A Flavor-Safe Composite Explanation of RK

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Nuclear and Particle Physics Proceedings 285–286 (2017) 93–98 www.elsevier.com/locate/nppp

A ﬂavor-safe composite explanation of RK

Adrian´ Carmona∗, Florian Goertz CERN, Theoretical Physics Department, CH-1211 Geneva 23, Switzerland

Abstract + − In these proceedings we discuss a flavor-safe explanation of the anomaly found in RK = B(B → Kμ μ )/B(B → Ke+e−) by LHCb, within the framework of composite Higgs models. We present a model featuring a non-negligible degree of compositeness for all three generations of right-handed leptons, which leads to a violation of lepton-flavor universality in neutral current interactions while other constraints from quark- and lepton-flavor physics are met. Moreoever, the particular embedding of the lepton sector considered in this setup provides a parametrically enhanded contribution to the Higgs mass that can weak considerably the need for ultra-light top partners. Keywords: Flavor physics, Composite Higgs models, Violation of Lepton Flavor Universality

1. Introduction are generated through such linear couplings after inte- grating out the corresponding composite counterparts, Composite Higgs models provide an elegant explana- it is usually thought that only third generation quarks tion to the hierarchy problem by protecting the Higgs will exhibit a sizable degree of compositeness and will mass by its finite size [1, 2]. In addition, a sizable be relevant for EWSB. However, the fact that neutri- mass gap between the electroweak (EW) and the com- nos may have Majorana masses, together with the ob- positeness scale Λ ≈ 4π fπ can be achieved if one as- served non-hierarchical mixing pattern in the PMNS sumes the Higgs to be a pseudo Nambu-Goldstone bo- matrix, can change this situation for the lepton sector, son (pNGB) of some global symmetry of the strong sec- see e.g. [6, 7]. In these proceedings we will discuss tor [3, 4, 5]. One typical assumption is that this global a very minimal implementation of leptons in compos- symmetry is only broken by the weak couplings of the ite Higgs models, where neutrino masses are generated elementary SM-like degrees of freedom, corresponding via a type-III seesaw mechanism and the RH lepton sec- to the SM fermions – with the possible exception of the tor is unified by embedding the RH charged leptons and right-handed (RH) top quark – and gauge bosons, which the RH neutrinos in a single representation of the global generates a Higgs potential radiatively and triggers the group G (for each generation) [8]. Linked to this uni- electroweak symmetry breaking (EWSB). Within the fication, our setup predicts a violation of lepton-flavor paradigm of partial compositeness, where one assumes universality (LFU) in neutral current interactions, while linear mixings of the SM-like fermions with their com- LFU is basically respected in charged currents, provid- posite counterparts, the light mass eigenstates become ing a natural and compelling explanation for the 2.6 σ mixtures of elementary and composite degrees of free- deviation observed by LHCb [9] in the very clean ratio dom, tying together the dynamics behind the observed [10, 11, 12] flavor pattern and EWSB. Since the Yukawa couplings

∗Speaker http://dx.doi.org/10.1016/j.nuclphysbps.2017.03.017 2405-6014/© 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 94 A. Carmona, F. Goertz / Nuclear and Particle Physics Proceedings 285–286 (2017) 93–98

