Probing new U(1) gauge symmetries via exotic Z→ Z ′ γ decays
MMITP/20-052
Probing new U (1) gauge symmetries via exotic Z → Z (cid:48) γ decays Lisa Michaels
1, a and Felix Yu
1, b PRISMA + Cluster of Excellence & Mainz Institute for Theoretical Physics,Johannes Gutenberg University, 55099 Mainz, Germany (Dated: October 2, 2020)New U (1) gauge theories involving Standard Model (SM) fermions typically require additionalelectroweak fermions for anomaly cancellation. We study the non-decoupling properties of these newfermions, called anomalons, in the Z − Z (cid:48) − γ vertex function, reviewing the connection between thefull model and the effective Wess-Zumino operator. We calculate the exotic Z → Z (cid:48) γ decay width in U (1) B − L and U (1) B models, where B and L denote the SM baryon and lepton number symmetries.For U (1) B − L gauge symmetry, each generation of SM fermions is anomaly free and the exotic Z → Z (cid:48) BL γ decay width is entirely induced by intragenerational mass splittings. In contrast, for U (1) B gauge symmetry, the existence of two distinct sources of chiral symmetry breaking enables aheavy, anomaly-free set of fermions to have an irreducible contribution to the Z → Z (cid:48) B γ decay width.We show that the current LEP limits on the exotic Z → Z (cid:48) B γ decay are weaker than previouslyestimated, and low-mass Z (cid:48) B dijet resonance searches are currently more constraining. We presenta summary of the current collider bounds on U (1) B and a projection for a TeraZ factory on the Z → Z (cid:48) B γ exotic decay, and emphasize how the Z → Z (cid:48) γ decay is emblematic of new anomalous U (1) gauge symmetries. I. INTRODUCTION
The Standard Model (SM) exhibits a chiral elec-troweak gauge symmetry under which bare mass termsfor the elementary quarks and charged leptons are forbid-den. Correspondingly, gauge anomaly cancellation im-poses conditions on the gauge quantum numbers of thefermions, without which the SM would suffer a failure ofrenormalizability [1]. Charging the SM fermions underan additional gauge symmetry, such as baryon numberor lepton number [2], imposes more anomaly cancellationconditions which necessitates introducing new fermionfields with electroweak quantum numbers. A prime ad-vantage of such a construction is the prediction of new Z (cid:48) gauge bosons, which give promising dijet and dileptonresonances at colliders [3–5]. New U (1) symmetries areubiquitous in beyond the Standard Model (BSM) physics,since they are motivated by numerous extensions of theSM, such as grand unified models that gauge the SMbaryon minus lepton number [4], the individual baryonnumber or lepton number global symmetries [2, 3, 5–7]or providing portals to dark sectors, as in Refs. [8–14].The focus of this work is exploring the phenomenol-ogy of new U (1) gauge extensions of the SM, particu-larly when the Z (cid:48) boson is light as well as when the U (1)symmetry is an anomalous global symmetry of the SM.Recent work on mapping out the possible U (1) gauge ex-tensions to the SM include Refs. [15–18]. When gaugingbaryon number, the anomaly coefficients SU (2) L × U (1) B and U (1) Y × U (1) B are nonzero after summing over theSM quark fields, requiring the introduction of new elec-troweak fields, called anomalons, for ultraviolet (UV) a [email protected] b [email protected] consistency. Correspondingly, the anomalons must bemassless in the unbroken phase when all spontaneousbreaking of all gauge symmetries is turned off. More-over, as a consequence of their chiral couplings, the vir-tual effects of the anomalons in loop processes is entirelyreminiscent of the behavior of the SM fermions in loop-induced Higgs processes. For instance, the heavy masslimit for loops of anomalon fields in the Z − Z (cid:48) − γ vertexdoes not decouple but instead approaches a nonzero con-stant. This non-decoupling behavior is familiar from theestablished exclusion of a pure four generation SM, asthe heavy fourth generation quarks would dramaticallyenhance the overall cross section of the 125 GeV Higgsboson by a factor of nine [19, 20].In the present context, where we consider the in-teraction of three distinct neutral current vectors me-diated at 1-loop by fermions with vector and axial-vector couplings, the non-decoupling limit of the vir-tual fermions gives a contribution to the effective Wess-Zumino term [21]. Specifically, we calculate the Z − Z (cid:48) − γ interaction vertex induced by fermions, where the Z and γ are the usual SM gauge bosons and the Z (cid:48) boson corre-sponds to a new, spontaneously broken U (1) gauge sym-metry. Using a UV complete model, we elucidate thematching condition from heavy anomalons and the effec-tive Wess-Zumino operator, particularly since both, the Z and the Z (cid:48) symmetries are spontaneously broken.Knowing the Z − Z (cid:48) − γ vertex is the critical ingredi-ent to calculating the exotic Z → Z (cid:48) γ decay, which is anew and attractive channel for studying light Z (cid:48) bosons.The corresponding vertex diagrams are reminiscent of thestandard Adler-Bell-Jackiw anomaly calculation [22, 23],but in order to calculate the decay width, we require thefull vertex structure and not only the divergences of eachcurrent [24].After performing the vertex calculation, we discuss thephenomenology of the anomalons in the UV-complete a r X i v : . [ h e p - ph ] S e p theory and the effective theory in matching to a Wess-Zumino term. Additionally, as the anomalon fields ex-hibit non-decoupling in loop-induced scalar decays, wecan also constrain the parameter space of realistic gauged U (1) models by measurements of the observed Higgs bo-son.In Sec. II, we review the two concrete models of U (1) B − L and U (1) B gauge symmetries and their respec-tive anomaly cancellation conditions. In Sec. III, we cal-culate the general Z − Z (cid:48) − γ vertex function by imposingthe appropriate Ward-Takahashi identities (WIs) on allexternal currents. In Sec. IV, we discuss the current col-lider constraints on electroweak charged anomalons, ap-plicable to general U (1) extensions of the SM where theanomalons form vector-like representations under the SMgauge groups. In Sec. V, we also present the current suiteof constraints in the g X vs. m Z (cid:48) B plane for the U (1) B model. We conclude in Sec. VI. We give a comparison toan effective operator approach and a critique of the Gold-stone boson equivalence implementation in the Z → Z (cid:48) B γ decay calculation in Appendix A. II. MODELS OF ADDITIONAL U (1) GAUGESYMMETRIES
Two particularly interesting models of additional U (1)gauge symmetries are the familiar U (1) B − L and theanomalous U (1) B symmetries, where B is baryon num-ber and L is lepton number. All SM quarks carry B charge of 1 / L charge of 1. Thegauged B − L symmetry has been studied extensively inthe literature to prevent proton decay in grand unifiedtheories [4]; moreover, with the introduction of right-handed neutrinos for neutrino masses, gauged B − L isalso anomaly-free.From a top-down view, the global symmetries of theSM with Dirac neutrino masses is U (1) B × U (1) L : we canthus gauge an arbitrary subgroup of this product sym-metry group without affecting the SM Yukawa structure.All distinct possibilities can be parametrized by U (1) L and U (1) B − xL , where x is a real multiplicative factor [3].For any choice of x (cid:54) = 1, the total contribution of SMfields to SU (2) L × U (1) B − xL and U (1) Y × U (1) B − xL anomalies is nonzero. For example, for x = 0, we have SU (2) L × U (1) B = 3 / U (1) Y × U (1) B = − / Z and Higgs decays.We emphasize that the essential distinction betweenthe first category of anomaly-free U (1) gauge symmetries(such as U (1) B − L ) and the second category of anomaloussymmetries (such as U (1) B ) is whether there are one ortwo scales of chiral symmetry breaking. In the first cat-egory, the Z (cid:48) boson mass is clearly independent of the SM masses: a simple Abelian Higgs model can serve asthe UV completion of the massive Z (cid:48) boson, where thecorresponding Higgs boson can be made heavy by a largequartic coupling and not appear in the mass spectrum.In the second category, the U (1)-breaking vacuum ex-pectation value (vev) of the underlying U (1)-chargedHiggs field plays a critical role by giving mass to boththe Z (cid:48) boson and the anomalon fields. In effect, hier-archies between the Yukawa couplings determining thephysical fermion masses and the gauge coupling deter-mining the Z (cid:48) boson mass dictate the resulting separa-tion between the particle species. Since these couplingsare renormalizeable, this mass splitting is stable underrenormalization group evolution, giving rise to the possi-bility that in a new sector of physics, the first kinemati-cally accessible state will be a Z (cid:48) boson, and the fermionicdegrees of freedom are further in the UV. Nevertheless,in an effective field theory (EFT) description where theanomalons are integrated out [1], the only possible scalesuppression of the higher dimension operators is simplythe U (1)-breaking vev [25, 26]. From this perspective,the non-decoupling behavior of the chiral anomalons isresponsible both for the Wess-Zumino term that arisesin loop-induced vertex functions of vectors [21], and thelow-energy theorem in Higgs physics [27]. A. Anomaly-free models: Gauged U (1) B-L symmetry
For the classic B − L gauge symmetry, the SM fieldcontent is augmented by three electroweak singlet right-handed neutrinos, which are required to cancel the (cid:80) U (1) L and U (1) L anomalies. We remark that eachgeneration of fermions satisfies the U (1) B − L anomalycancellation conditions independently. As we will seein Sec. III, this is why the net contribution of an en-tire generation of mass degenerate SM fermions vanishesin anomaly-induced processes. Hence, similarly to theGIM mechanism [28], the net effect of an anomaly-freeset of fermions in chiral anomaly-probing interactions isproportional to a charge-weighted mass difference of therespective fermions. B. Anomalous Models: Gauged U (1) B Symmetry
Our exemplary model for gauging anomalous globalsymmetries of the SM is gauged baryon number U (1) B .Again, all SM quarks carry charge 1 /
3, but this givesa nonzero anomaly coefficient to the mixed electroweakanomalies: A ( SU (2) L × U (1) B ) = 3 / A ( U (1) Y × U (1) B ) = − / SU (3) C × U (1) B is already zero, the new fermions donot need to carry color charge, but they must carry elec-troweak charges to cancel the mixed electroweak-baryonnumber anomalies.Following Ref. [29], we will hence consider a minimalset of colorless anomalons, denoted by L L , L R , E L , E R , N L , and N R , which mimic the SM leptons in their elec-troweak quantum numbers. The gauge charges for thenew fermions and the U (1) B scalar Higgs field Φ areshown in Table I. Since the new fermions come in vector-like pairs under the SU (2) L × U (1) Y gauge symmetry,there are no new pure electroweak gauge anomalies. The L L , L R , E L , and E R fields cancel the SU (2) L × U (1) B and U (1) Y × U (1) B mixed anomalies from the SM quarksbut introduce (cid:80) U (1) B and U (1) B anomalies, which arecorrespondingly cancelled by N L and N R . Note that N L and N R can alternatively carry U (1) B charges 1 and − q B Y ) = 0, summing over all fermions, which isnecessary to avoid large B − Z (cid:48) B mixing [29], where B isthe hypercharge gauge boson. We write the interactionsof the SM quark fields with the Z (cid:48) B vector as L ⊃ g X Z (cid:48) Bµ ( qγ µ q ) , (1)where g X is the U (1) B gauge coupling. The SM quarksalways have vector couplings to the Z (cid:48) B boson. Field SU (2) L U (1) Y U (1) B L L − / − L R − / E L − E R − − N L N R −
1Φ 1 0 3
TABLE I: Quantum numbers of colorless anomalonsand the U (1) B Higgs field Φ, from Ref. [29].The anomalon masses are protected by the SM elec- troweak chiral symmetry and the U (1) B chiral symmetry.Hence, at least one source of chiral symmetry breaking,either the Higgs vev or the vev of the U (1) B -breakingfield Φ, is needed in order to give mass to all of thefermions.For the model content in Table I, we write the La-grangian as L = L kin + L Yuk + L scalar , (2) L kin = ¯ L L i /DL L + ¯ L R i /DL R + ¯ E L i /DE L + ¯ E R i /DE R + ¯ N L i /DN L + ¯ N R i /DN R , (3) L Yuk = − y L ¯ L L Φ ∗ L R − y E ¯ E L Φ E R − y N ¯ N L Φ N R − y ¯ L L HE R − y ¯ L R HE L − y ¯ L L ˜ HN R − y ¯ L R ˜ HN L + h.c. , (4) L scalar = | D µ Φ | − µ | Φ | − λ Φ | Φ | − λ H Φ | H | | Φ | . (5)We will assume µ <
0, which will trigger spontaneousbreaking of U (1) B at a scale v Φ , where Φ = ( v Φ + φ ) / √ λ H Φ is negligible throughout this work. Similarly, wewill ignore a possible kinetic mixing term between thehypercharge field strength and the baryon-number fieldstrength in the calculations.The Yukawa Lagrangian in Eq. (4) exhibits two sourcesfor generating masses for the anomalons. The y L , y E ,and y N couplings become SM vector-like masses for theanomalons once Φ acquires a vev, since the anomalonscome in vector-like pairs under the electroweak gaugesymmetry. The y , y , y and y couplings correspond toSM-like Yukawa terms and mimic the lepton Yukawas ofthe SM (as well as the role of the SM leptons in cancel-lation of B − L gauge anomalies).It is straightforward to write the anomalon contribu-tions to the SM Z - and γ -mediated currents as well asthe U (1) B current. Adopting the electromagnetic charge Q = T + Y , and L L = ( ν L , e L ) T , L R = ( ν R , e R ) T , thecurrent interactions of the anomalons are L int = e EM A µ J µ EM + gc W Z µ J µZ + g X Z (cid:48) Bµ J µZ (cid:48) B , with (6) J µ EM = ¯ e L ( − γ µ e L + ¯ e R ( − γ µ e R + ¯ E L ( − γ µ E L + ¯ E R ( − γ µ E R , (7) J µZ = ¯ e L (cid:18) −
12 + s W (cid:19) γ µ e L + ¯ e R (cid:18) −
12 + s W (cid:19) γ µ e R + ¯ E L (cid:0) s W (cid:1) γ µ E L + ¯ E R (cid:0) s W (cid:1) γ µ E R + ¯ ν L γ µ ν L + ¯ ν R γ µ ν R , (8) J µZ (cid:48) B = ¯ e L ( − γ µ e L + ¯ e R (2) γ µ e R + ¯ E L (2) γ µ E L + ¯ E R ( − γ µ E R + ¯ ν L ( − γ µ ν L + ¯ ν R (2) γ µ ν R + ¯ N L (2) γ µ N L + ¯ N R ( − γ µ N R , (9)where c W and s W are the cosine and sine of the weak angle. As alluded to above, these Weyl fermions can bepaired into Dirac fermions in two limiting cases: (1) y L , y E , y N nonzero and the other Yukawas zero, or (2) y , y , y , y nonzero and the others zero.We note that if the dominant source of the fermionmasses follows the second case, then the fermions willgreatly impact the Higgs diphoton decay rate and bephenomenologically excluded from the observation of theSM-like nature of the 125 GeV Higgs boson [20]. Inparticular, the charged matter fields will behave as non-decoupling contributions to the h → γγ decay. On theother hand, if the dominant source of the fermion massesarises from the y L , y E , and y N couplings, the effects onHiggs observables are diluted and can be consistent withcurrent measurements of Higgs couplings. We discuss theconstraints from Higgs physics and direct searches of theelectroweak anomalons in Sec. IV.Correspondingly, we will mainly adopt the first case, assuggested by our naming convention, where the fermion mass eigenstates are vector-like under the SM gauge sym-metry and the axial-vector couplings to the Z boson van-ish. Explicitly, we have L mass = − m L (cid:18) φv Φ (cid:19) (¯ νν + ¯ ee ) (10) − m E (cid:18) φv Φ (cid:19) ¯ EE − m N (cid:18) φv Φ (cid:19) ¯ N N , where m L = y L v Φ / √ m E = y E v Φ / √
2, and m N = y N v Φ / √
2, and ν and e should not be confused with theSM neutrino and electron. For simplicity, our numeri-cal analysis in Secs. III and IV will consider degeneratecharged anomalons with mass M = m L = m E . The neu-tral gauge currents for these fermion mass eigenstatesbecome J µ EM = ¯ e ( − γ µ e + ¯ E ( − γ µ E , (11) J µZ = ¯ e (cid:18) −
