Lecture 12 Holomorphy: Gauge Theory

Lecture 12 Holomorphy: Gauge Theory

Outline SUSY Yang-Mills theory as a chiral theory: the holomorphic coupling and the holomorphic scale. Nonrenormalization theorem for SUSY YM: the gauge coupling runs perturbatively at one loop only, and nonperturbative corrections have a specific form. Nonperturbative instanton effects break R-symmetry explicitly to Z 2N. Gaugino condensation further breaks R-symmetry spontaneously to Z 2. Reading: Terning 8.4, 8.5

The Holomorphic Gauge Coupling Reminder: the chiral superfield for an SU(N) gauge supermultiplet (with adjoint index a = 1,..., N 2 1): W a α = iλ a α(y) + θ α D a (y) (σ µν θ) α F a µν(y) (θθ)σ µ D µ λ a (y), SUSY Yang Mills action as an integral over chiral superspace: Note: L SYM = 1 16πi d 4 x d 2 θ τ WαW a α a + h.c. = [ ] d 4 x 1 4g F aµν F a 2 µν θ YM 32π F aµν a F 2 µν + i g λ a σ µ D 2 µ λ a + 1 2g D a D a 2 the YM coupling g appears only through the holomorphic coupling: τ θ YM 2π + 4πi g 2. Gauge fields are not canonically normalized.

Running Coupling The one-loop running g is given by the RG equation: µ dg dµ = b 16π 2 g 3, where for an SU(N) gauge theory with F flavors and N = 1 SUSY, b = 3N F. The solution for the running coupling is 1 g 2 (µ) = b 8π 2 ln ( Λ µ. ) The integration constant Λ is called the intrinsic scale of the non- Abelian gauge theory

Holomorphic Intrinsic Scale The running of the holomorphic coupling at one loop: τ 1 loop = θ YM 2π + 4πi g 2 (µ) [ ( ) ] b = 1 2πi ln Λ µ e iθ YM ( ) = b 2πi ln Λ µ. We introduced the holomorphic intrinsic scale Λ Λ Λ e iθ YM/b = µe 2πiτ/b. The holomorphic intrinsic scale will play a central role in the analysis of SUSY gauge theory.

Recall: the θ YM -term CP Violating Term θ YM 32π 2 F aµν F a µν = θ YM 32π 2 4ɛ µνρσ µ Tr ( A ν ρ A σ + 2 3 A νa ρ A σ ), has no effect in perturbation theory, since it is a total derivative. Corollary: the coupling θ YM is not renormalized in perturbation theory.

Non-Perturbative Effects Nonperturbative effects: instantons have a nontrivial topological winding number, The path integral n = 1 32π 2 d 4 x F aµν F a µν. DA a Dλ a DD a e is, integrates over all field configurations, including those with nontrivial instanton number. The nonperturbative quantum theory thus depends on θ YM, but in a manner such that there remains an exact quantum symmetry: θ YM θ YM + 2π.

Instanton Action The Euclidean action of an instanton configuration saturates the bound 0 ( d 4 x Tr F µν ± F ) 2 ( µν = d 4 x Tr 2F 2 ± 2F F ) d 4 x TrF 2 d 4 x TrF F = 16π 2 n. Instanton effects are thus suppressed by e S int = e (8π2 n/g 2 (µ))+inθ YM = ( Λ µ ) bn. In summary: Instanton effects appear only through the holomorphic scale Λ. The power b in Λ b is required in order that the periodicity θ YM θ YM + 2π is preserved. The arbitrary positive integer n is the instanton number.

Effective Superpotential Starting from SUSY YM theory at a high scale, integrate down to the scale µ and find an effective (Wilsonian) superpotential with a renormalized coupling: W eff = τ(λ;µ) 16πi W a αw a α. Periodicity θ YM θ YM + 2π of θ YM is equivalent to Λ e 2πi/b Λ. So the general form of the effective holomorphic coupling: ( ) τ(λ; µ) = b 2πi ln Λ µ + f(λ; µ) ( ) = b 2πi ln Λ µ + ( ) bn n=1 a Λ n µ. Conclusion: the holomorphic gauge coupling only receives one-loop corrections and nonperturbative n-instanton corrections. So there is no perturbative running beyond one-loop.

