Bled Workshops in Physics Vol. 15, No. 1 p. 43
A
Proceedings of the Mini-Workshop
Quark Masses and Hadron Spectra
Bled, Slovenia, July 6 - 13, 2014

The constituent quark as a soliton in chiral quark models
Bojan Golli
Faculty of Education, University of Ljubljana, 1000 Ljubljana, Slovenia and JoZef Stefan Institute, 1000 Ljubljana, Slovenia
Abstract. We discuss the possibility that the soliton carrying the baryon number 1/3, obtained in the linear a-model and in the Nambu - Jona-Lasinio model can be identified with the constituent quark. In the linear a-model we have derived meson exchange potentials between two solitons which turn out to resemble potentials used in constituent quark models.
The mechanism in which a nonstrange constituent quark acquires its mass ~ 350 — 400 MeV is phenomenologically described via spontaneous breaking of chiral symmetry. Yet, such a structureless particle does not agree with a picture of the constituent quark as an extended object in which the quark is surrounded by a cloud of quark-antiquark (meson) and gluon excitations. The fact that the scale for chiral symmetry breaking appears at lower energies than the confinement scale supports a model in which the constituent quark is represented by a current quark surrounded by a chiral field rather than a gluon field, as first suggested by Georgi and Manohar [1], and further elaborated by Cheng and Li [2], and by Baumgartner, Pirner, Konigsmann and Povh [3] (see also the contribution of M. Rosina in these Proceedings [4]).
One of the simplest models describing the spontaneous breaking of chiral symmetry is the linear a-model (LSM). In the non-strange sector it involves u and d quarks, a triplet of pions and the a-meson [5-8]. The model possesses, for sufficiently strong pion-quark coupling constant g, soliton solutions obtained by putting three quarks in the lowest 1s orbit and allowing for nonzero pion field around the quark source. Below the critical coupling constant only free Dirac particles of mass M = gfn exist, fn being the pion decay constant. We found [9] another type of non-trivial solutions by putting only one quark in the lowest orbit which we identified with the constituent quark.
In figure 1a) the energy of such a quark soliton is displayed as a function of M = gfn. For comparison, the energy of the three quark soliton (the nucleon soliton) divided by 3 is also shown. The three quarks in the nucleon soliton generate a stronger chiral field than in the single quark soliton and the resulting attractive potential lowers more substantially the energy of the valence orbit (evai) producing a large gap between the two solutions. The energy of the quark soliton is higher than the popular value of the constituent quark mass. In our calculation
44 Bojan Golli
600		............................... ii^n. -I,-.:. Esoi(1q)
500		
400	-E = M	
300		\ £val(1q)
200		
100		\ £val(3q) \ \ 'vs,. - XN v V ...........*-................... . t tV
300 400 500 600 700
M
a)
300 400 500 600 700
M
b)
Fig.1. (a) (b)
we do project the solution onto the subspace of good angular momentum and isospin, but we do not perform the projection onto the states with good linear momentum. The solution is thus interpreted as a wave packet of states with good linear momenta; projecting out the zero-momentum state would further lower the energy of the soliton. Let us notice that for both solutions the valence orbit sinks into the Dirac sea at a sufficiently large coupling constant producing a topological soliton (Skyrmion).
From our solution it is possible to derive a potential between two quark solitons in the framework of the Born Oppenheimer approximation. The interesting part of the interaction is the pion exchange potential. The pion field around the quark soliton with good angular momentum and isospin can be written in the form:
nb(r) = 3«(r)t ■ L Tb ,	(1)
where L and T act on spin and isospin of the quark soliton, respectively. To obtain the potential between two such solitons, one at the origin and the other one at position r, we evaluate the quark-meson and the meson-meson interaction for such a configuration. For the quark-pion interaction we obtain
vq(r) = (QkcTclQ)
d3r'u(r>(r') n(|r - r'|) r' ■ La (r - r') ■ Lb TaTb , (2)
where cr0 and To now act on current quark spin and isospin. A similar expression is obtained from the meson self-interaction ("Mexican hat"). The potential (2) contains the scalar as well the tensor part. The scalar part is displayed in figure 2 and compared to a typical one-pion exchange potentials used in the constituent quark model calculations. The potential satisfies J" dr r2Vn(r) = 0, a constraint that has to be fulfilled for any pseudo-scalar exchange potentials. It has the correct asymptotic behavior leading to the appropriate form of the pion-exchange potential between two nucleons. The attractive part is too shallow and has a too large range which could be attributed to the spurious center-of-mass motion. We expect that
The constituent quark as a soliton in chiral quark models
45
Fig. 2. The pion exchange effective potential (multiplied by r2) between two quark solitons for M = 560 MeV (solid line) compared to a typical OPEP.
linear momentum projection would reduce its range and through the above integral constraint lower the depth of the attractive part, which could finally bring our prediction closer to a realistic OPEP.
