Flavor Physics and Flavor Anomalies in Minimal Fundamental Partial Compositeness
Francesco Sannino, Peter Stangl, David M. Straub, Anders Eller Thomsen
FFlavor Physics and Flavor Anomaliesin Minimal Fundamental Partial Compositeness
Francesco
Sannino ,
1, 2, ∗ Peter
Stangl , † David M.
Straub , ‡ and Anders Eller Thomsen § CP -Origins, University of Southern Denmark, Campusvej 55, 5230 Odense, Denmark Danish IAS, University of Southern Denmark, Odense, Denmark Excellence Cluster Universe, TUM, Boltzmannstr. 2, 85748 Garching, Germany
Partial compositeness is a key ingredient of models where the electroweak symmetry is bro-ken by a composite Higgs state. Recently, a UV completion of partial compositeness wasproposed, featuring a new strongly coupled gauge interaction as well as new fundamentalfermions and scalars. We work out the full flavor structure of the minimal realization ofthis idea and investigate in detail the consequences for flavor physics. While CP violation inkaon mixing represents a significant constraint on the model, we find many viable parameterpoints passing all precision tests. We also demonstrate that the recently observed hints fora violation of lepton flavor universality in B → K ( ∗ ) (cid:96)(cid:96) decays can be accommodated bythe model, while the anomalies in B → D ( ∗ ) τ ν cannot be explained while satisfying LEPconstraints on Z couplings. Preprint: CP -Origins-2017-058 DNRF90 I. Introduction
New composite dynamics is a long standing framework for electroweak (EW) symmetry breaking,providing a promising solution to the hierarchy problem by removing the Higgs boson as an ele-mentary scalar. Rather than the Higgs boson gaining a vacuum expectation value, the breaking ofthe EW symmetry is instead brought on by the formation of a condensate in a new, strongly inter-acting sector of the theory. In modern composite models the Higgs boson is realized as a pseudoNambu-Goldstone Boson (pNGB) keeping it light compared to the scale of the new dynamics [1].A major challenge in constructing a successful model of strong EW symmetry breaking is provid-ing masses to the Standard Model (SM) fermions. In this respect, the idea of partial compositenesshas proved popular [2]; here the SM fermions mix with composite fermions of appropriate quantumnumbers to gain their masses. Most of the phenomenological studies of composite Higgs modelshave focused on simplified models implementing the partial compositeness mechanism at low ener-gies, without specifying the UV completion. Constructing an explicit UV completion is importantnot only to lend credibility to the partial compositeness framework in general, but also since it maylead to specific correlations that can be tested in low-energy precision experiments. In a recentdevelopment, Fundamental Partial Compositeness (FPC) models were proposed that feature newfermions and scalars charged under a strong “technicolor” (TC) force. In these models, the SMfermions gain masses as a result of fundamental Yukawa interactions between SM fermions, TCfermions, and TC scalars [9]. This allowed for a controlled construction of the complete effective ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] Other almost UV completions of partial compositeness have also been proposed in the literature. See [3] forsupersymmetric constructions and [4–7] for purely fermionic constructions. It remains to be seen whether therequired large anomalous dimensions can be achieved in the purely fermionic constructions [8]. a r X i v : . [ h e p - ph ] D ec field theory (EFT) respecting all the symmetries of the Minimal FPC (MFPC) model [10]. Firstprinciple lattice simulations have begun to investigate this novel dynamics in [11] while the pio-neering work without techniscalars appeared first in [12, 13] and further developed in [14–17]. Theanalytic ultraviolet and perturbative conformal structure and fate of these type of theories hasbeen carefully analyzed in [18, 19].At the same time there has been a growing interest in the study of flavor physics as a means toprovide insight into new physics. Given the lack of direct evidence for new particles at LHC so far,flavor physics provides a unique opportunity to probe energy scales not accessible directly. Flavorobservables are also well known to impose stringent constraints on models with new compositedynamics [20–22]. Interestingly, several deviations from SM expectations have been observed inflavor physics in recent years. Most notably, hints for a violation of lepton flavor universality in b → s(cid:96) + (cid:96) − transitions with (cid:96) = e vs. µ [23, 24], and independent hints for a violation of leptonflavor universality in b → c(cid:96)ν transitions with (cid:96) = τ vs. µ or e [25–30]. If confirmed, these deviationswould constitute unambiguous evidence of physics beyond the SM. It is thus important to look formodels that can accommodate these anomalies.The aim of this paper is to perform a comprehensive study of flavor constraints on MFPC andto investigate whether it can explain the aforementioned “flavor anomalies”. The remainder ofthe paper is organized as follows. In section II, we review the MFPC model and fix our notation.In section III, we discuss all relevant low-energy precision constraints in our analysis and presentapproximate analytical formulae for the MFPC contributions. Section IV contains the descriptionof our strategy and the discussion of the results of our global numerical analysis of flavor in MFPC.Section V contains our conclusions. II. Minimal Fundamental Partial Compositeness
In the MFPC model, the SM is appended with a new fundamental sector featuring a strong TCforce. This sector contains both TC fermions, or technifermions, F , and TC scalars, or technis-calars, S , which are charged under TC and will form bound states below the TC confining scale.In particular, the Higgs boson will be realized as a bound state of technifermions, while the partialcompositeness mechanism is realized through a mixing between the SM fermions and fermionicbound states consisting of both technifermions and techniscalars. The full kinetic term of the new F (cid:108) ¯ F ↑ ¯ F ↓ S q S l G SM (1 , , (cid:0) , , − (cid:1) (cid:0) , , (cid:1) × (cid:0) , , − (cid:1) × (cid:0) , , (cid:1) TC symmetries 4 F ⊗ N TC S ⊗ N TC Table I. The table summarizes the new BSM states with their representation under G SM . Furthermore, itprovides their flavor symmetry in the absence of SM interactions and their representation of G TC . The leftpart lists the technifermions, F , and the right part the techniscalars, S .SM fermion Q ¯ u ¯ d L ¯ ν ¯ eG SM (cid:0) , , (cid:1) (cid:0) , , − (cid:1) (cid:0) , , (cid:1) (cid:0) , , − (cid:1) (1 , ,
0) (1 , , TC sector with both technifermions and techniscalars transforming in the fundamental, pseudorealrepresentation of G TC = Sp( N TC ) is then given by L kin = − G aµν G aµν + i F † ¯ σ µ D µ F − (cid:0) F T m F (cid:15) TC F + h . c . (cid:1) + ( D µ S ) † ( D µ S ) − S † m S S . (1)For there to be mixing between the fermionic bound states and SM fermions, the model includesfundamental Yukawa interactions between the TC and the SM sectors. This requires the TCparticles to carry SM charges and the technifermions are taken to be in a vector-like representationof G SM to avoid gauge anomalies. Given these constraints, the minimal content of new TC matteris given in Table I. The most general fundamental Yukawa interaction between the new TC sectorand the elementary (SM) fermions are then given by L yuk = y Q Q α S q (cid:15) TC F α (cid:108) − y ¯ u ¯ u S ∗ q ¯ F ↓ + y ¯ d ¯ d S ∗ q ¯ F ↑ + y L L α S l (cid:15) TC F α (cid:108) − y ¯ ν ¯ ν S ∗ l ¯ F ↓ + y ¯ e ¯ e S ∗ l ¯ F ↑ − ˜ y ¯ ν ¯ ν S l ¯ F ↑ + h . c . (2)where α is an SU(2) L index (here implicitly contracted using the SU(2)-invariant tensor). Forcompleteness, we list the quantum numbers of SM fermions in Table II, too. Giving masses to allthree generation of SM fermions, i.e. avoiding vanishing eigenvalues in the mass matrices, requiresthree generations of techniscalars, such that the total TC particle content is 12 N TC techniscalarsand 4 N TC technifermions. In this construction, the fundamental Yukawa couplings y f are tobe understood as 3 × y ¯ ν = ˜ y ¯ ν = 0 in the following. A. Global flavor symmetries and electroweak symmetry breaking
