Central exclusive diffractive production of axial-vector $f_{1}(1285)$ and $f_{1}(1420)$ mesons in proton-proton collisions

We present a study of the central exclusive diffractive production of the $f_{1}(1285)$ and $f_{1}(1420)$ resonances in proton-proton collisions. The theoretical results are calculated within the tensor-pomeron approach. Two pomeron-pomeron-$f_{1}$ tensorial couplings labelled by $(l,S) = (2,2)$ and $(4,4)$ are derived. We adjust the model parameters (coupling constants, cutoff constant) to the WA102 experimental data taking into account absorption effects. Both, the $(l,S) = (2,2)$ and $(4,4)$ couplings separately, allow to describe the WA102 differential distributions. We compare these predictions with those of the Sakai-Sugimoto model, where the pomeron-pomeron-$f_{1}$ couplings are determined by the mixed axial-gravitational anomaly of QCD. We derive an approximate relation between the pomeron-pomeron-$f_{1}$ coupling constants of this approach and the $(l,S) = (2,2)$ and $(4,4)$ couplings. Then we present our predictions for the energies available at the RHIC and LHC. The total cross sections and several differential distributions are presented. We find for the $f_{1}(1285)$ a total cross section $\sim 38$ $\mu$b for $\sqrt{s} = 13$ TeV and a rapidity cut on the $f_{1}$ of $|{\rm y_{M}}|<2.5$. We predict a much larger cross section for production of $f_{1}(1285)$ than for production of $f_{2}(1270)$ in the $\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ decay channel for the LHC energies. This opens a possibility to study the $f_{1}(1285)$ meson in experiments planned at the LHC.


I. INTRODUCTION
of [23]. In [34] a study of CEP of the f 2 (1270) meson was presented. The f 2 (1270) is expected to be abundantly produced in the pp → ppπ + π − reaction, and it was discussed in [34] how to extract the PP f 2 (1270) coupling from RHIC and LHC experimental results. We refer the reader to [35][36][37][38][39] for the latest measurements of central π + π − production in high-energy proton-(anti)proton collisions. In [39] a study of CEP of π + π − , K + K − , and pp pairs in pp collisions at a center-of-mass energy of √ s = 200 GeV by the STAR Collaboration at RHIC was reported. For the first (preliminary) STAR experimental results measured at √ s = 510 GeV see Ref. [40]. There are ongoing studies of CEP of the π + π − π + π − channel.
In this article we consider diffractive production of axial-vector f 1 -type mesons in the pp → pp f 1 reaction within the tensor-pomeron approach. As concrete examples we shall consider CEP of the f 1 (1285) and the f 1 (1420) via the pomeron-pomeron-fusion mechanism. We shall give a detailed discussion of various ways to write the PP f 1 couplings. In the calculations we include the absorptive corrections and show their role in describing the data measured by the WA102 Collaboration [3]. We will try to analyse whether our study could shed light on the nonperturbative PP f 1 couplings. In the future the corresponding PP f 1 couplings could be adjusted by comparison with precise experimental data from both RHIC and the LHC.
We also consider the PP f 1 couplings that follow from holographic models of QCD, in particular the Sakai-Sugimoto model based on type IIA superstring theory [41]. In the low energy regime this model is a gravitational dual to large-N c QCD, where glueballs are described by fluctuations of a confining geometry [42][43][44][45][46], and the pomeron can be represented by reggeization of the tensor glueball [13]. Quark degrees of freedom are introduced as probe branes in this background and their gauge field fluctuations are dual to mesons [47,48]. In [14] the PPη 0 couplings were derived from the bulk Chern-Simons term, which is uniquely fixed by requiring consistency of supergravity and the gravitational anomaly. Due to its universal form, the structure of the resulting couplings should be the same in all holographic models, although the strength of the couplings may vary. 1 In a similar calculation as was done in [14], we derive the PP f 1 couplings relevant for this study.
The four-pion channel, discussed in the past by the WA91 [52] and WA102 [1,5] Collaborations, seems to be a good candidate for an f 1 (1285) study in high-energy pp collisions. The intermediate states that should be considered are the J P = 1 + states ρ 0 ρ 0 and ρ 0 (π + π − ) P wave . The central π + π − π + π − system in proton-proton collisions was measured also by the ABCDHW Collaboration at √ s = 63 GeV at the CERN Intersecting Storage Rings (ISR); see Ref. [53]. A spin-parity decomposition of the 4π, ρππ, and ρρ states as a function of M 4π was performed with the assumption that the dominant contributions arise from J P = 0 + and 2 + states. Five contributions to the four-pion spectrum were identified: a 4π phase-space term with total angular momentum J = 0, two ρππ terms with J = 0 and J = 2, and two ρρ terms (J = 0, 2). Thus, an enhancement observed in the region M 4π ∼ 1300 MeV for the J P = 2 + ρρ and ρππ terms was assigned to the f 2 (1270) meson and for the J P = 0 + ρππ term to the f 0 (1370) meson (called f 0 (1400) in [53]). However, the J P = 1 + and J P = 0 − terms, possible in this process (e.g., via PP fusion), were not considered in the spin-parity analysis. This may invalidate the final conclusions of [53] where the enhancement in the four-pion invariant mass region around the PP f 1 (1285) and PP f 1 (1420) coupling constants, and in Appendix D we discuss subleading reggeon exchanges.
