Investigating convergence of the reaction $\gamma p\to\pi^\pm\Delta$ and tensor meson $a_2$ exchange at high energy

A Regge approach to the reaction processes $\gamma p\to\pi^-\Delta^{++}$ and $\gamma p\to\pi^+\Delta^0$ is presented for the description of existing data up to $E_\gamma= 16$ GeV. The model consists of the $t$-channel $\pi(139)+\rho(775)+a_2(1320)$ exchanges which are reggeized from the relevant Born amplitude. Discussion is given on the minimal gauge prescription for the $\pi$ exchange to render convergent the divergence of the $u$-channel $\Delta$-pole in the former process. A new Lagrangian is constructed for the $a_2N\Delta$ coupling in this work and applied to the process for the first time with the coupling constant deduced from the duality plus vector dominance. It is shown that, while the $\pi$ exchange dominates over the process, the role of the $a_2$ exchange is crucial rather than the $\rho$ in reproducing the cross sections for total, differential, and photon polarization asymmetry to agree with data at high energy.

Photoproduction of ∆-baryon is an example of the hadron reaction to study the formation of a composite particle from the πN interaction. Efforts in existing theories and experiments on this reaction process have been concentrated mainly on understanding of the dynamical and static properties of ∆-baryon around the ∆-resonance region [1][2][3]. But, it could also be of value to question how the production mechanism would proceed over the region, and it is, therefore, a challenging issue to establish a model for the γp → π − ∆ ++ process which is valid at high energy, because the ∆-propagation in the process would be highly divergent there.
In experiments evidences are clear for the formation of the ∆-baryon through the reaction γN → ππN in the photon energy below the threshold of ρN production. At high energies the reaction cross sections for the γp → π − ∆ ++ process were measured for the differential, spin observables for the photon beam and density matrix elements up to photon energy E γ = 16 GeV [4][5][6]. Only recently the total and differential cross sections for γp → π − ∆ ++ were obtained from threshold up to √ s = 2.6 GeV at the ELSA [7]. In particular, the total cross section for γp → π − ∆ ++ at the ELSA shows a sharp peak of σ max ≈ 70 µb in size around E γ ≈ 1. 6 GeV with the steep decrease following over the resonance region, and, hence, exhibits a typical feature of the nondiffractive two-body process.
In this work we will analyze the production mechanism of the reactions γp → π − ∆ ++ and γp → π + ∆ 0 at high energies with a focus on the convergence of the reaction processes there. From a theoretical point of view only the π exchange in the t-channel peripheral subprocess is expected to dominate at high energies and small momentum transfer. Hence the production amplitude should be M ∝ q · ǫ/(t − m 2 π ), where ǫ is photon polarization, and * E-mail: bgyu@kau.ac.kr † E-mail: kong@kau.ac.kr q and m π are pion momentum and mass. However, since the t-channel π exchange itself is not gauge invariant, an extension of the production amplitude is needed for gauge invariance. Furthermore this should be a specialized one for the extended amplitude has to be convergent at high energy, even if it includes highly divergent ∆ propagation for gauge invariance. For this requirement a theoretical speculation was suggested in Ref. [8] that the amplitude, thus extended, contain only the charge couplings of the s-channel proton-pole and u-channel ∆-pole coupling to photon field in the γp → π − ∆ ++ process. In other words, the transverse components in these s and u-channel poles should be removed in order for the convergence of the process to be ensured at high energy. This leads to the so-called the minimal gauge prescription in the sense that the proton-pole and ∆-pole terms are minimally introduced for gauge invariance of the t-channel π exchange. Moreover, such a scheme seems to be reasonable because the higher multipoles of the ∆-baryon and proton electromagnetic moments are defined uniquely in the static limit and such a uniqueness can no longer be valid at high energy.
Application of the minimal gauge condition is found in Ref. [9] in which case the dynamics of γN → π ± ∆ was investigated in the kinematical region, −t ≤ m 2 π and s → ∞ at forward angles. A more qualitative analysis of the reaction was made in Ref. [10] by using the π +b 1 +ρ+a 2 Regge-pole exchanges in the t-channel helicity amplitude with their residues and cuts considered to fit to data.
On the other hand, we note that the tensor-meson a 2 plays the role to significantly improve the cross sections for the differential and spin polarizations in the γN → π ± N process at high energy [11]. The significance of such a higher-spin interaction is confirmed in the cases of γp → K + Λ (Σ 0 ) as well by the role of the K * 2 [12]. Nevertheless, there are no attempts at present, however, to investigate the role of the tensor meson a 2 in the π∆ photoproduction utilizing the effective Lagrangian except for the case of the Regge-pole fit to data discussed above.
Typeset by REVT E X In this work our interest is to construct a model for the γp → π − ∆ ++ process at high energy where the reaction cross sections can be described without either fit parameters or any counter terms included in ad hoc fashion. For this purpose we consider to incorporate the two basic ingredients with the model, i.e., the minimal gauge-invariance and the role of the spin-2 tensor-meson exchange.
