Simulating spin-$\frac{3}{2}$ particles at colliders

Abstract

Support for interactions of spin-$\frac{3}{2}$ particles is implemented in the FeynRules and ALOHA packages and tested with the MadGraph 5 and CalcHEP event generators in the context of three phenomenological applications. In the first, we implement a spin-$\frac{3}{2}$ Majorana gravitino field, as in local supersymmetric models, and study gravitino and gluino pair-production. In the second, a spin-$\frac{3}{2}$ Dirac top-quark excitation, inspired from compositeness models, is implemented. We then investigate both top-quark excitation and top-quark pair-production. In the third, a general effective operator for a spin-$\frac{3}{2}$ Dirac quark excitation is implemented, followed by a calculation of the angular distribution of the s-channel production mechanism.

Appendices

Appendix A: Conventions

In this Appendix, we collect the conventions adopted in this manuscript. The metric is given by

$$ \eta_{\mu\nu} = \mathrm{diag}(1,-1,-1,-1) , $$

(A.1)

and the fully antisymmetric tensor of rank four is defined by ϵ ₀₁₂₃=1. The four-vectors built upon the Pauli matrices are given by their usual form,

$$ \sigma^\mu= \bigl( 1, \sigma^i \bigr) \quad\text{and} \quad\bar{\sigma}^\mu= \bigl(1, -\sigma^i \bigr) , $$

(A.2)

where the Pauli matrices σ ⁱ with i=1,2,3 read

$$ \begin{aligned} &\sigma^1 = \left( \begin{array}{c@{\quad}c} 0 & 1\\ 1 & 0 \end{array} \right) , \qquad \sigma^2 = \left( \begin{array}{c@{\quad}c} 0 & -i\\ i & 0 \end{array} \right) , \\ &\sigma^3 = \left( \begin{array}{c@{\quad}c} 1 & 0\\ 0 & -1 \end{array} \right) . \end{aligned} $$

(A.3)

This allows to write the generators of the Lorentz algebra in the (two-component) left-handed and right-handed spinorial representations as

$$ \sigma^{\mu\nu} = \frac{i}{4} \bigl( \sigma^\mu{\bar{\sigma}}^\nu- \sigma^\nu{\bar{\sigma}}^\mu\bigr) \quad\text{and} \quad{\bar{\sigma}}^{\mu\nu} = \frac{i}{4} \bigl ( { \bar{\sigma}}^\mu\sigma^\nu- {\bar{\sigma}}^\nu \sigma^\mu\bigr) , $$

(A.4)

respectively.

Moving to four-component spinors, Dirac matrices are defined in the Weyl representation by

$$ \gamma^\mu= \left( \begin{array}{c@{\quad}c} 0& \sigma^\mu\\ \bar{\sigma}^\mu&0 \end{array} \right) , $$

(A.5)

and span the Clifford algebra

$$ \bigl\{ \gamma^\mu, \gamma^\nu\bigr\} = 2 \eta^{\mu\nu} . $$

(A.6)

Additionally, the fifth Dirac matrix γ ₅ is given by

$$ \gamma_5 = i\gamma^0 \gamma^1 \gamma^2 \gamma^3 = \left( \begin{array}{c@{\quad}c} -1&0\\ 0&1 \end{array} \right) . $$

(A.7)

The γ-matrices allow to build the generators of the Lorentz algebra in the four-component spinorial representation,

$$ \gamma^{\mu\nu} = \frac{i}{4} \bigl(\gamma^\mu \gamma^\nu- \gamma^\nu\gamma^\mu\bigr), $$

(A.8)

as well as the quantity γ ^μνρ which may appear in some specific forms of the Rarita–Schwinger Lagrangian used in the literature,

$$ \begin{aligned}[b] \gamma^{\mu\nu\rho} &= \frac{1}{6} \bigl[ \gamma^\mu\gamma^\nu\gamma^\rho+ \gamma^\nu\gamma^\rho\gamma^\mu+ \gamma^\rho\gamma^\mu\gamma^\nu \\ & \quad{}- \gamma^\nu\gamma^\mu\gamma^\rho- \gamma^\mu\gamma^\rho\gamma^\nu- \gamma^\rho\gamma^\nu\gamma^\mu\bigr] . \end{aligned} $$

(A.9)

The γ ^μν and γ ^μνρ objects obey the important relations,

$$ \epsilon^{\mu\nu\rho\sigma} \gamma_5 \gamma_\sigma= -i \gamma ^{\mu\nu\rho} \quad\text{and}\quad\epsilon^{\mu\nu\rho\sigma} \gamma_{\mu\nu} \gamma_5 = 2 i \gamma^{\rho\sigma} , $$

(A.10)

helpful to prove the equivalence between all the spin-$\frac{3}{2}$ Lagrangian forms employed up to now.

Appendix B: Review of spin

In this appendix, we follow the approach of Weinberg to build up the properties of a spin-$\frac{3}{2}$ particle [85]. The spin of a particle determines its polarization vectors, the latter determining its propagation on-shell. Off-shell, the propagator can have further dependence which vanishes on-shell. Moreover, the on-shell propagator is related to the on-shell quadratic part of the Lagrangian whereas the off-shell pieces of the propagator match the model-dependent off-shell quadratic terms of the Lagrangian. We begin with a discussion on spin.

Both the vacuum and the dynamics of quantum field theory are symmetric under the Lorentz group, whose algebra $\mathfrak{so}(3,1)$ is isomorphic to $\mathfrak{sl}(2,\mathbb{R}) \oplus\overline{\mathfrak{sl}(2,\mathbb{R})}$, the representations of the two $\mathfrak{sl}(2,\mathbb{R})$ denoting the left-handed and right-handed chiral algebras. However, the four-momentum of the particles are further invariant under a subalgebra, called the little algebra. For a massive particle, this is easiest to see in the rest frame where its momentum is p ^μ=(M,0,0,0). Any Lorentz transformation that is a pure rotation in space leaves invariant a massive particle at rest. As a result, the little algebra for a massive particle is $\mathfrak{so}(3)$, the algebra of spatial rotations, which is isomorphic to $\mathfrak{su}(2)$, and each massive particle is classified according to its transformation properties under this little algebra or, in other words, according to its “spin”. This little algebra is the diagonal subalgebra of the left and right chiral Lorentz algebras, $\mathfrak{sl}(2,\mathbb{R})$ and $\overline{\mathfrak{sl}(2,\mathbb{R})}$. As a result, if a field transforms under the (a,b) representation of the Lorentz algebra, it has a spin between a+b and |a−b|.

As is well known, a scalar field transforms as (0,0) under the Lorentz algebra and is, therefore, a spin-0 field. Ordinary matter fermions transform as either $(\frac{1}{2},0)$ or $(0,\frac{1}{2})$ and are, therefore, spin-$\frac{1}{2}$ particle. Vector fields transform as $(\frac{1}{2},\frac{1}{2})$ and contain both a spin-1 and a spin-0 piece. On-shell, the spin-0 field is removed by ∂ _μ V ^μ=0. Rarita–Schwinger fields Ψ ^μ are formed as a direct product of $(\frac{1}{2},0)$ or $(0,\frac{1}{2})$ and $(\frac{1}{2},\frac{1}{2})$. Under the Lorentz symmetry, this contains

• $(0,\frac{1}{2})$ or $(\frac{1}{2},0)$ parts which are spin-$\frac{1}{2}$ fields and are removed by the on-shell equation γ _μ Ψ ^μ=0;

• $(1,\frac{1}{2})$ or $(\frac{1}{2},1)$ parts which contain both the spin-$\frac {3}{2}$ field that we are after and another spin-$\frac{1}{2}$ field which is removed by the on-shell equation ∂ _μ Ψ ^μ=0.

Higher spin states are formed in a similar way.

