Light scalar production from Higgs bosons and FASER 2

TL;DR

This paper investigates how the FASER 2 experiment can detect light scalar particles produced via Higgs decays, expanding the potential to explore new physics beyond current limits through modifications in experimental design.

Contribution

It provides a detailed analysis of FASER 2's sensitivity to Higgs portal scalars and suggests design modifications to enhance discovery potential.

Findings

FASER 2 can significantly extend the search for Higgs portal scalars.

Modest design changes can increase the explored parameter space by orders of magnitude.

Current limits allow up to 10% of Higgs bosons to decay into new scalars.

Abstract

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.

Tables

Table 1. 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 drastically increasing its reach towards the scalar portal. ℒ ℒ \mathcal{L} is the integrated luminosity of the corresponding LHC run. L 𝐿 L is the distance between the ATLAS interaction point and the entrance of the FASER decay vessel. R 𝑅 R is the radius of the decay vessel. l det subscript 𝑙 det l_{\text{det}} is the length of the detector and θ FASER = R / L subscript 𝜃 FASER 𝑅 𝐿 \theta_{\text{FASER}}=R/L is the angle, so that the solid angle subtended by the detector is given by Ω faser = π ​ θ faser 2 subscript Ω faser 𝜋 superscript subscript 𝜃 faser 2 \Omega_{\textsc{faser}}=\pi\theta_{\textsc{faser}}^{2} . For our investigation, we assume that the decay vessel is a cylinder, centered around the beam axis.

Phase	$ℒ$, ${\text{fb}}^{-1}$	$L$ [m]	$R$ [m]	$l_{\text{det}}$ [m]	${\theta }_{\text{faser}}$ [rad]
FASER	150	480	0.1	1.5	$2.1\cdot {10}^{-4}$
FASER 2	3000	480	1	5	$2.1\cdot {10}^{-3}$
FASER 2 (alternative configuration)	3000	480	1.5	5	$3.1\cdot {10}^{-3}$

Equations

$$\mathcal{L}={\mathcal{L}}_{SM}+\frac{1}{2}({\partial }_{\mu }S{)}^2+({\alpha }_{1}S+{\alpha }_2{S}^2)\left(H^{†}H-\frac{v^2}{2}\right)-\frac{m_S^2}{2}{S}^2,$$

$${\mathcal{L}}_{SM}^S=-\theta \frac{m_f}{v}S\bar{f}f+2\theta \frac{m_W^2}{v}S{W}^{+}{W}^{-}+\theta \frac{m_Z^2}{v}S{Z}^2+\frac{\alpha }{2}{S}^{2}h+\dots$$

$${\Gamma }_{h\to SS}=\frac{{\alpha }^2}{32\pi m_h}\sqrt{1-\frac{4m_S^2}{m_h^2}}$$

$$N_{\text{det}}=N_S\times {ϵ}_{\text{geom}}\times P_{\text{decay}}\times {ϵ}_{\text{det}}.$$

$$P_{\text{decay}}(l_{\text{decay}})=e^{-L/l_{\text{decay}}}-e^{-(L+l_{\text{det}})/l_{\text{decay}}},$$

$${\text{BR}}_{\text{fid}}(h\to SS)=\mathrm{0.05.}$$

$${ϵ}_{\text{geom}}^{\mathrm{naive}}=\frac{{\Omega }_{\text{textsc}faser}}{4\pi }\approx 1.1\times 10^{-6},$$

$$P_{\text{decay}}^{(\max )}\simeq \frac{l_{\text{det}}}{L}{e}^{-1}\approx 3.8\cdot 10^{-3},$$

$$\tan {\theta }_S=\frac{1}{{\gamma }_h}\frac{{\beta }_S^{\prime }\sin {\theta }_S^{\prime }}{{\beta }_S^{\prime }\cos {\theta }_S^{\prime }+{\beta }_h},$$

$${\beta }_S^{\prime }=\sqrt{1-\frac{4m_S^2}{m_h^2}}$$

$${ϵ}_{\text{geom}}\approx \int f_{p_L}\kappa (m_S,p_L)\frac{\Omega (p_L)}{4\pi }d{p}_L$$

$$\Omega =\{\begin{array}{l}{\Omega }_{\text{textsc}faser},{\theta }_{\text{textsc}faser}<{\theta }_{\text{max}},\\ \pi {\theta }_{\text{max}}^2(p_L),{\theta }_{\text{textsc}faser}>{\theta }_{\text{max}},\end{array}$$

