PHYSICAL REVIEW D 99, 015018 (2019)

Decay and detection of a light scalar mixing with the

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 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 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 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- 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 and . Our evaluation profits powerful search channel for scalars below the bottom mass from progress in the description of pion/ 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 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 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 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- 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 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 and 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 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 . 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 (u, d, s) and 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 . 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 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

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Even in the GeV range (and beyond), the L3 analysis can be considered robust since it merely relies on the domi- nance of hadronic decay modes, while the particular enhancement of the pionic decay rate does not play a role. Above the mass, LEP still sets the strongest constraints on light scalars (see Fig. 8). Turning to the LHC, light scalars are constrained by the search for -0 resonances in the dimuon channel. CMS σ andpffiffiffi LHCb provided constraints on pp→ϕ ×Brϕ→μμ¯ at s ¼ 7 TeV and 8 TeV respectively [65,66] (see also [67]). In the covered mass range mϕ ¼ 5.5–15 GeV, scalar production by B-meson decay is kinematically forbidden which makes gluon fusion the relevant process. We calculated the corresponding cross section with the tool SUSHI 1.6.1 [68] in order to translate the limits into exclusions on sθ for our considered mass range (see Fig. 8). Additional LHC constraints on light scalars arise from the nonobservation of exotic Higgs decays. These shall not be FIG. 6. LHCb constraints on light scalars derived from rare B considered in this work since they rely on the model- decays. Masses around the charmonium resonances have been dependent Higgs-scalar coupling and, furthermore, only masked by the collaboration or by us (see Sec. III D). lead to subdominant exclusions in the considered mass range [26]. For proposed detector concepts (MATHUSLA, momentum. The distribution of transverse momenta is CODEX-b, FASER) which would increase the LHC sensitivity to light scalars, we refer to [24,52,53]. again determined with PYTHIA. Below the threshold, the scalar typically escapes detection due to its long lifetime. It still leaves a trace in the C. Beam dump experiments þ → πþ þ νν¯ form of missing energy. The search for K by Beam dump experiments with detectors located E949 is used to set limits on BrKþ→πþϕ as a function of the Oð100 mÞ away from the interaction point provide a scalar mass and lifetime [63]. In this case, visible decays of sensitive laboratory to search for long-lived particles. ϕ are vetoed, and the sensitivity increases with the lifetime Light scalars are most efficiently generated by B and K of the scalar. We determine the corresponding exclusion in meson decays. For a beam impinging on a thick − the mϕ sθ plane. The standard model contribution to target which absorbs hadrons efficiently, the number of þ þ ¯ K → π þ νν is neglected for this purpose. produced scalars can be estimated as Figure 8 shows that rare decays set the strongest constraints on light scalars over wide regions of the −1 −3 −4 Nϕ ≃ N ðn Br → ϕ þhγ il n Γ →πϕÞð37Þ parameter space. Mixing angles down to sθ ¼ 10 –10 p:o:t: B B Xs K H K K are excluded for mϕ