As discussed in [10], in the absence of the mass terms m F , S , the technifermions satisfy an SU(4) F symmetry, while the techniscalars have an enhanced Sp(24) S global flavor symmetry. More gen-erally, the global symmetries of eq. (1) are explicitly broken both by SM interactions and by themass terms. It is, however, assumed that the strong dynamics will dominate the new physics atthe new composite scale Λ TC , while SM interactions remain subdominant. The symmetries of thestrong sector are thus expected to be approximately preserved in the low-energy effective theory.Therefore, the TC particles are conveniently arranged as F a ∈ F ⊗ N TC and Φ i = (cid:32) S− (cid:15) TC S ∗ (cid:33) ∈ S ⊗ N TC , (3)where a is an SU(4) F and i is an Sp(24) S index. In terms of F a and Φ i , the fundamental Yukawainteractions of eq. (2) are given by L yuk = − ψ ia (cid:15) ij Φ j (cid:15) TC F a + h . c . , (4)where the spurion field ψ consists of SM fermions and Yukawa matrices: ψ ia ≡ (Ψ y ) i a ∈ F ⊗ S . (5)As always, the benefit of the spurionic fields are that they may be included systematically in the lowenergy EFT to control the degree of breaking of the approximate flavor symmetries. In particular,the spurionic fields carry chiral dimension from the perspective of systematic power counting, sooperators with more insertions are suppressed. Note that the SM fermions only couple directlyto the strong sector through y f , and so they will always appear in the combination ψ . For thepurpose of this analysis, we will work in the limit of a flavor-trivial scalar mass matrix (proportionalto unity). More generally, a small but non-vanishing mass matrix, m S (cid:28) Λ , can be includedsystematically in the low-energy effective theory, but would not contribute to the order consideredin this work.The symmetry breaking of the model begins at the composite scale, Λ TC , of the TC dynamics.At this scale, the fermions are expected to form a condensate (cid:68) F a (cid:15) TC F b (cid:69) = Λ TC f Σ abθ , (6)thereby spontaneously breaking the global SU(4) F symmetry to an Sp(4) subgroup. Here Σ θ isan antisymmetric matrix determining the alignment of the Sp(4) stability group in SU(4), and f TC ∼ Λ TC / π is the decay constant of the Nambu-Goldstone Bosons (NGBs) associated to thespontaneous breaking. In the case of an exact global SU(4) F symmetry, making distinctions be-tween different alignments is pointless (and futile). However, in the realistic case, the EW gaugegroup is embedded into the SU(4) F group thus introducing a preferred direction for the vacuumalignment. The physical vacuum alignment is then parametrized using an angle θ such thatΣ θ = c θ (cid:32) iσ − iσ (cid:33) + s θ (cid:32) − (cid:33) , (7)where c θ = cos θ and s θ = sin θ [31]. Here c θ = 1 corresponds to a vacuum which preserves theEW gauge symmetry, whereas s θ = 1 leaves it maximally broken.The NGBs of the SU(4) F → Sp(4) symmetry breaking are parametrized by fluctuations aroundthe vacuum Σ θ in terms of the matrixΣ( x ) = exp (cid:34) i √ f TC Π i ( x ) X iθ (cid:35) Σ θ . (8)Here X iθ are the broken generators of SU(4) F , Π , , are identified with the EW NGBs, Π withthe Higgs boson, and Π is an SM singlet. As we will describe in more detail in the next section,physics at low energies can be described using an EFT. In this effective description, the NGBsappear through the leading-order (LO) kinetic term L EFT ⊃ f (cid:104) D µ Σ † D µ Σ (cid:105) , (9)which also gives rise to mass terms for the EW gauge bosons. In particular, recovering the experi-mental masses yields the relation v EW = s θ f TC .A radiatively generated potential promotes the NGBs to pNGBs and determines the actualalignment of the vacuum. These radiative effects are due to terms in the fundamental Lagrangianthat explicitly break the global symmetry: fundamental fermion masses, EW gauge couplings,and Yukawa couplings. Identifying the Higgs with the Π pNGB only makes sense in the case For the NGBs to parametrize the fluctuations around the actual θ -dependent vacuum Σ θ , the parametrization ofthe broken generators also depends on θ (cf. [32]). < s θ (cid:28) s θ can be realized. We therefore assume in the following that0 < s θ (cid:28) s θ in our numerics by varying f TC while keeping v EW fixed (cf. section IV A). Of the pNGB fields, only Π is new as compared to the SM. It genericallyhas a mass m = m h /s θ and does not have a Yukawa coupling to the SM fermions at leading order[31]. For this reason we will ignore it in our analysis. B. Effective theory at the electroweak scale
The TC condensation scale Λ TC is expected to be large compared to the EW scale, such thatthere is a clear hierarchy v EW (cid:28) Λ TC . The effects of the new composite dynamics on SM physicsat the EW scale can thus be described by an EFT in a controlled manner, where the effectivedegrees of freedom include the SM fermions and gauge bosons, and the pNGBs Π i . Meanwhile,the effects of physics above Λ TC are included in effective operators consistent with the symmetriesof the underlying dynamics. The resulting theory, which we will refer to as the MFPC-EFT, wasdetermined in detail in Ref. [10], and here we just present the operators of relevance for our analysis.The effective Lagrangian can be written as L EFT = L SM − Higgs + (cid:88) A C A O A + (cid:32)(cid:88) A C (cid:48) A O (cid:48) A + h . c . (cid:33) , (10)where the new physics is contained in the O ( (cid:48) ) A operators. The normalization of the effectiveoperators is due to symmetry factors and power counting for strongly interacting electroweakEFTs [33] . The strong coefficients C ( (cid:48) ) A are determined by the underlying TC dynamics, andexpected to be O (1) with the present choice of operator normalization.The leading-order operator with just two SM fermions in the effective theory is given by O Yuk = − f TC π ( ψ i a ψ i a ) Σ a a (cid:15) i i . (11)It is responsible for giving masses to the SM fermions and also provides a coupling to the Higgs bo-son (hence its name). In the flavor analysis of the model, this operator constrains the fundamentalYukawas y f to reproduce the SM masses and the Cabibbo–Kobayashi–Maskawa (CKM) matrix.Of particular relevance for the purpose of flavor physics are four-fermion operators induced bythe underlying dynamics. They are completely described by the set of self-hermitian operators O f = 164 π Λ ( ψ i a ψ i a )( ψ † i a ψ † i a )Σ a a Σ † a a (cid:15) i i (cid:15) i i , (12) O f = 164 π Λ ( ψ i a ψ i a )( ψ † i a ψ † i a ) (cid:0) δ a a δ a a − δ a a δ a a (cid:1) (cid:15) i i (cid:15) i i , (13) O f = 164 π Λ ( ψ i a ψ i a )( ψ † i a ψ † i a )Σ a a Σ † a a ( (cid:15) i i (cid:15) i i − (cid:15) i i (cid:15) i i ) , (14) O f = 164 π Λ ( ψ i a ψ i a )( ψ † i a ψ † i a ) (cid:0) δ a a δ a a (cid:15) i i (cid:15) i i + δ a a δ a a (cid:15) i i (cid:15) i i (cid:1) , (15) O f = 164 π Λ ( ψ i a ψ i a )( ψ † i a ψ † i a ) (cid:0) δ a a δ a a (cid:15) i i (cid:15) i i + δ a a δ a a (cid:15) i i (cid:15) i i (cid:1) , (16) In contrast to Ref. [10] we have not rescaled the fundamental Yukawas y f . E Λ QCD
160 GeV Λ TC L fund L EFT H weak Figure 1. Schematic representation of the theory descriptions employed in our analysis. The fundamentaltheory in the UV is in principle matched to the MFPC-EFT at the scale of compositeness, Λ TC , althoughwithout Lattice results we only posses naive estimates for the coefficients. Flavor physics is most convenientlydescribed by the Weak effective Hamiltonian at low energies. We match the MFPC-EFT with the WEH atthe scale 160 GeV. and the complex operators O f = 1128 π Λ ( ψ i a ψ i a )( ψ i a ψ i a )Σ a a Σ a a (cid:15) i i (cid:15) i i , (17) O f = 1128 π Λ ( ψ i a ψ i a )( ψ i a ψ i a ) (Σ a a Σ a a − Σ a a Σ a a ) (cid:15) i i (cid:15) i i , (18) O f = 1128 π Λ ( ψ i a ψ i a )( ψ i a ψ i a )Σ a a Σ a a ( (cid:15) i i (cid:15) i i − (cid:15) i i (cid:15) i i ) . (19)The TC sector is also responsible for modifying the couplings between SM fermions and SMgauge bosons. It induces the operator O Π f = i π ( ψ † i a ¯ σ µ ψ i a ) Σ † a a ←→ D µ Σ a a (cid:15) i i , (20)that modifies the couplings of the weak gauge bosons and is mainly constrained by LEP measure-ments of the Z branching ratios (cf. section III B 2). III. Low-energy signals from the Weak Effective Hamiltonian