The new and unknown main ingredient of the amplitude (2.4) is the pomeron-pomeronf 1 vertex Γ (PP f 1 ) which we want to study in the present article. In [21,23,[25][26][27][28]34] the following strategy for constructing pomeron-pomeron-meson (PPM) couplings was followed. First one looked at the possible couplings of two fictitious "real" pomerons to the meson M. This was easily done using elementary angular-momentum algebra; see Appendix A of [21]. Then PPM couplings were written down corresponding to the allowed values of orbital angular momentum l and total PP spin S for a given meson M in question. Finally these couplings were also used for the CEP reaction pp → pMp. We follow this strategy also for CEP of an f 1 meson. Thus, we investigate first the fictitious reaction where P are "real pomerons" of mass squared t > 0 and with polarisation tensors ǫ (1) and ǫ (2) . From the analysis of this type of reactions presented in Appendix A of [21] we find that for the f 1 with J P = 1 + there are two independent amplitudes for the reaction (2.5), labelled by (l, S) = (2, 2) and (4,4). Convenient covariant couplings leading to these amplitudes are easily constructed; see (A5) and (A7) in Appendix A. But these constructions are not unique. In the Sakai-Sugimoto model [47,48] the coupling of an I G = 0 + , J P = 1 + axial-vector meson to two tensor glueballs is determined by the gravitational Chern-Simons (CS) action describing axial-gravitational anomalies; see (59) of [14]. Identifying the tensor glueballs with the fictitious "real pomerons" of (2.5) we have derived corresponding bare coupling Lagrangians PP f 1 in (B3) and (B4) of Appendix B.
For the fictitious on-shell process (2.5) the sum of the Lagrangians of (A5) and (A7) is strictly equivalent to the sum of (B3) and (B4). The relation of the respective coupling constants is given in (B13). But for the realistic case where the pomerons have invariant masses t 1,2 < 0 and in general t 1 = t 2 this equivalence no longer holds. But we can expect that for small values |t 1 |, |t 2 | 0.5 GeV 2 the off-shell effects should not be drastic. And this, indeed, is confirmed by the explicit study presented in Appendix B.
In the following we shall present the formulas using the couplings (A5) and (A7) of Appendix A. The formulas using the couplings (B3) and (B4) of Appendix B are completely analogues. Results will be shown for both types of couplings.
From the coupling Lagrangians of Appendix A we obtain the following PP f 1 vertex The Γ ′ and Γ ′′ vertices in (2.6) correspond to (l, S) = (2, 2) and (4,4), respectively, as derived from the corresponding coupling Lagrangians (A5) and (A7) in Appendix A. The expressions for these PP f 1 vertices 2 are as follows: is defined in (A2), and g ′ PP f 1 , g ′′ PP f 1 are dimensionless coupling constants. The values of these coupling constants are not known and are not easy to obtain from first principles of QCD, as they are of nonperturbative origin. At the present stage the coupling constants, g ′ PP f 1 and g ′′ PP f 1 , should be fitted to experimental data.
For realistic applications we should multiply the 'bare' vertices (2.7) and (2.8) by a form factorF (PP f 1 ) which we take in the factorised ansatz as 3 For the on-shell meson we have F (PP f 1 ) (m 2 f 1 ) = 1. In (2.9) we use Alternatively, we use the exponential form given as where we have set k 2 = m 2 f 1 and the cutoff constant Λ E should be adjusted to experimental data.
In the high-energy and small-angle approximation, using (D. 18) in Appendix D of [21], the PP-fusion amplitude reads: (2.12) 2 Here the label "bare" is used for a vertex as derived from a corresponding coupling Lagrangian without a form-factor function. 3 We are taking in (2.6) the same form factor for each vertex Γ ′ and Γ ′′ . In principle, we could take different form factors for each of the vertices. We follow (3.43), (3.44) of [10] and take β PNN = 1.87 GeV −1 , and, for simplicity, we use for the pomeron-proton coupling the electromagnetic Dirac form factor F 1 (t) of the proton. The pomeron trajectory α P (t) is assumed to be of standard linear form, see e.g. [60,61], (2.14) For the PP f 1 vertex function we shall use in the following the form (2.6) with the bare vertices either from (2.7) and (2.8) (corresponding to the couplings discussed in Appendix A) or those from (B8) and (B9) from Appendix B.
Note that the vertices (2.7) and (2.8) derived from the coupling Lagrangians (A5) and (A7) automatically are divergence free, i.e., they satisfy For the vertices derived from (B3) and (B4) this does not hold. Thus, in calculations of cross sections with the vertices (B8) and (B9) one has to use for the f 1 spin sum since the k µ k ν term will give a non zero contribution. With the vertices from (2.7) and (2.8) the k µ k ν term does not contribute.
To give the full physical amplitudes we should include absorptive corrections to the Born amplitude. For the details how to include the pp-rescattering corrections in the eikonal approximation see e.g. Sec. 3.3 of [22]. In practice we work with the amplitudes in the high-energy approximation.

III. RESULTS
In this section we wish to present first results for the pp → pp f 1 (1285) and pp → pp f 1 (1420) reactions. We will first discuss the pp → pp f 1 reactions at the relatively low c.m. energy √ s = 29.1 GeV and compare our model results with the WA102 experimental data from [3]. We shall try to fix the parameters of our model including at first only the PP-fusion mechanism. Then we shall make predictions for the experiments at the RHIC and LHC. The secondary reggeon exchanges should give small contributions at high energies and in the midrapidity region. However, they may influence the absolute normalization of the cross section at low energies. Therefore, our predictions for the RHIC and LHC experiments, obtained in this way, should be regarded rather as an upper limit for the pp → pp f 1 reactions, but, as discussed in Appendix D, we expect that they should overestimate the cross sections by not more than a factor of four.

A. Comparison with the WA102 data
According to [3] the WA102 experimental cross sections are as quoted in Table I. 4 In [3] also the distributions in |t| and φ pp for the f 1 (1285) and f 1 (1420) meson production 4 Note that the cross sections for f 1 (1285) and f 1 (1420) mesons quoted in Table 1   at √ s = 29.1 GeV were presented. Here, t is the four-momentum transfer squared from one of the proton vertices [we have t = t 1 or t 2 , cf. (2.2)], and φ pp is the azimuthal angle between the transverse momentum vectors p t,1 , p t,2 of the outgoing protons.
