Extracting the pomeron-pomeron- f 2 (1270) coupling in the pp→pp π + π − reaction through angular distributions of the pions
aa r X i v : . [ h e p - ph ] D ec Extracting the pomeron-pomeron- f ( ) coupling in the p p → p p π + π − reaction through angular distributions of the pions Piotr Lebiedowicz, ∗ Otto Nachtmann,
2, † and Antoni Szczurek c1, §1
Institute of Nuclear Physics Polish Academy of Sciences,Radzikowskiego 152, PL-31342 Kraków, Poland Institut für Theoretische Physik, Universität Heidelberg,Philosophenweg 16, D-69120 Heidelberg, Germany
Abstract
We discuss how to extract the pomeron-pomeron- f ( ) ( PP f ( ) ) coupling within thetensor-pomeron model. The general PP f ( ) coupling is a combination of seven basic cou-plings (tensorial structures). To study these tensorial structures we propose to measure thecentral-exclusive production of a π + π − pair in the invariant mass region of the f ( ) me-son. An analysis of angular distributions in the π + π − rest system, using the Collins-Soper (CS)and the Gottfried-Jackson (GJ) frames, turns out to be particularly relevant for our purpose. Forboth frames the cos θ π + and φ π + distributions are discussed. We find that the azimuthal angledistributions in these frames are fairly sensitive to the choice of the PP f coupling. We showresults for the resonance case alone as well as when the dipion continuum is included. We showthe influence of the experimental cuts on the angular distributions in the context of dedicated ex-perimental studies at RHIC and LHC energies. Absorption corrections are included for our finaldistributions. c Also at
College of Natural Sciences, Institute of Physics, University of Rzeszów, Pigonia 1, PL-35310 Rzeszów,Poland . ∗ [email protected] † [email protected] § [email protected] . INTRODUCTION The pomeron ( P ) is an essential object for understanding diffractive phenomena inhigh-energy physics. Within QCD the pomeron is a color singlet, predominantly gluonic,object. The spin structure of the pomeron, in particular its coupling to hadrons, is, how-ever, not yet a matter of consensus. In the tensor-pomeron model for soft high-energyscattering formulated in [1] the pomeron exchange is effectively treated as the exchangeof a rank-2 symmetric tensor. The diffractive amplitude for a given process with softpomeron exchange can then be formulated in terms of effective propagators and verticesrespecting the rules of quantum field theory.It is rather difficult to obtain definitive statements on the spin structure of the pomeronfrom unpolarised elastic proton-proton scattering. On the other hand, the results frompolarised proton-proton scattering by the STAR Collaboration [2] provide valuable infor-mation on this question. Three hypotheses for the spin structure of the pomeron, tensor,vector, and scalar, were discussed in [3] in view of the experimental results from [2]. Onlythe tensor-ansatz for the pomeron was found to be compatible with the experiment. Alsosome historical remarks on different views of the pomeron were made in [3].In [4] further strong evidence against the hypothesis of a vector character of thepomeron was given. It was shown there that a vector pomeron necessarily decouplesin elastic photon-proton scattering and in the absorption cross sections of virtual photonson the proton, that is, in the structure functions of deep inelastic lepton-nucleon scatter-ing. A tensor pomeron, on the other hand, has no such problems and tensor-pomeronexchanges, soft and hard, give an excellent description of the absorption cross sectionsfor real and virtual photons on the proton at high energies.In the last few years we have undertaken a scientific program to analyse the produc-tion of light mesons in the tensor-pomeron and vector-odderon model in several ex-clusive reactions: pp → ppM [5], pp → pp π + π − [6, 7], pp → pn ρ π + ( pp ρ π ) [8], pp → ppK + K − [9], pp → pp ( σσ , ρ ρ → π + π − π + π − ) [10], pp → ppp ¯ p [11], pp → pp ( φφ → K + K − K + K − ) [12], and pp → pp ( φ → K + K − , µ + µ − ) [13]. Some azimuthalangle correlations between the outgoing protons can verify the PP M couplings for scalar f ( ) , f ( ) , f ( ) , f ( ) and pseudoscalar η , η ′ ( ) mesons [5, 9]. The cou-plings, being of nonperturbative nature, are difficult to obtain from first principles ofQCD. The corresponding coupling constants were fitted to differential distributions ofthe WA102 Collaboration [14–16] and to the results of [17]. As was shown in [5, 9], thetensorial PP f , PP η , and PP η ′ vertices correspond to the sum of two lowest orbital an-gular momentum - spin couplings, except for the f ( ) meson. The tensor meson caseis a bit complicated as there are, in