Chiral unitary theory of scalar mesons in a finite volume

Abstract

We develop a scheme for the extraction of the properties of the scalar mesons
$f_0(600)$, $f_0(980)$, and $a_0(980)$ from lattice QCD data. This scheme is
based on a two-channel chiral unitary approach with fully relativistic
propagators in a finite volume. In order to discuss the feasibility of finding
the mass and width of the scalar resonances, we analyze synthetic lattice data
with a fixed error assigned, and show that the framework can be indeed used for
an accurate determination of resonance pole positions in the multi-channel
scattering.

Full text

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arXiv:1108.3923v1 [hep-lat] 19 Aug 2011
Chiral unitary theory of scalar mesons in a finite
volume
E. Oset1, M. Döring2, U.-G. Meißner2,3and A. Rusetsky2
1Departamento de Física Teórica and IFIC, Centro Mixto
Universidad de Valencia-CSIC, Institutos de Investigación de Paterna, Aptdo. 22085, 46071
Valencia, Spain
2Helmholtz-Institut für Strahlen- und Kernphysik (Theorie) and Bethe Center for Theoretical
Physics, Universität Bonn, Nußallee 14-16, D-53115 Bonn, Germany
3JCHP, IAS-4, IKP-3, Forschungszentrum Jülich, D-52425 Jülich, Germany
We develop a scheme for the extraction of the propertiesof the scalar mesons f0(600), f0(980),
and a0(980) from lattice QCD data. This scheme is based on a two-channel chiral unitary
approach with fully relativistic propagators in a finite volume. In order to discuss the feasibil-
ity of finding the mass and width of the scalar resonances, we analyze synthetic lattice data
with a fixed error assigned, and show that the framework can be indeed used for an accurate
determination of resonance pole positions in the multi-channel scattering.
1 Introduction
We present a method to calculate energy levels for ππ and KK in a finite volume using for
this purpose the chiral unitary approach that produces the f0(600), f0(980), and a0(980)
resonances as dynamically generated in the continuum. After this we face the inverse
problem: assuming that the levels obtained in the finite box are lattice data we determine
the phase shifts in the continuum and extract the resonance properties [1].
2The formalism
In the chiral unitary approach we obtain the T-matrix for the ππ and KK coupled channels
by means of the Bethe Salpeter equation
(1)
T = [1 − VG]−1V
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XIV International Conference on Hadron Spectroscopy (hadron2011), 13-17 June 2011, Munich, Germany
where V is the 2x2 matrix of the transition potential and G, a diagonal matrix is the loop
function of the two meson propagators
Gj
=
|? q|<qmax
?
d3? q
(2π)3
1
2ω1(? q)ω2(? q)
ω1(? q) + ω2(? q)
E2− (ω1(? q) + ω2(? q))2+ iǫ,
ω1,2(? q) =
?
m2
1,2+? q2.(2)
In a box of length L the energy levels are given by the poles of the T matrix of eq. (1)
substituting G by˜G given by
˜Gj
=
1
L3
|? q|<qmax
∑
? q
1
2ω1(? q)ω2(? q)
ω1(? q) + ω2(? q)
E2− (ω1(? q) + ω2(? q))2,
? q =
2π
L? n,
? n ∈ Z3.(3)
As we can see, all we have done is to replace the integral by a discrete sum over the free
eigenvalues of the box given by the periodic boundary conditions.
In one channel, the poles of T in the box are obtained when
(4)
V−1(E) −˜G(E) = 0.
Thenthe T matrix in the continuum for the energieseigenvalues of the box can be obtained
by means of
(5)
T(E) =
?
V−1(E) − G(E)
?−1=?˜G(E) − G(E)?−1.
By changing the value of L one can achieve that different energies appear as eigenvalues
of the box and then obtain the T matrix in the continuum for any desired energy, via eq.
(5). As shown in [1] this equation is nothing else than Lüscher’s formula [2]. However,
for smaller values of L, differences emerge which are due to the presence of the relativistic
propagators in the loops (see Ref. [1] for a detailed discussion on this issue).
The main purpose of [1] was to extend the idea of Lüscher to two channels, following the
lines of [3]. For this purpose we propose several methods in [1], but we will outline only
one here, which is practical and at reach by present lattice calculations.
