Towards a complete study of central exclusive production of $K^{+}K^{-}$ pairs in proton-proton collisions within the tensor Pomeron approach

We present a study of the central exclusive production of the $K^{+} K^{-}$ pairs in proton-proton collisions at high energies. We consider diffractive mechanisms including the $K^{+} K^{-}$ continuum, the dominant scalar $f_{0}(980)$, $f_{0}(1500)$, $f_{0}(1710)$ and tensor $f_{2}(1270)$, $f'_{2}(1525)$ resonances decaying into the $K^{+} K^{-}$ pairs. We include also photoproduction mechanisms for the non-resonant (Drell-S\"{o}ding) and the $\phi(1020)$ resonance contributions. The theoretical results are calculated within the tensor-pomeron approach including both pomeron and reggeon exchanges. Predictions for planned or current experiments at RHIC and LHC are presented. We discuss the influence of the experimental cuts on the integrated cross section and on various differential distributions for outgoing particles. The distributions in two-kaon invariant mass, in a special `glueball filter variable', as well as examples of angular distributions in the $K^{+}K^{-}$ rest frame are presented. We compare the $\phi(1020)$ and continuum photoproduction contributions to the $f_{0}(980)$ and continuum diffractive contributions and discuss whether the $\phi(1020)$ resonance could be extracted experimentally. For the determination of some model parameters we also include a discussion of $K$-nucleon scattering, in particular total cross sections, and of $\phi(1020)$ photoproduction.


I. INTRODUCTION
Diffractive exclusive production of light mesons mediated by double pomeron exchange is expected to be an ideal process for the investigation of gluonic bound states (glueballs) due to the gluonic nature of the pomeron. Such processes were studied extensively at CERN starting from the Intersecting Storage Rings (ISR) experiments [1][2][3][4][5][6], later at the Super Proton Synchrotron (SPS) in fixed-target experiments by the WA76 and WA102 collaborations [7][8][9][10][11][12][13][14], and more recently by the COMPASS collaboration [15,16]. For reviews of experimental results see for instance [17][18][19]. The measurement of two charged pions in pp collisions was performed by the CDF collaboration at Tevatron [20]. Exclusive reactions are of particular interest since they can be studied in current experiments at the LHC by the ALICE, ATLAS, CMS [21], and LHCb collaborations, as well as by the STAR collaboration at RHIC [22,23]. In such experiments it is of great advantage for the theoretical analysis if the leading outgoing protons can be measured. There are several efforts to complete installation of forward proton detectors. The CMS collaboration combines efforts with the TOTEM collaboration while the ATLAS collaboration may use the ALFA sub-detectors. Also the STAR experiment at RHIC is equipped with detectors of similar type.
On the theoretical side, the main contribution to the central diffractive exclusive production at high energies can be understood as being due to the exchange of two pomerons between the external nucleons and the centrally produced hadronic system. We believe that the soft pomeron exchange can be effectively treated as an effective rank-2 symmetric-tensor exchange as introduced in [24]. In [25] it was shown that the tensorpomeron model is consistent with the experimental data on the helicity structure of proton-proton elastic scattering at √ s = 200 GeV and small |t| from the STAR experiment [26]. The paper [25] also contains some remarks on the history of the views of the pomeron spin structure. In [27] the central exclusive production of several scalar and pseudoscalar mesons in the reaction pp → ppM was studied for the relatively low WA102 energy. Then, in [28], the model was applied to the reaction pp → ppπ + π − at high energies including the π + π − continuum, the dominant scalar f 0 (500), f 0 (980) and tensor f 2 (1270) resonances decaying into the π + π − pairs. The resonant ρ 0 and nonresonant (Drell-Söding) π + π − photoproduction was studied in [29]. In [30], an extensive study of the reaction γp → π + π − p was presented. The ρ 0 meson production associated with a very forward/backward πN system in the pp → ppρ 0 π 0 and pp → pnρ 0 π + processes was discussed in [31]. Also the central exclusive π + π − π + π − production via the intermediate σσ and ρ 0 ρ 0 states in pp collisions was studied in [32]. Recently, in [33], the central exclusive production of the pp in the continuum and via scalar resonances in pp collisions was studied.
Some time ago two of us considered the exclusive pp → ppK + K − reaction in a simple Regge-like model [34]. The Born approximation is usually not sufficient and absorption corrections have to be taken into account, see e.g. [35,36]. In [34] the production of the diffractive K + K − continuum and of the scalar χ c0 meson decaying via χ c0 → K + K − was studied. For other related works see [37] for the pp → ppπ + π − reaction, [38] for the exclusive f 0 (1500), and [39] for χ c0 meson production.
In [40] a model for the exclusive diffractive meson production in pp collisions was discussed based on the convolution of the Donnachie-Landshoff parametrization of the pomeron distribution in the proton with the pomeron-pomeron-meson total cross sec-tion. In this approach the cross section is calculated by summing over the direct-channel contributions from the pomeron and two different f 1 and f 2 trajectories associated to the glueball candidate f 0 (980) and the f 2 (1270) resonances, respectively. Also the f 0 (500) resonance contribution dominating the small mass region and a slowly increasing background were taken into account. The absolute contribution of resonances, e.g. of the f 0 (980) and the f 2 (1270), to the total cross section cannot be derived within this approach, and must hence be deduced from experimental data. But the relative weights of the various resonances on one trajectory are correlated by the duality argument made in [40].
