Status of isospin splittings in mesons and baryons

TL;DR

This paper reviews recent precise measurements of isospin splittings in mesons and baryons, highlighting their role in estimating up-down quark mass differences and emphasizing the need for improved future measurements.

Contribution

It updates previous results on isospin splittings based on new data and discusses the importance of more accurate measurements of specific meson and baryon mass differences.

Findings

Updated estimates of quark mass differences from new measurements.

Identification of key observables needing more precise data.

Recognition of the impact of current measurement uncertainties.

Abstract

Current measurements of isospin splittings in mesons and baryons are sufficiently precise that they allow estimates of the mass difference between constituent up and down quarks. Some previous results are updated in the light of these new measurements, and the importance of better measurements of some observables such as $M(K^{*\pm})$, $M(B^{*0})-M(B^0)$, and isospin splittings in bottom baryons is noted.

Tables

Table 1. Table I: Results (in MeV) of a fit to mesons and baryons with additive quark masses (different for mesons and baryons), hyperfine terms, and a binding term B ​ ( s ​ s ) 𝐵 𝑠 𝑠 B(ss) . The label m ¯ ¯ 𝑚 \bar{m} denotes an average between m u subscript 𝑚 𝑢 m_{u} and m d subscript 𝑚 𝑑 m_{d} . The masses of ϕ italic-ϕ \phi , Ξ Ξ \Xi , and Ω Ω \Omega are corrected by terms − 2 ​ B ​ ( s ​ s ) , − B ​ ( s ​ s ) , 2 𝐵 𝑠 𝑠 𝐵 𝑠 𝑠 -2B(ss),~{}-B(ss), and − 3 ​ B ​ ( s ​ s ) 3 𝐵 𝑠 𝑠 -3B(ss) , respectively. We find m ¯ m = 307.5 superscript ¯ 𝑚 𝑚 307.5 \bar{m}^{m}=307.5 MeV, m s m = 487.6 subscript superscript 𝑚 𝑚 𝑠 487.6 m^{m}_{s}=487.6 MeV, m ¯ b = 362.1 superscript ¯ 𝑚 𝑏 362.1 \bar{m}^{b}=362.1 MeV, m s b = 543.9 subscript superscript 𝑚 𝑏 𝑠 543.9 m^{b}_{s}=543.9 MeV, b m / ( m ¯ m ) 2 = 79.4 superscript 𝑏 𝑚 superscript superscript ¯ 𝑚 𝑚 2 79.4 b^{m}/(\bar{m}^{m})^{2}=79.4 MeV, b b / ( m ¯ b ) 2 = 50.0 superscript 𝑏 𝑏 superscript superscript ¯ 𝑚 𝑏 2 50.0 b^{b}/(\bar{m}^{b})^{2}=50.0 MeV, B ​ ( s ​ s ) = 9.2 𝐵 𝑠 𝑠 9.2 B(ss)=9.2 MeV. The root-mean-square error of the fit is ∑ ( Δ ​ M 2 ) / 13 = 3.85 Δ superscript 𝑀 2 13 3.85 \sqrt{\sum(\Delta M^{2})/13}=3.85 MeV.

Meson	$\pi$	$\rho$	$K$	$K^{\ast }$	$\varphi$
Pred.	138.5	773.9	494.6	895.3	1019.9
Expt.	138.0	775.2	495.6	894.1	1019.5
$\Delta M^2$	0.2	1.8	1.0	1.4	0.2

Table 2. Table II: Isospin splittings of light-quark mesons [ 6 ] .

$\Delta M$	Label	Value (MeV)
${\pi }^{\pm }-{\pi }^0$	${\pi }_2$	4.5936$\pm$0.0005
${\rho }^{\pm }-{\rho }^0$	${\rho }_2$	0.15$\pm$0.42
$K^{+}-K^0$	$K_1$	–3.934$\pm$ 0.020
$K^{\ast +}-K^{\ast 0}$	$K_1^{\ast }$	–6.7$\pm$1.2a

Table 3. Table III: Experimental mass splittings between light-quark baryons [ 6 ] .

Splitting	Symbol	Value (MeV)
$M\left(p\right)-M\left(n\right)$	$N_1$	–1.2933
$M\left({\Sigma }^{+}\right)-M\left({\Sigma }^{-}\right)$	${\Sigma }_1$	$-8.08\pm 0.08$
$M\left({\Sigma }^{+}\right)-2M\left({\Sigma }^0\right)+M\left({\Sigma }^{-}\right)$	${\Sigma }_2$	$1.535\pm 0.090$
$M\left({\Sigma }^{\ast +}\right)-M\left({\Sigma }^{\ast -}\right)$	${\Sigma }_1^{\ast }$	$-4.40\pm 0.61$
$M\left({\Sigma }^{\ast +}\right)-2M\left({\Sigma }^{\ast 0}\right)+M\left({\Sigma }^{\ast +}\right)$	${\Sigma }_2^{\ast }$	$2.6\pm 2.1$
$M\left({\Xi }^0\right)-M\left({\Xi }^{-}\right)$	${\Xi }_1$	$-6.85\pm 0.21$
$M\left({\Xi }^{\ast 0}\right)-M\left({\Xi }^{\ast -}\right)$	${\Xi }_1^{\ast }$	$-3.20\pm 0.68$

Table 4. Table IV: Predicted and observed isospin splittings in light-quark baryons.

