$B$ decay anomalies and dark matter from vectorlike confinement

Lepton flavor universality violating $B\to K\ell\ell$ and $K^*\ell\ell$ decays tentatively observed by LHCb can be explained by leptoquark exchange. We explore a simple model for the $B$ anomalies with a composite leptoquark from new strong dynamics at the TeV scale, a confining SU($N_{\rm HC}$) hypercolor interaction. The new matter fields, fundamentals under SU($N_{\rm HC}$), are heavy vectorlike fermions $\Psi,\, S,$ and an inert scalar doublet $\phi$. $\Psi$ is colored under QCD while $S$ is neutral, and the hyperbaryon $S^N$ is a dark matter candidate. The model is tightly constrained by meson-antimeson oscillations, lepton flavor violation, and LHC searches for resonant production of the exotic bound states. The dark matter may be detectable through its magnetic dipole moment. If $m_S$ is sufficiently small, composite leptoquarks and heavy lepton partners can be pair-produced at an observable level at LHC.


INTRODUCTION
While the Standard Model (SM) continues to accurately describe particle interactions at the shortest probed distance scales, there is an encouraging hint of new physics from decays of neutral B mesons to K or K * and charged leptons. The LHCb collaboration measures the ratios of branching ratios [1,2] R ( * ) to be (20−30)% below the SM prediction for both K and K * final states. Uncertainties from the hadronic matrix elements cancel in the ratios, greatly reducing the theoretical errors and making these ratios particularly interesting probes of new physics that could violate lepton flavor universality. Although the discrepancies between theory and experiment in the individual measurements are below 3 σ, combining them increases the significance to 4 σ [3][4][5][6][7][8][9][10][11][12][13]. This conclusion remains true when other observables are included in the fit, like the branching ratio for B s → φµ + µ − , which also has a deficit, and angular decay distributions that are subject to hadronic uncertainties. Model-independent fits to the data show that the addition of the single effective operator to the effective Hamiltonian is sufficient to give a good fit to the observations, with [4] c The form (2) arises in the standard model from W exchange at one loop, and can be realized in models of new physics loop effects [14][15][16][17][18][19][20]. It could also be induced by exchange of a heavy Z vector boson, inspiring the construction of many models of this kind . By Fierz rearrangement, (2) takes the form suggestive of vector leptoquark exchange. Global analyses identify two possible vector leptoquarks U 1 and U 3 and one scalar S 1 as viable candidates, in the notation of ref. [55] where the subscript denotes the SU(2) L representation. Models involving leptoquark exchange have been studied in refs. [15,.
In the present work we offer a simple, UV complete model in which the vector U 1 leptoquark arises from confining dynamics of an SU(N HC ) gauge theory, referred to as hypercolor. It is an example of vectorlike confinement [88] in which the constituent fermions have bare masses that can be freely chosen. Our proposal differs from previous models for R K ( * ) with composite leptoquarks [57,71,78,81] where the constituent masses were assumed to be much lighter than the confinement scale Λ HC . In that case, approximate chiral symmetry is important, and the lightest bound states are pseudo-Nambu-Goldstone bosons (pNGBs).
Here we instead assume that the constituent masses M are larger than Λ HC (but not too much larger) so that chiral dynamics plays no role, yet the system is not quirky [89] as occurs if Λ HC M . One advantage of this choice is that the strong dynamics is somewhat more calculable, via a nonrelativistic potential model that works reasonably well for QCD -onium states like J/ψ and Υ. Moreover collider constraints on production of the new bound states are under better control in this regime, where resonant production is suppressed by the wave function of the constituents, and pair production is limited by the large bound state masses. We also thereby avoid complications associated with composite Higgs scenarios.
Another distinctive feature of our model is that it provides a dark matter candidate, which is intimately linked to the B decay phenomenology. In particular, one of the hyperquarks (denoted by S) is a singlet under the SM gauge group. It is a constituent of the composite leptoquark, and the baryon-like S N HC bound state is the dark matter. Exceptionally, S can be lighter than Λ HC without entailing a pNGB, since the approximate U(1) A flavor symmetry associated with S is anomalous. A further difference relative to previous models of composite leptoquarks is that we invoke a scalar hyperquark φ, a doublet under SU(2) L . It is motivated by the need to couple the new physics to left-handed quarks and leptons, eq. (2). Previous models achieved this by taking the fermionic hyperquarks to be doublets, but it is more straightforward to have a dark matter candidate if this is avoided, and the massless limit of S can be safely taken without introducing new relatively light scalars that would be easily produced in colliders. In our model both fermionic hyperquarks S, Ψ, are isosinglets, while Ψ carries color. The SΨ bound state is the leptoquark.
After defining the model in section 2 and reducing it to an effective theory suitable for addressing the B decay anomalies in section 3, we identify (without fully delineating) some regions of parameter space that are consistent with constraints on flavor changing neutral currents (FCNCs) in section 4, dark matter direct detection (section 5), and collider searches (section 6). The examples we focus on have potential for associated new signals in all of these categories of observables. A summary and conclusions are given. Appendices provide details of the potential model for estimating bound state properties, the formalism for computing dipole moments from partial compositeness of the light fermions, and the magnetic dipole moment of the dark matter constituent fermion.