The aim of the study presented here is the application of the tensor-pomeron model to central exclusive production of K + K − pairs in pp collisions. We wish to show first predictions in the tensor-pomeron approach for the production of the diffractive K + K − continuum, of the scalar f 0 (980), f 0 (1500), f 0 (1710), and the tensor f 2 (1270), f ′ 2 (1525) resonances decaying into K + K − pairs. This model, being formulated at the amplitude level, allows us also to calculate interference effects of the various contributions. In the following we wish to show differential distributions which can be helpful in the investigation of scalar and tensor resonance parameters. Therefore, we shall treat each resonance in its own right and shall not a priori suppose any correlations of the coupling parameters of different resonances. In addition the resonant φ(1020) and non-resonant (Drell-Söding) K + K − photoproduction mechanisms will be discussed. So far the cross sections for the exclusive pp → ppφ(1020) reaction were calculated within a pQCD k t -factorization approach [41], and in a color dipole approach [42,43].
We study central exclusive production of K + K − in proton-proton collisions at high energies where p a,b , p 1,2 and λ a,b , λ 1,2 ∈ {+1/2, −1/2}, indicated in brackets, denote the fourmomenta and helicities of the protons, and p 3,4 denote the four-momenta of the charged kaons, respectively. The full amplitude of K + K − production is a sum of the continuum amplitude and the amplitudes with the s-channel resonances: The amplitude for exclusive resonant K + K − production via the pomeron-pomeron fusion, shown by the diagram of Fig. 1, can be written as As indicated in Fig. 1 also contributions involving non-leading reggeons R: ρ R (ρ reggeon), ω R (ω reggeon), f 2R ( f 2 reggeon), a 2R (a 2 reggeon) can contribute. The relevant production modes via (C 1 , C 2 ) fusion 1 giving resonances are listed in Table II of [28].
FIG. 1: The Born diagram for double-pomeron/reggeon central exclusive scalar and tensor resonances production and their subsequent decays into K + K − in proton-proton collisions.
However, in the present paper we shall consider only resonance production by pomeronpomeron fusion in order not to be swamped by too many, essentially unknown, coupling parameters. Turning now to continuum diffractive K + K − production shown in Fig. 2 we have again pomeron and reggeon contributions. Here we will be able to extract all relevant coupling parameters from the kaon-nucleon total cross section data. Therefore, we shall include in the calculation pomeron and reggeon exchanges. In this way we will also get an estimate of the possible importance of the latter exchanges. In the following we treat the C = +1 pomeron and the reggeons R + = f 2R , a 2R as effective tensor exchanges while the C = −1 reggeons R − = ω R , ρ R are treated as effective vector exchanges.
In Table I I: A list of resonances, up to a mass of 1800 MeV, that decay into K + K − and/or π + π − . The meson masses, their total widths Γ and branching fractions are taken from PDG [44].  The generic diagrams for diffractive exclusive K + K − continuum production are shown in Fig. 2. At high energies the exchange objects to be considered are the pomeron P and the reggeons R. The amplitude can be written as the following sum: The PP-exchange amplitude on the Born level can be written as the sum: . Furthermore ∆ (P) and Γ (Ppp) denote the effective propagator and proton vertex function, respectively, for the tensorial pomeron. The propagator of the tensor-pomeron exchange is written as (see Eq. (3.10) of [24]): and fulfils the following relations Here the pomeron trajectory α P (t) is assumed to be of standard linear form, see e.g. [45,46], The pomeron-proton vertex function is written as (see Eq. (3.43) of [24]) (3.8) where β PNN = 1.87 GeV −1 . The PKK vertices in the amplitudes (3.3) and (3.4) can be written in analogy to the Pππ vertices (see (3.45) of [24]) but with the replacement The form factors, taking into account that the hadrons are extended objects, are chosen as where m p is the proton mass and m 2 D = 0.71 GeV 2 is the dipole mass squared and Λ 2 0 = 0.5 GeV 2 ; see Eqs. (3.29) and (3.34) of [24], respectively.
The off-shellness of the intermediate kaons is taken into account by the inclusion of form factors. The form factors are normalized to unity at the on-shell pointF K (m 2 K ) = 1 and parametrised here in the monopole form where Λ o f f ,M could be adjusted to experimental data. We take Λ o f f ,M = 0.7 GeV, that is, the same value as for the pion off-shell form factor in the reaction pp → ppπ + π − discussed in [28]. In [28] we fixed a parameter of the form factor for off-shell pion and a few parameters of the pomeron-pomeron-meson coupling constants to describe the CDF data [20]; see Fig. 9 of [28]. In our calculations we include both the tensor-pomeron and the reggeon R + and R − exchanges. In the following we collect the expressions for reggeon effective propagators and vertex functions in order to make our present paper self contained. For extensive discussions motivating the following expressions we refer to section 3 of [24].
