Weak Scale Leptogenesis, R-symmetry, and a Displaced Higgs
MMIT-CTP 4317
Weak Scale Leptogenesis, R -symmetry, and a Displaced Higgs Keith Rehermann
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Christopher M. Wells
Houghton College, Houghton, New York 14744 (Dated: October 30, 2018)We present a supersymmetric model of leptogenesis in which the right-handed neutrinos haveweak scale masses and O (10 − -10 − ) yukawa couplings. The model employs an R -symmetry at theweak scale that forces neutrino masses to be proportional to the gravitino mass. We predict thatthe lightest right-handed neutrino decays to a displaced Higgs and neutrino and that the NLSP islong-lived on collider scales. These could be striking signatures at the LHC if, for instance, theright-handed neutrino is produced in a cascade decay. INTRODUCTION
The dynamics that generate the observed baryon den-sity remain an unresolved problem. A number of modelshave been developed that address the baryon asymmetry,among which leptogenesis [1] stands out for its economyand simplicity. In the simplest version, right-handed neu-trinos with a large ( ∼ GeV) Majorana mass generatethe scale of neutrino masses via the Seesaw mechanism[2] and have CP violating decays which result in a leptonasymmetry. Despite their simplicity, such dynamics havelittle hope of ever being observed.In this letter we argue that leptogenesis with weakscale right-handed neutrinos [3–5] can be achieved ifSupersymmetry (SUSY) breaking is mediated in an R -symmetric fashion. Tying together R -symmetric SUSYbreaking and weak scale leptogenesis leads to a numberof predictions. First, the neutrino masses are controlledby the gravitino mass. Second, successful leptogenesispredicts that the decays of the lighest right-handed neu-trino produce a displaced Higgs and neutrino. This is arather novel signature. Third, the NLSP is long-lived oncollider scales. Together these lead to an intriguing con-nection between SUSY breaking, Electroweak SymmetryBreaking (EWSB) and neutrino masses, while providinga unique LHC signature in events with a right-handedneutrino.Turning an eye toward model building, we have twomore interesting observations. First, the R -symmetry al-lows the hypothesized dimensionless couplings in the lowenergy theory to be O (10 − -10 − ), avoiding the moreserious tunings that often appear in other attempts tomodel weak scale leptogenesis. Second, since the dynam-ics of leptogenesis are at the weak scale, the gravitinoproblem [6] that plagues SUSY models with high scaleright-handed neutrinos is easily avoided. MODEL
Our model is based on three hypotheses. First, we takethe superpotential to be that of the MRSSM [7] extendedto include right-handed neutrinos
W ⊃ µ u H u R u + µ d H d R d + y N ij R d L i N j + 12 M ij N i N j . (1)The fields R u,d are weak doublets with hypercharge op-posite the corresponding MSSM Higgs and N j are threegenerations of right-handed neutrinos. The right-handedneutrino mass term is consistent with a Z R R -symmetry[8]; charge assignments are shown in Table I. Our secondassumption is that D and F term SUSY breaking occurat a common scale. Finally, we assume that the domi-nant R -violation from SUSY breaking is the irreduciblecontribution from Supergravity (SUGRA) δL (cid:54) R ⊃ (cid:90) d θ Φ( µ d H d R d + µ u H u R u )= m / ( µ d H d R d + µ u H u R u + c.c.) (2)where Φ = 1 + m / θ is the conformal compensator [9].After EWSB, our assumptions lead to vevs for thescalars r d , r u (cid:10) r d (cid:11) = m / µ d v d µ d + m soft , (cid:10) r u (cid:11) = m / µ u v u µ u + m soft . (3)We see that the irreducible R -violation induced bySUGRA puts the r ’s at vevs proportional to m / .Consequently, the left-handed neutrinos have Majoranamasses controlled by m / /M N . We note, in passing,that if one enforces an exact U (1) R the Majorana massesin Eq. (1) are forbidden. In this case the neutrinos areDirac and m ν ∝ m / . This is an interesting possibilitywhich would imply low scale SUSY breaking [10].Our hypotheses necessitate Dirac masses for the gaug-inos [11]. We remain agnostic concerning the UV origin a r X i v : . [ h e p - ph ] O c t Field Z R Z H u,d N − R u,d , X Q, u c , d c , L, e c R -charges and parity assignments that can gener-ate the superpotential in Eq. (1) once F X (cid:54) = 0. of these masses, parameterizing them as (cid:90) d θ θ α D (cid:48) Λ W αi Φ i , (4)where θ α D (cid:48) a is D-type spurion with R -charge +1 andthe index i runs over the SM gauge groups. The Φ i are adjoint chiral superfields whose fermionic compo-nents ψ i marry the gauginos through Dirac mass terms ∝ ( D (cid:48) / Λ) λ i ψ i . Weak scale soft masses can be generatedif, for example, D (cid:48) ∼ F X ∼ GeV and Λ ∼ GeV.Studies of R -symmetric mediation suited to our modelcan be found in Ref’s. [12–14].We note that the µ terms in Eq. (1) need not be ex-plicit. Rather, they can be generated by the Giudice-Masiero mechanism [15] if we assume the existence of anadditional Z symmetry, as in Table I [16]. These chargeassignments allow the following operators up to dim 6 L ⊃ (cid:90) d θ X † Λ (cid:88) i = u,d c i R i H i + c XX † Λ H u H d + h.c. (5)For c , c , c of O (1) and F X Λ ∼ m soft these operators gen-erate the µ u , µ d , b h parameters that trigger EWSB.Conversely, given the field content in Table I M N can-not be generated in an analogous manner. In this sense M N is a free parameter, though as we shall see, success-ful leptogenesis requires M N ∼ < O (TeV). It would beinteresting to dynamically explain this scale [17]. LEPTOGENESIS, NEUTRINO MASSES AND R -SYMMETRY We briefly review the salient features of leptogenesis;see Ref. [18] for a thorough review. Ultimately the aimof leptogenesis is to explain the baryon asymmetry Y ∆ b ≡ n b − n ¯ b s ≈ × − (6)where n b , n ¯ b are the number densities of baryons andanti-baryons respectively, s is the entropy density and Y ∆ b is referred to as the baryon asymmetry yield. Inthe early universe an asymmetry in lepton number canbe converted to an asymmetry in baryon number bysphaleron processes. Schematically we have Y ∆ b (cid:39) C sph Y ∆ l (7) where Y ∆ l is defined in analogy to Y ∆ b and typically1 / ∼ > C sph ∼ > / Y ∆ l can be generated if the Sakharov con-ditions (lepton number violation, CP violation and outof equilibrium dynamics) [19] are satisfied. The leptonasymmetry yield can be parameterized by the degree of CP violation (cid:15) and the efficiency of out of equilibriumdynamics η : Y ∆ l (cid:39) kg ∗ (cid:15)η (8)where g ∗ is the number degrees of freedom in equilibrium,typically O (100), and k is an O (1) factor. Note that bydefinition (cid:15), η < Y ∆ l in the model presented above.Left-handed neutrino masses are obtained from Eq. (1)by integrating out the right-handed Majorana neutrinos:( m ν ) ij = (cid:88) k M N k ( y N y TN ) ij v r d (9)in the basis that diagonalizes M ij . As a direct conse-quence of the R -symmetry, Eq’s. (3) and (9) imply that m ν ∝ m / /M N . Said differently, the R -symmetry pro-tects (cid:10) r d (cid:11) thereby allowing weak scale M N without tun-ing the scalar