Asymmetric matter from $B-L$ symmetry breaking

The present matter content of our universe may be governed by a $U(1)_{B-L}$ symmetry---the simplest gauge completion of the seesaw mechanism which produces small neutrino masses. The matter parity results as a residual gauge symmetry, implying dark matter stability. The Higgs field that breaks the $B-L$ charge inflates the early universe successfully and then decays to right-handed neutrinos, which reheats the universe and generates both normal matter and dark matter manifestly.


I. INTRODUCTION
The standard model must be extended in order to account for small neutrino masses and flavor mixing [1,2] as well as dark matter component [3,4]. The seesaw mechanism is a compelling idea that realizes consistent neutrino masses, generated through the exchange of heavy righthanded neutrinos [5][6][7][8][9][10][11][12][13]. This mechanism can simply be realized in the gauge completion U (1) B−L for the standard model. Here, the right-handed neutrinos arise as a result of B − L anomaly cancellation, while their heavy Majorana masses (or seesaw scale) are set by B−L breaking scale. Several analyses of the model were presented in, e.g., [14][15][16][17] and dark matter may be recognized by modifying the symmetry and/or particle content [18][19][20][21][22][23][24]. In this work, we prove that the original proposal can supply dark matter stability naturally, without requiring any ad hoc modification 1 .
To be concrete, we reconsider the question of B − L anomaly cancellation. We show that right-handed neutrinos can be divided into two kinds: (i) dark matter includes N R fields that have even B − L number and (ii) normal matter contains ν R fields that possess odd B − L number. We prove that the matter parity arises naturally as a residual B − L gauge symmetry, derived by a B − L breaking scalar field. This scalar field inflates the early universe successfully [38][39][40] and defines the seesaw scale. The fields ν R obtain large Majorana masses in similarity to the often-studied right-handed neutrinos, which make observed neutrino masses small, whereas the fields N R have arbitrary masses providing a novel candidate for dark matter, stabilized by the matter parity conservation.
We point out that the inflaton decays to a pair of ν R or the Higgs field which reheats the universe. Then both normal and dark matter abundances observed today can be simultaneously generated due to CP-violating decays of the lightest ν R in the early universe, quite analogous to the standard leptogenesis [41][42][43]. This Abelian recognition for B − L symmetry is more simple than (and different from) our recent proposal [44]. Actually, it opens out a new direction in search for the dark matter candidate in connection to the baryon asymmetry production. Thus, both kinds of the matter relics originate from the same source, addressed in the common framework.
The rest of this work is arranged as follows. In Sec. II, we set up the model. In Sec. III we examine the potential minimization and scalar mass spectrum. In Sec. IV we discuss neutrino mass. In Sec. V we investigate cosmological inflation and reheating. In Sec. VI we obtain the dark and normal matter asymmetries. The other dark matter bounds are given in Sec. VII. Finally, we conclude this work in Sec. VIII.

II. THE MODEL
The gauge symmetry is given by where B −L is baryon minus lepton charge, while the rest is the ordinary gauge group. The electric charge operator is related to the hypercharge by The fermion content transfroms under the gauge symmetry as The nontrivial anomaly cancellation conditions are The solutions with the smallest M + N are x = y = −1 for M + N = 3 and (x, y) = (−4, 5) for (N, M ) = (2, 1), which were well-established in the literature, e.g., [18,20,23,45]. Such cases do not provide simultaneously dark matter candidates and successful leptogenesis. We consider the next solution for N + M = 4, or in other words, Here N R is a truly sterile neutrino under the gauge symmetry, which was actually omitted in the literature, e.g., [45]. Besides the standard model Higgs doublet, we introduce two scalar singlets, which are required to break the B − L symmetry, giving new fermion masses, as well as supplying asymmetric dark matter. The Lagrangian is where the first part defines kinetic terms and gauge interactions. Whereas, the Yukawa interactions and scalar potential are given, respectively, by Particle ν e u d gluons photon W Z Z φ ϕ N χ WP 1 1 1 1 We can choose the potential parameters so that φ and ϕ develop the vacuum expectation values (VEVs) such as while χ possesses vanishing VEV, i.e. χ = 0 2 . For consistency, one imposes The gauge symmetry is broken as where the first step implies the matter parity W P , while the second step yields the electric charge Q = T 3 +Y . The matter parity conserves the VEV of ϕ, i.e.
We obtain e i2α = 1, implying α = kπ for k = 0, ±1, ±2, · · · Considering the residual symmetry for k = 3, we get W P = (−1) 3(B−L) . The matter parity is conveniently rewritten as after multiplying the spin parity (−1) 2s , which is conserved by the Lorentz symmetry. The notation "W" means particles that have "wrong" B − L number and odd under the matter parity (i.e. W P = −1), say N R and χ, called wrong particles. All the other particles, including the standard model, ν R , ϕ, and U (1) gauge (called Z ) fields, are even under the matter parity (i.e. W P = 1), which have normal B − L number or differ from that number by even unit as ϕ does, called normal particles. They are summarized in Table I.
It is easily realized that the χ vacuum value vanishes, χ = 0, due to the matter parity conservation. Also, the lightest wrong particle (LWP) between N R and χ is absolutely stabilized responsible for dark matter. The new observation is that ν R couples both N χ and eφ, through the complex Yukawa couplings, h ν and y, respectively. Hence, the asymmetric dark and normal matter can be simultaneously produced by CP-violating decays of ν R , in the same manner of the standard leptogenesis. Of course, the ν R fields are generated after cosmic inflation derived by the inflaton ϕ-the scalar field that breaks B − L-which also induces the neutrino seesaw masses. Let us see.
Choosing the parameters as µ 2 The physical scalar fields with corresponding masses are given as Here H is identical to the standard model Higgs boson, while H is a new heavy Higgs boson associate to B − L symmetry breaking. G W , G Z , and G Z are massless Goldstone bosons eaten by W , Z, and Z gauge bosons, respectively. χ has an arbitrary mass m χ , but below the Λ scale.

