PHYSICAL REVIEW D 99, 015018 (2019)
Decay and detection of a light scalar boson mixing with the Higgs boson
Martin Wolfgang Winkler* Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 10 691 Stockholm, Sweden
(Received 10 September 2018; published 9 January 2019)
The simplest extension of the standard model consists in adding one singlet scalar field which mixes with the Higgs boson. OðGeVÞ masses of the new scalar carry strong motivation from relaxion, dark matter and inflation models. The decay of a GeV scalar is, however, notoriously difficult to address since, at this mass scale, the chiral expansion breaks down and perturbative QCD does not apply. Existing estimates of the GeV scalar decay rate disagree by several orders of magnitude. In this work, we perform a new dispersive analysis in order to strongly reduce these uncertainties and to address discrepancies in earlier results. We will update existing limits on light scalars and future experimental sensitivities which are in some cases strongly affected by the new-found decay rates. The meson form factors provided in this work, can be used to generalize our findings to non-universally coupled light scalars.
DOI: 10.1103/PhysRevD.99.015018
I. INTRODUCTION sensitivities to a light scalar thus crucially depend on its decay rate and decay pattern. Many prominent extensions of the standard model Since the chiral expansion breaks down shortly above the (SM) feature a gauge singlet scalar ϕ with a mass below two-pion threshold, while a perturbative QCD calculation or at the weak scale. Within the relaxion mechanism [1] becomes reliable for masses of a few GeV, the scalar decay the new scalar is introduced to cure the (little) hierarchy rate in the window mϕ ≃ 0.5–2 GeV suffers from notorious problem. In well-motivated dark matter models, a light uncertainties (see e.g., [8]). The problem already man- scalar emerges as the mediator which links the dark and ifested itself when a light SM Higgs was still considered the visible sector [2]. A light scalar appears in super- viable [9]. In the late 1980s, it was realized that the form symmetric theories such as the next-to-minimal super- factors determining the Higgs (or general scalar) decay rate symmetric standard model [3]. It has been identified with to meson final states are accessible through dispersion the field driving cosmic inflation [4,5] and it is present relations [10]. Unfortunately, the two most comprehensive in models which address the cosmological constant calculations based on this technique by Truong and Willey problem through radiative breaking of classical scale [11] and Donoghue et al. [12] disagree by orders of invariance [6]. magnitude at mϕ ∼ GeV. It was argued in [12] that Through mixing with the Higgs, the light scalar inherits Truong and Willey had obtained the wrong interference the Higgs couplings to SM matter reduced by a universal pattern between elastic and inelastic contributions to the suppression factor. While for scalar masses around the form factors due to a sign error. In this work, we will electroweak scale, LEP and LHC constraints on extended reinvestigate the discrepancy and recalculate the decay rate Higgs sectors apply, rare meson decays offer a particular of a light scalar to pions and kaons. Our evaluation profits powerful search channel for scalars below the bottom mass from progress in the description of pion/kaon phase shift threshold [7]. If the mixing is suppressed, the scalar may, data entering the dispersive integral. however, travel a macroscopic distance before decay. In After identifying the favored parameter regions for this case, searches including missing energy or displaced some of the most promising SM extensions with light vertices become relevant. Present and future experimental scalars, we will update the existing limits and future experimental sensitivities. These were previously based *[email protected] on varying sets of assumptions on the scalar decay. In several cases, we find the sensitivities to be substantially Published by the American Physical Society under the terms of alteredbyournew-founddecayrates.Thisholdsin the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to particular in the context of beam dump experiments which the author(s) and the published article’s title, journal citation, are very sensitive to the scalar decay length through the and DOI. Funded by SCOAP3. location of the detector.
