Hyperon-nucleon Scattering In A Covariant Chiral Effective Field Theory Approach

TL;DR

This paper applies a covariant chiral effective field theory to hyperon-nucleon interactions, achieving a good fit to scattering data and suggesting a more efficient potential construction method.

Contribution

It introduces a covariant approach with 12 low energy constants, differing from the heavy baryon method, and demonstrates its effectiveness in describing hyperon-nucleon scattering.

Findings

Achieves a chi-squared of about 17 for 36 data points.

Uses 12 low energy constants due to Lorentz invariance.

Provides a comparable or better fit than previous approaches.

Abstract

A recently proposed covariant chiral effective field theory approach is applied to study strangeness $S=-1$ hyperon-nucleon interactions at leading order. 12 low energy constants are introduced by Lorentz invariance, which is different from the heavy baryon approach, where only five appear. The Kadyshevsky equation is employed to iterate the chiral potentials. A quite satisfactory description of the 36 hyperon-nucleon scattering data is obtained with $\chi^2\simeq 17$, which is comparable with the next-to-leading order heavy baryon approach. The results hint at a more efficient way to construct the chiral potentials.

Tables

Table 1. Table 1: Low-energy constants (in units of 10 4 superscript 10 4 10^{4} GeV -2 ) at Λ F = 600 subscript Λ 𝐹 600 \Lambda_{F}=600 MeV in the covariant χ 𝜒 \chi EFT approach.

LECs	$C_{1S0}^{\Lambda \Lambda }$	$C_{1S0}^{\Sigma \Sigma }$	$C_{3S1}^{\Lambda \Lambda }$	$C_{3S1}^{\Sigma \Sigma }$	$C_{3S1}^{\Lambda \Sigma }$	${\hat{C}}_{1S0}^{\Lambda \Lambda }$
	$-0.0223$	$-0.0528$	$0.0032$	$0.0842$	$0.0232$	$3.9185$
LECs	${\hat{C}}_{1S0}^{\Sigma \Sigma }$	${\hat{C}}_{3S1}^{\Lambda \Lambda }$	${\hat{C}}_{3S1}^{\Sigma \Sigma }$	${\hat{C}}_{3S1}^{\Lambda \Sigma }$	$C_{3P1}^{\Lambda \Lambda }$	$C_{3P1}^{\Sigma \Sigma }$
	$4.0681$	$0.4190$	$-0.4132$	$0.7326$	$0.2044$	$0.2616$

Equations

$$u_{B}(\mbox{\boldmath$p$},s)=N_{p}\left(\begin{array}[]{c}1\\ \frac{\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$p$}}{\epsilon_{B}}\end{array}\right)\chi_{s},~{}~{}~{}~{}\qquad\qquad N_{p}=\sqrt{\frac{\epsilon_{B}}{2M_{B}}},$$

$${\mathcal{L}}_{\text{CT}}^1=\frac{C_i^1}{2}\text{tr}\left({\bar{B}}_a{\bar{B}}_b({\Gamma }_{i}B{)}_b({\Gamma }_{i}B{)}_a\right),{\mathcal{L}}_{\text{CT}}^2=\frac{C_i^2}{2}\text{tr}\left({\bar{B}}_a({\Gamma }_{i}B{)}_a{\bar{B}}_b({\Gamma }_{i}B{)}_b\right),$$

$${\mathcal{L}}_{\text{CT}}^3=\frac{C_i^3}{2}\text{tr}\left({\bar{B}}_a({\Gamma }_{i}B{)}_a\right)\text{tr}\left({\bar{B}}_b({\Gamma }_{i}B{)}_b\right),$$

$$\displaystyle\mathcal{L}_{\textrm{MB}}^{(1)}=\mathrm{tr}\Bigg{(}\bar{B}\big{(}i\gamma_{\mu}D^{\mu}-M_{B}\big{)}B-\frac{D}{2}\bar{B}\gamma^{\mu}\gamma_{5}\{u_{\mu},B\}-\frac{F}{2}\bar{B}\gamma_{\mu}\gamma_{5}[u_{\mu},B]\Bigg{)}\,,$$

