Precise determination of the mass of the Higgs boson and tests of compatibility of its couplings with the standard model predictions using proton collisions at 7 and 8 TeV

• The analysis combines CMS measurements in bb [21], WW [22], ZZ [16], ττ [23], γγ [18], and µµ [30], together with ttH production [29] and invisible-decay searches [28].The range of production and decay modes enables multiple studies of the Higgs boson’s couplings.Source: §1 · p. 3
• The Higgs-boson mass is determined by combining the γγ and ZZ → 4ℓ measurements [16] [18], while coupling strengths are tested against SM predictions using all measurements.The coupling analysis extracts ratios of observed coupling strengths to SM predictions.Source: §1 · p. 3
• The coupling-deviation formalism assumes JPC = 0++ and the narrow-width approximation, which factorizes the boson’s production and decay couplings.The analysis follows the formalism recommended by the LHC Higgs Cross Section Working Group [171].Source: §1 · p. 4
• The data comprise 2011–2012 proton-proton collisions, with up to 5.1 fb−1 at 7 TeV and 19.7 fb−1 at 8 TeV, processed with updated CMS alignment and calibrations.The γγ and ZZ → 4ℓ channels have high sensitivity and excellent mass resolution; WW has high expected yields but poorer mass resolution, while bb and ττ have large backgrounds and lower sensitivity.Source: §1 · pp. 3–4

2 Inputs to the combined analysis

• Table 1 summarizes the combined-analysis inputs by final state, targeted production and decay modes, signal composition, expected mass resolution, event categories, and integrated luminosity.The accompanying text defines the differing mass-resolution conventions and analysis notation.Source: §2 · p. 4, Table 1 · p. 5

2.1 H →γγ

• Diphoton events are divided into production-targeted categories, using large-rapidity-gap dijets for VBF and leptons, missing momentum, or characteristic jets for VH and ttH.For 7 TeV data, the leptonic and multijet ttH selections are combined, and VBF-tagged categories are further split with an MVA classifier.Source: §2.1 · p. 6
• Fewer than 1% of selected diphoton events are tagged by production mode; the remainder are categorized by an MVA using mass-resolution, photon-identification, and kinematic information.A separate photon-identification MVA uses shower shape and isolation variables to distinguish prompt photons from photons arising in jets.Source: §2.1 · p. 6
• Diphoton backgrounds are fit to the observed mass distribution, with discrete profiling over candidate functional forms incorporating background-model choice into the statistical procedure.Pseudo-data tests found correct coverage and negligible bias; the procedure yields confidence intervals at least as large as those from any single functional form.Source: §2.1 · p. 6

2.2 H →ZZ

• The four-lepton analysis uses a kinematic discriminant based on leading-order signal and background matrix elements, calculated from the dilepton masses and five angles.The 4μ, 2e2μ/2μ2e, and 4e categories are analyzed separately because their mass resolutions and reducible-background rates differ.Source: §2.2 · p. 6
• The four-lepton mass measurement combines m4ℓ, Dkin_bkg, and an event-by-event relative mass-resolution estimate derived from lepton momentum resolutions.The dominant irreducible ZZ background is estimated from simulation, while smaller reducible backgrounds, mainly from Z + jets, top pairs, and WZ + jets, are estimated from data.Source: §2.2 · p. 7
• The four-lepton sample is split by jet multiplicity; pT(4ℓ) helps distinguish VBF and VH from ggH below two jets, while a dijet discriminant separates VBF and ggH at higher multiplicity.The dijet discriminant uses the leading jets’ invariant mass and pseudorapidity difference.Source: §2.2 · p. 7

2.3 H →WW

• The H →WW analysis combines dilepton and multilepton channels, using category-specific mass and transverse-mass distributions or event counts to measure the signal.It also targets WH →WWW and ZH →Z + WW, with data-driven estimates for several backgrounds; up to 20% of the 3ℓ3ν signal is expected from H →ττ decays.Source: §2.3 · pp. 7–8
• 16–17% mass resolution was achieved for mH = 125 GeV in the dedicated H →WW eµ analysis using 0- and 1-jet categories.The analysis used observables also employed in searches for supersymmetric particles.Source: §2.3 · p. 8

