LHC and lepton avour violation phenomenology of a left-rightextension of the MSSMJ. N. Esteves∗ and J. C. Romao†Departamento de Físi a and CFTP, Instituto Superior Té ni oAv. Rovis o Pais 1, 1049-001 Lisboa, PortugalM. Hirs h‡ and A. Vi ente§AHEP Group, Instituto de Físi a Corpus ular C.S.I.C./Universitat de Valèn iaEdi io de Institutos de Paterna, Apartado 22085, E46071 Valèn ia, SpainW. Porod¶ and F. Staub∗∗Institut für Theoretis he Physik und Astronomie, Universität WürzburgAm Hubland, 97074 WuerzburgAbstra tWe study the phenomenology of a supersymmetri left-right model, assuming minimal supergrav-ity boundary onditions. Both left-right and (B-L) symmetries are broken at an energy s ale loseto, but signi antly below the GUT s ale. Neutrino data is explained via a seesaw me hanism. We al ulate the RGEs for superpotential and soft parameters omplete at 2-loop order. At low energieslepton avour violation (LFV) and small, but potentially measurable mass splittings in the hargeds alar lepton se tor appear, due to the RGE running. Dierent from the supersymmetri pureseesaw models, both, LFV and slepton mass splittings, o ur not only in the left- but also in theright slepton se tor. Espe ially, ratios of LFV slepton de ays, su h as Br(τR → µχ01)/Br(τL → µχ0

1)are sensitive to the ratio of (B-L) and left-right symmetry breaking s ales. Also the model predi tsa polarization asymmetry of the outgoing positrons in the de ay µ+ → e+γ, A ∼ [0, 1], whi h diersfrom the pure seesaw predi tion A = 1. Observation of any of these signals allows to distinguishthis model from any of the three standard, pure (mSugra) seesaw setups.Keywords: supersymmetry; neutrino masses and mixing; LHC; lepton avour violation∗joaomest ftp.ist.utl.pt†jorge.romaoist.utl.pt‡mahirs hi .uv.es§Avelino.Vi entei .uv.es¶porodphysik.uni-wuerzburg.de∗∗orian.staubphysik.uni-wuerzburg.de 1

I. INTRODUCTIONThe most popular explanation for the observed smallness of neutrino masses is ertainlythe seesaw me hanism [14. Literally hundreds of theoreti al papers based on the seesawhave been published sin e the dis overy of neutrino os illations [5. The seesaw an beimplemented at tree-level in exa tly three realizations [6: ex hange of a fermioni singlet,a.k.a. the right-handed neutrino (type-I) [1, 2; of a s alar triplet (type-II) [24; or of afermioni triplet (type-III) [7. In any of these seesaw me hanisms neutrino masses aregiven by mν ∼ v2/Λ, where v is the Higgs va uum expe tation value (vev) and Λ the s aleof the seesaw. For oe ients O(1) and Λ ∼ (1014 − 1015) GeV one nds neutrinos withsub-eV masses, just as experimental data demands. Unfortunately, attra tive as this ideamight appear from the theoreti al point of view, this estimate also implies that the seesawwill never be dire tly tested.This situation might hange slightly, if supersymmetry (SUSY) is found at the LHC,essentially be ause s alar leptons provide potentially additional information about seesawparameters. Assuming SUSY gets broken at a high energy s ale, the seesaw parameters leavetheir imprint on the soft parameters in the Renormalization Group Equation (RGE) running.Then, at least in prin iple, indire t tests of the seesaw be ome possible1. Indeed, this hasbeen pointed out already in [8, where it was shown that lepton avour violating (LFV)o-diagonal mass terms for sleptons are automati ally generated in seesaw (type-I), evenif SUSY breaking is ompletely avour blind at the GUT s ale as in minimal supergravity(mSugra)2.Motivated by the above arguments, many authors have then studied LFV in SUSY mod-els. For the seesaw type-I, low energy LFV de ays su h as li → ljγ and li → 3lj have been al ulated in [918; µ − e onversion in nu lei has been studied in [19, 20. The type-IIseesaw has re eived mu h less attention, although it has a tually fewer free parameters thantype-I. The latter implies that ratios of LFV de ays of leptons an a tually be predi tedas a fun tion of neutrino angles in mSugra, as has been shown in [21, 22. Finally, for ompleteness we mention that LFV in SUSY seesaw type-III has been studied in [23.Measurements at olliders, on e SUSY is dis overed, an provide additional information.LFV de ays of left sleptons within mSugra have been studied for type-I in [24 and for type-II in [22, 25. Pre ise mass measurements might also show indire t ee ts of the seesaw[2628. Most prominently, type-II and type-III seesaw ontain non-singlet superelds, sogauge ouplings run dierently from pure MSSM. One then expe ts that sparti le spe trashow a hara teristi deformation with respe t to mSugra predi tions. From dierent ombinations of masses one an form invariants, i.e. numbers whi h to leading order depend1 In the general minimal supersymmetri extension of the standard model (MSSM) all soft terms are freeparameters, xed at the ele troweak s ale and nothing an be learned about the high energy world.2 It might be te hni ally more orre t to all this setup the onstrained MSSM (CMSSM). We will sti kto the terminology mSugra. 2

only on the seesaw s ale [29, although there are important orre tions at 2-loop [22, 23,whi h have to be in luded before any quantitative analysis an be done. Experimentallyinteresting is also that at the LHC the mass splitting between sele trons and smuons maybe onstrained down to O(10−4) for 30 fb−1 of integrated luminosity [30. In mSugra, oneexpe ts this splitting to be unmeasurably tiny, whereas in mSugra plus seesaw signi antlydierent masses an be generated, as has been shown for type-I in [31.Interestingly, in pure seesaw models with avour blind SUSY boundary onditions all ofthe ee ts dis ussed above show up only in the left slepton se tor. Naturally one expe tsthat in a supersymmetri model with an intermediate left-right symmetri stage, also theright sleptons should ontain some indire t information about the high energy parameters.This simple observation forms the main motivation for the urrent paper. Before enteringin the details of our al ulation, let us rst briey dis uss left-right symmetri models.Quite a large number of dierent left-right (LR) symmetri models have been dis ussed inthe literature. Originally LR models were introdu ed to explain the observed left-handednessof the weak intera tion as a onsequen e of symmetry breaking [3234. However, LR modelsoer other advantages as well. First, the parti le ontent of LR models ontains automat-i ally the right-handed neutrino and thus the ingredients for generating a (type-I) seesawme hanism 3. Se ond, the gauge group SU(3)c × SU(2)L × SU(2)R ×U(1)B−L is one of thepossible hains through whi h SO(10) [35, 36 an be broken to the standard model gaugegroup 4. In addition, it has been shown that they provide te hni al solutions to the SUSYCP and strong CP problems [37 and they give an understanding of the U(1) harges of thestandard model fermions. Interesting only for the supersymmetri versions of LR models,(B-L) is gauged and thus, potentially, the low energy theory onserves R-parity [38, 39.This last argument requires possibly some elaboration. R-parity, dened as RP =

(−1)3(B−L)+2s (where B and L stand for baryon and lepton numbers and s for the spin of theparti le), is imposed in the MSSM to avoid dangerous baryon and lepton number violatingoperators. However, the origin of RP is not explained within the MSSM. In early LR modelsSU(2)R doublets were used to break the gauge symmetry. The non-supersymmetri modelproposed in referen es [33, 34 introdu ed two additional s alar doublets χL and χR, whereχL ≡ χL(1, 2, 1, 1) and χR ≡ χR(1, 1, 2,−1) under SU(3)c × SU(2)L × SU(2)R × U(1)B−L.Parity onservation implies that both, χL and χR, are needed. When the neutral ompo-nent of χR gets a vev, 〈χ0

R〉 6= 0, the gauge symmetry is broken down to the SM gaugegroup. However, χR is odd under U(1)B−L and thus, in the SUSY versions of this setup,RP is broken at the same time 5. A possible solution to this problem is to break the gaugesymmetry by SU(2)R elds with even harge under U(1)B−L, i.e. by triplets. For a SUSY3 Breaking the LR symmetry with triplets an generate also a type-II [2.4 Not all SO(10) breaking hains ontain a seesaw. Neither does SU(5). It is, of ourse, straightforward toadd a seesaw to SU(5).5 This ould be solved by imposing additional dis rete symmetries on the model that forbid the dangerousRp/ operators [40, but this annot be regarded as automati R-parity onservation.3

LR model, this was in fa t proposed in referen e [41, where four triplets were added to theMSSM spe trum: ∆(1, 3, 1, 2), ∆c(1, 1, 3,−2), ∆(1, 3, 1,−2) and ∆c(1, 1, 3, 2). Breaking thesymmetry by the vev of ∆c produ es at the same time a right-handed neutrino mass via theoperator Lc∆cLc, leading to a type-I seesaw me hanism. Depending on whether or not ∆gets a vev, also a type-II seesaw an be generated [42.However, whether R-parity is onserved in this setup is not lear. The reason is thatthe minimum of the potential might prefer a solution in whi h also the right-handed s alarneutrino gets a vev, thus breaking RP , as has been laimed to be the ase in [43. Later [44 al ulated some 1-loop orre tions to the s alar potential, on luding that RP onservingminima an be found. On the other hand, as rst noted in [45 and later showed by Aulakhand ollaborators [46, 47, by the addition of two more triplets, Ω(1, 3, 1, 0) and Ωc(1, 1, 3, 0),with zero lepton number one an a hieve LR breaking with onserved RP guaranteed alreadyat tree-level. La king a general proof that the model [41 onserves RP we will follow [46, 47as the setup for our numeri al al ulations.Finally, for ompleteness we mention the existen e of left-right models with R-parityviolation. For example, if the left-right symmetry is broken with the vevs of right-handedsneutrinos R-parity gets broken as well and the resulting phenomenology is totally dierent,as shown in [48, 49.Compared to the long list of papers about indire t tests of the seesaw, surprisingly littlework on the low-energy phenomenology of SUSY LR models has been done. One loopRGEs for two left-right SUSY models have been al ulated in [50. These two models are(with one additional singlet): (a) breaking LR by doublets a la [33, 34 and (b) by tripletsfollowing [41, but no numeri al work at all was done in this paper. The possibility thatright sleptons might have avour violating de ays in the left-right symmetri SUSY modelof [41 was mentioned in [51. A systemati study of all the possible signals dis ussed abovefor the seesaw ase is la king and to our knowledge there is no publi ation of any al ulationof these signals for the model of [46, 47. (For ompleteness we would like to mention that inGUTs based on SU(5) one an have the situation the LFV o urs only in the right sleptonse tor, as pointed out in [52. However, this model [52 is in a dierent lass from all themodels dis ussed above, sin e it does not ontain non-zero neutrino masses.)The rest of this paper is organized as follows. In the next se tion we dene the model[46, 47 and dis uss its parti le ontent and main features at ea h symmetry breaking s ale.We have al ulated the RGEs for ea h step omplete at the 2-loop level following the generaldes ription by [53 using the Mathemati a pa kage SARAH [5456. A summary is givenin the appendix, the omplete set of equations and the SARAH model les an be foundat [57. Neutrino masses an be tted to experimental data via a type-I seesaw me hanismand we dis uss dierent ways to implement the t. We then turn to the numeri al results.The output of SARAH has been passed to the program pa kage SPheno [58 for numeri alevaluation. We al ulate the SUSY spe tra and LFV slepton de ays, su h as τL/R → µχ01and τL/R → eχ0

1 and χ02 → eµχ0

1, as well as low-energy de ays li → ljγ for some samplepoints as a fun tion of the LR and (B-L) s ales. Potentially measurable signals are found4

