Derivation and analysis of bounds in dirac leptogenesis

Derivation and Analysis of Bounds in Dirac Leptogenesis by John A.BackusMayes A thesis submitted in partial fulfillment of the requirements for the Degree of Bachelor of Arts with Honors in Physics WILLIAMS COLLEGE Williamstown, Massachusetts May 23, 2005 Contents Introduction 1.1 Baryogenesis . . . . . . 1.2 Majorana Leptogenesis 1.3 Dirac Leptogenesis . . 1.4 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 12 C P Violation 2.1 Necessary Details . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 A Formula for . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Bounds on 3.1 Neutrino Masses . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Rewriting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Maximizing € . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 A Simplified Case . . . . . . . . . . . . . . . . . . . . . 3.3.2 A Different Simplification . . . . . . . . . . . . . . . . 17 17 19 20 20 21 Boltzmann Equations and Efficiency 23 . . . . . . . . . . . . . . . . . . . . . . . 4.1 Relevant Interactions 23 4.2 Boltzmann Equations . . . . . . . . . . . . . . . . . . . . . . . 24 4.3 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Characteristics of 5.1 Flavor . . . . . . . . . . 5.2 Bounds on m4, . . . . . 5.2.1 Motivations . . . 5.2.2 Preparatory Steps 5.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 30 33 33 34 37 Conclusions 43 6.1 Overview of Results . . . . . . . . . . . . . . . . . . . . . . . . 43 6.2 Suggestions for Further Research . . . . . . . . . . . . . . . . 44 Acknowledgements I would like t o express my sincere gratitude for the support and guidance given t o me by several people who, in doing so, have made this thesis possible. Most importantly, I thank my advisor, David Tucker-Smith, for taking me on as his thesis student and supporting me unerringly throughout the past year. I thank Sean OBrien, Professor Tucker-Smith's other thesis student this year, for his insights and camaraderie during the many late nights spent in our office. I thank Professor William Wootters for taking the time to read my nearly-completed thesis this spring; his comments were invaluable in producing this final version. I thank Sarah Croft for her companionship and relentless support in those times when finishing seemed slightly beyond the realm of possibility, and I thank my parents, Lois Backus and Robert Mayes, for their enthusiastic encouragement of my interest in science and physics throughout my life. Finally, I thank the physics department of Williams College for the excellent training I have received and for the opportunity t o conduct in-depth physics research as an undergraduate. Chapter Introduction 1.1 Baryogenesis Nearly everything on the surface of the Earth is made of matter, the only exceptions being at large particle accelerators, where minute amounts of antimatter are routinely made to be used in particle experiments. That the Earth exists testifies t o the fact that its interior is also composed entirely of matter, and the nature of cosmic rays from the Sun indicate that it is matter as well. Humans have visited the Moon, and mechanical probes have come in contact with eight of the nine planets in the solar system. That none of these events has resulted in a gargantuan explosion strongly implies that antimatter is effectively nonexistent in the solar system. Originating from outside the solar system, extrasolar cosmic rays allow us t o extend this complete matter-antimatter asymmetry t o cover our local galactic neighborhood. From the nature of these cosmic rays, it appears that baryons vastly outnumber anti-baryons in the Milky Way and nearby galaxies. On an even larger distance scale, the lack of any gamma radiation emanating from the intergalactic medium in galaxy clusters suggests that virtually no antimatter exists anywhere within several million light years of Earth. That this asymmetry persists over such large distances seems t o imply that the entire universe is made exclusively of matter, but the size of the observable universe is actually many orders of magnitude larger than any galaxy cluster. With that in mind, there is the possibility that we live in a pocket of matter, surrounded by antimatter pockets of similar size. This seems plausible, but it can be ruled out by cosmology and causality, following an argument in [I]. If pockets of matter and antimatter existed, then they would have to each contain at least the mass of an entire cluster of galaxies, about 1012 solar masses. In order t o avoid the "annihilation catastrophe," these pockets would have to have been separated by some physical mechanism before the temperature of the universe dropped low enough t o allow nearly complete annihilation of baryons with anti-baryons. At that temperature, however, the horizon only contained about lop7 solar masses. Causality, then, makes it impossible that such large pockets