The ansatz for the C = +1 reggeons R + = f 2R , a 2R is similar to (3.5) -(3.8). The R + propagator is obtained from (3.5) with the replacements (3.12) In (3.12) and in the following the parameters of the reggeon trajectories are taken from [46]. The f 2R -and a 2R -proton vertex functions are obtained from (3.8) with the replacements and respectively. In (3.13), (3.14) and in the following M 0 = 1 GeV is used in various places for dimensional reasons. The f 2R -and a 2R -kaon vertex functions are obtained from (3.9) with the replacements 16) respectively. For the C = −1 reggeons R − = ω R , ρ R we assume an effective vector propagator (see Eqs. (3.14) -(3.15) of [24]) The value of (3.19) is taken from [24] as default value for the parameter of the propagators for ω R and ρ R exchanges. For the R − -proton vertices we have (see Eqs. (3.59) -(3.62) of [24]) with g ω R pp = 8.65 , g ρ R pp = 2.02 , (3.22) respectively. Note that in (3.21) the vertex function for the isospin 1 ρ R reggeon changes sign when we replace protons by neutrons. This is also the case for the isospin 1 a 2R reggeon exchange; see (3.51) of [24]. The R − -kaon vertex (R − = ω R , ρ R ) can be written in analogy to the ρ R -pion vertex (see (3.63) of [24]) To obtain the pomeron/reggeon-kaon coupling constants we consider the following elastic scattering processes at high energies We treat (3.24) and (3.25) in analogy to the elastic π ± p scattering; see section 7 of [24]. For the case of the elastic kaon-nucleon scattering amplitudes we set for p and n also N(I 3 ) with I 3 = +1/2 and I 3 = −1/2, respectively. We obtain Here we have s = (p 1 + p 2 ) 2 and t = (p 1 − p 3 ) 2 and we work in the approximation s ≫ |t|, m 2 p . For the total cross sections we obtain from the optical theorem for large s Following Donnachie and Landshoff [45] we use a two component parametrisation for the total cross sections of kaon-nucleon scattering Here (a, b) = (K + , p), (K − , p), (K + , n), (K − , n), and M 0 = 1 GeV. The numbers X ab ≡ X and Y ab are where the values for the X, Y K + p and Y K − p are taken from Fig. 3.2 of [46] and the values for the Y K + n and Y K − n are from our fit to the world data from [44]. We compare now (3.27) with (3.28) taking into account the parameters of the pomeron and reggeon trajectories and of their vertices from [24] quoted above in Eqs. (3.5) to (3.22). We get then the following results for the couplings

B. Scalar mesons central production
The K + K − production amplitude through the s-channel exchange of scalar mesons, such as f 0 (980), f 0 (1370), f 0 (1500), and f 0 (1710), via the PP fusion can be written as The effective Lagrangians and the vertices for the fusion of two tensor pomerons into the f 0 meson were discussed in appendix A of [27]. The PP f 0 vertex, including a form factor, reads as follows (p 34 [27]. The vertex (3.33) contains two independent PP f 0 couplings corresponding to the lowest allowed values of (l, S), that is (l, S) = (0, 0) and (2, 2). We take the factorized form for the PP f 0 form factor In practical calculations we take There has been a long history of uncertainty about the properties of the f 0 (1710) meson, one of the earliest glueball candidates. This state was observed in the WA76 experiment at √ s = 23.8 GeV [7] in both the K + K − and K 0 S K 0 S channels in the dikaon invariant mass region around 1.7 GeV. By studying the K + K − angular distributions the authors of [7] found that the so called θ/ f J (1720) state has J PC = 2 ++ . In [11] a reanalysis of the K + K − channel from the WA76 experiment was performed. A partial wave analysis of the centrally produced K + K − system, as performed in [11] (see Fig. 4 there), shows in the S-wave a threshold enhancement and a structure in the 1.5 -1.7 GeV mass interval which has been interpreted as being due to the f 0 (1500) and f J (1710) with J = 0. The D-wave shows peaks in the 1.3 GeV and 1.5 GeV mass regions, presumably due to the f 2 (1270)/a 2 (1320) and f ′ 2 (1525) resonances. In the D-wave at higher masses there is no evidence for any significant structure in the 1.7 GeV mass region and only a wide structure around 2.2 GeV is seen that may be due to the f 2 (2150) meson. In the P-wave (P − 1 ) a peak corresponding to the φ(1020) is observed. These results are compatible with those coming from WA102 experiment [10] at √ s = 29 GeV. The f J (1710) with J = 2 state has been observed also in radiative J/ψ decays [47]. However, a new analysis of J/ψ → γK + K − and γK 0 S K 0 S [48] strongly demonstrates that the mass region around 1.7 GeV is predominantly 0 ++ from the f 0 (1710). 2 This conclusion is consistent with the latest central production data of WA76 and WA102 [10][11][12]. An important variable characterising the production mechanisms of the various f 0 mesons is the azimuthal angle φ pp between the outgoing protons, p(p 1 ) and p(p 2 ) in (2.1). As can be seen from the experimental results presented in [12,13,17] for the f 0 (980), f 0 (1500), and f 0 (1710) states the cross sections peak at φ pp = 0 in contrast to the f 0 (1370) meson. It was shown in [27] that the appropriate angular shapes for the central production of f 0 (980) and f 0 (1500) mesons could be obtained with the PP f 0 vertices corresponding to the sum of the two lowest values of (l, S) couplings, (l, S) = (0, 0) and (2, 2), with appropriate coupling constants g ′ PPM and g ′′ PPM .
