# Thermal corrections to π π scattering lengths in the linear sigma model

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(2) MARCELO LOEWE AND C. MARTINEZ-VILLALOBOS. 1 Z1 T‘I ðsÞ ¼ dðcos ÞP‘ ðcos ÞT I ðs; t; uÞ: 64 1. PHYSICAL REVIEW D 77, 105006 (2008). (6). Below the inelastic threshold the partial scattering amplitudes can be parametrized as [10] 1=2 s 1 2i I ðsÞ I T‘ ¼ ðe ‘ 1Þ; (7) 2i s 4m2 where ‘ is a phase shift in the ‘ channel. The scattering lengths are important parameters in order to describe low energy interactions. In fact, our last expression can be expanded according to 2 ‘ p p2 I I I <ðT‘ Þ ¼ (8) a‘ þ 2 b‘ þ : m2 m The parameters aI‘ and bI‘ are the scattering lengths and scattering slopes, respectively. In general, the scattering lengths obey ja0 j > ja1 j > ja2 j . . . . If we are only interested in the scattering lengths aI0 , it is enough to calculate the scattering amplitude T I in the static limit, i.e., when s ! 4m2 , t ! 0, and u ! 0 aI0 ¼. 1 I T ðs ! 4m2 ; t ! 0; u ! 0Þ: 32. (9). III. PION-PION SCATTERING AMPLITUDES The diagrams shown in Fig. 1, where the solid line denotes a pion, and the dashed line a sigma meson, contribute to the - scattering amplitude. The diagram with a sigma exchanged meson has to be considered also in the crossed t and u channels.. From these diagrams it is possible to get the results shown in Table I. The isospin dependent scattering amplitudes at the tree level have the form T 0 ðs; t; uÞ ¼ 102 . 124 v2 44 v2 44 v2 ; (10) s m2 t m2 u m2. T 1 ðs; t; uÞ ¼. 44 v2 44 v2 ; 2 u m t m2. T 2 ðs; t; uÞ ¼ 42 . 44 v2 44 v2 : t m2 u m2. (11). (12). Note that, the linear sigma model is in a better agreement with the experimental results than first order chiral perturbation theory. IV. ONE LOOP THERMAL CORRECTIONS FOR SCATTERING LENGTHS For our calculation of the thermal corrections to the scattering lengths, we will use the real time formalism. At the one loop level it is enough to use the Dolan-Jackiw propagators [12]. A general review about the real time formalism, beyond the one loop level can be found in [13,14]. In our case the most relevant thermal contributions will be related to the pion sector, due to the Boltzmann suppression in the case of the sigma meson and/or nucleons. Therefore, for the pion propagators we will use i ~ m Þ ¼ ðk0 ; k; 2 2 ~ k0 k m2 þ i þ 2nB ðjk0 jÞ ðk20 k~2 m2 Þ ; (13) where nB ðxÞ ¼ e x11 is the Bose-Einstein distribution. For the meson propagator, we will restrict ourselves to the T ¼ 0 case ~ m Þ ¼ ðk0 ; k;. FIG. 1. Tree level diagrams.. k20. i : 2 ~ k m2 þ i. TABLE I. Comparison between the experimental values [1], first order prediction from chiral perturbation theory [11], and our results at the tree level. Experimental Results a00. 0:26 0:05. b00 a20. 0:25 0:03 0:028 0:012. b20. 0:082 0:008. a11. 0:038 0:002. b11. . Chiral Perturbation Theory 7m2 32f2 m2 4f2 m2 16f2 m2 8f2 m2 24f2. ¼ 0:16 ¼ 0:18 ¼ 0:044 ¼ 0:089 ¼ 0:030 0. 105006-2. Linear Sigma Model 10m2 32f2 49m2 128f2 m2 16f2 m2 8f2 m2 24f2 m2 48f2. ¼ 0:22 ¼ 0:27 ¼ 0:044 ¼ 0:089 ¼ 0:030 ¼ 0:015. (14).