IV. NEUTRINO MASS
The ordinary fermions obtain appropriate masses as in the standard model. The new fermions get masses as follows where we define ν = (ν 1 ν 2 ν 3 ) T and The dark fermions N R get an arbitrary mass m N . The observed neutrinos (∼ ν L ) gain a mass via the seesaw mechanism due to v Λ to be Comparing to the neutrino data, m ν ∼ 0.1 eV, we get which is naturally at the inflation scale. Of course, the heavy neutrinos ∼ ν R have the mass, m R , proportional to the Λ scale.

V. INFLATION
We consider the inflation scheme derived by the B − L breaking scalar field, governed by the potential, where the inflaton field Φ ≡ √ 2 (ϕ) Λ + H is the real part of ϕ, while its imaginary part G Z was absorbed to the longitudinal component of Z gauge boson by a gauge transformation, ϕ → Φ/ √ 2 = e −iG Z /Λ ϕ. Note that this tree-level potential is disfavored by the current data [46]. Furthermore, Φ couples to the extra fields Z , ν R , φ, and χ which modify V (Φ) by a Coleman-Weinberg potential through quantum corrections [47]. However, the total potential does not naturally fit the data too, since the large-field inflation Φ > Λ simply mimics the tree-level one, while the small-field inflation Φ < Λ predicts a too large number of e-folds in contradiction to the standard cosmological evolution [44].
After inflation,Φ oscillates near the minimum which violates the scale symmetry. Expanding the potential and turning the soft µ 2 term on, we obtain whereΛ = ξ 3/2Λ 2 /m P and the inflaton mass is mΦ = λ 2 /3m P /ξ 2.8 × 10 13 GeV. Since the higher order correction is quickly vanished after the end of inflation, Φ <Φ e 0.4m P , the corresponding Klein-Gordon equation gives a solution,Φ (m P /mΦt) sin(mΦt) +Λ. Additionally, the inflaton field undergoes many oscillations after inflation to reach the minimumΦ =Λ Φ e due to the ξ constraint.
According to the following leptogenesis, one takes the x coupling to be flavor diagonal and Comparing withΦ e , the inflaton immediately decays after several oscillations, if x 11 < ∼ 10 −3 . Requiring x 22,33 ∼ g B−L > 10 −3 such that the inflaton cannot decay to ν 2,3R and Z , the channelΦ → ν 1R ν 1R sets the reheating temperature to be T R ∼ 5 × 10 11 GeV.
A common issue raised is that, comparing to the Hubble rate H 0.13mΦj −1 for j = mΦt/π, the inflaton undergoes 2j ∼ 10 11 oscillations in order for their products to thermalize, which implies a long stage of preheating. However, in this stage, the nonperturbative decayΦ → Z Z may happen through broad and narrow parametric resonances characterized by the gauge interaction 2g 2 B−L Z 2Φ2 . The effects of nonperturbative parametric resonance do not happen for fermion productŝ Φ → ν 2R ν 2R , ν 3R ν 3R .