potential.The degree of CP violation is governed by the param-eter (cid:15) = Γ( N → νh ) − Γ( N → ( νh ) † )Γ( N → νh ) + Γ( N → ( νh ) † ) . (10)where h is the physical Higgs [20]. Interference betweentree and one loop diagrams, examples of which are shownin Figures 1a and 1b, generate (cid:15) : (cid:15) ≈ − π y † N y N ) (cid:88) j I m (cid:20)(cid:16) ( y † N y N ) j (cid:17) (cid:21) M N M N j . (11)This is an adequate approximation for our purposes if M N , > M N and we take this to be the case in whatfollows. The scaling of (cid:15) is more obvious if we make thesimplifying assumption y N ij ≈ y N i ( i.e. entries along aparticular row are comparable) | (cid:15) | ≈ π (cid:88) j | y † N j y N j | sin θ j M N M N j (cid:39) M N πv r d m max ν sin δ (cid:39) − (cid:18) M N
300 GeV (cid:19)(cid:18) − GeV v r d (cid:19) (cid:18) m max ν . (cid:19) sin δ (12)where m max ν is the largest left-handed neutrino mass andsin δ is the associated CP violating phase. The scalingfrom Eq. (3), (cid:10) r d (cid:11) ∼ m / , allows a large CP asymme-try to be generated for a weak scale right-handed neu-trino, yukawa couplings of O (10 − -10 − ) and m / of O (1 MeV). Note that the scale of R -symmetry violation N R d L i H d N j N j R d L k (a) N R d L i H d N j N j R d L k (b) L i L k N R d R d (c) L i R d L k R d N (d) FIG. 1. The top row illustrates the diagrams contributing to (cid:15) . The bottom shows some of the diagrams that contributeto washout and suppress η . simultaneously determines the neutrino masses and themagnitude of CP violation.We now estimate the efficiency η of the out of equilib-rium dynamics. Our goal in this letter is not to map theentire parameter space of viable leptogenesis, but to showthat such a parameter space exists and that it is sizable.To this end we work in the “weak-washout” regime, η of O (1). This regime applies if N decays out of equilibriumand the rates for all other lepton number violating pro-cesses (such as inverse decay and 2 → (cid:15) . While successful leptoge-nesis may occur outside of the weak-washout regime - dueto a delicate interplay between the dynamics contributingto (cid:15) and η - this simplifying assumption is sufficient forour goal and in what follows we determine the parameterspace that satisfies η of O (1).The out of equilibrium condition requires that the de-cay rate of N is less than the Hubble rate at tempera-tures near M N ,Γ < g ∗ M N M pl (cid:39) − sec − (cid:18) M N
300 GeV (cid:19) . (13)It is generically difficult to achieve this for a weak scalemass because it implies that the width is many or-ders of magnitude smaller than the mass, Γ N /M N ∼ < − ( M N /
300 GeV). However, our model naturallysuppresses processes that violate the R -symmetry. Infact, the width N is protected by the R -symmetry if it islighter than the scalar r d . In this case, it decays throughthe R -violating channel N → νh , where h is the phys-ical Higgs. As with neutrino masses and the magnitudeof CP violation, the scale of R -violation sets this decay FIG. 2. This plot illustrates that a significant parameterspace exists for successful leptogenesis, even with the sim-plifying assumptions in the text. Shaded regions are Y ∆ l at T = 100 GeV, assuming maximal CP violation. Red dashed-dotted line are (cid:10) r d (cid:11) (GeV). Black dashed lines are values of˜ M/M N that result in suppressed washout, per Eq. (15). rate:Γ = ( y † N y N ) π b d m r d M N = ( y † N y N ) π (cid:18) v r d v h d (cid:19) M N (14)where the second expression follows directly from the tad-pole equation