To determine the effect of the MFPC model on low-energy observables, we follow the usual approachand derive its consequences on the Weak Effective Hamiltonian (WEH), H weak . As illustrated inFig. 1, we describe the physics at intermediate scales between Λ TC and the low energy regime usingthe effective theory L EFT as discussed above. At the the scale of 160 GeV, W , Z , t and the pNGBsΠ i are integrated out and L EFT is matched to the WEH H weak . The benefit of this procedure is acontrolled treatment of the (approximate) UV symmetries, which we can now trace to correlatedoperators in H weak . A. Matching the MFPC-EFT to the Weak Effective Hamiltonian
Among the MFPC-EFT operators, only O Yuk contains terms that are also present in the SM. Theseare the fermion-Higgs couplings and the fermion mass terms. In unitary gauge, we have C Yuk O Yuk = − (cid:88) f ∈{ u,d,e } C Yuk s θ f TC π ( y T f y ¯ f ) ij (cid:0) f i ¯ f j (cid:1) (cid:18) c θ hv EW + . . . (cid:19) (21)ignoring nonlinear terms in the pNGBs. We employ a compact notation where the fundamentalYukawa couplings of the SU(2) L doublets are labeled by the names of their doublet components,i.e. we use y Q = y u = y d and y L = y e = y ν . From the mass term, one may identify the massmatrices of the SM fermions m f,ij = C Yuk s θ f TC π (cid:0) y T f y ¯ f (cid:1) ij . (22)The WEH is defined in the mass basis where m f,ij has been diagonalized by a biunitary transfor-mation m diag f = U T f m f U ¯ f , f ∈ { u, d, e } , (23)which defines the unitary matrices U f and U ¯ f . These matrices appear in the Wilson coefficients ofthe WEH in the following combinations : • In the CKM matrix defined by V = U † u U d . (24) • In a product of two fundamental Yukawa matrices where one of them is complex conjugatedand the other is not: X f f = 14 π U † f y † f y f U f , X ∗ f f = X T f f . (25) • In a product of two fundamental Yukawa matrices where both of them are either unconju-gated or conjugated: Y f f = 14 π U T f y T f y f U f , Y ∗ f f = 14 π U † f y † f y ∗ f U ∗ f . (26)For the last two cases, f and f denote a SM fermion, i.e. f , f ∈ { u, d, e, ν, ¯ u, ¯ d, ¯ e } . Using thedefinition of Y f f , the fermion mass matrices in the mass basis can be written as m diag f = C Yuk s θ f TC Y f ¯ f (27)and the mass basis SM Yukawa couplings Y SM f can be identified as Y SM f = √ C Yuk Y f ¯ f . (28)Apart from O Yuk , all operators of L EFT describe pure NP effects not present in the SM. As such,they lead to deviations of the WEH Wilson coefficients with respect to the SM contributions.The four-fermion operators O i f can be readily matched to the WEH by summing over theglobal SU(4) F and Sp(24) S indices. For this purpose, we note that the spurion field ψ assumes thevalue ψ ia = y ¯ d ¯ d − y ¯ u ¯ u y ¯ e ¯ e y Q d − y Q u y L e − y L ν , (29) Since we treat neutrinos as massless, the charged lepton mass matrix can be chosen to be diagonal already in thegauge-basis, such that U e = U ν = U ¯ e = and the contribution of U e , U ν and U ¯ e to the Wilson coefficients istrivial. keeping the SU(3) c and SM generation part of the Sp(24) S index implicit. The spinors as well asthe fundamental Yukawa couplings are rotated to the mass bases via the unitary matrices definedin eq. (23). The resulting four-fermion operators are still expressed in the two-component chiralWeyl spinor notation employed in sec II B. We thus subsequently apply an assortment of Fierzidentities to match them to the WEH basis defined in terms of 4-component Dirac spinors.Besides the four-fermion operators in L EFT , an important role in our analysis is played bythe operator O Π f . Modifying the couplings of weak gauge bosons to SM fermions, it yields NPcontributions to four-fermion operators in the WEH when integrating out the W and Z bosons.For the matching, we first derive the W - and Z -couplings contained in O Π f . We then integrateout the W and Z bosons, yielding new four-fermion operators below the EW scale from tree-levelweak gauge boson exchange, where either one or both ends of the gauge boson propagator couplesto the SM fermions via the NP coupling induced by O Π f . These four-fermion operators are thenmatched to the WEH by applying the same steps as for the four-fermion operators O i f describedabove.Since the operator O Yuk will slightly modify the Higgs couplings to SM fermions, it leads toNP contributions to four-fermion operators in the WEH when integrating out the Higgs. However,these operators are always flavor-diagonal and subleading in an expansion in s θ and we will thereforeneglect their contributions. B. Constraints from EW scale physics
In addition to contributing to four-fermion operators in the WEH, the operators O Yuk and O Π f also affect observables at the EW scale. The modified Higgs couplings contained in the formerare constrained by measurements at the LHC and the new couplings of weak gauge bosons to SMfermions induced by the latter are constrained by Z -boson observables measured at LEP.
1. Higgs boson couplings
A pNGB Higgs boson in the SU(4) / Sp(4) breaking pattern has non-standard couplings to the SMparticles as compared to the SM Higgs [31, 32]. The modification of the Higgs coupling to fermionscan be read directly off eq. (21), and the single couplings to the weak gauge bosons may be foundby expanding the kinetic term of eq. (9). One finds g ffh = c θ g SM ffh , g ZZh = c θ g SM ZZh , g
W W h = c θ g SM W W h . (30)The resulting collider constraints have already been discussed in depth in the existing literature,see e.g. [34], so we will merely note that the strongest individual constraint comes from the Higgscoupling to the Z boson. The combined ATLAS and CMS analysis [35], using the Run I LHCdata, yields the bound s θ < .
44 @ 68% CL (31)just from the hZZ coupling. In our analysis, we will only consider points with f TC ≥ s θ < . Observable measurement R e R µ R τ R b R c Z boson partial width ratios used in our numerical analysis. Z boson couplings The NP couplings of the Z boson to SM fermions that are induced by O Π f can be expressed as C Π f O Π f ⊃ (cid:88) f ∈{ u,d,e,ν } gc W Z µ (cid:16) δg ijf L ¯ f iL γ µ f jL + δg ijf R ¯ f iR γ µ f jR (cid:17) , (32)where the deviations δg ijf L and δg ijf R from the SM Z couplings are given by δg iju L = + C Π f π s θ (cid:0) X uu (cid:1) ij , δg iju R = − C Π f π s θ (cid:0) X ∗ ¯ u ¯ u (cid:1) ij ,δg ijd L = − C Π f π s θ (cid:0) X dd (cid:1) ij , δg ijd R = + C Π f π s θ (cid:0) X ∗ ¯ d ¯ d (cid:1) ij ,δg ije L = − C Π f π s θ (cid:0) X ee (cid:1) ij , δg ije R = + C Π f π s θ (cid:0) X ∗ ¯ e ¯ e (cid:1) ij ,δg ijν L = + C Π f π s θ (cid:0) X νν (cid:1) ij , δg ijν R = 0 . (33)The flavor-diagonal terms modify the Z partial widths measured at LEP. To reproduce the correcttop quark mass, the fundamental Yukawa couplings of the third generation quark doublet are usu-ally large . This can yield a sizable contribution to the Zb L b L coupling and thus be in conflict withLEP data. In effective models of partial compositeness that satisfy all EW precision constraints,this problem is usually avoided by a custodial protection of the Zb L b L coupling [36, 37]. Sincethe MFPC model does not feature a protection of this kind , the LEP measurements of partialwidths of the Z boson are important constraints that have to be taken into account. To this end,we calculate the following observables for each parameter point, R b = Γ( Z → b ¯ b )Γ( Z → q ¯ q ) , R c = Γ( Z → c ¯ c )Γ( Z → q ¯ q ) , (34) R e = Γ( Z → q ¯ q )Γ( Z → e ¯ e ) , R µ = Γ( Z → q ¯ q )Γ( Z → µ ¯ µ ) , R τ = Γ( Z → q ¯ q )Γ( Z → τ ¯ τ ) , (35)where Γ( Z → q ¯ q ) implies a sum over all quarks except the top. We include higher order electroweakcorrections [38] as well as the leading order QCD correction [39] to reproduce the correct SMpredictions in the limit C Π f = 0. To some degree, large fundamental Yukawa couplings of the top quark singlet can ease the requirement of largedoublet couplings. However, even for singlet couplings of O (4 π ), the doublet couplings have to be O (1) and arethus never small. Possible FPC models that include a custodial protection of the Zb L b L coupling are discussed in [9].
