An exact solution of the Dirac equation with CP violation
Abstract
We consider Yukawa theory in which the fermion mass is induced by a Higgs like scalar. In our model the fermion mass exhibits a temporal dependence, which naturally occurs in the early Universe setting. Assuming that the complex fermion mass changes as a tanhkink, we construct an exact, helicity conserving, CPviolating solution for the positive and negative frequency fermionic mode functions, which is valid both in the case of weak and strong CP violation. Using this solution we then study the fermionic currents both in the initial vacuum and finite density/temperature setting. Our result shows that, due to a potentially large state squeezing, fermionic currents can exhibit a large oscillatory magnification. Having in mind applications to electroweak baryogenesis, we then compare our exact results with those obtained in a gradient approximation. Even though the gradient approximation does not capture the oscillatory effects of squeezing, it describes quite well the averaged current, obtained by performing a mode sum. Our main conclusion is: while the agreement with the semiclassical force is quite good in the thick wall regime, the difference is sufficiently significant to motivate a more detailed quantitative study of baryogenesis sources in the thin wall regime in more realistic settings.
pacs:
98.80.k, 04.62.+vDate: September 27, 2019
ITPUU12/50, SPIN12/47, NIKHEF2012027
I Introduction
Electroweak baryogenesis Morrissey:2012db is a very appealing idea, and yet the mechanism for dynamical baryon creation at the electroweak scale has suffered some serious blows. Firstly, in the mid 90s it was found that the electroweak phase transition in the standard model is a crossover Kajantie:1996qd ; Rummukainen:1998as ; Csikor:1998eu . While at first supersymmetric extensions looked promising, the most popular supersymmetric model  the MSSM  is almost ruled out on two grounds (a) it cannot give a strong enough phase transition for the observed Higgs mass Carena:2008vj and (b) it cannot produce enough baryons consistent with electric dipole moment Pospelov:2005pr bounds Carena:2012np ; Cirigliano:2009yd ; Blum:2008ym ; Konstandin:2005cd ; Cline:2000nw ; Huet:1995sh (albeit in some models resonance between fermionic flavors can be helpful to increase baryon production Kozaczuk:2012xv ; Cirigliano:2011di ; Cirigliano:2009yd ; Konstandin:2004gy ). The models that are still viable are the supersymmetric models with additional Higgs singlet(s) Huber:2000mg ; Kang:2004pp both because they allow for a stronger phase transition Carena:2011jy ; Konstandin:2006nd ; Huber:1998ck and generate more baryons Cline:2012hg ; Huber:2006ma ; Huber:2006wf ; Balazs:2007pf ; Menon:2004wv . In addition, general two Higgs doublet models Cline:2011mm ; Fromme:2006cm and composite Higgs models Espinosa:2011eu ; Konstandin:2011ds ; Cline:2008hr are still viable. Works on cold electroweak baryogenesis Tranberg:2012jp ; Tranberg:2012qu ; Tranberg:2009de are also worth mentioning. In summary, while electroweak baryogenesis has been a very attractive proposal, precisely because it is testable by contemporary accelerators, recent experiments have cornered it to models where most researchers have not focused their attention during the preLHC era. Hence, at this stage theoretical work, that will refine our ability to make a quantitative assessment of electroweak baryogenesis in different models, is still a worthy pursuit.
One of the most important unsolved problems in dynamical modeling of electroweak baryogenesis is a reliable calculation of the CPviolating sources that bias sphaleron transitions Manton:1983nd ; Klinkhamer:1984di , which at high temperatures violate baryon number. In the fermionic sector the most prominent CPviolating source is the fermionic axial vector current Prokopec:2003pj ; Prokopec:2004ic , since that current directly couples to sphalerons, and can thus bias baryon production. There are essentially two approximations used in literature to calculate axial vector currents:

the quantummechanical reflection Cohen:1990it ; Joyce:1994bi ; Joyce:1994zn used in the thin wall case, and

the semiclassical force Joyce:1994zt ; Joyce:1994fu ; Cline:2000nw ; Joyce:1999fw ; Prokopec:2003pj ; Prokopec:2004ic used in the thick wall case.
