Phenomenology of GeV-scale scalar portal

We review and revise the phenomenology of the scalar portal -- a new scalar particle with the mass in GeV range that mixes with the Higgs boson. In particular, we consider production channels $B\to S K_1(1270)$ and $B\to S K_0^*(700)$ and show that their contribution is significant. We extend the previous analysis by comparing the production of scalars from decays of mesons, of the Higgs bosons and direct production via proton bremsstrahlung, deep inelastic scattering and coherent scattering on nuclei. Relative efficiency of the production channels depends on the energy of the beam and we consider the energies of DUNE, SHiP and LHC-based experiments. We present our results in the form directly suitable for calculations of experimental sensitivities.

D Scalar production in proton bremsstrahlung 29 D. 1

Introduction
We review and revise the phenomenology of the scalar portal -a gauge singlet scalar particle S that couples to the Higgs boson and can play a role of a mediator between the Standard model and a dark sector (see e.g. [1][2][3]) or be involved in the cosmological inflation [4][5][6]. We focus here on the mass range 10 GeV (see however section 2.2 for a discussion of larger masses).
The interaction of the S particle with the Standard model particles is similar to the interaction of a light Higgs boson but is suppressed by a small mixing angle θ. Namely, the Lagrangian of the scalar portal is (1.1) After the electroweak symmetry breaking the Higgs doublet gains a non-zero vacuum expectation value v. As a result, the SHH interaction (1.1) provides a mass mixing between S and the Higgs boson h. Transforming the Higgs field into the mass basis, h → h + θS, one arrives at the following interaction of S with the SM fermions and gauge bosons: where α ≡ 2α 2 v. These interactions also mediate effective couplings of the scalar to photons, gluons, and flavor changing quark operators, see Fig. 1. Additionally, the effective proton-scalar interaction that originates from the interaction of scalars with quarks and gluons (see Fig. 2) will also be relevant for our analysis. The effective Lagrangian for these interactions is discussed in Appendix A.
The phenomenology of light GeV-like scalars has been studied in [1,5,8,[30][31][32][33], and in [34][35][36][37][38][39][40][41][42][43] in the context of a light Higgs boson. However, in the literature, there are still conflicting results, both for the scalar production and decay. In this work, we reanalyze the phenomenology of light scalars and present the results in the form directly suitable for experimental sensitivity estimates.   In this section, we will discuss the scalar production channels that are defined by the mixing between a scalar and the Higgs boson. W GG (m S = 2 GeV) 1.2 · 10 3 1.4 · 10 3 6.2 · 10 10 W GG (m S = 5 GeV) 1.5 · 10 −1 7.9 3.9 · 10 10 In proton-proton or proton-nucleus collisions, a scalar particle: (a) can be emitted by the proton, (b) produced from photon-photon, gluon-gluon or quark-antiquark fusion in proton-proton or proton-nucleus interactions or (c) produced in the decay of the secondary particles, see Fig. 3. Let us compare these three types of the scalar production mechanisms depending on the collision energy and the scalar mass. In the following we will present the results for three referent proton-proton center-ofmass energies: √ s pp ≈ 16 GeV (corresponding to the beam energy of the DUNE experiment), √ s pp ≈ 28 GeV (SHiP) and √ s pp = 13 TeV (LHC).
The proton bremsstrahlung (the case (a)) is a process of a scalar emission by a proton in proton-proton interaction. For small masses of scalars, m S < 1 GeV, this is a usual bremsstrahlung process described by elastic nucleon-scalar interaction with a coupling constant θg SN N ∼ θm p /v, see Appendix A.2. However, the probability of elastic interaction decreases with the scalar mass and we need to take into account inelastic processes. The probability for the bremsstrahlung is P brem = θ 2 g 2 SN N P brem (m S , s pp ), (2.1) where P brem is a proton bremsstrahlung probability for the case θ = g SN N = 1 (see Appendix D). This quantity varies from 10 −2 for DUNE and SHiP to 10 −1 for the LHC, see Appendix D. In this estimate, we neglect possible effects of QCD scalar resonances such as f 0 (500), f 0 (580), f 0 (1370), . . . , that could resonantly enhance the scalar production for some masses (see [44,45] and Appendix A.2, D for further details).
For the case (b), we have to distinguish the photon-photon fusion that can occur for an arbitrary transferred momentum and, therefore, an arbitrary scalar mass (as electromagnetic interaction is long-range), and gluons or quark-antiquarks fusion (the so-called deep inelastic scattering processes (DIS)), which is effective only for m S 1 GeV. The scalar production in the DIS process can be estimated as P DIS = (σ DIS,G + σ DIS,q )/σ pp , where σ pp is the total proton-proton cross section and σ DIS,X is the cross section of scalar production in the DIS process, Here, √ s pp is the center-of-mass energy of colliding protons and W XX given by Eq. (C.11) is a dimensionless combinatorial factor that, roughly, counts the num- ber of the parton pairs in two protons that can make a scalar. The values of W XX factors for some scalar masses and experimental energies are given in Table 1. In Fig. 4 we show the ratio between cross sections of gluon-gluon and quark-antiquark fusion. We see that quark fusion is relevant only for low scalar masses, while for m S 2 GeV the gluon fusion dominates for all collision energies considered. : The ratio of cross sections of the scalar production in deep inelastic scattering via gluon and quark fusion. The dashed line corresponds to a ratio equal to unity. "LHC", "SHiP" and "DUNE" denote correspondingly the results for the proton-proton center-of-mass energies √ s pp = 13 TeV, √ s pp = 28 GeV and √ s pp = 16 GeV.
