Probing New U (1) Gauge Symmetries Via Exotic Z→ Z Γ Decays

JHEP03(2021)120 γ B 0 B de- and Z γ gauge B U(1) vertex → models, L 0 BL γ − Springer Z decay are Z B B − U(1) 0 γ → 0 March 11, 2021 B Z U(1) : Z January 11, 2021 Z U(1) − : November 23, 2020 : Z → and Z L − B Published Accepted Received U(1) gauge symmetries. dijet resonance searches are currently exotic decay, and emphasize how the Published for SISSA by https://doi.org/10.1007/JHEP03(2021)120 0 B γ U(1) Z 0 B Z decay width in → γ [email protected] 0 , Z Z → Z Mainz Institute for Theoretical Physics, & gauge symmetries via exotic gauge theories involving Standard Model (SM) fermions typically . 3 U(1) U(1) 2010.00021 decays The Authors. denote the SM baryon and lepton number symmetries. For c Anomalies in Field and String Theories, Beyond Standard Model, Eﬀective γ L

New 0 Cluster of Excellence , [email protected] + decay is emblematic of new anomalous Z and γ 0 B Z → → E-mail: PRISMA Johannes Gutenberg University, 55099 Mainz, Germany Open Access Article funded by SCOAP ArXiv ePrint: a projection for aZ TeraZ factory on the Keywords: Field Theories, Spontaneous Symmetry Breaking cay width is entirelygauge induced symmetry, by the intragenerational existence mass ofa splittings. two heavy, distinct anomaly-free In sources set contrast, of of chiral for decay symmetry fermions width. breaking to enables have an Weweaker irreducible show than contribution that previously to estimated, the the andmore current low-mass constraining. LEP limits We on present a the summary exotic of the current collider bounds on decoupling properties of thesefunction, new reviewing fermions, the called connectionoperator. anomalons, between the in We full the calculate model thewhere and exotic the effective Wess-Zumino symmetry, each generation of SM fermions is anomaly free and the exotic Abstract: require additional electroweak fermions for anomaly cancellation. We study the non- Lisa Michaels and Felix Yu Probing new Z JHEP03(2021)120 10 11 16 19 ], the 6 , 4 symmetry ]. Charging 1 ] or providing U(1) 8 , gauge extensions 7 , 5 , U(1) 3 , 2 3 3 4 γ 0 B symmetries are ubiquitous in beyond the symmetry Z → symmetry B-L U(1) gauge bosons, which give promising dijet and – 1 – B B Z 0 14 ]. Z U(1) boson is light as well as when the 17 6 U(1) 0 – ]. New U(1) 9 Z 5 gauge symmetries – 3 20 U(1) 6 vertex 1 γ 18 0 L Z − B B → ], imposes more anomaly cancellation conditions which necessitates introducing 2 Z full vertex calculation U(1) U(1) The focus of this work is exploring the phenomenology of new A.1 Procedure for reproducing the Goldstone boson equivalence ansatz from the 3.3 2.1 Anomaly-free models:2.2 gauged Anomalous models: gauged 3.1 The generic3.2 vertex structure individual baryon number orportals lepton number to global dark sectors, symmetries as [ in refs. [ of the SM, particularlyis when an the anomalous global symmetry of the SM. Recent work on mapping out the possible number [ new fermion fieldsconstruction with is electroweak the quantumdilepton prediction numbers. resonances of at new colliders A [ Standard prime Model advantage (BSM) of physics,SM, since such such as they a grand are unified models motivated that by gauge the numerous SM extensions baryon minus of lepton the number [ The Standard Model (SM) exhibitsmass a terms chiral for electroweak the gauge elementary symmetrygauge quarks under and which anomaly charged bare leptons cancellation arefermions, forbidden. imposes without Correspondingly, conditions which on thethe SM the SM would gauge fermions suffer under quantum a an numbers failure additional of of gauge renormalizability the symmetry, [ such as baryon number or lepton 1 Introduction 6 Conclusions A Effective operator treatment for 4 Collider searches for anomalons 5 Coupling-mass mapping for 3 The Contents 1 Introduction 2 Models of additional JHEP03(2021)120 , B 5 decay U(1) γ 0 B models by Z bosons. The → 0 gauge symme- Z Z U(1) B , we calculate the boson corresponds 0 3 plane for the U(1) Z 0 B Z and m extensions of the SM where L ]. − interaction vertex induced by vs. B 23 γ are nonzero after summing over , ]. X U(1) g − B 22 27 U(1) 0 Z ]. When gauging baryon number, the U(1) − 21 – × Z 18 2 Y , we discuss the current collider constraints symmetries are spontaneously broken. 4 – 2 – U(1) 0 Z gauge symmetry. Using a UV complete model, we and and B Z U(1) . We give a comparison to an effective operator approach vertex is the critical ingredient to calculating the exotic U(1) 6 γ are the usual SM gauge bosons and the × γ 2 L − . 0 Z A and − SU(2) Z vertex function by imposing the appropriate Ward-Takahashi identities Z ], but in order to calculate the decay width, we require the full vertex γ ]. Specifically, we calculate the , we review the two concrete models of 26 − , 24 2 0 25 Z vertex does not decouple but instead approaches a nonzero constant. This non- decay, which is a new and attractive channel for studying light − γ Z γ 0 − 0 Z gauge extensions to the SM include refs. [ In section After performing the vertex calculation, we discuss the phenomenology of the anoma- Knowing the In the present context, where we consider the interaction of three distinct neutral Z → − the anomalons form vector-like representationswe under also the present SM themodel. gauge current groups. We suite conclude In in of section and section constraints a in critique of the thecalculation Goldstone in appendix boson equivalence implementation in the tries and their respectivegeneral anomaly cancellation conditions.(WIs) In on section all externalon electroweak currents. charged anomalons, In applicable section to general lons in the UV-completeterm. theory Additionally, and as the thedecays, effective anomalon we theory fields can in exhibitmeasurements matching non-decoupling also of to in constrain the a loop-induced observed the Wess-Zumino scalar Higgs parameter boson. space of realistic gauged Z corresponding vertex diagrams are reminiscentcalculation of [ the standard Adler-Bell-Jackiw anomaly structure and not only the divergences of each current [ fermions, where the to a new, spontaneouslyelucidate broken the matching conditionoperator, from particularly since heavy both anomalons and the effective Wess-Zumino current vectors mediated at 1-loopnon-decoupling by fermions limit with of vector andZumino the axial-vector term virtual couplings, [ the fermions gives a contribution to the effective Wess- loop processes is entirelyHiggs reminiscent processes. of the For behaviorZ instance, of the the SM heavy fermionsdecoupling mass in behavior limit loop-induced is for familiar loopsSM, from as of the the anomalon heavy established fieldssection fourth exclusion in of generation of the the quarks 125 a GeV would Higgs pure dramatically boson four enhance by generation a the factor overall of cross nine [ anomaly coefficients the SM quark fields, requiring thefor introduction ultraviolet of new (UV) electroweak fields,the consistency. called unbroken anomalons, phase Correspondingly, when theMoreover, all as anomalons spontaneous a breaking consequence must of of be their all chiral massless gauge couplings, the symmetries in virtual is effects turned of the off. anomalons in U(1) JHEP03(2021)120 . 3 L 1 xL and − B boson U(1) 2 boson, 0 / 0 is lepton Z -breaking Z U(1) ], the only and L 1 = 3 L × charge of anomaly can- U(1) B 2 Y L L − U(1) B U(1) boson mass dictate U(1) P 0 ]. For any choice of × Z 3 U(1) 2 L and ]. xL − and Higgs decays. 30 SU(2) B Z is baryon number and U(1) gauge symmetries are the familiar B × 2 L and SM leptons carry symmetry U(1) , we have 3 ) and the second category of anomalous / L B-L 1 SU(2) is also anomaly-free. − = 0 B L x – 3 – U(1) − ]; moreover, with the introduction of right-handed U(1) 4 B boson mass is clearly independent of the SM masses: gauge symmetries -breaking vacuum expectation value (vev) of the un- 0 is a real multiplicative factor [ charge of symmetries, where Z x B B U(1) U(1) U(1) ) is whether there are one or two scales of chiral symmetry ]. Thus, we must add new electroweak matter to cancel the B 2 [ , where 2 xL gauge symmetry, the SM field content is augmented by three elec- / U(1) − 3 symmetry has been studied extensively in the literature to prevent L B − L − = − B U(1) : we can thus gauge an arbitrary subgroup of this product symmetry group B -charged Higgs field plays a critical role by giving mass to both the L ], and the low-energy theorem in Higgs physics [ B 24 ]. From this perspective, the non-decoupling behavior of the chiral anomalons gauge symmetries (such as and and the anomalous U(1) U(1) U(1) L 29 L , × × − , the total contribution of SM fields to 28 boson, and the fermionic degrees of freedom are further in the UV. Nevertheless, in an U(1) 2 Y B B In the second category, the We emphasize that the essential distinction between the first category of anomaly- From a top-down view, the global symmetries of the SM with Dirac neutrino masses is 0 U(1) 6= 1 Z 2.1 Anomaly-free models:For gauged the classic troweak singlet right-handed neutrinos, which are requiredanomalies. to cancel We the remark that each generation of fermions satisfies the effective field theory (EFT) description wherepossible the anomalons scale are integrated suppression out of [ vev the [ higher dimensionis operators responsible is both simply for the of the vectors Wess-Zumino term [ that arises in