5.3.1. Simple alternative derivation

Without referring to the c-theorem, there is a more direct way to derive the enhancement from
scale invariance to conformal invariance in d= 2 dimension.^{43} For this purpose, we study two-point
functions of the energy-momentum tensor in momentum space. The point is that the conservation
and the canonical scaling of the energy-momentum tensor gives the unique structure of the two-point
functions so that the trace must vanish.

The assumption of the canonical scaling dimension in position space of the energy-momentum
tensor leads to the requirement that the momentum space energy-momentum tensor two-point function
must show (we use the complex momentumk=k_{σ}+ik_{τ} and ¯k=k_{σ}−ik_{τ})

⟨T(k)T(p)⟩=ck^{3}

¯kδ(k+p)

⟨T(k)Θ(p)⟩=ek^{2}log|k|^{2}δ(k+p)

⟨Θ(k)Θ(p)⟩=h|k|^{2}log|k|^{2}δ(k+p)

⟨T(k) ¯T(p)⟩=w|k|^{2}log|k|^{2}δ(k+p) , (5.16)

43The argument here is close to the historically original one presented in [181].

where we have neglected the contact terms that are polynomial in k and ¯k.

The appearance of log|k|^{2} is not in contradiction with the assumed scale invariance because the
scale transformation only gives the extra ultra local contact terms. These are related to the anomalous
contribution in the local Callan-Symanzik equations.

The conservation of the energy-momentum tensor (again up to the contact terms) requires

e= 0 , h= 0 , w= 0. (5.17)

From the Reeh-Schlieder theorem by going back to the position space, we conclude that Θ(x) = 0 as an operator identity. Thus, the scale invariance implies conformal invariance. If we had kept track of the contact terms, we could see ⟨Θ(x)Θ(0)⟩ contains the contact term proportional toc, which can be related to the Weyl anomaly from the second variation of the effective action with the metric.

We may attempt a similar derivation in higher dimensions [38][112]. However, we can immediately realize that the number of independent two-point functions of the energy-momentum tensor is larger than the constraint from conservation and unitarity even if we assumed the canonical scaling of the energy-momentum tensor, so we cannot derive the similar result in this way. We will explicitly see what happens in section 6.3. In retrospect, we have a good reason for this: two-point functions of the energy-momentum tensor do not seem to be a good barometer to show the higher dimensional analogue of Zamolodchikov’s c-theorem as we will see.

Let us briefly discuss what would happen if we relaxed the condition of the canonical scaling of the energy-momentum tensor. For the simplest example, suppose the energy-momentum tensor renormalization has the Jordan block form (see section 7.2 for more general situations)

d

dlogµTµν =η(∂µ∂ν −ηµν□)O d

dlogµO= 0 (5.18)

where O has the scaling dimension zero so that the scaling dimension matrix is not diagonalizable for non-zero η. We have relaxed the assumptions in Polchinski’s argument (see section 5.2) to allow dimension zero operator. This may happen in bosonic string theory with tachyon operators in the non-compact target space-time.

After imposing the conservation and the dilatation Ward-identity (with local anomalous sources), the momentum space two-point functions are given by

⟨O(k)O(p)⟩= cOO

|k|^{2}δ(k+p)

⟨O(k)T(p)⟩=(

c_{OT} −4ηc_{OO}log|k|^{2})k

¯kδ(k+p)

⟨O(k)Θ(p)⟩=ηc_{OO}log|k|^{2}δ(k+p)

⟨T(k)T(p)⟩=(

c−8ηc_{OT} log|k|^{2}+ 16η^{2}c_{OO}(log|k|^{2})^{2})k^{3}

¯kδ(k+p)

⟨T(k)Θ(p)⟩= (2ηcOT log|k|^{2}−4η^{2}cOO(log|k|^{2})^{2})k^{2}δ(k+p)

⟨Θ(k)Θ(p)⟩= (

−1

2ηc_{OT} log|k|^{2}+η^{2}c_{OO}(log|k|^{2})^{2}
)

|k|^{2}δ(k+p) (5.19)
up to contact terms. We may improve the energy-momentum tensor so that c_{OT} vanishes, but η
remains non-zero. Irrespective of the improvement, we see that the conservation of the

energy-momentum tensor alone does not lead to the conclusion Θ = 0 as an operator identity.^{44} Note
that Zamolodchikov’s argument still holds, but the (effective) central charge can decrease with an
arbitrary amount suggesting the violation of the unitary or the discreteness of the spectrum.^{45}

We have one technical comment on the Weyl anomaly. If there were no mixing due to η, the logarithmic term in the third equation in (5.19) would suggest that the Weyl anomaly contains the

“tachyon coupling”∫
d^{2}x√

|g|Φ, where Φ is the source for the dimension zero operatorO (see section 7.2 and [80]). However, the violation of the naive scaling symmetry in log here is cancelled by relating it to the other correlation functions from the non-diagonal scaling dimension matrix (i.e. wavefunction renormalization) rather than contributing to the Weyl anomaly ∫

d^{2}x√

|g|Φ. The similar comments apply to the other terms in (5.19) so that the Weyl anomaly is actually local.

5.3.2. Averaged c-theorem

Zamolodchikov’s argument can be presented in a slightly different way [184]. Define
c^{M}_{(2)}=−

∫

d^{2}xG_{(µ)}⟨Θ(x)Θ(0)⟩
Θ =B^{I}O_{I}

χ_{IJ} =− d
dlogµ

∫

d^{2}xG_{(µ)}(x)⟨O_{I}(x)O_{J}(0)⟩ (5.20)
where

G_{(µ)}(x) = 3πx^{2}θ(1−µ|x|) , (5.21)
withθ(x) being the step function so thatG_{(µ)}(x) has only support|x| ≥µ^{−1}. The metricχ_{IJ} is positive
definite from unitarity (no dangerous contact term will contribute because _{d}^{dG}_{log}^{(µ)}_{µ} has a support only
when O^{I} are separated). It can be easily shown that

dc^{M}_{(2)}

dlogµ =B^{I}χ_{IJ}B^{J} ≥0. (5.22)

We note that this c^{M}_{(2)}-function is equivalent to the one in section 5.1.

