Cross Ratios and Four-Point Kinematics

Four generic points in $d\geq2$ have two independent conformal invariants. Cross ratios compress the geometry, but they do not erase channel, prefactor, reality, or branch information. Euclidean configurations satisfy a conjugation condition; Lorentzian orderings arise as distinct boundary values in which the two variables can be independent. A complete kinematic convention must therefore name both the invariant pair and its analytic sheet.

Required background. Scalar Two- and Three-Point Functions fixes external normalization and conformal covariance. Conformal Geometry, Maps, and Compactification supplies finite-map domains. Helpful background. Branches, Sheets, Analytic Continuation, and Monodromy supplies continuation language.

Two invariants and one prefactor convention

Define

$$u=\frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2}, \qquad v=\frac{x_{14}^2x_{23}^2}{x_{13}^2x_{24}^2}, \qquad \Delta_{ij}=\Delta_i-\Delta_j.$$

A useful scalar four-point convention is

$$\begin{aligned} \langle\mathcal O_1\mathcal O_2\mathcal O_3\mathcal O_4\rangle ={}& \frac{1} {(x_{12}^2)^{(\Delta_1+\Delta_2)/2} (x_{34}^2)^{(\Delta_3+\Delta_4)/2}}\\ &\times \left(\frac{x_{24}^2}{x_{14}^2}\right)^{\Delta_{12}/2} \left(\frac{x_{14}^2}{x_{13}^2}\right)^{\Delta_{34}/2} \mathcal G(u,v). \end{aligned}$$

Every point has the required scaling weight. Other prefactors move powers of $u$ and $v$ between the kinematic factor and $\mathcal G$; crossing equations must be translated rather than compared term by term. Standard scalar conventions and their crossing transformations are reviewed in Poland, Rychkov, and Vichi 2019, §§ II.A–II.B.

Introduce $z,\bar z$ by

$$u=z\bar z, \qquad v=(1-z)(1-\bar z).$$

A conformal transformation places the points in a two-plane and, formally, at $(0,z,1,\infty)$. In a genuine Euclidean configuration, $\bar z=z^*$. For four collinear points in the standard ordered sector, $z=\bar z\in(0,1)$. The discriminant

$$(1+u-v)^2-4u=(z-\bar z)^2$$

vanishes for collinear configurations, where the conformal frame has an enhanced stabilizer; it is not a physical singularity by itself.

Channels and the six anharmonic images

The three pairings are:

$$\begin{array}{c|c|c} \text{channel}&\text{paired points}&\text{OPE limit}\\ \hline s&(12)(34)&z,\bar z\to0\\ t&(23)(14)&z,\bar z\to1\\ u&(13)(24)&z,\bar z\to\infty \end{array}$$

Point permutations generate six distinct images. Applying the same fractional-linear map to $z$ and $\bar z$ gives:

Map of $z$	Map of $(u,v)$	Distinguished limit
$z$	$(u,v)$	$s$ at $z\to0$
$1-z$	$(v,u)$	Exchanges $s$ and $t$
$1/z$	$(1/u,v/u)$	Exchanges zero and infinity
$1/(1-z)$	$(1/v,u/v)$	Sends $z=1$ to infinity
$z/(z-1)$	$(u/v,1/v)$	Generated by exchanging points 1 and 2
$(z-1)/z$	$(v/u,1/u)$	Remaining anharmonic image

These pairs follow directly from $u=z\bar z$ and $v=(1-z)(1-\bar z)$. They provide a round-trip test for a crossing implementation: composing the same transposition twice must return $(u,v)$, and the six images must close.

Euclidean region and Lorentzian sheets

The figure organizes channel choices and limits together with analytic continuations. Inspect the difference between changing pairings within the Euclidean region and winding around a branch point.

The $s$, $t$, and $u$ channels are centered at $z=0,1,\infty$. In Euclidean signature $\bar z=z^*$ and a chosen channel converges in its radial domain. Lorentzian continuation allows $z$ and $\bar z$ to approach independent real values with ordering-dependent $i\epsilon$ prescriptions. The ordinary discontinuity subtracts values on opposite sides of a named cut; the double discontinuity instead combines the Euclidean value with clockwise and counterclockwise continuations and external-dimension phases. The diagram is schematic; neither combination is defined without its continuation paths.

The accessible equivalent is:

Domain	Relation between $z$ and $\bar z$	Required extra data	Permitted statement
Separated Euclidean points	$\bar z=z^*$	Choice of prefactor and OPE channel	Single-valued Euclidean correlator away from coincidences
Euclidean collinear sector	$z=\bar z\in\mathbb R$ with an ordering interval	Ordering of the four points	Real boundary within one sector
Complexified configuration	$z,\bar z$ independent complex variables	Branch cuts and homotopy class of the path	Analytic continuation on a named sheet
Lorentzian Wightman ordering	Independent boundary values, often real after the limit	Strict $i\epsilon$ ordering	One ordered correlator
Time-ordered correlator	Sum of boundary values with step functions	Time-ordering prescription and contact terms	Distribution-valued time-ordered function
Coincident or null separation	At $0$, $1$, infinity, or another singular locus	Distributional extension and limiting procedure	Only a qualified limit

The principal Euclidean square root in

$$\rho(z)=\frac{z}{(1+\sqrt{1-z})^2}$$

uses a cut along $[1,\infty)$. Continuing around $z=1$ reverses the sign of the square root and changes $\rho$ to its reciprocal. Thus a radial series with $\lvert\rho\rvert<1$ cannot be carried around the branch point term by term and still be called the same convergent expansion. The cut-plane map and radial domain are derived in Hogervorst and Rychkov 2013, §§ 2–3.

Identical-scalar crossing check

For identical scalars with

$$\langle\phi_1\phi_2\phi_3\phi_4\rangle =\frac{\mathcal G(u,v)} {(x_{12}^2x_{34}^2)^{\Delta_\phi}},$$

exchange $x_1\leftrightarrow x_3$ sends $(u,v)\to(v,u)$. Equality of the correlator gives

$$\boxed{ v^{\Delta_\phi}\mathcal G(u,v) =u^{\Delta_\phi}\mathcal G(v,u). }$$

The powers arise from the prefactor; with another prefactor the reduced equation changes. The equality is first Euclidean and then extends to other sheets only through a declared analytic continuation.

Common pitfalls

$z$ and $\bar z$ are always complex conjugates. That is the Euclidean reality condition. Lorentzian boundary values generally treat them as independent before the limit.

A point permutation never changes the sheet. Its algebraic map is only part of the operation. A Lorentzian continuation path can wind around branch points.

The reduced correlator is convention independent. Only the full correlator is. Prefactor changes multiply $\mathcal G$ and its crossing equation by powers of $u,v$.

Exercises

Verify the map $z\mapsto z/(z-1)$ in the table.

Solution

Let $z'=z/(z-1)$ and $\bar z'=\bar z/(\bar z-1)$. Then

$$u'=z'\bar z'=\frac{u}{v}, \qquad v'=(1-z')(1-\bar z')=\frac1v.$$

Applying the map again returns $z$, so it represents a transposition.

References

• Hogervorst, Matthijs, and Slava Rychkov. “Radial Coordinates for Conformal Blocks.” Physical Review D 87 (2013): 106004. DOI; Open PDF
• Poland, David, Slava Rychkov, and Alessandro Vichi. “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications.” Reviews of Modern Physics 91 (2019): 015002. DOI; Open PDF