Baryogenesis in the Standard Model and its Supersymmetric Extension

In this work, we classify all the effective $U(1)$ symmetries and their associated Noether charges in the Standard Model (SM) and its minimal supersymmetric extension (MSSM) from the highest scale after inflation down to the weak scale. We then demonstrate that the discovery of the violation of baryon minus lepton number ($B-L$) which pinpoints to its violation in primordial Universe at any cosmic temperature above $30$ TeV will open up a new window of baryogenesis in these effective charges above this scale. While the fast SM baryon number violation in the early Universe could be the first piece to solve the puzzle of cosmic baryon asymmetry, $(B-L)$ violation could be the second important piece. In the background of expanding Universe, there is ample opportunity for out-of-equilibrium processes to generate an asymmetry in the numerous effective charges in the SM or the MSSM, making a baryon asymmetric Universe almost unavoidable. Finally we provide examples in the SM and the MSSM where baryogenesis can proceed through out-of-equilibrium dynamics without explicitly breaking baryon nor lepton number.


I. INTRODUCTION
Symmetry has been a guiding principle in physics. In the Standard Model (SM), the gauge symmetry SU (3) c × SU (2) L × U (1) Y fixes all the possible interactions among the fields. However, once one consider the same theory in an expanding Universe, new effective symmetries can arise if the rates of certain interactions are slower than the Hubble rate, i.e. out of thermal equilibrium. For instance, while baryon-number-violating interactions due to SU (2) L instanton-induced operators are suppressed to an unobservable rate today [1], they were in thermal equilibrium for cosmic temperature above the weak scale [2]. This baryon number violation in the primordial universe, among others, could be a key ingredient to understand the observed cosmic baryon asymmetry represented by the baryon number density asymmetry over cosmic entropic density Y ∆B ∼ 9 × 10 −11 [3].
Symmetry in the context of baryogenesis is a double-edged sword. On the one hand, if a symmetry remains exact, no asymmetry can be generated in the associated Noether charge.
On the other hand, it can protect an asymmetry generated from being erased. This was first elucidated by Weinberg [4] that as long as there is a linear combination of baryon and lepton number B + aL that is conserved, a baryon asymmetry is preserved even if baryonnumber-violating interactions are in equilibrium. The use of other types of symmetry as protective mechanism was further explored in the SM [5][6][7], in its extension [8,9], in its supersymmetric extension [10] and was generalized in [11].
In this work, we will categorize all the effective symmetries and their associated Noether charges in the SM and the Minimal Supersymmetric Standard Model (MSSM), from the highest temperature after reheating T ∼ 10 16 GeV down to the weak scale. Making use of (B − L)-violating interactions which are in equilibrium, we show that asymmetries can be generated in any of the effective charges through out-of-equilibrium processes which can, but do not have to violate baryon nor lepton number.

II. GENERALITY
Here we give a brief review of the formalism discussed in [11] which will be used in this work and also to fix the notations. For any particle species i that distinguishes from its antiparticle i i.e. a "complex particle", its number density asymmetry can be defined as n ∆i ≡ n i − n i where n i is the number density of particle i. This is equivalent to its charge density if we assign the particle (antiparticle) a charge 1(−1) under a global U (1) i .
Considering r species of complex particles i, we can define r number of such global U (1) i preserved by their kinetic terms (which include possible gauge interactions).
Next, let us put the system in an expanding Universe. Assuming fast gauge interactions are able to thermalize the particles, a common temperature T can be defined at each moment.
To scale out the effect due to cosmic expansion, we define the number asymmetry of the particle i as where s = 2π 45 g T 3 is the cosmic entropic density with g the total relativistic degrees of freedom. From r number of U (1) i , we can form any other r linear combinations U (1) x . A convenient basis is such that as the expanding Universe is cooling down, U (1) x is subsequently broken by interactions which get into thermal equilibrium.
The source of U (1) x can be due to additional interactions like the Yukawa interactions or those from instanton-induced effective operators due to the Bell-Jackiw anomaly [1]. To determine if a U (1) x is preserved by the latter type of interactions, we can look at the anomaly coefficient of the triangle diagram of the type U (1) x − SU (N ) − SU (N ) defined as where the sum is over particle i with degeneracy g i , charge q x i under U (1) x , and representation R i under SU (N ≥ 2) gauge group with c (R i ) = 1 2 in the fundamental representation and c 2 (R i ) = N in the adjoint representation. In general, each fermion i with representation where the product is over all left-handed chiral fields Ψ i with nontrivial representation R i under SU (N ). Notice that if A xN N = 0, U (1) x is conserved by the operator. 1 Due to the convention c (R i ) = 1 2 for fundamental representation, the factor of 2 is included such that the field enters in integer number.
