Dequantization of electric charge: Probing scenarios of cosmological multi-component dark matter

Since the electric charge in the standard model is theoretically not quantized, we may have a variant of it, called dark charge. Similar to the electric charge, the dark charge neither commutes nor closes algebraically with $SU(2)_L$. The condition of algebraic closure leads to a novel gauge extension, $SU(2)_L \otimes U(1)_Y \otimes U(1)_N$, where $Y$ and $N$ determine the electric and dark charges, respectively, apart from the color group. We argue that the existence of the dark charge, thus $N$, leads to novel scenarios of multi-component dark matter, in general. The dark matter stability is determined by a residual (or dark charge) gauge symmetry isomorphic to an even $Z_k$ discrete group, where $k$ is specified dependent on the value of the neutrino dark charge. This residual symmetry divides the standard model particles into distinct classes, which possibly accommodate dark matter, but each dark matter candidate cannot decay due to the color and electric charge conservation. We analyze in detail three specific models according to $k=2,4,6$ and determine the simplest dark matter candidates. For small $U(1)_N$ coupling, the two-component dark matter scenarios implied by the dark charge successfully explain the dark matter relic density and the recent XENON1T excess, as well as the beam dump, neutrino scattering, and astrophysical bounds. Otherwise, for large $U(1)_N$ coupling, we have multi-WIMPs coexisted beyond the weak scale.


I. INTRODUCTION
The standard model of particle physics [1] has been enormously successful in describing most of the observed phenomena. However, the explicit evidences of neutrino oscillation [2,3] and dark matter existence [4,5] cannot be explained within the framework of the standard model. This proves that the standard model must be extended in order to account for the new physics beyond.
The existing experiments of dark matter relic density, direct/indirect detections, and particle colliders have not revealed any detail of particle picture of dark matter. Obviously, many dedicated theories often assume dark matter to be composed of a single particle kind, such as a WIMP [6,7], while there is no any reason why only a single particle kind of dark matter is presented. Since the constituent of dark matter is still eluded, a scheme of multi-component dark matter seems to be naturally in comparison to the rich structure of normal matter within an atom. Indeed, the possibility that dark matter consists of more than one type of particle is very attractive , with intriguing results for galaxy structure [40,41], multiple gamma-ray line [42], boosted dark matter [43][44][45][46][47][48][49], and dark matter self-interaction [50,51], as well as those motivated with neutrino mass generation [52][53][54][55][56][57][58][59][60][61][62].
We wish to suggest that the structure of dark matter and the origin of neutrino mass can be simultaneously understood from a very fundamental aspect of the standard model, that is the electrodynamics. As shown in [63], since the electric charge (Q) is theoretically not fixed, there may exist a variant of it, interpreted to be a physical dark charge (D), which coexists with the electric charge, driving the model extension with implication for neutrino mass and dark matter.
As an indication, the most U (1) gauge extensions that mix with the hypercharge (Y ) actually imply such a dark charge to be a dequantization version of the usual electric charge. Analogous to the electric charge, the dark charge neither commutes nor closes algebraically with the SU (2) L weak isospin, T i (i = 1, 2, 3). By algebraic closure condition, or symmetry (flipping) principle, this leads to the full gauge symmetry, SU (3) C ⊗ SU (2) L ⊗ U (1) Y ⊗ U (1) N , where the last two U (1)'s factors determine the electric charge and the dark charge through the T 3 operator, i.e. Y = Q − T 3 and N = D − T 3 , respectively, where the QCD group has been included for completeness. Additionally, to cancel the U (1) N anomalies, three right-handed neutrino singlets must be included, which have a dark charge identical to that of the usual neutrinos, whereas both kinds of the neutrinos have vanished electric charge [64]. These right-handed neutrino singlets obtain large Majorana masses according to the N -charge breaking, while they couple to the usual neutrinos via the Higgs field. As a result, appropriate small neutrino masses are induced in terms of a canonical seesaw mechanism.

II. A MODEL OF DARK CHARGE
In the nature, electric charges of fundamental particles come in discrete amounts, equal to integer multiples of an elementary electric charge, called electric charge quantization. However, if the electric charge is theoretically not quantized, i.e. not discrete, it is dequantized, i.e. arbitrary or continuous. This term is eventually associated with the theories, such as the electrodynamics and the standard model, that do not explain the quantization of electric charge. The former theory obeys the charge quantization, if a magnetic monopole is presented, proposed by Dirac long ago [75].
But, the monopole has not been observed yet. The latter theory implies the charge quantization, if all hidden symmetries, e.g. L a − L b for a, b = e, µ, τ and B − L, are violated. Otherwise, they make the hypercharge, thus the electric charge, free, such that Y → Y + x ab (L a − L b ) + y(B − L) is always allowed, called dequantization effect [76,77].
Indeed, in the standard model with the gauge symmetry, the electric charge is related to the hypercharge through the weak isospin by The standard model does not predict the quantization of the electric charge because-while the value of T 3 is quantized due to the non-Abelian nature of SU (2) L Lie algebra-the value of Y is completely arbitrary on the theoretical ground of U (1) Y group, since [Y, Y ] = 0 for any Y . For phenomenological purpose, Y is often chosen to describe the observed electric charge values, while does not explain them. Additionally in the theoretical side, the value of Y can be constrained by the anomaly cancellation and the fermion mass generation for each family, but it is dequantized when including three families or right-handed neutrinos into account [78][79][80][81][82]. The reason is that the standard model always contains anomaly-free hidden symmetries, such as L a − L b for a, b = e, µ, τ or B − L for including the right-handed neutrinos, that subsequently make Y free, as mentioned.