In the case of the production of a scalar in photon fusion, the most interesting process is the coherent scattering on the whole nucleus, as its cross section is enhanced by a factor Z 2 , where Z is the charge of the nucleus. The electromagnetic process p + Z → p + Z + S involves the effective Sγγ vertex proportional to θα EM , see Appendix A.1. The probability of the process is P γ fus = σ γ fus /σ pZ , where the fusion cross section σ γ fus has a structure similar to that of gluon fusion (2.2): where √ s pZ is the CM energy of the proton and nucleus, and W coh given by Eq. (E.8) is a dimensionless combinatorial factor that counts the number of pair of photons that can form a scalar. It ranges from 10 6 for the DUNE energies to 10 14 for the LHC energies.
Let us compare the probabilities of photon fusion and proton bremsstrahlung, for all three energies considered. Here we used s pZ ≈ As pp , where A is the nucleus mass number. The proton-nucleus cross section σ pZ weakly depends on energy and can be estimated as σ pZ 50 mb A 0.77 [46,47]. This ratio is smaller than 10 −4 for all energies and scalar masses of interest. Next, comparing the probabilities of the production in photon fusion and in DIS, we obtain where we used that s pZ /s pp ≈ A and W coh /W DIS 1 for all three energies considered, see Appendix E. The proton-proton cross section also depends on the energy weakly, and we can estimate σ pZ /σ pp ∼ A 0.77 (see Appendix D).
We conclude that the scalar production in photon fusion is always sub-dominant for the considered mass range of scalar masses and beam energies.
Let us now compare gluon fusion and proton bremsstrahlung with the production from secondary mesons (type (c)). The latter can be roughly estimated using "inclusive production", i.e. production from the decay of a free heavy quark, without taking into account that in reality this quark is a part of different mesons with different masses. This is only an order of magnitude estimate that breaks down for m S m q − Λ QCD , so it can be used only for D and B mesons. We will see however that such an estimate is sufficient to conclude that for the energies of SHiP and LHC the production from mesons dominates and we need to study it in more details (see more detailed analysis below).
The inclusive branching BR incl (X Q i → X Q j S) can be estimated using the corresponding quark level process Q i → Q j S. To minimize QCD uncertainty we follow [16,43] and estimate the inclusive branching ratio as where Γ(Q i → Q eν e ) is the semileptonic decay width of a quark Q i calculated using the Fermi theory and BR incl (X → X Q eν e ) is the experimentally measured inclusive branching ratio. As both the quark decay widths in (2.7) get the QCD corrections, their total effect in (2.7) is expected to be minimal [43] For D and B mesons decays the inclusive production probabilities are [19,43]  1.0 · 10 −4 3.9 · 10 −3 2.9 · 10 −2 χ bb 1.0 · 10 −10 2.7 · 10 −7 8.6 · 10 −3 Table 2: Production fractions of the qq pair, χ qq = σ qq /σ pp , for the DUNE, SHiP and LHC experiments. We took the production fractions for the DUNE and SHiP experiments from [48,49]. To estimate the production fractions for the LHC, we calculated the total cross section of B and D production using FONLL [50] and took the total cross section of the pp collisions at the LHC energies from [46].
where χ qq is the production fraction of the qq pair in pp collisions, see Table 2. The difference in 10 −11 orders of magnitude is mostly coming from (m b /m t ) 4 ∼ 10 −7 and |V ub | 2 /|V ts | 2 ∼ 10 −2 (see Appendix A.3 for details). In fact, for D mesons the leptonic decay D → Seν with BR(D → Seν) ∼ 5 · 10 −9 θ 2 is more important than (2.8), see Appendix B.3 for details. We see that the production from D mesons may be important only if the number of B mesons is suppressed by 10 9 times, which is possible only if the center-of-mass energy of p-p collisions is close to the bb pair production threshold.
Let us compare the production from B mesons with the production from proton bremsstrahlung and DIS. Using Eqs. (2.1), (2.2) and (2.9) for masses of scalar below b quark kinematic threshold we get The ratios (2.11) and (2.10) depend on the center-of-mass energy of the experiment (see Tables 1 and 2).
We conclude that for the experiments with high beam energies, like SHiP or LHC, the most relevant production channel is a production of scalars from secondary mesons. For experiments with smaller energies like, e.g., DUNE the dominant channel is the direct production of scalars in proton bremsstrahlung and in DIS.