MODEL
Our model introduces three matter fields that are fundamentals of hypercolor (HC): a Dirac fermionic DM particle S, a vectorlike fermion Ψ that carries SM color, and a scalar φ that is an SU(2) L doublet. The quantum numbers are shown in table I. Gauge symmetry allows these fields to couple to left-handed SM quarks and leptons only through the interactions where a(A) is the SU(2) L (SU(N ) HC ) index and i is the flavor index. These interactions explicitly break the approximate flavor symmetries of the standard model (SM), and so one can anticipate that the matching of the observed b → sµ + µ − FCNC will give rise to other FCNC observables. The observed B decay anomaly is mediated by tree-level exchange of a composite leptoquark Φ = ΨS, while the other kinds of bound states ΨΨ and SS give analogous contributions to neutral meson mixing and flavor-violating decays of charged leptons, depicted in fig. 1.
To fully specify the model, we must define the flavor basis of the SM fields in (4). For simplicity we assume that the mass matrices of the charged leptons and downtype quarks are already diagonal in this basis, so that all the mixing giving rise to the CKM matrix is due to diagonalizing the up-type quark masses. Then going to the mass eigenbasis of the quarks, Thus the couplings to up-type quarks are given byλ i = V ijλj . It is worth noting that the individual lepton flavor symmetry for generation i becomes exact in the limit λ i → 0: these couplings are only multiplicatively renormalized, up to neutrino mass insertions. Therefore if we only wish to explain lepton flavor universality violation in the muon sector, it is technically natural to set the couplings λ 1 , λ 3 → 0.
If S is lighter than φ and Ψ, the hyperbaryon bound state Σ = S N is a dark matter candidate. Its stability is protected by the accidental Z 2 symmetry under which all the new matter fields are odd. It is a consequence of the SU(N HC ) gauge symmetry, analogous to baryon number conservation in the SM. Moreover the fields Ψ and S can consistently be assigned normal baryon and lepton number, respectively.

LOW-ENERGY EFFECTIVE THEORY
TheΨS bound states of our model have the quantum numbers of leptoquarks and contribute to the decays B → K ( * ) µ + µ − . The leptoquarks can be either pseudoscalar Π or vector ρ µ , with decay constants where λ labels the helicity state of the vector. These currents couple to the SM fields viaQγ µ L and so the interactions of the pseudoscalar are suppressed by small quark and lepton masses through the equations of motion, due to the momentum factor p µ Π = p µ q + p µ l . This can also be understood in terms of the helicity suppression of the amplitude for pseudoscalar decay to approximately chiral states in analogy to charged pion decay. Hence we are interested in the vector leptoquark for explaining the B decay anomalies.
We seek an effective description of the leptoquark interaction with the SM fields, The coupling g ij ρ can be determined by matching the decay rate for ρ µ → Q iLj in the effective theory and the underlying model. In the effective theory, the rate is neglecting the quark and lepton masses. In the UV theory, this rate can be computed as Γ = σv rel |ψ(0)| 2 , where ψ is the wave function for the ΨS bound state and σ is the cross section for ΨS → Q iLj annihilation, ignoring the initial velocities of the bound particles, and assuming only the spin-1 configuration of S and Ψ contributes in the sum over spins. This gives Once the effective interaction (7) is specified, we can integrate out the leptoquark to generate the dimension-6 operator shown in fig. 1(a), after Fierz transforming [90], where a, b are SU(2) L indices. This contribution must interfere destructively with the SM contribution to b → sµ + µ − , requiring g * 32 g 22 to be approximately real and positive.
In addition to the composite leptoquarks, there are compositeΨΨ ≡ ρ Ψ andSS ≡ ρ S vector bosons, whose exchanges are shown in fig. 1(b,c). The ensuing operators are where h ij and k ij are determined analogously to (10), withλ i λ * j → λ i λ * j andλ iλ * j , respectively.

FLAVOR CONSTRAINTS
The fit of ref. [4] implies that the R K ( * ) anomalies can be explained by taking To translate this into a constraint on parameters of the model, we must determine ψ(0) and the bound state mass. For this purpose we use a potential model that is described in appendix A. To simplify this preliminary analysis, we assume that m Ψ ∼ = m Φ . 1 We consider two possibilities for the dark constituent mass, Λ HC m S m φ ≡ M and m S Λ HC m φ , such that S is respectively nonrelativistic or relativistic within the bound state. Two version of the potential model are given to treat these cases. We avoid the quirky regime where Λ HC M [89] since collider constraints are expected to become more stringent (and difficult to quantify), as we will discuss in section 6.
In either case, the coefficients of the dimension-6 operators (12) that encodes the effects of the confining dynamics, where m R is the mass of the exchanged resonance. ζ is dimensionless and thus only depends upon ratios of mass or energy scales (and N HC ). In the case where all masses are approximately equal to scale M , there is just one such ratio, r = M/Λ HC . Using the nonrelativisitic potential model in the region M/Λ HC > 1, we numerically determine ζ(r) for N HC = 2, 3, 4, shown in fig. 2 (solid curves). It reaches a maximum value of ∼ 0.0037 for r ∼ = 3 if N HC = 3, 4, and is smaller if N HC = 2. For the case m S Λ HC , the low scale m S becomes irrelevant and we can can still express ζ as a function of the same r = M/Λ HC , using the relativistic version of the potential model. ζ is smaller for these models, ∼ 0.002 for r ∼ (2 − 3).
Since ζ is small, the best case for avoiding large couplings λ 2 ,λ 2,3 in the underlying model is to have m φ ∼ m Ψ ∼ M and Λ HC ∼ (0.3 − 0.4)M for N HC = 3 or 4. Then with ζ = 0.0037, eq. (13) implies From the potential model, the bound state mass m ρ at Λ/M = 0.4 and N HC = 3 is predicted to be 1.75 times the naive value of 2M that ignores the contribution from the hypercolor flux tube. For M = 1 TeV therefore, m ρ = 3.5 TeV.
1 Also this choice maximizes the wave function at the origin In the case where m S Λ HC , the couplings must be increased by ∼15%, to compensate for the smaller ζ factor. From the partially relativistic version of the potential model, the mass of the resonance is found to be approximately m ρ ∼ = (0.56 + 1.8 N HC ) Λ HC + m Ψ (16) valid for m Ψ /Λ HC 3; for smaller values m ρ tends toward a constant independent of m Ψ .