If a system possesses a U (1) x , its charge can be written as where i sum over all particle species with charge q x i under U (1) x . Assuming equilibrium phase space distribution for particle i characterized by a common temperature T , its energy E i and chemical potential µ i T , the formula above can be inverted as [11] where ζ i = 1(2) for i a massless fermion (boson) 2 and J is a symmetric matrix in charge space defined as From eq. (5), we can construct the baryonic charge as It is apparent that the cosmic baryon asymmetry is proportional to the U (1) x charges of the system. Next, our goal is to characterize all the U (1) x in the SM and MSSM.

III. THE STANDARD MODEL
Before the electroweak (EW) symmetry breaking, the SM kinetic terms respect a total of 16 U (1) Ψ j , each corresponds to the individual field rotation of the 15 fermionic fields and a scalar Higgs SU (2) L doublet H: with family index α = 1, 2, 3. Here Q α , α are respectively the quark and lepton SU (2) L doublets while U α , D α , E α are respectively the up-type quark, down-type quark and lepton SU (2) L singlets. 3 Since all the parameters of the SM have been measured, we can choose the following convenient basis according to the order in which the U (1) x symmetries are subsequently broken at T x as we go down in the cosmic temperature with U (1) Y is the hypercharge gauge symmetry which is only broken at T EW ∼ 160 GeV [12] while the rest are effective (global) symmetries which are broken at T t ∼ 10 15 GeV, violates U (1) B and the associated processes are in thermal equilibrium [2] from T B ∼ 2×10 12 GeV [13] down to T B− ∼ 130 GeV [12]. This is the source of baryon number violation for EW baryogenesis [14], though in the SM, two other Sakharov's conditions for baryogenesis [15], sufficient C and CP violation [16,17], and sufficiently out-of-equilibrium processes [18] are not fulfilled. where Assuming the EW symmetry is broken at T EW ∼ 160 GeV above T B− [12], we have c B(B−L) = 30 97 and c BY = − 7 97 assuming top quarks are nonrelativistic (cf. ref. [19]). Eq. (9) holds only down to temperature T B− below which the baryon number is frozen. If the Universe is always hypercharge neutral Y ∆Y = 0, this implies that U (1) B−L has to be broken above T B− to generate a nonzero baryon asymmetry, as utilized in leptogenesis [20], its variants [21,22] and SO(10) baryogenesis [23]. 4 With Y ∆Y = 0 and in the absence of new charges, 4 Extending the SM by new fields which carry nonzero B − L charges, baryogenesis can proceed with unbroken U (1) B−L [9,[24][25][26][27]. Some of this type of models can also accommodate the situation where compensating B − L charge remains in the hidden sector and can serve as dark matter [26,27].
if (B − L)-violating interactions remain in thermal equilibrium from T e down untill T B− , baryogenesis will fail. 5 It is usually required that (B − L)-violating interactions from new physics be out-ofequilibrium for a viable baryogenesis scenario (see for example refs. [19,32]). In this work, we will point out an orthogonal scenario. Rather, we argue that any in-equilibrium (B − L)violating interactions in fact facilitate baryogenesis and allow a new avenue of baryogenesis through out-of-equilibrium generation of asymmetry in the effective charges identified in eq.
In general, fast (B − L)-violating interactions are more than welcome since they will act as the source of nonzero B − L charge in eq. (9) [33]. The indication that U (1) B−L is broken from new physics is ubiquitous. If the SM is treated as an effective field theory at low energy, at mass dimension-5, we have the Weinberg operator α H β H which breaks B − L by two units and gives rise to Majorana neutrino mass at low energy [4,34]. It has been verified that all dimension-6 [4,34] and dimension-8 operators [35] conserve B − L while for the 18 dimension-7 [36,37] and 560 dimension-9 operators [38,39], B − L is violated by two units. If ∆B = −∆L = 1, they lead to nucleon decay channels on top of those from the operators that conserve B − L [4]. If ∆L = 2, these operators contribute to Majorana neutrino mass and neutrinoless double beta decay processes [4,34,40,41] while if ∆B = 2, they can lead to neutron-antineutron oscillation (see a review article [42]). Finally, a gauge U (1) B−L naturally arises from gauge symmetry SO(10) in grand unified theory and is broken spontaneously to the SM gauge group.
