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Physical reason why scale invariance implies conformal invariance in perturbative fixed

are inconsistent with the perturbativea-theorem. We have showed that the suitably defineda-function (whose particular realization is the dilaton scattering amplitudes) decreases at the rate dominantly proportional to the bilinear of B-functions in perturbation theory. In order to achieve the finite dilaton scattering amplitude, it must be accompanied with vanishing Bfunctions near the asymptotic IR region, and therefore it must show conformal invariance (up on possible improvement of the energy-momentum tensor).

If we perform an explicit computation, however, we may realize something a little bit more about the structure of the renormalization group flow. We realize that B functions in certain directions are identically zero irrespective of if the coupling constants are at the fixed point or not. In addition, within a few orders of perturbation theory, the zero-directions directly correspond to the “would-be”

virial current direction that may induce scale invariance without conformal invariance by using the equations of motion. We have seen this phenomena explicitly in section 3.5 when we discussed the conformal perturbation theory.

This may sound unexpected because we are solving the weaker equation Tµµ =∂µJµ rather than Tµµ= 0 by introducing extra parameters inJµ(as many as the number of non-conserved current). Why can’t we expect more solutions generically? The argument based on thea-theorem gives the constraint near the fixed point, but what is the origin of these zero directions during the entire renormalization group flow?

To see it in a simple example, we consider Yukawa interaction yϕψ2 in d = 4 dimension. The Yukawa interaction has the one-loopB function

Tµµ=BIOI = |y|2

16π2y(ψψϕ) +c.c.+O(ϕ4) . (8.28) We realize that the B function is orthogonal to the direction iy(ψψϕ) +c.c., which can be rewritten as the divergence of the non-conserved current (which is given by i∂µ( ¯ψγµψ)). In other words, the phase of the Yukawa coupling constants are not renormalized (as observed in [112][113]), and this is the reason why the “would-be” virial current does not appear in this one-loop computation. We can easily see that the phase of the Yukawa coupling is unphysical from the beginning because we can remove it by a field redefinition ofψ. Since the phase is not a parameter of the theory nor does it affect any observables of the theory, it had better not show any physical consequences in the renormalization group flow. Of course, the strong a-theorem dlogaµ16π1 |y|6 does tell that the ˜a-function must be decreasing along the renormalization group flow, but again note that the phase of the Yukawa coupling does not appear in the ˜a-function, either.

65Actually, the same subtlety appeared in Zamolodchikov’s argument in d = 2 dimension. His c-function is not manifestly positive definite in its definition away from the conformal fixed point, so it is logically possible that it decreases forever because there is no way to make the theory gapped by deformations.

As discussed in section 3.5, the B function that can be completely rewritten as a divergence of a virial current is related to redundant directions in the renormalization group flow. The physical reason why the scale invariant but non-conformal field theory is difficult to achieve in perturbation theory can be understood as the claim that redundant directions do not acquire B functions as can be checked directly within a first few orders of perturbation theory. At higher orders, B functions in the virial current direction can be non-zero, but they still possess the zero directions in the Bfunction flow, which essentially excludes the cyclic renormalization group flow.

The above observation can be made more precise when we can find the counterterm in which wI = 0 in local renormalization group flow discussed in section 7.66 In this situation, theB functions must have the same number of zero directions from the consistency condition BI =gIJJa˜ together with the fact that ˜a is a singlet under the “flavor” symmetries generated by the candidates of the virial current.

Let us labeli, jas the direction corresponding to the “would-be” virial current direction, andM, N as the direction that cannot be transformed to currents. The Wess-Zumino consistency condition of the local renormalization group and the “flavor” invariance of ˜arequires

0 =∂i˜a=giMβM +gijBj . (8.29) Suppose βM = 0 but Bi ̸= 0 so that we obtain scale invariant but non-conformal fixed point. This cannot occur because it is inconsistent with the expected zero directions in B function flow (8.29) as long asgij is non-degenerate (as expected becausegij must be isometric under the “flavor” symmetry group).

Even whenwI is not zero, we know that BI functions are not independent because∂i˜a= 0 gives the constraint along the renormalization group flow, which reflects the fact that there are as many redundant perturbations in renormalization group flow as the number of candidates for the virial current. The condition (8.29) is replaced by

0 =∂i˜a= (giM+wiMM + (gij +wij)Bj + (ˆρig)jwj (8.30) By demanding βM = 0 and contracting it with Bi, we obtain the same conclusion that Bi = 0 (and hence it must be conformal invariant) as long as gij is non-degenerate.

