Investigation of electroproduction of ϕ mesons off protons
PPKNU-NuHaTh-2020-01
Investigation of electroproduction of φ mesons off protons Sang-Ho Kim ∗ and Seung-il Nam
1, 2, † Department of Physics, Pukyong National University (PKNU), Busan 48513, Republic of Korea Asia Pacific Center for Theoretical Physics (APCTP), Pohang 37673, Republic of Korea (Dated: June 25, 2020)We investigate φ -meson electroproduction off the proton target, i.e., γ ∗ p → φp , by employing atree-level effective Lagrangian approach in the kinematical ranges of Q = (0 −
4) GeV , W = (2 − | t | ≤ . In addition to the universally accepted Pomeron exchange, we considervarious meson exchanges in the t channel with the Regge method. Direct φ -meson radiations in the s - and u -channels are also taken into account. We find that the Q dependence of the transverse ( σ T )and longitudinal ( σ L ) cross sections are governed by Pomeron and ( a , f ) scalar meson exchanges,respectively. Meanwhile, the contributions of ( π, η ) pseudoscalar- and f (1285) axial-vector-mesonexchanges are much more suppressed. The results of the interference cross sections ( σ LT , σ LT ) andthe spin-density matrix elements indicate that s -channel helicity conservation holds at Q = (1 − . The result of the parity asymmetry yield P (cid:39) .
95 at W = 2.5 GeV, meaning that natural-parity exchange dominates the reaction process. Our numerical results are in fair agreement with theexperimental data and thus the use of our effective Reggeized model is justified over the consideredkinematical ranges of Q , W , and t . PACS numbers: 13.40.-f, 13.60.Le, 14.40.CsKeywords: φ -meson electroproduction, Pomeron, ( a , f ) scalar meson, effective Lagrangian approach, Reggemethod. ∗ E-mail: [email protected] † E-mail: [email protected] a r X i v : . [ h e p - ph ] J un I. INTRODUCTION
Exclusive electroproduction of vector mesons is a suitable place to test model predictions in a kinematic region wherethe transition between the hadronic and partonic domains is involved according to the ranges of the photon virtuality Q and the photon center-of-mass (c.m.) energies W . The ZEUS and H1 Collaborations at HERA accumulated a lotof data for electroproductions of ρ - [1, 2], ω - [3], and φ - [2, 4, 5] light vector-mesons over wide ranges of Q and W (e.g., 2 . ≤ Q ≤
60 GeV and 35 ≤ W ≤
180 GeV at H1 [2]). The scale is large enough for perturbative quantumchromodynamics (pQCD) to be employed. Meanwhile, the Cornell group at the Laboratory of Nuclear Studies (LNS)at Cornell University [6–8], the CLAS Collaboration at Jefferson Lab [9–13], and the HERMES Collaboration atDESY [14–17] performed the experiments of vector-meson electroproductions at relatively low Q and W values (e.g.,1 . ≤ Q ≤ . and 3 . ≤ W ≤ . Q and W are more adequate for the hadronic or partonic descriptions?A series of works on electro- and photoproductions of light vector-mesons was carried out by Laget et al. previouslybased on the Regge phenomenology [18–21]. The exchanges of meson Regge trajectories and of a Pomeron trajectoryare considered in the t channel with phenomenological form factors for each vertex. The Q and t dependences on thecross sections at low photon energies ( W ≈ a few GeV) are reasonably described. Then Ref. [22] developed the workfor ρ -meson electroproduction by employing an effective Lagrangian approach with the updated CLAS data [9, 10].The transverse and longitudinal parts of the cross sections are examined in some detail. The separated componentshelp us to pin down the role of different meson exchanges and Pomeron exchange as well, which is difficult only withthe study of the unpolarized total cross section.In this paper, we take a similar approach for φ -meson electroproduction and test whether our hadronic descriptionis applicable or not in the kinematical ranges of Q = (0 −
4) GeV , W = (2 −
5) GeV, and | t | ≤ . We examinethe Q and t dependences on the transverse, longitudinal, and interference parts of the cross sections. The latterenables us to test s -channel helicity conservation (SCHC). The spin-density matrix elements of the produced φ mesonare also analyzed in the helicity frame which is in favor of SCHC. Parity asymmetry is calculated to check the relativestrengths of natural to unnatural parity exchanges in the t channel.For this purpose, we utilize our recent results for φ -meson photoproduction, γp → φp [23], where the relativecontributions among the Pomeron and various meson exchanges were discussed in detail by analyzing a vast amount ofCLAS data [24, 25]. The basic formalism used in Ref. [23] applied to the present work. However, the s -channel nucleonresonance contribution is excluded for brevity, although our kinematical range covers some high-mass resonanceregions. Indeed, in φ -meson photoproduction, the N ∗ contribution is found to be crucial only for the backward φ -meson scattering angles with small magnitudes and does not change much the integrated cross sections [23], whereasthe data for electro- and photoproductions of ρ - [22, 26] and ω - [12, 27] vector-mesons imply the necessity for the s -channel N ∗ contribution. We find that our hadronic approach provides a very successful description of the availableexperimental data over the considered kinematical ranges of Q , W , and t .The remaining part of this paper is organized as follows. In Sec. II, we define the kinematics of φ -meson electro-production process. In Sec. III, we explain the general formalism of the effective Lagrangian approach. We presentand discuss the numerical results in Sec. IV. The final section is devoted to the summary. II. KINEMATICS
