Contributions to ZZ V ∗ ( V=γ,Z, Z ′ ) couplings from CP violating flavor changing couplings
CContributions to
ZZV ∗ ( V = γ, Z, Z (cid:48) ) couplings from CP violating flavor changingcouplings A. I. Hern´andez-Ju´arez and G. Tavares-Velasco
Facultad de Ciencias F´ısico Matem´aticas, Benem´erita UniversidadAut´onoma de Puebla, Apartado Postal 1152, Puebla, Puebla, M´exico
A. Moyotl
Ingenier´ıa en Mecatr´onica, Universidad Polit´ecnica de Puebla, Tercer Carril del Ejido Serrano s/n,San Mateo Cuanal´a, Juan C. Bonilla, Puebla, Puebla, M´exico (Dated: February 4, 2021)The one-loop contributions to the trilinear neutral gauge boson couplings
ZZV ∗ ( V = γ, Z, Z (cid:48) ),parametrized in terms of one CP -conserving f V and one CP -violating f V form factors, are cal-culated in models with CP -violating flavor changing neutral current couplings mediated by the Z gauge boson and an extra neutral gauge boson Z (cid:48) . Analytical results are presented in terms ofboth Passarino-Veltman scalar functions and closed form functions. Constraints on the vector andaxial couplings of the Z gauge boson (cid:12)(cid:12) g tuV Z (cid:12)(cid:12) < . (cid:12)(cid:12) g tcV Z (cid:12)(cid:12) < .
011 are obtained from thecurrent experimental data on the t → Zq decays. It is found that in the case of the ZZγ ∗ vertexthe only non-vanishing form factor is f γ , which can be of the order of 10 − , whereas for the ZZZ ∗ vertex both form factors f Z and f Z are non-vanishing and can be of the order of 10 − and 10 − ,respectively. Our estimates for f γ and f Z are smaller than those predicted by the standard model,where f Z is absent up to the one loop level. We also estimate the ZZZ (cid:48)∗ form factors arising fromboth diagonal and non-diagonal Z (cid:48) couplings within a few extension models. It is found that in thediagonal case f Z (cid:48) is the only non-vanishing form factor and its real and imaginary parts can be ofthe order of 10 − − − and 10 − − − , respectively, with the dominant contributions arisingfrom the light quarks and leptons. In the non-diagonal case f Z (cid:48) can be of the order of 10 − , whereas f Z (cid:48) can reach values as large as 10 − − − , with the largest contributions arising from the Z (cid:48) tq couplings. PACS numbers:
I. INTRODUCTION
Trilinear gauge boson couplings (TGBCs) have long been the subject of considerable interest both theoreticallyand experimentally. In the experimental area, constraints on the corresponding form factors were first obtained atthe LEP [1–3] and the Tevatron [4–6] colliders, whereas the current bounds were extracted from the LHC data at 8TeV [7–9] and 13 TeV [10–12] by the ATLAS and CMS collaborations. Among TGBCs are of special interest the onesinvolving only neutral gauge bosons, namely, the trilinear neutral gauge boson couplings (TNGBCs)
ZZV ∗ and ZγV ∗ ( V = Z , γ ), which can only arise up to the one-loop level in renormalizable theories and have been widely studied withinthe standard model (SM) and beyond. The SM contributions to TNGBCs were studied in Refs. [13, 14], whereasnew physics contributions have been studied within several extension models, such as the minimal supersymmetricstandard model (MSSM)[13–15], the CP -violating two-Higgs doublet model (2HDM) [16–18], models with axial andvector fermion couplings [16], models with extended scalar sectors [19], and also via the effective Lagrangian approach[20]. In the theoretical side, the phenomenology of TNGBCs at particle colliders was widely studied long ago [15, 21–27] and also has been of interest lately [28–31]. Even more, study of the potential effects of TNGBCs at future collidershas been the source of renewed interest very recently [32–34]. TNGBCs, which require one off-shell gauge boson atleast to be non-vanishing due to Bose statistics, are induced through dimension-six and dimension-eight operators[13, 20, 35, 36] and can be parametrized in a model independent way by two CP -even and two CP -odd form factors.In the SM, only the CP -conserving form factors arise at the one-loop level of perturbation theory, whereas the CP -violating ones are absent at this order and require new sources of CP violation [13, 35]. In the SM, CP violation isgenerated via the Cabbibo-Kobayashi-Maskawa (CKM) mixing matrix, though the respective amount is not enoughto explain the invariance between matter and anti-matter in the universe, i.e. the so-called baryogenesis problem.Therefore, new sources of CP violation are required, which is in fact one of the three Sakharov’s conditions to explainthe baryon asymmetry of the universe [37]. In this work we are interested in the study of possible CP -violatingeffects in the TNGBCs via tree-level flavor changing neutral currents (FCNCs) mediated by the Z gauge boson [38],which are forbidden in the SM but can arise in several SM extensions [39, 40]. The possible effects of Z -mediatedFCNC couplings on CP -conserving TNGBCs have already been studied [13], nevertheless, possible contributions tothe CP -violating ones have not been reported yet to our knowledge. These new contributions are worth studying as a r X i v : . [ h e p - ph ] F e b they could shed some light in the path to a more comprehensive SM extension.Possible evidences of new heavy gauge boson have been searched for at the LHC by the CMS collaboration [41],which has been useful to set bounds on the masses of new neutral and charged heavy vector bosons. Such particlesare predicted by a plethora of SM extensions with extended gauge sector, for instance, little Higgs models [42, 43],331 models [44], left-right symmetric models [45], etc. Some of these models allow tree-level FCNCs mediated by anew neutral gauge boson, denoted from now on by Z (cid:48) [46], which means that CP -violating contributions to V ZZ (cid:48)∗ couplings ( V = γ , Z ) are possible. To our knowledge TNGBCs with new neutral bosons have not received muchattention in the literature up to now, though decays of the kind Z (cid:48) → V Z ( V = γ , Z ) [47] and Z (cid:48) → γA H [48] werealready studied. Here A H stands for a heavy photon.In this work we present a study of the one-loop contributions to the most general TNGBCs ZZV ∗ ( V = γ , Z , Z (cid:48) ) arising from a generic model allowing tree-level FCNCs mediated by the SM Z gauge boson and a new heavyneutral gauge boson Z (cid:48) . The rest of this presentation is as follows. In Sec. II we present a short review of theanalytical structure of TNGBCs along with the theoretical framework of the FCNCs Z and Z (cid:48) couplings via a modelindependent approach. Section III is devoted to the calculation of the one-loop contributions to the CP -conservingand CP -violating ZZV ∗ ( V = γ , Z , Z (cid:48) ) couplings, for which we use the Passarino-Veltman reduction scheme, withthe scalar functions given in closed form for completeness In Sec. IV we present the numerical analysis and discussion,whereas the conclusions and outlook are presented in Sec. V. II. THEORETICAL FRAMEWORKA. Trilinear neutral gauge boson couplings
We now turn to discuss the Lorentz structure of TNGBCs, which are induced by dimension-six and dimension-eightoperators. In this work we only focus on the contribution of dimension-six operators as it is expected to be thedominant one. In particular, the TNGBC
ZZV ∗ ( V = γ , Z ) coupling can be parametrized by two form factors:Γ αβµZZV ∗ ( p , p , q ) = i ( q − m V ) m Z (cid:104) f V (cid:0) q α g µβ + q β g µα (cid:1) − f V (cid:15) µαβρ ( p − p ) ρ (cid:105) , (1)where we have followed Ref. [13], with the notation for the gauge boson four-momenta being depicted in Fig. 1. FromEq. (1) it is evident that when the V ∗ gauge boson becomes on-shell ( q = m V ), Γ αβµZZV ∗ ( p , p , q ) vanishes, which isdue to Bose statistics and angular momentum conservation. The general form of this vertex for three off-shell gaugebosons can be found in [20, 35]. The form factor f V is CP -conserving, whereas f V is CP -violating. The former is theonly one induced at the one-loop level in the SM via a fermion loop since W ± boson loops give vanishing contributions[13]. It was found that f V decreases quickly as q becomes large [13]. The current bounds on the form factors f V and f V ( V = Z , γ ) were obtained by the CMS collaboration at √ s = 13 TeV [12]: − . ZZV ∗ ( V = γ , Z , Z (cid:48) ). B. FCNCs mediated by the Z and Z (cid:48) gauge bosons Beyond the SM, there are some extension theories that allow FCNC couplings mediated by the Z gauge boson[39, 40]. Such an interaction can be expressed by the following Lagrangian L = − e s W c W Z µ F i γ µ (cid:0) g iV Z − γ g iAZ (cid:1) F i − e s W c W Z µ F i γ µ (cid:16) g ijV Z − γ g ijAZ (cid:17) F j , (7)where F i,j are SM fermions in the mass eigenbasis. Here g iV Z,AZ are the diagonal SM couplings, whereas the non-diagonal couplings g ijV Z,AZ ( i (cid:54) = j ) will be