our approach, seven (!) possible pomeron-pomeron- f ( ) couplings in principle; see the list of possible PP f couplings in Appendix A of[7] and in Sec. II below.It was shown in [15, 18] that the cross section for the undisputed q ¯ q tensor mesons, f ( ) , f ′ ( ) , peaks at φ pp = π and is suppressed at small dP t in contrast to thetensor glueball candidate f ( ) ; see e.g. [19]. Here, φ pp is the azimuthal angle be-tween the transverse momentum vectors p t ,1 , p t ,2 of the outgoing protons and dP t (theso-called ’glueball-filter variable’ [20]) is defined by their difference dP t = p t ,2 − p t ,1 ,dP t = | dP t | . In [7] we gave some arguments from studying the φ pp and dP t distributions2hat one particular coupling PP f (denoted by j =
2) may be preferred. We roughlyreproduced the experimental data obtained by the WA102 Collaboration [15] and by theABCDHW Collaboration [21] with this coupling. It was demonstrated in [7] that the rel-ative contribution of resonant f ( ) and dipion continuum strongly depends on thecut on four-momentum transfer squared t in a given experiment. However, we mustremember that at low energies also the secondary (especially f R ) exchanges may play animportant role.Now, we ask the question whether and how the PP f couplings can be studied incentral-exclusive processes. In the present work we discuss such a possibility: analysisof angular distributions of pions from the decay of f , in two systems of reference, theCollins-Soper (CS) and the Gottfried-Jackson (GJ) systems. We will consider diffractiveproduction of the f ( ) resonance which is expected to be abundantly produced inthe pp → pp π + π − reaction; see e.g. [7]. We will try to analyse whether such a studycould shed light on the PP f ( ) couplings. In [22–24] the central exclusive produc-tion of two-pseudoscalar mesons in pp collisions at the COMPASS experiment at CERNSPS was reported. There, preliminary data of pion angular distributions in the π + π − restsystem using the GJ frame was shown. We refer the reader to [25–30] for the latest mea-surements of central π + π − production in high-energy proton-(anti)proton collisions. Inthe future the corresponding PP f ( ) couplings could be adjusted by comparison toprecise experimental data from both RHIC and the LHC. II. FORMALISM
We study central exclusive production of π + π − in proton-proton collisions p ( p a , λ a ) + p ( p b , λ b ) → p ( p , λ ) + π + ( p ) + π − ( p ) + p ( p , λ ) , (2.1)where p a , b , p and λ a , b , λ ∈ { + − } denote the four-momenta and helicities ofthe protons, and p denote the four-momenta of the charged pions, respectively.We are, in the present article, mainly interested in the region of the π + π − invari-ant mass in the f ( ) region. There we should take into account two main processesshown by the diagrams in Fig. 1. For the f ( ) resonance (the diagram (a)) we con-sider only the PP fusion. The secondary reggeons f R , a R , ω R , ρ R should give smallcontributions at high energies. We also neglect contributions involving the photon. Inthe case of the non-resonant continuum (the diagrams (b)) we include in the calculationsboth P and f R -reggeon exchanges. For an extensive discussion we refer to [6, 7].The kinematic variables for the reaction (2.1) are s = ( p a + p b ) , s = M ππ = ( p + p ) , q = p a − p , q = p b − p , t = q , t = q , p = q + q = p + p , s = ( p a + q ) = ( p + p ) , s = ( p b + q ) = ( p + p ) . (2.2)The PP -exchange (Born-level) amplitude for π + π − production via the tensor f -3 a) γ, IP, IRγ, IP, IR f (1270) p ( p a ) p ( p ) p ( p b ) p ( p ) π + ( p ) π − ( p ) (b) γ, IP, IRγ, IP, IR π + ( p ) π − ( p ) p ( p a ) p ( p ) p ( p b ) p ( p )ˆ tt t γ, IP, IRγ, IP, IR π − ( p ) π + ( p ) p ( p a ) p ( p ) p ( p b ) p ( p )ˆ ut t FIG. 1. The Born diagrams for the pp → pp π + π − reaction. In (a) we have the π + π − productionvia the f ( ) resonance, in (b) the continuum π + π − production. The exchange objects are thephoton ( γ ), the pomeron ( P ) and the reggeons ( R ). meson ( f ≡ f ( ) ) exchange can be written as M ( PP → f → π + π − ) λ a λ b → λ λ π + π − =( − 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 ππ ) αβ ( 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 ) . (2.3)Here ∆ ( P ) and Γ ( P pp ) denote the effective propagator and proton vertex function, respec-tively, for the tensor-pomeron exchange. For the explicit expressions, see Sect. 3 of [1].More details related to the amplitude (2.3) are given in [7]. ∆ ( f ) and Γ ( f ππ ) denote thetensor-meson propagator and the f ππ vertex, respectively. As was mentioned in [1]we cannot use a simple Breit-Wigner ansatz for the f propagator in conjunction withthe f ππ vertex from (3.37), (3.38) of [1] because the partial-wave unitarity relation isnot satisfied. We should use, therefore, a model for the f propagator considered in Eqs.