Amplitudes for astrophysicists: known knowns

Abstract

The use of quantum field theory to understand astrophysical phenomena is not new. However, for the most part, the methods used are those that have been developed decades ago. The intervening years have seen some remarkable developments in computational quantum field theoretic tools. In particle physics, this technology has facilitated calculations that, even ten years ago would have seemed laughably difficult. It is remarkable, then, that most of these new techniques have remained firmly within the domain of high energy physics. We would like to change this. As alluded to in the title, this paper is aimed at showcasing the use of modern on-shell methods in the context of astrophysics and cosmology. In this article, we use the old problem of the bending of light by a compact object as an anchor to pedagogically develop these new computational tools. Once developed, we then illustrate their power and utility with an application to the scattering of gravitational waves.

Appendices

A some group theory

A group, as everyone knows, is a set of objects closed under a group operation that satisfies the axioms of identity, associativity and invertibility. This definition is an abstract one, unmarried to any particular realization of the objects in the set. For our purposes, it will suffice to take a representation of G as a set of matrices. In this case, group multiplication is synonymous with standard matrix multiplication. In this article, we are mostly interested in Lie groups, G say, which are groups associated to some continuous set of parameters that in turn can be taken as coordinates in some continuous manifold, the Lie group manifold.

For an example, one need look no fourther than the group of rotations on the plane where a 2-vector $\vec {v}$, is transformed into the vector $\vec {v'}$ by the action of a $2\times 2$ matrix,

$$\begin{aligned} \vec {v}' = R\vec {v}, \end{aligned}$$

(A.1)

which depends on the (continuous, periodic) rotation angle $\theta$. This is the Lie group O(2), the set of $2\times 2$ orthogonal matrices. In general this group contains not only rotations but also reflections. We can restrict this group to one that does not contain reflections by demanding that $\det R = +1$. This restricted group is the special orthogonal group SO(2). It is not hard to see that this concept is easily generalised to rotations in N dimensions where,

$$\begin{aligned} SO(N) = \left\{ R_{N\times N}~\bigg |~ R^TR = 1 ~~\text {and}~~\det R = 1\right\} \,. \end{aligned}$$

(A.2)

As is often the case in physics, it is often more convenient to work with infinitesimal transformations i.e. to “Taylor expand” transformations about the identity. To this end let’s look at an infinitesimal rotation parameterized by an angle $\alpha$. In this case, the rotation matrix is “close” to the identity, so we can write:

$$\begin{aligned} R \simeq I + A\,. \end{aligned}$$

(A.3)

Since R is a rotation matrix, $R^TR = (I + A^T)(I + A) = I + A^T + A = I$. For this to hold, A must be an antisymmetric matrix, i.e. $A^T = -A$. In 2 dimensions, this means that

$$\begin{aligned} A = \alpha J = \alpha \begin{pmatrix} 0 &{}\quad 1 \\ -\,1 &{}\quad 0 \end{pmatrix}\,. \end{aligned}$$

(A.4)

The $2\times 2$ matrix J is known as the generator of the group.

In order to relate this back to a finite angle $\theta$, we replace $\alpha \rightarrow \theta /N$ and apply the rotation N times, finally taking the $N\rightarrow \infty$ limit,¹²

$$\begin{aligned} R(\theta ) = \lim _{N\rightarrow \infty }\left( I + \frac{\theta J}{N}\right) ^N = \exp (\theta J)\,. \end{aligned}$$

(A.5)

In the case of the N dimensional rotation group SO(N), this becomes $R = \exp (\theta _a J^a)$.

The number of real parameters n that characterise this group defines the dimension of the group, and is equal to the number of independent elements of the group generators. For SO(N), the number of independent elements in an antisymmetric $N\times N$ matrix

$$\begin{aligned} n = \dim {SO(N)} = \frac{N(N-1)}{2}\,. \end{aligned}$$

(A.6)

A closely related, and frequently occuring, group is the special unitary group SU(N), which is the set of $N\times N$ unitary matrices with unit determinant. Following the same arguments as for the special orthogonal group, we can show that this group has $N^2 -1$ generators. This group can be thought of as rotations in N complex dimensions.