$$V_{\text{CT}}^{B_1{B}_2\to B_3{B}_4}=C_i\left({\bar{u}}_3{\Gamma }_i{u}_1\right)\left({\bar{u}}_4{\Gamma }_i{u}_2\right),$$

$$V_{\mathrm{OME}}^{B_1{B}_2\to B_3{B}_4}=-N_{B_1{B}_3\varphi }{N}_{B_2{B}_4\varphi }\frac{({\bar{u}}_3{\gamma }^{\mu }{\gamma }_5{q}_{\mu }{u}_1)({\bar{u}}_4{\gamma }^{\nu }{\gamma }_5{q}_{\nu }{u}_2)}{q_0^2-q^2-m^2}{\mathcal{I}}_{B_1{B}_2\to B_3{B}_4},$$

$$\displaystyle T_{\rho\rho^{\prime}}^{\nu\nu^{\prime},J}(\mbox{\boldmath$p$}^{\prime},\mbox{\boldmath$p$};\sqrt{s})=V_{\rho\rho^{\prime}}^{\nu\nu^{\prime},J}(\mbox{\boldmath$p$}^{\prime},\mbox{\boldmath$p$})+\sum_{\rho^{\prime\prime},\nu^{\prime\prime}}\int_{0}^{\infty}\frac{dp^{\prime\prime}p^{\prime\prime 2}}{(2\pi)^{3}}\frac{2\mu_{\nu^{\prime\prime}}^{2}~{}V_{\rho\rho^{\prime\prime}}^{\nu\nu^{\prime\prime},J}(\mbox{\boldmath$p$}^{\prime},\mbox{\boldmath$p$}^{\prime\prime})~{}T_{\rho^{\prime\prime}\rho^{\prime}}^{\nu^{\prime\prime}\nu^{\prime},J}(\mbox{\boldmath$p$}^{\prime\prime},\mbox{\boldmath$p$};\sqrt{s})}{\left(\mbox{\boldmath$p$}^{\prime\prime 2}+4\mu_{\nu^{\prime\prime}}^{2}\right)\left(\sqrt{\mbox{\boldmath$q$}_{\nu^{\prime\prime}}^{2}+4\mu_{\nu^{\prime\prime}}^{2}}-\sqrt{\mbox{\boldmath$p$}^{\prime\prime 2}+4\mu_{\nu^{\prime\prime}}^{2}}+i\epsilon\right)},$$

Full text

1 Introduction

Baryon-baryon interactions play a crucial role in hypernuclear physics. During the past 20 years, chiral effective field theory ($\chi$EFT), first proposed by Weinberg [1, 2], has made great progress in deriving nucleon-nucleon ($NN$) [3, 4, 5], hyperon-nucleon ($YN$) [6, 7], hyperon-hyperon [8, 9, 10] interactions.

Nevertheless, current $\chi$EFT studies on baryon-baryon interactions, referred to as the heavy baryon (HB) approach, are based on a non-relativistic theoretical framework, which cannot be used for relativistic hypernuclear few- and many-body calculations. In addition, the HB approach is very sensitive to ultraviolet cutoffs, known as the renormalization group invariance problem [11, 12]. Recently, Epelbaum and Gegelia proposed a new $\chi$EFT approach [13, 14] (referred to as the EG approach) for the $NN$ interaction. At leading order (LO), the potential is the same as that of the HB approach but the scattering equation changes to the Kadyshevsky equation. It is shown that in the strangeness $S=-1$ $YN$ system [15], the description of the scattering data with the EG approach remains quantitatively similar to the HB approach, only the cutoff dependence is mitigated.

A new covariant power counting is recently proposed to study the $NN$ [16] and $YN$ [17] systems in $\chi$EFT, which is partly motivated by the successes of covariant chiral perturbation theory in heavy-light and one baryon systems [18, 19, 20, 21]. At the potential level, the baryons are treated in a covariant way to maintain all the symmetries and analyticities. In the present work, we report on the study of the strangeness $S=-1$ $\Lambda N-\Sigma N$ system in this covariant $\chi$EFT approach at LO.