	$N_1$	${\Sigma }_1$	${\Sigma }_1^{\ast }$	${\Sigma }_2$	${\Sigma }_2^{\ast }$	${\Xi }_1$	${\Xi }_1^{\ast }$
Pred.	-1.293	-8.087	-4.685	1.529	1.529	-6.794	-3.392
Expt.	-1.293	-8.080	-4.400	1.535	2.600	-6.850	-3.200
Error	$\sim 0$	0.080	0.610	0.090	2.100	0.210	0.680
${\chi }^2$	0.000	0.007	0.218	0.004	0.260	0.072	0.079

Table 5. Table V: Parameters describing isospin splittings in light-quark mesons and baryons.

	$\Delta$	$a$	$c/{\overline{m}}^2$
Meson	–4.12	2.20	–2.32
Baryon	–2.49	3.05	–1.52

Table 6. Table VI: Masses and isospin splittings of charmed mesons, in MeV.

$D^{+}$	$1869.65\pm 0.05$
$D^0$	$1864.83\pm 0.05$
$D_1\equiv M\left(D^{+}\right)-M\left(D^0\right)$	$4.822\pm {0.015}^a$
$D^{\ast +}$	$2010.26\pm 0.05$
$D^{\ast 0}$	$2006.85\pm 0.05$
$D_1^{\ast }\equiv M\left(D^{\ast +}\right)-M\left(D^{\ast 0}\right)$	$3.41\pm 0.07$

Table 7. Table VII: Masses and isospin splittings of charmed baryons, in MeV.

$M\left({\Sigma }_c^{++}\right)$	$2453.97\pm 0.14$
$M\left({\Sigma }_c^{+}\right)$	$2452.9\pm 0.4$
$M\left({\Sigma }_c^0\right)$	$2453.75\pm 0.14$
${\Sigma }_{c1}\equiv M\left({\Sigma }_c^{++}\right)-M\left({\Sigma }_c^0\right)$	$0.220\pm {0.013}^a$
${\Sigma }_{c2}\equiv M\left({\Sigma }_c^{++}\right)-2M\left({\Sigma }_c^{+}\right)+M\left({\Sigma }_c^0\right)$	$1.92\pm 0.82$
$M\left({\Sigma }_c^{\ast ++}\right)$	${2518.41}_{-0.19}^{+0.21}$
$M\left({\Sigma }_c^{\ast +}\right)$	$2517.5\pm 2.3$
$M\left({\Sigma }_c^{\ast 0}\right)$	$2518.48\pm 0.20$
${\Sigma }_{c1}^{\ast }\equiv M\left({\Sigma }_c^{\ast ++}\right)-M\left({\Sigma }_c^{\ast 0}\right)$	$0.01\pm {0.15}^b$
${\Sigma }_{c2}^{\ast }\equiv M\left({\Sigma }_c^{++}\right)-2M\left({\Sigma }_c^{\ast +}\right)+M\left({\Sigma }_c^{\ast 0}\right)$	$1.89\pm 4.61$
$M\left({\Xi }_c^{+}\right)$	$2467.93\pm 0.18$
$M\left({\Xi }_c^0\right)$	$2470.91\pm 0.25$
${\Xi }_{c1}\equiv M\left({\Xi }_c^{+}\right)-M\left({\Xi }_c^0\right)$	$-2.98\pm {0.22}^c$
$M\left({\Xi }_c^{\prime +}\right)$	$2578.4\pm 0.5$
$M\left({\Xi }_c^{\prime 0}\right)$	$2579.2\pm 0.5$
${\Xi }_{c1}^{\prime }\equiv M\left({\Xi }_c^{\prime +}\right)-M\left({\Xi }_c^{\prime 0}\right)$	$-0.8\pm {0.6}^d$
$M\left({\Xi }_c^{\ast +}\right)$	$2645.57\pm 0.26$
$M\left({\Xi }_c^{\ast 0}\right)$	$2646.38\pm 0.21$
${\Xi }_{c1}^{\ast }\equiv M\left({\Xi }^{\ast +}\right)-M\left({\Xi }^{\ast 0}\right)$	$-0.80\pm {0.26}^e$

Table 8. Table VIII: Predicted and observed isospin splittings in charmed baryons. Errors are experimental values, used in calculating χ 2 superscript 𝜒 2 \chi^{2} contributions.

	${\Sigma }_{c1}$	${\Sigma }_{c2}$	${\Sigma }_{c1}^{\ast }$	${\Sigma }_{c2}^{\ast }$	${\Xi }_{c1}$	${\Xi }_{c1}^{\prime }$	${\Xi }_{c1}^{\ast }$
Fit	0.221	1.916	-0.064	1.916	-2.827	-1.058	-1.200
Expt.	0.220	1.920	0.010	1.890	-2.980	-0.800	-0.800
Error	0.013	0.820	0.150	4.610	0.220	0.600	0.260
${\chi }^2$	0.003	0.000	0.243	0.000	0.484	0.185	2.369

Table 9. Table IX: Masses and isospin splittings of beauty mesons, in MeV.

