重子磁矩的协变手征有效理论研究

doi:10.3969/j.issn.1000-5641.2024.03.002

摘要/Abstract

摘要：

在SU(3)协变手征有效理论框架下, 使用扩展的极小减除(extended minimal subtraction,$ {\text{E}}\overline {{\text{MS}}} $)方案, 计算了至次领头阶(next-to-leading order, NLO)的重子质量和重子磁矩的一圈图修正贡献, 并使用实验数据与PACS-CS格点数据组的数据对解析结果进行了数值拟合. 结果表明, 在NLO重子质量和重子磁矩的修正贡献中, $ {\text{E}}\overline {{\text{MS}}} $方案可以给出不错的理论结果和数值拟合的结果; 该结果相较于重重子方法和红外正规化方法的结果更加优秀, 并且与EOMS (extended-on-mass-shell)方案的结果相近.

关键词: 手征有效场论, 扩展的极小减除方案, 重子质量, 重子磁矩, 数值拟合

Abstract:

In this paper, we calculate the next-to-leading order (NLO) corrections to the baryon mass and magnetic moment using the covariant chiral perturbation theory within the extended minimal subtraction$({\text{E}}\overline {{\text{MS}}})$scheme under SU(3). We also present a comparative analysis of the experimental data and lattice quantum chromodynamics data with the $ {\text{E}}\overline {{\text{MS}}} $ results, and extrapolate it into physical value. We show that $ {\text{E}}\overline {{\text{MS}}} $ provides a reasonable theoretical and numerical result at the NLO, better than those obtained from the heavy-baryon approach and infrared regularization, and close to that obtained by the extended-on-mass-shell (EOMS) scheme.

Key words: covariant chiral effective field theory, extended minimal subtraction scheme, baryon mass, baryon magnetic moment, numerical fitting

中图分类号:

O413.3

周海峰, 杨继锋. 重子磁矩的协变手征有效理论研究[J]. 华东师范大学学报（自然科学版）, 2024, 2024(3): 12-26.

Haifeng ZHOU, Jifeng YANG. Study of baryon magnetic moments in covariant chiral effective theory[J]. Journal of East China Normal University(Natural Science), 2024, 2024(3): 12-26.

图/表 13

图1

表1

图2

表2

重子质量NLO修正贡献中的系数"

系数	${\rm{N } }$	$\Lambda$	$\Sigma$	$\Xi$	系数	$\Delta$	${\Sigma ^*}$	${\Xi ^*}$	${\Omega ^ - }$
$\xi _{ {\rm{OO} }\text{π} }^{\left( 3 \right)}$	$ \dfrac{{3{{\left( {D + F} \right)}^2}}}{4} $	$ {D^2} $	$ \dfrac{{{D^2} + 6{F^2}}}{3} $	$ \dfrac{{3{{\left( {D - F} \right)}^2}}}{4} $	$\xi _{ {\rm{DO} }\text{π} }^{\left( 3 \right)}$	$ {\mathcal{C}^2} $	$ \dfrac{{5{\mathcal{C}^2}}}{6} $	$ \dfrac{{{\mathcal{C}^2}}}{2} $	$ 0 $
$\xi _{ {\rm{OO} }{\rm{K}}}^{\left( 3 \right)}$	$ \dfrac{{5{D^2} - 6DF + 9{F^2}}}{6} $	$ \dfrac{{{D^2} + 9{F^2}}}{3} $	$ {D^2} + {F^2} $	$ \dfrac{{5{D^2} + 6DF + 9{F^2}}}{6} $	$\xi _{ {\rm{DO} }{\rm{K}}}^{\left( 3 \right)}$	$ {\mathcal{C}^2} $	$\dfrac{ {4{\mathcal{C}^2} } }{6}$	$ {\mathcal{C}^2} $	$ 2{\mathcal{C}^2} $
$\xi _{ {\rm{OO} }\text{η} }^{\left( 3 \right)}$	$ \dfrac{{{{\left( {D - 3F} \right)}^2}}}{{12}} $	$ \dfrac{{2{D^2}}}{6} $	$ \dfrac{{2{D^2}}}{6} $	$ \dfrac{{{{\left( {D + 3F} \right)}^2}}}{{12}} $	$\xi _{ {\rm{DO} }\text{η} }^{\left( 3 \right)}$	$ 0 $	$ \dfrac{{{\mathcal{C}^2}}}{2} $	$ \dfrac{{{\mathcal{C}^2}}}{2} $	$ 0 $
$\xi _{ {\rm{OD} }\text{π} }^{\left( 3 \right)}$	$ {\mathcal{C}^2} $	$ \dfrac{{3{\mathcal{C}^2}}}{4} $	$ \dfrac{{{\mathcal{C}^2}}}{6} $	$ \dfrac{{{\mathcal{C}^2}}}{4} $	$\xi _{ {\rm{DD} }\text{π} }^{\left( 3 \right)}$	$ \dfrac{{5{\mathcal{H}^2}}}{6} $	$ \dfrac{{4{\mathcal{H}^2}}}{9} $	$ \dfrac{{{\mathcal{H}^2}}}{6} $	$ 0 $
$\xi _{ {\rm{OD} }{\rm{K}}}^{\left( 3 \right)}$	$ \dfrac{{{\mathcal{C}^2}}}{4} $	$ \dfrac{{{\mathcal{C}^2}}}{2} $	$ \dfrac{{5{\mathcal{C}^2}}}{6} $	$ \dfrac{{3{\mathcal{C}^2}}}{4} $	$\xi _{ {\rm{DD} }K}^{\left( 3 \right)} $	$ \dfrac{{{\mathcal{H}^2}}}{3} $	$ \dfrac{{8{\mathcal{H}^2}}}{9} $	$ {\mathcal{H}^2} $	$ \dfrac{{2{\mathcal{H}^2}}}{3} $
$\xi _{ {\rm{OD} }\text{η} }^{\left( 3 \right)}$	0	0	$ \dfrac{{{\mathcal{C}^2}}}{4} $	$ \dfrac{{{\mathcal{C}^2}}}{4} $	$\xi _{ {\rm{DD} }\text{η} }^{\left( 3 \right)}$	$ \dfrac{{{\mathcal{H}^2}}}{6} $	0	$ \dfrac{{{\mathcal{H}^2}}}{6} $	$ \dfrac{{2{\mathcal{H}^2}}}{3} $

