Study of the exclusive reaction pp \to pp K^{*0} \bar{K}^{*0}: f_{2}(1950) resonance versus diffractive continuum
aa r X i v : . [ h e p - ph ] F e b Study of exclusive reaction p p → p pK ∗ ¯ K ∗ ; f ( ) resonance vs. diffractive continuum Piotr Lebiedowicz ∗ Institute of Nuclear Physics Polish Academy of Sciences,Radzikowskiego 152, PL-31342 Kraków, Poland
Abstract
We present first predictions of the cross sections and differential distributions for the exclusivereaction pp → ppK ∗ ¯ K ∗ contributing to the K + K − π + π − channel. The amplitudes for the reactionare formulated within the nonperturbative tensor-pomeron approach. We consider separately the f ( ) s -channel exchange mechanism and the K ∗ t / u -channel exchange mechanism, focusingon their specificities. First mechanism is a candidate for the central diffractive production oftensor glueball and the second one is an irreducible continuum. We adjust parameters of ourmodel, assuming the dominance of pomeron-pomeron fusion, to the WA102 experimental data.We find that the continuum contribution alone one can describe the WA102 data reasonably well.We present predictions for the reaction pp → pp ( K ∗ ¯ K ∗ → K + K − π + π − ) for the ALICE, ATLAS,CMS and LHCb experiments including typical kinematic cuts. We find from our model a crosssections of σ ∼ = −
250 nb for the LHC experiments, depending on the assumed cuts. Theabsorption effects have been included in our analysis. ∗ Electronic address: [email protected] . INTRODUCTION Studies of the K ∗ ¯ K ∗ system have been carried out in two-photon interactions [1–3], inradiative J / ψ decay [4, 5], in K − p → K ∗ ¯ K ∗ Λ reaction [6, 7], and in central productionin proton-proton collisions [8–10]. It is known from the WA102 experiment [10] that al-though the K ∗ ¯ K ∗ final state is a major component of the K + K − π + π − channel it is notthe dominant component. In contrast, the φφ final state was found to be dominant com-ponent of the K + K − K + K − channel [11]. The cross section as a function of center-of-massenergy for the production of K ∗ ( ) ¯ K ∗ ( ) system was found [10] to be consistent withbeing produced via the double-pomeron-exchange mechanism.In hadronic proton-proton collisions [9, 10, 12] a broad low-mass enhancement in the K ∗ ¯ K ∗ and/or K + K − π + π − invariant mass distributions was seen. In [9] the authors statedthat the K ∗ ¯ K ∗ system is mainly produced as a non-resonant threshold enhancement.More recent analysis [13] give some evidence for the existence of f ( ) resonance inthe K ∗ ¯ K ∗ channel. In the radiative J / ψ decay [4, 5] the K ∗ ¯ K ∗ spectrum indicates twonarrow peaks at low mass. The analysis of angular distributions finds that the K ∗ ¯ K ∗ sys-tem in the radiative J / ψ decay show strong J PC = − + component whereas the hadronicproduction modes are all consistent with strong J PC = ++ component. The analysis ofthe partial wave structure of the K ∗ ¯ K ∗ state from the reaction γγ → K + K − π + π − [3]support the dominance of the ( J P , J z ) = ( + , ± ) wave.An interesting suggestion has been made for the broad isoscalar-tensor f ( ) res-onance to be the lightest tensor glueball, while the arguments are not yet fully settled.Namely, this state is occasionally discussed as a candidate for a tensor glueball as it ap-pears to have largely flavor-blind decay modes; see e.g. [14–16]. However, accordingto lattice-QCD simulations, the lightest tensor glueball has a mass between 2.2 GeV and2.4 GeV, see e.g. [17, 18]. Thus, the f ( ) and f ( ) states are good candidates to betensor glueball. The nature of these resonances is not understood at present and a tensorglueball has still not been clearly identified. It is also interesting to speculate whether thetensor states f ( ) , f ( ) , and f ( ) , observed by the WA102 Collaboration [19],are due to mixing between a tensor glueball and nearby q ¯ q states. Two of these states havesimilar φ pp and dP t dependencies and one the opposite; φ pp is the azimuthal angle be-tween the transverse momentum vectors of the outgoing protons, and dP t is the so-called“glueball-filter variable” [20] defined by the difference of the transverse momentum vec-tors of the outgoing protons. It is known from the WA102 analysis of various channelsthat all the undisputed q ¯ q states are suppressed at small dP t in contrast to glueball can-didates. Established q ¯ q states peak at φ pp = π whereas the f ( ) and f ( ) peakat φ pp = f ( ) probably contains large gluonic component and should be copiously producedvia the double-pomeron-exchange (i.e., PP -fusion) mechanism.However, the observation of f ( ) resonance in two-photon interaction processes,such as γγ → f ( ) → K ∗ ¯ K ∗ [1–3], and in other γγ -fusion processes [21–23], pre-cludes its interpretation as a pure gluonic state. In [23] a good description the Belle dataon γγ → p ¯ p including, in addition to the proton exchange, the f ( ) resonance wasobtained. One can observe there the dominance of the f ( ) resonance in the lowmass region M p ¯ p = W γγ < f ( ) -exchange amplitude only the