which indeed agrees with eq. (5.8) and (5.18). We can also compute $\tau_{\text{NL}}^{\text{PO}}$, which is given by $\sim B_4^\zeta/P_\zeta^3$, in terms of the observed power spectrum of $\zeta$, $\Delta_\zeta^2 \approx 4 \times 10^{-8}$. Using eq. (2.22) (the explicit choice of reference scale $\bar{k}$ is irrelevant due to scale invariance) we find

$$ \tau_{\text{NL}}^{\text{PO}} \approx -\frac{4 \times 10^{-7} (\Lambda^{28})^{\frac{1}{5}}}{\Lambda_{\text{PO}}^2 (\bar{\Lambda}^{18})} \qquad (5.21) $$

This can be made large for small $\Lambda_{\text{PO}}$, with the caveat that a too small $\Lambda_{\text{PO}}$ could lead to $\tau_{\text{NL}}^{\text{PO}} \times \Delta_\zeta^2$ becoming close to 1 which would jeopardize perturbativity. Given the fact that we propose GI only as an example of how (non-perturbative) deviations from the Bunch-Davies vacuum provide a counterexample to the results of section 3, we leave a more detailed investigation of naturalness constraints to future work. It would also be interesting to carry out the analysis of [10] using the template of eq. (5.18), in order to confirm whether current observational bounds from BOSS on $\tau_{\text{NL}}^{\text{PO}}$ are actually competitive with simple bounds coming from the requirement of perturbativity.

We have learned quite a lot from this calculation. Consider the operator in eq. (5.9) which only differs from eq. (5.10) by having one less time derivative. Adding time derivatives cannot alter the reality of the wavefunction coefficient because they come in the combination $\eta\partial_\eta$ by scale invariance. The final operator that we wrote, eq. (5.11), however, has a different number of spatial derivatives compared to the other two: its had seven while the others have nine. While the time derivatives do not alter the reality of the wavefunction coefficient, spatial derivatives can. Indeed, by removing two spatial derivatives we lose two powers of $\eta$ in the integrand so when we rotate we pick up a factor of $i$. This means that if an operator with nine spatial derivatives yields an imaginary wavefunction and therefore a non-zero correlator, an operator with seven spatial derivatives will yield a real wavefunction and therefore a vanishing correlator. We conclude that eqs. (5.9) and (5.10) yield a non-zero signal, while eq. (5.11) yields a vanishing signal.

More generally, $B_4^{\text{PO}} \neq 0$ in GI only if we have $5+4n$ spatial derivatives, regardless of the number of time derivatives.<fnref n="21" /> It is easy to convince oneself that in the case with five spatial derivatives, the necessary number of time derivatives required to find a non-zero operator ensures that the scaling dimension is never less than 3 i.e. it is never less than the scaling dimension of the operator in eq. (5.9). This is true for quartic interactions that come from four building block operators, and also those with fewer building blocks. With this in mind, let us also write down the trispectrum for curvature perturbations that comes from eq. (5.9). For a quartic action in conformal time given by

$$ S_{\pi\pi\pi\pi} = \frac{1}{M_{\text{PO}}} \int d^3x d\eta a^{-5}(\eta) \epsilon_{ijk} \partial_m \pi \partial_n \partial_i \pi \partial_m \partial_l \pi \partial_l \partial_k \pi, \qquad (5.22) $$

where we have introduced the scale $M_{\text{PO}}$ to distinguish this trispectrum from that coming

<footnote n="21">²¹One should be able to derive this result using the boostless bootstrap derivation in section 3.2, but using a modified COT that applies to Ghost Inflation. Such COT for Ghost Inflation was briefly discussed in [38].</footnote>
from eq. (5.10), we find

$$ \psi_4 = -\frac{8\pi^4 H^5}{\Gamma(\frac{3}{4})^4 M_{\text{PO}} \tilde{c}^6} \epsilon_{ijk} k_2^i k_3^j k_4^k k_1^m k_3^m k_1^n k_2^k k_3^l k_4^l k_1^{\frac{3}{2}} k_2^{\frac{3}{2}} k_3^{\frac{3}{2}} k_4^{\frac{3}{2}} \mathcal{T}(k_1, k_2, k_3, k_4) + 23 \text{ perms.}, \qquad (5.23) $$

where now we have

$$ \mathcal{T}(k_1, k_2, k_3, k_4) = \int_0^{+\infty} d\lambda \, \lambda^{11} H_{\frac{3}{4}}^{(1)}(2ik_1^2\lambda^2) H_{\frac{3}{4}}^{(1)}(2ik_2^2\lambda^2) H_{\frac{3}{4}}^{(1)}(2ik_3^2\lambda^2) H_{\frac{3}{4}}^{(1)}(2ik_4^2\lambda^2). \qquad (5.24) $$

One can check, using the integral representation of eq. (5.17), that $\mathcal{T}$ is purely imaginary, leading to an imaginary $\psi_4$. The final expression for the trispectrum of the comoving curvature perturbation is

$$ B_4^\zeta = \frac{128 i \pi^3 \Lambda^5 (H \tilde{\Lambda})^{1/2}}{M_{\text{PO}} \tilde{\Lambda}^5 \Gamma(\frac{3}{4})^2} (\Delta_\zeta^2)^3 \frac{(\mathbf{k}_2 \cdot \mathbf{k}_3 \times \mathbf{k}_4)(\mathbf{k}_1 \cdot \mathbf{k}_3)(\mathbf{k}_1 \cdot \mathbf{k}_2)(\mathbf{k}_3 \cdot \mathbf{k}_4)}{k_1^{\frac{3}{2}} k_2^{\frac{3}{2}} k_3^{\frac{3}{2}} k_4^{\frac{3}{2}}} \text{Im} \mathcal{T}(k_1, k_2, k_3, k_4) + 23 \text{ perms.} \qquad (5.25) $$

We can now estimate the size of the non-Gaussianities from this operator. We take the ratio between $B_4^\zeta$ from the $M_{\text{PO}}$ operator and the one from the $\Lambda_{\text{PO}}^2$ operator in the $\tau_{\text{PO}}^{\text{NL}}$ configuration of eq. (2.20), finding

$$ \frac{4 \Lambda_{\text{PO}}^2}{H M_{\text{PO}}} \approx \frac{3 \times 10^3 \Lambda_{\text{PO}}^2}{M_{\text{PO}} \Lambda_{\frac{5}{2}}^{\frac{2}{5}} \tilde{\Lambda}_{\frac{5}{2}}^{\frac{3}{5}}}, \qquad (5.26) $$

where we have used eq. (5.8) to express the Hubble rate in terms of $\Delta_\zeta^2 \approx 4 \times 10^{-8}$. We emphasize that we do this comparison only to have a vague idea of the size of the trispectrum from the operator (5.9): a proper analysis following [10, 11] is needed in order to assess the importance of the difference in the shapes. Nevertheless, it is important to stress that in terms of operators with four building blocks we expect that eq. (5.25) is the leading signal, and we will investigate other operators with the same scaling dimension in the future.

## 6 Yes-go 3: exchanging massive spinning fields

In this section we consider a different setup in which large parity violation can be obtained in the scalar trispectrum at tree level due to the exchange of a massive spinning field, which we denote by $\sigma$, with masses in the range $0 < m_\sigma/H < 3/2$. We go back to assuming a linear dispersion relation for the Goldstone mode, and consider cubic couplings within the EFTI following the set-up of [41]. We will first introduce this formalism. We will then show that the exchange diagram in the wavefunction calculation is purely real so it does not contribute to $\rho_4$ and therefore does not contribute to the parity-odd trispectrum. We show this by two complementary methods: using weight-shifting operators, and using a Wick rotation of the nested time integrals (see appendix A). Conversely, we show that the factorized contribution $B_4^{\text{PO}} \sim \rho_3\rho_3$ yields a non-zero signal.
from eq. (5.10), we find

$$ \psi_4 = - \frac{8\pi^4 H^5}{\Gamma(\frac{3}{4})^4 M_{PO} \tilde{c}^6} \epsilon_{ijk} k_2^i k_3^j k_4^k k_1^m k_3^m k_1^n k_2^n k_3^l k_4^l k_1^{\frac{3}{2}} k_2^{\frac{3}{2}} k_3^{\frac{3}{2}} k_4^{\frac{3}{2}} \mathcal{T}(k_1, k_2, k_3, k_4) + 23 \text{ perms.}, \qquad (5.23) $$

where now we have

$$ \mathcal{T}(k_1, k_2, k_3, k_4) = \int_0^{+\infty} d\lambda \, \lambda^{11} H_{\frac{3}{4}}^{(1)}(2ik_1^2\lambda^2) H_{\frac{3}{4}}^{(1)}(2ik_2^2\lambda^2) H_{\frac{3}{4}}^{(1)}(2ik_3^2\lambda^2) H_{\frac{3}{4}}^{(1)}(2ik_4^2\lambda^2). \qquad (5.24) $$

One can check, using the integral representation of eq. (5.17), that $\mathcal{T}$ is purely imaginary, leading to an imaginary $\psi_4$. The final expression for the trispectrum of the comoving curvature perturbation is

$$ B_4^\zeta = \frac{128i\pi^3\Lambda^5(H\tilde{\Lambda})^{1/2}}{M_{PO}\tilde{\Lambda}^5\Gamma(\frac{3}{4})^2} (\Delta_\zeta^2)^3 \frac{(\mathbf{k}_2 \cdot \mathbf{k}_3 \times \mathbf{k}_4)(\mathbf{k}_1 \cdot \mathbf{k}_3)(\mathbf{k}_1 \cdot \mathbf{k}_2)(\mathbf{k}_3 \cdot \mathbf{k}_4)}{k_1^{\frac{3}{2}} k_2^{\frac{3}{2}} k_3^{\frac{3}{2}} k_4^{\frac{3}{2}}} \text{Im} \mathcal{T}(k_1, k_2, k_3, k_4) + 23 \text{ perms.} \qquad (5.25) $$

We can now estimate the size of the non-Gaussianities from this operator. We take the ratio between $B_4^\zeta$ from the $M_{PO}$ operator and the one from the $\Lambda_{PO}^2$ operator in the $\tau_{PO}^{NL}$ configuration of eq. (2.20), finding

$$ \frac{4 \Lambda_{PO}^2}{H M_{PO}} \approx \frac{3 \times 10^3 \Lambda_{PO}^2}{M_{PO} \Lambda_{\frac{5}{2}}^{\frac{2}{5}} \tilde{\Lambda}_{\frac{5}{2}}^{\frac{3}{5}}}, \qquad (5.26) $$

where we have used eq. (5.8) to express the Hubble rate in terms of $\Delta_\zeta^2 \approx 4 \times 10^{-8}$. We emphasize that we do this comparison only to have a vague idea of the size of the trispectrum from the operator (5.9): a proper analysis following [10, 11] is needed in order to assess the importance of the difference in the shapes. Nevertheless, it is important to stress that in terms of operators with four building blocks we expect that eq. (5.25) is the leading signal, and we will investigate other operators with the same scaling dimension in the future.

## 6 Yes-go 3: exchanging massive spinning fields

In this section we consider a different setup in which large parity violation can be obtained in the scalar trispectrum at tree level due to the exchange of a massive spinning field, which we denote by $\sigma$, with masses in the range $0 < m_\sigma/H < 3/2$. We go back to assuming a linear dispersion relation for the Goldstone mode, and consider cubic couplings within the EFTI following the set-up of [41]. We will first introduce this formalism. We will then show that the exchange diagram in the wavefunction calculation is purely real so it does not contribute to $\rho_4$ and therefore does not contribute to the parity-odd trispectrum. We show this by two complementary methods: using weight-shifting operators, and using a Wick rotation of the nested time integrals (see appendix A). Conversely, we show that the factorized contribution $B_4^{PO} \sim \rho_3\rho_3$ yields a non-zero signal.
**Spinning fields in the EFTI.** To treat spinning fields in the Effective Field Theory of Inflation (EFTI) we follow the nice formalism of [41] and exemplify the discussion in the case of a spin-1 field. Within this set-up one can avoid constraints on the mass of spinning fields in the form of the Higuchi bound<sup>22</sup> [62] by allowing for sizeable couplings between spinning fields and the inflaton background. This is a natural set-up for us in this work as we are not assuming exact or approximate invariance under de Sitter boosts. We will be working with spin-1 where the mass range we choose does not actually violate the Higuchi bound, however this is primarily for illustrative purposes and our results are easily generalised to higher-spins where the allowed range of masses here is large than those allowed by the Higuchi bound. During inflation fields are classified according to their transformation under the unbroken group of rotations, but to couple them to four-dimensional fields within the EFTI, we introduce a fictitious four-vector that when coupled to gravity and the foliation ensures that the resulting EFT has the correct linear and non-linear symmetries. For spin-1, the object that transforms as a vector under all diffeomorphisms is [41]

$$ \Sigma^{\mu}(\Sigma^{i}, \pi) = \left( -\frac{\Sigma^{i} \partial_{i} \pi}{1 + \dot{\pi}}, \Sigma^{i} \right). \qquad (6.1) $$

It is convenient to make all scale factors manifest by contracting spatial indices with $\delta_{ij}$ as opposed to $g_{ij}$. To do so we introduce $\sigma^i = a\Sigma^i$ such that the free theory for the spin-1 field is [41]

$$ S_2 = \frac{1}{2} \int d^3x d\eta a^2(\eta) \left[ (\sigma_i')^2 - c_1^2(\partial_i \sigma_j)^2 - (c_0^2 - c_1^2)(\partial_i \sigma^i)^2 - a(\eta)^2 m_\sigma^2 (\sigma^i)^2 \right], \quad (6.2) $$

where $c_{0,1}$ are the speeds of sound of the longitudinal and transverse components of $\sigma^i$ and $m_\sigma$ is an arbitrary mass.<sup>23</sup> From now on all spatial indices are raised and lowered with $\delta_{ij}$. Crucially, in this set-up the kinetic term of the longitudinal mode is not related to the mass parameter which is why the Higuchi bound can be avoided. We note that the symmetries of EFTI dictate that this free action must also come with interactions of the form $\pi\sigma\sigma$ since $\Sigma\Sigma \supset \sigma\sigma + \pi\sigma\sigma$. Such interactions cannot contribute to the scalar trispectrum at tree level so they are not of interest for us in this work. Converting to Fourier space, we write $\sigma_i(\mathbf{k}, \eta) = \sum_h \sigma_k^{(h)}(\eta) \epsilon_i^h(\mathbf{k})$ where we normalise the polarisation vector according to $\epsilon_i^h(\mathbf{k})\epsilon_i^{h'}(-\mathbf{k}) = \delta_{hh'}$ (in this way eq. (6.2) is already the canonically-normalized quadratic action). We can easily see from the free theory that the mode functions will be equivalent to those of a massive scalar field in de Sitter. Assuming Bunch-Davies initial conditions, we have

$$ \sigma_k^{(h)} = \frac{H\sqrt{\pi}}{2} e^{i\pi(\nu+1/2)/2} (-\eta)^{3/2} H_\nu^{(1)}(-c_h k\eta), \quad \text{with} \quad \nu = \sqrt{\frac{9}{4} - \frac{m_\sigma^2}{H^2}}. \qquad (6.3) $$

