Type-II neutrino seesaw mechanism extension of NMSSM from SUSY breaking mechanisms
PPrepared for submission to JHEP
Type-II neutrino seesaw mechanism extension ofNMSSM from SUSY breaking mechanisms
Zhuang Li, a Fei Wang a, a School of Physics, Zhengzhou University,450000,ZhengZhou, P.R.China
E-mail: [email protected]
Abstract:
We propose to accommodate economically the type-II neutrino seesaw mech-anism in (G)NMSSM from GMSB and AMSB, respectively. The heavy triplets withinneutrino seesaw mechanism are identified to be the messengers. Therefore, the µ -problem,the neutrino mass generation, LFV as well the soft SUSY breaking parameters can be eco-nomically combined in a non-trivial way. General features of such extensions are discussed.The type-II neutrino seesaw-specific interactions can give additional Yukawa deflection con-tributions to the soft SUSY breaking parameters of NMSSM, which are indispensable torealize successful EWSB and accommodate the 125 GeV Higgs. Relevant numerical re-sults, including the constraints of dark matter and possible LFV processes l i → l j γ etc, arealso given. We find that our economical type-II neutrino seesaw mechanism extension ofNMSSM from AMSB or GMSB can lead to realistic low energy NMSSM spectrum, bothadmitting the 125 GeV Higgs as the lightest CP-even scalar. The possibility of the 125GeV Higgs being the next-to-lightest CP-even scalar in GMSB-type scenario is ruled outby the constraints from EWSB, collider and precision measurements. The possibility of the125 GeV Higgs being the next-to-lightest CP-even scalar in AMSB-type scenario is ruledout by dark matter direct detection experiments. Possible constraints from LFV processes l i → l j γ can give an upper bound for the messenger scale. Corresponding author. a r X i v : . [ h e p - ph ] A ug ontents A.1 Expressions from GMSB 30A.2 Expressions from deflected AMSB 31
TeV scale supersymmetry(SUSY) is one of the most promising candidates for new physicsbeyond the Standard Model(SM). It can prevent the Higgs boson mass from acquiringdangerous quadratic divergence corrections, realize successful gauge coupling unificationand provide viable dark matter(DM) candidates, such as the lightest neutralino assumingexact R-parity. Besides, the discovered 125 GeV Higgs [1, 2] lies miraculously in the narrow115-135 GeV (cid:48) window (cid:48) predicted by MSSM, which can also be seen as a triumph of lowscale SUSY. However, low energy SUSY confronts many challenges from LHC experiments,the foremost of which is the null search results of superpartners at LHC. Recent analysesbased on Run 2 of 13 TeV LHC and f b − of integrated luminosity constrain the gluinomass m ˜ g to lie above 2 TeV [3] and the top squark mass m ˜ t to lie above 1 TeV [4] in somesimplified models. In addition, the µ problem in MSSM needs an explanation.One of the major unresolved problems of particle physics now is the nature of tinyneutrino masses, which were discovered by neutrino oscillation experiments. It is known– 1 –hat Weinberg’s effective dimension-5 operator is the lowest one which can generate tinyMajorana neutrino masses. Such an operator can be ultraviolet(UV)-completed to obtainthree types of tree-level seesaw mechanism: type I seesaw [5], involving the exchange ofright-handed neutrinos; type II seesaw [6], involving the exchange of scalar triplet; typeIII [7], involving the exchange of fermion triplet. If the SUSY framework is indeed the newphysics beyond the SM, it should accommodate proper neutrino mass generation mecha-nisms. The seesaw mechanism extensions of low energy SUSY [8], which can provide typicalunified frameworks to solve all the remaining puzzles of SM together, are well motivatedtheoretically.However, simple seesaw mechanism extensions of MSSM still inherit the main difficul-ties of MSSM. The foremost one is the µ -problem, which is in general unsolved in suchextensions. Besides, to accommodate the 125 GeV Higgs, unnaturally heavy stop masses m ˜ t (cid:38) TeV are necessary unless large trilinear coupling A t is present, which on the otherhand may result in color breaking minimum for the scalar potential [9]. Next-to-minimalsupersymmetric standard model(NMSSM) [10] is the simplest gauge singlet extension ofMSSM, which can elegantly solve the µ -problem in MSSM by generating an effective µ -term after the singlet scalar acquires a vacuum expectation value (VEV). Furthermore, dueto possible new tree level contributions to the Higgs mass, NMSSM can easily accommodatethe 125 GeV Higgs boson without the needs of very large A t for light stops, ameliorating thecolor breaking minimum problem. Therefore, the seesaw mechanism extensions of NMSSMcan evade most of the difficulties that bother the seesaw mechanism extensions of MSSM.Attracting as the seesaw mechanism extensions of NMSSM are, there are too many freeparameters in such low energy SUSY models. To preserve their prediction power, we needto refer to their UV completion. It is known that the low energy SUSY spectrum can betotally determined by the SUSY breaking mechanism, which can predict the low energyparameters by very few UV inputs. So it is desirable to combine the seesaw mechanismextensions of (N)MSSM with the SUSY breaking mechanisms and survey which SUSYbreaking mechanism can give the favored low energy spectrum.Depending on the way the visible sector (cid:48) f eels (cid:48) the SUSY breaking effects in the hid-den sector, the SUSY breaking mechanisms can be classified into gravity mediation [11],gauge mediation [12](GMSB), anomaly mediation [13](AMSB) scenarios, etc. Both GMSBand AMSB are calculable, predictive, and phenomenologically distinctive. Especially, theywill not cause flavor and CP problems that bothers gravity mediation models. However,GMSB realization of MSSM can hardly explain the 125 GeV Higgs with TeV scale softSUSY breaking parameters because of the vanishing trilinear terms at the messenger scale.Although non-vanishing A t at the messenger scale can be obtained in GMSB with addi-tional messenger-matter interactions [14–16], it is rather ad hoc to include such interactionsin the superpotential. So it is interesting to see if certain types of messenger-matter interac-tions can arise naturally in an UV-completed model. Yukawa mediation contributions frommessenger-matter interactions can also possibly be present [17] in deflected AMSB [18, 19],which can elegantly solve the tachyonic slepton problem of minimal AMSB through thedeflection of the renormalization group equation (RGE) trajectory [20].It is fairly straightforward to accommodate SUSY breaking mechanism in the neutrino-– 2 –eesaw extended MSSM, for example, by introducing additional messenger sector in GMSBor (deflected) AMSB. However, to introduce as few new inputs as possible, it is morepredictive and economical to identify the messengers with the heavy fields that are in-tegrated out in the neutrino-seesaw mechanism. In such predictive models, the neutrinomass generation, lepton-flavor-violation(LFV) as well soft SUSY breaking parameters canbe related together. Besides, additional Yukawa couplings involving the heavy fields (inneutrino seesaw mechanism), which also act as the messengers, can be naturally present.Such messenger-matter type interactions can possibly give large contributions to trilinear A t term in GMSB (or deflected AMSB), which will play an important rule in obtaining the125 GeV Higgs with TeV scale soft SUSY breaking parameters.As noted previously, even though it is very predictive and well motivated to combineneutrino seesaw mechanism with SUSY breaking mechanism [21, 22] in an economical wayfor MSSM, the difficulties of MSSM mentioned previously, especially the µ -problem, are ingeneral not solved, making it interesting to turn instead to such realizations of NMSSM. Asthe case of MSSM, it is in general straightforward to accommodate SUSY breaking mech-anism in the neutrino-seesaw extended NMSSM by introducing an additional messengersector in GMSB(dAMSB). Nevertheless, it is still interesting to see if the neutrino massgeneration, LFV, the generation of µ -term as well the soft SUSY breaking parameters canbe combined in a non-trivial economical way by identifying the messengers with the heavyfields. Such an economical realization of Type I seesaw extension of NMSSM from GMSB,which introduce only gauge singlet neutrino superfields, can hardly generate soft SUSYbreaking parameters other than the left-handed sleptons and right-handed sneutrinos with-out additional non-singlet messengers . Similar extension in AMSB, however, cannot leadto positive squared masses for right-handed sleptons. The economical realization of Type IIneutrino seesaw extension of NMSSM, on the other hand, can generate realistic soft SUSYparameters without the need of an additional messenger sector other than the heavy fieldspresent in the seesaw mechanism. We will discuss the realization