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Identifying a New Gauge Structure of Dark Matter at the LHC
Myeonghun Park
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DOI: 10.22661/AAPPSBL.2018.28.5.39

Identifying a New Gauge Structure of Dark Matter at the LHC

MYEONGHUN PARK,1 & 2 *
1 INSTITUTE OF CONVERGENCE FUNDAMENTAL STUDIES AND SCHOOL OF LIBERAL ARTS,
SEOUL NATIONAL UNIVERSITY OF SCIENCE AND TECHNOLOGY
2 CENTER FOR THEORETICAL PHYSICS OF THE UNIVERSE, INSTITUTE FOR BASIC SCIENCE (IBS)

From power spectrum analyses of the cosmic microwave background observation, we know that particles in the Standard Model (SM) contribute about 16% out of the total matter density of our Universe. To explain the unknown 84%, non-baryonic dark matter was introduced. With extensive searches for dark matter in various experiments, it would be interesting to determine the structure of the gauge group in the dark sector and its phenomenological effects on observations. This raises another important question: how does dark matter gains its mass? As we know that particles in the SM get mass from interaction with the Higgs particle, it is natural to consider a new gauge structure in the dark sector in terms of dark matter mass. In this article, we cover distinct features, depending on dark gauge structures, of the dark matter signature at the Large Hadron Collider (LHC).1

INTRODUCTION

After the Higgs particle was discovered at the LHC, we were finally able to understand the Standard Model (SM) of a particle physics theory. The core of the SM is its gauge structure, an internal symmetry on the quantum mechanical description of underlying particles (Dirac fermions). Through experimental observations, weak gauge bosons that are responsible for the isospin violation should be massive. The Higgs mechanism was designed to break a weak gauge group of the SM, which makes corresponding gauge bosons become massive. Not only do weak gauge bosons become massive from the Higgs mechanism, but also Dirac fermions in the SM get their mass through an interaction with the Higgs field. Thus, we would be tempted to conclude that the Higgs would be the origin of SM particles' mass.

But in the SM, there are two other mechanisms for a particle to be massive. First, masses of baryons, like protons and neutrons, are not from the Higgs mechanism. If we add up the constituent particles' mass inside a proton, we only obtain just 1% of the proton's mass. The mass of baryons comes from the nature of strong gauge symmetry on a quark sector. The second Higgs-independent mass is the mass of neutrinos. In the SM, we don't have a mechanism to explain the nature of a neutrino's mass nor the existence of a neutrino's SU(2)L partner, a right-handed neutrino.

Thus, the origins of particles' mass depend on their gauge group structure: Dirac fermions and weak gauge bosons become massive due to the Higgs mechanism, which breaks a weak gauge symmetry; baryons are massive due to the nonperturbative nature of a strong gauge group; and the nature of the mass of neutrinos is still a mystery, as anti-neutrino behavior and neutrino behavior under SM gauge groups are the same.

As we mentioned earlier, particles in the SM are only responsible for a small portion of the total gravitational attractions in our Universe. Among various possibilities for these unknown gravitational effects, one of simplest solutions that has been strongly supported by various observations is found in dark matter particles. Thus, the problem of determining the unknown gravitational source of our Universe is associated with the task of identifying dark matter.

There have been various dark matter searches in all directions, including dark matter direct detection (DD) experiments [1, 2], indirect searches and collider experiments including the LHC [3]. Most DD experiments have focused on the case of weakly interacting massive particles (WIMPs) as a WIMP scenario has been supported strongly by supersymmetry theory. Thus, DD experiments will cover most of the mass range of WIMPs, from 𝒪(100) GeV up to a sensitivity limit of irreducible backgrounds from neutrino-nucleus coherent scattering [7].

Recently, a new possibility for light dark matter has been spotted [8, 9, 10]. As current DD experiments have very weak sensitivity in light dark matter cases of sub-GeV mass scale due to low nuclear recoil energy, ENR < 𝒪(0.1) KeV, over experimental resolutions and noises [4], detecting sub-GeV dark matter has become an interesting area for expansion in dark matter search programs. Here, we want to highlight the fact that the LHC experiment has good sensitivity in detecting light dark matter due to the precise understanding and use of quantum chromodynamics (QCD) in controlling experimental environments, including backgrounds. Theoretically, light dark matter phenomenology has generated various interesting ideas, including the possibility of interactions among dark matter particles to satisfy the observed relic density in our Universe. To consider interactions between dark matter, it is natural to think about a gauge group in the dark sector that introduces a force between "dark-charged" dark matter particles2.

