Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 312 / 1

Under spatial reflection, a scalar field obeys $P\varphi (\mathbf{x})P=\varphi (-\mathbf{x})$, and hence each spatial derivative changes sign. The interaction contains three such derivatives, so changing the integration variable from $\mathbf{x}$ to $-\mathbf{x}$ givesThus this is a parity-odd scalar interaction.

Put $O({\mathbf{k}}_a)={\prod }_{a=1}^n{\varphi }_a({\mathbf{k}}_a)$ and $A({\mathbf{k}}_a)=⟨H_{\mathrm{int}}O({\mathbf{k}}_a)⟩$. Hermitian conjugation and commutativity of equal-time scalar fields giveThe parity-invariant vacuum and the odd parity of $H_{\mathrm{int}}$ imply $A(-{\mathbf{k}}_a)=-A({\mathbf{k}}_a)$. ThereforeThere is no conflict with the usual $2i\mathrm{Im}A$ rule for two Hermitian operators: a momentum-space product at fixed ${\mathbf{k}}_a$ is generally not itself Hermitian, since its adjoint carries momenta $-{\mathbf{k}}_a$.

The first-order in-in formalism formula at observation time ${\tau }_0$ isFourier transforming the three derivatives giveswhere ${\int }_{\mathbf{p}}=\int d^{3}p/(2\pi {)}^3$. Because the four species are distinct, Wick contraction pairs each vertex field with the external field of the same species and introduces no permutation factor. IfthenCombining this with part i gives the requested time integral:The factor $\mathcal{E}$ is a pseudoscalar, so the resulting primordial trispectrum is parity odd and purely imaginary in this momentum-space convention.

The intended oscillatory factor is $e^{-ik\tau }$. Give the early-time endpoint the usual i-epsilon prescription and set $x=-k\tau$. For an integer $p\ge 0$,It is therefore purely imaginary. Equivalently, rotating the contour to the negative imaginary $\tau$ axis turns the remaining integral into a real Gamma integral and leaves one overall factor of $i$.

At late time $f(k,{\tau }_0)\to H/\sqrt{2k^3}$ and $a=-1/(H\tau )$. Define the elementary symmetric polynomialsThen part ii reduces towhere a prime removes the momentum-conserving Dirac delta distribution andExpanding the product givesRepeated integration by parts reduces every negative power to the supplied logarithmic integral. The power divergences are real and disappear when the imaginary part is taken. Writingone obtainsConsequently the late-time parity-odd primordial trispectrum isIts factor of $i$ is required by reality of a momentum-space scalar correlator: reversing all momenta complex-conjugates the correlator, while the scalar triple product changes sign.

Statistical homogeneity forces a scalar two-point function to have momenta $(\mathbf{k},-\mathbf{k})$. Statistical isotropy then makes it a function only of $|\mathbf{k}|$, so it is parity even without using perturbation theory.

For a scalar three-point function, momentum conservation gives ${\mathbf{k}}_1+{\mathbf{k}}_2+{\mathbf{k}}_3=0$, so all three vectors lie in one plane. A rotation by $\pi$ around the normal to that plane sends every ${\mathbf{k}}_a$ to $-{\mathbf{k}}_a$. Rotational invariance therefore identifies a triangle with its parity reverse, proving nonperturbatively that the scalar primordial bispectrum is parity even.

A rotationally invariant local parity-odd three-scalar vertex must contain a Levi-Civita symbol contracted with three spatial momenta. Its momentum-space factor is proportional tobecause momentum conservation makes the momenta linearly dependent. The equivalent position-space expression is a total divergence; its spatial integral vanishes under the stated vanishing boundary condition. Thus a parity-odd interaction of three scalar fields contributes nothing to the action.