of NMSSM through theeconomical combination of type-II neutrino seesaw mechanism with GMSB and deflectedAMSB, respectively.This paper is organized as follows. In Sec 2, we discuss the type-II neutrino seesawmechanism in SUSY. In Sec 3 and Sec 4, we discuss the economical realization of type-II The superpotential of type-I seesaw extension of NMSSM [23]can naively be embedded economically inYukawa mediation with W Type I ⊇ y Nij L i H u N j + XN j , which, however, can not generate realistic spectrum. Here N j the right-handed neutrino superfields and X the SUSY breaking spurion superfield with its VEV (cid:104) X (cid:105) = M + θ F X . The lowest component VEV of X can determine the N j thresholds. The inverse seesaw extension of NMSSM[24–28] can be written as W Inverse ⊇ y Nij L i H u N j + ˜ λSN j N + µ X N N . with the presence of a very small lepton number violating parameter µ X ∼ eV that is responsible for thesmallness of the light neutrino masses. N is the additional gauge singlet field. This extension does nothave a similar economical GMSB embedding and can be embedded in ordinary realization of GMSB withan additional messenger sector or (dAMSB). A successful realization can be seen in [29]. – 3 –eesaw mechanism extension of NMSSM from GMSB and AMSB, respectively. The softSUSY breaking parameters are given and numerical results for each scenarios are studied.Sec 5 contains our conclusions. In the ordinary type-II seesaw mechanism [6], the Lagrangian contains the coupling betweenthe scalar triplet to the Higgs doublet H as well as the Yukawa interaction between the SU (2) L doublet leptons to a very heavy SU (2) L triplet scalar with lepton number L = − and mass M ∆ L ⊃ − M | ∆ L | + y νij L TL ; i C ∆ L L L ; j + A T H T ∆ L H . (2.1)The third term, which contains a trilinear scalar coupling mass parameter A T , plays a keyrole in determining the minimum of the full scalar potential so as to give a tiny vacuumexpectation value(VEV) of ∆ L . Such a tiny VEV can in turn induce a Majorana mass forleft-handed neutrinos m ν ≈ y νij A T v M ∼ . , (2.2)with v ≈ . For y νij ∼ O (1) , M ∆ ∼ GeV in the case A T ∼ M ∆ and M ∆ ∼ GeV in the case A T ∼ v EW .The type-II neutrino seesaw mechanism extension of MSSM is non-trivial. There aretwo SU (2) L Higgs doublets in the MSSM, so the type-II neutrino seesaw mechanism ex-tension of MSSM can be seen as a special case of type-II neutrino seesaw extension of twoHiggs doublet model, which contains interactions between both scalar Higgs doublets tothe heavy scalar triplet. We can further extend the type-II seesaw mechanism to NMSSMby including the singlet sector.We need to introduce vector-like SU (2) L triplet superfields with U (1) Y quantum num-ber Y = ± in the superpotential W ⊇ W NMSSM + y Lij L j L i ∆ T + y d ∆ ∆ T H d H d + m T ∆ T ∆ T + y u ∆ ∆ T H u H u , (2.3)with general NMSSM superpotential W NMSSM = W MSSM /µ + λSH u H d + κ S + ξ S S + · · · . (2.4)The parameter m T , which is a free parameter in equation (2.3), will be determined by thespurion VEVs in GMSB (or deflected AMSB) if the triplets can act as components of themessengers. From the superpotential (2.3), we can obtain the F-terms of the triplets F ∆ T = ∂W ∂ ∆ T = y Lij L i L j + y d ∆ H d H d + m T ∆ T ,F ∆ T = ∂W∂ ∆ T = y u ∆ H u H u + m T ∆ T . (2.5) For Z -invariant NMSSM, the y d ∆ ∆ T H d H d term can be forbidden by proper Z charge assignments. We should note that it is consistent to generate m T also by the VEV of S for tiny coupling y νij inNMSSM. However, large fine tuning will be needed in general because of large effective µ . – 4 –e require the SUSY to be unbroken at the triplet scale m T . So the F-flat conditions F ∆ T = F ∆ T = 0 can give (cid:104) ∆ T (cid:105) = − y d ∆ v d m T , (cid:104) ∆ T (cid:105) = − y u ∆ v u m T . (2.6)The neutrinos will acquire tiny Majorana masses through the type-II seesaw mechanism (cid:104) m ν (cid:105) = − y Lij y u ∆ v u m T . (2.7)This result can be understood to arise from the scalar potential V ⊃ (cid:12)(cid:12)(cid:12) y Lij L i L j + y d ∆ H d H d + m T ∆ T (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) y u ∆ H u H u + m T ∆ T (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) λSH u + 2 y d ∆ ∆ T H d (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12) λSH d + 2 y u ∆ ∆ T H u (cid:12)(cid:12) + · · · . (2.8)The m T y u ∆ H ∗ u H ∗ u ∆ T term plays the role of the third term in (2.1).We should note that F-terms F H u for H u and F H d for H d can not vanish for solutionsin eqn (2.6) with non-negligible µ term. Therefore, tiny SUSY breaking effects of order | F H u | + | F H d | ∼ µ v will appear. In fact, the minimum conditions for H u and H d shouldalso involve the soft SUSY breaking terms.From the potential eqn.(2.8), we can see that the term involving µ with λ (cid:104) S (cid:105) y d ∆ ∆ T H d H ∗ u = 2 y d ∆ µ ∆ T H d H ∗ u , (2.9)also gives a subleading contribution to neutrino masses. Besides, there is an alternativecontribution to neutrino masses from the trilinear soft SUSY breaking term −L ⊇ A H d H d ∆ T y d ∆ ∆ T H d H d + · · · , (2.10)which will be generated after SUSY breaking. From the minimum conditions of the totalscalar potential, including the soft SUSY breaking terms, the triplet VEV can be approxi-mately given by (cid:104) ∆ T (cid:105) ≈ − y u ∆ v u m T − y d ∆ A H d H d ∆ T v d m T − y d ∆ µv d v u m T . (2.11)So the resulting neutrino masses are given by ( m ν ) ij = − y Lij (cid:20) y u ∆ v u m T + y d ∆ A H d H d ∆ T v d m T + y d ∆ µ tan βv d m T (cid:21) (cid:46) . . (2.12)The three terms can be destructive if A H d H d ∆ T or µ is negative. Besides, if | A H d H d ∆ T | (cid:38) m T for negative A H d H d ∆ T or similarly for µ , tiny neutrino masses can be generated by finetuning even if either terms in eqn (2.12) are not very small.Such a type-II neutrino seesaw mechanism extension of (N)MSSM can be nontriviallyembedded into SUSY breaking mechanisms. In this paper, the messenger threshold can be– 5 –dentified to be the heavy triplet scalar threshold in type-II seesaw mechanism. This possi-bility provide an economic unified framework to taking into account both SUSY extensionand neutrino masses. So m T is always much larger than the A H d H d ∆ T , which lies typicallyat the soft SUSY breaking scale. Successful EWSB requires µ to lie at the soft SUSYbreaking scale. Therefore, the second and third terms in eqn (2.12) are always subleadingunless the messenger scale is very low.The messenger threshold, which is just the heavy scalar triplet scale in type-II neutrinoseesaw mechanism, can possibly be constrained by the lepton flavor violation(LFV) pro-cesses, such as l i → l j γ . Detailed discussions on LFV constraints to SUSY seesaw modelscan be found in [31–33]. Especially, the LFV related discussions in scenario within whichthe triplet in type-II neutrino seesaw mechanism also account for the soft SUSY breakingmasses had been discussed in [21, 34].The branch ratio l i → l j γ can be generally written as [35] Br ( l i → l j γ ) = 48 π α e G F (cid:16) | A ijL | + | A ijR | (cid:17) Br ( l i → l j ν i ¯ ν j ) , (2.13)where A ijL ≈ (cid:16) m L (cid:17) ij m SUSY , A ijR ≈ (cid:16) m E cL (cid:17) ij m SUSY , (2.14)with m L and m E cL are the doublet and singlet slepton soft mass matrices, respectively. m SUSY is the typical SUSY mass scale. These estimations depend on the assumptionsthat (I) chargino/neutralino masses are similar to slepton masses and (II) left-right flavormixing induced by trilinear terms is negligible. As noted in [33], although the assumptionis not valid when large values of trilinear terms are considered, the above estimates cannevertheless be used to illustrate the dependence of the BRs on the low-energy neutrinoparameters.To avoid severe difficulties from SUSY flavor constraints, the soft sfermion masses(including the slepton masses) are universal at high energy input scale M U . The RGEsof the slepton soft terms, which contain non-diagonal contributions from neutrino-seesawspecific interactions, can possibly induce off-diagonal soft terms to slepton mass matrices.These contributions are decoupled at the characteristic scale of the heavy mediators m T .However, it is interesting to note that in our subsequent discussions with gauge mediationand (deflected) anomaly mediation, the trilinear couplings for slepton Yukawa (with A Eij ≈ ) and slepton masses are universal at the (input) messenger scale, which also act as theheavy triplet mediator scale. In the basis where the lepton Yukawa couplings are diagonal,all the LFV effects are encoded in the coupling Y Lij . From the RGE of the soft masses [31],one can obtain the leading-log approximation [31, 33] for the off-diagonal soft terms at lowenergy (cid:16) m L (cid:17) ij ≈ − π (cid:16) m L L (cid:17) (cid:104) Y L † ik Y Lkj (cid:105) log (cid:18) M U m T (cid:19) , (cid:16) m E cL (cid:17) ij ≈ , A Eij ≈ , (2.15)– 6 –ith Y L † ik Y Lkj = (cid:18) m T y u ∆ v u (cid:19) (cid:104) U ( m diagν ) U † (cid:105) ij . (2.16)Here U is the PMNS lepton mixing matrix U = c c s c s e − iδ − s c − c s s e iδ c c − s s s e iδ s c s s − c c s e iδ − c s − s c s e iδ c c · e iφ e iφ , (2.17)with s ij ≡ sin θ ij , c ij ≡ cos θ ij for the three mixing angles θ , θ and θ , respectively. Sothe BRs for rare lepton decays l i → l j γ , which are roughly given by Br ( l i → l j γ ) ≈ α e m l i (cid:107) ( m L ) ij (cid:107) m SUSY tan β ∝ (cid:104) U ( m diagν ) U † (cid:105) log (cid:18) M U m T (cid:19) , (2.18)will not receive large enhancement by the log factor with m T ∼ M U in the leading-logapproximation unless the sub-leading terms are sizeable. Therefore, unlike neutrino seesawmechanism extension of SUGRA-type mediation mechanism, in which the universal softparameter inputs are adopted at GUT scale with log( M U /m T ) (cid:29) , the BRs of l i → l j γ will give important but not too stringent constrains on the seesaw scale in our GMSB andAMSB type scenarios, in which the triplet mediator scale is identified with the messengerscale. Predictions for other LFV processes with best-fit values for the neutrino parameterscan be seen in [34]. It is known that additional settings are needed to solve the µ/Bµ problem in ordinaryGMSB realization of MSSM. Although such a problem can be naturally solved in NMSSM,additional Yukawa structures for superfield S are needed in GMSB because the soft pa-rameters involving only the singlet S can not receive any gauge mediation contributions.Besides, to accommodate the 125 GeV Higgs in MSSM, TeV scale stop masses with near-maximal stop mixing are necessary [30]. For O (10) TeV stops with small A t , although stillpossible to interpret the 125 GeV Higgs, exacerbate the (cid:48) little hierarchy (cid:48) problem arisingfrom the large mass gap between the measured value of the weak scale and the sparticlemass scale. As ordinary GMSB predicts vanishing A t at the messenger scale, it necessitatesthe introduction of additional large Yukawa deflection contributions from messenger-matterinteractions if we would like to reduce the fine tuning involved. NMSSM does not need toolarge A t for light stops to interpret the 125 GeV Higgs because of additional tree-level con-tributions, possibly avoiding the color breaking minimum problem of MSSM. An mildlylarge A t can, however, lead to reduced electroweak fine-tuning [36](EWFT) even with TeVscale stops. In our economical realization of the type-II neutrino seesaw mechanism exten-sion of NMSSM from GMSB, the Higgs sector can participate in new interactions involvingthe triplets, which leads to additional non-vanishing Yukawa mediation contributions to tri-linear couplings A t at the messenger scale, possibly reducing the EWFT involved. Besides,– 7 –dditional Yukawa couplings involving S and heavy fields can be naturally introduced,which will also give Yukawa mediation contributions to soft SUSY breaking parametersinvolving the gauge singlet S , making spontaneously symmetry breaking(SSB) possible togive correct range of the µ value. Therefore, our predictive type-II neutrino seesaw mecha-nism extension of NMSSM from GMSB, which can combine the solution to the µ problem,the neutrino mass generation, LFV, soft masses and EWSB, is very interesting. In GMSB, the VEV of spurion X is given by (cid:104) X (cid:105) = M + θ F X . (3.1)As emphasized in [10], successful electroweak symmetry breaking(EWSB) in NMSSM ne-cessitates non-vanishing soft SUSY masses for S and A κ . As the soft mass of the gaugesinglet S receives no contributions from ordinary GMSB, additional Yukawa mediation con-tributions should be included. It was noted in [37] that double species of messengers areneeded to avoid possible mixing between the spurion X and the gauge singlet S if we couplethe messengers to S . As the SU (2) L triplet superfields with SU (3) c × SU (2) L × U (1) Y quan-tum number ∆ i ( , , ) and ∆ i ( , , − ) and proper Z charge assigments are embeddedinto the messengers, the superpotential take the following form W mess ;∆ ⊇ y S S (cid:0) ∆ ∆ + ∆ ∆ (cid:1) + y X X ∆ ∆ , (3.2)To preserve gauge coupling unification, the ∆ i ( , , ) and ∆ i ( , , − ) messengersshould be embedded into complete SU(5) representations = ∆ S ( , ) − / ⊕ ∆ T ( , ) ⊕ ∆ D ( , ) / , = ∆ S ( ¯6 , ) / ⊕ ∆ T ( , ¯3 ) − ⊕ ∆ D ( ¯3 , ) − / . (3.3)So the superpotential (3.2) in terms of SU(5) representation is given by W mess ; A ⊇ n (cid:88) k =1 y X X a · a + n (cid:88) k =1 y S S k − · k . (3.4)Although such a double-messenger-species choice of superpotential is phenomenologicalviable, it had been utilized in our previous model buildings, see [38] for an example.In this work, we choose an alternative possibility to realize NMSSM spectrum with onemessenger species.Couplings of the form W mess ;∆ ⊇ y S S ∆ ∆ + y X X ∆ ∆ (3.5)in the superpotential will trigger mixing between X and S via messenger loops, generatingthe following Kahler potential after integrating out the messengers K = 3 y X y S SX † ln (cid:18) X † XM (cid:19) + h.c. , (3.6)– 8 –hich will give a tadpole term for S after SUSY breaking L ⊇ y X y S ( S + S ∗ ) M (cid:12)(cid:12)(cid:12)(cid:12) F X M (cid:12)(cid:12)(cid:12)(cid:12) . (3.7)Such a tadpole term can generate a suitable VEV for (cid:104) S (cid:105) . Therefore, we adopt this possi-bility for GMSB. The superpotential (3.5) can be embedded into the following form withcomplete SU(5) multiplets W mess ; B ⊇ y X X · + y S S · . (3.8)So the whole GMSB superpotential is given by W ⊇ y u ∆ · H · H + y d ∆ · H · H + y L ; ij ∆ · i · j + λS · H · H + κ S + · · · + y uij i · j · H + y dij i · j · H + W SB ( , · · · ) + W mess ; B , (3.9)with W SB ( H , · · · ) the SU(5) symmetry breaking sector, which possibly involving H Higgs etc. Besides, proper doublet-triplet(D-T) splitting mechanism is assumed so thatthe Higgs triplets in H and ¯ H will be very heavy and be absent from the low energyspectrum at the messenger scale M mess . After we integrating out the messengers, tinyMajorana neutrino masses will be generated by GNMSSM extension of type-II neutrinoseesaw mechanism, the superpotential (2.3). We should note that Z -invariant NMSSMcan be generated if we adopt the superpotential (3.4) instead of (3.8). From the superpotential (2.3), the general expressions for soft SUSY breaking parameters atthe messenger scale (which is also identified to be the scale of the triplets) can be calculatedwith the wavefunction renormalization approach [39]. • The expressions for gaugino masses M i = g i F X M ∂∂ ln | X | g i ( µ, | X | ) . (3.10)So we have M i = − F X M α i ( µ )4 π ∆ b i , (3.11)with ∆ b i = ( 7 , , . (3.12) There are many alternative model building possibilities. For example, it is possible to keep gaugecoupling unification by introducing only the vector-like octet and triplet superfields. In 5D orbifold GUTmodel, it is possible that only the triplets zero modes can survive the orbifolding boundary conditions,which can also naturally generate D-T splitting. – 9 –
The expressions for trilinear couplings A ijk ≡ A ijk y ijk = (cid:88) i F X M ∂∂ ln | X | Z ( µ ; | X | ) , = (cid:88) i F X M ∆ G i . (3.13)In our convention, the anomalous dimension are expressed in the holomorphic ba-sis [15] G i ≡ dZ ij d ln µ ≡ − π (cid:18) d ikl λ ∗ ikl λ jmn Z − ∗ km Z − ∗ ln − c ir Z ij g r (cid:19) , (3.14)with ∆ G ≡ G + − G − the discontinuity across the messenger threshold. Here (cid:48) G + ( G − ) (cid:48) denote respectively the value above (below) the messenger threshold.So we have the soft SUSY breaking trilinear couplings A t = − π F X M y u ) ,A b = − π F X M (cid:20) (cid:0) y L ; (cid:1) + 2 (cid:16) y d (cid:17) (cid:21) ,A τ = − π F X M (cid:20) (cid:0) y L ; (cid:1) + 2 (cid:16) y d (cid:17) (cid:21) ,A λ = − π F X M (cid:20)
15 ( y S ) + 2 ( y u ) + 2 (cid:16) y d (cid:17) (cid:21) ,A κ = − π F X M
45 ( y S ) ,m S (cid:48) = − µ (cid:48) π F X M
30 ( y S ) ,ξ S = − ξ F π F X M
15 ( y S ) ,m = − µ π F X M (cid:20) y u ) + 2 (cid:16) y d (cid:17) (cid:21) , (3.15)Here we neglect possible RGE effects of y H u H u ∆ etc between the GUT scale and themessenger scale. • The soft SUSY masses are given as m soft = − F X M ∂∂ (ln | X | ) ln [ Z i ( µ, X, T )] , = − F X M (cid:20) ∂∂ ln M ∆ G − ∂∂ ln M G − ( M, ln M ) (cid:21) . (3.16)The expressions for soft scalar masses are rather lengthy. So we collect their expres-sions in appendix A.1. – 10 –s discussed in section 2, the soft SUSY breaking trilinear term can give subleadingcontribution to Majorana neutrino mass via type-II seesaw mechanism. We require theknowledge of trilinear scalar coupling ˜∆ T − H d − H d in GMSB. However, there are nocontributions to trilinear couplings ˜∆ T − H d − H d at the M Mess scale in GMSB. So thetrilinear soft term contribution to type-II seeesaw neutrino masses will not play a role.