In the rest of this appendix, we review the spin algebra $\mathfrak {su}(2)$ (Sect. B.1), the polarization vectors (Sect. B.2), the propagators (Sect. B.3), the quadratic Lagrangian terms (Sect. B.4), and finally the Wigner d-functions (Sect. B.5). Further, for the rest of this Appendix, we consider particles which are not self-charge-conjugate. The special case when particles are self-conjugate can be recovered from our results by replacing antiparticle states by charge conjugates of the particle states.

2.1 B.1 Spin: $\mathbf{\mathfrak {so(3)}\sim\mathfrak{su(2)}}$

We begin by reviewing what it means to be a representation of the $\mathfrak{su}(2)$ spin algebra. By definition, in this algebra there are three generators that satisfy the commutator rule

$$ [ J_i , J_j ] = i \epsilon_{ijk} J_k . $$

(B.1)

Two of these generators can be combined to form the raising/lowering operators,

$$ J_\pm= J_1 \pm i J_2 , $$

(B.2)

which fulfill the commutator rules

$$ [J_+ , J_- ] = 2 J_3 \quad\text{and}\quad[J_3 , J_\pm] = \pm J_\pm. $$

(B.3)

These can easily be checked by means of Eq. (B.1). With these definitions, it is easy to show that the effects of the raising/lowering operators on a state of spin j represented by |j,σ〉 is to increase/decrease the eigenvalue of J ₃ denoted by σ,

$$ J_3 J_\pm|j,\sigma\rangle= (\sigma\pm1) J_\pm|j,\sigma\rangle. $$

(B.4)

A couple of general remarks are in order before we consider specific spin. Given the above definitions, we can easily see that the set of generators given by flipping the signs of all the generators and by switching J ₊ with J ₋ also forms a representation of the same algebra. We will show this by defining

$$ \bar{J}_3 = - J_3 \quad\text{and}\quad \bar{J}_\pm= - J_\mp. $$

(B.5)

With these, we easily obtain

$$ [ \bar{J}_+ , \bar{J}_- ] = 2 \bar{J}_3 \quad\text{and}\quad[ \bar{J}_3 , \bar{J}_\pm] = \pm\bar{J}_\pm. $$

(B.6)

This representation is called the complex representation and is very important in what follows. It finds its name due to the property

$$ \bigl[ -J^*_i , -J^*_j \bigr] = i \epsilon_{ijk} \bigl(-J^*_k \bigr). $$

(B.7)

Since we are taking our states as eigenvectors of J ₃, this representation is diagonal and real (the eigenvalues of a Hermitian operator are real), so that $\bar{J}_{3}=-J_{3}^{*}=-J_{3}$. Further, in this basis, J ₁ is real and symmetric while J ₂ is imaginary and antisymmetric, so $\bar {J}_{1}=-J^{*}_{1}=-J_{1}$ and $\bar{J}_{2}=-J^{*}_{2}=J_{2}$. As a result, $\bar {J}_{\pm}= (-J^{*}_{1} )\pm i (-J^{*}_{2} )=- (J_{1}\mp i J_{2} )=-J_{\mp}$.

Moreover, these raising/lowering operators can always be shifted by a phase and still satisfy the algebra. For example, if we define

$$ J_\pm^\prime= e^{\pm i\phi}J_\pm, $$

(B.8)

it is easy to show that

$$ \bigl[ J_+^\prime, J_-^\prime\bigr] = 2 J_3 \quad \text{and}\quad\bigl[ J_3 , J_\pm^\prime\bigr] = \pm J_\pm^\prime. $$

(B.9)

So that the set J ₃ with $J_{\pm}^{\prime}$ also forms a representation of $\mathfrak{su}(2)$. This will be important for understanding the Helas conventions for the spin-$\frac{1}{2}$ polarization vectors.

For the rest of this section, we consider the particle or antiparticle to have a mass M and a four-momentum

$$ p^\mu= (E,|\mathbf {p}|\sin\theta\cos\phi,|\mathbf {p}|\sin\theta\sin \phi,| \mathbf {p}|\cos\theta) $$

(B.10)

2.2 B.2 Polarization vectors

The polarization vectors for a particle u and its antiparticle v are determined by the spin of the particle. They transform under the spin representation of the particle and antiparticle, respectively, which is to say, the particle’s spin transforms according to J while the antiparticle’s spin transforms according to −J ^∗. On the other hand, they both transform according to the same representation of the Lorentz algebra whose generators are $\boldsymbol{\mathcal{J}}$. For the rest of this discussion, we will take J ₃ to measure the component of spin along the direction of motion. In other words, we will take J ₃ to be the helicity operator. J ₁ and J ₂ will be defined appropriately, with the direction 1 and 2 orthogonal to direction 3 and each other and satisfying Eq. (B.1).

As we mentioned, the spin algebra is a subalgebra of the Lorentz algebra. So, each spin transformation corresponds with a Lorentz transformation. For particles, we can write this as⁷

$$ \begin{aligned} J^j_3 u_\ell(j, \sigma) &= \mathcal{J}^{(3)}_{\ell,\ell'} u_{\ell '}(j,\sigma) = \sigma u_\ell(j,\sigma) , \\ J^j_\pm u_\ell(j,\sigma) &= \mathcal{J}^{(\pm)}_{\ell,\ell'} u_{\ell'}(j,\sigma) \\ &= \sqrt{(j\mp\sigma) (j\pm\sigma+1)} u_{\ell}(j,\sigma\pm1), \end{aligned} $$

(B.11)

where ℓ and ℓ′ are the field indices under the Lorentz transformations and ℓ′ is summed over. Furthermore, we have assumed a trivial phase for the raising and lowering operators. Other phase choices are discussed later. Antiparticles transform according to the same representation of the Lorentz algebra and, so, we use the same $\boldsymbol{\mathcal{J}}$ operators. However, their spin transforms under the conjugate representation. As a result, the Lorentz transformations that were used for particles have the following effect on antiparticles,

$$\begin{aligned} &{-\bar{J}^j_3} v_\ell(j,\sigma) = \mathcal{J}^{(3)}_{\ell,\ell'} v_{\ell'}(j,\sigma) = -\sigma v_\ell(j,\sigma), \\ &{-\bar{J}^j_\mp} v_\ell(j,\sigma) = \mathcal{J}^{(\pm)}_{\ell ,\ell'} v_{\ell'}(j,\sigma) \\ &\hphantom{{-\bar{J}^j_\mp} v_\ell(j,\sigma)} = -\sqrt{(j\pm\sigma) (j\mp\sigma+1)} v_{\ell}(j,\sigma\mp1). \end{aligned}$$

(B.12)

For the rest of this subsection, we review the polarization vectors for spin-$\frac{1}{2}$, spin-1, spin-$\frac{3}{2}$, and spin-2 fields. We use the Helas convention for the spin-$\frac {1}{2}$ and spin-1 polarization vectors and show that they satisfy the spin algebra and derive any relevant phases in the raising and lowering operators. We then construct the spin-$\frac {3}{2}$ and spin-2 polarization vectors from these.