taken as complex since we are interested in the CP -violating contribution.The latter must fulfil g ij ∗ V Z,AZ = g jiV Z,AZ because of their hermiticity. It is also customary to express the Lagrangianof Eq. (7) in terms of the left- and right-handed projectors P L and P R , with the chiral couplings denoted by (cid:15) ZL ij ,R ij ,which are given in terms of the vector and vector-axial couplings g ijV Z,AZ as follows g ijV Z,AZ = (cid:15) ZL ij ± (cid:15) ZR ij . (8)Below we will use both the g ijV Z,AZ and (cid:15) ZL ij ,R ij parametrizations. The former is useful for the purpose of comparisonwith previous works, whereas the latter is best suited for our numerical analysis.Possible phenomenological implications of FCNC couplings mediated by the Z gauge boson have been studiedwithin the SM [49–52], fourth-generation models [49, 50, 53–55], See-Saw models [56], etc. Such FCNC couplings havebeen constrained via the b → s transition [39, 40], Kaon decays [57, 58], B − B mixing [53], and B decays [59, 60].More, recently constraints on FCNC top quark decays t → qZ were reported by the ATLAS Collaboration at √ s = 13TeV [61].As for models with FCNC mediated by a new neutral gauge boson, which we generically have denoted by Z (cid:48) , theyhave been widely studied in the literature [62, 63]. In Table I we present a summary of some of the more popularmodels that predict a new Z (cid:48) gauge boson. TABLE I: Models in which there is a new neutral gauge boson with FCNC couplings [62].New heavy neutral gauge boson Model Gauge group Z h Sequential Z SU L (2) × U Y (1) × U (cid:48) (1) Z LR Left-right symmetric SU L (2) × SU R (2) × U Y (1) Z χ Gran Unification S → SU (5) × U (1) Z ψ Superstring-inspired E → SO (10) × U (1) Z η ≡ (cid:112) / Z χ − (cid:112) / Z ψ Superstring-inspired E → Rank-5 group To describe the FCNC Z (cid:48) couplings we follow the formalism presented in [46, 62] and introduce an effectiveLagrangian analogue to that of Eq. (7) in terms of the chiral couplings (cid:15) Z (cid:48) L ij ,R ij , where i and j now run over allthe SM fermions f SM = ν i , (cid:96) i , u i , d i . though there can also be new hypothetical fermions. We thus write L F CNCZ (cid:48) = − g Z (cid:48) (cid:88) i = f SM Z (cid:48) µ ¯ F i γ µ (cid:16) (cid:15) Z (cid:48) L i P L + (cid:15) Z (cid:48) R i P R (cid:17) F i , (9)where F i is a fermion triplet in the flavor basis, F T(cid:96) = ( e, µ, τ ), F Td = ( d, s, b ), and F Td = ( u, c, t ), with (cid:15) Z (cid:48) L i and (cid:15) Z (cid:48) R i being 3 × Z (cid:48) couplings. We will focus on the quark up-type sector since weexpect that the largest contribution to TNGBCs arise from the top quark, which will become evident in Sec. III.As it was pointed out in Ref. [46], we assume that the Z (cid:48) couplings to down-type quarks d , charged leptons (cid:96) andneutrinos ν are flavor-diagonal and family-universal, namely, (cid:15) Z (cid:48) L i ,R i = Q iL,R I × for i = d, (cid:96), ν , where I × is theidentity matrix and Q iL,R are the respective chiral charges. As far as the couplings of the Z (cid:48) gauge boson to up-typequarks are concerned, we assume that they are family non-universal and are given in the flavor basis as (cid:15) Z (cid:48) L u = Q uL x , (cid:15) Z (cid:48) R u = Q uR I × . (10)Thus, non-universal couplings are only induced through left-handed up-type quarks, with x a parameter that char-acterizes the size of the FCNCs and will be taken as x (cid:46) O (1). The chiral U (cid:48) (1) charges of the up-type quarks Q uL,R differ in each model as shown in Table II. TABLE II: Chiral charges for the models with a new heavy neutral gauge boson of Table I. A detailed discussion about thedetermination of these couplings can be found in Ref. [46].Sequential Z Z L,R Z χ Z ψ Z η Q uL − √ 10 1 √ − √ Q uR -0.1544 0.5038 √ − √ 24 22 √ Q dL -0.4228 -0.08493 − √ 10 1 √ − √ Q dR − √ − √ − √ Q eL -0.2684 0.2548 √ 10 1 √ 24 12 √ Q eR √ − √ 24 22 √ Q νL √ 10 1 √ 24 12 √ After rotating to the mass eigenstates, we obtain the left- and right-handed up quark fields in the mass eigenbasisvia the V L u and V R u matrices respectively. Thus the up-quark term of the Lagrangian of Eq. (9) reads L = − g Z (cid:48) Z (cid:48) µ ¯ F Mu γ µ (cid:16) V † L u (cid:15) Z (cid:48) L u V L u P L + V † R u (cid:15) Z (cid:48) R u V R u P R (cid:17) F Mu , (11)where the superscript M denotes the mass eigenbasis. For simplicity we will drop this superscript below and assumethat we are referring to the fermions in the mass eigenstate basis. In general B uL ≡ V † L u (cid:15) Z (cid:48) L u V L u will be non-diagonal.Since no mixing in the down-quark sector is assumed we will have V CKM = V † L u V L d = V † L u [46]. Therefore, theflavor mixing will be determined by the CKM matrix: B uL ≡ = V CKM (cid:15) Z (cid:48) L u V † CKM ≈ x − V ub V ∗ cb ( x − V ub V ∗ tb ( x − V cb V ∗ ub x − V cb V ∗ tb ( x − V tb V ∗ ub ( x − V tb V ∗ cb x , (12)where we have used the unitarity conditions of V CKM . As for the right-handed couplings B uR ≡ V † R u (cid:15) Z (cid:48) R u V R u , it iseasy to see that they are flavor-diagonal.The gauge coupling g Z (cid:48) is the same as that of the SM for the Z gauge boson in the sequential Z model, namely, g Z (cid:48) = e/ (2 s W c W ), whereas in the remaining models of Table I, it is given by g Z (cid:48) = (cid:114) ec W λ / g , (13)where λ g ∼ O (1). Below we will assume that λ g = 1. Constraints on FCNCs arise from D − D mixing [46, 64],single top-quark production at the LHC [65] and a simple ansatz analysis [66]. Implications of FCNC of a new neutralgauge boson Z (cid:48) have been studied in leptonic decays of the Higgs boson and the weak bosons [67], tZ (cid:48) production atthe LHC [68], Z (cid:48) decays [69], B s and B d decays [70], etc. III. ANALYTICAL RESULTS We now turn to present the calculation of the contribution to the TNGBCs ZZV ∗ ( V = Z , γ, Z (cid:48) ) arising fromcomplex FCNC couplings mediated by the SM Z gauge boson and a new neutral heavy gauge boson Z (cid:48) as shown inEqs. (7) and (11), respectively. This would allow non-vanishing CP -violating form factors. For our calculation wewill assume conserved vector currents and consider Bose symmetry [13]. This last condition will allows us to obtainall the Feynman diagrams contributing to the TNGBCs. We will see, however, that we only need to calculate thethree generic Feynman diagrams depicted in Fig. 2 since the amplitudes of the additional diagrams follow easily. Forthe calculation of the loop amplitudes we use the Passarino-Veltman reduction scheme with the help of the FeynCalcpackage [71]. V ∗ µ ( q ) Z α ( p ) Z β ( p ) m j m i m i (a) V ∗ µ ( q ) Z α ( p ) Z β ( p ) m i m j m i (b) V ∗ µ ( q ) Z α ( p ) Z β ( p ) m i m i m j (c) FIG. 2: Generic Feynman diagrams required for the contribution of FCNC couplings to TNGBCs ZZV ∗ and ZγV ∗ ( V = Z , γ, Z (cid:48) . A. ZZγ ∗ coupling In this case there are 4 contributing Feynman diagrams, but due to gauge invariance we only need to calculatethe amplitude of diagram 2(a) M αβµ since the amplitudes of the remaining diagrams are easily obtained as follows.There is an additional diagram that is obtained after the exchange f i ↔ f j so its amplitude follows from M αβµ afterexchanging the fermion masses M αβµ ( f i ↔ f j ). We also need to add a pair of diagrams where the Z gauge bosons areexchanged, which means that their total amplitude can be obtained from that of the two already described diagramsafter the exchange p µ ↔ p ν is done. We note that it is not possible to induce CP violation in the ZZγ ∗ coupling,which indeed was verified in our explicit calculation. Therefore there are only contributions to the form factor f γ ,which can be written as f γ = − (cid:88) i (cid:88) j (cid:54) = i N i Q i e m Z Re (cid:16) g ij ∗ AZ g ijV Z (cid:17) π s W c W ( q − m Z ) q R ij , (14)where m i , N f and Q i are the mass, color number and electric charge of the fermion f i . Note that Q j = Q i sincewe are considering neutral currents. The analytical expression for R ij is somewhat cumbersome and is presented inAppendix A in terms of Passarino-Veltman scalar functions and closed form functions. We have verified that Eq. (14)reduces to that reported in Ref. [13] for real FCNC couplings of the Z gauge boson. 