PHYSICAL REVIEW D 104, 035034 (2021)
Bounds on gauge bosons coupled to nonconserved currents
† ‡ Majid Ekhterachian ,1,* Anson Hook,1, Soubhik Kumar ,2,3, and Yuhsin Tsai 4,§ 1Maryland Center for Fundamental Physics, Department of Physics, University of Maryland, College Park, Maryland 20742, USA 2Berkeley Center for Theoretical Physics, Department of Physics, University of California, Berkeley, California 94720, USA 3Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 4Department of Physics, University of Notre Dame, Indiana 46556, USA
(Received 3 June 2021; accepted 5 August 2021; published 30 August 2021)
We discuss new bounds on vectors coupled to currents whose nonconservation is due to mass terms, such 1 as Uð ÞLμ−Lτ . Due to the emission of many final state longitudinally polarized gauge bosons, inclusive rates grow exponentially fast in energy, leading to constraints that are only logarithmically dependent on the symmetry breaking mass term. This exponential growth is unique to Stueckelberg theories and reverts back to polynomial growth at energies above the mass of the radial mode. We present bounds coming from the high transverse mass tail of monolepton þ MET events at the LHC, which beat out cosmological bounds to 1 place the strongest limit on Stueckelberg Uð ÞLμ−Lτ models for most masses below a keV. We also discuss a stronger, but much more uncertain, bound coming from the validity of perturbation theory at the LHC.
DOI: 10.1103/PhysRevD.104.035034
I. INTRODUCTION history, see, e.g., Refs. [23–26], and the famous application to the Higgs boson [27–30]. Recently, new light weakly coupled particles have Typically nonconservation of the currents comes from increasingly become a focus as either a mediator to a dark either anomalies or mass terms. In the first case, amplitudes sector [1–3], as dark matter itself [4–8], or to explain involving longitudinal modes are enhanced [15]. Later, potential anomalies [9–14]. A particularly well motivated Ref. [19,20] showed that this enhancement led to strong candidate is a vector boson. The currents that these vector constraints on anomalous gauge theories. In the second bosons couple to can either be conserved, e.g., Uð1Þ − B L case, inclusive amplitudes were shown to exhibit expo- with Dirac neutrino masses, or they can be nonconserved, nential growth [31]. We will show that this exponential e.g., Uð1Þ or Uð1Þ − . L Lμ Lτ growth leads to very strong constraints on these theories. In this paper we will continue a long line of research into In this letter we will focus on the case of currents that are the bounds that can be placed on vector bosons coupled to not conserved due to mass terms. Specifying this starting – nonconserved currents, see e.g., Refs. [15 22].Asiswell point locks us into considering the Stueckelberg limit of known, these models are nonrenormalizable field theories gauge theories1 as including the radial mode renders mass and as a result have amplitudes that grow with energy. terms gauge invariant. A Stueckelberg theory coupled to a Eventually these amplitudes grow so large that tree level nonconserved current can have inclusive rates that grow Λ amplitudes violate unitarity at the energy scale , indicat- exponentially fast in energy due to multiparticle emission ing that perturbation theory has broken down. Requiring [31]. This exponential growth is unique to the Stueckelberg Λ that new physics appears below the scale gives the limit and becomes polynomial at energies above the mass unitarity bound. Unitarity bounds have a long storied of the radial mode. The exponential growth of amplitudes in these models gives extremely strong unitarity bounds [31]. Redoing the *[email protected] † unitarity bound using our conventions, we find [email protected] ‡ [email protected] § [email protected] 4πm m Λ ≈ pffiffiffiffiffi X log3=2 X ; ð1Þ Published by the American Physical Society under the terms of 27gX gXm the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, 1The Stueckelberg limit is when the mass of the radial mode is and DOI. Funded by SCOAP3. taken to be heavier than the energy scale under consideration.
