In this talk, a unified model of scalar particles that addresses the flavour hierarchies, solves the strong CP problem, delivers a dark matter candidate, and radiatively triggers electroweak symmetry breaking is discussed. The recently proposed axiflavon is embedded together with an (elementary) Goldstone Higgs-sector in a single multiplet (and thereby also a model of flavour and strong CP conservation for the latter is provided). Bounds on the axion decay constant follow from requiring a SM-like Higgs potential at low energies and are confronted with constraints from flavour physics and astrophysics. In the minimal implementation, the axion decay constant is restricted to $f_a \approx (10^{11}-10^{12})$ GeV, while adding right-handed neutrinos allows for a heavy-axion model at lower energies, down to $f_a \sim 10$ TeV

Alanne, T.

Blasi, S.

Goertz, F.

English

MPG.PuRe

AXIFLAVON-HIGGS UNIFICATIONTOMMI ALANNE, SIMONE BLASI, FLORIAN GOERTZ aMax-Planck-Institut für Kernphysik, Saupfercheckweg 1,69117 Heidelberg, GermanyIn this talk, a unified model of scalar particles that addresses the flavour hierarchies, solvesthe strong CP problem, delivers a dark matter candidate, and radiatively triggers electroweaksymmetry breaking is discussed. The recently proposed axiflavon is embedded together withan (elementary) Goldstone Higgs-sector in a single multiplet (and thereby also a model offlavour and strong CP conservation for the latter is provided). Bounds on the axion decayconstant follow from requiring a SM-like Higgs potential at low energies and are confrontedwith constraints from flavour physics and astrophysics. In the minimal implementation, theaxion decay constant is restricted to fa ≈ (1011 − 1012) GeV, while adding right-handedneutrinos allows for a heavy-axion model at lower energies, down to fa ∼ 10 TeV.1 IntroductionAlthough being very successful in describing nature in many aspects, the Standard Model (SM)of particle physics lacks explanations for several experimental facts, such as the significantabundance of Dark Matter, the large hierarchies in fermion masses and mixings, and the excellentconservation of CP symmetry in strong interactions, the latter being in tension with in principlesizable sources of CP violation in the QCD/SM Lagrangian. Beyond that, although the SMallows to parameterize electroweak symmetry breaking (EWSB) via the Higgs mechanism, theorigin of the Higgs potential remains unexplained.While separate solutions to all these problems are well known in the literature, here weentertain a unified model, simultaneously addressing all these problems at a time via a singlescalar multiplet. To this end we show that it is possible to identify a radial component of thismultiplet with a Froggatt-Nielsen-like flavon, whose vacuum expectation value (vev) now breaksan appropriate enhanced global symmetry G – including a horizontal U(1)H flavour symmetry– down to H 6⊃ U(1)H. In turn, the corresponding (pseudo) Goldstone bosons of the G/H cosetcan be identified with the axion and a (Goldstone-)Higgs doublet, as will be discussed in detailbelow. This leads to a unified description of symmetry breaking via a single fundamental sourceand unites the so-far separate scalar particles that solve the issues of massive EW gauge bosonsaSpeakerarXiv:1905.07285v1  [hep-ph]  17 May 2019(via radiative EWSB), of strong CP conservation, and of fermion mass + mixing hierarchies inone multiplet.We start discussing our setup by reviewing briefly the Froggatt–Nielsen (FN) mechanism 1.This allows to address the fermion hierarchies by charging the chiral SM fermions differentlyunder a U(1)H flavour symmetry. In turn, they are only allowed to interact with each other viaa chain of new vector-like