loop-induced vertex functions and the anomalon fields.the In physical effect, fermion hierarchies masses between andthe the the gauge Yukawa resulting couplings coupling separation determining determining between the izeable, the particle this species. mass splitting Sincethe is these possibility stable that couplings under in are a renormalization renormal- newa group sector of evolution, physics, giving the rise first kinematically to accessible state will be a simple Abelian Higgswhere model the can corresponding Higgs servenot boson as appear can the in be UV the made completion mass heavy of spectrum. by the a massive largederlying quartic coupling and effective operators at low energies, which can be probedfree in symmetries (such as breaking. In the first category, the x anomalies is nonzero.U(1) For example,mixed for anomalies and alsointroduction, ensure since all the other gauge anomalon anomalies fields vanish. do not As mentioned obey in decoupling, the they will contribute to neutrinos for neutrino masses, gauged U(1) without affecting the SM Yukawaby structure. All distinct possibilities can be parametrized Two particularly interesting models ofU(1) additional number. AllThe SM gauged quarks carry proton decay in grand unified theories [ 2 Models of additional JHEP03(2021)120 , × R ]. L L and and 34 (2.1) , 1 2 L / L vector as SU(2) , summing 0 B is the hyper- ) = 3 Z charges B B , from ref. [ B ) = 0 Φ scalar Higgs field Y U(1) B B q U(1) × ( ], where 2 L U(1) 34 mixed anomalies from the , but this gives a nonzero Higgs field 3 B / B , this is why the net contribu- 1 (SU(2) B 3 A 1 1 1 U(1) 2 2 2 3 U(1) mixing [ − − − × , ], the net effect of an anomaly-free 0 U(1) B 2 Y ) Z 31 q − µ Y 2 2 / / 1 1 B U(1) qγ 0 0 0 1 1 ( − − anomalies, which are correspondingly can- − − symmetry U(1) can alternatively carry 0 3 B and Bµ B R Z B – 4 – L 1 3 N , which mimic the SM leptons in their electroweak U(1) X 2 2 1 1 1 1 1 R g U(1) U(1) N and SU(2) × and L 2 L L ⊃ B N , and L R L L R R ]. Any set of fermions introduced to cancel these anoma- Φ L L E L E N N 2 [ Field U(1) N SU(2) , 2 P ], we will hence consider a minimal set of colorless anomalons, R / 3 E gauge coupling. The SM quarks always have vector couplings to 34 , − – . Again, all SM quarks carry charge . Since the new fermions come in vector-like pairs under the L B . Note that B 32 1 E R , ) = N R U(1) B U(1) is already zero, the new fermions do not need to carry color charge, but they L , B and L fields cancel the U(1) L L Quantum numbers of colorless anomalons and the is the R N × U(1) E X boson. × 2 Y gauge symmetry, there are no new pure electroweak gauge anomalies. The g 2 C 0 B Y Following refs. [ Z , and , respectively. These charges also satisfy the trace condition, Tr are shown in table (U(1) 2 L Table 1. E SM quarks but introduce celled by − over all fermions, which is necessary tocharge avoid gauge large boson. We write the interactions of the SM quark fields with the denoted by quantum numbers. The gauge charges forΦ the new fermions and the U(1) where the anomaly coefficient to theA mixed electroweak anomalies: lies must necessarilySU(3) keep the othermust carry gauge electroweak charges anomalies to cancel zero. the mixed electroweak-baryon Since number anomalies. the mixed anomaly mass difference of the respective fermions. 2.2 Anomalous models:Our gauged exemplary modelbaryon for number gauging anomalous global symmetries of the SM is gauged cellation conditions independently. As wetion will see of in an section entire generationprocesses. of Hence, mass similarly degenerate to SMset the fermions of GIM fermions vanishes in in mechanism chiral anomaly-induced [ anomaly-probing interactions is proportional to a charge-weighted JHEP03(2021)120 , 0 Φ B N R Y y H Z ν λ 0 , B µ (2.7) (2.8) (2.9) (2.2) (2.3) (2.4) (2.5) + γ E Z 3 y T R , (2) E → -mediated at a scale L = R µ γ y ν γ Q B H + ¯ L 2 W ν s - and µ U(1) Z γ R . ¯ 1) 2 E | − + R ( Φ | , L L N 2 | ν R , E Φ E µ H + ¯ R L , | µ γ ¯ R Φ N γ / L E H N DN nonzero and the others zero. 1) h.c. µ i 2 W y λ s γ 4 R − / DE ( y − − ¯ + i 1) N R , L 4 L R | L 3 ¯ L − ¯ E + E ¯ ( y E E , is needed in order to give mass to all Φ | R L , + + Φ Φ , ˜ Φ 2 + ¯ HN HE E L L R y λ R R ¯ R e / E E with (2.6) , + DN ¯ ¯ µ L L 1 i µ − E 4 L 2 γ scalar y L γ / y 2 y DL y , | E ¯ L i – 5 – N 1) µ 0 B − Φ − R − | µ 2 W + γ Z − + ¯ R L s 2 Φ ( R J ) exhibits two sources for generating masses for the R couplings correspond to SM-like Yukawa terms and µ L L R , (2) + 4 + ∗ ¯ Yuk 0 , we write the Lagrangian as Bµ E L y 2.4 , the current interactions of the anomalons are ˜ R 1 − L Φ HE HN L ¯ Z 2 / 1 T E 2 DE + -breaking field L N − L L ) | i X + µ ¯ B ¯ ¯ R / L + L L R g R Φ DL couplings become SM vector-like masses for the anomalons and γ e 1 3 L i ¯ µ R R , E y y µ current. Adopting the electromagnetic charge 3 , e + y kin L e N 1) e y γ D R R . We will also assume the Higgs portal scalar coupling µ ¯ µ y Z L L − | U(1) − − + B , − ν ν 2 + ¯ γ J 1) ( 2 µ µ = = = = L y √ R , which will trigger spontaneous breaking of − γ e / Z , = ( (2) ( ¯ 0 1 2 L µ ) N 1 U(1) , and R R kin γ φ y R W g Yuk R e e + E c L < ν scalar L L y L + + ¯ + ¯ , L are the cosine and sine of the weak angle. As alluded to above, these , + + ¯ 2 Φ 2 W L L N Φ T L s L µ e e ) µ v y ν W µ µ µ EM L γ + s µ γ γ J 1 γ , e µ 2 (2) 1) 1) L 1 2 − A L ν − − Φ = ( L and ¯ ( ( N ν gauge anomalies). ]. Similarly, we will ignore a possible kinetic mixing term between the hypercharge EM L L L = ( W + ¯ e + e e e acquires a vev, since the anomalons come in vector-like pairs under the electroweak L chiral symmetry. Hence, at least one source of chiral symmetry breaking, either the 35 c L = ¯ = = ¯ = ¯ Φ − B L We note that if the dominant source of the fermion masses follows the second case, then It is straightforward to write the anomalon contributions to the SM The Yukawa Lagrangian in eq. ( For the model content in table The anomalon masses are protected by the SM electroweak chiral symmetry and the µ 0 Z B int µ B , where Z J µ EM L J Φ J where Weyl fermions can benonzero and paired the into other Dirac Yukawas zero, fermions or in (2) twothe limiting fermions cases: will greatly impact (1) the Higgs diphoton decay rate and be phenomenologically and gauge symmetry. The mimic the lepton Yukawas ofof the SM (as well as the role of thecurrents SM as leptons well in as cancellation the decay [ field strength and the baryon-number field strength in theanomalons. calculations. The once We will assume v is negligible throughout this work, although it can lead to an interesting U(1) Higgs vev or theof vev the of fermions. the JHEP03(2021)120 . E m ]. In (2.12) (2.13) (2.10) (2.11) = 23 L ν should not m e 5 , and as such, = γ . 4 µ y 4 γ M boson in this lim- and 3 2 to Z and a neutral gauge ν 1 boson decaying to a + y f µ Z γ ¯ NN, 1 2 , and 2 scattering, which is cured by Φ ν φ √ v / + ¯ Φ ν , VV v E µ 1 + N and vector-like under the EW sym- γ → y 5 B ¯ N γ f ) 1 2 = µ f m γ ee N 3 2 U(1) − ν + ¯ m − + ¯ ¯ νν µ EE (¯ E γ µ – 6 – 1 2 γ , and Φ Φ φ boson vanish. Explicitly, we have 2 φ v 2 W v ¯ E √ Z N. E, ¯ / Es µ + 1 + Φ 1 + γ v 5 + e γ E ]. Aside from the role of the axial-vector coupling in per- e 1) E L µ y µ 5 − 36 γ γ m γ m ( 2 3 = µ ¯ − E − γ E − 2 W 3 2 + = s couplings, the effects on Higgs observables are diluted and can be µ m e γ + µ , + N 2 1 vertex µ 2 γ y mass 1 γ 2 √ 1) L γ − 1 2 0 / ¯ N − Φ ( Z will consider degenerate charged anomalons with mass v e e e , and + L 4 E y = ¯ = ¯ = ¯ y → , µ 0 Z B = µ L Z J µ Z EM and decay. On the other hand, if the dominant source of the fermion masses arises y J leads to perturbative unitarity violation in L J 3 m V γγ -channel Higgs insertion [ We emphasize that the vanishing of the axial-vector coupling to the Correspondingly, we will mainly adopt the first case, as suggested by our naming con- s boson and a photon, where the loop is mediated by fermions. Note that for the partial → 0 We now calculate the partialZ width for anwidth exotic decay mediated of by thehas one SM to fermion, depend the onintermediate the anomaly fermions anomaly have is vector prescription. and certainly axial-vector We not couplings consider to cancelled the each general massive and gauge situation the boson, where result the mining the appropriate WIs in triangle diagram calculations of triple gauge3 boson vertices. The 3.1 The generic vertex structure iting case is intimately tiedthese to mass the eigenstate vanishing anomalons of are the chiralmetry. under Yukawa couplings In general, a nonzeroboson axial-vector coupling between a fermion the turbative unitarity violation, we will focus on the role of the axial-vector coupling in deter- The neutral gauge currents for these fermion mass eigenstates become where be confused with thesections SM neutrino and electron. For simplicity, our numerical analysis in vention, where the fermion massand eigenstates the are axial-vector vector-like under couplings the to SM the gauge symmetry particular, the charged matterH fields will behavefrom as the non-decoupling contributionsconsistent to with the current measurements ofHiggs Higgs physics couplings. and We direct discuss searches the of constraints the from electroweak anomalons in section excluded from the observation of the SM-like nature of the 125 GeV Higgs boson [ JHEP03(2021)120 1 p ′ ν are ρ Z γ (3.2) (3.1) a g , , v # g 2

