B decay anomalies and dark matter from strong dynamics
FFrascati Physics Series Vol. XXXX (yyyy)
Title
Date B decay anomalies and dark matter from strong dynamics James M. ClineNiels Bohr International Academy, Copenhagen, Denmark,andMcGill University, Physics Department, Montr´eal, Canada
Abstract
Indications of lepton flavor universality violation in semileptonic B decays to K or K ∗ and muons or electrons canbe explained by leptoquark exchange. I present a model in which the leptoquark is a bound state of constituentscharged under a new confining SU( N HC ) hypercolor interaction. The lightest neutral bound state in the theory is anasymmetric dark matter candidate, that might be directly detectable through its magnetic dipole moment interaction. Strong dynamics has been a useful idea for going beyond the standard model (SM) in the context of several tentativeexperimental anomalies from the past, such as the 750 GeV diphoton excess at LHC and the 130 GeV gamma rayexcess at the Fermi telescope. It has also proved useful for building models of composite dark matter arising from apossibly rich hidden sector. One motivation for such models is the hint of strong dark matter self-interactions fromcosmological N -body simulations versus observations of galactic structure.Recently the LHCb experiment at CERN has found tentative evidence for violation of lepton flavor universalityin the decays of B → K or K ∗ and e + e − or µ + µ − , , O b L µ L = c Λ (¯ s L γ α b L )(¯ µ L γ α µ L ) (1)in the effective Hamiltonian, with c Λ = 1 . × − TeV . (2)By a Fierz transformation, the operator (1) can be put into the form more suggestive of leptoquark exchange,(¯ s L γ α µ L )(¯ µ L γ α s L ). a r X i v : . [ h e p - ph ] M a r U(3) SU(2) L U(1) y U(1) em SU(N) HC Z Ψ 3 1 2 / / N − S N − φ − / , −
1) ¯ N − Our model 5) introduces three new particles: a vectorlike quark partner Ψ and right-handed neutrino partner S , andan inert Higgs doublet φ , all charged under SU(N) HC and an accidental Z , which are listed in table 1. The allowedcouplings to standard model left-handed quarks and leptons are L = ˜ λ f ¯ Q f,a φ aA Ψ A + λ f ¯ S A φ ∗ Aa L af (3)with f being the generation index. We work in a basis where the mass matrices of the charged leptons and down-likequarks are presumed to be diagonal, hence CKM mixing comes exclusively from diagonalization of the up-like quarkmass matrix. After going to the mass basis, the couplings to down-like quarks remain λ f , but those to up-like quarksare rotated, ˜ λ i ¯ Q i → ˜ λ j (cid:0) ¯ u L ,i V ij , ¯ d L ,j (cid:1) ≡ (cid:0) ˜ λ (cid:48) i ¯ u i , ˜ λ i ¯ d i (cid:1) . Because of confinement by the hypercolor interaction, there are various meson-like bound states ¯ SS , ¯ΨΨ and¯Ψ S , the last of which has leptoquark quantum numbers. All of these states can have either spin 0 or spin 1. Thespin-0 (pseudoscalar) leptoquark Π couples to SM fermions through a derivative interaction since the matrix element (cid:104) | ( ¯ Sγ µ γ Ψ) | Π (cid:105) = f Π p µ Π is analogous to that of the pion in QCD. When p µ Π is contracted with the ¯ qγ µ (cid:96) current of theSM fermions, it leads to the small masses m q and m (cid:96) following from the Dirac equation, which suppresses the matrixelement. For this reason the vector leptoquark Φ µ interacts more strongly with the SM fermions. Its matrix elementis (cid:104) | ( ¯ Sγ µ Ψ) | Φ λ (cid:105) = f Φ m Φ (cid:15) µλ for a state with polarization labeled by λ .To determine the effective coupling g fg Φ of Φ µ to the SM fermions, we can compare the decay rate computed inthe effective theory, Γ(Φ µ → L g ¯ Q f ) = | g fg Φ | π m Φ , to its prediction in terms of the constituents in the bound state 6),Γ(Φ µ → L g ¯ Q f ) = σv rel ( S ¯Ψ → L g ¯ Q f ) | ψ (0) | (4)where ψ (0) is the wave function of the bound state evaluated at the origin, and σ is the perturbative cross section forthe indicated scattering. This gives g fg Φ = (cid:18) N HC m Φ (cid:19) / ˜ λ f λ g ( m S + m Ψ ) ψ (0)( m φ + m s m Ψ ) (5)We still need to determine ψ (0). We will be interested in heavy constituent masses, of order the confinementscale Λ HC , for which a nonrelativistic potential model should give reasonable estimates. We take a Cornell potential V c = − α HC r (cid:18) N HC − N HC (cid:19) + 2( N HC − HC r (6)between fundamental and anti-fundamental states, and a hydrogen-like variational ansatz for the wave function, ψ ∼ e − µ ∗ r/ . The scale µ ∗ and the mass of the bound state are then found by minimizing the total energy. Thisallows us to make predictions for the Wilson coefficient of (1) in terms of the fundamental parameters of the theory.All of the nonperturbative physics is encoded in the dimensionless ratio ζ ≡ | ψ (0) | m , (7)It depends only on Λ HC /M , where M is the common mass scale for the new particles that we have assumed forsimplicity. As shown in fig. 1, ζ is always small and is maximized near ζ ∼ = 0 .
