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 , 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. as an important quantity in order to In peripheral heavy ion collisions, huge magnetic fields calculate low-energy interactions between and appear. In fact, the biggest fields existing in nature. The also in - 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 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 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 , 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 þ 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 0 02 7mπ 10mπ a0 . . 2 ¼ 0.16 2 ¼ 0.22 32πfπ 32πfπ 0 2 2 0 25 0 03 mπ 49mπ b0 . . 2 ¼ 0.18 2 ¼ 0.27 4πfπ 128πfπ 2 2 2 −0 0457 0 0125 −mπ −mπ a0 . . 2 ¼ −0.044 2 ¼ −0.044 16πfπ 16πfπ 2 2 2 −0 082 0 008 −mπ −mπ b0 . . 2 ¼ −0.089 2 ¼ −0.089 8πfπ 8πfπ 1 2 2 0 038 0 002 mπ mπ a1 . . 2 ¼ 0.030 2 ¼ 0.030 24πfπ 24πfπ 1 2 … mπ b1 0 2 ¼ 0.015 48πfπ

ð Þ2 Δð Þ¼ i − i qB i k 2 2 2 2 3 k − mπ þ iϵ ðk − mπ þ iϵÞ 2 ð Þ2 2 − i qB k⊥ ð Þ 2 2 4 : 14 ðk − mπ þ iϵÞ

Certainly, only the charged pions will receive a magnetic correction. There are many diagrams that contribute to the pion-pion 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. Explicit expressions will be given only for diagram (a) FIG. 2. Relevant one-loop diagrams in the s channel. and its equivalent diagram in the t channel. When it corresponds, symmetry factors, isospin index contractions, and multiplicity factors should be included. We work in the δ β ¼ð2 0⃗Þ center-of-mass momentum p mπ; . For our calcu- p p lation, since the sigma meson has a much bigger mass than the pions, its propagator will be contracted to a point, the so-called high mass limit. A numerical treatment, however, confirms the validity of such approximation in our case. For k k diagram (a) in Fig. 2,wehave

Z 4 4 d k ⃗ iM ¼ −2λ I iΔðk0; k; mπÞ a;s a ð2πÞ4 p p ⃗ α γ × iΔðk0 − 2mπ; k; mπÞ: ð15Þ

The corresponding t-channel diagram in Fig. 3, where no FIG. 3. Diagram (a) for the corresponding t channel. external momentum flows through the loop, is given by

Z 4 where the greek letters denote isospin indices. To get the 4 d k ⃗ 2 scattering lengths in the different isospin channels, we have iM ¼ −4λ I ½iΔðk0; k; mπÞ ; ð16Þ a;t a ð2πÞ4 used appropriate projectors, contracting them with the amplitudes [see Eq. (2)] that emerged from our calculation. where Ia is an isospin term associated to this diagram for the Notice that for the determination of the scattering lengths corresponding channel, which emerges from the contraction we only need the imaginary part of our diagrams. It is of the external pion isospin indices, which is given by natural in this context to choose the π-π center-of-mass frame of reference for carrying on the calculations in the s ¼ð7δ δ þ 2δ δ þ 2δ δ Þ ð Þ Ia αβ γδ αγ βδ αδ βγ ; 17 channel. In fact, the scattering lengths are defined in this

