Light Chiral Dark Sector

Keisuke Harigaya1, 2 and Yasunori Nomura1, 2 1Department of , University of California, Berkeley, California 94720, USA 2Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA An interesting possibility for dark is a scalar of of order 10 MeV – 1 GeV, interacting with a U(1) gauge (dark ) which mixes with the photon. We present a simple and natural model realizing this possibility. The dark matter arises as a composite pseudo Nambu- (dark ) in a non-Abelian gauge sector, which also gives a mass to the . For a fixed non-Abelian gauge group, SU(N), and a U(1) charge of the constituent dark , the model has only three free parameters: the dynamical scale of the non-Abelian , the gauge coupling of the dark photon, and the mixing parameter between the dark and . In particular, the gauge of the model does not allow any mass term for the dark quarks, and stability of the dark pion is understood as a result of an accidental global symmetry. The model has a significant parameter space in which thermal relic dark comprise all of the dark matter, consistently with all experimental and cosmological constraints. In a corner of the parameter space, the discrepancy of the g − 2 between experiments and the standard model prediction can also be ameliorated due to a loop contribution of the dark photon. Smoking-gun signatures of the model include a monophoton signal from the e+e− collision into a photon and a “dark rho .” Observation of two processes in e+e− collision, the mode into the dark photon and that into the dark , would provide strong evidence for the model.

I. INTRODUCTION mass term for the constituent “dark quarks.” The model, therefore, is fully (i.e. not only technically) The identity of the dark matter of the universe is un- natural—the of all the new arise known. An interesting possibility is that it is a relatively from the dynamics of new gauge interactions. light particle of mass of order 10 MeV – 1 GeV, interact- • Despite the lack of dark masses, all the new ing with the standard model particles through a U(1) particles of the model have nonzero masses. In par- (dark photon) that mixes with the pho- ticular, both the dark pion and dark photon obtain ton [1–3]. This avoids stringent constraints from dark masses of the same order, ≈ e Λ/4π, where e is matter direct detection experiments [4, 5] while still al- D D the gauge coupling associated with the dark pho- lows for understanding the current dark matter abun- ton. dance as the thermal relic from the early universe, if the mixing between the dark and standard model photons is • The stability of the dark pion is ensured as a result adequately suppressed. of an accidental symmetry of the model. This ac- This potentially elegant scenario, however, suffers from cidental symmetry is an extremely good symmetry the issue of naturalness. First of all, if the dark matter unless the cutoff scale of the theory is very low. is a , then the constraint on its late annihilations from observations of the cosmic microwave background We find that the model is phenomenologically viable excludes the scenario [6–8]. This (essentially) forces the and has interesting implications. In particular, dark matter to be a scalar, in which case the constraint is avoided because of the p-wave suppression of the anni- • There is a significant parameter region in which the hilation cross section [9]. This, however, raises the ques- dark pion comprises the dark matter of the uni- tion: why do we have such a light scalar? This is puzzling, verse. In this region, the dark pion mass is compa- arXiv:1603.03430v2 [hep-ph] 10 Aug 2016 especially given that a scalar mass is generally unstable rable but smaller than the dark photon mass, which under quantum corrections. A similar question can also is of order 10 MeV – 1 GeV. The mixing parameter be raised for the dark photon, whose mass must be in between the dark and standard model photons is in the same range as the scalar mass for phenomenological the range ≈ 10−4.5 – 10−2.5. reasons. In this paper, we present a simple model addressing • In a corner of the allowed parameter region, the dis- this issue, in which the dark matter arises as a “dark crepancy of the muon g −2 between the experimen- pion” of new gauge interactions with the dynamical scale tal result and standard model prediction [10, 11] is Λ ≈ O(10 MeV – 1 GeV). The dark photon is introduced ameliorated due to a loop contribution of the dark by gauging a U(1) subgroup of the flavor symmetry of photon. Clearly, the model may also ameliorate the this sector so that its mass is generated by the same muon g − 2 discrepancy even if the dark pion does dynamics as the one forming the dark pion. The model not comprise all of the dark matter. has the following salient features: • The model predicts a plethora of new resonances • Gauge symmetry of the model does not allow any around 10 MeV – 10 GeV, some of which may 2