VI. ASYMMETRIC MATTER
The Yukawa Lagrangian yields where m e ≡ −h e v/ √ 2 and m R was given. The gauge states ( a ) are related to mass eigenstates subscripted by ( i ), through mixing matrices, such that Without loss of generality, we assume m ν 1R < m ν 2,3R and V νR = 1, i.e. ν aR = ν iR are physical Majorana fields by themselves.
We rewrite (36) in mass bases, where all i, j indices are summed. The couplings z ij = −[V † eL h ν ] ij and y j are all complex, hence sources of CP violation. In the early universe, the lightest right-handed neutrino ν 1R decays simultaneously to normal matter: ν 1R → e i φ and dark matter: ν 1R → N χ, which produce the corresponding CP asymmetries through the diagrams supplied in Fig. 1 where Γ ν 1R is the total width of ν 1R .
Applying the Feynman rules we obtain where r j1 = (m ν jR /m ν 1R ) 2 and Let us note that during the electroweak sphaleron the largest interaction rate of charged lepton flavors would correspond to the tau flavor such as Γ τ 5×10 −3 (h τ ) 2 T [53]. However, this rate is slower than the Hubble rate H for T = m ν 1R ∼ 10 12 GeV. Hence, the lepton asymmetry FIG. 1. CP-violating decays of νR that produce both dark matter N χ and normal matter (eφ), respectively.
production does not depend on flavor, yielding the net contribution by summing [h ν † h ν ] 2 j1 g(r j1 ). (44) It is noteworthy that the sphaleron process only converts the lepton asymmetry to baryon asymmetry, given by η B = −(28/79)η NM . Such process cannot convert the dark matter asymmetry to baryon matter due to the matter parity conservation. The observation implies Ω DM 5Ω B . It follows that m DM /m p 5η B /η DM . Since ν 1R is directly produced by the inflaton decay Φ → ν 1R ν 1R , the lepton and dark matter asymmetries are related to the CP asymmetries by η NM,DM = 3 2 NM,DM × Br(Φ → ν 1R ν 1R ) × T R mΦ , leading to η NM /η DM = NM / DM . Even if ν 1R is thermally produced, the last equation is approximately hold, since the leptogenesis is flavor independent and very effective. All the cases imply It is evident that NM 10 −7 and g(r j1 ) −3m ν 1R /m ν jR as well as assuming m ν jR = 10 2 m ν 1R and y 2 = y 3 = y 1 e −iθ with real y 1 , we derive Provided that y 1 ∼ 10 −2 and sin 2θ ∼ 10 −3 -1, we have m DM ∼ 1-1000 GeV, as desirable.

VII. DARK MATTER DETECTION
The scalar dark matter χ can scatter off nuclei of a large detector via the Higgs portal L ⊃ −λ 5 vHχ * χ, in which H interacts with quarks confined in nucleons of the nuclei as usual. The χ-nucleon cross-section is easily evaluated as [54] σ χ−p/n λ 5 0.1 Provided that m χ at TeV and λ 5 similar to the Higgs coupling, the model predicts σ χ−p/n ∼ 6 × 10 −46 cm 2 , in good agreement with the current search [55]. Additionally, if the dark matter mass is at the weak scale, say m χ ∼ 100 GeV, comparing to the data σ χ−p/n ∼ 10 −46 requires λ 5 ∼ 0.004. The Higgs portal might reveal dark matter signals in form of missing energy at the LHC, governed by the effective Lagrangian L eff = (λ 5 α s /12πm 2 H )G nµν G µν n χ * χ. Studying the mono-gluon signatures, Ref. [56] shows λ 5 α s /12πm 2 H < (3 TeV) −2 , which is always satisfied for the above choice of the parameter.
Since the fermion candidate N R interacts very weakly with the normal matter, it easily escapes from the current experimental searches, analogous to the right-handed neutrino singlet often interpreted in the literature. In other words, the dark matter N R can have an arbitrary mass above keV.

VIII. CONCLUSION
We proved that the U (1) B−L gauge theory provides a manifest solution for the leading questions, such as the neutrino masses and cosmological issues of inflation, dark matter and baryon asymmetry. In fact, the B − L anomaly cancelation obeys the new degrees of freedom for dark matter, and the matter parity that is a residual B−L gauge symmetry makes these candidates stabilized. Additionally, the B−L dynamics determines the neutrino mass generation seesaw mechanism, new Higgs inflation scenario when including a nonminimal interaction with gravity, as well as reheating the early universe by inflaton decays to right-handed neutrinos. The lightest righthanded neutrino of which decays CP-asymmetrically to both the present-day observed dark matter and normal matter asymmetries.