Conformal Partial Wave Expansion for Four-Point Functions

1. From Correlators to Cross-Ratios: A Recap

As we've seen, conformal invariance fixes the form of correlators of primary operators. For scalars, we have:

• Two-point function:
  $⟨{\mathcal{O}}_1(x_1){\mathcal{O}}_2(x_2)⟩=\frac{C_{12}{\delta }_{{\Delta }_1,{\Delta }_2}}{|x_{12}{|}^{2{\Delta }_1}}$
• Three-point function:
  $⟨{\mathcal{O}}_1(x_1){\mathcal{O}}_2(x_2){\mathcal{O}}_3(x_3)⟩=\frac{f_{123}}{|x_{12}{|}^{{\Delta }_1+{\Delta }_2-{\Delta }_3}|x_{23}{|}^{{\Delta }_2+{\Delta }_3-{\Delta }_1}|x_{13}{|}^{{\Delta }_1+{\Delta }_3-{\Delta }_2}}$
  Here, $C_{12}$ and $f_{123}$ are constants (the OPE coefficients) that, along with the spectrum of dimensions $\Delta_i$ and spins $\ell_i$, constitute the "data" of a CFT.

For a four-point function of identical scalar primaries $\phi$ with dimension $\Delta_\phi$, the situation changes. Conformal symmetry only constrains its form to:

where $g(u, v)$ is an arbitrary function of the two independent, conformally invariant cross-ratios:

The central question of the bootstrap is: what constrains $g(u,v)$? The answer lies in the Operator Product Expansion.

2. Decomposing the Four-Point Function with the OPE

The OPE is not just a short-distance approximation; it's an exact operator statement that can be used inside correlation functions, provided the operator insertions are appropriately ordered. We can use it to systematically break down the four-point function. The key idea is to apply the OPE to the pair $\phi(x_1)\phi(x_2)$ and, separately, to $\phi(x_3)\phi(x_4)$.

To see how this works in detail, let's turn to a cornerstone resource for the modern bootstrap.

As derived in the notes, applying the OPE in the (12)(34) channel leads to the following expression:

Comparing this to our earlier formula, we see that the previously arbitrary function $g(u, v)$ is now expressed as a sum:

This is the conformal partial wave expansion (or conformal block decomposition). Let's break down its components:

• $\sum_{\mathcal{O}}$: The sum runs over all primary operators $\mathcal{O}$ in the theory's spectrum.
• $f_{\phi\phi\mathcal{O}}^2$: These are the squared OPE coefficients. Since we assume a unitary theory, these are real and positive ($f_{\phi\phi\mathcal{O}}^2 \ge 0$). This term is theory-dependent (dynamics).
• $g_{\Delta_{\mathcal{O}}, \ell_{\mathcal{O}}}(u, v)$: These are the conformal blocks or conformal partial waves. Each block $g_{\Delta, \ell}$ is a universal function of the cross-ratios, completely determined by the conformal symmetry. It represents the contribution from one entire conformal multiplet (the primary $\mathcal{O}$ with dimension $\Delta$ and spin $\ell$ and all its descendants) being exchanged between the two pairs of operators. This term is theory-independent (kinematics).

This decomposition separates the problem beautifully: the universal kinematic blocks are computable once and for all, while the theory-specific data are the spectrum $\{\Delta_{\mathcal{O}}, \ell_{\mathcal{O}}\}$ and the OPE coefficients $f_{\phi\phi\mathcal{O}}$.

3. The Block as a Projection onto a Conformal Multiplet

To gain a more profound understanding of the conformal block, it's helpful to view it from the perspective of radial quantization. In this picture, the OPE corresponds to replacing the state created by two operators with a sum over states corresponding to single operators at the origin.

The conformal block $g_{\Delta, \ell}$ can be interpreted as the result of inserting a projector $\mathbb{P}_{\Delta, \ell}$ onto the conformal multiplet of a single primary operator $\mathcal{O}_{\Delta, \ell}$ into the four-point function.

This perspective clarifies why the blocks $g_{\Delta, \ell}(u,v)$ must be functions of the cross-ratios alone. Since the projector $\mathbb{P}_{\Delta, \ell}$ commutes with all conformal generators by construction, the object $\langle \phi\phi \mathbb{P}_{\Delta, \ell} \phi\phi \rangle$ must transform under conformal transformations just like the original four-point function. Therefore, it must also be proportional to $g(u,v)/|x_{12}|^{2\Delta_\phi} |x_{34}|^{2\Delta_\phi}$, defining the function $g_{\Delta, \ell}(u,v)$.

4. The Casimir Equation: A Way to Compute the Blocks

We've established what conformal blocks are, but how do we compute them? Your background in representation theory will be particularly useful here. The key is the quadratic Casimir operator of the conformal algebra $\mathfrak{so}(d+1, 1)$.

Just as irreducible representations of $\mathfrak{su}(2)$ are labeled by eigenvalues of the Casimir $J^2$, irreducible representations of the conformal group (i.e., conformal multiplets) are labeled by the eigenvalues of its quadratic Casimir $\mathcal{C}$. This eigenvalue depends only on the dimension $\Delta$ and spin $\ell$ of the primary operator.

Since a conformal block, by definition, projects onto a single irreducible representation, it must be an eigenfunction of the Casimir operator.

The result is a second-order partial differential equation for the conformal block:

where $\mathcal{D}$ is a differential operator in the variables $u$ and $v$ (or $z, \bar{z}$ as used in the text, where $u=z\bar{z}$ and $v=(1-z)(1-\bar{z})$), and the eigenvalue is $\lambda_{\Delta, \ell} = \Delta(\Delta-d) + \ell(\ell+d-2)$.

This "Casimir equation," supplemented with the correct boundary conditions from the OPE limit ($u \to 0$), uniquely determines the functions $g_{\Delta, \ell}(u, v)$. While general closed-form solutions are not known in odd dimensions, they can be found in even dimensions (as shown in equations 166-168 of the notes) and are expressed in terms of hypergeometric functions. In any dimension, they can be computed as a power series.

Conclusion

This lesson is the first crucial step into the modern bootstrap. We have moved from a picture of a four-point function as an unknown $g(u,v)$ to a precise, structured decomposition.

Key Takeaways:

• The Operator Product Expansion allows us to decompose any four-point function into a conformal partial wave expansion.
• This expansion is a sum over all primary operators $\mathcal{O}$ in the theory, of the form $\sum_{\mathcal{O}} f_{\phi\phi\mathcal{O}}^2 g_{\Delta_{\mathcal{O}}, \ell_{\mathcal{O}}}(u, v)$.
• The expansion neatly separates dynamics (the theory-specific spectrum and OPE coefficients $f_{\phi\phi\mathcal{O}}$) from kinematics (the universal conformal blocks $g_{\Delta, \ell}$).
• A conformal block represents the total contribution from a single conformal multiplet.
• Conformal blocks are eigenfunctions of the conformal Casimir operator, which provides a powerful method for computing them via a differential equation.

Next Lesson Preview:
So far, we have only applied the OPE in one way: pairing operators (12) and (34). But what if we paired them differently, say (14) and (23)? The four-point function is a single, well-defined object and cannot depend on how we choose to compute it. The requirement that these different "channels" of OPE expansion must yield the same result is called crossing symmetry. This will give us the fundamental bootstrap equations, which form a powerful, self-consistent constraint on the allowed CFT data.