Vector Meson Photoproduction with a Linearly Polarized Beam
JPAC Collaboration V. Mathieu J. Nys C. Fernández-Ramírez A. Jackura A. Pilloni N. Sherrill A. Szczepaniak G. Fox





vmathieu@jlab.org
Theory Center, Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA


Theory Center, Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA
Department of Physics and Astronomy, Ghent University, Belgium
Center for Exploration of Energy and Matter, Indiana University, Bloomington, IN 47403, USA
Physics Department, Indiana University, Bloomington, IN 47405, USA






Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Ciudad de México 04510, Mexico


Center for Exploration of Energy and Matter, Indiana University, Bloomington, IN 47403, USA
Physics Department, Indiana University, Bloomington, IN 47405, USA








Theory Center, Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA


Center for Exploration of Energy and Matter, Indiana University, Bloomington, IN 47403, USA
Physics Department, Indiana University, Bloomington, IN 47405, USA


Theory Center, Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA
Center for Exploration of Energy and Matter, Indiana University, Bloomington, IN 47403, USA
Physics Department, Indiana University, Bloomington, IN 47405, USA


School of Informatics, Computing, and Engineering, Indiana University, Bloomington, IN 47405, USA

Joint Physics Analysis Center

JLAB-THY-18-2650






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 the spin-density matrix elements in photoproduction of  ω, ρ^0 and ϕ  mesons at E_γ∼ 8.5 GeV, which are soon to be measured at Jefferson Lab. 




Vector Meson Photoproduction with a Linearly Polarized Beam
    G. Fox
    December 30, 2023
===========================================================






§ INTRODUCTION

With the recent development of the 12 GeV electron beam at Jefferson Lab (JLab) <cit.>, new  precision measurements of  light meson photoproduction and electroproduction are expected in the near future. These will provide constraints on resonance production dynamics, including production of gluonic excitations. For example, the GlueX measurement of the photon beam asymmetry in the production of π^0 and η mesons <cit.> established the dominance of natural-parity t-channel exchanges for production in the forward direction <cit.>. This measurement seems to contradict earlier SLAC data <cit.> that suggests significant contribution from unnatural-parity exchanges. It was shown in  <cit.> 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. Nevertheless, more data from both GlueX and CLAS12 will be needed to refine our understanding of the production mechanisms. 

We consider the reaction γ(k,λ_γ) N(p,λ)→ V(q,λ_V) N'(p',λ'). At high energies, the amplitude in the forward direction is dominated by exchange of Regge poles (Reggeons).
As illustrated in Fig. <ref>, the Reggeon amplitude factorizes into a product of two vertices. The upper vertex describes the beam (photon) interactions, and the lower vertex describes the target (proton) interactions. The Mandelstam variables are s = (k+p)^2 and t = (k-q)^2.
  Factorization of Regge vertices follows from unitarity in the t-channel, where Regge pole is a common pole in all partial waves related by unitarity and its vertices determine residues of the poles <cit.>. Factorization of residues enables one to determine the helicity structure at the photon vertex independently from the target, and conservation of parity reduces the number of helicity  components at each vertex. In the center-of-mass frame, the net helicity transfer between the vector meson and photon |λ_γ - λ_V| can be 0, 1 or 2, which we refer to as helicity conserving, single and double helicity flip respectively. Measurement of the photon spin-density matrix elements (SDMEs) can be used to determine the relative strength of these components. 



Spin-density matrix elements can be reconstructed from the angular distributions of the vector meson decay products <cit.>. The first measurements of neutral vector meson SDMEs were performed at SLAC <cit.>, resulting in the following qualitative conclusions: the natural exchanges contributing to ρ^0, ω and ϕ production are predominantly helicity conserving, and the unnatural-parity contributions are negligible for ρ^0 production and consistent with a one-pion exchange for ω production. In this paper, we discuss the SLAC data in the context of a Regge-pole exchange model, which allows us to assess contributions of individual exchanges to the SDMEs. 


Various models have been proposed in the past <cit.>, with different descriptions of the momentum-transfer dependence of the helicity amplitudes. In general these models reproduce  
 the differential cross sections, but lack  a detailed discussion of the implication of the Regge pole model for the SDMEs.


 

The paper is organized as follows.  In Section <ref>, we define the Regge amplitudes and discuss model parameters. In Section <ref>, we discuss the fitting procedure. Specifically, we first 
 isolate the unnatural exchanges in  ρ^0 and ω production. We find that, within uncertainties, these components are consistent with  π and η exchanges so we neglect sub-leading trajectories. 
We  determine the residues of the dominant, natural exchanges by 
 the  γ p and γ d total cross sections. Using the SLAC data, the single and double helicity flip couplings are fitted to the three natural components of the SDMEs at the laboratory frame (target rest frame) photon beam energy of E_γ = 9.3. The model is extrapolated to E_γ = 2.8 and 4.7 GeV and compared  to the three natural components of the SLAC SDMEs at these energies. In Section <ref>, we compare the model to the nine ω and ρ^0 SDMEs obtained with a polarized beam at SLAC with E_γ = 9.3, to the nine ϕ SDMEs from LEPS <cit.> and Omega-Photon <cit.>, and to the three ω SDMEs obtained with a unpolarized beam from CLAS <cit.>, LAMP2 <cit.> and Cornell <cit.>.
 
 Furthermore, we test the Pomeron normalization for the ω and ρ^0 differential cross sections at E_γ > 50 GeV, and the Regge exchange normalization for the  ω, ρ^0 and ϕ differential cross sections at E_γ = 9.3 from Ref. <cit.>.  Lastly, we provide the predictions for the upcoming ω, ρ^0 and ϕ SDMEs measurements in JLab experiments. In Section <ref>, we summarize our findings and give conclusions.  Details regarding the relations between the frames (helicity, Gottfried-Jackson, s- and t-channel frames) are summarized in Appendix <ref>, the definition of the SDMEs are detailed in Appendix <ref>, and further details on the amplitude parametrization are given in Appendix <ref>.






