hep-ph/0411030

Testing the see-saw mechanism at collider energies

S. F. King and T. Yanagida

School of Physics and Astronomy, University of Southampton,

Southampton, SO17 1BJ, U.K

Department of Physics, University of Tokyo,

Tokyo 113-0033, Japan

Abstract

We propose a low energy extension of the Standard Model consisting of an additional gauged plus three right-handed neutrinos. The lightest right-handed neutrinos have TeV scale masses and may be produced at colliders via their couplings to the gauge boson whose mass and gauge coupling is constrained by the out-of-equilibrium condition leading to upper bounds on the right-handed neutrino and production cross-sections at colliders. We propose a brane-world scenario which motivates such TeV mass right-handed neutrinos. Our analysis opens up the possibility that the mechanism responsible for neutrino mass is testable at colliders such as the LHC or VLHC.

December 9, 2020

## 1 Introduction

The see-saw mechanism , is an attractive mechanism for accounting for light neutrino masses. The mechanism works by introducing right-handed neutrinos with “large” Majorana masses, which violate lepton number , and the Yukawa couplings between the left-handed leptons and right-handed neutrinos then results in small effective Majorana mass operators for left-handed neutrinos. Such a scenario has potentially important cosmological implications for the baryon asymmetry of the universe via a mechanism known as leptogenesis . It is clear that the see-saw mechanism satisfies the Sakharov conditions of lepton number violation, and CP violation due to the complex neutrino Yukawa couplings. Providing the out-of-equilibrium condition is also met when the right-handed neutrinos decay, a net lepton number may then be generated which may subsequently be converted into a net baryon number by sphaleron interactions which preserve .

Given the recent progress in neutrino physics, there has been much discussion concerning both the see-saw mechanism  and leptogenesis . In the simplest implementations of thermal leptogenesis the re-heat temperature after inflation must be in excess of the lower bound arising from gravitino production . 111In the type II see-saw mechanism this conflict can be somewhat ameliorated . Typically thermal leptogenesis works best when the lightest right-handed neutrino mass exceeds about GeV, while in order to avoid excessive thermal production of gravitinos the temperature of the universe after inflation must not exceed this value. Even ignoring the gravitino problem, such large values of right-handed neutrino masses make them inaccessible to experiment at planned or even imagined collider energies. Testing the see-saw mechanism experimentally in any direct way therefore seems virtually impossible under the thermal leptogenesis framework.

A possible solution to the gravitino problem is provided by the idea of resonant leptogenesis . The key observation of resonant leptogenesis is that, if the lightest two right-handed neutrinos are closely degenerate, then resonance effects can enhance the production of lepton number even for a right-handed neutrino mass scale as low as a TeV, allowing the reheat temperature to be low enough to avoid the gravitino problem. 222If supersymmetry (SUSY) is additionally assumed then TeV mass right-handed neutrinos allow the possibility that the soft SUSY breaking mass parameters may be at least partly responsible for leptogenesis and radiatively generated neutrino masses . The present analysis can be extended for this case.

With such light right-handed neutrinos one may think it possible to test the see-saw mechanism experimentally at collider energies. However in the absence of additional interactions this is not the case since, even if TeV mass right-handed neutrinos are kinematically accessible to high energy colliders such as the LHC, their Yukawa couplings are necessarily so weak as to render their production cross-section unobservable.

In this paper we propose a minimal scenario in which, in addition to having light right-handed neutrinos, there is also a low energy gauged symmetry . 333In general the extra could be some linear combination of and hypercharge . The vector space spanned by these generators includes the third generator of , , and the generator contained in the maximal subgroup . But here we focus on the case of an extra for simplicity and definiteness. It is natural that the scale of gauged symmetry breaking should be somewhat higher than the heaviest right-handed neutrino mass, since gauged symmetry forbids Majorana neutrino masses. Cancellation of gauged anomalies requires that there should be three right-handed neutrinos. If we assume the lightest pair of right-handed neutrinos to be degenerate, we would therefore expect that the third right-handed neutrino mass to be closer to the mass scale of symmetry breaking , which is somewhat higher than the lightest right-handed neutrinos mass scale . We are therefore led to propose a new low energy extension of the Standard Model consisting of an additional gauged plus three right-handed neutrinos with the following new mass scales: , with a possible accurate degeneracy involving the lightest right-handed neutrino pair.

