Vector Meson Photoproduction with a Linearly Polarized Beam

JLAB-THY-18-2650

V. Mathieu,1, ∗ J. Nys,1, 2, 3, 4 C. Fernández-Ram´ırez,5 A. Jackura,3, 4 A. Pilloni,1 N. Sherrill,3, 4 A. P. Szczepaniak,1, 3, 4 and G. Fox6 (Joint Physics Analysis Center) 1Theory Center, Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA 2Department of Physics and Astronomy, Ghent University, Belgium 3Center for Exploration of Energy and Matter, Indiana University, Bloomington, IN 47403, USA 4Physics Department, Indiana University, Bloomington, IN 47405, USA 5Instituto de Ciencias Nucleares, Universidad Nacional Autónomade México, Ciudad de México 04510, Mexico 6School of Informatics, Computing, and Engineering, Indiana University, Bloomington, IN 47405, USA We propose a model based on Regge theory to describe photoproduction of light vector mesons. We fit the SLAC data and make predictions for the energy and momentum transfer dependence of 0 the spin-density matrix elements in photoproduction of ω, ρ and φ mesons at Eγ ∼ 8.5 GeV, which are soon to be measured at Jefferson Lab.

I. INTRODUCTION independently from the target, and conservation of parity reduces the number of helicity components at With the recent development of the 12 GeV electron each vertex. In the center-of-mass frame, the net helicity beam at Jefferson Lab (JLab) [1, 2], new precision mea- transfer between the vector meson and photon |λγ − λV | surements of light meson photoproduction and electro- can be 0, 1 or 2, which we refer to as helicity conserving, production are expected in the near future. These will single and double helicity flip respectively. Measurement provide constraints on resonance production dynamics, of the photon spin-density matrix elements (SDMEs) including production of gluonic excitations. For example, can be used to determine the relative strength of these the GlueX measurement of the photon beam asymme- components. try in the production of π0 and η mesons [3] established the dominance of natural-parity t-channel exchanges for production in the forward direction [4]. This measurement seems to contradict earlier SLAC data [5] that suggests significant contribution from unnatural-parity exchanges. It was shown in [6] that the weak energy dependence of the axial-vector contributions suggested by the SLAC data is difficult to reconcile with predictions from Regge theory, while the GlueX data seem to be more in line with theory predictions. The GlueX measurement, however, was performed at fixed photon energy. Never- theless, more data from both GlueX and CLAS12 will FIG. 1. Schematic representation of the factorized amplitude be needed to refine our understanding of the production of a Regge exchange E in Eq. (3). The photon and nucleon mechanisms. vertices are denoted by T E and BE , respectively. The Regge We consider the reaction γ(k, λγ )N(p, λ) → propagator of the exchange E is RE . 0 0 0 V (q, λV )N (p , λ ). At high energies, the amplitude in the forward direction is dominated by exchange of Spin-density matrix elements can be reconstructed arXiv:1802.09403v1 [hep-ph] 26 Feb 2018 Regge poles (Reggeons). As illustrated in Fig. 1, the Reggeon amplitude factorizes into a product of two from the angular distributions of the vector meson de- vertices. The upper vertex describes the beam (photon) cay products [9]. The first measurements of neutral vec- interactions, and the lower vertex describes the target tor meson SDMEs were performed at SLAC [10], resulting in the following qualitative conclusions: the natural (proton) interactions. The Mandelstam variables are 0 s = (k + p)2 and t = (k − q)2. Factorization of Regge exchanges contributing to ρ , ω and φ production are predominantly helicity conserving, and the unnatural- vertices follows from unitarity in the t-channel, where 0 Regge pole is a common pole in all partial waves related parity contributions are negligible for ρ production and by unitarity and its vertices determine residues of the consistent with a one-pion exchange for ω production. poles [7, 8]. Factorization of residues enables one to In this paper, we discuss the SLAC data in the context determine the helicity structure at the photon vertex of a Regge-pole exchange model, which allows us to as- sess contributions of individual exchanges to the SDMEs. Various models have been proposed in the past [11–20], with different descriptions of the momentum-transfer de- ∗ [email protected] pendence of the helicity amplitudes. In general these 2

exchanges can be characterized by the quantum num- 0.2 U 0.2 U 0.2 U bers of the lowest spin meson on the trajectory, namely Ρ00 GJ Re Ρ10 GJ Ρ1-1 GJ à J 0.1 0.1 0.1 isospin I, naturality η = P (−1) (with the parity P ), æ à æ à à à à æ à à J à à à à à à à à signature τ = (−1) , charge conjugation C and G-parity 0.0 0.0 à à æ 0.0 à æ æ æ æ à à G = C(−1)I . The leading trajectories contributing to -0.1 -0.1 -0.1 æ Γp®Ωp vector meson photoproduction are -0.2 Γp®-Ρ0.2p -0.2

0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 Gητ PC Gητ PC Gητ PC -t HGeV2L -t HGeV2L -t HGeV2L I J I J I J −++ ++ −−+ −+ −−− ++ a2 : 1 2 π : 1 0 a1 : 1 1 0 +++ ++ +−+ −+ +−− ++ FIG. 2. Unnatural components of ω and ρ SDMEs at Eγ = f2 : 0 2 η : 0 0 f1 : 0 1 (1) 9.3 GeV. The dashed lines are the theoretical expectation for U U U a pseudoscalar exchange, ρ00 = ρ10 = ρ1−1 = 0. Data are In addition to the exchanges in Eq. (1), we also con- taken from Ref. [10]. sider the natural-parity Pomeron exchange, which domi- nates at high energies. In the ω photoproduction model from [20], a scalar exchange representing a σ meson tra- models reproduce the differential cross sections, but lack jectory was also considered. Since the σ meson trajectory a detailed discussion of the implication of the Regge pole is below the (leading) f2 trajectory, we do not include model for the SDMEs. it here. Among all unnatural exchanges, the π and η The paper is organized as follows. In Section II, we trajectories are expected to dominate, since they are the define the Regge amplitudes and discuss model param- closest to the scattering region. One can verify this by ex- U eters. In Section III, we discuss the fitting procedure. amining the SDMEs ρλ0λ, which in the Gottfried-Jackson Specifically, we first isolate the unnatural exchanges in (GJ) frame are determined by the unnatural exchanges ρ0 and ω production. We find that, within uncertain- (see Appendix A). The GJ frame is equivalent to the t- ties, these components are consistent with π and η ex- channel helicity frame where parity conservation implies changes so we neglect sub-leading trajectories. We de- a relation between helicity amplitudes and the natural- termine the residues of the dominant, natural exchanges ity of the exchanges. Inspecting the SLAC data [10], U U by the γp and γd total cross sections. Using the SLAC one finds that the matrix elements ρ00|GJ , Re ρ10|GJ and U data, the single and double helicity flip couplings are fit- ρ1−1|GJ for both ω and ρ production are all consistent ted to the three natural components of the SDMEs at with zero. Moreover, the unnatural component of the the laboratory frame (target rest frame) photon beam differential cross section is compatible with a π-exchange energy of Eγ = 9.3 GeV. The model is extrapolated to model [10]. Hence, we assume that the unnatural com- Eγ = 2.8 GeV and 4.7 GeV and compared to the three ponents of the SDMEs are dominated by either π or η natural components of the SLAC SDMEs at these ener- exchange. The η exchange is introduced to describe the gies. In Section IV, we compare the model to the nine ω SDMEs in production of the φ meson, while its contri- and ρ0 SDMEs obtained with a polarized beam at SLAC bution is negligible in ω and ρ0 production. As we will with Eγ = 9.3 GeV, to the nine φ SDMEs from LEPS [21] see in Sec. II, the normalization of these exchanges can and Omega-Photon [22], and to the three ω SDMEs ob- be determined by vector meson radiative decays. Re- tained with a unpolarized beam from CLAS [23, 24], garding axial vector exchanges, since the decay widths of 1 LAMP2 [25] and Cornell [26]. Furthermore, we test the f1, a1 → γV are not known, their contribution is diffi- Pomeron normalization for the ω and ρ0 differential cross cult to evaluate. Within a specific quark model [20], the sections at Eγ > 50 GeV, and the Regge exchange nor- contribution of the f1 to ω photoproduction is found to malization for the ω, ρ0 and φ differential cross sections at be negligible. As we will show, it is possible to saturate Eγ = 9.3 GeV from Ref. [10]. Lastly, we provide the pre- the unnatural components of the SDMEs by pseudoscalar dictions for the upcoming ω, ρ0 and φ SDMEs measure- exchanges. We therefore neglect the axial vector trajecto- ments in JLab experiments. In Section V, we summarize ries. In summary, we consider the s-channel amplitudes our findings and give conclusions. Details regarding the in the form relations between the frames (helicity, Gottfried-Jackson, X E s- and t-channel frames) are summarized in Appendix A, MλV ,λγ (s, t) = MλV ,λγ (s, t), (2) 0 the definition of the SDMEs are detailed in Appendix B, λ ,λ E λ0,λ and further details on the amplitude parametrization are given in Appendix C. where the sum extends over the following t-channel reggeons: E = π, η, P, f2, a2. From the s-channel helicity amplitudes in Eq. (2), one can compute the SDMEs in the II. REGGE MODEL FOR VECTOR MESON helicity or GJ frame using Eqs. (A1), and (B1), respec- PHOTOPRODUCTION