We consider the latter by introducing the three right-handed neutrinos, ν aR , to the standard  [63,64]. 2 We call δ to be the parameter of dequantization. Namely, for δ = 0, the hypercharge and the electric charge are properly recovered as in the normal sense, so we relabel Y = Y | δ=0 and Q = Q| δ=0 for convenience in reading. Whereas, for δ = 0, we obtain the new charges as corresponding variants of the electric charge and the hypercharge, by which we define D = Q| δ =0 , called dark charge, and N = Y | δ =0 , called hyperdark charge. One has a similar relation to Eq. (2), As the electric charge, the dark charge D neither commutes nor closes algebraically with the weak isospin, which indicates to a novel gauge extension, such as where N thus determines D in the same way that Y does so for Q, respectively.
It is easily realized that N was studied in the literature as a combination of xY + y(B − L) for x = 1 and y = −δ, since both Y, B − L are anomaly-free. However, the interpretation of the dark charge D as well as this combination to be a dequantization of the electric charge did not appear until our proposal [63,64]. An important result is that D implies dark matter stability; additionally, the dark matter can be unified with normal matter in SU (2) L multiplet, because the dark charge D is noncommutative, which differs from the most extensions. Let us stress that dark matter must be stabilized over the cosmological timescales. For instance, if a dark matter particle with a mass between the weak and TeV scales decays to two lighter particles, the relevant coupling strength that couples these fields must be around ∼ 10 −20 . Indeed, the current bound of dark matter lifetime is at least a billion times longer than our universe's age, leading to a corresponding coupling strength ∼ 10 −25 [83]. Such tiny coupling, that is much smaller than the Dirac neutrino coupling ∼ 10 −12 , is not naturally stabilized against quantum contributions, which might only be understood by symmetries.
The full fermion content supplied with the hyperdark charge under the gauge symmetry in Eq.
(4) is where a = 1, 2, 3 is a family index, and δ is arbitrarily nonzero, δ = 0. The left and right fermion f L,R have the same dark charge, which coincides with the hyperdark charge value of the right It is easily verified that all the anomalies are cancelled out for each family, independent of δ, as shown in Appendix of [64]. There, we also indicated that the anomalies are cancelled out δ → δ 1 , δ 2 , · · · , δ p , respectively. Multi-component dark matter is naturally recognized, since each U (1) N 's factor presents an independent stability mechanism (see below). However, within one factor, the U (1) N breaking also supplies the schemes of multi-component dark matter. That said, the multi-component dark matter is a generic result of the electric charge dequantization. Hereafter, only the latter case is taken into account.
To break the gauge symmetry and generate appropriate masses, the scalar content is introduced Here, the new field χ is necessarily to break U (1) N , producing Majorana ν R masses via the couplings χν R ν R . Additionally, φ is the usual Higgs doublet, which has the dark charge identical to the electric charge, such that both of these charges are not broken by the weak vacuum. The vacuum expectation values (VEVs) are given by for consistency with the standard model.

III. RESIDUAL DARK CHARGE
The gauge symmetry is broken via two stages, In the first stage, only U (1) N is broken by the VEV of χ, because of N χ = − √ 2δΛ = 0. But, this breaking is not completed. Indeed, a residual symmetry of N is defined as R N = e iαN , since it is a U (1) N transformation. R N conserves the vacuum, R N χ = χ , leading to e −i2αδ = 1, or α = kπ/δ for k integer. We deduce In the second stage, the VEV of φ breaks all SU (2) L ⊗ U (1) Y ⊗ R N , except for the color group, However, the breaking is also not completed. Indeed, a conserved charge after the breaking must take the form, that annihilates the weak vacuum, X φ = 0. This yields a 1 = a 2 = 0 and a 3 = b + c, hence where Q = T 3 + Y and D = T 3 + N are defined as before. Additionally, Q, D commute, [Q, D] = 0, and separately conserve the weak vacuum (similar to X), i.e. Q φ = D φ = 0. Hence, the residual symmetry X is Abelian by itself and factorized into according to a transformation, e iX = e ibQ e icD , multiplied by those of Q, D, respectively.
The first factor U (1) Q is well known to be the electromagnetic symmetry, which is the residual symmetry of SU (2) L ⊗ U (1) Y , obviously conserving the χ vacuum, since Q χ = 0. The second factor U (1) D is a residual symmetry of SU (2) L ⊗ R N , which similar to R N , must conserve the χ vacuum. In other words, e icD χ = χ leads to e −i2cδ = 1, implying c = α = kπ/δ. This restricts to be a residual symmetry of the dark charge D = T 3 + N , thus of SU (2) L ⊗ U (1) N .
It is clear that R D is shifted from R N by the weak vacuum, not commuted with the standard model symmetry, a consequence of the existence of noncommutative dark charge. R D would distinguish particles with different weak isospin values and give a potential unification of normal matter and dark matter in the same gauge multiplet [63]. This feature compares with supersymmetry, which does so for both kinds of matter (with different spin values) in supermultiplet. R D differs from the most U (1) extensions in the literature, which actually lead to commutative discrete symmetries, like R N , by contrast.
For comparison we collect the electric and dark charges as well as the R D values of all fields in   Table I, it is clear that if k = 0, then R D = 1 for all fields and every δ, which is the identity transformation. To search for the residual group structure, we find the minimal value of |k| = 0 that still satisfies R D = 1 for all fields. Such value of k depends on δ and obeys the following equations simultaneously, k(δ + 1)/3δ = 2n 4 , where n 1,2,3,4,5 = 0, ±1, ±2, · · · are integer. From the first equation, k must be even integer, i.e. k = ±2, ±4, ±6, · · · .
First, considering the case of |k| = 2, we get n 1 = ±1 and subsequent relations We also obtain a corresponding value of δ, denoted by Similarly, considering the next cases with |k| = 4 and 6, we obtain the corresponding values of δ, denoted by respectively. Notice that δ 4 contains in part the values of δ 2 which correspondingly reduce to the first case and must be omitted. Moreover, δ 6 also includes values of δ 2 and a part of δ 4 which reduce to the first or the second case and must be eliminated too.