Production from decays of different mesons. Let us discuss the production of scalars from decays of mesons in more details. The calculation of branching ratios for two-body decays of mesons is summarized in Appendix B.2. Above we made an estimate for the cases of D and B mesons that are the most efficient production channels for larger masses of S. Instead, for scalar masses m S < m K − m π the main production channel is the decay of kaons, Process BR(m S = 0)/θ 2 Closing mass [GeV] Appendix Table 3: Properties of the main production channels of a scalar S from kaons and B mesons through the mixing with the Higgs boson. First column: decay channels; Second column: branching ratios of 2-body meson decays evaluated at m S = 0 using formula (B.9) and normalized by θ 2 . For B mesons the numerical values are given for B ± mesons; in the case of B 0 meson all the given branching ratios should be multiplied by a factor of 0.93 that comes from the difference in total decay widths of B ± and B 0 mesons [46]. Uncertainties (where available) follow from uncertainties in meson transition form-factors; Third column: effective closing mass, i.e. a mass of a scalar at which the branching ratio of the channel decreases by a factor of 10; Fourth column: a reference to the appendix with details about form-factors used.
K → Sπ, see Table 3 with the relevant information about these production channels. Numerically, the branching ratio of the production from kaons is suppressed by 3 orders of magnitude in comparison to the branching ratio of the production from B mesons, but for the considered energies the number of kaons is at least 10 3 times larger than the number of B mesons.
For scalar masses m K − m π < m S < m B the main scalar production channel from hadrons is the production from B mesons. Inclusive estimate at the quark level, that we made above (see Eq. (2.9)), contains an unknown QCD uncertainty and completely breaks down for scalar masses m b − m S Λ QCD . Below we discuss therefore decays of different mesons containing the b quark B → X s/d S. We consider kaon and its resonances as the final states X s : • Pseudoscalar meson K; • Scalar mesons K * 0 (700), K * 0 (1430) (here assuming that K * 0 (700) is a di-quark state);  Figure 5: Branching ratios of the 2-body decays B + → SX s/d , where X q is a hadron that contains a quark q. By the K * channel, we denote the sum of the branching ratios for K * (892), K * (1410), K * (1680) final states, by K * 0 -for K * 0 (700), K * 0 (1430), and by K 1 -for K 1 (1270), K 1 (1400). The "Inclusive" line corresponds to the branching ratio (2.7) obtained using the free quark model.
• Vector mesons K * (892), K * (1410), K * (1680); • Axial-vector mesons K 1 (1270), K 1 (1400); • Tensor meson K * 2 (1430). We also consider the meson X d = π. Although the rate of the corresponding process B → πS is suppressed in comparison to the rate of B → X s S, it may be important since it has the largest kinematic threshold m S m B − m π .
We calculate the branching ratios BR(B + → X s/d S) at m S m K using Eq. (B.9) and give the results in Table 3. The main uncertainty of this approach is related to form factors describing meson transitions X Q i → X Q j , see Appendix F for details. They are calculated theoretically using approaches of light cone fum rules and covariant quark model, and indirectly fixed using experimental data on rare mesons decays [51][52][53][54][55]. The errors given in Table 3 result from uncertainties in the meson transition form-factors F BX s/d (see Appendix F). Since F BX s/d are the same for B + and B 0 mesons, the branching ratios BR The values of the branching ratios for the processes B → KS, B → K * S are found to be in good agreement with results from the literature [31,32]. We conclude that the most efficient production channels of light scalars with m S 3 GeV are decays B → K * 0 S, B → K 1 S and B → K * S; the channel B → KS, considered previously in the literature, is sub-dominant. Summing over all final K states, in the limit m S m B for the total branching ratio we have Using the estimate (2.9), for the ratio of the central value of the branching ratio (2.12) and the inclusive branching ratio at m S m B we find Provided that the inclusive estimate of the branching ration has a large uncertainty, we believe that Eq. (2.13) suggests that we have taken into account all main decay channels of this type.
Our results for decays of B mesons are summarized in Table 3 and Fig. 5. We have found that the channels with K * , K * 0 and K 1 give the main contribution to the production branching ratio for small scalar masses m S 3 GeV, while for larger masses the main channel is the decay B ±/0 → K ±/0 + S.
The comparison between the probability of the production from mesons and our estimates for bremsstrahlung and DIS for three center-of-mass energies are shown in Fig. 6. In order to indicate an uncertainty because of the unknown proton-scalar elastic form-factor, we show the production probability in bremsstrahlung process for the scalar masses m S > 0.5 GeV in gray (see, however, Fig. 16, where we show the bremsstrahlung probability with the form-factor from [44,45]).

