Conformal Standard Model, Leptogenesis and Dark Matter
The Conformal Standard Model (CSM) is a minimal extension of the Standard Model of Particle Physics based on the assumed absence of large intermediate scales between the TeV scale and the Planck scale, which incorporates only right-chiral neutrinos and a new complex scalar in addition to the usual SM degrees of freedom, but no other features such as supersymmetric partners. In this paper, we present a comprehensive quantitative analysis of this model, and show that all outstanding issues of particle physics proper can in principle be solved ‘in one go’ within this framework. This includes in particular the stabilization of the electroweak scale, ‘minimal’ leptogenesis and the explanation of Dark Matter, with a small mass and very weakly interacting Majoron as the Dark Matter candidate (for which we propose to use the name ‘minoron’). The main testable prediction of the model is a new and almost sterile scalar boson that would manifest itself as a narrow resonance in the TeV region. We give a representative range of parameter values consistent with our assumptions and with observation.
The conspicuous absence of any hints of ‘new physics’ at LHC, and, more pertinently,
of supersymmetric partners and exotics (1); (2) has prompted
a search for alternative scenarios beyond the Standard Model (SM)
based on the hypothesis that the SM could survive essentially as is
all the way to the Planck scale, modulo ‘minor’ modifications of the type discussed here,
see (3); (4); (5); (6); (7); (8); (9); (10); (11); (12); (13); (14); (15); (16); (17); (18); (19); (20); (21); (22)
for a (very incomplete) list of
references. In this paper we follow up on a specific proposal along these lines which is
based on our earlier work (6), and demonstrate that this proposal in principle allows
for a comprehensive treatment of all outstanding problems of particle physics proper.
This list includes perturbativity and stability of the model up to the Planck scale and an
explanation of leptogenesis and the nature of Dark Matter
Added motivation for the present investigation comes from very recent LHC results which indicate that the low energy supersymmetry paradigm which has dominated much of particle physics over the past three decades is close to failure, unless one resorts to the more exotic possibility that ‘low energy’ () supersymmetry is broken at a very high scale. In our opinion, however, the latter option would defeat the original purpose of solving the hierarchy problem, and thus lack the plausibility of the original Minimal Supersymmetric Standard Model (MSSM). One crucial question is therefore how the ‘naturalness’ of the electroweak scale can be explained without appealing to supersymmetric cancellations. In this paper we offer one possible such alternative explanation based on (25); another possibility which bears some resemblance to the present scheme as far as physics up to is concerned (but not beyond) is to invoke asymptotic safety, see e.g. (26); (13); (27).
In its original form the model proposed in (6) tried to exploit the fact that, with the exception of the scalar mass term that triggers spontaneous breaking of electroweak symmetry, the SM Lagrangian is classically conformally invariant. For this it relied on the Coleman-Weinberg (CW) mechanism (28) to break electroweak symmetry and to argue that mass scales can be generated purely by the quantum mechanical breaking of classical conformal invariance. In this paper a modified version of this model is presented which has explicit mass terms but which is still conformal in the sense that it postulates the absence of any intermediate scales between 1 TeV and the Planck scale – hence the name Conformal Standard Model. The model nevertheless achieves a stabilization of the electroweak hierarchy thanks to an alternative proposal for the cancellation of quadratic divergences presented in (25), and it is in this sense that we speak of softly broken conformal symmetry (SBCS). This term is meant to comprise three main assumptions, namely: (i) the avoidance of quadratic divergences, (ii) the smallness (w.r.t. the Planck scale) of all dimensionful quantities, and (iii) the smallness of all dimensionless couplings up to . With these assumptions the model is indeed ‘almost conformal’, and the quantum mechanical breaking of conformal invariance (as embodied in the CW correction to the effective potential) remains a small correction over this whole range of energies. Here, we extend our previous considerations towards a more complete picture, in an attempt to arrive at a minimal comprehensive solution to the outstanding problems of particle physics, in a way that remains compatible with all available LHC results. More specifically, we here focus on the question whether the model can offer a viable explanation for leptogenesis and the origin of Dark Matter. The main message of this paper, then, is that indeed these problems can be solved at least in principle within this minimal SBCS scheme. However, as we said, no attempt will be made towards a solution of cosmological constant problem, nor inflation or Dark Energy, as these probably require quantum gravity.