could have separated from each other at so early a time. It is therefore assumed that no pockets exist. The baryon asymmetry observed in the vicinity of the earth must extend throughout the universe. Now, a question arises: has this asymmetry between baryons and antibaryons always existed? In this thesis, we assume not. From observations made by the Wilkinson Microwave Anisotropy Probe (WMAP) and the Sloan Digital Sky Survey (SDSS), the ratio of the number of baryons t o the number of photons in the universe is (6.3 0.3) x lo-". The smallness of this ratio seems t o suggest that the initial baryon number of the universe was zero, but it is still a possibility that the universe began with a small but non-zero baryon number. With that said, the purpose of this thesis is t o investigate how a baryon asymmetry might develop in a universe with zero initial baryon number. Appropriately, the creation of a nonzero baryon number in the early universe is called "baryogenesis." There are three requirements for any model of baryogenesis. First, rather obviously, the model must include interactions that not conserve baryon number. One cannot hope to generate a baryon asymmetry if no violation of baryon number occurs. Second, there must be interactions that violate C P (charge conjugation combined with parity inversion) symmetry. This amounts t o preferentially producing baryons over anti-baryons, for example. Even with baryon-number violation, it is necessary to produce more baryons than anti-baryons in order to develop a nonzero total baryon number. Third, interactions must take place out of equilibrium. C P T (CP combined with time reversal) invariance, a symmetry required of all Lorentz-invariant theories, dictates that a particle must have the same mass as its antiparticle. Therefore, particle and antiparticle abundances in thermal equilibrium, determined by the Boltzmann factor e-"IT, are the same. If no particles venture out of equilibrium, then particle and antiparticle abundances are always equal, so no baryogenesis occurs. + 1.2 Majorana Leptogenesis Within the Standard Model, there are interactions that not conserve baryon number. The vacuum structure of the SU(2) gauge theory results in baryon-number-violating processes associated with transitions between different vacua. These processes are interpreted as interactions with massive, unstable particles, called "sphalerons." Letting B stand for baryon number and L for lepton number, sphaleron interactions conserve B - L while violating B+ L. They can thus create a nonzero baryon number from a pre-existing lepton asymmetry. Allowing this baryon number violation inherent in the Standard Model t o satisfy the first requirement of baryogenesis, we now ask how t o create a nonzero lepton number in the early universe. So-called "leptogenesis" models rely on the same requirements as baryogenesis, with lepton number violation instead of baryon number violation. Note, however, that although lepton number violation is required to create a nonzero net lepton number, it is not necessary in achieving a baryon asymmetry through sphalerons, a subtlety that is explained in the next section. One candidate leptogenesis model ernerged from an effort t o explain the observed smallness of left-handed neutrino masses. This model, as proposed by Fukugita and Yanagida [2], postulates the existence of right-handed neutrinos N that acquire a large mass through a Majorana mass term in the lagrangian, so I refer t o it as the "Majorana model." The existence of a Majorana mass term means, among other things, that the singlet N is its own antiparticle. Thus, N does not appear in the lagrangian: the terms concerning the right-handed neutrino are where L is the left-handed lepton doublet and H is the Higgs doublet. The constant matrix X dictates the strength of N's couplings t o L and H. Looking at equation 1.1, there is no way t o assign charges t o the particles involved so that the sum of the charges of each term in the lagrangian is zero. This is a result of the Majorana mass term; there are no conserved charges in the Majorana model. The smallness of left-handed neutrino masses is explained via the the "seesaw" mechanism, in which the masses of left-handed neutrinos are inversely proportional t o the right-handed neutrino masses. Integrating out the heavy N field from the Majorana lagrangian, we achieve the effective lagrangian for low energies, in which the left-handed neutrino mass matrix is apparent as the coefficient of the term coupling L t o L . It is found that where (H) is the vacuum expectation value of the Higgs field, about 174 GeV. For a more detailed explanation of the process of integrating out a heavy field from the lagrangian, please see the next section. For now, note that the right-handed neutrinos N acquire mass even in the absence of electroweak symmetry breaking, so there is nothing preventing their masses from being very large. If the nonzero