Baryogenesis from Symmetry Principle

In this work, a formalism based on symmetry which allows one to express asymmetries of all the particles in terms of conserved charges is developed. The manifestation of symmetry allows one to easily determine the viability of a baryogenesis scenario and also to identity the different roles played by the symmetry. This formalism is then applied to the standard model and its supersymmetric extension, which constitute two important foundations for constructing models of baryogenesis.


I. INTRODUCTION
The evidences that we live in a matter-dominated Universe are very well-established [1].
While the amount of antimatter is negligible today, the amount of matter (i.e. baryon) of the Universe has been determined with great precision by two independent methods. From the measurement of deuterium abundance originated from Big Bang Nucleosynthesis (BBN) when the Universe was about a second old (with temperature T BBN ∼ MeV), Ref. [2] quotes the baryon density normalized to entropic density as 10 11 Y BBN B = 8.57 ± 0.18. From the measurement of temperature anisotropy in the cosmic microwave background radiation imprinted by acoustic oscillation of photon-baryon plasma when the Universe was about 380000 years old (T CMB ∼ 0.3 eV), Planck satellite gives 10 11 Y CMB B = 8.66 ± 0.06 [3]. The impressive agreement between the two measurements is a striking confirmation of the standard cosmological model.
In order to account for the cosmic baryon asymmetry, baryogenesis must be at work before the onset of BBN. Although the Standard Model (SM) of particle physics (and cosmology) contains all the three ingredients for baryogenesis: baryon number violation, C and CP violation, and the out-of-equilibrium condition [4], it eventually fails and new physics is called for [5]. Clearly these ingredients are necessary but not sufficient. Moreover, the early Universe is filled with particles of different types that interact with each other at various rates, rendering it a daunting task to analyze them. In this work, I would like to advocate the use of symmetry as an organizing principle to analyze such a system. In particular, I will show that by identifying the symmetries of a system, one can relate the asymmetries of all the particles to the corresponding conserved charges without having to take into account details of how those particles interact. 1 This should not come as a surprise since symmetry dictates physics: when we specify a symmetry and how particles transform under it, the interactions are automatically fixed. I will first review the formalism in Sec. II. Then the roles of U (1) symmetries are clarified in Sec. III. In Sec. IV and V respectively, I will apply this formalism to the SM and its supersymmetric extension as they form important bases for constructing models of baryogenesis. Finally I conclude in Sec. VI. 1 It should be stressed immediately that the symmetries do not have to be exact. If a symmetry is approximate, the corresponding charge will be quasi-conserved with its evolution described by nonequilibrium formalism like Boltzmann equation. In other words, the description of the system boils down to identifying only the interactions related to approximate symmetries.

II. FORMALISM
Here I will review the formalism that we will use in this work. 2 For a system with s number of symmetries labeled U (1) x and consisting of r ≥ s distinct types of complex particles labeled i (i.e. not self-conjugate like real scalar or Majorana fermion) with corresponding chemical potentials µ i and charges q x i under U (1) x , the most general solution is given by where C x is some real constant corresponding to U (1) x . It is apparent that Eq. (1) is the solution for chemical equilibrium conditions for any possible in-equilibrium interactions since by definition, the interactions necessarily preserve the symmetry. Note that symmetry discussed in this work always refers to U (1) which characterizes the charge asymmetry between particles and antiparticles. The U (1) x can be exact (like gauge symmetry) or approximate (due to small couplings, and/or suppression by mass scale and/or temperature effects). The diagonal generators of a nonabelian group do not contribute as long as the group is not broken [6]. For instance one does not need to consider conservation of third component of weak isospin T 3 before electroweak (EW) phase transition. Now for each U (1) x , according to Noether's theorem there is a conserved current and the corresponding conserved charge density can be constructed as where n ∆i is the number density asymmetry for particle i. To proceed we need two further assumptions. Firstly, particle i is assumed to participate in fast elastic scatterings such that for fermion or boson respectively. Secondly, there are fast inelastic scatterings for particle i and its antiparticleī to gauge bosons (which have zero chemical potential) such that µī = −µ i . These two assumptions are justified for instance when the 2 The formalism was first introduced by Ref. [6] to prove that the generation of hypercharge asymmetry in a preserved sector implies nonzero baryon asymmetry. See also the relevant discussion in Chapter 3.3 of Ref. [7].
particles have gauge interactions. Now Eq. (2) can be related to its chemical potential for µ i T as follows 3 In the above g i specifies the number of gauge degrees of freedom and with z i ≡ m i /T . In the relativistic limit (T m i ), we have ζ i = 1(2) for i a fermion (boson) while in the nonrelativistic limit (T m i ), we obtain ζ i = 6 π 2 z 2 i K 2 (z i ) with K 2 (x) the modified Bessel function of type two of order two. Using Eqs. (1) and (3), Eq. (2) can be written as where we have defined the symmetric matrix J as follows We can invert Eq. (5) to solve for C y in terms of n ∆x and substituting it into Eq. (1) and then making use of Eq. (3), we obtain 4 Eventually one would like to relate this to baryon asymmetry i.e. the baryon charge density.
By substituting Eq. (7) into Eq. (2) for baryon charge density, we have 3 The expansion in µ i /T 1 is justified as long as the number asymmetry density is much smaller than its equilibrium number density. For instance with n ∆i the order of the observed baryon asymmetry, the expansion holds when the corresponding particle mass over temperature m i /T 20. 4 As long as r ≥ s and there are no redundant symmetries, in the sense that all the symmetries are linearly independent and there is no rotation in the s-dimensional symmetry space that can make all the r distinct particles uncharged under some U (1), J always has an inverse.