<footnote n="22"><sup>22</sup>In the limit where we have full de Sitter symmetries, the Higuchi bound sets a lower bound on the mass of spinning fields by demanding that their longitudinal mode is not a ghost thereby ensuring the fields adhere to unitary representations of the de Sitter group.</footnote>

<footnote n="23"><sup>23</sup>Note that our $m_\sigma$ differs from that in [41] because we wrote the action in terms of conformal time.</footnote>
**Spinning fields in the EFTI.** To treat spinning fields in the Effective Field Theory of Inflation (EFTI) we follow the nice formalism of [41] and exemplify the discussion in the case of a spin-1 field. Within this set-up one can avoid constraints on the mass of spinning fields in the form of the Higuchi bound<sup>22</sup> [62] by allowing for sizeable couplings between spinning fields and the inflaton background. This is a natural set-up for us in this work as we are not assuming exact or approximate invariance under de Sitter boosts. We will be working with spin-1 where the mass range we choose does not actually violate the Higuchi bound, however this is primarily for illustrative purposes and our results are easily generalised to higher-spins where the allowed range of masses here is large than those allowed by the Higuchi bound. During inflation fields are classified according to their transformation under the unbroken group of rotations, but to couple them to four-dimensional fields within the EFTI, we introduce a fictitious four-vector that when coupled to gravity and the foliation ensures that the resulting EFT has the correct linear and non-linear symmetries. For spin-1, the object that transforms as a vector under all diffeomorphisms is [41]

$$ \Sigma^{\mu}(\Sigma^{i}, \pi) = \left( -\frac{\Sigma^{i} \partial_{i} \pi}{1 + \dot{\pi}}, \Sigma^{i} \right). \qquad (6.1) $$

It is convenient to make all scale factors manifest by contracting spatial indices with $\delta_{ij}$ as opposed to $g_{ij}$. To do so we introduce $\sigma^i = a\Sigma^i$ such that the free theory for the spin-1 field is [41]

$$ S_2 = \frac{1}{2} \int d^3x d\eta a^2(\eta) \left[ (\sigma'_i)^2 - c_1^2(\partial_i \sigma_j)^2 - (c_0^2 - c_1^2)(\partial_i \sigma^i)^2 - a(\eta)^2 m_\sigma^2 (\sigma^i)^2 \right], \quad (6.2) $$

where $c_{0,1}$ are the speeds of sound of the longitudinal and transverse components of $\sigma^i$ and $m_\sigma$ is an arbitrary mass.<sup>23</sup> From now on all spatial indices are raised and lowered with $\delta_{ij}$. Crucially, in this set-up the kinetic term of the longitudinal mode is not related to the mass parameter which is why the Higuchi bound can be avoided. We note that the symmetries of EFTI dictate that this free action must also come with interactions of the form $\pi\sigma\sigma$ since $\Sigma\Sigma \supset \sigma\sigma + \pi\sigma\sigma$. Such interactions cannot contribute to the scalar trispectrum at tree level so they are not of interest for us in this work. Converting to Fourier space, we write $\sigma_i(\mathbf{k}, \eta) = \sum_h \sigma_k^{(h)}(\eta) \epsilon_i^h(\mathbf{k})$ where we normalise the polarisation vector according to $\epsilon_i^h(\mathbf{k})\epsilon_i^{h'}(-\mathbf{k}) = \delta_{hh'}$ (in this way eq. (6.2) is already the canonically-normalized quadratic action). We can easily see from the free theory that the mode functions will be equivalent to those of a massive scalar field in de Sitter. Assuming Bunch-Davies initial conditions, we have

$$ \sigma_k^{(h)} = \frac{H\sqrt{\pi}}{2} e^{i\pi(\nu+1/2)/2} (-\eta)^{3/2} H_\nu^{(1)}(-c_h k\eta), \quad \text{with} \quad \nu = \sqrt{\frac{9}{4} - \frac{m_\sigma^2}{H^2}}. \qquad (6.3) $$

<footnote n="22"><sup>22</sup>In the limit where we have full de Sitter symmetries, the Higuchi bound sets a lower bound on the mass of spinning fields by demanding that their longitudinal mode is not a ghost thereby ensuring the fields adhere to unitary representations of the de Sitter group.</footnote>

<footnote n="23"><sup>23</sup>Note that our $m_\sigma$ differs from that in [41] because we wrote the action in terms of conformal time.</footnote>
Therefore the bulk-to-boundary propagator for this spin-1 field is<fnref n="24" />

$$K_{\sigma}^{(h)} = \left(\frac{\eta}{\eta_0}\right)^{3/2} \frac{H_{\nu}^{(2)}(-c_h k \eta)}{H_{\nu}^{(2)}(-c_h k \eta_0)}, \qquad (6.4)$$

and the power spectrum is

$$P_{\sigma}^{(h)}(k) = \frac{\pi H^2}{4} (-\eta_0)^3 H_{\nu}^{(1)}(-c_h k \eta_0) H_{\nu}^{(2)}(-c_h k \eta_0), \qquad (6.5)$$

which has the correct mass dimension for a canonically-normalized field. This propagator simplifies for two special choices of the mass: for the massless case $m_{\sigma}^2 = 0$ ($\nu = 3/2$) and the conformally-coupled case $m_{\sigma}^2 = 2H^2$ ($\nu = 1/2$) we have

$$m_{\sigma}^2 = 0 \quad \Rightarrow \quad K_{\sigma} = (1 - ic_h k \eta) e^{ic_h k \eta}, \qquad (6.6)$$

$$m_{\sigma}^2 = 2H^2 \quad \Rightarrow \quad K_{\sigma} = \frac{\eta}{\eta_0} e^{ic_h k \eta}. \qquad (6.7)$$

In these limiting cases our results from section 3 show that no parity violation can occur due to the exchange of such a massive field so in the remainder of this section we will concentrate on more general light masses where $0 < m_{\sigma}/H < 3/2$.

Let's now turn our attention to interactions of the form $\pi\pi\sigma$ that can contribute to the scalar trispectrum via particle exchange. To realise a parity-odd trispectrum, we need one of the two vertices to have an odd number of spatial momenta, and the other to have an even number. To construct these actions we couple $\Sigma^{\mu}$ to the building blocks of the EFTI (see e.g. eqs. (2.3), (2.4), (2.5) for those with the lowest number of derivatives). Denoting schematically by $\mathcal{O}$ these building blocks, there are two ways we can construct $\pi\pi\sigma$ interactions: (i) operators of the form $\mathcal{O}\Sigma$ induce quadratic mixing terms where $\pi$ mixes with the longitudinal mode of $\sigma$, then the non-linearly realised symmetries also demands the presence of $\pi\pi\sigma$ couplings; (ii) operators of the form $\mathcal{O}\mathcal{O}\Sigma$ start at cubic order in fluctuations so the lowest order terms are just $\pi\pi\sigma$. For the purpose of this paper, where we are aiming to provide some yes-go examples for parity violation in the scalar trispectrum, we will work with the simplest set-up where there is no quadratic mixing. This ensures that there are no other Feynman diagrams to consider beyond the single-exchange diagram shown in figure 4, where we show only the s-channel for simplicity. Furthermore, we take $\sigma_i$ to be transverse since the longitudinal mode cannot contribute to parity-odd exchange. This means that we can take $\nabla_{\mu}\Sigma^{\mu} = 0$ and consequently there is no loss of generality in taking the Goldstone $\pi$ to appear only via $\partial_i\partial_j\pi$, $\bar{\pi}$ and $\partial_i\bar{\pi}$ at leading order in derivatives, since we can only use $\delta K_{\mu\nu}$ and $\nabla_{\mu}g^{00}$. In terms of symmetries, these are operators that are invariant under the leading part of the non-linearly realised symmetries. At zeroth order in fields, $\sigma^i$ does not transform which resonates with the fact that it can be thought of as a matter field in the CCWZ construct of effective actions [41]. Indeed, only Goldstones transform at this order and only $\pi$ is a Goldstone. The leading

<footnote n="24">²⁴Recall that the bulk-to-boundary propagator is fixed by $[\sigma_k^{(h)}]^*$ which is why it depends on the Hankel function of the second kind.</footnote>
Therefore the bulk-to-boundary propagator for this spin-1 field is<fnref n="24" />

$$K_{\sigma}^{(h)} = \left(\frac{\eta}{\eta_0}\right)^{3/2} \frac{H_{\nu}^{(2)}(-c_h k \eta)}{H_{\nu}^{(2)}(-c_h k \eta_0)}, \qquad (6.4)$$

and the power spectrum is

$$P_{\sigma}^{(h)}(k) = \frac{\pi H^2}{4} (-\eta_0)^3 H_{\nu}^{(1)}(-c_h k \eta_0) H_{\nu}^{(2)}(-c_h k \eta_0), \qquad (6.5)$$

which has the correct mass dimension for a canonically-normalized field. This propagator simplifies for two special choices of the mass: for the massless case $m_{\sigma}^2 = 0$ ($\nu = 3/2$) and the conformally-coupled case $m_{\sigma}^2 = 2H^2$ ($\nu = 1/2$) we have

$$m_{\sigma}^2 = 0 \quad \Rightarrow \quad K_{\sigma} = (1 - ic_h k \eta) e^{ic_h k \eta}, \qquad (6.6)$$

$$m_{\sigma}^2 = 2H^2 \quad \Rightarrow \quad K_{\sigma} = \frac{\eta}{\eta_0} e^{ic_h k \eta}. \qquad (6.7)$$

In these limiting cases our results from section 3 show that no parity violation can occur due to the exchange of such a massive field so in the remainder of this section we will concentrate on more general light masses where $0 < m_{\sigma}/H < 3/2$.

Let's now turn our attention to interactions of the form $\pi\pi\sigma$ that can contribute to the scalar trispectrum via particle exchange. To realise a parity-odd trispectrum, we need one of the two vertices to have an odd number of spatial momenta, and the other to have an even number. To construct these actions we couple $\Sigma^{\mu}$ to the building blocks of the EFTI (see e.g. eqs. (2.3), (2.4), (2.5) for those with the lowest number of derivatives). Denoting schematically by $\mathcal{O}$ these building blocks, there are two ways we can construct $\pi\pi\sigma$ interactions: (i) operators of the form $\mathcal{O}\Sigma$ induce quadratic mixing terms where $\pi$ mixes with the longitudinal mode of $\sigma$, then the non-linearly realised symmetries also demands the presence of $\pi\pi\sigma$ couplings; (ii) operators of the form $\mathcal{O}\mathcal{O}\Sigma$ start at cubic order in fluctuations so the lowest order terms are just $\pi\pi\sigma$. For the purpose of this paper, where we are aiming to provide some yes-go examples for parity violation in the scalar trispectrum, we will work with the simplest set-up where there is no quadratic mixing. This ensures that there are no other Feynman diagrams to consider beyond the single-exchange diagram shown in figure 4, where we show only the s-channel for simplicity. Furthermore, we take $\sigma_i$ to be transverse since the longitudinal mode cannot contribute to parity-odd exchange. This means that we can take $\nabla_{\mu}\Sigma^{\mu} = 0$ and consequently there is no loss of generality in taking the Goldstone $\pi$ to appear only via $\partial_i\partial_j\pi$, $\tilde{\pi}$ and $\partial_i\tilde{\pi}$ at leading order in derivatives, since we can only use $\delta K_{\mu\nu}$ and $\nabla_{\mu}g^{00}$. In terms of symmetries, these are operators that are invariant under the leading part of the non-linearly realised symmetries. At zeroth order in fields, $\sigma^i$ does not transform which resonates with the fact that it can be thought of as a matter field in the CCWZ construct of effective actions [41]. Indeed, only Goldstones transform at this order and only $\pi$ is a Goldstone. The leading

<footnote n="24">²⁴Recall that the bulk-to-boundary propagator is fixed by $[\sigma_k^{(h)}]^*$ which is why it depends on the Hankel function of the second kind.</footnote>
<figure><img src="image_4.png" /><figcaption>Figure 4. s-channel exchange diagram for the exchange of a massive spinning field.</figcaption></figure>

cubic action in conformal time is then given by

$$S_{\pi\pi\sigma} = \int d^3x d\eta \left[ \begin{aligned} & \lambda_1 \partial_i \pi' \partial_i \partial_j \pi \sigma^j + \lambda_2 \pi'' \partial_i \pi' \sigma^i + \lambda_3 a^{-1} \epsilon_{ijk} \partial_i \partial_l \pi \partial_j \partial_l \pi' \sigma^k \\ & + \lambda_4 a^{-1} \epsilon_{ijk} \partial_i \pi'' \partial_j \pi' \sigma^k \end{aligned} \right]. \quad (6.8)$$

In the presence of only the first and the second operators, the action would be invariant under parity if $\sigma^i$ transformed as a vector. Conversely, the third and fourth interactions are compatible with parity only if $\sigma^i$ is a pseudo-vector, $P\sigma^i(\mathbf{k})P = +\sigma^i(\mathbf{k})$. Hence, any process that involves both $\lambda_1$ or $\lambda_2$ and $\lambda_3$ or $\lambda_4$ leads to parity violation. For definiteness we will concentrate on the $O(\lambda_1\lambda_3)$ contribution to the trispectrum but many of the following results hold more generally. This action comes from the following unitary-gauge operators:

$$\lambda_1 \rightarrow (\nabla_\mu \delta g^{00}) \delta K^\mu_\nu \Sigma^\nu, \quad (6.9)$$

$$\lambda_2 \rightarrow \Sigma^\mu (n^\nu \nabla_\nu \delta g^{00}) \nabla_\mu \delta g^{00}, \quad (6.10)$$

$$\lambda_3 \rightarrow e^{\mu\nu\rho\sigma} n_\mu \delta K_{\nu\lambda} (n^\alpha \nabla_\alpha \delta K^\lambda_\rho) \Sigma_\sigma, \quad (6.11)$$

$$\lambda_4 \rightarrow e^{\mu\nu\rho\sigma} n_\mu \nabla_\nu (n^\alpha \nabla_\alpha \delta g^{00}) (\nabla_\rho \delta g^{00}) \Sigma_\sigma. \quad (6.12)$$

Let us stress that within this set-up where we look for parity violation due to particle exchange, it is not possible to have a large parity-odd signal in combination with a small parity-even one. Indeed, parity-even signals can come from both $O(\lambda_1^2)$ and $O(\lambda_3^2)$ contributions to the wavefunction and if the parity-odd signal is made large by having large $\lambda_1$ or $\lambda_3$, one of these parity-even signals will also be large. We therefore expect a detection of a parity-odd trispectrum due to particle exchange to be accompanied by a detection of a parity-even trispectrum, unless there is some kinematical configuration where the parity-even shape is small rather than the overall coupling. We will also discuss the constraints on the couplings coming from the requirements of perturbativity in a moment.

