Flavor changing neutral current decays $t\rightarrow c X$ ($X=\gamma ,\,g,\, Z,\, H$) and $t\rightarrow c{{\bar{\ell }}}\ell$ ($\ell =\mu ,\,\tau$) via scalar leptoquarks

Abstract

The flavor changing neutral current decays $t\rightarrow c X$ ($X=\gamma ,\,g,\, Z,\, H$) and $t\rightarrow c{{\bar{\ell }}}\ell$ ($\ell =\mu ,\,\tau$) are studied in a renormalizable scalar leptoquark (LQ) model with no proton decay, where a scalar SU(2) doublet with hypercharge $Y=7/6$ is added to the standard model, yielding a non-chiral LQ $\varOmega _{5/3}$. Analytical results for the one-loop (tree-level) contributions of a scalar LQ to the $f_i\rightarrow f_j X$ ($f_i\rightarrow f_j {\bar{f}}_m f_l$) decays, with $f_a=q_a, \ell _a$, are presented. We consider the scenario where $\varOmega _{5/3}$ couples to the fermions of the second and third families, with its right- and left-handed couplings obeying $\lambda _R^{\ell u_i}/\lambda _L^{\ell u_i}=O(\epsilon )$, where $\epsilon$ parametrizes the relative size between these couplings. The allowed parameter space is then found via the current constraints on the muon $(g-2)$, the $\tau \rightarrow \mu \gamma$ decay, the LHC Higgs boson data, and the direct LQ searches at the LHC. For $m_{\varOmega _{5/3}}=1$ TeV and $\epsilon =10^{-3}$, we find that the $t\rightarrow c X$ branching ratios are of similar size and can be as large as $10^{-8}$ in a tiny area of the parameter space, whereas ${\mathrm{Br}}(t\rightarrow c{{\bar{\tau }}}\tau )$ [${\mathrm{Br}}(t\rightarrow c{{\bar{\mu }}}\mu )$] can be up to $10^{-6}$ ($10^{-7}$).

1 Introduction

The conjecture that lepton number is the fourth color quantum number was put forward long ago in the context of an $SU(4)_R\times SU(4)_L\times SU(4')$ theory [1, 2], which requires the presence of new gauge and scalar bosons carrying both lepton and baryon number. Such particles, dubbed leptoquarks (LQs) since transform leptons into quarks and vice versa, appear naturally in grand unified theories [3,4,5,6,7], but they are also predicted in other well motivated theories, such as technicolor [8,9,10], models with composite fermions [11,12,13], superstring-inspired $\mathrm{E}_6$ models [14, 15], models with extended scalar sectors [16, 17], etc. However, LQs with lepton and baryon number violating interactions can give rise to dangerous effects such as large lepton flavor violating interactions (LFV) or tree-level-induced proton decay. The latter implies that, unless an extra symmetry is invoked to forbid diquark couplings, LQ masses must be as heavy as the Planck scale, thereby rendering unobservable effects on low energy processes. Therefore, only those theories with renormalizable lepton and baryon number conserving LQ interactions are phenomenologically appealing. LQ phenomenology and low-energy constraints on the parameter space of the most representative LQ models have been widely discussed (see [18] for instance). For a more up-to-date review on LQ physics we refer the reader to Ref. [19]. It turns out that vector LQ masses and couplings are tightly constrained by experimental data, therefore the study of scalar LQs has been favored in the literature. In this work we are interested in a simple renormalizable LQ model with no proton decay where the presence of relatively light scalar LQs can still be compatible with low energy constraints from experimental data [17, 20, 21]. In such a model, scalar LQs are introduced in the standard model (SM) via a doublet of SU(2).

In the SM flavor changing neutral currents (FCNCs) can arise up to the one-loop level or higher orders of perturbation theory, but are additionally suppressed by the so-called Glashow–Iliopoulos–Maiani (GIM) mechanism. On the other hand, the SM forbids LFV effects at any order, though experimental evidences hint that neutrinos are massive, thereby implying that LFV effects should be present in nature indeed. No evidences of large FCNCs transitions have yet been experimentally observed, so the search for this class of effects is a must in the physics program of any particle collider. While FCNC transition between fermions of the second and first family are considerably constrained by experimental data, transitions involving the fermions of the third and second generation have no such strong restrictions. In this regard, FCNC top quark transitions stand out among the most widely studied processes at the CERN LHC. This stems from the fact that in spite of having negligible rates in the SM, they can have a considerable enhancement in beyond the SM theories and could be at the reach of detection.

The scalar particle discovered at the LHC in 2012 seems to be consistent with the SM Higgs boson, but several of its properties remain to be tested more accurately, such as its couplings to light fermions. The LHC also offers great potential to search for some exotic Higgs decay channels that are highly suppressed or forbidden in the SM. Along these lines, there has been considerably interest in the study of the LFV Higgs boson decay $H\rightarrow {\bar{\ell }}_i\ell _j$, which was first studied in [22, 23] and has been the focus of great attention recently. Although an apparent excess of the $H \rightarrow \tau \mu$ branching ratio, with a significance of $2.4\,\sigma$, was observed at the LHC Run 1 [24], it was not confirmed in Run-2 data. The LFV Higgs boson decay $H\rightarrow {\bar{\ell }}_i\ell _j$ has been widely studied in several extension models. In particular, in the LQ model of Ref. [21], the $H \rightarrow \tau \mu$ rate is predicted to be at the reach of experimental detection in some regions of the allowed parameter space. On the other hand, in the quark sector, apart from the FCNC Higgs boson decay into light quarks $H\rightarrow q q'$, the FCNC top quark decay $t\rightarrow cH$ has also been widely studied along with other decays such as $t\rightarrow c\gamma$, $t\rightarrow c Z$ and $t\rightarrow cg$.

At hadron colliders, the dominant top quark production mechanisms are gluon fusion and quark annihilation. The former is the main top quark production process at the LHC (about 90 % at $\sqrt{s}=$ 14 TeV), with a small percentage due to quark annihilation. The millions of yearly top quark events at the LHC would allow experimentalist to search for its FCNCs decays such as $t\rightarrow cX$ ($X=g,\,\gamma ,\,Z,\,H$), whose rates are negligibly small in the SM [25,26,27]:

$$\begin{aligned} {\mathrm{Br}}^{\mathrm{SM}}(t\rightarrow c g)&= 10^{-8}, \end{aligned}$$

(1)

$$\begin{aligned} {\mathrm{Br}}^{\mathrm{SM}}(t\rightarrow c\gamma )&= 10^{-10}, \end{aligned}$$

(2)

$$\begin{aligned} {\mathrm{Br}}^{\mathrm{SM}}(t\rightarrow cZ)&=10^{-13}, \end{aligned}$$

(3)

$$\begin{aligned} {\mathrm{Br}}^{\mathrm{SM}}(t\rightarrow cH)&=10^{-13}. \end{aligned}$$

(4)

These rates however can be several orders of magnitude larger in several SM extensions, such as two-Higgs doublet models [25, 28], supersymmetric models [29,30,31,32], left-right supersymmetric models [33], extra dimensions [34], models with an extra neutral gauge boson [35], 331 models [36], etc. Therefore, any experimental evidence of these FCNCs top quark and Higgs boson decay channels may shed light on the underlying fundamental theory of particle interactions.

In this work we focus on the study of the FCNC decays $t\rightarrow cX$ ($X=\gamma ,\, g,\,Z,\,H$) in a simple renormalizable scalar LQ model in which there is no proton decay induced via tree-level LQ exchange, where these processes arise at one-loop level at the lowest order of perturbation theory. As a by-product, we present the exact results for the one-loop LQ scalar contribution to the $H\rightarrow \bar{f}_j f_i$ decay width, which follows easily from the $f_i\rightarrow f_j H$ decay width by crossing symmetry. For completeness, we also consider in our study the tree-level FCNC decay $t\rightarrow c{{\bar{\ell }}}\ell$ ($\ell =\mu ,\,\tau$), which in fact can have larger branching ratios than those of the one-loop induced decays.

The rest of this presentation is as follows. In Sect. 2 we briefly discuss the framework of the LQ model we are interested in. Section 3 is devoted to present the general calculation of the FCNC decay amplitudes and decay widths. For the one-loop induced decays we express the amplitudes in terms of both Passarino–Veltman scalar functions and Feynman parameter integrals. We present a discussion on the constraints on the LQ couplings from experimental data in Sect. 5, followed by the numerical analysis of the FCNCs Higgs boson transitions in Sect. 6. The conclusions and outlook are presented in Sect. 7. Finally, a few lengthy formulas for the loop integrals are presented in the Appendices.

2 A simple scalar LQ model

Rather than considering a specific theory, a convenient strategy to study the LQ phenomenology is via a model-independent approach through an effective lagrangian. One can thus focus on the low energy LQ interactions and, without loss of generality, disregard the complex framework of the ultraviolet completion, which is not relevant for the phenomenology below the TeV scale. The most general dimension-four $SU(3)_c\times SU(2)_L\times U(1)_Y$-invariant effective interactions of scalar and vector LQs, respecting both lepton and baryon number was first presented in [37] and has been analyzed recently in [19]. In this work we consider a simple renormalizable LQ model in which it is not necessary to invoke an extra symmetry to forbid the proton decay. A single SU(2) doublet with hypercharge 7/6 is added to the SM, giving rise to two LQs with electric charges 5/3e and 2/3e. The former one is a non-chiral LQ that couples to up quarks and charged leptons, thereby giving rise to FCNC top quark and Higgs boson decays at the one-loop level, but also to the $t\rightarrow c{{\bar{\ell }}}\ell$ decay at the tree-level. The phenomenology of this model was studied in [17] and bounds on its couplings to a lepton-quark pairs from the experimental constraints on the muon anomalous magnetic dipole moment and the LFV tau decay $\tau \rightarrow \mu \gamma$ were obtained in [38]. We first start by discussing the corresponding LQ couplings to quarks and leptons and afterwards we discuss the remaining interactions.

In the model we are interested in, a scalar LQ representation $R_2$ with $SU(3)\times SU(2)\times U(1)$ quantum numbers (3, 2, 7/6) is introduced. This LQ doublet has the following renormalizable zero-fermion-number interactions [37]

$$\begin{aligned} \mathcal{L}_{F=0}= h^{ij}_{2L}R_2^T \bar{u}^i_Ri\tau _2 L^j_L+h^{ij}_{2R}\bar{Q}^i_Le^j_R R_2+{\mathrm{H.c.}}, \end{aligned}$$

(5)

where ${L_L^i}$ and ${Q_L^i}$ are $SU(2)_L$ left-handed lepton and quark doublets, whereas $e^i_R$ and $q^i_R$ are singlets, with i and j being generation indices.

After rotating to the LQ mass eigenstates $\varOmega _{5/3}$ and $\varOmega _{2/3}$, where the subscript denotes the electric charge in units of e, we obtain the following interaction Lagrangian

$$\begin{aligned} \mathcal{L}_{F=0}&=\bar{e}^i\left( \lambda _{L}^{ij}P_L+ \lambda _{R}^{ij}P_R\right) u^j \varOmega _{5/3}^*\nonumber \\&\quad +\,\bar{e}^i \eta _R^{ij} P_R d^j\varOmega _{2/3}^*+{\mathrm{H.c.}}, \end{aligned}$$

(6)

where $P_{L,R}$ are the chiral projection operators. We are interested in the effects of the non-chiral LQ $\varOmega _{5/3}$ on the FCNC decays of the top quark and the Higgs boson. Since there are stringent constraints on the LQ couplings to the fermions of the two first families, in our study below we will consider that $\varOmega _{5/3}$ only couples to the second and third generation fermions.

Apart from the LQ interaction to up quarks and charged lepton pairs, which follow easily from the above expression, for our calculation we also need the LQ couplings to both the photon and the Z gauge boson, which are extracted from the LQ kinetic terms:

$$\begin{aligned} \mathcal{L}_S=\frac{1}{2}(D^\mu R_2)^\dagger D^\mu R_2, \end{aligned}$$

(7)

where the $SU(2)_L\times U(1)_Y$ covariant derivative is given by

$$\begin{aligned} D_\mu R_2= \left( \partial _\mu +ig\frac{\tau ^i}{2} W^i+i g'\frac{7}{6}B^\mu \right) R_2. \end{aligned}$$

(8)

Therefore, in the mass eigenstate basis we have

$$\begin{aligned}&\mathcal{L}_S\supset i\frac{5e}{3} \varOmega _{5/3}\overleftrightarrow {\partial _\mu } \varOmega _{5/3}^* A^\mu - \frac{ig}{c_W}g_{Z\varOmega _{5/3}\varOmega _{5/3}} {\varOmega _{5/3}} \overleftrightarrow {\partial _\mu } \varOmega _{5/3}^* Z^\mu \nonumber \\&\quad +{\mathrm{H.c.}}, \end{aligned}$$

(9)

where $g_{Z\varOmega _{5/3}\varOmega _{5/3}}=1/2-5/3s_W^2$.

Finally, we consider the following renormalizable effective LQ interactions to the SM Higgs doublet $\varPhi$

$$\begin{aligned} {\mathscr {L}} = \left( M_{R_2}^2+\lambda _{R_2} \varPhi ^\dagger \varPhi \right) \left( R_2^\dagger R_2 \right) , \end{aligned}$$

(10)

where $M_{R_2}$ is the LQ mass. From here we obtain the Higgs boson coupling to $\varOmega _{5/3}$:

$$\begin{aligned} {{\mathscr {L}}}\supset \lambda _{\varOmega _{5/3}} v H \varOmega _{5/3}^* \varOmega _{5/3}. \end{aligned}$$

(11)

For easy reference, we also present the SM Feynman rules for the interaction of the photon and the Z gauge boson with a fermion-antifermion pair:

$$\begin{aligned} \bar{f}f A_\mu&: -ie Q_f\gamma _\mu , \end{aligned}$$

(12)

$$\begin{aligned} \bar{f}f Z_\mu&:-i\frac{g}{2c_W}\gamma _\mu (g^f_L P_L+g^f_R P_R), \end{aligned}$$

(13)

where $g_L^f=2T^3_f-2Q_f s_W^2$ and $g_R^f=-2Q_fs_W^2$, with $T^3_f=1/2(-1/2)$ for up (down) fermions and $Q_f$ the fermion charge in units of that of the positron.

The corresponding Feynman rules follow straightforwardly from the above Lagrangians and are shown in Fig.  1. Below we present the calculation of the FCNC $t\rightarrow cX$ and $t\rightarrow c{{\bar{\ell }}}\ell$ decays.

Fig. 1

Feynman rules necessary for the calculation of the contribution of the scalar LQ $\varOmega _{5/3}$ to the $t\rightarrow c X$ ($X=\gamma ,\,Z,\,H$) decays. For completeness we also include the Feynman rules for the LQ singlet $\chi _{1/3}$ of the model discussed in [21] as our results are also valid for its contribution, thus S stands for $\varOmega _{5/3}$ and $\chi _{1/3}$, with $g_{ZSS}=\frac{1}{2}-\frac{10}{3}s_W^2\;(-\frac{2}{3}s_W^2)$ for $S=\varOmega _{5/3}$ ($\chi _{1/3}$)

3 LQ contribution to the FCNC $t\rightarrow cX$ decays

We now discuss the calculation of the FCNC $t\rightarrow cX$ decays, which in our scalar LQ model proceed at the one-loop level at the lowest order of perturbation theory. For the sake of completeness we present the most general expressions for the $f_i\rightarrow f_j X$ decays with $X=\gamma ,\,Z,\,H$ and $f_{i,j}$ quarks or leptons. From our result for the $q_i\rightarrow q_j\gamma$ decay, that for the $q_i\rightarrow q_j g$ decay will follow easily as discussed below.

