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Partial compositeness is a key ingredient of models where the electroweak symmetry is broken by a composite Higgs state. Recently, a UV completion of partial compositeness was proposed, featuring a new, strongly coupled gauge interaction as well as new fundamental fermions and scalars. We work out the full flavor structure of the minimal realization of this idea and investigate in detail the consequences for flavor physics. While

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 elementary scalar. Rather than the Higgs boson gaining a vacuum expectation value, the breaking of the EW symmetry is instead brought on by the formation of a condensate in a new, strongly interacting sector of the theory. In modern composite models the Higgs boson is realized as a pseudo-Nambu-Goldstone boson (pNGB), keeping it light compared to the scale of the new dynamics

A major challenge in constructing a successful model of strong EW symmetry breaking is providing masses to the Standard Model (SM) fermions. In this respect, the idea of partial compositeness has proved popular

Other almost UV completions of partial compositeness have also been proposed in the literature. See

At the same time, there has been a growing interest in the study of flavor physics as a means to provide insight into new physics. Given the lack of direct evidence for new particles at the LHC so far, flavor physics provides a unique opportunity to probe energy scales not accessible directly. Flavor observables are also well known to impose stringent constraints on models with new composite dynamics

The aim of this paper is to perform a comprehensive study of flavor constraints on MFPC and to investigate whether it can explain the aforementioned “flavor anomalies.” The remainder of the paper is organized as follows. In Sec.

In the MFPC model, the SM is appended with a new fundamental sector featuring a strong TC force. This sector contains both TC fermions, or technifermions,

The table summarizes the new BSM states with their representation under

A summary of the two-component SM spinors and their quantum numbers.

As discussed in

The symmetry breaking of the model begins at the composite scale,

The NGBs of the

For the NGBs to parametrize the fluctuations around the actual

A radiatively generated potential promotes the NGBs to pNGBs and determines the actual alignment of the vacuum. These radiative effects are due to terms in the fundamental Lagrangian that explicitly break the global symmetry: fundamental fermion masses, EW gauge couplings, and Yukawa couplings. Identifying the Higgs with the

The TC condensation scale

In contrast to Ref.

The leading-order operator with just two SM fermions in the effective theory is given by

Of particular relevance for the purpose of flavor physics are four-fermion operators induced by the underlying dynamics. They are completely described by the set of self-Hermitian operators

The TC sector is also responsible for modifying the couplings between SM fermions and SM gauge bosons. It induces the operator

To determine the effect of the MFPC model on low-energy observables, we follow the usual approach and derive its consequences on the weak effective Hamiltonian (WEH),

Schematic representation of the theory descriptions employed in our analysis. The fundamental theory in the UV is, in principle, matched to the MFPC-EFT at the scale of compositeness,

Among the MFPC-EFT operators, only

Since we treat neutrinos as massless, the charged lepton mass matrix can be chosen to be diagonal already in the gauge basis, such that

In the CKM matrix defined by

In a product of two fundamental Yukawa matrices where one of them is complex conjugated and the other is not:

In a product of two fundamental Yukawa matrices where both of them are either unconjugated or conjugated:

The four-fermion operators

Besides the four-fermion operators in

Since the operator

In addition to contributing to four-fermion operators in the WEH, the operators

A pNGB Higgs boson in the

The NP couplings of the

To some degree, large fundamental Yukawa couplings of the top quark singlet can ease the requirement of large doublet couplings. However, even for singlet couplings of

Possible FPC models that include a custodial protection of the

In addition to the above-described observables, the model is constrained by EW precision data in the form of the

Precision measurements of flavor-changing neutral current (FCNC) processes like meson-antimeson mixing and rare decays of

The part of the weak effective Hamiltonian responsible for meson-antimeson mixing in the

In our numerical analysis, we will also keep subleading terms.

inThe vanishing of this Wilson coefficient is in contrast to effective models of partial compositeness or extra-dimensional models based on flavor anarchy, where flavor-off-diagonal terms are induced by the exchange of heavy gluon resonances, i.e., massive spin one states transforming under the adjoint representation of