In other words, there are different values of δ resulting in R D = 1 for all fields according to the minimal solution of |k| = 2, 4, or 6, except the identity k = 0. This implies that the residual symmetry R D is automorphic to a discrete group, Z 2 , Z 4 , or Z 6 , respectively, providing dark matter candidates. To be concrete, in Table II, we show three specific cases, giving the results as expected. Notice that unlike the minimal value of k according to R D = 1 for all fields, we have restored k, without confusion, to be arbitrarily integer characterizing transformations of (18) within each residual group R D .
Of course, there are values of δ that lead to the residual symmetries higher than the given ones, such as Z 8 , Z 10 , Z 12 , and so forth. They would imply more the stabilized dark matter components than the chosen ones. Such generalization to a higher discrete symmetry is not trivial, but not discussed further in this work. The second column of Table II indicates R D = 1 for all fields and for the minimal value of |k| = 2, except k = 0. Hence, the residual symmetry R D is automorphic to where g ≡ e −iπD = (−1) −D and g 2 = 1.
Because the spin parity, P S = (−1) 2s , is always conserved by the Lorentz symmetry, we conveniently multiply the residual symmetry with the spin parity group S = {1, P S } with P 2 S = 1, to perform a new group Z 2 ⊗ S, which has an element satisfying g 2 S = 1. Therefore, we have to be an invariant subgroup of Z 2 ⊗ S, and decompose Since [(Z 2 ⊗ S)/Z 2 ] = {{1, g S }, {g, P S }} is conserved if g S is conserved, we can omit it, interpreting the Z 2 group as a residual symmetry instead of Z 2 .
The field representations under Z 2 are summarized in Table III. Notice that Z 2 has two onedimensional irreducible representations, 1 (1) and 1 (2) , corresponding to g S = 1 and −1, respectively, as usual.
From Table III, the model with δ = −1 implies scenarios of a single dark field, labeled Ψ 1 , which transforms nontrivially under Z 2 , i.e. g S = −1, such as for a fermion or Field for a scalar, where d 1 is integer. For simplicity, throughout this work, we assume dark matter candidates to be spin-0 bosonic and/or spin-1/2 fermionic fields, transforming as the standard model singlet. 3 This result is similar to the previous interpretation obtained in [63,64]. Therefore, we obtain the simplest dark matter candidate to be either a fermion or a scalar with d 1 = 0.
We would like to stress that Ψ 1 can have an arbitrary mass, which does not decay to the usual particles. First, all Ψ 1 , e, d, φ + , and W + are Z 2 odd. Next, Ψ 1 is color and electrically neutral, whereas the rest (e, d, φ + , W + ) carry a color and/or an electric charge. The color and electric charge conservation demands that Ψ 1 cannot decay to such a colored and/or electrically charged state. Indeed, by contrast, suppose that Ψ 1 decays to a final state. This state must be color and electrically neutral, as Ψ 1 is. Additionally, the Z 2 conservation requires that the final state must be combined of an odd number of the odd fields (e, d, φ + , W + ). Since φ + is eaten by W + and that W + decays to (e + , ν) or (d c , u), the final state includes only (e, d) as possible old fields. Thus, the decay process takes the form, where the dots stand for the remaining fields (if any) that are electrically and color neutral and Z 2 even. The conservation laws imply, i) x +x + y +ȳ = 2n + 1 (Z 2 odd), ii) −x +x − y/3 + y/3 + 2z/3 − 2z/3 = 0 (electrically neutral), and iii) y + z −ȳ −z = 3n (color neutral), for n, n integer. The last two equations lead to −x +x − y +ȳ + 2n = 0, which combined with the first equation, yields a contradiction to be 2(x +ȳ + n ) = 2n + 1. In other words, the symmetries SU (3) C , U (1) Q , and Z 2 jointly suppress the decay of the Ψ 1 dark matter, if it has a mass larger than the ordinary odd particles (e, d, φ + , W + ). Hence, the Ψ 1 dark matter can have an arbitrary mass, and its stability is different from the most extensions.
B. The model with δ = 2 From the third column of Table II, it is clear that R D = 1 for all fields with minimal value of |k| = 4, except k = 0. Thus, the residual symmetry R D is automorphic to where t ≡ e iπD/2 = (−1) D/2 and t 4 = 1.
For convenience, we multiply the residual symmetry with the spin-parity group S = {1, P S } to perform Z 4 ⊗ S, which has an invariant subgroup, where t S = t × P S = e iπ(D/2+2s) = (−1) D/2+2s and t 4 S = 1. Therefore, we decompose Since is conserved, we can consider Z 4 to be a residual symmetry instead of Z 4 (note that P S is always preserved). The field representations under Z 4 are given in Table IV (third column). Here, Z 4 has four one-dimensional irreducible representations, 1 (1) according to t S = 1, 1 (2) according to t S = i, 1 (3) according to t S = −1, and 1 (4) according to t S = −i, with i to be imaginary unit.
