In this article, we compute the two observables, impulse and waveform, in a

black hole scattering event for the Scalar-Tensor theory of gravity with a

generic scalar potential using the techniques of Worldline Quantum Field

Theory. We mainly investigate the corrections to the above mentioned

observables due to the extra scalar degree of freedom. For the computation of

impulse, we consider the most general scenario by making the scalar field

massive and then show that each computed diagram has a smooth massless limit.

We compute the waveform for scalar and graviton up to 2PM, taking the scalar as

massless. Furthermore, we discuss if the scalar has mass and how the radiation

integrals get more involved than the massless case. We also arrive at some

analytical results using stationary phase approximation. Interestingly, we also

show that the $lambda_4 varphi^4$ interaction vertex does not contribute to

the radiation by showing that the integral has no non-zero finite value.

## Impulse and Waveform in Black Hole Scattering Event for Scalar-Tensor Theory of Gravity

In this article, we explore the computation of two observables, impulse and waveform, in a black hole scattering event within the context of the Scalar-Tensor theory of gravity with a generic scalar potential. We utilize the techniques of Worldline Quantum Field Theory to investigate the corrections to these observables due to the presence of an extra scalar degree of freedom.

### Computing Impulse

To compute the impulse, we consider the most general scenario by introducing a mass for the scalar field. We then proceed to show that each computed diagram exhibits a smooth massless limit. This allows us to extract meaningful results in the massless case as well.

### Computing Waveform

In addition to impulse, we also calculate the waveform for both the scalar and graviton. We focus on computing the waveform up to 2PM, assuming that the scalar field is massless. We take into account the effects of the scalar’s mass and explore how the radiation integrals become more involved compared to the massless case. To aid our analysis, we employ the stationary phase approximation method, leading to some insightful analytical results.

### No Contribution from $lambda_4 varphi^4$ Interaction Vertex

An interesting finding in our investigation is that the $lambda_4 varphi^4$ interaction vertex does not contribute to the radiation. We demonstrate that the corresponding integral yields no non-zero finite value. This result provides valuable information about the nature of the radiation and the role of different interaction vertices in the Scalar-Tensor theory of gravity.

## Future Roadmap: Challenges and Opportunities

As we move forward, several challenges and opportunities lie ahead in the study of black hole scattering events within the Scalar-Tensor theory of gravity:

### Challenges

- Further understanding the effects of the extra scalar degree of freedom on the observables
- Exploring the implications of introducing mass for the scalar field on the waveform
- Investigating other potential interaction vertices and their contributions to the radiation
- Tackling the complexity of radiation integrals in scenarios beyond the massless case

### Opportunities

- Utilizing advanced computational techniques to handle intricate calculations
- Extending the analysis to higher orders in perturbation theory, beyond 2PM
- Exploring the impact of additional parameters on the observed scattering event
- Connecting the theoretical predictions to experimental observations and potential gravitational wave detections

By addressing these challenges and capitalizing on the opportunities, we can gain deeper insights into the Scalar-Tensor theory of gravity and its manifestations in black hole scattering events. These advancements have the potential to enhance our understanding of fundamental physics and contribute to the broader field of gravitational physics.