Scale and Conformal Invariance in d=4

Overview, N = 4 d = 2 Review d = 4 Argument Exceptions?Scale and Conformal Invariance in d = 4Joseph Polchinskiwork withMarkus Luty & Riccardo RattazziarXiv:1204xxxxN = 4 Super Yang-Mills Theory, 35 Years AfterCaltech, March 31, 2102

Overview, N = 4 d = 2 Review d = 4 Argument Exceptions?Overview: Scale vs. Conformal Invariance(a) original (b) scale (c) conformalIt is difficult to find quantum field theories that are scale invariantwithout being fully conformal invariant.In d = 2 there is a close connection between this question and themonotonicity of the RG flow (Zamolodchikov ’86; JP ’88)We will try to exploit recent progress in the understanding of the RGflow in d = 4 (Komargodski & Schwimmer 1107.3987;Komargodski 1112.4538).

Overview, N = 4 d = 2 Review d = 4 Argument Exceptions?Scale vs. Conformal InvarianceA general scale current has the form (Wess)S µ = T µ νx ν + V µ ,where T µ µ is the stress-energy tensor and V µ is often called the virialcurrent. Conservation of the scale current implies that0 = ∂ µ S µ ⇒ T µ µ = −∂ µ V µScale invariance allows any vector operator V µ , but in order toconstruct conformal generators we need the stronger conditionV µ = ∂ ν L µνfor some L µν . (Equivalently, there is an improved T µν whose tracevanishes.)

Overview, N = 4 d = 2 Review d = 4 Argument Exceptions?Candidate virialsIn many cases there simply is no nontrivial candidate for V µ . Forexample, in Banks-Zaks, a non-Abelian gauge field coupled to N ffundamentals and antifundamentals, with N f /N c = 112− ɛ: (1) Thefermionic vector current j µ V is conserved. (2) The fermionic axialcurrent j µ V is odd under CP; since we can rotate θ to zero the theory isCP-symmetric and the axial current cannot appear. (3) TheChern-Simons current V µ dx µ = ∗ Tr(AdA + 2A 3 /3) is not gaugeinvariant. So this theory is conformal.But this argument is not general enough: by adding scalars andYukawa couplings, we can get many cases where candidate virialsexist but do not actually appear in S µ .

Overview, N = 4 d = 2 Review d = 4 Argument Exceptions?N = 4 SYMBy the way, how do we know that the birthday theory is actuallyconformally invariant? One possible virial is the gradient of theKonishi operator, ∂ µ (Φ a Φ a ), but this can be improved away. But thefermionic U(1) currentV µ = ¯ψ i σ µ ψ i ?is not conserved (broken by Yukawa couplings), so is a candidate. It isCP-odd, so can’t appear at θ = 0, but could conceivably be generatedby instantons at nonzero θ.But of course, supersymmetry will impose strong constraints...

Overview, N = 4 d = 2 Review d = 4 Argument Exceptions?Howe, Stelle & Townsend, Mandelstam ’82 and Brink, Lindgren &Nilsson ’82 prove perturbative finiteness — does this implyconformal invariance? Multiplet of anomalies (Sohnius and West ’81)seems to show that T µ µ = 0, even nonperturbatively.Modern argument (Seiberg ’88): For scale invariance, go out on theCoulomb branch parameterized by Φ. Effective gauge Lagrangian is12g 2 (Φ) Tr F µνF µν .N = 4 forbids a field-dependent kinetic term, so g 2 must beindependent of scale. If there is a nontrivial virial, the effective actionmust reflect this through a term of the form1Φ ∂ µΦV µ .Again this is ∼ nonlinear two-derivative term, forbidden by N = 4.

Overview, N = 4 d = 2 Review d = 4 Argument Exceptions?Review of d = 2 argumentThe arguments thus far are rather special. In d = 2 there is a generalresult. From conservation of T µν ,x µwhere G µν,σρ = 〈0|T µν (x)T σρ (0)|0〉 and∂∂x µ c(x) = −24z2¯z 2 G z¯z,z¯z , (1)c(x) = 2z 4 G zz,zz − 4z 3¯zG z¯z,zz − 6z 2 G z¯z,z¯z . (2)We are using the improved energy-momentum tensor, which scalescanonically. In a scale invariant theory, c(x) is independent of theseparation, so 〈0|T z¯z (x)T z¯z (0)|0〉 = 0, implying that in a unitarytheory the trace T z¯z vanishes.

