Momentum-Space Correlators and Conformal Ward Identities

Fourier transformation turns conformal Ward identities into differential equations in momenta, but it does not erase distributions supported at coincident points. A correct momentum-space solution therefore specifies its Fourier convention, keeps the overall momentum delta function distinct from the reduced correlator, solves the nonlocal equations, and restores every semilocal, contact, counterterm, and anomaly contribution allowed by the Ward identities. The discussion is Euclidean in general dimension $d$; Lorentzian correlators require a stated boundary-value prescription and have different branch cuts.

Required background. Scalar Two- and Three-Point Functions supplies the separated-point position-space correlators. Tempered Distributions and Fourier Calculus supplies distributional Fourier transformation. Helpful background. Fourier Series, Fourier Transforms, and Plancherel Theory fixes the ordinary transform machinery. Contact Terms, Equal-Time Commutators, and Schwinger Terms explains contact contributions to Ward identities.

Fourier conventions and reduced correlators

Use

$$\widetilde{\mathcal O}(p) =\int d^d x\,e^{-ip\cdot x}\mathcal O(x), \qquad \mathcal O(x)=\int\frac{d^d p}{(2\pi)^d} e^{ip\cdot x}\widetilde{\mathcal O}(p).$$

Translation invariance gives

$$\left\langle\prod_{i=1}^{n}\widetilde{\mathcal O}_i(p_i)\right\rangle =(2\pi)^d\delta^{(d)}\!\left(\sum_{i=1}^{n}p_i\right) \left\langle\!\left\langle \prod_{i=1}^{n}\mathcal O_i(p_i) \right\rangle\!\right\rangle .$$

Double brackets mean that the momentum-conserving delta distribution has been removed; they do not mean that other distributions or contact terms have been discarded. Set $p_n=-\sum_{j=1}^{n-1}p_j$ only after differentiating the full distribution or after deriving the reduced Ward operator consistently.

For scalar primaries of dimensions $\Delta_i$, the homogeneous reduced dilatation equation is

$$\left[ \sum_{j=1}^{n-1}p_j^\mu\frac{\partial}{\partial p_j^\mu} -\left(\sum_{i=1}^{n}\Delta_i-(n-1)d\right) \right] \left\langle\!\left\langle\mathcal O_1\cdots\mathcal O_n \right\rangle\!\right\rangle=0.$$

The degree $\sum_i\Delta_i-(n-1)d$ includes the $-d$ carried by the removed delta distribution. After renormalization, the right-hand side may instead be a local anomaly. With the Fourier convention above, the scalar special-conformal generator is

$$\mathcal K_j^{\kappa} =p_j^{\kappa}\frac{\partial^2}{\partial p_j^\alpha\partial p_{j\alpha}} -2p_j^\alpha \frac{\partial^2}{\partial p_j^\alpha\partial p_{j\kappa}} +2(\Delta_j-d)\frac{\partial}{\partial p_{j\kappa}},$$

and the reduced homogeneous equation is

$$\sum_{j=1}^{n-1}\mathcal K_j^{\kappa} \left\langle\!\left\langle\mathcal O_1\cdots\mathcal O_n \right\rangle\!\right\rangle=0.$$

Changing the sign in the Fourier exponential changes intermediate factors of $i$ but not a consistently transformed final equation. Spinning primaries add the momentum-space Lorentz-generator term acting on their indices. The derivation and the reduction to scalar radial equations are given in Bzowski, McFadden, and Skenderis 2014, §§3–4.

A distributional scalar two-point transform

The position-space power law has the Euclidean Fourier transform

$$\int d^d x\,e^{-ip\cdot x}\frac1{(x^2)^\Delta} =2^{d-2\Delta}\pi^{d/2} \frac{\Gamma\!\left(\frac d2-\Delta\right)}{\Gamma(\Delta)} (p^2)^{\Delta-d/2}.$$

This formula is first defined as a Riesz distribution for $0<\operatorname{Re}\Delta<d/2$ and nonzero Euclidean $p$, with the endpoints and the rest of the $\Delta$-plane reached by analytic continuation. It is not an absolutely convergent ordinary integral on all of $\mathbb R^d$.