2.2.1 B.2.1 Spin-$\frac{1}{2}$ fields

According to the Helas conventions, the spin-$\frac{1}{2}$ polarization vectors are given by

$$ \begin{aligned} &u_{+1/2}(p) = \left( \begin{array}{c} \sqrt{E-|\mathbf {p}|}\left( \begin{array}{c}\cos\frac{\theta}{2} \\ \sin\frac{\theta }{2}e^{i\phi} \end{array} \right) \\ \sqrt{E+|\mathbf {p}|}\left( \begin{array}{c}\cos\frac{\theta}{2} \\ \sin\frac{\theta }{2}e^{i\phi} \end{array} \right) \end{array} \right), \\ &u_{-1/2}(p) = \left( \begin{array}{c} \sqrt{E+|\mathbf {p}|}\left( \begin{array}{c}-\sin\frac{\theta}{2}e^{-i\phi} \\ \cos\frac{\theta}{2} \end{array} \right) \\ \sqrt{E-|\mathbf {p}|}\left( \begin{array}{c}-\sin\frac{\theta}{2}e^{-i\phi} \\ \cos\frac{\theta}{2} \end{array} \right) \end{array} \right), \\ &v_{+1/2}(p) = \left( \begin{array}{c} -\sqrt{E+|\mathbf {p}|}\left( \begin{array}{c}-\sin\frac{\theta}{2}e^{-i\phi} \\ \cos\frac{\theta}{2} \end{array} \right) \\ \sqrt{E-|\mathbf {p}|}\left( \begin{array}{c}-\sin\frac{\theta}{2}e^{-i\phi} \\ \cos\frac{\theta}{2} \end{array} \right) \end{array} \right), \\ &v_{-1/2}(p) = \left( \begin{array}{c} \sqrt{E-|\mathbf {p}|}\left( \begin{array}{c}\cos\frac{\theta}{2} \\ \sin\frac{\theta }{2}e^{i\phi} \end{array} \right) \\ -\sqrt{E+|\mathbf {p}|}\left( \begin{array}{c}\cos\frac{\theta}{2} \\ \sin\frac{\theta }{2}e^{i\phi} \end{array} \right) \end{array} \right) . \end{aligned} $$

(B.13)

We begin with the spin-$\frac{1}{2}$ form of the rotation and boost generators

$$ \mathcal{J}_i = \frac{1}{2}\left( \begin{array}{c@{\quad}c}\sigma_i&0\\ 0&\sigma_i \end{array} \right) \quad\text{and}\quad\mathcal{K}_i = - \frac{i}{2}\left( \begin{array}{c@{\quad}c}\sigma_i&0\\ 0&-\sigma_i \end{array} \right) , $$

(B.14)

where σ _i denotes the Pauli matrices, and then combine $\mathcal{J}_{1}$ and $\mathcal{J}_{2}$ to form the ladder operators

$$ \mathcal{J}_\pm= \mathcal{J}_1\pm i \mathcal{J}_2 = \frac{1}{2}\left( \begin{array}{c@{\quad}c}\sigma_1\pm i \sigma_2&0\\ 0&\sigma_1\pm i \sigma_2 \end{array} \right) . $$

(B.15)

We then rotate and boost from the particle rest frame to the laboratory frame of reference by employing the operator

$$ \begin{aligned}[b] &\varLambda_{\frac{1}{2}} \\ &= \left( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} e^{-\frac{\eta}{2}-i\frac{\phi}{2}}c_{\frac{\theta}{2}}&-e^{\frac {\eta}{2}-i\frac{\phi}{2}}s_{\frac{\theta}{2}}&0&0\\ e^{-\frac{\eta}{2}+i\frac{\phi}{2}}s_{\frac{\theta}{2}}&e^{\frac {\eta}{2}+i\frac{\phi}{2}}c_{\frac{\theta}{2}}&0&0\\ 0&0&e^{\frac{\eta}{2}-i\frac{\phi}{2}}c_{\frac{\theta }{2}}&-e^{-\frac{\eta}{2}-i\frac{\phi}{2}}s_{\frac{\theta}{2}}\\ 0&0&e^{\frac{\eta}{2}+i\frac{\phi}{2}}s_{\frac{\theta }{2}}&e^{-\frac{\eta}{2}+i\frac{\phi}{2}}c_{\frac{\theta}{2}} \end{array} \right). \end{aligned} $$

(B.16)

In this expression, we have introduced the sine and cosine of half the polar angle c _θ/2=cos(θ/2) and s _θ/2=sin(θ/2) as well as the pseudorapidity

$$ \eta= \frac{1}{2}\ln\biggl(\frac{E+|\mathbf {p}|}{E-|\mathbf {p}|} \biggr) . $$

(B.17)

At the operator level, we hence get

$$ \begin{aligned} &\mathcal{J}^{(3)} = \varLambda_{\frac{1}{2}} \mathcal{J}_z\varLambda_{\frac{1}{2}}^{-1} = \left( \begin{array}{c@{\quad}c}B&0\\ 0&B \end{array} \right), \\ &\mathcal{J}^{(\pm)} = \varLambda_{\frac{1}{2}}\mathcal{J}_\pm \varLambda_{\frac{1}{2}}^{-1} = \left( \begin{array}{c@{\quad}c}e^{\mp\eta}A_\pm&0\\ 0&e^{\pm\eta}A_\pm \end{array} \right) , \end{aligned} $$

(B.18)

with

$$ \begin{aligned} &A_\pm= \left( \begin{array}{c@{\quad}c} -\frac{1}{2}\sin\theta&\pm\frac{1}{2} (1\pm\cos\theta )e^{-i\phi}\\ \mp\frac{1}{2} (1\mp\cos\theta)e^{i\phi}&\frac {1}{2}\sin\theta \end{array} \right) , \\ &B= \left( \begin{array}{c@{\quad}c}\frac{1}{2}\cos\theta&\frac{1}{2}\sin\theta e^{-i\phi}\\ \frac{1}{2}\sin\theta e^{i\phi}&-\frac{1}{2}\cos \theta \end{array} \right) . \end{aligned} $$

(B.19)

It can easily be shown that these operators satisfy the commutation properties of $\mathfrak{su}(2)$,

$$ \bigl[ \mathcal{J}^{(+)} , \mathcal{J}^{(-)} \bigr] = 2 \mathcal{J}^{(3)}\quad\text{and}\quad\bigl[ \mathcal{J}^{(3)} , \mathcal{J}^{(\pm)} \bigr] = \pm\mathcal{J}^{(\pm)} . $$

(B.20)

They correspond with the helicity operator, raising operator and lowering operator in spin space for particles. We can explicitly check that they have the following effect on the particle polarization vectors of Eq. (B.13),

$$ \begin{aligned} &J^{1/2}_3 u_{\pm1/2}(p) = \mathcal{J}^{(3)} u_{\pm1/2}(p) = \pm\frac{1}{2} u_{\pm1/2}(p), \\ &J^{1/2}_- u_{1/2}(p) = \mathcal{J}^{(-)} u_{1/2}(p) = e^{i\phi }u_{-1/2}(p) , \\ &J^{1/2}_- u_{-1/2}(p) = \mathcal{J}^{(-)} u_{-1/2}(p) = 0 , \\ &J^{1/2}_+ u_{-1/2}(p) = \mathcal{J}^{(+)} u_{-1/2}(p) = e^{-i\phi }u_{1/2}(p), \\ &J^{1/2}_+ u_{1/2}(p) = \mathcal{J}^{(+)} u_{1/2}(p) = 0 . \end{aligned} $$

(B.21)

From this, we learn that there is an extra phase associated with the ladder operators in the Helas conventions for spin-$\frac{1}{2}$,

$$ J^{1/2}_\pm u_\sigma(p) = e^{\mp i\phi}\sqrt{ \biggl(\frac{1}{2}\mp\sigma\biggr) \biggl(\frac{1}{2}\pm\sigma+1 \biggr)}\ u_{\sigma\pm1}(p) . $$

(B.22)

Since the generators formed in this way still form a representation of the spin algebra, this phase could have been absorbed into the definition of the polarization vectors.