One can now repeat the same analysis in d≥2. We define
c^{M}_{(d)}=−

∫

d^{d}xG^{(d)}_{(µ)}⟨Θ(x)Θ(0)⟩
Θ =T^{µ}_{µ}=B^{I}OI

χ_{IJ} =− d
dlogµ

∫

d^{d}xG^{(d)}_{(µ)}⟨O_{I}(x)O_{J}(0)⟩ (5.23)

44However, we may suspect that the appearance of (log|k|^{2})^{2} is inconsistent with the locality of the correlation
functions. If this additional locality constraint is imposed, either (cOO, cOT) orη must vanish at the scale invariant
fixed point. In the former case, the unitarity is violated as in log CFT, and in the latter case, the scaling dimension
matrix must be diagonalizable and reduced to the original argument. The author would like to thank A. Bzowski and
K. Skenderis for the related discussions.

45Physically, this is not unexpected because with a dimension zero operator at hand, one can make the effective central charge arbitrarily large by improvement (e.g. in Liouville theory), and what ηterm does is the renormalization of the improvement term, so the effective central charge should change with an arbitrary amount during the renormalization group.

where

G^{(d)}_{(µ)}(x) = 3πx^{d}θ(1−µ|x|) . (5.24)
The metric is again positive definite from unitarity. It can be easily shown that

dc^{M}_{(d)}

dlogµ =B^{I}χ_{IJ}B^{J} ≥0. (5.25)

Can we declare the proof of c-theorem in any dimension? Does it mean scale invariance implies conformal invariance in any dimension?

A related idea was explored in [187]. The integrated c^{M}_{(d)} is known as the averagedc-function. In
the later works [188][189][190], it was argued that the integral of the two-point functions of the trace
of the energy-momentum tensor (5.3.2) is directly related to the difference of ˜bcoefficient in the Weyl
anomaly in d = 4 dimension.^{46} It was also argued that although ˜b itself is scheme dependent, the
ambiguity cancels in the difference of the UV fixed point and IR fixed point, and the integral only
depends on the trajectory of the renormalization group flow. Since it depends on the trajectory, the
quantity has a very different nature than Zamolodchikov’sc-function ind= 2 dimension. We have no
direct way to connect the averaged c-function to the local correlation function so we cannot use the
Callan-Symanzik equation to trade the µdependence with beta functions.

What is special in d= 2 dimension is the identity [184]

∂^{µ}[(2x^{ν}x^{ρ}x^{σ}−2x^{2}x^{ν}η^{ρσ}−x^{2}x^{σ}η^{νρ})⟨Tµν(x)Tρσ(0)⟩] =−3x^{2}⟨Θ(x)Θ(0)⟩ . (5.26)
It enable us to integrate c^{M}_{(2)} by part to rewrite the averaged quantity (5.20) into the local form

c^{M}_{(2)}(µ) = 2π^{2}(2x^{µ}x^{ν}x^{ρ}x^{σ}−x^{2}x^{µ}x^{ν}η^{ρσ}−x^{2}x^{ρ}x^{σ}η^{µν}−x^{2}x^{µ}x^{σ}η^{νρ})⟨Tµν(x)Tρσ(0)⟩|µ|x|=1 , (5.27)
which is nothing but the one defined by Zamolodchikov. The application of the Callan-Symanzik
equation leads to the claim thatµ dependence is only through the running coupling constants.

We will come back to this point later when we discuss the renormalization scale dependence in the
proof of the higher dimensional analogue of Zamolodchikov’s c-theorem and its application to scale
invariance and conformal invariance. Here we only emphasize that the crucial distinction between
d= 2 andd >2 in relation to the argument of this section is that the so-defined averagedc-function is
not an intrinsic quantity of the fixed point, but it is a quantity of the flow. In particular, the equation
(5.25) by itself is consistent with the cyclic renormalization group flow with B^{I} ̸= 0 because there is
no reason why c^{M}_{(d)} should take a constant value when the theory is scale invariant. However, it is
remarkable to mention that within a few orders in perturbation theory when the theory is classically
conformal invariant, (5.25) gives the same renormalization scale dependence as that for the higher
dimensional analogue of Zamolodchikov’s c-function we will discuss in the next section, which only
depends on the running coupling constants at the scale µ.

46More precisely, we had to fine-tune local counterterms (see section 7) to achieve this claim.

6. Conjecture in d >2

6.1. Scale invariance vs Conformal invariance

Given a proof in d= 2 dimension reviewed in section 5, and various examples studied in section 4, we conjecture that any scale invariant quantum field theory (ind >2) is conformal invariant under the following assumptions

• unitarity

• Poincar´e invariance (causality)

• discrete spectrum in scaling dimension

• existence of scale current

• unbroken scale invariance

The necessity of these assumptions may be found in examples listed in section 4. Our focus in the following is d= 4 dimension, but we will add some remarks for the other dimensions in section 9.

In terms of the property of the energy-momentum tensor, the claim is that under the above assumptions, whenever the trace of the energy-momentum tensor is a divergence of the virial current

T^{µ}_{µ}=∂^{µ}J_{µ} , (6.1)

the virial current can be removed by the improvement. Or equivalently, it is a derivative of a certain local scalar operator

T^{µ}_{µ}=∂^{µ}∂_{µ}L . (6.2)