GD = SU(N) U(1)D U(1)B U(1)P quarks condense,1 Ψ1  1 1 1 D † †E D † †E Ψ2  −1 1 −1 Ψ Ψ¯ + Ψ Ψ¯ = Ψ Ψ¯ + Ψ Ψ¯ 6= 0, (1) ¯ ¯ 1 1 1 1 2 2 2 2 Ψ1  −a −1 −1 Ψ¯ ¯ a −1 1 2  breaking the axial part of the approximate SU(2)L × SU(2)R flavor symmetry. The U(1)D gauge symmetry is TABLE I. Charge assignment of the dark quarks under the spontaneously broken by this condensation, which can be taken to be real and positive without loss of generality GD and U(1)D gauge groups. Here, Ψ1,2 and Ψ¯ 1,2 are left- handed Weyl spinors. The charges under accidental global by a phase rotation of dark quark fields. The low energy symmetries U(1)B and U(1)P are also shown. physics below Λ is thus dictated by three (pseudo and would-be) Nambu-Goldstone πi(x)(i = 1, 2, 3). We define a non-linear sigma model field U(x) by

be detectable in future experiments. In particu- " 3 # i X lar, one of the lowest-lying -one C- and P -odd U(x) = exp π (x)σi . (2) f i states, the dark rho meson ρD3 , mixes with the dark i=1 photon and hence couples to the standard model Here, f is the decay constant, whose size is given by . This provides a monophoton signal, e.g. + − √ e e → γρD , which can be probed by the Belle II N 3 f ' m , (3) experiment. 4π ρD where m ∼ Λ represents the mass of the “dark rho The organization of this paper is as follows. In Sec- ρD .” The field π is the would-be Nambu-Goldstone tion II, we present our model and analyze its dynamics. 3 boson eaten by the U(1) dark photon. Note that In Section III, we study physics of the dark pion as dark D U(1) × U(1) remains unbroken by the condensation. matter. We present a parameter region in which the dark B P √ The combination φ ≡ (π + iπ )/ 2, which we call the pion is dark matter while avoiding constraints from ex- 1 2 dark pion, comprises a complex scalar field charged under isting experiments and observations. In Section IV, we the U(1) symmetry. discuss the muon g − 2. Finally, in Section V, we discuss P The kinetic term and gauge interactions of the dark the possibility of detecting the dark rho meson. pion are given by 2 f  µ † L = tr (DµU)(D U) , (4) II. THE MODEL 4 where The model has a gauge group G = SU(N), whose 1  −a  D D U = ∂ U −ie A U −ie A U , dynamical scale (the mass scale of generic low-lying res- µ µ D Dµ −1 D Dµ a onances) is Λ ≈ O(10 MeV – 1 GeV), and two flavors of (5) dark quarks transforming under it. In addition, we in- and ADµ is the dark photon field. The mass of ADµ is troduce an Abelian U(1)D gauge group under which the given by dark quarks are charged as in Table I while the standard √ model particles are singlet. We may take 0 ≤ a ≤ 1 N mA = eD(1 − a)f ' eD(1 − a)mρ , (6) without loss of generality. D 4π D We assume a 6= 1. This makes the theory chiral, which arises from the dark quark condensation. i.e. the mass terms of the dark quarks are forbidden by For a 6= 0, the spontaneously broken global symmetry the U(1)D gauge symmetry, so that the only free pa- corresponding to φ is also explicitly broken by U(1)D rameters in this sector are the dynamical scale of GD, gauge interactions. The dark pion φ, therefore, obtains Λ, and the gauge coupling of U(1)D, eD. (There are a mass from quantum corrections due to U(1)D gauge also θ parameters for GD and U(1)D, but they can be interactions in this case. At the lowest order in eD, the eliminated by phase rotations of dark quarks.) The mass is given by [12]2 SU(2)L × SU(2)R × U(1)B flavor symmetry of GD is 2 6a ln 2 2 2 explicitly broken by U(1)D gauge interactions. For a = 0 m ' e m . (7) φ 16π2 D ρD it is broken to U(1)D × SU(2)R × U(1)B, while for a 6= 0 the residual symmetry is U(1)D × U(1)B × U(1)P . (The U(1)B symmetry is anomalous with respect to U(1)D, 1 ¯ but this does not have any consequence for our discus- The condensation is expected to be in this direction, since Ψ1Ψ2 and Ψ2Ψ¯ 1 have larger U(1)D charges than Ψ1Ψ¯ 1 and Ψ2Ψ¯ 2. sion.) The charges of the dark quarks under U(1)B and 2 In Ref. [12], the electromagnetic current is given by a linear com- U(1)P are given in Table I. bination of a vector current and the number current. Here Let us first discuss the strong dynamics of the dark we need to include an axial current as well. A possible effect from the nonzero dark photon mass is suppressed by m /m 2. gauge group GD. Below the dynamical scale Λ, the dark AD ρD 3

