Eur. Phys. J. C (2015) 75:579 DOI 10.1140/epjc/s10052-015-3807-9

Regular Article - Theoretical Physics

Lepton flavour violation in composite Higgs models

Ferruccio Feruglio1,a, Paride Paradisi1,b, Andrea Pattori2,c 1 Sezione di Padova, Dipartimento di Fisica e Astronomia ‘G. Galilei’, INFN, Università di Padova, Via Marzolo 8, 35131 Padua, Italy 2 Physik-Institut, Universität Zürich, 8057 Zurich, Switzerland

Received: 8 October 2015 / Accepted: 23 November 2015 / Published online: 8 December 2015 © The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract We discuss in detail the constraints on the partial TeV scale, representing potential sources of flavour changing compositeness coming from flavour and CP violation in the neutral currents (FCNC) and CP violation. So far both direct leptonic sector. In the first part we present a formulation of searches at the LHC and indirect searches in the context of partial compositeness in terms of a flavour symmetry group precision tests and flavour physics have brought no conclu- and a set of spurions, whose background values specify the sive evidence of new physics at the TeV scale. The negative symmetry breaking pattern. In such a framework we con- outcome of the search for new physics in the flavour sector struct the complete set of dimension-six operators describing is particularly intriguing. Indeed a scale of new physics NP lepton flavour violation and CP violation. By exploiting the as large as 105 TeV [3] is required by an effective operator existing bounds, we derive limits on the compositeness scale analysis to preserve the good agreement between observa- in different scenarios, characterised by increasing restrictions tions and theory predictions, unless the new flavour sector on the spurion properties. We confirm that in the most general is highly non-generic and involves specific mechanisms to case the compositeness scale should lie well above 10 TeV. suppress FCNC and CP violation at the desired level. However, if in the composite sector the mass parameters and The latter possibility is empirically supported by the Yukawa couplings are universal, such a bound can be sig- huge hierarchies among fermion masses and fermion mixing nificantly lowered, without necessarily reproducing the case angles, which can only be explained by some special dynam- of minimal flavour violation. The most sensitive processes ics. An effective mechanism suppressing FCNC and CP vio- are decays of charged leptons either of radiative type or into lation can be introduced by observing that in the electroweak three charged leptons, μ → e conversion in nuclei and the theory the symmetry of the flavour sector is broken only by electric dipole moment of the electron. In the second part the Yukawa interactions. Minimal flavour violation (MFV) we explicitly compute the Wilson coefficients of the rele- [4] is defined by the assumption that, even including new vant dimension-six operators in the so-called two-site model, physics contributions, Yukawa couplings are the only source embodying the symmetry breaking pattern discussed in our of such symmetry breaking. In MFV flavour effects from new first part, and we compare the results with those of the general physics are controlled and damped by the smallness of the spurion analysis. fermion masses and mixing angles. In this framework data allow NP to be considerably smaller, close to the TeV scale. MFV provides a useful benchmark for the discussion of the 1 Introduction flavour sector, but it does not emerge as a unique framework from the known mechanisms aiming to explain the observed If the solution to the gauge hierarchy problem is based on fermion spectrum (for a review see [5]), or from the known a new symmetry and not on anthropic considerations (for a models providing a solution to the gauge hierarchy problem. review see [1]) or on special evolutions of the scalar sec- In this paper we reconsider the possibility that both the tor in the early universe [2], new physics at the TeV scale origin of fermion masses and the suppression of FCNC and is expected. In most of the existing models new degrees of CP violation are due to the mechanism of partial composite- freedom carrying flavour quantum numbers are present at the ness (PC) [6], as realised in the context of composite Higgs models (for reviews see [7,8]), and we perform a detailed analysis of flavour and CP violations in the leptonic sector. a e-mail: [email protected] According to PC there are no direct couplings between the b e-mail: [email protected] elementary fermions and the Higgs doublet. The Higgs dou- c e-mail: [email protected] 123 579 Page 2 of 26 Eur. Phys. J. C (2015) 75 :579 blet has potentially strong couplings to a composite sector, can be found in Ref. [33]. LFV and CP violation in the lep- including, in the simplest case, a set of vector-like fermions ton sector have been investigated also in SM extensions with with masses of the order of the compositeness scale. The SM extra vector-like heavy leptons [34,35] that can mimic the fermions are mostly elementary and get their masses through PC scenario, at least as far as the contributions from heavy mixing terms with operators of the composite sector, often fermions is concerned. modelled by vector-like fermions. In the present work we recast the framework of PC in terms An appealing realisation of this idea involves anarchic of a generalised flavour symmetry and a suitable set of rele- Yukawa couplings in the composite sector. In this case the vant spurions, much along the lines of Refs. [12,28,32,36]. observed hierarchies between SM fermion masses and mix- This involves a certain degree of model dependence, since ing angles are entirely due to the elementary–composite mix- both the flavour symmetry and the spurions are determined ing terms. This is of particular interest, especially for the lep- by the specific set of composite leptons that we choose by fol- ton sector, since the known pattern of neutrino masses and lowing the criterion of minimality. In particular we work in mixing angles as extracted from neutrino oscillation experi- the limit of vanishing neutrino masses, turning off the poten- ments [9] seems to support the idea of an underlying anarchic tial effects related to massive neutrinos. It is well known that dynamics [10,11]. Higher-dimensional operators describing within PC, LFV and CP violation in the lepton sector are low-energy FCNC and CP violations are depleted by both present also in the limit of massless neutrinos. By assum- inverse powers of the compositeness scale and by the mix- ing that the set of adopted spurions are the only irreducible ing terms, thus realising an efficient suppression mechanism sources of flavour and CP violation, we can construct an known as RS-GIM [12]. Quantitative studies in concrete exhaustive list of Wilson coefficients related to dimension models show that in the anarchic scenario limits from CP six operators describing LFV and CP-violating processes. violation in the quark sector lead to NP > 10 TeV [8], At variance with the previous studies, we include for the first while the existing bound on the rate of μ → eγ results in time all Wilson coefficients containing up to four powers NP > 25 TeV [8]. This strongly disfavours the anarchic of the spurions describing the elementary–composite mixing scenario when the compositeness scale is of the order of 1 and we discuss their role in deriving the bounds on the com- TeV. It is also known that PC at the TeV scale can satisfy positeness scale. We also provide a complete list of the LFV the bound from flavour physics if Yukawa couplings in the Wilson coefficients that can be constructed in the limit of van- composite sector are non-generic. For instance if we assume ishing “wrong” Yukawa couplings. “Wrong” Yukawas in the that such couplings are universal and, at the same time, that composite sector are allowed by gauge symmetry, but they the only irreducible sources of flavour symmetry breaking do not contribute to SM lepton masses, at the leading order. are proportional to the SM Yukawa couplings, we reproduce They directly contribute to the dipole operators describing exactly the MFV scheme [13]. radiative decays of the charged leptons and setting such In this paper we focus on the lepton sector. Several stud- Yukawas to zero can relax the bounds on the compositeness ies of lepton flavour violation (LFV) and CP violation in scale. the lepton sector of composite Higgs models have been The general scope of our analysis is to check whether there realised. Most of the analysis have been performed in the con- are alternative solutions, beyond MFV, to reconcile PC at the text of five-dimensional (5D) models with a warped space- TeV scale with the existing bounds on LFV and CP violation. time metric, weakly coupled duals to strongly coupled four- We also wish to verify if the anarchic scenario is completely dimensional conformal theories, believed to provide a calcu- ruled out or not. By exploiting the effective Lagrangian of our lable framework for composite Higgs models. Explicit com- construction and the existing experimental bounds we esti- putations with anarchic Yukawa couplings have been carried mate the limits on the new physics scale in several scenarios, out in Refs. [14–20]. They get a lower bound on the masses of where our spurions are subjected to a series of increasingly the first Kaluza–Klein modes of order 10 TeV. These bounds restrictive conditions. We confirm that in general the anar- can be relaxed by requiring discrete [21–24] or continuous chic scenario is not compatible with new physics at the TeV [25–27] flavour symmetries. The original motivation for dis- scale and we provide examples of how PC can be realised at crete symmetries, tailored to approximately reproduce the the TeV scale, without necessarily resorting to MFV. tri-bimaximal mixing pattern in the lepton sector, has weaken In the second part of our paper we consider as an explicit after the precise measurement of the θ13 angle and the mod- model realisation of the flavour symmetry and its breaking els considered in [21–24] require considerable corrections pattern the so-called two-site model, first introduced in Ref. now. Continuous non-abelian symmetries in the composite [37]. Such a realisation contains explicitly vector-like lep- sector, broken by the elementary–composite mixing terms tons, implementing partial compositness in the lepton sector, have been analysed in detail, especially in relation to FCNC as well as a set of spin-one resonances. By integrating out and CP violation in the quark sector [28–32]. A review on the states at the compositness scale, we evaluate the Wilson flavour physics in 5D models with warped space-time metric coefficients of the relevant LFV and CP-violating dimension 123 Eur. Phys. J. C (2015) 75 :579 Page 3 of 26 579 six operators and we compare the results with those of the heavy fermions in the limit of unbroken electroweak sym- general spurion analysis. metry. Finally the heavy fermion sector can interact with the ( ∗, ∗) Our paper has the following plan. In Sect. 2 we define the Higgs doublet and this implies additional spurions YL YR . flavour symmetry and the set of spurions of our setup and Notice that this set of spurions is the most general one com- we characterise the Wilson coefficients of the dimension- patible with our flavour group G f and with the assumption six operators relevant to LFV and CP violation in the lepton that the SM Higgs doublet directly couples only to the heavy sector. In Sect. 3 we perform a phenomenological analysis sector. From these considerations the following transforma- and we study the bounds on the new physics scale obtained tion properties for our spurions can be deduced: by making different types of assumptions on the available → † ,  →  † , ∗ → ∗ † , spurions. In Sect. 4 we recall the main aspects of the two- m VL mV V V Y VL L Y V ˜ L L R L R R R E R † ˜ ˜ † ∗ ∗ † site model, which explicitly incorporates the features of the m˜ → V ˜ mV˜ ,  → V˜V , → , E ˜ e ˜ YL VL R YL V ˜ flavour symmetry breaking defined in Sect. 2. In Sect. 5 we L E R EL EL collect our results on the dimension six operators obtained (3) from the model by integrating out heavy fermions and heavy ( ) gauge vector bosons. In Sect. 6 we present a phenomenolog- where it is evident from our notation which SU 3 fac- ical analysis of LFV in the two-site model. Finally we draw tors of G f are involved in each transformation. An equiv- our conclusion. In Appendix B we show the result of our alent set of spurions is obtained by replacing the dimen- (, )˜ computation of the full one-loop contribution to the electro- sionful quantities with dimensionless combinations ( , ˜ ) = ( −1, ˜ ˜ −1†) magnetic dipole operator in the two-site model. X X m m transforming as † X → V XV , L L ˜ ˜ † 2 Effective field theory for lepton flavour violation X → Ve˜ XV˜ . (4) E R

As a first step, we have to choose the flavour symmetry group In such a case at the leading order (LO) SM charged lepton of our effective theory and its breaking terms. Throughout masses are generated by the operator this paper we will work in the limit of massless neutrinos. O(0) = ( ϕ) ∗ ˜ † ˜ , The leptons are those of the SM, that is, three copies of SU(2) M L XYR X eR (5)  ˜ doublets and singlets e. In MFV the flavour symmetry group where ϕ is the Higgs electroweak doublet. ( ) × ( ) ∗ ∗ of the leptonic sector is SU 3  SU 3 e˜, corresponding to It is well known that, if all the spurions (X, X˜ ), (Y , Y )  ˜ L R independent transformations made on and e. In our anal- and (m, m˜ ) are present at the same time and if the Yukawa ysis we will instead assume a PC scenario. Charged leptons ( ∗, ∗) couplings YL YR are anarchic, a severe bound on the new have no direct coupling to the Higgs doublet and acquire physics scale applies. Indeed in concrete models belonging to masses via mixing with vector-like heavy fermions. In this the general class we are considering, the one-loop exchange framework it is natural to assume as flavour symmetry group of Higgs and heavy fermions leads to the following electro- (focussing only on the non-Abelian part): magnetic dipole operator:

= ( )6 1 ∗ −1 ∗† −1 ∗ ˜ † G f SU 3 (L ϕ)(σ · F)XY m˜ Y m Y X e˜R. (6) 16π 2 R L R = SU(3) × SU(3)˜ × SU(3)L × SU(3)L e L R If such a contribution is present, and the Yukawa couplings × ( ) × ( ) , ∗ ∗ SU 3 E˜ SU 3 E˜ (1) ( , ) O( ) L R YL YR are assumed to be anarchic of 1 , then the elec- tromagnetic dipole operator and the mass operator are not under which the lepton fields rotate in generation space in aligned in flavour space and the heavy fermion scale is the following way: bounded to be heavier than about 30 TeV,to respect the bound on BR(μ → eγ). One way to eliminate this dangerous con- Li → (V)ijLj, e˜Ri → (Ve˜)ije˜Rj, (2) tribution, while maintaining non-vanishing lepton masses, is ( ) ( ) ∗ = where V and Ve˜ are elements of SU 3  and SU 3 e˜, respec- to assume YL 0. We will come back to this assumption tively. In other words, the SM leptons only transform under later in this section. For the moment we will adopt it as a the SU(3)×SU(3)e˜ component of the flavour group G f , and working hypothesis. Our purpose is to analyse the flavour- they are invariant under the remaining SU(3)4 factor. Such a violating contributions surviving in this limit and to estimate factor will be used to specify the spurions of our effective field the corresponding bounds on the new physics scale. theory. We introduce three sets of spurions. We need spurions At variance with MFV, spurions with the dimension of a (, )˜ that mix SM leptons with heavy fermions. Moreover, mass are present in our setup and some additional prescrip- we allow for spurions (m, m˜ ) describing the masses of the tions are needed: 123 579 Page 4 of 26 Eur. Phys. J. C (2015) 75 :579

