Unitarity constraints on large multiplets of arbitrary gauge groups

We impose partial-wave unitarity on $2 \to 2$ tree-level scattering processes to derive constraints on the dimensions of large scalar and fermionic multiplets of arbitrary gauge groups. We apply our results to scalar and fermionic extensions of the Standard Model, and also to the Grand Unified Theories (GUTs) based on the groups $SU(5)$, $SO(10)$, and $E_6$. We find scenarios within the latter two GUTs that violate the unitarity condition; this may require a reevaluation of the validity of perturbation theory in those scenarios.


Introduction
The conservation of probability in quantum-mechanical processes implies unitarity of the S matrix.When this condition is phrased in terms of the partial waves of a scattering amplitude, one may derive constraints on the magnitude of the zeroth partial wave, if one wants perturbation theory to be trusted in the theory in question.Historically, a famous application of partialwave unitarity bounds was made by Lee, Quigg, and Thacker [1,2] to derive the upper bound m H ≲ 1 TeV on the mass m H of the Higgs boson.The same reasoning has been applied by Logan and her collaborators [3] to constrain the quantum numbers of large scalar multiplets in extensions of the Standard Model (SM); they have shown that one complex scalar multiplet with more than eight components (or one real multiplet with more than nine components) is enough to violate perturbation theory.They also found constraints on the hypercharges of such multiplets.More recently, by working with fields transforming in the trivial, fundamental, and adjoint representations of SU (N ) × U (1), upper bounds on Yukawa [4] and gauge [5] couplings have been derived by using perturbative unitarity.One-loop corrections to the Higgs-W W and Higgs-ZZ couplings have also been used to constrain the dimension of large electroweak representations of vector-like fermions [6].
The goal of this work is to extend Ref. [3] by using the unitarity bounds on the zeroth partial wave of 2 → 2 tree-level scattering amplitudes to constrain the dimensions of large scalar and fermionic multiplets that transform under arbitrary gauge groups.We compute both the fermion-pair and the scalar-pair annihilation into gauge-boson pairs in the high-energy limit.The calculations are performed in the unmixed basis of each gauge group and before spontaneous symmetry breaking.Therefore the scalars, the fermions, and the gauge bosons are taken to be massless, and the latter only have transverse polarizations.
In Section 2 we provide a brief review of the unitarity constraints on the zeroth partial wave of 2 → 2 tree-level scattering amplitudes; we then explicitly calculate the zeroth partial wave for a single fermion pair or scalar pair annihilating into a gauge-boson pair of a single gauge group.In Section 3 we introduce the technique of coupled-channel analysis to constrain the dimension of a single fermion or scalar multiplet under a single gauge group; we then generalize the reasoning to multiple multiplets transforming under multiple gauge groups.We apply our results to derive constraints on extensions of the SM and on the SU (5), SO (10), and E 6 Grand Unified Theories (GUTs) in Section 4, and draw our conclusions in Section 5. We use Appendix A to carefully derive the scattering amplitudes used in Section 3. In Appendix B we substantiate an assumption made about the largest eigenvalue of the coupled-channel matrix, by using a pedagogical example.In Appendix C we provide useful properties for the relevant irreducible representations of the various gauge groups used in this paper.

Interactions of the scalars and fermions with the gauge bosons
We consider a quantum field theory symmetric under a Lie Group G.In this theory, a fermion multiplet is placed in a representation n ψ of G, while a scalar multiplet is placed in a representation n φ of G: Without loss of generality, we let n denote both the dimension of n ψ and the one of n φ .The covariant derivative acting on either multiplet is defined as The T a are the n×n matrices that represent the generators of G; the V a µ are ñ gauge bosons which transform in the ñ-dimensional adjoint representation of G; g is the gauge coupling constant.
The interactions of the fermions with the gauge bosons derive from The scalars interact with gauge bosons through If n φ is in a real representation of G, then one must drop the †'s from Eq. ( 4) and add a factor 1/2 to its right-hand side.

