Leptonmediated electroweak baryogenesis
Abstract
We investigate the impact of the tau and bottom Yukawa couplings on the transport dynamics for electroweak baryogenesis in supersymmetric extensions of the Standard Model. Although it has generally been assumed in the literature that all Yukawa interactions except those involving the top quark are negligible, we find that the tau and bottom Yukawa interaction rates are too fast to be neglected. We identify an illustrative “leptonmediated electroweak baryogenesis” scenario in which the baryon asymmetry is induced mainly through the presence of a lefthanded leptonic charge. We derive analytic formulae for the computation of the baryon asymmetry that, in light of these effects, are qualitatively different from those in the established literature. In this scenario, for fixed CPviolating phases, the baryon asymmetry has opposite sign compared to that calculated using established formulae.
pacs:
I Introduction
Electroweak baryogenesis (EWB) is an attractive and testable explanation for the origin of the baryon asymmetry of the universe (BAU). Characterized by the baryonnumbertoentropyratio , the BAU has been measured through studies of Big Bang Nucleosynthesis (BBN) and the cosmic microwave background (CMB) to be in the following range
(1) 
at 95% C.L. Yao:2006px (); Dunkley:2008ie (). Assuming that the universe was matterantimatter symmetric at some initial time (e.g. at the end of inflation), the creation of the BAU requires three conditions (the Sakharov conditions Sakharov:1967dj ()): (1) violation of baryon number, (2) violation of C and CP, and (3) either a departure from equilibrium or a violation of CPT.
In EWB, these conditions are realized in the following way. First, a departure from equilibrium is provided by a strongly first order electroweak phase transition (EWPT) at temperature GeV Shaposhnikov:1987tw (); Shaposhnikov:1987pf (). During the EWPT, bubbles of broken electroweak symmetry nucleate and expand in a background of unbroken symmetry, filling the universe to complete the phase transition. Second, CPviolation may arise from complex phases. These phases induce CPviolating interactions at the walls of the expanding bubbles, where the Higgs vacuum expectation value (vev) is timedependent, leading the production of a CPasymmetric charge density. This is the socalled CPviolating source. This CPasymmetry, created for one species, diffuses ahead of the advancing bubble and is converted into other species through inelastic interactions in the plasma; in particular, some fraction is converted into lefthanded fermion charge density, denoted . Third, baryon number is violated by nonperturbative processes, which are unsuppressed outside the bubbles, in regions of unbroken electroweak symmetry 't Hooft:1976up (); 't Hooft:1976fv (); Manton:1983nd (); Klinkhamer:1984di (); Kuzmin:1985mm (). Following the common usage, we will refer to these as sphaleron processes. The presence of nonzero biases the sphaleron processes, resulting in the production of a baryon asymmetry Cohen:1994ss (). Electroweak sphalerons become quenched once electroweak symmetry is broken, as long as the EWPT is strongly first order; therefore, the baryon asymmetry becomes frozen in once it is captured inside the expanding bubbles.
In this work, we consider the charge transport dynamics during the EWPT: that is, how charge densities, induced by CPviolating sources, diffuse, interact, and get converted into , ultimately inducing . Although in the Standard Model (SM) this dynamics are insufficient to produce the observed BAU nocpvinsm (), supersymmetric extensions of the SM can readily include all the ingredients to make it successful^{1}^{1}1Another reason that EWB is not viable in the SM is that there is no EWPT; for a Higgs mass GeV, electroweak symmetry is broken through a continuous crossover Kajantie:1996mn ().. The most commonly accepted supersymmetic scenario is the following: the expanding bubble wall leads to a CPviolating source for charge density in the Higgs sector, which is then converted into third generation quarks through top Yukawa interactions, which in turn is converted into quark charge density of all generations through strong sphaleron processes Huet:1995sh (). The rate for baryon number production is proportional to ; in this picture, receives contributions from lefthanded quarks of all three generations.
However, as reported in a previous publication Chung:2008ay (), we have observed that bottom and tau Yukawa interactions, shown in Fig. 1, cannot in general be neglected from the computation of in supersymmetric EWB scenarios. While bottom and tau Yukawa couplings are small in the SM, in supersymmetric extensions they can be larger when the ratio of the vevs of the Higgs doublets, , is greater than unity. The inclusion of bottom and tau Yukawa interactions can change the EWB picture dramatically.

