On-Shell Electroweak Sector and the Higgs Mechanism

We take the first steps towards an entirely on-shell description of the bosonic electroweak sector of the Standard Model. We write down on-shell three particle amplitudes consistent with Poincare' invariance and little group covariance. Tree-level, four particle amplitudes are determined by demanding consistent factorization on all poles and correct UV behaviour. We present expressions for these $2\rightarrow 2$ scattering amplitudes using massive spinor helicity variables. We show that on-shell consistency conditions suffice to derive relations between the masses of the $W^\pm, Z$, the Weinberg angle and the couplings. This provides a completely on-shell description of the Higgs mechanism without any reference to the vacuum expectation value of the Higgs field.

B Amplitudes with one massless particle and 2 equal mass particles 26 1 Introduction Quantum fields, path integrals and Lagrangians have been a cornerstone of 20th century theoretical physics. They have been used to describe a variety of natural phenomena accurately. Yet, it is becoming increasingly apparent that these mathematical tools are both inefficient and insufficient. They obscure the presence of a deeper, underlying structure, particularly of scattering amplitudes in quantum field theories. The field of scattering amplitudes has undergone a paradigm shift in the past three decades. This was sparked by the discovery of the stunning simplicity of the tree-level gluon scattering amplitudes in [1,2]. The simplicity of these amplitudes was revealed due to the use of helicity spinors, (λ α ,λα) which correspond to the physical degrees of freedom of massless particles -helicity. The forbidding complexity of the Feynman diagram based calculation of tree level gluon scattering amplitudes is now understood to be an artefact of the unphysical degrees of freedom introduced by gauge redundancy. These unphysical degrees of freedom are necessary to package the physical degrees of freedom into local quantum fields in a manner consistent with Poincare' invariance [3]. The simplicity of these amplitudes fueled the development of a variety of "on-shell" techniques for computing scattering amplitudes involving massless particles. These methods do not rely on Feynman diagrams, do not suffer from gauge redundancies and do not invoke virtual particles. For an overview of these methods, see [4][5][6][7][8] and the references therein. However, most of this progress was limited to amplitudes involving only massless particles.
Since helicity spinors correspond to the physical degrees of freedom of massless particles, it is natural to attempt to find variables akin to these for massive particles. The physical degrees of freedom of massive particles correspond to the little group SU (2) [9]. Some early generalizations can be found in [10][11][12][13][14][15][16][17]. However, the little group covariance was not manifest in these generalizations until the introduction of Spin-spinors (or massive spinorhelicity variables) in [18]. These variables (λ I α ,λ Iα ) which carry both little group indices and Lorentz indices and make the little group structure of amplitudes manifest. Information about all the (2S +1) spin components of each particle is packaged into compact, manifestly Lorentz invariant expressions. Amplitudes written in terms of these variables are directly relevant to physics. This is in contrast to a Feynman diagram based computation which involves an intermediate object with Lorentz indices which must then be contracted with polarization tensors which carry the little group indices. For some interesting applications of these variables, ranging from black holes to supersymmetric theories see [19][20][21][22][23][24].
One of the biggest successes of path integrals and the Lagrangian formulation is the development of effective field theory and the Higgs mechanism. Recently, efforts have been made towards the development of effective field theory using on-shell methods [25][26][27][28][29][30]. A completely on-shell description of the Higgs mechanism was outlined in [18] for the abelian and non-abelian gauge theories. The conventional understanding of the Higgs mechanism involves a scalar field acquiring a vacuum expectation value and vector bosons becoming massive by "eating" the goldstone modes arising from spontaneously broken symmetry. However, the on-shell description has no mention of scalar fields, potentials and vacuum expectation values. Nevertheless, it reproduces all of the same physics. Additionally, well known results like the Goldstone Boson equivalence theorem become trivial consequences of the high energy limits of our expressions. From an amplitudes perspective, it is more natural to think of the Higgs mechanism as a unification of the massless amplitudes in the UV into massive amplitudes in the IR. In this paper, we will focus on computing scatter-ing amplitudes in the bosonic electroweak sector of the standard model and describing the Higgs mechanism and electroweak symmetry breaking using a completely on-shell language.
The paper is structured as follows. We begin with a brief review of the little group, spinspinors and their properties in Section [2]. We focus on constructing three particle amplitudes in the IR in Section [3.1] and the UV in Section [3.2]. In Section [3.4], we compute the high energy limits of the three particle amplitudes in the IR and demand that they are consistent with the three point amplitudes in the UV. This gives us the all the standard relations between the coupling constants, the masses of the Z and W ± and the Weinberg angle θ w . We also see the emergence of the custodial SO(3) symmetry in the limit in which the hypercharge coupling vanishes. Finally, in Section [4], we construct 4 point amplitudes in the IR by gluing together the three point amplitudes found before. We enunciate the details involved in the gluing process. We will also discover that demanding that these amplitudes have a well defined high energy limit imposes constraints on the structure of the theory.
2 Scattering amplitudes and the little group