where we have explicitly shown the decomposition un- × = H B(B → Kμ+μ−)exp der SU(2)L SU(2)R SO(4) (with the bidoublet R = = 0.745+0.090 ± 0.036 . × K B → + − −0.074 being represented by a 2 2 matrix on which the SU(2)L (B Ke e ) q2∈[1,6] GeV rotation acts vertically and the SU(2)R one horizon- (1) tally) and the signs in square brackets denote the bound- As we will see, this can be done in a completely flavor- ary conditions at the UV and IR branes. A Dirichlet save manner, due to the possibility of implementing a boundary condition for the RH/LH chirality is denoted very economical flavor symmetry, which avoids the ap- by [+/−], with LH/RH zero modes being present for pearance of new sources of flavor-changing neutral cur- fields with [+, +]/[−, −] boundary conditions. Finally, rents (FCNC) to very good approximation. Since the since the lepton sector will produce an additional non- lepton sector features a sizable degree of compositeness negligible contribution to the Higgs potential, we can and the RH lepton unification requires the presence of consider for the quark sector the previously disregarded non-minimal representations of G, it will provide a para- minimal model consisting of a fully composite tR and a 3 G metrically enhanced correction to the Higgs mass, such LH doublet qL embedded in a 5 of . More specifically, ξi ∼ ,ξi ∼ ,ξi ∼ ,ξi ∼ that the need for ultra-light top partners is weakened we consider 1 52/3 2 12/3 3 5−1/3 4 1−1/3, considerably, linking the mass of the latter with the size i = 1, 2, 3, or of the neutrino masses. Λ˜ i −, + i +, + ξi = [ ] u1[ ] ⊕ i −, + , 1 i −, + i +, + u1 [ ] u˜ [ ] d1[ ] 2. Setup ξi [−, −], (3) 2 Let us consider the so-called minimal composite i −, + ˜i −, + ξi = u3[ ] d [ ] ⊕ i −, + , Higgs model (MCHM), where the global symmetry 3 i −, + Ξ˜ i −, + d3 [ ] d3[ ] [ ] of the strong sector G = SO(5) is broken by the ξi [−, −]. strong dynamics to H = SO(4), delivering four Gold- 4 stone bosons that will be identified with the Higgs dou- This minimal realization of composite leptons natu- blet. We consider the minimal custodial embedding rally allows for a very strong flavor protection, requiring of the SM lepton sector including three RH fermion any lepton flavor violating (LFV) process to be medi- Σ = ,μ,τ triplets with zero hypercharge, R, with e .If ated by extremely suppressed neutrino-mass insertions these new degrees of freedom have Majorana masses and leading in particular to the absence of dangerous O of order (MGUT), the observed tiny neutrino masses FCNCs in the lepton sector to excellent approximation. can be explained with O(1) Yukawa couplings via the To this end, we promote the accidental SU(3)1 ×SU(3)2 (type-III) seesaw mechanism. In the framework of the flavor symmetry of the lepton sector in the decompact- MCHM, or its five dimensional (5D) holographic dual ified or conformal limit (arising from the arbitrary ro- [13, 14, 15, 16], this is realized by embedding every tation of ξ1 and ξ2 in the family space) to a 5D gauge generation of RH leptons in a symmetric representation group only broken at the UV brane (i.e., by the elemen- (14)ofSO(5), whereas every left-handed (LH) doublet tary sector) and the vacuum expectation value (vev) of G is embedded in a fundamental representation (5)of . some non-dynamical field Y [17, 18]. The bulk fields in ff In terms of the di erent 5D bulk fields transforming un- the lepton sector will thus transform as ζ ∼ (3, 1) and × 1 der SO(5) U(1)X, such embedding of the lepton sector ζ ∼ , Y∼ , ¯ 1 2 (1 3), whereas (3 3). Therefore, the corre- reads ζ ∼ 5− and ζ ∼ 14− , for = e,μ,τ, 1 1 2 1 sponding bulk masses will be given by ν [+, +] ˜1[−, +] † † ζ = [−, +] ⊕ 1 , c1 = η11 + ρ1YY , c2 = η21 + ρ2Y Y, (4) 1 1 [+, +] Y˜ [−, +] 1 1 ν [+, −] ˜ [+, −] whereas the IR brane masses will read ζ = [−, −] ⊕ 2 2 (2) 2 2 [+, −] Y˜ [+, −] 2 2 4 (1,1) (1,1) (2,2) (2,2) ⎛ ⎞ a ωS ζ¯ Yζ + ωB(ζ¯ Yζ ) + h.c., (5) ⎜ λˆ [−, −] ν[+, −] [+, −] ⎟ 1L 2R 1L 2R R ⎜ 2 2 2 ⎟ ⊕ ⎜ νˆ [−, −] [+, −] Y[+, −] ⎟ , ⎝⎜ 2 2 2 ⎠⎟ with η1,2,ρ1,2 ∈ R, ωS,B ∈ C, a(z) = R/z the warp factor, ˆ −, − +, − Θ +, − 2[ ] Y2 [ ] 2 [ ] z ∈ [R, R ] the coordinate of the extra dimension and the superscripts (1, 1) and (2, 2) denoting the singlet and the 1For simplicity, we will be rather schematic in the description of bidoublet components of the corresponding multiplets. the 5D setup. We thus refer the reader to Ref. [8] for further details. Since, as mentioned, the elementary sector represented A. Carmona, F. Goertz / Nuclear and Particle Physics Proceedings 285–286 (2017) 93–98 95 by the UV brane does not respect in general this sym- where the conformal sector becomes strongly coupled, metry, one can have general Majorana masses λ2 R 4 4 Ψ¯ − O O¯ − Ψ d p d q R( p) R(p) R( q) R(q) γ 1 c Λ2 R L ⊃− Σ¯ Σ + . ., UV MΣ Tr R h c (6) R z=R 2γ 2 μ R ∼ δ λ 2 4 Ψ¯ ∂Ψ . R d x R(x)i¡ R(x) where Λ √ (10) ν / λˆ ˆ2 2 2√ Σ = ,= e,μ,τ. (7) This leads, after canonical normalization, to the follow- −νˆ / 2 2 2 ing expressions for the physical masses,