2 Formalism

The details can be found in Refs. [16, 17]. In the following we briefly explain the procedures in obtaining the chiral potentials, calculating the scattering equation, and performing the fits. We start from the complete Dirac spinor so that Lorentz invariance is retained,

[TABLE]

where $\epsilon_{B}=E_{B}+M_{B}$ and E_{B}=\sqrt{\mbox{\boldmathp}^{2}+M_{B}^{2}}, while the small component of $u_{B}$ is omitted by employing a non-relativistic reduction in the HB approach. Here naive dimensional analysis is applied in obtaining the Feynman diagrams at LO, which consist of nonderivative four-baryon contact terms and one-pseudoscalar-meson exchanges. Specifically, following Refs. [6, 15], there are 3 for the former and 7 for the latter in the strangeness $S=-1$ $\Lambda N-\Sigma N$ system, as shown in Figure. 1.

The relevant chiral Lagrangians in this framework consist of the nonderivative four-baryon contact terms and the meson-baryon interactions

[TABLE]

where $\mathrm{tr}$ indicates trace in flavor space; $\Gamma_{i}$ are the elements of the Clifford algebra, and $C_{i}^{m}$ ($m=1,2,3$) are the low energy constants (LECs) corresponding to independent four-baryon operators. In addition, $D^{\mu}B=\partial_{\mu}B+[\Gamma_{\mu},B]$, $\Gamma_{\mu}=\frac{1}{2}\left[u^{\dagger}\partial_{\mu}u+u\partial_{\mu}u^{\dagger}\right]$, $u_{\mu}=i(u^{\dagger}\partial_{\mu}u-u\partial_{\mu}u^{\dagger})$ and $u^{2}=U=\exp\left(i\frac{\sqrt{2}\phi}{f_{0}}\right)$, where $f_{0}$ is the pseudoscalar meson decay constant in the chiral limit. The matrices $B$ and $\phi$ collect the octet baryons and pseudoscalar mesons. In numerical calculations, we use $D+F=1.26$, $F/(F+D)=0.4$ and $f_{0}=93$ MeV [7]. Potentials for the Feynman diagrams in Figure. 1 are derived from the above Lagrangians. Strict SU(3) symmetry is imposed on the contact terms thus 12 independent LECs remain: $C_{1S0}^{\Lambda\Lambda}$, $\hat{C}_{1S0}^{\Lambda\Lambda}$, $C_{1S0}^{\Sigma\Sigma}$, $\hat{C}_{1S0}^{\Sigma\Sigma}$, $C_{3S1}^{\Lambda\Lambda}$, $\hat{C}_{3S1}^{\Lambda\Lambda}$, $C_{3S1}^{\Sigma\Sigma}$, $\hat{C}_{3S1}^{\Sigma\Sigma}$, $C_{3S1}^{\Lambda\Sigma}$, $\hat{C}_{3S1}^{\Lambda\Sigma}$, $C_{3P1}^{\Lambda\Lambda}$, $C_{3P1}^{\Sigma\Sigma}$. SU(3) symmetry is only broken by the mass difference of the mesons in the one-pseudoscalar-meson exchanges. The potentials corresponding to the Feynmann diagrams shown in Fig. 1 can be expressed in the following way:

[TABLE]

[TABLE]

where the subscripts CT and OME denote the contact term and one-pseudoscalar-meson exchange, respectively. $N_{BB\phi}$ is the SU(3) coefficient and $\mathcal{I}_{B_{1}B_{2}\rightarrow B_{3}B_{4}}$ the isospin factor. Ddetails can be found in Refs. [6, 16, 17].