The main goal of this paper is to discuss the collider phenomenology of light right-handed neutrinos and gauge bosons, constrained by the out-of-equilibrium conditions as required by thermal leptogenesis. 444Note that above its breaking scale the presence of exact gauged means that the net of the universe must be exactly zero. Since is violated by sphaleron interactions this implies that baryogenesis or leptogenesis cannot occur above the scale . We shall require the effective leptogenesis neutrino mass parameter to be eV for efficient production and out-of-equibrium decay of the right-handed neutrinos . 555Our scenario should not be confused with a recently discussed scenario in which low energy leptogenesis occurs due to scattering from domain walls produced by the breaking of a discrete left-right symmetry, and the right-handed neutrinos are constrained not to erase the . The requirement that the lightest right-handed neutrino pair be out-of-equilibrium when they decay implies that the interactions must be sufficiently weak, leading to a lower bound on or an upper bound on the gauge coupling , depending on the mass ordering of and . These bounds in turn leads to upper limits on the production cross-section for right-handed neutrinos and gauge bosons at colliders, which we shall also discuss. We shall also explore the theoretical motivation for TeV mass right-handed neutrinos, and propose a specific brane-world scenario.

## 2 Out-of-equilibrium Condition

### 2.1 MB−L≫M1

We shall begin by assuming that , and show that the out-of-equilibrium condition leads to a lower bound on in this case.

The lightest right-handed neutrinos must decay while they are out-of-equilibrium in the early Universe. We have already assumed that the direct decays of right-handed neutrinos are always out-of-equilibrium, consistent with eV, so that we only need additionally ensure that the new interactions do not bring the lightest right-handed neutrinos back into thermal equilibrium. 666Note that the new interactions do not violate lepton number by themselves. However since the right-handed neutrinos are Majorana particles, such interactions (with unsuppressed L-violating mass insertions) would bring the right-handed neutrinos in the thermal equibrium with vanishing chemical potential unless they are out of equilibrium. The reaction rate of right-handed neutrinos is given by:

 Γ=<σannnv> (1)

where is the total annihilation cross-section of lightest right-handed neutrinos into three families of Standard Model fermions () and antifermions (), and is the number density of right-handed neutrinos. 777Note that the annihilation cross-section is the most relevant one for satisfying the out-of-equilibrium condition. We have also assumed that scalar fermions are heavier than the right-handed neutrinos and . The annihilation cross-section is given by:

 σann=σ(N1N1→∑ff¯f) (2)

due to the tree-level exchange of a gauge boson of mass with a gauge coupling ,

 σ(N1N1→∑ff¯f)∼3×133g4B−L48πE2M4B−L∼14πE2v4B−L (3)

where the mass of the gauge boson is given by . The number density of right-handed neutrinos given by:

 n=342.4042π2g(kTℏc)3∼1π2T3 (4)

setting .

To generate lepton number asymmetry, the right-handed neutrinos must be out-of-equilibrium when they decay. If they decay at a temperature then their reaction rate when they decay is given from Eqs.(1,3,4) by

 Γ=<σannnv>∼14πE2v4B−L.1π2T3∼14π3M51v4B−L. (5)

To be out-of-equilibrium the reaction rate be less than the Hubble constant whose square at a temperature is given by

 H2≈8π3GNρ∼4π345g∗T4M2P∼3g∗T4M2P (6)

setting where is the total number of degrees of freedom at the temperature and is the Planck mass GeV. The out-of-equilibrium condition is given by at , where , which from Eqs.(5,6) leads to the lower bound on :

 vB−L>(MP4π3a√3g∗)1/4M3/41∼106 GeV(M11 TeV)3/4. (7)

Eq.(7) tells us that for a degenerate pair of right-handed neutrinos of mass the scale of breaking must exceed . 888 A similar constraint was considered in left-right symmetric models .