(3.6)–(3.8) and (5.19)–(5.22) of [1]. The form factor F ( f ππ ) ( p ) for the f ππ vertex andfor the f propagator is taken to be the same as (2.7) below, but with Λ f ππ instead of Λ PP f .The main ingredient of the amplitude (2.3) is the pomeron-pomeron- f vertex 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 ) . (2.4)Here ˜ F ( PP f ) is a form factor for which we make a factorised ansatz (see (4.17) of [7])˜ F ( PP f ) ( q , q , p ) = F M ( q ) F M ( q ) F ( PP f ) ( p ) . (2.5)We are taking here the same form factor for each vertex with index j ( j =
1, ..., 7). Inprinciple, we could take a different form factor for each vertex. We take F M ( t ) = − t / Λ , Λ = ; (2.6) F ( PP f ) ( p ) = exp − ( p − m f ) Λ PP f ! , Λ PP f = Here the label “bare” is used for a vertex, as derived from a corresponding coupling Lagrangian inAppendix A of [7] without a form-factor function; see (2.8)–(2.14) below. i Γ ( PP f )( ) µν , κλ , ρσ = i g ( ) PP f M R µνµ ν R κλα λ R ρσρ σ g ν α g λ ρ g σ µ , (2.8) i Γ ( PP f )( ) µν , κλ , ρσ ( q , q ) = − iM g ( ) PP f (cid:16) ( q · q ) R µνρ α R ακλσ − q ρ q µ R µνµ α R ακλσ − q µ q σ R µνρ α R ακλµ + q ρ q σ R µνκλ (cid:17) R ρ σ ρσ ,(2.9) i Γ ( PP f )( ) µν , κλ , ρσ ( q , q ) = − iM g ( ) PP f (cid:16) ( q · q ) R µνρ α R ακλσ + q ρ q µ R µνµ α R ακλσ + q µ q σ R µνρ α R ακλµ + q ρ q σ R µνκλ (cid:17) R ρ σ ρσ ,(2.10) i Γ ( PP f )( ) µν , κλ , ρσ ( q , q ) = − iM g ( ) PP f (cid:16) q α q µ R µνµ ν R κλα λ + q α q µ R µνα λ R κλµ ν (cid:17) R ν λ ρσ ,(2.11) i Γ ( PP f )( ) µν , κλ , ρσ ( q , q ) = − iM g ( ) PP f (cid:16) q µ q ν R µνν α R ακλµ + q ν q µ R µνµ α R ακλν − ( q · q ) R µνκλ (cid:17) q α q λ R α λ ρσ , (2.12) i Γ ( PP f )( ) µν , κλ , ρσ ( q , q ) = iM g ( ) PP f (cid:16) q α q λ q µ q ρ R µνµ ν R κλα λ + q α q λ q µ q ρ R µνα λ R κλµ ν (cid:17) R ν ρ ρσ , (2.13) i Γ ( PP f )( ) µν , κλ , ρσ ( q , q ) = − iM g ( ) PP f q ρ q α q λ q σ q µ q ν R µνµ ν R κλα λ R ρσρ σ , (2.14)where R µνκλ = g µκ g νλ + g µλ g νκ − g µν g κλ . (2.15)In (2.8) to (2.14) the Lorentz indices of the pomeron with momentum q are denoted by µν ,of the pomeron with momentum q by κλ , and of the f by ρσ . Furthermore, M ≡ g ( j ) PP f ( j =
1, ..., 7) are dimensionless coupling constants. The values of the cou-pling constants g ( j ) PP f are not known and are not easy to be found from first principles ofQCD, as they are of nonperturbative origin. At the present stage these coupling constants g ( j ) PP f should be fitted to experimental data.Considering the fictitious reaction of two “real tensor pomerons” annihilating to the f meson, see Appendix A of [7], we find that we can associate the couplings (2.8)–(2.14)with the following ( l , S ) values (
0, 2 ) , (
2, 0 ) − (
2, 2 ) , (
2, 0 ) + (
2, 2 ) , (
2, 4 ) , (
4, 2 ) , (
4, 4 ) , (
6, 4 ) , respectively. Here, l and S denote orbital angular momentum and total spin of two fictitious “real pomerons” in therest system of the f meson, respectively.
5o give the full physical amplitudes we should include absorptive corrections to theBorn amplitudes. For the details how to include the pp -rescattering corrections in theeikonal approximation for the four-body reaction see e.g. Sec. 3.3 of [6]. Other rescatter-ing corrections, such as possible pion-proton [31, 32] and pion-pion [33] interactions inthe final state, and also so-called “enhanced” corrections [34], are neglected in the presentcalculations. In practice we work with the amplitudes in the high-energy approximation;see Eqs. (3.19)–(3.21) and (4.23) of [7].We are interested in the angular distribution of the π + in the center-of-mass systemof the π + π − pair. Various reference systems are commonly used; see e.g. [35] for adiscussion of such systems for the γ p → π + π − p reaction. For the Collins-Soper system[36, 37] for the reaction (2.1) we set the unit vectors defining the axes as follows: e , CS = ˆ p a − ˆ p b | ˆ p a − ˆ p b | , e , CS = ˆ p a × ˆ p b | ˆ p a × ˆ p b | , e , CS = ˆ p a + ˆ p b | ˆ p a + ˆ p b | . (2.16)These satisfy the condition e , CS = e , CS × e , CS . Here ˆ p a = p a / | p a | , ˆ p b = p b / | p b | ,where p a , p b are the three-momenta of the initial protons in the π + π − rest system. Therewe have p = p a + p b = p + p . Now we denote by θ π + , CS and φ π + , CS the polarand azimuthal angles of ˆ p (the π + meson momentum) relative to the coordinate axes(2.16). We have then e.g. cos θ π + , CS = ˆ p · e , CS , (2.17)where ˆ p = p / | p | .Alternatively, for the experiments that can measure at least