m 1 2

m p 2 p are generally − m b + m 2 b ]. From Lorentz 2 ) − gauge bosons, + . 2 + / p 2 k 0 38 p b ) 1 / and + b Z gauge symmetry, its +

vertex function.

+ / 2 a a

p + − p 1 a γ / k

, + b as possible shifts in each + k

b and + (

+ − #

/ k b U(1) k + ρ k 0 2 k Z ( γ Z 2 m 2 ρ m − and γ m breaking. The corresponding matrix m − 2 m m + a Z 2 B − − 1 ) m + + m 1 2 / p 2 ) 1 2 ) p − 1 / p 2 / p + U(1) + 2 1 p p + ) / a b / is the loop momentum, as depicted in − − a b µ − − / b a / + + Z k b a + + ) ) / k / k 2 2 + + k k + + p p ( ( / k / k ( ( ) ) k k ∗ ρ ∗ ρ + 5 5 ( ( ε ε – 7 – ) ) ) ) γ γ ]. The explicit choices of 5 5 1 1 ν ν p p γ γ γ γ ( ( 0 0 37 ′ µ µ ν ρ ∗ ν ∗ ν Z Z a a γ γ Z γ ε ε g g ) ) Z Z a a 2 2 g g + + p p ν ν + + + + γ γ µ µ 0 0 1 1 γ γ Z Z v v p p g g Z Z ( ( v v 2 1 ( ( p p g g µ µ a ( ( ε ε × × " " + X X k Tr Tr gg gg 4 4

) )

k k 1

p 4 4 − 2 π π

p EM EM d d

a + The triangle diagrams corresponding to the (2 (2

+ a

are the electric charge and the mass of the fermion in the loop, k Z Z + Qe Qe k − − × × m = = 1 2 . Since the photon mediates an unbroken, non-chiral and 0 are the external outgoing momenta and . We introduce the arbitrary constant four-vectors Z M M Figure 1. Q i i 2 1 p The corresponding matrix elements are given by We remark that, in the mass basis, the fermions may have flavor-changing neutral current couplings if 1 and µ Z and parity symmetry, the most general expression for the vertex function is the sum oftheir the masses arise fromelements both would then chiral involve sources mixing angles ofstates. and electroweak triangle and diagrams with two different fermions as intermediate and figure diagram because the finiteon result the from the choice cancellation offixed of these by the possible applying divergent external integrals shifts depends physical [ conditions, as first applied in ref. [ where the vector and axial coupling factors to the denoted massive coupling is necessarily vector-like. We show the two diagrams in figure Z JHEP03(2021)120 - 0 2 ], ], ρ 2 z G p | 37 and ν 1 2 with 39 (3.9) (3.3) (3.4) (3.5) (3.6) (3.7) (3.8) p , p | (3.10) (3.11) (3.12) || w 2 µ 2 1 and p p - and , || , 38 | , 0 1 1 w p w p )) )) µν | , , G 25 m ) + m µρ ( ( 2 α α 0 µ 1 0 ) 2 1 2 , p C p p C 1 0 z p 2 1 2 2 Z a , p p Z a p p . g 1 ( . , g 0 0 2 6 | | p = Z 2 Z v 2 ( v − − F G p p 3 g g µ || || α α ]. The WIs become 2 2 are all finite and hence F 1 + 1 1 2 a , p p p p | m | m 6 σ ρ 1 + 42 and ) ) 4 2 F , ], which makes the p µρ µν 2 2 | p ], µ 2 0 1 − 2 p p ) ) p 41 39 p ) )+4 · · to | , ) G 0 2 0 1 0 27 0 || 2 ) 1 1 1 1 p Z 0 Z a a G G µνρσ p p p || | g g Z F a ( ( 1 ) g Z + + Z v p v , µν | z g g Z | v 2 2 2 ; 2 ) to avoid a non-chiral anomaly [ νρ g p p 2 p + + 2 ), we calculate the finite form factors 0 · 6 || ) + a , m, m, m Z µνρα µνρα , p Z a 1 a 2 1 , p F by isolating the divergent piece of the 2 1 p g g Z Z 0 a 3.2 | 1 − p 0 0 p g , p 0 2 0 p − 4 ( Z 0 + − Z v v νρ 1 ( 2 | | g = g 2 F G , m Z 5 v p 2 2 p ) ( g G Z 2 F p p )( b 0 1 2 − · || || 1)( z 1 1 F + 2 1 1 + . In contrast, the two form factors G ) and ( p p , m – 8 – and − | | p p − 0 2 ] using Package-X [ σ ν 2 + 0 0 +1)( − 1 3 5 (0 0 1 G w w µν µν 3.1 p 40 p 0 z | 0 2 F F µ 1 , G 2 ( (( (( | C p | 0 1 | ρ 1 ρ 2 p G − − | 2 2 2 || p p 2 p p p G 1 p ≡ µνρσ || || || p . The six form factors || | + + 1 1 = ( = ( = ( 1 ) 1 ) z p p p | | | | p | µρ 2 2 | m w p p ( ; µρ µν νρ , etc., we set ) µνρ µνρ µνρ || || νρ 0 , and 2 2 β 1 1 ]. Γ Γ and 6 2 p p ) C | | )Γ ρ X X X ν , p 2 p , p 2 1 µ 43 1 w 1 α µρ µρ 2 W W W p to p – gg gg gg p ], 1 , p p c c c p ( 1 ( p − − 2 2 2 into redefinitions of the other form factors: we denote the redefined 2 ν 1 ν 1 41 4 p 27 EM EM EM p p π π π + 2 ( G F ], which implements the calculation procedure in refs. [ 4 4 4 = 3 1 µ F − − 1 νραβ Qe Qe Qe F i + + 27 p = = , ( ), the WIs are given by 0 = = = i | | and 2 2 = ) = F p p 1 3.3 | || || µνρ µνρ µνρ 2 1 1 F p p p Γ Γ | | w, z || ρ ν )Γ 1 ; 2 1 νρ νρ µ p 2 | p p 2 p µ 1 µ 2 . , p νρ − − p p has been reexpressed in terms of the external momenta, z 1 + p − − ( µ a From eq. ( Moving to the specific case in eqs. ( Following ref. [ 1 p and ( µνρ -dependent momentum shifts manifest in the definitions of Γ where is the usual Passarino-VeltmanPackage-X convention scalar [ loop function for the triangle loop, following the form factors as arise from the cancellation ofw divergent integrals, and their values depend on the choice of which absorb and constant scalar prefactors can be calculated inindependent. any Moreover, regularization because of prescriptionspace, the unambiguously: two linear of they dependence these of are can vectors be in eliminated a by four-dimensional using the identity [ where following form factors [ general three-vector vertex associated withcan the be axial evaluated vector using anomaly.z This a divergent momentum-shift piece integral identity [ of the vertex function in Mathematica [ we construct the ambiguous parts of JHEP03(2021)120 z w , and and and (3.14) (3.13) w , weak w c σ B w W b ρ W a ν ] to set ]. Isolating the W 38 -vertices vanish is 29 abc ρ , , the heavy fermion g 2 , 1 3 / 28 , ) 1 σ + − 12 - and Z a ν σ ρ ∂ W ν ) = ) contain a constant, fermion ρ ∂ A m a . The vertex function we study ν ( ) also determine the appropriate + z 0 3.11 W C σ 3.2 2 A 0 µ ρ and m ∂ Z ν w ), and ( Z →∞ ) and ( ( lim µνρσ m 0 µ 2 3.1 Z 3.10 g – 9 – X ), ( g µνρσ B to make the WIs on the 3.9 C X ], which results from the combination of choosing z − gg field, W 24 σ c 0 ) exhibits non-decoupling. EM B and Z e ρ ∂ B 3.10 w ν C , we get in the WIs. Since case, the SM fermions have axial-vector couplings only on the B γ 0 µ L Z − 0 − ) and ( L ⊃ − Z → ∞ B µνρσ anomaly is cancelled by two distinct chiral sectors of fermions. On the 3.9 − m 2 . Moreover, when all fermions are massless or otherwise degenerate, the 0 B Z z g , and a general a X g W U(1) B and C w = and taking z WZ Explicitly, we consider the Wess-Zumino term for the hypercharge gauge field In an effective theory where chiral fermions are taken heavy, the choice of Instead, with a UV-complete model, the WI on each vertex is independent of the At this point, we could naively adopt the method by Rosenberg [ Clearly, each of the WIs in eqs. ( L vertex, and thus choosing for each fermion such that the vector WIs are vanishing and the anomaly contributes , as long as they are chosen the same for all fermions in the UV-complete model. In other bosons to avoid breakingvertex the involving electromagnetic gauge symmetry [ where the coefficient of the weak gauge bosons is negative that of the hypercharge gauge mass limit in eqs. ( gauge fields prescription, as long as the momentum shift is applied consistently forto all parametrize fermions. the momentum shiftchoice in of eqs. the ( Wess-Zuminoand term [ WI on each vertex isanomaly-free also model vanishing. by In fact, requiringz we that can the provide total an WIs equivalentwords, condition are an for vanishing, an anomaly-free independent model of tum is shift insensitive intrinsic to to dimensional the regularization, ambiguity which introduced is by a the gauge-invariant regularization momen- other hand, for the µ consistent for all fermions. choice of z only the axial-vector divergence.the This loop would must be useis wrong, the the however, same first because consistent physical all choiceelectroweak- case fermions of where in this mistake would become apparent, because the mixed mass-independent anomaly piece.have Moreover, fermion mass-dependent the contributions, WIs but only foring when the axial-vector the fermion coupling. massive has the gauge correspond- Sinceeigenstate bosons fermion we also can calculate only in have the one flavor non-zero conserving axial-vector limit, coupling. a given mass JHEP03(2021)120 , the 2 , or