elements of r n are ~ much larger than (H) and the couplings in X are small, then the Majorana model correctly predicts that the left-handed neutrino masses are tiny. The most important result of Majorana leptogenesis is that it can actually produce the observed baryon asymmetry with certain parameter choices. In order t o so, however, it is shown in [8] that the mass of the lightest right-handed neutrino, Nl, must be greater than lo8 t o 10' GeV. In principle, such a large mass is not out of the question, but there could be a problem here. For Nl t o be thermally produced, the temperature of the universe immediately following inflation, called the reheat temperature TRH, niust be greater than m ~. Evidently, , TRH> 10' t o 10' GeV must hold if the Majorana model is the mechanism by which the current baryon asymmetry was created. Supersymmetric theories, however, require that the reheat 0'' GeV in order t o avoid disrupting the temperature be less than 10' to subsequent evolution of the universe by producing an excess of gravitinos and similar relics of supersymmetry. If supersymmetry is correct, then the bound on m ~is ,precariously close to this threshold. It is probable that r n ~ ~ actually exceeds the bound, so it's possible Majorana leptogenesis could be ruled out by supersymmetry. Partly because of the tension that exists between the Majorana model and supersymmetric theories, it is interesting t o study alternatives t o Majorana leptogenesis. The necessary reheat temperature could be lower in other models, perhaps far enough below the threshold for gravitino overproduction that supersymmetry would no longer be at odds with leptogenesis. Also, in part due t o the high energies at which Majorana leptogenesis occurs, researchers have been unable t o determine whether the Majorana model is correct. If the energy scales of any alternative models prove t o be much lower than those of the Majorana, it could be possible to access leptogenesis by experiment. With these possibilities in mind, we embark on an investigation of one such alternative. 1.3 Dirac Leptogenesis The Majorana model is so named because it proposes that neutrinos are Majorana fermions, acquiring mass through a Majorana Inass term in the Lagrangian. Similarly, in the framework of a Dirac leptogenesis model, neutrinos are Dirac fermions. A few different models of Dirac leptogenesis have been proposed; the purpose of this thesis is t o study the model conceived by Murayama and Pierce [7],which I will refer to as "the Dirac model." Its Lagrangian contains the terms Here, as before, N is the right-handed neutrino singlet, L is the left-handed lepton doublet, and H is the Higgs doublet. In addition t o those particles familiar from the Majorana model, however, the Dirac model contains two new fields: is a massive fermion doublet, and x is a scalar singlet. The constant matrices X and h govern the strengths of 4's couplings t o NH and Lx, respectively, and m+is the mass of the particle. Because of the absence of a Majorana mass term in the Lagrangian, it is possible t o identify a conserved charge in the Dirac model. Specifically, if we choose the lepton number of to be one, then lepton number is conserved: the sum of the lepton numbers of every particle in each term in the Lagrangian is zero. At first, this seems t o pose a problem. If lepton number is conserved, then no net lepton asymmetry is produced. Sphaleron interactions can convert a nonzero lepton number t o a nonzero baryon number, but how can a net baryon asymmetry be achieved through sphalerons if no lepton asymmetry develops? The answer is that sphalerons interact only with left-handed particles. (This is because sphalerons arise from the dynamics of the SU(2) gauge group, and only left-handed particles have SU(2) gauge interactions.) In the Dirac model, while no net lepton asymmetry develops, opposite asymmetries in left-handed and right-handed leptons are produced. It is the final nonzero left-handed lepton number that converts t o a nonzero baryon number via sphaleron interactions. Note that this mechanism fails if processes that convert left-handed neutrinos t o right-handed neutrinos and vice-versa occur rapidly in the early universe, because in that case the left and right-handed asymmetries are washed out. For instance, it is not possible t o store opposite asymmetries in the left and right-handed electrons because their coupling to the Higgs field, X,ELH, is strong enough t o ensure rapid e~ +-+ e~ conversion at high temperatures. We know this because the Yukawa coupling Xe is fixed by the known mass of the electron: me = Xe(H). Because of washout effects, in order for Dirac leptogenesis to work, we can't have the couplings X and h be too large, a constraint that will be made more precise in the course of this thesis. In the Dirac model, neutrinos acquire mass through a Dirac mass term in the Lagrangian, but careful inspection of (1.3) reveals no such