Supersymmetry – Exercises

Ben Hoare

These exercises accompany the Supersymmetry course held in the Epiphany term of the 2020-2021 academic year as part of the Particles, Strings and Cosmology MSc degree at Durham University.

Please send comments and corrections to Ben Hoare at ben.hoare-at-durham.ac.uk.

Durham, 11 January 2021

last updated 1 June 2021

1 Introduction

2 Poincaré and Lorentz groups, and spinor representations

Check that the transformations \[x^\mu \to \Lambda^\mu{}_\nu x^\nu + c^\mu ~, \qquad \Lambda^\mu{}_\rho \Lambda^\nu{}_\sigma \eta_{\mu\nu} = \eta_{\rho\sigma} ~,\] are indeed isometries of the metric \[ds^2 = \eta_{\mu\nu} dx^\mu dx^\nu ~, \qquad \eta_{\mu\nu} = \mathop{\mathrm{diag}}(+1,-1,-1,-1)_{\mu\nu} ~,\] and that they form a group.

Show that \[\Lambda^\mu{}_\nu = \frac12 \mathop{\mathrm{tr}}(N \sigma_\nu N^\dagger \bar \sigma^\mu) ~, \qquad N \in \mathrm{SL}(2,\mathbb{C}) ~,\] defines a group homomorphism \(\mathrm{SL}(2,\mathbb{C}) \to \mathrm{SO}^+(1,3)\).

Show that \[i(\sigma^{\mu\nu})_{\alpha}{}^\beta = \frac{i}{4}(\sigma^\mu\bar\sigma^\nu - \sigma^\nu\bar\sigma^\mu)_{\alpha}{}^\beta ~, \qquad i(\bar\sigma^{\mu\nu})^{\dot\alpha}{}_{\dot\beta} = \frac{i}{4}(\bar\sigma^\mu\sigma^\nu - \bar\sigma^\nu\sigma^\mu)^{\dot\alpha}{}_{\dot\beta} ~,\] satisfy the commutation relations of the Lorentz algebra.

Prove the following identities for anticommuting spinors: \[\begin{aligned} (i): & \quad \psi^\alpha\psi^\beta = -\frac{1}{2}\epsilon^{\alpha\beta} \psi\psi ~, & (ii): & \quad \bar\psi^{\dot\alpha}\bar\psi^{\dot\beta} = \frac{1}{2}\epsilon^{\dot\alpha\dot\beta} \bar\psi\bar\psi ~, \\ (iii): & \quad (\theta\phi)(\theta\psi) = -\frac{1}{2}(\theta\theta)(\phi\psi) ~, & (iv): & \quad (\bar\theta\bar\phi)(\bar\theta\bar\psi) = -\frac{1}{2}(\bar\theta\bar\theta)(\bar\phi\bar\psi) ~, \\ (v): & \quad \chi\sigma^\mu \bar\psi = -\bar\psi \bar\sigma^\mu \chi ~, & (vi): & \quad \chi\sigma^\mu \bar\sigma^\nu \psi = \psi \sigma^\nu \bar\sigma^\mu \chi ~, \\ (vii): & \quad (\chi\sigma^\mu\bar\psi)^\dagger = \psi\sigma^\mu\bar\chi ~, & (viii): & \quad (\chi\sigma^\mu \bar\sigma^\nu \psi)^\dagger = \bar\psi\bar\sigma^\nu\sigma^\mu\bar\chi ~, \\ (ix): & \quad (\theta\psi)(\theta\sigma^\mu\bar\phi) = -\frac{1}{2}(\theta\theta)(\psi\sigma^\mu\bar\phi) ~, & (x): & \quad (\bar\theta\bar\psi)(\bar\theta\bar\sigma^\mu\phi) = -\frac{1}{2}(\bar\theta\bar\theta)(\bar\psi\bar\sigma^\mu\phi) ~, \\ (xi): & \quad (\phi\psi)(\bar\chi\bar\theta) = \frac{1}{2}(\phi\sigma^\mu\bar\chi)(\psi\sigma_\mu\bar\theta) ~, & (xii): & \quad (\theta\sigma^\mu\bar\theta)(\theta\sigma^\nu\bar\theta) = \frac{1}{2}(\theta\theta)(\bar\theta\bar \theta)\eta^{\mu\nu} ~. \end{aligned}\]

