Bled Workshops in Physics Vol. 15, No. 1 p. 22
A
Proceedings of the Mini-Workshop
Quark Masses and Hadron Spectra
Bled, Slovenia, July 6 - 13, 2014

Mesonic Effects in Baryon Ground and Resonant States*
R. Kleinhappela, L. Cantonb, W. Plessasa, and W. Schweigera
a Theoretical Physics, Institute of Physics, University of Graz, Universitatsplatz 5, A-8010 Graz, Austria
b Istituto Nazionale di Fisica Nucleare, Via F. Marzolo 8, I-35131 Padova, Italy
Abstract. We investigate mesonic effects in baryon ground and resonant states by including meson loops in a relativistic coupled-channels approach. From calculations, so far done on the hadronic level, we obtain results for the dressed mass of the nucleon ground state and for dressed masses and decay widths of resonances, notably of the A, due to coupling to the pion channel. At this stage an improvement is found over the single-channel theory, the experimental data for decay widths, however, are still underestimated.
A proper description of hadron resonances still represents a big challenge in quantum chromodynamics (QCD), irrespective of the approach followed. Particularly in the framework of constituent-quark models, hadronic resonances are usually treated as excited bound states rather than as resonant states with finite widths. Calculations of strong decays have thus shown short-comings generally producing too small decay widths [1-4]. To remedy this situation we are investigating a coupled-channels (CC) approach taking into account explicit meson, especially pionic, degrees of freedom.
The CC approach has been tested before within a simple scalar toy model, leaving out all spin and flavor dependences. It turned out that the coupling to a mesonic channel shifts the ground-state mass down and generates the resonant state with a finite width, whose magnitude is dependent essentially on the coupling strength to the meson channel [5,6]. Recently we have obtained results for the nNN and the nNA systems, including all spin and flavor degrees of freedom.
Our theory relies on a relativistically invariant mass operator written in matrix form. It contains a bare baryon state i, here the N or A, coupled to the nNN and the nNA channels i + 1, respectively. After eliminating the latter by the Fesh-bach method one ends up with the following eigenvalue problem for the dressed baryon ground or resonant state >:
Evidently, it contains an optical potential, which becomes complex above the nN threshold. Herein, Mi and M1+i are the invariant mass operators of the i-th and
Mt - K (m - Mi+1 + i0) 1 K >= m|% >.
(1)
* Talk presented by R. Kleinhappel
Mesonic Effects in Baryon Ground and Resonant States 23
(i+1)-st channels and K describes the transition dynamics. It should be noted that the mass eigenvalue m appears also in the optical-potential term. Beyond the resonance threshold it acquires an imaginary part leading to a finite decay widths.
The transition dynamics contained in K is deduced from the following La-grangian densities
LNNn = -^^Y^® ,	(2)
Lan« = -W+ h.c..	(3)
m„
where ¥ and represent the N and A fields, which are coupled in pseudovector form by the n field ® with strengths fNNn and fANn, respectively. This leads to transition matrix elements from the bare N and A states to the channels including the explicit pions (with mass mn) for the cases of nNN
< N|L„nin (0)|Nk^ >= £ ^^u^^N)Y^u(kN,INHKiV (4)
and nNA
< AILnNNa(0)|Nk4'>= £ ^Mu^)u(kN,!n. (5)
Here u (kN, IN) are the spin-2 Dirac spinors of the N and u^ (kA, 1a) the spin-| Rarita-Schwinger spinors of the A. In the rest frame of the baryon B the eigenvalue equation (1) finally turns into the following explicit form
(r ¿3k"	1
mB + J (*# 2<2^N2mfi FnNB (kn') < B|LnNB (0)|N, >
^m2N + kn2 - ^mn + kn2 + tt) F^NBikn)
x <N,n;kn|LnNN(0)|B >) < B|^b >= m < B|^b > (6)
with B standing for N or A and all quantities with a tilde referring to bare particles. The wave functions of the baryon states < B|^b > are represented by free momentum eigenstates denoted by < B| or equivalently by free velocity states < B; v = 0| (for pertinent details see Ref. [5]). The processes corresponding to the optical potential in Eq. (6) are depicted in Fig. 1.
m- m
Fig. 1. Pion-loop diagrams for the nNN and nNA systems according to Eq. (6).