**Exchange contribution to the quartic wavefunction.** Let us first consider the exchange diagram wavefunction contribution $\psi_4$ to the trispectrum $B_4^{PO}$, before moving onto the factorised contribution. Within the range of masses that we are interested in, we are restricting to cases where $\nu$ is real cf. eq. (6.3), and throughout the remainder of this section
<figure><img src="image_5.png" /><figcaption>Figure 4. s-channel exchange diagram for the exchange of a massive spinning field.</figcaption></figure>

cubic action in conformal time is then given by

$$S_{\pi\pi\sigma} = \int d^3x d\eta \left[ \lambda_1 \partial_i \pi' \partial_i \pi \sigma^j + \lambda_2 \pi'' \partial_i \pi' \sigma^i + \lambda_3 a^{-1} \epsilon_{ijk} \partial_i \partial_l \pi \partial_j \partial_l \pi' \sigma^k \right. \\ \left. + \lambda_4 a^{-1} \epsilon_{ijk} \partial_i \pi'' \partial_j \pi' \sigma^k \right]. \quad (6.8)$$

In the presence of only the first and the second operators, the action would be invariant under parity if $\sigma^i$ transformed as a vector. Conversely, the third and fourth interactions are compatible with parity only if $\sigma^i$ is a pseudo-vector, $P\sigma^i(\mathbf{k})P = +\sigma^i(\mathbf{k})$. Hence, any process that involves both $\lambda_1$ or $\lambda_2$ and $\lambda_3$ or $\lambda_4$ leads to parity violation. For definiteness we will concentrate on the $O(\lambda_1\lambda_3)$ contribution to the trispectrum but many of the following results hold more generally. This action comes from the following unitary-gauge operators:

$$\lambda_1 \rightarrow (\nabla_{\mu} \delta g^{00}) \delta K^{\mu}_{\nu} \Sigma^{\nu}, \quad (6.9)$$

$$\lambda_2 \rightarrow \Sigma^{\mu}(n^{\nu}\nabla_{\nu}\delta g^{00})\nabla_{\mu}\delta g^{00}, \quad (6.10)$$

$$\lambda_3 \rightarrow \mathbf{e}^{\mu\nu\rho\sigma} n_{\mu} \delta K_{\nu\lambda} (n^{\alpha} \nabla_{\alpha} \delta K^{\lambda}_{\rho}) \Sigma_{\sigma}, \quad (6.11)$$

$$\lambda_4 \rightarrow \mathbf{e}^{\mu\nu\rho\sigma} n_{\mu} \nabla_{\nu} (n^{\alpha} \nabla_{\alpha} \delta g^{00}) (\nabla_{\rho} \delta g^{00}) \Sigma_{\sigma}. \quad (6.12)$$

Let us stress that within this set-up where we look for parity violation due to particle exchange, it is not possible to have a large parity-odd signal in combination with a small parity-even one. Indeed, parity-even signals can come from both $O(\lambda_1^2)$ and $O(\lambda_3^2)$ contributions to the wavefunction and if the parity-odd signal is made large by having large $\lambda_1$ or $\lambda_3$, one of these parity-even signals will also be large. We therefore expect a detection of a parity-odd trispectrum due to particle exchange to be accompanied by a detection of a parity-even trispectrum, unless there is some kinematical configuration where the parity-even shape is small rather than the overall coupling. We will also discuss the constraints on the couplings coming from the requirements of perturbativity in a moment.

**Exchange contribution to the quartic wavefunction.** Let us first consider the exchange diagram wavefunction contribution $\psi_4$ to the trispectrum $B_4^{PO}$, before moving onto the factorised contribution. Within the range of masses that we are interested in, we are restricting to cases where $\nu$ is real cf. eq. (6.3), and throughout the remainder of this section
we will fix the speed of the exchanged field to unity, $c_1 = 1$, while keeping the speed of the Goldstone mode, $c_s$, general. The s-channel exchange diagram at $O(\lambda_1\lambda_3)$ is given by

$$ \begin{aligned} \psi_{4,s} &= \sum_{h=\pm 1} (-i) \times [-i\lambda_1 k_1^i k_2^i k_2^j \epsilon_j^h(\mathbf{s})] \times [-\lambda_3 H \epsilon_{ijk} k_3^i k_3^l k_4^j k_4^l \epsilon_k^h(-\mathbf{s})] \\ & \qquad \times \int d\eta d\eta' \eta' K'_\pi(k_1, \eta) K_\pi(k_2, \eta) G_\sigma(\eta, \eta', s) K_\pi(k_3, \eta') K'_\pi(k_4, \eta') + 7 \text{ perms.} \\ &= \frac{\lambda_1 \lambda_3 H}{4} (s^2 - k_1^2 - k_2^2)(s^2 - k_3^2 - k_4^2) \epsilon_{ijk} k_3^i k_4^j k_2^m \sum_{h=\pm 1} \epsilon_m^h(\mathbf{s}) \epsilon_k^h(-\mathbf{s}) \\ & \qquad \times \int d\eta d\eta' \eta' K'_\pi(k_1, \eta) K_\pi(k_2, \eta) G_\sigma(\eta, \eta', s) K_\pi(k_3, \eta') K'_\pi(k_4, \eta') + 7 \text{ perms.}, \end{aligned} \quad (6.13) $$

where we have used momentum conservation at each vertex. Once we sum over the remaining two channels, we get the correct number of 4! permutations. We also sum over the helicities of the exchanged field, and we restrict to the transverse components since the longitudinal mode cannot give rise to a non-zero parity-odd wavefunction. Using known polarisation sums, see e.g. [16, 37, 63], we can eliminate all reference to the internal field. Using $\sum_{h=\pm 1} \epsilon_i^h(\mathbf{s})\epsilon_j^h(-\mathbf{s}) = \Pi_{ij}(\mathbf{s}) = \delta_{ij} - s_i s_j / s^2$, $^{25}$ we then have

$$ \psi_{4,s} = \frac{\lambda_1 \lambda_3 H}{4} (s^2 - k_1^2 - k_2^2)(s^2 - k_3^2 - k_4^2) \epsilon_{ijk} k_3^i k_4^j k_2^k \mathcal{I}_E(k_1, k_2, k_3, k_4, s) + 7 \text{ perms.}, \quad (6.14) $$

where we have defined

$$ \mathcal{I}_E(k_1, k_2, k_3, k_4, s) = \int d\eta d\eta' \eta' K'_\pi(k_1, \eta) K_\pi(k_2, \eta) G_\sigma(\eta, \eta', s) K_\pi(k_3, \eta') K'_\pi(k_4, \eta'). \quad (6.15) $$

We note that the second term in the polarisation sum does not contribute since $\epsilon_{ijk} k_3^i k_4^j s^k = 0$ by momentum conservation. It is useful to write the bulk-bulk propagator of the massive field explicitly in terms of the Hankel functions. We have

$$ G_{\sigma}(\eta, \eta', s) = \frac{i\pi H^2}{4} \left[ \theta(\eta - \eta')(\eta'\eta)^{3/2} H_{\nu}^{(2)}(-s\eta') [H_{\nu}^{(2)}(-s\eta) + H_{\nu}^{(1)}(-s\eta)] + (\eta \leftrightarrow \eta') \right], \quad (6.16) $$

where we have used

$$ \frac{P_{\sigma}(s)}{[(-\eta_0)^{3/2} H_{\nu}^{(2)}(-s\eta_0)]^2} = \frac{\pi H^2 H_{\nu}^{(1)}(-s\eta_0)}{4 H_{\nu}^{(2)}(-s\eta_0)} \rightarrow -\frac{\pi H^2}{4}, \quad (6.17) $$

$$ \mathrm{Im} K(s, \eta) = -\frac{i}{2} \frac{(-\eta)^{3/2}}{(-\eta_0)^{3/2} H_{\nu}^{(2)}(-s\eta_0)} [H_{\nu}^{(2)}(-s\eta) + H_{\nu}^{(1)}(-s\eta)], \quad (6.18) $$

and have assumed real $\nu$ which is the case for our masses of interest.

As we have explained a number of times, whether $\psi_{4,s}$ contributes to the trispectrum or not depends on the properties of $\mathcal{I}_E$: it must contain an imaginary part, such that $\psi_{4,s}$

<footnote>$^{25}$This projection tensor is transverse as it should be: $s_i \Pi_{ij}(\mathbf{s}) = s_j - s_j s^2 / s^2 = 0$, and has the correct normalisation since $\delta^{ij} \Pi_{ij}(\mathbf{s}) = 2$. The 2 comes from summing over the two helicities, using $\epsilon_i^h(\mathbf{s})\epsilon_i^{h'}(-\mathbf{s}) = \delta_{hh'}$.</footnote>
we will fix the speed of the exchanged field to unity, $c_1 = 1$, while keeping the speed of the Goldstone mode, $c_s$, general. The s-channel exchange diagram at $O(\lambda_1\lambda_3)$ is given by

$$ 
\begin{aligned}
\psi_{4,s} &= \sum_{h=\pm 1} (-i) \times [-i\lambda_1 k_1^i k_2^i k_2^j \epsilon_j^h(\mathbf{s})] \times [-\lambda_3 H \epsilon_{ijk} k_3^i k_3^l k_4^j k_4^l \epsilon_k^h(-\mathbf{s})] \\
&\quad \times \int d\eta d\eta' \eta' K'_\pi(k_1, \eta) K_\pi(k_2, \eta) G_\sigma(\eta, \eta', s) K_\pi(k_3, \eta') K'_\pi(k_4, \eta') + 7 \text{ perms.} \\
&= \frac{\lambda_1 \lambda_3 H}{4} (s^2 - k_1^2 - k_2^2)(s^2 - k_3^2 - k_4^2) \epsilon_{ijk} k_3^i k_4^j k_2^m \sum_{h=\pm 1} \epsilon_m^h(\mathbf{s}) \epsilon_k^h(-\mathbf{s}) \\
&\quad \times \int d\eta d\eta' \eta' K'_\pi(k_1, \eta) K_\pi(k_2, \eta) G_\sigma(\eta, \eta', s) K_\pi(k_3, \eta') K'_\pi(k_4, \eta') + 7 \text{ perms.},
\end{aligned}
\quad (6.13) $$

where we have used momentum conservation at each vertex. Once we sum over the remaining two channels, we get the correct number of 4! permutations. We also sum over the helicities of the exchanged field, and we restrict to the transverse components since the longitudinal mode cannot give rise to a non-zero parity-odd wavefunction. Using known polarisation sums, see e.g. [16, 37, 63], we can eliminate all reference to the internal field. Using $\sum_{h=\pm 1} \epsilon_i^h(\mathbf{s})\epsilon_j^h(-\mathbf{s}) = \Pi_{ij}(\mathbf{s}) = \delta_{ij} - s_i s_j / s^2$, $^{25}$ we then have

$$ \psi_{4,s} = \frac{\lambda_1 \lambda_3 H}{4} (s^2 - k_1^2 - k_2^2)(s^2 - k_3^2 - k_4^2) \epsilon_{ijk} k_3^i k_4^j k_2^k \mathcal{I}_E(k_1, k_2, k_3, k_4, s) + 7 \text{ perms.}, \quad (6.14) $$

where we have defined

$$ \mathcal{I}_E(k_1, k_2, k_3, k_4, s) = \int d\eta d\eta' \eta' K'_\pi(k_1, \eta) K_\pi(k_2, \eta) G_\sigma(\eta, \eta', s) K_\pi(k_3, \eta') K'_\pi(k_4, \eta'). \quad (6.15) $$

We note that the second term in the polarisation sum does not contribute since $\epsilon_{ijk} k_3^i k_4^j s^k = 0$ by momentum conservation. It is useful to write the bulk-bulk propagator of the massive field explicitly in terms of the Hankel functions. We have

$$ G_{\sigma}(\eta, \eta', s) = \frac{i\pi H^2}{4} \left[ \theta(\eta - \eta')(\eta'\eta)^{3/2} H_{\nu}^{(2)}(-s\eta') [H_{\nu}^{(2)}(-s\eta) + H_{\nu}^{(1)}(-s\eta)] + (\eta \leftrightarrow \eta') \right], \quad (6.16) $$

where we have used

$$ \frac{P_{\sigma}(s)}{[(-\eta_0)^{3/2} H_{\nu}^{(2)}(-s\eta_0)]^2} = \frac{\pi H^2 H_{\nu}^{(1)}(-s\eta_0)}{4 H_{\nu}^{(2)}(-s\eta_0)} \rightarrow -\frac{\pi H^2}{4}, \quad (6.17) $$

$$ \mathrm{Im} K(s, \eta) = -\frac{i}{2} \frac{(-\eta)^{3/2}}{(-\eta_0)^{3/2} H_{\nu}^{(2)}(-s\eta_0)} [H_{\nu}^{(2)}(-s\eta) + H_{\nu}^{(1)}(-s\eta)], \quad (6.18) $$

and have assumed real $\nu$ which is the case for our masses of interest.

