Light scalar production from Higgs bosons and FASER 2

The most general renormalizable interaction between the Higgs sector and a new gauge-singlet scalar $S$ is governed by two interaction terms: cubic and quartic. The quartic interaction is only loosely constrained by invisible Higgs decays. Given current experimental limits about $10\%$ of all Higgs bosons created at the LHC can be converted to new scalars with the mass up to $m_{\rm Higgs}/2$. This can significantly extend the reach of the LHC-based Intensity Frontier experiments. We analyze the sensitivity of the FASER experiment to this model and discuss modest changes in the FASER 2 design that would allow to explore an orders-of-magnitude wider part of the Higgs portal's parameter space.


Contents
1 Introduction: scalar portal and FASER experiment The Standard Model of particle physics (SM) is extremely successful in explaining accelerator data. Yet it fails to explain several observed phenomena: neutrino masses, dark matter and baryon asymmetry of the Universe. In order to explain these phenomena, we need to postulate new particles that should not nevertheless spoil extremely successful Standard Model predictions. New hypothetical particles can be heavy, thus evading detection and √ s = 13 TeV collision energy of the LHC. Such particles would induce higher-dimensional (non-renormalizable) interactions between SM fields and the signatures of such operators are being searched at the LHC (see e.g. [1] for review).
Alternatively, new particles can be light yet have very weak couplings to the Standard Model -feebly interacting particles or FIPs. In this case, their interaction with the SM can be governed even by the relevant (dimensions 3 and 4) operators with small couplings. Such models are generically called portals because through such interactions it is possible to mediate connection to some "dark sectors" -particles, that do not interact with the SM at all.
In this paper, we consider the most general form of the scalar (or Higgs) portal [2][3][4][5] that has been the subject of active analysis in the recent years, see e.g. [6][7][8][9] and refs. therein. Namely, we introduce a scalar particle S that carries no Standard Model charges and interacts with the Higgs doublet H via where v is the Higgs VEV and the model is parametrized by three new constants: α 1 , α 2 and the scalar mass m S . After the electroweak symmetry breaking, the SHH interaction (1.1) leads to a quadratic mixing between S and the Higgs boson h.
Transforming the Higgs field into the mass basis, h → h + θS (θ 1), one arrives at the following Lagrangian, describing interactions of the new boson S with the SM fermions, intermediate vector bosons and the Higgs boson: where . . . denote quartic and higher terms. The interactions (1.2) also mediate effective couplings of the scalar to photons, gluons, and flavor changing quark operators [10], opening many production channels at both LHC and Intensity Frontier experiments. The phenomenology of light GeV-like scalars has been worked out in [11][12][13][14][15][16][17][18][19][20] as well as in [21][22][23][24][25][26][27][28][29][30] in the context of the light Higgs boson. Most of these works concentrated on the Lagrangian with α 1 = 0 in which case the couplings θ and α in (1.2) become related. 1 In this work we consider α 1 = 0. Phenomenologically, this allows to decouple decay channels (controlled by θ) and production channels (controlled by α), c.f. [32] where phenomenology of such a model is also discussed. As we will see below, the parameter α is only weakly constrained by the invisible Higgs decays [33,34] and can be quite sizeable (if unrelated to θ). As a result, the production via h → SS process becomes possible and is operational for scalar masses up to m h /2 which allows to significantly extend the sensitivity reach of the LHC-based experiments.
We note that the production channel via the off-shell Higgs bosons (e.g. coming from neutral meson decays, such as B s → SS for 2m S < m B ) starts to dominate over production via flavour changing mixing for θ 2 < 10 −9 ÷ 10 −10 , see [10]. We will not consider this effect in the current work, mostly concentrating on m S 5 GeV.
Searches for light scalars have been previously performed by CHARM [35], KTeV [36], E949 [37,38], Belle [39,40], BaBar [41], LHCb [42,43], CMS [33,44,45] Figure 1. Left panel: branching ratios of decays of a scalar S as a function of its mass. We use perturbative decays into quarks and gluons (see [10] for details). Right panel: the lifetime of a scalar S as a function of its mass for the mixing angle θ 2 = 1. The lifetime is obtained using decays into quarks and gluons (and τ 's) within the framework of perturbative QCD.