In Fig. 2 we show the results for the f 1 (1285) meson production for √ s = 29.1 GeV and for the Feynman variable of the meson |x F,M | 0.2. 5 The WA102 data points from [3] and our model results have been normalised to the mean value of the total cross section σ exp. = (6919 ± 886) nb ; (3.1) see Table I. The experimental error of the total cross section is about 12.8 % (3.1) and is dominated by systematic effects. Correspondingly the error bars quoted in Fig. 2 are assumed to be 12.8 % of the cross section for each bin. We show the results for different PP f 1 couplings discussed in the present paper. The theoretical calculations in the top panels of Fig. 2 correspond to the (l, S) = (2, 2) term (2.7) while those in the bottom panels to the (4, 4) term (2.8). We can see from the left panels of Fig. 2 that the t-dependence of f 1 production is very sensitive to the form factor F (PP f 1 ) in the pomeron-pomeron-meson vertex. The results with the exponential form (2.11) and Λ E = 0.7 GeV describe the t dependence better than (2.9) with (2.10). The calculations with (2.11) give a sizeable decrease of the cross section at large |t|. Therefore, in the following we show the results calculated with (2.11). At t = 0 (here t = t 1 or t 2 ) all contributions vanish. Both the (l, S) = (2, 2) and (4, 4) couplings considered separately allow to describe the WA102 differential distributions.
In [62] 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 Fig. 3 we show the φ pp distribution of events from [62] for |t 1 − t 2 | 0.2 GeV 2 (left panel) and |t 1 − t 2 | 0.4 GeV 2 (right panel). Our model results have been normalised to the mean value of the number of events. The results for Λ E = 0.7 GeV in (2.11) are shown. We have checked that for Λ E = 0.6 GeV the shape of the φ pp distributions is almost the same. An almost "flat" distribution at large values of |t 1 − t 2 | can be observed. It seems that the (l, S) = (4, 4) term best reproduces the shape of the WA102 data. As we will show below in Fig. 4, the absorption effects play a significant role there. Note, that in [62] also the number of events for the f 1 (1285) meson for the two kinematical conditions (a) |t 1 − t 2 | 0.2 GeV 2 and (b) |t 1 − t 2 | 0.4 GeV 2 was given. The experimental ratio is R exp. = N a /N b ≃ 8.6, where N a and N b are the number of events  [62], respectively. Then, we define the ratio From our model using Λ E = 0.7 GeV in (2.11) we get for the (2, 2) term (2.7) the ratio R = 8.6, while for the (4, 4) term (2.8) we get R = 5.6. If we use Λ E = 0.6 GeV we get R = 15.9 and R = 10.3, respectively. Therefore, for the (2, 2) term, Λ E = 0.7 GeV is a good choice, while for the (4, 4) term we should use a bit smaller value. For the κ ′ term given by (B8) and Λ E = 0.7 GeV we get R = 13.2 while for Λ E = 0.8 GeV we get R = 8.8. For the (κ ′ , κ ′′ ) terms, respectively for κ ′′ /κ ′ = −(6.25, 3.76, 2.44, 1.0) GeV −2 and Λ E = 0.7 GeV we get R = (7.6, 10.5, 11.9, 13.2). In Fig. 4 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 [63] (see Figs. 3 (c), (d)) where the authors also included the absorption corrections.
In [3] also the dP t dependence for both the f 1 (1285) and f 1 (1420) mesons was presented. Here, dP t (the so-called 'glueball-filter variable' [6,64]) is defined as  The experimental values for the cross sections in three dP t intervals and for the ratio of f 1 production at small dP t to large dP t are given there. In Table II we show the WA102 data and our corresponding results for the different PP f 1 couplings. The small values of the experimental ratios for the f 1 (1285) and the f 1 (1420) as listed in the last column may signal that these two mesons are predominantly qq states [6]. From the comparison  TABLE II. Results of f 1 -meson production as a function of dP t expressed as percentage of the total contribution at the WA102 collision energy √ s = 29.1 GeV and for |x F,M | 0.2. In the last column the ratios of σ(dP t 0.2 GeV)/σ(dP t 0.5 GeV) are given. The experimental numbers are from [3]. The theoretical numbers correspond to the separate individual coupling terms (l, S) = (2, 2) and (4,4), see (2.7) and (2.8) respectively, for different Λ E parameters in the relevant type of the PP f 1 form factor. The κ ′ and κ ′′ results were calculated from (B8) and (B9), respectively. We show the results for the coupling range given by Eq. (3.4) and the result for κ ′′ /κ ′ = −1.0 GeV −2 from our fit to the WA102 data. The absorption effects have been included in our analysis within the one-channel-eikonal approach.
Meson dP t 0.2 GeV 0.2 dP t 0.5 GeV dP t 0.5 GeV Ratio of the first four rows we see again that the exponential form of the t dependences in the (l, S) = (2, 2) PP f 1 (1285) vertex is preferred. For the (4, 4) term an optimal value of the Λ E parameter is in the range of (0.6-0.7) GeV. There are also shown the results obtained for the couplings (B8) and (B9) and for the ratio of coupling constants from (3.4); see (B7) of Appendix B. For comparison, the results for κ ′′ /κ ′ = −1.0 GeV −2 are also presented. We use here the form factor (2.11) with Λ E = 0.7 GeV. Up to now, in Figs. 2, 3 and 4, we have shown the contributions of the individual (l, S) terms (couplings), calculated with the vertices (2.7) and (2.8), separately. In Fig. 5 we examine the combination of two PP f 1 couplings κ ′ and κ ′′ calculated with the vertices (B8) and (B9), respectively. We can see that the best fit is for the ratio κ ′′ /κ ′ ≃ −1.0 GeV −2 (see the red dotted lines on the top panels), which roughly agrees with the preliminary analysis performed in [65] (cf. Eq. (2.68) in [65]).
