Bled Workshops in Physics Vol. 16, No. 1 p. 91
A
Proceedings of the Mini-Workshop
Exploring Hadron Resonances
Bled, Slovenia, July 5 - 11, 2015

Resonances in the Nambu-Jona-Lasinio model
Mitja Rosinaa'b
aFaculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, P.O. Box
2964,1001 Ljubljana, Slovenia
bJ. Stefan Institute, 1000 Ljubljana, Slovenia
Abstract. We have designed a soluble model similar to the Nambu-Jona-Lasino model, regularized in a box with periodic boundary conditions, in order to explore the properties of resonances when only discrete eigenvalues are available. The study might give a lesson to similar problems in Lattice QCD.
1 The quasispin NJL-like model
It is very instructive to understand the key features of a simplified model containing the spontaneous chiral symmetry breaking. Some time ago we have constructed a soluble version of the Nambu-Jona-Lasino model [1,2]. Now we explore what it tells about the sigma meson. We make the following simplifications:
1.	We assume a sharp 3-momentum cutoff 0 < Ipi < A;
2.	The space is restricted to a box of volume V with periodic boundary conditions. This gives a finite number of discrete momentum states, N = NhNcNfVA3/6n2 occupied by N quarks. (Nh,N c and Nf are the number of quark helicities, colours and flavours.)
3.	We take an average value of kinetic energy for all momentum states: Ipi —»
4.	While in the NJL model the interaction conserves the sum of momenta of both quarks we assume that each quark conserves its momentum and only switches from the Dirac level to Fermi level.
5.	Temporarily, we restrict to one flavour of quarks, Nf = 1.
Let us repeat the "Quasispin Hamiltonian" [1,2].
P = 4 A.
k=1 v 7
92 Mitja Rosina
Here y5 and |3 are Dirac matrices, mo is the bare quark mass and g = 4G/V where G is the interaction strength in the original (continuum) NJL.
We introduce the quasispin operators which obey the spin commutation relations
1 „ • 1 1
jx = 2 I , jy = 2 ilY5 , jz = 2 Y5 ,
n 1 + h(k) n 1 h(k)	n
Ra = Y_ -2-ja(k) , La = Y_ -2-j«^) ' J« = Ra + La = H j«(k) .
k=1 k=1 k=1
The model Hamiltonian can then be written as
H = 2P(Rz - Lz)+ 2moJx - 2g(jX + jy) . The three model parameters
A = 648 MeV, G = 40.6 MeV fm3, m0 = 4.58 MeV have been fitted (in a Hartree-Fock + RPA approximation) to the observables
M = \l ( Eg(N) — Eg(N - 1)) - P2 = 335MeV
Q = < g|= V( gl X 9 ) = 1 ( 9l Jxl9 > = 2503 MeV3 i
mn = Ei(N) - Eg (N) = 138MeV.
The values of our model parameters are very close to those of the full Nambu-Jona Lasinio model used by the Coimbra group [3] and by Buballa [4].
2 The spectrum of 0 and 0+ excitations -Emergence of the a meson
It is easy to evaluate the matrix elements of the quasispin Hamiltonian using the angular momentum algebra. If N is not too large the corresponding sparse matrix can be diagonalized using Mathematica.
Excited levels of the ground state band (R=L=N/4) in Fig. 1 are almost equidistant and are suggestive of n-pion states (in s-state). The level spacings AE are slightly decreasing with the assumed number of pions nn due to the attractive interaction between pions. Inbetween appear also "intruder states" which can be interpreted as sigma excitations. The interpretation as ct meson is further supported by the large value of the matrix element of Jx between the ground state and the "intruder state". (Odd "multipion states" have zero value and even ones have a rather small value.)
Resonances in the Nambu-Jona-Lasinio model
93
np	parity	E [MeV]	DE [MeV]	Intruder
8	+	866	63	
-		816		s(667)+p(136)+13 MeV
7 -		803	93	
6	+	710	99	
+		667		s(667)
5 -		611	108	
4	+	503	115	
3 -		388	123	
2	+	265	129	
1 -		136	136	
0	+	0	0	
Fig. 1. Levels of the ground state band (R=L=N/4), level spacing between opposite parity states, and the assumed number of pions nn pions
3 The width of the a meson
In the attempt to describe resonances when only discrete eigenvalues are available we get a discrete sigma resonance energy, but not its width. We are trying to get the complex pole. For that purpose, we explore the method of analytic continuation from the bound state [5]. For this purpose, we vary one of the model parameters, the bare quark mass m from the region where the ct meson would be bound (Ect < E2n) down to the physical value of m —» m0 (where ECT > E2n).
At m > 64 MeV there are two positive parity states between the first and second negative parity states (the one-pion and three-pion excitations); the lower one is the intruder (ct meson) and the upper one is the correlated two-pion state. At m = 64 MeV both positive parity states coincide - the threshold for ct —» 2n. When we decrease m further, the energy of the ct meson decreases slower than the 2n energy and it appears at higher multipion states. For the physical value m = m0 = 4.58 MeV ct is already the sixth excited state, next to the six-pion state. It is obviously in the continuum, prompt to decay into 2n, in a more complete choice of interaction.
The method consists of the following steps:
•	Determine the threshold value mth and calculate e = ECT — E2n as a function of m for m > mth.
•	Introduce a variable x = a/m — mth; calculate k(x) = i^f—e in the bound state region (Fig. 2).
•	Fit k(x) by a polynomial k(x) = i(c0 + cix + c2x2 + ... + c2Mx2M) .
94 Mitja Rosina
•	Construct a Pade approximant:
k( . _ . ao + aix + ... + aMxM . 1 + bix + ... + bMXM .