015018-9 MARTIN WOLFGANG WINKLER PHYS. REV. D 99, 015018 (2019) Z η η Γ d2 geom recmϕ ϕ TABLE I. Comparison between the CHARM, NA62 and SHiP P ¼ −mϕΓϕz=pϕ ð Þ ϕ dz e : 38 beam dump experiments. d1 pϕ ¯ Np:o:t d1 − d2 [m] ηgeom The decay vertex z of the scalar needs to be located within CHARM 2.4 × 1018 480–515 0.001–0.002 (K) the distance d1 − d2 from the target to be detected. The 0.002–0.01 (B) η 18 – – geometric efficiency geom accounts for the probability that NA62 10 95 160 0.002 0.005 (K) the decay products of ϕ pass through the detector. It 0.002–0.02 (B) 20 – – depends on the angular coverage of the detector and varies SHiP 2 × 10 69 120 0.05 0.08 (K) – with the scalar’s momentum and the location z of the decay 0.2 0.5 (B) η vertex. The factor rec is the reconstruction efficiency for final states of a certain type. In order to determine the total number of events, we need to integrate the product NϕPϕ us is substantially weaker than in previous evaluations over the momentum distribution of ϕ. The latter is again [8,24,26,50–53,62]. We believe that in these references, determined with PYTHIA by creating large samples of B and kaon absorption in the thick copper target—which drasti- K mesons which are then decayed further to scalars. Kaon cally reduces Nϕ from kaon decays—has been neglected. events are properly weighted to account for the fact that Sensitivity projections for NA62 and SHiP in Fig. 8 highly boosted kaons are more likely to be absorbed due to again correspond to 3 events. They should be considered their longer decay length. The geometric efficiency is as optimistic since a negligible background level was determined from the momentum spectrum of ϕ by decaying assumed. While the number of produced scalars in the scalars and selecting events with all final states passing NA62 is similar as in CHARM, NA62 is sensitive to through the detector. higher masses since it can reconstruct pion final states. We consider the CHARM beam dump (which operated SHiP will cover a huge parameter region not previously in the 1980s), the upcoming run of NA62 in dump mode accessible to any experiment. For SHiP and NA62, we and the planned SHiP experiment. All three detectors have again find deviations from the semi-official sensitivity been/ will be located at the CERN SPS and employ a 15 estimates [75,76] (see Fig. 7). 400 GeV proton beam. The meson multiplicities are In this case, the discrepancy can be traced back to the ≃ 3 2 10−7 ≃ 0 9 16 estimated as nB . × [70] and nK . [71]. The assumptions on the scalar decay rates. While we relied target materials copper (CHARM, NA62) and molybdenum on a dispersive analysis in the non-perturbative QCD l ≃ 15 3 (SHiP) share a hadronic absorption length H . cm regime (see Sec. III), the perturbative spectator model with [72]. Locations and coverage of the detectors are described in [70,73,74]. CHARM is sensitive to leptonic final states with efficiency 0.5 [73]. SHiP and NA62 should be η ¼ 0 4 sensitive to all sorts of final states with rec . η ¼ 0 7 ( rec . ) below (above) the two-muon threshold for η ≃ 1 SHiP [70] and rec for NA62 [75]. We summarize the luminosities, locations of the decay volumes and mean η¯ geometric efficiencies geom (for detection of B- and K-induced scalars) in Table I.17 SHiP will be a factor Oð104Þ more sensitive compared to its predecessors due to the larger beam intensity and the better detector coverage. CHARM did not observe any signal events which translates to an upper limit of 3 expected events (at 95% confidence level). The corresponding exclusion −4 on light scalars reaches down to sθ ∼ 10 (see Fig. 8). We note that the CHARM constraint obtained by

15A search for long-lived scalars could potentially also be performed at the Fermilab SeaQuest Experiment after minor modifications of the setup [69]. 16 ¼ 0 62 ≃ We extracted nK . from [71] and estimated nK L 0.28 by taking the KL=K ratio from PYTHIA. FIG. 7. SHiP sensitivity to light scalars found in this work 17 η¯ The mean geometric efficiency geom was derived by aver- compared to [76]. The blue shaded region is obtained for the aging ηgeom over the momentum distribution and the location of scalar decay rates derived in Sec. III and represents our preferred the decay vertex within d1 − d2. The stated ranges are obtained estimate. The yellow region is obtained if we treat the scalar by varying the scalar mass between 0.01 GeV and mB − mK. decay in the perturbative spectator model.

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FIG. 8. Constraints on light scalars mixing with the Higgs. The filled regions with solid boundaries correspond to model-independent constraints. Sensitivity projections are indicated by the dashed boundary. The hatched regions refer to model-dependent exclusions which apply to the relaxion model (cyan) and the dark matter model (red, ocher) discussed in Sec. II. Masses around the charmonium resonances have been masked in some probes. adjusted quark masses [9] has been employed in [75,76].In (BBN), the hadronic energy injection would have spoiled Fig. 7 it can be seen that our sensitivity estimate approx- the light element abundances. The resulting upper limit on imately reproduces the SHiP projection from [76] if we also the scalar lifetime ranges from 1=100s − 1 s in the con- switch to the same spectator model. The same observation sidered mass range [77].18 It was converted to a constraint is made in the case of NA62. We emphasize, however, that on sθ by using the decay rate from Fig. 4. our dispersive analysis provides a much more realistic Finally, astrophysical processes can be affected by light description of the scalar decay properties in the GeV range scalars. Most importantly, scalar emission could carry away compared to the spectator model. significant amounts of energy in supernova explosions We finally comment that the sensitivity of NA62 to light [78,79]. This would lead to a shortening of the scalars could be significantly improved: the present esti- pulse which is constrained by observations of SN1987a. mate refers to the experiment running in dump mode. This We determine the corresponding exclusions on light scalars means that the beryllium target is lifted and the collimator is following the treatment described in [24,51]. closed such that it acts as dump for the proton beam. The While accelerator searches exclude large mixing angles, disadvantage of this layout is that most produced kaons are cosmology constrains sθ from below (see Fig. 8). −3 −5 absorbed in the thick collimator before they can decay. It For mϕ ≲ 5 GeV, a window of sθ ∼ 10 –10 and sθ ∼ appears preferential to leave the (thin) beryllium target in 10−4–10−8 remains viable below and above the two-muon the beam line and keep the collimator closed. The latter threshold respectively. In models, where the light scalar is would then still filter hadronic backgrounds. But since it is identified with the relaxion (Sec. II B) or the mediator located 20 m downstream the target, a significant fraction connecting to dark matter (Sec. II A), additional constraints of the kaons created in the target could decay before apply which close parts of this window. Nevertheless, there reaching the collimator. This would increase the number of remains an exciting discovery potential for the next gen- light scalars from kaon decay by a factor 10–100 compared eration of experiments. to dump mode. V. CONCLUSION D. Cosmology and astrophysics We have reinvestigated the decay properties of a light Light scalars can also be constrained by requiring that scalar boson mixing with the Higgs. A special focus was they do not spoil the cosmological evolution. In the hot early universe, the light scalars are copiously produced in 18The constraint mildly depends on the (model-dependent) the thermal bath. Due to their small coupling to SM matter, Higgs-scalar coupling and was shown for three different choices their freeze-out abundance is significant. If their decay in [77]. To be conservative we used the weakest of the three happens after the onset of primordial nucleosynthesis constraints at each mass.

