PHYSICAL REVIEW D 97, 056023 (2018) Magnetic corrections to π-π scattering lengths in the linear sigma model † ‡ M. Loewe,1,2,3,* L. Monje,1, and R. Zamora4,5, 1Instituto de Física, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile 2Centre for Theoretical and Mathematical Physics and Department of Physics, University of Cape Town, Rondebosch 7700, South Africa 3Centro Científico Tecnológico de Valparaíso-CCTVAL, Universidad T´ecnica Federico Santa María, Casilla 110-V, Vaparaíso, Chile 4Instituto de Ciencias Básicas, Universidad Diego Portales, Casilla 298-V, Santiago, Chile 5Centro de Investigación y Desarrollo de Ciencias Aeroespaciales (CIDCA), Fuerza A´erea de Chile, Santiago 8020744, Chile (Received 5 January 2018; published 27 March 2018) In this article, we consider the magnetic corrections to π-π scattering lengths in the frame of the linear sigma model. For this, we consider all the one-loop corrections in the s, t, and u channels, associated to the insertion of a Schwinger propagator for charged pions, working in the region of small values of the magnetic field. Our calculation relies on an appropriate expansion for the propagator. It turns out that the leading scattering length, l ¼ 0 in the S channel, increases for an increasing value of the magnetic field, in the isospin I ¼ 2 case, whereas the opposite effect is found for the I ¼ 0 case. The isospin symmetry is valid because the insertion of the magnetic field occurs through the absolute value of the electric charges. The channel I ¼ 1 does not receive any corrections. These results, for the channels I ¼ 0 and I ¼ 2, are opposite with respect to the thermal corrections found previously in the literature. DOI: 10.1103/PhysRevD.97.056023 I. INTRODUCTION the projection of the scattering lengths in the isospin I ¼ 0 channel grows, whereas it diminishes in the I ¼ 2 channel Scattering lengths were introduced a long time ago in for an increasing temperature. nuclear physics as an important quantity in order to In peripheral heavy ion collisions, huge magnetic fields calculate low-energy interactions between nucleons and appear. In fact, the biggest fields existing in nature. The also in pion-nucleon systems. The scattering lengths of interaction between the produced pions in those collision two-pion systems are relevant in order to explore QCD may be strongly affected by the magnetic field. In this predictions in the low-energy sector. They were first article, we analyze, in the frame of the linear sigma model, measured by Rosellet et al. [1]. New measurements have the influence of the magnetic field on the π-π scattering been reported using the formation of pionium atoms in lengths. For this purpose, we will use the weak field the DIRAC experiment [2], establishing for the S-wave expansion of the bosonic Schwinger propagator [6].We π-π scattering lengths a 4% difference between scattering ¼ 0 ¼ 2 present in detail the different analytical techniques we have lengths in the isospin channels I and I . Another used for our calculations. experimental measurement can be explored in the heavy 0 0 quarkonium π -π transitions [3]. In the past, thermal II. LINEAR SIGMA MODEL effects on scattering lengths have been considered by many AND π-π SCATTERING authors in the literature, invoking effective approaches as the Nambu–Jona-Lasinio model [4] or the linear sigma The linear sigma model was introduced by Gell-Mann model [5]. A common result, at least qualitatively, is that and L´evy [7] as an effective approach for describing chiral symmetry breaking via explicit and spontaneous mecha- nism. In the phase in which the chiral symmetry is broken, *[email protected] the model is given by † [email protected] ‡ μ [email protected] L ¼ ψ¯ ½iγ ∂μ − mψ − gðs þ iπ⃗· τγ⃗5Þψ 1 2 2 2 1 2 2 2 Published by the American Physical Society under the terms of þ ½ð∂π⃗Þ þ mππ⃗þ ½ð∂σÞ þ mσs the Creative Commons Attribution 4.0 International license. 2 2 Further distribution of this work must maintain attribution to 2 2 2 2 λ 2 2 2 2 the author(s) and the published article’s title, journal citation, − λ vsðs þ π⃗Þ − ðs þ π⃗Þ þðεc − vmπÞs: ð1Þ and DOI. Funded by SCOAP3. 