A Dirac spinor is made up of one left-handed and one right-handed Weyl spinor. In the Weyl basis we have \[\Psi = \begin{pmatrix} \psi_\alpha \\ \bar\chi^{\dot\alpha} \end{pmatrix} ~, \qquad \gamma^\mu = \begin{pmatrix} \mathbf{0}& \sigma^\mu \\ \bar \sigma^\mu & \mathbf{0}\end{pmatrix} ~.\] Check that the gamma matrices \(\gamma^\mu\) satisfy the Clifford algebra relations \(\{\gamma^\mu,\gamma^\nu\} = 2\eta^{\mu\nu}\mathbf{1}\). Express the Dirac Lagrangian \[\mathcal{L}= -i\bar\Psi\gamma^\mu\partial_\mu \Psi - m \bar\Psi\Psi ~,\] in terms of the Weyl spinors \(\psi\) and \(\bar\chi\). What happens when \(\Psi\) is taken the be a Majorana spinor?

Can you think of any fields that you have encountered that transform in the real irreducible representations \((1,0)\oplus(0,1)\) and \((1,1)\)?

A complex second rank antisymmetric tensor \(f^{\mu\nu} = - f^{\nu\mu}\) can be decomposed into a self-dual part \(f_+^{\mu\nu}\) and an anti-self-dual part \(F_-^{\mu\nu}\) \[f^{\mu\nu} = f_+^{\mu\nu} + f_-^{\mu\nu} ~, \qquad f_\pm^{\mu\nu} = \frac12 \Big(f^{\mu\nu} \pm \frac{i}{2}\epsilon^{\mu\nu\rho\sigma}f_{\rho\sigma}\Big) ~,\] where \(\epsilon^{\mu\nu\rho\sigma}\) is the Levi-Civita symbol with \(\epsilon^{0123} = 1\). The self-dual and anti-self-dual parts are therefore constrained such that \[f_\pm^{\mu\nu} = \pm \frac{i}{2}\epsilon^{\mu\nu\rho\sigma}f_\pm{}_{\rho\sigma} ~.\] Check that \(\sigma^{\mu\nu} = \frac{1}{4}(\sigma^\mu\bar\sigma^\nu - \sigma^\nu\bar\sigma^\mu)\) and \(\bar\sigma^{\mu\nu} = \frac{1}{4}(\bar\sigma^\mu\sigma^\nu - \bar\sigma^\nu\sigma^\mu)\) are self-dual and anti-self-dual respectively. Raising and lowering indices with \(\epsilon_{\alpha\beta}\), \(\epsilon_{\dot\alpha\dot\beta}\) and their inverses, show that \((\sigma^{\mu\nu})_{\alpha\beta}\) and \((\bar\sigma^{\mu\nu})^{\dot\alpha\dot\beta}\) realise the isomorphism between self-dual and anti-self-dual second-rank antisymmetric tensors and the \((1,0)\) and \((0,1)\) representations of \(\mathrm{SL}(2,\mathbb{C})\).

Show that \([W^2,P_\mu] = 0\) where \(W^\mu\) is the Pauli-Lubański vector \(W^\mu = \frac{1}{2}\epsilon^{\mu\nu\rho\sigma}P_\nu M_{\rho\sigma}\).

3 Supersymmetry algebras and supermultiplets

Using commutators (\([X,Y] = XY-YX\)) and anticommutators (\(\{X,Y\} = XY+YX\)), write down the four types of Jacobi identity.

Supersymmetric quantum mechanics with two supercharges (sometimes called \(\mathcal{N}=2\) or \(\mathcal{N}=(1,1)\)) has the superalgebra \[\begin{gathered} \{Q,\bar Q\} = 2H ~, \qquad \{Q,Q\} = Z ~, \qquad \{\bar Q,\bar Q\} = \bar Z ~, \\ \phantom{}[R,Q] = -Q ~, \qquad [R,\bar Q] = \bar Q ~, \qquad [R,Z] = -2Z ~, \qquad [R,\bar Z] = 2\bar Z ~, \end{gathered}\] where \(H\) is the Hamiltonian and all other commutators vanish. Check the graded Jacobi identity for this superalgebra.

Write down the content of the \(\mathcal{N}=2\) gravitino (\(\lambda_0 = \frac{1}{2}\)) and graviton (\(\lambda_0 = 1\)) multiplets. Decompose these supermultiplets in terms of \(\mathcal{N}=1\) massless supermultiplets.

