Derivation and analysis of bounds in dirac leptogenesis

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 uni

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

6 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

as well Humans have visited the Moon, and mechanical probes have come

in contact with eight of the nine planets in the solar system T h a t 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

T h a t 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 a t 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 t h a t 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 a t 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 anti- baryons 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 do 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 ( C P combined with time reversal) invariance, a symmetry required of all Lorentz-invariant the- ories, dictates that a particle must have the same mass as its antiparticle Therefore, particle and antiparticle abundances in thermal equilibrium, de- termined by the Boltzmann factor e-"IT, are the same If no particles ven-

ture 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 do not conserve baryon number The vacuum structure of the SU(2) gauge theory results in baryon-number-violating processes associated with transitions between dif- ferent 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 violat- ing 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 a n 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 neu- trinos N t h a t 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

a t 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 "see- saw" 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 t h a n (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 actu-

ally produce the observed baryon asymmetry with certain parameter choices

In order t o do 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 prin-

ciple, 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 asymme-

try was created Supersymmetric theories, however, require that the reheat

temperature be less than 10' to 1 GeV in order t o avoid disrupting the

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 mod-

els, perhaps far enough below the threshold for gravitino overproduction that

supersymmetry would no longer be a t odds with leptogenesis Also, in part

due t o the high energies a t 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, neu- trinos 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: 4 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 4 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 4 to be one, then lepton number is con- served: 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 interac- tions 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 lep- ton 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 neu- trinos 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 a t 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 ap- pears only in the effective Lagrangian, a t energy scales well below mq At such low energies, the dynamics of the massive 4 field are essentially frozen,

so we can remove 4 from our effective theory, using the equations dCldq5 = 0

and dC/d$ = 0 Solving these for 4 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 four- component Dirac fermion with mass

where ( H ) and ( 2 ) are the vacuum expectation values of H and X, respec- tively 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 vac- uum expectation value of the same order as the weak scale, so ( x ) N ( 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,

4 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 4 is much larger than the weak scale, then the observed smallness of neutrino masses is a t 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

We encode this difference using a dimensionless CP-violation parameter,

where represents the decay rate indicated in parentheses The physical meaning of E can be understood by considering the following scenario: if a collection of 4 and 4 particles spontaneously decays without back-reactions through the four channels of the Dirac model, the resulting left-handed lepton number will be E multiplied by the initial number of 4 particles

T h e maximum asymmetry in left-handed leptons is generated if three conditions on the q5 particles are met First, they must start out at high tem- peratures T >> mq in thermal equilibrium Second, they must decouple from the thermal bath before the temperature drops low enough for their equi- librium abundance t o become suppressed by the Boltzmann factor e P r n d T

Third, they must eventually decay a t T << m4, when back-reactions, for the most part, don't occur In this case, defining, for example, Y4 as the number density of 4 particles divided by the entropy density, we would get (YL - = Y i q ( T >> m4) E, where Yiq refers t o the number density for

4 particles in thermal equilibrium More realistically, back-reactions tend to decrease the right-handed and left-handed lepton asymmetries As a result, the final left-handed asymmetry can be expressed as the maximum asymme- try multiplied by an efficiency factor, 0 _< 7 < 1:

While the first baryogenesis requirement is met via sphaleron interac- tions in leptogenesis models, E and 7 measure the extent t o which the second and third requirements for baryogenesis, respectively, are satisfied If either

is zero, then no lepton or baryon number asymmetry develops As each in- creases, so does the final left-handed lepton number and, through sphalerons, the baryon number of the universe The CP-violation parameter 6 clearly shows how much the symmetry of charge-conjugation with parity-inversion

is broken by the decays of the Dirac model The efficiency factor 7 is zero if $ particles remain in thermal equilibrium as they decay: as stated before, out- of-equilibrium effects are necessary for the production of any lepton or baryon asymmetry As the decays occur further from equilibrium, 7 increases

7 as well

Throughout this thesis, an 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 temperature could

be resolved if m4, can be somewhat smaller than the Majorana m ~ , If Dirac leptogenesis can indeed improve upon the Majorana model by having

a smaller bound on m+, , we ask exactly how small m4, can be, and whether

might be detectable in particle colliders

The complex decay amplitude of 4 - LX is exactly given by

It is impossible t o calculate these complex decay amplitudes exactly, so

a n approximation is in order In this thesis, we use perturbation theory t o approximate every relevant M If X and h are roughly one or smaller, then 4

couples weakly in the Dirac model, and e"d4xcint(x) is closely approximated

by t h e first few terms in its Taylor expansion In this manner, a n approximate form for the decay amplitudes is achieved; for example,

Because 14) and J L X ) are orthogonal, the zeroth-order term does not appear The second-order term, however, vanishes for less obvious reasons

