Scattering Amplitudes For All Masses and Spins

# Scattering Amplitudes For All Masses and Spins

###### Abstract

We introduce a formalism for describing four-dimensional scattering amplitudes for particles of any mass and spin. This naturally extends the familiar spinor-helicity formalism for massless particles to one where these variables carry an extra SU(2) little group index for massive particles, with the amplitudes for spin S particles transforming as symmetric rank 2S tensors. We systematically characterise all possible three particle amplitudes compatible with Poincare symmetry. Unitarity, in the form of consistent factorization, imposes algebraic conditions that can be used to construct all possible four-particle tree amplitudes. This also gives us a convenient basis in which to expand all possible four-particle amplitudes in terms of what can be called “spinning polynomials”. Many general results of quantum field theory follow the analysis of four-particle scattering, ranging from the set of all possible consistent theories for massless particles, to spin-statistics, and the Weinberg-Witten theorem. We also find a transparent understanding for why massive particles of sufficiently high spin can not be “elementary”. The Higgs and Super-Higgs mechanisms are naturally discovered as an infrared unification of many disparate helicity amplitudes into a smaller number of massive amplitudes, with a simple understanding for why this can’t be extended to Higgsing for gravitons. We illustrate a number of applications of the formalism at one-loop, giving few-line computations of the electron (g-2) as well as the beta function and rational terms in QCD. “Off-shell” observables like correlation functions and form-factors can be thought of as scattering amplitudes with external “probe” particles of general mass and spin, so all these objects–amplitudes, form factors and correlators, can be studied from a common on-shell perspective.

institutetext: {}^{1} School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USAinstitutetext: {}^{2} Walter Burke Institute for Theoretical Physics , California Institute of Technology, Pasadena, CA 91125, USA institutetext: {}^{3}Department of Physics and Astronomy, National Taiwan University, Taipei 10617, Taiwaninstitutetext: {}^{4}Physics Division, National Center for Theoretical Sciences, National Tsing-Hua University, No.101, Section 2, Kuang-Fu Road, Hsinchu, Taiwan

NCTS-TH/1714

## 1 Scattering Amplitudes in the Real World

Recent years have seen an explosion of progress in our understanding of scattering amplitudes in gauge theories and gravity. Infinite classes of amplitudes, whose computation would have seemed unthinkable even ten years ago, can now be derived with pen and paper on the back of an envelope using a set of ideas broadly referred to as “on-shell methods”. This has enabled the determination of scattering amplitudes of direct interest to collider physics experiments, while at the same time opening up novel directions of theoretical research into the foundations of quantum field theory, amongst other things revealing surprising and deep connections of this basic physics with areas of mathematics ranging from algebraic geometry to combinatorics to number theory.

Almost all of the major progress in this field has been in understanding scattering amplitudes for massless particles. There are seemingly good reasons for this, both technically and conceptually. Technically, almost all treatments of the subject, especially in four dimensions, involve the introduction of special variables (such as spinor-helicity, twistor or momentum-twistor variables) to trivialise the kinematical on-shell constraints for massless particles (see Reviews for a comprehensive review). And conceptually, while it is clear that the conventional field-theoretic description of massless particles with spin, which involves the introduction of huge gauge redundancy, leaves ample room for improvement–provided by on-shell methods that directly describe particles, eliminating any reference to quantum fields and their attendant redundancies–the advantage of “on-shell physics” seems to disappear for the case of massive particles where no gauge redundancies are needed.

As we will see, the technical issue about massless kinematics is just that–the transition to describing massive particles is a triviality–while the conceptual issue is not an obstacle but rather an invitation to understand the both the physics of “infrared deformation” of massless theories (by the Higgs mechanism and confinement), as well that of UV completion (such as with perturbative string theory), from a new on-shell perspective.

But before getting too far ahead of ourselves it suffices to remember that the only exactly massless particles we know of in the real world are photons and gravitons; even the spectacular success of on-shell methods applied to collider physics are for high-energy gluon collisions, which are ultimately confined into massive hadrons at long distances. Even if we consider the weakly coupled scattering amplitudes for Standard Model particles above the QCD scale, almost all the particles are massive. If the amazing structures unearthed in the study of gauge and gravity scattering amplitudes are indeed an indication of a radical new way of thinking about quantum particle interactions in space-time, they must naturally extend beyond photons, gravitons and gluons to electrons, W,Z particles and top quarks as well.

