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=kσ+ikτ and ¯k=kσ−ikτ)
⟨T(k)T(p)⟩=ck3
¯kδ(k+p)
⟨T(k)Θ(p)⟩=ek2log|k|2δ(k+p)
⟨Θ(k)Θ(p)⟩=h|k|2log|k|2δ(k+p)
⟨T(k) ¯T(p)⟩=w|k|2log|k|2δ(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|2δ(k+p)
⟨O(k)T(p)⟩=(
cOT −4ηcOOlog|k|2)k
¯kδ(k+p)
⟨O(k)Θ(p)⟩=ηcOOlog|k|2δ(k+p)
⟨T(k)T(p)⟩=(
c−8ηcOT log|k|2+ 16η2cOO(log|k|2)2)k3
¯kδ(k+p)
⟨T(k)Θ(p)⟩= (2ηcOT log|k|2−4η2cOO(log|k|2)2)k2δ(k+p)
⟨Θ(k)Θ(p)⟩= (
−1
2ηcOT log|k|2+η2cOO(log|k|2)2 )
|k|2δ(k+p) (5.19) up to contact terms. We may improve the energy-momentum tensor so that cOT 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”∫ d2x√
|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 ∫
d2x√
|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 cM(2)=−
∫
d2xG(µ)⟨Θ(x)Θ(0)⟩ Θ =BIOI
χIJ =− d dlogµ
∫
d2xG(µ)(x)⟨OI(x)OJ(0)⟩ (5.20) where
G(µ)(x) = 3πx2θ(1−µ|x|) , (5.21) withθ(x) being the step function so thatG(µ)(x) has only support|x| ≥µ−1. The metricχIJ is positive definite from unitarity (no dangerous contact term will contribute because ddGlog(µ)µ has a support only when OI are separated). It can be easily shown that
dcM(2)
dlogµ =BIχIJBJ ≥0. (5.22)
We note that this cM(2)-function is equivalent to the one in section 5.1.
One can now repeat the same analysis in d≥2. We define cM(d)=−
∫
ddxG(d)(µ)⟨Θ(x)Θ(0)⟩ Θ =Tµµ=BIOI
χIJ =− d dlogµ
∫
ddxG(d)(µ)⟨OI(x)OJ(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) = 3πxdθ(1−µ|x|) . (5.24) The metric is again positive definite from unitarity. It can be easily shown that
dcM(d)
dlogµ =BIχIJBJ ≥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 cM(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[(2xνxρxσ−2x2xνηρσ−x2xσηνρ)⟨Tµν(x)Tρσ(0)⟩] =−3x2⟨Θ(x)Θ(0)⟩ . (5.26) It enable us to integrate cM(2) by part to rewrite the averaged quantity (5.20) into the local form
cM(2)(µ) = 2π2(2xµxνxρxσ−x2xµxνηρσ−x2xρxσηµν−x2xµxσηνρ)⟨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 BI ̸= 0 because there is no reason why cM(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
Tµµ=∂µJµ , (6.1)
the virial current can be removed by the improvement. Or equivalently, it is a derivative of a certain local scalar operator
Tµµ=∂µ∂µL . (6.2)