I • First of all we require that our operators are local in the Wμν and Bμν are the field strengths for the gauge vector spurions (X, X˜ ). These are mixing parameters that are bosons of SU(2) andU(1), respectively. The flavour structure generically treated as small and provide one set of expan- of these two operators is the same and we will focus on the sion parameters for our spurion analysis. combinations • We also assume that the operators are local in the Yukawa ∗ ≤| ∗|≤ ( ) = θ ( ) − θ ( ) , couplings YR, which we restrict in the range 1 YR Qeγ ij cos W QeB ij sin W QeW3 ij π 4 . (Q ) = sin θ (Q ) + cos θ (Q ) , (10) • ∗ eZ ij W eB ij W eW3 ij Each power of YR occurs accompanied either by an Higgs electroweak doublet ϕ or by a factor 1/4π. where θW is the weak mixing angle and QeW3 denotes the • Masses of the composite sector are described by the spu- 1,2 contribution to QeW obtained setting to zero Wμν . The elec- rions (m, m˜ ) to which we add an additional parameter tromagnetic dipole operator Qeγ is the only operator that M, singlet under the flavour symmetry group, to describe gives a tree-level contribution to the radiative decays of masses of other composite particles, such as for instance charged leptons, when i = j. The diagonal elements, i = j, a new set of vector boson resonances. These additional contribute to the anomalous magnetic moments and to the states are coupled to the Higgs doublet and to the heavy EDM of the charged leptons. We have operators bilinear in fermions with a strong coupling constant g∗, in the range the Higgs doublet 1 ≤ g∗ ≤ 4π. ↔ ( (1)) = (ϕ† ϕ)(¯ γ μ ), • In our operators masses will always appear in negative Qϕl ij iDμ Li Lj ↔ powers, to allow for decoupling of all the operators with ( ) (Q 3 ) = (ϕ†iD I ϕ)(¯ τ I γ μ ), the scale of new physics. ϕl ij μ Li Lj ↔ † μ (Qϕe)ij = (ϕ iDμ ϕ)(e˜Riγ e˜Rj). (11) With this set of assumptions, the LO operator describing charged lepton masses is still OM in Eq. (5). We stress that After the breaking of the electroweak symmetry these oper- ∗ this operator provides the definition of the spurion YR.Any ators modify the couplings of the Z boson to leptons, poten- polynomial of the type tially violating both universality, through the diagonal terms,   and lepton flavour, through the non-diagonal ones. There is ∗ c ∗ ∗ c ∗ ∗ Y 1 + 1 (Y †Y ) + 2 (Y †Y )2 +··· (7) a unique operator trilinear in the Higgs doublet R 16π 2 R R (16π 2)2 R R † ¯ (Qeϕ)ij = (ϕ ϕ)(Lie˜Rjϕ), (12) ∗ O could replace YR in M . With no loss of generality we can ∗ which contributes, with a different weight, to both the masses redefine as YR the particular combination of Eq. (7) occurring in OM . and the Higgs couplings of the charged leptons. Finally, we The physical processes we are interested in concern lepton have the four-lepton operators ¯ ¯ μ flavour violation in the charged lepton sector as well as the (Qll)ijmn = (LiγμLj)(Lmγ Ln), magnetic dipole moments and the electric dipole moments μ (Qee)ijmn = (e˜Riγμe˜Rj)(e˜Rmγ e˜Rn), (EDM) of the charged leptons. To this purpose it is conve- ( ) = (¯ γ  )(˜ γ μ ˜ ), nient to adopt an effective field theory description where the Qle ijmn Li μ Lj eRm eRn (13) SM Lagrangian is extended by an appropriate set of gauge invariant operators depending on the SM fields [38]: which can contribute to muon and tau decays into three  charged leptons. We also consider dimension-six operators 1 μ → L = L + C Q +··· (8) of the type llqq that can contribute to e conversion in SM 2 i i 1 i nuclei: where we have restricted our attention to the lowest- (u) ¯ μ (Q ) = ( γμ )(u¯ γ u ) dimensional operators relevant to the processes we are inter- q ij Li Lj Lm Ln ( (d)) = (¯ γ  )( ¯ γ μ ) ested in, namely those of dimension six. Dots denote higher- Qq ij Li μ Lj dLm dLn ¯ μ ¯ ¯ μ dimensional operators. We list below a complete set of (Qu,d )ij = (LiγμLj)(u¯ Rmγ u Rn), (LiγμLj)(dRmγ dRn), dimension six operators depending on lepton fields and on the μ (Q ) = (e˜ γμe˜ )(q¯ γ q ) q = (u, d), scalar electroweak doublet ϕ [39]. We start with the dipole eq ij Ri Rj Lm Ln ( ) = (˜ γ ˜ )( ¯ γ μ ), (˜ γ ˜ )( ¯ γ μ ). operators Qeu,d ij eRi μeRj u Rm u Rn eRi μeRj dRm dRn (14) ¯ μν I I (QeW )ij = (Liσ e˜Rj)τ ϕWμν, ( ) ( ) 1 The operators (Q u ) and (Q d ) are linear combinations of the ¯ μν q ij q ij (QeB)ij = (Liσ e˜Rj)ϕBμν. (9) ( (1)) ( (3)) operators Qq ij and Qq ij of Ref. [39]. 123 Eur. Phys. J. C (2015) 75 :579 Page 5 of 26 579

There are other three independent operators of this type [39], Table 1 Spurion combination CLR¯ , in a matrix notation, for the lepton ¯ ( ) ˜ but the chosen subset is sufficiently general for the purposes bilinear Li CLR¯ ijeRj. NY and NX are the orders of the expansion in ∗ ( , ˜ ) ≤ ≤ of the present discussion. Notice that each operator carries YR and X X , respectively. We restrict the list to NY 3andNX 4. flavour indices. Hermiticity of the effective Lagrangian is For convenience we distinguish spurion combinations depending on composite fermion matrices c and c˜ (column HB) from combinations guaranteed either by appropriate symmetry properties of the not involving c and c˜ (column HF) Wilson coefficients under transposition of the family indices N N HF HB or by addition of the hermitian conjugate operator. Y X ∗ ˜ † ∗(˜† ˜)−1 ˜ † Our aim is to estimate the Wilson coefficients of these 12 XYR X XYR c c X operators, by expressing them in terms of the spurions using ( †)−1 ∗ ˜ † X cc YR X the set of rules described above. Weexpand each Wilson coef- 14 XY∗ X˜ † X˜ X˜ † X (cc†)−1Y ∗ X˜ † X˜ X˜ † ficient in powers of the mixings (X, X˜ ) and the Yukawa cou- R R XX† XY∗ X˜ † XY∗(c˜†c˜)−1 X˜ † X˜ X˜ † pling Y ∗. Since the spurions (X, X˜ ) control lepton masses, R R R ∗ ˜ † ˜ (˜† ˜)−1 ˜ † ( . / ∗) 2 XYR X X c c X they are expected to be small, of order 0 1 YR at most. X (cc†)−1 X † XY∗ X˜ † We will stop the expansion in (X, X˜ ) at the fourth order. In R ∗ † ( †)−1 ∗ ˜ † XX X cc YR X the expansion we will go up to the third order in YR, since † ∗(˜† ˜)−1 ˜ † in the anarchic scenario trilinear combinations of Yukawa XX XYR c c X ∗ ∗† ∗ ˜ † ( †)−1 ∗ ∗† ∗ ˜ † couplings are in general misaligned with respect to linear 32 XYRYR YR X X cc YRYR YR X ∗(˜† ˜)−1 ∗† ∗ ˜ † ones. Higher orders in Yukawa couplings do not bring any XYR c c YR YR X new qualitative feature in our analysis. Finally the correct ∗ ∗†( †)−1 ∗ ˜ † XYRYR cc YR X dimension is provided by negative powers of M or (m, m˜ ). ∗ ∗† ∗(˜† ˜)−1 ˜ † XYRYR YR c c X Since the operators under study have dimension six, in prac- ∗ ∗ ∗ − ∗ ∗ ∗ 34 XY X˜ † XY˜ †Y X˜ † X (cc†) 1Y X˜ † XY˜ †Y X˜ † tice we have two classes of operators, those suppressed by R R R R R R XY∗Y ∗† X † XY∗ X˜ † ... 1/M2 and those suppressed by the heavy fermion masses. R R R XY∗Y ∗†Y ∗ X˜ † X˜ X˜ † X (cc†)−1Y ∗Y ∗† X † XY∗ X˜ † Each of these two classes refers to a specific decoupling limit. R R R R R R † ∗ ∗† ∗ ˜ † When M |m|, |˜m|, the heavy bosons decouple first and XX XYRYR YR X ... the operators are suppressed by negative powers of (m, m˜ ). We call this the heavy boson (HB) case. In the opposite limit, M |m|, |˜m|, we have a fast heavy fermion decoupling and ing either at tree level or at one loop, and we will keep explicit the operators are suppressed by the smaller scale M. We call track of such a dependence. this the heavy fermion (HF) case. When the two scales M To facilitate the identification of the relevant Wilson coef- and (m, m˜ ) are comparable, the Wilson coefficient can be a ficients, it is useful to identify the combinations of spurions generic function of the ratio of the two scales. For the present that fit the lepton bilinears discussion the two limiting cases are sufficient to capture the ¯ ¯  (C ¯ ) e˜ ,  (C ¯ )  , e˜ (C ¯ ) e˜ . (16) behaviour of the system. For our spurion analysis it is con- Li LR ij Rj Li LL ij Lj Ri RR ij Rj venient to rewrite the mass matrices m, m˜ in this way In Tables 1, 2 and 3 we collect such combinations, con- structed with the rules outlined above. m = m0 c, m˜ = m0 c˜, (15) where m0 is a flavour-independent mass parameter, while the 2.1 Dipole operators flavour dependence is carried by the dimensionless matrices c, c˜. The dipole operators in Eq. (9) involve the lepton bilinear ¯ ( ) ˜ Concerning the dependence of the operators on the strong Li CLR¯ ijeRj and their possible Wilson coefficients are the ( ) coupling constant g∗, we observe that g∗ is flavour blind; CLR¯ ij combinations listed in Table 1. We expect that the it only affects the overall strength of the operators and can dipole operators are loop generated in perturbation theory. be absorbed in a redefinition of the NP scale. Therefore we The naive loop suppression factor 1/(16π 2) is not present in will simply set g∗ = 1 for the rest of this model-independent Table 1 and should be included in the Wilson coefficient, analysis. In the specific model discussed in Sect. 4, we will 1 (Ceγ,Z )ij = (C ¯ )ij. (17) focus on the LO contributions to the relevant operators, aris- 16π 2 LR The new physics scale  associated to dipole operators is 2 Actually, it is possible to elude such conclusion in particular sce- M in the HF case (M |m|, |˜m|) and m0 in the HB one narios, where one of the two fermion chiralities has a large degree of (M |m|, |˜m|). compositeness. In these cases, the perturbative expansion in either X or X˜ breaks down. In this paper, we will mainly focus on the anarchical The first important outcome of our analysis is that there scenario, assuming that both X and X˜ are small. is a large set of potentially lepton flavour-violating combi- 123 579 Page 6 of 26 Eur. Phys. J. C (2015) 75 :579

Table 2 Spurion combination CLL¯ , in a matrix notation, for the lepton directly read from Table 1 the list of possible Wilson coef- ¯ ( )  bilinear Li CLL¯ ij Lj. NY and NX are the orders of the expansion in ficients, provided we have at least three Yukawa couplings, ∗ ( , ˜ ) ≤ ≤ YR and X X , respectively. We restrict the list to NY 2andNX 4. NY ≥ 3, since this operator contains three Higgs doublets. For convenience we distinguish spurion combinations depending on Up to an overall flavour-independent coefficient, which is composite fermion matrices c and c˜ (column HB) from combinations not involving c and c˜ (column HF) expected to be of order one, we have