III. THE ROLES OF U (1) SYMMETRIES
In general, the reaction rate of a process γ in the early Universe is temperature-dependent Γ γ (T ). At each range of temperature T * , by comparing Γ γ (T * ) to the expansion rate of the Universe H(T * ), we can categorize the reactions into three types [9,10] . The reactions of type (i) are fast enough to establish chemical equilibrium and are implicitly 'resummed' in the J matrix in Eq. (6).
The reactions of type (ii) either do not occur or proceed slow enough. The former is due to exact symmetry like gauge symmetry while the latter is due to small couplings, and/or suppression by mass scale and/or temperature effects. Finally the reactions of type (iii) should be described by nonequilibrium formalism like Boltzmann equation in order to obtain quantitative prediction. In this work, the effective symmetries concern both reactions of types (ii) and (iii). In particular gauge symmetry always belongs to type (ii) and can play an interesting role as 'messenger '. If an approximate symmetry belongs to type (ii), it can acquire a role as a 'messenger ' or 'preserver ' while if it is of type (iii), it can act as 'creator/destroyer '.
To understand the roles of U (1) alluded to above, it is illuminating to group the charges as follows. Among all the charges U = {n ∆x }, there is a subset U 0 = {n ∆a } where the net charges vanish n ∆a = 0. In this case, we can remove them from the beginning and left with U = U −U 0 = {n ∆m } to describe the system. From Eq. (5), we have a set of linear equations , which allows us to solve for C a in terms of C m . 5 After eliminating C a , the number density asymmetry for particle i can be expressed as where we have definedq The equation above can be succinctly written as n ∆B = m,nJ Bm J −1 mn n ∆n but it is elucidating to keep it as it is: the two terms in the square bracket of Eq. (10) represent two different types of contributions to the baryon asymmetry. The first term is the direct contribution ofŨ sector to the baryon asymmetry while the second term is the contribution ofŨ sector through U 0 (the messenger sector). Hence even ifŨ sector does not carry baryon charge J Bm = 0, as long as it carries charges in the messenger sector J bm = 0, and some baryons also carry charges in the messenger sector J Ba = 0, we will have n ∆B = 0. Herẽ U sector can play two roles: as creator/destroyer or preserver of asymmetries depending on their rates as discussed in the beginning of this section. In short, the roles of U (1) symmetries in baryogenesis can be concisely stated as follows: 1. Creator/destroyer : type (iii) reaction with an approximate U (1) m . The dynamical violation of U (1) m results in the development of n ∆m = 0 from n ∆m = 0. As mentioned earlier, quantitative prediction requires one to solve dynamical equation like 5 We use a, b, ... to label the charges in U 0 and m, n, ... to label the charges inŨ .
Boltzmann equation for n ∆m and the generated asymmetry depends on the rates of creation and washout.

2.
Preserver : type (ii) reaction with U (1) m and n ∆m = 0. The symmetry prevents the asymmetry from being washed out. The lightest electrically neutral particle in this sector can be a good (asymmetric) dark matter candidate. shows that a preserved sector which carries nonzero hypercharge asymmetry implies nonzero baryon asymmetry (set a = b = Y in the second term in Eq. (10)). We can readily extend this result to post-EW-sphaleron baryogenesis scenario [11] where U (1) Q plays the role of messenger. In this case, baryon asymmetry cannot be erased by fast B-violating interactions as long as there is a preserved sector carrying nonzero electric charge asymmetry. Of course, phenomenological constraint will require that the electric charge asymmetry to decay away before BBN.