Let us first specify kinematics of the φ -electroproduction process ep → eφp drawn in Fig. 1 graphically. Thefour-momenta of the involved particles described in the hadron production plane are given by γ ∗ ( k ) + p ( p ) → φ ( k ) + p ( p ) , (1)in parentheses, where k = ( (cid:112) k − Q , , , k ) , k = ( (cid:113) p + M φ , p sin θ φ , , p cos θ φ ) ,p = ( (cid:113) k + M N , , , − k ) , p = ( (cid:113) p + M N , − p sin θ φ , , − p cos θ φ ) . (2)They are defined in the γ ∗ p center-of-mass (c.m.) frame where the z axis is set to be parallel to the direction ofthe virtual photon and the y axis normal to the hadron production plane along (cid:126)k × (cid:126)k . Here, the magnitudes of thethree-momenta of the initial- and final-particles are given by k = λ ( − Q , M N , W ) / (2 W ) = M N (cid:112) ν + Q /W,p = λ ( M φ , M N , W ) / (2 W ) , (3) e e ′ Production Plane (c . m . )Electron Plane (lab) γ ∗ pφp Φ θ φ θ e FIG. 1. Graphical representation of the electron scattering ( ee (cid:48) γ ∗ )- and hadron production ( γ ∗ φp )-planes for the ep → e (cid:48) φp reaction defined in the laboratory (lab) and γ ∗ p center-of-mass (c.m.) frames, respectively. where the K¨all´en function is defined as λ ( x, y, z ) ≡ x + y + z − xy + yz + zx ).We define some relevant variables as follows: • Q = − k >
0, the negative four-momentum squared of the virtual photon, i.e., photon virtuality; • W = ( k + p ) = M N + 2 M N ν − Q , the square of the invariant mass of the γ ∗ p system, where ν = E e − E e (cid:48) is the energy transfer from the incident electron to the virtual photon in the laboratory (lab) frame; • t = ( k − k ) , the squared four-momentum transfer from the γ ∗ to the φ . t (cid:48) = | t − t min | , t min being the minimalvalue of t at fixed Q and W ; • θ e , the angle between the incident and scattered electrons; • Φ, the angle between the electron scattering ( ee (cid:48) γ ∗ ) and hadron production ( γ ∗ φp ) planes; • θ φ , the c.m. φ -meson angle relative to the virtual photon direction; III. THEORETICAL FRAMEWORK
We employ an effective Lagrangian approach here. The production mechanisms under consideration are drawnwith the relevant Feynman diagrams in Fig. 2, which includes Pomeron ( P ) ( a ), Reggeized f (1285) axial-vector(AV)-meson, ( π , η ) pseudoscalar (PS)-meson, and ( a , f ) scalar (S)-meson exchanges in the t channel ( b ), and direct φ -meson radiations via the proton in the s and u channels ( c and d ).We can write the invariant amplitude as M = ε ∗ ν ( λ )¯ u N (cid:48) ( λ f ) M µν u N ( λ i ) (cid:15) µ ( λ γ ) , (4)the helicities of the particles being given in parentheses. The forms of the invariant amplitudes are in completeanalogy to the φ -meson photoproduction case [23] for the corresponding diagrams. The Dirac spinors of the incomingand outgoing nucleons are designated by u N and u N (cid:48) , respectively. (cid:15) µ and ε ν denote the polarization vectors of thevirtual photon and the φ meson, respectively. The extension of the photo- to electro-production of mesons entails anadditional longitudinal component ( λ γ = 0) for the virtual-photon polarization vector in addition to the transverse( λ γ = ±
1) ones: (cid:15) ( ±
1) = 1 √ , ∓ , − i, , (cid:15) (0) = 1 (cid:112) Q ( k, , , E γ ∗ ) , (5)where E γ ∗ = (cid:112) k − Q = ( M N ν − Q ) /W . The polarization vectors of the virtual photon and the φ meson satisfythe conventional completeness relations [22] (cid:88) λ γ =0 , ± ( − λ γ (cid:15) µ ( λ γ ) (cid:15) ∗ ν ( λ γ ) = g µν − k µ k ν k , (cid:88) λ =0 , ± ε µ ( λ ) ε ∗ ν ( λ ) = − (cid:34) g µν − k µ k ν M φ (cid:35) . (6) Pomeron f (1285) ,γ ∗ ( k ) φ ( k ) p ( p ) p ( p ) γ ∗ φp pγ ∗ φp p γ ∗ φp ppp ( a ) ( b )( c ) ( d ) π, η, a , f FIG. 2. Tree-level Feynman diagrams for γ ∗ p → φp , which include Pomeron ( a ), Reggeized f (1285) axial-vector-meson, ( π , η )pseudoscalar-meson, and ( a , f ) scalar-meson exchanges in the t channel ( b ), and direct φ -meson radiations via the proton inthe s and u channels ( c and d ). A. Pomeron exchange
Figure 2(a) draws the Pomeron exchange that governs the scattering process in the high-energy and small t regions.We follow the Donnachie-Landshoff (DL) model [28] where a microscopic description of the Pomeron exchange invector meson photo- and electroproduction is given in terms of nonperturbative Reggeized-two-gluon exchange basedon the Pomeron-isoscalar-photon analogy (see Fig. 3) [29, 30]. Pomeron γ ∗ φp p Γ ν Γ µ F N FIG. 3. Quark diagram for Pomeron exchange in the DL model, based on the Pomeron-isoscalar-photon analogy.
