A facility to search for hidden particles at the CERN SPS: the SHiP physics case

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——————————————————————————–

CERN-SPSC-2015-017 (SPSC-P-350-ADD-1)

A facility to Search for Hidden Particles at the CERN SPS: the SHiP physics case

Sergey Alekhin,

^1,2

Wolfgang Altmannshofer,

³

Takehiko Asaka,

⁴

Brian Batell,

⁵

Fedor Bezrukov,

^6,7

Kyrylo Bondarenko,

⁸

Alexey Boyarsky?,

⁸

Nathaniel Craig,

⁹

Ki-Young Choi,

¹⁰

Crist´ obal Corral,

¹¹

David Curtin,

¹²

Sacha Davidson,

^13,14

Andr´ e de Gouvˆ ea,

¹⁵

Stefano Dell’Oro,

¹⁶

Patrick deNiverville,

¹⁷

P. S. Bhupal Dev,

¹⁸

Herbi Dreiner,

¹⁹

Marco Drewes,

²⁰

Shintaro Eijima,

²¹

Rouven Essig,

²²

Anthony Fradette,

¹⁷

Bj¨ orn Garbrecht,

²⁰

Belen Gavela,

²³

Gian F. Giudice,

⁵

Dmitry Gorbunov,

^24,25

Stefania Gori,

³

Christophe Grojean§,

^26,27

Mark D. Goodsell,

^28,29

Alberto Guffanti,

³⁰

Thomas Hambye,

³¹

Steen H. Hansen,

³²

Juan Carlos Helo,

¹¹

Pilar Hernandez,

³³

Alejandro Ibarra,

²⁰

Artem Ivashko,

^8,34

Eder Izaguirre,

³

Joerg Jaeckel§,

³⁵

Yu Seon Jeong,

³⁶

Felix Kahlhoefer,

²⁷

Yonatan Kahn,

³⁷

Andrey Katz,

^5,38,39

Choong Sun Kim,

³⁶

Sergey Kovalenko,

¹¹

Gordan Krnjaic,

³

Valery E. Lyubovitskij,

^40,41,42

Simone Marcocci,

¹⁶

Matthew Mccullough,

⁵

David McKeen,

⁴³

Guenakh Mitselmakher ,

⁴⁴

Sven-Olaf Moch,

⁴⁵

Rabindra N. Mohapatra,

⁴⁶

David E. Morrissey,

⁴⁷

Maksym Ovchynnikov,

³⁴

Emmanuel Paschos,

⁴⁸

Apostolos Pilaftsis,

¹⁸

Maxim Pospelov§,

^3,17

Mary Hall Reno,

⁴⁹

Andreas Ringwald,

²⁷

Adam Ritz,

¹⁷

Leszek Roszkowski,

⁵⁰

Valery Rubakov,

²⁴

Oleg Ruchayskiy?,

²¹

Jessie Shelton,

⁵¹

Ingo Schienbein,

⁵²

Daniel Schmeier,

¹⁹

Kai Schmidt-Hoberg,

²⁷

Pedro Schwaller,

⁵

Goran Senjanovic,

^53,54

Osamu Seto,

⁵⁵

Mikhail Shaposhnikov?, §,

²¹

Brian Shuve,

³

Robert Shrock,

⁵⁶

Lesya Shchutska§,

⁴⁴

Michael Spannowsky,

⁵⁷

Andy Spray,

⁵⁸

Florian Staub,

⁵

Daniel Stolarski,

⁵

Matt Strassler,

³⁹

Vladimir Tello,

⁵³

Francesco Tramontano§,

^59,60

Anurag Tripathi,

⁵⁹

Sean Tulin,

⁶¹

Francesco Vissani,

^16,62

Martin W. Winkler,

⁶³

Kathryn M. Zurek

^64,65

Abstract: This paper describes the physics case for a new fixed target facility at CERN SPS. The SHiP (Search for Hidden Particles) experiment is intended to hunt for new physics in the largely unexplored domain of very weakly interacting particles with masses below the Fermi scale, inacces- sible to the LHC experiments, and to study tau neutrino physics. The same proton beam setup can be used later to look for decays of tau-leptons with lepton flavour number non-conservation, τ → 3µ and to search for weakly-interacting sub-GeV dark matter candidates. We discuss the evidence for physics beyond the Standard Model and describe interactions between new particles and four differ- ent portals — scalars, vectors, fermions or axion-like particles. We discuss motivations for different models, manifesting themselves via these interactions, and how they can be probed with the SHiP experiment and present several case studies. The prospects to search for relatively light SUSY and composite particles at SHiP are also discussed. We demonstrate that the SHiP experiment has a unique potential to discover new physics and can directly probe a number of solutions of beyond the Standard Model puzzles, such as neutrino masses, baryon asymmetry of the Universe, dark matter, and inflation.

?

Editor of the paper

§

Convener of the Chapter

arXiv:1504.04855v1 [hep-ph] 19 Apr 2015

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1

Deutsches Elektronensynchrotron DESY, Platanenallee 6, D–15738 Zeuthen, Germany

2

Institute for High Energy Physics, 142281 Protvino, Moscow region, Russia

3

Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, Canada

4

Department of Physics, Niigata University, Niigata 950-2181, Japan

5

Theory Division, Physics Department, CERN, CH-1211 Geneva 23, Switzerland

6

Physics Department, University of Connecticut, Storrs, CT 06269-3046, USA

7

RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA

8

Instituut-Lorentz for Theoretical Physics, Universiteit Leiden, Niels Bohrweg 2, Leiden, The Netherlands

9

Department of Physics, University of California, Santa Barbara, CA 93106, USA

10

Korea Astronomy and Space Science Institute,Daejon 305-348, Republic of Korea

11

Departamento de F´ısica, Universidad T´ ecnica Federico Santa Mar´ıa and Centro Cient´ıfico Tecnol´ ogico de Val- para´ıso, Casilla 110-V, Valpara´ıso, Chile

12

Maryland Center for Fundamental Physics, University of Maryland, College Park, MD 20742, USA

13

IPNL, CNRS/IN2P3, 4 rue E. Fermi, Universit´ e Lyon 1,69622 Villeurbanne cedex, France

14

Universit´ e de Lyon, F-69622, Lyon, France

15

Northwestern University, Department of Physics & Astronomy, Evanston, IL 60208-3112, USA

