# Viable Supersymmetry and Leptogenesis with Anomaly Mediation

###### Abstract

The seesaw mechanism that explains the small neutrino masses comes naturally with supersymmetric (SUSY) grand unification and leptogenesis. However, the framework suffers from the SUSY flavor and CP problems, and has a severe cosmological gravitino problem. We propose anomaly mediation as a simple solution to all these problems, which is viable once supplemented by the -terms for and . Even though the right-handed neutrino mass explicitly breaks and hence reintroduces the flavor problem, we show that it lacks the logarithmic enhancement and poses no threat to the framework. The thermal leptogenesis is then made easily consistent with the gravitino constraint.

^{†}

^{†}thanks: On leave of absence from Department of Physics, University of California, Berkeley, CA 94720.

The past several years have seen revolutionary progress in neutrino physics. The atmospheric neutrinos showed the first convincing evidence for neutrino oscillation in the SuperKamiokande experiment Fukuda:1998mi . The long-standing solar neutrino problem since 1960s was shown to be due to the neutrino flavor conversion by the SNO experiment Ahmad:2002jz , and all possibilities other than neutrino oscillation have been excluded by the KamLAND experiment Eguchi:2002dm . All these experiments suggest finite but extremely small neutrino mass in the sub-electronvolt range, more than a million times smaller than the smallest particle mass known before, namely the electron mass.

The standard framework to understand the newly discovered neutrino
masses and mixings is the seesaw mechanism seesaw , which comes
naturally with the supersymmetric (SUSY) grand-unified theories (GUT).
Here, SUSY plays a dual role: it stabilizes the hierarchy and makes
the gauge coupling constants unify. Furthermore, the seesaw mechanism
predicts new heavy particles, right-handed neutrinos, whose decay can
potentially produce the baryon asymmetry of the universe
Fukugita:1986hr . This possibility is called leptogenesis,
which requires GeV to generate the observed
baryon asymmetry Giudice:2003jh ^{1}^{1}1A high degree of
degeneracy among right-handed neutrinos may resonantly enhance the
baryon asymmetry and may allow for lighter right-handed neutrinos
Pilaftsis:2003gt .. The combination of seesaw, SUSY GUT, and
leptogenesis is further supplemented by the possibility of SUSY dark
matter, which predicts the cosmic abundance of the dark matter
particle in the right ballpark.

There are, however, severe problems with this attractive framework. SUSY tends to induce unacceptably large flavor-changing and CP-violating effects; SUSY flavor and CP problems gabbiani . Generic SUSY parameters imply a lower limit on the masses in excess of 100 TeV, making supersymmetry not a viable mechanism to stabilize the electroweak scale GeV. It is customary to make an ad hoc assumption of the universal scalar mass at the GUT- or Planck-scale to avoid the flavor problem. However, the rates of Lepton Flavor Violation (LFV) processes are typically predicted to be too high in SUSY-GUT models based on the seesaw mechanism and flavor symmetries even with the universal scalar mass Sato:2000zh . At the same time, SUSY predicts the existence of the gravitino, the superpartner of the graviton. Once it is produced in early universe, it decays only by the gravitational interaction and hence slowly, upsetting the success of BBN Pagels:ke . The gravitino yield is larger for higher reheating temperatures. A detailed analysis including the hadronic decay of gravitino shows a very tight upper limit GeV Kawasaki:2004yh for –1 TeV as commonly assumed in the literature. Therefore the leptogenesis appears incompatible with the gravitino problem.

There had been suggestions to achieve leptogenesis at a relatively low temperature (for a compilation of proposals, see Davidson:2002qv ). In particular, coherent oscillation of right-handed scalar neutrino Murayama:1993em had been considered a natural possibility, and it may even be the inflaton Murayama:1992ua . However, this proposal requires the reheating temperature to be above GeV Hamaguchi:2001gw which is still in conflict with the gravitino problem. More recent suggestions include the gravitino LSP Fujii:2003nr , but even this case is getting tightly constrained.