§ REGGE MODEL FOR VECTOR MESON PHOTOPRODUCTION
 



At high energies, vector meson photoproduction is dominated by Pomeron and Regge exchanges. 

Regge exchanges can be characterized by the quantum numbers of the lowest spin meson on the trajectory, namely  
  isospin I, naturality η = P(-1)^J (with the parity P), signature τ = (-1)^J, charge conjugation C and G-parity G = C(-1)^I. The leading trajectories contributing to vector meson photoproduction are 

    I^GητJ^PC        I^GητJ^PC        I^GητJ^PC
    
    a_2    : 1^-++2^++   π   : 1^–+0^-+    a_1   : 1^—1^++
    
    f_2    : 0^+++2^++   η   : 0^+-+0^-+    f_1   : 0^+–1^++

In addition to the exchanges in Eq. (<ref>), we also consider the natural-parity Pomeron exchange, which dominates at high energies. 
In the ω photoproduction model from <cit.>,
a scalar exchange representing a σ meson trajectory was also considered. Since the σ meson trajectory is below the (leading) f_2 trajectory, we do not include it here.
Among all unnatural exchanges, the π and η trajectories are expected to dominate, since they are the closest to the scattering region. One can verify this by examining the SDMEs ρ_λ'λ^U, which in the Gottfried-Jackson (GJ) frame are determined by the unnatural exchanges (see Appendix <ref>). The 
GJ frame is equivalent to the t-channel helicity frame where parity conservation implies a relation between  helicity amplitudes and the naturality of the exchanges.
Inspecting the SLAC data <cit.>,  one finds that the matrix elements ρ^U_00|_GJ, ρ^U_10|_GJ and ρ^U_1-1|_GJ for both ω and ρ production are all consistent with zero. Moreover, the unnatural component of the differential cross section is compatible with a π-exchange model <cit.>. Hence, we assume that the unnatural components of the SDMEs are dominated by either π or η exchange. The η exchange is introduced to describe the SDMEs in production of the ϕ meson, while its  contribution is negligible in ω and ρ^0 production. As we will see in Sec. <ref>, the normalization of these exchanges can be determined by vector meson radiative decays. Regarding axial vector exchanges, since the decay widths of f_1, a_1 →γ V are not known,[The only exception is for Γ(f_1→γϕ) ∼ 18 keV.] their 
 contribution is difficult to evaluate. Within a specific quark model  <cit.>, the contribution of the f_1 to ω photoproduction is found to be negligible. As we will show, it is possible to saturate the unnatural components of the SDMEs by pseudoscalar exchanges. We therefore neglect the axial  vector trajectories. 
In summary, we consider the s-channel amplitudes in the form 

    M_λ_V, λ_γ
    λ', λ(s,t)     = ∑_E M^E_λ_V, λ_γ
    λ', λ(s,t),

where the sum extends over the following t-channel reggeons: E= π, η, ℙ, f_2, a_2. 
  From the s-channel helicity amplitudes in Eq. (<ref>), one can compute the SDMEs in the helicity or GJ frame  using Eqs. (<ref>), and (<ref>), respectively. Assuming a factorized form for each exchange, 

    M^E_λ_V, λ_γ
    λ', λ(s,t)     = 
    T^E_λ_V λ_γ(t)  R^E(s,t)  B^E_λ' λ(t),

where the top and bottom vertices  T^E and B^E  describe the helicity transfer from the photon to the vector meson and between the nucleon target and recoil, respectively. 
According to Regge theory <cit.>, the energy dependence factorizes into a power-law dependence s^α_E(t). The phase of the amplitude is determined by the signature factor 1+e^-iπα_E(t), which is contained in R^E, 


    R^U(s,t)     =  1+e^-iπα_U(t)/sinπα_U(t)ŝ^α_U(t)    U     = π, η
    
    R^N(s,t)     =  α_N(t)/α_N(0)1+e^-iπα_N(t)/sinπα_N(t)ŝ^α_N(t)    N     = ℙ, f_2, a_2 .


We defined ŝ = s/s_0 with the scale chosen as s_0 = 1 GeV^2. We use a linear trajectory α_E(t) = α_E(0) + α_E' t for 
all exchanges.  The signature factor eliminates contributions from spin-odd poles induced by the denominator sinπα_E(t). The factor α_E(t)/α_E(0) simply removes the unphysical pole at α_E(t)=0 that arises in the scattering region for the f_2 and a_2 exchanges. For consistency, we also include this factor for the Pomeron exchange, although the point α_ℙ(t)=0 is far from the region of interest -t≤ 1 GeV^2. For the pseudoscalar exchanges, the pole at  α_π,η(t)=0 is physical. 




 §.§ Unnatural exchanges
 
For unnatural exchanges U=π,η, the helicity structure of the photon vertex T^U and the nucleon vertex B^U 
 can be obtained by comparison with the high-energy limit of a single-particle exchange model. We obtain  (see Appendix <ref>), 


    T^U_λ_V λ_γ(t)     =  
    β^U_γ V(    λ_γδ_λ_V, λ_γ 
    - √(2)√(-t)/m_Vδ_λ_V,0+ 
     -t/m_V^2λ_γδ_λ_V,-λ_γ),
    
    B^U_λ' λ(t)    = β^U_pp(  δ_λ,-λ'√(-t)/2m_p),




with m_V and m_p being the vector meson and nucleon masses, respectively. 
The residues β^U_γ V and β^U_pp are determined from the radiative decay widths Γ(V→γπ), Γ(V→γη) and the nucleon couplings g_π pp, g_η pp, respectively. The overall nonflip couplings of the reaction are written  β^U_0,V = β^U_γ Vβ^U_pp.[The index 0 stands for the helicity difference at the top vertex, |λ_γ-λ_V|=0.] The details of the calculation are given in Appendix <ref>. The unnatural trajectory is α_U(t) = α'_U(t-m_π^2) with α_U' = 0.7 GeV^-2.
The parameters for the unnatural exchanges are summarized in Table <ref>.
The photon vertex T^U_λ_V λ_γ involves all possible helicity structures, with each unit of helicity flip contributing a factor of √(-t).
Because of charge conjugation, there is only one helicity structure at the the nucleon vertex, the helicity flip, which corresponds to the factor δ_λ,-λ'√(-t)/2m_p. 