### 2.2 MB−L\vbox<\nointerlineskip∼2M1

We now consider the case that , and show that the out-of-equilibrium condition leads to an upper bound on in this case.

In this case the estimate in Eq.(3) becomes

 σ(N1N1→∑ff¯f)∼3×133g4B−L48π116E2∼14πg4B−L16E2. (8)

The resulting reaction rate at in Eq.(5) becomes modified to

 Γ=<σannnv>∼14π3(gB−L2)4M1. (9)

The out-of-equilibrium condition is given by at , where , which from Eqs.(9,6) leads to the upper bound on :

 gB−L<2×[4π3a√3g∗(M1MP)]1/4∼2×10−3(M11 TeV)1/4. (10)

Other reactions are important at for the case , for example real pair production and annihilation: whose cross-section will be of similar magnitude to that in Eq.(8).

For at it becomes possible to have single production and decay: (including ). This special case leads to a quite different bound on the gauge coupling since the decay rate of is given by:

 Γ(ZB−L→∑ff¯f)∼3×133g2B−L48πM1∼14πg2B−LM1. (11)

The out-of-equilibrium condition in this case is obtained by comparing the decay rate in Eq.(11) to the Hubble expansion rate at , where , which from Eqs.(11,6) leads to the upper bound on :

 gB−L<√4πa(3g∗)1/4(M1MP)1/2∼10−6(M11 TeV)1/2. (12)

In this special case when the bound on the gauge coupling in Eq.(12) is much more stringent than the bound on the gauge coupling in Eq.(10) for the more general case .

Finally we should note that the upper bound on is obtained even for the case, . For the inverse decay has a Boltzmann suppression with . The Boltzmann factor is only for and hence we may have still a strigent bound on but weaker than the case . So the bound on is given when since the Boltzmann factor is and the inverse decay of becomes negligible compared with the gauge exchanges.

## 3 Phenomenology

For the case of a heavy , , the typical cross-section for production of the lightest right-handed neutrinos at colliders is given from Eq.(3) and bounded from Eq.(7) by

 σ(e+e−→N1N1)<148πE21024 GeV4(1 TeVM1)3∼3×10−9 fb(E1 TeV)2(1 TeVM1)3 (13)

where we have used the conversion . Unfortunately the cross-section may be too small to enable right-handed neutrinos to be produced at TeV energies such as the LHC or CLIC. For efficient production of right-handed neutrinos a collider with an energy approaching the mass of the gauge boson would be required, such as the VLHC for example. Although the right-handed neutrinos are produced in pairs via their couplings, they will decay singly via their small Yukawa couplings into either (left-handed) neutrino plus Higgs , or charged lepton plus (longitudinal) , with a characteristic signature in both cases. The expected decay rate for TeV mass right-handed neutrinos with a Yukawa coupling about would be GeV corresponding to a lifetime of about s.

Turning to the other possibility of a lighter , with , from Eq.(10) we see that the gauge coupling must be smaller than about for TeV scale right-handed neutrinos. This also leads to a cross-section for right-handed neutrinos at colliders bounded by

 σ(e+e−→N1N1)<4×10−7 fb(1 TeVE)2(M11 TeV). (14)

Unlike the cross-section in Eq.(13), the cross-section in Eq.(14) is suppressed at higher energies. However, in this case one might hope to produce real on-shell with mass below the TeV scale. The may be produced singly by mechanisms which involve only two powers of in the rate, and so the typical production cross-sections are expected to be enhanced relative to that in Eq.(14) by a factor of about a million. For example the may be produced singly by the Drell-Yan process, and may be discovered via its decays into , , or jet-jet. The discovery limits for such a at the LHC have been studied by ATLAS, and for example a TeV mass may be discovered at the 5 level with 100 providing the square of the gauge coupling is about . With the small gauge coupling bounded in Eq.(10), discovery at the LHC will be clearly difficult, but since this bound is independent of the mass of the , which may be as light as the ordinary for example, its discovery may be possible in principle at the LHC. Note that the couplings of the gauge boson to leptons and quarks are 1:1/3, which may be tested if its decay is observed.

