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In Gubser's famous paper on breaking Abelian gauge symmetry near a black hole horizon, he talks about how to connect the asymptotic behavior of the scalar field $\psi$ to the scaling dimension $\Delta$ of the dual operator. Solving the equation of motion for $\psi$ (Eqn. 9 in the text),

$$\psi''+\frac{-1+(8r-4)k+4(4r^3-1)/L^2}{(r-1)(-1+4kr+4r(r^2+r+1)/L^2)}\psi'+\frac{m_{eff}^2}{(r-1)(-1+4kr+4r(r^2+r+1)/L^2)}\psi=0$$

he finds that

$$\psi \sim \frac{A_\psi}{r^{3-\Delta}}+\frac{B_{\psi}}{r^\Delta}$$

where $A_\psi$ and $B_\psi$ are constants. I'm a little confused how he gets this expansion; i.e., how he gets this specific $r$ dependence. A similar calculation is done in "Exact Gravity Dual of a Gapless Superconductor", by Koutsoumbas et. al., where an exact form of the hair is given in terms of the greatly simplified MTZ solution:

$$\psi(r)=-\sqrt{\frac{3}{4\pi G}}\frac{r_0}{r+r_0}$$

The asymptotic solution is given in equation 5.12:

$$\psi\sim \frac{\psi^1}{r}+\frac{\psi^2}{r^2}+...$$

If these two expansions are equal, then $\Delta=2$. This agrees with Gubser's result (below Eqn. 17), but I am not sure if this is intentional or not.

Ultimately, I have three interconnected questions that can be summed up as follows:

1) How, exactly, does the conformal dimension come out of Gubser's calculation? Is it connected to k?

2) Is the asymptotic expansion performed by Gubser and Koutsoumbas equivalent?

3) What is the physical significance of having $\Delta=2$ in both cases?

Any explanation or clarifying references would be appreciated.

EDIT: Let me clarify the first question. Taking the asymptotic limits of the above expressions, the differential equation for $\psi$ can be simplified to

$$ \psi''+\frac{4}{r}\psi'+\frac{1}{4}m_{eff}^2\left(\frac{L}{r}\right)^4\approx 0$$

From this, we can then solve for $\psi$ and take a further expansion to get

$$A+B\frac{1}{L^2 m_{eff}^2 r^2}(L^4m_{eff}^4\alpha-\beta \sqrt{-L^2m_{eff}^2})+...$$

and so on, where A, B, $\alpha$, and $\beta$ are constants. Now I know that we can relate the mass to the conformal dimension by $L^2m^2=\Delta(\Delta -3)$ in AdS$_4$, but my confusion with Gubser's calculation is the following:

1a) Why does he get an expansion in terms of r's to the power of $\Delta$? Shouldn't it be in integer powers of $r$ (like Koutsoumbas' calculation), with the conformal dimension multiplying each term?

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  • $\begingroup$ I might attempt a solution at some point if no other does it before me. A couple of questions for you. Have you tried to expand the equations of motion in the UV and obtain an asymptotic solution? Or, strip off the $x^{\mu}$ dependence and try to solve the rest. There are lecture notes explaining these steps in simpler examples.This gives you the power law. Since I have not studied both papers I cannot address the other two questions. However, there are general statements that are true since $\Delta$ is the conformal dimension of the operators (unitarity bounds, etc). $\endgroup$ Apr 29 '20 at 16:26
  • $\begingroup$ @Konstantinos Sorry if I wasn't clear; just expanded my question to clarify 1a) and show my (possible) solution. $\endgroup$ Apr 29 '20 at 17:03
  • $\begingroup$ @Konstantinos If you expand upon your comment and include some external references in an answer, I'll give you the bounty assuming no one supplies an answer before you. $\endgroup$ May 5 '20 at 2:11
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    $\begingroup$ Hi Joshua, I have not forgotten about this post. I just wanted to see if someone could provide a full-fledged answer to the post. I will start a write-up with potentially useful stuff today after I finish some computations for a project $\endgroup$ May 5 '20 at 7:43
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I will be doing a similar example that is quite simpler for illustrative purposes. The following example has been analyzed in various places in the literature. I will be giving them at the end.