From now on, we only consider a 6= 0. 1. mφ < mAD /2 We next discuss couplings between the dark and stan- dard model sectors. These couplings are induced by The case mφ < mAD /2 occurs if the U(1)D charge a is a kinetic mixing between the , U(1)Y , and sufficiently small or if N is sufficiently large. The thermal U(1)D gauge bosons, defined by abundance of dark pions in this case is determined by the annihilation into standard model particles through 1 µν 1 µν 1  µν L = − Bµν B − ADµν A + Bµν A , (8) s-channel dark photon exchange. The annihilation cross 4 4 D 2 cos θ D W section into a pair of standard model fermions f with the where Bµν and ADµν are the U(1)Y and U(1)D gauge qf and mass mf is given by field strengths, and θW is the Weinberg angle. In the limit  → 0, the Z boson and the photon couple to the 2q2(1 + a)2e2e2 m2 σv = f D φ v2 standard model particles as usual, and the dark photon 6π (m2 − 4m2 )2 couples to φ through interactions in Eq. (4): AD φ 3/2 m2 ! m2 ! L = (D φ)(Dµφ)†, (9) f f µ × 1 − 2 1 − 2 , (13) mφ 4mφ where where v is the relative velocity between the initial two D φ = ∂ φ + ie (1 + a)A φ. (10) µ µ D Dµ dark pions. We find that the annihilation cross section For  6= 0, after solving the kinetic and mass mixings, we is suppressed by the relative velocity. This is because find that the dark photon also couples to standard model the intermediate dark photon has an odd C and particles as [13] hence the initial state dark pions must be in p-wave. This implies that the velocity suppression exists regardless of µ L = −eADµJem, (11) the final state as long as the annihilation is through the s-channel dark photon exchange.3 where e and J µ are the electromagnetic coupling and em In Fig. 1, we show the constraints on (m , ) for current, respectively. Here, we have assumed m  AD AD m = 0.4m and a = 1/2. Here, we determine the m and   1, and neglected interactions suppressed by φ AD Z U(1) gauge coupling, e , so that the thermal relic of m /m or higher powers of . The standard model Z D D AD Z dark pions explains the observed dark matter abundance. boson also couples to the U(1) charged particles as D The blue shaded region in the upper part is excluded µ by the search for the process e+e− → γ + A at the L = eD tan θW ZµJD, (12) D BaBar experiment [14–16]. The brown shaded region is µ + + where JD is the U(1)D current. excluded by the K → π νν¯ searches at E787 [17] and + + Finally, we mention that spontaneous breaking of the E949 [18], which constrain the K → π AD process [16]. U(1)D symmetry generates cosmic strings. Observa- The purple shaded region in the upper-left part is elim- tional constraints on them, however, are very weak and inated because of too large contributions to the do not affect the phenomenology discussed below. g − 2 [19, 20]. The gray shaded region in the left is 0 excluded by the search for the process π → γAD fol- ¯ lowed by AD → φφ with subsequent scatterings of φ in III. DARK PION AS DARK MATTER the LSND detector [21, 22]. In the yellow shaded re- gion, dark pions annihilate after decouple from As we have seen, the model has two accidental sym- the thermal bath in the early universe, so that the neu- metries U(1)B and U(1)P . These symmetries guarantee trino energy density becomes too small [8]. The green the stability of dark and the dark pion φ, re- shaded region is excluded by the search for the process − − ¯ spectively. In the early universe, dark baryons effectively e N → e NAD followed by AD → φφ with subsequent annihilate into dark pions, so that their thermal abun- scatterings of φ in the E137 detector [23, 24]. Other con- dance is negligible. We are therefore left with the dark straints, such as the one in Ref. [25], do not reduce the pion as our dark matter candidate. allowed parameter space further. The U(1)P symmetry, in fact, is an extremely good In the region below the gray dashed line, which runs symmetry. The GD and U(1)D gauge symmetries forbid diagonally from the lower left to the upper right, the interactions that explicitly break the U(1)P symmetry dark photon coupling is large (eD > 1 around the dark up to dimension eight. Therefore, unless the scale sup- pion mass scale), so that eD hits the Landau pole below pressing higher dimension operators is very low, the dark typical unification scales of 1014 – 1017 GeV. If we want pion is stable at cosmological timescales. The phenomenology of the dark particles depends sig- nificantly on the ratio of the masses of the dark pion, mφ, 3 and dark photon, mAD . Below we discuss three cases Diagrams with two intermediate dark photons lead to s-wave annihilation. Such processes are suppressed by  and a loop or mφ < mAD /2, mAD /2 < mφ < mAD , and mφ > mAD in turn. extra two-body phase space factor, and hence negligible. 4

