npsm 새물리 New Physics : Sae Mulli

pISSN 0374-4914 eISSN 2289-0041


Research Paper

New Phys.: Sae Mulli 2022; 72: 291-295

Published online April 29, 2022

Copyright © New Physics: Sae Mulli.

Determination of Astrophysical S Factor for 15N(p,γ)16O at Low-energies Within Effective Field Theory

Sangyeong Son1∗, Shung-Ichi Ando2†, Yongseok Oh1‡

1Department of Physics, Kyungpook National University, Daegu 41566, Korea
2Department of Display and Semiconductor Engineering, Sunmoon University, Asan 31460, Korea

Correspondence to:*E-mail:

Received: January 3, 2022; Revised: March 1, 2022; Accepted: March 4, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License( which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

The radiative proton capture on 15N targets, i.e., 15N(p, γ)16O, is an important thermonuclear reaction of the CNO cycle in stellar environments. This reaction is a key process in the cycle because it initiates the NO cycle by producing 16O. In this study, we investigated the 15N(p, γ)16O reaction within the framework of cluster effective field theory. We construct an effective Lagrangian to describe the reaction at stellar energies and calculate the astrophysical S factor. The parameters of our calculations, such as low-energy constants, resonance energy, and decay width of the 16O resonance are determined by fitting empirical data of the S factor at the range of 72.8 keV ≤ Ecm ≤ 368.3 keV. The S factor value for this reaction at extremely low energy is then determined to be S(E=0) = 30.4 keV·b, which is consistent with the estimates of R-matrix approaches.

Keywords: Nuclear reaction, Effective field theory, CNO cycle

A star, such as the sun, burns by converting four protons into an alpha particle through pp chain reactions. The conversion starts through the CNO cycles in stars heavier than the sun, namely, M1.5M where M is the mass of the sun. The radiative proton capture on 15N reaction, 15N(p, γ)16O, is a key reaction that connects some types of CNO cycles, forming a bridge from type I of the CNO cycle (or the CN cycle) to type II of the CNO cycle (or the NO cycle) and type III. Therefore, a description of this reaction is required for a deep understanding of the CNO cycle.

The cross-sections of the 15N(p, γ)16O reaction were measured in laboratories at relatively high proton kinetic energies compared to that in the stellar environment[1-6]. Then, the experimental data are used to fit the parameters of theoretical models, primarily the R-matrix analyses[1, 2, 4, 6-10], and the astrophysical S factor is estimated by extrapolating the formula to extremely low energy. Based on this approach, Hebbard estimated the astrophysical S factor to be S(E=0) = 32 keV·b[1]. The data show two broad resonances at Ep=338 and 1028 keV, where Ep is the proton energy in the laboratory frame, and Rolfs and Rodney[2] investigated the inclusion of direct capture contribution in the R-matrix analysis. These authors obtained a value of the S factor larger almost by a factor of two than that of Ref.[1]. However, Mukhamedzhanov et al. estimated the asymptotic normalization coefficient of the proton in the ground state of 16O for the direct capture contribution using the 15N(3He, d)16O reaction[8], and they obtained a value of the S factor that agrees with the value of Ref.[1]. Therefore, more detailed and elaborated investigations into this reaction are strongly recommended. For example, it is important to constrain the parameters of R-matrix analyses using various reactions[10]. However, it is beneficial to study the reaction using other theoretical approaches.

In this study, we estimate the S factor of the 15N(p, γ)16O reaction using cluster effective field theory[11] by focusing on the first resonance region of 16O in the data. When the reaction has a clear separation scale between relevant degrees of freedom in low energy and irrelevant degrees of freedom that may play a nontrivial role only in high energy, the effective Lagrangian in terms of the relevant degrees of freedom in low energies and perturbatively expanded by counting the number of derivatives can be constructed. The reaction amplitude is then perturbatively calculated using propagators and vertex functions obtained from the Lagrangian. The amplitude obtained preserves the gauge symmetry and is determined by a small number of coupling constants up to a given order.

This study is organized as follows. In the following section, we construct an effective Lagrangian required to describe the 15N(p, γ)16O reaction at low energies. In Sec. III, the radiative capture amplitude is obtained and its explicit form is derived. In Sec. IV, we present our numerical results for this reaction and deduce the S factor of the 15N(p, γ)16O reaction. Section V contains a summary.