004 for M ∼ . HC . As benchmarkvalues we will adopt M = 1 TeV , Λ HC = 400 GeV (8) r = M / Λ HC | ψ ( ) | / m R N=2 N=3 N=4m S ≈ m Ψ m S ≈ 0 Figure 1: The function ζ = | ψ (0) | /m (eq. (7)) Solid curves correspond to both constituents (and the inert doublet φ ) having the same mass M , while dashed ones show the case of m S (cid:28) M . L L QLLQ Q L L Q(a) (b) (c)S ψ S Q QS ψψ Figure 2: Flavor changing neutral currents mediated by the three different kinds of bound states.
We can fit the anomalies in B → K(cid:96) ¯ (cid:96) decays by imposing | λ ˜ λ ˜ λ | ∼ = 0 . (cid:18) M TeV (cid:19) (cid:18) N HC (cid:19) . (9)hence the relevant couplings can be reasonably small. However it is not trivial to find value that satisfy otherflavor constraints. This is because analogous exchanges of ¯ΨΨ bound states give rise to meson-antimeson mixing, asillustrated in fig. 2(c). Especially for B s mixing, the same combination of quark couplings ˜ λ ˜ λ as in (9) is relevant. Tokeep them sufficiently small, we must take λ in (9) to be sizable. An example of values that can satisfy all constraintsis ˜ λ = − . , ˜ λ = 0 . , ˜ λ = 0 . , λ = 2 . λ (cid:48) = 0 . , ˜ λ (cid:48) = 0 . , ˜ λ (cid:48) = 0 . . (10)The predicted values of products of couplings relevant to mixing of the neutral mesons is shown in table 2. We chooseto saturate the B s mixing constraint 7).In addition to meson mixing, there are radiative FCNCs like b → sγ , coming from transition magnetic momentsbetween heavy bound state quark partners Ψ φ and the SM quarks. Because there is mass mixing induced by theinteraction (3) between these states, in the mass basis the transition moment between heavy quark partners and SMquarks induces transition moments between different flavors of SM quarks, notably b and s . However the amplitudeturns out to be well below the current limit.meson quantity upper limit(units M /TeV) fiducial value(units M /TeV) K | ˜ λ ˜ λ | . × − × − D | ˜ λ (cid:48) ˜ λ (cid:48) | × − × − B | ˜ λ ˜ λ | .
026 0 . B s | ˜ λ ˜ λ | .