2470-0010=2021=104(3)=035034(10) 035034-1 Published by the American Physical Society EKHTERACHIAN, HOOK, KUMAR, and TSAI PHYS. REV. D 104, 035034 (2021) where gX and mX are respectively the coupling and the coming from mono-lepton+MET events at the LHC and the mass of the gauge boson, and m is the symmetry breaking total decay width of the W boson respectively. In the high mass term. To show how strong this unitarity bound is, let mass limit, this is only a factor of ∼100 weaker than the 1 − 2 us consider the Stueckelberg limit of Uð ÞLμ . Typically one extremely precise measurements of the g of the muon ≲ 1 assumes that the strongest unitarity bound on this model [32,33]. Meanwhile for mX keV, this is the strongest comes from the fact that this current is anomalous. Redoing constraint on these models beating out even cosmological 2 the unitarity bound (for details see the Appendix) using our bounds. conventions leads to a slightly stronger result than the standard result [15] II. UNITARITY BOUND pffiffiffiffiffiffi 4π 32π2 In this section, we derive the unitarity bound on these Λ ≈ mX a 2 ; ð2Þ models, which was also done in Ref. [31] using slightly gX g2 different techniques.3 In order to demonstrate the expo- 2 nential growth of amplitudes, we will consider the toy where g2 is the SUð ÞW gauge coupling. Comparing this scenario of a gauge boson coupled to only the left handed ≃ 1 with Eq. (1) taking mX=gX GeV, motivated by our later piece of a Dirac fermion ν, whose current is not conserved results, and m ≃ 0.05 eV, as the mass of the neutrino, we due to explicit breaking by a small Dirac mass term mν. Λ ∼ Λ 10 find that a= . This shows that despite the extremely Namely we consider a theory with small neutrino mass, its nonzero value still gives a unitarity bound almost an order of magnitude more stringent than the 1 1 L − 2 ν¯ ∂ − ν − νν¯ 2 2 anomaly does. Thus before considering moving to an ¼ 4 FX þ i ð igX=AXPLÞ mν þ 2 mXAX; ð4Þ 1 anomaly free theory such as Uð ÞLμ−Lτ , one should first UV complete the symmetry breaking neutrino mass terms. where FX is the field strength of the gauge boson AX. In the 1 limit of small mν, the scale at which perturbation theory In contrast, for a Higgsed Uð ÞLμ one can naturally have the breaks down, Λ, is much larger than the mass of the gauge exact opposite scenario where Λa ≪ Λ. Unitarity in these models is restored by the inclusion of bosons. In this limit, things simplify as the high energy the radial mode. However in these UV completions, the behavior of AX can be obtained from the Goldstone boson small fermion mass is the result of a higher dimensional equivalence theorem. Namely, the matrix element obeys M L ≈ M ϕ L operator so that scattering of the radial mode has its own ðAX þ Þ ð þ Þ, where AX is a longitudinally ϕ unitarity bound. Constraints on the UV completion are polarized AX and is the Goldstone boson. fairly model dependent, however since it is likely that the We obtain the theory with Goldstone bosons by leaving UV completion only couples to the SM via the neutrino unitarity gauge via a chiral gauge transformation, ν → ϕ ν mass term, bounds on it are plausibly fairly weak. On the L expðigX =mXÞ L. As the mass term is not gauge other hand, for Stueckelberg gauge bosons we obtain invariant, it transforms into robust, model-independent bounds that do not rely on X n ϕ mν igXPLϕ the dynamics of the radial mode. While these bounds can ν¯ igXPL =mX ν ν¯ ν V ¼ mν e ¼ ! : ð5Þ be evaded by going away from the Stueckelberg limit by n n mX including the radial mode at sufficiently low energies, the radial mode should then be taken into account whenever From this, it is clear that the theory is nonrenormalizable processes at energies larger than its mass are studied. and has a UV cutoff. The Goldstone boson equivalence In this paper we will focus on gauge bosons coupled to theorem lets us compute the probability of emitting many the lepton number currents, which are only nonconserved longitudinal gauge bosons by calculating the much simpler because of the extremely small neutrino masses. To find process of emitting many ϕ particles using the above explicit bounds, we consider the total decay width of the W interaction. ϕ boson and the high transverse mass tail of mono-lepton þ As one can see, considering processes involving n s ∼ n MET events at the Large Hadron Collider (LHC). The gives a matrix element mνðgXE=mXÞ that becomes more bounds we obtain are essentially independent of what the and more insensitive to the small mass parameter mν as n exact model under consideration is, thus in the introduction becomes larger and larger. This causes the best unitarity 1 bound to come from taking n larger and larger. However, in we will only list our bounds on Uð ÞLμ−Lτ with Dirac neutrino masses. A somewhat uncertain bound of mX=gX ≳ 2 24 GeV comes from requiring that physics is perturbative If the gauge boson instead coupled to Le, the strongest bounds at low mass are from neutrino oscillations in the earth [34] and it at the LHC. We also find −16 is only for masses mX ≲ 10 eV that our bounds win out. 3 m m We use a slightly different definition of the unitarity bound X > 1.3 GeV; X > 54 MeV ð3Þ and we consider the process ν þ nϕ → ν þ nϕ rather than gX gX ν þ ν¯ → nϕ, which leads to stronger unitarity bounds.