1 ν (3.20) (3.18) (3.19) (3.15) (3.16) (3.17) — and − !# z w ) f = and , m z 0 2 w 2 Z . This provides 3 m ( , − 0 ]. B w ν 43 charges under EM and − = ) , z f . 0 BL,µ | , 2 | 2 | Z 2 p 2 p U(1) , m µ || p , || 1 || 2 Z 1 ) p 1 | p νγ f | p m | the charged leptons, and µρ ( − µν 0 νρ ` , m ` f B X are the X X gg 0 W , m 0 , is gg gg BL,µ decay for these two complete models in 2 Z c W W 2 V BL f f c c Z EM m γ µ Q e Q EM EM m 0 2 Z ( e e for down-type quarks and charged leptons, − B Z 0 `γ m B B C ! 2 Z B 1 − C C 0 and corresponding to the restriction → − 4 Z m – 10 – 4 Z q ≡ e f = = 2 = Z m ) m Q f 6= 0 ) + 0 BL,µ 0 − µνρ µνρ µνρ f . Z a , m Z Γ Γ 1 m )Γ ρ µ ν 2 V including anomalons, the dependence on (

, g 2 1 µ 3.3 0 2 0 p p Z m v qγ B Z p ( C 2 Z g − 3 1 corresponding to the restriction − 0 2 f m m + B and m and electric charge decay width, summing over all contributions for the diﬀerent µ U(1) 2 for leptons, and 1 BL 6= 0 f γ BL 2 W p

3.2 1 g ( c m Z a 2 αα 0 BL = BL f = 0 π , g Z 0 or for Q Z a EM Z v 96 e f BL L g g α → − L Q for up-type quarks and gauge theories, and we provide a full discussion on the eﬀective approach = f B c Z 0 L . ) = N and Z v gauge symmetry, the SM fermion content (including three right handed neu- gauge boson are given by f γ − 3 g A U(1) L T B with masses U(1) L = +1 0 " BL − = 0 f includes all up-type and down-type quarks, − for quarks and B Z f 0 3 f Z gauge symmetries, and we also abbreviate q a X T B

U(1) g → L The exotic In a full theory with a complete, anomaly-free set of fermions, e.g. for the case of the SM = 3 U(1) × Z = − f c Z v Γ( N B which is the usual Passarino-Veltman bubble scalar loop function [ where neutrinos. fermions of the where the following subsections 3.2 For trinos) is anomaly free and there is no need to introduce new chiral matter. The interactions in appendix fermions in thus the shift dependenceWe — calculate drops the out induced and width the for vertex the can be calculated unambiguously. We see that thesecouplings WIs of are identicalg to the contributions ofa a concrete given matching heavy conditionanomalous fermion for with the effective field theory of a heavy chiral fermion in which then results in the following WI structure: JHEP03(2021)120 . . ] 1 2 45 , mass. 44 Z boson and vector Z compared to keeping 80 . The top quark dominates ), the analytic behavior of 10% BL 2.1 g share the same chiral symme- 0 Z light necessarily reintroduces the . 0 B-L 60 2 Z . The residual finite mass splittings =1 case. In particular, since the anomaly 2 BL g L − =0.5 B [GeV] i U(1) in γ) BL g ). BL BL 40 Z' U(1) ) and figure =0.3 m 3.19 for various choices of – 11 – masses: they are visible as bumps in figure BL g γ 0 3.19 =0.2 Z 0 BL Z' → Br(Z =0.1 BL Z ), it is clear that a mass-degenerate generation of SM g GeV, however, the threshold behavior from the lighter BL 20 g → 10 3.19 Z . 0 Z m 0 vanishes for a degenerate set of SM quarks and leptons. Hence, -7 -8 -9 -10 -11

10 10 10

10 10

vertex can be non-decoupling regardless of the scale set by the

γ → ) ( BL boson. Correspondingly, the expected Landau-Yang behavior [ Z' Z Br BL f γ 0 symmetry, with quark interactions as in eq. ( Q − Z e f 0 B GeV, they have an overall effect of less than Q Z f c is markedly different from the is also self-evident in eq. ( − 20 ) N U(1) B 0 Z γ f Branching fraction for 3 & 0 B T → 0 Z f 0 Z U(1) Z P → m We emphasize that the cancellation of an entire generation of mass-degenerate SM From the structure of eq. ( m Z cancelling fermions can become massive independentlytion of to the the SM Higgs vev,On their the contribu- other hand, intry such breaking a scale, case, and the the anomalons infraredanomalons and limit into the the of spectrum making too. the The anomalon fields we consider are charged as in table 3.3 For gauged Γ( fermions only occurs in thissymmetry case because structure the which SM dictates fermionscouplings share the to the axial-vector same the couplings underlying chiral tofor the become more significantFor for smaller them degenerate. Below fermions can give an enhancementtop of quark the contribution only. decay width by a factor of two compared to the fermions will havethat a vanishing contribution tothe largest the contribution stems partial fromand width, the the intragenerational by bottom hierarchy virtue between and the tau of top leptons, quark the as fact shown in figure Figure 2. the decay width, while the lighterthe SM lighter fermions SM cause fermion small peaks mass when effects they as go in on-shell. eq. We include ( JHEP03(2021)120 # 0 B ) Z f m signal (3.21) ( decays , 0 2 0 B C

. Hence, γγ 2 f Z E ! ]. The ], when the y ) m → M 47 48 , H , ( +2 symmetry, their 0 arise solely from 5 and [ 46 ) C B for various choices 0 L f π M 2 Z y 2 Z 0 B m ,m m Z m 0 . U(1) 2 & . For concreteness, we 2 Z m 0 . If the L3 and ALEPH Φ ) 0 B m M v 5 , but then the contribution ( Z → − 0 X X m g 0 +2 B g B Z (10 ) − m ) O = 3 branching ratio, where f . Such light, electrically-charged ,M 0 2 B for down-type quarks. We remark 0 γ . First, since the anomaly cancelling / ] as decreases: the cusp behavior at the Z ,m 2 Z dependence in the physical width. 0 B branching fraction maximizes around 1 3 Z ] on new electrically charged fermions, MeV masses. We also remark that, as 2 Z m 0 m B Z z − γ 45 ( for a given m Z m as a function of , 0 0 50 B ( , ﬁnding that anomalons whose dominant (1) , or → 0 m γ B 44 boson. Z M = 4 O is 0 B B 49 Z w − 0 B ]. Z γ → ) 0 Z M 0 and B 51 2 Z Z – 12 – → Z 0 B ,M m 2 Z Z 2 Z Z → − ! m m m 0 2 Z ( 4 Z Z 4 Z 0 is necessarily at most enjoy an open parameter space to induce a branching m m m B " γ Φ e f − 0 v 0 B 1 2 Z breaking vev set by Q Z )

m f B 2 Z 2 ( Z 0 Z → then the exotic − for up-type quarks and 3 as a ﬁxed value for the Yukawa couplings m m m T 2 2 Z Z π 3 U(1) 4 m X SM ]. Thus, we indicate the LEP direct search bound on the charged 2 W X

c ∈ . Finally, the turnover feature of the anomalons is also necessary to αα 2 f ) 50 π − 5 , ) = +1 +3