The utility of infinitesimal transformations has already manifest in our discussion. Geometrically, restricting to transformations close to the identity corresponds to working in the tangent vector space at the origin of the group manifold defines. For a given Lie group G, this defines the associated¹³ Lie algebra, g with the group generators forming a basis for the vector space. Equivalently, the algebra is defined by the commutation relation satisfied by the group generators,

$$\begin{aligned}{}[T^a,T^b] = f^{ab}_cT^c\,. \end{aligned}$$

(A.7)

Here, the $f^{ab}_c$ are the structure constants of the Lie algebra. Given a Lie algebra, an element of the associated group is found by exponentiating an element of the algebra, as for the rotation group. Schematically,

$$\begin{aligned} \hbox {Lie Algebra }~~\xrightarrow {exp}~~\hbox { Lie Group}\,. \end{aligned}$$

For the groups SO(3) and SU(2), note that while the generators are different, the structure constants of each of these groups is the same, $f_{abc} = \epsilon _{abc}$, the completely antisymmetric tensor in 3 dimensions. Consequently, the structure of each group must be connected in some way. In fact, both SO(3) and SU(2) are rotation groups with 3 real parameters.

One dimension down, the group SO(2) acts on 2-vectors $\vec {v}$. However we can obviously rotate higher dimensional vectors in the same 2-plane as well.¹⁴ This is where the idea of a group representation enters. Matrices that act on a particular space that have the same structure as the group, but where the matrices can have a rank different from N. In practice, this means that the group generators will be matrices of rank M that obey the group algebra (and have the same structure constants). We can classify some interesting representations using N and M:

• The trivial representation is one where $M = 1$. The generators are scalars, and nothing happens to vectors under group transformations.

• The fundamental representation is where $M = N$. In this case, the generators are themselves the group matrices, i.e. the SO(N) rotations acting on $\mathbb {R}^N$. Elements of the group are vectors.

• The adjoint representation is where $M = \dim {G}$. In this representation the structure constants generate the group and group elements are represented as matrices.

Taking SO(3) as an example, Eq. (A.6) tells us that the group acts naturally on a 3 dimensional vector space spanned by the 3 generators $T^a$ of the adjoint representation.

1.1 SO(3) and SU(2)

Let’s expand a little on some of the groups that feature prominently in this article. We have already seen that SO(3) is the group of orthogonal $3\times 3$ matrices with unit determinant. The group SU(2) is defined similarly, as $2\times 2$ complex, unitary matrices with unit determinant,

$$\begin{aligned} SU(2) = \left\{ \begin{pmatrix} \alpha &{}\quad \beta ^*\\ \beta &{}\quad -\,\alpha \end{pmatrix}~~\Bigg |~~ |\alpha |^2 + |\beta |^2 = 1\right\} \,. \end{aligned}$$

(A.8)

Here $\alpha$ is a real number and $\beta$ is complex. Consequently, each SU(2) transformation depends on 3 real parameters. Elements of SU(2) can be neatly expanded in a basis of Pauli matrices,

$$\begin{aligned} \sigma _1 = \begin{pmatrix} 0 &{}\quad 1\\ 1 &{}\quad 0 \end{pmatrix}, \qquad \sigma _2 = \begin{pmatrix} 0 &{}\quad -\,i\\ i &{}\quad 0 \end{pmatrix}, \qquad \sigma _3 = \begin{pmatrix} 1 &{}\quad 0\\ 0 &{}\quad -\,1 \end{pmatrix}. \end{aligned}$$

(A.9)

This means that any SU(2) matrix can be written as $M = \theta ^a\sigma _a/2$, and therefore a group element can be written $U=\exp (i\theta ^a\sigma _a/2)$. The three real components of an element of SU(2) uniquely determine a unit 3-vector $\vec {v} = (x,y,z)$, and transformations under SU(2) lead to another unit 3-vector $\vec {v}' = (x',y',z')$. This means that SU(2) constitutes a rotation in 3 dimensions, exactly like SO(3). In fact, SU(2) is the so-called (double) covering group of SO(3).