16

Gran Sasso Science Institute, Viale Crispi 7, 67100 L’Aquila, Italy

17

Department of Physics and Astronomy, University of Victoria, Victoria, BC, V8P 5C2, Canada

18

Consortium for Fundamental Physics, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom

19

Bethe Center for Theoretical Physics & Physikalisches Institut der Universit¨ at Bonn, 53115 Bonn, Germany

20

Physik-Department, Technische Universit¨ at M¨ unchen, James-Franck-Straße, 85748 Garching, Germany

21

Ecole Polytechnique F´ ed´ erale de Lausanne, FSB/ITP/LPPC, BSP, CH-1015, Lausanne, Switzerland

22

C. N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY 11794, USA

23

Departamento de F´ısica Te´ orica and Instituto de F´ısica Te´ orica, IFT-UAM/CSIC, Universidad Aut´ onoma de Madrid, Cantoblanco, 28049, Madrid, Spain

24

Institute for Nuclear Research of the Russian Academy of Sciences, Moscow 117312, Russia

25

Moscow Institute of Physics and Technology, Dolgoprudny 141700, Russia

26

ICREA at IFAE, Universitat Aut` onoma de Barcelona, E-08193 Bellaterra, Spain

27

Deutsches Elektronen-Synchroton (DESY), Notkestrasse 85, D-22607 Hamburg, Germany

28

Sorbonne Universit´ es, UPMC Univ Paris 06, UMR 7589, LPTHE, F-75005, Paris, France

29

CNRS, UMR 7589, LPTHE, F-75005, Paris, France

30

Niels Bohr International Academy and Discovery Center, Niels Bohr Institute, University of Copenhagen, Bleg- damsvej 17, DK-2100 Copenhagen, Denmark

31

Service de Physique Th´ eorique, Universit´ e Libre de Bruxelles,Bld du Triomphe, CP225, 1050 Brussels, Belgium

32

Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, 2100 Copen- hagen, Denmark

33

Instituto de F´ısica Corpuscular (IFIC), CSIC-Universitat de Val` encia Apartado de Correos 22085,E-46071 Va- lencia, Spain

34

Department of Physics, Kiev National Taras Shevchenko University, Glushkov str. 2 building 6, Kiev, 03022, Ukraine

35

Institut f¨ ur theoretische Physik, Universit¨ at Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany

36

Department of Physics and IPAP, Yonsei University, Seoul 120-749, Korea

37

Massachusetts Institute of Technology, Cambridge, MA 02139, USA

38

Universit´ e de Gen` eve, Department of Theoretical Physics and Center for Astroparticle Physics (CAP), 24 quai E. Ansermet, CH-1211 Geneva 4, Switzerland

39

Department of Physics, Harvard University, Cambridge, MA 02138, USA

40

Institut f¨ ur Theoretische Physik, Universit¨ at T¨ ubingen, Kepler Center for Astro and Particle Physics, Auf der Morgenstelle 14, D-72076 T¨ ubingen, Germany

41

Department of Physics, Tomsk State University, 634050 Tomsk, Russia

42

Mathematical Physics Department, Tomsk Polytechnic University, Lenin avenue 30, 634050 Tomsk, Russia

43

Department of Physics, University of Washington, Seattle, Washington 98195, USA

44

University of Florida, Gainesville, USA

45

II. Institut f¨ ur Theoretische Physik, Universit¨ at Hamburg, Luruper Chaussee 149, D–22761 Hamburg, Germany

46

Maryland Center for Fundamental Physics and Department of Physics, University of Maryland, College Park, Maryland 20742, USA

47

TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T 2A3, Canada

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48

Department of Physics, Technical University of Dortmund, D-44221, Dortmund, Germany

49

University of Iowa, Iowa City, Iowa, 52242, USA

50

National Centre for Nuclear Research, Hoza 69, 00-681 Warsaw, Poland

51

1110 West Green Street Urbana, IL 61801, Dept of Physics, University of Illinois at Urbana-Champaign, USA

52

LPSC, Universit´ e Grenoble-Alpes, CNRS/IN2P3, 53 avenue des Martyrs, 38026 Grenoble, France

53

Theory Group, Gran Sasso Science Institute, Viale Crispi 7, 67100 L’Aquila, Italy

54

ICTP, Trieste, Italy

55

Department of Life Science and Technology,Hokkai-Gakuen University,Sapporo 062-8605, Japan

56

C. N. Yang Institute for Theoretical Physics, Stony Brook University Stony Brook, NY 11794 USA

57

Institute for Particle Physics Phenomenology, Department of Physics, Durham University, Durham DH1 3LE, United Kingdom

58

ARC Centre of Excellence for Particle Physics at the Terascale, School of Physics, The University of Melbourne, Victoria 3010, Australia

59

INFN, sezione di Napoli, Complesso di Monte Sant’Angelo, via Cintia, I-80126, Napoli, Italy

60

Universit´ a di Napoli “Federico II”, Complesso di Monte Sant’Angelo, via Cintia, I-80126, Napoli, Italy

61

Department of Physics and Astronomy, York University 4700 Keele Street, Toronto, Ontario, M3J 1P3, Canada

62

INFN, Laboratori Nazionali del Gran Sasso, Assergi, L’Aquila, Italy

63

Bethe Center for Theoretical Physics and Physikalisches Institut der Universit¨ at Bonn Nussallee 12, 53115 Bonn, Germany

64

Theory Group Lawrence Berkeley National Laboratory, Berkeley, CA 94709, USA

65

Berkeley Center for Theoretical Physics University of California, Berkeley, CA 94709, USA

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We are extremely grateful to Walter Bonivento, Annarita Buonaura, Geraldine Conti, Hans Dijk- stra, Giovanni De Lellis, Antonia Di Crescenzo, Andrei Golutvin, Elena Graverini, Richard Jacobs- son, Gaia Lanfranchi, Thomas Ruf, Nicola Serra, Barbara Storaci, Daniel Treille for their invaluable contributions during all stages of work on this document and in particular for their help with models’

sensitivity estimates.

We are grateful to all the members of SHiP for their dedicated work to make this experiment

possible.

