FERMILAB-PUB-22-670-T Freezing In Vector Dark Matter Through Magnetic Dipole Interactions Gordan Krnjaic1 2 3Duncan Rocha4yand Anastasia Sokolenko1 3z

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FERMILAB-PUB-22-670-T

Freezing In Vector Dark Matter Through Magnetic Dipole Interactions

Gordan Krnjaic,1, 2, 3, ∗Duncan Rocha,4, †and Anastasia Sokolenko1, 3, ‡

1Theoretical Physics Department, Fermi National Accelerator Laboratory, Batavia, IL 60510

2Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637

3Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637

4Department of Physics, University of Chicago, Chicago, IL 60637

(Dated: October 14, 2022)

We study a simple model of vector dark matter that couples to Standard Model particles via

magnetic dipole interactions. In this scenario, the cosmological abundance arises through the freeze-

in mechanism and depends on the dipole coupling, the vector mass, and the reheat temperature.

To ensure cosmological metastability, the vector must be lighter than the fermions to which it

couples, but rare decays can still produce observable 3γﬁnal states; two-body decays can also

occur at one-loop with additional weak suppression, but are subdominant if the vector couples

mainly to light fermions. For suﬃciently heavy vectors, induced kinetic mixing with the photon

can also yield additional two body decays to lighter fermions and predict indirect detection signals

through ﬁnal state radiation. We explore the implications of couplings to various ﬂavors of visible

particles and emphasize leptophilic dipoles involving electrons, muons, and taus, which oﬀer the

most promising indirect detection signatures through 3γ,e+e−γ, and µ+µ−γdecay channels. We

also present constraints from current and past telescopes, and sensitivity projections for future

missions including e-ASTROGAM and AMEGO.

I. INTRODUCTION

While the evidence for the existence dark matter (DM)

is overwhelming, its microscopic properties remain elu-

sive (see [1] for a historical review). Since there are few

clues about its non-gravitational interactions, it is cur-

rently not known how DM was produced in the early uni-

verse or when that production took place. Thus, there is

great motivation to identify and test all predictive mech-

anisms for this key epoch in the history of the universe.

Cosmological “freeze-in” is among the simplest and

most predictive DM production mechanisms [2,3]. In

this scenario, DM is initially not present at reheating

when the hot radiation bath is ﬁrst established. Rather,

its density builds up through ultra-feeble interactions

with Standard Model (SM) particles and production

halts when this process becomes Boltzmann suppressed.

Since these reaction rates are sub-Hubble, the DM never

equilibrates with the SM, so unlike freeze-out, there is no

need to deplete the large thermal entropy with additional

DM annihilation when equilibrium is lost. Freeze-in pro-

duction ends when the temperature of the universe cools

below either the mass of either the DM or the SM species

to which it couples, whichever is greater.

It is well known that dark photons A0can be produced

via freeze-in through a kinetic mixing interaction with

the SM photon [4]. Since A0are also unstable and decay

through this same interaction, only the mA0<2memass

∗krnjaicg@fnal.gov

†drocha@uchicago.edu

‡sokolenko@kicp.uchicago.edu

range can be cosmologically metastable to provide a DM

candidate. In this range, the DM decays via A0→3γre-

actions and predicts a late time X-ray ﬂux uniquely spec-

iﬁed by the A0mass, once the kinetic mixing parameter

is ﬁxed to obtain the observed DM abundance. However,

this tight relationship between abundance and ﬂux has

been used to sharply constrain this simple model with ob-

servations of X-ray lines, extragalactic background light,

and direct detection via absorption [4,5]. Collectively,

these probes have eliminated nearly all viable parame-

ter space for vector DM produced through freeze-in via

kinetic mixing interactions.

In this paper we generalize dark photon freeze-in to

allow for the possibility that its main interaction with

visible matter is a magnetic dipole coupling to charged

fermions, instead of kinetic mixing with the photon.

Since the dipole operator has mass dimension 5, the

freeze-in abundance is sensitive to the reheat tempera-

ture. Thus, unlike kinetic mixing, there is a parametric

separation between the production rate at early times

and the decay rate at late times; the former depends on

the reheat temperature and the latter does not, so it is

possible to achieve the observed DM abundance with a

much feebler coupling to SM particles and, thereby, open

up viable parameter space for direct and indirect detec-

tion.