Assume a five-dimensional AdS spacetime in the following parameterization

$ \begin{equation} ds^2=\frac{1}{x_0^2} (\eta_{\mu \nu} dx^{\mu} dx^{\nu} + dx^2_0) \end{equation} $

In this parameterization, the conformal boundary of space is reached for $x_0 \rightarrow 0$. Another frequent choice is the one corresponding to the change of variables $x_0 \rightarrow \frac{1}{r}$.

We want to study a massive scalar and its dynamics that is governed by the action

$$ \begin{equation} S = \int d^5x \sqrt{-g} (g^{AB} \partial_A \phi \partial_B \phi + m^2 \phi^2 ) \end{equation} $$

where $\phi$ is the scalar field under consideration and the capital letters are indices in the bulk of the theory. The field, of course, can depend on any of the coordinates and so we abbreviated essentially the formally written $\phi(x_0,x_{\mu})$ by $\phi$ in the above.

From standard techniques, it is easy to show that the equations of motion

$$ \begin{equation} \begin{split} \frac{1}{\sqrt{g}} \partial_{A} (\sqrt{-g} g^{AB} \partial_B \phi) - m^2 \phi &= 0 \\ \partial_{x_0} \left( \frac{1}{x_0^3} \partial_{x_0} \phi \right) + \partial_{\mu} \left( \frac{1}{x_0^3} \partial^{\mu} \phi \right) = \frac{m^2}{x_0^5} \phi \end{split} \end{equation} $$

The important thing to understand is that from the above equation, the $x_0$ dependence will yield the relation to the conformal dimension associated to the boundary operator.

Focusing on the $x_0$ part of the above differential equation yields to power like solutions. In other words, assume an ansatz $\phi = x_0^{\Delta}$ and obtain

$$ \begin{equation} x_0^{\Delta} (-m^2+\Delta(\Delta-4)) = 0 \end{equation} $$

and a little cute Mma "hack" for the above

x0^5 D[1/x0^3 D[f[x0], x0], x0] - m^2 f[x0] /. 
  f -> (#^\[CapitalDelta] &) // Factor

From which you obtain the infamous relation between the bulk AdS mass of the field and the conformal dimension of the operator. It is a very straightforward generalization to obtain the equivalent for a $(d+1)$-dimensional AdS spacetime.

Now one can start to think about what kind of values the dimension can get and what that means for the operator. I am skipping the discussion here, but you can find details in all the references at the end of the answer.

A next step of the analysis would be to decompose the scalar field (separate variables) by performing a Fourier decomposition. That is

$$ \phi = e^{i ~ k^{\mu} ~ x_{\mu}} f(x_0) $$

A brief comment: The difference that I see between the expansions of Gubser and Koutsoubas is that it seems that the latter author has specified the scaling dimension of the scalar operator. I have not studied the papers, but I am taking your word that we are dealing with the same gravity construction in both works. I also don't see anything wrong with Gubser's expression. He has integer powers.

Regarding the physical significance/ special meaning of that particular value of the conformal dimension, I have no idea. Maybe it is related to superconductors and their properties. Maybe they wanted a particular relevant operator of the theory (???) -see on page 47 of the first reference for some discussion on (ir)relevant and marginal operators.

A common practice here is to not include links to pdf but rather abstract pages so I am choosing to present the references in the following way as I cannot find a link to an abstract page for the first one.

A place where you can find a neat analysis that is concise is the first result after you google search "alberto zaffaroni lectures ads/cft"

There are many formal analyses. The SUGRA book by Freedman and Van-Proeyen would be such a place, as well as the famous review by D'Hoker and Freedman, but pretty much all lecture notes on the AdS/CFT contain that example and discussion. For more applied matters and discussion you might want to have a loot at the book by Ammon and Erdmenger.

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