Searching for the odderon in $pp \to pp K^{+}K^{-}$ and $pp \to pp \mu^{+}\mu^{-}$ reactions in the $\phi(1020)$ resonance region at the LHC

We explore the possibility of observing odderon exchange in the $pp \to pp K^{+}K^{-}$ and $pp \to pp \mu^{+}\mu^{-}$ reactions at the LHC. We consider the central exclusive production (CEP) of the $\phi(1020)$ resonance decaying into $K^{+} K^{-}$ and $\mu^{+}\mu^{-}$. We compare the purely diffractive contribution (odderon-pomeron fusion) to the photoproduction contribution (photon-pomeron fusion). The theoretical results are calculated within the tensor-pomeron and vector-odderon model for soft reactions. We include absorptive corrections at the amplitude level. In order to fix the coupling constants for the photon-pomeron fusion contribution we discuss the reactions $\gamma p \to \omega p$ and $\gamma p \to \phi p$ including $\phi$-$\omega$ mixing. We compare our results for these reactions with the available data, especially those from HERA. Our coupling constants for the pomeron-odderon-$\phi$ vertex are taken from an analysis of the WA102 data for the $p p \to p p \phi$ reaction. We show that the odderon-exchange contribution significantly improves the description of the $pp$ azimuthal correlations and the $dP_{t}$ ``glueball-filter variable'' dependence of $\phi$ CEP measured by WA102. To describe the low-energy data more accurately we consider also subleading processes with reggeized vector-meson exchanges. However, they do not play a significant role at the LHC. We present predictions for two possible types of measurements: at midrapidity and with forward measurement of protons (relevant for ATLAS-ALFA or CMS-TOTEM), and at forward rapidities and without measurement of protons (relevant for LHCb). We discuss the influence of experimental cuts on the integrated cross sections and on various differential distributions. With the corresponding LHC data one should be able to get a decisive answer concerning the presence of an odderon-pomeron fusion contribution in single $\phi$ CEP.


I. INTRODUCTION
So far there is no unambiguous experimental evidence for the odderon (O), the charge conjugation C = −1 counterpart of the C = +1 pomeron (P), introduced on theoretical grounds in [1,2]. A hint of the odderon was seen in ISR results [3] as a small difference between the differential cross sections of elastic proton-proton (pp) and proton-antiproton (pp) scattering in the diffractive dip region at √ s = 53 GeV. Recently the TOTEM Collaboration has published data from high-energy elastic proton-proton scattering experiments at the LHC. In [4] results were given for the ρ parameter, the ratio of real to imaginary part of the forward scattering amplitude. This is a measurement at t = 0. In [5] the differential cross section dσ/dt was measured for 0.36 GeV 2 < |t| < 0.74 GeV 2 . The interpretation of these results is controversial at the moment. Some authors claim for instance that the ρ measurements show that there must be an odderon effect at t = 0 [6,7]. But other authors find that no odderon contribution is needed at t = 0 [8][9][10][11][12]. For a general analysis of pp and pp elastic scattering see e.g. [13].
As was discussed in [14] exclusive diffractive J/ψ and φ production from the pomeron-odderon fusion in high-energy pp and pp collisions is a direct probe for a possible odderon exchange. The photoproduction mechanism (i.e. pomeron-photon fusion) constitutes a background for pomeron-odderon exchanges in these reactions. Other sources of background involve secondary reggeon exchanges, for instance pomeron-φ Rreggeon exchanges. Exclusive production of heavy vector mesons, J/ψ and Υ, from the pomeron-odderon and the pomeron-photon fusion in the pQCD k t -factorization approach was discussed in [15]. The exclusive pp → ppφ reaction via the (pQCD-pomeron)photon fusion in the high-energy corner was studied in [16]; see also [17] for the exclusive photoproduction of charmonia J/ψ and ψ ′ and [18] for the exclusive ω production.
Another interesting possibility is to study the charge asymmetry caused by the interference between pomeron and odderon exchange. This was discussed in diffractive cc pair photoproduction [29], in diffractive π + π − pair photoproduction [30][31][32][33], and in the production of two pion pairs in photon-photon collisions [34]. However, so far in no one of the exclusive reactions a clear identification of the odderon was found experimentally. For a more detailed review of the phenomenological and theoretical status of the odderon we refer the reader to [35,36]. In this context we would also like to mention the EMMI workshop on "Central exclusive production at the LHC" which was held in Heidelberg in February 2019. There, questions of odderon searches were extensively discussed. Corresponding remarks and the link to the talks presented at this workshop can be found in [37].
Recently, the possibility of probing the odderon in ultraperipheral proton-ion collisions was considered [38,39]. In [40] the measurement of the exclusive η c production in nuclear collisions was discussed. The situation of the odderon in this context is also not obvious and requires further studies.
In [41] the tensor-pomeron and vector-odderon concept was introduced for soft re-actions. In this approach, the C = +1 pomeron and the reggeons R + = f 2R , a 2R are treated as effective rank-2 symmetric tensor exchanges while the C = −1 odderon and the reggeons R − = ω R , ρ R are treated as effective vector exchanges. For these effective exchanges a number of propagators and vertices, respecting the standard rules of quantum field theory, were derived from comparisons with experiments. This allows for an easy construction of amplitudes for specific processes. In [42] the helicity structure of small-|t| proton-proton elastic scattering was considered in three models for the pomeron: tensor, vector, and scalar. Only the tensor ansatz for the pomeron was found to be compatible with the high-energy experiment on polarized pp elastic scattering [43]. In [44] the authors, using combinations of two tensor-type pomerons (a soft one and a hard one) and the R + -reggeon exchange, successfully described low-x deep-inelastic leptonnucleon scattering and photoproduction.
Applications of the tensor-pomeron and vector-odderon ansatz were given for photoproduction of pion pairs in [33] and for a number of central-exclusive-production (CEP) reactions in proton-proton collisions in [45][46][47][48][49][50][51][52][53]. Also contributions from the subleading exchanges, R + and R − , were discussed in these works. As an example, for the pp → pppp reaction [50] the contributions involving the odderon are expected to be small since its coupling to the proton is very small. We have predicted asymmetries in the (pseudo)rapidity distributions of the centrally produced antiproton and proton. The asymmetry is caused by interference effects of the dominant (P, P) with the subdominant (O + R − , P + R + ) and (P + R + , O + R − ) exchanges. We find for the odderon only very small effects, roughly a factor 10 smaller than the effects due to reggeons.
In this paper we consider the possibility of observing odderon exchange in the pp → ppφ, pp → pp(φ → K + K − ), and pp → pp(φ → µ + µ − ) reactions in the light of our recent analysis of the pp → ppφφ reaction [52]. In the diffractive production of φ meson pairs it is possible to have pomeron-pomeron fusion with intermediatet/û-channel odderon exchange. Thus, the pp → ppφφ reaction is a good candidate for the odderonexchange searches, as it does not involve the coupling of the odderon to the proton. By confronting our model results, including the odderon, the reggeized φ exchange, and the f 2 (2340) resonance exchange contributions, with the WA102 data from [54] we derived an upper limit for the POφ coupling. Taking into account typical kinematic cuts for LHC experiments in the pp → ppφφ → ppK + K − K + K − reaction we have found that the odderon exchange contribution should be distinguishable from other contributions for large rapidity distance between the outgoing φ mesons and in the region of large four-kaon invariant masses. At least, it should be possible to derive an upper limit on the odderon contribution in this reaction.