termwith a f ( ) γγ coupling and g ( ) f ( ) p ¯ p coupling was used. There, a - and b -type couplingparametrise the so-called helicity-zero and helicity-two γγ → f amplitudes, respec-2ively; see e.g. [24]. For instance, for the γγ → f ( ) process the helicity-2 contri-bution ( b -type coupling) is dominant. As will be presented in this work, for diffractiveprocesses shown in Fig. 1, the b -type coupling in the f ( ) K ∗ ¯ K ∗ and P K ∗ K ∗ verticeswill be more preferred than the a -type coupling.The study of φφ and K ∗ ¯ K ∗ systems could provide also helpful information for search-ing for the fully-strange ( ss ¯ s ¯ s ) tetraquark. In the relativistic quark model based on thequasipotential approach in QCD [25], the f ( ) and f ( ) states are considered asa candidates for the ground state ( h L i =
0) light tetraquarks as diquark-antidiquark(composed from an axial vector diquark and antidiquark), qq ¯ q ¯ q and ss ¯ s ¯ s , respectively. In[26] the f ( ) is assigned to be ss ¯ s ¯ s tetraquark state. These two states f ( ) and f ( ) are close in mass within errors [27]. Very recently, in [28] it was stated that the f ( ) resonance may be assigned to 1 S -wave tetraquark T ss ¯ s ¯ s ( ) in the frameworkof a nonrelativistic potential quark model without the diquark-antidiquark approxima-tion. The f ( ) state may have large decay rates into the φφ and ηη final states throughquark rearrangements, and/or into K ∗ ¯ K ∗ final state through the annihilation of s ¯ s andcreation of a pair of nonstrange q ¯ q . To confirm this assignment, the above decay modesand such as ηη ′ , η ′ η ′ should be investigated in experimental searches. On the other hand,flavor mixings could be important for the light flavor systems and pure ss ¯ s ¯ s states maynot exist; see e.g. [29].With the idea of bringing more information on the topic, in the present work, we studythe diffractive PP → f ( ) fusion mechanism in the reaction pp → ppK ∗ ¯ K ∗ and the K ∗ ¯ K ∗ continuum which is a background for diffractively produced resonances. But weemphasize that in the following we make no assumptions if the f ( ) resonance isglueball or tetraquark.In the tensor-pomeron model for soft high-energy scattering formulated in [24], on thebasis of earlier work [30], the pomeron exchange is effectively treated as the exchangeof a rank-2 symmetric tensor. In the last few years a scientific program was undertakento analyse the central exclusive production of mesons in the tensor-pomeron model inseveral reactions: pp → ppM [31], where M stands for a scalar or pseudoscalar meson, pp → pp π + π − and pp → pp ( f ( ) → π + π − ) [32–34], pp → pn ρ π + ( pp ρ π )[35], pp → ppK + K − [36], pp → pp ( σσ , ρ ρ → π + π − π + π − ) [37], pp → ppp ¯ p [38], pp → pp ( φφ → K + K − K + K − ) [39], pp → pp ( φ → K + K − , µ + µ − ) [40], pp → pp f ( ) and pp → pp f ( ) [41]. The present paper aims to underline the importance of thestudy of the pp → pp ( K ∗ ¯ K ∗ → K + π − K − π + ) reaction.Some effort to measure exclusive production of higher-multiplicity central systems atthe energy √ s =
13 TeV has been initiated by the ATLAS Collaboration; see, e.g., [42]. Wethink that a study of CEP of the K ∗ ¯ K ∗ pairs decaying into K + π − K − π + should be quiterewarding for experimentalists. Our analysis are designed to facilitate the study of suchprocesses at the LHC, for instance, by investigating in detail the continuum and tensorresonance production.The paper is organized as follows. In Sec. II we describe our theoretical frame-work. In Sec. III we show and discuss our numerical results. We determine themodel parameters from the comparison to the WA102 experimental data for the re-action pp → ppK ∗ ¯ K ∗ . We also predict the total and differential cross sections for pp → pp ( K ∗ ¯ K ∗ → K + K − π + π − ) including typical kinematic cuts for the LHC experi-ments. The final section is devoted to the conclusions.3 a) IPIP f (1950) p pp p K + π − K − π + K ∗ ¯ K ∗ (b) K ∗ IPIP ¯ K ∗ p pp pK ∗ K + π − K − π + FIG. 1: The “Born level” diagrams for double pomeron central exclusive K ∗ ¯ K ∗ production andtheir subsequent decays into K + π − K − π + in proton-proton collisions: (a) K ∗ ¯ K ∗ production viathe f ( ) resonance; (b) continuum K ∗ ¯ K ∗ production. II. THEORETICAL FRAMEWORK
In the present paper we consider two processes shown in Fig. 1 that may contribute tothe K + π − K − π + final state via an intermediate K ∗ ¯ K ∗ ≡ K ∗ ( ) ¯ K ∗ ( ) . Figure 1(a)shows the process with intermediate production of f ( ) resonance, pp → pp ( PP → f ( ) → K ∗ ¯ K ∗ ) → pp K + π − K − π + (2.1)In Fig. 1(b) we have the continuum process pp → pp ( PP → K ∗ ¯ K ∗ ) → pp K + π − K − π + (2.2)with the K ∗ ( ) t / u -channel exchanges.The processes (2.1) and (2.2) are expected to be most important ones at high energiessince they involve pomeron exchange only. We can replace one or two pomerons byone or two f R reggeons. However, for the LHC collision energies and central K ∗ ¯ K ∗ production (midrapidity region) such f R f R -, f R P -, and P f R -fusion contributions areexpected to be small and we shall not consider them in our present paper.We treat effectively the 2 → pp → ppK ∗ ¯ K ∗ reaction. The general cross-section formula can be written approximately as σ → = Z max { m X } m K + m π Z max { m X } m K + m π σ → ( ..., m X , m X ) f K ∗ ( m X ) f K ∗ ( m X ) dm X dm X . (2.3)We use for the calculation of decay processes K ∗ → K π the spectral function f K ∗ ( m X i ) = C K ∗ − ( m K + m π ) m X i ! π m X i m K ∗ Γ K ∗ ( m X i − m K ∗ ) + m K ∗ Γ K ∗ , (2.4)where i =
