Exploring the cosmological dark matter coincidence using infrared xed points Alexander C. Ritter1and Raymond R. Volkas1y 1ARC Centre of Excellence for Dark Matter Particle Physics School of Physics

2025-05-06 0 0 938.12KB 13 页 10玖币

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Exploring the cosmological dark matter coincidence using infrared ﬁxed points

Alexander C. Ritter1, ∗and Raymond R. Volkas1, †

1ARC Centre of Excellence for Dark Matter Particle Physics, School of Physics,

The University of Melbourne, Victoria 3010, Australia

The asymmetric dark matter (ADM) paradigm is motivated by the apparent coincidence between

the cosmological mass densities of visible and dark matter, ΩDM '5ΩVM. However, most ADM

models only relate the number densities of visible and dark matter, and do not motivate the similarity

in their particle masses. One exception is a framework introduced by Bai and Schwaller, where the

dark matter is a conﬁned state of a dark QCD-like gauge group, and the conﬁnement scales of

visible and dark QCD are related by a dynamical mechanism utilising infrared ﬁxed points of the

two gauge couplings. We build upon this framework by properly implementing the dependence of

the results on the initial conditions for the gauge couplings in the UV. We then reassess the ability

of this framework to naturally explain the cosmological mass density coincidence, and ﬁnd a reduced

number of viable models. We identify features of the viable models that allow them to naturally

relate the masses of the dark baryon and the proton while also avoiding collider constraints on the

new particle content introduced.

I. INTRODUCTION

Determining the particle nature of dark matter (DM)

is one of the deepest tasks facing particle physics today.

This goal is hampered by the purely gravitational nature

of the observational evidence for DM, which has allowed

for a cornucopia of DM candidates to be proposed over

a wide range of mass scales [1]. To focus our model-

building pursuits, it is instructive to see what clues may

lie in the existing astrophysical observations.

One such result is the similarity between the present-

day cosmological mass densities of dark and visible mat-

ter (VM), which we refer to as the cosmological coinci-

dence [2]:

ΩDM '5ΩVM,(1)

where ΩXis the mass density ρXfor species Xdivided

by the critical density ρc.

We take the cosmological coincidence to be a hint to-

wards an underlying link between the origins of the abun-

dances of VM and DM. Such a connection is not present

in the majority of prominent DM candidates. Consider,

for example, the WIMP: it is a GeV–TeV scale thermal

relic species with an abundance generated through ther-

mal freezeout. This stands in contrast to VM, whose

number density is due to the baryon asymmetry of uni-

verse and whose mass arises from the conﬁnement en-

ergy of QCD. With such distinct generation mechanisms,

there is no a priori reason why the cosmological mass

densities of these species should fall in the same order of

magnitude.

The main model-building paradigm that seeks to ex-

plain the cosmological coincidence problem is that of

asymmetric dark matter (ADM) [3,4]. In this frame-

work, the dark matter candidate is charged with a dark

∗rittera@student.unimelb.edu.au (corresponding author)

†raymondv@unimelb.edu.au

particle number that develops an asymmetry related to

the visible baryon asymmetry. Models of ADM thus nat-

urally generate similar number densities for VM and DM

–nVM ∼nDM. However, as a mass density is given

by ΩX=nXmX/ρc, to satisfactorily explain the cos-

mological coincidence problem we must also motivate

the similarity in the particle masses of VM and DM –

mVM ∼mDM. This problem is not addressed in the

majority of the ADM literature, which mostly treats the

DM mass as a free parameter and thus merely shifts the

cosmological mass-density coincidence to a particle mass

coincidence.

Our goal then is to construct a theory where DM and

VM naturally have similar particle masses. By analogy

with the proton, this leads us to consider DM candidates

that are conﬁned states of a dark, QCD-like gauge group

[5–25], where some mechanism is present to relate the

conﬁnement scale ΛdQCD of this gauge group to that of

visible QCD, ΛQCD.

Previous eﬀorts to relate the visible and dark conﬁne-

ments scales have followed one of two general directions:

1. introduce a symmetry between the gauge groups.