1. Asymptotic behavior It is straightforward to obtain the high-energy limit q (cid:29) m i , m j , m Z f γ ≈ − (cid:88) i (cid:88) j (cid:54) = i e Q i N i m Z Re (cid:16) g ij ∗ AZ g ijV Z (cid:17) π q c W s W , (15)which agrees up to terms of the order q − with the one reported in [13] for the CP -conserving case ( m i = m j ), thoughwe must consider a factor of 1 / f i ↔ f j ). It is evident that f γ → m i (cid:29) q , m Z , m j must be worked out morecarefully as the expansion of the two- and three-point scalar Passarino-Veltman functions around small m j diverge.This scenario could arise in 331 model [72] or little Higgs models [43], for instance, where new heavy quarks andneutrinos are predicted. B. ZZZ ∗ coupling The calculation of this coupling is more intricate than the previous one since there are 36 contributing Feynmandiagrams, though we only need to calculate the three generic Feynman diagrams of Fig. 2, which by Bose symmetrymust be complemented with the diagrams obtained by performing six permutations of four-momenta and Lorentzindices as well as the exchange of the fermions running into the loops. In this case there are both CP -violating and CP -conserving form factors. The former is due to the fact that the virtual boson is assumed to have complex FCNCcouplings. As for the CP -conserving form factor f Z , it can be written as f Z = − (cid:88) i (cid:88) j (cid:54) = i e N i m Z π c W s W ( q − m Z ) (cid:26) g iAZ (cid:18)(cid:12)(cid:12)(cid:12) g ijAZ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) g ijV Z (cid:12)(cid:12)(cid:12) (cid:19) ( R ij + R ij )+2 g iV Z Re (cid:16) g ij ∗ AZ g ijV Z (cid:17) ( R ij − R ij ) + g iAZ (cid:20)(cid:12)(cid:12)(cid:12) g ijAZ (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) g ijV Z (cid:12)(cid:12)(cid:12) (cid:21) R ij + ( i ↔ j ) (cid:27) , (16)where the R kij ( k = 1 , , 3) functions are presented in Appendix A in terms of Passarino-Veltman scalar functionsand closed form functions. This results is in agreement with that reported in Ref. [13] for real FCNC couplings ofthe Z gauge boson.As for the CP -violating form factor f Z , it reads f Z = − (cid:88) i (cid:88) j (cid:54) = i N i e m i m j m Z Im (cid:16) g ij ∗ AZ g ijV Z (cid:17) g iAZ π c W s W ( q − m Z ) ( q − m Z ) q S ij , (17)where S ij is presented in Appendix A. It is easy to see that f Z vanishes for real couplings, which is also true ifwe consider the same fermion running into the loop ( i = j ). Thus, a non-vanishing f Z ij requires complex FCNCcouplings. Furthermore, we can also see that we need different complex phases for g ijV Z and g ijAZ to obtain a non-vanishing CP -violating form factor. Since f Z is proportional to m i m j we expect that the main contribution comesfrom FCNC couplings associated with the top quark. We would like to stress that the result of Eq. (17) has neverbeen reported in the literature. 1. Asymptotic behavior As we did it with the ZZγ ∗ vertex, we study the high-energy limit q (cid:29) m i , m j , m Z f Z ≈ − (cid:88) i (cid:88) j (cid:54) = i e N i m Z π q c W s W (cid:18) g iAZ (cid:18)(cid:12)(cid:12)(cid:12) g ijAZ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) g ijV Z (cid:12)(cid:12)(cid:12) (cid:19) + 2 g iV Z Re (cid:16) g ij ∗ AZ g ijV Z (cid:17)(cid:19) (18)Our result for f Z also reproduces the one reported in Ref. [13] for the CP -conserving case ( m i = m j ), though thistime we must consider a factor of 1/6 as we are considering twice the three kinds of Feynman diagrams of Fig. 2 bythe exchange m i ↔ m j . In ths case we also observe that f ij → CP -violating form factor f Z , which behaves in the high-energy limit as f Z ∼ /q .In the case m i (cid:29) q , m Z , m j , both form factors diverge similar as in the case of the ZZγ ∗ vertex. C. ZZZ (cid:48)∗ coupling For the sake of completeness we now consider a Z (cid:48) gauge boson with complex FCNCs couplings and calculatethe corresponding contributions to the TNGBC ZZZ (cid:48)∗ . We first present the diagonal case, where there is no flavorviolation. Since m i = m j , there is only one independent diagram in Fig. 2 and we only need to add one extra diagramobtained after the exchange p µ ↔ p ν . After some algebra, the CP -conserving form factor f Z (cid:48) reads f Z (cid:48) = − (cid:88) i eN i m Z (cid:48) π c W s W q ( q − m Z ) (cid:110) g iAZ (cid:48) (cid:104) ( g iV Z ) L i + ( g iAZ ) L i (cid:105) + g iV Z (cid:48) g iV Z g iAZ L i (cid:111) , (19)where the L ji ( j = 1, 2, 3) functions are presented in Appendix A. The CP -violating form factor f Z (cid:48) is not inducedat the one-loop level in this scenario.As far as the non-diagonal case with complex FCNCs couplings, it requires more effort. Apart from the three genericFeynman diagram of Fig. 2, we must add those diagrams obtained after the exchanges p µ ↔ p ν and f ↔ f , sothere are 12 contributing Feynman diagrams in total. However, we only need to calculate the amplitudes of the threegeneric diagrams. In this scenario both f Z (cid:48) and f Z (cid:48) form factors are non-vanishing. The CP -conserving form factorcan be written as f Z (cid:48) = − (cid:88) i (cid:88) j (cid:54) = i eN i m Z (cid:48) π c W s W q ( q − m Z ) (cid:110) g iAZ (cid:104) Re (cid:16) g ijV Z g ij ∗ V Z (cid:48) (cid:17) U ij + Re (cid:16) g ijAZ (cid:48) g ij ∗ AZ (cid:17) U ij (cid:105) +2 g iV Z (cid:48) Re (cid:16) g ijV Z g ij ∗ AZ (cid:17) U ij + 2 g iV Z (cid:104) Re (cid:16) g ijV Z g ij ∗ AZ (cid:48) (cid:17) U ij + Re (cid:16) g ijV Z (cid:48) g ij ∗ AZ (cid:17) U ij (cid:105) + g iAZ (cid:48) (cid:104) (cid:12)(cid:12)(cid:12) g ijAZ (cid:12)(cid:12)(cid:12) U ij + (cid:12)(cid:12)(cid:12) g ijV Z (cid:12)(cid:12)(cid:12) U ij (cid:105)(cid:111) , (20)whereas the CP -violating one reads f Z (cid:48) = (cid:88) i (cid:88) j (cid:54) = i eN i m Z (cid:48) π c W s W q ( q − m Z ) (cid:110) g iV Z (cid:104) Im (cid:16) g ijV Z (cid:48) g ij ∗ V Z (cid:17) T ij + Im (cid:16) g ijAZ (cid:48) g ij ∗ AZ (cid:17) T ij (cid:105) + g iAZ (cid:48) Im (cid:16) g ijV Z g ij ∗ AZ (cid:17) T ij + g iAZ (cid:104) Im (cid:16) g ijAZ (cid:48) g ij ∗ V Z (cid:17) T ij + Im (cid:16) g ijV Z (cid:48) g ij ∗ AZ (cid:17) T ij (cid:105)(cid:111) , (21)where the U kij ( k = 1 . . . 7) and T kij ( k = 1 . . . 5) functions are presented in Appendix A. We note that f Z (cid:48) ( f Z (cid:48) )depends only on the real (imaginary) part of the combinations of products of the vector and axial couplings. Wealso note that it is not necessary that both Z and Z (cid:48) gauge bosons have simultaneously complex FCNC couplings toinduce the CP -violating form factor. 1. Asymptotic behavior In the diagonal case the form factor f Z (cid:48) can be written in the high-energy limit q (cid:29) m i , m j , m Z as f Z (cid:48) (cid:39) − (cid:88) i e m Z (cid:48) N f π c W s W q (cid:110) g iAZ (cid:48) (cid:0) ( g iAZ ) + ( g iV Z ) (cid:1) + 2 g iAZ g iV Z g iV Z (cid:48) (cid:111) , (22)whereas in the non-diagonal case we obtain f Z (cid:48) (cid:39) − (cid:88) i (cid:88) j (cid:54) = i e m Z (cid:48) N f π c W s W q (cid:110) g iAZ (cid:48) (cid:18)(cid:12)(cid:12)(cid:12) g ijAZ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) g ijV Z (cid:12)(cid:12)(cid:12) (cid:19) + 2 (cid:104) g iAZ (cid:16) Re (cid:16) g ijAZ (cid:48) g ij ∗ AZ (cid:17) + Re (cid:16) g ijV Z g ij ∗ V Z (cid:48) (cid:17) (cid:17) + g iV Z (cid:16) Re (cid:16) g ijV Z (cid:48) g ij ∗ AZ (cid:17) + Re (cid:16) g ijV Z g ij ∗ AZ (cid:48) (cid:17) (cid:17) + g iV Z (cid:48) Re (cid:16) g ijV Z g ij ∗ AZ (cid:17) (cid:105)(cid:111) . (23)We note that in both scenarios f Z (cid:48) ∼ m Z (cid:48) /q , thereby decreasing quickly when q (cid:62) m Z (cid:48) . However this effect isattenuated for q (cid:46) m Z (cid:48) due to the mass of the heavy Z (cid:48) boson. We also note that Eq. (23) reduces to Eq. (22)except by a factor of 6, which is due to the fact that the CP -violating form factor receives the contribution of twelveFeynman diagrams in the diagonal scenario instead of two as in the diagonal case.On the other hand, the CP -violating form factor f Z (cid:48) is of the order of m Z (cid:48) /q in the high energy limit and decreasesquickly as q increases.Furthermore, when m i (cid:29) q , m Z , m j both form factors also show the same behavior observed in the case of thevertices ZZγ ∗ and ZZZ ∗ . IV. CONSTRAINTS ON FCNC Z COUPLINGS We would like to assess the magnitude of the new contributions to the f V and f V ( V = γ, Z, Z (cid:48) ) form factors. It isthus necessary to obtain constraints on the FCNC Z couplings to obtain an estimate of the numerical values of suchform factors. Since we expect that the main contributions arise from the FCNC couplings of the top quark, we usethe current bounds on the branching ratios of the FCNC decays t → qZ , where q = c , u [61] to constrain the g tqV Z,AZ couplings. A. Constraints on the FCNC Z couplings from t → qZ decay A comprehensive compilation of the branching ratios of top FCNC decays within the SM and several extensionmodels can be found in [52]. In the case of tree-level FCNC Z couplings, the decay width t → qZ can be written interms of the vector and axial couplings for negligible m q as followsΓ t → Zq = e m t πc W m Z s W (cid:16)(cid:12)(cid:12) g tqAZ (cid:12)(cid:12) + (cid:12)(cid:12) g tqV Z (cid:12)(cid:12) (cid:17) (cid:18) − m Z m t (cid:19) (cid:18) m Z m t (cid:19) (24)The current upper limits