term. The explanation for this inconsistency is that the Dirac neutrino mass term appears only in the effective Lagrangian, at energy scales well below mq. At such low energies, the dynamics of the massive field are essentially frozen, so we can remove from our effective theory, using the equations dCldq5 = and dC/d$ = 0. Solving these for and 4, respectively, and substituting the solutions into equation (1.3) produces the effective Lagrangian, Here is the Dirac mass term for neutrinos, coupling the right-handed N t o the left-handed L. The two-component fermions N and L form a fourcomponent Dirac fermion with mass where ( H ) and ( )are the vacuum expectation values of H and X, respectively. In the model of Murayama and Pierce, the x field arises from the supersymmetry-breaking sector of the theory; it is expected t o have a vacuum expectation value of the same order as the weak scale, so ( x ) ( H ) . From equation (1.5), it is apparent that the Dirac model, like the Majorana model, incorporates the see-saw mechanism. Like N in the Majorana model, obtains mass even in the absence of electroweak symmetry breaking, so there is no reason why it can't be very heavy. If the mass of is much larger than the weak scale, then the observed smallness of neutrino masses is at least partially explained by the fact that m, is suppressed by mq. For leptogenesis, the most important interactions in the Dirac model are the decays of the massive particles, q5 and 6. Each particle has two decay N + Upon rewriting ISliI2 IRliI2, and since m,, is defined t o be less than or equal t o m, , Finally, minimizing f i by letting ISli - RliI2 = 0, and using the condition SIR = I, a lower bound is acheived: The lowest possible m,,, given the data fromobserving neutrino oscillations, is = x 10-~eV. Assuming ( x ) -- (H),if 6z x 10-~eV,then the'efficiency has an upper bound: > loe2, approximately. (5.3) Since the neutrino masses must be hierarchical t o approach this bound on q, and because m,, = 0, the results of section 3.3 are applicable here. From equation (3.11), The final lepton asymmetry per comoving volume is, from equation (1.7), and Y4(T>> m41) M x 10V3. Therefore, To generate the observed baryon number asymmetry, Y L - ~must be greater than 2.5 x lo-'', approximately. In order t o meet this requirement, the inequality must be satisfied. Since x is expected to have a vacuum expectation value of the same order as the Higgs, m i , must be greater than GeV. In general, such a large mass is not at all outside the realm of possibility. In supersymmetric theories, however, this range of m+, is problematic: in this case, we have excessive gravitino production if the reheat temperature of the universe is higher than 10' to 10" GeV. The reheat temperature would have t o be higher than this bound to produce enough q5 particles with only two q5 flavors. If supersymmetry is correct, and with ( x ) (H),then q5 must have more than two flavors. In general, though, having three flavors simply allows for more efficient leptogenesis, which in turn provides more freedom t o lower m4,. It is for these reasons that this thesis assumes q5 has three flavors. N 5.2 5.2.1 Bounds on md, Motivations The other characteristics of $ithat profoundly impact leptogenesis are its masses, particularly m4,. Finding a bound on m+, is of special interest for two reasons. First, as mentioned above, supersymmetric theories impose an upper limit of about 10' t o 0'' GeV for the reheat temperature TRHof the universe. Above that bound, excessive production of supersymmetric relics such as gravitinos would dramatically alter the subsequent evolution of the universe. This scenario is incompatible with currently successful cosmological models, so if supersymmetry exists, then TRH< 10' t o 101° GeV must hold. In order for leptogenesis t o occur, the mass of the decaying particle must be less than the reheat temperature, so supersymmetry imposes an upper bound on m+,. Interestingly, in the Majorana model, the mass of the decaying right-handed neutrino is very close to this bound, potentially even exceeding it t o achieve the necessary lepton asymmetry. It is possible that the Dirac model could allow TRHt o be more comfortably below the temperature where gravitino production becomes problematic. Second, if the q5 masses were very small relative to the Majorana mw, then experimental tests of the Dirac model could be possible. For example, if m4, were less than a few TeV, then the q51 particle could be detected a t future particle accelerators. If a detection actually occurred, then leptogenesis could be studied experimentally, albeit with significant difficulty. Because this possibility exists, it is important t o investigate how small m+, can be. 5.2.2 Preparatory Steps Before embarking on this investigation, it is necessary t o keep in mind the observational results that will ultimately determine whether Dirac leptogenesis works. From WMAP [3] and the Sloan