Bled Workshops in Physics Vol. 17, No. 1
p. 33
A
Proceedings of the Mini-Workshop Quarks, Hadrons, Matter
Bled, Slovenia, July 3 - 10, 2016

Soliton model analysis of SU(3) symmetry breaking for baryons with a heavy quark*
H. Weigel, J. P. Blanckenberg
Physics Department, Stellenbosch University, Matieland 7602, South Africa
Abstract. In these proceedings we review the construction of the collective coordinate Hamiltonian that describes the spectrum of baryons with a single heavy quark and up, down or strange degrees of freedom in the context of chiral soliton models.
1	Introduction
This presentation is based on Ref. [1] that describes the numerical results of this soliton model analysis in detail. The derivation of the Hamilton for the (light) flavor degrees of freedom (up,down, strange) and, in particular, the origin of the constraint that projects onto certain flavor SU(3) representations are discussed only by the way in Ref. [1]. We therefore provide more details of the derivation here.
2	Collective rotations in flavor symmetric SU(3)
The approach builds up from a chiral soliton generated from light flavors and heavy meson fields that are bound to the soliton. Both acquire strangeness components by collectively rotating in flavor SU(3). Without symmetry breaking this corresponds to approximating time dependent configurations by large zero-mode fluctuations.
2.1 Chiral soliton
The major building block for the chiral soliton is the non-linear representation of the pseudoscalar mesons in form of the chiral field U but also vector mesons p and œ may be included. In a first step we construct the stable static soliton (with winding number one). Subsequently we approximate time dependent solutions and introduce collective coordinates for the flavor orientation A(t) G SU(3). Generically we write this as
U(r,t) = A(t)Uc(r)At(t),
(1)
* Talk delivered by H. Weigel
34 H. Weigel, J. P. Blanckenberg
where U0(r) represents the classical (static) soliton. The time dependence is most conveniently parameterized via eight angular velocities Ha
i ^	—A(t)
-^HaAa = At(t) -A-^ .	(2)
a=1
The resulting collective coordinate Lagrange function has the structure
Li(Qa) =-Ecl + 2a2 L n? + l^2 H - :N|n8 . (3)
i=1	a=4	*
The term linear in the time derivative originates from the Wess-Zumino-Witten action [3] and therefore carries an explicit factor Nc (number of colors). The coefficients a2 and p2 are radial integrals of the profile functions and represent moments of inertia for rotations in isospace and the strangeness subspace of flavor SU(3), respectively. The form of Eq. (3) is generic. The particular numerical values for the classical energy and the moments of inertia are, of course, subject to the particular model. They are reviewed in Ref. [2].
2.2 Heavy meson bound states
In the heavy flavor limit the pseudoscalar and vector meson components become degenerate [4]. In contrast to the light sector it is hence inevitable to include both components. Since the soliton configuration itself has non-zero orbital angular momentum the most strongest coupling to the solution dwells in the P-wave channel [5] (P and Q^ are SU(3) flavor spinors):
eio>t
P = ^n^Wr ■ Tx,
eio>t
Qo = ^4=Vo(r)x,	(4)
eio>t
Qi = [i¥i (r)ti + 2V2(r)eijktjxk] x,
where x = x(^) is a three component spinor that is constant in space but should be viewed as the Fourier amplitude of the heavy meson wave-function. Since the coupling to the light mesons occurs via a soliton in the isospin subspace, only the first two components of x are non-zero. The parameterization that emerges by left multiplication with t ■ T has different profile functions and leads to the S-wave bound states.
The field equations for heavy mesons turn into coupled linear differential equations for the profile functions in eq. (4) with the soliton generating a binding potential. This (so-called bound state) approach assumes the soliton as infinitely heavy and corresponds to (formally) assuming the large N C limit. Normalizable solutions to these differential equations only exist for certain frequencies w below the heavy meson mass.
Soliton model analysis of SU(3) symmetry breaking
35
To quantize the Fourier amplitudes, x as harmonic oscillators it is necessary to properly normalize the bound state profiles. The normalization condition is that occupying the bound produces one unit of heavy charge (charm or bottom). This heavy charge arises from the Noether current associated with an infinitesimal phase transformation of the heavy field. We write the Lagrange function for the heavy meson as
LhM =
d3r
M cp + wÇÂ cp + 9tft
2
Ç
xfMxM,
(5)
where cpt = (®,Y0,Yi ,Y2) contains the bound state profiles while the soliton determines the matrices M, A and A are matrices that also contain differential operators. Since the phase transformation in Eq. (4) can be modeled as w —» w + 6w, the normalization condition reads
d3r [w^Acp + çtÂÂcp + çtAcp]
= 1 .
