15.2. Supersymmetric Gauge Theories

Transcription

1 LINEAR SIGMA MODELS 15.. Supersymmetric Gauge Theories We will now consider supersymmetric linear sigma models. First we must introduce the supersymmetric version of gauge field, gauge transformation, gauge invariant Lagrangian, etc. We consider here only the case where the gauge group is abelian Vector Multiplet. We first recall how we introduced gauge symmetries and gauge fields. Let us consider a field theory of a complex scalar field φ(x with the Lagrangian (15.18 L = µ φ. This Lagrangian is invariant under the phase rotation (15.19 φ(x e iα φ(x, where α is a constant. If we let α depend on x, the derivative µ φ transforms to e iα ( µ +i µ αφ and the Lagrangian shown in Eq. (15.18 is not invariant under the phase rotation. However, if we introduce a vector field (oneform field v µ that transforms as v µ v µ µ α, the modified derivative D µ φ := ( µ + iv µ φ transforms under the phase rotation as (15.0 D µ φ(x e iα(x D µ φ(x and the Lagrangian (15.1 L = D µ φ is invariant. Now let us consider a supersymmetric field theory of a chiral superfield Φ with the Lagrangian (15. L = d 4 θ ΦΦ, which is invariant under the constant phase rotation Φ e iα Φ. If we replace α by a chiral superfield A = A(x µ, θ ±, θ ±, the transformation (15.3 Φ e ia Φ sends a chiral superfield to a chiral superfield. However, ΦΦ transforms to Φ e ia+ia Φ and the Lagrangian is not invariant. Now, as in the case above, we introduce a real superfield V = V (x µ, θ ±, θ ± that transforms as (15.4 V V + i(a A.

2 15.. SUPERSYMMETRIC GAUGE THEORIES 349 Then the modified Lagrangian (15.5 L = d 4 θ Φ e V Φ is invariant under the transformation Eqs. (15.3 (15.4. A real scalar superfield V that transforms as in Eq. (15.4 under a gauge transformation is called a vector superfield. Using the gauge transformations one can eliminate the lower components of the theta-expansion of V and express it in the form (15.6 V =θ θ (v 0 v 1 + θ + θ + (v 0 + v 1 θ θ + σ θ + θ σ + iθ θ + (θ λ + θ + λ + + iθ + θ (θ λ + θ + λ + + θ θ + θ + θ D. Exercise Show that by a suitable gauge transformation V can be expressed as above. Since V is a Lorentz singlet, v 0 and v 1 define a one-form field, σ defines a complex scalar field, λ ± and λ ± define a Dirac fermion field, and D is a real scalar field. The gauge in which V is represented as Eq. (15.6 is called the Wess Zumino gauge. The residual gauge symmetry (gauge transformations that keep the form Eq. (15.6 is the one with A = α(x µ which transforms (15.7 v µ (x v µ (x µ α(x, with all the other component fields unchanged. The supersymmetry variation is given by δ = ϵ + Q ϵ Q + ϵ + Q + ϵ Q + where Q ± and Q ± are the differential operators given in Eqs. (1.6 (1.7. The Wess Zumino gauge is not in general preserved by this variation. In order to find the supersymmetry transformation of the component fields σ, λ ±, v µ and D, we need to amend it with a gauge transformation that brings δv back into the Wess Zumino gauge. It turns out that the required gauge transformation is the one with (15.8 A =iθ + (ϵ + σ + ϵ (v 0 + v 1 iθ (ϵ σ + ϵ + (v 0 v 1 + θ + θ ( ϵ λ + ϵ + λ +, where + are the derivative terms to make A chiral. In this way we find the following supersymmetry transformation for the component fields

3 LINEAR SIGMA MODELS in Wess Zumino gauge: For the vector multiplet fields it is δ v ± = iϵ ± λ ± + iϵ ± λ ±, δσ = iϵ + λ iϵ λ +, δd = ϵ + λ + ϵ + λ + ϵ + λ + + ϵ + λ, δλ + = iϵ + (D + iv 01 + ϵ + σ, δλ = iϵ (D iv 01 + ϵ + σ. For the charged chiral multiplet fields it is δφ = ϵ + ψ ϵ ψ +, δψ + = iϵ (D 0 + D 1 φ + ϵ + F ϵ + σφ, δψ = iϵ + (D 0 D 1 φ + ϵ F + ϵ σφ, δf = iϵ + (D 0 D 1 ψ + iϵ (D 0 + D 1 ψ +ϵ + σψ + ϵ σψ + + i(ϵ λ + ϵ + λ φ, where D µ φ and D µ ψ ± are the covariant derivatives (15.9 D µ := µ + iv µ, with respect to the connection defined by v µ. The superfield (15.30 Σ := D + D V is invariant under the gauge transformation V V + i(a A. twisted chiral superfield It is a (15.31 D + Σ = D Σ = 0 which is expressed as (15.3 Σ = σ(ỹ + iθ + λ + (ỹ iθ λ (ỹ + θ + θ [D(ỹ iv 01 (ỹ], in terms of the component fields in the Wess Zumino gauge as shown by Eq. (15.6. In the above expressions ỹ ± := x ± iθ ± θ ± and v 01 is the field-strength of v µ (or the curvature (15.33 v 01 := 0 v 1 1 v 0. The superfield Σ is called the super-field-strength of V.