2.4 H →ττ

• The H →ττ search uses multiple final states and jet-based categories, with broad reconstructed-mass distributions or paired MVA discriminants to distinguish signal and backgrounds.Expected H →WW contributions are 17–24% in the ee and µµ categories, 23–45% in eµ categories, and 10–15% and 70% in the ZH →Z + ℓτh and ZH →Z + eµ channels, respectively; reducible WH and ZH backgrounds are estimated from data.Source: §2.4 · p. 8

2.5 VH with H →bb

• The VH, H →bb analysis uses b-tagged dijets, pT(V) categories, and MVA classifiers, with a dijet mass resolution of about 10% for mH = 125 GeV.The results include an NLO prediction for gg →ZH, particularly important in the analysis’s most sensitive categories.Source: §2.5 · p. 9

2.6 ttH production

• The ttH searches target H →bb, τhτh, and leptonic decays, classifying events by jets, b tags, and leptons and using MVA discriminants in several channels.The main backgrounds include top-quark pairs with extra jets and nonprompt leptons; rates are estimated from simulation or data and constrained using control samples.Source: §2.6 · pp. 9–10

2.7 Searches for Higgs boson decays into invisible particles

• The invisible-decay search combines VBF-tagged and ZH →ℓℓ channels, using a one-bin count for VBF and fitted kinematic distributions for ZH.The VBF selection uses 8 TeV data, requires dijet mass above 1100 GeV and missing transverse energy above 130 GeV, and estimates the largest Z(νν) + jets background from visible Z →µµ events. These searches contribute to later probes of invisible and undetected partial widths.Source: §2.7 · p. 10

2.8 H →µµ

• The H →µµ search fits dimuon-mass spectra in mutually exclusive categories defined by event topology, kinematics, and dimuon resolution.The background-model choice contributes a systematic uncertainty about 75% of the overall statistical uncertainty; the analysis informs a later study of coupling scaling with particle mass.Source: §2.8 · pp. 10–11

3 Combination methodology

• The combination simultaneously analyzes all selected measurements, accounting for statistical and systematic uncertainties and their correlations, using a profile-likelihood-ratio test statistic and frequentist nuisance parameters.Results use asymptotic formulae, including routines in ROOSTATS.Source: §3 · p. 11

3.1 Characterizing an excess of events: p-value and significance

• The local p-value is the background-only probability of observing an excess test statistic at least as large as the data; its significance uses a one-sided Gaussian-tail convention.The excess statistic compares the background-only hypothesis with a signal-plus-background fit, with the signal strength constrained to be positive; very small p-values warrant caution because model biases and uncertainties are known only to finite precision.Source: §3.1 · pp. 11–12

3.2 Extracting signal model parameters

• Best-fit signal-model parameters come from profile-likelihood scans; 68% and 95% confidence intervals use q(ai) = 1.00 and 3.84, while 2D regions use q(ai, aj) = 2.30 and 5.99.Intervals are approximate without large non-Gaussian uncertainties; at physical boundaries, the method's theoretical basis is lacking, although results here were numerically similar to the alternative method in Ref. [189].Source: §3.2 · p. 12

3.3 Grouping of channels by decay and production tags

• Categories are grouped by the decay or production mode expected to dominate their sensitivity, and all contributions within a group are treated as signal and scaled accordingly.Selections can be mixed: VBF-tagged categories can include up to 50% ggH, and VH tags commonly mix WH and ZH; some ttH categories also contain substantial signal from multiple decay modes.Source: §3.3 · pp. 12–13

3.4 Expected differences with respect to the results of input analyses

• Evaluating the combination at mH = 125.0 GeV rather than 125.6 GeV changes expected branching fractions and production cross sections; the ττ-tag analysis also treats both H → ττ and H → WW as signal, increasing sensitivity to both decays.For the mass change, the H → ZZ and H → WW branching-fraction ratios are 0.95 and 0.96, respectively, while the total production cross-section ratio is approximately 1.01; treating H → WW as signal differs from Ref. [23], which treated it as background.Source: §3.4 · pp. 13–14
• Adding gg → ZH raises expected signal yields by 10%–30% in the specified high-pT(Z) channels and increases expected VH, H → bb sensitivity by about 10%; excluding overlapping invisible-decay events does not significantly affect overall sensitivity.The bbH contribution is neglected because it is smaller than uncertainties in inclusive selections and has low selection efficiency in exclusive categories, with no sensitivity loss reported.Source: §3.4 · p. 14

