Maximally self-interacting dark matter:models and predictions

Ayuki Kamadaa, Hee Jung Kimb, and Takumi Kuwaharaa

a Center for Theoretical Physics of the Universe, Institute for Basic Science (IBS),Daejeon 34126, Korea

b Department of Physics, KAIST, Daejeon 34141, Korea

Abstract

We study self-interacting dark matter (SIDM) scenarios, where the s-wave self-scattering cross section almost saturates the Unitarity bound. Such self-scatteringcross sections are singly parameterized by the dark matter mass, and are featured bystrong velocity dependence in a wide range of velocities. They may be indicated byobservations of dark matter halos in a wide range of masses, from Milky Way’s dwarfspheroidal galaxies to galaxy clusters. We pin down the model parameters thatsaturates the Unitarity bound in well-motivated SIDM models: the gauged Lµ−Lτmodel and composite asymmetric dark matter model. We discuss implications andpredictions of such model parameters for cosmology like the H0 tension and dark-matter direct-detection experiments, and particle phenomenology like the beam-dump experiments.

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1 Introduction

Collisionless cold dark matter (CDM) is a minimal hypothesis for missing mass of theUniverse. The CDM hypothesis works well in explaining the large-scale structure of theUniverse from cosmic microwave background (CMB) anisotropies to galaxy clusterings.On the other hand, there have been tensions between naıve CDM predictions and theobserved small-scale structure of the Universe. Such tensions are collectively dubbed asthe small-scale issues, and challenge our understanding of how Milky Way (MW)-size anddwarf galaxies form (see Ref. [1] for a review).

The small-scale issues may indicate the nature of DM, e.g., self-interacting dark matter(SIDM) [2] (see Ref. [3] for a review). It is intriguing that we can probe self-interactionof DM, which is not accessible in terrestrial experiments, in cosmological observations.Such gravitational probes of DM have been attracting growing interests [4], in light of nulldetection of DM interactions with standard-model (SM) particles in collider searches andDM direct-detection and indirect-detection experiments [5, 6]. Since traditional weaklyinteracting massive particle (WIMP) DM behaves as CDM on galactic scales, the small-scale issues may call for a new paradigm, i.e., beyond-WIMP DM.

DM self-interaction is not necessarily constant in collision energy. Ref. [7] illustrateshow we can probe velocity dependence of the self-scattering cross section by combiningobservations of DM halos in a wide range of masses. The velocity dependence may bealready indicated. To form cores of galaxy clusters [8] and evade constraints from bulletclusters [9] and merging clusters [10], where the collision velocity is v ∼ 1000 km/s,the self-scattering cross section per mass is preferred to be σ/m ∼ 0.1 cm2/g (see alsoRef. [11] for a recent reanalysis of galaxy clusters and a new constraint from galaxygroups [12]). On the other hand, to explain the diversity of galactic rotation curves [13],where v ∼ 100 km/s, it is preferred to be σ/m ∼ 1 cm2/g [14,15]. This velocity dependencecall for particle-physics model buildings: DM with a light mediator [16–24], DM with aresonant mediator [25, 26], and composite DM [27–34]. Hereafter, we focus on an elasticscattering, while an inelastic scattering may also introduce the scale-dependence of coreformation [35–39].

MW’s dwarf spheroidal satellite galaxies, where v ∼ 30 km/s, are the subject of activedebate. They show a large diversity in the central density profiles [40]. The preferred self-scattering cross section also vary as σ/m ∼ 0.1-40 cm2/g [41]. The high central densityof Draco actually provides a tight constraint of σ/m < 0.57 cm2/g [42] (see also therecent analysis of ultra-faint dwarf galaxies [43]). However, it was recently pointed outthat while the above discussions focus on the so-called core expansion (formation) phase,the situation changes in the core contraction phase (but long before the core collapsetime). There, SIDM halo may experience a gravothermal core collapse [44], leading toa cuspy profile. The gravothermal core collapse may be further accelerated by the tidalstripping [45]. In addition, the core collapse may explain the anti-correlation between thecentral DM densities and their orbital pericenter distances [46].

If the MW’s dwarf spheroidal satellite galaxies are in the core contraction phase, theself-scattering cross section may need to be as large as σ/m ∼ 30-200 cm2/g to have asufficient core collapse [47]. This implies a strong velocity dependence of σ/m. Meanwhile,such large σ/m seem not compatible with the inferred values of σ/m from dwarf galaxies

1

in the field [3], σ/m ∼ 0.1-10 cm2/g, though they have similar collision velocities, v ∼20-60 km/s. It might be because Ref. [3] assumes the halos to be in the core expansionphase. If we repeated their analyses in the core contraction phase, we might find a largervalue of σ/m for the field dwarf galaxies as well. On the other hand, the inferred σ/mfrom the MW’s satellites changes with different initial conditions of halos and modelingof tidal stripping. With higher initial concentrations of halos, the inferred cross sectionsmay be appreciably lower than in Ref. [47], σ/m ∼ 3 cm2/g [48], mitigating the tensionbetween the MW’s dwarf spheroidal satellite galaxies and the field dwarf galaxies. In thispaper, we take the inferred values from Ref. [47] at face values and study the implicationsof the strong velocity dependence of σ/m (see Appendix A for the changes in conclusionsif the inferred σ/m in the MW’s satellites was lowered).

Furthermore, the above values of σ/m are inferred under the assumption of a constantcross section. If we reanalyze the astronomical data by taking velocity-dependent crosssections, the tension between the MW’s satellites and the field dwarf galaxies could bemitigated. For velocity-dependent cross sections, we also have to take an average over localvelocity distribution (see Appendix B for more discussion). In this paper, we comparethe velocity-dependent cross section without the distribution averaging with the inferredvalues of σ/m mentioned above. The velocity-dependent cross section considered in thispaper may give a benchmark with which the astronomical data will be reanalyzed.

As seen in the next section, this strong velocity dependence is realized when the self-scattering cross section almost saturates the s-wave Unitarity bound, i.e., on the quantum(zero-energy) resonance [16, 17]. On the quantum resonanace, the self-scattering crosssection is singly parameterized by the DM mass. We take a closer look at the quantumresonance by using the effective-range theory. The quantum resonance allows us to pindown the model parameters as well as the DM mass. In Section 3, we consider DM witha light mediator based on the gauged Lµ − Lτ model. We identify the mediator and DMmasses that explain the strong velocity dependence of DM self-interaction. Interestingly,we find that the discrepancy in muon anomalous magnetic moment g − 2 and tension inH0 are mitigated in the very parameter points. The light Lµ − Lτ gauge boson are alsosubject to various experimental searches such as non-standard interactions of neutrinosand missing-energy events in colliders. In Section 4, we consider composite asymmetricdark matter (ADM) based on the dark quantum chromodynamics (QCD) and electro-dynamics (QED). The dark QCD scale and dark pion masses are constrained so thatdark nucleon DM can have the indicated velocity-dependent cross section. The bindingenergies of two nucleons and vector resonances are also predicted, which have importantimplications for the dark nucleosynthesis in the early Universe and dark spectroscopymeasurements in lepton colliders, respectively. The dark photon, which is required tomake cosmology viable, is subject to intensive experimental efforts such as beam-dumpexperiments. The dark proton may be found in DM direct-detection experiments. Wegive concluding remarks in Section 5.

2

2 Effective-range theory

The effective-range theory is first developed in the attempt to understand low-energyscatterings of nucleons [49,50]. See Ref. [51] for a review of the effective-range theory in thecontext of effective-field theory. Here we refer to Ref. [52] that revisits the effective-rangetheory in the context of SIDM. The 2-body scattering cross section can be decomposedinto the partial-wave (multipole `) contributions:

σ =∑`

σ` =∑`

4π

k2(2`+ 1) sin2 δ` . (1)

Here we ignore a possible quantum interference for the identical particles.1. The momen-tum k = µvrel is given by the reduced mass µ (µ = m/2 for the identical particles withthe mass of m) and relative velocity vrel. The analytic properties of the wave functiondetermine the low-energy behavior of δ` as

k2`+1 cot δ` → −1

a2`+1`

+1

2r2`−1e`

k2 (k → 0) . (3)

Here a` and re` are called the scattering length and effective range, respectively.In the following, we focus on the s-wave (` = 0), which is expected to be dominant at

the low energy.2 Then, the low-energy cross section is given by

σ =4πa2

1 + k2(a2 − are) + a2r2ek

4/4. (4)

When |a| |re|, in the range of 1/|re| > k > 1/|a|, the cross section saturates theUnitarity bound,

σmax =4π

k2, (5)

which is singly parameterized by the DM mass, and thus gives a good benchmark forstudying halos in velocity-dependent SIDM. For the DM mass of m ∼ 10 GeV, σmax/msimultaneously explains the inferred σ/m’s from MW’s dwarf spheroidal satellite galaxiesand galaxy clusters, as we will see shortly. A large |a| indicates the existence of theshallow bound state (a > 0) or virtual state (a < 0; bound state with a non-normalizable

1One may also choose to use the transfer cross section σT , which may be a more suitable quantityto parameterize the momentum-transfer effect among DM particles through elastic scattering. For thescattering of identical particles, σT is written as [53]

σT =

∫dΩ(1− | cos θ|) dσ

dΩ, (2)

where σ is the standard cross section, σ =∫dΩ(dσ/dΩ). Since we focus on the s-wave scattering, this

amounts to an additional factor of 1/2, σT = σ/2.2One prominent counterexample is the Rutherford scattering via the exchange of an (almost) massless

mediator. Hereafter we omit the subscript of ` = 0 for notational simplicity.