As we have explained a number of times, whether $\psi_{4,s}$ contributes to the trispectrum or not depends on the properties of $\mathcal{I}_E$: it must contain an imaginary part, such that $\psi_{4,s}$

<footnote>$^{25}$This projection tensor is transverse as it should be: $s_i \Pi_{ij}(\mathbf{s}) = s_j - s_j s^2 / s^2 = 0$, and has the correct normalisation since $\delta^{ij} \Pi_{ij}(\mathbf{s}) = 2$. The 2 comes from summing over the two helicities, using $\epsilon_i^h(\mathbf{s})\epsilon_i^{h'}(-\mathbf{s}) = \delta_{hh'}$.</footnote>
is itself imaginary, for there to be a non-vanishing contribution. We will now study the properties of this nested time integral. It turns out that $\mathcal{I}_E$ can be related to a simpler building block, namely the four-point function of a conformally-coupled field that interacts with a massive scalar field $\sigma$ through the trilinear vertex $\varphi^2\sigma$. The Lagrangian of the $\varphi-\sigma$ theory is given by

$$S = \int d^3x d\eta a^4(\eta) \left( -\frac{1}{2}(\partial_\mu \varphi)^2 - H^2 \varphi^2 - \frac{1}{2}(\partial_\mu \sigma)^2 - \frac{1}{2}m_\sigma^2 \sigma^2 - g\varphi^2\sigma \right), \quad (6.19)$$

where $\varphi$ is a conformally-coupled field and $\sigma$ is a scalar field with the same mass as $\sigma^i$. The four-point wavefunction coefficient of $\varphi$ induced by the exchange of $\sigma$ is given by

$$\psi_{4,s}^{\varphi} = -\frac{4ig^2}{\eta_0^4} \int \frac{d\eta d\eta'}{\eta^2 \eta'^2} e^{i(k_1+k_2)\eta} e^{i(k_3+k_4)\eta'} G_{\sigma}(s, \eta, \eta'). \quad (6.20)$$

This four-point coefficient is convergent in the $\eta_0 \to 0$ limit iff $0 \le \nu < \frac{1}{2}$. Let us define $F(u, v) = -\frac{\eta_0^4 s}{4g^2} \psi_4$ which only depends on the dimensionless quantities $u = \frac{s}{k_1+k_2}$ and $v = \frac{s}{k_3+k_4}$. A similar quantity was bootstrapped in [64]. Here we follow the conventions laid out in section 4.2 of [38], where $F(u, v)$ was found to be

$$F(u, v) = \begin{cases} \sum_{m,n=0}^{\infty} c_{mn} u^{2m+1} \left(\frac{u}{v}\right)^n + \frac{\pi}{2 \cos(\pi \nu)} g(u, v), & |u| \le |v| \\ \sum_{m,n=0}^{\infty} c_{mn} v^{2m+1} \left(\frac{v}{u}\right)^n + \frac{\pi}{2 \cos(\pi \nu)} g(v, u), & |v| \le |u| \end{cases} . \quad (6.21)$$

Here, the $c_{mn}$'s are a set of real numbers given by

$$c_{mn} = \frac{(-1)^n (n+1)(n+2)\dots(n+2m)}{[(n+\frac{1}{2})^2 - \nu^2][(n+\frac{5}{2})^2 - \nu^2]\dots[(n+\frac{1}{2}+2m)^2 - \nu^2]}, \quad (6.22)$$

and

$$g(u, v) = \kappa(\nu) P_{\nu-\frac{1}{2}}\left(\frac{1}{u}\right) - Q_{\nu-\frac{1}{2}}\left(\frac{1}{u}\right) + Q_{-\nu-\frac{1}{2}}\left(\frac{1}{u}\right), \quad (6.23)$$

with $P_\nu$ and $Q_\nu$ the associated Legendre functions of the first and second type, respectively, and

$$\kappa(\nu) = i\pi + \frac{\pi}{\cos(\pi\nu)} \frac{\sigma(s, \eta_0)}{\sigma^*(s, \eta_0)} = \pi \tan(\pi\nu). \quad (6.24)$$

Above, $\sigma(s, \eta)$ is the positive-frequency mode function of the massive field, which is the same as (6.3) with $c_h = 1$. Putting everything together, we see that for physical configurations, namely for $0 \le u \le 1$, the entire $\psi_{4,s}^\varphi$ is a real quantity. $\mathcal{I}_E$ will inherit this reality as it is related to $\psi_4$ through a Hermitian weight-shifting operator.<fnref n="26" /> This operator can be reconstructed from the knowledge of the type of interactions inserted at each vertex. The

<footnote n="26">²⁶Weight-shifting operators are not always Hermitian, but for our interaction vertices the weight-shifting procedure does not alter the reality of the time integral.</footnote>
is itself imaginary, for there to be a non-vanishing contribution. We will now study the properties of this nested time integral. It turns out that $\mathcal{I}_E$ can be related to a simpler building block, namely the four-point function of a conformally-coupled field that interacts with a massive scalar field $\sigma$ through the trilinear vertex $\varphi^2\sigma$. The Lagrangian of the $\varphi-\sigma$ theory is given by

$$S = \int d^3x d\eta a^4(\eta) \left( -\frac{1}{2}(\partial_\mu \varphi)^2 - H^2 \varphi^2 - \frac{1}{2}(\partial_\mu \sigma)^2 - \frac{1}{2}m_\sigma^2 \sigma^2 - g\varphi^2\sigma \right), \quad (6.19)$$

where $\varphi$ is a conformally-coupled field and $\sigma$ is a scalar field with the same mass as $\sigma^i$. The four-point wavefunction coefficient of $\varphi$ induced by the exchange of $\sigma$ is given by

$$\psi_{4,s}^{\varphi} = -\frac{4ig^2}{\eta_0^4} \int \frac{d\eta d\eta'}{\eta^2 \eta'^2} e^{i(k_1+k_2)\eta} e^{i(k_3+k_4)\eta'} G_{\sigma}(s, \eta, \eta'). \quad (6.20)$$

This four-point coefficient is convergent in the $\eta_0 \to 0$ limit iff $0 \le \nu < \frac{1}{2}$. Let us define $F(u, v) = -\frac{\eta_0^4 s}{4g^2} \psi_4$ which only depends on the dimensionless quantities $u = \frac{s}{k_1+k_2}$ and $v = \frac{s}{k_3+k_4}$. A similar quantity was bootstrapped in [64]. Here we follow the conventions laid out in section 4.2 of [38], where $F(u, v)$ was found to be

$$F(u, v) = \begin{cases} \sum_{m,n=0}^{\infty} c_{mn} u^{2m+1} \left(\frac{u}{v}\right)^n + \frac{\pi}{2 \cos(\pi \nu)} g(u, v), & |u| \le |v| \\ \sum_{m,n=0}^{\infty} c_{mn} v^{2m+1} \left(\frac{v}{u}\right)^n + \frac{\pi}{2 \cos(\pi \nu)} g(v, u), & |v| \le |u| \end{cases}. \quad (6.21)$$

Here, the $c_{mn}$'s are a set of real numbers given by

$$c_{mn} = \frac{(-1)^n (n+1)(n+2)\dots(n+2m)}{[(n+\frac{1}{2})^2 - \nu^2][(n+\frac{5}{2})^2 - \nu^2]\dots[(n+\frac{1}{2}+2m)^2 - \nu^2]}, \quad (6.22)$$

and

$$g(u, v) = \kappa(\nu) P_{\nu-\frac{1}{2}}\left(\frac{1}{u}\right) - Q_{\nu-\frac{1}{2}}\left(\frac{1}{u}\right) + Q_{-\nu-\frac{1}{2}}\left(\frac{1}{u}\right), \quad (6.23)$$

with $P_\nu$ and $Q_\nu$ the associated Legendre functions of the first and second type, respectively, and

$$\kappa(\nu) = i\pi + \frac{\pi}{\cos(\pi\nu)} \frac{\sigma(s, \eta_0)}{\sigma^*(s, \eta_0)} = \pi \tan(\pi\nu). \quad (6.24)$$

Above, $\sigma(s, \eta)$ is the positive-frequency mode function of the massive field, which is the same as (6.3) with $c_h = 1$. Putting everything together, we see that for physical configurations, namely for $0 \le u \le 1$, the entire $\psi_{4,s}^\varphi$ is a real quantity. $\mathcal{I}_E$ will inherit this reality as it is related to $\psi_4$ through a Hermitian weight-shifting operator.<fnref n="26" /> This operator can be reconstructed from the knowledge of the type of interactions inserted at each vertex. The

<footnote n="26">²⁶Weight-shifting operators are not always Hermitian, but for our interaction vertices the weight-shifting procedure does not alter the reality of the time integral.</footnote>
result is given by

$$ \mathcal{I}_E(k_1, k_2, k_3, k_4, s) = -\frac{k_1^2 k_4^2}{c_s^3} \left( \frac{\partial^3}{\partial(k_1+k_2)^3} - k_2 \frac{\partial^4}{\partial(k_1+k_2)^4} \right) \left( \frac{\partial^4}{\partial(k_3+k_4)^4} - k_2 \frac{\partial^4}{\partial(k_1+k_2)^5} \right) \times F \left( \frac{s}{c_s(k_1+k_2)}, \frac{s}{c_s(k_3+k_4)} \right). \qquad (6.25) $$

In this relation, $s$ is held fixed when the partial derivatives $\frac{\partial}{\partial(k_1+k_2)}$ and $\frac{\partial}{\partial(k_3+k_4)}$ operate. The indicated weight-shifting operator in front of $F(u, v)$ turns the external conformally-coupled, massive states into massless ones, hence the name (see [16] for the systematic study of such operators in de Sitter-isometric situations and [65, 66], when de Sitter boosts are broken). It follows from the reality of this weight-shifting operator and that of the seed function $F(u, v)$ that $\mathcal{I}_E$ has to be real. Two technical comments are in order: (i) The expression for $F(u, v)$ in (6.21) on the upper (lower) line contains a series expansion that converges only for $|u| < 1$ ($|v| < 1$). However, for a subluminal $c_s < 1$, both ratios $u = \frac{s}{c_s(k_1+k_2)}$ and $v = \frac{s}{c_s(k_3+k_4)}$ can take values beyond the unit disk where the aforementioned series expansion is not applicable.<fnref n="27" /> Nevertheless, the reality of $\mathcal{I}_E$ across the region defined by $0 < u < 1$ and $0 < v < 1$ carries over to the entire $u, v > 0$ region because $F(u, v)$ is analytic around $u = 1$ (for arbitrary $v$) and $v = 1$ (for arbitrary $u$).<fnref n="28" /> (ii) Our proof so far only applies to $0 < v < 1/2$ for which the four-point $\psi_{4,s}$ is IR-convergent. Nevertheless, for lighter states, namely $\frac{1}{2} \le v < \frac{3}{2}$, $\mathcal{I}_E$ is still convergent and analytic as a function of $v > 0$. Consequently, the reality of $\mathcal{I}_E$ follows for an arbitrary positive $v$ from its reality across $0 < v < 1/2$. In appendix A we present a complementary proof that the time integral $\mathcal{I}_E$ is purely real by performing Wick rotations on the two time variables.

**Factorised contribution to the trispectrum.** Having shown that there is no exchange contribution to the trispectrum for our range of masses, we now move on to the factorised contribution for which we need to compute the two cubic wavefunction coefficients. We are considering the $O(\lambda_1\lambda_3)$ trispectrum. The cubic wavefunction coefficient due to the $\lambda_1$ vertex is given by (throughout we are suppressing the integration limits which are always the same)

$$ \begin{aligned} \psi_{3,1} &= -\lambda_1 k_1^i k_2^i k_2^j \epsilon_j(\mathbf{k}_3) \int d\eta K'_\pi(k_1, \eta) K_\pi(k_2, \eta) K_\sigma(k_3, \eta) + (1 \leftrightarrow 2) \\ &= -\frac{\lambda_1}{2} (k_3^2 - k_1^2 - k_2^2) k_2^i \epsilon_i(\mathbf{k}_3) [\mathcal{I}_1(k_1, k_2, k_3) - \mathcal{I}_1(k_2, k_1, k_3)], \end{aligned} \qquad (6.26) $$

where the overall factor of $(-i)$ in the Feynman rules combines with the factor of $(-i)$ we get from converting the three spatial derivatives to momentum space to give an overall factor of $(-1)$, and in the final line we have used momentum conservation and transversality

<footnote n="27">²⁷This situation was studied in depth in [65], and an explicit expression for the four-point function was found as a series expansion that converges outside the corresponding unit disk $|u| < 1$ (or $|v| < 1$).</footnote>

<footnote n="28">²⁸This is a direct consequence of having a Bunch-Davies initial condition which implies regularity at the collinear limit (i.e. $u = 1$ or $v = 1$).</footnote>
result is given by

$$ \mathcal{I}_E(k_1, k_2, k_3, k_4, s) = -\frac{k_1^2 k_4^2}{c_s^3} \left( \frac{\partial^3}{\partial(k_1+k_2)^3} - k_2 \frac{\partial^4}{\partial(k_1+k_2)^4} \right) \left( \frac{\partial^4}{\partial(k_3+k_4)^4} - k_2 \frac{\partial^4}{\partial(k_1+k_2)^5} \right) \times F \left( \frac{s}{c_s(k_1+k_2)}, \frac{s}{c_s(k_3+k_4)} \right). \qquad (6.25) $$

In this relation, s is held fixed when the partial derivatives $\frac{\partial}{\partial(k_1+k_2)}$ and $\frac{\partial}{\partial(k_3+k_4)}$ operate. The indicated weight-shifting operator in front of $F(u, v)$ turns the external conformally-coupled, massive states into massless ones, hence the name (see [16] for the systematic study of such operators in de Sitter-isometric situations and [65, 66], when de Sitter boosts are broken). It follows from the reality of this weight-shifting operator and that of the seed function $F(u, v)$ that $\mathcal{I}_E$ has to be real. Two technical comments are in order: (i) The expression for $F(u, v)$ in (6.21) on the upper (lower) line contains a series expansion that converges only for $|u| < 1$ ($|v| < 1$). However, for a subluminal $c_s < 1$, both ratios $u = \frac{s}{c_s(k_1+k_2)}$ and $v = \frac{s}{c_s(k_3+k_4)}$ can take values beyond the unit disk where the aforementioned series expansion is not applicable.<fnref n="27" /> Nevertheless, the reality of $\mathcal{I}_E$ across the region defined by $0 < u < 1$ and $0 < v < 1$ carries over to the entire $u, v > 0$ region because $F(u, v)$ is analytic around $u = 1$ (for arbitrary $v$) and $v = 1$ (for arbitrary $u$).<fnref n="28" /> (ii) Our proof so far only applies to $0 < v < 1/2$ for which the four-point $\psi_{4,s}$ is IR-convergent. Nevertheless, for lighter states, namely $\frac{1}{2} \le v < \frac{3}{2}$, $\mathcal{I}_E$ is still convergent and analytic as a function of $v > 0$. Consequently, the reality of $\mathcal{I}_E$ follows for an arbitrary positive $v$ from its reality across $0 < v < 1/2$. In appendix A we present a complementary proof that the time integral $\mathcal{I}_E$ is purely real by performing Wick rotations on the two time variables.