表2

图3

表3

图4

表4

八重态重子磁矩NLO修正贡献中的系数"

系数	$\rm{p}$	$\rm{n }$	$\Lambda$	${\Sigma ^ - }$	${\Sigma ^ + }$	${\Sigma ^0}$	${\Xi ^ - }$	${\Xi ^0}$	$\Lambda {\Sigma ^0}$
$\xi _{ {\rm{O} }\text{π} \gamma }^{\left( a \right)}$	$ - {\left( {D + F} \right)^2} $	$ {\left( {D + F} \right)^2} $	$ 0 $	$ \dfrac{2}{3}\left( {{D^2} + 3{F^2}} \right) $	$ - \dfrac{2}{3}\left( {{D^2} + 3{F^2}} \right) $	$ 0 $	$ {\left( {D - F} \right)^2} $	$ - {\left( {D - F} \right)^2} $	$ - \dfrac{4}{{\sqrt 3 }}DF $
$\xi _{{\rm{OK}}\gamma }^{\left( a \right)}$	$ - \dfrac{2}{3}\left( {{D^2} + 3{F^2}} \right) $	$ - {\left( {D - F} \right)^2} $	$ 2DF $	$ {\left( {D - F} \right)^2} $	$-(D + F)^{2}$	$ - 2DF $	$ \dfrac{2}{3}\left( {{D^2} + 3{F^2}} \right) $	$ {\left( {D + F} \right)^2} $	$ - \dfrac{2}{{\sqrt 3 }}DF $
$\xi _{ {\rm{O\text{π}} } \gamma }^{\left( b \right)}$	$ - \dfrac{1}{2}{\left( {D + F} \right)^2} $	$-\left(D + E\right)^{2}$	$ 0 $	$ 2{F^2} $	$ - 2{F^2} $	$ 0 $	$ \dfrac{1}{2}{\left( {D - F} \right)^2} $	$ {\left( {D - F} \right)^2} $	$ \dfrac{4}{{\sqrt 3 }}DF $
$\xi _{{\rm{OK}}\gamma }^{\left( b \right)}$	$ - {\left( {D - F} \right)^2} $	$ {\left( {D - F} \right)^2} $	$ 2DF $	$ {\left( {D + F} \right)^2} $	$ - {\left( {D - F} \right)^2} $	$ 2DF $	$ {\left( {D + F} \right)^2} $	$ - {\left( {D + F} \right)^2} $	$ \dfrac{2}{{\sqrt 3 }}DF $
$\xi _{ {\rm{O\text{η}} } \gamma }^{\left( b \right)}$	$ - \dfrac{1}{6}{\left( {D - 3F} \right)^2} $	$ 0 $	$ 0 $	$ \dfrac{2}{3}{D^2} $	$ - \dfrac{2}{3}{D^2} $	$ 0 $	$ \dfrac{1}{6}{\left( {D + 3F} \right)^2} $	$ 0 $	$ 0 $
$\xi _{ {\rm{D\text{π}} } \gamma }^{\left( c \right)}$	$ - \dfrac{8}{9}{\mathcal{C}^2} $	$ \dfrac{8}{9}{\mathcal{C}^2} $	$ 0 $	$ - \dfrac{2}{9}{\mathcal{C}^2} $	$ \dfrac{2}{9}{\mathcal{C}^2} $	$ 0 $	$ - \dfrac{4}{9}{\mathcal{C}^2} $	$ \dfrac{4}{9}{\mathcal{C}^2} $	$ - \dfrac{4}{{3\sqrt 3 }}{\mathcal{C}^2} $
$\xi _{{\rm{DK}}\gamma }^{\left( c \right)}$	$ \dfrac{2}{9}{\mathcal{C}^2} $	$ \dfrac{4}{9}{\mathcal{C}^2} $	$ \dfrac{2}{9}{\mathcal{C}^2} $	$ - \dfrac{4}{9}{\mathcal{C}^2} $	$ - \dfrac{8}{9}{\mathcal{C}^2} $	$ - \dfrac{2}{3}{\mathcal{C}^2} $	$ - \dfrac{2}{9}{\mathcal{C}^2} $	$ \dfrac{8}{9}{\mathcal{C}^2} $	$ - \dfrac{2}{{3\sqrt 3 }}{\mathcal{C}^2} $