For the production of f 0 (1370) meson the (l, S) = (0, 0) coupling alone already describes the azimuthal angular correlation reasonably well. In [27] we determined the corresponding (dimensionless) PP f 0 coupling constants by approximately fitting the theoretical results to the WA102 data for the angular distributions and the total cross sections given in Table 1 of [17]. The following "preferred" values for the couplings were obtained, see Table 3 of [27] GeV. The experimental data points from [12] have been normalized to the total cross section σ = 245 nb from [17]. Plotted is the cross section dσ/dφ pp for 0 < φ pp < π. We show the individual contributions to the cross section with (l, S) = (0, 0) (the long-dashed line), (l, S) = (2, 2) (the short-dashed line), and their coherent sum (the solid line).
In Fig. 3 we show the distribution in azimuthal angle φ pp between the outgoing protons for the central exclusive production of the f 0 (1710) meson at √ s = 29.1 GeV with the data measured by the WA102 collaboration in [17]. Similarly as for the f 0 (980) and f 0 (1500) mesons (see Figs. 5 and 6 in [27], respectively) also for the f 0 (1710) meson both (l, S) contributions are necessary to describe the φ pp distribution accurately. For the f 0 (1710) we obtain the coupling constants as (g ′ PP f 0 (1710) , g ′′ PP f 0 (1710) ) = (0.45, 2.6). The scalar-meson propagator in (3.32) is parametrized as with a constant decay width.
where the dimensionless coupling constant The analogous relation for In (3.37) we assume that F ( f 0 KK) (p 2 34 ) has the same form as F (PP f 0 ) (p 2 34 ), see (3.35). In order to estimate the coupling constants g f 0 K + K − for the various f 0 states from (3.38) we need data for the partial decay rates Γ( f 0 → K + K − ). Since the Particle Data Group [44] does not give these decay rates explicitly we shall estimate them in the following using the available information.
The f 0 states have isospin I = 0. Assuming isospin invariance in the decays we get Let us now consider the various f 0 states in turn. The f 0 (980) has only the ππ, KK and the electromagnetic γγ decays. Therefore we have, to very good approximation, for the total decay rate In [49] the ratio (3.43) was found from the B meson decays. To obtain g f 0 (980)K + K − we assume the approximate relation 3 where σ( f 0 (980) → π + π − , K + K − ) are the integrated cross sections for the pp → pp( f 0 (980) → π + π − , K + K − ) processes via the PP fusion at √ s = 13 TeV. We get from (3.40) - (3.44), and with m f 0 (980) = 980 MeV, Γ f 0 (980) = 50 MeV, assuming g f 0 (980)K + K − > 0 and g f 0 (980)π + π − > 0, The error bars in (3.45) were obtained using only error bars in (3.44). Uncertainties of the rather poorly known Γ f 0 (980) are similar. For the f 0 (1370) meson we take the following input: from [44], and from [50]. From (3.40) and (3.42) we get then, assuming again g f 0 (1370)K + K − > 0 and g f 0 (1370)π + π − > 0, For diffractive K + K − production through the s-channel f 2 -meson exchange the amplitude is more complicated to treat. The f 2 (1270) and f ′ 2 (1525) mesons could be considered as potential candidates. The amplitude for the PP fusion is given by The PP f 2 vertex can be written as (3.60) A possible choice for the iΓ µν,κλ,ρσ | bare terms j = 1, ..., 7 is given in appendix A of [28]. In [28] we found that the j = 2 coupling for g (2) PP f 2 (1270) = 9.0 is optimal to describe the main characteristics measured in the WA102 and ISR experiments and by the CDF collaboration [20] including e.g. a gap survival factor S 2 = 0.1 for the CDF.
The f 2 (1270) and f ′ 2 (1525) have similar φ pp and dP t dependences [12]. dP t is the socalled "glueball-filter variable" [54] defined by the difference of the transverse momentum vectors of the outgoing protons in (2.1) dP t = q t,1 − q t,2 = p t,2 − p t,1 , dP t = |dP t | . (3.61) It has been observed in Ref. [9] that all the undisputed qq states (i.e. η, η ′ , f 1 (1285) etc.) are suppressed when dP t → 0, whereas the glueball candidates, e.g. f 0 (1500), survive. As can be seen in Refs. [9,12] the f 2 (1270) and f ′ 2 (1525) states have larger dP t and their cross sections peak at φ pp = π in contrast to the "enigmatic" f 0 (980), f 0 (1500) and f 0 (1710) states. Note, that at √ s = 29.1 GeV the experimental cross section for the production of the f 2 (1270) meson, whose production has been found to be consistent with double pomeron/reggeon exchange, is more than 48 times greater than the cross section of the f ′ 2 (1525) meson [17]. For the f ′ 2 (1525) we assume also only the j = 2 coupling with g (2) PP f ′ 2 (1525) = 2.0 fixed to the experimental total cross section from [17]. With this we roughly reproduced the shapes of the differential distributions of the WA102 data [12]. In the future the corresponding PP f 2 coupling constants could be adjusted by comparison with precise experimental data.
