1 Introduction

We study perturbative unitarity constraints on general models by considering the high energy behavior of fermion scattering into gauge bosons. In most cases we survey, a boson with a comparable mass must be present for the theory to be consistent, with fixed couplings to the standard model gauge bosons and fermions. Applying these results to a class of models which explains the top quark forward–backward asymmetry observed at the Tevatron, we find that a must exist with a mass below 78 TeV and sizable coupling to the light quarks. While such a is strongly constrained by existing experiments, we show that the LHC can explore the entire mass range up to the unitarity limit. We also show how it is possible, by raising the mass consistent with unitarity, to explain the CDF excess in terms of a light , without generating an excess in events.


November 21, 2011

Perturbative unitarity constraints on general models

and collider implications

K.S. Babu***Email: babu@okstate.edu,  J. JulioEmail: julio.julio@okstate.edu

Department of Physics and Oklahoma Center for High Energy Physics,

Oklahoma State University, Stillwater, OK 74078, USA

Yue ZhangEmail: yuezhang@ictp.it

Abdus Salam International Centre for Theoretical Physics,

Strada Costiera 11, Trieste 34014, Italy

1 Introduction

It is a well established fact that in a weakly coupled renormalizable theory, the high-energy behavior of scattering amplitudes should not violate unitarity under perturbative calculations [1, 2]. In the standard model (SM), an upper limit on the Higgs boson mass of about a TeV has been inferred based on unitarity in the gauge boson () scattering process [3, 4]. Prior to the discovery of the boson, it was shown, from the unitarity in the high energy scattering of various types that the coupling of the boson to the SM fermions must be unique [5]. Perturbative unitarity has also proved to be a useful guide for studying beyond the SM physics, often in a model–independent fashion.

A charged gauge bosons, , arises in a variety of extensions of the SM. It is present in theoretical frameworks such as left-right symmetric models – as the right–handed parity partner of the gauge boson [6], in Kaluza-Klein theories of extra dimensions – as the excitation of the  [7, 8], in little Higgs theories – as the gauge bosons of the extended symmetry [9], and in several other well-motivated extensions of the SM. With the running of the Large Hadron Collider (LHC), models have drawn renewed interest [10] as it is on the list of particles subject to direct searches. Furthermore, recently there have been new phenomenological motivations for having a light charged gauge boson. A boson with mass below a TeV and coupling to top and down quarks has been introduced as one of the leading explanations [11, 12, 13, 14] for the anomalous top-quark forward-backward asymmetry observed at the Tevaron [15, 16]. There are also proposals in the literature which utilize a light with a mass of order 150 GeV to explain the plus dijet event excess [17] observed by the CDF collaboration [18].

In this paper we study the theoretical and phenomenological constraints on models assuming that it is the remnant of spontaneous symmetry breaking through the Higgs mechanism. We take a model-independent approach in the sense that the mass of and its coupling to SM fermions are the only input information that will be used. We study the implication of perturbative unitarity on this setup from two body scattering of fermions into gauge boson final states. Throughout this paper, we consider weakly coupled theories, since we focus on the weak interaction sector where there is no evidence for strong dynamics. Violation of perturbative unitarity in low-energy effective theory does not necessarily imply inconsistency in a strongly coupled theory, but would rather indicate appearance of resonances or other such non-perturbative objects.

Our main conclusion is that generically a gauge boson should also be present along with the in a consistent theory. Unitarity of the theory fixes the couplings of the to SM gauge bosons and fermions, and implies an upper bound on its mass. We derive general formulas that relate the various couplings and masses and apply them to specific models with purely left-handed or purely right-handed couplings to the fermions. In the class of models that have been suggested for asymmetry, a light (sub-TeV) has been postulated with flavor-changing coupling of the type . We point out that the same coupling also leads to the scattering of a pair of fermions (e.g., ) into . Although such could evade all current constraints with the particular choice of quark flavors, the boson is subject to more severe direct search limits. Unitarity sets an upper bound on its mass which we find to be less than about 78 TeV. The LHC is capable of probing such a for most part of this mass window, as we show. We also note that unitarity restoration would imply significant destructive interference when each diagram contributing to a given process individually has a bad high energy behavior. Therefore, a large hierarchy between and cross sections can be achieved by consistently raising the mass of which unitarizes the process. This could be interesting for models explaining the CDF event excess. For completeness, we also classify realistic models that contain the SM gauge symmetry and has a charged gauge boson (in a more general sense) but no gauge boson, and show how unitarity is preserved in this class of models.