In this Letter, we revisit these problems and propose a simple solution: anomaly-mediated supersymmetry breaking Randall:1998uk ; Giudice:1998xp . It predicts a heavy gravitino that decays before BBN and makes leptogenesis viable. It solves the flavor and CP problems automatically if supplemented by the UV-insensitive -terms Arkani-Hamed:2000xj . Even though the strict version of the UV-insensitive anomaly mediation requires conservation, it was pointed out that the seesaw mechanism can be used with minimal flavor-changing effects Arkani-Hamed:2000xj . However, the details of reintroduced flavor-changing effects had not been worked out. We present consequences of the seesaw mechanism on LFV. We find that the LFV effects lack logarithmic enhancements unlike in conventional supergravity-based scenarios and hence are easily compatible with current limits. Therefore this framework preserves all virtues of anomaly mediation to solve the SUSY flavor and CP problems and makes the thermal leptogenesis viable. On the other hand, the small LFV may lead to a signature observable in the near future.

Let us set up notations to discuss seesaw mechanism with SUSY GUT. The relevant part of the superpotential is

(1) |

We use the basis where the right-handed neutrino masses are diagonal and real positive. The light neutrino mass is then obtained by integrating out , . Therefore the light neutrino masses are suppressed relative to the other quark and lepton masses by the inverse power of . In order to obtain the heaviest mass eV, we find GeV, which can be induced from the grand-unification scale GeV.

How do we solve the SUSY flavor and CP problems, while make the leptogenesis consistent with the gravitino problem? There is a promising mechanism to make the whole framework consistent. Anomaly mediation of SUSY breaking Randall:1998uk ; Giudice:1998xp induces the SUSY breaking effects from the superconformal anomaly, and hence they are determined solely by physics at the energy scale of interest. When applied to the SUSY standard model, it automatically solves the serious flavor and CP problems. Because the anomaly is a quantum effect and hence loop-suppressed, the typical SUSY masses are smaller than the gravitino mass by , implying that the gravitino is heavy, TeV. Such a large mass allows the gravitino to decay before BBN, and hence the gravitino is harmless. The only constraint is that stable Lightest Supersymmetric Particle (LSP) from the gravitino decay does not provide too much dard matter of the universe. It requires Kawasaki:1994af

(2) |

This constraint can be satisfied together with the requirement for the thermal leptogenesis GeV. The upper bound on the LSP mass, GeV, is derived from the consistency between those two requirements, by assuming there is no significant annihilation after the gravitino decays.

Despite these attractive features, anomaly mediation has not been used widely in the literature because of several initial problems. The slepton mass-squared comes out negative, breaking the electromagnetism spontaneously. Many fixes proposed in the literature Pomarol:1999ie ; Chacko:1999am ; Katz:1999uw ; Bagger:1999rd ; Kaplan:2000jz ; Chacko:2001jt ; Okada:2002mv ; Nelson:2002sa ; Anoka:2003kn unfortunately spoils the UV insensitivity and hence reintroduces the flavor and CP problems, unless -parity is violated Allanach:2000gu . On the other hand, the addition of -terms for and can make the slepton mass-squared positive Jack:2000cd , and furthermore the UV insensitivity is preserved Arkani-Hamed:2000xj . The viable electroweak symmetry breaking was demonstrated only recently Kitano:2004zd which goes extremely well with the low-energy limit of the Minimal Fat Higgs Model Harnik:2003rs . Even though the original setting relied on extra dimensions Randall:1998uk , it can now be constructed in a purely four-dimensional setting Luty:2001jh together with the required -terms Harnik:2002et . Therefore, anomaly mediation can be finally regarded as a consistent and viable framework of supersymmetry breaking.

The gauge invariance is broken at some high scale, and its only remnant is its -term and an accidental global non-anomalous symmetry to ensure the UV insensitivity. The Kähler potential for a matter field of charge gives a contribution to its mass-squared (there is also Fayet–Illiopoulos term for that is not relevant in this paper). On the other hand, the seesaw mechanism breaks explicitly by the Majorana mass of right-handed neutrinos, and reintroduces the flavor violation in a highly controlled fashion. This effect had not been worked out quantitatively in the original work Arkani-Hamed:2000xj , and we study it in detail in this Letter.