12 + s W (cid:19) γ µ e + ¯ Es W γ µ E + ¯ ν (cid:18) (cid:19) γ µ ν , (12) J µZ (cid:48) B = ¯ e (cid:18) γ µ + 32 γ µ γ (cid:19) e + ¯ E (cid:18) γ µ − γ µ γ (cid:19) E + ¯ ν (cid:18) γ µ + 32 γ µ γ (cid:19) ν + ¯ N (cid:18) γ µ − γ µ γ (cid:19) N . (13)We emphasize that the vanishing of the axial-vectorcoupling to the Z boson in this limiting case is intimatelytied to the vanishing of the Yukawa couplings y to y ,and as such, these mass eigenstate anomalons are chi-ral under U (1) B and vector-like under the EW symme-try. In general, a nonzero axial-vector coupling betweena fermion f and a neutral gauge boson V leads to pertur-bative unitarity violation in f ¯ f → V V scattering, whichis cured by the s -channel Higgs insertion [30]. Aside fromthe role of the axial-vector coupling in perturbative uni-tarity violation, we will focus on the role of the axial-vector coupling in determining the appropriate WIs intriangle diagram calculations of three gauge boson ver-tices. III. THE Z → Z (cid:48) γ VERTEXA. The generic vertex structure
We now calculate the partial width for an exotic decayof the SM Z boson decaying to a Z (cid:48) boson and a photon,where the loop is mediated by fermions. Note that forthe partial width mediated by one fermion, the anomalyis certainly not cancelled and the result has to dependon the anomaly prescription. We consider the generalsituation where the intermediate fermions have vectorand axial-vector couplings to each massive gauge boson, Z and Z (cid:48) . Since the photon mediates an unbroken, non-chiral U (1) gauge symmetry, its coupling is necessarilyvector-like. We show the two diagrams in Fig. 1.The corresponding matrix elements are given by We remark that, in the mass basis, the fermions may have flavor-changing neutral current couplings if their masses arise from bothchiral sources of electroweak and U (1) B breaking. The corre- sponding matrix elements would then involve mixing angles andtriangle diagrams with two different fermions as intermediatestates. Z ′ ν γ ρ Z µ Z µp p p p Z ′ ν γ ρ + k + a − p k + a k + a + p k + b − p k + b k + b + p FIG. 1: The triangle diagrams corresponding to the Z − Z (cid:48) − γ vertex function. i M = − Qe EM gg X (2 π ) ε µ ( p + p ) ε ∗ ν ( p ) ε ∗ ρ ( p ) × (cid:90) d k Tr (cid:34) ( g Zv γ µ + g Za γ µ γ ) /k + /a − /p + m ( k + a − p ) − m ( g Z (cid:48) v γ ν + g Z (cid:48) a γ ν γ ) /k + /a + m ( k + a ) − m γ ρ /k + /a + /p + m ( k + a + p ) − m (cid:35) (14) i M = − Qe EM gg X (2 π ) ε µ ( p + p ) ε ∗ ν ( p ) ε ∗ ρ ( p ) × (cid:90) d k Tr (cid:34) ( g Zv γ µ + g Za γ µ γ ) /k + /b − /p + m ( k + b − p ) − m γ ρ /k + /b + m ( k + b ) − m ( g Z (cid:48) v γ ν + g Z (cid:48) a γ ν γ ) /k + /b + /p + m ( k + b + p ) − m (cid:35) (15)where Q and m are the electric charge and the mass ofthe fermion in the loop, g v , g a are the vector and axialcoupling factors to the denoted massive Z and Z (cid:48) gaugebosons, p and p are the external outgoing momentaand k is the loop momentum, as depicted in Fig. 1. Weintroduce the arbitrary constant four-vectors a and b aspossible shifts in each diagram because the finite result from the cancellation of the divergent integrals dependson the choice of these possible shifts [31]. The explicitchoices of a and b are generally fixed by applying exter-nal physical conditions, as first applied in Ref. [32]. FromLorentz and parity symmetry, the most general expres-sion for the vertex function is the sum of the followingform factors [24],Γ µνρ ( p , p ; w, z ) = (16) F ( p , p ) (cid:15) νρ | p || p | p µ + F ( p , p ) (cid:15) νρ | p || p | p µ + F ( p , p ) (cid:15) µρ | p || p | p ν + F ( p , p ) (cid:15) µρ | p || p | p ν + F ( p , p ) (cid:15) µν | p || p | p ρ + F ( p , p ) (cid:15) µν | p || p | p ρ + G ( p , p ; w ) (cid:15) µνρσ p σ + G ( p , p ; z ) (cid:15) µνρσ p σ , where (cid:15) νρ | p || p | = (cid:15) νραβ p α p β , etc., we set b = − a to avoid a non-chiral anomaly [31], and a has beenreexpressed in terms of the external momenta, a µ = z p µ + w p µ with constant scalar prefactors w and z . Thesix form factors F to F are all finite and hence can becalculated in any regularization prescription unambigu-ously: they are w - and z -independent. Moreover, becauseof the linear dependence of vectors in a four-dimensionalspace, two of these can be eliminated by using the iden- tity [24], − p µ (cid:15) νρ | p || p | = − p ν (cid:15) µρ | p || p | + p ρ (cid:15) µν | p || p | + (cid:15) µνρα (cid:0) ( p · p ) p α − p p α (cid:1) , (17) − p µ (cid:15) νρ | p || p | = − p ν (cid:15) µρ | p || p | + p ρ (cid:15) µν | p || p | − (cid:15) µνρα (cid:0) ( p · p ) p α − p p α (cid:1) , (18)which absorb F and F into redefinitions of the otherform factors: we denote the redefined form factors as F (cid:48) i , i = 3 to 6, and G (cid:48) , G (cid:48) . In contrast, the two formfactors G (cid:48) and G (cid:48) arise from the cancellation of divergentintegrals, and their values depend on the choice of w and z . From Eq. (16), the WIs are given by( p µ + p µ )Γ µνρ = ( G (cid:48) − G (cid:48) ) (cid:15) νρ | p || p | , (19) − p ν Γ µνρ = ( − F (cid:48) p − F (cid:48) p · p + G (cid:48) ) (cid:15) µρ | p || p | , (20) − p ρ Γ µνρ = ( − F (cid:48) p · p − F (cid:48) p + G (cid:48) ) (cid:15) µν | p || p | . (21) Following Ref. [24], which implements the calculationprocedure in Refs. [22, 32, 33], we construct the ambigu-ous parts of G (cid:48) and G (cid:48) by isolating the divergent pieceof the general three-vector vertex associated with the ax-ial vector anomaly. This divergent piece can be evalu-ated using a momentum-shift integral identity [33], whichmakes the w and z -dependent momentum shifts manifestin the definitions of G (cid:48) and G (cid:48) .Moving to the specific case in Eqs. (14) and (15), wecalculate the finite form factors of the vertex functionin Mathematica [34] using Package-X [35, 36]. The WIsbecome( p µ + p µ ) Γ µνρ = Qe EM gg X π c W (cid:15) νρ | p || p | (( w − z )( g Z (cid:48) v g Za + g Zv g Z (cid:48) a ) + 4 m g Z (cid:48) v g Za C ( m )) , (22) − p ν Γ µνρ = Qe EM gg X π c W (cid:15) µρ | p || p | (( w − g Z (cid:48) v g Za + g Zv g Z (cid:48) a ) − m g Zv g Z (cid:48) a C ( m )) , (23) − p ρ Γ µνρ = Qe EM gg X π c W (cid:15) µν | p || p | ( z + 1)( g Z (cid:48) v g Za + g Zv g Z (cid:48) a ) , (24)where C ( m ) ≡ C (0 , m Z , m (cid:48) Z , m, m, m ) (25)is the usual Passarino-Veltman scalar loop functionfor the triangle loop, following the Package-X conven-tion [35–37].Clearly, each of the WIs in Eqs. 22, 23, and 24 containa constant, fermion mass-independent anomaly piece.Moreover, the WIs for the massive gauge bosons also havefermion mass-dependent contributions, but only whenthe fermion has the corresponding axial-vector coupling.Since we calculate in the flavor conserving limit, a givenmass eigenstate fermion can only have one non-zero axial-vector coupling.At this point, we could naively adopt the method byRosenberg [32] to set w and z for each fermion suchthat the vector WIs are vanishing and the anomaly con-tributes only the axial-vector divergence. This would bewrong, however, because all fermions in the loop must usethe same consistent choice of w and z . The vertex func-tion we study is the first physical case where this mistakewould become apparent, because the mixed electroweak- U (1) B anomaly is cancelled by two distinct chiral sectorsof fermions. On the other hand, for the B − L case, theSM fermions have axial-vector couplings only on the µ vertex, and thus choosing w and z to make the WIs onthe ν - and ρ -vertices vanish is consistent for all fermions.Instead, with a UV-complete model, the WI on eachvertex is independent of the choice of w and z . More-over, when all fermions are massless or otherwise degen-erate, the WI on each vertex is also vanishing. In fact,we can provide an equivalent condition for an anomaly-free model by requiring that the total WIs are vanishing, independent of w and z , as long as they are chosen thesame for all fermions in the UV-complete model. In otherwords, an anomaly-free model is insensitive to the ambi-guity introduced by the momentum shift intrinsic to di-mensional regularization, which is a gauge-invariant reg-ularization prescription, as long as the momentum shiftis applied consistently for all fermions.In an effective theory where chiral fermions are takenheavy, the choice of w and z to parametrize the momen-tum shift in Eqs. (14) and (15) also determine the appro-priate choice of the Wess-Zumino term [21], which resultsfrom the combination of choosing w and z and taking m → ∞ in the WIs. Since lim m →∞ m C ( m ) = − /