3

wave function), depending on its sign. The bound (virtual) state energy is given by thepositive (negative) imaginary pole kpole:

Eb = −k2

pole

2µ, kpole =

i

re

(1−

√1− 2re/a

). (6)

In the left panel of Fig. 1a, we show the velocity dependence of σ/m (black) in theeffective-range theory. The data points show the preferred values of σ/m as a function ofthe relative velocity; they are inferred from the observations on central DM densities ofMW’s dwarf spheroidal satellite galaxies (cyan), field dwarf spheroidal galaxies (red)/lowsurface brightness spiral galaxies (blue), and galaxy clusters (green). There can be seenthat the cyan data points disagree with the red data points at v ∼ 30 km/s, as discussedin the previous section. Hereafter we take the former seriously, while being ignorant tothe latter; we give further discussion focusing on the red data points in Appendix A.We take the DM mass m = 20 GeV so that the s-wave Unitarity bound σmax/m (dot-dashed) crosses the inferred σ/m’s from the MW’s satellites (cyan) and the galaxy clusters(green). The effective-range theory parameter sets took are (a, a/re) = (−292 fm,−152)(solid) and (a, a/re) = (−292 fm,−22.5) (dashed). Both data sets exhibit |a/re| 1.This indicates that they are on the quantum (zero-energy) resonance. The Unitaritybound is saturated from k & 1/|a| (unfilled circle) to k . 1/|re| (filled circle); in thisregime, the cross section is singly determined by the DM mass. While the both data setshave the same a, the former data set has smaller |re| and hence it saturates the Unitaritybound in a wider range of k. Indeed, the solid curve saturates the Unitarity bound all theway up to vrel ∼ 3000 km/s, while the dashed curve desaturates for vrel & 400 km/s. Thesolid curve exhibits σ/m ∝ 1/v2

rel at the presented velocity range. Such strong velocitydependence simultaneously explains the preferred σ/m ∼ 0.1 cm2/g at vrel ∼ 1000 km/sand σ/m ∼ 100 cm2/g at vrel ∼ 30 km/s. Interestingly, such velocity dependence providesa good fit to the inferred σ/m’s from the MW’s satellites (cyan) at low-velocities. Aswe take smaller |a/re|, σ/m desaturate like the dashed curve, but may still explain thecyan data points. Meanwhile, taking a larger |a| than the data sets would extend therange of k where the Unitarity bound is saturated; it would retreat the unfilled circlesto smaller velocities. We remark that while an arbitrarily larger |a| would still providea good fit to the cyan data points, it is not clear how such velocity dependence of σ/maffects the structure of DM halos as small as (or smaller than) the MW’s satellites. Thisis because the SIDM evolution of such small halos may transit into the short mean freepath regime [44,54] due to the enhanced σ/m at low velocities.

It is illustrating to consider the scattering under the Hulthen potential:

V (r) = − αδe−δr

1− e−δr .(7)

The Hulthen potential approximates the Yukawa potential,

V (r) = −αe−mφr

r, (8)

with a proper choice of δ. Here mφ is the mass of a mediator and we take δ =√

2ζ(3)mφ

4

101 102 10310-3

10-2

10-1

100

101

102

103

101 102 10310-3

10-2

10-1

100

101

102

103

fm = 20GeV

( ) (a, a/re) = (292 fm, 22.5)( ) (a, a/re) = (292 fm, 152) f

m = 20GeV( ) (↵, m) = (3.09 102, 0.4 GeV)( ) (↵, m) = (4.58 103, 0.06 GeV)

(a)

100 101

-600

-400

-200

0

200

400

600

100 101100

101

102

(b)

Figure 1: (a) (Left): Velocity dependence of σ/m (in black) in the effective-range theoryfor given DM mass and the effective-range theory parameters (a, re). The dot-dashedcurve is the Unitarity bound for the DM self-scattering cross section, σmax/m. σ/msaturates the Unitarity bound from k & 1/|a| (unfilled circle) to k . 1/|re| (filled circle),where the cross section is singly parametrized by the DM mass. The data points arethe inferred values of σ/m from the observations on the field dwarf (red)/LSB (blue)galaxies, and galaxy clusters (green) [7]. The data points in cyan are the values that mayexplain the anti-correlation between the central DM densities and their orbital pericenterdistances [47]. The effective-range theory deviates from the Hulthen potential (see theright panel) results for k & 1/|re| (in brown) as the effective-range expansion fails. (Right):Same as the left panel but for the s-wave contribution of σ/m in the Hulthen potential.The Hulthen-potential parameter sets took are in one-to-one correspondence with theeffective-range theory parameter sets in the left panel. The displayed velocity dependenceof σ/m is not reliable in the classical regime (in gray), i.e., k & mφ, as the higher partial-wave contributions become important; the filled circle denotes the point k = mφ. (b)s-wave scattering length (left) and effective range (right) in units of m−1

φ as a functionof εφ for the Hulthen potential. The stars indicate the benchmark parameters took inFig. 1a, which are near the first quantum resonance, εφ = 1.

5

with the ζ(z) being the Riemann zeta function.3 The Hulthen potential enjoys an analyticexpression of the phase shift:

δ0 = arg

(iΓ(λ+ + λ− − 2)

Γ(λ+)Γ(λ−)

). (9)

Here Γ(z) is the gamma function and

λ± = 1 + iεvεφ ±√εφ − ε2vε2φ , εv =

vrel

2α, εφ =

2αµ

δ. (10)

This phase shift results in the scattering length and effective range of

a =ψ(0)(1 +

√εφ) + ψ(0)(1−√εφ) + 2γE

δ,

re =2a

3− 1

3δ3√εφa2

3[ψ(1)(1 +

√εφ)− ψ(1)(1−√εφ)

]+√εφ[ψ(2)(1 +

√εφ) + ψ(2)(1−√εφ) + 16ζ(3)

],

(11)

respectively. Here ψ(n)(z) is the polygamma function of order n and γE = 0.577 . . . isthe Euler–Mascheroni constant. Above we consider an attractive potential, by takingpositive α. One can use the same expressions with an analytic continuation to negative

α: λ± = 1 + iεvεφ ± i√εφ + ε2vε

2φ, with εv =

vrel

2|α| and εφ =2|α|µδ

. Note that for a given

DM mass m, the effective-range theory parameter set (a, re) and the Hulthen-potentialparameter set (α,mφ) has a one-to-one correspondence.

While effective-range theory reproduces well the analytic results of the Hulthen poten-tial in the k → 0 limit, it starts to deviate for k−1 . re; in other words, when the de Brogliewavelength of the incoming particles is shorter than the effective range. Such regime isdepicted in brown in the left panel of Fig. 1a. This is expected since the low-momentumeffective-range expansion fails at high k. This feature can be seen by comparing the leftand the right panel of Fig. 1a; the right panel is the analytic result of the Hulthen poten-tial with the parameter sets that give the same effective-range theory parameter sets asin the left panel.

Meanwhile, the velocity dependence of σ/m in the Hulthen potential is not reliablein the classical regime, k−1 . m−1

φ , i.e., where the de Broglie wavelength of the incomingparticles is shorter than the range of the Hulthen potential. This is because the higherpartial-wave contributions become important in the classical regime. In the right panelof Fig. 1a, the classical regime is depicted in gray. Note that the classical regime in theHulthen potential roughly coincides with the regime of k−1 . re in the effective-rangetheory.

Fig. 1b shows a (left) and re (right) as a function of εφ in the units of m−1φ . The

resonances appear at εφ = n2 (n = 1, 2, . . . ), where |a| → ∞ for finite re. In the following,we focus on the first resonance, n = 1, while discussing higher resonances in Appendix C.

3The coefficient of√

2ζ(3) = 1.55 . . . is obtained by equating the Born cross sections with the Hulthenand Yukawa potentials [17]. The relation of δ = 2ζ(3)mφ in Ref. [52] would be a typo. A similarprescription for the Sommerfeld-enhancement factor leads to the coefficient of ζ(2) = π2/6 = 1.64 . . . [55].

6

The points depicted by stars corresponds to the parameters took in Fig. 1a; they are bothnear the first resonance, nearly saturating the Unitarity bound.

Above we consider the elastic scattering and see that we need a light mediator andinduced relatively long-range force to reproduce the strong velocity dependence. TheSommerfeld enhancement for DM annihilation [55–63] is controlled by the same potential.If the Sommerfeld enhancement is also almost on the quantum resonance, this would havemultiple implications for the cosmology: strong constraints from the indirect-detectionexperiments if DM annihilation ends up with the electromagnetic energy injection [64]; andthe second stage of the DM freeze-out (i.e., re-annihilation) [63,65–68]. The enhancementfactor in the Hulthen potential is given by [21]4

Ss-wave =π

εv

sinh(2πεvεφ)

cosh(2πεvεφ)− cos(

2π√εφ − ε2vε2φ

) ,Sp-wave = Ss-wave

(εφ − 1)2 + 4ε2vε2φ

1 + 4ε2vε2φ

,

(13)

for s-wave and p-wave annihilation, respectively. For annihilation, we take δ = ζ(2)mφ.The left panels of Fig. 2 shows the Sommerfeld-enhancement factor as a function of the

relative velocity for s-wave (top) and p-wave (bottom). The Hulthen-potential parameterstook corresponds to the parameters depicted as stars in Fig. 1b, which are close to thequantum resonance for elastic scattering. For these parameters, the s-wave Sommerfeld-enhancement factor is also resonantly enhanced towards low-velocity as ∝ 1/v2

rel andsaturates; as shown in the left panel of Fig. 2a, the parameters are close to the s-wavequantum resonance for annihilation. This may be dangerous in terms of the constraintsfrom observations on the CMB and indirect-detection experiments. The constraints maybe evaded if we consider neutrinos as the final DM annihilation product or consider thep-wave annihilation [22, 64]. Indeed, the p-wave Sommerfeld-enhancement factors for theparameters are not huge, since they are sufficiently far from the p-wave quantum resonancefor DM annihilation (see the right panel of Fig. 2b).

In Section 3, we consider a model with a light mediator based on the gauged Lµ −Lτmodel, where DM annihilation is p-wave and ends up with neutrinos. ADM is anothergood candidate, since it involves a light mediator to deplete the thermal relic abun-dance of the symmetric component. ADM may also be expected to be safe from theindirect-detection bounds at the first sight. In Section 4, nevertheless, we see that evena (inevitable induced) tiny DM-anti DM oscillation leads to a significant annihilation inthe late Universe.

4Note that our εφ is different from Ref. [63]. This expression coincides with Ref. [55]:

S` =

∣∣∣∣ Γ(λ+`)Γ(λ−`)Γ(λ+` + λ−` − 1 + `)Γ(1 + `)

∣∣∣∣2 , λ±` = `+ λ± . (12)

7

v = 103

v = 102

v = 101

v = 1

10-7 10-6 10-5 10-4 10-3 10-2 10-110-1

100

101

102

103

104

100 101 102100

101

102

103

104

105

106

fm = 20GeV

( ) (↵, m) = (3.09 102, 0.4 GeV)( ) (↵, m) = (4.58 103, 0.06 GeV)

(a)

v = 103

v = 102

v = 101

v = 1100 101 102

100101102103104105106107108

10-7 10-6 10-5 10-4 10-3 10-2 10-1

100

101

fm = 20GeV

( ) (↵, m) = (3.09 102, 0.4 GeV)( ) (↵, m) = (4.58 103, 0.06 GeV)

(b)

Figure 2: (a) (Left): Velocity dependence of the s-wave Sommerfeld-enhancement fac-tor. The Hulthen-potential parameters took corresponds to the benchmark parametersin Fig. 1a. The s-wave annihilation is near the quantum resonance for the benchmarkparameters, as shown in the right panel. The Sommerfeld-enhancement factor grows as∝ 1/v2

rel (brown) towards low velocity and saturates. (Right): εφ-dependence of the s-wave Sommerfeld-enhancement factor. We take εv = 1, · · · , 10−3 from bottom to top.The dashed lines show the Sommerfeld-enhancement factors in the Coulomb potentialthat correspond to εφ → ∞. The points depicted by stars correspond to the benchmarkparameters took in Fig. 1a for fixed εv = 10−3. (b) Same as Fig. 2a, but for the p-waveSommerfeld-enhancement factor.