fermions, ξj (the FN messengers), connected via insertions of a newcomplex scalar field Φ, the flavon, carrying away the U(1)H-charge difference. After acquiringa vev, 〈Φ〉 6= 0, the scalar spontaneously breaks the U(1)H symmetry, generating hierarchicalYukawa couplings, suppressed by powers of its vev over the mass of the FN messengers, whichare assumed to be somewhat heavier and thus integrated out in the IR theory, leaving us witheffective interactions between the SM-fermion chiralities (and the Higgs boson).In fact, it has been shown already2,3 that the angular component of Φ – the axiflavon – whichplays no vital role in the original FN mechanism, can be identified with the QCD axion 4,5,6,thereby addressing two more issues of the SM, namely the strong CP problem and the DMpuzzle, in a unified scenario. This increases the predictivity of the axion solution, since thecouplings of the latter are now dictated by the flavour structure, leading to interesting signalsin flavour experiments 2. Subsequently, it was shown that a further unification with the Higgssector can be achieved 7, as sketched above, increasing even more the predictivity of the scenario– making this axion-like solution to the strong CP problem fully testable in the near future, whileadding a dynamics to EWSB. Before discussing this framework in detail in the following, wefinally add that, from a different perspective, it can also be seen as naturally including a modelof flavour (and strong CP conservation) in the recently proposed elementary-Goldstone-Higgsscenario 8,9, furnishing a compelling renormalizable alternative to partial compositeness 10.2 Explicit Model and Mass HierarchiesWe embed the Higgs, the axion, and the flavon in a single multiplet, Σ, transforming under theenlarged symmetry group G ⊃ U(1)H×GEW, with GEW ≡ SU(2)L×U(1)Y . Since the Higgs massshall be much smaller than the U(1)H breaking scale, setting the flavon mass, we envisage boththe axion and the Higgs boson components to correspond to pseudo-Nambu–Goldstone bosons(pNGBs), providing an example of axion-Higgs unification 11 and allowing for dynamical EWSBvia the Coleman–Weinberg potential for the emerging Goldstone Higgs. In fact, the vanishingof the quartic Higgs coupling in the SM around 1010 GeV, just about at the natural scale foraxiflavon dark matter, might hint both to a Goldstone nature of the Higgs and to a connectionbetween the latter and the axion.We thus formulate a linear sigma model for the field Σ at the scale f , featuring Yukawainteractions with the FN messengers and the SM fields, with the (radial) flavon componentdeveloping a vev breaking G 〈Σ〉−−→ H ⊃ GEW, while the axion and the Higgs reside in the G/Hcoset. A particular simple choice for a viable breaking pattern is 7[SO(5)×U(1)H]×U(1)X → SO(4)×U(1)X , (1)where the U(1)X factor is introduced to reproduce the fermion hypercharges as Y = T3 + X.The breaking above is obtained via a Σ field living in the fundamental representation, 5, ofSO(5) and with U(1)H flavour charge HΣ = 1, featuring a potentialV (Σ,Σ∗) = λ1(Σ†Σ)2− λ2 ΣTΣ Σ†Σ∗ − µ2Σ†Σ , (2)which is bounded from below if λ1 > λ2 > 0. The EW gauge group is embedded in SO(5) viathe usual SU(2)L × SU(2)R ∼= SO(4) generators 7 T aL,Rij , a= 1, .., 3, i, j= 1, .., 4. The SO(4)-preserving minimum is then given by 〈Σ〉 = (0, 0, 0, 0, f/√2), with µ2 = (λ1 − λ2)f2.After the breaking of Eq. (1), the scalar sector can be parametrized asΣ = ei(√2hâT̂â+a)/f(H̃(f + σ)/√2), (3)with the broken generatorsT̂ âij = −i√2[δâi δ5j − δâj δ5i], (4)where â= 1, .., 4, and i, j= 1, .., 5. The physical states contained in the radial Σ-components area heavy Higgs doublet, H̃, with mass m2H̃= 2λ2f2, and a heavy flavon fluctuation σ around thevev f , with mass m2σ = 2(λ1−λ2)f2, while the SM-like Higgs doublet, hâ, and the axion, a, areinstead pNGBs.We assume that the explicit breaking of G originates from the SM sector only, namely fromthe QCD anomaly, EW gauging, and via the SM fermions not filling full G representations,while the FN messengers always enter as complete representations. The latter are denoted byξj , where the subscript refers to the U(1)H charge, Hξj = j. They transform in the spinorialrepresentation, 4, of SO(5) (which allows for a very minimal FN messenger sector 7) and arenon-chiral under SO(5) × U(1)H. The SM fermions, qiL, uiR, and diR (with i = 1, 2, 3), come inincomplete SO(5) representations in the spinorialΨiqL = ∆TLqiL, ΨiuR= ∆TuuiR, ΨidR= ∆Td diR, (5)where the spurions, ∆L,u,d, feature background values∆L =(1 0 0 00 1 0 0), ∆u = (0, 0, 1, 0) , ∆d = (0, 0, 0, 1) , (6)that parametrize the explicit SO(5) breaking (while U(1)H remains exact at the Lagrangianlevel). The rows of these 2 × 4 matrices correspond to the fundamental of the weak gaugegroup, while the columns label the components of 4 of SO(5). The U(1)H charge of each Ψifis chosen such that the correct pattern of masses and mixings is reproduced – the larger thecharge difference between the left- and right-handed components of a given SM fermion, themore suppressed is the resulting mass term, see below.The corresponding Lagrangian includes renormalizable operators made out of Ψif , ξj , and Σallowed by symmetries and reads−L =∑j(aj ξ̄j+1 Γα Σα ξj + h.c.)+mj ξ̄j ξj ,+∑i,f=qL,uR,dRzfi Ψ̄if Γα Σα ξ̄ + z̃fi ξ̄̄+2 Γα Σα Ψif + x Ψ̄3qLΓα Σα Ψ3uR+ h.c. ,(7)where Γa are the matrices defining the spinorial representation and the dimensionless coefficientsaj , zfi , z̄fi , x are all assumed to be of O(1). The first line contains the interactions of the FNmessengers with the Σ-field and their (vector-like) mass terms, while the second line consists ofYukawa couplings involving the SM fermions and the FN messengers, with ̄ ≡ ̄(f, i) = Hf i−1such that the terms are U(1)H invariant, where the last term allows for an unsuppressed top mass.It is easy to see how this Lagrangian leads to FN-like mass hierarchies. For two chiral SMfermions ΨiqL and ΨjuR with a U(1)H charge difference of δij ≡ HqiL − HqjR , a chain of at least|δij | − 1 FN messengers ξi together with |δij | insertions of the Σ field is required to couplethem. After integrating out the ξi at the tree level (and suppressing O(1) factors), one finds thecorresponding effective leading order Lagrangian−Leff ∼1m|δij |−1Ψ̄juR(ΓαΣα)|δij |ΨiqL + h.c. , (8)4 6 8 10 12 14 16-0.010-0.0050.0000.005log10(m/GeV)λ(m)4 6 8 10 12 14 160.000.020.040.06log10(m/GeV)λ(m)Figure 1 – Left: Matching of the Higgs quartic coupling at the scale m in the SM (red band) with the predictionof Eq. (11), considering a Yukawa coupling spread of δ = ±0.6 (light blue band), δ = ±0.3 (light gray band), andδ = 0 (dashed black line). The intersection corresponds to the allowed range for m. Right: Same plot for the lowscale model, Eq.(19). See text for details.which is non-vanishing for odd |δij |, as assumed in the following, while for even |δij | it is zerodue to Γ-matrix properties.Below the symmetry-breaking scale and after integrating out the flavon and the second Higgsdoublet, the Σ-field can be written via the Goldstone parametrization in the unitary gauge asΣ =f√2eia/f (0, 0, sinh/f, 0, cosh/f)T , (9)where h represents the Higgs field, and a is the axion. With this, we can single out the leadingcontribution to the mass matrix−Leff ⊃ mij q̄iLujR + h.c. , mij ∼v√2(f22m2) |δij |−12, (10)where v ≡ f sin(〈h〉/f) is the EW scale. This exhibits a typical FN suppression in ε ≡(f2/2m2),which we identify with the Cabibbo angle, ε = sin θC ' 