SU(3)重子质量和1/Nc扩展下的σ项研究

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"这篇研究文章发表在Physics Letters B 781 (2018) 719–722上，由I.P.Fernando, J.M.Alarcón和J.L.Goity共同撰写。文章探讨了SU(3)味手性有效理论结合1/Nc扩展在研究重子质量和核子σ项中的应用。" 文章中提到的 SU(3) BChPT×1/Nc 是一种理论框架，它结合了手征对称性（Chiral Perturbation Theory, ChPT）和1/Nc展开（1 over N_c expansion），其中Nc是色荷的数量（在QCD中，Nc=3）。这种理论方法用于研究包含三个基本夸克口味（u, d, s）的重子系统，例如核子。 Gell-Mann-Okubo质量公式是描述强相互作用下重子质量的一种方式，它基于SU(3)味对称性。在理想情况下，这个公式可以给出精确的关系，但在实际中，由于味对称性的破缺，公式会有偏差。研究指出，这种偏差与核子的σ项有关，σ项是与标量密度u'd + d-s相关的量。标量密度涉及夸克的无味组合，这里的u'd + d-s表示上夸克与下夸克的对称组合减去奇异夸克的贡献。 σ项是一个重要的物理量，它反映了夸克之间强相互作用的强度，特别是在低能区域。在本文中，σπN 表示π介子与核子间的σ项，而σs则涉及到奇异夸克的贡献。研究显示，尽管核子的σ项显示出良好的低能量展开行为，即随着能量降低，其行为可预测，但与σ项相关的质量组合却受到了夸克质量的非分析性高阶修正的影响，这些修正可能相当大。通过将QCD重子质量引入分析，并利用晶格QCD数据，研究人员估计σπN = 69（10）MeV，这在一定的不确定性范围内处于自然量级。对于σs，其值也具有类似的自然规模，但不确定性较大。这些结果对于理解核子的内部结构、强相互作用的性质以及解决所谓的“σ项难题”（σ-term puzzle）具有重要意义，因为σ项的低值与理论预测存在显著差异。这项研究深入探讨了SU(3)手性QCD与1/Nc扩展在重子质量和核子σ项计算中的应用，揭示了σ项与味对称性破缺之间的联系，并提供了σπN 和σs的初步估计，这对于进一步理解强相互作用的细节和QCD动力学具有重要价值。

Physics Letters B 781 (2018) 719–722

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Baryon masses and σ terms in SU(3) BChPT ×1/N

I.P. Fernando

, J.M. Alarcón

, J.L. Goity

a,b,∗

Department of Physics, Hampton University, Hampton, VA 23668, USA

Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA

a r t i c l e i n f o a b s t r a c t

Article history:

Received

9 April 2018

Received

in revised form 20 April 2018

Accepted

25 April 2018

Available

online 27 April 2018

Editor:

B. Grinstein

Keywords:

Sigma

terms

Nucleon

mass

Baryon

masses

Gell-Mann–Okubo

mass formula

Baryon masses and nucleon σ terms are studied with the effective theory that combines the chiral and

1/N

expansions for three ﬂavors. In particular the connection between the deviation of the Gell-Mann–

Okubo

relation and the σ term associated with the scalar density

uu +

dd −2

ss is emphasized. The latter

is at lowest order related to a mass combination whose low value has given rise to a σ term puzzle. It

is shown that while the nucleon σ terms have a well behaved low energy expansion, that mass combi-

nation

is affected by large higher order corrections non-analytic in quark masses. Adding to the analysis

lattice QCD baryon masses, it is found that σ

π N

= 69(10) MeV and σ

has natural magnitude within its

relatively large uncertainty.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

Baryon mass dependencies on quark masses, quantiﬁed by

the different σ -terms, are among the fundamental observables in

baryon chiral dynamics. In particular, they give information on the

baryon matrix elements of scalar quark densities, for which there

is no alternative way for their determination. The deﬁnition of

σ terms is through the Feynman–Hellmann theorem,

which, for

three ﬂavors, through the physical baryon masses gives access to

only two such terms, namely those associated with the SU(3) octet

quark mass combinations m

= m

− m

and m

√

(

m − m

where

m is the average of the u and d quark masses. The σ terms

associated with the singlet component m

m + m

) require

knowledge of baryon masses for unphysical quark masses, which

is made possible through lattice QCD (LQCD) calculations. On the

other hand, the pion–nucleon σ term σ

π N

≡

N |

uu +

dd | N

is accessible through its connection to pion–nucleon scattering via

a low energy theorem [1–3]. Such a determination of σ

π N

had

a long evolution through the availability of increasingly accurate

data and the development of combined methods of dispersion the-

ory

and chiral perturbation theory [4–11]. The values obtained for

Corresponding author.

E-mail

address: goity @jlab .org (J.L. Goity).

The following notation will be used: σ

(B) = m

∂

∂m

, where m

indicates a

quark mass (i = u, d, s) or combination thereof (i =0, 3, 8), and B is a given baryon.

When B is not explicitly indicated it is assumed to be a nucleon.

π N

range from 45 MeV [4–6]to 64 MeV [7–12], where the dif-

ference

between the results of the different dispersive analyses

resides mostly in the different values of the S-wave π N scatter-

ing

lengths a

1/2,3/2

used in the subtractions, cf. [12]. In addition to

the results from the analyses of π N scattering, LQCD calculations

extrapolated to or at the physical point obtain different results,

with values consistent with the recent π N results [13] and smaller,

π N

≈ 40 MeV [14–17]. The relatively large range of values ob-

tained

for σ

π N

keeps it as an active topic of study, and in part

motivates the present work. An additional motivation is the rele-

vance

of scalar quark operator matrix elements, quantities that are

relevant in studies of direct dark matter detection [18–20], and of

lepton ﬂavor violation through μ–e conversion in scattering with

nuclei [21].

A puzzle that has been emphasized for a long time [22]is the

relation between σ

π N

in the isospin symmetry limit and the nucle-

on’s

σ ≡

√

, namely σ

π N

σ + 2

, which for a natural

size value of σ

should give σ

π N

∼

σ . The origin of the puzzle

is the relation: σ

(2m

− m



− m



) (or other combinations

related via the Gell-Mann–Okubo (GMO) relation) valid at linear

order in quark masses, which gives

σ ∼ 25 MeV. If that relation

is a reasonable approximation to the value of

σ , the implication

is that, contrary to expectations, m

must give a very large con-

tribution

to the nucleon mass even for the smaller values of σ

π N

The puzzle is particularly striking for the larger values that have

been obtained for σ

π N

, which would imply σ

∼500 MeV. Indeed,

this is clearly impossible if one considers that σ

is OZI suppressed

with respect to σ

π N

https://doi.org/10.1016/j.physletb.2018.04.054

0370-2693/

SCOAP

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