one of the outgoing pro-tons, the Gottfried-Jackson (GJ) system could be used as well. For the GJ system [38] weset e , GJ = q | q | , e , GJ = q , c.m. × q , c.m. | q , c.m. × q , c.m. | , e , GJ = e , GJ × e , GJ . (2.18)Here q is the three-momentum of the pomeron (emitted by the proton with positive p z ) in the π + π − rest system. The second axis of the GJ coordinate system is fixed bythe normal to the production plane ( P - P - π + π − plane) in the pp center-of-mass (c.m.)system. q , c.m. and q , c.m. are three-momenta defined in the pp c.m. frame.For some further remarks on this GJ system see Appendix A.Having defined these angles we can now examine the differential cross sections d σ / ( d cos θ π + , CS d φ π + , CS ) , d σ / d cos θ π + , CS , d σ / d φ π + , CS , and the corresponding distri-butions in the GJ system. 6 II. RESULTS
As discussed in the introduction, very good observables which can be used for visu-alizing the role of the PP f couplings, given by Eqs. (2.8)–(2.14) (cf. also Appendix A of[7]), could be the differential cross sections d σ / d cos θ π + and d σ / d φ π + , both in the CS andthe GJ systems of reference; see (2.16) and (2.18), respectively. In Figs. 2–5 and 7–10 weshow such angular distributions for the π + meson in the π + π − rest frame.In Fig. 2 we collected angular distributions for all (seven) independent PP f ( ) couplings for √ s =
13 TeV, p t , π > | η π | < | η π | < ( p x , p + ) + p y , p < ,0.2 GeV < | p y , p | < p x , p > − | η π | . From the left top panelin Fig. 2 we see that the condition | η π | < θ π + , CS ≈ ± | η π | < d σ / d φ π + , CS ≈ A ± B cos ( n φ π + , CS ) , (3.2)for | cos θ π + , CS | < A and B depend on experimentalconditions. For most of the couplings n = j = n = j = φ π + , CS distributions depends also on the cuts on | η π | . Therefore, we expectthese differences to be better visible when one compares the results related to differentregions of pion pseudorapidity. Let us note that the LHCb Collaboration can measure π + π − production for 2.0 < η π < φ π + , CS , cos θ π + , CS ) for √ s =
13 TeV and | η π | < pp → pp π + π − reaction. We show results for the individual PP f ( ) coupling terms and for the con-tinuum π + π − production. Different tensorial couplings generate very different patternswhich should be checked experimentally.Some preliminary low-energy COMPASS results [22, 23] suggest the presence of twomaxima in the φ π + , GJ distribution. So far there are no official analogous data for high-energy scattering either from STAR or the LHC experiments. Nevertheless we have askedourselves the question if and how we can get a similar structure (two maxima at φ π + , GJ = π /2, 3/2 π ) in terms of our PP f couplings (2.8) to (2.14).In Fig. 5 we show the azimuthal angle distributions using the CS (2.16) and the GJ(2.18) frames. Here we examine the combination of two PP f couplings: j = j = j =
2, 5 coupling terms and for theircoherent sum. For this purpose, we fixed the j = g ( ) PP f = CS + π θ cos − − b ) µ ( , C S + π θ / d c o s σ d − − − −
10 110 ) - π + π → (1270) pp (f → pp > 0.1 GeV π t, | < 1, p π η = 13 TeV, |s (deg) ,CS + π φ b ) µ ( , C S + π φ / d σ d − − − −
10 110 ) - π + π → (1270) pp (f → pp > 0.1 GeV π t, | < 1, p π η = 13 TeV, |s ,CS + π θ cos − − b ) µ ( , C S + π θ / d c o s σ d − − −
10 110 ) - π + π → (1270) pp (f → pp > 0.1 GeV π t, | < 2.5, p π η = 13 TeV, |s (deg) ,CS + π φ b ) µ ( , C S + π φ / d σ d − − −
10 110 ) - π + π → (1270) pp (f → pp > 0.1 GeV π t, | < 2.5, p π η = 13 TeV, |s FIG. 2. The distributions in cos θ π + , CS (the left panels) and in φ π + , CS (the right panels) for the pp → pp ( f ( ) → π + π − ) reaction. The calculations were done for √ s =
13 TeV with differentcuts on | η π | and p t , π > PP f couplings from(2.8) to (2.14) are shown: j = j = j = j = j = j = j = g ( j ) PP f = assumed various values for g ( ) PP f . In the top and bottom panels, the red and green linescorrespond to the results when both couplings have opposite signs and the same signs,respectively. Different interference patterns can be seen there depending on the ratio ofthe two couplings, R = g ( ) PP f / g ( ) PP f .Now we discuss whether the absorption effects (the pp -rescattering corrections) maychange the angular distributions discussed so far in the Born approximation. We havechecked that a slightly different size of absorption effects may occur for the j =
1, ..., 78 CS + π θ cos − − b ) µ ( , C S + π θ / d c o s σ d − − − − − − −
10 1 ) - π + π → (1270) pp (f → pp > 0.2 GeV π t, | < 0.7, p π η = 200 GeV, |s (deg) ,CS + π φ b ) µ ( , C S + π φ / d σ d − − − − − − ) - π + π → (1270) pp (f → pp > 0.2 GeV π t, | < 0.7, p π η = 200 GeV, |s ,GJ + π θ cos − − b ) µ ( , G J + π θ / d c o s σ d − − − − − − −
