Electron and Muon Anomalous Magnetic Moments in the Inverse Seesaw Extended NMSSM
EElectron and Muon Anomalous Magnetic
Moments in the Inverse Seesaw Extended
NMSSM
Junjie Cao, Yangle He, Jingwei Lian, Di Zhang and Pengxuan Zhu
Department of Physics, Henan Normal University, 453007, China
E-mail: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] Abstract:
The recently improved observation of the fine structure constant has ledto a negative 2 . σ anomaly of electron g −
2. Combined with the long-existing positive3 . σ discrepancy of the muon anomalous magnetic moment, it is interesting anddifficult to explain these two anomalies with a consistent model without introducingflavor violations. We show that they can be simultaneously explained in the inverseseesaw extended next-to-minimal supersymmetric standard model (ISS-NMSSM) bythe Higgsino–sneutrino contributions to ( g − e and ( g − µ . The spectrum featuresprefer light µ , which can predict m Z naturally, and it is not difficult to obtain a τ -type sneutrino dark matter candidate that is compatible with the observed darkmatter relic density and the bounds from dark matter direct detection experiments.Due to the compressed spectra and the undetectable decay mode of selectrons, theycan evade the current Large Hadron Collider (LHC) constraints. a r X i v : . [ h e p - ph ] F e b ontents g − ∆ a e and ∆ a µ in ISS-NMSSM 84 Dark matter phenomenology 16 Since Schwinger showed that a (cid:96) ≡ ( g (cid:96) − / α π [1], the anomalous magneticmoments of charged leptons have survived rigorous tests of the quantum electro-dynamics (QED) and the later Standard Model (SM) of particle physics for morethan half a century. Recently, an improvement of the measurement of the fine struc-ture constant α , via the recoil frequency of cesium-133 atoms, has yielded the mostaccurate measurement [2]: α − (Cs) = 137 . . (1.1)As a result, there is a negative 2 . σ discrepancy between the theoretical prediction a SM e [3] and the existing experimental measurement a exp e [4, 5] of the electron anoma-lous magnetic moment,∆ a e ≡ a exp e − a SM e = ( − ± × − . (1.2)– 1 –eanwhile, the long-standing discrepancy of the muon anomalous magnetic mo-ment [6] between the SM prediction a SM µ [7–26] and the observed value in the E821experiment of Brookhaven National Laboratory (BNL) a exp µ [27, 28] is∆ a µ ≡ a exp µ − a SM µ = (279 ± × − , (1.3)corresponding to a 3 . σ discrepancy.There is insufficient evidence to show that these two anomalies are indeed signsof new physics (NP). The Discovery level confirmation of ∆ a µ , for example, requiresefforts from the currently running E989 experiment at the Fermilab and the fu-ture J-PARC experiment and also progress in reducing the theoretical uncertainty.Providing a common explanation to these two anomalies in an NP model is very chal-lenging. In general, in a complete renormalizable model, a (cid:96) can only be a quantumloop effect, because it comes from a dimension-5 operator. In a generic NP modelwithout flavor violation, the new contribution to the anomalous magnetic moment a NP (cid:96) is proportional to the mass square of the lepton times an NP factor R NP (cid:96) . Takingthe central values of the two anomalies in Eq. (1.2) and (1.3), one can easily findthat there needs to be a difference of about −
14 between R NP e and R NP µ , R NP e R NP µ = m µ m e ∆ a e ∆ a µ ∼ − , (1.4)which is difficult to achieve from a common physical origin.At present, there have already been several discussions offering combined ex-planations of the experimental results for electron and muon anomalous magneticmoments [29–63]. Among these discussions, the supersymmetry (SUSY) frameworkincludes a chiral enhancement factor tan β , which has shown promising results [47].Ref. [29] argued that the combined explanation in the SUSY framework needs rela-tively large non-universal trilinear A -terms and also requires a flavor violation (fora more detailed discussion, see Ref. [64] and [48]). Due to the constraint from thelepton flavor violating process, Ref. [46] examined the minimal flavor violation withinthe minimal supersymmetric standard model (MSSM) and found its compatibilitywith the Higgs mediation scenario. However, since the value of parameter µ needs tobe at O (100 TeV), the parameter space of the explanation in Ref. [46] is unattractive.More recently, Ref. [43] argued that in the MSSM without any flavor violation, acombined explanation can be achieved by setting the conditions that sgn( M µ ) < M µ ) >
0. The corresponding result features very light selectrons andwino-like charginos, which avoid the Large Hadron Collider (LHC) constraint due totheir degenerate spectra. The solution of Ref. [43] is impressive, but it also has twounsatisfactory characteristics. One is that the solution prefers heavy Higgsinos withmasses µ ∼ O (1 TeV). This leads to a relatively fine-tuned electroweak sector. Ingeneral, µ should be close to the Z boson mass m Z to avoid large cancellation when– 2 –redicting the observed value of m Z = 91 . µ ∼ / m Z . The other is that wino-like particlesare too light due to the current restrictions of the LHC direct SUSY searches. Thewino exclusion planes reported by ATLAS and CMS within the simplified modelframework are appropriate for the scenario of Ref. [43]. According to Fig. 8 in theCMS report [70], for example, the benchmark points in Ref. [43] with M ∼
200 GeVare on the verge of being excluded by the multi-lepton plus E missT signal via theelectroweakino channel pp → ˜ χ ± ˜ χ .In our previous work [71], we investigated the observation that in the inverse-seesaw mechanism extended next-to-minimal supersymmetric standard model (ISS-NMSSM), due to the O (0 .
1) level Yukawa coupling Y ν of the Higgs field to theright-handed neutrino, the Higgsino–sneutrino (HS) loop can be a new source of a µ to explain ∆ a µ . Unlike the MSSM, the newly introduced HS contribution a HS (cid:96) in theISS-NMSSM prefers a light µ . The sign of a HS (cid:96) is determined by the kinematic mixingof sneutrino fields ˜ ν (cid:96)L , ˜ ν (cid:96)R , and ˜ ν (cid:96)x for a given flavor (cid:96) , not by the mixing of charginosor neutralinos. In the ISS-NMSSM explanation, the masses of wino-like particles canbe much heavier than the current LHC bounds. One can also assume one generationof sneutrinos to be the lightest supersymmetric particles (LSPs), which act as adark matter (DM) candidate co-annihilating with Higgsinos to achieve the observedrelic density. Due to the singlet nature, the DM-nucleus scattering cross sectionis naturally suppressed below the current experimental detection limits [71–73]. Inthis case, the neutral Higgsinos are the next-to-lightest supersymmetric particles(NLSPs) that decay into the invisible final states of the collider ( ˜ H → ν ˜ ν ). Thecharged Higgsino decays into a soft charged lepton and DM ( ˜ H ± → (cid:96) ± ˜ ν ). Due tothe lower production rate than that of winos and the degenerate mass spectrum, thecurrent LHC data still allow a low Higgsino mass of around 100 GeV. Thus, comparedwith the MSSM framework, the ISS-NMSSM is more natural for providing commonexplanations for ∆ a e and ∆ a µ .In this work, we investigate this issue by applying the ISS-NMSSM to explain∆ a e and ∆ a µ . The remainder of this paper is organized as follows. First, we brieflyintroduce the ISS-NMSSM and the properties of leptonic g − g − τ -typesneutrino, which co-annihilates with a Higgsino to achieve the observed DM relicdensity and does not conflict with the current DM direct search observations. InSection 5, we show the impact of the current LHC SUSY particle direct searches.Finally, we draw conclusion in Section 6.