As discussed in Appendix B, the prediction for κ ′′ /κ ′ obtained in the Sakai-Sugimoto model is for M KK = (949 . . . 1532) MeV. This agrees with the above fit (κ ′′ /κ ′ = −1.0 GeV −2 ) as far as the sign of this ratio is concerned, but not in its magnitude. Other than a simple inadequacy of Sakai-Sugimoto model, this could indicate that the Sakai-Sugimoto model needs a more complicated form of reggeization of the tensor glueball propagator as indeed discussed in [14] in the context of CEP of η and η ′ mesons. It could also be an indication of the importance of secondary reggeon exchanges.
Fitting the mean value of the total cross section (3.1) we find In the bottom right panel of Fig. 5 we show results for the total φ pp distribution for the individual κ ′ and κ ′′ coupling terms and for their coherent sum. Here we take (κ ′ , κ ′′ ) = (−8.88, 8.88 GeV −2 ). The interference effect of the κ ′ and κ ′′ terms is clearly seen there. As we see from (B14) the κ ′′ term corresponds (approximately) to a superposition of the (l, S) = (2, 2) and (4, 4) terms with opposite signs. We expect then destructive interference of the two (l, S) terms, and indeed, the κ ′′ contribution shows such a behaviour, i.e. there is a complete cancellation of the two (l, S) terms for φ pp ≃ 90 • . Hence, the option κ ′ = 0, κ ′′ = 0 is clearly ruled out by the data for the φ pp distribution. In fact, this option is also incompatible with the result (B7) obtained in the Sakai-Sugimoto model, since it would correspond to the limit M KK → 0 where the holographic model ceases to have large-N c QCD as its infrared limit.
Summarizing our findings for f 1 (1285) CEP, we have obtained a reasonable description of the WA102 data with either a pure (l, S) = (2, 2) or a pure (l, S) = (4, 4) coupling, as well as with the κ ′ , κ ′′ couplings with parameters It is also interesting to compare the results (3.6) and (3.8) with the approximate relation (B14) for κ ′′ = 0 and k 2 = m 2 f 1 . We note that we see no way to fix the overall sign of the f 1 couplings from experiment. The states | f 1 and − | f 1 are clearly equivalent from quantum mechanics. Of course, relative signs of couplings have physical significance, Results for the (κ ′ , κ ′′ ) term calculated with the vertices (B8) and (B9) are shown. We use here the form factor (2.11) with Λ E = 0.7 GeV. In the top panels the theoretical results have been normalised to the mean value of the number of events from [62]. In the bottom panels we compare the theoretical curves with the WA102 data from [3]. Here the results have been normalised to the mean value of the total cross section (3.1) and the error bars on the data have been calculated as in Fig. 2. In the bottom right panel we show the results for (κ ′ , κ ′′ ) = (−8.88, 8.88 GeV −2 ) for the individual κ ′ and κ ′′ coupling terms and for their coherent sum. The κ ′′ contribution has been enhanced by a factor 10 for better visibility. The absorption effects are included in the calculations.
for instance, the relative sign of g ′ Keeping this in mind we compare the absolute values of the l.h. and r.h. sides of (B14). With m f 1 = (1281.9 ± 0.5) MeV [4] we This shows that the approximate relation (B14) is here satisfied to an accuracy of around 10 %. Using (B14) we can also see to which values of g ′ PP f 1 and g ′′ PP f 1 the (κ ′ , κ ′′ ) values of (3.9) roughly correspond. With (3.9) and setting t 1 = t 2 = −0.1 GeV 2 in (B14) we get Thus, (κ ′ , κ ′′ ) from (3.9) corresponds practically to a pure (l, S) = (4, 4) term and the values for g ′′ PP f 1 from (3.7) and (3.11) agree to within 5 % accuracy. Now we present a comparison of our theoretical results also for the f 1 (1420) meson with relevant data from the WA102 experiment [3]. In Fig. 6 we show the |t| (left panels) and φ pp (right panels) distributions for √ s = 29.1 GeV and |x F,M | 0.2. The WA102 data points from [3] and our model results have been normalised to the mean value of the total cross section σ exp. = (1584 ± 145) nb ; (3.12) see Table I. The experimental error bars are assumed to be 9.2 % corresponding to the error of σ exp. in (3.12). From Fig. 6 we can see that the (l, S) = (2, 2) term is sufficient to describe the WA102 data. We have checked that the shape of φ pp distributions almost does not depend on the choice of the cutoff parameter Λ E , in particular for the (l, S) = (2, 2) term. Taking into account the results listed in Table II we conclude that Λ E = 0.7 GeV is an optimal choice. To get the mean value of the total cross section (3.12) we find (assuming positive values of the coupling constants): g ′ PP f 1 (1420) = 2.06 in (2.7) for Λ E = 0.8 GeV, 2.39 for Λ E = 0.7 GeV, 2.94 for Λ E = 0.6 GeV, g ′′ PP f 1 (1420) = 4.20 in (2.8) for Λ E = 0.7 GeV, 5.24 for Λ E = 0.6 GeV, κ ′ = 5.08 in (B8) for Λ E = 0.7 GeV, and 4.39 for Λ E = 0.8 GeV.
