1. Scale vs Conformal invariance
Yu Nakayama　（IPMU & Caltech）

2. Scale invariance is ubiquitous
in nature

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

4. 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)

5. 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?

6. 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

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

8. 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

9. 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?

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

11. 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)

12. 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

13. 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

14. 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:

15. (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.

16. 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)

17. 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:

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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.

24. 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

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

26. 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??

27. 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?

28. 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

29. 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

30. 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

31. 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

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