– 3 – Inverse seesaw mechanism extended next-to-minimal su-persymmetric standard model (ISS-NMSSM) and the lep-ton g − The complete definition of the ISS-NMSSM Lagrangian, such as quantum numbersetting, can be found in Ref. [74]. Here, we only briefly introduce the basic idea ofthe ‘inverse-seesaw’ extension and the neutrino sector.The ‘inverse-seesaw’ mechanism is added to the NMSSM framework by introduc-ing two gauge singlet superfields ˆ ν and ˆ X with opposite lepton numbers L = − L = 1, respectively. With the assumptions of R -parity conservation, not introducingthe ∆ L = 1 lepton number violation, the superpotential W is given as follows: W = Y u ˆ Q · ˆ H u ˆ u + Y d ˆ H d · ˆ Q ˆ d + Y e ˆ H d · ˆ L ˆ e + λ ˆ S ˆ H u · ˆ H d + κ S + 12 µ X ˆ X ˆ X + λ N ˆ S ˆ ν ˆ X + Y ν ˆ L · ˆ H u ˆ ν. (2.1)The first line of Eq. (2.1) is the standard NMSSM superpotential. The soft breakingterms of the ISS-NMSSM are given as follows: V soft = V NMSSM + M ν ˜ ν R ˜ ν ∗ R + M X ˜ x ˜ x ∗ + (cid:18) B µ X ˜ x ˜ x + λ N A N S ˜ ν ∗ R ˜ x + Y ν A ν ˜ ν ∗ R ˜ L · H u + h . c . (cid:19) , (2.2)where V NMSSM is the NMSSM soft breaking term, and ˜ ν R and ˜ x are the scalar partsof superfields ˆ ν and ˆ X , respectively. The dimensional parameter µ X is a small X -type neutrino mass term, which often is treated as an effective mass parameter toobtain the tiny masses of active neutrinos. The introduction of µ X violates the leptonnumber due to the ∆ L = 2 term µ X ˆ X ˆ X and the Z symmetry of the superpotential.Numerically, both µ X and the soft mass term B µ x are extremely small, and theyslightly split the complex sneutrino field into the CP -even part and the CP -oddpart. In the numerical calculation of a (cid:96) , the influence of these two non-vanishingparameters can be ignored. For simplicity, we simply set their value to zero. In this Actually, µ X can be determined by the active neutrino experimental data and other parameters,which is discussed elsewhere [75, 76], as follows: µ X = M T R m TD − U ∗ PMNS m diag ν U † PMNS m D − M R , (2.3)where m D = √ Y ν v u and M R = √ λ N v s are the 3 × U PMNS is the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix extracted from low-energy ex-periments [6]. – 4 –ork, all the non-SM parameters, such as Y ν , λ N , and A N , are flavor diagonal. Thatis, there are no flavor mixing parameters.The SUSY particles of particular importance to a (cid:96) are sleptons ˜ (cid:96) , (cid:96) -type sneutri-nos, neutralinos ˜ χ i , and charginos ˜ χ ± i . The neutralino and chargino sector in the ISS-NMSSM were same as that of the NMSSM. On the basis of φ = ( ˜ B, ˜ W , ˜ H d , ˜ H u , ˜ S ) T ,the symmetric neutralino mixing matrix M is as follows: M = M − g √ v d g √ v u M g √ v d − g √ v u − µ − λv u − λv d κv s , (2.4)where the Higgsino mass µ = λv s is an effective µ -term after the electroweak symme-try breaking, and v u , v d , and v s represent the vacuum expectation values (vevs) of theHiggs field H u , H d , and S, respectively. The mass eigenstates ˜ χ i = N ij φ j are arrangedin ascending order of mass. With the basis φ + = ( ˜ W + , ˜ H + u ) and φ − = ( ˜ W − , ˜ H − d ),the chargino mass term is given by φ − M ± φ + + h.c. with the mass matrix M ± = (cid:18) M g v u g v d µ (cid:19) . (2.5)The corresponding mass eigenstates are defined by˜ χ + i = V ij φ + j , ˜ χ − i = U ij φ − j . (2.6)The symmetric mass matrix M (cid:96) for slepton ˜ (cid:96) for each flavor (cid:96) in the (˜ (cid:96) L , ˜ (cid:96) R ) basis is M (cid:96) = (cid:18) M L (cid:96) + m (cid:96) + m Z cos 2 β (sin θ W − ) m (cid:96) ( A E (cid:96) − µ tan β ) M E (cid:96) − sin θ W m Z cos 2 β (cid:19) , (2.7)and the corresponding rotation matrix is represented by X (cid:96) .In terms of the particle composition, the ISS-NMSSM differs from the NMSSM [77]only in the neutrino sector. In the neutrino part, the observation on the unitarity ofactive neutrino mass rotation matrix can be derived from the non-diagonality of µ X .As a result, the unitary constraint can be read in a flavor diagonal form, λ N e Y ν e µλv u > . , λ N µ Y ν µ µλv u > . , λ N τ Y ν τ µλv u > . . (2.8)These inequalities indicate that, for given λ N and the Higgs sector parameters λ ,tan β , and µ , this unitary constraint allows Y ν e to be greater than Y ν µ , which isgood for explaining ∆ a e . For each generation (cid:96) = e, µ, τ , the sneutrino fields arethe mixtures of left-handed sneutrino ˜ ν (cid:96)L , right-handed sneutrino ˜ ν (cid:96)R , and x -type– 5 –neutrino ˜ ν (cid:96)x . On the basis of φ (cid:96)ν = (˜ ν (cid:96)L , ˜ ν (cid:96)R , ˜ ν (cid:96)x ), the symmetric mass matrix M ν isgiven by M ν = 12 M L (cid:96) + Y ν (cid:96) v u + m Z cos 2 β Y ν (cid:96) ( √ A ν (cid:96) v u − λv d v s ) Y ν (cid:96) λ N (cid:96) v u v s M ν (cid:96) + Y ν (cid:96) v u + λ N (cid:96) v s λ N (cid:96) ( √ A N (cid:96) v s + κv s − λv u v d )2 M X (cid:96) + λ N (cid:96) v s . (2.9) The mass eigenstate of one generation sneutrino is ˜ ν (cid:96)i = (cid:80) j Z (cid:96)ij φ (cid:96)ν,j , with Z (cid:96) denotingthe unitary matrix to diagonalize M ν . From Eq. (2.9), one can easily find that thediagonal elements can be adjusted by the soft breaking parameters M L (cid:96) , M ν (cid:96) , and M X (cid:96) . For the off-diagonal elements, Higgs vev terms, such as Y ν (cid:96) v u and λ N (cid:96) v s , providea scale for the mixing of the three fields of left-handed, right-handed, and x -typesneutrinos. The relative signs and the magnitude of different sneutrino componentscan be adjusted by two A -term soft breaking parameters A ν (cid:96) and A N (cid:96) . The lepton anomalous magnetic moment a (cid:96) always corresponds to lepton chirality-flipping interactions. In the ISS-NMSSM, the chirality of the (cid:96) -lepton number can beflipped by Y e (cid:96) or Y ν (cid:96) . All the SM-like diagrams (the (cid:96) -lepton number is carried onlyby lepton (cid:96) and/or neutrino ν (cid:96) ) involve only SM-particles, so their contribution to a (cid:96) is identical to the SM prediction a SM (cid:96) . Therefore, the SUSY contribution a SUSY (cid:96) , inwhich the (cid:96) -lepton number is carried also by a scalar lepton ˜ (cid:96) and/or (cid:96) -type sneutrino˜ ν (cid:96) , provides the source of the observed anomaly ∆ a (cid:96) .The one-loop SUSY contribution to a (cid:96) in the ISS-NMSSM is given as follows: a SUSY (cid:96) = a ˜ χ ˜ (cid:96)(cid:96) + a ˜ χ ± ˜ νµ ,a ˜ χ ˜ (cid:96)(cid:96) = m (cid:96) π (cid:88) i,l (cid:40) − m (cid:96) m (cid:96) l (cid:0) | n L il | + | n R il | (cid:1) F N1 ( x il ) + m ˜ χ i