Digital Sky Survey [4], the ratio of baryons t o photons in the universe is It can be proven (see [I],for example) that n,/s = 0.14, so Sphalerons can convert a left-handed lepton asymmetry to a nonzero baryon number. This effect amounts t o the relation YB = YL- (0.35), so in order t o satisfy observation, YL- = (2.6 2~ 0.1) x lo-''. For the purposes of this section, YL- is required t o be greater than 2.5 x lo-''. It should also be noted tha,t Ml~rayamaand Pierce expect that (x)2 (H) because of x's relation t o supersymmetry breaking. Although their research is the source of the Dirac model in this thesis, in the interest of fully exploring how small the masses can be, we consider cases with (x) < (H). Lower bounds for mdl are achieved using the formula for the final lefthanded lepton asymmetry, equation (1.7): In all cases, Y T ( T > mil) e x lop3. Applying the observationally-derived bound on YL-, equation (5.5) yields To produce sufficient lepton asymmetry, we have only to ensure that this inequality is satisfied. The form of the CP-violation parameter was found in chapter two, and bounds are readily available from chapter three, so need not be closely examined in this chapter. Instead, an analysis of the efficiency will lead us t o the desired m+,bounds. The efficiency depends sharply on whether x and N are initially in equilibrium. This is a result of 4's gauge couplings. Because is an SU(2) doublet, it couples t o the SU(2) gauge field. This means that there are interactions in which $ and $ annihilate t o produce particle-antiparticle pairs of other SU(2) doublets, such as LL or HH , and vice-versa. Really, for the Boltzmann equation in Y++to be complete, it should have a term for this interaction. Without actually calculating the size of that term, it is possible t o deduce its form. The result is ~ P L -g 4f (x) $ other terms. m41 Here, g is the strength of 4's gauge couplings, mPl is the Planck mass, and f (x) is an unknown function of x = m4,/T. Four powers of g are necessary since the gauge interactions are 2-to-2 processes. The Planck mass appears from dividing by the Hubble constant t o have only Yi+on the left-hand side. The right-hand side must then be divided by m+;because Yi+is dimensionless. In general, the gauge couplings tend t o force particles into equilibrium until their abundance becomes so small that the 2-to-2 gauge interactions are suppressed. Decreasing m4, causes the gauge term to increase in magnitude, so $ stays in equilibrium longer for small m+, than for large m4,. If x and N are initially in equilibrium, they will stay in equilibrium as long as $ does. Leptogenesis becomes less and less efficient as the length of time spent in equilibrium increases, so efficiency is reduced for small m+,. This is confirmed by figure 5.1, the contour plot of efficiency for the case when x and N are in equilibrium. If x and N are not initially in equilibrium, however, inverse decays are very unlikely, so there is no cancellation of the asymmetries produced by and $ decays. In this case, large lepton asymmetries are possible no matter how long $ remains in equilibrium. If x and N are initially out of equilibrium, efficiencies as high as a third are possible even for m+, I TeV. This is explicitly shown in figure 5.2, the contour plot of efficiency for the case when x and N are out of equilibrium. Now, an interesting question arises. How is it that x and N could initially be in equilibrium? Certainly no processes outside the Dirac model can produce them, so how can they even exist before Dirac leptogenesis begins? The answer has t o with the word "initially." In this context, "initially" means before the decays of begin. With this definition, it's possible that nonzero initial abundances of x and N are produced in the decays of $2 if the reheat temperature is high enough t o allow $2 production by gauge pro- Figure 5.1: Contours of constant efficiency for x and N in equilibrium. Range is from = at the bottom t o = a t the top. Interval size is a factor of 10. cesses. More precisely, if TRH> m4,, then we can treat x and N as being initially in equilibrium only if m2 > lop3 eV. This is confirmed by explicit is below lop3 eV, the couplings of 42 are calculations from [12]. When not strong enough t o bring x and N into equilibrium. For TRH> mq,, the case with x and N out of equilibrium seems t o be a better candidate for finding a lower bound on md,, but consider the CPviolation parameter. An illuminating bound on E follows simply from the fact that Im(z) (xI2for any complex number x: where Al and A2 represent, roughly, the largest couplings of q!q and Clearly, the Al's cancel, and A;/m+, is related t o m , so that 42. lo8 M4 / GeV 10 Figure 5.2: Contours of constant efficiency for x and N out of equilibrium. Values start at q~ = lop5 at the right and go t o rj = lo-' in the middle. Interval size is a factor of 10. Now, if m2 < lop3 eV, the observatiol~alrequirement that YL-be greater than 2.5 x 10-lo can be applied t o find that m4, 2.5 x 10'' GeV must hold. It is somewhat counterintuitive, but the dependence of E on f i 2actually forces the m4, bound t o be larger if x and N are not brought into equilibrium by the decays of qb2 Therefore, the case of TRH> m4, with x and N being initially out of equilibrium should be neglected in the search for a lowest bound on m4,. 