NY NX HF HB ( ) = ( ) ( ≥ ). Ceϕ ij CLR¯ ij NY 3 (18) 00l1 Notice that there is no loop suppression in this case, since 02 XX† X (cc†)−1 X † this operator can be generated at tree level. As for the dipole 04 XX† XX† X (cc†)−1 X † XX† operators, the scale of new physics  is M in the HF case XX† X (cc†)−1 X † and m0 in the HB one. ∗ ∗† † ( †)−1 ∗ ∗† † 22 XYRYR X X cc YRYR X This operator, together with the mass operator of Eq. (5), ∗(˜† ˜)−1 ∗† † XYR c c YR X contributes to both masses and Yukawa couplings of charged ∗ ∗†( †)−1 † XYRYR cc X leptons. If the Wilson coefficients are not exactly aligned in ∗ ˜ † ˜ ∗† † ( †)−1 ∗ ˜ † ˜ ∗† † 24 XYR X XYR X X cc YR X XYR X flavour space, we have flavour-violating decays of the Higgs. ∗ ∗† † † Even in the case where there is a perfect alignment among XYRYR X XX ... † ∗ ∗† † ( †)−1 ∗ ∗† † † all Wilson coefficients, the overall strength of Yukawa cou- XX XYRYR X X cc YRYR X XX ... plings is altered compared to the case of the SM, where it is completely fixed by the fermion mass and by the electroweak VEV. We will discuss the impact of these modifications in Table 3 Spurion combination CRR¯ , in a matrix notation, for the lepton Sect. 3. ˜ ( ) ˜ bilinear eRi CRR¯ ijeRj. NY and NX are the orders of the expansion in ∗ ( , ˜ ) ≤ ≤ YR and X X , respectively. We restrict the list to NY 2andNX 4. For convenience we distinguish spurion combinations depending on 2.3 Vector operators composite fermion matrices c and c˜ (column HB) from combinations not involving c and c˜ (column HF) ( (1,3)) ( ) In the operators Qϕl ij, Qϕe ij of Eq. (11) new flavour NY NX HF HB structures arise. The corresponding Wilson coefficients can (1,3) be read from Table 2,for(Q ) and Table 3,for(Qϕ ) , 0011 ϕl ij e ij − but excluding the case NY = 0, since the vector operators 02 X˜ X˜ † X˜ (c˜†c˜) 1 X˜ † are bilinear in the Higgs doublets and this requires at least ˜ ˜ † ˜ ˜ † ˜ (˜† ˜)−1 ˜ † ˜ ˜ † 04 X X X X X c c X X X two powers of the Yukawa couplings. We have X˜ X˜ † X˜ (c˜†c˜)−1 X˜ † , ˜ ∗† ∗ ˜ † ˜ (˜† ˜)−1 ∗† ∗ ˜ † ( 1 3) = ( ) ,( ) = ( ) ,( ≥ ). 22 XYR YR X X c c YR YR X Cϕl ij CL¯ L ij Cϕe ij CR¯ R ij NY 2 ˜ ∗†( †)−1 ∗ ˜ † XYR cc YR X (19) ˜ ∗† ∗(˜† ˜)−1 ˜ † XYR YR c c X 24 XY˜ ∗† X † XY∗ X˜ † X˜ (c˜†c˜)−1Y ∗† X † XY∗ X˜ † There is no loop suppression, in general, and the new physics R R R R scale is identified as for the previous operators. We see that, XY˜ ∗†Y ∗ X˜ † X˜ X˜ † ... R R in general, the Wilson coefficients for the vector operators X˜ X˜ † XY˜ ∗†Y ∗ X˜ † X˜ (c˜†c˜)−1Y ∗†Y ∗ X˜ † X˜ X˜ † R R R R are not diagonal in the mass basis. This leads to violation of ... flavour in the couplings of the Z boson, with consequences that we discuss in Sect. 3. nations, beyond that of Eq. (6). The actual appearance of 2.4 Contact operators these combinations in concrete models containing our set of spurions will depend on the specific dynamics of the model Finally we consider the contact operators (Qll)ijmn, under consideration. We will discuss the phenomenological (Qee)ijmn and (Qle)ijmn given in Eq. (13). In these operators implications of the new structures for the Wilson coefficients we recognise combinations of the flavour structures already ( ) Ceγ,Z ij in Sect. 3. discussed. We expect the following possible factorisations:  ( ) × ( ) 2.2 Scalar operator ( ) = CLL¯ ij CLL¯ mn , Cll ijmn ( ) × ( ) (20) CLL¯ in CLL¯ mj ¯  The scalar operator (Q ϕ) = (ϕ†ϕ)( e˜ ϕ) has exactly ( ) × ( ) e ij Li Rj ( ) = CR¯ R ij CR¯ R mn , Cee ijmn ( ) × ( ) (21) the same flavour structure of the dipole operators and we can CRR¯ in CRR¯ mj 123 Eur. Phys. J. C (2015) 75 :579 Page 7 of 26 579  ( ) × ( ) 2 m2 CL¯ L ij CR¯ R mn ∗ g W,Z −1 ∗ −1 (Cle)ijmn = † . (22) Y ≈ c Y c˜ , (C ¯ ) × (C ) L π 2 2 R LR in LR¯ mj 16 m0 2 2 ∗ g m , − ∗ ∗ ∗ − Since the contact terms contain no Higgs doublets, we have Y ≈ k W Z c 1Y Y †Y c˜ 1. (27) L 16π 2 m2 R R R no restrictions on NY . For the same reason, each bilinear 0 in the Yukawa coupling should be accompanied by a loop These contributions are naturally suppressed by the ratio /( π 2) 3 suppression factor 1 16 . Concerning the new physics m2 /m2 and can easily be kept at the percent level.  W,Z 0 scale , when the two factors C come from the HF column In the other regime, v  M  m , beyond the combina-  = 0 we have M. When one the factors comes from the HF tions of Eq. (27), we can also consider column and the other from the HB column, we have  = m0. 2 2 Similarly, for the llqq operators we have M − ∗ − M − ∗ ∗† ∗ − c 1Y c˜ 1, c 1Y Y Y c˜ 1, (28) m2 R m2 R R R (u,d) 0 0 (C ) = (C ¯ ) ,(C , ) = (C ¯ ) , q ij LL ij u d ij LL ij (23) ∗ ( ) = ( ) ,( ) = ( ) . which also transform as the spurion YL . These combinations Ceq ij CR¯ R ij Qeu,d ij CR¯ R ij (24) formally decouple in the limit of infinitely large m0. There are, however, also combinations that do not decouple, such with no restrictions on NY . as ∗ = −1 ∗ ˜†, −1 ∗ ∗† ∗ ˜†. 2.5 Stability of the solution YL 0 c YRc c YRYR YR c (29) ∗ We have checked, through the one-loop computation in the Since we are interested in the scenario where YL is nearly van- ishing, a legitimate question is whether and to which extent two-site model that we will consider in Sect. 4, that this kind of contributions are generated. They cannot be parametrically such a limit is stable under quantum corrections. To this pur- ∗ pose it is better to distinguish the two regimes v  m  M suppressed by mass ratios and the corresponding YL can be 0 , ∗ ≈ and v  M  m . When v  m  M, we can consider depleted only in the semi-perturbative regime g∗ YR 1. 0 0 ≈ the following spurion combinations: Notice that for m0 M, the effective Yukawa couplings arising from Eqs. (26) and (28), are insensitive to the over- m2 m2 all mass scale. In summary, in the regime of heavy gauge 0 † ∗ ˜†, 0 † ∗ ∗† ∗ ˜†, c YRc c YRYR YR c (25) ∗ M2 M2 bosons much heavier than heavy fermions, the effective YL ∗ can remain close to the percent level even in a strongly that behave as effective Yukawas of type YL since they have ∗ coupled regime, while in the opposite regime the solution ∗ the same transformation properties as YL . We expect that ≈ YL 0 is typically stable under quantum corrections (up to termslikethosein(25) arise, in perturbation theory, through the % level) only if we assume the semi-perturbative regime ∗ threshold corrections induced by the one-loop exchange of g∗, Y ≈ 1. heavy gauge bosons and heavy fermions, from which we R estimate 3 Model-independent bounds on the scale of new 2 2 2 2 g∗ m g∗ m Y ∗ ≈ 0 c†Y ∗c˜†, Y ∗ ≈ k 0 c†Y ∗Y ∗†Y ∗ c˜† , physics L 16π 2 M2 R L 16π 2 M2 R R R (26) Model-independent studies of LFV processes have already been done in the literature [53–55]. In this section we collect where g∗ is a coupling constant of the strong sector and k the bounds on the Wilson coefficients of the Lagrangian of / π 2 is either an additional loop factor 1 16 stemming from a Eq. (8) and discuss their impact on our spurion analysis. The Higgs loop, or a factor coming from the electroweak VEV, main bounds on the coefficients of the dipole operators Oeγ v2/M2, if the two Yukawas are attached to external Higgs and OeZ are given in Table 5, showing the results obtained legs. The contributions of Eq. (26) are small if there is an in Refs. [54,55] and derived from Ref. [56] in the case of hierarchy between heavy fermions and heavy gauge boson μ−Au → e−Au. They come from the present limits reported masses. They are also suppressed in the semi-perturbative in Table 4. , ∗ ≈ regime g∗ YR 1. Similarly, exchange of ordinary gauge ∗ The amplitudes for these processes get a tree-level contri- bosons and heavy fermions lead to effective YL couplings of bution from the off-diagonal elements of the electromagnetic the type ij ij dipole operator Oeγ , while OeZ contributes at one loop. Each bound has been derived by assuming a single non-vanishing 3 If contact terms originate from dimension eight operators, the loop Wilson coefficient at the time. This also applies to all the factor can be effectively replaced by v2/2. bounds discussed in this section. The bounds on the coef- 123 579 Page 8 of 26 Eur. Phys. J. C (2015) 75 :579

Table 4 Present and future experimental sensitivities for relevant low- we would need energy observables   2 μμ 1TeV − LFV process Present bound Future sensitivity (C ) = . × 5 Re eγ  1 2 10 (32) μ → eγ 5.7 × 10−13 [40] ≈6 × 10−14 [41] to account for the central value of the discrepancy. μ → 3e 1.0 × 10−12 [42] ≈10−16 [43] Also the scalar operator Oeϕ is mostly bounded by the μ− → − . × −13 Au e Au 7 0 10 [44]? limits on radiative lepton decays [54,55,59–61]. The scalar μ− → − . × −12 Ti e Ti 4 3 10 [45]? operator contributes to lepton masses and to higgs couplings − − −16 μ Al → e Al −≈10 [46,47] with a different weight: τ → eγ 3.3 × 10−8 [48] ∼10−8–10−9 [49] τ → μγ 4.4 × 10−8 [48] ∼10−8–10−9 [49] LY = Mij e¯Lie˜Rj + Yij h e¯Lie˜Rj + h.c., (33) τ → 3e 2.7 × 10−8 [50] ∼10−9–10−10 [49] − − − where τ → 3μ 2.1 × 10 8 [50] ∼10 9–10 10 [49]   v2 v Lepton EDM Present bound Future sensitivity SM Mij = −y + (Ceϕ)ij √ , ( ) . × −29 ij 22 de ecm 8 7 10 [51]?  2 ( ) . × −19 v2 dμ ecm 1 9 10 [52]? 1 SM 3 Yij = √ −y + (Ceϕ)ij , (34) 2 ij 22 ij 2 Table 5 The bounds on off-diagonal Wilson coefficients Ceγ / and SM and yij are the Standard Model Yukawa couplings. Radia- Cij /2 from Refs. [54,55]. The bounds from μ−Au → e−Au have eZ tive charged lepton decays constrain the off-diagonal ele- been derived from Ref. [56]. In the second column we list the upper ij ments of Yij in the basis where the mass matrix Mij is diag- bound on |C γ, | assuming  = 1 TeV, while in the third column we e Z onal. To convert these constraints in bounds on the Wilson fix |Cij |=1 and we list the corresponding lower bound on ,in eγ,Z ( ) ji coefficients Ceϕ ij it is convenient to work in the basis where TeV. The bounds on the coefficients Ceγ,Z are equal to the bounds on SM the SM couplings yij are diagonal and expand the unitary the coefficients Cij eγ,Z matrices that diagonalise M in powers of v2/2. We found |C| ( = 1TeV)  (TeV) (|C|=1) LFV process that, to first order in this parameter, the off-diagonal elements Y (i = j) in the lepton mass basis are given by: μe −10 4 ij Ceγ 2.5 × 10 6.3 × 10 μ → eγ μe − C γ 4.0 × 10 9 1.6 × 104 μ → 3e v2 e Y = √ ( ) ( = ). μe . × −9 . × 4 μ− → − ij Ceϕ ij i j (35) Ceγ 5 2 10 1 4 10 Au e Au 22 τe . × −6 . × 2 τ → γ Ceγ 2 4 10 6 5 10 e Y ( = ) τμ −6 2 By translating the bounds on ij i j given in Refs. [59– Ceγ 2.7 × 10 6.1 × 10 τ → μγ μ 61] into bounds on (Ceϕ)ij, we get the results shown in Table C e 1.4 × 10−7 2.7 × 103 μ → eγ [1-loop] eZ 6. The bounds on the coefficients (Ceϕ) ji are equal those on Cτe 1.3 × 10−3 28 τ → eγ [1-loop] eZ ( ϕ) τμ the coefficients Ce ij. These bounds are dominated by two- . × −3 τ → μγ 1-loop] CeZ 1 5 10 26 [ loop contributions of the corresponding operator to the radia- tive lepton decay, through Barr-Zee type diagrams, assuming that the top Yukawa coupling is as in the SM. One-loop con- ji ficients Ceγ,Z are equal to the bounds on the coefficients tributions to charged lepton radiative decays and tree-level ij contributions to  → 3 decays lead to less severe bounds Ceγ,Z . The diagonal elements of the dipole operators contribute than the ones given in Table 6. to electric and magnetic dipole moments of the charged lep- tons. From the present bounds reported in Table 4 we have Table 6 The bounds on off-diagonal Wilson coefficients (Ceϕ)ij from [54] | ij |   Refs. [59–61]. In the second column we list the upper bound on Ceϕ 2 ij 1TeV assuming  = 1 TeV, while in the third column we fix |C ϕ|=1and (Cee ) < . × −12, e Im eγ  3 9 10 we list the corresponding lower bound on , in TeV. The bounds on the ji ij   coefficients C ϕ are equal to the bounds on the coefficients C ϕ 2 e e μμ 1TeV − (C ) < . × 3. | |  =  | |= Im eγ  8 4 10 (30) C ( 1TeV) (TeV) ( C 1) LFV process μe −5 EXP SM Ceϕ 8.4 × 10 109 μ → eγ [2-loop] Given the current deviation aμ = aμ − aμ in the muon Cτe 0.33 1.7 τ → eγ [2-loop] μ = ( − )μ/ eϕ anomalous magnetic moment a g 2 2[57,58] τμ Ceϕ 0.37 1.6 τ → μγ [2-loop] −10 aμ = (29 ± 9) × 10 , (31) 123 Eur. Phys. J. C (2015) 75 :579 Page 9 of 26 579