Here we will try to understand the pp → ppφ reaction at relatively low center-of-mass energy √ s = 29.1 GeV by comparing our model results with the WA102 experimental data from [55]. We shall calculate the photoproduction mechanism. For this purpose we have to consider also low-energy photon-proton collisions in the γp → φp reaction where the corresponding mechanism is not well established yet; see, e.g., [56][57][58][59][60][61][62][63][64][65]. The amplitude for γp → φp cannot be realised by the C = −1 odderon exchange. Therefore, the corresponding results of [65] which include the odderon contribution cannot correspond to reality. In addition to the γP-fusion processes we shall estimate also subleading contributions, e.g. the γ-pseudoscalar-meson-fusion, the φP-fusion, the ωP-fusion, the ω f 2R -fusion, and the ρπ 0 -fusion, to determine their role in the pp → ppφ reaction. Our aim is to see how much room is left for the OP-fusion which is the main object of our studies.
Our paper is organized as follows. In Sec. II we consider the pp → pp(φ → K + K − ) reaction. Section III deals with µ + µ − production. For both reactions we give analytic expressions for the resonant amplitudes. Sec. IV contains the comparison of our results for the pp → ppφ reaction with the WA102 data. We discuss the role of different contributions such as γP, OP, φP, ωP, and ω f 2R fusion processes. Then we turn to high energies and show numerical results for total and differential cross sections calculated with typical experimental cuts for the LHC experiments. We discuss our predictions for the K + K − channel for √ s = 13 TeV. In addition, we present our predictions for the µ + µ − production also at √ s = 13 TeV which is currently under analysis by the LHCb Collaboration. We briefly discuss and/or provide references to relevant works for the continuum contributions. Sec. V presents our conclusions and further prospects. In Appendices A and B we discuss useful relations and properties concerning the photoproduction of ω and φ mesons. In Appendix C we give the definition of the Collins-Soper (CS) frame used in our paper.
In our paper we denote by e > 0 the proton charge. We use the γ-matrix conventions of Bjorken and Drell [66]. The totally antisymmetric Levi-Civita symbol ε µνκλ is used with the normalisation ε 0123 = 1.
The full amplitude of the reaction (2.1) is a sum of the continuum amplitude and the amplitudes through the s-channel resonances as was discussed in detail in [51]. Here we focus on the limited dikaon invariant mass region i.e., the φ ≡ φ(1020) resonance region, (a) The Born-level diagrams for diffractive production of a φ meson decaying to K + K − in proton-proton collisions with odderon exchange: (a) odderon-pomeron fusion; (b) pomeronodderon fusion.
That is, we consider the reaction The kinematic variables are (2.4) For high energies and central φ production we expect the process (2.3) to be dominated by diffractive scattering. The corresponding diagrams are shown in Figs. 1 and 2. That is, we consider the fusion processes γP → φ and OP → φ. For the first process all couplings are, in essence, known. For the odderon-exchange process we shall use the ansätze from [41] and we shall try to get information on the odderon parameters and couplings from the reaction (2.3). The amplitude for (2.3) gets the following contributions from these diagrams To give the full physical amplitude, for instance, for the pp → ppK + K − process (2.1) we should include absorptive corrections to the Born amplitudes. For the details how to include the pp-rescattering corrections in the eikonal approximation for the four-body reaction see e.g. Sec. 3.3 of [46] and [67].
The measurement of forward protons would be useful to better understand absorption effects. The GenEx Monte Carlo generator [68,69] could be used in this context. We refer the reader to [70] where a first calculation of four-pion continuum production in the pp → ppπ + π − π + π − reaction with the help of the GenEx code was performed.

A. γ-P fusion
The Born-level amplitude for the γ-P exchange, see diagram (a) in Fig. 1, reads The γpp vertex and the photon propagator are given in [41] by formulas (3.26) and (3.1), respectively. The γ → φ transition is made here through the vector-mesondominance (VMD) model; see (3.23)-(3.25) of [41]. ∆ (P) and Γ (Ppp) denote the effective propagator and proton vertex function, respectively, for the tensorial pomeron. The corresponding expressions, as given in Sec. 3 of [41], are as follows where t = (p ′ − p) 2 and β PNN = 1.87 GeV −1 . For simplicity we use for the pomeronnucleon 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. [71,72], Our ansatz for the Pφφ vertex follows the one for the Pρρ in (3.47) of [41] with the replacements a Pρρ → a Pφφ and b Pρρ → b Pφφ . This was already used in Sec. IV B of [51]. The Pφφ vertex function is taken with the same Lorentz structure as for the f 2 γγ coupling defined in (3.39) of [41]. With k ′ , µ and k, ν the momentum and vector index of the outgoing and incoming φ, respectively, and κλ the pomeron indices the Pφφ vertex reads with form factors F M andF (φ) and two rank-four tensor functions, (2.14) For details see Eqs. (3.18)-(3.22) of [41]. In (2.12) the coupling parameters a Pφφ and b Pφφ have dimensions GeV −3 and GeV −1 , respectively. In [51] we have fixed the coupling parameters of the tensor pomeron to the φ meson based on the HERA experimental data for the γp → φp reaction [73,74]. However, the ω-φ mixing effect was not taken into account there. In the calculation here we include the ω-φ mixing and we take the coupling parameters found in Appendix B. The full form of the vector-meson propagator is given by (3.2) of [41]. Using the properties of the tensorial functions (2.13) and (2.14), see (3.18)-(3.22) of [41], we can make for the φ-meson propagator the following replacement where we take the simple Breit-Wigner expression, as discussed in [51], For the φKK vertex we have from (4.24)-(4.26) of [51] iΓ with g φK + K − = 8.92 and F (φKK) a form factor. In the hadronic vertices we take into account corresponding form factors. We insert in the Pφφ vertex (2.12) the form factor F M (k 2 ) to take into account the extended nature of φ mesons andF (φ) (k 2 ) since we are dealing with two off-shell φ mesons; see (4.27) of [51] and (B.85) of [33]. Convenient forms are Fig. 32 of Appendix B. In practical calculations we include also in the φKK vertex the form factor [see (4.28) of [51]] Inserting all this in (2.7) we can write the amplitude for the γP-exchange as follows Here γ φ is the γ-φ coupling constant; see (3.23)-(3.25) of [41].
For the Pγ-exchange we have the same structure as for the above amplitude with In the following we shall also consider the single φ CEP in pp collisions In (2.24) ǫ (φ) denotes the polarisation vector of the φ and we have p 2 34 = m 2 φ . The amplitude for the γP fusion contribution to the reaction (2.24) is obtained from (2.7) by making the replacement (2.25) The same replacement holds for the Pγ fusion contribution. Analogous replacements hold for all other diagrams when going from the reaction (2.3) to (2.24).

B. O-P fusion
The amplitude for the diffractive production of the φ(1020) via odderon-pomeron fusion, see diagram (a) in Fig. 2, can be written as (2.26) Our ansatz for the C = −1 odderon follows (3.16), (3.17) and (3.68), (3.69) of [41]: where η O is a parameter with value η O = ±1; M 0 = 1 GeV is inserted for dimensional reasons; α O (t) is the odderon trajectory, assumed to be linear in t: The odderon parameters are not yet known from experiment. In our calculations we shall choose as default values The coupling of the odderon to the proton, β Opp , in (2.28) has dimension GeV −1 . For our study here we shall assume which is not excluded by the data of small-t proton-proton high-energy elastic scattering from the TOTEM experiment [4,5].