m (cid:96) l Re( n L il n R il ) F N2 ( x il ) (cid:41) ,a (cid:101) χ ± ˜ ν(cid:96) = m (cid:96) π (cid:88) j,n (cid:40) m (cid:96) m ν (cid:96),n (cid:0) | c L jn | + | c R jn | (cid:1) F C1 ( x jn ) + 2 m (cid:101) χ ± j m ν (cid:96),n Re( c L jn c R jn ) F C2 ( x jn ) (cid:41) . (2.10)Here, i = 1 , · · · , j = 1 , l = 1 , n = 1 , , n L il = 1 √ g N i + g N i ) X (cid:96), ∗ l − Y e (cid:96) N i X (cid:96), ∗ l , n R il = √ g N i X (cid:96)l + Y e (cid:96) N i X (cid:96)l ,c L jn = − g V j Z (cid:96), ∗ n + Y ν (cid:96) V j Z (cid:96), ∗ n , c R jn = Y e (cid:96) U j Z (cid:96)n . (2.11)– 6 –he kinematic loop function F ( x ) is normalized with condition F (1) = 1, whichis given as follows: F N1 ( x ) = 2(1 − x ) (cid:0) − x + 3 x + 2 x − x ln x (cid:1) ,F N2 ( x ) = 3(1 − x ) (cid:0) − x + 2 x ln x (cid:1) ,F C1 ( x ) = 2(1 − x ) (cid:0) x − x + x + 6 x ln x (cid:1) ,F C2 ( x ) = − − x ) (cid:0) − x + x + 2 ln x (cid:1) , (2.12)with definitions x il ≡ m χ i /m (cid:96) l and x jn ≡ m χ ± j /m ν n .We compare the differences between the MSSM contribution and the ISS-NMSSMcontribution with a (cid:96) . The full expression of a SUSY (cid:96) in the MSSM is very similar tothat in the ISS-NMSSM; the difference is that c L in Eq. (2.11) of MSSM does notcontain the Y ν (cid:96) V j term [78]. There are two main reasons that it is difficult for theMSSM to explain the two anomalies ∆ a µ and ∆ a e . The first reason is that most ofthe relevant parameters are irrelevant to the lepton flavor. The parameter µ gov-erns the Higgsino ˜ H u − ˜ H d transition and the dominant part of the scalar leptonmixing term, so that all the contributions acquire a common factor µ tan β . Otherparameters, e.g., M and M , affect the sign of the corresponding contributions in a SUSY (cid:96) . Compared with µ tan β , the trilinear soft parameter A e is usually ignored inthe left-handed and right-handed scalar lepton mass mixing. Thus, in general, a SUSY e and a SUSY µ are highly correlated in the MSSM. The second reason is the correlationbetween a ˜ χ ˜ (cid:96)(cid:96) and a ˜ χ ± ˜ ν(cid:96) . In the MSSM, the mass of the left-handed slepton ˜ (cid:96) L isslightly larger than the mass of sneutrino ˜ ν (cid:96) by about 15 GeV for large tan β . As aresult, the wino–Higgsino loop in a ˜ χ ˜ (cid:96)(cid:96) is about − / a ˜ χ ± ˜ ν(cid:96) .In ISS-NMSSM, the correlation between the contributions coming from thechargino–sneutrino loop and from the neutralino–slepton loop is relatively weak.Moreover, the ISS-NMSSM contribution a SUSY (cid:96) includes an extra HS term, which isdepicted in Fig. 1 via the mass insertion technique. Accordingly, this HS contribution a HS (cid:96) can be expressed as a HS (cid:96) ≈ m (cid:96) × π v Y ν (cid:96) µ cos β (cid:40)(cid:88) n Z (cid:96)n Z (cid:96)n × x n F C ( x n ) (cid:41) , x n ≡ µ m ν (cid:96),n , (2.13)where v = (cid:112) v u + v d = 173 GeV, and xF C ( x ) increases monotonically with x , withits value ranging from 0 to 1.5. Eq. (2.13) shows that a HS (cid:96) is enhanced by the factorcos β ≈ / tan β for large tan β . A relatively large Y ν (cid:96) is expected to achieve a larger a HS (cid:96) . The lepton chirality flipping in the HS contribution is reflected in the left–right mixing term Z n Z n for each sneutrino ˜ ν n . If the x -field-dominated sneutrino– 7 – L ‘ R γ ˜ ν R ˜ ν L ˜ H ± u ˜ H ± d Figure 1 . One-loop diagram of Higgsino–sneutrino contribution, the additional contribu-tion to a SUSY (cid:96) in the inverse-seesaw mechanism extended next-to-minimal supersymmetricstandard model (ISS-NMSSM) compared to the minimal supersymmetric standard model(MSSM). is too heavy, the contribution from the left-handed dominated sneutrino and theright-handed dominated state to a (cid:96) cancel each other because Z (cid:96) Z (cid:96) ≈ − Z (cid:96) Z (cid:96) .As a result, the way to avoid this cancellation is to mix enough x -field componentsin one of the lighter sneutrinos. As shown in Eq. (2.9), the mixture of the x -fieldand other fields is dominated by λ N v s . A light µ is favored for large a HS (cid:96) , so thesmall λ at O (0 .
01) often has a larger a HS (cid:96) . However, the leptonic unitary conditionin Eq. (2.8) also prefers λ to have small values. The most attractive property of theHS explanation is that the left-right mixing Z n Z n is positively related to A ν . Themagnitude and sign of the HS contribution can be adjusted by A ν . Numerically, an | A ν | at O (100 GeV) to O (1 TeV) is sufficient for a HS (cid:96) to explain both anomalies. ∆ a e and ∆ a µ in ISS-NMSSM The relevant SUSY particle masses have different contributions to a SUSY (cid:96) . To revealthe detailed features of a SUSY (cid:96) in the ISS-NMSSM, which is covered up by the complex-ity of the loop functions F ( x ) in Eq. (2.12), we use the MultiNest technique [79, 80]to scan the ISS-NMSSM parameter space with the following parameter values andranges: – 8 – < λ < . , | κ | < . ,
100 GeV < µ <
500 GeV , tan β = 60 ,A λ = 2 TeV , − < A κ < , − < A t = A b < , | M | < , | M | < . ,
100 GeV < M L e < ,
100 GeV < M E e < , A E e = 00 < Y ν e < . , | A ν e | < , < λ N e < . , | A N e | < , | M ν e | < , | M X e | < ,
100 GeV < M L µ < ,
100 GeV < M E µ < , A E µ = 00 < Y ν µ < . , | A ν µ | < , < λ N µ < . , | A N µ | < , | M ν µ | < , | M X µ | < , (3.1)where all parameters are defined at the scale of 1 TeV. All the other soft breakingparameters, like those first two generation squark, are fixed at a common value of3 TeV. The prior probability distribution function (PDF) of these inputs are set asuniform distributions, and the n live parameter, which indicates the number of theactive points to determine the iso-likelihood contour in the MultiNest algorithmiteration, is set at 10000. The likelihood function χ in the scan is taken as χ = χ a (cid:96) + χ + χ , (3.2)where χ a (cid:96) = (cid:16) a SUSY e +87 × − × − (cid:17) + (cid:16) a SUSY µ − × − × − (cid:17) is a standard Gaussian formof two anomalies. χ requires the sample in the parameter space given by Eq. (3.1)to predict an SM-like Higgs boson compatible with current experimental observationsusing the HiggsSignals-2.2.3 code [81–83] and to satisfy the constraints of a directsearch of the Higgs boson using the
HiggsBounds-5.3.2 code [84]. χ is introducedto ensure that the LSP is a Higgsino, the electroweak vacuum is stable, sneutrinofields have not developed non-zero vevs, and the unitary constraint of Eq. (2.8) issatisfied.In the numerical calculation, an ISS-NMSSM model file is generated by the SARAH-4.14.3 package [85, 86], the particle spectra of the parameter samples are cal-culated by the
SPheno-4.0.3 [87, 88] and
FlavorKit [89] packages, and electroweakvacuum stability and sneutrino stability are tested by the
Vevacious [90, 91] pack-age, in which the tunneling time from the input electroweak potential minimum tothe true vacuum is obtained via the