For the calculation of the loop integrals, we use both the Feynman parameter technique and the Passarino–Veltman reduction scheme, which allows one to cross-check the results. For the algebra we used the Mathematica software routines along with the FeynCalc package [39]. It is worth mentioning that our results are also valid for the contribution of the LQ singlet $\chi _{1/3}$ of the model of Ref. [21], where the LQ contribution to the $H\rightarrow \mu \tau$ was discussed. The Feynman rules for such a LQ are of Majorana-like type (there are two fermion-flow arrows clashing into a vertex as shown in Fig. 1) and they require a special treatment. We have followed the approach of Ref. [40] and found that the results for the contribution of LQ $\varOmega _{5/3}$ to the $f_i\rightarrow f_j X$ decays are also valid for the contributions of LQ $\chi _{1/3}$ after replacing the respective coupling constants. A similar result was found for the contribution of single and doubly charged scalars to the muon anomalous MDM [40].

3.1 $f_i\rightarrow f_j V$ ($V=\gamma ,\,Z$) decays

This decay proceeds at the lowest order via the Feynman diagrams of Fig. 2, where the internal fermion is a lepton (quark) provided that the external fermions are quarks (leptons).

Fig. 2

LQ contribution to the decay $f_i\rightarrow f_j V$ ($V=\gamma ,Z$) , where $f_i$ and $f_j$ are charged leptons (quarks) and $f_k$ is a quark (lepton). Here $Q_{i-k}$ is the LQ charge in units of e. Analogue diagrams give rise to the $f_i\rightarrow f_j H$ decay with the V gauge boson replaced by the Higgs boson. As far as the $q_i\rightarrow q_j g$ decay is concerned, there is only contribution from diagrams b) to d) as the internal fermion is a lepton

The ultraviolet divergences cancel out when summing over all the partial amplitudes. The most general invariant amplitude can be written as

$$\begin{aligned} \mathcal{M}(f_i\rightarrow f_j V)= & {} \bar{f}_j\Big (\frac{iL^V}{m_i}\,\sigma _{\mu \nu }P_L q^\nu +\frac{iR^V}{m_i}\,\sigma _{\mu \nu }P_R q^\nu \nonumber \\&+L^{'V}\,\gamma ^\mu P_L +R^{'V}\,\gamma ^\mu P_R\Big )f_i\epsilon (q)^\mu ,\nonumber \\ \end{aligned}$$

(14)

where the monopole terms $L^{'\gamma }$ and $R^{'\gamma }$ vanish for the $f_i \rightarrow f_j \gamma$ decay due to gauge invariance: the bubble diagrams only give contributions to the monopole terms, which are canceled out by those arising from the triangle diagrams. The corresponding $L^V$, $R^V$, $L^{'Z}$, and $R^{'Z}$ form factors are presented in terms of Passarino–Veltman scalar functions in Appendix A.

After averaging (summing) over polarizations of the initial (final) fermion and gauge boson, we use the respective two-body decay width formula, which reduces to

$$\begin{aligned}&\varGamma (f_i\rightarrow f_j V)=\frac{\lambda ^{1/2}(m_i^2,m_V^2,m_j^2) }{32 \pi m_i^3} \Bigg (f_{ij}\left( |L^{V}|^2+|R^{V}|^2\right) \nonumber \\&\quad +\, g_{ij}\left( |L^{'V}|^2+|R^{'V}|^2\right) \nonumber \\&\quad +\, 3 \left( m_j^2-m_i^2+m_V^2\right) \left( L^{'V}R^{V*}+L^{V}R^{'V*}\right) \nonumber \\&\quad +\, \frac{3 m_j}{m_i} \left( m_i^2-m_j^2+m_V^2\right) \left( L^{V}L^{'V*}+R^{V}R^{'V*}\right) \nonumber \\&\quad -\, \frac{3 m_j m_V^2}{m_i}{\mathrm{Re}}\left( L^V R^{V*}\right) \nonumber \\&\quad -\, 12 m_i m_j {\mathrm{Re}}\left( L^{'V} R^{'V*}\right) \Bigg ), \end{aligned}$$

(15)

with $f_{ij}=\frac{1}{m_i^2}(2(m_i^2-m_j^2)^2 - (m_i^2+m_j^2)m_V^2-m_V^4)$ and $g_{ij}=\frac{1}{m_V^2}((m_i^2-m_j^2)^2+(m_i^2+m_j^2))m_V^2-2 m_V^4)$. The so-called triangle function is given by

$$\begin{aligned} \lambda (x,y,z)=x^2+y^2+z^2-2(xy+xz+yz). \end{aligned}$$

(16)

For the $f_i\rightarrow f_j\gamma$ decay, Eq. (15) reduces to

$$\begin{aligned} \varGamma (f_i\rightarrow f_j \gamma )&=\frac{ m_i}{16 \pi } \left( 1-\left( \frac{m_j}{m_i}\right) ^2\right) ^3\left( |L^\gamma |^2+|R^\gamma |^2\right) . \end{aligned}$$

(17)

3.2 $q_i\rightarrow q_j g$ decay

This one-loop FCNC process is induced by Feynman diagrams similar to those shown in Fig. 2, except that there is no contribution from Feynman diagram of type a) as the internal fermion is a lepton. The Feynman rules necessary for the calculation are presented in Fig. 3.

Fig. 3

Feynman rules that are required for the calculation of the contribution of a scalar LQ to the $t\rightarrow c g$ decay. Here $T^a$ are the $SU(3)_c$ generators in the fundamental representation

The $q_i\rightarrow q_j g$ amplitude can be written as

$$\begin{aligned} \mathcal{M}(q_i\rightarrow q_j g)&=\bar{q}_jT^a\Big (\frac{iL^g}{m_i}\,\sigma _{\mu \nu }P_L q^\nu +\frac{iR^g}{m_i}\,\sigma _{\mu \nu }P_R q^\nu \Big )q_i\nonumber \\&\quad \times \, \epsilon (q)_\mu , \end{aligned}$$

(18)

where the $L^g$ and $R^g$ coefficients can be obtained from Eqs. (A.11) and (A.30) of Appendix A once the replacements $Q_k\rightarrow 0$, $Q_S\rightarrow 1$, and $N_c e\rightarrow g_s$ are done. After averaging (summing) over initial (final) polarizations and colors, we obtain the average square amplitude and thereby the corresponding decay width, which has the same form of Eq. (17), though we must multiply the right-hand side by the color factor $C_F=4/3$.

3.3 $f_i \rightarrow f_j H$ decay

We now present the invariant amplitude for the LQ contribution to the $f_i \rightarrow f_j H$ decay, which is induced at the one-loop level by Feynman diagrams analogue to those shown in Fig. 2, but with the gauge boson V replaced by the Higgs boson H. We have found that while the amplitude of Feynman diagram (d) is ultraviolet finite, that of Feyman diagram (a) has ultraviolet divergences, but they are canceled out by those arising from the bubble diagrams (b) and (c). After some algebra, the invariant amplitude can be cast in the form

$$\begin{aligned} \mathcal{M}(f_i \rightarrow f_j H)=\bar{f}_j\left( F_L P_L+F_R P_R\right) f_i, \end{aligned}$$

(19)

where the $F_L$ and $F_R$ form factors are presented in Appendix B in terms of Passarino–Veltman scalar functions and Feynman parameter integrals.

After summing (averaging) over the polarizations of the final (initial) fermion, we plug the average squared amplitude into the two-body decay width formula to obtain

$$\begin{aligned} \varGamma (f_i \rightarrow f_j H)&= \frac{\lambda ^{1/2}(m_i^2,m_H^2,m_j^2)}{8 m_i^3 \pi } \Big (\left( |F_L|^2+|F_R|^2\right) p_i \cdot p_j \nonumber \\&\quad +2 m_i m_j{\mathrm{Re}}\left( F_L F_R^*\right) \Big ), \end{aligned}$$

(20)

with $p_i \cdot p_j=(m_i^2+m_j^2-m_H^2)/2$.

3.4 $H\rightarrow \bar{f}_j f_i$ decay

As a by-product we present the $H\rightarrow \bar{f}_j f_i$ decay width, which follows straightforwardly from the above results by crossing symmetry. Although the scalar LQ contribution to the LFV decay $H\rightarrow \tau \mu$ has been already presented in the zero lepton mass approximation [21, 55,56,57], we now present the exact one-loop calculation for the $H\rightarrow \bar{f}_j f_i$ decay width. It reads

$$\begin{aligned} \varGamma (H\rightarrow f_i f_j)&= \frac{\lambda ^{1/2}(m_H^2,m_i^2,m_j^2)}{16 \pi m_H^3} \Big (\left( |F_L|^2+|F_R|^2\right) p_i \cdot p_j\nonumber \\&\quad -\, 2 m_i m_j {\mathrm{Re}} (F_L F_R^*)\Big ), \end{aligned}$$

(21)

where $\varGamma (H\rightarrow f_i f_j)=\varGamma (H\rightarrow \bar{f}_i f_j)+\varGamma (H\rightarrow \bar{f}_j f_i)$. Also $p_j\cdot p_j=(m_H^2-(m_i^2+m_j^2))/2$, and the $F_L$ and $F_R$ form factors are the same as those presented in Appendix B for the $f_i\rightarrow f_j H$ decay as discussed in Appendix C.

4 Three-body tree-level decay $f_i\rightarrow f_j{{\bar{f}}}_m f_l$

Fig. 4

Feynman diagram for the tree-level FCNC decay $f_i\rightarrow f_j {{\bar{f}}}_mf_l$ induced by a scalar LQ. Here $f_l$ and $f_m$ are leptons (quarks) if $f_i$ and $f_j$ are quarks (leptons)

Finally we discuss the calculation of the three-body decay $t\rightarrow c{{\bar{\ell }}}\ell$. Following our calculation approach, we consider the general decay $f_i\rightarrow f_j {{\bar{f}}}_mf_l$, where $f_l$ and $f_m$ are leptons (quarks) if $f_i$ and $f_j$ are quarks (leptons). This process is induced by a scalar LQ at the tree-level via the Feynman diagram of Fig. 4. We denote the four-momentum of fermion $f_a$ ($a=i, j,l,m$) by $p_a$. The corresponding decay width can be written as

$$\begin{aligned} \varGamma (f_i\rightarrow f_j \bar{f}_m f_l)=\frac{m_i}{256 \pi ^3}\int _{x_{ji}}^{x_{jf}}\int _{x_{li}}^{x_{lf}}|\mathscr {{\overline{M}}}|^2dx_j dx_l. \end{aligned}$$

(22)

In the center-of-mass frame of the decaying fermion, the scaled variables $x_{a}$ ($a=j,l,m$) are given as $x_{a}=2 E_{a}/m_{a}$. From energy conservation, these variables obey $x_j+x_l+x_m=2$. The kinematic limits in Eq. (22) are in turn

$$\begin{aligned} x_{jf}= & {} 2\sqrt{\mu _j}, \end{aligned}$$

(23)

$$\begin{aligned} x_{ji}= & {} 1+\mu _j-\mu _l-\mu _m-2\sqrt{\mu _l\mu _m}, \end{aligned}$$

(24)

$$\begin{aligned} x_{li,lf}= & {} \frac{1}{2(1-x_j+\mu _j)}\Bigl [(2-x_j)(1+\mu _j+\mu _l-\mu _m-x_j)\nonumber \\&\mp \sqrt{x_j^2-4\mu _j}\lambda ^{1/2}(1+\mu _j-x_i,\mu _l,\mu _m)\Bigr ], \end{aligned}$$

(25)

where $\mu _a=m_a^2/m_i^2$ $(a=j,l,m)$.

The square average amplitude can be expressed as

$$\begin{aligned} |\mathscr {{\overline{M}}}|^2= & {} \frac{2 \Bigl (p_j\cdot p_m (|\lambda ^{jm}_L|^2+|\lambda ^{jm}_R|^2)-2 m_m m_j \lambda ^{jm}_L \lambda ^{jm}_R\Bigl )}{\Bigl (m_m^2+m_j^2-m_S^2-2 p_j\cdot p_m\Bigl )^2}\nonumber \\&\times \, \Bigl ( \Bigl (m_l^2+p_l\cdot p_m+p_j\cdot p_l\Bigl )\Bigl (|\lambda ^{il}_L|^2+|\lambda ^{il}_R|^2\Bigl )\nonumber \\&+\, 2 m_i m_l \lambda ^{il}_L \lambda ^{il}_R \Bigr ), \end{aligned}$$

(26)

where the scalar products can be written as

$$\begin{aligned} p_l\cdot p_m= & {} \frac{m_i^2}{2}(1 + \mu _j - \mu _l - \mu _m - x_j), \end{aligned}$$

(27)

$$\begin{aligned} p_j\cdot p_m= & {} \frac{m_i^2}{2}(1-\mu _j+\mu _l-\mu _m-x_l), \end{aligned}$$

(28)

$$\begin{aligned} p_j\cdot p_l= & {} \frac{m_i^2}{2}(1-\mu _j+\mu _m-\mu _l-x_m). \end{aligned}$$

(29)

The integration of Eq. (22) can be performed numerically.

5 Constraints on the parameter space of the scalar LQ models

We now consider the LQ model introduced above and present an analysis of the constraints on the LQ couplings to SM fermions and the Higgs boson. While the LQ couplings to fermions can be obtained from the muon anomalous MDM and LFV tau decays, the LQ coupling to a Higgs boson pair can be extracted from the constraint on the $H\gamma \gamma$ and Hgg couplings obtained by the ATLAS and CMS collaborations [41].

5.1 Constraints on scalar LQ masses

The phenomenology of the scalar LQ doublet $R_2$ has been long studied in the literature [17, 18, 38, 42,43,44], and constraints on their mass and couplings have been derived from the $Z\rightarrow b\bar{b}$ decay, the muon anomalous MDM, and LFV decays. Since low energy physics strongly constrains the LQ couplings to the first-generation fermions, it is usually assumed that the only non-negligible couplings are those to the fermions of the second and third generations. The most stringent current constraint on the mass of the scalar LQ doublet $R_2$ masses is $m_{\varOmega _{2/3}}\gtrapprox 1$ TeV, which was obtained by the ATLAS [45] and CMS [46] collaborations from the LHC data at $\sqrt{s}=13$ TeV under the assumption that $\varOmega _{2/3}$ is a third-generation LQ that decays mainly as $\varOmega _{2/3}\rightarrow {\bar{\tau }}b$, though such a bound relaxes up to 800 GeV when it is assumed that $\varOmega _{2/3}$ decays into both the ${\bar{\tau }}b$ and $t\nu _\tau$ channels. Also, the LQ search via pair production [47] gives a very stringent upper bound of 1500 GeV on the mass of second-generation LQs, which we do not consider here as we are interested in a LQ that couples to both second and third-generation fermions. We will then assume the less stringent bound $m_{\varOmega _{5/3}}\ge 800$ GeV in our analysis below since $m_{\varOmega _{2/3}}$ and $m_{\varOmega _{5/3}}$ are mass degenerate, cf. Eq. (10). In fact, a non-degenerate scalar LQ doublet could give dangerous contributions to the oblique parameters [48].