The present paper thus puts together different ideas most of which have already appeared in different forms in the literature (in particular, conformal symmetry, extra sterile scalars and ‘Higgs portals’, low mass heavy neutrinos, resonant leptogenesis, and quantum gravity induced violations of the Goldstone theorem), though, to the best of our knowledge, never in the combination proposed and elaborated here. Let us therefore summarize the distinguishing special features and assumptions underlying the present work:
There are no large intermediate scales between the TeV scale and the Planck mass; in particular there is no grand unification nor GUT scale physics.
There is no low energy supersymmetry; instead the electroweak hierarchy is stabilized by the alternative mechanism for the cancellation of quadratic divergences proposed in (25).
The consistency of the model up to the Planck scale is ensured by demanding absence of Landau poles and of instabilities or meta-stabilities up to that scale. Possible pathologies that might appear if the model is extrapolated beyond that scale are assumed to be taken care of by quantum gravity, hence are not relevant for the present analysis.
The model naturally incorporates resonant leptogenesis (29); (30); (31); (32) with low mass heavy neutrinos, where we show that a range of parameters exists which meets all requirements. Furthermore, the predictions of the model do not in any way affect the SM tests that have so far confirmed the SM as is.
The Majoron, i.e. the Goldstone boson of spontaneously broken lepton number symmetry, is assumed to acquire a small mass eV due to a (still conjectural) folklore theorem according to which there cannot exist unbroken continuous global symmetries in quantum gravity, as a consequence of which it becomes a possible Dark Matter candidate (whose abundance comes out with the right order of magnitude subject to our assumptions). The ensuing violation of the Goldstone Theorem entails calculable couplings to SM particles from radiative corrections, which are naturally very small.
The main testable prediction of the model is a new scalar resonance at or even below that is accompanied by a (in principle measurable) reduction of the decay width of the SM-like Higgs boson. The couplings of the new scalar to SM particles are strongly suppressed in comparison with those of the SM Higgs boson by a factor , where the angle parametrizes the mixing between the SM Higgs boson and the new scalar. The only new fermionic degrees of freedom are three right-chiral neutrinos.
Because our model contains no new scalars that carry charges under SM gauge symmetries it can be easily discriminated against many other models with an enlarged scalar sector, such as two doublet models.
We note that a comprehensive ‘global’ and quantitative analysis of the type performed here would be rather more cumbersome, or even impossible, for more extensive scenarios beyond the SM with more degrees of freedom and more free parameters. For instance, even with a very restricted minimal set of new degrees of freedom and parameters as in the present setup, closer analysis shows that in order to arrive at the desired physical effects such as resonant leptogenesis with the right order of magnitude for the lepton asymmetry a very careful scan over parameter space is required, as the physical results can depend very sensitively on all parameters of the model, so some degree of fine-tuning may be unavoidable.
The structure of this paper is as follows. In section II we describe the basic properties of the model, and explain how to maintain perturbativity and stability up to the Planck scale. Section III is devoted to a detailed discussion of leptogenesis in the CSM, and shows that a viable range of parameters exists for which resonant leptogenesis can work. In section IV we discuss breaking and the possible role and properties of the associated pseudo-Goldstone boson (‘minoron’) as a Dark Matter candidate. Although we present a representative range of parameters consistent with all our assumptions and with observations, we should emphasize that our numerical estimates are still quite preliminary. Of course, these estimates could be much improved if the new scalar were actually found and its mass value measured. For the reader’s convenience we have included an appendix explaining basic properties of neutrino field operators in Weyl spinor formalism.