1.2 The Lorentz group

The Lie group of most interest to us is the Lorentz group of rotations and boosts in 4-dimensional Minkowski space. For our purposes though, it will suffice to consider the subset of proper orthochronus Lorentz transformations, that exclude the discrete parity and time-reversal transformations. These are elements of a restricted Lorentz group $SO^+(1,3)$, where the plus means ’restricted’, and is written here only once and implied for the rest of this section. Specifically, Lorentz transformations act on the spacetime metric $g_{\mu \nu }$ as,

$$\begin{aligned} \Lambda ^T g_{\mu \nu } \Lambda = g_{\mu \nu }\,. \end{aligned}$$

(A.10)

Close to the identity, the transformations can be expanded as

$$\begin{aligned} \Lambda = I + \frac{1}{2}\omega _{\mu \nu }M^{\mu \nu }\,, \end{aligned}$$

(A.11)

where $M_{\mu \nu }$ is the generator of the group, defined as $M_{\mu \nu } = \partial _\mu x_\nu - \partial _\nu x_\mu$ and $\omega _{\mu \nu }$ parameterize the transformations. The generators satisfy the Lie algebra:

$$\begin{aligned}{}[M_{\mu \nu },M_{\rho \sigma }] = -i(g_{\mu \rho }M_{\nu \sigma } - g_{\mu \sigma }M_{\nu \rho } + g_{\nu \sigma }M_{\mu \rho } - g_{\mu \rho }M_{\nu \sigma }). \end{aligned}$$

(A.12)

These generators as defined above are a convenient notation that captures both the usual rotation generators $J_i$ that we have encountered already, together with the generators $K_i$ of Lorentz boosts through

$$\begin{aligned} J_i = \frac{1}{2} \epsilon _{ijk}M_{jk},\quad \text {and}\quad K_i = M_{0i}. \end{aligned}$$

(A.13)

These generators satisfy their own algebras, one of which we know as the SO(3) algebra,

$$\begin{aligned}{}[J_i,J_j] = i\epsilon _{ijk}J_k, \end{aligned}$$

(A.14)

while

$$\begin{aligned}{}[J_i,K_j] = i\epsilon _{ijk}K_k\,,~~~~~~[K_i,K_j] = -i\epsilon _{ijk}J_k. \end{aligned}$$

(A.15)

This furnishes one representation of the Lorentz group. It will prove instructive to look for another by changing basis. To this end, let’s define two new operators from linear combinations of boosts and rotations,

$$\begin{aligned} J^\pm _i = \frac{1}{2} (J_i \pm iK_i). \end{aligned}$$

(A.16)

Each of these are independent, a fact easily demonstrated by checking that $[J_i^+,J_j^-] = 0$. Further, by substituting in the relevant operators, it can be shown that

$$\begin{aligned}{}[J^\pm _i,J^\pm _j] = \frac{1}{2} \epsilon _{ijk}J_k^\pm . \end{aligned}$$

(A.17)

This is precisely the algebra of the rotation group and SU(2), except now there are two copies, one for the ‘$+$’ and one for the ‘−’ generators. In other words, we have arrived at another representation of the Lorentz group where evidently

$$\begin{aligned} SO(1,3)\simeq SU(2)\otimes SU(2)\,. \end{aligned}$$

(A.18)

To finish off this lightning review of group theory, let’s discuss one more important group that is a cover of the Lorentz group. This is the complex special linear group $SL(2,\mathbb {C})$. This group is very similar to SU(2), except that all entries of the $2\times 2$ matrices are complex,

$$\begin{aligned} SL(2,\mathbb {C}) = \left\{ \begin{pmatrix} a &{}\quad b\\ c &{}\quad d \end{pmatrix}~~\Bigg |~~ ad - bc = 1\right\} . \end{aligned}$$

(A.19)

where a, b, c, d are now complex numbers. Clearly SU(2) is a subgroup of $SL(2,\mathbb {C})$.

Any vector $V^\mu$ in $\mathbb {R}^{1,3}$ can be represented as a matrix in $SL(2,\mathbb {C})$, since we can encode the information of a Lorentz 4 vector as:

$$\begin{aligned} V^\mu = (t,x,y,z) \longrightarrow W = \begin{pmatrix} t + z &{}\quad x - iy\\ x + iy &{}\quad t - z \end{pmatrix} = tI_{2\times 2} + ix^i\sigma _i. \end{aligned}$$

(A.20)

If we define the sigma matrices as,

$$\begin{aligned} \sigma ^\mu = (I,\sigma ^i),\quad \overline{\sigma }^\mu = (I,-\sigma ^i)\,, \end{aligned}$$

(A.21)

then we can define the standard conversion from a Lorentz four vector to a $2\times 2$ matrix of $SL(2,\mathbb {C})$ via the new basis¹⁵:

$$\begin{aligned} V^\mu \longrightarrow W = V^\mu \sigma _\mu \end{aligned}$$

(A.22)

The square of a four vector in $\mathbb {R}^{1,3}$ is equal to the determinant of W.