This generally leads to models of mirror matter,

where the mirror symmetry is either exact [5–9]

or judiciously broken [10–22] (For other models of

mirror matter, see Refs. [26–35]).

2. introduce new ﬁeld content so that the running

gauge couplings αsand αdapproach infrared ﬁxed

points with similar magnitudes. This has not been

widely discussed in the literature, with the origi-

nal idea introduced by Bai and Schwaller [24] and

expanded upon by Newstead and TerBeek [25].

In this paper we focus on the latter approach, develop-

ing upon the framework introduced by Bai and Schwaller

by properly implementing the dependence of the conﬁne-

ment scale ΛdQCD on the initial values of the running

gauge couplings in the UV, αUV

sand αUV

d. We then ask

how readily this approach generates conﬁnement scales

arXiv:2210.11011v1 [hep-ph] 20 Oct 2022

of the same order of magnitude, and so reassess the va-

lidity of this framework as a natural explanation of the

cosmological coincidence.

Compared to the work of Bai and Schwaller, we ﬁnd

that in our analysis there are a smaller number of mod-

els that naturally generate similar conﬁnement scales for

visible and dark QCD, where models in this framework

are deﬁned by the new ﬁeld content we introduce. This

reduced set of models is due to our deﬁnition of ‘natu-

ralness’ now being more stringent, as it must take into

account the dependence of the dark conﬁnement scale on

the initial gauge coupling values in the UV.

In general, we ﬁnd that models whose infrared ﬁxed

points have small values for the gauge couplings are bet-

ter at generically generating similar visible and dark con-

ﬁnement scales. However, we also ﬁnd that these models

generally require the mass scale of the new particle con-

tent to be sub-TeV for most selections of the initial cou-

plings in the UV, and thus are subject to strong collider

constraints.

Looking for models where the new physics mass scale

is on the order of a few TeV, we do ﬁnd a number of such

models that can fairly generically generate related visible

and dark conﬁnement scales. We identify these models as

the most promising candidates within this framework for

naturally explaining the cosmological coincidence prob-

lem.

The paper is organised at follows: in Section II we de-

scribe the dark QCD framework of Bai and Schwaller. In

Section III we describe the threshold corrections to the

running of the gauge couplings as implemented by New-

stead and TerBeek. In Section IV we analyse the eﬀect

of the values of the gauge couplings in the UV on ΛdQCD.

In Section Vwe discuss the ability of this framework to

explain the cosmological coincidence, before presenting

our results in Section VI and concluding in Section VII.

II. THE BAI-SCHWALLER FRAMEWORK

The scheme of Bai and Schwaller [24] utilises a dark

QCD-like gauge group SU(Nd)dQCD, along with a selec-

tion of new particle content charged under SU(3)QCD ×

SU (Nd)dQCD. As in the original paper, we only consider

Nd= 3, and limit the new ﬁeld content to fermions and

scalars in the fundamental representations of either one

or both of the QCD gauge groups. The multiplicities of

the new particles are given in Table I, along with their

masses. In addition to the given multiplicities, we also

deﬁne nfd=nfd,h +nfd,l as the total number of dark

fermion species that are fundamentals of SU (3)dQCD, and

nfc=nfc,h + 6 as the multiplicity of visible fermions

that are fundamentals of SU(3)QCD (including the 6 SM

quarks).

The majority of the new ﬁeld content is, for simplicity,

taken to exist at a common heavy mass scale M, except

for the nfd,l light dark fermions. These latter particles,

which we refer to as ‘dark quarks’, have masses much

Field SU(3)QCD ×SU (3)dQCD Mass Multiplicity

Fermion

(3,1)M nfc,h

(1,3)<ΛdQCD nfd,l

M nfd,h

(3,3)M nfj

Scalar

(3,1)M nsc

(1,3)M nsd

(3,3)M nsj

TABLE I. The new particle content in a model. The given

multiplicities are for Dirac fermions and complex scalars. A

subscript containing l(h) indicates a multiplicity for light

(heavy) particles in cases where this is a relevant distinction

to make.

lighter than the dark conﬁnement scale ΛdQCD and are

conﬁned into the dark baryons that serve as the DM can-

didate.