In this article, we review our recent works in the LHC collider phenomenology of a new gauge group for the dark sector [5, 6]. We divide new gauge groups into two categories, depending on whether there would be a non-perturbative physics. If a new gauge group develops non-perturbative physics like QCD in the SM, there would be the possibility for dark matter to be a composite particle. This would introduce two important characteristics for dark matter: 1) that the stability of dark matter would be due to "baryon number" conservation; and 2) the mass of dark matter would arise from the nature of strong dynamics. To illustrate our point of combining collider phenomenology with non-perturbative physics, we consider an SU(3)d gauge group. In a case of perturbative physics, we take the example of a U(1)d abelian gauge group with a dark Higgs to break a gauge symmetry.

Once light dark matter particles are produced at the LHC, they will be highly accelerated due to the large production energy from the LHC. As dark matter is charged under a new gauge group, they will radiate corresponding gauge particles, similar to the case of accelerated electrons that radiate photons. We will show that by examining the energy deposit patterns from the radiation, we can see structures of gauge interactions among dark matter, which eventually will allow us to understand the origins of dark matter mass.

U(1)d ABELIAN DARK GAUGE

The observation of parity violation in the decay of cobalt-60 led to our ability to understand the nature of a weak gauge symmetry; particles that are charged under weak gauge symmetry are chiral. Fermions, which are building blocks of the SM, are chiral under a weak gauge symmetry; the charge of a right-handed fermion is different from the charge of the "same" left-handed fermion. The Higgs mechanism provides a natural way to endow chiral fermions with mass. If a fermion is vector-like, where the right-handed fermion has the same charge as a left-handed fermion, there is no need to introduce a Higgs field for a fermion to be massive.

When we consider the possibility of U(1)d gauge symmetry in the dark sector3, it is natural to check whether dark matter is chiral or vector-like under this gauge symmetry. If dark matter is chiral, then a Higgs field would be necessary for dark matter to be massive.

A corresponding a U(1)d gauge boson, called a dark photon, is preferable in terms of creating mass because otherwise dark matter would have to be electrically charged from a charge under the SM U(1)y. This is due to kinematic mixing between SM U(1)y and dark U(1)d. As an electric charge on dark matter is highly constrained based on observations, here we consider a massive dark photon. Interaction Lagrangians in a dark sector are described below:

(1)

(2)

with Dμ μ+ig'Q'A'μ where A'μ is the quantum field of a dark photon γd, g' is the gauge coupling of the dark gauge symmetry, and Q' is the dark U(1)d charge. Fμν and F'μν are the field strengths of the SM photon and dark photon respectively, and ε is the kinetic mixing parameter [11]. Here, we introduce a dark Higgs, Φ, to break U(1)d. The Yukawa terms in eq. (2) dictate the relations of the dark U(1)d charges:

(3)

The non-zero charge Q'Φ of a dark Higgs induces the chiral nature of dark matter. When we rewrite the interaction between dark matter and a dark photon,

(4)

the axial Q'A and vector coupling Q'v are written as

(5)

(6)

The chirality of a dark matter particle (Q'A≠0) results in different dark photon showering and corresponding signatures at the LHC. With a dark Higgs, the mass of a dark photon is given by mγd=g'Q'ΦvS and the mass of dark matter with mχ=yχvS /√2, where vS is a vacuum expectation value of the dark Higgs. When dark matter is vector-like; Q'χR=Q'χL, an axial coupling Q'A becomes 0 and the physics of dark matter can be simply given without a dark Higgs:

(7)

A. The pattern of dark photon showering at the LHC
A showering process that describes a radiated gauge boson from an accelerated charged particle is characterized by a splitting function Pχχγd. In a collinear region, the differential probability of the splitting process χχγd is

(8)

Here, α'=g'2/4π, t is the virtuality of incoming χ, and x is the energy fraction taken by outgoing χ. As we focus on the phenomenology of energetic dark matter production at the LHC, we take the limit of the splitting kernel for vector-like dark matter without m2χ /t or m2rd /t. In the case of vector-like dark matter, a splitting kernel is written as

(9)

This shower pattern is similar to the familiar QED shower where a photon has only transverse polarizations.