The Kadyshevsky equation is used to iterate the potentials due to the infrared enhancement in two-baryon propagations following Ref. [15]:

[TABLE]

where $\sqrt{s}$ is the total energy of the baryon-baryon system in the center-of-mass frame, \mbox{\boldmathq}_{\nu^{\prime\prime}} is the relativistic on-shell momentum. The labels $\rho$ and $\nu$ denote the partial waves and particle channels, respectively. Relativistic kinematics is used for the relation of the laboratory momenta and the center-of-mass momenta. We solve the Kadyshevsky equation in particle basis in order to account for the physical thresholds properly. The Coulomb force is treated with the Vincent-Phatak method [22]. To regularize the integration in the high-momentum region, the chiral potentials are multiplied with an exponential form factor f_{\Lambda_{F}}(\mbox{\boldmathp},\mbox{\boldmathp}^{\prime})=\exp\left[-\left(\mbox{\boldmathp}/\Lambda_{F}\right)^{4}-\left(\mbox{\boldmathp}^{\prime}/\Lambda_{F}\right)^{4}\right]. One should note that although the potentials are fully covariant, the scattering amplitude is not because of the use of the Kadyshevsky equation and the cutoff regularization.

The LECs can be pinned down by fitting to the 36 $YN$ scattering data, as it has been done in Ref. [7]. In actual calculations we introduce the following constraints, i.e. the $\Lambda p$ ${}^{1}S_{0}$, ${}^{3}S_{1}$ and $\Sigma^{+}p$ ${}^{3}S_{1}$ scattering lengths. The $\Lambda$ hypertriton is not taken into account because we are not able to perform 3-body calculations at present.

3 Results and discussion

The fit was done by varying the cutoff value $\Lambda_{F}$ around the $\rho$ meson mass. Cutoff dependence of the $\chi^{2}$ in describing the 36 $YN$ data is shown in Figure. 2, in comparison with other $\chi$EFT approaches. It can be seen that the LO covariant $\chi$EFT approach yields a smaller $\chi^{2}$ value and a milder cutoff dependence of the $\chi^{2}$, compared with the LO HB and EG approaches. They are similar to the next-to-leading (NLO) HB approach, which has 23 LECs, however. Nevertheless, renormalization group invariance is still not achieved, as exhibited by the variation of $\chi^{2}$ with the cutoff (more clearly shown in the insert).

The minimum value of the $\chi^{2}$ is 16.7, located at $\Lambda_{F}=600$ MeV, and the corresponding LECs are listed in Table 1.

Note that the LECs in the $\Lambda p$ ${}^{1}S_{0}$ partial wave cannot be precisely determined for the $\Lambda$ hypertriton binding energy is not used as a constraint. We show only a typical case here. Moreover, the description of the experimental data are shown in Fig. 3.

The LO HB approach [6] and two phenomenological models [23, 24] are taken as references. Clearly the covariant $\chi$EFT approach can better reproduce the data compared with the LO HB approach, especially for the $\Lambda p\rightarrow\Lambda p$ and $\Sigma^{+}p\rightarrow\Sigma^{+}p$ reactions.

The improvements in our new scheme mainly come from the contact terms. In the LO HB approach, contact terms only contribute to central and spin-spin potentials, and they only appear in $S$-waves. In our new scheme, tensor, spin-orbit, asymmetric spin-orbit and quadratic spin-orbit terms also appear in the contact interactions. These terms are momentum dependent and improve the description. These are parallel to the $NN$ case. Detailed discussions can be found in Ref. [16].

4 Summary

We have applied a new covariant chiral effective field theory approach to the strangeness $S=-1$ $\Lambda N-\Sigma N$ system at leading order. Starting from the covariant chiral Lagrangians, the full baryon spinor is retained to calculate the potentials. As such, Lorentz invariance is preserved. We use strict SU(3) symmetry in the contact terms thus only 12 low energy constants are independent. SU(3) symmetry is broken in the one-pseudoscalar-meson exchanges due to the mass difference of the exchanged mesons. The Kadyshevsky equation is employed to iterate the chiral potentials. A very satisfactory description of the 36 hyperon-nucleon scattering data is obtained, as well as a mitigated cutoff dependence. Our approach yield results similar to the next-to-leading order heavy baryon approach but with 11 fewer low energy constants. It hints at a more efficient way to construct the baryon-baryon potentials.

Acknowledgements

This work is partly supported by the National Natural Science Foundation of China under Grants No. 11375024, No. 11522539, and No. 11375120, the China Postdoctoral Science Foundation under Grant No. 2016M600845, and the Fundamental Research Funds for the Central Universities.