for r d , Eq. (3). The time scale in Eq. (13)suggests that N decays are displaced on collider scales.We revisit this point in the next section.Lepton number violating processes, such as those il-lustrated in Fig. 1c and Fig. 1d, can also washout theasymmetry, suppressing η . However, if all scalars areheavy except for a light Higgs, then the washout pro-cesses decouple because their rates are Boltzmann sup-pressed. Assuming that the left handed sleptons and r d are heavy with common mass ˜ M , we can estimate the ra-tio ˜ M /M N necessary to suppress washout. In analogy toEq. (13), the washout rate must be less than the Hubblerate. The rate for non-relativistic scalar scattering witha relativistic fermion is [21]Γ Washout ≡ n (cid:104) σv (cid:105) ≈ ( ˜ M T ) / e − ˜ M/M N α y ( s − M N ) (15)where α y = y / (4 π ).To determine the lepton asymmetry yield we solve thefollowing Boltzmann equation˙ Y ∆ l = (cid:15) Γ Decay ( Y N − Y eq N ) − Γ Washout Y ∆ l (16)where we have defined˙ Y ≡ sH | T = M N z dYdz , z ≡ M N T . (17)Figure 2 plots solutions to Eq. (16) where we assumemaximal CP violation in Eq. (11), fix v h d = v/ β = 10) and enforce Eq. (13). We also plot con-tours of constant (cid:10) r d (cid:11) and ˜ M /M N , while omitting theregion ˜ M /M N < Y ∆ l ∼ − by moving away from the assumptionof maximal CP violation and η of O (1).We note that there exists some viable parameter spacein the regime ˜ M ∼ < M N . In this case only the smallnessof y N can prevent washout, which in turn implies smaller (cid:15) . We estimate the constraint on y N by using the rela-tivistic form of the rate in Eq. (15) and by requiring it tobe less than the Hubble rate at T = M N y N ∼ < − (cid:18) g ∗ M N
300 GeV (cid:19) / ⇒ (cid:15) ∼ < − (18)where the (cid:15) bound follows from Eq. (11).Since there exist viable solutions in both regimes,we conclude that the interpolating regime of moderateyukawa couplings and comparable masses offers the pos-sibility that the yukawa couplings and the mass rationeed not be tuned against each other to the extent inour simple estimates. We leave a more detailed explo-ration of this regime for later work. A DISPLACED HIGGS AT THE LHC?
A spectrum that is consistent with our assumptionsis heavy gauginos and scalar superpartners ˜ M ∼ TeV,moderate right-handed neutrinos and higgsinos M N ∼ µ u,d and a light Higgs. In this regime the lightest right-handed neutrino decays to the Higgs and a neutrino. Theout of equilibrium condition in Eq. (13) sets the lifetimeof the lightest right-handed neutrino. Intriguingly, thiscondition necessitates that N → νh is a displaced decay.This follows from demanding that N has a weak scalemass and that it decays before the sphaleron processesturn off at T ∼
100 GeV. The lifetime of N is then inthe range( H | T =100 GeV ) − ∼ > τ N ∼ > (cid:0) H | T = M N (cid:1) − ⇒∼ − sec ∼ > τ N ∼ > − (cid:18)
300 GeV M N (cid:19) sec (19)This lifetime is comparable to those of B and D mesonsif M N ∼ < TeV. ATLAS, CMS, and LHCb all have theability to resolve decays on this time scale. Therefore, our model has the striking possibility of being observedvia displaced Higgs events.We note that although the direct production cross sec-tion of N is vanishingly small at the LHC, in the spec-trum noted above the sneutrino width can be dominatedby ˜ ν i → N ˜ r d . This occurs if the yukawa couplings y N are larger than the yukawa couplings in the operator y L H d Le c . Generating a sufficient asymmetry requires y N ∼ > − , and at minimum we expect ˜ ν e to dominantlydecay through ˜ ν e → N ˜ r d . Therefore, cascade decays in-volving ˜ ν e terminate with a displaced Higgs. The samemay be true for the other flavors depending on the exactvalues of the yukawa couplings.Finally, we draw attention to the fact that success-ful leptogenesis puts the gravitino mass at O (1 MeV).Therefore the NLSP, χ , is long-lived on collider scales: τ χ ≈ π F m χ ≈ − s (cid:16) m / (cid:17) (cid:18) m χ (cid:19) . (20)where we have neglected phase space suppression. TheNLSP must then appear either as missing energy, aCHAMP [22], or an R -hadron [23]. This gives us an ad-ditional distinctive collider signature accompanying theprediction of a displaced Higgs. We intend to explorethese possibilities in future work. A NOTE ON DIRAC GAUGINOS
For the sake of completeness, we recall a few well-known complications to the minimal Dirac gaugino storyand simple extensions that address them. The hyper-charge adjoint Φ Y , which gives mass to the Bino, is acomplete SM and Z R singlet. As such, our effective the-ory should contain terms like L ⊃ (cid:90) d θ X † X Λ Φ Y + (cid:90) d θθ α θ α D (cid:48) Λ Φ Y + h.c. . (21)If Φ Y receives a soft mass ˜ m , then the tadpole abovepushes Φ Y to a vev on the order of | (cid:104) Φ Y (cid:105) | ∼ F X + D (cid:48) Λ ˜ m ∼ F X Λ (cid:0) F X Λ (cid:1) = Λ . (22)This cutoff scale vev introduces a D-term for hyperchargeof order D (cid:48) via Eq. (4). Also, because Φ Y is an R -symmetry singlet, its vev introduces cutoff size correc-tions to µ u,d . Both of these effects can spoil EWSB.Furthermore, singlet couplings to the supergravity mul-tiplet generically reintroduce quadratic divergences anddestabilize the hierarchy [24].There are a number of UV solutions to these problemscompatible with our low energy phenomenology: Ref. [12]utilizes an unbroken discrete symmetry, Ref. [14] exploitsa clever set of couplings between the messenger fields andthe adjoints and Ref. [25] forbids these terms by embed-ding the theory into SU (5). DISCUSSION
We have presented a model of leptogenesis which isconnected to the physics of the weak scale through SUSYbreaking and R -symmetry. The dynamics of neutrinomasses and leptogenesis are both controlled by the scaleof R -violation in the low energy theory. The model tiesthe neutrino mass scale to the gravitino mass, which inconjunction with demanding a large enough lepton asym-metry, predicts m / of O (MeV). Furthermore, thereare striking LHC signatures: cascade decays through thelightest right-handed neutrino terminate with a displacedHiggs and the NLSP is long-lived on collider scales. Fi-nally, the gravitino problem is avoided because all of theinteresting visible sector dynamics are at the weak scalewhich allows for reheating temperatures of O (TeV).Of course, nothing is free, and our dynamics rely on R -symmetric mediation of SUSY breaking. This intro-duces hurdles to, for example, unification. However, wefind the interplay of supersymmetry breaking, neutrinomasses and the potential observability of leptogenesis dy-namics at the LHC a tantalizing possibility. There are anumber of directions for further study including a refinednumerical study of the dependence of the asymmetry onthe spectrum of soft masses, a dedicated study of the col-lider phenomenology, and more detailed considerations ofUV model building. We intend to pursue these topics infuture publications. ACKNOWLEDGMENTS