0.23, to reproduce the flavour hierarchies.3 Higgs Potential and Constraints on the Axion Decay ConstantTo calculate the Higgs potential, we consider the top sector, with a charge assignment Hq3L= 1,Hu3R= 2, compatible with the top mass, and two (mass-degenerate) FN messengers, ξ0,1 coupledto them. This allows for a minimal non-trivial chain of messengers and captures the leadingeffect 7. The potential is then computed by matching at one loop the SM Coleman–Weinbergpotential, renormalized at the mass scale of the FN messengers m0 = m1 ≡ m, with the onein the axiflavon-Higgs scenario, induced by explicit SO(5) breaking interactions involving thefields mentioned above, see Ref. 7 for details. Requiring the quadratic term to reside at theelectroweak scale v (f) allows us to make a prediction for the quartic coupling at the scale min terms of the top Yukawa, which turns out to be very small and negative. It reads 7λ(m) = − Nc2π2f22m2y6t (m)(1 + δ)6 , (11)where following the assumptions of the FN mechanism, all dimensionless coupling parametershave been chosen to be of the order of the (O(1)) top Yukawa, which is explicitly realized byreplacing them with an average Yukawa coupling yt(m)(1 + δ), including a spread of |δ| < 1.Since below the threshold of the FN messengers λ(m) is fully predicted via SM running,Eq. (11) can be used to determine the scale m at which a successful matching is achieved. Inthe left panel of Fig. 1 we show the SM running of λ(m) (including uncertainties) via the redband and the RHS of Eq. (11) for δ = ±0.6 (light blue band), δ = ±0.3 (light gray band), andδ = 0 (dashed black line). The matching is possible only for negative values of λ(m), whichselects 109 GeV . m . 1014 GeV. This translates, via f2/2m2 ' 0.23 and fa = f/N , withN =∑i2HΨiqL−HΨiuR −HΨidR≈ 50, (12)to a axion decay constant of107 GeV . fa . 1012 GeV. (13)This can be confronted with constraints due to the flavour-violating couplings of our axion b.Limits from searches for K+ → π+a lead to fa & 7.5 × 1010 GeV at 90% C.L. 2, leaving arelatively thin stripe offa ≈ (1011 − 1012) GeV , (14)which will almost entirely be tested by the NA62 experiment, that just started operation 12.The combined exclusions from requiring a consistent matching of the Higgs potential andsatisfying flavour bounds are visualized in Fig. 2 as red and blue shaded regions, respec-tively, which shows the axion parameter space 2, where gaγγ is the axion-photon coupling andma = 5.7µeV (1012 GeV/fa)13.We finally discuss the impact of including right-handed (RH) neutrinos NR, which enter,together with the left-handed lepton doublet lL as SO(5) spurions (see Eq. (5))ΨL = ∆TL lL, ΨN = ∆TuNR. (15)We assign flavour charge to ΨN such that the term−LN =1√2yN Ψ̄NΣ′CΨ̄TN + h.c. = −12yNf cos(h/f)N̄RCN̄TR eıa/f + h.c. (16)is allowed, leading to Majorana and Dirac masses (the latter via a FN-chain)m2NR(h) = y2Nf2 cos2(h/f) , mD ∼ mtε|δν |−12 , (17)where δν = HlL −HNR . The light neutrino mass is then given bymν ∼ mtε|δν |−1mtmNR, (18)which shows a double suppression, originating from the type-I seesaw and from the FN mech-anism. Including the impact of three almost degenerate RH neutrinos, described by Eq. (16)with a typical coupling parametrized as yN = (1 + δ)yt, on the Higgs potential we arrive now(similar as before) at a positiveλ(m) =38π2log(12y2t (m)(1 + δ)2ε)(1 + δ)4y4t (m). (19)In the right panel of Fig. 1 we confront the SM running of λ(m) in red and the RHS ofEq. (19) for δ = ±0.6 (light blue band), δ = ±0.3 (light gray band) and δ = 0 (dashed blackline). The matching is now possible for smaller values of m with respect to the case without RHneutrinos, leading to6 TeV . fa . 