Soft supersymmetry breaking in the nonlinear sigma model

In this work we discuss the dynamical generation of mass in a deformed ${\cal N}=1$ supersymmetric nonlinear sigma model in a two-dimensional ($D=1+1$) space-time. We introduce the deformation by imposing a constraint that softly breaks supersymmetry. Through the tadpole method, we compute the effective potential at leading order in $1/N$ expansion showing that the model exhibit a dynamical generation of mass to the matter fields. Supersymmetry is recovered in the limit of the deformation parameter going to zero.


I. INTRODUCTION
While the Nonlinear Sigma model (NLSM) has applications as a theory for the interaction between pions and nucleons [1] and, in lower dimensional systems, it can also describe several aspects of condensed matter physics (for example, applications to ferromagnets [2][3][4][5]), the model is also appealing for purely theoretical investigations. In particular, it possesses an interesting phase structure and at the same time it shares some special features with more realistic theories, being a simple example of an asymptotically free theory [6,7].
The action for the NLSM in D space-time dimensions may be written as where the field σ is a Lagrange multiplier that constraints the fields φ a to satisfy φ 2 a = N g , such that the model has an O(N ) symmetry (the index a assumes the values 1, 2, ..., N ).
The phase structure and the renomalizability of the NLSM in (2+1) dimensions was established by the late 1970s, showing that this model possesses two phases [8,9]. One phase is O(N ) symmetric and exhibits a spontaneous generation of mass due to a non-vanishing vacuum expectation value (VEV) of the Lagrange multiplier field σ, i.e., σ = 0. The other phase is characterized by a nonvanishing VEV of the fundamental bosonic field φ, so that the O(N ) symmetry is spontaneously broken to O(N − 1), and there's no generation of mass. Several extensions of this model was later studied showing no changing in its phase structure [10][11][12][13][14][15][16][17][18]. Unlike the two-phase structure of the 3D model, in two dimensions we have supersymmetry and O(N ) symmetry both unbroken, in agreement with a theorem by Coleman that states that in two dimensions Goldstone's theorem does not end with two alternatives (either manifest symmetry or Goldstone boson) but with only one: manifest symmetry [19].
Although the supersymmetric counterpart of (1) presents a similar phase structure in (2 + 1) dimensions, it was pointed out in [20] that there's no soft transition to the non-SUSY model for the mass acquired by the fields in the symmetric phase. To understand their point, consider the N = 1 SUSY NLSM, described by the action where 2D 2 = D α D α , D α = ∂ α + iθ β ∂ αβ is the covariant supersymmetric derivative 1 and Σ is the Lagrange multiplier superfield that constraints Φ a to satisfy Φ 2 a (z) = N g .
If we write the superfields components as: we can integrate over d 2 θ, and eliminate the auxiliary field F a using its equation of motion, to express the action of the model as and see that the auxiliary field σ acts as the Lagrange multiplier associated to the constraint φ 2 a = N g so that the usual (bosonic) model (1) is obtained setting ψ = ρ = χ = 0, and σ = 0.
From (4) it is easy to see that if exist a phase where mass is generated to the fundamental fields φ and ψ, their masses will be given by the VEV of the fields ρ and σ as from which we observe that, for σ = 0 and a non-vanishing VEV of ρ, the fundamental bosonic and fermionic fields acquire the same squared mass 4 ρ 2 , indicating generation of mass in a supersymmetric phase as is well-known [12][13][14][15][16]. This acquired mass, however, is due to ρ, while in the non-SUSY model the spontaneous generation of mass occurs due to σ acquiring a nonvanishing vacuum expectation value. Therefore, we may say that we do not have anything that we can interpret as a non-SUSY limit of the spontaneous generation of mass from the SUSY model (since ρ is not present in the bosonic model).
The aim of the present paper is to use this more general constraint on the two-dimensional SUSY NLSM and to discuss the dynamical generation of mass in this model.