 §.§ Natural exchanges
 
The trajectories of the natural exchanges are known and we use <cit.>


    α_ℙ(t)    = 1.08 + 0.2  t/GeV^2, 
    α_f_2 , a_2(t)    = 0.5 + 0.9  t/GeV^2.


For natural exchanges, N = ℙ,f_2, a_2. The top vertex involves three  helicity components: a helicity nonflip, single flip and double flip. As for unnatural exchanges, each of these comes with an appropriate power of the factor √(-t)/m_V,

    T^N_λ_V λ_γ(t)     =  β^N_γ V e^b_N t
    ×(    δ_λ_V,λ_γ + β^N_1 √(-t)/m_Vλ_γ/√(2)δ_λ_V,0 +β^N_2
     -t/m^2_Vδ_λ_V,-λ_γ).

To be consistent with factorization, and to reduce the number of parameters, we assume that the couplings β_1^N and β_2^N are the same for all vector mesons.  

The steep falloff of the forward differential cross section is well described by exponential factors, gamma functions <cit.> or dipole form factors <cit.>. All of these models can be approximated by an exponential function of the form e^b_N t <cit.>.  We obtain b_ℙ=3.6 GeV^-2 by approximating the form factors from <cit.>, and b_a_2=0.53 GeV^-2 and b_f_2=0.55 GeV^-2 by approximating the t-dependence of the a_2 and f_2 poles with a Breit-Wigner line shape as described in Appendix <ref>. 
For the nucleon vertex we include the two possible helicity combinations, a nonflip and single flip, 

    B^N_λ' λ(t)     = β^N_pp( δ_λ,λ'  + 2 λκ_N √(-t)/2m_pδ_λ,-λ').

The SDMEs probe the helicity structure of the photon vertex. They are weakly dependent on the helicities  
 at the  nucleon vertex. On the contrary, The helicity flip couplings κ_N thus play a minor role in our analysis. Moreover isoscalar exchanges, e.g., the f_2 and Pomeron, are empirically helicity nonflip at the nucleon vertex <cit.>. Therefore, we set κ_f_2=κ_ℙ=0. The isovector exchanges are empirically helicity flip dominant. We model this feature by  using κ_a_2 = 8.0 <cit.>. 


The special nature of the Pomeron prevents us from computing its overall normalization β_0,V^ℙ = β^ℙ_γ Vβ^ℙ_pp by using radiative decays. 

We thus determine the normalization β_0,V^N = β^N_γ Vβ^N_pp by fitting the γ p and γ d total cross sections and invoking vector meson dominance (VMD). We first relate the overall normalizations β_0,V^N to the γ p and γ d total cross section. Using the optical theorem, our Regge parametrization in (<ref>) leads to









    σ(γ p )     = 1/2m_pE_γ(β^γγ_ℙŝ^α_ℙ(0) + β^γγ_f_2ŝ^α_f_2(0)  + β^γγ_a_2ŝ^α_a_2(0)), 
    σ(γ d)     = 1/2m_pE_γ(2β^γγ_ℙŝ^α_ℙ(0) +2 β^γγ_f_2ŝ^α_f_2(0)).
 



The factors  β^γγ_N represent couplings of the natural exchange N in the forward scattering direction γ p →γ p.  We need to relate these factors, via VMD, to the factors β_0,V^N appearing in vector meson photoproduction. In order to use VMD, we use the following interaction between photon field A_μ and the vector meson fields <cit.>:

    L    = - eA^μ( m^2_ρ/γ_ρρ_μ 
    +  m^2_ω/γ_ωω_μ 
    + m^2_ϕ/γ_ϕϕ_μ).

From this interaction,[The γ_V couplings can be cast in terms of the vector meson decay constants ⟨ 0| ∑_q =u,d,s e_q q̅γ_μ q(0) | V(ϵ,P)⟩ = f_V ϵ^V_μ(P)= (m_V^2/γ_V) ϵ^V_μ(P).] and neglecting the electron mass, one finds for the electronic decay width Γ(V → e^+ e^-) = m_V (α^2/3) (4π/γ^2_V), which determines the couplings γ_V that we tabulate in Table <ref>. 
 
The SU(3) quark model predictions γ_ω/γ_ρ=3 and γ_ω/γ_ϕ=-√(2) compare well with the VMD predictions, γ_ω/γ_ρ=3.4(6) and γ_ω/γ_ϕ=-1.3(1). However, it is well known that the ϕ meson differential cross section produces a value of γ_ϕ that is twice as large as the one obtained from the leptonic decay width <cit.>. For consistency, we will use the γ_ϕ value obtained from the leptonic decay width, but we keep an eye on this  discrepancy when comparing to the data.

Assuming that the Pomeron has a gluonic nature and therefore has couplings which are independent of the quark flavor <cit.>, we derive the relation between the total cross section couplings in Eq. (<ref>) and the overall normalization of the Pomeron β^ℙ_0,V in our model for vector meson photoproduction,
 

    β^ℙ_0,V    = β_ℙ^γγe/γ_V×( e^2/γ_ρ^2 + e^2/γ_ω^2 + e^2/γ_ϕ^2)^-1.

We note that by increasing γ_ϕ by a factor of two, the ω and ρ^0 couplings of the Pomeron would change by only 10%. 
For the Regge exchanges, we assume ideal mixing for vector and tensors mesons and extract the remaining couplings using vector meson dominance:

    β^f_2_0,ω/ρ    =  β_f_2^γγe /γ_ω/ρ×(e^2/γ_ρ^2 + e^2/γ_ω^2)^-1,  
    
    β^a_2_0,ω/ρ    =  β_f_2^γγγ_ω/ρ/2 e, 
    β_γϕ^f_2   =β_γϕ^a_2=0.