obtained by ATLAS collaboration at √ s = 13 TeV are: B ( t → uZ ) < . × − and B ( t → cZ ) < . × − with 95% C.L. [61]. Previous results at √ s = 7 TeV are also available [73]. The SMcontribution to the t → cZ branching ratio is negligible: B ( t → cZ ) (cid:39) − [52]. We thus obtain for the contributionof the FCNC couplings of the Z gauge boson: B ( t → qZ ) = 0 . (cid:16)(cid:12)(cid:12) g tqAZ (cid:12)(cid:12) + (cid:12)(cid:12) g tqV Z (cid:12)(cid:12) (cid:17) , (25)which allows us to obtain the following limits (cid:12)(cid:12) g tuAZ (cid:12)(cid:12) + (cid:12)(cid:12) g tuV Z (cid:12)(cid:12) < . × − , (26)and (cid:12)(cid:12) g tcAZ (cid:12)(cid:12) + (cid:12)(cid:12) g tcV Z (cid:12)(cid:12) < . × − , (27)Eqs. (26) and (27) can also be written in terms of the chiral couplings of Eq. (8).We show in Fig. 3 the allowed areas on the (cid:12)(cid:12) g tqAZ (cid:12)(cid:12) vs (cid:12)(cid:12) g tqV Z (cid:12)(cid:12) and | (cid:15) ZR tq | vs | (cid:15) ZL tq | planes. The blue-solid (green-dashed)line corresponds to the Ztc ( Ztu ) couplings. We observe that the FCNC couplings of the Z gauge boson can be aslarge as 10 − . In fact, if we assume (cid:12)(cid:12) g tqAZ (cid:12)(cid:12) (cid:39) (cid:12)(cid:12) g tqV Z (cid:12)(cid:12) , we obtain (cid:12)(cid:12) g tuV Z (cid:12)(cid:12) < . , (cid:12)(cid:12) g tcV Z (cid:12)(cid:12) < . , (28)which in terms of the chiral coupling read | (cid:15) ZR tu | < . , | (cid:15) ZR tc | < . . (29)Thus, our bounds are of the order of 10 − − − , which are similar to the constraints on FCNC couplings of downquarks obtained from B and Kaon meson decays. For instance the constraint on the (cid:12)(cid:12) g bdV Z (cid:12)(cid:12) coupling is at the10 − − − level [39, 40, 53, 57, 60], whereas the (cid:12)(cid:12) g bsV Z (cid:12)(cid:12) coupling is constrained to be below 10 − [39]. In someextension models these couplings can be of the order of 10 − − − [58, 59]. B. Constraints on the lepton flavor violating Z couplings from Z → (cid:96) i (cid:96) j Following the above approach, we now obtain constraints on the lepton flavor violating (LFV) couplings of the Z gauge boson from the experimental limits on the Z → (cid:96) ± (cid:96) ∓ decays, which have been obtained by the ATLAS andCMS collaborations: B ( Z → eτ ) < . × − , B ( Z → µτ ) < . × − at √ s = 14 TeV [74] and B ( Z → eµ ) < . × − − . × − at √ s = 8 TeV [75, 76]. The decay width Z → (cid:96) i (cid:96) j is given byΓ Z → (cid:96) i (cid:96) j = e m Z πc W s W (cid:18)(cid:12)(cid:12)(cid:12) g (cid:96) i (cid:96) j AZ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) g (cid:96) i (cid:96) j V Z (cid:12)(cid:12)(cid:12) (cid:19) , (30) FIG. 3: Allowed area with 95% C.L. in the (cid:12)(cid:12) g tqAZ (cid:12)(cid:12) vs (cid:12)(cid:12) g tqV Z (cid:12)(cid:12) (left) and | (cid:15) ZR tq | vs | (cid:15) ZL tq | (right) planes from the experimental boundson t → Zq decays, for the Ztc and (solid-line boundaries) and Ztu (dashed-line boundaries) couplings. and the corresponding branching ratio is B ( Z → (cid:96) i (cid:96) j ) = 0 . (cid:18)(cid:12)(cid:12)(cid:12) g (cid:96) i (cid:96) j AZ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) g (cid:96) i (cid:96) j V Z (cid:12)(cid:12)(cid:12) (cid:19) . (31)If we assume that (cid:12)(cid:12)(cid:12) g (cid:96) i (cid:96) j AZ (cid:12)(cid:12)(cid:12) (cid:39) (cid:12)(cid:12)(cid:12) g (cid:96) i (cid:96) j V Z (cid:12)(cid:12)(cid:12) , we obtain with 95 % C.L. | g τµV Z | < . , | g τeV Z | < . , | g µeV Z | < . . (32)We note that the constraints on | g τeV Z | and | g µeV Z | are less competitive to those obtained through the µ → eee and τ − → e − µ + µ − decays [55], which yield | g τeV Z | < . × − and | g µeV Z | < . × − . As for the bound on | g τµV Z | , itis of the same order than the one obtained from the τ − → µ − µ + µ − decay, namely, | g τµV Z | < . × − [55]. In ouranalysis below we consider the most stringent bounds, thus we will use the values reported in Ref. [55].As far as the LFV Z couplings to neutrinos are concerned, there are no experimental data to obtain reliableconstraints, so to obtain a rough estimate of these contributions we can assume couplings of the same order ofmagnitude than those used for the charged leptons. Nevertheless, the f V and f V form factors are mainly dominatedby the contribution of the heaviest quarks, whereas the lepton contributions are negligibly.Finally, in Table III we summarize the constraints on the FCNC Z gauge boson couplings that we will use in ournumerical analysis, in terms of the corresponding chiral couplings. TABLE III: Bounds on the FCNC couplings of the Z gauge boson, with 95 % C.L., from the current experimental limits onFCNC Z decays. The second row stands for the limit when either (cid:15) ZR ij or (cid:15) ZL ij is taken as vanishing and the other non-vanishing. tc tu cu d i d j (cid:96) i (cid:96) j ν i ν j (cid:12)(cid:12)(cid:12) (cid:15) ZL ij (cid:12)(cid:12)(cid:12) (cid:39) (cid:12)(cid:12)(cid:12) (cid:15) ZR ij (cid:12)(cid:12)(cid:12) − − − − (cid:12)(cid:12)(cid:12) (cid:15) ZL ij ,R ij (cid:12)(cid:12)(cid:12) − − − − V. NUMERICAL ANALYSIS We now turn to present the numerical evaluation of the TNGBCs. We first analyze the case of the ZZV ∗ ( V = γ, Z )couplings. As a matter of convenience, we write the complex chiral FCNC Z couplings as (cid:15) ZL ij ,R ij = (cid:15) ZL ij ,R ij + i ˜ (cid:15) ZL ij ,R ij , (33)where the bar (tilde) denotes the real (imaginary) part of each coupling. The CP -violating phase can then be writtenas arctan (cid:0) φ L ij ,R ij (cid:1) = ˜ (cid:15) ZL ij ,R ij (cid:15) ZL ij ,R ij . (34)We thus can write the real and imaginary terms that enter into the f V and f V ( V = γ, Z, Z (cid:48) ) form factors [Eqs.(14)-(17)] as follows (cid:12)(cid:12)(cid:12) g ijV Z,AZ (cid:12)(cid:12)(cid:12) = 14 (cid:18)(cid:16) (cid:15) ZL ij ± (cid:15) ZR ij (cid:17) + (cid:16) ˜ (cid:15) ZL ij ± ˜ (cid:15) ZR ij (cid:17) (cid:19) , (35)2Re (cid:16) g ij ∗ AZ g ijV Z (cid:17) = 12 (cid:18)(cid:16) (cid:15) ZL ij (cid:17) − (cid:16) (cid:15) ZR ij (cid:17) + (cid:16) ˜ (cid:15) ZL ij (cid:17) − (cid:16) ˜ (cid:15) ZR ij (cid:17) (cid:19) , (36)2Im (cid:16) g ij ∗ AZ g ijV Z (cid:17) = (cid:16) (cid:15) ZL ij ˜ (cid:15) ZR ij − (cid:15) ZR ij ˜ (cid:15) ZL ij (cid:17) . (37)Below we will analyze the behavior of the f V , form factors as functions of the (cid:15) ZL ij ,R ij and ˜ (cid:15) ZL ij ,R ij parameters as wellas the transfer momentum q of the V gauge boson. A. ZZγ ∗ coupling It is convenient to assume small phases of the FCNC Z couplings, namely, we consider that the imaginary parts ofthe left-handed couplings are smaller than ten percent of their real parts: φ L ij (cid:39) ˜ (cid:15) ZL ij (cid:15) ZL ij ≤ O (cid:0) − (cid:1) , (38)whereas for the right-handed couplings we assume by simplicity that φ R ij = 0 (˜ (cid:15) ZR ij = 0). As far as the size of thechiral couplings ˜ (cid:15) ZL ij ,R ij we consider the bounds shown in Table III to obtain an estimate of f γ , which is the onlynon-vanishing ZZγ ∗ form factor .We show the behavior of the FCNC contributions to f γ as a function of the photon transfer momentum q in Fig.4, where we only plot the non-negligible imaginary and real parts arising from each fermion loop as well as their totalsum. We find that the only non-negligible contributions arise from the up and down quarks, though the former arethe only ones yielding a non-negligible imaginary part, which thus coincides with the total imaginary contribution.We have considered the parameter values of Table III, but the curves shown in Fig. 4 exhibit a similar behavior forother parameter values: there is a shift upwards (downwards) when the chiral couplings values increase (decrease) as f γ is proportional to Re (cid:16) g ij ∗ AZ g ijV Z (cid:17) . We then conclude that the contributions to f γ arising from FCNCs Z couplingsare expected to be considerably smaller than the SM contribution, which is of the order of 10 − . B. ZZZ ∗ coupling In this case both f Z and f Z are non-vanishing. For our analysis we find it convenient to consider two scenarios:1 FIG. 4: Behavior of the FCNC contributions to the f γ form factor as a function of the momentum of the photon for φ L ij = 0 . φ R ij = 0, | (cid:15) ZR ij | = 0 . | (cid:15) ZL ij | and the | (cid:15) ZL ij | values shown in Table III. Only the non-negligible contributions are shown: upquarks ( tc , tu and cu ) and down quarks ( bs , bd and sd ). The total imaginary contribution coincides with the respective upquark contribution since the down quark contribution (not shown in the plot) is negligible. • Scenario I (Left- and right-handed couplings of similar size): | (cid:15) ZR ij | = 0 . | (cid:15) ZL ij | , φ L ij = 0 . 1, and φ R ij = 0. • Scenario II (Dominating left-handed couplings): | (cid:15) ZR ij | (cid:39) − × | (cid:15) ZL ij | , φ L ij = 0 . 