ЯДЕРНАЯ ФИЗИКА, 2004, том 67, № 4, с. 783-793
= ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
GROUND-STATE BARYONS IN NONPERTURBATIVE QUARK
DYNAMICS
© 2004 I. M. Narodetskii, M. A. Trusov*
Institute of Theoretical and Experimental Physics, Moscow, Russia Received January 17, 2003; in final form, September 17,2003
We have studied the three-quark systems in an Effective Hamiltonian (EH) approach, which is derived from QCD. The EH has the form of the nonrelativistic three-quark Hamiltonian with the perturbative Coulomb-like and nonperturbative string interactions and the specific mass term. After outlining the approach, methods of calculations of the baryon eigenenergies and some simple applications are explained in details. With only two parameters: the string tension a = 0.15 GeV2 and the strong coupling constant as = 0.39 we obtain a good description of the ground-state light and heavy baryons. The prediction of masses of the doubly heavy baryons not discovered yet are also given. In particular, a mass of 3660 MeV for the lightest Scc baryon is found by employing the hyperspherical formalism to the three-quark confining potential with the string junction.
1. INTRODUCTION
Doubly heavy baryons are baryons that contain two heavy quarks, either cc, bc, or bb. Their existence is a natural consequence of the quark model of hadrons, and it would be surprising if they did not exist. In particular, data from the BaBar and Belle Collaborations at the SLAC and KEK ^-factories would be good places to look for doubly charmed baryons. Recently the SELEX, the charm hadropro-duction experiment at Fermilab, reported a narrow state at 3519 ± 1 MeV decaying in A+K-n+, consistent with the weak decay of the doubly charmed baryon H+ [1]. The candidate is 6.3a signal.
The SELEX result was recently critically discussed in [2]. Whether or not the state that SELEX reports turns out to be the first observation of doubly charmed baryons, studying their properties is important for a full understanding of the strong interaction between quarks.
Estimations for the masses and spectra of the baryons containing two or more heavy quarks have been considered by many authors [3]. The purpose of this paper is to present a consistent treatment of the results of the calculation1) of the masses and wave functions of the doubly heavy baryons obtained in a simple approximation within the nonperturbative QCD.
The paper is organized as follows. In Section 2 we briefly review the Effective Hamiltonian (EH) method. In Section 3 we discuss the hyperspherical approach
E-mail:trusov@heron.itep.ru
!)The preview of this calculation has been done in [4].
which is a very effective numerical tool to solve this Hamiltonian. In Section 4 our predictions for the ground-state spectra of doubly heavy baryons are reported and a detailed comparison with the results of other approaches is given. Section 5 contains our conclusions.
2. THE EFFECTIVE HAMILTONIAN IN QCD
Starting from the QCD Lagrangian and assuming the minimal-area law for the asymptotics of the Wilson loop, the Hamiltonian of the 3q system in the rest frame has been derived. The methodology of the approach has been reviewed recently [5] and so will be sketched here only briefly. The Y-shaped baryon wave function has the form
By (xi,X2,X3,X) = (1)
= e.^ qa(xi ,X)q3 (X2,X)qY (x3,X),
where q(xiyX) is the extended operator of the ith quark at a point xi, a, 3, y are the color indices, and X = (0, X) is the equilibrium junction position (see below). This is the only gauge-invariant configuration possible for baryons. The starting point of the approach is the Feynman—Schwinger representation for the gauge-invariant Green function of the three quarks propagating in the nonperturbative QCD vacuum
3 00
G(x, y) = ^ fdsif Dzi exp(-Ki)(W)B, (2)
Fig. 1. Three-lobes Wilson loop.
where x = {xi,x2,x3}, y = {yi,V2,V3}, Zi = Zi(si) are the quark trajectories with zi(0) = xi, zi(T) = = yi, while si is the Fock—Schwinger proper time of the «th quark. Angular brackets mean averaging over background field. The quantities Ki are the kinetic energies of quarks, and all the dependence on the vacuum background field is contained in the generalized Wilson loop W:
with
VV — —^ijk^imnUfui171^71
Uk = P exp ( «g J A^(x)dx^
(3)
(4)
Ci
k = 1,2,3.
Here, P denotes the path-ordered product along the path Ci in Fig. 1, where the contours run over the classical trajectories of static quarks. In this figure three quark lines start at junction X at time zero, run in the time direction from 0 to T with the spatial position of quarks fixed and join again in the junction Y at time T. There are three planes that are bounded, respectively, by one quark line, two lines connecting junction and quark at t = 0 and t = T, and the connection line of two junctions. Under the minimal-area-law assumption, the Wilson loop configuration takes the form
{W)b « exp(-a(S1 + S2 + S3)),
In contrast to the standard approach of the constituent quark model the dynamical masses mi are no longer free parameters. They are expressed in
terms of the running masses mf\Q2) defined at the appropriate hadronic scale of Q2 from the condition of the minimum of the baryon mass as a function of mi.
