / ′ obtained in the Sakai-Sugimoto model is for M KK = (949 · · · 1532) MeV. Usually [27] M KK is fixed by matching the mass of the lowest vector meson to that of the physical ρ meson, leading to M KK = 949 MeV. However, this choice leads to a tensor glueball mass which is too low, M T ≈ 1.5 GeV. The pomeron trajectory [α (t) = α (0) + α ′ t, α (0) = 1.0808, α ′ = 0.25 GeV −2 ] corresponds to M T ≈ 1.9 GeV, whereas lattice predictions correspond to M T 2.4 GeV.

Comparison with the WA102 data
The WA102 collaboration obtained for the pp → pp f 1 (1285) reaction the total cross section of σ exp = (6919 ± 886) nb at s = 29.1 GeV and for a cut on the central system |x F | 0.2 [2]. The WA102 collaboration also gave distributions in t and in φ pp (0 φ pp π), the azimuthal angle between the transverse momenta of the two outgoing protons. We are assuming that the reaction (1) is dominated by pomeron exchange already at s = 29.1 GeV. In [4] an interesting behaviour of the φ pp distribution for f 1 (1285) meson production for two different values of |t 1 − t 2 | was presented. In Figure 2 we show some of our results [1] which include absorptive corrections; see Eqs. (5), (6). We show the φ pp distribution of events from [4] for |t 1 − t 2 | 0.2 GeV 2 (left panels) and |t 1 −t 2 | 0.4 GeV 2 (right panels). From the top panels, it seems that the (l, S) = (4, 4) term (8) best reproduces the shape of the WA102 data. The absorption effects play a significant role there. In the bottom panels of Fig. 2 we examine the combination of two f 1 couplings ′ and ′′ calculated with the vertex (9). The ratio (12) agrees with the fit ′′ / ′ = −1.0 GeV −2 as far as the sign of this ratio is concerned, but not in its magnitude. This could indicate that the Sakai-Sugimoto model needs a more complicated form of reggeization of the tensor glueball propagator as indeed discussed in [29] in the context of CEP of η and η ′ mesons. It could also be an indication of the importance of secondary contributions with reggeon exchanges, i.e.
-, -, and -fusion processes. We get a reasonable description of the WA102 data with Λ E = 0.7 GeV and the following possibilities: (l, S) = (2, 2) term only : (l, S) = (4, 4) term only : CS terms : Now we can use our equivalence relation (10) in order to see to which (l, S) couplings (15) corresponds. Replacing in (10) m 2 by t 1 = t 2 = −0.1 GeV 2 and k 2 by m 2 Thus, the CS couplings of (15) correspond to a nearly pure (l, S) = (4, 4) coupling (14). In Figure 3 we show the results for the φ pp distributions for different cuts on |t 1 − t 2 | without and with the absorption effects included in the calculations. The results for the two (l, S) couplings  are shown. The absorption effects lead to a large reduction of the cross section. We obtain the ratio of full and Born cross sections, the survival factor, as 〈S 2 〉 = 0.5-0.7. Note that 〈S 2 〉 depends on the kinematics. We can see a large damping of the cross section in the region of φ pp ∼ π, especially for |t 1 − t 2 | 0.4 GeV 2 . We notice that our results for the (4, 4) term have similar shapes as those presented in [30] [see Figs. 3(c) and 3(d)] where the authors also included the absorption corrections.
Having fixed the parameters of the model in this way we will give predictions for the LHC experiments. Because of the possible influence of nonleading exchanges at low energies, these predictions for cross sections at high energies should be regarded rather as an upper limit. The secondary reggeon exchanges should give small contributions at high energies and in the midrapidity region. As discussed in Appendix D of [1] we expect that they should overestimate the cross sections by not more than a factor of 4.

Predictions for the LHC experiments
Now we wish to show (selected) results for the pp → pp f 1 (1285) reaction for the LHC; see [1] for many more results. In Figure 4 we show our predictions for the distributions of φ pp and the transverse momentum of the f 1 (1285) for s = 13 TeV, |y M | < 2.5, and for the cut on the leading protons of 0.17 GeV < |p y,p | < 0.50 GeV. The results for the (l, S) = (2, 2) term (7), the (4, 4) term (8), and for the ′ plus ′′ terms calculated with (9) for (12) obtained in the Sakai-Sugimoto model (see Appendix B of [1]) are shown. For comparison, the results for our fit to WA102 data ( ′′ / ′ = −1.0 GeV −2 ) are also presented. The contribution with ′′ / ′ = −6.25 GeV 2 gives a significantly different shape. This could be tested in experiments, such as ATLAS-ALFA [26], when both protons are measured. We obtain the ratio of full and Born cross sections as 〈S 2 〉 ≃ 0.3 for s = 13 TeV. The four-pion decay channel seems well suited to measure the CEP of the f 1 (1285) at the LHC [26]. We predict a large cross section for the exclusive axial-vector f 1 (1285) → 4π production compared to the CEP of the tensor f 2 (1270) → 4π [16,18]. The 4π continuum for the pp → pp4π reaction was studied in [17,31].

Conclusion
• The calculations for the pp → pp f 1 (1285) reaction have been performed in the tensorpomeron approach [6]. We have discussed in detail the forms of the f 1 coupling. Detailed tests of the Sakai-Sugimoto model are possible.
• We obtain a good description of the WA102 data at s = 29.1 GeV [2,4] assuming that the pp → pp f 1 (1285) reaction is dominated by pomeron-pomeron fusion.
• We obtain a large cross section for CEP of the f 1 (1285) of σ ∼ = 6 − 40 µb for the ALICE, ATLAS-ALFA, CMS, and LHCb experiments, depending on the assumed cuts (see Table III of [1]). Predictions for the STAR experiments at RHIC are also given in [1]. In all cases the absorption effects were included.
• Experimental studies of single meson CEP reactions will allow to extract many M coupling parameters. The holographic methods applied to QCD already give some predictions [1,29].
• Detailed analysis of the distributions in φ pp , the azimuthal angle between the transverse momenta of the outgoing protons, can help to solve several important problems for soft processes, to check/study the real pattern of the interaction (absorption models), to understand the difference in the dynamics of production of qq mesons and glueballs (or more accurately, states which are believed to have a large glueball component), to disentangle f 1and η-type resonances contributing to the same final channel.
• Such studies could be extended, for instance by the COMPASS experiment where presumably one could study the influence of reggeon-pomeron and reggeon-reggeon fusion terms. Future experiments available at the GSI-FAIR with HADES and PANDA should provide new information about the ρρ f 1 and ωω f 1 couplings [32].