Meson-antimeson mixing
Diagrams of the type fig. 1(c) contribute to the oscillations of neutral mesons: s -B 0 s , placing constraints on different combinations of theλ i couplings by similar reasoning as led to eq. (15). For the fiducial model parameters with ζ = 0.0037 and N HC = 3, we find the upper limits given in table II. These are inferred by comparing the operator coefficients k 2 ij /2m ρ 2 Ψ with the upper limits from ref. [16] for B s mixing and refs. [91? ] for K, D and B d mixing (which gives the most stringent bounds in each case). The bound on K and D mixing are conservative in the sense that we have assumed that the contribution to the imaginary part of the amplitude, which is much more strongly constrained, is as large as that of the real part.
As an example of parameters that can satisfy these constraints, we take M = 1 TeV and λ 1 = −0.01,λ 2 = 0.1,λ 3 = 0.66, λ 2 = 2.1 (λ 1 = 0.014,λ 2 = 0.13,λ 3 = 0.66) The primed couplings (pertaining to the up-type quarks) are determined by the unprimed ones through eq. 5. The resulting contributions to the products of couplings relevant to meson mixing are shown in table II (last column). The relative minus sign betweenλ 1 andλ 2 leads to an accidental cancellation in V ijλj such that the D 0 mixing contribution is just below the limit; without this sign, we would getλ 1λ 2 = 0.004, saturating the limit. The fact that all the mixing constraints are close to being saturated, with the exception of B s , can be understood from the lack of any approximate flavor symmetry such as minimal flavor violation [94] in this model. By requiring a large enough b → sµ + µ − transition and trying to avoid couplings much larger than 1, we are pushed into this restricted region of parameter space.
One way to relieve the tension is to take smaller values of M , say 800 GeV. Then the couplings could on average be smaller by a factor of √ 0.8 than in eq. (17). If we keep the leptonic coupling large, λ 2 = 1.1, the quark couplings can all be reduced by 0.8, and the last column in table II is reduced by 0.64, leading to a 60% reduction in the actual amplitude. Allowing for larger leptonic coupling can further ameliorate the situation, which is thus not as marginal as the limiting case makes it appear. M cannot  (14)) that encodes the strong dynamics dependence of the exchange of bound state resonances between SM fermion bilinears. Solid curves are for all constituents having same mass M , while dashed are for the case of mS ΛHC.
meson quantity upper limit (units M /TeV) be made lower than 800 GeV because of LHC constraints, as we will see in section 6. Our main goal in this paper is to establish some region of viability for the model. A thorough exploration of the parameter space will become more strongly motivated if the B decay anomalies are confirmed with greater significance.

Lepton flavor violating decays
As mentioned above, nothing obliges us to turn on the couplings λ 1 , λ 3 to e and τ leptons in this model, apart from aesthetic considerations. If these couplings are nonvanishing, then the diagram of fig. 1(b) contributes to lepton-flavor violating (LFV) decays τ → 3 (where is µ or e) and µ → 3e. We can derive upper bounds on |λ 1 |, |λ 3 | by comparing the amplitude from exchange of the ρ S = SS vector meson, to the SM amplitude 2 for the corresponding allowed decay, giving a ratio of branching ratios of R ij = 1.8 × 10 −9 |λ i | 2 |λ j | 6 . From τ → 3µ we require R τ µ < 1.2 × 10 −7 and from µ → 3e, R µe < 10 −12 ,  giving This implies that the anomalous contribution to the b → se + e − amplitude (which goes as |λ 1 | 2 ) is less than 0.01 times that of b → sµ + µ − .