Unitarity bounds
We want to consider scattering processes of the type The amplitude of each process may be decomposed into partial-wave amplitudes according to the formula [8] where θ is the scattering angle and J (which is a non-negative integer) is the total orbital angular momentum of the final state.The a J are numerical coefficients.The functions P J (cos θ) are the Legendre polynomials; in particular, The Legendre polynomials satisfy the orthogonality relation Therefore, the zeroth partial wave is given by Tree-level zeroth partial-wave unitarity dictates that [8,9]: In this work, we want to constrain the dimension of large fermionic or scalar multiplets by using condition (9a).In order to find the most stringent constraint we employ the technique of coupledchannel analysis.The zeroth partial waves will be calculated for each process ψ i ψ j → V a V b and/or φ i φ † j → V a V b , and will then be assembled in a coupled-channel matrix for various pair of indices (i, j) and (a, b).The unitarity bound arises from applying condition (9a) to the largest (in modulus) eigenvalue of the coupled-channel matrix [9,10,11].It should be pointed out that the partial wave for a process involving two identical initial-state or final-state particles receives an extra factor 1 √ 2; if the two initial-state particles are identical and the two final-state particles are also identical, then the extra factor is 1/2.

Fermionic Scattering
We want to calculate the zeroth partial wave of the process Three channels contribute to the corresponding amplitude: We work in the high-energy limit and we therefore neglect the masses of the fermions and gauge bosons.In that limit, fermions become chiral, meaning ψ j has opposite chirality to ψ i .Consequently, if the two final-state gauge bosons have opposite polarizations (either left-right or right-left), then the sum of the three diagrams in Eq. ( 11) vanishes.On the other hand, if the two final-state gauge bosons have the same polarization (either left-left or right-right), then the third diagram (the s channel) in Eq. ( 11) vanishes and only the first two are nonzero.The amplitudes for the t-and u-channels read respectively, where u i and v j are the the spinors of the initial-state ψ i and ψ j , respectively, ε (p 3 ) and ε (p 4 ) are the polarization four-vectors of the final-state V a and V b , respectively, t ≡ (p 1 − p 3 ) 2 and u ≡ (p 1 − p 4 ) 2 are the Mandelstam variables.
Since the gauge bosons have the same polarization, either V a and V b are both right-handed (RR configuration), or they are both left-handed (LL configuration).Using the results of Appendix A.1, the total amplitude for each scenario is found to be We use Eq. ( 8) for a 0 and to obtain Note that if V a = V b , then the results (15) must be divided by √ 2.

Scalar scattering
In this section, we calculate the zeroth partial-wave of the process Four diagrams contribute to the corresponding amplitude: We work in the high-energy limit and we therefore neglect the masses of the scalars and gauge bosons.In that limit, if the two final-state gauge bosons have opposite polarizations, then the sum of the four diagrams in Eq. ( 17) vanishes.On the other hand, if the two final-state gauge bosons have the same polarization, then both the third and fourth diagrams in Eq. ( 11) vanish and only the first two diagrams are nonzero.The amplitudes for those two diagrams read respectively.Once again, either V a and V b are both right-handed (RR configuration), or they are both (LL configuration).For each scenario, using the results of Appendix A.2 we arrive at the following total amplitudes: Therefore, using Eq. ( 8) and the corresponding zeroth partial waves read These partial waves must be divided by √ 2 when V a = V b .Furthermore, if n φ is a real representation and φ i = φ j , then one must once again divide by √ 2. Notice the presence of π in the denominator of Eq. ( 21), while π is absent from Eqs. (15).

