The detailed dependences of central meson production on the azimuthal angle
phi, t and the meson J^P are shown to be consistent with the hypothesis that
the soft Pomeron transforms as a non-conserved vector current. Further tests
are proposed. This opens the way for a quantitative description of q-qbar and
glueball production in p p -> p M p.Comment: 12 pages, latex, 4 figure

Arens

Castoldi

Chauvat

Close

Frank E. Close

Gerhard A. Schuler

Physics Letters B

English

arXiv

Crossref

Evidence that the Pomeron transforms as a non-conserved vector current

The detailed dependences of central meson production on the azimuthal angle phi, t and the meson J^P are shown to be consistent with the hypothesis that the soft Pomeron transforms as a non-conserved vector current. Further tests are proposed. This opens the way for a quantitative description of q-qbar and glueball production in p p -> p M p.The detailed dependences of central meson production on the azimuthal angle phi, t and the meson J^P are shown to be consistent with the hypothesis that the soft Pomeron transforms as a non-conserved vector current. Further tests are proposed. This opens the way for a quantitative description of q-qbar and glueball production in p p -> p M p.The detailed dependences of central meson production on the azimuthal angle phi, t and the meson J^P are shown to be consistent with the hypothesis that the soft Pomeron transforms as a non-conserved vector current. Further tests are proposed. This opens the way for a quantitative description of q-qbar and glueball production in p p -> p M p.The detailed dependences of central meson production on the azimuthal angle phi, t and the meson J^P are shown to be consistent with the hypothesis that the soft Pomeron transforms as a non-conserved vector current. Further tests are proposed. This opens the way for a quantitative description of q-qbar and glueball production in p p -> p M p.The detailed dependences of central meson production on the azimuthal angle φ , t and the meson J P are shown to be consistent with the hypothesis that the soft Pomeron transforms as a non-conserved vector current. Further tests are proposed. This opens the way for a quantitative description of q q ̄ and glueball production in p p → p M p 

Close, Frank E.

Schuler, Gerhard A.