Quartic coupling
Above we discussed the production of scalars only through the mixing with the Higgs boson. One more interaction term in the Lagrangian (1.2), (the so-called "quartic coupling" that originates from the term S 2 H † H in the Lagrangian (1.1)) affects the production of scalars from decays of mesons and opens an additional production channel -production from Higgs boson decays, see Fig. 7.
The production from the Higgs boson (case (a)) can be important for highenergy experiments like LHC. The branching ratio is where p S is a momentum of a scalar in the rest frame of the Higgs boson and we used the SM decay width of the Higgs boson Γ h ≈ 4 MeV [56]. If the decay length of the scalar is large enough cγτ S 1 m this decay channel manifests itself at ATLAS and CMS experiments as an invisible Higgs boson decay. The invisible Higgs decay is constrained at CMS [9], the 2σ upper bound is This puts an upper bound α < 3.1 GeV for the scalar masses m S < m h /2.

The production probability
where χ h is a production fraction of the Higgs bosons. Comparing with the production from B mesons for a scalar mass below the B threshold estimated by the inclusive production (2.9) we get where we used χ h ∼ 10 −9 for the LHC energy [57] and χ bb ∼ 10 −2 (see Table 2).
Also, the quartic coupling generates new channels of scalar production in meson decays (case (b)). In addition to the process X → X S shown in Fig. 3 (c) the quartic coupling enables also additional processes X → SS and X → X SS shown in Fig. 7 (b) [1,31,[58][59][60][61].
First, let us make a simple comparison between the branching ratios for the scalar production from mesons in the case of mixing with the Higgs boson and quartic coupling. Comparing Feynman diagrams in Figs. 3 (c) and 7 (b) we see that for the case m S m X where f X is a meson's decay constant (see Appendix G).
The channel X → X SS is very similar to the channel X → X S from Fig. 3 (c). By the same reasons, this process is strongly suppressed for D-mesons and is efficient only for kaons and B mesons. Table 4: Properties of the main production channels of a scalar S from kaons and B mesons through quartic coupling (2.14). First column: decay channels; Second column: branching ratios of 2-body meson decays evaluated at m S = 0 and α = 1 GeV, see Eqs. (G.7), (G.8). For B mesons the numerical values are given for B ± mesons; in the case of decays of B 0 mesons, all the given branching ratios should be multiplied by a factor 0.93 that comes from the difference in total decay widths of B ± and B 0 mesons [46]. Uncertainties (where available) follow from uncertainties in meson transition form-factors; Third column: effective closing mass, i.e. a mass of a scalar at which the branching ratio of the channel decreases by a factor of 10. Fourth column: a reference to appendix with details about form-factors used.
The decay X → SS is possible only for K 0 , D 0 , B 0 and B 0 s due to conservation of charges. The production from D 0 mesons is strongly suppressed by the same reason as above (small Yukawa constant and CKM elements in the effective interaction, see Appendix A.3).
Our results for the branching ratio of the scalar production from mesons decays in the case of the quartic coupling are presented in Table 4  Br(B→XSS) where X q is a hadron that contains a quark q, versus the scalar mass. By the K * channel, we denote the sum of the branching ratios for K * (892), K * (1410), K * (1680) final states, by K * 0 -for K * 0 (700), K * 0 (1430), and by K 1 -for K 1 (1270), K 1 (1400). The values of the branching ratios are given at α = 1 GeV.

Scalar decays
The main decay channels of the scalar are decays into photons, leptons and hadrons, see Appendix H. In the mass range m S 2m π the scalar decays into photons, electrons and muons, see Appendix H.1.
For masses small enough compared to the cutoff Λ QCD ≈ 1 GeV, ChPT (Chiral Perturbation Theory) can be used to predict the decay width into pions [34]. For masses of order m S Λ QCD a method making use of dispersion relations was employed in [35][36][37] to compute hadronic decay rates. As it was pointed out in [33] and later in [30], the reliability of the dispersion relation method is questionable for m S 1 GeV because of lack of the experimental data on meson scattering at high energies and unknown contribution of scalar hadronic resonances, which provides significant theoretical uncertainties. To have a concrete benchmark -although in the light of the above the result should be taken with a grain of salt -we use the decay width into pions and kaons from [32], see Fig. 9, which combines the next-to-leading order ChPT with the analysis of dispersion relations for the recent experimental data. For the ratio of the decay widths into neutral and charged mesons we have For scalar masses above f 0 (1370) the channel S → 4π becomes important and should be taken into account [62]. The decay width of this channel is currently not known; its contribution can be approximated by a toy-model formula [32] where a dimensional constant C is chosen so that the total decay width is continuous at large m S that will be described by perturbation QCD, see Fig. 10.  Figure 9: The ratio of the decay widths of a light scalar into pions, kaons and into muons obtained in [32]. We summed over all final meson states, i.e. for decays into pions we summed over π + π − , π 0 π 0 . A peak in the decay width corresponding to the channel S → ππ around m S 1 GeV is caused by the narrow f 0 (980) resonance.
For m S ≥ Λ pert S 2 − 4 GeV hadronic decays of a scalar can be described perturbatively using decays into quarks and gluons, see Appendix H.2. Currently, the value of Λ pert S is not known because of lack of knowledge about scalar resonances which can mix with S and enhance the scalar decay width. In [32] the value of Λ pert S is set to 2 GeV, in [5] it is Λ pert S = 2.5 GeV, while in [19] its value is Λ pert S = 2m D . This scale certainly should above the mass of the heaviest known scalar resonance f 0 (1710), so in this work we choose Λ pert S = 2 GeV.
The summary of branching ratios of various decay channels of the scalar and the total lifetime of the scalar is shown in Fig. 10. Table 5: Relevant scalar decay channels. Only channels with the branching ratio above 1% covering the scalar mass range up to 10 GeV are shown. The gray line corresponds to the fake multi-meson channel, see discussion in text. Columns: (1) the scalar decay channel. (2) The scalar mass at which the channel opens; (3) The scalar mass starting from which the channel becomes relevant (contributes larger than 10%); (4) The scalar mass above which the channel contributes less than 10%; "-" means that the channel is still relevant at m S = 10 GeV; (5) The maximal branching ratio of the channel for m S < 10 GeV; (6) Reference to the formula (or figure in case of decays into pions and kaons) for the decay width of the channel.