At two-loop level, the β-functions for gcand gdare cou-

pled, thanks to the presence of the ‘joint’ ﬁelds charged

under both SU(3)QCD and SU(3)dQCD. The two-loop

β-function for gc,βc(gc, gd) = dgc

d(log(µ)) , is given by

βc=g3

16π22

3nfc+ 3nfj+1

6nsc+ 3nsj−11

+g5

(16π2)238

3nfc+ 3nfj+11

3nsc+ 3nsj−102

+g3

cg2

(16π2)28nfj+ 8nsj,

(2)

and the β-function for gd,βd(gc, gd)≡dgd

d(log(µ)) , is ob-

tained by exchanging the indices c↔d[36]. Note that

these β-functions are given for Nc=Nd= 3; the β-

functions for general Ncand Ndcan be found in Ref. [24].

Depending on the particle content, there may be non-

trivial couplings α∗

sand α∗

dfor which both β-functions

are zero, where αs=g2

c/4π,αd=g2

d/4π, and

βc(g∗

c, g∗

d) = βd(g∗

c, g∗

d)=0.(3)

The couplings α∗

sand α∗

ddenote an infrared ﬁxed

point (IRFP) of the renormalization group running sim-

ilar to a Banks-Zaks ﬁxed point for a single gauge

coupling [37]. The couplings at the IRFP only de-

pend on the multiplicities of the new particle content,

(nfc,h , nfd,l , nfd,h , nfj, nsc, nsd, nsj); we refer to a given

selection of multiplicities as a ‘model’.

So, for a given model, if there exists a nontrivial pertur-

bative IRFP, the coupled gauge couplings evolve toward

α∗

sand α∗

dregardless of the initial values for the gauge

couplings in the UV. This ﬁxed point feature is broken

when the new heavy ﬁelds decouple at the mass scale M;

below this scale, the gauge couplings run independently

until they become non-perturbative at the conﬁnement

scales ΛQCD and ΛdQCD.

The process to calculate ΛdQCD for given model is then,

ideally, as follows:

•Calculate the IRFP gauge couplings α∗

sand α∗

•Determine Mby running the visible coupling up

from the measured value αs(MZ) = 0.11729 [38] to

αs(M) = α∗

•Evolve the dark coupling down from αd(M) = α∗

until the scale ΛdQCD at which it becomes large

enough to trigger conﬁnement. As a rough pertur-

bative condition, the Cornwall-Jackiw-Tomboulis

bound αs> π/4 for Nd= 3 [39] is used to deﬁne the

dark conﬁnement scale through αd(ΛdQCD) = π/4.

So, for a given model with a nontrivial IRFP, the dark

conﬁnement scale ΛdQCD can be uniquely determined.

The argument is then that a model with related IRFP

values α∗

sand α∗

dcould lead to compatible conﬁnement

scales for visible and dark QCD of a similar order of

magnitude.

However, this framework relies upon a number of

simplifying assumptions that should be investigated in

greater detail. The ﬁrst is that the heavy ﬁelds decouple

at the mass scale Mwithout any threshold corrections,

which was addressed in Ref. [25]; we summarise their ap-

proach in the next section. The second key assumption

is that the couplings run all the way to their IRFP val-

ues by the decoupling scale Mregardless of the initial

coupling values in the UV. However, this is not true in

general, and in a later section we analyse how the value

of ΛdQCD depends on the UV coupling values αUV

sand

αUV

III. THRESHOLD CORRECTIONS

In MS-like mass-independent renormalization schemes,

the Appelquist-Carazzone decoupling theorem [40] does

not apply in its naive form; heavy ﬁelds can continue to

inﬂuence coupling constants and β-functions at energy

scales below their masses. In these schemes there also

arises the related issue of large logarithms when working

at energy scales much smaller than the mass Mof the

heavy ﬁelds.