Let us assume that we have the lattice data of fig. 1, which have been obtained from the
chiral unitary approach in the finite box, and we want to obtain the ππ and KK phase
shifts in the continuum from these data. For this purpose we take the levels 2 and 3 of Fig.
1 and a few energies from them, associating an error of 10 MeV to these energies. Then we
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XIV International Conference on Hadron Spectroscopy (hadron2011), 13-17 June 2011, Munich, Germany
2
2.5
L [Mπ
3
3.5
-1]
300
400
500
600
700
800
900
1000
1100
1200
E [MeV]
1
2
3
4
5
6
0
Figure 1: Energy levels as functions of the cubic box size L, derived from the chiral
unitary approach of Ref. [4] and using ˜G from Eq. (3).
make a fit to these data assuming that we have a potential, suggested by the chiral unitary
approach, of the type
(6)
Vij= aij+ bij(s − 4M2
K).
The levels in the box with two channels are obtained from the poles of the T matrix which
come from the condition that the determinant of (1− VG) is zero,
(7) det(1− V˜G) = 1− V11˜G1− V22˜G2+ (V11V22− V2
12)˜G1˜G2= 0.
As one can see in fig. 2, the method shows that one can reconstruct the phase shifts in the
continuum with an acceptable accuracy and from there the f0(980) resonance. The same
can be done for the a0(980) and the f0(600). As lessons that one draws from the work we
can say that the two channel method is necessary for the description of the f0(980) and
a0(980) resonances, the analysis with only the ππ channel leading to inaccurate results
and incorrect conclusions. Another lesson learned is that the flattening of a level in the
box as a funtion of L is not a guarantee that this energy corresponds to a resonance. The
flattening can occur around the threshold of a new channel without it corresponding to a
resonance in the continuum. The method developed in [1] should be very helpful in the
analysis of future lattice data for groups looking for hadron spectra from lattice QCD [5].
References
[1] M. Döring, U. -G. Meißner, E. Oset, A. Rusetsky, [arXiv:1107.3988 [hep-lat]].
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XIV International Conference on Hadron Spectroscopy (hadron2011), 13-17 June 2011, Munich, Germany
Figure 2: Left: Generated data points (10 MeV error, 13 points) with periodic b.c.; Fits
that fulfill the χ2
best+ 1 criterion are also shown (bands). Right: Extracted phase shifts
corresponding to the set shown in the figure to the left. Dashed line: The calculated
phase shifts by using the approach of Ref. [4]; Bands: Reconstruction of these phase shift.
[2] M. Lüscher, Commun. Math. Phys. 105 (1986) 153 (1986).
[3] V. Bernard, M. Lage, U.-G. Meißner and A. Rusetsky, JHEP 1101 (2011) 019.
[4] J. A. Oller, E. Oset, Nucl. Phys. A620, 438-456 (1997).
[5] Y. Nakahara, M. Asakawa, T. Hatsuda, Phys. Rev. D60 (1999) 091503; K. Sasaki,
S. Sasaki and T. Hatsuda, Phys. Lett. B 623 (2005) 208; N. Mathur, A. Alexandru,
Y. Chen et al., Phys. Rev. D76 (2007) 114505; S. Basak, R. G. Edwards, G. T. Fleming et
al., Phys.Rev.D76 (2007) 074504; J.Bulava, R.G. Edwards,E.Engelsonetal., Phys.Rev.
D82 (2010) 014507; C. Morningstar, A. Bell, J. Bulava et al., AIP Conf. Proc. 1257 (2010)
779; J. Foley, J. Bulava, K. J. Juge et al., AIP Conf. Proc. 1257 (2010) 789; M. G. Alford
and R. L. Jaffe, Nucl. Phys. B 578 (2000) 367; T. Kunihiro, S. Muroya, A. Nakamura,
C. Nonaka, M. Sekiguchi and H. Wada [SCALAR Collaboration], Phys. Rev. D 70
(2004) 034504; H. Suganuma, K. Tsumura, N. Ishii and F. Okiharu, PoS LAT2005 (2006)
070; Prog. Theor. Phys. Suppl. 168 (2007) 168; C. McNeile and C. Michael [UKQCD
Collaboration], Phys. Rev. D 74 (2006) 014508; A. Hart, C. McNeile, C. Michael and
J. Pickavance [UKQCD Collaboration], Phys. Rev. D 74 (2006) 114504; H. Wada, T. Ku-
nihiro, S. Muroya, A. Nakamura, C. Nonakaand M. Sekiguchi, Phys. Lett.B 652 (2007)
250; S. Prelovsek, C. Dawson, T. Izubuchi, K. Orginos and A. Soni, Phys. Rev. D 70
(2004) 094503; S. Prelovsek, T. Draper, C. B. Lang, M. Limmer, K. F. Liu, N. Mathur
and D. Mohler, arXiv:1002.0193 [hep-ph].
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