It is noted that Z 4 contains an invariant subgroup Z 2 = {1, t 2 S } with the corresponding quotient group consists of two elements of Z 4 , the characteristic x and the other t 2 S x. 4 However, Z 4 is not isomorphic to Z 2 ⊗ Z 2 . Additionally, one can verify that the fields transform the same under Z 2 and Z 2 , identical to the is always isomorphic to t 2 S . They are also included in Table IV as the last three columns, for clarity. 5 That said, the Z 4 basis is necessarily to supply multi-component dark matter, while the Z 2 ⊗Z 2 group is basically reducible to 4 Hereafter, because the models with δ = −1, 2, 1 are distinct, the notations used for labeling discrete groups and their presentations may be identical among the models, without confusion. 5 Hereafter, although the representations denoted by corresponding dimensions, as underlined, may appear the same across different groups, they must be distinguished as always supplied under their own group, which should not be confused.
the single dark matter. Extra comment is that if one allows two factors U (1) N 1 ⊗ U (1) N 2 according to δ → δ 1,2 , as mentioned, they yield a residual Klein group Z (1) 2 with the factors determined by independent dark charges D 1,2 , respectively. This supplies a scheme of two-component dark matter whose candidates transform nontrivially under the Z 2 's factors, respectively, similar to the present Z 4 with only one δ. Fermion Scalar From Table IV, the model with δ = 2 implies three kinds of dark field, as determined in Table   V, where d 2,3,4 are integer. It is able to show that Z 4 is the smallest (i.e. Z k ) cyclic symmetry that allows a scenario of two-component dark matter [13,18,39]. Among the solutions, we consider here the two scalar dark fields to be the simplest candidates of dark matter, as resupplied in Table VI.
In this case of the model, the usual fields e, u, d, φ + , and W + transform nontrivially under Z 4 , analogous to the dark fields. Since φ + is eaten by W + , while W + decays to νe + or ud c , a potential decay of dark matter need not consider these bosons in the product, similar to the previous model. and neutral under the electric and color charges. On the other hand, dark matter that transforms as 1 (2) or 1 (4) cannot decay to usual particles, since it is odd under t 2 S = g S , thus suppressed by the t 2 S conservation, similar to the previous model. Dark matter that transforms as 1 (3) cannot decay to a state that contains a single field of e, u, d, because of the electric and color charge conservation. Also, it cannot decay to a state that contains any two of e, u, d, since the charge conservation restricts such state to e + e − , u c u, and d c d, but this is not allowed by Z 4 . Moreover, a final state that includes three of them is only eud c or udd, but not allowed by Z 4 . And, a final state that has four of them is only euud, e + ddd, or others as combined of two states of two (e, u, d) fields above, but discarded by Z 4 . Finally, a final state that contains more than four of them must be eeuuu or combine the above states due to the charge conservation, which are all suppressed by Z 4 . Hence, SU (3) C , U (1) Q , and Z 4 do not permit any dark matter decaying to ordinary particles.
Notice that the dark fields may have self-interactions; for example, the case of the simplest dark fields above includes a Lagrangian term, Φ † 3 Φ 2 2 . The two dark fields are simultaneously stabilized responsible for two-component dark matter, if the mass of Φ 3 is smaller than twice that of Φ 2 . C. The model with δ = 1 From the last column of Table II, we derive that R D = 1 for all fields with the minimal value of |k| = 6, except for k = 0. Hence, the residual symmetry is automorphic to where p ≡ e iπD = (−1) D and p 6 = 1. Following [63,72], we factorize Z 6 into where Z 2 is the invariant subgroup of Z 6 , while Z 3 is the quotient group of Z 6 by Z 2 , given by Here each (coset) element of Z 3 , denoted by [y], contains two elements of Z 6 , the characteristic y and the other p 3 y as multiplied by where ω ≡ e i2π/3 defines the cube root of unity. Whereas, Z 2 is generated by the element, It is noted that 3D is always integer, since p 6 = 1.
The representations of all fields under the Z 2,3 groups are computed, given in Table VII. It is clear that the 1 (3) representation of Z 3 is not presented for the existing fields. But, let us note that the antiquarks transform as 1 (3) , since 1 (2) * = 1 (3) , and vice versa.
The necessary condition for F 5 and F 6 to be co-stabilized is that the net mass of F 5 and F 6 must be smaller than that of Φ 7 , i.e. m 5 + m 6 < m 7 , where m 5,6,7 are the mass of F 5,6 and Φ 7 , respectively.
The phenomenology of this model is completely different from the U (1) B−L extension [72], since the ordinary particles transform nontrivially under P M in the same with the dark fields (cf . Table   VIII). Additionally, it happens similarly for Z 3 , where the quarks are nontrivial as the dark fields.
The electric and color charge conservation is necessarily to ensure dark matter stability. And, the dark matter components can have arbitrary masses, which need not be smaller than the e, u, d masses. Let us see.
Since P M is identical to g S , the stability of the dark matter component F 5 is always ensured by the SU (3) C , U (1) Q , and P symmetries, independent of its mass, analogous to the case of the dark matter Ψ 1 in the model with δ = −1. Whereas, the SU ( Substituting the VEVs, ν R obtain a Majorana mass matrix, They are given through the mass Lagrangian, Since v Λ, we have m D m R . Hence, the observed neutrino (∼ ν L ) masses are given by the seesaw mechanism, while the heavy neutrinos (∼ ν R ) have a mass proportional to m R at Λ scale.
The breaking of the hyperdark charge N is necessarily to generate small neutrino masses, proportional to m ν ∼ v 2 /Λ, where Λ is large, compared with the weak scale. This breaking (or seesaw) scale, Λ, is constrained by the particle colliders below.

B. Collider searches for Z
The U (1) N gauge boson, Z , obtains a mass, where g N is the U (1) N gauge coupling, and that small contributions due to mixing effects from the kinetic mixing term and the weak breaking are neglected.
The field Z couples to the usual fermions and ν R via the Lagrangian, where we define This applies for usual fermions, except for neutrinos. By the seesaw mechanism, the usual neutrinos ν ν L and the sterile neutrinos ν ν R are distinct (Majorana) particles. Hence, g Z V,A (ν) are computed by suppressing N (ν R ), whereas g Z V,A (ν ) are achieved by omitting N (ν L ). If m Z is smaller than the highest collision energy of the LEPII experiment, √ s = 209 GeV, the particle Z may be resonantly produced at the LEPII. The agreement between the experimental measurement and the standard model indicates that g N 10 −2 , up to a factor of δ [84].
