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Schwinger Model À La Very Special Relativity ∗ Jorge Alfaro, Alex Soto

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Schwinger Model À La Very Special Relativity ∗ Jorge Alfaro, Alex Soto Physics Letters B 797 (2019) 134923 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Schwinger model à la Very Special Relativity ∗ Jorge Alfaro, Alex Soto Instituto de Física, Pontificia Universidad de Católica de Chile, Av. Vicuña Mackenna 4860, Santiago, Chile a r t i c l e i n f o a b s t r a c t Article history: In this work, we show that Lorentz invariant theories in 1 + 1dimensions admit new terms inspired by Received 14 July 2019 Very Special Relativity (VSR) theories. We have studied the Schwinger model in VSR. We show the axial Received in revised form 29 August 2019 current is classically conserved in the presence of a mass term coming from the VSR invariant terms but Accepted 4 September 2019 without standard Lorentz invariant mass. Furthermore, it is shown that both the vector current as well as Available online 9 September 2019 the axial current are modified with respect to the free case when the fermion is coupled to an external Editor: M. Cveticˇ electromagnetic field due to the nonlocal operator present in the theory. The axial anomaly is computed, and we found the same standard topological invariant with a modification in the coefficient. © 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license 3 (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP . 1. Introduction Lorentz subgroup SIM(2) [14]. This theory, studied in four di- mensions and called Very Special Relativity (VSR), does not have Quantum Electrodynamics in 1 + 1dimensions (QED2) has invariant tensors, and it has the same important features of Spe- been studied, and it has an exact solution discovered by Schwinger cial Relativity, like time dilation, velocity addition, and maximum [1] when the fermion remains massless. In this model, called attainable velocity. Under this framework, a fixed null vector n · · Schwinger Model, the photon acquires a mass e2/π . This model transforms with a phase. Hence, new invariant terms like n p/n q can be constructed in the lagrangian. As a consequence of this, has been studied and reviewed extensively (see for example the neutrino gets mass without new particles or violation of lep- [2–5]) since this model presents confinement, because in two- tonic number [15]. In four dimensions VSR has been studied in dimensional space-time the Coulomb potential increases linearly, electrodynamics [16], and an important feature is the possibility and it prevents the fermions to become free [6,7]. Interesting prop- and new consequences of a non-violating gauge invariant photon erties as the nontrivial vacuum structure were studied in the work mass [17]. Also, the electroweak model under this formalism has of Lowenstein and Swieca [8]. Moreover, instantons in this model been reviewed [18]. In addition, we found VSR studies in locally have been analyzed by Smilga [9]. In addition, despite the mas- anisotropic cosmology [19], considering two-time physics [20], sive Schwinger model is not exactly solvable, it has been reviewed curved space-time [21] and using gaugeon formalism [22]. too [10,11]. Another essential feature in the Schwinger model is In this work, we will focus our attention on two dimensions the presence of the chiral anomaly, which is easier to compute instead of four, and we will analyze the chiral anomaly in the VSR- than in four dimensions. The axial vector current, which classically QED without the standard Lorentz invariant mass. The anomaly is conserved, gets a new term after radiative corrections. Thus, computation is easier than in four dimensions and it is simpler to test the new VSR-like terms here before we apply them in four- μ5 e μν ∂μ j = Fμν. (1) dimensional models. Meanwhile we have to keep in mind that in 2π lower dimensions we have less degrees of freedom than in real life. For good reviews of this anomaly using a perturbative treatment In fact there are not true dynamical degrees of freedom associated see [12,13]. with the electromagnetic field in two dimensions [11]. Anomalies have not been studied yet in the context of SIM(2) Thus, the outline of this work is as follows. In section 2 we will invariant theories. These theories began with the claim of Co- analyze the Lorentz group for two dimensions and the connec- hen and Glashow that nature could be described