We start by considering the effective Lagrangian for the radiative proton capture of 15N(p, γ)16O at stellar energy scales. The Gamow peak energy region, where the thermonuclear reaction in a stellar environment most likely occurs, is approximately EG26 keV at a relevant temperature of 1.5 × 107 K. The length scale of this reaction is much larger than the size of the 15N nucleus and the reaction cannot probe the internal structure of the nucleus. In this study, we treat the nucleon and nuclei involved in the 15N(p, γ)16O reaction as point-like particles, and the resonance state of the 16O nucleus as a bounded system of the proton and the 15N nucleus, as described by the cluster effective field theory (EFT).

The first resonance state in the cluster EFT is described by introducing a di-field as an auxiliary composed of the proton and the 15N nucleus. This approach leads to the effective Lagrangian as

L=ψpivD+12mp(vD)2D2ψp+ψNivD+12mN(vD)2D2ψN+ n=0 n max Cndi ivD+ 1 2( m p + m N ) (vD) 2 D 2 ndi[ysythϕO(iDi)di+ytdiψpTPi(3 S1)ψN+ysϕOψpTP(3 P0)ψN+(H.c)],

where ψp, ψN, ψO, and d are the fields for the proton, 15N nucleus, ground state of the 16O nucleus, and the resonance state of the 16O nucleus with Jπ=1, respectively. The masses of the proton and the 15N nucleus are represented by mp and mN, respectively, and vμ is the four-velocity vector chosen as vμ=(1, 0). The covariant derivative is Dμ=μ+ieQ^Aμ, where e, Q^, and Aμ are the elementary charge, charge operator, and the photon field, respectively. The coefficient Cn can be determined by the effective range parameters of the effective range expansion (ERE), and nmax denotes the number of effective range parameters considered in the ERE. The couplings, ys, yt, and h, are the low-energy constants (LECs).

Particularly, the interaction term containing h is introduced as a counter-term to renormalize divergences from loop calculations. The projection operators are defined as


for s-wave and p-wave interactions, respectively.

The Feynman diagrams for the reaction of 15N(p, γ)16O derived from the effective Lagrangian of Eq. (1) are shown in Fig. 1. %As labeled on external legs of each diagram in Fig. 1, The wavy, thin solid, thick solid, and dotted lines represent the out-going photon, in-coming proton, 15N, and the ground state of 16O, respectively, as shown on the external legs of each diagram in Fig. 1. The double solid line with a black circle represents the intermediate resonance state of 16O. The Coulomb repulsion between the proton and 15N is represented by shaded bubbles. The vertex with a crossed dot in diagram (g) corresponds to the counterterm, which absorbs the loop divergences from the diagrams in Figs. 1(d,e,f).

Figure 1. Feynman diagrams for the radiative capture process of 15N(p, γ)16O.

The fully-dressed propagator of the di-field is expressed as an infinite series of the bare propagator and the self-energy (Fig. 2). The self-energy is denoted by the shaded bubble diagram which can be evaluated using Coulomb Green's function[12, 13] and can be written as

Figure 2. Fully-dressed propagator of the di-field.


where H(η) and J0div are defined by


where κ=ZpZNμα and μ is the reduced mass of the two nuclei. The last term of Eq. (5), which is linear in the regularization scale Λ, comes from the power divergence subtraction scheme[14]. In Eqs. (4) and (5), ψ(x) represents the digamma function, γE represents the Euler constant, α represents the fine structure constant, and ϵ is introduced by performing dimensional regularization in d=4ϵ space-time dimensions. In terms of the magnitude of the relative momentum p in the center of mass (c.m.) frame, E=p2/2μ is the total kinetic energy, and η=κ/p is the Sommerfeld parameter. Thus, the fully-dressed propagator of the di-field in the c.m. frame reads


and the effective range expansion leads to


where aR and r are the scattering length and the effective range, respectively. The divergent J0div of Eq. (3) is absorbed by the scattering length aR.

Following Refs.[12, 13, 15, 16], we obtain the radiative capture amplitudes for the 15N(p, γ)16O process as


where ϵγ* is the out-going photon polarization vector, χp (χN) is the spinor for the incident proton (15N). Factor X can be written in the c.m. frame as

X=ysCηeiσ0{i8γ3p3Γ(2+κ/γ)L(a+b)+(ZNmNZpmp)N(16γ3p23Γ(1+iη)Γ(2+κ/γ)L(d+e)+ (ZNmNZpmp)2κH(η)4πZOμhR)},



where Cη is the Sommerfeld factor, Γ(x) is the Gamma function, and σ0=arg{Γ(1+iη)}. The loop integration give L(a+b) and L(d+e) as