In Appendix C we derive the ratio of the coupling constants for the two axial-vector mesons resulting from the assumption that the pomeron couples only to the flavour-SU(3) singlet components, which would be the case in the chiral limit for couplings that are exclusively determined by the axial-gravitational anomaly (as in the Sakai-Sugimoto model). For f 1 -mixing angles that are often considered in the literature, namely ideal mixing (φ f = 0 • ) and φ f 20 • , the ratio of all couplings for f 1 (1420) over those for f 1 (1285) would then be given uniformly by a factor 1/ √ 2 = 0.71 and 1.44, respectively. However, from (3.6) and (3.14), (3.7) and (3.15), (3.9) and (3.17), we get 20) respectively. If at the WA102 energy of √ s = 29.1 GeV only PP fusion contributes to the CEP of both f 1 mesons, this means that pomerons do not couple predominantly to the flavour-SU(3) singlet components which are involved in the axial-gravitational anomaly. However, if the breaking of the SU(3) flavour symmetry by the strange quark mass has a large effect for PP f 1 couplings, this presents a problem for the chiral Sakai-Sugimoto model. The discrepancy could, however, be partly due to important contributions from subleading reggeon exchanges at WA102 energies.
To summarize, we have seen in this section that PP fusion with suitable PP f 1 couplings can give a reasonable description of the WA102 data. We have also seen that with the distributions explored it is very hard to discriminate between the various possible couplings, that is, to see which combination of coupling constants is preferred experimentally. In addition we have the problem that at the relatively low c.m. energy of √ s = 29.1 GeV subleading reggeon exchanges may still be rather important. This topic will be dealt with in Appendix D.
In the next sections we shall show our results for RHIC and LHC energies where subleading reggeon exchanges should be negligible, at least, for the midrapidity region. For these results we shall use the PP f 1 couplings as determined in the present section. But we must emphasize that our results for the RHIC and LHC obtained in this way should be considered as upper limits of the cross sections. If at the WA102 energies there are important contributions from subleading reggeon exchanges the cross sections at the RHIC and LHC energies could be significantly smaller. As we discuss in Appendix D, we estimate that the reduction could be by a factor of up to four relative to the predictions given below. The theoretical results and the WA102 data points from [3] have been normalised to the mean value of the total cross section (3.12). The meaning of the lines is as in Fig. 5.

B. Predictions for the LHC experiments
Now we wish to show our results (predictions) for the LHC.
Here we consider only the PP fusion with the coupling parameters found in Sec. III A from the comparison with the WA102 data.
In Table III we have collected cross sections in µb for the reactions pp → pp f 1 (1285) and pp → pp f 1 (1420) at √ s = 13 TeV. We show results for some kinematical cuts on the rapidity of the mesons, |y M | < 2.5, and also with an extra cut on momenta of leading protons 0.17 GeV < |p y,p | < 0.50 GeV that will be applied when using the ALFA sub-detector on both sides of the ATLAS detector. We also show results for larger (forward) rapidities and without a measurement of outgoing protons relevant for the LHCb experiment. The calculations have been done in the Born approximation and with the absorption corrections included. For the f 1 (1285) we show the individual results for the (l, S) = (2, 2) and (4, 4) terms with g ′ PP f 1 (1285) = 4.89 in (2.7) and g ′′ PP f 1 (1285) = 10.31 in (2.8); see (3.6) and (3.7), respectively. For the (κ ′ , κ ′′ ) terms, (B8) plus (B9), we use (3.5). We have taken here the form factor (2.11) with Λ E = 0.7 GeV. For the f 1 (1420) we show the results for the (l, S) = (2, 2) term with g ′ PP f 1 = 2.39, see (3.14), and the (κ ′ , κ ′′ ) option from (3.13). As we see from comparing the last two columns of Table III the absorption effects lead to a sizeable reduction of the cross sections compared to the Born results.
In Fig. 8 we show our predictions for the pp → pp f 1 (1285) reaction for √ s = 13 TeV, |y M | < 2.5, and for the cut on the leading protons of 0.17 GeV < |p y,p | < 0.
TeV for some kinematical cuts on the rapidity y M of the meson, and also when limitations on the outgoing protons are imposed. The results for the (l, S) = (2, 2) and (4, 4) terms calculated from (2.7) and (2.8), respectively, and for the κ ′ plus κ ′′ terms calculated with the vertices (B8) plus (B9) are shown. The parameter values for (κ ′ , κ ′′ ) are taken from (3.5) for the f 1 (1285) and from (3.13) for the f 1 (1420). We have taken here the form factor (2.11) with Λ E = 0.7 GeV. The results without and with absorption effects are presented.  In all cases the absorption effects are included. Inclusion of absorption effects modifies the differential distributions because their shapes depend on the kinematics of outgoing protons. We have checked numerically that the absorption effects decreases the distributions mostly at higher values of the variables φ pp and dP t and at smaller values of p t,M and |t|. The measurement of such distributions would allow to better understand absorption effects. This could be tested in future in experiments at the LHC, when both protons are measured, such as ATLAS-ALFA and CMS-TOTEM.  (4,4), and (κ ′ , κ ′′ ) contributions are shown. Here we use for the (2, 2) and (4, 4) terms (3.6) and (3.7), respectively. For the (κ ′ , κ ′′ ) terms we use (3.5). The absorption effects are included in all the calculations. Now we discuss one of the most prominent decay modes of the f 1 (1285), the decay f 1 (1285) → π + π − π + π − . This four-pion decay channel seems well suited to measure the f 1 (1285) meson in CEP. However, the f 1 (1285) is rather close in mass to the f 2 (1270) which also decays into four pions. In principle, the f 1 (1285) and f 2 (1270) decays will interfere in the four-pion final state. Note that this interference could be used to determine the relative sign of the f 1 and f 2 production times decay amplitudes. But the interference terms will drop out in the total decay rates.