5.2 Constraints from the LHC data on the Higgs boson

LHC data indicate that the 125 GeV Higgs boson couplings are compatible with those predicted by the SM, which provides a useful approach to constrain the parameter space of SM extension models by means of the so-called Higgs boson coupling modifiers, which are defined as

$$\begin{aligned} \kappa _i^2=\frac{\varGamma ^{}(H\rightarrow i)}{\varGamma ^{\mathrm{SM}}(H\rightarrow i)}, \end{aligned}$$

(30)

where $\varGamma ^{\mathrm{SM}}(H\rightarrow i)$ is the SM Higgs boson decay width and $\varGamma (H\rightarrow i)$ is the one including new physics effects. Bounds on the Higgs boson coupling modifiers were obtained by fitting the combined data of the ATLAS and CMS collaborations [41]. Since LQs contribute at the one-loop level to the $H\rightarrow \gamma \gamma$ and $H\rightarrow gg$ decays, to constrain the LQ couplings to a Higgs boson pair $H\varOmega _{5/3}\varOmega _{5/3}$, we use $\kappa _\gamma$ and $\kappa _g$, which are given as [19]

$$\begin{aligned} \kappa _\gamma \simeq \dfrac{\left| F_1\left( \tau _W\right) +\frac{4}{3}F_{1/2}\left( \tau _t\right) +\sum \limits _i \dfrac{3Q_{S_i}^2\lambda _{S_i}\upsilon ^2}{2m_{S_i}^2}F_0(\tau _{S_i}) \right| }{\left| F_1\left( \tau _W\right) +\frac{4}{3}F_{1/2}\left( \tau _t\right) \right| },\nonumber \\ \end{aligned}$$

(31)

and

$$\begin{aligned} \kappa _g\simeq \dfrac{\left| \frac{1}{2}F_{1/2}\left( \tau _t\right) + \sum \limits _{i}\dfrac{\lambda _{S_i}\upsilon ^2}{4m_{S_i}^2} F_{0}\left( \tau _{S_i}\right) \right| }{\left| \frac{1}{2}F_{1/2}\left( \tau _t\right) \right| }, \end{aligned}$$

(32)

where the sum is over the LQs $S_i$, $\tau _a= 4 m_{H}^2/m_a^2$, and the $F_{s}(\tau _a)$ function is given by

$$\begin{aligned} F_{s}(\tau )=\left\{ \begin{array}{lcl} -2\tau (1+(1-\tau )f(\tau )) &{}&{}\quad s=1/2,\\ \\ 2+3\tau +3\tau (2-\tau )f(\tau )&{}&{}\quad s=1,\\ \\ \tau (1-\tau f(\tau ))&{}&{}\quad s=0, \end{array}\right. \end{aligned}$$

(33)

where

$$\begin{aligned} f(x)=\left\{ \begin{array}{cr} \left[ \arcsin \left( \frac{1}{\sqrt{x}}\right) \right] ^2&{}x\ge 1,\\ -\frac{1}{4}\left[ \log \left( \frac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right) -i\pi \right] ^2&{}x<1. \end{array} \right. \end{aligned}$$

(34)

Although in our model FCNCs top quark decays receive contribution from $\varOmega _{5/3}$ only, $\varOmega _{2/3}$ also contribute to the decays $H\rightarrow \gamma \gamma$ and $H\rightarrow gg$. As already mentioned, these LQs are mass degenerate: $m_{\varOmega _{5/3}}= m_{\varOmega _{2/3}}$. We show in the left plot of Fig.  5 the area allowed by the experimental constraints on $\kappa _\gamma$ and $\kappa _g$ in the $\lambda _{\varOmega _{5/3}}$ vs $\lambda _{\varOmega _{2/3}}$ plane for two values of $m_{\varOmega _{5/3}}$. In general, values of the order of O(10) are allowed for either $\lambda _{\varOmega _{2/3}}$ or $\lambda _{\varOmega _{5/3}}$, with the largest allowed values obtained for either large $m_{\varOmega _{5/3}}$ or $\lambda _{\varOmega _{2/3}}=-\lambda _{\varOmega _{5/3}}$. We also show the allowed area in the $m_{\varOmega _{5/3}}$ vs $\lambda _{\varOmega _{5/3}}$ plane in several $\lambda _{\varOmega _{2/3}}$ scenarios. We observe that for a particular $m_{\varOmega _{5/3}}$ value, the strongest constraints are obtained when $\lambda _{\varOmega _{2/3}}=\lambda _{\varOmega _{5/3}}$, whereas the less stringent constraints are obtained when $\lambda _{\varOmega _{2/3}}=-\lambda _{\varOmega _{5/3}}$.

Fig. 5

Allowed regions with 95 % C.L. of the parameter space of our LQ model from the experimental bounds on the Higgs boson multipliers $\kappa _\gamma$ and $\kappa _g$ for $m_{\varOmega _{2/3}}=m_{\varOmega _{5/3}}$. The left plot shows the allowed region in the $\lambda _{\varOmega _{5/3}}$ vs $\lambda _{\varOmega _{2/3}}$ plane for three values of $m_{\varOmega _{5/3}}$: 1000 GeV (solid-line boundary), and 1500 GeV (dashed-line boundary). The right plot shows the allowed area in the $\lambda _{\varOmega _{5/3}}$ vs $m_{\varOmega _{5/3}}$ plane in the following scenarios of $\lambda _{\varOmega _{2/3}}$: $\lambda _{\varOmega _{2/3}}=\lambda _{\varOmega _{5/3}}$ (solid-line boundary) and $\lambda _{\varOmega _{2/3}}=-\lambda _{\varOmega _{5/3}}$ (dot-dashed-line boundary). The vertical and horizontal lines correspond to the bounds from perturbativity $|\lambda _{\varOmega _{2/3,\,5/3}}|\le 4\pi$

In summary the $\kappa _\gamma$ and $\kappa _g$ constraints are satisfied for $\lambda _{\varOmega _{5/3}}$ of the order of O(10), with the largest values allowed for a heavy LQ. In our analysis below we will use however the conservative value $\lambda _{\varOmega _{5/3}}\simeq 1$ as a very large value would violate the perturbativity of the LQ coupling.

5.3 Constraints from the muon anomalous magnetic moment and the LFV decay $\tau \rightarrow \mu \gamma$

The experimental bounds on the muon anomalous magnetic dipole moment (MDM) $a_\mu$ and the LFV tau decays provide an useful tool to constrain LFV effects [49]. In particular, $a_\mu$ can be useful to constrain the LQ couplings $\lambda ^{\mu u_i}_{L,R}$ ($u_i=c,t$), whereas the decay $\tau \rightarrow \mu \gamma$ allow us to constrain the $\lambda ^{\tau u_i}_{L,R}$ ones.

5.3.1 Muon anomalous magnetic dipole moment

Currently there is a discrepancy between the experimental and theoretical values of the muon anomalous MDM $\varDelta a_\mu =a_\mu ^{\mathrm{Theo.}}-a_\mu ^{\mathrm{Exp.}}=268\, (63)\,(43)\times 10^{-11}$ [49]. We assume that this discrepancy is due to the LQ contribution, though such a puzzle could be settled in the future once new experimental measurements and more accurate evaluations of the hadronic contributions were available.

The contribution of scalar LQs $\varOmega _{5/3}$ to the muon anomalous magnetic dipole moment $a_{\mu }^{\mathrm{LQ}}$ arises at the one-loop level from the triangle diagrams of Fig. 2 with $f_j=f_i=\mu$ and $f_k=u_k$. It can be written as [38]

$$\begin{aligned} a_{\mu }^{\mathrm{LQ}}&=-\sum _ {u_k=c,\,t}\frac{3\,\sqrt{x_{\mu }}}{32\pi ^2} \Bigg (\sqrt{x_\mu }\left( \left| \lambda ^{\mu u_k}_L\right| ^2+ \left| \lambda ^{\mu u_k}_R\right| ^2\right) F\left( x_\mu ,x_{u_k}\right) \nonumber \\&\quad +2\,\sqrt{x_{u_k}}\,{\mathrm{Re}}\left( \lambda ^{\mu u_k}_L {\lambda ^{\mu u_k }_R}^*\right) G\left( x_\mu ,x_{u_k}\right) \Bigg ), \end{aligned}$$

(35)

where $x_a=m_a^2/m^2_{\varOmega _{5/3}}$. The F(x, y) and G(x, y) functions are presented in Appendix D in terms of Feynman parameter integrals and Passarino–Veltman scalar functions. Since $x_\mu \ll x_{u_k}$, we have the following approximate expression

$$\begin{aligned} a_{\mu }^{\mathrm{LQ}}&\simeq -\sum _ {u_k=c,\,t}\frac{3\,\sqrt{x_{\mu }}\sqrt{x_{u_k}}}{16\pi ^2}\,{\mathrm{Re}}\left( \lambda ^{\mu u_k}_L {\lambda ^{\mu u_k }_R}^*\right) G\left( x_\mu ,x_{u_k}\right) , \end{aligned}$$

(36)

Since $\varOmega _{2/3}$ is a chiral LQ, its contribution to $a_\mu$ is proportional to the muon mass and is thus subdominant. We now consider that the $\varDelta a_\mu$ discrepancy is due to the LQ contribution $a_{\mu }^{\mathrm{LQ}}$ and show in Fig. 6 the allowed area in the ${\mathrm{Re}}\left( \lambda _{L}^{\mu c}\lambda _{R}^{\mu c}\right)$ vs ${\mathrm{Re}}\left( \lambda _{L}^{\mu t}\lambda _{R}^{\mu t}\right)$ plane for three values of $m_{\varOmega _{5/3}}$. We note that a positive contribution from LQs to $a_\mu$ is required to explain the discrepancy, therefore there are three possible scenarios:

• 1. ${\mathrm{Re}}\left( \lambda _{L}^{\mu c}\lambda _{R}^{\mu c}\right)$ and ${\mathrm{Re}}\left( \lambda _{L}^{\mu t}\lambda _{R}^{\mu t}\right) <0$.

• 2. ${\mathrm{Re}}\left( \lambda _{L}^{\mu c}\lambda _{R}^{\mu c}\right) <0$ and ${\mathrm{Re}}\left( \lambda _{L}^{\mu t}\lambda _{R}^{\mu t}\right) >0$.

• 3. ${\mathrm{Re}}\left( \lambda _{L}^{\mu c}\lambda _{R}^{\mu c}\right) >0$ and ${\mathrm{Re}}\left( \lambda _{L}^{\mu t}\lambda _{R}^{\mu t}\right) <0$.

In the first scenario (left plot of Fig. 6) we observe that while ${\mathrm{Re}}\left( \lambda _{L}^{\mu c}\lambda _{R}^{\mu c}\right)$ can range between $10^{-4}$ and $10^{-3}$ for negligible ${\mathrm{Re}}\left( \lambda _{L}^{\mu c}\lambda _{R}^{\mu c}\right)$, the latter can range between $10^{-3}$ and $10^{-2}$ for negligible ${\mathrm{Re}}\left( \lambda _{L}^{\mu t}\lambda _{R}^{\mu t}\right)$, with the largest allowed values corresponding to heavy $m_{\varOmega _{5/3}}$. On the other hand, more large values of the LQ couplings are allowed when ${\mathrm{Re}}\left( \lambda _{L}^{\mu c}\lambda _{R}^{\mu c}\right)$ and ${\mathrm{Re}}\left( \lambda _{L}^{\mu t}\lambda _{R}^{\mu t}\right)$ are of opposite sign (right plot) as there is a cancellation between the contributions of the c and t quarks. In particular, there is a very narrow band where ${\mathrm{Re}}\left( \lambda _{L}^{\mu c}\lambda _{R}^{\mu c}\right) \simeq O(10)$ and ${\mathrm{Re}}\left( \lambda _{L}^{\mu t}\lambda _{R}^{\mu t}\right) \simeq O(1)$.

Fig. 6

Allowed regions with 95 % C.L. of the parameter space of our LQ model assuming that the contributions of LQ $\varOmega _{5/3}$ along with the c and t quarks are responsible for the muon anomalous MDM discrepancy $\varDelta a_\mu$. We show the allowed region in the ${\mathrm{Re}}\left( \lambda _{L}^{\mu c}\lambda _{R}^{\mu c}\right)$ vs ${\mathrm{Re}}\left( \lambda _{L}^{\mu t}\lambda _{R}^{\mu t}\right)$ plane for two values of $m_{\varOmega _{5/3}}$: 1000 GeV (solid-line boundary), 1500 GeV (dashed-line boundary), and 2000 GeV (dot-dashed-line boundary). In the left plot we assume that both ${\mathrm{Re}}\left( \lambda _{L}^{\mu c}\lambda _{R}^{\mu c}\right) <0$ and ${\mathrm{Re}}\left( \lambda _{L}^{\mu t}\lambda _{R}^{\mu t}\right) <0$, whereas in the right plot we consider that they are of opposite sign

5.3.2 Decay $\tau \rightarrow \mu \gamma$

The LQ couplings $\lambda _{L,R}^{\tau u_i}$ and $\lambda _{L,R}^{\mu u_i}$ can be constrained by the experimental bound on the LFV tau decay $\tau \rightarrow \mu \gamma$, which can receive the contributions of loops with $\varOmega _{5/3}$ accompanied by the up quarks. Such contributions follow straightforwardly from our result for the $f_i\rightarrow f_j\gamma$ decay width given in Eq. (15) after the proper replacements are made. The result is in agreement with previous calculations of the $\ell _i\rightarrow \ell _j\gamma$ decay width [38].

If the LQ couples to both the c and t quarks, the $\tau \rightarrow \mu \gamma$ decay width acquires the form

$$\begin{aligned} \varGamma (\tau \rightarrow \mu \gamma )&\sim \Big \Vert \sum _{u_i=c,t}\left( \alpha _{LL}^{\mu u_i}\lambda _L^{\mu u_i} \lambda _L^{\tau u_i} +\alpha _{RR}^{\mu u_i}\lambda _R^ {\mu u_i} \lambda _R^{\tau u_i}\right. \nonumber \\&+\left. \alpha _{LR}^{\mu u_i}\lambda _L^{\mu u_i} \lambda _R^{\tau u_i}\right) \Big \Vert ^2+\left( L\leftrightarrow R\right) , \end{aligned}$$

(37)

where $\alpha _{LL}^{\mu u_i}$, etc. stand for the loop integrals. To simplify our analysis we assume the scenario where $\lambda _R^{\ell u_i}/\lambda _L^{\ell u_i}=O(\epsilon )$ ($\ell =\mu ,\,\tau$), with $\epsilon =10^{-3}$(predominantly left-handed couplings), $10^{-1}$ (small right handed-couplings), and 1 (purely scalar couplings). We do not analyze the case when $\epsilon >1$ as a similar situation is observed as in the $\epsilon <1$ case but with $\lambda _L^{\ell u_i}$ replaced by $\lambda _R^{\ell u_i}$. Thus the parameter $\epsilon$ is a measure of the relative size between the right and left-handed LQ couplings. Under this assumption, the $\tau \rightarrow \mu \gamma$ decay width becomes a function of the products $\lambda _L^{\mu c} \lambda _L^{\tau c}$ and $\lambda _L^{\mu t}\lambda _L^{\tau t }$. We thus show in Fig. 7 the allowed area in the $\lambda _L^{\mu c} \lambda _L^{\tau c}$ vs $\lambda _L^{\mu t}\lambda _L^{\tau t }$ plane for three values of $m_{\varOmega _{5/3}}$. We observe that for $m_{\varOmega _{5/3}}=1000$ GeV the largest allowed area is obtained in the scenario with $\epsilon =10^{-3}$, which allow $\lambda _L^{\mu u_i} \lambda _L^{\tau u_i}$ values as large as $O(10^{-1})$, whereas the smallest area is obtained when $\epsilon =1$, which allows $\lambda _L^{\mu u_i} \lambda _L^{\tau u_i}$ values of the order of $O(10^{-4})$. Such bounds are slightly relaxed when $m_{\varOmega _{5/3}}$ increases up to 2 TeV.