056023-3 M. LOEWE, L. MONJE, and R. ZAMORA PHYS. REV. D 97, 056023 (2018) ∂ 2 frame. Let us consider first the s-channel diagram (a). The ∝ ð Þ2 Lgeneral term qB 2 idea is to calculate the loop using the weak expansion for ∂μπ the propagators. Then, we get expressions that involve free Z 4 d k ð 2−⃗2− 2 þ ϵÞ propagators or powers of them. We found it quite useful to is1 k0 k mπ i × ds1ds2 ð2πÞ4 e use the following technique, introduced in Ref. [21], where 2 ⃗2 2 a proper time representation for the free propagators is × eis2ððk0−2mπÞ −k −μπ þiϵÞ: ð24Þ used. The one-loop diagram, at the lowest order, i.e., using normal free propagators Several of the diagrams that contribute to the s channel Z have this form where we have to take derivatives with d4k respect to a pion mass parameter, taking then, after the iLðpÞ¼ DðkÞDðp − kÞ; ð Þ ð2πÞ4 18 derivation, all masses as the pion mass. Using the identities shown above, we can see that at the threshold these can be written in terms of a proper time representation for contributions vanish. So, the s-channel diagram, calculated ⃗ each propagator, at the center-of-mass momentum p ¼ð2mπ; 0Þ, does not Z contribute to the scattering lengths. ∞ 2 2 DðkÞ¼ dseisðk −mπ þiϵÞ; ð19Þ Now, we will proceed with the calculation of the 0 equivalent t-channel diagram. At the end, we have to also take into account the u-channel contribution, which is, as however essentially the same as in the t channel. For Z Z this calculation, we will invoke some properties of the 4 ∞ d k 2 ð Þ¼ −iðs1þs2Þmπ Hurwitz-ζ function. If we consider (16), including the iL p 4 ds1ds2e ð2πÞ 0 coupling, the isospin term, and the magnetic field, integrat- 2 2 × eis1ðp−kÞ eis2k : ð20Þ ing the transverse momenta and the proper times, we get ∞ Z 8λ4πqB X After integrating the Gaussian term in the loop momen- iM ¼ − dk0dk3 a;t ð2πÞ4 tum and introducing the variables l¼0 ð−1Þ ð Þ 1 − v 1 þ v × 2 2 2 : 25 ¼ ¼ ðqBð2l þ 1Þ − k þ mπ − iϵÞ s1 s 2 and s2 s 2 ; k In the above expression, k2 ¼ k2 − k2. Using the mass we find that the imaginary part, ℑL, is given by k 0 3 ð ϵÞ−1 ¼ Z Z derivative, and the Plemelj decomposition a i 1 ∞ ð1 Þ ∓ πδð Þ 1 ds ð1ð1− 2Þ 2− 2 Þ P =a i a , where P is the Cauchy principal value, ℑL ¼ dv eis 4 v p mπ : ð21Þ 0 2πi −∞ s we get 8λ4π ∂ ¼ − qB Using the integral representation of the Heaviside iMa;t 4 2 function, we obtain ð2πÞ ∂mπ X∞ Z Z 2 2 1 1 × dk0dk3iπδðqBð2l þ 1Þ − k þ mπÞ: ℑ ¼ θ ð1 − 2Þ 2 − 2 k L dv v p mπ l¼0 0 4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð26Þ 2 4mπ 2 2 ¼ 1 − θðp − 4mπÞ; ð22Þ p2 After some change of variables, the integration in k0 gives us where p is the total momentum that goes into the loop. 8λ4π2 ∂ ¼ − ¼ð2 0⃗Þ iMa;t 4 i 2 If we choose p mπ; , we see that this contribution ð2πÞ ∂mπ vanishes. For higher powers in the denominators, which Z 2 2 qB 1 1 k3 þ mπ appear when magnetic field terms are introduced, we will × dk3 pffiffiffiffiffiffiffiffiffi ζ ; þ ; ð27Þ use the following identity [22]: 2qB 2 2 2qB where we have used the Hurwitz-ζ function 1 i∂ n Δ ¼ Δnþ1 ð Þ 2 : 23 X∞ N! ∂μ 1 ζðs; qÞ¼ : ð28Þ ð þ Þs ¼0 q n In this way, we found that all the contributions to order n ðqBÞ2 have the general form We may use the identity [23]