In the following, for CEP of the f 1 (1285) meson, we assume the (l, S) = (2, 2) coupling and Λ E = 0.7 GeV; see (3.6) and σ abs. in Table III. For CEP of the f 2 (1270) meson the cross section is σ pp→pp f 2 (1270) = 11.25 µb with the parameters from Ref. [34]: The absorption effects are taken into account in the calculation. We obtain the integrated cross sections for √ s = 13 TeV and |y M | < 2.5, including the PDG branching fractions (3.21) and (3.22), as follows and respectively. Thus we predict a large cross section for the exclusive axial-vector f 1 (1285) production compared to the production of the tensor f 2 (1270) meson in the π + π − π + π − channel. Even if we scale down the f 1 cross section by a factor 4 it will still be larger than our result for the f 2 cross section. In addition Γ( f 2 (1270)) ≫ Γ ( f 1 (1285)), so f 1 (1285) will be seen as a sharp peak on top of a smaller bump corresponding to the f 2 (1270).

C. Predictions for the STAR experiment at RHIC
The STAR experiments at RHIC measure CEP reactions at √ s = 200 GeV [39] and at √ s = 510 GeV [40]. It has the possibility to observe the outgoing protons at least in a certain phase space region. We shall present the predictions of our model for the cut on the rapidity of the meson |y M | < 0.7 and for limitations on the outgoing protons, for √ s = 200 GeV: (3.25) as specified in Eq. (6.1) of [39], and for √ s = 510 GeV: (p x,p + 0.6 GeV) 2 + p 2 y,p < 1.25 GeV 2 , 0.4 GeV < |p y,p | < 0.8 GeV , p x,p > −0.27 GeV , (3.26) as specified in [40].
In Table IV we give the analog of Table III but for the STAR experiments. In Fig. 9 we show as an example various predictions for f 1 (1285) CEP at √ s = 200 GeV, |y M | < 0.7, and with extra cuts on the leading protons (3.25). The experimental cuts have crucial influence on the shape of the differential distributions. In particular the result Eq. (3.14) 30.7 6.8 and Eq. (3.26) ( that the distributions (nearly) vanish for certain values of the variables φ pp , p t,M and dP t is caused by the specific cuts (3.25). We have made also a similar analysis for the √ s = 510 GeV option but decided not to show it here. The results can be obtained on request. The general situation is similar but there are also some noticeable differences.

IV. CONCLUSIONS
In this paper, we have discussed in detail the exclusive central production of the pseudovector f 1 (1285) and f 1 (1420) mesons in proton-proton collisions. The calculations for the pp → pp f 1 (1285) and pp → pp f 1 (1420) reactions have been performed in the tensorpomeron approach [10]. In general, two PP f 1 couplings with different orbital angular momentum and spin of two "pomeron particles" are possible, namely (l, S) = (2, 2) and (4,4). We have presented explicitly amplitudes and formulae for the PP f 1 vertices as derived from corresponding coupling Lagrangians. Two different approaches for the PP f 1 coupling have been considered.
(1) In the first approach, two independent PP f 1 coupling constants, g ′ PP f 1 and g ′′ PP f 1 that correspond to the (l, S) = (2, 2) and (l, S) = (4, 4) couplings [see Eqs. (2.7) and (2.8), respectively], not known a priori as they are of nonperturbative origin, have been fitted to existing data from the WA102 experiment. A reasonable agreement with the WA102 data can be obtained with either a pure (l, S) = (2, 2) or a pure (l, S) = (4, 4) coupling.
(2) The second approach is based on holographic QCD, namely the (chiral) Sakai-Sugimoto model, where the pomeron-pomeron-f 1 couplings (B3)-(B4) are obtained from a Chern-Simons action representing the mixed axial-gravitational anomaly of QCD. This also involves two coupling constants, with a prediction for their ratio in terms of the Kaluza-Klein mass scale of the model as given by (3.4). Comparing the φ pp distribution for different values of this ratio confirms the sign of this ratio as predicted by the Sakai-Sugimoto model, but not its magnitude. However, freely fitting the magnitude of the couplings, reasonable agreement with the WA102 data is again obtained.
Assuming that the WA102 data are already dominated by pomeron exchanges, we have presented various predictions for experiments at the RHIC and the LHC. The total cross sections and several differential distributions for the pp → pp f 1 (1285) reaction have been presented. In our opinion the π + π − π + π − channel seems the best to observe f 1 (1285) both for RHIC and the LHC experiments. We have shown that independent of the PP f 1 coupling decomposition the cross section for the pp → pp( f 1 (1285) → π + π − π + π − ) reaction is much larger than for the pp → pp( f 2 (1270) → π + π − π + π − ) reaction. As the f 1 (1285) has a much narrower width than the f 2 (1270) it would be seen in the mass distribution as a narrow peak on a somewhat broader bump corresponding to the f 2 (1270).
Our predictions can be tested by the STAR Collaboration at RHIC and by all collaborations (ALICE, ATLAS, CMS, LHCb) working at the LHC.
In all cases considered we have included absorption effects. We have found that the absorption effects strongly depend on kinematics, i.e. also on experimental cuts, as well as on the type of the PP f 1 coupling used in the calculation. Different tensorial couplings discussed in the present paper lead to different dependences on t 1 and t 2 which are crucial for the size of absorption effects. The effect of absorption was not the primary aim of this study, therefore, the discussion of this point was kept rather short in our present paper.
To summarize: we think that a study of CEP of the axial vector mesons f 1 should be quite rewarding for experimentalists. We have analysed in detail the results of the WA102 experiment which worked at √ s = 29.1 GeV and we have shown that we get a good description of the results with the pomeron-pomeron fusion mechanism. Such studies could be extended, for instance by the COMPASS experiment [66,67], where presumably one could study the influence of reggeon-pomeron and reggeon-reggeon fusion terms. At high energies, at RHIC and LHC, pomeron-pomeron fusion is expected to dominate. We have given predictions for CEP of f 1 mesons there. Comparing them with future experimental results should allow a good determination of the PP f 1 coupling constants. These are nonperturbative QCD parameters. Their theoretical calculation is a challenge. The holographic methods applied to QCD already give some predictions here, as we have shown in our paper. We can envisage a fruitful interplay of experiment and theory in this field in the future leading finally to a satisfactory picture of the couplings of two pomerons to the axial vector f 1 mesons studied here and, quite generally, to single mesons.