Ii The CSM Model
The Conformal Standard Model (CSM) is a minimal extension of the Standard Model that incorporates right-chiral neutrinos and an additional complex scalar field, which is charged under SM lepton number, like the right-chiral neutrinos, and generates a Majorana mass term for the right-chiral neutrinos after spontaneous breaking of lepton number symmetry. In keeping with our basic SBCS hypothesis of softly broken conformal symmetry, that is, the absence of large intermediate scales between the TeV scale and the Planck scale , this mass is here assumed to be of . To ensure the stability of the electroweak scale it makes use of a novel mechanism to cancel quadratic divergences (25), relying on the assumed existence of a Planck scale finite theory of quantum gravity, as a consequence of which the cutoff is a physical scale that is not taken to infinity. The phase of the new scalar is a Goldstone boson that within the framework of ordinary quantum field theory remains massless to all orders due to the vanishing anomaly, but will be assumed to acquire a tiny mass by a quantum gravity induced mechanism, as a result of which it acquires also small and calculable non-derivative couplings to SM matter. The viability of the model up to the Planck scale will be ensured by imposing the consistency requirements listed above. In particular, the extra degrees of freedom that the CSM contains beyond the SM are essential for stability: without these extra degrees of freedom the SM does suffer from an instability (or rather, meta-stability) because the running scalar self-coupling becomes negative around (33).
The field content of the model is thus almost the same as for the SM (see e.g. (34); (35); (36) for further details, and (37) for a more recent update). For the fermions we will mostly use SL(2,) Weyl spinors in this paper, together with their complex conjugates , see e.g. (38) for an introduction. The quark and lepton doublets are thus each composed of two SL(2, spinors
where indices label the three families. In addition we have their -singlet partners , and . The new fermions in addition to the ones present in the SM are made up of a family triplet of gauge singlet neutrinos. The scalar sector of the model consists of the usual electroweak scalar doublet and a new gauge-sterile complex singlet scalar , which carries lepton number. This field couples only to the sterile neutrinos and, via the ‘Higgs portal’, to the electroweak doublet .
Apart from the SM-like BRST-exact terms required for gauge fixing (34), the CSM Lagrangian takes the form
with gauge invariant kinetic terms
where we only display the kinetic term of the Higgs doublet and the kinetic terms of the new fields, while takes the standard form that can be found in any textbook, see (34); (35); (36). The scalar potential reads
Exploiting the symmetries of the action we assume that the vacuum expectation
values take the form
with non-negative and . Clearly, we are interested in a situation in which both the electroweak symmetry and lepton number symmetry are broken, and therefore we assume that and are non-zero. The values (4) correspond to the stationary point of (3), provided that the mass parameters are chosen as follows
The tree-level potential (3) is bounded from below provided that the quartic couplings obey
If, in addition to (5), , then (4) is the global minimum of . The physical spin-zero particles are then
two CP-even scalars and , and one CP-odd scalar .
The two heavy scalar bosons are thus described as mixtures of the two real scalar fields with non-vanishing vacuum expectation values (, ),
with masses and . The angle thus measures the mixing between the SM Higgs boson and the new scalar. In order not to be in conflict with existing data the angle must obviously be chosen small, and furthermore such that can be identified with the observed SM-like Higgs boson with (39). Introducing the tree-level SM quartic coupling
one can conveniently parametrize the tree level values of unknown parameters , and in terms of the five parameters as follows
As a consequence, the model predicts the appearance of a ‘heavy brother’ of the usual Higgs boson, which would manifest itself as a narrow resonance in or below the TeV region (see Table 1 below; the narrowness of the resonance is due to the small mixing and the relatively large scale). The SM-like Higgs boson can, in principle, decay into a pair of pseudo-Goldstone bosons. The corresponding branching ratios are, however, very small (for all exemplary points in Table 1 they do not exceed ). Thus, the decay width of is decreased with respect to the SM value by a factor . In the numerical analysis we assume that ; we note that available LHC data leave enough room for such a modification of SM physics (40); (41).