Explicitly construct the most general massless supermultiplet in a SQFT with \(\mathcal{N}=3\) supersymmetry. What is the content of this supermultiplet? Show that it coincides with the content of the \(\mathcal{N} =4\) vector multiplet.

\(\mathcal{N}=8\) is special because there is a single massless supermultiplet for which all states have helicity \(|\lambda| \leq 2\). This is the \(\mathcal{N}=8\) graviton multiplet. Find the content of the \(\mathcal{N}=8\) graviton multiplet. Decompose the \(\mathcal{N}=8\) graviton multiplet in terms of \(\mathcal{N}=1\) massless supermultiplets.

Starting from the super-Poincaré algebra with \(\mathcal{N}=2\), fixing the Lorentz frame \(p_\mu = k_\mu = (m,0,0,0)_\mu\) and setting \(Z^{IJ} = \big(\begin{smallmatrix} 0 & z \\ -z & 0 \end{smallmatrix}\big)^{IJ}\), \(z\in\mathbb{R}\) and \(z\geq0\), check that the operators \[a_\alpha = \frac{1}{\sqrt{2}}\big(Q_\alpha^{1} + \epsilon_{\alpha}{}^{\beta} \bar Q_{2\beta}\big) ~,\qquad b_\alpha = \frac{1}{\sqrt{2}}\big(Q_\alpha^{1} - \epsilon_{\alpha}{}^{\beta} \bar Q_{2\beta}\big) ~, \qquad\] satisfy the anticommutation relations \[\{a_\alpha,a_\beta^\dagger\} = (2m+z)\delta_{\alpha\beta} ~, \qquad \{b_\alpha,b_\beta^\dagger\} = (2m-z)\delta_{\alpha\beta} ~,\] with all other anticommutators vanishing. Note that \(\epsilon_{1}{}^2 = - \epsilon_{2}{}^1 = - 1\), \(\epsilon_1{}^1 = \epsilon_2{}^2 = 0\) and we have dropped the distinction between dotted and undotted indices. Working in an eigenbasis of \(J_3 = M_{12}\), determine the effect of the fermionic annihilation and creation operators on the \(z\)-component of the spin (the eigenvalue of \(J_3\)).

Construct the physical states of the long and short \(\mathcal{N}=2\) massive vector multiplets in terms of oscillators.

4 Superspace and superfields

Starting from the differential operator realisation of \(P_\mu\), \(Q_\alpha\) and \(\bar Q_{\dot \alpha}\) \[P_\mu = -i\partial_\mu ~, \qquad Q_\alpha = - i \big(\partial_\alpha -i c (\sigma^\mu\bar\theta)_\alpha\partial_\mu\big) ~, \qquad \bar Q_{\dot\alpha} = i \big(\bar\partial_{\dot\alpha} - i \bar c (\theta \sigma^\mu)_{\dot\alpha}\partial_\mu\big) ~,\] where \(c\) and \(\bar c\) are free real constants, show that the anticommutation relations \[\{Q_\alpha,\bar Q_{\dot\alpha}\} = 2 (\sigma^{\mu})_{\alpha\dot\alpha} P_\mu ~, \qquad \{Q_\alpha, Q_{\beta}\} = \{\bar Q_{\dot\alpha}, \bar Q_{\dot\beta}\} = 0 ~,\] together with \((Q_\alpha)^\dagger = \bar Q_{\dot\alpha}\), imply that \(c=\bar c=1\). Show that the anticommutation relations are equivalent to \([\varepsilon_1 Q,\bar \varepsilon_2 \bar Q] = 2 \varepsilon_1\sigma^\mu\bar\varepsilon_2 P_\mu\) and \([\varepsilon_1 Q,\varepsilon_2 Q] = [\bar\varepsilon_1\bar Q,\bar\varepsilon_2\bar Q] = 0\).

Show that \(\partial^\alpha = -\epsilon^{\alpha\beta}\partial_\beta\) and \(\bar\partial^{\dot\alpha} = - \epsilon^{\dot\alpha\dot\beta} \bar\partial_{\dot\beta}\).