B Scattering cross-section

In Sect. 2.2 we relate the scattering amplitude to the differential cross-section and this in turn to the impact parameter in order to calculate the scattering angle. Here, we give a brief argument to illustrate how one arrives at these equations. To relate the cross-section to the scattering amplitude we consider a scattering event with states $\phi _i(\bar{p}_i)$. For simplicity, we limit ourselves to a two to two scattering process with labels as used in the rest of this article: $\phi _1 \phi _4 \rightarrow \phi _2 \phi _3$. We specify no particle content for the derivation that follows and institute the appropriate limits at the end, closely following the treatment given in [23]. All the states are appropriately normalised and we suppose that the initial states are very closely distributed, in momentum space, around some central momenta, say $p'_1$ and $p'_4$. The initial state of our system is given by

$$\begin{aligned} \vert \phi _1(p_1) \phi _4(p_4) \rangle = \int \frac{d^3 \bar{p}_1 d^3 \bar{p}_4}{(2 \pi )^6 \sqrt{4 E_1 E_4}} \phi _1(\bar{p}_1) \phi _4(\bar{p}_4) e^{-i \bar{b}\cdot \bar{p}_1} \vert \bar{p}_1 \bar{p}_4 \rangle . \end{aligned}$$

(B.1)

Of course any experiment requires repetition so we can consider many states of type $\phi _1$ in a cylindrical beam along the z-axis incident on a single target state $\phi _4$ situated in the centre of the beam. We will assume that the incident particles are uniformly distributed about the z-axis and that the transverse offset is given by the impact parameter b radially from the centre of the beam. This means that the exponential factor in the above equation will account for this offset. Since the differential cross-section is an infinitesimal quantity, we only want to consider final states that fall within some small measure of momentum space, i.e. $d^3\bar{p}_3 d^3\bar{p}_2$. We are interested in the scattering probability

$$\begin{aligned} \mathcal {P}(p_1,p_4\rightarrow p_3,p_2) = \frac{d^3 \bar{p}_3 d^3 \bar{p}_2}{(2 \pi )^6 4 E_3 E_2} \left| \langle \bar{p}_3 \bar{p}_2 \vert \phi _1(p_1) \phi _4(p_4) \rangle \right| ^2 . \end{aligned}$$

(B.2)

The cross-section $\sigma$ is defined as the ratio

$$\begin{aligned} \sigma = \frac{N}{n_i\times n_p} \end{aligned}$$

(B.3)

Where N is the number density of scattering events, $n_i$ is the number of incident particles and $n_p$ is the number of particles at the scattering centre, in our case $n_p = 1$. The number of scattering events can be written as the number of incident particles times the probability that a scattering event will occur within some region of the beam,

$$\begin{aligned} N = \int d^2b\, n_i \mathcal {P}(\bar{b}) . \end{aligned}$$

(B.4)

The cross-section is then

$$\begin{aligned} \sigma = \frac{N}{n_i \times 1} = \int d^2 b\, \mathcal {P}(\bar{b}), \end{aligned}$$

(B.5)

and using the previous expressions we write the infinitesimal cross-section as

$$\begin{aligned} d\sigma = \frac{d^3 \bar{p}_3 d^3 \bar{p}_2}{(2 \pi )^6 4 E_3 E_2} \int d^2 e^{i \bar{b}\cdot (\bar{p}_1^*-\bar{p}_1)} \left| \int \frac{d^3 \bar{p}_1 d^3 \bar{p}_4}{(2 \pi )^6 \sqrt{4 E_1 E_4}} \phi _1(\bar{p}_1) \phi _4(\bar{p}_4) \langle \bar{p}_3 \bar{p}_2 \vert \bar{p}_1 \bar{p}_4 \rangle \right| ^2 . \end{aligned}$$

(B.6)