As a consequence, the invariant amplitude for the Pomeron exchange can be expressed as M µν P = − M P ( s, t ) (cid:20)(cid:18) g µν − k µ k ν k (cid:19) / k − (cid:18) k ν − k ν k · k k (cid:19) γ µ − (cid:18) γ ν − / k k ν k (cid:19) k µ (cid:21) . (7)One find that the last term in the square bracket on the right-hand side (r.h.s.) breaks the gauge invariance and thefollowing modification is performed to make it gauge invariant [31]: k µ → k µ − ( p + p ) µ k · k ( p + p ) · k . (8)Other prescriptions for the spin structure for conserving the gauge invariance are detailed in Ref. [32] but are notused in this work, because qualitative descriptions of φ -meson electroproduction with them are found to be very poor.The scalar function in Eq. (7) is given by M P ( s, t ) = C P F φ ( t ) F N ( t ) 1 s (cid:18) ss P (cid:19) α P ( t ) exp (cid:20) − iπ α P ( t ) (cid:21) . (9)The strength factor is determined to be C P = 3 . s P = ( M N + M φ ) . F N ( t )and F φ ( t ) stand for the nucleon isoscalar electromagnetic (EM) form factor [18, 33] and the form factor of the γ P φ coupling [34, 35], respectively, and take the forms F N ( t ) = 4 M N − . t (4 M N − t )(1 − t/ . , F φ ( t ) = 2 µ Λ φ ( Q + Λ φ − t )(2 µ + Q + Λ φ − t ) , (10)The mass scale Λ φ is proportional to the quark mass of the loop diagram in Fig. 3 and is chosen to be Λ φ = M φ as done previously. The momentum scale is given by µ = 1 . and the Pomeron trajectory is known to be α P ( t ) = 1 .
08 + 0 . t . B. f (1285) axial-vector meson exchange We first consider Reggeized f (1285) AV meson among meson exchanges depicted in Fig. 2(b). Its importance isindicated in elastic p - p scattering and elastic photoproduction of ρ and φ mesons due to its special relation to the axialanomaly through the matrix elements of the flavor singlet axial vector current [36] and is confirmed more specificallyin φ -meson photoproduction done by the authors [23]. Thus we include f (1285) AV-meson exchange in φ -mesonelectroproduction.The effective Lagrangian for the AV V vertex is obtained from the hidden gauge approach [37] L γφf = g γφf (cid:15) µναβ ∂ µ A ν ∂ λ ∂ λ φ α f β , (11)where f denotes the f (1285) field with its quantum number I G ( J P C ) = 0 + (1 ++ ). The experimental data for thebranching ratio (Br) Br f → φγ = 7 . × − and the decay width Γ f = 22.7 MeV [38] lead to g γφf = 0 .
17 GeV − , (12)from Eq. (11).The effective Lagrangian of the AV meson interaction with the nucleon takes the form L f NN = − g f NN ¯ N (cid:20) γ µ − i κ f NN M N γ ν γ µ ∂ ν (cid:21) f µ γ N. (13)The coupling constant g f NN is obtained to be [39] g f NN = 2 . ± . , (14)and we use the maximum value g f NN = 3 .
0. Although the tensor term can have an effect on φ -meson electroproduc-tion, we take the value of κ f NN to be zero in this work for brevity.The corresponding invariant amplitude reads M µνf = i M φ g γφf g f NN t − M f (cid:15) µναβ (cid:34) − g αλ + q tα q tλ M f (cid:35) (cid:20) γ λ + κ f NN M N γ σ γ λ q tσ (cid:21) γ k β , (15)where q t = k − k . We substitute the exchange of the entire f (1285) Regge trajectory for the above single f (1285)meson exchange as [40] P Feyn f ( t ) = 1 t − M f → P Regge f ( t ) = (cid:18) ss f (cid:19) α f ( t ) − πα (cid:48) f sin[ πα f ( t )] 1Γ[ α f ( t )] D f ( t ) , (16)such that the spin structures of the interaction vertices are kept and the Regge propagator effectively interpolatesbetween small- and large-momentum transfers and can contribute to the high energy region properly. The Reggetrajectory is determined to be α f ( t ) = 0 .
99 + 0 . t [36] and the energy-scale factor s f = 1 GeV . The odd signaturefactor is given by [36] D f ( t ) = − − iπα f ( t ))2 . (17)The invariant amplitude is modified by introducing the following form factors g γφf → g γφf F γφf ( Q , t ) , g f NN → g f NN F f NN ( t ) , (18)where F γφf ( t, Q ) = Λ f − M f Λ f − t Λ q Λ q + Q , F f NN ( t ) = Λ f − M f Λ f − t , (19)which are normalized at t = M f and Q = 0 as F γφf = F f NN = 1. C. Pseudoscalar- and scalar-meson exchanges
Figure 2(b) also includes the contributions of the t -channel ( π , η ) PS- and ( a , f ) S-meson exchange diagrams. TheEM interaction Lagrangians for the PS- and S-meson exchanges, respectively, can be written as L γ Φ φ = eg γ Φ φ M φ (cid:15) µναβ ∂ µ A ν ∂ α φ β Φ , L γSφ = eg γSφ M φ F µν φ µν S, (20)where Φ = π (135 , − ) , η (548 , − ) and S = a (980 , + ) , f (980 , + ). The field-strength tensors for the photon and φ -meson are given by F µν = ∂ µ A ν − ∂ ν A µ and φ µν = ∂ µ φ ν − ∂ ν φ µ , respectively, and e the unit electric charge. TheEM coupling constants are deduced from the φ → Φ γ and φ → Sγ decay widthsΓ φ → Φ γ = e π g γ Φ φ M φ (cid:32) M φ − M M φ (cid:33) , Γ φ → Sγ = e π g γSφ M φ (cid:32) M φ − M S M φ (cid:33) . (21)With the φ -meson branching ratios of Br φ → πγ = 1 . × − , Br φ → ηγ = 1 . × − , Br φ → a γ = 7 . × − , andBr φ → f γ = 3 . × − and the value of Γ φ = 4 .