Supereld generations SU(3)c SU(2)L SU(2)R U(1)B−L

Q 3 3 2 1 13

Qc 3 3 1 2 −13

L 3 1 2 1 -1Lc 3 1 1 2 1Φ 2 1 2 2 0∆ 1 1 3 1 2∆ 1 1 3 1 -2∆c 1 1 1 3 -2∆c 1 1 1 3 2Ω 1 1 3 1 0Ωc 1 1 1 3 0TABLE I. Matter ontent between the GUT s ale and the SU(2)R breaking s ale.in both, left and right slepton se tors, if (a) the seesaw s ale is above (very roughly) 1013GeV and (b) if the s ale of LR breaking is signi antly below the GUT s ale. Sin e we ndsizable LFV soft masses in both slepton se tors, also the polarization in µ → eγ is dierentfrom the pure seesaw expe tation. We then lose with a short summary and outlook.II. LEFT-RIGHT SUPERSYMMETRIC MODELIn this se tion we dene the model, its parti le ontent and give a des ription of thedierent symmetry breaking steps. The t to neutrino masses and its onne tion to LFVviolation in the slepton se tor is dis ussed in some detail, to prepare for the numeri al resultsgiven in the next se tion. We summarize briey the free parameters of the theory.The model essentially follows [46, 47. We have not attempted to nd a GUT ompletion.We will, however, assume that gauge ouplings and soft SUSY parameters an be unied,i.e. impli itly assume that su h a GUT model an indeed be onstru ted.A. Step 1: From GUT s ale to SU(2)R breaking s aleJust below the GUT s ale the gauge group of the model is SU(3)c×SU(2)L ×SU(2)R ×

U(1)B−L. In addition it is assumed that parity is onserved, see below. The matter ontentof the model is given in table I. Here Q, Qc, L and Lc are the quark and lepton supereldsof the MSSM with the addition of (three) right-handed neutrino(s) νc.Two Φ superelds, bidoublets under SU(2)L × SU(2)R, are introdu ed. They ontainthe standard Hd and Hu MSSM Higgs doublets. In this model, two opies are needed for anon-trivial CKM matrix. Although there are known attempts to build a realisti LR modelwith only one bidoublet generating the quark mixing angles at the loop level [59, we will5

not rely on su h a me hanism. Finally, the rest of the superelds in table I are introdu edto break the LR symmetry, as explained above.Table I shows also the gauge harges for the matter ontent in the model. In parti ular,the last olumn shows the B − L value for the dierent superelds. However, the followingdenition for the ele tri harge operator will be used throughout this paperQ = I3L + I3R +

B − L

2(1)and thus the U(1)B−L harge is a tually B−L

2.With the representations in table I, the most general superpotential ompatible with thegauge symmetry and parity is

W = YQQΦQc + YLLΦLc − µ

2ΦΦ + fL∆L+ f ∗Lc∆cLc

+ a∆Ω∆ + a∗∆cΩc∆c + αΩΦΦ + α∗ΩcΦΦ

+ M∆∆∆ +M∗∆∆

c∆c +MΩΩΩ +M∗ΩΩ

cΩc . (2)Note that this superpotential is invariant under the parity transformations Q ↔ (Qc)∗,L ↔ (Lc)∗, Φ ↔ Φ†, ∆ ↔ (∆c)∗, ∆ ↔ (∆c)∗, Ω ↔ (Ωc)∗. This dis rete symmetry xes, forexample, the Lc∆cLc oupling to be f ∗, the omplex onjugate of the L∆L oupling, thusredu ing the number of free parameters of the model.Family and gauge indi es have been omitted in eq. (2), more detailed expressions anbe found in [46. YQ and YL are quark and lepton Yukawa ouplings. However, with twobidoublets there are two opies of them, and thus there are four 3 × 3 Yukawa matri es.Conservation of parity implies that they must be hermitian. µ is a 2× 2 symmetri matrix,whose entries have dimensions of mass, f is a 3 × 3 (dimensionless) omplex symmetri matrix, and α is a 2×2 antisymmetri matrix, and thus it only ontains one (dimensionless) omplex parameter, α12. The mass parameters MΩ and M∆ an be ex hanged for vR andvBL, the va uum expe tation values of the s alar elds that break the LR symmetry, seebelow.The soft terms of the model are

−Lsoft = m2QQ

†Q +m2QcQc

†Qc +m2

LL†L+m2

LcLc†Lc

+ m2ΦΦ

†Φ +m2∆∆

†∆+m2∆∆

†∆ +m2∆c∆c†∆c +m2

∆c∆c †∆c

+ m2ΩΩ

†Ω +m2ΩcΩc†Ωc +

1

2

[

M1B0B0 +M2(WLWL + WRWR) +M3gg + h.c.

]

+[

TQQΦQc + TLLΦLc + Tf L∆L+ T ∗f L

c∆cLc

+ Ta∆Ω∆ + T ∗a∆

cΩc∆c + TαΩΦΦ + T ∗αΩ

cΦΦ + h.c.]

+[

BµΦΦ +BM∆∆∆ +BM∆

∗∆c∆c +BMΩΩΩ +BMΩ

∗ΩcΩc + h.c.]

. (3)Again, family and gauge indi es have been omitted for the sake of simpli ity. The LRmodel itself does not, of ourse, x the values of the soft SUSY breaking terms. In thenumeri al evaluation of the RGEs we will resort to mSugra-like boundary onditions, i.e.6

m20I3×3 = m2

Q = m2Qc = m2

L = m2Lc , m2

0I2×2 = m2Φ, m2

0 = m2∆ = m2

∆= m2

∆c = m2∆c = m2

Ω =

m2Ωc , M1/2 = M1 = M2 = M3, TQ = A0YQ, TL = A0YL, Tf = A0f, Ta = A0a, Tα = A0α,

Bµ = B0, BM∆= B0M∆, BMΩ

= B0MΩ. The superpotential ouplings f , YQ and YL arexed by the low-s ale fermion masses and mixing angles. Their values at the GUT s ale areobtained by RGE running. This will be dis ussed in more detail in se tion IID.The breaking of the LR gauge group to the MSSM gauge group takes pla e in two steps:SU(2)R × U(1)B−L → U(1)R × U(1)B−L → U(1)Y . In the rst step the neutral omponentof the triplet Ω takes a vev:

〈Ωc 0〉 = vR√2

(4)whi h breaks SU(2)R. However, sin e I3R(Ωc 0) = 0 there is a U(1)R symmetry left over.Next, the group U(1)R × U(1)B−L is broken by

〈∆c 0〉 = vBL√2, 〈∆c 0〉 = vBL√

2. (5)The remaining symmetry is now U(1)Y with hyper harge dened as Y = I3R + B−L

2.The tadpole equations do not link Ωc, ∆c and ∆c with their left-handed ounterparts,due to supersymmetry. Thus, the left-handed triplets an have vanishing vevs [46 and themodel produ es only a type-I seesaw.Although a hierar hy between the two breaking s ales may exist, vBL ≪ vR, one annotnegle t the ee ts of the se ond breaking stage on the rst one, sin e mass terms of Ω and

∆ enter in both tadpole equations. If we assume vBL = vBL the tadpole equations of themodel an be written∂V

∂vR= 4|MΩ|2vR +

1

2|a|2v2BLvR − 1

2v2BL [a

∗(M∆ +MΩ) + c.c] = 0 , (6)∂V

∂vBL

= |M∆|2vBL +1

4|a|2(v2BL + v2R)vBL − 1

2vBLvR [a∗(M∆ +MΩ) + c.c] = 0 . (7)In these equations (small) soft SUSY breaking terms have been negle ted. Similarly, atthis stage there are no ele troweak symmetry breaking vevs vd and vu. From equations(6) and (7) one sees that, in fa t, there is an inverse hierar hy between the vevs and thesuperpotential masses M∆, MΩ, given by

vR =2M∆

a, vBL =

2

a(2M∆MΩ)

1/2 . (8)And so, vBL ≪ vR requires M∆ ≫ MΩ, as has already been dis ussed in [46.B. Step 2: From SU(2)R breaking s ale to U(1)B−L breaking s aleAt this step the gauge group is SU(3)c×SU(2)L×U(1)R×U(1)B−L. The parti le ontentof the model from the SU(2)R breaking s ale to the U(1)B−L breaking s ale is given in tableII. 7

Supereld generations SU(3)c SU(2)L U(1)R U(1)B−L

Q 3 3 2 0 13

dc 3 3 1 12 −1

3

uc 3 3 1 −12 −1

3

L 3 1 2 0 −1

ec 3 1 1 12 1

νc 3 1 1 −12 1

Hd 1 1 2 −12 0

Hu 1 1 2 12 0

∆c 0 1 1 1 1 -2∆c 0 1 1 1 -1 2Ω 1 1 3 0 0Ωc 0 1 1 1 0 0TABLE II. Matter ontent from the SU(2)R breaking s ale to the U(1)B−L breaking s ale.Some omments might be in order. Despite M∆ being of the order of vR (or larger), seeeq.(8), not all omponents of the ∆ superelds re eive large masses. The neutral omponentsof ∆c and ∆c lie at the vBL s ale. One an easily he k that the F-term ontributions totheir masses vanish in the minimum of the s alar potential eq. (8). Moreover, Ωc does notgenerate D-terms ontributions to their masses. Therefore, ontrary to the other omponentsof the ∆ triplets, they only get masses at the vBL s ale. On the other hand, one might guessthat all omponents in the Ω,Ωc superelds should be retained at this stage, sin e theirsuperpotential mass MΩ is required to be below vBL. However, some of their omponents get ontributions from SU(2)R breaking, and thus they be ome heavy. The harged omponentsof Ωc do develop large masses, in the ase of the s alars through D-terms, while in the aseof the fermions due to their mixing with the harged gauginos W±

R , whi h have massesproportional to vR. However, the neutral omponents of Ωc do not get SU(2)R breaking ontributions, sin e they have I3R(Ωc 0) = 0, and then they must be in luded in this energyregime. See referen e [47 for a more quantitative dis ussion.After SU(2)R breaking the two bidoublets Φ1 and Φ2 get split into four SU(2)L dou-blets. Two of them must remain light, identied with the two Higgs doublets of the MSSM,responsible for EW symmetry breaking, while, at the same time, the other two get massesof the order of vR. This strong hierar hy an be only obtained by imposing a ne-tuning ondition on the parameters involved in the bidoublet se tor.The superpotential terms mixing the four SU(2)L doublets an be rewritten asWM = (Hf

d )TMHH

fu (9)where Hf

d = (H1d , H

2d) and Hf

u = (H1u, H

2u) are the avour eigenstates. In this basis reads

8

the matrixMH =

(

µ11 µ12 + α12MR

µ12 − α12MR µ22

)

, (10)where the relations µij = µji and αij = −αji have been used and MR = vR2has been dened.In order to get two light doublets we impose the ne-tuning ondition [47

Det(MH) = µ11µ22 − (µ212 − α2

12M2R) = 0 . (11)The result of eq. (11) is to split the two Higgs bidoublets into two pairs of doublets (Hd, Hu)Land (Hd, Hu)R, where (Hd, Hu)L is the light pair that appears in table II, and (Hd, Hu)Ra heavy pair with mass of order of vR. In pra ti e, equation (11) implies that one of thesuperpotential parameters must be hosen in terms of the others. Sin e this ne-tuning ondition is not prote ted by any symmetry, the RGEs do not preserve it, and one mustimpose it at the SU(2)R breaking s ale. In our omputation we hose to ompute µ11 interms of the free parameters µ12, µ22, α12 and vR.In order to ompute the resulting ouplings for the light Higgs doublets one must rotatethe original elds into their mass basis. Sin eMH is not a symmetri matrix (unless α12 = 0)one has to rotate independently Hf

d and Hfu , i.e. Hf

d = DHmd , Hf

u = UHmu , where D and Uare orthogonal matri es and Hm

d = (HLd , H

Rd ) and Hm

u = (HLu , H

Ru ) are the mass eigenstates.This way one nds

WM = (Hfd )

TMHHfu = (Hm

d )TDTMHUHmu = (Hm

d )TMHHmu (12)where MH is a diagonal matrix, with eigenvalues

M2H,1 = 0 ,

M2H,2 =

1

µ222

(

α412M

4R + 2α2

12M2R(µ

222 − µ2

12) + (µ222 + µ2

12)2)

. (13)The D and U rotations are, in general, dierent and we parametrize them asD =

(

cos θ1 sin θ1− sin θ1 cos θ1

)

, U =

(

cos θ2 sin θ2− sin θ2 cos θ2

) (14)and getH1

d = cos θ1HLd + sin θ1H

Rd ,

H2d = − sin θ1H

Ld + cos θ1H

Rd , (15)and similar forHu. In general the angles θ1 and θ2 are dierent. However, they are onne tedto the same matrix MH and an be al ulated by diagonalizing MH(MH)

T or (MH)TMHand one nds

tan θ1,2 =µ12 ± α12MR

µ22. (16)9

In these expressions Det(MH) = 0 has been used to simplify the result. Exa t Det(MH) = 0implies that the µ-term of the MSSM is zero, so this ondition an only be true up to small orre tions, see the dis ussion below. Note that there are two interesting limits. First,µ12 ≫ α12MR : this implies tan θ1 = tan θ2 and therefore D = U . This is as expe ted, sin ethat limit makes MH symmetri . And, se ond, µ12 ≪ α12MR : this implies tan θ1 = − tan θ2and therefore D = UT .The superpotential at this stage is

W = YuQHuuc + YdQHdd

c + YeLHdec + YνLHuν

c + µHuHd

+ f 1c ν

cνc∆c 0 +M1∆c∆c 0∆c 0 + a1c∆

c 0∆c 0Ωc 0

+ bΩHdHu + bcΩc 0HdHu +MΩΩΩ +MΩcΩc 0Ωc 0. (17)Parti les belonging to the same SU(2)R gauge multiplets split due to their dierent U(1)R harges. At this stage both the LR group, that symmetrizes the SU(2)L and SU(2)R gaugeintera tions, and the dis rete parity symmetry that we imposed on the ouplings are broken.The soft terms are

−Lsoft = m2QQ

†Q+m2uc uc†uc +m2

dc dc†dc +m2

LL†L+m2

ec ec†ec +m2

νc νc†νc

+ m2Hu

H†uHu +m2

HdH†

dHd +m2∆c 0∆c 0†∆c 0 +m2

∆c 0∆c 0 †∆c 0

+ m2ΩΩ

†Ω+m2Ωc 0Ωc 0 †Ωc 0 +

1

2

[

M1B0B0 +MLWLWL +MRW 0

RW0R +M3gg + h.c.