$B^{+}$	$5279.33\pm 0.13$
$B^0$	$5279.64\pm 0.14$
$B_1\equiv M\left(B^{+}\right)-M\left(B^0\right)$	$-0.31\pm 0.07$
$M\left(B^{\ast +}\right)-M\left(B^{+}\right)$	$45.37\pm 0.21$
$M\left(B^{\ast +}\right)$	$5324.70\pm 0.27$
$M\left(B^{\ast 0}\right)$	–
$B_1^{\ast }\equiv M\left(B^{\ast +}\right)-M\left(B^{\ast 0}\right)$	–

Table 10. Table X: Masses and isospin splittings of beauty baryons, in MeV.

$M\left({\Sigma }_b^{+}\right)$	$5810.56\pm 0.25$
$M\left({\Sigma }_b^{-}\right)$	$5815.64\pm 0.27$
${\Sigma }_{b1}\equiv M\left({\Sigma }_b^{+}\right)-M\left({\Sigma }_b^{-}\right)$	$-5.06\pm {0.18}^a$
$M\left({\Sigma }_b^{\ast +}\right)$	$5830.32\pm 0.27$
$M\left({\Sigma }_b^{\ast -}\right)$	$5834.74\pm 0.30$
${\Sigma }_{b1}^{\ast }\equiv M\left({\Sigma }_b^{\ast +}\right)-M\left({\Sigma }_b^{\ast -}\right)$	$-4.37\pm {0.33}^a$
$M\left({\Xi }_b^0\right)$	$5791.8\pm {0.5}^b$
$M\left({\Xi }_b^{-}\right)$	$5797.0\pm {0.9}^c$
${\Xi }_{b1}\equiv M\left({\Xi }_b^0\right)-M\left({\Xi }_b^{-}\right)$	$-5.9\pm 0.6$
$M\left({\Xi }_b^{\ast 0}\right)$	$5952.3\pm 0.9$
$M\left({\Xi }_b^{\ast -}\right)$	$5955.33\pm 0.13$
${\Xi }_{b1}^{\ast }\equiv M\left({\Xi }_b^{\ast 0}\right)-M\left({\Xi }_b^{\ast -}\right)$	$-3.03\pm 0.91$

Table 11. Table XI: Predicted and observed isospin splittings in beauty baryons. Errors are experimental values, used in calculating χ 2 superscript 𝜒 2 \chi^{2} contributions.

	${\Sigma }_{b1}$	${\Sigma }_{b2}$	${\Sigma }_{b1}^{\ast }$	${\Sigma }_{b2}^{\ast }$	${\Xi }_{b1}$	${\Xi }_{b1}^{\prime }$	${\Xi }_{b1}^{\ast }$
Fit	-5.015	0.410	-4.522	0.410	-5.979	-3.095	-2.848
Expt.	-5.060	–	-4.370	–	-5.900	–	3.030
Error	0.180	–	0.330	–	0.600	–	0.910
${\chi }^2$	0.063	–	0.211	–	0.017	–	0.040

Table 12. Table XII: Masses (in MeV) contributing to relation ( 48 ) between charmed and beauty meson hyperfine splittings.

State	Mass
${\overline{B}}_s^{\ast }$	${5415.4}_{-1.5}^{+1.8}$
${\overline{B}}_s$	$5366.88\pm 0.17$
${\overline{B}}_s^{\ast }-{\overline{B}}_s$	${48.6}_{-1.5}^{+1.8}$
${\overline{B}}^{\ast 0}$	$5324.70\pm {0.22}^a$
${\overline{B}}^0$	$5279.63\pm 0.15$
${\overline{B}}^{\ast 0}-{\overline{B}}^0$	$45.07\pm {0.21}^b$
$D_s^{\ast }$	$2112.2\pm 0.4$
$D_s$	$1968.34\pm 0.07$
$D_s^{\ast }-D_s$	$143.86\pm 0.41$
$D^{\ast +}$	$2010.26\pm 0.05$
$D^{+}$	$1869.65\pm 0.05$
$D^{\ast +}-D^{+}$	$140.603\pm 0.015$

Table 13. Table XIII: Comparison of parameters governing isospin splittings in quark models.