Channel
Opens

Conclusion
In this paper, we have reviewed and revised the phenomenology of the scalar portal, a simple extension of the Standard Model with a scalar S that is not charged under the SM gauge group, for masses of scalar m S 10 GeV. We considered three examples of experimental setup that correspond to DUNE (with proton-proton center of mass energy √ s pp ≈ 16 GeV), SHiP ( √ s pp ≈ 28 GeV) and LHC based experiments Interactions of a scalar S with the Standard Model can be induced by the mixing with the Higgs boson and the interaction Sh 2 (the "quartic coupling"), see Lagrangian (1.1). The mixing with the Higgs boson is relevant for a scalar production and decay, while the quartic coupling could be important only for the scalar production.
For the scalar production through the mixing with the Higgs boson, we have explicitly compared decays of secondary mesons, proton bremsstrahlung, photon-photon fusion, and deep inelastic scattering. For the energy of the SHiP experiment, the most relevant production channel is the production in decays of secondary mesons, specifically kaons and B mesons. For smaller energies (corresponding in our examples to the DUNE experiment) the situation is more complicated, and direct production channels from p − p collisions (proton bremsstrahlung, deep inelastic scattering) give the main contribution to the production of scalars, see Fig. 6.
Our results for various channels of the scalar production from mesons via mixing with the Higgs boson are summarized in Table 3 and in Fig. 5. The results for decays B → KS, B → K * (892)S agree with the references [1,2,5,8,31,32], while other decay channels have not been studied in these papers.
For the LHC-based experiments, an important contribution to the production of scalars is given by the production in decays of Higgs bosons that may be possible due to non-zero quartic coupling. This production channel, when allowed by the energy of an experiment, allows to search for scalars that are heavier than B mesons. It may also significantly increase the experimental sensitivity in the region of the lower bound of the sensitivity curve, where production through the mixing with the Higgs boson is less efficient.
The quartic coupling also gives rise to meson decay channels X → SS and X → X SS, that are important for scalar masses m S m B /2. Our results for these channels are presented in Table 4 and in Fig. 8.
The description of scalar decays is significantly affected by two theoretical uncertainties: (i) the decay width of a scalar into mesons like S → ππ and S → KK (that may be uncertain more than by an order of magnitude for masses of a scalar around 1 GeV) and (ii) the uncertainty in the scale Λ pert S at which perturbative QCD description can be used. As a benchmark, for decays into mesons we use results of [32] and choose Λ pert s = 2 GeV, but we stress that the correct result is not really known for such masses. The main properties of scalar decays are summarized in Table 5 and Fig. 10. The effective lagrangian of the interaction of S with photons and gluons is generated by the diagrams 11. It reads Here the effective vertices F γ , F G are [5,63] Their behavior in dependence on the scalar mass is shown in Fig. 12. The values of the constants C SGG and C Sγγ vary in the literature. Namely, in [5] they are C Sγγ = 1, C SGG = 1/ √ 8. From the other side, in [63] predicts |C Sγγ | = 1/2, |C SGG | = 1/4. Calculating the decay branching ratio of the Higgs boson into two photons, we found that the value C Sγγ = 1/2 is consistent with experimental results for the signal strength of the process p + p → h + X, h → γγ [64]. 1 The gluon loop factor in the triangle diagram 11 differs from the photon loop factor by the factor tr[t a t a ] = 1 2 , where t a is the QCD gauge group generators, and therefore C Sgg = 1/4.

A.2 Nucleons
Consider the low-energy interaction Lagrangian between the nucleons N and the scalar: The coupling g SN N is defined as For energy scales of order of the nucleon mass, the u, d, s quarks are light, while the c, b, t quarks are heavy. Therefore, the latter can contribute to the effective coupling (A.6) only through effective interactions involving the lighs quarks and gluons. The latter can be obtained using the heavy quarks expansion [65,66]. Keeping only the leading 1/m q heavy term, for the effective interaction operator we obtain [67] q=c,b,t Here α s is the QCD interaction constant evaluated on the scale of the hadronic mass, G a µν is the gluon strength tensor and n heavy = 3 is the number of the heavy quarks. Therefore, in the leading order of 1/m q heavy expansion the coupling (A.6) takes the form [67,68] The last expression we can written in terms of the nucleon mass m N , where θ µµ is the trace of the stress-energy tensor in the QCD [67] θ µµ = q=u,d,s where β s is the QCD β function, in the leading order by α s . Therefore, we get [67,68] g SN N = 2 9 The numerical value is g SN N ≈ 1.2 · 10 −3 [69].
In order to incorporate effects of non-zero momentum transfer q 2 in the SN N vertex, we need to take into account an scalar nucleon form-factor F N,S (q 2 ): From general grounds, we expect that the form-factor F N,S (q 2 ) incorporates a mixing with scalar J P C = 0 ++ resonances, which causes the resonant enhancement of the production at q 2 = m 2 f 0 . A model for the form-factor is discussed in the works [44,45]. According to it, the scalar incorporates mixing through a sum of Breit-Wigner components of the lightest scalar resonances f 0 (500), f 0 (980), f 0 (1370): The constants are extracted from the criteria F N,S (0) = 1 and F N,S (q 2 ) ∝ q −4 at q 2 m 2 f 0 , with the latter following from the quark counting rules. Their central numerical values are f 0 = {0.28, 1.8, −0.99}.