2470-0010=2019=99(1)=015018(15) 015018-1 Published by the American Physical Society MARTIN WOLFGANG WINKLER PHYS. REV. D 99, 015018 (2019)
II. STANDARD MODEL EXTENSIONS WITH by the mixing angle sθ. The annihilation cross section σ ¼ σ 2 LIGHT SCALARS times relative velocity vrel is of the size vrel 1vrel A new scalar can connect to the SM at the renormalizable with [13,15] level via the Higgs portal κ4 9 4 − 8 2 2 þ 2 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mχ mχ mχmϕ mϕ 2 2 σ1 ≃ mχ − m ; ð4Þ 2 † 24π ð2 2 − 2 Þ4 ϕ L ⊃ ðg1ϕ þ g2ϕ ÞðH HÞ: ð1Þ mχ mϕ
1 2 where we assumed a vanishing trilinear scalar self-coupling Once electroweak symmetry is broken, the couplings g ; 1 induce mixing between the scalar and the Higgs. We will for simplicity. Since the annihilation cross section is p-wave focus on the case where the scalar mass is considerably suppressed, strong indirect dark matter detection constraints below the electroweak scale. In the low energy effective are avoided. The fermion relic density is approximated theory, the Higgs can then be integrated out and it arises the as [16] coupling of the new scalar to SM fermions 2 2 −11 −2 mχ Ωχh ¼ 2.8 × 10 GeV pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð5Þ sθmf ¯ ð Þσ 2 L ⊃− ϕff; ð2Þ g TF 1TF v where g denotes the number of relativistic degrees where sθ denotes the sine of the Higgs-scalar mixing angle of freedom and TF the freeze-out temperature which and v the Higgs vacuum expectation value (vev). With we take from [17]. For a given set of masses, the coupling 2 regard to experimental searches, the light scalar behaves as κ is fixed by requiring that Ωχh matches the observed dark a light version of the Higgs boson with universally sup- pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi matter relic density. We find κ ¼ð0.03–0.05Þ × mχ=GeV pressed couplings. In concrete models, a more complicated 2 for mχ ¼ 10 MeV–10 TeV. coupling pattern may emerge if they feature e.g., more than one Higgs doublet. While we focus on the simplest case We have implicitly assumed a standard thermal freeze- given above, many of our results can be applied to more out of the singlet fermion. This is justified if the dark sector general couplings after simple rescaling. In order to identify was in thermal equilibrium with the SM bath prior to freeze-out. We, therefore, require that the thermalization the most promising parameter space for the mixing angle, Γ we shall briefly discuss some well-motivated SM exten- rate therm of the dark sector exceeds the Hubble rate of expansion H at freeze-out, i.e., sions with light scalars. rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4π3 ð Þ 2 Γ ð Þ ð Þ¼ g TF pffiffiffiffiffiffiTF ð Þ A. Connection to dark matter therm TF >H TF 45 : 6 8πMP New particles with a weak scale annihilation cross Γ 2 section have been considered among the leading dark Since therm scales with sθ, (6) puts a lower limit on the matter candidates since—within the thermal production mixing angle. Notice that thermal decoupling of the dark mechanism—their relic density naturally matches the sector is not a strict exclusion criterion. It would, however, observed dark matter density. The absence of a signal in invalidate the simple connection between mχ, κ and Ωχh, direct detection experiments, however, suggests even fee- making the relic density a UV sensitive quantity. bler interactions between dark matter and nuclei. An At the same time, large mixing angles are excluded due appealing possibility is that dark matter resides within a to direct dark matter detection. The dark matter-nucleon dark sector of particles which do not directly feel the strong cross section reads3 [14] or electroweak forces [2]. In this scenario, a scalar boson 2 2 could be the mediator which communicates between dark 4μχ sθκ σ ≃ n ð Þ n 2 mnf 7 and visible matter. In the simplest realization, dark matter π 2vmϕ is identified with a gauge singlet Majorana fermion χ which is stable due to a (discrete) symmetry and couples to the with scalar via the Yukawa term [13,14] 6 κ fn ¼ fn þ fn þ fn þ f : ð8Þ L ⊃ ϕχχ¯ ð Þ u d s 27 G 2 : 3
1 We will assume mχ >mϕ, such that a hierarchy between The general expression for the annihilation cross section for nonvanishing trilinear coupling can be found in [15]. the annihilation cross section and the dark matter nucleus 2 This holds unless for very degenerate cases mχ − mϕ < cross section can naturally be realized: the fermions 0.01mχ. annihilate into scalars via the (unsuppressed) coupling κ, 3The formula is valid for scalar masses substantially larger than while dark matter nucleus interactions are suppressed the momentum transfer, i.e., mϕ ≳ 100 MeV.