$$\kappa (m_S,p_L)\approx \{\begin{array}{l}\frac{2{\gamma }_h^2({\beta }_S^{{}^{\prime }2}+{\beta }_h^2)}{{\beta }_S^{{}^{\prime }2}},{\beta }_h>{\beta }_S^{{}^{\prime }},\\ \frac{{\gamma }_h^2({\beta }_S^{{}^{\prime }}+{\beta }_h{)}^2}{{\beta }_S^{{}^{\prime }2}},{\beta }_h<{\beta }_S^{{}^{\prime }}\end{array}$$

$$f_{h\to \text{textsc}faser}\approx {ϵ}_{\text{geom}}^{\max }=\frac{1}{2}\underset{0}{\overset{\infty }{\int }}{f}_{p_L}d{p}_L\underset{0}{\overset{p_L^h{\theta }_{\text{textsc}faser}}{\int }}{f}_{p_T}d{p}_T\approx 1.1\cdot 10^{-3},$$

$$E_S=\frac{E_h}{2}(1+{\beta }_S^{\prime }{\beta }_h\cos ({\theta }_S))\approx \frac{E_h}{2}(1+{\beta }_S^{\prime }{\beta }_h),$$

$$⟨P_{\text{decay}}^{(\text{max})}⟩\approx \int \kappa (m_S,p_L)\cdot f_{p_L}\cdot P_{\text{decay}}(m_S,{\theta }^2,E_S)d{p}_L$$

$$⟨P_{\text{decay}}^{(\text{max})}⟩\simeq 3.2\cdot 10^{-3}$$

$$N_{\text{events}}^{(\text{max})}=N_h\cdot {\text{BR}}_{\text{fid}}(h\to SS)\cdot {ϵ}_{\text{geom}}\cdot ⟨P_{\text{decay}}^{(\text{max})}⟩$$

$$\tan ({\theta }_S)=\frac{1}{{\gamma }_h}\frac{{\beta }_S^{{}^{\prime }}\sin ({\theta }_S^{{}^{\prime }})}{{\beta }_S^{{}^{\prime }}\cos ({\theta }_S^{{}^{\prime }})+{\beta }_h}$$

$$f_{\pm }({\theta }_S)=-\frac{{\beta }_h{\gamma }_h^2{\tan }^2({\theta }_S)\pm \sqrt{{\beta }_S^{{}^{\prime }2}+({\beta }_S^{{}^{\prime }2}-{\beta }_h^2){\gamma }_h^2{\tan }^2({\theta }_S)}}{{\beta }_S^{{}^{\prime }}(1+{\gamma }_h^2{\tan }^2({\theta }_S))},$$

$${\theta }_{S,\text{max}}=\arctan \left[\frac{{\beta }_S^{{}^{\prime }}}{{\gamma }_h\sqrt{{\beta }_h^2-{\beta }_S^{{}^{\prime }2}}}\right]$$

$$\begin{gathered} \cos ({\theta }_S^{{}^{\prime }})=\{\begin{array}{l}{f}_{-}({\theta }_S),0<{\theta }_S<\pi /2,\\ f_{+}({\theta }_S),\pi /2<{\theta }_S<\pi \end{array}= \\ =-\frac{{\beta }_h{\gamma }_h^2{\sin }^2({\theta }_S)-\cos ({\theta }_S)\sqrt{{\beta }_S^{{}^{\prime }2}{\cos }^2({\theta }_S)+({\beta }_S^{{}^{\prime }2}-{\beta }_h^2){\gamma }_h^2{\sin }^2({\theta }_S)}}{{\beta }_S^{{}^{\prime }}{\cos }^2({\theta }_S)+{\gamma }_h^2{\sin }^2({\theta }_S)} \end{gathered}$$

$$\kappa =\left|\frac{d{f}_{+}({\theta }_S)}{d\cos ({\theta }_S)}\right|+\left|\frac{d{f}_{-}({\theta }_S)}{d\cos ({\theta }_S)}\right|=\frac{dg({\theta }_S)}{d\cos ({\theta }_S)},$$

$$g({\theta }_S)=\left|2\cos ({\theta }_S)\frac{\sqrt{{\beta }_S^{{}^{\prime }2}{\cos }^2({\theta }_S)+({\beta }_S^{{}^{\prime }2}-{\beta }_h^2){\gamma }_h^2{\sin }^2({\theta }_S)}}{{\beta }_S^{{}^{\prime }}({\cos }^2({\theta }_S)+{\gamma }_h^2{\sin }^2({\theta }_S))}\right|$$

$$g({\theta }_S)\approx 2-\frac{{\theta }_S^2({\beta }_h^2+{\beta }_S^{{}^{\prime }2})}{{\beta }_S^{{}^{\prime }2}}\Rightarrow \kappa \approx \frac{2{\gamma }_h^2({\beta }_S^{{}^{\prime }2}+{\beta }_h^2)}{{\beta }_S^{{}^{\prime }2}}$$