This paper is organized as follows: in Sec. II we de-

scribe the model, in Sec. III we calculate the A0abun-

dance via freeze in, in Sec. IV we present structure for-

mation limits, in Sec. Vwe explore the indirect detection

constraints and future projections for this model, and in

Sec. VI we oﬀer some concluding remarks.

arXiv:2210.06487v1 [hep-ph] 12 Oct 2022

µc

µL



µc

hHi

µL



µc

hHi

µL



{ i}

µc

hHi

µL



{ i}



µc

µL



A

B

FcF

µL



A

B

FcF

µc

µL



FcF

µc

µL



vFB



vFB

µc

µL



⌫e

¯⌫e

FIG. 1: Feynman diagrams representing the A0→3γ

(top) and the A0→¯νν decay (bottom) channels for

mA0<2m`. In both processes, the gray dot at the A0-`

vertex is the magnetic dipole interaction from Eq. (1)

for charged leptons `=e, µ, τ . There are corresponding

diagrams involving quarks for which the electroweak

loop yields decays to lighter quark ﬂavors instead of

neutrinos.

II. THEORY OVERVIEW

A. Model Description

We extend the SM with a hidden U(1)Hgroup with

corresponding gauge boson A0, which doesn’t couple to

any SM particles through renormalizable operators. The

leading infrared interaction between A0and SM particles

is taken to be a magnetic dipole coupling

Lint =df

2F0

µν ¯

fσµν f , (1)

where fis a charged SM fermion, dfis the corresponding

magnetic dipole moment, and F0

µν is the A0ﬁeld strength

tensor. Such an interaction can arise if the U(1)His

unbroken at high energies and heavy particles charged

appropriately under U(1)Hand the SM are integrated

out at low energies.1Since the magnetic dipole coupling

is a dimension-5 operator, Eq. (1) is only valid at energy

(or temperature) scales that satisfy Ed−1

If the A0is initially massless, any potential kinetic mix-

ing between U(1)Hand U(1)Ygauge bosons can be ro-

tated away, so the operator in Eq. (1) can be the domi-

nant interaction with SM particles [6]. However, for A0

1See Ref. [6] for an explicit construction involving two-loop di-

agrams with virtual exchange of both U(1)Hcharged and SM

charged particles. In this example, it is important that the

new states are not bifundamentals under the SM and the hidden

group so that kinetic mixing doesn’t arise at lower loop order.

to be a viable dark matter candidate, it must acquire a

mass at some lower energy scale, at which point kinetic

mixing of the form 

2Fµν F0

µν can arise from loops of SM

particles through their dipole interactions, where

∼edfmf

4π2≈2×10−15 mf

medf·GeV

10−8,(2)

which is derived in Appendix VI. In Ref.[4], it was found

that vector freeze-in through kinetic mixing could ac-

count for the full DM abundance for ∼10−11 −10−12

over the ∼keV-MeV mass range. However, from Eq. (2),

it is clear that the induced kinetic mixing can easily be

subdominant to dipole production through the operator

in Eq. (1); throughout this paper, we will only consider

parameter space for which this requirement holds.

B. Leptonic Couplings

In this section we consider the decay channels that

arise from coupling A0to charged leptons with f=`

in Eq. (1), where `=e, µ, τ is the ﬂavor of the dipole

interaction. For mA0>2m`, the dominant decay chan-

nel is A0→`¯

`, which is generically too prompt for the

dark photon to serve as a viable dark matter candidate.

However, for mA0<2m`, the A0→3γchannel shown at

the top of Fig. 1can be cosmologically metastable due

to phase-space suppression, so throughout this paper, we

will only consider this mass ordering. In the mA0m`

limit, the 3γdecay width is

ΓA0→3γ=α3d2

`m9

155520 π4m6

,(3)

which corresponds to a vector lifetime of

τA0≈1018 Gyr 10−10

d`·GeV 210 keV

mA09m`

me6

,(4)

so the A0can easily be metastable on cosmological

timescales if there are no faster decay channels.

The A0can also decay to neutrinos through one-loop

diagrams involving virtual Wexchange, as shown in the

bottom of Fig. 1. The partial width for this process is

ΓA0→¯νν =d2

`G2

Fm2

`m5

4π3log2mW

m`,(5)

and the ratio of partial widths satisﬁes

ΓA0→¯νν

ΓA0→3γ≈3×10−3m`

me810 keV

mA04

,(6)

where GF= 1.16 ×10−2GeV−2is the Fermi constant

and we have set m`=meinside the log of Eq. (5). Since

avoiding cosmologically prompt A0→¯

`` decays requires

mA0<2m`, saturating this inequality maximizes the

dominance of the photon channel

ΓA0→¯νν

ΓA0→3γ≈1.5×10−2m`

mµ4mA0=2m`

.(7)

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FERMILAB-PUB-22-670-TFreezingInVectorDarkMatterThroughMagneticDipoleInteractionsGordanKrnjaic,1,2,3,DuncanRocha,4,yandAnastasiaSokolenko1,3,z1TheoreticalPhysicsDepartment,FermiNationalAcceleratorLaboratory,Batavia,IL605102DepartmentofAstronomyandAstrophysics,UniversityofChicago,Chicago,IL606373Kavl...

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