Pis'ma v ZhETF, vol. 99, iss. 2, pp. 61 - 66
© 2014 January 25
Magnetic moments of negative parity baryons from effective hamiltonian approach to QCD
I. M. Narodetskii+*, M. A. Trusov+1)
+ State Research Center, Institute of Theoretical and Experimental Physics, 117218 Moscow, Russia * Institute of Theoretical and Experimental Physics, 117218 Moscow, Russia Submitted 9 December 2013
Magnetic moments of S11(1535) and S11 (1650) baryons are studied in the framework of the relativistic three-quark Hamiltonian derived in the Field Correlation Method. The baryon magnetic moments are expressed via the average current quark energies which are defined by the fundamental QCD parameters: the string tension a, the quark masses, and the strong coupling constant as. Resulting magnetic moments for the JP = 1/2" nucleons are compared both to model calculations and to those from lattice QCD.
DOI: 10.7868/S0370274X14020015
1. Introduction. The dipole magnetic moment encodes information about the leading-order response of a bound system to a soft external magnetic field. In particular, baryon magnetic moments are dynamical characteristics which provide valuable insights into baryon internal structure in terms of quark and gluon degrees of freedom. In this paper we shall explore the magnetic moments of negative parity resonances employing the QCD dynamics of a baryon in the form of the three-quark Effective Hamiltonian (EH). The EH is derived from the QCD path integral (see, e.g., [1]), and was already used in the studies of baryon spectra without external fields [2-5]. (The extension of the EH to the case of external magnetic field has been done recently in Ref. [6], where the nucleon spectrum as a function of magnetic field was calculated.) Within this method the magnetic moments of the 1/2+ octet baryons have been studied analytically in Ref. [7]. The model was shown to agree with experiment within 10% accuracy. The same accuracy was achieved for the baryon magnetic moments in Ref. [8], where the QCD string dynamics was investigated from another point of view.
Negative parity partners of the baryon octet arise from excitation of one unit of orbital angular momentum. Although the magnetic moments of the 1/2+ baryon octet are well-known both experimentally and theoretically, little is known about their 1/2" counterparts. Experimentally, magnetic moments of these states can be extracted through bremsstrahlung processes in photo- and electro-production of mesons at intermediate energies. For # (1535) a similar process
1)e-mail: trusov@itep.ru
YP ^ YIP can be used [9], but to date no such measurements have been made.
There exist limited number of theoretical studies of the magnetic moments of negative parity baryons based on constituent quark model [9], unitarized chi-ral perturbation theory (UxPT) [10], chiral constituent quark model (\CQM) [11], Bethe-Salpeter approach [12], and on the lattice [13] where magnetic moments of the baryon resonances have been obtained from the mass shifts. Comparison study of magnetic moments for positive- and negative-parity states offers insight into underlying quark-gluon dynamics. Given that the mass spectrum of the 1/2+ and 1/2" states has been reasonably well established from the EH, it is instructive to investigate the magnetic moments of these states. In this paper we extend the results of Ref. [7] to calculate the magnetic moments of the negative parity ^n(1535) and Sii(1650) resonances. The paper builds on the previous work presented in Ref. [4] where the EH contains the three quark string junction confined interaction and the Coulomb potential with the fixed strong coupling constant.
In Section 2 we briefly discuss the theoretical formalism of EH method for baryons, including the techniques required to extract the average quark energies wi which are cornerstones of the present calculation. As a result one obtains the resonance magnetic moments without introduction of any fitting parameters. Details of calculation of the magnetic moments for excited 1/2" nucleons are given in Section 4. In this Section, we also report the magnetic moments of the 1/2+ and 3/2+ octet baryons. Section 5 contains the summary of the obtained results.
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2. Effective Hamiltonian for Baryons. The key
ingredient of the EH method is the use of the auxiliary fields (AF) initially introduced in order to get rid of the square roots appearing in the relativistic Hamiltonian [14]. Using the AF formalism allows one to derive a simple local form of the EH for the three-quark system which comprises both confinement and relativistic effects, and contains only universal parameters: the string tension a, the strong coupling constant as, and the bare (current) quark masses mi. Neglecting the spin-dependent forces responsible for the fine and hyperfine splittings of baryon states the EH has the form
H
3
E
i=i
m„
2
+
2
+ Ho + V.