10 1 ) - π + π → (1270) pp (f → pp > 0.2 GeV π t, | < 0.7, p π η = 200 GeV, |s (deg) ,GJ + π φ b ) µ ( , G J + π φ / d σ d − − − − − − ) - π + π → (1270) pp (f → pp > 0.2 GeV π t, | < 0.7, p π η = 200 GeV, |s FIG. 3. The same as in Fig. 2 but for √ s =
200 GeV and the STAR experimental cuts from [28]: | η π | < p t , π > π + π − rest system using the CS frame (2.16). In the bottompanels, we show the results using the GJ frame (2.18). resonant terms. The absorption effects lead to a significant reduction of the cross section.However, the shapes of the polar and azimuthal angle distributions are practically notchanged. This indicates that the absorption effects should not disturb the determinationof the type of the PP f ( ) coupling. However, the continuum and the resonant termsmay be differently affected by absorption. This will have to be taken into account whenone tries to extract the strengths of the couplings from such distributions.The measurement of forward protons would be useful to better understand absorptioneffects. The GenEx
Monte Carlo generator [40, 41] could be used in this context. Werefer the reader to [42] where a first calculation of four-pion continuum production in the pp → pp π + π − π + π − reaction with the help of the GenEx code was performed.Clearly, by a comparison of our model results to high-energy experimental data we9 a) (deg) ,CS + π φ , C S + π θ c o s − − ), j = 1 - π + π → (1270) pp (f → pp b) µ ( ,CS + π φ d ,CS + π θ /dcos σ d (b) (deg) ,CS + π φ , C S + π θ c o s − − ), j = 2 - π + π → (1270) pp (f → pp b) µ ( ,CS + π φ d ,CS + π θ /dcos σ d (c) (deg) ,CS + π φ , C S + π θ c o s − − ), j = 3 - π + π → (1270) pp (f → pp b) µ ( ,CS + π φ d ,CS + π θ /dcos σ d (d) (deg) ,CS + π φ , C S + π θ c o s − − ), j = 4 - π + π → (1270) pp (f → pp b) µ ( ,CS + π φ d ,CS + π θ /dcos σ d (e) (deg) ,CS + π φ , C S + π θ c o s − − ), j = 5 - π + π → (1270) pp (f → pp b) µ ( ,CS + π φ d ,CS + π θ /dcos σ d (f) (deg) ,CS + π φ , C S + π θ c o s − − , continuum - π + π pp → pp b) µ ( ,CS + π φ d ,CS + π θ /dcos σ d FIG. 4. The two-dimensional distributions in ( φ π + , CS , cos θ π + , CS ) for the pp → pp π + π − reac-tion. The calculations were done for √ s =
13 TeV and with cuts on | η π | < f ( ) → π + π − contributions for five PP f couplings, (a) j =
1, (b) j =
2, (c) j =
3, (d) j = j =
5, and (f) the π + π − continuum term are presented. The results for π + π − production via f ( ) resonance were obtained with coupling constants g ( j ) PP f =
100 200 300 (deg) ,CS + π φ − − −
10 1 ( nb ) , C S + π φ / d σ d = 200 GeVs), - π + π → (1270) pp (f → pp STAR cuts = 0 (5) = 1, g (2) g = 1 (5) = 0, g (2) g = 1 fixed (2) , g (5) /g (2) R = -0.25 R = gR = -0.5R = -1R = -2 (deg) ,GJ + π φ ( nb ) , G J + π φ / d σ d − − −
10 1 = 200 GeVs), - π + π → (1270) pp (f → pp STAR cuts (deg) ,CS + π φ − − −
10 1 ( nb ) , C S + π φ / d σ d = 200 GeVs), - π + π → (1270) pp (f → pp STAR cuts = 0 (5) = 1, g (2) g = 1 (5) = 0, g (2) g = 1 fixed (2) , g (5) /g (2) R = 0.25 R = gR = 0.5R = 1R = 2 (deg) ,GJ + π φ ( nb ) , G J + π φ / d σ d − − −
10 1 = 200 GeVs), - π + π → (1270) pp (f → pp STAR cuts FIG. 5. The distributions in azimuthal angle in the π + π − rest system using the CS frame (leftpanels) and the GJ frame (right panels), respectively. The calculations were done for the STARkinematics (see the caption of Fig. 3). No absorption effects were included here. shall be able to determine or at least set limits on the parameters of the PP f ( ) cou-pling. At the moment, however, this is not yet possible since only some, mostly prelimi-nary, experimental distributions were presented [25–29].In Fig. 6 we show the dipion invariant mass distributions for different experimentalconditions specified in the legend. One can see the recent high-energy data from theSTAR, CDF, and CMS experiments, as well as predictions of our model. Panels (a) and(b) show the preliminary STAR data from [25] and [28], respectively. Panels (c) and (d)show the CDF experimental data from [26]. Panel (e) shows a very recent result obtainedby the CMS Collaboration [29]. In the calculations we include both the nonresonant con-tinuum and f ( ) terms. The panels (b) and (f) show the results including extra cutson the outgoing protons. For the STAR experiment we take the cuts (3.1) and for theATLAS-ALFA experiment we take 0.17 GeV < | p y , p | < pp -rescattering corrections only) were taken into account at the amplitude level. Thetwo-pion continuum was fixed by using the monopole form of the off-shell pion formfactor with the cut-off parameter Λ off,M = f ( ) contribution, in order to