We first derive the expression of the threshold corrections from the right-handed neutrinos. The correction to the slepton mass matrices due to the breaking in right-handed Majorana masses can be worked out using the “analytic continuation into superspace” Arkani-Hamed:1998kj used extensively in anomaly mediation Pomarol:1999ie ; Arkani-Hamed:2000xj ; Boyda:2001nh . in Eq. (1) are the only violating parameters in the theory. Using the fictitious gauge invariance , , and , it is clear that the only combination that can appear in the low-energy theory is . It appears in the Kähler potential for the left-handed leptons and at the one-loop level

(3) |

is the charge of the superfield. is the renormalization scale. The wave-function renormalization factor is for the case of massless right-handed neutrinos and exact invariance, and hence gives a fully UV insensitive supersymmetry breaking. The second term subtracts the contribution of the right-handed neutrinos between their mass thresholds and the renormalization scale . Here, the vector superfield contains the required -term, and makes dependence invariant under the (spurious) symmetry. It induces the correction to the left-handed slepton mass-squared matrix,

(4) |

Note that there is no flavor-violating correction to the right-handed sleptons. Corresponding formula in the minimal supergravity (mSUGRA) that assumes the universal scalar mass is Hisano:1995nq

(5) |

where is the universal scalar mass at the ultraviolet cutoff and the universal trilinear coupling.

There are several remarkable aspects in Eq. (4). First, there is no logarithmic enhancement. In contrast, the usual supergravity theories are UV sensitive and hence the contributions of the right-handed neutrinos are enhanced by as seen in Eq. (5). Therefore the size of possible LFV is under a much stronger control in the anomaly mediation. Related to the aspect, Eq. (4) is independent of physics above the right-handed mass scale since the contribution originates from the threshold effect of as we see below in the explicit diagrammatic calculation. In the mSUGRA, in contrast, we need to know the theory above , especially above , to calculate by carrying out the integration of the renormalization group equations with initial conditions given at the Planck scale. Second, the corrections do not depend on the mass of the right-handed neutrinos explicitly (but implicitly through once the light neutrino masses are held fixed). Once we observe LFV processes and measure the branching ratios, the simple structure of in Eq. (4) enables us to extract easily the the Yukawa matrix which has important information on the origin of the large mixing among neutrinos, i.e., whether the large mixing comes from the left-handed lepton sector, the right-handed neutrino sector, or both sectors. Third, because the corrections are suppressed by the one-loop factor relative to the leading anomaly-mediated contributions thanks to the absence of the logarithmic enhancement, the leading-order trajectory of the soft SUSY breaking parameters is still strictly that of the UV insensitive anomaly mediation. Finally, even if there is a flavor-violating interaction between quarks and leptons below (i.e., leptoquarks), the induced flavor-violation in quarks is at most of the order of above, while the squarks are heavier than the typical size of by a factor of . Therefore it will not pose a serious threat.

There is a corresponding correction to the Higgs mass

(6) |

However, this is a loop-suppressed effect relative to the leading anomaly-mediated contribution, and does not lead to flavor-violating effects. It can be safely ignored for all practical purposes.

Because the absence of the logarithmic enhancement is a striking result, it is useful to examine it with the conventional Feynman diagrams shown in Fig. 1. We use regularization by dimensional reduction (DRED) in dimensions to perform loop integrals. Because the anomaly-mediated pieces due to the neutrino Yukawa couplings are canceled exactly by the threshold corrections Arkani-Hamed:1998kj ; Boyda:2001nh , we only consider -term contributions to the scalar masses-squared, for , for , and for (of course , but we retain it for the clarity of presentation). The sum of all boson loops at gives

(7) | |||||

while the fermion loop gives the correction at ,

(8) |

Finally the leading dependence of the two-point function is

(9) |

which gives the wave function renormalization factor. Putting them together, the final correction to the slepton masses-squared is

(10) | |||||

Because the -term contributions satisfy due to the conservation of the Yukawa coupling, the logarithmic piece automatically cancels, and the result agrees with the spurion method.