Radiative transitions
Another potentially important class of FCNCs is the radiative decays such as b → sγ, τ → µγ. They arise through the mass mixing of SM fermions with heavy composite fermions from the confining sector. These are vectorlike doublet leptons and quarks, φS ≡ F and φΨ ≡ F q , that naturally have magnetic moments coming from their charged constituents. They contribute to dipole operators of the light fermions in the mass eigenbasis, including transition magnetic moments.
The mass mixing in this kind of model was previously considered in ref. [95]. Here we make the connection to transition moments more precise, using methods that were previously applied to the rare decays of Z → Υγ in the SM in ref. [96]. The contributions look like loop diagrams ( fig. 3) but the integral is over the relative momentum of the bound constituents and is weighted by the wave function in momentum space. Details are given in appendix B. There we show that the mass-mixing between the SM and heavy fermions takes the form where ψ is the wave function of the bound state and M is the mass of the constituents, that we continue to take to be equal for simplicity. For unequal masses, M is replaced by twice the reduced mass.
In addition, as shown in appendix B, the diagrams of fig. 3 give rise to dimension-6 transition moment operators of the form where q f is the electric charge of particle f , a quark or lepton. Ref. [97] noted that these lead to transition magnetic moments among the light fermion states upon diagonalization of the mass matrices, which take the form where f, F denote the SM and heavy fermions, respectively, with m f being the SM mass matrix and M F the heavy composite mass. To leading order for small mixing, (22) is diagonalized by separate left-and right-handed with mixing angles that are vectors in flavor space, After transforming to the mass eigenbasis, a transition moment for the light fermions is generated, It is proportional to |ψ(0)| 2 /M 4 F = ζ/M F rather than just ζ (recall eq. (14)). The same potential model can be used to compute M F as we used for the bosonic bound states, since quantum mechanics is insensitive to the spins of the constituent particles, and we are ignoring the spinspin interactions in our simple treatment. Then for the fiducial parameters with Λ HC = 0.4 M , |ψ(0)| 2 /M 4 F = ζ/(3.5 M ).

FCNC quark transitions
From the general formula (24), the operator contribut- This implies that the conventionally normalized Wilson coefficient of the dipole operator O 7 is using the parameter values (17) with Λ HC = 400 GeV, well below the experimental limit of ∼ 0.02 [98]. Under the scaling of parameters described at the end of section 4.1, this prediction remains unchanged. Similarly for c → uγ, we predict a transition magnetic moment like (25) withλ i →λ i , eq. (5). Using the values (17), the contribution to the c → uγ amplitude is ∼ 10 −2 smaller than the SM contribution. Likewise, the contribution to d → sγ is orders of magnitude below the limit from K S → π 0 + − decays (see section 7.7 of [99]).

FCNC leptonic decays
The radiative decays i → j γ proceed analogously to those of the quarks, due to mixing with the heavy composite lepton whose mass is M F ∼ = 3.5 M for our fiducial parameter choice. The induced transition magnetic moment is and the partial decay width is δΓ = µ 2 ij m 3 i /8π [100]. This gives limits respectively, from µ → eγ and τ → µγ that are more stringent than (19).

Muon anomalous magnetic moment
The contribution to the muon anomalous magnetic moment is For our benchmark values, we get a µ ∼ 10 −11 , which is 300 times too small [101]. It will be seen below that LHC constraints prevent taking M F to be smaller than the TeV scale, leaving little room for increasing a µ .

COMPOSITE DARK MATTER
The η -like SS bound state is unstable to selfannihilation into µ + µ − through φ exchange, but the hyperbaryonic bound state Σ ≡ S N is a stable dark matter candidate. Determination of its thermal relic density can depend upon the annihilation of the constituent SS particles above the SU(N HC ) confinement transition, as well as ΣΣ annihilations following confinement. For the parameters of interest, with Λ HC ∼ 400 GeV and m S 1 TeV, it has been shown [95,102] that the resulting relic density is too small (by a factor of ∼ 1000). However since hyperbaryon number is conserved, 2 it is possible that an asymmetry between S andS was produced in the early universe, so that Σ can be asymmetric dark matter. We make this assumption here, without attempting to account for the origin of the asymmetry.
From the nonrelativistic potential model, the DM mass can be numerically determined as a function of m S . In the range 1 < m S /Λ HC < 10, we find the approximate fit The dependence on m S is shown in fig. 4(left), including the limit m S → 0 (heavy dots on y-axis), obtained from the relativistic version of the potential model.
Although S has no direct interactions with nuclei, it gets a magnetic moment at one loop (with φ ± or µ ± in the loop), depending upon If Σ has spin, it inherits a magnetic moment from its S constituents. This will be the case if N HC is odd. If N HC = 3, the wave function of S a S b S c is abc in hypercolor space, and if the spatial wave function is s-wave, total antisymmetry demands that the spins be aligned, like the ∆ baryon in QCD. The quark model then predicts that Σ has a magnetic moment that is three times that of S.
The gyromagnetic ratio of Σ is given (for odd N HC ) by g Σ = 4N HC µ s M Σ /e. Ref. [103] derived an upper limit on g versus DM mass based upon early XENON100 data [104]. We update their limit by comparing the relative sensitivities of that search to the recent PandaX-II result [105] in the high mass regime, to constrain our model. Allowing m S to vary between Λ HC ∼ = 400 GeV (the value that maximizes ζ if M = 1 TeV) and 1 TeV, we rescale λ 2 2 by (ζ max /ζ) 1/2 where ζ max = 0.0037 for the fiducial model parameters, and ζ is smaller for m S < M . This keeps the predicted B decay signals constant while varying m S , if all the couplings scale in the same way. Using (30), g Σ and m Σ are determined as a function of m S . The result for N HC = 3 is shown in fig. 30(right). It translates to an upper limit m S 800 GeV. (ζ is not appreciably decreased at this value of m S .)