To properly account for these problems, the decoupling

is treated explicitly by constructing an eﬀective ﬁeld the-

ory in which the heavy ﬁelds have been integrated out.

Consistency is ensured by matching the full theory onto

the eﬀective ﬁeld theory at a matching scale µ0, where

µ0∼Mto avoid large logarithms. This procedure leads

to a ‘consistency condition’ relating the coupling con-

stants in the full theory with the coupling constants in

the eﬀective ﬁeld theory [41,42],

αEFT

s(µ0) = ζ2

c(µ0, αs(µ0))αs(µ0),(4)

αEFT

d(µ0) = ζ2

d(µ0, αs(µ0))αd(µ0).(5)

The decoupling functions ζ2

cand ζ2

dhave been deter-

mined to three- and four-loop order in Refs. [43,44] for

the case of integrating out one heavy quark from QCD

with nfﬂavours. Since we are working with two-loop β-

functions, we only require the one-loop decoupling func-

tions; in Ref. [25], the results of Ref. [43] were adapted

for integrating out the full selection of heavy ﬁeld content

at mass M, with the one-loop decoupling functions given

ζ2

c(µ, α(µ)) = 1 −αs(µ)

6π˜nclog µ2

M2,(6)

ζ2

d(µ, α(µ)) = 1 −αd(µ)

6π˜ndlog µ2

M2,(7)

where the coeﬃcients ˜ncand ˜ndaccount for the degrees

of freedom of the relevant heavy ﬁelds,

˜nc=nfc,h + 3nfj+1

4nsc+ 3nsj,(8)

˜nd=nfd,h + 3nfj+1

4nsd+ 3nsj.(9)

We note that, for a given M, the values of the couplings

at low energies now depend on the choice of decoupling

scale µ0; this renormalization scale dependence is a non-

physical artifact of our perturbative analysis being trun-

cated at ﬁnite loop order. As a result, the procedure

of Bai and Schwaller (matching αs(MZ) to its experi-

mental value) no longer uniquely determines the value

of Mwhen we incorporate the threshold corrections to

the coupling constants. Instead, we obtain a relationship

between Mand µ0, and can solve for one of these values

if we specify the other.

In Ref. [25], values of ΛdQCD were determined for

a given model and value of M. By using the consis-

tency condition and again matching the strong coupling

to its measured value at MZ, they solved for the de-

coupling scale µ0, and then ran the dark coupling until

αd(ΛdQCD) = π/4. However, this process does not en-

sure that µ0is on the order of M. We take a diﬀerent

approach to implementing threshold corrections, as is de-

tailed in the following section.

IV. DEPENDENCE ON INITIAL UV

COUPLINGS

As mentioned earlier, the framework in the previous

sections assumes that the couplings run to their IRFP

values by the decoupling scale µ0regardless of their ini-

tial values in the UV, αUV

sand αUV

d. However, this

is not the case, as illustrated in Fig. 1for the model

(nfc,h , nfd,l , nfd,h , nfj, nsc, nsd, nsj) = (3,3,3,3,3,2,0)

with IRFP (α∗

s, α∗

d) = (0.045,0.077), where we show the

running of a grid of couplings (αs, αd) from the UV scale

ΛUV = 1019 GeV down to a typical infrared scale ΛIR = 1

TeV. While the couplings evolve towards their ﬁxed point

values, they do not precisely reach them, and so the low

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ExploringthecosmologicaldarkmattercoincidenceusinginfraredxedpointsAlexanderC.Ritter1,andRaymondR.Volkas1,y1ARCCentreofExcellenceforDarkMatterParticlePhysics,SchoolofPhysics,TheUniversityofMelbourne,Victoria3010,AustraliaTheasymmetricdarkmatter(ADM)paradigmismotivatedbytheapparentcoincidencebetwee...

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Exploring the cosmological dark matter coincidence using infrared xed points Alexander C. Ritter1and Raymond R. Volkas1y 1ARC Centre of Excellence for Dark Matter Particle Physics School of Physics

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