Quark charge density may be supressed: For , bottom Yukawa interactions become nonnegligible, leading to two important effects: (1) strong sphaleron processes no longer induce charge densities of first and second generation quarks, and (2) the third generation lefthanded quark charge density vanishes when the masses of righthanded bottom and top squarks are equal, or when their masses are large compared to the temperature .

Lepton charge density generated: For , tau Yukawa interactions also are nonnegligible, leading to the conversion of Higgs charge density into third generation lepton charge density^{2}^{2}2Measurements of the muon anomalous magnetic moment favors large ; see, e.g., Ref. Stockinger:2006zn ()..
These novel effects, which come into play for moderate , can lead to qualitatively different situations from those previously considered for supersymmetric EWB. In the present paper, we focus on a new scenario, where can be purely leptonic. We call this “leptonmediated electroweak baryogenesis.” This scenario occurs when , and the righthanded bottom and top squark masses are, either, approximately equal (), or large compared to ( GeV). (In other regions of parameter space where the quark contribution to is not quenched, the lepton contribution may still provide an additional enhancement or suppression to the total .)
In the leptonmediated EWB scenario, the value of has opposite sign compared to the value of computed when neglecting the bottom and tau Yukawa rates. The ingredients of this scenario will be tested in the near future at the Large Hadron Collider and by precision electric dipole moment (EDM) searches Pospelov:2005pr (). Clearly, to the extent that these experiments can determine the supersymmetric spectrum, and the signs and magnitudes of relevant CPviolating phases, inclusion of these Yukawa rates may be essential for testing the consistency of supersymmetric EWB with observation.
In Sec. II, we present the system of Boltzmann equations, generalized from previous work Huet:1995sh (); Lee:2004we () to include bottom and tau Yukawa interactions. In Sec. III, we provide an analytic estimate of the baryon asymmetry in detail. We solve the Boltzmann equations analytically in the limit that , such that bottom and tau Yukawa interactions are in chemical equilibrium. A new qualitative feature of our analysis is our treatment of lepton diffusion; we argue analytically how the lefthanded lepton charge density (and therefore ) is enhanced by virtue of righthanded leptons diffusing more efficiently in the plasma.
In Sec. IV, we verify these conclusions numerically. First, we calculate the bottom and tau Yukawa interaction rates, showing in what regimes they are sufficiently fast to induce chemical equilibrium. Next, after defining the parameters of our leptonmediated EWB scenario, we solve the system of Boltzmann equations numerically. We illustrate all of the aforementioned new effects and verify the agreement between our numerical and analytical solutions. In Sec. V, we summarize our results. The Appendix summarizes some additional numerical inputs used for this work.
Ii Boltzmann Equations
ii.1 Preliminaries
The transport dynamics leading to CPasymmetric charge densities during the EWPT are governed by a system of Boltzmann equations. These Boltzman equations have been derived using the closedtimepath formulation of nonequilibrium quantum field theory Chou:1984es (), leading to a system of equations of the form
(2) 
where is the charge current density of the species . The density , induced by CPviolating source , is coupled to other species via coefficients that describe the rate for a process to occur. (We have explicitly factored out of , for reasons that will become clear below.) The chemical potentials are denoted by . Chemical equilibrium, occuring when
(3) 
is maintained when when the interaction rate is sufficiently large.
Following previous work Huet:1995sh (); Cline:2000nw (); Lee:2004we (); Cirigliano:2006wh (), we simplify Eq. (2) in three ways. First, we assume a planar bubble wall profile, so that all charge densities are functions only of , the displacement from the moving bubble wall in its rest frame. Second, we apply Fick’s law Fick:1855a (); Fick:1855b (); Joyce:1994bi (); Cohen:1994ss (); Joyce:1994zn (), which allows us to replace on the LHS of Eq. (2), with charge density . The diffusion constant is the mean free path of particle in the plasma. Third, the chemical potentials appearing in Eq. (2) are related to their corresponding charge densities by
(4) 
where we have performed an expansion assuming . In the above, the statistical weight is defined by
(5) 
in which counts the number of internal degrees of freedom, the () sign is taken for fermions (bosons), and the mass of the th particle is taken to be the effective mass at temperature . In our analysis to follow, these factors are ubiquitous; they essentially count the degrees of freedom of a species in the plasma, weighted by a Boltzmann suppression.
