Muon Beam Experiments to Probe the Dark Sector

TL;DR

This paper proposes muon beam-dump experiments at CERN and Fermilab to search for new light scalar particles that could explain anomalies in muon physics, exploring unexplored parameter space.

Contribution

It introduces a novel experimental approach using muon beam-dump techniques to detect light scalars with specific couplings, addressing current gaps in searches.

Findings

Muon beam experiments can probe scalar masses below di-muon threshold.

Both missing momentum and decay vertex searches are effective.

Significant parameter space relevant to muon g-2 anomaly can be explored.

Abstract

A persistence of several anomalies in muon physics, such as the muon anomalous magnetic moment and the muonic hydrogen Lamb shift, hints at new light particles beyond the Standard Model. We address a subset of these models that have a new light scalar state with sizable couplings to muons and suppressed couplings to electrons. A novel way to search for such particles would be through muon beam-dump experiments by (1) missing momentum searches; (2) searches for decays with displaced vertices. The muon beams available at CERN and Fermilab present attractive opportunities for exploring the new scalar with a mass below the di-muon threshold, and potentially covering a range of relevant candidate models. For the models considered in this paper, both types of signals, muon missing momentum and anomalous energy deposition at a distance, can probe a substantial fraction of the unexplored parameter space of the new light scalar, including a region that can explain the muon anomalous magnetic moment discrepancy.

Tables

Table 1. Table 1: Parameters for the proposed muon beam-dump experiments at NA64 and Fermilab.

	NA64-type	Fermilab
Incident muon beam energy, $E_{\mu ,\text{beam}}$	160 GeV	3 GeV
Total number of incident muons, $N_{\mu }$	$8\times {10}^{12}$	$3\times {10}^{14}$
Target material	Lead (Pb)	Tungsten (W)
Atomic number density, $n_{\text{atom}}$	$3.3\times {10}^{22}$/cm3	$6.3\times {10}^{22}$/cm3
Muon energy loss per unit length, $⟨\text{d}{E}_{\mu }/\text{d}y⟩$	$12.7\times {10}^{-3}$ GeV/cm	$22.1\times {10}^{-3}$ GeV/cm
Target Length, $L_{\text{tg}}$	0.2 m	1.5 m
Detector Length, $L_{\text{dec}}$	5 m	3 m
Min fiducial range for the decay, $z_{\min }$	$L_{\text{tg}}+L_{\text{dec}}-\Delta y\left(E_{\mu }\right)$	$L_{\text{tg}}-\Delta y\left(E_{\mu }\right)$
Max fiducial range for the decay, $z_{\max }$	$\infty$	$L_{\text{tg}}+L_{\text{dec}}-\Delta y\left(E_{\mu }\right)$

Equations

$$\mathcal{L}\supset \frac{1}{2}({\partial }_{\mu }S{)}^2-\frac{1}{2}{m}_S^2{S}^2-\sum _{\ell =e,\mu ,\tau }{g}_{\ell }S\bar{\ell }\ell ,$$

$${\mathcal{O}}_5=\frac{1}{\Lambda }(\bar{L}E)HS,$$

$$\Delta a_{\mu }=\frac{g_{\mu }^2}{8{\pi }^2}{\int }_0^1\text{d}z\frac{(1-z{)}^2(1+z)}{(1-z{)}^2+z(m_S/m_{\mu }{)}^2}.$$

$${\Gamma }_{e^{+}{e}^{-}}=\frac{m_S}{8\pi }{g}_e^2{\left(1-\frac{4m_e^2}{m_S^2}\right)}^{3/2}.$$

$${\Gamma }_{\gamma \gamma }$$

$$\displaystyle f(\tau)=\left\{\begin{array}[]{ll}{\rm arcsin}^{2}\left({\tau^{-1/2}}\right),&\hskip 14.22636pt\tau>1\\ -\frac{1}{4}\left[\ln\left(\frac{1+\sqrt{1-\tau}}{1-\sqrt{1-\tau}}\right)-i\pi\right]^{2},&\hskip 14.22636pt\tau\leq 1\end{array}\right.\,.$$

$${\Gamma }_S^{\text{A}}={\Gamma }_{e^{+}{e}^{-}}+{\Gamma }_{\gamma \gamma }\approx {\Gamma }_{e^{+}{e}^{-}}$$

$${\Gamma }_S^{\text{B}}=\frac{{\alpha }^2{m}_S^3}{64{\pi }^3}\frac{g_{\mu }^2}{m_{\mu }^2}{\left|{\tau }_{\mu }\left[1+(1-{\tau }_{\mu })f({\tau }_{\mu })\right]\right|}^2$$

$$L_S=\frac{E_S}{m_S}\frac{{\beta }_S}{{\Gamma }_S},$$

$$L_S=25\mathrm{cm}\times {\left(\frac{5\times 10^{-4}}{g_{\mu }}\right)}^2\times {\left(\frac{100\mathrm{MeV}}{m_S}\right)}^2,\mathrm{Model}\mathrm{A},$$

$$L_S=20\mathrm{m}\times {\left(\frac{5\times 10^{-4}}{g_{\mu }}\right)}^2\times {\left(\frac{100\mathrm{MeV}}{m_S}\right)}^4,\mathrm{Model}\mathrm{B}.$$