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Here, mn denotes the nucleon mass and μχ the reduced loop, is sufficiently suppressed and does not trap the mass of the dark matter-nucleon system. The scalar relaxion before electroweak symmetry breaking [25,26]. n ϕ ∼ coefficients fu;d;s and fG define the quark and gluon The relaxion slowly rolls down its potential and, at ¼ 1 − n − n − n M=g triggers electroweak symmetry breaking. As soon as content of the nucleon. While fG fu fd fs n the Higgs field is displaced, the cosine term induces derives from the QCD trace anomaly, fu;d;s have been determined in lattice-QCD and in chiral perturbation wiggles on the relaxion potential which ultimately stop theory. We employ the value fn ¼ 0.30 from [18] which its motion. The required dissipation mechanism is provided is consistent with other recent evaluations [19,20]. The dark by the Hubblep frictionffiffiffiffiffiffiffiffiffiffiffiffiffiffi of inflation. If HI exceeds a critical 3 matter direct detection constraints can now be mapped into value HI;c ∼ gM =f, the relaxion immediately stops in the scalar mass-mixing plane. Besides the constraints of one of its first minima. Otherwise, it continues rolling and XENON1T [21], we also include those of CRESST-III [22] later settles in one of the steeper minima, further down the and DarkSide-50 [23] which dominate at mχ ≲ 5 GeV. potential. The difference between both cases manifests in Since the most conservative (weakest) bounds are the phase factor χ ϕ obtained if is just slightly heavier than , we fix 2 vϕ H f mχ ¼ 1.1mϕ. In this case, the thermalization rate is ∼ 1 I ð Þ sin Min ; 3 ; 10 dominated by the inverse decay of the scalar [24] and f gM Γ ≃ Γ we have to apply (6) with therm ϕ. As shown in Fig. 8, the parameter space, where the correct relic density can be where we introduced the relaxion vev vϕ. The sine is of achieved via standard freeze-out and the direct detection order unity if HI >HI;c, while it can be substantially constraints are satisfied, spans several orders of magnitude suppressed for a low inflationary scale. The Higgs vev in sθ. Further experimental constraints on this window will emerges as be discussed in Sec. IV. gM3f v2 ≃ : ð11Þ Λ2 sinðvϕÞ B. Relaxion f The relaxion mechanism constitutes a dynamical sol- Validity of the effective theory (9) without further light ution to the (little) hierarchy problem of the standard model degrees of freedom requires f ≫ v ≳ Λ. This implies that [1]. It provides another motivation for the existence of a the relaxion is lighter than the Higgs and the mixing effect light scalar boson. While the phenomenology of Higgs- 6 on mh is negligible. The relaxion mass and the Higgs- relaxion mixing has been comprehensively studied [25,26], relaxion mixing angle can be approximated as [26] we wish to include the additional possibility of a low 2 2 2 inflationary Hubble scale HI. Λ v vϕ 2Λ vϕ ϕ m2 ≃ cos − sin2 ; The evolution of the relaxion reduces the initially ϕ 2f2 f m2 f M ≫ v h large Higgs boson mass to the observed mass 2 ¼ Oð Þ 4 Λ v vϕ mh v . This is achieved via the potential ≃ ð Þ sθ 2 sin : 12 fmh f 2 2 3 2 2 ϕ 4 V ¼ðM − gMϕÞh − gM ϕ − Λ h cos þ λh ; ð9Þ The relaxion couples to SM matter via its Higgs admixture f and via pseudoscalar couplings which are generically present but model-dependent. Requiring that the mixing- where g is a dimensionless coupling and h denotes the induced couplings dominate leads to the constraint neutral component of the Higgs doublet. Since the relaxion 2 7 sinðvϕ=fÞ ≳ 1=ð16π Þ. The resulting theory exclusion settles in a CP breaking minimum, it is not identified with on the parameter space (requiring also f>v) is depicted the QCD axion in the basic model. Instead, the periodic in Fig. 8. Compared to [25,26], we obtain a larger relaxion potential may stem from the instantons of a new strongly window since suppression of sθ by small H has not been coupled gauge group [1].5 The scale Λ must not exceed the I considered in these references. electroweak scale since, otherwise, the Higgs vev is driven up to Λ. This constraint also ensures that a constant term in 6 front of the cosine, which is generated by closing the Higgs More precisely, we are referring to the mass of the relaxion- like scalar mass eigenstate. 7 2 For sinðvϕ=fÞ ≳ 1=ð16π Þ, the CP violating scalar relaxion 4We neglect an Oð1Þ coefficient in front of the gM3 term which couplings can still dominate since pseudoscalar couplings may does not play a role for the following discussion. suffer additional loop suppression [26]. We note that viable 5 For concreteness, we assumed that the new strongly coupled relaxion models with smaller sinðvϕ=fÞ may exist. The con- sector does not break electroweak symmetry such that odd straint, however, singles out the parameter region in which the powers of h are absent in front of the cosine. The phenomenology relaxion can be described as a minimal singlet scalar mixing with is, however, hardly sensitive to this assumption (see [26]). the Higgs.