In general thin wall baryogenesis is more efficient in producing baryons. Its main drawback is that the calculational methods used are unreliable: one calculates the CPviolating reflected current ignoring the plasma, and then inserts it into a transport equation in an intuitive (but otherwise rather arbitrary) manner Joyce:1994zn . How bad the situation can get is witnessed by the controversy that developed around the work of Farrar and Shaposhikov Farrar:1993hn (who used a quantum mechanical reflection to calculate the source). The subsequent works Gavela:1994ds ; Gavela:1994dt ; Huet:1994jb came up with an orders of magnitude smaller answer for baryon production. And yet these latter works used unreliable methods that e.g. violate unitarity, such that the issue remained unsettled ^{1}^{1}1Research on the topic subsided not because the problem was resolved, but because standard model baryogenesis was ruled out based on equilibrium considerations alone Kajantie:1996qd .. So, the problem of the source calculation in the thin wall regime remains still to a large extent open.
In the thick wall case the situation became much more satisfactory after the works of Joyce, Kainulainen, Prokopec, Schmidt and Weinstock Joyce:1999fw ; Kainulainen:2001cn ; Kainulainen:2002th . It was shown that one can calculate the semiclassical force (which rather straightforwardly sources the axial vector current) from first principles and in a controlled approximation from the KadanoffBaym (KB) equations for Wightman functions. These KB equations are the quantum field theoretic generalization of kinetic equations. The positive and negative frequency Wightman functions represent the quantum field theoretic generalization of the Boltzmann distribution function, that provide statistical information on both onshell and offshell phase space flow. In a certain limit, when integrated over energies, the Wightman functions yield Boltzmann’s distribution function. When written in a gradient approximation, the KB equations can be split into the constraint equations (CE) and the kinetic equations (KE). The authors of Refs. Kainulainen:2001cn ; Kainulainen:2002th have rigorously shown that, in the presence of a moving planar interface, in which fermions acquire a mass that depends on one spatial coordinate, single fermions live on a shifted energy shell, which to first order in gradients (linear in ) and in the wall frame equals
(1) 
where is the fermion mass, which varies in the direction in which the wall moves, is particle’s momentum, is the momentum orthogonal to the wall, and is the corresponding spin. This energy shift acts as a pseudogauge field (also known from condensed matter studies), which lowers or increases particle’s energy. Relation (1) clearly shows that particles with a positive spin orthogonal to the wall and a positive frequency (as well as particles with a negative spin and a negative frequency) will feel a semiclassical force that is proportional to the gradient of . Particles with a negative spin and a positive frequency will feel an opposite force. This force appears in the kinetic equation for the Boltzmannlike distribution functions , and reads
(2) 
It was also shown that this force then sources an axial vector current, which in turn can bias sphalerons.
The work of Huet:1994jb ; Huet:1995sh ; Riotto:1997vy ; Carena:1997gx ; Carena:2000id has shown that, in the case that fermions mix through a mass matrix, there is an additional CPviolating source resulting from flavor mixing. This was put on a more formal ground by Konstandin:2004gy , where a flavor independent formalism was developed, and where it was shown that flavor nondiagonal source is subject to flavor oscillations induced by a commutator term of the form , not unlike the famous flavor (vacuum) oscillations of neutrinos. This idea was further developed by Cirigliano:2009yt . Since we do not deal with flavor mixing in this work, we shall not further dwell on this mechanism, which should not diminish its importance. In passing we just mention that in most of the relevant parameter space of e.g. the chargino mediated baryogenesis in the MSSM, the semiclassical force induces the dominant CPviolating source current Konstandin:2005cd .
We shall now present a qualitative argument which suggests that in many situations thin wall sources can dominate over the thick wall sources (calculated in a gradient approximation). If true this means that any serious attempt to make a quantitative assessment of baryon production cannot neglect the thin wall contribution. To see why this is so, recall that a gradient approximation applies for those plasma excitations whose orthogonal momentum, , satisfies:
(3) 
where is the typical thickness of the bubble wall. On the other hand, the thin wall approximation belongs to the realm of momenta which satisfy
(4) 
Typical momenta of particles in a plasma (per direction) is . Now, unless , we have a larger or comparable number of particles in the thin and thick wall regimes! But, since the thin wall source is typically stronger, unless thermal scattering significantly suppress the thin wall source, it will dominate over the thick wall source. It is often incorrectly stated in literature that the number of particles to which thin wall calculation applies is largely phase space suppressed, i.e. that their number is small when compared to the number of particles to which the semiclassical treatment applies. So, to conclude, it is of essential importance to get the thin wall source right if we are to claim that we can reliably calculate baryon production at the electroweak transition in a model.