Keeping this central motivation in mind, in this paper we initiate a systematic exploration of the physics of scattering amplitudes in four dimensions, for particles of general masses and spins. We proceed with an on-shell formalism where the amplitude is manifestly covariant under the massive SU(2) little group. This approach allows us to cleanly categorize all distinct three-couplings for a given set of helicities or masses and spins. When constructing four-point amplitudes, this formalism sharply pinpoints the tension between locality and consistent factorization, which, in turn provides a portal into the difficulty of having higher-spin massive particles that is fundamental. As we will see, everything that is typically taught in an introductory courses on QFT and the Standard Model–including classic computations of the electron (g-2) and the QCD \beta function-can be transparently reproduced from an on-shell perspective directly following from the physics of Poincare invariance, locality and unitarity, without ever encountering quantum fields, Lagrangians, gauge and diff invariance, or Feynman rules.

There are a number of other motivations for developing this formalism. For instance, much of the remarkable progress in our understanding of the dynamic of supersymmetric gauge theories came from exploring their moduli spaces of vacua. From this point of view the study of massless scattering amplitudes has been stuck on a desert island at the origin of moduli space; we should now be able to study how the S-matrix varies on moduli space in general supersymmetric theories, especially beginning with the Coulomb branch of {\cal N}=4 SYM in the planar limit.

Another motivation, alluded to above, is the physics of UV completion for gravity scattering amplitudes. It is easy to show on general grounds that any weakly coupled UV completion for gravity amplitudes must involve an infinite tower of particles with infinitely increasing spins (as of course seen in string theory). This raises the possibility that string theory might be derivable from the bottom-up, as the unique weakly-coupled UV completion of gravity. But it has become clear that consistency conditions for massless graviton scattering alone are not enough to uniquely fix amplitudes–deformations of the graviton scattering amplitudes compatible with all the standard rules have been identified stringpaper. This is not surprising, since the most extreme tension in this physics is the coexistence of gravitons with massive higher-spin particles. Indeed (as we will review from an on-shell perspective) the presence of gravity makes the existence of massless higher-spin particles impossible. We should therefore expect the strongest consistency conditions on perturbative UV completion to involve the scattering of massless gravitons and massive higher-spin particles, the study of which calls for a good general formalism for treating amplitudes for general mass and spin.

Finally, an understanding of amplitudes for general mass and spin removes the distinction between “on-shell” observables like scattering amplitudes and “off-shell” observables like correlation functions. After all, loosely speaking the way experimentalists actually measure correlation functions of some system is to weakly couple the system to massive detectors, and effectively measure the scattering amplitudes for the detectors thought of as massive particles with general mass and spin! More precisely, to compute the correlation functions for (say) the stress tensor (in momentum-space), we need only imagine weakly coupling a continuum of massive spin 2 particle to the system with a universal (and arbitrarily weak) coupling; the leading scattering amplitudes for these massive particles is then literally the correlation function for the stress tensor in momentum space. This should allow us to explore both on- and off- shell physics in a uniform “on-shell” way.

## 2 The Little Group

Much of the non-trivial physics of scattering amplitudes traces back to the simple question–“what is a particle?”–and the attendant concept of Wigner’s “little group” governing the kinematics of particle scattering. Let us review this standard story. Following Wigner (and Weinberg’s exposition and notation) Wigner; Weinberg:1995mt, we think of “particles” as irreducible unitary representations of the Poincare group. We diagonalize the translation operator by labelling particles with their momentum p^{\mu}; any other labels a particle state can carry are labelled by \sigma. In order to systematically label all one-particle states, we start with some reference momentum k_{\mu} and the states |k,\sigma\rangle. Now, we can write any momentum p as a specified Lorentz-transformation L(p;k) acting on k, i.e. p_{\mu}=L_{\mu}^{\nu}(p;k)k_{\nu}. Note that L(p;k) is not unique since there are clearly Lorentz transformations that leave p invariant–these “little group” transformations will figure prominently in what follows, for now we simply emphasize that we pick some specific L(p;k) for which p=L(p;k)k. We also assume that we have a unitary representation of the Lorentz group, i.e. for every Lorentz transformation \Lambda there is an associated unitary operator U(\Lambda) acting on the Hilbert space, such that U(\Lambda_{1}\Lambda_{2})=U(\Lambda_{1})U(\Lambda_{2}). Then we simply define one-particle states |p,\sigma\rangle as

 |p,\sigma\rangle\equiv U(L(p;k))|k,\sigma\rangle\,. (1)