CERN Document Server

CERN{TH/99{131hep-ph/9905305Evidence that the Pomeron transformsas a non-conserved vector currentFrank E. CloseaandGerhard A. SchulerbTheoretical Physics Division, CERN, CH-1211 Geneva 23AbstractThe detailed dependences of central meson production on the azimuthal angleφ, t and the meson JP are shown to be consistent with the hypothesis that thesoft Pomeron transforms as a non-conserved vector current. Further tests areproposed. This opens the way for a quantitative description of qq and glueballproduction in pp ! p M p.CERN{TH/99{131May 1999a On leave from Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire, OX11 0QX,UK; supported in part by the EEC-TMR Programme, Contract NCT98-0169.b Heisenberg Fellow; supported in part by the EU Fourth Framework Programme \Trainingand Mobility of Researchers", Network \Quantum Chromodynamics and the Deep Structureof Elementary Particles", contract FMRX-CT98-0194 (DG 12-MIHT).1 IntroductionIn order to understand the dynamics of the proposed glueball lter[1] and to separate glueballsfrom qq states in central production [2] it is necessary to establish the transformation propertiesof the Pomeron at low momentum transfers. The observations [3] of non-trivial dependence onthe azimuthal angle φ of the outgoing protons in pp ! pp + M shows that the eective spin ofthe Pomeron cannot be simply zero [4, 5]. Recently we have shown [4] that many features ofthe central production of several established qq mesons in pp ! pp+M , in both the azimuthal,φ, and glueball lter, kT, dependence can be understood if the Pomeron behaves as a conservedvector current. However, there are both empirical [4] and theoretical reasons [5, 6] to believethat this cannot be the whole story and that current non-conservation is important, especiallyat the central meson production-vertex.In this letter we show that the data clearly require that the Pomeron transforms eectivelyas a non-conserved vector with a behaviour of a specic type. We propose further tests of thishypothesis and discuss the practicalities of dierentiating glueballs from qq in 0++, 2++ mesonproduction.As in our study of the conserved vector-current (CVC) case [4] we consider the centralproduction of a JP+ meson M in high-energy proton{proton scattering. We have shown, withcurrent conservation at the proton{Pomeron vertex (see also [5]), that the cross section maybe written asdσ/dt1 dt2 dφ dxF  t1 t2 (σ2 + σ1 + σ0) , (1)where the subscripts i denote the helicity states of the meson and the suppressed pre-factorhas the following properties: it is concentrated at xF  0, steeply falling with decreasing four-momentum transfers, ti, and nite for ti approaching tmini  0 (i.e. approximately proportionalto expfb(t1 + t2)g).In the kinematic regime of interest, jtij  M2 and xi  (√x2F + 4M2/s  xF)/2  1, wehaveσ2 =12A2+−σ1 = A2+L + A2L+ − 2 η ξ2 A+L AL+ cos φσ0 = (ALL − ξ1 A++ cos φ)2 (η = +1)= A2++ sin2 φ (η = −1) , (2)where the subscripts  and L refer to the Pomeron helicities, ξi are sign factors, and η is theproduct of the naturality of the meson and the two currents, η = η1 η2 ηM = 1. The generalstructure of the φ dependence as a function of JP is then as follows, from which we will abstractspecic tests:dσ[0−]  t1 t2 A2++ sin2 φ (3)dσ[0+] (pt1 t2 ALL − ξ1pt1 t2 A++ cos φ)2(4)dσ[1+]  t1 t2 A2++ sin2 φ + (α ~q1T − β ~q2T)2(α = q2T A+L, β = ηξ2 q1T AL+ or β = q1T A+L, α = ηξ2 q2T AL+) (5)dσ[2+] (pt1 t2 ALL − ξ1pt1 t2 A++ cos φ)2+ (α ~q1T − β ~q2T)2 + t1 t2 12A2+− (6)1dσ[2−]  t1 t2 A2++ sin2 φ+ (α ~q1T − β ~q2T)2 + t1 t2 12A2+− . (7)φdσ/dφ02505007501000125015001750200022500 20 40 60 80 100 120 140 160 180Figure 1: The sin2 φ prediction compared to η0 production at WA102 [3].(i) JP=0−The pseudoscalar cross sections behaves as dσ  t1 t2 A2++ sin2 φ. The φ dependence is henceindependent of the 0− being a glueball or a quarkonium state [7, 8] and thereby providesan immediate test of the Pomeron transformation properties. The observed η and η0 indeedhave the sin2 φ behaviour (see, for example, Figure 1). They also exhibit the t eb t behaviour[3]. There is the interesting possibility that glueball production might have a compensating(t1 t2)−1 Pomeron{Pomeron{glueball vertex that could give a nite cross section as t1,2 ! 