Physics Letters B 781 (2018) 719–722
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Baryon masses and σ terms in SU(3) BChPT ×1/N
c
I.P. Fernando
a
, J.M. Alarcón
b
, J.L. Goity
a,b,∗
a
Department of Physics, Hampton University, Hampton, VA 23668, USA
b
Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA
a r t i c l e i n f o a b s t r a c t
Article history:
Received
9 April 2018
Received
in revised form 20 April 2018
Accepted
25 April 2018
Available
online 27 April 2018
Editor:
B. Grinstein
Keywords:
Sigma
terms
Nucleon
mass
Baryon
masses
Gell-Mann–Okubo
mass formula
Baryon masses and nucleon σ terms are studied with the effective theory that combines the chiral and
1/N
c
expansions for three flavors. In particular the connection between the deviation of the Gell-Mann–
Okubo
relation and the σ term associated with the scalar density
¯
uu +
¯
dd −2
¯
ss is emphasized. The latter
is at lowest order related to a mass combination whose low value has given rise to a σ term puzzle. It
is shown that while the nucleon σ terms have a well behaved low energy expansion, that mass combi-
nation
is affected by large higher order corrections non-analytic in quark masses. Adding to the analysis
lattice QCD baryon masses, it is found that σ
π N
= 69(10) MeV and σ
s
has natural magnitude within its
relatively large uncertainty.
© 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
Baryon mass dependencies on quark masses, quantified by
the different σ -terms, are among the fundamental observables in
baryon chiral dynamics. In particular, they give information on the
baryon matrix elements of scalar quark densities, for which there
is no alternative way for their determination. The definition of
σ terms is through the Feynman–Hellmann theorem,
1
which, for
three flavors, through the physical baryon masses gives access to
only two such terms, namely those associated with the SU(3) octet
quark mass combinations m
3
= m
u
− m
d
and m
8
=
1
√
3
(
ˆ
m − m
s
),
where
ˆ
m is the average of the u and d quark masses. The σ terms
associated with the singlet component m
0
=
1
3
(2
ˆ
m + m
s
) require
knowledge of baryon masses for unphysical quark masses, which
is made possible through lattice QCD (LQCD) calculations. On the
other hand, the pion–nucleon σ term σ
π N
≡
ˆ
m
2m
N
N |
¯
uu +
¯
dd | N
is accessible through its connection to pion–nucleon scattering via
a low energy theorem [1–3]. Such a determination of σ
π N
had
a long evolution through the availability of increasingly accurate
data and the development of combined methods of dispersion the-
ory
and chiral perturbation theory [4–11]. The values obtained for
*
Corresponding author.
E-mail
address: goity @jlab .org (J.L. Goity).
1
The following notation will be used: σ
i
(B) = m
i
∂
∂m
i
m
B
, where m
i
indicates a
quark mass (i = u, d, s) or combination thereof (i =0, 3, 8), and B is a given baryon.
When B is not explicitly indicated it is assumed to be a nucleon.
σ
π N
range from 45 MeV [4–6]to 64 MeV [7–12], where the dif-
ference
between the results of the different dispersive analyses
resides mostly in the different values of the S-wave π N scatter-
ing
lengths a
1/2,3/2
used in the subtractions, cf. [12]. In addition to
the results from the analyses of π N scattering, LQCD calculations
extrapolated to or at the physical point obtain different results,
with values consistent with the recent π N results [13] and smaller,
σ
π N
≈ 40 MeV [14–17]. The relatively large range of values ob-
tained
for σ
π N
keeps it as an active topic of study, and in part
motivates the present work. An additional motivation is the rele-
vance
of scalar quark operator matrix elements, quantities that are
relevant in studies of direct dark matter detection [18–20], and of
lepton flavor violation through μ–e conversion in scattering with
nuclei [21].
A puzzle that has been emphasized for a long time [22]is the
relation between σ
π N
in the isospin symmetry limit and the nucle-
on’s
ˆ
σ ≡
√
3
ˆ
m
m
8
σ
8
, namely σ
π N
=
ˆ
σ + 2
ˆ
m
m
s
σ
s
, which for a natural
size value of σ
s
should give σ
π N
∼
ˆ
σ . The origin of the puzzle
is the relation: σ
8
=
1
3
(2m
N
− m
− m
) (or other combinations
related via the Gell-Mann–Okubo (GMO) relation) valid at linear
order in quark masses, which gives
ˆ
σ ∼ 25 MeV. If that relation
is a reasonable approximation to the value of
ˆ
σ , the implication
is that, contrary to expectations, m
s
must give a very large con-
tribution
to the nucleon mass even for the smaller values of σ
π N
.
The puzzle is particularly striking for the larger values that have
been obtained for σ
π N
, which would imply σ
s
∼500 MeV. Indeed,
this is clearly impossible if one considers that σ
s
is OZI suppressed
with respect to σ
π N
.
https://doi.org/10.1016/j.physletb.2018.04.054
0370-2693/
© 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.