When dark matter is chiral under a gauge group, it couples to a dark Higgs. This leads to an interesting difference compared to vector-like dark matter. Through the Goldstone boson equivalence theorem (GBET) [12], the longitudinal polarization of a dark photon can be described with a dark Higgs. Thus, the coupling between dark Higgs and dark matter provides an interaction between the longitudinal polarization of a dark photon with dark matter. In return, accelerated dark matter radiates a longitudinal polarization of dark photons on top of transverse modes. The corresponding splitting kernel would be

(10)

In short, vector-like dark matter only radiates the transverse polarization of a dark photon while longitudinal polarization of a dark photon will be emitted from chiral dark matter. This difference can be maximized in the limit when a coupling constant α' is small. In this limit, the dark photon showering probability described by a splitting kernel in vector-like dark matter becomes negligible:

(11)

Compared to vector-like dark matter, chiral dark matter radiates dark photons even in the limit of a coupling constant α'→0:

(12)

since chiral dark matter radiates longitudinal polarization of dark photons, which is equivalent to the emission of dark Higgs from the highly accelerated dark matter at the LHC.

B. Numerical studies at the LHC
From the recent focus on dark photon search experiments [13, 14, 15, 8, 10], we take very light dark matter with mrd=0.4 GeV to avoid constraints. A dark photon will decay into SM particles through a mixing between the SM U(1)y gauge boson. In our study point, the branching ratios of the dark photon are BR(γd μ+ μ-) ≃0.45,
BR(γd e+ e-)≃0.45, and BR(γd π+ π-)≃0.10.

To examine differences in signatures at the LHC from distinct showering patterns, we checked to see how many changes occured in the number of dark photons produced at the LHC depending on the chirality of dark matter. For a numerical illustration, we took a study point of mχ=1.0 GeV with a coupling constant α'=0.15. In the upper of Fig. 2, we plotted histograms of the number of showered dark photons per event. We observed that we had more dark photon radiations in the case of chiral dark matter, as all three polarization modes of dark photon radiated off from dark matter.



Fig. 1: A generic event process diagram for accelerated dark matter particles to shower dark photons at the LHC.

Fig. 2: To show a different dark photon shower pattern, we demonstrated the number of showered dark photons and corresponding HT(ℓ) distributions with Monte Carlo simulations. The upper plot compares the dark photon number distribution between vector-like dark matter and chiral dark matter. The red line and blue line correspond to a vector-like model and a chiral model, respectively. The lower plot is for the comparison of the HT(ℓ) distribution with the same color code as cases in the upper plot.

Once a dark photon is produced from a dark showering process, it decays to SM particles through the kinetic mixing ε in Eq. (1). To quantify those signatures, we use the scalar sum of light leptons (muons and electrons) transverse momentum pT :

(13)

In the case of vector-like dark matter, there is a soft singularity with x≃1 in Eq. (11). Thus, dark photons from vector-like dark matter tends to be softer compared to dark photons from chiral dark matter. This can be observed numerically in the lower figure of Fig. 2. As leptons from the decays of dark photons inherit energy from dark photons, most of the leptons become soft in vector-like dark matter.

Fig. 3: χ2 distance of HT(ℓ) distributions between a vector-like dark matter scenario and a chiral dark matter case with 100 signal events with background, after cuts.

Finally, we checked to see if differences in dark photon showering at the LHC could be detected using integrated beam luminosity of L = 3,000fb-1. To deal with the finite luminosity of the LHC, we generated 100 reconstructed signal events, after cuts, and 100 pseudo-experiments to reduce statistical uncertainties coming from the finite statistics of simulations. We calculate a binned-χ2 of HT() distributions from a vector-like case and chiral case, including corruptions from backgrounds as we described above. Fig. 3 shows our results from the χ2 comparison. The LHC can tell us the origin of the mass of a dark matter particle by discriminating between vector-like and chiral dark matter models of more than a 2σ significance level, when the mass ratio mχ /mγd is large enough. As explained earlier, differences in signatures can be maximized as α'→0.

