Bled Workshops in Physics Vol. 10, No. 1 p. 39
Quadrupole polarizabilities of the pion in the Nambu-Jona-Lasinio model*
Brigitte Hillera, Wojciech Broniowskib,c, Alexander A. Osipova,d, Alex H. Blina
a Centro de Física Computacional, Departamento de Física da Universidade de Coimbra, 3004-516 Coimbra, Portugal
b The H. Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, PL-31342 Cracow, Poland
c Institute of Physics, Jan Kochanowski University, PL-25406 Kielce, Poland d Dzhelepov Laboratory of Nuclear Problems, JINR, 141980 Dubna, Russia
Abstract. We present the results for the pion electromagnetic quadrupole polarizabilities, calculated within the Nambu-Jona-Lasinio model. We obtain the sign and magnitude in agreement with the respective experimental analysis based on the Dispersion Sum Rules. At the same time the dipole polarizabilities are well reproduced. Comparison is also made with the results from the Chiral Perturbation Theory.
The neutral and charged pion dipole and quadrupole polarizabilities have been recently analyzed using the Dispersion Relations (DR) and the Dispersion Sum Rules (DSR) [1,2], as displayed in Tables 2 and 3, together with the results of the Chiral Perturbation Theory (xPT) [3-5]. The first row shows our results based on the Nambu-Jona-Lasinio model (NJL) [6] model; for that purpose we have extended [7] the study of Ref. [8], where the dipole polarizabilities have been calculated, to the quadrupole case. We refer to these papers for details.
Our leading-Nc calculations are done according to the Feynman diagrams of Fig. 1. The amplitude is a function of the Mandelstam variables related to the Y(pi, ) + y(P2, e2) —> na(p3) + nb(p4) reaction for the on-shell pions and photons,
T (pi,p2,p3) = e2efeW, V = A(s,t,u)£^v + B(s,t,u)£^, with the Lorentz tensors
¿T =P2MP7 -	= -	+tlP£PJ +ulP£Pi +	.
Terms that vanish upon the conditions e1 • p1 = e2 • p2 = 0 are omitted and the notation £ = £ — m^, £ = s,t,u is used. The scalar quantities A and B enter the amplitudes H++ = —(A + m^B) and H+_ = (u1t1/s — m^)B for the equal-helicity and helicity-flipped photons. The dipole, a}, p1, and quadrupole,
* Talk delivered by Brigitte Hiller
a2> P2, polarizabilities are obtained in the t-channel and extracted from the first two coefficients of the Taylor expansion of the amplitudes aAl(s,t,u)/(2mn) and -amnBl(s,t,u) around s = 0 with u = t = m^ [9],
(AMO.m^m2) +sAAi(0,m2,m^ = p} + Apt,
-am« ^BMO.m^m^+s^BMO.m^m2)^ = (ai + Pi )l + ^j(<x2 + p2)\
where the superscripts i = N, C denote the neutral and charged pions, respectively, and a ~ 1/137 is the fine structure constant. The Born term, arising in the case of the charged pion, is removed from the amplitudes.
The NJL Lagrangian used in this work contains pseudoscalar isovector and scalar isoscalar four-quark interactions and is minimally coupled to the electromagnetic field.
The diagrams of Fig. 1 provide polarizabilities which scale as N0 .Besides the quark one-loop diagrams, it is expected that the pion loops yield important contributions, mainly in the case where the tree-level results are absent (in the NJL model the chiral counting of meson tree-level results are classified in Refs. [8,10]). We include the lowest model-independent pion-loop diagram at the p4 order, calculated within xPT in Refs. [11-13], and known to be the only non-vanishing contribution to the amplitude A at this order in the neutral channel. The pion loop in the charged mode contributes only to the quadrupole polarizabilities, with half the strength of the neutral quadrupole case. The pion loop as well as the CT-exchange diagram of Fig. 1 enter only the amplitude A.
The amplitude B for the neutral (charged) mode is completely determined by the quark box (quark box + pion exchange) diagrams, starting from the p6 order for the dipole and from the p8 order for the quadrupole polarizabilities. Thus the combinations (a, + (3,)1, (j = 1,2), to which the B amplitude leads, provide a genuine test of the dynamical predictions of the NJL model at the leading order in 1 /Nc, as they are insensitive to the lowest-order xPT corrections.
All quark one-loop integrals are regularized using the Pauli-Villars prescription with one regulator A and two subtractions, which is consistent with the requirements of gauge invariance.
Table 1. The NJL model parameters for the charged and neutral channels, with the input marked by * and f*n = 93.1 MeV.
M* [MeV]	m* [MeV]	m [MeV]	G [GeV~2]	A [MeV]
300	139	7.5	13.1	827
300	136	7.2	13.1	827
Table 2. The dipole (in units of 1 0 4 fm3) and quadrupole (in units of 10 4 fm5) neutral pion polarizabilities. The first row shows our model prediction at M = 300 MeV.