We choose to determine the helicity couplings β_1^N and β_2^N through a fit to the SLAC data. 

Since our formalism is based on a high-energy expansion, we determine the parameters only with the highest energy bin. Specifically, we inspect the natural components of the SDMEs at E_γ = 9.3 GeV. Assuming only one natural exchange N, our form in Eq. (<ref>) for the top vertex leads to
 

    ρ^N_00(s,t)     = (β_1^N)^2/A(t)-t/m_V^2, 
    ρ^N_10(s,t)     = β^N_1/2A(t)√(-t)/m_V(1+β^N_2  -t/m_V^2),  
    ρ^N_1-1(s,t)     =(β_2^N)^2/A(t)-t/m_V^2,


with A(t) = 1-(β_1^N)^2 t/m_V^2 + (β_2^N)^2 t^2/m_V^4. The factorization hypothesis in Eq. (<ref>) and the conservation of angular momentum implies the vanishing of these SDMEs in the forward direction. This is indeed observed in all of the ρ^0 SDMEs, but is inconsistent with the ρ^N_1-1 elements for ω photoproduction as seen in Fig. <ref>. The expressions in Eq. (<ref>) also tell us that we should expect |ρ^N_00|  < |ρ^N_10| for small t. Again, this relation is satisfied for ρ^0 photoproduction but seems to be violated for ω photoproduction. The element ρ^N_00 is significantly larger for ω photoproduction compared to ρ^0 photoproduction, suggesting a larger single-helicity flip for the isovector exchange. The deviation from zero observed in the elements ρ^N_10 and ρ^N_1-1 for ρ^0 photoproduction suggests a nonzero single and double helicity flip for the isoscalar exchanges. We associate these couplings with the f_2 exchange and keep the Pomeron helicity conserving as is often assumed. This hypothesis could be checked with ϕ photoproduction as we will discuss later. According to our discussion we impose β_1^ℙ = β_2^ℙ= β_2^a_2=0 and thus need to fit the helicity couplings β_1^f_1, β_1^f_2, β_1^a_2.





§ FITTING PROCEDURE
 


We determine the six couplings β^γγ_ℙ, β^γγ_f_2, β^γγ_a_2, β_1^f_1, β_1^f_2, β_1^a_2 using  a combined fit of the γ p and γ d total cross sections from the Review of Particle Physics <cit.> for E_γ>2 GeV,  the three ρ^0 natural exchange SDMEs (ρ^N_00, ρ^N_10 and ρ^N_1-1) and the element ρ^N_00 for ω photoproduction at E_γ= 9.3 GeV obtained at SLAC <cit.>. We do not include the two other natural components of the SDMEs in ω photoproduction as they are inconsistent with our 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) data points and six fit parameters. The other model parameters (b_N, κ_N, γ_V and the π- and η-exchange couplings) are kept fixed at values discussed in the previous section.  The expressions for the natural components of the SDMEs used in the fit is given in Eqs (<ref>) and (<ref>). The fit results in the reduced χ^2/d.o.f. of 1.96 (1.84 for the total cross sections and 0.12 for the SDMEs),  and the fitted parameters are



    β_ℙ^γγ    = 0.187(1)    β_1^f_2    =  0.95(19)
    β_f_2^γγ    = 0.164(2)    β_2^f_2    =  -0.56(17)
    β_f_2^γγ    = 0.045(3)    β_1^a_2    =  0.83(34) .


The photon couplings are extracted from Eqs (<ref>). The parameters of the exchanges calculated from Eq. (<ref>) for vector meson photoproduction are summarized in Table <ref>. 













§ COMPARISON WITH DATA
 
As we discussed above, the SDMEs for ρ^0 photoproduction are more consistent with our model for diffractive production than for ω photoproduction. This can be observed in Fig. <ref>. The bands on the figures represent one standard deviation from our model.  The wider band in the ω model originates from the stronger dominance of the Regge exchanges, whose normalizations are less constrained by the total cross sections. The Pomeron normalization is indeed more constrained and yields a smaller uncertainty in the ρ^0 model. We have also included the data at E_γ= 4.7 and 2.8 GeV from SLAC in Fig. <ref>. They compare well to our model evaluated at these lower energies. 



In Fig. <ref>,  we present the comparison between the ω and ρ^0 models and the SLAC data at 9.3 GeV for all nine SDMEs. There is a general agreement between the model and the data, but we wish to discuss some inconsistencies. The elements in the bottom panels ρ_1-1^1, ρ^2_10 and ρ^2_1-1 were not included in the fitting but are nevertheless well described by the model. In particular, we note the dominance of the natural exchanges in ρ_1-1^1 and ρ^2_1-1 in the case of ρ^0 photoproduction with small deviation for the ω case, as expected from the stronger π exchange. The main noticeable discrepancy arises in ρ^1_11 for ω photoproduction. Since the pseudoscalar exchanges are smaller than the natural exchanges, we would expect ρ_11^1 ∼ρ_1-1^0. The data does not display this feature and thus our model does not describe ρ^1_11 well. Furthermore, since the contribution from the π exchange to ρ^1_11 is negative (see Appendix <ref>), we would expect ρ_11^1 < ρ_1-1^0, which is featured in our ω model but not in the SLAC data. The sign of the element ρ^1_11 would be an important check for our model when GlueX data becomes available. 



Although our model has been constrained at E_γ = 9 GeV, we present in Fig. <ref> the comparison between our model and the unpolarized SDMEs at lower energies. The extrapolation to lower energies is in principle not in the range of applicability of the Regge-pole approximation. Despite the significant uncertainties in all the presented data sets, we conclude that our extrapolated model describes the lower energies data sets fairly well. It is also worth noting that the data from Ref. <cit.> at E_γ=8.9 GeV are consistent with our factorization hypothesis, i.e., ρ^0_1-1∼ -t in the forward direction. We conclude that the SLAC data may suffer from large errors. The forthcoming measurement by the GlueX collaboration could confirm the factorization of the vector meson production, i.e., ρ_1-1^0(t) ∼ -t in the forward direction at high energies. 




