Isospin splittings in baryons with two heavy quarks

TL;DR

This paper calculates isospin mass splittings in doubly heavy baryons using simplified methods, providing specific numerical predictions with associated uncertainties.

Contribution

The authors apply previously proposed simple methods to compute isospin splittings in baryons with two heavy quarks, offering new quantitative results.

Findings

Mass difference for $ ext{Xi}_{cc}$ baryons: 1.41 MeV

Mass difference for $ ext{Xi}_{bb}$ baryons: -4.78 MeV

Mass difference for $ ext{Xi}_{bc}$ baryons: -1.69 MeV

Abstract

Isospin splittings in baryons with two heavy quarks and a $u$ or $d$ quark are calculated using simple methods proposed previously by the authors. The results are $M(\Xi_{cc}^{++})-M(\Xi_{cc}^+) = 1.41 \pm 0.12^{+0.76}$ MeV, $M(\Xi_{bb}^0) - M(\Xi_{bb}^-) = - 4.78 \pm 0.06^{+0.03}$ MeV, and $M(\Xi_{bc}^+)-M(\Xi_{bc}^0)=-1.69 \pm0.07 ^{+0.39}$ MeV, where the statistical errors reflect uncertainties in input mass splittings, and the systematic errors are associated with the choice of constituent-quark masses.

Tables

Table 1. Table I: Experimental mass splittings between octet baryons [ 49 ] .

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

Table 2. Table II: Contributions to isospin splittings (MeV) using universal constituent-quark masses in mesons and baryons.

	$N_1$	${\Sigma }_1$	${\Sigma }_2$	${\Xi }_1$	${\Xi }_{cc,1}$	${\Xi }_{bb,1}$	${\Xi }_{bc,1}$
$m_u-m_d$	–2.68	–5.36	0.00	–2.68	–2.68	–2.68	–2.68
Coulomb	0.94	–0.94	2.83	–1.89	3.77	-1.89	0.94
StrHF	0.88	-0.24	0.00	-1.12	-0.33	–0.11	–0.22
EMHF	–0.43	–1.54	–1.30	–1.11	0.64	–0.11	0.27
Total	–1.293	–8.086	1.535	-6.793	1.409	–4.783	-1.687
					$\pm 0.116$	$\pm 0.058$	$\pm 0.067$

Table 3. Table III: Contributions to isospin splittings (MeV) using separate constituent-quark masses in mesons and baryons.

	$N_1$	${\Sigma }_1$	${\Sigma }_2$	${\Xi }_1$	${\Xi }_{cc,1}$	${\Xi }_{bb,1}$	${\Xi }_{bc,1}$
$m_u-m_d$	–2.48	–4.95	0.00	–2.48	–2.48	–2.48	–2.48
Coulomb	1.02	–1.02	3.05	–2.04	4.07	–2.04	1.02
StrHF	0.67	–0.24	0.00	–0.91	–0.29	–0.10	–0.19
EMHF	–0.51	–1.88	–1.52	–1.37	0.86	–0.15	0.36
Total	–1.293	–8.086	1.535	–6.793	2.167	–4.754	–1.293
					$\pm 0.109$	$\pm 0.058$	$\pm 0.062$

Table 4. Table IV: Comparison of predictions for isospin splittings (MeV) in doubly heavy baryons.

Reference	${\Xi }_{cc,1}$	${\Xi }_{bb,1}$	${\Xi }_{bc,1}$
This work	$1.41\pm {0.12}^{+0.76}$	$-4.78\pm {0.06}^{+0.03}$	$-1.69\pm {0.07}^{+0.39}$
[23]	4.7
[46]	$1.5\pm 2.7$	$-6.3\pm 1.7$	$-0.9\pm 1.8$
[52]a	$2.3\pm 1.7$	$-5.3\pm 1.1$	$-1.5\pm 0.9$
[53]	$2.16\pm 0.11\pm 0.12$
[54]	4.7
[55]	1.11
[56]	–9

Equations

$$\Delta E_{ij\mathrm{HFs}}=\mathrm{const}\times \frac{|{\Psi }_{ij}(0){|}^2⟨{\sigma }_i\cdot {\sigma }_j⟩}{m_i{m}_j},$$

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

$$N_1$$

$${\Sigma }_1$$

$${\Sigma }_2$$

$${\Xi }_1$$

$${\Xi }_{cc,1}\equiv M({\Xi }_{cc}^{++})-M({\Xi }_{cc}^{+})$$

$${\Xi }_{bb,1}\equiv M({\Xi }_{bb}^0)-M({\Xi }_{bb}^{-})$$

$${\Xi }_{bc,1}\equiv M({\Xi }_{bc}^{+})-M({\Xi }_{bc}^0)$$

Full text

EFI 17-14

TAUP 3019/17

arXiv:1706.06961

Isospin splittings in baryons with two heavy quarks

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

Isospin splittings in baryons with two heavy quarks and a $u$ or $d$ quark are calculated using simple methods proposed previously by the authors. The results are $M(\Xi_{cc}^{++})-M(\Xi_{cc}^{+})=1.41\pm 0.12^{+0.76}$ MeV, $M(\Xi_{bb}^{0})-M(\Xi_{bb}^{-})=-4.78\pm 0.06^{+0.03}$ MeV, and $M(\Xi_{bc}^{+})-M(\Xi_{bc}^{0})=-1.69\pm 0.07^{+0.39}$ MeV, where the statistical errors reflect uncertainties in input mass splittings, and the systematic errors are associated with the choice of constituent-quark masses.

