Contact, Exchange, and Seed Solutions

Seed solutions turn abstract Ward and singularity constraints into normalized boundary coefficients. A contact seed represents a local vertex in a chosen field basis; an exchange seed carries nonlocal propagation for a specified internal mass and spin. Differential operators can generate broader families, but they also generate boundary and contact terms that must be retained.

Required background. The object dictionary fixes the coefficient convention, cosmological Ward identities fix the differential constraints, and cubic interactions and bispectrum shapes supplies explicit inflationary vertices.

Helpful background. Scalar contact and exchange amplitudes provide the flat-space comparison, and derivative interactions and contact terms clarify boundary contributions.

A normalized contact seed

Let $u_k$ denote the usual positive-frequency Bunch–Davies field mode. In the ket-wavefunction convention used here, the bulk-to-boundary solution is

$$K_k(\eta)=\frac{u_k^*(\eta)}{u_k^*(\eta_0)},$$

normalized to one at the late cutoff and damped on the ket $i\epsilon$ contour. (The bra branch uses the conjugate solution.) For a local $n$-field vertex with time-dependent weight $W(\eta)$, the stripped contact coefficient is

$$\mathcal C_n(\{k_a\}) =\int_{-\infty(1-i\epsilon)}^{\eta_0} d\eta\,W(\eta)\prod_{a=1}^nK_{k_a}(\eta).$$

The coupling, overall branch phase, polarization tensors, and momentum delta function are restored according to the object convention. Integration by parts changes $\mathcal C_n$ by endpoint pieces unless the boundary conditions make them vanish.

An elementary benchmark is a conformally rescaled scalar $\chi$ on the conformal half-space with a constant quartic interaction. After stripping the common coupling and branch phase, $K_k=e^{ik\eta}$ and

$$\mathcal C_4(k_1,k_2,k_3,k_4) =\int_{-\infty(1-i\epsilon)}^0d\eta\,e^{ik_T\eta} =\frac1{ik_T}, \qquad k_T=\sum_{a=1}^4k_a.$$

Thus the reduced real seed is proportional to $1/k_T$. It has the expected degree $-1$, only a total-energy singularity, and no exchange-channel pole. The statement applies to this conformal variable and interaction; converting to a covariant de Sitter field can introduce scale factors, derivatives, and boundary terms.

Massive scalar exchange

For an interaction $g\phi^2\sigma$, with exchanged comoving momentum

$$p=\lvert\mathbf k_1+\mathbf k_2\rvert,$$

the $s$-channel wavefunction seed is

$$\begin{aligned} \mathcal E_s={}&g^2 \int d\eta\,d\eta'\, a(\eta)^4a(\eta')^4 K_{k_1}(\eta)K_{k_2}(\eta)\\ &\times G_p^{\Psi}(\eta,\eta') K_{k_3}(\eta')K_{k_4}(\eta'). \end{aligned}$$

$G_p^{\Psi}$ is not an arbitrary Feynman propagator: it obeys the Bunch–Davies condition in the far past and the wavefunction boundary condition at $\eta_0$. These conditions fix homogeneous additions.

For a scalar of mass $m$ in four-dimensional de Sitter, define

$$\nu=\sqrt{\frac94-\frac{m^2}{H^2}}.$$

The two late-time behaviors are $(-\eta)^{3/2-\nu}$ and $(-\eta)^{3/2+\nu}$. In the collapsed limit, the exchange seed contains nonanalytic powers determined by these weights. For $m>3H/2$, $\nu=i\mu$ and the pair becomes oscillatory in $\log p$. Local contact interactions produce analytic powers in $p^2$ and cannot remove this nonanalytic exchange signal.

Near a partial-energy singularity on a specified complex sheet,

$$E_L=k_1+k_2+p\longrightarrow0,$$

the singular part factorizes into the normalized left and right three-point data times the appropriate internal two-point kernel. The power, phase, and whether one takes a residue or discontinuity depend on the mass and branch. The check is best performed by inserting a spectral representation of $G_p^\Psi$ and comparing with the direct double integral.

Baumann, Duaso Pueyo, Joyce, Lee, and Pimentel construct the conformally coupled scalar exchange seed and weight-shifting maps to massless and spinning correlators (Baumann et al. 2020, §§ 3–5).

Differential generation and helicity

Weight-raising, spin-raising, and momentum differential operators can map a scalar seed to other external weights or exchanged spins:

$$\mathcal E_{\Delta_a,s}^{(\lambda)} =\mathcal D_{\Delta_a,s}^{(\lambda)} \mathcal E_{\mathrm{seed}} +\mathcal C_{\mathrm{induced}}.$$

The helicity label $\lambda$ is defined using transverse polarization tensors, and the induced contact term is part of the result. The operator must map both the Ward equation and its source; commuting it through a regulated integral may produce boundary contributions. A generated expression is accepted only after checking its weights, transversality, soft limits, channel residue, and absence of spurious singularities.

Equation-of-motion adversarial test

Add an interaction proportional to a free equation of motion,

$$\delta S =\alpha\int d^4x\,\sqrt{-g}\, F(\phi)(\Box+m^2+\xi R)\phi .$$

A local field redefinition removes the bulk operator, but integration by parts and the finite late boundary generate contact and boundary vertices. Compute the exchange coefficient in the original and redefined actions. After translating the external field and adding every induced term:

• the nonanalytic massive-exchange powers and factorization residues must agree;
• analytic contact coefficients may differ;
• an equal-time correlator agrees only after the operator redefinition is included.

Reject a seed basis that declares the two actions physically different because it omitted induced contacts. Maldacena’s treatment of inflationary cubic terms gives a canonical example of equation-of-motion operators moved into a nonlinear field redefinition (Maldacena 2003, Eqs. (3.9)–(3.12)).

The structure map shows contact and exchange seeds as different inputs to later singularity constraints. Inspect the nonlocal internal line and the explicit local-completion branch.

Contact, exchange, and differential seed construction. The diagram is schematic and not to scale; state, boundary conditions, normalization, helicity, and induced contact terms are part of each seed.

The failure map identifies basis artifacts. Inspect the stops for a wrong propagator, dropped endpoint term, unnormalized external mode, or differential operator that introduces a spurious pole.

Failure conditions for cosmological seed solutions. The diagram is schematic and not to scale; nonlocal exchange data survive allowed basis changes, while local representatives need not.

These conditions refine the chapter’s domain and failure conditions. Their singularities are classified in total-energy and factorization singularities.

Exercise

Evaluate the regulated conformal contact integral

$$I_\epsilon=\int_{-\infty}^{0}d\eta\, e^{(ik_T+\epsilon)\eta}, \qquad \epsilon>0,$$

and take $\epsilon\to0^+$ for $k_T>0$.

Solution

Direct integration gives

$$I_\epsilon =\frac1{ik_T+\epsilon}.$$

The lower endpoint vanishes because $\epsilon\eta\to-\infty$. Therefore $I_\epsilon\to1/(ik_T)=-i/k_T$. Restoring the interaction’s branch factor determines whether the final wavefunction coefficient is real or imaginary in a given convention.

References

• Baumann, D., C. Duaso Pueyo, A. Joyce, H. Lee, and G. L. Pimentel. “The Cosmological Bootstrap: Weight-Shifting Operators and Scalar Seeds.” Journal of High Energy Physics 2020, no. 12 (2020): 204. DOI. Open PDF.
• Maldacena, J. “Non-Gaussian Features of Primordial Fluctuations in Single Field Inflationary Models.” Journal of High Energy Physics 2003, no. 05 (2003): 013. DOI. Open PDF.