For a heuristic introduction of the minimal gauge prescription, let us begin with the Born amplitude for the process, where the production amplitude for the γp → π − ∆ ++ process consists of the proton-pole in the s-channel, the ∆ ++ -pole in the u-channel, and the π − exchange in the tchannel to respect the charge conservation, e N −e π −e ∆ = 0, with the charges of nucleon, ∆, and π denoted by e N , e ∆ , and e π , respectively. These are summarized in diagrams in Fig. 1.
With the contact term further the reggeized π exchange which is gauge invariant is, thus, given by [11] where with the masses of nucleon, ∆, and π denoted by M N , M ∆ , m π , respectively. Here, the off-shell effect in the πN ∆ vertex is neglected for simplicity. For the charge coupling of the γ∆∆ vertex we use and the spin projection for the spin-3/2 ∆ baryon, The t-channel Regge pole in Eq. (2) is given by written collectively for ϕ(= π, ρ, a 2 )-meson of spin J with s 0 = 1 GeV 2 . For the phase of the Regge pole the canonical form of 1 2 ((−1) J + e −iπαϕ(t) ) is generally assigned to the exchange non-degenerate meson [13]. In the case of the exchange-degenerate (EXD) trajectories π-b 1 and ρ-a 2 pairs the determination of the phases will be discussed later.
The minimal gauge prescription for π exchange As discussed above the reaction amplitude converging at high energy should couple to the Coulomb component of photon field, and such a condition should be taken into account in the gauge invariant amplitude in Eq. (2).
We recall that the u-channel ∆-pole as well as the schannel proton-pole term is introduced merely to preserve gauge invariance for the t-channel π exchange at high energy, as discussed in Ref. [8]. We, then, consider only the charge couplings of the s-, and u-channel amplitudes that are indispensable to restore gauge invariance, but remove all the transverse ones which are redundant by gauge invariance. In the proton-pole term in Eq. (3), for instance, only the 2p · ǫ term has to be there, whereas such a gauge invariant term, / k/ ǫ from the (/ p + / k + M N )/ ǫ as well as from the magnetic moment term is redundant and, hence, excluded.
Similarly, in the case of the u-channel amplitude in Eq.
(4) which is written in a fully expended form as only the first term proportional to 2ǫ · p ′ is not invariant and should be reserved, whereas all the others are excluded by redundancy. It should be noted that this simplification is possible only when the M u(∆) term can be written in such an antisymmetric form as in Eq. (12) with respect to photon polarization ǫ and momentum k, and this can only be done by the benefit of the second term −ǫ ν γ α in the γ∆∆ coupling in Eq. (4). This signifies the validity of the Ward identity Eq. (8) at the photon coupling vertex. The invariant amplitude M s(N ) + M u(∆) + M t(π) + M c in Eq. (2) in this minimal gauge is, therefore, written as [8,9] In the numerical analysis, the effect of the minimal gauge on the cross section will be examined as the photon energy increases up to 16 GeV. The value for the coupling constant f π − p∆ ++ is scattered in various reactions, i.e., from the quark model prediction f πN ∆ = 6 √ 2 5 f πN N ≈ 1.7 with f πpp = 1 for the NN interaction [14] to f π − p∆ ++ = 2.16 from the decay width Γ ∆→πN = 120 MeV for the π∆ photoproduction [15]. We consider the one within the range of 1.7 ≤ f πN ∆ ≤ 2.16 that is better to agree with experimental data.
ρ and a2 exchanges The reggeized ρ exchange in the t-channel is expressed as [15], where we use the interaction Lagrangian ≈ 5.05/m ρ is estimated from the one-boson-exchange in the NN potential [14] by using g ρN N = 2.6 and κ ρ = 3.7 [16,17]. The radiative decay constant is determined from the decay width Γ ρ→γπ = 67.1 keV and we obtain g γρπ = ±0.224, by using the Lagrangian, It is expected that the exchange of tensor meson a 2 (1320) of spin-2 plays a role at high energy from the previous studies of the charged pion photoproduction [9][10][11]. However, no information is available at present either for the interaction Lagrangian or for the coupling constants of the a 2 N ∆ interaction. In this work we construct a new Lagrangian for the a 2 N ∆ coupling for application.
By considering parities and spins, but neglecting the off-shell effect in the meson-∆ coupling for simplicity, we write the Lagrangian for the a 2 N ∆ coupling as 1 in favor of using the identity to determine the f a2N ∆ coupling constant, which follows the duality and vector meson dominance [10]. Here ← → ∂ ν = ( − → ∂ ν − ← − ∂ ν )/2. The γπa 2 coupling was derived in Ref. [18] and the interaction Lagrangian is given by whereF αβ = 1 2 ǫ µναβ F µν is the pseudotensor of photon field. The decay of the a 2 → πγ is reported to be Γ a ± 2 →π ± γ = (311 ± 25) keV in the Particle Data Group (PDG), which gives g γπa2 = ±0.276. 1 The interaction form of the Lagrangian in Eq. (17), of course, might not be unique but one of the possible couplings between a 2 and N ∆ baryon transition-current, and one could also consider the interaction of form L a 2 N∆ = f a 2 N∆∆ν γµγ 5 N a µν 2 by replacing the ρ νµ with a νµ 2 in Eq. (15), for instance, though unnatural to identify the ρ νµ with a νµ 2 . In this case, however, he(she) cannot use the identity in Eq. (18) to determine the f a 2 N∆ , but has to consider it as a parameter to fit to data, because of the different mass dimensions between the f ρN∆ and f a 2 N∆ .