**Factorised contribution to the trispectrum.** Having shown that there is no exchange contribution to the trispectrum for our range of masses, we now move on to the factorised contribution for which we need to compute the two cubic wavefunction coefficients. We are considering the $O(\lambda_1\lambda_3)$ trispectrum. The cubic wavefunction coefficient due to the $\lambda_1$ vertex is given by (throughout we are suppressing the integration limits which are always the same)

$$ \begin{aligned} \psi_{3,1} &= -\lambda_1 k_1^i k_2^i k_2^j \epsilon_j(\mathbf{k}_3) \int d\eta K'_\pi(k_1, \eta) K_\pi(k_2, \eta) K_\sigma(k_3, \eta) + (1 \leftrightarrow 2) \\ &= -\frac{\lambda_1}{2} (k_3^2 - k_1^2 - k_2^2) k_2^i \epsilon_i(\mathbf{k}_3) [\mathcal{I}_1(k_1, k_2, k_3) - \mathcal{I}_1(k_2, k_1, k_3)], \end{aligned} \qquad (6.26) $$

where the overall factor of $(-i)$ in the Feynman rules combines with the factor of $(-i)$ we get from converting the three spatial derivatives to momentum space to give an overall factor of $(-1)$, and in the final line we have used momentum conservation and transversality

<footnote n="27">²⁷This situation was studied in depth in [65], and an explicit expression for the four-point function was found as a series expansion that converges outside the corresponding unit disk $|u| < 1$ (or $|v| < 1$).</footnote>

<footnote n="28">²⁸This is a direct consequence of having a Bunch-Davies initial condition which implies regularity at the collinear limit (i.e. $u = 1$ or $v = 1$).</footnote>
of the spin-1 field to write $k_1^i \epsilon_i(\mathbf{k}_3) = -k_2^i \epsilon_i(\mathbf{k}_3)$. We have also defined

$$ \mathcal{I}_1(k_1, k_2, k_3) = \int d\eta K'_{\pi}(k_1, \eta) K_{\pi}(k_2, \eta) K_{\sigma}(k_3, \eta). \qquad (6.27) $$

Similarly, for the cubic wavefunction coefficient due to the $\lambda_3$ vertex we have

$$ \psi_{3,3} = \frac{iH\lambda_3}{2}(k_3^2 - k_1^2 - k_2^2)\epsilon_{ijk}k_1^i k_2^j \epsilon_k(\mathbf{k}_3)[\mathcal{I}_3(k_1, k_2, k_3) - \mathcal{I}_3(k_2, k_1, k_3)], \qquad (6.28) $$

where we have defined

$$ \mathcal{I}_3(k_1, k_2, k_3) = \int d\eta \eta K_{\pi}(k_1, \eta) K'_{\pi}(k_2, \eta) K_{\sigma}(k_3, \eta). \qquad (6.29) $$

Note that there is an additional factor of $\eta$ in $\mathcal{I}_3$ that distinguishes it from $\mathcal{I}_1$. Now it is the combination $\rho_3 = \psi_3(\{k\}, \{\mathbf{k}\}) + \psi_3^*(\{k\}, \{-\mathbf{k}\})$ that contributes to the trispectrum and for these two wavefunction coefficients we have

$$ \rho_{3,1} = -i\lambda_1(k_3^2 - k_1^2 - k_2^2)k_2^i \epsilon_i(\mathbf{k}_3) \mathrm{Im}[\mathcal{I}_1(k_1, k_2, k_3) - \mathcal{I}_1(k_2, k_1, k_3)], \qquad (6.30) $$

$$ \rho_{3,3} = -H\lambda_3(k_3^2 - k_1^2 - k_2^2)\epsilon_{ijk}k_1^i k_2^j \epsilon_k(\mathbf{k}_3) \mathrm{Im}[\mathcal{I}_3(k_1, k_2, k_3) - \mathcal{I}_3(k_2, k_1, k_3)]. \qquad (6.31) $$

As expected, we see that for the interaction vertex with an odd number of spatial momenta, we have an imaginary $\rho_3$, while the interaction with an even number of spatial momenta yields a real $\rho_3$.

Now, the s-channel contribution to this factorised part of the trispectrum takes the form

$$ B_{4,s}^{\pi} = \prod_{a=1}^{4} P_{\pi}(k_a) \sum_{h=\pm 1} P_{\sigma}^{h}(s) \rho_{3,3}(\mathbf{k}_1, \mathbf{k}_2, \mathbf{s}) \rho_{1,3}(\mathbf{k}_3, \mathbf{k}_4 - \mathbf{s}) + [(1,2) \leftrightarrow (3,4)], \qquad (6.32) $$

where we only add one permutation since we have already explicitly summed over some permutations in arriving at the above expressions for $\rho_{3,1}$ and $\rho_{3,3}$, and as always we don't include the spin zero “exchange” since this will always give a vanishing result. We use an equal sign for this s-channel exchange since as we showed above, the quartic wavefunction coefficient does not contribute to the correlator. Computing this product and summing over the helicities of the exchanged field (recalling that only the $\delta_{ij}$ in the polarisation sum contributes) yields

$$ B_{4,s}^{\pi} = \left( \prod_{a=1}^{4} P_{\pi}(k_a) \right) iH\lambda_1\lambda_3 P_{\sigma}(s)(s^2 - k_1^2 - k_2^2)(s^2 - k_3^2 - k_4^2)\epsilon_{ijk}k_3^i k_4^j k_2^k \\ \times \mathrm{Im}[\mathcal{I}_3(k_1, k_2, s) - \mathcal{I}_3(k_2, k_1, s)] \mathrm{Im}[\mathcal{I}_1(k_3, k_4, s) - \mathcal{I}_1(k_4, k_3, s)] + [(1,2) \leftrightarrow (3,4)]. \qquad (6.33) $$

Let us first convince ourselves that this contribution is indeed non-zero. By construction it is invariant under $(1 \leftrightarrow 2)$ and $(3 \leftrightarrow 4)$. This is manifest for $(3 \leftrightarrow 4)$ due to the anti-symmetric nature of $\epsilon_{ijk}$, while it can be made manifest for $(1 \leftrightarrow 2)$ by writing $\epsilon_{ijk}k_3^i k_4^j k_2^k = \epsilon_{ijk}k_3^i k_4^j (k_2^k - k_1^k)/2$. The final sum over replacing $(1,2) \leftrightarrow (3,4)$ cannot
of the spin-1 field to write $k_1^i \epsilon_i(\mathbf{k}_3) = -k_2^i \epsilon_i(\mathbf{k}_3)$. We have also defined

$$ \mathcal{I}_1(k_1, k_2, k_3) = \int d\eta K'_{\pi}(k_1, \eta) K_{\pi}(k_2, \eta) K_{\sigma}(k_3, \eta). \qquad (6.27) $$

Similarly, for the cubic wavefunction coefficient due to the $\lambda_3$ vertex we have

$$ \psi_{3,3} = \frac{iH\lambda_3}{2}(k_3^2 - k_1^2 - k_2^2)\epsilon_{ijk}k_1^i k_2^j \epsilon_k(\mathbf{k}_3)[\mathcal{I}_3(k_1, k_2, k_3) - \mathcal{I}_3(k_2, k_1, k_3)], \qquad (6.28) $$

where we have defined

$$ \mathcal{I}_3(k_1, k_2, k_3) = \int d\eta \eta K_{\pi}(k_1, \eta) K'_{\pi}(k_2, \eta) K_{\sigma}(k_3, \eta). \qquad (6.29) $$

Note that there is an additional factor of $\eta$ in $\mathcal{I}_3$ that distinguishes it from $\mathcal{I}_1$. Now it is the combination $\rho_3 = \psi_3(\{k\}, \{\mathbf{k}\}) + \psi_3^*(\{k\}, \{-\mathbf{k}\})$ that contributes to the trispectrum and for these two wavefunction coefficients we have

$$ \rho_{3,1} = -i\lambda_1(k_3^2 - k_1^2 - k_2^2)k_2^i \epsilon_i(\mathbf{k}_3) \mathrm{Im}[\mathcal{I}_1(k_1, k_2, k_3) - \mathcal{I}_1(k_2, k_1, k_3)], \qquad (6.30) $$

$$ \rho_{3,3} = -H\lambda_3(k_3^2 - k_1^2 - k_2^2)\epsilon_{ijk}k_1^i k_2^j \epsilon_k(\mathbf{k}_3) \mathrm{Im}[\mathcal{I}_3(k_1, k_2, k_3) - \mathcal{I}_3(k_2, k_1, k_3)]. \qquad (6.31) $$

As expected, we see that for the interaction vertex with an odd number of spatial momenta, we have an imaginary $\rho_3$, while the interaction with an even number of spatial momenta yields a real $\rho_3$.

Now, the s-channel contribution to this factorised part of the trispectrum takes the form

$$ B_{4,s}^{\pi} = \prod_{a=1}^{4} P_{\pi}(k_a) \sum_{h=\pm 1} P_{\sigma}^{h}(s) \rho_{3,3}(\mathbf{k}_1, \mathbf{k}_2, \mathbf{s}) \rho_{1,3}(\mathbf{k}_3, \mathbf{k}_4 - \mathbf{s}) + [(1,2) \leftrightarrow (3,4)], \qquad (6.32) $$

where we only add one permutation since we have already explicitly summed over some permutations in arriving at the above expressions for $\rho_{3,1}$ and $\rho_{3,3}$, and as always we don't include the spin zero “exchange” since this will always give a vanishing result. We use an equal sign for this s-channel exchange since as we showed above, the quartic wavefunction coefficient does not contribute to the correlator. Computing this product and summing over the helicities of the exchanged field (recalling that only the $\delta_{ij}$ in the polarisation sum contributes) yields

$$ B_{4,s}^{\pi} = \left( \prod_{a=1}^{4} P_{\pi}(k_a) \right) iH\lambda_1\lambda_3 P_{\sigma}(s) (s^2 - k_1^2 - k_2^2)(s^2 - k_3^2 - k_4^2) \epsilon_{ijk} k_3^i k_4^j k_2^k \\ \times \mathrm{Im}[\mathcal{I}_3(k_1, k_2, s) - \mathcal{I}_3(k_2, k_1, s)] \mathrm{Im}[\mathcal{I}_1(k_3, k_4, s) - \mathcal{I}_1(k_4, k_3, s)] + [(1,2) \leftrightarrow (3,4)]. \qquad (6.33) $$

Let us first convince ourselves that this contribution is indeed non-zero. By construction it is invariant under $(1 \leftrightarrow 2)$ and $(3 \leftrightarrow 4)$. This is manifest for $(3 \leftrightarrow 4)$ due to the anti-symmetric nature of $\epsilon_{ijk}$, while it can be made manifest for $(1 \leftrightarrow 2)$ by writing $\epsilon_{ijk} k_3^i k_4^j k_2^k = \epsilon_{ijk} k_3^i k_4^j (k_2^k - k_1^k)/2$. The final sum over replacing $(1,2) \leftrightarrow (3,4)$ cannot
yield a vanishing result since $\mathcal{I}_1(k_1, k_2, s) \neq \mathcal{I}_3(k_1, k_2, s)$. So generically we expect this
contribution to the trispectrum to be non-zero.

Let's now study the time integrals in more detail. First consider the time integral for
the $\lambda_1$ interaction, and we remind the reader that throughout this discussion we take $\nu$ to
be real. We have

$$
\mathcal{I}_1(k_1, k_2, s) = - \frac{1}{(-\eta_0)^{3/2} H_{\nu}^{(2)}(-s\eta_0)} \int d\eta (-\eta)^{5/2} c_s^2 k_1^2 (1 - ic_s k_2\eta) e^{ic_s k_{12}\eta} H_{\nu}^{(2)}(-s\eta), \quad (6.34)
$$

and therefore

$$
\begin{equation}
\begin{split}
\mathcal{I}_1(k_1, k_2, s) - \mathcal{I}_1(k_2, k_1, s) = & \\
& - \frac{c_s^2}{(-\eta_0)^{3/2} H_\nu^{(2)}(-s\eta_0)} \int d\eta (-\eta)^{5/2} [k_1^2(1 - ic_s k_2\eta) - k_2^2(1 - ic_s k_1\eta)] e^{ic_s k_{12}\eta} H_\nu^{(2)}(-s\eta).
\end{split}
\tag{6.35}
\end{equation}
$$

Here we have defined $k_{12} = k_1 + k_2$. A very similar integral,<fnref n="29" /> that we can use to compute
this one, was computed in appendix B of [15]. Indeed in that work it was shown that the
integral<fnref n="30" />

$$
I_n(a, b) = \frac{H\sqrt{\pi}}{2} e^{-\frac{i\pi}{2}(\nu+1/2)} \int_{-\infty}^{0} d\eta (-\eta)^{n-1/2} e^{ia\eta} H_{\nu}^{(2)}(-b\eta) \quad (6.36)
$$

is given by

$$
I_n(a, b) = (-1)^{n+1} \frac{H}{\sqrt{2b}} \left(\frac{i}{2b}\right)^n \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(1+n)} \times {}_2F_1\left(\alpha, \beta; 1+n; \frac{1}{2} - \frac{a}{2b}\right), \quad (6.37)
$$

where we have defined $\alpha = \frac{1}{2} + n - \nu$ and $\beta = \frac{1}{2} + n + \nu$. We can therefore write

$$
\mathcal{I}_1(k_1, k_2, s) - \mathcal{I}_1(k_2, k_1, s) = - \frac{2c_s^2}{H\sqrt{\pi}} \frac{e^{\frac{i\pi}{2}(\nu+1/2)}(k_1-k_2)}{(-\eta_0)^{3/2}H_\nu^{(2)}(-s\eta_0)} [k_{12}I_3(c_s k_{12}, s) + ic_s k_1 k_2 I_4(c_s k_{12}, s)], \quad (6.38)
$$

which neatly gives us a closed-form expression. Similarly, for the $\lambda_3$ interaction we have

$$
\mathcal{I}_3(k_3, k_4, s) - \mathcal{I}_3(k_4, k_3, s) = - \frac{2c_s^2}{H\sqrt{\pi}} \frac{e^{\frac{i\pi}{2}(\nu+1/2)}(k_3-k_4)}{(-\eta_0)^{3/2}H_\nu^{(2)}(-s\eta_0)} [k_{34}I_4(c_s k_{34}, s) + ic_s k_3 k_4 I_5(c_s k_{34}, s)]. \quad (6.39)
$$

We can now write a more compact expression for eq. (6.33), the full s-channel trispectrum,
which is given by

$$
\begin{equation}
\begin{split}
B_{4,s}^{\pi} = - \left( \prod_{a=1}^{4} P_{\pi}(k_a) \right) & c_s^4 H \lambda_1 \lambda_3 (s^2 - k_1^2 - k_2^2) (s^2 - k_3^2 - k_4^2) (k_1 - k_2) (k_3 - k_4) \epsilon_{ijk} k_3^i k_4^j k_2^k \\
& \times [k_{12} I_3(c_s k_{12}, s) + ic_s k_1 k_2 I_4(c_s k_{12}, s)] [k_{34} I_4(c_s k_{34}, s) + ic_s k_3 k_4 I_5(c_s k_{34}, s)] \\
& \times \sin\left(\frac{\pi}{2}(\nu + 1/2)\right) \cos\left(\frac{\pi}{2}(\nu + 1/2)\right) + [(1, 2) \leftrightarrow (3, 4)],
\end{split}
\tag{6.40}
\end{equation}
$$

<footnote n="29"><sup>29</sup>It would be interesting to use the recently developed techniques in [67] to extract the cosmological collider oscillations implied by this signal.</footnote>

<footnote n="30"><sup>30</sup>We are actually working with the complex conjugate of the integral that was computed in [15] since our integral contains $H_{\nu}^{(2)}$ rather than $H_{\nu}^{(1)}$.</footnote>
yield a vanishing result since $\mathcal{I}_1(k_1, k_2, s) \neq \mathcal{I}_3(k_1, k_2, s)$. So generically we expect this
contribution to the trispectrum to be non-zero.

