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 þÁÁÁÞ ð þÁÁÁÞ, 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.