Production of f2(1270) meson in pp collisions at the LHC via gluon-gluon fusion in the k-factorization approach

Abstract We calculate inclusive cross section for f2(1270) tensor meson production via color singlet gluon-gluon fusion in the kt-factorization approach with unintegrated gluon distribution functions (UGDFs). The process is maybe interesting in the context of searches for saturation effects. The energy-momentum tensor, equivalent to helicity-2 coupling, and helicity-0 coupling are used for the g∗g∗ → f2(1270) vertex. Some parameters are extracted from γγ → f2(1270) → ππ reactions. Different modern UGDFs from the literature are used. The results strongly depend on the parametrization of the g∗g∗ → f2(1270) form factor. Our results for transverse momentum distributions of f2 are compared to preliminary ALICE data. We can obtain agreement with the data only at larger f2(1270) transverse momenta only for some parametrizations of the g∗g∗ → f2(1270) form factor. No obvious sign of the onset of saturation is possible. At low transverse momenta one needs to include also the ππ final-state rescattering. The agreement with the ALICE data can be obtained by adjusting probability of formation and survival of f2(1270) in a harsh quark-gluon and multipion environment. The pomeron-pomeron fusion mechanism is in addition discussed and results are quantified.


I. INTRODUCTION
The mechanism of f 2 (1270) meson production in proton-proton collisions at high energies was not carefully studied so far. Till very recently in the PYTHIA event generator f 2 (1270) is not produced in a primary fragmentation process but occurs only in decays, e.g. J/ψ → f 2 (1270)ω, D ± s → f 2 (1270)π ± , B ± → τ ± ν τ /ν τ f 2 (1270). The corresponding branching fractions are rather small so one cannot expect large contributions. On the other hand it is rather difficult to observe f 2 (1270) experimentally. The dominant, and relatively easy, decay channel is f 2 (1270) → π + π − . Then the signal is on a huge non-reduceable π + π − background. So far only STAR [1] and ALICE [2] undertook experimental efforts. Some time ago [3] it was suggested that the gluon-gluon fusion could be the dominant production mechanism. Certainly this interesting working hypothesis requires further elaboration and experimental confirmation.
In the present paper we follow the idea from [3] and try to shed new light on the situation. We will apply the k t -factorization approach successfully used for χ c quarkonium production [4,5], for η c (1S, 2S) production [6,7], and recently for f 0 (980) production [8].
In this study, we focus on the production of f 2 (1270) meson. The g * g * → f 2 (1270) vertex is not known a priori. We will use modern unintegrated gluon distributions from the literature. We will try to verify the hypothesis of the dominance of the helicity-2 component, the coupling of spin-2 mesons to the energy-momentum tensor [3], by comparing our results to the preliminary ALICE data [2].
To obtain a f 2 γγ and b f 2 γγ in (2.1) we use the experimental value of the radiative decay width as quoted for the preferred solution III in Table 3 of [13]. Using the decay rate from (5.28) of [10] Γ( f 2 → γγ) = m f 2 80π and assuming a f 2 γγ > 0 and b f 2 γγ > 0, we find where α em = e 2 /(4π) ≃ 1/137 is the electromagnetic coupling constant.
We can express the transition form factors as In the limit Q 2 1,2 → 0 only T (0,T) and T (2) contribute and their values at Q 2 1,2 → 0 determine the two-photon decay width of f 2 (1270) meson.
Comparing two approaches given by (2.1) and (2.8)-(2.10) at both real photons (Q 2 1 = Q 2 2 = 0) and at √ s = m f 2 we found the correspondence We will apply the formalism for the γ * γ * → f 2 vertices discussed in Sec. II A. Because f 2 (1270) is extended, finite size object one can expect in addition a form factor F(Q 2 1 , Q 2 2 ) associated with the gluon virtualities for the g * g * → f 2 vertex. In the present letter the form factor is parametrized in different ways as: (2.14) where Λ is a parameter whose value is expected to be close to the resonance mass [23].
In the calculations for the factorised monopole (2.13) and dipole (2.14) forms we take  Fig. 3. The results strongly depend on the parametrization chosen and the value of the corresponding parameter.

C. k t -factorization approach
In Fig. 1 we show a generic Feynman diagram for f 2 (1270) meson production in proton-proton collision via gluon-gluon fusion. This diagram illustrates the situation adequate for the k t -factorization calculations used in the present paper. The differential cross section for inclusive f 2 (1270) meson production via the g * g * → f 2 (1270) fusion in the k t -factorization approach can be written as: Here q 1 , q 2 and p denote the transverse momenta of the gluons and the f 2 (1270) meson. The f 2 meson is on-shell and its momentum satisfies p 2 = m 2 f 2 . M g * g * → f 2 is the matrix element for off-shell gluons for the hard subprocess and F g are the gluon unintegrated distribution functions (UGDFs) for both colliding protons. The UGDFs depend on gluon longitudinal momentum fractions x 1,2 = m T exp(±y)/ √ s and q 2 1 , q 2 2 entering the hard process. In principle, they can depend also on factorization scales µ 2 The δ (2) function in Eq. (2.17) can be easily eliminated by introducing q 1 + q 2 and q 1 − q 2 transverse momenta [4].