A.3 Flavor changing effective Lagrangian
A light scalar S can be produced from a hadron via flavor changing quarks transitions (see diagrams in Fig. 13). The flavor changing amplitude was calculated using different techniques in many papers [38][39][40][41][42][43]. The corresponding effective Lagrangian of flavor changing quark interactions with the S particle is Figure 13: Diagrams of the production of the scalar S in flavor changing quarks transitions in the unitary gauge.
where Q i and Q j are both upper or lower quarks and P R ≡ (1 + γ 5 )/2 is a projector on the right chiral state. The effective coupling ξ ij is defined as where Q k are the lower quarks if Q i and Q j are the upper and vice versa, V ij are the elements of the CKM matrix, and G F is the Fermi constant. One power of the quark mass in the expression for ξ comes from the hqq coupling, while another one comes from the helicity flip on the quark line in the diagrams in Fig. 13. Because of such behavior, the quark transition generated by the Lagrangian (A.15) is more probable for lower quarks than for upper ones, since the former goes through the virtual top quark. Numerical values of some of ξ ij constants are given in Table 6.

B Scalar production from mesons
In the scalar production from hadron decays, the main contribution comes from the lightest hadrons in each flavor, which are mesons. 2 The list of the main hadron candidates is the following (the information is given in the format "Hadron name (quark contents, mass in MeV)"): The production of a scalar from mesons is possible through the flavor changing neutral current A.3, so the production from D s , J/ψ, B s , B c and Υ mesons does not have any advantage with respect to the production from D 0 , D + , B − and B 0 , while their amount at any experiment is significantly lower. Therefore, we will discuss below only production from later mesons.

B.1 Inclusive production
The decay widths for the processes where p S is the S particle momentum at the rest frame of the meson X, the integration limits are and is the phase space factor.

B.2 Scalar production in two-body mesons decays
Let us consider exclusive 2-body decay of a meson corresponding to the transition Q i → Q j + S. Here and below, X Q denotes a meson which contains a quark Q.
The Feynman diagram of the process is shown in Fig. 3 (c). Using the Lagrangian (A.15), for the matrix element we have is the matrix element of the transition X Q i → X Q j . Expressions for these matrix elements for different initial and final mesons are given in Appendix F. So, we can calculate the branching fraction of the corresponding process by the formula [46] BR( where Γ X is the decay width of the meson X. We use the lifetimes of mesons from [46].
For the kaons, the only possible 2-body decay is the process There are 3 types of the kaons -K ± , K 0 L , K 0 S . Although the decay width for each of them is by given by the same loop factor, ξ sd , the branching ratios differ. The first reason is that these kaons have different decay widths. The second reason is that the K 0 S is approximately the CP -even eigenstate. Therefore the decay K 0 S → πS is proportional to the CKM CP -violating phase and is strongly suppressed [41]. Further we assume that the corresponding branching ratio vanishes. See Table 3 for the branching ratios of K 0 L , K ± .

B.3 Scalar production in leptonic decays of mesons
Consider the process X → Seν. Its branching ratio is [70,71] BR( The values of the branching ratios for different types of the mesons are shown in Table 7. However, although for the D this channel enhances the production in O(100) times, the production from D is still sub-dominant. Meson

C DIS
The scalar production in the DIS is driven by the interaction with the quarks and gluons: where F G is a loop factor being of order of |F G | 2 10 − 20 for the scalars in the mass range m S 10 GeV (see Appendix A.1). Processes of the scalar production in DIS are quark and gluon fusions: Corresponding diagrams are shown in Fig. 14 The differential cross section is given by where Y denotes a quark/antiquark or a gluon, |M(GG → S)| 2 is the squared matrix element averaged over gluon or quark polarizations and The hard cross sections for the gluon and quark fusions are thus Using hard cross sections (C.8) and (C.7), one can calculate the total cross section of the production in DIS as Here, f Y (Q, x) is the parton distribution function (pdf) of the parton Y carrying the momentum fraction x at the scale Q. g q = 2, g G = 1; g q is a combinatorial factor taking into account that the quark/antiquark producing a scalar can be stored in both of colliding protons.
The result is Here, s denotes the pp CM energy, m q is the MS quark mass at the scale m S , and is the partonic weight of the process. Since the partonic model breaks down at scales Q 1 GeV, the description of the scalar production in DIS presented in this section is valid only for scalars with masses m S 1 GeV. For numerical estimates we have used LHAPDF package [72] with CT10NLO pdf set [73].
The main contribution to the DIS cross section comes from gluons. To see this, let us compare the gluon cross section σ DIS,G with the s-quark cross section σ DIS,S , which is the largest quark cross section. 3 Their ratio is The product |F G | 2 α 2 s changes with m S relatively slowly, and therefore the ratio (C.12) is determined by the product (m S /1 GeV) 2 W GG /W ss . It is larger than one for the masses m S 2 GeV in broad CM energy range, see Fig. 4.
Having the cross sections (C.10), we calculate the DIS probability as where for the total proton-proton cross section σ pp we used the data from [46].