Existing bounds
The up to date experimental constraints in the m S -θ plane can be found in the scalar portal section of [60]. The strongest experimental constraints on the parameter α come from the invisible Higgs decay. In the Standard Model the decay h → ZZ → 4ν has the branching ratio O(10 −3 ). Current limits on the Higgs to invisible are BR inv < 0.19 at 95% CL [33]. Future searches at LHC Run 3 and at the High-Luminosity (HL) LHC (HL-LHC, Run 4) are projected to have sensitivity at the level BR inv ∼ 0.05 -0.15 at 95% CL [61] maybe going all the way to a few percents [62]. In what follows we will assume that the branching ratio BR inv is saturated by the h → SS decay. Using we obtain the corresponding value of α 2 ∼ 5 GeV 2 for m S m h .  mass ranging between O(10) GeV and m h /2. The obtained constraints, however, do not restrict the parameters relevant for the FASER 2 experiment as they search for prompt decays of the scalars, while in our model the cτ S ∼ O(100) meters. Fig. 2), is an Intensity Frontier experiment dedicated to searching for light, extremely weakly-interacting particles that may be produced in the LHC's high-energy collisions in the far-forward region and then travel long distances without interacting [65]. FASER is approved to collect data in 2021-2023 during the LHC Run 3. If FASER is successful, FASER 2, a much larger successor, could be constructed in Long Shutdown 3 and collect data during the High-Luminosity Run 4 in 2026-2035. The relevant parameters of FASER and FASER 2 are shown in Table 1. We also list the alternative configuration of FASER 2 which we will use for comparison in this work.

The FASER experiment
While the design of the first phase is fixed, the FASER 2 is not finalized yet. We demonstrate therefore how the parameters of the future FASER 2 experiment will affect its sensitivity.
The paper is organized as follows: -In Section 2 we estimate the number of decay events in the FASER detectors. This Section allows for easy cross-check of our main results and gives the feeling of the main factors that affect the sensitivity. -In Section 3 we outline our Monte-Carlo simulations based on which the final conclusion is drawn. We also demonstrate that increase of the geometric acceptance  Table 1. Parameters of the FASER experiment: Prototype detector (FASER) is approved to collect data during the LHC Run 3. FASER 2 is planned for HL-LHC phase, but its configuration is not finalized yet. In the third line, we propose an alternative configuration of FASER 2 that would allow to drastically increase its reach towards the scalar portal. L is the integrated luminosity of the corresponding LHC run. L is the distance between the ATLAS interaction point and the entrance of the FASER decay vessel. R is the radius of the decay vessel. l det is the length of the detector and θ FASER = R/L is the angle, so that the solid angle subtended by the detector is given by Ω faser = πθ 2 faser . For the purposes of our investigation we assume that the decay vessel is a cylinder, centered around the beam axis.
by the factor ∼ 2 (e.g. via increase of the radius of the decay vessel of FASER 2 from 1 m to 1.5 m) would allow a wide region of the parameter space to be probed. -Appendices provide some details of our computations that would permit the interested reader to reproduce them. Before running MC simulations (and in order to have a way to verify the simulation results) we start with analytic estimates of the sensitivity of FASER 2. The number of detected events is given by the following formula [50]: Here, N S is the number of scalars produced at the LHC experiment; in our case , N h -the number of produced Higgs bosons, geom is the geometric acceptance -the fraction of scalars whose trajectories intersect the decay volume, so that they could decay inside it. The decay probability is given by the well-known formula where L is the distance from the interaction point to the entrance of the fiducial volume, l det is the detector length, and l decay = cτ S γ S is the decay length. Finally, det ≤ 1 is the detection efficiency -a fraction of all decays inside the decay volume for which the decay products could be detected. In the absence of detector simulations, we optimistically assume detector efficiency of FASER to be det = 1.
The high luminosity LHC phase is expected to deliver 1.7 · 10 8 Higgs bosons (the Higgs boson production cross-section at √ s = 13 TeV is σ h ≈ 55 pb [66], going to 60 pb at 14 TeV). Further, we assume the fiducial Higgs decay to scalars equal to the lower bound of HL-LHC reach [61]: For the initial estimate of the number of produced scalars, we consider these Higgs bosons decaying at rest. In this case we estimate the number of scalars flying into the solid angle of FASER 2 as Plugging in the numbers we get N naive For l det L (as it is the case for FASER/FASER 2) the probability of decay (2.2) reaches its maximum for l decay ≈ L. The maximum is purely geometric, not related to the parameters of the scalar S and numerically it is equal to Multiplying Eqs. (2.4) and (2.5) we find O(0.1) detectable events. Given that this was a (strong) underestimate -we see that more careful analysis is needed. It will proceed as follows: 1. We start by assuming that all Higgs bosons travel along the beam axis, which allows for a much simplified analytic treatment. Then we comment on the effect of p T distribution of the Higgs bosons. 2. We determine the realistic geometrical acceptance geom naive geom , since the actual angular distribution of scalars is peaked in the direction of the FASER detector. 3. Finally, as scalars have non-trivial distribution in energy, for most of the scalars the decay probability is not equal to the maximal value, thus determining the width of the sensitivity area in the θ direction for a given mass.