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III. SCALAR DECAY RATES with the purpose of reproducing the rate of Donoghue et al., it was not meant as a test of previous results. A calculation It is straightforward to evaluate the scalar decay rates into of the hadronic decay rates in an independent two-channel leptonic final states. One finds dispersive analysis is still missing. It will be performed 2 in the next sections, before matching the result to the ¯ sθGFmϕ 2 3 Γðϕ → llÞ ≡ Γll¯ ¼ pffiffiffi mlβl; ð13Þ perturbative spectator model at higher mass. 4 2π l ¼ μ τ with qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie, , . Here, GF denotes the Fermi constant and B. Chiral perturbation theory 2 2 βl ¼ 1 − 4ml=mϕ the velocity of the final state leptons. We first consider scalar masses below the charm thresh- Hadronic decay rates require a more careful treatment due old. The Lagrangian describing the interaction of the scalar to the strong final state interactions. This holds in particu- with light quarks (u, d, s) and gluons reads ð980Þ lar, if the scalar mass resides in the vicinity of the f0 ϕ 3α s a aμν ¯ resonance. L ⊃ sθ GμνG − m uu¯ − m dd − m ss¯ v 12π u d s A. Status of hadronic decay rates ϕ 2 μ 7 ¯ ¼ −sθ Θμ þ ðmuuu¯ þ mddd þ msss¯ Þ ; ð14Þ Figure 1 shows that different evaluations of the scalar v 9 9 decay rate to pions disagree by several order of magnitude at mϕ ∼ GeV. The result of Voloshin was obtained at where the effective coupling to gluons origins from heavy leading order in chiral perturbation theory (ChPT) [27]. quark (c, b, t) loops. In the second step, we used the trace In the “Higgs Hunter’s guide” the perturbative spectator identity model is extrapolated into the nonperturbative regime. 9α Quark masses were adjusted such as to (approximately) μ s a aμν ¯ Θμ ¼ − GμνG þ m uu¯ þ m dd þ m ss;¯ ð15Þ reproduce Voloshin’s decay rate at low mass [9]. Both 8π u d s evaluations are frequently used to describe GeV scalar decays although they do not apply to this mass range due to of the energy-momentum tensor which results from the its proximity to the chiral symmetry breaking scale. Raby conformal anomaly [29,30]. The decay rates of the scalar and West [10] introduced the use of dispersion relations to into pion and kaon pairs read access the GeV regime and predicted a huge enhancement 3 2 7 7 2 2 of the scalar decay rate to pions close to the f0ð980Þ sθGF Γππ ¼ pffiffiffi βπ Γπ þ Δπ þ Θπ ; resonance. However, they treated f0ð980Þ as an elastic 16 2πmϕ 9 9 9
ππ-resonance which leads to an overestimation of the rate. 2 s G 7 7 2 2 A full two-channel analysis including KK and ππ was Γ ¼ pθffiffiffi F β Γ þ Δ þ Θ ð Þ KK K 9 K 9 K 9 K ; 16 finally performed by Truong and Willey [11] and 4 2πmϕ Donoghue et al. [12]. Their results are incompatible with one another, which was related to a sign error in Truong and where we introduced the form factors Willey’s T-matrix parametrization in [12]. Monin et al. [28] Γ ¼hππj ¯ þ ¯ j0i recently performed a modified one-channel analysis in π muuu mddd ; order to provide an analytic expression for Γππ in terms of Δπ ¼hππjm ss¯ j0i; ππ s the -scattering phase. Since free parameters were chosen μ Θπ ¼hππjΘμj0i; ð17Þ
for pions and analogous for kaons. The pion form factors have been determined to lowest order in ChPT in [27]. A ChPT calculation of the kaon form factors may seem pointless since the scalar decay to kaons only opens in the regime, where chiral symmetry is strongly broken. However, the low-momentum kaon form factors will later define the matching conditions for the dispersive analysis. Therefore, we briefly outline the computation using the (strangeness-conserving part of the) 3-flavor chiral Lagrangian which reads8 FIG. 1. Evaluations of the light scalar decay rate to pions by Voloshin [27], Raby and West [10], the Higgs Hunter’s Guide [9], 8An analogous determination of the kaon form factors can be Truong and Willey [11], Donoghue et al. [12] and Monin et al. found in [12]. For a review on the application of ChPT techniques [28]. In this figure sθ has been set to unity. to Higgs physics, see [9,31].