3. Electroweak precision tests
In addition to the above described observables, the model is constrained by EW precision datain form of the S and T parameters [41]. There are contributions to the S , T parameters due tonon-standard couplings between the SM particles, which will result in contributions to the EWvacuum polarizations different from the SM prediction. At leading order in the MFPC-EFT, onlythe Higgs coupling is different from the SM, cf. eq. (30), and so only the pNGB loops will give loopcontributions to S , T . It was shown in Ref. [42] that this results in contributions S IR = S MFPCpNGB − S SMHiggs = s θ π (cid:20) f ( m z /m h ) + log Λ m h + 56 , (cid:21) , (36) T IR = T MFPCpNGB − T SMHiggs = − s θ πc w log Λ m h . (37)Here the divergences have been replaced with Λ TC , as they will be absorbed into counter termsat next-to-leading order (NLO). Additionally, the S , T parameters will receive contributions fromphysics at energies higher than Λ TC . In the MFPC-EFT such contributions show up as NLOoperators which have been described in Ref. [10]: S UV = s θ C W W π , (38) T UV = s θ ( C D + C D )16 πc w + s θ ( C y Π D + C y Π D )64 π α (3Tr[ X ¯ u ¯ u − X ¯ d ¯ d ] − Tr[ X ¯ e ¯ e ]) − s θ ( C y Π D + C y Π D )64 π α Tr [3( X ¯ u ¯ u X ¯ u ¯ u − X ¯ u ¯ d X ¯ d ¯ u + X ¯ d ¯ d X ¯ d ¯ d ) + X ¯ e ¯ e X ¯ e ¯ e ] . (39)The strong coefficients appearing in these contributions are the coefficients of the relevant NLOcorrections to the kinetic terms (the terms have been included in appendix A for completeness).Combining the contributions from the changed Higgs sector and those coming from UV physicsthrough new effective operators, the total deviation from the SM prediction of the oblique param-eters are S = S UV + S IR and T = T UV + T IR . (40)The uncertainty in the strong coefficients will make it difficult to make a true prediction as to the S and T parameters. Since in addition these coefficients are independent of the ones appearing inthe flavor observables that are in the focus of the present study, we will not consider them in ournumerical analysis. C. Low-energy probes of flavor and CP violation
Precision measurements of flavor-changing neutral current (FCNC) processes like meson-antimesonmixing and rare decays of K and B mesons are well known to be important constraints on modelswith new strong dynamics. But also flavor-changing charged currents, mediated by the W bosonat tree level in the SM, are relevant since models with partial compositeness can violate leptonflavor universality or the unitarity of the CKM matrix. We use the open source package flavio [43] for our numerics. f ( x ) = x + x − x +(9 x + x ) log x (1 − x ) is a loop function.
1. Meson-antimeson mixing
The part of the weak effective Hamiltonian responsible for meson-antimeson mixing in the K , B ,and B s systems reads H ∆ F =2weak = − (cid:88) i C i O i , (41)where the sum runs over the following operators, O ijV LL = ( ¯ d jL γ µ d iL )( ¯ d jL γ µ d iL ) , O ijV RR = ( ¯ d jR γ µ d iR )( ¯ d jR γ µ d iR ) , O ijV LR = ( ¯ d jL γ µ d iL )( ¯ d jR γ µ d iR ) ,O ijSLL = ( ¯ d jR d iL )( ¯ d jR d iL ) , O ijSRR = ( ¯ d jL d iR )( ¯ d jL d iR ) , O ijSLR = ( ¯ d jR d iL )( ¯ d jL d iR ) , (42) O ijT LL = ( ¯ d jR σ µν d iL )( ¯ d jR σ µν d iL ) , O ijT RR = ( ¯ d jL σ µν d iR )( ¯ d jL σ µν d iR ) , where ij = 21 , ,
32 for K , B , and B s , respectively. In the MFPC model, new physics con-tributions to all eight operators are generated from the operators in section II B. There are twocontributing mechanisms: direct contributions from the four-fermion operators O i f that containthe operators in H ∆ F =2weak , and Z -mediated contributions from flavor-changing Z couplings inducedby the operator O Π f . In the limit of small s θ , the latter are however subleading. To leading order in s θ , only four operators are generated, C ijV LL = (cid:0) X ∗ dd (cid:1) ij (cid:0) X ∗ dd (cid:1) ij C f + C f Λ , (43) C ijV RR = (cid:0) X ¯ d ¯ d (cid:1) ij (cid:0) X ¯ d ¯ d (cid:1) ij C f + C f Λ , (44) C ijV LR = (cid:0) X ∗ dd (cid:1) ij (cid:0) X ¯ d ¯ d (cid:1) ij C f Λ , (45) C ijSLR = (cid:0) Y d ¯ d (cid:1) ij (cid:0) Y ∗ ¯ dd (cid:1) ij C f Λ . (46)The combination of fundamental Yukawa couplings in C ijSLR turns out to be proportional to thesquare of the down quark mass matrix, which is diagonal in the mass basis by definition. Thus,the operator O SLR is flavor-diagonal and does not contribute to meson-antimeson mixing. Thevanishing of this Wilson coefficient at leading order in s θ is in contrast to effective models of partialcompositeness or extra-dimensional models based on flavor anarchy and is a consequence of ourassumption of a flavor-trivial mass matrix for the elementary scalars. However, even for a vanishing C SLR at the electroweak scale – which is where we match the MFPC-EFT onto the WEH – theQCD renormalization group (RG) running down to the hadronic scale of the order of a few GeVinduces a sizable contribution to C SLR proportional to C V LR .The two left-right operators are well-known to be most problematic in models based on partialcompositeness, in particular in the kaon sector where their QCD matrix elements are stronglychirally enhanced in addition to the RG enhancement of the Wilson coefficients. We thus expectthe strongest bound from meson-antimeson mixing observables to come from (cid:15) K , measuring indirectCP violation in K - ¯ K mixing. Although the Wilson coefficients C f and C f are real, a sizableCP-violating phase in the mixing amplitude can be induced by the fundamental Yukawa couplings. In our numerical analysis, we will keep also subleading terms.
2. Rare semi-leptonic B decays Decays based on the b → s(cid:96)(cid:96) transition, such as B → K ∗ (cid:96)(cid:96) or B → K(cid:96)(cid:96) with (cid:96) = e or µ , areprobes of flavor violation that are complementary to meson-antimeson mixing. On the one hand,since they only involve one flavor change, they are much more sensitive to contributions mediatedby flavor-changing Z couplings induced by O Π f . On the other, recent hints for violation of leptonflavor universality (LFU) between the electronic and muonic B → K ∗ (cid:96)(cid:96) and B → K(cid:96)(cid:96) rates raisethe question whether – and to what level – LFU can be violated in MFPC. To leading order in s θ , the Z -mediated contributions are lepton flavor universal, but direct contributions from thefour-fermion operators O i f containing two quarks and two leptons are in fact expected to violateLFU and enter at the same order in s θ as the Z -mediated effects.The effective Hamiltonian for b → s(cid:96)(cid:96) transitions can be written as H b → s(cid:96)(cid:96) weak = − (cid:88) i,(cid:96) ( C (cid:96)i O (cid:96)i + C (cid:48) (cid:96)i O (cid:48) (cid:96)i ) + h.c. (47)The most important operators for our discussion read O (cid:96) = (¯ s L γ µ b L )(¯ (cid:96)γ µ (cid:96) ) , O (cid:48) (cid:96) = (¯ s R γ µ b R )(¯ (cid:96)γ µ (cid:96) ) , (48) O (cid:96) = (¯ s L γ µ b L )(¯ (cid:96)γ µ γ (cid:96) ) , O (cid:48) (cid:96) = (¯ s R γ µ b R )(¯ (cid:96)γ µ γ (cid:96) ) . (49)The direct four-fermion contributions to their Wilson coefficients, to leading order in s θ , reads C (cid:96) ⊃ − (cid:0) X ∗ dd (cid:1) bs (cid:0) X ¯ e ¯ e (cid:1) (cid:96)(cid:96) C f Λ + 14 (cid:0) X ∗ dd (cid:1) bs (cid:0) X ee (cid:1) (cid:96)(cid:96) C f + C f Λ , (50) C (cid:48) (cid:96) ⊃ − (cid:0) X ¯ d ¯ d (cid:1) bs (cid:0) X ee (cid:1) (cid:96)(cid:96) C f Λ + 14 (cid:0) X ¯ d ¯ d (cid:1) bs (cid:0) X ¯ e ¯ e (cid:1) (cid:96)(cid:96) C f + C f Λ , (51) C (cid:96) ⊃ − (cid:0) X ∗ dd (cid:1) bs (cid:0) X ¯ e ¯ e (cid:1) (cid:96)(cid:96) C f Λ − (cid:0) X ∗ dd (cid:1) bs (cid:0) X ee (cid:1) (cid:96)(cid:96) C f + C f Λ , (52) C (cid:48) (cid:96) ⊃ + 14 (cid:0) X ¯ d ¯ d (cid:1) bs (cid:0) X ee (cid:1) (cid:96)(cid:96) C f Λ + 14 (cid:0) X ¯ d ¯ d (cid:1) bs (cid:0) X ¯ e ¯ e (cid:1) (cid:96)(cid:96) C f + C f Λ , (53)while the Z -mediated contributions can be written as C (cid:96) ⊃ π (cid:0) X ∗ dd (cid:1) bs (4 s w − C Π f Λ , (54) C (cid:48) (cid:96) ⊃ − π (cid:0) X ¯ d ¯ d (cid:1) bs (4 s w − C Π f Λ , (55) C (cid:96) ⊃ π (cid:0) X ∗ dd (cid:1) bs C Π f Λ , (56) C (cid:48) (cid:96) ⊃ − π (cid:0) X ¯ d ¯ d (cid:1) bs C Π f Λ . (57) In particular, we neglect dipole operators [44], which always conserve LFU. Scalar operators are flavor-diagonal inthe mass basis and thus do not contribute. In our numerical analysis, we will keep also subleading terms.
