Four-Point Functions and Cross-Ratios

Two- and three-point functions teach us the kinematics of conformal symmetry. Four-point functions are where conformal field theory becomes genuinely dynamical.

For scalar primaries, conformal symmetry fixes the two-point function up to normalizations and the three-point function up to OPE coefficients. But for four operators, conformal symmetry leaves a nontrivial function of two variables. That function is not a nuisance. It is the first place where the full interacting content of the CFT appears.

For AdS/CFT, this is a crucial transition:

$$\boxed{ \text{two-point data} \leftrightarrow \text{bulk spectrum and kinetic terms}, \qquad \text{three-point data} \leftrightarrow \text{bulk cubic couplings}, }$$

while

$$\boxed{ \text{four-point functions} \leftrightarrow \text{bulk exchange diagrams, contact interactions, loops, and locality constraints}. }$$

So this page is not merely about introducing $u$ and $v$. It is about learning the language in which crossing symmetry, OPE associativity, conformal blocks, Witten diagrams, and eventually bulk locality all speak to each other.

The first nontrivial correlator

Let $\mathcal O_i(x_i)$ be scalar primary operators in a Euclidean CFT on $\mathbb R^d$. We write

$$x_{ij}^2=(x_i-x_j)^2.$$

The two-point and three-point functions are highly constrained:

$$\langle \mathcal O_1(x_1)\mathcal O_2(x_2)\rangle = \frac{C_{12}\delta_{\Delta_1,\Delta_2}}{(x_{12}^2)^{\Delta_1}},$$

and

$$\langle \mathcal O_1(x_1)\mathcal O_2(x_2)\mathcal O_3(x_3) \rangle = \frac{C_{123}} {(x_{12}^2)^{\frac{\Delta_1+\Delta_2-\Delta_3}{2}} (x_{23}^2)^{\frac{\Delta_2+\Delta_3-\Delta_1}{2}} (x_{31}^2)^{\frac{\Delta_3+\Delta_1-\Delta_2}{2}}}.$$

The constants $C_{12}$ and $C_{123}$ are CFT data after a choice of normalization. No arbitrary function appears.

For four points, the situation changes. There are conformally invariant combinations of the positions, so conformal symmetry cannot determine the full answer. Schematically,

$$\langle \mathcal O_1(x_1)\mathcal O_2(x_2)\mathcal O_3(x_3)\mathcal O_4(x_4) \rangle = \text{fixed powers of }x_{ij}^2 \times \text{an arbitrary function of cross-ratios}.$$

That arbitrary function contains infinitely many pieces of CFT data. Its singular limits encode OPEs. Its consistency under permutations gives crossing equations. Its large-$N$ structure tells us whether the theory can look like weakly coupled gravity in AdS.

Why only cross-ratios can remain

A conformal transformation is a composition of translations, rotations, dilatations, and special conformal transformations. Translation invariance says a correlator can depend only on differences $x_i-x_j$. Rotation invariance says scalar correlators can depend only on squared distances $x_{ij}^2$. Scale invariance then says only dimensionless ratios of these distances may survive after extracting the required powers.

Special conformal transformations are the final constraint. Under

$$x^\mu \mapsto x^{\prime\mu} = \frac{x^\mu-b^\mu x^2}{1-2b\cdot x+b^2x^2},$$

one finds

$$\boxed{ {x_{ij}^{\prime}}^2 = \frac{x_{ij}^2}{\sigma_i\sigma_j}, \qquad \sigma_i=1-2b\cdot x_i+b^2x_i^2. }$$

Therefore the following two quantities are invariant:

$$\boxed{ u=\frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2}, \qquad v=\frac{x_{14}^2x_{23}^2}{x_{13}^2x_{24}^2}. }$$

The factors of $\sigma_i$ cancel between numerator and denominator. These are the standard conformal cross-ratios.

For four generic points in $d\geq 2$, there are two independent conformal invariants. In two dimensions, it is often more natural to use one complex cross-ratio $z$ and its antiholomorphic partner $\bar z$. In higher-dimensional Euclidean CFTs, one can also introduce $z,\bar z$ by

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

For Euclidean configurations, $z$ and $\bar z$ are complex conjugates when the four points lie in a real two-plane. After Lorentzian continuation, they should be treated as independent variables living on different analytic sheets.