EM − using f . We include the limit on the 96 49 ( α × 3 3 2 Φ (10 T v √ , the anomalons become lighter as ) = O π 3 X 4 γ g 0 B = Z breaking, the decay width of in figure B → Applying the L3 and ALEPH constraints [ We remark on many interesting features in figure We show the branching fraction of Assuming the charged anomalons are degenerate and their masses M X For example, see the model considered in ref. [ 2 g Z Γ( exhibit the well-known Landau-Yang behavior [ the exotic branching of constraints were relaxed, the direct scaling relationship between to the anomalon massesstrength. from the We evaluate SM this Higgs constraintmass in Yukawa would section contribution also comes affect from the fraction of collaborations [ anomalons as a solidparameter circle space on for each the curve, branchingis marking fractions thus left where excluded of their these by masses circles,masses direct cross indicated could searches 90 by also GeV. for thinner receive The lines, large the contributions charged from anomalons. the SM Of Higgs course, vev, the which would anomalon weaken masses scale with the set for fixed maximum of each curve thento marks the when the loop anomalonsanomalons function develop imaginary are by contributions already going excluded, on-shell however, at by searches at LEP by the L3 and ALEPH hadronically, which has beenboson probed will dominantly at decay LEPanomalons to by introduced a the are dijet heavier resonance L3 than for collaboration the masses [ fermions obtain mass from the spontaneous breaking of the chiral that it is anfive flavors excellent of approximation SM (to fermionsrequired as better by degenerate anomaly than with cancellation, 1%) there to is no set theof masses of the first where again U(1) JHEP03(2021)120 ) ], ]. 5 47 − 47 , (10 46 O search [ γ + res ) 250 jj ( =0.1=0.1 → 80 xx gg . The degenerate anomalon Z . . X 0 B g Φ Z decay probe by L3 [ 200 v m γ =0.1) 0 B 0.2 X = . B B x Z Φ 60 g versus =1 v versus x (g L3 γ g and a photon could also be expected → γ 0 are more difficult to reconstruct at the 0.1 B 150 0 0

= B x Z 0 B Z Z g Z =0.2 Z → i U(1) in γ) x i U(1) in γ) [GeV] → g B B Z [GeV] B 3exclusion L3 Z 40 Φ Z' =0.5 v ]. x m g 100 – 13 – for various choices of 52 γ =0.3 x 0 B g Z' → Br(Z Z' → Br(Z boson make this a promising avenue to probe possible Z Z → 20 =0.5 =0.3 x x 50 . The dots mark the point where the anomalon mass equals g g Z 2 √ =1) X / Φ (b) Branching fraction =1 v x (a) Branching fraction (g L3 g 3) 0 0 to a leptonically decaying -5 -6 -7 -8 -3 -4 -5 -6 -7 -8 -3 -4 π/

10 10 10 10 10 10 10 10 10 10 10 10

B B Z' Z Br Z' Z Br (4 γ → γ → ) ( ) ( Z dependent when plotted versus the vev X , in competition with the exotic g 4 − Branching fraction for are set to 10 × M ) resonance reconstructing the γ few + ( jj anomalous gauge symmetries at the LHC.exotic We decay note a of similar the sensitivityfrom improvement in the the LHC experiments,level are where set the by current the branching OPAL collaboration fraction [ limits at the O While the hadronic decays ofLHC the compared relatively light to LEP, the immense statistics and the additional coincident feature of the Figure 3. masses 90 GeV. The grey (purple,and it red) becomes shaded region is excluded by the L3 JHEP03(2021)120 , = Φ v We E (4.1) (4.2) (4.3) y 3 (3.22) , while the , H y . h.c. ± ! 0 + 4 Z 4 Z boson, and the new R m m E W L − ¯ e 1 H v

2 2 H . ) y 0 √ Φ 0 2 Z v , 2 Z . − ) is the ﬁrst possible non-trivial m ) | m ]. For illustration, a hypothetical R H − Z e H v 27 v 3.21 L 2 Z m ¯ H H E y m y H ( v + − X 2 Φ H Φ and the vev 2 W v y √ v c partial width. αα Φ 2 Φ Φ y − y y π – 14 – ( | EM , γγ R 2 2 α 32 H 1 1 E 3 y √ √ → L ¯ E = = = H Φ 1 2 v breaking vev, gives a non-decoupling decay width of 2 M M Φ B y √ − , their collider phenomenology also mimics the electroweakino non-anom. . The charged anomalon mass Lagrangian becomes U(1) Φ ) R v H γ e y L 0 B ¯ e Z ≡ Φ v 2 → y 2 ), we assume the Yukawa couplings are real and for simplicity set Φ y √ Z = 2.4 1 Γ( y ⊃ − and mass Φ ]. L y From eq. ( When the anomalon masses receive contributions from the electroweak vev, they induce Finally, we remark that the expression in eq. ( 53 For a discussion of the Landau-Yang theorem and its applications to non-Abelian gauge bosons, see ≡ 3 L The mass eigenstates coupleDirac to masses the depend SM on Higgs the values with of the Yukawa coupling ref. [ and the two Dirac masses are then they will exhibit decoupling in the y of the 1-loop radiative electroweak corrections. corrections to the observed 125 GeVthis Higgs correction boson as decay into a two coherentanomalons. photons. sum Since We of can we calculate the assume top the dominant quark, source bottom of quark, the anomalon masses comes from have the samephenomenological SM signatures gauge as quantumcontribution fourth-generation numbers arises as leptons. from leptons,sector Given they from their share supersymmetry, dominant many whereelectrically mass neutral of the anomalons charged the and anomalons exhibit same are a compressed slightly mass heavier spectrum, than as a the consequence theless a curious fact thatwith the two decoupling sources of of anomaly free chiral sets symmetry is breaking. not guaranteed in theories 4 Collider searches forIn anomalons this section, we discuss the phenomenology of the anomalon sector. Since the anomalons masses arise solely by the Of course, the SM-like nature of the 125 GeV Higgs precludes this scenario, but it is never- decay width of a massive,also neutral note gauge that boson heavy into sectors twomixed of further electroweak anomaly neutral anomaly free gauge sets bosons. is ofcontribution carried fermions, to in by the virtue two of distinct partialcomplete the vertices, set width, fact can of that as heavy, give the mass-degenerate noted a SM in non-decoupling fermions ref. and anomalons, [ where the anomalon JHEP03(2021)120 γγ γγ (4.8) (4.4) (4.5) (4.6) (4.7) → → H H is dominant , are the two ], the 2 Φ 4

y as a function of ) 54 2 M γγ τ , ( ) GeV. The open region f → )) W A boson, and the two new are only slightly lighter τ W 1 1 H 2 ( τ H ( N √ W v W ≤ f 2 = 300 → H A ] on the signal strength of 1) y Φ M , gg v and 55 − that is allowed by the , τ > ) + − 15 14 b . . 2 ν ) 0 τ W H 1 . ( τ y +0 − , f M iπ where τ )) 2 99 H A 2 f,W . ( ) f − 1 3 M f τ m Φ 1 1 + 3(2 ( 4 A − − f , y 2 τ τ W ) + couplings of the anomalons would guarantee = ) = 0 H H t 1) − − τ √ v y τ , τ but high 1 1 1 ( ( 2 γγ − H √ √ ) f Φ f,W y + 3 M − – 15 – f τ y τ A 1+ 1 τ → SU(2) 2 + √ 4 3 W is the Higgs mass of 125 GeV, and we include the

τ H + ( log 3 (2 H H f 3 → τ 1 4 2 π W M 1 M ( ]. The τ 2 2 (arcsin − f gg 2 τ √ − 2 EM ( 58    – µ α F 128 56 ) = ) = ) = G f τ W τ ( τ ( f being ( f ) = ) are the mass ratios ]. We see that the charged anomalons can have a mild effect on the A W τ γγ τ ( A f 50 , → 49 , we plot contours of the signal strength for uncertainties are given. The hatched grey regions in figure H 4 σ Γ( is the Fermi constant, 1 rate, which is well within the experimental uncertainty when GeV [ F . This is a direct result of their vector-like SM gauge representations. As a result, G 90 γγ H y In our model, if the electrically neutral anomalons In figure Adapting the expression for the Higgs decay to two photons from ref. [ → to the electroweakino and sleptondifficult searches from signatures supersymmetry, for which are the onecompressed LHC mass of splittings experiments the [ more because ofelectroweak the Drell-Yan production presence rates, of butdecay soft for from leptons small the from mass splittings, heavy the charged charged anomalon current to the electrically neutral anomalon would give we expect the bestsearches improvement for in anomalons, testing subject thisthat to parameter there the space is model would also dependence comerate, a in but from their region we direct decay at expect signatures. this very to small Note be excluded bythan electroweak the precision electrically measurements. charged anomalons, we would have a completely analogous situation where the sigma exclusion limits. We alsobelow show the exclusion from LEPH searches on charged particles over the two Yukawa couplings of the anomalons, shows the parameter space allowed by the ATLAS limits [ and the arguments with the function where dominant contributions from the topfermions. quark, bottom The quark, loop the functions are defined as partial width including the two additional charged fermions becomes JHEP03(2021)120 σ 2 ], the most 58 gives two values signal strength ), setting the mass X γγ g → 3.21 breaking Higgs. The vev H B 1.0 and using the SM value for 1.1 1.2 1.3 1.05 U(1) 3 ] or jets, with a subsequent jet π/ 62 , with numerous searches carving 4 0.8 5 for the value of 0 Z 0.6 m 1.01 the one to the H y B Φ y 9 GeV 90 < M 0.4 – 16 – plane. boson searches in the mass region accessible by the U(1) oe Mvalue SM over Br(H→γγ) 0 B 0 B Z ]. It is calculated following eq. ( Z m is marked “L3” and is derived from the search for the 0.2 GeV. The dashed regions are the 47 5 vs. X = 300 g 0.0