$$d\text{Br}(h\to SS)=\frac{1}{2}\frac{1}{(2\pi {)}^2}\frac{|\mathcal{M}{|}^2}{2{\Gamma }_{h,\text{rest}}{m}_h}\frac{d^3{\mathbf{p}}_{S_1}}{2E_{S_1}}\int \frac{d^3{\mathbf{p}}_{S_2}}{2E_{S_2}}{\delta }^4(p_h-p_{S_1}-p_{S_2}),$$

$$\sin ({\theta }_{12,\text{min}})=\frac{2m_h^2{\beta }_h\sqrt{{\gamma }_h^2-1}}{m_h^2{\gamma }_h^2-4m_S^2}$$

$$d\text{Br}(h\to SS)=\frac{d^3{\mathbf{p}}_S}{8(2\pi {)}^2}\frac{|\mathcal{M}{|}^2}{4{\Gamma }_{h,\text{rest}}{m}_h{E}_S}\delta (m_h^2-2E_S{E}_h+2|{\mathbf{p}}_S||{\mathbf{p}}_h|\cos (\alpha )),$$

$$\cos (\alpha )=\cos ({\varphi }_h)\sin ({\theta }_h)\sin ({\theta }_S)+\cos ({\theta }_h)\cos ({\theta }_S)$$

$$\begin{gathered} f_{{\theta }_S,E_S}=\frac{1}{{\mathrm{BR}}_{h\to SS}}\frac{d\mathrm{BR}(h\to SS)}{d{\theta }_{S}d{E}_S}= \\ =2\pi \frac{\sin ({\theta }_S)E_S\sqrt{E_S^2-m_S^2}}{\text{Br}(h\to SS)}\int \frac{d{\varphi }_h}{2\pi }d{E}_{h}d{\theta }_h{f}_{{\theta }_h,E_h}\frac{d^3\text{Br}(h\to SS)}{d^3{\mathbf{p}}_S}= \\ =\frac{m_h\sqrt{E_S^2-m_S^2}}{|{\mathbf{p}}_{S,\text{rest}}|}\sin ({\theta }_S)I[{\theta }_S,E_S], \end{gathered}$$

Full text

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. To explain these phenomena, we need to postulate new particles that should not nevertheless spoil extremely successful Standard Model predictions. These new hypothetical particles can be heavy, thus evading detection at $\sqrt{s}=13$ TeV collision energy of the LHC. Such particles would induce higher-dimensional (non-renormalizable) interactions with SM fields, the signatures of such operators are being searched at the LHC (see e.g. Brivio:2017vri for a 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 relevant (dimensions 3 and 4) operators with small couplings. Such models are generically called portals because trough such operators FIPs can mediate interactions with some “dark sectors” – other new particles that otherwise are inaccessible.

In this paper, we consider the most general form of the scalar (or Higgs) portal McDonald:1993ex ; Burgess:2000yq ; Patt:2006fw ; OConnell:2006rsp that has been the subject of active analysis in the recent years, see e.g. Djouadi:2012zc ; Curtin:2013fra ; Alekhin:2015byh ; Arcadi:2019lka and refs. therein. Namely, we introduce a scalar particle $S$ that carries no Standard Model charges and interacts with the Higgs doublet $H$ via

[TABLE]

where $v$ is the Higgs VEV and the model is parametrized by three new constants: $\alpha_{1},\alpha_{2}$ and the scalar mass $m_{S}$. After electroweak symmetry breaking, the $SHH$ interaction (1) leads to a quadratic mixing between $S$ and the Higgs boson $h$. Transforming the Higgs field into the mass basis, $h\to h+\theta S$ ($\theta\ll 1$), one arrives at the following Lagrangian, describing interactions of the new boson $S$ with the SM fermions, intermediate vector bosons and the Higgs boson:

[TABLE]