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1 ! c † the large n limit, the =n coming from dealing with is ν MdU νF where Md is a diagonal matrix of the neutrino identical final state particles penalizes taking n too large. masses m1;2;3. To leave unitary gauge, the flavor-basis SM Thus the optimal unitarity bound comes from taking an neutrinos are rotated by, intermediate value of the number of gauge bosons nopt. þigXϕ=mX −igXϕ=mX A calculation done in the Appendix shows that the νF → PνF P ¼ diagð1;e ;e Þ: ð9Þ amplitude Mˆ of the process ν þ nϕ → ν þ nϕ is ˆ Thus the neutrino mass term involving the Goldstone jMðν þ nϕ → ν þ nϕÞj ϕ bosons becomes 2n−1 gXmν gXE ¼ ð6Þ D c † 2 1 ! ! − 1 ! 4π Lν ¼ ν M U Pν þ H:c: mXðn þ Þ n ðn Þ mX mass d F X 1 ϕ n igX c † n † ≫ Mˆ 1 ⊃ ν M ðU νμ þð−1Þ U ντÞ: in the limit that E nmX. Unitarity requires that j j < . n! m j d;j jμ jτ In order to obtain the strongest bounds, we choose the n n;j X that maximizes Eq. (6) to obtain in the large n limit From this, we see that any 1-flavor process involving n high gX E 2=3 gXmν 3ð4π Þ energy gauge bosons can be converted into the 3-flavor Mˆ ν ϕ → ν ϕ ∼ mX j ð þ nopt þ nopt Þj e : ð7Þ 2mX result by replacing
This maximum value of jMˆ j is obtained for X3 2 2 2 2 ≈ 4π 2=3 mν → ðjUμ j þjUτ j Þm : ð10Þ nopt ðgXE= mXÞ . Requiring unitarity holds for j j j 1 Eq. (7) gives the leading logarithmic behavior j¼ The other gauge theory we will consider is Uð1Þ with 4πmX 3 2 mX L E ¼ Λ ≈ pffiffiffiffiffi log = : ð8Þ Majorana neutrino masses. In this case, after leaving 27g g mν X X unitary gauge the mass term is
From this calculation we see the behavior claimed in the 2 ϕ LM igX =mX νT ν Introduction. Amplitudes have an exponential growth in νmass ¼ e MMd M þ H:c: ð11Þ energy and the strongest growth comes from emitting multiple gauge bosons. In this calculation, we made the From this, the 1-flavor results can be generalized using the approximation that the energy carried by each of the AX substitution gauge bosons is much larger than the mass when we X X3 utilized the Goldstone boson equivalence theorem. 2 2 2 g → 2g ;mν → jU j m : ð12Þ Combining the expression for nopt with Eq. (8) and the X X lj j ≫ l¼e;μ;τ j¼1 requirement that E noptmX, we find that our massless approximationpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the unitarity bound is valid when ≲ 4π 3 gX logðmX=gXmνÞ= . In the large log limit, the IV. CONSTRAINTS massless limit is always a valid approximation. We now present exclusions coming from the emission of many final state gauge bosons. We will consider III. MODELS three different constraints coming from the total decay We now briefly describe the models under consideration width of the W boson, the tail of high transverse mass and set up some notation. The results in the next section mono-lepton þ MET events, and a rough estimate of when will be given in the 1-flavor approximation so we will also perturbativity breaks down at the LHC. Our results will be discuss how to easily take into account the standard 3- presented in the simplified 1-flavor model. Equations (10) flavor set up. For ease of expression, in this section we will and (12) can be used to convert the results into the two use Weyl notation for fermions. specific models of interest. In particular, in Figs. 1 and 2 we As mentioned before, we will be considering the show the results after incorporating the full 3-flavor set-up. Stueckelberg limit of different Uð1Þ gauge theories. We W Boson decay: We will consider the partial decay width 1 first consider Uð ÞLμ−Lτ. We assume Dirac neutrinos and of the W boson into leptons and n gauge bosons that the right-handed neutrinos are neutral under Lμ − Lτ. Γ − → − ν¯ The flavor basis is related to the mass basis by νF ¼ UνM, nðW L þ þ nAXÞ: ð13Þ where νF (νM) are the flavor (mass) basis left handed neutrinos and U is the Pontecorvo-Maki-Nakagawa-Sakata The constraint we will impose is that the sum of these decay (PMNS) matrix. In the flavor basis the neutrino mass term widths is less than the total decay width