Through these three simplifications, the Boltzmann equations become a system of coupled, second order, ordinary differential equations for the set of charge densities . Ultimately, it is the total lefthanded fermionic charge density
(6) 
that biases weak sphaleron transitions, thereby determining .
While in principle there is an interaction coefficient for every interaction in the MSSM Lagrangian, we can determine which ones need to be taken into account for the computation of by considering the relevant time scales. After a time , charge densities created at the bubble wall will have diffused on average a distance (with the effective diffusion constant to be defined below). At the same time, the moving bubble wall advances a distance . The diffusion time scale, defined by , gives the time that it takes for charge, having been created at the bubble wall and having diffused into the unbroken phase, to be recaptured by the advancing bubble wall and be quenched through CPconserving scattering within the phase of broken electroweak symmetry. This time scale is
(7) 
Numerically, we have (shown in Sec. IV). To this, we compare , the interaction time scale. If , then the process is slow and may be neglected from the Boltzmann equations. Physically speaking, charge density is recaptured by the advancing bubble wall before conversion processes can occur. On the other hand, if , then these interactions are rapidly occuring as the charge density is diffusing ahead of the advancing wall, leading to chemical equilibrium (3). Expressed in terms of charge densities, the chemical equilibrium condition is
(8) 
In this case, the interaction must be included in the Boltzmann equations.
ii.2 Setting up the Boltzmann equations
We now derive the Boltzmann equations within the context of the Minimal Supersymmetric Standard Model (MSSM). In principle, the complete system of Boltzmann equations encompasses one equation for each species of particle. However, the assumption that certain interactions are in chemical equilibrium (such that ) implies relations among the relevant chemical potentials (and therefore among their corresponding charge densities), allowing one to reduce the system. First, we assume that weak interactions (neglecting flavor mixing) are in chemical equilibrium, so that particles in the same isodoublet have equal chemical potential. Second, we assume that gaugino interactions (involving SM particles and their superpartners) are also in chemical equilibrium, so that a particle and its superpartner have equal chemical potential superequilibrium ().
Under these assumptions, the complete set of charge densities relevant for the computation of is
(9)  
where labels the generations. Furthermore, we define the following additional notation: , , , , and .
The system of Boltzmann equations contains, in principle, a coefficient for every interaction in the MSSM. However, interactions that satisfy may be neglected. In particular, we neglect interactions induced by first and second generation quark and lepton Yukawa couplings. The weak sphaleron rate may also be neglected, since Bodeker:1999gx (). Therefore, baryon and lepton number are conserved in the collision terms of the Boltzmann equations.
Not all of the densities in Eq. (9) are independent. Neglecting electroweak sphalerons from the Boltzmann equations, baryon and lepton number are individually conserved:
(10) 
Because the left and righthanded (s)lepton have different gauge quantum numbers, they have different diffusion constants in the plasma. Even though lepton number is globally conserved, regions of net lepton number can develop since diffuses more easily than since righthanded (s)leptons do not undergo SU(2) gauge interactions. For quarks and squarks, this does not occur since the left and righthanded (s)quark diffusion constants, dominated by strong interactions, are approximately equal Joyce:1994zn (). Therefore, baryon number is locally conserved:
(11) 
Other simplifications arise since we neglect first and second generation Yukawa couplings. There is no production of first and second generation lepton charge, so . Next, first and second generation quark charge can only be produced through strong sphaleron processes, e.g., , changing the number of left and righthanded quarks by one unit per flavor. Since first and second generation quarks are produced in equal numbers, we have
(12) 
Together, Eqs. (11,12) imply that
(13) 
Therefore, we may consider a reduced set of Boltzmann equations involving only the densities , , , , , ; the remaining densities are then determined by Eqs. (12,13). The equations are of the form of Eq. (2), where we use the relation given in Eq. (4) to express the chemical potentials in terms of charge densities. For the quarks and squarks, we obtain
(14c) 
and for Higgs bosons and Higgsinos we have
and lastly for leptons and sleptons we have
(16a)  
(16b) 
The relevant interaction coefficients in Eqs. (1416) are as follows:

The coefficients , where , denote the interaction rates arising from third generation Yukawa couplings . (The top Yukawa interaction rate has been denoted in previous work.)

The strong sphaleron rate is , where is the strong coupling and Moore:1997im ().

The coefficients and , where , denote the CPconserving scattering rates of particles with the background Higgs field within the bubble Lee:2004we ().