Matrix of zeroth partial-waves
As stated before, the strongest constraints are achieved by enforcing the unitarity condition (9a) on the largest (in modulus) eigenvalue of a coupled-channel matrix of zeroth partial waves.In general, that largest eigenvalue 3 corresponds to an eigenvector that couples symmetrically, i.e. in group-invariant fashion, both pairs of initial states and pairs of final states.The concept is illustrated by the example in Appendix B. We define the symmetric linear combinations of states as The factor 1 √ 2 in the second Eq. ( 22) accounts for the identical gauge bosons in the final state.We assume-based on our experience in particular cases, like the one in Appendix B-that the process with the largest zeroth partial wave is 4 RR/LL .In order to find the corresponding eigenvalues, we move to the basis in which the coupled-channel matrix takes the form where We derive explicit expressions for these matrix elements by resorting to the group-theoretical relation [12] ñ a=1 where C (n) is the quadratic Casimir invariant of the n-dimensional representation n and 1 n×n is the n × n identity matrix.From Eqs. (15a) and ( 25) we derive In a similar fashion, the other matrix elements are and We plug Eqs. ( 27) and ( 28) into Eq.( 24) and we find the matrix of zeroth partial waves to be Its largest eigenvalue of course is Note that considering simultaneously the final states [V V ] sym RR and [V V ] sym LL is effectively equivalent to multiplying by a factor √ 2 the zeroth partial wave of either.Henceforward, we use this trick to account for both polarization configurations and we drop the subscripts RR and LL.
For a scalar multiplet, we have instead The difference between the scalar and fermionic case is just a factor π/2: The factor π arises from the integrations in Eqs. ( 14).

Multiple Particles and Multiple Symmetry Groups
The above analysis has assumed the existence of only one matter (either fermionic or scalar) multiplet transforming non-trivially under only one symmetry group G.We extend this situation to a scenario where there are N F fermionic multiplets F 1 , F 2 , . . .and N S scalar multiplets S 1 , S 2 , . .., that transform under a single symmetry group G.In this scenario, the matrix of zeroth partial waves is defined in the basis 5 : The largest eigenvalue of the corresponding matrix of zeroth partial-waves is then simply given by where we have used the results of Eqs. ( 31) and (32) for a ψ 0 max and (a φ 0 ) max , respectively.In Eq. (35b), n F i is the number of fermions in the irreducible representation n F i of the gauge group, and n S i is the number of fermions in the irreducible representation n S i of the gauge group.
This formalism may be further extended to a scenario where there are multiple matter (either fermionic or scalar) multiplets A, B, C, D, . . .transforming under the direct product of multiple symmetry groups We work in the basis where is the symmetric linear combination of gauge bosons of the group G i , as defined in Eq. (22). 6If the matter multiplet A transforms non-trivially under the gauge group G i , then there are copies of that multiplet of the group G i which interact with [V V ] sym G i .We keep the original definition of [AA] sym provided in Eq. ( 22), but multiply the corresponding zeroth partial wave by the square root of the factor in Eq. (38).In the basis (37), the full matrix of zeroth partial waves is written as Each matrix element of M ′ is defined as where g G i is the gauge coupling constant of G i , ñG i the dimension of its adjoint representation, and C(n G i A ) the quadratic Casimir invariant of the representation of the matter field A under G i .Notice that the product in Eq. ( 40) runs over all k, not just over k ̸ = i as in Eq. (38); the extra factor n G i A comes from Eqs. (31) or (32).