4.1 Mass of the observed state

• 125.02 +0.26 -0.27 (stat) +0.14 -0.15 (syst) GeV is the measured mass from the combined H →γγ and H →ZZ →4ℓ analysis.The mass scan profiles three independent signal-strength modifiers and other nuisance parameters; the result includes statistical and systematic uncertainties.Source: §4 · p. 14, §4.1 · pp. 15–16, Figure 2 · p. 16
• The H →γγ and H →ZZ →4ℓ mass measurements agree at the 1.6σ level.The compatibility is assessed by scanning the test statistic against the difference between the two measured masses.Source: §4.1 · p. 16
• Alternative mass measurements differ from the main result by less than 0.1 GeV in both best-fit value and uncertainty.The alternatives allow a common signal-strength modifier to float or constrain relative production cross sections and branching fractions to the SM predictions (µ = 1).Source: §4.1 · pp. 16–17

4.2 Direct limits on the width of the observed state

• 1.7 (2.3) GeV is the combined observed (expected) 95% CL upper limit on the natural width.The individual observed (expected) limits are 2.4 (3.1) GeV for H →γγ and 3.4 (2.8) GeV for H →ZZ →4ℓ; the mass, common signal strength, and nuisance parameters are profiled.Source: Figure 3 · p. 17, §4.2 · p. 17

5 Significance of the observations in data

• Significance is close to or above 5σ in the three diboson decay-mode tags and above 3σ in the ττ tag at mH = 125.0 GeV.Table 2 reports median expected and observed local significance; expected values use post-fit background rates and signal rates expected from the SM.Source: §5 · p. 18
• Differences from individual publications reflect the decay-mode grouping, the mass used for evaluating H → ZZ → 4ℓ and H → WW significance, and the H → WW treatment in the H → ττ analysis.H → WW is treated as signal rather than background in the H → ττ analysis.Source: §5 · p. 18
• The H → γγ and H → ZZ → 4ℓ observations indicate that the particle is a boson, and diphoton decay implies spin different from unity.Other observations beyond this paper's scope disfavor spin-1 and spin-2 hypotheses; assuming zero spin, they are consistent with a pure scalar and disfavor a pure pseudoscalar [16] [22] [41].Source: §5 · p. 18

6.1 Overall signal strength

• The combined signal strength was 1.00 ± 0.09 (stat) +0.08 -0.07 (theo) ± 0.07 (syst), consistent with the SM expectation of unity.The fit assumes mH = 125.0 GeV; the theory component includes scale, parton-distribution, branching-fraction, and underlying-event uncertainties.Source: §6.1 · p. 19, Figure 4 · p. 21

6.2 Grouping by predominant decay mode and/or production tag

• Grouped fits were SM-compatible: production tags gave χ2/dof = 5.5/4 (p = 0.24), decay tags 1.0/5 (p = 0.96), and joint tags 10.5/16 (p = 0.84).The ttH-tagged excess was compatible with the SM at 2.0σ; the tagged samples are not pure production or decay modes.Source: §6.2 · p. 20, Figure 4 · p. 21

6.3 Fermion- and boson-mediated production processes and their ratio

• For the full combination, the observed best fit for µVBF,VH/µggH,ttH was 1.25 +0.62 −0.44, compatible with the SM Higgs expectation, µVBF,VH/µggH,ttH = 1.The SM point (1, 1) was within the 68% CL confidence regions for all predominant decay groups; the H →bb group constrained µVBF,VH more than µggH,ttH.Source: §6.3 · pp. 20, 22

6.4 Individual production modes

• The best-fit strengths were µggH = 0.85 +0.19 −0.16, µVBF = 1.16 +0.37 −0.36, and µttH = 2.90 +1.08; VBF expected and observed significances were above 3σ.VH had expected and observed significances of 2.9σ and 2.7σ, respectively; data were compatible with µttH = 1 at 2.2σ.Source: §6.4 · pp. 24–25