(3) THERMAL CORRECTIONS TO - SCATTERING . . .. PHYSICAL REVIEW D 77, 105006 (2008). There are many diagrams that contribute to the pionpion scattering amplitude at the one loop level. For each one of these diagrams we have to add also the corresponding crossed t and u channel diagrams. In Fig. 2 we have shown only the s channel contribution. We will give the analytic expression only for the first two diagrams (2(a) and 2(b)) shown in Fig. 2. It should be noticed that, when it corresponds, symmetry factors, isospin index contractions, and multiplicity factors should be included iMa ¼ 24 ð7 þ 2 þ2 Þ Z d4 k ~ m Þðk0 2m ; k; ~ m Þ; ðk0 ; k; ð2Þ4 (15). Z d4 k i 2 2 4m m ð2Þ4 ~ m Þðk0 þ m ; k; ~ m Þ ½ðk0 ; k;. iMb ¼ 168 v4. . ~ m Þ: ðk0 m ; k;. (16). In some of these diagrams we have to deal with terms and ðkÞ k . In those cases the following identity is useful [15] 2 ðkÞ. 1 i@ n ¼ nþ1 : N! @m2. (17). Actually this expression can be proved in the full matrix formalism of thermofield dynamics. However, for some diagrams, we checked our results using also the Matsubara (imaginary time) formalism. When dealing with integrals of the following shape, @ nB ðjk0 jÞ I ¼ 2i dk0 Fðk0 Þ ð ðk0 wk Þ 2wk @m2 n ðjk jÞ @ þ ðk0 þ wk ÞÞ þ B 0 2wk @m2 ½ ðk0 wk Þ þ ðk0 þ wk Þ ; Z. a. c. b. qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where Fðk0 Þ is an arbitrary function and wk ¼ k~2 þ m2 . We can integrate by parts getting I ¼ Iwk þ Iwk , where. d. Iwk ¼ 2 lim. iðdFðk0 ÞÞn ðjk jÞ B 0 dk0. k0 !wk. e. (18). f. i. 4w2k. Fðk0 Þsgnðk0 Þðcschðjk2T0 jÞÞ2 Fðwk ÞnB ðwk Þ i 16w2k T 4w3k (19). and g. h. Iwk ¼ 2 lim. k0 !wk. i. iðdFðk0 ÞÞn ðjk jÞ B 0 dk0 4w2k. Fðk0 Þsgnðk0 Þðcschðjk2T0 jÞÞ2 Fðwk ÞnB ðwk Þ þi : i 16w2k T 4w3k (20). j. FIG. 2. The ten relevant one loop diagrams (s channel only) for the determination of the - scattering lengths. The solid (dashed) lines denote pion (sigma meson) states.. After taking all the diagrams into account, we find the following expression for the one loop thermal corrections to the - scattering amplitudes. 105006-3.