$$\frac{\text{d}}{\text{d}x}\sigma ({\mu }^{+}+N\to {\mu }^{+}+N+S)\simeq \frac{g_{\mu }^2{\alpha }^2}{12\pi }\chi {\beta }_{\mu }{\beta }_S\frac{x^3\left[m_{\mu }^2(3x^2-4x+4)+2m_S^2(1-x)\right]}{{\left[m_S^2(1-x)+m_{\mu }^2{x}^2\right]}^2},$$

$$\chi ={\int }_{t_{\min }}^{t_{\max }}\text{d}t\frac{t-t_{\min }}{t^2}{G}_2(t)\simeq {\int }_{m_S^4/\left(4E_{\mu }^2\right)}^{m_S^2+m_{\mu }^2}\text{d}t\frac{t-m_S^4/\left(4E_{\mu }^2\right)}{t^2}{G}_2(t),$$

$$N_S=$$

$$=$$

$$\Delta y\equiv y-y_{\min }=\frac{E_{\mu ,\text{beam}}-E_{\mu }}{⟨\text{d}{E}_{\mu }/\text{d}y⟩},$$

$$N_S\approx \frac{N_{\mu }{n}_{\mathrm{atom}}}{⟨\text{d}{E}_{\mu }/\text{d}y⟩}{\int }_{E_{\mu ,\min }}^{E_{\mu ,\text{beam}}}\text{d}{E}_{\mu }{\int }_{x_{\min }}^1\text{d}x\frac{\text{d}{\sigma }_{2\to 3}}{\text{d}x}\left(e^{-\frac{z_{\min }}{L_S}}-e^{-\frac{z_{\max }}{L_S}}\right),$$

$$z_{\min }(E_{\mu })=L_{\text{tg}}+L_{\text{det}}-\Delta y(E_{\mu }),z_{\max }=\infty .$$

$$\tilde{\sigma }(E_{\mu })={\int }_{x_{\min }}^1\text{d}x\frac{\text{d}{\sigma }_{2\to 3}}{\text{d}x}\left(e^{-\frac{z_{\min }}{L_S}}-e^{-\frac{z_{\max }}{L_S}}\right)$$

$$z_{\min }(E_{\mu })=L_{\text{tg}}-\Delta y(E_{\mu }),z_{\max }(E_{\mu })=L_{\text{det}}+z_{\min }.$$

$$\text{d}\sigma (\mu +N\to \mu +K^0+X)\sim \sigma (N+\gamma \to K^0+X)\text{d}{n}_{\gamma },$$

$$N_{\text{kaon}}$$

$$G_2^{\text{el}}(t)={\left(\frac{a^{2}t}{1+a^{2}t}\right)}^2{Z}^2{\left(\frac{1}{1+t/d}\right)}^2,$$

$$G_2^{\text{in}}(t)={\left(\frac{a^{\prime 2}t}{1+a^{\prime 2}t}\right)}^{2}Z{\left(\frac{1+t({\mu }_p^2-1)/(4m_p^2)}{{\left(1+t/(0.71{\text{GeV}}^2)\right)}^4}\right)}^2,$$

Full text

I Introduction

New Physics (NP) at low-mass, treated in all generality, has become an actively pursued topic of the intensity frontier physics Hewett:2012ns ; Essig:2013lka ; Alexander:2016aln given the abundant evidence for NP in the neutrino and dark matter sectors, coupled with the lack of NP signal at the Large Hadron Collider (LHC). Motivations for searches of low-mass, weakly-coupled particles can come from top-down theoretical arguments (see e.g. Jaeckel:2010ni ). But a bigger role is played by the existing anomalous observations in particle experiments, astrophysics, and cosmology, which might find their explanations in models with NP at low-mass (see e.g. Boehm:2003hm ; Pospelov:2007mp ; ArkaniHamed:2008qn ). The current $\sim 3.5\sigma$ discrepancy between the predicted and observed value of the muon anomalous magnetic moment Bennett:2006fi , $a_{\mu}$, will be scrutinized in the upcoming experiments at Fermilab and JPARC Grange:2015fou ; Mibe:2010zz . It is not clear that the current tension is a result of experimental errors or theoretical errors or a combination of the two. With new measurements of muon $g-2$ and improved Standard Model (SM) calculations based on lattice QCD Benayoun:2014tra ; Jin:2016rmu , one hopes to clarify the origin of the existing discrepancy. Lamb shifts of muonic atoms, such as muonic hydrogen and deuterium Pohl:2010zza ; Antognini:1900ns ; Pohl1:2016xoo , present another formidable puzzle. When interpreted in terms of the charge radius of the proton, $r_{p}$, these measurements disagree with the electron scattering and hydrogen spectroscopy extracted values of $r_{p}$ by $\sim 7\sigma$ Mohr:2015ccw .