3, 4, Γ K ∗ is the total width of the K ∗ ( ) resonance and m K ∗ its mass takenfrom [43], the factor C K ∗ is found from the condition Z max { m X } m K + m π f K ∗ ( m X ) dm X = K ∗ ( ) mesons isotropically in the K ∗ -meson rest frame and then use relativistictransformations to the overall center-of-mass frame.Now we discuss the production of K ∗ ¯ K ∗ in proton-proton collisions, p ( p a , λ a ) + p ( p b , λ b ) → p ( p , λ ) + p ( p , λ ) + K ∗ ( p , λ ) + ¯ K ∗ ( p , λ ) , (2.6)where p a , b , p and λ a , b , λ = ± denote the four-momenta and helicities of the protonsand p and λ = ± K ∗ mesons,respectively.The amplitude for the 2 → M λ a λ b → λ λ K ∗ ¯ K ∗ = (cid:16) ǫ ( K ∗ ) κ ( λ ) (cid:17) ∗ (cid:16) ǫ ( ¯ K ∗ ) κ ( λ ) (cid:17) ∗ M κ κ λ a λ b → λ λ K ∗ ¯ K ∗ , (2.7)where ǫ ( K ∗ ) κ are the polarisation vectors of the K ∗ mesons. Taking into account summationover the K ∗ polarisations we get for the amplitudes squared [to be inserted in σ → inEq. (2.3)]14 ∑ spins (cid:12)(cid:12)(cid:12) M λ a λ b → λ λ K ∗ ¯ K ∗ (cid:12)(cid:12)(cid:12) = ∑ λ a , λ b , λ , λ (cid:16) M σ σ λ a λ b → λ λ K ∗ ¯ K ∗ (cid:17) ∗ M ρ ρ λ a λ b → λ λ K ∗ ¯ K ∗ g σ ρ g σ ρ .(2.8)We take into account two main processes shown by the diagrams in Fig. 1. The full am-plitude is then the sum of the f ( ) resonance term and the K ∗ -exchange continuumterm: M κ κ λ a λ b → λ λ K ∗ ¯ K ∗ = M ( PP → f → K ∗ ¯ K ∗ ) κ κ λ a λ b → λ λ K ∗ ¯ K ∗ + M ( K ∗ − exchange ) κ κ λ a λ b → λ λ K ∗ ¯ K ∗ . (2.9)To give the full physical amplitude for the reaction (2.6) we include absorptive correc-tions to the Born amplitudes in the one-channel eikonal approximation; see e.g. Sec. 3.3of [32]. In practice we work with the amplitudes in the high-energy approximation, i.e.assuming s -channel helicity conservation for the protons. A. f ( ) resonance contribution Now we consider the amplitude representing by the diagram in Fig. 1(a) but limitingto the final state ppK ∗ ¯ K ∗ .The Born-level amplitude for the PP -fusion process through the s -channel f ( ) -meson exchange is given by M ( PP → f → K ∗ ¯ K ∗ ) κ κ λ a λ b → λ λ K ∗ ¯ K ∗ = ( − i ) ¯ u ( p , λ ) i Γ ( P pp ) µ ν ( p , p a ) u ( p a , λ a ) i ∆ ( P ) µ ν , α β ( s , t ) × i Γ ( PP f ) α β , α β , ρσ ( q , q ) i ∆ ( f ) ρσ , αβ ( p ) i Γ ( f K ∗ ¯ K ∗ ) αβκ κ ( p , p ) × i ∆ ( P ) α β , µ ν ( s , t ) ¯ u ( p , λ ) i Γ ( P pp ) µ ν ( p , p b ) u ( p b , λ b ) , (2.10)5here s = ( p + p + p ) , s = ( p + p + p ) , q = p a − p , q = p b − p , t = q , t = q , and p = q + q = p + p . Here Γ ( P pp ) and ∆ ( P ) denote the effectiveproton vertex function and propagator, respectively, for the tensor-pomeron exchange.The corresponding expressions, as given in Sec. 3 of [24], are as follows i Γ ( P pp ) µν ( p ′ , p ) = − i β P NN F ( t ) (cid:26) (cid:2) γ µ ( p ′ + p ) ν + γ ν ( p ′ + p ) µ (cid:3) − g µν ( p / ′ + p / ) (cid:27) , (2.11) i ∆ ( P ) µν , κλ ( s , t ) = s (cid:18) g µκ g νλ + g µλ g νκ − g µν g κλ (cid:19) ( − is α ′ P ) α P ( t ) − , (2.12)where β P NN = − and F ( t ) is the Dirac form factor of the proton. For exten-sive discussions of the properties of these terms we refer to [24]. In (2.12) the pomerontrajectory α P ( t ) is assumed to be of standard linear form, see e.g. [44, 45], α P ( t ) = α P ( ) + α ′ P t , α P ( ) = α ′ P = − . (2.13)The PP f vertex, including a form factor, can be written as i Γ ( PP f ) µν , κλ , ρσ ( q , q ) = i Γ ( PP f )( ) µν , κλ , ρσ | bare + ∑ j = i Γ ( PP f )( j ) µν , κλ , ρσ ( q , q ) | bare ! ˜ F ( PP f ) ( q , q , p ) .(2.14)A possible choice for the i Γ ( PP f )( j ) µν , κλ , ρσ coupling terms j =
1, ..., 7 is given in Appendix A of[33]. The couplings j =
1, ..., 7 can be associate to the following orbital angular momen-tum and spin of the two “real pomerons” ( l , S ) values: (
0, 2 ) , (
2, 0 ) − (
2, 2 ) , (
2, 0 ) + (
2, 2 ) , (
2, 4 ) , (
4, 2 ) , (
4, 4 ) , (
6, 4 ) , respectively. In the following we shall, for the purpose of orien-tation, assume that only the j = ( l , S ) , that is ( l , S ) = (