Our model simplifies for ϕ photoproduction. In this case we simply neglect the f_2 and a_2 Regge exchanges, as they are not expected to couple to γϕ if one assumes perfect mixing. The relevant exchange would then be the f_2', the hidden strangeness  partner of the f_2. However, its intercept, and therefore its overall strength, is smaller due its higher mass. We neglect this contribution and assume that the only relevant natural contribution is provided by the Pomeron. Since our Pomeron is purely helicity conserving, the SDMEs are very simple at high energies. The only non-zero components are ρ^1_1-1 = - ρ^2_1-1 = 1/2. This picture is consistent with the SLAC measurement at 9.3 GeV <cit.>. In Fig. <ref>, we compare our model to the data from the Omega-Photon collaboration <cit.>. Their data are taken in the energy range E_γ = 20 - 40 GeV. They are consistent with the SLAC data but have somewhat smaller uncertainties. We also extrapolated our model to E_γ = 2.27 GeV to compare with the data from the LEPS collaboration <cit.>. At lower energies, we observe deviations from pure helicity conservation, i.e., deviation from ρ^1_1-1 = - ρ^2_1-1 = 1/2. This is triggered by unnatural exchanges. Since the π couples weakly to γϕ, we included η exchange in our model. The very small coupling g_ϕγπ, inferred from radiative decays, cannot solely explain the deviation from helicity conservation in the elements ρ^1_1-1 and ρ^2_1-1 at E_γ = 2.27 GeV. The inclusion of η exchange increases the relative importance of unnatural exchange. We should also note that we considered the η degenerate with the π. With the η pole being further from the scattering region, the factor α' π / sinπα_η(t) ∼ 1/(m_η^2-t) is not strong enough to trigger the depletion close to the forward direction in ρ^1_1-1 and  ρ^2_1-1. Nevertheless, the SDMEs from the LEPS collaboration indicate an even larger relative strength of unnatural vs. natural exchanges than in our model. As we pointed out, the Pomeron coupling g_ℙ^γϕ from the ϕ meson leptonic width and VMD is overestimated. The relative strength of the unnatural exchanges in the SDMEs are thus underestimated. We illustrate the effect of reducing the Pomeron coupling by a factor of two in Fig. <ref>. The dashed red line, obtained with β_0,ϕ^ℙ = 1/2 × 0.932, leads to a better agreement with the data. Alternatively, we could have increased the coupling g_η NN. As we discussed in Ref. <cit.>, the η coupling to the nucleon is not known very precisely.  From the investigation of ϕ SDMEs at E_γ=2.27 GeV, we conclude that the ratio of natural and unnatural component is β_0,ϕ^N/β_0,ϕ^U =  0.266.  





Our prediction for ω, ρ^0 and ϕ vector meson photoproduction at GlueX is displayed in Fig. <ref>. We used E_γ = 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 SDMEs are very small. The bending of the curves as |t| increases in our ϕ model originate from the pseudoscalar exchanges. We have not included an exponential falloff in their parametrization. Therefore, their effects can be observed away from the forward direction where the natural exchanges are exponentially suppressed. If the ϕ SDMEs remain flat in a larger t range, one would just need to incorporate an exponential falloff in the η exchange. 



Our model has been designed to describe the SDMEs, but it is also interesting to compare it with high-energy unpolarized differential cross-section data. We first compare our model to high-energy data in Fig. <ref>. At energies above 50 GeV, the Regge exchanges contribute less that 1% of the differential cross section. The data therefore gives a very good indication of the validity of our Pomeron model. We observe that the overall normalization at t=0 is in fairly good agreement with the data. Our phenomenological intercept α_ℙ (0)= 0.08 produces a small rise of the differential cross section in the forward direction. At very high energies, E_γ > 1 TeV, the data seems to display a slower growth at t=0, in agreement with the unitarity bound. However, these energies are far from our region of interest. The t-dependence was approximated by a simple exponential falloff, which describes the falloff of the differential cross section in the range 0< -t/m_V^2 ≲ 1. We observe deviations from this simple picture at |t|>0.3 GeV^2.



Unfortunately, our model does not compare very well with the ω and ϕ differential cross sections at 9.3 GeV, as shown in Fig. <ref>. Although the ρ^0 differential cross section is roughly in agreement with our model, the ϕ differential cross section is overestimated. We already explained that the leptonic width of the ϕ meson led to a Pomeron coupling to γϕ much stronger than the experimental value. This was already observed in the original experimental publication <cit.>. It has been argued in Ref. <cit.> that the large ϕ mass needs to be taken into account. The authors of Refs. <cit.> corrected the differential cross section by the ratio of the ϕ and photon momenta, (k_ϕ/k_γ)^2 ≈ 0.87 at E_γ = 9.3 GeV. This factor is nevertheless not small enough to reproduce the experimental normalization of the ϕ differential cross section. As we did for the SDMEs, we reduce the Pomeron coupling β_0,ϕ^ℙ by a factor of two. The resulting normalization at t=0 seems more in agreement with the data.








§ CONCLUSIONS
 
We presented a model describing the SDMEs of light vector meson photoproduction. Our model includes π and η exchanges, whose parameters are fixed. We incorporated the leading natural exchanges: the Pomeron, f_2 and a_2 exchanges. Their normalizations were determined from the total cross section using the VMD hypothesis. We paid special attention to the t-dependence of the various exchanges. We proposed a flexible and intuitive ansatz for the t-dependence of each natural exchange. The helicity structure of these exchanges was then inferred from the data on photoproduction of ω and ρ^0 at E_γ = 9.3 GeV from SLAC. The joint inspection of these two reactions allowed us to assume that the f_2 isoscalar exchange must have a small double helicity flip coupling, in addition to a single helicity flip coupling. The a_2 isovector exchange was consistent with only a single flip and no double helicity flip coupling. 
 