The off-shell matrix element can be written as (we restore the color-indices a and b) where ǫ ( f 2 ) is the polarisation tensor for the f 2 (1270) meson.
In the k t -factorization approach in [3] the matrix element squared (for energy-momentum tensor coupling) was written as: where N c is the number of colors, V αβµν ab is the gg → f 2 vertex 1 (see Eq. (A1) of [3]), and κ ≈ O(0.1 GeV) is to be fixed by experiment. The explicit forms for the spin-2 projector P (2) and H αβ ⊥ functions (with transverse components) are given in [3]. In the above formula (2.20) α s is not explicit but is hidden in the normalization constant. In our calculation we will make α s explicit, i.e. include its running with relevant scales. We have checked that the approach in [3] is equivalent to the approach with the helicity-2 EMN vertex function (2.3) when ignoring running of α s and vertex form factor is crucial for description of transverse momentum distribution of f 2 (1270) as will be discussed in the result section.
The g * g * → f 2 (1270) coupling entering in the matrix element squared can be obtained from that for the γ * γ * → f 2 (1270) coupling by the following replacement: (2.21) 1 Please note that the order of Lorentz indices here (and in Ref. [3]) is different than in Eq. (2.3).
Here (< e 2 q >) 2 = 25/162 for the 1 √ 2 uū + dd flavour structure assumed for f 2 (1270). In realistic calculations the running of strong coupling constants must be included. In our numerical calculations presented below the renormalization scale is taken in the form: The Shirkov-Solovtsov prescription [24] is used to extrapolate down to small renormalization scales relevant for the f 2 (1270) production for the ALICE kinematics. As will be shown in the present paper the g * g * → f 2 (1270) mechanism is insufficient at low f 2 (1270) transverse momenta therefore we consider also a final-state rescattering of produced pions. The general diagram representing the ππ rescattering is shown in Fig. 2. Both π + π − and π 0 π 0 rescatterings may lead to the production of the f 2 (1270) meson as an effect of final state resonance interactions.
The spectrum of pions will be not calculated here but instead we will use a Lévy parametrization of the inclusive π 0 cross section proposed in [25] for √ s = 7 TeV. At the ALICE energies and midrapidities we assume the following relation: to be valid. Our approach here is similar in spirit to color evaporation approach considered, e.g., in [8,26]. In our approach here we do not include possible ππ correlation functions. They are discussed usually at very small | p 1 − p 2 |/2. For identical particles (π 0 π 0 in our case) this is discussed usually in the context of Bose-Einstein correlations. The nonidentical particle correlations (π + π − in our case) is less popular but also very interesting [27,28]. To form the resonance the two pions must be produced in an invariant mass window corresponding to the f 2 (1270) meson and close in space one to each other. Including explicitly the second condition would require knowledge of the space-time development of the hadronization process and goes far beyond the present study devoted to the g * g * → f 2 (1270) mechanism. Instead we write the number of produced f 2 (1270) per event as N = dy 1 dp 1t dy 2 dp 2t dφ 1 2π dN π dy 1 dp 1t dN π dy 2 dp 2t P ππ→ f 2 , (2.24) where dN π /(dydp t ) is number of pions per interval of rapidity and transverse momentum. Here we use the Tsallis parametrization of π 0 at √ s = 7 TeV from [25]; see Eq.
(2) of [25] and fit parameters in Table 3 therein. In Eq. (2.24) P ππ→ f 2 parametrizes probability of the π + π − and π 0 π 0 formation of f 2 (1270) as well as probability of its survival in a dense hadronic system. It will be treated here as a free parameter adjusted to the f 2 (1270) data from [2]. The distributions dN π /(dydp t ) are obtained then by calculating y and p t of the f 2 (1270) meson and binning in these variables.
The effect of hadronic rescattering in high-energy pp collisions is also discussed in [29] and the application is being developed and will be implemented to PYTHIA event generator.