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FIG. 1. Low mass constraints on a Stueckelberg Uð1Þ − 1 Lμ Lτ FIG. 2. High mass constraints on a Stueckelberg Uð ÞLμ−Lτ gauge boson with Dirac neutrino masses. The purple region gauge boson with Dirac neutrino masses. The solid and dot- shows the constraint derived in Eq. (19) coming from the high dashed purple lines are the same as in Fig. 1. Other constraints are transverse mass tail of mono-lepton+MET events at the LHC. The from: ΔNeff during BBN due to thermalization of AX and entropy dot-dashed purple line denotes the constraint obtained from dump during its decay [35,40,41] (solid: TRH > 4 MeV, see, e.g., demanding the validity of perturbation theory at the LHC, given [42,43]; dashed: TRH > 10 MeV), neutrino trident process at the Δ þ − þ − by Eq. (20). Other constraints are from: Neff during BBN CCFR experiment [44,45], and a search for e e → μ μ AX, þ − through thermalized AX [35], black hole superradiance (BHSR) AX → μ μ at BABAR [46]. In the shaded green region, the muon Δ νν → instability [36], rare K decays [22], Neff through AXAX (g − 2) anomaly [47,48] can be explained by AX at 2σ [9,49]. [22,35], constraints on ν decay through terrestrial experiments [22,37,38]. We also update the cosmological constraints on ν −6 decay as discussed in the main text, based on the recent result of smaller than 10 of the total decay width of the W boson Ref. [39]. Given the ν decay bounds from cosmology are model- (roughly 1 over the number of W bosons produced at LEP) dependent and can be relaxed, we show it via dashed lines. 1 tightens the above Uð ÞLμ−Lτ bound to 76 MeV. monolepton þ MET events : In the next two subsec- X∞ tions, we will discuss bounds that stem from the breaking of Γ ≡ Γ Γ BSM n < W: ð14Þ perturbativity at the LHC. From the decay of the W boson, n¼2 we see that we are faced with a theory which is becoming dominated by a large number of final state particles. For ≥ 2 For simplicity we will take n to avoid the soft black holes [50], when the cross section becomes domi- divergence. nated by a large number of final states, the cross section for Using the Goldstone boson equivalence theorem, we find s-channel two-to-two scattering becomes highly sup- the result to leading order in mν to be pressed. The nonobservation of this suppression leads to
2 2 a constraint. As illustrated below, it is plausible that a Γ g2mν 4 1 1 2 3 3 5 2 similar feature exists in our case when perturbation theory BSM ¼ x 2F4ðf ; g; f ; ; ; g;x Þð15Þ 1536πMW breaks down. The case that we will consider is pp → W⋆ → lν via an where x ¼ gXMW , F is the generalized hypergeometric off-shell W boson. At high enough energies, the tree level off- 4πmX p q 2 shell width of the W boson becomes larger than the function and g2 is the SUð ÞW gauge coupling. In the limit of large argument, it simplifies as momentum flowing through the propagator, suppressing the high transverse mass tail of monolepton þ MET events.4 pffiffiffi 16 3 We obtain a bound by requiring that this suppression is small 1 1 2 3 3 5 2 ≈ 3x2=3 enough that the suppression was not observed in Ref. [51]. 2F4ðf ; g; f ; ; ; g;x Þ 20=3 e ð16Þ πx At high energies, the propagator of the W boson is given by showing the exponential growth of the amplitudes men- tioned earlier. i Bounds are obtained by requiring that this decay width is Σ s M Γ s ; − 2 Σ Imð ð ÞÞ ¼ W Wð Þ ð17Þ smaller than the total decay width of the W boson. For s MW þ ðsÞ 1 1 54 Uð ÞLμ−Lτ (Uð ÞL) we find that mX=gX > MeV 108 (mX=gX > MeV). Because of the exponential, our 4Perturbation theory has broken down when this occurs, so that results are insensitive to the details with which we constrain our calculation is only really an estimate of what happens in the the model. For example, requiring that this decay width is full nonperturbative theory.
