Poster presented at the Computational Sciences 2013 Conference (Winner of poster competition). We present results on the masses of all forty light, strange and charm baryons from Lattice QCD simulations, focusing particularly on the computational aspects and requirements of such calculations.

1. Computing the masses of hyperons and charmed baryons from lattice QCD
C. Kallidonis1
with C. Alexandrou1,2, V. Drach3, K. Hadjiyiannakou2, K. Jansen3, G. Koutsou1
[1] Computation-based Science and Technology Research Center, The Cyprus Institute
[2] Department of Physics, University of Cyprus
[3] Deutsches Elektronen-Synchrotron (DESY), Zeuthen, Germany
References
[1] J. Beringer et al., (PDG), Phys.Rev. D86, 010001 (2012)
[2] H. Na and S. A. Gottlieb, PoS LAT2007, 124 (2007), 0710.1422
[3] H. Na and S. A. Gottlieb, PoS LATTICE2008, 119 (2008), 0812.1235
[4] R.A. Briceno et al., (2011), 1111.1028
[5] L. Liu et al., Phys. Rev. D81, 094505 (2010), 0909.3294
Introduction
• Lattice Quantum Chromodynamics (LQCD) is the only method so far which solves
the fundamental theory of strong interactions non-perturbatively
• Petascale computer resources available nowadays through PRACE make lattice
QCD simulations at the physical pion mass possible
• Most of the simulations and our calculations were performed on HPC facilities
such as the JUGENE, JUROPA and JUQUEEN at JSC
• Part of the analysis was performed on the Cy-Tera machine at
The Cyprus Institute, provided through the project LinkSCEEM
In this work:
• we present results on the hyperon and charmed baryon masses, including results
from simulations at the physical pion mass
• we compare with the known spectrum and predict the mass of charmed baryons
Hyperons and charmed baryons
Baryons consist of three quarks. There is a total of 6 quarks
and they come in three generations. According to the
theory of Strong Interactions, known as Quantum
Chromodynamics (QCD), quarks cannot be observed freely.
Nuclear interactions confine them into bound states, the
hadrons. The mediators of the strong force are the gluons.
Hadrons can have half integer spin in which case they are
called baryons, or integer spin and are known as mesons. In
this work we focus on baryons consisting of up, down,
strange and charm quarks.
hadrons
mesons
baryons
Light baryons are made of up and down
quarks. Hyperons have at least one strange
quark and charmed baryons at least one
charm quark.
s
u
d
hyperon
c
d
c
charmed
baryon
u
d
u
proton
gN
A = F + D
g⌃
A = 2F
g⌅
A = −D + F
⇒ gN
A − g⌃
A + g⌅
A = 0
SU(3) = gN
A − g⌃
A + g⌅
A
x = (m2
K − m2
⇡)4⇡2
f2
⇡
C(tf − ti) =
me↵(t) = log
C(t)
C(t + 1)
→
t→∞
M
Gµ(tf − ti,Aµ(x,t)) =
4 ⊗ 4 ⊗ 4 = 20 ⊕ 20 ⊕ 20 ⊕ ¯4
gA = lim
tf −ti→∞
t−ti→∞
Gµ(tf − ti,Aµ)
C(tf − ti)
Aµ(x,t) = ¯q(x) µ 5q(x)
1
Baryons consisting of a combination of three of up, down
strange and charm quarks are perceived as SU(3) subgroups
of SU(4) flavor symmetry. They are grouped into two 20-plets, one with spin 1/2
baryons and one with spin 3/2 as shown below.