Reference	$\Delta$ or $m_u-m_d$	$a$	Comments
	(MeV)	(MeV)
This work	${\Delta }^m=-4.117$	$a^m=2.119$	Light-quark meson octet
	${\Delta }^b=-2.491$	$a^b=3.052$	Light-quark baryons
[2]	${\Delta }^b=-{2.57}^a$	$a^b={3.06}^a$	Neglecting kinetic term $K$
[3]		$a^m=3.18\pm 0.48$	Eq. (15) and Appendix A
[4]	${\Delta }^b=-{2.48}^a$	$a^b={3.05}^a$
	${\Delta }^b=-{2.67}^b$	$a^b={2.83}^b$
[6]	$m_u-m_d=-2.55\pm 0.25$		$\overline{\text{MS}}$, ${\mu }_{\text{renorm.}}=2$ GeV
[15]	$m_u-m_d=-6$
[24]	$m_u-m_d=-3.8$
[25]	$m_u-m_d=-2.54\pm 0.04$		J. Franklin, priv. commun.
[26]	$m_u-m_d=-2.66$	$a^m=1.5\pm 0.5$	Baryon $a$ unclear
[27]	$m_u-m_d=-4.12$		MIT bag model
[28]	$m_u-m_d=-6.7$		MIT bag model
[29]	$m_u-m_d=-4.4$	$a^b=2.9$
[30]	$m_u-m_d=-2.4$	Ignored	“Photon cloud” effects
[33]	$m_u-m_d=-11$	3.39	Potential models
[35]	$m_u-m_d=-1.88$	3.52	Including 3-body terms
[36]	$m_u-m_d=-1.82$
[38]	${\Delta }^b=-1.84\pm 0.16$
[39]	$m_u-m_d=-2.5$
[42]	$m_u-m_d=-5.7$		$\overline{\text{MS}}$, ${\mu }_{\text{renorm.}}=100$ MeV
	$m_u-m_d=-4.7$		$\overline{\text{MS}}$, ${\mu }_{\text{renorm.}}=200$ MeV

Table 14. Table XIV: Observables needed to refine understanding of isospin breaking.

Observable	Value, if known
$M\left(B^{\ast 0}\right)$	–
$M\left(K^{\ast \pm }\right)$	$891.76\pm 0.25$ MeV, hadroproduction
	$895.5\pm 0.8$ MeV, $\tau$ decay
$M\left(B_s^{\ast }\right)$	${5415.4}_{-1.5}^{+1.8}$ MeV
${\Sigma }_2^{\ast }\equiv M\left({\Sigma }^{\ast +}\right)-2M\left({\Sigma }^{\ast 0}\right)+M\left({\Sigma }^{\ast +}\right)$	$2.6\pm 2.1$MeV
$M\left({\Xi }_{cc}^{++}\right)-M\left({\Xi }_{cc}^{+}\right)$	Predicted in Ref. [4]

Equations

$$\Delta E_{ij,\mathrm{HFs}}=b⟨{\sigma }_i\cdot {\sigma }_j⟩/(m_i{m}_j).$$

$$\Delta E_{ij\mathrm{em}}=\alpha Q_i{Q}_j⟨1/r_{ij}⟩$$

$$\Delta E_{ij\mathrm{HFe}}=-\frac{2\pi \alpha Q_i{Q}_j|{\Psi }_{ij}(0){|}^2⟨{\sigma }_i\cdot {\sigma }_j⟩}{3m_i{m}_j}.$$

$$\sum _{i<j}\Delta E_{ij}=⟨\Delta ⟩+a\sum _{i<j}⟨Q_i{Q}_j⟩+b\sum _{i<j}⟨{\sigma }_i\cdot {\sigma }_j/(m_i{m}_j)⟩+c\sum _{i<j}⟨Q_i{Q}_j{\sigma }_i\cdot {\sigma }_j/(m_i{m}_j)⟩.$$

$${\pi }_2$$

$${\rho }_2$$

$$K_1$$

$$K_1^{\ast }$$

$$N_1$$

$${\Sigma }_1$$

$${\Sigma }_1^{\ast }$$

$${\Sigma }_2$$

$${\Xi }_1$$

$${\Xi }_1^{\ast }$$

$${\Sigma }_1-{\Xi }_1(=-1.23\pm 0.22\mathrm{MeV})={\Sigma }_1^{\ast }-{\Xi }_1^{\ast }(=-1.20\pm 0.91\mathrm{MeV}).$$

$${\Sigma }_2(=1.535\pm 0.090\mathrm{MeV})={\Sigma }_2^{\ast }(=2.6\pm 2.1\mathrm{MeV})$$

$$D_1$$

$$D_1^{\ast }$$

$$-{\Delta }_c^m+\frac{2a_c^m}{3}=3.76\mathrm{MeV}.$$

$$-{\Delta }^m+\frac{2a^m}{3}=(4.12+1.41)\mathrm{MeV}=5.53\mathrm{MeV}.$$

$${\Sigma }_{c1}$$

$${\Sigma }_{c1}^{\ast }$$

$${\Sigma }_{c2}$$

$${\Xi }_{c1}$$

$${\Xi }_{c1}^{\prime }$$

$${\Xi }_{c1}^{\ast }$$

$${\Sigma }_{c2}\equiv M({\Sigma }_c^{++})-2M({\Sigma }_c^{+})+M({\Sigma }_c^0)={\Sigma }_2,$$

$${\Sigma }_{c1}-2{\Xi }_{c1}^{\prime }={\Sigma }_1^{\ast }-2{\Xi }_1^{\ast },$$

$${\Sigma }_{c1}^{\ast }-2{\Xi }_{c1}^{\ast }={\Sigma }_1^{\ast }-2{\Xi }_1^{\ast },$$

$$\bar{m}=362.1\mathrm{MeV},m_s=543.9\mathrm{MeV},b_c/({\bar{m}}^2)=b/(\bar{m}{)}^2=50.0\mathrm{MeV},m_c=1710.5\mathrm{MeV}.$$

$$B_1$$

$$B_1^{\ast }$$

$${\Delta }_b^m+\frac{a_b^m}{3}=\frac{1}{4}(B_1+3B_1^{\ast }).$$

$${\Sigma }_{b1}$$

$${\Sigma }_{b1}^{\ast }$$

$${\Sigma }_{b2}$$

$${\Xi }_{b1}$$

$${\Xi }_{b1}^{\prime }$$

$${\Xi }_{b1}^{\ast }$$

$$\bar{m}=362.1\mathrm{MeV},m_s=543.9\mathrm{MeV},b_b/({\bar{m}}^2)=b/(\bar{m}{)}^2=50.0\mathrm{MeV},m_b=5043.5\mathrm{MeV}.$$