8

3 Dark matter with a light mediator: gauged Lµ−Lτmodel

In this section, we consider DM with a light mediator based on the gauged Lµ − Lτmodel [22]. In this model, a new gauge boson Z ′ couples to the muon and tau lepton (andtheir neutrinos) [69, 70]. Interestingly, this new force contributes to the muon (g − 2)µ,alleviating a possible tension between the SM prediction and measurement [71–73] (Seealso the latest review on (g − 2)µ in the SM, Ref. [74]).5 After other constraints aretaken into account, the (g − 2)µ discrepancy is ameliorated for the gauge coupling ofg′ ' 4× 10−4-10−3 and the mass of the new gauge boson of mZ′ ' 8-200 MeV.

Ref. [22] extends this model with the a vector-like pair of fermions, N and N . Weassume that the Lµ−Lτ breaking Higgs Φ carries a unit charge andN (N) carries a (minus)half charge in units of the muon and tau-lepton charges.6 The resultant Z2 symmetry,which is unbroken by the Higgs vacuum expectation value (VEV), guarantees the stabilityof the lightest mass eigenstate N1 among N and N , namely, N1 being DM. We considerthe pseudo-Dirac dark matter, whose Dirac mass mN is larger than the Majorana massinduced by the VEV of Φ. The masses are given by the Yukawa coupling y > 0:

mN1(2)= mN ±

∆m

2, ∆m =

√2y

g′mZ′ mN . (14)

Hereafter we assume the CP symmetry to restrict the Yukawa coupling. For more generaldiscussion, please see Ref. [22]. The relevant interactions are

Lint ⊃ −y

2√

2ϕ(−N1N1 +N2N2) + i

g′

2Z ′µN2γ

µN1 . (15)

Here N1 and N2 denote Majorana fermions and ϕ is a real scalar resulting from Φ.The thermal relic abundance of N1 is predominantly determined by co-annihilation of

N1N1, N2N2 → Z ′Z ′, ϕϕ and N1N2 → Z ′ϕ. The thermally averaged annihilation crosssections are given by the dimensionless temperature x = mN/T :

〈σv〉11 = 〈σv〉22 =9y4

64πm2N

1

x, 〈σv〉12 =

y4

64πm2N

. (16)

Here we drop terms proportional to the gauge coupling g′, because it is subdominantcompared to those with Yukawa coupling y. Note that self-annihilation channels are p-wave, while the co-annihilation channel is s-wave. The effective annihilation cross section

5Two new experiments would shed light on the longstanding tension in (g − 2)µ in the near future:E989 experiment at Fermilab [75] and the E34 experiment at J-PARC [76]. The lattice studies of hadroniccontributions to (g − 2)µ may also resolve the tension in the future. The latest study on the hadronicvacuum polarization contribution [77] claims that the discrepancy between the SM prediction and theexperimental result disappears. Their result is also in tension with the other result (such as the R-ratiodetermination), and thus it is expected to scrutinize their result in detail by the other lattice groups.

6The charge of Φ determines the neutrino mass matrix by the see-saw mechanism [78–81]. Ourchoice is a minimal viable one [82] (see also Appendix D) and may explain the baryon asymmetry of theUniverse [83] through the leptogenesis [84–87].

9

is given by the relative yield r [88]:7

〈σv〉eff = (1− r)2〈σv〉11 + r2〈σv〉22 + 2r(1− r)〈σv〉12 , r =e−∆m/T

1 + e−∆m/T. (17)

We compare the effective cross section at x = 20 with the canonical cross section, (σv)can =3 × 10−26 cm3/s. We fix the Yukawa coupling in this way. The above expression of theeffective cross section is derived under the assumption that N1 and N2 are in chemicalequilibrium. We check that in parameters of interest, the chemical equilibrium is achievedby N2 ↔ N1 + Z ′.

Again since the gauge coupling g′ is tiny, the 2-body potential of DM is also dominatedby the Yukawa coupling:

V = −αe−mϕr

r, (18)

where α = y2/(8π).8 We approximate the Yukawa potential by the Hulthen potential.We assume the Higgs mass mϕ to be larger than the gauge boson mass mZ′ , namely,mϕ > mZ′ ; otherwise the Higgs lifetime exceeds 1 s and cause cosmological problems.The left panels of Fig. 3 shows the parameter regions for σ/m = 100-200 cm2/g in MW’sdwarf spheroidal satellite galaxies (cyan), 0.1-1 cm2/g (light blue) and 1-10 cm2/g (blue)in field dwarf galaxies, 0.1-1 cm2/g in MW-size galaxies (red), and > 0.1 cm2/g (green) ingalaxy clusters, where we take vrel = 20, 30, 200, and 1000 km/s, respectively. There, wefocus on g′ and mZ′ that ameliorates the (g−2)µ discrepancy. We use the analytic resultsof the Hulthen potential [16], Eq. (9), to calculate the elastic scattering cross section ofN1; we multiply additional 1/2 factor to Eq. (1) since we consider the elastic scatteringamong identical N1 particles. It should be noted that the analytic results of the Hulthenpotential is valid only in the resonant regime, i.e., αmN1/mϕ & 1 and mN1vrel/mϕ . 1;the analytic formula for the cross section in the classical regime, i.e., αmN1/mϕ & 1 andmN1vrel/mϕ 1, is given in Ref. [63], and the Born approximation can be used in theBorn regime, i.e., αmN1/mϕ 1. However, we remark that using the correspondingformulas in each regimes do not change our conclusion. We indicate the classical andBorn regimes (brown) in the left panels of Fig. 3; for the classical regime, we require thecondition for the velocity range that covers the MW’s satellites, vrel . 100 km/s.

From Fig. 3a and Fig. 3b, we see that with a light mediator, i.e., mϕ . 20 MeV, σ/mcan be as large as ∼ 100 cm2/g in the MW’s satellites, while diminishing towards thegalaxy clusters. As an example, in the right panels of Fig. 3, we show the velocity depen-dence of σ/m (black) for the depicted benchmark parameters (as a star or a diamond) inthe left panels. In order to exhibit such large cross section at low-velocities, the param-eters should lie very close to the quantum resonance (yellow region in the left panels).The velocity dependence near the quantum resonance, σ/m ∝ 1/v2

rel, fits well the cyandata points. The presented velocity dependence of σ/m’s saturates the Unitarity bound

7As we see below, we consider an O(10) MeV ϕ, by taking a O(10−6) quartic coupling. This maydelay the phase transition of the Lµ − Lτ breaking, possibly inducing a small thermal inflation. Pleasesee Ref. [23] for a further discussion.

8α = y2/(4π) in Ref. [22] is an error, though it does not change the results much.

10

101 102 10310-3

10-2

10-1

100

101

102

103

10-2 10-1100

101

102

mZ0 = 10MeVg0 = 5.0 104

m'

<m

Z0

f( ) : (mN1 , m') = (17.8 GeV, 10.0 MeV)

( ) : (mN1 , m') = (21.9 GeV, 14.2 MeV)

dw : 0.1-1 cm2/g

dw : 1-10 cm2/g

MW : 0.1-1 cm2/g

cl : > 0.1 cm2/g

MW sat. : 100-200 cm2/g

Born regime

Classic

alreg

ime

forfor

(a)

101 102 10310-3

10-2

10-1

100

101

102

103

10-2 10-1100

101

102

dw : 0.1-1 cm2/g

dw : 1-10 cm2/g

MW : 0.1-1 cm2/g

cl : > 0.1 cm2/g

MW sat. : 100-200 cm2/g

f( ) : (mN1 , m') = (24.8 GeV, 20.0 MeV)

mZ0 = 20MeV

g0 = 5.6 104

m'

<m

Z0Born regim

e

Classic

alreg

ime

(b)

101 102 10310-3

10-2

10-1

100

101

102

103

10-2 10-1100

101

102

dw : 0.1-1 cm2/g

dw : 1-10 cm2/g

MW : 0.1-1 cm2/g

cl : > 0.1 cm2/g

MW sat. : 100-200 cm2/g

f

m'

<m

Z0

Bornregim

e

Classic

alreg

ime

mZ0 = 50MeV

g0 = 7.1 104

( ) : (mN1 , m') = (37.3 GeV, 50.0 MeV)

(c)

Figure 3: (Left): Parameter regions for σ/m at the velocity scales of MW’s dwarfspheroidal satellite galaxies (cyan), field dwarf galaxies (blue), MW-size galaxies (red),and galaxy clusters (green) for given (mZ′ , g′). The region where mϕ < mZ′ (gray) maybe disfavored by the BBN observations [22]; lifetime of ϕ is longer than 1 s. The yellowregion depicts the parameters close to the first quantum resonance for elastic scattering,i.e., 0.85 < εφ < 1.15 where εφ = αmN1/(

√2ζ(3)mϕ); this range includes the benchmark

parameters shown as stars in Fig. 1b. (Right): The velocity dependence of σ/m (black)for the benchmark parameter points near the quantum resonance (depicted in the leftpanel). The classical regime, i.e., mN1vrel/mϕ & 1, is depicted in gray, where the higherpartial-wave contributions become important. The dot-dashed curves are the Unitaritybound, σmax/m. The data points are the same as in Fig. 1a.

11

(dot-dashed) at low velocities, vrel = 20-200 km/s, where σ/m is singly parameterized bythe DM mass. The σ/m’s desaturate from the Unitarity bound from the onset of the clas-sical regime (filled circle) where the higher partial-wave contributions become significant.Parametrically, we may extend the maximally self-interacting regime by taking a largermϕ (but with a smaller α), so that σ/m ∝ 1/v2

rel all the way up to the velocity scales ofthe galaxy clusters as in Fig. 1a. However, this is not possible since the Yukawa couplingis fixed by requiring a correct DM relic abundance; therefore, we have no freedom to varymϕ for a given εφ.

With Z ′ heavier than mZ′ & 20 MeV, it is impossible to achieve such large crosssections (σ/m ∼ 100 cm2/g) at the MW’s satellites even at the quantum resonance (ifwe took the error bars at face values). This is demonstrated in Fig. 3c. There, we takemZ′ = 50 MeV as an example. The cosmological constraints restricts mϕ & mZ′ . Thus,with heavier Z ′, the constraint also restricts N1 to be heavier near the quantum resonance(yellow). Eventually, it becomes impossible to achieve ∼ 100 cm2/g in the MW’s satelliteseven if σ/m saturates the Unitarity bound; see the right panel of Fig. 3c.

