SEARCH FOR DARKONIUM IN ELECTRON-POSITRON COLLISIONS

6.1 Introduction

The existence of dark matter is overwhelmingly supported by astrophysical and cosmological observations, and it is one of the most important tasks of contemporary physics to understand its nature and properties. The concept of dark matter can be traced back to 1880s, when Lord Kelvin estimated the number of dark bodies in the Milky Way from the observed velocity dispersion of the stars orbiting the center of the galaxy. From his results, he concluded that many of our stars may be dark bodies. In 1933, Fritz Zwicky reached a similar conclusion by applying the virial theorem to the Coma Cluster in an attempt to estimate its mass, finding a result 400 times larger than the visible mass. During the last decades, much evidence for dark matter has been accumulated from measurements of galaxy rotation curves, velocity dispersions, galaxy clusters, and gravitational lensing.

Many dark matter candidates have been proposed during the last decades. A long- leading paradigm is in the form of Weakly Interacting Massive Particles (WIMPs), as the dark matter self-annihilation cross section is roughly consistent with a weak- scale massive particle interacting via the electroweak force. Many beyond Standard Model theories, such as Supersymmetry [4] or models containing extra dimensions, include WIMP-like candidates. However, the absence of observation of WIMPs so far has motivated the exploration of alternative models, such as light dark matter and the existence of a dark sector.

The dark sector (also known as hidden sectors) is a collection of particles that do not interact directly with SM particles, i.e. the dark sector fields are singlets under the SM gauge groups. They are motivated by many BSM theories, for example, in string theory constructions and type-II compactifications. The dark sector has its own symmetries, which could be arbitrarily complex. If the dark sector contains an extra U(1) symmetry, the dark and visible sectors are indirectly connected via the so-called vector portal, namely they can interact via kinetic mixing between the dark gauge field (A⁰_µ) and the SM hypercharge fieldF^µν:

F^µν,YF_µν,D F_µν,D = ∂µA⁰_ν−∂νA⁰_µ,

with denoting the mixing strength, as shown in Fig. 6.1. Values of mixing strength in the 10⁻¹²−10⁻³range have been predicted in the literature [5, 6, 7]. The kinetic mixing term can be removed by redefining the vector field

A⁰_µ → A⁰_µ, Aµ→ Aµ+A⁰_µ,

so that SM fermions pick up a small dark charge ∼ e leading to "milli-charged"

interaction between the two sectors. The dark photon can be massive, and its mass can arise via the Higgs or the Stueckelberg mechanisms. Its mass range is predicted to be in the MeV-GeV range [8, 9, 10, 11], though much smaller (sub-eV) masses are also possible [12].

Besides the vector portal, there are a few other indirect interactions that can connect the dark sector to the SM, including the scalar and Fermion portals. In this analysis we focus on the vector portal.

Dark sector

DM

!(1)′

SM

!" 3 ×!" 2 ×"(1)

&′

!"_!"#^#!"

Figure 6.1: The dark sector scenario under study: The dark sector has an additional U(1)⁰group and the corresponding dark photon A⁰(γ⁰in the picture) plays a role as intermediate messenger between SM and dark sector (hidden sector).

6.2 Dark Matter Bound States Model

The possibility of dark sector bound states was first proposed by An et al.[13]. A specific model contains a Dirac dark matter field (χ) charged under an additional U(1) gauge group in the dark sector, with the corresponding vector boson acting as a mediator between the dark sector and SM via kinetic mixing. The mass of dark matter particles isO(few GeV). The Lagrangian for this model is

L =L_SM+ χiγ¯ ^µ(∂µ−ig_DA⁰_µ)χ−mχχ χ¯

− 1

4A⁰_µνA^0µν−

2FµνA^0µν + 1

2m²_A⁰A⁰_µA^0µ, (6.1) where A⁰_µ is the vector boson mediator with field strength A⁰_µν, Fµν is the SM hypercharge field, is the kinetic mixing strength, and α_D = ^g₄^2D_π is the strength of the dark electromagnetic interaction. When g_D is sufficiently large, the force between the dark fermions mediated becomes attractive, resulting in the formation of dark matter bound states (χχ¯).

The two lowest (1S) bound states,¹S₀and³S₁, are respectively denotedη_D andΥ_D in analogy with the SM. The critical conditions for the existence of stable bound states has been determined numerically [14],

1.68m_A⁰ ≤ α_Dmχ.

If we assumeα_D = 0.5 andmχ = 3.5 GeV, the dark photon must be lighter than 1 GeV, as shown in Fig. 6.2. The value ofα_D has to be large enough to produce a bound state;mχhigher than 3.5 GeV has been excluded by other measurements [15, 16].

The quantum numbers of the two lowest (1S) bound states suggest the following production mechanisms at electron-positron colliders:

e⁺e⁻ → η_D +A⁰ e⁺e⁻ →Υ_D+γ.

These production processes are mediated by a mixedγ−A⁰propagator, as shown in Fig. 6.3. Theη_D(Υ_D) decays into 2 (3) pairs of dark photons, leading to multi-lepton and/or multi-quark final states.