Full text

EFI 18-19

TAUP 3034/18

arXiv:1906.07799

Status of isospin splittings in mesons and baryons

Marek Karlinera†††marek@proton.tau.ac.il and Jonathan L. Rosnerb‡‡‡rosner@hep.uchicago.edu

a School of Physics and Astronomy

Raymond and Beverly Sackler Faculty of Exact Sciences

Tel Aviv University, Tel Aviv 69978, Israel

b Enrico Fermi Institute and Department of Physics

University of Chicago, 5620 S. Ellis Avenue, Chicago, IL 60637, USA

ABSTRACT

Current measurements of isospin splittings in mesons and baryons are sufficiently precise that they allow estimates of the mass difference between constituent up and down quarks. Some previous results are updated in the light of these new measurements, and the importance of better measurements of some observables such as $M(K^{*\pm})$, $M(B^{*0})-M(B^{0})$, and isospin splittings in bottom baryons is noted.

PACS codes: 14.20.Lq, 14.20.Mr, 12.40.Yx

I INTRODUCTION

Isospin-violating mass differences among hadrons are treated in the quark model as a combination of effects. The $u$ and $d$ quarks have an intrinsic mass difference, expressed as a direct contribution to hadron masses and via differing kinetic energies in bound states. Coulomb interactions between quarks depend on the product of their charges times the expectation value of the inverse of their separation. Strong hyperfine interactions between quarks depend on the inverse product of their masses, and electromagnetic hyperfine interactions depend both on that inverse product and on the product of quark charges. One can then write meson and baryon isospin-violating mass differences in terms of a few parameters, yielding sum rules for masses in the limit of small values of these parameters. These were exploited, for example, for mesons with heavy quarks in Ref. [1] and for baryons in Ref. [2]. Isospin splittings in baryons with two heavy quarks were examined in Refs. [3] and [4].

The experimental status of isospin splittings continues to improve. There has been a relatively new measurement of $M(D^{*+})-M(D^{+})$ [5]. Information on masses of individual charge states of charmed and bottom hadrons continues to grow, with exceptional progress in the past year for $\Xi^{c}$, $\Sigma_{b}$, and $\Xi_{b}$ states [6] (compare PDGLive with the 2018 print version). An update of Ref. [1] was performed about ten years ago [7]. Even the light-quark sector has seen improvements since the analysis of Ref. [2], driven by the improved precision in the $\Xi^{0}$ mass measured by the NA48 Collaboration at CERN [8]. An analysis of the present status of isospin splittings in hadrons thus seems appropriate.

We set forth our assumptions, including the interpretation of quarks as constituents with masses of several hundred MeV, in Section II. In Sec. III we update analyses of light-quark mesons and baryons. We treat charmed hadrons in Sec. IV, beauty hadrons in Sec. V, and the relation between the two heavy sectors in Sec. VI. We compare our results with those of several other approaches in Sec. VII, and conclude in Sec. VIII.

II ASSUMPTIONS

In a constituent-quark framework, hadron masses are governed by the sum of their quark masses, the hyperfine interactions among those quarks, and — for hadrons with more than one heavy quark ($c$ or $b$) — an additional binding term between heavy quarks. This approach [9, 10] successfully describes the masses of light-quark hadrons [11], those with a single charm or bottom quark [12], and the mass of the recently observed baryon with two charmed quarks [13].

When the masses of mesons and baryons are fitted with constituent-quark masses and hyperfine interactions, the quark masses in baryons are about 55 MeV heavier than those in mesons [10]. This scheme was used in Ref. [12] to predict $M(\Xi_{cc})=(3627\pm 12)$ MeV, in satisfactory agreement with the observed value [13] $M(\Xi_{cc})=(3621.40\pm 0.78)$ MeV. An alternative scheme explains the mass difference by adding a “string-junction” term of 165 MeV, allowing one to fit mesons and baryons with a universal set of quark masses [14]. However, this scheme predicts $M(\Xi_{cc})$ about 40 MeV higher, so for definiteness we shall stay with the picture of separate quark masses for mesons and baryons.

Quark masses in [12], from Ref. [11], did not include a small binding term for a pair of $s$ quarks, which we now take into account. The results are shown in Table I. Here the strong hyperfine term is parametrized as

[TABLE]

Superscripts $m$ and $b$ will refer to values in mesons and baryons, respectively. The quark masses differ only slightly from those in Ref. [12].