In this paper, we are interested in the scenarios where the deficit of theoretical predictions for $a_{\mu}$ is compensated by a contribution from NP. Although the overall size of the $a_{\mu}$ discrepancy, $a_{\mu}^{\rm obs}-a_{\mu}^{\rm th}\approx+3\times 10^{-9}$, is on the order of the corresponding contributions from the weak sector of the SM, the NP states correcting the anomalous magnetic moment do not have to reside at the weak scale. Indeed it is well known that the existing theoretical deficit can be compensated by loop contributions from new light particles Gninenko:2001hx ; Fayet:2007ua ; Pospelov:2008zw . One such candidate model, the dark photon, has been searched for in a variety of experiments, with recent results ruling out the most minimal version as a possible explanation of the $a_{\mu}$ discrepancy. Some other candidate models still survive the existing constraints, including the $L_{\mu}-L_{\tau}$ gauged model and its variations Altmannshofer:2014pba ; Altmannshofer:2016brv .

Here we would like to examine the models with a new light scalar, $S$, tuned to explain the $a_{\mu}$ discrepancy Chen:2015vqy ; Batell:2016ove . We will employ a simplified framework, with a relevant Lagrangian given by

[TABLE]

where $g_{\ell}$ is the coupling between $S$ and leptons. Notice that Eq. (1) is an effective Lagrangian that does not respect the full gauge symmetry of the SM. Its $SU(2)\times U(1)$ generalization is given by the following dimension-five effective operator,

[TABLE]

where $H$ is the SM Higgs doublet, and $L$, $E$ are the lepton doublets and singlets respectively. The effective operator, $\mathcal{O}_{5}$, can be embedded into a full model in a variety of ways. Refs. Chen:2015vqy ; Batell:2016ove discuss the phenomenology of such a model and choose different types of UV completion of $\mathcal{O}_{5}$ with vector-like fermions or multiple Higgs states respectively. The latter UV completion generates strong constraints for the mass range $m_{S}>2m_{\mu}$ due to recent searches of unexpected peaks in the di-muon mass spectrum for $B\to K(\mu^{+}\mu^{-})$ decays at the LHCb Aaij:2015tna . The mass range of $m_{S}<2m_{\mu}$ remains largely unexplored. In this mass range, the new light particles can be relatively long-lived, and thus amenable to beam-dump experiments and fixed-target searches.

In this paper, we investigate the potential of experiments where the light scalar, $S$, is sourced by the collision of muons with nuclei. Subsequent displaced decays of $S$ present an opportunity for both the missing momentum and the anomalous energy deposition searches. We take the existing CERN and Fermilab muon111“Muon” or “$\mu$” refers to $\mu^{+}$ for the muon beam-dump experiments. sources as an example and illustrated our main idea in Fig. 1. For a NA64-type setup at CERN Andreas:2013lya ; Gninenko:2014pea , the dark emission of $S$ states with $S$ decaying outside of the detector would cause the anomalous loss of muon energy, which can be detected in the muon scattering experiment. The muon beam with beam energy around a few GeV at Fermilab would also provide a great opportunity. Here muons are stopped in the dense material, and subsequent anomalous energy deposition is searched directly behind it. In what follows, we demonstrate that both approaches allow probing unexplored parts of the parameter space of the simplified model potentially responsible for the $a_{\mu}$ discrepancy. We use the bremsstrahlung, $\mu^{+}+N\to\mu^{+}+N+S$, as the main production mechanism as illustrated in Fig. 2, where an incident muon, $\mu^{+}$, interacts with a target nucleus, $N$, by exchanging a photon, $\gamma$, and radiates the exotic scalar, $S$. The two muon beam-dump experiments considered in this paper can be easily implemented with only modest modifications/additions to the existing experimental infrastructure. Looking into more distant future, the proton beam-dump facilities, such as SHiP Alekhin:2015byh , would also provide strong sensitivities to muon-coupled light states.

The paper is organized as follows: We first show two specifications of the simplified model in Sec. II and analyze the two proposed muon beam-dump experiments in Sec. III. We show the resulting expected sensitivities in Sec. IV, conclude, and discuss other related experiments in Sec. V.

II Models with new light scalars

For the simplified model introduced in the previous section, Eq. (1), the couplings $g_{\ell=e,\mu,\tau}$ are free parameters. The muon anomalous magnetic moment, $a_{\mu}$, receives corrections due to the one-loop contribution of $S$,

[TABLE]

Requiring this correction to reduce current tension between measurement and the SM calculation of $a_{\mu}$, one arrives at the preferred values of $\{g_{\mu},m_{S}\}$ parameters. For example, given $g_{\mu}=5\times 10^{-4}$ and $m_{S}=100$ MeV, $\Delta a_{\mu}$ is around $1.6\times 10^{-9}$. This would bring the theoretical and experimental values for the muon anomalous magnetic moment within $2\sigma$. Other couplings, $g_{e}$ and $g_{\tau}$, still remains free. Here we will consider two models with further specifications on the couplings of $g_{e}$ and $g_{\tau}$:

• •

Model A: Mass proportionality, $g_{\ell}\propto m_{\ell}$. In particular, it implies that the couplings between the scalar $S$ and electrons are $\sim 200$ times smaller than those with muons. Despite this, the dominant decay channel for $S$ below the di-muon threshold is $S\to e^{+}e^{-}$.