We believe that this represents a good motivation for what follows: a complete analytic treatment of fermion treelevel dynamics for a time dependent mass. The time dependence has been chosen such to correspond to a kink wall, because it is known that this represents a good approximation to a realistic bubble wall Moore:1995ua ; Moore:1995si , and equally importantly, in this case one can construct exact solutions for mode functions. Before we begin our quantitative analysis, we recall that a related study for the CP even case and planar wall has been conducted by Ayala, JalilianMarian, McLerran and Vischer Ayala:1993mk , while a semianalytic, perturbative treatment of the CPviolating case has been conducted in Ref. Funakubo:1994yt . The main advantage of the latter study is that it allows for a general profile of the CPviolating mass parameter, the drawbacks are that the method is semianalytic (the final expression for the source is in terms of an integral), and furthermore it is perturbative, such that it can be applied to small CP violation only. To conclude, an exact treatment of fermion dynamics in the presence of a strong CP violation is highly desirable, and this is precisely what we do in this paper.
Ii The model
Here we consider the free fermionic lagrangian of the form,
(5) 
where and are the left and right handed single fermionic fields, and are the left and right handed projectors, and and are the Dirac gamma matrices. We shall assume that the fermion mass is complex and spacetime dependent. This can be generated e.g. when a Yukawa interaction term, , is approximated by , where stands for a Higgslike scalar field condensate which can generate a spacetime dependent fermion mass,
(6) 
where is a (complex) Yukawa coupling. The Dirac equation implied by (5) is
(7) 
In this paper we consider the simplest case: a single fermion in a time dependent, but spatially homogeneous, background. Such situations can occur, for example in expanding cosmological backgrounds Garbrecht:2006jm , or during second order phase transitions and crossover transitions in the early Universe. In this case helicity is conserved Garbrecht:2002pd ; Garbrecht:2003mn ; Garbrecht:2005rr . We shall perform the usual canonical quantization procedure, according to which the spinor operator satisfies the following anticommutator (),
(8) 
In the free case under consideration, the Dirac equation (7) is linear, and consequently can be expanded in terms of the creation and annihilation operators, which in the helicity basis reads,
(9) 
where and are particle and antiparticle fourspinors. and are the annihilation operators that destroy the fermionic vacuum state , , while and are the creation operators that create a particle and an antiparticle with momentum and helicity . These operators obey the following anticommutator algebra,
(10) 
where all mixed anticommutators are zero. The momentum space quantization conditions (10) and the position space quantization rule (8) have to be mutually consistent. This imposes the following consistency condition on the positive and negative frequency spinors,
(11) 
This is usually supplied by the mode orthogonality conditions,
(12) 
and by the mode normalization conditions,
(13) 
which – as we will see below – are chosen to be consistent with the more general requirement (11). Although the orthogonality condition (12) is usually met, it is however not a necessity. Important is that the mode functions span all of the Hilbert space, which is true in this case. Because we consider a system which is timetranslationally invariant, helicity is conserved, and it is thus convenient to work with helicity conserving spinors
(14) 
where is the helicity two eigenspinor, satisfying , where is the helicity operator and are its eigenvalues.
We shall work here with the Dirac matrices in the chiral representation, in which
(15) 
where the last equalities follow from the usual direct product (Bloch) representation of the Dirac matrices. Here and are the Pauli matrices obeying, and . The left and right projectors are then,
(16) 
which can be used to write, and as it is done in (5–7). Now, making use of Eqs. (9–16) in the Dirac equation (7) one gets the following four equations for the component functions
(17) 
and
(18) 
where the mass can be complex and time dependent, , and the modes are normalized to unity,
(19) 
The equations of motion for and can be decoupled, resulting in the second order equations,
(20) 
where . For the case at hand a better way of proceeding is to go to the positive and negative frequency basis, defined by:
(21) 
since then the equation of motion can be reduced to the Gauss’ hypergeometric equation. Indeed, from (17–18) and (21) it follows,
(22) 
which, when decoupled, yields a second order equation,
(23) 
So far our analysis has been general, in the sense that we have assumed no special time dependence in . In order to make progress however, we have to make a special choice for , which is what we do next.