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Chapter 1

Introduction

The Standard Model of elementary particle physics has provided a consistent description of Nature’s fundamental constituents and their interactions. Its predictions have been tested and confirmed by numerous experiments. The Large Hadron Collider’s runs at 7 and 8 TeV culminated in the discovery of a Higgs boson-like particle with the mass of about 126 GeV – the last critical Standard Model component [1–5]. Thus for the first time we are in the situation when all the particles, needed to explain the results of all previous accelerator experiments have been found. At the same time, no significant deviations from the Standard Model were found in direct or in indirect searches for new physics (see e.g. the summary of the recent search results in [6–25] and most up-to-date information at [26–29]). For this particular value of the Higgs mass it is possible that the Standard Model remains mathematically consistent and valid as an effective field theory up to a very high energy scale, possibly all the way to the scale of quantum gravity, the Planck scale [30–32].

However, it is clear that the SM is not a complete theory. It fails to explain a number of observed phenomena in particle physics, astrophysics and cosmology. These major unsolved challenges are commonly known as “beyond the Standard Model” problems:

B Neutrino masses and oscillations: what makes neutrinos disappear and then re-appear in a different form? Why do neutrinos have mass?

B Baryon asymmetry of the Universe (BAU): what mechanism created the tiny matter- antimatter imbalance in the early Universe?

B Dark Matter (DM) : what is the most prevalent kind of matter in our Universe?

B Cosmological inflation: What drives the accelerated expansion of the universe during the early stages of its evolution?

B Dark Energy: What drives the accelerated expansion of the universe during the present stage of its evolution?

Some yet unknown particles or interactions would be needed to explain these puzzles and to answer these questions. But in that case, why haven’t they yet been observed?

One possible answer is that the hypothetical particles are heavy and require even higher collision energy to be observed, the so-called “energy frontier ” research. Major particle physics experiments of the last few decades, including LEP and LHC at CERN, and Tevatron in the US, have followed this path.

Another possibility is that our inability to observe new particles lies not in their heavy mass, but rather in their extremely feeble interactions. If true, this would imply that a different approach to detect them should be used: an experiment needs to cross the “intensity frontier ”, rather than the “energy frontier” (Figure 1.1).

An example when a part of beyond-the-Standard Model phenomena mentioned above is resolved

by introducing relatively light new particles only is given by the νMSM (discussed in Section 4.8).

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Seminar at University of Berlin, Germany, June 6, 2014 R. Jacobsson

What about solutions to (some) these questions below Fermi scale?

Must have very weak couplings Hidden Sector

9 Inter ac tion str ength

Energy scale Known physics

Unknown physics

Energy Frontier

SUSY, extra dim.

Composite Higgs LHC, FHC

Intensity Frontier

Hidden Sector

Fixed target facility

Figure 1.1: New physics that can be explored at intensity frontier experiments and its compli- mentarily with the energy frontier

Alternatively, some of the new particles, responsible for the resolution of the BSM puzzles, can be heavy or do not interact directly with the SM sector. These “hidden sectors” may nevertheless be accessible to the intensity frontier experiments via few sufficiently light particles, which are cou- pled to the Standard Model sectors either via renormalizable interactions with small dimensionless coupling constants (“portals”) or by higher-dimensional operators suppressed by the dimensionful couplings Λ

⁻ⁿ

, corresponding to a new energy scale of the hidden sector.

For the Standard Model, renormalizable portals can be classified into the following 3 types, depending on the mass dimension of the SM singlet operator.

Dimension GeV

²

, Vector portal: new particles are Abelian fields, A

⁰_µ

with the field strength F

_µν⁰

, that couple to the hypercharge field F

_Y^µν

via

L

Vector portal

= F

_µν⁰

F

_Y^µν

, (1.0.1) where is a dimensionless coupling characterising the mixing between the new vector field with the Z-boson and the photon. The phenomenology of the vector portal is discussed in Section 2.

Dimension GeV

²

, Scalar portal: new particles are neutral singlet scalars, S

i

that couple to the square of the Higgs field |Φ|

²

:

L

Scalar portal

= (λ

i

S

_i²

+ g

i

S

i

)(Φ

^†

Φ) , (1.0.2) where λ

i

are dimensionless and g

i

are dimensionful couplings. The phenomenology of the scalar portal is discussed in Section 3.

Dimension GeV

⁵²

, Neutrino portal: the singlet operators ( ¯ L

α

· ˜ Φ) couple to new neutral singlet fermions N

I

L

Neutrino portal

= F

αI

( ¯ L

α

· ˜ Φ)N

I

.

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Here L

α

is one of the SU (2) lepton doublets, and ˜ Φ

a

=

ab

Φ

b

, F

αI

are dimensionless Yukawa couplings, for other notations see Appendix B. The phenomenology of the neutrino portal will be discussed in Section 4.

Of course, higher dimensional, non-renormalizable couplings of new particles to the SM oper- ators are also possible. An important example is provided by pseudo-scalar axion-like particles A, that couple to a dimension 4 two photon operator (Section 5) via

L

^A

= A 4f

A

^µνλρ

F

µν

F

λρ

. (1.0.3)

Yet another example is a Chern-Simons like (parity odd) interaction of electroweak gauge bosons with a new vector field V

µ

, [33]:

L

cs

=

^µνλρ

c

Y

V

µ

(Φ

^†

D

ν

Φ)(F

Y

)

λρ

+ . . . (1.0.4) (see Section 2.1.4 for details).

The goal of this paper is demonstrate the capability of high intensity proton fixed target exper- iments to discover relatively light new particles. In particular, we will show that such experiments can probe an interesting parameter space for a number of BSM models representing the portals described above. This will potentially allow for direct experimental checks of the mechanisms of matter-antimatter asymmetry of the Universe, the origin of neutrino masses, and the particle physics nature of dark matter.

This paper was prepared together with the companion document, Technical Proposal [34], that describes a concrete experiment, SHiP (Search for Hidden Particles)

¹

that was proposed in 2013 [35].

Both documents together are submitted to the SPS and PS experiments Committee (SPSC) at CERN. Therefore, in this paper we use the characteristics of the SHiP experiment (summarised in Appendix A) when estimating the potential to detect new particles. This document gives an overview of the physics, while [34] provides sensitivity estimates for selected models.

In addition we describe the sensitivity of the SHiP facility to discover new interactions between the known Standard Model particles by searching for rare processes such as τ → 3µ decays, and to study the physics of the τ -neutrino sector (detect ¯ ν

τ

, measure cross-sections and form-factors, etc.).