For antiparticles, we find

$$ \begin{aligned} &\bar{J}^{1/2}_3 v_{\pm1/2}(p) = -\mathcal{J}^{(3)} v_{\pm1/2}(p) = \pm \frac{1}{2}v_{\pm1/2}(p) , \\ &\bar{J}^{1/2}_- v_{1/2}(p) = -\mathcal{J}^{(+)} v_{1/2}(p) = e^{-i\phi}v_{-1/2}(p) , \\ &\bar{J}^{1/2}_- v_{-1/2}(p) = -\mathcal{J}^{(+)} v_{-1/2}(p) = 0, \\ &\bar{J}^{1/2}_+ v_{-1/2}(p) = -\mathcal{J}^{(-)} v_{-1/2}(p) = e^{i\phi}v_{1/2}(p), \\ &\bar{J}^{1/2}_+ v_{1/2}(p) = -\mathcal{J}^{(-)} v_{1/2}(p) = 0 , \end{aligned} $$

(B.23)

and as a result,

$$ \bar{J}^{1/2}_\pm v_\sigma(p) = e^{\pm i\phi} \sqrt{ \biggl(\frac {1}{2}\mp\sigma\biggr) \biggl(\frac{1}{2}\pm \sigma+1 \biggr)}\ v_{\sigma\pm 1}(p) . $$

(B.24)

2.2.2 B.2.2 Spin-1 fields

The polarization vectors associated with a spin-1 particle are given, following the Helas conventions, by

$$ \begin{aligned} \epsilon^\mu_+(p) &= \frac{1}{\sqrt{2}} \left( \begin{array}{c} 0\\ -\cos\theta\cos\phi+i\sin\phi\\ -\cos\theta\sin\phi-i\cos\phi\\ \sin\theta \end{array} \right), \\ \epsilon^\mu_0(p) &= \frac{1}{M}\left( \begin{array}{c}|\mathbf {p}|\\ E\sin\theta\cos\phi\\ E\sin\theta\sin \phi\\ E\cos\theta \end{array} \right), \\ \epsilon^\mu_-(p) &= \frac{1}{\sqrt{2}}\left( \begin{array}{c} 0\\ \cos\theta\cos\phi+i\sin\phi\\ \cos\theta\sin\phi-i\cos\phi\\ -\sin\theta \end{array} \right) , \end{aligned} $$

(B.25)

whereas those associated with a spin-1 antiparticle read

$$ \begin{aligned} \bar{\epsilon}_+^\mu(p) &= \epsilon^{\mu*}_+(p), \\ \bar{\epsilon}_0^\mu(p) &= \epsilon^{\mu*}_0(p), \\ \bar{\epsilon}_-^\mu(p) &= \epsilon^{\mu*}_-(p) . \end{aligned} $$

(B.26)

In the vectorial representation, the spatial pieces of the rotation generators are given by

$$ (\mathcal{J}_i)^j{}_k = i \epsilon_i{}^j{}_k, $$

(B.27)

where ϵ _ijk =1 and i,j,k=1,2,3, while the time components are zero. On the other hand, the spatial components of the boost generators vanish while their time components are given by

$$ (\mathcal{K}_i)^0{}_\mu= i \eta_{i\mu} \quad\text{and}\quad(\mathcal{K}_i)^\mu{}_0 = -i \delta^\mu{}_i . $$

(B.28)

As a result, in the rest frame, we have

$$ \begin{aligned} &\mathcal{J}_3 = \left( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c}0&0&0&0\\ 0&0&-i&0\\ 0&i&0&0\\ 0&0&0&0 \end{array} \right)\quad\text{and} \\ &\mathcal{J}_\pm=\left( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c}0&0&0&0\\ 0&0&0&\mp1\\ 0&0&0&-i\\ 0&\pm1&i&0 \end{array} \right) . \end{aligned} $$

(B.29)

To get these generators in the laboratory frame of reference, we boost along the z-direction and rotate using

$$\begin{aligned} \varLambda_1 =& \left( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c}1&0&0&0\\ 0&c_\phi&-s_\phi&0\\ 0&s_\phi&c_\phi&0\\ 0&0&0&1 \end{array} \right) \left( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c}1&0&0&0\\ 0&c_\theta&0&s_\theta\\ 0&0&1&0\\ 0&-s_\theta&0&c_\theta \end{array} \right) \\ &{}\times \frac{1}{M}\left( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c}E&0&0&p\\ 0&1&0&0\\ 0&0&1&0\\ p&0&0&E \end{array} \right) , \end{aligned}$$

(B.30)

where the sine and cosine of the azimuthal (polar) angle are denoted by s _ϕ (s _θ ) and c _ϕ (c _θ ). One can now derive the rotation generators

$$ \begin{aligned} &\mathcal{J}^{(3)} = \varLambda_1 \mathcal{J}_3 \varLambda_1^{-1} \\ &\hphantom{\mathcal{J}^{(3)}}= \left( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 0&0&0&0\\ 0&0&-ic_\theta& i s_\theta s_\phi\\ 0&ic_\theta&0&-is_\theta c_\phi\\ 0&-is_\theta s_\phi&is_\theta c_\phi&0 \end{array} \right) , \\ &\mathcal{J}^{(\pm)}= \varLambda_1 \mathcal{J}_\pm \varLambda_1^{-1} \\ &\hphantom{\mathcal{J}^{(\pm)}}= \left( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 0& \frac{p}{M}C_{1\pm} & \frac{p}{M}C_{2\pm} & \mp\frac{ps_\theta}{M}\\ \frac{p}{M}C_{1\pm} & 0& \frac{iEs_\theta}{M} & -\frac{E}{M}C_{3\pm} \\ \frac{p}{M}C_{2\pm} & -\frac{iEs_\theta}{M} & 0& -\frac{E}{M}C_{4\pm} \\ \mp\frac{ps_\theta}{M} & \frac{E}{M}C_{3\pm} & \frac{E}{M}C_{4\pm} & 0 \end{array} \right) , \end{aligned} $$

(B.31)

where we have introduced the quantities

$$ \begin{aligned} C_{1\pm} &= \pm\cos\theta\cos\phi-i\sin\phi, \\ C_{2\pm} &= \pm\cos\theta\sin\phi+i\cos\phi, \\ C_{3\pm} &= -i\cos\theta\sin\phi\pm\cos\phi, \\ C_{4\pm} &= i\cos\theta\cos\phi\pm\sin\phi. \end{aligned} $$

(B.32)

The commutators of these generators satisfy the usual relations,

$$ \bigl[ \mathcal{J}^{(+)} , \mathcal{J}^{(-)} \bigr] = 2 \mathcal{J}^{(3)} \quad\text{and}\quad\bigl[ \mathcal{J}^{(3)} , \mathcal{J}^{(\pm)} \bigr] = \pm\mathcal{J}^{(\pm)} . $$

(B.33)

Acting on the spin-1 polarization vectors, we get

$$ \begin{aligned} &J_3 \epsilon_\pm^\mu(p) =\mathcal{J}^{(3)\mu}_{\nu} \epsilon_\pm^\nu(p) = \pm\epsilon_\pm^\mu(p), \\ &J_3 \epsilon_0^\mu(p) ={\mathcal{J}^{(3)\mu}}_{\nu} \epsilon_0^\nu(p) = 0 , \\ &J_- \epsilon_+^\mu(p) ={\mathcal{J}^{(-)\mu}}_{\nu } \epsilon_+^\nu(p) = \sqrt{2}\epsilon_0^\mu(p), \\ &J_- \epsilon_0^\mu(p) ={\mathcal{J}^{(-)\mu}}_{\nu } \epsilon_0^\nu(p) = \sqrt{2}\epsilon_-^\mu(p), \\ &J_- \epsilon_-^\mu(p) ={\mathcal{J}^{(-)\mu}}_{\nu } \epsilon_-^\nu(p) = 0 , \\ &J_+ \epsilon_-^\mu(p) ={\mathcal{J}^{(+)\mu}}_{\nu } \epsilon_-^\nu(p) = \sqrt{2}\epsilon_0^\mu(p), \\ &J_+ \epsilon_0^\mu(p) ={\mathcal{J}^{(+)\mu}}_{\nu } \epsilon_0^\nu(p) = \sqrt{2}\epsilon_+^\mu(p), \\ &J_+ \epsilon_+^\mu(p) ={\mathcal{J}^{(+)\mu}}_{\nu } \epsilon_+^\nu(p) = 0, \end{aligned} $$

(B.34)

so that we observe that no additional phase is required and the particle spin raising and lowering operators satisfy

$$ J^1_\pm\epsilon_\sigma^\mu(p) = \sqrt{(1\mp\sigma) (1\pm\sigma+1)}\ \epsilon_{\sigma\pm1}^\mu (p) . $$