( , ) ( 1 3 ) ( ) ( (1,3)) Coming to the vector operators Qϕl ij, Qϕe ij,they Table 7 The bounds on off-diagonal Wilson coefficients Cϕl ij and ij − − lead to lepton flavour-violating Z decays, but the correspond- Cϕe from Refs. [54,55]. The bounds from μ Au → e Au have been ing limits on the Wilson coefficients, assuming  = 1TeV, derived from Ref. [56]. In the second column we list the upper bound  = are of order 10 % [54]. Through one-loop diagrams they also on the Wilson coefficients assuming 1 TeV, while in the third column we set to unity the coefficients and we list the corresponding contribute to radiative decays of the charged leptons [54,55]. lower bound on ,inTeV It turns out that the most restrictive bounds come from the |C| ( = 1TeV)  (TeV) (|C|=1) LFV process processes μ−Au → e−Au and  → 3 whose branching ratios satisfy the experimental limits of Table 4. ( (1,3)) . × −5 μ → Cϕl μe 3 7 10 164 3e We collect the corresponding bounds in Table 7.Alsothe (1,3) − − − (C )μ 5.0 × 10 6 447 μ Au → e Au contact operators can contribute to μ−Au → e−Au,  → 3 ϕl e (1,3) − (C )τ . × 2 . τ → e and, through one-loop diagrams, to the radiative decays of ϕl e 1 5 10 8 3 3 ( (1,3)) . × −2 . τ → μ the charged leptons. The most significant bounds are given Cϕl τμ 1 2 10 9 0 3 μe −5 in Table 8. Cϕe 3.9 × 10 160 μ → 3e μe −6 − − We can translate the bounds collected in Tables 5, 6, 7 and Cϕe 5.0 × 10 447 μ Au → e Au 8 into limits on the masses M and m0 of our spurion analysis. τe . × −2 . τ → Cϕe 1 5 10 8 1 3e To do this we should determine or make some assumptions on τμ − Cϕ 1.3 × 10 2 8.8 τ → 3μ ˜ ˜ ∗ ∗ e the parameters c, c, X, X, YR and YL , through which we can express all the Wilson coefficients, as explained in Sect. 2. ijkl By exploiting the flavour symmetry of our setup, we see that Table 8 The bounds on coefficients Cll,ee,le from Ref. [54]. The bounds it is not restrictive to work in the basis where the mixing from μ−Au → e−Au have been derived from Ref. [56]. In the second matrices X and X˜ are diagonal, real and non-negative and column we list the upper bound on the Wilson coefficients assuming  = 1 TeV, while in the third column we set to unity the coefficients we will adopt this choice, unless otherwise stated. The LO and we list the corresponding lower bound on ,inTeV mass matrix of the charged fermions |C| ( = 1TeV) (TeV) (|C|=1)LFV process ∗ v = ˜ † √ , μ ml XYR X (36) C eee 2.3 × 10−5 207 μ → 3e 2 ll,ee eτee . × −3 . τ → Cll,ee 9 2 10 10 4 3e is diagonalised by a bi-unitary transformation μτμμ . × −3 . τ → μ Cll,ee 7 8 10 11 3 3 μ , μ † C eee ee e 3.3 × 10−5 174 μ → 3e (Lml R )ij = mi δij . (37) le μμμ , μμμ C e e 2.1 × 10−4 69 μ → eγ [1-loop] ∗ le Assuming that Y is anarchic, it is straightforward to find μττe,eττμ . × −5 μ → γ R Cle 1 2 10 289 e [1-loop] τ , τ that Ce ee eee 1.3 × 10−2 8.8 τ → 3e ⎛ ⎞ le μτμμ,μμμτ . × −2 . τ → μ 1 X / X X / X Cle 1 1 10 9 5 3 1 2 1 3 μ ⎝ ⎠ (u)e . × −6 μ− → − L ≈ X1/ X2 1 X2/ X3 , C q 2 0 10 707 Au e Au ( )eμ − − − X1/ X3 X2/ X3 1 C d 1.8 × 10 6 745 μ Au → e Au ⎛ ⎞ q ˜ ˜ ˜ ˜ eμ . × −7 . × 3 μ− → − 1 X1/X2 X1/X3 Ceq 9 2 10 1 0 10 Au e Au ⎝ ˜ ˜ ˜ ˜ ⎠ eμ . × −6 μ− → − R ≈ X1/X2 1 X2/X3 , (38) Cu,eu 2 0 10 707 Au e Au ˜ ˜ ˜ ˜ eμ . × −6 μ− → − X1/X3 X2/X3 1 Cd,ed 1 8 10 745 Au e Au where factors of order one have been omitted from the matrix elements. Similarly the lepton masses are approximately ˜ ∗ ∗ given by If c, c, YR and YL are all generic matrices, the dipole operator, Eq. (6), leads to the well-known limit v ≈ ( ∗) ˜ √ . mi Xi YR ii Xi (39) c 2 m > 33 TeV, (41) 0 Y  The most favorable scenario to minimise FCNC effects is ˜ where Y  denotes a suitable average of the Y ∗ couplings realised when Xi = Xi , which we assume, for the time being. L,R  We have and similarly for c . Such a bound can be evaded only if spe- ˜ ∗ ∗  cial features of the flavour parameters c, c˜, X, X, Y and Y √ 1/2 R L ˜ 2mi are adopted. One possibility is to recover the framework of Xi = Xi = ∗ , (40) v(Y )ii minimal flavour violation (MFV), as proposed in Ref. [13]. R ˜ ˜ ∗ ∗ This can be done by assuming c, c, X, X, YR and YL all which will be used in our estimates. proportional to the identity matrix, except either X (right- 123 579 Page 10 of 26 Eur. Phys. J. C (2015) 75 :579 handed compositeness) or X˜ (left-handed compositeness). In At variance with MFV,in IFV the sources of flavour break- this case, when neutrino masses are neglected, the only spu- ing are both X and X˜ . In the basis defining IFV the mixing rion that breaks the flavour symmetry becomes proportional matrices X and X˜ are not diagonal, in general. They are to the charged lepton Yukawa couplings, exactly as in MFV. generic complex matrices. At the LO, the lepton mass matrix A choice of basis where such a spurion is diagonal is always is proportional to the product X X˜ †. When we move to the possible and LFV is only present when neutrino masses are charged lepton mass basis, in general, only the product X X˜ † turned on. In the latter case one can reconcile LFV with a becomes diagonal, not X and X˜ individually and they both scale of new physics close to the TeV scale, however, we can lead to LFV. think it is interesting to explore other options allowing for a By exploiting the freedom related to our flavour symmetry, TeV scale . we can provide an alternative, but equivalent, description of ∗ = ˜ We start by taking YL 0. To guarantee the stability the IFV scenario. We can choose a basis where X and X are of this condition we should also assume either a perturba- diagonal, real and non-negative. In this case by means of the | ∗|≤  tive regime, where YR 1, or the hierarchy M m0, symmetry transformations of Eqs. (3) and (4) we can still as explained in Sect. 2.5. In this case the dipole operator is maintain c and c˜ proportional to the identity, but the matrix ( , ) = ( , ) ∗ dominated by the contribution NY NX 1 2 (see Table YR becomes a generic unitary matrix. In such a basis the ∗ ˜ † 1) lepton mass matrix XYR X is non-diagonal and LFV is now ⎧ ascribed to the interplay between Y ∗ and X, X˜ . ∗ − ˜ R ⎪ XY (c˜†c˜) 1 X † ˜ ⎨⎪ R Within IFV the special case X = X, which was previously C γ 2 e m0 assumed to minimise FCNC effects, forbids any LFV effect. = − ∗ (42) 2 ⎪ X(cc†) 1Y X˜ † = ˜ ⎩⎪ R Indeed, if X X, there exist a basis where both X and the 2 ∗ ˜ † m0 mass matrix XYR X are diagonal at the same time. As long as neutrino masses are neglected there is no source of LFV and and the bound on μ → eγ leads to m0 c > 33 TeV. As all the Wilson coefficients are diagonal in flavour space. The ∗ = a next step we consider the case of YL 0 and univer- only difference with respect to MFV is that the spurion X is sal heavy fermion masses, namely c and c˜ proportional to not proportional to the charged lepton Yukawa couplings, but the unit matrix. With this assumption the Wilson coefficients to their square root. Therefore in discussing IFV we consider (NY , NX ) = (1, 2) of the dipole operator are diagonal in the the general case where X and X˜ are not equal. mass basis, to LO. This, however, is not the case, in general, In IFV no LFV is generated from the Wilson coefficients ( , ) = ( , ) for the coefficients NY NX 3 2 , which are compara- of the dipole and scalar operators with NX = 2, to LO. These ble in size to those of Eq. (6). Therefore, even working in the coefficients are of the type XAX˜ †, where A is any combi- ∗ ∗ limit of vanishing “wrong” Yukawa coupling YL , and univer- nation of c, c˜ and Y , and they are automatically aligned ∗ R sal heavy fermion masses, we expect that with anarchic YR to the charged lepton mass matrix, to LO. To estimate the the bound of Eq. (41) still applies. To allow for a lower new other Wilson coefficients we should specify the choice of X physics scale , we are lead to postulate that also the Yukawa and X˜ . Rather than scanning for the most general possibility, ∗ couplings YR are universal. More precisely we define a new here we provide an example. We assume that, in the basis scenario, which we call intermediate flavour violation (IFV), where X and X˜ are diagonal, their elements are given, up to by the following assumption: coefficients of order one, by IFV scenario: ∗ 1 1 The Yukawa couplings YL are negligible and a choice X = √ diag(λ4,λ3,λ3), X˜ = √ diag(λ5,λ2, 1), ˜ ∗ of basis exists where c, c and YR are simultaneously y y proportional to the identity matrix. (44) From the discussion in Sect. 2.5, we know that a non- ∗ ∗ λ ≈ . vanishing YL can be generated. Since such a YL will only where 0 22 is a small parameter of the order of the ˜ ∗ depend on c, c and YR, it will be universal too. Admitting Cabibbo angle. The choice of elements of nearly the same ∗ X a universal YL would not change our conclusions and, for order of magnitude for is motivated by a possible role ∗ = X simplicity, we keep the assumption YL 0. Moreover, from that the matrix can play in describing large lepton mixing now on we set to unity the matrices c and c˜, absorbing their angles, once neutrino masses are turned on [5]. The value λ3 ÷ λ4 effect in the overall scale m0 of the heavy fermion masses. is chosen here for convenience, to adequately sup- ∗ X˜ The remaining coupling YR is described by a single parameter press LFV. The choice for the elements of is fixed, at the y: level of orders of magnitude, by the relation (39). The trans- formations needed to diagonalise the lepton mass matrix are ∗ = l. YR y (43) given in Eq. (38). 123 Eur. Phys. J. C (2015) 75 :579 Page 11 of 26 579 √ Now we can complete our discussion concerning the Wil- corresponds to a bound >(0.6/ y) TeV. Similarly, from son coefficients that are bilinear in the spurions X, X˜ . Coef- ( , ) = ( , ) (Ceγ )τμ,μτ − − ficients NX NY 2 2 of the vector operators are of the < 2.7 × 10 6 TeV 2 (54) † ˜ ˜ ˜ † ˜ ∗ 2 type XAX or X AX where A depends on c, c and YR.From ∗ √ Tables 2 and 3, we see that YR enters A only in the combina- we get >(1.1/ y) TeV. We have checked that all the ∗ ∗† ∗† ∗ tions YRYR or YR YR. Therefore, up to an overall coefficient other limits listed in Tables 5, 6, 7 and 8 do not lead to more ˜ of order one, in the basis where X and X are diagonal and restrictive bounds on . For instance, the bound on the con- real we have tact operators, (1,3) 2 2 2 ˜ 2 ττμ (C ) = y X δ ,(Cϕ ) = y X δ . (45) e ϕl ij i ij e ij i ij C − − le < 1.2 × 10 5 TeV 2, (55) When we move to the charged lepton mass basis, we get 2 ( (1,3)) ≈ 2 ,( ) ≈ 2 ˜ ˜ . produces the bound >(0.3/y) TeV. The bounds from the Cϕl ij y Xi X j Cϕe ij y Xi X j (46) √ vector operators scale as y, while those√ from the dipole The most stringent limits of Table 7 arise from μ−Au → or contact operators scale as 1/y or 1/ y. Therefore a new e−Au and require physics scale  around few TeV is still acceptable provided

˜ ˜ 7 the Yukawa coupling y is close to one. Xμ X Xμ X λ − − y2 e ≈ y2 e ≈ y < 5.0 × 10 6 TeV 2, (47) A new physics scale  ≈ 1 TeV is too large to allow, 2 2 2 √ in the present framework, for an explanation of the central which translate, respectively, into > y 2.2TeV. value of the aμ anomaly. Indeed, in the lepton mass basis Here and in the rest of this section  stands for either M we have or m0. The limit on the decay τ → 3μ gives rise to a similar √ μμ 1 2mμ − bound: Re(C ) ≈ = 3.8 × 10 6 (56) eγ π 2 v ˜ ˜ 2 16 Xτ Xμ λ − − y2 ≈ y < 1.3 × 10 2 TeV 2, (48) 2 2 and to match the required value, Eq. (32), we need  = √ 0.56 TeV. Since the contribution to aμ scales with the resulting in > y 1.9 TeV. Coefficients of the type inverse square of , choosing  = 1 TeV gives aμ = (N , N ) = (2, 0) can arise for the contact operators of − X Y 9 × 10 10, less than one third of the central value of the cur- the type llqq and we expect, in the mass basis: rent anomaly. Concerning the electron EDM, we assume as ( (u,d)) ,( ) ≈ ,( ) ,( ) ≈ ˜ ˜ . in MFV that the sources of CP violation and LFV are the Cq ij Cu,d ij Xi X j Ceq ij Ceu,d ij Xi X j (49) same. Since the Wilson coefficients of the dipole operators with NX = 2 are aligned in flavour space with the mass oper- − − The limits of Table 8 from μ Au → e Au require ator, we identify in (51) the dominant coefficients that can ˜ ˜ 7 contain non-trivial phases. We estimate Xμ X Xμ X λ − − e ≈ e ≈ < 1.0 × 10 6 TeV 2, (50) √ 2 2 y2 ( ee ) ≈ 1 2me , √ Im Ceγ Xe Xe (57) and we get >5.0/ y TeV, respectively. 16π 2 v √ We are left with the coefficients that are quadrilinear in >(. / ) ˜ which, for 0 16 y TeV, respects the bound of X, X. In the IFV scenario the off-diagonal elements of the Eq. (30). ( ) dipole operator Qeγ ij are dominated by the coefficients In summary, even in the limit of “wrong” Yukawa cou- pling negligibly small, an anarchic Y ∗ requires a scale of 1 ∗ ˜ † ˜ ˜ † R (Ceγ )ij = (XY X X X )ij, 16π 2 R new physics well above 10 TeV.One way to lower this bound consists in mimicking the case of MFV, where there is a sin- 1 † ∗ ˜ † (Ceγ )ij = (XX XY X )ij, (51) ˜ 16π 2 R gle non-universal spurion, either X or X. In this case, when neutrino masses are neglected, LFV is absent. By discussing which, in the lepton mass basis are approximately given by a large class of possible LFV effects, we have shown that √ √ MFV is not the only possibility to reconcile LFV bounds 1 2mi ˜ ˜ 1 2m j (Ceγ )ij ≈ Xi X j ,(Ceγ )ij ≈ Xi X j . with a new physics scale around few TeV. In the example 16π2 v 16π2 v ˜ (52) analyzed here, both X and X are non-universal and represent potential sources of LFV, which is allowed also in the limit The limit of vanishing neutrino masses. The corresponding bounds on  approach the TeV scale, provided the Yukawa couplings (Ceγ )eμ,μe − − < 2.5 × 10 10 TeV 2 (53) ∗ 2 YR are close to one. 123 579 Page 12 of 26 Eur. Phys. J. C (2015) 75 :579

Table 9 Gauge subgroups and their associated generators, boson fields Table 10 Particle content and quantum numbers of the two-site min- and couplings. The normalisation of the Bcomp and B˜ comp generators imal model. The index i = 1, 2, 3 runs over three families for each has been chosen to match the SO(10) GUT normalisation of the hyper- representation. Lower case letters denote elementary fields, capital let- 3 ters denote composite fields. The ‘tilde’ apex denotes SU(2)L singlets, charge, YGUT = Y 5 in order to distinguish them from the doublets Subgroup Generator(s) Field(s) Coupling Elementary Composite ( ) aL, el, = , , a, el el el SU 2 L T a 1 2 3 Wμ g1 SU(2)L U(1)Y SU(2)L SU(2)R U(1)X U(1) YBel gel Y μ 2  − 1 , a, comp comp Li 2 2 110 ( ) T aL comp, a = , , Wμ g comp SU 2 L 1 2 3 1 ˜ −     eRi 1 1 110 3 , 2 comp comp  SU(2) × U(1) T 3R comp + X Bμ g R X 5 3 2 − 1 · 3 Li 1 0 21 2 2 , , ˜ 1,2, comp comp  T 1R comp, T 2R comp Wμ g   2 ˜ 3   Ei 1 0 11−1 · 3 , 2 ˜ comp comp 2 T 3R comp − X Bμ g 5 3 2 (ϕ,ϕ)˜ 1 0 220