In (3.60)F (PP f 2 ) is a form factor for which we takẽ Here, for qualitative calculations only, one may use the tensor-meson propagator with the simple Breit-Wigner form whereĝ µν = −g µν + p 34µ p 34ν /p 2 34 . In (3.64) Γ f 2 is the total decay width of the f 2 resonance and m f 2 its mass.
The f 2 KK vertex can be written as (see Eq. (3.37) of [24] for the analogous f 2 ππ vertex) where g f 2 K + K − can be obtained from the corresponding partial decay width. We assume that In analogy to the f 2 → ππ decay, treated in section 5.1 of [24], we can write We assume further that isospin symmetry holds, that is, With Γ( f 2 → KK)/Γ f 2 from [44] (see Table I) we get, assuming g f 2 K + K − > 0, For the π + π − decay channel, based on (5.6) of [24] and the numerical values from Table I, we have g f 2 (1270)π + π − = 9.28 , (3.71)

IV. PHOTOPRODUCTION CONTRIBUTIONS
For the φ resonance production we consider the diagrams shown in Fig. 4. In these diagrams all vertices and propagators will be taken here according to Ref. [24]. The diagrams to be considered for the non-resonant (Drell-Söding) contribution are shown in Fig. 5. In the following we collect formulae for the amplitudes for the pp → ppK + K − reaction within the tensor-pomeron approach [24].
The central exclusive φ meson production and its subsequent decay into P-wave K + K − in proton-proton collisions. (a) FIG. 5: The diagrams for photon-induced central exclusive continuum K + K − production in proton-proton collisions. There are also 3 additional diagrams with the role of (p(p a ), p(p 1 )) and (p(p b ), p(p 2 )) exchanged.

A. K + K − continuum central production
The amplitude for photoproduction of the K + K − continuum can be written as the following sum: The γP-exchange amplitude can be written as the sum: Here the γ and P propagators and the γpp, Ppp vertices are given in section 3 of [24]. The K propagator is standard and given after (3.4) above. The γKK, PKK and PγKK vertices are as the corresponding vertices for pions, see appendix B of [30], but with β Pππ replaced by β PKK (3.30).
In order to assure gauge invariance and "proper" cancellations among the three terms (4.3) to (4.5) we have introduced, somewhat arbitrarily, one common energy dependence on s 2 defined as: This is satisfied as we see easily be replacing in (4.3) -(4.5) Γ (γpp) µ (p 1 , p a ) by q 1µ . The formulas (4.3) -(4.5) do not include hadronic form factors for the inner subprocess γP → K + K − . A possible way to include form factors for the inner subprocesses is to multiply the amplitude obtained from (4.3) to (4.5) with a common factor, see [29,[55][56][57], A common form factor for all three diagrams is chosen in order to maintain gauge invariance, and a convenient form is given in [58] with the exponential parametrizations (4.12) The parameter Λ K should be fitted to the experimental data. We expect it in the range of 0.8 to 1 GeV. For the Pγ-exchange the amplitude has the same structure with p(p a ), p(p 1 ) ↔ p(p b ), p(p 2 ), t 1 ↔ t 2 and s 2 ↔ s 1 . We shall consider also contributions involving nonleading reggeons. For the f 2R exchange the formulae have the same tensorial structure as for pomeron exchange and are obtained from (4.3) to (4.5) with the corresponding effective f 2R pp, f 2R KK and f 2R γKK vertices and the f 2R reggeon propagator; see [24,30]. Analogous statements hold for the case for the a 2R reggeon exchange. The relevant reggeon-kaon coupling constants are given in Eq. (3.31). The contributions involving C = −1 reggeon exchanges are different. We recall that R − = ω R , ρ R exchanges are treated as effective vector exchanges in our model; see Sec. 3 of [24]. The vertex for ρ R γKK is in analogy to the vertex ρ R γππ given in (B.81) of [30]. The ω R exchange is treated in a similar way.

B. Photoproduction of φ meson
Since the proton contains no valence s quarks we shall assume that the amplitude for the γp → φp reaction at high energies includes only the pomeron exchange contribution. In contrast, in the amplitudes for the γp → ρ 0 p reaction [29,30] and for the γp → ωp reaction [59] also reggeon exchanges play an important role.
