Using Graviton EFT and Massive Gravity to Compute Gravitational Potentials for Black Hole Inspirals

Introduction

Effective Field Theory

Background

Graviton EFT

From Scattering Amplitudes to Black Hole Inspirals

A Higher Order Calculation

Massive Gravity

Introduction

An important consequence of the change from a massless graviton theory to a massive graviton theory is the addition of 3 additional degrees of freedom. As a caveat, these extra degrees of freedom cause many of the issues with the theory of massive gravity, so this will come up again and again in this chapter.

Motivations for a Massive Graviton

One of the most pressing issues in cosmology today is the explanation of the small but non-zero value of the cosmological constant. However, massive gravity is able to provide a natural explanation for the small value of the cosmological constant.

History

We also note that due to the symmetry of the diagram, the symmetry factor is 1/2!. We note again that the vertex factor must be 1/2. due to the symmetry of the diagrams.

Feynman Diagrams

The Basic Feynman Rules

Conservation of energy/momentum: For each vertex, write a delta function in the form (2𝜋)4𝛿4(𝑘1+𝑘2+𝑘3), where positive momentum is defined as coming into the vertex. Cancel the delta function: Finally, there should be one remaining delta function of the form (2𝜋)4𝛿4(𝑝1 + 𝑝2 + .. − 𝑝𝑛), which contains the overall momentum conservation for the chart.

Changes in the Feynman Rules due to Massive Gravity

Vertex factors: For each vertex between propagators, a vertex factor corresponding to the vertex's particle type is written down. The delta functions from momentum conservation should cancel out most of the integrals unless there are loops. We can see that in the limit 𝑚 → 0 the massive propagator does not return smoothly to the massless propagator.

This is because for this project we chose to use the Fierz-Pauli theory of massive gravity instead of the Stückelberg theory.

The Nonanalytic Component of the Scattering Matrix

Through this procedure we were able to derive each of the contributions to the gravitational potential from the diagrams. Thus we were able to find and use expressions for each of the scalar integrals that we needed. When we tried to calculate the box and cross box plots, two problems arose.

Given the change to a massive graviton, the new uncontracted amplitude for the double seagull diagram (see 6.3) takes the form. because of the symmetry of the diagram.

Tree Level Amplitudes

Massless Graviton Interactions

With Feynman's rules in hand, calculating the tree-level diagram (see 5.1) is straightforward. To confirm this as a known result, we take a nonrelativistic and zero velocity limit to obtain the Newtonian potential. However, we are only interested in long-range nonlocal classical interactions, so we can neglect the delta function (localized at a single point) to obtain a final solution that matches the Newtonian potential.

Massive Graviton Interactions

However, we are able to simplify the expression considerably by using the identity. We can add up all the contributions to the potential from each of the diagrams to find the full gravitational potential at one loop sequence. 6.41). We therefore had to calculate all the necessary integrals to find the correct expressions for the amplitudes.

Next, we examine two nodal correction diagrams (see 5.cd in 6.4) that contribute only to the quantum part of the amplitude and potential.

One-loop Modification (Massless)

Techniques

We also reproduced the integral results from the two papers using Feyncalc's integral solution techniques. We also used some identities given in [5] that helped us to eliminate many analytical terms from the beginning. In all cases where the two previous papers we referred to agreed, our result also exactly confirmed theirs [5, 21].

In the only case where there was one difference between the results of the two papers, our result was consistent with that of [21], the newer paper.

Results

These simplifications are a simple example of the Passerino-Veltman decomposition described in detail in Section 7.2. This paper explains its presence from the second order Born approximation and must be subtracted from it to find the correct correction to the Newtonian potential, which produces. These diagrams have the second most propagators (second only to the box and crossed box diagrams) and are therefore among the more difficult to solve.

Finally, we investigate the effect of the two vacuum polarization diagrams, both of a pure graviton loop as well as a ghost loop (see fig. 6.5).

Figure 6.2: The set of triangle diagrams contributing to the scattering amplitude.

Analysis of Full Potential

All the integrals we solved were coded in the Mathematica notebook for future use. We can first start by adding together all the different contributions to the classical potential. 𝑟3, and is therefore a quantum term. In addition, most of the terms have an exponential decay term as seen in the tree-level potential for the massive graviton.