The amplitude for the OP-exchange can now be written as For the PO-exchange we have the same structure as for the above amplitude with the replacements (2.23). 1 Here we assume thatF M (q 2 1 ) andF M (q 2 2 ) have the same form (2.19) with the same Λ 2 0, POφ parameter. In principle, we could take different form factors with different Λ 2 0 parameters. (a)

C. Subleading contributions
At the relatively low c.m. energy of the WA102 experiment, √ s = 29.1 GeV, we have to include also subleading contributions with meson exchanges. In this section we discuss the following processes contributing to pp → ppφ. The fusion processes γ-π 0 , γ-η, γ-η ′ , and γ-f 0 , γ-a 0 , and fusion processes involving vector mesons φ-P, ω-P, ω-f 2R , ρ-π 0 , ωη, and ω-η ′ . We can have also ω-f 0 and ω-f ′ 2 contributions. But these contributions are expected to be very small since the φ is nearly a pure ss state, the ω nearly a pure uū + dd state. In the following we shall, therefore, neglect such contributions.

γ-pseudoscalar-meson contributions
First we consider processes with pseudoscalar meson M = π 0 , η, η ′ exchanges. The generic diagrams for these contributions are shown in Fig. 3 (a), (b). We have for the total γ-pseudoscalar-meson-exchange contribution The γ-M amplitude can be written as For the M-proton vertex we have (see (3.4) of [49]) We take g π pp = √ 4π × 14.0, g η pp = √ 4π × 0.99; see Eqs. (28) and (29) of [58]. An effective Lagrangian for the φγ M coupling is given in (22) of [58] with A β the photon field and g φγ M a dimensionless coupling constant. From this we get the φγ M vertex, including a form factor, as follows We use a factorised ansatz for the φγ M form factor Based on considerations of the vector-meson-dominance model (VMD) we write theF (γ) form factor asF with V = ρ 0 for M = π 0 and V = ω for M = η, η ′ . For the form factorsF (V) we choose the form as forF (φ) in (2.20) replacing φ by V = ρ 0 , ω. The effective coupling constant g φγ M is related to the decay width of φ → γ M, see (31) of [58], (2.42) Using the most recent values from [75], and taking the negative signs as in [58], we have found g φγπ 0 = −0.137, g φγη = −0.705, and |g φγη ′ | = 0.726. Note that |g φγη ′ | > |g φγη |. But the contribution of η ′ exchange is suppressed relative to the η exchange because of the heavier mass occurring in the propagator and of the smaller value of g η ′ pp ≃ g η pp /2, where we follow [58]. However, we note that there is no consensus on this latter relation in the literature. In [76] g η ′ pp ∼ = 6.1 and g η pp = 6.14 are given.
To examine uncertainties of the photoproduction contribution in the pp → ppφ reaction we intend to show also the result with Λ MNN = 1.2 GeV and Λ φγ M = 1.2 GeV in (2.43) and (2.44), respectively, which are slightly different from the values given in [58]. This choice of parameters was used in [62]; see Sec. II B there.
In Appendix B we discuss the γp → φp reaction. There we compare our model calculations for different parameter sets with the experimental data.

γ-scalar-meson contributions
Next we turn to the amplitudes for φ production through the fusion of γ with scalar mesons S = f 0 (500), f 0 (980), and a 0 (980). Their contribution is The generic diagrams for these contributions are as in Fig. 3 with M replaced by S. The same applies to the analytic expressions. We get M (γS) from M (γ M) in (2.36) replacing , and Γ (Spp) , respectively. We use the following expressions for the S-proton and for the φγS effective coupling Lagrangians, see (34) and (35), respectively, of [58], From these we get the vertices including form factors, as follows, where the momentum flow and the indices are chosen as for the Mpp and φγ M vertices, respectively, see (2.37) and (2.39), For the contributions of scalar exchanges we take the parameters found in Appendix C of [58]: g φγ f 0 (500) = 0.047, g f 0 (500)pp = √ 4π × 8.0, g φγ f 0 (980) = −1.81, g f 0 (980)pp = 0.56, g φγa 0 (980) = −0.16, g a 0 (980)pp = 21.7. For f 0 (500) the monopole form of the form factors as in (2.43) and (2.44) with M replaced by f 0 (500) and Λ f 0 (500)NN = Λ φγ f 0 (500) = 2 GeV is used. For the heavier mesons ( f 0 (980) and a 0 (980)) the following compact form is used [58]: (2.51) The final expression for the γS-exchange amplitude in (2.46) reads M (γS) pp→ppK + K − we have to make the replacements (2.23).

φ-P and ω-P contributions
Here we discuss two approaches, reggeized-vector-meson-exchange approach (I) and reggeon-exchange approach (II). For the second approach the corresponding diagrams are shown in Fig. 4.
First we consider the contributions through the vector mesons V = φ and ω: The amplitude for the VP-exchange can be written as with the tensor-to-vector coupling ratio, Following [76] we assume κ φ = κ ω to be in the range ≃ ±0.5, g φpp = −0.6 and g ω pp = 9.0; see also [77]. Thus, the tensor term in (2.55) is small and in the calculation we take the vectorial term only with g φpp = −0.6 and g ω pp = 8.65. This latter value was determined in Sec. 6.3 of [41] and, as discussed there, we assume g ω pp = g ω R pp . We also make the assumption that the t-dependence of the V-proton coupling can be parametrised in a simple exponential form This form factor is normalized to unity when the vector meson V is on its mass shell, i.e., when t = m 2 V . The amplitude for the VP-exchange can now be written as For the Pφφ and Pωφ coupling vertices and constants see the discussion in the Appendices A and B.
For small values of s 1 = (p 1 + p 34 ) 2 the standard form of the vector-meson propagator factor ∆ (V) For higher values of s 1 we must take into account the reggeization. We do this, following (3.21), (3.24) of [52], by making in the amplitude M (PV) (2.57) the replacement where s thr is the lowest value of s 1 (2.4) possible here: Note, that in (2.58) we take s 1 α ′ V instead of s 1 /s thr as in (3.21) of [52]. We assume for the Regge trajectories [78]. Alternatively, we shall consider the exchange of the reggeons φ R and ω R instead of the mesons φ and ω as discussed above. We recall that C = −1 exchanges (ω R , φ R ) are treated as effective vector exchanges in our model. In order to obtain the ω R P-exchange amplitude we make in (2.54) the following replacements: We take the corresponding terms (2.64) and (2.65) from (3.59)-(3.60) and (3.14)-(3.15) of [41], respectively. In (2.66) we use the relations (A13) and (B9) and we take the factorised form for the Pω R φ form factor with F M (q 2 ) as in (2.19) but with Λ 2 0 = 0.5 GeV 2 and F (φ) (p 2 34 ) = F (φKK) (p 2 34 ); see (2.21). Then, the ω R P-exchange amplitude can be written as M (ω R P) We use for the parameter M − in the ω R propagator the value found in (3.14), (3.15) of [41] M − = 1.41 GeV . (2.69) In a similar way we obtain the φ R P-exchange amplitude. We assume that g φ R pp = g φpp . The M (PV) and M (PV R ) amplitudes are obtained from (2.57) and (2.68), respectively, with the replacements (2.23).
For the WA102 energy, √ s = 29.1 GeV, also the secondary f 2R exchange may play an important role. Setting √ s 1 ≈ √ s 2 ( √ s 1 and √ s 2 are the energies of the subprocesses , respectively) and using the relation GeV. Therefore, in interpreting the WA102 data it is necessary to take possible contributions from ω-f 2R and ω R -f 2R exchanges into account, in addition to the ω-P and ω R -P exchanges. In a way similar to (2.54)-(2.68) we can write the amplitudes for the ω-f 2R and ω Rf 2R exchanges, since both, P and f 2R exchange, are treated as tensor exchanges in our model. The effective f 2R -proton vertex function and the f 2R propagator are given in [41] by Eqs. (3.49) and (3.12), respectively. As an example, the ω R f 2R -exchange amplitude can be written as in (2.68) with the following replacements: [41] and g f 2R pp = 11.04, M 0 = 1 GeV from (3.50) of [41]. For the f 2R ω R φ coupling parameters we assume that In addition, we could have also the ρ-a 2R and ρ R -a 2R exchanges, but the couplings of ρ R and a 2R to the protons are much smaller than those of ω R and f 2R ; see (3.62), (3.52), (3.60), and (3.50) of [41]. Therefore, we neglect the ρ-a 2R and ρ R -a 2R terms in our considerations.