Helicity spinors and spin-spinors
In this section, we briefly review some aspects of the on-shell approach to constructing scattering amplitudes. We will review the formalism of spin-spinors introduced in [18] whilst highlighting some features important for this paper. One particle states are irreducible representations of the Poincare' group. They are labeled by their momentum and a representation of the little group. If the particle is charged under any global symmetry group, appropriate labels must be appended to these. In (3+1) spacetime dimensions, the little groups for massless and massive particles are SO(2) and SO(3) respectively.
Representations of the massless little group, SO(2) = U (1) can be specified by an integer corresponding to the helicity of the massless particle. A massless one particle state is thus specified by its momentum and helicity. Under a Lorentz transformation Λ, where σ are labels of any global symmetry group and w has the same meaning as in [3] and [18]. It is useful to introduce elementary objects λ α ,λα which transform under the little group as We can use these objects to build representations with any value of h. The natural candidates for these elementary objects are the spinors which decompose the null momentum p αα ≡ p µ σ µ αα . We have Throughout the paper we will find it convenient to make use of the following notation, For any two null momenta p 1 , p 2 , we can form two Lorentz invariant combinations of these spinors, The massive little group is SO(3) = SU (2). It representations are well known and can be specified by the value of the Casimir operator which is restricted to values S(S + 1), where S is defined as the spin of the particle. The spin S representation is 2S + 1 dimensional. A massive one-particle state of spin S thus transforms as a tensor of rank 2S under SU (2).
The elementary objects in this case are the spinors of SU (2). These transform as Higher representations can be built by taking tensor products of these. A decomposing the rank 2 momentum, similar to eq.(2.3) yields the requisite spinors.
Note that we have det(p) = det(λ) det(λ) = m 2 . For the rest of the paper, we will set det(λ) = det(λ) = m 1 . We will find it convenient to suppress the little group indices on the spin-spinors. We do this according to the convention in eq. (A.2). Finally, we can construct Lorentz invariants out of spin-spinors corresponding to two massive momenta p 1 , p 2 similar to eq.(2.4).
As an example, we display the transformation law for a 4-particle amplitude where particle 1 is massive with spin 1, particle 2 is massless with helicity 5/2, particle 3 is massless with helicity −2 and particle 4 is massive with spin 0. Using these, one can show that the three point amplitudes can only take the following form.
In cases involving one or more massive particles, Lorentz invariance and little group covariance are not sufficient to completely fix the amplitude. However, they narrow down the form of the amplitude to a finite number of terms. For an exhaustive analysis, we refer the reader to [18] and [31,32]. In this paper, we will discuss only the amplitudes relevant to us.

The high energy limit of spin-spinors
When particle energies are much higher than their masses, it is intuitive to treat them as massless. We can formalize this by expanding the spin-spinors in a convenient basis in little group space.
where λ,λ are the helicity spinors, ζ ±I are eigenstates of spin 1/2 along the momentum. We give explicit expressions for all objects involved are in Appendix [A]. Here, we just note that η α ,ηα ∝ √ E − m = m + O(m 2 ). Taking the high energy limit corresponds to taking m/E → 0. In this limit, the spin-spinors reduce to massless helicity ones. Finally, it should be pointed out that we must take special care while taking the high energy limit of 3-point amplitudes. Owing to the special three point kinematics, factors like 12 or [12] can tend to zero in the high energy limit.
3 Three particle amplitudes 3-particle amplitudes the fundamental building blocks of scattering amplitudes. In this paper, we are interested in analyzing the bosonic content of the standard model both in the UV and IR. The spectrum in the UV is comprised solely of massless particles. The three point amplitudes are completely determined by Poincare' invariance and little group scaling as outlined in Section [2]. The form of these amplitudes was given in eq.(2.12). We will use this formula to write down all the relevant 3-particle amplitudes in Section [3.2]. All the amplitudes in the UV obey the SU (2) L × U (1) Y symmetry.
The spectrum in the IR consists of massive particles and a single massless vector. 3-particle amplitudes involving massive particles aren't completely fixed. They can have several contributing structures. In Section [3.1], we will write down all the relevant amplitudes. The amplitudes in the IR obey only a U (1) EM symmetry.
Finally, we will demand that the high energy limit of the IR amplitudes is consistent with the amplitudes in the UV. We will find that this consistency is possible only if the masses of particles in the IR are related in a specific way. These turn out to be the usual relations involving the Weinberg angle.