Iii Mode functions for the kink profile
In Ref. Ayala:1993mk an exact solution of the Dirac equation was found for a wall of arbitrary thickness with a kink wall profile , where characterizes the wall thickness. Here we generalize this solution to include CP violation. While in this paper we consider only a time dependent mass profile, the generalization to the planar (dependent) case is straightforward, and will be considered separately. Constructing an exact solution is important for baryogenesis since one can then consider in detail how the CPodd quantities that source baryogenesis (directly or indirectly) depend on the mass profile, and in particular investigate what is the optimal profile and its duration. Unfortunately, analytic solutions cannot include plasma scattering and width effects, whose treatment will be therefore typically left to numerical simulations.
Here we assume the following ‘wall’ profile
(24) 
where represents the time scale over which the wall varies (for convenience we shall use the terms ‘wall’ and ‘profile’ interchangeably). Both and are complex mass parameters. In the case when a single Higgs field is responsible for the phase transition, one expects that both real and imaginary part of exhibit a similar behaviour, which is reflected in the Ansatz (24). Moreover, we do not know how to construct exact solutions when different time scales govern the rate of change of the real and imaginary masses. Nevertheless, we believe that the Ansatz (24) represents quite well realistic walls for a wide variety of single stage phase transitions, cf. Refs. Moore:1995ua ; Moore:1995si .
Note that the thin wall limit is (). In that limit the mass function becomes the step function Ansatz (97), whereby . In appendix A we construct the normalized fundamental solutions of Eqs. (17) for a constant mass. The thin wall case is treated explicitly in appendix B. The thin wall results serve as a check for the kink wall case in the appropriate limits. Moreover it allows for a quantitative comparison of the thick wall to the thin wall results.
Since the ratio of the real and imaginary parts of the mass is time dependent, the Ansatz (24) contains CP violation (which can be either small or large, depending on how much the ratio changes. Since the physical CPviolating phase is in the relative phase between and , one can perform a global rotation of the left and righthanded spinors that does not affect CP violation. It turns out that the equations of motion simplify if one performs a global rotation that removes the imaginary part of . The constant rotation that does that is
(25) 
In that case
This rotation is important, because the mode equations (23) significantly simplify to become
(26) 
where . Furthermore, from (22) one can infer that obey the same equations as . In what follows, we show that these equations can be reduced to the Gauss’ hypergeometric equation.
To show this, it is instructive to introduce a new variable,
(27) 
in terms of which
with . Eq. (26) becomes,
(28) 
Now, performing a rescaling,
(29) 
and choosing
(30) 
where
(31) 
yields the following Gauss’ hypergeometric equation for ,
(32) 
where
(33) 
Note that the rescaling (29) was chosen such to remove the terms and from Eq. (32). Since are noninteger, the two independent solutions for are the usual ones. A detailed normalization procedure is provided in Appendix D and the result are the following normalized early time mode functions
(34) 
These functions are valid of course for all times. They are called early time mode functions because at early times () they reduce to the positive frequency mode functions (118), and they are normalized as, , which follows from Eqs. (19) and (21), see also Eq. (134).
For completeness, we also quote the second pair (117) of early time solutions,
(35) 
Just as before, at early times () these solutions reduce to the negative frequency mode functions (118), and they are also normalized as, .
An analogous procedure as above yields the following normalized fundamental solutions suitable for late times,
(36) 
and
(37) 
while the late time solutions (37) reduce at asymptotically late times to positive and negative frequency solutions , respectively, see Eq. (123).
Now, a general early time solution can be written as a linear combination of the fundamental solutions (34–35); for simplicity we shall take here (34) for the early time solutions. Similarly, general late time solutions are a linear combination of the fundamental late time solutions (36–37),
(38) 
where and are complex functions of (for spatially homogeneous systems they are functions of the magnitude only) that satisfy the standard normalization condition,
(39) 
Now, upon choosing (34) as the early time solutions and making use of the matching between the general early and late time solutions
(40) 
and of the relation for the Gauss’ hypergeometric functions (119) one gets,
(41) 
It can be shown that
(42) 
Useful identities here are
(43) 
Because and are functions of , and , (just as in the thin wall case (102–103)) there are no CP odd contributions in the mode mixing (Bogoliubov) coefficients (41). and are indeed the usual Bogoliubov coefficients that transform an asymptotically early time vacuum state to a late time vacuum state. Hence is the particle number observed by a late time observer, in the late time state that evolves from the early time positive frequency vacuum state.