U(1) R Symmetry of SUSY YM U(1) R symmetry: a chiral rotation of the superspace coordinate. For pure SUSY YM, the gaugino is the only chiral particle so U(1) R is equivalent to the chiral rotation λ a e iα λ a This is a classical symmetry but, because of the chiral anomaly, in SU(N) SYM it is equivalent to the shift θ YM θ YM 2Nα. The 2N is because the gaugino λ a is in the adjoint representation and so the anomaly index 2T (Ad) = 2N. Thus: chiral rotation is only a symmetry of SUSY YM when α = kπ N U(1) R is broken to a discrete Z 2N subgroup!

Instanton Breaking Alternative view: a chiral rotation transforms θ YM as θ YM θ YM 2Nα, because there are 2N zero modes in an instanton background. Thus: it is instantons that break U(1) R to its Z 2N subgroup.

Spurion Analysis Treat τ as a spurion chiral superfield and define spurious symmetry λ a e iα λ a, τ τ + Nα π. Assume that SUSY YM has no massless particles: then effective low energy superpotential is just some function of the coupling. As always, the superpotential has charge 2 under the spurious R-symmetry W eff e 2iα W eff, and it must be a holomorphic function of τ. This determines the effective superpotential uniquely (constant a still to be determined): W eff = aµ 3 e 2πiτ/N = aλ 3.

Gaugino Condensation The SUSY YM Lagrangian shows that F τ spurion field τ) acts as a source for λ a λ a : (the F component of the L SYM = 1 16πi d 2 θ τ W a αw a α + h.c. ; W a α = iλ a α +.... The gaugino condensate in the effective low energy theory (recall b = 3N): λ a λ a = 16πi F τ ln Z = 16πi F τ d 2 θw eff = 16πi τ W eff = 16πi 2πi N aµ3 e 2πiτ/N = 32π2 N a(µe2πiτ/b ) 3 = 32π2 N aλ3. Thus: the ground state of pure SUSY YM has a gaugino condensate!

Spontaneous Symmetry Breaking Reminder: for SUSY YM, the chiral anomaly breaks U(1) R to a discrete Z 2N subgroup. However, among the 2N values, k = 0,..., 2N 1, the gaugino condensate is only invariant for k = 0 or k = N λ a λ a e 2iα λ a λ a, α = kπ N. Thus, there is spontaneous symmetry breaking: the vacuum does not respect the discrete Z 2N symmetry of the quantum theory. Performing a chiral transformation by e iπ/n sweeps out the N vacua when repeated N consecutive times (each time is equivalent to θ YM θ YM + 2π. Physical distinction between N degenerate vacua: there are N different phases for λ a λ a.

Domain Walls Physical consequence of spontaneously broken discrete symmetry: domain walls. Suppose the phase of λ a λ a 0 for x and π N These are two distinct vacua. as x +. At intermediate positions the field configuration must be nontrivial: there is energy localized near x = 0. This configuration is a domain wall solution. Domain walls are completely stable because relaxation to vacuum requires tunneling of infinite volume at one of the asymptotic spaces.

The Glueball Field Another perspective: introduce the glueball field with VEV The superpotential W Λ 3 b log Λ) W (S) = S S = W α W α, S = λ a λ a. has Legendre transform (with source [ ( ) ] Λ log 3N + N S N This is the Veneziano-Yankielowicz superpotential. The VY superpotential encodes the vacuum structure in that. W S = 0 S = e 2πik N Λ 3

The Witten Index In any SUSY theory it is illuminating to compute the Witten index: I W = Tr[( ) F ]. Fermion-Boson degeneracy: any boson B has fermion partner F = Q B, except for SUSY ground states. So the Witten index counts the number of SUSY ground states! Compute the Witten index by exploiting β-independence of I W = Tr[e βh ( ) F ]. For β : projection onto the SUSY ground states. For β 0: high temperature phase well approximated by path integral in the Gaussian approximation. The result for pure SYM: I W = N.