Search for exotic decays of the Higgs boson into bb̅ and missing transverse momentum in pp collisions at √s = 13 TeV with the ATLAS detector

: A search for the exotic decay of the Higgs boson ( H ) into a b ¯ b resonance plus missing transverse momentum is described. The search is performed with the ATLAS detector at the Large Hadron Collider using 139 fb − 1 of pp collisions at √ s = 13 TeV. The search targets events from ZH production in an NMSSM scenario where H → ˜ χ 02 ˜ χ 01 , with ˜ χ 02 → a ˜ χ 01 , where a is a light pseudoscalar Higgs boson and ˜ χ 01 , 2 are the two lightest neutralinos. The decay of the a boson into a pair of b -quarks results in a peak in the dijet invariant mass distribution. The ﬁnal-state signature consists of two leptons, two or more jets, at least one of which is identiﬁed as originating from a b -quark, and missing transverse momentum. Observations are consistent with Standard Model expectations and upper limits are set on the product of cross section times branching ratio for a three-dimensional scan of the masses of the ˜ χ 0 2 , ˜ χ 0 1 and a boson.


Introduction
Since its introduction, the Standard Model (SM) has successfully predicted several new particles, culminating in the discovery of the Higgs boson ( ) [1,2]. The boson, discovered by ATLAS and CMS at the CERN Large Hadron Collider (LHC) [3], appears to have properties consistent with those predicted by the SM within current experimental uncertainties [4][5][6]. The 95% confidence level (CL) upper limit on the branching ratio for boson decays to beyond-the-SM (BSM) particles, from a combined ATLAS and CMS measurement of the Higgs boson couplings, is 34% [6], although this limit comes with some assumptions. A more recent ATLAS measurement, based on 80 fb −1 of 13 TeV data [4], set a 95% CL upper limit of 21% on the branching ratio for boson decays via undetected modes. Given the magnitude of these limits and the assumptions that go into deriving them, direct searches for exotic decays of the Higgs boson remain a high priority [7]. Among the many final states discussed in Ref. [7], this analysis most closely considers a scenario arising in the next-to-minimal supersymmetric SM (NMSSM) [8], a generalization of the minimal supersymmetric SM (MSSM) [9][10][11][12][13].
The MSSM is the simplest extension to the SM that incorporates supersymmetry (SUSY). It predicts four additional Higgs bosons, generally assumed to be heavier than the boson: two neutral states, the 0 and 0 bosons, as well as two charged states, the ± bosons. The measured mass of the Higgs boson [14,15] close to 125 GeV results in the reintroduction of a 'little hierarchy' problem [16] in the MSSM. This hierarchy is alleviated in the NMSSM by allowing for additional contributions to the mass of the Higgs boson from new scalar particles. The NMSSM contains an additional pseudoscalar Higgs boson ( ), generally assumed to be less massive than the boson since its mass is protected by a Peccei-Quinn (PQ) symmetry [8].
Previous searches for exotic Higgs boson decays involving the production of the boson have focused on the -symmetry limit [8] of the NMSSM, where the dominant decay channel is → [17-27]. As proposed in Refs. [28,29], the present analysis considers instead the region of parameter space near the PQ symmetry limit of the NMSSM. Near this limit, the decay →˜0 2˜0 1 →˜0 1˜0 1 dominates over → , where˜0 2 and˜0 1 are the two lightest neutralinos, which are admixtures of the supersymmetric partners of the Higgs and gauge bosons of the SM. If the decay → is kinematically allowed, it is typically highly favoured. The˜0 1 is assumed to be stable, as obtained in R-parity conserving SUSY models [11].
The analysis presented in this paper uses 139 fb −1 of collisions at √ = 13 TeV collected with the ATLAS detector. It targets the →˜0 2˜0 1 →˜0 1˜0 1 cascade decay, where the →¯decay dominates, and the boson is produced in association with a Z boson. The Z boson is required in the analysis selection to decay into a pair of electrons or muons (hereafter referred to as leptons), which provide a signature to trigger upon and which reduce the multĳet background. 1 The resulting final state consists of a pair of oppositely charged leptons, two jets, each containing a -hadron ( -jets), and missing transverse momentum ( miss T ) from the two˜0 1 neutralinos. The search is performed for a range of values, and for a few sets of fixed values of˜0 1 and˜0 2 .