If m Z > 209 GeV, the process e + e − → ff gets an off-shell contribution of Z , described by the effective Lagrangian, Taking f = µ, τ , the LEPII searched for such chiral structures, yielding a strong limit for Z mass per coupling to be which translates to Λ ≥ 1.5|(2δ − 1)/δ| TeV [85].
The LHC searched for dilepton signals through pp → ff for f = e, µ, τ , mediated by Z , which provide a mass bound about m Z = 4 TeV for Z coupling similar to Z. This indicates to a bound for Λ ∼ m Z /2g ∼ 3 TeV, where g is the SU (2) L coupling [86].
All the above searches imply a bound on the new physics scale to be Λ ∼ m Z /g N > O(1) TeV.
Note that Z has not been considered, decaying to the dark fields. By contrast, if Z decays to some dark fields, such bound is more relaxed.
Let us remind the reader that although Λ is constrained at TeV, the Z mass, m Z ∼ g N Λ, may be small, as g N is small, and vice versa.
Notice that the interactions of dark fields with U (1) N gauge portal, as well as dark field selfcouplings, govern the dark matter observables. Here, such interaction of a dark field, called X, with Z is given by the Lagrangian, either L ⊃X(iγ µ D µ − m X )X or L ⊃ (D µ X) † (D µ X) − m 2 X X † X, for X to be a dark fermion or scalar, respectively, where D µ = ∂ µ + ig N N X Z µ is the covariant derivative, and N X is the hyperdark charge of the dark field. Due to the cyclic property of discrete symmetry, N X may be arbitrary, similar to the ones given via d 1,2,··· ,7 above. One may wonder that the theory is not well defined for large N X .
First of all, opposite to non-Abelian charges, Abelian charges are completely arbitrary, as also mentioned from outset. Additionally, the Lagrangian of a U (1) N gauge theory always conserves a scaling symmetry, where g N and N are the coupling and charge after transformation by any value of c. Thus, one may work in a basis, such that g N is infinitesimal, while N is rarely big, leaving the physics unchanged.
Further, the running of g N along with energy scale µ satisfies the RG equation, where the beta function, is summed over the left and right chiral fermions and scalars, respectively. The RG equation preserves the above scaling symmetry as a result. This implies that a theory with large N behaves similarly to the normal ones, when the energy scale slides. A special feature of the Abelian theory is b N < 0 for every N . Hence, the coupling g N decreases when µ decreases. Given that the theory is well defined at the Planck or grand unification scale, i.e. g N N ∼ 1, it works perturbatively and is predictive below such a large scale. 6 As a consequence, we divide the dark matter phenomenology into two cases. If g N is small, the relevant dark matter may have a large hyperdark charge. This would imply a large signal strength for dark matter when scattering with electrons, as shown below (see also [72] for B − L model). 6 An exception is that the hyperdark charge may not be unified with the normal charges in a GUT framework, since it necessarily results from a dequantization of electric charge. However, such charge possibly originates from a string [87] or a flipped GUT [88].
By contrast, if g N is large, the charge N should be small as of usual particles. In this case, the Z portal effectively governs the WIMP components, co-existed beyond the weak scale, also shown below (see also [71] for noncommutative B − L model). order [90][91][92][93][94][95][96][97][98]. Additionally, the U (1) N coupling constant, g N , is small enough to evade the low energy experimental bounds, simultaneously this enhances the charge of boosted dark matter, thus its signal strength as measured by the XENON1T [72].
In the model with δ = 2, both the dark matter components Φ 2,3 populate our galaxy, possessing densities set by their annihilation to the usual leptons and quarks through the U (1) N gauge boson portal (Z ), as well as the self-annihilation between them by the self-couplings, Above, Z has a mass to be smaller than those of Φ 2,3 due to the small g N . The boosted dark matter is either both Φ 2,3 or only Φ 3 , depending on their relative masses.
In the model with δ = 1, the dark matter component F 5 contributes to the whole cold dark matter, set by its annihilation to F 6 via the self-interaction F 5 F 6 Φ 7 , where m 6 < m 5 . The component F 6 has vanished relic density due to its annihilations to Z as well as the usual particles.
However, F 6 is presently produced, boosted in the annihilation of the cold dark matter F 5 due to the F 5 F 6 Φ 7 self-interaction, similar to the thermal process. In this model, only the boosted dark matter component is F 6 .
In all of the mentioned models, the boosted dark matter subsequently scatters on electrons in the XENON1T experiment, causing the observed effect, as enhanced by the large charge of the boosted dark matter. The next phenomenology happens quite the same. Hence, we take only the model with δ = 1 into account, hereafter.
The Lagrangian of the considering model includes where D µ = ∂ µ + ig N N Z µ is the covariant derivative. Note that F 5,6 are vectorlike, while the electron e is chiral (cf.
for n integer, which differ from the minimal basic charges in Table X by even numbers, similar to   the ones in Table IX. This is always allowed by the cyclic property of residual Z 3 and P groups and obviously that they do not change the representations of the fields.
The cold dark matter F 5 obtains a correct density by its t-channel annihilation to the second dark matter, i.e. F 5 F 5 → F 6 F 6 , exchanged by the Φ 7 dark field, as depicted by the left diagram in Fig. 1. F 6 strongly couples to the U (1) N gauge boson Z , by which it has a vanished density given by the right diagram in Fig. 1. Note that F 6 also annihilates to the usual particles via Z portal, but it is small, compared to the given channel to Z , since its hyperdark charge is large.