The Yukawa couplings are given by
with the usual Yukawa matrices and of the SM, where is the antisymmetric metric. The matrix mediates the coupling of the SM fields to the sterile neutrino components, while the complex symmetric matrix describes the interactions of the latter with the new scalar . By fermionic field redefinitions that preserve Eq. (2), one can assume that , and are diagonal and non-negative, and that differs from a positive and diagonal by the (inverse of) unitary CKM matrix . In more customary notation the fermionic fields are described by 4-component Dirac spinors of charged leptons and up-type quarks
together with the analogous 4-spinor field
for the down quarks, with a -induced rotation
of the upper components (34).
After spontaneous symmetry breaking the neutrino mass terms are
Introducing unitary matrices and and the block matrix
with the submatrices
one has , and
with complex symmetric matrices
Observe that up to the matrix achieves the diagonalization of the matrix in (15) into the two blocks of matrices exhibited above, but that the latter are not necessarily in diagonal form yet. Employing the Casas-Ibarra parametrization (46) of the Dirac mass matrix (or equivalently the Yukawa matrix ; as explained above, can be assumed positive diagonal)
with the unitary Maki-Nakagawa-Sakata matrix (see (46) and references therein) and a complex orthogonal Casas-Ibarra matrix () one has
which shows that are light neutrino masses at the tree level. The main advantage of the Casas-Ibarra parametrization, and the reason we use it here, is that it provides a clear separation of the parameters of into the ones that are relevant for neutrino oscillations, namely and the CKM-like unitary matrix , and the ones describing heavy neutrinos and their properties ().
The matrix is now chosen such as to make a positive diagonal matrix (note that differs from , cf. (19)!). The matrix in Eq. (17) is then diagonal provided that ; however, we will be mainly interested in light neutrino states that participate in specific fast interactions during the leptogenesis, i.e. that are approximate eigenstates of weak interactions. Therefore, we take , and change the basis in the field space so that in Eq. (17) is a new mass matrix (in other words, we are using interaction eigenstates rather than mass eigenstates for the light neutrinos). Henceforth, and denote neutrino fields in this new basis, unless stated otherwise, and are referred to as light and heavy neutrinos. It is sometimes convenient to assemble these 2-component Weyl spinors into Majorana 4-spinors
Note that, as a result of the rotation with , the (new) ’s do couple to the massive gauge bosons already at the tree-level
where for clarity we also give the result in standard 4-spinor notation, and where are the usual chiral projectors. The matrices follow immediately from Eq. (16)
To avoid confusion we also use calligraphic letters to denote the couplings between the new fields and the scalars , to wit,
where the leading terms in read
(as said, ). Because a main postulate behind the CSM is the presumed absence of any intermediate scales between the electroweak scale and the Planck scale , the scale of lepton number symmetry breaking is assumed to lie in the range. With , the masses of heavy neutrinos are relatively small, and the light neutrino data (39) indicate that is of order . To allow for baryon number generation despite the low masses of heavy neutrinos, the mechanism of ‘resonant leptogenesis’ was proposed and explored in (29); (30); (31); (32). This mechanism is based on the observation that CP-violation (a crucial ingredient in dynamically generated baryon asymmetry (47)) is enhanced whenever the masses of heavy neutrinos are approximately degenerate. Accordingly, we assume that the Yukawa Majorana matrix is in fact proportional to the unit matrix, that is,
with . Consequently there is an approximate SO(3) symmetry in the heavy neutrino sector, which is only very weakly broken by the Yukawa couplings . For definiteness, we assume Eq. (28) to hold at the electroweak scale, for the renormalization scale . In turn, the mass splitting of heavy neutrinos is entirely due to the seesaw mechanism, Eq. (19). (As emphasized in (48) the SO(3) symmetry ensures that (28) is stable against quantum corrections in a good approximation; nonetheless, when (28) holds instead at high RG scale , then -induced RG-splitting of ’s yields splitting of heavy neutrino masses that is of similar order as the seesaw one, see e.g. (49)). It should be stressed here that, due to the degeneracy (28), the matrix in Eq. (17) is clearly not an perturbation of the identity matrix; this is technically similar to (though physically different from) the Dashen’s vacuum realignment condition (50) (see also (34)).
ii.2 Cancelling Quadratic Divergences
We stress again the presence of explicit scalar mass terms in (3), in contrast to the original model of (6) which relied on the CW mechanism (28) to break electroweak symmetry. Our main reason for this is that the CW mechanism does not eliminate quadratic divergences, and thus the low energy theory would remain sensitive to Planck scale corrections.