The IR
The spectrum in the IR consists of the following particles.
• Three massive spin 1 bosons (W + , W − , Z) which have masses (m W , m W , m Z ) and charges (+, −, 0) respectively under a global symmetry group U (1) EM . Note that W + and W − are eigenstates of the U (1) EM generator. They must have equal mass as they are related by charge conjugation.
• One massless spin 1 boson, the photon, γ which is not charged under the U (1) EM .
• One massive scalar, the higgs, h which is also uncharged under U (1) EM .
The only symmetry of the IR is the U (1) EM . We will now discuss all the relevant three point amplitudes in the IR. Owing to the existence of various identities amongst the spin-spinors, each amplitude can be written in a multitude of different ways. In many of the cases below, we have chosen particularly convenient ways of writing them. Different form three point amplitudes lead to different expressions for four point amplitudes. The difference between these are contact terms that can be fixed by imposing other constraints on the amplitude. While the form of the contact term will depend on the form of the three point amplitudes used, the final amplitude will be the same. We will elaborate on these comments in the appropriate places below.
This is a form of the three point amplitude that is chosen to suit our needs. It should be noted that it can be reduced to a combination of < > and [ ]. As an example, consider the first term in the above equation which can be re-written as follows. 12 where we made use of the Schöuten identity 12 3 + 23 1 + 31 2 = 0.
We discuss other forms of writing the same vertex in Appendix [B].
We will set N 1 = N 2 = 0 in what follows as yield four point amplitudes which grow as E 2 where E is the center-of-mass energy.

The UV
The UV spectrum of the electroweak sector of the standard model consists of the following • One massless spin-1 particle B with charge 1 2 under a global U Y (1) symmetry group. • Three massless spin-1 particles (W 1 , W 2 , W 3 ), in the adjoint representation of SU(2) L .
These are not charged under the group U (1) Y . In order to facilitate easy comparison to the massive particles in the IR, we will work with particle states W ± = 1 √ 2 W 1 ± iW 2 which are eigenstates of the U (1) EM symmetry in the IR. The electroweak sector has an SU (2) L × U (1) Y symmetry. The generators of these symmetries are related to the generators of SO(4) listed in Appendix [D] as follows.
The generator of U (1) EM , which we denote by Q can be written as a linear combination of the generators of the UV.
where e is U (1) EM coupling. Since T ± = 1 √ 2 T 1 ± iT 2 are eigenstates of Q, we are free to work with the states W ± in the UV. We will now list all the relevant amplitudes in the UV. The superscripts on the particles indicate the corresponding helicities. .
The above list doesn't contain any amplitudes which involve W s and B since the W 's are not charged under the U (1) Y . Note that all the above amplitudes involve particles whose helicities, h 1 , h 2 , h 3 are such that h i > 0. The amplitudes with h i < 0 are given by flipping < >→ [ ].

The HE Limit of the IR
All the amplitudes in the IR listed above have one or more factors of 1 m . At first glance, this seems to suggest that they blow up in the UV and cannot be matched onto any 3-particle amplitude of massless particles. However, we will see that all these factors of inverse mass drop out when we take the special 3 particle kinematics into account and carefully take the high energy limit. Many of these high energy limits are worked out in [18] and [31]. We present them here in a form compatible with our conventions. For each massive leg, in order to take the high energy limit we must first specify the component which we are interested in.
Amplitudes with one longitudinal mode and two transverse vanish in the high energy limit. 23 31 12 (3.13) where X = W, Z. Amplitudes involving only one transverse mode vanish in the high energy limit.