At high energies, vector meson photoproduction is 1 dominated by Pomeron and Regge exchanges. Regge The only exception is for Γ(f1 → γφ) ∼ 18 keV. 3

0 2 0 −2 tively. Assuming a factorized form for each exchange, is αU (t) = αU (t − mπ) with αU = 0.7 GeV . The E E E E parameters for the unnatural exchanges are summarized Mλ ,λ (s, t) = Tλ λ (t)R (s, t)Bλ0λ(t), (3) U V γ V γ in Table I. The photon vertex Tλ λ involves all pos- λ0,λ V γ sible helicity structures,√ with each unit of helicity flip where the top and bottom vertices T E and BE describe contributing a factor of −t. Because of charge con- the helicity transfer from the photon to the vector meson jugation, there is only one helicity structure at the the and between the nucleon target and recoil, respectively. nucleon vertex,√ the helicity flip, which corresponds to the According to Regge theory [27], the energy dependence factor δλ,−λ0 −t/2mp. factorizes into a power-law dependence sαE (t). The phase of the amplitude is determined by the signature factor TABLE I. Model parameters for the unatural exchanges. The −iπαE (t) E 1 + e , which is contained in R , 0 −2 parameters αU are expressed in GeV . 1 + e−iπαU (t) U π η RU (s, t) = sˆαU (t) U = π, η (4a) sin παU (t) U β0,ω 3.11 0.36 −iπαN (t) N αN (t) 1 + e α (t) U R (s, t) = sˆ N N = , f , a . β0,ρ 1.11 0.10 α (0) sin πα (t) P 2 2 N N βU 0.30 0.27 (4b) 0,φ αU (0) −0.013 −0.013 We defineds ˆ = s/s0 with the scale chosen as s0 = 0 2 0 αU 0.7 0.7 1 GeV . We use a linear trajectory αE(t) = αE(0) + αEt for all exchanges. The signature factor eliminates contributions from spin-odd poles induced by the denominator sin πα (t). The factor α (t)/α (0) simply removes the E E E B. Natural exchanges unphysical pole at αE(t) = 0 that arises in the scattering region for the f2 and a2 exchanges. For consistency, we also include this factor for the Pomeron exchange, The trajectories of the natural exchanges are known and we use [27, 28] although the point αP(t) = 0 is far from the region of 2 interest −t ≤ 1 GeV . For the pseudoscalar exchanges, α (t) = 1.08 + 0.2 t/GeV2, (6a) the pole at α (t) = 0 is physical. P π,η 2 αf2,a2 (t) = 0.5 + 0.9 t/GeV . (6b) For natural exchanges, N = , f , a . The top vertex in- A. Unnatural exchanges P 2 2 volves three helicity components: a helicity nonflip, single flip and double flip. As for unnatural exchanges, each For unnatural exchanges U = π, η, the helicity struc- of these comes with an appropriate power of the factor ture of the photon vertex T U and the nucleon vertex √ −t/mV , BU can be obtained by comparison with the high-energy N N bN t limit of a single-particle exchange model. We obtain (see Tλ λ (t) = βγV e V γ √ Appendix C), −t λ −t × δ + βN √γ δ + βN δ . U λV ,λγ 1 m λV ,0 2 m2 λV ,−λγ Tλ λ (t) = V 2 V V γ √ √ (7) U −t −t βγV λγ δλV ,λγ − 2 δλV ,0 + 2 λγ δλV ,−λγ , mV m To be consistent with factorization, and to reduce the V N (5a) number of parameters, we assume that the couplings β1 √ N and β2 are the same for all vector mesons. The steep U U −t falloff of the forward differential cross section is well de- Bλ0λ(t) = βpp δλ,−λ0 , (5b) 2mp scribed by exponential factors, gamma functions [18, 28] or dipole form factors [17, 19, 20, 29, 30]. All of these with mV and mp being the vector meson and nucleon U U models can be approximated by an exponential func- masses, respectively. The residues β and βpp are de- γV tion of the form ebN t [12, 14–16]. We obtain b = 3.6 termined from the radiative decay widths Γ(V → γπ), P GeV−2 by approximating the form factors from [29], and Γ(V → γη) and the nucleon couplings g , g , re- πpp ηpp b = 0.53 GeV−2 and b = 0.55 GeV−2 by approxi- spectively. The overall nonflip couplings of the reaction a2 f2 mating the t-dependence of the a and f poles with a are written βU = βU βU .2 The details of the calcula- 2 2 0,V γV pp Breit-Wigner line shape as described in Appendix C. For tion are given in Appendix C. The unnatural trajectory the nucleon vertex we include the two possible helicity combinations, a nonflip and single flip, √ 2 The index 0 stands for the helicity difference at the top vertex, N N −t Bλ0λ(t) = βpp δλ,λ0 + 2λκN δλ,−λ0 . (8) |λγ − λV | = 0. 2mp 4

The SDMEs probe the helicity structure of the photon width [10]. For consistency, we will use the γφ value vertex. They are weakly dependent on the helicities at obtained from the leptonic decay width, but we keep an the nucleon vertex. On the contrary, The helicity flip eye on this discrepancy when comparing to the data. couplings κN thus play a minor role in our analysis. Assuming that the Pomeron has a gluonic nature and Moreover isoscalar exchanges, e.g., the f2 and Pomeron, therefore has couplings which are independent of the are empirically helicity nonflip at the nucleon vertex [28]. quark flavor [34], we derive the relation between the to-