As the only unknown coupling constant f a2N ∆ is determined, once the f ρN ∆ is given, there are, therefore, no free parameters in the present calculation. We choose the signs of the coupling constants g γπρ and g γπa2 to agree with photon polarization asymmetry Σ of γp → π − ∆ ++ and γp → π + ∆ 0 .
The reggeized a 2 exchange is given by where P = (p + p ′ )/2 and the spin-2 projection is (21) with η βρ = −g βρ + Q β Q ρ /m 2 a2 . In the Regge framework the predictions for the physical observables are strongly dependent on the phase and sensitive to the intercept of the trajectory as well. Therefore, to use the right phase and trajectory is of importance, though there is no rigorous theory for this purpose. One feasible scheme for this is the addition of the phases of the EXD pairs π-b 1 and ρ-a 2 as discussed in Ref. [13].

Results and discussion
Given the trajectories for π, ρ, and a 2 Regge-poles as respectively, we calculate the total cross section for γp → π − ∆ ++ and present the result in Fig. 2, where the blue dashed line denotes the total cross section σ from the case of the π exchange with the EXD phase, 1, as determined in Eq. (23), and the solid line is from the case of the canonical phase for the π exchange, respectively. For a better agreement with data as shown in the resonance region we choose the canonical phase for the dominating π exchange in the absence of b 1 exchange in the present calculation. I. Meson-baryon coupling constants for (a) γp → π − ∆ ++ and (b) γp → π + ∆ 0 processes. The superscript indicates the phase taken for the process denoted.
A summary of the coupling constants and the phases of the exchanged mesons used for the present calculation is given in Table I. Figure 2 shows the convergence of the total cross section up to E γ = 16 GeV depending on the treatment of the ∆-pole, i.e., by using Eq. (2) for the full propagation of ∆, or by the minimal gauge as in Eq. (13). We also examine the validity of the approximation, Π µν ∆ ≈ −g µν widely used for hadron reactions involved in the ∆ coupling. Both the cases of the ∆-pole with the full propagation (red dashed line) and with the approximation of Π µν ∆ ≈ −g µν in Eq. (10) (red dotted line) are highly divergent as the photon energy increases, whereas a good behavior of the cross section for convergence is obtained by using the minimal gauge as shown by the solid and the blue dashed lines. These findings confirm the validity of the minimal gauge for the high energy behavior of the π∆ photoproduction. The role of the tensor meson a 2 in the reaction process is evident in the differential cross section dσ/dt and the photon polarization asymmetry which is defined as in the c. m. frame of the pion production plane. We present the cross sections for the differential and the photon polarization asymmetry from the SLAC data for the γp → π − ∆ ++ at E γ = 16 GeV in Fig. 3, and for the γp → π + ∆ 0 in Fig. 4, respectively. While our model predictions are remarkably in good agreement with data, the tensor meson a 2 exchange gives the contribution crucial to agree with existing data in comparison to the dotted lines resulting from the ρ + π exchanges without a 2 in the minimal gauge.
To summarize, we have investigated the γp → π − ∆ ++ process up to E γ = 16 GeV with our focus on the production mechanism of the reaction process at high energy. For this purpose we constructed a Born term model where the π exchange is reggeized in the t-channel with the u-channel ∆-pole included for gauge invariance in addition to the s-channel proton-pole and the contact term. In order to make convergent the energy-dependence of the reaction cross section against the divergence of the ∆-pole at high energy we utilized the minimal gauge prescription to simplify the ∆-pole, which is possible for the antisymmetric form of the charge coupling terms in the γ∆∆ vertex due to the Ward identity at the vertex.
We further showed that the tensor meson a 2 exchange plays the key role to agree with the existing data on the differential and photon polarization asymmetry for γp → π − ∆ ++ and γp → π + ∆ 0 at high energy. For doing this we constructed a new effective Lagrangian for the tensor meson-nucleon-∆ coupling in this work and demonstrate its validity by obtaining quite improved differential cross section as well as photon polarization asymmetry with the coupling constant fa 2 N ∆ ma 2 = −3 fρN∆ mρ which is deduced from the duality plus vector dominance.
While revealing the well-known approximation Π µν ∆ ≈ −g µν to be valid only in the limited energy region near threshold, we understand the production mechanism in this minimal gauge as the dominance of the π exchange incorporating with the a 2 exchange rather than the ρ.