Note that the fact of having just two SO(5) lepton mul- 2 −1 Me ∼ δ vL Mν ∼ v LR (MΣ) LR (11) tiplets and thus being able to use only one SU(3)1 × Y SU(3)2 spurion, , allows us to diagonalize at the same where v is the Higgs vev and we have deﬁned L,R = γ time (4) and (5) via the rotation λ μ/Λ L,R L,R( ) . It is then clear that in order to have si- multaneously hierarchical charged lepton masses and a ζ →Uζ ,ζ→Uζ , 1 1 1 2 2 2 (8) non-hierarchical neutrino mass matrix one needs

† U YU = , , ≡ μ τ 1 and ∼ constant (12) where 1 2 diag(yee yμμ yττ) y. In this par- eL L L L R ticular basis, the whole Lagrangian will be flavor diag- and thus onal with the exception of the Majorana mass in (6), T which becomes U MΣ U . Therefore, any potentially 2 2 0 τR μR eR. (13) induced FCNC will be suppressed by large Majorana masses. Regarding the quark sector, we consider the This has several interesting consequences. First of more general case of arbitrary sources of flavor break- all, since the degree of compositeness of RH leptons is ing, see [8] for more details. non-negligable and their contribution to the Higgs quar- 2 4 tic scale with R, instead of the usual R for smaller representations of OR, leptons give a sizable contri- 3. EWSB, lepton non-universality and RK bution to the Higgs potential. As already mentioned, this is interesting since it allows to replace the role of In order to make the discussion simpler, it will be 3 tR in EWSB, making possible for R to cancel the qL useful in the following to use the language of the dual contribution with a moderate value of R. In order to four dimensional (4D) strongly coupled theory. Very get a more quantitative idea of the impact of the lep- schematically, we consider an elementary sector, con- ton sector on the Higgs potential and the Higgs mass, sisting of the would-be SM with the exeption of the we show in Figure 1 the mass of the lightest top part- Higgs sector and the addition of the corresponding RH ner versus the Higgs mass evaluated at the composite neutrinos Σ needed for the see-saw mechanism, and a scale O( fπ), with fπ = 1 TeV and the yellow band cor- composite sector mixing linearly with the elementary responding to the high-scale value of the actual Higgs one. Focusing on the lepton sector, this mixing will be mass mH( fπ) = 105 GeV (1 ± 7.5%), after accounting given by the following Lagrangian for the uncertainties of the running in a conservative way. We also display the Barbieri-Giudice (BG) mea- λ λ lep L R sure of the tuning ΔBG, through the color of each point L = l¯ O + Ψ¯ O + h.c. (9) mix γ L L γ R R − min Λ L Λ R in the mH m2/3 plane. We can see from the figure that top-partner masses up to 5 TeV are allowed with a more where Λ=O(PPl) is the UV cut-off scale, γ , = than reasonable amount of tuning. L R ff ff [OL,R] − 5/2 are the different anomalous dimensions, Secondly, since di erent RH leptons exhibit a di er- λ ent degree of compositeness, see eq. (13), diagrams L,R are order one dimensionless parameters and all RH leptons have been embedded in ΨR ∼ 14. Since we with a tree-level exchange of neutral heavy vector reso- γ < nances like the ones schematically depicted in Figure 2 expect R 0 due to the smallness of the neutrino masses (since otherwise they would require an elemen- will lead to a violation of LFU. 2 In particular, they will tary Majorana mass MΣ much smaller than Λ), ΨR will Ψ be rather composite and a large contribution to the R 2See Refs. [19, 20, 21] for different examples in the context of kinetic term will be generated at the scale μ = O(TeV) CHMs and Refs. [22, 23, 24, 25, 26, 27, 28] for other Z models. 96 A. Carmona, F. Goertz / Nuclear and Particle Physics Proceedings 285–286 (2017) 93–98