]

+[

TuQHuuc + TdQHddc + TeLHdec + TνLHuνc (18)+ T 1

fc νcνc∆c 0 + T 1

ac∆c 0Ωc 0∆c 0 + TbΩHdHu + TbcΩ

c 0HdHu + h.c.]

+[

BµHuHd +BM1∆c∆c 0∆c 0 +BMΩ

ΩΩ +BMcΩΩc 0Ωc 0 + h.c.

]

.Again we suppress gauge and family indi es.We must impose mat hing onditions at the SU(2)R breaking s ale. These are for super-potential parameters given byYd = Y 1

Q cos θ1 − Y 2Q sin θ1 , Yu = −Y 1

Q cos θ2 + Y 2Q sin θ2 ,

Ye = Y 1L cos θ1 − Y 2

L sin θ1 , Yν = −Y 1L cos θ2 + Y 2

L sin θ2 ,

f 1c = −f ∗ , a1c = − a∗√

2,

M1∆c = M∗

∆ , MΩc = M∗Ω ,

b = 2αR , bc =√2α∗R , (19)where R = sin(θ1 − θ2). For the soft masses we have

m2uc = m2

dc = m2Qc , (20)

m2ec = m2

νc = m2Lc ,

m2∆c 0 = m2

∆c ,

m2∆c 0 = m2

∆c ,

m2Ωc 0 = m2

Ωc ,

ML = MR = M2 .10

Soft trilinears mat hing follow orresponding onditions. In addition, one hasm2

Hd= cos2 θ1(m

2Φ)11 + sin2 θ1(m

2Φ)22 − sin θ1 cos θ1

[

(m2Φ)12 + (m2

Φ)21]

,

m2Hu

= cos2 θ2(m2Φ)11 + sin2 θ2(m

2Φ)22 − sin θ2 cos θ2

[

(m2Φ)12 + (m2

Φ)21]

,as obtained when the operator m2ΦΦ

†Φ is proje ted into the light Higgs doublets operators(HL

d )†HL

d and (HLu )

†HLu . Gauge ouplings are mat hed as gL = gR = g2.C. Step 3: From U(1)B−L breaking s ale to EW/SUSY s aleWemention this stage only for ompleteness, sin e the last regime is just the usual MSSM.We need mat hing onditions in the gauge se tor. Sin e U(1)R ×U(1)B−L breaks to U(1)Y ,the MSSM gauge oupling g1 will be a ombination of gR and gBL. The resulting relationshipis

g1 =

√5gRgBL

√

2g2R + 3g2BL

. (21)Analogously, the following ondition holds for gaugino massesM1(MSSM) =

2g2RM1 + 3g2BLMR

2g2R + 3g2BL

. (22)Note that in the last two equations the gauge ouplings are GUT-normalized. Ele troweaksymmetry breaking o urs as in the MSSM. We take the Higgs doublet vevs〈H0

d〉 =vd√2, 〈H0

u〉 =vu√2, (23)as free parameters and then solve the tadpole equations to nd µMSSM and Bµ. µMSSM mustbe dierent from zero, that is Det(MH) an not be exa tly zero. Instead the tuning must beexa t up to Det(MH) = O(µ2

MSSM). As usual tanβ = vuvd

is used as a free parameter. Alsothe sign of µMSSM is not onstrained as usual.D. Neutrino masses, LFV and Yukawa ouplingsNeutrino masses are generated after U(1)B−L breaking through a type-I seesaw me ha-nism. The matrix f 1c leads to Majorana masses for the right-handed neutrinos on e ∆c 0gets a vev. We dene the seesaw s ale as the lightest eigenvalue of

MS ≡ f 1c vBL . (24)As usual, we an always rotate the elds to a basis where MS is diagonal. However,this will introdu e lepton avour violating entries in the YLi

Yukawas, see dis ussion below.As mentioned above, ontrary to non-supersymmetri LR models [2, there is no type-II ontribution to neutrino masses. 11

parameter best t 2-σ∆m2

21[10−5eV2] 7.59+0.23

−0.18 7.22 − 8.03

|∆m231|[10−3eV2] 2.40+0.12

−0.11 2.18 − 2.64

sin2 θ12 0.318+0.019−0.016 0.29 − 0.36

sin2 θ23 0.50+0.07−0.06 0.39 − 0.63

sin2 θ13 0.013+0.013−0.009 ≤ 0.039TABLE III. Best-t values with 1-σ errors and 2-σ intervals (1 d.o.f.) taken from the referen e [60,whi h is updated ontinuously on the web.Global ts to all available experimental data provide values for the parameters involvedin neutrino os illations, see table III for updated results and ref. [61, 62 for experimentalresults. As rst observed in [63, these data imply that the neutrino mass matrix an bediagonalized to a good approximation by the so- alled tri-bimaximal mixing pattern:

UTBM =

√

23

√

13

0

− 1√6

1√3

− 1√2

− 1√6

1√3

1√2

. (25)The matrix produ t Yν · (f 1c )

−1 · Y Tν is onstrained by this parti ular stru ture. LFV entries an be present in both Yν and f 1

c , see also the dis ussion about parameter ounting in thenext subse tion. However, in the numeri al se tion we will onsider only two spe i kindsof ts:• Yν-t: avour stru ture in Yν and diagonal f 1

c .• f -t: avour stru ture in f 1

c and diagonal Yν .While at rst it may seem either way of doing the t is equivalent, f 1c and Yν in our setup anleave dierent tra es in the soft slepton mass parameters if vBL ≪ vR. This last onditionis essential to distinguish between both possibilities, be ause otherwise one obtains thestraightforward predi tion that LFV entries in left and right slepton are equal, due to theassumed LR symmetry above vR.These two types of t were already dis ussed in referen e [64, whi h investigates lowenergy LFV signatures in a supersymmetri seesaw model where the right-handed neutrinomass is generated from a term of the form f∆cνcνc. When the s alar omponent of ∆ca quires a vev a type-I seesaw is obtained, generating masses for the light neutrinos. There-fore, this model has the ingredients to a ommodate a Yν-t, named as Dira LFV in [64,or a f -t, named as Majorana LFV. Note, however, that the left-right symmetry, entral inour work, is missing in this referen e, thus implying dierent signatures at the ele troweaks ale.The dieren e in phenomenology of the two ts an be easily understood onsideringapproximated expressions for the RGEs for m2

L and m2ec . In the rst step, from the GUT12

s ale to the vR s ale RGEs at 1-loop order an be written in leading-log approximation as[51∆m2

L = − 1

4π2

(

3ff † + Y(k)L Y

(k) †L

)

(3m20 + A2

0) ln

(

mGUT

vR

)

,

∆m2Lc = − 1

4π2

(

3f †f + Y(k) †L Y

(k)L

)

(3m20 + A2

0) ln

(

mGUT

vR

)

. (26)Of ourse, also the A parameters develop LFV o-diagonals in the running. We do not givethe orresponding approximated equations for brevity. After parity breaking at the vR s alethe Yukawa oupling YL splits into Ye, the harged lepton Yukawa, and Yν , the neutrinoYukawa. The later ontributes to LFV entries in the running down to the vBL s ale. Thus,∆m2

L ∼ − 1

8π2YνY

†ν

(

3m2L|vR + A2

e|vR)

ln

(

vRvBL

)

,

∆m2ec ∼ 0 , (27)where m2

L|vR is the matrix m2L at the s ale vR and A2

e|vR is dened as Te = YeAe and also hasto be taken at vR. In order to understand the main dieren e between the two ts, let us rst onsider the f -t. This assumes that Yν is diagonal at the seesaw s ale and thus the observedlow energy mismat h between the neutrino and harged lepton se tors is due to a non-trivialavour stru ture in f 1c . Of ourse, non-diagonal entries in f generate in the running alsonon-diagonal entries in Yν and Ye, but these an be negle ted in rst approximation. In this ase, equations (26) and (27) show that the LR symmetry makes m2

L and m2ec run with thesame avour stru ture and the magnitudes of their o-diagonal entries at the SUSY s aleare similar. If, on the other hand, Yν is non-trivial (Yν-t), while f is diagonal, the runningfrom the GUT s ale to the vR s ale indu es again the same o-diagonal entries in m2

L andm2

Lc . However, from vR to vBL the o-diagonals entries in m2L ontinue to run, while thosein m2

ec do not. This ee t, generated by the right-handed neutrinos via the Yν Yukawas,indu es additional avour violating ee ts in the L se tor ompared to the R se tor. SeeingLFV in both left and right slepton se tors thus allows us to indire tly learn about the highenergy theory. We will study this in some detail in the numeri al se tion below.E. Parameter ountingLet us briey summarize the free parameters of the model. With the assumption ofmSugra (or better: mSugra-like) boundary onditions, in the SUSY breaking se tor weonly have the standard parameters m0, M1/2, A0, tan β, sign(µMSSM). Thus, we ount 4+1parameters in the soft terms. We note in passing that the soft terms of the heavy se tor,of ourse, do not have to follow stri tly the onditions outlined in equation (3), as long asthese parameters are small ompared to vBL there are no hanges ompared to the abovedis ussion.In the superpotential we have a, α, µ,M∆ andMΩ. This leaves, at rst sight, 7 parametersfree. However, we an redu e them to 4+2 parameters as follows. Sin e αij = −αji, α only13

ontains one free parameter: α12. The matrix µ has 3 entries, but one of them, µ11, is xedby the ne-tuning ondition Det(MH) = O(µ2MSSM). This leaves two free parameters, µ12,

µ22. We have traded M∆ and MΩ for the vevs vR, vBL, sin e ln( vRvBL

) and ln(vGUT

vR) enterinto the RGEs and thus an, at least in prin iple, be determined from low-energy spe tra.There are then in summary 6 parameters, four independent of low-energy onstraints andtwo whi h ould be xed from LFV data, see below.In addition, in the superpotential we have the Yukawa matri es YQi

, YLiand f . Let's onsider the quark se tor rst. Sin e we an always go to a basis in whi h one of the YQi

isdiagonal with only real entries, there are 12 parameters. Ten of them are xed by six quarkmasses, three CKM angles and the CKM phase, leaving two phases undetermined.In the lepton se tor we have the symmetri matri es, YL1and YL2

. As with the quarkse tor, a basis hange shows that there are only 12 free parameters. f is symmetri and thus ounts as another 9 parameters. Going to a basis in whi h f is diagonal does not redu e thenumber of free parameters, sin e in this basis we an no longer assume one of the YLito bediagonal. In summary there are thus free 21 parameters in these three matri es.In the simple, pure seesaw type-I with three generations of right-handed neutrinos thenumber of free parameters is 21. Only 12 of them an be xed from low-energy data: threeneutrino and three harged lepton masses, three leptoni mixing angles and three phases (twoMajorana and one Dira phase). However, as pointed out in [11, in prin iple, m2