For the ρ-π 0 amplitude we have The ρ-proton vertex is given by (2.55) and (2.56) with V = ρ. The φρπ 0 vertex is as the φγ M vertex in (2.39) with the replacements (2.76) The proton-π 0 vertex is given in (2.37).
In principle we can also have ω-η and ω-η ′ fusion contributions. g φωη and g φωη ′ cannot be obtained from mesonic decays. Then one could rely only on models. Due to these model uncertainties of the coupling constants for the ω-η and ω-η ′ fusion processes we neglect these contributions in our present study.
In this section we will focus on the exclusive reaction where p a,b , p 1,2 and λ a,b , λ 1,2 = ± 1 2 denote the four-momenta and helicities of the protons and p 3,4 and λ 3,4 = ± 1 2 denote the four-momenta and helicities of the muons, respectively.
The amplitudes for the reaction (3.1) through φ resonance production can be obtained from the amplitudes discussed in Sec. II with iΓ The standard φ-γ coupling (see e.g. (3.23), (3.24) of [41]) gives The decay rate φ → µ + µ − is calculated from the diagram Fig. 5 (neglecting radiative corrections) as From the experimental values [75] m φ = (1019.461 ± 0.016) MeV , we get and using (3.4) On the other hand, using (3.3) directly with the standard range for γ φ quoted in (3.24) of [41], 4π/γ 2 φ = 0.0716 ± 0.0017, we get Within the errors the two values obtained in (3.7) and (3.8) are compatible. In the following we shall take (3.8) for our calculations.

IV. RESULTS
In this section we wish to present first results for three cases pp → ppφ(1020), and with φ decaying to K + K − or µ + µ − , corresponding to the processes discussed in Secs. II and III.

A. Comparison with the WA102 data
The φ-meson production in central proton-proton collisions was studied by the WA102 Collaboration at √ s = 29.1 GeV. The experimental cross section quoted in Table 1 of [55] is In [55] also the dP t dependence of φ production and the distribution in φ pp were presented. Here dP t is the "glueball-filter variable" [79,80] defined as: and φ pp is the azimuthal angle between the transverse momentum vectors p t,1 , p t,2 of the outgoing protons. For the kinematics see e.g. Appendix D of [45]. In Fig. 6 (left panel) we compare our theoretical predictions for the φ pp distribution to the WA102 experimental data for the pp → ppφ reaction normalised to the central value of the total cross section σ exp = 60 nb from [55]; see (4.1). We consider the two photoproduction contributions: γP plus Pγ and γ M plus Mγ with M = π 0 , η. For the photon-pomeron fusion we show the results for the two parameter sets, set A and set B, discussed in Appendix B (see Fig. 32). For the estimation of an upper limit of the γ-M plus M-γ contribution we take Λ MNN = Λ φγ M = 1.2 GeV in (2.43) and (2.44); see the discussion in Appendix B and Fig. 33. We find that the γ-M plus M-γ contribution is much smaller than the γ-P plus P-γ contribution. It constitutes about 15 % of γP plus Pγ in the integrated cross section. The γ-S (S = f 0 (500), f 0 (980), a 0 (980)) contribution terms are expected to be even smaller than the γ-M ( M = π 0 , η) ones; see Fig. 33. Therefore, we neglect the γ M and γS fusion contributions in the further considerations. Clearly, we see that the photoproduction mechanism is not enough to describe the WA102 data, at least if we take the central value of σ exp quoted in (4.1) for normalising the data for the φ pp distribution. In Fig. 6 (right panel) we show the distributions in rapidity of the φ meson. The photoproduction mechanisms with P exchange (γP and Pγ) dominate at midrapidity. The γ-M and M-γ components are separated and contribute in backward and forward region of y φ , respectively. The separation in rapidity means also the lack of interference effects between the γ-M and M-γ components.
In Fig. 6 we denote, for brevity, the sum of the contributions γP plus Pγ by γP, the sum of γ M plus Mγ by γ M. The analogous notation will be used for these and all other contributions in the following.
It is a known fact that absorption effects due to strong proton-proton interactions have an influence on the shape of the distributions in φ pp , dP t , |t 1 | and |t 2 |. Thus, absorption effects should be included in realistic calculations. In the calculations presented we have included the absorptive corrections in the one-channel eikonal approximation as was discussed e.g. in Sec. 3.3 of [46]. The absorption effects lead to a large damping of the cross sections for purely hadronic diffractive processes and a relatively small reduction of the cross section for the photoproduction mechanism. We obtain the ratio of full and Born cross sections S 2 (the gap survival factor) at √ s = 29.1 GeV and without any cuts included as follows S 2 ∼ = 0.8 for the photoproduction contribution and S 2 ∼ = 0.4 for the purely hadronic diffractive contributions discussed below. However, the absorption The distributions in φ pp and in y φ for the φ photoproduction processes in the pp → ppφ reaction at √ s = 29.1 GeV. The data points have been normalized to the central value for σ exp (4.1) from [55]. The results for the photon-pomeron fusion are presented for the two parameter sets, set A and set B, as defined in Appendix B, see the caption of Fig. 32, (the bottom and top solid lines, respectively). We also show the contribution from the γ-M ( M = π 0 , η) fusion (the dashed lines). The absorption effects are included here. strongly depends on the kinematic cuts on |t 1 | and |t 2 |. This will be discussed in detail when presenting our predictions for the LHC; see Sec. IV B below.
In Fig. 7 we show results for the γP and the subleading fusion processes discussed in Sec. II C. We present results for the two approaches as follows. In the top panels (approach II) we show results for the reggeon-pomeron (φ R -P, ω R -P) and the reggeonreggeon (ω R -f 2R ) contributions, (2.64)-(2.68), and in the bottom panels (approach I) we show results for the reggeized-φ/ω-meson exchanges (2.57)-(2.63). The ρ-π 0 fusion contribution is calculated in the approach I, i.e., for the reggeized ρ 0 -meson exchange.
In Figs. 8-10 we present several differential distributions for the γ-P and the O-P fusion processes corresponding to the diagrams shown in Figs. 1 and 2, respectively, and for the subleading processes ω-P, φ-P, ω-f 2R and ρ-π 0 fusion. For the O-P contribution we take the following parameters, see (2.27)-(2.33), and we choose different values for a POφ and b POφ : The results shown in panels : Distributions in proton-proton relative azimuthal angle φ pp (left panels) and in dP t (4.2), the "glueball filter" variable (right panels), for the pp → ppφ reaction at √ s = 29.1 GeV. The data points have been normalized to the central value of the total cross section (4.1) from [55]. The results for the fusion processes γP (the blue solid lines), ωP (the black dashed line), ω f 2R (the black dotted line), φP (the green dash-dotted line), and ρπ 0 (the violet dotted line) are presented. In the top panels the ω-P, φ-P and ω-f 2R exchanges are treated, respectively, as reggeon-pomeron and reggeon-reggeon exchanges (approach II) while in the bottom panels these contributions were calculated in the reggeized-vector-meson approach (2.58) (approach I). The coherent sum of these contributions is shown by the two black solid lines. The lower line is for the parameter set A (B7) and the upper line is for the parameter set B (B8). The absorption effects are included here.
We have checked that these parameters are compatible with our analysis of the WA102 data for the pp → ppφφ reaction in [52]. Comparing the results shown in Fig. 7 with those in Figs. 8 and 9 we can see that the complete results indicate a large interference effect between the γP, OP, ωP, ω f 2R , and φP terms.