We also allow for new CPviolating sources , although in the present work we do not evaluate their magnitudes. In the MSSM, the most viable CPviolating source is , arising from CPviolating HiggsinoBino mixing within the expanding bubble wall Li:2008ez (); in our work, we take this as the sole source of CPviolation. The constant is the velocity of the expanding bubble wall bubbles (). The factors, e.g.
(17) 
follow the same notation as in Eqs. (9).
After solving the system of Boltzmann equations (1416) for each density, the lefthanded fermion charge density is
(18) 
where , , etc. The three terms in Eq. (18) correspond to the contributions to from third generation quarks, first/second generation quarks, and third generation leptons, respectively. If the masses of all lefthanded squarks and sleptons are much above the temperature of the phase transition, only fermions contribute to the lefthanded density and we have
(19) 
Finally, we show how our Boltzmann equations reproduce those given in previous work in the limit . In this limit, we can neglect the rates and , and CPviolating sources . First, since there is no source for lepton charge, we have . Second, the only source for density is strong sphaleron processes; therefore, we have
(20) 
in analogy with Eq. (12). Thus, Eqs. (13,20) imply that . Therefore, by Eq. (19), we have the oftenused relation ; this relation is no longer valid for . In addition, the Boltzmann equations of Refs. Huet:1995sh (); Lee:2004we (); Cirigliano:2006wh () follow from Eqs. (14,b,II.2); they too are no longer valid for .
Iii Analytic results
In this section, we estimate the solution to the Boltzmann equations, Eqs. (1416), of which the endpoint is an expression for , the lefthanded fermion charge density that biases weak sphalerons. We assume that top, bottom, and tau Yukawa interactions, in addition to strong sphaleron and gaugino interactions, are all in chemical equilibrium. These assumptions lead to a series of conditions relating the chemical potentials, and therefore the number densities, of various species. By exploiting these relations, we will express all quark and lepton densities in terms of the Higgs density ; then, we will simplify the full system of Boltzmann equations to a single equation for , which is analytically solvable Huet:1995sh ().
iii.1 Lepton charge densities
When tau Yukawa interactions are in chemical equilibrium condition, the relation
(21) 
is satisfied. The sum of the Boltzmann equations for and (16) is
(22) 
Since the left and righthanded lepton diffusion constants are not equal, there is no simple relation that would allow us to relate to . However, in the static limit (where ), Eq. (22) implies that
(23) 
(We have assumed the boundary conditions .) Therefore, we have
(24a)  
(24b) 
where and are the corrections to these relations, derived below.
Let us now describe the physics of Eqs. (24) through two limiting cases. Case (i): set . In this limit, Eq. (22) implies that lepton number is locally conserved: . The Higgs density , created by the CPviolating source, is converted into through tau Yukawa interactions, until chemical equilibrium (21) is reached, when
(25) 
Case (ii): take , keeping finite. Any density created by tau Yukawa interactions instantly diffuses away to ; therefore, we set . Now, tau Yukawa chemical equilibrium (21) implies
(26) 
In other words, tau Yukawa interactions will enforce chemical equilibrium locally. Since the RH lepton density is diffusing away, reducing the local , more conversion of into and occurs to compensate, thereby resulting in more LH lepton density. This conversion ceases when Eq. (26) is reached. Therefore, a large diffusion constant for RH leptons enhances the density for LH leptons. This enhancement, maximized for , is at most a factor of
(27) 
Both cases agree with Eqs. (24), setting .
Next, consider the case of physical relevance, where , but keeping both finite. Close to the bubble wall, LH lepton density will be enhanced, as argued above. However, far from the bubble wall, an additional effect occurs: RH lepton density, having diffused far into the unbroken phase, is converted into and by tau Yukawa interactions. This effect suppresses . Close to the bubble wall, Higgsinos created by the CPviolating source () will be converted into LH leptons () and RH antileptons (), and then, far from the wall, the RH antileptons will be converted into LH antileptons, thereby suppressing . This physics is incorporated in the nonlocal corrections and , which we now consider. Using Eqs. (21, 22, 24), we can derive differential equations for these densities:
(28a)  
(28b) 
where . With the boundary conditions , the solutions to these equations are
(29a)  
(29b) 
These terms describe how regions of net lepton number can develop when . Using Eqs. (24,29), it is straightforward to show that these solutions for and satisfy
(30) 
conserving lepton number.