In the high-energy small-angle approximation we get, using (D. 19) in appendix D of [27], where the explicit tensorial functions Γ (i) µνκλ (p φ , −q), i = 0, 2, are given in Ref. [24], formulae (3.18) and (3.19), respectively. In Eq. (4.15) 4π/γ 2 φ = 0.0716, (∆ (φ) The form factors F 1 (t) and F M (t) are chosen in (4.15) as the electromagnetic form factors (3.10) only for simplicity. Here, it seems reasonable to assume rather Λ 2 0 ≈ m 2 φ than Λ 2 0 = 0.5 GeV 2 from (3.10). This will be discussed in Fig. 6. Alternatively, we can take a common form factor with the slope parameter B (P) φp obtained from comparison to the experimental data. In order to get estimates for the Pφφ coupling constants a Pφφ and b Pφφ we make the assumption based on the additive quark model [60][61][62][63][64] (see also chapter II of [65]): for transversely polarised φ mesons (m = ±1). In analogy to the ρp scattering discussed in section 7.2 of [24] the total cross section for the φp scattering at high energies is obtained from (4.15) as With the pomeron parts of the Kp and πp total cross sections from (3.27) above and (7.6) of [24], respectively, we obtain from (4.17) and (4.18) In the left panel of Fig. 6 we show the integrated cross section for the γp → φp reaction, calculated from (4.13), as a function of the center-of-mass energy together with the experimental data. The experimental point at W γp = 70 GeV was obtained by extrapolating the differential cross section to t = 0 assuming a simple exponential t dependence and integrating over the range |t| < 0.5 GeV [66]. In our calculation we also integrate over the same t range. We see that our model calculation including only the pomeron exchange describes the total cross section for the γp → φp reaction fairly well. 4 The right panel of Fig. 6 shows the differential cross section for elastic φ photoproduction. The calculations, performed for two energies, W γp = 70 GeV and 94 GeV, are compared with ZEUS data, [66] and [71], respectively. We show results for two parameters of the form factor F M (t), Λ 2 0 = 0.5 GeV 2 and 1 GeV 2 , represented by the bottom lines and the top lines, respectively. We can see that the results for Λ 2 0 = 1 GeV 2 with the relevant values of coupling constants a and b describe more accurately the slope of the t distribution.

C. φ(1020) meson central production
The φ photoproduction is dominated by diffractive scattering via pomeron exchange. The amplitude for the γP-exchange, see diagram (a) in Fig. 4, reads as (3.5) of [29] with appropriate modifications: Here we use the φ propagator with the simple Breit-Wigner expression as defined in (3.7), (3.10) and (3.11) of [29] with In (4.22) the running (energy-dependent) width is approximately parametrized as A more accurate parametrization of ∆ (φ) T (s) and Γ φ (s) must take into account also non KK decay channels of the φ, in particular, the 3π decays which amount to (15.32 ± 0.32)% of all decays, see [44]. For such a program one could use the methods explained for the ρ propagator in [80]; see also [24,30].
For the φKK vertex we have (4.24) The g φK + K − coupling constant can be determined from the partial decay width Γ(φ → K + K − ), With the parameters of Table I, assuming g φK + K − > 0, we get In the diagram of Fig. 4 at the Pφφ vertex the incoming φ is always off shell, the outgoing φ also may be away from the nominal "mass shell" p 2 34 = m 2 φ . As suggested in [30], see (B.82) there, we insert, therefore, in the Pφφ vertex extra form factors. A convenient form, given in (B.85) of [30] (see also (3.9) of [29]) is withΛ φ a parameter close to 2 GeV andñ φ > 0. In practical calculations we also include in (4.24) the form factor (4.28)

V. RESULTS
In this section we present results for integrated cross sections of the reaction pp → ppK + K − and dikaon invariant mass distributions. For convenience of the reader we collect in Table II the numerical values of default parameters of our model used in calculations. There are also the parameters of pomeron/reggeon-kaon couplings, see (3.30) and (3.31), not shown in Table II, obtained from fits to kaon-nucleon total cross-section  data as discussed in section III A. Our attempts to determine the parameters of pomeronpomeron-meson couplings as far as possible from experimental data have been presented in sections III B and III C, and in Refs. [27,28]. Note that we take here somewhat smaller values of the pomeron-pomeron-meson coupling parameters than in our previous paper [27] because there they were fixed at the WA102 energy where we expect also large contributions to the cross sections from the reggeon exchanges. We have checked for the central K + K − continuum contribution calculated at √ s = 13 TeV and for three different cuts on pseudorapidities |η K | < 1, |η K | < 2.5, and 2 < η K < 4.5, that adding the exchange of secondary reggeons increases the cross section by 2.4 %, 2.9 %, and 6.5 %, respectively. We expect a similar role of secondary reggeons for production of resonances at √ s = 13 TeV. For continuum K + K − photoproduction we find even less effect on the cross sections from secondary reggeons than for the purely diffractive production above. Recently, in Ref. [33], we also discussed the role of reggeons for the pp → pppp reaction.
Many of the parameters listed in Table II were obtained from fits to available data but they are still rather uncertain and some are only our educated guess. Clearly, it would be desirable to experimentally test our predictions obtained with our default parameters and then adjust these if necessary. Such an adjustment of the model parameters will be possible with high-energy experimental data for the purely exclusive reactions pp → ppπ + π − and pp → ppK + K − which are expected to become available soon. The GenEx Monte Carlo generator [81] could be used in this context.