In [55] experimental values for the cross sections in three dP t intervals and for the ratio  of φ production at small dP t to large dP t are given. We show our corresponding results in Table I for the two approaches, I and II, with appropriate POφ coupling constants (4.5), (4.6), (4.7). Here we take the parameter set B (B8) for the γP contribution. Now we discuss our results concerning the WA102 data. As already mentioned we find that the γ-P fusion processes alone cannot describe the WA102 data for the φ pp distribution. This holds even if we scale down the experimental data by about 30 % cor- responding to the quoted error on the total cross section in (4.1). Thus, we need other contributions, subleading ones or maybe odderon-pomeron fusion. From the subleading ones we find that the γ-π 0 and γ-η contributions are very small; see Fig. 6. Also the ρ-π 0 -fusion contribution turns out to be very small. According to our results, the important subleading contributions are ω-P, ω-f 2R and φ-P fusion. We have treated them with two methods of reggeization, I and II. The reggeized vector-meson approach I, see (2.58), (2.59), almost certainly overestimates these contributions. The reggeization means that we replace the vector-meson exchange by a coherent sum of exchanges with spin 1 + 3 + 5 + .... The higher the spin the higher the mass of the exchanged particle. In (2.58) this increase of mass is not taken into account leading to the overestimate. Also, the distribution in φ pp in this approach I is too flat and does not fit the data; see the ωP contribution in the left bottom panel in Fig. 7. The approach II, on the other hand, assumes reggeon exchanges, ω R and φ R . This approach maybe underestimates the contributions if s 1 or s 2 are small, but should be very reasonable for large s 1 or s 2 . But note that in our reaction the threshold for s 1 and s 2 is already quite large s thr ≈ 4 GeV 2 ; see (2.60). We see clearly from Fig. 7 that in this approach the sum of the γP, γ f 2R , ω R P, ω R f 2R , φ R P and ρπ 0 contributions 2 , added coherently, cannot explain the φ pp data. This gives a hint that the missing contribution could be the odderon-pomeron fusion. And, indeed, with suitable odderon parameters we arrive at a decent description of the φ pp and the dP t data from WA102; see Figs. 8, 9, and Table I, respectively. However, we have to remember that the φ pp distributions have a large normalisation uncertainty due to the relatively large dσ/d(dP t 0.5 GeV) are given. The experimental numbers are from Table 2 of [55]. The theoretical numbers correspond to the total results including all terms contributing; see the upper black lines in Fig. 9. dP t 0.2 GeV 0.2 dP t 0.5 GeV dP t 0. error on σ exp (4.1). Therefore, we emphasise that our fits to the WA102 data on single φ CEP only give a hint that this reaction could be very interesting for a search of odderon effects. From Fig. 10 we see that the odderon-pomeron contribution dominates at larger |y φ | and p t,φ compared to the photon-pomeron contribution. As we shall see this also holds at LHC energies and should help in searches for odderon effects there.

B. Predictions for the LHC experiments
1. The pp → ppK + K − reaction In this subsection we wish to show our predictions for the LHC experiments. We start with the presentation of the differential distributions for the pp → pp(φ → K + K − ) reaction (2.3) which we integrate in the φ resonance region (2.2). First we show, for orientation purposes, results for the γP-fusion and the OP-fusion contributions separately (see the diagrams shown in Figs. 1 and 2, respectively). For the final results we shall, of course, add these contributions coherently and calculate absorption corrections at the amplitude level. We have checked that in the kinematic regimes discussed in the following the subleading contributions (see Sec. II C) can be safely neglected.
In Figs. 11-17 we show the results for √ s = 13 TeV, and |η K | < 2.5, p t,K > 0.1 GeV and sometimes with extra cuts on the leading protons of 0.17 GeV < |p y,1 |, |p y,2 | < 0.50 GeV as will be the proton momentum window for the ALFA detectors placed on both sides of the ATLAS detector. Figure 11 shows the Born-level distributions in |t 1 | (top panels) and in transverse momentum p t,1 = |p t,1 | of the proton p (p 1 ) (bottom panels). In the left panels the photoproduction contributions are plotted while in the right panels we show the results for the odderon contributions. The results for the parameter set B (B8) for the photoproduction term and for the parameters quoted in (4.3), (4.4), (4.6) for the OP-fusion are presented. We show results for two diagrams separately and for their coherent sum (total). The interference effects between the two diagrams are clearly visible, especially for the OP-fusion mechanism. A different behaviour is seen at small |t 1 | for the γP and the OP components. Due to the photon exchange the protons are scattered only at small angles and the γP distribution has a singularity for |t 1 | → 0. Of course, t 1 = 0 cannot be reached here from kinematics. In contrast, the OP distribution shows a dip for |t 1 | → 0. An explanation of this type of behaviour is given in Appendix C of [33]. In the bottom panels we show the p t distributions for proton p (p 1 ). Here these differences are also clearly visible. The distributions in four-momentum transfer squared |t 1 | (top panels) and in transverse momentum p t,1 of the proton p (p 1 ) (bottom panels) for the pp → pp(φ → K + K − ) reaction at √ s = 13 TeV and for |η K | < 2.5, p t,K > 0.1 GeV. Absorption effects are not included here. In the left panels we show the results for the photoproduction mechanism obtained with the parameter set B (B8). The results for the γP and Pγ fusion contributions are presented. Their coherent sum is shown by the blue solid thick line. In the right panels we present the results for the odderon-pomeron-fusion mechanism obtained with the parameters quoted in (4.3), (4.4), and (4.6). Again, we show the OP-and PO-fusion contributions separately and their coherent sum (red long-dashed thick line).
In Fig. 12 we show results for the hadronic diffractive contribution for the two type of couplings in the POφ vertex (2.32) separately and when both couplings are taken into account. We can see that the complete result indicates a large interference effect of the a and b coupling contributions in the amplitudes.  Figure 13 shows the correlations in the relative azimuthal angle between the outgoing protons (see the top panels) and the rapidity distance between the two centrally produced kaons, y diff = y 3 − y 4 (see the bottom panels) without (the left panels) and with (the right panels) limitations on the leading protons. The thin lines represent the results for one of the two diagrams separately (γP or Pγ as well as OP or PO) and the thick lines represent their coherent sum (γP plus Pγ, OP plus PO). The influence of kinematic cuts on the leading protons is also shown. The reader is asked to note a reversed interference behaviour for the photon-pomeron and odderon-pomeron mechanisms. The odderonpomeron contribution dominates at larger |y diff | compared to the photon-pomeron contribution. In Fig. 14 we show the kaon angular distributions in the K + K − rest system using the Collins-Soper (CS) frame; see Appendix C. The Collins-Soper frame which we use here is defined as in our recent paper on extracting the PP f 2 (1270) couplings in the pp → ppπ + π − reaction [53] with K + and K − in the place of π + and π − , respectively. For the pp → pp(φ → K + K − ) reaction we can observe interesting structures in the φ K + , CS (top panel) and in the cos θ K + , CS (bottom panel) distributions. The distributions in φ K + , CS for the hadronic diffractive contribution (OP + PO) are relatively flat. The photoproduction term, in contrast, shows pronounced maxima and minima which are due to the interference of the γP and Pγ terms. The cuts on leading protons considerably change the shape of the φ K + , CS distributions for the photon-exchange contribution. The angular distribution dσ/d cos θ K + , CS looks promising for a search of odderon effects as it is very different for the γP fusion and the OP fusion processes. In Fig. 15 we compare results without (the thin lines) and with (the thick lines) absorption effects. The absorption effects have been included in our analysis within the onechannel-eikonal approach. For the ATLAS-ALFA kinematics the absorption effects lead to a large damping of the cross sections both for the hadronic diffractive and for the pho-toproduction mechanisms. We find a suppression factor of the cross section S 2 ≃ 0.3; see Table II. From Fig. 15 we see that the absorption effects also modify the shape of the distributions.