0, 2 ) , is unequal to zero. The expressions for j = i Γ ( PP f )( ) µν , κλ , ρσ | bare = i g ( ) PP f M R µνµ ν R κλα λ R ρσρ σ g ν α g λ ρ g σ µ , (2.15) R µνκλ = g µκ g νλ + g µλ g νκ − g µν g κλ , (2.16)see (A12) of [33]. In (2.15), M ≡ g ( ) PP f is dimensionless coupling constantwhich should be fitted to experimental data. We take the factorized form for the PP f form factor in (2.14): ˜ F ( PP f ) ( q , q , p ) = ˜ F M ( q ) ˜ F M ( q ) F ( PP f ) ( p ) ; (2.17)˜ F M ( t ) = − t / ˜ Λ . (2.18) Here the label “bare” is used for a vertex, as derived from a corresponding coupling Lagrangian inAppendix A of [33] without a form-factor function. F ( PP f ) is normalized to unity at the on-shell point F ( PP f ) ( m f ) = F ( PP f ) ( p ) = exp − ( p − m f ) Λ f , E ! , (2.19) F ( PP f ) ( p ) = Λ f , P Λ f , P + ( p − m f ) . (2.20)The cutoff parameters ˜ Λ , Λ f , E and Λ f , P in (2.18), (2.19) and (2.20), respectively, aretreated as free parameters which could be adjusted to fit the experimental data.We use in (2.10) the tensor-meson propagator with the simple Breit-Wigner form; see(3.35) of [39]. A better representation for the propagator could be constructed using themethods of [24, 46], used there for the ρ and f ( ) propagators. In our calculationswe take the nominal values for the f ( ) from [27]: m f = ±
12 MeV , Γ f = ±
24 MeV . (2.21)For the f K ∗ ¯ K ∗ vertex function we take the same ansatz as for the f φφ vertex definedin (3.32) of [39]. The f K ∗ ¯ K ∗ vertex is as follows: i Γ ( f K ∗ ¯ K ∗ ) µνκλ ( p , p ) = i M g ′ f K ∗ ¯ K ∗ Γ ( ) µνκλ ( p , p ) F ′ ( f K ∗ ¯ K ∗ ) ( p ) − i M g ′′ f K ∗ ¯ K ∗ Γ ( ) µνκλ ( p , p ) F ′′ ( f K ∗ ¯ K ∗ ) ( p ) (2.22)with two rank-four tensor functions, Γ ( ) µνκλ ( p , p ) = h ( p · p ) g µν − p µ p ν ih p κ p λ + p κ p λ − ( p · p ) g κλ i , (2.23) Γ ( ) µνκλ ( p , p ) = ( p · p )( g µκ g νλ + g µλ g νκ ) + g µν ( p κ p λ + p κ p λ ) − p ν p λ g µκ − p ν p κ g µλ − p µ p λ g νκ − p µ p κ g νλ − [( p · p ) g µν − p µ p ν ] g κλ ; (2.24)see Eqs. (3.18) and (3.19) of [24]. The coupling parameters g ′ f K ∗ ¯ K ∗ and g ′′ f K ∗ ¯ K ∗ are dimen-sionless. Different form factors F ′ and F ′′ are allowed a priori . We assume, in the presentexploratory study, that F ′ ( f K ∗ ¯ K ∗ ) ( p ) = F ′′ ( f K ∗ ¯ K ∗ ) ( p ) = F ( PP f ) ( p ) (2.25)and for the cutoff parameters to be the same, Λ ′ f = Λ ′′ f = Λ f , E or Λ f , P ; see (2.19) and(2.20).One has to keep in mind that relative signs of couplings have physical significance,for instance, the relative sign of g ′ and g ′′ . However, for orientation purposes, in thecalculation we treat them separately and do not fix the sign of the f couplings. With our7hoice to keep only one PP f ( ) coupling from (2.14), namely (2.15) with g ( ) PP f , theresults will depend on the product of the couplings g ( ) PP f g ′ f K ∗ ¯ K ∗ and g ( ) PP f g ′′ f K ∗ ¯ K ∗ with g ′ f K ∗ ¯ K ∗ and g ′′ f K ∗ ¯ K ∗ given in (2.22). In the following we assume that only either the firstor the second of the above products of couplings is non zero. B. Continuum contribution
The diagram for the K ∗ ¯ K ∗ production with an intermediate K ∗ exchange is shownin Fig. 1 (b). The Born-level amplitude can be written as the sum M ( K ∗ − exchange ) κ κ λ a λ b → λ λ K ∗ ¯ K ∗ = M ( ˆ t ) κ κ λ a λ b → λ λ K ∗ ¯ K ∗ + M ( ˆ u ) κ κ λ a λ b → λ λ K ∗ ¯ K ∗ (2.26)with the ˆ t - and ˆ u -channel amplitudes: M ( ˆ t ) κ κ = ( − i ) ¯ u ( p , λ ) i Γ ( P pp ) µ ν ( p , p a ) u ( p a , λ a ) i ∆ ( P ) µ ν , α β ( s , t ) × i Γ ( P K ∗ K ∗ ) κ κ α β ( ˆ p t , − p ) i ∆ ( K ∗ ) κ κ ( ˆ p t ) i Γ ( P K ∗ K ∗ ) κ κ α β ( p , ˆ p t ) × i ∆ ( P ) α β , µ ν ( s , t ) ¯ u ( p , λ ) i Γ ( P pp ) µ ν ( p , p b ) u ( p b , λ b ) , (2.27) M ( ˆ u ) κ κ = ( − i ) ¯ u ( p , λ ) i Γ ( P pp ) µ ν ( p , p a ) u ( p a , λ a ) i ∆ ( P ) µ ν , α β ( s , t ) × i Γ ( P K ∗ K ∗ ) κ κ α β ( p , ˆ p u ) i ∆ ( K ∗ ) κ κ ( ˆ p u ) i Γ ( P K ∗ K ∗ ) κ κ α β ( ˆ p u , − p ) × i ∆ ( P ) α β , µ ν ( s , t ) ¯ u ( p , λ ) i Γ ( P pp ) µ ν ( p , p b ) u ( p b , λ b ) , (2.28)where ˆ p t = p a − p − p , ˆ p u = p − p a + p , s ij = ( p i + p j ) .Our ansatz for the P K ∗ K ∗ vertex follows the one for the P ρρ in (3.47) of [24] withthe replacements a P ρρ → a P K ∗ K ∗ and b P ρρ → b P K ∗ K ∗ ; see also Eqs. (3.12)–(3.14) of [39].With k ′ , µ and k , ν the momentum and vector index of the outgoing and incoming K ∗ ,respectively, and κλ the tensor-pomeron indices, the P K ∗ K ∗ vertex reads i Γ ( P K ∗ K ∗ ) µνκλ ( k ′ , k ) = iF M (( k ′ − k ) ) h a P K ∗ K ∗ Γ ( ) µνκλ ( k ′ , − k ) − b P K ∗ K ∗ Γ ( ) µνκλ ( k ′ , − k ) i . (2.29)Here the coupling parameters a and b have dimensions GeV − and GeV − , respectively.We take for F M ( t ) the form given in (2.18) but with ˜ Λ → Λ , F M ( t ) = − t / Λ . (2.30)The amplitudes (2.27) and (2.28) also contain a form factors for the off-shell dependenciesof the intermediate K ∗ mesons, ˆ F K ∗ ( ˆ p t ) and ˆ F K ∗ ( ˆ p u ) , respectively. These form factors areparametrised in the exponential formˆ F K ∗ ( ˆ p ) = exp ˆ p − m K ∗ Λ ! . (2.31)8e assume that only one coupling in (2.29) contributes, that is, a P K ∗ K ∗ = b P K ∗ K ∗ =