24
R. Kleinhappel, L. Canton, W. Plessas, and W. Schweiger
In Eq. (6) we have inserted form factors g for the extended meson-baryon vertices. They are taken from three different models, namely, a relativistic constituent-quark model (RCQM) [7,8] as well as two phenomenological meson-nucleon models, namely, the one by Sato and Lee (SL) [9] and the one by Polinder and Rijken (PR) [10]. The corresponding parametrizations are all given in Ref. [11] according to the form
F(q2) = —, ,2 , ,4■	(7)

AW + I A2
The cut-off parameters occurring in Eq. (7) and the values of the coupling constants are summarized in Tab. 1. The functional dependences of the various vertex form factors are shown in Figs. 2 and 3.
RCQM SL PR
f2
0.0691 0.08 0.013 N At 0.451 0.453 0.940 A2 0.931 0.641 1.102
f2
0.188	0.334 0.167
A At 0.594	0.458 0.853
A2 0.998	0.648 1.014
Table 1. nNN and nNA coupling constants as well as cut-off parameters entering into Eq. (7) for the three different form-factor models used in the present work (cf. Ref. [11]).
By solving the eigenvalue equation (6) with the physical nucleon mass mN = 939 MeV as input for m we find the bare nucleon mass m^ and thus the influence of the pion loop. Tab. 2 contains the results for the pion dressing of the nucleon ground state. It is seen that all three different form-factor models lead to very similar magnitudes for the mass differences mN — mi of about 100 MeV.
_RCQM SL PR
mi	1.067 1.031 1.051
mN — mN -0.128 -0.092 -0.112
Table 2. Mesonic effects on the nucleon mass mN from coupling to the nNN channel.
In the case of the A resonance we are interested in the mesonic effects on both the mass as well as the decay width. In the first instance, we employ a bare intermediate nucleon N as is shown in the graph on the r.h.s. of Fig. 1. The corresponding results are given in Tab. 3. Again the pionic effects on the masses are quite similar for the three different form-factor models. The n-decay widths,
Mesonic Effects in Baryon Ground and Resonant States 
25
kn2 [GeV2]
Fig. 2. Momentum dependences of the three different form-factor models in case of the nNN system.
kn2 [GeV2]
Fig. 3. Momentum dependences of the three different form-factor models in case of the nNA system.
however, show bigger variations. Still, they are all too small as compared to the phenomenological value.
A more realistic description of the A —» Nn decay width is obtained by replacing the bare N with mass m^ in the intermediate state by the physical nucleon N with mass mN = 939 MeV, as depicted in Fig. 4.
The corresponding results are given in Tab. 4. It is immediately seen that the decay widths get much enhanced, while the effects on the masses are only slightly changed. We expect the larger phase space for the pionic decay to be responsible for the enhancement of the decay widths.
At this stage an open problem is left with regard to dressing the vertex form factors and the coupling strengths in our work. Corresponding studies have ear-
26
R. Kleinhappel, L. Canton, W. Plessas, and W. Schweiger
_RCQM SL PR
mN	1.067 1.031 1.051
mN	1.300 1.295 1.336
Re (m^) — mA -0.068 -0.062 -0.104 r = 2Im (mA ) 0.0026 0.017 0.0048
Table 3. Mesonic effects on the A mass Re (mA) and n-decay width r from coupling to the nNA channel, according to the loop diagram on the r.h.s. of Fig. 1. The bare nucleon masses m^ are the same as in Tab. 2.
n
Fig. 4. Pion-loop diagram for the nNA system with an intermediate physical nucleon with mass mN = 939 MeV.
_RCQM SL PR
mN	0.939 0.939 0.939 '
m	1.318 1.306 1.358
Re (mA) — mA -0.086 -0.073 -0.125 r = 2Im (mA ) 0.042 0.069 0.039
Table 4. Mesonic effects on the A mass Re (mA) and n-decay width F from coupling to the nNA channel, according to the loop diagram in Fig. 4.
lier been undertaken, e.g., by both Sato and Lee [9] as well as Polinder and Ri-jken [10]. We may expect a further improvement of our results by following a similar way, but it constitutes a difficult task to realize such a framework consistently in our approach.
In summary we are encouraged by the results obtained so far. We have identified the magnitudes of the pionic effects on the N ground state as well as the A resonance. In addition, we could demonstrate, how the pionic A decay width comes about by explicitly including the n-decay channel. Analogous investigations are presently under way for the N* resonances.
Acknowledgment
This work was supported by the Austrian Science Fund, FWF, through the Doctoral Program on Hadrons in Vacuum, Nuclei, and Stars (FWF DK W1203-N16).