2× 106 TeV . (20)Although such low values of fa are excluded for the usual QCD axion, by disentangling the axionmass and decay constant, low-fa models can become viable14,15. Supernova cooling and flavourconstraints can then be avoided by pushing the axion mass to the GeV or TeV scale. While thisaxion cannot be a dark matter candidate anymore, it still solves the strong CP problem and theRH neutrinos can add a link to the matter-antimatter asymmetry via leptogenesis 16.bNote that, for fixed fa, the axion couplings to fermions and to photons remain basically unchanged comparedto the original axiflavon model 2 (becoming exactly equal in the limit of large U(1)H charges). For example,due to the different U(1)H charge assignment7 (|δij | → 2|δij | + 1), the color anomaly, Eq. (12), changes byN → 2N + 2 ≈ 2N . This translates (for constant fa) into the same change in f , while also the electromagneticanomaly changes by approximately a factor of two, canceling in the coupling to photons.  Figure 2 – Constraints on the axion parameter space, i.e. axion mass vs. photon coupling, in the scenario consid-ered. The axiflavon prediction is depicted by the thin brown band. The exclusions from various axion experiments,summarized in the legend, are given as grey numbered regions. The dashed colored lines show the projected reachof future experiments. On the other hand, the blue shaded region corresponds to the estimated bound fromcurrent flavour experiments, while the dashed blue line gives the expected reach of NA62. Finally, the red shadedarea depicts the region that is excluded since a consistent matching of the Higgs potential is impossible (for theminimal model), which furnishes a new constraint due to axiflavon-Higgs unification.4 ConclusionsA unified model that addresses the flavour puzzle, solves the strong CP problem, provides a darkmatter candidate, and radiatively triggers electroweak symmetry breaking has been presented,employing a single scalar multiplet which also contains the Higgs boson. The model is highlypredictive, leading to the constraints on the axion decay constant presented in Eqs. (14) and(20) for the minimal incarnation and the model including right handed neutrinos, respectively,with the relevant (current and projected) bounds for the former summarized in Fig. 2.AcknowledgmentsFG would like to thank the organizers of the Rencontres de Moriond 2019 EW for the invitationand the wonderful atmosphere during the conference.References1. C. D. Froggatt and H. B. Nielsen, Nucl. Phys. B 147 (1979) 277.2. L. Calibbi, F. Goertz, D. Redigolo, R. Ziegler and J. Zupan, Phys. Rev. D 95 (2017) no.9,0950093. Y. Ema, K. Hamaguchi, T. Moroi and K. Nakayama, JHEP 1701 (2017) 0964. R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38 (1977) 1440.5. F. Wilczek, Phys. Rev. Lett. 40 (1978) 279.6. S. Weinberg, Phys. Rev. Lett. 40 (1978) 223.7. T. Alanne, S. Blasi and F. Goertz, Phys. Rev. D 99 (2019) no.1, 0150288. T. Alanne, H. Gertov, F. Sannino and K. Tuominen, Phys. Rev. D 91 (2015) no.9, 0950219. T. Alanne, A. Meroni, F. Sannino and K. Tuominen, Phys. Rev. D 93 (2016) no.9, 09170110. D. B. Kaplan, Nucl. Phys. B 365 (1991) 259.11. M. Redi and A. Strumia, JHEP 1211 (2012) 10312. B. Döbrich [NA62 Collaboration], Frascati Phys. Ser. 66 (2018) 31213. G. Grilli di Cortona, E. Hardy, J. Pardo Vega and G. Villadoro, JHEP 1601 (2016) 03414. T. Gherghetta, N. Nagata and M. Shifman, Phys. Rev. D 93 (2016) no.11, 11501015. M. K. Gaillard, M. B. Gavela, R. Houtz, P. Quilez and R. Del Rey, Eur. Phys. J. C 78(2018) no.11, 97216. T. Alanne, A. Meroni and K. Tuominen, Phys. Rev. D 96 (2017) no.9, 095015

Axiflavon-Higgs Unification

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Axiflavon-Higgs Unification

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