The model we employ takes into account the intrinsic difference $\Delta=(1-\frac{K}{m})(m_{u}-m_{d})$ between $u$ and $d$ quarks, where $K$ is a one-body kinetic energy term [15]; Coulomb interactions

[TABLE]

between quarks; strong hyperfine (HF) interactions $\Delta E_{ij~{}{\rm HFs}}$ as mentioned above; and electromagnetic HF interactions

[TABLE]

Symbols are defined in Ref. [2]. We may thus write the total isospin splitting as

[TABLE]

Separate parameters, labeled by superscripts $m$, will be used for mesons. Henceforth parameters without superscripts will refer to quantities for baryons.

III LIGHT-QUARK HADRONS

A Mesons

The isospin splittings of light-quark mesons, based on masses quoted in Ref. [6], are summarized in Table II. Labels denote the change in isospin associated with each mass splitting. The conflict between the quoted $K^{*}$ mass splitting (which we use) and the individual $K^{*}$ masses needs to be resolved before we can take our analysis as definitive.

Each splitting can be written as the sum of terms depending on the parameters $\Delta^{m}$, $a^{m}$, $b^{m}$, and $c^{m}$:

[TABLE]

Here we have substituted $\Delta^{m}$ for $m^{m}_{u}-m^{m}_{d}$ in the terms for hyperfine splittings. We then see that four observables are expressed in terms of the three unknowns $\Delta^{m}$, $a^{m}$, and $c^{m}$. (We use the value of $b^{m}/(\bar{m}^{m})^{2}=79.4$ MeV from the fit in Table I.) A fit to the observables yields the values $\pi_{2}=4.593$ MeV, $\rho_{2}=-0.065$ MeV, $K_{1}=-3.935$ MeV, and $K^{*}_{1}=-3.200$ MeV, with $\Delta^{m}=-4.117$ MeV, $a^{m}=2.119$ MeV, and $c^{m}/(\bar{m}^{m})^{2}=-2.315$ MeV. The $\chi^{2}$ for this fit is 8.771, nearly all (8.508) contributed by the $K^{*}_{1}$. In view of the non-unanimity of the Particle Data Group’s values for $M(K^{*\pm})$ [6], we urge further study of this state.

B Baryons

We express the observed mass splittings among the octet baryons and the $\Sigma^{*}$ and $\Xi^{*}$ resonances [6], labeled with subscripts denoting their $\Delta I$ values, summarized in Table III, as functions of $\Delta$ ($u-d$ mass difference with effect on kinetic energies), $a$ (Coulomb interaction), $b$ (strong HF interaction), and $c$ (electromagnetic HF interaction). We have neglected effects of two-body kinetic energy operators and additional small corrections [15]. This decomposition is summarized below, where we have linearized expressions from Ref. [4] in $\Delta$. Here $\bar{m}$ is the average of $m_{u}$ and $m_{d}$.

[TABLE]

The predicted isospin splittings are very close to the observed ones, since the Coleman-Glashow relation [16] $\Sigma_{1}=N_{1}+\Xi_{1}$ is very close to being obeyed by those quantities with the smallest experimental errors. The derived parameters are $\Delta=-2.491$ MeV, $a=3.052$ MeV, and $c/\bar{m}^{2}=-1.523$ MeV. The predicted isospin splittings are compared with the observed ones in Table IV. The $\chi^{2}$ for the fit is 0.642, driven mainly by the $\Sigma^{*}$ splittings.

In comparison with Ref. [2], the following relation is satisfied to greater accuracy:

[TABLE]

On the other hand, the relation

[TABLE]

is still plagued with a large experimental error on the right-hand side.

C Meson-baryon comparison

The parameters $\Delta$, $a$, and $c/\bar{m}^{2}$ derived from fits to isospin splittings in mesons and baryons are compared in Table V. The signs are consistent, but central values are rather different. Slightly different parameters are obtained if one adopts a model in which quark masses are universal for mesons and baryons [4]. The difference between parameters obtained from mesons and baryons is not surprising in view of the large spread of values for $m_{u}-m_{d}$ obtained in various models (see Sec. IV in [2]).

IV CHARMED HADRONS

A Mesons

The states at our disposal are summarized in Table VI. There is not enough information to derive a set of parameters describing these mass differences. However, the total spin-dependent terms contribute in a manner proportional to $\langle\sigma_{i}\cdot\sigma_{j}\rangle$, so one may write

[TABLE]

where the superscript denotes charmed mesons. Eliminating the hyperfine contribution $h^{m}_{c}$, one finds

[TABLE]

This is to be compared with the corresponding value for light-quark mesons,

[TABLE]

As for the hyperfine term $h^{m}_{c}=0.35$ MeV, it contains both strong $b^{m}_{c}$ and electromagnetic $c^{m}_{c}$ contributions, which cannot be separated from one another without further assumptions.

B Baryons

In analogy for the light-quark baryons, we write expressions for isospin splittings of charmed baryons:

[TABLE]

We update a couple of relations, noted in Ref. [17], which follow from our assumptions. In 1998 the relation

[TABLE]

appeared to be violated [2], with the left-hand side giving $-2.0\pm 1.3$ MeV while the right-hand side gave $1.71\pm 0.18$ MeV. The present status of charmed baryon masses and isospin splittings is summarized in Table VII. The sum rule is now satisfied, with the left-hand side giving $1.92\pm 0.82$ MeV while the right-hand side gives $1.535\pm 0.090$ MeV.