D Scalar production in proton bremsstrahlung
p p S p X Figure 15: A diagram of the production of a scalar in the proton bremsstrahlung process.
A scalar S can be produced through the SN N vertex (see Sec. A.2) in proton-proton bremsstrahlung process p + p → S + X, (D.1) with the diagram of the process shown in Fig. 15. Corresponding probability can be estimated using generalized Weizsacker-Williams method, allowing to express the cross section of the given process by the cross section of its sub-process [74][75][76][77][78][79][80]. Namely, let us denote the momentum of the incoming proton in the rest frame of the target proton by p p , the fraction of p p carried by S as z and the transverse momentum of S as p T . Then, under conditions the differential production cross section of S production can be written as (see Appendix D.1) dσ brem ≈ σ ppt (s ) × P p→pS (p T , z)dp 2 T dz, where we denoted a target proton as p t , σ ppt is the total p-p cross section, s = 2m p p p (1 − z) + 2m 2 p and the differential splitting probability of the proton to emit a scalar is with g SN N being low-energy proton-scalar coupling, and F pS the scalar-proton formfactor, see Appendix A.2 and Eq. (A.14).
For the total pp cross section, we use the experimental fit where Z = 35.45 mb, B = 0.308 mb, C 1 = 42.53 mb, C 2 = 33.34 mb, √ s 0 = 5.38 GeV, √ s 1 = 1 GeV, η 1 = 0.458 and η 2 = 0.545 [46]. This cross section is shown in Fig. 16, from we see that it is almost constant for a wide range of energies.
The total cross section can be written in the form where P brem (s, m S ) = 1 g 2 SN N θ 2 dp 2 The domain of the definition of p T and z is determined by the conditions (D.2). For definiteness, we fix the domain of integration by the requirement The probability of a scalar production in proton bremsstrahlung is where s is the CM energy of two protons. We show its dependence on the scalar mass and the incoming beam energy in Fig. 16.

D.1 Splitting probability derivation
Following the approach described in [74], let us consider the process (D.1) within the old-fashioned perturbation theory. The corresponding diagrams are shown in Fig. 17. The matrix element has the form (D.10) Here, M denotes Lorentz-invariant amplitude of the processes. There exists a kinematic domain at which |M b | |M a |. Namely, let us consider an ultrarelativistic Neglecting the difference in the energy conservation arguments in the delta-functions that are of order O(m 2 p/S /p 2 p , p 2 T /p 2 p ), we can relate these two cross sections as dσ ppt→SX = dP p→p S (z, p T )dσ p pt→X (p T , z), (D. 18) where we introduced differential splitting probability dP p→p S : Here a factor of 2 is combinatorial factor taking into account that a scalar can be produced from both the legs of colliding protons.
Integrating the differential cross section (D.18) over the momenta of the final states particles X and summing over all possible sets {X}, we finally arrive at where s ≈ 2m p p p (1 − z) + 2m 2 p 4 and σ pp (s ) is the total proton-proton cross section.
Let us now find explicit expression for the splitting probability (D. 19). Using the expressions (D.2), we find Finally, we arrive at (D.22)

E Scalar production in photon fusion
A scalar can be produced elastically in pp collisions through the Sγγ vertex (see Appendix A.1). The production process is with the corresponding diagram shown in Fig. 19. To find the number of produced scalars in the photon fusion, we will use the equivalent photon approximation (EPA), which provides a convenient framework for studying processes involving photons emitted from fast-moving charges [81][82][83]. The basic idea of the EPA is a replacement of the charged particle Y in the initial and final state, that interacts through the virtual photon carrying the virtuality q and the fraction of charge's energy x, by the almost real photon with a distribution n Y (x; q 2 ) that depends on the type of the Z Z p S p γ Figure 19: A diagram of the production of a scalar in photon fusion. Figure 20: The idea of the equivalent photon approximation. If a charge with the momentum k, emitting the virtual photon with the virtuality q, is ultrarelativistic, then the cross section of the process (a) can be expressed in terms of the cross section of the process (b). The remained effect of the charge is the distribution function n charge (x, q 2 ), where x is the energy fraction carried by photon.
charged particle, see Fig. 20. The magnitude of the momentum transfer carried by the virtual photon can be approximated as where q t is the transverse component of the spatial momentum of the photon with respect to the spatial momentum of the particle Y , and m Y is the mass of Y . Conditions for validity of the EPA are x 1 and q t xm Y [83]. The distribution n Y (x, q 2 t ) of the emitted photons can be described by where C(q 2 ), D(y 2 ) are appropriate form-factors. We take the proton and nucleus form-factors from [83].
Within the EPA, we approximate the cross section of the process (E.1) by Here Let us discuss the boundaries of integration in Eq. (E.4). Following [83], for the upper limit of q we choose q max = 1 GeV for the maximal virtuality of a photon emitter by the proton and q max = 4.49/R 1 for a photon emitted by the nucleus.
Using (E.2), we get x p,max ≈ 0.63, x Z,max = 0.018. The lower bound on q, it is given by the kinematic threshold for the S particle production. For the nucleus, there is additional constraint q 2 r −2 s , where r s 10 keV is the inverse radius of the electron shell (at larger scales the nucleus is screened by electrons).
Substituting the photon fusion cross section (E.5) into (E.4), for the pZ cross section we get where (E.8) is the integrated form-factor. Here we simplified the integration domain for p t assuming q 1t,max , q 2t,max = 1 GeV, since the integrand is nonzero only in some region of parameters within the integration area, and therefore by increasing of integration limits we will not affect the result.
The production probability is calculated using the cross section (E.7) as where σ pZ ≈ 53 A 0.77 mb is the total pZ cross section, with A being the mass number of the nucleus target [47].
The dependence of P γ fusion on the scalar mass and collision energy is shown in Fig. 21 describing transitions of mesons X(Q i ) → X (Q j ) in the case of the same and opposite parities P , P correspondingly. These matrix elements can be related to the matrix elements describing the weak charged current mediating mesons transition X → X . To derive the relation, we follow [84] in which a relation for pseudoscalar transition X was obtained. We generalize this approach to the arbitrary final-state meson. We first notice that where p µ Q ≡ p µ Q i − p µ Q j and we used the Dirac equation for free quarks. Using then the identityQ whereP µ ≡ i∂ µ is the momentum operator, we find where q µ ≡ p X µ − p Xµ ; for deriving the expression we have acted byP µ on the meson states |X , |X . Similarly, for P = P we find Further we will assume that X is a pseudoscalar, and therefore transitions in pseudoscalar, pseudovector mesons X are parity even, while transitions in scalar, vector and tensor mesons are parity odd.
Contracting it with q µ , we obtain We take the expression for the form-factor f XP 0 (q 2 ) from [52]: The values of the parameters m X fit , F XP 0 for different X, P are summarized in Table 8.