DM annihilates into Z ′ and ϕ predominantly in the p-wave, which subsequently de-cay predominantly into neutrinos. It seems to follow that this DM is safe from strin-gent constraints from indirect-detection experiments. On the other hand, considering thelarge Sommerfeld-enhancement factor for the s-wave, we need to be careful about thesubdominant-modes of annihilation. A leading s-wave annihilation channel is N1N1 →Z ′Z ′:

(σv)s-wave =g′4

128πm2N

' 4.6× 10−36 cm3/s

(g′

5× 10−4

)4(20 GeV

mN

)2

. (19)

Considering the current (future) bound of DM annihilation into neutrinos mainly fromSuper-Kamiokande (Hyper-Kamiokande) [89], (σv) . 10−24 (10−25) cm3/s for 20 GeV DM,the Sommerfeld-enhancement factor of S & 1011 needs attention. The electromagneticdecay of Z ′ → e+e− via the kinetic mixing between Z ′ and the SM hypercharge gaugeboson needs another attention. Its natural value is ε = g′/70 from the muon-tau leptonloop. The electromagentic branching ratio is Br ∼ (εe/g′)2 ' 2× 10−5. We may compare(σv)Br with the constraints on DM electromagentic annihilation, e.g., from the CMBmeasurements [90], (σv) . 10−26 cm3/s for 20 GeV DM.

Before concluding this section, we describe the implications and prospects of our re-sults. We again assume the natural value of ε = g′/70 for the kinetic mixing, unlessotherwise noted. Since the light Lµ − Lτ gauge boson predominantly decays into neutri-nos, it heats only neutrinos after the neutrino decoupling, increasing the effective numberof neutrino degrees of freedom Neff [22,91]. Interestingly, its slight deviation from the SMvalue, ∆Neff ' 0.2-0.5 mitigates the tension in the Hubble expansion rate H0 betweenthe local measurements (i.e., local ladder) [92–94] and CMB-based measurements (i.e.,inverse ladder) [90, 95].9 The corresponding Z ′ mass is mZ′ = 10-20 MeV [98], whichcoincides with our “prediction” surprisingly. Such a sizable ∆Neff can be examined by

9Here we take ∆Neff = 0.2-0.5 by naıvely following [90], though it is not a perfect solution to theH0 tension [96, 97]. ∆Neff > 0 leads to a smaller size of the sound horizon at the time of the decou-pling (approximate to the last scattering), preferring a larger H0 to keep the angular scale fixed. Itcannot reproduce the “redshift dependence” of the inferred H0 in the time-delay of the strongly lensed

12

CMB-S4 experiments [99, 100]. The non-standard interactions of solar neutrinos withelectrons and nuclei, in neutrino experiments (e.g., COHERENT) and DM experiments(e.g., LZ and Darwin), examine this light Z ′ region [101, 102]. The Z ′-resonant non-standard interactions of high-energy neutrinos with cosmic neutrino background lead to a“dip” in the high-energy neutrino spectrum [91,103,104], which may explain (though notstatistically significant) the null detection of astrophysical neutrinos with 200-400 TeVin IceCube [105–107]. The light Z ′ can be probed as missing-energy events in collid-ers [108,109].

4 Composite dark matter: asymmetric dark matter

In this section, we consider composite ADM based on the dark QCD×QED dynamicswith light dark quarks, up U ′(+2/3,+1/3) and down D′(−1/3,+1/3) (dark QED charge,B −L charge) [33]. DM consists of dark nucleons, whose asymmetry has the same originas the SM baryon asymmetry (i.e., co-genesis). As a consequence, the mass of the darknucleons is similar to but slightly heavier than that of SM nucleons as indicated from thecoincidence of the mass densities Ωdm ∼ 5Ωb. In other words, the dynamical scales ofdark QCD and SM QCD are similar to each other: ΛQCD′ ∼ (10/N ′g)ΛQCD, where N ′g isthe number of the generations of U ′ and D′.

The simple model of Ref. [33] is featured by the (intermediate-scale) neutrino portaland (low-scale) dark photon portal. Non-equilibrium decay of the right-handed neutrinoswith the masses of MR > 109 GeV generates the whole baryon asymmetry (i.e., thermalleptogenesis [84–87]). The asymmetry is shared among the dark and SM sectors viathe higher-dimensional portal operator. The portal operator originates from the right-handed neutrinos and dark colored Higgs, H ′C , with the mass of MC . The dark photonis assumed to be the lightest particle in the dark sector. It carries all the dark sectorentropy in the late Universe, and transfers it through decay into SM particles via a kineticmixing with the SM photon. To this end, the dark photon mass is to be mA′ & 10 MeVand the kinetic mixing parameter to be ε & 10−10 [33, 110]. This phenomenologicalmodel enjoys compelling ultraviolet physics such as SU(5)SM×SU(4)dark [111] and mirrorSU(5)SM×SU(5)dark [112]. They involve an intermediate-scale supersymmetry and arediscussed in Appendix E.

To discuss the phenomenology, we follow the simplified version of the model given byRef. [113]. The intermediate-scale portal operators are

L ⊃ 1

M3∗

(U ′D′D′)(LH) +1

M3∗

(U ′†D′†D′)(LH) + h.c. ,1

M3∗

=yNYNYC2M2

CMR

. (20)

Hereafter we assume the CP symmetry to restrict the Yukawa coupling. YC is the Yukawacoupling of the dark colored Higgs to dark U ′D′ and U ′D′. yN and YN are the Yukawacouplings of the right-handed neutrinos to SM LH and dark H ′CD

′, respectively. Theformer is related with the observed tiny neutrino masses mν through the see-saw mecha-

systems [94] (though not statistically significant). ∆Neff > 0 may also be not preferred in light of theweak Silk damping measured in Ref. [95].

13

nism [78–81]:

y2N ∼ 10−5

( mν

0.1 eV

)1/4(

MR

109 GeV

). (21)

The generated asymmetry is washed out when the other operators that carry different B−L charges are also relevant after the asymmetry is generated. Such harmful operators arenot generated in ultraviolet physics based on a unification model discussed in Refs. [111,112]. These portal operators are to be relevant at least around the time of the thermalleptogenesis: xB−L = MR/TB−L ' 5–10 [84–87]:

MR < MC .10

x5/4B−L

( mν

0.1 eV

)1/4√YNYCMR . (22)

The first inequality is to prohibit direct decay of the right-handed neutrinos into the darkcolored Higgs, which may affect the thermal leptogenesis [114].

As shown in Ref. [33,115,116], if the asymmetry is fully shared among the dark and SMsectors, i.e., if the portal interactions are relevant in the course of the thermal leptogenesis,the DM mass is to be mN ′ ' 8.5/N ′g GeV. This is because the B − L asymmetries of theSM and dark sectors follow

ADM =N ′g(20Ng + 6m)

3Ng(22Ng + 13m)ASM , (23)

where Ng is the generation of SM fermions and m is the number of light Higgs doublets.The relation between the present B and B−L asymmetries within the SM sector dependson the nature of the electroweak symmetry breaking (EWSB) [117,118]:

AB =4(2Ng +m)

22Ng + 13mASM , sphaleron decoupling after the EWSB ,

AB =8Ng + 4(m+ 2)

24Ng + 13(m+ 2)ASM , rapid sphaleron after the EWSB ,

AB =2(2Ng − 1)(2Ng + (m+ 2))

24N2g + 14Ng − 4 +m(13Ng − 2)

ASM , rapid sphaleron + top decoupling after the EWSB .

(24)

In the first relation, we assume that the B − L number and hypercharge are conserved,while in the second and third, we assume that the B − L number and electromagneticcharge are conserved. Hereafter, we use the third relation with Ng = 3, m = 1, and

N ′g = 1, namely, ADM =44

237

97

30AB.

Since we consider the QCD-like dark sector, we may employ the QCD-calculationresults in the DM phenomenology. The 2-body dark nucleon scattering is described by

the effective-range theory as σ =1

2

(1

4σs +

3

4σt

)with

as =0.58/Λ

mπ/Λ− 0.57, res =

0.63/Λ

mπ/Λ+ 2.5/Λ ,

at =0.39/Λ

mπ/Λ− 0.49, ret =

0.0015/Λ

(mπ/Λ)3+ 2.2/Λ ,

(25)

14

for the spin singlet and triplet (deuteron) channels, respectively (in SM QCD).10 We onlyconsider the dark neutron-dark proton (n′-p′) scattering even though we also have theother scattering processes: n′-n′ and p′-p′ scattering.11 We assume that the half of thetotal DM consists of dark protons. The DM can interact with both of dark proton anddark neutron, and thus we divide each scattering cross sections by two. Here we followRef. [29] and fit the QCD-calculation results given in Ref. [119] with Λ = 250 MeV.Thisis valid for mπ/Λ ' 0-1.4. We approximate the nucleon mass as mN ′ ' 1.25NcΛ, wherethe color number is Nc = 3 in the present model [125]. We show the velocity dependenceof dark nucleon self-scattering cross section in Fig. 4 for fixed mN ′ near the quantumresonance.

As seen in Fig. 4, mN ′ ' 8.5 GeV has a constant cross section below vrel . 100km/s,barely reproducing the velocity dependence of σ/m inferred by the MW’s dwarf spheroidalsatellite galaxies. The binding energies are

Ebs ' 0 , Ebt ' 60 MeV for resonant singlet scattering ,

Ebt ' 0 for resonant triplet scattering ,(26)

where Eb ' 0 means a shallow bound/virtual state. Meanwhile, mN ′ ' 16 GeV has aσ/m ∝ 1/v2

rel in the low-velocity region, providing a better fit. The binding energies are

Ebs ' 0 , Ebt ' 110 MeV for resonant singlet scattering ,

Ebt ' 0 for resonant triplet scattering .(27)

These 2-body binding energies are important inputs for the bound-state formation in theUniverse (i.e., the dark nucleosynthesis) [126–133]. In the present model, the dark photonis typically heavier than the binding energy, and thus the 2-body bound-state formationproceeds only through the electron/positron emission and is suppressed by the kineticmixing parameter. Though it is beyond the scope of this paper, we may also be ableto predict the strength of the dark QCD phase transition. If it is the strong first order,we have a chance to examine this model in near-future gravitational-wave detectors [134]and DM may consist of dark-quark nuggets [135], changing the search strategy (e.g.,direct/indirect-detection signals) in the late Universe.

To make mN ′ ' 16 GeV consistent with Ωdm ∼ 5Ωb, we need to make portal operatorspartially irrelevant. We leave more general analysis, including the direct decay of right-handed neutrinos into the dark colored Higgs, for a future work (see Ref. [114] for a similar

10The binding energy increases with a larger pion mass [119,120]. Depending on the assumptions, onealso finds the opposite [121–123]. These results are very delicate in light of the perturbative calculationsof the effective field theory [124].