3. Tree-level semi-leptonic decays
Charged-current semi-leptonic decays based on the q → q (cid:48) (cid:96)ν transition are mediated at tree levelby the W boson in the SM and are used to measure the elements of the CKM matrix withoutpollution by loop-induced new physics effects. In MFPC however, these processes can receive newphysics contributions from the operators in the MFPC-EFT. Similarly to the semi-leptonic FCNCdecays, there are contributions from modified W couplings to quarks induced by O Π f that arelepton flavor universal to leading order in s θ , as well as direct four-fermion contributions from O i f that are expected to violate LFU. In addition, in charged-current decays, O Π f induces contributionsfrom modified W couplings to leptons that are also expected to violate LFU.Decays where (cid:96) is a light lepton, i.e. an electron or muon, must be taken into account as constraints in our analysis. They are important for two reasons: they constrain the amount ofLFU that can potentially be observed in FCNC decays with light leptons, and they are necessaryto consistently compare the CKM matrix obtained from diagonalizing the quark mass matriceswith the CKM measurements .In addition, we consider the semi-tauonic decays based on the b → cτ ν transition. The worldaverages for the ratios R D ( ∗ ) of the B → D ( ∗ ) τ ν over the B → D ( ∗ ) (cid:96)ν ( (cid:96) = e, µ ) branching ratioscurrently deviate from the SM prediction at a combined level of 4 σ [45]. Assessing whether theMFPC model can account for these deviations is an important goal of our study.The effective Hamiltonian for d i → u j (cid:96)ν transitions can be written as H d i → u j (cid:96)ν weak = (cid:88) i C (cid:96) ( (cid:48) ) i O (cid:96) ( (cid:48) ) i + h.c. , (58)where the sum runs over the following operators, O d i u j (cid:96)V = (¯ u jL γ µ d iL )(¯ (cid:96) L γ µ ν (cid:96)L ) , O d i u j (cid:96) (cid:48) V = (¯ u jR γ µ d iR )(¯ (cid:96) L γ µ ν (cid:96)L ) , (59) O d i u j (cid:96)S = m b (¯ u jL d iR )(¯ (cid:96) R ν (cid:96)L ) , O d i u j (cid:96) (cid:48) S = m b (¯ u jR d iL )(¯ (cid:96) R ν (cid:96)L ) , (60) O d i u j (cid:96)T = (¯ u jR σ µν d iL )(¯ (cid:96) R σ µν ν (cid:96)L ) . (61)In the SM, C u i d j (cid:96)V = 4 G F V ij / √ in s θ , reads C d i u j (cid:96)V ⊃ (cid:0) X ∗ du (cid:1) ij (cid:0) X eν (cid:1) (cid:96)(cid:96) C f − C f Λ , (62) C d i u j (cid:96) (cid:48) V ⊃ , (63) C d i u j (cid:96)S ⊃ (cid:0) Y ∗ ¯ du (cid:1) ij (cid:0) Y ¯ eν (cid:1) (cid:96)(cid:96) C f Λ , (64) C d i u j (cid:96) (cid:48) S ⊃ (cid:0) Y d ¯ u (cid:1) ij (cid:0) Y ¯ eν (cid:1) (cid:96)(cid:96) C ∗ f − C ∗ f Λ , (65) C d i u j (cid:96)T ⊃ (cid:0) Y d ¯ u (cid:1) ij (cid:0) Y ¯ eν (cid:1) (cid:96)(cid:96) C ∗ f Λ , (66) In our numerical analysis, we will keep also subleading terms. W -mediated contributions read C d i u j (cid:96)V ⊃ − π (cid:16)(cid:0) X ∗ du (cid:1) ij + V ji (cid:0) X eν (cid:1) (cid:96)(cid:96) (cid:17) C Π f Λ , (67) C d i u j (cid:96) (cid:48) V ⊃ π (cid:0) X ¯ d ¯ u (cid:1) ij C Π f Λ . (68)As constraints , we consider the following processes sensitive to these Wilson coefficients: • For d → u(cid:96)ν , the branching ratio of π + → eν (which is sensitive to e - µ LFU violation sincethe branching ratio of the muonic mode is almost 100%), • For s → u(cid:96)ν , the branching ratio of K + → µν and the ratio of K + → (cid:96)ν branching ratioswith (cid:96) = e and µ , • For b → c(cid:96)ν , the branching ratios of B → D(cid:96)ν and B → D ∗ (cid:96)ν with (cid:96) = e and µ .As predictions , we further consider: • For b → cτ ν , the ratios R D and R D ∗ .Table IV lists all the experimental values and SM predictions according to flavio v0.23 used inour analysis. Note that the uncertainties on the SM prediction shown in this table include (andin many cases are dominated by) the parametric uncertainties due to the limited knowledge ofCKM elements. In our numerical scan, as detailed in the following section, CKM parameters arepredicted as functions of the model parameters, such that only the non-CKM uncertainties arerelevant for the χ in any given parameter point. IV. Numerical analysis
To investigate possible NP effects of the MFPC model on the low-energy observables discussedabove, we calculate predictions for these observables, depending on the position in the parameterspace of the MFPC-EFT. To avoid strong constraints from charged lepton flavor violation (see e.g.[49]), we assume that the fundamental Yukawa coupling matrices y L and y ¯ e can be diagonalized inthe same basis at the matching scale . A. Parameters
The observables in our analysis depend on the following MFPC-EFT parameters: • The new strong coupling scale Λ TC = 4 πf TC . We vary f TC between 1 TeV and 3 TeV. • The six real Wilson coefficients C f , C f , C f , C f , C f and C Π f . We vary their absolutevalues logarithmically between 0 . Note that this assumption is not renormalization group invariant in the presence of lepton flavor universalityviolation [50]. Observable measurement SM prediction∆ M s (17 . ± .
02) ps [46] (19 . ± .
7) ps∆ M d (0 . ± . . ± .
09) ps S ψφ (3 . ± . × − [46] (3 . ± . × − S ψK S . ± .
020 [46] 0 . ± . | (cid:15) K | (2 . ± . × − [47] (1 . ± . × − BR( B + → D (cid:96) + ν (cid:96) ) (2 . ± . × − [46] (2 . ± . × − BR( B → D ∗− (cid:96) + ν (cid:96) ) (4 . ± . × − [46] (5 . ± . × − BR( π + → e + ν ) (1 . ± . × − [48] (1 . ± . × − BR( K + → µ + ν ) 0 . ± . . ± . R eµ ( K + → (cid:96) + ν ) (2 . ± . × − [47] (2 . ± . × − R D . ± .
049 [45] 0 . ± . R D ∗ . ± .