L ontains9 observable entries and thus, if the normalization (i.e. m0, A0, tanβ et .) is known fromother sfermion measurements, one ould re- onstru t the type-I seesaw parameters 6.How does the SUSY LR model ompare to this? We have, as dis ussed above, also21 parameters in the three oupling matri es, but neutrino masses depend also on vBL.However, in prin iple, we have 9 more observables in m2ec , assuming again that the softSUSY breaking terms an be extra ted from other measurements. Sin e in the RGEs also

vR appears we have in total 23 parameters whi h need to be determined. The number ofobservables, on the other hand is xed to 30 in total, as we have 12 (low-energy leptonse tor) plus 9 (left sleptons) plus 9 (right sleptons) possible measurements.III. NUMERICAL RESULTSA. Pro edure for numeri sAll ne essary, analyti al expressions were al ulated with SARAH. For this purpose, twodierent model les for the model above the two threshold s ales were reated and usedto al ulate the full set of 2-loop RGEs. SARAH al ulates the RGEs using the generi expressions of [53 in the most general form respe ting the omplete avour stru ture. TheseRGEs were afterwards exported to Fortran ode and implemented in SPheno. As starting6 Of ourse, this dis ussion is slightly a ademi , sin e at least one of the Majorana phases will never bemeasured in praxis. 14

point for the RGE running, the gauge and Yukawa ouplings at the ele troweak s ale areused. In the al ulation of the gauge and Yukawa ouplings we follow losely the pro eduredes ribed in ref. [58: the values for the Yukawa ouplings giving mass to the SM fermionsand the gauge ouplings are determined at the s ale MZ based on the measured values forthe quark, lepton and ve tor boson masses as well as for the gauge ouplings. Here, wehave in luded the 1-loop orre tions to the mass of W- and Z-boson as well as the SUSY ontributions to δV B for al ulating the gauge ouplings. Similarly, we have in luded the omplete 1-loop orre tions to the self-energies of SM fermions [65. Moreover, we haveresummed the tan β enhan ed terms for the al ulation of the Yukawa ouplings of the b-quark and the τ -lepton as in [58. The va uum expe tation values vd and vu are al ulatedwith respe t to the given value of tanβ at MZ . Sin e we are working with two distin tthreshold s ales, not all heavy elds are integrated out at their mass and the orresponding 1-loop boundary onditions at the threshold s ales are needed. It is known that these parti les ause a nite shift in the gauge ouplings and gaugino masses. The general expressions are[66gi → gi

(

1± 1

16π2g2i I

i2(r) ln

(

M2

M2T

))

, (28)Mi → Mi

(

1± 1

16π2g2i I

i2(r) ln

(

M2

M2T

))

. (29)I i2(r) is the Dynkin index of a eld transforming as representation r with respe t to thegauge group belonging to the gauge oupling gi, M is the mass of this parti le and MT is thethreshold s ale. When evaluating the RGEs from the low to the high s ale, the ontributionis positive, when running down, it is negative. The dierent masses used for al ulatingthe nite shifts are the eigenvalues of the full tree-level mass matrix of the harged, heavyparti les removed from the spe trum. The orre t mass spe trum is al ulated in an iterativeway. The GUT s ale is dened as the s ale at whi h gBL = g2 = gGUT holds. Generally,there is dieren e with g3 to gGUT in the per ent range, the a tual numeri al mismat hdepending on the s ales vBL and vR and being larger for lower values of vBL and vR. Ithas been stressed in parti ular in [67 that within supersymmetri LR models, the LRsymmetry breaking s ale has to be lose to the GUT s ale, otherwise this mismat h willgrow too large. However, several solutions are known. In [68 it was pointed out that GUTthresholds - unknown unless the GUT model, in luding the omplete Higgs se tor used tobreak the GUT symmetry, is spe ied - an lead to important orre tions, a ounting for thisapparent non-uni ation (for a dis ussion of these ee ts in the ontext of SU(5) see [69).Another possibility is the addition of new parti les to the spe trum. For example, referen e[70 pointed out that new oloured superelds, harged under SU(3)c but singlet underthe other gauge subgroups, an easily lead to gauge oupling uni ation. Nevertheless,we simply use gBL = g2 = gGUT and attribute departures from omplete uni ation to(unknown) thresholds and/or the existen e of additional oloured parti les below mGUT .After applying the GUT s ale boundary onditions, the RGEs are evaluated down to thelow s ale and the mass spe trum of the MSSM is al ulated. The MSSM masses are, in15

1012 1013 1014 1015

150

200

250

m(χ

0 1),

m(χ

0 2),

m(µ

L),

m(µ

R)

vBL [GeV]1014 1015 1016

140

160

180

200

220

240

260

280

m(χ

0 1),

m(χ

0 2),

m(µ

L),

m(µ

R)

vR [GeV]FIG. 1. Example of spe tra at the SUSY s ale and its dependen e on vBL (left side) and vR (rightside). The masses of four states are shown: χ01 (blue line), χ0

2 (blue dashed line), µR (red line)and µL (red dashed line). In both panels the mSugra parameters have been taken as in the SPS3ben hmark point.general, al ulated at the 1-loop level in the DR s heme using on-shell external momenta.For the Higgs elds also the most important 2-loop ontributions are taken into a ount.We note that the orresponding Fortran routines are also written by SARAH but they areequivalent to the routines in luded in the publi version of SPheno based on [65. Theiteration stops when the largest hange in the al ulation of the SUSY and Higgs bosonmasses at mSUSY is below one per-mille between two iterations.B. Mass spe trumThe appearan e of harged parti les at s ales between the ele troweak s ale and the GUTs ale leads to hanges in the beta fun tions of the gauge ouplings [21, 29. This does notonly hange the evolution of the gauge ouplings but also the evolution of the gaugino ands alar mass parameters [22, 29. The LR model ontains additional triplets, and similar towhat is observed in the seesaw models [23 the mass spe trum at low energies is shifted withrespe t to mSugra expe tations. Two examples of this behaviour are shown in gure 1. Inthis gure we show the two lightest neutralino masses and the masses of the left and rightsmuons versus vBL (left side) and vR (right side). We note that also all other sfermion andgaugino masses show the same dependen e and in general smaller values are obtained forlower values of vBL and vR. One nds that gaugino masses depend stronger on vBL and vRthan sfermion masses and that right sleptons are the sfermions for whi h the sensitivity tothese vevs is smallest.The hange in the low energy spe trum, however, maintains to a good degree the standard16

1012 1013 1014 10150.49

0.50

0.51

0.52

0.53

0.54

0.55

M1

M2

vBL [GeV]1012 1013 1014 1015

0.26

0.28

0.30

0.32

0.34

M2

M3

vBL [GeV]FIG. 2. Gaugino mass ratios as a fun tion of vBL for the xed value vR = 1015 GeV. To the left,M1/M2, whereas to the right M2/M3. In both gures the three oloured lines orrespond to threemSugra ben hmark points: SPS1a' (blue), SPS3 (green) and SPS5 (red). Note the small variationin the numbers on the Y axis.mSugra expe tation for the ratios of gaugino asses, as shown in gures 2 and 3. Here, gure2 shows the ratios M1/M2 and M2/M3 versus vBL, while gure 3 shows the same ratiosversus vR. Shown are the results for three dierent SUSY points, whi h in the limit ofvR, vBL → mGUT approa h the standard SPS points SPS1a' [71, SPS3 and SPS5 [72. Forexample, the ratio M1/M2 is expe ted to be (5/3) tan2 θW ≃ 0.5 at 1-loop order in mSugra.The exa t ratio, however, depends on higher order orre tions, and thus on the SUSYspe trum. The LR model will thus appear rather mSugra like, if these ratios are measured.Only with very high pre ision on mass measurements, possible only at a linear ollider, anone hope to nd any (indire t) dependen e on vBL and vR.C. LFV of leptonsLepton avour violation in harged lepton de ays has attra ted a lot of attention forde ades. Pro esses like µ → eγ are highly suppressed in the standard model (plus non-zeroneutrino masses) due to the GIM me hanism [73, and thus the observation of these rarede ays would imply new physi s. The MEG experiment [74 is urrently the most advan edexperimental setup in the sear h for µ+ → e+γ. This rare de ay will be observed if itsbran hing ratio is above the MEG expe ted sensitivity, around Br(µ → eγ) ∼ 10−13.LFV de ays like li → ljγ are indu ed by 1-loop diagrams with the ex hange of neutralinosand sleptons. They an be des ribed by the ee tive Lagrangian, see for example the review[75,

Leff = emi

2liσµνF

µν(AijLPL + Aij

RPR)lj + h.c. . (30)17

1014 1015 1016

0.515

0.520

0.525

0.530

0.535

0.540

0.545

0.550

M1

M2

vR [GeV]1014 1015 1016

0.29

0.30

0.31

0.32

0.33

M2

M3

vR [GeV]FIG. 3. Gaugino mass ratios as a fun tion of vR for the xed value vBL = 1014 GeV. To the left,M1/M2, whereas to the right M2/M3. In both gures the three oloured lines orrespond to threemSugra ben hmark points: SPS1a' (blue), SPS3 (green) and SPS5 (red). Note the small variationin the numbers on the Y axis.Here PL,R = 1

2(1 ∓ γ5) are the usual hirality proje tors and therefore the ouplings ALand AR are generated by loops with left and right sleptons, respe tively. In our numeri al al ulation we use exa t expressions for AL and AR. However, for an easier understandingof the numeri al results, we note that the relation between these ouplings and the sleptonsoft masses is very approximately given by

AijL ∼ (m2

L)ijm4

SUSY

, AijR ∼ (m2

ec)ijm4

SUSY

, (31)where mSUSY is a typi al supersymmetri mass. Here it has been assumed that (a) hargino/neutralino masses are similar to slepton masses and (b) A-terms mixing left-right transitions are negligible. Therefore, due to the negligible o-diagonal entries in m2ec ,a pure seesaw model predi ts AR ≃ 0.The bran hing ratio for li → ljγ an be al ulated from the previous formulas. The resultis

Br(li → ljγ) =48π3α

G2F

(

|AijL |2 + |Aij

R|2)

Br(li → ljνiνj) . (32)Figure 4 shows two examples for Br(µ → eγ) in the m0,M1/2 plane. Here, we have xedvBL = 1014 GeV and vR = 1015 GeV and show to the left MS = 1012 GeV, whereas to theright MS = 1013 GeV. Here we have assumed a degenerate spe trum right-handed neutrinoswhi h we denote by MS = MRi. On e Yukawas are tted to explain the observed neutrinomasses, the bran hing ratio shows an approximately quadrati dependen e on the seesaws ale, with lower MS giving smaller Br(µ → eγ). As expe ted, the bran hing ratio alsostrongly de reases as m0 and/or M1/2 in rease. This is be ause the superparti les in the18

MSeesaw= 1012 (GeV) tanβ=10, A0=0 (GeV)

1.2x10-11

10-12

10-13

10-14 10-15

250 500 750 1000 1250 1500M1/2 (GeV)

0

250

500

750

1000

1250

1500

m0

(GeV

)

MSeesaw= 1013 (GeV) tanβ=10, A0=0 (GeV)

1.2x10-11

10-12 10-13

2x10-14

250 500 750 1000 1250 1500M1/2 (GeV)

0

250

500

750

1000

1250

1500

m0

(GeV

)FIG. 4. Contours of Br(µ → eγ) in the m0,M1/2 plane for vBL = 1014 GeV and vR = 1015 GeV.To the left MS = 1012 GeV, whereas to the right MS = 1013 GeV. Neutrino os illation data havebeen tted with the Yν t.loops leading to µ → eγ be ome heavier in these dire tions, suppressing the de ay rate. Infa t, from equations (31) and (32) one easily nds the dependen e

Br(µ → eγ) ∼ 48π3α

G2F

(m2L,ec)

2ij

m8SUSY

, (33)whi h shows that Br(µ → eγ) de reases as m−8SUSY .It is also remarkable that for a given seesaw s ale, Br(µ → eγ) is sizeably larger in theLR model than in a pure seesaw type-I model, see for example [25. The explanation of thisis that right sleptons ontribute signi antly in the LR model to Br(µ → eγ) and these ontributions are absent in seesaw models.As already dis ussed, a pure seesaw model predi ts simply AR ≃ 0. However, in the LRmodel we expe t a more ompli ated pi ture. Left-right symmetry implies that, above theparity breaking s ale, non-negligible avour violating entries are generated inm2

ec . Therefore,AR 6= 0 is obtained at low energy. The angular distribution of the outgoing positron at, forexample, the MEG experiment ould be used to dis riminate between left- and right-handedpolarized states [76, 77. If MEG is able to measure the positron polarization asymmetry,dened as