where $\dots$ denote quartic and higher terms. The interactions (2) also mediate effective couplings of the scalar to photons, gluons, and flavor changing quark operators Boiarska:2019jym , opening many production channels at both LHC and Intensity Frontier experiments. The phenomenology of light GeV-like scalars has been worked out in Bird:2004ts ; Batell:2009jf ; Bezrukov:2009yw ; Clarke:2013aya ; Schmidt-Hoberg:2013hba ; Evans:2017lvd ; Bezrukov:2018yvd ; Monin:2018lee ; Winkler:2018qyg ; Frugiuele:2018coc ; Helmboldt:2016zns as well as in Voloshin:1985tc ; Raby:1988qf ; Truong:1989my ; Donoghue:1990xh ; Willey:1982ti ; Willey:1986mj ; Grzadkowski:1983yp ; Leutwyler:1989xj ; Haber:1987ua ; Chivukula:1988gp in the context of the light Higgs boson. Most of these works concentrated on the Lagrangian with $\alpha_{1}=0$ in which case the couplings $\theta$ and $\alpha$ in (2) become related.111Alternative class of models has super-renormalizable interaction only between the Higgs boson and the scalar ($\alpha_{2}=0$), see Fradette:2018hhl and refs. therein. In this case, of course, there is no $S^{2}h$ term in the Lagrangian (2). In this work we consider $\alpha_{1}\neq 0$. Phenomenologically, this allows to decouple decay channels (controlled by $\theta$) and production channels (controlled by $\alpha$), c.f. Fradette:2017sdd where phenomenology of such a model is also discussed. As we will see below, the parameter $\alpha$ is only weakly constrained by the invisible Higgs decays Sirunyan:2018owy ; Aaboud:2018sfi and can be quite sizeable (if unrelated to $\theta$). As a result, the production via $h\to SS$ 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 $B_{s}\to SS$ for $2m_{S}<m_{B}$) starts to dominate over production via flavour changing mixing for $\theta^{2}<10^{-9}\div 10^{-10}$, see Boiarska:2019jym . We will not consider this effect in the current work, mostly concentrating on $m_{S}\gtrsim 5\,\mathrm{GeV}$.

Searches for light scalars have been previously performed by CHARM Bergsma:1985qz , KTeV AlaviHarati:2000hs , E949 Artamonov:2008qb ; Artamonov:2009sz , Belle TheBelle:2015mwa ; Seong:2018gut , BaBar Lees:2013kla , LHCb Aaij:2015tna ; Aaij:2016qsm , CMS Sirunyan:2018owy ; Sirunyan:2018mot ; Sirunyan:2018mgs and ATLAS Aad:2015txa ; Aaboud:2018sfi ; Aaboud:2018esj ; Aaboud:2018iil experiments. Significant progress in searching for light scalars can be achieved by the proposed and planned intensity-frontier experiments such as SHiP Alekhin:2015byh ; Anelli:2015pba ; Bondarenko:2019yob , CODEX-b Gligorov:2017nwh , MATHUSLA Chou:2016lxi ; Evans:2017lvd ; Curtin:2018mvb ; Bondarenko:2019yob , FASER Feng:2017uoz ; Feng:2017vli , SeaQuest Berlin:2018pwi , NA62 Mermod:2017ceo ; CortinaGil:2017mqf ; Drewes:2018gkc and a number of other experiments (see Beacham:2019nyx for an overview). The summary of the current experimental status of the light scalar searches is provided in the Physics Beyond Collider report Beacham:2019nyx .

1.1 Existing bounds

The up to date experimental constraints in the $m_{S}$-$\theta$ plane can be found in the scalar portal section of Beacham:2019nyx . The strongest experimental constraints on the parameter $\alpha$ come from the invisible Higgs decay. In the Standard Model the decay $h\to ZZ\to 4\nu$ has the branching ratio $\mathcal{O}(10^{-3})$. Current limits on the Higgs to invisible are $\text{BR}_{\text{inv}}<0.19$ at $95\%$ CL Sirunyan:2018owy . 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 $\text{BR}_{\text{inv}}\sim 0.05\text{ --- }0.15$ at $95\%$ CL Bechtle:2014ewa maybe going all the way to a few percents deBlas:2019rxi . In what follows we will assume that the branching ratio $\text{BR}_{\text{inv}}$ is saturated by the $h\to SS$ decay. Using

[TABLE]

we obtain the corresponding value of $\alpha^{2}\sim 5\,\mathrm{GeV^{2}}$ for $m_{S}\ll m_{h}$.

Apart from the invisible Higgs decays, the ATLAS and CMS collaborations have previously performed studies of the $h\to SS\to 4b$, $h\to SS\to 2b2\mu$, $h\to SS\to 2\tau 2\mu$, $h\to SS\to 2\tau 2b$, etc. Aaboud:2018esj ; Aaboud:2018iil ; Sirunyan:2018mot ; Sirunyan:2018mgs ; Sirunyan:2018pzn ; Sirunyan:2018mbx for the light (pseudo)scalar in the mass ranging between $\mathcal{O}(10)$ 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 $c\tau_{S}\sim\mathcal{O}(100)$ meters.

1.2 The FASER experiment

FASER (ForwArd Search ExpeRiment, 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 Feng:2017uoz ; Ariga:2018pin ; Ariga:2018uku ; Ariga:2018zuc ; Ariga:2019ufm . 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.

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 estimates based on which the conclusion is drawn. We also demonstrate that an increase of the geometric acceptance by the factor $\sim 2$ (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.