066 0 . γ SS µ log (m Σ / GeV) -4-3-2-10 l og g Σ XENON100 limitPandaX-II(Banks et al.) composite Σ dark matter Figure 3: Left: diagram generating a magnetic moment for the S fermion. Right: direct detectioon constraint on thedark matter S N HC gyromagnetic ratio.The previous processes have counterparts involving leptons, from fig. 2(b). They can be avoided by assuming λ = λ = 0 (the couplings to first and third generation leptons), which is radiatively stable since to generate themfrom λ at one loop requires a neutrino mass insertion. But in general one finds upper bounds on λ and λ from µ → e , τ → µ , and radiative transitions. The most stringent constraint arises from µ → eγ and τ → µγ , | λ | (cid:46) . × − , | λ | (cid:46) . , (11)The new contribution to ( g − µ is much smaller (by a factor of 300) than needed to explain the outstandingdiscrepancy. The new S particle is neutral under SM interactions, and stable by virtue of the accidental Z symmetry, if it isthe lightest of the new particles. The baryon-like bound state Σ = S N HC is therefore a stable dark matter (DM)candidate. The nonrelativistic potential model predicts its mass to be several TeV, given (8). Previous studies ofcomposite baryon-like DM in this mass and coupling range show that its thermal relic density is highly suppressed byannihilations to hypergluons at temperature above the confinement scale 8 , N HC is odd. The S fermion gets amagnetic moment at one loop from the diagram in fig. 3(a), µ S = e | λ | m S π m φ f ( R ) , (12)where R ≡ m S /m φ and the loop function f ( R ) ∼
1. If N HC is odd, Σ is fermionic and inherits a magnetic momentfrom its constituents of order N HC µ S . Updating older constraints on dark matter with a magnetic moment 10), weobtain fig. 3(b) where the predicted curve (dashed line) is parametrized by m S . Since m Σ varies rather weakly with m S , due to the large contribution to its mass from the hypergluons, the curve is steep as a function of m Σ . It is onlybelow current limits for m S (cid:46)
800 GeV.
Bound states can be produced resonantly at a hadron collider through the processes shown in fig. 4(left). The partonlevel cross sections can be computed in analogy to those for producing QCD bound states like J/ Ψ at an electroncollider. For example the cross section to produce the vector meson ρ Ψ = ¯ΨΨ from q ¯ q is σ ( q ¯ q → ρ Ψ ) = N HC π α s | ψ (0) | m ρ Ψ δ ( s − m B ) (13) _ qq_ g qq_g ψ gq q gqq* φψψ_ψ gg gg (γ)(γ) qq_ * φγ, Z,W γ, Z,W φ _ll (a) (b)(c) (d) ψ_ qq_ * φγ, Zgq q * φψ q_ ψ_ q_ ψψ_ψψ_ψ gggg ψ_ qq_ * φγ, Zgq q * φψ q_ q_ L_ φ * φφ * φφ * φ W + W − ψ_ qq_ * φγ, Zgq q * φψ L_S_S_ L_q_ q_S_ L_ ψ g φ (b)(e)(h)q qgg qggg ψ g φ (c)(f)(i)q qL γ q γγγγψ g φ (a)(d)(g) S L γγ S Lq qS L γ q Figure 4: Left: Resonant production of HC bound states leading to dileptons, dijets or diphotons; right: pair productionof bound states. m R (TeV) -6 -5 -4 -3 -2 -1 σ × A ( pb ) CMS ggATLAS Λ HC = 0.42 m Ψ gg → Π Ψ → jj, qq _ → ρ Ψ → jj √ s = 13 TeV, N HC = 3 p s e udo s ca l a r Π Ψ A = 50 % v ec t o r ρ Ψ h ea vy qu a r k F q
200 400 600 800 1000 1200 1400 1600 m ρ (GeV) σ × β ( pb ) N = N = N = N = N = N = C M S ττ C M S µµ Λ HC = 100 GeV β = 1/4 Figure 5: Left: LHC dijet limits; right: leptoquark search limitswhose nonperturbative component resides in the same ratio ζ as in (7), given that we have approximated all the boundstate masses and wave functions as being approximately the same. Thus we can predict the cross sections for theseprocesses at LHC with no extra freedom from adjusting parameters.The processes in fig. 4(left) produce dileptons, dijets or diphotons. ATLAS and CMS dijet constraints turn outto give the most stringent limits on the model 11 , ρ Ψ vector meson mass must exceed2.8 TeV, which does not yet rule out our fiducial model where all the resonances have mass ∼ = 3 . N HC ) sector, as shown in fig. 4(right). This requires more energy and leads to large phase-spacesuppression of the cross sections. If m S (cid:28) Λ h c then this effect is mitigated for the states containing S , including theleptoquarks. Then leptoquark searches can be used to constrain the model, where final states with two leptons andtwo jets are scrutinized 13 , m S = 0, areless constraining than those from the resonant production searches. Ours is not the first model of composite leptoquarks that has been proposed to account for the B → K(cid:96) ¯ (cid:96) decayanomalies, but we believe it is considerably simpler than others 15 , , , N HC ) bound state properties, that we have estimated in a roughmanner. eferences
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