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

I Introduction

Baryons with more than one heavy quark have proved to be elusive. The SELEX collaboration has presented evidence for several states [1, 2, 3], but other experiments have not confirmed them [4, 5, 6, 7, 8]. Simple constituent-quark models incorporating effective quark masses, hyperfine interactions, and estimates of binding energies [9, 10] have proved remarkably successful in reproducing the masses of known hadrons with accuracies of several MeV. In agreement with most other estimates [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35] including ones using lattice gauge theory [36, 37, 38, 39, 40, 41, 42, 43, 44], this method [45] gives masses of $ccq$ ($q=u,d$) about 100 MeV above the SELEX values, and close to the most recent lattice estimates [44].

The capability of the LHCb experiment to identify hadrons containing heavy quarks makes it a prime instrument for determining the masses of the lowest $\Xi_{cc}^{++}=ccu$ and $\Xi_{cc}^{+}=ccd$ states. As a benchmark, Ref. [45] predicts $M(\Xi_{cc})=3627\pm 12$ MeV for their isospin average. Their isospin splitting is then of interest, both as a theoretical question and as a guide to further observation. In particular, the SELEX Collaboration reports large splittings whose values depend on which of several bumps are assigned to the lowest $\Xi_{cc}$ states [46]. In the present paper we apply some simple methods, used with previous success, to estimate isospin splittings in the ground-state $\Xi_{cc}$, $\Xi_{bb}$, and $\Xi_{bc}$ baryons. We describe the methods in Sec. II, present an alternative set of input parameters in Sec. III, quote results in Sec. IV, and conclude in Sec. V.

II Methods

The impending improvement in the mass of the $\Xi^{0}$ baryon by the NA48 experiment at CERN [47] and the KTeV experiment at Fermilab led one of us [48] to consider improved tests of relations for baryon isomultiplet splittings. A simple model was adopted which took into account the intrinsic difference $\Delta=m_{u}-m_{c}$ between $u$ and $d$ quarks, Coulomb interactions $\Delta E_{ij~{}{\rm em}}=\alpha Q_{i}Q_{j}\langle 1/r_{ij}\rangle$ between quarks, strong hyperfine (HF) interactions

[TABLE]

and electromagnetic HF interactions

[TABLE]

where symbols are defined in Ref. [48]. We use the observed mass splittings among the octet baryons [49], labeled with subscripts denoting their $\Delta I$ values, summarized in Table I, to define the relative strengths of each contribution.

Each of these splittings may be expressed as a function of four unknowns $\Delta$ (intrinsic $u-d$ mass difference), $a$ (Coulomb interaction), $b$ (strong HF interaction), and $c$ (electromagnetic HF interaction), where we have simplified the notation of Ref. [48] and neglected effects of two-body kinetic energy operators:

[TABLE]

where $\bar{m}$ is the average of $m_{u}$ and $m_{d}$, and we have neglected a term of second order in $\Delta$ in $\Sigma_{2}$. We have written a shorthand for $\Sigma_{1}$ since under the present assumptions it satisfies the Coleman-Glashow relation $\Sigma_{1}=N_{1}+\Xi_{1}$ [50] and is not independent. Given quark masses and an estimate of strong hyperfine structure from the splitting between the $\Delta$ resonance and the nucleon (fixing $b$), one can determine the three free parameters $\Delta$, $a$, and $\gamma\equiv c/\bar{m}^{2}$.

Similar methods lead to estimates for isospin splittings in baryons with two heavy quarks. The results, after neglecting terms of second order in $\Delta$, and defining $\beta\equiv b/\bar{m}^{2}$, are

[TABLE]

In order to specify $\Delta,~{}a,$ and $\gamma$ we must choose a set of constituent-quark masses. This was done in Ref. [51], in two models, depending on whether or not a universal set of masses was chosen for mesons and baryons. In this section we shall consider quark masses which fit both baryons and mesons simultaneously, with an added “string-junction” contribution $S=161.5$ MeV for baryons. Such an additive constant does not affect mass differences, with which we are concerned here. (The alternative set is considered in the next section.) Thus we take $\bar{m}=308.5$ MeV, $m_{s}=482.2$ MeV, $\beta=50.4$ MeV, $m_{c}=1655.6$ MeV, and $m_{b}=4988.6$ MeV. A fit to octet baryon masses then give $\Delta=-2.681$ MeV, $a=2.830$ MeV, $\gamma=-1.295$ MeV, and contributions summarized in Table II. Here we have fixed $N_{1}$ at its measured value of -1.2933 MeV, as its experimental error is negligible. The uncertainties are those generated by varying each octet-baryon splitting by $1\sigma$ and adding the errors in quadrature.