In summary, we have two physical intuitions for non-existence of (perturbative) scale invariant but non-conformal field theories in power-counting renormalization scheme. The first one is thea-theorem:

the dilaton scattering amplitude (or Osborn’s ˜a-function) must be bounded and it cannot decrease forever from non-trivial B functions in scale invariant but non-conformal field theories. The second one is that B functions in the directions that might be used to construct a non-trivial virial current are actually redundant directions in perturbation theory. Even at higher orders, we expect to keep the same number of zero directions in theBfunction flow as the number of redundant directions. The both intuitions are beautifully realized by the generalized “gradient formula” if true. In holographic computations, we will precisely see both obstructions if we try to construct the gravitational dual of scale invariant but non-conformal field theories.

In the above argument, we have not talked about the importance of dimension 2 operators in d = 4 dimension, and the argument is very similar to the one in d = 2 dimension. Within power-counting renormalization scheme, the dimension 2 operators do not show any essential contributions

66This is shown to be possible within the first few orders in perturbation theory. The superpotential flows in holography also suggest this is the case in explicit holographic models. In general this may not be true unlesswI is exact, but the author does not know any explicit examples that showwI is not exact.

to the structure of the local renormalization group in relation to the a-theorem, but the mixing with the energy-momentum tensor may give additional subtlety in our discussions on the relation between scale invariance and conformal invariance beyond perturbation theory as we have mentioned a few times in this section. Clearly, the possibility of the renormalization of the energy-momentum tensor is one novel complexity in higher dimensions that did not appear ind= 2 dimension, and it has given a small gap between the establishment of a-theorem and the enhancement of conformal invariance in higher dimensions compared with the situations ind= 2. It would be very interesting to obtain more intuitive physical understanding of the role of these dimension 2 operators.

9. Other dimensions or less symmetry

We have discussed the problem of the possible enhancement of conformal invariance from scale invariance mainly ind= 2 andd= 4 dimensions. In this section, we would like to review the situations in the other dimensions, which is less understood. Also we would like to review the extension of our program in less symmetric situations e.g. without assuming the full Poincar´e invariance.

9.1. Summary of the situations in other dimensions

Let us first summarize the situations of scale invariance vs conformal invariance in various dimen-sions other than d= 2 andd= 4 dimensions.

• In d = 1, due to the lack of Poincar´e invariance, we cannot use the Reeh-Schlieder theorem.

This is a major drawback. If we assume its validity then scale invariance implies conformal invariance [226]. Similarly, the boundary g-theorem [227], which claims that boundary entropy of the two-dimensional system is monotonically decreasing along the renormalization group flow, can be proved [228][229]. On the other hand, a quantum field theory ind= 1 is equivalent to a simple quantum mechanical system, and there are examples of cyclic renormalization group flow [230] realized in non-relativistic field theories [231] as well as the system with scale invariance without conformal invariance [226]. In these cases, the Reeh-Schlieder theorem does not hold so the formal argument does not apply.

• Ind= 3 dimension, a candidate of Zamolodchikov’sc-function is the finite part of theS3partition function F =−logZS3|reg as we will elaborate a little more in section 9.2.1. This is equivalent to the finite part of the entanglement entropy of the half S3 when the theory is at the conformal fixed point. It is an interesting open question if there is a strong version of theF-theorem that would imply enhancement from scale invariance to conformal invariance ind= 3 dimension. We have more to say in section 9.2.1.

• In even dimensions, Cardy’s conjecture (or a-theorem) has a natural generalization: the coeffi-cient in front of the Euler density in the Weyl anomaly must be monotonically decreasing along the renormalization group flow. Ind= 6 dimension, so far we have not been successful in using the dilaton-scattering argument to show the weak version of the a-theorem. A reported prob-lem [232] is that it is hard to show the positivity of the dilaton scattering amplitudes in d= 6 dimension. On the other hand, there is no counterexample of a-theorem reported and there is no known scale invariant but non-conformal invariant field theories (with gauge invariant scale current). Within perturbation theory, the argument similar to the one presented in section 7 can be found in [248] and will be reviewed in the following. In d = 6 dimension, a possible non-trivial (super)conformal fixed point may be related to the so-called little string theory67

• In higher dimension d≥7, it is likely that there is no interacting unitary conformal field theory, but there is no proof of it. Certainly, there is no classical scale invariant Lagrangian with two-derivative kinetic terms other than free field theories. The reason why higher dimensional free Maxwell theory cannot be conformal invariant is consistent with the fact that there is no superconformal algebra ind≥7, but we know supersymmetric Maxwell theories exist ind≤10.

67We believe that the little string theory with (2,0) supersymmetry becomes superconformal in the IR limit. On the other hand, the little string theory with (1,1) supersymmetry becomes free super Yang-Mills theory in the IR limit, which is scale invariant but not conformal invariant. We refer [233] for a review of the little string theory.

If it were conformal, it would be inconsistent with the non-existence of the superconformal algebra [234] (unless it breaks supersymmetry). We note that Nahm’s classification is based on the assumption of the existence of the S-matrix, so it does not exclude the possibility of superconformal membrane field theories in higher dimension than 6.