The densities of F 5,6 are explicitly computed at the end of this section. Presently, F 6 is boosted via a process similar to the left diagram in Fig. 1, and subsequently this fast dark matter scatters on electrons in the XENON1T experiment via the diagram in Fig. 2, by the Z portal. The mildly boosted (i.e., v ∼ 0.1c) phenomenon happens when m 5 ≈ m 6 , and we also assume m 5,6 > m e to satisfy the BBN and CMB bounds [99]. By the process in Fig. 2, the electronic recoil energy approaches E R < 2m e v 2 for m 6 > m e and agrees with the XENON1T if F 6 is boosted with v 2 = (m 2 5 − m 2 6 )/m 2 5 ∼ 1% [90]. The transferred momentum is |q| = E R /v ∼ 40 keV. In the limit of the Z mass to be much larger than |q|, the F 6 -e scattering cross section is given by [100] Since m Z ∼ g N Λ, the cross-section is enhanced for large boosted dark matter charge, as expected.
In the literature, two potential sources that produce the flux of boosted dark matter have been discussed, namely (i) dark matter annihilation in the current galaxy center/halo and (ii) dark matter capture and their annihilation in the sun. However, the second source does not significantly contribute to the flux in our model, because the cold dark matter F 5 ∼ (1, 1, 0, 0) is a gauge singlet which does not interact with the sun matter, such as nuclei, at tree level. Through 1-loop effects that include both F 6 , Φ 7 contributions, the field F 5 effectively couples to Z , such as with g N = |1/3 + 2n|g N , in agreement to [43]. This induces a spin-independent scattering cross-section of F 5 on nucleons via Z exchange in the sun, where we take m 7 1 GeV, the log function is of one order, and the squared momentum transfer is t = −2m p,n E R ∼ −m 2 Z , since the recoil energy typically E R < 2m p,n v 2 ∼ 1 keV and m Z ∼ 1 MeV, as shown below. Additionally, we have the reduced mass µ p,n = m p,n m 5 /(m p,n + m 5 ) 1 GeV, the Z vector couplings to nucleons g p(n) = 1(3)g N /4 ∼ 10 −7 (see below for g N value and g N 1), and y O(1) due to the perturbative limit. The sun will reach a capture-annihilation equilibrium, such that the scattering cross-section sets the flux of boosted dark matter to be [91] Φ Sun which is much smaller than the following galaxy source and is neglected.
Hence, the dark matter is not captured/accumulated in the sun core over time, i.e. the captureannihilation phenomenon is not enhanced by the sun. The boosted dark matter flux in our model only comes from the first source (i). We assume that the dark matter obeys an NFW profile [101].
The F 6 flux from full sky is given by [43] Φ F 6 = 1.6 × 10 −4 cm −2 s −1 σv rel F 5 F 5 →F 6 F 6 5 × 10 −26 cm 3 s −1 1 GeV where σv rel F 5 F 5 →F 6 F 6 is the thermally-averaged dark matter annihilation cross-section at present time, computed below in (89), and 1 GeV belongs to the dark matter mass range investigated, but not necessarily indicated to a benchmark. The flux of the fast dark matter F 6 that comes from all the sky due to the F 5 cold dark matter annihilations in our galaxy would cause a signal rate at the XENON1T detector. The total number of signal events can be estimated as in agreement to [91]. Here, Z eff ∼ 40 is the effective number of recoil electrons in a Xenon atom, which is governed by electrons on N, O shells; n Xe V = M/m Xe is the number density of Xenons in the detector, multiplied by the fiducial volume of the detector, which equals the detector-to-Xenon mass ratio; T is the total operation time; and notice that M T = 0.65 tonne-years relevant to the current XENON1T data.
As shown below, the present-day dark matter is dominated by F 5 , set by its dominant annihilation to F 6 in the early universe, hence constraining the cross-section σv rel F 5 F 5 →F 6 F 6 5 × 10 −26 cm 3 s −1 for the correct abundance. With this condition, from Eqs. (66) and (67), the relation between the number of signal events and the scattering cross section is derived to be Required to explain the number of signal events, N sig ∼ 100, the hyperdark charge of dark matter is large, where both Λ and m 5 have a lower bound, Λ > 3 TeV and m 5 > m e , as mentioned. Additionally, from Eq. (70), the dark matter charge required is increased, when the dark matter mass m 5 and/or the new physics scale Λ are larger than the given bounds.
In this case, we approximate the beta function through b N −(5/3)(1/3 + 2n) 2 , yielding the RG equation, where g N = |1/3 + 2n|g N slides as the usual couplings. Defining the hyperdark charge coupling strength, α N = g 2 N /4π, we obtain the solution, where G indicates to those at the large (GUT or Planck) scale. Imposing a perturbative limit for the hyperdark charge interaction at the large scale, say α N G = 1 at µ G = 10 16 GeV, it leads to α N 0.08 at scale µ ∼ 1 MeV of interest, or in other words, Combining this equation with the condition N sig = 100 required to explain the excess, we get From Eq. (73) and Eq. (70), we obtain a bound for the U (1) N gauge coupling, since m 5 > m e . This bound is very close to the sensitive limit of beam dump experiments, which detect the signal of the decay Z → e + e − , that requires m Z > 2m e 1 MeV [102,103]. That said, Z is always viable below such mass limit since the beam dump experiments are not able to find it, in spite of what size g N is.
Notice that m Z and g N are further constrained by the neutrino-electron scattering, horizontal branch stars, and supernova 1987A experiments, similar to the B −L model [72], which are signified as follows. The neutrino-electron scattering due to the Z exchange yielding a bound g N 10 −6 for m Z ∼ MeV is obviously suppressed by the condition (75). The experiment with horizontal branch stars studies energy loss, carried by Z , implying a bound m Z > 0.34 MeV for g N satisfying (75), but being above the excluded regime g N 10 −8 -10 −11 , as constrained by the supernova 1987A. The BBN supplies a lower limit on g N , which would exclude the viable lower region with g N 10 −11 , according to m Z ∼ MeV. In summary, in addition to the XENON1T constraints in (76) and (75), the astrophysical and cosmological experiments restrict m Z > 0.34 MeV and g N 10 −8 .