Four scalar insertions have two independent conformal cross-ratios, $u$ and $v$. Different OPE limits organize the same four-point function into different channels: $(12)(34)$, $(14)(23)$, and $(13)(24)$. Crossing symmetry says that these decompositions describe the same correlator.

The general scalar four-point function

Let

$$\Delta_{ij}=\Delta_i-\Delta_j.$$

A convenient conformally covariant form for four scalar primaries is

$$\boxed{ \begin{aligned} &\left\langle \mathcal O_1(x_1)\mathcal O_2(x_2) \mathcal O_3(x_3)\mathcal O_4(x_4) \right\rangle \\[4pt] &\quad = \frac{1} {(x_{12}^2)^{\frac{\Delta_1+\Delta_2}{2}} (x_{34}^2)^{\frac{\Delta_3+\Delta_4}{2}}} \left(\frac{x_{24}^2}{x_{14}^2}\right)^{\frac{\Delta_{12}}{2}} \left(\frac{x_{14}^2}{x_{13}^2}\right)^{\frac{\Delta_{34}}{2}} \mathcal G(u,v). \end{aligned} }$$

The prefactor is conventional. It is chosen so that all nontrivial dependence is packaged into a dimensionless function $\mathcal G(u,v)$. Other prefactors are equally valid, but they move simple powers of $u$ and $v$ between the prefactor and the reduced correlator.

For identical scalar primaries $\phi$ of dimension $\Delta$, the formula simplifies to

$$\boxed{ \left\langle \phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4) \right\rangle = \frac{\mathcal G(u,v)}{(x_{12}^2x_{34}^2)^\Delta}. }$$

This is the most important normalization for first contact with the conformal bootstrap.

The identity operator contribution in the $12\to 34$ channel is then simply

$$\mathcal G(u,v)=1+\cdots, \qquad u\to 0, \quad v\to 1.$$

Here the dots are contributions from nontrivial operators in the $\phi\times\phi$ OPE.

Conformal frames

Cross-ratios become more transparent in a conformal frame. The conformal group can move four generic points into a two-dimensional plane and then fix three of them. A standard choice is

$$x_1\to 0, \qquad x_3\to 1, \qquad x_4\to \infty, \qquad x_2\to z.$$

Then the two cross-ratios become

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

This frame is extremely useful, but it hides some symmetry. The four original points are all on equal footing. Choosing a conformal frame is like fixing a gauge: it is convenient, but the final answer must be expressible in terms of invariant data.

The main OPE limits are:

$$\begin{array}{ccl} \text{$s$ channel: } x_1\to x_2 &\Longleftrightarrow& u\to 0,\quad v\to 1,\quad z,\bar z\to 0,\\[4pt] \text{$t$ channel: } x_2\to x_3 &\Longleftrightarrow& v\to 0,\quad u\to 1,\quad z,\bar z\to 1,\\[4pt] \text{$u$ channel: } x_1\to x_3 &\Longleftrightarrow& z,\bar z\to \infty\quad \text{after a suitable frame choice.} \end{array}$$

The names $s,t,u$ are borrowed from scattering theory. In a CFT, the channels are not scattering channels in flat spacetime, but the analogy is powerful: each channel corresponds to grouping the four operators into two OPE pairs.

Crossing symmetry

For identical scalar operators, the four-point function must be invariant under permutations of the insertions. This gives functional equations for $\mathcal G(u,v)$.

Using

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

exchange $x_1\leftrightarrow x_3$. The cross-ratios transform as

$$(u,v)\mapsto (v,u).$$

The prefactor changes, and equality of the correlator gives

$$\boxed{ \mathcal G(u,v) = \left(\frac{u}{v}\right)^\Delta \mathcal G(v,u). }$$

Another useful permutation is $x_1\leftrightarrow x_2$, under which

$$(u,v) \mapsto \left(\frac{u}{v},\frac1v\right),$$

and for a bosonic identical scalar one obtains

$$\boxed{ \mathcal G(u,v)=\mathcal G\left(\frac{u}{v},\frac1v\right). }$$

These equations are simple to write and hard to solve. Their meaning is deep: the OPE must be associative.