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1 μ † 1 2 † projection of the ππ state has been absorbed into the L ¼ fπTr∂μΣ∂ Σ þ fπðTrμMΣ þ H:c:Þ; ð18Þ 4 2 definition of F [12]. Below the kaon threshold, the phase of the pion form factors coincides with the isoscalar s-wave with ππ phase shift according to Watson’s theorem [35]. Its 8 0 19 generalization to two channels is expressed in form of the π0 η þ þ > pffiffi þ pffiffi π K > unitary relation <>pffiffiffi B 2 6 C=> 2i B 0 η C Σ ¼ B π− − pπ ffiffi þ pffiffi K0 C ð Þ exp> @ 2 6 A> 19 ¼ β θð − 4 2ÞðÞ > fπ > ImFi Tij jFj s mj 23 : − ¯ − p2ηffiffi ; K K0 6 with β1;2 ¼ βπ;K. The (isoscalar s-wave projection of the) T-matrix for ππ, KK → ππ, KK scattering is parametrized and M ¼ diagðmu;md;msÞ. Here fπ denotes the pion decay constant. The mass parameters in the chiral in terms of two phases δ, ψ and an inelasticity parameter g Lagrangian are related to the physical meson masses as 2 δ ! ηe i −1 iψ 2 β ge 2 ¼ μð þ Þ ¼ i π ð Þ mπ mu md ; T 2 ðψ−δÞ ; 24 iψ ηe i −1 2 ge 2 β ¼ μð þ Þ i K mK0 md ms ; 2 ¼ μð þ Þ ð Þ mK mu ms : 20 where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi One can now use the Feynman-Hellmann theorem η ¼ 1 − 4β β 2θð − 4 2 Þ ¯ ¼ − ∂L ∂ π Kg s mK ; [32,33] mqqq mq = mq and the trace of the rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi energy-momentum tensor 4m2 β ¼ 1 − i : ð25Þ i s Θμ ¼ fπ ∂ Σ∂μΣ† − μL ð Þ μ 2 Tr μ gμ ; 21 The parameters of the T matrix are efficiently determined ππ to evaluate the form factors at lowest order in the chiral by invoking , KK scattering data and theoretical con- expansion (denoted by the superscript 0). One finds straints in form of the Roy-Steiner equations. We extract the phases and inelasticity parameter from the analysis of 1 0 2 0 2 Hoferichter et al.p[36]ffiffiffiffiffi which incorporates earlier results Γπ ¼ mπ; Γ ¼ mπ; K 2 [37,38].Above s0 ¼ 1.3 GeV, the correct asymptotic 1 behavior of the T-matrix is ensured by guiding δ, ψ Δ0 ¼ 0 Δ0 ¼ 2 − 2 π ; K mK 2 mπ; smoothly to 2π. We follow [39] and assume that the 0 2 0 2 difference between the phases and their asymptotic values Θπ ¼ s þ 2mπ; Θ ¼ s þ 2m ; ð22Þ K K in this regime decreases as ¼ where we setpffiffiffimu md. The form factors have to be 2 ¼ pffiffi ð Þ evaluated at s mϕ. Higher orders are suppressed by m ; 26 Λ ∼ 1 1 þ pffiffiffis powers of the chiral symmetry breaking scale χ GeV. s0 The lowest order does, hence, not provide a realistic ≳ 0 5 estimate of the form factors for mϕ . GeV. pwhereffiffiffi pmffiffiffiffiffiis set to 3. We have verified that form factors at s < s0 are rather insensitive to the particular function C. Dispersive analysis by whichpffiffiffiffiffi the phases approach their asymptotic values. Fortunately, form factors at higher mass are accessible Above s0 the form factors obtained from the two-channel through dispersion relations. These employ analyticity and analysis are anyway less trustable since further channels 4π ηη unitarity conditions without relyingp onffiffiffi any details of the such as , become relevant. microscopic interaction theory. For s ≲ 1.3 GeV a two- Form factors satisfying the unitary relation (23) can be channel approximation in terms of ππ and KK can be expressed as [40,41] applied. This is becausepffiffiffi scalar decays are controlled by the f0ð980Þ resonance at s ∼ GeV which mainly couples to Ω11 Ω12 P1 ð500Þ F ¼ ; ð27Þ these states [34]. At even lower mass, where also f0 Ω21 Ω22 P2 contributes, ππ is the only relevant decay channel due to kinematics. where P1;2 are polynomials and ðΩ11; Ω21Þ, ðΩ12; Ω22Þ are ¼ð p2ffiffi Þ ¼ Γ Δ Θ We define F Fπ; 3 FK (F , , ), where the the two linear independent solution-vectors fulfilling the Clebsch-Gordan coefficient occurring in the isoscalar dispersion relation