Lacking gauge interactions for S , ordinary GMSB predicts vanishing trilinear couplings A κ , A λ and vanishing m S . Therefore, it can not predict realistic low energy NMSSM spec-trum unless additional Yukawa mediation contributions are present [40]. Fortunately, be-cause of the new interactions involving H u , H d , S and triplets, additional Yukawa deflectioncontributions related to type-II neutrino seesaw can lead to new contributions to trilinearcouplings and soft scalar masses. Therefore, phenomenological viable parameters can bepossible in our scenario.In ordinary setting, the spurion X is normalized so that y X = 1 . Due to possiblemixing between X and S through messengers in representation of SU(5), tadpole termsin the scalar potential of S can be generated as ξ S = 15 y X y S F X M . (3.17)So general NMSSM soft SUSY breaking parameters will appear in the GMSB scenario.Besides, we set ξ F = µ (cid:48) = 0 to keep the predictive power of the scenario.The free parameters in this scenario are given as F X M mess , M mess , y L ; a , y d , y u , y S , λ, κ. (3.18)For simplicity, we adopt the universal inputs for the new Yukawa couplings at themessenger scale y L ; a = λ , y d = y u = λ , y S = λ . The soft SUSY masses m H u , m H d , m S can be reformulated into µ, tan β, M Z by theminimum conditions of the scalar potential M A = 2 µ eff sin 2 β B eff , µ eff ≡ λ (cid:104) s (cid:105) , B eff = ( A λ + κ (cid:104) s (cid:105) ) . (3.19)In our numerical study, κ is a free parameter while tan β is not. This choice is differ-ent to ordinary numerical setting in NMSSM in which tan β is free while κ is a derivedquantity [41]. Such a choice can be convenient for those predictable NMSSM models fromtop-down approach. A guess of tan β is made to obtain the relevant Yukawa couplings y t , y b at the EW scale. After RGE evolving up to the messenger scale, the whole soft SUSYbreaking parameters at the messenger scale can be determined. Low energy tan β can beobtained iteratively with such a spectrum from the minimization conditions of the Higgs– 11 –otential. Obtaining an iteratively stable tan β indicates that the EWSB conditions aresatisfied by the model input.It can be calculated that n generations of , superfields of SU(5) will contribute ∆ b i = 14 n to the gauge beta functions. Perturbativity of the gauge coupling at the unifi-cation scale requires the combination [42] δ = − n π ln M GUT M mess , (3.20)to satisfy | δ | (cid:46) . , (3.21)with M GUT and M mess the GUT scale and messenger scale, respectively. So the messengerscale need to satisfy M mess (cid:38) GeV for n = 1 and M mess (cid:38) GeV for n = 2 .We use NMSSMTools 5.5.0 [43] to scan the whole parameter space. Randomly scanin combine with MCMC method are used. We interest in relatively large values of λ inorder to increase the tree-level mass of the 125 GeV CP-even Higgs boson. Besides, thecouplings λ , λ , λ should be pertubative and λ, κ should satisfy the perturbative bound λ + κ (cid:46) . . The parameters are chosen to lie the following range GeV < M mess ≡ m T < . × GeV ,
10 TeV < F X M <
500 TeV , < λ , λ , λ < √ π , . < λ, κ < . . (3.22)The coupling λ , which is just the y u ∆ in equation (2.3), should not be too small. Oth-erwise, very large ( m T /y u ∆ ) factor will lead to large (cid:16) m L (cid:17) ij , which may exceed the cur-rent bounds on Br ( l i → l j γ ) even if the leading-log contributions are not enhanced bythe log factor. Conservative bound ( m T /y u ∆ ) (cid:46) . × GeV, obtained numericallyin [21], can be imposed in subsequent numerical results to ensure that our scenarios canbe safely compatible with µ → eγ constraints etc. Therefore, we have an upper bound m T ≡ M mess (cid:46) . × GeV for y u ∆ ∼ √ π . This upper bound of M mess also safelylie below the GUT scale. So we choose the conservative upper bound of M mess to be . × GeV in our numerical scan. The lower bound of M mess comes from the perturba-tive requirements of gauge couplings below M GUT , which can be seen from the discussionbelow eqn(3.20).In addition to the constraints from neutrino masses (cid:12)(cid:12)(cid:12) ( m ν ) ij (cid:12)(cid:12)(cid:12) ≈ y Lij (cid:20) y u ∆ v u m T (cid:21) (cid:46) . , (3.23)we also impose the following constraints in our numerical scan • (I) The conservative lower bounds from current LHC constraints on SUSY parti-cles [44, 45]: – Light stop mass: m ˜ t (cid:38) . TeV.– 12 –
Gluino mass: m ˜ g (cid:38) . ∼ . TeV. – Light sbottom mass m ˜ b (cid:38) . TeV. – Degenerated first two generation squarks m ˜ q (cid:38) . ∼ . TeV. • (II) We impose the following lower bounds for neutralinos and charginos, including theinvisible decay bounds for Z -boson. The most stringent constraints of LEP [46] require m ˜ χ ± > . and the invisible decay width Γ( Z → ˜ χ ˜ χ ) < .
71 MeV , which isconsistent with the σ precision EW measurement constraints Γ non − SMinv < . . • (III) Recent flavor constraints from rare B meson decays [47]: . × − < Br ( B + → τ + ν ) < . × − , . × − < Br ( B s → µ + µ − ) < . × − , . × − < Br ( B S → X s γ ) < . × − . (3.24) • (IV) The CP-even component S in the Goldstone- (cid:48) eaten (cid:48) combination of H u and H d doublets corresponds to the SM Higgs boson. The S dominated CP-even Higgsshould lie in the combined mass range for the Higgs boson: ± from ATLASand CMS data, where the width of the band is given by the theoretical uncertainty ofthe Higgs mass calculation. The uncertainty is 3 GeV instead of default 2 GeV becauselarge λ may induce additional O (1) GeV correction to m h at two-loop level [48], whichis not included in the NMSSMTools.It is known that gravitino ˜ G will be much lighter in GMSB than that in mSUGRAand in general will be the lightest supersymmetric particle(LSP). Such a light gravitino isalso motivated by cosmology since it can evade the gravitino problem. The interaction ofgoldstino component of gravitino is /F X instead of /M P l . If gravitinos are in thermalequilibrium at early times and freeze out at the temperature T f , their relic density is [49] Ω ˜ G h = m ˜ G keV 100 g ∗ ( T f ) . (3.25)In order to obtain the required dark matter(DM) relic density, one needs to adjust thereheating temperature as a function of the gravitino mass. Besides, it is shown in [50] thatthe late decay of the lightest messenger to visible sector particles can induce a substantialamount of entropy production which would result in the dilution of the predicted gravitinoabundance. As a result, one would obtain suitable gravitino dark matter for arbitrarilyhigh reheating temperatures. Due to the flexibility of the theory, we do not impose the DMrelic density constraints in our GMSB scenario.To illustrate the constraints from LFV processes l i → l j γ , we show in the right panelsof fig.1 the survived points with additional LFV bounds ( m T /y u ∆ ) (cid:46) . × GeV (left)and . × GeV(right), respectively. We have the following discussions related to ournumerical results – 13 –
It is fairly nontrivial to check if successful EWSB condition is indeed fulfilled. Thesurvived points after imposing the EWSB constraints and the bounds from (I) to (IV)are shown in fig.1. As shown in upper left panel of fig.1, numerical results indicatethat the non-trivial couplings λ , λ , λ , especially λ ∈ [1 . , . , are required toobtain realistic low energy NMSSM spectrum. Non-vanishing λ , which determinesthe couplings between S and the messengers, is necessary to give sizable contributionsto the trilinear couplings A κ and m S , which receive no additional contributions frompure GMSB. Such Yukawa mediation contributions, whose sizes need to be of orderthe EW scale, are indispensable to satisfy the EWSB conditions and could determinethe size of λ to be of O (1) . The couplings λ can also contribute to the Higgs masses.Constraints from the neutrino masses on λ , λ and M mess are fairly mild because thecombination eqn(3.23) can easily be satisfied in the allowed range of the parameters.It can be seen that the scale of the triplet in GMSB scenario are constrained to lieabove GeV. We checked that lower value of M mess can not survive the boundsfrom Higgs mass and LHC data. No additional upper bounds for M mess (other than . × GeV ) are found from constraints (I) to (IV). From the upper right panels offig.1, we can see that LFV bounds can be fairly restrictive. Many otherwise survivedpoints are ruled out by the m T /y u ∆ bound. If we choose ( m T /y u ∆ ) (cid:46) . × , anupper bound for the messenger scale M mess (cid:46) . × GeV can be obtained fromour numerical results. • Without the constraints on m T /y u ∆ , the values of κ should lie between 0.54 to 0.66(see the left panel in the second row of fig.1). It is also clear that the allowed rangesof λ and the iteratively obtained (from EWSB conditions) tan β , are found to liebetween [0.1,0.2] and [8,16], respectively. The value of F X /M determines the wholescale of the soft SUSY breaking spectrum, including the top squark masses and thescale of the trilinear coupling A t . From the left panel in the third row of fig.1, wecan see that F X /M should take the values between TeV to
TeV to generatesparticles masses of order ∼ TeV.Again, it is clear from the middle right panels that the bounds from m T /y u ∆ canimpose stringent constraints on the otherwise survived parameters. If we choose ( m T /y u ∆ ) (cid:46) . × , the values of κ are constrained to lie between . and . while the values of F X /M should lie between TeV to