First let us define the
is the quadratic Casimir operator in representation R of SU (N ) with c 2 (R) = 1 2 in the fundamental representation and c 2 (R) = N in the adjoint representation. Here g i is the degeneracy of particle i of charge q x i in a given representation. In the following, for N = 2, we always refer to weak SU (2) L while for N = 3, color SU (3) c .
The SM Yukawa sector is described by operator O EWsp = α (QQQ ) α (here onwards, these interactions will be referred to as EW sphalerons) [12]. Nonetheless, one can form an anomaly-free charge combination U (1) (B−L) α respected by O EWsp . Although exponentially suppressed today [12], the EW sphalerons are in thermal equilibrium in the temperature range T −  GeV and T + EWsp ∼ 10 12 GeV [13,14]. For most of the epoch in the early Universe, quark intergeneration mixing violates baryon flavors and hence the exact symmetries are instead 6 Outside this temperature window, B and L α are effectively conserved. In the SM before EWPT, U (1) ∆α can act as both creator and preserver while U (1) Y is a messenger. After EWPT, the role of U (1) ∆α is taken over by U (1) B while the messenger becomes U (1) Q . More often than not, when one considers scenario beyond the SM, there are new symmetries (see Sec. V) which can play the roles of U (1) discussed before.
The symmetries of the SM in the early Universe is summarized in Fig. 1.

A. Some specific cases
Let us define the vectors q T i ≡ q ∆α i , q Y i and n T ≡ (n ∆α , n ∆Y ). Consider first the temperature regime T ∼ 10 4 GeV when all Yukawa interactions are in thermal equilibrium and all particles are relativistic. In this case, the J matrix is easily determined from Eq. (6) to be (here we express in its inverse) where N H −1 is number of extra pairs of Higgses H with the assumption that they maintain chemical equilibrium with the SM Higgs H. Using the matrix above, n ∆i can be expressed in terms of conserved charge densities through Eq. (7). In particular, setting N H = 1 and n ∆Y = 0, we obtain lepton asymmetries n ∆ α and Higgs asymmetries n ∆H in terms of n ∆α which are in agreement with Ref. [16]. We can further consider cases at higher temperature when e Yukawa interactions are out-of-equilibrium in which we gain a chiral U (1) e . Formally we can create another conserved charge n ∆e and determine J which is now a 5 × 5 matrix.
However if we were to take n ∆e = 0, in practice, we can just set ζ e = 0 from the beginning to For simplicity, we will assume that all particles are relativistic although particle decoupling effects [18,19] can be straightforwardly taken into account by considering generic form of Setting n ∆Y = 0, we have from Eq. (8) On the other hand, assuming T − EWsp < T EWPT , we need to consider the components of SU (2) L doublets and use Q in place of Y as in Table II. Doing so we obtain Now setting n ∆Q = 0, we have from Eq. (8) The results above agree with Ref. [20] albeit obtained from simpler derivation based on symmetry principle.

V. THE MINIMAL SUPERSYMMETRIC STANDARD MODEL
Here we consider a well-motivated extension to the SM which is the minimal supersymmetric SM (MSSM). The MSSM superpotential is given by where all the fields above stand for left-chiral superfields. One observes that the superpotential has an R symmetry U (1) R for e.g. with q R (H d ) = q R ( α ) = q R (U c α ) = −q R (E c α ) = 2 and the rest of the fields having zero charges. 7 The R symmetry has mixed anomalies there is only A R22 = −1 anomaly. Thus one can form an anomaly-free charge combination as follows with c BL ≡ c B + c L any number. Notice that R is exactly conserved by Eq. (17). Further setting µ H = 0, we gain an anomalous global U (1) P Q for e.g. with −q P Q (Q α ) = q P Q ( α ) = q P Q (H u ) = q P Q (H d ) = 1, q P Q (E c α ) = −2 and the rest of the fields having zero charges. One can verify that U (1) P Q is anomalous with A P Q33 = −N f and A P Q22 = −N f + N H . With N f = 3 and N H = 1, the A P Q22 anomaly-free charge combination is In order to cancel the A P Q33 anomaly, we need another mixed SU (3) c anomalous symmetry.
For instance, when the u Yukawa interactions are out-of-equilibrium, we gain an anomalous chiral symmetry U (1) u c with A u c 33 = q u c /2. The anomaly-free charge combination is The U (1) charges of the superfields are listed in Table III. The anomalous U (1) R and U (1) P Q discussed above were first studied in Ref. [21] and were shown to be effective at T 10 7 GeV when the interactions mediated by weak scale µ H , soft trilinear couplings and gaugino masses are out-of-equilibrium. While these symmetries impart only order of one effects in the standard supersymmetric leptogenesis [9], it significantly enhances the efficiency of soft leptogenesis [10]. 7 Note that the R-symmetry is preserved also with R-parity violating terms as well as in supersymmetric type-I seesaw with right-handed neutrino chiral superfields

VI. CONCLUSIONS
The use of symmetry principle in analyzing the early Universe system allows all the particle asymmetries to be expressed in terms of conserved charges corresponding to the symmetries. These charges form the appropriate basis to describe the system. Besides its simplicity i.e. without having to resort to details of how the particles interact, this method serves as a powerful tool in accessing the viability of a baryogenesis scenario. In addition, the roles of U (1) symmetries as creator/destroyer, preserver or messenger become apparent, rendering it easier to construct interesting models of baryogenesis.