019 [45] 0 . ± . R [1 , K . +0 . − . [23] 1 . ± . R [0 . , . K ∗ . +0 . − . [24] 0 . ± . R [1 . , . K ∗ . +0 . − . [24] 0 . ± . flavio v0.23) of flavor observables used inour analysis. The first two blocks are the meson-antimeson mixing and charged current observables used as constraints , while the observables in the last block are considered as predictions . • The four complex Wilson coefficients C f , C f , C f and C Yuk . We vary their absolute valueslogarithmically between 0 . π . • The four fundamental Yukawa coupling matrices y Q , y L , y ¯ u , y ¯ d . For parameterizing them,we first introduce the effective Yukawa matrices˜ y f = (cid:112) C Yuk y f , (69)which allow for expressing the SM fermion mass matrices independently of C Yuk . Eachcomplex matrix ˜ y f can in general be written in terms of one positive real diagonal and twounitary matrices. One of those two unitary matrices can always be absorbed in a redefinitionof the SM fields. For two of the matrices ˜ y f , the second unitary matrix can be absorbed intothe techniscalar fields, and thus two effective Yukawa matrices can be chosen to be positivereal diagonal. We choose˜ y Q = diag( y Q , y Q , y Q ) , ˜ y L = diag( y L , y L , y L ) . (70)Parameterizing the two remaining unitary matrices that enter ˜ y ¯ u , ˜ y ¯ d by in total six angles Since we assume y ¯ e to be diagonal in the same basis as y L , its entries are fixed by requiring that the product of y L and y ¯ e yields the correct masses for the charged leptons. t u , t u , t u , t d , t d , t d and four phases δ d , δ u , a d , b d , we get˜ y ¯ u = unitary( t u , t u , t u , δ u ) · diag( y u , y u , y u ) , ˜ y ¯ d = unitary( t d , t d , t d , δ d , a d , b d ) · diag( y d , y d , y d ) . (71)We vary the diagonal entries logarithmically between .
002 and 4 π and the angles andphases linearly between 0 and 2 π .We thus have in total 14 real parameters for the Wilson coefficients, 22 real parameters for thefundamental Yukawa matrices and one real parameter for the new strong scale. The Wilsoncoefficients as well as the fundamental Yukawa matrices are defined at the matching scale, i.eat 160 GeV. B. Strategy
Given the high dimensionality of the parameter space, a naive brute-force scan by randomly choos-ing each of the parameters is not applicable. We observe, however, that the quark masses andCKM elements only depend on the effective Yukawa matrices ˜ y Q , ˜ y ¯ u and ˜ y ¯ d (see section III A).This can be used in a first step to find a region in parameter space where the predictions for thequark masses and CKM elements are close to experimental observations. In this step, only theeffective quark Yukawa matrices have to be varied. The lepton Yukawa matrix ˜ y L , all MFPC-EFTWilson coefficients and the new strong scale do not enter. In a second step, one can then randomlychoose the remaining parameters while preserving the predictions of SM fermion masses andCKM elements.For predicting the quark masses, we construct the mass matrix in eq. (21) from the effectiveYukawa matrices and numerically diagonalize it via eq. (23). We interpret each quark mass asMS running mass at 160 GeV and run it to the scale where it can be compared to its PDGaverage. The numerical diagonalization also yields the rotation matrices from which we calculatethe CKM elements via eq. (24). However, the CKM elements cannot be directly compared to theexperimental values, as the observables are affected by dimension-six operators too. Contrary tothe quark masses, we consequently cannot impose the constraints on the CKM elements alreadyin the first step of the scan. The CKM elements are therefore constrained in the second step bythe charged-current semi-leptonic decays discussed in section III C 3. This is done after taking thecontributions from dimension-six operators into account. In the first step, however, we require theCKM elements to be close to certain input values that we have found to yield many points thatpass the constraints imposed in the second step. To compare the predictions for the masses totheir PDG averages and the predictions for the CKM elements to our input values, we constructa χ -function χ . This function only depends on the 19 parameters of ˜ y Q , ˜ y ¯ u and ˜ y ¯ d . Wethen proceed in the following way: • Starting from a randomly chosen point in the 19-dimensional parameter-subspace where χ lives, we numerically minimize χ to find a viable point that predictscorrect quark masses and CKM elements close to our input values. A general 3 × y ¯ u and ˜ y ¯ d can beabsorbed by field redefinitions, leaving four phases in total. The lower boundary for the diagonal entries of ˜ y L is adjusted such that the diagonal entries of y ¯ e stay below 4 π when requiring the correct charged lepton masses. As described above, by adjusting ˜ y ¯ e , the charged lepton masses are always fixed to their experimental value andare thus unaffected by varying ˜ y L . • Starting from this viable point, we use a Markov-Chain for an efficient sampling of theparameter space, as first proposed in [51] and also applied in [52, 53]. This is done byemploying the Markov-Chain-Monte-Carlo implementation from the pypmc package [54]. Thechain samples the region around the previously found minimum and generates 10 k viablepoints with a low value of χ . • We reduce the auto-correlation of the 10 k viable points generated by the Markov-Chain byselecting only 1000 points.The above steps are repeated 100 k times to yield 100 M points from 100 k local minima of χ that all predict CKM elements close to our input values and correct quark masses.For these points we then randomly choose the remaining 18 parameters and calculate all theobservables discussed in sections III C and III B 2 using the open source package flavio [43]. Wesubsequently construct χ -functions for three classes of constraints: • χ Z compares the experimental values shown in table III to our predictions of Z -decay ob-servables discussed in section III B 2. • χ F =2 compares the meson-antimeson mixing constraints from table IV to the predictionsof the observables discussed in section III C 1. • χ compares the constraints from semi-leptonic charged-current decays from table IV tothe predictions of the observables discussed in section III C 3.These χ functions are then used to apply the various experimental constraints on the parameterpoints. C. Results
1. Meson-antimeson mixing
As discussed in section III C 1, the constraints from meson-antimeson mixing, in particular the neu-tral kaon sector, are expected to be very important in case of “flavor anarchic” fundamental Yukawacouplings. This is confirmed by our numerical findings, where many parameter points that have thecorrect quark masses and CKM mixing angles predict an order-of-magnitude enhancement of (cid:15) K .This “ (cid:15) K problem”, that plagues all models with partial compositeness (or its extra-dimensionaldual description) without additional flavor symmetries [20, 55, 56], is often phrased as requiring ascale Λ TC in excess of 15 TeV, based on a naive estimate C V LR ∼ C SLR ∼ m d m s / ( v Λ ). How-ever, the exact result depends strongly on the precise form of the fundamental Yukawa couplingsand can deviate from this naive estimate by orders of magnitude in either direction. In fact, wedo find a significant number of points where (cid:15) K is within the experimentally allowed range. Toget a feeling of the size of the new physics contributions to (cid:15) K , we present the histogram infig. 2. It includes a representative subset of all the points that have the correct fermion masses Here we are referring to the genuine dimension-6 NP contributions but remind the reader that CKM elements arevaried during our scan, so also the SM prediction itself differs from point to point. − − − − − − − − − − − − − × (cid:15) NP K . < × (cid:15) K < . Figure 2. Histogram showing the NP contribution to (cid:15) K for a representative subset of all points that featurethe right masses and CKM elements, compared to the points among those that pass the experimentalconstraint. A positive NP contribution corresponds to constructive interference with the SM. and CKM matrix, along with the points surviving the (cid:15) K constraint . This histogram shows thatthe new physics contribution varies over many orders of magnitude. Our variation of the Wilsoncoefficients, which enter linearly, between 0 . B and B s meson mixing are generated as well, even though theeffects are less problematic than in K mixing since the chiral enhancement of the LR operatorsis absent. In figure 3, we show the predictions for the mass differences ∆ M d and ∆ M s for allour allowed points as well as for the points excluded by constraints other than meson-antimesonmixing. We emphasize again that the CKM parameters are varied during our scan. Consequently,the allowed ranges for ∆ M d and ∆ M s for a given parameter point, with fixed CKM elements, aredetermined by the experimental measurements smeared by the uncertainties of the matrix elementsfrom lattice QCD [58]. The elliptic outline visible in the left-hand panel of figure 3 corresponds tothese allowed ranges imposed at 3 σ in our scan. The reason for the allowed (blue) points clusteringin the lower part of this ellipse is that the maximal values of ∆ M s are most easily accessed forhigh values of V cb , that are however disfavored by the B → D(cid:96)ν branching ratio imposed in ourscan. To disentangle the shifts in ∆ M d and ∆ M s due to variation of CKM parameter vs. genuinedimension-6 new physics contributions, it is instructive to plot the total contribution divided bythe SM contribution for the given value of the CKM parameters at each point. The result isshown in the right-hand panel of figure 3. The allowed points show relative modifications of bothobservables of up to 40% with respect to the SM; this is possible since the modifications can bepartially compensated by shifts in the CKM parameters. Both observables can be enhanced orsuppressed. We further observe three clusters of points with sizable new physics effects: wheremostly ∆ M d is affected, where mostly ∆ M s is affected, and where both are affected in the sameway. An interesting feature of the histogram is the fact that there are more allowed points with a NP contribution to (cid:15) K interfering constructively with the SM. The reason is that, as discussed above, we used the exclusive semi-leptonicdecays B → D ( ∗ ) (cid:96)ν as constraints in our scan. They currently prefer a lower value of V cb compared to the inclusivesemi-leptonic decay. Since the SM prediction of (cid:15) K is highly sensitive to the value of V cb , this tends to lead to avalue that is on the low side of the measurement [57], favoring constructively interfering NP. . . . . . . ∆ M d [ps − ] ∆ M s [ p s − ] excluded by other constraintsallowed 0 . . . . . ∆ M d / ∆ M SM d . . . . . ∆ M s / ∆ M S M s excluded by other constraintsallowed Figure 3. Predictions for ∆ M d and ∆ M s . Gray points are excluded by constraints other than ∆ F = 2.Blue points are allowed by all constraints. Apart from modifying the mass differences in the B and B s systems, also new CP-violatingphases can be generated in the mixing amplitudes. These can be probed in the mixing induced CPasymmetries in B → J/ψK S and B s → J/ψφ . The predictions for these observables are shownin figure 4. The left-hand panel again shows the allowed points due to variation of CKM elementsand new physics contributions, while the right-hand panel shows the shift in the asymmetries dueto genuine dimension-6 new physics contribution by subtracting the SM contribution for the givenvalues of CKM elements in each point. We observe that the shift in both asymmetries can be oforder 0 .