2 Scalars from Higgs bosons

2.1 Naive estimate: what can be expected?

Before running MC simulations (and 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 Bondarenko:2019yob :

[TABLE]

Here, $N_{S}$ is the number of scalars produced at the LHC experiment; in our case $N_{S}=2N_{h}\mathop{\mathrm{BR}}(h\to SS)$, $N_{h}$ – the number of produced Higgs bosons, $\epsilon_{\text{geom}}$ 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

[TABLE]

where $L$ is the distance from the interaction point to the entrance of the fiducial volume, $l_{\text{det}}$ is the detector length, and $l_{\text{decay}}=c\tau_{S}\beta_{S}\gamma_{S}$ is the decay length. Finally, $\epsilon_{\text{det}}\leq 1$ 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 $\epsilon_{\text{det}}=1$.

The high luminosity LHC phase is expected to deliver $1.7\cdot 10^{8}$ Higgs bosons (the Higgs boson production cross-section at $\sqrt{s}=13$ TeV is $\sigma_{h}\approx 55$ pb Cepeda:2019klc , going to $60\,\mathrm{pb}$ at 14 TeV). Further, we assume the fiducial Higgs decay to scalars equal to the lower bound of HL-LHC reach Bechtle:2014ewa :

[TABLE]

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

[TABLE]

where $\Omega_{\textsc{faser}}=\pi\theta_{\textsc{faser}}^{2}$, see Table 1. Plugging in the numbers we get $N_{S}^{\rm naive}=2N_{h}\times\epsilon_{\text{geom}}^{\rm naive}\times\text{BR}_{\text{inv}}\approx 33$ scalars. As most of the Higgs bosons fly along the beam axis, Eq. (7) is a strong underestimate and we should expect a lot of scalars flying through the FASER fiducial volume.

For $l_{\text{det}}\ll L$ (as it is the case for FASER/FASER 2) the probability of decay (5) reaches its maximum for $l_{\text{decay}}\approx L$. The maximum is purely geometric, not related to the parameters of the scalar $S$ and numerically it is equal to

[TABLE]

see also Fig. 3.222The independence of the value (8) of the mass $m_{S}$ can be understood in the following way. Since the production of the scalar is independent on the coupling $\theta^{2}$ while the decay length depends on $\theta^{2}$, we can always adjust it for a fixed mass $m_{S}$ in a way such that $l_{\text{decay}}(m_{S},\theta^{2})=L$. As we demonstrate below the values of $\theta^{2}$ for masses of interest (from few GeV to $m_{h}/2$) correspond to the region of the scalar parameter space that is currently unprobed by existing experiments. Multiplying Eqs. (7) and (8) we find $\mathcal{O}(0.1)$ detectable events. Given that this was a (strong) underestimate – we see that more careful analysis is needed. It will proceed as follows:

1. 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. 2.

We determine the realistic geometrical acceptance $\epsilon_{\text{geom}}\gg\epsilon_{\text{geom}}^{\rm naive}$, since the actual angular distribution of scalars is peaked in the direction of the FASER detector. 3. 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 $\theta$ direction for a given mass.

2.2 Geometrical acceptance

Most Higgs bosons are traveling along the beam axis and therefore have $p_{T}\ll p_{L}$ (see Appendix A). Therefore, we perform the analytic estimates based on the purely longitudinal distribution of the Higgs bosons $f_{p_{L}}\equiv\frac{1}{N_{h}}\frac{dN_{h}}{dp_{L}}$ shown in Fig. 3.

The angle $\theta_{S}$ between the scalar and Higgs boson directions in the laboratory frame is related to the scalar direction in the Higgs rest frame via

[TABLE]

where

[TABLE]

is the velocity of a scalar in the rest frame of the Higgs boson, $\gamma_{h}$ and $\beta_{h}$ are Higgs boson’s gamma factor and the velocity in the laboratory frame.333Although two scalars originate from each Higgs decay, the angle between the scalars in the laboratory frame is larger than $\theta_{\textsc{faser}}$ unless $m_{S}$ is very close to $\frac{m_{h}}{2}$. In Appendix B.2 we provide detailed estimates.