-2.0 then adopted the upper bound on the annihilation cross m =0.4×m K +→π+A ϕ AD D + - e e →γA D section at the maximum temperature at which the BBN a=1/2 could be affected by the annihilations. This treatment 2.5 - (g-2)e most likely gives a too aggressive bound. For - conversion, we assumed the maximum tempera- ture of 1 MeV, where the proton-neutron ratio is fixed in - - -3.0 e N→e NAD, the standard BBN. A precise estimate of the constraint ϵ AD→ϕϕ eD>1

10 from the BBN requires a dedicated calculation, which is

2 2 2 beyond the scope of this paper. log -3.5 cm cm cm In addition to the bounds discussed above, the anni- -38 -40 -42 hilation cross section of dark pions is also constrained = 10 = 10 = 10 e e e

σ σ σ by the hadro- and photo-dissociation processes dur- -4.0 Nν,eff 0 ing the BBN, effect on fluctuations of the cosmic mi- π →γAD, BBN crowave background, and production of gamma rays in AD→ϕϕ pn the present universe. Because of the p-wave suppression -4.5 of the annihilation, these constraints are weak and do not 2.5 2.0 1.5 1.0 0.5 0.0 0.5 - - - - - exclude the parameter region further. log (m /GeV) 10 AD Since the dark pion mass is much smaller than the mass, it is difficult to directly detect dark pi- FIG. 1. Constraints on the dark photon mass, m , and the AD ons through collisions with nuclei. However, scattering mixing parameter  for m = 0.4m and a = 1/2. The φ AD of dark pions with may be detectable [29]. In U(1) gauge coupling, e , is determined so that the thermal D D Fig. 1, we show the contours of the scattering cross sec- abundance of dark pions reproduces the observed dark matter abundance. The blue, brown, purple, gray, yellow, and green tion between the dark pion and the electron, given by shaded regions are excluded by the mono-photon search at the + + BaBar experiment, K → π νν¯ searches at E787 and E949, 2 2 2 2 2 0  (1 + a) e eD me the electron g − 2, the search for π → γAD → γφφ¯ with σ = , (14) φe 2π m4 subsequent scatterings of φ in the LSND detector, dark pions AD annihilating after the decoupling, and the search for e−N → e−NA → e−Nφφ¯ with subsequent scatterings of φ D in the nonrelativistic and m  m limit. We find that in the E137 detector, respectively. In the region below the φ e the model gives gray dashed line, eD > 1 around the dark pion mass scale, so that eD hits a Landau pole below the unification scale. Between the two vertical green dashed lines, the annihilation σ ∼ 10−42–10−39 cm2 for m = 0.4m . (15) of dark pions during the BBN may result in too much 4He. φe φ AD The calculation of this bound, however, is subject to large theoretical uncertainties; see the text. A projected