get distinct maxima at φ π + , GJ = π /2, 3/2 π , we take a combina-tion of two PP f couplings, ( g ( ) PP f , g ( ) PP f ) = ( − ) (set A) and ( − ) (set B),which correspond to the solid and long-dashed lines, respectively. The complete resultsindicate an interference effect of the continuum and the f ( ) term. For comparisonwe show also the contributions of the individual terms separately.We can see from Fig. 6, that in the f mass region we describe fairly well the prelimi-nary STAR and CMS data but we overestimate the CDF data [26]. In the CMS and CDFmeasurements there are possible contributions of proton dissociation. The continuumcontribution underestimates the data in the region M π + π − < f ( ) , f ( ) , and ρ production not included inthe present analysis; see e.g. [6, 7]. Also other effects, such as the rescattering correctionsdiscussed in [31–33], can be very important there.We emphasize, that in our calculation of the π + π − -continuum term we include notonly the leading pomeron exchanges ( PP → π + π − ) but also the P f R , f R P , and f R f R exchanges. There is interference between the corresponding amplitudes. Their role isvery important especially at low energies (COMPASS, WA102, ISR) but even for theSTAR kinematics their contribution is not negligible. Adding the f R reggeon exchangesincreases the cross section by 56% and 45% for the kinematical conditions shown inFigs. 6 (a) and (b), respectively. A similar role of secondary reggeons can be expectedfor the production of resonances. This means that our results for the f ( ) resonance(roughly matched to the STAR data) should be treated rather as an upper estimate. Thismay be the reason why our result for f ( ) is well above the CDF data.We summarize this part by the general observation that it is very difficult to describeall available data with the same set of parameters. High-energy central exclusive dataexpected from CMS-TOTEM and ATLAS-ALFA will allow a better understanding of thediffractive production mechanisms.In Figs. 7 and 8 we show the two-dimensional angular distributions for the STAR andATLAS-ALFA kinematics, respectively. In the left panels the results for the CS systemand in the right panels for the GJ system are presented. In the top panels we show resultsfor the continuum term, in the center panels for the f ( ) term, and in the bottompanels for their coherent sum. Here we take the set A with the PP f coupling parameters ( g ( ) PP f , g ( ) PP f ) = ( − ) . Figures 9 and 10 show that the complete results indicatean interference effect of the continuum and the f ( ) term calculated for the sets Aand B, see the solid and long-dashed lines, respectively. The interference effect dependscrucially on the choice of the PP f ( ) coupling. A combined analysis of the M π + π − and angular distributions in the π + π − rest frames would, therefore, help to pin downthe underlying reaction mechanism. 12 a) (GeV) - π + π M ( nb / G e V ) - π + π / d M σ d = 200 GeVs, - π + π pp → pp > 0.15 GeV π t, | < 2, p - π + π η | < 1, | π η | < 0.03 GeV , -t < -t continuum(1270) ftotal, set Atotal, set B (b) (GeV) - π + π M ( nb / G e V ) - π + π / d M σ d −
10 110 = 200 GeVs, - π + π pp → pp > 0.2 GeV π t, | < 0.7, p π η |with cuts on protons STAR preliminary data continuum(1270) ftotal, set Atotal, set B (c) (GeV) - π + π M b / G e V ) µ ( - π + π / d M σ d − − −
10 110 = 1.96 TeVs, - π + π p p → pp > 0.4 GeV π t, | < 1, p - π + π | < 1.3, |y π η |CDF data continuum(1270) ftotal, set Atotal, set B (d) (GeV) - π + π M b / G e V ) µ ( - π + π / d M σ d − − −
10 110 = 1.96 TeVs, - π + π p p → pp > 0.4 GeV π t, | < 1, p - π + π | < 1.3, |y π η | > 1 GeV - π + π t, pCDF data continuum(1270) ftotal, set Atotal, set B (e) (GeV) - π + π M b / G e V ) µ ( - π + π / d M σ d = 13 TeVs, - π + π pp → pp > 0.2 GeV π t, | < 2.4, p π η |CMS preliminary data continuum(1270) ftotal, set Atotal, set B (f) (GeV) - π + π M b / G e V ) µ ( - π + π / d M σ d − −
10 110 = 13 TeVs, - π + π pp → pp > 0.1 GeV π t, | < 2.5, p π η | | < 0.50 GeV y,p continuum(1270) ftotal, set Atotal, set B FIG. 6. Two-pion invariant mass distributions with the relevant kinematical cuts for (a), (b) STAR,(c), (d) CDF, (e) CMS, and (f) ATLAS-ALFA experiments. The STAR preliminary data from [25, 28],the CDF data from [26], and the CMS preliminary data from [29] are shown. The calculationsfor the STAR and ATLAS-ALFA experiments were done with extra cuts on the leading protons.The short-dashed lines represent the nonresonant continuum contribution, the dotted lines rep-resent the results for the f ( ) contribution, while the solid and long-dashed lines representtheir coherent sum for the two parameter sets A and