11The quantum resonance of these scattering processes may require a different dark pion mass fromthat for n′-p′ scattering since the dark isospin symmetry, among n′ and p′, is broken by U(1)dark and thecurrent dark quark masses. Therefore, we expect that we can ignore the other dark nucleon scatteringon the quantum resonance of n′-p′ scattering. When the quantum resonance of p′-p′ and n′-n′ scattering

coincides with that of n′-p′, we have to take into account each scattering cross section,1

2

2

4σp

′p′s and

1

2

2

4σn

′n′s . Here the additional factor of 2 originates from the difference in the decomposition of the

scattering amplitude into the isospin irreducible representations and from the symmetric factor of thefinal states. It is beyond the scope of this paper to make a further study of the dark nucleon scatteringin detail.

15

101 102 10310-3

10-2

10-1

100

101

102

103

101 102 10310-3

10-2

10-1

100

101

102

103

fmN 0 = 8.5 GeVm0 = 1.292 GeVm0 = 1.110 GeV

fmN 0 = 16GeVm0 = 2.432 GeVm0 = 2.090 GeV

“naive”

singlet

triplet

“naive”

singlet

triplet

Figure 4: Velocity dependence of the 2-body dark nucleon scattering for given dark nucleonmass mN ′ and dark pion mass mπ′ . The solid (dashed) curves represent the resonantsinglet (triplet) scattering case. The dot-dashed curves are the Unitarity bounds for thenucleon-nucleon scattering cross section. The “naive” Unitarity bound σmax/m wouldbe fully saturated only if both singlet and triplet scatterings were simultaneously onresonance for a given parameter set (mN ′ and mπ′). For the presented σ/m’s, only oneof the channels, i.e., singlet or triplet scattering, is on resonance and saturates its ownUnitarity bound, σs,max/m = 1/(2 · 4) × σmax/m or σt,max/m = 3/(2 · 4) × σmax/m;the unfilled circles depict the point k = 1/|a| and σ/m saturates the Unitarity boundfor higher scattering velocities. The effective-range expansion is expected to be a goodapproximation, i.e., k < 1/|re| in the presented velocity range. The data points are thesame as in Fig. 1a.

16

analysis). Hereafter, instead, we assume that the both cases (mN ′ ' 8.5 GeV, 16 GeV)

are realized with MC '(

10/x5/4B−L

)(mν/0.1 eV)1/4√YNYCMR.

In the present model, the dark photon plays a key role in releasing the dark sectorentropy to the visible sector via the kinetic mixing ε with the SM photon. Fig. 5 sum-marizes the current constraints on the dark-photon parameters (mA′ , ε) and the futureprospects in beam-dump and collider experiments. In collider experiments, dark photonscan be produced through the kinetic mixing and subsequently decay into SM particles.For mA′ < 5 GeV, BaBar [140, 141], LHCb [142], and KLOE [143–146] place the upperlimit on ε. Belle-II [147,148] will place the upper limit on ε mainly for mA′ = 10-5000 MeV.The displaced vertex searches at the LHCb experiment will also explore the region be-tween the collider and beam-dump experiments [149, 150]. The beam-dump experimentsexclude a broad parameter space for mA′ < 1 GeV: current constraints are mainly fromCHARM [151, 152], LSND [153, 154], and U70/νCal [155, 156]. The dark-photon param-eters in this region will be further explored by projected facilities like SHiP [157, 158],FASER [159], and SeaQuest [160,161].

The dark photon decay rate is given by

Γ(A′ → e+e−) =1

3αε2mA′

√1− 4m2

e

m2A′

(1 +

2m2e

m2A′

),

Γ(A′ → hadrons) =1

3αε2mA′

√1− 4m2

µ

m2A′

(1 +

2m2µ

m2A′

)R(√s = mA′) ,

(28)

where R(√s) =

σ(e+e− → hadrons)

σ(e+e− → µ+µ−)takes into account the hadronic resonances (i.e.,

vector meson dominance). In the following, we consider the latter for mA′ > 100 MeVtaken from Ref. [162]. Since the late-time decay of the dark photon heats only the electron-photon plasma after the neutrino decoupling, Neff is lowered and the upper bound is puton the dark photon lifetime. The dark photon lifetime τA′ = 1/ΓA′ exceeds 1 s in the graycrosshatched region of Fig. 5.

In the right side of the vertical solid lines of Fig. 5, the dark photon is no longer thelightest particle in the dark sector (the blue-solid vertical line for mN ′ = 16 GeV, whilethe red-solid vertical line for mN ′ = 8.5 GeV). In particular, the dark photon is heavierthan the dark pion, whose mass is indicated by the strong velocity dependence of the2-body dark nucleon scattering. The constraints on the dark photon decay would changein this region. In the left side of vertical dashed lines, the dark photon mass is less thanthe binding energy Ebt when the resonant singlet scattering is realized (we use the samecolor scheme for the vertical lines as that in the right region). In this region, the darknucleosynthesis would proceed by emitting the dark photon, and then the DM wouldconsists of the dark nuclei.

When the DM consists of the dark proton charged under U(1)dark, DM interacts withSM nuclei through the kinetic mixing between the SM photon and dark photon. Thedifferential cross section between the dark proton and target nucleus is

dσp′

dq2=

4πααDε2Z2

(q2 +m2A′)2

1

v2F 2T (q2) . (29)

17

Beam dump

SN

Collider

Future Prospects

Figure 5: Constraints on the dark-photon parameters: dark photon mass mA′ and ki-netic mixing ε. The gray meshed region is the current bounds from supernova 1987Aconstraints (Refs. [136, 137]), beam-dump experiments, and collider experiments (takenfrom Ref. [138]). The green dot-dashed line shows the future prospects for beam-dumpand collider experiments. The diagonal blue (red) lines show the upper limits on ε fromthe PandaX-II experiment [139] for mN ′ = 16 GeV (mN ′ = 8.5 GeV) and fixed U(1)dark

coupling, αD = g2D/(4π). The right side of vertical solid lines indicate the region where the

dark photon is heavier than the dark pion for mN ′ = 16 GeV (blue) and mN ′ = 8.5 GeV(red); the constraints on the dark photon decay would change in the indicated regions.The left side of vertical dashed lines indicate the region where dark photon mass is lessthan the binding energy Ebt for mN ′ = 16 GeV (blue) and mN ′ = 8.5 GeV (red); the darknucleosynthesis would proceed by emitting the dark photon, and the DM would consistof the dark nuclei.

18

Here the momentum transfer is q2 = 2µ2Tv

2rel(1 − cos θcm), where µT is the reduced mass

of DM and the target nucleus and θcm is the center-of-mass scattering angle. FollowingRef. [163], we set q2 = 2µ2

Tv2, where v = 232 km/s is the typical DM speed at the rest

frame of the Earth [164]. FT is the nuclear form factor. We discuss the dark neutron-target nucleus scattering through the magnetic moment in Appendix F, which is negligible.We place the upper limit on the kinetic mixing parameter ε from the direct-detectionconstraints by the 54 ton-day exposure of the PandaX-II experiment [139] (diagonal redand blue lines in Fig. 5). We assume that the half of the total DM consists of darkprotons. The DM mass is set to be 16 GeV (blue lines) and 8.5 GeV (red lines), and wetake U(1)dark coupling to be αD = αEM = 1/137 (solid lines) and αD = αEM/100 (dashedlines).

The intermediate-scale portal operators Eq. (20) lead to the DM decay into the darkmeson and anti-neutrino [116]. Such decaying DM can be explored in indirect ways: one isa neutrino signal from the DM decay at the Super-Kamiokande, the others are constraintson a flux from eventual decay of the decay product to the SM particles. No excess of aneutrino signal has been found at the Super-Kamiokande experiment [165], and this setsthe lower limit on the DM lifetime [166]. The decay rate is given by

ΓDM '1

64π

v2HmN ′

M6∗|W |2 . (30)

Here, vH is a VEV of the SM Higgs and W is a matrix element for DM to a darkmeson. We assume the matrix element to be W ' 0.1 GeV2(mN ′/mn)2, to which a latticecalculation result [167] is naıvely scaled. For the decaying DM with a mass of 16 GeV,the lower limit is roughly τDM = 1/ΓDM & 1023 s. We naıvely extrapolate the results ofRef. [166] below the DM mass of 10 GeV. The black-shaded region in Fig. 6 is excluded byno excess of a neutrino signal in the Super-Kamiokande experiment. The decay product,neutral dark mesons, eventually decays into e+e− through kinetic mixing. We naıvelyutilize a constraint from the e+e− flux observations (Voyager-1 [168] and AMS-02 [169])on the DM lifetime given in Ref. [170]. The e+e− flux from the DM decay constrains theDM lifetime to be larger than τDM = 1/ΓDM & 1027 s for mN ′ & 3 GeV. The gray-shadedregion in Fig. 6 shows the constraint from the e+e− flux from the DM decay. The AMS-02observation gives a dominant constraint for mN ′ & 3 GeV, while the e+e− flux constraintscomes from the Voyager-1 observation for mN ′ . 3 GeV.

The same ultraviolet origin of the operator in Eq. (20) induces the Majorana massterm for the dark neutrons [113]:

L ⊃ 1

2mM n

′n′ + h.c. , mM 'Y 2NY

2CΛ6

QCD′

4MRM4C

. (31)

Some portion of the dark neutron DM is converted into its anti-particle at the late timet through the dark neutron-anti neutron oscillation: Pn ∼ (mM t)

2. Then dark nucleonsn′ and p′ annihilate with n′ into dark pions with the effective cross section,

(σv)eff =4π

m2N ′SeffPn . (32)

19

Figure 6: Indirect-detection constraints on (mN ′ ,MR) plane. The black-shaded region isexcluded by no excess of a neutrino signal at the Super-Kamiokande experiment. Thegray-shaded region is constrained by e+e− flux from eventual decay of the dark mesonfrom DM decay (constraints by the AMS-02 for mN ′ & 3 GeV and the Voyager-1 formN ′ . 3 GeV). The green-hatched region is excluded by γ-ray constraints by the Fermi-LAT, and the blue-hatched region is excluded by e+e− flux constraints (by the AMS-02 formN ′ & 1.5 GeV and the Voyager-1 for mN ′ . 1.5 GeV). The blue dashed lines show thee+e− flux constraints with taking a Sommerfeld-enhancement factor of the annihilationcross section to be Seff = 103 and 105.