Geometrical acceptance
Most Higgs bosons are traveling along the beam axis and therefore have p T p L (see Appendix A). Therefore, we perform the analytic estimates based on the purely longitudinal distribution of the Higgs bosons, Fig. 4.  [67] and following [68]. See Appendix A for details.
The angle θ S between the scalar and Higgs boson directions in the laboratory frame is related to the scalar direction in the Higgs rest frame via where is the velocity of a scalar in the rest frame of the Higgs boson, γ h and β h are Higgs boson's gamma factor and velocity in the laboratory frame.
Taking into account that angular distribution is isotropic in the rest frame of Higgs boson, we can estimate the average angle θ S for a given Higgs energy E h = γ h m h : where we used β h ≈ 1. The average θ S as a function of scalar mass is shown in Fig. 5. We see that as long as m S is not very close to the threshold mass, m h /2, the average angle θ S θ faser . Moreover, the distribution of scalars in the direction θ faser is sufficiently flat, see Fig. 5, right panel.
Based on these considerations, we can calculate the geometric acceptance (once again assuming that all Higgs bosons fly in the direction of the beam). The geometric acceptance as a function of γ h is given by Here, dP/dp L is the distribution of the Higgs bosons in p L (neglecting the p T distribution) as seen in Fig. 4; Ω is the solid angle of FASER 2 available for scalars - and Ω = πθ 2 max (p L ) otherwise; finally, the function κ defines how collimated is the beam of scalars as compared to the isotropic distribution: L /m h and in the last equality we took β h 1). The acceptance grows with the mass since the maximal angle θ S decreases; when the mass of the scalar is very close to m h /2, the acceptance reaches its maximum equal to the fraction of Higgs bosons flying into the direction of the FASER 2 decay volume, f h→faser . With p L distribution only, obviously, f h→faser = 1. To make realistic estimates, we need to take into account the p T distribution of the Higgs bosons. The fraction f h→faser under an assumption that p L and p T distributions of Higgs boson are independent is where factor 1/2 comes from the fact that we do not take into account Higgs bosons that fly in the opposite direction to FASER. This number represents a maximally possible geometric acceptance.
Finally, multiplying the number of Higgs boson produced by the fiducial branching ratio (2.3), by the maximal decay probability (2.5) and acceptance ∼ 5 · 10 −5 , we get that the expected number of scalars' decays is O(1). The number of events increases as m S → m h /2 due to the increased geometric acceptance. These estimates warrant a more detailed sensitivity study, using the realistic distribution of Higgs bosons, etc.

Decay of scalars
The decay width and branching S → visible is determined based on the (extended) results of Ref. [10] (see Fig. 1). At these masses all major decays are visible with a number of charged tracks 2, and therefore it is reasonable to assume that BR visible = 100% and that every decay is reconstructable with 100% efficiency.
So far we have kept the decay probability at its maximum (corresponding to l decay = L). This condition would give a line in the (m S , θ) plane. In order to determine the transversal shape of the sensitivity region, we need to vary θ and take into account the γ factor of the scalar, γ S . The energy of a scalar is proportional to the energy (p L ) of the corresponding Higgs boson: The average energy of the scalar is determined by weighting the Higgs distribution dP/dp L with the function κ, defined in Eq. (2.10). In this way, only the energies of scalars flying into the FASER 2 solid angle are considered. The resulting E S as a function of mass is shown in Fig. 7 (central panel). One can see that the γ factor ranges from O(100) for small masses down to O(10) for m S ≈ m h /2.
Finally, for proper analytic estimates one should take into account that the decay probability depends non-linearly on l decay , and therefore P l decay = P l decay . The averaging should be done using the function κf p L (shown in the right panel of Fig. 6). The resulting probability can be estimated to be Substituting this value for the decay probability, as well as geom (Fig. 6