6.5 Ratios between decay modes

• The SM ratio λyy,xx = 1 lay within the 68% confidence interval for every tested decay-mode ratio, with statistical uncertainty dominating.Ratios can cancel some production, decay-prediction, and correlated experimental uncertainties; production modifiers were allowed to float independently to reduce dependence on expected production rates.Source: §6.5 · pp. 25–26, Table 8 · p. 26, Table 9 · p. 26

6.6 Search for mass-degenerate states with different coupling structures

• Mass observables cannot distinguish states closer than experimental resolution, but production-and-decay tags can test differing coupling structures through the signal-strength matrix rank.A single state implies rank(M) = 1; multiple mass-degenerate states with different interaction strengths may produce rank(M) > 1.Source: §6.6 · p. 27
• For the single-state parameterization, the SM expectation is λi = μj = 1.This parameterization is nested within the more general model and was used for the numerator of the test statistic.Source: Table 11 · p. 28, §6.6 · p. 28
• The test compared a general rank-1 matrix, expected for a single particle, with a saturated model that allows each production-decay matrix element to vary independently.A large qλ favors the saturated model; under the single-particle hypothesis, the rows are multiples of one another and qλ is not expected to be large.Source: §6.6 · p. 28
• qobsλ = 12.2; the SM-tail probability was (7.9 ± 0.3)%, corresponding to about 1.4σ.The small tension was due to the dijet-tagged H → ZZ channel; without H → ZZ inputs, the p-value was about 33%.Source: §6.6 · p. 28, Figure 7 · p. 29

7 Compatibility of the observed data with the SM Higgs boson couplings

• The combined coupling framework models production and decay together, with each mode’s yield depending on its production cross section and partial and total decay widths.Production processes include ggH, VBF, WH, ZH, and ttH; decay modes include WW, ZZ, bb, ττ, gg, and γγ.Source: §7 · p. 30
• The coupling tests assume a single JPC = 0++ particle with a narrow enough width for production and decay to be treated independently, following Ref. [171].The paper states that these assumptions are supported by results on additional particles, alternative spin-parity assignments and mixtures, and the particle’s width.Source: §7 · p. 30
• Coupling modifiers κi test rate deviations for tree-level particles, particle classes, and loop-induced gluon and photon couplings, while BSM, invisible, and undetectable decays are also considered.Loop modifiers can be expressed through tree-level modifiers assuming the SM loop structure or treated as effective modifiers; electron and up- and down-quark couplings are neglected. Undetectable decays include unstudied BSM decays and decays that may evade detection under the experimental trigger conditions.Source: §7 · pp. 30–31
• The combined fits fix mH to 125.0 GeV and vary different numbers of parameters, setting others to SM expectations or profiling them with nuisance parameters.The analyses examine vector-boson and fermion couplings, differences among fermion classes, tree-level couplings versus particle masses, loop effects, and BSM, invisible, and undetectable decays.Source: §7 · p. 31

7.1 Relation between the coupling to the W and Z bosons

• λWZ = 0.94^+0.22_-0.18 in the WW/ZZ combination, consistent with the Standard Model expectation of 1.The ratio of ggH-dominated yields in WW 0/1-jet and untagged ZZ channels provides a nearly model-independent measurement, assuming SM fermion couplings.Source: §7.1 · p. 32, Figure 8 · p. 32
• λWZ = 0.92^+0.14_-0.12 in the all-channel combination, consistent with the Standard Model expectation of 1.The fit profiles κZ and a common fermion coupling modifier κf, with ΓBSM set to zero.Source: §7.1 · p. 32, Figure 8 · p. 32
• Subsequent measurements assume λWZ = 1 and use a common κV to modify the W- and Z-boson couplings.This convention follows from the consistency of both ratio measurements with unity.Source: §7.1 · p. 32