TeV. • From the lower left panel of fig.1, the gluinos are constrained to lie between 4.5 TeVto 8 TeV, which can be accessible only in the future VLHC with √ s = 100 TeV. Itis also clear that our scenario can successfully account for the 125 GeV Higgs bosonin the case that the 125 GeV Higgs is the lightest CP-even scalar.The Higgs mass inNMSSM can be approximately given by [10] m h (cid:39) M Z cos β + λ v sin β − λ κ v ( λ − κ sin 2 β ) + 3 m t π v (cid:34) ln (cid:32) m T m t (cid:33) + A t m T (cid:32) − A t m T (cid:33)(cid:35) , (3.26)– 14 – igure 1 . Survived points that can satisfy the EWSB conditions and the constraints from (I) to(V) in type-II neutrino seesaw mechanism extension of NMSSM from GMSB. The 125 GeV Higgsis found to be the lightest CP-even scalar for all the survived points. The BG fine tuning measuresare also shown in different colors. In the right panels, we show the survived points with additionalLFV bounds ( m T /y u ∆ ) (cid:46) . × GeV (left) and . × GeV(right), respectively. – 15 –ith v ≈
174 GeV , m T = m U and A t the stop trilinear coupling. As the survivedpoints require large κ , small λ and intermediate tan β , the NMSSM specific tree-levelcontribution λ v sin β to Higgs mass is always small. Besides, the mixing with thesinglet scalar will provide destructive contributions to Higgs mass, which can be seenin eqn.(3.26). Therefore, large A t or heavy stop masses are still needed in this scenarioto accommodate 125 GeV Higgs. Fortunately, due to the new contributions to A t fromtype-II neutrino seesaw specific interactions, the 125 GeV Higgs can be successfullyobtained by some portion of input parameters. The survived ranges of F X /M canjust lead to such TeV scale stops and A t term. We also note that the value A t /m ˜ T liestypically away from the maximal mixing value A t /m ˜ T (cid:39) ±√ . So, the contributionfrom the second term of second line in eqn.(3.26) is very small, necessitating largecontributions from the ln( m ˜ T /m t ) term with relatively heavy stops. Although the125 GeV Higgs boson can be either the lightest or the next-to-lightest CP-even scalar,our numerical results indicate that it can only be the lightest CP-even scalar in thisscenario. A benchmark point is given in Table 1 to illustrate the soft spectrum of oureconomical type-II neutrino seesaw mechanism extension of NMSSM from GMSB. Table 1 . Benchmark point for our economical type-II neutrino seesaw mechanism extension ofNMSSM from GMSB. All mass parameters are in the unit of GeV. F X /M mess . M mess . × λ . λ . λ . λ . κ . β . A λ − . A κ − . A t . A b . A τ . m h . m h . m h . m a . m a . m ± h . m ˜ d L . m ˜ d R . m ˜ u L . m ˜ u R . m ˜ s L . m ˜ s R . m ˜ c L . m ˜ c R . m ˜ b . m ˜ b . m ˜ t . m ˜ t . m ˜ e L . m ˜ e R . m ˜ ν e . m ˜ µ L . m ˜ µ R . m ˜ ν µ . m ˜ τ . m ˜ τ . m ˜ ν τ . m ˜ χ − . m ˜ χ − . m ˜ χ . m ˜ χ − . m ˜ χ . m ˜ χ ± − . m ˜ χ ± . µ eff . m ˜ g . • The Barbieri-Giudice fine-tuning(FT) measure with respect to certain input param-eter (cid:48) a (cid:48) is defined as [51] ∆ a ≡ (cid:12)(cid:12)(cid:12)(cid:12) ∂ ln M Z ∂ ln a (cid:12)(cid:12)(cid:12)(cid:12) , (3.27)– 16 –hile the total fine-tuning is defined to be ∆ = max a (∆ a ) with { a } the set of param-eters defined at the input scale.The Barbieri-Giudice FT measures of our scenario are shown in the the lower panels.Without the constraints on m T /y u ∆ , the BGFT satisfies (cid:46) ∆ (cid:46) . Especially,in the most interesting region where m h (cid:38) , the BGFT are of order .The BGFT can be as low as 100 in low gluino mass regions. As the gluino mass isdetermined by F X /M , which set all the soft SUSY mass scale, lighter m ˜ g in generalindicates lighter stop, reducing the FT involved. The survived points with LFVbounds are shown in the lower right panels of fig.1. The predicted Higgs mass can notexceed . GeV in such cases. If we choose ( m T /y u ∆ ) (cid:46) . × , the predictedHiggs mass should lie near 122 GeV with the BGFT of order 1000.We should note BGFT in general will overestimate the fine-tuning [52]. In fact, evenif the low energy effective theory looks fine-tuned, the high scale correlations presentin the ultimate theory lead to little or no fine-tuning. • Although it is possible for lightest stau to be the next-to-lightest supersymmetricparticle(NLSP) in ordinary GMSB, we checked that the NLSP in our GMSB scenariowill always be the lightest neutralino. Its dominant decay mode is ˜ χ → γ + ˜ G .We know that the triplet messenger scale should be very high (of order GeV)to accommodate the type-II neutrino seesaw mechanism. Therefore, to obtain TeVscale SUSY particle, the SUSY breaking √ F X should be of order ∼ GeV.As the parameter /F X determines the lifetime of the NLSP decaying into gravitino,the average distance traveled by neutralino NLSP can be large so as that it decaysoutside the detector and therefore behaves like a stable particle. So the collidersignatures closely resemble those of the ordinary supersymmetric scenarios with astable neutralino. To generate realistic EWSB in NMSSM, soft SUSY breaking parameters relating to singlet S are necessarily present. As the gauge singlet receives no contributions from pure gaugemediation, additional Yukawa mediation contributions should be present in addition to pureGMSB contributions, complicating the relevant model building. Besides, the numericalresults in previous section indicate that it is still hard to interpret the 125 GeV Higgs masseven in (G)NMSSM because of the small NMSSM-specific tree-level contributions to Higgsmass constrained from EWSB conditions.To simplify the previous problems in GMSB, we can move to the predictive AMSBscenario, in which AMSB contributions to m S , A κ , A λ are naturally present. Unfortunately,the minimal AMSB scenario predicts negative slepton square masses and must be extended.The most elegant solution to tachyonic slepton is the deflected AMSB [18] scenario in whichadditional messenger sectors are introduced to deflect the AMSB trajectory and lead topositive slepton mass by additional gauge or Yukawa mediation contribution. The triplets,– 17 –hich are required in type-II neutrino seesaw mechanism, can naturally be fitted into themessenger sector. In AMSB, the GMSB contributions of the messengers will cancel the change of AMSB con-tributions if simple mass thresholds for messengers are present. To evade such a difficulty, apseudo-moduli field X can be introduced with its VEV (cid:104) X (cid:105) = M + θ F X to determine themessenger threshold as well as the SUSY breaking order parameter. A deflection parameter (cid:48) d (cid:48) , which characterizes the deviation from the ordinary AMSB trajectory, can be intro-duced and its concrete value will depend on the form of the pseudo-moduli superpotential W ( X ) . Positive slepton masses can be achieved with either sign of deflection parameter (cid:48) d (cid:48) . The superpotential in this scenario can also be written as the form in eqn(3.9). The W mess ; B is replaced by W = y X X · + W ( X ) , (4.1)within which the coupling between S and , is absent, simplifying the AMSB modelbuilding. So Z -invariant NMSSM can be adopted here without the needs of double messen-ger species. Expression of W ( X ) can be fairly generic and leads to a deflection parameterof either sign given by d ≡ F X M F φ − . (4.2)After integrating out the messengers, the type-II neutrino seesaw mechanism extension ofNMSSM can be obtained with the NMSSM superpotential taking the Z invariant form.We can calculate the soft SUSY breaking parameters following the approach in our previousworks [17]. • The soft gaugino mass is given at the messenger scale by M i ( M mess ) = g i (cid:18) F φ ∂∂ ln µ − dF φ ∂∂ ln | X | (cid:19) g i ( µ, | X | , T ) . (4.3)So the gaugino masses are given as M i = − F φ α i ( µ )4 π ( b i − d ∆ b i ) , (4.4)with ( b , b , b ) = ( 335 , , − , ∆( b , b , b ) = ( 7 , , . (4.5)– 18 – The trilinear soft terms will be determined by the superpotential after replacing canon-ical normalized superfields. They are given by A ijk ≡ A ijk y ijk = (cid:88) i (cid:18) − F φ ∂∂ ln µ + dF φ ∂∂ ln X (cid:19) ln [ Z i ( µ, X, T )] , = (cid:88) i (cid:18) − F φ G − i + dF φ ∆ G i (cid:19) . (4.6)Similarly, we can obtain the m S (cid:48) and ξ S terms. The trilinear soft terms etc are givenby A t = F φ π (cid:104) ˜ G y t − d ( y u ) (cid:105) ,A b = F φ π (cid:26) ˜ G y b − d (cid:20) (cid:0) y L ; (cid:1) + 2 (cid:16) y d (cid:17) (cid:21)(cid:27) ,A τ = F φ π (cid:26) ˜ G y τ − d (cid:20) (cid:0) y L ; (cid:1) + 2 (cid:16) y d (cid:17) (cid:21)(cid:27) ,A λ = F φ π (cid:26) ˜ G λ − d (cid:20) y u ) + 2 (cid:16) y d (cid:17) (cid:21)(cid:27) ,A κ = F φ π (cid:104) ˜ G κ (cid:105) ,m S (cid:48) = µ (cid:48) F φ π
23 ˜ G κ ,ξ S = ξ F F φ π
13 ˜ G κ ,m = µ F φ π (cid:26) ˜ G H u ,H d − d (cid:20) y u ) + 2 (cid:16) y d (cid:17) (cid:21)(cid:27) , (4.7)with ˜ G λ = 4 λ + 2 κ + 3 y t + 3 y b − (3 g + 35 g ) , ˜ G κ = 6 λ + 6 κ , ˜ G y t = λ + 6 y t + y b − ( 163 g + 3 g + 1315 g ) , ˜ G y b = λ + y t + 6 y b − ( 163 g + 3 g + 715 g ) , ˜ G y τ = λ + 3 y b − (3 g + 95 g ) , ˜ G H u ,H d = 2 λ + 3 y t + 3 y b − (3 g + 35 g ) . (4.8) • The soft scalar masses are given by m soft = − (cid:12)(cid:12)(cid:12)(cid:12) − F φ ∂∂ ln µ + dF φ ∂∂ ln X (cid:12)(cid:12)(cid:12)(cid:12) ln [ Z i ( µ, X, T )] , (4.9) = − (cid:32) F φ ∂ ∂ (ln µ ) + d F φ ∂∂ (ln | X | ) − dF φ ∂ ∂ ln | X | ∂ ln µ (cid:33) ln [ Z i ( µ, X, T )] , Although Z invariant NMSSM is adopt in this scenario, we list the expressions of the most generalGNMSSM spectrum. The Z invariant results can be obtained by setting ξ S etc to vanish. – 19 –etails of the expression involving the derivative of ln X can be found in our previousworks [53–55].Expressions for scalars can be parameterized as the sum of each contributions m soft = δ A + δ I + δ G , (4.10)with δ A the anomaly mediation contributions, δ G the general gauge(Yukawa) media-tion contributions and δ I the interference contributions, respectively. Because of eachterm is rather lengthy, we collect their expressions in appendix A.2.If the y d ∆ ∆ T H d H d term is present in the superpotential of GNMSSM, the soft SUSY break-ing trilinear term can give subleading contribution to Majorana neutrino mass via type-IIseesaw mechanism. The trilinear scalar coupling ˜∆ T − H d − H d can be obtained before weintegrate out the messengers involving ∆ T A H d H d ∆ T = y H d H d ∆ T F φ π G H d H d ∆ T , (4.11)with G H d H d ∆ T = y X + (cid:88) c (cid:0) y L ; c (cid:1) + 5 (cid:16) y d (cid:17) + 6 y b − (cid:18) g + 95 g (cid:19) , (4.12)the corresponding Yukawa beta function between M mess and M GUT . In our scenario with Z invariant NMSSM, such trilinear term vanishes because of vanishing y d ∆ ∆ T H d H d term. The free parameters for our economical type-II neutrino seesaw mechanism extension ofNMSSM from deflected AMSB are given as F φ , M mess , d, y L ; a , y d , y u , λ, κ. (4.13)The spurion X is also normalized so that y X = 1 . We also adopt y L ; a = λ , y d = y u = λ , to reduce further the free parameters of this scenario.In ordinary AMSB realization of NMSSM, large A λ , A κ needs large λ and κ so as toinduce large positive m S , suppressing the singlet VEV [56]. In our scenario, as can beseen in eqn(4.7) and eqn(4.10), new interactions involving H u , H d and triplets will lead toadditional contributions to A λ , possibly ameliorating the previous difficulties.We still use NMSSMTools 5.5.0 [43] to scan the whole parameter space. Randomlyscan in combine with MCMC method are used. Similar to the choice in GMSB, the rangeof the free parameters are chosen as GeV < M mess < . × GeV ,