Based on these considerations, we can calculate the geometric acceptance (once again assuming that all Higgs bosons fly in the direction of the beam):

[TABLE]

Here, $\Omega$ is the solid angle of FASER 2 available for scalars:

[TABLE]

with $\theta_{\text{max}}=\arctan\left[\frac{\beta_{S}^{{}^{\prime}}}{\gamma_{h}\sqrt{\beta_{h}^{2}-\beta_{S}^{{}^{\prime}2}}}\right]$ if $\beta_{h}>\beta_{S}$ and $\theta_{\text{max}}=\pi$ otherwise. Finally, the function $\kappa=|d\Omega^{\prime}/d\Omega|$, where $\Omega$ is the solid angle in the lab frame corresponding to the solid angle $\Omega^{\prime}$ in the Higgs rest frame. It defines how collimated is the beam of scalars as compared to an isotropic distribution. For the details of the derivation of the explicit expression of $\kappa$ see Appendix 2.2. In the case $\theta=0$ it becomes

[TABLE]

The resulting acceptance (see Fig. 5, left panel) grows with the mass since the maximal angle $\theta_{S}$ 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, $f_{h\to\textsc{faser}}$. Even for the light scalars the acceptance $\epsilon_{\text{geom}}\approx 4\cdot 10^{-5}$ is an order of magnitude larger than the naive estimate (7). The reason for this is that most of the Higgs bosons have large energies, so the resulting angular distribution of scalars is peaked in the direction of small angles, see Fig. 4.

With $p_{L}$ distribution only, obviously, $f_{h\to\textsc{faser}}=1$. To make realistic estimates, we need to take into account the $p_{T}$ distribution of the Higgs bosons. The fraction $f_{h\to\textsc{faser}}$ under the assumption that $p_{L}$ and $p_{T}$ distributions of Higgs boson are independent is

[TABLE]

where a 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.

2.3 Decay of scalars

The decay width and branching $S\to\text{visible}$ is determined based on the (extended) results of Ref. Boiarska:2019jym (see Fig. 1). For these masses, all major decay channels have $\geqslant 2$ charged tracks and therefore it is reasonable to assume that $\mathop{\mathrm{BR}}_{\text{visible}}=100\%$ and that every decay is reconstructable with $100\%$ efficiency. The verification of this assumption requires detailed studies beyond the scope of this paper.

So far we have kept the decay probability at its maximum (corresponding to $l_{\text{decay}}=L$). This condition would give a line in the $(m_{S},\theta)$ plane. To determine the transversal shape of the sensitivity region, we need to vary $\theta$ and take into account the $\gamma$ factor of the scalar, $\gamma_{S}$. The energy of a scalar is proportional to the energy ($p_{L}$) of the corresponding Higgs boson:

[TABLE]

where we have taken into account that the FASER detector is almost co-aligned with the beam axis and therefore $\theta_{S}\approx 0$ and neglected the $p_{T}$ distribution of the Higgs boson. The average energy of the scalar is determined by weighting the Higgs distribution $f_{p_{L}}$ with the function $\kappa$, defined in Eq. (13). In this way, only the energies of scalars flying into the FASER 2 solid angle are considered. The resulting $\langle E_{S}\rangle$ as a function of the scalar mass is shown in Fig. 7 (central panel). One can see that the $\gamma$ factor ranges from $\mathcal{O}(100)$ for small masses down to $\mathcal{O}(10)$ for $m_{S}\approx m_{h}/2$.

Let us now improve the estimate (8) of the maximally possible value of the decay probability $P_{\text{decay}}^{(\text{max})}$. The value (8) is obtained using the average energy $\langle E_{S}\rangle$. Taking into account the continuous scalar spectra leads to a decrease of $P_{\text{decay}}^{(\text{max})}$. The averaging over the spectrum can be done using the function $\kappa f_{p_{L}}$ (shown in the right panel of Fig. 5):

[TABLE]

As is demonstrated by Fig. 5, $\kappa\cdot f_{p_{L}}$ have similar flat shape for wide range of momenta for all possible scalar masses. We can always adjust the appropriate $\theta^{2}$ value to maximize the probability, and independently on the mass we get

[TABLE]

Substituting this value for the decay probability, as well as the number of Higgs bosons produced by the fiducial branching ratio (6), $\epsilon_{\text{geom}}$ (Fig. 5, left panel) into Eq. (4), one can compute the improved analytic estimate for the maximal number of decay events inside the FASER 2 detector:

[TABLE]

It is shown in Fig. 6. The behavior of $N_{\text{events}}^{\text{(max)}}$ with the scalar mass is completely determined by $\epsilon_{\text{geom}}$. Namely, the masses $m_{S}\lesssim 30\text{ GeV}$ it is a constant of the order of $\mathcal{O}(1)$, while for larger masses increases due to the behavior of the geometric acceptance.

However, these estimates warrant a more detailed sensitivity study using the realistic distribution of Higgs bosons.

3 Results

We simulated Higgs boson production at the LHC using MadGraph5_aMCNLO Alwall:2014hca and following Hirschi:2015iia , see Appendix A for details. Using the $p_{L}$ and $p_{T}$ distributions of the Higgs bosons, we derived the energy distribution of scalars $f_{E_{S}}=\frac{1}{N_{S}}\frac{dN_{S}}{dE_{S}}$ and computed the geometric acceptance $\epsilon_{\text{geom}}$, see Appendix B.2.