It is therefore important to finish running all quantum diagrams through our Mathematica pipeline.

Integration with a Massive Graviton

Scalar Integrals

Thus, we were able to use these solutions and separate the non-analytical part of each of them. Feyncalc unfortunately did not have the expressions for 𝐶0or𝐷0, so we had to look elsewhere. We found several resources that had solved these integrals for different cases, but none of these solved the integrals in the most general case, which is what we needed for our project.

2- (

Passarino-Veltman Decomposition

Now, since this is an integral of a tensor, we know that the solution to the integral must also be a tensor, using only the moment available to it from the integral as. We can repeat this process for each of the 𝑝𝑖 to obtain a system of n equations and n variables and thus solve for all coefficients A. The only difference is that we must consider𝜂𝜇 𝜈 on both sides of the equation.

This extended method lowers the rank of the resulting integrals by one each time (it does not always produce scalar integrals immediately).

Tensor Integrals

After confirming all the massless results, we begin the massive case for diagrams contributing to the classical potential term. We now specifically select two nodal correction diagrams (see 5.ab in 6.4) that contribute to the classical term in the potential. This expression for the classical potential has similarities and differences with the original massless graviton potential.

One can then Fourier transform the expression to obtain the quantum contribution to the potential. 8.28).

One-loop Modification (Massive)

Classical vs Quantum

Since we expect the diagrams to still produce the same q-dependence just with additional factors, we expect that the diagrams should still be split between those that contribute to both the classical term and the quantum term, and those that only contribute to the quantum term. In particular, we expect that the triangle plots (Fig. 6.2) and the first two vertex correction plots (Fig. 6.4) will still contribute to the classical part. Therefore, we decided to concentrate on deriving the results for the more relevant classical part.

However, since we have already made some progress on the other diagrams, we are still connecting what we did in Section 8.5.

Techniques

Moreover, although we have not been able to solve the term 𝐶𝜇 𝜈𝛼 𝛽𝛾, it is only used for the box and crossed box charts.

Results for Classical Diagrams

Once this amplitude is passed through the Mathematica pipeline, we get the following expression for the contracted amplitude. 8.4) By adding the two differential equations together, we get the full amplitude from the two triangle diagrams. After contracting and simplifying in the Mathematica pipeline, we get the following non-relativistic amplitudes. 8.11).

Let's now add two amplitudes and determine which terms can cancel or add.

Classical Discussion

All terms have 𝐺2 and mass dependence typical of second-order terms. There are also many conditions for mass modification, again similar to those seen in the tree-level potential. Now these are still clearly second-order contributions due to their dependence on mass and G, and their magnitude is suppressed by the small graviton mass factors, but it is still unexpected to find this dependence on r.

This is somewhat expected given the form of the integrals, but presents another problem.

Results for Quantum Diagrams

In addition, we realized that our method of solving these integrals was for integrals in a specific form that was valid for all integrals except those used in the box and crossed box diagrams. Unfortunately, due to the extra complication of the different integrals required by these two diagrams, we were unable to find the full contracted and simplified amplitudes. Under the non-relativistic approach, the amplitude takes the form 8.21) With a Fourier transform we derive the contribution to the potential for the double gull diagram.

All other non-scalar triangle integrals used were decomposed in terms of the 𝐶0, 𝐵0 and 𝐴0 integrals.

Future Work

Massive Scalar Propogator

Massless Graviton Propogator

Massive Graviton Propagator

Massless Ghost Propagator

We recreate the integrals used in the massless case here due to some copying errors observed in the integrals given by [5].

Bubble Integrals

Triangle Integrals

Box and Cross Box Integrals

Tadpole and Bubble Integrals

Triangle Integrals

We have Taylor expanded the expressions in terms of the graviton mass𝑚 up to order 2 and the exchanged momentum𝑞 up to order 4.

Box and Cross Box Integrals

At one point in this thesis we considered exploring the utility of using color kinematic duality as a means of checking our answers. In practice, this means that the amplitudes of various particle interactions can be "squared" to produce an equivalent amplitude for interactions in other theories [24]. We hoped that enough research would be done in the field of massive gluons that it would be a fairly simple method to check our conclusions by an independent process.

We anticipate that this process could be the source of an entirely new project or thesis and suggest it as a possibility for any interested readers.