1http://ship.web.cern.ch

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Chapter 2

Vector portal

We review the main features and physics motivations behind vector portals between the Standard Model (SM) and dark sectors. Several case studies for the SHiP experiment are presented.

2.1 Classification of vector portals

The gauge structure of the Standard Model, the celebrated SU (3) × SU(2) × U(1) combination, is a minimal choice compatible with the chiral nature of fermions and spontaneous electroweak symmetry breaking that renders the weak bosons and fermions massive. Establishing this structure, through a combination of theoretical and experimental efforts, is one of the major achievements of 20

^th

century physics.

It is possible that the gauge structure of the SM descends from a larger gauge group, as is the case in Grand Unified Theories, or GUTs. In that case one expects that at least several of the new vector states are very heavy, e.g. m

V

∼ 10

¹⁶

GeV, well beyond the direct reach of accelerators. It is also possible that the SM is accompanied by additional gauge structures that allow for (sub-)TeV gauge bosons, as is the case of multiple U (1)’s, SU (3) × SU(2) × [U(1)]

ⁿ

[36]. The high-energy LHC experiments place very strong bounds on the possible existence of new vector states associated with new U (1) gauge groups, provided that the coupling of such vector states to the SM is sizable [37, 38]. An alternative possibility, relatively light vector states (e.g. in the GeV mass range) with small couplings to the SM, is poorly constrained by the LHC experiments and represents instead an attractive physics target for many experiments at the intensity frontier [39].

The fixed target experiment SHiP proposed at the CERN SPS [35] is a powerful tool for studying extensions of the SM based on new vector particles. This chapter discusses the physics motivations and phenomenology for such particles, provides a classification of vector portals, and points to several promising models that can be probed with proton fixed target (and beam dump) experiments. The specific examples identified as attractive physics opportunities for SHiP to be discussed in this chapter are:

• Kinetically mixed dark photons in the GeV mass range with mixing angles ∼ O(10

⁻⁶

), as well as gauge bosons coupled to baryons, V

^(B)

.

• Signatures of “dark Higgs” states produced through the vector portal and the dark Higgsstral- hung process.

• Signatures of heavy neutral leptons (HNL) produced via the vector portal.

• Signatures of sub-GeV (meta)stable states (e.g. light dark matter) via its production and

subsequent scattering via the vector portal in a neutrino-like detector.

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Throghout this chapter, we denote the new vector state as V

µ

, or simply V . We will also often employ a superscript to indicate the SM current coupled to the new vector state, e.g. V

^(B−L)

, V

^(B)

etc. Furthermore, the new (beyond the SM) states that couple to V

µ

will be generically called χ, while the new dark Higgs states associated with U (1)

⁰

group will be called h

⁰

.

2.1.1 Kinetic mixing

Consider a QED-like theory with one (or several) extra vector particle(s), coupled to the electro- magnetic current. A mass term, or in general a mass matrix for the vector states, is protected against additive renormalization due to the conservation of the electromagnetic current. If the mass matrix for such vector states has a zero determinant, det(M

_V²

) = 0, then the theory contains one massless vector, to be identified with a photon, and several massive vector states.

This is the model of ‘para-photons’, introduced by Okun in early 1980s [40], that can be reformulated in an equivalent language using the kinetic mixing portal. Following Holdom [41], one writes a QED-like theory with two U (1) groups, supplemented by the cross term in the kinetic Lagrangian and a mass term for one of the vector fields,

L = L

^ψ,A

+ L

^χ,A⁰

−

2 F

µν

F

_µν⁰

+ 1

2 m

²_A⁰

(A

⁰_µ

)

²

. (2.1.1) L

^ψ,A

and L

^χ,A⁰

are the standard QED-type Lagrangians,

L

^ψ,A

= − 1

4 F

_µν²

+ ¯ ψ[γ

µ

(i∂

µ

− eA

^µ

) − m

^ψ

]ψ, L

^χ,A⁰

= − 1

4 (F

_µν⁰

)

²

+ ¯ χ[γ

µ

(i∂

µ

− g

⁰

A

⁰_µ

) − m

^χ

]χ, (2.1.2) with F

µν

and F

_µν⁰

standing for the field strength tensors. States ψ represent the QED electron fields while states χ are charged under the ”dark” U (1)

⁰

. In the limit of → 0, the two sectors become completely decoupled. In eq. (2.1.1), the mass term for A

⁰

explicitly breaks the second U (1) but is protected from additive renormalization and hence is technically natural

¹

. Using the equations of motion, ∂

µ

F

µν

= eJ

_ν^EM

, the interaction term can be rewritten as

−

2 F

µν

F

_µν⁰

= A

⁰_µ

× (e)J

µ^EM

, (2.1.3) showing that the new vector particle couples to the electromagnetic current with strength, reduced by a small factor . The generalization of (2.1.1) to the SM is straightforward, by subsituting the QED U (1) with the hypercharge U (1) of the SM.

There are a multitude of notations and names referring to one and the same model. We shall refer to the A

⁰

state as the ”dark photon”. It can also be denoted as V

^{(Y )}

, a vector state coupled to the hypercharge current. We choose to call the mixing angle , and throughout this chapter assume 1. In contrast, one does not have to assume a smallness of g

⁰

coupling, which can be comparable to the gauge couplings of the SM, g

⁰

∼ g

^SM

.

Although the model of this type is an exceedingly simple and minimal extension of the SM, one can already learn a number of instructive features:

1. The mixing parameter is dimensionless, and therefore can retain information about loops of charged particles at some heavy scale M without power-like decoupling. In the simplest example, a new fermionic field charged under both U (1)’s will generate an additional contri- bution to the mixing angle that scales as ∆ ∼ g

⁰

e/(12π

²

) × log(Λ

²U V

/M )

²

. Alternatively, the

1

When breaking of U (1)

⁰

is triggered by a Higgs mechanism, there can be an additional ”gauge hierarchy” issue

related to the naturalness of the h

⁰

mass term.

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γ ǫ

A

^′

e

χ

Figure 2.1: The interaction through the exchange by a mixed γ − A

⁰

propagator between the SM particles and particles χ charged under new U (1)

⁰

group. In the limit of m

A⁰

→ 0 the apparent electromagentic charge of χ is e.

mixing can be generated at two or higher loop order, so that one can entertain a wide range of mixing angles.