Count the number of bosonic and fermionic degrees of freedom of a general superfield \[\begin{split} Y(x,\theta,\bar\theta) & = y(x) + \theta\psi(x) + \bar\theta \bar\chi(x) + (\theta\theta) m(x) + (\bar\theta\bar\theta)\bar n(x) \\ & \quad + (\theta\sigma^\mu\bar\theta) v_\mu(x) + (\theta\theta) (\bar\theta\bar\lambda(x)) + (\bar\theta\bar\theta) (\theta \rho(x)) + (\theta\theta)(\bar\theta\bar\theta) D(x) ~. \end{split}\] Derive the supersymmetry transformations of the component fields by equating \[\delta_{\varepsilon,\bar\varepsilon} Y(x,\theta,\bar\theta) = i (\varepsilon Q + \bar\varepsilon \bar Q) Y(x,\theta,\bar\theta) ~,\] and \[\delta_{\varepsilon,\bar\varepsilon} Y(x,\theta,\bar\theta) = \delta_{\varepsilon,\bar\varepsilon} y(x) + \theta \delta_{\varepsilon,\bar\varepsilon} \psi(x) + \bar\theta \delta_{\varepsilon,\bar\varepsilon} \bar\chi(x) + \dots + (\theta\theta)(\bar\theta\bar\theta) \delta_{\varepsilon,\bar\varepsilon} D(x) ~,\] where \[Q_\alpha = - i \big(\partial_\alpha -i(\sigma^\mu\bar\theta)_\alpha\partial_\mu\big) ~, \qquad \bar Q_{\dot\alpha} = i \big(\bar\partial_{\dot\alpha} - i (\theta \sigma^\mu)_{\dot\alpha}\partial_\mu\big) ~.\] What is special about the variation of \(D\)?

Starting from the differential operators \[\begin{aligned} Q_\alpha & = -i\big( \partial_\alpha - i (\sigma^\mu\bar\theta)_\alpha \partial_\mu \big)~, \qquad & \bar Q_{\dot\alpha} & = i\big(\bar\partial_{\dot\alpha} - i (\theta\sigma^\mu)_{\dot\alpha} \partial_\mu \big) ~, \\ D_\alpha & = \partial_\alpha + i (\sigma^\mu\bar\theta)_\alpha \partial_\mu ~, \qquad & \bar D_{\dot\alpha} & = \bar\partial_{\dot\alpha} + i (\theta\sigma^\mu)_{\dot\alpha} \partial_\mu ~, \end{aligned}\] show that \[\begin{gathered} \{D_\alpha,Q_\beta\} = \{D_\alpha,\bar Q_{\dot\alpha}\} = \{\bar D_{\dot\alpha},Q_\alpha\} = \{\bar D_{\dot\alpha},\bar Q_{\dot\beta} \} = 0 ~, \\ \{D_\alpha,\bar D_{\dot\alpha}\} = 2 i (\sigma^\mu)_{\alpha\dot\alpha} \partial_\mu ~, \qquad \{D_\alpha,D_\beta\} = \{\bar D_{\dot\alpha},\bar D_{\dot\beta} \} = 0 ~. \end{gathered}\]

Show that \[\bar D_{\dot\alpha} y^\mu = D_{\alpha} \bar y^\mu = 0 ~,\] where \[\begin{aligned} y^\mu & = x^\mu + i \theta \sigma^\mu\bar \theta ~, & \qquad D_\alpha & = \partial_\alpha + i (\sigma^\mu\bar\theta)_\alpha \partial_\mu ~, \\ \bar y^\mu & = x^\mu - i \theta \sigma^\mu \bar \theta ~, & \qquad \bar D_{\dot\alpha} & = \bar\partial_{\dot\alpha} + i (\theta\sigma^\mu)_{\dot\alpha} \partial_\mu ~. \end{aligned}\]

Express the supercovariant derivatives \(D_\alpha\) and \(\bar D_{\dot\alpha}\) and the supercharges \(Q_\alpha\) and \(\bar Q_{\dot\alpha}\), realised as differential operators, \[\begin{aligned} D_\alpha & = \partial_\alpha + i (\sigma^\mu\bar\theta)_\alpha \partial_\mu ~, \qquad & \bar D_{\dot\alpha} & = \bar\partial_{\dot\alpha} + i (\theta\sigma^\mu)_{\dot\alpha} \partial_\mu ~, \\ Q_\alpha & = -i\big( \partial_\alpha - i (\sigma^\mu\bar\theta)_\alpha \partial_\mu \big)~, \qquad & \bar Q_{\dot\alpha} & = i\big(\bar\partial_{\dot\alpha} - i (\theta\sigma^\mu)_{\dot\alpha} \partial_\mu \big) ~, \end{aligned}\] in terms of derivatives with respect to \((y,\theta,\bar\theta)\) and \((\bar y,\theta,\bar\theta)\) where \[y^\mu = x^\mu + i \theta \sigma^\mu\bar \theta ~, \qquad \bar y^\mu = x^\mu - i \theta \sigma^\mu \bar \theta ~.\]