249 MeV [38], we obtain g γπφ = − . , g γηφ = − . , g γa φ = − . , g γf φ = − . . (22)The strong interaction Lagrangians for the PS- and S-meson exchanges read L Φ NN = − ig Φ NN ¯ N Φ γ N, L SNN = − g SNN ¯ N SN, (23)respectively. We use the following strong coupling constants determined by the Nijmegen potential [41, 42]: g πNN = 13 . , g ηNN = 6 . , g a NN = 4 . , g f NN = − . . (24)We obtain the invariant amplitudes for PS- and S-meson exchanges as M µν Φ = i eM φ g γ Φ φ g Φ NN t − M (cid:15) µναβ k α k β γ , M µνS = − eM φ g γSφ g SNN t − M S + i Γ S M S ( k · k g µν − k µ k ν ) , (25)respectively, where we use M a = 980 MeV, M f = 990 MeV, and Γ a ,f = 75 MeV [38].Here we also consider the form factors F γMφ ( t, Q ) and F MNN ( t ) for each vertex describing the dependence on the t and Q similar to Eqs. (18) and (19): F γMφ ( t, Q ) = Λ M − M M Λ M − t Λ q Λ q + Q , F MNN ( t ) = Λ M − M M Λ M − t , (26)where M = (Φ , S ). D. Direct φ -meson radiation term It is argued that the direct φ -meson radiation term drawn in Figs. 2(c) and 2(d) gives a small contribution to theunpolarized cross sections but a very distinct contribution to some polarization observables in φ -meson photoproduc-tion [23, 43]. Thus it is interesting to include this term in φ -meson electroproduction.The effective Lagrangians for the direct φ -meson radiation contributions can be written as L γNN = − e ¯ N (cid:20) γ µ − κ N M N σ µν ∂ ν (cid:21) N A µ , L φNN = − g φNN ¯ N (cid:20) γ µ − κ φNN M N σ µν ∂ ν (cid:21) N φ µ , (27)where the anomalous magnetic moment of the proton is κ p = 1 .
79 [38] and the vector and tensor coupling constantsfor the φ -meson to the nucleon are determined to be g φNN = − .
24 and κ φNN = 0 . φ -radiation invariant amplitudes are computed as M µνφ rad ,s = eg φNN s − M N (cid:18) γ ν − i κ φNN M N σ να k α (cid:19) (/ q s + M N ) (cid:18) γ µ F p + iF p κ N M N σ µβ k β (cid:19) , M µνφ rad ,u = eg φNN u − M N (cid:18) γ µ F p + iF p κ N M N σ µα k α (cid:19) (/ q u + M N ) (cid:18) γ ν − i κ φNN M N σ νβ k β (cid:19) , (28)for the s and u channels, respectively, with the EM form factors being involved. q s,u are the four momenta of theexchanged particles, i.e., q s = k + p and q u = p − k .Note that the Ward-Takahashi identity (WTI) is violated when a different form for the form factor F p is used forthe electric terms of the two invariant amplitudes. Thus we use the same form and can check the sum of them restoresthe WTI as M elec φ rad ,s ( (cid:15) → k ) = eg φNN k · p − Q ¯ u N (cid:48) (cid:18) / ε ∗ + κ φNN M N (/ ε ∗ / k − / k / ε ∗ ) (cid:19) (2 k · p − Q ) F p u N , M elec φ rad ,u ( (cid:15) → k ) = − eg φNN k · p + Q ¯ u N (cid:48) F p (2 k · p + Q ) (cid:18) / ε ∗ + κ φNN M N (/ ε ∗ / k − / k / ε ∗ ) (cid:19) u N , (29)such that M elec φ rad ( (cid:15) → k ) ∝ ( F p − F p ) = 0. We follow the suggestion given by David and Workman [45] for the formfactors: M φ rad = ( M elec φ rad ,s + M elec φ rad ,u ) F c ( s, u ) F p ( Q ) + M mag φ rad ,s F N ( s ) F p ( Q ) + M mag φ rad ,u F N ( u ) F p ( Q ) . (30)Here a common form factor is introduced which conserves the on-shell condition and the crossing symmetry: F c ( s, u ) = 1 − [1 − F N ( s )][1 − F N ( u )] , (31)with F N ( x ) = Λ N Λ N + ( x − M N ) , x = ( s, u ) . (32)Since the magnetic terms are self-gauge-invariant, the form of Eq. (32) is just used for them.Now, we give some details for the Dirac ( F ) and Pauli ( F ) form factors by using their relations with the Sachsone ( G E,M ) [46]: G E ( Q ) = F ( Q ) − κ N τ F ( Q ) , G M ( Q ) = µ N G E ( Q ) = F ( Q ) + κ N F ( Q ) , (33)where τ = Q / M p and correspondingly, F ( Q ) = G E ( Q ) + τ G M ( Q )1 + τ , F ( Q ) = G M ( Q ) − G E ( Q ) κ N (1 + τ ) . (34)The Sachs form factors are parametrized for the proton and the neutron in the literature by G pE ( Q ) (cid:39) G D ( Q ) , G pM ( Q ) (cid:39) µ p G D ( Q ) , G nE ( Q ) (cid:39) − aµ n τ bτ G D ( Q ) , G nM ( Q ) (cid:39) µ n G D ( Q ) , (35)with the dipole-type of form factor G D ( Q ) = (cid:20)