In the $s$ channel one expands the product $\phi(x_1)\phi(x_2)$ and separately $\phi(x_3)\phi(x_4)$. In the $t$ channel one instead expands $\phi(x_2)\phi(x_3)$ and $\phi(x_1)\phi(x_4)$. Both procedures compute the same function. Crossing symmetry is the equality of these two expansions.

This is the conceptual heart of the conformal bootstrap.

OPE interpretation

The OPE of two identical scalars has the schematic form

$$\phi(x_1)\phi(x_2) \sim \sum_{\mathcal O} \lambda_{\phi\phi\mathcal O} C_{\mathcal O}(x_{12},\partial_{x_2}) \mathcal O(x_2),$$

where the sum runs over primary operators and their descendants. Plugging this OPE into the four-point function gives an expansion of $\mathcal G(u,v)$.

For identical scalars in a unitary CFT, the schematic conformal block expansion is

$$\boxed{ \mathcal G(u,v) = \sum_{\mathcal O\in \phi\times\phi} \lambda_{\phi\phi\mathcal O}^2 G_{\Delta_{\mathcal O},\ell_{\mathcal O}}(u,v). }$$

The functions $G_{\Delta,\ell}(u,v)$ are conformal blocks. They are fixed by symmetry and encode the contribution of one primary operator together with all of its descendants. The coefficients $\lambda_{\phi\phi\mathcal O}^2$ are nonnegative in a reflection-positive Euclidean CFT when $\phi$ is real and the operators are normalized appropriately.

The identity operator contributes

$$G_{0,0}(u,v)=1.$$

A scalar primary of dimension $\Delta_{\mathcal O}$ begins at small $u$ roughly as

$$G_{\Delta_{\mathcal O},0}(u,v) \sim u^{\Delta_{\mathcal O}/2} \times \text{a function of }v, \qquad u\to 0,$$

while a spin-$\ell$ primary carries angular dependence controlled by $SO(d)$ representation theory.

We will study conformal blocks systematically later. For now, the point is this:

$$\boxed{ \text{the four-point function is a generating function for the spectrum and OPE coefficients.} }$$

The crossing equation becomes a nontrivial constraint on those data:

$$\sum_{\mathcal O\in s\text{-channel}} \lambda_{\phi\phi\mathcal O}^2 G_{\Delta,\ell}(u,v) = \left(\frac{u}{v}\right)^\Delta \sum_{\mathcal O\in t\text{-channel}} \lambda_{\phi\phi\mathcal O}^2 G_{\Delta,\ell}(v,u).$$

For identical scalars, the set of exchanged operators is the same on both sides, but the functions are evaluated in different channels.

Generalized free fields

A very important benchmark is a generalized free field. Let $\phi$ be a scalar operator with unit-normalized two-point function

$$\langle \phi(x)\phi(0)\rangle=\frac1{(x^2)^\Delta}.$$

A generalized free field has Wick-like factorization of higher-point functions, even though it need not come from a free local Lagrangian. Its four-point function is

$$\langle \phi_1\phi_2\phi_3\phi_4 \rangle_{\mathrm{GFF}} = \frac1{(x_{12}^2x_{34}^2)^\Delta} + \frac1{(x_{13}^2x_{24}^2)^\Delta} + \frac1{(x_{14}^2x_{23}^2)^\Delta}.$$

Therefore, in our reduced normalization,

$$\boxed{ \mathcal G_{\mathrm{GFF}}(u,v) = 1+u^\Delta+\left(\frac{u}{v}\right)^\Delta. }$$

This formula is one of the fastest ways to see why generalized free fields are central to holography. In a large-$N$ CFT, single-trace operators often behave at leading order like generalized free fields:

$$\mathcal G(u,v) = \mathcal G_{\mathrm{GFF}}(u,v) + \frac1{N^p}\mathcal G_{\mathrm{conn}}(u,v) + \cdots.$$

The disconnected terms describe free propagation of noninteracting bulk particles. The connected term is where bulk interactions begin. In a semiclassical AdS dual, $\mathcal G_{\mathrm{conn}}$ is computed by tree-level Witten diagrams: exchange diagrams plus contact diagrams.