表4

表5

表6

图5

表7

表8

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系数	$\rm{ N}$	$\Lambda$	$\Sigma$	$\Xi$	$\Delta$	${\Sigma ^*}$	${\Xi ^*}$	${\Omega ^ - }$
$\xi _{B{\text{π} } }^{\left( 2 \right)}$	$ 2{b_0} + 4{b_F} $	$2{b_0} - \dfrac{4}{3}{b_D}$	$ 2{b_0} + 4{b_F} $	$ 2{b_0} - 4{b_F} $	$ {t_0} + 3{t_D} $	$ {t_0} + {t_D} $	$ {t_0} - {t_D} $	$ {t_0} - 3{t_D} $
$\xi _{B{\rm{K}}}^{\left( 2 \right)}$	$ 4{b_0} + 4{b_D} - 4{b_F} $	$4{b_0} + \dfrac{ {16} }{3}{b_F}$	$ 4{b_0} $	$ 4{b_0} + 4{b_D} + 4{b_F} $	$ 2{t_0} $	$ 2{t_0} + 2{t_D} $	$ 2{t_0} + 4{t_D} $	$ 2{t_0} + 6{t_D} $

系数	$\rm{p}$	$\rm{n}$	$\Lambda$	${\Sigma ^ - }$	${\Sigma ^ + }$	${\Sigma ^0}$	${\Xi ^ - }$	${\Xi ^0}$	$\Lambda {\Sigma ^0}$
$ {\alpha _B} $	$ \dfrac{1}{3} $	$ - \dfrac{2}{3} $	$ - \dfrac{1}{3} $	$ \dfrac{1}{3} $	$ \dfrac{1}{3} $	$ \dfrac{1}{3} $	$ \dfrac{1}{3} $	$ - \dfrac{2}{3} $	$ \dfrac{1}{{\sqrt 3 }} $
$ {\beta _B} $	1	0	0	–1	1	0	–1	0	0

方案或方法	${M_{\rm{N}}}/ { {\text{MeV} } }$	${M_\Lambda }/ { {\text{MeV} } }$	${M_\Sigma }/ { {\text{MeV} } }$	${M_\Xi }/{ {\text{MeV} } }$	$M_{B0}^{\rm{eff} }/ { {\text{MeV} } }$	${b_D}/ { {\text{Ge} }{ {\text{V} }^{ - 1} } }$	${b_F}/ { {\text{Ge} }{ {\text{V} }^{ - 1} } }$
GMO	942	1115	1188	1325	1192	0.060	–0.213
HB	939	1116	1195	1315	2422	0.412	–0.781
EOMS	941	1116	1190	1322	1840	0.199	–0.530
$ {\text{E}}\overline {{\text{MS}}} $	941	1115	1193	1319	1671	0.115	–0.454
Expt	940	1116	1193	1318

方案或方法	${M_\Delta }/ { {\text{MeV} } }$	${M_{ {\Sigma ^*} } } /{ {\text{MeV} } }$	${M_{ {\Xi ^*} } }/ { {\text{MeV} } }$	${M_{ {\Omega ^ - } } }/ { {\text{MeV} } }$	$M_{T0}^{\rm{eff} }/ { {\text{MeV} } }$	${t_D}/ { {\text{Ge} }{ {\text{V} }^{ - 1} } }$
GMO	1233	1380	1526	1672	1215	–0.326
HB	1235	1372	1518	1673	1763	–0.960
EOMS	1234	1376	1523	1673	1519	–0.694
$ {\text{E}}\overline {{\text{MS}}} $	1225	1393	1539	1665	1501	–0.736
Expt	1232	1385	1533	1672