10 TeV < F φ <
500 TeV , − < d < , < λ , λ < √ π , . < λ, κ < . λ + κ (cid:46) . . (4.14)– 20 –ounds for λ and M mess from LFV are also similar to that of GMSB. In our scan, inaddition to the constraints of the neutrino masses eqn(3.23), constraints from (I) to (IV)in GMSB scenario are also imposed here. Besides, we also impose the following constraints • (V) The purpose of deflection in AMSB is to solve the notorious tachyonic sleptonproblem. So non-tachyonic sleptons should be obtained after RGE running to theSUSY scale. • (VI) The relic density of cosmic DM should satisfy the Planck data Ω DM = 0 . ± . [57] in combination with the WMAP data [58](with a theoretical uncer-tainty). We impose only the upper bound of Ω DM in our numerical studies becauseother DM species can also possibly contribute to the relic abundance of DM.In NMSSM, the 125 GeV Higgs boson in general can be either the lightest or the next-to-lightest CP-even scalar. Depending on the nature of the 125 GeV Higgs, we have thefollowing discussions related to our numerical results • A ) 125 GeV Higgs is the lightest CP-even scalar.Our numerical scan indicates that EWSB conditions alone can already ruled out alarge portion of the total parameter space. Combing with the constraints from (I)to (VI), we can obtain the survived points that lead to realistic SUSY spectrum atlow energy, which are shown in fig.2. In the left panel of fig.2, the allowed κ versus λ regions are given. We can see that κ should lie between . to . while λ should liebetween . to . . The tan β , which is obtained iteratively from the minimizationcondition of the Higgs potential for EWSB, are constrained to lie between and for
40 TeV ≤ F φ ≤
140 TeV .It is interesting to note that the messenger scale, which is just the heavy tripletscalar scale in type-II seesaw mechanism, are constrained to lie in a small band, from . × GeV to . × GeV. Lower values of M mess are ruled out by Higgsmasses and LHC data. The allowed ranges of λ , which is just the coupling y u ∆ ,can be seen to lie in a very narrow band centered at λ ≈ . . Neutrino massesbounds alone allow light M mess with tiny y u ∆ . However, successful EWSB in NMSSMas well as non-tachyonic slepton requirements etc forbid too small y u ∆ , as relativelylarge couplings are needed to give non-negligible Yukawa mediation contributions tothe soft SUSY breaking parameters. We also show the possible exclusion lines from l i → l j γ LFV processes, which give an upper bounds for M mess /λ . For example, theconservative requirement M mess /λ (cid:46) (0 . × GeV) will set an upper bound formessenger scale to be . × GeV . We left the detailed discussions on LFV boundsin AMSB scenarios in our future works.The plot of Higgs mass m h versus A t or lighter stop mass ˜ t are shown in the middleleft panel of fig.2. It is clear from the panel that our scenario can successfully accountfor the 125 GeV Higgs boson. From the allowed values of λ and tan β , it can beseen that the NMSSM specific λ v sin β contributions to the Higgs mass m h issmall, which is estimate to be
49 GeV for tan β = 10 . We note that this small– 21 – igure 2 . Survived points that can satisfy the EWSB conditions and the constraints from (I) to(V) in case the 125 GeV Higgs is the lightest CP-even scalar in AMSB-type scenario. The lowerright panel shows the survived points with additional LFV bound M mess /λ (cid:46) (0 . × GeV) . contribution to Higgs mass is still much larger than the case of GMSB. So, large A t or heavy stop is necessary to give the 125 GeV Higgs. Fortunately, A t receivesadditional contributions in our scenario, which will increase A t for negative deflectionparameters. Besides, unlike our GMSB case, the ratio A t /m ˜ T lies much nearer tothe maximal mixing value A t /m ˜ T (cid:39) ±√ , making the second term of the secondline of eqn.(3.26) to give important contributions to Higgs mass. So the radiative– 22 –orrection from ln( m ˜ T /m t ) term needs not be too large, making light m ˜ T possible.As m ˜ T , which is typically determined by F φ , characterizes the mass scale for coloredsparticles, the SUSY breaking spectrum of other colored sparticles can be relativelylight.The middle right panel of fig.2 shows that the preferred deflection parameters liebetween − . and − . , which indeed increase the value of A t . It can also be seenfrom this panel that the allowed regions require non-vanishing λ , λ couplings, whichmeans that Yukawa deflection in AMSB by the triplet messengers etc is indispensableto obtain realistic low energy SUSY spectrum. We can see that stop as light as almost2 TeV can interpret the observed 125 GeV Higgs. It can also be seen that larger A t or ˜ t can predict larger Higgs mass as expected. The values of F φ can determine thewhole scales of the soft spectrum. We can see from the lower left panel of fig.2 thatthe gluino are constrained to lie upon 3.5 TeV. Such a heavy gluino can evade thecurrent LHC bounds. The upper bounds for gluino is 10 TeV, which is not accessiblein the near future experiment. We also show the allowed Higgs mass with additionalLFV constraints M mess /λ (cid:46) (0 . × GeV) in the lower right panel of fig.2. We cansee that the predicted Higgs mass can not exceed 124 GeV because lower messengerscale will lead to smaller RGE contributions to soft scalar masses. The gluino masswill also be bounded to be less than 5.8 TeV with this constraint.The Barbieri-Giudice FT measures of our scenario are shown in the middle left panelof fig.2. In the allowed region, the BGFT satisfies ∆ (cid:46) . The FT can be as lowas 30 in the low gluino mass regions. We know that the gluino mass is determined by F φ , which set all the soft SUSY mass scale. So, lighter m ˜ g in general indicates lighterstop. We can see that lighter A t or ˜ t will lead to smaller BGFT for fixed Higgs mass.On the other hand, increasing ˜ t while at the same time increasing A t can possiblymake the involved FT unchanged. This conclusion agrees with conclusions from theelectroweak FT measure ∆ EW [36]. As the BGFT in general will overestimate theFT involved, the region with intermediate BGFT can still be natural. To illustratethe spectrum of this scenario, we show a benchmark point in Table 2.In AMSB, the gravitino mass will be of order F φ , which is heavier than ordinary softSUSY breaking parameters. Therefore, the lightest neutralino can act as the DMcandidate. We can see from the upper left panel of fig.3 that in most of the previousallowed parameter space, the neutralino will lead to under-abundance of DM, althoughfull abundance of DM is still possible for a small portion of the parameter space. Thiscan be understood from the ingredients of the neutralino, which is shown in the upperright panel. We can see that DM is singlino-dominant in most of the parameter space.The almost pure singlino-like DM is a distinctive feature of NMSSM. Its relic densitycan be compatible with WMAP bounds if it can annihilate via s-channel CP-even (orCP-odd) Higgs exchange when such Higgs has sufficient large singlet component fornot too small κ . Under abundance of DM will not cause a problem as other specie ofDM, such as axion or axino, can possibly contribute to the remaining abundances ofDM. – 23 – able 2 . Benchmark point for type-II neutrino seesaw mechanism extension of NMSSM fromAMSB in the case that the 125 GeV Higgs is the lightest CP-even scalar. All mass parametersare in the unit of GeV. ∆ a µ denotes additional SUSY contributions to muon anomalous magneticmomentum. F φ . M mess . × λ . λ . d − . λ . κ . β . A λ . A κ − . A t . A b − . A τ . m h . m h . m h . m a . m a . m ± h . m ˜ d L . m ˜ d R . m ˜ u L . m ˜ u R . m ˜ s L . m ˜ s R . m ˜ c L . m ˜ c R . m ˜ b . m ˜ b . m ˜ t . m ˜ t . m ˜ e L . m ˜ e R . m ˜ ν e . m ˜ µ L . m ˜ µ R . m ˜ ν µ . m ˜ τ . m ˜ τ . m ˜ ν τ . m ˜ χ . m ˜ χ − . m ˜ χ . m ˜ χ − . m ˜ χ − . m ˜ χ ± . m ˜ χ ± − . µ eff . m ˜ g . χ h . σ SIP . × − pb ∆ a µ − . × − The direct detection experiments, such as LUX [59],Xenon [60],PandaX [61], will setupper limits on the WIMP-nucleon scattering cross section. The spin-independent(SI)and spin-dependent(SD) scattering cross section of the neutralino DM is displayed inthe left and right lower panels in fig.3, respectively. As the SI neutralino-nucleoninteraction arises from s-channel squark, t-channel Higgs (or Z) exchange at the treelevel and neutralino-gluon interactions from the one-loop level involving quark loops,singlino-like DM can evade the direct detection constraints if the t-channel exchangedHiggs is not too light for heavy squarks. The SD neutralino-nucleon interaction isdominated by Z exchange for heavy squarks with the corresponding cross sectionproportional to the difference of the Higgsino components σ SD ∝ | N − N | . Ifthe two Higgsino components are large but similar, the SD cross section can becomesmall, which however will lead to large σ SI as σ SI ∝ | N + N | . We can see fromthe middle panels that although some portion of the allowed parameter space is ruledout by DM direct detection experiments, especially by σ SI in case the singlino-likeDM provides full abundance of DM(the green points), a large portion of parameterspace is still not reached by current experiments if there are other DM componentsother than the lightest neutralino. • B ) The 125 GeV Higgs is the next-to-lightest CP-even scalar.