The primary Standard Model backgrounds in this search are bosons produced with heavy-flavour (bottom and charm) jets, hereafter labelled Z+HF, and tt events; their contributions are estimated from the data in control regions enhanced in these backgrounds. The dĳet invariant mass is used as the final discriminant in a binned likelihood fit. 1 The contribution from → + − decays with the subsequent decays of the -leptons into light leptons is included in the signal definition and simulation but is suppressed by a factor of at least 2000 due to the event selection requirements.
The search, which compares an expected background shape in the signal region with the measured shape, is also sensitive in principle to other distortions of the dĳet invariant mass spectrum arising from BSM physics effects.

The ATLAS detector
The ATLAS experiment [30] at the LHC is a multipurpose particle detector with a forward-backward symmetric cylindrical geometry and a near 4 coverage in solid angle. 2 It consists of an inner tracking detector (ID) surrounded by a thin superconducting solenoid providing a 2 T axial magnetic field, electromagnetic and hadron calorimeters, and a muon spectrometer. The inner tracking detector covers the pseudorapidity range | | < 2.5. It consists of silicon pixel, silicon microstrip, and transition radiation tracking detectors. The insertable B-layer, installed before Run 2 [31,32], typically provides the innermost hit on a track. Lead/liquid-argon (LAr) sampling calorimeters provide electromagnetic (EM) energy measurements with high granularity. An iron/scintillator hadron calorimeter covers the central pseudorapidity range (| | < 1.7). The endcap and forward regions are instrumented with LAr calorimeters for EM and hadronic energy measurements up to | | = 4.9. The muon spectrometer surrounds the calorimeters and is based on three large air-core toroidal superconducting magnets with eight coils each. The field integral of the toroids ranges between 2.0 and 6.0 T m across most of the detector. The muon spectrometer includes a system of precision tracking chambers and fast detectors for triggering. A two-level trigger system [33] is used to select events. The first-level trigger is implemented in hardware and uses a subset of the detector information to keep the accepted event rate below 100 kHz. This is followed by a software-based trigger that reduces the accepted event rate to 1 kHz on average depending on the data-taking conditions. An extensive software suite [34] is used for real and simulated data reconstruction and analysis, for operation and in the trigger and data acquisition systems of the experiment.

Data and simulated event samples
The data used in this analysis, corresponding to the full Run 2 dataset for collisions, were collected at a centre-of-mass energy of 13 TeV during the 2015-2018 running periods using unprescaled single-lepton triggers with a threshold of 26 GeV (transverse energy, T , for electrons and transverse momentum, T , for muons) [35,36]. Events are selected for analysis only if they are of good quality [37] and if all the relevant detector components are known to have been in good operating condition, which corresponds to a total integrated luminosity of 139.0 ± 2.4 fb −1 [38]. The recorded events contain an average of 34 inelastic collisions per bunch-crossing.
Although the dominant SM backgrounds are modelled using a data-driven technique, Monte Carlo (MC) simulated events provide input to these techniques and are used to model the subdominant backgrounds and the , →˜0 2˜0 1 signal process. A summary of all the MC event generator programs used in 2 ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector and the -axis along the beam pipe. The -axis points from the IP to the centre of the LHC ring, and the -axis points upwards. Cylindrical coordinates ( , ) are used in the transverse plane, being the azimuthal angle around the -axis. The pseudorapidity is defined in terms of the polar angle as = − ln tan( /2). Angular distance is measured in units of Δ ≡ √︃ (Δ ) 2 + (Δ ) 2 . The transverse momentum and transverse energy, T and T , are defined as sin and sin , respectively. the analysis is provided in Table 1. Samples produced with alternative generators are used to estimate systematic uncertainties in the event modelling, as described in Section 6.
In the →˜0 2˜0 1 simulated samples, a Higgs boson is produced in association with a Z boson, using P B , while P 8.210 is used to force the decay chain: →˜0 2˜0 1 and˜0 2 →˜0 1 . The Z boson is forced to decay into + − or + − or + − . Both the Higgs and bosons have narrow widths, with the Higgs boson width set to its SM value and the width set to its mass (in GeV) times 10 −5 . The boson is then required to decay into a pair of -quarks. The width is narrow enough that the experimental resolution dominates the reconstructed dĳet invariant mass width for all masses considered here.