The model compares well with the nine SDMEs for ρ^0, ω and ϕ photoproduction in a wide energy range E_γ∼ 3-9 GeV, as well as with the unpolarized data in the same energy range. Except for ρ^0_1-1 in ω production, the SDME are consistent the factorization of Regge residues.  We made predictions for the future measurements of light meson photoproduction at JLab. Our predictions and our model are available online on the JPAC website <cit.>. With the online version of the model, users have the possibility to vary the model parameters and generate the SDMEs for ρ^0, ω and ϕ photoproduction. The code can also be downloaded.

The differential cross section at very high energies, E_γ > 50 GeV, is well reproduced by our Pomeron exchange. However, the effect of the high-energy approximation led to non negligible deviation in normalization from the data at E_γ = 9.3 GeV. These deviations appear only in the differential cross section, since they cancel in the ratio of the SDMEs. 




We thank C. Meyer for pointing out the future measurements by the GlueX collaboration.
This work was supported by BMBF,
by the U.S. Department of Energy under grants No. DE-AC05-06OR23177 and No. DE-FG02-87ER40365, by
PAPIIT-DGAPA (UNAM, Mexico) grant No. IA101717,
CONACYT (Mexico) grant No. 251817, by
Research Foundation – Flanders (FWO), by
U.S. National Science Foundation under award numbers PHY-1415459 and PHY-1205019,
and by Ministerio de Economía y Competitividad (Spain) through grant
No. FPA2016-77313-P.





§ FRAMES


The properties of helicity amplitudes are best described in two popular frames: the s-channel and the t-channel frames. The s-channel corresponds to the center-of-mass of the reaction γ p → V p. The t-channel corresponds to the center-of-mass of the reaction γV̅→ p p̅. These channels are illustrated on Fig. <ref>.

The angular distribution of a vector meson is analyzed in its rest frame. In the rest frame, the beam, target and recoil form the reaction plane xz. The y-axis is defined as the cross product between the target and the recoil momenta. For the z-axis, the two common choices are the opposite direction of the recoil in the helicity frame, and the beam direction in the GJ frame <cit.>.

The helicity amplitudes in these four frames are different. For instance, a boost along the recoil momentum between the s-channel and the helicity frames rotates the helicities of the beam, target and recoil. It also transforms the helicity of the vector meson in the s-channel into its spin projection along the direction opposite to the recoil in the helicity frame. The summation over beam, target and recoil helicities in the SDMEs is not affected by these rotations. Hence, the SDMEs in the s-channel and helicity frames are equivalent. 

Similarly, a boost along the beam direction between the t-channel and the GJ frames brings the helicity of the vector in the t-channel to its spin projection along the beam direction in the GJ frame. The helicities of the other particles undergo a rotation which does not affect the SDMEs, as demonstrated in Ref <cit.>. 

Finally, from the SDMEs in the GJ frame, the SDMEs in the helicity frame are obtained by a rotation of angle θ_q, the angle between the opposite direction of the recoil and the beam direction (see Fig. <ref>)

    ρ_MM'|_GJ = ∑_λ_V, λ'_V d^1_M,λ_V(θ_q) ρ_λ_V, λ'_V|_H d^1_M',λ'_V(θ_q),

with cosθ_q = (β- cosθ_s)/(βcosθ_s-1) and β = λ^1/2(s,m_p^2,m_V^2)/(s-m_p^2+m_V^2). The leading s expression is simply cosθ_q → (m_V^2+t)/(m_V^2-t). 







§ SPIN-DENSITY MATRIX ELEMENTS
 
The relation between SDMEs and helicity amplitudes are well known <cit.>. For completeness, we provide the expressions for the nine SDMEs accessible with a linearly polarized photon beam:


    ρ^0_00   = 1/N∑_λ, λ' M_1, 0 
    λ, λ' M^*_1, 0 
    λ, λ',
    
    
    Re ρ^0_10   = 1/2N∑_λ, λ'(  M_1, 1 
    λ, λ' -  M_1, -1 
    λ, λ')  M^*_1, 0 
    λ, λ', 
    
    ρ^0_1-1   = 1/N∑_λ, λ' M_1, 1 
    λ, λ' M^*_1, -1 
    λ, λ', 
    
    ρ^1_11   = 1/N∑_λ, λ' M_-1, 1 
    λ, λ' M^*_1, 1 
    λ, λ', 
    
    ρ^1_00   = 1/N∑_λ, λ' M_-1, 0 
    λ, λ' M^*_1, 0 
    λ, λ',  
    
    ρ^1_1-1 + ρ^2_1-1    = 1/N∑_λ, λ' M_-1, 1 
    λ, λ' M^*_1, -1 
    λ, λ', 
    
    
    
    ρ^1_1-1 - ρ^2_1-1    = 1/N∑_λ, λ' M_1, 1 
    λ, λ' M^*_-1, -1 
    λ, λ',
    
    ρ^1_10 + ρ^2_10    = 1/N∑_λ, λ' M_-1, 1 
    λ, λ' M^*_ 1,0 
    λ, λ',    
    
    ρ^1_10 - ρ^2_10    = 1/N∑_λ, λ' M_1, 1 
    λ, λ' M^*_-1,0 
    λ, λ'.
 

Of course, the SDMEs and the helicity amplitudes need to be define in the same frame, or in equivalent frames, as explained in the previous section. 
The frame-independent normalization is 

    N     = 1/2∑_λ_γ, λ_V, λ, λ' 
    | M_λ_γ, λ_V 
    λ, λ'|^2.

The implication of helicity conservation at the photon vertex, i.e., M_λ_γ, λ_V 
λ, λ'∝δ^λ_V_λ_γ can easily be checked in the SDMEs. As can be readily verified with Eqs (<ref>), this hypothesis leads to vanishing SDMEs except for ρ^1_1-1 and ρ^2_1-1. The SDMEs also provide other useful information concerning the helicity structure of the photon vertex. For instance, the elements ρ^0_00 and ρ^0_1-1 give indications about the magnitude of the single-flip contribution and the interference between the nonflip and the double-flip amplitudes. Moreover, they can be used to separate the contributions from natural and unnatural exchanges. Indeed, at high energies, an exchange with positive naturality (N) or negative naturality (U), satisfies

    M^N
     U_-λ_γ, -λ_V 
    λ, λ'    = ± (-1)^λ_γ-λ_V M^N
     U_λ_γ, λ_V 
    λ, λ'.