2 General framework

We will focus on the class of processes in which a pair of fermions scatter into a pair of gauge bosons, , and study its unitary properties at very high center-of-mass energy. As pointed out in Ref. [2], the asymptotic form of a four-particle amplitude should not grow with energy. The general process is shown in Fig. 1. In the following we will consider tree-level process only. At very high energies, in a spontaneously broken gauge theory, due to the Goldstone boson equivalence theorem, the contributions to the scattering amplitude is dominated by the longitudinal components of the final state gauge bosons.

At the tree–level, the processes above can happen through the following channels: a) t-channel fermion exchange, b) u-channel fermion exchange, c) s-channel gauge boson exchange, and d) s-channel scalar exchange. The corresponding topologies are shown in Fig. 2.

We define the relevant couplings which will be used in our analysis. The couplings between gauge boson with two fermions , can be written in the general form as


where are the chirality projection operators. The coupling between three gauge bosons , and can be written in the general gauge and Lorentz invariant form as




The coupling between a scalar field with fermions , can be written as


The coupling between a scalar field with two gauge bosons and is written as

Figure 1: A pair of fermions scattering into a pair of gauge bosons.
Figure 2: Different topologies of the scattering . All the internal particles must have masses not far above , otherwise their effects will decouple.

Perturbative unitarity constraints are twofold. First, the coherent sum of all relevant amplitudes should be finite when the energy scale goes to infinity. Second, in the partial wave expansion [19], each finite partial wave of the amplitude should satisfy , where


and where is the helicity difference between two initial state fermions. are the Wigner -functions [20]. The leading functions relevant for our study are


The first requirement implies sum rules among different couplings involved in the scattering process, while the second one leads to constraints on the masses of the states being exchanged in different channels.

According to the initial state fermion helicity, the scattering can be divided into two classes. In the first class, the helicity of initial state fermions is opposite to that of (or the same as ), which implies even number of fermion mass insertions in the process. We call this the helicity violating case, since the initial state has nonzero total helicity while the final state has . Such processes can involve t- and u-channels fermion exchange as well as s-channel gauge boson exchange. In this case, the amplitude can be expanded in terms of the center-of-mass energy squared as


The finiteness of implies and therefore a sum rule among the relevant couplings:


There is a similar sum rule with in the couplings in Eq. (9), which arises from . If these conditions are not satisfied, the partial wave amplitude would increase with energy, and one should expect new states to appear with the proper couplings around the scale with


When the rum rule Eq. (9) is satisfied, , and the theory can remain perturbative. The finite amplitude at very high energy can be written as


A similar expression for the finite amplitude holds for with the replacement in in the couplings in Eq. (11). Demanding the partial waves of to satisfy unitarity, one can obtain an upper bound on the masses of the various particles involved in the scattering process.

The singularities at in Eq. (11) are not physical. They simply reflect the breakdown of the expansion in term of in the t-channel amplitude. The integral in Eq. (6) can be performed to . Here can be taken infinitesimal because for any , one can always find sufficiently large such that the above expansion is valid. In fact, all of the partial wave amplitudes above are finite, so are the total cross sections. As we noted already, the total helicities of initial and final states differ by one unit. Therefore, in the massless fermion limit the scattering must happen in the p-wave, i.e., there is an additional angular factor which removes any singularity in the partial wave expansion.

The second class of scattering involves and () with the same (opposite) helicity. We call it the helicity conserving case because both initial and final states have vanishing total helicity. In this case, there must be odd number of mass insertions along the fermion line. In this helicity conserving case, the amplitude can be expanded in terms of “half-integer” powers of ,


The Feynman diagrams generally include t- and u-channels fermion exchange and s-channel scalar boson exchange and vanishing implies a sum rule


There is a similar one as Eq. (2) with arising from . One can also rewrite Eq. (2) in a simpler form by using Eq. (9),


If the sum rule Eq. (2) is not satisfied, , relevant new states must appear below the scale


When the rum rule Eq. (2) is satisfied, . All the remaining terms go to zero at infinite energy. Therefore, when discussing finite amplitudes at high energy below, we will only talk about the chirality preserving cases, which has the general asymptotic form given Eq. (11).