1, and φ R ij = 0.We do not consider the scenario with dominating right-handed couplings since there is no substantial change in themagnitude of the ZZZ ∗ form factors as that observed in Scenario II. We show in Fig. 5 the behavior of the FCNCcontributions to f Z as a function of the virtual Z transfer momentum q in the two scenarios described above. Againwe only show the real and imaginary parts arising from the up and down quarks along with the total contribution,though the imaginary part of the down quark contribution is negligible and is not shown in the plots. We observethat the largest values of f Z are of the order of 10 − , which are reached for smaller q but decreases by one orderof magnitude as q becomes large. Again, the contribution to f Z from FCNC Z couplings is smaller than the SMcontribution. 200 400 600 800 100005. × - × - × - × - || q || [ GeV ] | f Z | Scenario I Up quarks ( Re ) Down quarks ( Re ) Up quarks ( Im ) Total ( Re ) 200 400 600 800 100005. × - × - × - || q || [ GeV ] Scenario II FIG. 5: Behavior of the FCNC contributions to the f Z form factor as a function of the momentum of the virtual Z gaugeboson in the two scenarios discussed in the text. Only the non-negligible contributions are shown: up quarks ( tc , tu and cu )and down quarks ( bs , bd and sd ). The total imaginary contributions coincide with the respective up quark contributions sincethe down quark contribution (not shown in the plot) is negligible. We now analyze the f Z form factor, which is absent in the SM up to the one-loop level, which means that anysizeable excess can be attributed to new physics effects. We find that the only non-negligible contribution to f Z arise2from the loops induced by the Ztc coupling, with the remaining up and down quark contributions being several ordersof magnitude smaller. We thus show in Fig. 6 the tc contribution to the f Z form factor as a function of the virtual Z four-momentum in the right plot, whereas is in the left plot we show the Z (cid:48) bs and Z (cid:48) bd contributions. We haveextracted the factor Im (cid:16) g qq (cid:48) ∗ AZ g qq (cid:48) V Z (cid:17) , so for the Z (cid:48) tc contribution f Z is of the order of f Z (cid:39) Im (cid:0) g tc ∗ AZ g tcV Z (cid:1) × − , (39)for relatively small || q || ∼ 200 GeV, but there is a decrease of up to two orders of magnitude as || q || becomes of theorder of a few TeVs. All the remaining contributions are considerably suppressed. tc ( Re ) tc ( Im ) 200 400 600 800 100010 - - - - || q || [ GeV ] | f Z | bs ( Re ) bs ( Im ) bd ( Re ) bd ( Im ) 200 400 600 800 100010 - - - - - || q || [ GeV ] FIG. 6: Behavior of the f Z form factor as a function of the momentum | q | of the virtual Z gauge boson. We have extracteda factor of Im (cid:16) g tc ∗ AZ g qq (cid:48) V Z (cid:17) from the respective contribution. All other contributions not shown in the plots are well below the10 − level. C. ZZZ (cid:48) ∗ coupling We now turn to the analysis of the CP -conserving f Z (cid:48) and the CP -violating f Z (cid:48) form factors for an off-shell Z (cid:48) boson. For the FCNCs couplings mediated by the Z gauge boson we consider the same scenarios analyzed in the caseof the ZZZ ∗ vertex. We thus use the constraints presented in Table III, which were obtained from the data on Zf i f j decays. Furthermore, for the Z (cid:48) couplings we use the interaction of Eq. (9), with the values of Table II for the chiralcharges, along with x = 0 . φ L (cid:48) ij = 0 . x stands for the parameter characterizing the size of Z (cid:48) FCNCcouplings and φ L (cid:48) ij is the CP -violating phase of the Z (cid:48) couplings to left-handed up quarks. Since all the modelssummarized in Table II give rise to similar results, we will only present the numerical results for the Z η model.We first analyze the behavior of the CP -conserving form factor f Z (cid:48) in the scenario with no FCNCs (diagonal case).We show in Fig. 7 the behavior of f Z (cid:48) as a function of the heavy Z (cid:48) gauge boson transfer momentum q (left plot)and the m Z (cid:48) mass (right plot). We observe that the dominant contributions arise from the light quarks and leptons,whereas the top quark contribution is the smaller one as its coupling with the Z (cid:48) gauge boson is proportional tothe x parameter, which is taken of the order of 10 − . This behavior is also observed in the CP -conserving ZZZ ∗ form factor in the diagonal case [13]. We also note that f Z (cid:48) decreases for increasing transfer momentum | q | , but itincreases for large values of m Z (cid:48) . Since f Z (cid:48) is proportional to m Z (cid:48) [see Eq. (19)], a similar behavior is expected inthe non-diagonal case. In Fig. 8 we present the contour lines of the total real (left plot) and imaginary (right plot)3parts of f Z (cid:48) in the | q | vs m Z (cid:48) plane. It is observed that at high energy, both real and imaginary parts of f Z (cid:48) areconsiderably small, of the order of 10 − and 10 − , respectively, which is true even if the mass of the heavy boson isvery large. For m Z (cid:48) (cid:29) | q | , the value of the real part of f Z (cid:48) can be of the orderof O (1), whereas the imaginary part is of the order of 10 − . FIG. 7: Behavior of the f Z (cid:48) form factor as a function of the transfer momentum | q | (left plot) and the m Z (cid:48) mass (right plot)in the diagonal case .FIG. 8: Contour lines of the f Z (cid:48) form factor in the | q | vs m Z (cid:48) plane in the diagonal case. We now show in Fig. 9 the form factor f Z (cid:48) as function of the transfer momentum | q | of the Z (cid:48) gauge boson inthe non-diagonal case. For the FCNCs couplings of the Z gauge boson, we consider both scenario I (left plot) andscenario II (right plot), which were also considered in the analysis of the ZZZ ∗ vertex. As we are assuming only flavorviolation in the up quark sector for the FCNCs mediated by the Z (cid:48) boson, we only plot this class of contributions.As in the analysis of the ZZZ ∗ form factor, we only show the non-negligible contributions, which are those where thetop quark runs into the loops. We observe that in both scenarios the real parts of the Z (cid:48) tc and Z (cid:48) tu contributionsare of similar size, although the largest contribution is distinct in each case. We also find that in scenario I the realparts of the Z (cid:48) tc and Z (cid:48) tu contributions are of the same sign, but they are of opposite sign in scenario II. Thus, theytend to cancel each other out. As for the imaginary parts of the partial contributions, they exhibit a similar behaviorin both scenarios, nevertheless there is a peak in the 600 GeV < | q | < 900 GeV region, which is present in a distinctcontribution in each scenario. We also show in Fig 10 the contour lines in the | q | vs m Z (cid:48) plane of the real (left plot)and imaginary (right plot) parts of f Z (cid:48) in scenario II, where it is manifest the cancellation effect between the real4parts of the Z (cid:48) tc and Z (cid:48) tu contributions for | q | around 900 GeV. In this scenario the form factor f Z (cid:48) can be of theorder of 10 − , though for intermediate | q | and large m Z (cid:48) it can reach values one order of magnitude larger. As forscenario I, the real and imaginary parts of f Z (cid:48) are of the order 10 − in general, but they could be larger for smallenergies and an ultra heavy Z (cid:48) . FIG. 9: Behavior of the f Z (cid:48) form factor in the non-diagonal case as a function of the transfer momentum | q | of the Z (cid:48) gaugeboson.FIG. 10: Contour lines in the | q | vs m Z (cid:48) plane of the real and imaginary parts of the f Z (cid:48) form factor in the non-diagonal caseand scenario II. It is also possible to induce the CP -violating form factor f Z (cid:48) via FCNC Z and Z (cid:48) couplings. We find that the onlynon-negligible contributions arise from the Z (cid:48) tc and Z (cid:48) tu couplings, though the dominant contribution to both realand imaginary parts of f Z (cid:48) is the Z (cid:48) tc one, which is one order of magnitude larger than the Z (cid:48) tu contribution. Wepresent in Fig. 11 the form factor f Z (cid:48) as a function of | q | . We observe that the real and imaginary parts behave ina rather similar way. As was the case for the ZZZ ∗ CP -violating form factor, there is no considerable distinctionbetween the results for scenario I and scenario II of the FCNC Z couplings, thus we only consider scenario I in ouranalysis. We also show in Fig. 12 the contour lines of the real part of f Z (cid:48) in the | q | vs m Z (cid:48) plane. The behavior ofthe imaginary part is similar as already stated. We note that at high energy f Z (cid:48) can be of the order of 10 − − − ,though it can be one order of magnitude larger at low energy and for an ultra heavy Z (cid:48) gauge boson. In