2, theheavy fermion mass limit in Eqs. 22 and 23 exhibits non-decoupling.Explicitly, we consider the Wess-Zumino term for thehypercharge gauge field B , weak gauge fields W a , and ageneral Z (cid:48) field, L WZ = C B g X g (cid:48) (cid:15) µνρσ Z (cid:48) µ B ν ∂ ρ B σ − C B g X g (cid:15) µνρσ Z (cid:48) µ (cid:18) W aν ∂ ρ W aσ + 13 g(cid:15) abc W aν W bρ W cσ (cid:19) , (26)where the coefficient of the weak gauge bosons is negativethat of the hypercharge gauge bosons to avoid breakingthe electromagnetic gauge symmetry [11, 25, 26]. Isolat-ing the vertex involving Z − Z (cid:48) − γ , we get L ⊃ − C B e EM gg X c W (cid:15) µνρσ Z (cid:48) µ ( Z ν ∂ ρ A σ + A ν ∂ ρ Z σ ) , (27)which then results in the following WI structure:( p µ + p µ )Γ µνρ = C B e EM gg X c W (cid:15) νρ | p || p | , (28) − p ν Γ µνρ = 2 C B e EM gg X c W (cid:15) µρ | p || p | , (29) − p ρ Γ µνρ = C B e EM gg X c W (cid:15) µν | p || p | . (30)We see that these WIs are identical to the contributionsof a given heavy fermion with couplings of g Z (cid:48) v = g Za = 0and g Zv , g Z (cid:48) a (cid:54) = 0 corresponding to the restriction 2 z = w − g Zv = g Z (cid:48) a = 0 and g Z (cid:48) v , g Za (cid:54) = 0 correspond-ing to the restriction 2 z = w −
3. This provides a con-crete matching condition for the effective field theory ofa heavy chiral fermion in anomalous U (1) gauge theories,and we provide a full discussion on the effective approachin Appendix A.In a full theory with a complete, anomaly-free set offermions, e.g. for the case of the SM fermions in U (1) B − L or for U (1) B including anomalons, the dependence on w and z – and thus the shift dependence – drops out andthe vertex can be calculated unambiguously. We calcu- late the induced width for the Z → Z (cid:48) γ decay for thesetwo complete models in the following subsections III Band III C. B. U (1) B − L For U (1) B − L gauge symmetry, the SM fermion content(including three right handed neutrinos) is anomaly freeand there is no need to introduce new chiral matter. Theinteractions of the B − L gauge boson are given by L BL = g BL (cid:18) qγ µ Z (cid:48) BL,µ q − (cid:96)γ µ Z (cid:48) BL,µ (cid:96) − νγ µ Z (cid:48) BL,µ ν (cid:19) , (31)where q includes all up-type and down-type quarks, (cid:96) thecharged leptons, and ν the neutrinos.The exotic Z → Z (cid:48) BL γ decay width, summing over allcontributions for the different fermions f with masses m f and electric charge Q f , isΓ( Z → Z (cid:48) BL γ ) = α EM αα BL π c W m Z (cid:48) m Z (cid:18) − m Z (cid:48) m Z (cid:19) × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) f (cid:20) T f N fc Q ef Q BLf (cid:18) m f C ( m f ) + m Z m Z − m Z (cid:48) (cid:0) B ( m Z , m f ) − B ( m Z (cid:48) , m f ) (cid:1)(cid:19)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (32)where T f = +1 for up-type quarks and − N fc = 3 for quarks and 1 forleptons, Q ef and Q BLf are the U (1) charges under EM and B − L gauge symmetries, and we also abbreviate B ( m V , m f ) ≡ B ( m V , m f , m f ) , (33)which is the usual Passarino-Veltman bubble scalar loopfunction [37].From the structure of Eq. (32), it is clear that a mass-degenerate generation of SM fermions will have a van-ishing contribution to the partial width, by virtue ofthe fact that (cid:80) f T f N fc Q ef Q BLf vanishes for a degener-ate set of SM quarks and leptons. Hence, the largestcontribution stems from the intragenerational hierarchybetween the top quark and the bottom and tau leptons,as shown in Fig. 2. The residual finite mass splittingsbecome more significant for smaller Z (cid:48) masses: they arevisible as bumps in Fig. 2. For m Z (cid:48) (cid:38)
20 GeV, theyhave an overall effect of less than 10% compared to keep-ing them degenerate. Below m Z (cid:48) (cid:46)
10 GeV, however,the threshold behavior from the lighter fermions can givean enhancement of the decay width by a factor of twocompared to the top quark contribution only. We emphasize that the cancellation of an entire gener- - - - - - m Z' BL [ GeV ] B r ( Z → Z ' B L γ ) Br ( Z → Z' BL γ ) in U ( ) B - L g BL = g BL = g BL = g BL = g BL = FIG. 2: Branching fraction for Z → Z (cid:48) BL γ for variouschoices of g BL . The top quark dominates the decaywidth, while the lighter SM fermions cause small peakswhen they go on-shell. We include the lighter SMfermion mass effects as in Eq. (32).ation of mass-degenerate SM fermions only occurs in thiscase because the SM fermions share the same underlyingchiral symmetry structure which dictates the axial-vectorcouplings to the Z boson and vector couplings to the Z (cid:48) boson. Correspondingly, the expected Landau-Yang be-havior [38, 39] for m Z (cid:48) → C. U (1) B For gauged U (1) B symmetry, with quark interactionsas in Eq. (1), the analytic behavior of Γ( Z → Z (cid:48) B γ ) is markedly different from the U (1) B − L case. In particu-lar, since the anomaly cancelling fermions can becomemassive independently of the SM Higgs vev, their contri-bution to the Z − Z (cid:48) − γ vertex can be non-decouplingregardless of the scale set by the Z mass. On the otherhand, in such a case, the anomalons and the Z (cid:48) share thesame chiral symmetry breaking scale, and the infraredlimit of making the Z (cid:48) light necessarily reintroduces theanomalons into the spectrum too. The anomalon fieldswe consider are charged as in Table I.Assuming the charged anomalons are degenerate andtheir masses M arise solely from U (1) B breaking, thedecay width of Z → Z (cid:48) B γ isΓ( Z → Z (cid:48) B γ ) = α EM αα X π c W m (cid:48) Z m Z (cid:18) − m Z (cid:48) m Z (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:88) f ∈ SM T ( f ) Q ef (cid:20) m Z m Z − m Z (cid:48) (cid:0) B ( m Z , m f ) − B ( m Z (cid:48) , m f ) (cid:1) + 2 m f C ( m f ) (cid:21) + 3 (cid:18) m Z m Z − m Z (cid:48) (cid:0) B ( m Z , M ) − B ( m Z (cid:48) , M ) (cid:1) + 2 M m Z m Z (cid:48) C ( M ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (34)where again T ( f ) = +1 for up-type quarks and − O (1) MeV masses. We also remark that, as required byanomaly cancellation, there is no w or z dependence inthe physical width.We show the branching fraction of Z → Z (cid:48) B γ as afunction of m Z (cid:48) B for various choices of g X in Fig. 3.We include the limit on the Z → Z (cid:48) B γ branching ratio,where Z (cid:48) B decays hadronically, which has been probedat LEP by the L3 collaboration [40, 41]. The Z (cid:48) B bosonwill dominantly decay to a dijet resonance for masses m Z (cid:48) B (cid:38) m π [5, 42], when the anomalons introduced areheavier than the Z (cid:48) B boson.We remark on many interesting features in Fig. 3.First, since the anomaly cancelling fermions obtain massfrom the spontaneous breaking of the chiral U (1) B sym-metry, their masses scale with the U (1) B breaking vev setby m Z (cid:48) B = 3 g X v Φ . For concreteness, we set M = π v Φ √ using π as a fixed value for the Yukawa couplings y L and y E . Hence, for fixed g X , the anomalons become lighteras m Z (cid:48) B decreases: the cusp behavior at the maximumof each curve then marks when the anomalons developimaginary contributions to the loop function by goingon-shell at M = m Z /