2. If both groups are unbroken, m

V

→ 0, the states χ are “millicharged particles” with electric charge q

χ

= e. For m

V

6= 0, at |q

²

| < m

²V

, the particles χ can be thought of as neutral particles with a non-vanishing electric charge radius, r

_χ²

' 6m

⁻²V

. The diagram describing the basic interaction between the two sectors is shown in Fig. 2.1.

3. If there are no states charged under U (1)

⁰

(or they are very heavy), and m

V

is taken to be zero, then the two sectors decouple even at non-zero . This leads to the suppression of all interactions for a dark photon inside a medium. If m

V

becomes smaller than the characteristic plasma frequency all processes with emission or absorption of dark photons decouple as ∼ m

²V

[42].

4. The new vector boson interacting with the SM via the electromagnetic current conserves several approximate symmetries of the SM, including parity, flavour, and CP . Moreover, A

⁰

does not couple directly to neutrinos. As a consequence of these two features, the interaction strength due to the exchange of A

⁰

can be taken to be stronger than that of weak interactions, (e)

²

/m

²_A0

; (eg

⁰

)/m

²_A0

G

^F

. This property proves very useful in constructing light dark matter models with the use of vector portal.

Although this model was known to theorists and well-studied over the years (e.g. Refs. [43, 44]), there has been a revival of scenarios involving a kinetically-mixed A

⁰

during the last decade. Much of this activity has been in response to various astrophysical anomalies which can be interpreted as a sign of dark matter interacting with the SM through a kinetically mixing vector. Renewed interest in dark photons has triggered new analyses of past or existing experiments [45–54], and generated proposals for new dedicated experiments, which are currently at various stages of im- plementation [55–58]. In this chapter, we will demonstrate that the SHiP proposal is capable of probing new domains of the parameter space for this model, with and without light dark matter.

2.1.2 Anomaly-free gauge groups (B − L, L

^µ

− L

^τ

etc)

The kinetically-mixed portal described above represents the simplest way to couple a new vector particle to the SM, without charging any of the SM fields under the new gauge group. There is also an interesting alternative route in which certain combinations of the SM fields are charged under the new U (1)

⁰

.

The most prominent example of this type is V

^(B^−L)

, which, provided the SM is supplemented

with three right-handed neutrinos, is anomaly free. While the multitude of scales for the mass

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of B − L gauge boson is possible, the value of the coupling is quite constrained by the fact that neutrinos acquire a new interaction with electrons and baryons, so that broadly speaking we require (g

_(B−L)

)

²

/m

²_V(B−L)

< G

F

.

The only model-building complication that arises in this construction is related to neutrino masses. One can consider Dirac masses for the neutrinos, in which case this problem does not exist.

On the other hand, the Majorana masses of right-handed (RH) neutrinos, and more generally the effective operator of dimension five that generates light neutrino masses, (LH)

²

, are incompatible with B − L gauge symmetry. One can solve this problem with additional model ingredients. For example, the right-handed neutrino mass can be associated with the condensation of an additional Higgs field, Φ, that has a charge −2 under this gauge group. so that the mass term in the Lagrangian y

N

N N Φ is gauge-invariant. Then the ratio of masses of the U (1)

B−L

gauge boson and the right- handed neutrino mass would scale as

m

_V(B−L)

m

N

∼ g

^(B^−L)

y

N

. (2.1.4)

It is possible then that the masses of the RH neutrinos and the B −L gauge boson can be comparable, and the lightness of V

^(B^−L)

may in turn imply a relative lightness of N . This will be important for the phenomenological signals of the B − L interaction in fixed target experiments.

Individual lepton flavours can also be gauged in specific anomaly-free combinations. Thus, the B − 3L

ⁱ

combinations, where i is an individual flavour, have been considered in the past, and their phenomenology is very similar to the B − L case discussed above. One specific group, based on the L

µ

− L

^τ

combination, is an exception: since neither electrons nor quarks are charged under this group it is very difficult to constrain experimentally, and the strength of the new interaction can be comparable with the weak strength. For example, (g

(Lµ−Lτ)

)

²

/m

²_V

∼ G

^F

is not excluded. There is an extensive theoretical literature dedicated to such symmetries, see e.g. Refs. [59–64].

2.1.3 Other froms of vector portals.

The examples of vector portals described above are special in that they are UV complete and do not require new physics at the weak scale. In this subsection we discuss other possibilities that require additional steps for UV completion.

One example with a distinct phenomenology is based on gauged baryon number, U (1)

B

, [65–68].

This symmetry is anomalous (in particular there are mixed electroweak-B anomalies) and therefore a gauged U (1)

B

construction requires a UV completion. Such a completion can be obtained, for instance, with new electroweak charged chiral fermions that cancel the anomaly and obtain a weak scale mass through spontaneous symmetry breaking. Therefore, from the low-energy point of view, the U (1)

B

portal is legitimate, leading to rather distinct “leptophobic” phenomenology.

Individual quark flavours can also be gauged in anomaly-free combinations. For example, gauging two quark flavour q

1

− q

²

in a manner similar to L

µ

− L

^τ

is possible. The difficulty lies with quark mass sector, where additional Higgses must be introduced in order to generate mass terms consistent with the gauge groups of the SM and of the new U (1). While this may be possible from a model-building point of view, one has to address the rather severe phenomenological challenge of new tree-level flavour-changing neutral currents.

Finally, even anomalous gauge groups can be considered as low energy effective field theories.

In general, even in the presence of gauge anomalies gauge invariance can be restored at the expense

of non-renormalizability of the theory [69]. This leads to an associated energy scale at which one

loses calculability and requires a UV completion. For our purposes, it is sufficient to assume that

this scale is at the weak scale or above and then consider the effective theory of vector particles,

under the assumption that some UV completion may be found.

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On pure phenomenological grounds, one can classify the possible vector couplings to the SM fields using the framework of the effective Lagrangian where for simplicity we assume the minimal flavour violation (MFV) ansatz. The vector interactions of the MFV type [70] are given by the following combinations of the SM fields, Yukawa matrices, and unknown coefficients a

I

and b

IJ

:

L

^int

= V

µ

Qγ ¯

µ

(a

Q

1 + b

QU

Y

_U^†

Y

U

+ b

QD

Y

_D^†

Y

D

+ ...)Q + V

µ

U γ ¯

µ

(a

U

1 + b

U U

Y

U

Y

_U^†

+ ...)U

+ V

µ

Dγ ¯

µ

(a

D

1 + b

DD

Y

D

Y

_D^†

+ ...)D (2.1.5) + V

µ

Lγ ¯

µ

(a

L

1 + b

LE

Y

_E^†

Y

E

+ ...)L

+ V

µ

Eγ ¯

µ

(a

E

1 + b

EE

Y

E

Y

_E^†

+ ...)E.