A(µ+ → e+γ) =|AL|2 − |AR|2|AL|2 + |AR|2

, (34)there will be an additional observable to distinguish from minimal seesaw models. In a pureseesaw model one expe ts A ≃ +1 to a very good a ura y. However, the LR model typi ally19

MSeesaw= 1012 (GeV) tanβ=10, A0=0 (GeV)

0.5

0.3

0.75

0.9

0.95

0.99

250 500 750 1000 1250 1500M1/2 (GeV)

0

250

500

750

1000

1250

1500

m0

(GeV

)

MSeesaw= 1013 (GeV) tanβ=10, A0=0 (GeV)

0.50.3

0.75

0.9

0.95

0.99

250 500 750 1000 1250 1500M1/2 (GeV)

0

250

500

750

1000

1250

1500

m0

(GeV

)FIG. 5. Contours of A(µ+ → e+γ) in the m0,M1/2 plane. To the left MS = 1012 GeV, whereas tothe right MS = 1013 GeV. The parameters have been hosen as in gure 4.leads to signi ant departures from this expe tation, giving an interesting signature of thehigh energy restoration of parity.Figure 5 shows ontours for A(µ+ → e+γ) in the m0,M1/2 plane. For the orrespondingbran hing ratios see gure 4. Note the rather strong dependen e on m0. The latter an beunderstood as follows. Sin e vBL in these examples is one order of magnitude smaller thanvR, and the Yν t has been used, the LFV mixing angles in the left slepton se tor are largerthan the orresponding LFV entries in the right sleptons. At very large values of m0, werethe masses of right and left sleptons are of omparable magnitude, therefore left LFV ismore important and the model approa hes the pure seesaw expe tation. At smaller valuesof m0, right sleptons are lighter than left sleptons, and due to the strong dependen e ofµ → eγ on the sfermion masses entering the loop al ulation, see eq. (31), AR and AL anbe ome omparable, despite the smaller LFV entries in right slepton mass matri es. In thelimit of very small right slepton masses the model then approa hes A ∼ 0. We have notexpli itly sear hed for regions of parameter spa e with A < 0, but one expe ts that negativevalues for A are possible if vBL is not mu h below vR and sleptons are light at the sametime, i.e. small values of m0 and M1/2. Note that, again due to the LR symmetry above tovR, the model an never approa h the limit A = −1 exa tly.The positron polarization asymmetry is very sensitive to the high energy s ales. Figure6 shows A as a fun tion of vR for MS = 1013 GeV, vBL = 1014 GeV and the mSugraparameters as in the SPS3 ben hmark point. The plot has been obtained using the Yν t.This example shows that as vR approa hesmGUT the positron polarizationA approa hes +1,whi h means AL dominates the al ulation. This is be ause, in the Yν t, the right-handedLFV soft slepton masses, and thus the orresponding AR oupling, only run from mGUT to20

1014 1015 1016

0.3

0.4

0.5

0.6

0.7

A(µ

+→

e+γ)

vR [GeV]FIG. 6. Positron polarization asymmetry A(µ+ → e+γ) as a fun tion of vR for the parameter hoi e MS = 1013 GeV and vBL = 1014 GeV. The mSugra parameters have been taken as in theSPS3 ben hmark point and neutrino os illation data have been tted with the Yν t, assumingdegenerate right-handed neutrinos.vR.

A(µ+ → e+γ) also has an important dependen e on the seesaw s ale. This is shownin gure 7, where A is plotted as a fun tion of the lightest right-handed neutrino mass.This dependen e an be easily understood from the seesaw formula for neutrino masses. Itimplies that larger MS requires larger Yukawa parameters in order to t neutrino masseswhi h, in turn, leads to larger avour violating soft terms due to RGE running. However,note that, for very small seesaw s ales all lepton avour violating ee ts are negligible andno asymmetry is produ ed, sin e AL ∼ AR ∼ 0.In addition, gure 7 shows again the relevan e of vR, whi h determines the parity breakings ale at whi h the LFV entries in the right-handed slepton se tor essentially stop running.Lighter olours indi ate larger vR. As shown already in gure 6 for a parti ular point, thepositron polarization approa hes +1 as vR is in reased.Below the SU(2)R breaking s ale parity is broken and left and right slepton soft massesevolve dierently. The approximate solutions to the RGEs in equations (26) and (27) showthat, if neutrino data is tted a ording to the Yν t, the left-handed ones keep runningfrom the SU(2)R breaking s ale to the U(1)B−L s ale. In this ase one expe ts larger avourviolating ee ts in the left-handed slepton se tor and a orrelation with the ratio vBL/vR,whi h measures the dieren e between the breaking s ales. This orrelation, only present inthe Yν t, is shown in gure 8. On the one hand, one nds that as vBL and vR be ome verydierent, vBL/vR ≪ 1, the positron asymmetry approa hes A = +1. On the other hand,when the two breaking s ales are lose, vBL/vR ∼ 1, this ee t disappears and the positronpolarization asymmetry approa hes A = 0. Note that the Yν t does not usually produ e a21

A(µ

+→

e+γ)

MS [GeV]FIG. 7. Positron polarization asymmetry A(µ+ → e+γ) as a fun tion of the seesaw s ale, denedas the mass of the lightest right-handed neutrino, for the parameter hoi e vBL = 1015 GeV andvR ∈ [1015, 1016] GeV. Lighter olours mean higher values of vR. The mSugra parameters have beentaken as in the SPS3 ben hmark point and neutrino os illation data have been tted with the Yνt, assuming degenerate right-handed neutrinos.negative value for A sin e the LFV terms in the right slepton se tor never run more than the orresponding terms in the left-handed se tor. The only possible ex eption to this generalrule is, as dis ussed above, in the limit of very small m0 and vBL/vR ∼ 1.The determination of the ratio vBL/vR from gure 8 is shown to be very ina urate.This is due to the fa t that other parameters, most importantly mGUT (whi h itself has animportant dependen e on the values of vBL and vR), have a strong impa t on the results.Therefore, although it would be possible to onstrain the high energy stru ture of the theory,a pre ise determination of the ratio vBL/vR will require additional input. Figure 9, on theother hand, shows that the polarization asymmetry A(µ+ → e+γ) is mu h better orrelatedwith the quantity log(vR/mGUT )/ log(vBL/mGUT ). This is as expe ted from equations (26)and (27) and onrms the validity of this approximation.We lose our dis ussion on the positron polarization asymmetry with some omments onthe f t. Sin e this type of t leads to ∆m2

L ∼ ∆m2ec ∼ 0 in the vBL − vR energy region,there is little dependen e on these symmetry breaking s ales. This is illustrated in gure 10,where the asymmetry A is plotted as a fun tion of vR for three dierent mSugra ben hmarkpoints: SPS1a' (blue line), SPS3 (green line) and SPS5 (red line). One learly sees that thedependen e on vR is quite weak ompared to the Yν t. In fa t, the variations in this gureare mostly due to the hanges in the low energy supersymmetri spe trum due to dierent

vR values. In the ase of the f -t one then typi ally nds A ∈ [0.0− 0.3].22

A(µ

+→

e+γ)

vBL

vRFIG. 8. Positron polarization asymmetry A(µ+ → e+γ) as a fun tion of the ratio vBL/vR. Theseesaw s ale MS has been xed to 1013 GeV, whereas vBL and vR take values in the ranges vBL ∈[1014, 1015] GeV and vR ∈ [1015, 1016] GeV. Lighter olours indi ate larger vBL. The mSugraparameters have been taken as in the SPS3 ben hmark point and neutrino os illation data havebeen tted with the Yν t, assuming degenerate right-handed neutrinos.D. LFV at LHC/ILCLepton avour violation might show up at ollider experiments as well. Although thefollowing dis ussion is fo used on the LHC dis overy potential for LFV signatures, let usemphasize that a future linear ollider will be able to determine the relevant observableswith mu h higher pre ision.Figure 11 shows Br(τi → χ0

1 e) and Br(τi → χ01 µ) as a fun tion of the seesaw s ale. Thedashed lines orrespond to τ1 ≃ τR and the solid ones to τ2 ≃ τL. As in the ase of µ → eγ,see gure 4, lower seesaw s ales imply less avour violating ee ts due to smaller Yukawa ouplings. Moreover, gure 11 presents the same results for two dierent ben hmark points,SPS1a' and SPS3. As already shown in gure 4, µ → eγ is strongly dependent on the SUSYspe trum. For lighter supersymmetri parti les, as in the ben hmark point SPS1a', µ → eγis large, setting strong limits on the seesaw s ale and thus on the possibility to observe LFVat olliders. In the ase of heavier spe trums, as in SPS3, µ → eγ is still the most stringent onstraint, but larger values of the seesaw s ale and thus LFV violating bran hing ratiosbe ome allowed. Whether de ays su h as Br(τi → χ0

1 e) and Br(τi → χ01 µ) are observableat the LHC or not, thus depends very sensitively on the unknown m0, M1/2 and MS.Furthermore, the right panel of gure 11 also shows that right staus an also have LFVde ays with sizable rates. Of ourse, as emphasized already above, this is the main noveltyof the LR model ompared to pure seesaw models. This is dire t onsequen e of parityrestoration at high energies. 23

0.2 0.4 0.6 0.8 1.0

0.3

0.4

0.5

0.6

0.7

A(µ

+→

e+γ)

log(vR/mGUT )

log(vBL/mGUT )FIG. 9. Positron polarization asymmetry A(µ+ → e+γ) as a fun tion oflog(vR/mGUT )/ log(vBL/mGUT ). The parameters have been hosen as in gure 8.

1014 1015 1016

0.10

0.15

0.20

0.25

0.30

A(µ

+→

e+γ)

vR [GeV]FIG. 10. Positron polarization asymmetry A(µ+ → e+γ) as a fun tion of vR for three dierentmSugra ben hmark points: SPS1a' (blue line), SPS3 (green line) and SPS5 (red line). In this gurea xed value vBL = 1014 is taken. Neutrino os illation data have been tted with the f t.Moreover, as in our analysis of the positron polarization asymmetry, one expe ts to ndthat if the dieren e between vR and vBL is in reased, the dieren e between the LFVentries in the L and R se tors gets in reased as well. This property of the Yν t is shown ingure 12, whi h shows bran hing ratios for the LFV de ays of the staus as a fun tion of vBLfor vR ∈ [1015, 5 · 1015] GeV. As the gure shows, the theoreti al expe tation is onrmednumeri ally: the dieren e between Br(τL) and Br(τR) strongly depends on the dieren e24

1011 101210-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

Br(τ

i→

χ0 1e)

,Br(τ

i→

χ0 1µ)

MS [GeV]

Excluded by

µ → eγ

1012 101310-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

Br(τ

i→

χ0 1e)

,Br(τ

i→

χ0 1µ)

MS [GeV]

Excluded by

µ → eγ

FIG. 11. Br(τi → χ01 e) and Br(τi → χ0

1 µ) as a fun tion of the seesaw s ale, dened as the massof the lightest right-handed neutrino, for the parameter hoi e vBL = 1015 GeV and vR = 5 · 1015GeV. The dashed lines orrespond to τ1 ≃ τR and the solid ones to τ2 ≃ τL. To the left, the mSugraparameters have been taken as in the SPS1a' ben hmark point, whereas to the right as in the SPS3ben hmark point. In both gures neutrino os illation data have been tted a ording to the f t,with non-degenerate right-handed neutrinos. The blue shaded regions are ex luded by µ → eγ.between vR and vBL.The question arises whether one an determine the ratio vBL/vR by measuring bothBr(τL) and Br(τR) at olliders. Figure 13 attempts to answer this. Here the ratio Br(τR →χ01 e)/Br(τL → χ0

1 e) is plotted as a fun tion of vBL/vR. A measurement of both bran hingratios would allow to onstrain the ratio vBL/vR and in rease our knowledge on the highenergy regimes. For the sake of brevity we do not present here the analogous plots forother LFV slepton de ays and/or other lepton nal states, sin e they show very similar orrelations with vBL/vR. For example, in prin iple, one ould also use the ratio Br(µR →χ01 τ)/Br(µL → χ0

1 τ) to determine the ratio between the two high s ales.However, as observed also in the polarization asymmetry for µ → eγ there is an importantdependen e on other parameters of the model, espe ially the exa t value of mGUT . Thisimplies a theoreti al un ertainty in the determination of vBL/vR. Again, as for A, a mu hbetter orrelation with log(vR/mGUT )/ log(vBL/mGUT ) is found, see gure 14.In on lusion, to the determine vBL and vR individually more theoreti al input is needed,su h as the GUT s ale thresholds, whi h are needed to x the value ofmGUT . Re all, that wedid not spe ify the exa t values of these thresholds in our numeri al al ulation. This leadsto a oating value of mGUT when vR and vBL are varied. Also more experimental data isneeded to make more denite predi tions. Espe ially SUSY mass spe trum measurements,whi h may or may not be very pre ise at the LHC, depending on the SUSY point realizedin nature, will be of great importan e. Re all that, if in rea h of a linear ollider, slepton25

1014 1015

10-4

10-3

Br(τ

R→

χ0 1µ),

Br(τ

L→

χ0 1µ)

vBL [GeV]FIG. 12. Br(τL → χ01 µ) and Br(τR → χ0

1 µ) as a fun tion of vBL for MS = 1013 GeV andvR ∈ [1015, 5 · 1015] GeV. Red dots orrespond to τ1 ≃ τR, whereas the blue ones orrespond toτ2 ≃ τL. The mSugra parameters have been taken as in the SPS3 ben hmark point and neutrinoos illation data have been tted with the Yν t, assuming degenerate right-handed neutrinos.

