Resonance dynamics in the coherent $\eta$ meson production in the $(p,p^\prime)$ reaction on the spin-isospin saturated nucleus

For the forward going proton and $\eta$ meson, the coherent $\eta$ meson production in the $(p,p^\prime)$ reaction on the spin-isospin saturated nucleus occurs only due to the $\eta$ meson exchange interaction between the beam proton and nucleus. In this process, the nucleon in the nucleus can be excited to resonances $N^*$ and the $\eta$ meson in the final state can arise due to $N^* \to N\eta$. We investigate the dynamics of resonances, including nucleon Born terms, and their interferences in the coherently added cross section of this reaction. We discuss the importance of $N(1520)$ resonance and show the sensitivity of the cross section to the hadron nucleus interaction.

≈ 63 MeV [1]. Therefore, this resonance is considered to study the coherent η meson production in the proton nucleus reaction [12]. N(1535) is shown as a sensitive probe to study the non-local effects in the coherent η meson photoproduction reaction [13]. The importance of N(1535) is elucidated in context of the η meson production in the hadron nucleus reaction [14]. In addition to N(1535), other resonances, e.g., N(1520), N(1650) ..... etc, (whose branching ratio in the Nη channel is much less than that of N(1535)) and nucleon Born terms are also considered in the study of η meson production in the photonuclear reaction below 1 GeV [15]. The change in the cross section because of the interference of N(1520) and N(1535) resonances is described in Ref. [16].
Sometime back, Alvaredo and Oset studied the coherent η meson production in the (p, p ′ ) reaction on the spin-isospin saturated nucleus [12]: p + A(gs) → p ′ + A(gs) + η. The elementary reaction in the nucleus is assumed to proceed as pN → p ′ N * ; N * → Nη (presented in Fig. 1). The resonance N * considered in the intermediate state is N(1535), since it has large branching ratio (as mentioned earlier) in the Nη channel. This resonance is produced due to the η meson (a pseudoscalar-isoscalar meson) exchange interaction only, specifically, for the forward going proton and η meson. The contributions from other meson exchange interactions vanish in this reaction [12]. The projectile excitation in this reaction is null for the spin saturated nucleus.
It may be argued that though the resonance N(1520) has very small decay width (at its pole mass) in the Nη channel, there are enough reasons (mentioned below) not to neglect this resonance in the η meson production reaction. (i) The mass of N(1520) is close to that of N(1535), and therefore there could be interference effect in the η meson production reaction [16], as quoted above. (ii) The earlier value for Γ N (1520)→N η (m = 1520 MeV), as reported in Ref. [17], is 0.12 MeV, which corresponds to the coupling constant: f ηN N (1520) = 6.72 [13]. According to the recent result: Γ N (1520)→N η (m = 1520 MeV) ≃ 0.265 MeV [1], the value of f ηN N (1520) is equal to 9.98. The later value of f ηN N (1520) is about 1.5 times larger than its previous value. Due to this enhancement, the coherent η meson production cross section because of N(1520) is increased by a factor about 5. (iii) The decay width Γ N * →N η (m) varies with the mass m of the resonance N * [18] as m N * in this equation is the pole mass of N * . The value of Γ N * →N η (m N * ) is already mentioned for N(1520) and N(1535) resonances. The suffix l in the phase-space factor Φ l represents the angular momentum associated with the decay. Φ l is given by Φ l (m) = k m B 2 l (kR), wherek is the relative momentum of the decay products (i.e., N and η) in their c.m. frame. B l (kR) is the Blatt-Weisskopf barrier-penetration factor, listed in Table 1. R (= 0.25 fm) is the interaction radius. Using Eq. (1), we show in Fig. 2 that Table 1: Blatt-Weisskopf barrier-penetration factor B l (kR) [18].
the decay probability of N(1520) → Nη rises sharply over that of N(1535) → Nη with the increase in resonance mass m, i.e., ηN invariant mass. Therefore, though the decay width of N(1520) → Nη at the pole mass, as quoted earlier, is much less than that of N(1535) → Nη, the previous (as shown in Fig. 2) supersedes the later at higher values of m. The steep rise in Γ N (1520)→N η (m) with m can shift the peak position of the η meson production cross section due to N(1520) towards the larger value of m.