All simulated processes are normalized using the most accurate theoretical cross-section predictions currently available and were generated at least to next-to-leading-order QCD accuracy. All samples of simulated background events were passed through the ATLAS detector simulation [70] based on G 4 [71], while signal samples were passed through a fast simulation [72] based on a parameterization of showers in the ATLAS calorimeters and employing G 4 elsewhere. The effects of multiple interactions in the same and nearby bunch crossings (pile-up) were modelled by overlaying the hard-scatter events with minimum-bias events simulated using the soft QCD processes of P 8.186 [69] with the A3 [73] set of tuned parameters (tune) and NNPDF2.3LO [47] parton distribution functions (PDF). The minimum-bias samples were reweighted such that the pile-up distribution matches that in the data. For all samples of simulated events, except for those generated using S [55], the E G 1.6.0 program [74] was used to describe the decays of bottom and charm hadrons.

Object reconstruction
Tracks measured in the ID [75] are used to reconstruct interaction vertices [76], of which the one with the highest sum of squared transverse momenta of associated tracks is selected as the primary vertex of the hard interaction.
Electrons are reconstructed from clusters of energy deposits [77] in the electromagnetic calorimeter and matched to a track in the ID [78]. One electron must satisfy the Tight identification criteria with T > 30 GeV to be on the trigger efficiency plateau, and be matched to the trigger electron, while the second is required to satisfy the Medium identification criteria with T > 20 GeV [78]. In addition, these electrons must have | | < 2.47, be outside the transition region between the barrel and endcap calorimeters (1.37 < | | < 1. 52), and have small impact parameters: 3 | 0 / 0 | < 5 and | 0 sin( )| < 0.5 mm. Finally, these electrons are required to pass the Gradient isolation requirements [78].
Muons are reconstructed as described in Refs. [79,80] and required to have | | < 2.7. The leading T muon is required to have T > 30 GeV so as to be on the trigger efficiency plateau, and must be matched to the trigger muon, while the second muon is required to have T > 20 GeV. These muons must satisfy the Medium identification criteria, pass the Gradient isolation requirements [79], and have | 0 / 0 | < 3 and | 0 sin( )| < 0.5 mm.
In order to veto events with additional leptons, looser selection criteria are employed. Electrons are required to pass the LooseAndBLayerLLH requirements with T > 10 GeV and | | < 2.47. Muons are required to satisfy the Medium criteria with T > 4 GeV and | | < 2.7.
Jets are reconstructed from energy deposits in clusters of calorimeter cells [77] using the anti-algorithm [81,82] with radius parameter = 0.4. Jet cleaning criteria are used to identify jets arising from non-collision backgrounds or noise in the calorimeters [83] and events containing such jets are removed. Jets are calibrated using the standard energy scale corrections [84]. Jets are required to have T > 20 GeV and | | < 2.4. A jet vertex tagger [85] at the Medium working point is used to remove jets with 20 < T < 60 GeV and | | < 2.4 which are identified as not being associated with the primary vertex of the hard interaction. Jets containing a -hadron are identified as -jets ( -tagged) using the MV2 multivariate discriminant [86], with the selection tuned to produce an average efficiency of 77% for -jets, with corresponding light-flavour ( -, -, -quark and gluon) and -jet misidentification efficiencies of 0.9% and 25% respectively, as measured in simulated tt events.
Jets and leptons are reconstructed independently. To prevent double counting of these reconstructed objects, an overlap removal procedure for leptons and jets is applied [87]. The looser selection criteria for leptons, as described above, are employed for the overlap removal.
The missing transverse momentum, with magnitude miss T , is calculated as the negative vector sum of the transverse momenta of all calibrated selected objects, such as electrons and jets, and is corrected to take into account the transverse momentum of muons. Tracks with T > 500 MeV, compatible with the primary vertex but not matched to any reconstructed object, are included in the miss T reconstruction to take into account the soft-radiation component that does not get clustered into any hard object [88].
To account for small efficiency differences between simulation and data, simulated events are corrected with scale factors covering lepton reconstruction, identification, isolation and trigger efficiencies, as well as jet pile-up rejection and flavour tagging efficiencies.

Event selection
To select events consistent with the decay of the Higgs boson into a¯pair plus miss T , this analysis focuses on production in which the leptonic decay of the Z boson into electrons or muons provides the trigger signature for the event.
Events are required to have two leptons of the same flavour and opposite charge; events are rejected if any additional leptons are found. Events containing muons that are poorly reconstructed, with ( / )/| / | > 0.2 where / is the charge-to-momentum ratio, or muons from cosmic-ray background, with | 0 | > 0.2 mm or | 0 | > 1 mm, are also rejected.
Events in the signal region (SR) are required to satisfy the following: • Dilepton invariant mass in the range 81 < ℓℓ < 101 GeV • Dilepton T with ℓℓ T > 40 GeV • At least two jets with T > 20 GeV • Dĳet invariant mass in the range 20 < < 120 GeV, based on the two jets with the highest T in the event, at least one of which must be -tagged , defined [28] as the scalar sum of the T of the dĳet system and miss T divided by the dilepton T , in the range: The requirement on frac T is especially useful in reducing the tt background. Requiring only one -tagged jet is a trade-off between signal acceptance and background rejection. Although the dĳet resonance search is mainly sensitive to decays of particles with masses in the range 20-65 GeV, the window for the dĳet invariant mass extends to higher values to take into account the tail in the dĳet invariant mass distribution that results from choosing a jet that does not come from → , and also to better constrain the background shape.
In addition to the SR, two control regions are defined. A control region for Z+HF (CRZ) is defined with the same requirements as the SR except with 60 < miss T < 100 GeV. A tt control region (CRTop) is defined with the same criteria as the SR but with the ℓℓ requirement inverted (and ℓℓ > 50 GeV). To validate the modelling of miss T for the Z+HF processes, a validation region VRMET is defined with the same criteria as the SR except that the miss T requirement is loosened to miss T > 50 GeV and the dĳet invariant mass requirement is > 150 GeV.
The selection criteria for the signal, control and validation regions are summarized in Table 2. The acceptance times efficiency (for →˜0 2˜0 1 →˜0 1˜0 1 →¯˜0 1˜0 1 ) of the selection varies across the three dimensions of ,˜0 1 , and˜0 2 , but is primarily a function of and varies from approximately 0.4% to 1.1%, depending on the configuration of the three masses.

Background and statistical model
The background in the SR is primarily composed of Z+HF and tt, with a small contribution from plus light-flavour jets ( +light), single-top and diboson events. The background from multĳet production was studied using same-sign lepton pairs and found to be negligible. The cross sections for Z+HF and tt production in the simulation are scaled by normalization factors, Z+HF and tt , respectively, determined from a simultaneous fit to the number of events in data and simulation in CRZ and CRTop. Subdominant backgrounds in these regions are normalized to the theoretical cross sections specified in Section 3, while the Z+HF and tt components are allowed to float. The calculated scale factors are Z+HF = 0.955 ± 0.032 and tt = 0.798 ± 0.033 where the uncertainties are statistical. Figure 1 shows comparisons between data and simulation in the control and validation regions for a few characteristic observables after applying the Z+HF and tt scale factors. The contribution of signal events is less than 1% in CRTop and at most 3% in CRZ for all signal points.
As described below, the background in the SR is modelled using a combination of the distribution shapes from data in CRZ and CRTop to model Z+HF and tt, respectively, with only a small dependence on the MC simulation. The relatively small contributions of non-Z+HF events to CRZ (approximately 30%) and of non-tt events to CRTop (approximately 20%) are derived from MC simulation and subtracted, after applying the scale factors Z+HF and tt to the simulated data. The resulting distributions, assumed to correspond to pure Z+HF and tt, are referred to as the 'CRZ Z+HF' and 'CRTop tt' distributions.
The shape of the distribution for Z+HF in CRZ is found to differ slightly from that in the SR, according to the predictions of the S MC generator. Therefore, the bin-by-bin ratio of the distributions in CRZ and the SR, obtained from MC simulation and denoted by U Z+HF , is applied to the CRZ Z+HF distribution from the data when assembling the background model. The correction is linear as a function of and ranges from a factor of 0.6 at 20 GeV to just under 1.2 at 120 GeV. A similar comparison via MC simulation between the shapes in CRTop and the SR shows no statistically significant shape difference. Therefore, no shape correction is made for CRTop.