035034-4 BOUNDS ON GAUGE BOSONS COUPLED TO NONCONSERVED … PHYS. REV. D 104, 035034 (2021) where the earlier calculation of the decay width of the W came from tree level calculations, they require that pertur- boson gives bation theory is valid. As such, strictly speaking, they only apply if the stronger calculability constraint has misesti- 2 2 g2mν 4 2 mated the breakdown of perturbation theory by a factor M Γ s x F4 1; 1 ; 2; 3; 3; 5 ;x 18 W Wð Þ¼1536π 2 ðf g f g ÞðÞ of few. pffiffi We show our constraints visually in Figs. 1 and 2 for gX s ≪ Γ 1 with x ¼ . When s M ðsÞ, monolepton þ MET Uð ÞLμ−Lτ . We take normal ordering of the neutrino masses 4πmX W W production is highly suppressed. with the lightest neutrino being massless. As explained in Ref. [51] observes the high MT tail of the monolepton þ the text, the bounds are not sensitive to the precise value of MET events5 to be the expected SM result when the mass. For other neutrino mixing parameters, we use the pffiffiffi MT ≲ 2 TeV. Given MT ≤ s by definition, it implies results from Table III of Ref. [53]. that at center of mass energies of at least 2 TeV, we do not In Fig. 1 we have updated the cosmological bounds on see any deviation from the SM expectation. The observed these models coming from the decay of the heavier neutrinos, ⋆ 5−6 5 events dominantly come from pp → W → lν. The off- which requires τ0 ≳ 4 × 10 secðm3=50 meVÞ [39].The ν shell W contributionffiffiffi would be suppressed by at least a two decay (cosmology) bounds represent the uncertainty in p the bound on the ν lifetime. After refining the calculation of factor of 2 if 2s ¼ MWΓWðsÞ and would have been ≲ 2 the damping of anisotropic stress due to neutrino decay and observed for anyffiffiffi MT TeV. We obtain a bound by p inverse decay, this new bound was obtained and found to be requiring that 2s ≳ M Γ ðsÞ at 2 TeV. For Uð1Þ − we W W Lμ Lτ several orders of magnitude weaker than previous work find that Δ (e.g., [54,55]). For the Neff constraints, we take TRH
5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiThe transverse mass MT is defined as MT ¼ showing that unitarity did indeed predict the correct scale of 2 l miss 1 − Δϕ Δϕ pT pT ð cos Þ, where is the azimuthal angle new physics in these models. In this UV theory, one can ⃗l ⃗miss between pT and pT . easily show that scattering involving gauge bosons no
035034-5 EKHTERACHIAN, HOOK, KUMAR, and TSAI PHYS. REV. D 104, 035034 (2021) longer grows. However, the price is that scattering involv- ACKNOWLEDGMENTS ing Φ does grow. Thus this partial UV completion will itself We thank Simon Knapen and Maxim Pospelov for require a UV completion at the scale Λ0. As this particular comments on the draft and Zhen Liu for useful discussions. higher dimensional operator is identical to those seen in M. E. and A. H. were supported in part by the NSF Grants Froggatt-Nielsen models [56], its UV completion proceeds No. PHY-1914480, No. PHY-1914731, and by the along a manner completely analogous to those models and Maryland Center for Fundamental Physics (MCFP). will not be expounded upon here. For a full UV completion S. K. was supported in part by the NSF Grant No. PHY- see Ref. [57]. 