0. With this assumption, the sign of a or b does not matter as the corresponding couplingoccurs twice in the amplitude. The P K ∗ K ∗ coupling parameters ( a , b ) and the cutoffparameters ( Λ , Λ off,E ) could be adjusted to experimental data.For the K ∗ -meson propagator ∆ ( K ∗ ) κ κ using the properties of tensorial functions we canmake the replacement ∆ ( K ∗ ) κ κ ( ˆ p ) → − g κ κ ∆ ( K ∗ ) T ( ˆ p ) . We take for ˆ p < ( ∆ ( K ∗ ) T ( ˆ p )) − = ˆ p − m K ∗ .We should take into account the reggeization of intermediate K ∗ meson. In [47] it wasargued that the reggeization should not be applied when the rapidity distance betweentwo centrally produced mesons, Y diff = Y − Y , tends to zero (i.e. for | ˆ p | ∼ s ). Wefollow (3.25) of [39] and use a formula for the K ∗ propagator which interpolates continu-ously between the regions of low Y diff , where we use the standard K ∗ propagator, and ofhigh Y diff where we use the reggeized form: ∆ ( K ∗ ) κ κ ( ˆ p ) → ∆ ( K ∗ ) κ κ ( ˆ p ) F ( Y diff ) + ∆ ( K ∗ ) κ κ ( ˆ p ) [ − F ( Y diff )] (cid:18) exp ( i φ ( s )) s s thr (cid:19) α K ∗ ( ˆ p ) − , F ( Y diff ) = exp (cid:0) − c y | Y diff | (cid:1) , φ ( s ) = π (cid:18) s thr − s s thr (cid:19) − π s = M K ∗ ¯ K ∗ , s thr = m K ∗ , and c y is an unknown parameter which measures howfast one approaches to the Regge regime. Here we take c y =
2. This choice is motivatedby Fig. 6 of [39].We assume for the K ∗ Regge trajectory a simple linear form [see (5.3.1) of [48]] α K ∗ ( ˆ p ) = α K ∗ ( ) + α ′ K ∗ ˆ p , (2.33)with the intercept and slope of the trajectory α K ∗ ( ) = α ′ K ∗ = − , respec-tively. We will also show the results using a nonlinear Regge trajectory for the K ∗ mesons,the so-called “square-root” trajectory, parametrised as [49] α K ∗ ( ˆ p ) = α K ∗ ( ) + γ (cid:18)p T K ∗ − q T K ∗ − ˆ p (cid:19) , (2.34)where γ governs the slope of the trajectory and T K ∗ denotes the trajectory terminationpoint. The parameters are fixed to be α K ∗ ( ) = γ = − , √ T K ∗ = III. NUMERICAL RESULTS AND DISCUSSIONS
In this section we wish to present first results for the pp → ppK ∗ ( ) ¯ K ∗ ( ) reac-tion and for the pp → ppK + π − K − π + reaction corresponding to the diagrams in Fig. 1. A. Comparison with the WA102 data
It was noticed by the WA102 Collaboration [10] that the cross section for the produc-tion of a K ∗ ( ) ¯ K ∗ ( ) system slowly rises with rising the center-of-mass energy √ s .9he experimental results, for same interval on the central K ∗ ¯ K ∗ system | x F | σ exp = ±
16 nb at √ s = σ exp = ±
14 nb at √ s = σ exp = ±
10 nb at √ s = pp → ppK ∗ ¯ K ∗ reaction in above energy range.A similar behaviour of the cross section as a function √ s was observed experimentallyalso for the φφ production [11]. In the following we neglect, therefore, secondary reggeonexchanges.In Fig. 2 we show the invariant mass distributions for the PP → f ( ) mechanismtogether with the WA102 experimental data from Fig. 2 of [10]. The data points have beennormalised to the mean value of the total cross section σ exp = ±
10 nb from [10]. For thepurpose of orientation, we have assumed, that in the PP f ( ) vertex (2.14) only g ( ) coupling constant is unequal to zero. We have checked that for the distributions studiedhere the choice of PP f coupling is not important. This is similar to what was found in[39] for the reaction pp → pp ( PP → f ( ) → φφ ) . In the calculation we take only one PP f coupling [ g ( ) PP f from (2.14)] and only one f K ∗ ¯ K ∗ coupling [ g ′ f K ∗ ¯ K ∗ or g ′′ f K ∗ ¯ K ∗ from(2.22)]. The results shown in the left panel correspond to the product of the couplings | g ( ) × g ′ | = | g ( ) × g ′′ | = f -meson off-shell form factor (2.25) can be determined. We have checked that in both casesthe results for the product of the form factors F ( PP f ) ( p ) × F ( f K ∗ ¯ K ∗ ( p ) assuming thesame type of form factors, (2.19) or (2.20), are similar. In the following we choose in thecalculation only the power form (2.20) with the cutoff parameter Λ f , P (2.22). It is clearlyseen from the left panel that the result without these form factors, i.e. for p = m f ,is well above the WA102 experimental data for M K ∗ ¯ K ∗ > Λ f , P decreasesthen mainly the right flank of the resonance is reducing and it becomes narrower. For Λ f , P = − | g ( ) × g ′′ | = M K ∗ ¯ K ∗ ∈ ( ) GeVcan be obtained.In Figs. 3 and 4 we show different differential observables in Y diff , the rapidity differ-ence between the two K ∗ mesons, in | t | , the transferred four-momentum squared fromone of the proton vertices ( t = t or t ), and in φ pp , the azimuthal angle between thetransverse momentum vectors p t ,1 and p t ,2 of the outgoing protons. We present the re-sults obtained separately for different couplings taking into account the absorptive cor-rections. In the left panel of Fig. 4 we show results for the individual j coupling terms g ( j ) PP f × g ′ f K ∗ ¯ K ∗ (only for five terms), while in the right panel for g ( j ) PP f × g ′′ f K ∗ ¯ K ∗ . For il-lustration, the results have been obtained with coupling constants | g ( j ) × g ′ | = | g ( j ) × g ′′ | = diff distribution dependson the choice of the f ( ) K ∗ ¯ K ∗ coupling. It can be expected that this variable will bevery helpful in determining the f K ∗ ¯ K ∗ coupling using data from LHC measurements,in particular, if they cover a wider range of rapidities; see the discussion in Sec. IV B ofRef. [39]. The shapes of the distributions in Y diff within each group are similar except of j = g ′′ coupling.We have