Appendix A: The coupling of an f 1 -type meson to two pomerons
Here we study the coupling of a meson f 1 with I G J PC = 0 + 1 ++ to two tensor pomerons. We use the relations for the tensor pomeron from [10,21].
In Appendix A of [21] the fictitious reaction of two "real spin-2 pomerons" annihilating to a meson was studied. This was done in order to get an idea what type of pomeronpomeron-meson (PPM) couplings we would have to expect. Looking at Table 6 of [21] we see that for the production of a J P = 1 + meson we can have the following values of angular momentum l and total spin S of the two pomerons: (l, S) = (2, 2) , (4,4) . (A1) We find only these two possibilities.

Appendix B: Different forms for the PP f 1 coupling as obtained in holographic QCD
In (A5) and (A7) of Appendix A we have given a possible form for the PP f 1 couplings. In the holographic framework another form is obtained. In the Sakai-Sugimoto model [47,48], the coupling of singlet pseudoscalar and axial-vector mesons to two tensor glueballs is determined by the gravitational Chern-Simons (CS) action (describing axial-gravitational anomalies), as given in Eq. (59) of [14] S CS ⊃ N c The (singlet component of the) axial-vector meson is contained in Tr where Z refers to the holographic direction.
Five-dimensional gravitons correspond to four-dimensional tensor glueballs and their coupling to f 1 is obtained by expanding this term to second order in transverse-traceless metric perturbations and integrating over radial wave functions. Using the same notation as in Appendix A we derive the coupling Lagrangians where and N is a normalization constant that we leave undetermined because of the ambiguities [14] in the reggeization of the tensor glueball into pomerons (it would be unity for a purely flavour-singlet axial-vector meson when P µν was replaced by the tensor glueball T µν normalized as in [43]). The Sakai-Sugimoto model has two free parameters, a Kaluza-Klein mass scale M KK and the dimensionless 't Hooft coupling λ at this scale. Both, λ and the normalization N , drop out of the ratio between the two PP f 1 couplings, (B7) Usually [47,48] 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 ≈ 1487 MeV. The standard pomeron trajectory (2.13) corresponds to a tensor glueball mass of M T ≈ 1917.5 MeV, whereas lattice gauge theory [68] indicates a tensor glueball mass M T ≃ 2400 MeV (or even higher [69] Here we define the tensor [unrelated to the Riemann tensor in (B1)] In (B8) and (B9) we have taken out explicitly the traces in (κλ) and (ρσ). The momenta and vector indices for these vertices are oriented and distributed as in (2.7) and (2.8).
Now we consider the reaction (2.5), the fusion of two "real pomerons" (or two glueballs) of mass m giving an f 1 meson of mass squared k 2 : P(q 1 , ǫ (1) ) + P(q 2 , ǫ (2) Here q 1 , q 2 , and ǫ (1) , ǫ (2) are the momenta and the polarisation tensors of the two "real pomerons", k and ǫ are the momentum and the polarisation vector of the f 1 . We know from the results of Table 6 in Appendix A of [21] that there are two independent amplitudes for the reaction (B11). Thus, for the reaction (B11) we expect to find an equivalence relation of the form between the Lagrangians (B3), (B4), and (A5), (A7). Of course, the respective coupling parameters must then satisfy certain relations which we determined as The proof of (B12), (B13) is given at the end of this Appendix. We note that the relation (B13) involves k 2 , the invariant mass squared of the resonance f 1 . For a narrow resonance of mass m f 1 we can set k 2 = m 2 f 1 = const. Then (B13) gives a linear relation of the couplings (κ ′ , κ ′′ ) and (g ′ PP f 1 , g ′′ PP f 1 ) with constant coefficients. For a broad resonance k 2 varies. Then we see from (B13) that for constants (κ ′ , κ ′′ ) the couplings g ′ PP f 1 and g ′′ PP f 1 contain additional form factors depending on k 2 and vice versa. The strict equivalence relation (B12) does not hold any more for the scattering process (2.1) where two pomerons with invariant masses t 1 < 0 and t 2 < 0, and in general t 1 = t 2 , collide to give an f 1 meson; see Fig. 1. But for small |t 1 | and |t 2 | we can expect the following approximate equivalence to hold: The reverse reads Again, taking e.g., g ′ PP f 1 and g ′′ PP f 1 as constants κ ′ and κ ′′ will contain suitable form factors and vice versa.
We have made a numerical investigation of the above equivalence relations, (B14) and (B15), for the case g ′′ PP f 1 (1285) = 0 setting In Fig. 10 we show, in two-dimensional plot, the ratio R(p t,1 , p t,2 ) = d 2 σ κ ′ /dp t,1 dp t,2 d 2 σ (2,2) /dp t,1 dp t,2 for the pp → pp f 1 (1285) reaction at √ s = 13 TeV and |y M | < 2.5. The ratio 1 occurs at p t,1 = p t,2 . In the limited range of transverse momenta of the outgoing protons, p t,1 0.6 GeV and p t,2 0.6 GeV, both approaches give similar contributions. The deviations from the ratio 1 are here less than about 15 %. The same remains true for larger p t,1 , p t,2 , provided p t,1 − p t,2 0.4 GeV. But clear differences can be seen if one p t is large and the other one is small. We note that adjusting the t 1,2 dependent form factors we could, presumably, obtain the ratio R(p t,1 , p t,2 ) in (B17) even closer to 1 for a larger range of p t,1 and p t,2 . At the end of this Appendix we give the proof of (B12) and (B13). For this we study the reaction (B11) in the rest system of f 1 (k, ǫ) choosing the direction of q 1 as z axis. We have then We shall evaluate the T -matrix elements for (B11) in the basis where ǫ (M) and ǫ are polarisation vectors and tensors, respectively, corresponding to definite eigenvalues of the angular momentum operator J z . In detail we choose for the f 1 Here the J z eigenvalues are M. For the pomeron (1) we define the four-vectors and the polarisation tensors ǫ For the pomeron (2) we define the four-vectors can be different from zero. The calculations are straightforward but a bit lengthy. We shall only give the results. For this we define two "reduced" amplitudes M|T (2,2) |M 1 , M 2 and M|T (4,4) |M 1 , M 2 ; see Table V. From the Lagrangians (A5) plus (A7), respectively the vertices (2.7) plus (2.8), we obtain for the matrix elements (B19) Note that the (l, S) = (2, 2) coupling gives an amplitude proportional to |q 1 | 2 , the (l, S) = (4, 4) term an amplitude proportional to |q 1 | 4 , as it should be for these values of the orbital angular momentum l. Now we consider the Lagrangians (B3) plus (B4) giving the vertices (B8) and (B9), respectively. Here we get for the matrix elements (B19) collaboration measured f 1 (1285) and f 1 (1420) also at the significantly lower energy √ s = 12.7 GeV.