In Fig. 7 we present the K + K − invariant mass distribution at √ s = 13 TeV and |η K | < 1. Here we take into account the non-resonant continuum, including both pomeron and reggeon exchanges, and the scalar f 0 (980) resonance created here only by the pomeronpomeron fusion. We show results for different values of the relative phase φ f 0 (980) in the coupling constant (3.45) not known a priori g f 0 (980)K + K − → g f 0 (980)K + K − e iφ f 0 (980) . (5.1) We can see that the complete result indicates a large interference effect of the continuum and the f 0 (980) terms. It should be recalled that the f 0 (980) resonance appears as a sharp drop around the 1 GeV region in the π + π − mass spectrum. The black solid line corresponds to the calculations with the phase used for π + π − exclusive production. The phase for K + K − does not need to be the same as the production of ππ and KK systems may be a complicated coupled-channel effect not treated here explicitly. In some of the following figures we show predictions for two representative values for this phase, φ f 0 (980) = 0 and π/2. We must leave it to the experiments to determine this phase from data. As can be clearly seen from the left panel of Fig. 8 the resonance contributions generate a highly structured pattern. In the calculations we include the non-resonant continuum, and the dominant scalar f 0 (980), f 0 (1500), f 0 (1710), and tensor f 2 (1270), f ′ 2 (1525), resonances decaying into the K + K − pairs. In principle, there may also be a contribution from the broad scalar f 0 (1370) meson. The right panel of Fig. 8 shows the photoproduction contributions without and with some form factors included in the amplitudes. The narrow φ(1020) resonance is visible above the continuum term. It may in principle also be visible on top of the broader f 0 (980) resonance. This will be discussed in Fig. 10.
In Figs. 9 and 10 we show the invariant mass distributions for centrally produced π + π − (the black lines) and K + K − (the blue lines) pairs imposing cuts on pseudorapidities and transverse momenta of produced particles that will be measured in the RHIC and LHC experiments. The pp → ppπ + π − reaction was discussed within the  tensor-pomeron model in [28]. The short-dashed lines represent the purely diffractive continuum term. The solid lines represent the coherent sum of the diffractive continuum, and the scalar f 0 (980), f 0 (1500), f 0 (1710), and tensor f 2 (1270), f ′ 2 (1525) resonances. For the pp → ppK + K − reaction we show predictions for φ f 0 (980) = 0 and π/2 in (5.1), see the solid and long-dashed blue lines, respectively. The f 0 (980) resonance term in the pp → ppK + K − reaction is calculated with the upper limit for the coupling, g f 0 (980)K + K − = 3.48; see (3.45). The lower red lines show the photoproduction contributions. The diffractive and photoproduction contributions to K + K − production must be added coherently at the amplitude level and in principle could interfere. However, this requires the inclusion of absorption effects (at the amplitude level) that are different for both classes of processes, see e.g. [29]. In [28] we found that for the reaction pp → ppπ + π − a similar interference effect is below 1%. The reader is asked to note different shapes of the π + π − and K + K − invariant mass distributions for different experimental setups. In the left panel of Fig. 9 we show distributions for the STAR experiment. In the right panel we show results for the CDF experimental conditions together with data for the pp → ppπ + π − reaction [20]. The limited CDF acceptance, in particular the p t > 0.4 GeV condition on centrally produced K + and K − mesons, causes a reduction of the cross sections in the region M 34 < 1.3 GeV; see e.g. the clearly visible minimum for the photoproduction term there.
The calculations were done at Born level and the absorption corrections were taken into account by multiplying the cross section for the corresponding collision energy by a common factor S 2 obtained from [36] and [34]. For the purely diffractive contribution the gap survival factors S 2 = 0. In the right panel the photoproduction terms are shown without and with some form factors included in the amplitudes. The lower lines correspond to results for the φ(1020) photoproduction with form factor (4.28) and for the non-resonant term with form factors (4.9) -(4.12). Absorption effects have been included here by the gap survival factors S 2 = 0.1 for the diffractive contributions and S 2 = 0.9 for the photoproduction contributions.
were taken. For the photoproduction contribution the Born calculation was multiplied by the factor S 2 = 0.9; see [29]. The absorption effects lead to a huge damping of the cross section for the purely diffractive contribution and a relatively small reduction of the cross section for the φ(1020) photoproduction contribution. Therefore we expect that one could observe the φ resonance term, especially when no restrictions on the leading protons are included. This situation is shown in Fig. 10, see the top left and right panels for the ALICE and LHCb experimental conditions, respectively. However, the final answer can only be given considering the experimental mass resolution of a given experiment. Here, for √ s = 13 TeV, we take S 2 = 0.1 for the purely diffractive contribution and S 2 = 0.9 for the photoproduction contribution. In the bottom panel of Fig. 10 we show results with extra cuts on the leading protons of 0.17 GeV < |p y,1 |, |p y,2 | < 0.5 GeV as will be the momentum window for ALFA on both sides of the ATLAS detector [82]. Here the φ(1020) resonance is not so-well visible.