To simplify the above expression we first perform the integral over b to get a delta function $(2\pi )^2\delta ^{(2)}(\tilde{p}^*_1-\tilde{p}_1)$, where the tilde indicates the directions transverse to the z-axis. We also write the correlation function in terms of the scattering amplitude

$$\begin{aligned} \langle \bar{p}_3 \bar{p}_2 \vert \bar{p}_1 \bar{p}_4 \rangle = (2\pi )^4 \delta ^{(4)} (p_1+p_4 - p_3-p_2) A(p_1,p_2,p_3,p_4), \end{aligned}$$

(B.7)

providing some more delta functions. This allows us to do all the integrals in the complex conjugate part of equation (B.6), after which we can fix the incident states’ momenta to the central momenta mentioned at the start of the appendix. Therefore any smooth functions dependent on $\lbrace p_1, p_4 \rbrace$ can be evaluated at $\lbrace p'_1, p'_4 \rbrace$ and removed from the integral. If we also assume that the resolution of the detector is insufficient to resolve the fluctuations of the initial state momenta from the central value we can set all the remaining momenta to the central value. This gives the expression

$$\begin{aligned} d\sigma&= \frac{1}{4 E_1 E_4 \vert v_1-v_4 \vert }\frac{d^3 \bar{p}_3 d^3 \bar{p}_2}{(2 \pi )^6 4 E_3 E_2} \vert A(p'_1,p_2,p_3,p'_4) \vert ^2 (2\pi )^4 \delta ^{(4)} (p'_1+p'_3 - p_3-p_2) \nonumber \\&\quad \times \int \frac{d^3 p_1 d^3 p_3}{(2\pi )^6} \vert \phi _1(\bar{p}_1) \vert ^2 \vert \phi _4(\bar{p}_4) \vert ^2 \nonumber \\ d\sigma&= \frac{1}{4 E_1 E_4 \vert v_1-v_4 \vert }\frac{d^3 \bar{p}_3 d^3 \bar{p}_2}{(2 \pi )^6 4 E_3 E_2} \vert A(p'_1,p_2,p_3,p'_4) \vert ^2 (2\pi )^4 \delta ^{(4)} (p'_1+p'_4 - p_3-p_2), \end{aligned}$$

(B.8)

where $v_1,v_4$ are the velocities of the initial states along the z-axis. To get the above expression in a form we can use we need to partially evaluate the phase-space integrals on the right hand side. We can compute the integral over $\bar{p}_3$ using the delta function corresponding to the 3-momentum and we can switch to spherical coordinates in order to do the integral over $\bar{p}_2$, where the solid angle is $d\Omega = \sin (\theta ) d\theta d\phi$, to find in the end

$$\begin{aligned} I = \int \frac{ d\Omega }{16 \pi ^2} \frac{\vert \bar{p}_2\vert }{E_1+E_4}. \end{aligned}$$

(B.9)

Inserting this into the latest expression for the cross-section and rearranging we have

$$\begin{aligned} \frac{d\sigma }{d\Omega } = \frac{1}{4 E_1 E_4 \vert v_1-v_4 \vert } \frac{\vert \bar{p}_2\vert }{16 \pi ^2(E_1+E_4)} \vert A(p'_1,p_2,p_3,p'_4) \vert ^2 \end{aligned}$$

(B.10)

Pulling everything together, we can now impose the particle content and limits used in Sect. 2.2. Recall that states $\lbrace \phi _3,\phi _4 \rbrace$ are scalars of mass m, and $\lbrace \phi _1,\phi _2 \rbrace$ are photons. Also suppose the scalar is stationary, but with $m \gg E_1$, corresponding to the centre of mass frame, meaning that $E_3=E_4=m$. Next, we note that $\vert v_1-v_4 \vert =1$ since the scalar is stationary and the photon propagates at the speed of light, which we all know is one. Lastly we recall that the change in energy of the photon is small, allowing us to write $E_1 \approx E_2 = \vert \bar{p}_2\vert$. Plugging all of this into (B.10) we get

$$\begin{aligned} \frac{d\sigma }{d\Omega } = \frac{1}{64 \pi ^2 s_{14}} \vert A(p'_1,p_2,p_3,p'_4) \vert ^2 . \end{aligned}$$

(B.11)

This is a useful expression, however it is still not enough, since next we need to find some relation between this and the impact parameter. To this end consider very much the same set-up as before but with all particles classical. We have a cylindrical beam of equally distributed particles, with number density $N_i$, incident on some other spherical target particle at the centre of the beam. The translational offset of the incident particles from the centre of the beam is measured by the impact parameter b.