COLLIDER CONSTRAINTS
Because of hypercolor confinement, only HC-neutral bound states of the new particles φ, Ψ, S can be produced in high-energy collisions. We expect the dominant mode for discovery to be resonant production of vector and scalar HC mesons, through the diagrams in fig. 5. We work in a region of parameter space where Λ HC is not too much lower than the constituent masses M , which avoids the complications of quirks [89]. In the quirky regime, there are many excited quantum states to scatter into, that can efficiently radiate hypergluons and settle to the ground state before self-annihilating. Then partons with energy above that of the resonance could still be effective for production.
On the other hand for Λ HC M , the resonances are well-separated and can only radiate QCD gluons or electroweak gauge bosons since the HC glueball mass is too heavy, ∼ 7Λ HC [106] for N HC = 3. In this regime, the parton center of mass energy must match the mass of the resonance, and we can use similar techniques for computing resonant production of bound states as for J/ψ and Υ in e + e − collisions. This approximation misses events in which the bound state is produced by partons above the threshold for the resonance, plus QCD radiation to take away the excess energy. We defer study of such higher-order corrections.

Production cross section
In general, the production cross section from colliding protons to form a narrow resonance R from partons p i , p j can be expressed as where f i is the parton distribution (PDF) for p i , 2s i + 1 and N ci are the number of spin states and colors respectively of the incoming particles, 2J + 1 and N c likewise for the resonance, and Γ is the partial width for the decay that is inverse to the production process. (In the case of incoming gluons, the doubling of PDFs in the second line is correct, compensating for the factor of 1/2 in the phase space for decay of R into identical particles.) The generalization to resonances carrying SU(2) L rather than QCD quantum numbers is obvious.

Predicted widths
The decay widths of bound states have been quantitatively addressed in regimes where the constituent masses are either much heavier than the confinement scale [89] or much lighter [88,107]. We are interested in the case where Λ HC ∼ 0.4 M (with the possible exception of m S Λ HC , but this is not relevant here because the SS bound states cannot be produced at LHC). For N HC = 3, the potential model predicts that the kinetic energy of the constituent is 0.3 of its mass energy, so the nonrelativistic approximation is not very bad. Ultimately, lattice calculations should be done to make more quantitative predictions.
We ignore decays of bound states into hypergluons, since we work in a regime where any hyperhadrons are too heavy to be produced. Even if m S < Λ HC , the Π S = SS pseudoscalar is not a light pseudo-Nambu Goldstone boson, because the approximate chiral U(1) flavor symmetry is anomalous, like for the η of QCD. Likewise the glueballs of the SU(N HC ) are also too heavy.
Ref. [89] calculates (in terms of |ψ(0)| 2 ) the decay widths assuming that the bound state constituents do not form a resonance of definite spin, but these are straightforward to rescale for the physical eigenstates of spin. 3 Doing this gives results in agreement with the treatment of bound state decays in ref. [108]. In our model, the ΨΨ states can be in the QCD color singlet or octet representations. Given that Λ HC Λ QCD , this distinction is unimportant for the dynamics since the lifetime of the bound state is much shorter than the QCD hadronization time scale.
The last one, unlike the previous two, involves only the color singlet state so it does not entail an average over QCD colors. We include it because it is potentially relevant for the dilepton final state. However counting channels, it predicts a branching ratio of B l + l − = (α/α s ) 2 /2 ∼ = 2.5 × 10 −7 into electrons or muons, implying that the dilepton channel is actually unimportant for this resonance.

φφ resonance
There are four kinds of ρ φ = φφ * vector bound states, three components in an SU(2) L triplet and one singlet. Since the SU(2) L dynamics are not important at the TeV scale, we average over the isospins of the bound state constituents. The angular momentum of ρ φ is purely orbital, so the constituents are in a relative p-wave for which ψ(0) = 0. Therefore the width depends upon the derivative of ψ at the origin. Following [109], we find that the SU(2) L contribution to the width is for decay into a single generation of approximately massless quarks. For decays into leptons, the result is 1/3 times smaller (from lack of QCD color), giving a branching ratio of 3/4 to quarks and 1/4 to leptons. Hence this process leads to both dijet and dilepton signals. For decays into muons, there is an additional, potentially larger contribution from S exchange (that interferes with the W exchange contribution, but we ignore this interference for purposes of estimation), given by

Composite fermion
For decay of the composite fermion to qg, taking m φ ∼ = m Ψ as before, we find where i is the generation index of the quark and as usual the prime is for couplings to up-type quarks. The squared coupling must be absorbed into sum over quark flavors in the parton luminosity factor for production via qg fusion.

Vertex correction
For all of the annihilation decays, a more quantitative estimate can be made by taking account of perturbative corrections that dress the annihilation vertex. For charmonium and upsilon, the correction from a gluon loop is known to be important, and we adopt the analogous correction for the HC gluon exchange, with α HC evaluated at the scale of the constituent mass. The correction factors depend upon the annihilation process, and in analogy with QCD we take [110]  with α HC evaluated at the scale µ * , twice the inverse Bohr radius. We have assumed that the known N HC = 3 results generalize to other values by rescaling by the number of hypergluons.