There is a correspondence between the terms of any M and the many Feynman diagrams representing the specified interaction In every term, each space-time integral represents a distinct vertex in the corresponding Feyn- man diagram In the case of 4 -+ Lx, the first-order term corresponds to

a diagram with only one vertex, the so-called "tree-level" diagram Simi- larly, the third-order term corresponds t o diagrams with three vertices The second-order term is zero because in the Dirac model, there is no Feynman diagram connecting 4, L and x with only two vertices

To illustrate these ideas, consider the following The first term in (2.3) is really the complex amplitude of the interaction represented by the tree-level Feynman diagram,

in which 4 decays directly into L and X In the perturbative regime, the first term of any M is dominant, so nearly all of 4's decays are direct If I were interested only in the decay rate I'(4 -+ Lx), then the first term in ( 2 3 )

would probably be a good enough approximation; however, this tree-level interaction has no CP-violation To achieve a nonzero c , it is necessary t o include terms of higher order

T h e higher-order Feynman diagrams are less direct than the tree-level, in- volving many intermediate "virtual" particles that exist only inst ant aneously

to facilitate the decay Of these, the diagram with the largest contribution

t o the amplitude is one with a single loop:

Here, q5 decays into N and H , tvhich immediately annihilate t o produce another 4 that decays into L and X The loop is the closed curve formed by the paths of N and H Actually, this sort of indirect decay can also occur with L and x as the intermediate particles, but that interaction, like the tree-level, does not violate C P symmetry It can be ignored because its effect

is only to add a 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

T h e constants X and h are revealed t o be matrices, containing the coupling strengths of every q5 flavor with every 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 knowl- edge of quantum field theory, the complex amplitudes of the decays a t 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 one-loop amplitudes is considerably more complicated than the tree-level The results are that

and

Here, p is the four-momentum of the initial 4 particle, and q is the four- niornelit u ~ n associated with the intermediate particles The symbol j/ is shorthand for ypq,, where yp are the Dirac spin matrices As the flavor

of the intermediate q5 particle, i is a a index that must be summed over, since any 4 flavor could potentially mediate this decay

2.2 A Formula for 6

To simplify calculations, let

where C1 holds the coupling constants that appear in the formula for M1 (4 -+ L x ) ,

equation (2.5), and MI holds the rest of the formula Similarly, let

and

With these simplified forms, we first calculate the numerator of E Note that /MI2 = lM1 + M3I2; therefore,

After summing over spins, one finds M2Ml = MI M f , M4M$ = M3M3*, and M2M$ = MIM,* Making these substitutions, and subtracting the two equations, it is found that

The CP-violation is fully encoded in this proportionality, but t o achieve equality it is necessary t o include the denominator Here, we calculate only

t o first order, since the third-order correction would make only a miniscule difference, and the CP-violation has already been encoded in the numerator

We obtain

Summing over spins, and choosing as the decaying particle, the denomi- nator becomes

where p~ and pd are the 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 E

The neutrino mass matrix rn, and its formula (1.5) in the Dirac model provide

a link between leptogenesis and observation Oscillations have been observed

in solar and atmospheric neutrinos as the mechanism by which they change flavor during travel A large fraction of the electron neutrinos emitted by the sun are found t o have changed into muon and tau neutrinos by the time they are detected on Earth Similarly, more energetic neutrinos produced

by cosmic rays colliding with atoms in the atmosphere tend t o change flavor

as they travel t o detectors on the other side of the planet The existence of these oscillations impacts Dirac leptogenesis in two important ways

By measuring the frequency of neutrino oscillations, researchers have ar- rived a t nonzero estimates of the differences in mass squared between the three neutrino mass eigenstates From the solar neutrino measurements, they find Am;ol z 8 x e v 2 , and from the atmospheric neutrinos, Arnitm z

2.5 x e v 2 It is unknown to which neutrino mass splittings these ob- servations correspond, but in the course of this thesis I will suggest that

2

Dmiol = rn:, - rnEl and = rn, - rn:, Clearly, the existence of two different nonzero differences in mass squared implies that no more than one neutrino can be massless, so the neutrino mass matrix, from equation (1.5), must have a t least two nonzero eigenvalues The specific values for the mass differences can be used as a guide in exploring the parameter space of the Dirac model

Second, neutrino oscillations are only possible if the neutrino eigenstates

Going one step further, we arrive a t formulae relating the hermitian objects hht and XXt t o the neutrino masses; in particular,

With these results, it is possible to directly relate the CP-violation parameter and the neutrino masses From that, an upper bound on /el can be achieved