We have elucidated that the cross section due to N(1520) in the η meson production reaction could be large. This can change the shape and magnitude of the coherently added cross section arising due to nucleon Born terms and resonances. To disentangle it, we revisit the coherent η meson production in the (p, p ′ ) reaction on the scalar-isoscalar nucleus where Born terms, N(1520) and other resonances of Nη branching ratio ≥ 4%, i.e., N(1535), N(1650), N(1710), N(1720), are considered. In this reaction, the virtual η meson (emitted by projectile) is elastically scattered to its real state by the nucleus which remains in its ground state. Since 4 He does not have excited state, this nucleus is preferred to study the mechanism of this reaction. In the experiment both coherent production and breakup will occur, but the coherent channel can in principle be indentified.
The Lagrangian L for the coupling of η meson to a particle depends on their spin and parity [13,16]. For 1 2 + particle, i.e., N(940) and N * ≡ N(1710), the form for L is g η (ηNN coupling constant) ≃ 7.93 [20], and g * η (ηNN(1710) coupling constant) ≃ 4.26. For 1 2 − resonance N * , i.e., N(1535) and N(1650), L is given by g * η ≃ 1.86 for N(1535) and g * η ≃ 0.67 for N(1650). For N(1520) 3 2 − , L can be written as f * η = 9.98. For 3 2 + resonance [19], i.e., N * ≡ N(1720), the expression for L is The coupling constants are extracted from the measured decay width of the resonances, i.e., N * → Nη [1]. F η and F * η appearing in Lagrangians are the ηNN and ηNN * form factors respectively [20]: The T -matrix for the coherent η meson production in the (p, p ′ ) reaction on a nucleus can be written as T f i = T B + T N * , where T B represents the T -matrix for nucleon Born terms (described later), and the resonance term, i.e., T N * , is given by ̺(r) in this equation is the matter density distribution of the nucleus.Γ N * →N η denotes the vertex factor for the decay: N * → Nη. The spin S dependent part of N * propagator, i.e., Λ(S), is expressed in Table-2. V η (q) in above equation represents the η meson exchange interaction between the beam proton and the nucleon in the nucleus (see Fig. 1):Ṽ η (q) =Γ ηN N * G η (q 2 )Γ ηN N . TheΓs are described by the Lagrangians given in Eqs. (2) -(5).G η (q 2 ) denotes the virtual η meson propagator, given byG η (q 2 ) = − 1 The distorted wave functions for proton and η meson, denoted by χs in Eq. (6), are evaluated by using Glauber model [21]. For the beam proton p, it can be written as For outgoing particles, i.e., p ′ and η meson, χ is given by v X is the velocity of the particle X which is either proton or η meson appearing in Eqs. (7) and (8). V OX denotes the particle X nucleus optical potential. This potential, in fact, describes the initial and final state interactions. The proton nucleus optical potential V Op(p ′ ) (r) in Eqs. (7) and (8) is calculated using the "t̺(r)" approximation [21], i.e., where α pN denotes the ratio of the real to imaginary part of the proton nucleon scattering amplitude f pN . σ pN t represents the corresponding total cross section. To evaluate this potential, we use the energy dependent experimentally determined values for α pN and σ pN t [22]. The η meson nucleus optical potential V Oη (r) in Eq. (8), following Alvaredo and Oset [12], is evaluated from the η meson self-energy Π η (r) in the nucleus: The prefactor C(N * ) depends on the resonance N * used to calculate Π η (r). Γ N * (m) represents the total width of N * for its mass equal to m. It is composed of partial widths of N * decaying into various channels, listed duly along with the physical parameters in Ref [1]. The resonance mass m dependence of these widths are worked out following Eq. (1). Values of Γ N * (m) at the pole mass, i.e., m = m N * , are given in Table-3. V ON * is the N * nucleus interacting potential, described latter. The nucleon potential energy in the nucleus is taken as V ON (r) = −50̺(r)/̺(0) MeV [12]. Π η (r) arising due to nucleon-hole pair is evaluated following that for π 0 meson, see page no. 157 in Ref. [3]. The scalar part of the resonance propagator, denoted by G N * (m, r) in Eq. (6), is given by where E N * denotes the energy of N * . V ON * (r) in Eqs. (10) and (11), which describes the N * nucleus interaction, is also evaluated by using the "t̺(r)" approximation, as given in Eq. (9). In this case, the measured values of the N * nucleon scattering parameters, i.e., α N * N and σ N * N t , are not available. To estimate them, we take α N * N ≃ α N N and σ N * N el ≃ σ N N el since the elastic scattering dynamics of N * can be assumed not much different from that of a nucleon. For the reactive part of σ N * N t , we consider the dynamics of N * is same as that of a nucleon at its kinetic energy enhanced by ∆m, i.e., σ N * N r (T N * N ) ≈ σ N N r (T N * N + ∆m) [23]. Here, ∆m is the mass difference between the resonance and nucleon. T N * N is the total kinetic energy in the N * N center of mass system [23].