III. NUMERICAL RESULTS
To convert to the number of f 2 (1270) mesons per event, as was presented in Ref. [2], we use the following relation: The inelastic cross section for √ s = 7 TeV was measured at the LHC and is: 2) σ inel = 71.34 ± 0.36 (stat.) ± 0.83 (syst.) mb , (3.3) as obtained by the TOTEM [30] and ATLAS [31] collaborations, respectively. In our calculations we take σ inel = 72.5 mb. In Fig. 3 we present the f 2 (1270) meson transverse momentum distributions at √ s = 7 TeV and |y| < 0.5 together with the preliminary ALICE data from [2]. Here, for the color-singlet gluon-gluon fusion mechanism, we used JH UGDF from [32]. 2 We compare results for the monopole and dipole form factors for two different g * g * → f 2 vertices discussed in Sec. II A. The result strongly depends on the parametrization of the form factors according to Eqs. (2.13)-(2.16) and (2.10). Fig. 4 shows that there is some difference in the role of Λ = 0, 2 contributions for the EMN and PPV vertices. In the formalism of [19] [see the PPV vertex (2.8)] there is no interference between so-called Λ = 0, T and Λ = 2 terms while the naive use of the formalism from [10] [see the EMN vertex (2.1)] generates some interference effects. Different couplings (independent invariant amplitudes) lead to different shapes of the transverse momentum distributions. The shape could be verified by experimental data.      √ s = 7 TeV and |y| < 0.5 together with the preliminary ALICE data from [2]. Shown are the results calculated in the two approaches, EMN (left panel) and PPV (right panel) vertices, and the helicity-0 and -2 components separately and their coherent sum (total). In this calculation we used dipole form factor parametrization (2.14) with Λ D = m f 2 . The dotted line corresponds to the Born-level result for the pp → pp f 2 (1270) process via pomeron-pomeron fusion.
In the left panel of Fig 5 we show results for the KMR UGDF. 3 The KMR UGDF (dashed lines) gives smaller cross section than the JH UGDF (solid lines). The results for both UGDFs coincide for large p t . The larger the f 2 (1270) transverse momentum the larger the range of gluon transverse momenta q 1t and/or q 2t are probed. This means that at larger gluon transverse momenta one enters a more perturbative region.
To better illustrate the dependence of UGDFs on q 2 we present in the right panel of Fig 5 the results with the Gaussian smearing of collinear GDF, often used in the context of TMDs, for different smearing parameter σ 0 = 0.25, 0.5, 1.0 GeV. The GJR08VFNS(LO) collinear GDF [40] was used for this purpose. As expected the shape of dσ/dp t strongly depends on the value of the smearing parameter σ 0 used in the calculation. The speed of dσ/dp t approaching to zero for p t → 0 strongly depends on the value of σ 0 . It is impossible to describe simultaneously p t < 1 GeV and p t > 1 GeV regions with the same value of σ 0 . This illustrates the generic situation with all UGDFs.   √ s = 7 TeV and |y| < 0.5 together with the preliminary ALICE data from [2]. In the left panel results for two different UGDFs, JH (solid lines) and KMR (dashed lines), are shown. In the right panels we show the dependence on the Gaussian smearing parameter σ 0 for the GJR08VFNS(LO) GDF [40]. Here the EMN vertex discussed in Sec. II A 1 and the dipole form factor (2.14) with Λ D = m f 2 were used.
In Fig. 6 we present d 2 σ/dq 1t dq 2t for the EMN (left panel) and PPV (right panel) g * g * → f 2 (1270) vertices. Here the JH UGDF was used. The maximal contributions come from the region of rather small gluon transverse momenta q 1t , q 2t < 1 GeV. It is easy to check (numerically) that the larger-p t region (p t > 2 GeV) is sensitive to q 1t , q 2t > 1 GeV where perturbative methods apply. At low p t there is a nonnegligible contribution from the nonperturbative region of UGDFs which is not under full theoretical control. Here the gluon saturation effects may be potentially important. We have checked that i.e. the two vertices are equivalent for both on-shell photons.
In Fig. 7 we show auxiliary (normalized) distributions to discuss a possible role of the remaining terms in the g * g * f 2 PPV vertex (2.8) corresponding to helicities (Λ = 0, L) and (Λ = 1). Here we assumed the same Q 2 dependence of the form factor functions for all Λ terms; see Eqs. (2.10) and (2.14). The dominance of Λ = 2 term over Λ = 0 and Λ = 1 terms is certainly maintained at small values of Q 2 ave and of p t . However, the situation changes drastically at large gluon virtualities, i.e., the (Λ = 1) and (Λ = 0, L) structures of the g * g * f 2 vertex become equally important for p t > 2 GeV.
Note that from the analysis of the γ * (Q 2 1 )γ * (Q 2 2 ) → ππ processes performed in [21,22] it is clear that in the f 2 (1270) resonance region the helicity-(0, T) amplitude gives the dominant contribution and the other helicity projections become increasingly important for larger virtualities. From Fig. 5 of [21] and Fig. 3 of [22] we can see that for Q 2 1 fixed the helicity-1 contribution increases with increasing Q 2 2 while helicity-(0, L) contribution only slightly decreases. The situation changes when both photon virtualities are identical Q 2 1 = Q 2 2 and large, i.e. then the helicity-(0, L) component increases with increasing virtualities and becomes even larger than the helicity-1 component. This observation is consistent with our results presented in Fig. 7.
