Scale vs Conformal invariance Yu Nakayama　（IPMU & Caltech）

Scale invariance is ubiquitous in nature

In relativistic QFT Massless fields are scale invariant Asymptotic freedom Renormalization group fixed point  scale invariance Scale + Poincare = Closed algebra

From Polyakov 1970 to BPZ 1984 Pointed out scale invariance in Euclidean system may (must?) show conformal invariance Checked in 2d critical Ising model BPZ “classified” 2d critical phenomena with conformal invariance (completed?) Since then, the assumption of conformal invariance in critical system seems to be considered as a fact (or folklore)

Scale = Conformal? In many textbooks (both high energy and condensed matter), you can find the folklore argument: If your theory is local, then scale transformation is allowed locally (why?), and you get conformal invariance (Zumino’s theorem) At least I felt cheated How can I get extra symmetry for free?

The response to Weyl transform in QFT Scale vs Conformal 101 The response to Weyl transform in QFT Require Weyl (local scale) invariance: Instead require only constant Weyl inv: May be improved to be traceless when : virial current Obviously conformal is stronger than scale inv

Flat space-time Conformal invariance Scale transformation  EM tensor is divergence of virial current Conformal transformation  EM tensor is traceless

Scale = Conformal? What is surprising is almost all scale invariant field theories we know are actually conformal invariant (e.g. 3D Ising model) !!! Why? Lack of imagination? Must be deep reason behind it Structure of the renormalization group Causality and unitarity (notion of time) May tell how/why AdS/CFT works

A disclaimer The same questions can be addressed in non-relativistic system e.g. Scale + Galilei  Schrodinger? No conformal extension for Lifshitz Existence of limit cycles (Effimov effect) No c-therem? Interesting but difficult. No universality?

Part 1. Field theory analysis scale = conformal conjecture Counterexample Local renormalization group Perturbative proof

A scale invariant QFT is conformal invariant when Scale = conformal conjecture: A scale invariant QFT is conformal invariant when 1. It is unitary 2. It is Poincare invariant (causal) 3. It has a discrete dilatation spectrum 4. It has a well-defined energy-momentum tensor and dilatation current (Noether assumption)

Simple counterexample: Maxwell theory Consider U(1) free Maxwell theory in d >4. EM tensor and Virial current Virial current 　　　　 is not a derivative  cannot improve EM tensor to be traceless Does not satisfy conformal Ward identity Dilatation current is not gauge invariant, but charge is gauge invariant

Formulation of the problem Compute trace of EM-tensor Composite operator renormalization (EM tensor is renormalized)  vector beta functions… Use (renormalized) operator identities e.g. Demand scale invariance 4. If it is total derivative  conformal

One word about improvement In flat space-time, EM tensor is not unique For any (dim d-2) scalar operator O. Equivalent to adding For scale invariance: For conformal invariance:

(1+1) d proof (Zamolodchikov-Polchinski) According to Zamolodchikov, we define At RG fixed point,　　　 , which means   C-theorem! Causality and unitarity is used here.

General strategy? Constraint on RG-flow Construct monotonically decreasing function along the renormalization group (at least valid near scale invariant “fixed point”). Show that it is a constant if and only if it is conformal invariant (up to improvement) This works in perturbation theory and holography (strong a-theorem in d=4)

Schwinger functional and local RG constraint We had to deal with composite op renormalization Introduce space-time dependent source and renormalize vacuum energy functional Local renormalization group operator There are anomalies in curved background:

A flat space RG ambiguity Flat space-time renormalization may hiddenly generate vector beta functions Schwinger functional is invariant under the background gauge transformation for source Operator identity Gauge invariant B-function Flat space-time beta function is ambiguous Neglecting vector beta functions led to alleged “counter-examples” of scale but non-conformal in d=4

Wess-Zumino consistency condition 1 Renormalization group (Weyl transform) is Abelian Class 1 consistency on beta functions Note: this must hold irrespective of anomaly so valid for general QFTs in any dimension

Wess-Zumino consistency condition 2 Renormalization group (Weyl transform) is Abelian Class 2 consistency on anomaly (d=4: Osborn)  Gradient formula and a-theorem in power-counting renormalization

No constraint from WZ condition Positivity? No constraint from WZ condition Perturbatively it is positive (Zamolodchikov’s metric) in unitary QFT  perturbative proof In d=2, there is a scheme in which it is always positive in unitary QFT  non-perturbative proof In holography it must be positive from unitarity

Comment In conformally flat space-time, Schwinger functional for IR fixed point is essentially dilaton effective action of Schwimmer and Komargodski From (subtracted) sum rule gives the weak a-theorem between two CFTs (non-perturbatively) 　Probably one more step is needed to show scale = conformal beyond perturbation theory

d = 3 ? In odd dim, there is no trace anomaly, so we consider S3 partition function instead CFT one-point function must vanish on S3, so near conformal fixed point, we obtain the gradient formula It will imply the (strong) F-theorem The question is whether “metric” is positive. If it is positive, scale  conformal. In perturbation theory the metric is positive.

Not enough to compute beta functions for conformal invariance Lessons Not enough to compute beta functions for conformal invariance Can we compute B function directly? With manifest SUSY preserving regularization, B = beta to all order in perturbation theory Scale = conformal in perturbation theory, but non-perturbative regime remains open

Part 2. Holographic verdict Holographic c-theorem revisited Gauge artifact Scale  Conformal NEC violation?

Holographic c-theorem revisited According to holographic c-theorem: Null energy-condition leads to strong c-theorem Suppose the matter is given by NSLM Strict null energy-condition demands the positivity of the metric (unitarity needs positivity of the metric) Is scale  conformal??

More precise version 1 Implement the operator identity: Achieved by gauge transformation In Poincare patch We may have the non-zero beta functions (B gauge) But this can be gauge equivalent to the vector condensation (virial gauge) In both cases, the field configuration is scale inv but non-conformal (holographic cyclic RG). Is it possible?

More precise version 2 Reconsider holographic RG-flow Null energy-condition leads to strong c-theorem Matter is given by gauged NLSM Assume positive metric from strict NEC Scale inv but non-conformal config forbidden

General theorem Scale inv  Conformal inv in Holography when Gravity with full diff (cf Horava gravity) Matter: must satisfy strict NEC When saturate the NEC, matter must be trivial configuration (no net energy propagation) Sufficient condition for strong a-theorem Sufficient to protect unitarity in NLSM In any space-time dimension Scale inv  Conformal inv in Holography

NEC violation? Quantum effect can violate NEC Quantum NEC violation potentially makes holographic a-theorem fail Example: in d= 3+1 dimensional bulk, 1-loop trace anomaly induced quantum energy-momentum tensor can violate NEC However, consistent EOM + redefinition of a-function makes (strong) a-theorem valid

Lessons and outlooks A lot of confusions in field theory, but crystal-clear in holography Debate over perturbative cyclic RG flow and scale but non-conformal counterexample was gauge artifact Violation of NEC?  OK in reasonable quantum 1-loop corrections. In general? Notion of time (e.g. NEC) seems crucial

Reference I have omitted many references But most comprehensive reference list can be found in my lecture note  arXiv:1302.0884 If you find any comments, suggestions, counterexamples or proof, please let me know!