11 + Q (cid:104) r (cid:105) pE / (cid:21) , (36)where the electric root-mean-squared charge radius of the proton is given by (0.863 ± IV. NUMERICAL RESULTS AND DISCUSSIONS
We now discuss our numerical results from the present work. The remaining model parameters are the cutoff massesinvolved in the form factors. The cutoff masses for the t dependent form factors for meson exchanges are determinedto be Λ f ,a ,f = 1 . π,η = 0 . φ -meson radiations in Eq. (32) to be Λ N = 1 . φ -meson photoproduction at theconsidered energy region W = (2 −
3) GeV and at even much higher one W (cid:46)
10 GeV as well in our recent work [23].The cutoff masses for the Q dependent form factors [Λ q / (Λ q + Q )] for all meson exchanges are chosen to be Λ q = 0 . dσd Φ = 12 π ( σ + εσ TT cos 2Φ + (cid:112) ε (1 + ε ) σ LT cos Φ) , (37)where σ = σ T + εσ L . We refer to Appendix A for the explicit expressions for the T-L separated differential crosssections. If helicity is conserved in the s channel (SCHC), then the second and third terms vanish. The virtual-photonpolarization parameter ε is defined by ε = (cid:20) k Q tan θ e (cid:21) − . (38)In all our calculations, we fix it to be ε = 0 . W [GeV] -3 -2 -1 σ [ µ b ] Q =0Q =0.5Q =1.0Q =2.2Q =3.0 FIG. 4. Total cross sections for γ ∗ p → φp are plotted as a function of W for five different photon virtualities Q =(0 , . , . , . , .
0) GeV . The φ photoproduction ( Q = 0) data are from Refs. [48] (diamond), [49] (circle), and [50] (triangle).The Cornell [8] (star) and CLAS [13] (square) data correspond to the results at Q = 2.2 GeV . Figure 4 displays the results of the total cross sections as a function of W for five different photon virtualities Q .The slowly rising total cross sections with increasing W are kept for all values of Q due to the dominant Pomeroncontribution. The agreement with the experimental data [48–50] is good at the real photon limit Q = 0 over thewhole energy range. The magnitude of the total cross section when Q = 0 . reaches the level around 40%relative to that when Q = 0. The results get more suppressed for higher values of Q .It is essential to investigate the separated components of the cross sections to clarify the different role of consideredmeson exchanges. The CLAS Collaboration [13] extracted the interference cross sections to be σ TT = − . ± . σ LT = 2 . ± . dσ/d Φ data measured in the range of W = (2.0 − Q = (1.4 − using the relation Eq. (37). The matrix element is extracted as well to be r = 0 . ± .
12 from five bins ofthe polar angular distribution W (cos θ H ). With the additional assumption of SCHC, the ratio of the longitudinal totransverse cross section is obtained to be R = σ L /σ T = 1 . ± .
38. Also r − = 0 . ± .
23 and R = 0 . ± . W ( ψ = φ H − Φ) under the SCHC approximation. Lastly, thelongitudinal cross section is calculated to be σ L ( Q = 2 .
21 GeV ) = 4 . ± . R = 0 . ± .
24 and of the average cross section σ = 6 . ± . σ T ) and longitudinal ( σ L ) cross sections as functions of Q in theupper and lower panels, respectively, for three different c.m. energies W . We find that a predominant mechanism -1 σ T [ nb ] -1 -2 -1 σ L [ nb ] Q [GeV ] -2 -1 (a) W = 2.5 GeV (b) W = 2.8 GeV (c) W = 4.7 GeV(d) W = 2.5 GeV (e) W = 2.8 GeV (f) W = 4.7 GeV FIG. 5. Transverse σ T [(a)-(c)] and longitudinal σ L [(d)-(f)] cross sections are plotted as functions of Q for three different c.m.energies labeled on each subplot. The green dotted, cyan dot-dot-dashed, blue dot-dashed, and red dashed curves stand forthe contributions from the individual Pomeron, AV-meson, PS-meson, and S-meson exchanges, respectively. The black solidcurves indicate the total contribution. The CLAS data in panels (a) and (d) are extracted from Ref. [13]. that contributes to the transverse cross section σ T is the Pomeron exchange. The individual AV-, PS-, and S-mesonexchanges all have little influences on σ T for three W = (2 . , . , .