experiment with semiconductor targets can −42 2 probe the cross section down to σφe ∼ 10 cm [29]. to avoid this, we are left with the region centered around −3.5 −2.5 mAD ≈ O(100 MeV – 1 GeV) and  ≈ O(10 – 10 ). We note that the region between the two vertical green dashed lines may be excluded by the big bang nucleosyn- 2. mAD /2 < mφ < mAD thesis (BBN) because annihilations of dark pions into in the BBN era may lead to proton-nucleon con- If m /2 < m < m , the dark photon does not version that yields too many 4He [26, 27]. However, this AD φ AD bound, derived naively by extrapolating the bound in decay into dark pions. It decays only into standard model Ref. [27], might be too strong. In Ref. [27], the dark fermions. For the annihilation of dark pions, the three- body channel φφ¯ → A A∗ → A ff¯ is now open, whose matter mass was taken to be larger than 10 GeV, and D D D the annihilation was assumed to be s-wave. To convert cross section is given by this to our bound, we first assumed that the upper bound on the dark matter annihilation cross section scales as 2q2(1 + a)4e2e4  m2  3/2 ¯ ¯ f D φ 2 σ(φφ → ADff) v = f . (16) σv ∼ mφ [28]. The factor of mφ comes from the square 384π3m2 m2 −1/2 φ AD of the number density of φ, and mφ from the multi- plicity of particles produced by the annihilations.4 We

1 GeV, but we also use it below 1 GeV, since the resulting error 4 Strictly speaking, the latter scaling is valid only down to mφ ' is expected to be insignificant in the region of interest. 5

-2.0 followed by A → e+e−, µ+µ− at the BaBar experi- m =0.6×m D ϕ AD + - + - 0 e e →γe e ment [30], the search for the process π → γAD followed a=1/2 + − by AD → e e at the NA48/2 [31], and the search for 2.5 − − − + - (g-2)e the process e N → e NAD followed by AD → e e in beam dump experiments [32], respectively. The mean- π0→γe +e- ings of the other shaded regions and lines are the same -3.0 as those in Fig. 1. This leaves us the region around ϵ N −4.5 −3 10 ν,eff mAD ≈ O(10 MeV – 100 MeV) and  ≈ O(10 – 10 ) BBN as the bulk of the viable parameter space. The scattering log pn -3.5 eD>1 cross section between the dark pion and the electron in

2 2 2 this region is cm cm cm

-37 -39 -41 −40 −37 2 -4.0 σφe ∼ 10 –10 cm for mφ = 0.6mAD . (18) = 10 = 10 = 10 e e e σ σ σ For larger mφ, the three body annihilation mode is - - + - CMB e N→e e e N more effective because of larger phase space of the final -4.5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 state. As a result, the Planck data becomes more con- straining, excluding the model up to larger values of . In log10(mAD /GeV) particular, for mφ & 0.8mAD no viable parameter region remains.

