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 = iψσ¯ Dµψ Notation: • ψ¯ = ψ† 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] = DψDψ e = DψDψ 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 xEψ Dψ6 . More precisely: expand ψ, ψ¯ on complete bases

P ¯ P † ψ(x) = n anfn(x) , ψ(x) = n bngn(x) . Orthogonality and completeness:

P † P † n fn(x)fn(y) = δ(x − y)I2 , n gn(x)gn(y) = δ(x − y)I2 .

R 4 † R 4 † 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] = DψDψe = n,m dandbme = n λn . More on eigenvalues λn and bases {fn}, {gn}. 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 † D6 is a hermitean operator on the positive chirality space so there is a complete basis of eigenfunctions: † 2 D6 Df6 n = λnfn , 2 with λn real and non-negative. The operator with opposite ordering has eigenfunctions: † 2 D6 D6 gn = λngn .

1 1 † 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: † Df6 n = 0 or D6 gn = 0 . The Chiral Current The Noether current for classical symmetry under ψ(x) → eiαψ(x), ψ¯(x) → e−iαψ¯(x). Remark: this is a chiral transformation because ψ(x) is a Weyl fermion (upper two components only). At the quantum level: change variables in the path integral as ψ(x) → eiα(x)ψ(x) , ψ¯(x) → e−iα(x)ψ¯(x) .

The classical action transforms as

R 4 ¯ −iα(x) µ iα R 4 ¯ µ
SE → SE − d xEψe iσ¯ (∂µe )ψ = SE − d xEα(x)∂µ ψσ¯ ψ . The classical action is invariant under arbitrary variations so the current is conserved µ µ ¯ µ ∂µjA = 0 , jA(x) = ψσ¯ ψ . 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 ψ(x) = n anfn(x) → n ane fn(x) = n anfn(x) ,

0 R 4 † 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 † † (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 † † A(x) = Tr (fn(x)fn(x) − gn(x)gn(x)) n 2 2 2 2 X † −λn/Λ † −λn/Λ = limΛ→∞Tr [fn(x)e fn(x) − gn(x)e gn(x)] n 4 Z d p † 2 † 2 = lim Tr [e−6D 6D/Λ − e−6D6D /Λ ] Λ→∞ (2π)4 4 Z d p µ 2 1 1 = lim e−DµD /Λ Tr[ (σµν F )2 − (¯σµν F )2] Λ→∞ (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 † 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 † = 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 † 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 † † d xA(x) = Tr d x n(fn(x)fn(x) − gn(x)gn(x)) = nψ − nψ¯ . † The number of zero modes of D6 , D6 : nψ, nψ¯. The Atiyah-Singer index theorem: 1 R 4 ˜µν nψ − nψ¯ = 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 † 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 nψ − nψ¯. 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 ψ → eiαψ, ψ¯ → e−iαψ¯ 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 DψDψe¯ → DψDψe¯ = DψDψe¯ 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 d4yψ/Aψ hjA(x)iint = hψ(x)γ γ ψ(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. This is not possible in any regu- larization scheme. Explicit computations in several regularization schemes

µ e2 µναβ ∂µhjAiint = − 16π2  Fµν Fαβ , The Gauge Anomaly Setting: a chiral gauge theory with matter in a chiral representation (the couplings are chiral). Sample Lagrangian:

† µ † a a L = ψLiσ DµψL + ψRiσ∂µψR ,Dµ = ∂µ − igAµt

Noether current of interest: the gauge current (the symmetry is not chiral): µa µ a j = ψLσ t ψL

Gauge anomaly: two triangle diagrams fail to cancel (but details quite different from the chiral anomaly). The significance: the gauge current must be exactly conserved or else the decoupling of longitudinal gauge particles (and ghosts) fail. Gauge theory is inconsistent at the quantum level if there is a gauge anomaly. Evaluation: 1) explicit computation of triangle diagram. or 2) topological argument (reducing path integral to zero modes). Result at linear order (non-linear order not just field strength!):

µa g2 µναβ b c abc ∂µhj i = − 16π2  Fµν FαβA , where the anomaly coefficient: Aabc = tr[ta, {tb, tc}] .

Consistency condition on gauge theory: the anomaly coefficient must vanish (when summed over all matter representations). Anomaly Cancellation in SM

Remarkable fact: SM spectrum is such that all gauge anomalies vanish.

2 Example: anomaly of U(1)EM × SU(2)L: a b ab 2 1 tr[Q, {τ , τ }] = δ tr[Q] = 3 · ( 3 − 3 ) + (0 − 1) = 0 . Anomaly Matching Important application of anomalies: ’t Hooft’s anomaly matching. Setting: a UV theory with some global symmetry G. Anomaly matching gives information about the IR behavior of the theory. Promote G to a gauge symmetry, with some weak coupling g.

The anomaly coefficients AUV may fail to vanish. Add some spectator matter fields that only couple to G, designed such that AUV + AS = 0. The gauge group G is just a redundancy of the theory so it must remain in the IR theory. The anomaly coefficients (including the contribution from the spectators) of the IR theory therefore must vanish AIR +AS = 0. Thus AUV = AIR.

The gauge bosons and the spectator fields decouple as g → 0 but main- tain the three point functions of the currents so that AUV = AIR re- mains.