B, respectively. Here we take, in set A ( g ( ) PP f , g ( ) PP f ) = ( − ) and, in set B ( g ( ) PP f , g ( ) PP f ) = ( − ) ; see (2.5), (2.9), (2.12). Theabsorption effects are included here. (deg) ,CS + π φ , C S + π θ c o s − − continuum - π + π pp → pp (nb) ,CS + π φ d ,CS + π θ /dcos σ d (deg) ,GJ + π φ , G J + π θ c o s − − continuum - π + π pp → pp (nb) ,GJ + π φ d ,GJ + π θ /dcos σ d (deg) ,CS + π φ , C S + π θ c o s − − (1270) f - π + π pp → pp (nb) ,CS + π φ d ,CS + π θ /dcos σ d (deg) ,GJ + π φ , G J + π θ c o s − − (1270) f - π + π pp → pp (nb) ,GJ + π φ d ,GJ + π θ /dcos σ d (deg) ,CS + π φ , C S + π θ c o s − − - π + π pp → pp (nb) ,CS + π φ d ,CS + π θ /dcos σ d (deg) ,GJ + π φ , G J + π θ c o s − − - π + π pp → pp (nb) ,GJ + π φ d ,GJ + π θ /dcos σ d FIG. 7. The distributions in ( φ π + , CS , cos θ π + , CS ) (the left panels) and in ( φ π + , GJ , cos θ π + , GJ ) (theright panels) for the pp → pp π + π − reaction. The calculations were done in the dipion invariantmass region M π + π − ∈ ( ) GeV for √ s =
200 GeV and the STAR experimental cuts from [28]: | η π | < p t , π > π + π − continuumterm, in the center panels, for the f ( ) resonance term (set A), and in the bottom panels, forboth the contributions added coherently. Here we took ( g ( ) PP f , g ( ) PP f ) = ( − ) as discussedin the main text. The absorption effects are included here. (deg) ,CS + π φ , C S + π θ c o s − − continuum - π + π pp → pp b) µ ( ,CS + π φ d ,CS + π θ /dcos σ d (deg) ,GJ + π φ , G J + π θ c o s − − continuum - π + π pp → pp b) µ ( ,GJ + π φ d ,GJ + π θ /dcos σ d (deg) ,CS + π φ , C S + π θ c o s − − (1270) f - π + π pp → pp b) µ ( ,CS + π φ d ,CS + π θ /dcos σ d (deg) ,GJ + π φ , G J + π θ c o s − − (1270) f - π + π pp → pp b) µ ( ,GJ + π φ d ,GJ + π θ /dcos σ d (deg) ,CS + π φ , C S + π θ c o s − − - π + π pp → pp b) µ ( ,CS + π φ d ,CS + π θ /dcos σ d (deg) ,GJ + π φ , G J + π θ c o s − − - π + π pp → pp b) µ ( ,GJ + π φ d ,GJ + π θ /dcos σ d FIG. 8. The same as in Fig. 7 but for √ s =
13 TeV and the ATLAS-ALFA experimental cuts: | η π | < p t , π > < | p y , p | < M π + π − ∈ ( ) GeV. The absorption effects are included here. (deg) ,CS + π φ ( nb ) , C S + π φ / d σ d = 200 GeVs, - π + π pp → pp > 0.2 GeV π t, | < 0.7, p π η |with cuts on protons (1.0, 1.5) GeV ∈ - π + π M continuum(1270) ftotal, set Atotal, set B (deg) ,GJ + π φ ( nb ) , G J + π φ / d σ d = 200 GeVs, - π + π pp → pp > 0.2 GeV π t, | < 0.7, p π η |with cuts on protons (1.0, 1.5) GeV ∈ - π + π M continuum(1270) ftotal, set Atotal, set B ,CS + π θ cos − − ( nb ) , C S + π θ / d c o s σ d = 200 GeVs, - π + π pp → pp > 0.2 GeV π t, | < 0.7, p π η |with cuts on protons (1.0, 1.5) GeV ∈ - π + π M continuum(1270) ftotal, set Atotal, set B ,GJ + π θ cos − − ( nb ) , G J + π θ / d c o s σ d = 200 GeVs, - π + π pp → pp > 0.2 GeV π t, | < 0.7, p π η |with cuts on protons (1.0, 1.5) GeV ∈ - π + π M continuum(1270) ftotal, set Atotal, set B FIG. 9. The angular distributions for the pp → pp π + π − reaction. The calculations were donefor √ s =
200 GeV in the dipion invariant mass region M π + π − ∈ ( ) GeV and for the STARexperimental cuts specified in [28]. The results for the π + π − continuum term (the short-dashedline), for the f ( ) resonance term (the dotted line), and for their coherent sum (the solid andlong-dashed lines corresponding to sets A and B, respectively) are presented. We have takenhere A ( g ( ) PP f , g ( ) PP f ) = ( − ) , and B ( g ( ) PP f , g ( ) PP f ) = ( − ) as parameter sets. Theabsorption effects are included here. (deg) ,CS + π φ b ) µ ( , C S + π φ / d σ d = 13 TeVs, - π + π pp → pp > 0.1 GeV π t, | < 2.5, p π η | | < 0.50 GeV y,p ∈ - π + π M continuum(1270) ftotal, set Atotal, set B (deg) ,GJ + π φ b ) µ ( , G J + π φ / d σ d = 13 TeVs, - π + π pp → pp > 0.1 GeV π t, | < 2.5, p π η | | < 0.50 GeV y,p ∈ - π + π M continuum(1270) ftotal, set Atotal, set B ,CS + π θ cos − − b ) µ ( , C S + π θ / d c o s σ d = 13 TeVs, - π + π pp → pp > 0.1 GeV π t, | < 2.5, p π η | | < 0.50 GeV y,p ∈ - π + π M continuum(1270) ftotal, set Atotal, set B ,GJ + π θ cos − − b ) µ ( , G J + π θ / d c o s σ d = 13 TeVs, - π + π pp → pp > 0.1 GeV π t, | < 2.5, p π η | | < 0.50 GeV y,p ∈ - π + π M continuum(1270) ftotal, set Atotal, set B FIG. 10. The same as in Fig. 9 but for the ATLAS-ALFA kinematics: √ s =
13 TeV, | η π | < p t , π > < | p y , p | < M π + π − ∈ ( ) GeV. The absorption effects are included here. V. CONCLUSIONS