(B.35)

For antiparticles, we find

$$ \begin{aligned} &\bar{J}_3 \epsilon_\pm^{\mu*}(p) = -{\mathcal{J}^{(3)\mu}}_{\nu} \epsilon_\pm^{\nu*}(p) = \pm\epsilon_\pm^{\mu*}(p), \\ &\bar{J}_3 \epsilon_0^{\mu*}(p) = - {\mathcal{J}^{(3)\mu}}_{\nu} \epsilon_0^{\nu*}(p) = 0 , \\ &\bar{J}_- \epsilon_+^{\mu*}(p) = -{\mathcal{J}^{(+)\mu}}_{\nu} \epsilon_+^{\nu*}(p) = \sqrt{2}\epsilon_0^{\mu*}(p), \\ &\bar{J}_- \epsilon_0^{\mu*}(p) = -{\mathcal{J}^{(+)\mu}}_{\nu} \epsilon_0^{\nu*}(p) = \sqrt{2}\epsilon_-^{\mu*}(p), \\ &\bar{J}_- \epsilon_-^{\mu*}(p) = -{\mathcal{J}^{(+)\mu}}_{\nu} \epsilon_-^{\nu*}(p) = 0 , \\ &\bar{J}_+ \epsilon_-^{\mu*}(p) = -{\mathcal{J}^{(-)\mu}}_{\nu} \epsilon_-^{\nu*}(p) = \sqrt{2}\epsilon_0^{\mu*}(p), \\ &\bar{J}_+ \epsilon_0^{\mu*}(p) = -{\mathcal{J}^{(-)\mu}}_{\nu} \epsilon_0^{\nu*}(p) = \sqrt{2}\epsilon_+^{\mu*}(p), \\ &\bar{J}_+ \epsilon_+^{\mu*}(p) = -{\mathcal{J}^{(-)\mu}}_{\nu} \epsilon_+^{\nu*}(p) = 0, \end{aligned} $$

(B.36)

which again shows us there is no extra phase associated with these polarization vectors. The raising and lowering operators satisfy in this case

$$ \bar{J}^1_\pm\epsilon_\sigma^{\mu*}(p) = \sqrt{(1\mp\sigma) (1\pm\sigma+1)}\ \epsilon_{\sigma\pm 1}^{\mu*}(p) . $$

(B.37)

2.2.3 B.2.3 Spin-$\frac{3}{2}$ fields

The spin-$\frac{3}{2}$ polarization vectors can be formed as a direct product of the spin-$\frac{1}{2}$ and spin-1 polarization vectors. The generators are then simply the sum of those of the spin-$\frac {1}{2}$ and spin-1 representations,

$$ J^{3/2}_3= J^{1/2}_3 + J^1_3 \quad\text{and}\quad J^{3/2}_\pm= J^{1/2}_\pm+ J^1_\pm. $$

(B.38)

It is easily seen that these satisfy the $\mathfrak{su}(2)$ algebra.

We begin with the highest weight of the spin-$\frac{3}{2}$ representation which has helicity $\frac{3}{2}$ and is given by the product of the highest polarization of spin-$\frac{1}{2}$ and spin-1

$$ u^\mu_{3/2,3/2}(p) = \epsilon^{\mu}_{+}(p)u_{+1/2}(p) . $$

(B.39)

We then lower the J ₃-eigenvalue by using the $J^{3/2}_{-}$ operator to get the other states, $J^{1/2}_{-}$ only acting on spin-$\frac{1}{2}$ polarization vectors and $J^{1}_{-}$ on spin-1 polarization vectors. We now have the choice of the overall phase for the raising and lowering operators. We take the trivial choice with no additional phase,

$$ J^{3/2}_\pm u^\mu_{3/2,\sigma}(p) = \sqrt{ \biggl(\frac{3}{2}\mp\sigma\biggr) \biggl(\frac{3}{2}\pm\sigma+1 \biggr)}\ u^\mu_{3/2,\sigma\pm1}(p) . $$

(B.40)

We now work out the first step explicitly,

$$\begin{aligned} &J^{3/2}_- u^\mu_{3/2,3/2}(p) \\ &\quad= \sqrt{3} u^\mu_{3/2,1/2}(p) \\ &\quad= \bigl(J^1_-\epsilon^\mu_+(p) \bigr)u_{+1/2}(p)+\epsilon^\mu_+ \bigl(J^{1/2}_- u_{+1/2}(p) \bigr) \\ &\quad= \sqrt{2}\epsilon^\mu_0(p)u_{+1/2}(p)+e^{i\phi} \epsilon^\mu_+(p)u_{-1/2}(p), \end{aligned}$$

(B.41)

so that

$$ u^\mu_{3/2,1/2}(p) = \sqrt{\frac{2}{3}} \epsilon^\mu_0(p)u_{+1/2}(p)+ \frac{1}{\sqrt{3}}e^{i\phi}\epsilon^\mu_+(p)u_{-1/2}(p) . $$

(B.42)

Following the same procedure for the other states gives

$$ \begin{aligned} u^\mu_{3/2,-1/2}(p) &= \sqrt{ \frac{2}{3}}e^{i\phi}\epsilon^\mu_0(p)u_{-1/2}(p) \\ &\quad{}+\frac{1}{\sqrt{3}}\epsilon^\mu_-(p)u_{+1/2}(p) , \\ u^\mu_{3/2,-3/2}(p) &= e^{i\phi}\epsilon^{\mu}_{-}(p)u_{-1/2}(p) . \end{aligned} $$

(B.43)

Antiparticle polarization vectors are derived in a similar fashion using the conjugate lowering operators and starting with the highest weight of the antiparticle polarization vector,

$$ \begin{aligned} &v^\mu_{3/2,3/2}(p) = \epsilon^{\mu*}_{+}(p)v_{+1/2}(p), \\ &v^\mu_{3/2,1/2}(p) = \sqrt{\frac{2}{3}} \epsilon^{\mu *}_0(p)v_{+1/2}(p) \\ &\hphantom{v^\mu_{3/2,1/2}(p) =}{}+\frac{1}{\sqrt{3}}e^{-i\phi}\epsilon^{\mu*}_+(p)v_{-1/2}(p), \\ &v^\mu_{3/2,-1/2}(p) = \sqrt{\frac{2}{3}}e^{-i\phi} \epsilon^{\mu *}_0(p)v_{-1/2}(p) \\ &\hphantom{v^\mu_{3/2,-1/2}(p) =}{}+\frac{1}{\sqrt{3}}\epsilon^{\mu *}_-(p)v_{+1/2}(p), \\ &v^\mu_{3/2,-3/2}(p) = e^{-i\phi}\epsilon^{\mu*}_{-}(p)v_{-1/2}(p) , \end{aligned} $$

(B.44)

where we have again not included any extra phase in the raising and lowering operators.

2.2.4 B.2.4 Spin-2 fields

Just as in the case of the spin-$\frac{3}{2}$ fields, the spin-2 polarization vectors can be formed as the direct product of two spin-1 polarization vectors. The generators are hence the sum of the spin-1 generators for each polarization vector,

$$ J^{2}_3= J^1_3 + J^1_3 \quad\text{and}\quad J^{2}_\pm= J^1_\pm+ J^1_\pm, $$

(B.45)

where it is understood that the first generator only acts on the first polarization vector and the second generator only acts on the second polarization vector. It is easily seen that these satisfy the $\mathfrak{su}(2)$ algebra.