UV-IR consistency
Thus far, we have specified the structure of the IR which consists of the interactions among the W ± , Z, γ and h which preserve the U (1) EM symmetry and the structure of the UV which consists of the interactions among the W a , B, Φ which preserve the SU (2) L × U (1) Y symmetry. We must now ensure that they are compatible with each other. We take the high energy limit of the IR amplitudes and demand that they are equal to the appropriate amplitudes in the UV. We refer to this process as 'UV-IR matching'. This imposes many constraints and determines the couplings in the IR in terms of those in the UV. Furthermore, it also imposes constraints on the masses of the particles in the IR. To begin with, we must relate the degrees of freedom in the IR to the ones in the UV. We assume that they are related by the following orthogonal transformation This is a result of working with the same states in the UV and IR. Orthogonality demands that the matrix be block diagonal, and so we have the simpler relation for some unknown angle θ w . All the massive particles in the IR have longitudinal components which must be generated by some some linear combination of the scalars in the UV. We assume that The remaining linear combination of the components of Φ, h = U hi Φ i has an independent existence. Indeed, it is well known that its presence is crucial for the theory to have a good UV behaviour. The high energy limit of each of the three point amplitudes in the IR must be equal to some combination of the amplitudes in the UV. This determines the masses in the IR in terms of the couplings in the UV. It also imposes some constraints on the couplings in the UV. All the constraints arising from eq.(3.12) -eq.(3.14) are determined below.
There are a total of 27 components to the W + W − Z amplitude corresponding to the (+, −, 0) spin component of each particle. Amplitudes with just one longitudinal mode all vanish in the high energy limit. This is consistent with the fact that there are no W W Φ, W BΦ, BBΦ amplitudes in the UV. The independent constraints arising from the remaining components are given below. Recall that the superscript on the particle is its helicity. These are also listed at the top of each diagram.
(+ + −) Using the expressions from eq.(3.12) and eq.(3.7), we get The absence of a W + W − B interaction in the UV means that there is no term proportional to O ZB on the RHS.] Using eq.(3.12), eq.(3.10) and eq.(3.11) in the above gives, Again, eq.(3.12) and eq. (3.8) give Since the photon is massless in the IR, the W + W − γ amplitude only has 18 components. This leads to the following constraints.
(+ + −) Conservation of the U (1) EM charge in the IR must be imposed. This is achieved by setting the W + Zγ amplitude to zero. A similar equation is given by setting the W − Zγ amplitude to zero.
This set of equations can be solved by the ansatz Note that despite the similarity of this equation with the usual Lagrangian based description of the Higgs mechanism, V does not have the interpretation as the vacuum expectation value of scalar field here. The solution is We get the exact solutions as the Standard Model because we have restricted the form of the three point amplitude in eq.(3.1). Allowing for other structures will generalize the relation between m Z and m W . Further note that when g → 0, we have θ w = 0 and m Z = m W . Here, we see the emergence of the custodial SU (2) = SO (3). The three particles W ± , Z all have equal mass in the limit where the hypercharge coupling vanishes.