In the present work we have considered the possibility to extract the PP f ( ) cou-plings from the analysis of pion angular distributions in the π + π − rest system, using theCollins-Soper (CS) and the Gottfried-Jackson (GJ) frames. We have considered the tensor-pomeron model for which there are 7 possible PP f ( ) couplings; see Eqs. (2.8)–(2.14)and Appendix A of [7]. We have shown that the shape of such distributions stronglydepends on the functional form of the PP f ( ) coupling. In particular, we haveshown that the azimuthal angle distributions may have different numbers of oscillations.The corresponding distributions can be approximately represented by the formula (3.2): A ± B cos ( n φ π + , CS ) , where n =
2, 4. Two-dimensional distributions in the CS system( φ π + , CS , cos θ π + , CS ), ( M π + π − , φ π + , CS ), ( M π + π − , cos θ π + , CS ), and respectively in the GJ sys-tem, will give even more information and could also be useful in understanding the roleof experimental cuts. Can such distributions be used to fix the PP f ( ) coupling? Theanswer will require dedicated experimental studies by the STAR, ALICE, ATLAS-ALFA,CMS-TOTEM, and LHCb Collaborations. This requires comparisons of our model resultswith precise ’exclusive’ experimental data simultaneously in several differential observ-ables.We have shown how to select linear combinations of the different PP f ( ) couplingconstants to get two maxima in φ π + , GJ (or φ π + , CS ) as observed at low energies by theCOMPASS Collaboration; see [22, 23].In the diffractive process considered the f ( ) resonance cannot be completely iso-lated from the continuum background as the corresponding amplitudes strongly inter-fere [7]. We have discussed how the interference of the resonance and the continuumbackground may change the angular distributions d σ / d cos θ π + , CS and d σ / d φ π + , CS . Theabsorption effects change the overall normalization of such distributions but leave theshape essentially unchanged. This is in contrast to the d σ / d φ pp distributions where ab-sorption effects considerably modify the corresponding shapes; see e.g. [10, 31].In the present analysis we have concentrated on the pronounced f ( ) resonance,clearly seen in the π + π − channel. We have discussed methods how to pin down thepomeron-pomeron- f ( ) coupling. The analysis presented may be extended also toother resonances seen in different final state channels. We strongly encourage experimen-tal groups to start such analyses. We think that this will bring in a new tool for analysingexclusive diffractive processes and will provide new inspirations in searching for moreexotic states such as glueballs, for instance. The exclusive diffractive processes were al-ways claimed to be a good area to learn about the physics of glueballs. The extension ofour methods to the production of glueballs, to be identified in suitable decay channels,should shed light on the pomeron-pomeron-glueball couplings. These represent veryinteresting quantities: the coupling of three (mainly) gluonic objects. Appendix A: Remarks on the transformation from the c.m. to the ππ rest system In this section we discuss the relation between quantities in the c.m. system and the ππ rest system. Momenta in the c.m. system will be denoted by p c.m. , k c.m. , etc., momentain the ππ rest system by p R , k R , etc. We assume that the transformation from the c.m. to18he ππ rest system is made by a boost, that is, by a rotation free Lorentz transformation Λ ( − p , c.m. ) = (cid:16) Λ µ ν ( − p , c.m. ) (cid:17) = p M ππ − p j
34, c.m. M ππ − p i
34, c.m. M ππ δ ij + (cid:16) p M ππ − (cid:17) p i
34, c.m. p j
34, c.m. ( p , c.m. ) , (A1)where i , j ∈ {
1, 2, 3 } .We have then for any four vector l = ( l µ ) Λ ( − p , c.m. ) l c.m. = l R . (A2)The reverse transformation is Λ − ( − p , c.m. ) = Λ ( p , c.m. ) Λ − ( − p , c.m. ) l R = Λ ( p , c.m. ) l R = l c.m. . (A3)In particular we get Λ ( − p , c.m. ) p
34, c.m. = p
34, R = (cid:18) M ππ (cid:19) , (A4) Λ ( − p , c.m. ) M ππ √ s ( p a + p b ) c.m. = M ππ √ s ( p a + p b ) R = (cid:18) p − p , c.m. (cid:19) , (A5) q
1, R = Λ ( − p , c.m. ) q
1, c.m. = ( p · q ) M ππ q , c.m. + p , c.m. M ππ − q
01, c.m. + ( p − M ππ ) ( p , c.m. · q , c.m. )( p , c.m. ) ! . (A6)With these relations we can now express the unit vectors of the Gottfried-Jackson (GJ)system of (2.18) entirely by vectors defined in the ππ rest system. We have p = q + q and therefore from (A5) and (A6) q , c.m. × q , c.m. = q , c.m. × p , c.m. = − M ππ √ s q , R × ( p a + p b ) R ; (A7) e , GJ = q , R | q , R | , e , GJ = − q , R × ( p a + p b ) R | q , R × ( p a + p b ) R | , e , GJ = e , GJ × e , GJ . (A8)Note that for setting up this GJ system only the momentum of one of the outgoingprotons in the reaction (2.1) has to be measured, plus, of course the momenta of π + and π − giving p . 19 CKNOWLEDGMENTS
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