015018-11 MARTIN WOLFGANG WINKLER PHYS. REV. D 99, 015018 (2019) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ≃ 0 5–2 pffiffiffi placed on the mass range mϕ . GeV in which 27 2 hadronic decay modes are affected by strong final state 2 ¼ j 0 ð0Þj ð Þ mϕχ sθ GFmχ 0 Rχ 0 : A1 c0 π c c interactions. We performed an updated dispersive analysis and derived the decay rates of the scalar to pions and kaons. These were confronted with two earlier evalua- The derivative of the radial at the origin can be extracted from the measured decay tions [11,12] which are inconsistent with one another. 0 2 rate Γχ ð1 Þ→γγ ¼ 2.0 keV [80]. We find jR ð0Þj ≃ Our calculation confirms the result of Donoghue et al. c0 P χc0ð1PÞ [12] to within Oð1Þ precision. The remaining difference 0.06 GeV5 and assume the same value for the heavier state. can be explained by our updated input of pion-kaon The angles characterizing the mixing of the scalar with the phase shift data. Truong and Willey’s decay rate [11] two charmonium states are approximated as [48] differs substantially from ours due to a sign error in their 2 T-matrix parametrization, first pointed out in [12].We mϕχ ϑ ≃ c0 ð Þ flipped the sign in their calculation and showed that this, 1;2 2 2 : A2 mϕ − mχ þ imχ Γχ indeed, brings their result into qualitative agreement with our c0 c0 c0 findings. By matching the dispersive calculation to the The mixing-induced contributions to the scalar decay perturbative spectator model at higher mass, we then amplitude derive from the meson decay amplitudes obtained—with the mentioned caveats—a realistic estimate multiplied by ϑ1;2. These are added to the perturbative of scalar decay rates over the full mass range (Fig. 4). We amplitudes from which the charm part is subtracted in also provided the hadronic form factors which allow to order to avoid double counting. The interference between generalize our result to nonuniversally coupled light perturbative and mixing-induces terms depends on the scalars (Fig. 2). assumptions regarding the charmonium branching ratios Finally, we rederived the accelerator-, cosmological and (see [48]). We set Brðχ 0ð1PÞ → ggÞ¼1 for the lighter – c theoretical constraints on light scalars in the MeV GeV resonance which is located below the DD-threshold. For mass window (Fig. 8). We covered the model-independent the heavier state, we assume Brðχ 0ð2PÞ → DDÞ¼0.95, case as well some of the most prominent explicit models c Brðχc0ð2PÞ → ggÞ¼0.05 (such that the decay rate to with light scalars. Sensitivity projections for future key gluons is similar for both resonances). The resulting searches were also provided. The strongest deviations hadronic scalar decay rate for this estimate is shown compared to previous evaluations occur for beam dump in Fig. 4. experiments. In the case of CHARM, previous exclusions were too restrictive since they had neglected kaon absorp- APPENDIX B: SCALAR IN RARE DECAYS tion in the target. In addition, our new-found decay rates strongly impact the sensitivity window of beam dumps by 1. Radiative ϒ decays affecting the decay length of light scalars. A light scalar can emerge in the radiative decay ϒ → γϕ and induce a meson or pair [81]. It is convenient to ACKNOWLEDGMENTS express the corresponding branching ratio in the form   I would like to thank Kai Schmidt-Hoberg, Felix 2 2 2 Brϒ→γϕ mϕ Kahlhöfer, Katherine Freese, Luca Visinelli and Sebastian ¼ sθpGffiffiffiFmb F 1 − ð Þ 2 ; B1 Baum for helpful discussions. Brϒ→ee¯ 2πα mϒ