At this point, it is more appropriate to think of V

µ

as of “ordinary” Maxwell-Proca field, rather than a gauge boson. Written in this form, it is easy to see that the mass m

V

is not protected at loop level, and indeed will receive additive corrections proportional to the cutoff scale. Within effective field theories, the coefficients a

I

and b

IJ

cannot be fixed from first principles, but instead can be constrained directly from experiment. Their smallness guarantees that the cutoff can be taken at the TeV scale or higher.

So far we have primarily discussed dimension four vector portals, but the non-renormalizable part of (2.1.5) essentially descends from higher-dimensional operators. It is clear that at dim> 4, one can construct many new forms of the higher-dimensional operators. For example, one can have a fully gauge invariant dipole portal,

L

^dipole

= X

ij

f ¯

i

(1 × µ

^ij

+ γ

5

× d

^ij

)σ

αβ

f

j

V

αβ

+ (h.c.). (2.1.6)

Here, f

i

represents different SM fermions, V

αβ

is the field strength of the exotic vector state, and µ

ij

and d

ij

are complex-valued Wilson coefficients. This generalizes the dipole operator discussed in [71]. Due to the explicit gauge invariance, the mass of the vector boson V does not receive corrections from (2.1.6) and can be small. Also, the SU (2) × U(1) properties of the SM fermions implies that the actual dimension of such operators is six, with µ and d scaling as ∼ v

^EW

/Λ

²_UV

. The phenomenology of this portal, including the production and decay of dipole-coupled vectors, has not been studied in any detail in the literature. We note in passing that the production and decay of V via (2.1.6) can look similar to the case of the axion-like particle, (∂

µ

a) ¯ f

i

γ

µ

f , due to the same scaling with the momentum of the exotic particle.

2.1.4 Chern-Simons portal

Motivation. Another way to couple a new vector particle V

µ

to the SM is given by the so-called effective Chern-Simons interaction (or the Chern-Simons portal ):

L

^cs

= c

Z

^µνλρ

Z

µ

V

ν

∂

λ

Z

ρ

+ c

γ

^µνλρ

Z

µ

V

ν

∂

λ

A

ρ

+ c

W

^µνλρ

W

µ

V

ν

∂

λ

W

ρ

(2.1.7) where A

µ

, Z

µ

, W

µ

stand for the photon, W

^±

and Z-boson fields, and c

Z

, c

γ

, c

W

are some dimension- less coefficients. In terms of fields A, Z and W the operators (2.1.7) are “dimension 4 operators”.

The SU (2) gauge invariant form of these interaction (where Z

µ

= Φ

^†

D

µ

Φ, etc.) requires, however, to consider them as higher dimension operators (dimension-6):

L

^cs

= 1

Λ

²_Y

^µνλρ

Φ

^†

D

µ

ΦV

ν

F

_Y^λρ

+ 1

Λ

²_su(2)

^µνλρ

ΦF

_W^λρ

D

µ

Φ

^†

V

ν

, . . . (2.1.8)

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where F

Y

and F

W

is the hypercharge and the SU(2) field strengths correspondingly and the Λ’s are new scales that depend on the origin of the effective operators (2.1.8). The coefficients in (2.1.7) are expressed as the ratios of these scales to the Higgs vev: c

γ

= cos θ

W v²

Λ²_Y

, c

Z

= sin θ

W v² Λ²_Y

and c

W

=

_Λ^v2²

su(2)

which fixes c

Z

/c

γ

= tan θ

W

. It is possible however to write an operator of dimension 8 that would also lead to V Z∂Z term. Therefore in what follows we consider c

Z

and c

γ

as two independent dimensionless parameters.

Such generalized Chern-Simons interactions appear in various models (see e.g. [33, 72–81]).

They can appear, for example, if a mixed gauge anomaly with respect to the SM gauge field and a new gauge symmetry related to V , is cancelled in a non-trivial way between new chiral heavy particles. This is similar to the way the U (1)

Y

× SU(2)

²

and U (1)

³_Y

gauge anomalies are cancelled between quark and leptonic sectors. In this respect, the Chern-Simons interaction is similar to the so-called D’Hoker-Farhi interaction [82, 83], describing the contribution of the top quark at the energies below its mass (but above the masses of all other SM fermions). As without the top quark the gauge current in the SM would not be conserved, there is a contribution at E < m

top

that is not suppressed by the mass of the top (in fact, does not depend on it at all). The interactions (2.1.7) are analogs of the DF term , but involving also a new vector field V . If the UV model that leads to the effective interactions (2.1.8) becomes ill-defined as |Φ| → 0 (see e.g. [ 33]), one can write Λ

²_Y

→

^|Φ|²c^{cos θ}_γ ^W

(similarly for Λ

²_su(2)

) and the terms (2.1.7) become true operators of dimension- 4. Therefore, similarly to the case of kinetic mixing, the “non-decoupling” of the anomalies can be viewed as an additional window into the deep UV physics. Other mechanisms leading to the effective Chern-Simons interaction can appear in the models with extra dimensions, models with

“anomaly inflow”, string theory inspired models.

