Lecture 12 Holomorphy: Gauge Theory Outline • SUSY Yang-Mills theory as a chiral theory: the holomorphic cou- pling and the holomorphic scale. • Nonrenormalization theorem for SUSY YM: the gauge coupling runs perturbatively at one loop only, and nonperturbative correc- tions have a specific form. • Nonperturbative instanton effects break R-symmetry explicitly to Z2N . • Gaugino condensation further breaks R-symmetry spontaneously to Z2. 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): a a a µν a µ ay Wα = −iλα(y) + θαD (y) − (σ θ)αFµν (y) − (θθ)σ Dµλ (y) ; SUSY Yang{Mills action as an integral over chiral superspace: 1 R 4 R 2 a a LSYM = d x d θ τ W W + h:c: h16πi α α i R 4 1 aµν a θYM aµν a i ay µ a 1 a a = d x − 4g2 F Fµν − 32π2 F Feµν + g2 λ σ Dµλ + 2g2 D D Note: • the YM coupling g appears only through the holomorphic coupling: θYM 4πi τ ≡ 2π + g2 : • Gauge fields are not canonically normalized. Running Coupling The one-loop running g is given by the RG equation: dg b 3 µ dµ = − 16π2 g ; 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 b jΛj g2(µ) = − 8π2 ln µ : The integration constant jΛj is called the intrinsic scale of the non- Abelian gauge theory Holomorphic Intrinsic Scale The running of the holomorphic coupling at one loop: θYM 4πi τ1−loop = 2π + g2(µ) b 1 jΛj iθYM = 2πi ln µ e b Λ = 2πi ln µ : We introduced the holomorphic intrinsic scaleΛ Λ ≡ jΛjeiθYM=b = µe2πiτ=b : The holomorphic intrinsic scale will play a central role in the analysis of SUSY gauge theory. CP Violating Term Recall: the θYM-term θYM aµν a θYM µνρσ 2 32π2 F Feµν = 32π2 · 4 @µTr Aν @ρAσ + 3 Aν AρAσ ; has no effect in perturbation theory, since it is a total derivative. Corollary: the coupling θYM is not renormalized in perturbation theory. Non-Perturbative Effects Nonperturbative effects: instantons have a nontrivial topological winding number, 1 R 4 aµν a n = 32π2 d x F Feµν : The path integral R DAaDλaDDa eiS ; integrates over all field configurations, including those with nontrivial instanton number. The nonperturbative quantum theory thus depends on θYM, but in a manner such that there remains an exact quantum symmetry: θYM ! θYM + 2π : Instanton Action The Euclidean action of an instanton configuration saturates the bound 2 R 4 R 4 2 0 ≤ d x Tr Fµν ± Feµν = d x Tr 2F ± 2F Fe R 4 2 R 4 2 ) d x TrF ≥ j d x TrF Fej = 16π jnj : Instanton effects are thus suppressed by 2 2 bn −Sint −(8π n=g (µ))+inθYM Λ e = e = µ : In summary: • Instanton effects appear only through the holomorphic scale Λ. b • The power b in Λ is required in order that the periodicity θYM ! θYM + 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 renormal- ized coupling: τ(Λ;µ) a a Weff = 16πi Wα Wα : Periodicity θYM ! θYM + 2π of θYM is equivalent to Λ ! e2πi=bΛ : So the general form of the effective holomorphic coupling: b Λ τ(Λ; µ) = 2πi ln µ + f(Λ; µ) bn b Λ P1 Λ = 2πi ln µ + n=1 an µ : Conclusion: the holomorphic gauge coupling only receives one-loop cor- rections 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 λa ! eiαλa This is a classical symmetry but, because of the chiral anomaly, in SU(N) SYM it is equivalent to the shift θYM ! θYM − 2Nα : The \2N" is because the gaugino λa is in the adjoint representation and so the anomaly index 2T (Ad) = 2N. Thus: chiral rotation is only a symmetry of SUSY YM when kπ α = N ) U(1)R is broken to a discrete Z2N subgroup! Instanton Breaking Alternative view: a chiral rotation transforms θYM as θYM ! θYM − 2Nα ; because there are 2N zero modes in an instanton background. Thus: it is instantons that break U(1)R to its Z2N subgroup. Spurion Analysis Treat τ as a spurion chiral superfield and define spurious symmetry a iα a Nα λ ! e λ ; τ ! τ + 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 2iα Weff ! e Weff ; and it must be a holomorphic function of τ. This determines the effective superpotential uniquely (constant a still to be determined): 3 2πiτ=N 3 Weff = aµ e = aΛ : Gaugino Condensation The SUSY YM Lagrangian shows that Fτ (the F component of the spurion field τ) acts as a source for λaλa: 1 R 2 a a a a LSYM = 16πi d θ τ Wα Wα + h:c: ; Wα = −iλα + :::: The gaugino condensate in the effective low energy theory (recall b = 3N): hλaλai = −16πi @ ln Z @Fτ @ R 2 = 16πi d θWeff @Fτ @ = 16πi @τ Weff 2πi 3 2πiτ=N = 16πi N aµ e 32π2 2πiτ=b 3 = − N a(µe ) 32π2 3 = − N aΛ : 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 Z2N subgroup. However, among the 2N values, k = 0;:::; 2N − 1, the gaugino conden- sate is only invariant for k = 0 or k = N a a 2iα a a kπ hλ λ i ! e hλ λ i ; α = N : Thus, there is spontaneous symmetry breaking: the vacuum does not respect the discrete Z2N symmetry of the quantum theory. Performing a chiral transformation by eiπ=N sweeps out the N vacua when repeated N consecutive times (each time is equivalent to θYM ! θYM + 2π. Physical distinction between N degenerate vacua: there are N different phases for hλaλai. Domain Walls Physical consequence of spontaneously broken discrete symmetry: do- main walls. a a π Suppose the phase of hλ λ i ! 0 for x ! −∞ and ! N as x ! +1. These are two distinct vacua. 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 re- quires tunneling of infinite volume at one of the asymptotic spaces. The Glueball Field Another perspective: introduce the glueball field α S = −WαW ; with VEV hSi = hλaλai : The superpotential W ∼ Λ3 has Legendre transform (with source ∼ b log Λ) h Λ3N i W (S) = S log SN + N : This is the Veneziano-Yankielowicz superpotential. The VY superpotential encodes the vacuum structure in that 2πik @W N 3 @S = 0 $ S = e Λ The Witten Index In any SUSY theory it is illuminating to compute the Witten index: F IW = Tr[(−) ] : Fermion-Boson degeneracy: any boson jBi has fermion partner jF i = QjBi, 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 −βH F IW = Tr[e (−) ] : For β ! 1: 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: IW = N..