20

Here Seff is the Sommerfeld-enhancement factor Ss-wave times a fudge factor. When thedark photon is the lightest particle in the dark sector, dark pions decay into dark photons,and then dark photons eventually decay into e+e− pair. The final state radiation ofthis process can emit visible photons. The late-time annihilation is constrained by theobservations of the γ-ray from the dwarf spheroidal galaxies by the Fermi-LAT [171] andfrom the interstellar e+e− flux by the Voyager-1 [168] and the AMS-02 [169]. We usethe γ-ray limits on the oscillation time scale given by Ref. [113] with Seff = 1. Eventhough the Fermi-LAT observation constrains the effective annihilation cross section onlyfor mN ′ & 3 GeV, we use a naıve extrapolation of the constraint below 3 GeV. Thegreen-hatched region in Fig. 6 shows the constraints from the observations of the γ-rayflux. We utilize the e+e− flux constraint on the dark matter annihilation cross section inRef. [170] as a constraint on (σv)eff .

The constraints from the e+e− flux and γ-ray flux observations depend on the annihi-lation cross section of dark nucleons. The strong velocity dependence of the self-scatteringcross section implies a sizable Sommerfeld-enhancement factor as seen in Fig. 2a, and thuswe also show the lower limit on MR from the e+e− flux when we take the factor to beSs-wave = 103 (long-dashed line) and Ss-wave = 105 (short-dashed line). Though the con-straints from γ-ray flux change when we take the enhancement factor, we do not depictthem in Fig. 2a.

The lightest dark ρ meson mass is mρ′ ' 7 GeV (13.1 GeV) for mN ′ = 8.5 GeV(mN ′ = 16 GeV) from a naıve scaling of our ρ meson. From a naıve scaling of our ρmeson’s mass spectrum, we also expect that higher vector resonances appear as m2

ρn ∼76n GeV2 (270n GeV2) (n = 1, 2, . . . ) for mN ′ = 8.5 GeV (mN ′ = 16 GeV). With a siz-able kinetic mixing parameter ε & 10−3, we have a chance of spectroscopic measurementsof such resonances in lepton colliders (e.g., Belle II and BES-III) [172,173].

5 Conclusion

We have studied the possibility that self-interacting dark matter has a maximal self-scattering cross section from the Unitarity point of view. When the self-scattering crosssection nearly saturates the Unitarity bound, the cross section is singly parameterizedby the DM mass. Interestingly, for the DM mass of m ∼ 10 GeV, maximally self-interacting dark matter can explain the observed structure of the Universe ranging fromσ/m ∼ 0.1 cm2/g at galaxy clusters to ∼ 100 cm2/g at the MW’s dwarf spheroidal satellitegalaxies. We have demonstrated that a general requirement of the quantum (zero-energy)resonant scattering, by employing the effective-range theory and analytic results with theHulthen potential. It requires a light force mediator and model parameters to be thespecial (fine-tuned) values. We have also demonstrated that DM annihilation tends to belargely boosted by the Sommerfeld enhancement with the same parameters.

It requires DM annihilation to end up with neutrinos or to be the p-wave. As such amodel, we have considered the gauged Lµ−Lτ model extended with Dirac DM. The DiracDM is slightly split into two Majorana DM by the VEV of the gauged Lµ − Lτ Higgs.We have taken the Higgs mode to be light so that it mediates the (relatively) long-rangeforce between DM particles. We have pinned down the parameter points that lead to

21

the DM self-scattering saturating the Unitarity bound. With them, we can explain thediscrepancy in the muon g − 2, predict the sizable ∆Neff mitigating the tension in H0,and have rich implications for non-standard neutrino interactions and collider searches.

ADM is another option to evade the indirect-detection bounds from the boosted DMannihilation (at the first sight). Since ADM requires a lighter state to deplete its sym-metric components through efficient annihilation, such a light state is a natural candidatefor a force mediator. We have considered dark nucleon ADM with dark pions being theforce mediator, based on the dark QCD×QED dynamics. The dark pion mass is verypredictive to explain the quantum-resonant self-scattering between dark nucleon DM. Forviable cosmology, we have assumed that dark photon is the lightest state in the darksector, which provide various ways to probe the dark sector: collider and beam-dumpexperiments and DM direct-detection experiments. Furthermore, the binding energies oftwo nucleon states are also predicted, which are important for the dark nucleosynthesisthat proceeds with a light dark photon. The predicted dark vector resonances can bemeasured by lepton colliders.

It is worthwhile to note that we have compared the velocity-dependent cross sectionswithout the local velocity distribution averaging with the inferred values of σ/m fromastronomical observations. This is partially because the inferred values of σ/m wereinterpreted by assuming a constant cross section inside halos of interest. It would beencouraging to reanalyze the astronomical data with a proper distribution averaging.The velocity-dependent cross section of maximally self-interacting dark matter may givea benchmark for the reanalyses.

We have shown that the recent astronomical data for structure formation of the Uni-verse are already interesting and have rich implications for DM physics. Though it is tobe clarified which data are robust or suffer from astrophysical uncertainties such as super-nova feedback, we are optimistic that the situation will get quickly improved with the fastdevelopment of hydrodynamic simulations and more precise observations. Gravitationalprobes of DM and related beyond-WIMP DM scenarios enjoy an exciting era.

Acknowledgement

The work of A. K. and T. K. is supported by IBS under the project code, IBS-R018-D1.A. K. thanks Keisuke Yanagi for former collaborations, without which this work was notinitiated.

22

A MW’s dwarf spheroidal galaxies vs field dwarf galax-

ies

As seen in Fig. 1a, the inferred values of σ/m, in the MW’s dwarf spheroidal satellitegalaxies (cyan) [47] and the field dwarf galaxies (red) [3], seem not compatible with eachother, though they have similar collision velocities, vrel ∼ 20-60 km/s. In the main text, wediscuss implications for DM phenomenology, taking the former seriously. In this section,instead, we discuss how our conclusions changed, if the inferred values of σ/m from theMW’s satellites were lowered.

In the left panel of Fig. 7a, we take the effective-range theory parameters so that σ/mcrosses the inferred values from the field dwarf galaxies (red), rather than the ones fromMW’s dwarf spheroidal satellite galaxies (cyan) as in Fig. 1a. In this case, we do notneed a strong velocity dependence of σ/m as in Fig. 1a. Notice that the |a/re|’s tookare relatively smaller than the ones in Fig. 1a; this indicates that the parameters arerelatively away from the quantum resonance. This is explicitly shown in Fig. 7b, wherethe points depicted by stars represent the Hulthen-potential parameters that correspondto the effective-range theory parameters in left panel of Fig. 7a. Notice that the εφ’s donot need to be so close to the first quantum resonance, i.e., εφ ' 1, to cross the reddata points in Fig. 7a. This is in contrast to the parameters took in Fig. 1a, where theeffective-range theory parameters need to be fine-tuned to be near the quantum resonanceto cross the cyan data points, as shown in Fig. 1b.

At the same time, the parameters of Fig. 7a are also relatively away from the quan-tum resonance for annihilation; compare the points depicted by stars in Fig. 8 with theones in Fig. 2. Consequently, they exhibit relatively smaller Sommerfeld-enhancementfactors than the ones of Fig. 1a, making them less constrained from the indirect-detectionexperiments.

The lack of need for the strong velocity dependence amounts to relaxation of the“prediction” of the narrow model parameter regions of the gauged Lµ − Lτ model in themain text. In the left panels of Fig. 9, we depict the example parameters of the modelthat explains the inferred values of σ/m from the field dwarf galaxies, i.e., σ/m ∼ 1 cm2/gat vrel ∼ 20 km/s. We show the corresponding velocity dependence of σ/m in the rightpanels. In Fig. 9a, we present the case of mZ′ = 10 MeV, which lies in the Z ′ massrange preferred to explain the inferred σ/m from the MW’s satellites, mZ′ . 20 MeV,as discussed in the main text. Contrary to the parameter points depicted in Fig. 3a, wesee that the parameters here do not need to be so fine-tuned to be close to the quantumresonance (yellow) to explain the red data points. Furthermore, the viable Z ′ mass rangeextends towards larger values, compared to the case discussed in the main text. Forheavier Z ′, e.g., mZ′ = 50 MeV, it is possible to find parameters to explain the inferredσ/m from the field dwarf galaxies, while they need to lie near the quantum resonance.

For the ADM based on the dark QCD×QED dynamics, the notable change wouldbe that the dark nucleon mass of mN ′ ' 8.5 GeV also becomes preferable. In Fig. 4, weshowed thatmN ′ ' 8.5 GeV poorly fits the inferred velocity dependence of σ/m inferred bythe MW’s dwarf spheroidal satellite galaxies. But if we focus on σ/m inferred by the fielddwarf galaxies, mN ′ ' 8.5 GeV also provides a good fit to the inferred velocity dependence

23

(see Fig. 10), where we do not need to consider additional entropy production to dilutethe DM abundance or consider asymmetric generation of B −L number. Meanwhile, thepreferred dark pion masses in both Fig. 4 and Fig. 10 do not differ much, i.e., the pionmass (or mπ′/Λ) still has to be reasonably tuned near the quantum resonance for elasticscattering. Similarly, the binding energy of nucleons, Ebt, decrease by a factor of ∼ 0.9as we take pion mass slightly away from the quantum resonance to explain the velocitydependence of σ/m inferred by the field dwarf galaxies. This would slightly retreat theleft shaded region in Fig. 5 to the further left, in which the dark nucleosynthesis mayoccur by emitting dark photons.

B Velocity averaging

In this section, we comment on the velocity averaging of the self-scattering cross sectionσ/m.

The horizontal axis of Fig. 1a, 〈vrel〉, is the average relative velocity between two DMparticles. Consider the collision between particle 1 and 2. Assuming the Maxwellianvelocity distribution for both particles, but with different 1d velocity dispersion: σ2

1 andσ2

2, respectively. After integrating out the center-of-mass velocity, whose 1d velocity

dispersion is σ2cm =

σ21σ

22

σ21 + σ2

2

, we obtain the distribution function for the relative velocity:

d3vrel1

(2πσ2rel)

3/2exp

(− v2

rel

2σ2rel

), (33)

where the 1d velocity dispersion is related with the average relative velocity as

σ2rel = σ2

1 + σ22 =

π

8〈vrel〉2 . (34)

In the gravitationally interacting system, we expect σ1 = σ2 = σ and thus 〈vrel〉 = 4σ/√π

as found in the literature [3]. Note that this differs from the kinetically equilibrium system,where m1σ

21 = m2σ

22.

Meanwhile, the inferred values of σ/m at given 〈vrel〉 in Fig. 1a are achieved by as-suming a constant self-scattering cross section inside a halo of interest. In the case ofvelocity-dependent σ/m, a fairer comparison may be done by taking local distributionaverage for the cross section. In the main text, we do not take the distribution averagefor the cross section. This is partially just for the simplicity. Another reason is that whenastronomical data are interpreted by SIDM, a self-scattering cross section is assumed tobe constant. Thus, to be consistent, we need to reanalyze the data by taking into accountthe velocity (dispersion) dependence of the (distribution averaged) cross section.