Starting from a chiral superfield expressed in terms of components as \[\Phi(y,\theta) = \phi(y) + \sqrt{2} \theta \psi(y) - (\theta\theta) F(y) ~,\] derive the component field expansion in terms of the coordinates \((x,\theta,\bar\theta)\) \[\begin{split} \Phi(x,\theta,\bar\theta) & = \phi(x) + i (\theta\sigma^\mu\bar\theta)\partial_\mu\phi(x) - \frac{1}{4}(\theta\theta)(\bar\theta\bar\theta) \partial_\mu\partial^\mu \phi(x) \\ & \quad + \sqrt{2}\theta\psi(x) -\frac{i}{\sqrt{2}}(\theta\theta)(\partial_\mu\psi(x) \sigma^\mu\bar\theta) -(\theta\theta) F(x) ~, \end{split}\] where \(y^\mu = x^\mu + i \theta \sigma^\mu\bar \theta\).

5 SQFTs of chiral multiplets

Compute the expansion in component fields of the superfield \(W(\Phi)\), where \(\Phi\) denotes a set of superfields \(\Phi^i\), and show that the coefficient of \(-\theta\theta\) is \[F_W = \partial_i W(\phi) F^i + \frac{1}{2}\partial_i\partial_jW(\phi)\psi^i\psi^j ~,\] where \(\partial_i = \frac{\partial}{\partial \Phi^i}\).

Show that the component of the canonical Kähler potential \(\bar\Phi_i\Phi^i\) proportional to \((\theta\theta)(\bar\theta\bar\theta)\) is \[D_K = \partial_\mu\bar\phi_i \partial^\mu\phi^i - i\bar\psi_i \bar\sigma^\mu \partial_\mu \psi^i + \bar F_i F^i + \text{total derivative} ~.\] Determine the total derivative.

Consider the Wess-Zumino model of a single chiral superfield \(\Phi\) with \[K(\Phi,\bar\Phi) = \bar\Phi\Phi ~, \qquad W(\Phi) = \frac{m}{2}\Phi^2 + \frac{\lambda}{3}\Phi^3 ~.\] Argue that this \(W(\Phi)\) is the most general renormalisable superpotential and find the supersymmetric vacua of the theory.

Express the action of the Wess-Zumino model in terms of component fields both before and after integrating out the auxiliary fields. Show that the effective physical mass of \(\phi\) is equal to that of \(\psi\) and is given by \(m_{\text{eff}}(\langle\phi\rangle) = m +2 \lambda\langle\phi\rangle\) when the action is expanded around one of the supersymmetric vacua. How is the quartic coupling in the scalar potential related to the Yukawa coupling? Interpret these two results.

Derive the equations of motion for the component fields \(\phi\), \(\psi\) and \(F\) directly from the action expressed in terms of component fields. Finally, expand the equation of motion for the chiral superfield \(\Phi\) and show it is equivalent to the equations of motion for the component fields.

Consider the Wess-Zumino model of three chiral superfields \(X\), \(Y\) and \(Z\) with the homogeneous trilinear superpotential \(W(X,Y,Z) = \lambda XYZ\). Write down the scalar potential of the theory. Find the moduli space of supersymmetric vacua.

Argue that the superpotentials \(W(\Phi) = \sum_{n=2}^N \lambda^n \Phi^n\) and \(W(\Phi) = \frac{1}{2} m_{ij} \Phi^i\Phi^j + \frac{1}{3} \lambda_{ijk}\Phi^i\Phi^j\Phi^k\) are not renormalised in an SQFT of chiral multiplets.

6 Supersymmetric gauge theories

Starting from an abelian vector superfield \(V\) and its supersymmetric gauge transformation \(V \to V + \Lambda + \bar\Lambda\), where \(\Lambda\) is a chiral superfield, determine the gauge transformations of the component fields of \(V\).

Fixing the Wess-Zumino (WZ) gauge \(V = V_{\text{WZ}}\), compute the supersymmetry transformation of \(V_{\text{WZ}}\) and determine the compensating gauge transformation that is needed to restore the WZ gauge.