2. Such light, electrically-chargedanomalons are already excluded, however, by searchesat LEP by the L3 and ALEPH collaborations [43, 44].Thus, we indicate the LEP direct search bound on thecharged anomalons as a solid circle on each curve, mark- ing where their masses cross 90 GeV. The parameterspace for the branching fractions left of these circles, indi-cated by thinner lines, is thus excluded by direct searchesfor the charged anomalons. Of course, the anomalonmasses could also receive large contributions from theSM Higgs vev, which would weaken the direct scalingrelationship between m Z (cid:48) B and M for a given g X , butthen the contribution to the anomalon masses from theSM Higgs Yukawa would also affect the h → γγ signalstrength. We evaluate this constraint in Sec. IV, find-ing that anomalons whose dominant mass contributioncomes from v Φ enjoy an open parameter space to inducea branching fraction of O (10 − ). Finally, the turnoverfeature of the anomalons is also necessary to exhibit thewell-known Landau-Yang behavior [38, 39] as m Z (cid:48) B → Z → Z (cid:48) B γ is necessarily at most O (10 − ). If theL3 and ALEPH constraints were relaxed, then the ex-otic Z → Z (cid:48) B γ branching fraction maximizes around O (few) × − , in competition with the exotic Z → Z (cid:48) B γ decay probe by L3 [40, 41]. While the hadronic decays ofthe relatively light Z (cid:48) B are more difficult to reconstruct atthe LHC compared to LEP, the immense statistics andthe additional coincident feature of the jj + γ resonancereconstructing the Z boson make this a promising av-enue to probe possible anomalous gauge symmetries at For example, see the model considered in Ref. [45].
L3 exclusiong x = x = x = - - - - - - m Z' B [ GeV ] B r ( Z → Z ' B γ ) Br ( Z → Z' B γ ) in U ( ) B g x = . g x = . (a) Branching fraction Z → Z (cid:48) B γ versus m Z (cid:48) B . g x = ( g X = ) L3 ( g X = ) g x = x = x = x = x = - - - - - - v Φ [ GeV ] B r ( Z → Z ' B γ ) Br ( Z → Z' B γ ) in U ( ) B (b) Branching fraction Z → Z (cid:48) B γ versus v Φ . FIG. 3: Branching fraction for Z → Z (cid:48) B γ for variouschoices of g X . The degenerate anomalon masses M areset to (4 π/ v Φ / √
2. The dots mark the point where theanomalon mass equals 90 GeV. The grey (purple, red)shaded region is excluded by the L3 Z → ( jj ) res + γ search [41], and it becomes g X dependent when plottedversus the vev v Φ .the LHC. We note a similar sensitivity improvement inthe exotic decay of the Z to a leptonically decaying Z (cid:48) and a photon could also be expected from the LHC ex-periments, where the current branching fraction limits atthe O (10 − ) level are set by the OPAL collaboration [46].Finally, we remark that the expression in Eq. (34)is the first possible non-trivial decay width of a mas-sive, neutral gauge boson into two further neutral gaugebosons . We also note that heavy sectors of anomaly freesets of fermions, by virtue of the fact that the mixed elec-troweak anomaly is carried in two distinct vertices, can For a discussion of the Landau-Yang theorem and its applicationsto non-Abelian gauge bosons, see Ref. [47]. give a non-decoupling contribution to the partial width,as noted in Ref. [24]. For illustration, a hypotheticalcomplete set of heavy, mass-degenerate SM fermions andanomalons, where the anomalon masses arise solely bythe U (1) B breaking vev, gives a non-decoupling decaywidth ofΓ( Z → Z (cid:48) B γ ) non-anom. =3 α EM αα X π c W ( m Z − m Z (cid:48) ) m Z m Z (cid:48) (cid:18) − m Z (cid:48) m Z (cid:19) . (35)Of course, the SM-like nature of the 125 GeV Higgs pre-cludes this scenario, but it is nevertheless a curious factthat the decoupling of anomaly free sets is not guaranteedin theories with two sources of chiral symmetry breaking. IV. COLLIDER SEARCHES FOR ANOMALONS
In this section, we discuss the phenomenology of theanomalon sector. Since the anomalons have the sameSM gauge quantum numbers as leptons, they sharemany of the same phenomenological signatures as fourth-generation leptons. Given their dominant mass contri-bution arises from v Φ , their collider phenomenology alsomimics the electroweakino sector from supersymmetry,where the charged anomalons are slightly heavier thanthe electrically neutral anomalons and exhibit a com-pressed mass spectrum, as a consequence of the 1-loopradiative electroweak corrections.When the anomalon masses receive contributions fromthe electroweak vev, they induce corrections to the ob-served 125 GeV Higgs boson decay into two photons. Wecan calculate this correction as a coherent sum of the topquark, bottom quark, W boson, and the new anomalons.Since we assume the dominant source of the anomalonmasses comes from v Φ , they will exhibit decoupling inthe H → γγ partial width.From Eq. (4), we assume the Yukawa couplings are realand for simplicity set y E = y L ≡ y Φ and y = y ≡ y H .The charged anomalon mass Lagrangian becomes L mass ⊃ − y Φ √ v Φ ¯ e L e R − y Φ √ v Φ ¯ E L E R − y H √ v H ¯ E L e R − y H √ v H ¯ e L E R + h.c. , (36)and the two Dirac masses are then M = 1 √ y Φ v Φ + y H v H ) , (37) M = 1 √ | y Φ v Φ − y H v H | . (38)The mass eigenstates couple to the SM Higgs with theYukawa coupling ± y H , while the Dirac masses dependon the values of y H , y Φ and the vev v Φ .0Adapting the expression for the Higgs decay to twophotons from Ref. [48], the H → γγ partial width in-cluding the two additional charged fermions becomesΓ( H → γγ ) = G F α M H √ π (cid:12)(cid:12)(cid:12)(cid:12) A f ( τ t ) + 13 A f ( τ b ) + A W ( τ W )(39)+ y H v H M √ A f ( τ M ) − y H v H M √ A f ( τ M ) (cid:12)(cid:12)(cid:12)(cid:12) , where G F is the Fermi constant, M H is the Higgs massof 125 GeV, and we include the dominant contributionsfrom the top quark, bottom quark, the W boson, and thetwo new fermions. The loop functions are defined as A f ( τ f ) = 2 τ f ( τ f + ( τ f − f ( τ f )) , (40) A W ( τ W ) = − τ W (2 τ W + 3 τ W + 3(2 τ W − f ( τ W )) , (41)with the function f ( τ ) being f ( τ ) = (arcsin √ τ ) , τ ≤ − (cid:16) log √ − τ − −√ − τ − − iπ (cid:17) , τ > τ are the mass ratios τ f,W = M H m f,W .In Fig. 4, we plot contours of the signal strength for gg → H → γγ as a function of the two Yukawa couplingsof the anomalons, ( y H , y Φ ) where v Φ = 300 GeV. Theopen region shows the parameter space allowed by theATLAS limits [49] on the signal strength of µ ( gg → H → γγ ) = 0 . +0 . − . , (43)where the 1 σ uncertainties are given. The hatched greyregions in Fig. 4 are the two sigma exclusion limits. Wealso show the exclusion from LEP searches on chargedparticles below 90 GeV [43, 44]. We see that the chargedanomalons can have a mild effect on the H → γγ rate,which is well within the experimental uncertainty when y Φ is dominant over y H . This is a direct result of theirvector-like SM gauge representations. As a result, weexpect the best improvement in testing this parameterspace would come from direct searches for anomalons,subject to the model dependence in their decay signa-tures. Note that there is also a region at very small y Φ but high y H that is allowed by the H → γγ rate, butwe expect this to be excluded by electroweak precisionmeasurements.In our model, if the electrically neutral anomalons ν and N are only slightly lighter than the electricallycharged anomalons, we would have a completely analo-gous situation to the electroweakino and slepton searchesfrom supersymmetry, which are one of the more difficultsignatures for the LHC experiments because of the pres-ence of soft leptons from compressed mass splittings [50–52]. The SU (2) couplings of the anomalons would guar-antee electroweak Drell-Yan production rates, but for y H y Φ Br ( H →γγ ) over SM value M <
90 GeV1.01 1.051.11.21.3