– 24 – igure 3 . Relic density of the neutralino DM versus the DM mass is given in the upper left panel.The ingredients of the neutralino DM is shown in the upper right panel. The spin-independentcross section σ SI (left) and the spin-dependent cross section σ SD (right) versus DM mass for DMdirect detection experiments are given in the lower panels, respectively. The green points denotethe parameters that can provide full DM relic abundances. It can be seen in the panels of fig.4 that the nature of SM Higgs as the next-to-lightestCP-even scalar in addition to EWSB conditions and bounds from (I) to (VI) can ruleout most of the points in the parameter space. We can give similar discussions asCase A.Numerical results indicate that the non-trivial deflection parameter (cid:48) d (cid:48) and couplings λ , λ are absolutely necessary to obtain realistic low energy NMSSM spectrum. Fromthe upper left panel of fig.4, we can see that the central value of (cid:48) d (cid:48) is − . and thecouplings λ , λ are constrained to take non-vanishing values. The values of λ lie in anarrow band centered at λ ≈ . . From the upper and middle panels of fig.4, we cansee that the allowed κ should lie between . to . while λ should lie between . to . with the iteratively obtained tan β lying between and for
50 TeV ≤ F φ ≤
130 TeV . We also show the possible exclusion lines from LFV, which give an upperbounds for M mess /λ . The conservative requirement M mess /λ (cid:46) (0 . × GeV) will set an upper bound for messenger scale to be . × GeV . From the left panelin the second row of fig.4, the gluino can be seen to be constrained to lie between– 25 – igure 4 . Survived points that can satisfy the EWSB conditions and the constraints from (I) to(IV) in the case the 125 GeV Higgs is the next-to-lightest CP-even scalar in AMSB scenario. Othernotations are the same as that in Fig.1 except the right panel in the second row, which shows thelightest CP-even scalar mass m h versus the lightest CP-odd scalar mass m a . – 26 –.5 TeV to 6.5 TeV, which maybe accessible in the HE-LHC. It is also obvious fromthis panel that the 125 GeV Higgs mass can readily act as the next-to-lightest CP-even scalar. As the width of the SM-like Higgs boson is quite narrow, the masses ofthe lightest CP-even scalar and the lightest CP-odd scalar can not be too light soas that the 125 GeV Higgs decaying into h h and a a are kinetically suppressed.Otherwise, such exotic decay modes may have sizable branching ratios and in turnsuppress greatly the visible signals of the SM-like Higgs boson at the LHC. We showthe masses of the lightest CP-even scalar versus the lightest CP-odd scalar in themiddle right panel of fig.4. All the survived points can pass the constraints from thepackage HiggsBounds 5.3.2 [62]. A benchmark point is shown in Table 3 to illustratethe typical spectrum of this scenario. Table 3 . Benchmark point for type-II neutrino seesaw mechanism extension of NMSSM from AMSBin the case that the 125 GeV Higgs is the next-to-lightest CP-even scalar. All mass parameters arein the unit of GeV. F φ . M mess . × λ . λ . d − . λ . κ . β . A λ . A κ − . A t . A b − . A τ . m h . m h . m h . m a . m a . m ± h . m ˜ d L . m ˜ d R . m ˜ u L . m ˜ u R . m ˜ s L . m ˜ s R . m ˜ c L . m ˜ c R . m ˜ b . m ˜ b . m ˜ t . m ˜ t . m ˜ e L . m ˜ e R . m ˜ ν e . m ˜ µ L . m ˜ µ R . m ˜ ν µ . m ˜ τ . m ˜ τ . m ˜ ν τ . m ˜ χ . m ˜ χ − . m ˜ χ . m ˜ χ − . m ˜ χ − . m ˜ χ ± . m ˜ χ ± − . µ eff . m ˜ g . χ h . σ SIP . × − pb ∆ a µ − . × − The lightest neutralino can be the DM candidate, which can provide full abundance ofcosmic DM only in a small region. Even though the singlino-like DM can not accountfor the full DM relic abundance in a large portion of the allowed parameter space,direct DM detection bounds from spin-independent cross section σ SI can rule outthe majority of the survived points (see the panels in the bottom of fig.4). Besides,the spin-dependent cross section σ SD can rule out the whole parameter space of thisscenario. This can be understood from the ingredients of neutralino (shown in the– 27 –ight panel of the third row), in which the difference of the Higgsino components canbe sizable. We propose to accommodate economically the type-II neutrino seesaw mechanism in NMSSMfrom GMSB and AMSB, respectively. The heavy triplets within neutrino seesaw mecha-nism are identified to be the messengers. Therefore, the µ -problem, the neutrino massgeneration, lepton-flavor-violation as well the soft SUSY breaking parameters can be eco-nomically combined in a non-trivial way. General features related to the type-II neutrinoseesaw mechanism extension of NMSSM are discussed. The type-II neutrino seesaw-specificinteractions can give additional Yukawa deflection contributions to the soft SUSY breakingparameters of NMSSM, which are indispensable to realize successful EWSB and accom-modate the 125 GeV Higgs. Relevant numerical results, including the constraints of darkmatter and possible LFV processes l i → l j γ etc, are also given. We find that our economicaltype-II neutrino seesaw mechanism extension of NMSSM from AMSB or GMSB can leadto realistic low energy NMSSM spectrum, both admitting the 125 GeV Higgs as the lightestCP-even scalar. The possibility of the 125 GeV Higgs being the next-to-lightest CP-evenscalar in GMSB-type scenario is ruled out by the constraints from EWSB, collider andprecision measurements. The possibility of the 125 GeV Higgs being the next-to-lightestCP-even scalar in AMSB-type scenario is ruled out by dark matter direct detection exper-iments. Possible constraints from LFV processes l i → l j γ can give an upper bound for themessenger scale.It is interesting to distinguish between the two scenarios of type-II neutrino seesawmechanism extension of NMSSM generated by GMSB and (d)AMSB, respectively. It isobvious from the expressions of the gaugino masses that GMSB predicts the mass ratio forgauginoes M : M : M = α ( M mess ) : α ( M Mess ) : α ( M Mess ) . (5.1)at the messenger scale M mess ∼ GeV , which will lead to the approximate mass ratio M : M : M ≈ , (5.2)at the TeV scale. For our AMSB scenario, the mass ratio of gauginoes are predicted to be M : M : M = α ( M mess ) ( − − d ) : α ( M mess ) (1 − d ) : α ( M mess ) (cid:18) − d (cid:19) , (5.3)with the values of deflection parameter d centered approximately at − . by our numericalresults. So we can get the approximate mass ratio M : M : M ≈
45 : 23 : 18 . , (5.4)at TeV scale for AMSB case. If gluino can be discovered by LHC, the mass of the lightestneutralino in GMSB, whose dominant component( (cid:38) ) is the bino for most survived– 28 –arameter space, can be predicted by such a gaugino mass ratio. The lightest neutralinoin our AMSB scenario, on the other hand, is mostly singlino-dominant and its mass cannotbe determined simply by such mass relation unless µ is known.As noted previously, the LSP in GMSB is always the gravitino ˜ G , which could actas the DM candidate. The long-lived neutralino, predicted by our scenario with M mess determined by heavy triplet threshold, behaves like a stable particle in the detector and itscollider signatures closely resemble those of the ordinary supersymmetric scenarios with astable neutralino. As the lightest neutralino decays into photons outside the detector, thediscovery of additional high energy photon sources near the detector can be an evidenceof this GMSB scenario. The AMSB scenario, however, will lead to stable neutralino. Theneutralino DM of our AMSB scenario can possibly be discovered by future DM directdetection experiments, such as LUX, Xenon1T or PandaX. The gravitino DM of GMSB,which is very light, is impossible to be discovered by such experiments.We should also brief note the differences between our scenario and ordinary deflectedAMSB (GMSB). In our scenario, we need to introduce new interaction terms involving thecouplings of the triplet to leptons as well as to the Higgs doublets to generate tiny neutrinomasses via type-II seesaw mechanism, which will lead to new contributions to the discon-tinuity of the anomalous dimensions across the triplet thresholds. That is, the soft SUSYbreaking parameters at the messenger scale take a different form in our scenario in contrastto that of ordinary deflected AMSB (GMSB). With these new contributions to sleptonmasses and A t etc, our scenarios can lead to realistic spectrum and accommodate the 125GeV Higgs more easily than ordinary deflected AMSB (GMSB). From our numerical results,it is also clear that there is a lower bound on the scale of messenger. Such a lower boundorigin from the 125 GeV Higgs and the difficulty to generate realistic NMSSM spectrum.Besides, possible LFV bounds from l i → l j γ can set an upper bound for messenger scale. Ifwe set the conservative requirement M mess /λ (cid:46) (0 . × GeV) , the messenger scale willhave an upper bound to be . × GeV ( . × GeV ) in the case of dAMSB (GMSB),respectively.
Acknowledgments
We are very grateful to the referee for helpful discussions and useful comments. Thiswork was supported by the Natural Science Foundation of China under grant numbers11675147,11775012.