Another sum rule [17],

[TABLE]

is beginning to be tested, with the left-hand side yielding $1.8\pm 1.2$ MeV while the right-hand side is $2.0\pm 1.5$ MeV. The large errors are associated both with $\Xi^{\prime}_{c1}$ and $\Xi^{*}_{1}$. A further relation is

[TABLE]

where the left-hand side is $1.61\pm 0.54$ MeV. The sum rule is satisfied, with the main uncertainty coming from the right-hand side.

The information about charmed baryons is complete enough that one can perform a fit to their isospin splittings, determining parameters $\Delta_{c}$, $a_{c}$, and $c_{c}/\bar{m}^{2}$ which may be compared with their light-quark counterparts. Fixed parameters in this fit (see the caption to Table I, with $m_{c}$ taken from [12]) are

[TABLE]

The results of this fit are summarized in Table VIII. The derived parameters are $\Delta_{c}=-2.494$ MeV, $a_{c}=2.769$ MeV, and $c_{c}/\bar{m}^{2}=-0.853$ MeV. The first two are rather close to those obtained for light-quark baryons. while the last is of the same sign but only about half as large as $c/\bar{m}^{2}$. The $\chi^{2}$ for the fit is $3.284$, driven mainly by $\Xi^{*}_{c1}$.

V BEAUTY HADRONS

A Mesons

The information on beauty mesons relevant for analysis of isospin splittings is summarized in Table IX. An analysis parallel to that for charmed mesons is not possible in the absence of a value of $M(B^{*0})$. Thus in analogy to Eq. (17) all we can write is

[TABLE]

Eliminating the spin-dependent term $h^{m}_{b}$, one finds

[TABLE]

Now, $h^{m}_{b}$ contains quark charges different from those in $h^{m}_{c}$, but is smaller in magnitude by about a factor of $m^{m}_{b}/m^{m}_{c}\simeq 3$. Thus we probably make an error of only about 0.1 MeV in neglecting it. In that case we would predict $B^{*}_{1}\simeq B_{1}\simeq-0.31\pm 0.07$ MeV. This is consistent with the Particle Data Group’s charge-averaged value $M(B^{*})-M(B)=45.22\pm 0.21$ MeV, to be compared with $M(B^{*+})-M(B^{+})=45.37\pm 0.21$ MeV [6], implying that $B^{*}$ and $B$ isospin splittings are not too different from one another. Definitive conclusions await the measurement of $M(B^{*0})$. For light-quark mesons, the combination $\Delta^{m}+\frac{a^{m}}{3}$ is equal to $(-4.12+0.71)$ MeV = –3.41 MeV.

B Baryons

The relevant masses of beauty baryons are summarized in Table X. Here we have only information on $\Delta I=1$ splittings, as the neutral $\Sigma_{b}$ and $\Sigma_{b}^{*}$ masses are still unmeasured. The decomposition of isospin splittings in terms of $\Delta_{b}$, $a_{b}$, $b_{b}$, and $c_{b}$ is:

[TABLE]

One may perform a fit to these quantities, varying $\Delta_{b}$, $a_{b}$, and $c_{b}/\bar{m}^{2}$. Fixed parameters in this fit (see the caption to Table I, with $m_{b}$ taken from [12]) are

[TABLE]

The results are shown in Table XI. The associated $\chi^{2}$ is 0.331, so a consistent set of parameters is obtained. However, they differ from those fitting the light-quark or charmed baryons:

[TABLE]

Two relations analogous to those for charmed baryons are predicted:

[TABLE]

with the second holding only for equal light-quark baryon and beauty baryon parameters. The right-hand side of this relation is

[TABLE]

whether for light-quark, charmed, or beauty baryons. In Sec. IV we found $\Sigma^{*}_{1}-2\Xi^{*}_{1}=2.0\pm 1.5$ MeV. However, the large splitting between neutral and charged $\Xi_{b}$ states leads the middle term of this sum rule to the value

[TABLE]

The violation of this sum rule is further evidence that one cannot always assume equal values of $\Delta,a,c$ for bottom and lighter-quark systems.

VI CHARM – BEAUTY RELATIONS

A Universal parameters?

The comparison of isospin-violating parameters among light-quark, charmed, and beauty hadrons shows that one cannot regard them as universal. Suppose, first of all, that one took $\Delta^{m}=\Delta^{m}_{c}=\Delta^{m}_{b}$. With this assumption one could solve Eqs. (19) and (31) to obtain $a^{m}_{c}=-0.54$ MeV, $a^{m}_{b}=11.42$ MeV. This makes little sense because the parameter $a^{m}_{c}$ should be positive.