3 2 1 0 4 Φ

Φ y v ]. ]. 63 61 – exotic decay [ branching fraction exclusion plot including both charged anomalons, where 59 ]. The grey band denotes the 90 GeV exclusion limit on the anomalon masses jj γγ 55 → → ]. 0 B H 50 Z decay. Our summary plot is shown in figure , , γ γ 49 0 0 B B , where the limit is given by the smaller one. In the case of a discovery, however, Z Z X The first constraint in figure g is the coupling to the SM Higgs boson and → → H for this uncertainty has toconstraint be is resolved largely by supplanted an byresonances the independent triggered more measurement. using recent initial We CMS state see searchessubstructure radiation targeting that analysis photons low-mass the [ [ dijet L3 Z of the anomalons bythe their top nominal quark Yukawa coupling mass.solving of the The other branching SM fraction quark at masses a are given set mass to be of order MeV. Note that 5 Coupling-mass mapping for We now discuss the overall statusZ of out excluded regions in the and we can only adoptthe collider the phenomenology limit of fromneutral direct the anomalon LEP anomalon searches searches. as and a We thelines reserve dark suitability include a matter of refs. dedicated candidate the [ study for lightest of future work. Related work along these exclusion limits [ by LEP [ very soft leptons orpessimistic pions choice and missing of transverse mass energy. splitting As means shown there in is ref. no [ limit from the LHC experiments, Figure 4. y of the new Higgs is fixed to JHEP03(2021)120 ] , ]. γ → 62 0 for B 69 ], our Z 5 34 (in red). → 3 Z = Γ(Υ as a dashed π/ . For masses Υ is taken from = 4 5 R 0 = 1 bosons produced Z Φ bosons produced , we can turn on y Φ m Z y Z 4 12 ] is plotted for two differ- vs. Γ(Z) 10 in figure mixing induced at one- decays, 80 ] is shown in lighter blue. 50 X ) , × 0 g B 63 Υ 3 search by the ARGUS col- Z 49 Z ] is also shown in figure Γ( Υ − 50 , . For this projection, we perform Z B 5 ] (in solid green) and a TeraZ factory 49 60 47 IRj ISR + jj CMS ] for (in orange, dashed) and 34 plane including the limit from the decay of the ] over the number of boson (in gray), both taken from ref. [ [GeV] 0 B ' 70 = 1 Z Z B Z

Φ m )

y γ ISR + jj CMS

3 , m

TeraZ

/ – 17 – that we use for the anomalon masses in deriving

π X 4

Φ y

y L3 ( (1) U for limit Exclusion

. Importantly, as shown in ﬁgure GeV in the

) . The corresponding limit in ]. We also simply assume the improvement in sensitivity 90 B 1

1 < = 0

= .

mixing bound become the leading existing bounds on these

]. This constraint is labeled

Φ 71 M

y 2 [ ( H 0 B 68 U(1) Υ 7 y – Z < 10 ) − 65 − , × 0 1 Z µ 7

0.01 0.50 0.10 0.05 34

. + X g 1 µ data from the L3 experiment at LEP [ → , we also show the constraint based on the jjγ ], where they measured the hadronic ratio of 5 → 64 ]. We also take the limit from ref. [ to increase the anomalon masses, which would make the collider dijet searches, Γ(Υ , which is the same value of Exclusion limits for Z / H decay as a dotted green outlined region in figure 3 GeV, a fuller discussion of constraints from kinetic mixing can be found in ref. [ bosons. 65 ) y , 1 0 B π/ jj bound, and the 34 . Z Finally, we also show a projection for a TeraZ factory searching for the exotic The LEP constraint from the L3 and ALEPH collaborations that new electrically In figure Υ → 0 B = 4 (in purple) and the limit from the width of the Z 0 B Φ hadrons ground or systematic uncertainties. We reserve a such a sensitivity study for future work. a simple extrapolation accounting for theduring increased four statistics of years ofat runtime LEP, of which was theonly FCC-ee arises [ from the increased statistics,should but also we account recognize for that a possible realistic improvements collider in projection mass reach as well as problematic back- nonzero the light Z charged particles must be heaviery than at least 90 GeVthe [ L3 limit. We alsoline. show We the remark LEP that constraintthese for these a curves constraints different are have choice subject of also to set additional model dependence, since laboration [ refs. [ loop by SM quarksm [ The low-mass dijet resonance limitis from shown CMS in based darker on blue, a while dijet the resonance jet + substructure ISR + photon ISR search [ jet search [ Figure 5. Υ limit from projection in lighter green. Theent direct maximum bound values on for the the anomalon mass Yukawa coupling, [ JHEP03(2021)120 B L − U(1) -pole decay model U(1) B Z Z B U(1) case study and B vertex. We find that U(1) decay. This appendix γ γ − 0 Z 0 B mass are the only continuous Z model when this exotic decay 0 → gauge symmetry. In the B − Z vertex function, where the vari- Z B Z decay width. γ decay, searches for the electroweak U(1) γ − γ U(1) 0 0 symmetries is exceedingly rich in both 0 B BL Z Z Z symmetries with the − → → rate, since the nonzero anomaly coefficients dijet resonance searches, and shifts in U(1) Z Z γ 0 Z B 0 U(1) – 18 – Z Z decay is an irreducible probe of the exotic decay search are strongly motivated. → γ γ 0 B Z 0 B to the discussion of the non-decoupling nature of the Z Z rate also crucially retains information about the anomaly A ], is directly related to the fact that two sets of fermions γ → → 0 B 27 gauge symmetry and Z Z Z L symmetry. Moreover, a degenerate set of anomaly-free fermions → − mixing, are all different manifestations of the underlying scale of symmetries of the SM fermions are necessarily discrete. Hence, L B 0 Z B − B Z , including the exotic U(1) U(1) − B gauge couplings that can be probed at current and near future colliders. Z U(1) , shows that all of these diverse probes with their separate considerations decay width does not vanish if all fermions are degenerate. This feature, 5 decay can prove an exciting test of the U(1) γ decay compared to any approach that neglects the SM fermions, especially for U(1) γ 0 B γ 0 0 B Z Z Z boson and a photon. Our calculation, while reminiscent of textbook calculations → breaking for their respective particle content. The corresponding state of the art, 0 → → Z B Z We see that the phenomenology of new We also dedicated appendix In contrast, the SM fermions give non-vanishing electroweak anomalies when gaug- Importantly, the exotic Z Z the is kinematically accessible. realistic new the theoretical complexity and theparticularly phenomenological special predictions. behavior We of have anomalous highlightedthe the non-decoupling of anomalons in the corresponding of backgrounds and systematic uncertainties are nonetheless generally competitive. anomalon fields and theof corresponding induced the Wess-Zumino Goldstone term asdemonstrates boson well that equivalence as an a ultraviolet treatment discussion completethe of theory the predicts markedly different behavior for dictions from charged anomalons at colliders, low-mass observables from U(1) shown in figure ing baryon number. Correspondingly,breaking, when the the anomaly anomalon cancellationotic masses between SM arise fermions only and from which anomalons was ensures also the notedcan ex- in have ref. distinct [ chiral symmetry breaking scales. As a result, the phenomenological pre- gauge symmetries: case, since the SM fermionsentations content under is anomaly-free, they necessarilywould have have vector-like a repre- vanishing contribution to the to a of anomaly coefficients, focusesous Ward-Takahashi on identities the are full obtainedexternal by currents. taking the Wedecoupling appropriate illustrated divergences behavior the of in different the types the of Ward-Takahashi identities anomaly by contributions and considering non- two distinct for different global future improvements on the 6 Conclusions In this work, we have presented the calculation and phenomenology of the exotic compared to the anomalon searches, whichby depend the anomalons. on the The specific decaycoefficient channels contribution employed from the heavythe fermions, SM which fermion matches the content. anomalyparameters In coefficient that fact, of the dictate gauge the coupling exotic and JHEP03(2021)120 , ]. 0 72 and and → (A.1) (A.2) ) -EXC 0 B 2 0 B + , Z π Z m exotic de- c m σ (16 γ ]. Note that / W 2 b 0 B ρ Z BB W 0 a ν Z → A W Z = abc g B 1 3 C + a σ , we obtain the longitudinally ) 0 W ) by using the Goldstone boson ρ Z , f ∂ γ a ν X A.1 0 µν is the pseudoscalar decay constant, B g ˜ W ( boson with the derivatively coupled, F Z will also establish an upper bound on X ), and analyze the validity of assuming 0 µν ϕ/ 0 µ /g Z ), reiterate the discussion about the re- X µ Z → Z 0 0 g B ∂ 3.11 Z 3.2 boson. We will derive the effective operator gg Z → · m )–( µνρσ 0 B 0 0 µ 2 Z – 19 – = Z ϕ 3.9 Z g f 0 X B Z ) and ( g f and recovering the Landau-Yang limit for C B 3.1 3 C − L ⊃ σ B ρ ). By replacing the ∂ ν B 0 µ 3.14 is the dual electromagnetic field tensor, Directions”). Feynman diagrams were generated using jaxodraw [ Z ν ρσ F µνρσ we deduced the contribution of the Wess-Zumino Lagrangian to the boson and the gauge coupling 2 is the baryon number anomaly coefficient from SM fermions [ 0 ], where the authors show a result based on an effective operator inducing g 0 µνρσ B 2 3.1 X / Z g 3 12 2) is the Wilson coefficient from decoupling the anomalons. We remark that the , B − / vertex in eq. ( 10 C B ) is a factor of 2 too small, since the anomaly contribution from the massless SM = C γ = – = (1 0 A.2 We proceed with the Wess-Zumino terms in eq. ( We begin by integrating out the anomaly cancelling physics and replacing their effects BB 0 Z WZ , keeping the anomalons decoupled can be inconsistent: in fact, it is responsible for the Z µν – L X ˜ after an integration byF parts, where A eq. ( equivalence (GBE) theorem to calculate the longitudinallyIn enhanced parts subsection of the amplitude. Z linearly realized Goldstone pseudoscalar, equivalent Lagrangian, the perturbative masses ofg the anomalons.peak Hence, behavior for already seen a in givenwhich figure is combination the of regime of validity motivated by the Goldstone boson equivalence approach. where Yukawa couplings of the chiralmass anomalons of cannot the be arbitrarily large, and thus fixing the from our matrix elementssulting Ward-Takahashi in identities in eqs. eq. ( ( longitudinal dominance. by Wess-Zumino terms in the Lagrangian, In this appendix, wecay perform calculation. an effective Our operatorin inspiration treatment refs. for of [ our thisan approach enhanced came longitudinal from coupling the to calculation the performed under Grants No.Council KO (ERC) 4820/1–1, under the and European No. Union’s(Grant Horizon No. 2020 FOR 637506, research “ 2239, and innovation and program from the EuropeanA Research Effective operator treatment for FY would like to acknowledge helpful discussionsToby with Opferkuch. Bogdan Dobrescu, The Andrey Katz, authorsand and suggestions would on like the to manuscript.“Precision thank This Physics, research Joachim is Fundamental Kopp supported Interactions for by and2118/1). the helpful Structure Cluster The discussions of of Excellence work Matter” of (PRISMA LM is also supported by the German Research Foundation (DFG) Acknowledgments JHEP03(2021)120 : ϕγ (A.3) → Z mass and Z boson. Taking , we find the net 0 B 1 Z boson is replaced in 0 Z 3 ! , this treatment breaks down 2 ϕ 2 Z 0 B m m Z − 1 ) and consider anomalons with pure