Because Z is light, potentially affecting the BBN measurement, in what follows we will discuss the BBN bound, for clarity.
Since the dark matter components F 5,6 are above the BBN bound given at MeV, we need only examine two cases of Z mass. If the Z mass is also above this BBN bound, no sub-MeV hidden states exist. Hence, no new relativistic degrees of freedom are presented during the BBN. Only constraint from the BBN is that Z decays into the standard model fast enough, larger than the Hubble rate, in order to avoid late dissociation process. Thus, the decay Z → e + e − , νν c before If the Z mass is below the BBN bound, the field Z does not significantly contribute to the allowed number of relativistic degrees of freedom during the BBN at the standard model temperature. Indeed, after the neutrino decoupling, a significant population of Z may be generated through inverse decays, such as νν c → Z , known as a freeze-in mechanism. Note that in this regime, a population of Z derived through the reaction γe ± ↔ Z e ± or e + e − ↔ Z γ is ineffective, because such process is suppressed by a factor α 1/137 relative to the present process νν c ↔ Z and therefore is safely neglected. 7 Once the Z population is in equilibrium with the neutrino population, it will maintain the distribution through the active decay Z → νν c and the inverse decay νν c → Z . Until the temperature drops to T ν ∼ m Z /3, at which time Z decays out of equilibrium, as the inverse decay is kinetically suppressed, the active decay Z → νν c potentially produces a net contribution to the effective number of neutrinos, N eff . This contribution is viable, since the decay products of Z have an energy E ν ∼ m Z /2, generally larger than 3T ν , implying a final neutrino distribution to be more energetic than that found in thermal equilibrium.
The momentum distribution functions of Z and ν, labeled f Z and f ν respectively, obey the Boltzmann equations, such as [104] ∂f where is the rest frame width, and H =ȧ/a is the cosmic expansion rate with a the scale factor. The energy densities of the decay particle and the products are obtained by where g Z = 3 and g ν = 2 are the number of spin states of Z and ν, respectively. When the temperature drops below m Z /3, the inverse decays are forbidden and the entropy of Z population is transferred to other species. If only neutrinos and photons exist, the effective number of neutrinos is N eff = 8 7 where ρ ν differs from the SM value due to the entropy transferred from Z decays. The deviation of the effective number is which is integrated over comoving momentum of ν, y ν = ap ν /MeV, where f FD ν = (1 + e yν ) −1 is the distribution function of a free-streaming decoupled neutrino. The numerical investigation of the 7 The process e + e − ↔ Z does not occur for cosmic temperatures and m Z below 1 MeV, so it would not be included.
Also, similar processes that thermalize Z with the light quarks (u, d) are ineffective in comparison to νν c ↔ Z .
Boltzmann equations (79) and (80) in the fast decay regime Γ Z ≥ H(m Z ) was already done in [105], with noticing that at very high temperatures f Z = 0, and its result would apply to our model without change. That said, the deviation of the effective number of neutrinos is ∆N eff 0.21. This value is suitable to the Hubble tension between local measurements and temperature anisotropies of CMB, reported in [106]. Notice that the fast decay regime Γ(Z → νν c ) ≥ H(m Z ) under consideration requires different from the previous case of the Z mass. Hence, the new light state Z may be presented, which does not alter the BBN measurement.
It is noted that F 5,6 are populated and follow a thermal distribution before the freeze-out of the dark matter. Hence, Z must also be populated, since it is annihilated in by F 6 , and should have enough interactions with the standard model sector to keep the dark matter in thermal equilibrium until freeze-out. However, the freeze-in mechanism discussed above is valid only if g N is sufficiently small such that the Z population will not reach equilibrium with the standard model thermal bath in the very early universe, by contrast. 8 The condition for Z not to reach equilibrium with the heavy leptons (µ, τ ) and the heavy quarks (c, b, t) at early times can be considered as n µ σv µ + µ − →γZ < H(m µ ), because the annihilations of τ, c, b, t to γZ are quite smaller than that of µ due to the suppressions of larger fermion masses and/or smaller γ, Z couplings. Correspondingly, the condition requires where σv µ + µ − →γZ ∼ αg 2 N /16m 2 µ is approximately given in non-relativistic limit near T ∼ m µ and is independent of m Z due to m Z m µ . From Eqs. (84) and (85), there is a sizable range of g N , i.e. 0.53 × 10 −9 g N 10 −8 , under which the Z population does not reach equilibrium with the heavy fermions at early times, but Z will equilibrate with neutrinos, leading to a N eff deviation through its decay Z → νν c appropriately. Unfortunately, this range of g N that ensures the freeze-in mechanism is obviously excluded by the condition in Eq. (77) which comes from the supernova 1987A and is not favored by the WIMP criteria, as mentioned. Nonetheless, this study of the BBN bounds given in Eqs. (78) and (84) is aimed at ruling out the bound g N 10 −11 as hinted from the supernova 1987A too. It is noteworthy that g N is finally bounded by Eq. (77) and in this case the condition in Eq. (85) is not satisfied. Hence, g N 10 −8 ensures Z to be in kinetic equilibrium with the thermal plasma at early times, protecting the freeze-out mechanism of the dark matter from outset, and this dark matter setup also independently excludes g N 10 −11 .
Last, but not least, let us evaluate the dark matter relic densities as stated. It is suitably to assume m 6 > m Z , as above mentioned. Hence, F 6 is obviously annihilated to Z via the right diagram in Fig. 1. The annihilation cross section is since g N 1 and that m 6 is radically below the weak scale. The F 6 density is negligible, for m F 6 < 10 GeV. Here the factor 1 2 associated with the annihilation cross-section arises because F 6 is a Dirac fermion and that the F 6 density refers to the sum of the abundances for F 6 andF 6 .
Such factor and density definition also apply to every Dirac fermion candidate below, which should be understood.