## 4 A Brane Scenario

In this section we address the following questions:

1. Why are the Yukawa couplings inducing the Dirac neutrino masses so small ? To get neutrino masses of order eV with 1 TeV right-handed Majorana mass we need GeV. Thus the Yukawa coupling should be .

2. Is there a natural inflation model with reheat temperature a few TeV ? In other words, why do we assume 1 TeV in the estimates above? The constraint from the gravitino problem tells us only GeV and hence could be as large as GeV.

The answer to the second question is straighforward: lower-energy scale inflation may produce lower reheating temperature, in general. The new inflation model is a candidate for the low-energy scale inflation. One of us has constructed a new inflation model in SUGRA and found that the reheating temperature is given by few TeV .

Moreover, the low-energy scale inflation is favored in string landscape. The string theory may have a number of vacuua which may have many inflaton candidates. The vacuum which has many inflations is favored, since many inflations make larger universe. So it is better to have as lower-energy scale inflation as possible and it is likely that a new inflation is the last inflation we can see. 999T.Y. thanks K.-I. Izawa for the discussion of inflation models on the string landscape.

The answer to the first question is not difficult also. Small right-handed neutrino masses at the TeV scale which require small Dirac Yukawa couplings in the range . To account for this we propose the following extra-dimensional set up consisting of two parallel 3-branes where the standard model fermions and Higgs are on one brane, and the right-handed neutrinos are on the other brane, and the gauge multiplet is in the bulk which contains extra dimensions. The gauge interaction has 4 dimensional anomaly on the each branes, but it is cancelled by bulk Chern-Simons term . The is broken by a Higgs vacuum expectation value located on one of the 3-branes. The Yukawa coupling constant in this case is exponentially suppressed where is the cut-off (string) scale and is the compactification scale of the extra dimension(s). Assuming can account for Dirac Yukawa couplings in the range .

In such a scenario the gauge coupling of the is given by

 gB−L∼g0(M∗L)−n/2 (15)

Assuming and we find for which implies the gauge boson mass TeV for GeV. For n=6 we find , which is in the appropriate range for the light scenario

## 5 Conclusion

We have proposed a low energy extension of the Standard Model consisting of an additional gauged plus three right-handed neutrinos where the lightest right-handed neutrino mass scale is TeV. In the absence of SUSY, this requires resonant leptogenesis with the lightest right-handed neutrino pair being approximately degenerate.

We have discussed the collider phenomenology of light right-handed neutrinos and gauge bosons, constrained by the out-of-equilibrium conditions. We find that for there is a lower limit on the symmetry breaking scale , while for there is an upper limit on the gauge coupling . Although the TeV mass right-handed neutrinos may be produced at colliders via their couplings to the gauge bosons, the above limits severely constain the production cross-sections of both right-handed neutrinos and gauge bosons at colliders.

We have also considered the theoretical motivation for TeV scale right-handed neutrinos coming from brane-world set-ups and string theory. We have proposed a particular brane-world scenario in which small Yukawa couplings emerge, with a gauge coupling and mass in the appropriate ranges consistent with the bounds from thermal leptogenesis, depending on the number of extra dimensions. For two extra dimensions the mass of the gauge boson is expected to have a mass TeV. The cross-section for production of right-handed neutrinos is expected to become observable in this case when the centre of mass energy of the collider approaches , which motivates a future collider such as the VLHC. For six extra dimensions the gauge boson could be as light as the ordinary boson with a cross-sections for production that will make its discovery at the LHC challenging.

In conclusion, the possibility of TeV mass right-handed neutrinos, together with additional , is cosmologically consistent from the point of view of leptogenesis, and has some theoretical motivation from string theory and extra dimensions. It would open up the possibility of testing the mechanism responsible for neutrino mass experimentally at collider energies corresponding to the LHC or a future VLHC.

Acknowledgments

We would like to thank the organisers of the Nobel Symposium 129 on Neutrino Physics where this work was begun. S.K. thanks Stefan Antusch for useful comments.
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