checked that with the g ′ coupling for these observables the shapes of the distri-butions are very similar. Compared to the WA102 data from [10] that will be presented10 (GeV) *0 K *0 K M ( nb / G e V ) *0 K *0 K / d M σ d *0 K *0 pp K → pp | < 0.2 F = 29.1 GeV, |xsWA102 data(1950) f → IP IP and g’ couplings (1) with g form factors no f = 2.0 GeV ,P f Λ = 1.6 GeV ,P f Λ = 1.4 GeV ,P f Λ = 1.0 GeV ,P f Λ (GeV) *0 K *0 K M ( nb / G e V ) *0 K *0 K / d M σ d *0 K *0 pp K → pp | < 0.2 F = 29.1 GeV, |xsWA102 data(1950) f → IP IP and g’’ couplings (1) with g form factors no f = 2.0 GeV ,P f Λ = 1.6 GeV ,P f Λ = 1.4 GeV ,P f Λ = 1.0 GeV ,P f Λ FIG. 2: The distributions in K ∗ ¯ K ∗ invariant mass compared to the WA102 data [10] for the PP → f ( ) contribution. The calculations were done for √ s = | x F | K ∗ ¯ K ∗ system. The data points have been normalized to the total cross section σ exp =
85 nb.We show results for the two sets of coupling constants (left panel) | g ( ) PP f g ′ f K ∗ ¯ K ∗ | = | g ( ) PP f g ′′ f K ∗ ¯ K ∗ | = Λ f , P = f meson. We have takenhere ˜ Λ = (2.18). In addition, we show also a naive results that corresponds to thecalculations without these form factors. The absorption effects are included. later (Fig. 6), it can be concluded that the terms j = j =
1, 3 and 4, give similar characteristics to the WA102 data. In thefollowing considerations, for simplicity, we assume only one set of couplings, namely, j = g ( ) PP f and g ′′ f K ∗ ¯ K ∗ .Now we turn to the diffractive continuum mechanism. In Fig. 5 we show the resultsfor the continuum process via the K ∗ -meson exchange including the reggeization effectgiven in (2.32), (2.33). In our calculation we take Λ off,E = P K ∗ K ∗ coupling, a or b , to the WA102 exper-imental data for the M K ∗ ¯ K ∗ distribution. Our model calculation with only the b -typecoupling ( a = | b | = − ) describes the experimental data reasonably well,although, because of large experimental error bars, a small contribution from the a -typecoupling cannot be ruled out.The option | a | = − and b = f ( ) resonance. We wish to point out that the interference effectspossible between these terms may also play an important role; see [39]. This requires fur-ther analysis and will only be meaningful once experiments with better statistics becomeavailable. Hopefully this will be the case at the LHC.In Fig. 6 we show the results for the | t | and φ pp distributions together with the experi-mental data from Fig. 3 of [10]. The data points have been normalised to the mean valueof the total cross section ( σ exp = ±
10 nb) from [10]. We present results only for thecontinuum K ∗ -exchange contribution without (the top lines) and with (the bottom lines)11 − − diff Y − − − −
10 110 ( nb ) d i ff / d Y σ d | < 0.2 F = 29.1 GeV, |xs, *0 K *0 pp K → pp g’| = 1.0 × (j) for |g − − diff Y − − − −
10 110 ( nb ) d i ff / d Y σ d | < 0.2 F = 29.1 GeV, |xs, *0 K *0 pp K → pp g’’| = 1.0 × (j) for |g j = 12345 FIG. 3: The distributions in Y diff for the process PP → f ( ) → K ∗ ¯ K ∗ . The calculationswere done for √ s = | x F | K ∗ ¯ K ∗ system. We show theindividual contributions of the different PP f couplings (2.14) with index j . We have taken here˜ Λ = and Λ f , P = | g ( j ) PP f g ′ f K ∗ ¯ K ∗ | = | g ( j ) PP f g ′′ f K ∗ ¯ K ∗ | = ) |t| (GeV − − −
10 110 ) / d t ( nb / G e V σ d | < 0.2 F = 29.1 GeV, |xs, *0 K *0 pp K → pp g’’| = 1.0 × (j) for |g j = 12345 (deg) pp φ − − −
10 110 ( nb ) pp φ / d σ d | < 0.2 F = 29.1 GeV, |xs, *0 K *0 pp K → pp g’’| = 1.0 × (j) for |g FIG. 4: . The | t | (left panel) and φ pp (right panel) distributions for the pp → pp ( PP → f ( ) → K ∗ ¯ K ∗ ) reaction for the WA102 kinematics. The meaning of the lines is the same as in the rightpanel of Fig. 3. The calculation was done for | g ( j ) PP f g ′′ f K ∗ ¯ K ∗ | = the absorption effects included in the calculations. We have checked that the f ( ) -exchange contribution (with g ( ) and g ′ or g ′′ couplings) has a very similar shape of thesedistributions. The absorption effects lead to a large reduction of the cross section. We12 (GeV) *0 K *0 K M ( nb / G e V ) *0 K *0 K / d M σ d *0 K *0 pp K → pp | < 0.2 F = 29.1 GeV, |xsWA102 dataContinuum, a-type couplingContinuum, b-type coupling FIG. 5: The same as in Fig. 2 but here we show the theoretical results for the continuum mech-anism. We show results for the two type of P K ∗ K ∗ coupling considered separately, a P K ∗ K ∗ and b P K ∗ K ∗ , that occurring in (2.29). The results are normalized to the same value σ =
85 nb. The reddashed line corresponds to | a | = − and b =
0, while the black solid line corresponds to a = | b | = − . The absorption effects are included. can see a larger damping of the cross section in the region of φ pp ∼ π . The ratio offull (including absorption) and Born cross sections h S i , the gap survival factor, for theWA102 kinematics ( √ s = | x F , K ∗ ¯ K ∗ | h S i ∼ = f contribution. ) |t| (GeV ) / d t ( nb / G e V σ d *0 K *0 pp K → pp | < 0.2 F = 29.1 GeV, |xsWA102 dataContinuum, BornContinuum, with absorptiononly b-type coupling (deg) pp φ ( nb ) pp φ / d σ d Bornwith absorption