FIG. 4: H → γγ branching fraction exclusion plotincluding both charged anomalons, where y H is thecoupling to the SM Higgs boson and y Φ the one to the U (1) B breaking Higgs. The vev of the new Higgs isfixed to v Φ = 300 GeV. The dashed regions are the H → γγ signal strength 2 σ exclusion limits [49]. Thegrey band denotes the 90 GeV exclusion limit on theanomalon masses by LEP [43, 44].small mass splittings, the charged current decay fromthe heavy charged anomalon to the electrically neutralanomalon would give very soft leptons or pions and miss-ing transverse energy. As shown in Ref. [52], the mostpessimistic choice of mass splitting means there is nolimit from the LHC experiments, and we can only adoptthe limit from the LEP searches. We reserve a dedicatedstudy of the collider phenomenology of direct anomalonsearches and the suitability of the lightest neutral anoma-lon as a dark matter candidate for future work. V. COUPLING-MASS MAPPING FOR U (1) B We now discuss the overall status of Z (cid:48) B boson searchesin the mass region accessible by the Z → Z (cid:48) B γ decay. Oursummary plot is shown in Fig. 5, with numerous searchescarving out excluded regions in the g X vs. m Z (cid:48) B plane.The first constraint in Fig. 5 is marked “L3” and isderived from the search for the Z → Z (cid:48) B γ , Z (cid:48) B → jj ex-otic decay [41]. It is calculated following Eq. (34), settingthe mass of the anomalons by their nominal Yukawa cou-pling of 4 π/ m Z (cid:48) for the value of g X gives two values for g X ,1where the limit is given by the smaller one. In the caseof a discovery, however, this uncertainty has to be re-solved by an independent measurement. We see that theL3 constraint is largely supplanted by the more recentCMS searches targeting low-mass dijet resonances trig-gered using initial state radiation photons [53] or jets,with a subsequent jet substructure analysis [54].In Fig. 5, we also show the constraint based on theΥ search by the ARGUS collaboration [55], where theymeasured the hadronic ratio of Υ decays, R Υ = Γ(Υ → hadrons) / Γ(Υ → µ + µ − ) < .
1. The corresponding limitin g X vs. m Z (cid:48) is taken from Refs. [29, 56]. We also takethe limit from Ref. [29] for Z − Z (cid:48) B mixing induced atone-loop by SM quarks [29, 56–59]. This constraint islabeled Γ( Z ) in Fig. 5.The LEP constraint from the L3 and ALEPH col-laborations that new electrically charged particles mustbe heavier than at least 90 GeV [43, 44] is also shownin Fig. 5 for y Φ = 4 π/
3, which is the same value of y Φ that we use for the anomalon masses in deriving the L3limit. We also show the LEP constraint for a differentchoice of y Φ = 1 as a dashed line. We remark that theseconstraints are subject to additional model dependence,since these curves have also set y H = 0. Importantly, asshown in Fig. 4, we can turn on nonzero y H to increasethe anomalon masses, which would make the collider di-jet searches, the Υ bound, and the Z − Z (cid:48) B mixing boundbecome the leading existing bounds on these light Z (cid:48) B bosons.Finally, we also show a projection for a TeraZ factorysearching for the exotic Z → Z (cid:48) B γ , Z (cid:48) B → jj decay asa dotted green outlined region in Fig. 5. For this pro-jection, we perform a simple extrapolation accountingfor the increased statistics of 3 × Z bosons pro-duced during four years of runtime of the FCC-ee [60]over the number of Z bosons produced at LEP, whichwas 1 . × [61]. We also simply assume the improve-ment in sensitivity only arises from the increased statis-tics, but we recognize that a realistic collider projectionshould also account for possible improvements in massreach as well as problematic background or systematicuncertainties. We reserve a such a sensitivity study forfuture work.Importantly, the exotic Z → Z (cid:48) B γ decay is an ir-reducible probe of the U (1) B model compared to theanomalon searches, which depend on the specific decaychannels employed by the anomalons. The Z → Z (cid:48) B γ rate also crucially retains information about the anomalycoefficient contribution from the heavy fermions, whichmatches the anomaly coefficient of the SM fermion con-tent. In fact, the gauge coupling and Z (cid:48) mass are the onlycontinuous parameters that dictate the exotic Z → Z (cid:48) γ rate, since the nonzero anomaly coefficients for differ-ent global U (1) symmetries of the SM fermions are nec-essarily discrete. Hence, future improvements on the Z → Z (cid:48) B γ exotic decay search are strongly motivated. Υ Γ ( Z ) CMS jj + ISR jCMS jj + ISR γ TeraZL3 m Z B ' [ GeV ] g X Exclusion limit for U ( ) B M < G e V ( y Φ = π / ) ( y Φ = ) FIG. 5: Exclusion limits for U (1) B in the g X , m Z (cid:48) B plane including the limit from the decay of the Υ (inpurple) and the limit from the width of the Z boson (ingray), both taken from Ref. [29], our limit from Z → jjγ data from the L3 experiment at LEP [41] (insolid green) and a TeraZ factory projection in lightergreen. The direct bound on the anomalon mass [43, 44]is plotted for two different maximum values for theYukawa coupling, y Φ = 1 (in orange, dashed) and y Φ = 4 π/ VI. CONCLUSIONS
In this work, we have presented the calculation andphenomenology of the exotic Z decay to a Z (cid:48) boson anda photon. Our calculation, while reminiscent of text-book calculations of anomaly coefficients, focuses on thefull Z − Z (cid:48) − γ vertex function, where the various Ward-Takahashi identities are obtained by taking the appro-priate divergences of the external currents. We illus-trated the different types of anomaly contributions andnon-decoupling behavior in the Ward-Takahashi identi-ties by considering two distinct U (1) gauge symmetries: U (1) B − L gauge symmetry and U (1) B gauge symmetry.In the B − L case, since the SM fermion content isanomaly-free, they necessarily have vector-like represen-tations under U (1) B − L symmetry. Moreover, a degener-ate set of anomaly-free fermions would have a vanishingcontribution to the Z → Z (cid:48) BL γ decay width.In contrast, the SM fermions give non-vanishing elec-troweak anomalies when gauging baryon number. Cor-respondingly, when the anomalon masses arise only from U (1) B breaking, the anomaly cancellation between SMfermions and anomalons ensures the exotic Z → Z (cid:48) B γ decay width does not vanish if all fermions are degen-erate. This feature, which was also noted in Ref. [24],2is directly related to the fact that two sets of fermionscan have distinct chiral symmetry breaking scales. Asa result, the phenomenological predictions from U (1) B ,including the exotic Z → Z (cid:48) B γ decay, searches for theelectroweak charged anomalons at colliders, low-mass Z (cid:48) B dijet resonance searches, and shifts in Z -pole observablesfrom Z − Z (cid:48) B mixing, are all different manifestations ofthe underlying scale of U (1) B breaking for their respec-tive particle content. The corresponding state of the art,shown in Fig. 5, shows that all of these diverse probeswith their separate considerations of backgrounds andsystematic uncertainties are nonetheless generally com-petitive.We also dedicated Appendix A to the discussion of thenon-decoupling nature of the anomalon fields and the cor-responding induced Wess-Zumino term as well as a dis-cussion of the Goldstone boson equivalence treatment ofthe Z → Z (cid:48) γ decay. This Appendix demonstrates thatan ultraviolet complete theory predicts markedly differ-ent behavior for the Z → Z (cid:48) γ decay compared to anyapproach that neglects the SM fermions, especially forrealistic new U (1) gauge couplings that can be probed atcurrent and near future colliders.We see that the phenomenology of new U (1) symme-tries is exceedingly rich in both the theoretical complex-ity and the phenomenological predictions. We have high-lighted the particularly special behavior of anomalous U (1) symmetries with the U (1) B case study and the non-decoupling of anomalons in the corresponding Z − Z (cid:48) B − γ vertex. We find that the Z → Z (cid:48) B γ decay can prove anexciting test of the U (1) B model when this exotic decayis kinematically accessible. ACKNOWLEDGMENTS
FY would like to acknowledge helpful discussions withBogdan Dobrescu, Andrey Katz, and Toby Opferkuch.The authors would like to thank Joachim Kopp for help-ful discussions and suggestions on the manuscript. Thisresearch is supported by the Cluster of Excellence “Pre-cision Physics, Fundamental Interactions and Structureof Matter” (PRISMA + -EXC 2118/1). The work of LMis also supported by the German Research Foundation(DFG) under Grants No. KO 4820/1–1, and No. FOR2239, and from the European Research Council (ERC)under the European Union’s Horizon 2020 research andinnovation program (Grant No. 637506, “ ν Directions”).Feynman diagrams were generated using jaxodraw [62].