4 2470-0010=2018=97(5)=056023(6) 056023-1 Published by the American Physical Society M. LOEWE, L. MONJE, and R. ZAMORA PHYS. REV. D 97, 056023 (2018) I In this expression, v ¼hσi is the vacuum expectation in the scattering lengths a0, it is enough to calculate the value of the scalar field σ. The idea is to define a new field s scattering amplitude TI in the static limit, i.e., when σ ¼ þ h i¼0 ψ 2 such that s v. Obviously, s . corresponds to s → 4mπ, t → 0, and u → 0: an isospin doublet associated to the nucleons, π⃗denotes the pion isotriplet field, and cσ is the term that breaks explicitly 1 ð2Þ ð2Þ ϵ I ¼ Ið → 4 2 → 0 → 0Þ ð Þ the SU × SU chiral symmetry. is a small dimension- a0 32π T s mπ;t ;u : 9 less parameter. It is interesting to remark that all fields in the model have masses determined by v. In fact, the following 2 2 2 2 2 relations are valid: mψ ¼ gv, mπ ¼ μ þ λ v ,andmσ ¼ 2 2 2 III. ONE-LOOP MAGNETIC CORRECTIONS μ þ 3λ v . Perturbation theory at the tree level allows us to FOR π-π SCATTERING LENGTHS identify the pion decay constants as fπ ¼ v. This model has been considered in the context of finite temperature by The tree-level diagrams shown in Fig. 1, where the several authors, discussing the thermal evolution of masses, continuum line denotes a pion and the dashed line denotes a fπðTÞ, the effective potential, etc. [8–15]. sigma meson, contribute to the π-π scattering amplitude. Since our idea is to use the linear sigma model for The diagram with a sigma exchanged meson has to be calculating π-π scattering lengths, let us remind the reader considered also in the crossed t and u channels. From these briefly of the formalism. A scattering amplitude has the diagrams, it is possible to get the results shown in Table I. general form [16,17] The isospin-dependent scattering amplitudes at tree level have the form Tαβ;δγ ¼ Aðs; t; uÞδαβδδγ þ Aðt; s; uÞδαγδβδ 4 2 4 2 4 2 þ Aðu; t; sÞδαδδβγ; ð2Þ 12λ 4λ 4λ 0ð Þ¼−10λ2 − v − v − v ð Þ T s; t; u 2 2 2 ; 10 s − mσ t − mσ u − mσ where α, β, γ, and δ denote isospin components. By using appropriate projection operators, it is possible to find the isospin-dependent scattering amplitudes 4λ4 2 4λ4 2 1ð Þ¼ v − v ð Þ T s; t; u 2 2 ; 11 u − mσ t − mσ T0 ¼ 3Aðs; t; uÞþAðt; s; uÞþAðu; t; sÞð3Þ 1 T ¼ Aðt; s; uÞ − Aðu; t; sÞ; ð4Þ 4λ4 2 4λ4 2 2ð Þ¼−4λ2 − v − v ð Þ T s; t; u 2 2 : 12 T2 ¼ Aðt; s; uÞþAðu; t; sÞ; ð5Þ t − mσ u − mσ where TI denotes a scattering amplitude in a given isospin Note that the linear sigma model is in better agreement channel. at tree level with the experimental results than first-order As is well known [16], the isospin-dependent scattering chiral perturbation theory. I amplitude can be expanded in partial waves Tl , The magnetic corrections to the scattering lengths will be Z 1 1 calculated using an appropriate expansion of the Schwinger I I TlðsÞ¼ dðcos θÞPlðcos θÞT ðs; t; uÞ: ð6Þ bosonic propagator, which is given by 64π −1 Z ∞ 2 2 tanðqBsÞ 2 Below the inelastic threshold, the partial scattering ds isðk −k −mπ þiϵÞ iΔðkÞ¼ e k ⊥ qBs : ð13Þ amplitudes can be parametrized as [18] 0 cosðqBsÞ 1 s 2 1 2 δI ð Þ I ¼ ð i l s − 1Þ ð Þ In the above expression, kk and k⊥ represent the parallel Tl 2 2 e ; 7 s − 4mπ i and perpendicular components of the momentum k with respect to the external magnetic field B. In general, as is where δl is a phase shift in the l channel. The scattering well known, this propagator includes a phase factor, which, lengths are important parameters in order to describe low- however, does not play any role in our calculation. We energy interactions. In fact, our last expression can be proceed by taking the weak field expansion [6] expanded according to p2 l p2 ℜð I Þ¼ I þ I þÁÁ ð Þ Tl 2 al 2 bl : 8 mπ mπ I I The parameters al and bl are the scattering lengths and scattering slopes, respectively. In general, the scattering lengths obey ja0j > ja1j > ja2j… If we are only interested FIG. 1. Tree-level diagrams. 056023-2 MAGNETIC CORRECTIONS TO π-π SCATTERING … PHYS. REV. D 97, 056023 (2018) TABLE I. Comparison between the experimental values [19], first-order prediction from chiral perturbation theory [20], and our results at tree level. Experimental Chiral perturbation Linear sigma results theory model 0 2 2 0 218 Æ 0 02 7mπ 10mπ a0 . 2 ¼ 0.16 2 ¼ 0.22 32πfπ 32πfπ 0 2 2 0 25 Æ 0 03 mπ 49mπ b0 . 2 ¼ 0.18 2 ¼ 0.27 4πfπ 128πfπ 2 2 2 −0 0457 Æ 0 0125 −mπ −mπ a0 . 2 ¼ −0.044 2 ¼ −0.044 16πfπ 16πfπ 2 2 2 −0 082 Æ 0 008 −mπ −mπ b0 . 2 ¼ −0.089 2 ¼ −0.089 8πfπ 8πfπ 1 2 2 0 038 Æ 0 002 mπ mπ a1 .