At one loop the coefficients of the quadratic divergences for the two scalar fields are (25)
Here and are the gauge couplings, while is the top quark Yukawa coupling. For simplicity (and without much loss in precision) we neglect all other Yukawa couplings. Note that Eqs. (II.2) are independent of the details of the cutoff regularization, as long as the regulator (here assumed to be provided by the quantum theory of gravity) acts in the same way on all fields. Of course, another crucial assumption here is that we can neglect contributions of graviton loops to (II.2); this assumption is based on the hypothesis that the UV finite theory of quantum gravity effectively screens these contributions from low energy physics.
An obvious question at this point is the following. One would at first think that
Eqs. (II.2) depend on the renormalization scale via the RG running of
the couplings, a well-known issue in the context of Veltman’s conditions (51).
This is, however, only apparent, since when all higher corrections are included,
the functions obey appropriate renormalization group equations, in such a way
that the implicit scale dependence is exactly canceled by the explicit presence of introduced by higher loop corrections. Therefore, the all-order coefficients are in fact
-independent (and -independent) functions of the bare couplings (which
themselves depend on the cutoff , as the latter is varied). Thus, the couplings appearing on the
right-hand-side of (II.2) are etc., rather than
see also (55) for a discussion of the issues appearing in cutoff regularized gauge theories.
The appearance of bare couplings in (II.2) can also be motivated and understood from the point of view of constructive QFT (see e.g. (56)), although we are, of course, aware that there is no rigorous construction of the SM. There one attempts to rigorously construct a functional measure for interacting QFTs. This requires the introduction of both UV and IR (i.e. finite volume) regulators. For the regularized theory one then introduces counterterms as functions of the bare parameters and tries to adjust the latter as functions of the UV cutoff in such a way that the theory gives well defined physical answers in the limit (in which the bare couplings usually assume singular values). In particular, for a given value of the cutoff one can thus impose the vanishing of the coefficient of the quadratic divergence as a single condition on the bare parameters. In that framework running couplings play no role; they are merely an auxiliary device to conveniently parametrize the scale dependence of correlation functions.
In summary, the coefficients of quadratic divergences (II.2) are calculable functions of the cutoff scale , provided that all low energy parameters etc. are fixed by experiment. To determine the evolution of the couplings from up to (where they are identified with the bare couplings) in the leading logarithmic (LL) approximation we need only the one-loop beta functions (25) (we use the notation ; furthermore we make use of (28))
which show in particular how the gauge coupling affects the evolution of so no Landau pole develops for . This effect is also seen in the other expressions where bosonic and fermionic contributions balance each other in such a way that the theory remains perturbatively under control up to (with appropriate initial values).
At this point it should be stressed that all the ingredients necessary to find the coefficients with resummed next-to-leading logarithms are at our disposal. In particular, the two-loop beta functions in -MS together with the two-loop coefficients in a generic renormalizable model are given in (55) (one can also find there the generic one-loop relation between renormalized parameters in -MS and their counterparts in the conventional scheme of dimensional regularization). However, as most of the parameters of CSM are still unknown, we are content here with resummation of the leading logarithms only. The rationale behind this restriction, is that the one-loop RG evolution in gauge-Yukawa sector is independent of quartic scalar couplings, which significantly simplifies the scan over the parameter space; in particular and gauge couplings at the Planck scale are known. Recall that the one-loop beta functions reflect the structure of non-local terms in one-particle-irreducible effective action , and thus are universal across different regularizations (at least in the class of mass independent renormalization schemes (57), to which -MS belongs). Therefore the RG-improved coefficients (II.2) at the LL order are independent of the details of cutoff regularization as well.