To make contact with the thin wall case (102), we take the limit in (41) to get,
(44) 
It can be checked that Eq. (44) satisfies . Moreover, since , the and are always positive. One can show that and given in (41) obey , as they should. This equality follows from,
(45) 
from which it also follows that . Now, taking a thin wall limit in (45) yields
(46) 
where we made use of Given that . This expression agrees with the thin wall particle number Eq. (103) derived in appendix B.
It is interesting to note that, although particle number agrees, the Bogoliubov coefficient in the thin wall limit (44) appears very different from the one derived explicitly for the thin wall (102). For instance, the coefficients in (102) are complex and depend explicitly on helicity, whereas the limiting coefficient (44) is real and helicity independent. A similar situation occurs for , see (149). The apparent discrepancy is caused by an overall phase factor by which the coefficients in the thin wall limit differ from those directly computed for the thin wall. This phase factor does not affect particle number and can be removed by a global rotation of the (anti)particle spinors. In appendix E we show explicitly how the kink wall case and thin wall case are connected.
The particle production can also be analyzed in the opposite limit, . In this thick wall regime particle production is exponentially suppressed as,
(47) 
which is also what one expects. However, note that when , the suppression is not large. This is demonstrated in figures 2 and 2, where the particle number is shown as a function of for several different wall thicknesses. In figure 2 the mass parameters are and . In this case CP violation is weak. For these mass parameters the thin wall particle number (46), represented by the dashed line, reaches the maximal particle number as . For thicker walls (decreasing ) the particle number is exponentially suppressed with respect to the thin wall. For very small the suppression is much smaller, since .
In figure 2 the mass parameters are chosen such that . In this case CPviolation is maximal for the thin wall in the limit , see also (105). The maximal particle number in this limit is 1, which indicates an inverse population. This inverse population, induced by large CP violation, is a novel result and, as far as we know, not noticed in literature before. For thicker walls the particle number is still suppressed, but much less than for the mass parameters in figure 2. In fact, for the particle number is unsuppressed in the limit .
A large late time Bogoliubov particle number for a free fermionic system indicates large squeezing. It is interesting to see what effect such a large squeezing may have on the fermionic currents. In particular, we are interested in the CPodd fermionic axial vector current that couples to sphalerons. The next section is devoted to computing these currents in the setting of a tanhkink wall.
Iv The currents and CP violation
In this section we consider the evolution of the two point Wightman functions, defined as the expectation values Prokopec:2003pj ; Garbrecht:2002pd
(48) 
and which satisfy the homogeneous Dirac equations (7)
(49) 
For the problem at hand, when written in a Wigner mixed representation
(50) 
the fermionic Wightman function can be written in a helicity blockdiagonal form
(51) 
where () are the Pauli matrices and are the (offshell) distribution functions measuring the vector, scalar, pseudoscalar and pseudovector phase space densities of fermions, respectively. Their onshell version
(52) 
satisfy the following equations of motion Prokopec:2003pj ; Garbrecht:2002pd ,
(53) 
where here . To make the connection with section III and Appendix B, we note that one can express in terms of or and as follows ^{2}^{2}2Note that, due to difference in conventions, there are sign differences when compared with Ref. Garbrecht:2002pd .:
(54) 
such that . From Eq. (95) and (54) we immediately obtain that for (),
(55) 
where we took account of , (as ) and of . Inserting Eqs. (55) into the particle number definition Garbrecht:2002pd ,
(56) 
yields that for , as it should be since we have prepared the initial state to be in the pure free vacuum.
One can also consider the statistical particle number Prokopec:2012xv ,
(57) 
A statistical particle number is defined as the particle number associated with the basis in which the density operator is diagonal Prokopec:2012xv . Statistical particle numbers can be used as a quantitative measure of state impurity, i.e. of how much a state deviates from a pure state. From previous work we have learned that the statistical particle number is constant in the absence of interactions. This can also be seen from the kinetic equations (53), which give