L(a+b)=0dre(γ+ip)rr3U(2+κ/γ,4,2γr)×ZN mN j0μmN krZp mp j0μmp kr×[M(1+iη,2,2ipr)(1+iη)M(2+iη,3,2ipr)],

where γ=2μB is the binding momentum for the binding energy B of 16O from the p-15N breakup threshold, k is the out-going photon momentum, ZA is the charge number of the nuclei A, and j0(x) is the spherical Bessel function. Here M(a,b,c) and U(a,b,c) are the first and second types of confluent hypergeometric functions, respectively. The effective range parameters of the denominator in the di-field propagator can be written in terms of the resonance energy ER and the width ΓR(ER)[17]. The divergent parts are absorbed by hR and the cutoff rC is introduced in the integration of Eq. (12) to evaluate the finite part numerically.

We obtain the astrophysical S factor of the 15N(p, γ)16O reaction at the energy region relevant to the stellar environment in this study. The astrophysical S factor is defined as a scaling factor of the total cross-section of the thermonuclear reaction in stars using


where exp(2πη) is the factor of penetrating the Coulomb barrier between two charged nuclei. Based on the obtained radiative capture amplitude of Eq. (8), the total cross-section for the 15N(p, γ)16O reaction can be written as


which can read the S factor.

In order to extrapolate the S factor to extremely low energy regions for estimating S(E=0), we determine the parameters of our model to reproduce the experimental data of S(E) reported in Ref.[5]. Because of the paucity of other experimental data, we use the LECs, resonance energy, and resonance width as free parameters in this exploratory study. We fit the parameters using the minimum χ2 fitting using the empirical data of the S factor for the 15N(p, γ)16O reaction at c.m. energies in the range of 72.8 keVEcm368.3 keV, as reported in Ref.[5], and our results for S(E) are compared with the experimental data (Fig. 3). This figure shows that our model successfully reproduces the experimental data in the considered energy region. However, it should be mentioned that there are uncertainties in the experimental values at energies below 0.2 MeV. This shows the difficulties of experiments at low energies and introduces uncertainties in our estimates. The values obtained from the parameters are presented in Table 1, which leads to aR=2.174×104 fm and r=1.458 fm.

Table 1 . Numerical values of the parameters, χ2, and S(E=0) with N = 34 of data points. The cutoff rC = 1.0 fm is used.

ys (MeV-1/2)hR (MeV)ER (keV)ΓR(ER) (keV)χ2/NS(0) (KeV · b)
6.735 × 10-30.1006376.4403.413.5830.4

Figure 3. (Color online) The astrophysical S factor as a function of the c.m. energy Ecm. The solid curve is our result with the cut-off rC = 1.0 fm. The experimental data are obtained from Ref.[5].

We can extrapolate the results based on the determined parameters to estimate the value of S(0). The obtained value of S(0) is presented in Table 1. For comparison, the values of S(0) obtained in previous works are shown in Table 2, which is demonstrated in Fig. 4. This shows that our result is consistent with other predictions except that of Ref.[2].

Table 2 . S(0) values for the 15N(p, γ)16O reaction in units of keV·b.

This workRef. [1]Ref. [2]Ref. [4]Ref. [7]-IRef. [7]-IIRef. [8]Ref. [10]Ref. [9]
S(0)30.43264 ± 639:6 ± 2:6355033.1-40.136:0 ± 640 ± 0:3

Figure 4. (Color online) Estimates of the S(0) value for the 15N(p, γ)16O reaction.

In this study, we describe the S factor of the 15N(p, γ)16O reaction based on the cluster EFT and effective range expansion. The R-matrix model was used for most theoretical calculations in this reaction, and this study is the first calculation for this reaction performed in the EFT scheme. This model could successfully describe the obtained experimental data from Ref. [5], and the astrophysical S factor obtained from this reaction is consistent with the values estimated from the previous study except that from Ref.[2]. As shown in Fig. 4, our estimate, S(0)=30.4 keV·b, supports the conclusion that S(0) value is less than 40 keV·b, as in most R-matrix analyses.

This estimate is based on the experimental data of Ref. [5] and our analyses, therefore, focus on the first resonance region of the data. Thus, it would be interesting to extend this analysis to the second resonance region for investigating the dependence of S(0) on the number of resonances considered for the intermediate state. Work in this direction is in progress and will be reported elsewhere.

This work was supported by the National Research Foundation of Korea (NRF) under Grants No. NRF-2019R1F1A1040362, No. NRF-2020R1A2C1007597, and No. NRF-2018R1A6A1A06024970 (Basic Science Research Program).

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