Large-$N$ interpretation

Suppose $\mathcal O$ is a single-trace scalar primary in a large-$N$ CFT, normalized so that

$$\langle \mathcal O(x)\mathcal O(0)\rangle\sim N^0.$$

Large-$N$ factorization says

$$\langle \mathcal O_1\mathcal O_2\mathcal O_3\mathcal O_4 \rangle = \langle\mathcal O_1\mathcal O_2\rangle \langle\mathcal O_3\mathcal O_4\rangle + \langle\mathcal O_1\mathcal O_3\rangle \langle\mathcal O_2\mathcal O_4\rangle + \langle\mathcal O_1\mathcal O_4\rangle \langle\mathcal O_2\mathcal O_3\rangle + \langle\mathcal O_1\mathcal O_2\mathcal O_3\mathcal O_4\rangle_{\mathrm{conn}}.$$

Usually

$$\langle\mathcal O\mathcal O\mathcal O\mathcal O\rangle_{\mathrm{conn}} \ll \langle\mathcal O\mathcal O\rangle\langle\mathcal O\mathcal O\rangle$$

at large $N$. The leading disconnected answer is generalized free. The first correction encodes anomalous dimensions and OPE coefficients of double-trace operators

$$[\mathcal O\mathcal O]_{n,\ell} \sim \mathcal O\,\partial^{2n}\partial_{\{\mu_1}\cdots\partial_{\mu_\ell\}}\mathcal O - \text{traces},$$

with approximate dimensions

$$\Delta_{n,\ell}^{(0)}=2\Delta+2n+\ell.$$

Interactions shift these dimensions:

$$\Delta_{n,\ell} = 2\Delta+2n+\ell+\frac1{N^p}\gamma_{n,\ell}+\cdots.$$

In AdS, these double-trace operators are two-particle states. Their anomalous dimensions are boundary measurements of bulk interactions.

Singular limits and physics

The most important information in a four-point function often appears in its singular limits.

OPE limit

The $s$-channel OPE limit is

$$x_1\to x_2, \qquad u\to 0, \qquad v\to 1.$$

This limit reveals which operators appear in $\phi\times\phi$. Operators of smaller dimension dominate. For example, the identity contribution gives $1$, while a scalar primary $\mathcal O$ of dimension $\Delta_{\mathcal O}$ contributes at order $u^{\Delta_{\mathcal O}/2}$.

Lightcone limit

In Lorentzian signature, one can take one cross-ratio small while keeping the other fixed or small in a different way. A basic lightcone limit is

$$u\to 0, \qquad u \ll 1-v \ll 1.$$

Such limits are the entry point to the lightcone bootstrap. They imply the existence of large-spin double-twist operators and are closely related to the emergence of locality in AdS.

Regge and chaos limits

After analytic continuation around branch points in $z,\bar z$, one obtains Lorentzian Regge limits. These limits probe high-energy bulk scattering in AdS and are constrained by causality and chaos bounds. The same Euclidean function $\mathcal G(u,v)$ therefore contains much more than Euclidean geometry: its analytic continuation knows about Lorentzian dynamics.

AdS/CFT checkpoint

Four-point functions are the natural meeting point of CFT bootstrap and AdS dynamics.

A tree-level AdS four-point function has two broad types of contributions:

$$\boxed{ \text{contact Witten diagram} \quad\Longleftrightarrow\quad \text{local bulk quartic interaction}, }$$

and

$$\boxed{ \text{exchange Witten diagram} \quad\Longleftrightarrow\quad \text{bulk field exchanged between two pairs of boundary insertions}. }$$

On the CFT side, the same answer must admit an OPE expansion in every channel. Thus a single AdS diagram is not merely a Feynman-like object; it must be compatible with conformal block decompositions and crossing symmetry.