•	Analytically continue k(x) to the region m < mth (i.e. to imaginary x) where k(x) becomes complex.
•	Determine the position and the width of the resonance as analytic continuation in m (Fig. 3 and Fig. 4):
ETes _ Re (contm_>mok2), r _ -2Im (contm_>mok2) .
k [MeV1/2] 20
15
10 -
5 -
10
12
x [MeV1/2]
Fig. 2. The fit of k(x) with quadratic(lower middle) and quartic polynomial (upper middle) and with Pade approximants of order 1 (below) and 2 (above)
2
4
6
8
We notice that the results for ETes and r in Fig. 3 and 4 deviate strongly for first and second order Pade approximants. This is due to the large stretch for the analytic continuation so that convergence at higher orders cannot be expected. Nevertheless, it is rewarding that the physical values for ETes and r lie somewhere in the middle between both curves.
To conclude, the method of analytic continuation in this case is just a game, but it is instructive. Intentionally, we have plotted the energy and width of the d meson as a function of the corresponding pion mass rather than as a function of the model parameter m. This is reminiscent of the extrapolation of pion mass from about 500 Mev towards its physical value the way tha lattice people have to struggle.
References
1. B. T. Oblak and M. Rosina, Bled Workshops in Physics 7, No. 1 (2006) 92; 8, No. 1 (2007) 66; 9, No. 1 (2008) 98 ; also available at http://www-f1.ijs.si/BledPub.
[MeV] Ere
Resonances in the Nambu-Jona-Lasinio model
95
900
800
700
600
200
300
400
500
mn [MeV]
Fig. 3. The resonance energy ETes of the a meson as a function of the pion mass - extrapolation using Pade approximants of order 1 (below) and 2 (above)
T [MeV]
Fig. 4. The width F of the a meson as a function of the pion mass - extrapolation using Pade approximants of order 1 (below) and 2 (above)
2.	M.Rosina and B.T.Oblak, Few-Body Syst. 47 (2010) 117-123.
3.	M. Fiolhais, J. da Providencia, M. Rosina and C. A. de Sousa, Phys. Rev. C 56 (1997) 3311.
4.	M. Buballa, M.: Phys. Reports 407 (2005) 205.
5.	V.M. Krasnopolsky and V.I.Kukulin, Phys. Lett, 69A (1978) 251, V.M. Krasnopolsky and V.I.Kukulin, Phys. Lett, 96B (1980) 4, N. Tanaka et al. Phys. Rev. C59 (1999) 1391.
106 Povzetki v slovenščini
Vektorske in skalarne resonance carmonija v kromodinamiki na mreži
Luka Leskoveca, C. B. Langc, Daniel Mohlerd in Saša Prelovšeka'b
a Institut JoZef Stefan, Ljubljana, Slovenija b Univerza v Ljubljani, Ljubljana, Slovenija c Institute of Physics, University of Graz, Graz, Austria d Fermi National Accelerator Laboratory, Batavia, Illinois, USA
Proučujemo sipanje mezonov D na D s kromodinamiko na mrezi, da bi določili mase in razpadne sirine vektorskih in skalarnih resonanc čarmonija nad pragom za razpad v carobne mezone. V vektorskem kanalu dobimo znano resonanco ^(3770). Simulacija pri vrednosti pionove mase mn = 156 MeV da maso in raz-padno sirino resonance, ki se ujema z eksperimentalnimi podatki znotrajvelike statisticne negotovosti. V skalarnem kanalu proucujemo prvo vzbujeno stanje Xco(1P), za katero ni zaenkrat nobenega sprejetega kandidata. Za sipanje O na D v s-valu raziskujemo razne scenarije. Simulacija nakazuje se neopazzeno ozko resonanco z maso malo pod 4 GeV. Potrebne so nadaljnje raziskave, da bi osvetlili uganke pri skalarnih vzbujenih stanjih carmonija.
Resonance v modelu Nambuja in Jona-Lasinia
Mitja Rosina
Fakulteta za matematiko in fiziko, Univerza v Ljubljani, Jadranska 19, p.p.2964,1001 Ljubljana, Slovenija in Institut J. Stefan, 1000 Ljubljana, Slovenija
Pred leti smo sestavili resljivo verzijo modela Nambuja in Jona-Lasinia, ki se vedno ustrezno opise spontani zlom kiralne simetrije in pojav mezonov pi. Njeni znacilnosti sta regularizacija polja v skatli s periodicnimi robnimi pogoji ter poenostavljena kineticna energija in interakcija. Sedajnas pa zanima opis resonanc, kadar so na voljo le diskretne laste vrednosti energije. Kot zgled navajamo mezon d. Raziskava je lahko poucna za podobne probleme pri kromodinamiki na mrezi.
Roperjeva resonanca — ignoramus ignorabimus?
S. ¡Sirca
Fakulteta za matematiko in fiziko, Univerza v Ljubljani, Jadranska 19, p.p.2964,1001 Ljubljana, Slovenija in Institut J. Stefan, 1000 Ljubljana, Slovenija
V tem prispevku ponudimo kratek pregled nekaterih zadnjih dosezkov na po-drocju raziskav Roperjeve resonance. Nastejemo nekajnajboljrazburljivih eksperimentalnih rezultatov iz centrov MAMI in Jefferson Lab ter drugih laboratorijev; osvetlimo nekajposkusov, da bi razločili naravo te zagonetne strukture v okviru modelov s kvarkovskimi in mezonskimi ali barionskimi in mezonskimi prostost-nimi stopnjami; in odpremo vpogled v znaten napredek, ki so ga v zadnjih letih naredili kromodinamski racuni na mrezi.