II. SOFT BROKEN SUPERSYMMETRY IN (1+1) DIMENSIONAL SUSY NLSM
We start with a slight deformation of the SUSY NLSM, introducing a more general constraint for the superfields Φ a : where Σ(z) is a Lagrange multiplier for the modified constraint Φ 2 a constant superfield which possesses the θ-expansion H(z) = 1 − θ 2 g η. Note that H(z) breaks SUSY explicitly and we recover the supersymmetric action for the NLSM, Eq. (2), for η = 0.
The new constraints to the components of the fundamental superfields Φ a are: In order to study the phase structure of the model, let us start assuming that the N-th component Φ N (x, θ) and Σ both have constant non-trivial VEVs given by Let us also make a shift in these superfields by redefining Σ → (Σ + Σ cl ) and Φ N → so that we can rewrite the action (6) in terms of the new fields as We can immediately see that Σ cl (i.e., the VEV of the superfield Σ) gives mass to the fundamental superfields Φ a , and that this "mass" is θ-dependent, therefore generating different masses to the bosonic and fermionic components of Φ a , showing a possible solution where supersymmetry is broken.
At the leading order, the propagator of Φ a must satisfy where δ (4) By solving (10) using the methods described in [23,24], we get the propagator for the superfield Φ a : which reduces to the usual propagator of a massive scalar superfield for σ cl = 0.
From Eq.(9) we can see that there exists a mixing between Φ N and Σ, but this mixing only contributes to the next-to-leading order in the 1/N expansion. For now, we can neglect this mixing, since we will deal with the SUSY NLSM only at leading order in 1/N .
With the propagator of Φ a , we can evaluate the effective potential through the tadpole method [25][26][27]. At leading order, the tadpole equation for the superfield Φ N can be cast as where U ef f is the superfield effective superpotential.
On the other hand, the tadpole equation for Σ is (cf. Fig. 1): Substituting the expression for ∆(k), and using the fact that D 2 δ (2) (θ − θ) = 1 and δ (2) (θ − θ) = 0, we obtain where λ = 1 g is the renormalized coupling and µ is a mass scale introduced by the regularization by dimensional reduction (i.e., In the tadpole equations, each term of the θ expansion has to vanish independently, i.e., the classical fields have to satisfy With the tadpole equations in hands, the effective potential V ef f = d 2 θU ef f is obtained by integrating Eq.(12) over Φ cl and Eq.(14) over Σ cl as As we did for the classical action, we can eliminate the auxiliary field F cl using its equation of motion, allowing us to write the effective potential as Since σ cl is an auxiliary field we may use its equation ( ∂V ef f ∂σ cl = 0) to find σ cl = 2ρ 2 cl − µ 2 2 e −4πλ and write V ef f (φ cl , ρ cl ), from which we derive the conditions that extremize the effective potential: Solving these equations, we determine two critical points: where β = ηπ µ e 2πλ and W (β) is the Lambert's W -function, and we have used W (±β) = ±βe −W (β) . In order to determine if those critical points are minima, we compute the Hessian of V ef f (φ cl , ρ cl ): At the critical point, φ cl = 0, so we have: We can see that the condition det H > 0 is satisfied for 1 + 2πλ + ln Such solution is O(N ) symmetric, presenting a dynamical generation of mass to the fundamental matter fields φ and ψ, which are given by The Lambert's W -function assumes its lowest real value for β = −1/e, where the mass rate becomes M 2 ψ /M 2 φ = e −2 . In the Figure 2 we plot the mass rate M 2 φ /M 2 ψ as function of β. It is easy to see that if we take the limit η → 0 (β → 0), we recover the supersymmetric solution with M 2 φ = M 2 ψ .

III. FINAL REMARKS
Summarizing, we study the dynamical generation of mass in a deformed D = (1 + 1) supersymmetric nonlinear sigma model, where the deformation is introduced by imposing a supersymmetry breaking constraint. We showed that the generated masses to the matter fields φ and ψ are M 2 φ = µ 2 e −4πλ and M 2 ψ = e 2W (±β) M 2 φ , respectively. We see that supersymmetry is broken for a nonvanishing deformation parameter η, while O(N ) symmetry is kept manifest. Supersymmetry is restored in the limit η → 0.
As we have mentioned, in the manifest supersymmetric solution, the generated mass is due to ρ, which is not present in the non-supersymmetric model, while in the ordinary non-supersymmetric case the spontaneous generation of mass occurs due to σ acquiring a non-vanishing vacuum expectation value. We may say that we do not have anything that we can interpret as a non-supersymmetric limit of the spontaneous generation of mass from the supersymmetric model. In fact, even in the deformed model such limit is absent, however, the generated mass of fermionic fields are dependent of a nonvanishing σ .