Results
In the following sections, we will calculate the largest eigenvalue of the matrix of zeroth partial waves for the gauge groups SU (3) × SU (2) × U (1) (i.e. the SM), SU (5), SO (10), and E 6 .
We shall impose the unitarity bound (9a) to constrain the dimension of additional fermionic or scalar multiplets in various scenarios.We follow the group-theory conventions of Mathematica's package GroupMath [13].The Dynkin indices and the quadratic Casimir invariants for the relevant irreducible representations of the various gauge groups are listed in Appendix C. A more comprehensive compilation of group-theory data may be found in Ref. [14].
The SM is invariant under the gauge group G SM ≡ SU (3) colour × SU (2) L × U (1) Y .Using the convention in Eq. (36), the matter fields of the SM transform as There are three copies of each of the multiplets (41a)-(41e), which we label through subscripts f = 1, 2, 3.An appropriate basis for the scattering states is The corresponding matrix M ′ , in the notation of Eq. (39), is 3 × 16 and may be written in terms of a 3 × 5 sub-matrix M ′ F and a 3 × 1 sub-matrix M ′ H as with Each matrix element of M ′ F and M ′ H has been computed by using Eq.(40).We use the gauge coupling constants measured at the Fermi scale √ s ≈ 10 2 GeV [15]: We then numerically7 calculate the largest eigenvalue of the matrix a 0 of Eq. (39) to be We thus reassuringly conclude that the SM is perturbative, since a 0 = 0.269 is well compatible with the unitarity condition |Re a 0 | ≤ 1/2 of Eq. (9a).
If we suppose the existence of an additional fermionic multiplet , and Y ψ we use Eqs.( 39) and (40) to construct the matrix a 0 and we then compute its largest eigenvalue.In order to fulfill the unitarity condition |Re (a 0 ) max | ≤ 1  2 , when the dimensions of the representations of SU (3) colour and SU (2) L that ψ sits in are fixed, the hypercharge Y ψ cannot be arbitrarily large.We thus constrain the absolute value of Y ψ ; the results are shown in the left panel of Table 1.Obviously, smaller n , the upper bound on |Y φ | is presented in the right panel of Table 1.We conclude that scalar multiplets are less constrained; they must satisfy n The first column of the right panel of Table 1 corresponds to the scenario studied in Ref. [3]; our constraints are, however, more stringent and accurate, because we consider not only the contributions from the extra scalar multiplet φ but also the contributions from the SM matter fields of Eqs.(41).
A general feature of GUTs is the over-abundance of particles in their scalar sectors.But, if perturbativity is to be preserved, the largest eigenvalue of the matrix of zeroth partial-wavesgiven in Eq. ( 35)-must satisfy the unitarity condition (9a).As a result, the number of fermions N F , the number of scalars N S , and the dimensions of their corresponding SU (5) irreducible representations ('irreps') are constrained by ≈ 29, 243.( This implies an upper bound on the size of the scalar sector.Note that, for the most usual set of fermionic multiplets in (51), the second term in the left-hand side of Eq. (52) takes the value We firstly consider four minimal scenarios found in the literature [18,19,20], all of which employ a 24-dimensional scalar multiplet to spontaneously break SU (5) into G SM , but differ in the set of scalars used to further break G SM into the gauge group G 31 ≡ SU (3) colour ×U (1) electromagnetism and to generate fermion masses.For each scenario below, we use Eq. ( 35) and the results of Appendix C.4 to compute the largest eigenvalue of the matrix of zeroth partial waves: We find that every scenario satisfies the unitarity condition (9a).
A natural extension of these four scenarios is the addition of either another fermionic multiplet ψ ∼ n ψ or another scalar multiplet φ ∼ n φ .Constraints on the dimensionality of such multiplets are attained through Eq. ( 52).Regarding the former possibility, one can immediately see that unitarity is violated for the irrep 70' and for any fermionic multiplet with dimension n ψ ≥ 105.For scalars the bounds are less strict: in the first three scenarios, unitarity is violated for the irreps 126', 175" and for any multiplet with dimension n φ ≥ 200; in the fourth scenario, unitarity is furthermore violated for the irreps 160 and 175'.
The five-dimensional representation of SU (5) contributes very little to (a 0 ) max .We checked that, in order to break perturbative unitarity, one must add to the minimal SU (5) GUT close to one thousand 5-plets of scalars.
We summarize the results of this subsection in Table 2.