4 Two-site model We are ready now to introduce the Lagrangian of the two- site model: In the previous sections, we focussed on PC scenarios for L = L + L + L , charged leptons from a general perspective, exploiting a el comp mix (59) spurionic analysis. Now, instead, we consider a specific with simplified composite Higgs model, the so-called two-site model [37]. Its relevant features are: 3 1 a 2 ¯ ¯ Lel =− (Fμν) + (LiiD/ Li + e˜RiiD/ e˜Ri), (60) 4 = (i) The gauge group is Ggauge = Gel × Gcomp where i 1

Gel =[SU(2) × U(1) ]el, L Y L =−1(ρa )2 +| ϕ|2 − (ϕ) comp comp comp μν Dμ V G =[SU(2)L × SU(2)R × U(1)X ] . (58) 4 3  ¯ + (L¯ (iD/ − m )L + E˜ (iD/ −˜m )E˜ ) In Table 9 is summarised our notation for generators, i i i i i i i=1 boson fields and coupling constants associated with 3 each simple subgroup. The group for Gcomp has been ∗ ¯ ˜ ∗ ¯ ˜ − (Y L RiϕELj + Y L LiϕE Rj) + h.c., chosen in order to provide a custodial symmetry. PC L ij Rij i, j=1 is assumed to arise from an unknown dynamical spon- (61) taneous symmetry breaking mechanism, which takes el × comp place at higher energies and which breaks G G   2 el 2 el 2 into the diagonal group (which can be recognised as the M∗ a 2 2 g μ ∗ M∗ g L = (ρμ) − M∗ A ρμ + Aμ electroweak gauge group). This spontaneous symmetry mix 2 gcomp 2 gcomp breaking mechanism, which in turns triggers the PC sce- 3   ¯ ˜ ¯ ˜ nario, will be effectively described by linear couplings − ijLi L Rj + ije˜Ri ELj + h.c., (62) between elementary and composite bosons. i, j=1 (ii) The fermionic sector of the model includes three fam- ilies of chiral fermions charged under Gel and three where lower (upper) case letters denote elementary (com- comp families of vector-like fermions charged under G . posite) fields and the ‘tilde’ denotes SU(2)L singlets. More- PC for fermions is realised through linear mass-mixing over, among the heavy vector bosons, we distinguish between ∗ ∗ ∗ terms between elementary and composite fermions. those mixing with the SM gauge bosons, ρμ ={Wμ, Bμ}, (ϕ,ϕ)˜ ˜ ˜ (iii) The Higgs sector consists of a real bidoublet and those that do not, ρ˜μ ={Wμ, Bμ}. [ ( ) × ( ) ]comp a charged under SU 2 L SU 2 R , which will be In Lel we can recognise a SM-like Lagrangian (with Fμν identified with the composite Higgs field, interacting collectively denoting the field strength tensors for the ele- only with the composite fermions. mentary gauge bosons) without the Higgs doublet, which Table 10 summarises the quantum numbers for fermions is now a composite particle. Lcomp contains mass terms and Higgs doublet. for composite leptons, kinetic terms for composite leptons 123 Eur. Phys. J. C (2015) 75 :579 Page 13 of 26 579 and bosons (we have collectively denoted the field strength Using the rotations (63) and (65) one can recast the a tensors with ρμν), their interactions, the Higgs sector and Lagrangian (59) in terms of lepton mass eigenstates. Using an Yukawa interactions. We have allowed the Yukawa couplings approximate expression for the rotation matrices VL,R, UL,R, to explicitly break the SU(2)R symmetry, which applies to assuming universal masses for the heavy leptons, (mi = the bosonic part of the composite sector and protects the m, m˜ i =˜m), and retaining the leading terms relevant for model against large contributions to the ρ parameter. Such FC neutral-current (FCNC), one gets more general Yukawa interactions are technically acceptable, since the cutoff-dependent, loop-induced contributions to the h ∗ ˜ ∗ ˜ † ¯ LFCNC = √ ((XY )ije¯Li E Rj + (Y X )ijELie˜Rj) + h.c. ρ parameter from heavy fermion exchange are negligible, at R R 2  least in the perturbative regime, as explicitly shown in [37]. v2 v2 h ∗ ∗† † SM SM ˜ ∗† L × √ XY Y X y + y XY Finally, mix contains linear mass-mixing terms among ele- 2m˜ 2 R R 2m2 R 2  mentary and composite particles. We have also included in v2 L a mass terms for composite vector bosons. L explic- ×Y ∗ X˜ † − XY∗Y ∗†Y ∗ X˜ † e¯ e˜ + mix mix R ˜ R L R Li Rj h.c. itly breaks Gel × Gcomp down to the diagonal subgroup, and mm ij  v it effectively reproduces the mechanism of PC. 1 g ∗ μ ˜ + √ Zμ (XY ) e¯ γ E ˜ R ij Li Lj From the above Lagrangian we can easily recognise the 2 2 cW m  v ∗ μ flavour symmetry group of Eq. (1) and convince ourselves − (Y X˜ †) E¯ γ e˜ + h.c. ∗ ∗ ˜  ˜ R ij Ri Rj that promoting YL , YR, m, m, and to spurions with the m  v2 transformation properties of Eq. (3) we actually restore G f . 1 g ∗ ∗† † μ + Zμ (XY Y X )ije¯Liγ eLj The Lagrangians of Eqs. (60)–(62) are expressed in the 4 c m˜ 2 R R W  elementary/composite basis. For the bosons, in order to v2 ˜ ∗† ∗ ˜ † ¯ μ switch to the mass basis (before EWSB), we diagonalise the − (XY Y X )ije˜Riγ e˜Rj m2 R R mass mixings in Lmix by the following field transformations: g∗ ∗ ˜ ∗, 3 † μ − (B − Bμ + W )(XX ) e¯ γ e      μ μ ij Li Lj el 2 Aμ cos θ − sin θ Aμ g ∗ ˜ ˜ ˜ † ¯ μ ∗ → ∗ tan θ = , + g∗(Bμ − Bμ)(X X )ije˜Riγ e˜Rj ρμ sin θ cos θ ρμ gcomp g∗ ∗ ˜ ∗, 3 μ ρ˜ →˜ρ , + (Bμ − Bμ + Wμ ) Xije¯Liγ ELj + h.c. μ μ (63) 2 ∗ ˜ ˜ ¯ μ ˜ + g∗(B − Bμ) X e˜ γ E + h.c. (66) ∗ μ ij Ri Rj At this stage the fields Aμ are massless, while ρμ and ρ˜μ ∗ have masses of order M . Taking into account EWSB, the From Lagrangian (66) one can easily read some of the diagonalisation of the mass terms becomes rather involved. most relevant features of FCNC in this class of models: To a good approximation, Eq. (63) still diagonalises the boson fields, while the mass matrices for the fermions read • The first line of Eq. (66) accounts for FCNC interactions among light and heavy leptons and the Higgs. They con- e˜ E E˜ ⎛ R R R ⎞ tribute to low-energy processes such as i →  j γ at loop NR 0  0 eL   level. v ∗ ,  ν . ME =⎝ 0 m √ Y ⎠ E L (64) 2 R L MN = • The second line of Eq. (66) refers to FCNC interac- v mN ˜ † √ Y ∗† m˜ E˜ L tions among light leptons and the Higgs. As we know 2 L L from previous sections, FCNC effects can arise in this With this notation, one can then perform a suitable rotation case through the dimension six operator (Qeϕ)ij = † ¯ that transforms to the mass basis: (ϕ ϕ)(Lie˜Rjϕ) and this explains the appearance of the factors v2/m˜ 2, v2/m2 and v2/mm˜ after EW symme- ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ try breaking. Such Yukawa interactions generate FCNC eL eL e˜R e˜R ⎝ ⎠ → † ⎝ ⎠ , ⎝ ⎠ → † ⎝ ⎠ , Higgs decays like ϕ → τμ, tree-level contributions to EL VL EL E R VR E R ˜ ˜ ˜ ˜ processes sensitive to four fermion operators such as EL EL E R E R     μ → 3e and μN → eN and loop induced effects on ν ν L → † L , → ,  →  γ . UL NR UR NR i j NL NL • ThethirdlineofEq.(66) describes the interactions among (65) light and heavy leptons and the Z boson. Since the oper- ˜ ¯ ators e¯LiZ/ ELj and E RiZ/e˜Rj are not SU(2)L invariant, , † where VL,R UL,R are defined in order to have VL ME VR and they can be generated only after the EW symmetry break- † ( ) UL MN UR diagonal. ing through the SU 2 L invariant dimension five opera- 123 579 Page 14 of 26 Eur. Phys. J. C (2015) 75 :579

¯ ˜ ¯ tors LiZ/ ELjϕ and L RiZ/e˜Rjϕ. This explains the factors the spurions as dynamical fields, this procedure defines a of v/m˜ and v/m. These interactions will contribute to (non-unique) embedding of the effective theory in a possible i →  j γ at loop level. UV completion. The spurions include two types of degrees • The fourth line of Eq. (66) contains FCNC interactions of freedom: the would-be Goldstone bosons, eaten by the among light leptons and the Z bosons. As discussed in heavy gauge vector bosons of the composite sector through previous sections, such effects can arise by means of the Higgs mechanism, and physical scalars i . By working dimension six operators bilinear in the Higgs doublet like in the unitary gauge, which we adopt in our computations, a ↔ † ¯ μ generic amplitude comprises two separate contributions: one (Qϕe)ij = (ϕ iDμ ϕ)(e˜Riγ e˜Rj). As a result, after EW coming only from the exchange of the physical polarisations symmetry breaking, we generate the operators e¯LiZe/ Lj and e˜¯ Z/e˜ , which are suppressed by v2/m2 and v2/m˜ 2 of the gauge vector bosons and one including the exchange Ri Rj  factors, respectively. The leading effects induced by these of some physical scalar degrees of freedom i . Clearly our computation retains only the first one, while the second is terms are the tree-level FCNC decay modes Z → i  j as well as  → 3 and μN → eN. missing and makes our results sensitive to the details of the i j  • ThefifthlineofEq.(66) describes SU(2) invariant inter- UV completion. If the masses of the extra scalars i are close L  actions between heavy gauge bosons and light-fermions. the cut-off scale UV, we expect that the contributions we Also in this case we can induce FCNC processes at tree are neglecting in our estimates of the Wilson coefficients are generically of order (M/ )2 and/or (m, m˜ / )2. level such as i → 3 j and μN → eN and loop induced UV UV The embedding of the model in a UV completion also effects on i →  j γ . • The sixth line of Eq. (66) refers to FC interactions among shows that it is not consistent to deal with the HF case, heavy gauge bosons, heavy and light leptons. Effects on within the effective theory. Indeed to keep the masses of low-energy observables are induced by the loop exchange the elementary leptons non-vanishing in the HF limit, we (, )˜ of heavy gauge bosons and leptons. If the heavy leptons have to consider at the same time large , such that ( , ˜ ) ,˜ of different generations were degenerate, FCNC effects the ratios X X remain constant. In a UV completion would vanish according to the GIM-mechanism. How- are proportional to VEVs that break the gauge symmetry el × comp ever, the GIM cancellation is broken by non-universal G G and contribute to the masses of the heavy gauge mass splittings of order X † X and X˜ † X˜ . The latter point vector bosons of the composite sector. Therefore, barring , ˜  has been overlooked in the literature so far. tuning of the parameters, we cannot make m m M.

Concerning the flavour structure of the various interaction 5 Lepton flavour violation in the two-site model terms, we remember that each Higgs electroweak doublet ϕ or its vacuum expectation value v occur accompanied by a In this section we will present our results for the Wilson coef- ∗ ∗ ˜ power of YR or YL , while the lepton fields eL and eR come ficients of the various LFV operators considered in Sect. 2, with X and X˜ , respectively. in the context of the two-site model introduced above. We Before discussing LFV in this model we comment on the work at the leading order in the loop expansion and we pay issues of renormalizability, gauge-invariance and UV sensi- particular attention to the spurionic structure of the coeffi- tivity. The two-site model is a non-renormalizable effective cients. Throughout this section we assume universal masses theory and therefore it is valid only up to energies of the order for the heavy leptons: mi = m and m˜ i =˜m. of an UV cut-off UV. Such a cut-off has been estimated in ∗ Ref. [37], where it has been found that UV ≈ 8π M /g∗, 5.1 Dipole operators by analysing self-interactions in the composite sector. The non-renormalizability is a consequence of the explicit break- Dipole operators are among the most important operators in ing of the gauge symmetry Gel × Gcomp, by the heavy vector the leptonic sector as they generate LFV radiative decays ∗ ∗ μ → γ ( − ) boson masses, by the Yukawa couplings YL and YR in the such as e , leptonic EDMs and g 2 s. We will con- composite sector and by the mixing between elementary and sider the one-loop contribution to dipole operators in the two- composite fermions  and ˜ . site model, assuming the validity of the perturbative expan- This explicit breaking raises the question of the reliabil- sion. The result of our computation of the full one-loop con- ity and consistency of our results. We can always promote tribution to the electromagnetic dipole operator is shown in our effective Lagrangian to a gauge-invariant theory by inter- Appendix B. Given the fact that the composite Higgs scenario preting the sources of gauge symmetry breaking as spurions. is naturally characterised by a strongly interacting regime, the Providing the spurions with suitable transformation proper- use of perturbation theory can be questioned. Here we will ties under the gauge group we can recover the gauge invari- restrict ourselves to the portion of the parameter space that is ance under the full local group Gel × Gcomp. Once we treat compatible with perturbation theory. This allows us to con- 123 Eur. Phys. J. C (2015) 75 :579 Page 15 of 26 579