), we critically evaluate the simplifica- 0 0 3 Z 3 Z 2 2 Z Z , 3.11 f f m m 0 A.3 ) to the approximation above. In sequence, 3 Z gauge boson in the sum over polarization 2 Z 2 2 )–( Goldstone boson only couples to the heavy g g m m 0 B 0 2 W 2 W 3.21 c c 3.9 Z 2 X 2 EM 2 EM Z 2 W – 20 – e e g c 2 5 2 2 B B g ]. We present this intermediate result because we π C C ), we now can calculate the exotic width of π π 2 EM 12 1 1 e , 8192 24 3 24 masses. A.2 10 = = ≈ 0 B boson is replaced at the operator level, when the Goldstone Z 0 . boson and axial-vector couplings to the 2 Z GBE / and Γ 3 Z and summing over the anomalon fields in table − Z 2 each go to zero, with their ratio remaining constant. By construction, = / 1 X − g BB 0 Z ) = A and 0 m ( Z mass grows. We start with the WIs in eqs. ( 0 m C 0 B 2 Z the full vertex calculation vectors. compared to the a Wess-Zumino term. m Approximate all SM fermion masses, including the top quark mass,Neglect as the negligible transverse mode of the “Integrate out” the anomalon field content and substitute their loop contribution by Given the interaction in eq. ( 2. 3. 1. →∞ lim Step 1. vector couplings to the m these simplifications are A.1 Procedure for reproducing the Goldstone boson equivalence ansatzHaving from derived the resolved GBEtions needed result to in reduce eq. our ( full result in eq. ( where the second line approximates thethen Goldstone we mass used small compared to the since the GBE alsoas neglects the the transverse modes of the distinguish two interpretations ofthe the resolved GBE triangle approach: vertex,internal either whereby anomalons, the the or the coupling also then includes thethe limit SM fermions. In either case, the GBE approach is valid in fermions must also be included [ JHEP03(2021)120 1 1) + νβ , is g − − 2 Z νβ − (A.9) (A.4) (A.5) (A.6) (A.7) (A.8) w g w ( (A.10) /m − , 0 = 2) 2 Z z ). This dis- 2 − m ) = z 1 p − ( 3.14 ∗ β . w ), setting all SM z ( ) . ). 1 3 p 3.2 boson, as expected. ( , we need and , , A.2 ν , , 0 , B !! ) . 0 ) 1) w z Z = 0 2 Z z 2 Z P − − + 1) Z m a m + 1) − + 1) ) and ( g z w w

( z boson by neglecting the w ( w ( | | | ( ( ( 3.1 2 | 2 | 2 | O 0 2 p B 2 p 2 p p p || p || || Z 1 || 1 || 1 || 1 p 1 p 1 p | | | 1 + p p p | | |

µν νρ µρ νρ µρ µν 0 3 Z 2 Z X X X ), which takes into account the massless X X X W W W m m W W W c c c gg gg gg 2 c c c X 2 2 2 gg gg gg 2 W 2 2 2 g π π π c 2 EM EM EM . Summing over the WZ term and SM fermion π π π A.10 5 ), as expected. Crucially, our procedure does EM EM EM g e e e z 16 16 16 π e e e 3 3 3 16 16 16 – 21 – 3 3 3 2 EM − − − A.3 e given in the Wess-Zumino term eq. ( = = = 3 2048 = = = B ), introducing a Wess-Zumino effective interaction in boson. In the typical sum, C µνρ µνρ µνρ µνρ µνρ µνρ ) = Γ Γ 0 B Γ Γ ρ γ 3.17 ν )Γ Z 2 ρ 1 ν )Γ 4 µ 2 1 p p 2 µ p )–( p , with 2 long p − − ) p − − 2 0 + momentum coupling that remains is exactly the derivatively- B, π + 3.15 µ Z 0 1 µ B 1 p (16 Z p ( → / ( Z +1) Γ( z = 3( After evaluating the matrix elements in eqs. ( Correspondingly, the SM fermions give as the coefficients instead. B C + 1) , we can discard the transverse component of the β z The resulting longitudinal decay width, where we also expand in powers of In this case, with the anomalon mass eigenstates having 0 1 ( In the case that the anomalons are SM-like in their Yukawa couplings, we have p 2 Z 4 ν m 1 p We see that ourSM longitudinal fermions width as in well eq.tained as ( by the the heavy GBE anomalons, assumption is in a eq. factor ( and of 4 larger than the result ob- coupled Goldstone boson that leads to the Lagrangian term in eq. ( Step 3. fermion masses to zero andfinal all state anomalon polarizations masses to of infinity, we the next consider the sumterm. over Moreover, the where we have set alland SM anomalon fermions contributions to to be the massless.gardless Ward identities of We to the remark the choice that of photon the the vertexcontributions sum shift always to parameter cancels, of the re- SM WIs, fermion the only non-vanishing WI is from the requires cussion makes it manifest that amentum particular shift Wess-Zumino term relevant for is the not SM independent fermions, of the which remain mo- inStep the effective 2. theory description. As discussed below eqs.order ( to integrate out the anomalons dictates a specificto choice match of the Wess-Zumino interaction contributions to the WIs. The matching condition anomalon WI contributions are JHEP03(2021)120 . ) 2 0 B ”, . 2 Z for Z 0 0 B m /m A.10 = 0 Z 80 0 → 2 Z m t X g m m 0 =0.01 → t x ), to a GBE Mfrin,ln.pol. long. fermions, SM o fermions SM o m / / / 60 against ” (purple, small 0 o g for γ GBE w EFT w EFT EFT w full result full 3.21 B 0 B → but all SM fermion Z t [GeV] 3 B m → Z' 40 π/ 4 m i U(1) in γ) Z B as explained in “Step 1” in the . The red curve takes the width 20 z X MeV (“full result w/ Z' → Br(Z g (1) and . for a rather large value of O w 01 ) where the SM fermions are massless, . γ 0 0 -5 -6 -7 -8 -9 B . The curve “EFT w/o SM fermions, long. ), setting the anomalon masses to their -10 ). The curve “full w/