1915314 and the U.S. DOE Contract No. DE-AC02- It is worth mentioning that while the partial 05CH11231. Y. T. was also supported in part by the UV completion presented above can easily obtain NSF Grant No. PHY-2014165. mΦ ≫ 4πm =g , it cannot saturate Λ ∼ mΦ. In this model, pffiffiffi X X mΦ ∼ λf which can satisfy mΦ >m =g ∼ f=q while X X —n n perturbation theory is still under control λq ≲ 1. However APPENDIX: UNITARITY BOUNDS -TO- pffiffiffi SCATTERING in this limit Λ ∼ qf>mΦ, so that saturating the unitarity bound is not possible. Even when one allows λq>1 and When discussing our unitarity bounds, we will follow the instead uses semiclassical methods [58], one still finds that conventions and discussions of Ref. [59,60] and for sim- it is impossible to saturate the unitarity bound in this model. plicity we will be working in the limit of massless particles. The importance of multiparticle emission in Stueckelberg Any S matrix can be decomposed into S ¼ 1 þ iT.The theories could have been anticipated from this partial UV identity matrix describes the situation where particles pass by completion. In the UV completion, the symmetry breaking without interacting while the transition matrix T describes mass term arises from a higher dimensional operator of nontrivial processes. The states we will be considering have a dimension ∼q, see Eq. (21). Discovering the bad high energy continuous label P, the total momentum, and discrete labels behavior of a higher dimensional operator of dimension ∼q α. These states are normalized as requires ∼q states. The IR Stueckelberg theory should match 0 0 4 4 0 the UV Higgs theory at the scale of the radial mode. However, hP ; α jP; αi¼ð2πÞ δ ðP − P Þδαα0 : ðA1Þ the IR Stueckelberg theory does not know which UV Higgs ˆ theory to match onto. Thus all it can do is at a given energy The amplitude M is defined by scale E, match onto the UV Higgs theory which has 0 α0 α 2π 4δ4 − 0 Mˆ mΦ ¼ E þ ϵ. As the energy scale E increases, the IR theory hP ; jTjP; i¼ð Þ ðP P Þ αα0 0 0 4 4 0 has to match onto different UV theories with larger and larger hP ; α jSjP; αi¼ð2πÞ δ ðP − P ÞSαα0 : ðA2Þ mΦ and hence larger and larger q. Thus when scattering ˆ particles at higher and higher energies, the dominant final The main result that we will use is that jMαα0 j ≤ 1 for all α state involves an ever increasing number of particles. and α0 at tree level. When α ≠ α0, this statement follows directly from the conservation of probability. For α ¼ α0 we VI. CONCLUSION use unitarity X X † ˆ ˆ 2 In this paper, we considered Stueckelberg gauge bosons 1 ¼ δαα ¼ SαγSγα ¼ 1 − 2ImMαα þ jMγαj : ðA3Þ coupled to nonconserved currents broken by mass terms γ γ and calculated the bounds on these models coming from mono-lepton+MET events at the LHC. For large gauge From this we have boson masses, these constraints are only a bit weaker than X ˆ ˆ 2 ˆ 2 even the strongest of bounds, such as g − 2 experiments. 2ImMαα ¼ jMγαj ≥ jMααj ðA4Þ For most masses below a keV, these constraints are the γ strongest bounds on these models. 1 ≥ Mˆ 2 The strength of these bounds comes from the exponential which can be massaged into the form jRe ααj þ ˆ 2 ˆ growth of inclusive rates, a feature only present in the jImMαα − 1j . From this we see that jReMααj ≤ 1.Since ˆ ˆ Stueckelberg limit [31]. This growth is a double edged Mαα is real at tree level, we have jMααj ≤ 1. ˆ sword. On one hand, it allows one to use very crude We now have the unitarity bound jMαα0 j ≤ 1 so we can measurements to place extremely stringent constraints. On apply it to the theory described in Eq. (4). We will be the other hand, it does not benefit from precision mea- considering the initial and final states each with a neutrino ν surements so that it is not easy to improve on the constraints and n Goldstone bosons ϕ. We define our states as without access to a higher energy environment. For Z example, a 100 TeV collider would improve upon the 4 −iPx ð−Þ n ð−Þ jP; n; αi¼Cn d xe ϕ ðxÞ να ðxÞj0iðA5Þ LHC bounds by about an order of magnitude.