FIG. 2. Constraints on the dark photon mass, mAD , and the mixing parameter  for mφ = 0.6mAD and a = 1/2. The gray shaded region at the bottom is excluded by the con- 3. mφ > mAD straint from the Planck experiment on the annihilation mode ¯ ∗ + − φφ → ADAD → ADe e . The blue, green, and brown shaded + − regions are excluded by the search for the process e e → For mφ > mAD , the thermal abundance of dark pions + − 0 + − ¯ γAD → γ` ` (` = e, µ), the process π → γAD → γe e , is determined by the s-wave annihilation mode of φφ → − − − − + and the process e N → e NAD → e Ne e , respectively. ADAD. Since the same annihilation mode is effective The meanings of the other shaded regions and lines are the in the eras of the BBN and recombination, this mass same as those in Fig. 1. spectrum is excluded by the constraints from the BBN and the cosmic microwave background.

Here, IV. MUON g − 2 1  f(x) = 4(1 − 4x)(1 − 2x)(1 − 3x + 4x2) x(1 − 2x)3 √ √ The coupling between the muon and the dark photon − 4x − 1(1 − 4x)(3 − 2x) Arctan 4x − 1 given by Eq. (11) contributes to the muon g − 2 [1, 33]:

1 2 2 + √ (3 − 12x + 8x )(5 − 10x + 8x ) α Z 1 2x(1 − x)2m2 1 − x ∆a = 2 dx µ . (19) √ √ µ,AD 2π xm2 + (1 − x)2m2  r x x − 1/2 x 0 AD µ × Arctan + Arctan √ 1 − x 1 − x In the absence of this contribution, there is a discrepancy  of the muon g − 2 between the experimental result and + (3 − 4x)(1 − 2x)3 ln(4x) , (17) standard model prediction, aexp − aSM = (26.1 ± 8.0) × 10−10 [10, 11]. In Figs. 1 and 2, we depict the region and we have neglected the mass of the standard model of (mAD , ) in which the experimental and theoretical fermion f. Being the three-body channel, the process values become consistent at the 1σ (2σ) level because of is subdominant in determining the thermal abundance this contribution by the red (pink) shaded bands. We of dark pions. It can, however, give dominant effects in find that the regions in which the discrepancy can be later stages of the evolution of the universe, since it is ameliorated are excluded by other experiments. s-wave. In particular, it is effective around the recom- This conclusion, however, does depend on the ratio bination era and affects the fluctuations of the cosmic between the dark pion and dark photon masses. In par- microwave background. ticular, if mφ is slightly smaller than mAD /2, we find

Fig. 2 shows the constraints on (mAD , ) for mφ = a small parameter region in which the muon g − 2 dis-

0.6mAD and a = 1/2. The gray shaded region at the crepancy can be ameliorated. This is because the value bottom is excluded by the constraint on the annihilation of eD required to obtain the correct thermal relic abun- ¯ ∗ + − mode φφ → ADAD → ADe e from the measurement dance becomes smaller for a fixed (mAD , ), so that the of the cosmic microwave background by the Planck ex- scattering rates of φ with the detectors become smaller, periment [7]. The blue, green, and brown shaded regions weakening constraints. In Fig. 3 we show the same plot + − are excluded by the search for the process e e → γAD as Fig. 1 except that we take mφ = 0.45mAD , instead of 6

-2.0 g − 2 discrepancy without requiring that the dark pion m =0.45×m K +→π+A ϕ AD D + - e e →γA D comprises all of the dark matter. In Fig. 4, we show a=1/2 the constraints on (mAD , eD) for mφ = 0.45mAD and 2.5 - (g-2)e a = 1/2, with the value of  determined so that the discrepancy of the muon g − 2 between the experimen- tal and standard model values is ameliorated by the -3.0 e-N→e-NA , D dark photon contribution to the 2σ level, ∆aµ,AD = ϵ −10 AD→ϕϕ

10 (26.1 − 16.0) × 10 . In the red shaded region at the bottom, the thermal relic abundance of dark pions ex- log -3.5 eD>1 ceeds the observed dark matter abundance. The mean- 0 π →γAD, ings of the other shaded regions are the same as those in AD→ϕϕ 2 2 2 Fig. 1. We find that the discrepancy of the muon g−2 can