We begin with the highest weight of the spin-2 representation which has helicity 2 and is given by the product of the highest polarization of the two spin-1 polarization vectors,

$$ \epsilon^{\mu\nu}_{2,2}(p) = \epsilon^{\mu}_{+}(p) \epsilon^{\nu }_{+}(p) . $$

(B.46)

We then lower the J ₃-eigenvalue by using the $J^{2}_{-}$ operator to get the other states. Again, we choose to not include an additional phase,

$$ J^{2}_\pm\epsilon^{\mu\nu}_{2,\sigma}(p) = \sqrt{ (2\mp\sigma) (2\pm\sigma+1 )}\ \epsilon^{\mu\nu }_{2,\sigma\pm1}(p) . $$

(B.47)

We now work out the first step explicitly,

$$ \begin{aligned}[b] J^{2}_- \epsilon^{\mu\nu}_{2,2}(p) &= 2 \epsilon^{\mu\nu }_{2,1}(p) \\ &= \bigl(J^1_-\epsilon^\mu_+(p) \bigr) \epsilon^\nu_{+}(p)+\epsilon^\mu_+ \bigl(J^{1}_- \epsilon^\nu_{+}(p) \bigr) \\ &= \sqrt{2}\epsilon^\mu_0(p)\epsilon^\nu_{+}(p)+ \sqrt{2}\epsilon^\mu_+(p)\epsilon^\nu_{0}(p), \end{aligned} $$

(B.48)

so that

$$ \epsilon^{\mu\nu}_{2,1}(p) = \frac{1}{\sqrt{2}} \bigl(\epsilon ^\mu_0(p)\epsilon^\nu_{+}(p)+ \epsilon^\mu_+(p)\epsilon^\nu_0(p) \bigr) . $$

(B.49)

Following the same procedure for the other states gives

$$ \begin{aligned} &\epsilon^{\mu\nu}_{2,0}(p) = \frac{1}{\sqrt{6}} \bigl(\epsilon^\mu_{-}(p) \epsilon^\nu_+(p)+2\epsilon^\mu_0(p) \epsilon^\nu_0(p) \\ &\hphantom{\epsilon^{\mu\nu}_{2,0}(p) =}{}+\epsilon^\mu_+(p)\epsilon^\nu_-(p) \bigr) , \\ &\epsilon^{\mu\nu}_{2,-1}(p) = \frac{1}{\sqrt{2}} \bigl(\epsilon ^\mu_-(p)\epsilon^\nu_0(p)+ \epsilon^\mu_0(p)\epsilon^\nu_-(p) \bigr) , \\ &\epsilon^{\mu\nu}_{2,-2}(p) = \epsilon^{\mu}_{-}(p) \epsilon^\nu_{-}(p) . \end{aligned} $$

(B.50)

Antiparticle polarization vectors are derived in a similar fashion using the conjugate lowering operators and starting with the highest weight of the antiparticle polarization vector, which gives

$$ \bar{\epsilon}^{\mu\nu}_{2,\sigma}(p) = \epsilon^{\mu\nu *}_{2,\sigma}(p) . $$

(B.51)

We have again not included any extra phase in the raising and lowering operators.

2.3 B.3 Propagators

On-shell, a propagator is fully determined by the spin of the particle. In particular, the numerator of the propagator is given by the sum over the polarization vectors in the following way

$$ \lim_{p^2\to M^2}\bigl(p^2-M^2+i\epsilon \bigr)\varDelta^j(p) = \sum_{\sigma} u_{j,\sigma}(p) u^*_{j,\sigma}(p) , $$

(B.52)

where Δ ^j(p) is the full propagator for a spin-j particle. Off-shell, the propagator can be different as long as the difference vanishes when evaluated on-shell. This difference is due to the presence of lower spin components in the propagator which are removed on-shell by the classical equations of motion for the field. These extra components are determined by the associated terms in the quadratic part of the Lagrangian, which is model-dependent. In this section, we show that the on-shell relation above holds in the spin-$\frac{1}{2}$, spin-1, spin-$\frac{3}{2}$, and spin-2 cases.

2.3.1 B.3.5 Spin-$\frac{1}{2}$ fields

Using the polarization vectors given in Sect. B.2.1, we find

$$ \sum_{\sigma=-1/2}^{1/2} u_\sigma(p) \bar{u}_\sigma(p) = \left( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} M & 0 & E-|\mathbf {p}|\cos\theta& -e^{-i\phi}|\mathbf {p}|\sin\theta\\ 0 & M & -e^{i\phi}|\mathbf {p}|\sin\theta& E+|\mathbf {p}|\cos\theta\\ E+|\mathbf {p}|\cos\theta& e^{-i\phi}|\mathbf {p}|\sin\theta& M & 0\\ e^{i\phi}|\mathbf {p}|\sin\theta& E-|\mathbf {p}|\cos\theta& 0 & M \end{array} \right) . $$

(B.53)

This is to be compared with the spin-$\frac{1}{2}$ propagator, which is exactly the same both on-shell and off-shell since there is no possible spin lower by an integer that can contribute,

(B.54)

So, the spin-$\frac{1}{2}$ propagator is fixed entirely by its spin.

We have also checked that antiparticle polarization vectors satisfy a similar relation,

(B.55)

2.3.2 B.3.6 Spin-1 fields

Again, using the particle polarization vectors given in Sect. B.2.2, we find

$$ \sum_{\sigma=-1}^1 \epsilon_\sigma^\mu \epsilon_\sigma^{\nu*} = \frac{1}{M^2}\left( \begin{array}{c@{\quad}c} |\mathbf {p}|^2 & \cdots\\ E|\mathbf {p}|\sin\theta\cos\phi& \cdots\\ E|\mathbf {p}|\sin\theta\sin\phi& \cdots\\ E|\mathbf {p}|\cos\theta& \cdots \end{array} \right) . $$

(B.56)

The propagator numerator is given by

$$ \varPi_1^{\mu\nu}=-\eta^{\mu\nu}+\frac{p^\mu p^\nu}{M^2} = \frac{1}{M^2}\left( \begin{array}{c@{\quad}c} E^2-M^2 & \cdots\\ E|\mathbf {p}|\sin\theta\cos\phi& \cdots\\ E|\mathbf {p}|\sin\theta\sin\phi& \cdots\\ E|\mathbf {p}|\cos\theta& \cdots \end{array} \right) . $$

(B.57)

In these last two equations, we have omitted the matrix elements of the second, third and fourth column for brevity. We have however checked that they all agree with each other on-shell, so that

$$ \lim_{p^2\to M^2} \biggl(-\eta^{\mu\nu}+\frac{p^\mu p^\nu }{M^2} \biggr) = \sum_{\sigma=-1}^1 \epsilon_\sigma^\mu\epsilon_\sigma^{\nu*} . $$

(B.58)

Similar results can be obtained starting from antiparticle polarization vectors.

In the off-shell case, we find that the propagator numerator differs from the sum over the polarization vectors due to the presence of spin-0 components controlled by the quadratic part of the Lagrangian. It can be noted that for spin-1 fields, unitarity also plays a role in fixing the propagator.

2.3.3 B.3.7 Spin-$\frac{3}{2}$ fields

In this case, there are four sets of indices so that we omit to display the entire results in this paper for brevity. However, we have checked every element and find that

$$ \lim_{p^2\to M^2}\varPi_{RS}^{\mu\nu} = \sum _{i=-3/2}^{3/2} u_{3/2,i}^\mu \bar{u}_{3/2,i}^\nu, $$

(B.59)

where the numerator of the propagator reads

(B.60)

This is the propagator used by both CalcHEP and MadGraph 5. We have also checked that the related antiparticle relation

$$ \lim_{p^2\to M^2} \varPi_{RS\mathrm{anti}}^{\mu\nu} = \sum _{i=-3/2}^{3/2} v_{3/2,i}^\mu \bar{v}_{3/2,i}^\nu, $$

(B.61)

holds, where

(B.62)

Many terms are different in the off-shell case due to the presence of spin-$\frac{1}{2}$ components. These are model-dependent, as is the exact form of the propagator. For example, another spin-$\frac{3}{2}$ propagator that has been used in the literature [41] has its numerator given by

(B.63)

The difference between the numerators of the propagators of Eqs. (B.60) and (B.63) is

(B.64)

which vanishes on-shell.