Therefore, we set κf2 = κP = 0. The isovector exchanges tal cross section couplings in Eq. (9) and the overall nor- P are empirically helicity flip dominant. We model this malization of the Pomeron β0,V in our model for vector feature by using κa2 = 8.0 [28]. meson photoproduction, The special nature of the Pomeron prevents us from P P P !−1 computing its overall normalization β0,V = βγV βpp by e e2 e2 e2 using radiative decays. We thus determine the normal- βP = βγγ × + + . (11a) 0,V P 2 2 2 N N N γV γρ γω γφ ization β0,V = βγV βpp by fitting the γp and γd total cross sections and invoking vector meson dominance (VMD). N We note that by increasing γφ by a factor of two, the ω We first relate the overall normalizations β0,V to the γp 0 and γd total cross section. Using the optical theorem, and ρ couplings of the Pomeron would change by only our Regge parametrization in (3) leads to 10%. For the Regge exchanges, we assume ideal mixing for vector and tensors mesons and extract the remaining couplings using vector meson dominance: 1 γγ α (0) γγ α (0) γγ α (0) σ(γp) = β sˆ P + β sˆ f2 + β sˆ a2 , P f2 a2 2mpEγ 2 2 −1 f2 γγ e e e 1 γγ α (0) γγ α (0) β = β × + , (11b) σ(γd) = 2β sˆ P + 2β sˆ f2 . (9) 0,ω/ρ f2 2 2 f2 γω/ρ γρ γω 2mpEγ P γω/ρ βa2 = βγγ , (11c) γγ 0,ω/ρ f2 2e The factors βN represent couplings of the natural ex- f2 a2 change N in the forward scattering direction γp → γp. βγφ = βγφ = 0. (11d) We need to relate these factors, via VMD, to the factors βN appearing in vector meson photoproduction. In or- N 0,V We choose to determine the helicity couplings β1 and der to use VMD, we use the following interaction between N β2 through a fit to the SLAC data. Since our formal- photon field Aµ and the vector meson fields [31–33]: ism is based on a high-energy expansion, we determine the parameters only with the highest energy bin. Specif- 2 2 2 ! µ mρ mω mφ ically, we inspect the natural components of the SDMEs L = −eA ρµ + ωµ + φµ . (10) γρ γω γφ at Eγ = 9.3 GeV. Assuming only one natural exchange N, our form in Eq. (7) for the top vertex leads to From this interaction,3 and neglecting the electron mass, + − N 2 one finds for the electronic decay width Γ(V → e e ) = N β1 −t 2 2 ρ00(s, t) = 2 , (12a) mV (α /3)(4π/γV ), which determines the couplings γV A(t) m √ V that we tabulate in Table II. The SU(3) quark model N N β1 −t N −t Re ρ10(s, t) = 1 + β2 2 , (12b) 2A(t) mV mV TABLE II. Vector meson dominance parameters. N 2 N β2 −t + − 2 ρ (s, t) = , (12c) V Γ(V → e e ) 4π/γV 1−1 2 A(t) mV ρ0 7.04(6) keV 0.506(4) N 2 2 N 2 2 4 ω 0.60(2) keV 0.044(1) with A(t) = 1− β1 t/mV + β2 t /mV . The factorization hypothesis in Eq. (3) and the conservation of an- φ 1.26(1) keV 0.070(1) gular momentum implies the vanishing of these SDMEs √ in the forward direction. This is indeed observed in all 0 N predictions γω/γρ = 3 and γω/γφ = − 2 compare well of the ρ SDMEs, but is inconsistent with the ρ1−1 el- with the VMD predictions, γω/γρ = 3.4(6) and γω/γφ = ements for ω photoproduction as seen in Fig. 3. The −1.3(1). However, it is well known that the φ meson expressions in Eq. (12) also tell us that we should expect N N differential cross section produces a value of γφ that is |ρ00| < | Re ρ10| for small t. Again, this relation is satis- twice as large as the one obtained from the leptonic decay fied for ρ0 photoproduction but seems to be violated for ω N photoproduction. The element ρ00 is significantly larger for ω photoproduction compared to ρ0 photoproduction, suggesting a larger single-helicity flip for the isovector 3 The γV couplings can be cast in terms of the vector meson exchange. The deviation from zero observed in the ele- P V N N 0 decay constants h0| q=u,d,s eqqγ¯ µq(0)|V (, P )i = fV µ (P ) = ments Re ρ and ρ for ρ photoproduction suggests 2 V 10 1−1 (mV /γV )µ (P ). a nonzero single and double helicity flip for the isoscalar 5

for vector meson photoproduction are summarized in Ta- TABLE III. Model parameters for the natural exchanges. 0 −2 ble III. The parameters bN and αN are expressed in GeV . The N β{0,1,2},V parameters are calculated using the ﬁt discussed in Section III; the other parameters are estimated or discussed 0.2 N 0.2 N 0.2 N Ρ00 H Re Ρ10 H Ρ1-1 H in Section II. æ æ 0.1 0.1 0.1 æ æ æ æ æ N f2 a2 æ æ æ æ P æ æ æ æ æ æ æ 0.0 æ 0.0 æ æ 0.0 æ æ æ æ æ æ æ N æ β 0.739(1) 0.730(10) 1.256(85) -0.1 -0.1 -0.1 0,ω æ Γp®Ωp N -0.2 EΓ = 2.8 GeV Γp®Ρp-0.2 -0.2 β0,ρ 2.506(5) 2.476(34) 0.370(25) 0.0 0.2 0.4 0.6 0.80.0 0.2 0.4 0.6 0.80.0 0.2 0.4 0.6 0.8 0.2 0.2 0.2 N æ β 0.932(2) 0 0 æ æ 0,φ æ æ 0.1 0.1 0.1 æ æ æ æ æ N æ æ æ æ æ æ β 0 0.95(19) 0.83(34) 0.0 æ 0.0 æ 0.0 æ æ 1 æ æ æ æ æ æ æ æ æ æ N -0.1 -0.1 -0.1 β2 0 −0.56(17) 0 -0.2 EΓ = 4.7 GeV -0.2 -0.2 κN 0 0 8.0 0.0 0.2 0.4 0.6 0.80.0 0.2 0.4 0.6 0.80.0 0.2 0.4 0.6 0.8 0.2 0.2 0.2

à æ b 3.60 0.55 0.53 0.1 æ 0.1 à 0.1 æ N æ æ à æ à à à à à à æ 0.0 à à 0.0 à 0.0 à æ à à à à æ à à αN (0) 1.08 0.5 0.5 à

-0.1 -0.1 -0.1 à α0 0.2 0.9 0.9 N -0.2 EΓ = 9.3 GeV -0.2 -0.2 0.0 0.2 0.4 0.6 0.80.0 0.2 0.4 0.6 0.80.0 0.2 0.4 0.6 0.8 -t HGeV2L -t HGeV2L -t HGeV2L exchanges. We associate these couplings with the f2 exchange and keep the Pomeron helicity conserving as is FIG. 3. Natural components of ω and ρ0 photoproduction often assumed. This hypothesis could be checked with φ SDMEs at Eγ = 2.8, 4.7 and 9.3 GeV. The lines are our model, photoproduction as we will discuss later. According to determined by the 9.3 GeV data only and extrapolated to P P a2 our discussion we impose β1 = β2 = β2 = 0 and thus lower energies. The dashed points are not included in the f1 f2 a2 need to ﬁt the helicity couplings β1 , β1 , β1 . ﬁtting procedure. The data are taken from Ref. [10].