Br(

τ R→

χ0 1e)

/Br(

τ L→

χ0 1e)

vBL

vRFIG. 13. Br(τR → χ01 µ)/Br(τL → χ0

1 µ) as a fun tion of vBL/vR. The seesaw s ale MS hasbeen xed to 1013 GeV, whereas vBL and vR take values in the ranges vBL ∈ [1014, 1015] GeV andvR ∈ [1015, 1016] GeV. Lighter olours indi ate larger vBL. The rest of the parameters have been hosen as in gure 12. 26

0.2 0.4 0.6 0.8 1.0

0.10

0.15

0.20

0.25

0.30

Br(

τ R→

χ0 1e)

/Br(

τ L→

χ0 1e)

log(vR/mGUT )

log(vBL/mGUT )FIG. 14. Br(τR → χ01 e)/Br(τL → χ0

1 e) as a fun tion of log(vR/mGUT )/ log(vBL/mGUT ). Theparameters have been hosen as in gure 13.mass and bran hing ratio measurements an be highly pre ise.So far only slepton de ays have been dis ussed. This served to illustrate the most inter-esting signatures of the model, namely, lepton avour violation in the right slepton se tor.However, LHC sear hes for lepton avour violation usually on entrate on the de ay hain[7880χ02 → l±l∓ → χ0

1l±l∓ .This well known signature has been widely studied due to the a urate information it anprovide about the parti le spe trum [8185. Note that in this de ay one assumes usually thatthe χ0

2 themselves stem from the de ay hain qL → qχ02. If the mass orderingmχ0

2> ml > mχ0

1is realized, the dilepton invariant mass [83, 86, dened as m2(l+l−) = (pl+ + pl−)2, has anedge stru ture with a prominent kinemati al endpoint at

[

m2(l+l−)]

max≡ m2

ll =(m2

χ02

−m2l)(m2

l−m2

χ01

)

m2l

, (35)where the masses of the harged leptons have been negle ted. The position of this edge anbe measured with impressively high pre ision at the LHC [8183, implying also an a uratedetermination of the intermediate slepton masses.In fa t, if two dierent sleptons l1,2 have su iently high event rates for χ02 → l±1,2l

∓j →

χ01l

±i l

∓j and their masses allow these hains to be on-shell, two dierent dilepton edge dis-tributions are expe ted [30, 87. This presents a powerful tool to measure slepton masssplittings, whi h in turn allows to dis riminate between the standard mSugra expe tation,with usually negligible mass splittings for the rst two generations, and extended modelswith additional sour es of avour violation.27

1010 1011 1012 1013 1014

10-5

10-4

10-3

∆m

ll

mll

MS [GeV]

1010 1011 1012 1013 1014

10-4

10-3

10-2

∆m

l

ml

MS [GeV]FIG. 15. ∆mll

mll(left-hand side) and ∆m

l

ml(right-hand side) as a fun tion of the seesaw s ale, denedas the mass of the lightest right-handed neutrino, for the parameter hoi e vBL = 1015 GeV and

vR ∈ [1015, 1016] GeV. Blue dots orrespond to the mass distribution generated by intermediate leftsleptons whereas red dots orrespond to the mass distribution generated by the right ones. ThemSugra parameters have been taken as in the SPS3 ben hmark point and neutrino os illation datahave been tted a ording to the Yν t, with degenerate right-handed neutrinos.The relation between the slepton mass splitting and the variation in the position of thekinemati al is edge is found to be [30∆mll

mll=

∆ml

ml

m2χ01

m2χ02

− m4l

(m2l−m2

χ01

)(m2l−m2

χ02

). (36)Here ∆mll(i, j) = mlili −mlj lj is the dieren e between two edge positions, ∆ml = mli

−mljthe dieren e between slepton masses and mll and ml average values of the orrespondingquantities. Note that higher order ontributions of ∆ml

ml

have been negle ted in equation(36).A number of studies about the dilepton mass distribution have been performed [8183, on luding that the position of the edges an be measured at the LHC with an a ura y upto 10−3. Moreover, as shown in referen e [30, this an be generally translated into a similarpre ision for the relative e − µ mass splitting, with some regions of parameter spa e wherevalues as small as 10−4 might be measurable. Sin e this mass splitting is usually negligiblein a pure mSugra s enario, it is regarded as an interesting signature of either lepton avourviolation or non-universality in the soft terms. Furthermore, in the ontext of this paper, itis important to emphasize that pure seesaw models an have this signature only in the leftslepton se tor [31.Figure 15 shows our results for the observables ∆mll

mlland ∆m

l

ml

as a fun tion of the seesaws ale. Large values for MS lead to sizable deviations from the mSugra expe tation, with a28

10-4 10-3 10-2

10-16

10-15

10-14

10-13

10-12

10-11

10-10

Br(

µ→

eγ)

(

∆mll

mll

)

L

10-3 10-2

10-16

10-15

10-14

10-13

10-12

10-11

10-10

Br(

µ→

eγ)

(

∆mll

mll

)

RFIG. 16. Br(µ → eγ) as a fun tion of (∆mll

mll

)

L(left-hand side) and (∆mll

mll

)

R(right-hand side).The parameters are hosen as in gure 15.distin tive multi-edge stru ture in the dilepton mass distribution. Moreover, this ee t isfound in both left- and right- mediated de ays. Observing this ae t would learly pointtowards a non-minimal seesaw model, su h as the LR model we dis uss.As expe ted, these observables are orrelated with other LFV signals [31, 88. Figure16 shows Br(µ → eγ) as a fun tion of (∆mll

mll

)

L(mass distribution with intermediate Lsleptons) and (∆mll

mll

)

R(mass distribution with intermediate R sleptons). Again, the mainnovelty with respe t to the usual seesaw implementations is the orrelation in the rightse tor, not present in the minimal ase [31.Furthermore, the pro ess χ0

2 → χ01l

+i l

−j might provide additional LFV signatures if therate for de ays with li 6= lj is su iently high. Referen e [89 has investigated this possibilityin great detail, performing a omplete simulation of the CMS dete tor in the LHC for thede ay χ0

2 → χ01eµ. The result is given in terms of the quantity

Keµ =Br(χ0

2 → χ01eµ)

Br(χ02 → χ0

1ee) +Br(χ02 → χ0

1µµ), (37)whi h parametrizes the amount of avour violation in χ0

2 de ays. The study, fo used onthe CMS test point LM1 (m0 = 60 GeV, M1/2 = 250 GeV, A0 = 0 GeV, tanβ = 10,sign(µ) = +) [84, on ludes that LFV an be dis overed at the LHC at 5σ level with anintegrated luminosity of 10fb−1 if Keµ ≥ Kmin

eµ = 0.04.Figure 17 shows our omputation ofKeµ as a fun tion of the lightest right-handed neutrinomass, for the parameter hoi e vBL = 1015 GeV and vR = 5 · 1015 GeV. The results areshown splitting the ontributions from intermediate left (blue) and right (red) sleptons.Although the sele ted mSugra parameters belong to the SPS3 point, and not to LM1 as in29

1011 1012 101310-5

10-4

10-3

10-2

10-1

100

KL,R

eµ

MS [GeV]

Excluded by

µ → eγ

FIG. 17. Keµ as a fun tion of the lightest right-handed neutrino mass, for the parameter hoi evBL = 1015 GeV and vR = 5·1015 GeV. The blue urve orresponds to ontributions from intermedi-ate L sleptons, whereas the red one orresponds to intermediate R sleptons. The mSugra parametershave been taken as in the SPS3 ben hmark point, whi h satises m(χ0

2) > m(li) > m(χ01), and thusthe intermediate L and R sleptons an be produ ed on-shell. Neutrino os illation data have beentted a ording to the f t, with non-degenerate right-handed neutrinos. The blue shaded regionis ex luded by µ → eγ.referen e [89, a similar sensitivity for Kmin

eµ is expe ted7. This is be ause the redu tion inthe ross-se tion due to the slightly heavier supersymmetri spe trum is possibly partially ompensated by the orresponding redu tion in the SM ba kground and thus a limiting valueKmin

eµ of a similar order is expe ted. Moreover, [89 uses 10 fb−1 and with larger integratedluminosities even smaller Kmineµ should be ome a essible at the LHC.The main result in gure 17 is that for large MS values the rates for LFV χ0

2 de aysare measurable for both left and right intermediate sleptons. In fa t, for MS & 1012 GeVthe parameter Keµ is above its minimum value for the 5σ dis overy of χ02 → χ0

1eµ. Seereferen es [89, 90 for more details on the LHC dis overy potential in the sear h for LFV inthis hannel.7 Moreover, the LM1 point, being very similar to SPS1a', is strongly onstrained by µ → eγ.30

IV. CONCLUSIONSWe have studied a supersymmetri left-right symmetri model. Our motivation for study-ing this setup was twofold. First, LR models are theoreti ally attra tive, sin e they ontainall the ne essary ingredients to generate a seesaw me hanism, instead of adding it by handas is so often done. And, se ond, in a setup where the SUSY LR is supplemented by avourblind supersymmetry breaking boundary onditions, dierent from all pure seesaw setups,lepton avour violation o urs in both, the left and the right slepton se tors.We have al ulated possible low-energy signals of this SUSY LR model, using full 2-loopRGEs for all parameters. We have found that low-energy lepton avour violating de ays,su h as µ → eγ are (a) expe ted to be larger than for the orresponding mSugra pointsin parameter spa e of seesaw type-I models and (b) the polarization asymmetry A of theoutgoing positron is found to dier signi antly from the pure seesaw predi tion of A = +1in large regions of parameter spa e. We have also dis ussed possible ollider signatures of theSUSY LR model for LHC and a possible ILC. Mass splittings between smuons and sele tronsand LFV violating slepton de ays should o ur in both the left and the right slepton se tor,again dierent from the pure seesaw expe tations.We think therefore that the SUSY LR model is a good example of a beyond minimal,pure seesaw and oers many interesting novelties. For example, the impa t of the interme-diate s ales on dark matter reli density and on ertain mass ombinations and the inuen eof the right-handed neutrino spe trum on low energy observables, are topi s that ertainlydeserve further studies.ACKNOWLEDGEMENTSW.P. thanks IFIC/C.S.I.C. for hospitality. This work was supported by the SpanishMICINN under grants FPA2008-00319/FPA, by the MULTIDARK Consolider CSD2009-00064, by Prometeo/2009/091, by the EU grant UNILHC PITN-GA-2009-237920 andFPA2008-04002-E/PORTU. A.V. thanks the Generalitat Valen iana for nan ial supportand the people at CFTP in Lisbon for hospitality. The work of J. N. E. has been supportedby Fundação para a Ciên ia e a Te nologia through the fellowship SFRH/BD/29642/2006.J. N. E. and J. C. R. also a knowledge the nan ial support from Fundação para a Ciên ia ea Te nologia grants CFTP-FCT UNIT 777 and CERN/FP/109305/2009. W.P. is partiallysupported by the German Ministry of Edu ation and Resear h (BMBF) under ontra t05HT6WWA and by the Alexander von Humboldt Foundation. F.S. has been supported bythe DFG resear h training group GRK1147.Appendix A: RGEsWe present in the following appendi es our results for the RGEs of the model above theU(1)B−L breaking s ale. We will only show the β-fun tions for the gauge ouplings and the31

anomalous dimensions of all hiral superelds. We briey dis uss in this se tion how theseresults were al ulated. Furthermore, we show how they an be used to al ulate the otherβ-fun tions of the models and give as example the 1-loop results for the soft SUSY breakingmasses of the sleptons. The omplete results are given online on this sitehttp://theorie.physik.uni-wuerzburg.de/~fnstaub/supplementary.htmlIn addition, the orresponding model les for SARAH are also given on this web page.1. Cal ulation of supersymmetri RGEsFor a general N = 1 supersymmetri gauge theory with superpotential