In our numerical study, we find that the impact from and on the analytic computation of (given below) is only . Since there are much larger uncertainties in the analytic computation, it is safe to neglect the nonlocal terms and from Eqs. (24).
iii.2 Quark charge densities
When top and bottom Yukawa interactions are in chemical equilibrium, the relations
(31a)  
(31b) 
are satisfied; cf. Eqns. (3, 4). These equations imply that
(32) 
First and second generation quark densities only couple to third generation densities, via strong sphaleron interactions, through the linear combination , as can be seen from Eqns. (14). Since this combination vanishes, third generation quark densities do not induce 1st/2nd generation quark densities. Mathematically, if we impose Eq. (32), the Boltzmann equation (14c) becomes
(33) 
which, with the boundary conditions , implies . According to Eq. (11), we have , for . Therefore, we conclude that all first and second generation quark and squark charge densities vanish in the presence of fast top and bottom Yukawa interations. Strong sphalerons only induce first and second generation densities in order to wash out an asymmetry between left and righthanded quark chemical potentials; when bottom Yukawas are active, this asymmetry vanishes and strong sphalerons have no effect.
The contribution to from third generation LH quarks is
(35) 
while that from first and second generation LH quarks vanishes. Let us contrast these results to previous work that neglected bottom Yukawa interactions Huet:1995sh ():
(36a)  
(36b) 
The formulae are completely different. Whereas in previous work significant baryon asymmetry could arise from first and second generation LH quarks, the presence of bottom Yukawa interactions completely changes the picture: no first and second generation quark density is created. In addition, with fast bottom Yukawa interactions, the third generation quark charge vanishes when , or equivalently ; without them, this cancellation never occurs.
Let us explain the physical origin of this cancellation. Suppose that the CPviolating source creates positive Higgs/Higgsino density, such that . Due to hypercharge conservation, top Yukawa interactions convert Higgsinos and Higgs bosons into LH quark and squark antiparticles (driving ) and RH top quark and squark particles (driving ), while bottom Yukawa interactions drive and . Which effect wins is determined by whether or has more degrees of freedom available, according to the equipartition theorem. This is determined by the statistical weights and , which are governed by the masses and . When the masses are equal, we have , suppressing . Similarly, the sign of is positive or negative, depending on whether is greater or less than unity, respectively. In the leptonmediated scenario, we suppress the quark contribution by choosing . In scenarios beyond the MSSM, it is also suppressed for .
iii.3 Solving the Boltzmann equation
In terms of , the lefthanded fermion charge density (18) becomes
(37) 
where is given in Eq. (24a). The first term is the contribution to from third generation quarks, while the second and third terms are contributions from third generation leptons. The lepton contribution is predominantly given by the second term only; the third term, as discussed above, is suppressed for . This equation is the main result of this paper; from it, we infer several conclusions:

The lepton contribution is enhanced for , when is largest and smallest; (cf. Eqs. (4,17)). It is also enhanced for . Its sign is fixed with respect to , which in turn is fixed by the sign of the CPviolating source, as we show below. Therefore, in a leptonmediated EWB scenario, where is predominantly leptonic, the sign of the CPviolating phase most relevant for EWB uniquely fixes the sign of , in contrast with the quarkmediated scenarios.

Lefthanded charge arises from third generation quarks and leptons, and not first and second generation quarks and leptons. The form of is qualitatively different than in previous treatments that neglected and , where lefthanded charge came from quarks of all generations, and not from leptons.

Furthermore, the quark contribution to vanishes for , since . Its sign is opposite to that of the leptonic contribution for and the same for .
We explore these implications in more detail numerically in Sec. IV.
We emphasize that our conclusions are quite general, although it appears that our Boltzmann equations (1416) have been specialized to the MSSM. In any extention of the MSSM, Eq. (37) and its conclusions remain valid if the following conditions hold: (i) third generation Yukawa interaction rates are faster than the diffusion rate, and (ii) CPviolation is communicated to the first and second generation quark sectors solely through strong sphalerons.