7.2 Test of the couplings to massive vector bosons and fermions

• The fit assumes ΓBSM = 0 and uses κV = κW = κZ and κf = κb = κt = κτ as common scaling factors for vector-boson and fermion couplings.At leading order, partial widths scale as κ²f except Γγγ, which depends on both κV and κf through W-boson and top-quark loops.Source: §7.2 · p. 33
• The analyses determine whether κV and κf have the same or different signs, but cannot distinguish the degenerate quadrants; associated production with a single top quark could lift that degeneracy.H →γγ is the only combined channel sensitive to their relative sign.Source: §7.2 · p. 33
• The SM point (κV, κf) = (1, 1) lies within the data-defined 68% CL confidence region, and the data are compatible with the standard model Higgs boson.Figure 9 shows the 68%, 95%, and 99.7% CL regions.Source: §7.2 · p. 33, Figure 9 · p. 34
• H →γγ drives the 95% CL exclusion of part of the opposite-sign (+, −) region through destructive W-loop and top-quark-loop interference for same-sign couplings.The region is excluded at 95% CL but not at 99.7% CL; vector-boson decay analyses constrain κV better, while fermion decay analyses constrain κf better.Source: §7.2 · p. 33, Figure 10 · p. 34
• The 95% CL confidence intervals are [0.87, 1.14] for κV and [0.63, 1.15] for κf, with the other parameter floated.Source: §7.2 · p. 33

7.3 Test for asymmetries in the couplings to fermions

• Motivated by possible two-Higgs-doublet modifications, the combinations test ratios of down-type to up-type fermion couplings and lepton to quark couplings.In more general 2HDMs, leptons can virtually decouple from a Higgs boson that otherwise behaves in an SM-like way toward W and Z bosons and quarks.Source: §7.3 · p. 35
• For positive λdu and λℓq, the 95% CL intervals are [0.65, 1.39] and [0.62, 1.50], respectively, with no evidence of different scaling factors across fermion classes.Figure 11 shows the likelihood scans for both ratios, with the other specified coupling modifiers and nuisance parameters profiled.Source: Figure 11 · p. 35, §7.3 · p. 35
• The λdu scan profiles κV and κu, while the λℓq scan profiles κV and κq, with all other nuisance parameters profiled in both.The ratios are defined as λdu = κd/κu and λℓq = κℓ/κq.Source: Figure 11 · p. 35, §7.3 · p. 35

7.4 Test of the scaling of couplings with the masses of SM particles

• The fit of κW, κZ, κb, κτ, κt, and κµ showed no significant deviations from the SM expectation.Loop-induced processes were expressed using the tree-level κ parameters and scaled according to their SM loop structure; the small κt uncertainty reflects the ggH description in terms of κt and κb, with a small κb contribution.Source: §7.4 · p. 35
• The (M, ϵ) fit found no significant deviation from the SM, with 95% CL intervals of [217, 279] GeV for M and [−0.054, 0.100] for ϵ.The model recovers the SM at (M, ϵ) = (v, 0), while ϵ = −1 corresponds to couplings independent of particle mass [205, 206].Source: §7.4 · p. 36
• Measurements spanning more than three orders of magnitude in particle mass support couplings that differ among particles and are related to their masses, while the muon scaling-factor measurement remains imprecise.Assuming SM production cross sections, the 95% CL upper limit is B(H →ee) < 1.9 × 10^-3 [30].Source: §7.4 · p. 36

7.5 Test for the presence of BSM particles in loops

• Because H→γγ and ggH are loop-induced and sensitive to new particles, the analysis jointly fits κγ and κg while assuming tree-level rates match the SM.Source: §7.5 · p. 36
• With ΓBSM=0, the best-fit (κγ, κg) is (1.14, 0.89); (1, 1) lies within the 68% CL region, and their 95% CL intervals are [0.89, 1.40] and [0.69, 1.11].Source: §7.5 · p. 36
• The observed 95% CL interval for BRBSM is [0.00, 0.32], narrower than the SM-hypothesis expected interval [0.00, 0.42].The observed limit also reflects the data’s preference for κg below unity.Source: §7.5 · p. 38
• With H(inv) search data and BRundet=0, expected and observed 95% CL intervals for BRinv are [0.00, 0.29] and [0.00, 0.32], with a shallow observed minimum at 0.03.The expected interval is 31% narrower than without H(inv) data; the observed H(inv) upper limit was 0.58 (expected 0.44) at 95% CL [28].Source: §7.5 · p. 38
• With H(inv) data, BRundet=0, and κi=1, expected and observed 95% CL intervals for BRinv are [0.00, 0.21] and [0.00, 0.27], with a shallow minimum at 0.06.The expected interval is 28% narrower than when the total width is not fixed to the SM expectation.Source: §7.5 · p. 38
• Combining H(inv) searches with visible-decay measurements makes the expected 95% CL upper limit on BRinv up to a factor of two smaller than either alone, depending on assumptions.Source: §7.5 · p. 38