W (φ) =1

2µijφiφj +

1

6Y ijkφiφjφk (A1)the soft SUSY-breaking s alar terms are given by

Vsoft = (1

2bijφiφj +

1

6hijkφiφjφk + . .)+ (m2)ijφiφ

∗j . (A2)The anomalous dimensions are given by [53

γ(1)ji =

1

2YipqY

jpq − 2δji g2C2(i) , (A3)

γ(2)ji =− 1

2YimnY

npqYpqrYmrj + g2YipqY

jpq[2C2(p)− C2(i)]

+ 2δji g4[C2(i)S(R) + 2C2(i)

2 − 3C2(G)C2(i)] , (A4)and the β-fun tions for the gauge ouplings are given byβ(1)g =g3 [S(R)− 3C2(G)] , (A5)

β(2)g =g5

−6[C2(G)]2 + 2C2(G)S(R) + 4S(R)C2(R)

− g3Y ijkYijkC2(k)/d(G) . (A6)Here, C2(i) is the quadrati Casimir for a spe i supereld and C2(R), C2(G) are thequadrati Casimirs for the matter and adjoint representations, respe tively. d(G) is thedimension of the adjoint representation.The β-fun tions for the superpotential parameters an be obtained by using supereld te h-nique. The obtained expressions are [91, 92.βijkY = Y p(ijγp

k) , (A7)βijµ = µp(iγp

j) . (A8)The (..) in the supers ripts denote symmetrization. Most of the β-fun tions of the models an be derived from these results using the pro edure given in [93 based on the spurionformalism [94. In the following, we briey summarize the basi ideas of this al ulation for32

ompleteness.The exa t results for the soft β-fun tions are given by [93:βM = 2O

[

βg

g

]

, (A9)βijkh = hl(jkγi)

l − 2Y l(jkγ1i)l ,

βijb = bl(iγj)

l − 2µl(iγ1j)l ,

(βm2) ij = ∆γi

j . (A10)where we denedO = Mg2

∂

∂g2− hlmn ∂

∂Y lmn, (A11)

(γ1)ij = Oγi

j , (A12)∆ = 2OO∗ + 2MM∗g2

∂

∂g2+

[

Y lmn ∂

∂Y lmn+ . .]+X

∂

∂g. (A13)Here, M is the gaugino mass and Y ijk = (m2)ilY

jkl + (m2)j lYikl + (m2)klY

ijl. Eqs. (A9)-(A10) hold in a lass of renormalization s hemes that in ludes the DRED′-one [95. We takethe known ontributions of X from [96:XDRED′(1) = −2g3S , (A14)XDRED′(2) = (2r)−1g3tr[WC2(R)]− 4g5C2(G)S − 2g5C2(G)QMM∗ , (A15)where

S = r−1tr[m2C2(R)]−MM∗C2(G) , (A16)W j

i =1

2YipqY

pqn(m2)jn +1

2Y jpqYpqn(m

2)ni + 2YipqYjpr(m2)qr + hipqh

jpq − 8g2MM∗C2(R)j i .(A17)With Q = T (R)− 3C2(G), and T (R) = tr [C2(R)], r being the number of group generators.2. From GUT s ale to SU(2)R breaking s aleIn the following se tions we will use the denitionsY ijQk

= Y ijkQ , Y ij

Lk= Y ijk

L (A18)and in the same way T ijQk

and T ijLk. We will also assume summation of repeated indi es.

33

a. Anomalous Dimensionsγ(1)

Q= 2Y ∗

QkY TQk

− 1

12

(

18g22 + 32g23 + g2BL

)

1 (A19)γ(2)

Q= +

1

144

(

− 128g43 + 2052g42 + 289g4BL + 36g22

(

32g23 + g2BL

)

+ 64g23g2BL

)

1

+ Y ∗Qm

(

6g22δmn +27

4Tr(αα∗

)

δmn − 2Tr(Y ∗LnY TLm

)

− 6Tr(Y ∗Qn

Y TQm

))

Y TQn

− 32Y ∗Qm

Y †Qn

YQnY TQm

(A20)γ(1)

Qc= 2Y †

QkYQk

− 1

12

(

18g22 + 32g23 + g2BL

)

1 (A21)γ(2)

Qc= +

1

144

(

− 128g43 + 2052g42 + 289g4BL + 36g22

(

32g23 + g2BL

)

+ 64g23g2BL

)

1

+ Y †Qm

(

6g22δmn +27

4Tr(αα∗

)

δmn − 2Tr(Y ∗LnY TLm

)

− 6Tr(Y ∗Qn

Y TQm

))

YQn

− 32Y †Qm

YQnY TQm

Y ∗Qn

(A22)γ(1)

L= 2(

3f †f + Y ∗LkY TLk

)

− 3

4

(

2g22 + g2BL

)

1 (A23)γ(2)

L=

3

16

(

12g22g2BL + 76g42 + 99g4BL

)

1 + 3f †f(

− 3|a|2 − 4Tr(ff †)

+ 6g2BL + 8g22

)

+ Y ∗Lm

(

6g22δmn +27

4Tr(αα∗

)

δmn − 2Tr(Y ∗LnY TLm

)

− 6Tr(Y ∗Qn

Y TQm

)

− 11f †fδmn

)

Y TLn

− 4Y ∗Lm

Y †LnYLn

Y TLm

− 2f †(

17ff † + 3YLkY †Lk

)

f

− 6f †YLkfY ∗

Lk(A24)

γ(1)

Lc= 2(

3ff † + Y †LkYLk

)

− 3

4

(

2g22 + g2BL

)

1 (A25)γ(2)

Lc=

3

16

(

12g22g2BL + 76g42 + 99g4BL

)

1 + 3ff †(

− 3|a|2 − 4Tr(ff †)

+ 6g2BL + 8g22

)

+ Y †Lm

(

6g22δmn +27

4Tr(αα∗

)

δmn − 2Tr(Y ∗LnY TLm

)

− 6Tr(Y ∗Qn

Y TQm

)

− 11ff †δmn

)

YLn− 4Y †

LmYLn

Y TLm

Y ∗Ln

− 2f(

17f †f + 3Y TLkY ∗Lk

)

f †

− 6Y †LkYLk

ff † (A26)(γ

(1)

Φ)ij = −3g221− 3

2

(

αα∗ + α∗α)

+ δimδjn

(

3Tr(Y ∗Qm

Y TQn

)

+ Tr(Y ∗Lm

Y TLn

)) (A27)(γ

(2)

Φ)ij = 33g421− 9

(

2(ααα∗α∗ + α∗α∗αα) + 3(αα∗αα∗ + α∗αα∗α))

− 24(αα∗ + α∗α)(

2g22 + 2Tr(αα∗)

− |a|2)

− 3

2

(

αjmα∗in + α∗

jmαin

)(

3Tr(Y ∗Qm

Y TQn

)

+ Tr(Y ∗Lm

Y TLn

))

− 1

2δimδin

(

− 3g2BLTr(Y ∗Lm

Y TLn

)

− (32g23 + g2BL)Tr(Y ∗Qm

Y TQn

)

34

+ 2(

5Tr(ff †YLnY †Lm

)

+ Tr(fY ∗LnY TLm

f †)

+ 6Tr(fY ∗Lm

Y TLnf †))

+ 4(

2Tr(Y †Lm

YLnY TLnY ∗Ln

)

+ Tr(Y †Lm

YLmY TLnY ∗Lm

)

+ Tr(Y †Lm

YLnY TLm

Y ∗Lm

))

+ 12(

2Tr(Y †Qm

YQnY TQn

Y ∗Qn

)

+ Tr(Y †Qm

YQmY TQn

Y ∗Qm

)

+ Tr(Y †Qm

YQnY TQm

Y ∗Qm

))) (A28)γ(1)

∆= 2Tr(ff †

)

− 3g2BL − 4g22 +3

2|a|2 (A29)

γ(2)

∆= 48g42 + 24g22g

2BL + 81g4BL

+3

2|a|2(

4g22 + Tr(αα∗)

− 7

2|a|2)

−(

2g22 + 3g2BL

)Tr(ff †)

− 24Tr(ff †ff †)

− 6Tr(ff †YLkY †Lk

)

− 2Tr(fY ∗LkY TLkf †) (A30)

γ(1)ˆ∆

= −3g2BL − 4g22 +3

2|a|2 (A31)

γ(2)ˆ∆

=3

4

(

4(

16g42 + 27g4BL + 8g22g2BL

)

+ |a|2(

2Tr(αα∗)

− 3Tr(ff †)

− 7|a|2 + 8g22

)) (A32)γ(1)

∆c= 2Tr(ff †

)

− 3g2BL − 4g22 +3

2|a|2 (A33)

γ(2)

∆c= 48g42 + 24g22g

2BL + 81g4BL

+3

2|a|2(

4g22 + Tr(αα∗)

− 7

2|a|2)

−(

2g22 + 3g2BL

)Tr(ff †)

− 24Tr(ff †ff †)

− 8Tr(fY TLkY ∗Lkf †) (A34)

γ(1)ˆ∆c

= −3g2BL − 4g22 +3

2|a|2 (A35)

γ(2)ˆ∆c

=3

4

(

4(

16g42 + 27g4BL + 8g22g2BL

)

+ |a|2(

2Tr(αα∗)

− 3Tr(ff †)

− 7|a|2 + 8g22

)) (A36)γ(1)

Ω= 2|a|2 − 4g22 (A37)

γ(2)

Ω= 3Tr(α(αα∗ − α∗α)α∗

)

+ 48g42 + |a|2(

12g2BL − 3Tr(ff †)

− 6|a|2 + 8g22

) (A38)γ(1)

Ωc= 2|a|2 − 4g22 (A39)

γ(2)

Ωc= 3Tr(α(αα∗ − α∗α)α∗

)

+ 48g42 + |a|2(

12g2BL − 3Tr(ff †)

− 6|a|2 + 8g22

) (A40)Note that the previous formulas are totally general and an be applied with any num-ber of bidoublets. Nevertheless, if two bidoublets are onsidered αα∗ = α∗α and furthersimpli ations are possible.35

b. Beta fun tions for soft breaking masses of sleptonsUsing the pro edure explained in se . A 1, we an al ulate the soft breaking masses forthe sleptons. The results are16π2 d

dtm2

L = 6ff †m2L + 12fm2

Lf† + 6m2

Lff† + 12m2

∆ff†

+2YLkY †Lkm2

L + 2m2LYLk

Y †Lk

+ 4YLkm2

LcY†Lk

+4(m2Φ)mnY

(m)L Y

(n) †L + 12TfT

†f + 4TLk

T †Lk

−(3g2BL|M1|2 + 6g22|M2|2 +3

2g2BLS1)1 (A41)

16π2 d

dtm2

Lc = 6f †fm2Lc + 12f †m2

Lcf + 6m2Lcf †f + 12m2

∆cf †f

+2Y †LkYLk

m2Lc + 2m2

LcY†LkYLk

+ 4Y †Lkm2

LYLk

+4(m2Φ)mnY

(m) †L Y

(n)L + 12T †

fTf + 4T †LkTLk

−(3g2BL|M1|2 + 6g22|M2|2 −3

2g2BLS1)1 (A42)where

S1 = 3(m2∆ −m2

∆ −m2∆c +m2

∆c)