Four point amplitudes in the Electroweak sector
As we explained in the previous section, the structure of three point amplitudes is is severely restricted by Poincare' invariance and little group constraints. The construction of four point amplitudes from the three point ones requires more work. Translation invariance is assured by the delta function in eq.(2.8) and Lorentz invariance is guaranteed if we build the amplitude from the invariants in eq.(2.4) and eq.(2.7). These amplitudes must be little group tensors of the appropriate rank (or in the case of massless particles have appropriate little group weights). This still leaves open a multitude of possibilities. But beyond three points, we have new constraints arising from unitarity. The amplitude must factorize consistently on all the poles, i.e. when some subset of the external momenta goes on shell, the residue on the corresponding pole must factorize into the product of appropriate lower point amplitudes. In particular, if the exchanged particle is massless, we must have Here and below, a is an index for the intermediate particle. For the rest of this section, we will work with four particle amplitudes with particles 1 and 2 incoming and 3 and 4 outgoing. Diagrammatically, At four points, there are only three possible factorization channels defined by We must ensure that the four point amplitude factorizes into appropriate three point amplitudes on all these channels. We do this by computing the residues in the s, t and u channels and where m s , m t , m u are the masses of the particles exchanged in the s, t, u channels respectively. This procedure will yield local amplitudes for almost all cases. Only in the case of the W + W − γ amplitude, which has one massless particle and two particles of equal mass, this yields a four point amplitude with x factors which must be eliminated to get a local expression. We will go into more details in the corresponding section.
This represents only the factorizable part of the four point amplitude. We will find that these need to be supplemented by contact terms which depend on the specific form of the three point vertices. We can determine these by specifying the UV behaviour of the four point amplitudes. For the case of the Standard Model, we demand that they do not have any terms which grow with energy. This lets us determine the required contact terms. The complete four point amplitude is then written as where P is a Lorentz invariant polynomial in the spin spinors corresponding to the four particles with the appropriate number of little group indices.
In this section, we analyze the scattering of W + W − → W + W − . For the sake of explicit calculations, we make the following choice for the 4 particle kinematics (with particles 1, 2, 3 and 4 corresponding to W − , W + , W + , W − respectively. We will see that, based on the three point amplitudes listed in Section [3.1], the scattering can occur in the s and t channels via the exchange the Z, A or h. p2 p1 p3 p4 We can glue together two W + W − Z three point amplitudes and construct the residue in the s -channel.
Here I = p 1 + p 2 is the momentum exchanged and we have suppressed the little group indices corresponding to the external particles. The residue on the s -channel is Evaluating this expression yields The extra minus sign that accompaniesλ Iα in the equations defining x ± 34 is because the momentum I is incoming. The residue corresponding to the photon exchange is a sum over both the possibilities in eq.(4.12).
We must now eliminate the x -factors in order to obtain a local expression for this residue.
There are multiple ways to achieve this and they generally result in different expressions for the residue. It is important to emphasize that while these forms are precisely equal on the factorization channel, they all lead to different expressions away from the pole. Since the physical amplitude must be the same, they yield different contact terms. The complete details of the calculation are delegated to Appendix [C]. Here, we present two different expressions for the residue on the s−channel. (4.12) This expression can be manipulated to look identical to eq.(4.9). This requires the use of the following Schöuten identities and where I = p 1 + p 2 . These identities are true only on the factorization channel I 2 = 0 on which we can write R γ s = R Z s (m Z = 0). This is not true away from the factorization channel. Consequently the contact terms that must be added to achieve the correct UV behaviour differ. This explicitly demonstrates the dependence of contact terms on the specific form of the three point amplitudes.
• Higgs exchange This is the simplest to compute. We just glue together the following amplitudes. (4.14) The complete contribution of the s− channel is The computation of the t -channel residues is very similar to that of the s -channel. In fact, we can obtain them from the s− channel ones by the replacement p 2 ↔ −p 4 . The results are presented below with I = p 1 − p 4 .
• Z exchange • Photon exchange The residue on the t− channel resulting from gluing together two W + W − γ amplitudes is [32] (4.16) • Higgs exchange The total contribution from the t−channel is

Contact terms
The quantity M ≡ M s + M t has been constructed to have the correct factorization properties. As explained before, the behaviour away from the factorization channels depends on the specific forms of the three point amplitudes. We can impose further constraints on the amplitude to fix it completely. It is evident that the high energy limit of the amplitude is ill defined due to the presence of the 1 m 4 W poles which leads to amplitudes which grow with energy as E 4 . This violates perturbative unitarity. If we insist that the theory has a well defined high energy limit, we must add contact terms (which by definition have 0 residue on the factorization poles) to cancel this E 4 growth. The form of the contact terms can be deduced by figuring out which components of the amplitude grow in the UV. Plugging in the 4-particle kinematics in 4.5, we find that only the all longitudinal component grows as The following contact term serves to kill these high energy growths Adding these contact terms, we find that the amplitude still grows as E 2 /m 2 W . Demanding that the coefficient of this growing term vanishes enforces e 2 W W H = e 2 + e 2 W .