Results
We simulated Higgs boson production at the LHC using MadGraph5 aMCNLO [67] and following [68]. The details are provided in Appendix A.
The resulting distribution of scalars, traveling into the solid angle of FASER 2, over energies E S are shown in Fig. 7 (left panel).
In Fig. 7 (right panel) we compare the geometric acceptance obtained in simulations with the analytic prediction from Fig. 6. The simulation results lie slightly below the analytic estimate due to the p T distribution of Higgs bosons. The smallness of the discrepancy is related to the smallness of the ratio p T / p L for the Higgs bosons.
Next, we compute the number of scalars traveling through the FASER 2 fiducial volume and estimate the number of decay events, using Eq. (2.1) with the decay probability P decay averaged over the energies of scalars flying in the direction of the experiment. The resulting sensitivity region is shown in Fig. 8. We assume background free experiment and therefore determine the sensitivity as a region that includes at least 2.3 events. With the current configuration of FASER 2 one can expect to see any events only in the region around 50 − 60 GeV. The blue contours are based on the full MC simulations. The green line is an analytic estimate obtained using the geometric acceptance from Fig. 6. The slight difference between the green and blue line is the difference (2.5).  Fig. 6). The discrepancies are due to the p T distribution of the Higgs bosons in the realistic simulations.
However, a very modest increase of the geometric acceptance (by a factor 2 − 3) leads to a drastic change of the situation, as the blue dashed line demonstrates. This increase can be achieved by scaling the radius of the FASER 2 from 1 meter to 1.5 meters, which is allowed by the size of the TI12 tunnel where FASER experiment will be located.

Conclusion
In this work, we presented the analytic estimates for the sensitivity of the FASER 2 experiment for the most general scalar portal model including renormalizable operators only. The estimates were verified by the MadGraph simulations, showing a very good agreement. Majority of previous works on the subject [10-14, 17, 19] considered the models of the scalar where the term α 1 SH † H was absent in the Lagrangian (1.1) (assuming a Z 2 symmetry S → −S). In this case, two scalar couplings θ and α in the effective Lagrangian (1.2) become related (and should both be small to satisfy bounds from the previous experiments).
However, if cubic and quartic couplings (α 1 and α 2 in the Lagrangian (1.1)) are independent and both non-zero, the resulting triple coupling between Higgs and two scalars can be quite sizeable. Indeed, the main experimental bound on its value is the branching fraction of the invisible Higgs decay (assuming it is saturated by the h → SS process). The current bound on the invisible branching ration BR inv < 0.19 (at 95%CL, [69]). Future runs of the LHC are expected to probe this branching at the level 0.1 or slightly below.
As a result, for the experimentally admissible values of the parameter α the production of scalars at the LHC from the decays of the Higgs boson (h → SS) dominates  Figure 8. Sensitivity of the FASER 2 to scalars produced in decays of Higgs bosons. Blue solid line encloses the region where one expects to observe at least 2.3 events, given the current configuration of the experiment (the radius of the decay vessel R = 1 m). A modest increase of the geometric acceptance (by changing the radius to R = 1.5 m) allows to probe an order-of-magnitude-wide stripe for all masses (between blue dashed lines). The black solid line shows parameters for which l decay = L (used for our analytic estimates). Gray dashed line shows upper and lower regions of the MATHUSLA200 experiment where similar production from the Higgs bosons is possible (partially based on [60]). The green line is an analytic estimate, see text for details. Sensitivity estimates assume the 100% efficiency of the reconstruction of decay products but take into account geometric acceptance. The branching ratio BR(h → SS) is taken at the level of 5%.
significantly over all other production channels. This makes the production and decay of a scalar controlled by independent coupling constants. This independence qualitatively changes the behavior of the sensitivity curves of the LCH-based intensity frontier experiments (MATHUSLA, FASER, CODEX-b). Indeed, normally the sensitivity of the intensity frontier experiments has a lower bound, defined by the minimal number of events in the detector, depends both on the production and decay, and an upper bound, defined by the requirement that new particles should not decay before reaching the detector (the lifetime gets smaller with mass). Their intersection often defines the maximal mass of scalar that can be probed [50]. In our case, the maximal mass is determined solely by the kinematics (m S ≤ m h /2). However, as the geometrical acceptance drops with the decrease of the scalar's mass (see left panel of Fig. 6) while the number of produced scalars is mass-independent, for a given geometry there can be a minimal mass that can be probed (c.f. the blue solid line in Fig. 8).
For our analysis, we assumed that the invisible Higgs decay has a significant contribution from h → SS and, as an example, adopted a fiducial branching fraction BR(h → SS) at the level of 5%. We show that in this case, even if the HL-LHC does not discover invisible Higgs decay, the FASER 2 experiment is capable of discovering dark scalars with masses 40 GeV m S m h /2. Moreover, if its geometric acceptance is increased by a factor ∼ 2, FASER 2 will have sensitivity for all scalar masses from m h /2 down to a few GeV and even lower, where the production from B mesons starts to contribute. This can be achieved, for example, by scaling the radius of the detector from 1 meter to 1.5 meters.
Another possibility would be to put the detector closer to the interaction point, in which case the number of particles, counterintuitively, increases as L 3 (L 2 dependence comes from the increase of the solid angle Ω FASER and an extra factor comes from the L-dependence on the maximal decay probability, Eq. (2.5)). The latter effect is due to the independence of the decay probability on the coupling α controlling production, and is specific for the model in question. As suggested e.g. in the original FASER paper [54], another possibility would be to put the detector at 150 meters behind the TAN neutral particle absorber [70]. Such a position, however, would suffer from a high background and therefore our estimates (performed under the background-free assumption) will not be valid. Another option suggested in [54] does not increase acceptance. Indeed, it was proposed to use a hollow cylinder around the beam axis, with inner angle around 1 mrad (the size being dictated by the position of TAS quadrupole magnets shield) and the outer size of about 2 mrad. Such a detector would have a factor of few lower geometric acceptance. Of course, such a detector would be too complicated and cumbersome, so its realistic version, occupying only a small sector in the azimuthal angle ∆φ, would have its geometric acceptance further reduced by ∆φ/2π.