To start, it is necessary t o approximate 6's dependence on the q5 masses

Equation (2.9) states that

Although m4, or m4, could be close t o m4,, this thesis focuses on the hierarchical case, in which m4, << m+, , m4, This is the case most often studied in the literature, partly because it allows the easiest calculation of the efficiency, 7 One should keep in mind that by choosing nearly degenerate

4 particles, it is possible t o increase E beyond the bounds we will obtain in the hierarchical case Note, however, that as m4, or m4, is lowered towards

m4,, the enhancement of E does not arise as quickly as one might expect from the mj, - mat term in the denominator This is because, for fixed neutrino

masses, the couplings h and X also depend on the q5 masses, with smaller

masses corresponding to smaller couplings

With the hierarchical approximation,

The condition that i # 1 is satisfied: the i = 1 term is zero 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

upon substitution of h for X and r n d for m ~ Also, because of the relationship between A' and h', R = S* Making this simplification, but allowing ( x ) to have any value, the formula for the CP-violation parameter becomes

T h e first step in maximizing I E / is t o focus on the denominator As is the case with any complex number, lSliI2 > 11m(S&) 1, so

Because R = S* in this case, the requirement that SIR = I simplifies t o

STS = I In other words, S is a complex orthogonal matrix In particular,

Using this substitution,

Now, consider the numerator Because of the orthogonality of Si

This is maximized if 1m(S12~) and I ~ ( s ? , ) are chosen t o be either both positive

or both negative Thus,

since mu, > mu, Similarly,

Placing these results into equation (3.9)) a bound on (€1 is achieved:

I - 1 m4,

This bound is independent of the coupling constants No matter the choice

of values for h, it must hold if A' = h'* and ( x ) = ( H ) To actually reach the bound, however, two conditions must be met First, ,512, and S122 must

be purely imaginary and have the same complex phase Second, mu, must equal m,, but this is impossible since Amit, # Amiol Realistically,

mu, -mu, << m, -mu, should hold in order to approach the bound Thus, /€I

the previous simplification, however, that parameter space was effectively reduced t o the same size as the Majorana model's It seems as though the bound achieved by letting A' = h'* should not apply to the entire Dirac parameter space, only t o a small subset of it The CP-violation parameter might actually exceed the bound for certain choices of parameters outside of this subset

In the interest of exploring the Dirac parameter space more fully, it would

be prudent t o search for a CP-violation bound without restricting (x) or relating X and h in any way beyond their connection through the neutrino masses In the most general case, obtaining an analytic bound on 161 appears difficult, but it is straightforward t o examine the case where neutrino masses are hierarchical, with m.,, = 0

Immediately, the formula for 6 is dramatically simplified In the hierar- chical limit, mu, << m,, so equation (3.8) reduces to

Now, (Im(S13RT3) ( I (S13((R131, SO

This is maximized if 1 S13 1 = I R13 1 Finally,

This was a completely different simplification than the previous one, but the resultant CP-violation bound is the same Since mu, was taken t o be zero in this case, the inequalities (3.10) and (3.11) are exactly equivalent Clearly, this bound does not depend on requiring that A' = h'*, so it actually applies t o a large extent of the Dirac model's parameter space However, both simplifying caxes have depended on the 4 and neutrino masses being hierarchical Numerical studies [I 21 indicate that relaxing our assumptions about the neutrino masses does not allow for larger values of 6 than that given

in equation (3.10) On the other hand, we stress again that the assumption of hierarchical q5 masses is important here For a nearly degenerate d, spectrum,

161 can be much larger

Chapter 4

Boltzmann Equations and

4.1 Relevant Interact ions

The decays and inverse decays of 4 and $ are the most important processes for Dirac leptogenesis They directly produce the left-handed lepton asymmetry Really, t o achieve the observed YBWB, the decays are the only necessary reactions from the Dirac lagrangian In order for the Dirac model t o be self-consistent, however, other interactions allowed by the lagrangian should

be considered These extra reactions could have effects on the final lepton asymmetry, and these effects must be investigated

It makes sense t o start with the next least complicated reactions after the decays The 2-to-2 processes allowed by the interaction lagrangian tend t o force particles into equilibrium, decreasing the efficiency of the Dirac model For instance, the interactions LX c-t NH and NH t+ tend t o equalize the net left-handed and right-handed lepton numbers Other examples are the pair-annihilation processes, in which 4 and 6 annihilate to produce LL,

NN, X X or HH Clearly, pair-annihilations and their inverse reactions keep

4 and $ from straying very far from equilibrium

It is true that 2-to-2 processes decrease the efficiency and final left-handed lepton asymmetry, but this effect is actually quite small The amount that the efficiency changes upon the inclusion of 2-to-2 reactions is negligible

To account for this, recall that in order t o satisfy the out-of-equilibrium leptogenesis requirement, the couplings in h and X must be relatively small