Existing bounds. The bounds on c

Z

, c

W

, c

γ

in the range of masses m

V

< few GeV come from the possible contributions of the Chern-Simons interactions to the Z or W total width. This contributions is dominated by the longitudinal component of V -boson:

Γ(Z → γV ) = c

²_γ

cos θ

W

M

W

96π

M

_Z²

m

²_V

+ 1

; Γ(W

⁺

→ Xu ¯ d) ≈ c

²_W

α

W

M

W

432π

²

M

_W²

m

²_V

(2.1.9) where α

W

is the weak coupling constant and θ

W

is the Weinberg’s angle. Similar formula exists for Γ(Z → Z

^∗

X). We are interested in the V bosons that can travel cτ

V

60 m (decay volume of the SHiP detector) and having mass m

V

< 5 GeV. For m

V

M

^Z

the bounds go c

²_V

/m

²_V

< const where c

V

is one of the c

γ

, c

Z

or c

W

constants. These bounds are roughly at the level c

²_Z

, c

²_W

. 10

⁻³ _{1 GeV}^m^V

2

. In case of c

γ

a significantly stronger bound comes from the measuring of the single photon events at LEP [84]. There the branching at the level Br < 10

⁻⁶

was established for photons with the mass above 15 GeV. This leads to the strong bound c

²_γ

. 10

⁻⁹ _{1 GeV}^m^V

2

The main production modes of V

µ

depending on each of the three terms in (2.1.7) are listed below:

• For m

^V

< m

D

the dominant source of production is via weak decays of the mesons (such as D

^±

→ W

^{∗ cs}

→ ρ

^±

+ V or D → W

^{∗ cs}

→ ` + ¯ν + V where ρ

^±

is the vector meson, ` is one of the leptons (e, µ) and ¯ ν is the corresponding flavour of neutrino. These processes are controlled by the c

W

constant (the matrix element for the process D

^±

→ W

^{∗ cs}

→ π

^±

+ V is zero due to antisymmetric nature of the vertex (2.1.7))

• At masses m

^V

< m

J/ψ

channels of production via Z bosons J/ψ → Z

^∗

→ γ + V (controlled

by the c

γ

constant) or J/ψ → Z

^∗

→ Z + V (controlled by the c

^Z

constant) with the subsequent

decay of Z to a fermion-antifermion pair. This is the main channel for m

D

< m

V

< m

J/ψ

or in the

situations when c

W

c

^Z

, c

γ

.

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V W

e ⁺

e ⁻ ν _e

W

Figure 2.2: Loop decay of V boson to a pair of fermions, mediated by the c

W

term in the Chern- Simons interaction (2.1.7). A similar process with c

Z

coupling and two Z bosons running in the loop also exist.

• Additionally, for m

^V

below the masses of light electromagnetically decaying mesons π

⁰

, η, ω new channels open, governed by the c

γ

vertex: π

⁰

→ γγ

^∗

with the off-shell photon γ

^∗

decaying to V and a fermion-antifermion pair.

The decay modes of V boson again depend on the relative size of c

γ

, c

Z

, c

W

coefficients.

• Naively, the decay via c

^γ

vertex is only singly weak suppressed and therefore should dominate.

There are many interesting experimental signatures for decays V → γh

⁰

where h

⁰

is a neutral meson (π

⁰

, η, ρ

⁰

). The same vertex governs the decay modess V → γ + f + ¯ f , suppressed by the extra phase space factor). However, due to the strong bounds from LEP on the c

γ

the number of events due to c

γ

vertex is expected to be well below 1 at all masses, even for maximally efficient production (maximal values of c

W

, c

Z

constants allowed based on the LEP results).

• The decay via c

^W

or c

Z

are double weak suppressed. Among the tree-level processes the dominant one is V → π + ρ (where π is a pseudo-scalar meson and ρ is a vector meson. (Decay of V to two pions is impossible due to the antisymmetric structure of the interaction (2.1.7)). Additional 3-body decays are strongly suppressed by the phase-space factor. However, a loop-mediated decay V → f + ¯ f (Fig. 2.2) dominates over the tree-level processes due to “compensation” of one of the W or Z propagators in the loop.

The resulting number of “detector events” can be as large as few thousands (via the loop-mediated process producing a pair of fermions) for masses O(1) GeV and c

^Z

, c

W

of the order of their maximal values, allowed from the LEP bounds.

2.2 Matter states charged under new U (1)

The new vector portal is likely to be accompanied by new states charged under U (1)

⁰

. In this section we review different theoretical constructions behind the finite mass for the new vector bosons and discuss the possibility that the new U (1)

⁰

sector is supersymmetric.

2.2.1 Higgs mechanism in the dark sector

There are several theoretical ways of breaking the U (1)

⁰

symmetry. In all UV-complete models, the new vector particle couples to the conserved current, which allows for the introduction of a

”hard” or ”Stuckelberg” mass term, L

^m

=

¹₂

m

²_V

V

_µ²

. This represents the most minimal possibility.

If, however, the additional gauge symmetry is broken in the same way as in the SM, a new Higgs

field in the dark sector will be introduced.

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The “Higgsing” of the U (1)

⁰

introduces extra interactions in the dark sector. Consider the simplest possibility in which the new scalar field is charged under U (1)

⁰

[85],

L

^φ

= |D

^µ

φ |

²

− λ|φ|

⁴

+ µ

²

|φ|

²

, (2.2.1) where D

µ

is the covariant derivative. After the spontaneous symmetry breaking, φ = (v

⁰

+ h

⁰

)/ √

2, the vector field becomes massive with m

V

= g

⁰

v

⁰

. Besides the mass terms, several interaction terms appear:

L

^h⁰

= 1

2 m

²_V

V

_µ²

+ 1

2 (∂

µ

h

⁰

)

²

− 1

2 m

²_h⁰

h

⁰²

+ m

²_V

v

⁰

h

⁰

V

_µ²

+ m

²_V

v

⁰²

h

⁰²

V

_µ²

− m

²_h⁰

2v

⁰

h

⁰³

− m

²_h⁰

8v

⁰²

h

⁰⁴

, (2.2.2) where m

h⁰

is the mass for the dark Higgs particle. Depending on the relative sizes of the quartic coupling λ and the square of the gauge coupling g

⁰²

the dark Higgs can be lighter or heavier than the vector. This will become important for the potential signals of dark sector as m

h⁰

≤ m

^V

opens the possibility for a macroscopic decay length of the dark Higgs [85, 86].

A more complicated Higgs sector can lead to an additional important effect for a dark photon beyond the kinetic mixing. In particular, a mass mixing between V and the SM Z boson becomes possible [87], opening a host of additional phenomenological consequences such as parity violation mediated by V exchange and the coupling of V to the SM neutrinos.