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FIG. 2. Left: Constraint on the dark photon parameter space from the B A B AR dark Higgsstrahlung searches, adapted to the production and decay of dark bound states ⌘

D

and ⌥

D

. The solid purple curve corresponds to the current B A B AR limit for the parameters ↵

D

= 0.5, m = 3.5 GeV. The dashed purple curve shows the future reach of B-factories. Right: Current constraints on the m m

V

plane for the SIDM scenario are shown with 

²

= 10

⁷

and different values of ↵

D

. The green (blue) region is favored for SIDM solving the galactic small-scale structure problems [3] for ↵

D

= 0.3 (0.5). The combined constraints via the e

⁺

e ! (⌘

D

V, ⌥

D

) ! 3V channels are shown in thick purple curves, and the constraints via the e

⁺

e ! ¯ + 3V channel are shown in thin blue curves. Allowed regions are in the arrow direction. Assuming no SM background, the constraints via the e

⁺

e ! ¯ + 2V channel are shown in dot-dashed black curves for ↵

D

= 0.3, 0.4, 0.5 (bottom-up). The brown region is excluded by CDMSlite [37] and LUX [38]. The region m

V

. 30 MeV is ruled out by the XENON10 electron recoil analysis [39]

for ↵

D

= 0.3.

beams, the most important production channel is from the quark-anti-quark fusion, q q ¯ ! ⌥

_D

. Generalizing cal- culations of [42], the production cross section is given by

pp(n)!⌥D

= 4⇡

²

↵

²



²_D

s

X

q

Q

²_q

Z

₁

⌧

dx x

⇥ h

f

_q/p

(x)f

_q/p(n)¯

⇣ ⌧ x

⌘ + f

_q/p¯

(x)f

_q/p(n)

⇣ ⌧ x

⌘i , (10)

where ⌧ = m

²_V

/s , f

_q/p(n)

and f

_q/p(n)¯

are the relevant structure functions for this process, and Q

_q

is the quark charge in units of e . Unlike B -factories, only muonic de- cays of dark bound states, such as ⌥

_D

! 3V ! 3(µ

⁺

µ ) , constitute a useful signature, as backgrounds in other channels are likely to be too large. The multi-dark pho- ton FSR channels can also be relevant for the proton beam experiments.

Among the possible candidates of proton-on-target ex- periments, we focus our discussion on SeaQuest [43] and the planned SHiP [44] facilities. Note that only a fixed target mode of operation, rather than a beam dump mode that would try to remove prompt muons, is suit- able for the search of ⌥

_D

. Taking a point in the param- eter space, m = 2 GeV, 

²

= 10

⁷

, m

_V

= 300 MeV,

↵

_D

= 0.5 and the energy of incoming proton beam of 400 GeV, we estimate a probability of producing a

⌥

_D

decaying to 3(µ

⁺

µ ) for a 1 mm tungsten target, P = n ` ⇠ 2 ⇥ 10

¹⁷

. With O(10

²⁰

) particles on tar- get, one could potentially expect up to 2 ⇥ 10

³

six muon events. The large multiplicity of signal events gives some hope that this signal could be extracted from large num- ber of muons produced per each proton spill. Given the

current uncertainties in estimating the background, we refrain from showing the potential reach of proton ex- periments in Fig. 2, noting that in any case, it would not cover the most interesting region for SIDM, namely m

_V

< 200 MeV.

Outlook. Among the various probes of dark sectors sug- gested and conducted in recent years, only a few are sensitive to both the dark force and dark matter at the same time. We have pointed out that in case of relatively strong self-interaction, the presence of dark force greatly facilitates the discovery of the entire sector, as it leads to the formation of dark bound states, and causes dark FSR radiation that decay into multiple charged parti- cles of the SM. The existing searches at B A B AR and Belle already limit this possibility, and further advance in sen- sitivity can be made by searching for the missing energy plus pairs of charged particles.

Acknowledgement. We would like to thank Clifford Che- ung, Ying Fan, Ming Liu, Mark Wise and Hai-bo Yu for useful discussions. H.A. is supported by the Wal- ter Burke Institute at Caltech and by DOE Grant de- sc0011632. B.E. is supported by the U.S. Department of Energy (DoE) under grant DE-FG02-92ER40701 and DE-SC0011925. Y.Z. is supported by the Gordon and Betty Moore Foundation through Grant #776 to the Caltech Moore Center for Theoretical Cosmology and Physics, and by the DOE Grant DE-FG02-92ER40701, and also by a DOE Early Career Award under Grant No.

DE-SC0010255. H.A., M.P. and Y.Z. acknowledge the hospitality from the Aspen Center for Physics and the support from NSF Grant #PHY-1066293.

𝜖

𝑚

_!^!

(GeV)

Figure 6.2: Constraints on the dark photon parameter space from a re-interpretation of theBABAR dark Higgsstrahlung searches. The solid purple curve corresponds to the currentBABAR limit for the parametersα_D = 0.5, mχ = 3.5 GeV. The figure is taken from [13].

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