I am studying this paper of Luty, Polchinski and Rattazzi about the $a-$theorem in $d=4$ and the possibly allowed RG flow between fixed points of a theory with metric $g_{\mu\nu}$.

First of all, they cosider the theory on a metric $\hat{g}_{\mu\nu}$ which depends on $g_{\mu\nu}$ in terms of a conformal factor which embed the dilaton field $\tau(x)$ $$ \hat{g}_{\mu\nu} = e^{-2\tau}g_{\mu\nu}. $$

We define in the following $$ \Omega = e^{-\tau} = 1+\frac{\varphi}{f} $$ where $\varphi$ is the physical dilaton.

Under Weyl transformation

$$g_{\mu\nu} \rightarrow e^{2\sigma} g_{\mu\nu}\\ \mathcal{O} \rightarrow e^{-\sigma\Delta}\mathcal{O} $$

where $\mathcal{O}$ is a primary operator of dimension $\Delta$ and this transformation is realized in the theory with the redressed metric $\hat{g}_{\mu\nu}$ just by letting $\tau$ transform as a Goldston boson, namely

$$ \tau \rightarrow \tau - \sigma $$

provided we also redress the primary operators, that is we construct the theory of $\hat{\mathcal{O}}=e^{\tau\Delta}\mathcal{O}$.

The authors write the most general effective theory of the $CFT_{IR}$ + dilaton sector. They notice that in the IR, there may be relevant operators which can dominate dilaton-dilaton scattering amplitude (as well as CFT dynamics). Hence, they study these possible relevant deformations.

It is easy to see that only two possible deformations can take place are

$$ S_1[\mathcal{O}] = \int d^4 x \sqrt{-g}(m\Omega)^{4-\Delta} \mathcal{O}\\ S_2[\mathcal{O}] = \int d^4 x \sqrt{-g}(m\Omega)^{2-\Delta}\left( R(g)-\Omega^{-1}\square \Omega \right) \mathcal{O} $$

where $m$ is the mass scale responsible for the deformation of the $CFT_{UV}$

  • First question At page 8, they say that any $CFT_{IR}$ has at least the identity operator with dimension $\Delta=0$. This implies that $S_1[\mathcal{O}=1]$ is a cosmological constant term that gives contributions to the dilaton-dilaton scattering that are larger than the Wess-Zumino term of the effective action. They then decide to fine-tune this relevant deformation to vanish. Why do they do this fine-tuning? How can you fine-tune this term to zero? Can anyone give me an example? I see that the cosmological term can give a contribution to the $2\rightarrow 2$ scattering from the term $m^4/f^4 \varphi^4$. What's the problem with this contribution?
  • Second question They say that if $\Delta <2$, then $S_2[\mathcal{O}]$ gives a relevant coupling of the dilaton to the CFT. I agree with that. However, they state this contribution cannot be fine-tuned to zero.Actually, the presence of this term is not a problem because on-shell this contribution goes away (I agree) and the $CFT_{IR}$ dynamics in the flat space is not affected. Why can $S_2[\mathcal{O}]$ not be fine-tuned to zero?? Why is the $CFT_{IR}$ dynamics in flat-space not affected?
  • Third question Look page 9 and consider we have an operator with $\Delta< 2$; then the leading contribution of $S_2[\mathcal{O}]$ to the $2\rightarrow 2$ scattering amplitude of dilatons is $$ A(p_1,...,p_4) \sim \left( \frac{m^{2-\Delta}}{f} \right)^4 p_1^2...p_4^2 \langle \mathcal{O(p_1)}...\mathcal{O(p_4)}\rangle $$ where $\langle ... \rangle$ is to be computed with the unperturbed CFT (I agree). The authors state that this amplitude is singular in the IR but vanishes on-shell. While I agree with the fact that $A(p_1,...,p_4)=0$ on-shell, I don't see why in the IR (and off-shell) we have a singularity. Can you elucidate this point?


  • 2
    $\begingroup$ About the first question: the identity operator (which has dimension 0, not 1 as you write) must be tuned to zero because otherwise the IR theory is not a conformal theory in Minkowski space, or in fact you haven't in fact flowed at all to any IR and stayed in the UV. Indeed, this operator gives rise to a quartic potential $\Omega^4$. Insisting in a Minkowski vacuum and minimizing the potential you get $\Omega=0$ (for positive CC) or $\Omega=\infty$ (for negative CC), i.e. respectively to $f=0$ or $f=\infty$. More physically, one doesn't actually end in Mink. inv. vacuum when there is a CC $\endgroup$ – TwoBs Dec 15 '17 at 21:26
  • $\begingroup$ Then, also $S_2$would be tuned to zero, otherwise still the CFT IR is not conformal. But actually they say they cannot generally tune this contribution to zero. Why is $S_2$ so special? $\endgroup$ – apt45 Dec 15 '17 at 22:08
  • 2
    $\begingroup$ That $S_2$ could or should then be tuned to zero is a non-sequitur. The $S_2$ contains just deriivative interactions and it is perfectly consistent in Minkowski for any value of $f$. Besides, if you tune it to zero you would introduce a ghost in the IR theory, which can't happen if the UV theory was healthy and unitary. $\endgroup$ – TwoBs Dec 15 '17 at 23:40
  • $\begingroup$ @TwoBs Can you explain why you would introduce a ghost? $\endgroup$ – apt45 Dec 16 '17 at 0:18
  • $\begingroup$ Then, I don't see why you are talking about the value of $f$ which is not of interest, doesn't? $\endgroup$ – apt45 Dec 16 '17 at 0:31

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