• •

Model B: Coupling exclusively to muons, $g_{\mu}\neq 0$ and $g_{e}=g_{\tau}=0$. This is the most collider and electron/proton beam-dumps unfriendly case, that still can be relevant for the muon $g-2$. Due to the longer lifetime of $S$, the missing energy search at NA64-type experiments would be particularly useful to constrain the parameter space of this model.

Model A can be explicitly constructed using the leptonic Higgs doublet model completion of Ref. Batell:2016ove . In that model, the lepton flavor conservation and $g_{\ell}\propto m_{\ell}$ proportionality are guaranteed, as only one Yukawa matrix determines the lepton masses and their couplings to $S$. Model B is in some sense more artificial but phenomenologically minimal. Only one coupling is introduced that is necessary to correct $a_{\mu}$ in this model.

In both models, $S$ dominantly decays to $\mu^{+}\mu^{-}$ for $2m_{\mu}<m_{S}<2m_{\tau}$. For the mass range that we concentrate on, $2m_{e}<m_{S}<2m_{\mu}$, the total decay width, $\Gamma_{S}$, is a sum of the decay widths of $S\to e^{+}e^{-}$ and $S\to\gamma\gamma$ channels. They are respectively given by

[TABLE]

and

[TABLE]

where $\tau_{\ell}\equiv{4m_{\ell}^{2}/m_{S}^{2}}$ and the loop function $f(\tau)$ reads,

[TABLE]

For model A, the decay is dominantly through $S\to e^{+}e^{-}$,

[TABLE]

For model B, the only decay channel is $S\to\gamma\gamma$ via $\mu$ loop, i.e.,

[TABLE]

Given $\Gamma_{S}$, the decay length of the scalar $S$ is expressed by

[TABLE]

where $E_{S}$ is the energy of the scalar and the boost factor $\beta_{S}=\sqrt{1-{m_{S}^{2}/E_{S}^{2}}}$. In particular, taking a fiducial choice of parameters, and the energy of the scalar $E_{S}=3$ GeV, we find

[TABLE]

These sizable decay lengths give a good motivation to search for $S$ in the beam-dump experiments.

A previous study of the muon-beam-initiated emission of axion-like particles, Essig:2010gu , shares several common features with our scalar model. Light dark vector particles emitted from a muon beam, in the context of the NA64 experiment, have been also studied in Gninenko:2014pea . Our paper aims to extend these previous works to the scalar case, and explore the sensitivity reach on the $m_{S}$ –$g_{\mu}$ parameter space.

III Signals and backgrounds

III.1 Signals

III.1.1 NA64-type muon beam-dump experiment

In this subsection we investigate the constraints from the NA64-type experiment at CERN. NA64 is a fixed-target experiment searching for dark sector particles and kaon invisible decays at the CERN Super Proton Synchrotron (SPS). The detailed setup of the NA64 experiment can be found in Refs. Andreas:2013lya ; Gninenko:2014pea , and the experiment has reported its first results from the $2.75\times 10^{9}$ electrons on target in 2017 Banerjee:2016tad . We adopt similar geometries of the target and detector as suggested in Refs. Andreas:2013lya ; Gninenko:2014pea and sketch the setup in Fig. 1. The target is made of lead (Pb) with a thickness of $\sim$ 20 cm. The length of the detector is $\sim$ 5 meters. As pointed out in Ref. Gninenko:2014pea , the muon beam has a maximum luminosity of $10^{6}$ muons per second in order to evade loss of the signal efficiency due to the pileup effect. We assume a three-month run of the experiment at the maximum luminosity, which yields $8\times 10^{12}$ muons in total on target. The incident muon beam energy, $E_{\mu,\text{beam}}$, is assumed to be around 160 GeV.

To estimate the reach, we need to calculation of the differential cross section of the beam-dump process $\mu^{+}+N\to\mu^{+}+N+S$ as shown in Fig. 2. Given the large muon beam energy of NA64, we use the improved Weizsacker-Williams (IWW) approximation Kim:1973he in the evaluation.

When the beam energy is far greater than the mass of beam particle and the mass of produced particle, $E_{\text{beam}}\gg m_{\text{beam}},m_{S}$, virtual photons generated by the highly-boosted beam particle are nearly transverse and behave as plane waves. Then the virtual photon can be approximate as a real photon, and the phase space integration of a $2\to 3$ ($\mu^{+}+N\to\mu^{+}+N+S$) process is simplified to a $2\to 2$ ($\mu^{+}+\gamma\to\mu^{+}+S$) process. This is the so-called Weizsacker-Williamsuses (WW) approximation Kim:1973he . The WW approximation can be further refined by using the fact that the production of $S$ is dominantly collinear when the energy of $S$ approaches the energy of $\mu$. This yields the improved Weizsacker-Williamsuses (IWW) approximation where the integration limits on the virtuality, $t$, are further simplified Kim:1973he ; Bjorken:2009mm . The original IWW derivation Kim:1973he ; Bjorken:2009mm regards the beam particle to be massless. A detailed derivation of IWW with massive beam particles is presented in Liu:2016mqv . The resulting differential cross section is given by

[TABLE]

where $x\equiv E_{S}/E_{\mu}$ is the ratio between the energy of the exotic scalar, $E_{S}$, and the energy of the muon, $E_{\mu}$, inside the material. The boost factor for muon and the new scalar are given respectively by $\beta_{\mu}=\sqrt{1-m_{\mu}^{2}/E_{\mu}^{2}}\approx 1$ and $\beta_{S}=\sqrt{1-m_{S}^{2}/(xE_{\mu})^{2}}$. The effective photon flux, $\chi$, is given by

[TABLE]

where $G_{2}$ is the combined atomic and nuclear form factor. The explicit expression of $G_{2}$ is given in the appendix A. The simplified integration on $t$ from the IWW approximation is implemented in the second approximation of Eq. (15).