Using our result Eq. (4), we list in Table 1
the branching ratios of the decay, the
conversion process in Al nuclei Kitano:2002mt , and the decay using representative parameter sets point I and II
worked out in Kitano:2004zd ^{2}^{2}2Strictly speaking, the Fat
Higgs model Harnik:2003rs requires an mixing between the
composite and elementary Higgs fields, and the Yukawa couplings
need to be scaled by the mixing angle..
For other parameter sets, the branching ratios can be estimated by the
scaling of and .

Point I | Point II | |
---|---|---|

TeV | TeV | |

To get a sense on the size of LFV, we consider a simple neutrino mass model based on flavor symmetry that is consistent with leptogenesis. It assigns the flavor charges , , , , , . The flavor is assumed to be broken by an order parameter . The Yukawa matrix is

(11) |

while the right-handed neutrino masses are . The top Yukawa coupling is approximately for . The light neutrino masses from the seesaw mechanism are

(12) |

This type of model was considered in Buchmuller:1996pa , and can successfully produce the observed baryon asymmetry from the decay of . Note that the mass of is about GeV and is allowed by the gravitino constraint Eq. (2) from the overclosure by the LSP. We find in this model Sato:2000zh . Thus the branching ratios are roughly estimated to be (point I) and (point II). By taking into account the ambiguity in the model parameters, the predictions are comparable to the current experimental upper bound Brooks:1999pu . In both points, observation of the decay in the planned experiments PSI ; PRISM is quite promising. Also, the values for and conversion are in the interesting range for on-going Abe:2003sx ; Brown:2002mp , or future MECO ; PRISM experiments, respectively.

The corresponding analyses have been done in the mSUGRA, and stringent bounds on the model parameters are obtained since the logarithmic factor in Eq. (5) gives enhancement in . The situation is particularly severe in models with Yukawa unification and flavor symmetries Sato:2000zh . With fixing and the SU(2) gaugino mass to be and GeV, the lower bound on the scalar electron mass is found to be more than 1 TeV for the same Yukawa couplings as above. The bound is more severe for larger values of . Too large LFV is a generic feature in the mSUGRA with the seesaw model when the right-handed neutrino scale is relatively high, unless we assume a specific texture of the Yukawa matrix Ellis:2002fe or a cancellation among the diagrams Ellis:2001xt . It is interesting to recall that the universal scaler mass is introduced by hand to avoid too large FCNC in the mSUGRA. However, the prescription is insufficient in the seesaw model. The situation is significantly improved in the anomaly mediation because of the absence of the logarithmic factor.

In conclusion, we have presented a framework where SUSY flavor and CP problems are automatically solved and the thermal leptogenesis is made consistent with the gravitino constraint. It relies on UV insensitive anomaly-mediated supersymmetry breaking supplemented by the -terms for and . The right-handed neutrino mass explicitly breaks and reintroduces the lepton flavor violation (LFV), but it lacks the logarithmic enhancement unlike in mSUGRA. Therefore the size of LFV is easily consistent with the current limits while it is already threatening in the mSUGRA for the same neutrino parameters. Also, calculation of the LFV processes is less ambiguous in the framework, since the corrections to the slepton masses are independent of physics above the mass scale of the right-handed neutrinos.

###### Acknowledgements.

HM and TY thank the organizers of the seesaw 04 workshop at KEK, especially Kenzo Nakamura, where the initial idea of the project emerged. MI thanks the hospitality of the Institute for Advanced Study where the actual work started. RK is the Marvin L. Goldberger Member at the Institute for Advanced Study. HM was supported by the Institute for Advanced Study, funds for Natural Sciences. This work was also supported in part by the DOE under contracts DE-FG02-90ER40542 and DE-AC03-76SF00098, and in part by NSF grant PHY-0098840.## References

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