We thank Matthew McCollough and Jesse Thaler forinsightful conversions and Matthew McCollough, DavidE. Kaplan, and Gordan Krnjaic for comments on themanuscript. This work is supported by the U.S. Depart-ment of Energy under cooperative research agreementDE-FG02-05ER-41360. CMW was partially supportedby the Houghton College Summer Research Institute. [1] V. Kuzmin, V. Rubakov, and M. Shaposhnikov,Phys.Lett.,
B155 , 36 (1985); M. Fukugita andT. Yanagida, ibid ., B174 , 45 (1986); M. Luty, Phys.Rev.,
D45 , 455 (1992).[2] T. Yanagida, Prog.Theor.Phys., , 1103 (1980); R. N.Mohapatra and G. Senjanovic, Physical Review Letters, , 912 (1980).[3] M. Flanz, E. A. Paschos, U. Sarkar, and J. Weiss,Phys.Lett., B389 , 693 (1996), arXiv:hep-ph/9607310[hep-ph]; L. Covi, E. Roulet, and F. Vissani, ibid ., B384 ,169 (1996), arXiv:hep-ph/9605319 [hep-ph]; A. Pilaft-sis, Phys.Rev.,
D56 , 5431 (1997), arXiv:hep-ph/9707235 [hep-ph]; A. Pilaftsis and T. E. Underwood, Nucl.Phys.,
B692 , 303 (2004), arXiv:hep-ph/0309342 [hep-ph].[4] S. Blanchet, P. Dev, and R. Mohapatra, Phys.Rev.,
D82 ,115025 (2010), arXiv:1010.1471 [hep-ph].[5] N. Haba and O. Seto, Prog.Theor.Phys., , 1155(2011).[6] L. M. Krauss, Nucl.Phys.,
B227 , 556 (1983); C. Cheung,G. Elor, and L. J. Hall, (2011), arXiv:1104.0692 [hep-ph].[7] G. D. Kribs, E. Poppitz, and N. Weiner, Phys.Rev.,
D78 , 055010 (2008), arXiv:0712.2039 [hep-ph].[8] H. M. Lee, S. Raby, M. Ratz, G. G. Ross, R. Schieren,et al., Phys.Lett.,
B694 , 491 (2011), arXiv:1009.0905[hep-ph]; Nucl.Phys.,
B850 , 1 (2011), arXiv:1102.3595[hep-ph].[9] W. Siegel and J. Gates, S.James, Nucl.Phys.,
B147 ,77 (1979); S. Gates, M. T. Grisaru, M. Rocek,and W. Siegel, Front.Phys., , 1 (1983), arXiv:hep-th/0108200 [hep-th]; V. Kaplunovsky and J. Louis,Nucl.Phys., B422 , 57 (1994), dedicated to the memoryof Brian Warr, arXiv:hep-th/9402005 [hep-th].[10] KR thanks Matthew McCullough for an insightful dis-cussion on this point.[11] L. Hall and L. Randall, Nucl.Phys.,
B352 , 289 (1991);L. Randall and N. Rius, Phys.Lett.,
B286 , 299 (1992);P. J. Fox, A. E. Nelson, and N. Weiner, JHEP, ,035 (2002), arXiv:hep-ph/0206096 [hep-ph].[12] S. D. L. Amigo, A. E. Blechman, P. J. Fox, and E. Pop-pitz, JHEP, , 018 (2009), arXiv:0809.1112 [hep-ph].[13] A. E. Blechman, Mod.Phys.Lett.,
A24 , 633 (2009),arXiv:0903.2822 [hep-ph].[14] K. Benakli and M. Goodsell, Nucl.Phys.,
B840 , 1 (2010),arXiv:1003.4957 [hep-ph].[15] G. F. Giudice and A. Masiero, Physics Letters B, ,480 (1988).[16] R. Davies, J. March-Russell, and M. McCullough, JHEP, , 108 (2011), arXiv:1103.1647 [hep-ph].[17] One possibility would be if there existed X (cid:48) with R =-2and F X (cid:48) ∼ F X .[18] S. Davidson, E. Nardi, and Y. Nir, Phys.Rept., , 105(2008), arXiv:0802.2962 [hep-ph].[19] A. Sakharov, Pisma Zh.Eksp.Teor.Fiz., , 32 (1967).[20] The next section explains why it is the physical Higgs.[21] E. W. Kolb and M. S. Turner, Front.Phys., , 1 (1990).[22] M. Dine, A. E. Nelson, Y. Nir, and Y. Shirman,Phys.Rev., D53 , 2658 (1996), arXiv:hep-ph/9507378[hep-ph]; M. Dine, A. E. Nelson, and Y. Shirman,
D51 , 1362 (1995), arXiv:hep-ph/9408384 [hep-ph]; J. L.Feng, T. Moroi, L. Randall, M. Strassler, and S.-f. Su,Phys.Rev.Lett., , 1731 (1999), arXiv:hep-ph/9904250[hep-ph].[23] S. Raby, Phys.Rev., D56 , 2852 (1997), arXiv:hep-ph/9702299 [hep-ph]; H. Baer, K.-m. Cheung, andJ. F. Gunion, Phys. Rev.,
D59 , 075002 (1999),arXiv:hep-ph/9806361; G. Giudice and A. Romanino,Nucl.Phys.,
B699 , 65 (2004), arXiv:hep-ph/0406088[hep-ph]; N. Arkani-Hamed and S. Dimopoulos, JHEP, , 073 (2005), arXiv:hep-th/0405159 [hep-th].[24] J. Bagger, E. Poppitz, and L. Randall, Nucl.Phys.,
B455 , 59 (1995), arXiv:hep-ph/9505244 [hep-ph].[25] S. Abel and M. Goodsell, JHEP,1106