III. FITTING PROCEDURE 0.4

We determine the six couplings βγγ , βγγ , βγγ , βf1 , βf2 , P f2 a2 1 1 a2 0.3 β1 using a combined ﬁt of the γp and γd total cross sections from the Review of Particle Physics [35] for Eγ > 2 GeV, the three ρ0 natural exchange SDMEs (ρN , Re ρN 00 10 L N N

mb 0.2 and ρ1−1) and the element ρ00 for ω photoproduction at H Eγ = 9.3 GeV obtained at SLAC [10]. We do not in- Σ clude the two other natural components of the SDMEs in ω photoproduction as they are inconsistent with our 0.1 working hypothesis. The fit of the total cross sections and the fit of the SLAC SDMEs are combined in a single fit. There are 308 (total cross sections) plus 24 (SDMEs) 0.0 data points and six fit parameters. The other model pa- 1 2 3 4 5 Log p GeV rameters (bN , κN , γV and the π- and η-exchange cou- 10 lab plings) are kept fixed at values discussed in the previous section. The expressions for the natural components of FIG. 4. Total cross section σ(γp) (blue) and σ(γd) (red). The the SDMEs used in the fit is given in Eqs (B1) and (B4). 2 black lines are the results of our fit (the thickness of the lines The fit results in the reduced χ /d.o.f. of 1.96 (1.84 for represent the error band). The data are taken from Ref. [35]. the total cross sections and 0.12 for the SDMEs), and the fitted parameters are

γγ f2 β = 0.187(1) β1 = 0.95(19) (13a) P IV. COMPARISON WITH DATA βγγ = 0.164(2) βf2 = −0.56(17) (13b) f2 2 βγγ = 0.045(3) βa2 = 0.83(34). (13c) 0 f2 1 As we discussed above, the SDMEs for ρ photoproduction are more consistent with our model for diﬀrac- The photon couplings are extracted from Eqs (11). The tive production than for ω photoproduction. This can be parameters of the exchanges calculated from Eq. (13) observed in Fig. 3. The bands on the ﬁgures represent 6 one standard deviation from our model. The wider band 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.4 0.4 0.4 Ρ10 H 0.4 0.4 0.4 æ æ æ æ æ in the ω model originates from the stronger dominance æ LAMP2 EΓ=2.8-3.5 æ 0 æ Ρ1-1 H æ SLAC EΓ=2.8 æ æ 0.2 æ 0.2 0.2 0.2 0.2 0.2 æ CLAS EΓ=2.8 æ of the Regge exchanges, whose normalizations are less æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ 0.0 æ 0.0 0.0 0.0 0.0æ 0.0 æ æ æ æ constrained by the total cross sections. The Pomeron æ æ æ æ æ æ æ æ Ρ0 æ æ æ -0.2 00 H -0.2 -0.2 -0.2 -0.2 -0.2 normalization is indeed more constrained and yields a æ 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.40 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 Ρ10 H smaller uncertainty in the ρ model. We have also in- 0 0.4 0.4 0.4 æ æLAMP2 E =3.5-4.20.4 æ0.4 Ρ 0.4 æ Γ æ 1-1 H æ CLAS EΓ=3.81 cluded the data at E = 4.7 and 2.8 GeV from SLAC æ γ æ 0.2 0.2 0.2 0.2 0.2 æ æ 0.2 ææ æ æ æ in Fig. 3. They compare well to our model evaluated at æ æ æ æ æ æ æ æ æ æ æ æ 0.0 æ æ 0.0 æ0.0 0.0 0.0æ æ æ 0.0 æ æ æ æ æ 0 æ these lower energies. Ρ00 H æ æ æ -0.2 0 0.1 0.2 0.3 0.4 0.5 0.6-0.2 -0.2 -0.2 0-0.2 0.1 0.2 0.3 0.4 0.5 0.6 -0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.20 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.6 0.6 0.6 Ρ10 H 0.6 0.6 0 0 Ρ00 H LAMP2 EΓ=4.2-4.8 Ρ1-1 H 0.4 0.4 0.4 0.4 0.4 SLAC EΓ=4.7

0 0 0.2 æ æ 0.2 0.2 0.2 0.2 0.2 à 0.2 0.2 æ Re Ρ Ρ æ H H æ æ æ 10 à 1-1 æ æ æ æ æ æ æ ææ æ ææ æ æ 0.0 æ æ 0.0 0.0 æ æ 0.0 æ æ 0.0 æ æ æ æ 0.1 0.1 æ 0.1 æ æ æ æ æ æ à -0.2 -0.2 -0.2 -0.2 æ -0.2 æ æ æ æ æ æ æ à à à æ 0.0 à æ æ 0.0 æ 0.0 -0.4 -0.4 -0.4 -0.4 -0.4 æ 0 à æ 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 æ æ à Ρ00 H æ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.1 -0.1 -0.1 æ 0 0.4 Ρ0 0.4 0.4 Ρ10 H 0.4 0.4 Ρ0 0.4 Γp®Ωp æ 00 H 1-1 H -0.2 -0.2 -0.2 Cornell EΓ=8.9 Γp®Ρp SLAC EΓ=9.3 0.2 æ 0.2 0.2 0.2 0.2 0.2 æ -0.3 -0.3 -0.3 æ æ æ æ æ æ æ æ æ æ ææ æ 0.0 æ 0.0 æ0.0æ æ æ 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.80.0 0.2 0.4 0.6 0.80.0 0.2 0.4 0.6 0.8 æ æ æ æ 0.3 0.3 0.3 æ æ æ 0.2 0.2 0.2 1 Re Ρ10 H -0.2 -0.2 -0.2 -0.2 -0.2 -0.2 à 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.1 à 0.1 0.1 æ æ 2 2 2 à æ æ æàæ -t HGeV L -t HGeV L -t HGeV L æ æ 0.0 æ 0.0 à 0.0 æ æ æ à æ æ æ æ æ æ æ -0.1 -0.1 à -0.1 à æ à æ -0.2 1 -0.2 1 -0.2 Ρ11 H Ρ00 H -0.3 -0.3 -0.3 FIG. 6. Unpolarized SDMEs for ω photoproduction. The

0.6 æ 0.6 0.6 0.0 0.2 à0.4 0.6 0.80.0 0.2 0.4 0.6 0.80.0 0.2 0.4 0.6 0.8 æ æ æ æ lines are our model. Eγ is the beam energy in the laboratory 0.4 æ 0.4 2 0.4 2 àæ Im Ρ10 H Im Ρ1-1 H à frame in GeV. The data are taken from SLAC [10], CLAS [23, 0.2 0.2 0.2 à à æ æ 0.0 0.0 àæ æ æ æ 0.0 24], LAMP2 [25] and Cornell [26]. æ 1 -0.2 Ρ -0.2 -0.2 à à 1-1 H à