CosmoTransitions package [92].The one-dimensional profile likelihood (PL) plots of χ a (cid:96) with the values normal-ized to the maximum and marginal posterior probability density function (PDF) ofthe input parameters are shown in Fig. 2, 3, and 4. The value of PL for a fixed valueof the parameter of interest, θ = θ , is the profiled value of nuisance parameters thatmaximized L . Therefore, in a multi-dimensional model, the PL can be treated as– 9 – o s t e r i o r P D F
12 10 8 6 4 2 0 a SUSY e (GeV) P r o f il e L i k e li hood ×10 Profile LikelihoodBest-fit point, & confidence interval Posterior PDFPDF mode, & credible region Test Test P o s t e r i o r P D F a SUSY (GeV) P r o f il e L i k e li hood ×10 Test Test P o s t e r i o r P D F M (GeV) P r o f il e L i k e li hood Test Test P o s t e r i o r P D F M (GeV) P r o f il e L i k e li hood Test Test P o s t e r i o r P D F P r o f il e L i k e li hood Test Test P o s t e r i o r P D F P r o f il e L i k e li hood Test Test P o s t e r i o r P D F eff (GeV) P r o f il e L i k e li hood Test Test P o s t e r i o r P D F A (GeV) P r o f il e L i k e li hood Test Test
Figure 2 . One-dimensional profile likelihood L and posterior probability density function(PDF) distributions of a SUSY (cid:96) and electroweakino input parameters. Regions of orange areascolored blue show the 1 σ (2 σ ) confidence interval, and the best-point is marked by a blackvertical line. Regions of yellow areas colored green represent the 1 σ (2 σ ) credible region. – 10 – o s t e r i o r P D F Y e P r o f il e L i k e li hood Test Test P o s t e r i o r P D F A e (GeV) P r o f il e L i k e li hood Test Test P o s t e r i o r P D F N e P r o f il e L i k e li hood Test Test P o s t e r i o r P D F A N e (GeV) P r o f il e L i k e li hood Test Test P o s t e r i o r P D F M L e (GeV) P r o f il e L i k e li hood Test Test P o s t e r i o r P D F M E e (GeV) P r o f il e L i k e li hood Test Test P o s t e r i o r P D F M e (GeV) P r o f il e L i k e li hood Test Test P o s t e r i o r P D F M X e (GeV) P r o f il e L i k e li hood Test Test
Figure 3 . One-dimensional profile likelihood L and posterior probability density function(PDF) distributions of the input parameters of e -type slepton and sneutrino. Regions oforange areas colored blue show the 1 σ (2 σ ) confidence interval, and the best-point is markedby a black vertical line. Regions of yellow areas colored green represent the 1 σ (2 σ ) credibleregion. – 11 – o s t e r i o r P D F Y P r o f il e L i k e li hood Test Test P o s t e r i o r P D F A (GeV) P r o f il e L i k e li hood Test Test P o s t e r i o r P D F N P r o f il e L i k e li hood Test Test P o s t e r i o r P D F A N (GeV) P r o f il e L i k e li hood Test Test P o s t e r i o r P D F M L (GeV) P r o f il e L i k e li hood Test Test P o s t e r i o r P D F M E (GeV) P r o f il e L i k e li hood Test Test P o s t e r i o r P D F M (GeV) P r o f il e L i k e li hood Test Test P o s t e r i o r P D F M X (GeV) P r o f il e L i k e li hood Test Test
Figure 4 . One-dimensional profile likelihood L and posterior probability density function(PDF) distributions of the µ -flavor related input parameters. Regions of orange areascolored blue show the 1 σ (2 σ ) confidence interval, and the best-point is marked by a blackvertical line. Regions of yellow areas colored green represent the 1 σ (2 σ ) credible region. – 12 – eL eR eX e L e R L R X L R M a ss ( G e V ) Test Test
Figure 5 . Violin plots showing the mass distributions of supersymmetry (SUSY) par-ticles. Sleptons and sneutrinos are labeled by their dominated components. The violinsare scaled by count. The thick vertical bar in the center indicates the interquartile rangewith the white dot representing the median, and the long vertical line represents the 95%confidence interval. an indicator of how well the theory can explain the experiments in the subspace of θ = θ . Complementarily, the one-dimensional marginal posterior PDF reflects thevolume of subspace θ = θ .These figures show that ISS-NMSSM has a large parameter space to give acommon explanation of ∆ a e and ∆ a µ . The results are summarized as follows: • In the view of the PL, the smooth bell-shaped PL curves of a SUSY e and a SUSY µ indicate the ability of the ISS-NMSSM to fit the two anomalies. The missingpart of the PL curve where a SUSY e < − × − indicates that a large a SUSY e isdifficult to achieve in theory. • The distributions of the input parameters M , λ , µ , Y ν e , Y ν µ , and A ν e showsome trends that verify the discussion in the previous section. The narrowPL picks of 100 GeV (cid:46) µ (cid:46)
150 GeV and Y ν e ≥ . a e . The wider credible region ofthe µ -type input parameter than that of e -type parameter also confirms thisfeature. – 13 – For most input parameters, the credible regions are smaller than the confidenceinterval. The PL values are very close to 1 almost in the entire space. Thisimplies that the value of these parameters are not affected by the two anomalies. • µ ∼
110 GeV and 0 . (cid:46) λ (cid:46) .
05 cause the masses of all the singlet Higgsparticles to be on the order of several TeV. • M and M are often much greater than µ to reduce the correlations in theMSSM contributions to a SUSY e and a SUSY µ . The distributions of M shows thatthe bino–slepton loop contribution is far from dominant. The distribution of M shows that the wino-related contributions are suppressed in this explanation.There were 8311 samples obtained that explained two anomalies within the 2 σ range in total. In Fig. 5, we plot the mass distributions of the SUSY particlesvia violin plots , where sleptons and sneutrinos are labeled by their dominatingcomponents. The masses of the Higgsino triplets were around 110 GeV, and thewino particles were often heavier than 700 GeV. The masses of left-handed selectron˜ e L and ˜ µ L were distributed around 400 GeV, and the right-handed sleptons wererelatively heavy. In contrast to the MSSM spectrum, the mass of ˜ ν eL can be muchlower than that of ˜ e L in the ISS-NMSSM. M a ss ( G e V ) h h a e L e RLR eLLa
SUSY e = 8.70×10 a SUSY =2.69×10 M = 512 GeVtan =60 Test Test 050010001500 M a ss ( G e V ) h h a e L e RLR eLLRa
SUSY e = 8.59×10 a SUSY =2.68×10 M =184 GeVtan =60 Test Test
Figure 6 . Higgs and sparticle spectra for the typical samples in the ISS-NMSSM. Sleptonsand sneutrinos are labeled by their dominant components, the decay paths are shown withbranching ratios > The Higgs and sparticle spectra for two typical parameter points are shown inFig. 6. M may play a crucial role in the mass splitting of Higgsinos. Taking themixing terms as a perturbation and calculating the neutralino and chargino massesto the first order in perturbation theory, the mass splitting between Higgsinos areapproximately given as follows: A violin plot is similar to a box plot; it shows the probability density smoothed by kerneldensity estimation [93]. – 14 – .1 1 10 100 m ±1 m (GeV) m m ( G e V ) M ( G e V ) Test Test