F.1.2 Scalar
For the scalar meson X =S we have [54] [54]). Similarly to the case h = P , Consider the transition B → K * 0 S. There is an open question whether hypothetical K * 0 (700) is a state formed by two or four quarks, see, e.g. [87], discussions in [54,88] and references therein. We assume that K * 0 (700) is a di-quark state and K * 0 (1430) is its excited state. There are no experimentally observed decays B → K * 0 (700)X, and therefore there is quite large theoretical uncertainty in determination of the form-factors (see a discussion in [89]). We will use [54], where there are results for B → K * 0 (700) and B → K * 0 (1430), and the results for the latter are in good agreement with the experimental data for B → K * 0 (1430)η decay. We fit the q 2 dependence of f BK * 0 + from [54] by the standard pole-like function that is used in the literature discussing the B → K * 0 transitions (see, e.g., [88]): where m B = 5.3 GeV is the mass of the B + meson. The fit parameters are given in Table 9.
where µ (p V ) is the polarization vector of the vector meson, and A i , V are the formfactors. The form-factor A 3 is related to A 1 and A 2 as Contracting (F.15) and (F.16) with q µ , we obtain that the vector part of the matrix element vanishes, while for the axial-vector part we find where we used the relation (F.17). Consider a scalar product ( * (p V ) · p X ) in the rest frame of the meson X. In this case only longitudinal polarization of * µ (p V ) contributes.
For the case B → K * (892), we follow [51] and parametrize the form-factor as (F.20) The values of parameters are given in Table 10.
For the case B → V = K * (1410), K * (1680), we use an expression for the formfactors [55,90]: The values of the parameters are given in Table 11.    Table 11: Values of the parameters in the vector form-factors (F.21) from [55,90].

F.2.2 Pseudo-vector
For the pseudo-vector mesons, X = A, the expansion of the matrix elements is similar to (F.15), (F.16), but the expressions for the vector and axial-vector matrix elements are interchanged [91,92], with the same relation between V i as for A i in the case of vector mesons (F.17). We therefore obtain We will consider two lightest pseudo-vector resonances K 1 (1270), K 1 (1400), each of which is the mixture of unphysical K 1A and K 1B states [91], |K 1 (1270) |K 1 (1400) = sin(θ K 1 ) cos(θ K 1 ) cos(θ K 1 ) − sin(θ K 1 ) |K 1A |K 1B , (F.26) The form-factors V BK 1 0 can be related to the form-factors V A/B 0 of the K 1A , K 1B as The values of all relevant parameters are given in Tables 12, 13.  Table 12: Values of the parameters in the vector form-factors (F.29) from [91].

F.3 Tensor final meson state
For the tensor meson, X = T , the expansion of the matrix element is [53,93] T (p T )|Q i γ µ γ 5 Q j |X(p X ) = (m X + m T ) µ * ,s Here, s T µ (p T ) is a vector defined by Repeating the same procedure as in the previous section, we find that to q µ M µ,s XT contributes only the polarization s = 0, and therefore The parametrization of the form-factor A XT 0 is [53,93] A XT 0 (q 2 ) =  Table 14: Values of meson decay constants. We use [94] and references therein.

G Production from mesons through quartic coupling
The quartic coupling L quartic = α 2 hS 2 (G.1) generates new production channels from the mesons that are described by Feynman diagrams in Fig. 7 (b).
The matrix element for decays X Q i → X Q j SS can be written in terms of the matrix element M XX of hadronic transitions given by Eq. (B.8): where q 2 is invariant mass of scalars pair, M XX (q 2 ) is the matrix element of hadronic transitions X Q i → X Q j given by Eq. (B.8).
The matrix element for a process X Q i Q j → SS can be expressed in terms of the decay constant f X of the meson X. Namely, f X is defined by Contracting it with p µ and using the same trick as in Eq. (F.6), we obtain The values of f X are summarized in Table 14. For the decay width of the process and the running mass [95] m q (m S ) is given by m q (m S ) = m q (Q) c(α s (m S )/π) c(α s (Q)/π) , (H. 6) with the coefficient c, which is equal to c(x) = 9 2 x 4/9 (1 + 0.895x + 1.371x 2 + 1.