015018-5 MARTIN WOLFGANG WINKLER PHYS. REV. D 99, 015018 (2019) Z ∞ 0 1 0 ImFðs Þ ReFðsÞ¼ − ds 0 : ð28Þ π 2 − 4mπ s s
The Ωij (which are found as described in [39]) are conveniently normalized such that Ω11ð0Þ¼Ω22ð0Þ¼1, Ω12ð0Þ¼Ω21ð0Þ¼0. The form factors Γi, Δi are expected to vanish at high energy due to the composite nature of mesons. Since Ωij ∝ s−1 for large s, the polynomial prefactors in (27) need to be constants. Their values can be determined by matching (27) to the lowest order result in chiral perturbation theory (22) at s ¼ 0. In the case of the energy-momentum form factors, Lorentz-invariance and four-momentum conservation require the structure [42] 3 Θ ¼ Θ þ 2 2 − s Θ ð ¼ π Þ ð Þ i 2 s S;i mi 2 T;i i ;K ; 29 where ΘS;i and ΘT;i refer to the scalar and tensor parts of Θi. In order to match the chiral result at s ¼ 0, one needs to FIG. 2. Modulus (upper panel) and phase (lower panel) of the require that ΘS;i, ΘT;i (rather than Θi) vanish asymptotically (see also [12]). We thus obtain pion and kaon form factors. 1 2 due to the opening of further hadronic decay channels Γπ ¼ mπ Ω11 þ pffiffiffi Ω12 ; 3 beyond ππ and KK are not included. 2 As can be seen in the same figure, our result on the decay 2 2 mπ Δπ ¼ pffiffiffi m − Ω12; rate agrees reasonably well with that of Donoghue et al. 3 K 2 [12]. Differences reside within a factor of ∼3 and follow 2 2 2 from our updated phase shift input [36]. The decay rates Θπ ¼ð2mπ þ psÞΩ11 þ pffiffiffi ð2m þ qsÞΩ12; 3 K found by us are, however, incompatible with those in [11]. 2 pffiffiffi mπ The reason for the discrepancy is indeed a sign error in Γ ¼ ð 3Ω þ Ω Þ ’ K 2 21 22 ; Truong and Willey s parametrization of the T-matrix. Their 2 choice leads to a negative sign of T12 at low energy which is 2 mπ Δ ¼ m − Ω22; inconsistent with ChPT [12]. In Fig. 3 we also depict the K K 2 λ pffiffiffi decay rate after flipping the sign of their parameter . It can 3 be seen that this correction puts Truong and Willey’s rate Θ ¼ ð2 2 þ ÞΩ þð2 2 þ ÞΩ ð Þ K 2 mπ ps 21 mK qs 22; 30 into qualitative agreement with our result. where we introduced D. Perturbative spectator model 4m2 We now turn to the hadronic decays at higher energy, p ¼ 1 − 2m2Ω0 ð0Þ − pffiffiffiK Ω0 ð0Þ; π 11 3 12 where the perturbative spectator model can be applied. The pffiffiffi decay rates to quarks are given as9 ¼ 1 − 3 2Ω0 ð0Þ − 2 2 Ω0 ð0Þ ð Þ q mπ 21 mK 22 : 31 ¼ 0 73 ¼ 0 52 Γ ∶Γ ∶Γ ¼ 2 β3∶3 2β3 ∶3 2β3 ð Þ Numerically, we find p . and q . . The most ll¯ ss¯ cc¯ ml μ ms K mc D 32 relevant SU(3) breaking effect (concerning the impact on decay rates) applies to the value of q which can get changed ¯ by up to 0.2 [12]. In Fig. 2 we depict the pion and kaon and analogous for the bb-channel. The kinematic threshold form factors resulting from our dispersive analysis. The is set by the lightest meson containing an s or c quark corresponding scalar decay rate to pions is shown in Fig. 3. respectively [9]. In addition, we need to consider the loop- We also provide an estimate of the uncertainties related to induced decay rate into gluon pairs [43] the phase extrapolation and to the matching conditions. The 9 error band was obtained by varying m in (26) between 2 We set ms ¼ 95 MeV, mc ¼ 1.3 GeV [34] and neglect the and 4, and q in the range 0.32–0.72. Note that uncertainties tiny decay rate into u, d quarks.