Unlike earlier studies for the dark sector searches via electron beam-dumps, here we keep the beam particle mass to be non-zero. As a consequence, the expected energy spectrum of $S$ varies quite significantly as $m_{S}$ increases. In Fig. 3, we demonstrate this point by comparing $x$ spectrum for $m_{S}=10$ MeV and $m_{S}=100$ MeV. The emission of light particles ($m_{S}\ll m_{\mu}$) clearly favors the low $x$ region, while for $m_{S}\approx m_{\mu}$, the outgoing $S$ takes more significant portion of the muon energy.

The total number of events, $N_{S}$, can be obtained as a convolution of the production cross section and the decay probability. It is given by

[TABLE]

where $N_{\mu}$ is the total number of incident muons and $\sigma_{2\to 3}$ is a shorthand for $\sigma(\mu^{+}+N\to\mu^{+}+N+S)$. $n_{\rm atom}$ is the number density of target nuclei and its integration over the muon penetration length, $y$, accounts for the number of target nuclei that a incident muon encounters. $P(z)$ stands for the decay probability density (per length) of $S$ decaying within the fiducial decay range from $z_{\min}$ to $z_{\max}$. $L_{S}$ is the decay length given by Eq. (11) with $E_{S}=E_{\mu}x$. We also impose a $x_{\min}$ in the integration over $x$ to satisfy the specific requirements for the search of the signal.

Muon beams penetrate the target to a much longer depth compared to that of the electron beams. It can loose energy through multiple mechanisms Olive:2016xmw . For the energy range we are interested in, from a few GeV to $\sim$100 GeV, the muon energy loss is dominantly through the ionization and the stopping power $\langle\text{d}E_{\mu}/\text{d}y\rangle$ is relatively flat with respect to the muon momentum Olive:2016xmw . Hence we approximate the muon energy loss per unit length $\langle\text{d}E_{\mu}/\text{d}y\rangle$ to be a constant. Consequently $E_{\mu}$ is related to the penetration length $y$ via

[TABLE]

where $E_{\mu,\text{beam}}$ is the initial energy of the incident muon beam. Eq. (18) can be used to simplify Eq. (17) into

[TABLE]

where the fiducial range for the decay, $z_{\min}$ and $z_{\max}$, are respectively given by

[TABLE]

$L_{\text{tg}}$ and $L_{\text{det}}$ represent the lengths of the target and detector, respectively. For a lead target, $n_{\rm atom}=3.3\times 10^{22}$/cm3 and $\langle\text{d}E_{\mu}/\text{d}y\rangle\approx 12.7\times 10^{-3}$GeV/cm for the relevant beam energy range pdgmat . For the thin target ($L_{\text{tg}}=0.2$ m), the muon energy after penetrating the target, $E_{\mu,\min}=159.75$ GeV, is close to the incident beam energy $E_{\mu,\text{beam}}=160$ GeV.

The signature of the signal at NA64-type experiment for the incoming electron mode is a single EM shower in the target corresponding to the final state electron with significant missing energy. The required missing energy, $E_{\text{miss}}$, should be above expected background values. In the muon mode, the detection strategy would be modified somewhat, as one would need to detect the energy of the final state muon. As suggested by Gninenko:2014pea ; Banerjee:2016tad , here we require $E_{\rm miss}>E_{\mu,\rm bearm}/3\approx 53$ GeV, which is equivalent to setting $x_{\min}=1/3$ given $E_{\mu}$ is close to $E_{\mu,\text{beam}}$ for the thin target of NA64. We further assume the efficiency for the signal reconstruction $\approx 100\%$ and leave a more detailed detector modeling and study for the future.

III.1.2 Fermilab muon beam-dump experiment

Fermilab have capabilities of producing a more intense source of muons, albeit at a smaller muon beam energy. We suggest the simplest muon beam dump experiment, where a muon beam is fully stopped in a dense target with a thickness of several meters. Similar to the NA64-type of experiments, we can estimate the number of signal events using existing setup for the Fermilab muon beam. The incident muon beam energy we propose for the experiment, $E_{\mu,\text{beam}}$, is $\sim 3$ GeV, as the accelerator complex is already tuned to this energy for the muon $g-2$ experiment Chapelain:2017syu . Such a beam will be completely stopped in 1.5 m thickness tungsten target ($n_{\rm atom}=6.3\times 10^{22}$/cm3 and $\langle\text{d}E_{\mu}/\text{d}y\rangle\approx 22.1\times 10^{-3}$GeV/cm for the beam energy range pdgmat ). Hence we propose a setup for the Fermilab muon beam-dump experiment as shown in Fig. 1. The lengths of the target (tungsten) and detector are around 1.5 m and 3 m, respectively. The total exposure taken for this proposed experiment is $10^{7}$ muons per second for 1 year of running, or $3\times 10^{14}$ muons in total on target.