We can then use six SDMEs to get information about the helicity structure of natural and unnatural components:


    ρ^N
     U_00   = 1/2( ρ^0_00∓ρ^1_00), 
    
    
    
    
    Re ρ^N
     U_10   = 1/2(  ρ^0_10∓ρ^1_10),  
    
    
    
    
    ρ^N
     U_1-1   = 1/2(  ρ^1_1-1±ρ^1_11).










§ HIGH-ENERGY LIMIT
 
At high energies, models for reaction amplitudes simplify.  In this section, we perform the high-energy limit of single-meson exchange interaction and keep the leading-order dependence in s, the total energy squared. Our goal is to derive the t-dependence arising from the factorization of Regge poles. We consider the reaction γ (k,λ_γ) p(p,λ) → V(q,λ_V) p (p',λ') in the center-of-mass frame (s-channel frame). Let m_p and m_V be the nucleon and vector meson masses, respectively.



 §.§ Unnatural exchanges

Let us first focus on the pseudoscalar exchanges. According to the factorization theorem for Regge poles, the interaction is a product of a γ V P vertex, a Regge factor and a P NN vertex. At the photon vertex we use

    T_λ_γλ_V    =  -i g_V Pγ ε_αβμνϵ^α(λ_γ) ϵ^*β(λ_V) k^μ q^ν.

The polarization vectors, in the s-channel, are


    ϵ^α(λ_γ)     = -λ_γ/√(2) (0,1,λ_γ i, 0), 
    ϵ^*β(λ_V)     = λ_V/√(2) (0,-cosθ_s,λ_V i, sinθ_s) 
        + 1-λ_V^2/m_V(q_s, E^V_s sinθ_s,0,E^V_s cosθ_s),


where E_s^V and q_s are the energy and momentum of the vector meson in the s-channel frame, respectively, and θ_s is the scattering angle. The expression of the kinematical quantities can be found in the appendix of Ref. <cit.>.  
In the center-of-mass frame, the angular dependence of the interaction (<ref>) is instructive:

    T_λ_γλ_V   ∝( cosθ_s/2)^|λ_γ+λ_V|( sinθ_s/2)^|λ_γ-λ_V|,

with θ_s the scattering angle in the s-channel frame.
This factor, known as the half-angle factor, encodes all the t-dependence of the interaction. At large energies, the t-dependence of the half-angle factor becomes very intuitive,[In what follows, we will denote the leading term in s by an arrow.] 
 sinθ_s/2 →√(-t/s) and cosθ_s/2 → 1. Throughout this paper, we neglect the difference between t and t', where t' = t-t_min, since in the kinematical region of interest t_min/m_V^2 → -(m_V/2p_lab)^2 is on the order of 10^-3 at p_lab = 9 GeV. 

Keeping only the leading term in s of the interaction in Eq. (<ref>), we obtain

    T_λ_γλ_V   → g_V Pγm_V^2/2
    ×   (
    λ_γδ_λ_V,λ_γ
    - √(2)√(-t)/m_Vδ_λ_V,0 + -t/m_V^2λ_γδ_λ_V,-λ_γ).

This example illustrates a general statement: each helicity flip “costs" a factor of √(-t)/m_V. The mass scale associated to the factor √(-t) can only be m_V. For completeness, we derive the decay width from the interaction (<ref>):

    Γ(V→γ P)     = g_V Pγ^2/96π( m_V^2-m_P^2/m_V)^3.

We use Eq. (<ref>) to extract the couplings from the decay widths. The relevant couplings are summarized in Table <ref>.


The considerations at the photon vertex apply equally well at the nucleon vertex. For an unnatural spin-zero exchange, there is only one possible structure at the nucleon vertex:

    g_P NNu̅ (p',λ) γ_5 u (p,λ)    → g_P NN√(-t)δ_λ',-λ.

There is one unit of helicity flip associated with the factor √(-t). In this case the scale factor (nucleon mass) is implicitly removed by our spinor normalization u̅(p,λ) u(p,λ) = 2m. For the π-nucleon and η-nucleon couplings, we take g^2_π NN/4π = 14 
<cit.>, and g^2_η NN/4π = 0.4 is the value we used in our fixed-t dispersion relation analysis of η photoproduction <cit.>
 based on the available literature <cit.>.

The couplings we determined are normalized at the pseudoscalar pole. We then add a factor πα'/2 to the Regge factor in Eq. (<ref>) such that

    lim_t→ m_P (t-m_P)πα_P'/2 R^P(s,t) =1.

The Regge trajectory is α_P(t) = α_P'(t-m_π^2) with α_P' = 0.7 GeV^-2. We choose the same trajectory for both π and η exchange. As explained in Sec. <ref>, this enhances the η pole to compensate for the Pomeron normalization in the ϕ photoproduction SDMEs. 
Finally, collecting all the pieces, we arrive to the amplitude in Eq. (<ref>) for a π or η exchange in the high-energy limit with the normalization β^P_0,V = (1/4)πα' m_V^2 g_VPγ g_PNN. 


It is instructive to derive the SDMEs for only a π exchange in both the GJ and helicity frames. The SDMEs induced by a π exchange take a simple form in the GJ frame, i.e., all SDMEs are zero except for ρ^1_1-1 = - ρ^2_1-1 = -1/2. This is of course expected since the π in its rest frame only has the spin projection zero. We can easily get the SDMEs for a π exchange in the helicity frame from the rotation in Eq. (<ref>): 
 

    ρ^0_00    = ρ^1_00 = -2 t /m_V^2/(1-t/m_V^2 )^2, 
    ρ^0_1-1    = -ρ^1_11 = -t /m_V^2/(1-t/m_V^2 )^2, 
    ρ^0_10    = ρ^1_10 = -1/√(2)√(-t)/m_V1+ t /m_V^2/(1-t/m_V^2 )^2, 
    ρ^0_11    = -1/21+ t /m_V^2/(1-t/m_V^2 )^2, 
    ρ^2_1-1    = -1/21- t /m_V^2/(1-t/m_V^2 )^2, 
    ρ^2_10    = -1/√(2)√(-t)/m_V1/(1-t/m_V^2 )^2.