Up to now we have shown results including the ATLAS-ALFA experimental cuts for concrete set of parameters, set B (B8) for the photoproduction term and (4.3), (4.4), (4.6) for the OP-fusion contribution.
In Figs. 16 and 17 we show distributions in several variables for different choices of parameters. We show results for the γPand OP-fusion contributions separately (see the blue and red lines, respectively) and when both terms are added coherently at the amplitude level (the black line). The absorption effects are included in the calculations. The upper blue solid line is for the parameter set B (B8) and the lower blue solid line is for the parameter set A (B7). The red long-dashed line corresponds to the odderon parameters quoted in (4.3), (4.4), and the POφ coupling parameters (b) (4.6), the red dash-dotted line is for the choice of POφ coupling parameters (a) (4.5), and the red dotted line is for (4.7). The black solid line represents the coherent sum of γP-fusion and OP-fusion contributions for the coupling parameters (B8) and (4.6), respectively. We can see that the complete result indicates a large interference effect of γPand OP-fusion terms. The odderonpomeron contribution dominates clearly at larger |y K |, |y diff |, p t,K + K − , the transverse momentum of the K + K − pair, and cos θ K + , CS = ±1, compared to the photon-pomeron contribution. We encourage the experimentalists associated to the ATLAS-ALFA experiment to prepare such distributions, especially dσ/dy diff , dσ/d cos θ K + , CS , and dσ/dφ K + , CS . Observation of the pattern of maxima and minima would be interesting by itself as it is due to interference effects. It is worth adding that much smaller interference effects are predicted when no cuts on the outgoing protons are required; see the results in Table II and Figs. 19, 20 below. When cuts on transverse momenta of the outgoing protons are imposed then the γPand OP-fusion contributions become comparable and large interference effects are in principle possible. Now we shall discuss results for the LHCb experimental conditions. In Fig. 18 we show the two-dimensional distributions in (p t,K + , p t,K − ) for √ s = 13 TeV, 2.0 < η K < 4.5, and p t,K > 0.1 GeV. In the left panel we show the result for γ-P fusion obtained with the parameter set B (B8). In the right panel we show the result for O-P fusion for the parameters quoted in (4.3), (4.4), and (4.6). We can see that the γP-fusion contribution is larger at smaller p t,K than the OP-fusion contribution. Therefore, a low-p t,K cut on transverse momenta of the kaons can be helpful to reduce the γP-fusion contribution; compare the left and right panels in Figs. 19 and 20 below.
In Figs. 19 and 20 we show several distributions for γP-fusion and OP-fusion contributions for the LHCb experimental conditions, √ s = 13 TeV, 2.0 < η K < 4.5, p t,K > 0.3 GeV (left panels) or p t,K > 0.5 GeV (right panels). Our predictions for different choices of parameters are shown. The absorption effects were included in the calculations. For larger kaon transverse momenta (or transverse momentum of the K + K − pair) the odderon-exchange contribution is bigger than the photon-exchange one.
As in the previous (ATLAS-ALFA) case the angular distributions in the K + K − Collins-Soper rest system seem interesting. In Fig. 21 we show the two-dimensional distributions in (φ K + , CS , cos θ K + , CS ) for 2.0 < η K < 4.5 and p t,K > 0.3 GeV.

The pp → ppµ + µ − reaction
The φ meson can also be observed in the µ + µ − channel. In this subsection we wish to show our predictions for the pp → ppµ + µ − reaction for the LHCb experiment at The differential cross sections for the pp → pp(φ → K + K − ) reaction. Calculations were done for √ s = 13 TeV, 2.0 < η K < 4.5, and p t,K > 0.3 GeV (left panels) or p t,K > 0.5 GeV (right panels). Results for the photoproduction (blue solid lines) and the OP-fusion (red lines) contributions are shown separately. The meaning of the lines is the same as in Fig. 16. The black solid line corresponds to the coherent sum of the γP and OP fusion reactions with the coupling parameters (B8) and (4.6), respectively. The absorption effects are included here.  Fig. 19. Also the meaning of the lines is as in Fig. 19. √ s = 13 TeV for the 2.0 < η µ < 4.5 pseudorapidity range. Here we require no detection of the leading protons.
In Fig. 22 we present the µ + µ − invariant mass distributions in the φ(1020) resonance region. We show the contributions from the γPand OP-fusion processes and the continuum γγ → µ + µ − term. The dimuon-continuum process (γγ → µ + µ − ) was discussed e.g. in [81] in the context of the ATLAS measurement [82]. In our analysis here we are looking at the dimuon invariant mass region M µ + µ − ∈ (1.01, 1.03) GeV. In Fig. 23 we show two-dimensional distributions in (p t,µ + , p t,µ − ) for three different processes. The result in the panel (a) corresponds to the continuum contribution without the cut on M µ + µ − . Here the maximum of the cross section is placed along the p t,µ + = p t,µ − line which is due to the predominantly small transverse momenta of the photons in this photon-exchange process. The results in the panels (b), (c), and (d) correspond to the continuum term, the γPand OP-fusion processes, respectively, including the limitation on M µ + µ − .
Note, that in the continuum term, γγ → µ + µ − , the µ + µ − are in a state of charge conjugation C = +1. For φ → µ + µ − we have a state of C = −1. Thus, the interference of the continuum and the φ-production reactions will lead to µ + -µ − asymmetries. We have checked, however, that the interference in µ + µ − channel is smaller then numerical precision, definitely smaller than 2%.
In Figs. 24 and 25, we show the predictions for the pp → ppµ + µ − reaction for typical experimental conditions on the transverse momentum of the muons, p t,µ > 0.1 GeV and p t,µ > 0.5 GeV, respectively. In contrast to dikaon production here there is for both the γPand the OP-fusion contributions a maximum at y diff = 0 (or cos θ µ + ,CS = 0). In Fig. 24 the continuum contribution is large. Imposing a larger cut on the transverse momenta of the muons reduces the continuum contribution which, however, still remains sizeable at y diff = 0. Such a cut reduces the statistics of the measurement; see the results in Table II. The µ + µ − channel seems to be less promising in identifying the odderon exchange at least when only the p t,µ cuts are imposed. Eventually, the absolute normalization of the cross section and detailed studies of shapes of distributions should provide a clear answer whether one can observe the odderon-exchange mechanism here.
In Fig. 26 we present the distributions in transverse momentum of the µ + µ − pair. We can see that the low-p t,µ + µ − cut can be helpful to reduce the continuum (γγ → µ + µ − ) and photon-pomeron-fusion contributions.
In Fig. 27 we show the results when imposing in addition a cut p t,µ + µ − > 0.8 GeV. The γγ → µ + µ − contribution is now very small. We can see from the y diff distribution that the photon-pomeron term gives a broader distribution than odderon-pomeron term. At y diff = 0 the odderon-exchange term is now bigger than the photoproduction terms.