We note that the cutoff dependence of coefficients of quadratic divergences in the pure SM was already analyzed in (58) where it was found that they cancel for , and thus (logarithmically speaking) not so far from the Planck scale. This observation motivated our proposal that the vanishing of quadratic divergences at the Planck scale, and thus stabilization of the electroweak scale, may be achieved by means of a ‘small’ modification of the SM like the one proposed here. Ultimately, the cancellation of quadratic divergences would be due to a still unknown quantum gravity induced mechanism which is different from low energy supersymmetry (but which could still involve Planck scale supersymmetry in an essential way).
From the perspective of effective field theory (EFT), valid for energies , we have a clear distinction between low energy () supersymmetry and the present proposal. In supersymmetric models, the underlying mechanism of quantum gravity appears via the (super)symmetry of the EFT itself, and thus the cancellation holds independently of the value of the cutoff
By contrast, in the present context the absence of quadratic divergences (and thus the stabilization of the electroweak scale) manifests itself via the existence of a distinguished value of the cutoff (close to the Planck scale) such that . Importantly, the question whether or not such a scale exists for which both coefficients (II.2) vanish, can in principle be answered provided that all CSM parameters can be measured with sufficient accuracy.
We therefore assume that such a distinguished value close to exists, so we can impose the conditions
on the running couplings with equal to the (reduced) Planck scale; from a
low-energy perspective these can be considered as an RG-improved version
of Veltman’s conditions (51). Disregarding
the other SM couplings this condition restricts the four-dimensional space of parameters
, cf. Eqs. (8), to a two-dimensional
ii.3 Stability of electroweak vacuum
One of the very few ‘weak spots’ of the pure SM is the meta-stability of the electroweak vacuum (33). Namely, the effective potential of the SM (with appropriately resummed large logarithms) develops a new deeper minimum for , thus implying an instability of the electroweak vacuum via quantum mechanical tunneling. This can be seen also more heuristically, by following the RG evolution of the scalar self-coupling and noticing that for the function dips below zero due to the large negative contribution from the top quark (33) (but becomes positive again for yet larger values of ). For values of the fields that are much larger than the electroweak scale, the full effective potential of the SM is well approximated by the quartic term
However, here one cannot simply substitute the self-coupling at the electroweak scale; rather, in order to avoid huge logarithmic corrections on the right-hand-side, the correct value of the quartic coupling in the above formula is obtained by substituting the running coupling evaluated at the appropriate energy scale of the order of , i.e.
rather than . For theories like the SM, in which the effective potential depends only on a single field (up to the orbits of symmetry group), this somewhat heuristic reasoning can be put on firmer grounds, by resumming large logarithms via the renormalization group improvement (52); (53); (54).
For the CSM there are now two scalar fields (up to symmetries of ) and the situation is more complicated, basically because with more than one scalar field, the RG-improvement cannot simultaneously determine the resummation of logarithms in all directions in field space. For this reason we have to rely on the more heuristic argument, by demanding that the positivity conditions (5) be satisfied not only at the electroweak scale , but also for the running couplings at all intermediate scales . This provides a strong indication that the electroweak vacuum (4) in the CSM remains the global minimum of the full effective potential, at least in the region , , in which EFT is valid. Thus, following the RG evolution from the Planck scale, where the conditions (33) are imposed, down to the electroweak scale we impose the inequalities
in addition to the conditions enunciated at the end of the foregoing subsection. These extra stability conditions lead to further restrictions on the parameters. It is therefore a non-trivial fact that parameter ranges exist which satisfy all these conditions and restrictions.
