Lecture 11 Anomalies and Instantons Outline • The chiral anomaly: detailed discussion. • Zero-modes and the index theorem. • Triangle diagrams: the qualitative story. • The gauge anomaly. • Anomaly matching. Reading: Terning 7.2, 7.3, 7.4, 7.5. The Chiral Anomaly Setting: massless chiral fermions coupled to a classical gauge field. Comment: for massless particles chiralities make sense { we can focus on two component Weyl spinors ("the upper two components" of a Dirac four-spinor). Lagrangian: ¯ µ Lfermion = i σ¯ Dµ Notation: • ¯ = y for two component spinors. • σ¯µ = (1; −~σ). a a • Dµ = @µ + iBµ with Bµ = BµT a gauge field with values in a Lie algebra. Consider the fermion path integral as function of a background gauge field: R ¯ iSfermion R ¯ −SE Z[Bµ] = D D e = D D e = det(D6 ) : Euclidean continuation: µ µ •6 D = iσ¯E(@µ + iBµ) withσ ¯E = (iI2; −~σ). 4 4 • id x = d xE . R 4 ¯ • SE = − d xE D6 . More precisely: expand , ¯ on complete bases P ¯ P y (x) = n anfn(x) ; (x) = n bngn(x) : Orthogonality and completeness: P y P y n fn(x)fn(y) = δ(x − y)I2 ; n gn(x)gn(y) = δ(x − y)I2 : R 4 y R 4 y Tr d xfn(x)fm(x) = δnm ; Tr d xgn(x)gm(x) = δnm : Choose basis so for some real λn's: Df6 n = λngn : Then the path-integral is a diagonal Gaussian: P R ¯ −SE R Q n λnbnan Q Z[Bµ] = D D e = n;m dandbme = n λn : More on eigenvalues λn and bases ffng, fgng. Issue: the Dirac operator D6 maps positive chirality spinors to negative chirality spinors. Therefore it can have no eigenfunctions: input and output is not even on the same space! Resolution: D6 y D6 is a hermitean operator on the positive chirality space so there is a complete basis of eigenfunctions: y 2 D6 Df6 n = λnfn ; 2 with λn real and non-negative. The operator with opposite ordering has eigenfunctions: y 2 D6 D6 gn = λngn : 1 1 y The two orderings are related as gn = 6Dfn (or fn = 6D gn) so λn λn the operators have the same spectrum except for zero-modes, ie. the eigenvalues λn = 0: y Df6 n = 0 or D6 gn = 0 : The Chiral Current The Noether current for classical symmetry under (x) ! eiα (x), ¯(x) ! e−iα ¯(x). Remark: this is a chiral transformation because (x) is a Weyl fermion (upper two components only). At the quantum level: change variables in the path integral as (x) ! eiα(x) (x) ; ¯(x) ! e−iα(x) ¯(x) : The classical action transforms as R 4 ¯ −iα(x) µ iα R 4 ¯ µ SE ! SE − d xE e iσ¯ (@µe ) = SE − d xEα(x)@µ σ¯ : The classical action is invariant under arbitrary variations so the current is conserved µ µ ¯ µ @µjA = 0 ; jA(x) = σ¯ : The chiral anomaly: quantum correction to classical current conserva- tion. The chiral transformation acts on the basis coefficients (Tr=trace over spin and group indices): P P iα(x) P 0 (x) = n anfn(x) ! n ane fn(x) = n anfn(x) ; 0 R 4 y iα(x) an ! an = Cnmam = Tr d xEfn(x)e fm(x)am : The path integral measure transforms accordingly: Q ¯ −1 Q n;m dandbm ! (detCC) n;m dandbm : where R 4 ¯ −1 −i d xE α(x)A(x) P y y (detCC) = e ;A(x) = Tr n(fn(x)fn(x) − gn(x)gn(x)) : Quantum equation for anomalous current µ @µjA = A(x) : 2 Compute anomaly using completeness (regulate by damping large λN ): X y y A(x) = Tr (fn(x)fn(x) − gn(x)gn(x)) n 2 2 2 2 X y −λn=Λ y −λn=Λ = limΛ!1Tr [fn(x)e fn(x) − gn(x)e gn(x)] n 4 Z d p y 2 y 2 = lim Tr [e−6D 6D=Λ − e−6D6D =Λ ] Λ!1 (2π)4 4 Z d p µ 2 1 1 = lim e−DµD =Λ Tr[ (σµν F )2 − (¯σµν F )2] Λ!1 (2π)4Λ4 2 E µν 2 E µν 2 2 2 Z π p dp 2 1 1 = e−p · µνρλTrF F = TrF F~µν : (2π)4 2 µν ρλ 16π2 µν Explicit form of Dirac operators and Pauli matrices: i D6 y D6 = D Dµ + (σµ σ¯ν − σν σ¯µ )i[D ;D ] = D Dµ − σµν F µ 4 E E E E µ ν µ E µν i D6 D6 y = D Dµ + (¯σµ σν − σ¯ν σµ )i[D ;D ] = D Dµ − σ¯µν F µ 4 E E E E µ ν µ E µν µν ρλ µν ρλ µνρλ Tr[σE σE − σ¯E σ¯E ] = Summary: the operator form of the chiral anomaly: µ 1 ~µν @µjA = A(x) = 16π2 TrFµν F : The important point: current conservation at the classical level is vio- lated by a quantum effect of a very specific form. The Atiyah-Singer Index Theorem Instructive alternative computation of A(x): 1 1 y gn = Df6 n ; fn = D6 gn ; λn