where α is the Sommerfeld constant and F a correction APPENDIX A: MIXING WITH THE function taken from [82]. It accounts for higher order QCD CHARMONIUM RESONANCES processes [83,84] as well as effects appearing We estimate the impact of the charmonium resonances close to the kinematic endpoint [85,86]. on the scalar decay rates in a nonrelativistic potential approach [48]. For this purpose, we consider the 2. Rare B decays χ ð1 Þ χ ð2 Þ c0 P state and the c0 P candidate. Masses and The scalar appears in an effective flavor violating widths are set to the central values listed in [46]. Notice, coupling ϕ-s-b. By integrating out the W-t-loop one however, that the uncertainty on Γχ ð2 Þ reaches almost an c0 P obtains [87] order of magnitude. Two higher χc0 resonances are listed as candidates for exotic structure [46] and can likely not be Lϕsb ¼ gϕsbϕs¯LbR þ H:c:; described within simple potential models. pffiffiffi 3 2 2 The off-diagonal mass matrix element between ϕ and sθmb GFmt VtsVtb gϕ ¼ ; ð Þ sb 16π2 B2 χc0ðnPÞ is estimated as [48], v

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ð 2 − 2 Þ2 where Vts and Vtb denote the Cabibbo-Kobayashi- 2 mB mϕ Γ → ϕ ¼jgϕ j : ðB7Þ Maskawa matrix elements. The above Lagrangian triggers B Xs sb 32πm3 the decay B → KðÞϕ for which the rate reads B This estimate is not valid close to the kinematic endpoint, λ1=2 where the spectator model breaks down. In this regime, the 2 ðÞ 2 B;KðÞϕ Γ → ðÞϕ ¼jgϕ j jhK js¯ b jBij ; ðB3Þ inclusive rate should, however, converge towards Γ → ϕ B K sb L R 16πm B K B since this is the only available final state. In order to obtain a smooth function with the correct asymptotic where we introduced 4 7 Γ ¼ behavior, we use (B7) for mϕ < . GeV and set B→Xsϕ 2 2 2 2 Γ → ϕ above. m − ðm − m Þ m − ðm þ m Þ B K λ ¼ x y z x y z : ðB4Þ x;yz m2 m2 x x 3. Rare K decays The matrix elements can be approximated as [88,89] The scalar can also induce rare decays of lighter mesons, for instance K → πϕ. The corresponding decay rate is 4 19 λ ðÞ again dominated by the W-t-loop. One finds [91] 2 1 mB B;K ϕ 2 jhK js¯ b jBij ¼ A ; L R 4 ð þ Þ2 K 1 2 mb ms λ = 2 2 K;πϕ 2 2 2 Γ ≃ j j jhπj¯ j ij ð Þ 1 ðm − m Þ K →π ϕ gϕds dLsR K 16π ; B8 jh j¯ j ij2 ¼ B K 2 ð Þ mK K sLbR B 4 2 fK B5 ðmb − msÞ and Γ 0 ≃ Γ . The effective coupling gϕ is KL→π ϕ K →π ϕ ds with obtained from (B2) by the replacement ðb; sÞ → ðs; dÞ. The matrix element reads [92] 1.36 0.99 ¼ − 2 2 AK 2 2 2 2 ; 1 ðm − mπÞ 1 − q =27.9 GeV 1 − q =36.8 GeV jhπjd¯ s jKij ≃ K : ðB9Þ L R 2 m − m 0.33 s d f ¼ : ðB6Þ K 1 − q2=37.5 GeV2 Since the corresponding rate for the KS decays is proportional to the small CP violating phase in the 2 2 The transferred momentum is set to q ¼ mϕ. In the case of Cabibbo–Kobayashi–Maskawa matrix, it suffers a stronger K we already took the sum over polarizations. suppression [90]. For cases where the nature of the strange (s) in the final state is not of relevance, one can define 19We neglect subleading contributions due to scalar brems- → ϕ the inclusive decay rate B Xs . The spectator model strahlung and the charm loop which would amount to a correction predicts [7] ≲10% [90].

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