The two left-right operators are well known to be most problematic in models based on partial compositeness, in particular, in the kaon sector where their QCD matrix elements are strongly chirally enhanced in addition to the RG enhancement of the Wilson coefficients. We thus expect the strongest bound from meson-antimeson mixing observables to come from

Decays based on the

The effective Hamiltonian for

In particular, we neglect dipole operators

^{8}

Charged-current semileptonic decays based on the

Decays where

In addition, we consider the

The effective Hamiltonian for ^{8}

For

For

For

For

To investigate possible NP effects of the MFPC model on the low-energy observables discussed above, we calculate predictions for these observables, depending on the position in the parameter space of the MFPC-EFT. To avoid strong constraints from charged lepton flavor violation (see e.g.,

Note that this assumption is not renormalization group invariant in the presence of lepton flavor universality violation

The observables in our analysis depend on the following MFPC-EFT parameters:

The new strong coupling scale

The six real Wilson coefficients

The four complex Wilson coefficients

The four

Since we assume

A general

The lower boundary for the diagonal entries of

Given the high dimensionality of the parameter space, a naive brute-force scan by randomly choosing each of the parameters is not applicable. We observe, however, that the quark masses and CKM elements only depend on the effective Yukawa matrices

As described above, by adjusting

For predicting the quark masses, we construct the mass matrix in Eq.

Starting from a randomly chosen point in the 19-dimensional parameter subspace where

Starting from this viable point, we use a Markov chain for an efficient sampling of the parameter space, as first proposed in

We reduce the autocorrelation of the 10 k viable points generated by the Markov chain by selecting only 1000 points.

For these points, we then randomly choose the remaining 18 parameters and calculate all the observables discussed in Secs.

Experimental values of

Measurements and SM predictions (computed with flavio v0.23) of flavor observables used in our analysis. The first two blocks are the meson-antimeson mixing and charged current observables used as

As discussed in Sec.

Here we are referring to the genuine dimension-6 NP contributions but remind the reader that CKM elements are varied during our scan; thus, the SM prediction itself also differs from point to point.

we present the histogram in Fig.An interesting feature of the histogram is the fact that there are more allowed points with a NP contribution to

Histogram showing the NP contribution to

New physics contributions to

Predictions for

Apart from modifying the mass differences in the

Predictions for the mixing induced

The precise measurements of

Histogram showing the distribution of the predictions for two observables probing

Lepton-flavor universality in charged currents is also tested in the decays

Predictions for lepton flavor universality tests in

Measurements by the LHCb experiment of the ratios

An alternative explanation with partial compositeness using NP both in the electronic and muonic channels has been suggested as well

Predictions for

We have performed a comprehensive numerical analysis of flavor physics in minimal fundamental partial compositeness. To the best of our knowledge, this is the first numerical analysis of a UV completion of partial compositeness with a realistic flavor structure in the quark sector. Our main findings can be summarized as follows.

Indirect

For the points allowed by the

While we impose the absence of charged lepton flavor violation for simplicity, the violation of lepton flavor

LFU violation in

The MFPC model can explain hints for both LFU violation in

Our explorative study can be generalized in several ways. There are additional low-energy precision tests that we have not considered, e.g., the anomalous magnetic moment of the muon or electric dipole moments. We have also not attempted to construct a realistic lepton sector explaining the origin of neutrino masses or the absence of lepton flavor violation. In contrast to effective models of partial compositeness, the form factors of the new strong interaction, which we have simply scanned here, could also be computed in principle, boosting the predictiveness of the model.

P. S. would like to thank Christoph Niehoff for useful discussions. The work of P. S. and D. S. was supported by the DFG cluster of excellence “Origin and Structure of the Universe.” F. S. and A. E. T. acknowledge partial support from the Danish National Research Foundation Grant No. DNRF:90.

Reference