For practical purposes, we also present the sum rules Eqs. (9) and (2) in the limit of massless initial state fermions , . In this case, one does not have to distinguish between helicity and chirality eigenstates and the sum rules simplify to




Meanwhile, the finite amplitude Eq. (11) also simplifies to


Before closing this section, we illustrate how the above sum rules work in two explicit examples where unitarity has been well established.

2.1 Application to standard model

The first example is the SM. We consider amplitudes for massless quark-antiquark scattering into gauge bosons. The final gauge bosons include , and . The scattering of occurs through exchange in both t-, u-channels, and without chirality flip. Since all coupling in t- and u-channels are identical, the sum rule Eq. (16) is satisfied. The scattering can occur through t-channel exchange, u-channel exchange and s-channel exchange. The sum rule is realized since . Finally, the scattering occurs through t-channel exchange and s-channel exchange. The sum rule is realized as .

2.2 Application to seesaw models of neutrino mass

The second application we discuss is the seesaw models for neutrino mass. We will consider the lepton number violating process , which ought to be proportional to the Majorana mass [21]. Thus it belongs to the class of scatterings with chirality flip, i.e., it has the general form of Eq. (12). If neutrino Majorana mass is the only source of lepton number violation, the process can occur through t-channel or u-channel neutrino exchange, with a Majorana mass insertion. However, as shown in the sum rule Eq. (2), in this case the t- and u-channel amplitudes have the same sign, thus no cancellation occurs among the two. In this case, the asymptotic form of the scattering amplitude would increase with energy . Using Eq. (15), unitarity tells us high scale physics for neutrino mass generation must appear below the scale  [21]. In fact, gauge invariant Majorana neutrino masses in the minimal SM must arise from the Weinberg operator . With this operator, there is also an s-channel SM Higgs boson exchange contribution to the scattering, which restores unitarity.

Next, we show how unitarity is restored in three types of seesaw theories. In type I and type III seesaw models, neutrino masses arise from the mixing of the light neutrino with a heavy Majorana fermion . The mixing angle is . Because of this mixing, when scattering energy is higher than these heavy fermion mass, it also contributes to the above lepton number violating process. The corresponding amplitude is . The type I seesaw mass formula contains a minus sign which facilitates the cancellation with t- and u-channel contributions and restores unitarity. On the other hand, in the case of type II seesaw, neutrino mass arises from coupling of the light neutrino to the VEV of a complex Higgs triplet, . Due to gauge invariance, the above scattering can also happen through s-channel exchange of doubly-charged component from the Higgs triplet, , which also exactly cancels the t- and u-channel contributions.

Now that we have verified the validity of the general sum rules in two known cases, we turn to theoretical constraints for standard model extensions with additional gauge boson.

3 Unitarity constraints on general models

In this section we consider simple extensions of the standard model with an additional gauge boson which couples to SM fermions. We focus on the scattering amplitudes for fermions to gauge bosons. In particular, we point out that generically a gauge boson also has to be present in order to preserve perturbative unitarity of all the amplitudes. The coupling of to the fermions are fixed by the sum rules with its mass bounded from above, as we shall see.

We start by considering the scattering , where is assumed to possess the interaction with , being standard model fermions with electric charges . With this coupling, such processes can take place through t-channel exchange. This amplitude itself would violate unitarity at high energy. One possible solution, according to the sum rule, is to introduce corresponding u-channel process. In order to preserve electric charge, the new fermion to be exchanged in the u-channel process must have electric charge and an identical coupling strength in However, in this case one can also consider the scattering and new fermion with higher electromagnetic charge must also be introduced. Therefore, unless we introduce infinite chain of fermions whose electric charge differ by one unit, unitarity cannot be restored.111The above argument is based on the assumption where the fermions , and are all SM fermions, which is the case in the conventional models. We will show that the infinite tower of particles can be avoided in generalized models in Section 5 .

Therefore, in order to find a finite solution, one must rely on s-channel processes. In fact, there could be and couplings which also contribute to the above process. We shall see that such couplings are fixed by considering various gauge boson final states and additional s-channel contribution is generally necessary.