To avoid the background from soft muons, we adopt a lower limit on the muon beam energy $E_{\mu,\min}=0.5$ GeV. To estimate the number of signal events, we need to account for the muon energy loss inside the target via an integration over $E_{\mu}$ from 0.5 GeV to 3 GeV. Here the IWW approximation is not applicable since the muon energy around $E_{\mu,\min}$ is not much greater than $m_{\mu}$. Instead we use MadGraph 5 aMC@NLO Alwall:2014hca to obtain the cross sections of $\sigma(\mu^{+}+N\to\mu^{+}+N+S)$ for various $E_{\mu}$. The combined atomic and nuclear form factor, $G_{2}(t)$, is implemented in the model file (see Appendix A for more details) and the decay probability is implemented by reweighting the generated events. The procedure yields numerical values of the reweighed cross section

[TABLE]

as a function of $E_{\mu}$. The fiducial range of the decay here is given by

[TABLE]

To estimate the number of signal events, we interpolate over the samplings of $\tilde{\sigma}(E_{\mu})$ and preform the integration over $E_{\mu}$ according to Eq. (19).

The signature of the signal at Fermilab experiment is a decay with a sizable displaced vertex reconstructed from $e^{+}e^{-}$ or $\gamma\gamma$. A typical electron or photon tracker/calorimeter requires a minimum momentum/energy of the particle around 10 MeV. This threshold is much smaller than the momentum of the decayed electrons or photons in the lab-frame, $p_{e,\gamma}^{\text{lab}}\approx\gamma_{S}m_{S}/2=xE_{\mu}/2$, and hence can be easily satisfied given a small $x$. Therefore we approximate $x_{\text{min}}\approx 0$ in the estimation of $N_{S}$. Like the NA64 case, we further assume the efficiency for the signal reconstruction $\approx 100\%$ and leave a more careful detector modeling for future experimental studies.

III.1.3 Difference between setups of NA64-type and Fermilab muon beam-dump experiments

Tab. 1 summarizes the setups of NA64-type and Fermilab muon beam-dump experiments. To illustrate the kinematic difference between the two setups, in Fig. 4 we show the decay probabilities of $S$ within the geometrical acceptance with $m_{s}=100$ MeV in model A (black) and model B (red dashed) for the NA64 (left) and Fermilab (right), respectively. For the NA64-type experiment (Fig. 4, right panel), the curve corresponding to model B (red dashed) rises earlier as we gradually turn off the coupling. This is due to the fact that the decay length of $S$ in model B is much longer compared to that in model A. Since the fiducial volume is from $\sim 5$ m to infinity, we find that the decay probabilities remain close to one for very small $g_{\mu}$, corresponding to the region of parameter space where the scalars are very long-lived. This tells us that the lower contours of NA64 in Fig. 5 is set by the production rate. Since the production rate of the light scalar is identical in both models A and B as illustrated in Fig. 2, the lower limits for both models for NA 64 in Fig. 4 are the same. For the muon beam-dump experiment at Fermilab, the width of the peak is due the finite size of the detector. Similar to the NA64 case, the curve corresponding to model B (red dashed) peaks earlier as the decrease of the coupling $g_{\mu}$.

III.2 Potential backgrounds

The NA64 experiment has addressed the issues of potential background for the missing energy/momentum search, and many components of the experiment are tuned to reject various sources of backgrounds Gninenko:2014pea ; Banerjee:2016tad . As described above, those studies suggest a missing energy cut $E_{\text{miss}}\gtrsim 50$ GeV, which is equivalent to requiring $1/3<x<1$. A more detailed analysis on potential backgrounds requires knowledge of the detector and is beyond the scope of this paper.

Next, we focus on possible backgrounds to the proposed Fermilab muon beam-dump experiment. All charged particles can be efficiently stopped inside the target given the target length we adopted. Potential backgrounds may arise from the neutral long-lived kaons. They are produced by muons and decay after reaching the detector. They can mimic the long-lived scalars $S$. For model B, $K_{L}\to\pi^{+}\pi^{-}\pi^{0}$ and $K_{L}\to 3\pi^{0}$ decays are particularly worrisome, as they produce photon-like energy deposition. It is instructive to estimate how many $K_{L}$ reach the end of the target without suffering the degradation in energy. To do that we consider the muon-initiated kaon production cross section $\sigma(\mu+N\to\mu+K^{0}+X)$, where $X$ is a baryonic state with an open strangeness. The cross section size can be estimated using the WW approximation, and related to the underlying kaon photoproduction cross section,

[TABLE]

where $n_{\gamma}$ is the number of quasi-real (or equivalent) photons carried by the muon. The total number of $K_{L}$ is then given by