In Figs. 11 and 12 we present differential observables for the ALICE kinematics ( √ s = 13 TeV, |η K | < 1, p t,K > 0.1 GeV) and for two regions: M 34 ∈ (1.45, 1.60) GeV (the left panels) and M 34 ∈ (1.65, 1.75) GeV (the right panels). Fig. 11 shows the distributions in the "glueball filter" variable dP t ; see (3.61). We see that the maximum for the qq state f ′ 2 (1525) is around of dP t = 0.6 GeV. On the other hand, for the scalar glueball candidates The short-dashed lines represent the purely diffractive continuum term. The solid and longdashed blue lines correspond to the results for φ f 0 (980) = 0 and π/2 in (5.1), respectively. The lower red line represents the φ(1020) meson plus continuum photoproduction contribution. The CDF experimental data from [20] in the right panel for the pp → ppπ + π − reaction are shown for comparison. Absorption effects were taken into account effectively by the gap survival factors.
f 0 (1500) and f 0 (1710) the maximum is around dP t = 0.25 GeV, that is, at a lower value than for the f ′ 2 (1525). This is in accord with the discussion in section III C and in Ref. [9]. Angular distributions in the dimeson rest frame are often used to study the properties of dimeson resonances. Fig. 12 shows the distribution of the cosine of θ r. f . K + , the polar angle of the K + meson with respect to the beam axis, in the K + K − rest frame. 5 It should be emphasized that our predictions were done with our choice of parameters collected in Table II. From the partial wave analysis, performed by the WA76/102 collaborations [10,11], more amount of the S-wave than of the D-wave in the mass region around 1.5 GeV was observed. This observation was confirmed by the E690 experiment [83] at the Fermilab Tevatron at √ s = 40 GeV in the pp → p slow K 0 S K 0 S p fast reaction. This would change the behavior of invariant mass distribution around M KK = 1.5 GeV. Note that the relative phase between K + K − -continuum and f 0 (1500) amplitudes is not so well determined and a constructive interference cannot be excluded. Here we do not show explicitly the corresponding result.

VI. CONCLUSIONS
We have discussed central exclusive production (CEP) of K + K − pairs in proton-proton collisions at high energies. We have taken into account purely diffractive and diffractive photoproduction mechanisms. For the purely diffractive mechanism we have included the continuum and the dominant scalar f 0 (980), f 0 (1500), f 0 (1710) and tensor f 2 (1270), f ′ 2 (1525) resonances decaying into K + K − pairs. The amplitudes have been calculated using Feynman rules within the tensor-pomeron model [24]. The effective Lagrangians and the vertices for PP fusion into the scalar and tensor mesons were discussed in [27] and [28], respectively. The model parameters of the pomeron-pomeron-meson couplings have been roughly adjusted to recent CDF data [20] and then used for predictions for the STAR, ALICE, CMS and LHCb experiments. For the photoproduction of K + K − pairs we have discussed the dominant φ(1020) meson contribution and the non-resonant (Drell-Söding) contribution. Similar mechanisms were discussed in [29] for the π + π − photoproduction. The coupling parameters of the tensor pomeron to the φ meson have been fixed based on the HERA experimental data for the γp → φp reaction [66,71].
In the present study we have focused mainly on the invariant mass distributions of centrally produced K + K − . In Fig. 9 we also presented, for comparison, the purely diffractive contribution previously developed in [28] for the central production of π + π − pairs. The pattern of visible structures in the invariant mass distributions is related to the scalar and tensor isoscalar mesons and it depends on experimental kinematics. One can expect, with our default choice of parameters, that the scalar f 0 (980), f 0 (1500), f 0 (1710) and the tensor f 2 (1270), f ′ 2 (1525) mesons will be easily identified experimentally in CEP. The φ-photoproduction and purely diffractive contributions have different dependences on the proton transverse momenta. Furthermore, the absorptive corrections for the K + K − photoproduction processes lead to a much smaller reduction of the cross section than for the diffractive ones. It can therefore be expected that the φ-photoproduction will be seen in experiments requiring only a very small deflection angle for at least one of the outgoing protons. However, we must keep in mind that other processes can contribute in experimental studies of exclusive φ production where only large rapidity gaps around the centrally produced φ meson are checked and the forward and backward going protons are not detected. Recently, experimental results for this kind of processes have been published by the CDF [20] and CMS [21] collaborations. We refer the reader to Ref. [31] in which ρ 0 production in pp collisions was studied with one proton undergoing diffractive excitation to an nπ + or pπ 0 system.
In addition, we have presented distributions in the so-called glueball filter variable, dP t (3.61), which shows different behavior in the K + K − invariant mass windows around glueball candidates with masses ∼ 1.5 GeV and ∼ 1.7 GeV than in other regions. Also examples of angular distributions in the K + K − rest frame were shown. The dP t distribution may help to interpret the relative rates between the f 0 (1500) and f ′ 2 (1525) resonances and to resolve the controversial discussion about the existence of the supernumerous resonances in the scalar sector [84].
Finally we note that central exclusive φ production in pp collisions offers the possibility to search for effects of the elusive odderon, as was pointed out in [85]. The odderon was introduced on theoretical grounds in [86,87]. For a review of odderon physics see e.g. [88]. The experimental status of the odderon is still unclear even if there seems to be some evidence for it from the recent TOTEM result [89]. For recent discussions of possible odderon effects in pp elastic scattering at LHC energies see [90][91][92][93]. Using the methods and results of the present paper it would be straightforward to include also φ production by odderon-pomeron fusion and to discuss odderon effects, e.g. in K + -K − distributions, in a way analogous to the program presented in [30]. But this is beyond the scope of our present paper.
To summarize: we have given a consistent treatment of central exclusive K + K − continuum and resonance production in an effective field-theoretic approach. Our studies could help in understanding the production mechanisms of some light resonances and their properties in the pp → ppK + K − reaction. A rich structure has emerged which will give experimentalists interesting challenges to check and explore it.