Acknowledgements.
We thank J. Boyd, O. Mattelaer, and S. Trojanowski for fruitful discussions. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (GA 694896), from the Netherlands Science Foundation (NWO/OCW). O.R. also acknowledges support from the Carlsberg Foundation.

A Higgs boson distribution
For our estimate we used a number of Higgs bosons for HL LHC N h = 1.7 · 10 8 . To find Higgs bosons momentum distribution, we simulated Higgs boson production at the LHC using MadGraph5 aMCNLO [67] and following [68]. Using the generated events, we find that the p T distribution depends only weakly on p L , see Fig. 9. Therefore, the correlations between p T and p L distributions can be neglected, and the double distribution of Higgs bosons in p T , p L can be approximated by the product of p T and p L single distributions.
We validated our simulation by comparing the p T spectrum of the Higgs bosons with the theoretical spectra from [68] and [71], in which the spectrum was obtained using POWHEG, see Fig. 9. Our results agree well with [68], while there is a discrepancy with [71] in the domain of high p T . However, the discrepancy is not significant; in particular, the amounts of Higgs bosons flying in the direction of FASER 2 experiment calculated using our distribution and the distribution from [71] differs by no more than 30%.
For each simulated event we calculated κ(θ h , γ h ) and the energy E S (θ h , γ h ) of a scalar traveling into the solid angle of FASER 2. The κ is then obtained as the arithmetic mean, while the energy distribution is obtained as the weighted distribution, where the energy E S (θ h , γ h ) has the corresponding weight κ(θ h , γ h ).

B Distributions of scalars over energies and polar angles
We calculated the double distribution f E S ,θ S of scalars produced in the decay h → SS in the following way. The differential branching ratio to produce a scalar flying in the direction θ S from the Higgs boson flying in the direction θ h , φ h is given by where M is the invariant matrix element of the transition h → SS, p S,2 is the momentum of the second of the two produced scalars, and cos(α) = cos(φ h ) sin(θ h ) sin(θ S ) + cos(θ h ) cos(θ S ) (B.2) Rewriting the scalar phase space volume as d 3 p S = sin(θ S )dθ S E S E 2 S − m 2 S dE S dφ S , for the distribution of the scalar in the energy and polar angle is given by where f θ h ,E h is the double differential distribution of the Higgs bosons, and