± h, ϕZ,ϕ B∗, B,W˜ ∗ eL e˜R eL e˜R

† XY∗ Y ∗ Y ∗ X˜ † ∗ ˜ † R L R XYR,L X γ γ

Fig. 1 Feynman diagrams for LO contributions to i →  j γ with loop exchange of SM bosons h, Z, W and heavy fermions (left) and heavy gauge bosons and heavy fermions (right) sider moderately large coupling constants of the composite g − 2. However, we point out here that the second term of ( ) sector, with loop factors still providing a sizeable suppres- Ceγ B˜ +B∗ , not discussed in the literature to our knowledge, sion. The same considerations apply to processes receiving is not aligned with the mass operator and therefore generates non-vanishing contributions already at the tree level. Within flavour and CP-violating effects. the two-site model, dipole amplitudes receive leading con- From Eq. (67) one can easily find that, in the anar- tributions from the virtual exchange of (i) light SM bosons √chic scenario, the bound from BR(μ → eγ) imposes that (h, Z, W) and heavy fermions and (ii) heavy gauge bosons mm˜ / Y   10 TeV. In order to relax such a strong bound ∗ and heavy fermions. Examples of diagrams for the above while keeping YR anarchic, we analyze here in great detail ∗ = classes of contributions are shown in Fig. 1. the solution with YL 0, as already done in the model- In particular, summing over the h, Z and W amplitudes, independent analysis of Sect. 2. ∗ = we find that the dominant effects for class (i) are given by Setting YL 0, the NLO effects stemming from class (i) are given by: (Ceγ )h+Z+W e 1 ∗ ∗ ∗ = XY Y †Y X˜ †, (67)  2 π 2 ˜ R L R h 64 mm (C γ ) e f (x) e h =− 1 XY∗Y ∗† X † XY∗ X˜ † 2 π 2 ˜ 2 R R R in agreement with the result of Refs. [35,62]. As we can 256 m see, the amplitude of Eq. (67) has precisely the same spuri- f h(x−1) v2 + 1 XY∗ X˜ † XY˜ ∗†Y ∗ X˜ † + onic structure as the one of Eq. (6). In particular, according m2 R R R (m2 −˜m2)2  to our spurionic classification of Sect. 2, it turns out that ∗ ∗† ∗ ∗† ∗ ˜ † , (Ceγ )h+Z+W is of order (NY , NX ) = (3, 2), where NY and XYRYR YRYR YR X (70) ( ∗, ∗) ( , ˜ ) NX are the orders of the expansion in YR YL and X X ,  (C γ ) e f Z (x) respectively. Interestingly, we observe that contributions of e Z =− 1 XY∗Y ∗† X † XY∗ X˜ † ( , ) = ( , ) ∗ 2 256π 2 m˜ 2 R R R order NY NX 3 2 containing only the Yukawa YR, which would be allowed by the flavour symmetries of our f Z (x) + 2 XY∗ X˜ † XY˜ ∗†Y ∗ X˜ † model, are absent, to one-loop order. m2 R R R The leading effects for class (ii) start from the order v2 ∗ ∗† ∗ ∗† ∗ ˜ † (N , N ) = (1, 2) and read + XY Y Y Y Y X Y X (m2 −˜m2)2 R R R R R  2 2 ∗ ∗ ∗ M (Ceγ ) ˜ ∗ e g + † ˜ † Z , B+B = ∗ [ ∗ ˜ † B ( , ) + ∗ ˜ † B ( , )], 8 XYRYR YR X (71) XYR X f1 y z XYL X f2 y z m2m˜ 2 2 256π 2 M2  (68) ( ) Ceγ W e 20 ∗ ∗† † ∗ ˜ † 2 = XY Y X XY X (Ceγ )W ∗ e g∗ ∗ ∗ 2 π 2 ˜ 2 R R R =− XY X˜ † f W (y), (69) 256 3m 2 256π 2 M2 R 1  + 4 ∗ ˜ † ˜ ∗† ∗ ˜ † . XYR X XYR YR X (72) where M stands for a common heavy boson mass, y = 3m2 m2/M2, z =˜m2/M2 and the loop functions are defined ( ) = 2/ ˜ 2 ( ) , in the appendix. Notice that the first term of Ceγ B˜ +B∗ where x m m . The first (second) terms of Ceγ h and (Ceγ )W ∗ are aligned with the mass operator as long as (Ceγ )Z and (Ceγ )W arise from diagrams such as those shown m, m˜ ∝ 1, at least. Therefore, as already pointed out in the in the left (right) plot of Fig. 2. These contributions are of literature [12], they cannot induce neither flavour nor CP- order (NY , NX ) = (3, 4) and occur through a chirality flip violating effects after switching to the mass basis for the SM implemented in the internal heavy-fermion line (left plot) and leptons. Yet, we stress that they contribute to the leptonic external light-fermion line (right plot). On the other hand, the 123 579 Page 16 of 26 Eur. Phys. J. C (2015) 75 :579

± h, ϕZ,ϕ ± h, ϕZ,ϕ

eL e˜R eL e˜R

XY∗ Y ∗† X† XY∗ X˜ † ∗ ˜ † ˜ ∗† ∗ ˜ † R R R XYR X XYR YR X γ γ

Fig. 2 NLO Feynman diagrams for i →  j γ of order (NY , NX ) = (3, 4) with loop exchange of h, Z, W and heavy fermions

Z, B∗, B,W˜ ∗ h, ϕZ e e˜ eL e˜R L R

∗ ∗† ∗ ∗† ∗ ˜ † XY∗ Y ∗† Y ∗ X˜ † XYR YR YR YR YR X R R R γ γ

Fig. 3 NLO Feynman diagrams for i →  j γ arising from dim-8 operators. The diagram on the left is of order (NY , NX ) = (5, 2), the one on the right of order (NY , NX ) = (3, 2)

 ( ) 2 last term of (C γ ) as well as the third terms of (C γ ) stem Ceγ ˜ + ∗ e g∗ ∗ e h e Z B B =− f B(y, z) XX† XY X˜ † from dimension eight operators such as 2 256π 2 M2 3 R + B( , ) ∗ ˜ † ˜ ˜ † ϕ†ϕ f4 y z XYR X X X ∗ − ∗† ∗ ∗† ∗  ( ϕ)(σ · ) ( ˜ † ˜ ) 2 ˜ † ˜ , 2 L F XYR m m YR YRYR YR X eR v ∗ ∗ ∗ 16π 2 + f B(y, z) XY Y †Y X˜ † , (75) (73) 5 M2 R R R  2 (C γ ) ∗ e g∗ ∗ e W =− f W (y, z)XX† XY∗ X˜ † and the corresponding diagrams are shown in the left plot of 2 256π 2 M2 2 R Fig. 3. Such contributions are of order (N , N ) = (5, 2) Y X W ∗ ∗ ˜ † ˜ ˜ † v2/ ˜ 2 + f (y, z)XY X X X and turn out to be suppressed by a factor of m com- 3 R  2 pared to dimension six contributions. As a result, the para- ∗ v ∗ ∗ ∗ + f W (y, z) XY Y †Y X˜ † . (76) metric ratio between dimension six and dimension eight 4 M2 R R R 2 ˜ 2/ ∗2v2 operators is of order X m YR showing that both kind of operators might provide the dominant effects depending ( ) ( ) ∗ on the model parameters. Let us stress that contributions The first two terms of Ceγ B˜+B∗ and Ceγ W are of order ( , ) = ( , ) of order (NY , NX ) = (5, 2) can arise also at dimension NY NX 1 4 and arise from the one-loop exchange of six level if the extra Higgses of Eq. (73) close in a loop heavy fermions and bosons. The relevant Feynman diagrams instead of getting a vacuum expectation value. Therefore, are shown in Fig. 4. On the other hand, the last contributions ( ) ( ) ∗ ( , ) = ( , ) we expect that one-loop induced dimension eight operators to Ceγ B˜+B∗ and Ceγ W are of order NY NX 3 2 with (NY , NX ) = (5, 2) dominate over two-loop induced and stem from dimension eight operators such as, for exam- dimension six operators with (NY , NX ) = (5, 2) provided ple, 2 2 2 v /m˜  1/16π . The last term of (Ceγ )Z stems from 2 ϕ†ϕ dimension eight operators of the form g∗ ∗ † −1 ∗† ∗ ˜ † (L ϕ)(σ · F)XY (m˜ m˜ ) Y Y X e˜R. (77) M2 16π 2 R R R ϕ†ϕ 2 ∗ † −2 ∗† ∗ ˜ † g (L ϕ)(σ · F)XY (m˜ m˜ ) Y Y X e˜R, (74) 16π 2 R R R The relevant Feynman diagrams are shown in the right plot of Fig. 3. and the relevant diagram is shown in the right plot of Fig. 3. Finally, we observe that there are not effects for (Ceγ )W from dimension eight operators since our model doesn’t include 5.2 Scalar operator heavy right-handed neutrinos. We discuss now NLO effects arising from class (ii) which The Yukawa interactions for charged leptons are modified are given by: after we integrate out at tree level the heavy fermions. Setting 123 Eur. Phys. J. C (2015) 75 :579 Page 17 of 26 579

B∗, B,W˜ ∗ B∗, B,W˜ ∗

eL e˜R eL e˜R

∗ ˜ † ˜ ˜ † † ∗ ˜ † XYR X X X XXXYR X γ γ

Fig. 4 NLO Feynman diagrams for i →  j γ of order (NY , NX ) = (1, 4) with loop exchange of heavy gauge bosons and heavy fermions

∗ = (1) (3) YL 0, we find that: Switching to the coefficients Cϕl , Cϕl and Cϕe we find that: h ( ) −L = √ ¯  + , 1 Y eL y eR h.c. (78) Cϕ 1 ∗ ∗ ( ) 2 l =− XY Y † X †, C 3 = 0, 2 2m˜ 2 R R ϕl Cϕ 1 ∗ ∗ where in the mass basis for the charged leptons we have e = XY˜ †Y X˜ †. (84) 2 2m2 R R v2 SM ∗ ∗† † SM y = y − XY Y X y 5.4 Contact operators 2m˜ 2 R R v2 − SM ˜ ∗† ∗ ˜ †, Contact operators induced by the exchange of heavy gauge y XYR YR X (79) 2m2 bosons can be easily derived starting from the coefficients (ρ) (ρ) CLL and CRR entering the interaction Lagrangian of heavy with gauge bosons and SM leptons which we report in the SM = ∗ ˜ †. y XYR X (80) appendix. Integrating out the heavy gauge bosons at tree level, we find that Therefore, the coefficient Ceϕ of the dimension six oper-   = (ϕ†ϕ)(¯ ϕ) ( ) 2 2 ator Qeϕ Ceϕ L eR reads Cll eμee † g1 g2 = (XX ) μ + , 2 e 2 2 4MB∗ 4MW ∗   3 Ceϕ 1 ∗ ∗ 1 ∗ ∗ = XY Y † X † ySM + ySM XY˜ †Y X˜ †. ( ) 2 2 ˜ 2 R R   2 R R Cee eμee ˜ ˜ † g1 2m 2m = (X X ) μ , 2 e 2 (81)  M ∗  B  ( ) 2 Cle eμee † g1 ∗ SM = (XX ) μ , Notice that for Y = 0 the corrections to y are pro- 2 e 2 L   2M ∗ SM B portional to y itself, as in scenarios where the Higgs is a   ( ) 2 pseudo-Goldstone boson [63–65]. Cle eeeμ ˜ ˜ † g1 = (X X ) μ , (85) 2 e 2 2MB∗ 5.3 Vector operators and   (u) The Z boson interactions with charged leptons are also mod- (C ) μ 2 2 q e 1 † g1 g2 =− (XX ) μ + , ified after we integrate out at tree level the heavy fermions. 2 e 2 2  4 ∗ ∗ 3MB MW In particular, we find that  3  (d) (C ) μ 2 2 q e 1 † g1 g2 =− (XX ) μ − , g 2 e 2 2  4 ∗ ∗ LZ = e¯ ( gL PL + gR PR )Ze/ , (82) 3MB MW cW   3 ( ) 2 Ceu eμ ˜ ˜ † 2g1 =−(X X ) μ , 2 e 2 where gL and gR are defined as follows:  3M ∗  B ( ) 2 Ced eμ ˜ ˜ † g1 v2 = (X X ) μ , 1 2 ∗ ∗† † 2 e 2 gL =− + s + XY Y X ,  3M ∗ 2 W 4m˜ 2 R R  B  v2 ( ) 2 2 ˜ ∗† ∗ ˜ † Cu eμ † g1 g = s − XY Y X . (83) =−(XX ) μ , R W 2 R R 2 e 2 4m 3MB∗ 123 579 Page 18 of 26 Eur. Phys. J. C (2015) 75 :579   ( ) 2 Cd eμ † g1 Radiative LFV transitions are tightly related to the mag- = (XX ) μ , 2 e 2 netic and electric leptonic dipole moments, which are also 6MB∗   extremely sensitive probes of new physics. In particular, one ( ) 2 Ceq eμ ˜ ˜ † g1 =−(X X ) μ , can find that 2 e 2 (86)  6M ∗ √ B √ v   v   ee 2 2 mμ μμ d =− 2 Im C γ , aμ = Re C γ . e 2 e e 2 e where higher-order effects suppressed by additional factors (88) 2 2 2 2 2 ˜ 2 of (gSM/g∗) , v /m (m˜ ) and X (X ) have been neglected. In concrete scenarios as our two-site model, a, d and BR( →  γ)are expected to be correlated. However, such 6 Phenomenological analysis correlations depend on the flavour and CP structure of the couplings which are unknown. In our discussion, we assume ∗ In this section, we evaluate the most relevant low-energy pro- order one CP-violating phases and anarchic YR. μ → γ cesses in the charged lepton sector in the context of the two- Here we provide the dominant contribution to e site model by making use of the Wilson coefficients derived which arises from dimension eight operators. Focussing on  , ˜ ∗ = ˜ in the previous section. One of the most interesting features the HB scenario (M m m) with anarchic YR and X X, of the two-site model is the absence of the Wilson coefficients it turns out that (NY , NX ) = (3, 2) from dimension six dipole operators, at α v8| ∗|8 3 em YR me least in the one-loop approximation. Such coefficients are BR(μ → eγ)≈ , (89) 64π (m2 −˜m2)4 mμ particularly dangerous for LFV. As we saw in Sect. 3, their ∗ presence is not compatible with anarchic Yukawas YR if the where we made use of Eq. (40) to eliminate X, X˜ and where compositeness scale is close to 1 TeV. We do not know if this ∗ YR now stands for an average element of the anarchic matrix feature of the model is an accident of the one-loop approx- ∗ YRij. Notice that the above expression is valid only in the imation or if it persists at higher loop orders. In the follow- mass insertion approximation which requires that m, m˜ , |m− ( , ) = ( , ) ∗ ing we will neglect the NY NX 3 2 contribution to m˜ |vY . As an example, if |m2 −˜m2|≈m2, one can find dimension six dipole operators, assuming that higher loops R   8 provide a sufficient suppression. Our purpose is to check if − 1.5TeV ∗ BR(μ → eγ)≈ 3 × 10 13 |Y |8, (90) under these conditions anarchic Yukawas are still viable or m R not when we consider a compositeness scale around 1 TeV. Among the most interesting LFV channels are μ → eγ , implying that μ → eγ saturates its current experimental μ → μ → τ ≈ . ∗ ≈ 3e, e conversion in nuclei as well as LFV pro- bound for m 1 5 TeV and YR 1. cesses. However, hereafter, we focus on processes with an This is confirmed by our numerical results shown in Fig. 5 underlying μ → e transition since they are the best probes where we have reported the predictions for BR(μ → eγ)as ∗ of composite Higgs models with anarchic YR. Concerning a function of the heavy fermion mass m.InFig.5,aswellas ∗ ∗ flavour-conserving processes, we are interested in the elec- in all other plots, we have assumed anarchic YR and YL and − ˜ tron EDM and the muon g 2. The current status and future X = X so that the relevant flavour mixing angles entering ˜ ˜ experimental sensitivities for the above processes are col- μ → e transitions are Xe/ Xμ = Xe/Xμ ∼ me/mμ.The lected in Table 4. As recalled in Sect. 3, there is a ∼3.5σ lower (upper) red line in the left plot refers to the case where SM ∗ = ∗ = ∗ = ∗ = discrepancy between the prediction and the experimental YR YL 1(YR YL 2) while the lower (upper) black g − ∗ = ∗ = ∗ = value of the muon 2. line corresponds to YL 0 and YR 1(YR 2). The most prominent features emerging by this plot are: (i) in the case Y ∗ = Y ∗ = μ → eγ 6.1  →  γ of L 0( L 0), the bound on is satisfied for i j ≥ ≥ ∗ = (μ → γ)∼ m 1TeV(m 10 TeV), (ii) for YL 0, BR e ∗8/ 8 ∗ = (μ → γ) ∼ ∗2 ∗2/ 4 The dipole transition  →  γ is responsible for both LFV YR m while for YL 0, BR e YL YR m i j ∗, ∗ radiative decays (when i = j) like μ → eγ and flavour- and this explains the different growth with YR YL and the conserving processes like the electron EDM and the muon decoupling properties with m of the various curves. In the μ → γ g −2 when i = j = e or μ, respectively. The branching ratio right plot of Fig. 5 we show the branching ratio of e ∗ = μ → γ as a function of the heavy fermion mass m for YL 0 and for the process e can be written as ∗ various values of YR.        A quite similar behaviour is expected for the electron π 2 v2 2 2 ∗ 24  eμ  μe EDM. Indeed, if Y = 0 and assuming O(1) CP-violating BR(μ → eγ)= C γ  +C γ  . (87) L 2 4 2 e e G F mμ phases, it turns out that 123 Eur. Phys. J. C (2015) 75 :579 Page 19 of 26 579