0and provide a dynamical discrimination.The t1 t2 factor in (3) originates from the fusion of two transversely-polarized currents (TT)and is hence independent of whether the current is conserved or not. The question of currentconservation becomes testable if longitudinal polarization (L) can contribute, as in the case ofall other states,2The 1+ state is interesting because Bose symmetry (A++ / (t1− t2)2 in (5)) suppresses theTT part (with helicity zero) and leaves TL (with helicity one) dominant. As such this becomessensitive to (non)-conserved vector eects.In order to exploit current (non)-conservation we consider three scenarios:i) Current conservation (Model C):q M = 0 . (8)ii) Non-conservation (Model B):q M = O(1) , (9)or (Model A):q M = O(p−t) . (10)We shall show how data discriminate among these alternatives and provide a consistent solution.In (8{10) q (t = q2) denotes the four-momentum of the current (i.e. q = q1 or q2). Since thelongitudinal polarization vector L  q/p−t as t ! 0 we obtain L  M  q  M/p−t incontrast to   M = O(1) at small jtj. Model C thus corresponds to the conserved-vectorhypothesis (CVC): ML/MT p−t/µ, where µ is a mass scale. Model B is what was argued [5]to correspond to the soft (\Donnachie{Landsho") Pomeron, ML/MT  µ/p−t where µ ’ M ,the mass of the produced meson. Contrary to [5], we anticipate that µ is a rather small massscale, related to constituent binding or to instanton size. Finally, model A is a possible furtheralternative where longitudinal and transverse amplitudes have similar strengths, ML/MT  1.We now illustrate this in the case of 1++.(ii) JP=1+Here we concentrate on the fusion of two identical currents (i.e. photon-photon or Pomeron-Pomeron). Then σ0 is as for the pseudoscalars but obeying Bose symmetryt1 t2 σ0 = t1 t2 µ2 A^2++(t1 − t2)2µ4sin2 φ . (11)Here and in the following the hatted quantities denote the residual, dimensionless amplitudes,which, in general, are of order one1. As for the pseudoscalars, the helicity-zero part is indepen-dent of the model assumption.Bose symmetry allows us to determine the sign of ξ2 in σ1, (2,5), and we ndt1 t2 σ1 = µ2 t1 t2 4 sin2 φ2A^2+L model Aµ4 k2T A^2+L model Bt1 t2 k2T A^2+L model C . (12)Recall that kT ! 0 implies φ ! 0 but φ ! 0 yields kT ! 0 only if t2/t1 ! 1. We can makethis manifest by introducing the following quantities: =pt1 t2µ2, r =√t2t1(13)so thatt1 − t2µ2= 1− r2r,k2Tµ2= 4 (sin2φ2+(1− r)24 r). (14)1That is we generally assume that Aˆλλ′  1 at small jtij. Additional dynamics might change this and yield,for example, AˆLL = c kT/µ.3This allows us to write the 1+ cross section asσ  (2 µ3 A^+L)2p1{sin2φ2+ p2 + p3 sin2 φ}, (15)whereVariableModel p1 p2 p3A  0 2 FB 1 κ 3 FC 2 κ  Fκ =(1− r)24 rF =14(1− r2r)2 (A^++A^+L)2. (16)Note that p3 characterizes the strength of the helicity-zero part relative to the helicity-onecomponent.The kinematics of the WA102 experiment is such that in average r  0.6, i.e. close to unity.We make the following observations. Helicity-one dominance in all cases for integrated cross sections. This is seen in the WA102data [3]. It is amusing that a dominant helicity-one component was observed already atthe ISR 15 years ago [9]. This was regarded as an \unusual feature" and, to the best ofour knowledge, has remained a puzzle until now. We predict an enhanced helicity-zeropart for φ around pi/2 and asymmetric t values (implying small F ). Vanishing cross sections for kT ! 0 in all cases, also in agreement with data [3, 10] (recallthat kT ! 0 implies r ! 1). A strong t1 t2 suppression for model A and an even stronger one for model C, which isnot observed in the data. There are, however, indications [11] that data do not simplyfollow an eb t distribution but exhibit a weak turn-over at small t, precisely as is predictedin model B. Hence we favour model B. Owing to the smallness of the helicity-zero part (p3) we expect a dominant sin2(φ/2)distribution for model A. Indeed, since r is close to unity, this is also the dominantbehaviour for models B and C, modulated by an additional isotropic term.Figure 2 shows a comparsion with the f1(1420) data from WA102 exhibiting the dominanceof the sin2(φ/2) term. An excellent description is described in both model