The background model is constructed from the data as a weighted sum of the CRZ Z+HF and CRTop tt distributions to which is added the distribution of the subdominant backgrounds in the SR obtained from simulation. Signal contamination in the CRs is neglected. The weights for the CRZ and CRTop components, denoted Z+HF and tt respectively, are the ratios of the integral event counts from simulation between = 20 GeV and 120 GeV in the SR to those in the respective control region. The background model can be described as bkg model SRf where is the total event count of sample in region , andf is the probability density function as a function of of sample in region , both obtained from data in the respective control regions. In particular,f data Z+HF CRZ-corr is the normalized product of U Z+HF andf data Z+HF CRZ , where U Z+HF is the bin-by-bin shape correction factor described above. The superscript 'subdom.' on the last term in Eq. (1) indicates the subdominant backgrounds ( +light, single-top, and diboson events) that are derived directly from the Monte Carlo simulation in the signal region, as indicated by the subscript 'SR-MC'.
Because and U + are ratios of quantities from the MC samples, they are less sensitive to detector and theory systematic uncertainties. The factor bkg is a background normalization parameter whose expected value is close to unity, given that the left-hand side is normalized to model SR , the number of background events expected from simulation in the SR, after having applied the Z+HF and tt scale factors, Z+HF and tt , respectively. To smooth out statistical fluctuations in the background shape, the histogram resulting from the superposition in Eq. (1) is fit with a fourth-order polynomial over the range ∈ (20 GeV, 120 GeV).
Hypothesis tests for the presence of an exotic Higgs boson decay signal are performed with a multi-bin fit of the distribution in the SR, based on a product of Poisson likelihood terms in a profile likelihood-ratio [89] test statistic. Systematic uncertainties are included as additional terms in the likelihood, assuming Gaussian auxiliary measurements for the nuisance parameters. Upper limits are obtained at 95% CL, based on the CL s prescription [90].
The model that is fit to the data is the sum of two templates: • The SM background, whose shape comes from the parameterization described earlier in this section.
The normalization is controlled by the nuisance parameter bkg .
• The signal, whose shape comes directly from simulation. The signal normalization is controlled by the fit parameter-of-interest, sig , which is constrained to be positive. The parameter is scaled to correspond to the branching ratio for the exotic Higgs boson decay, →˜0 2˜0 1 →˜0 1˜0 1 , without constraining the branching ratio to be less than unity, and assuming the Standard Model cross section for associated production and a branching ratio BR( →¯) = 100%.

Systematic uncertainties
Systematic uncertainties affect our ability to model the shape of the background dĳet invariant mass spectrum from simulation, and therefore the background model used in the final fit. Since the background model is built primarily using the data distributions in CRZ and CRTop, the impact of several sources of theoretical and experimental uncertainty is reduced. Nevertheless, the simulated distributions enter through the subdominant backgrounds in both control regions and in the SR, through the predicted yields of Z+HF and tt in the three regions, and through the shape correction for Z+HF events, all of which enter into Eq. (1). In addition, the signal acceptance and line shape are subject to theoretical and experimental uncertainties.
Systematic uncertainties are evaluated for detector effects, for theoretical modelling and for variations in the parameterized background model shape arising from the statistical uncertainties in the polynomial fit parameters. In addition, a systematic uncertainty is evaluated to account for the choice of background functional form by comparing the results obtained using the fourth-order polynomial with those from a perfect fit to the data, i.e. the background-model data points themselves. The uncertainty associated with a systematic effect is taken into account by producing a new background distribution that includes the systematic variation. Scale factors for Z+HF and tt and the shape correction for Z+HF are recomputed for each systematic variation and the background analysis is fully re-run to determine the size of the effect on the shape of the background model.
The procedure to derive the background model introduces statistical uncertainties when subtracting the subdominant backgrounds. These uncertainties are accounted for in the statistical uncertainty of the background model. However, to avoid double counting of these statistical uncertainties when evaluating the impact of systematic variations, the subtraction of the subdominant backgrounds is omitted; both the nominal distribution and the systematically varied distribution are evaluated without subtracting the subdominant backgrounds. Comparing the shape of the background model with and without the subtraction of the subdominant backgrounds, the difference, measured as the maximum absolute difference between the two background curves, was found to be at most 3%. Uncertainties in the theoretical modelling of the dominant background processes (Z+HF and tt) are evaluated as follows. For Z+HF, the effects of varying the factorization and renormalization scales were evaluated using seven variations in S [94] each of which changed the renormalization and factorization scales by a factor of 0.5 or 2, 4 leading to a maximum shape difference in the distribution of 4%. Results are also evaluated using an alternative sample based on M G 5_aMC@NLO [68] and their difference from those of the nominal S samples, at most 5%, is assigned as an additional systematic uncertainty.