Note that the \sigma index is the same on the left and the right, this is the sense in which we are defining |p,\sigma\rangle. Having made this definition, we can ask how |p,\sigma\rangle transforms under a general Lorentz transformation

 U(\Lambda)|p,\sigma\rangle=U(\Lambda)U(L(p;k))|k,\sigma\rangle=U(L(\Lambda p;k% ))U(L^{-1}(\Lambda p;k)\Lambda L(p;k))|k,\sigma\rangle\,. (2)

Now, W(\Lambda,p,k)=L^{-1}(\Lambda p;k)\Lambda L(p;k) is not in general a trivial Lorentz transformation, it is only a transformation that leave k invariant since clearly (Wk)=k. This subgroup of the Lorentz group is the “little group”. Thus, we must have that

 U(W(\Lambda,p;k))|k,\sigma\rangle=D_{\sigma\sigma^{\prime}}(W(\Lambda,p;k))|k,% \sigma^{\prime}\rangle\,, (3)

where D_{\sigma\sigma^{\prime}}(W) is a representation of the little group. We have therefore found the desired transformation property

 U(\Lambda)|p,\sigma\rangle=D_{\sigma\sigma^{\prime}}(W(\Lambda,p;k))|\Lambda p% ,\sigma^{\prime}\rangle\,. (4)

We conclude that a particle is labeled by its momentum and transforms under some representation of the little group.

Scattering amplitudes for n particles are thus labeled by (p_{a},\sigma_{a}) for a=1,\cdots,n. The Poincare invariance of the S-matrix –translation and Lorentz invariance–then tells us that

 \displaystyle{\cal M}(p_{a},\sigma_{a}) \displaystyle= \displaystyle\delta^{D}(p_{a_{1}}^{\mu}+\cdots p_{a_{n}}^{\mu})M(p_{a},\sigma_% {a}) \displaystyle M^{\Lambda}(p_{a},\sigma_{a}) \displaystyle= \displaystyle\prod_{a}\left(D_{\sigma_{a}\sigma_{a}^{\prime}}(W)\right)M((% \Lambda p)_{a},\sigma_{a}^{\prime})\,. (5)

In D spacetime dimensions, the little group for massive particles is SO(D{-}1). For massless particles the little group is the the group of Euclidean symmetries in (D{-}2) dimensions, which is SO(D{-}2) augmented by (D{-}2) translations. Finite-dimensional representations require choosing all states to have vanishing eigenvalues under these translations, and hence the little group is just SO(D{-}2).

So much for the basic kinematics of particle scattering amplitudes. It is when we come to dynamics, and in particular to the crucial question of guaranteeing that the physics of particle interactions is compatible with the most minimal notion of locality encoded in the principle of cluster decomposition, that a fateful decision is made to choose a particular description of particle scattering, introducing the idea of quantum fields. Beyond particles of spin zero (and their associated scalar fields), there is a basic kinematical awkwardness associated with introducing fields: fields are manifestly “off-shell”, and transform as Lorentz tensors (or spinors), while particle states transform instead under the little group. The objects we compute directly with Feynman diagrams in quantum field theory, which are Lorentz tensors, have the wrong transformation properties to be called “amplitudes”. This is why we introduce the idea of “polarisation vectors”, that are meant to transform as bi-fundamentals under the Lorentz and little group, to convert “Feynman amplitudes” to the actual “scattering amplitudes”. For instance in the case of spin 1 particles, we introduce \epsilon_{\sigma}^{\mu}(p), with the property that \epsilon_{\sigma}^{\mu}(\Lambda p)=\Lambda^{\mu}_{\nu}\epsilon^{\nu}_{\sigma^{% \prime}}(p)D_{\sigma\sigma^{\prime}}(W), so that \epsilon^{\mu}_{\sigma}(p)M_{\mu}(p,\cdots) transforms properly. For massive particles, such polarization vectors certainly exist, though they have to satisfy constraints. For instance we must have p_{\mu}\epsilon^{\mu}_{\sigma}=0 for massive spin 1, or for massive spin 1/2, we use a Dirac spinor \Psi_{\sigma}^{A} with (\Gamma^{\mu}p_{\mu}-m)_{B}^{A}\Psi^{B}=0. These constraints are an artifact of using fields as auxiliary objects to describe the interactions of the more fundamental particles. For massless particles with spin \geq 1 the situation is worse, since “polarisation vectors” transforming as bi-fundamentals under the Lorentz and little groups don’t exist. Say for massless particles in four dimensions, if we make some choice for the \epsilon^{\mu}_{\pm} for photons of helicity \pm 1, we find that for Lorentz transformations (\Lambda p)=p, (\Lambda\epsilon_{\pm})^{\mu}=e^{\pm i\theta}\epsilon_{\pm}^{\mu}+\alpha(% \Lambda,p)p^{\mu}. So polarisation vectors don’t genuinely transform as vectors under Lorentz transformations, only the “gauge equivalence class” \{\epsilon_{\pm}^{\mu}|\epsilon_{\pm}^{\mu}+\alpha p^{\mu}\} is invariant under Lorentz transformations. This infinite redundancy is hard-wired into the usual field-theoretic description of scattering amplitudes for gauge bosons and gravitons, and is largely responsible for the apparent enormous complexity of amplitudes in these theories, obscuring the remarkable simplicity and hidden infinite-dimensional symmetries actually found in the physics.