Grand Unified Theory SO(10)
In the SO(10) GUT, the three SM coupling constants unify at the energy scale √ s ≈ 10 16 GeV at the value [21] g SO(10) ≈ 0.60.
We use where N S is the total number of scalars in representations n S i and N F is the total number of fermions in representations n F i ; thus, if there are three fermionic 16 representations, The condition (61) may be used to evaluate the validity of perturbation theory in each scenario presented in Eqs. ( 57) and ( 59).The results are shown in Table 3, wherein the scalar contents of the grey-shaded scenarios violate the partial-wave unitarity condition |Re (a 0 ) max | ≤ 1/2.Since (a 0 ) max is an increasing function of the number of scalar multiplets, we omit all other scenarios for which a subset of scalars already violates unitarity.Thus, we advocate a reexamination of the validity of perturbation theory in SO(10) models that have, at least, one of the following sets of scalar irreps: One may want to expand the scalar sector of the remaining SO(10) models by incorporating low-dimensional irreps.Specifically, we consider adding either N 10 copies of ten-dimensional scalar multiplets 10, or (but not simultaneously) N 16 copies of sixteen-dimensional scalar multiplets 16.According to Eq. (61), N 10 and N 16 must be bounded from above.In Table 4 we present upper bounds on N 10 and N 16 for some SO(10) models that do not violate tree-level partial-wave unitarity, yet have rather low (smaller than 10) maximal N 10 and/or maximal N 16 .

Grand Unified Theory E 6
We now study the E 6 Grand Unified Theory.At the GUT energy scale √ s ≈ 10 16 GeV, we adopt the conservative estimate for the unified coupling constant [24] g E 6 ≈ 0.55. (63) The fermionic sector is composed of three generations of a 27-dimensional multiplet ψ f 27 ∼ 27, f ∈ {1, 2, 3}.Each 27 accommodates the 15 SM fermions of Eqs.(41a)-(41e) along with 12 new fermions, out of which one is a right-handed neutrino [25].
The tensor product of two 27's decomposes as under E 6 .Therefore, fermion masses can be generated through a Higgs-like mechanism by introducing any linear combination of 27, 351, and 351' scalar multiplets.However, we expect the masses of the 12 fermions beyond the SM to be orders of magnitude larger than the masses of the SM fermions.This splitting of mass scales can only be achieved by including at least one scalar multiplet 351' [26,27,28].With this in mind, before specifying the scalar multiplets needed to break E 6 into G SM , we compute the largest eigenvalue of the matrix of zeroth partial-waves for a model composed of three 27 fermionic multiplets and one 351' scalar multiplet.We define the matrix of zeroth partial-waves in the basis: which obviously violates the unitarity condition (9a).For this reason, we raise questions about the viability of any perturbative Grande Unified Theory with E 6 gauge symmetry, capable of generating the correct mass hierarchy in the fermion sector.

Conclusion
We have computed the largest (in modulus) eigenvalue of the matrix of zeroth partial waves in the scattering of a fermion-antifermion pair and of a scalar-scalar pair into two gauge bosons.We have made the educated guess that the largest eigenvalue is always obtained when the fermionantifermion, scalar-scalar, and gauge boson-gauge boson pairs are in symmetric combinations under the gauge group; if this assumption fails in some cases-which we do not expect to happenthen the bounds that we have obtained will still be valid, even if they are not the strongest possible ones.We have found that no problem of violation of the unitarity requirement occurs in most extensions of the SM and/or of the SU (5) Grand Unified Theory.On the other hand, many schemes of SO(10) symmetry breaking display unitarity violation, and this is also true of all the E 6 -based GUTs.In those GUTs, perturbative calculations seem to be unwarranted.