Figure 7 . Mass splitting of Higgsinos for the scanned samples with the color bar indicatingthe value of M . ∆ m ( ˜ χ , ˜ χ ) ≈ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g v ( M + µ sin 2 β )2( M − µ ) + g v ( M + µ sin 2 β )2( M − µ ) + λ v ( m ˜ S − µ sin 2 β ) m S − µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≈ (cid:12)(cid:12)(cid:12)(cid:12) g v M M − µ ) + g v M M − µ ) (cid:12)(cid:12)(cid:12)(cid:12) , ∆ m ( ˜ χ ± , ˜ χ ) ≈ (cid:12)(cid:12)(cid:12)(cid:12) g v (1 + sin 2 β )2( M − µ ) + g v (1 + sin 2 β )2( M − µ ) + λ v (1 − sin 2 β ) m ˜ S − µ + λ ( g + g ) v M − µ )( M ˜ S − µ ) µ − g v µ (cid:18) ( m cos β + µ sin β ) m − µ − cos β (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≈ (cid:12)(cid:12)(cid:12)(cid:12) g v M − µ ) + g v M − µ ) − g v µ M − µ ) (cid:12)(cid:12)(cid:12)(cid:12) , (3.3)where m ˜ S = 2 κv s is the singlino mass. These formulae indicate that the effect of M isnegligibly small when | M | is extremely large, while as it approaches zero from above(below), it can enhance (decrease) the splitting significantly. This characteristic isshown in Fig. 7, where we projected the scanned samples on the ∆ m ( ˜ χ ± , ˜ χ ) − ∆ m ( ˜ χ , ˜ χ ) plane with the color bar denoting the M value. This figure revealsthat ∆ m ( ˜ χ ± , ˜ χ ) (cid:39) m ( ˜ χ , ˜ χ ) (cid:39)
10 GeV when | M | = 1 TeV. When M (cid:39)
200 GeV, the mass splittings increase to several tens of GeV, and when M (cid:39) −
200 GeV, they decrease to less than 1 GeV.– 15 –
Dark matter phenomenology
In the ISS-NMSSM explanation of the two leptonic anomalous magnetic moments∆ a e and ∆ a µ , the Higgsino mass is less than 200 GeV and acts as the LSP. Sucha light Higgsino is not a good DM candidate, due to its spin-dependent and spin-independent scattering rates with nuclei larger than the current experimental lim-its [66, 94–98]. In the ISS-NMSSM, a right-handed-field- or x -field-dominated sneu-trino can also serve as the DM candidate. The sizable neutrino Yukawa coupling Y ν contributes significantly to the DM-nucleon scattering rate, so that e -type and µ -type sneutrinos cannot act as DM candidates. Fortunately, flavor mixing is forbid-den in our model, and this enables us to take the lightest τ -type sneutrinos, ˜ ν τ , andits charge conjugate sate, ˜ ν τ ∗ , as feasible DM candidates [72–74] . The candidatesshould be lighter than ˜ χ and dominated in components by the right-handed field˜ ν R , the ˜ x -field, or their mixture. In addition, given the preferred parameters in thelast section, one can further restrict their properties. First, we consider the annihilation of τ -type sneutrinos, whose thermally averagedcross-section at the freeze-out temperature should satisfy (cid:104) σv (cid:105) F ∼ × − cm / sto obtain the measured abundance. As was pointed out in Ref. [73], the DM hasthree types of popular annihilation channels: ˜ ν τ ˜ ν τ ∗ → h s h s , A s A s , where h s and A s denote the singlet-dominated CP-even and CP-odd Higgs bosons, respectively,˜ ν τ ˜ ν τ ∗ → ¯ ν h ν h , ˜ ν τ ˜ ν τ → ν h ν h , and ˜ ν τ ∗ ˜ ν τ ∗ → ¯ ν h ¯ ν h , where ν h represents any of the heavyneutrinos, and the co-annihilation with the Higgsinos. Since the singlet-dominatedparticles are much heavier than the DM, ˜ ν τ ˜ ν τ ∗ → h s h s , A s A s are kinematically for-bidden. The process ˜ ν τ ˜ ν τ ∗ → ¯ ν h ν h proceeds mainly by the s -channel exchange of h s and the t -channel exchange of the singlino-dominated neutralino. Evidently, it issuppressed by the mediator mass. This situation applies to the channels ˜ ν τ ˜ ν τ → ν h ν h and ˜ ν τ ∗ ˜ ν τ ∗ → ¯ ν h ¯ ν h . Thus, only the co-annihilation may play a crucial role. In thiscase, the effective annihilation rate at temperature T takes the following form [99]: σ eff = (cid:88) a,b σ ab g a g b g (1 + ∆ a ) / (1 + ∆ b ) / × exp[ − x (∆ a + ∆ b )] , (4.1)where σ ab = σ ( ab → XY ), a, b = ˜ ν τ , ˜ ν τ ∗ , ˜ χ , ˜ χ , ˜ χ ± , X and Y denote any possible SMparticles, ∆ i ≡ ( m i − m ˜ ν τ ) /m ˜ ν τ for i = a, b represents the mass splitting between theinitial particle i and ˜ ν τ , x ≡ m ˜ ν τ /T , g i represents the i th particle’s internal degreesof freedom, with g ˜ χ i = 2, g ˜ χ ± = 4, and g ˜ ν τ = g ˜ ν τ ∗ = 2, and the effective degree of In this study, we take ˜ ν τ as a complex field by setting B ν X = 0. Consequently, ˜ ν τ ∗ also actsas a DM candidate. The ISS-NMSSM is then a two-component DM theory [73]. We add that DMphysics requires non-trivial configurations in the theory’s parameter space. – 16 –reedom g eff is g eff ≡ (cid:88) a g a (1 + ∆ a ) / exp( − x ∆ a ) . (4.2)Since the Higgsino pair annihilation cross-section is much larger than 3 × − cm /s for µ (cid:39)
100 GeV, these formulae indicate that, even in the case of σ ˜ ν τ ˜ ν τ ∗ (cid:39) σ ˜ ν τ ˜ ν τ = σ ˜ ν τ ∗ ˜ ν τ ∗ (cid:39)
0, the co-annihilation can explain the abundance by choosing anappropriate ∆ i s. For the samples in the last section, we verified that this is feasibleby setting Y ν τ (cid:39) λ N τ (cid:39) M ν τ and M X τ to change∆ i s. We also found that the measured abundance requires ∆ m ( ˜ χ , ˜ ν τ ) ∼ m ( ˜ χ , ˜ χ ) (cid:39) m ( ˜ χ ± , ˜ χ ) (cid:39) m ( ˜ χ , ˜ χ ) and∆ m ( ˜ χ ± , ˜ χ ), ∆ m ( ˜ χ , ˜ ν τ ) decreases monotonically to maintain the abundance.Next, we consider the DM-nucleon scattering proceeded by the t -channel ex-change of the CP-even Higgs bosons and Z boson. The spin-dependent cross sectionis vanishing, and the spin-independent (SI) cross section is [73] σ SI N = 12 (cid:16) σ SI˜ ν τ − N + σ SI˜ ν τ ∗ − N (cid:17) = σ hN + σ ZN , (4.3)where σ hN and σ ZN with N = p, n denote the Higgs-mediated and the Z -mediatedcontribution, respectively. For the preferred parameters in the last section, σ hN and σ ZN are approximated by σ hN / cm (cid:39) . × − × C ν τ ∗ ˜ ν τ (cid:60) [ H u ] m ν τ ,σ Zn / cm = 7 . × − × ( Z τ ) , σ Zn / cm = 4 . × − × ( Z τ ) , (4.4)where C ˜ ν τ ∗ ˜ ν τ (cid:60) [ H u ] represents the coupling of the DM pair to the CP-even H u fieldand takes the following form: C ˜ ν τ ∗ ˜ ν τ Re[ H u ] (cid:39) −√ λ N τ A Y ντ Z τ Z τ − λ N τ Y ν τ v s Z τ Z τ − Y ν τ v u Z τ Z τ . (4.5)Noting that Z τ is proportional to Y ν τ , one can conclude that the scattering is sup-pressed in the case of a small Y ν τ and λ N τ . This case is favored by current and futureDM direct detection experiments [100]. In the ISS-NMSSM, the coupling of the DM to electroweakinos are g ˜ ν τ ¯ τ ˜ χ ± i = Y e τ U ∗ i Z τ P L − ( g V i Z τ − Y ν τ V i Z τ ) P R ,g ˜ ν τ ¯ ν τ ˜ χ i = 1 √ g N i − g N i ) P Z τ P R − ( Y ν τ N i P Z τ + λ N τ N i P Z τ ) P R − ( λ N τ N ∗ i P ∗ Z τ + Y ν τ N ∗ i P ∗ Z τ ) P L , – 17 – M a ss ( G e V ) h e L e RLR eLeReXLRXa
SUSY e = 7.18×10 a SUSY =2.38×10 M =143 GeVtan =60 Test Test 050010001500 M a ss ( G e V ) h e L e RLR eLeReXLRXX
Test Test