The modern on-shell approach to scattering amplitudes departs from the conventional approach to field theory already at this early kinematical stage, by directly working with objects that transform properly under the little group (and so at least kinematically deserve to be called “scattering amplitudes”) from the get-go. Auxiliary objects such as “quantum fields” are never introduced and no polarization vectors are needed. It is maximally easy to do this in the D=4 spacetime dimensions of our world, where the kinematics is as simple as possible. Here the little groups are SO(2)=U(1) for massless particles, and SO(3)=SU(2) for massive particles, which are the simplest and most familiar Lie groups.

In four dimensions, we label massless particles by their helicity h. Massive particles transform as some spin S representation of SU(2). The conventional way of labelling spin states familiar from introductory quantum mechanics is by picking a spin axis \hat{z}. and giving the eigenvalue of J_{z} in that direction. This is inconvenient for our purposes, since the introduction of the reference direction \hat{z} breaks manifest rotational (not to speak of Lorentz) invariance. We will find it more convenient instead to label states of spin S as a symmetric tensor of SU(2) with rank 2S; this entirely elementary group theory is reviewed in appendix LABEL:SU(2)App. Let’s illustrate the labelling of states by considering a four-particle amplitudes where particles 1,2 are massive with spin 1/2 and 2, and particles 3,4 are massless with helicities +3/2 and -1. This would be represented as an object

 M^{\{I_{1}\},\{J_{1},J_{2},J_{3},J_{4}\},\{+\frac{3}{2}\},\{-1\}}(p_{1},p_{2},% p_{3},p_{4}) (6)

where \{I_{1}\},\{J_{i}\} are the little group indices of particle 1 and 2 respectively, and the amplitude transforms as

 M^{\{I_{1}\},\{J_{1},J_{2},J_{3},J_{4}\},\{+\frac{3}{2}\},\{-1\}}\to(W_{1K_{1}% }^{I_{1}})(W_{2L_{1}}^{J_{1}}\cdots W_{2L_{4}}^{J_{4}})(w_{3})^{3}(w_{4})^{-2}% M^{\{K_{1}\},\{L_{1},L_{2},L_{3},L_{4}\},\{+\frac{3}{2}\},\{-1\}} (7)

where the W matrices are SU(2) transformation in the spin 1/2 representation and w=e^{i\theta} is the massless little group phase factor for helicity +1/2.

### 2.1 Massless and Massive Spinor-Helicity Variables

Our next item of business is to find variables for the kinematics that hardwires these little group transformation laws, this will be simultaneously associated with convenient representations of the on-shell momenta. As usual we will use the \sigma^{\mu}_{\alpha\dot{\alpha}} matrices to convert between four-momenta p^{\mu} and the 2\times 2 matrix p_{\alpha\dot{\alpha}}=p_{\mu}\sigma^{\mu}_{\alpha\dot{\alpha}}. Note that detp_{\alpha\dot{\alpha}}=m^{2}, so that there is an obvious difference between massless and massive particles.

