One-point function in CFT on an infinite strip through scaling analysis

Question

In Philippe Di Francesco's book on Conformal Field Theory in section 11.2.3 on the Infinite Strip, the one point function of a primary operator (with scaling dimension $\Delta$) is calculated by considering a conformal mapping from the upper half plane. For an infintie strip of width L this is found to be: $$ \langle \Phi (w,\bar{w}) \rangle_{strip} = \left(\frac{2iL}{\pi} \right)^{\Delta} \frac{1}{[\sin(\pi v /L)]^{\Delta}} $$

With $w = u + iv$ and $u$ being the longitudinal coordinate and $v$ the transverse. In the limit $v << L$ we have $$\langle \Phi (v) \rangle_{strip} \propto \frac{1}{v^\Delta} [1 + \frac{1}{6}\pi^2 \Delta (v/L)^2 + ... ] $$

The book then states that this is compatible with the more general result of Fisher and de Gennes obtained through a scaling analysis in dimension $d$:

$$\langle \Phi (v) \rangle_{strip} \sim \frac{1}{v^\Delta} [1 + const.(v/L)^d + ... ] $$

My question is what is the derivation of this general result. The only Fisher and de Gennes paper that I was able to find was this paper from 1978 written in French. I am unable to understand the text and the equations there don't appear to be very relevant. I would be grateful if someone could provide a detailed derivation of this general result.

Answer 1

I believe Fisher and de Gennes in the paper you link only conjecture the result (and only explicitly for d = 3). See below:

I'm not sure one can give a general proof. Even some more recent papers, such as this one: https://arxiv.org/abs/1503.00254 still state the Fisher-de Gennes scaling as an ansatz. Although, like in the proof you gave for d = 2, maybe one can use some universal form of the one-point function?