Mass inequalities for baryons with heavy quarks
Marek Karliner
1,*
and Jonathan L. Rosner
2,†
1
School of Physics and Astronomy, Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
2
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
u
¼ m
d
, satisfied for M ¼ m
b
or M ¼ m
c
but
not for M ¼ m
s
, unless care is taken to remove effects of spin-spin interactions. Thus in the quark-level analog
of nuclear fusion, the reactions Λ
b
Λ
b
→ Ξ
bb
N and Λ
c
Λ
c
→ Ξ
þþ
cc
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
cc
¼ ccu
state in the decay modes Λ
c
K
−
π
þ
π
þ
[1] and Ξ
þ
c
π
þ
[2],ata
mass MðΞ
þþ
cc
Þ¼3621.24 0.65 0.31 MeV, very close
to that predicted in Ref. [3]. (Hints of a possible isospin
partner Ξ
þ
cc
decaying to Λ
c
K
−
π
þ
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
u
¼ m
d
, 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
m
b
or M ¼ m
c
but not for M ¼ m
s
. Thus, in the quark-level
analog of nuclear fusion [17], the reactions Λ
b
Λ
b
→ Ξ
bb
N
and Λ
c
Λ
c
→ Ξ
þþ
cc
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,
m
x
¯
y
>
1
2
ðm
x
¯
x
þ m
y
¯
y
Þ;m
xyy
>
1
2
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
H
ij
¼
p
2
2μ
ij
þ VðxÞ; μ
ij
≡
m
1
m
2
m
1
þ m
2
: 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
3
.
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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