QCD on the lattice
Lattice QCD is a discrete formulation of QCD on a finite
4-dimensional space-time lattice that enables numerical
simulation of QCD using Monte-Carlo methods. Lattice
sites are separated by a lattice spacing, a. Quarks are
defined on lattice sites while gluons are links
connecting neighboring sites. gN
A = F + D
g⌃
A = 2F
g⌅
A = −D + F
⇒ gN
A − g⌃
A + g⌅
A = 0
SU(3) = gN
A − g⌃
A + g⌅
A
x = (m2
K − m2
⇡)4⇡2
f2
⇡
C(tf − ti) =
me↵(t) = log
C(t)
C(t + 1)
→
t→∞
M
G(t − t ,A (x,t)) =
At the large time limit, the two-point functions
yield the energy of the low-lying hadrons. A
constant fit on a plateau region is performed to
extract the value.
gN
A = F + D
g⌃
A = 2F
g⌅
A = −D + F
⇒ gN
A − g⌃
A + g⌅
A = 0
SU(3) = gN
A − g⌃
A + g⌅
A
x = (m2
K − m2
⇡)4⇡2
f2
⇡
C(tf − ti) =
me↵(t) = log
C(t)
C(t + 1)
→
t→∞
M
G(tf − ti,Aµ(x,t)) =
4 ⊗ 4 ⊗ 4 = 20 ⊕ 20 ⊕ ¯4
Computer resources requirements
As can be seen from the strong scaling plot of Fig. 1
the behavior of our inversion algorithm for the
BlueGene/Q is most optimal when running on
1024 nodes.
The inversion of the Dirac operator for the four quarks
on a 483x96 lattice requires
9600 JUQUEEN core-hours per gauge configuration
Quark propagators are inverses of the Dirac operator. For a typical lattice size this
operator is a sparse matrix of dimensions ~108x108. To invert the Dirac operator we
use iterative methods such as the conjugate gradient method on high-performance
computing facilities.
The extraction of hadron masses requires the evaluation of correlators. We
developed hybrid MPI-OpenMP codes to evaluate the correlators for the 40 low-
lying baryons. These codes are imported on JUQUEEN, JUROPA and Cy-Tera and
for a 483x96 lattice they require
304 JUROPA core-hours per gauge configuration
Recent algorithmic improvements include deflation
methods that reduce the time required for inversions.
This is particularly beneficial when multiple inversions
are performed for a single gauge configuration.
Masses of hyperons and charmed baryons
The lattice results are in units of the lattice spacing. Before comparing with
experiment, the lattice spacing needs to be fixed. This is done by using a physical
quantity whose value is known from experiment. In our case we use the nucleon
mass.
Thus, an analysis with 1000 gauge configurations would require 9,600,000 JUQUEEN
core-hours for quark propagators and 304,000 JUROPA core-hours for two-point
functions.
Conclusions
• Lattice QCD provides a method of calculating key observables using high-
performance computing resources. In particular, Petascale computers enable
simulations with physical pion mass which otherwise would not be possible. Thus,
the results are more accurate and the systematic errors are reduced.
• Our results for the masses of the hyperons and charmed baryons produced at the
physical pion mass are in very good agreement with experiment and other lattice
calculations.
• We provide predictions for the masses of the charmed baryons which are sought
experimentally.
In Figs. 2, 3 and 4 we present our results
extracted directly from simulations at
the physical pion mass. Comparison with
experiment [1] as well as with other
lattice calculations [2-5] shows an
overall agreement. Our prediction for the
masses of the charmed baryons that
have not yet been measured
experimentally are also displayed.
Fig. 1
Fig. 2
Fig. 3 Fig. 4
The Project GPUCW (ΤΠΕ/ΠΛΗΡΟ/0311(ΒΙΕ)/09) is co-financed by the European Regional Development Fund and
the Republic of Cyprus through the Research Promotion Foundation.
In order to calculate observables on the lattice, such as the
mass of a baryon, one needs to obtain information on the
probability of a quark to transport from one lattice site to the other. This is
encompassed in objects called quark propagators, which in our calculations are
the most computationally intensive objects to evaluate. These propagators are
appropriately combined to form correlation functions and from those the various
observables can be extracted. Baryon masses are obtained from two-point
correlation functions. The blue lines in the figure represent the propagator for
each quark.