Finally, in Fig. 28 we show two-dimensional distributions in (η µ + , η µ − ). The results for three different processes are shown. One can see quite different distributions for the (γP plus Pγ) and the (OP plus PO) contributions. The odderon exchange contribution shows an enhancement at η µ − ∼ η µ + ∼ (4.0 − 4.5). In Table II we have collected integrated cross sections in nb for √ s = 13 TeV and with different experimental cuts for the exclusive pp → ppK + K − and pp → ppµ + µ − reactions including the γPor OP-fusion processes separately. We also show the results for the coherent sum of the γP and OP fusion reactions including absorption corrections. Here we take for the γP and OP fusion contributions the coupling parameters (B8) and (4.6), respectively. γ /dp σ The two-dimensional distributions in (p t,µ + , p t,µ − ) for the pp → ppµ + µ − reaction. The calculations were done for √ s = 13 TeV and 2.0 < η µ < 4.5. The results in the panels (a) and (b) correspond to the µ + µ − continuum without and with the cut on M µ + µ − ∈ (1.01, 1.03) GeV, respectively. The results in the panels (c) and (d) correspond to the φ production via γP fusion and via OP fusion, respectively. No absorption effects are included here.  The integrated cross sections in nb for the central exclusive production of single φ mesons in proton-proton collisions with the subsequent decays φ → K + K − or φ → µ + µ − . The results have been calculated for √ s = 13 TeV in the dikaon/dimuon invariant mass region M 34 ∈ (1.01, 1.03) GeV and for some typical experimental cuts. The ratios of full and Born cross sections S 2 (the gap survival factors) are shown in the last column.

V. CONCLUSIONS
In the present paper we have discussed the possibility to search for odderon exchange in the pp → ppφ reaction with the φ meson observed in the K + K − or µ + µ − channels. There are two basic processes: relatively well known (at the Born level) photon-pomeron fusion and rather elusive odderon-pomeron fusion. In our previous analysis on two φmeson production in proton-proton collisions [52] we tried to tentatively (optimistically) fix the parameters of the pomeron-odderon-φ vertex to describe the relatively large φφ invariant mass distribution measured by the WA102 Collaboration [54]. The calculation for the pp → ppφ process requires in addition knowledge of the rather poorly known coupling of the odderon to the proton. The latter can be fixed, in principle, by a careful study of elastic proton-proton scattering. The present estimates suggest β Opp ≃ 0.1 β PNN (see Eq. 2.31). In the present study we therefore fixed the odderon coupling to the proton at this reasonable value and tried to make predictions for central exclusive φ-meson production. Our results also depend on the assumptions made for the Regge trajectory of the odderon, Eqs. (2.29) and (2.30). In this context the photon-pomeron fusion is a background for the odderon-pomeron fusion. The parameters of photoproduction were fixed to describe the HERA φ-meson photoproduction data; see Appendices A and B. There, we pay special attention to the importance of the φ-ω mixing effect in the description of the γp → φp and γp → ωp reactions. To fix the parameters of the pomeron-odderon-φ vertex (coupling constants and cut-off parameters) we have considered several subleading contributions and compared our theoretical predictions for the pp → ppφ reaction with the WA102 experimental data from [55].
Having fixed the parameters of the model we have made estimates of the integrated cross sections as well as shown several differential distributions for pp → ppφ at the WA102 energy √ s = 29.1 GeV. In addition we have discussed in detail exclusive production of single φ mesons at the LHC, both in the K + K − and µ + µ − observation channels, for two possible distinct types of measurements: (a) at midrapidity and with forward measurement of protons (relevant for ATLAS-ALFA or CMS-TOTEM), (b) at forward rapidities and without measurement of protons (relevant for LHCb). In contrast to low energies, where several processes may compete, at the large LHC energies the odderonexchange contribution competes only with the photoproduction mechanism. We have considered different dedicated observables. Some of them seem to be promising. The distributions in y diff (rapidity difference between kaons) and angular distributions of kaons in the Collins-Soper frame seem particularly interesting for the K + K − final state. Increasing the value of the cut on the transverse momenta of kaons improves the signal (pomeron-odderon fusion) to the background (photon-pomeron fusion) ratio. Of course, in this way the rates are reduced; see Table II. In general, the µ + µ − channel seems to be less promising in identifying the odderon exchange. In this case detailed studies of shapes of dσ/dy diff or/and dσ/d cos θ µ + ,CS would be very useful in understanding the general situation. To observe a sizeable deviation from photoproduction a p t,µ + µ − > 0.8 GeV cut on the transverse momentum of the µ + µ − pair seems necessary. Such a cut reduces then the statistics of the measurement considerably. A combined analysis of both the K + K − and the µ + µ − channels should be the ultimate goal in searches for odderon exchange. We are looking forward to first experimental results on single φ CEP at the LHC.
In summary, we have presented results for single φ CEP both at the Born level as well as including absorption effects in the eikonal approximation. We have argued that the WA102 experimental results at c.m. energy √ s = 29.1 GeV leave room for a possible odderon-exchange contribution there. Then we have turned to LHC energies where single φ CEP can be studied by experiments such as: ATLAS-ALFA, CMS-TOTEM, ALICE, and LHCb. Using our results it should be possible to see experimentally if odderon effects as calculated are present, if our odderon parameters have to be changed, or if it is only possible to derive limits on the odderon parameters. We present distributions which are sensitive to the odderon-exchange contribution. We are looking forward also to relevant data from the lower energy COMPASS experiment. At high energies the deviations from the γP-fusion contribution can be treated as a signal of odderon exchange. In our opinion several distributions should be studied to draw a definite conclusion on the odderon exchange. So far the odderon exchange was not unambiguously identified in any reaction. In the present paper we have shown that for the odderon search the study of central exclusive production of single φ mesons is a valuable addition and alternative to the study of elastic proton-proton scattering or production of two φ mesons in the pp → ppφφ reaction discussed by us very recently; see [52].
Inserting this in (A1) and defining ω 0 = 1 √ 2 uū + dd and φ 0 = −ss the mixing equation reads: The reverse reads It is well known that experimentally the angle ∆θ V is small. Thus, the physical ω and φ are nearly equal to ω 0 and φ 0 , respectively. Now we consider the Pω R ω, Pω R φ, Pφ R ω, and Pφ R φ vertices for which we assume a structure as in (2.12) with appropriate coupling constants a and b. In our case (CEP of φ meson in proton-proton collisions) the ω R (ω reggeon) is, however, off-mass shell and we neglect the rather unknown mixing in this Regge-like state and include mixing in the on-shell φ only. We shall argue, therefore, that in the Pω R ω and Pω R φ vertices only the ω 0 will couple. In this way we get for our coupling constants a and b In an analogous way we shall assume that in the Pφ R ω and Pφ R φ vertices only the φ 0 will couple. This gives In Sec. II we consider also the couplings of the pomeron to reggeized vector mesons and vector mesons. In Appendix B below we need the couplings of the pomeron to the offshell vector mesons at q 2 = 0 and the vector mesons. We denote here, for clarity, these reggeized or off-shell mesons by V. In the following we shall assume that From (A7) to (A14) we obtain the coupling constants to be inserted in (2.68) and (2.57). The deviation ∆θ V from the ideal mixing in (A9) can be estimated through the decay widths of φ → π 0 γ and ω → π 0 γ (π 0 is assumed not to have any ss component); see Eq. (B2) of [58]. Using the most recent values from [75] we have 3 and ∆θ V = arctan(0.076) = 4.35 • . In Refs. [83][84][85] a smaller value was found, ∆θ V ≃ 3.7 • . In the following we shall use this latter value for ∆θ V .
In order to estimate the relevant coupling parameters we shall assume that the f 2R ωω couplings are similar to the f 2R ρρ ones. Then we take the default values for the f 2R ρρ and a 2R ρω couplings estimated from VMD in Sec. 7.2, Eqs. (7.31), (7.32), (7.36), and (7.43), of [41]: In (B2) we assume that both coupling constants are positive. To estimate the Pωω coupling constants we use the relation: in analogy to the corresponding one for the ρ meson; see (7.27) of [41] and (2.13) of [46]. Note that a Pωω must be positive in order to have a positive ωp total cross section for all ω polarisations. This follows from (7.21) of [41] replacing there the ρ by the ω meson.