A set of exemplary points consistent with all our restrictions is given in Table 1. denote decay width of the Higgs particle and its ‘heavy brother’ . Br([SM]) is the branching ratio for SM-like decay channels of , while non-SM-like decay channels of are negligible for all points in the Table. denotes the current baryon number density to entropy density ratio calculated on assumptions specified in Sec. III. In particular, the Table displays a viable range of mass values for both the new scalar and the heavy neutrinos. For all points the heavy neutrinos are heavier than the new scalar field , and thus their decays are the main source of lepton asymmetry. Note also the relatively large values of which are necessary for successful leptogenesis. This comes about because must remain sufficiently small so as to allow for the departure of heavy neutrinos from thermal equilibrium, while their masses should be large enough so that the departure takes place when baryon-number violating processes are still fast. Importantly, the values of dimensionless couplings corresponding to all points in the Table are small while masses of new states are comparable to the electroweak scale; thus one can trust that radiative corrections to the tree-level masses etc. are small.
Iii Resonant Leptogenesis
By assumption the lepton number symmetry of the CSM is spontaneously
broken by the non-vanishing vacuum expectation value .
The proper quantity to study is therefore the lepton number density of the SM
under which heavy neutrinos have vanishing charges.
(where denotes the number density of photons)
is produced by non-perturbative SM interactions that break baryon and lepton number
symmetries down to the non-anomalous combination , and generate baryons
from non-vanishing lepton number density via the usual sphaleron mechanism (63).
Thus the problem can be reduced to that of explaining the lepton asymmetry ,
which itself is produced in lepton number and CP violating out-of-equilibrium
decays of heavy neutrinos, as they occur in the CSM. In this way
all the Sakharov conditions (47) can be satisfied.
To achieve the correct order of CP-violation despite small values, we rely on the mechanism of “resonant leptogenesis” (29); (30); (31); (32), which can be naturally realized within the present scheme as a consequence of the assumed degeneracy of the Yukawa matrix , cf. Eq. (28). The baryon number density can then be calculated by solving the relevant Boltzmann equations (see e.g. (64)).
The CP-asymmetries relevant for calculation of are
together with their counterparts with additional scalars (or -bosons and charged leptons)
in the final states. The tree-level contributions to the decay widths in the formulae above
follow immediately from the vertices in Eqs. (27)
In fact, calculation of ‘external’ line corrections (especially in the resonant regime) requires some care, since the incoming states correspond to unstable particles. In (66), see also (29); (67); (68), the CP-asymmetry
calculated without any references to the external lines of unstable states. Instead, the amplitudes of associated scattering processes in which unstable heavy neutrinos appear only as internal lines were studied; the resulting prescription for can be summarized as follows (66). Consider the interaction (27) between a hermitian scalar field , heavy neutrinos (described in terms of Majorana fields ), and (approximately) massless SM (anti)neutrinos, described in terms of Majorana fields . Suppose that the matrix of propagators of heavy neutrino Majorana fields has
the following form ( is the charge-conjugation matrix)
where the matrix of pole masses
is diagonal with positive real parts , while its imaginary part gives the total decay widths. The residue matrices can be written as
with 33 matrices carrying only family indices, and chiral projections ; clearly, at tree level, in the basis of mass eigenstates one has . If these matrices are known, the CP-asymmetry (38) can then be calculated with the aid of the following formula (66)
where the ellipses indicate contributions of corrections to external lines of and fields, as well as loop corrections to the 1PI vertices (which are negligible in -scale leptogenesis). If heavy neutrinos were stable, the matrix would be the complex conjugate of . In that case Eqs. (44)-(45) are nothing more than the ordinary LSZ-reduction rules for calculating the S-matrix elements, see e.g. (69). Similarly, the CP-asymmetry can be calculated with the aid of Eq. (43), with the following replacements
(the change of chirality is caused by ). The enhancement effect that underlies resonant leptogenesis is due to the matrices which contain the factors etc. (see below).
To find the matrices we use the prescription given in (70), to which we also refer for further details. Adopting some renormalization scheme, let be the matrix of renormalized 1PI two-point functions (inverse propagators) of the Majorana fields
where matrices and carry only family indices. Now let be the following matrix (with )
Then the propagator of has the form (40) where the (complex) pole masses are solutions to