Appendix A: Effective operator treatment for Z → Z (cid:48) B γ In this appendix, we perform an effective operatortreatment of our Z → Z (cid:48) B γ exotic decay calculation.In particular, we compare to the calculation performedin Refs. [9, 11], where the authors show a result based on an effective operator inducing an enhanced longitudi-nal coupling to the Z (cid:48) B boson. We will derive the effec-tive operator from our matrix elements in Eqs. (14) and(15), reiterate the discussion about the resulting Ward-Takahashi identities in Eq. (22) − Eq. (24), and analyzethe validity of assuming longitudinal dominance.We first outline the effective ansatz being made inRefs. [9, 11]. The calculation starts by integrating outthe anomaly cancelling physics and replacing their effectsby Wess-Zumino terms in the Lagrangian, L WZ = C B g X g (cid:48) (cid:15) µνρσ Z (cid:48) µ B ν ∂ ρ B σ (A1) − C B g X g (cid:15) µνρσ Z (cid:48) µ ( W aν ∂ ρ W aσ + 13 g(cid:15) abc W aν W bρ W cσ ) , where C B is the Wilson coefficient from decoupling theanomalons. We remark that the Yukawa couplings of thechiral anomalons cannot be arbitrarily large, and thusfixing the mass of the Z (cid:48) B boson and the gauge coupling g X will also establish an upper bound on the perturbativemasses of the anomalons. Hence, for a given combinationof m Z (cid:48) B and g X , keeping the anomalons decoupled can beinconsistent: in fact, it is responsible for the peak behav-ior already seen in Fig. 3 and recovering the Landau-Yanglimit for m Z (cid:48) B →
0, which is the regime of validity moti-vated by the Goldstone boson equivalence approach.Refs. [9, 11] proceed with the Wess-Zumino termsin Eq. (A1) by using the Goldstone boson equivalence(GBE) theorem to calculate the longitudinally enhancedparts of the amplitude. In Subsec. III A we deducedthe contribution of the Wess-Zumino Lagrangian to the Z – Z (cid:48) – γ vertex in Eq. (27). By replacing the Z (cid:48) bosonwith the derivatively coupled, linearly realized Goldstonepseudoscalar, Z (cid:48) µ → ∂ µ ϕ/ ( g X f Z (cid:48) ), we obtain the longitu-dinally equivalent Lagrangian, L ⊃ C B ϕf Z (cid:48) · gg (cid:48) Z µν ˜ F µν , (A2)after an integration by parts, where f Z (cid:48) = m Z (cid:48) B /g X is thepseudoscalar decay constant, ˜ F µν = (1 / (cid:15) µνρσ F ρσ is thedual electromagnetic field tensor, C B = A Z (cid:48) BB / (16 π )and A Z (cid:48) BB = − / m Z (cid:48) and g X each go to zero, with their ratio re-maining constant. By construction, since the GBE alsoneglects the transverse modes of the Z (cid:48) B , this treatmentbreaks down as the Z (cid:48) B mass grows.Given the interaction in Eq. (A2), we now can calculatethe exotic width of Z → ϕγ :Γ GBE = 124 π C B e g c W m Z f Z (cid:48) (cid:32) − m ϕ m Z (cid:33) ≈ π C B e g c W m Z f Z (cid:48) = 3 e g g X π c W m Z m Z (cid:48) , (A3)3where the second line approximates the Goldstone masssmall compared to the Z mass and then we used A Z (cid:48) BB = − /
1. Procedure for reproducing the Goldstone bosonequivalence ansatz from the full vertex calculation
Having derived the GBE result in Eq. (A3), we crit-ically evaluate the simplifications needed to reduce ourfull result in Eq. (34) to the approximation above. Insequence, these simplifications are1. “Integrate out” the anomalon field content and sub-stitute their loop contribution by a Wess-Zuminoterm.2. Approximate all SM fermion masses, including thetop quark mass, as negligible compared to the Z and Z (cid:48) B masses.3. Neglect the transverse mode of the Z (cid:48) B gauge bosonin the sum over polarization vectors. Step 1:
We start with the WIs in Eqs. (22) − (24)and consider anomalons with pure vector couplings tothe Z boson and axial-vector couplings to the Z (cid:48) B boson.Taking lim m →∞ m C ( m ) = − / ( p µ + p µ ) Γ µνρ = 3 e EM gg X π c W (cid:15) νρ | p || p | ( w − z ) , (A4) − p ν Γ µνρ = 3 e EM gg X π c W (cid:15) µρ | p || p | ( w + 1) , (A5) − p ρ Γ µνρ = 3 e EM gg X π c W (cid:15) µν | p || p | ( z + 1) . (A6)As discussed below Eqs. (28) − (30), introducing a Wess-Zumino effective interaction in order to integrate out theanomalons dictates a specific choice of w and z .In this case, with the anomalon mass eigenstates hav-ing g Za = 0, we need 2 z = w − C B = 3( z + 1) / (16 π ), with C B given in the Wess-Zumino term Eq. (27). This dis-cussion makes it manifest that a particular Wess-Zuminoterm is not independent of the momentum shift relevantfor the SM fermions, which remain in the effective theorydescription. In the case that the anomalons are SM-like in their Yukawa cou-plings, we have ( w − z − w −
1) and ( z + 1) as the coefficientsinstead. Step 2:
Correspondingly, the SM fermions give( p µ + p µ ) Γ µνρ = − e EM gg X π c W (cid:15) νρ | p || p | ( w − z ) , (A7) − p ν Γ µνρ = − e EM gg X π c W (cid:15) µρ | p || p | ( w − , (A8) − p ρ Γ µνρ = − e EM gg X π c W (cid:15) µν | p || p | ( z + 1) , (A9)where we have set all SM fermions to be massless. Weremark that the sum of SM fermion and anomalon con-tributions to the Ward identities to the photon vertexalways cancels, regardless of the choice of the shift pa-rameter z . Summing over the WZ term and SM fermioncontributions to the WIs, the only non-vanishing WI isfrom the Z (cid:48) B boson, as expected. Step 3:
After evaluating the matrix elementsin Eqs. (14) and (15), setting all SM fermion masses tozero and all anomalon masses to infinity, we next considerthe sum over final state polarizations of the Z (cid:48) B boson.In the typical sum, (cid:80) (cid:15) ν ( p ) (cid:15) ∗ β ( p ) = − g νβ + p ν p β m Z (cid:48) , wecan discard the transverse component of the Z (cid:48) B bosonby neglecting the − g νβ term. Moreover, the Z (cid:48) B momen-tum coupling that remains is exactly the derivatively-coupled Goldstone boson that leads to the Lagrangianterm in Eq. (A2).The resulting longitudinal decay width, where we alsoexpand in powers of m Z (cid:48) /m Z , isΓ( Z → Z (cid:48) B, long γ ) = 3 e g g X m Z π c W m Z (cid:48) (cid:18) O (cid:18) m Z (cid:48) m Z (cid:19)(cid:19) . (A10)We see that our longitudinal width in Eq. (A10), whichtakes into account the massless SM fermions as well as theheavy anomalons, is a factor of 4 larger than the resultobtained by the GBE assumption in Eq. (A3). Crucially,our procedure does reproduce the GBE-derived widthonly when the SM fermions are completely neglected.In that case, the momentum shift parameters are fixedto the relations specified above to appropriately matchto a Wess-Zumino operator description and the normal-ization set by the anomaly coefficient A Z (cid:48) BB = − / m Z (cid:48) /m Z . Thus, we have established that the GBE widthis an incorrect result for the exotic decay of Z → Z (cid:48) B γ , ifthe SM fermions are neglected in the calculation.We show a comparison between the GBE result andour full result after applying successive approximationsin Fig. 6, including illustrative intermediate effective the-ory results.Having