C Higher resonances

As seen in Fig. 1b, the quantum resonances for elastic scattering appear at εφ = n2

(n = 1, 2, . . . ). In the main text, we focused on the parameters near the first one, n = 1.

24

101 102 10310-3

10-2

10-1

100

101

102

103

101 102 10310-3

10-2

10-1

100

101

102

103

fm = 20GeV

( ) (↵, m) = (3.02 102, 0.4 GeV)( ) (↵, m) = (3.93 103, 0.06 GeV)

fm = 20GeV

( ) (a, a/re) = (23.9 fm, 12.1)( ) (a, a/re) = (23.9 fm, 1.53)

(a)

100 101-600

-400

-200

0

200

400

600

100 101100

101

102

(b)

Figure 7: (a) Same as Fig. 1a, but for the effective-range theory parameters, (a, re), thatexplains the inferred values of σ/m from the observations on the field dwarf galaxies (red).The Hulthen-potential parameter sets took in the right panel exhibits the same effective-range theory parameter sets as the left panel. (b) Same as Fig. 1b, but the stars indicatethe benchmark parameters took in Fig. 7a. Compared to the parameters took in Fig. 1a,the parameters here are relatively away from the quantum resonance at εφ = 1.

25

v = 103

v = 102

v = 101

v = 1

100 101 102100

101

102

103

104

105

106

10-7 10-6 10-5 10-4 10-3 10-2 10-110-1

100

101

102

103

104

fm = 20GeV

( ) (↵, m) = (3.02 102, 0.4 GeV)( ) (↵, m) = (3.93 103, 0.06 GeV)

(a)

v = 103

v = 102

v = 101

v = 1

10-7 10-6 10-5 10-4 10-3 10-2 10-1

100

101

100 101 102

100101102103104105106107108

fm = 20GeV

( ) (↵, m) = (3.02 102, 0.4 GeV)( ) (↵, m) = (3.93 103, 0.06 GeV)

(b)

Figure 8: Same as Fig. 2, but with the Hulthen-potential parameters took in Fig. 7a. Thes-wave annihilation is relatively away from the quantum resonance compared the onesin 1a; the enhancement factors are smaller, and the s-wave enhancement factors growtowards low velocity slower than the resonant behavior, i.e., ∝ 1/v2

rel (brown).

26

101 102 10310-3

10-2

10-1

100

101

102

103

10-2 10-1100

101

102

f( ) : (mN1 , m') = (13 GeV, 10 MeV)

( ) : (mN1 , m') = (28 GeV, 16 MeV)

dw : 0.1-1 cm2/g

dw : 1-10 cm2/g

MW : 0.1-1 cm2/g

cl : > 0.1 cm2/g

MW sat. : 100-200 cm2/g

mZ0 = 10MeVg0 = 5.0 104

m'

<m

Z0

Born regime

Classic

alreg

ime

forfor

(a)

101 102 10310-3

10-2

10-1

100

101

102

103

10-2 10-1100

101

102

dw : 0.1-1 cm2/g

dw : 1-10 cm2/g

MW : 0.1-1 cm2/g

cl : > 0.1 cm2/g

MW sat. : 100-200 cm2/g

f( ) : (mN1 , m') = (52 GeV, 100 MeV)

( ) : (mN1 , m') = (35 GeV, 50 MeV)

m'

<m

Z0

Bornregim

e

Classic

alreg

ime

mZ0 = 50MeV

g0 = 7.1 104

forfor

(b)

Figure 9: Same as the left panel of Fig. 3, but the with different benchmark parameterpoints; here, the depicted points (as a star and a diamond) explain the red data points,rather than the cyan ones as in Fig. 3.

101 102 10310-3

10-2

10-1

100

101

102

103

101 102 10310-3

10-2

10-1

100

101

102

103

fmN 0 = 16GeV

fmN 0 = 8.5 GeVm0 = 1.289 GeVm0 = 1.108 GeV

m0 = 2.430 GeVm0 = 2.088 GeV

“naive”

singlet

triplet“naive”

singlet

triplet

Figure 10: Same as the Fig. 4, but for the dark pion masses that explain the inferredvalues of σ/m from the field dwarf galaxies.

27

101 102 10310-3

10-2

10-1

100

101

102

103

101 102 10310-3

10-2

10-1

100

101

102

103

fm = 20GeV

( ) (a, a/re) = (292 fm, 22.5)( ) (a, a/re) = (292 fm, 152) f

m = 20GeV( ) (↵, m) = (3.09 102, 0.4 GeV)( ) (↵, m) = (1.85 102, 0.06 GeV)

(a)

100 101-600

-400

-200

0

200

400

600

100 101100

101

102

(b)

Figure 11: Same as Fig. 1a, but comparing the parameter points close to the first (cyanstar) and the second (blue star) quantum resonance for elastic scattering, i.e., n = 1 andn = 2 for εφ = n2, respectively. In the right panel of Fig. 11a, for the parameter pointnear the second resonance (dashed), the analytic results for the for the Hulthen potentialexhibits diminishing σ/m at vrel ∼ 2000 km/s.

28

v = 103

v = 102

v = 101

v = 1

100 101 102100

101

102

103

104

105

106

10-7 10-6 10-5 10-4 10-3 10-2 10-110-1

100

101

102

103

104

fm = 20GeV

( ) (↵, m) = (3.09 102, 0.4 GeV)( ) (↵, m) = (1.85 102, 0.06 GeV)

(a)

10-7 10-6 10-5 10-4 10-3 10-2 10-1100

101

102

103

104

105

106v = 103

v = 102

v = 101

v = 1

100 101 102100101102103104105106107108

fm = 20GeV

( ) (↵, m) = (3.09 102, 0.4 GeV)( ) (↵, m) = (1.85 102, 0.06 GeV)

(b)

Figure 12: Same as Fig. 2, but for the parameter points depicted in Fig. 11b, which areclose to the first (cyan star) and the second quantum resonance (blue star) for elasticscattering. In the left panels, the brown curve corresponds to ∝ 1/v2

rel. Contrary to thecyan-star parameter set, which was discussed in Fig. 2, the blue-star parameter set isclose to both s-wave and p-wave resonances for annihilation.

29

While it is certainly encouraging to discuss the higher resonances as well, there are acouple of reasons we try not to. In this section, we discuss these reasons and remark onthe possible differences and difficulties in investigating higher resonances.

The first aspect is that at higher resonances for elastic scattering, σ/m could exhibitdiminishing values at specific momentum. This is demonstrated in the right panel ofFig. 11a, where we show a velocity dependence of σ/m in the Hulthen potential. There,we see that for the Hulthen-potential parameter set near the second resonance, i.e., theblue star of Fig. 11b, σ/m is diminishing around vrel ∼ 2000 km/s. This is related tothe behavior of the phase shift δ0 at the second resonance. At the vicinity of the secondresonance, δ0 starts out from 0 at high-k limit and approaches ' 3π/2 in the k → 0 limit;between the two limits, δ0 = π for some specific value of k and σ/m diminishes at such k.One may expect that the existence of this “dip” generally makes the parameters hard tobe compatible with the data points in Fig. 11a. However, the “dip” appears in the classicalregime (depicted in gray), i.e., k & mφ, where the higher partial-wave contributions aresignificant. After taking into account the higher partial-wave contributions, the “dip”may become relaxed. We could also consider a larger mφ (but with the same εφ) to pushthe onset of the classical regime (filled gray circle) to higher velocities, so that the “dip”does not overlap with the velocity scales of the astronomical data.

The second aspect is related to the Sommerfeld enhancement of DM annihilation. Aswe consider the resonances beyond the first one for elastic scattering, higher partial waveresonances for annihilation may become important. As an example, in Fig. 12, we showthe velocity dependence of the Sommerfeld-enhancement factors for the parameter pointnear the second resonance (blue star), which corresponds to the parameter took in theright panel of Fig. 11a. Coincidentally, the parameter lies near the s-wave and p-waveresonance for annihilation. For this parameter point, taking p-wave annihilating DMto evade the constraints from indirect-detection experiments (as for the cyan parameterpoint) may not help much. Likewise, the implications on the constraints may highlydepend on the value of the took εφ for higher resonances of elastic scattering. A dedicatedinvestigation on this aspect for higher resonances may be done elsewhere, while we try tofocus on the simplest case in this work.

D Neutrino masses in the gauged Lµ − Lτ model

The renormalizable Lagrangian density can be written as

L =LSM + g′Z ′µ

(L†2σ

µL2 − L†3σµL3 − µ†σµµ+ τ †σµτ)− 1

4Z ′µνZ

′µν − 1

2ε Z ′µνB

µν

+ (DµΦ)†DµΦ− V (Φ, H) + iN †i σµDµNi

− λeL1HNe − λµL2HNµ − λτL3HNτ −1

2MeeNeNe −Mµτ NµNτ − yeµΦ∗NeNµ − yeτΦNeNτ + h.c.

(35)

Here, Li (i = 2, 3) denotes the left-handed leptons in the flavor basis, while µ and τdenote the right-handed charged leptons. The discussion below is applicable to any Z ′

mass, though we focus on mZ′ ∼ 10 MeV in the main text. With a different Z ′ mass,

30

phenomenology in colliders and cosmology also changes (e.g., see Ref. [174] for very lightZ ′).

After the SM Higgs H and Lµ − Lτ breaking Higgs Φ develop the VEVs, vH/√

2 andvΦ/√

2, the resultant neutrino mass sector takes a form of

−Lmass =(νe νµ ντ

)MD

Ne

Nµ

Nτ

+1

2

(Ne Nµ Nτ

)MN

Ne

Nµ

Nτ

+ h.c. ,

MD =

Ye 0 00 Yµ 00 0 Yτ

, MN =

Mee Meµ Meτ

Meµ 0 Mµτ

Meτ Mµτ 0

.

(36)

Here (Ye, Yµ, Yτ ) =vH√

2(λe, λµ, λτ ) and (Meµ,Meτ ) =

vΦ√2

(yeµ, yeτ ). We fix the phases of

Nl so that Meµ,Meτ ,Mµτ > 0, while leaving the phases of νl for the PMNS parametriza-tion [175, 176]. The see-saw mechanism [78–81] provides the neutrino mass term at lowenergy as

−Lmass '(νe νµ ντ

)Mν

νeνµντ

=(ν1 ν2 ν3

)m1 0 00 m2 00 0 m3

ν1

ν2

ν3

,

Mν = −MDM−1N MT

D ,

νeνµντ

= Uν

ν1

ν2

ν3

,

(37)

where the PMNS matrix is parametrized by

Uν =

1 0 00 cos θ23 sin θ23

0 − sin θ23 cos θ23

cos θ13 0 sin θ13e−iδ

0 1 0− sin θ13e

iδ 0 cos θ13

cos θ12 sin θ12 0− sin θ12 cos θ12 0

0 0 1

1 0 00 eiα2/2 00 0 eiα3/2

,

(38)

with θij ∈ [0, π/2] and δ, αi ∈ [0, 2π). and the mass eigenvalues are written as

m22 = m2

1 + δm2 , m23 = m2

1 + ∆m2 + δm2/2 , (39)

for the normal ordering, while m3 < m1 < m2 for the inverted ordering. The neutrinooscillation parameters are summarized in Table 1.