Resonant search constraints
Combining the decay widths with the general formula for the production cross section (33), we find that it depends upon the same combination ζ, eq. (14) that appeared in the dimension-6 FCNC operators. One finds a production cross section for the color octet plus singlet states Π Ψ given by where L gg is the parton luminosity factor for gluon fusion. The perturbative correction (40) is C = 0.5 at m Ψ = 2.5Λ HC , similar to the value that occurs in QCD for J/ψ, and ζ = 0.0037 as in the previous sections. In fig. 6 we plot this prediction along with current constraints from ATLAS [111] and CMS [112] resonant dijet searches, assuming an acceptance of events passing experimental cuts of A = 50%, comparable to that of various models tested in the searches. The resonance mass must exceed 2.3 TeV, which is satisfied for our fiducial model with m Π = 3.6 TeV. Keeping m Ψ /Λ HC fixed, this limit would allow m Ψ to be no lower than 650 GeV. Similarly for production of the color octet vector ρ Ψ resonance for N HC = 3 we find  The loop correction factor (40) is 0.7 for this process. The corresponding prediction is also plotted in fig. 6. 4 We infer a limit of m Ψ > 820 GeV for the fiducial model.
At larger values of m Ψ /Λ HC = 5, 10, this limit becomes more stringent, m Ψ 1 TeV, approximately independent of the value of m Ψ /Λ HC . In addition to the Π Ψ → gg channel, the color singlet pseudoscalar can decay to two photons. Its decay width into gluons is smaller than the color average (34) by a Figure 9: Pair production of composite states factor of 16/9, and its branching ratio into photons is B γγ = 9α 2 q 4 ψ /2α 2 s [113]. The cross section for pp → Π Ψ → γγ is therefore (9/128)B γγ times eq. (41). The predicted cross section is shown along with the ATLAS diphoton limit [114] in fig. 7; it constrains m Π > 1.7 TeV.
The cross section for heavy quark F q production is The result assuming couplings (17) and 100% branching into dijets is plotted in fig. 6, giving a weaker constraint than those from ρ Ψ and Π Ψ resonant production.

Pair production of bound states
The wave function at the origin (the ζ factor) suppresses the predicted resonant production signficantly below the cross section that would govern the creation of free pairs of hypercolored constituents. This penalty is avoided if bound states are pair-produced by hadronization in the HC sector, as depicted in fig. 9. Then each possible kind of final state will occur with a relative probability not much smaller than 1 since for every pair there are only three kinds of particles (Ψ, φ, S) that can appear from the vacuum. The only other bottleneck is that the incoming partons must have at least enough energy to make both bound states. Then the cross section aver-  Figure 10: Cross section for pair production of composite leptoquarks times branching ratio squared for decays into µ or τ plus jet (j). Solid is CMS upper limit for jjµµ [117] or jjτ τ [118] final states, dotted is prediction from qq → ρρ * , dashed is prediction from gg → ρρ * , assuming low compositeness scale ΛHC = 100 GeV and massless S constituents.
aged over PDFs takes the form where m c is the mass of the composite state. The largest cross sections are for qq → ΨΨ and gg → ΨΨ, with χ = 1 − 4m 2 Ψ /ŝ andŝ = x 1 x 2 s. We find that the composite mass m c must be as small as possible to get interesting constraints from this process. This occurs for m S Λ HC in the ΨS bound states, for which the potential model gives These are leptoquark states that decay into q , and have been searched for by ATLAS [116] and CMS [117,118]. The searches look for two jets and two leptons, either µ + µ − or τ + τ − respectively, for second or third generation leptoquarks. The limits are on the production cross section times β 2 where β is the branching ratio into the final state searched for. In our model, β ≤ 1/2 since ρ can decay into either a charged lepton or neutrino with equal probability. Moreover the branching ratio into muons is |λ 2 | 2 /(|λ 2 | 2 + |λ e | 2 , and that into tau is |λ 3 | 2 /(|λ 2 | 2 + |λ e | 2 , typically giving a further reduction in β. (Recall that LFV constraints on λ 1 make the branching into e + e − negligible in our model.) The predicted cross sections and observed limits are shown in fig. 10 for a model with Λ HC = 100 GeV and m S Λ HC , which is a limiting case for lowering the masses of bound states containing S. We have assumed λ 3 ∼ λ 2 so that β ∼ = 1/4. The most stringent limits come from the search for final state muons, with production via the gg → ΨΨ cross section (47). It requires the leptoquark mass to be 1.1, 1.2, 1.26 TeV for N HC = 2, 3, 4, hence m Ψ 670, 590, 470 GeV respectively.
Other bound states not involving S as a constituent, although they have potentially interesting signals, are too heavy to be produced at a significant level relative to current constraints from LHC. The composite lepton F l = Sφ * arises from electroweak qq → φφ * pair production, with parton-level cross section in the simplifying approximation g g (sin θ W → 0). CMS has searched for excited muons and taus decaying to the normal state plus photon [119]. Figure 11 shows the limit on excited µ production, which is more stringent than that for excited τ , versus the predicted cross sections (100% branching of F l → µγ is assumed). Again we take the extreme choice with Λ HC = 100 MeV and m S = 0 to obtain light enough m F l to fall within the currently probed mass range; larger values of Λ HC and m S are not excluded. Comparison with fig. 10 shows that F l is excluded up to similar masses as the leptoquarks, m F l 1.3 − 1.5 TeV, depending upon N HC .
The remaining mechanism to produce the lightest states is diagram (g) of fig. 9, which has partonic cross sectionσ neglecting the quark mass (a good approximation since the top quark PDF is very small). This produces the heavy lepton F l in conjunction with a leptoquark, and would be probed by the CMS search [119]. The predicted cross section (assuming 100% F l → µγ) is also shown in fig. 11 and happens to be close to that coming from F l pair production. Since the leptoquark is produced singly, it does not produce the same signal as for the existing searches that assume pair production and would require a new analysis, searching for the unusual final state of two leptons, a jet and a photon.