The dictionary is:

$$\begin{array}{ccl} \text{single-trace exchange} &\leftrightarrow& \text{single-particle bulk field},\\[3pt] \text{double-trace tower} &\leftrightarrow& \text{two-particle bulk states},\\[3pt] \text{anomalous dimensions }\gamma_{n,\ell} &\leftrightarrow& \text{bulk interactions},\\[3pt] \text{crossing symmetry} &\leftrightarrow& \text{consistent factorization of the same bulk process},\\[3pt] \text{large-spin behavior} &\leftrightarrow& \text{bulk locality and long-distance propagation}. \end{array}$$

This is why four-point functions are a central object in modern holography. A CFT with a sparse large-$N$ spectrum and appropriately behaved four-point functions can look like a local gravitational theory in AdS. A CFT whose four-point functions violate the expected analyticity, positivity, or large-spin constraints cannot.

Common conventions

There are several common notational choices. They are all equivalent, but mixing them carelessly causes many wrong factors.

One convention uses $x_{ij}=|x_i-x_j|$ and writes powers of $x_{ij}$ directly. This page uses $x_{ij}^2$ and therefore many exponents include factors of $1/2$.

Another convention defines

$$\langle\phi_1\phi_2\phi_3\phi_4\rangle = \frac{1}{(x_{12}^2x_{34}^2)^\Delta} \mathcal G(u,v),$$

while another defines a symmetrized object $\mathcal F(u,v)$ by extracting additional powers of $u$ and $v$. The crossing equation changes form under this redefinition, but the physical content is unchanged.

In two-dimensional CFT, one often writes

$$\langle \phi_1(z_1,\bar z_1)\cdots\phi_4(z_4,\bar z_4) \rangle = \text{fixed powers} \times \mathcal F(z,\bar z),$$

where

$$z=\frac{z_{12}z_{34}}{z_{13}z_{24}}, \qquad \bar z=\frac{\bar z_{12}\bar z_{34}}{\bar z_{13}\bar z_{24}}.$$

For a 2D CFT, $z$ and $\bar z$ are especially powerful because the global conformal group factorizes into holomorphic and antiholomorphic pieces, and the local conformal symmetry enhances this further to Virasoro symmetry.

Summary

The four-point function is the first CFT correlator with an undetermined function:

$$\boxed{ \langle \mathcal O_1\mathcal O_2\mathcal O_3\mathcal O_4\rangle = \text{kinematic prefactor}\times \mathcal G(u,v). }$$

The two cross-ratios are

$$\boxed{ u=\frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2}, \qquad v=\frac{x_{14}^2x_{23}^2}{x_{13}^2x_{24}^2}. }$$

The OPE decomposes $\mathcal G(u,v)$ into conformal blocks. Crossing symmetry equates different decompositions. In large-$N$ holographic CFTs, the disconnected part is generalized free, while the connected part encodes bulk interactions.

So the real lesson is:

$$\boxed{ \text{Four-point functions are where CFT data becomes dynamical geometry.} }$$

Exercises

Exercise 1. Invariance of $u$ and $v$

Use

$${x_{ij}^{\prime}}^2 = \frac{x_{ij}^2}{\sigma_i\sigma_j}$$

under a special conformal transformation to show that

$$u=\frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2}, \qquad v=\frac{x_{14}^2x_{23}^2}{x_{13}^2x_{24}^2}$$

are invariant.

Solution

Under the transformation,

$${x_{12}^{\prime}}^2{x_{34}^{\prime}}^2 = \frac{x_{12}^2x_{34}^2}{\sigma_1\sigma_2\sigma_3\sigma_4},$$

and

$${x_{13}^{\prime}}^2{x_{24}^{\prime}}^2 = \frac{x_{13}^2x_{24}^2}{\sigma_1\sigma_3\sigma_2\sigma_4}.$$

The same product $\sigma_1\sigma_2\sigma_3\sigma_4$ appears in numerator and denominator, so it cancels:

$$u' = \frac{{x_{12}^{\prime}}^2{x_{34}^{\prime}}^2}{{x_{13}^{\prime}}^2{x_{24}^{\prime}}^2} = \frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2} =u.$$

The same calculation gives

$$v' = \frac{{x_{14}^{\prime}}^2{x_{23}^{\prime}}^2}{{x_{13}^{\prime}}^2{x_{24}^{\prime}}^2} = \frac{x_{14}^2x_{23}^2}{x_{13}^2x_{24}^2} =v.$$

Thus $u$ and $v$ are conformal invariants.