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1 a1−s Zðs; aÞ IV. RESULTS AND CONCLUSIONS ζð Þ¼ −s þ þ ð Þ s; a 2 a − 1 Γð Þ ; 29 s s The magnetic corrections were calculated analytically. The different parameters in our expressions are renor- ð Þ where Z s; a in the large-a (Poincar´e) asymptotic expan- malized at B ¼ 0. The linear sigma model, excluding the sion takes the form 2 2 nucleons, has three parameters: mπ, fπ,andλ .Thefirst 2 X∞ two parameters, mπ and fπ,aregivenbyexperiments, B2k Γð2k þ s − 1Þ Zðs; aÞ ∼ : ð30Þ and the third one is a free parameter. Notice that fπ is ð2kÞ! a2kþs−1 k¼1 related to the vacuum expectation value v. In fact, at tree level, fπ ¼ v. The three parameters are not independent. 1 1 qB In our case, a ¼ 2 þ 2 , where x ¼ 2 2. Notice that a large- If instead of fπ we use the vacuum expectation value v x k3þmπ a value corresponds to a small magnetic field. Expanding and consider a mass of the sigma meson mσ ¼ 700 MeV, 2 2 around x ¼ 0, we get we have λ , v ¼ 90 MeV; if λ ¼ 5.6, v ¼ 120 MeV [24]. We know, however, that the mass of the sigma 4 2 8λ π ∂ meson is about mσ ¼ 550 MeV [25]. Therefore, we need ¼ − pqBffiffiffiffiffiffiffiffiffi iMa;t 4 i 2 λ ð2πÞ 2qB ∂mπ to find new values for and v associated to the new Z pffiffiffi lower mass of the sigma meson. We found λ2 ¼ 4.26 and 2 3=2 x 7=2 ¼ 89 × dk3 − pffiffiffi − pffiffiffi þ O½x : ð31Þ v MeV, following the philosophy presented in x 12 2 Ref. [24]. The scattering lengths associated to the isospin channels I ¼ 0 and I ¼ 2, including our magnetic ð Þ2 Keeping only the magnetic contribution qB , after corrections, are given by integrating in k3, finally, we find for the t-channel contribution 3 ð Þþ2 ð Þ 0ð Þ¼0 217 þ AB s; t; u AB t; s; u ð Þ a0 B . ; 35 8λ4π2 ∂ ð Þ2 λ4 ð Þ2 32π ¼ qB ¼ − i qB ð Þ iMa;t 4 i 2 2 2 4 : 32 ð2πÞ ∂mπ 12mπ 24π mπ 2 2ABðt; s; uÞ a0ðBÞ¼−0.041 þ : ð36Þ All the other diagrams reduce to one of the previous 32π cases, once the sigma propagator is cut, i.e., when the 0ð Þ a0 B ⃗ i The behavior of the normalized scattering lengths 0 approximation Δσðk0; k; mσÞ ≈− 2 is used. a0 mσ 2 a0ðBÞ and 2 is shown in Fig. 4. After taking all the diagrams into account for the one- a0 loop magnetic corrections to the π-π scattering amplitudes, The channel I ¼ 2 corresponds to the most symmetric we get the amplitude in the s channel, state for a two-pion state in the isospin space. The fact that     the scattering length in this channel increases, due to 1 2 qB 2 6 2 2 2 ðλ v Þ magnetic effects, shows that the interaction between pions 4mπ −mσ mπ A ðs; t; uÞ¼ becomes more intense. This, in turn, can be associated to a B 4π2   proximity effect between the pions. In a different context, qB 2 8 4 2 2 ðλ v Þ we have found recently similar effects when computing the − mπ ð Þ 2 4 ; 33 6π mσ and for the t and u channels, a.u 1.5

A ðt; s;uÞ¼A ðu;t; sÞ B B     2 2 qB 8 4 qB 8 4 1.0 2 2 ðλ v Þ 2 2 ðλ v Þ mπ mπ ¼ − − 0 2 4 2 4 a0 qB 2π mσ 6π mσ     0 2 2 a0 qB 8 4 qB 6 2 0.5 2 2 ðλ v Þ 2 2 ð5λ v Þ 2 þ mπ − mπ a0 qB 2 4 2 4 6π mσ 12π mσ a 2       0 2 2 2 qB 6 2 4 qB qB 6 2 Normalized Scattering Lenghts 0.0 2 2 ðλ v Þ 2λ 2 2 ðλ v Þ − mπ − mπ þ mπ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 2 2 2 2 4 : 2 12π mσ 24π 4π mσ qB m ð Þ 34 FIG. 4. Scattering lengths normalized to B ¼ 0.

056023-5 M. LOEWE, L. MONJE, and R. ZAMORA PHYS. REV. D 97, 056023 (2018) correlation distance for in the - plasma. ACKNOWLEDGMENTS This magnitude increases as a function of an increasing M. Loewe acknowledges support from FONDECYT external magnetic field [26], with the effect of temperature being exactly the opposite. A similar result was found, this (Chile) under Grants No. 1170107, No. 1150471, and time in the context of QCD sum rules, in which the effects No. 1150847 and ConicytPIA/BASAL (Chile) Grant of the magnetic field increase the continuum threshold [27], No. FB0821; L. Monje acknowledges support from whereas temperature induces the opposite effect. We see FONDECYT (Chile) under Grant No. 1170107; and R. that the results found in this article are consistent with this Zamora would like to acknowledge support from CONICYT general picture. FONDECYT Iniciación under Grant No. 11160234.

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