The η meson emitted because of the nucleon Born terms is addressed by T B above Eq. (6). It is similar to the expression appearing in this equation except the interaction vertices and propagator of N * to be replaced by those of nucleon. The previous is described by L ηN N in Eq. (2) and the later, i.e., propagators G N , are discussed in Ref. [2]. However, the calculated cross section due to Born terms, as shown later, is negligibly small. We calculate the differential cross section, i.e., dσ dE p ′ dΩ p ′ dΩη , for the forward going coherent η meson energy E η distribution in the (p, p ′ ) reaction on 4 He nucleus. The calculated results are presented in Figs. 3 and 4. On the upper x-axis of these figures, we mention the resonance mass m corresponding to E η . The spatial density distribution ̺(r) of this nucleus is ̺(r) = ̺ 0 1+w(r/c) 2 1+e (r−c)/z ; w = 0.445, c = 1.008 fm, z = 0.327 fm [24]. We discuss the contribution of each resonance and nucleon Born terms (including their interferences) to the cross section of this reaction. We focus on two aspects of N(1520) which can change the η meson production cross section considerably: (i) measured decay width Γ N (1520)→N η (m = 1520 MeV) and (ii) decay probability N(1520) → Nη specifically at high energy.
The calculated plane wave results at T p = 1.2 GeV are illustrated in Fig. 3(a), where the N * nucleus interaction is not included, i.e., V ON * = (0, 0). The cross sections arising due to each resonance and Born terms along with their coherent contribution are shown in this figure. For N(1520) resonance, the measured value of its decay width Γ N (1520)→N η (m = 1520 MeV) is taken equal to 0.12 MeV (earlier value). This figure elucidates the cross section due to N(1535), shown by short-long-short dash curve, is the largest. The dot-dot-dash curve represents the second largest cross section (which is 24.7% of the previous at the peak) arises because of N(1520). The peak cross section due to it appears at higher value of E η (which corresponds to larger m) compared to that because of N(1535). This occurs, as described in Fig. 2, due to the sharp rise in N(1520) → Nη decay probability with m. Fig. 3(a) also shows that the cross sections arising because of other resonances and Born terms are negligibly small. The coherently added cross section (dot-dash curve) shows that the dominant contribution to it arises due to N(1535). The effect of interference in the cross section is distinctly visible in this figure. In Fig. 3(b), we present the distorted wave results where V ON * is also incorporated. It shows that the cross section is reduced drastically, i.e., by a factor of 9.6 at the peak, and the peak position is shifted by 32 MeV towards the lower value of E η due to the inclusion of distortions (both initial and final states) and V ON * . These are the features usually reported in the low energy η meson production reaction.
As stated earlier, the calculated cross section due to N(1520) can go up by a factor of ∼ 5 because of the use of the latest measured value of Γ N (1520)→N η (m = 1520 MeV), i.e., 0.265 MeV. This can change the η meson energy E η distribution of the coherently added cross section. We illustrate those in Fig. 3(c). It is remarkable that the cross section due to N(1520) for Γ N (1520)→N η (m = 1520 MeV) = 0.265 MeV (dot-dot-dash curve) is comparable with that because of N(1535) (short-long-short curve). In fact, the previous is about 18% larger than the later. Due to this, an additional peak in the spectrum of the coherently added cross section (dot-dash curve) appears close to the peak arising because of N(1520). The magnitude of the cross section due to the inclusion of distortions and V ON * , as shown in Fig. 3(d), is reduced drastically, but the change in the shape of spectrum due to them is insignificant. Two peaks are distinctly visible in this figure.