SUMMARY AND CONCLUSIONS
We have presented a particularly simple realization of composite leptoquarks to explain the anomalous B decay ratios R K ( * ) , as well as providing a composite dark matter candidate. The new ingredients are an SU(N HC ) confining gauge sector under which three types of matter fields Ψ, S, φ transform as fundamentals. Only two kinds of interactions with SM fermions are allowed, and these break the approximate flavor symmetries of the SM through the six couplings λ i to leptons andλ i to quarks. The leptoquark is the ΨS vector bound state.
There is a limited region of parameter space allowed by FCNC constraints on meson-antimeson oscillations (mediated by the ΨΨ vector bound states), direct dark matter searches, and LHC searches for resonant dijets, which require the Ψ and φ constituents to be near the TeV scale, and the new confinement scale Λ HC must be 100 GeV. (Our benchmark model takes Λ HC = 400 GeV.) The dark matter is a baryon-like bound state Σ = S N HC with mass ∼ (1 − 5) TeV. If N HC is odd, Σ can have a magnetic dipole moment that requires m Σ < 2.5 TeV to avoid direct detection.
For very light dark matter constituent and low confinement scale with m S Λ HC ∼ 100, the new bound states (leptoquarks and heavy lepton partners) are light enough (∼ 1 TeV) to be pair-produced and significantly constrained by ATLAS or CMS searches for leptoquarks and excited leptons. The model also predicts production of the leptoquark and heavy lepton in association, leading to the unusual signal of two muons, a jet and a photon. The dark matter dipole moment is below direct detection constraints for small m S .
The predictions are sensitive to nonperturbative aspects of confinement, especially bound state masses and wave functions. We have used a simple potential model that works reasonably well for known examples from QCD, but which is not necessarily reliable for the parameters of interest here, and neglects spin interactions between the fermionic consituents. Lattice studies of the masses and decay constants would allow for a useful improvement. Nevertheless it seems likely that the kind of model presented here should give rise to other anomalous signals in FCNC, dark matter or collider searches. It is encouraging that the experimental status of the B decay anomalies should become more clear from future data at LHCb and Belle II during the next few years [120].
We first consider bound states where all the constituents are heavier than Λ HC and hence nonrelativistic. Following [109,121], we estimate the size of hypercolor mesonic bound states using a variational method, with hydrogen-like ansätze for the wave functions. The Coulomb-like contribution to the potential from hypergluon exchange is [122] for a hyperquark-antiquark pair in the hypercolor singlet state. There is a linear confining potential where σ ∼ = 2(N HC − 1)Λ 2 HC is the string tension and Λ HC is identified with the Λ parameter of the 0-flavor running coupling in the MS scheme. The coefficient is deduced for the case of N HC = 3 from lattice studies [123] and its generalization to other values of N HC is inferred from large-N scaling. The energy of an s-wave state with wave function where m r is the reduced mass of the constituent particles, and α HC implicitly depends on µ * since it is the running coupling evaluated at that scale. We thus minimize the energy numerically. We use the 4-loop, 0-flavor expression for the running coupling given in [124,125]. For µ * < Λ HC the perturbative coupling diverges and we cut it off at α HC = 2. (Larger cutoffs lead to numerical artifacts in the minimization.) This model gives reasonable results for J/ψ and Υ using m c = 1.3 GeV, m b = 4.2 GeV for the quark masses, and Λ QCD = 200 MeV. For these systems ψ(0) is measurable through the electromagnetic decays via γ * to e + e − . For J/ψ we find ψ(0) 2 = 1/(0.5 fm) 3 , close to the measured value 1/(0.53 fm) 3 , while the mass is predicted to be 3.0 GeV, compared to the measured 3.1 GeV. The model predicts ψ(0) 2 Υ /ψ(0) 2 J/ψ = 4.6, while experimentally one finds 4.7, by comparing the decay rate formula (including the correction (40)) to the observed partial widths. The worst agreement is for m Υ , predicted to be 8.6 GeV versus the measured value 9.5 GeV.
For p-wave states with we find the energy For baryons in an s-wave state, we take the ansatz ψ ∼ e −µ * (r1+r2+··· ) and the energy is (note that the Coulomb attraction between qq in an antisymmetric hypercolor state is 1/(N HC − 1) times weaker than that between qq). Comparison to the nucleon of QCD motivates the correction factor f = 0.065, indicating that the string tension is much smaller between qq than qq states.