2.2.2 Supersymmetric U (1)

⁰

models

Supersymmetric extensions of vector portal models are also well-studied. For a SUSY version of the dark photon model, see e.g. [88–91]. One can imagine a variety of scenarios vis-a-vis the origin of supersymmetry breaking that could lead to different mass patterns between the observable and dark sectors. One of the most interesting possibilities includes a scenario with an approximately supersymmetric dark sector in which the breaking of supersymmetry in the U (1)

⁰

sector is mediated by the vector portal coupling from the SM. In the simplest scenarios of this kind, one expects the following relations to hold true:

m

²_A0

∼

^1/2

M

Z

; m

A⁰

= m

h⁰

. (2.2.3) The first relation arises because of the induced D-term in the U (1)

⁰

sector that scales as ∼ [ 90].

The second relation is the consequence of the minimal Higgs sector in the SUSY version of the dark photon model [91]. Equal masses for h

⁰

and A

⁰

particles forbid the decays of h

⁰

to A

⁰

and prolong the lifetime of h

⁰

, thus making supersymmetric dark Higgses an interesting target for the studies at SHiP. More details on supersymmetric scenarios in conjunction with the light vector portal can be found in the SUSY chapter of this white paper, Chapter 6.

2.3 Physics motivation for light mass (less than weak scale) vector par- ticles

The interest in vector portals comes from several sources. The appearance of such light vector portals is quite common in top-down constructions, such string-inspired models and GUT theories.

At the same time, several variants of the vector portal models, especially with relatively light

mediators, have been invoked as a remedy for a number of observational anomalies, both in particle

physics and astrophysics. This section reviews a subset of the problems that vector portal models

help to resolve.

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γ

A

^′

e

ǫ ǫ

Figure 2.3: One-loop correction to the muon anomalous magnetic moment due to the exchange of the dark photon.

2.3.1 Putative solution to the muon g − 2 discrepancy

The persistent discrepancy between the measured muon anomalous magnetic moment and the SM prediction at the level of ∼3σ [ 92] has generated significant experimental and theoretical activity aiming for a possible explanation. The intense scrutiny of the SM contributions has not produced any obvious candidate for an extra contribution ∆a

µ

∼ +3 × 10

⁻⁹

that would cover a theoretical shortfall and match the observed value. Among the new physics explanations for this discrepancy are weak scale solutions [93], as well as possible new contributions from light and very weakly coupled new particles (see, e.g., [47, 86, 94]). With the LHC continuously squeezing the available parameter space for the weak-scale g − 2-relevant new physics, solutions with light particles appear as an attractive alternative.

It is easy to see that a light vector particle coupled to muons via the vector portal provides an upward correction to g − 2. In most models the new vector particle does not have an axial-vector coupling to charged leptons, and a simple one-loop diagram, Fig. 2.3, gives a positive correction to the magnetic anomaly,

a

^V_l

= α 2π

g

⁰

e

2

× Z

1

0

dz 2m

²_l

z(1 − z)

²

m

²_l

(1 − z)

²

+ m

²_V

z = α 2π

g

⁰

e

2

×

1 for m

l

m

^V

,

2m

²_l

/(3m

²_V

) for m

l

m

^V

. (2.3.1) In this expression g

⁰

/e is the strength of the V

µ

coupling to the muon vector current in units of electric charge. For the kinetically-mixed dark photon A

⁰

, we have g

⁰

/e = . For the choice of

∼ few × 10

⁻³

at m

V

∼ m

^µ

, the new contribution brings theory and experiment into agreement.

Since 2008, significant experimental advances have been made towards testing this possibility, while at the same time various theoretical extensions of the simplest kinetic mixing portal explanation have been proposed. The following picture has emerged:

• The minimal dark photon model, with no light particles charged under U(1)

⁰

is excluded (or at least very close to being excluded) by a complementary array of experiments. The most difficult part of the parameter space, the vicinity of m

A⁰

∼ 30 MeV, has finally been ruled out only recently as a solution to the g − 2 puzzle, [ 52, 54].

• A slightly extended model of the dark photon can still offer a solution to the g −2 discrepancy.

For example, if A

⁰

→ χ ¯ χ in the dark sector the visible A

⁰

→ e

⁻

e

⁺

decays will be diluted. In any case, it appears that m

A⁰

< 200 MeV is required [95].

• Finally, the least constrained model is based on the gauged L

^µ

− L

^τ

vector portal [61, 62, 64],

and the vector mass below m

V

∼ 400 MeV can still be considered as a potential solution to

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A

^′

ǫ γ χ

χ

^∗

e

⁻

e

⁺

Figure 2.4: Light (m

χ

∼ few MeV) scalar dark matter annihilating to electron-positron pairs due to the mixed γ − A

⁰

propagator. The annihilation occurs in the p-wave.

the muon g − 2 discrepancy [ 96, 97].

To summarize, a light vector particle remains an attractive solution to the muon g − 2 discrep- ancy, and more experimental work is required to exclude this possibility in a model-independent way.

2.3.2 Mediator of interaction with DM and possible connection to astrophysical positron excess

Vector portals offer a means to connect the SM to dark matter. In the last few years, the di- rect searches for dark matter have intensified, paralleled by broad investigations of the theoretical opportunities for dark matter. The weakly interacting massive particle (WIMP) paradigm offers perhaps the largest number of opportunities for the experimental discovery of dark matter via its non-gravitational interaction. In the standard WIMP paradigm, known from 1970s [98, 99], the correct cosmological abundance of dark matter is achieved via its self annihilation at high temper- atures, T ∼ m

^χ

, where m

χ

is the WIMP mass. Simple calculations show that the required WIMP abundance is achieved if

σ

annih

(v/c) ∼ 1 pbn =⇒ Ω

^DM

' 0.25, (2.3.2) where v/c is the approximate relative velocity at the time of annihilation. The nature of the interaction responsible for the self-annihilation of WIMPs to the SM states is important. It sets the size of the self-annihilation cross section and ultimately the abundance of WIMP dark matter. If the interactions are mediated by forces that have weak strength and operate through the exchange of weak scale particles, then for small and large masses one expects the following scaling with the WIMP mass,

σ(v/c) ∝

G

²_F

m

²_χ

for m

χ

M

^W

,

1/m

²_χ

for m

χ

M

^W

. = ⇒ few GeV < m

^χ

< few TeV (2.3.3) This famously determines the so-called ”Lee-Weinberg window”, or the mass range for the DM under the assumption of weak-scale mediators. According to this logic thermal relic MeV-GeV scale dark matter is disfavored.

The crucial assumption in the argument above is the link between the weak scale and the

mass of the mediator particles. As was argued in the previous sections certain vector portals do

allow interaction strengths in excess of G

F