A Explicit calculation of the scattering amplitudes
As we will be considering scattering amplitudes of processes of the type A i (p 1 )A † j (p 2 ) → V a (p 3 )V b (p 4 ) (with A = ψ, φ), it is important to define the four-momenta of each particle.In the x − z plane these can be written as: where the center-of-mass frame of reference was adopted and the high-energy limit ( √ s ≫ m) was taken.In this limit the corresponding chiral spinors associated with ψ i and ψ j (i.e.u i , v j ) and gauge boson transverse polarization vectors (ε µ L/R ) can be written in the helicity basis as where c θ and s θ are the cosine and sine of the scattering angle, respectively.The numerator of where The coupled-channel matrix of zeroth partial-waves then reads where M is the following 12 × 4 matrix: The 16 × 16 matrix a ψ 0 of Eq. (81) has eigenvalues The eigenvalue with the largest modulus is and the corresponding (transposed) eigenvector is Table 6: Irreducible representations and their Casimir invariants of SU (2).
This means that the largest zeroth partial-waves arise from the processes We have found this feature to be a general property of any representation of any symmetry group.

C Group Representations and Casimir Invariants C.1 U (1) Gauge Group
The dimension of the U (1) adjoint representation is ñU(1) = 1.All U (1) representations are onedimensional, and given by a charge Y .Table 5 shows these properties alongside the corresponding quadratic Casimir invariant (C(Y )).

C.3 SU (3) Gauge Group
The dimension of the adjoint representation is ñSU(3) = 8.Table 7 shows the lowest-dimensional SU (3) irreducible representations (n), alongside their Dynkin indices, dimensions (n), and quadratic Casimir invariants (C(n)).For a more intuitive understanding, in Figure 1 we plot the quadratic Casimir invariant as a function of the dimension of the irreducible representations found in Table 7.The primed marker •' represents a primed representation n'.

C.4 SU (5) Gauge Group
The dimension of the adjoint representation is ñSU(5) = 24.Casimir invariants (C(n)).In Figure 2 we plot the quadratic Casimir invariant as a function of the dimension of irreducible representations found in Table 8.The primed (double primed) marker •' (•") represents a primed (double primed) representation n' (n").

C.5 SO(10) Gauge Group
The dimension of the adjoint representation is ñSO(10) = 45.Table 9 shows the lowest-dimensional SO (10) irreducible representations (n), alongside their Dynkin indices, dimensions (n), and quadratic Casimir invariants (C(n)).In Figure 3 we plot the quadratic Casimir invariant as a function of the dimension of irreducible representations found in Table 9.The primed marker •' represents a primed representation n'.

Figure 1 :
Figure 1: SU (3) Casimir invariant as a function of the dimension of the irreducible representation.

Figure 2 :
Figure 2: SU (5) Casimir invariant as a function of the dimension of the irreducible representation.

Figure 3 :
Figure 3: SO(10) Casimir invariant as a function of the dimension of the irreducible representation.

Figure 4 : E 6
Figure 4: E 6 Casimir invariant as a function of the dimension of the irreducible representation.

Table 1 :
Upper bounds on |Y ψ | (left panel) and on |Y φ | (right panel) imposed by the tree-level unitarity condition |Re a 0 | ≤ 1 2 , in an extension of the Standard Model through either a fermionic multiplet ψ (left panel) or a scalar multiplet φ (right panel).The symbol (-) stands for a negative, hence unphysical number.

Table 2 :
Summary of the SU (5) extensions considered.

Table 3 :
Largest zeroth partial waves of minimal SO(10) GUTs.Every scenario assumes a fermionic sector composed of three families of 16 multiplets.

Table 4 :
(10)r bounds on N 10 and N 16 for some allowed SO(10) models.We always assume the fermionic content to be three 16 of SO(10).
[29]situation can only worsen with the introduction of further scalar or fermionic multiplets.To give one out of many examples found in the literature, Ref.[29]considers a model with three 27 fermionic multiplets and a scalar sector composed of 27 ⊕ 78 ⊕ 351'.By running the three SM couplings up to the GUT energy scale, where they unify, Ref.[29]obtains g E 6 ≈ 0.70.

Table 10 :
Lowest-dimensional E 6 irreducible representations and their Casimir invariants.