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Mass inequalities for baryons with heavy quarks

Marek Karliner

1,*

and Jonathan L. Rosner

2,†

School of Physics and Astronomy, Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel

Enrico Fermi Institute and Department of Physics, University of Chicago,

5640 South Ellis Avenue, Chicago, Illinois 60637, USA

(Received 13 December 2019; accepte d 6 February 2020; published 18 February 2020)

Baryons with one or more heavy quarks have been shown, in the context of a nonrelativistic description, to

exhibit mass inequalities under permutations of their quarks, when spin averages are taken. These inequalities

sometimes are invalidated when spin-dependent forces are taken into account. A notable instance is the

inequality 2EðMmmÞ >EðMMmÞþEðmmmÞ, where m ¼ m

¼ m

, satisfied for M ¼ m

or M ¼ m

but

not for M ¼ m

, unless care is taken to remove effects of spin-spin interactions. Thus in the quark-level analog

of nuclear fusion, the reactions Λ

→ Ξ

N and Λ

→ Ξ

þþ

n are exothermic, releasing respectively 138

and 12 MeV, while ΛΛ → ΞN is endothermic, requiring an input of between 23 and 29 MeV. Here we explore

such mass inequalities in the context of an approach, previously shown to predict masses successfully, in

which contributions consist of additive constituent-quark masses, spin-spin interactions, and additional

binding terms for pairs each member of which is at least as heavy as a strange quark.

DOI: 10.1103/PhysRevD.101.036015

I. INTRODUCTION

Quantum chromodynamics (QCD) predicts the existence

of baryons containing not only one but two or three heavy

quarks (c or b). The LHCb Collaboration at CERN has

discovered the first doubly heavy baryon, a Ξ

þþ

¼ ccu

state in the decay modes Λ

−

[1] and Ξ

[2],ata

mass MðΞ

þþ

Þ¼3621.24  0.65  0.31 MeV, very close

to that predicted in Ref. [3]. (Hints of a possible isospin

partner Ξ

decaying to Λ

−

have also been found

[4,5].) In this approach, one adds up constituent-quark

masses and spin-spin interactions [6–8] and corrects for the

additional binding in any quark pair both of whose

members are at least as heavy as a strange quark [3].

Under some circumstances hadrons satisfy mass inequal-

ities associated with permutations of their quarks [9–16].

For example, under some conditions one expects

2EðMmmÞ >EðMMmÞþEðmmmÞ, where E denotes

the mass, m ¼ m

¼ m

, and M is the mass of a heavy

quark, to apply to spin-averaged states [cf. Eq. (3) in

Ref. [15]]. This inequality may be invalidated when spin-

dependent forces are taken into account; it holds for M ¼

or M ¼ m

but not for M ¼ m

. Thus, in the quark-level

analog of nuclear fusion [17], the reactions Λ

→ Ξ

and Λ

→ Ξ

þþ

n are exothermic, releasing respectively

138 and 12 MeV, while ΛΛ → ΞN is endothermic, requir-

ing energy input of between 23 and 29 MeV, depending on

which members of the N and Ξ doublets one uses.

Here we give some examples of inequalities involving

baryon masses in our constituent-quark approach. We

outline in Sec. II some of the relations and their origin.

In Sec. III we treat light-quark systems in which the strange

quark plays the role of a heavy quark. Baryons with one and

two heavy quarks (c, b) are described in Secs. IV and V,

respectively, while Sec. VI concludes.

II. INEQUALITIES AND THEIR ORIGIN

A number of mass inequalities involving ground-

state mesons and baryons were noted by Nussinov [11],

respectively,

ðm

þ m

Þ;m

xyy

ðm

xxy

þ m

yyy

Þ: ð1Þ

We shall motivate these relations in a simple case where

quark masses enter through their nonrelativistic kinetic

energy, but they are much more general (see many of the

references quoted above, in particular [15]). Consider

systems governed by the Hamiltonians

2μ

þ VðxÞ; μ

≡

þ m

: ð2Þ

Then, since

marek@proton.tau.ac.il

†

rosner@hep.uchicago.edu

Published by the American Physical Society under the terms of

the Creative Commons Attribution 4.0 International licens e.

Further distribution of this work must maintain attribution to

the author(s) and the published article’s title, journal citation,

and DOI. Funded by SCOAP

PHYSICAL REVIEW D 101, 036015 (2020)

2470-0010=2020=101(3)=036015(4) 036015-1 Published by the American Physical Society

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