It is remarkable that the U(1)Lµ−Lτ symmetry restricts MD to be a diagonal matrix.Thus if MD has a inverse matrix, the following relation holds:

M−1ν = −M−1

D MN (M−1D )T . (40)

Since M−1D is also diagonal, the flavor structure of M−1

ν should follow the structure ofMN. Two zero entries of MN (see Eq. (36)) provides four constraints for the PMNSparameters [82]: δ ' π/2 or 3π/2, m ' 0.05-0.1 eV, α2/π ' 0.6, and α3/π ' 1.4 fromδm2, ∆m2, and θij given in Table 1. The other entries provide 8 constraints, while there

31

Parameter best fit 1σ range 2σ rangeδm2/(10−5 eV2) 7.37 7.21–7.54 7.07–7.73∆m2/(10−3 eV2) 2.525 2.495–2.567 2.454–2.606

sin2 θ12/10−1 2.97 2.81–3.14 2.65–3.34sin2 θ23/10−1 4.25 4.10–4.46 3.95–4.70sin2 θ13/10−2 2.15 2.08–2.22 1.99–2.31

δ/π 1.38 1.18–1.61 1.00–1.90

Table 1: Neutrino oscillation parameters from Ref. [82,177].

are 11 model parameters (6 from Yl and 5 from Mll′). This is because Eq. (40) is invariantunder the following transformation:

Ye →ab

cYe , Yµ →

ac

bYµ , Yτ →

bc

aYτ ,

Mee →a2b2

c2Mee , Meµ → a2Meµ , Meτ → b2Meτ , Mµτ → c2Mµτ ,

with a, b, c being real. We obtain Table 2 by fitting the other eight model parameters inEq. (40). The mass matrices are rewritten as

MD =

Y ′eab

c0 0

0 Y ′µac

b0

0 0 Y ′τbc

a

, MN =

M ′ee

a2b2

c2M ′

eµa2 M ′

eτb2

M ′eµa

2 0 M ′µτc

2

M ′eτb

2 M ′µτc

2 0

. (41)

Parameter best fit in Table 1Y ′e/(1 eV) 0.360− 0.0819 iY ′µ/(1 eV) 0.153 + 0.189 iY ′τ/(1 eV) −0.180− 0.219 iM ′

ee/(1 eV) −1.98 + 0.0522 iM ′

eµ/(1 eV) 1M ′

eτ/(1 eV) 1M ′

µτ/(1 eV) 1

Table 2: Model parameters from Eq. (40). We fix M ′eµ = M ′

eτ = M ′µτ = 1 eV (prime

denotes this fixing) by using Eq. (41).

E Supersymmetric realization of composite ADM

A supersymmetric realization of composite ADM scenario [111, 112] may cause the fastDM decay as in nucleon decay in supersymmetric grand unified theories [178,179]. Heavy

32

particles in ultraviolet physics induce the intermediate-scale portal operators with themass dimension-six rather than the mass dimension-seven,

L ⊃∫d2θ

yNYNYCMCMR

(U ′D′D′)(LH) + h.c. (42)

The equilibrium condition now reads

MR < MC .100

x3/2B−L

( mν

0.1 eV

)1/2

YNYCMR . (43)

The decay rate is given by

ΓDM 'L2

32π

mN ′

M4∗

v2HM

21/2

M4S

|W |2 , 1

M2∗

=yNYNYCMCMR

. (44)

MS and M1/2 denote typical mass scales of supersymmetric scalar particles and super-symmetric fermionic particles, respectively. L denotes a typical one-loop factor. Theintermediate-scale portal operator Eq. (42) vanishes when the generation N ′g = 1 due toanti-symmetrization over color indices. Hereafter, we assume N ′g = 2.

The supersymmetric particles change the relation between the B − L asymmetries inthe SM and dark sectors:

ADM =2N ′g(20Ng + 3m)

3Ng(13m+ 44Ng)ASM , (45)

with the full supersymmetric particles are available. On the other hand, they do notchange the relation between the present relation between the B and B − L asymmetries

in the SM sector, AB =30

97ASM, with the decoupling limit of supersymmetric particles

and heavy Higgses. As a result, mN ′ = 4.3 GeV for m = 2 (MSSM), Ng = 3, and N ′g = 2.As with non-supersymmetric realization, we also have the Majorana mass term for the

dark neutrons in the supersymmetric realization. The corresponding superpotential hasthe mass dimension seven, and is given by

L ⊃ − Y 2NY

2C

2MRM2C

∫d2θ(U ′D′D′)2 + h.c. (46)

This operator induces the dimension nine operator at the mass scale of supersymmetricparticles with two-loop diagrams [180], and gives the Majorana mass term for the darkneutrons. A typical size of the Majorana mass is

mN 'Y 2NY

2C

2MRM2C

L2M21/2

M4S

Λ6QCD′ . (47)

Fig. 13 shows the indirect-detection constraints on the MR-MS plane from cosmic rayobservations. In this figure, we assume that the loop factor is L = 0.01, and M1/2 =10−4MS that is a spectrum motivated by split supersymmetry scenario [181–183]. Weshow only the constraint from the e+e− flux observations in this figure (the blue-hatched

33

Figure 13: Constraints on MR-MS plane in a supersymmetric realization of compositeADM. The blue-hatched region and the blue-dashed lines are the same as in Fig. 6. TheLSP abundance from the gravitino decay exceeds the observed DM abundance in themeshed-purple region in thermal leptogenesis scenario.

region), and we ignore the other bounds from the neutrino signal from the DM decay, thee+e− flux from the DM decay, and the γ-ray flux from the ADM annihilation. As we seein Fig. 6, the γ-ray constraint on the annihilation cross section is comparable with thee+e− flux constraint.

In supersymmetric realization, the lightest supersymmetric particle (LSP) can alsobe stable due to the R-parity. We have the LSPs in each of visible and dark sectors,and then the heavier state can decay into the lighter one through portal interactions atthe late time. The supersymmetric counterpart of the kinetic mixing between the darkphoton and SM hypercharge gauge boson leads the prompt decay of the heavier stateto the lighter one when the higgsinos are the lightest particles in each of sectors [111].Here, the higgsino in the visible sector, called visible higgsino, is the supersymmetricpartner of the SM Higgs, while the one in the dark sector, called dark higgsino, is that ofthe U(1)dark breaking Higgs. Even though the dark higgsino is less constrained than thevisible higgsino, the LSP abundance from the gravitino decay would cause the overclosureof the Universe [184,185]. The purple-meshed region in Fig. 13 shows the LSP abundanceexceeds the observed DM abundance when TR 'MR and the LSP with a mass of M1/2.

The constraints from the LSP abundance may be naturally relaxed in the modelswith an intermediate-scale dark grand unification: SU(5)SM×SU(4)dark [111] and mirrorSU(5)SM×SU(5)dark [112]. They associate an intermediate-scale dark monopole in theSU(4)dark →SU(3)dark× U(1)dark phase transition. Their abundance is determined by the

34

pair-annihilation as [186,187]12

nms∼ 1

B

2g4D

π

√45

4π3g∗

Mm

MPl

= 6.9× 10−14

(αDαEM

)3(Mm

1010 GeV

). (48)

Here we use g∗ = g∗MSSM + g∗dark + g∗N and g∗MSSM = 228.75 (full MSSM multiplets),g∗dark = 131.25 (2 generations of U ′/U ′ and D′/D′, g′, A′, and φD/φD multiplets), andg∗N = 11.25 (3 generations of N multiplets). Mm denotes the mass of the dark monopole,

and MPl the reduced Planck mass. B is a dimensionless quantity defined by B = q2∗ζ(3)

4π2g2D

,

where q2∗ =

91

3is the summation of the dark charges squared with a weight of 1 (3/4) for

bosons (fermions).The entropy production factor is

Safter

Sbefore

=Teq

Tann

=

(g∗(Tann)T 3

eq

g∗(Tc)T 3c

)1/4

' 0.002

(αDαEM

)21/8(1 GeV

mA′

)3/4(Mm

1010 GeV

)3/2

.

(49)

Note that this expression is valid only for Safter/Sbefore > 1; otherwise, Safter/Sbefore = 1(i.e., no entropy production). Here Tc ∼ mA′/gD is the critical temperature of the U(1)dark

phase transition. Tann is the temperature at which monopole annihilates. In the secondequality, we take g∗(Tc) = g∗(Tann). Teq is the temperature at which monopole dominationbegins:

Teq =4

3Mm

nms. (50)

In the above discussion, we assume that the SM and dark sectors are in equilibriumin the course of the monopole domination, while it may not be valid for a tiny kineticmixing. Monopole annihilation may also produce the LSP [188], while we leave furtheranalysis for a future work.

F Direct detection through the magnetic moment of

the dark neutron

Similarly to the dark proton scattering with SM nuclei, the dark neutron carries themagnetic moment under U(1)dark:

L ⊃ µn′

2n′σµνn′FA′

µν . (51)

Here we take the magnetic moment to be µn′ = (gn′/2)(eD/2mN ′) with the g-factors beinggn′ = −3.83 (gp′ = 5.59 for the dark proton) in analogy to the SM nucleons. The matrixelement for the dark neutron-SM proton scattering is

Mn′ =µn′/eD2mN ′

[q2 + 4mN ′ONR

5 + 2gpmN ′

mN

(ONR

4 q2 −ONR6

)]Mp′ , (52)

12Note that Ref. [187] considers SO(3)→U(1) and thus, the monopole charge is 4π/gD, while in thepresent case, it is 2π/gD.

35

where Mp′ is the matrix element for the dark proton-SM proton scattering, and

ONR4 = ~sn′ · ~sp , ONR

5 = i~sn′ · (~q × ~v⊥) , ONR6 = (~sn′ · ~q)(~sp · ~q) . (53)

We refer readers to Ref. [189] for exchange of SM massless photon, rather than massivedark photon. ~sp(n′) is the spin vector of the SM proton (dark neutron, i.e., DM) and~v⊥ = ~v − ~q/(2µN). We can further evaluate the matrix element for the dark neutron-target nucleus scattering with the form factors of FN1N2

i,j [189,190], while we leave it for afuture work. In orders of magnitude, the direct-detection bounds on ε from dark-neutronscattering is weakened by a factor of 1/v2 ∼ 106 compared to the dark proton-targetnuclei scattering.

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