In the following, we also take into account possible deviations from theoretical predictions of SM couplings, due to possible and mixings. We do not include possible small deviations from the SM in coupling due to the precise measurement of and , or deviations in , couplings which have their forms guaranteed by electromagnetic gauge invariance. We parametrize the and couplings as follows.


Here parametrizes the deviation of coupling to RH fermions, due to mixing. We will present our results to leading order in the small parameters , which are allowed to be at most 5% [22].

3.1 General left-handed

We first consider the case where couples to left-handed SM quark doublets, in a way similar to the -boson:


We will also include a gauge boson and consider all possible couplings among gauge bosons as well as general couplings of the to SM fermions. We show the relevance of due to its non-vanishing couplings demanded by the sum rules. Throughout the paper, we do not consider the coupling of to leptons and assume the gauge anomalies are canceled by heavy fermion spectators [23], when embedded in a complete theory.

The processes we consider are summarized in Table 1.

Process Exchanged particle Sum rule t-channel quark s-channel , , t-channel quark u-channel quark s-channel , t-channel quark u-channel quark s-channel , t-channel quark u-channel quark s-channel , t-channel quark u-channel quark s-channel ,                t-channel quark if LH s-channel , ,

Table 1: Processes considered and sum rules, in the case of with couplings to left-handed fermions. Here , etc. Solutions to the sum rules are given in Eq. (3.1).

A crucial point to note is, there are enough sum rules to solve for all the unknown couplings of . We summarize them here, taking into account the leading order corrections.


along with and , in order to balance the coefficients of , and in the sum rules. These structures are the common predictions of left-handed models. Recall that , thus and are only tiny couplings, arising from and/or mixings.

Since the must possess non-zero couplings to fermions and other gauge bosons, it cannot be simply decoupled from a consistent theory. We would like to emphasize that when the -fermion coupling is given, the -fermion coupling is fixed automatically. The model studied in Ref. [24] is one of the incarnations of this generic structure. In particular, the model starts from where the two ’s break down to the diagonal subgroup which is identified as the SM . The SM fermion doublets are charged only under the first . After this stage of symmetry breaking, the , couplings to fermions has a strength given by , where are gauge couplings corresponding to and , respectively.

Next, we proceed to calculate the finite amplitude at very high energy, given the above sum rules. We are interested in finding the upper bound on the mass scale required by perturbative unitarity. Among the above processes, the most important one is . Using the general formula Eq. (11), we get


Here we neglect all the quark masses. In this case, the amplitude is a pure p-wave. The partial wave amplitude can be calculated with Eq. (6).

As an example, we consider the initial quarks being and and the fermion being exchanged in the t-channel being the quark. Demanding the partial wave , the allowed regions in the parameter space for different masses equal to 150, 500 and 800 GeV are shown in the left panel of Fig. 3.

The first point we note from the plot is the upper bound on mass is most stringent for very large coupling . This will be important in obtaining the mass bound in top quark asymmetry models studied in the next section.

There is also a possibility where one can have the much heavier than the . This can be understood because in this case couples exactly in the same way as the SM boson, and the SM itself is a consistent theory from the unitarity point of view. Perturbative unitarity of the finite part of and still gives a finite but very mild upper bound


3.2 Comment on CDF event excess

It has been argued that the CDF excess can be explained via the associated production of and extra gauge bosons, such as , with GeV and a relatively small . The needs to be left-handed in order to avoid chiral suppression and to give large enough cross section  4 pb. In the model, unitarity implies that there is a destructive interference between t-channel quark exchange and s-channel exchange. Here we point out that for the model presented in Ref. [17] to be complete, there must exist some other resonances at low energy, or the coupling should be modified.

From Fig. 3, we learn that the appearance of can be postponed up to  TeV for . If mass is close to this upper bound, one can achieve a large hierarchy between the cross sections and . The latter process does not show any excess as reported by the CDF collaboration. It happens through t- and u-channel quark exchange as well as s-channel exchange. Since here the mass is fixed, the destructive interference for associated production is much more significant, which leads to a much smaller cross section. In fact, we find for the parameters GeV, , and a heavy inaccessible at the Tevatron energy, the leading-order cross sections calculated using MadGraph [25] to be


while the two cross sections are similar for . Alternatively, since the CDF analysis does not veto the possibility of being reconstructed into a resonance, one can obtain a large ratio by taking a relatively light