(4) MARCELO LOEWE AND C. MARTINEZ-VILLALOBOS. PHYSICAL REVIEW D 77, 105006 (2008). 2 1 1 4 v4 Aðs; t; uÞT ¼ 4 14 þ 802 v2 þ 48 Aðwk ; TÞ 84 Bðwk ; TÞ 4m2 m2 4m2 m2 2 1 1 6 2 2 4 126 v2 Cðw ; TÞ þ þ 32 v 16v ½D1 ðm ; m ; wk ; TÞ k 4m2 m2 4m2 m2. þ D1 ðm ; m ; wk ; TÞ þ 326 v2 ½E10 ðm ; m ; wk ; TÞ þ E10 ðm ; m ; wk ; TÞ 1 þ 328 v4 ½E11 ðm ; m ; wk ; TÞ þ E11 ðm ; m ; wk ; TÞ þ 86 v2 ½F01 ðm ; m ; wk ; TÞ 4m2 m2 1 1 6 2 4 2 þ F0 ðm ; m ; wk ; TÞ þ 96 v (21) þ 16v F11 ðm ; m ; wk ; TÞ; 4m2 m2 1 1 2 1 4 v4 6 v2 Aðt; s; uÞT ¼ 44 Aðwk ; TÞ 4 11 þ 802 v2 ; TÞ þ 4 þ 48 Bðw k m2 m2 m2 4 3 þ 2 Cðwk ; TÞ þ 166 v2 ½D1 ðm ; m ; wk ; TÞ þ D1 ðm ; m ; wk ; TÞ 2 2 m m m 1 þ 168 v4 ½D2 ðm ; m ; wk ; TÞ þ D2 ðm ; m ; wk ; TÞ þ 6 32v2 þ 322 v4 ½E10 ðm ; m ; wk ; TÞ m2 1 2 þ E10 ðm ; m ; wk ; TÞ þ 168 v4 ½E20 ðm ; m ; wk ; TÞ þ E20 ðm ; m ; wk ; TÞ þ 6 482 v4 þ 8v m2 ½F02 ðm ; m ; wk ; TÞ þ F02 ðm ; m ; wk ; TÞ ¼ Aðu; t; sÞT ;. (22). where we have introduced the following definitions: 1 Z ~ nB ðwk Þ Aðwk ; TÞ ¼ djkj ; (23) wk ð2Þ2 Z d3 k nB ðwk Þ ðcschðwk ÞÞ2 2T þ ; Bðwk ; TÞ ¼ ð2Þ3 2w3k 8Tw2k Cðwk ; TÞ ¼. Z d3 k nB ðwk Þ ; ð2Þ3 wk. Z d3 k nB ðwk Þ 1 n f ðx; yÞ ð2Þ3 wk 1 ; 4x2 þ 4xwk. n ðx; y; wk ; TÞ Fm. fn ðx; yÞ ¼ ð2x2 þ 2xwk y2 Þn :. (29). V. NUMERICAL RESULTS (24). (25). Dn ðx; y; wk ; TÞ ¼ . Enm ðx; y; wk ; TÞ ¼. and where. (26). Z d3 k ðx þ wk ÞnB ðwk Þ ð2Þ3 2w2k fnþ1 ðx; yÞfm ðx; yÞ k 2 ðcschðw nB ðwk Þ 2T ÞÞ þ 4w3k 16Tw2k 1 n ; (27) f ðx; yÞfm ðx; yÞ. Z d3 k nB ðwk Þ 1 ; ¼ fn ðx; yÞfm ðx; yÞ ð2Þ3 wk (28). The thermal corrections must be calculated numerically. The different parameters in our expressions are renormalized at T ¼ 0, since thermal corrections do not add new ultraviolet divergencies. The linear sigma model, excluding the nucleons, has three parameters: m2 , f , and 2 . The first two parameters, m2 and f , are given by experiments and the third one is a free parameter. Notice that f is related to the vacuum expectation value v. In fact, at the tree level f ¼ v. The three parameters are not independent. If instead of f we use the vacuum expectation value v and consider a mass of the sigma meson m ¼ 700 MeV, we have 2 ¼ 7, v ¼ 90 MeV; if 2 ¼ 5:6, v ¼ 120 MeV [16]. We know, however, that the mass of the sigma meson is about m ¼ 550 MeV [17]. Therefore, we need to find new values for and v associated to the new lower mass of the sigma meson. We found 2 ¼ 4:26 and v ¼ 89 MeV, following the philosophy presented in [16]. The authors used the Padé approximant method, but they also suggested an analytic approach in order to find the variation of one parameter if the other two change as a consequence of one loop corrections. The values given above for 2 and v were found following this procedure. The scattering lengths, including our thermal corrections, are given by. 105006-4.