Nν,eff cm cm cm -4.0 be ameliorated in the region around mAD ' 150 MeV, -38 -40 -42 BBN consistently with all the other experiments. In the fig- = 10 = 10 = 10

e e pn e ure, we also show the contours of the effective scattering -4.5 σ σ σ cross section between the dark pion and the electron, -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 σe,eff = σφe × (Ωφ,th/ΩDM).

log10(mAD /GeV) V. DARK RHO MESONS FIG. 3. The same as Fig. 1 except that mφ = 0.45mAD , instead of 0.4mAD . The model predicts a plethora of resonance states com- posed of the dark quarks and GD gauge bosons. The 0.5 masses of these states depend on parameters of the model mϕ =0.45m AD , a=1/2 -10 but are generally in the range of 10 MeV – 10 GeV. These Δaμ =(26.1-16)×10 0.4 states may be detectable in various experiments. Here we discuss a possible way to detect the lowest-lying spin-one

(g-2) C- and P -odd composite states: the dark rho mesons ρD. 0.3 The masses of ρD are expected to be larger than the

e

π 2 2

0

e D A →γ mass of the dark photon by about an order of magnitude - cm cm e

N→e

A

-40 -40

D

D

→ϕϕ

A ,

- 4π

0.2 NA

D mρD ≈ √ mAD . (20)

→ϕϕ =3×10 =2×10

D

, NeD(1 − a)

e,eff e,eff

σ σ 0.1 One of the dark rho mesons, which we call ρD3 , has the same charge as π3 under the vectorial subgroup of Ω >Ω ϕ,th DM SU(2)L × SU(2)R and mixes with the dark photon. The 0.0 mixing induces the coupling of ρ with the standard 50 100 150 200 D3 model fermions mAD /MeV 0 µ L =  ρD3µJem, (21) FIG. 4. Constraints on the dark photon mass, m , and the AD where U(1)D gauge coupling, eD, for mφ = 0.45mAD and a = 1/2. √ The value of the mixing parameter  is determined so that 0 N −10  ≈ eD . (22) ∆aµ,AD = (26.1−16.0)×10 . In the red shaded region, the 4π thermal relic abundance of dark pions exceeds the observed This coupling leads to processes such as e+e− → γρ , dark matter abundance. The meanings of the other shaded D3 regions are the same as those in Fig. 1. The dotted lines are which yield monophotons. Adopting the analysis for the the contours of the effective scattering cross section between dark photon in Ref. [16], we find that the Belle II ex- the dark pion and the electron. periment will be sensitive to the dark rho meson with 0 10−4 for 1 GeV < m < 10 GeV.5 Observation of & ρD3 + − the two processes e e → γAD and γρD3 in future ex- periments would provide strong evidence for the model. 0.4mAD . We see a small region in which the experimental and theoretical values of the muon g − 2 become consis- tent within 2σ. The weakening of the constraints also enlarges the parameter space for dark pion dark matter. 5 In Ref. [16] the width of the dark photon is assumed to be negli- gible, while our ρ has a decay width of ∼ m /N. For small In particular, there is now a sliver of the allowed region D3 ρD N, this width is not negligible, so that the sensitivity to 0 would along the eD ∼ 1 line, which goes down to smaller dark be somewhat weaker. Performing spectroscopy of a dark sector photon masses. connected with the standard model through U(1) kinetic mixing We may consider the model to ameliorate the muon has been discussed recently in Ref. [34]. 7

ACKNOWLEDGMENTS ence, Office of High Energy and , of the U.S. Department of Energy under Contract DE-AC02- 05CH11231, by the National Science Foundation under We thank Hitoshi Murayama for discussion. This work grants PHY-1316783 and PHY-1521446, and by MEXT was supported in part by the Director, Office of Sci- KAKENHI Grant Number 15H05895.

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