Hence, the cold dark matter abundance is approximated by the F 5 contribution, Applying the Feynman rules for the left diagram in Fig. 1, we obtain the annihilation cross-section Using the approximation m 6 ≈ m 5 , the data for dark matter density [1] is recovered if σv rel F 5 F 5 →F 6 F 6 1.67 pb, 9 or The dark matter self-coupling depends on m 7 /m 5 and m 5 /m e , having a lower bound |y| > 10 −3 , for m 7 > 2m 5 and m 5 > m e , where note that δm 2 /m 2 5 ≡ (m 2 5 − m 2 6 )/m 2 5 ∼ 1%. As an instance, we make a contour of the dark matter self-coupling y and the F 5 mass m 5 in Fig. 3. Note that F 5 may annihilate to Z via 1-loop diagrams with both Φ 7 and F 6 running in the loop. The amplitude is then evaluated as M rad ∼ y 2 m 5 m 2 6,7 g 2 N 16π 2 , which is substantially smaller than the given tree-level amplitude, M tree ∼ y 2 m 5,6 m 2 7 , due to the loop factor suppression ∼ 1/16π 2 and g N 1. The tree-level annihilation of F 5 to F 6 sets the relic density.
E. Large U (1) N coupling: Implication for co-existed WIMPs WIMP is a natural solution of the thermal dark matter paradigm, which in the past it contributed to the thermal bath of the early universe, then decoupled, i.e. freeze-out, with a presentday density, when its annihilation rate into usual particles matches the Hubble expansion rate.
Hence, it is sufficiently to investigate one of them, say the model with δ = 1, with the candidates in Table X, which have the minimal basic hyperdark charges and a large U (1) N gauge coupling.
We also assume the F 5,6 masses to be beyond the weak scale.
Dark matter pair annihilation in the present case includes those given similarly to Fig. 1, plus the extra s-channel diagrams (not depicted) for F 6 annihilation to the standard model particles (leptons, quarks, Higgs and gauge bosons) exchanged by Z , namely F 6 F c 6 → f f c , ZH where f indicates to the usual leptons and quarks. Notice that F 5 does not annihilate to the usual particles, since its dark charge vanishes.
It is straightforward to determine the annihilation cross section of F 6 , such as where g Z V,A (f ) were supplied in Table XI, while the HZZ coupling is given by g Z ZH −g N v/2. Since the dark matter components F 5,6 now have arbitrary masses above the weak scale, the conversion between F 5,6 is either The former process is described by the left diagram in Fig. 1, in which the result in Eq. (89) properly applies for this case. The latter, or inverse process, is given by a t-channel diagram, similar to the previous process by replacement F 5 ↔ F 6 with the same Φ 7 mediator, leading to an annihilation cross section, It is noted that for the case m 6 > m 5 , the annihilation cross sections in Eqs. (92) and (93) set the F 6 density. However, F 5 never annihilates after the temperature of the universe falls below the F 6 mass. Thus, it has a large density, overpopulating the universe. This case should be discarded.
Hence, we consider only m 5 > m 6 . And, the F 5 density is governed by its annihilation to F 6 , i.e.
with the annihilation cross-section given by Eq. (89), while the F 6 density is determined by its annihilation to the Z and/or standard model particles, with the annihilation cross-section obtained by Eq. (92). The total dark matter density is Ω DM h 2 = Ω F 5 h 2 + Ω F 6 h 2 , where both the components contribute, unlike the above model for XENON1T.
The parameters that significantly govern the dark matter observables are the dark field self- resonance in the density, at which m 6 = 1 2 m Z , which reduces the density to zero, thus suppressed for correct abundance.
Since F 5 ∼ (1, 1, 0, 0) is sterile, it does not interact with detectors in direct detection experiment. Effect of dark matter in the direct detection comes only from a potential scatter of the F 6 component with nuclei in a large detector. 10 Indeed, the effective Lagrangian describing the F 6 -quark interaction can be derived from the t-channel exchange diagram by the new gauge boson Z to be L eff ⊃F 6 γ µ F 6 [qγ µ (α q P L + β q P R )q] , where P L,R = (1 ∓ γ 5 )/2, and α u,d = −β u /2 = β d /4 = −g 2 N /18m 2 Z .
Hence, we obtain the spin-independent (SI) cross-section for the scattering of F 6 on a nucleon as where λ p = [2(α u + β u ) + α d + β d ]/8, Above, Z is the nucleus charge and A is the total number of nucleons in the nucleus. Because the dark matter components partially contribute to the total density, the effective SI cross-section of F 6 is given by Taking A = 131 and Z = 54 according to the Xe nucleus and m nucleon 1 GeV, in Fig. 5 we plot the SI cross-section of F 6 corresponding to the previous choices of Λ = 3 and 3.5 TeV, y = 0.95, x = 1.1, and g N = 0.2, which resulted in Fig. 4 lower panel. We always assume that the two dark matter components give the correct relic density. The XENON1T experimental bounds [119,120] have also been included to the plot. Combining the results in Fig. 4 lower panel and This setup not only yields suitable neutrino masses, but also implies the novel schemes of multicomponent dark matter. The dark matter stability is ensured by the residual dark charge identical to an even Z k symmetry, a remnant of the gauge symmetry after symmetry breaking. Additionally, the dark matter components can have an arbitrary mass, despite the fact that they and the normal particles transform nontrivially under the discrete symmetry, that cannot decay as a consequence of the color and electric charge conservation, combined with Z k . With this mechanism, we have pointed out the simplest models for two-component dark matter. Additionally, we have proved that they can address the XENON1T anomaly recently observed, as the second dark matter component is boosted in annihilation of the first dark matter component. Alternatively, these models can contain two WIMPs with relevant masses above the weak scale, satisfying the relic density and direct detection bounds.