5.2.3 Results Because nonzero but out-of-equilibrium initial x and N abundances can be neglected, there are only two cases t o consider in finding the lowest bound on m4,: when x and N are initially in equilibrium due to $2 decay, and when x and N are initially nonexistent. In both cases, the CP-violation bound from Figure 5.3: Contours of constant (x)for x and N in equilibrium. Top contour is for (x) = (H);middle is (x) = (H)/lO;bottom is (x),~, chapter applies, so Using equation (5.6), we arrive at a formula for (x) as a function of m+,and the efficiency. The vacuum expectation value required of x in order to match observation with maximum CP-violation is given by Although the analytic form of the efficiency is unknown, q is defined point-wise as a function of m+, and ?Elin figures 5.1 and 5.2. This allows for a point-wise definition of (x) as well. Using the data from which the efficiency contour plots were generated, similar plots can be made for (x). Figure 5.3 is the contour plot of (x) for the case when x and N are initially in equilibrium, while figure 5.4 is for x and N initially out of equilibrium. With these plots, lower bounds on m+,are readily apparent. For example, if (x) = (H),then m4, must be greater than or equal t o the minima of the Figure 5.4: Contours of constant (x) for x and N out of equilibrium. Top contour is for (x) = (H);rrliddle is (x) = (H)/10; bottom is (x),~, top contours. For the equilibrium case, in which TRH> md2, we find that m?, > x 10'' GeV, approximately. This is actually a larger bound than in the Majorana model, so the Dirac framework does not help to alleviate the tension between leptogenesis and excessive gravitino production in this case. The CP-violation parameter in the Dirac model is one-third as large as it is in the Majorana, and efficiency is reduced in the Dirac model by gauge interactions, so a higher bound for the mass of the decaying particle is what Tive expect. Moving on to the other plot, in the out-of-equilibrium case the bound on m?, is about x 10' GeV. While this is a smaller bound than in the equilibrium case, it is comparable in size to the Majorana bound. Again, the tension between supersymmetry and leptogenesis is not relieved. The choice of (x) = (H) offers no relief from the problem of excessive gravitino production, and it certainly does not allow for any particle to have a chance of being detected at modern accelerators. However, it is clear from figures 5.3 and 5.4 that the Dirac model can better if (x) is lowered. The bound on m?, generally seems t o approach zero as (x) approaches zero. If leptogenesis works with ( x )arbitrarily close to zero, then there is no bound on how small md, can be. It would certainly be interesting if this were the case, but, as might be expected, there are complicating factors that prevent ( x )from decreasing beyond a certain minimum value. The program used to produce all of the contour plots doesn't take into account exchange in 2-to-2 interactions such as LX -7- N H , and these interactions have the potential to decrease the efficiency dramatically. The reaction LX i N H specifically tends t o decrease any left-handed lepton asymmetry by converting left-handed leptons into right-handed leptons. To avoid this washout effect, the rate of these 2-to-2 interactions must be less than the rate of expansion of the universe around the time when is decaying. More concisely, we want I'(T m m4,)< H(T FZ m4,), where I' is the rate per particle of the 2-to-2 interactions mediated by $2. H is the Hubble constant, given by H m 20T2/mpl. As might be expected from classical mechanics, the reaction rate r is proportional t o the cross-section of the $2-mediated 2-to-2 interactions multiplied by the equilibrium velocities and number densities of the reactants. Since I? and H are being compared at temperatures roughly equivalent t o the $1 mass, the reactants and products of the 2-to-2 interactions are essentially moving at the speed of light because their masses are all much smaller the m+,. The form of the equilibrium number densities can be computed from thermodynamics (see Kolb and Turner), resulting in n,, = T3/7r2. The cross sections of the relevant 2-to-2 processes can be approximated by = (1/16n)(At/m$,), where A2 represents the largest of $ ' ~ couplings. Putting all of this together, we obtain A"; T T~ < 203 16n rng2 m ~ l for T = md,. Solving for A2, an approximate bound on the strength of the couplings of A2 is found: F'roni [12], inclusion of the q52-mediated2-to-2 reactions in the Boltzmann equations results in washout being avoided only if This is a more accurate result than was found by approximation above, but the similarity of the two bounds shows that the approximation scheme was valid. Substituting the result of the Boltzmann equations into (5.7)) we require that [...]