2.3.4 B.3.8 Spin-2 fields

For spin-2, one form of the propagator numerator is given by

$$ \varPi_2^{\mu\nu\alpha\beta}=\frac{1}{2}\varPi_1^{\mu\alpha} \varPi_1^{\nu\beta}+\frac{1}{2}\varPi_1^{\mu\beta} \varPi_1^{\nu\alpha }-\frac{1}{3}\varPi_1^{\mu\nu} \varPi_1^{\alpha\beta} , $$

(B.65)

which is the propagator used by both CalcHEP and MadGraph 5. Other forms differ in their off-shell effects. The resulting tensor is much too long for every term to be included. We have however checked every term and find exact agreement with the sum of the polarization vectors as in

$$ \lim_{p^2\to M^2}\varPi_2^{\mu\nu\alpha\beta}=\sum _{\sigma =-2}^{2}\epsilon_{2,\sigma}^{\mu\nu} \epsilon_{2,\sigma}^{\alpha \beta*} . $$

(B.66)

2.4 B.4 Quadratic Lagrangian

In this subsection, we have followed the approach suggested by Weinberg, noting that the spin determines the on-shell propagator without any discussion of the Lagrangian. So, in this way, we learn that the spin also fixes the on-shell quadratic terms of the Lagrangian. The off-shell piece is model-dependent, as we have mentioned with the propagator being the inverse of the quadratic Lagrangian terms off-shell as well as on-shell.

In this section, we show that the quadratic Lagrangian pieces correspond to the inverse of the propagator for spin-$\frac{1}{2}$, spin-1, spin-$\frac {3}{2}$ and spin-2 fields.

2.4.1 B.4.9 Spin-$\frac{1}{2}$ fields

In this case, there is no ambiguity for either the propagator or the quadratic Lagrangian which reads

(B.67)

for a spin-$\frac{1}{2}$ fermionic field ψ. We can easily verify that this operator is the inverse of the propagator, since

(B.68)

2.4.2 B.4.10 Spin-1 fields

Although, in general, the spin-1 Lagrangian is ambiguous, the unitarity condition requires massive vector bosons to obtain their mass from spontaneous symmetry breaking. Therefore, the Lagrangian for a vector boson field V _μ is required to be gauge invariant. After spontaneous symmetry breaking, the quadratic piece is of the form

$$ \mathcal{L} = \frac{1}{2}V^\mu\bigl( \bigl(-p^2+M^2 \bigr)\eta_{\mu\nu}+p_\mu p_\nu \bigr)V^\nu, $$

(B.69)

where we have inserted a factor of 1/2 since gauge bosons are real fields. Multiplying this operator by the associated propagator gives

$$ \lim_{\epsilon\to0} \biggl[ \frac{-\eta^{\mu\nu}+\frac{p^\mu p^\nu}{M^2}}{p^2-M^2+i\epsilon } \bigl( \bigl(-p^2+M^2\bigr)\eta_{\nu\omega}+p_\nu p_\omega\bigr) \biggr] = \delta^\mu{}_\omega. $$

(B.70)

This proves the original statement in the spin-1 case.

2.4.3 B.4.11 Spin-$\frac{3}{2}$ fields

The Lagrangian describing the dynamics of a free spin-$\frac{3}{2}$ field Ψ _μ of mass M can be casted under several forms, all widely used in the literature. The one associated with the equations of motion presented in Eq. (1) read

$$ \mathcal{L} = \epsilon^{\mu\nu\rho\sigma} \bar{\varPsi}_\mu \gamma_5 \gamma_\sigma\partial_\nu \varPsi_\rho+ 2 i M \bar{\varPsi}_\mu\gamma^{\mu\nu} \varPsi_\nu, $$

(B.71)

our conventions on the Dirac matrices and antisymmetric tensors being collected in Appendix A. Equivalently, the original form introduced by Rarita and Schwinger [57],

(B.72)

can be retrieved after employing the property of the Dirac matrices of Eq. (A.10), together with the Clifford algebra relation of Eq. (A.6). It can be easily checked, after making use of Eq. (A.6), that the operator appearing in Eq. (B.72) is inverted by the propagator of Eq. (B.60),

$$ \begin{aligned} \lim_{\epsilon\to0} \biggl[ \frac{\varPi^{\mu\nu }_{RS}}{p^2-M^2+i\epsilon} \varLambda_{\nu\rho} \biggr] = \delta ^\mu{}_\rho. \end{aligned} $$

(B.73)

2.4.4 B.4.12 Spin-2 fields

The standard Lagrangian for a massive graviton field h _μν with a mass M is obtained starting from the Fierz–Pauli Lagrangian [86]. In this case, it is given by

$$ \begin{aligned}[b] \mathcal{L} &= \frac{1}{2} \partial_\rho h_{\mu\nu} \partial^\rho h^{\mu\nu} - \frac{1}{2} \partial_\rho h^\mu{}_\mu\partial^\rho h^\nu{}_\nu \\ &\quad{} + \partial_\rho h^\mu{}_\mu \partial_\nu h^{\rho\nu} - \partial_\rho h_{\mu\nu} \partial^\nu h^{\mu\rho} \\ &\quad{} -\frac{M^2}{2} \bigl(h_{\mu\nu} h^{\mu\nu} - h^\mu{}_\mu h^\nu{}_\nu\bigr) , \end{aligned} $$

(B.74)

where the structure of the mass term follows the conventions of the so-called Fierz–Pauli tuning [87] as usually employed to prevent ghost contributions from being required. After integrating the previous Lagrangian by parts, it can be rewritten as

$$ \mathcal{L} = \frac{1}{2} h_{\mu\nu} \mathcal{O}^{\mu\nu,\alpha \beta} h_{\alpha\beta} , $$

(B.75)

where the operator $\mathcal{O}^{\mu\nu,\alpha\beta}$ is given by

$$ \begin{aligned}[b] \mathcal{O}^{\mu\nu,\alpha\beta} &= \frac{1}{2} \bigl[ \eta^{\mu\alpha}\eta^{\nu\beta} + \eta^{\mu \beta}\eta^{\nu\alpha} - 2 \eta^{\mu\nu} \eta^{\alpha\beta} \bigr] \bigl[p^2 - M^2 \bigr] \\ & \quad {}+ p^\mu p^\nu\eta^{\alpha\beta} + p^\alpha p^\beta\eta^{\mu\nu} - \frac{1}{2} p^\mu p^\alpha\eta^{\nu\beta} \\ & \quad {}- \frac{1}{2} p^\mu p^\beta\eta^{\nu\alpha} - \frac{1}{2} p^\nu p^\alpha\eta^{\mu\beta} - \frac{1}{2} p^\nu p^\beta\eta^{\mu\alpha} . \end{aligned} $$

(B.76)

Under this form, it can be easily verified that this operator is the inverse of the massive spin-two propagator defined in Sect. B.3.8,

$$ \lim_{\epsilon\to0} \biggl[ \frac{\varPi^{\mu\nu\alpha\beta }_2}{p^2-M^2+i\epsilon} \mathcal{O}_{\alpha\beta, \rho\sigma} \biggr] = \frac{1}{2} \bigl[ \delta^\mu{}_\rho \delta^\nu{}_\sigma+ \delta^\mu {}_\sigma\delta^\nu{}_\rho\bigr] . $$

(B.77)

2.5 B.5 Wigner d-functions

Since total angular momentum is conserved in any system, if the angular momentum of the initial state is described by a |j,m〉 state with m being along an initial direction, then so will the final state. However, if we measure the final state along a different direction, we need to calculate the corresponding wave function overlap between the two directions under consideration. We suppose that the final state is described by |j,m′,θ〉, where m′ is the component of the angular momentum along the final direction which makes an angle θ with respect to the initial direction. The wavefunction overlap between these two bases is given by the product

$$ \bigl\langle j,m',\theta\big| j,m \bigr\rangle. $$

(B.78)

These different bases are related by the rotation operator J, so that

$$ \bigl| j,m',\theta\bigr\rangle= e^{-i\theta J_\perp} \bigl| j,m' \bigr\rangle, $$