Overview, N = 4 d = 2 Review d = 4 Argument Exceptions?In a nonunitary theory there might be exceptions, as the vanishing ofthe two-point function of the trace need not imply the vanishing of thetrace: we will see an example later.Exceptions can also arise if the field theory does not have anormalizable ground state, so that the two-point function does notexist. One class consists of nonlinear sigma models whose targetspace geometry satisfiesR ij = ∇ (i ξ j)(Hull and Townsend; Tseytlin ’86) Conformal invariance requiresfurther that ξ i be a gradient ∂ i f . Such manifolds are necessarilynoncompact, there is no normalizable ground state.

Overview, N = 4 d = 2 Review d = 4 Argument Exceptions?d = 4: Brief KS reviewKomargodsky and Schwimmer monotonicity argument in d = 4: puta QFT in a background metric ĝ µν (x). Splitĝ µν (x) = e −2τ(x) g µν (x) .Expand around g µν = η µν and look at the 4-dilaton S-matrix.On-shell condition is□ϕ = 0 , ϕ/f = 1 − e −τ ;(dilaton = background field). If the QFT is conformal (over somerange of scales), ϕ decouples from it and the ϕϕ → ϕϕ amplitudecomes only from the effective local Wess-Zumino term aS WZ =

Overview, N = 4 d = 2 Review d = 4 Argument Exceptions?The anomaly σE 4 anomaly underKS review∫aS WZ = a d 4 x √ gτE 4 + . . .∫→ −2a d 4 x (∂ µ ϕ∂ µ ϕ) 2 .τ → τ + σ ,g µν e 2σis the same at all scales (zero, here) and equal to a CFT − a at scaleswhere the theory is conformal.By a dispersive argument, a decreases toward the IR, soa CFT,IR < a CFT,UV .

Overview, N = 4 d = 2 Review d = 4 Argument Exceptions?Evaluateon∫ dss 3 M ϕϕ→ϕϕThe integral on I 2 is nonnegative and (positive if the cross-section isnonzero).Now, use these ingredients to address scale vs. conformal.

Overview, N = 4 d = 2 Review d = 4 Argument Exceptions?Scale vs. conformal(i) In a conjectured SFT (scale but not conformal theory), the dilatonwill not decouple (Nakayama, 1110.2586):τT µ µ = −τ∂ µ V µ .The ττ → SFT cross section is then generically nonzero, and by scaleinvariance ∝ f −4 s. The dispersive argument then implies alogarithmic running of a.

Overview, N = 4 d = 2 Review d = 4 Argument Exceptions?Scale vs. conformal(i) In a conjectured SFT (scale but not conformal theory), the dilatonwill not decouple (Nakayama, 1110.2586):τT µ µ = −τ∂ µ V µ .The ττ → SFT cross section is then generically nonzero, and by scaleinvariance ∝ f −4 s. The dispersive argument then implies alogarithmic running of a.(ii) Such a logarithmic running is inconsistent, so an SFT cannot exist.

Overview, N = 4 d = 2 Review d = 4 Argument Exceptions?Inconsistency of running of aExpand on (ii): consider the theory constructed in the background ĝ µνand defined by a UV cutoff (or flow from a UV CFT) at scale Λ.At a much lower scale µ, the effective lagrangian will contain a termC log(Λ/µ)(∂ µ ϕ∂ µ ϕ) 2 .The UV-divergent part must be completed to an interaction that is (a)local and (b) a function only of ĝ µν . But it can’t: it comes only from alocal term that has an anomalous variation.

Overview, N = 4 d = 2 Review d = 4 Argument Exceptions?Perturbative fixed points(i) It remains to show that the log divergence is actually there. Forweakly coupled CFT’s we can do this explicitly. The leading couplingof the dilaton is throughτT µ µ = ( ϕ + 1 2 ϕ2 + . . . ) T µ µHereT µ µ = ∑ Aβ A O ASFT= g A Q AB O B .Next order is of the formτ 2 (∂ 2 µg A )O A ∼ ϕ 2 β B ∂ gB β A O Aand is suppressed by ∂ gB β A ≪ 1 at a weakly coupled fixed point.