Let's now study the time integrals in more detail. First consider the time integral for
the $\lambda_1$ interaction, and we remind the reader that throughout this discussion we take $\nu$ to
be real. We have

$$
\mathcal{I}_1(k_1, k_2, s) = - \frac{1}{(-\eta_0)^{3/2} H_\nu^{(2)}(-s\eta_0)} \int d\eta (-\eta)^{5/2} c_s^2 k_1^2 (1 - ic_s k_2\eta) e^{ic_s k_{12}\eta} H_\nu^{(2)}(-s\eta), \quad (6.34)
$$

and therefore

$$
\begin{equation}
\begin{split}
\mathcal{I}_1(k_1, k_2, s) - \mathcal{I}_1(k_2, k_1, s) = & \\
& - \frac{c_s^2}{(-\eta_0)^{3/2} H_\nu^{(2)}(-s\eta_0)} \int d\eta (-\eta)^{5/2} [k_1^2(1 - ic_s k_2\eta) - k_2^2(1 - ic_s k_1\eta)] e^{ic_s k_{12}\eta} H_\nu^{(2)}(-s\eta).
\end{split}
\tag{6.35}
\end{equation}
$$

Here we have defined $k_{12} = k_1 + k_2$. A very similar integral,<fnref n="29" /> that we can use to compute
this one, was computed in appendix B of [15]. Indeed in that work it was shown that the
integral<fnref n="30" />

$$
I_n(a, b) = \frac{H\sqrt{\pi}}{2} e^{-\frac{i\pi}{2}(\nu+1/2)} \int_{-\infty}^{0} d\eta (-\eta)^{n-1/2} e^{ia\eta} H_{\nu}^{(2)}(-b\eta) \quad (6.36)
$$

is given by

$$
I_n(a, b) = (-1)^{n+1} \frac{H}{\sqrt{2b}} \left(\frac{i}{2b}\right)^n \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(1+n)} \times {}_2F_1\left(\alpha, \beta; 1+n; \frac{1}{2} - \frac{a}{2b}\right), \quad (6.37)
$$

where we have defined $\alpha = \frac{1}{2} + n - \nu$ and $\beta = \frac{1}{2} + n + \nu$. We can therefore write

$$
\mathcal{I}_1(k_1, k_2, s) - \mathcal{I}_1(k_2, k_1, s) = - \frac{2c_s^2}{H\sqrt{\pi}} \frac{e^{\frac{i\pi}{2}(\nu+1/2)}(k_1-k_2)}{(-\eta_0)^{3/2}H_\nu^{(2)}(-s\eta_0)} [k_{12}I_3(c_s k_{12}, s) + ic_s k_1 k_2 I_4(c_s k_{12}, s)], \quad (6.38)
$$

which neatly gives us a closed-form expression. Similarly, for the $\lambda_3$ interaction we have

$$
\mathcal{I}_3(k_3, k_4, s) - \mathcal{I}_3(k_4, k_3, s) = - \frac{2c_s^2}{H\sqrt{\pi}} \frac{e^{\frac{i\pi}{2}(\nu+1/2)}(k_3-k_4)}{(-\eta_0)^{3/2}H_\nu^{(2)}(-s\eta_0)} [k_{34}I_4(c_s k_{34}, s) + ic_s k_3 k_4 I_5(c_s k_{34}, s)]. \quad (6.39)
$$

We can now write a more compact expression for eq. (6.33), the full s-channel trispectrum,
which is given by

$$
\begin{equation}
\begin{split}
B_{4,s}^{\pi} = - \left( \prod_{a=1}^{4} P_{\pi}(k_a) \right) & c_s^4 H \lambda_1 \lambda_3 (s^2 - k_1^2 - k_2^2)(s^2 - k_3^2 - k_4^2)(k_1 - k_2)(k_3 - k_4) \epsilon_{ijk} k_3^i k_4^j k_2^k \\
& \times [k_{12} I_3(c_s k_{12}, s) + ic_s k_1 k_2 I_4(c_s k_{12}, s)][k_{34} I_4(c_s k_{34}, s) + ic_s k_3 k_4 I_5(c_s k_{34}, s)] \\
& \times \sin\left(\frac{\pi}{2}(\nu + 1/2)\right) \cos\left(\frac{\pi}{2}(\nu + 1/2)\right) + [(1, 2) \leftrightarrow (3, 4)],
\end{split}
\tag{6.40}
\end{equation}
$$

<footnote n="29"><sup>29</sup>It would be interesting to use the recently developed techniques in [67] to extract the cosmological collider oscillations implied by this signal.</footnote>

<footnote n="30"><sup>30</sup>We are actually working with the complex conjugate of the integral that was computed in [15] since our integral contains $H_{\nu}^{(2)}$ rather than $H_{\nu}^{(1)}$.</footnote>
where, as we have seen before, all $\eta_0$ dependence has cancelled out to leave us with an expression that is IR-finite. There are a few quick checks we can do on this result. We notice that it is purely imaginary, as it should be, due to the $I_3$ and $iI_4$ terms, and we see that for $\nu = 1/2$ and $\nu = 3/2$ this trispectrum vanishes thereby reproducing our results in section 3. We also see that the result is non-zero for the intermediate range of masses that we are interested in here. Finally, we see that the overall scaling with momenta is $k^{-9}$ as it should be for external massless scalars. Once we sum over the remaining two permutations (the $t$ and $u$ channels), and we divide by $H^4$ this gives us our final result for the parity-odd trispectrum of the comoving curvature perturbation $\zeta$ due to the exchange of a massive (but light) spin-1 field during inflation:

$$B_4^\zeta = - \left( \prod_{a=1}^4 P_\zeta(k_a) \right) \frac{c_s^4 \lambda_1 \lambda_3}{H^3} (s^2 - k_1^2 - k_2^2)(s^2 - k_3^2 - k_4^2)(k_1 - k_2)(k_3 - k_4) \epsilon_{ijk} k_3^i k_4^j k_2^k \\ \times [k_{12} I_3(c_s k_{12}, s) + i c_s k_1 k_2 I_4(c_s k_{12}, s)][k_{34} I_4(c_s k_{34}, s) + i c_s k_3 k_4 I_5(c_s k_{34}, s)] \\ \times \sin\left(\frac{\pi}{2}(\nu + 1/2)\right) \cos\left(\frac{\pi}{2}(\nu + 1/2)\right) + [(1, 2) \leftrightarrow (3, 4)] + t + u. \quad (6.41)$$

Let us now discuss the constraints on the couplings $\lambda_1, \lambda_3$ from the requirements of perturbativity. For $c_s$ close to 1, i.e. in the case that the spinning field and the Goldstone move with approximately the same speed, we can use the fact that $I_n \sim H$ (up to dimensionful functions of $k_a$ which are fixed by scale invariance and an innocuous hypergeometric function). From this we get that $B_4^\zeta \sim \lambda_1 \lambda_3 \Delta_\zeta^8/H$, which gives $\tau_{\text{NL}}^{\text{PO}} \sim B_4^\zeta / P_\zeta^3$ of order$^{31}$

$$\tau_{\text{NL}}^{\text{PO}} \sim \frac{\Delta_{\zeta}^2 \lambda_1 \lambda_3}{H}. \quad (6.42)$$

Let us then estimate the unitarity cutoff of the theory. For $c_s$ close to 1 this is straightforward. Once we canonically normalize $\pi$ we find that the $\lambda_1$ interaction is a dimension 7 operator suppressed by $\Lambda_1 \sim H^{4/3}/(\Delta_\zeta^2 \lambda_1)^{1/3}$, while the $\lambda_3$ one is a dimension 8 operator suppressed by $\Lambda_3 \sim H/(\Delta_\zeta^2 \lambda_3)^{1/4}$. Requiring that at crossing we are below the cutoff, $H \ll \Lambda_{1,3}$ leads to

$$\lambda_3 \Delta_\zeta^2 \ll 1, \qquad \lambda_1 \Delta_\zeta^2 \ll H. \quad (6.43)$$

To relate this to the overall size $\tau_{\text{NL}}^{\text{PO}}$, let's focus on the regime $\Lambda_1 \sim \Lambda_3$. Using this we can re-write $\lambda_1$ in terms of $\lambda_3$ and obtain

$$\tau_{\text{NL}}^{\text{PO}} \sim \lambda_3 (\Delta_\zeta^2 \lambda_3)^{\frac{3}{4}} \quad (\text{assuming } \Lambda_1 \sim \Lambda_3). \quad (6.44)$$

From this we see that it is possible to choose $\lambda_3$ in order for the theory to be weakly coupled at horizon crossing and to have $\tau_{\text{NL}}^{\text{PO}} \gg 1$.

Let us then see what happens if $c_s \gg 1$ or $c_s \ll 1$. In both cases we need to keep track of the hypergeometric function in eq. (6.41). Let us consider first the limit $c_s \gg 1$. We are interested in $2F_1(\alpha, \beta; \gamma; z)$ at large negative $z$. In order to isolate the scaling with $c_s$ we can focus for simplicity on the case $\nu = 3/2$, disregarding the fact that the overall

<footnote>$^{31}$Notice that $\lambda_3$ is dimensionless while $\lambda_1$ has dimension of mass.</footnote>
where, as we have seen before, all $\eta_0$ dependence has cancelled out to leave us with an expression that is IR-finite. There are a few quick checks we can do on this result. We notice that it is purely imaginary, as it should be, due to the $I_3$ and $iI_4$ terms, and we see that for $\nu = 1/2$ and $\nu = 3/2$ this trispectrum vanishes thereby reproducing our results in section 3. We also see that the result is non-zero for the intermediate range of masses that we are interested in here. Finally, we see that the overall scaling with momenta is $k^{-9}$ as it should be for external massless scalars. Once we sum over the remaining two permutations (the t and u channels), and we divide by $H^4$ this gives us our final result for the parity-odd trispectrum of the comoving curvature perturbation $\zeta$ due to the exchange of a massive (but light) spin-1 field during inflation:

$$ B_4^\zeta = - \left( \prod_{a=1}^4 P_\zeta(k_a) \right) \frac{c_s^4 \lambda_1 \lambda_3}{H^3} (s^2 - k_1^2 - k_2^2)(s^2 - k_3^2 - k_4^2)(k_1 - k_2)(k_3 - k_4) \epsilon_{ijk} k_3^i k_4^j k_2^k \\ \times [k_{12} I_3(c_s k_{12}, s) + i c_s k_1 k_2 I_4(c_s k_{12}, s)][k_{34} I_4(c_s k_{34}, s) + i c_s k_3 k_4 I_5(c_s k_{34}, s)] \\ \times \sin\left(\frac{\pi}{2}(\nu + 1/2)\right) \cos\left(\frac{\pi}{2}(\nu + 1/2)\right) + [(1, 2) \leftrightarrow (3, 4)] + t + u. \quad (6.41) $$

Let us now discuss the constraints on the couplings $\lambda_1, \lambda_3$ from the requirements of perturbativity. For $c_s$ close to 1, i.e. in the case that the spinning field and the Goldstone move with approximately the same speed, we can use the fact that $I_n \sim H$ (up to dimensionful functions of $k_a$ which are fixed by scale invariance and an innocuous hypergeometric function). From this we get that $B_4^\zeta \sim \lambda_1 \lambda_3 \Delta_\zeta^8 / H$, which gives $\tau_{\text{NL}}^{\text{PO}} \sim B_4^\zeta / P_\zeta^3$ of order$^{31}$

$$ \tau_{\text{NL}}^{\text{PO}} \sim \frac{\Delta_{\zeta}^2 \lambda_1 \lambda_3}{H}. \quad (6.42) $$

Let us then estimate the unitarity cutoff of the theory. For $c_s$ close to 1 this is straightforward. Once we canonically normalize $\pi$ we find that the $\lambda_1$ interaction is a dimension 7 operator suppressed by $\Lambda_1 \sim H^{4/3}/(\Delta_\zeta^2 \lambda_1)^{1/3}$, while the $\lambda_3$ one is a dimension 8 operator suppressed by $\Lambda_3 \sim H/(\Delta_\zeta^2 \lambda_3)^{1/4}$. Requiring that at crossing we are below the cutoff, $H \ll \Lambda_{1,3}$ leads to

$$ \lambda_3 \Delta_\zeta^2 \ll 1, \qquad \lambda_1 \Delta_\zeta^2 \ll H. \quad (6.43) $$

To relate this to the overall size $\tau_{\text{NL}}^{\text{PO}}$, let's focus on the regime $\Lambda_1 \sim \Lambda_3$. Using this we can re-write $\lambda_1$ in terms of $\lambda_3$ and obtain

$$ \tau_{\text{NL}}^{\text{PO}} \sim \lambda_3 (\Delta_\zeta^2 \lambda_3)^{\frac{3}{4}} \quad (\text{assuming } \Lambda_1 \sim \Lambda_3). \quad (6.44) $$

From this we see that it is possible to choose $\lambda_3$ in order for the theory to be weakly coupled at horizon crossing and to have $\tau_{\text{NL}}^{\text{PO}} \gg 1$.