References
1. T. Melde, W. Plessas, and R. F. Wagenbrunn, Phys. Rev. C 72, 015207 (2005); ibid. C 74, 069901 (2006).
Mesonic Effects in Baryon Ground and Resonant States 
27
2.	T. Melde, W. Plessas, and B. Sengl, Phys. Rev. C 76, 025204 (2007).
3.	B. Sengl, T. Melde, and W. Plessas, Phys. Rev. D 76, 054008 (2007).
4.	T. Melde, W. Plessas, and B. Sengl, Phys. Rev. D 77,114002 (2008).
5.	R. Kleinhappel, Diploma Thesis, University of Graz, 2010.
6.	R. Kleinhappel and W. Schweiger, in: Dressing Hadrons (Proceedings of the MiniWorkshop, Bled, Slovenia, 2010), ed. by B. Golli, M. Rosina, and S. Sirca, DMFA, Ljubljana (2010), p. 33; arXiv:1010.3919.
7.	L.Y. Glozman, W. Plessas, K. Varga, and R.F. Wagenbrunn, Phys. Rev. D 58, 094030 (1998).
8.	L.Y. Glozman, Z. Papp, W. Plessas, K. Varga, and R.F. Wagenbrunn, Phys. Rev. C 57, 3406 (1998) .
9.	T. Sato and T. S. H. Lee, Phys. Rev. C 54, 2660 (1996).
10.	H. Polinder and T. A. Rijken, Phys. Rev. C 72, 065210 (2005); ibid. C 72, 065211 (2005).
11.	T. Melde, L. Canton, and W. Plessas, Phys. Rev. Lett. 102,132002 (2009).
60 Povzetki v slovenščini
Kvarkovska snov v močnih magnetnih poljih
Debora Peres Menezes
Universidade Federal de Santa Catarina, Departamento de Fisica-CFM-CP 476, Campus Universitario-Trindade, CEP 88040-900, Florianopolis-SC, Brazil
V pričujočem prispevku skusamo razumeti razne lastnosti kvarkovske snovi, kot jih opisuje model Nambuja in Jona-Lasinia v prisotnosti močnih magnetnih polj. Najprejanaliziramo raznovrstne fazne diagrame. Potem raziskujemo razlike, ki nastanejo zaradi različnih vektorskih interakcijv Lagrangeovi gostoti in uporabimo izsledke za opis zvezdne snovi. Nato se ozremo na značilnosti dekonfinačije in vzpostavitve kiralne simetrije pri kemičnem potenčialu nič v okviru prepletene Polyakovove verzije modela Nambuja in Jona-Lasinia. Končno proučimo lego kritične točke za različne izbire kemičnega potenčiala in gostote.
Schwinger-Dysonov pristop h kvantni kromodinamiki razloži nastanek oblečenih mas kvarkov
D. Klabučara in D. Kekezb
a Physičs Department, Fačulty of Sčienče, Zagreb University, Bijenička č. 32,10000 Zagreb, Croatia
b Rugjer Boskovič Institute, Bijenička č. 54,10000 Zagreb, Croatia
Poleg drugih uspehov Sčhwinger-Dysonov pristop k neperturbativni kvantni kromodinamiki razlozi tudi to, zakajso v efektivnih kvarkovih modelih oblečene mase kvarkov zelo različne od golih mas. Ce pa interakčijsko jedro vsebuje tudi perturbativni delez kromodinamske interakčije, poda Sčhwinger-Dysonov pristop tudi znano visokoenergijsko obnasanje kvarkovih mas, tako kot jih napoveduje perturbativna kvantna kromodinamika.
Mezonski učinki pri osnovnih in resonančnih stanjih barionov
R. Kleinhappela, L. Cantonb, W. Plessasa in W. Sčhweigera
a Theoretičal Physičs, Institute of Physičs, University of Graz, Universitatsplatz 5, A-8010 Graz, Austria
b Istituto Nazionale di Fisiča Nučleare, Via F. Marzolo 8, I-35131 Padova, Italy
Za raziskavo mezonskih učinkov pri osnovnih in resonančnih stanjih barionov smo vključili mezonske zanke v relativistični pristop s sklopljenimi kanali. Iz računov, ki so bili doslejnapravljeni na hadronskem nivoju, smo dobili rezultate za oblečene mase osnovnega stanja in resonanč nukleona. S sklopitvijo na pionski kanal smo dobili tudi sirine resonanč, zlasti resonanče A. Za zdajsmo sičer izboljsali rezultate v primerjavi z računi z enim samim kanalom, vendar so razpadne sirine se vedno premajhne v primerjavi z meritvami.