In the case of a single exchange, the SDMEs depend only on the details of the photon vertex. The only scale that arises is the mass of the vector meson.



 §.§ Natural exchanges

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 general form in Eq (<ref>). Since the use of effective Lagrangians is very popular, it is instructive to compare our model in Eqs (<ref>) and (<ref>) to these types of interactions. 

Let us start with the standard interaction for a Pomeron exchange <cit.>

    M^ℙ_λ_γ, λ_V 
    λ, λ'(s,t)     = ϵ^*_ν(q,λ_V)  [k^μ ϵ^ν(k,λ_γ)  
     - k^νϵ^μ(k,λ_γ)  ] 
       ×u̅(p',λ')γ_μ  u(p,λ).

At leading order in s, we have k^μ→ n_+^μ√(s)/2, with n_±^μ = (1,0,0,± 1), and the helicity structure at the nucleon vertex is simply 

    u̅(p',λ')γ^μ  u(p,λ)→√(s) δ_λ,λ'  n_-^μ.
 
Only the first term in the bracket in Eq. (<ref>) survives. From the result

    ϵ^*(q,λ_V)·ϵ(k,λ_γ)     → - δ_λ_V,λ_γ + δ_λ_V,0λ_γ/√(2)√(-t)/m_V,

we conclude that this model for the Pomeron implicitly includes a single helicity flip structure and is not therefore purely helicity conserving. A more flexible model can be obtained with more general interactions. In order to determine all of the possible structures at both the photon and the nucleon vertices, let us first observe that in a factorizable model, the top and bottom vertices are linked by a propagator transverse to the momentum transferred. The propagator removes the x component since (q-k)^μ→ (0,√(-t),0,0) at leading order in s. Secondly, the general structures at the nucleon vertex are easily obtained. In addition to Eq. (<ref>), we can have an nucleon helicity-flip interaction

    u̅(p',λ') (2p^μ- γ^μ) u (p,λ)    → 2 λ√(-t)δ_λ',-λ  n_-^μ.

Note that any p' momentum can be substituted by p since the difference is orthogonal to the propagator. We summarized the two possible structures at the nucleon vertex in Eqs  (<ref>) and (<ref>) in Eq. (<ref>). 

At the photon vertex, the only tensorial structures that connect to the nucleon vertex and survive at leading order in s are k^μ→ n_+^μ√(s)/2  and ϵ^*( q,λ_V) → (1-λ^2_V) (q/m_V) n_+^μ. We can then form a single helicity-flip coupling at the photon vertex with the interaction

    ϵ^μ *( q,λ_V) q·ϵ(k,λ_γ)    →1/2λ_γ/√(2)√(-t)/m_Vδ_λ_V,0  n_+^μ.

Finally, since the maximum helicity difference between a photon and a vector meson in their center of mass is two, a tensor exchange should involve all possible relevant structures at the photon vertex. Indeed, we find that a double-flip structure can arise with the interaction between a photon, vector and tensor <cit.>:

    ϵ^*(q,λ_V)· k ϵ(k,λ_γ) · q →λ_γλ_V t/2.

We then conclude that, in addition to the nonflip interaction in Eq. (<ref>), the general structure with a photon, vector and natural exchange also includes the single-flip interaction in Eq. (<ref>) and the double-flip interaction in Eq. (<ref>). To leading order in s, we summarize these interactions with the intuitive vertex in Eq. (<ref>). 

In our model we added a helicity-independent exponential factor b_N to reproduce the energy-independent shrinkage of the differential cross section. 
This feature is generally described by exponential factors <cit.>, gamma functions <cit.> or dipole form factors <cit.>. This t-dependence originates from the energy dependence of the nearest cross-channel singularity. For the given Regge exchanges, these are the f_2(1270) and a_2(1320) tensor mesons. The energy dependence of these singularities in the cross channel can be described by Breit-Wigner line shape in t, the relevant energy variable in the cross channel:

    BW_E(t)     = m_E Γ_E/m_E^2-t-i m_E Γ_E,

where m_E and Γ_E are the mass and the width of the f_2 and a_2 tensor mesons. Its effect in the physical region of the direct channel can be modeled by a simple exponential falloff, i.e., |BW_E(t)|^2 ≈  |BW_E(0)|^2 e^2b_E t for t ∈ [-m_ω^2, 0]. We determine b_E at t = - m_ω^2/2, the middle point of the interval t ∈ [-m_ω^2, 0]. We find b_f_2 = 0.55 GeV^-2 and b_a_2 = 0.53 GeV^-2. 

The t-dependence of the Pomeron is often described by the following dipole form factors <cit.>
 

    F_1(t)     = 4m_p^2-2.8 t/(4m_p^2-t)(1-t/t_0)^2 ,
    
    F_V(t)     = 1/1-t/m_V^22μ_0^2+ m_V^2/2μ_0^2+ m_V^2-t,


with μ_0^2 = 1.1 GeV^2 and t_0=0.7 GeV^2. The form factor F_1(t) is the dipole approximation of the nucleon Dirac form factor <cit.>, and F_V(t) is an empirical form factor at the photon vertex.[We have chosen the normalization such that F_V(0) = 1.] As for the Regge exchanges, we approximate this form factor by an exponential falloff at t_0 = -m_ω^2/2, F_1(t_0)F_V(t_0) =  e^b_ℙ t_0. Under this approximation, we obtain b_ℙ = 3.60 GeV^-2. 

apsrev4-1