λn so contributions to A(x) cancel in pairs except for λn = 0: R 4 R 4 P y y d xA(x) = Tr d x n(fn(x)fn(x) − gn(x)gn(x)) = n − n ¯ : y The number of zero modes of D6 , D6 : n ; n ¯. The Atiyah-Singer index theorem: 1 R 4 ~µν n − n ¯ = 16π2 d xTrFµν F : Mathematical significance: a relation between a topological quantity (the difference in the number of zero-modes cannot change continuously) to an analytical quantity (an integral over a smooth function). Towards the physical significance: there would be no chiral anomaly were it not for the zero modes of the Dirac operator (and those make the path integral vanish). Instantons R 4 µν The Euclidean action S ∼ d xTrFµν F of a gauge field configuration diverges at infinity, unless the gauge field is asymptotically pure gauge y Aµ ! U(x)@µU(x) . µν TrFµν F~ is a total derivative so its integral vanishes except for a surface term at the asymptotic Euclidean 3-sphere S3. This surface term counts the wrapping number ν of the U(x) around the asymptotic S3. Example: G = SU(2) ∼ S3 as a group, so the obvious embedding G ! S3 has winding number ν = 1. Field configurations with winding number have non-trivial field strength in bulk. Their Euclidean action is finite but large for small coupling 2 SE ∼ 1=g . These configurations are localized in Euclidean space, including in time. They are called instantons. The explicit integral over a winding configuration gives 1 R 4 ~ 16π2 d xTrFµν Fρλ = ν : Remark: the winding number ν is just the integral difference of zero modes n − n ¯. The integral above is in the fundamental representation (denoted ) where the gauge matrices U(x) are just the defining matrices. 1 The index T () = 2 in a general representation r is Tr[T aT b] = T (r)δab : The anomaly for a theory with n(r) chiral fermions in representation r: R 4 1 R 4 ~µν P d xA(x) = 16π2 d xTrFµν F = r n(r)2T (r) : The θ-term The gauge theory action generally includes the CP-violating term θYM R 4 µν ~ SCP = − 16π2 d xTrF Fµν : The term is a total derivative but it does not vanish: SCP = −θYMν : The path integral sums over all field configurations, with different in- stanton sectors weighed by the phase factor e−iθYMν . Theories with different θYM are generally physically distinct. Specifically they have different spectrum. Exception: shifting the phase θYM ! θYM + 2πZ is inconsequential. The chiral transformation ! eiα , ¯ ! e−iα ¯ shifts the path integral as: R 4 α R 4 ~µν R iS R iS−iα d xA(x) R iS−i d xTrFµν F D D ¯e ! D D ¯e = D D ¯e 16π2 : Equivalently: P θYM ! θYM − α r nr2T (r) : Example: in pure SYM the gaugino is the only chiral fermion. A chiral rotation λa ! eiαλa is equivalent to θYM ! θYM − 2Nα : because the index of the adjoint is T (Ad) = N. Thus the chiral anomaly breaks the continuous U(1)R in the classical k theory to the Z2N generated by α = N π (with k = 0;:::; 2N − 1). The Euclidean action of instantons is bounded: R 4 2 R 4 µν µν 0 ≤ d xTr(Fµν ± F~µν ) = 2 d x Fµν F ± Fµν F~ ; so R 4 µν R 4 ~µν 2 d xFµν F ≤ d xFµν F = 16π ν : Instantons in fact saturate this bound. Instanton contributions to the Euclidean path integral are suppressed by the characteristic nonperturbative factor: 2 ν − 8π +iθ e g2 : Such effects are qualitatively important because they are the leading contributions to some processes. For example, we would like to break SUSY in a non-perturbative manner, where the breaking is heavily suppressed. Triangle Diagrams Alternative derivation of chiral anomaly: Triangle diagrams. Chiral current (in the interacting theory): µ ¯ ν 5 −ie R d4y =A hjA(x)iint = h (x)γ γ (x) e i0 Expanding to second order, contract all fermions: find triangle diagram with current insertion at one angle and gauge field on two others: The divergence of the chiral current vanishes formally: µ qµhjA(q)iint = 0 (formally) : The reason: explicit integrals for the two diagrams cancel upon linear shifts in the loop momentum variables. The catch: shift of loop momentum is only justified if the regulated integral preserves the shift symmetry.