Fig. 7

Allowed area with 95% C.L. in the $\lambda _L^{\mu c} \lambda _L^{\tau c}$ vs $\lambda _L^{\mu t}\lambda _L^{\tau t }$ plane from the experimental bound on the $\tau \rightarrow \mu \gamma$ decay for three values of $m_{\varOmega _{5/3}}$ in the scenario with $\lambda _R^{\ell u_i}/\lambda _L^{\ell u_i}=O(\epsilon )$, for $\epsilon =10^{-3}$ (solid-line boundary), $\epsilon =0.1$ (dashed-line boundary), and $\epsilon =1$ (dot-dashed-line boundary)

As far as constraints on the $| \lambda _{L,R}^{\ell u_i}|$ couplings from direct LQ searches at the LHC via the Drell–Yan process [50], single production [51], and pair production [47], an up-to-date discussion is presented in Ref. [52]. A restricted scenario (minimal LQ model) is considered where each LQ is allowed to couple to just one lepton-quark pair. In particular, a 95% C.L. limit on $| \lambda ^{\mu c}|$ of the order of O(1) is obtained for a LQ with a mass above the 1 TeV level from the Drell–Yan process [50], whereas the bounds obtained from the LHC Run 1 and Run 2 data on single production [51] yield less stringent bounds. Although such limit could be relaxed in a more general scenario where the LQ is allowed to couple to more than one fermion pair, below we assume a conservative scenario and consider the bound $| \lambda ^{\mu c}|\le O(1)$, whereas for the remaining couplings we impose the $|\lambda _{L,R}^{\mu t}|,\,|\lambda _{L,R}^{\tau u_i}|< 4\pi$ bound to avoid the breakdown of perturbativity.

We are interested in the region of the parameter space where the largest $t\rightarrow c X$ and $t\rightarrow c{{\bar{\ell }}}\ell$ branching ratios can be reached, which is the area where either $\lambda _L^{\tau t} \lambda _L^{\tau c}$ or $\lambda _L^{\mu t} \lambda _L^{\mu c}$ reaches their largest allowed values. Again we consider the scenario with $\lambda _R^{\ell u_i}/\lambda _L^{\ell u_i}=O(\epsilon )$, with four $\epsilon$ values, and perform a scan of $(\lambda _L^{\mu c},\,\lambda _L^{\mu t}, \, \lambda _L^{\tau c},\,\lambda _L^{\tau t})$ points consistent with both the $\varDelta a_\mu$ discrepancy (Fig. 6) and the constraint on the $\tau \rightarrow \mu \gamma$ decay (Fig. 7) for two values of $m_{\varOmega _{5/3}}$: we consider a large mass splitting to observe how the LQ couplings get constrained by the experimental data. As already discussed, we also impose the bound $| \lambda _{L,R}^{\mu c}|\le O(1)$ from the direct LQ search at the LHC and, to avoid perturbativity violation, we impose the extra constraint $| \lambda _{L,R}^{\mu t}|,\,| \lambda _{L,R}^{\tau u_i}|< 4\pi$. The corresponding allowed areas in the $\lambda _L^{\mu t } \lambda _L^{\mu c}$ vs $\lambda _L^{\tau t }\lambda _L^{\tau c}$ plane are shown in Fig. 8. We observe that, for $m_{\varOmega _{5/3}}=1000$ GeV, the scenario with $\epsilon =10^{-3}$ (top left plot) allows values of $\lambda _L^{\tau c}\lambda _L^{\tau t}$ as large as O(1) for $\lambda _L^{\mu t} \lambda _L^{\mu c}$ of the order of $10^{-1}$, but values of the order of O(1) are allowed for $\lambda _L^{\mu c}\lambda _L^{\mu t}$ for $\lambda _L^{\tau t} \lambda _L^{\tau c}$ of the order of $10^{-2}$. For fixed $\epsilon$, the allowed area expands slightly when the LQ mass increases, which is expected as the loop functions become suppressed for large LQ mass, thereby allowing larger couplings. On the other hand, for fixed $m_{\varOmega _{5/3}}$, the allowed areas shrink significantly in the $\lambda _L^{\mu t} \lambda _L^{\mu c}$ direction and slightly in the $\lambda _L^{\tau t} \lambda _L^{\tau c}$ direction as $\epsilon$ increases. For instance, in the scenario when $\epsilon =1$ (bottom right plot), the largest allowed $\lambda _L^{\mu t}\lambda _L^{\mu c}$ values for $m_{\varOmega _{5/3}}=1000$ GeV are of the order of $O(10^{-3})$ for small $\lambda _L^{\tau t} \lambda _L^{\tau c}$, whereas the latter can be as large as $O(10^{-1})$ for very small $\lambda _L^{\mu t}\lambda _L^{\mu c}$. We conclude that the scenario with predominantly dominant left-handed couplings ($\epsilon =10^{-3}$) is the one that allows the largest values of the LQ couplings.

Fig. 8

Allowed areas with 95% C.L. in the $\lambda _L^{\mu t}\lambda _L^{\mu c}$ vs $\lambda _L^{\tau t }\lambda _L^{\tau c}$ plane consistent with both the $\varDelta a_\mu$ discrepancy and the experimental bound on the $\tau \rightarrow \mu \gamma$ decay for $m_{\varOmega _{5/3}}=1000$ (dark points) and 5000 GeV (light points) in the scenarios where $\lambda _R^{\ell u_i}/\lambda _L^{\ell u_i}= O(\epsilon )$, for four $\epsilon$ values. For the $\lambda ^{\mu c}$ coupling we use the constraint $| \lambda ^{\mu c}|\le O(1)$ from direct searches at the LHC [52], whereas for the remaining couplings we use the additional constraint $| \lambda _{L,R}^{\ell u_i}|\le 4\pi$ to avoid perturbativity violation. Here $\ell =\mu ,\,\tau$ and $u_i=c,\,t$

6 Numerical analysis of the $t\rightarrow c X$ and $t\rightarrow c{{\bar{\ell }}}\ell$ branching ratios

We now turn to analyze the behavior of the $t\rightarrow c X$ and $t\rightarrow c{{\bar{\ell }}}\ell$ branching ratios in the allowed area of the parameter space. For the numerical evaluation of the one-loop induced decays $t\rightarrow c X$ we have made a cross-check by evaluating the Passarino–Veltman scalar functions via the LoopTools package [53, 54] and then comparing the results with those obtained by numerical integration of the parametric integrals. For the tree-level induced decay $t\rightarrow c{{\bar{\ell }}}\ell$ we have used the Mathematica numerical integration routines to solve the two-dimensional integral of Eq. (22).

6.1 $t\rightarrow c X$ branching ratios

We first consider two $\epsilon$ values and present in Table 1 a few sets of allowed $(\lambda _L^{\mu c}$, $\lambda _L^{\mu t }$, $\lambda _L^{\tau c}$, $\lambda _L^{\tau t })$ points where the $t\rightarrow cX$ decays can reach their largest branching ratios for three LQ masses. In the scenario where $\epsilon =10^{-3}$ we observe that there is a small area where all of the $t\rightarrow c X$ branching ratios can be as large as $10^{-9}-10^{-8}$ for $m_{\varOmega _{5/3}}=1000$ GeV, though they get suppressed by one order of magnitude when $m_{\varOmega _{5/3}}$ increases up to 2000 GeV. In such an area, the LQ couplings $\lambda _L^{\mu u_i}$ are rather small, whereas the $\lambda _L^{\tau t}$ one is very close to the perturbative limit, which means that this possibility would require a large amount of fine-tuning. As for the $\epsilon = 10^{-1}$ scenario, we observe that the $t\rightarrow cX$ branching ratios are much smaller than in the $\epsilon = 10^{-3}$ scenario: they can be of the order of $10^{-9}-10^{-10}$ at most for $m_{\varOmega _{5/3}}=1000$ GeV, and decrease by one order of magnitude as $m_{\varOmega _{5/3}}$ increases up to 2000 GeV. We refrain from presenting the results for the $\epsilon =1$ scenario as the $t\rightarrow cX$ branching ratios are two orders of magnitude than in the $\epsilon =10^{-3}$ scenario.

Table 1 Branching ratios of the $t\rightarrow cX$ decays for a few $(\lambda _L^{\mu c}, \lambda _L^{\mu t },\lambda _L^{\tau c}, \lambda _L^{\tau t })$ points inside the area allowed by the $\varDelta a_\mu$ discrepancy and the experimental bound on the $\tau \rightarrow \mu \gamma$ decay. We consider the scenario where $\lambda _R^{\ell u}/\lambda _L^{\ell u_i}=O(\epsilon )$ for $\epsilon =10^{-3}$ and $10^{-1}$. For the coupling of the Higgs boson to a LQ pair we use $\lambda _{\varOmega _{5/3}}=1$. The $t\rightarrow cX$ branching ratios are given in units of $10^{-8}$ for $\epsilon =10^{-3}$ and $10^{-9}$ for $\epsilon =10^{-1}$

We also observe in Table 1 that all the branching ratios ${\mathrm{Br}}(t\rightarrow c X)$ are of similar order of magnitude, with ${\mathrm{Br}}(t\rightarrow cZ)$ slightly larger. It seems surprising that ${\mathrm{Br}}(t\rightarrow c g)$ is about the same size than ${\mathrm{Br}}(t\rightarrow c \gamma )$, whereas in the SM and other of its extensions it is one or two orders of magnitude larger. To explain this result, let us examine the case of the SM, where the $t\rightarrow c \gamma$ decay proceeds via a Feynman diagram where the photon emerges off a down-type quark and so the squared amplitude for the analogue $t\rightarrow cg$ diagram has an enhancement factor of $c_F \left[ g_S/(-1/3 e)\right] ^2\simeq O(10^2)$, where $c_F=4/3$ is the color factor. On the other hand, in our LQ model the photon emerges off the charge 5 / 3e LQ, which means that the enhancement factor for the squared $t\rightarrow cg$ amplitude is just $c_F \left[ g_S/(5/3 e)\right] ^2\simeq O(1)$. Furthermore, in our LQ model the Feynman diagram where the photon emerges off the LQ gives a smaller contribution than that where it emerges off the lepton, which is absent in the $t\rightarrow cg$ decay. These two facts conspire to yield ${\mathrm{Br}}(t\rightarrow c g)\gtrsim {\mathrm{Br}}(t\rightarrow c \gamma )$. It is also worth mentioning that the $t\rightarrow c H$ decay receives its main contribution from the diagram where the Higgs boson is emitted off the LQ line, and thus its decay width is very sensitive to the magnitude of the $\lambda _{\varOmega _{5/3}}$ coupling.

Fig. 9

Contours of the branching ratios of the $t\rightarrow c X$ decays for $m_{\varOmega _{5/3}}=1000$ (top plots) and 2000 GeV (bottom plots) in the $\lambda _L^{\mu t}\lambda _L^{\mu c}$ vs $\lambda _L^{\tau t }\lambda _L^{\tau c}$ plane for $(\lambda _L^{\mu c}, \lambda _L^{\mu t },\lambda _L^{\tau c}, \lambda _L^{\tau t })$ values inside the area allowed by the $\varDelta a_\mu$ discrepancy and the experimental bound on the $\tau \rightarrow \mu \gamma$ in the scenario with $\lambda _R^{\ell u}/\lambda _L^{\ell u_i}=O(10^{-3})$

Finally we show in Fig. 9 the contours of the $t\rightarrow c X$ branching ratios in the allowed area of the $\lambda _L^{\mu t}\lambda _L^{\mu c}$ vs $\lambda _L^{\tau t }\lambda _L^{\tau c}$ plane in the scenario with $\lambda _R^{\ell u_i}/\lambda _L^{\ell u_i}=O(10^{-3})$, where the largest values of the $t\rightarrow c X$ branching ratios are reached. As already noted, when $m_{\varOmega _{5/3}}=1000$ GeV the largest $t\rightarrow c X$ branching ratios, of the order of $10^{-9}-10^{-8}$, are obtained in a tiny area where $\lambda _L^{\mu t}\lambda _L^{\mu c}$ is very small and $\lambda _L^{\tau t }\lambda _L^{\tau c}$ reaches its largest allowed values (top-left corner of the upper plots), but they decrease as the allowed area expands. It means that the largest branching ratios are obtained in the region where the main contribution arises from the loops with an internal tau lepton, which is due to the fact that the LQ couplings to the tau lepton are less constrained than those to the muon. We also observe that the $t\rightarrow c X$ branching ratios decrease by one or two orders of magnitude as $m_{\varOmega _{5/3}}$ reaches the 2 TeV level, where they can be as large as $10^{-9}$–$10^{-10}$. The behavior of the $t\rightarrow cX$ branching ratios in the scenarios with $\epsilon =10^{-1}$ and $\epsilon =1$ is rather similar to that observed in Fig. 9, but they are one or two orders of magnitude below: they can only be as large as $10^{-9}$ for $\epsilon =10^{-1}$ and $10^{-10}$ for $\epsilon =1$.

6.2 $t\rightarrow c{{\bar{\ell }}}\ell$ branching ratios

We now perform the corresponding analysis for the $t\rightarrow c{{\bar{\ell }}}\ell$ ($\ell =\mu ,\,\tau$) branching ratios in the area allowed by the experimental constraints discussed above. In Fig. 10 we show the contours of ${\mathrm{Br}}(t\rightarrow c{{\bar{\ell }}}\ell )$ in the $\lambda _L^{\ell t}$ vs $\lambda _L^{\ell c}$ plane in the scenario with $\lambda _R^{\ell u_i}/\lambda _L^{\ell u_i}=O(10^{-3})$ for two values of the LQ mass. We observe that for $m_{\varOmega _{5/3}}=1000$ GeV, ${\mathrm{Br}}(t\rightarrow c{{\bar{\tau }}}\tau )$ can be of as large as $10^{-6}$, whereas ${\mathrm{Br}}(t\rightarrow c{{\bar{\mu }}}\mu )$ is one order of magnitude below, which is due to the fact that the $\lambda _L^{\mu q}$ couplings are more constrained than the $\lambda _L^{\tau q}$ ones. When $m_{\varOmega _{5/3}}$ increases up to 2000 GeV, the $t\rightarrow c{{\bar{\ell }}}\ell$ branching ratios decrease by about one order of magnitude. As for the $t\rightarrow c{{\bar{\mu }}}\tau$ decay, its branching ratio is also suppressed as involves the $\lambda _L^{\mu q}$ couplings. In conclusion, the three-body tree-level decay $t\rightarrow c{{\bar{\ell }}}\ell$ can have larger branching ratios than the two-body one-loop decays $t\rightarrow cX$.

Fig. 10

Contours of the branching ratios of the $t\rightarrow c {{\bar{\tau }}}\tau$ and $t\rightarrow c {{\bar{\mu }}}\mu$ decays for $m_{\varOmega _{5/3}}=1000$ GeV (left plots) and 2000 GeV (right plots) in the $\lambda _L^{\ell t}$ vs $\lambda _L^{\ell c}$ plane for $(\lambda _L^{\mu c}, \lambda _L^{\mu t },\lambda _L^{\tau c}, \lambda _L^{\tau t })$ values inside the area allowed by the $\varDelta a_\mu$ discrepancy and the experimental bound on the $\tau \rightarrow \mu \gamma$ in the scenario with $\lambda _R^{\ell u}/\lambda _L^{\ell u_i}=O(10^{-3})$

7 Summary and outlook

The FCNC decays of the top quark $t\rightarrow cX$ ($X=\gamma ,\,g,\,Z,\,H$) and $t\rightarrow c{{\bar{\ell }}}\ell$ ($\ell =\mu ,\,\tau$) were calculated in a simple LQ model with no proton decay, where the SM is augmented by a SU(2) scalar LQ doublet with hypercharge $Y=7/6$. In such a model there is a non-chiral LQ with electric charge $Q=5/3e$ that couples to charged leptons and up quarks and contribute to the FCNC decays of the top quark.