Overview, N = 4 d = 2 Review d = 4 Argument Exceptions?Perturbative fixed pointsEach vertex brings in one factor of T µ µ ∝ β, so (a) dominates at weakcoupling,∫d 4 x e ip·x β 2 A〈0|O A (x)O A (0)|0〉 = c A β 2 A(Komargodsky 1112.4538). QED.∫d 4 x eip·xx −8 ∝ c Aβ 2 As 2 ln Λ s .

Overview, N = 4 d = 2 Review d = 4 Argument Exceptions?Perturbative fixed pointsWe can make an independent argument based on (c), because (a) and(b) constribute only to s-wave scattering. For l ≥ 1, (c) makes anunambiguously positive contribution to the cross section, e.g.

Overview, N = 4 d = 2 Review d = 4 Argument Exceptions?Perturbative fixed pointsEither way, weakly coupled SFT’s are ruled out.Theories with discrete scale invariance also excluded, as is any flowfor which∫ dµµ β2diverges in either the IR or the UV.

Overview, N = 4 d = 2 Review d = 4 Argument Exceptions?Strongly coupled fixed pointsCould these cancel, for all intermediate states? It seems unlikely. Itrequires, e.g.〈0|T T µ µ(x 1 )T ν ν(x 2 )T σ σ(x 3 )T ρ ρ(x 4 )|0〉const=(x 1 − x 2 ) 4 + 2 perms. + just-right contact terms ,(x 3 − x 4 ) 4Impossible?

Overview, N = 4 d = 2 Review d = 4 Argument Exceptions?Exceptions? I. Nonrenormalizable scalar theoryColeman and Jackiw ’71:L = ∂Φ∂Φ + (∂Φ∂Φ)2Φ 4 .Nonrenormalizable, but could be treated as an effective Lagrangianbelow the scale Φ. The background Φ breaks scale invariance, soσ ϕϕ→SFT ∝s5f 4 Φ 8(rather than s/f 4 ). Octic divergence rather than a log, canceled byΦ −8 , leaving O(1) which can be canceled by UV thresholdcontribution: no contradiction.Φ ≠ 0: there is no scale invariant background.

Overview, N = 4 d = 2 Review d = 4 Argument Exceptions?Exceptions? II. ElasticityTheory of elasticity (Callan, Coleman and Jackiw ’70; Riva andCardy hep-th/0504197). Displacement u µ (x) with action∫d d x √ [ 1g4 (∂ µu ν − ∂ ν u µ )(∂ µ u ν − ∂ ν u µ ) + h ]2 (∇ µu µ ) 2 .Not reflection positive/unitary. Scale but not conformally invariant inany d. In d = 4 the couplings to the dilaton are2h ( ∂ µ u µ u ν ∂ ν ϕ + ∂ µ u µ ϕ u ν ∂ ν ϕ + (u µ ∂ µ ϕ) 2) .Even though it is nonunitary, it provides a check on argument (ii): thelog divergence should not appear.

Overview, N = 4 d = 2 Review d = 4 Argument Exceptions?Exceptions? II.Individual graphs have log divergences proportional to a + bh + ch 2 ,but the sum is zero. Exclusive amplitudes are nonzero and have bothlocal and contact pieces, but the total ‘cross section’ is zero.

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Overview, N = 4 d = 2 Review d = 4 Argument Exceptions?Exceptions? IV. d = 4 BZFGSFGS also have at least one weakly coupled candidate example ind = 4, using a Banks-Zaks sector to provide a small parameter.This example has a potential unbounded below, but this should notinvalidate our argument based on UV divergences.There is the possibility that three-loop terms will make V µ cancel.Our argument predicts that this will occur.

Overview, N = 4 d = 2 Review d = 4 Argument Exceptions?ConclusionsThe arguments, both for monotonicity and nonexistence of SFT’s, arevery dimension-dependent. Is there a general principle?We have illuminated one small corner of the space of possiblequantum field theories. Thanks in large part to lessons learned fromN = 4, quantum field theory still retains a primary role in the searchfor a ‘theory of everything.’