... lepton The index A controls which components interact in the SU(2) doublets 4, L and H PR and PL are right and left-handed projection operators Applying them t o N and L, respectively, allows the convenience of using four-component Dirac spinors while requiring that 4 couple separately to right-handed and left-handed leptons With the precise interaction lagrangian (2.4) and an elementary knowledge of quantum... attempt is made to determine whether Dirac leptogenesis is capable of producing a realistic baryon asymmetry Assuming that it is, the parameters of the Dirac model are examined in an effort t o find what observational constraints require of them The mass of the lightest $ particle is of particular interest, since the tension between supersymmetric gravitino overproduction and a necessarily large reheat... because (hht),, and ( x x ~ are real Therefore, applying the hierarchical 4 approximation, ) ~ ~ Moving l/mdi between (hht)li and (XX+)il ,and using the formula for the diagonalized neutrino mass matrix, Now, making use of the results of the previous section, h and X are eliminated from the formula for E , in favor of the neutrino masses and elements of the S and R matrices: 3.3 3.3.1 Maximizing 161 A Simplified... YN_, YN+, Yx+, Ym- and Y++ functions of z = m+,/T The as decay densities are given by and as is discussed in [ll].Here, K1 and K2 are modified Bessel functions of the second kind, and rLand I'R are the decay rates into left-handed and righthanded leptons, respectively The coupling constants h and X are chosen to have a range of values The range of the couplings corresponds to a range of values for the... amplitudes of the decays at tree level are straightforward t o calculate We find that (with lower-case indices representing flavor and upper-case representing the SU(2) component) and where M1 denotes the first-order (tree-level) term of the complex amplitude The symbols u and v are fermion and antifermion Dirac spinors Their subscripts so and sl label, respectively, the spins of q5 and L Calculation of the... relatively tiny amount t o the CP-conserving first term The interaction terms relevant to leptogenesis in the Dirac lagrangian are more precisely written, Here, the full structure of the Dirac model starts t o become apparent The flavors of 4, N, and L are encoded with the indices i, j , and k , respectively The constants X and h are revealed t o be matrices, containing the coupling strengths of every... Maximizing 161 A Simplified Case In the Majorana model, the left-handed neutrino mass matrix is given by equation (1.2): x'~;~x(H)~, where X is the coupling strength of L and H with the massive right-handed mu = neutrino An interesting simplification of the Dirac model is to reduce the number of free parameters by identifying A' = h'* In this case, if ( x )= (H), the form of the neutrino mass matrix becomes... process, so far unsuccessful, are ongoing In light of the fundamental differences between the two models, and given the extensive study of leptogenesis in the Majorana model, it seems worthwhile t o carry out a careful analysis of leptogenesis in the Dirac model As argued in the previous section, this amounts to a careful calculation of the parameters E and 7.The main purpose of this thesis is t o analyze... define YL- = YL - YL and subtract equations (4.11) and (4.12) t o obtain, finally, + Here, it has been assumed that YL+ = YL YE = 2Yiq, and likewise for H Unlike L and H, however, the particles N and x are only produced in the interactions of the Dirac model, so we cannot assume that they are in equilibrium The number 6 may appear out of place, but it comes from assuming YL- = Yx- (there are no interactions... four-momenta of the L and 4 particles Although the numerator of c is more complicated, it can be simplified similarly t o the denominator, using methods from quantum field theory Bringing the two results together, we have a formula for the CP-violation warameter: Chapter 3 Bounds on 3.1 E Neutrino Masses The neutrino mass matrix rn, and its formula (1.5) in the Dirac model provide a link between leptogenesis and . Derivation and Analysis of Bounds in Dirac Leptogenesis by John A. BackusMayes A thesis submitted in partial fulfillment of the requirements for the Degree of Bachelor of Arts. to determine whether Dirac leptogenesis is capable of producing a realistic baryon asymmetry. Assuming that it is, the parameters of the Dirac model are examined in an effort to find what. Lagrangian. Similarly, in the framework of a Dirac leptogenesis model, neu - trinos are Dirac fermions. A few different models of Dirac leptogenesis have been proposed; the purpose of this thesis