The momentum power has degree $2\Delta-d$, exactly the reduced two-point degree. At

$$\Delta=\frac d2+k, \qquad k=0,1,2,\ldots,$$

$\Gamma(d/2-\Delta)$ has a pole. After subtracting a local polynomial counterterm, the renormalized nonlocal part is proportional to

$$(p^2)^k\log\frac{p^2}{\mu^2},$$

plus a scheme-dependent polynomial in $p^2$. Its scale derivative is local:

$$\mu\frac{\partial}{\partial\mu} \left[(p^2)^k\log\frac{p^2}{\mu^2}\right] =-2(p^2)^k.$$

Thus a logarithm and a local scale anomaly are two descriptions of the same renormalized distribution. Deleting the polynomial terms before applying the Ward identity can erase precisely the anomaly one wants to determine.

Scalar three-point functions and triple-K integrals

Rotational invariance makes a scalar reduced three-point function depend on the magnitudes $p_i=\sqrt{p_i^2}$. Away from exceptional momentum configurations, the special-conformal equations reduce to

$$(K_i-K_j)\left\langle\!\left\langle \mathcal O_1(p_1)\mathcal O_2(p_2)\mathcal O_3(p_3) \right\rangle\!\right\rangle=0,$$

where

$$K_i=\frac{\partial^2}{\partial p_i^2} +\frac{d+1-2\Delta_i}{p_i}\frac{\partial}{\partial p_i}.$$

A solution of the homogeneous primary Ward identities is

$$\left\langle\!\left\langle \mathcal O_1\mathcal O_2\mathcal O_3 \right\rangle\!\right\rangle =C_{123}\,I_{\alpha\{\beta_1\beta_2\beta_3\}}(p_1,p_2,p_3),$$

with

$$\begin{aligned} I_{\alpha\{\beta_1\beta_2\beta_3\}} &=\int_0^\infty dx\,x^\alpha \prod_{i=1}^{3}p_i^{\beta_i}K_{\beta_i}(p_i x),\\ \alpha&=\frac d2-1, \qquad \beta_i=\Delta_i-\frac d2. \end{aligned}$$

Here $K_\nu$ is the modified Bessel function, not the conformal generator. The Bessel equation proves $(K_i-K_j)I=0$. Rescaling every momentum by $\lambda$ and then $x\mapsto x/\lambda$ gives

$$I(\lambda p_1,\lambda p_2,\lambda p_3) =\lambda^{\Delta_1+\Delta_2+\Delta_3-2d}I(p_1,p_2,p_3),$$

which matches the reduced dilatation equation and the Fourier transform of the position-space three-point function. This representation, including convergence and renormalization, is derived in Bzowski, McFadden, and Skenderis 2014, §§4.1–4.2.

The integral is singular when, for some independent choices $\sigma_i\in\{+1,-1\}$ and some integer $k\geq0$,

$$\alpha+1+\sigma_1\beta_1+\sigma_2\beta_2+\sigma_3\beta_3=-2k.$$

One then regulates $d$ and the $\Delta_i$ in a symmetry-compatible way, expands the integral, adds local counterterms, and takes the regulator to zero. Different sign patterns encode different endpoint singularities and support types; they must not all be interpreted as an ordinary ultraviolet divergence. At special dimensions, the renormalized answer can contain logarithms and inhomogeneous conformal Ward identities. The systematic current and stress-tensor treatment appears in Bzowski, McFadden, and Skenderis 2018, §§2–4.