7.6 Test of a model with scaling factors for SM particles

• The model has six independent modifiers: a common κV for W and Z, independent κb, κτ, and κt, free loop-induced κg and κγ, and ΓBSM = 0.The scan profiles each modifier against the other five and nuisance parameters.Source: §7.6 · pp. 38–39
• All probed 95% CL intervals contain κi = 1; the SM fit gives χ2/dof = 7.5/6 (p-value 0.28), while κt = 1.60+0.34−0.32 is constrained by ttH measurements.In this model, ggH-production measurements constrain κg, leaving ttH measurements to constrain κt.Source: §7.6 · p. 39

7.7 Test of a general model without assumptions on the total width

• The model leaves the total width as a free parameter embedded in κgZ = κgκZ/κH, rather than rescaling it according to the κi values.This is the general model proposed in Ref. [171].Source: §7.7 · p. 39
• The model expresses all other parameters of interest as coupling-scaling ratios λij = κi/κj.The likelihood scans cover κgZ, λWZ, λZg, λbZ, λγZ, λτZ, and λtg.Source: §7.7 · p. 39
• The likelihood scans for seven parameters are in line with the results of Section 7.6.Each parameter is scanned while profiling the other six and all nuisance parameters.Source: §7.7 · p. 39

7.8 Constraints on BRBSM in a scenario with free couplings

• The free-coupling model fits κV, κb, κτ, κt, κg, κγ, and BRBSM, imposing κV ≤1 to avoid degeneracy between total width and coupling scaling.This bound holds in models with arbitrary numbers of Higgs doublets, with or without additional Higgs singlets [171]; the analysis can combine H(inv) search data.Source: §7.8 · p. 40
• The 95% CL interval for BRBSM is [0.00, 0.57] in data and [0.00, 0.52] expected under the SM hypothesis.The likelihood scan profiles the other coupling modifiers and nuisance parameters, with κV ≤1.Source: §7.8 · p. 40, Figure 18 · p. 42
• With BRundet = 0, the 95% CL interval for BRinv is [0.00, 0.49] in data and [0.00, 0.32] expected under the SM hypothesis.The observed and expected intervals differ in light of the H(inv) search upper limits of 0.58 and 0.44, respectively, at 95% CL [28].Source: §7.8 · p. 40, Figure 18 · p. 42
• In the simultaneous BRinv and BRundet fit, the 95% CL interval for BRundet is [0.00, 0.52] in data and [0.00, 0.51] expected under the SM hypothesis.H(inv) searches constrain BRinv, while visible decays constrain BRBSM = BRinv + BRundet.Source: §7.8 · p. 40, Figure 19 · p. 42

7.9 Summary of tests of the compatibility of the data with the SM Higgs boson couplings

• All results in the fewest-parameter benchmark scenarios are compatible with SM expectations.Figure 20 summarizes likelihood scans for the benchmark-model parameters, with inner and outer bars denoting 68% and 95% CL intervals.Source: §7.9 · p. 43, Figure 20 · p. 43
• Table 12 provides a comprehensive overview of searches for deviations from the SM Higgs boson expectation and summarizes all results from this section.Source: §7.9 · p. 43
• No statistically significant deviations from the SM Higgs boson expectation are observed.Source: §7.9 · p. 43

8 Summary

• 125.02 +0.26 −0.27 (stat) +0.14 −0.15 (syst) GeV is the Higgs boson mass determined from the high-resolution γγ and ZZ channels.The measurements used proton-proton collision data collected by CMS at 7 and 8 TeV.Source: §8 · p. 43
• 1.00 ± 0.09 (stat) +0.08 −0.07 (theo) ± 0.07 (syst) is the combined best-fit signal relative to the standard model expectation; event yields are consistent with standard model predictions.The signal strength is evaluated at the measured mass.Source: §8 · p. 43
• No significant deviations from standard model coupling predictions were found in tests of coupling magnitudes, including searches for invisible and undetected decays.Compatibility tests of the data with standard model Higgs couplings are summarized in Table 12.Source: §8 · p. 43, Table 12 · p. 44