Let us then see what happens if $c_s \gg 1$ or $c_s \ll 1$. In both cases we need to keep track of the hypergeometric function in eq. (6.41). Let us consider first the limit $c_s \gg 1$. We are interested in ${}_2F_1(\alpha, \beta; \gamma; z)$ at large negative $z$. In order to isolate the scaling with $c_s$ we can focus for simplicity on the case $\nu = 3/2$, disregarding the fact that the overall

<footnote>$^{31}$Notice that $\lambda_3$ is dimensionless while $\lambda_1$ has dimension of mass.</footnote>
<figure><img src="image_5.png" /><figcaption>Figure 5. $\tau_{\text{NL}}^{\text{PO}}$ for the curvature perturbation $\zeta$ from the exchange of a spin-1 field with mass $m_{\sigma}$ and interactions $\lambda_1$ and $\lambda_3$ in eq. (6.8). Notice that the trispectrum vanishes for the conformally-coupled mass $m_{\sigma}/H = \sqrt{2}$, in agreement with our general no-go theorems.</figcaption></figure>

trispectrum is zero in this case: the scaling with $c_s$ is unaffected by $\sigma$ having a non-zero mass. Hence, we have $\beta - \alpha = 3$, and we can use the relation

$$ {}_2F_1(\alpha, \beta; \gamma; z) \sim \frac{2(1-z)^{-\alpha}}{\Gamma(\beta)\Gamma(\gamma-\alpha)}, \qquad (6.45) $$

valid for $|z-1| > 1$, $|\arg(1-z)| < \pi$ and $\beta - \alpha \in \mathbb{N}$. From this we see that the square brackets in the second line of eq. (6.41) yield a factor of $H^2/c_s^5$ in the limit of large $c_s$, giving

$$ \tau_{\text{NL}}^{\text{PO}} \sim \frac{\Delta_{\zeta}^2 \lambda_1 \lambda_3}{H c_s}. \qquad (6.46) $$

In the case of $c_s \ll 1$, given the regularity of the hypergeometric function at $z = 1/2$, we see that

$$ \tau_{\text{NL}}^{\text{PO}} \sim \frac{\Delta_{\zeta}^2 c_s^4 \lambda_1 \lambda_3}{H}. \qquad (6.47) $$

It is more difficult to estimate, in this case, the unitarity cutoff of the theory without doing an explicit calculation. We leave this, and a more detailed discussion on how large $\tau_{\text{NL}}^{\text{PO}}$ must be in order to explain the measurement of refs. [10, 11], to a future work: we only notice that having a slow $\sigma^i$ suppresses non-Gaussianities at fixed $\lambda_1$ and $\lambda_3$. In figure 5 we plot $\tau_{\text{NL}}^{\text{PO}}$ as defined by our normalization condition of eq. (2.22) at varying mass of the exchanged spinning particle and $c_s$.
<figure><img src="image_6.png" /><figcaption>Figure 5. $\tau_{\text{NL}}^{\text{PO}}$ for the curvature perturbation $\zeta$ from the exchange of a spin-1 field with mass $m_{\sigma}$ and interactions $\lambda_1$ and $\lambda_3$ in eq. (6.8). Notice that the trispectrum vanishes for the conformally-coupled mass $m_{\sigma}/H = \sqrt{2}$, in agreement with our general no-go theorems.</figcaption></figure>

trispectrum is zero in this case: the scaling with $c_s$ is unaffected by $\sigma$ having a non-zero mass. Hence, we have $\beta - \alpha = 3$, and we can use the relation

$$ {}_2F_1(\alpha, \beta; \gamma; z) \sim \frac{2(1-z)^{-\alpha}}{\Gamma(\beta)\Gamma(\gamma-\alpha)}, \qquad (6.45) $$

valid for $|z - 1| > 1$, $|\arg(1 - z)| < \pi$ and $\beta - \alpha \in \mathbb{N}$. From this we see that the square brackets in the second line of eq. (6.41) yield a factor of $H^2/c_s^5$ in the limit of large $c_s$, giving

$$ \tau_{\text{NL}}^{\text{PO}} \sim \frac{\Delta_{\zeta}^2 \lambda_1 \lambda_3}{H c_s}. \qquad (6.46) $$

In the case of $c_s \ll 1$, given the regularity of the hypergeometric function at $z = 1/2$, we see that

$$ \tau_{\text{NL}}^{\text{PO}} \sim \frac{\Delta_{\zeta}^2 c_s^4 \lambda_1 \lambda_3}{H}. \qquad (6.47) $$

It is more difficult to estimate, in this case, the unitarity cutoff of the theory without doing an explicit calculation. We leave this, and a more detailed discussion on how large $\tau_{\text{NL}}^{\text{PO}}$ must be in order to explain the measurement of refs. [10, 11], to a future work: we only notice that having a slow $\sigma^i$ suppresses non-Gaussianities at fixed $\lambda_1$ and $\lambda_3$. In figure 5 we plot $\tau_{\text{NL}}^{\text{PO}}$ as defined by our normalization condition of eq. (2.22) at varying mass of the exchanged spinning particle and $c_s$.
# 7 Summary and future directions

In this work we have studied signatures of parity violation from the inflationary primordial universe. Until tensor modes are detected, our best hope is to study the statistics of scalar fluctuations imprinted in primordial curvature perturbations. Parity violation cannot appear in the power spectrum and bispectrum, even at the full non-perturbative level, hence in this paper we have focused here on the scalar, parity-odd trispectrum $B_4^{PO}$. We have re-derived and extended previous no-go results [1] that rule out parity violation at tree-level in the presence of any number of scalars of any mass, or fields of any spin with massless de Sitter mode functions. Crucially, these no-go results assume scale invariance and a Bunch-Davies initial state. The *raison d’être* of these no-go theorems is to help us identify the classes of models in which we can get a non-vanishing $B_4^{PO}$ and to determine the physical implications of a possible discovery of this signal. It is particularly timely to pursue this goal, in light of the recent measurements of the parity-odd four-point function of BOSS galaxies of refs. [10, 11]. While it is important to keep in mind that it is possible that the signal detected (to different levels of significance) by these two groups could be due to systematics instead of fundamental physics, it is equivalently important to stress that this is the first example in a long time where it is data that drives the theory, as far as the study of the inflationary universe is concerned.

To the end of identifying models where a non-vanishing parity-odd trispectrum can be obtained, we have then relaxed these assumptions and derived explicitly the parity-odd trispectra that one can generate in more general classes of models. In section 4, we showed that general deviations from scale invariance, which show up in an explicit time dependence of the coupling constants for perturbations, leads to a non-vanishing $B_4^{PO}$. We stress that, for a time-dependence that is logarithmic in conformal time, the resulting $B_4^{PO}$ is actually scale-invariant, see e.g. eq. (4.11). In this case, deviations from scale invariance are present in the wavefunction, but drop out when we compute the trispectrum, which for parity-odd interactions is the imaginary part of the wavefunction. In section 5 we discussed a different scenario where a non-zero parity-odd trispectrum can arise: while maintaining scale invariance, we deviate from the massless de Sitter mode functions and consider those of Ghost Inflation (GI). In the infinite past, the GI mode functions do not reduce to the standard $e^{\pm ikt}$ mode functions in Minkowski, and as such they avoid the no-go theorems we derived in section 3. We concentrated on self-interactions for the Goldstone boson that are “invariant” under the leading part of the non-linear boost symmetry (as opposed to Wess-Zumino terms), and presented two examples that lead to a non-zero signal. The resulting $B_4^{PO}$ are given in (5.18) and (5.25), for two different parity-odd interactions. Despite the fact that we concentrated on these operators that arise from four building block covariant ones, we have shown that other operators cannot yield interactions with a lower scaling dimension than that of eq. (5.9). Finally, we have considered the case in which massive fields are present. For scalars of any mass the primordial parity-odd trispectrum vanishes at tree-level, but in the presence of massive spinning fields it can be non-vanishing. As an example, we have computed $B_4^{PO}$ for the exchange of massive vector and the result is in eq. (6.41) and plotted as function of mass for a specific configuration in figure 5.
# 7 Summary and future directions

In this work we have studied signatures of parity violation from the inflationary primordial universe. Until tensor modes are detected, our best hope is to study the statistics of scalar fluctuations imprinted in primordial curvature perturbations. Parity violation cannot appear in the power spectrum and bispectrum, even at the full non-perturbative level, hence in this paper we have focused here on the scalar, parity-odd trispectrum $B_4^{PO}$. We have re-derived and extended previous no-go results [1] that rule out parity violation at tree-level in the presence of any number of scalars of any mass, or fields of any spin with massless de Sitter mode functions. Crucially, these no-go results assume scale invariance and a Bunch-Davies initial state. The *raison d’être* of these no-go theorems is to help us identify the classes of models in which we can get a non-vanishing $B_4^{PO}$ and to determine the physical implications of a possible discovery of this signal. It is particularly timely to pursue this goal, in light of the recent measurements of the parity-odd four-point function of BOSS galaxies of refs. [10, 11]. While it is important to keep in mind that it is possible that the signal detected (to different levels of significance) by these two groups could be due to systematics instead of fundamental physics, it is equivalently important to stress that this is the first example in a long time where it is data that drives the theory, as far as the study of the inflationary universe is concerned.

To the end of identifying models where a non-vanishing parity-odd trispectrum can be obtained, we have then relaxed these assumptions and derived explicitly the parity-odd trispectra that one can generate in more general classes of models. In section 4, we showed that general deviations from scale invariance, which show up in an explicit time dependence of the coupling constants for perturbations, leads to a non-vanishing $B_4^{PO}$. We stress that, for a time-dependence that is logarithmic in conformal time, the resulting $B_4^{PO}$ is actually scale-invariant, see e.g. eq. (4.11). In this case, deviations from scale invariance are present in the wavefunction, but drop out when we compute the trispectrum, which for parity-odd interactions is the imaginary part of the wavefunction. In section 5 we discussed a different scenario where a non-zero parity-odd trispectrum can arise: while maintaining scale invariance, we deviate from the massless de Sitter mode functions and consider those of Ghost Inflation (GI). In the infinite past, the GI mode functions do not reduce to the standard $e^{\pm ikt}$ mode functions in Minkowski, and as such they avoid the no-go theorems we derived in section 3. We concentrated on self-interactions for the Goldstone boson that are “invariant” under the leading part of the non-linear boost symmetry (as opposed to Wess-Zumino terms), and presented two examples that lead to a non-zero signal. The resulting $B_4^{PO}$ are given in (5.18) and (5.25), for two different parity-odd interactions. Despite the fact that we concentrated on these operators that arise from four building block covariant ones, we have shown that other operators cannot yield interactions with a lower scaling dimension than that of eq. (5.9). Finally, we have considered the case in which massive fields are present. For scalars of any mass the primordial parity-odd trispectrum vanishes at tree-level, but in the presence of massive spinning fields it can be non-vanishing. As an example, we have computed $B_4^{PO}$ for the exchange of massive vector and the result is in eq. (6.41) and plotted as function of mass for a specific configuration in figure 5.
There are other ways to generate a non-vanishing $B_4^{PO}$, which we don't discuss here. One possibility is a process in which four external scalars exchange a spinning field whose power spectrum is chiral, i.e. the power in one helicity is different from the power in the other. Such a situation arises in models of axions coupled to a U(1) gauge sector in the presence of an $f(\phi)(FF + F\tilde{F})$ coupling, as discussed e.g. in [5], however sometimes with the added feature of the breaking of statistical isotropy. Other examples of parity-odd scalar trispectra were given in [1].

Our work could be extended in a number of directions:

* Loop corrections can contribute to $B_4^{PO}$ already in single-clock inflation with massless de Sitter mode functions. In the scale invariant limit this is an interesting setup where loop effects can be larger than tree-level ones. We will discuss this in detail in an upcoming paper.

* It would be interesting to assess what happens when we go beyond the decoupling limit i.e. where we include the effects of dynamical gravity. This can yield new exchange diagrams where curvature perturbations exchange the transverse, traceless part of the graviton, but also new contact diagrams where now the self-interactions can include inverse Laplacians which arise when we integrate out the non-dynamical parts of the metric. Such gravitational effects are expected to be too small to aid with an explanation of the signal potentially found in [10, 11], yet it is interesting to understand if such corrections can avoid our no-go theorems.

* In section 6 where we considered the exchange of a massive spinning field, we choose EFTI interactions between the Goldstone mode and the new massive field that did not come with any quadratic mixing terms. However, it would be interesting to extend our analysis to include such mixings that will introduce new exchange diagrams as was considered in [68] for parity-even bispectra. Such quadratic mixing terms come with a number of spatial derivatives that is dictated by the spin of the massive field. This leads to distinctive signatures in the squeezed limit of cosmological correlators, and potentially new shapes of parity-violating trispectra.

In summary, the no-go results and yes-go examples in our work contribute to establish the parity-odd trispectrum as a particularly sensitive probe of new physics during inflation.

## Acknowledgments

We thank Harry Goodhew, Jiamin Hou, Misha Ivanov, Sachin Jain, Mehrdad Mirbabayi, Oliver Philcox, Sébastien Renaux-Petel, Fabian Schmidt, Marko Simonović, Jakub Supeł and Matias Zaldarriaga for useful discussions. G.C. acknowledges support from the Institute for Advanced Study. The work of G.C. was initiated at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. E.P. has been supported in part by the research program VIDI with Project No. 680-47-535, which is (partly) financed by the Netherlands Organisation for Scientific Research (NWO). This work has been partially supported by STFC consolidated grant ST/T000694/1. S.J. is
There are other ways to generate a non-vanishing $B_4^{PO}$, which we don't discuss here. One possibility is a process in which four external scalars exchange a spinning field whose power spectrum is chiral, i.e. the power in one helicity is different from the power in the other. Such a situation arises in models of axions coupled to a U(1) gauge sector in the presence of an $f(\phi)(FF + F\tilde{F})$ coupling, as discussed e.g. in [5], however sometimes with the added feature of the breaking of statistical isotropy. Other examples of parity-odd scalar trispectra were given in [1].

Our work could be extended in a number of directions:

* Loop corrections can contribute to $B_4^{PO}$ already in single-clock inflation with massless de Sitter mode functions. In the scale invariant limit this is an interesting setup where loop effects can be larger than tree-level ones. We will discuss this in detail in an upcoming paper.

* It would be interesting to assess what happens when we go beyond the decoupling limit i.e. where we include the effects of dynamical gravity. This can yield new exchange diagrams where curvature perturbations exchange the transverse, traceless part of the graviton, but also new contact diagrams where now the self-interactions can include inverse Laplacians which arise when we integrate out the non-dynamical parts of the metric. Such gravitational effects are expected to be too small to aid with an explanation of the signal potentially found in [10, 11], yet it is interesting to understand if such corrections can avoid our no-go theorems.

* In section 6 where we considered the exchange of a massive spinning field, we choose EFTI interactions between the Goldstone mode and the new massive field that did not come with any quadratic mixing terms. However, it would be interesting to extend our analysis to include such mixings that will introduce new exchange diagrams as was considered in [68] for parity-even bispectra. Such quadratic mixing terms come with a number of spatial derivatives that is dictated by the spin of the massive field. This leads to distinctive signatures in the squeezed limit of cosmological correlators, and potentially new shapes of parity-violating trispectra.

In summary, the no-go results and yes-go examples in our work contribute to establish the parity-odd trispectrum as a particularly sensitive probe of new physics during inflation.

