Study of baryon magnetic moments in covariant chiral effective theory

doi:10.3969/j.issn.1000-5641.2024.03.002

Abstract

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

CLC Number:

O413.3

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.

Figures/Tables 13

Fig.1

Table 1

Fig.2

Table 2

Coefficients of the contribution to the NLO corrections for the baryon masses"

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} $

Table 2

Fig.3

Table 3

Fig.4

Table 4

Coefficients of the contribution to the NLO corrections for the baryon-octet magnetic moments"

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} $

Table 4

Table 5

Table 6

Fig.5

Table 7

Table 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