The theoretical results for the color-singlet gluon-gluon fusion contribution underestimate the ALICE data especially for low-p t region, p t < 2 GeV. Does it mean that other mechanism(s) is (are) at the game? Results for different Λ = 0, 1, 2 helicity terms in the g * g * f 2 vertex (2.8)-(2.10) using the same form of vertex form factors F (Λ) (Q 2 1 , Q 2 2 ) (2.14) with Λ D = m f 2 are shown. In the calculation the JH UGDF was used.
In Fig. 8 we show the ππ rescattering contribution. Clearly the ππ → f 2 (1270) rescattering effect cannot describe the region of p t > 2 GeV, where the gg-fusion mechanism is a possible explanation. In addition, we present the Born result (without absorptive corrections important only when restricting to purely exclusive processes) for the pp → pp f 2 (1270) process proceeding via the pomeron-pomeron fusion mechanism calculated in the tensor-pomeron approach. For details regarding this approach we refer to [10,[41][42][43]. In the calculation we take the pomeron-pomeron-f 2 (1270) coupling parameters from [43]. Results for the ππ rescattering mechanism (long-dashed line), for the gg-fusion mechanism (solid lines), and for the pomeron-pomeron fusion mechanism (dotted line) together with the preliminary ALICE data from [2]. We show maximal contribution from the ππ rescattering as described in the main text. The results for the gg-fusion contributions were calculated for the JH UGDF and for the PPV vertex [helicity-2 plus helicity-(0, T) terms] and for two form factor functions (2.15) (top solid line) and (2.14) (bottom solid line).

IV. CONCLUSIONS
In the present paper we have discussed production of f 2 (1270) tensor meson in proton-proton collisions. Two different approaches for the γ * γ * → f 2 (1270) coupling have been considered. We have discussed their equivalence for both on-shell photons. The coupling constants have been fixed by the Belle data for γγ → f 2 (1270) → ππ. The g * g * → f 2 (1270) vertex has been obtained by replacing electromagnetic coupling constant by the strong coupling constant, modifying colour factors and assuming a simple flavour structure of the f 2 (1270) isoscalar meson. We have shown that the energy-momentum tensor vertex, proposed in [3], is equivalent to the dominant in γγ → f 2 (1270) Γ (2) tensorial structure [see Eq. (2. 3)] discussed in [10].
We have performed our calculation of the cross section for pp → f 2 (1270) + X within the k t -factorization approach. Different unintegrated gluon distributions from the literature have been used. We have discussed corresponding uncertainties.
Our results have been compared to preliminary ALICE data presented in [2]. We have taken into account only the case when both photons are transverse. At low f 2 (1270) transverse momenta the helicity-2 (Λ = 2) contribution dominates, while the helicity-0 (Λ = 0, T) is small, almost negligible, but competes with the Λ = 2 and even dominates at larger transverse momenta of f 2 (1270). In the PPV formalism there could be also Λ = 0, L and Λ = 1 contributions which are difficult to fix by available data.
It has been shown that the results strongly depend on the form of the vertex form factor. With the GVDM form factor used previously in γ * γ → f 2 (1270) fusion [14,15] one cannot describe the preliminary ALICE data. We have tried also other choices. With plausible form factor [e.g., dipole ansatz (2.14) with Λ D ≃ m f 2 , factorized VDM ansatz (2.15) with Λ 1 ≃ 1 GeV] one can describe the data for p t > 2 GeV but it seems impossible to describe the low-p t data. Clearly some mechanism at low-p t must be in the game there.
We have shown that the final state rescattering may be the missing candidate. A simple empirical model has been proposed. Adjusting corresponding probability for the ππ → f 2 (1270) rescattering and the Λ D parameter in the dipole form factor for the g * g * → f 2 vertex we have been able to describe the preliminary ALICE data.
The gluon saturation is expected at low x 1 and x 2 i.e. automatically rather low transverse momenta of f 2 (1270) where most probably the ππ rescattering dominates, which enables observation of saturation.
We have calculated also the exclusive production of f 2 (1270) meson via the pomeronpomeron fusion mechanism with the parameters found in our previous analysis where they were fixed to 'purely' exclusive data for the reaction pp → ppπ + π − . This contribution is concentrated at small f 2 (1270) transverse momenta but its role is rather marginal.
Our calculation suggest that the gluon-gluon fusion may be the dominant mechanism of the f 2 (1270) production at larger transverse momenta, p t > 3 GeV. Other mechanisms are of course not excluded but it is clear that the gluon-gluon fusion is a very important mechanism which cannot be ignored in the analysis.