7) GeV energy values. The contribution ofthe Pomeron exchange at W = 2.5 GeV and Q = 2.21 GeV is in very good agreement with the CLAS dataas shown in Fig. 5(a). However, it is quite the opposite in the case of ρ -meson electroproduction. That is, thePS-meson exchange governs the transverse cross section σ T due to the M γ ∗ T + π ( η ) → ρ andPomeron exchange is relatively much more suppressed at low c.m. energies ( W ∼ a few GeV) and low photonvirtualities ( Q ∼ a few GeV ) [22]. In this work, employing a strong form factor for the PS-meson exchange obviouslyoverestimates the available φ -meson electroproduction CLAS data. Thus we use a rather small value of the cutoffmass as Λ π,η = 0 . σ L is an order ofmagnitude smaller than that for σ T at W = 2.5 GeV and Q = (1 −
4) GeV as shown in Fig. 5(d). The differencebecomes larger for higher c.m. energies W = 2 . , . σ L . That is how the S-meson exchange form factor is determined,i.e., Λ a ,f = 1 . σ L is highlyenhanced relative to σ T and even prevails over that of the Pomeron exchange over all the ranges of W and Q . Allthe amplitudes for the longitudinal photons must vanish at the limit Q = 0 and this behavior is imposed explicitlyin our calculation. For both σ T and σ L cases, the contribution of the AV-meson exchange is comparable to that ofthe PS-meson exchange or even more suppressed.We present the results of the interference cross sections σ TT and σ LT in the upper and lower panels in Fig. 6,respectively. The overall results are the decrease of the absolute magnitudes with increasing W for all contributions,indicative of SCHC at relatively higher values of W and Q . The meson-exchange contributions are all close to zerofor σ TT and the total contribution is entirely dependent on the Pomeron exchange which is the strongest at Q = 0and gradually decreases with increasing Q . At Q ≥ , the results are consistent with zero for all values of W ,which are within the CLAS data at W = 2.5 GeV and Q = 2.21 GeV as shown in Fig. 6(a).Different patterns are observed for the results of the individual σ LT cross sections in comparison to σ TT as seenin Fig. 6(d)-6(f). The overall positive sign applies to the Pomeron contribution for σ LT . It is peaked at about 0.3GeV and falls off with increasing Q . The signs of the PS- and S-meson contributions are the same each otherbut are opposite to that of the Pomeron contribution. The CLAS data shown in Fig. 6(d) is close to the Pomeroncontribution. However, the inclusion of PS- and S-meson exchanges pulls down σ LT and finally the total contributionreaches zero at Q = 2.21 GeV . That is one more reason why PS-meson exchange should be suppressed in φ -meson0 σ TT [ nb ] σ LT [ nb ] Q [GeV ] (a) W = 2.5 GeV (b) W = 2.8 GeV (c) W = 4.7 GeV(d) W = 2.5 GeV (e) W = 2.8 GeV (f) W = 4.7 GeV FIG. 6. The same as in Fig. 5 but for the interference cross sections σ TT (upper panels) and σ LT (lower panels), respectively. electroproduction. The small increase of the cutoff masses for both the PS- and S-meson form factors from the presentones makes the total results of σ LT worse. The AV-meson exchange contributes almost negligibly to both σ TT and σ LT . However, note that its contribution to σ TT is much more sensitive than that to σ LT under the variation of thecutoff mass Λ f . -1 -1 -1 -1 (a) W = 2.3 GeV(2.0-2.6 GeV) (b) W = 2.5 GeV(2.0-3.0 GeV)(c) W = 2.8 GeV(2.0-3.7 GeV) (d) W = 4.7 GeV (4-6 GeV) Q [GeV ] σ ( Q ) [ nb ] FIG. 7. Total cross sections are plotted as functions of Q for four different c.m. energies labeled on each subplot. The curvesare defined in the caption of Fig. 5. The circle [11] and square [13] data are from the CLAS Collaboration. The star andtriangle data from the Cornell [8] and HERMES Collaboration [14], respectively. (b),(c),(d) The photoproduction points at Q = 0 are from Refs. [48, 49]. Figure 7 depicts the results of the unpolarized total cross sections as functions of Q for four different c.m. energies W . The model parameters are all constrained previously from the study of the separated cross sections althoughthe available data on a Rosenbluth separation [52] are too poor to be a reliable basis for verifying the φ -mesonelectroproduction mechanism. It is interesting that the unpolarized cross sections are also well described over the whole1 -2 -1 -1 -1 -1 (a) 〈 Q 〉 = 2.2 GeV (b) 〈 Q 〉 = 0.23 GeV (c) 〈 Q 〉 = 0.43 GeV (d) 〈 Q 〉 = 0.97 GeV 〈 W 〉 = 2.9 GeV 〈 W 〉 = 2.9 GeV 〈 W 〉 = 2.9 GeV 〈 W 〉 = 2.5 GeV ( Q + M φ _______ M φ ) ( Q + M φ _______ M φ ) d σ ___d t [ nb_____ G e V ] d σ ___d t [ nb_____ G e V ] d σ ___d t ( Q + M φ _______ M φ ) [ nb_____ G e V ] d σ ___d t -t [GeV ] [ nb_____ G e V ] -t [GeV ]-t [GeV ]t ′ [GeV ] FIG. 8. (a) Differential cross section dσ/dt is plotted as a function of t (cid:48) = | t − t min | and compared with the CLAS data [13].(b),(c),(d) dσ/dt multiplied by the factor [( Q + M φ ) /M φ ] is plotted as functions of − t for three different photon virtualitieslabeled on each subplot at W = 2 . kinematical ranges of W and Q , since the Pomeron exchange is mainly responsible for describing the experimentaldata. Although small, the effects of meson exchanges are revealed at larger values of Q mostly due to the milderslope of the S-meson contribution than the Pomeron one. Our effective hadronic model accounts for the points at thereal photon limit Q = 0 as expected from Fig. 4,The results of the differential cross section dσ/dt are displayed in Fig. 8(a) at Q = 2.2 GeV and W = 2.5 GeVas a function of t (cid:48) ≡ | t − t min | where t min stands for the minimum value of t at fixed values Q and W . Figures 8(b)-8(d) depict the results of dσ/dt , which are multiplied by the factor [( Q + M φ ) /M φ ] to eliminate the φ -propagatordependence, as functions of − t for the three different photon virtualities Q at W = 2 . t dependence is properly described.The strength of the S-meson exchange becomes larger than that of the Pomeron exchange at | t | (cid:38) . . Itshould be mentioned that the a - and π -meson contributions are more important than those of the f and η mesons,respectively. Q [GeV ] R FIG. 9. The ratio of cross sections for longitudinally and transversely polarized photons R = σ L /σ T is plotted as a functionof Q at W = 2.5 GeV. The green dotted and black solid curves stand for the Pomeron and total contributions, respectively.The data are from the Cornell [7] (circle) and CLAS Collaboration [13] (square). Q [GeV ] r Re r r r r Re r r Im r
Im r