Figure 8 . Higgs and sparticle spectra for the benchmark sample with large Higgsino masssplitting before (left) and after (right) being inserted into τ -type sneutrino dark matter(DM). Sleptons and sneutrinos are labeled by their dominant components, the decay pathsare shown with branching ratios > where P is a unitary matrix to diagonalize the neutrino mass matrix in one-generationbases ( ν τL , ν τR , x τ ). These expressions show that the coupling strengths are determinedby Y ν τ and λ N τ , and they vanish when Y ν τ = 0. This characteristic has crucialapplications in the phenomenology at the LHC. Concretely speaking, in the casewhere Y ν τ and λ N τ are tiny and the DM achieves the measured abundance by theco-annihilation mechanism, the collider signal of the sparticle is roughly identicalto that of the NMSSM with ˜ χ acting as the LSP [72]. This conclusion can beunderstood from the following aspects: • The co-annihilation mechanism is effective only when m ˜ χ (cid:39) m ˜ ν τ . • Since ˜ χ of the NLSP decays as ˜ χ → ˜ ν τ ν τ , it appears as a missing track inthe LHC detectors. • ˜ χ may decay by ˜ χ → ˜ ν τ ν τ and ˜ χ → ˜ χ Z ∗ → ˜ χ f ¯ f . The two-body decayis always suppressed by the tiny coupling strength, while the phase space maysuppress the three-body decay. We checked that in the case of Y ν τ = λ N τ = 10 − and m ˜ χ − m ˜ χ (cid:46) m ˜ χ − m ˜ χ > χ mainly decaysby the three-body channel, so its signal is identical to that of the NMSSM with˜ χ as the LSP. • The situation of ˜ χ ± is quite similar to that of ˜ χ except that it decays by˜ χ ± → ˜ ν τ τ and ˜ χ ± → ˜ χ W ∗ → ˜ χ f ¯ f (cid:48) .– 18 – For the other sparticles, their interactions with ˜ ν τ are weak, and thus, theirdecay chains do not change significantly.Table 1 shows the details of a sample obtained in the last section, including themass spectra and decay modes of some moderately light sparticles. In Fig. 8, weshow the decay path of these sparticles to illustrate their properties further. Thisresult indicates that the collider signature will not change after being embedded intothe DM, which suggests that the constraints from the DM experiments should notbe considered when studying the collider phenomenology. Parameters Value Particles Mass Before M χ Decays Branching ratio[%] M χ -149.4 GeV ˜ χ (LSP) - µ χ χ → ˜ χ q ¯ q/ ˜ χ (cid:96)(cid:96)/ ˜ χ νν/ ˜ χ ± qq (cid:48) / ˜ χ ± (cid:96)ν λ , κ χ χ ± → ˜ χ qq (cid:48) / ˜ χ (cid:96)ν Y ν e , λ N e χ χ → ˜ ν eR ν/ ˜ χ ± qq (cid:48) / ˜ χ ± (cid:96)ν A ν e χ ± χ → ˜ χ ± W ∓ / ˜ χ i h/ ˜ χ i Z/ ˜ e L e/ ˜ νν M (cid:96) e χ ± χ ± → ˜ χ ± h/ ˜ χ ± Z/ ˜ χ i W ± / ˜ ν(cid:96)/ ˜ eν M E e e L After Y ν µ , λ N µ e R Decays Branching ratio[%] A ν µ -1943 GeV ˜ µ L χ → ˜ ν τX ν M (cid:96) µ µ R χ → ˜ χ q ¯ q/ ˜ χ (cid:96)(cid:96)/ ˜ χ νν/ ˜ χ ± qq (cid:48) / ˜ χ ± (cid:96)ν M E µ ν eL χ ± → ˜ χ qq (cid:48) / ˜ χ (cid:96)ν Y ν τ , λ N τ ν eR χ → ˜ ν eR ν/ ˜ χ ± qq (cid:48) / ˜ χ ± (cid:96)ν Z τ . × − ˜ ν eX χ → ˜ χ ± W ∓ / ˜ χ i h/ ˜ χ i Z/ ˜ e L e/ ˜ νν m ˜ ν τX ν µL χ ± → ˜ χ ± h/ ˜ χ ± Z/ ˜ χ i W ± / ˜ ν(cid:96)/ ˜ eν h ν µR σ SI˜ ν − p . × − cm ˜ ν µX Table 1 . Input parameters, mass spectrum, and decay modes of the sample in Fig. 8before and after inserting τ -type sneutrino DM. Fig. 5 shows that to simultaneously explain ∆ a e and ∆ a µ , very light electroweakinosand sleptons are necessary in many samples. Generally, such light sparticles arestrongly constrained by the current LHC SUSY searches.Using the data taken at √ s = 7 ,
8, and 13 TeV, searches for SUSY particleshave been conducted for several years. First, we perform analyses at 8 and 13 TeVin
SModelS v1.2 [102, 103] to refine the samples. After this, any point that passes
SModelS is further tested by analyses in
CheckMATE-2.0.26 [104–106]. The physicalprocesses considered in our work are as follows: pp → ˜ χ i ˜ χ ± j , i = 2 , , , j = 1 , pp → ˜ χ ± i ˜ χ ∓ j , i = 1 , j = 1 , pp → ˜ χ i ˜ χ j , i = 2 , , , j = 2 , , , pp → ˜ (cid:96) i ˜ (cid:96) i , i = e, µ. (5.1)– 19 –he cross section of √ s = 8 ,
13 TeV were normalized at the NLO using the
Prospino2 package [107]. The Monte Carlo events were generated by
MadGraph aMC@NLO [108,109] with the
PYTHIA8 package [110] for parton showering and hadronization. Theevent files were then input into
CheckMATE for analysis with
Delphes [111] for detectorsimulation.In addition to the analysis that has been implemented before, we added thefollowing newly released LHC analyses to
CheckMATE . • ATLAS search for chargino and neutralino production using recur-sive jigsaw reconstruction in three-lepton final states [112]: This anal-ysis is optimized for signals from ˜ χ ˜ χ ± production with on-shell W Z decaymodes. The signal regions (SRs) are split into a low-mass region (jet-veto) andthe initial state radiation (ISR) region (contains at least one energetic jet) usinga variety of kinematic variables, including the dilepton invariant mass m (cid:96)(cid:96) , thetransverse mass m T , and variables arising from the application of the emulatedrecursive jigsaw reconstruction technique. The smallness of the mass splittingslead to events with lower- p T leptons or smaller E missT in the final state. Cuts inthe low-mass SR are designed to reduce the W Z background and the numberof fake or non-prompt leptons, and cuts in the ISR region requiring large E miss T to identify events have a real E missT source. This search is sensitive to sampleswith relative light winos. • ATLAS search for chargino and slepton pair production in two leptonfinal states [101]: This analysis targets pair production of charginos and/orsleptons decaying into final states with two electrons or muons. Signal eventsare required to have an exactly opposite-sign (OS) signal lepton pair witha large invariant mass m (cid:96)(cid:96) >
100 GeV to reduce diboson and Z + jets back-grounds. SRs are separated into same-flavor and different-flavor categories withvariables m (cid:96)(cid:96) , the stransverse mass m T2 [113], E missT and E missT significance, andthe number of non- b -tagged jets. The sensitivity of this analysis to the sleptonmass can reach 700 GeV, and that to the chargino mass can reach about 1 TeV(420 GeV) of the decay mode ˜ χ ± → ˜ (cid:96)ν/(cid:96) ˜ ν → (cid:96)ν ˜ χ ( ˜ χ ± → W ± ( → (cid:96)ν ) ˜ χ ). • ATLAS search for electroweak production of supersymmetric parti-cles with compressed mass spectra [114]: This was optimized on a simpli-fied model of mass-degenerated Higgsino triplets that assumed ˜ χ ˜ χ ± productionfollowed by the decays ˜ χ ± → W ∗ ˜ χ and ˜ χ → Z ∗ ˜ χ . It is also sensitive to thedegenerate slepton-LSP mass spectrum. The selected events have exactly twoOS same-flavor leptons or one lepton plus at least one OS track, and at leastone jet is required. The pre-selection requirements include the requirementsthat the invariant mass m (cid:96)(cid:96) is derived from the J/ψ meson mass window, that E missT is greater than 120 GeV, and that the p T of the leading jet is larger– 20 –han 100 GeV. After applying the pre-selection requirements, SRs are furtheroptimized for the specific SUSY scenario into three categories: SR-E (for elec-troweakino recoiling against ISR), SR-VBF (electroweakino produced throughvector boson fusion (VBF)), and SR-S (sleptons recoiling against ISR). A vari-ety of kinematic variables and the recursive jigsaw reconstruction technique areused to identify the SUSY signals. Assuming Higgsino production, this searchoccurs at the minimum mass of ˜ χ at 193 GeV at a mass splitting of 9.3 GeV. • ATLAS search for chargino and neutralino pair production in finalstate with three-leptons and missing transverse momentum [115]: Thissearch targets chargino–neutralino pair production decaying via