Derive the component field expansion of the gaugino superfield \(W_\alpha = -\frac{1}{4} \bar D^2 D_\alpha V\) in terms of the coordinates \((y,\theta,\bar\theta)\) using that \(W_\alpha\) is gauge invariant.

The field content of an abelian supersymmetric gauge theory consists of \(r\) abelian vector superfields \(V_a\), \(a=1,\dots,r\), and \(N\) chiral superfields \(\Phi^i\), \(i=1,\dots,N\), with charges \([\Phi^i]_{_{Q_a}} = q^i_a\) under the \(\mathrm{U}(1)^r\) gauge group.

Write down the component field expansion of the most general supersymmetric action \[\begin{split} \mathcal{S}& = \mathcal{S}_{\text{Maxwell}} + \mathcal{S}_{\text{FI}} + \mathcal{S}_{\text{matter}} + \mathcal{S}_{\text{W}} ~, \\ \mathcal{S}_{\text{Maxwell}} & = \sum_{a=1}^r \mathop{\mathrm{Im}}\Big(\int d^4x d^2\theta \, \frac{\tau_a}{8\pi} W_a^{\alpha} W_{a\alpha} \Big) ~, \qquad \tau_a = \frac{\theta_a}{2\pi} + \frac{4\pi i}{g_a^2} ~, \\ \mathcal{S}_{\text{FI}} & = -2 \sum_{a=1}^r \xi_a \int d^4x d^2\theta d^2\bar\theta \, V_a ~, \\ \mathcal{S}_{\text{matter}} & = \sum_{i=1}^N \int d^4x d^2\theta d^2\bar\theta \, \bar \Phi_i e^{2\sum_{a=1}^r q_a^i V_a} \Phi^i ~, \\ \mathcal{S}_{\text{W}} & = \int d^4x d^2\theta \, W(\Phi) + \int d^4x d^2\bar \theta \, \bar W(\bar\Phi) ~, \end{split}\] where the superpotential \(W(\Phi)\) is a gauge invariant holomorphic function of \(\Phi^i\), i.e. \([W]_{_{Q_a}} = 0\) for all \(a=1,\dots r\).

Integrate out the auxiliary fields \(F^i\), \(\bar F_i\) and \(D_a\) in the chiral and abelian vector superfields, and derive the scalar potential \[\begin{split} V(\phi,\bar\phi) & = \sum_{i=1}^N \bar F_i F^i + \sum_{a=1}^r \frac{1}{2g_a^2} (D_a)^2 \\ & = \sum_{i=1}^N \partial_i W(\phi) \bar\partial^i \bar W(\bar\phi) + \sum_{a=1}^r \frac{g_a^2}{2}(\mu_a(\phi,\bar\phi)-\xi_a)^2 ~, \qquad \mu_a(\phi,\bar\phi) = \sum_{i=1}^N q_a^i \bar\phi_i\phi^i ~. \end{split}\]

Supersymmetric quantum electrodynamics (SQED) with one flavour is a supersymmetric \(\mathrm{U}(1)\) gauge theory with matter fields, i.e. chiral superfields, \(Q\) of charge \(1\) and \(\tilde Q\) of charge \(-1\). Its action is given by \[\begin{split} \mathcal{S}& = \mathop{\mathrm{Im}}\Big(\int d^4 x d^2\theta \, \frac{\tau}{8\pi} W^\alpha W_\alpha \Big) + \int d^4x d^2\theta d^2\bar\theta \, \big(\bar Q e^{2V} Q + \bar{\tilde{Q}} e^{-2V} \tilde{Q} - 2\xi V\big) \\ & \quad + \int d^4 x d^2\theta \, m \tilde{Q} Q + \int d^4x d^2\bar\theta \, \bar m \bar{\tilde{Q}} \bar{Q} ~, \end{split}\] where \(m\) is a complex mass parameter, \(\tau = \frac{4\pi i}{g^2} + \frac{\theta}{2\pi}\) is the complexified gauge coupling, \(\xi\) is the real Fayet-Iliopoulos (FI) parameter, and \(W_\alpha = - \frac{1}{4} \bar D^2 D_\alpha V\) is the gaugino (or photino) superfield.

Derive the scalar potential of this theory in terms of the component fields of the chiral superfields \(Q\) and \(\tilde{Q}\). Determine the moduli space of supersymmetric vacua for

Determine the vacuum expectation values of the gauge invariant chiral operator \(M = \tilde{Q} Q\) in these supersymmetric vacua. You may use that unbroken supersymmetry implies that \(\langle XY \rangle = \langle X\rangle\langle Y \rangle\) for any two chiral superfields \(X\) and \(Y\).