So, some particles pass through a ring in the beam of width db situated at impact parameter b. These particles proceed to scatter off the target particle into some solid angle region $d\Omega = 2\pi \sin (\theta )d\theta$ situated at $\theta$. From this, we can define the number of particles that pass through the ring as

$$\begin{aligned} dn_i = 2\pi b \,db N_i . \end{aligned}$$

(B.12)

Since all the particles incoming through db scatter into the solid angle $d\Omega$, the number of particles scattered through this region is simply $dn_s=dn_i$. The cross-section as defined before is the ratio of the number of scattered particles to the number of incident particles. Therefore the cross section is given by

$$\begin{aligned} d\sigma = \frac{dn_s}{N_i} = 2\pi b\, db, \end{aligned}$$

(B.13)

which is simply the area the incoming particles pass through. This quantity per unit solid angle through which the scattered particles pass is then

$$\begin{aligned} \frac{d\sigma }{d\Omega } = \frac{b\, db}{ \sin \theta \, d\theta }, \end{aligned}$$

(B.14)

and easily manipulated into the form necessary to calculate the scattering angle

$$\begin{aligned} \left( \frac{d\sigma }{d\Omega } \right) \sin \theta \, d\theta = b \,db. \end{aligned}$$

(B.15)

C A guide to the literature

In appreciation of the fact that a substantial part of learning any new subject is to know where to find the information and, wanting to offer a more helpful answer than “the arXiv”, in this appendix, we collect some of the references that we have found useful in learning the subject ourselves. Our list is by no meant complete or even authoritative but we hope it will help draw the interested reader further into an exhilarating field.

• Scattering amplitudes in gauge theory and gravity by Elvang and Huang. By now, this has evolved into the standard reference on amplitudes. Based on a set of graduate lectures given at the University of Michigan, it contains plenty worked examples and useful exercises. While it focuses mainly on developments in gauge theory, it contains enough material on gravity to bring the reader up to speed on the frontiers of the field. It is available from Cambridge University Press, with a preprint available at https://arxiv.org/abs/1308.1697.

• Tales of 1001 gluons by Stefan Weinzierl. A more modern introduction, slightly more up to date and covering some aspects that are omitted in Elvang and Huang. In addition to and excellent section on perturbative gravity, it contains, for example, new material on the scattering equations and CHY representations. As a bonus, it also contains numerous exercises and solutions. making it an excellent guide for students. It can be found on the arXiv at https://arxiv.org/abs/1610.05318.

• A brief introduction to modern amplitude methods notes by Lance Dixon. Another modern (although slightly older now) general introduction to the topic. Mostly focusses on gluon scattering/loops. https://arxiv.org/abs/1310.5353.

• A first course on twistors, integrability and gluon scattering amplitudes is based on a set of lectures given by Martin Wolf at Cambridge University. It is decidedly for the more mathematically inclined readership but pedagogical and with some excellent references for anyone wanting to fall further down this particular rabbit hole. It can be found on the arXiv at https://arxiv.org/abs/1001.3871.

• Lectures on Twistor theory and scattering amplitudes and Wilson loops in Twistor space are also an excellent sets of pedagogical notes. They can both be found on the arXiv at https://arxiv.org/abs/1104.2890 and https://arxiv.org/abs/1712.02196.

• Quantum field theory and the standard model by Mathew Schwartz. This is primarily a text on quantum field theory geared toward the standard model of particle physics. As such, its focus is more on practical calculations rather than excessive formality. This makes it a very useful concise alternative introduction to QFT. Of particular interest to us is its introduction to spinor helicity methods which, although not as complete as the above texts, is certainly a useful suppliment.

• Quantum field theory by Mark Srednikci. Another contemporary introduction to QFT. In addition to some slightly different topics covered from other canonical texts, this one offers some useful examples of calculating cross sections and using the spinor helicity formulation.

• Quantum field theory in a nutshell by Anthony Zee. Now in its second edition, Zee’s book has developed somewhat of a cult following among graduate students of many branches of physics. A very readable, if somewhat colloquial introduction to QFT, Zee develops the subject from the very basics and takes the reader all the way up to the frontiers of the field with many stops along the way to admire the scenery. While both editions are superb, it is the second edition in particular which contains a section on spinor helicity methods and a section on gravitational waves (both in ’Part N’). What it lacks in rigor, Zee’s text more than makes up for in building intuition.