For tt production, uncertainties are estimated by comparing different matrix-element calculations (P B vs aMC@NLO), different choices of parton-showering model (P 8 vs H 7), and different amounts of initial-state (ISR) and final-state (FSR) radiation within P 8, while leaving all other parameters for each comparison unchanged. The uncertainty due to ISR is estimated by comparing the nominal tt sample with two additional samples to simulate higher/lower parton radiation [95]. The impact of FSR is evaluated by varying the renormalization scale for emissions from the parton shower up and down by a factor of two. All of these variations lead to differences compared to the nominal choice of less than 2%. The systematic uncertainty in the cross section is taken from Ref. [65] and includes the effect of QCD scale, PDF, and s uncertainties. The impact of theoretical uncertainties on the Higgs boson production kinematics were neglected, based on the studies described in Ref. [96].

Results
The distribution in the SR is shown in Figure 2. The observation is consistent with the background model for the SM hypothesis. The long tail on the example signal distribution comes from selecting a jet that does not come from the →¯decay. The dominant uncertainty limiting the final result is statistical and comes from the limited number of events in the signal region. The dominant systematic uncertainty is again statistical in nature and comes from the normalization of the background shape function to the data observed in the SR; for signal models where this analysis has sensitivity, this uncertainty has a 6%-10% effect on the fitted sig . Subdominant systematic uncertainties in this analysis include the theoretical uncertainty of the Z+HF shape correction (2%-3%), the jet energy resolution (1%-3%), the flavour dependence of the jet energy scale (1%-2%), and the statistical uncertainties of the background shape parameters (1%-3%), which arise from the limited number of data events in CRZ and CRTop.
Upper limits at 95% CL on the → cross section times branching ratio for → ℓ + ℓ − (where ℓ = , or ) and →˜0 2˜0 1 →˜0 1˜0 1 →¯˜0 1˜0 1 are shown in Figure 3 as a function of for several fixed values of˜0 1 and˜0 2 . Assuming the SM value for production, these results can be interpreted as upper limits on the branching ratio for →˜0 2˜0 1 →˜0 1˜0 1 →¯˜0 1˜0 1 . In the region of highest sensitivity, a branching ratio upper limit of 31% is obtained. For comparison between the results at different˜0 2 ,˜0 1 mass points, Figure 4 shows the six 95% CL upper limits of Figure 3 together, without the uncertainty bands.

Conclusion
A search for the exotic decay of the Higgs boson ( ) into a bb resonance plus missing transverse momentum has been performed with the ATLAS detector at the Large Hadron Collider in 139 fb −1 of pp collisions at √ = 13 TeV . The search was designed to target Higgs bosons produced in association with a Z boson and was conducted in events with two leptons, two or more jets, at least one of which must be -tagged, and missing transverse momentum. The analysis was optimized on a model in the Peccei-Quinn symmetry limit of the NMSSM where →˜0 2˜0 1 , with˜0 2 →˜0 1 , and is a new, light pseudoscalar Higgs boson. The decay of the boson into a pair of -quarks results in a resonance in the dĳet invariant mass. Such models differ from those considered in past searches for exotic Higgs boson decays involving the production of , where the NMSSM was considered in the -symmetry limit in which the dominant decay channel is → . Observations are consistent with SM expectations and upper limits on the → cross section times the branching ratio for → ℓ + ℓ − and →˜0 2˜0 1 →˜0 1˜0 1 →¯˜0 1˜0 1 have been obtained for a three-dimensional scan of masses of the˜0 1 ,˜0 2 and boson. Assuming the SM cross section for production, and assuming 100% branching ratios for the decays˜0 2 →˜0 1 and → , an upper limit on the branching ratio BR( →˜0 2˜0 1 ) of 31% is obtained at 95% confidence level for a range of values between 35 and 55 GeV for fixed values of˜0