(5) THERMAL CORRECTIONS TO - SCATTERING . . .. a00 ðTÞ ¼ 0:217 þ. 3Aðs; t; uÞT þ 2Aðt; s; uÞT ; 32. a20 ðTÞ ¼ 0:041 þ. 2Aðt; s; uÞT : 32. PHYSICAL REVIEW D 77, 105006 (2008). (30). (31) a0 ðTÞ. The behavior of the normalized scattering lengths 0a0 0 a2 ðTÞ and 0a2 are shown in Fig. 3. Notice that a10 ðTÞ vanishes 0 identically. We normalized with the zero temperature values of the scattering lengths, according to a two loop perturbation calculation [18]. It is interesting to remark that a00 ðTÞ grows with temperature and the same occurs with the a20 ðTÞ. Similar calculations have been done in the context of the Nambu-Jona-Lasinio model [19] and in the frame of chiral perturbation theory [20]. The results from the Nambu-Jona-Lasinio analysis do not agree with our conclusions. In this approach both scattering lengths remain almost constant, but with the same growing tendency. The results from chiral perturbation theory agree quite well with our conclusions for a20 ðTÞ. Nevertheless, a00 ðTÞ is almost constant in this approach. In [21], a different determination of a00 ðTÞ, based on the heat kernel expansion technique applied to the linear sigma model, is presented. We agree with their results.. We acknowledge support from FONDECYT under Grant No. 1051067. M. L. acknowledges also support from the Centro de Estudios Subatómicos. We also would like to thank Professor A. Ayala and Professor A. Das for helpful correspondence. We thank Jorge Ruiz for several discussions.. [1] L. Rosselet et al., Phys. Rev. D 15, 574 (1977). [2] J. L. Uretsky, arXiv:0712.4162v1. [3] M. Gell-Mann and M. Lévy, Nuovo Cimento 16, 705 (1960). [4] B. Lee, Chiral Dynamics (Gordon and Breach, New York, 1972). [5] C. Contreras and M. Loewe, Int. J. Mod. Phys. A 5, 2297 (1990). [6] A. Larsen, Z. Phys. C 33, 291 (1986). [7] N. Bilic and H. Nikolic, Eur. Phys. J. C 6, 513 (1999). [8] H. Mao, N. Petropoulos, and W-K. Zhao, J. Phys. G 32, 2187 (2006); N. Petropoulos, arXiv:hep-ph/0402136 and references therein. [9] P. D. B. Collins, Introduction to Regge Theory (Cambridge University Press, Cambridge, England, 1977). [10] J. Gasser and H. Leutwyler, Ann. Phys. (N.Y.) 158, 142 (1984). [11] S. Weinberg, Phys. Rev. Lett. 17, 336 (1966).. [12] L. Dolan and R. Jackiw, Phys. Rev. D 9, 3320 (1974). [13] A. Das, Finite Temperature Field Theory (World Scientific Publishing, Singapore, 1997). [14] M. Le Bellac, Thermal Field Theory (Cambridge University Press, Cambridge, England, 1996). [15] Y. Fujimoto, H. Matsumoto, and H. Umezawa, Phys. Rev. D 30, 1400 (1984). [16] J. L. Basdevant and B. W. Lee, Phys. Rev. D 2, 1680 (1970). [17] E. M. Aitala et al., Phys. Rev. Lett. 86, 765 (2001); Phys. Rev. Lett. 86, 770 (2001); J. M. de Miranda and I. Bediaga, in Proceedings of the Eleventh International Conference on Hadron Spectroscopy, AIP Conf. Proc. No. 814 (AIP, New York, 2006). [18] J. Bijnens et al., Phys. Lett. B 374, 210 (1996). [19] E. Quack et al., Phys. Lett. B 348, 1 (1995). [20] N. Kaiser, Phys. Rev. C 59, 2945 (1999). [21] B. J. Schaefer and H. J Pirner, Nucl. Phys. A627, 481 (1997).. FIG. 3. Scattering lengths normalized to T ¼ 0.. ACKNOWLEDGMENTS. 105006-5.

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