Since is determined by , all that remains is to solve for the Higgs charge density . We can reduce the Boltzmann equations (1416) into a single equation for by taking the appropriate linear combination of equations
(38) 
such that the Yukawa and strong sphaleron rates all cancel, and expressing the densities in terms of using Eqs. (24,34). This master Boltzmann equation equation is an integrodifferential equation for , due to the presence of the term. Therefore, for simplicity, we treat perturbatively: first, we neglect in our solution for , and then, given our solution , we include the contribution in Eq. (37) for . Neglecting , the master Boltzmann equation is
(39) 
where
(40a)  
(40b)  
(40c) 
Although the expressions in Eq. (39) are identical to that in the established literature Huet:1995sh (); Lee:2004we (), the form of Eqs. (40) is dramatically different. We note that there is no dependence on the first/second generation quark sector, owing to the fact that they do not participate in the dynamics which determines .
To solve Eq. (39) analytically, we follow Ref. Huet:1995sh () making the approximations (a) that the true spatial dependence of the chiral relaxation rates may be replaced by a stepfunction, so that we may write ; and (b) that for . For the symmetric phase, where , we obtain
(41) 
where
(42) 
Furthermore, we have defined
(43) 
We reiterate that although the form of Eqns. (4143) is similar to that in previous work Huet:1995sh (), our results for , , and are different, due to the modified structure of the Boltzmann equations in the presence of fast bottom and tau Yukawa rates.
We now ask: was it safe to neglect in solving for ? Substituting our solution for into Eq. (29), we find that , in the limit that . In short, in the physical limit where large RH lepton diffusion has the biggest impact upon , our solution is most accurate. There may be situations in general in which the impact of nonlocal corrections is not suppressed; we show how the Boltzmann equations may be solved in this case in future work chung ().
Iv Leptonmediated electroweak baryogenesis: Numerical Results
We now consider an MSSM scenario that illustrates some of the novel features discussed in Sec. III. As we will see, the picture here is that the BAU is induced predominantly by leptonic lefthanded charge: hence, leptonmediated. The key parameters that govern the behavior of this scenario are (i) and pseudoscalar Higgs mass (at zero temperature) GeV, ensuring , and (ii) righthanded top and bottom squarks with approximately equal mass, thereby suppressing the quark contribution to . Here, we take both squarks to be light, with masses, since a strong first order phase transition requires a light top squark.
Although we work within the context of the MSSM, many of our conclusions are much more general. In EWB scenarios beyond the MSSM, light squarks are not required for a strong first order phase transition (see e.g. Refs. Huber:2000mg (); Menon:2004wv (); Huber:2006wf ()). Even if the squarks are very heavy, EWB is still mediated by leptons occurs as long as the previous two conditions are met.
In this section, we first summarize the parameters of the leptonmediated EWB scenario. Next, we compute the bottom and tau Yukawa interaction rates and , showing for what regions of parameter space they are fast compared to . Last, we numerically solve the system of Boltzmann equations (1416) and compute the lefthanded fermion charge density that generates . Our main result is Fig. 3: it illustrates how arises from leptons instead of quarks, how our analytic and numerical results agree, and how this scenario differs dramatically from previous work neglecting and .
iv.1 Input parameters
The computation of relies upon many numerical inputs, some described here and others described in the Appendix. We have evaluated the masses of particles during the EWPT assuming that electroweak symmetry is unbroken. This approximation is motivated by the fact that most of the charge transport dynamics take place outside the bubble in the region of unbroken symmetry. These masses receive contributions from the mass parameters in Table 1 and from finite temperature corrections, listed in the Appendix. The righthanded stop, sbottom, and stau SUSYbreaking masssquared parameters are , , and , respectively. The RH stop is required to be light to achieve a strong first order phase transition Carena:2008vj (); taking the RH sbottom and stau to be light as well ensures that the quark contribution to is suppressed, while the lepton contribution is enhanced, in accord with Eq. (37). We take all other squark and slepton mass parameters to be 10 TeV.
For Higgs bosons, the story is more complicated. Again, we study the degrees of freedom assuming unbroken electroweak symmetry. The mass term in the Lagrangian is
(44) 
and the same for but with . The finite temperature corrections that restore electroweak symmetry are given by (see Table 2)
(45a)  
(45b) 
(In the high limit, there are no offdiagonal thermal corrections, since these corrections are proportional to dimensionful parameters.) We can reexpress this mass matrix using the minimization conditions for electroweak symmetry breaking at Martin:1997ns ():
(46a)  
(46b)  
(46c) 
where and are the and pseudoscalar Higgs boson masses at . The approximations in Eqs. (46) follow assuming . Therefore, in this limit, the Higgs boson mass matrix (44) is diagonal, with eigenvalues
(47a)  