In Fig. 30 we show the cross sections for the γp → ωp reaction together with the experimental data. From the comparison of our results to the experimental data, taking first only the diagrams of Fig. 29 (a) into account, we found that even a small (and positive) value of the a Pωω coupling leads to a reduction of the cross section. Therefore, for simplicity, we choose a Pωω = 0 in (B3). The black solid line corresponds to the calculation including only the terms shown in the diagram (a) of Fig. 29. We used here the Pωω coupling constants a Pωω = 0 , b Pωω = 7.04 GeV −1 (B4) and the parameters (B1) and (B2) for the reggeon exchanges. We recall that for all exchanges participating in the diagram (a) we take Λ 2 0 = 0.5 GeV 2 in the form factor F M (t); see (3.34) of [41]. Now we include the off-diagonal terms from the diagram of Fig. 29 (b). For estimating the coupling constants a P φω and b P φω we use (A14) and the determination of a Pφφ and b Pφφ from the discussion of the γp → φp reaction below. We get with the sets A and B, respectively, with ∆θ V = 3.7 • set A : a P φω = 0.05 GeV −3 , b P φω = 0.23 GeV −1 , Λ 2 0, P φω = 1.0 GeV 2 ; (B5) set B : a P φω = 0.07 GeV −3 , b P φω = 0.19 GeV −1 , Λ 2 0, P φω = 4.0 GeV 2 . (B6) In a similar way the coupling parameters for f 2R exchange, a f 2R φω and b f 2R φω , can be obtained. However, the f 2R φφ couplings are expected to be very small. In practice, we do not consider an f 2R -exchange contribution from the diagram of Fig. 31 (a). Here, we neglect also the f 2R exchange from the diagram of Fig. 29 (b). The blue solid line in Fig. 30 corresponds to the calculation including in addition to the processes from diagram (a) of Fig. 29 the φ-ω mixing effect for the P exchange [see diagram (b) of Fig. 29]. Our model calculation describes the total cross section fairly well 4 for energies W γp > 10 GeV. At low γp energies there are other processes contributing, such as the π 0 -meson exchange, and the ω bremsstrahlung; see e.g. [18,86] for reviews and details concerning the exclusive ω production. We nicely describe also the differential cross section dσ/d|t|. We have checked that the complete results including the φ-ω mixing effect with sets A (B5) and B (B6) differ only marginally.  [18] for more references). The black solid line corresponds to results with both the pomeron and reggeon ( f 2R , a 2R ) exchanges. The black long-dashed line corresponds to the pomeron exchange alone while the black short-dashed line corresponds to the reggeon term. In the calculation we used the parameters of the coupling constants given by (B1), (B2) and (B4). The blue solid line corresponds to the complete result including the φ-ω mixing effect (for the P exchange) with the parameter set A (B5). Right panel: The differential cross section for the γp → ω p reaction at W γp = 80 GeV. Our complete results, without (the black line) and with (the blue line) the mixing effect, are compared to the ZEUS data [87].
Next, we discuss the γp → φp reaction. At high γp energies the pomeron exchange contribution, shown by the diagram (a) of Fig. 31, is the dominant one; see Sec. IV B of [51]. As was mentioned in Sec. I, in the low-energy region the corresponding production mechanism is not well established yet. There the nondiffractive processes of the pseudoscalar π 0 -and η-meson exchange are known to contribute and are not negligible due to constructive η-π 0 interference; see, e.g., [58,59]. In addition, many other processes, e.g., direct φ meson radiation via the sand u-channel proton exchanges [58,65], ss-cluster knockout [57], t-channel σ-, f 2 (1270)and f 1 (1285)-exchanges [64] were considered. In [64] no vertex form factors were taken into account for the reggeized meson exchange contributions and instead of the f 2 (1270)-exchange there one should consider f ′ 2 -exchange with appropriate parameters. However, a peak in the differential cross sections (dσ/dt) t=t min at forward angles around E γ ∼ 2 GeV (W γp ∼ 2.3 GeV) observed by the LEPS [88,89] and CLAS [90] collaborations cannot be explained by the processes mentioned above. To explain the near-threshold bump structure the authors of [61,62,65] propose to include exchanges with the excitation of nucleon resonances. In [60,63] another explanation, using the coupled-channel contributions with the Λ(1520) resonance, was investigated. In [63] the hadronic box diagrams with the dominant KΛ(1520) rescattering amplitude in the intermediate state were treated only approximately in a coupledchannel formalism neglecting the real part of the transition amplitudes.
Implementation of the box diagrams in our four-body calculation is rather cumbersome. On the other hand, we expect that they do not play a crucial role for the pp → ppφ reaction at the high energies of interest to us here.
In Fig. 32 we show the elastic φ photoproduction cross section as a function of the center-of-mass energy W γp (left panel) and the differential cross section dσ/d|t| (right panel). We show results for two parameter sets, set A and set B,  Fig. 31 only with the parameter set (B7). Right panel: The differential cross section dσ/d|t| for the γp → φp process. We show the ZEUS data at low |t| (at W γp = 70 GeV and the squared photon virtuality Q 2 = 0 GeV 2 , solid marks, [73]) and at higher |t| (at W γp = 94 GeV and Q 2 < 0.01 GeV 2 , open circles, [74]). Again, the results for the two parameter sets, set A (B7) and set B (B8), are presented.
Note that the parameter set (B7) for Λ 2 0, Pφφ = 1.0 GeV 2 is different than found by us in Sec. IV B of [51] (see Fig. 6 where the ω-φ mixing effect was not included. For comparison, the red lower line represents the result without the ω-φ mixing, i.e., it contains only the terms represented by the diagram (a) of Fig. 31. We can see from Fig. 32 (right panel) that the parameter set B (B8) for Λ 2 0, Pφφ = 4.0 GeV 2 with the relevant values of the coupling constants a and b describes more accurately the t distribution.
In Fig. 33 we show the integrated cross section for the γp → φp reaction at low W γp energies. We can see that the diffractive pomeron and reggeon exchanges, even including the pseudoscalar and scalar meson exchange contributions, are not sufficient to describe the low-energy data. Here we want to examine the uncertainties of the photoproduction contribution due to the meson exchanges in the t channel. In the left panel, for the meson exchanges, we use the values of the coupling constants and the cut-off parameters from [58] while in the right panel we choose Λ MNN = Λ φγ M = 1.2 GeV in (2.43) and (2.44).
Our extrapolations of the cross section, using the theory applicable at high energies, represents the experimental data roughly on the average. But the scatter of the experimental data is quite considerable. Thus, it is impossible for us to draw any further conclusions concerning these low-energy results at the moment.
(GeV) p γ The elastic φ photoproduction cross section as a function of W γp integrated over t min < |t| < 1 GeV 2 . The theoretical results are compared with a compilation of low-energy experimental data from [91][92][93], and [56]. The open data points are taken from [64] (data was obtained there by integrating over the differential cross sections given in [90]). The solid lines correspond to a coherent sum of pomeron, f 2R reggeon, pseudoscalar, and scalar exchanges. For the diffractive component (P + R) we take the set A of parameters from Fig. 32. The results for the pseudoscalar and scalar exchanges shown in the left panel were obtained with the parameters from [58]; see Sec. II C 1. In the right panel, for comparison, we show results obtained for different values of the cut-off parameters in the pseudoscalar term. Here we take Λ MNN = Λ φγ M = 1.2 GeV in (2.43) and (2.44).
The CS frame is then defined by the coordinate-axes unit vectors e 1, CS =p a +p b |p a +p b | , The angles θ K + , CS and φ K + , CS are the polar and azimuthal angles of the momentum vectorp 3 in this system. We have cos θ K + , CS =p 3 · e 3, CS . (C3)