2. Tree-level decays and lepton flavor universality
The precise measurements of BR( π → eν ) and R eµ ( B → K(cid:96)ν ) = BR( K → eν ) / BR( K → µν ), thatwe impose as constraints in our analysis, lead to a strong restriction of e - µ universality violation.This is important since we are interested in the allowed size of e - µ universality violation in flavor-changing neutral currents, as indicated by LHCb measurements. In our scan, we find points wherethe deviations in these two observables are much larger than allowed by experiments, but we findthe ratio of the two to always be SM-like. This can be easily understood since the dominant effectsin these transitions involving light quarks, u → d(cid:96)ν or s → u(cid:96)ν , is through a modified W couplingto leptons induced by the operator O Π f , while the direct four-fermion contributions induced bythe operators O i f are suppressed by the small fundamental Yukawa couplings of the light quarkgenerations. By SU(2) L symmetry, this lepton flavor non-universal modification of W couplingsimplies a corresponding modification of Z couplings that is constrained by Z pole measurementsat LEP. In figure 5, we show a histogram of the values for the two observables of interest for all the0 .
600 0 .
625 0 .
650 0 .
675 0 .
700 0 .
725 0 . S ψK S − . − . . . . . S ψ φ excluded by other constraintsallowed − . − . − . . . . S ψK S − S SM ψK S − . − . − . . . . . S ψ φ − S S M ψ φ excluded by other constraintsallowed Figure 4. Predictions for the mixing induced CP asymmetries in B → J/ψK S and B s → J/ψφ , sensitiveto the B and B s mixing phases. Gray points are excluded by constraints other than ∆ F = 2. Blue pointsare allowed by all constraints. points passing the meson-antimeson mixing constraints. We distinguish points excluded by LEP,excluded by flavor (i.e. one of the charged-current decays imposed as constraints in the analysis),excluded by both, and allowed by all constraints. These plots demonstrate that LEP and flavorconstraints are both relevant to constrain e - µ universality violation in Z couplings and that theresulting constraint is at the per cent level.Lepton-flavor universality in charged currents is also tested in the decays B → D ( ∗ ) τ ν based onthe b → cτ ν transition, that are experimentally more challenging than the B → D ( ∗ ) (cid:96)ν decays with (cid:96) = e or µ that are used to measure the CKM element V cb . In recent years, several measurementsby BaBar, Belle, and LHCb [25–30] have consistently shown higher values for the ratios R D ( ∗ ) = Γ( B → D ( ∗ ) τ ν )Γ( B → D ( ∗ ) (cid:96)ν ) (72)than predicted, with small uncertainties, in the SM. A global combination by the HFLAV collabo-ration finds a combined significance of around 4 σ [45]. In figure 6, we show our predictions for R D and R D ∗ for all allowed points. The dominant effects lead to a simultaneous increase (or decrease)of both ratios, as observed by experiment, since they are generated by a vector operator withleft-handed quarks and leptons. But although there are some points in parameter space where thetension with experiment can be reduced compared to the SM, the overall size of the effects is toosmall to accommodate the experimental central values. The main reason for this is the limit on thesize of the τ lepton fundamental Yukawa coupling coming from Z → τ τ decays at LEP. Switchingoff the LEP constraints, we find huge effects in both R D and R D ∗ , as shown by the light graypoints in figure 6. An interesting question is whether a non-minimal FPC model with a vanishingWilson coefficient for the operator O Π f or some other protection of the Zτ τ coupling exists thatcould accommodate a large violation of LFU in R D and R D ∗ . We leave the investigation of this1 .
21 1 .
22 1 .
23 1 .
24 1 .
25 1 . BR( π + → e + ν ) p o i n t s p e r b i n allowedexcluded by LEPexcluded by flavorexcl. by LEP & flavor 2 .
44 2 .
46 2 .
48 2 .
50 2 .
52 2 . R eµ ( K + → ‘ + ν ) allowedexcluded by LEPexcluded by flavorexcl. by LEP & flavor Figure 5. Histogram showing the distribution of the predictions for two observables probing e - µ universalityviolation in Z couplings for all points passing the meson-antimeson mixing constraints. Points labeled“excluded by LEP” are excluded by the partial Z width measurements at LEP, while points labeled “excludedby flavor” are excluded by one of the charged-current decays imposed as constraints. .
20 0 .
25 0 .
30 0 .
35 0 .
40 0 .
45 0 . R D . . . . . . . . . R D ∗ HFLav 1 σ excluded by LEPallowed Figure 6. Predictions for lepton flavor universality tests in B → Dτ ν and B → D ∗ τ ν compared to the SMprediction and the experimental world averages for all allowed points (dark blue) as well as for all pointsexcluded by LEP Z pole constraints (light gray). question to a future analysis.2 . . . . . . . . . . R [0 . , . K ∗ . . . . . . . . . . R [ . , . ] K ∗ SMLHCb 1 σ . . . . . . . . . . R [1 . , . K . . . . . . . . . . R [ . , . ] K ∗ SMLHCb 1 σ Figure 7. Predictions for µ - e universality tests in B → K ∗ (cid:96) + (cid:96) − and B → K(cid:96) + (cid:96) − compared to the SMprediction and the LHCb measurements for all allowed points.
