The sensitivity of the Higgs boson branching ratios to the W boson width

The Higgs boson branching ratio into vector bosons is sensitive to the decay widths of those vector bosons because they are produced with at least one boson significantly off-shell. Gamma(H to V V ) is approximately proportional to the product of the Higgs boson coupling and the vector boson width. Gamma Z is well known, but Gamma W gives an uncertainty on Gamma(H to W W ) which is not negligible. The ratio of branching ratios, BR(H to W W )/BR(H to ZZ) measured by a combination of ATLAS and CMS at LHC is used herein to extract a width for the W boson of Gamma W = 1.8+0.4-0.3 GeV by assuming Standard Model couplings of the Higgs bosons. This dependence of the branching ratio on Gamma W is not discussed in most Higgs boson coupling analyses.


Introduction
The Higgs boson discovered at LHC [1,2] has been the subject of combined mass [3] and couplings [4] analyses by the ATLAS and CMS collaborations. The couplings analysis uses the so-called κ framework of the LHC Higgs cross-section working group [5,6], and relies upon the cross-section and branching ratio calculations contained therein. This includes the properties of the vector bosons, W and Z, for which the masses reported in the RPP [7], are used to extract pole masses of m Z = 91.15349 GeV and m W = 80.36951 GeV in Ref. [6]. In addition, and especially relevant for this note, the vector boson widths are calculated from their masses and assuming the Standard Model(SM), to be Γ Z = 2495.81 MeV and Γ W = 2088.56 MeV.
The use of the theoretically expected W boson width is not discussed in Ref. [6], it is merely stated. It is not obvious that this is the best motivated assumption when looking for beyond the Standard Model (SM) effects in Higgs boson properties. The primary purpose of this document is to highlight that assumption.
The widths of the Z and W bosons have also been measured experimentally. The Z boson width is measured via the scan of the Z resonance at LEP [8] to be 2495.2 ± 2.3 MeV.
The W boson width has been measured using mass reconstruction at LEP 2 [9] and with better precision using the transverse mass distribution at the Tevatron [10]. These different approaches agree well and are combined in the RPP [7] to give Γ W = 2085 ± 42 MeV, an error a factor twenty times larger than that for the Z. In consequence, effects due to the vector boson width uncertainties are dominated by those from the W boson.
The Higgs boson partial widths and branching ratios are not experimentally accessible at the LHC, where only products of production and decay can be studied. However, the ratio of the branching ratios to W W and ZZ, is measurable, and it is presented in Ref. [4]. The measured value of BR W W /BR ZZ is 6.8 +1.7 −1.3 . It is also accurately calculable, using just m H and the masses and widths of the W , Z and H bosons. The SM value given in Ref. [6] is 8.09 This ratio is not the only test of the H → W W width which could be made. Most obviously the measured rate into diphotons could be included in the analysis. However, further assumptions about the interaction strengths of all particles entering the decay loop would be needed, and there could even be unknown particles. The analysis using the vector bosons alone is easier to justify.

Analysis of the widths
The full calculation of the Higgs boson partial widths in the SM is rather complex. However, the results are tabulated in Ref. [6], and the approach taken here is to use a leading-order approximation [11], and then scale its results to those in Ref. [6] for the nominal input parameters. This captures the dependence on the W boson width to a very good approximation.
The calculation is reproduced below.
In this formula Γ 0 is where λ(x, y, z) = (1 − x/z − y/z) 2 − 4xy/z 2 and δ V has different values depending upon the vector boson: δ W = 2 and δ Z = 1 [11]. By performing the integration the partial width can be found. This calculation assumes the SM coupling strengths to the W and Z boson. The numerical evaluation uses the parameters from the LHC Higgs cross-section working group as given in the introduction and was done using root [12]. To check the calculation it is first evaluated at m H = 126 GeV because Ref. [6] provides partial widths at this mass. The The ratio of the partial widths gives directly the ratio of the branching ratios, 7.99. This is about 1% lower then the 8.09 contained in Ref. [6] and the difference is assumed to come from the more complete calculation used in that document. The 2-3% changes in the WW to the full calculation. The equation is numerically inverted to find the range of widths which corresponds to the measured branching ratio range. This is: An alternative presentation would be to invert the assumptions, and say that the 2% uncertainty on Γ W repsrensts a 2% uncertainty on Γ(H → W W ) which should be allowed for in the analysis.

Errors from the extraction procedure
The extraction of the ratio of branching ratios from the LHC data currently has limited precision, mostly for statistical reasons, but also with many systematic errors. These are not the subject of this note, which considers that input as a given. Only the errors discussed below affect the interpretation.
The Higgs boson mass of 125.09 ± 0.21 ± 0.11 GeV has the largest mass uncertainty in the formula. It changes the extracted value of Γ W by around 0.2 MeV, which is clearly negligible, and similarly the W and Z boson masses contribute negligible uncertainty.
The Z boson width is known to 2 per mille, and this translates to a 1 per mille or 2 MeV uncertainty on the prediction of Γ(H → ZZ). This is far below the precision achievable at LHC and is ignored here.
The width of the Higgs boson could also influence this result by changing the relative suppression of W W and ZZ states. The tightest model-independent upper limit on the H boson width is 3.4 GeV from the CMS studies in the llll final state. [13] An integration over the Higgs boson width has not been made, but its magnitude is estimated by changing the mass by 3.4 GeV, which gives a 3 MeV shift in the extracted Γ W . This is again negligible.
There is a 1% correction made in the double ratio between the first order calculation used here and the full calculation. However, the measured value is compatible with the SM expectation, and so the calculation has been corrected to the full calculation at least in some part of the range. The total calculational error is expected to be dominated by the uncertainty with which both the W W and ZZ partial widths are calculated, 0.5% [6]. A pessimistic combination of these, 1%, gives the largest uncertainty on Γ W , 20 MeV.
In summary, the total error of the extraction is estimated to be 20 MeV, which is negligible in comparison with the experimental error.

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The partial width Γ(H → V V ) is proportional to the full width of the vector boson involved.
While it is possible to impose the SM expectation, this seems to this author a restrictive way of testing the SM. The alternative, of using the experimentally measured value. should at least be considered. The 2% uncertainty on Γ(H → W W ) from the limited experimental knowledge of the W boson width is currently well below to 20% uncertainty from the Higgs boson couplings.
The alternative presentation, discussed here, treats the Higgs boson physics as known and the W as unknown and is perhaps extreme, but under this assumption Γ W = 1800 +400 −300 MeV has been extracted. A conservative 20 MeV error on the W boson width is estimated due to uncertainties on the calculation of the partial widths to W W and ZZ.
The uncertainty on this derivation of Γ W is thus dominated by the errors on the Higgs boson W W and ZZ measurements and will remain so at HL-LHC. Various projections for these in the future exist. For example, ATLAS concluded [14] that 5% and 4% errors on the H → W W and H → ZZ signal strength, respectively, were possible using 3000 fb −1 if theoretical systematic errors are ignored. Some of these theoretical errors will cancel in the ratio, so an error approaching 7% error might be achievable, and presumably a combination of two experiments will be better. At this point a 2% error on Γ W would have a significant impact on the physics interpretation.