方案或方法	${M_{\rm{N} } }/ $$ { {\text{MeV} } }$	${M_\Lambda }/ $$ { {\text{MeV} } }$	${M_\Sigma }/ $$ { {\text{MeV} } }$	${M_\Xi }/ $$ { {\text{MeV} } }$	${M_\Delta }/ $$ { {\text{MeV} } }$	${M_{ {\Sigma ^*} } }/ $$ { {\text{MeV} } }$	${M_{ {\Xi ^*} } }/ $$ { {\text{MeV} } }$	${M_{ {\Omega ^ - } } }/ $$ { {\text{MeV} } }$	${m_B}/ $$ { {\text{MeV} } }$	${b_0}/ $$ { {\text{Ge} }{ {\text{V} }^{ - 1} } }$	${b_D}/ $$ { {\text{Ge} }{ {\text{V} }^{ - 1} } }$	${b_F}/ $$ { {\text{Ge} }{ {\text{V} }^{ - 1} } }$	${m_B}/ $$ { {\text{MeV} } }$	${b_0}/ $$ { {\text{Ge} }{ {\text{V} }^{ - 1} } }$	${b_D}/ $$ { {\text{Ge} }{ {\text{V} }^{ - 1} } }$	$\chi _{\rm{d.o.f}}^2$
GMO	971 (22)	1115 (21)	1165 (23)	1283 (22)	1319 (28)	1433 (27)	1547 (27)	1661 (27)	900	–0.264	0.042	–0.174	1245	–0.12	–0.254	0.63
HB	764 (21)	1042 (20)	1210 (22)	1392 (21)	1264 (22)	1466 (22)	1622 (23)	1733 (25)	508	–1.656	0.368	–0.824	1117	–0.709	–0.739	9.2
EOMS	893 (19)(39)	1088 (20)(14)	1178 (24)(7)	1322 (24)(20)	1222 (24)(49)	1376 (24)(29)	1531 (25)(8)	1686 (28)(13)	756	–0.978	0.190	–0.519	954	–1.05	–0.682	2.1
$ {\text{E}}\overline {{\text{MS}}} \left( 1 \right) $	914 (15)(28)	1103 (15)(6)	1196 (13)(15)	1331 (6)(24)	1238 (9)(41)	1397 (10)(21)	1534 (9)(16)	1650 (13)(17)	1143	–0.563	0.150	–0.496	1335	–0.363	–0.718	1.81
$ {\text{E}}\overline {{\text{MS}}} \left( 2 \right) $	909 (15)(30)	1105 (9)(5)	1219 (10)(27)	1349 (6)(33)	1234 (5)(42)	1394 (4)(22)	1534 (7)(14)	1654 (10)(16)	1077	–0.622	0.148	–0.488	1298	–0.413	–0.714	1.41
Expt	940	1116	1193	1318	1232	1385	1533	1672

方案或方法	$\rm{p}$	$\rm{n }$	$\Lambda$	${\Sigma ^ - }$	${\Sigma ^ + }$	${\Sigma ^0}$	${\Xi ^ - }$	${\Xi ^0}$	$\Lambda {\Sigma ^0}$	$ b_6^D $	$ b_6^F $	$ {\tilde \chi ^2} $
Tree level	2.56	–1.60	–0.80	–0.97	2.56	0.80	–1.60	–0.97	1.38	2.40	1.77	0.50
HB	3.01	–2.62	–0.42	–1.35	2.18	0.42	–0.70	–0.52	1.68	4.71	2.48	1.01
IR	2.08	–2.74	–0.64	–1.13	2.41	0.64	–1.17	–1.45	1.89	4.81	0.012	1.86
$ {\text{EOMS}}\left( {\text{O}} \right) $	2.58	–2.10	–0.66	–1.10	2.43	0.66	–0.95	–1.27	1.58	3.82	1.20	0.18
$ {\text{E}}\overline {{\text{MS}}} \left( {\text{O}} \right) $	2.61	–2.29	–0.61	–1.16	2.38	0.61	–0.92	–1.14	1.63	4.27	2.39	0.26
$ {\text{EOMS}}\left( {{\text{O + D}}} \right) $	3.18	–2.51	–0.29	–1.26	1.84	0.29	–0.78	–1.05	1.88	5.76	1.03	1.06
$ {\text{E}}\overline {{\text{MS}}} \left( {{\text{O + D}}} \right) $	2.43	–2.20	–0.71	–1.13	2.54	0.71	–0.96	–1.20	1.56	7.31	2.39	0.33
Expt	2.79	–1.91	–0.61	–1.16	2.45	—	–0.65	–1.25	±1.61