035034-6 BOUNDS ON GAUGE BOSONS COUPLED TO NONCONSERVED … PHYS. REV. D 104, 035034 (2021) where Cn is a normalization constant, (−) are the part of the fields that contain the creation operators, and α is the spinor index of the fermion. These states are normalized as
αα_ 0 0 4 4 0 P hP ;n; α_jP; n; αi¼ð2πÞ δ ðP − P Þδ 0 ðA6Þ nn E where E is the total energy. This normalization was chosen to reproduce Eq. (A1) in the center of mass frame, which we will be using from here on. From this, we have the normalization constant FIG. 3. The decay W → nϕ þ l þ ν. 1 1 E 2n−1 2 ¼ 2 1 − 1 ! 4π : ðA7Þ jCnj ðn þ Þðn Þ
We can finally calculate the amplitude of interest using have the same helicity. A short calculation gives the Eq. (5) to give amplitude Z 2n 2 2 mν ig P ϕðxÞ s g g2A 0 α − 4 ν¯ X L ν α Mˆ W1 W1 → W1 W1 X hP ;n; jð iÞ d x ðxÞ ðxÞjP;n; i j ð þ þ Þj ¼ 4π 32π2 ðA12Þ ð2nÞ! mX mX ¼ð2πÞ4δ4ðP − P0ÞiMˆ ðν þ nϕ → ν þ nϕÞðA8Þ Requiring the unitarity bound jMˆ j < 1 be saturated at the center of mass energy Λ , we arrive at the result jMˆ ðν þ nϕ → ν þ nϕÞj a ffiffiffiffiffiffi 2n−1 p 2 gXmν gXE 4π 32π ¼ ðA9Þ Λ mX 2m ðn þ 1Þ!n!ðn − 1Þ! 4πm a ¼ 2 : ðA13Þ X X gX g2A Imposing jMˆ j ≤ 1 gives the results shown in the text. For Mˆ reference, the amplitude is related to the more familiar W matrix element M by normalization constants and phase 2. Decay width of the boson space integrals Here we compute the decay width of the W boson into a Z lepton, a neutrino and n gauge bosons. The decay is ˆ ⋆ dominated by the decay into longitudinal modes which Mαα0 ¼ CαCα0 dΦαdΦα0 Mαα0 : ðA10Þ we calculate using the Goldstone boson equivalence theorem. The phase space differentials dΦα will be given below. The relevant process is shown in Fig. 3. We denote the four momenta of the W boson, the outgoing lepton l,the U 1. Unitarity bounds on an anomalous ð1Þ intermediate neutrino and the outgoing neutrino as pW, In this section we calculate the unitarity bounds on an pl, q and pν, respectively. We also denote the collec- anomalous gauge theory in a manner analogous to what we tive momenta of the n Goldstone bosons ϕ as did for n-to-n scattering. We leave unitarity gauge by a pϕ ¼ p1 þ p2 þ þpn. We will ignore the masses of ϕ ν gauge transformation ϕ=mX. Because the theory is anoma- , and l throughout our calculation, except those appearing lous, an anomaly term is added to the Lagrangian in the neutrino-Goldstone boson coupling. Wewill consider a single neutrino flavor, and later generalize to the standard g2A g ϕ three-flavor structure. The matrix element is L ⊃ 2 X a ˜ a 2 W W ; ðA11Þ 32π mX M − g2ffiffiffi ¯ γ 1 − γ ϵα i=q κ ˜ μν 1 μνρσ a i ¼ p uðplÞ αð 5Þ 2 nvðpνÞ; ðA14Þ where W ¼ 2 ϵ Wρσ and W are the gauge bosons with 2 2 q 1 A which Uð ÞX is anomalous and is the anomaly coefficient. where κn denotes the coupling of neutrinos to n-Goldstone 1 We will consider 2-to-2 scattering of W gauge bosons bosons, obtained from Eq. (5). We see that κn ∝ γ5 for odd via the Goldstone boson ϕ, which contains all of the values of n. However, it can be checked that iM is the same leading high energy behavior in Feynman-’t Hooft gauge. for both even and odd values of n. The amplitude can be The largest amplitude occurs when all four gauge bosons squared to give
035034-7 EKHTERACHIAN, HOOK, KUMAR, and TSAI PHYS. REV. D 104, 035034 (2021) X 2 1 2 2 g2κn β α 1 jMj ¼ pffiffiffi Tr½γ p γ qpνq ð1 þ γ5Þ Παβ; ðA15Þ 3 3 2 2 l q4
pWαpWβ where Παβ ¼ð−ηαβ þ 2 Þ. The factor of 1=3 comes from averaging over initial W polarizations. After some algebra this MW can be reduced to, X 2 1 2 g2κn 1 αβ 2 α β 2 α β jMj ¼ pffiffiffi ð8g ½q ðp · pνÞ − 2ðq · pνÞðq · p Þ þ 32ðq · pνÞp q − 16q p pνÞΠαβ: ðA16Þ 3 2 2 3q4 l l l l
Let us now integrate over the phase space of the n ϕ and the outgoing neutrino. To this end, we will use the following identities involving k-body phase space of massless particles [60], Z Z 3 1 3 1 1 2k−4 Φ Φ ≡ d p1 d pk 2π 4δ4 − E kðPÞ¼ d kðPÞ 3 3 ð Þ ðp1 þ þpk PÞ¼ ; ðA17Þ ð2πÞ 2E1 ð2πÞ 2Ek 8πðk − 1Þ!ðk − 2Þ! 4π Z μ 1 dΦ ðPÞp ¼ Φ ðPÞPμ; ðA18Þ k 1 k k Z 1 Φ Φ 2 d kðPÞp1 · p2 ¼ k kðPÞP ; ðA19Þ 2ð2Þ ffiffiffiffiffiffi p 2 with E ¼ P being the center of mass energy. Utilizing the above identities and doing the contraction with Παβ we get, Z 1 X 8 κ 2 2 Φ 2 g2 n ðpW · qÞðpl · pWÞ nþ1ðqÞ dΦ 1ðqÞ jMj ¼ pffiffiffi ðp · qÞþ ; nþ 3 3q2 2 2 l M2 ðn þ 1Þ W 2 2n−4 2 g2κ q 1 2ðp · p Þ ¼ pffiffiffin 3ðp · p Þ − W l : ðA20Þ 3π 1 ! − 1 ! 4π 2n−2 W l 2 2 2 ðn þ Þ ðn Þ ð Þ MW As a final step, we go to the rest frame of the W and integrate over the lepton momenta to get,
2 2n−1 2 g2M κ 1 ΓðW → l þ ν þ nϕÞ¼ W n for n>1: ðA21Þ ð4πÞ2n 16πðn!Þ2ðn þ 2Þ!ðn − 1Þ
Here we have also multiplied by a factor of 1=n! to account for the n identical ϕs in the final state. The total decay width of the W-boson into a final state containing an arbitrary number of ϕs is then obtained after summing over n, X 1 2 2 4 2 Γ ≡ Γ → ν ϕ g2mν MWgX 1 1 2 3 3 5 MWgX BSM ðW l þ þ n Þ¼16π 96 4π 2F4 f ; g; f ; ; ; g; 4π : ðA22Þ n>1 × MW mX mX
The n ¼ 1 case is special as there is a soft divergence leading to a log enhancement of the form log ðmW=mXÞ. As this log is only present for n ¼ 1, we conservatively neglect the n ¼ 1 contribution to the decay width.
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