The decay probability of N(1520) → Nη, as shown in Fig. 2, rises sharply at higher values of the resonance mass m. Therefore, this resonance could be more significant in the high energy η meson production reaction. To investigate it, we calculate the plane wave (without V ON * included) cross sections at T p = 2.5 GeV (taking the earlier value of Γ N (1520)→N η (m = 1520 MeV), i.e., 0.12 MeV), and present those in Fig. 4(a). The cross sections arising because of each resonance and Born terms along with the coherently added cross section are distinctly visible in this figure. Compare to the spectra presented in Fig. 3(a), there are considerable changes in those at T p = 2.5 GeV which are noticeable in Fig. 4(a). Three distinct peaks in the later figure arise due to N(1520) (dotdot-dash curve), N(1535) (short-long-short dash curve) and N(1710) (short dash curve). Amongst them, the cross section because of N(1520) is the largest. The cross sections due to resonances other than these three resonances and Born terms are insignificant. The separation between the peaks of the cross sections due to N(1520) and N(1535) at T p = 2.5 GeV is larger than that found at T p = 1.2 GeV (see Figs. in 3(a) and 4(a)). This occurs, as illustrated in Fig. 2, because of the sharp increase in N(1520) → Nη decay probability with m.
Because of the large η meson production cross section due to N(1520) at T p = 2.5 GeV (even for Γ N (1520)→N η (m = 1520 MeV) = 0.12 MeV), the shape of the η meson energy E η distribution spectrum of the coherently added cross section is significantly different from that calculated at T p = 1.2 GeV (see dot-dash curves in Fig. 3(a) and Fig. 4(a)). We show the distorted wave (V ON * included) result at T p = 2.5 GeV in Fig. 4(b). Both Figs. 4(a) and 4(b) show the coherently added cross sections possess multiple peaks appearing in the peak regions of the cross sections due to N(1520), N(1535) and N(1710) resonances. In addition, these figures elucidate that the largest peak arises in the N(1520) excitation region. These results are unlike to those presented in Figs. 3(a) and 3(b).
The calculated plane wave, without V ON * included, cross section at T p = 2.5 GeV due to N(1520) for the latest value of Γ N (1520)→N η (m = 1520 MeV), i.e., 0.265 MeV, is presented by the dot-dot-dash curve in Fig. 4(c). Along with it, the cross sections because of other resonances and Born terms are also presented for comparison. Due to the increase in Γ N (1520)→N η (m = 1520 MeV), in addition to large N(1520) → Nη decay probability at higher energy, the η meson production (as shown in this figure) dominantly occurs because of the N(1520) resonance. In fact, the peak cross section due to it is distinctly largest. The coherently added cross section (dot dash curve in Fig. 4(c)) shows a small peak close to that because of N(1535) and a large peak distinctly visible near to that due to N(1520). We illustrate the distorted wave (V ON * included) result for it in Fig. 4(d).
The dominant contribution to the coherently added cross section, as shown in Figs. 4(c) and 4(d), distinctly arises because of the resonance N(1520).
We have calculated the differential cross sections for the coherent η meson energy E η distribution in the (p, p ′ ) reaction on 4 He nucleus. At lower beam energy, i.e., 1.2 GeV, the cross section because of N(1535) is distinctly largest if we consider the earlier value of the decay width Γ N (1520)→N η (m = 1520 MeV), i.e., 0.12 MeV. The cross section due to N(1520) is drastically increased because of the use of the latest value of Γ N (1520)→N η (m = 1520 MeV), i.e., 0.265 MeV. Due to this, an additional peak appears in the shape of the coherently added cross section. At higher beam energy, i.e., 2.5 GeV, the peak cross section because of N(1520), even for Γ N (1520)→N η (m = 1520 MeV) = 0.12 MeV, supersedes that due to N(1535). The shift of N(1520)-peak towards the higher η meson energy (which corresponds to larger resonance mass) depends on the beam energy. This occurs because of the sharp increase in N(1520) → Nη decay probability with the resonance mass. The cross section due to N(1520) is further increased by a factor ∼ 5 because of the increase in Γ N (1520)→N η (m = 1520 MeV) from 0.12 MeV to 0.265 MeV. The coherently added cross section shows the contribution from N(1520) is distinctly dominant amongst the resonances and Born terms at higher energy. These features are unlike those reported in the previous studies where N(1535) is shown to contribute dominantly in the η meson production reaction.
The author gratefully acknowledges the referee for giving valuable comments on this work.