FIG. 10. Various matrix elements are plotted as functions of Q at W = 2.5 GeV. The extracted data for r and r − arefrom the CLAS Collaboration [13]. We present the results of the ratio of the longitudinal to transverse cross section R = σ L /σ T in Fig. 9 at W = 2.5 GeValthough the Cornell data (circle) correspond to W = 2.9 GeV. Our results yield R ≤ . Q . The agreement is noticeablybetter when the S-meson exchange is additionally included. Note that as the fixed c.m. energy W increases, the riseof R with respect to Q becomes smaller as indicated in Fig. 5 where the transverse and longitudinal components ofthe total contribution exhibit the opposite pattern with W .Finally, we make our predictions of various matrix elements, denoted by r αij and related to the φ -meson spin-densitymatrix elements (SDMEs) [51]. When the experiments cannot conduct a σ L /σ T separation, r αij can be represented as r ij = ρ ij + εRρ ij εR ,r αij = ρ αij εR , for α = (0 − ,r αij = √ R ρ αij εR , for α = (5 − . (39)We refer to Appendix B for the definitions of the SDMEs ρ αij . The matrix elements are described in the helicity frame,where the φ meson is at rest and its quantization axis is chosen to be antiparallel to the momentum of the outgoingproton in the c.m. frame of the hadron production process.The results are shown in Fig. 10 as functions of Q for nine different matrix elements labeled on each subplot. IfSCHC holds, all presented matrix elements becomes zero except for r , r − , and Im r − . It turns out that theSCHC approximation indeed applies at Q = (1 −
4) GeV . This conclusion is consistent with the results of Fig. 6where SCHC is verified from the interference cross sections σ TT and σ LT . A good agreement with the CLAS data isobtained for r and r − .It is useful to examine the relative contribution between the natural ( N ) and unnatural ( U ) parity exchangeprocesses for the transverse cross section. The parity asymmetry is defined by [40, 51] P ≡ σ NT − σ UT σ NT + σ UT = (1 + εR )(2 r − − r ) , (40)and our results yield P (cid:39) W = 2.5 GeV and Q = (0 −
4) GeV . This observation indicates that the dominant mechanism of the transverse cross section is the3natural-parity exchange. Also when a purely natural-parity exchange is considered, e.g., the S-meson exchange, wehave the following relation [12, 53]:1 − r + 2 r − − r − r − = 0 . (41)For the Pomeron contribution, our results yield the values close to zero. The results from the total contribution yield (cid:39) V. SUMMARY
We have investigated the reaction mechanism of φ -meson electroproduction off the proton target based on the gauge-invariant effective Lagrangians in the tree-level Born approximation. Each contribution of the Pomeron, Reggeized f (1285) AV-meson, ( π , η ) PS-meson, and ( a , f ) S-meson exchanges is scrutinized in the t channel diagram employingthe CLAS and Cornell data. The direct φ -meson radiations via the proton are also taken into account in the s and u channels simultaneously to conserve gauge invariance. We summarize the essential points on which our modelcalculations have performed. • The unpolarized cross sections σ show slow rising with increasing W . The results for the real photon limit Q = 0 match with the available data very well. The magnitudes of σ become smaller with increasing Q and the main contribution turns out to be Pomeron exchange. The cross sections σ give us no insight intowhat meson exchanges contributes to φ -meson electroproduction and thus it is necessary to examine the T-Lseparated cross sections. • The Pomeron and S-meson exchanges dominate the transverse ( σ T ) and longitudinal ( σ L ) cross sections, respec-tively, at Q = (0 −
4) GeV for three considered W = 2 . , . , . σ T at W = 2.5 GeV and Q = 2.21 GeV . Other mesoncontributions are more suppressed and the difference between the Pomeron and meson contributions becomeslarger as W increases. Also the CLAS data on σ L at the same W and Q values are accounted for solely by theS-meson exchange. The Pomeron contribution for σ T falls off faster than the S-meson contribution for σ L as Q increases. • The interference cross sections σ TT and σ LT are found to be useful to clarify the role of AV- and PS-mesonexchanges which is difficult from the study of σ T and σ L . First, for σ TT , all meson-exchange contributions arecompatible to zero and the total contribution is determined entirely by the Pomeron exchange which is negativein sign and the strongest at Q = 0 and falls off steadily with increasing Q . However, note that the AV-mesonexchange is relatively more sensitive than other meson exchanges under the variation of each cutoff mass andits role can be more clearly verified when more experimental data on σ TT are produced. Next, for the case of σ LT , the patterns of each contribution are totally different from σ TT . The Pomeron contribution is peaked atabout Q = 0 . and falls off gradually with increasing Q . The S-meson and PS-meson contributions alsoshow similar peak positions. The signs of these two contributions are the same each other but are opposite tothat of the Pomeron contribution. If the S-meson and PS-meson contributions increase from the present ones,then the total contribution will become negative and deviate from the CLAS data at W = 2.5 GeV and Q =2.21 GeV . Thus the S-meson and PS-meson contributions must be small from the present results. The effectof the AV-meson exchange is almost negligible. For both σ TT and σ LT cases, the total contribution is close tozero at Q = (1 −
4) GeV , indicating SCHC. • The t dependence of the differential cross sections dσ/dt is also well described in the range of | t | = (0 − . The Pomeron exchange dominates over the whole t regions. Although small, the role of the mesonexchanges are significant only at | t | (cid:38) mostly from the S-meson exchange. The ratio of the longitudinalto transverse cross section R = σ L /σ T rises linearly with increasing Q . R for the total contribution is about7 times larger than that for the Pomeron contribution due to the S-meson contribution and is good agreementwith the experimental data. • We examine various matrix elements defined in helicity frame which is in favor of SCHC. We find that thevalues of Re r , r − , r , r , Re r , and Im r are close to zero at Q = (1 −
4) GeV and thus SCHC holds.Our results match with the CLAS data on r and r − . When we come to ρ -meson electroproduction, inthe similar low Q and W ranges, it is difficult to draw a firm conclusion concerning SCHC although mostphysical observables seem to support SCHC [10]. On the contrary, SCHC is known to be broken in w -mesonelectroproduction because of the different contributions of Pomeron and various meson exchanges [12]. Also a4small but significant violation of SCHC is found in φ -meson electroproduction in the high ranges of Q and W [4], where the generalized parton distributions (GPD) and factorization of scales will become relevant. • The parity asymmetry provides us with the information of the relative strength of the natural to unnaturalparity exchanges in the t channel for σ T . Our results yield P (cid:39) φ -meson radiationsvia the proton is found to be almost negligible.The currently available data on φ -meson electroproduction are very limited and new experiments at current or futureelectron facilities are strongly called for. We can gain a deeper understanding of φ -meson electroproduction mechanismby comparing our numerical results with those of the GPD-based model. It is valuable to extend the present work toelectroproductions of ρ , ω , and J/ψ mesons. The corresponding work is underway.
