Abstract
We consider Lorentzian CFT Wightman functions in momentum space. In particular, we derive a set of reference formulas for computing two- and three-point functions, restricting our attention to three-point functions where the middle operator (corresponding to a Hamiltonian density) carries zero spatial momentum, but otherwise allowing operators to have arbitrary spin. A direct application of our formulas is the computation of Hamiltonian matrix elements within the framework of conformal truncation, a recently proposed method for numerically studying strongly-coupled QFTs in real time and infinite volume. Our momentum space formulas take the form of finite sums over 2F1 hypergeometric functions, allowing for efficient numerical evaluation. As a concrete application, we work out matrix elements for 3d ϕ4-theory, thus providing the seed ingredients for future truncation studies.
Highlights
The IR limit of a UV conformal field theory (CFT) that has been deformed by one or more relevant operators OR
To study any theory with conformal truncation, one needs to construct the Hamiltonian matrix elements for the relevant deformation(s) OR, which can be written in the general schematic form
The Hamiltonian is built from two key ingredients: the OPE coefficients COO OR of the UV CFT and the kinematic function of external momenta MOORO (P, P ), which is completely fixed by conformal symmetry and the operator dimensions and spins
Summary
In 3d, CFT Euclidean two-point functions are linear combinations of terms of the general form. If we analytically continue to Lorentzian signature, the corresponding terms are e−. In order to obtain the contribution of such a term to the momentum space Wightman function, we need to compute the function. Inner products of (general-spin) primary operators will be linear combinations of the functions IA(P ). The first step in evaluating IA is to perform the integral over x⊥. Since we are working in the frame P⊥ = 0, the formula we need is xn. In this frame, the Fourier transform vanishes for odd a⊥.
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