FIG. 6: The | t | (left panel) and φ pp (right panel) distributions for the pp → ppK ∗ ¯ K ∗ reaction at √ s = | x F , K ∗ ¯ K ∗ | σ exp =
85 nb. We show results for the continuum contribution obtained with the b -type coupling only in the Born approximation and with absorption.
13n [10] also the dP t dependence for the K ∗ ¯ K ∗ system was presented. Here, dP t (theso-called “glueball-filter variable” [20, 50]) is defined as dP t = q t ,1 − q t ,2 = p t ,2 − p t ,1 , dP t = | dP t | . (3.1)In Table I we show the WA102 experimental values for the fraction of K ∗ ¯ K ∗ produc-tion in three dP t intervals and for the ratio of production at small dP t to large dP t andour corresponding results for the f ( ) meson and continuum contributions. The cal-culations have been done with the absorption effects included. From the comparison tothe WA102 results we see that smaller values of cutoff parameter, ˜ Λ = in (2.18)and Λ = in (2.30), are preferred. We can conclude that both the continuumcontribution with the b -type P K ∗ K ∗ coupling and the f contribution with the g ( ) and g ′′ couplings have similar characteristics as the WA102 data. TABLE I: Results of K ∗ ¯ K ∗ production as a function of dP t (3.1), in three dP t intervals, expressedas a percentage of the total contribution at the WA102 collision energy √ s = | x F , K ∗ ¯ K ∗ | σ ( dP t ) / σ ( dP t > ) are given.The experimental numbers are from [10]. The theoretical numbers correspond to the f ( ) production mechanism with the g ( ) × g ′ and g ( ) × g ′′ couplings, Λ f , P = Λ = in (2.18). For the continuum mechanism, we show the results only with the b -type P K ∗ K ∗ coupling, Λ off,E = Λ = in (2.30). The absorptioneffects have been included in our analysis.dP t dP t t > ± ± ± ± f ( ) , g ( ) and g ′ couplings˜ Λ = Λ = f ( ) , g ( ) and g ′′ couplings˜ Λ = Λ = b -type coupling Λ = Λ = By comparing the theoretical results and the differential cross sections obtained by theWA102 Collaboration we fixed the parameters of our model. With them we will provideour predictions for the LHC. For the continuum term we take | b P K ∗ K ∗ | = − , a P K ∗ K ∗ = Λ off,E = Λ = , and for the f ( ) term we take | g ( ) PP f g ′′ f K ∗ ¯ K ∗ | = Λ f , P = Λ = .In the future the model parameters (coupling constants, form-factor cutoff parameters)could be verified or, if necessary, adjusted by comparison with precise experimental datafrom the LHC experiments. 14 . Predictions for the LHC experiments We shall give our predictions for the reaction pp → ppK + π − K − π + represented by thediagrams in Fig. 1. The results were obtained in the calculations with the tensor-pomeronexchanges including the absorptive corrections within the one-channel-eikonal approach.In Fig. 7 we present the K + π − K − π + invariant mass distributions for the continuum K ∗ -exchange contribution and the f ( ) -exchange contribution for the parametersfixed by the WA102 data. According to the same strategy as in the previous section bothcontributions are considered separately, i.e. without possible interference effect of bothterms. The calculations were done for √ s =
13 TeV with typical experimental cuts on η M (pseudorapidities) and p t , M (transverse momenta) of centrally produced pions andkaons. There are shown the results with an extra cut on momenta of leading protons0.17 GeV < | p y , p | < M K + K − π + π − ≃ f ( ) resonance. A clear difference is visible at higher val-ues of the invariant mass of the K + K − π + π − . The invariant mass distributions for thecontinuum contribution is broader compared to the f contribution which we show fortwo cutoff parameters Λ f , P = K ∗ Regge trajectory: the simple linear trajectory(2.33) (see the lower solid lines) and the “square-root” trajectory (2.34) (see the uppersolid lines).In Table II we have collected integrated cross sections in nb for different experimentalcuts for the exclusive K + K − π + π − production via the intermediate K ∗ ¯ K ∗ states includ-ing the contributions shown in Fig. 1. The ratio of the full and Born cross sections at √ s =
13 TeV is approximately h S i ∼ = K ∗ -exchange continuum contributionand h S i ∼ = f ( ) -exchange contribution. For the continuum, we used(2.33). For the f case we show the results for Λ f , P = TABLE II: The integrated cross sections in nb for the reaction pp → pp ( K ∗ ¯ K ∗ → K + π − K − π + ) for the f ( ) contribution (for Λ f , P = − K ∗ -exchange continuum con-tribution; see Fig. 1. The results have been calculated for √ s =