For a complete theoretical discussion of all results of the WA102 experiment we should consider also the lower energy and include subleading reggeon-exchange contributions to f 1 CEP. We list here the possible fusion reactions leading to an f 1 meson and involving such reggeons: Let us now discuss the effective couplings for these processes, taking as model the results of Appendix A; see (A2)-(A7). Following [10] the f 2R and a 2R reggeons will be treated as effective second rank symmetric traceless tensors, the ω R , ρ R , and φ R as effective vectors. Our coupling Lagrangians for (D2) and (D3) are then as in (A5) and (A7) but with the replacements and respectively. All these couplings must be real. For the process (D1) there are more coupling possibilities than the analogs of (A5) and (A7), since P and f 2R are distinct. Indeed, using the methods of Appendix A of [21], we find here six independent couplings. For the process (D4) we can rely on the general analysis of two real vector particles giving an f 1 with J P = 1 + in Appendix B of [21]. From Table 8 there we find that there is only one possible coupling, (l, S) = (2, 2), for this on shell process. A convenient coupling Lagrangian is easily written down where and g ω R ω R f 1 is a dimensionless coupling constant. Similar coupling ansätze apply to the processes (D5) and (D6). The vertex following from (D9) reads as follows: This vertex function satisfies the relations We shall use in the following the coupling (D9) and the vertex function (D11) for ω reggeons as well as ω mesons.
As for the case of the PP f 1 coupling we find it useful to consider the analog of the reaction (B11) here, the fusion of two real ω mesons giving an f 1 state ω(q 1 , ǫ For our purpose we consider fictitious ω mesons of arbitrary mass m 0 and a fictitious f 1 of mass √ k 2 2m. We shall work again in the rest system of the f 1 and choose the kinematics as in (B18). The polarisation vectors ǫ (M) (M = ±1, 0) for the f 1 are taken as in (B20). The polarisation vectors for the ω mesons are taken as follows Note that the amplitude (D15) is proportional to |q 1 | 2 as it should be since it is derived from the (l, S) = (2, 2) coupling (D9). Furthermore, the amplitude (D15) vanishes for m = 0 as it must be due to the Landau-Yang theorem [73,74]. Indeed, we can consider the production of an f 1 meson by two virtual photons of mass squared q 2 0. For this we use the standard vector-meson-dominance (VMD) ansatz for the coupling of the photons to the ω mesons (see e.g., (3.23) of [10]), which then fuse to give the f 1 . In this case we get for the amplitude the same expression as in (D15) with m replaced by q 2 and multiplied with the appropriate VMD factor e m 2 ω γ ω ∆ (ω) where ∆ (ω) T (q 2 ) is the transverse part of the ω meson propagator (cf. (3.2)-(3.4) of [10]). All gauge invariance relations for these amplitudes are satisfied due to (D12) and the amplitudes vanish for q 2 → 0 in accord with the Landau-Yang theorem.
A different ansatz for the ω R ω R f 1 coupling is obtained in the holographic approach [75]: with κ ω a dimensionless parameter. For the vertex function (D18) we find the relations For the process (D13) we find here With constant κ ω , these amplitudes diverge for m → 0. Here we cannot use the usual VMD relations to relate these amplitudes to the ones for the fusion of two virtual or real photons giving an f 1 meson. Due to (D20) the corresponding amplitudes for γ * γ * → f 1 would not satisfy the necessary gauge invariance relations.
Vector-meson dominance is in fact realized in holographic QCD (for an extensive discussion in the Sakai-Sugimoto model see [48]). The coupling to virtual or real photons involves bulk-to-boundary propagators which correspond to sums over an infinite tower of massive vector mesons. In place of the constant κ ω one obtains an asymmetric transition form factor that does satisfy the Landau-Yang theorem and which has been studied in [50], where good agreement with available data from the L3 experiment [76,77] on single-virtual γγ * → f 1 has been found.
Clearly the inclusion of all these subleading exchanges (D1)-(D6) would introduce many new coupling parameters and form factors and would make a meaningful analysis of the WA102 data practically impossible. However, for the analysis of data from the COMPASS experiment, which operates in the same energy range as previously the WA102 experiment, it could be very worthwhile to study all the above subleading exchanges in detail. In addition one also has to keep in mind that there should be a smooth transition from reggeon to particle exchanges when going to very low energies. Clearly, all these topics deserve careful analyses, but they go beyond the scope of the present paper.
From some rough estimates of subleading contributions at the WA102 energy of √ s = 29.1 GeV we make the educated guess that the true PP f 1 couplings at this energy may be up to a factor 2 smaller than given in (3.5)-(3.9) and (3.13)- (3.17).