A non-abelian vector superfield \(V\) transforms under supersymmetric gauge transformations as \[e^{2V} \to e^{i\bar\Delta} e^{2V} e^{-i\Delta} ~,\] where \(\Delta\) is a chiral superfield valued in the Lie algebra of the gauge group, so that \(e^{i\Delta}\) is a chiral superfield valued in the gauge group itself. \(\bar\Delta = \Delta^\dagger\) is the conjugate antichiral superfield.

Check that the gaugino superfield \[W_\alpha = -\frac{1}{8} \bar D^2(e^{-2V}D_\alpha e^{2V}) ~,\] transforms as \(W_\alpha \to e^{i\Delta} W_\alpha e^{-i \Delta}\) under supersymmetric gauge transformations.

Derive the component field expansion of \(W_\alpha\) in the Wess-Zumino gauge \[W_{\text{WZ}}{}_\alpha(y,\theta) = - i \lambda_\alpha(y) + \theta_\alpha D(y) + i (\sigma^{\mu\nu}\theta)_\alpha F_{\mu\nu}(y) + (\theta\theta)(\sigma^\mu D_\mu \bar \lambda(y))_\alpha ~,\] where \[F_{\mu\nu}(y) = \hat\partial_\mu A_\nu(y) - \hat\partial_\nu A_\mu(y) - i [A_\mu(y),A_\nu(y)] ~, \qquad D_\mu \bar\lambda(y) = \hat\partial_\mu \bar\lambda(y) - i [A_\mu(y),\bar\lambda(y)] ~,\] are the non-abelian field strength and the gauge covariant derivative in the adjoint representation of \(\bar\lambda\) respectively. Recall that \(\hat\partial_\mu\) just denotes the partial derivative with respect to \(y^\mu\).

7 Spontaneous supersymmetry breaking

The O’Raifeartaigh model is a Wess-Zumino model of three chiral superfields with superpotential \[W(X,Y,Z) = Y(a^2 - X^2) + b XZ ~,\] where \(a\) and \(b\) are non-zero complex parameters. Show that supersymmetry is spontaneously broken. Assuming \(2|a|^2<|b|^2\), find the global minimum (or minima) of the scalar potential and determine the vacuum expectation values of the scalar potential.

The Polonyi model is a Wess-Zumino model of a single chiral superfield \(X\) with linear superpotential \[W(X) = f X ~,\] where \(f\) is a non-zero complex parameter. The Polonyi model can be modified by adding a small cubic term to the superpotential \[W_\epsilon (X) = f X + \epsilon \frac{\lambda}{3} X^3 ~, \qquad 0 < \epsilon \ll 1 ~,\] where \(\lambda\) is a non-zero complex parameter. Show that supersymmetry is broken spontaneously in the Polonyi model and compute the mass spectrum of bosons and fermions. Show that supersymmetry is restored in the modified Polonyi model and that \(\langle \phi^X \rangle = 0\) is an unstable vacuum, i.e. a stationary point of the potential that is not a minimum. What happens to the supersymmetric vacua and to the scalar potential of the modified Polonyi model in the limit \(\epsilon \to 0\)?

Consider massive SQED with matter fields \(Q\) of charge 1 and \(\tilde{Q}\) of charge \(-1\), and a non-zero FI term. Assume that the complex mass \(m \neq 0\) and that the FI parameter \(\xi > 0\). Show that \(F^Q\), \(F^{\tilde{Q}}\) and \(D\) cannot be set to zero at the same time and explain why this implies that supersymmetry is broken spontaneously.

Let \(X = \frac{g^2\xi}{|m|^2}\). Find the global minimum of the scalar potential and determine the vacuum expectation values \(\langle Q\rangle\) and \(\langle\tilde{Q}\rangle\), up to gauge freedom, as functions of \(m\), \(g\) and \(X\). For \(X \leq 1\) and \(X > 1\) determine whether D-term or F-term spontaneous supersymmetry breaking occurs, or both.

Calculate the mass spectra of bosons and fermions and identify the Goldstino, i.e. the massless Goldstone fermion associated to spontaneously broken supersymmetry, in the two regimes \(X \leq 1\) and \(X > 1\).