Exercise 2. Crossing under $x_1\leftrightarrow x_3$

For an identical scalar $\phi$ of dimension $\Delta$, define

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

Show that exchanging $x_1$ and $x_3$ gives

$$\mathcal G(u,v)=\left(\frac{u}{v}\right)^\Delta\mathcal G(v,u).$$

Solution

Under $x_1\leftrightarrow x_3$, the cross-ratios exchange:

$$(u,v)\mapsto (v,u).$$

The correlator can therefore also be written as

$$\langle\phi_3\phi_2\phi_1\phi_4\rangle = \frac{\mathcal G(v,u)}{(x_{32}^2x_{14}^2)^\Delta} = \frac{\mathcal G(v,u)}{(x_{23}^2x_{14}^2)^\Delta}.$$

Since the fields are identical bosonic scalars, the correlator is unchanged by the permutation. Therefore

$$\frac{\mathcal G(u,v)}{(x_{12}^2x_{34}^2)^\Delta} = \frac{\mathcal G(v,u)}{(x_{23}^2x_{14}^2)^\Delta}.$$

Multiplying by $(x_{12}^2x_{34}^2)^\Delta$ gives

$$\mathcal G(u,v) = \left( \frac{x_{12}^2x_{34}^2}{x_{23}^2x_{14}^2} \right)^\Delta \mathcal G(v,u).$$

But

$$\frac{x_{12}^2x_{34}^2}{x_{23}^2x_{14}^2} = \frac{u}{v}.$$

Hence

$$\mathcal G(u,v)=\left(\frac{u}{v}\right)^\Delta\mathcal G(v,u).$$

Exercise 3. Generalized free four-point function

Assume

$$\langle\phi(x)\phi(0)\rangle=\frac1{(x^2)^\Delta}$$

and Wick-like factorization. Show that

$$\mathcal G_{\mathrm{GFF}}(u,v) = 1+u^\Delta+\left(\frac{u}{v}\right)^\Delta.$$

Solution

Wick-like factorization gives three pairings:

$$\langle\phi_1\phi_2\phi_3\phi_4\rangle = \frac1{(x_{12}^2)^\Delta(x_{34}^2)^\Delta} + \frac1{(x_{13}^2)^\Delta(x_{24}^2)^\Delta} + \frac1{(x_{14}^2)^\Delta(x_{23}^2)^\Delta}.$$

Factor out $(x_{12}^2x_{34}^2)^{-\Delta}$:

$$\langle\phi_1\phi_2\phi_3\phi_4\rangle = \frac1{(x_{12}^2x_{34}^2)^\Delta} \left[ 1+ \left(\frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2}\right)^\Delta + \left(\frac{x_{12}^2x_{34}^2}{x_{14}^2x_{23}^2}\right)^\Delta \right].$$

The second term is $u^\Delta$. For the third term,

$$\frac{x_{12}^2x_{34}^2}{x_{14}^2x_{23}^2} = \frac{u}{v}.$$

Thus

$$\mathcal G_{\mathrm{GFF}}(u,v) = 1+u^\Delta+\left(\frac{u}{v}\right)^\Delta.$$

Exercise 4. Cross-ratios in the conformal frame

In the conformal frame

$$x_1=0, \qquad x_2=z, \qquad x_3=1, \qquad x_4=\infty,$$

show that

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

Solution

Treat the point at infinity by taking $x_4=R$ and sending $R\to\infty$. In complex notation,

$$x_{12}^2=|z|^2=z\bar z, \qquad x_{13}^2=1,$$

and for large $R$,

$$x_{34}^2\sim R^2, \qquad x_{24}^2\sim R^2.$$

Therefore

$$u = \frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2} \to z\bar z.$$

Similarly,

$$x_{23}^2=|z-1|^2=(1-z)(1-\bar z), \qquad x_{14}^2\sim R^2,$$

while

$$x_{13}^2=1, \qquad x_{24}^2\sim R^2.$$

Thus

$$v = \frac{x_{14}^2x_{23}^2}{x_{13}^2x_{24}^2} \to (1-z)(1-\bar z).$$