only with the tion with VSR theories. In section 3 we will review the classical conservation of the free vector and axial current under the VSR formalism. In section 4 we will derive the vector and axial cur- * Corresponding author. rent when the fermion is coupled with an external electromagnetic E-mail addresses: jalfaro@fis.uc.cl (J. Alfaro), [email protected] (A. Soto). field. Section 5 presents the path integral view of the lagrangian https://doi.org/10.1016/j.physletb.2019.134923 0370-2693/© 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 2 J. Alfaro, A. Soto / Physics Letters B 797 (2019) 134923 10 reviewed in section 4, and we will compute the vacuum polariza- γ 5 = . (9) tion of the photon. In section 6, we will compute the expectation 0 −1 value of the vector current. In section 7, we present the result of In addition m is a mass parameter in the SIM(2) theory which the axial anomaly in the VSR two dimensional model, and finally, in section 8, we will summarize and highlight the results. parametrizes the deviations of the Lorentz Symmetry. Also, we can associate this parameter with the neutrino mass (see [15]). We no- → 2. Lorentz group in 1 + 1dimensions tice if m 0, we recover the standard result. In addition, M is the standard Lorentz invariant fermion mass. We will consider from We recall the Lorentz group as the group of transformations here M = 0to study the Schwinger model in VSR. Nevertheless, that left invariant the metric under this assumption, our model is a fermion with a small mass m. Although there is not standard mass, our fermion is massive μν = ρσ μ ν g g ρ σ . (2) due to the VSR term. Thus, in VSR we have a slightly different massive Schwinger model. Also, ψ is a two component spinor. As In two dimensions we choose the metric as g = diag(1, −1). If we μ μ μ M = 0, the lagrangian is consider an infinitesimal transformation ρ = δρ + ωρ , it is easy μν μ ρν to see that ω = ω g is antisymmetric. Thus, due to the an- 2 ρ ¯ m n/ tisymmetry, the Lorentz group in 1 + 1dimensions has only one L0 = ψ i∂/ + i ψ. (10) 2 n · ∂ parameter. It means we have one generator of the Lorentz group. We define it as ¯ From the equation (10)we compute the equations of motion for ψ 01 and ψ K = . (3) 10 ← 2 ¯ m n/ We notice that we can construct the Lorentz transformations ap- ψ −i ∂/ − i ← = 0 (11) 2 n · ∂ plying successive transformations to the identity m2 n/ (θ) = exp (K θ), (4) i∂/ + i ψ = 0 (12) 2 n · ∂ where θ is a parameter. With this, the transformation reads ← where ∂ indicates the derivative acts over the object in the left. If cosh θ sinh θ ¯ = . (5) we multiply by ψ in the right in (11)an by ψ in the left in (12) sinh θ cosh θ and we sum both we get We can check effectively satisfies the identity (2). Moreover, is 2 easy to see there are not invariant tensors under this transforma- ¯ μ m 1 ¯ μ 1 ∂μ ψγ ψ + ψ nn/ ψ = 0. (13) tion. However, if we observe the following null vector 2 n · ∂ n · ∂ 1 Therefore, we define the expression in the bracket as the free vec- n = , (6) 1 tor current 2 it transforms with a phase under this transformation, n = eθ n. μ m 1 1 j = ψ¯ γ μψ + ψ¯ nn/ μ ψ . (14) Therefore, we can add to the lagrangian terms with fractions which free 2 n · ∂ n · ∂ contain n as in the numerator as in the denominator, because they are invariant under the Lorentz transformation. This kind of terms We can proceed in an analogue way to define the axial current jμ5 have been studied in four dimensional VSR theories (see for in- as stance [15–18]), where the null vector (1, 0, 0, 1) transforms in 2 the same way under SIM(2) group transformations. Nevertheless, μ5 m 1 1 j = ψ¯ γ μγ 5ψ + ψ¯ n/γ 5nμ ψ . (15) these terms have not been incorporated in two dimensional works free 2 n · ∂ n · ∂ in Lorentz invariant theories and they could be added. μ = μ5 = We observe that ∂μ j free 0 and ∂μ j free 0; thus, both currents 3. Classical axial and vector currents for free VSR fermions are conserved at the classical level. μ 5 μν In two dimensions the relation γ γ =− γν holds, where Since the two dimensional Lorentz theories admit terms with μν is the two-dimensional Levi-Civita symbol. Hence, we apply the null vector n = (1, 1) we proceed as in the VSR theories. The this relation in the equation (15) and we use the equation (14)to general VSR lagrangian for a free fermion is given by get m2 n/ L = ¯ − + 2 ψ i∂/ M i ψ. (7) μ5 μν free m 1 ¯ μν 5 μ 1 2 n · ∂ j =− jν + ψ nn/ ν + n/γ n ψ .
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