If any of the (B − L)-violating processes discussed above are in thermal equilibrium in certain temperature regime, we can construct B − L charge asymmetry as where processes get out of equilibrium shortly after, the final baryon asymmetry will be given by (9).
To recapitulate, this new class of baryogenesis can be realized by extending the SM with the following: • some in-equilibrium (B − L)-violating interactions to enforce eq. (A9); • some out-of-equilibrium processes that violate the effective symmetries in eq. (8) where we have defined the CP parameter as ψ→j ≡ Γ(ψ→j)−Γ(ψ→j) Γ ψ with Γ ψ the total decay width of ψ and Γ (ψ → j) and Γ ψ → j the partial decay widths. The charge Y ∆x generated 6 For simplicity, we assume the decay channels are flavor diagonal.
from the decays of ψ can be parametrized by where ψ(x) refers to the decay process which violates x charge, η x ≤ 1 is the efficiency for x charge production through out-of-equilibrium dynamics and Y eq ψ = n eq ψ /s with n eq ψ the relativistic equilibrium number density of ψ. If ψ particles start from a thermal abundance and decay far from equilibrium when T m ψ , we have η u−d = η u−s = η u−b = 1. Making use of eqs. (A9) and (11) and with all conserved charges remain zero, after all ψ particles have decayed above T ∼ 10 12 GeV, we end up with Below T B−L , B − L is conserved and the final baryon asymmetry is given by eq. (9) with Y ∆Y = 0. In order to obtain Y ∆B ∼ 10 −10 in accordance with observation [3], since Y eq ψ ∼ 10 −3 , one would need a reasonable CP violation of ψ→bb ∼ 10 −6 .

IV. THE MINIMAL SUPERSYMMETRIC SM
In the MSSM, all the SM fermionic fields are promoted to superfields. For anomaly cancellation, two Higgs superfields H u and H d are introduced and we can choose the additional U (1) P Q conserved by all the superpotential terms 7 except µH u H d with the following charge assignments The mixed anomaly coefficients of U (1) P Q with SU (3) c and SU (2) L are respectively given by The anomaly-free choice −3q P Q D c α + q P Q E c α = 0 is the hypercharge and hence we will consider only the solutions with −3q P Q D c α +q P Q E c α = 0. Since both B and L have the same anomaly coefficient A B22 = A L22 = 3 2 , an SU (2) L mixed anomaly-free charge can be formed [11]  where c BL = c B + c L with c B and c L any numbers. As for the SU (3) c mixed anomaly, we can cancel it with any of the chiral symmetries of the quark fields U (1) Qα , U (1) U c α and U (1) D c α with respective anomaly coefficients A Qα33 = 1 and A U c α 33 = A D c α 33 = 1 2 (all the chiral charges are fixed to be 1). For instance, a completely anomaly-free combination will be P = P + 9 2 u c .
Since P is violated explicitly only by µH u H d term, by comparing the interaction rate Γ P ∼ µ 2 /T to the Hubble rate H = 1.66 √ g T 2 /M Pl (g = 228.75 for the MSSM), we obtain [10] T P ∼ 2 × 10 7 µ 100 GeV Above this temperature, U (1) P is preserved by all the MSSM interactions.
In a supersymmetric theory, gauginos can carry nonvanishing chemical potentials and, scalar and fermionic components of a chiral superfield do not necessarily carry the same chemical potentials. This is captured by the R symmetry in which the superspace coordinate transforms as θ → e iφ θ where we fix its R charge to be 1. Requiring the superpotential in eq. (14) to have an R charge equals to 2, we have The R symmetry only has SU (2) L mixed anomaly with anomaly coefficient A R22 = −1 and an SU (2) L mixed anomaly-free R charge can be constructed [11] Gaugino masses m g break the R symmetry explicitly and by comparing the associated interaction rate Γ R ∼ m 2 g /T to the Hubble rate H, we obtain [10] T R ∼ 8 × 10 7 m g 1 TeV Above this temperature, U (1) R is preserved by all the MSSM interactions.
In any extension to the MSSM, one has the freedom choose c B and c L such that P and/or R are conserved by the new interactions. 8 Let us consider a simple model of baryogenesis which breaks P and/or R without explicitly breaking B and L. We introduce a new chiral superfield S uncharged under the SM gauge symmetry with the following superpotential λSH u H d + 1 2 M SS. Taking q P S = 0 and q R S = 1, both P and R are broken by nonzero λ respectively by ∆P = 3 2 and ∆R = 3 − 2 = 1 while respecting all the U (1) x symmetries in eq. (8). In this case, there is still an exactly conserved R charge Nonzero charges can develop in both R and P (related by R c ) through CP-violating decays Let us consider B − L violation from the dimension-7 operator D c α U c β D c γ δ H u which can induce nucleon decay such as n → e − π + . In order to sufficiently suppress this process, the effective scale of the operator should be of the order of 10 11 GeV [4]. With large couplings, the processes mediated by the operator can be in equilibrium at temperature 10 10 GeV T 10 11 GeV. We will take c B = −2c L such that the operator conserves R. 9 This operator violates U (1) P and hence R c is no longer conserved. Assuming the operator involves all generations of quarks, all effective symmetries related to quarks are violated. With the remaining conserved charges (e, µ, Y ) being zero, after all S particles have decayed above T ∼ 10 10 GeV, we have Below T B−L , B − L is conserved and the final baryon asymmetry will be given by eq. (9) with Y ∆Y = 0 and c B(B−L) = 30 97 assuming at T B− , the thermal bath has the same relativistic degrees of freedom as in the SM. Future direction in this exploration includes identifying new effective symmetries and explicit sources of B − L violation that come out from more fundamental theories like grand unified theories. In studying these more specific models to realize the new baryogenesis proposed here, one will be able to have more definite predictions in the rate of (B − L)violating processes.
Note added: While this work is being written up, a similar idea appears on arXiv [43], The interactions induced by this operator get into thermal equilibrium at T u ∼ 2 × 10 13 GeV [13]. It is then convenient to consider the following combinations of U (1) x with All the charges besides t, u and B are free from SU (2) L and SU (3) c mixed anomalies. t is the first to be broken at T t ∼ 10 15 GeV by top Yukawa mediated interactions, u is broken Next we will estimate when the rest of the symmetries are broken as cosmic temperature decreases. In order to do so, we first have to determine the appropriate basis of quarks in the thermal bath. The general quark Yukawa interactions are given by In the thermal bath, all the quarks would acquire chirality-conserving thermal mass due to interactions and one should consider the thermal mass basis. The contributions from gauge interactions are flavor blind while those from the Yukawa interactions are m 2 In general, the Yukawa couplings can be diagonalized as followsŶ where V CKM = U u U † d is identified with the Cabibbo-Kobayashi-Maskawa mixing matrix. The fact that V CKM = I 3×3 results in the breaking of U (1) Bα . By going to a basis Q = U Q Q, m 2 Q can be diagonalized as well. We can split V CKM = I 3×3 + δV and U Q = U u + δU . Since the elements of δV are in general much smaller than unity (the largest elements being δV 12 ∼ δV 21 ∼ 0.23), we can solve for δU perturbatively. At the leading order, we obtain In the thermal mass basis where m 2 U , m 2 D and m 2 Q are all diagonal, we have where for the last two terms which violate U (1) Bα , we have denoted y Qu ≡ δU U † uŶ u and y Qd ≡ δV + δU U † u Ŷ d . Keeping only the leading terms considering y u < y d < y s < y c < y b < y t , we have For the rate of quark-Yukawa-coupling-mediated interactions, we will use the result of [13] Γ y ≈ 10 −2 c (T ) y 2 T where y refers to elements ofŶ u ,Ŷ d , y Qu and y Qd and we will make an extrapolation in c(T ) to take into account the running of strong coupling. We will also consider the running of quark Yukawa couplings [44] but ignore the running of mixing angles.
For completeness, we will also estimate the rate of charged-lepton-Yukawa-couplingmediated interactions with Γ y ≈ 5 × 10 −3 y 2 T from ref. [13]. For convenience of the readers, we collect here the T x in the order when U (1) x is broken as we go down in cosmic tempera-ture: T t ∼ 10 15 GeV, T u ∼ 2 × 10 13 GeV, T µ ∼ 10 9 GeV is spontaneously broken at T EW ∼ 160 GeV. The smooth transition between the regime can be described in a unified manner in the density matrix formalism [45][46][47][48] and will be explored in an upcoming publication. In the MSSM, besides modification to the running couplings, and the change of relativistic degrees of freedom, the transition temperatures for down type quark and charged lepton will be modified by an overall factor of 1 + tan 2 β where If (B − L)-violating processes are in thermal equilibrium in certain temperature regime, we can construct B − L charge as 10 where the explicit coefficients are collected in Table I.