• Introduction to the effective field theory description of gravity by John Donoghue. Very good guide to perturbative gravity and effective field theories of gravity. https://arxiv.org/abs/gr-qc/9512024.

• Perturbative quantum gravity and its relation to gauge theory by ’t Hooft. Very concise introduction to Perturbative gravity by one of the masters in the field. ’t Hooft’s take on the subject, as always, brings with it a different perspective. An excellent reference source, it can be downloaded from http://bit.ly/2m1rvsX.

• EPFL lectures on general relativity as a quantum field theory A good introduction to various aspects of GR as viewed through the lens of quantum field theory. https://arxiv.org/abs/1702.00319.

D Glossary of terms

Amplitude The probability amplitude for a certain interaction process of particles. In quantum field theory it is calculated by summing all the possible ways the interaction can be take place, which in Feynman diagram language are all the possible diagrams for the process allowed by the Feynman rules. An n-point amplitude is the amplitude of a process involving n physical particles.

BCFW recursion relation The BCFW recursion relations are a set of relations that allows one to construct multiple particle amplitudes from sub-amplitudes (amplitudes with a lower number of particles). That is, if one were to calculate an n-point amplitude one could write it as the summed product of i-point and $(n-i)$-point amplitudes with $i=2,\ldots ,n-2$.

Cross-section The classical cross-section is the area transverse to the relative motion of two particles within which they need to be in order to interact. For hard spheres the cross section is the area of overlap of the objects in order to collide. In instances where the interaction is mediated by a potential the cross section is generally bigger than the actual particle.

Differential cross-section The differential cross-section, $d \sigma /d \Omega$, is the ratio of particles scattered into a certain direction per unit time per unit solid angle divided by the the number of incident particles. This is equivalent to taking the normalized spin sum of a scattering process and can be related to the total cross section by integrating over all solid angles.

Feynman rules The set of interaction rules for particles derived from the action of an appropriate theory. Includes vertices , propagators and external particle state contraction.

Helicity The helicity of a particle is the projection of the spin onto the linear momentum of the particle.

Impact parameter Commonly denoted by b, the classical impact parameter is defined as the perpendicular distance between a particle and the center of a potential field the particle scatters off of.

Little group The little group is the set of transformations that leaves the momentum in a given direction of an on-shell particle invariant.

Mandelstam variable These are gauge invariant quantities that are constructed from the four-momenta of the physical particles in a scattering process. More precisely they are the square of the sum of the momenta of all particles as a given vertex containing a propagator. And are generally denoted $s_{ijk\ldots } = -\,(p_i^\mu + p_j^\mu + p_k^\mu + \cdots )^2$.

On/off-shell Particles that are on-shell adhere to the equations of motion, hence satisfying a physical constraint that relates their momentum and energy. For a particle of mass m, this is the well-known relation in special relativity $p^{\mu }p_{\mu }= -\,m^2$. An off-shell particle does not satisfy this constraint, so that $p^{\mu }p_{\mu } \ne -\,m^2$.

Propagator In the language of quantum field theory the propagator is a virtual particle that transfers momentum between two particles that are interacting.

Scalar particle Scalars are particles with spin 0. In this paper we use this particles to approximate stellar bodies such as the Sun.

Spin sum The spin sum is the sum of the complex square of an amplitude with the sum ranging over all the possible spin-state configurations of the amplitude.

Spinor particle Spinors are particles with spin 1 / 2, commonly called fermions. Fermions can be represented as four component Dirac Spinors or as is used in the spinor-helicity formalism as two component Weyl spinors.

Vector particle Vectors bosons, or gauge bosons, are particles with spin 1, like photons or gluons. They are represented mathematically using Lorentz vectors.

Vertex In Feynman diagrams the vertex is a point in which three or more particles interact in the diagram, and momentum is conserved at all vertices in the given diagram. The vertex is represented mathematically by the vertex expression that is derived from the the interaction Lagrangian.

Virtual particle A virtual particle is an off-shell particle used in Feynman diagrams to mediate the interaction between two or more interacting physical particles, i.e. to act as a way to transfer momentum/information between physical particles. In scattering amplitudes, it is always represented by a propagator.