(1)
In Eq. (1) Hq is the nonrelativistic kinetic energy operator for the constant AF wi, the spin-independent potential V is the sum of the string potential
Vy(ri, Г2, Гз) = arn
(2)
with rm;n being the minimal string length corresponding to the ^-shaped configuration, and a Coulomb interaction term
Vc(ri, Г2, Гз)
i<j
(3)
ij
arising from the one-gluon exchange. In Eq. (3) CF = = 2/3 is the color factor. The constant as was treated either as a fixed parameter, as = 0.39 [4] or as the running coupling constant with the freezing value ~ 0.5 [5]. The results for wi coincide with the accuracy better than 1% (compare Tables 1 and 2 of Refs. [4] and [5], respectively). In what follows we use wi taken from Ref. [4].
3. The auxiliary field formalism. The EH depends explicitly on both bare quark masses mi and the constants AF wi that finally acquire the meaning of the dynamical quark masses. These quantities with a good accuracy coincide the average kinetic energies of the current quarks ^ \J~pi + m? ^ [4]. As the first step the eigenvalue problem is solved for each set of wi; then one has to minimize (H) with respect to wi. Although being formally simpler the EH is equivalent to the relativistic Hamiltonian up to elimination of AF.
The formalism allows for a very transparent interpretation of AF wi: starting from bare quark masses mi, we arrive at the dynamical masses wi that appear due to the interaction and can be treated as the dynamical masses of constituent quarks. These have obvious quark model analogs, but are derived directly using the AF
formalism. Due to confinement u>i ~ л/а ~ 400 MeV or higher, even for the massless current quarks. The baryon mass is given by
Mb = Mo + C + AM string,
Mo =
3
E
a=1
2wn
+
2
+ Eo(Wa),
(4)
(5)
where E0(wa) is an eigenvalue of the Schrodinger operator H0 + V, and the wa are defined by minimization condition
dMo(ma,Wa )
dc
0.
(6)
The right-hand side of Eq. (4) contains the perturbative quark self-energy correction C that is created by the color magnetic moment of a quark propagating through the vacuum background field [15]. This correction adds an overall negative constant to the hadron masses. Finally, AMstring in Eq. (4) is the correction to the string junction three-quark potential in a baryon due to the proper moment of inertia of the QCD string [16]. We stress that both corrections, C and AMstring are added perturbatively and do not influence the definition of wa.
The confinement Hamiltonian contains three parameters: the current quark masses mn and ms and the string tension a. Let us underline that they are not the fitting parameters. In our calculations we used a = 0.15 GeV2 found in the SU(3) QCD lattice simulations [17]. We employed the current light quark masses mu = md = 9 MeV and the bare strange quark mass ms = 175 MeV.
4. Magnetic moments of S,n(1535) and ^ii (1650) resonances. To calculate the nucleon magnetic moment one introduces a vector potential A and calculate the energy shift AMB due the Hamiltonian H = H(A) + Ha where H is defined by Eq. (1) with the substitution pa ^ pa — ea Aa and
Hn
-E-
2Ша
(7)
where B is an external magnetic field. The magnetic moment operator consists of contributions from both intrinsic spins of the constituent quarks that make up the bound state fiS and angular momentum of the three-quark system fiL with the center of mass motion removed. Straightforward calculation using the London gauge A = -|(B x r) yields
A = A s + A l .
(8)
2
m
Ш
a
a
e и
Письма в ЖЭТФ том 99 вып. 1-2 2014
Taking the constituent quarks to be Dirac point particles the spin contribution in Eq. (8) is determined by the effective quark masses wa
As
E
ea cra 2w„
The orbital contribution in Eq. (8) reads
a l = e
2w,
■ra X Pa-
(9)
(10)
In what follows instead the usual prescription which is to symmetrize the nucleon wave function between all three quarks we symmetrize only between equal-charge (up or down) quarks. In other words for the proton we use the uud basis in which the d quark is singled out as quark 3 but in which the quarks uu are still antisym-metrized. In the same way, for the neutron we use the basis in which the u quark is singled out as quark 3. The uud basis state diagonalizes the confinement problem with eigenfunctions that correspond to separate excitations of the quark 3 (p and A excitations, respectively). In particular, excitation of the A variable unlike excitation in p involves the excitation of the "odd" quark (d for uud or u for ddu). The physical P-wave states are not pure p or A excitations but linear combinations of all states with a given total momentum J. Most physical states are, however, close to pure p or A states [18]. In terms of the Jacobi variables
P
Eq. (10), reads
ri - r2
V2
A
ri + r2 - 2r3
V6 •
(11)
Al
where
^(mi + + ^ (MI + M2 + 4/X3)1A +
+
Ml - M2 2a/3
(p X PA + A X Pp),
lp = P x Pp, 1a = A X PA
(12)
(13)
and where the quark magnetic moments fj,u,fj,d are expressed in terms of parameters
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