-0.4 -0.4 -0.4 æ æ æ æ æ æ -0.6 -0.6 -0.6 æ 0.0 0.2 0.4 0.6 0.80.0 0.2 0.4 0.6 0.80.0 0.2 0.4 0.6 0.8 mation. Despite the significant uncertainties in all the -t HGeV2L -t HGeV2L -t HGeV2L presented data sets, we conclude that our extrapolated model describes the lower energies data sets fairly well. FIG. 5. Comparison between our model and ω and ρ0 SDMEs It is also worth noting that the data from Ref. [26] at at Eγ = 9.3 GeV. The data are taken from Ref. [10]. Eγ = 8.9 GeV are consistent with our factorization hy- 0 pothesis, i.e., ρ1−1 ∼ −t in the forward direction. We In Fig. 5, we present the comparison between the ω conclude that the SLAC data may suffer from large er- and ρ0 models and the SLAC data at 9.3 GeV for all rors. The forthcoming measurement by the GlueX col- nine SDMEs. There is a general agreement between the laboration could confirm the factorization of the vector 0 model and the data, but we wish to discuss some inconsis- meson production, i.e., ρ1−1(t) ∼ −t in the forward di- 1 2 rection at high energies. tencies. The elements in the bottom panels ρ1−1, Im ρ10 2 and Im ρ1−1 were not included in the fitting but are nev- Our model simplifies for φ photoproduction. In this ertheless well described by the model. In particular, we case we simply neglect the f2 and a2 Regge exchanges, as 1 note the dominance of the natural exchanges in ρ1−1 and they are not expected to couple to γφ if one assumes per- 2 0 0 Im ρ1−1 in the case of ρ photoproduction with small de- fect mixing. The relevant exchange would then be the f2, viation for the ω case, as expected from the stronger π the hidden strangeness partner of the f2. However, its in- 1 exchange. The main noticeable discrepancy arises in ρ11 tercept, and therefore its overall strength, is smaller due for ω photoproduction. Since the pseudoscalar exchanges its higher mass. We neglect this contribution and assume are smaller than the natural exchanges, we would expect that the only relevant natural contribution is provided by 1 0 ρ11 ∼ ρ1−1. The data does not display this feature and the Pomeron. Since our Pomeron is purely helicity con- 1 thus our model does not describe ρ11 well. Furthermore, serving, the SDMEs are very simple at high energies. The 1 1 2 since the contribution from the π exchange to ρ11 is neg- only non-zero components are ρ1−1 = − Im ρ1−1 = 1/2. 1 0 ative (see Appendix C), we would expect ρ11 < ρ1−1, This picture is consistent with the SLAC measurement at which is featured in our ω model but not in the SLAC 9.3 GeV [10]. In Fig. 7, we compare our model to the data 1 data. The sign of the element ρ11 would be an important from the Omega-Photon collaboration [22]. Their data check for our model when GlueX data becomes available. are taken in the energy range Eγ = 20 − 40 GeV. They Although our model has been constrained at Eγ = 9 are consistent with the SLAC data but have somewhat GeV, we present in Fig. 6 the comparison between our smaller uncertainties. We also extrapolated our model model and the unpolarized SDMEs at lower energies. to Eγ = 2.27 GeV to compare with the data from the The extrapolation to lower energies is in principle not LEPS collaboration [21]. At lower energies, we observe in the range of applicability of the Regge-pole approxi- deviations from pure helicity conservation, i.e., deviation 7

and unnatural component is βN /βU = 0.266. 0 0.05 0.1 0.15 0.20 0.05 0.1 0.15 0.20 0.05 0.1 0.15 0.2 0,φ 0,φ

0.4 0 0.4 0.4 0 0.4 0.4 0 0.4 Ρ00 H Re Ρ10 H Ρ1-1 H

0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.3 0.3 à à à à 0.2 0.2 0.2 à à à à 0.0 0.0 0.0 0.0 0.0à 0.0 à à à 0.1 0.1 0.1 -0.2 -0.2 -0.2 -0.2 -0.2 -0.2 0.0 0.0 0.0 -0.1 Γp®Ωp -0.1 -0.1 -0.4 0 0.05 0.1 0.15-0.40.20-0.40.05 0.1 0.15-0.40.20-0.40.05 0.1 0.15 0.2-0.4 Γp®Ρp -0.2 0 -0.2 0 -0.2 0 0 0.05 0.1 0.15 0.20 0.05 0.1 0.15 0.20 0.05 0.1 0.15 0.2 Ρ Γp®Φp Re Ρ Ρ 1 1 1 -0.3 00 H -0.3 10 H -0.3 1-1 H 0.4 Ρ H 0.4 0.4Ρ H 0.4 0.4 Re Ρ H 0.4 11 00 10 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

0.0 à 0.0 0.0à à à0.0 0.0à à à à 0.0 0.1 0.1 0.1 à à à à 0.0 0.0 0.0 -0.2 -0.2 -0.2 -0.2 -0.2 -0.2 -0.1 -0.1 -0.1 -0.2 -0.2 -0.2 -0.4 0 0.05 0.1 0.15-0.40.20-0.40.05 0.1 0.15-0.40.20-0.40.05 0.1 0.15 0.2-0.4 Ρ1 Ρ1 Re Ρ1 -0.3 11 H -0.3 00 H -0.3 10 H 0.6 0 0.05 0.1 0.15 0.60.200.6 0.05 0.1 0.15 0.60.200.6 0.05 0.1 0.15 0.20.6 0.60.0 0.2 0.4 0.6 0.80.6 0.0 0.2 0.4 0.6 0.80.6 0.0 0.2 0.4 0.6 0.8 0.4 0.4 0.4 0.4 0.4 Im Ρ2 0.4 à 1-1 H 0.4 0.4 2 0.4 2 0.2 à à à0.2 0.2 0.2 0.2 0.2 Im Ρ10 H Im Ρ1-1 H 0.2 0.2 0.2 0.0 0.0 0.0à à à à0.0 0.0 0.0 à 0.0 0.0 0.0 -0.2 -0.2 -0.2 -0.2 -0.2 à à -0.2 à 1 -0.4 1 -0.4 -0.4 2 -0.4 -0.4 -0.4 -0.2 Ρ -0.2 -0.2 Ρ1-1 H Im Ρ10 H 1-1 H -0.6 -0.6 -0.6 -0.6 -0.6 -0.6 -0.4 -0.4 -0.4 0 0.05 0.1 0.15 0.20 0.05 0.1 0.15 0.20 0.05 0.1 0.15 0.2 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 2 2 2 -t HGeV2L -t HGeV2L -t HGeV2L -t HGeV L -t HGeV L -t HGeV L

FIG. 8. The SDMEs of ω, ρ0 and φ photoproduction at E = FIG. 7. The SDMEs of φ photoproduction at E = 2.17 − γ γ 8.5 GeV, the average polarized beam energy in the laboratory 2.37 GeV (red squares) from Ref. [21] and at E = 20 − 40 γ frame. GeV (gray band) from Ref. [22] (SDMEs integrated over t represented as a band over the t range). The lines are our 0 P Our prediction for ω, ρ and φ vector meson photo- model at Eγ = 2.27 GeV (solid red with β0,φ = 0.932 and P production at GlueX is displayed in Fig. 8. We used dashed red with β0,φ = 1/2×0.932 for the Pomeron coupling) and Eγ = 30 GeV (black). Eγ = 8.5 GeV, the average beam energy with polarization. As already commented, the bulk of the uncertainties in our model come from Regge exchanges. It is therefore not surprising that the uncertainties in the φ meson 1 2 from ρ1−1 = − Im ρ1−1 = 1/2. This is triggered by un- SDMEs are very small. The bending of the curves as |t| natural exchanges. Since the π couples weakly to γφ, we increases in our φ model originate from the pseudoscalar included η exchange in our model. The very small cou- exchanges. We have not included an exponential falloff in pling gφγπ, inferred from radiative decays, cannot solely their parametrization. Therefore, their effects can be ob- explain the deviation from helicity conservation in the served away from the forward direction where the natural 1 2 elements ρ1−1 and Im ρ1−1 at Eγ = 2.27 GeV. The in- exchanges are exponentially suppressed. If the φ SDMEs clusion of η exchange increases the relative importance remain flat in a larger t range, one would just need to of unnatural exchange. We should also note that we incorporate an exponential falloff in the η exchange. considered the η degenerate with the π. With the η Our model has been designed to describe the SDMEs, pole being further from the scattering region, the factor but it is also interesting to compare it with high-energy 0 2 α π/ sin παη(t) ∼ 1/(mη −t) is not strong enough to trig- unpolarized differential cross-section data. We first com- 1 ger the depletion close to the forward direction in ρ1−1 pare our model to high-energy data in Fig. 9. At ener- 2 and Im ρ1−1. Nevertheless, the SDMEs from the LEPS gies above 50 GeV, the Regge exchanges contribute less collaboration indicate an even larger relative strength of that 1% of the differential cross section. The data there- unnatural vs. natural exchanges than in our model. As fore gives a very good indication of the validity of our we pointed out, the Pomeron coupling gγφ from the φ me- Pomeron model. We observe that the overall normaliza- P son leptonic width and VMD is overestimated. The rel- tion at t = 0 is in fairly good agreement with the data. ative strength of the unnatural exchanges in the SDMEs Our phenomenological intercept αP(0) = 0.08 produces a are thus underestimated. We illustrate the effect of re- small rise of the differential cross section in the forward ducing the Pomeron coupling by a factor of two in Fig. 7. direction. At very high energies, Eγ > 1 TeV, the data P The dashed red line, obtained with β0,φ = 1/2 × 0.932, seems to display a slower growth at t = 0, in agreement leads to a better agreement with the data. Alternatively, with the unitarity bound. However, these energies are we could have increased the coupling gηNN . As we dis- far from our region of interest. The t-dependence was cussed in Ref. [4], the η coupling to the nucleon is not approximated by a simple exponential falloff, which de- known very precisely. From the investigation of φ SDMEs scribes the falloff of the differential cross section in the 2 at Eγ = 2.27 GeV, we conclude that the ratio of natural range 0 < −t/mV . 1. We observe deviations from this 8