our numericalanalysis we did not extract the complex phases as in the ZZZ ∗ case, since the f Z (cid:48) factors depends on five distinct5combinations of all of the involved phases [see Eq. (21)]. FIG. 11: Behavior of the f Z (cid:48) form factor as a function of | q | in the non-diagonal case and scenario I.FIG. 12: Contour lines of the real part of f Z (cid:48) in the q vs m Z (cid:48) plane in the non-diagonal and scenario I. VI. CONCLUSIONS AND OUTLOOK We have presented a calculation of the TNGBCs ZZV ∗ ( V = γ , Z , Z (cid:48) ) in models where FCNCs couplings mediatedby the Z and Z (cid:48) gauge bosons are allowed. These TNGBCs are given in terms of one CP -conserving form factor f V and another CP -violating one f V , for which we present analytical results in terms of both Passarino-Veltmanscalar functions and closed form functions. Such results reduce to the contributions with diagonal Z couplings already6studied in the literature. To asses the behavior of f V and f V , for the numerical analysis we obtain constraints onthe FCNCs couplings of the Z gauge boson to up quarks, which are the less constrained by experimental data: it isfound that the current constraints on the t → qZ branching ratios obtained at the LHC translate into the followingconstraints on the vector and axial Z couplings | g tuV Z | < . | g tcV Z | < . ZZγ ∗ coupling isconcerned, it is found that the only non-vanishing form factor is the CP -conserving one f γ , whose real and imaginaryparts are of the order of 10 − , with the dominant contributions arising from the heavier up and down quarks. On theother hand, as for the ZZZ ∗ coupling, both the CP -conserving and the CP violating form factors are non-vanishing.We consider two scenarios for the FCNC Z couplings (scenario I and scenario II) and find that the magnitude ofthe real and imaginary parts of these form factors are of the order of | f Z | ∼ − and | f Z | ∼ | Im ( g tc ∗ AZ g tcV Z ) | − ,with the dominant contributions arising from the non-diagonal top quark couplings. Our estimates for the FCNCcontributions to the CP -conserving f γ and f Z form factors are smaller than the prediction of the SM, whereas the f Z form factor is not induced in the SM up to the one loop level.We also consider the case of a new heavy neutral Z (cid:48) gauge boson with FCNCs and obtain the TNGBC ZZZ (cid:48)∗ , forwhich we present analytical results in the case of both diagonal and non-diagonal Z (cid:48) couplings in terms of Passarino-Veltman scalar functions and closed form functions. In the diagonal case we find the following numerical estimate forthe CP -conserving f Z (cid:48) form factor, which is the only non-vanishing, Re | f Z (cid:48) | ∼ − − − and Im | f Z (cid:48) | ∼ − − − ,with the dominant contributions arising from the light quarks and leptons. In the non-diagonal case we also considertwo scenarios for the FCNC couplings of the Z gauge boson (scenario I and scenario II). It is found that both thereal and imaginary parts of f Z (cid:48) are of the order of 10 − in scenario I, whereas in scenario II Re | f Z (cid:48) | ∼ − andIm | f Z (cid:48) | ∼ − . In general, in the non-diagonal case the magnitude of both real and imaginary parts of the f Z (cid:48) form factor are one order of magnitude larger for moderate energies and an ultra heavy Z (cid:48) gauge boson than for highenergies, with the dominant contributions arising from the Z (cid:48) tu and Z (cid:48) tc couplings. The real (imaginary) part of thenon-diagonal contributions to f Z (cid:48) are at least two (one) orders of magnitude smaller than the real (imaginary) partsof the diagonal contributions.As far as the CP -violating form factor | f Z (cid:48) | is concerned, we obtain similar estimates for its real and imaginaryparts, of the order of 10 − − − in both scenarios of the FCNC couplings of the Z gauge boson. In closing we wouldlike to stress that FCNC couplings can also yield CP violation in the TNGBCs of a new neutral gauge boson, whichmay be on interest. Acknowledgments We acknowledge support from Consejo Nacional de Ciencia y Tecnolog´ıa and Sistema Nacional de Investigadores(Mexico). Partial support from Vicerrector´ıa de Investigaci´on y Estudios de Posgrado de la Ben´emerita UniversidadAut´onoma de Puebla is also acknowledged. Appendix A: Analytical form of the TNGBCs ZZV ∗ ( V = γ, Z, Z (cid:48) ) In this appendix we present the analytical expressions for the loop functions appearing in the contributions to theTNGBCs ZZV ∗ ( V = γ, Z, Z (cid:48) ) arising from the FCNC couplings mediated by the Z gauge boson and a new heavyneutral gauge boson Z (cid:48) . For the calculation we use the Passarino-Veltman reduction scheme and also present theresults for the two- and three-point scalar functions in closed form. 1. Passarino-Veltman results a. ZZγ ∗ coupling There are only contribution to the f γ form factor, which is given in Eq. (14), where R ij reads R ij = 4 (cid:0) q − m Z (cid:1) (cid:0) m j − m i (cid:1) (cid:0) B ii (cid:0) q (cid:1) − B jj (cid:0) q (cid:1)(cid:1) + 2 (cid:0) q − m Z (cid:1) (cid:0) q − m Z (cid:1) + 2 m Z (cid:0) q + 2 m Z (cid:1) (cid:0) B ii (cid:0) q (cid:1) + B jj (cid:0) q (cid:1) − B ij (cid:0) m Z (cid:1)(cid:1) + 4 (cid:0) q − m Z (cid:1) (cid:0) m i + m j + m Z − m i m j − m i m Z − m j m Z (cid:1) (cid:0) C iji (cid:0) q (cid:1) + C jij (cid:0) q (cid:1)(cid:1) + 2 q (cid:0) q + 2 m Z (cid:1) (cid:0) m j C iji (cid:0) q (cid:1) + m i C jij (cid:0) q (cid:1)(cid:1) , (A1)7where we have introduced the shorthand notation B ij ( c ) = B (cid:0) c , m i , m j (cid:1) ,C ijk (cid:0) q (cid:1) = C (cid:0) m Z , m Z , q , m i , m j , m k (cid:1) , (A2)with B and C being the usual two- and three-point Passarino-Veltman scalar functions. It is useful observe thefollowing symmetry relations B ij ( c ) = B ji ( c ) ,C ijk (cid:0) q (cid:1) = C kji (cid:0) q (cid:1) ,C iij (cid:0) q (cid:1) = C jji (cid:0) q (cid:1) ,C iji (cid:0) q (cid:1) = C jij (cid:0) q (cid:1) . (A3)In Eq. (A1) it is evident that ultraviolet divergences cancel out. We have also verified that R ij vanishes for anon-shell photon. b. ZZZ ∗ coupling There are contributions to both the CP-conserving form factor f Z and the CP-violating one f Z . They are given inEqs. (17) and (16), with the R kij , and S ij functions given in terms of Passarino-Veltman scalar functions as follows R ij =1 + 2 (cid:0) m i − m j (cid:1) (cid:0) m Z + q (cid:1) q ( q − m Z ) (cid:0) B ij (cid:0) m Z (cid:1) − B ii (cid:0) m Z (cid:1)(cid:1) − q − m Z ) ( q − m Z ) (cid:34) q C iij (cid:0) q (cid:1) (cid:0) − q (cid:0) m i (cid:0) m i − m j (cid:1) − m Z (cid:0) m i + 3 m j − m Z (cid:1)(cid:1) +4 q m Z (cid:0) m i m j − m i m Z − m j + m Z (cid:1) + 4 m Z (cid:0) m i − m j (cid:1) − q m i (cid:17) − C iji (cid:0) q (cid:1) (cid:0) q (cid:0) − m i m j − m i m Z + m i − m j m Z + 2 m j + 2 m Z (cid:1) +2 m Z (cid:0) m i + 2 m j m Z − m j − m Z (cid:1) + q m j (cid:1) + (cid:0) B ii (cid:0) q (cid:1) − B ii (cid:0) m Z (cid:1)(cid:1) (cid:0)(cid:0) m i − m Z (cid:1) (cid:0) m Z + q (cid:1) − m j (cid:0) q − m Z (cid:1)(cid:1) + (cid:0) B ij (cid:0) q (cid:1) − B ij (cid:0) m Z (cid:1)(cid:1) (cid:0)(cid:0) m j − m Z (cid:1) (cid:0) m Z + q (cid:1) − m i (cid:0) q − m Z (cid:1)(cid:1) (cid:35) , (A4) R ij = m i q − m Z (cid:104) (cid:0) m i − m j − m Z (cid:1) (cid:0) C iij (cid:0) q (cid:1) − C iji (cid:0) q (cid:1)(cid:1) + (cid:0) q − m Z (cid:1) C iij (cid:0) q (cid:1) + B ii (cid:0) q (cid:1) − B ii (cid:0) m Z (cid:1) + 2 (cid:0) B ij (cid:0) q (cid:1) − B ij (cid:0) m Z (cid:1)(cid:1) (cid:105) , (A5) R ij = m i m j q − m Z (cid:104) (cid:0) m i − m j + q (cid:1) (cid:2) C iij (cid:0) q (cid:1) − C iji (cid:0) q (cid:1)(cid:3) + 2 (cid:0) q − m Z (cid:1) C iji (cid:0) q (cid:1) + 2 (cid:2) B ii (cid:0) q (cid:1) − B ii (cid:0) m Z (cid:1)(cid:3) + 4 (cid:2) B ij (cid:0) q (cid:1) − B ij (cid:0) m Z (cid:1)(cid:3) (cid:105) , (A6)and S ij = (cid:0) m Z − q (cid:1) (cid:2) B ii (cid:0) m Z (cid:1) − B jj (cid:0) m Z (cid:1)(cid:3) − q (cid:2) B ii (cid:0) q (cid:1) − B jj (cid:0) q (cid:1)(cid:3) − (cid:0) m Z − q (cid:1) (cid:0) m i − m j + q (cid:1) C iij (cid:0) q (cid:1) − q (cid:2) q − (cid:0) m i − m j + m Z (cid:1)(cid:3) C iji (cid:0) q (cid:1) + (cid:0) q − m Z (cid:1) (cid:0) m i − m j − q (cid:1) C jji (cid:0) q (cid:1) + q (cid:2) q − (cid:0) m j − m i + m Z (cid:1)(cid:3) C jij (cid:0) q (cid:1) . (A7)8 c. ZZZ (cid:48) ∗ coupling The contributions to the f Z (cid:48) and f Z (cid:48) form factors are given in Eqs. (19)-(21), with the L i T kij and U kij functionsgiven as follows L i = 2 m Z (cid:0) (cid:0) q (cid:0) m i + m Z (cid:1) − m Z (cid:0) m i + m Z (cid:1)(cid:1) C iii − (cid:0) m Z + q (cid:1) (cid:0) B ii (cid:0) m Z (cid:1) − B ii (cid:0) q (cid:1)(cid:1) (cid:1) − q m Z + 8 m Z + q , (A8) L i = − m Z − q ) ( m Z + q ) (cid:0) m i (cid:0) q − m Z (cid:1) + m Z (cid:1) C iii − (cid:0) m i (cid:0) q − m Z (cid:1) + m Z (cid:0) m Z + q (cid:1)(cid:1) (cid:0) B ii (cid:0) m Z (cid:1) − B ii (cid:0) q (cid:1)(cid:1) − q m Z + 8 m Z + q , (A9) L i = 2 (cid:0) (cid:0) m i (cid:0) − q m Z + 8 m Z + q (cid:1) − m Z (cid:0) m Z − q (cid:1)(cid:1) C iii − m Z (cid:0) m Z + q (cid:1) (cid:0) B ii (cid:0) m Z (cid:1) − B ii (cid:0) q (cid:1)(cid:1) − q m Z + 8 m Z + q (cid:1) . (A10) U ij = − q (cid:104) B ii (cid:0) m Z (cid:1) (cid:0) q (cid:0) m i ( m i − m j ) + m Z (cid:1) + 2 q m Z (cid:16) m Z − m i − m j ) (cid:17) − m Z (cid:0) m i − m j (cid:1) (cid:1) + q (cid:0) B ij (cid:0) m Z (cid:1) − B ij (cid:0) q (cid:1)(cid:1) (cid:0) q (cid:16) ( m i − m j ) + m Z (cid:17) − m Z ( m i − m j ) + 2 m Z (cid:1) + C iij (cid:0) q (cid:1) (cid:0) q m i ( m j − m i ) + q (cid:0) m Z (cid:0) m i (13 m i − m j ) + 3 m j (cid:1) − m i ( m i − m j ) ( m i + m j ) − m Z (cid:1) + 4 q m Z (cid:16) ( m i − m j ) ( m i + m j ) − m i m Z + m Z (cid:17) + 4 m Z (cid:0) m i − m j (cid:1) (cid:1) + (cid:0) q m Z − q m Z − q (cid:1) (cid:105) + ( i ↔ j ) , (A11) U ij = U ij ( m j → − m j ) , (A12) U ij = 2 (cid:104) B ii (cid:0) q (cid:1) (cid:0) q (cid:0) m j − m j ) + m Z (cid:1) + 2 m Z (cid:0) m i − m j + m Z (cid:1)(cid:1) − m Z (cid:0) m Z + q (cid:1) B ij (cid:0) m Z (cid:1) + C iji (cid:0) q (cid:1) (cid:0) q (cid:16) m Z − m Z (cid:0) m i + m j (cid:1) + (cid:0) m i − m j (cid:1) (cid:17) − m Z (cid:0) ( m i + m j ) − m Z (cid:1) (cid:0) ( m i − m j ) − m Z (cid:1) + q m j (cid:1) − q m Z + 4 m Z + 12 q (cid:105) + ( i ↔ j ) , (A13) U ij = − q (cid:104) m Z B ii (cid:0) m Z (cid:1) (cid:0) q (cid:0) m i − m j + m Z (cid:1) − m Z (cid:0) m i − m j (cid:1) + q (cid:1) + q (cid:0) m Z (cid:0) m Z + q (cid:1) (cid:0) B ij (cid:0) m Z (cid:1) − B ij (cid:0) q (cid:1)(cid:1) + (cid:0) q m Z − m Z + q (cid:1)(cid:1) + C iij (cid:0) q (cid:1) (cid:0) q m i ( m j − m i ) − q m Z (cid:0) m i m j − m i − m j + 4 m Z (cid:1) − q m Z (cid:0) ( m i − m j ) − m Z (cid:1) (cid:16) ( m i + m j ) + m Z (cid:17) + 4 m Z (cid:0) m i − m j (cid:1) (cid:1)(cid:105) + ( i ↔ j ) , (A14) U ij = U ij ( m j → − m j ) , (A15) U ij = − (cid:104) B ii (cid:0) q (cid:1) (cid:0) − q (cid:0) m j ( m i + m j ) + m Z (cid:1) + 2 m Z ( m i + m j ) (3 m i + m j ) − m Z (cid:1) + B ij (cid:0) m Z (cid:1) (cid:16) q (cid:16) ( m i + m j ) + m Z (cid:17) − m Z ( m i + m j ) + 2 m Z (cid:17) + C iji (cid:0) q (cid:1) (cid:0) − q m j ( m i + m j ) + 2 q (cid:0) m Z (cid:0) m i m j + m i + m j (cid:1) + m j (cid:0) m i − m j (cid:1) ( m i + m j ) − m Z (cid:1) − m Z (cid:0) ( m i + m j ) − m Z (cid:1) (cid:0) ( m i − m j ) (3 m i + m j ) + m Z (cid:1) (cid:1) + 3 q m Z − m Z − q (cid:105) + ( i ↔ j ) , (A16)9and U ij = U ij ( m j → − m j ) . (A17) T ij = (cid:104) m i − m j ) (cid:0) q ( m i + m j ) (cid:0) B ij (cid:0) q (cid:1) (cid:0) q − m Z (cid:1) − m Z B ij (cid:0) m Z (cid:1)(cid:1) + C iij (cid:0) q (cid:1) (cid:0) q (cid:0) m i ( m i + m j ) − m Z (3 m i + m j ) (cid:1) − q m Z ( m i + m j ) (cid:0) m i ( m i + m j ) − m Z (cid:1) + 2 m Z ( m i − m j ) ( m i + m j ) + q m i (cid:1)(cid:1) + B ii (cid:0) m Z (cid:1) (cid:16) q m Z (cid:0) m i m j + 7 m i − m j − m Z (cid:1) − m Z (cid:0) m i − m j (cid:1) + q (cid:0) m Z − m i (cid:1)(cid:17) − q B ii (0) m i (cid:0) q − m Z (cid:1) (cid:105) − ( i ↔ j ) , (A18) T ij = T ij ( m j → − m j ) , (A19) T ij = 6 q m i m j (cid:104) C iji (cid:0) q (cid:1) (cid:0) (cid:0) m i − m j + m Z (cid:1) − q (cid:1) − B ii (cid:0) q (cid:1) (cid:105) − ( i ↔ j ) , (A20) T ij = (cid:104) m i + m j ) (cid:0) C iij (cid:0) q (cid:1) (cid:0) q (cid:0) m Z ( m i + m j ) + 2 m i m j ( m j − m i ) (cid:1) − q m Z ( m i − m j ) × (cid:0) m i ( m i − m j ) − m Z (cid:1) + 2 m Z ( m i − m j ) (cid:0) m i − m j (cid:1) − q m i (cid:1) − q ( m i − m j ) (cid:0) B ij (cid:0) q (cid:1) (cid:0) m Z + q (cid:1) + (cid:0) m Z − q (cid:1) B ij (cid:0) m Z (cid:1) (cid:1)(cid:1) + B ii (cid:0) m Z (cid:1) (cid:0) q (cid:0) m i ( m i + 3 m j ) + m Z (cid:1) − q m Z (cid:0) m i m j − m i + 3 m j + 4 m Z (cid:1) − m Z (cid:0) m i − m j (cid:1) (cid:1) − q B ii (0) m i (cid:0) q − m Z (cid:1) (cid:105) − ( i ↔ j ) , (A21) T ij = T ij ( m j → − m j ) , (A22) 2. Closed form results We now present the closed form of the TGNBCs presented above. We only expand the two-point scalar functions interms of transcendental functions as the three-point functions are too cumbersome to be expanded. We first introducethe following auxiliary functions: η ( x , y ) = (cid:112) x − x y , (A23) β ( x , y , z ) = (cid:112) − x ( y + z ) + x + ( y − z ) , (A24) χ ( x , y ) = log (cid:32) (cid:112) x − x y + 2 y − x y (cid:33) , (A25)and ξ ( x , y , z ) = log (cid:32) (cid:112) − x ( y + z ) + x + ( y − z ) + x + y − z xy (cid:33) . (A26)0 a. ZZγ ∗ coupling The R ij function of Eq.(14) can be written as follows R ij = − q (cid:104) − q (cid:0) q (cid:0) − m Z (cid:0) m i + m j (cid:1) + (cid:0) m i − m j (cid:1) + m Z (cid:1) − m Z ( − m i − m j + m Z ) ( m i − m j + m Z ) × ( − m i + m j + m Z ) ( m i + m j + m Z ) + q m j (cid:1) C iji (cid:0) q (cid:1) − q (cid:0) m i (cid:0) − q (cid:0) m j + m Z (cid:1) + 4 m Z (cid:0) m j + m Z (cid:1) + q (cid:1) + 2 m i (cid:0) q − m Z (cid:1) − (cid:0) m j − m Z (cid:1) (cid:0) m Z − q (cid:1) (cid:1) C jij (cid:0) q (cid:1) + 3 q log (cid:32) m i m j (cid:33) (cid:0) m j − m i (cid:1) − η ( q , m j ) log ( χ ( q , m j )) (cid:0) q (cid:0) m i − m j + m Z (cid:1) + 2 m Z (cid:0) − m i + m j + m Z (cid:1)(cid:1) + 2 η ( q , m i ) log ( χ ( q , m i )) (cid:0) m i − m j (cid:1)(cid:0) q − m Z (cid:1) + 6 q m Z + 4 β ( m i , m j , m Z ) q m Z log ( ξ ( m i , m j , m Z )) − η ( q , m i ) q m Z log ( χ ( q , m i )) − q m Z − η ( q , m i ) m Z log ( χ ( q , m i )) − q + 2 β ( m i , m j , m Z ) q × log ( ξ ( m i , m j , m Z )) (cid:105) . (A27) b. ZZZ ∗ coupling The R kij and S ij functions of Eqs. (17) and (16) read R ij = 1( q − m Z ) q m Z (cid:34) − q − m Z ) (cid:110) − q m Z (cid:0) q (cid:0) − m i (cid:0) m j + m Z (cid:1) + m i + 2 (cid:0) − m j m Z + m j + m Z (cid:1)(cid:1) − m Z (cid:0)(cid:0) m j − m Z (cid:1) − m i (cid:1) + q m j (cid:1) C iji (cid:0) q (cid:1) + m Z (cid:0) q (cid:0) m i (cid:0) m j + 7 m Z (cid:1) − m i + 3 m j m Z − m Z (cid:1) + 4 q m Z (cid:0) m i (cid:0) m j − m Z (cid:1) − m j + m Z (cid:1) + 4 m Z (cid:0) m i − m j (cid:1) − q m i (cid:1) C iij (cid:0) q (cid:1) − m Z q (cid:0) q − q m Z + m Z (cid:1) + η ( q , m i ) m Z (cid:0) q (cid:0) m i − m j − m Z (cid:1) + 2 m Z (cid:0) m i + m j − m Z (cid:1)(cid:1) log ( χ ( q , m i ))+ η ( m Z , m i ) (cid:0) m i (cid:0) q − m Z (cid:1) + m Z (cid:0) m j (cid:0) m Z − q (cid:1) + 2 q m Z + q (cid:1)(cid:1) log ( χ ( m Z , m i ))+ β ( m i , m j , m Z ) (cid:0) q (cid:0) m i + m j + 2 m Z (cid:1) + 4 m Z (cid:0) m i − m j (cid:1) + 4 q m Z (cid:0) m Z − m i (cid:1)(cid:1) log ( ξ ( m i , m j , m Z ))+ β ( m i , m j , q ) m Z (cid:0) m i (cid:0) m Z − q (cid:1) + (cid:0) m j − m Z (cid:1) (cid:0) m Z + q (cid:1)(cid:1) log ( ξ ( m i , m j , q )) (cid:111) − (cid:0) m i − m j (cid:1) (cid:110) m i (cid:0) m Z − q (cid:1) − (cid:0) m j + 4 m Z (cid:1) (cid:0) m Z + q (cid:1) (cid:111) log (cid:32) m i m j (cid:33) (cid:35) , (A28) R ij = m i q m Z (cid:34) − m Z − q ) (cid:110) q m Z (cid:0) m i − m j − m Z + q (cid:1) C iij (cid:0) q (cid:1) + q m Z (cid:0) − m i + m j + m Z (cid:1) C iji (cid:0) q (cid:1) + m Z (cid:0) β ( m i , m j , q ) log ( ξ ( m i , m j , q )) + η ( q , m i ) log ( χ ( q , m i )) (cid:1) − q (2 β ( m i , m j , m Z ) log ( ξ ( m i , m j , m Z )) + η ( m Z , m i ) log ( χ ( m Z , m i ))) (cid:111) + (cid:0) m i − m j (cid:1) log (cid:32) m i m j (cid:33) (cid:35) , (A29) R ij = m i m j q m Z (cid:34) − m Z − q ) (cid:110) q m Z (cid:0) m i − m j + q (cid:1) C iij (cid:0) q (cid:1) + q m Z (cid:0) q − (cid:0) m i − m j + m Z (cid:1)(cid:1) C iji (cid:0) q (cid:1) + 2 m Z (cid:0) β ( m i , m j , q ) log ( ξ ( m i , m j , q ))+ η ( q , m i ) log ( χ ( q , m i )) (cid:1) − q (cid:0) β log ( ξ ( m i , m j , m Z )) + η ( m Z , m i ) log ( χ ( m Z , m i )) (cid:1)(cid:111) + 2 (cid:0) m i − m j (cid:1) log (cid:32) m i m j (cid:33) (cid:35) , (A30)1and S ij = − C iij (cid:0) q (cid:1) (cid:0) m Z − q (cid:1) (cid:0) m i − m j + q (cid:1) + q C iji (cid:0) q (cid:1) (cid:0) − (cid:0) q − (cid:0) m i − m j + m Z (cid:1)(cid:1)(cid:1) + C ijj (cid:0) q (cid:1) (cid:0) q − m Z (cid:1) (cid:0) m i − m j − q (cid:1) + q C jij (cid:0) q (cid:1) (cid:0) m i − (cid:0) m j + m Z (cid:1) + q (cid:1) − η ( q , m i ) log ( χ ( q , m i )) + 2 η ( q , m j ) log ( χ ( q , m j )) + 4 (cid:0) q − m Z (cid:1) log (cid:32) m i m j (cid:33) + 2 (cid:0) m Z − q (cid:1) m Z (cid:110) η ( m Z , m i ) log ( χ ( m Z , m i )) − η ( m Z , m j ) log ( χ ( m Z , m j )) (cid:111) . (A31) c. ZZZ (cid:48)∗ coupling Finally, the L i T