## Acknowledgments

We thank Harry Goodhew, Jiamin Hou, Misha Ivanov, Sachin Jain, Mehrdad Mirbabayi, Oliver Philcox, Sébastien Renaux-Petel, Fabian Schmidt, Marko Simonović, Jakub Supeł and Matias Zaldarriaga for useful discussions. G.C. acknowledges support from the Institute for Advanced Study. The work of G.C. was initiated at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. E.P. has been supported in part by the research program VIDI with Project No. 680-47-535, which is (partly) financed by the Netherlands Organisation for Scientific Research (NWO). This work has been partially supported by STFC consolidated grant ST/T000694/1. S.J. is
supported by the European Research Council under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 758792, project GEODESI). D.S. is supported by a UKRI Stephen Hawking Fellowship [grant number EP/W005441/1] and a Nottingham Research Fellowship from the University of Nottingham. For the purpose of open access, the authors have applied a CC BY public copyright licence to any Author Accepted Manuscript version arising.

## A Wick rotation for $\psi_4$ due to massive exchange

In this appendix we study the exchange time integral (6.15) and show that it is purely real. In turn this tells us that the associated $\psi_{4,s}$ is real and therefore does not contribute to the trispectrum.

We will actually work with a more general time integral of the form

$$ \mathcal{I}_E(k_a, s) = \int d\eta d\eta' \eta^{n_L-4} \eta'^{n_R-4} f_L(k_1, k_2, \eta) f_R(k_3, k_4, \eta') G_\sigma(\eta, \eta', s), \quad (A.1) $$

arising from two vertices with $n_L$ and $n_R$ derivatives. Here the function $f_L(f_R)$ depends on $K_\pi(k_1, \eta)$, $K_\pi(k_2, \eta)$ ($K_\pi(k_3, \eta)$, $K_\pi(k_4, \eta)$) and their time derivatives. Using the relation

$$ \frac{d^n}{d\eta^n} K_\pi(k, \eta) = e^{ic_s k\eta} (ic_s k)^n (1 - n - ic_s k\eta) \quad (A.2) $$

we can, for example, write

$$ f_L(k_1, k_2, \eta) = (ic_s k_1)^{n_1} (ic_s k_2)^{n_2} \text{poly}_L(ic_s k_1 \eta, ic_s k_2 \eta) e^{ic_s k_{12} \eta}, \quad (A.3) $$

where $k_{12} = k_1 + k_2$, poly$_L$ is some polynomial, and $n_1 + n_2 \le n_L$. A similar expression can be derived for $f_R$, with a corresponding poly$_R$ that depends on $\eta'$. We will assume that $n_{L,R} \ge 4$ such that there are no IR divergences at $\eta \to 0$. We can confirm this by focusing on the case of a massless exchange, $\nu = 3/2$. In this case, we have that the bulk-bulk propagator is IR-finite (see eq. (A.4) below), hence as long as we have a number of derivatives equal to or higher than 4 on each vertex the integral is guaranteed to converge at late times.

Now recall that for real $\nu$, which we assume throughout this appendix, we can write the bulk-bulk propagator as

$$ G_{\sigma}(\eta, \eta', s) = \frac{i\pi H^2}{2} \left[ \theta(\eta - \eta')(\eta')^{3/2}(-\eta)^{3/2} H_{\nu}^{(2)}(-s\eta')J_{\nu}(-s\eta) + (\eta \leftrightarrow \eta') \right]. \quad (A.4) $$

Let's concentrate on the case where $\eta > \eta'$ meaning that we pick up the first Heaviside theta function in the bulk-bulk propagator. We integrate $\eta'$ from the far past (where we Wick rotate) up to $\eta$, then integrate $\eta$ from the far past to the late-time boundary. To aid with assessing the reality of this integral, we Wick rotate to the path $\eta(\lambda_1) = i\lambda_1$, $\eta'(\lambda_1, \lambda_2) = \eta(\lambda_1) + i\lambda_2$, with $\lambda_1, \lambda_2$ from $+\infty$ to 0 (the path for the second contribution in the bulk-bulk propagator is readily obtained by $\eta \leftrightarrow \eta'$). With this rotation we have
supported by the European Research Council under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 758792, project GEODESI). D.S. is supported by a UKRI Stephen Hawking Fellowship [grant number EP/W005441/1] and a Nottingham Research Fellowship from the University of Nottingham. For the purpose of open access, the authors have applied a CC BY public copyright licence to any Author Accepted Manuscript version arising.

## A Wick rotation for $\psi_4$ due to massive exchange

In this appendix we study the exchange time integral (6.15) and show that it is purely real. In turn this tells us that the associated $\psi_{4,s}$ is real and therefore does not contribute to the trispectrum.

We will actually work with a more general time integral of the form

$$ \mathcal{I}_E(k_a, s) = \int d\eta d\eta' \eta^{n_L-4} \eta'^{n_R-4} f_L(k_1, k_2, \eta) f_R(k_3, k_4, \eta') G_\sigma(\eta, \eta', s), \quad (A.1) $$

arising from two vertices with $n_L$ and $n_R$ derivatives. Here the function $f_L(f_R)$ depends on $K_\pi(k_1, \eta)$, $K_\pi(k_2, \eta)$ ($K_\pi(k_3, \eta)$, $K_\pi(k_4, \eta)$) and their time derivatives. Using the relation

$$ \frac{d^n}{d\eta^n} K_\pi(k, \eta) = e^{ic_s k\eta} (ic_s k)^n (1 - n - ic_s k\eta) \quad (A.2) $$

we can, for example, write

$$ f_L(k_1, k_2, \eta) = (ic_s k_1)^{n_1} (ic_s k_2)^{n_2} \text{poly}_L(ic_s k_1 \eta, ic_s k_2 \eta) e^{ic_s k_{12} \eta}, \quad (A.3) $$

where $k_{12} = k_1 + k_2$, poly$_L$ is some polynomial, and $n_1 + n_2 \le n_L$. A similar expression can be derived for $f_R$, with a corresponding poly$_R$ that depends on $\eta'$. We will assume that $n_{L,R} \ge 4$ such that there are no IR divergences at $\eta \to 0$. We can confirm this by focusing on the case of a massless exchange, $\nu = 3/2$. In this case, we have that the bulk-bulk propagator is IR-finite (see eq. (A.4) below), hence as long as we have a number of derivatives equal to or higher than 4 on each vertex the integral is guaranteed to converge at late times.

Now recall that for real $\nu$, which we assume throughout this appendix, we can write the bulk-bulk propagator as

$$ G_{\sigma}(\eta, \eta', s) = \frac{i\pi H^2}{2} \left[ \theta(\eta - \eta')(\eta')^{3/2}(-\eta)^{3/2} H_{\nu}^{(2)}(-s\eta')J_{\nu}(-s\eta) + (\eta \leftrightarrow \eta') \right]. \quad (A.4) $$

Let's concentrate on the case where $\eta > \eta'$ meaning that we pick up the first Heaviside theta function in the bulk-bulk propagator. We integrate $\eta'$ from the far past (where we Wick rotate) up to $\eta$, then integrate $\eta$ from the far past to the late-time boundary. To aid with assessing the reality of this integral, we Wick rotate to the path $\eta(\lambda_1) = i\lambda_1$, $\eta'(\lambda_1, \lambda_2) = \eta(\lambda_1) + i\lambda_2$, with $\lambda_1, \lambda_2$ from $+\infty$ to 0 (the path for the second contribution in the bulk-bulk propagator is readily obtained by $\eta \leftrightarrow \eta'$). With this rotation we have
exponential convergence at infinity and one does not have to worry about the inner limit of integration depending on the outer integration variable. We now have

$$ \begin{aligned} \mathcal{I}_E(k_a, s) = & (c_s k_1)^{n_1} (c_s k_2)^{n_2} (c_s k_3)^{n_3} (c_s k_4)^{n_4} i^{n_1+n_2+n_3+n_4+n_L+n_R} \\ & \times \int_0^{+\infty} \int_0^{+\infty} d\lambda_1 d\lambda_2 F(k_1, k_2, k_3, k_4, c_s, \lambda_1, \lambda_2) \\ & \times H_\nu^{(2)}(-is\lambda_1) J_\nu(-is\lambda_1) + (\eta \to \eta'), \end{aligned} \quad (A.5) $$

where $\lambda_{12} = \lambda_1 + \lambda_2$ and $F$ is a real function. Crucially, we then see that the reality of $\mathcal{I}_E(k_a, s)$ is determined by the combination of Hankel and Bessel functions, and the number of derivatives in each vertex. Using the following integral representations of the Hankel and Bessel functions:

$$ \begin{aligned} H_\nu^{(2)}(z) = & -\frac{e^{\frac{i\pi\nu}{2}}}{i\pi} \int_{-\infty}^{+\infty} dt e^{-iz \cosh t - \nu t} \quad \text{and} \\ J_\nu(-iz) = & \frac{2^{1-\nu}(-iz)^\nu}{\sqrt{\pi}\Gamma(\nu + \frac{1}{2})} \int_0^1 dt (1-t^2)^{\nu-\frac{1}{2}} \cosh(zt), \end{aligned} \quad (A.6) $$

valid for $-\pi < \arg z < 0$ and $\text{Re}\,\nu > -\frac{1}{2}$,<fnref n="32" /> respectively, we see that the product $H_\nu^{(2)}(-is\lambda_{12})J_\nu(-is\lambda_1)$ is purely imaginary. Indeed, using fact that $\nu$ is real, we have $H_\nu^{(2)}(-is\lambda_{12}) = ie^{\frac{i\nu\pi}{2}} \times (\text{Real})$ and $J_\nu(-is\lambda_1) = e^{-\frac{i\nu\pi}{2}} \times (\text{Real})$. We therefore have

$$ \mathcal{I}_E(k_a, s) = i^{n_1+n_2+n_3+n_4+n_L+n_R+1} \times (\text{Real}). \quad (A.7) $$

Now to find a parity-odd $\psi_4$, we need one vertex to have an even number of spatial momenta and the other to have an odd number. Let's take $n_L$ to contain an odd number and $n_R$ to contain an even number. It follows that $n_1 + n_2 + n_L$ is odd while $n_3 + n_4 + n_R$ is even. It then follows that $\mathcal{I}_E(k_a, s)$, and therefore $\psi_4$, are purely real. This means that the parity odd quartic wavefunction coefficient due to the exchange of a massive spinning field with real $\nu$ does not contribute to the trispectrum. This proof complements the one we have in section 6 using weight-shifting operators.

**Open Access.** This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. SCOAP³ supports the goals of the International Year of Basic Sciences for Sustainable Development.

## References

[1] T. Liu, X. Tong, Y. Wang and Z.-Z. Xianyu, Probing P and CP Violations on the Cosmological Collider, JHEP 04 (2020) 189 [arXiv:1909.01819] [INSPIRE].

[2] P. Creminelli, J. Gleyzes, J. Noreña and F. Vernizzi, Resilience of the standard predictions for primordial tensor modes, Phys. Rev. Lett. 113 (2014) 231301 [arXiv:1407.8439] [INSPIRE].

<footnote n="32">³²Both of these conditions are satisfied for us since for $z = -is\lambda_{12}$ we have $\arg z = -\frac{\pi}{2}$ and for $0 < m_\sigma^2 < 2H^2$ we always have $\text{Re}\,\nu > -\frac{1}{2}$.</footnote>
exponential convergence at infinity and one does not have to worry about the inner limit of integration depending on the outer integration variable. We now have

$$ \begin{aligned} \mathcal{I}_E(k_a, s) = & (c_s k_1)^{n_1} (c_s k_2)^{n_2} (c_s k_3)^{n_3} (c_s k_4)^{n_4} i^{n_1+n_2+n_3+n_4+n_L+n_R} \\ & \times \int_0^{+\infty} \int_0^{+\infty} d\lambda_1 d\lambda_2 F(k_1, k_2, k_3, k_4, c_s, \lambda_1, \lambda_2) \\ & \times H_{\nu}^{(2)}(-is\lambda_{12}) J_{\nu}(-is\lambda_1) + (\eta \rightarrow \eta'), \end{aligned} \quad (A.5) $$

where $\lambda_{12} = \lambda_1 + \lambda_2$ and $F$ is a real function. Crucially, we then see that the reality of $\mathcal{I}_E(k_a, s)$ is determined by the combination of Hankel and Bessel functions, and the number of derivatives in each vertex. Using the following integral representations of the Hankel and Bessel functions:

$$ \begin{aligned} H_{\nu}^{(2)}(z) = & -\frac{e^{\frac{i\pi\nu}{2}}}{i\pi} \int_{-\infty}^{+\infty} dt e^{-iz \cosh t - \nu t} \quad \text{and} \\ J_{\nu}(-iz) = & \frac{2^{1-\nu}(-iz)^{\nu}}{\sqrt{\pi}\Gamma(\nu + \frac{1}{2})} \int_{0}^{1} dt (1-t^2)^{\nu-\frac{1}{2}} \cosh(zt), \end{aligned} \quad (A.6) $$

valid for $-\pi < \arg z < 0$ and $\text{Re}\nu > -\frac{1}{2}$,<fnref n="32" /> respectively, we see that the product $H_{\nu}^{(2)}(-is\lambda_{12})J_{\nu}(-is\lambda_1)$ is purely imaginary. Indeed, using fact that $\nu$ is real, we have $H_{\nu}^{(2)}(-is\lambda_{12}) = ie^{\frac{i\nu\pi}{2}} \times (\text{Real})$ and $J_{\nu}(-is\lambda_1) = e^{-\frac{i\nu\pi}{2}} \times (\text{Real})$. We therefore have

$$ \mathcal{I}_E(k_a, s) = i^{n_1+n_2+n_3+n_4+n_L+n_R+1} \times (\text{Real}). \quad (A.7) $$

Now to find a parity-odd $\psi_4$, we need one vertex to have an even number of spatial momenta and the other to have an odd number. Let's take $n_L$ to contain an odd number and $n_R$ to contain an even number. It follows that $n_1 + n_2 + n_L$ is odd while $n_3 + n_4 + n_R$ is even. It then follows that $\mathcal{I}_E(k_a, s)$, and therefore $\psi_4$, are purely real. This means that the parity odd quartic wavefunction coefficient due to the exchange of a massive spinning field with real $\nu$ does not contribute to the trispectrum. This proof complements the one we have in section 6 using weight-shifting operators.

**Open Access.** This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. SCOAP³ supports the goals of the International Year of Basic Sciences for Sustainable Development.

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<footnote n="32">³²Both of these conditions are satisfied for us since for $z = -is\lambda_{12}$ we have $\arg z = -\frac{\pi}{2}$ and for $0 < m_{\sigma}^2 < 2H^2$ we always have $\text{Re}\nu > -\frac{1}{2}$.</footnote>
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