13 TeV and some typical exper-imental cuts on pseudorapidities and transverse momenta of produced pions and kaons. Theabsorption effects were included here. Cross sections (nb) √ s (TeV) Cuts f ( ) Continuum13 | η M | < p t , M > | η M | < p t , M > | η M | < p t , M > | η M | < p t , M > < | p y | < < η M < p t , M > < η M < p t , M > Let us complete our analysis with the following remark. We are assuming that the15 (GeV) + π - K - π + K M b / G e V ) µ ( + π - K - π + K / d M σ d ) + π - K - π + K → *0 K *0 pp (K → pp > 0.1 GeV t,M | < 2.5, p M η = 13 TeV, |s Continuum(1950) f (GeV) + π - K - π + K M b / G e V ) µ ( + π - K - π + K / d M σ d ) + π - K - π + K → *0 K *0 pp (K → pp > 0.2 GeV t,M | < 2.5, p M η = 13 TeV, |s Continuum(1950) f (GeV) + π - K - π + K M b / G e V ) µ ( + π - K - π + K / d M σ d ) + π - K - π + K → *0 K *0 pp (K → pp > 0.1 GeV t,M < 4.5, p M η = 13 TeV, 2.0 < s Continuum(1950) f (GeV) + π - K - π + K M b / G e V ) µ ( + π - K - π + K / d M σ d ) + π - K - π + K → *0 K *0 pp (K → pp > 0.2 GeV, t,M | < 2.5, p M η = 13 TeV, |s | < 0.5 GeV y f FIG. 7: Invariant mass distributions for the central K + π − K − π + system via the K ∗ ¯ K ∗ states cal-culated for √ s =
13 TeV with the kinematical cuts specified in the figure legends. The results forthe two mechanisms are presented. For the f ( ) term we show the results for Λ f , P = K ∗ Regge trajectory, linear (2.33) (the lower solid lines) and“square-root” trajectory (2.34) (the upper solid lines). The absorption effects are included. reaction pp → ppK ∗ ¯ K ∗ is dominated by pomeron exchange, for both the f ( ) andcontinuum mechanisms, already at √ s = f R f R -, f R P -, P f R -fusion processes)can also participate. The inclusion of these subleading exchanges would introduce manynew coupling parameters and form factors and would make a meaningful analysis of theWA102 data practically impossible. However, for the analysis of data from the COM-PASS experiment, which operates in the same energy range as previously the WA10216xperiment, it could be very worthwhile to study all the above subleading exchanges indetail. Keep in mind that at high energies and in the midrapidity region the subleadingexchanges should give small contributions. However, they may influence the absolutenormalization of the cross section at low energies. In general, our two mechanisms mayhave different production modes, and therefore also different energy dependence of thecross section. Therefore, our predictions for the LHC experiments should be regardedrather as an upper limit. IV. CONCLUSIONS
In this paper, we have discussed diffractive production of K ∗ ¯ K ∗ system in proton-proton collisions within the tensor-pomeron approach. Two different mechanism havebeen considered, central exclusive production of the f ( ) resonance and the contin-uum with the intermediate K ∗ -meson exchange. By comparing the theoretical resultsand the WA102 experimental data [10] we have fixed some coupling parameters and off-shell dependencies of an intermediate mesons.We have shown that the continuum contribution alone, taking into account the domi-nance of the b -type P K ∗ K ∗ coupling, describes the invariant mass spectrum obtained bythe WA102 Collaboration reasonably well. This is not the case for the f ( ) meson forwhich an agreement with the WA102 data for the preferred type of couplings g ( ) PP f and g ′′ f K ∗ ¯ K ∗ in the limited invariant mass range was found. We have found that, in both cases,the model results are in better agreement with the WA102 data taking into account thetensor-vector-vector vertices (couplings) with the Γ ( ) function rather than Γ ( ) one. Thisobservation in our tensor-pomeron approach should be verified by future experimentalresults. Hopefully this will be the case at the LHC.In the calculation the absorptive corrections calculated at the amplitude level and re-lated to the proton-proton nonperturbative interactions have been included. It is knownthat the absorption effects change considerably the shape of the distribution for φ pp , theazimuthal angle between the outgoing protons, and the shape of the distribution for dP t ,the difference of the transverse momentum vectors of the outgoing protons. The φ pp anddP t dependences for the K ∗ ¯ K ∗ system was measured by the WA102 Collaboration. Wehave reproduced these results fairly well in our model including the absorption effects.The final distributions in these variables for both mechanisms considered are very similarto each other.Assuming that the WA102 data are already dominated by pomeron exchange, we havecalculated the cross sections for experiments at the LHC imposing the cuts on pseudo-rapidities and transverse momenta of the pions and kaons from the decays K ∗ ¯ K ∗ → ( K + π − )( K − π + ) . The distributions of the invariant mass of the K + π − K − π + system havebeen presented. We should keep in mind that both considered mechanisms have a maxi-mum around M K + K − π + π − ≃ f ( ) resonance. We have found in this paperthat the diffractive processes leads to a cross section for the K ∗ ¯ K ∗ → K + π − K − π + pro-duction more than one order of magnitude larger than the corresponding cross sectionfor the φ ( ) φ ( ) → K + K − K + K − processes considered in [39]. Our predictions canbe tested by all collaborations (ALICE, ATLAS, CMS, LHCb) working at the LHC. A mea-surable cross section for the exclusive process pp → pp ( K ∗ ¯ K ∗ → K + π − K − π + ) should17rovide experimentalists with an interesting challenge to check and explore. Acknowledgments
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