For massless particles, we have detp_{\alpha\dot{\alpha}}=0 and thus the matrix p_{\alpha\dot{\alpha}} has rank 1. Thus we can write it as the direct product of two, 2-vectors \lambda,\tilde{\lambda} as SpinorHelicity

 p_{\alpha\dot{\alpha}}=\lambda_{\alpha}\tilde{\lambda}_{\dot{\alpha}} (8)

For general complex momenta the \lambda_{\alpha},\tilde{\lambda}_{\dot{\alpha}} are independent two-dimensional complex vectors. For real momenta in Minkowski space p_{\alpha\dot{\alpha}} is Hermitian and so we have \tilde{\lambda}_{\dot{\alpha}}=\pm(\lambda_{\alpha})^{*}, (with the sign determined by whether the energy is taken to be positive or negative).

Often the introduction of these “spinor-helicity” variables is motivated by the desire to explicitly represent the (on-shell constrained) four-momentum p_{\alpha\dot{\alpha}} by the unconstrained \lambda_{\alpha},\tilde{\lambda}_{\dot{\alpha}}. But the spinor-helicity variables also have another conceptually important role to play: they are the objects that transform nicely under both the Lorentz and Little groups. Thus while amplitudes for massless particles are not functions of momenta and polarization vectors (or better yet, are only redundantly represented in this way), they are directly functions of spinor-helicity variables.

The relation to the little group is clearly suggested by the fact that it is impossible to uniquely associate a pair \lambda_{\alpha},\tilde{\lambda}_{\dot{\alpha}} with some p_{\alpha\dot{\alpha}}, since we can always rescale \lambda_{\alpha}\to w^{-1}\lambda_{\alpha},\tilde{\lambda}_{\dot{\alpha}}\to w% \tilde{\lambda}_{\dot{\alpha}} keeping p_{\alpha\dot{\alpha}} invariant. The connection can be made completely explicit by attempting to give some specific prescription for picking \lambda_{\alpha}^{(p)},\tilde{\lambda}^{(p)}_{\dot{\alpha}}, which leads us through an exercise completely parallel to our discussion of the little group. We first choose some reference massless momentum k_{\alpha\dot{\alpha}} and also choose some fixed \lambda_{\alpha}^{(k)},\tilde{\lambda}_{\dot{\alpha}}^{(k)} so that k_{\alpha\dot{\alpha}}=\lambda^{(k)}_{\alpha}\tilde{\lambda}^{(k)}_{\dot{% \alpha}}. For every other null momentum, we choose a Lorentz transformation {\cal L}(p;k)_{\alpha}^{\beta},\tilde{{\cal L}}(p;k)_{\dot{\alpha}}^{\dot{% \beta}} such that p_{\alpha\dot{\alpha}}={\cal L}(p;k)_{\alpha}^{\beta}\tilde{{\cal L}}(p;k)_{% \dot{\alpha}}^{\dot{\beta}}k_{\beta\dot{\beta}}, and we then define \lambda_{\alpha}^{(p)}\equiv{\cal L}(p;k)_{\alpha}^{\beta}\lambda^{(k)}_{\beta% },\tilde{\lambda}^{(p)}_{\dot{\alpha}}\equiv\tilde{{\cal L}}(p;k)_{\dot{\alpha% }}^{\dot{\beta}}\tilde{\lambda}^{(k)}_{\dot{\beta}}. Having now picked a way of associating some \lambda_{\alpha}^{(p)},\tilde{\lambda}^{(p)}_{\dot{\alpha}} with p_{\alpha\dot{\alpha}}, we can ask for the relationship between e.g. \lambda_{\alpha}^{(\Lambda p)} and \lambda^{(p)}_{\alpha} for some Lorentz transformation \Lambda; what we find is

 \lambda_{\alpha}^{(\Lambda p)}=w^{-1}(\Lambda,p,k)\,\Lambda_{\alpha}^{\beta}% \lambda_{\beta}^{(p)} (9)

For general complex momenta w is simply a complex number and we have the action of GL(1), for real Lorentzian momenta we must have w^{-1}=\pm(w)^{*} so w=e^{i\theta} is a phase representing the U(1) little group. Most obviously we can perform a Lorentz transformation W for which Wk=k, we simply find \lambda\to w^{-1}\lambda. To be explicit, let

 k_{\alpha\dot{\alpha}}=\left(\begin{array}[]{cc}2E&0\\ 0&0\cr\omit\span\@@LTX@noalign{ }\omit\\ \end{array}
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