3. Lepton flavor universality tests in FCNC decays
Measurements by the LHCb experiment of the ratios R [ a,b ] K ( ∗ ) = (cid:82) ba dq d Γ dq ( B → K ( ∗ ) µ + µ − ) (cid:82) ba dq d Γ dq ( B → K ( ∗ ) e + e − ) (73)show tensions with the theoretically very clean SM prediction at the level of 2–3 σ [23, 24]. Severalanalyses have shown that these tensions can be consistently explained by physics beyond the SM,in particular by a vector operator with left-handed quarks and muons [59–64]. As seen fromthe discussion in section III C 2, such an operator is generated in MFPC as well, along with theanalogous operator with right-handed muons. In effective models of partial compositeness, it hasbeen shown that the deviation in R K ( ∗ ) can be explained if left-handed muons have a significantdegree of compositeness [67] (see also [68, 69] for extra-dimensional constructions), correspondingto a sizable fundamental Yukawa coupling in MFPC. In figure 7, we show our predictions for R K and R K ∗ for all allowed points in the bins measured by LHCb, compared to the SM prediction andthe experimental measurement. We find a significant number of points where all three observationscan be explained within 1–2 σ , demonstrating that the MFPC model can explain all R K ( ∗ ) anomaliesin terms of new physics. Since this comes about by means of an operator involving left-handedmuons, the model also fits the global fit to b → sµ + µ − observables, where additional tensions arepresent (see e.g. [70]), much better than the SM. Alternative explanations with partial compositeness mostly using NP in the electronic channels have been suggestedas well, but cannot explain additional tensions present in b → sµ + µ − transitions [65, 66]. V. Conclusions
We have performed a comprehensive numerical analysis of flavor physics in Minimal FundamentalPartial Compositeness (FPC). To the best of our knowledge, this is the first numerical analysis ofa UV completion of partial compositeness with a realistic flavor structure in the quark sector. Ourmain findings can be summarized as follows. • Indirect CP violation in kaon mixing (measured by the parameter (cid:15) K ) is larger than observedin large parts of the parameter space, but we also find a large number of points where it issmall enough. • For the points allowed by the (cid:15) K constraints, sizable effects in B and B s mixing are ob-served for many points, including non-standard CP-violating mixing phases close to the levelcurrently probed in precision experiments. • While we impose the absence of charged lepton flavor violation for simplicity, the violationof lepton flavor universality (LFU) is unavoidable with partial compositeness. We find LFUtests like the ratios of π or K → eν vs. µν to constitute important constraints on theparameter space. • LFU violation in B → D ( ∗ ) τ ν , as currently indicated by several experiments at the levelof 4 σ , cannot be generated at a sufficient size to reproduce the experimental central valuesdue to LEP constraints on the Zτ τ couplings. The tensions can however be amelioratedcompared to the SM. • The MFPC model can explain both hints for LFU violation in B → Kee vs. µµ ( R K ) and B → K ∗ ee vs. µµ ( R K ∗ ) simultaneously, as shown in figure 7.To summarize, Minimal Fundamental Partial Compositeness is a predictive UV complete modelwith a realistic flavor sector that can be tested by present and future flavor physics experiments. Ifthe anomalies in R D and R D ∗ are confirmed to be due to NP, a non-minimal model with protected Z couplings to tau leptons might be preferred. If the deviations in R K and R K ∗ are confirmed,they could be first indications of technifermions and techniscalars coupling strongly to muons.Our explorative study can be generalized in several ways. There are additional low-energyprecision tests that we have not considered, e.g. the anomalous magnetic moment of the muonor electric dipole moments. We have also not attempted to construct a realistic lepton sectorexplaining the origin of neutrino masses or the absence of lepton flavor violation. In contrast toeffective models of partial compositeness, also the form factors of the new strong interaction, thatwe have simply scanned here, could be computed in principle, boosting the predictiveness of themodel. Acknowledgments
PS would like to thank Christoph Niehoff for useful discussions. The work of PS and DS wassupported by the DFG cluster of excellence “Origin and Structure of the Universe”. FS and AETacknowledge partial support from the Danish National Research Foundation grant DNRF:90.4
AppendixA. Next-to-leading order operators for the kinetic terms
Ref. [10] listed all operators that modify the kinetic terms of the EW gauge bosons and thepNGBs at NLO. For completeness we refer here the operators which contribute to the EW precisionparameters, S and T . The leading operator contributing to the S parameter is O W W = 132 π A Iµν A Jµν Tr (cid:104) T I F Σ( T J F ) T Σ † (cid:105) . (A1)There are two kinds of operators contributing to the T parameter. Two operators are due tocorrections from the EW gauge interactions, O D = 132 f π Tr (cid:104) (Σ ←→ D µ Σ † ) T I F (Σ ←→ D µ Σ † ) T I F (cid:105) , (A2) O D = 132 f π Tr (cid:104) (Σ ←→ D µ Σ † ) T I F (cid:105) Tr (cid:104) (Σ ←→ D µ Σ † ) T I F (cid:105) , (A3)and four operators are due to corrections from SM fermions, O y Π D = 132 f (4 π ) ( y ∗ f y f ) a a i i ( y ∗ f (cid:48) y f (cid:48) ) a a i i (Σ † ←→ D µ Σ) a a (Σ † ←→ D µ Σ) a a (cid:15) i i (cid:15) i i , (A4) O y Π D = 132 f (4 π ) ( y ∗ f y f ) a a i i ( y ∗ f (cid:48) y f (cid:48) ) a a i i (Σ † ←→ D µ Σ) a a (Σ † ←→ D µ Σ) a a (cid:15) i i (cid:15) i i , (A5) O y Π D = 132 f (4 π ) ( y ∗ f y f ) a a i i ( y ∗ f (cid:48) y f (cid:48) ) a a i i (Σ † ←→ D µ Σ) a a (Σ † ←→ D µ Σ) a a (cid:15) i i (cid:15) i i , (A6) O y Π D = 132 f (4 π ) ( y ∗ f y f ) a a i i ( y ∗ f (cid:48) y f (cid:48) ) a a i i (Σ † ←→ D µ Σ) a a (Σ † ←→ D µ Σ) a a (cid:15) i i (cid:15) i i . (A7)We have normalized these operators corresponding to the normalization of the decay constant inthe LO kinetic terms, such that the corresponding strong coefficients are expected to be O (1). [1] D. B. Kaplan and H. Georgi, Phys. Lett. , 183 (1984).[2] D. B. Kaplan, Nucl. Phys. B365 , 259 (1991).[3] F. Caracciolo, A. Parolini, and M. Serone, JHEP , 066 (2013), arXiv:1211.7290 [hep-ph].[4] J. Barnard, T. Gherghetta, and T. S. Ray, JHEP , 002 (2014), arXiv:1311.6562 [hep-ph].[5] G. Ferretti and D. Karateev, JHEP , 077 (2014), arXiv:1312.5330 [hep-ph].[6] G. Ferretti, JHEP , 142 (2014), arXiv:1404.7137 [hep-ph].[7] L. Vecchi, JHEP , 094 (2017), arXiv:1506.00623 [hep-ph].[8] C. Pica and F. Sannino, Phys. Rev. D94 , 071702 (2016), arXiv:1604.02572 [hep-ph].[9] F. Sannino, A. Strumia, A. Tesi, and E. Vigiani, JHEP , 029 (2016), arXiv:1607.01659 [hep-ph].[10] G. Cacciapaglia, H. Gertov, F. Sannino, and A. E. Thomsen, (2017), arXiv:1704.07845 [hep-ph].[11] M. Hansen, T. Janowski, C. Pica, and A. Toniato, (2017), arXiv:1710.10831 [hep-lat].[12] R. Lewis, C. Pica, and F. Sannino, Phys. Rev. D85 , 014504 (2012), arXiv:1109.3513 [hep-ph].[13] A. Hietanen, R. Lewis, C. Pica, and F. Sannino, JHEP , 116 (2014), arXiv:1404.2794 [hep-lat].[14] R. Arthur, V. Drach, M. Hansen, A. Hietanen, C. Pica, and F. Sannino, Phys. Rev. D94 , 094507(2016), arXiv:1602.06559 [hep-lat]. [15] E. Bennett, D. K. Hong, J.-W. Lee, C. J. D. Lin, B. Lucini, M. Piai, and D. Vadacchino, (2017),arXiv:1712.04220 [hep-lat].[16] J.-W. Lee, B. Lucini, and M. Piai, JHEP , 036 (2017), arXiv:1701.03228 [hep-lat].[17] V. Drach, T. Janowski, and C. Pica, in (2017) arXiv:1710.07218 [hep-lat].[18] F. F. Hansen, T. Janowski, K. Langaeble, R. B. Mann, F. Sannino, T. G. Steele, and Z.-W. Wang,(2017), arXiv:1706.06402 [hep-ph].[19] M. B. Einhorn and D. R. T. Jones, Phys. Rev. D96 , 055035 (2017), arXiv:1705.00751 [hep-ph].[20] C. Csaki, A. Falkowski, and A. Weiler, JHEP , 008 (2008), arXiv:0804.1954 [hep-ph].[21] K. Agashe, A. Azatov, and L. Zhu, Phys. Rev. D79 , 056006 (2009), arXiv:0810.1016 [hep-ph].[22] R. Barbieri, D. Buttazzo, F. Sala, D. M. Straub, and A. Tesi, JHEP , 069 (2013), arXiv:1211.5085[hep-ph].[23] R. Aaij et al. (LHCb), Phys. Rev. Lett. , 151601 (2014), arXiv:1406.6482 [hep-ex].[24] R. Aaij et al. (LHCb), JHEP , 055 (2017), arXiv:1705.05802 [hep-ex].[25] J. P. Lees et al. (BaBar), Phys. Rev. D88 , 072012 (2013), arXiv:1303.0571 [hep-ex].[26] M. Huschle et al. (Belle), Phys. Rev.
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