W
The 4 particle kinematics appropriate to this situation is This configuration automatically satisfies momentum conservation. We can rewrite E 2 in terms of E 1 by using the on-shell constraint as E 2 = E 2 1 − m 2 W + m 2 Z . We can build this amplitude by gluing together two W + W − Z amplitudes in two ways and by gluing two W + W − h amplitudes.
We present the final expressions below. The calculations are very similar to those involved in  • Contact terms We are again in the familiar situation where the quantity factorizes correctly on all the factorization channels. However, the all longitudinal component again grows with energy as can be seen by evaluating this using the kinematics in eq. (4.20). We find that the following contact term is needed to fix this and have a well behaved theory in the UV,
We can build this amplitude by gluing together (W + W − Z, Zhh) on the s−channel and by gluing together (W + W − h, ZZh) in the u and t channels as shown Using the familiar procedure, we get To compute this amplitude, we can glue together two W + W − h amplitudes in the t and u channels.

Conclusions and Outlook
We have presented a completely on-shell description of the higgs mechanism within the Standard Model. We see that all the physics is reproduced by demanding consistent factorization, correct ultraviolet behaviour and consistency of the UV and IR. The precise relations between the masses of the W ± , Z and θ w depend on the structures that have been included in the three point W + W − Z amplitude. Our choice of the three point amplitude in eq.(3.1) ensured that we reproduced the usual result. We have constructed four particle, tree-level amplitudes from three particle amplitudes. The construction of higher point amplitudes and extensions to loop amplitudes are the obvious next questions. We have also restricted the particle content of the scalar sector to a single, real scalar transforming under an SO(4) global symmetry. We have studied the Higgs mechanism for SU (2) L × U (1) Y breaking to U (1) EM , relevant to electroweak symmetry breaking. It would be interesting to extend this analysis to completely general theories.
This work is a preliminary step in connecting modern methods in scattering amplitudes to the real world. There have been many developments in new ways of thinking about scattering amplitudes. It has proved useful to think of them as differential form on kinematic space [33]. These differential forms are associated to geometric structures in many cases. The physics of scattering amplitudes emerges from simple properties of the underlying geometry as seen in the few known cases [34][35][36][37][38]. It would therefore be useful to rewrite amplitudes in the Standard Model as differential forms. This would lay the groundwork for an attempt to look for hidden geometric structure within these amplitudes. correct helicity weight for particle three and is invariant under the little groups for particles 1 and 2. Unfortunately, the obvious candidates vanish, The x-factors defined in [18] solve this problem. In our paper, we adopt a slightly different definition and notation which we explain below. For all outgoing momenta, we define Under little group scaling of particle 3, the helicity spinors scale as λ 3 → t −1 λ 3 and λ 3 → tλ 3 . It follows that x + 12 → t −2 x + 12 and x − 12 → t 2 x − 12 . An object with helicity h transforms as t −2h under a little group scaling. This justifies the ± signs on the x-factors.
We can obtain explicit expressions for the x-factors by contracting eq.(B.2) with reference spinors ξ α orξα.
These are the same as the conventional expressions for polarization vectors of massless particles upto a factor of 1 m . It is crucial that the x-factors are independent of the reference spinor. To see this, consider two different definitions of x + 12 with reference spinors ξ 1 and ξ 2 . Their difference, where the first equality follows from a Schöuten identity and the second from eq.(B.1).
We can build three point amplitudes using the x-factors. Here, we will focus on the amplitude involving two spin 1 particles of mass m and a massless particle of helicity ±1. We pick our amplitudes to be 12 2 x − 12 and [12] 2 x + 12 . This corresponds to minimal coupling. For more details about this and amplitudes corresponding to multipole moments, see [22]. We can also compare these with the vertices that we get from the usual Feynman rules (for a photon with positive helicity).

C Computation of 4-particle amplitudes
In Sections [4.1] and [4.2], we need to glue together two three point amplitudes to construct the four point amplitude. In cases in which the exchanged particle has spin 1, the following identities are useful. Note that the second equality can be obtained by using a property of the two dimensional Levi Civita tensor. I 1 J 1 I 2 J 2 + I 1 I 2 J 2 J 1 + I 1 J 2 J 1 I 2 = 0 (C.2) The following identities are useful in the computation of the 4 point amplitude in Section [4.  The combinations satisfy two copies of SU (2), i.e X + , X − = 2X 3 , X + , X 3 = X + , We will associate the generators X ± , X 3 with the symmetry SU (2) L . These are referred to as T ± , T 3 in the paper. The U (1) Y is a subgroup of the SU (2) formed by Y ± , Y 3 and we will set Y 3 ≡ T B .