SU(3)d NON-ABELIAN GAUGE GROUP WITH NON-PERTURBATIVE PHYSICS

As contributions from protons and neutrons play a major part in the mass of our visible Universe, it is natural to check whether an unknown source for the invisible and massive part of our Universe comes from composite particles. For various theoretical models for composite dark matter, asymmetric dark matter models are interesting as they can also answer the baryon asymmetry of our Universe [16, 17]. In an asymmetric dark matter model, the light dark matter of 𝒪(1) GeV is preferred to address the problem of baryon genesis to the relic density of our Univese, ΩDM ≃ 5ΩB, where ΩB is the abundance of SM baryonic matters in our universe.


Nd

nf

Λd (GeV)

q' (GeV)

mπd (GeV)

πd Decay Mode

A

3

2

15

20

10

πd cc̅

B

3

6

2

2

2

πd ss̅

C

3

2

15

20

10

πd γ'γ' with mγ' = 4.0 GeV

D

3

6

2

2

2

πd γ'γ' with mγ' = 0.7 GeV


Table 1: All dark hadrons are assumed to decay promptly. We consider 2 cases: a high Λd case (A and C), and a low Λd case (B and D). Parameters are in the dark sector for A and C; B and D are the same except the decay channel of a dark pion, πd.

The SM sector can be connected to the dark sector by a mediator [18]. Due to null results at the LHC, a mediator should be massive enough to avoid current heavy resonance searches with a dijet signature [19, 20]. Through this heavy mediator, dark partons can be produced at the LHC. Once dark partons are produced at the LHC, a process of "dark" hadronization will take place and dark hadrons will be formed. Dark hadrons will decay into SM particles through a heavy mediator. Depending on the parameters of a heavy mediator, the signature of dark hadrons will be a QCD-jet like signature [6] or an exotic signature with various displaced vertices [21]. Collider phenomenology with dark hadrons is more challenging as most of the signatures are similar to the QCD background. Separation from the QCD background is made even more complicated when the lifetimes of dark mesons are too short to develop displaced vertices. Here, we point out that 1) there are interesting physics with prompt dark meson decays, and 2) it is possible to suppress SM QCD backgrounds by combining various jet substructure techniques [22].

Fig. 4: Running coupling of dark-QCD depending on models [orange and blue] and a running coupling of SM QCD with five flavor nf =5 without top-quark effect [green].

A QCD-like dark sector ("dark-QCD") can be described with dark quarks, q', and SU(Nd) gauge bosons:

(14)

where G'μν is the field strength of a dark gauge boson; here, we call it a dark gluon. A heavy mediator that connects the dark sector and the SM sector can have bi-fundamental representation under SU(NdSU(3)SM or be a singlet. For the bi-fundamental particle X, the interaction Lagrangian can be written as;

(15)

For this minimal model, where a mediator connects a dark-QCD to the SM QCD, dark mesons decay into the SM quarks, where a signature forms a QCD jet. Unlike the process of tagging leptons with various sub-detectors at the LHC, probing the substructure of a QCD jet is difficult due to low resolution in the hadron calorimeter and underlying events. Thus, previously exotic signatures have been considered.

If a dark pion can decay into heavy flavor as in Table 1, the proper decay length of dark pion could be shorter than 1mm. This decay length is not long enough to be captured with displaced vertex techniques. In the case where a dark quark is charged under U(1)d, a dark pion decays into two dark photons similar to the case where a pion decays into two photons in the SM. Thus, depending on the details of dark-QCD models, tagging signatures from dark matter over the SM would be a nontrivial task.

A. Features in jet substructures from dark-QCD
The most important factors in forming a jet signature from the hadronization process are the running coupling strength α of the gauge group and a energy scale, Λd, from which a non-perturbative physics emerge below.

At a short distance smaller than 1/Λd, parton distribution inside a jet is controlled by a parton shower with a running coupling α. Thus, the number of showered particles inside a jet is proportional to α. The renormalization group equation (RGE) of α is the following:

(16)

with a boundary condition α-1d (Λd)=0, which describes the breakdown of perturbativity. Here Nd is a number for the SU(Nd) gauge group and nf is the number of dark quarks. When Nd=2, the ratio of stable dark baryons inside a jet can be significant, which makes a signature more distinctive as compared to the SM QCD jet. Thus, for phenomenological reasons we take Nd=3 when identifying a signature is challenging.