As far as the analytical results are concerned, we perform a general calculation of the FCNC fermion decays $f_i\rightarrow f_j X$ and $f_i\rightarrow f_j{\bar{f}}_m f_l$. The loop amplitudes of the $f_i\rightarrow f_j X$ decays are presented in terms of both Passarino–Veltman scalar functions and Feynman parameter integrals, which can be useful to calculate the contributions of other scalar LQs. On the other hand, an analytical expression is presented for the $f_i\rightarrow f_j{\bar{f}}_m f_l$ decay width, which can be numerically evaluated.

As for the numerical analysis, to obtain bounds on the parameter space of the model we assumed that the LQ only couples to the fermions of the last two families and used the experimental constraints on the LHC Higgs boson data, the muon anomalous magnetic dipole moment $a_\mu$, the LFV decay of the tau lepton $\tau \rightarrow \mu \gamma$, as well as the direct LQ searches at the LHC via the Drell–Yan process, single production, and double production. For the LQ couplings to charged leptons and up quarks $\lambda _{L,R}^{\ell u_i}$, a scenario was considered where $\lambda _R^{\ell u}/\lambda _L^{\ell u_i}=O(\epsilon )$, with $\epsilon$ being a measure of the relative size between the right- and left-handed LQ couplings. Afterwards, the $t\rightarrow cX$ and $t\rightarrow c{{\bar{\ell }}}\ell$ branching ratios were evaluated in the allowed region of the parameter space. In particular, we find that in the scenario where the LQ couplings are predominantly left-handed, $\epsilon =O(10^{-3})$, there is a tiny region of the parameter space where the branching ratios of the one-loop induced $t\rightarrow cX$ decay can be as large as $10^{-8}$ for $m_{\varOmega _{5/3}}=1000$ GeV, with the main contribution arising from the loops with an internal tau lepton, although a large amount of fine-tuning between the LQ couplings would be required. However, for $\epsilon =10^{-1}$ ($\epsilon =1$), the main part of the allowed region yields $t\rightarrow cX$ branching ratios of the order of $10^{-9}$ ($10^{-10}$) at most. For $m_{\varOmega _{5/3}}\ge 2000$ GeV, the largest $t\rightarrow cX$ branching ratios are of the order of $10^{-10}$ in all the scenarios analyzed in this work. Although the $t\rightarrow cX$ branching ratios are larger in our LQ model than in the SM, such contributions would be out of the reach of detection in the near future. As for the tree-level induced decays $t\rightarrow c{{\bar{\ell }}}\ell$, the $t\rightarrow c{{\bar{\tau }}}\tau$ branching ratio can be as large as $10^{-6}$ for $m_{\varOmega _{5/3}}=1000$ GeV in the scenario with $\epsilon =O(10^{-3})$, but ${\mathrm{Br}}(t\rightarrow c{{\bar{\mu }}}\mu )$ is one order of magnitude below. These branching ratios decrease by about one order of magnitude when the LQ mass increases up to 2000 GeV.

It is worth noting that experimental constraints on the LQ mass and couplings obtained from the direct search at the LHC are very stringent, but they rely on several assumptions and may be relaxed, which would yield a slight enhancement of the LQ contribution to the top quark FCNC top quark decays. The magnitude of the $t\rightarrow cX$ branching ratios is similar to that recently found for the contributions from a scalar LQ with charge $-1/3e$, which arises in a model with a scalar LQ singlet [55]. We do not consider this scenario in our analysis as we are interested in LQ models where no further symmetries must be invoked to forbid the proton decay [17].

Appendices

Appendix A: Loop integrals for the $f_i \rightarrow f_j V$ decay

We now present the contribution of the scalar LQ S to the $f_i \rightarrow f_j V$ loop amplitudes. Although in these Appendices S will stand for the $\varOmega _{5/3}$ LQ, as already explained, our results are also valid for the contribution of any other scalar LQ.

1.1 Appendix A.1: Passarino–Veltman results

We first define the following sets of ultraviolet finite Passarino–Veltman scalar function combinations

$$\begin{aligned}&\varDelta _0=B_0(m_Z^2,m_k^2,m_k^2)-B_0(m_j^2,m_k^2,m_S^2), \end{aligned}$$

(A.1)

$$\begin{aligned}&\varDelta _1=B_0(m_Z^2,m_k^2,m_k^2)- B_0(0,m_k^2,m_S^2), \end{aligned}$$

(A.2)

$$\begin{aligned}&\varDelta _2=B_0(m_i^2,m_k^2,m_S^2) -B_0(m_j^2,m_k^2,m_S^2), \end{aligned}$$

(A.3)

$$\begin{aligned}&\varDelta _3= B_0(m_j^2,m_k^2,m_S^2) -B_0(0,m_k^2,m_S^2), \end{aligned}$$

(A.4)

$$\begin{aligned}&\varDelta _4=B_0(0,m_k^2,m_k^2) -B_0(0,m_S^2,m_S^2), \end{aligned}$$

(A.5)

$$\begin{aligned}&\varDelta _5= B_0(0,m_S^2,m_S^2) -B_0(0,m_k^2,m_S^2), \end{aligned}$$

(A.6)

$$\begin{aligned}&C_{kSk}=m_S^2C_0(m_i^2, m_j^2, 0, m_k^2, m_S^2,m_k^2), \end{aligned}$$

(A.7)

$$\begin{aligned}&C_{SkS}=m_S^2C_0(m_i^2, m_j^2, 0, m_S^2, m_k^2,m_S^2), \end{aligned}$$

(A.8)

$$\begin{aligned}&C_{ZkSk}=m_S^2C_0(m_i^2, m_j^2, m_Z^2, m_k^2, m_S^2,m_k^2), \end{aligned}$$

(A.9)

$$\begin{aligned}&C_{ZSkS}=m_S^2C_0(m_i^2, m_j^2, m_Z^2, m_S^2, m_k^2,m_S^2). \end{aligned}$$

(A.10)

The form factors of Eq. (14) are given in terms of these ultraviolet-finite functions as follows.

1.1.1 Appendix A.1.1: $f_i \rightarrow f_j \gamma$ decay

There are only dipole form factors as the monopole ones must vanish due to electromagnetic gauge invariance. Although each Feynman diagram has ultraviolet divergences, they cancel out when summing over all the contributions. The results read

$$\begin{aligned} L^\gamma&=\frac{N_c e}{16\pi ^2}\frac{\sqrt{x_i}\xi _{ij} }{8}\Bigg (\frac{\lambda _L^{{ik}} \lambda _L^{{kj}}}{\sqrt{x_i}} \Big (x_i(Q_S-Q_k)\nonumber \\&\quad +\, \frac{1}{\xi _{ij}} \left( Q_S \left( x_i \left( x_j-2 \xi _{k}\right) +x_j \xi _{k}\right) \right. \nonumber \\&\quad \left. -\quad Q_k \left( x_i \left( x_j+2\xi _{k}\right) -x_j\xi _{k}\right) \right) \varDelta _2\nonumber \\&\quad - \, \xi _{k} \left( Q_k+Q_S\right) \varDelta _3-2 x_i \left( Q_k x_k C_{{kSk}}- Q_S C_{{SkS}}\right) \Big )\nonumber \\&\quad +\, \frac{\lambda _R^{{ik}} \lambda _R^{{kj}}}{\sqrt{x_j}}\Big ( \frac{ x_j }{\xi _{ij}}\left( Q_k \left( \xi _{k}+x_i\right) +Q_S \left( \xi _{k}-x_i\right) \right) \varDelta _2\nonumber \\&\quad + \, 2 \xi _{k} \left( Q_k+Q_S\right) \varDelta _3 +2 x_j \left( Q_k x_k C_{{kSk}}\right. \nonumber \\ {}&\quad \left. - \quad Q_S C_{{SkS}}\right) +x_j(Q_k-Q_S)\Big )\nonumber \\&\quad -\, \frac{\sqrt{x_k} \lambda _L^{{kj}} \lambda _R^{{ik}}}{\xi _{ij}^3 }\Big (\xi _{{ij}} \left( Q_k+Q_S\right) \varDelta _2\nonumber \\&\quad +\, Q_k \left( x_j^2-x_i \left( \eta _{{jk}}+\xi _{{jk}}\right) +x_i^2\right) C_{{kSk}}\Big )\Bigg ), \end{aligned}$$

(A.11)

where $N_c$ is the color number of the internal fermion and we introduced the following definitions $x_a=m_a^2/m_S^2$, $\xi _{ab}=x_a-x_b$, $\eta _{ab}=x_a+x_b$, $\xi _a=x_a-1$, and $\eta _a=x_a+1$. In addition, the right handed form factors can be obtained from the left-handed ones as follows

$$\begin{aligned} R^\gamma =L^\gamma \left( \lambda ^{lm}_L\leftrightarrow \lambda ^{lm}_R, Q_S\rightarrow -Q_S\right) . \end{aligned}$$

(A.12)

1.1.2 Appendix A.1.1.2: $f_i \rightarrow f_j Z$ decay

The amplitude for this decay contains both dipole and monopole form factors. Again the ultraviolet divergences cancel when summing over partial contributions. The $L^Z$ and $L^{'Z}$ form factors are too lengthy and can be written as a sum of partial terms arising from each contributing diagram as follows

$$\begin{aligned} L^Z=\frac{N_cg}{32\pi ^2c_W}\frac{\sqrt{x_i}}{8}\sum _{j={\mathrm{a,b}}}\sum _{i=1}^3L^{\mathrm{(j)}}_i, \end{aligned}$$

(A.13)

and

$$\begin{aligned} L^{'Z}=\frac{N_cg }{32\pi ^2 c_W}\frac{\sqrt{x_i}}{8}\sum _{j= {\mathrm{a,b,cd}}}\sum _{i=1}^3 L^{{'\mathrm (j)}}_{i}, \end{aligned}$$

(A.14)

where the superscript stands for the Feynman diagram of Fig. 2 out of which the corresponding term arises, with (cd) standing for the sum of the contributions of diagrams (c) and (d).

The contributions of diagram (a) are given by

$$\begin{aligned} L_1^{(\mathrm a)}&=\frac{\sqrt{x_j} g_R^k \lambda _R^{{ik}} \lambda _R^{{kj}}}{2\delta ^2 }\Bigg (\Big (x_Z^3 +x_Z^2 \left( 4 x_i-2 x_j-6 {\xi _{k}}\right) \nonumber \\&\quad -\, \xi _{{ij}} x_Z \left( x_j+6 {\xi _{k}}+5 x_i\right) \Big )\varDelta _1\nonumber \\&\quad +\, \Big (x_Z^2\left( \xi _{k}-5 x_i\right) +2 x_Z \left( 2x_i\eta _{ij} +5 x_i \xi _{k}-x_j \xi _{k}\right) \nonumber \\&\quad +\, \xi _{{ij}}^2 \left( \eta _{{ik}}-1\right) \Big )\varDelta _2\nonumber \\&\quad +\, \frac{1}{x_j}\Big (x_Z^3\left( 1-\eta _{{jk}}\right) \nonumber \\&\quad +\, x_Z^2 \left( x_i \left( 3 {\xi _{k}}-4x_j\right) +x_j \left( 2 x_j+7 {\xi _{k}}\right) \right) \nonumber \\&\quad +\, \xi _{{ij}} x_Z \left( x_i \left( 5 x_j-3 {\xi _{k}}\right) \right. \nonumber \\&\quad \left. +\, x_j \left( x_j+5 {\xi _{k}}\right) \right) +\xi _{{ij}}^3 {\xi _{k}} \Big )\varDelta _3\nonumber \\&\quad +\, 2\Big (x_Z^3\left( \xi _{{ik}}+2\right) +x_Z^2 \left( x_i \left( x_j-3 x_k+2\right) \right. \nonumber \\&\quad \left. +\, x_j \left( x_k-4\right) +3 {\xi _{k}}^2-2 x_i^2\right) \nonumber \\&\quad +\, \xi _{{ij}} x_Z \left( x_i \left( 2 x_j+3 x_k-4\right) \right. \nonumber \\&\quad \left. -\, x_j \left( x_k+2\right) +3 {\xi _{k}}^2+x_i^2\right) +\xi _{{ij}}^3 x_k \Big )C_{ZkSk} \nonumber \\&\quad +\, \delta \left( \xi _{{ij}}+x_Z\right) \Bigg ), \end{aligned}$$

(A.15)

$$\begin{aligned} L_2^{(\mathrm a)}&=\frac{g_L^k \lambda _L^{{ik}} \lambda _L^{{kj}}}{2\sqrt{x_i}\delta ^2 }\Bigg (x_i\Big (x_Z^3 \nonumber \\&\quad -\, 2 x_Z^2 \left( 3\xi _{k}-2 x_j+x_i\right) +\xi _{{ij}} x_Z \left( 5 x_j+6 \xi _{k}+x_i\right) \Big )\varDelta _1\nonumber \\&\quad +\, \Big (x_Z^3\left( 1-\eta _{{ik}}\right) \ +\left( x_j+2 x_i\right) \left( 3 \xi _{k}+x_i\right) x_Z^2 \nonumber \\&\quad -\, x_Z \left( x_i^2 \left( 8 x_j+3 \xi _{k}\right) \right. \nonumber \\&\quad \left. -\, x_i x_j \left( x_j-2 \xi _{k}\right) +3 x_j^2 \xi _{k}+x_i^3\right) \nonumber \\&\quad -\, \xi _{{ij}}^2 \left( x_i \left( x_j+2 \xi _{k}\right) -x_j\xi _{k}\right) \Big )\varDelta _2\nonumber \\&\quad +\, \Big ( x_Z^3\left( 1-\eta _{{ik}}\right) \nonumber \\&\quad +\, x_Z^2 \left( x_i \left( 7 \xi _{k}-4 x_j\right) +3 x_j \xi _{k}+2 x_i^2\right) \nonumber \\&\quad -\, \xi _{{ij}} x_Z \left( x_i \left( 5 \eta _{{jk}}+x_i-5\right) -3 x_j \xi _{k}\right) -\xi _{{ij}}^3 \xi _{k} \Big )\varDelta _3\nonumber \\&\quad +\, 2x_i\Big (x_Z^3\left( \xi _{{jk}}+2\right) +x_Z^2 \left( x_j \left( 2-3 x_k\right) \right. \nonumber \\&\quad \left. -\, 2 x_j^2+x_i \left( \eta _{{jk}}-4\right) +3 \xi _{k}^2\right) \nonumber \\&\quad -\, \xi _{{ij}} x_Z \left( \left( 3 x_j-x_i-6\right) x_k+\xi _{j} \left( x_j+2 x_i-3\right) +3 x_k^2\right) \nonumber \\&\quad -\, \xi _{{ij}}^3 x_k \Big )C_{ZkSk}+\delta x_i\left( x_Z-\xi _{{ij}}\right) \Bigg ), \end{aligned}$$

(A.16)

$$\begin{aligned} L_{3}^{\mathrm{(a)}}&=\frac{\sqrt{x_k} \lambda _L^{{kj}} \lambda _R^{{ik}}}{\delta }\Bigg (g_L^k\bigg ( \Big (x_Z-\xi _{{ij}}\Big )\varDelta _0\nonumber \\&\quad +\, \Big (\eta _{{ij}}-x_Z\Big )\varDelta _2+\Big (\left( \xi _{{jk}}+1\right) x_Z+\xi _{{ij}} \left( \eta _{{jk}}-1\right) \Big )C_{{kSk}}\bigg ) \nonumber \\&\quad +\, g_R^k\Bigg ( \Big (x_Z+\xi _{{ij}}\Big )\varDelta _0-2 x_i\varDelta _2+\Big (x_Z \left( x_i-\xi _k\right) \nonumber \\&\quad +\, x_j \xi _k+x_i \left( \xi _{{jk}}-\xi _i\right) \Big )C_{{kSk}}\Bigg ), \end{aligned}$$