4 15.. SUPERSYMMETRIC GAUGE THEORIES Supersymmetric Lagrangians. Let us present a supersymmetric Lagrangian for the vector multiplet V and the charged chiral multiplet Φ. The gauge invariant Lagrangian in Eq. (15.5 is supersymmetric. In terms of the component fields it is written as L kin = d 4 θ Φ e V Φ (15.34 = D µ φd µ φ + iψ (D 0 + D 1 ψ + iψ + (D 0 D 1 ψ + + D φ + F σ φ ψ σψ + ψ + σψ iφλ ψ + + iφλ + ψ + iψ + λ φ iψ λ + φ. This contains the kinetic terms for the fields φ and ψ ±. They are minimally coupled to the gauge field v µ via the covariant derivative in Eq. (15.9. They are also coupled to the scalar and fermionic components of the vector mutiplet. The kinetic terms for the vector multiplet fields can be described in terms of the super-field-strength Σ as (15.35 L gauge = 1 e d 4 θ ΣΣ = 1 e ( µ σ µ σ + iλ ( λ + iλ + ( 0 1 λ + + v 01 + D. Here e is the gauge coupling constant and has dimensions of mass. One can also write twisted F-terms for twisted superpotentials involving Σ. The twisted superpotential that will be important later is the linear one (15.36 WFI,θ = tσ where t is a complex parameter (15.37 t = r iθ. The twisted F-term is written as (15.38 L FI,θ = 1 ( t d θ Σ + c.c. = rd + θv 01. The parameter r is called the Fayet Iliopoulos parameter and θ is called the theta angle. These are dimensionless parameters.

5 LINEAR SIGMA MODELS Now let us consider a supersymmetric and gauge invariant Lagrangian which is simply the sum of the above three terms ( (15.39 L = d 4 θ Φ e V Φ 1 e ΣΣ + 1 ( t d θ Σ + c.c.. This Lagrangian is invariant under the vector and axial R-rotations under assigning the U(1 V U(1 A charges (0, to Σ. Thus the classical system has both U(1 V and U(1 A R-symmetries. The fields D and F have no kinetic term and can be eliminated using the equation of motion. After elimination of these auxiliary fields we obtain the Lagrangian for the other component fields L = D µ φd µ φ + iψ (D 0 + D 1 ψ + iψ + (D 0 D 1 ψ + (15.40 e ( φ r σ φ ψ σψ + ψ + σψ iφλ ψ + + iφλ + ψ + iψ + λ φ iψ λ + φ + 1 ( µ e σ µ σ + iλ ( λ + iλ + ( 0 1 λ + + v01 + θv 01. In particular the potential energy for the scalar fields φ and σ is given by (15.41 U = σ φ + e (. φ r It is straightforward to generalize the above construction to the cases where there are many U(1 gauge groups and many charged matter fields. Suppose the gauge group is U(1 k = k a=1 U(1 a, and there are N matter fields Φ i, i = 1,..., N, with charges Q ia under the group U(1 a (meaning Φ i e iq iaa a Φ i. Then the generalization of the above Langrangian is N L = d 4 θ Φ i e Q iav a 1 Φ i Σ a Σ b (15.4 i=1 + 1 ( d θ e a,b=1 a,b ( t a Σ a + c.c., a=1 where in the exponent of the Φ i kinetic term the sum over a = 1,..., k is assumed. This is invariant under U(1 V U(1 A R-rotations under the charge assignment (0, to each Σ a. If one can find a polynomial W (Φ i of Φ i which is invariant under the gauge transformations, one can also find an

6 15.3. RENORMALIZATION AND AXIAL ANOMALY 353 F-term (15.43 L W = d θ W (Φ i + c.c. The Lagrangian L + L W is still U(1 A -invariant but U(1 V -invariance holds only if W (Φ i is quasi-homogeneous. After eliminating the auxiliary fields D a and F i, we obtain the Lagrangian with the potential energy for the scalar fields being U = (15.44 N Q ia σ a φ i + i=1 + W φ i i=1, a,b=1 (e a,b ( Qia φ i ( r a Qjb φ j r b where (e a,b is the inverse matrix of 1/e a,b and the summations over a and i, j are implicit Renormalization and Axial Anomaly Let us consider the simplest model U(1 gauge theory with a single chiral superfield of charge 1. We consider here the effective theory at a high but finite energy scale µ. This is obtained by integrating out the modes of the fields with the frequencies in the range µ k Λ UV, where Λ UV is the ultraviolet cut-off. Let us look at the terms in the Lagrangian involving the D field ( e D + D( φ r 0. Here r 0 is the FI parameter at the cut-off scale. Integrating out the modes of φ, the term D φ is replaced by D φ, where φ is the one-point correlation function of φ. The φ-propagator can be read from the φ- quadratic term in the action 4 1 π d xφ D µ D µ φ and is given by φ(xφ d k π (y = (π k. Thus, the one-point function in question is (15.46 φ d k π = (π k = log µ k Λ UV ( ΛUV µ. 4 We choose the action here to be related to the Lagrangian by S = 1 π d xl.