Jet substructure variables that are sensitive to parton evolution from a parton shower become effective in discriminating dark-QCD with a large coupling, as compared to SM QCD. For example, a linear radial geometric moment (girth) that measures energy deposits inside a jet according to the distance from a jet axis is known to identify a gluon-jet (a jet initiated from a gluon) over quark-jet (a jet emerging from a quark), as the color factor of gluon CA=3 is larger compared to the color factor of the quark CF=4/3. The multiplication of color factor with a running coupling that enters into a splitting kernel describes the parton shower process. Girth is defined as:

(17)

Here, ri is the distance between a component i of a jet J and a jet axis. The performance of girth depends on the confinement scale of dark-QCD as a dark-QCD jet from a higher confinement scale (model A and model C), which is easier to be distinguished than cases from a low confinement scale (model B and model D). By comparing model A and C (and also model B and D) we find that girth is not sensitive to the different decay channel of a dark meson.



Fig. 5: Girth distribution of a jet with different kinds of dark-QCD models and SM QCD for 180 GeV<pT<220 GeV.

To further discriminate different decay patterns of dark-QCD compared to SM-QCD, charged track multiplicity can be utilized due to the high resolution and trigger efficiency of the track reconstruction at the LHC. As this variable is IRC (Infra-red and collinear) unsafe, we rely on results from a Monte Carlo Simulation as in FIG. 6.



Fig. 6: Charged track multiplicity distribution of different kinds of jets with pT ∈ (180 GeV, 220 GeV).

In our analysis, we utilize various jet substructure variables, including the above two variables. We are not showing the details of the jet substructure variables as they are too technical for this report. Instead of displaying the individual performances, we summarize the results using a BDT (Boosted Decision Tree) to maximize performance. Here we only show results with pT ∈ (180 GeV, 220 GeV). For larger pT, the performance in dark-QCD jet discrimination would be enhanced as differences in the showering pattern become significant with the large energy of a jet. For all of our model settings and the jet's pT 200 GeV, we can either exclude 99% of the background gluon jets while reserving more than 30% of the signal dark jet, or exclude 99% the background quark jet while reserving more than 50% of the signal dark jet.

Fig. 7: ROC curves from BDT analysis: dark jet vs. gluon jet ROC curves for all four models and different variables combination, with pT ∈ (180 GeV, 220 GeV).

Our results demonstrate that by considering the information inside a jet, we will achieve much better understanding of dark jets and we will be able to enhance collider search sensitivity to identify signatures of the dark-QCD model at the LHC.

DISCUSSIONS AND OUTLOOK

The core of the Standard Model is its gauge structure. Depending on the gauge structure on a particle, a different mechanism for the mass of a particle is given; for example, the Higgs mechanism for a chiral fermion and non-perturbative origin as in baryons. Thus, identifying a gauge structure on a dark matter particle will lead to understanding dark matter's mass, which remains a large and unknown portion of our Universe. Here, we utilize the dependence of showering patterns in accelerated dark matter at the LHC. By examining patterns in the showering process, (1) we can identify the chirality of dark matter for fermionic dark matter or (2) we can enhance search sensitivity over SM QCD background and detailed feature of dark-QCD in composite dark matter case.

Acknowledgments : The research projects which have been summarized here are supported by the National Research Foundation of Korea (NRF) Basic Science Research Program (NRF-2017R1C1B5075677).



* parc.seoultech@seoultech.ac.kr
1 Here, we consider a charge from a new gauge group. Thus, dark matter is still neutral under the SM gauge groups
2 Here, we consider a charge from a new gauge group. Thus, dark matter is still neutral under the SM gauge groups.
3 Here, we define a dark sector as the collection of non-SM particles including dark matter.

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Myeonghun Park is an assistant professor at Seoul National University of Science and Technology (Seoultech), and a visiting research fellow at IBS-CTPU. After receiving a Ph.D. from University of Florida, he worked at CERN theory department, Kavli IPMU, APCTP and IBS-CTPU before joining Seoultech in 2017. His research field is the phenomenology of physics beyond the Standard Model.