+∑

m,n

[

(m2Q)mn − (m2

Qc)mn − (m2L)mn + (m2

Lc)mn

] (A43) . Beta fun tions for gauge ouplingsβ(1)gBL

= 24g3BL (A44)β(2)gBL

=1

2g3BL

(

− 192|a|2 − 93Tr(ff †)

+ 2(

115g2BL + 162g22

− 2Tr(Y ∗QkY TQk

)

− 6Tr(Y ∗LkY TLk

)

+ 8g23

)) (A45)β(1)g2

= 8g32 (A46)β(2)g2

=1

6g32

(

660g22 + 144g23 + 162g2BL − 192|a|2 + 108Tr(αα∗)

− 73Tr(ff †)

− 24Tr(Y ∗LkY TLk

)

− 72Tr(Y ∗QkY TQk

)) (A47)β(1)g3

= −3g33 (A48)β(2)g3

= g33

(

14g23 + 18g22 − 8Tr(Y ∗QkY TQk

)

+ g2BL

) (A49)36

3. From SU(2)R breaking s ale to U(1)B−L breaking s alea. Anomalous Dimensionsγ(1)

Q= − 1

12

(

18g2L + 32g23 + g2BL

)

1+ Y ∗d Y

Td + Y ∗

u YTu (A50)

γ(2)

Q= +

1

144

(

109g4BL − 128g43 + 36g2BLg2L + 64g23

(

18g2L + g2BL

)

+ 972g4L

)

1

− 2(

Y ∗d Y

Td Y ∗

d YTd + Y ∗

u YTu Y ∗

u YTu

)

+ Y ∗d Y

Td

(

− 3Tr(YdY†d

)

− |bc|2 −3

2|b|2 − Tr(YeY

†e

)

+ g2R

)

+ Y ∗u Y

Tu

(

− 3Tr(YuY†u

)

− |bc|2 −3

2|b|2 − Tr(YνY

†ν

)

+ g2R

) (A51)γ(1)

dc= 2Y †

d Yd −1

12

(

32g23 + 6g2R + g2BL

)

1 (A52)γ(2)

dc= − 1

144

(

− 109g4BL + 128g43 − 12g2BLg2R − 64g23

(

6g2R + g2BL

)

− 684g4R

)

1

− 2(

Y †d YdY

†d Yd + Y †

d YuY†uYd

)

− Y †d Yd

(

2|bc|2 + 2Tr(YeY†e

)

+ 3|b|2 − 6g2L + 6Tr(YdY†d

)) (A53)γ(1)uc = 2Y †

uYu −1

12

(

32g23 + 6g2R + g2BL

)

1 (A54)γ(2)uc = − 1

144

(

− 109g4BL + 128g43 − 12g2BLg2R − 64g23

(

6g2R + g2BL

)

− 684g4R

)

1

− 2(

Y †uYdY

†d Yu + Y †

uYuY†uYu

)

− Y †uYu

(

2|bc|2 + 2Tr(YνY†ν

)

+ 3|b|2 − 6g2L + 6Tr(YuY†u

)) (A55)γ(1)

L= −3

4

(

2g2L + g2BL

)

1+ Y ∗e Y

Te + Y ∗

ν YTν (A56)

γ(2)

L= +

9

16

(

12g4L + 13g4BL + 4g2BLg2L

)

1− 2(

Y ∗e Y

Te Y ∗

e YTe + Y ∗

ν YTν Y ∗

ν YTν

)

− Y ∗ν FcY

Tν

+ Y ∗e Y

Te

(

− 3Tr(YdY†d

)

− |bc|2 −3

2|b|2 − Tr(YeY

†e

)

+ g2R

)

+ Y ∗ν Y

Tν

(

− 3Tr(YuY†u

)

− |bc|2 −3

2|b|2 − Tr(YνY

†ν

)

+ g2R

) (A57)γ(1)ec = 2Y †

e Ye −1

4

(

2g2R + 3g2BL

)

1 (A58)γ(2)ec = +

1

16

(

117g4BL + 12g2BLg2R + 76g4R

)

1− 2(

Y †e YeY

†e Ye + Y †

e YνY†ν Ye

)

− Y †e Ye

(

2|bc|2 + 2Tr(YeY†e

)

+ 3|b|2 − 6g2L + 6Tr(YdY†d

)) (A59)γ(1)νc = 2Y †

ν Yν −1

4

(

2g2R + 3g2BL

)

1+ F ∗c (A60)37

γ(2)νc =

1

16

(

117g4BL + 12g2BLg2R + 76g4R

)

1− (f 1†c + f 1∗

c )Fc(f1c + f 1T

c )

− 2Y †ν

(

YeY†e + YνY

†ν

)

Yν − 2(f 1†c + f 1∗

c )Y Tν Y ∗

ν (f1c + f 1T

c )

− Y †ν Yν

(

2(

− 3g2L + 3Tr(YuY†u

)

+ |bc|2 + Tr(YνY†ν

))

+ 3|b|2)

+ F ∗c

(

2g2R + 3g2BL − |a1c |2 − Tr(f 1c f

1†c

)

− Tr(f 1†c f 1T

c

)) (A61)γ(1)

Hd

= 3Tr(YdY†d

)

− 1

2g2R +

3

2|b|2 − 3

2g2L + |bc|2 + Tr(YeY

†e

) (A62)γ(2)

Hd=

1

4

(

27g4L + 6g2Lg2R + 19g4R + 2

(

32g23 + g2BL

)Tr(YdY†d

)

+ 6g2BLTr(YeY†e

)

+ |b|2(

− 18Tr(YuY†u

)

− 15|b|2 − 19|bc|2 + 24g2L − 6Tr(YνY†ν

))

− 4|bc|2(

3|bc|2 + 3Tr(YuY†u

)

+ |a1c |2 + Tr(YνY†ν

))

− 36Tr(YdY†d YdY

†d

)

− 12Tr(YdY†d YuY

†u

)

− 12Tr(YeY†e YeY

†e

)

− 4Tr(YeY†e YνY

†ν

)) (A63)γ(1)

Hu= 3Tr(YuY

†u

)

− 1

2g2R +

3

2|b|2 − 3

2g2L + |bc|2 + Tr(YνY

†ν

) (A64)γ(2)

Hu=

1

4

(

27g4L + 6g2Lg2R + 19g4R

+ |b|2(

− 18Tr(YdY†d

)

− 15|b|2 − 19|bc|2 + 24g2L − 6Tr(YeY†e

))

− 4|bc|2(

3|bc|2 + 3Tr(YdY†d

)

+ |a1c |2 + Tr(YeY†e

))

+ 2(

32g23 + g2BL

)Tr(YuY†u

)

+ 6g2BLTr(YνY†ν

)

− 4Tr(f 1c f

1†c Y T

ν Y ∗ν

)

− 4Tr(f 1c Y

†ν Yνf

1†c

)

− 12Tr(YdY†d YuY

†u

)

− 4Tr(YeY†e YνY

†ν

)

− 36Tr(YuY†uYuY

†u

)

− 4Tr(Yνf1†c f 1T

c Y †ν

)

− 12Tr(YνY†ν YνY

†ν

)

− 4Tr(f 1†c Y T

ν Y ∗ν f

1Tc

)) (A65)γ(1)

∆c0= −2g2R − 3g2BL + |a1c |2 + Tr(f 1

c f1†c

)

+ Tr(f 1†c f 1T

c

) (A66)γ(2)

∆c0=

1

2

(

72g4BL + 24g2BLg2R + 44g4R − 4|a1c |2

(

|a1c |2 + |bc|2)

−(

2g2R + 3g2BL

)(Tr(f 1c f

1†c

)

+ Tr(f 1†c f 1T

c

))

− 4Tr(f 1c f

1†c f 1

c f1†c

)

− 8Tr(f 1c f

1†c f 1T

c f 1†c

)

− 8Tr(f 1c f

1†c f 1T

c f 1∗c

)

− 4Tr(f 1c f

1†c Y T

ν Y ∗ν

)

− 4Tr(f 1c Y

†ν Yνf

1†c

)

− 4Tr(Yνf1†c f 1T

c Y †ν

)

− 4Tr(f 1†c f 1T

c f 1†c f 1T

c

)

− 8Tr(f 1†c f 1T

c f 1∗c f 1T

c

)

− 4Tr(f 1†c Y T

ν Y ∗ν f

1Tc

)) (A67)γ(1)ˆ∆c0

= −2g2R − 3g2BL + |a1c |2 (A68)γ(2)ˆ∆c0

= 12g2BLg2R + 22g4R + 36g4BL

− |a1c |2(

2|a1c |2 + 2|bc|2 + Tr(f 1c f

1†c

)

+ Tr(f 1†c f 1T

c

)) (A69)γ(1)

Ω= −4g2L + |b|2 (A70)38

γ(2)

Ω= 28g4L − |b|2

(

2|bc|2 + 3|b|2 + 3Tr(YdY†d

)

+ 3Tr(YuY†u

)

− g2R + g2L + Tr(YeY†e

)

+ Tr(YνY†ν

)) (A71)γ(1)

Ωc0= 2|bc|2 + |a1c |2 (A72)

γ(2)

Ωc0= −2|a1c |4 − |a1c |2

(

− 4g2R − 6g2BL + Tr(f 1c f

1†c

)

+ Tr(f 1†c f 1T

c

))

− 2|bc|2(

2|bc|2 + 3|b|2 − 3g2L + 3Tr(YdY†d

)

+ 3Tr(YuY†u

)

− g2R + Tr(YeY†e

)

+ Tr(YνY†ν

)) (A73)In these expressions we have denedFc = f 1

c f1†c + f 1

c f1∗c + f 1T

c f 1†c + f 1T

c f 1∗c (A74)b. Beta fun tions for soft breaking masses of sleptonsAgain, the results for the slepton soft SUSY breaking masses at 1-loop are shown. Thebeta fun tions read

16π2 d

dtm2

L = 2Yem2ecY

†e + 2m2

HdYeY

†e + 2m2

HuYνY

†ν +m2

LYeY†e

+YeY†e m

2L +m2

LYνY†ν + YνY

†ν m

2L + 2Yνm

2νcY

†ν

+2TeT†e + 2TνT

†ν − (3g2BL|M1|2 + 6g2L|ML|2 +

3

4g2BLS2)1 (A75)

16π2 d

dtm2

ec = 2Y †e Yem

2ec + 2m2

ecY†e Ye + 4m2

HdY †e Ye + 4Y †

e m2LYe

+4T †eTe − (3g2BL|M1|2 + 2g2R|MR|2 −

3

4g2BLS2 −

1

2g2RS3)1 (A76)where

S2 = 2(m2∆c 0 −m2

∆c 0) + Tr[

m2dc −m2

ec +m2uc −m2

νc + 2m2L − 2m2

Q

] (A77)S3 = 2(m2

∆c 0 −m2∆c 0 −m2

Hd+m2

Hu) + Tr

[

3m2dc +m2

ec − 3m2uc −m2

νc

] (A78) . Beta fun tions for gauge ouplingsβ(1)gBL

= 9g3BL (A79)β(2)gBL

=1

2g3BL

(

16g23 + 50g2BL + 18g2L + 30g2R − 12|a1c |2 − 9Tr(f 1c f

1c†)

− 2Tr(YdY†d

)

− 6Tr(YeY†e

)

− 2Tr(YuY†u

)

− 6Tr(YvY†v

)

− 9Tr(f 1c†f 1cT)) (A80)

β(1)gL

= 3g3L (A81)39

β(2)gL

=1

3g3L

(

− 28|b|2 + 3(

24g23 − 2|bc|2 − 2Tr(YeY†e

)

− 2Tr(YvY†v

)

+ 3g2BL + 49g2L

− 6Tr(YdY†d

)

− 6Tr(YuY†u

)

+ g2R

)) (A82)β(1)gR

= 9g3R (A83)β(2)gR

= g3R

(

24g23 + 15g2BL + 3g2L + 15g2R − 4|a1c |2 − 4|b|2 − 2|bc|2 − 3Tr(f 1c f

1c†)

− 6Tr(YdY†d

)

− 2Tr(YeY†e

)

− 6Tr(YuY†u

)

− 2Tr(YvY†v

)

− 3Tr(f 1c†f 1cT)) (A84)

β(1)g3

= −3g33 (A85)β(2)g3

= g33

(

14g23 + 3g2R − 4Tr(YdY†d

)

− 4Tr(YuY†u

)

+ 9g2L + g2BL

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