Tensor decomposition must retain longitudinal terms

For a current or stress tensor, rotations permit a transverse decomposition using, for $p^2\neq0$,

$$\pi_{\mu\nu}(p)=\delta_{\mu\nu}-\frac{p_\mu p_\nu}{p^2}.$$

The transverse form factors satisfy the primary conformal equations, while longitudinal and trace pieces are fixed by conservation and trace Ward identities. But a current insertion generally obeys

$$p^\mu\left\langle\!\left\langle J_\mu(p) \mathcal O_1(p_1)\cdots\mathcal O_n(p_n) \right\rangle\!\right\rangle =\text{contact terms from the transformations of the insertions},$$

not zero. Projecting every leg transversely before recording this equation loses those terms. A safe order is: derive the full distributional Ward identities, solve their longitudinal and trace parts, decompose the remaining transverse tensor, and only then solve its form-factor equations.

At $p^2=0$, at zero momentum, or on collinear boundaries of the Euclidean momentum triangle, $\pi_{\mu\nu}$ or a chosen tensor basis becomes singular or loses rank. Statements derived for generic momenta require a separate distributional limit there. In Lorentzian signature the continuation of $(p^2)^\nu$ and the triple-K representation depends on whether the target is Wightman, time-ordered, retarded, or another correlator; an $i0$ prescription and sheet must be specified.

Support, counterterms, and anomaly poles

Position-space support determines what may be changed without altering a separated-point correlator.

Momentum-space term	Position-space support	What fixes it
Polynomial in all external momenta	All insertions coincident (ultralocal)	Local counterterms and scheme choice
Polynomial in one momentum times a nonanalytic lower-point function	A proper subset of insertions coincident (semilocal)	Ward identities and lower-point normalization
Nonanalytic dependence on all independent invariants	Separated points (nonlocal)	CFT three-point data and boundary conditions
Scale derivative of a logarithmic term	Local or semilocal anomaly	Counterterm residue and consistency conditions

Tensor decompositions can make an anomaly appear as a factor such as $1/p^2$ in an individual form factor. Such an anomaly pole can be forced by a local Ward identity because the tensor basis itself contains powers of momentum. Its residue and allowed contact terms are meaningful; the pole alone does not establish a propagating massless particle. One must reconstruct the full correlator and examine its spectral and Lorentzian boundary conditions before making that interpretation.

The position–momentum match is therefore threefold: the nonlocal momentum dependence must transform back to the separated-point power law; polynomial ambiguities must transform to derivatives of delta functions; and scale or conformal anomalies must match the variation of local counterterms. Agreement of only the homogeneity degree is necessary but not sufficient.

Check: the triple-K scaling degree

Derive the scaling degree of $I_{\alpha\{\beta_1\beta_2\beta_3\}}$ directly and specialize to the conformal values of $\alpha$ and $\beta_i$.

Solution

Under $p_i\mapsto\lambda p_i$, set $y=\lambda x$. The three factors $p_i^{\beta_i}$ contribute $\lambda^{\sum_i\beta_i}$, while $dx\,x^\alpha$ contributes $\lambda^{-\alpha-1}$. Hence $I(\lambda p_i)=\lambda^{\sum_i\beta_i-\alpha-1}I(p_i)$. With $\beta_i=\Delta_i-d/2$ and $\alpha=d/2-1$, the exponent is $\sum_i\Delta_i-2d$, exactly the degree of a reduced three-point function.

Handoff

For separated Euclidean points, Scalar Two- and Three-Point Functions gives the position-space answer that the nonlocal terms must reproduce. For four-point spectral decompositions and crossing, return to Crossing Equations and Positivity; contact and anomaly polynomials are not additional exchanged conformal blocks.

References

• Bzowski, A., McFadden, P., and Skenderis, K. (2014). “Implications of Conformal Invariance in Momentum Space.” Journal of High Energy Physics 03 (2014), 111. doi:10.1007/JHEP03(2014)111. Open version.
• Bzowski, A., McFadden, P., and Skenderis, K. (2018). “Renormalised CFT 3-point Functions of Scalars, Currents and Stress Tensors.” Journal of High Energy Physics 11 (2018), 159. doi:10.1007/JHEP11(2018)159. Open version.