The resulting energy distribution of scalars of particular masses traveling into the solid angle of FASER 2 is shown in Fig. 7 (left panel). In the same figure (middle and right panels) we compare the geometric acceptance and average energy for scalars obtained in simulations with the analytic prediction from Fig. 5. 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 $\langle p_{T}\rangle/\langle p_{L}\rangle$ 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. (4) with the decay probability $P_{\text{decay}}$ 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 green line follows from the analytic estimate (4) in which the geometric acceptance and average energy from Fig. 7, whereas the blue contours are based on the more accurate estimate using the scalar energy spectrum (see Appendix B.2). A slight difference between these estimates is caused by the difference between the value of $P_{\text{decay}}(\langle l_{\text{decay}}\rangle))$ and $\langle P_{\text{decay}}\rangle$ where in the former case $l_{\text{decay}}$ is evaluated for $\langle E_{S}\rangle$ and in the latter case one averages $P_{\text{decay}}$ over the energy distribution.

Our results lead to an important conclusion regarding a configuration of the FASER 2. Fig. 6 shows that the FASER 2 in its current configuration (as shown in Table 1) will not detect any events for $m_{N}\lesssim 40\text{ GeV}$ (region to the left of the blue solid line). However, a modest (factor of 2) increase in the geometrical acceptance would allow probing the whole mass range few GeV${}\lesssim m_{S}\lesssim m_{h}/2$, as demonstrated by the blue dashed line in Fig. 8. This increase can be achieved for example by increasing the radius of the FASER 2 from $1\,\mathrm{meter}$ to $1.5\,\mathrm{meters}$, which is allowed by the size of the TI12 tunnel where the experiment will be located. The angular distribution of scalars is flat for relevant angles, see Fig. 4, which provides the desired conclusion.

4 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 MadGraph simulations, showing a very good agreement. Majority of previous works on the subject Bird:2004ts ; Batell:2009jf ; Bezrukov:2009yw ; Clarke:2013aya ; Bezrukov:2018yvd ; Winkler:2018qyg ; Boiarska:2019jym considered the models of the scalar where the term $\alpha_{1}SH^{\dagger}H$ was absent in the Lagrangian (1) (assuming a $\mathbbm{Z}_{2}$ symmetry $S\to-S$). In this case, two scalar couplings $\theta$ and $\alpha$ in the effective Lagrangian (2) become related (and should both be small to satisfy bounds from the previous experiments).

However, if cubic and quartic couplings ($\alpha_{1}$ and $\alpha_{2}$ in the Lagrangian (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 $h\to SS$ process). The current bound on the invisible branching ratio $\text{BR}_{\text{inv}}<0.19$ (at 95%CL, Sirunyan:2018mtv ). 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 $\alpha$, the production of scalars at the LHC from the decays of the Higgs boson ($h\to SS$) dominates 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 LHC-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 Bondarenko:2019yob . In our case, the maximal mass is determined solely by the kinematics ($m_{S}\leq m_{h}/2$). However, as the geometrical acceptance drops with the decrease of the scalar’s mass (see left panel of Fig. 5) 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 $h\to SS$ and, as an example, adopted a fiducial branching fraction $\mathop{\mathrm{BR}}(h\to SS)$ 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 of $40\text{ GeV}\lesssim m_{S}\lesssim m_{h}/2$. Moreover, if its geometric acceptance is increased by a factor $\sim 2$, 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 $\Omega_{\text{FASER}}$ and an extra factor comes from the $L$-dependence on the maximal decay probability, Eq. (8)). The latter effect is due to the independence of the decay probability on the coupling $\alpha$ controlling production and is specific for the model in question. As suggested e.g. in the original FASER paper Feng:2017uoz , another possibility would be to put the detector at 150 meters behind the TAN neutral particle absorber Adriani:2015iwv . 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 Feng:2017uoz does not increase acceptance. Indeed, it was proposed to use a hollow cylinder around the beam axis, with an 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 a 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 $\Delta\phi$, would have its geometric acceptance further reduced by $\Delta\phi/2\pi$.

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.

Appendix A Higgs boson distribution

For our estimate we used a number of Higgs bosons for HL LHC $N_{h}=1.7\cdot 10^{8}$. To find Higgs bosons momentum distribution, we simulated Higgs boson production at the LHC using MadGraph5_aMCNLO Alwall:2014hca and following Hirschi:2015iia . 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 Hirschi:2015iia and Bagnaschi:2015bop , in which the spectrum was obtained using POWHEG, see Fig. 9.