W Z , W ∗ Z ∗ ,or W h into three-lepton final states. This analysis uses the full LHC Run IIdataset. The simplified model has an ˜ χ mass of up to 640 GeV for on-shell W Z decay mode with massless ˜ χ , up to 300 GeV for the off-shell W Z decaymode, and up to 185 GeV for the
W h decay mode with an ˜ χ mass below 20GeV. • ATLAS search for supersymmetric states with a compressed massspectrum [116]: This analysis uses the OS lepton pair and large E missT , search-ing for the electroweakino and slepton pair production with a compressed massspectrum. Two sets of SRs are constructed separately for the production ofelectroweakinos and sleptons. The electroweakino SRs require the invariantmass of the lepton pair m (cid:96)(cid:96) to be less than 60 GeV, and the slepton SRsrequire the stransverse mass m m χ T2 to be greater than 100 GeV, where the hy-pothesized mass of the LSP m χ is equal to 100 GeV. The most sensitive locationof the mass splitting is at about 5–10 GeV. The 95% confidence level exclusionlimits of the Higgsino, wino, and slepton are up to 145, 175, and 190 GeV,respectively. • CMS combined search for charginos and neutralinos [70]: Various sim-plified models of the SUSY are used in this combined search to interpret theresults. Related to our work, the simplified model scenario interpretation of˜ χ ˜ χ ± with decays ˜ χ → Z ( ∗ ) ˜ χ /h ˜ χ and ˜ χ ± → W ( ∗ ) ˜ χ represents the moststringent constraints from the CMS to date for electroweakino pair production.Compared with the results of individual analyses, this interpretation improvesthe observed limit in ˜ χ ± to about 650 GeV for W Z topology. • ATLAS search for electroweakino production in
W h final states [117]:This was optimized on a simplified model that assumed ˜ χ ˜ χ ± production withdecay modes ˜ χ ± → W ± ˜ χ and ˜ χ → h ˜ χ . Signal events were selected withexactly one lepton, two b -jets requiring 100 GeV < m bb <
140 GeV and E missT >
240 GeV, a transverse mass of the lepton- E missT system m T greater than 100– 21 –eV, and a “contransverse mass” m CT [118, 119] greater than 180 GeV. Massesof the winos up to 740 GeV are excluded at 95% confidence level for the masslessLSP. • CMS search in final states with two OS same-flavor leptons, jets, andmissing transverse momentum [120]: This search is sensitive to the on-shelland off-shell Z boson from BSM processes and to direct slepton production.Search regions are split into on- Z SRs, off- Z SRs, and slepton SRs via variouskinematic variables, including the invariant mass of the lepton pair m (cid:96)(cid:96) , M T2 ,the scalar sum of jet p T , the missing transverse momentum E missT , the numberof jets n j , and the number of b -tagging jets n b . The result interpretation usingthe simplified model assuming direct slepton pair production with 100% decayinto dilepton final states shows that the probing limit of the slepton mass m ˜ (cid:96) is up to 700 GeV. Certainly, this search is sensitive to the sparticles in theISS-NMSSM interpretation of ∆ a (cid:96) .Appendix A shows a part of the validation table of the analyses above. We usedthe R values obtained from CheckMATE to apply the LHC constraints. Here, R ≡ max { S i /S obs i, } for individual analysis, in which S i represents the simulated eventnumber of the i th SR or bin of the analysis, and S obs i, is the 95% confidence levelupper limit of the event number in the corresponding SR or bin. The combinationprocedure of the CMS electroweakino search [70] was also performed though the CL s method [121] with RooStats [122] using the likelihood function described previously[70].The impacts of LHC constraints on the samples were relatively strong, as 6915samples were excluded, and only 1396 samples survived, corresponding to a totalposterior probability of 11.2%. In Fig. 9, we plot the SUSY particle mass of thesurviving samples and excluded samples via a split violin plot . The LHC constraintshave not significantly changed the distributions of the SUSY particle mass. In Fig. 10,we plot the samples on the m ˜ χ − ∆ m ( ˜ χ , ˜ χ ± ) plane and the m ˜ e L − m ˜ µ L plane with M valued color. The detection ability of the LHC was mainly affected by the decaymodes of SUSY particles.1. For the light left-handed slepton pair production processes, • when ˜ ν e is very light, as shown by the samples in Fig. 8, the dominantdecay mode was ˜ e L → W ˜ ν e (the sample in Fig. 8 for example), where˜ ν e contains large left-handed ingredients . Such samples are difficult to Similar to a violin plot, the split violin plot splits the violins in half to see the difference betweentwo sample groups. Note that the widths of both sides of the violin are fixed, so that the ratio ofthe two widths does not represent the relative probability or relative number. There is only one left-handed sneutrino ˜ ν eL in the MSSM or NMSSM, and its mass is slightlylighter than that of the left-handed selectron ˜ e L – 22 – eL eR eX e L e R L R X L R M a ss ( G e V ) Test Test
Figure 9 . Split violin plot showing the SUSY particle mass distributions of samplesbased on the Large Hadron Collider (LHC) result. The left colorful KDEs are the massdistributions of samples that survive the LHC search results, and the right gray KDEsindicate samples excluded by the LHC results. The medians, interquartile ranges, and 95%confidence intervals are the same as those in Fig. 5. detect by the current LHC experiments. In the right plane of Fig. 10 andFig. 9, we find that the m ˜ e L values of the surviving samples can reach200 GeV, and such a light ˜ e L is good for explaining ∆ a e . • when | M | is lighter than m ˜ e L or m ˜ ν L , if ˜ e L → W ˜ ν e is kinematicallyforbidden, as shown in the right panel of Fig. 6, the dominated decaymode of ˜ (cid:96) L is into (cid:96) plus a bino-like neutralino ˜ χ . In this case, theLHC slepton pair production searches provide the most sensitivity to theslepton mass. Fig. 10 shows that samples with ˜ µ L <
400 GeV and small | M | have difficulty escaping the LHC constraints. • when | M | is very large and ˜ ν e is very heavy, the left-handed slepton willdecay into (cid:96) ˜ χ , (cid:96) ˜ χ , and ν ˜ χ ± , as depicted in the left panel of Fig. 6.Because the decay products of ˜ χ ± are too soft, the LHC constraints forthese samples are weaker than those in the light bino case.2. For light right-handed slepton pair production processes, the production crosssection is about 2.7 times smaller than that of left-handed slepton. The right-handed sleptons mainly decay into (cid:96) ˜ χ and (cid:96) ˜ χ . Only a few samples contain– 23 –ight ˜ µ R or ˜ e R , so the right-handed slepton had little effect on the result.3. For the Higgsino pair ˜ χ ˜ χ ± production process, the analysis in Ref. [114] pro-vides a strong constraint on the samples featured by the small positive M , asshown in the left plane of Fig. 10.4. For the wino-dominated neutralino/chargino pair production process, the ex-planation of the two lepton anomalies require M >
400 GeV, and in aboutmore than 97% samples, the wino-like particle mass is greater than 700 GeV.Therefore, in this study, because the decay modes of the winos are more com-plicated, the winos did contribute to the results, but they were not the maineffect.