Note that the $\Delta I=2$ mass difference is fitted exactly. The $\chi^{2}$ for this fit is 0.083, of which 0.010 comes from $\Sigma_{1}$ and 0.073 comes from $\Xi_{1}$. This is just the extent to which the Coleman-Glashow relation is obeyed.

III Alternative parameters

In a model in which mesons and baryons are described by separate constituent-quark masses [51], the parameters are $\bar{m}=363.7$ MeV, $m_{s}=536.3$ MeV, $\beta=49.3$ MeV, $m_{c}=1710.5$ MeV, and $m_{b}=5043.3$ MeV. The fit gives $\Delta=-2.476$ MeV, $a=3.053$ MeV, and $\gamma=-1.518$ MeV. The results are shown in Table III. The uncertainties are those generated by varying each octet-baryon splitting by $1\sigma$ and adding the errors in quadrature.

The fit again reproduces the value of $\Sigma_{2}$ exactly, obtains the same values for $\Sigma_{1}$ and $\Xi_{1}$, and thus has the same individual and overall $\chi^{2}$ values.

IV Results

A slight preference for the string-based constituent-quark masses was expressed in Ref. [51]. Hence we shall quote predictions for isospin splittings based on that model, with a systematic error associated with the possible choice of independent constituent-quark masses for mesons and baryons. The results are: $M(\Xi_{cc}^{++})-M(\Xi_{cc}^{+})=1.41\pm 0.12^{+0.76}$ MeV, $M(\Xi_{bb}^{0})-M(\Xi_{bb}^{-})=-4.78\pm 0.06^{+0.03}$ MeV, and $M(\Xi_{bc}^{+})-M(\Xi_{bc}^{0})=-1.69\pm 0.07^{+0.39}$ MeV. The first error is the greater of two very similar statistical errors in Tables II and III.

Some approaches give values consistent with ours. Ref. [52] finds $\Xi_{cc,1}=2.3\pm 1.7$ MeV, $\Xi_{bb,1}=-5.3\pm 1.1$ MeV, and $\Xi_{bc,1}=-1.5\pm 0.9$ MeV. Ref. [46] finds $1.5\pm 2.7$ MeV, $-6.3\pm 1.7$ MeV, and $-0.9\pm 1.8$ MeV for these quantities, while a lattice-QCD-based approach [53] finds $\Xi_{cc,1}=(2.16)(11)(17)$ MeV, slightly favoring our set of independent quark masses for mesons and baryons. These results, along with some others, are compared in Table IV.

V Discussion and conclusions

We have estimated isospin mass splittings in baryons $\Xi_{cc}$, $\Xi_{bb}$, and $\Xi_{bc}$ containing two heavy quarks. A major source of systematic error, particularly in $\Xi_{cc,1}\equiv M(\Xi_{cc}^{++})-M(\Xi_{cc}^{+})$, is uncertainty in the choice of constituent-quark mass, giving $\Xi_{cc,1}=1.41$ MeV for our favored model of universal quark masses in mesons and baryons, while separate quark masses for mesons and baryons yield $\Xi_{cc,1}=2.17$ MeV.

One assumption we have made concerns the universality of the expectation value $\langle r_{ij}\rangle$ in evaluating the Coulomb self-energy. It is possible that two heavy quarks are more tightly bound to one another than a light quark and a heavy one or two light quarks. To lowest order, this should not affect isospin splittings. However, the difference between binding of two light quarks from binding of a heavy quark with a light one remains to be tested. A start on this program was made in Sec. VI of Ref. [48]. A relation $\Sigma_{c2}\equiv M(\Sigma_{c}^{++})-2M(\Sigma_{c}^{+})+M(\Sigma_{c}^{+})=\Sigma_{2}$ was found there to be poorly obeyed, but now reads $(1.92\pm 0.82)$ MeV $=(1.535\pm 0.090)$ MeV, in satisfactory agreement with the predicted equality.

It is worth recalling predicted lifetimes of baryons with two heavy quarks, as the states with longer lifetimes are likely to be easier to distinguish from background in a hadron collider. Predictions by the authors are given in Table XVI of Ref. [45], including $\tau(\Xi_{cc}^{++})=185$ fs and $\tau(\Xi_{cc}^{+})=53$ fs. Most other predictions quoted there are about three times as large, while preserving the ratio $\tau(\Xi_{cc}^{++})/\tau(\Xi_{cc}^{+})\simeq 3$. The reason for the shorter lifetime of $\Xi_{cc}^{+}=ccd$ is that the internal $W$ exchange process $cd\to su$ is permitted, while it cannot occur for $\Xi_{cc}^{++}=ccu$. For a similar reason, one expects $\tau(\Xi_{bc}^{+})>\tau(\Xi_{bc}^{0})$ whereas $\tau(\Xi_{bb}^{0})\simeq\tau(\Xi_{bb}^{-})$.

We hope that these estimates prove of use in discovery of such states.

Acknowledgements

The work of J.L.R. was supported by the U.S. Department of Energy, Division of High Energy Physics, Grant No. DE-FG02-13ER41958.