Heavy-light or relativistic systems
In the case where m S < Λ HC , one or more constituents is relativistic. Then in the center of mass system of a heavy-light meson we have mass plus kinetic energy m 1 + p 2 /(2m 1 ) + p 2 + m 2 in the Hamiltonian [126]. Still using the wave function ansatz ψ ∼ e −µ * r/2 for swave states, the expectation value of p 2 + m 2 can be found by Fourier transforming ψ and evaluting the integral in the momentum eigenstate basis. The result is a complicated analytic function that can be fit to the simpler form (At m = 0 the exact result is 8µ * /(3π).) We find that this approximation is good to better than 1% for m < µ * . The kinetic energy in (A4,A6) is then replaced by For baryons with some constituents light, the appropriate replacement is obvious. The coupling α HC should be evaluated taking into account running with the appropriate number of light constituents.

Appendix B: Transition moments from compositeness
Here we semiquantitatively estimate the transition magnetic moment interactions between heavy composite fermions F R and their elementary SM counterparts, f R , following the same formalism used by ref. [96] for radiative decays of the Z boson to quarkonium bound states. The formalism is appropriate for nonrelativistic systems, which is not a very good approximation in our case since we prefer the constituent masses and confinement scale to be of the same order. Hopefully it gives a reasonable estimate, which would require a dedicated lattice study to improve upon.
The amplitude for F → f i γ can be written as where the momenta are indicated in fig. 3. Q is the momentum of they decaying heavy F , k is that of the photon, q is the relative momentum between the heavy fermion constituents and χ(Q, q) is the Bethe-Salpeter wave function for the bound state. The operator O γ comes from the part of the diagram that is introduced by insertion of the photon vertex, which depends upon what kind of particle F (f ) is. For down-like quarks, the photon attaches to both Ψ and φ ± ; for up-like quarks it attaches only to Ψ (since φ 0 is the other constituent), while for leptons it attaches only to φ ± since S is neutral.
For simplicity we take the constituent masses to be equal so that in the absence of exchanged momenta, each carries half of the momentum Q µ of the bound state. The general form of O can then be written as where P R projects onto right-handed chirality. The spinor wave function takes the form where µ is the reduced mass of the bound state constituents, u ψ(S) is the Dirac spinor for the fermionic member, andψ is the Fourier transform of the spatial wave function. For nonrelativistic systems, the wave function is strongly peaked at small q and it is a good approximation to set q = 0 in O γ . Then the factor d 3 q/(2π) 3ψ (q) = ψ(0), the spatial wave function evaluated at the origin, and the amplitude becomes fi O γ (0, k, Q) P R u ψ(S) (B4) It is pertinent to compare this to the corresponding amplitude with no photon, which is just the mass mixing amplitude, λ i ψ(0) √ 2µū fi P R u ψ(S) = −iµ i fūfi P R u ψ(S) (B5) This allows us to infer that the transition moment is related to the mass mixing in a definite way.
For leptons, the decay amplitude from diagram (a) of fig. 3 is where in the second line we used the Gordon identity (p + p ) µū L u R = m ū R γ µ u R + mū L γ µ u L +ū L iσ µν k ν u R (B7) and took m φ m F /2. The extra terms going as γ µ in the Gordon identity are canceled by diagrams (c) and (d) of fig. 3, which serves as a check on the sign. We ignored terms going as k µ that do not contribute to the amplitude because of transversality of the photon polarization vector.
Similarly for up-and down-type quarks respectively we obtain where µ i u =λ i Λ HC withλ i given in eq. (5), and µ i d = λ i Λ HC . After rotating the heavy and light fields to eliminate mass mixing, these lead to magnetic moments involving just the light states.

Appendix C: DM dipole moment
We give details for the one-loop contribution to the magnetic dipole moment of the elementary dark matter particle S, with diagrams shown in fig. 12. They turn out to be infrared convergent in the limit m µ → 0 for the internal muon, hence we make this simplification. The 1PI diagrams (a,b) respectively give e|λ 2 | 2 2 where, defining u = x + y, As usual, one shifts the loop momentum variable to simplify the denominators. Only the finite parts contribute to the dipole operator. To extract the dipole moment, one makes the replacements P L / pγ µ → P L ( / p + / q)γ µ → (P R m S + P L/ q)γ µ γ µ / p P R → γ µ ( / p − / q)P R → γ µ (m S P L − / qP R ) (C2) in diagram (a), using the Dirac equation (and similarly letting / p γ µ → m S γ µ , γ µ / p → m S γ µ ), and keeping the terms linear in / q. Both of these can be put into the form γ µ / q up to terms going as q µ , that give a vanishing contribution when contracted with the external photon polarization vector. These are directly related to the magnetic moment since −iσ µν q ν → γ µ / q in this way. In diagram (b), we again use the Dirac equation to rewrite / pP R → m S P L , P L / p → m S P R , and reexpress p µ , p µ as linear combinations of (p+p ) µ and q µ , of which only the former contribute to the amplitude. Using the Gordon identity, (p + p ) µ → iσ µν q ν → −γ µ / q. We find that both diagrams (a,b) contribute with the same sign. A check on this relative sign is provided by gauge invariance: the couplingSγ µ SA µ is forbidden, and must not be generated by loops. The divergent parts of diagrams (a,b) contribute with the same magnitude and sign to the vector current, and these cancel the contributions from (c,d). Such a cancellation would not occur if (a) and (b) came with opposite signs.
Defining R = m 2 S /m 2 φ , the final expression for the magnetic moment (which is the coefficient ofū p γ µ / qu p ) is In the limit of light dark matter, f a (0) + f b (0) = 1/2.