10 10 10 10 10

= 0

. 0

B 10

B 3.21 Z' Z Br γ → ) ( 0 Z X (b) Branching fraction g m 3.21 → → → ∞ 0 Z B – 22 – for for two values of M Z Z γ 0 B m . In addition, the longitudinal width in eq. ( m 0 Z 80 B 2 , the top quark mass is set to its SM value, while for Z / . Both of these curves show a turnover at small m 2 3 0 B √ → Z − / 0 =0.2 m → Φ Z x t = MeV masses. We can see that it is similar to the case where v Mfrin,ln.pol. long. fermions, SM o fermions SM o m / / / 60 against · and we thus have to set o g for 3 γ (1) BB GBE w EFT w EFT EFT w full result full B and other SM fermion masses of order MeV. 0 0 B O . We want to begin the discussion from the “full result” (black, Z t π/ Z A m [GeV] 01 → ∞ . B → = 4 Z' 40 and approximates ). It is calculated from a WZ term and neglects SM fermions, the transverse i U(1) in γ) m Z M B = 0 0 B M A.3 Z X ), we want to illustrate the impact of the different approximations in fig- g ) and anomalons (with masses 20 Z' → Br(Z 0 A.3 , if the SM fermions are neglected in the calculation. γ Branching fraction for → 0 B q Z . , the SM value for m 2 . 3 . It shows the branching fraction of 0 → Having explained the different steps from the full calculation, eq. ( -8 -5 -6 -7 -3 -4 π/ 6

= 0

10 10 10 10 10 10

B Z' Z Br γ → ) ( X the other quarks we chose we also set thepurple, top small quark dashing). mass The tothe large be branching value fraction of negligible, for the i.e. large and top recover quark the mass Landau-Yang leads limit to for a small increase of ansatz, eq. ( ure and for smaller dotted curve). Itmaximum takes value the of width from eq. ( the anomaly coefficient is also the limiting casethe of anomalon our masses full are result decoupled,Thus, in and we eq. we ( have perform established a that seriesof expansion the for GBE small width is an incorrect result for the exotic decay reproduce the GBE-derived widthIn only that when case, the theappropriately SM momentum match shift fermions to parameters are a are completely Wess-Zumino fixed operator neglected. to description the and relations the specified normalization above set to by text, and we take theadds the longitudinal full polarisation polarisation only. hereto.masses “EFT In w/o “EFT” SM (blue, fermions” mediumdashing) dashed) (green, we sets dot-dashed) include the the SM anomalonmasses quarks including Yukawa (with couplings the top toof to a order maximum MeV. The value “full of result” (black, dotted) takes anomalon Yukawas Figure 6. from GBE, see eq. ( polarisation of the pol.” (yellow, largeanomalons dashing) only is with an effective approximation in our calculation, where we include (a) Branching fraction g JHEP03(2021)120 and + 1 z ]. The EFT masses. 47 = 2 0 B w Z are set as before. . z 80 → ∞ and ). This becomes a good , and set . 0 0 Z M w . It becomes clear that the boson mass, preventing the → → Z . q 0 B Z Z → ∞ m B m m 60 M , when comparing to the EFT result 0 Z branching fraction from L3 [ m plane, where the limit resulting from our full γ ) 0 B [GeV] jj Z ' and over the whole range of ( B , up to a small deviation from the top quark 40 Z m approaches X , X 0 → m – 23 – g g X Z g Z EFT m L3 (1) U for limit Exclusion . The curve labelled “GBE” (red, solid) additionally 0 in the 20 Z B m to be small compared to the 0 Z U(1) . Note, however, that the EFT curve does not turn over but 0 m B Z 0 m . Therefore, the EFT approximation that decouples the anomalons

0.0 1.0 0.8 0.6 0.4 0.2

0 boson and vector-like couplings to the SM X g 0 B → Z 0 B Z m . Thus, we take the contribution from the two anomalons that have axial-vector z Exclusion limit for ) and assume all SM quark masses to vanish ( , which we calculated to be the necessary settings to reproduce the correct Wess- 2 / and 3 In summary, the most important conclusion is that the constant piece from the SM The next simplification we apply is to take the longitudinal polarisation only of the When we neglect the SM fermions entirely, i.e. not onlyw assume their masses to vanish In the curve labelled “EFT” (blue, medium dashing) we decouple the anomalons → ∞ − (“EFT w/o SM fermions, long. pol.”, yellow, large dashing). Still, all SM fermions = 0 B M here, it is validtakes at the Goldstone energies mass branching fraction to fall so quickly as quarks cannot be neglected. We have also seen that the EFT approach is generally accept- Z are neglected, the anomalonThe branching masses fraction are becomes sent smallerwithout at to SM larger infinity, fermions and including the full polarisation. We expect the GBE to break down couplings to the z Zumino term. For thedot-dashed) we branching furthermore fraction decouple curvecontribution the named from anomalons, “EFT the SM w/o fermions SMfor cannot be small fermions” neglected, as (green, since well they as give large large gauge contributions couplings diverges for is valid in an increasing interval, but it breaksbut down also as neglect their constant anomalyters piece, we have to fix the values of the shift parame- ( approximation for small gaugemass couplings visible at higher Figure 7. calculation is shown in solidblue), green, both comparing using to the theincludes, measurements limit as on from before, the an massless effective SM calculation quarks (“EFT”, and in anomalons with masses JHEP03(2021)120 ]. Phys. , , needed z . SPIRE symmetry. 0 B IN Z L and ][ branching frac- − w (1991) 323 γ ) B (2009) 1199 jj ( 210 81 → range by about 10-20%, 0 B Z Z m arXiv:1002.1754 ]. gauge bosons at the Tevatron 0 and the mass of the Annals Phys. ] on the . Hence we advocate experimental Z , ) for phenomenological studies. X 0 B 47 SPIRE Rev. Mod. Phys. g Z (green solid, “L3”), and the approxi- , IN couplings, as long as the SM fermions 3.21 m 7 ][ X ) (blue shaded, “EFT”). We see that, over- g and the width has to be calculated adding ) to represent the anomaly cancelling physics (2010) 079901] [ B A.10 C A.2 82 – 24 – reach in the lighter gauge bosons 0 X g Z Baryon and lepton number as local gauge symmetries hep-ph/0408098 [ ), which permits any use, distribution and reproduction in , the effective treatment can be valid independent of the ]. L masses and small have to be set in the SM fermion induced matrix element to Erratum ibid. − 0 B z [ B Z SPIRE m theory. IN and symmetry breaking scale is independent of the mass generation of 0 U(1) ][ 0 CC-BY 4.0 w (2004) 093009 Z U(1) This article is distributed under the terms of the Creative Commons The physics of heavy 70 Gauge anomalies in an effective field theory (2010) 011901 ]. 82 SPIRE arXiv:0801.1345 IN Rev. D Phys. Rev. D [ [ Having analyzed the limitations of the effective Wess-Zumino and furthermore the GBE An analogous prescription applies when integrating out parts of an anomaly-free set Taking a Wess-Zumino term as in eq. ( M. Carena, A. Daleo, B.A. Dobrescu and T.M.P. Tait, P. Langacker, J. Preskill, P. Fileviez Perez and M.B. Wise, [3] [4] [1] [2] References results should use the full decay expression inOpen eq. Access. ( Attribution License ( any medium, provided the original author(s) and source are credited. tion. We plot the L3couplings limit and from the the correct full topmate width result quark using including mass the anomalons EFT in with expression figure all, in maximal eq. the Yukawa ( EFT result overestimatesand the this overestimate is generally worse for larger parameters of the ansatz, we finally present the practical differencesion between for using the an exotic effective operator decay expres- widthpreting versus the the constraint from full the expression L3 with experiment heavy at anomalons LEP when [ inter- The Wilson coefficient ofthe remaining the fermions, Wess-Zumino and termto it is calculate must given be the matched by contributiona to the of theory the anomaly where the shift the coefficient light parameters the fermions of fermions, which such remain as in the EFT. Hence, in the shift parameters match the choice ofthe the contributions Wilson of coefficient the Wess-Zumino term and the SMof fermions fermions coherently. which share theby same a position Wess-Zumino of term, the axial-vector such coupling as and the replacing case them of the top quark in gauged are included. Numerically, theThe omission large of top the quark constantof mass piece the from actually calculation the has independent SM a of fermions, the rather however, gauge leads small coupling to impact. ais therefore break only down valid if the SM fermions are still being considered in the calculations. Then able for a wide range of JHEP03(2021)120 , , ]. , , , 88 47 Phys. ]. , ’s and (2018) (2019) SPIRE IN ]. Phys. Rev. 04 Phys. Rev. U(1) , 100 ][ SPIRE , IN ]. ]. ][ SPIRE (2019) 082 JHEP IN model , Phys. Rev. D 02 ][ , J. Math. Phys. SPIRE anomaly equations SPIRE with light mediators , IN U(1) IN R ][ portal to Chern-Simons 3 ][ Phys. Rev. D × 0 ]. 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