Our model of the constituent quark is further supported by our finding that similar quark solitons exist also in a more fundamental chiral model, the Nambu - Jona-Lasinio (NJL) model. In this model the sigma and pion fields are related to the quark-antiquark excitations of the Dirac sea by
a(r) = Y- qj qj R, n(r) = Y_ qR
£j<0 £j<0
where R denotes a regulator which is needed to regularize the ultraviolet divergences, and introduces a new parameter, the cut-off. In our calculation [10,11] we used a version of the model in which the interaction between quarks is induced by the instantons [12] and has a finite range. The mass of the "bare" constituent quark M which is equal to gfn in the linear a-model is now substituted by the 4-momentum-dependent mass M —» MR(k2), k2 = k2 — E2; M remains in the model as a (free) parameter measuring the strength of the a-field in the
vacuum. The pole of the quark propagator is obtained by solving the condition k2
+ M2R(k2) = 0. The solution exists only below a certain value of M. Using R(k2) = e-k /A and fixing the cut-off parameter A to reproduce the pion decay constant, the critical value of M is around 300 MeV. Above this value only solutions with non-trivial values of chiral fields exist. Similarly as in the linear a-model, putting three quarks in the valence orbit a soliton corresponding to the nucleon emerges; if we put only one quark in the valence orbit, we obtain a solitonic solution which we identify with the constituent quark.
The energies of both solutions as functions of M are displayed in figure 1b) in the same way as the analogous solutions in the linear a-model. The energy of the " bare" constituent quark is denoted by Mq; in contrast to the LSM this
46 Bojan Golli
solution smoothly continues into the soliton solution. Interestingly, the solutions of both models have similar energies, however, the energies in the NJL model raise with M while those in the LSM lower. This is a consequence of the regular-ization of the valence orbit which is not performed in the LSM as well in other versions of the NJL model. The regularization used in our approach prevents the orbit to shrink below a certain size and thus makes the soliton absolutely stable without any further ad hoc constraint. The energy of the valence orbit remains almost constant with M and does not sink into the Dirac sea. The presence of the time variable in the regulator does not allow us to perform the exact angular and linear momentum projection which would lower the soliton energy further, and eventually bring it in the ball park of values used in the constituent quark model calculations.
References
1.	A. Manohar and H. Georgi, Nucl. Phys. B234 (1984) 189.
2.	T. P. Cheng and Ling-Fong Li, Phys. Rev. Lett. 74 (1995) 2872.
3.	S. Baumgartner, H. J. Pirner, K. Konigsmann and B. Povh, Z. Phys. A 353 (1996) 397.
4.	M. Rosina, these Proceedings.
5.	M. Gell-Mann and M. Levy, Nuovo Cim. 16 (1960) 705.
6.	M. C. Birse and M. K. Banerjee, Phys. Lett. B 136 (1984) 284; Phys. Rev. D 31 (1985) 118.
7.	S. Kahana, G. Ripka and V. Soni, Nucl. Phys. A 415 (1984) 351.
8.	B. Golli and M. Rosina, Phys. Lett. B 165 (1985) 347.
9.	B. Golli and M. Rosina, Phys. Lett. B 393 (1997) 161.
10.	B. Golli, W. Broniowski, and G. Ripka: Phys. Lett. B437, 24 (1998)
11.	W. Broniowski, B. Golli, and G. Ripka: Nucl. Phys. A703, 667 (2002)
12.	D. I. Diakonov and V. Y. Petrov, Nucl. Phys. B 272 (1986) 457
62 Povzetki v slovenščini
Izbrani spektroskopski rezultati kolaboracije Belle
Marko Bračkoa'b
a University of Maribor, Smetanova ulica 17, SI-2000 Maribor, Slovenia b JoZef Stefan Institute, Jamova cesta 39, SI-1000 Ljubljana, Slovenia
V	prispevku smo poročali o izbranih rezultatih iz spektroskopskih eksperimentov, pred kratkim izvedenih s spektrometrom Belle, ki deluje na energijsko asimetričnem trkalniku elektronov in pozitronov KEKB v laboratoriju KEK, Tsukuba, Japonska.
Konstituentni kvark kot soliton v kiralnih kvarkovskih modelih
Bojan Golli
Pedagoska fakulteta, Univerza v Ljubljani, 1000 Ljubljana in Institut Jozzef Stefan, 1000 Ljubljana, Slovenija
Obravnavamo mozznost, da lahko soliton z barionskim stevilom 1/3, dobljenim v linearnem modelu sigma in v modelu Nambuja in Jona-Lasinija, identificiramo s konstituent-nim kvarkom. V linearnem modelu sigma smo izpeljali potenčial med dvema solitonoma, ki je podoben potencialom, ki se uporabljajo v modelih s konstituentnimi kvarki.
Mezoni Ds s pozitivno parnostjo in Z+ v kromodinamiki na mreZi
Luka Leskovec
Institut Jozef Stefan, Jamova 39,1000 Ljubljana, Slovenija
Predstavljena sta dva se posebej zanimiva kanala: Ds mezoni s pozitivno parnostjo ter ek-soticni hadron Z+. V kanalu z Ds je bilo nekaj napetosti med eksperimentom ter teorijo, saj je eksperiment nasel stanji D*0(2317) in Dsi (2460) pod pragom za sipanje mezonov DK in D*K, medtem ko je teorija napovedala mase teh mezonov nad tem istim pragom.
V	kromodinamiki na mrezi smo simulirali doticni kanal tako, da smo uporabili operatorje cs ter tudi D(*'K. Upostevajoc pojave na pragu sipanja smo izlocili mase mezonov Ds s pozitivno parnostjo, ki se nahajajo pod pragom za sipanje in se v okviru napak ujemajo z eksperimentalnimi. Simulirali pa smo tudi eksoticni kanal v katerem se nahaja Z+. Uporabili smo vse relevantne dvomezonske sipalne operatorje J/^rt,r|cp, DD*, ^(2S)rt, D*D*, ^(3770)n, —n, kot tudi dodatne operatorje tipa dikvark anti-dikvark. Identificirali smo vse diskretne energijske nivoje, a nismo nasli prepoznavnega kandidata za Z+.