kij and U kij functions of Eqs. (19)-(21) are given as follows L = 1 q (cid:110) q m Z C iii (cid:0) q (cid:1) (cid:0) q (cid:0) m i + m Z (cid:1) − m Z (cid:0) m i + m Z (cid:1)(cid:1) − q m Z + 2 (cid:0) m Z + q (cid:1) (cid:0) η ( q , m i ) m Z log ( χ ( q , m i )) − η ( m Z , m i ) q log ( χ ( m Z , m i )) (cid:1) + 8 q m Z + q (cid:111) , (A32) L = 1 q m Z (cid:110) − q m Z C iii (cid:0) q (cid:1) (cid:0) m Z − q (cid:1) (cid:0) m i (cid:0) q − m Z (cid:1) + m Z (cid:1) + 2 (cid:0) m i (cid:0) q − m Z (cid:1) + m Z (cid:0) m Z + q (cid:1) (cid:1) (cid:0) η ( q , m i ) m Z log ( χ ( q , m i )) − η ( m Z , m i ) q log ( χ ( m Z , m i )) (cid:1) + q m Z − q m Z + 8 q m Z (cid:111) , (A33) L = 2 q (cid:110) q C iii (cid:0) q (cid:1) (cid:0) m i (cid:0) − q m Z + 8 m Z + q (cid:1) − m Z (cid:0) m Z − q (cid:1)(cid:1) − q m Z + 2 (cid:0) m Z + q (cid:1) (cid:0) η ( q , m i ) m Z log ( χ ( q , m i )) − η ( m Z , m i ) q log ( χ ( m Z , m i )) (cid:1) + 8 q m Z + q (cid:111) , (A34) U ij = − q (cid:110) C iij (cid:0) q (cid:1) (cid:0) q m i ( m j − m i ) + q (cid:0) m Z (cid:0) − m i m j + 13 m i + 3 m j (cid:1) − m i ( m i − m j ) ( m i + m j ) − m Z (cid:1) + 4 q m Z (cid:0) ( m i − m j ) ( m i + m j ) − m i m Z + m Z (cid:1) + 4 m Z (cid:0) m i − m j (cid:1) (cid:1) + C jji (cid:0) q (cid:1) (cid:0) q m j ( m i − m j )+ q (cid:0) m Z (cid:0) − m i m j + 3 m i + 13 m j (cid:1) − m j ( m i − m j ) ( m i + m j ) − m Z (cid:1) + 4 q m Z (cid:0) − ( m i − m j ) ( m i + m j ) − m j m Z + m Z (cid:1) + 4 m Z (cid:0) m i − m j (cid:1) (cid:1) − (cid:0) − q m Z + 8 q m Z + q (cid:1) (cid:111) − m Z (cid:110) β ( m i , m j , m Z ) × log ( ξ ( m i , m j , m Z )) (cid:0) q (cid:0) ( m i − m j ) + m Z (cid:1) − m Z ( m i − m j ) + 2 m Z (cid:1) (cid:111) + 4 q (cid:110) β ( m i , m j , q ) × log ( ξ ( m i , m j , q )) (cid:0) q (cid:0) ( m i − m j ) + m Z (cid:1) − m Z ( m i − m j ) + 2 m Z (cid:1) (cid:111) + (cid:0) m i − m j (cid:1) (cid:0) m Z − q (cid:1) q m Z log (cid:32) m i m j (cid:33) (cid:110) q (cid:0) − (cid:0) ( m i − m j ) + 2 m Z (cid:1)(cid:1) − m Z (cid:0) m Z − ( m i − m j ) (cid:1) (cid:111) + η ( m Z , m i ) log ( χ ( m Z , m i )) q m Z (cid:110) q (cid:0) − (cid:0) m i ( m i − m j ) + m Z (cid:1)(cid:1) − q m Z (cid:0) m Z − m i − m j ) (cid:1) + 4 m Z (cid:0) m i − m j (cid:1) (cid:111) − η ( m Z , m j ) log ( χ ( m Z , m j )) q m Z (cid:110) q (cid:0) m j ( m j − m i ) + m Z (cid:1) + 2 q m Z (cid:0) m Z − m i − m j ) (cid:1) + 4 m Z (cid:0) m i − m j (cid:1) (cid:111) , (A35)2 U ij = U ij ( m j → − m j ) , (A36) U ij = 2 (cid:110) ( C iji (cid:0) q (cid:1) (cid:0) q (cid:0) − m Z (cid:0) m i + m j (cid:1) + (cid:0) m i − m j (cid:1) + m Z (cid:1) − m Z (cid:0) − ( m i − m j ) + m Z (cid:1) × (cid:0) − ( m i + m j ) + m Z (cid:1) + q m j (cid:1) + C jij (cid:0) q (cid:1) (cid:0) m i (cid:0) − q (cid:0) m j + m Z (cid:1) + 4 m Z (cid:0) m j + m Z (cid:1) + q (cid:1) (A37)+ 2 m i (cid:0) q − m Z (cid:1) − (cid:0) m j − m Z (cid:1) (cid:0) m Z − q (cid:1) (cid:1) − q m Z + 8 m Z + q (cid:111) + 2 η ( q , m i ) log ( χ ( q, m i )) q (cid:110) q (cid:0) − m i + 2 m j + m Z (cid:1) + 2 m Z (cid:0) m i − m j + m Z (cid:1) (cid:111) + 2 η ( q, m j ) log ( χ ( q, m j )) q (cid:110) q (cid:0) m i − m j + m Z (cid:1) + 2 m Z (cid:0) − m i + m j + m Z (cid:1) (cid:111) + 6 q (cid:0) m i − m j (cid:1) log (cid:32) m i m j (cid:33) − β ( m i , m j , m Z ) log ( ξ ( m i , m j , m Z )) (cid:0) m Z + q (cid:1) , (A38) U ij = − q (cid:110) C iij (cid:0) q (cid:1) (cid:0) q m i ( m j − m i ) + q m Z (cid:0) − m i m j + m i + 3 m j − m Z (cid:1) + 4 q m Z (cid:0) − ( m i − m j ) + m Z (cid:1) × (cid:0) ( m i + m j ) + m Z (cid:1) + 4 m Z (cid:0) m i − m j (cid:1) (cid:1) + C jji (cid:0) q (cid:1) (cid:0) q m j ( m i − m j ) + q m Z (cid:0) − m i m j + 3 m i + m j − m Z (cid:1) + 4 q m Z (cid:0) − ( m i − m j ) + m Z (cid:1) (cid:0) ( m i + m j ) + m Z (cid:1) + 4 m Z (cid:0) m i − m j (cid:1) (cid:1) − (cid:0) − q m Z + 8 q m Z + q (cid:1)(cid:111) − η ( m Z , m i ) log ( χ ( m Z , m i )) q (cid:110) q (cid:0) m i − m j + m Z (cid:1) − m Z (cid:0) m i − m j (cid:1) + q (cid:111) − η ( m Z , m j ) log ( χ ( m Z , m j )) q (cid:110) q (cid:0) − m i + 2 m j + m Z (cid:1) + 4 m Z (cid:0) m i − m j (cid:1) + q (cid:111) + (cid:0) m i − m j (cid:1) (cid:0) q m Z − m Z + q (cid:1) q log (cid:32) m i m j (cid:33) − β ( m i , m j , m Z ) log ( ξ ( m i , m j , m Z )) (cid:0) m Z + q (cid:1) + log ( ξ ( m i , m j , q )) (cid:110) β ( m i , m j , q ) m Z q + 4 β ( m i , m j , q ) m Z (cid:111) , (A39) U ij = U ij ( m j → − m j ) , (A40) U ij = 2 (cid:110) C iji (cid:0) q (cid:1) (cid:0) q m j ( m i + m j ) − q (cid:0) m Z (cid:0) m i m j + m i + m j (cid:1) + m j ( m i − m j ) ( m i + m j ) − m Z (cid:1) − m Z (cid:0) − ( m i + m j ) + m Z (cid:1) (cid:0) ( m i − m j ) (3 m i + m j ) + m Z (cid:1) (cid:1) + C jij (cid:0) q (cid:1) (cid:0) q m i ( m i + m j )+ 2 q (cid:0) − m Z (cid:0) m i m j + m i + m j (cid:1) + m i ( m i − m j ) ( m i + m j ) + m Z (cid:1) − m Z (cid:0) − ( m i + m j ) + m Z (cid:1) × (cid:0) m Z − ( m i − m j ) ( m i + 3 m j ) (cid:1) (cid:1) − q m Z + 8 m Z + q (cid:111) − β ( m i , m j , m Z ) log ( ξ ( m i , m j , m Z )) m Z (cid:110) q (cid:0) ( m i + m j ) + m Z (cid:1) − m Z ( m i + m j ) + 2 m Z (cid:111) + 2 η ( q, m i ) log ( χ ( q, m i )) (cid:0) q (cid:0) m j ( m i + m j ) + m Z (cid:1) − m Z ( m i + m j ) (3 m i + m j ) + 2 m Z (cid:1) q + 2 η ( q, m j ) log ( χ ( q, m j )) (cid:0) q (cid:0) m i ( m i + m j ) + m Z (cid:1) − m Z ( m i + m j ) ( m i + 3 m j ) + 2 m Z (cid:1) q − (cid:0) m i − m j (cid:1) m Z log (cid:32) m i m j (cid:33) (cid:110) m Z (cid:0) ( m i + m j ) − m Z (cid:1) − q (cid:0) ( m i + m j ) + 2 m Z (cid:1) (cid:111) , (A41) U ij = U ij ( m j → − m j ) , (A42)3 T ij = ( m i − m j ) (cid:110) C iij (cid:0) q (cid:1) (cid:0) q (cid:0) m i ( m i + m j ) − m Z (3 m i + m j ) (cid:1) − q m Z ( m i + m j ) (cid:0) m i ( m i + m j ) − m Z (cid:1) + 2 m Z ( m i − m j ) ( m i + m j ) + q m i (cid:1) + 3 C jji (cid:0) q (cid:1) (cid:0) q (cid:0) m j ( m i + m j ) − m Z ( m i + 3 m j ) (cid:1) − q m Z ( m i + m j ) (cid:0) m j ( m i + m j ) − m Z (cid:1) + 2 m Z ( m i − m j ) ( m i + m j ) + q m j (cid:1) + 4 q ( m i + m j ) (cid:0) q − m Z (cid:1) (cid:111) + 6 β ( m i , m j , m Z ) q log ( ξ ( m i , m j , m Z )) (cid:0) m j − m i (cid:1) − β ( m i , m j , q ) log ( ξ ( m i , m j , q )) (cid:0) m i − m j (cid:1) (cid:0) m Z − q (cid:1) + η ( m Z , m i ) log ( χ ( m Z , m i )) m Z (cid:110) q m Z (cid:0) m i m j + 7 m i − m j − m Z (cid:1) − m Z (cid:0) m i − m j (cid:1) + q (cid:0) m Z − m i (cid:1) (cid:111) + η ( m Z , m j ) log ( χ ( m Z , m j )) m Z (cid:110) q m Z (cid:0) − m i m j + 3 m i − m j + 4 m Z (cid:1) + 6 m Z (cid:0) m i − m j (cid:1) + q (cid:0) m j − m Z (cid:1) (cid:111) + log (cid:32) m i m j (cid:33) (cid:110) q (cid:0) (cid:0) m i + m j (cid:1) − m Z (cid:1) − q m Z (cid:0) m i + m j ) − m Z (cid:1) + 12 m Z (cid:0) m i − m j (cid:1) (cid:111) , (A43) T ij = T ij ( m j → − m j ) , (A44) T ij = 6 q m i m j (cid:110) C iji (cid:0) q (cid:1) (cid:0) (cid:0) m i − m j + m Z (cid:1) − q (cid:1) + C jij (cid:0) q (cid:1) (cid:0) m i − (cid:0) m j + m Z (cid:1) + q (cid:1) (cid:111) + 12 q m i m j log (cid:32) m i m j (cid:33) − η ( q, m i ) q m i m j log ( χ ( q, m i )) + 12 η ( q, m j ) q m i m j log ( χ ( q, m j )) , (A45) T ij = ( m i + m j ) (cid:110) C iij (cid:0) q (cid:1) (cid:0) q (cid:0) m Z ( m i + m j ) + 2 m i m j ( m j − m i ) (cid:1) − q m Z ( m i − m j ) (cid:0) m i ( m i − m j ) − m Z (cid:1) + 2 m Z ( m i − m j ) ( m i + m j ) + q ( − m i ) (cid:1) + 3 C jji (cid:0) q (cid:1) (cid:0) − q (cid:0) m Z ( m i + m j ) + 2 m i m j ( m i − m j ) (cid:1) + 2 q m Z ( m i − m j ) (cid:0) m j ( m i − m j ) + m Z (cid:1) + 2 m Z ( m i − m j ) ( m i + m j ) + q m j (cid:1) + 4 q ( m i − m j ) (cid:0) q − m Z (cid:1) (cid:111) + 6 β ( m i , m j , m Z ) q log ( ξ ( m i , m j , m Z )) m Z (cid:0) m i − m j (cid:1) (cid:0) q − m Z (cid:1) − β ( m i , m j , q ) log ( ξ ( m i , m j , q )) (cid:0) m i − m j (cid:1) (cid:0) m Z + q (cid:1) + η ( m Z , m i ) log ( χ ( m Z , m i )) m Z (cid:110) q (cid:0) m i ( m i + 3 m j ) + m Z (cid:1) − q m Z (cid:0) m i m j − m i + 3 m j + 4 m Z (cid:1) − m Z (cid:0) m i − m j (cid:1) (cid:111) + η ( m Z , m j ) log ( χ ( m Z , m j )) m Z (cid:110) q (cid:0) − (cid:0) m j × (3 m i + m j ) + m Z (cid:1)(cid:1) + q m Z (cid:0) m i m j + 3 m i − m j + 4 m Z (cid:1) + 6 m Z (cid:0) m i − m j (cid:1) (cid:111) + 1 m Z log (cid:32) m i m j (cid:33) (cid:110) q (cid:0) − (cid:0) m i m j m Z + 3 (cid:0) m i − m j (cid:1) + m Z (cid:1)(cid:1) + 2 q m Z (cid:0) − m Z ( m i − m j ) + 3 (cid:0) m i − m j (cid:1) + 2 m Z (cid:1) + 12 m Z (cid:0) m i − m j (cid:1) (cid:111) , (A46)and T ij = T ij ( m j → − m j ) , (A47) [1] M. 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