ACKNOWLEDGMENTS
The authors are grateful to A. Hosaka (RCNP) for fruitful discussions. This work is supported in part by theNational Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (MSIT)(NRF-2018R1A5A1025563). The work of S.H.K. was supported by NRF-2019R1C1C1005790. The work of S.i.N. wasalso supported partly by NRF-2019R1A2C1005697.
Appendix A: APPENDIX A: T-L SEPARATED DIFFERENTIAL CROSS SECTIONS
The separated components of the differential cross sections in the Rosenbluth formula [52] take the forms1 N dσ T dt = 12 (cid:88) λ γ = ± |M ( λ γ ) | , N dσ L dt = |M ( λ γ =0) | , N dσ TT dt = − (cid:88) λ γ = ± M ( λ γ ) M ( − λ γ ) ∗ , N dσ LT dt = − √ (cid:88) λ γ = ± λ γ ( M (0) M ( λ γ ) ∗ + M ( λ γ ) M (0) ∗ ) , (A1)for vector-meson electroproduction. The common kinematical factor N is defined by N = [32 π ( W − M N ) W k ] − . (A2)Here the squared invariant amplitude is expressed as |M ( λ γ ) | = 12 (cid:88) λ i ,λ f ,λ M λ f λ ; λ i λ γ M ∗ λ f λ ; λ i λ γ , (A3)where the averaging over the incoming nucleon ( λ i ) helicity and the summation over the outgoing φ meson ( λ ) andnucleon ( λ f ) helicities are indicated. M ∗ λ f λ ; λ i λ γ stands for the complex conjugate of the amplitude M λ f λ ; λ i λ γ . Notethat the differential cross sections have the following relations: d Ω φ dt = π | k || p | . (A4) Appendix B: APPENDIX B: SPIN-DENSITY MATRIX ELEMENTS
We obtain nine components of the spin-density matrix elements (SDMEs) of the φ meson if those of the virtualphoton are decomposed into the standard set of nine matrices Σ α ( α = 0 −
8) [51]: ρ αλλ (cid:48) = 12 N α (cid:88) λ γ ,λ (cid:48) γ ,λ i ,λ f M λ f λ ; λ i λ γ Σ αλ γ λ (cid:48) γ M ∗ λ f λ (cid:48) ; λ i λ (cid:48) γ , (B1)5where the normalization factors N α are defined by N α = N T = 12 (cid:88) λ γ = ± ,λ,λ i ,λ f |M λ f λ ; λ i λ γ | for α = (0 − ,N α = N L = (cid:88) λ,λ i ,λ f |M λ f λ ; λ i | , for α = 4 ,N α = (cid:112) N T N L for α = (5 − . (B2)We finally obtain the SDMEs in terms of the helicity amplitudes: ρ λλ (cid:48) = 12 N T (cid:88) λ γ = ± M λλ γ M ∗ λ (cid:48) λ γ ,ρ λλ (cid:48) = 12 N T (cid:88) λ γ = ± M λ − λ γ M ∗ λ (cid:48) λ γ ,ρ λλ (cid:48) = i N T (cid:88) λ γ = ± λ γ M λ − λ γ M ∗ λ (cid:48) λ γ ,ρ λλ (cid:48) = 12 N T (cid:88) λ γ = ± λ γ M λλ γ M ∗ λ (cid:48) λ γ ,ρ λλ (cid:48) = 1 N L M λ M ∗ λ (cid:48) ,ρ λλ (cid:48) = 1 √ N T N L (cid:88) λ γ = ± λ γ M λ M ∗ λ (cid:48) λ γ + M λλ γ M ∗ λ (cid:48) ) ,ρ λλ (cid:48) = i √ N T N L (cid:88) λ γ = ±
12 ( M λ M ∗ λ (cid:48) λ γ − M λλ γ M ∗ λ (cid:48) ) ,ρ λλ (cid:48) = 1 √ N T N L (cid:88) λ γ = ±
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