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¼ 2 ð4 2Þ with xi mϕ= mi and 8 pffiffiffi < arcsin2 x;x≤ 1 pffiffiffiffiffiffiffiffiffi ð Þ¼ 2 ð Þ f x : 1 1þpffiffiffiffiffiffiffiffiffi1−1=x 34 − 4 log − iπ ;x>1: 1− 1−1=x
We take αsðmϕÞ from [44]. Following [45] we assume that the perturbative spectator model is valid at mϕ > 2 GeV. The dispersive analysis holds for mϕ ≲ 1.3 GeV, where ππ and KK dominate the hadronic decay rate. In the regime mϕ ¼ 1.3–2 GeV, significant corrections are expected. We will use the dispersive results up to 2 GeV, but include an additional contribution
2 3 Γ4π;ηη;ρρ;… ¼ Csθmϕβ2π; ð35Þ
to account for the increasing number of hadronic channels opening above the 4π threshold. The mass scaling is leaned upon the gluon channel. Setting C ¼ 5.1 × 10−9 GeV−2, the hadronic decay rate transits smoothly into the rate of the spectator model at mϕ ¼ 2 GeV. In reality, peaks may occur in different hadronic channels due to further unflavored scalar resonances including f0ð1370Þ, f0ð1500Þ, f0ð1710Þ, f0ð2020Þ, f0ð2100Þ, FIG. 3. Light scalar decay rate into pions from this work, from f0ð2200Þ and f0ð2300Þ [46]. The strong increase of the Truong & Willey [11] and from Donoghue et al. [12]. We also decay rate around GeV, however, appears since f0ð980Þ is ’ show Truong and Willey s decay rate after correcting a sign error narrow and located just below the kaon threshold to which in their T-matrix parametrization (see text). it strongly couples [11]. Since a comparable situation does not seem to arise for the listed heavier f0-resonances, less
2α2 3 X þð − 1Þ ð Þ 2 pronounced enhancements are expected at higher mass. sθ smϕ xi xi f xi Γ ¼ ; ð33Þ Therefore, the hadronic decay rates we obtain at mϕ ¼ gg 32π3 2 quarks 2 v xi 1.3–2 GeV may at least provide a valid order-of-magnitude estimate. On the other hand, presently unconfirmed
FIG. 4. Hadronic and leptonic decay rates of a light scalar mixing with the Higgs. These were obtained from our dispersive analysis (mϕ < 2 GeV) and from the perturbative spectator model (mϕ > 2 GeV). The possible impact of the charmonium resonances on the 2 hadronic decay rate is illustrated by the gray line. All decay rates scale with sθ which was set to unity in this plot.
015018-7 MARTIN WOLFGANG WINKLER PHYS. REV. D 99, 015018 (2019) resonances including potential glueball states could still have significant impact [47]. Around the two-charm threshold, the scalar mixes with the CP even quarkonia χc0ðnPÞ. In Appendix A, we assess the effect on the scalar decay rates due to χc0ð1PÞ and χc0ð2PÞ within a nonrelativistic potential approach [48]. Since nonperturbative corrections to the simple quantum mechanical picture are unknown10 and due to the sparse- FIG. 5. Radiative ϒ decays and flavor changing B decays ness of experimental data on the heavier charmonium mediated by a light scalar. resonances, this should only be seen as a very qualitative estimate. For masses mϕ ∼ 10 GeV, which are not in the þ þ main focus of this work, the bottomonium resonances cause ratio B → K þ μμ¯ in several bins of dilepton invariant analogous mixing effects [48]. mass [56]. The corresponding upper limit on the ϕ-induced In Fig. 4, we depict the leptonic and hadronic decay rates branching ratio in each bin is determined as in [14]. It must of the light scalar below the bb¯ -threshold. These were be taken into account that LHCb triggered on prompt decays obtained from the dispersive results matched to the spec- in this search. Following [14], we estimate that events with a ≃ 5 tator model as described above. The possible distortion of (boosted) scalar decay length d