(A.17)

$$\begin{aligned} L_{1}^{'(\mathrm a)}&=\frac{ g_R^k \lambda _R^{{ik}}\lambda _R^{kj}}{2\delta ^2}\Bigg (\Big (-x_Z^4 \nonumber \\&\quad +\, 2x_Z^3 \left( 2 \eta _{ij}- \xi _{k}\right) -x_Z^2 \left( 4x_i x_j+2\eta _{ij} \xi _{k}+5 \xi _{ij}^2\right) \nonumber \\&\quad +\, 2 \xi _{{ij}}^2 x_Z \left( \eta _{{ij}}+2 \xi _{k}\right) \Big )\varDelta _1 \nonumber \\&\quad +\, \Big (x_Z^2 \left( x_j\xi _{k}-x_i \left( 5 (\eta _{j}-x_k)+x_i\right) \right) \nonumber \\&\quad +\, 2 \xi _{{ij}} x_Z \left( x_j \xi _{k}+x_i \left( 2(1- \eta _{{jk}})+x_i\right) \right) \nonumber \\&\quad -\, \xi _{{ij}}^3 \left( \eta _{{ik}}-1\right) \Big )\varDelta _2 \nonumber \\&\quad +\, \Big (x_Z^2 \left( 2 x_i \left( 3\xi _{k}-5 x_j\right) -x_j \left( x_j-6 \xi _{k}\right) -x_i^2\right) \nonumber \\&\quad +\, 2 \xi _{{ij}}^2 x_Z \left( \eta _{{ij}}-3 \xi _{k}\right) -\xi _{{ij}}^4 \Big )\varDelta _3\nonumber \\&\quad +\, 2\Big (x_Z^3 \left( 2 \eta _{ij}-\eta _{{ij}} x_k+x_i x_j+\xi _{k}^2\right) \nonumber \\&\quad +\, x_Z^2 \left( x_i \left( x_j \left( 4-6 x_k\right) +x_j^2+\xi _{k}^2\right) \right. \nonumber \\&\quad \left. +\, x_j \left( x_j \left( x_k-4\right) +\xi _{k}^2\right) \right. \nonumber \\&\quad \left. +\, x_i^2 \left( \eta _{{jk}}-4\right) \right) +\xi _{{ij}}^2 x_Z \left( x_j \left( x_k+2\right) \right. \nonumber \\ {}&\left. +x_i \left( x_k-2 \xi _{j}\right) -2 \xi _{k}^2\right) \nonumber \\&\quad -\, \xi _{{ij}}^4 x_k \Big )C_{ZkSk}+\delta x_Z \left( x_Z-\eta _{{ij}}\right) -\delta ^2 B_{{kS}}\Bigg )\nonumber \\&\quad +\, g_L^k \lambda _R^{{ik}} \lambda _R^{{kj}} x_k C_{ZkSk}, \end{aligned}$$

(A.18)

$$\begin{aligned} L_2^{'(\mathrm a)}&=\frac{g_L^k \lambda _L^{{ik}} \lambda _L^{{kj}}}{2\sqrt{x_i} \sqrt{x_j} \delta ^2}\Bigg (2x_i x_j \delta x_Z+6x_i x_j \Big ( x_Z^3 \nonumber \\&\quad -\, x_Z^2 \left( \eta _{{ij}}+2\xi _{k}\right) \Big )\varDelta _1-x_j\Big (x_Z^3\left( \xi _{k}+3 x_i\right) \nonumber \\&\quad +\, x_Z^2 \left( x_i \left( 7(1- \eta _{{jk}})+x_i\right) -3 x_j \xi _{k}\right) \nonumber \\&\quad -\, \xi _{{ij}} x_Z \left( x_i \left( 5 x_j+7\xi _{k}\right) +3 x_j \xi _{k}+5 x_i^2\right) \nonumber \\&\quad -\, \xi _{{ij}}^3 \left( 1-\eta _{{ik}}\right) \Big )\varDelta _2 \nonumber \\&\quad -\, \Big (x_Z^3 \left( \eta _{{ij}} \xi _{k}+6 x_i x_j\right) \nonumber \\&\quad -\, x_Z^2 \left( 3 x_i^2 \left( 2 x_j+\xi _{k}\right) +2 x_i x_j \left( 3 x_j+7 \xi _{k}\right) +3 x_j^2 \xi _{k}\right) \nonumber \\&\quad +\, 3 \eta _{{ij}} \xi _{{ij}}^2 \xi _{k} x_Z -\xi _{{ij}}^4 \xi _{k} \Big )\varDelta _3\nonumber \\&\quad +\, 2x_i x_j\Big (x_Z^4 - 2 x_Z^3 \left( \eta _{{ij}}+2 x_k-3\right) \nonumber \\&\quad +\, x_Z^2 \left( 2 x_i x_j+2 \eta _{ij} \left( x_k-3\right) +\eta _{ij}^2+6 \xi _{k}^2\right) \nonumber \\&\quad +\, 2 \xi _{{ij}}^2 x_k x_Z \Big )C_{ZkSk}\Bigg ), \end{aligned}$$

(A.19)

$$\begin{aligned} L_3^{'(\mathrm a)}&=\frac{ \sqrt{x_k} g_L^k}{\delta } \left( \sqrt{x_i} \lambda _L^{{ik}} \lambda _R^{{kj}}+\, \sqrt{x_j} \lambda _L^{{kj}} \lambda _R^{{ik}}\right) \Bigg (2 x_Z\varDelta _0-\Big (x_Z \nonumber \\&\quad +\, \xi _{{ij}}\Big )\varDelta _2+\Big (x_Z^2-x_Z \left( \eta _{{ij}}+2 \xi _{k}\right) \Big )C_{ZkSk}\Bigg ), \end{aligned}$$

(A.20)

where $\delta =x_i^2-2 (x_j+x_Z) x_i+(x_j-x_Z)^2$. The contributions of diagram (b) are

$$\begin{aligned} L_1^{(\mathrm b)}&=\frac{g_{{VSS}} \sqrt{x_j} \lambda _R^{{ik}} \lambda _R^{{kj}}}{2\delta ^2 }\Bigg (\Big (-x_Z^3 \nonumber \\&\quad +\, 2 x_Z^2 \left( x_j-3 \xi _{k}-2 x_i\right) +\xi _{{ij}} x_Z \left( x_j-6 \xi _{k}+5 x_i\right) \Big )\varDelta _1 \nonumber \\&\quad +\, \Big (x_Z^2\left( \xi _{k}+5 x_i \right) \nonumber \\&\quad -\, 2 x_Z \left( x_i \left( 2 \eta _{ij}-5 \xi _{k}\right) +x_j \xi _{k}\right) \nonumber \\&\quad -\, \xi _{{ij}}^2 \left( \xi _{{ik}}+1\right) \Big )\varDelta _2 \nonumber \\&\quad +\, \frac{1}{x_j}\Big (x_Z^3\left( \xi _{{jk}}+1\right) \nonumber \\&\quad +\, x_Z^2 \left( x_i \left( 4 x_j+3 \xi _{k}\right) +x_j \left( 7\xi _{k}-2 x_j\right) \right) \nonumber \\&\quad -\, \xi _{{ij}} x_Z\left( x_j \left( x_j-5\xi _{k}\right) +x_i \left( 5 x_j+3 \xi _{k}\right) \right) \nonumber \\&\quad +\, \xi _{{ij}}^3\xi _{k} \Big )\varDelta _3 -2\Big ( x_Z^3\left( \xi _{k}+\eta _{ik}\right) \nonumber \\&\quad +\, x_Z^2 \left( x_i \left( \xi _j+2 (\xi _k-x_i)\right) -x_j \left( 4 x_k-1\right) +3 \xi _{k}^2\right) \nonumber \\&\quad -\, \xi _{{ij}} x_Z \left( 2 \left( x_j+2 x_i\right) x_k+x_j-x_i \left( 2 x_j+x_i+3\right) -3 \xi _{k}^2\right) \nonumber \\&\quad +\, \xi _{{ij}}^3 \Big )C_{ZSkS}-\delta \left( \xi _{{ij}}+x_Z\right) \Bigg ), \end{aligned}$$

(A.21)

$$\begin{aligned} L_2^{(\mathrm b)}&=\frac{ g_{{VSS}} \lambda _L^{{ik}} \lambda _L^{{kj}}}{2 \sqrt{x_i}\delta ^2 }\Bigg (x_i\Big (-x_Z^3 \nonumber \\&\quad +\, 2 x_Z^2 \left( x_i-2 x_j-3 {\xi _{k}}\right) +\xi _{{ij}} x_Z \left( 6 {\xi _{k}}-5 x_j-x_i\right) \Big )\varDelta _1\nonumber \\&\quad +\, \Big ( x_Z^3\left( \xi _{{ik}}+1\right) +x_Z^2\left( x_j+2 x_i\right) \left( 3 {\xi _{k}}-x_i\right) \nonumber \\&\quad +\, x_Z \left( x_i^2 \left( 8 x_j-3 {\xi _{k}}\right) -x_i x_j \left( x_j+2 {\xi _{k}}\right) -3 x_j^2 {\xi _{k}}+x_i^3\right) \nonumber \\&\quad +\, \xi _{{ij}}^2 \left( x_i \left( x_j-2 {\xi _{k}}\right) +x_j {\xi _{k}}\right) \Big )\varDelta _2\nonumber \\&\quad +\, \Big (x_Z^3\left( \xi _{{ik}}+1\right) +x_Z^2 \left( x_i \left( 4 x_j+7 {\xi _{k}}\right) +3 x_j {\xi _{k}}-2 x_i^2\right) \nonumber \\&\quad +\, \xi _{{ij}} x_Z \left( 3 x_j {\xi _{k}}+x_i \left( 5 \xi _{{jk}}+x_i+5\right) \right) \nonumber \\&\quad -\, \xi _{{ij}}^3 {\xi _{k}} \Big )\varDelta _3-2x_i\Big (x_Z^3 \left( \xi _{j}+2 x_k\right) \nonumber \\&\quad -\, x_Z^2 \left( x_j \left( 3-2 x_k\right) -x_i \left( \eta _{j}-4 x_k\right) +2 x_j^2-3 {\xi _{k}}^2\right) \nonumber \\&\quad +\, x_Z \xi _{{ij}} \left( 3-2 \left( 2 x_j+x_i+3\right) x_k\right. \nonumber \\ {}&\quad \left. +\quad x_j(x_j+2 x_i +3)+3 x_k^2-x_i\right) \nonumber \\&\quad +\, \xi _{{ij}}^3\Big )C_{ZSkS}+x_i\delta \left( \xi _{{ij}}-x_Z\right) \Bigg ), \end{aligned}$$

(A.22)

$$\begin{aligned} L_3^{(\mathrm b)}&=\, \frac{g_{{VSS}} \sqrt{x_k} \lambda _L^{{kj}} \lambda _R^{{ik}}}{\delta }\Bigg (2 x_Z\varDelta _0-\Big (x_Z +\xi _{{ij}}\Big )\varDelta _2 \nonumber \\&\quad +\, \Big (x_Z^2 +x_Z \left( 2 \xi _{k}-\eta _{{ij}}\right) \Big )C_{ZSkS}\Bigg ), \end{aligned}$$

(A.23)

$$\begin{aligned} L_1^{'(\mathrm b)}&=\frac{g_{{VSS}} \lambda _R^{{ik}} \lambda _R^{{kj}}}{2 \delta ^2}\Bigg (\Big (-x_Z^4 \nonumber \\&\quad +\, 2 x_Z^3 \left( \eta _{{ij}}-\xi _{k}\right) \nonumber \\&\quad -\, x_Z^2 \left( 2 x_i \left( 5 x_j+\xi _{k}\right) +x_j \left( x_j+2\xi _{k}\right) +x_i^2\right) \nonumber \\&\quad +\, 4 \xi _{{ij}}^2 \xi _{k} x_Z \Big )\varDelta _1\nonumber \\&\quad +\, \Big (2 x_i x_Z^3 +x_Z^2 \left( x_i \left( 3 x_j+5 \xi _{k}\right) +x_j \xi _{k}-5 x_i^2\right) \nonumber \\&\quad +\, 2 \xi _{{ij}} x_Z \left( x_i \left( 3 x_j-2 \xi _{k}\right) +x_j \xi _{k}+2 x_i^2\right) \nonumber \\&\quad +\, \xi _{{ij}}^3 \left( 1-\eta _{{ik}}\right) \Big )\varDelta _2\nonumber \\&\quad +\, \Big (2 \eta _{{ij}} x_Z^3 -x_Z^2 \left( x_j \left( 5 x_j-6 \xi _{k}\right) \right. \nonumber \\&\quad \left. +\, x_i \left( 6-6 \eta _{{jk}}+5 x_i\right) \right) \nonumber \\&\quad +\, 2 \xi _{{ij}}^2 x_Z \left( 2 \eta _{ij}-3 \xi _{k}\right) -\xi _{{ij}}^4 \Big )\varDelta _3\nonumber \\&\quad +\, \Big (-2 x_k x_Z^4 +x_Z^3 \left( 2 x_k \left( \eta _{{ij}}-x_k+2\right) \right. \nonumber \\&\quad \left. +\, x_i \left( 4-6 x_j\right) +4 x_j-2\right) \nonumber \\&\quad +\, 2 x_Z^2 \left( x_i^2 \left( 3 x_j+x_k-4\right) \right. \nonumber \\&\quad \left. +\, x_i \left( x_j \left( 4-6 x_k\right) +3 x_j^2-\xi _{k}^2\right) \right. \nonumber \\ {}&\quad \left. +\, x_j \left( x_j \left( x_k-4\right) -\xi _{k}^2\right) \right) \nonumber \\&\quad +\, 2 \xi _{{ij}}^2 x_Z \left( 2 \xi _{k}^2-\eta _{{ij}} \left( x_k-2\right) \right) \Big )C_{ZSkS}\nonumber \\&\quad +\, \delta x_Z \left( \eta _{{ij}}-x_Z\right) -\delta ^2 B_{{kS}} \Bigg ), \end{aligned}$$

(A.24)

$$\begin{aligned} L_2^{'(\mathrm b)}&=\frac{g_{{VSS}}\sqrt{x_i} \sqrt{x_j} \lambda _L^{{ik}} \lambda _L^{{kj}}}{\delta ^2}\nonumber \\ {}&\quad \Bigg (\Big (-x_Z^3 -x_Z^2 \left( \eta _{{ij}}+6 \xi _{k}\right) +2 \xi _{{ij}}^2 x_Z \Big )\varDelta _1\nonumber \\&\quad +\, \frac{1}{2 x_i}\Big (\left( \xi _{{ik}}+1\right) x_Z^3 \nonumber \\&\quad +\, x_Z^2 \left( x_i \left( 7 \xi _{k}-x_j\right) +3 x_j \xi _{k}+3 x_i^2\right) \nonumber \\&\quad +\, \xi _{{ij}} x_Z \left( x_i \left( x_j+7\xi _{k}\right) +3 x_j \xi _{k}-3 x_i^2\right) \nonumber \\&\quad +\, \xi _{{ij}}^3 \left( 1-\eta _{{ik}}\right) \Big )\varDelta _2+\frac{1}{2 x_i x_j}\Big (x_Z^3 \left( 2 x_i x_j-\eta _{{ij}} \xi _{k}\right) \nonumber \\&\quad +\, x_Z^2 \left( x_i^2 \left( 2 x_j+3\xi _{k}\right) +2 x_i x_j \left( x_j+7\xi _{k}\right) +3 x_j^2 \xi _{k}\right) \nonumber \\&\quad -\, \xi _{{ij}}^2 x_Z \left( 3 \eta _{{ij}} \xi _{k}+4 x_i x_j\right) \nonumber \\&\quad +\, \xi _{{ij}}^4 \xi _{k} \Big )\varDelta _3+\Big (x_Z^3 \left( 2-\eta _{{ij}}-4 x_k\right) \nonumber \\&\quad +\, 2 x_Z^2 \left( \eta _{{ij}}\eta _{k}+x_i x_j+\xi _{ij}^2-3 \xi _{k}^2\right) \nonumber \\&\quad +\, \xi _{{ij}}^2 x_Z \left( 2x_k-\eta _{{ij}}-4\right) \Big )C_{ZSkS}-\delta x_Z\Bigg ), \end{aligned}$$