100 110 120 130 140 150 m (GeV) m ± m ( G e V ) M ( G e V ) Test Test 0 200 400 600 800 1000 m e L (GeV) m L ( G e V ) M ( G e V ) Test Test
Figure 10 . Samples projected on m ˜ χ − ∆ m ( ˜ χ , ˜ χ ± ) plane and m ˜ e L − m ˜ µ L plane, wherethe color indicates the value of M . In each plane, the gray points indicate the samplesexcluded by the LHC results. In conclusion, except for some samples where ˜ e L decayed to W ˜ ν e , the samples withpositive M were strongly restricted by the current LHC experiments. The searchesfor the compressed electroweakino spectrum and the searches for sleptons at the LHCare complementary in detecting samples via leptonic final states. Finally, we plot the a SUSY e and a SUSY µ values of the samples in Fig. 11. After considering the constraintsfrom the LHC, there were still a large number of samples that could simultaneouslyfit ∆ a e and ∆ a µ at the 1 σ level.Before we conclude this section, we briefly comment that the signal of the spar-ticles may differ significantly from that of the NMSSM in the case of sizable Y ν τ and λ N τ [72–74]. In general, because the decay chain becomes lengthened and the signalcontains at least two τ leptons, the sparticles of the ISS-NMSSM are more challeng-ing to detect at the LHC than those of the NMSSM. We drew this conclusion byglobally fitting the ISS-NMSSM with various experimental data and studying the– 24 – a SUSY e a S U S Y ×10 ×10 M ( G e V ) Test Test
Figure 11 . Similar to Fig. 10 but on a SUSY e − a SUSY µ plane with the M values representedby the color. scanned samples, similar to our previous work [73]. Therefore, we may overestimatethe LHC constraints in this work, but this does not affect the main conclusion thatthe ISS-NMSSM can easily explain both anomalies . We proposed scenarios in the ISS-NMSSM that explain both electron and muon g − When this work was about to be finished, a new measurement of fine structure constant, viathe rubidium atom, is reported [123]: α − (Rb) = 137 . , (5.2)which leads to a +1 . σ discrepancy of ∆ a e :∆ a e (Rb) = a exp e − a SM , Rb e = (48 ± × − . (5.3)There are two points to be clarified here. The first is that there is a 5 . σ discrepancy betweentwo measurements α − (Cs) and α − (Rb) [123]. It is currently suspected that this difference maybe caused by speckle or by a phase shift during the measurement, which requires further study.Another point is that ISS-NMSSM can also interpret ∆ a e (Rb) and ∆ a µ simultaneously. On the onehand, since ∆ a e (Rb) and ∆ a µ have the same sign, the correlation between different contributions to a SUSY (cid:96) and the correlation between a SUSY e and a SUSY µ are no longer strongly restricted, as discussedin Section 2. On the other hand, the absolute value of ∆ a e (Rb) is about half smaller than that of∆ a e in Eq. (1.2), so it is easier to explain ∆ a e (Rb) in ISS-NMSSM. – 25 –s follows: • The HS loop induced by the neutrino Yukawa coupling Y ν plays an importantrole in the explanation. The advantage of this explanation is that the signof the HS contribution to a (cid:96) can be determined by the sign of A ν (cid:96) , whichgreatly reduces the correlation between a e and a µ . A larger HS contributioncorresponds to a light µ , which is natural for predicting m Z . • The features of the sparticle spectrum preferentially had large tan β , light Hig-gsino µ (cid:39)
110 GeV, and heavy wino M (cid:38)
600 GeV. Moreover, the massesof the left-handed selectron and smuon were around 400 and 500 GeV, respec-tively, and the singlet Higgs-dominated particles are often heavier than 1 TeV. M affects the mass splitting of Higgsino triplets and the decay mode of thesleptons. • The mass spectrum can be introduced into a right-handed or x -field-dominated τ -type sneutrino as a proper DM candidate, which co-annihilated with Higgsi-nos to obtain the observed DM relic density. As indicated by the previousdiscussion, choosing λ N τ and Y ν τ to be less than about 0.001 can avoid thechange of the LHC signal caused by the introduction of the DM, and the corre-sponding DM direct detection cross section is much smaller than the detectionability of the current experiment. • The signals of the electroweakino/slepton pairs produced at the LHC are sen-sitive to the parameter space explaining the two anomalies, especially for apositive M . However, due to the compressed mass spectrum, the insensitivedecay mode of ˜ e L → W ˜ ν e , and the very heavy wino, the surviving samples cansatisfy the current LHC constraints. Acknowledgments
Junjie Cao and Pengxuan Zhu wish to thank Jia Liu for the helpful discussion of theexperimentally measuring the fine structure constant. This work is supported by theNational Natural Science Foundation of China (NNSFC) under Grant No. 11575053,No. 11905044 and No. 12075076.
A Validations of LHC analyses
This appendix verifies the correctness of our implementation of the necessary analysesin the package
CheckMATE . For the sake of brevity, we only provide validation of thelatest analyses. Table 2 shows the cut-flow validation of the analysis in Ref. [101]for chargino pair production channel. Table 3 shows the cut-flow validation for– 26 –he slepton pair production channel of analysis [114]. All the cut-flow data wereprovided by experimental groups. The results indicate that our simulations were ingood agreement with the analysis of the experimental groups.
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50) GeV in search channel of ˜ χ ± ˜ χ ∓ production with W -boson-mediated decays. Theyields in “Baseline” of “CheckMATE” are normalized to “Baseline” of “ATLAS”. “Effi-ciency” is defined as the ratio of the event number passing though the cut-flow to the eventnumber of the previous event. Process pp → ˜ χ +1 ˜ χ − , ˜ χ ± → W ± ˜ χ Point m ( ˜ χ ± , ˜ χ ) (300, 50) GeVGenerated events 500,000Selection ATLAS CheckMATEevents efficiency events efficiency Preselection
Baseline 1144.0 · · · · · ·
Trigger 793.0 69.3% 766.0 66.9%OS signal leptons 661.0 83.4% 725.1 94.7% p (cid:96) ,(cid:96) T >
25 GeV 565.0 85.5% 548.6 75.7% m (cid:96)(cid:96) >
25 GeV 559.0 98.9% 542.5 98.9% n b − jet = 0 526.0 94.1% 507.8 93.6% SR-DF-0J
Different flavor & n jets = 0 122.7 23.3% 134.3 26.4% m (cid:96)(cid:96) >
100 GeV 94.2 76.8% 96.1 71.6% E missT >
110 GeV 46.5 49.4% 43.4 45.2% E missT significance >
10 42.2 90.8% 39.5 90.9% m T2 >
100 GeV 26.4 62.6% 26.3 66.7%
SR-DF-1J
Different flavor & n jets = 1 81.9 15.6% 84.6 16.7% m (cid:96)(cid:96) >
100 GeV 62.3 76.1% 57.7 68.2% E missT >
110 GeV 33.8 54.3% 30.3 52.5% E missT significance >
10 27.2 80.5% 28.2 93.3% m T2 >
100 GeV 15.3 56.3% 14.7 52.0%
SR-SF-0J
Same flavor & n jets = 0 138.7 26.4% 111.1 21.9% m (cid:96)(cid:96) > . E missT >
110 GeV 47.1 50.0% 40.4 54.5% E missT significance >
10 42.9 92.3% 35.9 88.7% m T2 >
100 GeV 25.4 61.9% 21.8 60.7%
SR-SF-1J
Same flavor & n jets = 1 88.8 16.9% 82.4 16.2% m (cid:96)(cid:96) > . E missT >
110 GeV 32.6 55.4% 25.4 52.8% E missT significance >
10 26.9 82.5% 24.9 98.0% m T2 >
100 GeV 14.0 52.0% 11.2 45.1% – 36 – able 3 . Similar to Table 2, but for the cut-flow validation of ATLAS analysis [112] in thesearch channel of the slepton pair production with mass point m (˜ (cid:96), ˜ χ ) = (150 , Process pp → ˜ (cid:96) ˜ (cid:96), ˜ (cid:96) → (cid:96) ˜ χ Point m ˜ (cid:96) = 150 GeV; m ˜ χ = 140 GeVGenerated events 100,000Selection ATLAS CheckMATEevents efficiency events efficiency E missT trigger 2355.37 · · · · · · < m (cid:96)(cid:96) < . φ (jet , p missT )) > φ ( j , p missT ) > < m (cid:96)(cid:96) <
60 GeV 827.86 86.3% 883.55 86.0%∆ R ee > R µµ > R eµ > p (cid:96) T > n jets ≥ p j T >
100 GeV 705.86 87.1% 702.58 79.8% n b − jets = 0 611.05 86.6% 643.78 91.6% m ττ < m ττ >
160 GeV 533.29 87.3% 569.78 88.5%same flavor 532.33 99.8% 569.01 99.9%
SR-highMass E missT >
200 GeV 229.81 43.2% 265.83 46.7%max(0 . , . − . × m ) < R ISR < . p (cid:96) T > min(20 . , . . × ( m T − m <
140 GeV 70.71 100.0% 72.51 100.0% m <
130 GeV 70.71 100.0% 72.51 100.0% m <
120 GeV 70.71 100.0% 72.31 99.7% m <
110 GeV 70.71 100.0% 72.23 99.9% m <
105 GeV 53.72 76.0% 57.10 79.1% m <
102 GeV 20.21 37.6% 23.77 41.6% m <
101 GeV 9.38 46.4% 9.90 41.6% m < . SR-lowMass
150 GeV < E missT <
200 GeV 146.36 27.5% 167.63 29.5%0 . < R ISR < . p (cid:96) T > min(15 . , . . × ( m T − m <
140 GeV 52.74 100.0% 42.29 100.0% m <
130 GeV 52.74 100.0% 42.29 100.0% m <
120 GeV 52.74 100.0% 42.29 100.0% m <
110 GeV 52.64 99.8% 41.65 98.5% m <
105 GeV 38.05 72.3% 29.09 69.9% m <
102 GeV 16.66 43.8% 11.24 38.6% m <
101 GeV 8.70 52.2% 5.60 49.8% m < .5 GeV 4.39 50.5% 2.29 40.9%