[TABLE]

where $\sigma(\omega)$ is the kaon photoproduction cross section with an incident photon energy $\omega$ on a tungsten nucleus. We assume that the cross section is given by an incoherent sum of the production cross section on individual nucleons, $\propto A\;\sigma_{\text{kaon}}$ with $A$ being the number of nucleons. In turn, $\sigma_{\text{kaon}}$ receives contributions from several subprocesses, $\sigma(\gamma+n\to K^{0}+\Lambda)$, $\sigma(\gamma+p\to K^{0}+\Sigma^{+})$, and $\sigma(\gamma+n\to K^{0}+\Sigma^{0}).$ Each of these individual cross sections is about 0.8 $\mu$b on average, and becomes very small for $\omega<1.5$ GeV Goers:1999sw ; Tsukada:2007jy ; Kaon . The neutral kaons are produced predominantly in the upstream part of the target, and then propagate through about a meter of dense material. In the process of doing so their energy is degraded, and an idealized factor $e^{-{z_{\min}\over L_{N}}}$ in Eq. (24) is to account for the probability of the produced kaons reaching the end of the target without interaction with the material. $L_{N}$ represents the nuclear collision length, which is $\simeq$ 6 cm for tungsten. Estimated that way, for one year of running, the number of kaons produced is around $\mathcal{O}(1)$ before any cuts. We expect that those background events can be further rejected by applying selection criteria, such as absence of charged pions and/or invariant mass cuts. The kaon background can also be reduced by lowering the energy of the incident muon beam. Therefore we neglect the kaon background for Fermilab muon beam-dump in the projection below. More detailed background estimations can be achieved via specialized simulations for a concrete experimental design.

IV Results

Based on the estimations of the number of signal events for the muon beam-dump experiments at NA64 and Fermilab, we project sensitivities for the two proposed experiments for various models. Fig. 5 shows the resulting exclusions with 95% confidence level (CL) on the $m_{S}$ vs. $g_{\mu}$ plane for model A where $S$ dominantly decays through $e^{+}e^{-}$ (left) and model B where $S$ decays into $\gamma\gamma$ via a $\mu$-loop (right). The orange and cyan contours on the plots represent constraints from NA64 and Fermilab respectively. The projected constraints from the muon beam-dump experiments are compared with current constraints and the favored parameter space from other experiments. The red band is the favored region to solve the $a_{\mu}$ discrepancy with 2$\sigma$ CL. The blue region are contributions of $S$ to $a_{\mu}$ that excluded with 5$\sigma$ CL. We have also show the limits from electron beam-dump experiments, Orsay Davier:1989wz (purple) and E137 Bjorken:1988as (gray) for model A. Those limits are not relevant for model B since there $S$ does not couple to electrons. The BaBar collaboration search through the process $e^{+}e^{-}\to\mu^{+}\mu^{-}S$ TheBABAR:2016rlg and excludes the upper right conner of the parameter space for both models.

For model A, the projected constraints from NA64 and Fermilab is largely covered by existing E137 constraint for small $m_{S}$ and $g_{\mu}$. Nevertheless, the muon beam-dump experiments will cover new grounds for the range of $m_{S}$ from 50 MeV to 210 MeV and $g_{\mu}$ from several of $10^{-5}$ to $10^{-3}$. They can effectively exclude the $a_{\mu}$ favored region for the $m_{S}$ range, which is not reached by E137. The importance of this region of parameter space has also been pointed out by Liu:2016qwd in the context of a more general model, where the authors attempt to address both the muon g-2 anomaly and the proton charge radius puzzle. As for model B, both muon beam-dump experiments at Fermilab and NA64 can place strong limits and completely exclude the parameter space favored by muon $g-2$ experiment below the di-muon threshold. Again the NA64-type experiment is more sensitive to smaller $g_{\mu}$ region comparing to the Fermilab experiment.

V Conclusions and Discussions

Muon beams have many applications in particle physics. Among fundamental physics applications, they have been used to study nuclear structure and perform precision measurements of $g-2$. The latter presents an intriguing 3$\sigma$–4$\sigma$ deficit of theoretical predictions relative to experimental observations. It could be a sign of low-mass new physics coupled to muons. All attempts to find such particles so far has rendered only exclusions on masses and couplings of such particles. The majority of those searches have concentrated on hadron or electron-initiated production. In light of the main discrepancy coming from the muon sector, it makes sense to explore the possibility of light particles coupled predominantly to muons, and try to use a muon beam as a source of such particles.

We have shown that muon beam-dump experiments at NA64 and Fermilab can effectively explore the light scalars that are predominantly coupled to muons. Full UV-complete models with such scalars can be built. In this paper we have explored a simplified low-energy version of such models without going into details of the UV completion. The scalar $S$ can have a small, or vanishing, coupling to electrons. This also make the exotic scalar, $S$, long-lived, leading to the displaced decays in the beam-dump experiments. We have found that the experiments with muon beams indeed extend the reach to the parameter space of the exotic scalars. In particular, the favored parameter space to explain the $a_{\mu}$ discrepancy can be effectively probed.

Below we would like to discuss additional aspects of the low-mass new physics experiments with muon beams and the models they can explore:

• •

Practical aspects of muon beam-dump at Fermilab. The beam-dump experiment with the anomalous energy deposition downstream from the dump is among the simplest particle physics experiments. The muon beam energies available at Fermilab allow to make this setup relatively compact, with the total length of a few meters. As such this proposed experiment could go into the $g-2$ experimental hall. Moreover, depending on the availability of protons, the proposed beam-dump can be run in parallel with the $g-2$ experiment.