200

70.0 æ 100 æ ô 50.0 æ ô Γp®Ρp æ ô 30.0 æ ò ô 50 ôà æ 4.7 TeV 20.0 æ æ L L 15.0 2 ò à 2.8 TeV 2 10.0 æ à æ 20 ò 1.6 TeV 7.0 æ æ

GeV 5.0 ò ô 111.5 GeV GeV ò æ æ à b 10 b 3.0 Μ Μ æ

H ò H 2.0 ò æ 1.5 æ æ

dt 5 òà dt 1.0 æ æ Σ æ Σ d 2 d æ æ Γp®Ρp ò 1 ò ò Γp®Ωp Γp®Φp 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6 -t HGeV2L -t HGeV2L 20.0

10.0 ò æ ò 0 5.0 ò æ FIG. 10. γp → (ρ , ω, φ)p diﬀerential cross section at 9.3 GeV

æ L ò æ in solid blue, green and red lines respectively. The dashed red 2 2.0 æ

æ line is obtained with a Pomeron coupling reduced by a factor æ

GeV 1.0 ò æ

b æ two. The data are taken from Ref [10].

Μ 0.5 æ H Γp®Ωp æ dt 0.2 æ Σ d ¥ 50 - 130 GeV 0.1 æ ¥ 60 - 225 GeV V. CONCLUSIONS ¥ 2.6 - 4.3 TeV æ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 We presented a model describing the SDMEs of light -t HGeV2L vector meson photoproduction. Our model includes π and η exchanges, whose parameters are fixed. We in- corporated the leading natural exchanges: the Pomeron, 0 FIG. 9. γp → ρ p (top) and γp → ωp (bottom) differential f2 and a2 exchanges. Their normalizations were deter- cross sections at high energies. Left panel: Data from Ref [36] mined from the total cross section using the VMD hy- (green triangles), Ref [37] (black triangles), Ref [38] (red cir- pothesis. We paid special attention to the t-dependence cles) and Ref [39] (blue squares). The higher energies curves of the various exchanges. We proposed a flexible and overestimate data at low t, as expected from the saturation of intuitive ansatz for the t-dependence of each natural ex- the unitarity bound. Right panel: Data from Ref [40] (black change. The helicity structure of these exchanges was circles), Ref [41] (blue circles) and Ref [42] (red circles). then inferred from the data on photoproduction of ω and 0 ρ at Eγ = 9.3 GeV from SLAC. The joint inspection of these two reactions allowed us to assume that the f2 isoscalar exchange must have a small double helicity flip coupling, in addition to a single helicity flip coupling.

2 The a2 isovector exchange was consistent with only a simple picture at |t| > 0.3 GeV . single flip and no double helicity flip coupling. The model compares well with the nine SDMEs for ρ0, Unfortunately, our model does not compare very well ω and φ photoproduction in a wide energy range Eγ ∼ with the ω and φ differential cross sections at 9.3 GeV, 3 − 9 GeV, as well as with the unpolarized data in the 0 0 as shown in Fig. 10. Although the ρ differential cross same energy range. Except for ρ1−1 in ω production, the section is roughly in agreement with our model, the φ SDME are consistent the factorization of Regge residues. differential cross section is overestimated. We already We made predictions for the future measurements of light explained that the leptonic width of the φ meson led to meson photoproduction at JLab. Our predictions and a Pomeron coupling to γφ much stronger than the ex- our model are available online on the JPAC website [46, perimental value. This was already observed in the orig- 47]. With the online version of the model, users have the inal experimental publication [10]. It has been argued in possibility to vary the model parameters and generate Ref. [43] that the large φ mass needs to be taken into the SDMEs for ρ0, ω and φ photoproduction. The code account. The authors of Refs. [44, 45] corrected the dif- can also be downloaded. ferential cross section by the ratio of the φ and photon The differential cross section at very high energies, 2 momenta, (kφ/kγ ) ≈ 0.87 at Eγ = 9.3 GeV. This factor Eγ > 50 GeV, is well reproduced by our Pomeron ex- is nevertheless not small enough to reproduce the exper- change. However, the effect of the high-energy approx- imental normalization of the φ differential cross section. imation led to non negligible deviation in normalization As we did for the SDMEs, we reduce the Pomeron cou- from the data at Eγ = 9.3 GeV. These deviations appear P pling β0,φ by a factor of two. The resulting normalization only in the differential cross section, since they cancel in at t = 0 seems more in agreement with the data. the ratio of the SDMEs. 9

ACKNOWLEDGMENTS into its spin projection along the direction opposite to the recoil in the helicity frame. The summation over beam, We thank C. Meyer for pointing out the future mea- target and recoil helicities in the SDMEs is not aﬀected surements by the GlueX collaboration. This work was by these rotations. Hence, the SDMEs in the s-channel supported by BMBF, by the U.S. Department of Energy and helicity frames are equivalent. under grants No. DE-AC05-06OR23177 and No. DE- Similarly, a boost along the beam direction between FG02-87ER40365, by PAPIIT-DGAPA (UNAM, Mex- the t-channel and the GJ frames brings the helicity of ico) grant No. IA101717, CONACYT (Mexico) grant the vector in the t-channel to its spin projection along No. 251817, by Research Foundation – Flanders (FWO), the beam direction in the GJ frame. The helicities of the by U.S. National Science Foundation under award num- other particles undergo a rotation which does not aﬀect bers PHY-1415459 and PHY-1205019, and by Ministerio the SDMEs, as demonstrated in Ref [48]. de Econom´ıa y Competitividad (Spain) through grant Finally, from the SDMEs in the GJ frame, the SDMEs No. FPA2016-77313-P. in the helicity frame are obtained by a rotation of angle θq, the angle between the opposite direction of the recoil and the beam direction (see Fig. 11) Appendix A: Frames X 1 1 ρMM 0 |GJ = d (θq) ρλ ,λ0 |H d 0 0 (θq), M,λV V V M ,λV 0 λV ,λV (A1)

with cos θq = (β − cos θs)/(β cos θs − 1) and β = 1/2 2 2 2 2 λ (s, mp, mV )/(s−mp +mV ). The leading s expression 2 2 is simply cos θq → (mV + t)/(mV − t).