(6)
We require absolute values because bound states with w < 0 have opposite heavy charge and eventually describe heavy pentaquarks with a heavy anti-quark. The heavy meson fields are spinors in SU (3) flavor space and thus subject to the collective flavor rotation from Eq. (1),
A(t)P
and
Qk
A(t)Qk
(7)
where the right hand sides contain the bound state profile functions. It is then very instructive to compute the time derivative
P =
A(t) [-= iA(t)
because it shows that enforces
9L
iw + Af(t)A(t)] ^=n°(r) 1 1 7
+ -^Os + -Y QaAa
2V3 2
9Lh(W) 9Os
1
f • *x'

O(r)
f • Tx' 0
1 9Lh(W) 2V3 9w
. Then the normalization condition
9Os
(Nc - sign(w)xfx) = zm (Nc - N) ,
(8)
2^3 c &	7 2^
where we have also identified the charge of the heavy quark. Finally, the collective rotation of the bound state yields the hyperfine coupling [6]
Lhf = pxf (O • T
x,
(9)
where p is an integral involving all profile functions, including those of the classical soliton. The bound state also contributes to the moments of inertia, a2 and p2, but numerically that contribution is negligible since the bound state is localized at the center of the soliton.
P
0
v^t
36 H. Weigel, J. P. Blanckenberg
3 Symmetry breaking and mass formula
Though it is appropriate to work with mu = md, the deviation ms > mu is substantial and requires to add terms like
Lsb--^— Tr
n o o\
0 1 0 1 (U + U - 2)
lOOx/
+... where x — 2ms— > 1. (10) mu + ma
to the effective chiral Lagrangian that describe different masses and decay constants of strange and non-strange mesons1. These symmetry breaking terms yield an explicit A dependence of the collective coordinate Lagrange function
Lsb = -Xy [1 - D88(A)] with Dab = ?Tr [AaAAbAt] .	(11)
Again, y is a radial integral2 over all profile functions. Collecting Eqs. (3,9) and (11) and Legendre transforming to the right SU(3) generators Ra = yields the Hamilton operator whose eigenvalues are the baryon masses that are expressed in the mass formula
E - E + ( 1 1 ^ T(T + 1] + £(x) 1 (N n)2
E = Ecl H02 - P^J^^ + 2p2 - 24p2 (Nc - N)
+ MN + 202 [j(j +1)-r(r + 1)]N.	(12)
The moments of inertia are the same for the light and heavy degrees of freedom as they result from a single local Lagrangian. In Eq. (12) e(x) is the eigenvalue of Osb — Y.a=i Ra + xp2Y [1 - D88(A)] according subject to the constraint R8 — (Nc - N)/2\/3. For odd Nc and N — 1 this constraint requires diquark SU(3) representations. The total spin is j and r(r + 1) is the eigenvalue of Y.i=1 R?. It is zero and one for the anti-symmetric and the symmetric diquark wave-functions, respectively.
Obtaining the eigenvalues e(x) of the operator Osb amounts to a non-pertur-bative treatment of light flavor symmetry breaking. Yet, the approach can be illuminated in the language of perturbation theory as it corresponds to linearly combining states that belong to different SU(3) representations, but otherwise have identical quantum numbers. Possible representations are subject to the constraint on R8: For the physical value Nc — 3 representations with the lowest eigenvalue of the quadratic Casimir operator, Y_ a=1 Ra are the anti-triplet and the sextet with r — 0 and r — 1, respectively. That is, these are the quark model representations. With symmetry breaking added an anti-fifteen-plet and a 24 dimensional representation follow suit [7]. Increasing to the next odd value, Nc — 5, an anti-sextet, a mixed- and a fully symmetric fifteen-plet are allowed by the constraint. The latter has r — 2 and does not have a counterpart for Nc — 3. Hence the Nc counting effects (heavy) baryon masses via modified eigenvalues of Osb.
1	Symmetry breaking for the heavy mesons, proportional to e.g. Mgs - Mg, is also included.
2	The notation is chosen to distinguish it from y — xy in the literature [2].
Soliton model analysis of SU(3) symmetry breaking
37
4 Summary
In these short proceedings we have explained the origin of the collective coordinate Hamiltonian from treating baryons with heavy quark as a heavy meson bound to a chiral soliton. The resulting spectrum and its comparison with empirical data has been discussed at length elsewhere [1]. We stress that light baryons are simultaneously described in this approach by setting N = 0 in Eq. (12) and in the constraint on R8. In particular, we have unique moments of inertia and symmetry breaking coefficients regardless of the value for N. This is in contrast to the approach of Ref. [8] that employs different Lagrangians in the light and heavy sectors.
Acknowledgments
One of us (HW) is very grateful to the organizers for providing this worthwhile workshop. This work supported in parts by the NRF under grant 77454.
References
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2.	H. Weigel, Lect.Notes Phys. 743,1 (2008).
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4.	M. Neubert, Phys.Rept. 245, 259 (1994).
5.	J. Schechter, A. Subbaraman, S. Vaidya, H. Weigel, Nucl.Phys. A590, 655 (1995).
6.	M. Harada, A. Qamar, F. Sannino, J. Schechter, H. Weigel, Nucl.Phys. A625, 789 (1997).
7.	A. Momen, J. Schechter, A. Subbaraman, Phys.Rev. D49, 5970 (1994).
8.	G. S. Yang, H. C. Kim, M. V. Polyakov, M. Praszalowicz, arXiv:1607.07089 [hep-ph].