B, or in C, with asmall isotropic term (p2 = 0.15) and a very small helicity-zero contribution (p3 = 0.005). Ifwe x p2 to its theoretical value using h−t1i = 0.145 GeV2 and h−t2i = 0.240 GeV2, we candetermine the mass scale µ by taking A^++/A^+L = 1. We nd µ  0.34 GeV for f1(1285) andµ  0.40 GeV for f1(1420) based on the t values p3 = 0.27 and 0.11, respectively.To test our prediction further, we propose that experimental data on the φ distributionsshould be analyzed for various r bins. Lowering r from 1 to 0 we predict a change from sin2(φ/2)to a distribution essentially flat in φ, see Figure 3.We now turn to scalars and tensors; these are interesting as both potential glueballs as wellas established quarkonia are known to arise. Further it is known that, both in the scalar andtensor sector, there are some mesons that are suppressed at low kT and others that do not showthis feature [3]. Therefore it is interesting to separate the Pomeron from the meson dynamics.4φdσ/dφ020040060080010001200140016000 20 40 60 80 100 120 140 160 180Figure 2: The φ distribution of f1(1420) production at WA102[3] tted to (15). Solid line:the form of models B and C with p2 and p3 as free parameters (t values are p2 = 0.15 andp3 = 5  10−3); Dashed line: ditto but with p2 = κ = 0.03 xed to the theoretical value (16)for r = 0.78 (t value is p3 = 0.11); Dotted line: the form of model A, i.e. p2 = 0, with p3 asfree parameter (t value is p3 = 0.15).5φ(180/pi) 1/σ dσ/dφ00.10.20.30.40.50.60 20 40 60 80 100 120 140 160 180Figure 3: Predictions for the normalized φ distribution of 1+ production for various values oft2/t1: 0.5 (solid), 0.1 (dashed), 0.05 (dotted), 0.01 (dashed-dotted). The helicity-zero contri-bution has been neglected.6(iii) JP=0+Obviously σ2 = σ1 = 0 but σ0 = (ALL − ξ1 A++ cos φ)2 depends on the model:dσ  µ2 t1 t2 A^2++ (R− cos φ)2 model Aµ6 A^2LL(1− Rcos φ)2model Bµ2 t1 t2 A2++ (R − cos φ)2 model C . (17)For all cases,R = ξ1A^LLA^++(18)is, in general2, a number with absolute value of order one, jRj = O(1). Note that the sign of Rcannot be xed from rst principles.Experimentally the ti distributions seem to continue to grow at small jtij [12] indicating thatnon-conserving parts must be present in the cross section. In model B we have the possibilityto compensate the factor t1 t2 in (1) through the 1p−ti enhanced longitudinal amplitude.Focussing now on model B we note that the φ dependence is very sensitive to the ratioδ  /R = pt1 t2/(R µ2). The φ distribution changes from isotropic at small δ to cos2 φ atlarge δ. Data thus allow the determination of the size of µ. The interesting regime is whenjδj  1. If, as suggested by the 1+ analysis, µ is of the order of QCD, the constituent-quarkmass, the average kT, or the inverse instanton size, then this would occur for the typical ti  0.2of WA102. In such a case, depending on whether δ = 1, one expects cos4(φ/2) or sin4(φ/2),see Figure 4.Data on f0(980) and f0(1500) show no suppression at small kT [3]. This implies δ is negativeand we predict a dominant cos4(φ/2) dependence. Conversely, the f0(2000), which is observedto vanish as kT ! 0 [3], will have δ positive and hence be maximum as φ ! pi. To furtherprobe the dynamics of scalar production we propose that the φ distributions of the data beanalyzed for various bins ofpt1 t2, see Figure 4.(iv) JP=2+The cross section in the various models is given bydσB  µ6 A^2LL(1− Rcos φ)2+k2Tµ2(A^+LA^LL)2+122(A^+−A^LL)2 (19)dσA  µ2 t1 t2{A^2++ (R− cos φ)2 + A^2+L 4 sin2φ2+12A^2+−}(20)dσC  µ2 t1 t2{A^2++ ( R− cos φ)2 +k2Tµ2A^2+L +12A^2+−}. (21)The three terms for each case correspond to meson helicity zero, one, and two, respectively.Here we have again made use of Bose symmetry to x the sign in σ1.In CVC, the residual amplitude A^+− is naturally of order one; hence a large helicity-twocomponent is expected for quarkonium states. The central-production data [3] do not agree withthis even though helicity-two dominance is well established and understood for e+e− [13, 14].This marked dierence was commented on in our earlier paper and is another motivation fornon-CVC dynamics.2See footnote 1.7φ(180/pi) 1/σ dσ/dφ00.10.20.30.40.50.60.70.80.90 20 40 60 80 100 120 140 160 180Figure 4: Predictions for the normalized φ distribution of 0+ production in model B, dσ/dφ (1 − δ cos φ)2, for various values of δ = /R: −10 (solid), −1 (dashed), 0 (solid), +1 (dotted),+10 (dashed-dotted).As for scalar-meson production, experimental hints [12] for a continuous growth of the tdistributions at small jtj favour the current non-conserving alternative B. Let us see whetherwe can nd a consistent picture.Well-established quarkonia are known to be suppressed at small kT [3]. Referring to (19)this implies that2 (A^+−/A^LL)2  1 and δ = /R  +1 . (22)If this was the case we would predict a helicity hierarchy, namely σ0  σ1  σ2. This looksconsistent since preliminary WA102 data on f2(1270) and f02(1525) [15] seem to support thisordering. Moreover, the condition jδj  1 is also what we already found from our analysis ofscalars.If δ  +1 (in accord with the suppression as kT ! 0) then it is sensible to expand in powersof sin2(φ/2)dσB  4 µ6 A^2LL{p2 sin4 φ2+ p1 sin2 φ2+ p0}, (23)wherep2 = δ2p1 = (1− δ) + (A^+LA^LL)28p0 = (1− δ)2 + 4(1− r)2r(A^+LA^LL)2+28(A^+A^LL)2. (24)We see that helicity zero contributes to all pi, helicity one to p1 and p0, and helicity two to p0only. Hence we predict that both the helicity structure and the φ dependence should vary witht.If, as has been suggested elsewhere [1], 2+ glueballs survive as kT ! 0, then we wouldexpect a φ distribution more similar to f0(980, 1500) than to f2(1270).(v) JP=2−The cross section in model B is given bydσ[2−]  t1 t2 µ2A^2++ sin2 φ + µ4 p2TA^2+L +t1 t2 (t1 − t2)22 µ2A^2+− , (25)where the three terms on the rhs correspond to helicity zero, one, and two, respectively. Theexpression for models A and C are given by replacing the helicity-one term by (12) with thesubstitutions kT 7! pT and sin(φ/2) 7! cos(φ/2). We can write (25) asdσ[2−]   µ6{ A^2++ sin2 φ + 4 A^2+L[cos2φ2+(1− r)24 r]+22(1− r2r)2A^2+− . (26)Experimentally the 2− states are known to be suppressed at small kT [3]. This implies that thereduced helicity-one amplitude A^+L(kT ! 0) is suppressed. This happens in the non-relativisticquark model coupling to two photons, where A^+L is identically zero. If this is also true in theQCD case, then we predict: Cross sections that behave as t eb t at small t, similar to the 0− ones. A small helicity-one contribution. A helicity-zero contribution that behaves as sin2 φ. A helicity-two contribution that is isotropic in φ. A ratio σ2/σ0 that vanishes for t2 = t1 and increases with increasing dierence jt2 − t1j.In summary, we eagerly await new results on the φ and t dependences for central mesonproduction enabling the parameters, as described in this paper, to be determined. Once theseparameters are determined, the dynamical nature of the mesons will become clear.Acknowledgements:We thank A. Donnachie, A. Kaidalov, A. Kirk, N. Kochelev and P. Landsho for discussions.References[1] F.E. Close and A. Kirk, Phys. Lett. B397 (1997) 333.9[2] D. Robson, Nucl. Phys. B130 (1977) 328;F.E. Close, Rept. Prog. Phys. 51 (1988) 833.[3] WA102 Collaboration (A. Kirk et al.), hep-ph/9810221;WA102 Collaboration (D. Barberis et al.), Phys. Lett. B440 (1998) 225, Phys. Lett. B432(1998) 436, Phys. Lett. B427 (1998) 398, Phys. Lett. B397 (1997) 339.[4] F.E. Close and G.A. Schuler, hep-ph/9902243, Physics Letters B in press.[5] T. Arens, O. Nachtmann, M. Diehl, P.V. Landsho, Z. Phys. C74 (1997) 651.[6] A.B. Kaidalov (private communications).[7] P. Castoldi, R. Escribano and J.M. Frere, Phys. Lett. B425 (1998) 359.[8] N.I. Kochelev, hep-ph/9902203;N.I. Kochelev, T. Morii and A.V. Vinnikov, hep-ph/9903279.[9] P. Chauvat et al., Phys. Lett. B148 (1984) 382.[10] F.E. Close, Phys. Lett. B419 (1998) 387.[11] A. Kirk (private communications).[12] WA76 Collaboration (T.A. Armstrong et al.), Z. Phys. C51 (1991) 351.[13] See, for example, M. Poppe, Int. J. Mod. Phys. A1 (1986) 612.[14] F.E. Close, Nucl. Phys. B (proc. suppl.) 21 (1991) 423;T. Barns, F.E. Close and Z. Li, Phys. Rev. D43 (1991) 2161;G.A. Schuler, F.A. Berends and R. van Gulik, Nucl. Phys. B523 (1998) 423.[15] WA102 Collaboration (D. Barberis et al.), hep-ph/9903042, hep-ph/9903043, to be pub-lished in Phys. Lett. B.10

Evidence that the Pomeron transforms as a non-conserved vector current

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