Appendix B: Spin-Density Matrix Elements

The relation between SDMEs and helicity amplitudes are well known [9]. For completeness, we provide the expressions for the nine SDMEs accessible with a linearly polarized photon beam:

0 1 X ∗ ρ00 = M 1,0 M 1,0 , (B1a) N 0 0 λ,λ0 λ,λ λ,λ 0 1 X ∗ Re ρ10 = Re M 1,1 − M1,−1 M 1,0 , 2N 0 0 0 λ,λ0 λ,λ λ,λ λ,λ FIG. 11. Illustration of the frame deﬁned in Appendix A. (B1b)

0 1 X ∗ ρ1−1 = Re M 1,1 M1,−1, (B1c) The properties of helicity amplitudes are best de- 0 0 N 0 λ,λ λ,λ scribed in two popular frames: the s-channel and the t- λ,λ channel frames. The s-channel corresponds to the center- 1 1 X ∗ ρ11 = Re M−1,1M 1,1 , (B1d) of-mass of the reaction γp → V p. The t-channel corre- N 0 0 λ,λ0 λ,λ λ,λ sponds to the center-of-mass of the reaction γV¯ → pp¯. 1 1 X ∗ These channels are illustrated on Fig. 11. ρ00 = Re M−1,0M 1,0 , (B1e) N 0 0 The angular distribution of a vector meson is analyzed λ,λ0 λ,λ λ,λ in its rest frame. In the rest frame, the beam, target and 1 2 1 X ∗ recoil form the reaction plane xz. The y-axis is deﬁned ρ1−1 + Im ρ1−1 = M−1,1M1,−1, (B1f) N 0 0 as the cross product between the target and the recoil λ,λ0 λ,λ λ,λ momenta. For the z-axis, the two common choices are 1 2 1 X ∗ ρ1−1 − Im ρ1−1 = M 1,1 M−1,−1, (B1g) the opposite direction of the recoil in the helicity frame, 0 0 N 0 λ,λ λ,λ and the beam direction in the GJ frame [48]. λ,λ The helicity amplitudes in these four frames are dif- 1 2 1 X ∗ Re ρ10 + Im ρ10 = Re M−1,1M 1,0 , (B1h) N 0 0 ferent. For instance, a boost along the recoil momentum λ,λ0 λ,λ λ,λ between the s-channel and the helicity frames rotates the 1 2 1 X ∗ helicities of the beam, target and recoil. It also trans- Re ρ10 − Im ρ10 = Re M 1,1 M−1,0. (B1i) N 0 0 forms the helicity of the vector meson in the s-channel λ,λ0 λ,λ λ,λ 10

Of course, the SDMEs and the helicity amplitudes The polarization vectors, in the s-channel, are need to be define in the same frame, or in equivalent α −λγ frames, as explained in the previous section. The frame- (λγ ) = √ (0, 1, λγ i, 0), (C2a) independent normalization is 2 ∗β λV 1 X 2 (λV ) = √ (0, − cos θs, λV i, sin θs) N = |Mλγ ,λV | . (B2) 2 0 2 0 λγ ,λV ,λ,λ λ,λ 2 1 − λV V V + (qs,Es sin θs, 0,Es cos θs), (C2b) The implication of helicity conservation at the photon mV vertex, i.e., M ∝ δλV can easily be checked in the λγ ,λV λγ V λ,λ0 where Es and qs are the energy and momentum of the SDMEs. As can be readily verified with Eqs (B1), this vector meson in the s-channel frame, respectively, and θs 1 is the scattering angle. The expression of the kinematical hypothesis leads to vanishing SDMEs except for Im ρ1−1 2 quantities can be found in the appendix of Ref. [6]. In and Im ρ1−1. The SDMEs also provide other useful information concerning the helicity structure of the pho- the center-of-mass frame, the angular dependence of the 0 0 interaction (C1) is instructive: ton vertex. For instance, the elements ρ00 and ρ1−1 give indications about the magnitude of the single-flip con- |λγ +λV | |λγ −λV | tribution and the interference between the nonflip and θs θs Tλ λ ∝ cos sin , (C3) the double-flip amplitudes. Moreover, they can be used γ V 2 2 to separate the contributions from natural and unnat- with θ the scattering angle in the s-channel frame. This ural exchanges. Indeed, at high energies, an exchange s factor, known as the half-angle factor, encodes all the with positive naturality (N) or negative naturality (U), t-dependence of the interaction. At large energies, the satisfies t-dependence of the half-angle factor becomes very intu- N N 4 p MU = ±(−1)λγ −λV MU . (B3) itive, sin θs/2 → −t/s and cos θs/2 → 1. Throughout −λγ ,−λV λγ ,λV 0 0 0 this paper, we neglect the difference between t and t , λ,λ λ,λ 0 where t = t − tmin, since in the kinematical region of in- We can then use six SDMEs to get information about the 2 2 −3 terest tmin/mV → −(mV /2plab) is on the order of 10 helicity structure of natural and unnatural components: at plab = 9 GeV. Keeping only the leading term in s of the interaction N in Eq. (C1), we obtain U 1 0 1 ρ00 = ρ00 ∓ ρ00 , (B4a) 2 m2 N 1 T → g V U 0 1 λγ λV V P γ 2 Re ρ10 = Re ρ10 ∓ Re ρ10 , (B4b) √ 2 √ −t −t N 1 × λ δ − 2 δ + λ δ . ρU = ρ1 ± ρ1 . (B4c) γ λV ,λγ m λV ,0 m2 γ λV ,−λγ 1−1 2 1−1 11 V V (C4)

Appendix C: High-Energy Limit This example illustrates a general√ statement: each helicity ﬂip “costs” a factor√ of −t/mV . The mass scale associated to the factor −t can only be mV . For com- At high energies, models for reaction amplitudes sim- pleteness, we derive the decay width from the interac- plify. In this section, we perform the high-energy limit of tion (C1): single-meson exchange interaction and keep the leading- order dependence in s, the total energy squared. Our 2 2 2 3 gV P γ m − m goal is to derive the t-dependence arising from the fac- Γ(V → γP ) = V P . (C5) 96π m torization of Regge poles. We consider the reaction V 0 0 γ(k, λγ )p(p, λ) → V (q, λV )p(p , λ ) in the center-of-mass We use Eq. (C5) to extract the couplings from the de- frame (s-channel frame). Let mp and mV be the nucleon cay widths. The relevant couplings are summarized in and vector meson masses, respectively. Table IV. The considerations at the photon vertex apply equally well at the nucleon vertex. For an unnatural spin-zero 1. Unnatural exchanges exchange, there is only one possible structure at the nucleon vertex: Let us ﬁrst focus on the pseudoscalar exchanges. Ac- 0 √ cording to the factorization theorem for Regge poles, the gPNN u¯(p , λ)γ5u(p, λ) → gPNN −tδλ0,−λ. (C6) interaction is a product of a γV P vertex, a Regge factor and a PNN vertex. At the photon vertex we use

α ∗β µ ν 4 Tλγ λV = −igV P γ εαβµν (λγ ) (λV )k q . (C1) In what follows, we will denote the leading term in s by an arrow. 11