explained the different steps from the full cal-culation, Eq. (34), to a GBE ansatz, Eq. (A3), we wantto illustrate the impact of the different approximationsin Fig. 6. It shows the branching fraction of Z → Z (cid:48) B γ GBEEFT w / o SM fermions, long. pol.EFT w / o SM fermionsEFTfull w / m t → - - - - - - m Z' B [ GeV ] B r ( Z → Z ' B γ ) Br ( Z → Z' B γ ) in U ( ) B for g x = (a) Branching fraction Z → Z (cid:48) B γ against m Z (cid:48) for g X = 0 . GBEEFT w / o SM fermions, long. pol.EFT w / o SM fermionsEFTfull w / m t → - - - - - - m Z' B [ GeV ] B r ( Z → Z ' B γ ) Br ( Z → Z' B γ ) in U ( ) B for g x = (b) Branching fraction Z → Z (cid:48) B γ against m Z (cid:48) for g X = 0 . FIG. 6: Branching fraction for Z → Z (cid:48) B γ for two values of g X . The red curve takes the width from GBE,see Eq. (A3). It is calculated from a WZ term and neglects SM fermions, the transverse polarisation of the Z (cid:48) B andapproximates m Z (cid:29) m Z (cid:48) . The curve “EFT w/o SM fermions, long. pol.” (yellow, large dashing) is an effectiveapproximation in our calculation, where we include anomalons only with M → ∞ and we thus have to set w and z as explained in “Step 1” in the text, and we take the longitudinal polarisation only. “EFT w/o SM fermions”(green, dot-dashed) adds the full polarisation hereto. In “EFT” (blue, medium dashed) we include the SM quarks(with masses m q →
0) and anomalons (with masses M → ∞ ). The curve “full w/ m t →
0” (purple, small dashing)sets the anomalon Yukawa couplings to a maximum value of 4 π/ π/
3, the SM value for m t and other SMfermion masses of order MeV.for a rather large value of g X = 0 . g X = 0 .
01. We want to begin the discussion from the“full result” (black, dotted curve). It takes the widthfrom Eq. (34), setting the anomalon masses to theirmaximum value of M = 4 π/ · v Φ / √
2, the top quarkmass is set to its SM value, while for the other quarks wechose O (1) MeV masses. We can see that it is similar tothe case where we also set the top quark mass to be neg-ligible, i.e. O (1) MeV (“full result w/ m t → m Z (cid:48) . Both of these curves show a turnover at small m Z (cid:48) and recover the Landau-Yang limit for m Z (cid:48) → M → ∞ ) and assume allSM quark masses to vanish ( m q → g X , upto a small deviation from the top quark mass visible athigher m Z (cid:48) B . Note, however, that the EFT curve does notturn over but diverges for m Z (cid:48) B →
0. Therefore the EFTapproximation that decouples the anomalons is valid inan increasing interval, but it breaks down as m Z (cid:48) B → i.e. notonly assume their masses to vanish but also neglect theirconstant anomaly piece, we have to fix the values of theshift parameters w and z . Thus, we take the contributionfrom the two anomalons that have axial-vector couplingsto the Z (cid:48) B boson and vector-like couplings to the SM Z , and set w = 2 z + 1 and z = − /
2, which we calculated tobe the necessary settings to reproduce the correct Wess-Zumino term. For the branching fraction curve named“EFT w/o SM fermions” (green, dot-dashed) we further-more decouple the anomalons, M → ∞ . It becomes clearthat the contribution from the SM fermions cannot beneglected, since they give large contributions for smallas well as large gauge couplings g X masses and over thewhole range of Z (cid:48) B masses.The next simplification we apply is to take the lon-gitudinal polarisation only of the Z (cid:48) B (“EFT w/o SMfermions, long. pol.”, yellow, large dashing). Still, all SMfermions are neglected, the anomalon masses are sent toinfinity, and w and z are set as before. The branchingfraction becomes smaller at larger m Z (cid:48) , when comparingto the EFT result without SM fermions including the fullpolarisation. We expect the GBE to break down here, itis valid at energies (cid:29) m Z (cid:48) . The curve labelled “GBE”(red, solid) additionally takes the Goldstone mass m Z (cid:48) tobe small compared to the Z boson mass, preventing thebranching fraction to fall so quickly as m Z (cid:48) approaches m Z .In summary, the most important conclusion is that theconstant piece from the SM quarks cannot be neglected.We have also seen that the EFT approach is generallyacceptable for a wide range of m Z (cid:48) B masses and small g X couplings, as long as the SM fermions are included. Nu-merically, the large top quark mass actually has a rathersmall impact. The omission of the constant piece from5 L3 ( ) L3 m Z B ' [ GeV ] g X Exclusion limit for U ( ) B FIG. 7: Exclusion limit for U (1) B in the g X , m Z (cid:48) B plane, where the limit resulting from our calculation isshown in solid green, comparing to the limit calculatedin Refs. [9, 11] in transparent green, both using themeasurements on the Z → ( jj ) γ branching fractionfrom L3 [41]. Note that in Refs. [9, 11] the SM fermionswere omitted in the calculation of the width and wefurthermore obtain a factor 16 difference to their resultwhen reproducing the GBE ansatz.the SM fermions, however, leads to a break down of thecalculation independent of the gauge coupling g X and themass of the Z (cid:48) B .Taking a Wess-Zumino term as in Eq. (A2) to rep- resent the anomaly cancelling physics is therefore onlyvalid if the SM fermions are still being considered in thecalculations. Then the shift parameters w and z haveto be set in the SM fermion induced matrix element tomatch the choice of the Wilson coefficient C B and thewidth has to be calculated adding the contributions ofthe Wess-Zumino term and the SM fermions coherently.We remark that if the anomalons were heavy copiesof the SM leptons, then their axial-vector coupling lieson the Z vertex instead of the Z (cid:48) B vertex and the corre-sponding Z → Z (cid:48) B γ decay width is entirely driven by in-tragenerational mass splittings. The corresponding effec-tive operator treatment is then driven by non-decouplingeffects of heavy fermions, and since all fermions share thesame chiral symmetry breaking scale distinct from U (1) B breaking, the EFT treatment is valid for generic choicesof U (1) B parameters.Having analyzed the limitations of the effective Wess-Zumino and furthermore the GBE ansatz, we need tocomment on the limit calculation in the g X versus m Z (cid:48) B plane that was obtained in Refs. [9, 11], where they usethe measurements from the L3 experiment at LEP [41]on the Z → ( jj ) γ branching fraction. We plot the L3limit from the full width including anomalons with max-imal Yukawa couplings and the correct top quark massin Fig. 5 (green solid, “L3”). Comparing our L3 limitwith the one obtained in Refs. [9, 11] in Fig. 7, we seethat for the full calculation the limit becomes signifi-cantly weaker. We note that when we take the widthfor the GBE we obtained in Eq. (A3), the limit is tooweak to appear on Fig. 7 . [1] J. Preskill, Gauge anomalies in an effective field theory , Annals Phys. (1991) 323–379.[2] P. Fileviez Perez and M. B. Wise,
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