Fig. 5 Branching ratio of μ → eγ as a function of the heavy fermion mass m

Fig. 6 Electron EDM de as a function of the heavy fermion mass m

  |d | m Y ∗Y ∗ 2 e ≈ e R L ≈ × −29 20√ TeV ∗ ∗, In Fig. 6 we show our numerical results for de. Indeed, 8 10 cm YRYL e 32π 2 mm˜ mm˜ Fig. 6 represents the analogous of Fig. 5 for the electron (91) EDM case and similar conclusions drawn for μ → eγ apply here too. and therefore the electron EDM poses a severe constraint on the heavy fermion scale at the same level of μ → eγ . ∗ = Setting YL 0, the EDM bound is satisfied with much lighter 6.2 μ → 3e masses. In particular, we find that   | | v2 ∗4 4 de me Y − 3TeV ∗4 The process μ → 3e receives more contributions than μ → ≈ R ≈ 10 28 cm Y , e 64π 2 (m2 −˜m2)2 m R eγ as it is sensitive also to four-fermion operators. In par- (92) ticular, starting from the general expression of Refs. [53,66] and neglecting scalar and non-dipole photonic contributions where in the last equality we have assumed that |m2 −˜m2|≈ which are negligible in our model, we find the following m2. expression for its branching ratio: 123 579 Page 20 of 26 Eur. Phys. J. C (2015) 75 :579

Fig. 7 Branching ratio of μ → 3e (left)andμ− Au → e− Au (right) as a function of the heavy fermion mass m

BR(μ → 3e) ∗ =  the two-site model with YL 0 can be disentangled among 1 other models if both μ → 3e and μ → eγ will be observed. = 2|C |2 + 2|C |2 +|C |2 +|C |2 4 2 VLL VRR VLR VRL  , ˜ 8 G F In particular, in the HB scenario (M m m) with anar- ∗ =˜ = ˜ 8ve eμ μe chic YR, m m and X X, we find + √ Re((2CVLL +CVLR) Ceγ +(2CVRR +CVRL) Ceγ ) 2mμ     ∗ 2   v2 m2 3 |Y | memμ + 2 μ − 11 (| eμ|2 +| μe|2) , BR(μ → 3e)  R (4s4 + (2s2 −1)2) 16 e log Ceγ Ceγ (93) 2 4 v2 W W mμ2 m2 4 16G m e F   4 − 1TeV ∗ ≈ 5 × 10 13 |Y |2, (96) where m R =( 2 − ) (1)eμ + eμee, = 2 eμ + eμee, CVLL 2sW 1 Cϕ C CVRR 2sW Cϕe Cee implying that μ → 3e saturates its current experimental = 2 (1)eμ + eμee, =( 2 − ) eμ + eeeμ, bound for m ≈ 1 TeV and Y ∗ ≈ 1. This result is well repro- CVLR 2sW Cϕ Ce CVRL 2sW 1 Cϕe Ce R (94) duced by our numerical analysis shown in the left plot of Fig. 7 where the behaviour of BR(μ → 3e) as a function of the heavy fermion mass m is shown for different values and the explicit expression for the Wilson coefficients Cab ∗ (μ → ) ∼ ∗2/ 4 of YR. Notice that BR 3e YR m and therefore it can be found in Sect. 5. In the above equations, we have ∗ neglected scalar four-fermion operators, since they are very has a much milder growth with YR and a slower decoupling (μ → γ) ∼ ∗8/ 8 suppressed in our model. We remind the reader that, when- with m compared to BR e YR m (in the case ∗ = ever the dipole operator is dominant in μ → 3e, there exists of YL 0). a model-independent correlation between the branching ratio of μ → 3e and μ → eγ given by 6.3 μN → eN   2 BR(μ → 3e) α mμ 11 −  el log − ≈6.6 × 10 3. (95) As the last process governed by an underlying μ → e tran- (μ → γ) π 2 BR e 3 me 4 sition we consider μ − e conversion in nuclei. As in the As a result, the current MEG bound BR(μ → eγ) ≤ μ → 3e case, this process receives effects from both dipole 5.7×10−13 already implies that BR(μ → 3e)  3.8×10−15. and four-fermion operators. In particular, the μ − e conver- However, as shown in our numerical analysis and illustrated sion branching ratio is defined as in Fig. 8, μ → 3e turns out to be dominated by non-dipole − − ωconv operators and the correlation of Eq. (95) is significantly vio- BR(μ N → e N) ≡ , (97) ωcapt lated. This is a relevant result, as within composite Higgs ∗ = ω ω models with YL 0, as well as in supersymmetric scenar- where capt is the muon capture rate while conv is given by ios, Eq. (95) holds to an excellent approximation. Therefore, the expression 123 Eur. Phys. J. C (2015) 75 :579 Page 21 of 26 579

Table 11 Overlap integrals from Ref. [56] and the correlation of Eq. (105) is not at work. This is shown  , ˜ Nucleus DV(p) V (n) ω (106 s−1) in Fig. 8. In particular, in the HB scenario (M m m) with capt ∗ =˜ = ˜ anarchic YR, m m and X X, we find Au 0.189 0.0974 0.146 13.07 . 5 ∗ 2   Ti 0.0870 0.0399 0.0495 2 59 mμ |Y | m mμ (μ− Au →e− Au) R e |V (n)|2 Al 0.0362 0.0161 0.0173 0.7054 BR 4 2 m ωcapt v   4 5 −13 3TeV ∗ 2 mμ ( ) ( ) ≈4×10 |Y | , (106) ω = (|A∗ D + g p V (p) + g n V (n)|2 m R conv 4 R LV LV +| ∗ + (p) (p) + (n) (n)|2), AL D gRV V gRV V (98) and therefore the current experimental bound is saturated for m ≈ 3 TeV and Y ∗ ≈ 1. which we have obtained starting from the general expression R These expectations are fully confirmed numerically as of Ref. [56] and setting to zero the scalar and non-dipole shown by the right plot of Fig. 7 where we report operator contributions which are negligible in our model. (μ− → − ) (p,n) (p,n) BR Au e Au as a function of the heavy fermion ∗ The coupling constants AL , AR, gLV and gRV in Eq. (98) mass m for different values of YR. Since both are defined as BR(μ− Au → e− Au) and BR(μ → 3e) are proportional ∗2/ 4 μ → v to YR m , the consideration done for the case of =− √1 eμ∗, AR Ceγ (99) 3e hold here too. However, we point out that in our 2 2 mμ (μ− → − ) v model BR Au e Au is much enhanced with respect =− √1 μe, (μ → ) AL Ceγ (100) to BR 3e as opposite to scenarios with dipole- 2 2 mμ dominance where BR(μ− Au → e− Au) ≈ 0.6 × BR(μ → μ μ (p) = (1)eμ( 2 − )− ( (u)e + eμ)−( (d)e + eμ), ) gLV Cϕ 4sW 1 2 Cq Cu Cq Cd 3e . As a result, the simultaneous observation of these pro- (101) cesses would enable us to disentangle among the underlying theory at work. (p) = eμ( 2 − ) − ( eμ + eμ)−( eμ + eμ), gRV Cϕe 4sW 1 2 Ceq Ceu Ceq Ced (102) μ μ 7 Conclusion (n) = (1)eμ − ( (u)e + eμ)− ( (d)e + eμ), gLV Cϕ Cq Cu 2 Cq Cd (103) We have reanalysed the bounds on the compositeness scale ( ) μ μ μ μ μ derived from LFV and CP violation in the lepton sector, g n = Ce − (Ce + Ce )−2(Ce + Ce ). (104) RV ϕe eq eu eq ed in the framework of PC. In the generic case of anarchic Yukawa couplings, such bounds are known to be quite severe, The flavour-conserving interactions of the heavy vector stronger than the analogous bounds from the quark sec- resonances with the light quarks have been derived from the tor and requiring a compositeness scale above 30 TeV. In respective couplings with the light leptons simply rescaling this work we focussed on the case of vanishing neutrino them according to the different charges of the quarks under Y masses and vanishing “wrong” Yukawa couplings, in the and T . Finally, the quantities D and V p,n refer to the overlap 3 hope of minimising FCNC and CP-violating effects. We per- integrals between the wave functions and the nucleon densi- formed a general effective operator analysis, where the Wil- ties [56] which we report for few relevant nuclei in Table 11. son coefficients of the relevant dimension-six operators are An inspection of Eqs. (97), (98) and Table 11 shows that determined by a flavour symmetry group and by a set of at present μ− Au → e− Au is the most sensitive probe of spurions. We have considered all lowest-dimensional oper- new physics among the various μ − e conversion in nuclei ators leading to LFV violation and CP violation that can processes. In scenarios with dipole dominance, the following be constructed with the Higgs doublet and the SM leptons. model-independent relation holds: We have formally expanded the Wilson coefficients in pow- ers of the Yukawa couplings of the composite sector and (μ− → − ) 2 m5 G2 BR Au e Au = D μ F ≈ . × −3 in powers of the elementary–composite mixing terms. Our 2 3 8 10 BR(μ → eγ) 192π ωcapt expansion includes, for the first time, terms quadrilinear in (105) the elementary–composite mixing (X, X˜ ). By exploiting the known limits on the Wilson coefficients we have shown that, and therefore BR(μ− Au → e− Au)  2.2 × 10−15 after even in the case of vanishing “wrong” Yukawa couplings, imposing the bound BR(μ → eγ)≤ 5.7×10−13. However, the anarchic scenario is not compatible with a compositeness as in the case of μ → 3e, in turns out that in our model scale of 1 TeV, barring accidental cancellations not incorpo- BR(μ− Au → e− Au) is dominated by non-dipole operators rated in our general spurion analysis. We have also shown 123 579 Page 22 of 26 Eur. Phys. J. C (2015) 75 :579