Our results agree well with Hirschi:2015iia , while there is a discrepancy with Bagnaschi:2015bop 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 Bagnaschi:2015bop differs by no more than 30%.

For each simulated event we calculated $\kappa(\theta_{h},\gamma_{h})$ and the energy $E_{S}(\theta_{h},\gamma_{h})$ of a scalar traveling into the solid angle of FASER 2. The $\langle\kappa\rangle$ is then obtained as the arithmetic mean, while the energy distribution is obtained as the weighted distribution, where the energy $E_{S}(\theta_{h},\gamma_{h})$ has the corresponding weight $\kappa(\theta_{h},\gamma_{h})$.

Appendix B Distributions

B.1 Kinematics in laboratory frame

Consider the relation between the laboratory frame angle $\theta_{S}$ and the rest frame angle $\theta^{{}^{\prime}}_{S}$:

[TABLE]

Let us introduce two functions

[TABLE]

representing the solution of Eq. (19) in terms of $\cos(\theta_{S}^{{}^{\prime}})$ for given parameters $\beta_{h},\beta_{S}$. In order to express $\cos(\theta_{S}^{{}^{\prime}})$ from Eq. (19), we find first the values of $\theta_{S}$ where the functions (20) become complex. These are $\theta_{S,\text{max}}<\theta_{S}<\pi-\theta_{S,\text{max}}$, defined as

[TABLE]

They are always real as long as $\beta_{h}/\beta_{S}^{{}^{\prime}}<1$. Next, we can construct the physical solution $\cos(\theta_{S}^{{}^{\prime}})$ requiring the solutions (20) to cover all the domain of the definition of the cosine, $\cos(\theta_{S}^{{}^{\prime}})\in[-1,1]$. For $\beta_{h}/\beta_{S}^{{}^{\prime}}<1$ it is

[TABLE]

For $\beta_{h}>\beta_{S}^{{}^{\prime}}$ both the solutions $f_{\pm}$ exist in the domain $\theta_{S}<\theta_{S,\text{max}}$.

Let us now find the function $\kappa$. By the definition, $\kappa=|d\cos(\theta_{S}^{{}^{\prime}})/d\cos(\theta_{S})|$. In the case $\beta_{h}<\beta_{S}^{{}^{\prime}}$ it is simply given by the derivative of (22), while for the case $\beta_{h}>\beta_{S}^{{}^{\prime}}$ it reads

[TABLE]

where

[TABLE]

In particular, in the domain $\theta_{S}\ll\theta_{S,\text{max}}$ for $\beta_{h}>\beta_{S}^{{}^{\prime}}$ we have

[TABLE]

B.2 Distribution of scalars over energies and polar angles

The double differential distribution $f_{E_{S},\theta_{S}}$ of scalars produced in the decay $h\to SS$ has been calculated in the following way. Consider a differential branching ratio for a Higgs bosons flying in the direction $\theta_{h},\phi_{h}$:

[TABLE]

where $\mathcal{M}$ is the invariant matrix element of the process $h\to SS$ (independent on momenta for $1\to 2$ process), $\bm{p}_{S_{1,2}}$ are momenta of two produced scalars.

Two scalars are indistinguishable (extra factor $1/2$ in Eq. (26)) and after phase space integration we would lose the information about the relative distribution of the two scalars. In particular we cannot trace whether one or both scalars simultaneously could enter the FASER 2 decay volume which could lead to underestimate of the number of events by as much as a factor of 2. However, because of the small angular size of the FASER 2 experiment, the fraction of events with two $S$s flying into the detector’s fiducial volume is negligibly small. Indeed, the minimal angle $\theta_{12,\text{min}}$ between two scalars produced in the decay $h\to SS$ is given by

[TABLE]

It is larger than $\theta_{\textsc{faser}}\approx 2.6\cdot 10^{-3}$ for all values of $\gamma_{h}$ reachable at the LHC for $m_{S}\lesssim 62\text{ GeV}$, see Fig. 10.

After the integration over $\mathbf{p}_{S_{2}}$, replacing $S_{1}\to S$ we get

[TABLE]

where

[TABLE]

is the angle between the Higgs boson and the scalar. Rewriting the scalar phase space volume as $d^{3}\bm{p}_{S}=\sin(\theta_{S})d\theta_{S}E_{S}\sqrt{E_{S}^{2}-m_{S}^{2}}dE_{S}d\phi_{S}$, for the distribution in the energy and polar angle is given by

[TABLE]

where $f_{\theta_{h},E_{h}}$ is the double differential distribution of the Higgs bosons obtained in simulations, and

[TABLE]

Having the distribution function (30), the number of events may be determined as

[TABLE]