Technically, this has been done using the einbein (auxiliary fields) approach, which is proven to be rather accurate in various calculations for relativistic systems. Einbeins are treated as c-number varia-tional parameters: the eigenvalues of the EH are minimized with respect to einbeins to obtain the physical spectrum. Such procedure, first suggested in [6, 7], provides the reasonable accuracy for the meson ground states [8].
This method was already applied to study baryon Regge trajectories [6] and very recently for computation of magnetic moments of light baryons [9]. The essential point adopted in [4] is that it is very reasonable that the same method should also hold for hadrons containing heavy quarks. As in [9], we take as the universal QCD parameter the string tension a. We also include the perturbative Coulomb interaction with the frozen strong coupling constant as.
From experimental point of view, a detailed discussion of the excited QQ'q states is probably premature. Therefore we consider the ground-state baryons without radial and orbital excitations, in which case tensor and spin—orbit forces do not contribute pertur-batively. Then only the spin—spin interaction survives in the perturbative approximation. In what follows we disregard the spin—spin interaction, then the EH has the following form:
H
£
i=1
m
(0)2
2mi
+
m
2
+ Ho + V.
(6)
Here, H0 is the nonrelativistic kinetic energy operator and V is the sum of the perturbative one-gluon-exchange potential Vq:
t/ 2^1
Vc = V—,
(5)
i<3
' v
where Si are the minimal areas inside the contours formed by quarks and the string-junction trajectories and a is the QCD string tension.
In Eq. (2) the role of the time parameter along the trajectory of each quark is played by the Fock— Schwinger proper time si. The proper and real times for each quark are related via a new quantity that eventually plays the role of the dynamical quark mass. The final result is the derivation of the EH, see Eq. (6) below.
where rij are the distances between quarks, and the string potential Vstring. The baryon mass is given by formula
Mb = min{H) + C,
(7)
where C is the quark self-energy correction calculated in [10]:
c = V —
n ¿-r1 mi
2
with n = 1 for q quark2), n = 0.88 for s quark, n = = 0.234 for c quark, and n = 0.052 for b quark.
The string potential calculated in [6] as the static energy of the three heavy quarks was shown to be consistent with that given by a minimum length configuration of the strings meeting in a Y-shaped configuration at a junction X:
Vstring(ri, r2, r3) = almin, (9)
where lm;n is the sum of the three distances |r^| between quarks and the string-junction point X. The Y-shaped configuration was suggested long ago [11], and since then was used repeatedly in many dynamical calculations [12].
3. SOLVING THE THREE-QUARK EQUATION
3.1. Jacobi Coordinates
The baryon wave function depends on the three-body Jacobi coordinates
(ri - Yj),
A
Pij
ßij,k ( miYi + mjYj
mi + mj
Yk
(10)
(11)
ßij,k
mi + mj (rrij + m,j)mk mi + mj + mk '
Ho = — ——
1 2/j, ( d2
2 n
dp2
&2 \
+ d\2)
(13)
__K2(Q)
OR2 + R OR + R2
where R is the six-dimensional hyperradius,
R
p2 + A2
(14)
Q furnishes five residuary angular coordinates, and K2(Q) is angular momentum operator whose eigen-functions (the hyperspherical harmonics) are
K 2{ti)Y[K ] = -K (K + 4)Y[k ]
(15)
with K being the grand orbital momentum. In terms of Y[K] the wave function ^(p, A) can be written in a symbolical shorthand as
^(p, A) ^k(R)Y[k](Q).
K
In the hyperradial approximation, which we shall use below, K = 0 and ^ = ^(R). Such wave function is completely symmetric under quark permutations obviously. Note that the centrifugal potential in the Schrodinger equation for the reduced radial function X(R) = R5/2^k(R) with a given K
(K + 2)2 — 1/4 R2
is not zero even for K = 0.
The Coulomb potential can be directly expressed in terms of Jacobi coordinates:
(i, j, k cyclic), where ¡iij and Hij>k are the appropriate reduced masses
^ = (12)
20s
3
E
i<j
ßij 1 V \pij\
(16)
and n is an arbitrary parameter with the dimension of mass which drops off in the final expressions. The coordinate pij is proportional to the separation of quarks i and j, and coordinate Aij is proportional to the separation of quarks i and j, and quark k. There are three equivalent ways of introducing the Jacobi coordinates, which are related to each other by linear transformations with the coefficients depending on quark masses, with Jacobian equal to unity. In what follows we omit indices i, j.
In terms of the Jacobi coordinates the kinetic energy operator H0 is written as
while for the string potential the situation is not so simple. We will c
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