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In Di Francesco's book, the four point function in minimal model is studied in eq(9.88), $$ \langle \phi_{(r_1, s_1)}(0) \phi_{(2,1)}(z) \phi_{(r_3, s_3)}(1) \phi_{(r_4, s_4)}(\infty) \rangle\\ \sim |z|^{\#} |1-z|^{\#} \Big(\frac{s(b)\sin(a+b+c)}{s(a+c)} |I_1(z)|^2 + \frac{s(a)\sin(c)}{s(a+c)} |I_2(z)|^2 \Big) \ . $$

In particular, the limit $z \to 0$ of the above is analyzed in eq(9.89), giving $$ \langle \phi_{(r_1, s_1)}(0) \phi_{(2,1)}(z) \phi_{(r_3, s_3)}(1) \phi_{(r_4, s_4)}(\infty) \rangle\\ \sim |z|^{\#} |1-z|^{\#} \Big(\frac{s(b)\sin(a+b+c)}{s(a+c)} N_1^2 + \frac{s(a)\sin(c)}{s(a+c)} N_2^2 |z|^{2(1 + a + c)} + ... \Big) \ . $$

This computation is then computed against the the result (eq (9.61)) following from OPEs, for example, $$ \frac{s(a)s(a+b+c)}{s(a+c)} N_1^2 \sim C_{r_1, s_1; 2,1}^{r_1+1 , s_1}C_{r_3, s_3; r_4, s_4}^{r_1 + 1, s_1} \ . $$

But I'm confused by this relation. On the right, the $C$'s are structure constants, in particular, it can vanish if the $r$'s and $s$'s do not get along, $$ C_{r_1, s_1; r_2, s_2}^{r_3, s_3} = 0 \ \text{when} \ r_3 \ge \min(r_1 + r_2 - 1, 2p' - r_1 - r_2 - 1) . $$

However, with the given expression for $N_1, N_2$ in terms of $\Gamma$ functions, I fail to see the left hand side of the $\sim$ vanishes when (for example) $$ r_1 + 1 = \min(r_3 + r_4 - 1, 2p' - r_3 - r_4 - 1) \ . $$ Explicitly, $$ \frac{s(a)s(a+b+c)}{s(a+c)} N_1^2 = \frac{\sin\pi a \sin\pi(a+b+c)}{\sin\pi(a+c)} \frac{\Gamma(-1 - a - b - c)^2\Gamma(b + 1)^2}{\Gamma(- a - c)^2} \ , $$ where $a = 2 \alpha_+\alpha_{r_1, s_1}$, $b = 2 \alpha_+\alpha_{r_3, s_3}$, $c = 2\alpha_+ \alpha_{2,1}$, and $$ a+b+c = -2 + s_1 + s_3 - \frac{p(r_1 + r_3 - 1)}{p'} , \\ b + 1 = \frac{p - pr_3 + p' s_3}{p'}, \\ a+c = -1 + s_1 - \frac{p r_1}{p'} \ . $$

Assuming $p'$ large and $r_3, r_4$ are small, and therefore $r_1 + 1 = r_3 + r_4 - 1$, $$ a + b + c = -2 + s_1 + s_3 - \frac{p(-2 + 2r_3 + r_4)}{p'}, \\ b + 1 = \frac{p - p r_3 + p' s_3}{p'},\\ a + c = -1 + s1 - \frac{p(-1 + r_3 + r_4)}{p'} \ . $$ I don't think these values makes the $s(a)s(a+b+c)/s(a+c) N_2^2 = 0$. Am I misunderstanding the book?

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Your calculations look OK but you are making a conceptual mistake. The structure constants $C$ do not have to vanish when fusion rules are violated: rather, they are not defined. If you have a formula for structure constants (or combinations thereof) that can be analytically continued to situations that violate fusion rules, nothing guarantees that it will vanish.

In your case, your combination of sines is basically a product of two structure constants $CC$, and you are computing it in a case when fusion rules are violated. But it does not have to be zero. See for instance https://arxiv.org/abs/hep-th/0505063 by Al. Zamolodchikov.

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  • $\begingroup$ But at eq (9.56) in the book says "This coefficient vanishes if the fusion rule is not allowed", which sounds very natural. Also in section 8.4 where the author discuss fusion rules, it seems that the structure constant $g(h_0, h_1, h) \ne 0$ leads to allowed fusion (8.78). Maybe the textbook statement should be taken with a grain of salt? $\endgroup$
    – user31415926
    Commented Nov 29, 2022 at 17:02
  • $\begingroup$ Besides, eq (3.12) in Zamolodchikov seems to be saying that outside the triangle-relation (specified by 3-inequalities), the structure constant always vanishes (the "otherwise" scenario). This sort of matches my intuition as well: outside some range (or, when the fusion rule is violated), the structure constant automatically vanishes. Unless we are talking about different structure constant, since I notice that there's a continuous one, $C_M(\alpha, ...)$. $\endgroup$
    – user31415926
    Commented Nov 29, 2022 at 17:13
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    $\begingroup$ Indeed, the statement around (9.56) in the Big Yellow Book sounds natural but it is probably wrong in general. The statement of vanishing in Zamolodchikov (3.12) is basically a definition of the GMM structure constants: the question is to which extent these structure constants are values of the three-point function $C_M$, which is a smooth function of three variables. In (3.16) we see that $C_M$ does not vanish when we would like it to vanish, which is why we also need the prefactors $f$. $\endgroup$
    – Sylvain Ribault
    Commented Nov 30, 2022 at 10:12
  • $\begingroup$ Ok this is quite different from what I previously believed. But let me try to summarize: the minimal model structure constants (as appeared in the fusion rules and OPEs) are different from the 3-point functions, where the latter is somewhat "more continuous", and the former is "more vanishing/discrete". Is this the correct summary of your point of view? $\endgroup$
    – user31415926
    Commented Dec 1, 2022 at 16:21
  • $\begingroup$ If so, then I would still be puzzled: what happens if one discovers 3-point functions forbidden by the fusion rules? Should one change the fusion rules to incorporate the discovery, or just proceed with the old rules? Or are both valid actions that lead to "different theories"? More generally, what is the rule of determining the fusion rules from studying the correlators? $\endgroup$
    – user31415926
    Commented Dec 1, 2022 at 16:27

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