In the case of a single exchange, the SDMEs depend only TABLE IV. Vector meson radiative decay widths and pseu- on the details of the photon vertex. The only scale that doscalar exchange couplings. arises is the mass of the vector meson. 0 V Γ(V → γπ ) gV πγ Γ(V → γη) gV ηγ ω 703 keV 0.696 GeV−1 44.8 KeV 0.479 GeV−1 2. Natural exchanges ρ0 89.6 keV 0.252 GeV−1 3.91 KeV 0.136 GeV−1 −1 −1 φ 5.41 keV 0.040 GeV 56.8 KeV 0.210 GeV The two guiding√ rules, the factorization of Regge poles and the factor of −t for each unit of helicity flip, equally apply to natural exchanges. We can then postulate the There is√ one unit of helicity flip associated with the general form in Eq (3). Since the use of effective La- factor −t. In this case the scale factor (nucleon grangians is very popular, it is instructive to compare mass) is implicitly removed by our spinor normaliza- our model in Eqs (7) and (8) to these types of interac- tionu ¯(p, λ)u(p, λ) = 2m. For the π-nucleon and η- tions. 2 nucleon couplings, we take gπNN /4π = 14 [49–56], and Let us start with the standard interaction for a 2 gηNN /4π = 0.4 is the value we used in our fixed-t disper- Pomeron exchange [30, 63] sion relation analysis of η photoproduction [4] based on MP (s, t) = ∗(q, λ )[kµ ν (k, λ ) − kν µ(k, λ )] the available literature [57–62]. λγ ,λV ν V γ γ λ,λ0 The couplings we determined are normalized at the 0 0 pseudoscalar pole. We then add a factor πα0/2 to the × u¯(p , λ )γµu(p, λ). (C9) √ Regge factor in Eq. (4a) such that µ µ At leading order in s, we have k → n+ s/2, with nµ = (1, 0, 0, ±1), and the helicity structure at the nu- 0 ± παP P cleon vertex is simply lim (t − mP ) R (s, t) = 1. (C7) t→mP 2 0 0 µ √ µ u¯(p , λ )γ u(p, λ) → s δλ,λ0 n−. (C10) The Regge trajectory is α (t) = α0 (t − m2 ) with α0 = P P π P Only the first term in the bracket in Eq. (C9) survives. 0.7 GeV−2. We choose the same trajectory for both π and From the result η exchange. As explained in Sec. IV, this enhances the √ η pole to compensate for the Pomeron normalization in ∗ λγ −t (q, λV ) · (k, λγ ) → −δλV ,λγ + δλV ,0 √ , (C11) the φ photoproduction SDMEs. Finally, collecting all the 2 mV pieces, we arrive to the amplitude in Eq. (5) for a π or η exchange in the high-energy limit with the normalization we conclude that this model for the Pomeron implicitly P 0 2 includes a single helicity flip structure and is not there- β0,V = (1/4)πα mV gV P γ gPNN . It is instructive to derive the SDMEs for only a π fore purely helicity conserving. A more flexible model exchange in both the GJ and helicity frames. The can be obtained with more general interactions. In or- SDMEs induced by a π exchange take a simple form der to determine all of the possible structures at both in the GJ frame, i.e., all SDMEs are zero except for the photon and the nucleon vertices, let us first observe 1 2 1 that in a factorizable model, the top and bottom vertices ρ1−1 = − Im ρ1−1 = − 2 . This is of course expected since the π in its rest frame only has the spin projection are linked by a propagator transverse to the momentum zero. We can easily get the SDMEs for a π exchange in transferred. The propagator removes the x component µ √ the helicity frame from the rotation in Eq. (A1): since (q − k) → (0, −t, 0, 0) at leading order in s. Sec- ondly, the general structures at the nucleon vertex are −2t/m2 easily obtained. In addition to Eq. (C10), we can have ρ0 = ρ1 = V , (C8a) an nucleon helicity-flip interaction 00 00 2 2 (1 − t/mV ) 0 0 µ µ √ µ −t/m2 u¯(p , λ ) (2p − γ ) u(p, λ) → 2λ −tδλ0,−λ n−. (C12) ρ0 = −ρ1 = V , (C8b) 1−1 11 2 2 0 (1 − t/mV ) Note that any p momentum can be substituted by p √ 2 since the difference is orthogonal to the propagator. We 0 1 −1 −t 1 + t/mV Re ρ10 = Re ρ10 = √ 2 , (C8c) summarized the two possible structures at the nucleon 2 mV 2 (1 − t/mV ) vertex in Eqs (C10) and (C12) in Eq. (8). 1 1 + t/m2 At the photon vertex, the only tensorial structures ρ0 = − V , (C8d) 11 2 2 2 that connect to the nucleon vertex and survive at lead- (1 − t/mV ) µ µ √ ∗ ing order in s are k → n+ s/2 and (q, λV ) → 1 1 − t/m2 2 µ Im ρ2 = − V , (C8e) (1 − λV )(q/mV )n+. We can then form a single helicity- 1−1 2 2 2 flip coupling at the photon vertex with the interaction (1 − t/mV ) √ √ −1 −t 1 2 µ∗ 1 λγ −t µ Im ρ10 = √ . (C8f) (q, λ ) q · (k, λ ) → √ δ n . (C13) m 2 2 V γ λV ,0 + 2 V (1 − t/mV ) 2 2 mV 12

Finally, since the maximum helicity difference between where mE and ΓE are the mass and the width of the a photon and a vector meson in their center of mass is f2 and a2 tensor mesons. Its effect in the physical re- two, a tensor exchange should involve all possible rel- gion of the direct channel can be modeled by a simple 2 2 2bE t evant structures at the photon vertex. Indeed, we find exponential falloff, i.e., |BWE(t)| ≈ |BWE(0)| e for 2 2 that a double-flip structure can arise with the interaction t ∈ [−mω, 0]. We determine bE at t = −mω/2, the mid- 2 between a photon, vector and tensor [64]: dle point of the interval t ∈ [−mω, 0]. We find bf2 = 0.55 −2 −2 GeV and ba = 0.53 GeV . t 2 ∗(q, λ ) · k (k, λ ) · q → λ λ . (C14) V γ γ V 2 The t-dependence of the Pomeron is often described by We then conclude that, in addition to the nonflip inter- the following dipole form factors [17, 19, 20, 29, 30] action in Eq. (C11), the general structure with a photon, vector and natural exchange also includes the single-flip interaction in Eq. (C13) and the double-flip interaction in Eq. (C14). To leading order in s, we summarize these interactions with the intuitive vertex in Eq. (7). 4m2 − 2.8t F (t) = p , (C16a) In our model we added a helicity-independent expo- 1 2 2 (4mp − t)(1 − t/t0) nential factor b to reproduce the energy-independent N 1 2µ2 + m2 shrinkage of the differential cross section. This feature 0 V FV (t) = 2 2 2 , (C16b) is generally described by exponential factors [12, 14–16], 1 − t/mV 2µ0 + mV − t gamma functions [18, 28] or dipole form factors [17, 19, 20, 29, 30]. This t-dependence originates from the energy dependence of the nearest cross-channel singularity. For the given Regge exchanges, these are the f2(1270) 2 2 2 and a2(1320) tensor mesons. The energy dependence of with µ0 = 1.1 GeV and t0 = 0.7 GeV . The form factor these singularities in the cross channel can be described F1(t) is the dipole approximation of the nucleon Dirac by Breit-Wigner line shape in t, the relevant energy vari- form factor [29], and FV (t) is an empirical form factor able in the cross channel: at the photon vertex.5 As for the Regge exchanges, we approximate this form factor by an exponential falloff at mEΓE 2 b t0 BWE(t) = , (C15) t0 = −mω/2, F1(t0)FV (t0) = e P . Under this approxi- 2 −2 mE − t − imEΓE mation, we obtain bP = 3.60 GeV .

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