• •

*Model dependence. * The simplified model of one scalar particle considered in this paper is an example of a physics goal that muon beam-dump experiments may pursue. An interesting variation of this is when the multiplicity of exotic states $N_{d}$ is large, as may occur in the models with extra dimensions where the dark forces are allowed to live McDonald:2010iq ; Wallace:2013eya ; Jaeckel:2014eba , or in models with some conformal dynamics, where the new states are continuously spread over the invariant mass Georgi:2007ek . It is easy to see the qualitative difference in the phenomenology of such models compared to an exotic single state models. The effects of virtual dark force particles (such as corrections to $g-2$) can be enhanced by large multiplicity. Therefore, smaller individual couplings can be responsible for the same size of the corrections. Moreover, the mass step, $\Delta m_{S}$, can lead to overlapping resonances within a detector mass resolution, undermining the “bump hunt” searches. This type of models with, e.g. a tower of dark photons, will escape current direct searches at NA48/2, BaBar etc, but can be a source of sizeable corrections in $g-2$. It is easy to see that such models generically lead to longer lifetimes of individual states, and therefore can be subjected to tighter displaced decay bounds. Such models can also be probed in the muon beam-dump experiments.

• •

*The advantage of running NA64 in the muon mode. * NA64 experiment currently occupies a unique niche (which can be followed up by a similar experiment in North America Izaguirre:2014bca ). In this paper we have argued that a muon run in NA64 is warranted, as it provides a very strong sensitivity to models (model B) where the decay of $S$ happens well outside the detector. This adds to an important case of $L_{\mu}-L_{\tau}$ gauge boson with mass $m_{Z^{\prime}}<2m_{\mu}$, where the final state of decay is always neutrinos Gninenko:2014pea .

• •

Neutrino sources, SHiP. In this paper, we have concentrated on considering dedicated experiments with muon beams. Two other possibilities involve proton beam-dumps, which also creates a lot of muons, as well as beams of mesons used to source the neutrino beams. None of these possibilities is suitable for the missing energy or missing momentum studies. However, the anomalous energy deposition at the distance can indeed be probed, as is well known. Perhaps a very powerful probe of new physics coupled to muons can be achieved at a proposed SHiP facility Alekhin:2015byh . There, a large number of muons created in the target propagates through tens of meters of material before getting stopped or deflected. The decay products of the light particles produced in the collision of muons with nuclei can be detected downstream, and a relatively short distance to detector (compared to past proton beam-dump experiments), as well as large boosts, may significantly increase the reach to unexplored parts of parameter space. Constraints on the muonic forces from the proton beam-dumps deserve a separate dedicated study.

• •

*Tau-initiated production. * Finally, model A with $g_{\ell}\propto m_{\ell}$ can be probed at high-luminosity $e^{+}e^{-}$ colliders through the process $e^{+}e^{-}\to\tau^{+}\tau^{-}S\to\tau^{+}\tau^{-}e^{+}e^{-}$ Batell:2016ove . In light of the discussions in this paper, the decay of $S$ can also be displaced, producing a rather unique signature that is easy to be distinguished from the SM processes. Both Belle and BaBar collaborations could perform corresponding analyses.

Acknowledgements

We would like to thank R. Essig, E. Izaguirre, Y.-S. Liu, G. Magill, D. McKeen, R. Plestid, M. Schmaltz, and N. Toro for helpful discussions. YZ thank N. Blinov, R. Essig, E. Izaguirre, V. Khachatryan, O. Mattelaer, A. Simonyan, and M. Takashi for setting up and testing the modified MadGraph 5 event generator for fixed-target experiments. The work of C.-Y.C and M.P. is supported by NSERC, Canada. Research at the Perimeter Institute is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MEDT. Y.Z. is supported through DoE grant DE-SC0015845.

Appendix A Atomic and nuclear form factors

The combined atomic and nuclear form factor, $G_{2}$, is presented in Kim:1973he ; Tsai:1973py ; Tsai:1986tx ; Bjorken:2009mm . Two components contribute to $G_{2}$: (1) the elastic part is given by

[TABLE]

where the virtuality $t$ represents the momentum transfer squared. $a=111Z^{-1/3}/m_{e}$ under Thomas-Fermi model and $d=0.164~{}\text{GeV}^{2}A^{-2/3}$ where $A$ and $Z$ stand for the mass number and atomic number of the target material, respectively. $m_{e}$ is the electron mass Kim:1973he ; Tsai:1973py ; Tsai:1986tx ; (2) the inelastic part, in the limit $t/m_{p}^{2}$ is small, is given by

[TABLE]

where $m_{p}$ is the proton mass, $a^{\prime}=773Z^{-2/3}/m_{e}$, and $\mu_{p}=2.79$ Kim:1973he .

For the parameter space of $m_{S}$ we are interested in, the expression for $G_{2}^{\text{in}}$ is valid, and we do not have to include the inelasticity at the nucleon level. The form factors dress the nucleus-nucleus-photon vertex with $G_{2}^{1/2}=(G_{2}^{\text{el}}+G_{2}^{\text{in}})^{1/2}$, which we implement in MadGraph 5 aMC@NLO according to MG5FF .