One could, instead, assume that the heavy-quark parameters

[TABLE]

are equal for charmed and beauty mesons. (As we shall see, this is approximately true for baryons.) Then solving Eqs. (19) and (33), assuming $h^{m}_{b}=0$, one finds

[TABLE]

to be compared with the light-quark meson value (see Sec. III A)

[TABLE]

The larger value of $a$ makes sense, because of deeper binding of charmed and bottom hadrons (hence a larger expectation value of $1/r$). However, the difference between $\Delta^{m}_{Q}$ [close to the value in Eq. (41)] and $\Delta^{m}$ is somewhat puzzling. Note that in Table IV we found $\Delta=-2.49$ MeV for light-quark baryons, considerably different from the value $\Delta^{m}$.

B Relations between hyperfine splittings

Although it is not an isospin splitting, a relation between charmed meson and beauty meson hyperfine splittings makes used of the relatively new result from the BaBar Collaboration [5] which enters the Particle Data Group compilation. The relation [1] (updated in Ref. [7] to account for QCD corrections) is

[TABLE]

The left- and right-hand sides of this equation, based on heavy-quark symmetry, are related to one another by $b\leftrightarrow c$. The present status of its terms is summarized in Table XII ([6]).

The left-hand side of Eq. (48) is $3.5\pm 1.7$ MeV, while the right-hand side is $(m_{c}/m_{b})(3.26\pm 0.41)$ MeV $\simeq(1.09\pm 0.14)$ MeV. A decisive test of this relation awaits separate measurements of the masses of $\bar{B}^{*+}$ and $\bar{B}^{*0}$, and a reduced error on the mass of $B^{*}s$.

VII COMPARISON WITH OTHER APPROACHES

Thanks to improvements in computing power, lattice quantum chromodynamics (LQCD) is beginning to be able to take into account isospin splittings in masses and decay constants. (For some references on the latter, see [20].) For LQCD approaches to light-quark splittings see Refs. [21, 22] (octet baryons) and [23] (octet mesons and baryons). We look forward to LQCD calculations of isospin splittings in mesons and baryons containing at least one heavy quark.

Within quark models there is a long history of tackling isospin splittings in hadrons [2, 3, 4, 15, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41]. (The second-to-last reference, though using chiral perturbation theory, gives an extensive list of works based on quark models.) The parameters $\Delta$ (or $m_{u}-m_{d}$) and $a$, when given, are compared in Table XIII. We show there also the latest estimate of $m_{u}-m_{d}$ in the current-quark picture [6].

The relation between current-quark masses (see the mini-review No. 66 in Ref. [6], and the formalism set forth in Ref. [42]) and the constituent-quark masses we are using has been discussed in [15]. However, it has been pointed out in [6] that this relation (and hence the definition of constituent-quark masses) is model-dependent. We note that many of our determinations of $m_{u}-m_{d}$ in the constituent-quark picture are not that far from the current-quark value of $\sim-2.5$ MeV,§§§At a scale of 2 GeV, one recent lattice QCD determination [43] finds $m_{u}=2.130(41)$ MeV, $m_{d}=4.675(56)$ MeV, while another [44] finds $m_{u}=2.50\pm 0.17$ MeV, $m_{d}=4.88\pm 0.20$ MeV. suggesting that in those cases the QCD “dressing” of current quarks may act linearly on their masses. (An exception is presented by the light-quark mesons, for which $|m_{u}-m_{d}|$ is considerably larger, and by the strange-quark mass, which is about 90 MeV heavier than the average non-strange mass in the current-quark picture [6, 43] but 180 MeV heavier than the average non-strange mass in our constituent-quark picture (see the caption of Table I).

VIII CONCLUSIONS

Within a constituent-quark picture, we have updated predictions of isospin splittings in hadrons with at most one $c$ or $b$ quark. Effects considered included an intrinsic $u$–$d$ mass difference and its effect on kinetic energies (parameter $\Delta$), Coulomb interactions among the constituent quarks (parameter $a$), and quark mass dependence on strong and electromagnetic hyperfine splittings (parameters $b$ and $c$, respectively). The parameter $\Delta$ is found to have a non-universal value, ranging from $-4.1$ MeV in light-quark mesons to $-1.5$ MeV in heavy-quark mesons and possibly in $b$-quark baryons. This latter conclusion is preliminary in the absence of a direct measurement of the masses of both $B^{*}$ charge states. A value of $\Delta$ near –2.5 MeV seems consistent with isospin splittings in light-quark and charmed baryons, but more negative than in bottom baryons. Most estimates of the Coulomb interaction term $a$ lie between 2 and 3 MeV.

Quantities whose measurement would help to test relations in the present analysis include improved masses of $K^{*\pm}$ and $B_{s}^{*}$; some isospin splittings in beauty baryons; and $M(\Xi_{cc}^{++})-M(\Xi_{cc}^{+})$, predicted in Ref. [4] to be $(2.17\pm 0.11)$ MeV under the present set of assumptions [and $(1.49\pm 0.12)$ MeV in a model with universal quark masses for mesons and baryons.] We look forward to these developments, summarized in Table XIV, in the data.

ACKNOWLEDGMENTS

We thank Tom DeGrand for a question leading to the present investigation; Mike Sokoloff for alerting us to Refs. [5] and [7] and for a useful conversation; and Davide Giusti and Feng-Kun Guo for helpful comments on the manuscript.