(B.79)

where J _⊥ is the component of the rotation operator that is perpendicular to the plane spanned by the initial direction and the final direction. A rotation of angle θ hence rotates the initial basis into the final basis. As a result, the wavefunction overlap is given by

$$ d^j_{m'm} (\theta) = \bigl\langle j, m', \theta\big| j,m \bigr\rangle= \bigl\langle j, m'\bigr| e^{i \theta J_\perp} | j,m \rangle. $$

(B.80)

These functions of θ are called the Wigner d-functions. Since they only depend on the conservation of angular momentum, they can be calculated independently of the internal dynamics defined by the Lagrangian. For an illustrative derivation of the Wigner d-functions for j=0,1, we refer to the Appendix C of Ref. [88]. The full set of Wigner d-functions for j≤2 can be found in the Particle Data Group review [89] and satisfy the relations

$$ \begin{aligned}[b] d^j_{mm'} (\theta) &= (-1)^{m-m'} d^j_{m'm} (\theta) \\ &= d^j_{-m',-m}(\theta) \\ &= (-1)^{j-m} d^j_{m,-m'} (\pi-\theta) . \end{aligned} $$

(B.81)

Although the Wigner d-functions apply to any decay or collision, they are often discussed in the context of a 2→2 process in the center-of-momentum frame. For this case, we define the initial spin component direction to be along the direction of motion of one of the initial particles. Since the other initial particle has momentum equal in size but opposite in direction, we find that the initial spin component is equal to the difference of the initial helicities. We will define the final spin component direction to be along the direction of motion of one of the final state particles. As in the initial particle case, we find that the final spin component is equal to the difference of the helicities of the final states. Although the orbital angular momentum can contribute to the total angular momentum j, since it is perpendicular to the plane of the collision, it does not contribute to the component of either the initial angular momentum or the final angular momentum. So, if the total angular momentum is j (after including orbital angular momentum), the initial difference of helicity is m, and the final difference of helicity is m′, we find that the angular dependence of the collision must be given by $d^{j}_{m,m'}(\theta)$.

One consequence of the conservation of angular momentum is that any scattering amplitude can be expanded in terms of the Wigner d-functions (this is called a partial wave expansion).

$$ \mathcal{M}_{m'm} = \sum_{j} M_{jm'm} d^j_{m'm} (\theta) . $$

(B.82)

This equation only determines which Wigner d-functions are allowed for a certain scattering process, independent of the internal dynamics. However, on the other hand, the values of the M _jmm′ are determined by a combination of the initial spins, the final spins and the internal dynamics. For example, an electron-positron collider can be polarized so that the contribution to m=1 is larger than the contribution to m=−1. For another example, if we know that at any moment in the scattering process, there is only one particle in the s-channel and it is on its mass shell, then we know that the total angular momentum must equal the spin of that particle. Since there is only one particle, it is at rest in the collision center-of-momentum frame, so there is no orbital angular momentum at that moment. This means that only the M _jm′m with j equal to the spin of this particle are non-zero,

$$ \mathcal{M}_{m'm} = M_{jm'm} d^j_{m'm} (\theta) . $$

(B.83)

Appendix C: Helicity amplitudes for gravitino pair-production

In this Appendix, we present analytical expressions for the helicity amplitudes $\mathcal{M}_{\lambda_{1}\lambda_{2},\lambda_{3}\lambda_{4}}$ associated with the process

$$ g(p_1,\lambda_1)+g(p_2, \lambda_2)\to\tilde{G}(p_3,\lambda_3)+\tilde{G}(p_4,\lambda_4), $$

(C.1)

where the four-momentum (p _i ) and the helicity (λ _1,2=±1 and $\lambda_{3,4}=\pm\frac{3}{2},\pm\frac{1}{2}$) are explicitly indicated. The helicity amplitudes can be expressed as a sum upon s-, t- and u-channel contributions,

$$ \mathcal{M}_{\lambda_1\lambda_2,\lambda_3\lambda_4}=\mathcal {M}^s+\mathcal{M}^t+ \mathcal{M}^u, $$

(C.2)

with the helicity dependence of the right-hand side of the equation being understood for clarity. We derive, starting from the Feynman rules extracted from the Lagrangians of Eq. (14),

(C.3)

where the Mandelstam variables are defined by s=(p ₁+p ₂)², $t_{\tilde{g}}=(p_{1}-p_{3})^{2}-m_{\tilde{g}}^{2}$ and $u_{\tilde{g}}=(p_{1}-p_{4})^{2}-m_{\tilde{g}}^{2}$, $m_{\tilde{g}}$ being the gluino mass. The Lorentz structure of each amplitude has been embedded into the functions

$$ \begin{aligned} &B^{\alpha\beta,\gamma\delta}= \frac{1}{2}\bigl( \eta^{\alpha\gamma }\eta^{\beta\delta}+\eta^{\alpha\delta}\eta^{\beta\gamma}- \eta^{\alpha\beta}\eta^{\gamma\delta}\bigr) , \\ &\varGamma_g^{\mu\nu,\alpha\beta} = (p_1\cdot p_2)\,C_{\alpha\beta,\mu\nu} +D_{\alpha\beta,\mu\nu}+E_{\alpha \beta,\mu\nu}, \\ &\varGamma_{\tilde{G}}^{\gamma\delta,\rho\sigma} = \frac{1}{4} \epsilon^{\rho\sigma\lambda\gamma}\gamma^5\gamma^{\delta }(p_3-p_4)_{\lambda}+( \gamma\leftrightarrow\delta) \\ &\hphantom{\varGamma_{\tilde{G}}^{\gamma\delta,\rho\sigma} =}{} +\frac{i}{8}\epsilon^{\rho\sigma\lambda\gamma}\gamma ^5\bigl\{ \gamma_{\lambda},\sigma^{\delta\tau}\bigr\}(p_3+p_4)_{\tau} \\ &\hphantom{\varGamma_{\tilde{G}}^{\gamma\delta,\rho\sigma} =}{} +( \gamma\leftrightarrow\delta) \\ &\hphantom{\varGamma_{\tilde{G}}^{\gamma\delta,\rho\sigma} =}{} -im_{3/2}\bigl(\eta^{\rho\gamma}\eta^{\delta\sigma}+ \eta^{\sigma \gamma}\eta^{\delta\rho}-\eta^{\gamma\delta}\eta^{\rho\sigma} \bigr), \end{aligned} $$

(C.4)

with

$$ \begin{aligned} &C^{\alpha\beta,\mu\nu} = \eta^{\alpha\mu} \eta^{\beta\nu}+\eta^{\alpha\nu}\eta^{\beta\mu}-\eta^{\alpha \beta} \eta^{\mu\nu}, \\ &D^{\alpha\beta,\mu\nu} = p_{1}^{\nu}p_{2}^{\mu} \eta^{\alpha\beta} +\bigl(p_{1}^{\alpha}p_{2}^{\beta} \eta^{\mu\nu} - p_{1}^{\nu }p_{2}^{\beta}\eta^{\alpha\mu} \\ &\hphantom{D^{\alpha\beta,\mu\nu} =}{} - p_{1}^{\beta}p_{2}^{\mu} \eta^{\alpha\nu} + (\alpha\leftrightarrow\beta)\bigr), \\ &E^{\alpha\beta,\mu\nu} = \eta^{\alpha\beta}\bigl(p_1^{\mu }p_1^{\nu}+p_2^{\mu}p_2^{\nu }+p_1^{\mu}p_2^{\nu} \bigr) \\ &\hphantom{E^{\alpha\beta,\mu\nu} =}{} -\bigl(p_1^{\mu}p_1^{\beta} \eta^{\nu\alpha}+p_2^{\nu}p_2^{\beta} \eta^{\mu\alpha}+(\alpha\leftrightarrow\beta)\bigr). \end{aligned} $$

(C.5)