(A.25)

$$\begin{aligned} L_3^{'(\mathrm b)}&=\frac{g_{{VSS}} \sqrt{x_k}}{\delta }\left( \sqrt{x_i}\lambda _L^{{ik}} \lambda _R^{{kj}}+\sqrt{x_j}\lambda _L^{{kj}} \lambda _R^{{ik}} \right) \Bigg (2 x_Z\varDelta _0\nonumber \\&\quad -\, \Big (x_Z+\xi _{{ij}} \Big )\varDelta _2 +x_Z\Big ( x_Z+ 2 \xi _{k}-\eta _{{ij}}\Big )C_{ZSkS}\Bigg ). \end{aligned}$$

(A.26)

Finally Feynman diagrams (c) and (d) only contribute to monopole terms. The corresponding contribution of both diagrams is

$$\begin{aligned} \sum _{i=1}^{3} L_i^{'(\mathrm cd)}&=\frac{g_L^j \lambda _R^{{ik}} \lambda _R^{{kj}}}{2\xi _{{ij}}}\Bigg (\Big (\eta _{{ik}}-1 \Big )\varDelta _2+\xi _{{ij}}\varDelta _3-2x_k\varDelta _4 \nonumber \\&\quad - \, 2\xi _{k}\varDelta _5+ \xi _{{ij}}B_{{kS}}-2\xi _k\Bigg )\nonumber \\&\quad +\, \frac{g_L^j \lambda _L^{{ik}} \lambda _L^{{kj}}}{2 \sqrt{x_i} \sqrt{x_j}}\Bigg (\frac{1}{\xi _{{ij}}}\left( \eta _{{ik}}-1\right) x_j \varDelta _2 - \xi _{k}\varDelta _3\Bigg )\nonumber \\&\quad +\, \frac{ \sqrt{x_k} g_L^j}{\xi _{{ij}}} \left( \sqrt{x_i} \lambda _L^{{ik}} \lambda _R^{{kj}}+\sqrt{x_j} \lambda _L^{{kj}} \lambda _R^{{ik}}\right) \varDelta _2. \end{aligned}$$

(A.27)

We can observe that the ultraviolet divergent term $B_{kS}\equiv B_0(0,m_k^2,m_S^2)$, which appears only in the monopole terms, is canceled out when summing over all the contributions.

Furthermore, the form factors associated with the right-handed terms are given by

$$\begin{aligned} R^{Z}=L^Z\left( \lambda ^{lm}_L\leftrightarrow \lambda ^{lm}_R,g_{ZSS}\rightarrow -g_{ZSS}\right) , \end{aligned}$$

(A.28)

and

$$\begin{aligned} R^{'Z}=-L^{'Z}\left( \lambda ^{lm}_L\leftrightarrow \lambda ^{lm}_R,g_{ZSS}\rightarrow -g_{ZSS}\right) . \end{aligned}$$

(A.29)

1.2 Appendix A.2: Feynman parameter results

1.2.1 Appendix A.2.1: $f_i\rightarrow f_j \gamma$ decay

The $L^\gamma$ form factor of Eq. (14) is ultraviolet finite and is given in terms of Feynman parameter integrals as follows

$$\begin{aligned} L^\gamma&=\frac{N_cg^2e\sqrt{x_i}}{62c_W^2\pi ^2}\frac{1}{2}\int _0^1 dx \int _0^{1-x}dy\Bigg (\frac{Q_k}{\zeta _1} \left( x y \sqrt{x_i} \lambda _L^{{ik}} \lambda _L^{{kj}}\right. \nonumber \\&\quad \left. -\, \lambda _R^{{ik}} \left( x \sqrt{x_j} (x+y-1) \lambda _R^{{kj}}+(x-1) \sqrt{x_k} \lambda _L^{{kj}}\right) \right) \nonumber \\&\quad +\, \frac{Q_S}{\zeta _2} (x+y-1) \left( x \sqrt{x_i} \lambda _L^{{ik}} \lambda _L^{{kj}}+y \sqrt{x_j} \lambda _R^{{ik}} \lambda _R^{{kj}}+\sqrt{x_k} \lambda _L^{{kj}} \lambda _R^{{ik}}\right) \Bigg ), \end{aligned}$$

(A.30)

where

$$\begin{aligned} \zeta _1&=x \left( y \xi _{ji}+x_j (x-1)-\xi _k\right) +x_k, \end{aligned}$$

(A.31)

$$\begin{aligned} \zeta _2&=x y \eta _{ij}-x \left( \eta _{ik}-1\right) +x^2 x_i-y \left( \eta _{jk}-y x_j\right) +x_k+y. \end{aligned}$$

(A.32)

The $R^\gamma$ form factor is given by Eq. (A.12), whereas monopole terms $L^{'\gamma }$ and $R{'\gamma }$ are zero as already mentioned (one must consider electric charge conservation).

1.2.2 Appendix A.2.2: $f_i\rightarrow f_j Z$ decay

The dipole terms of Eq. (A.13), which only arise from diagrams (a) and (b), are ultraviolet finite and are given by

$$\begin{aligned} L^{\mathrm{(a)}}= & {} \int _0^1 dx\int _0^{1-x}dy\frac{1}{\zeta '_1} \left( x y \sqrt{x_i} g_L^k \lambda _L^{{ik}} \lambda _L^{{kj}}\right. \nonumber \\&-\, x \sqrt{x_j} (x+y-1) g_R^k \lambda _R^{{ik}} \lambda _R^{{kj}}\nonumber \\&\left. +\, \sqrt{x_k} \lambda _L^{{kj}} \lambda _R^{{ik}} \left( y g_L^k-(x+y-1) g_R^k\right) \right) , \end{aligned}$$

(A.33)

$$\begin{aligned} L^{\mathrm{(b)}}= & {} g_{{VSS}}\int _0^1 dx\int _0^{1-x}dy\frac{(x+y-1) }{\zeta '_2}\left( x \sqrt{x_i} \lambda _L^{{ik}} \lambda _L^{{kj}}\right. \nonumber \\&\left. +\, y \sqrt{x_j} \lambda _R^{{ik}} \lambda _R^{{kj}}+\sqrt{x_k} \lambda _L^{{kj}} \lambda _R^{{ik}}\right) , \end{aligned}$$

(A.34)

whereas the partial contributions to the monopole terms of Eq. (A.14) are ultraviolet divergent and read

$$\begin{aligned} L^{'\mathrm (a)}&=\int _0^1 dx\int _0^{1-x}dy\frac{1}{\zeta '_1}\Bigg (x (y-1) \sqrt{x_i} \sqrt{x_j} g_L^k \lambda _L^{{ik}} \lambda _L^{{kj}}\nonumber \\&\quad +\, (y-1) \sqrt{x_i} \sqrt{x_k} g_L^k \lambda _L^{{ik}} \lambda _R^{{kj}}\nonumber \\&\quad +\, \lambda _R^{{ik}} \lambda _R^{{kj}} \left( g_R^k \left( (x+y-1) \left( y (x_Z-\xi _{{ij}})+x x_j\right) \right. \right. \nonumber \\&\quad \left. \left. +\, \zeta '_1 \left( 1+ \log \left( \zeta '_1\right) \right) \right) -x_k g_L^k\right) \nonumber \\&\quad +\, \sqrt{x_j} \sqrt{x_k}\lambda _R^{{ik}} \lambda _L^{{kj}} \left( (x+y-1) g_R^k-x g_L^k\right) \Bigg )\nonumber \\&\quad -\, \frac{g_R^k}{2} \lambda _R^{{ik}}\lambda _R^{{kj}}\varDelta _{\mathrm{UV}}, \end{aligned}$$

(A.35)

$$\begin{aligned} L^{{'\mathrm (b)}}&=\, \frac{g_{{VSS}}}{2\zeta '_2}\int _0^1 dx\int _0^{1-x}dy(2 x-1)\nonumber \\ {}&\Bigg (\sqrt{x_i} \lambda _L^{{ik}} \left( \sqrt{x_j} (x+y) \lambda _L^{{kj}}+\sqrt{x_k} \lambda _R^{{kj}}\right) \nonumber \\&\quad +\, \lambda _R^{{ik}} \left( \left( x x_i+y x_j+\frac{2\zeta '_2}{2x-1} \log \left( \zeta '_2\right) \right) \lambda _R^{{kj}}\right. \nonumber \\&\quad \left. +\, \sqrt{x_j} \sqrt{x_k} \lambda _L^{{kj}}\right) \Bigg )-\frac{g_{{VSS}}}{2}\lambda _R^{{ik}}\lambda _R^{{kj}}\varDelta _{\mathrm{UV}}, \end{aligned}$$

(A.36)

$$\begin{aligned} L^{{'\mathrm (cd)}}&=\int _0^1 dx\frac{g_L^j}{\xi _{{ij}}} \Bigg ( \left( \sqrt{x_i} \lambda _L^{{ik}} \left( x \sqrt{x_j} \lambda _L^{{kj}}+\sqrt{x_k} \lambda _R^{{kj}}\right) \right. \nonumber \\&\quad \left. +\, \sqrt{x_j} \sqrt{x_k} \lambda _L^{{kj}} \lambda _R^{{ik}}\right) \left( \log \left( \zeta '_{32}\right) -\log \left( \zeta '_{31}\right) \right) \nonumber \\&\quad +\, x \lambda _R^{{ik}} \lambda _R^{{kj}}\left( x_j\log \left( \zeta '_{32}\right) \right. \nonumber \\&\quad \left. -\, x_i\log \left( \zeta '_{31}\right) \right) \Bigg )+ \frac{g_L^j}{2} \lambda _R^{{ik}}\lambda _R^{{kj}}\varDelta _{\mathrm{UV}}. \end{aligned}$$

(A.37)

where $\varDelta _{\mathrm{UV}}$ stands for the ultraviolet divergence, which cancels out when summing over the partial contributions as it is proportional to $g_L^k-g_R^j-g_{VSS}$. We also have defined the following functions

$$\begin{aligned} \zeta '_1&=x y \left( x_Z-\xi _{{ij}}\right) +x^2 x_j-x\left( \eta _{{jk}}-1\right) \nonumber \\ {}&\quad +x_k+(y-1)y x_Z,\nonumber \\ \zeta '_2&=x^2 x_i+x y \left( \eta _{{ij}}-x_Z\right) -x\left( \eta _{{ik}}-1\right) \nonumber \\ {}&\quad -y\left( \eta _{{jk}}-yx_j\right) +x_k+y,\nonumber \\ \zeta '_{3a}&=x^2 x_a-x \left( \eta _{{ak}}-1\right) +x_k. \end{aligned}$$

(A.38)

Appendix B: Loop integrals for the $f_{i}\rightarrow f_j H$ decay

The $F_L$ and $F_R$ form factors of Eq. (20) are given by

$$\begin{aligned} F_{L,R}= \frac{N_cg m_S}{32\pi ^2 m_W} \sum _{k=\mathrm{(a), (b), (c)}}^3f_{L,R}^{(k)}+\frac{3}{16\pi ^2}\frac{\lambda _S v}{m_S}f_{L,R}^{\mathrm{(d)}},\nonumber \\ \end{aligned}$$

(B.39)

with $f_{L,R}^{(k)}$ (${\mathrm{k=a,b,c,d}}$) being the contributions of the Feynman diagram analogue to the diagram (k) of Fig. 2, with the V gauge boson replaced by the Higgs boson. Again we present our results in terms of Passarino–Veltman scalar functions and Feynman parameter integrals.

1.1 Appendix B.1: Passarino–Veltman results

The sum of the contributions of the triangle and bubble diagrams (a), (b) and (c) is ultraviolet finite and reads

$$\begin{aligned}&\sum _{k={\mathrm{(a), (b), (c)}}}f_L^{(k)} =\frac{1}{2\chi }\Bigg (\sqrt{x_i}\Big (2 x_k (x_H+ \xi _{ji}) (\xi _{Hj}-x_i-2\xi _{k})C'_1\nonumber \\&\quad +\, (\zeta x_{j} x_i(x_{j} +\xi _k)-8 x_{j} x_k)\varDelta '_1\nonumber \\&\quad +\, (4 x_k (x_i-\xi _{Hj})-\zeta x_{j}^2 (x_i+\xi _k))\varDelta '_2\nonumber \\&\quad +\, \zeta x_{j} \xi _{ji}\xi _k\varDelta '_3 \Big )\lambda _L^{kj} \lambda _L^{ik}\nonumber \\&\quad +\, \sqrt{x_{j}}\Big (2 x_k (x_H-\xi _{ji}) (\xi _{Hj}-x_i-2 \xi _k)C'_1\nonumber \\&\quad + \, (4 x_k (x_i-\xi _{Hj})+\zeta x_i^2 (x_{j} +\xi _k))\varDelta '_1\nonumber \\&\quad -\, (\zeta x_{j} x_i (x_i+\xi _k)+8 x_i x_k)\varDelta '_2\nonumber \\&\quad +\, \zeta x_i\xi _{ji} \xi _k\varDelta '_3 \Big )\lambda _R^{kj} \lambda _R^{ik}\nonumber \\&\quad +\, 2\sqrt{x_k}\Big ( \left( x_k \left( 2 \xi _{ji}^2+x_H^2-3 x_H \xi _{ij}\right) \right. \nonumber \\&\quad \left. +\, x_H (x_H+x_{j} (2 x_i-1)-x_i)\right) C'_1\nonumber \\&\quad +\, (\zeta x_i- x_{j} (x_H-\xi _{ji}))\varDelta '_1 \nonumber \\&\quad -(\zeta x_{j}+ x_i (x_H+\xi _{ji}) )\varDelta '_2 \Big )\lambda _L^{kj} \lambda _R^{ik}\nonumber \\&\quad +\, 2\sqrt{x_{j}} \sqrt{x_i} \sqrt{x_k}\Big ( x_H (\xi _{Hj}-x_i-2 \xi _k)C'_1 \nonumber \\&\quad -\, (\xi _{ji}+x_H-\zeta )\varDelta '_1 + \left( \xi _{ji}-x_H-\zeta \right) \varDelta '_2 \Big )\lambda _R^{kj} \lambda _L^{ik}\Bigg ), \end{aligned}$$