μ → μ− → − μ → γ ∗ = Fig. 8 Branching ratio of 3e (left)and Au e Au (right) versus the branching ratio of e for YL 0. The case of dominance of the dipole operator is shown in yellow that there are schemes where a TeV scale can be accommo- PITN-GA-2011-289442). The research of P.P. is supported by the ERC dated, without necessarily reproducing the case of MFV. In Advanced Grant No. 267985 (DaMeSyFla), by the research grant TAsP the example we have provided this is achieved in a semi- (Theoretical Astroparticle Physics), and by the Istituto Nazionale di Fisica Nucleare (INFN). P.P. thanks G. Buchalla, G. Isidori, U. Nierste perturbative regime, where the universal Yukawa coupling y and J. Zupan for the invitation to the MIAPP workshop Flavour 2015: of the composite sector is close to unity. In this respect it is New Physics at High Energy and High Precision, where part of his work  was performed. The research of A.P. is supported by the Swiss National interesting to note that while the bound on√ NP coming from operators bilinear in (X, X˜ ) scales with y, the limit from Science Foundation (SNF) under Contract 200021-159720. ˜ the operators quadrilinear in (X, X) scales with 1/y, thus Open Access This article is distributed under the terms of the Creative making essentially impossible to lower NP below the TeV. Commons Attribution 4.0 International License (http://creativecomm This also shows the importance of keeping terms quadrilinear ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit in the mixing in the expansion of the Wilson coefficients. to the original author(s) and the source, provide a link to the Creative We have also derived the low-energy effective Lagrangian Commons license, and indicate if changes were made. relevant to LFV and CP violation in the lepton sector in the Funded by SCOAP3. two-site model, that includes heavy vector-like fermions as well as heavy spin-one particles. We focussed on the case ∗ = A Contact operators YL 0, finding a strong suppression of LFV effects com- pared to a generic composite Higgs scenario where Y ∗ = 0. L In the following, we provide the full results for the coeffi- In a perturbative analysis where we evaluated the relevant (ρ) (ρ) cients C and C entering the interaction Lagrangian of amplitudes at the LO, we found that the μ → eγ transition LL¯ RR¯ is dominated by dimension eight operators, at variance with heavy gauge bosons and SM leptons, ¯ μ (ρ) (ρ) our spurion analysis where the most important contribution L = ρμ γ (C P + C P ), (107) LL¯ L RR¯ R comes from operators of dimension six. As a result the best probes of our scenario are μ → e conversion in nuclei and ρ ={ ∗, ˜ , ∗} with μ B B W3 .FortheCL¯ L coefficient, we find the electron EDM. Al least in that portion of the parameter  ∗ space where perturbation theory is applicable, LFV allows (B ) g∗ 2 2 † C ¯ = tan(θ2) − (tan(θ2) + 1)XX a compositeness scale close to the TeV, even in the case of LL 2 2 † † anarchic Yukawas. In this regime there are interesting rela- + (tan(θ2) + 1)XX XX tions among the various LFV transitions, which allow one to 2 1 v ∗ ∗ disentangle the model from other possibilities. − (tan(θ )2 + 2) XY Y † X † + h.c. 2 ˜ 2 R R 4 m  2 Acknowledgments 1 v ∗ ∗ We would like to thank warmly Andrea Wulzer − (tan(θ )2 + 1) XY Y † X † + h.c. , (108) and Robert Ziegler for useful discussions. This work was supported 2 2 mm˜ R L in part by the MIUR-PRIN Project 2010YJ2NYW and by the Euro-  ( ˜ ) g∗ pean Union network FP7 ITN INVISIBLES (Marie Curie Actions, C B = XX†(1− XX†) LL¯ 2 cos θ2 123 Eur. Phys. J. C (2015) 75 :579 Page 23 of 26 579   v2 v2 ∗ ∗† † 1 ∗ ∗† † We summarise them in Fig. 10, where the expressions for + XY Y X + XY Y X +h.c. , ρ (∗) ˜ 2 R R ˜ R L ϒ,  , m 2 mm L,R L,R are (109)     ( ∗) ∗ 00 000 W3 g 2 2 †  = , C =− − tan(θ ) + (tan(θ ) + 1)XX R † VR LL¯ 1 1 0 U 010 2  R  100 − (tan(θ )2 + 1)XX† XX†  = U † V , (116) 1 L L 010 L 1 v2    + (θ )2 ∗ ∗† † + ∗ 00 000 tan 1 XYRYR X h.c.  = V , ˜ 2 R † 0cotθ 0 R 4 m  0 UR 1 2   1 v ∗ ∗ − θ + ( (θ )2 + ) † † + . ∗ = † tan 1 00 , tan 1 1 XYRYL X h.c. L UL VL (117) 2 mm˜ 0cotθ1 0 ⎡ ⎤ (110) 0 Z g 2 † ⎣ g ⎦  = s − V VR, R c W R 2cW W 0 For the CRR¯ coefficient, we find ⎡ ⎤    0 ∗ g 1 (B ) 2 2 ˜ ˜ † Z = 2 − + † ⎣ 0 ⎦ , C = g∗ tan(θ ) − (tan(θ ) + 1)X X L sW VL VL (118) RR¯ 2 2 cW 2 g ⎡ ⎤ 2cW + ( (θ )2 + ) ˜ ˜ † ˜ ˜ † tan2 θ tan 2 1 X X X X ∗ 2 B = g V † ⎣ − 1 ⎦ V , 2 R ∗ R 2 R 1 v ∗ ∗ − − ( (θ )2 + ) XY˜ †Y X˜ † + 1 2tan 2 1 2 R R h.c. ⎡ ⎤ 8 m 1 2  tan θ2 2 ∗ 2 1 v ∗ ∗ B = g V † ⎣ − 1 ⎦ V , (119) − (tan(θ )2 + 1) XY˜ †Y X˜ † +h.c. , (111) L ∗ L 2 L 2 2 mm˜ R L −1  ⎡ ⎤ ˜ v2 0 (B) g∗ ˜ ˜ † ˜ ˜ † ˜ ˜ † 1 ˜ ∗† ∗ ˜ † ˜ C = X X − X X X X + XY Y X B = g sec θ V † ⎣ 1 ⎦ V , RR¯ cos θ 4 m2 R R R ∗ 2 R 2 R  2  1 2 ⎡ ⎤ 1 v ∗ ∗ + XY Y † X † +h.c. , (112) 0 ˜ R L B˜ = θ † ⎣ 1 ⎦ , 2 mm L g∗ sec 2 VL 2 VL (120) ∗ 2 1 (W ) g∗ v ∗ ∗ C 3 =− XY˜ †Y X˜ †. (113) ⎡ ⎤ RR¯ 2 R R 0 4 m W ∗,3 g∗ † ⎣ ⎦  = V −1 VR, R 2 R 0 ⎡ ⎤ 2 tan θ1 B One-loop contributions to the dipole operator W ∗,3 g∗ † ⎣ ⎦  = V −1 VL , (121) L 2 L 0 In the following, all lepton fields are assumed to be in the ⎡ ⎤ 00 0 mass basis. We introduce a compact notation to collectively ϒ = † ⎣ ∗ ⎦ . VL 00YR VR (122) denote such fields, using an upper case Latin index to distin- ∗† 0 YL 0 guish among them:

    ˜ ˜ where g∗ = g cot θ and g = g cot θ . Remember that (EL )A = L , EL , EL ,(ER)A = e˜R, ER, ER , A =1, 2, 3, 1 ∗ 2 this is a compact notation in which the flavour indices are (114) omitted. Then each element of the above matrices is really a (N ) = {ν , } , N = , = , . L B L NL R NR B 1 2 3x3 matrix in flavour space (such as Y ∗, Y ∗). (115) R L We can now summarise the leading results for the different 2 2 m m The flavour index, when necessary, will be made explicit loop diagrams. In these expressions, terms of order v2 or 2 with a lower case Latin letter. For most of the expressions, (where  is either an heavy lepton or boson mass) have been however, this index will be omitted to simplify the notation. neglected. The final results are  In those cases one should interpret the expressions as 3x3 e 1 ( h ) =− √ 1 ϒ 1 ϒ + 1 ϒ 1 ϒ Ceγ ij 12 21 13 31 matrices in flavour space. 2 32 2π 2v 2 m 2 m˜ In Fig. 9 we see the different kinds of loop diagrams con- SM SM mi 1 † mi 1 † ij + ϒ12 ϒ + ϒ13 ϒ tributing to Ceγ . From the Lagrangian (59), after perform- 12 m2 21 12 m˜ 2 31 SM SM  ing the rotations (63) and (64) to the mass basis, one can m 1 m 1 + j ϒ† ϒ + j ϒ† ϒ , (123) 12 2 21 13 ˜ 2 31 read the Feynman rules of interest for these computations. 12 m 12 m ij 123 579 Page 24 of 26 Eur. Phys. J. C (2015) 75 :579

h Z, B∗, B,W˜ ∗,3 ei NB ej E ei A ej ei EA ej W ±,W∗, ± γ γ γ

(1) (2) (3)

Fig. 9 One-loop diagrams contributing to the dipole operator in our model. In diagrams (1)and(2)wehaveA = 1, 2, 3, in diagram (3)wehave B = 1, 2

E A EA h i † ρ = −√ (ΥPR +Υ PL)CA , μ ρ ρ , = iγ (ΛLPL +ΛRPR)CA 2 μ EC EC

EA NA W ±,W∗,± W ±,W∗,± √g μ (∗) (∗) , g μ (∗)† (∗)† , = i γ (ΣL PL +ΣR PR)BA = i√ γ (ΣL PL +ΣR PR)BA μ 2 μ 2 NB EB

Fig. 10 Feynman rules needed for the computations. Here A, C = 1, 2, 3andB = 1, 2, while ρ ={Z, B∗, B˜ , W ∗,3}. The expressions for ϒ, ρ ,(∗) L,R L,R can be found in the appendix

e g2 ρ H ρ ρ H ρ SM 1 ( Z ) =− √ 1 +  mf (y) +  mf˜ (z) + m Ceγ ij L 12 1 R 21 L 13 1 R 31 i 2 16 2π 2v c2 M2  W Z   2 ρ ρ ρ ρ  × −   +  mf H (y) 1 M2 3 L 11 L 11 L 12 2 L 21 × 2Z mSMZ + Z m+4 Z Z  L 11 R 11 L 12 R 21 ρ ρ 2 m + mf˜ H (z)    L 13 2 L 31 2  1 M 2 + Z m˜ + Z Z − mSM Z Z SM 2 ρ ρ ρ H ρ L 13 4 ˜ R 31 i L 11 L 11 + m −   +  mf (y) 2 m 3 j 3 L 11 L 11 L 12 2 L 21       2 2 M M ρ H ρ + 5 Z 1 + Z Z + 5 Z 1 + Z + mf˜ (z) , (126) L 12 L 21 L 13 L 13 2 L 31 6 2 m2 6 2 m˜ 2 ij     2  2 1 W ∗ 1 g∗ ∗ † SM ∗ Z SM 2 Z Z 5 Z 1 MZ (C γ )ij =− √ − 2  m  × − m   +  + 2 e π 2v M2 L 11 R 11 L 31 j 3 R 11 R 11 6 R 12 2 m2 32 2 H     + 1 ∗ † H ( )∗ 5 1 M2 L 12mf3 y R 21 × Z + Z + Z Z , (124) 2   R 21 6 R 13 2 m˜ 2 R 31 ij + SM 5 ∗ † ∗ +∗ † H ( )∗  mi L 11 L 11 L 12mf4 y L 21 e g2 6 1 ( W ) =− √ × − † SM + 1 †   Ceγ ij 2 L m R 11 L 2 π 2v 2 11 12 SM 5 ∗ † ∗ ∗ † H ∗ 32 2 MW 2 + m   +  mf (y) ,    j 6 R 11 R 11 R 12 4 R 21 M2 M2 ij × −m + 3 W 3 + 2log W  (127) m m R 21   = m2 , = m˜ 2 ρ = where y 2 z 2 , while in Eq. (126) SM 5 † † 1 M M + m  L +  ρ ρ i 6 L 11 11 L 12 3    {B∗, B˜ , W ∗,3} and the expression is the same for the three M2 M2 − W 11 + 3 W  heavy bosons. Notice that the result is expressed in a block 2 log L 21 m 4 2 m matrix notation. For example, taking Eq. (123) and recall-    2 ϒ × ϒ SM 5 † † 1 MW 11 ing that is a 9 9 matrix, 12 should be read as + m  R +  − j 6 R 11 11 R 12 3 m2 4 the 3 × 3 block of ϒ in position (1, 2). In this notation,    ϒ 1 ϒ ) M2 the expression ( 12 21 ij stands for the matrix element: + 3 W  , " m log R 21 (125) 3 (ϒ ) ( / )(ϒ ) 2 m ij k=1 12 ik 1 mk 12 kj.  The expressions for the loop functions used in Eqs. (126) 1 ρ 1 1 ρ SM ρ (C γ )ij =− √ 2 m  2 e 2 2 L 11 R 11 16 2π v Mρ and (127)are

123 Eur. Phys. J. C (2015) 75 :579 Page 25 of 26 579 √   − − 3 + − − 3 + B yz 4 3z z 6z log z H 4 3x x 6x log x f (y, z) = − (y ↔ z) , f (x) = , (128) 2 3 1 (1 − x)3 (y − z) (1 − z) − + − 2 + 3 − 4 + 2 (133) H ( ) = 8 38x 39x 14x 5x 18x log x , f2 x (129) (1 − x)4 ∗ 4 − 27y + 24y2 − y3 − 6y(1 + 2y) log y f W (y) = . −4 + 15x − 12x2 + x3 + 6x2 log x 1 (1 − y)3 f H (x) = , (130) 3 2(1 − x)3 (134) 10 − 43x + 78x2 − 49x3 + 4x4 + 18x3 log x f H (x) = . (131) 4 12(1 − x)4 The loop functions for dipoles at the next-to-leading order read − h 4 7x Loop functions f (x) = , (135) 1 3(1 − x) −8 + 16s2 (−1 + x) + 11x The loop functions for dipoles at the LO read f Z (x) = W , (136) 1 3(1 − x) 2(y + z)(−4 + 3y + y3 − 6y log y) 19 + 16s2 (−1 + x) − 16x f B(y, z) = f Z (x) = W , (137) 1 (1 − y)3(y − z) 2 3(1 − x) ( − )(− + + 3 − ) + 4 z 2y 4 3z z 6z log z , (1 − z)3(y − z) (132)

y4(79 + 58z + 7z2) + 2z(32 − 37z + 23z2) + 3y2(74 + 41z + 14z2 + 3z3) f B(y, z) =− 3 3(1 − y)3(y − z)(1 − z)2 ( + + 2 + 3) + 3( + + 2 + 3) + 2y 20 53z 5z 12z y 279 85z 19z 13z 3(1 − y)3(y − z)(−1 + z)2 6y(5y3 + 2y2(−2 + z) + 4z2 − yz(4 + 3z)) 12z(4y2 − 3yz + z2) + log y + log z, (138) (1 − y)4(y − z)2 (1 − z)3(y − z)2 5y4(1 − z)2 + 4z2(1 + 7z − 2z2) − y3(−5 + 52z + 19z2 + 6z3) f B(y, z) =− 4 2(1 − y)2(y − z)3(1 − z)2 − ( + − 2 + 3) + 2( + + 2 + 3 + 4) + yz 24 95z 10z 11z y 20 62z 45z 40z z 2(1 − y)2(y − z)3(1 − z)2 3y(5y2 + 6yz − 3z2) 12(3y − z)z2 − log y + log z, (139) (1 − y)3(y − z)3 (1 − z)3(y − z)3 4z(31 − 33z + 36z2 − 16z3) + y3 − 157 + 15z + 45z2 + 25z3 f B(y, z) =− 5 3(1 − y)2(y − z)(1 − z)3 2( + − 2 − 3 − 4) + (− − + 2 + 3 − 4) − y 323 58z 60z 86z 19z y 148 233z 75z 97z 7z 3(1 − y)2(y − z)(1 − z)3 6(y3 − 5yz2) 24z(y(7z − 5z2) + (−2 + z)z2 + y2(−4 + 3z)) − log y − log z, (140) (1 − y)3(y − z)2 (1 − z)4(y − z)2 ∗ 3(4 − 36y + 55y2 − 24y3 + y4 + 2y(−4 − 5y + 6y2) log y) f W (y, z) =− , (141) 2 2(1 − y)4 ∗ −4 + 27y − 24y2 + y3 + 6y(1 + 2y) log y f W (y, z) = , (142) 3 2(1 − y)3 ∗ 4 + y2(−1 + z)2 + 10z − 2z2 + y(1 − 8z − 5z2) f W (y, z) = 4 4(1 − y)2(y − z)(1 − z)2 3y(y + z) 3z2 + log(y) − log(z). (143) 2(1 − y)2(y − z)2 (y − z)2(1 − z)3

123 579 Page 26 of 26 Eur. Phys. J. C (2015) 75 :579

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