	(«1 + Pi)no	(«1 - Pi )no	(0C2 + p2)n0	(0C2 - p2)7t°
M = 300 MeV	0.73	-1.56	-0.14	36.1
DR fit [1] DSR [1] XPT [5,3]	0.98 ±0.03 0.802 ±0.035 1.1 ±0.3	-1.6 ±2.2 -3.49 ±2.13 -1.9 ±0.2	-0.181 ±0.004 -0.171 ±0.067 0.037 ±0.003	39.70 ±0.02 39.72 ±8.01 37.6 ±3.3
Table 3. Same as in Table 2 for the dipole and quadrupole charged pion polarizabilities.
	(«1 + Pi)„±	(«1 - Pi )n±	(«2 + P2)7t±	(«2 - p2)7t±
M = 300 MeV	0.19	9.4	0.20	17.5
DR fit [2] DSR [2] XPT [5,4]	0.166 ±0.024 0.16	13.0+f;g 13.60 ±2.15 5.7 ± 1.0	0.133 ±0.015 0.121 ±0.064 -0.001	25.0+0^3 25.75 ± 7.03 16.2
In Table 1 we collect the model parameters, obtained by fitting the physical pion mass and the weak decay constant. The parameters of the model are the four-quark coupling constant G, the cutoff A, and the current quark mass m. These are determined by the choice of fn = 93.1 MeV, mn = 139 MeV (charged mode) or mn = 136 MeV (neutral mode), and the constituent quark mass, M. In Table 4 we present the anatomy of our result for the case M = 300 MeV. We display separately all the gauge invariant contributions to the polarizabilities: the box (for neutral polarizabilities), box + pion exchange diagram (for the charged polarizabilities), the ct exchange, and the pion loop. The pion exchange diagram arises only for the charged channel and builds together with the box a gauge invariant amplitude. Let us first comment on the channels involving the difference of the electric and magnetic polarizabilities:
•	(a1 — (31 )no: here the box contribution is largely canceled by the scalar exchange. At the p4-order of the chiral counting they cancel exactly [8]. The higherorder contributions have a slow convergence rate, at p8-order one reaches only about 50% of the full sum.
•	(a1 — (i )n±: contrary to the neutral channel, the size of the c-exchange diagram for this combination is about an order of magnitude larger than the box + pion exchange diagram, and it becomes the most important contribution. The pion loops are absent.
Table 4. Contribution of various diagrams for M = 300 MeV. Units as in Table 2.
		box + 7t-exchange	ff-exchange	pion-loop	total
(«1 + Pi	lu»	0.73	0	0	0.73
(«1 - Pi		-11.13	10.57	-1.0	-1.56
(«2 + P2	lu»	-0.144	0	0	-0.144
(0i2 - p2		5.09	9.07	21.97	36.13
(<X1 + Pi	)n±	0.189	0	0	0.189
(<X1 - Pi	)n±	-0.977	10.36	0	9.39
(«2 + p2	)n±	0.198	0	0	0.198
(a2 - p2	)n±	-1.63	8.87	10.29	17.54
•	(a2 — p2)n±: the pattern observed for the charged dipole difference repeats itself for the quadrupole polarizabilities. However in this case the subleading in the 1 /Nc counting pion-loop diagram has the same magnitude as the a-exchange term.
•	(a2 — p2)no: the pion loop dominates, ~ 2 x a-exchange.
•	Finally, regarding the sums of polarizabilities, we stress that the values (including the overall signs) of (a2 + p2)no>n± are totally determined by the gauge-invariant quark box or the box + pion exchange contribution. This is a key result of the presented calculation. The sign is stable when the model parameters are changed. The magnitude depends on the value chosen for the constituent quark mass, but the best overall fit to the other empirical data, typically yielding M ~ 300 MeV, also yields the optimum values for the polarizabilities [7]. Moreover, the main part of the box contribution comes from the first non-vanishing p8-order term in the chiral expansion. Based on this fact we expect that the contact term of the p8 3-loop calculation in xPT may also play an important role in reversing the signs of the 2-loop order results for these quantities.
Acknowledgement We are very grateful to the organizers of the "Mini-Workshop Bled 2009: Problems in multi-quark states", for the kind invitation to present this work. This research is supported by the Polish Ministry of Science and Higher Education, grants N202 034 32/0918 and N202 249235, by Fundacao para a Ciencia e Tecnologia, grants FEDER, OE, POCI 2010, CERN/FP/83510/2008, and by the European Community-Research Infrastructure Integrating Activity Study of Strongly InteractingMatter (Grant Agreement 227431) under the Seventh Framework Programme of the EU.
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