(B.40)

whereas the contribution of triangle diagram (d), which is ultraviolet finite by itself, can be written as

$$\begin{aligned} f_L^{\mathrm{(d)}}&=\frac{1}{\chi }\Bigg ( \sqrt{x_i}\Big ( (x_H (x_{j} +\xi _k)-\xi _{ji} (\xi _{ik}+1))C'_2 \nonumber \\&\quad -\, 2 x_{j} \varDelta '_1 + (x_i-\xi _{Hj})\varDelta '_2+(x_H+\xi _{ji})\varDelta '_4 \Big )\lambda _L^{kj} \lambda _L^{ik}\nonumber \\&\quad +\, \sqrt{x_{j} }\Big ( (x_H (x_i+\xi _k)+\xi _{ji} (\xi _{ik}+1))C'_2\nonumber \\&\quad +\, (x_i-\xi _{Hj})\varDelta '_1-2 x_i\varDelta '_2+ (x_H-\xi _{ji})\varDelta '_4 \Big )\lambda _R^{kj} \lambda _R^{ik}\nonumber \\&\quad -\, \sqrt{x_k} \chi C'_2 \lambda _L^{kj} \lambda _R^{ik}\Bigg ), \end{aligned}$$

(B.41)

where we have introduced the auxiliary variable $\chi =\xi _{ji}^2+x_H^2-2 x_H \xi _{ij}$, and $\zeta =\chi /(x_{j} x_i \xi _{ji})$. As for the $C'_{j}$ and $\varDelta '_{j}$ functions, they are given by

$$\begin{aligned} \varDelta '_1&=B_0(m_{j}^2,m_k^2,m_S^2)-B_0(m_H^2,m_k^2,m_k^2), \end{aligned}$$

(B.42)

$$\begin{aligned} \varDelta '_2&= B_0(m_i^2,m_k^2,m_S^2)- B_0(m_H^2,m_k^2,m_k^2), \end{aligned}$$

(B.43)

$$\begin{aligned} \varDelta '_3&=B_0(0,m_k^2,m_S^2)-B_0(m_H^2,m_k^2,m_k^2), \end{aligned}$$

(B.44)

$$\begin{aligned} \varDelta '_4&= B_0(m_H^2,m_S^2,m_S^2)-B_0(m_H^2,m_k^2,m_k^2), \end{aligned}$$

(B.45)

$$\begin{aligned} C'_1&=m_S^2 C'_0(m_H^2, m_{j}^2, m_i^2, m_k^2, m_k^2, m_S^2), \end{aligned}$$

(B.46)

$$\begin{aligned} C'_2&=m_S^2 C'_0(m_H^2, m_{j}^2, m_i^2, m_k^2, m_k^2, m_S^2). \end{aligned}$$

(B.47)

It is thus evident that ultraviolet divergences cancel out. As far as the right-handed terms are concerned, they obey

$$\begin{aligned} f_R^{\mathrm{(k)}}&=f_L^{\mathrm{(k)}}(L\leftrightarrow R)\quad (k={\mathrm{a,\, b,\, c,\, d}}). \end{aligned}$$

(B.48)

1.2 Appendix B.2: Feynman parameter results

Feynman parametrization yield the following results for the $f_{L,R}^{\mathrm{(k)}}$ coefficients:

$$\begin{aligned}&f_L^{\mathrm{(a)}} = \sqrt{x_k}\int _{0}^{1}dx\int _{0}^{1-x}dy\frac{1}{\varrho _1}\Bigg (\sqrt{x_i} \sqrt{x_k} (2 y-1)\lambda _L^{kj} \lambda _L^{ik}\nonumber \\&\,\quad \qquad +\, \sqrt{x_{j} } \sqrt{x_k} (1-2 (x+ y))\lambda _R^{kj} \lambda _R^{ik}\nonumber \\&\,\quad \qquad -\, \Big (2 \varrho _1 \log \left( \varrho _{1}\right) +x y (\xi _{ji}+x_H)+(x-1) x x_{j} \nonumber \\&\,\quad \qquad +\, \varrho _1+x_Hy(y-1)+x_k\Big )\lambda _L^{kj} \lambda _R^{ik}\nonumber \\&\,\quad \qquad -\, x \sqrt{x_{j} } \sqrt{x_i}\lambda _R^{kj} \lambda _L^{ik}\Bigg )\nonumber \\&\,\quad \qquad +\, \sqrt{x_k} \lambda _L^{kj} \lambda _R^{ik}\varDelta _{\mathrm{UV}}, \end{aligned}$$

(B.49)

$$\begin{aligned}&\sum _{k=\mathrm{(b), (c)}}f_L^{\mathrm{(k)}}=\, \frac{1}{\xi _{ji}}\int _{0}^{1}dx\Bigg ( \sqrt{x_{j}}\sqrt{x_i}\left( x\left( \sqrt{x_{j}}\lambda _L^{kj} \lambda _L^{ik}+ \sqrt{x_i } \lambda _R^ {kj} \lambda _R^{ik}\right) \right. \nonumber \\&\,\quad \qquad \,\qquad \qquad \left. +\, \sqrt{x_k} \lambda _R^{kj} \lambda _L^{ik}\right) (\log (\varrho _{2j})-\log (\varrho _{2i}))\nonumber \\&\,\quad \qquad \,\qquad \qquad +\, \sqrt{x_k} (x_{j} \log (\varrho _{2j})-x_i \log (\varrho _{2i}))\lambda _L^{kj} \lambda _R^{ik} +\Bigg ) \nonumber \\&\,\quad \qquad \, \qquad \qquad -\, \sqrt{x_k} \lambda _L^{kj} \lambda _R^{ik}\varDelta _{\mathrm{UV}}, \end{aligned}$$

(B.50)

where it is evident that the ultraviolet divergence $\varDelta _{\mathrm{UV}}$ cancels out when summing over the partial contributions. We also use the following auxiliary variables

$$\begin{aligned} \varrho _{1}=x^2 x_{j} +x (y (x_H+\xi _{ji})+ \xi _{kj}+1)+x_H (y-1) y+x_k, \end{aligned}$$

and

$$\begin{aligned} \varrho _{2a}=x x_a(x-1) -x (x_k-1)+x_k. \end{aligned}$$

As far as the contribution of Feynman diagram (d) is concerned, it is given by

$$\begin{aligned} f_L^{\mathrm{(d)}}&=\int _{0}^{1}dx\int _{0}^{1-x}dy\frac{1}{ \varrho _{3}}\left( \sqrt{x_k}\lambda _L^{kj} \lambda _R^{ik}\right. \nonumber \\&\quad \left. -\, \sqrt{x_i} (x+y-1)\lambda _L^{kj} \lambda _L^{ik}+\sqrt{x_{j} } y\lambda _R^{kj} \lambda _R^{ik}\right) , \end{aligned}$$

(B.51)

with

$$\begin{aligned} \varrho _{4}=x^2 x_i+x (y (x_H-\xi _{ji})+\xi _{ki}-1)+x_H (y-1) y+1.\nonumber \\ \end{aligned}$$

(B.52)

Appendix C: Loop integrals for the $H\rightarrow \bar{f}_j f_i$ decay

As already mentioned, the form factors $F_L$ and $F_R$ for the $f_i\rightarrow f_j H$ decay width are also valid for the $H\rightarrow f_j f_i$ decay width given in (21). It is interesting to obtain the approximate results in the limit of small $x_{j}$ and $x_i$. In the case of the Passarino–Veltman results $\varDelta '_ 1\rightarrow \varDelta '_3+O(x_{j})$ and $\varDelta '_ 2\rightarrow \varDelta '_3+O(x_i)$ for small $x_i$ and $x_j$, which means that in the limit of vanishing external fermion masses we have

$$\begin{aligned} \sum _{k={\mathrm{(a), (b), (c)}}}f_L^{(k)}\simeq \lambda _L^{kj} \lambda _R^{ik} \sqrt{x_k}\left( (x_k+1)C'_1-\varDelta '_3\right) , \end{aligned}$$

(C.53)

and

$$\begin{aligned} f_L^{\mathrm{(d)}}=- \lambda _L^{kj} \lambda _R^{ik}\sqrt{x_k} C'_2, \end{aligned}$$

(C.54)

which means that in this scenario the $H\rightarrow f_i f_j$ decay width can be written as

$$\begin{aligned}&\varGamma (H\rightarrow f_i f_j)\simeq \frac{m_H}{32 \pi } \left( |\lambda _L^{kj} \lambda _R^{ik}|^2\right. \nonumber \\&\left. +|\lambda _R^{kj}\lambda _L^{ik}|^2\right) |f(m_k,m_S,m_i,m_j)|^2. \end{aligned}$$

(C.55)

where

$$\begin{aligned}&f(m_k,m_S,m_i,m_j) \simeq \frac{3m_k}{16\pi ^2 m_S}\nonumber \\ {}&\Bigg (\left( (m_k^2+m_S^2)C_0(m_H^2, 0, 0, m_k^2, m_k^2, m_S^2)\right. \nonumber \\&\quad \left. -\, \left( B_0(0,m_k^2,m_S^2) -B_0(m_H^2,m_k^2,m_k^2)\right) \right) \nonumber \\&\quad -\, m_S^2 C_0(m_H^2, 0, 0, m_k^2, m_k^2, m_S^2)\Bigg ), \end{aligned}$$

(C.56)

This result agrees with the one presented in [21, 55,56,57].

As far as the Feynman parameter results, in the vanishing limit of $m_{j}$ and $m_i$ one can obtain

$$\begin{aligned} f_L^{\mathrm{(a)}}&=-\sqrt{x_k}\int _{0}^{1}dx\int _{0}^{1-x}dy\frac{1}{\varrho '_{1}}\left( 2 \varrho '_{1} \log (\varrho '_{1})\right. \nonumber \\&\quad \left. +\, x_H y (x+y-1)+\varrho '_{1}+x_k\right) \lambda _L^{kj} \lambda _R^{ik}\nonumber \\&\quad +\, \sqrt{x_k} \lambda _L^{kj} \lambda _R^{ik}\varDelta _{\mathrm{UV}}, \end{aligned}$$

(C.57)

$$\begin{aligned} \sum _{k=\mathrm{(b), (c)}} f_L^{(k)}&=\sqrt{x_k} \int _{0}^{1}dx\log (\varrho '_{2})\lambda _L^{kj} \lambda _R^{ik}- \sqrt{x_k} \lambda _L^{kj} \lambda _R^{ik}\varDelta _{\mathrm{UV}}, \end{aligned}$$

(C.58)

and

$$\begin{aligned} f_L^{\mathrm{(d)}}= \sqrt{x_k}\int _{0}^{1}dx\int _{0}^{1-x}dy \frac{1}{\varrho '_{3}}\lambda _L^{kj} \lambda _R^{ik}, \end{aligned}$$

(C.59)

with

$$\begin{aligned} \varrho '_{1}&=x (x_k+1)+x_H y(x+y-1) +x_k, \end{aligned}$$

(C.60)

$$\begin{aligned} \varrho '_{2}&=x_k-x (x_k-1), \end{aligned}$$

(C.61)

$$\begin{aligned} \varrho '_{3}&=x (x_k-1)+x_H y (x+y-1)+1. \end{aligned}$$

(C.62)

Appendix D: Lepton anomalous magnetic dipole moment

The F and G functions of Eq. (35) read

$$\begin{aligned} F (z_1,z_2)= & {} \, Q_k F _1(z_1,z_2)+Q_{S} F _2(z_1,z_2), \end{aligned}$$

(D.63)

$$\begin{aligned} G (z_1,z_2)= & {} \, Q_j G _1(z_1,z_2)+Q_{S_k} G _2(z_1,z_2), \end{aligned}$$

(D.64)

with the $F_a$ and $G_a$ functions given in terms of Feynman parameter integrals by

$$\begin{aligned} F _a(z_1,z_2)=2\int _0^1\frac{(1-x) x \xi _a(x)}{(1-x)(z_2- x z_1)+x}dx, \end{aligned}$$

(D.65)

$$\begin{aligned} G _a(z_1,z_2)=2\int _0^1\frac{(1-x) \xi _a(x)}{(1-x)(z_2- x z_1)+x}dx, \end{aligned}$$

(D.66)

where $\xi _1(x)=1-x$ and $\xi _2(x)=x$. The integration is straightforward in the limit of a light external fermion and heavy internal fermion and LQ: $x_i\ll x_k$

$$\begin{aligned}&F (x_i\simeq 0,x_k)= \frac{Q_k}{3(1-x_k)^4}\left( 2+3x_k-6x_k^2+ x_k^3+6 x_k \log (x_k)\right) \nonumber \\&\quad +\, \frac{Q_S}{3(1-x_k)^4}\left( 1-6x_k+3x_k^2+ 2x_k^3-6 x_k^2 \log (x_k)\right) . \end{aligned}$$

(D.67)

$$\begin{aligned}&G (x_i\simeq 0,x_k)=-\frac{Q_k}{(1-x_k)^3}\left( 3-4x_k+x_k^2+2 \log (x_k)\right) \nonumber \\&\quad +\, \frac{Q_S}{(1-x_k)^3}\left( 1-x_k^2+2 x_k \log (x_k)\right) . \end{aligned}$$

(D.68)

For completeness we also present the results in terms of Passarino–Veltman scalar functions:

$$\begin{aligned}&F_1 (z_1,z_2)=- \frac{1}{z_1^2 \zeta (z_1,z_2)}\left( 2 z_2 \left( 2 z_1+\zeta (z_1,z_2) \right) \varDelta _6(z_1,z_2)\right. \nonumber \\&\quad \left. -\, 2 \left( z_1 \left( 1-z_1+z_2\right) +\zeta (z_1,z_2) \right) \varDelta _7(z_1,z_2)\right. \nonumber \\&\quad +\, \left. z_1 \left( 4 z_2+\zeta (z_1,z_2) -4\right) +2 \left( z_2-1\right) \zeta (z_1,z_2) +4 z_1^2\right) , \end{aligned}$$

(D.69)

$$\begin{aligned}&F_2 (z_1,z_2)=-\frac{1}{z_1^2 \zeta (z_1,z_2) } \left( 2 z_2 \left( z_1 \left( z_2+1\right) -\left( z_2-1\right) {}^2\right) \varDelta _6(z_1,z_2)\right. \nonumber \\&\quad \left. +\, 2 \left( \left( z_2-1\right) {}^2+\left( z_1-2\right) z_1\right) \varDelta _7(z_1,z_2)\right. \nonumber \\&\quad +\, \left. z_1^3+\left( z_2-1\right) \left( z_2+3\right) z_1-2 \left( z_2-1\right) {}^3\right) , \end{aligned}$$

(D.70)

$$\begin{aligned}&G_1 (z_1,z_2)=- \frac{1}{z_1 \zeta (z_1,z_2)}\left( 2 \left( z_1^2-\left( 2 z_2+1\right) z_1\right. \right. \nonumber \\&\quad \left. +\, \left( z_2-1\right) z_2\right) \varDelta _6(z_1,z_2)+2 \left( z_1-z_2+1\right) \varDelta _7(z_1,z_2)\nonumber \\&\quad +\, \left. 2 \left( z_1-z_2+1\right) {}^2\right) , \end{aligned}$$

(D.71)

$$\begin{aligned}&G_2 (z_1,z_2)=- \frac{1}{z_1 \zeta (z_1,z_2) }\left( 2 \left( z_1-z_2+1\right) z_2\varDelta _6(z_1,z_2)\right. \nonumber \\&\quad \left. +\, 2 \left( z_1+z_2-1\right) \varDelta _7(z_1,z_2)+2 \left( z_1^2-\left( z_2-1\right) {}^2\right) \right) , \end{aligned}$$

(D.72)

with

$$\begin{aligned} \varDelta _6(x,y)= & {} B_0(0, y\, m_S^2, y \,m_S^2)- B_0(x\, m_S^2, y \,m_S^2, m_S^2),\nonumber \\ \end{aligned}$$

(D.73)

$$\begin{aligned} \varDelta _7(x,y)= & {} B_0(0, m_S^2, m_S^2)- B_0(x\, m_S^2, y \,m_S^2, m_S^2),\nonumber \\ \end{aligned}$$

(D.74)

and $\zeta (x,y)=\left( 1+y-x\right) ^2-4 y$.