A The soft SUSY breaking scalar masses from type-II neutrino seesawmechanism extension of NMSSM
We collect the expressions for the soft SUSY breaking scalar masses in the appendix.– 29 – .1 Expressions from GMSB
For later convenience, we list the discontinuity of various Yukawa beta functions across themessenger threshold ∆ β y t = 116 π (cid:104) y u ) (cid:105) , ∆ β y b = 116 π (cid:20) (cid:16) y d (cid:17) (cid:21) , ∆ β λ = 116 π (cid:20) y u ) + 2 (cid:16) y d (cid:17) + 15 ( y S ) (cid:21) , ∆ β κ = 116 π (cid:104)
45 ( y S ) (cid:105) , (A.1)and define ∆ ˜ G y t ≡ π ∆ β y t , ∆ ˜ G y b ≡ π ∆ β y b , ∆ ˜ G λ ≡ π ∆ β λ , ∆ ˜ G κ ≡ π ∆ β κ . (A.2)The soft scalar masses are given as m Q L,a = (cid:18) F X M (cid:19) π ) (cid:20) − y t ∆ ˜ G y t δ a, − y b ∆ ˜ G y b δ a, + (cid:18) g + 32 g + 130 g (cid:19) (cid:21) ,m U cL,a = (cid:18) F X M (cid:19) π ) (cid:20) − y t ∆ ˜ G y t δ a, + (cid:18) g + 815 g (cid:19) (cid:21) ,m D cL,a = (cid:18) F X M (cid:19) π ) (cid:104) y DD ∆ se ; a ) ˜ G + DD ∆ se ; a + 2 (cid:0) y LD ∆ , ; a (cid:1) ˜ G + LD ∆ , ; a − y b ∆ ˜ G y b δ a, + (cid:18) g + 215 g (cid:19) (cid:21) ,m L L,a = (cid:18) F X M (cid:19) π ) (cid:104) y LL ∆ T ; a ) ˜ G + LL ∆ T ; a + 3 (cid:0) y LD ∆ , ; a (cid:1) ˜ G + LD ∆ , ; a + (cid:18) g + 310 g (cid:19) (cid:21) ,m E cL,a = (cid:18) F X M (cid:19) π ) (cid:20)(cid:18) g (cid:19) (cid:21) ,m S = (cid:18) F X M (cid:19) π ) (cid:20) (cid:16) y S ∆ T ∆ T (cid:17) ˜ G + S ∆ T ∆ T + 6 (cid:0) y S ∆ se ∆ se (cid:1) ˜ G + S ∆ se ∆ se + 6 (cid:16) y S ∆ , ∆ , (cid:17) ˜ G + S ∆ , ∆ , − λ ∆ ˜ G λ − κ ∆ ˜ G κ (cid:21) ,m H u = (cid:18) F X M (cid:19) π ) (cid:20) (cid:16) y H u H u ∆ T (cid:17) ˜ G + H u H u ∆ T − y t ∆ ˜ G y t − λ ∆ ˜ G λ + (cid:18) g + 310 g (cid:19) (cid:21) ,m H d = (cid:18) F X M (cid:19) π ) (cid:20) y H d H d ∆ T ) ˜ G + H d H d ∆ T − y b ∆ ˜ G y b − λ ∆ ˜ G λ + (cid:18) g + 310 g (cid:19) (cid:21) , (A.3)with ˜ G + DD ∆ se ; a = 10 (cid:0) y L ; a (cid:1) + (cid:88) c (cid:0) y L ; c (cid:1) + ( y X ) + 4 y b δ a, − g − g , ˜ G + LD ∆ , ; a = 10 (cid:0) y L ; a (cid:1) + (cid:88) c (cid:0) y L ; c (cid:1) + ( y X ) + 2 y b δ a, − g − g − g , – 30 – G + LL ∆ T ; a = 10 (cid:0) y L ; a (cid:1) + (cid:16) y d (cid:17) + (cid:88) c (cid:0) y L ; c (cid:1) + ( y X ) − g − g , ˜ G + S ∆ T ∆ T = ( y u ) + (cid:16) y d (cid:17) + (cid:88) c (cid:0) y L ; c (cid:1) + 2 ( y X ) + 2 λ + 2 κ − g − g , ˜ G + S ∆ se ∆ se = (cid:88) c (cid:0) y L ; c (cid:1) + 2 ( y X ) + 2 λ + 2 κ − g − g , ˜ G + S ∆ , ∆ , = (cid:88) c (cid:0) y L ; c (cid:1) + 2 ( y X ) + 2 λ + 2 κ − g − g − g , ˜ G + H u H u ∆ T = 3 ( y u ) + ( y X ) + 6 y t − g − g , ˜ G + H d H d ∆ T = (cid:88) c (cid:0) y L ; c (cid:1) + 3 (cid:16) y d (cid:17) + ( y X ) + 6 y b − g − g , (A.4)and y S ∆ T ∆ T = y S ∆ se ∆ se = y S ∆ , ∆ , = y S ,y LL ∆ T ; a = y DD ∆ se ; a = y LD ∆ , ; a = y L ; a ,y H u H u ∆ T = y u , y H d H d ∆ T = y d . (A.5) A.2 Expressions from deflected AMSB
The expressions of δ G can be obtained by the following replacement F X M → dF φ , y S = 0 , (A.6)in eqn(A.3). The expressions of δ A are given by ordinary AMSB predictions δ AH u = F φ π (cid:20) G α + 310 G α (cid:21) + F φ (16 π ) (cid:104) λ ˜ G λ + 3 y t ˜ G y t (cid:105) ,δ AH d = F φ π (cid:20) G α + 310 G α (cid:21) + F φ (16 π ) (cid:104) λ ˜ G λ + 3 y b ˜ G y b (cid:105) ,δ A ˜ Q L ; a = F φ π (cid:20) G α + 32 G α + 130 G α (cid:21) + δ a, F φ (16 π ) (cid:104) y t ˜ G y t + y b ˜ G y b (cid:105) ,δ A ˜ U cL ; a = F φ π (cid:20) G α + 815 G α (cid:21) + δ a, F φ (16 π ) (cid:104) y t ˜ G y t (cid:105) ,δ A ˜ D cL ; a = F φ π (cid:20) G α + 215 G α (cid:21) + δ a, F φ (16 π ) (cid:104) y b ˜ G y b (cid:105) ,δ A ˜ L L ; a = F φ π (cid:20) G α + 310 G α (cid:21) ,δ A ˜ E cL ; a = F φ π G α ,δ AS = F φ (16 π ) (cid:104) λ ˜ G λ + 2 κ ˜ G κ (cid:105) , (A.7)– 31 –ith G i = − b i , ( b , b , b ) = ( 335 , , − . (A.8)The expressions of the gauge-anomaly interference terms are δ I ˜ Q L,a = − dF φ (8 π ) (cid:20) δ a, y t ∆ ˜ G y t + δ a, y b ∆ ˜ G y b − (cid:18) g + 32 g + 130 g (cid:19)(cid:21) , δ I ˜ U cL,a = − dF φ (8 π ) (cid:20) δ a, y t ∆ ˜ G y t − (cid:18) g + 815 g (cid:19)(cid:21) , δ I ˜ D cL,a = − dF φ (8 π ) (cid:20) δ a, y b ∆ ˜ G y b − (cid:18) g + 215 g (cid:19)(cid:21) , δ I ˜ L L,a = − dF φ (8 π ) (cid:20) − (cid:18) g + 310 g (cid:19)(cid:21) , δ I ˜ E cL,a = − dF φ (8 π ) (cid:20) − (cid:18) g (cid:19)(cid:21) , δ IH u = − dF φ (8 π ) (cid:20) λ ∆ ˜ G λ + 3 y t ∆ ˜ G y t − (cid:18) g + 310 g (cid:19)(cid:21) , δ IH d = − dF φ (8 π ) (cid:20) λ ∆ ˜ G λ + 3 y b ∆ ˜ G y b − (cid:18) g + 310 g (cid:19)(cid:21) , δ IS = − dF φ (8 π ) (cid:16) λ ∆ ˜ G λ + 2 κ ∆ ˜ G κ (cid:17) , (A.9)with ∆ ˜ G y b , ∆ ˜ G y b , ∆ ˜ G λ , ∆ ˜ G κ given in eqn (A.2). References [1] G. Aad et al.(ATLAS Collaboration), Phys. Lett. B710, 49 (2012).[2] S. Chatrachyan et al.(CMS Collaboration), Phys. Lett.B710, 26 (2012).[3] G. Aad et al. (ATLAS collaboration), Phys. Lett. B710 (2012) 67 (2011); Phys. Rev. D 87(2013) 012008.[4] S. Chatrchyan et al. (CMS collaboration), Phys. Rev. Lett. 107 (2011) 221804;[5] P. Minkowski, Phys. Lett. 67B (1977)421;R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44 (1980) 912;T. Yanagida, Conf. Proc. C7902131 (1979)95;M. Gell-Mann, P. Ramond, and R. Slansky, Conf. Proc. C790927 (1979) 315.[6] M. Magg and C. Wetterich, Phys. Lett. B 94 (1980) 61;J. Schechter and J. W. F. Valle, Phys. Rev. D 22 (1980) 2227;C. Wetterich, Nucl. Phys. B187 (1981) 343;G. Lazarides, Q. Shafi, and C. Wetterich, Nucl. Phys. B181 (1981) 287;R. N. Mohapatra and G. Senjanovic, Phys. Rev. D 23 (1981) 165.[7] R. Foot, H. Lew, X. G. He, and G. C. Joshi, Z. Phys. C44 (1989) 441;E. Ma, Phys. Rev. Lett. 81 (1998) 1171;E. Ma and D. P. Roy, Nucl. Phys. B644 (2002) 290. – 32 –
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