0.012

0.010

0.008 ] 2 / F 0.006 G 4 [ ee c

0.004

0.002

0.000 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20

f [TeV ]

Figure 3: Value of the c Wilson coeﬃcient as a function of fπ. The Figure 1: Mass of the lightest top partner versus the Higgs mass as ee blue curve shows the best ﬁt to the data while the yellow line corre- a function of the tuning Δ , with lighter points corresponding to BG sponds to the upper bound at 95% C.L.. smaller values of ΔBG, for fπ = 1 TeV. The yellow band corresponds to mH( fπ) = 105 GeV (1 ± 7.5%).

Concerning flavor, the most relevant operators will be 32 2 3 μ lead to four-fermions operators O = q¯ γμq ¯ γ , (15) qe L L R R B 2 3 2 μ 3 O∼ ψ¯ γ ψ χ γμχ , ∼ / 2, O s = q¯ γμq q¯ γ q , (16) c c ( 2 μ 1)( ¯ 2 1) c ψ1 ψ2 χ1 χ2 fπ (14) 1 L L L L where the first of them will provide the leading contri- that, besides flavor, will be relevant also for electroweak bution to R and we expect the latter to appear unavoid- precision data (EWPD). K able if we generate the first one. Instead of performing a complete flavor analysis of the quark sector, we prefer to ψ χ 1 1 focus on the possible correlations between Bs − B¯ s mixing and RK. On the other hand, note already that a large class of potentially dangerous constraints, coming from χ ψ¯ 2 ¯ 2 limits on LFU violation in charged current interactions, mediating e.g. K, π, and μ decays [30, 31], is fulfilled Figure 2: Relevant diagrams for the generation of four-fermion oper- in this model by construction. In fact, the LH charged ators. ¯ γ ν current L μ L is mostly elementary and the light neutrino mass eigenstates contain only a negligible amount of RH fields. Thus, charged currents respect LFU to According to eq. (13), the most important of excellent approximation. O = these operators regarding EWPD will be ee We evaluate RK by computing the Wilson coefficients γ γμ / ffi (eR μeR)(eR eR) 2, whose Wilson√ coe cient cee is of the constrained to be c ∈ 4G / 2 · [−1.8, +2.8] · 10−3 ee F O = γ γ¯ α γ , at 95% C.L. [29]. We present in Figure 3 the value of 9(10) s¯ αPLb ( 5) (17) cee as a function of fπ, where the blue curve corresponds O = O → 9(10) 9(10)[PL PR] (18) to the best fit to the data. We also show the 95% C.L. upper bound on cee by a yellow line. One can see from operators from the usual |ΔB| = |ΔS | = 1 Hamiltonian this plot that values of fπ 1 TeV give already a rea- [32]. Note that, even though we are also generating con- . O O sonable agreement with the data, while for fπ 1 2TeV tributions to 10 and 10 that could in principle lead to the EWPD impose no significant constraint. Therefore, large deviations with respect to the SM predictions in + − in order to provide a conservative assessment of the fla- Bs → decays [33], we expect the largest effect to + − vor predictions of the model, we consider fπ = 1.2TeV arise in the poorly measured Bs → e e decay, rather + − henceforth. than in Bs → μ μ [34]. A. Carmona, F. Goertz / Nuclear and Particle Physics Proceedings 285–286 (2017) 93–98 97

0.00007 Therefore, if conﬁrmed, the ﬁnal observation of viola-

0.00006 tion of LFU could provide an unexpected ﬁrst probe of the dynamics solving the hierarchy problem.

0.00005 Acknowledgments. The research of A.C. has been sup- ] 2 0.00004 ported by a Marie Skłodowska-Curie Individual Fel- TeV /

1 lowship of the European Community’s Horizon 2020 |[ s

B 1 0.00003 c | Framework Programme for Research and Innovation under contract number 659239 (NP4theLHC14). The 0.00002 research of F.G. is supported by a Marie Curie Intra Eu-

0.00001 ropean Fellowship within the EU FP7 (grant no. PIEF- GA-2013-628224). 0 0.5 0.6 0.7 0.8 0.9 1.0 1.1

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