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I am not understanding what the advantage of using only two specific decay channels in combined measurement of the Higgs boson mass. This is how it is stated in a published paper titled "Combined Measurement of the Higgs Boson Mass in pp Collisions at sqrt(s) = 7 and 8 TeV with the ATLAS and CMS Experiments":

"The combination is performed using only the H → γγ and H → ZZ → 4l decay channels, because these two channels offer the best mass resolution. Interference between the Higgs boson signal and the continuum background is expected to produce a downward shift of the signal peak relative to the true value of mH. The overall effect in the H → γγ channel [18–20] is expected to be a few tens of MeV for a Higgs boson with a width near the SM value, which is small compared to the current precision. The effect in the H → ZZ → 4l channel is expected to be much smaller [21]. The effects of the interference on the mass spectra are neglected in this Letter"

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The two channels are combined means that two measurements are combined. The combination of two measuremts increases the accuracy you can claim on the quantity you are measuring, the mass of the Higgs boson in this case, therefore it is advantageous to combine two measurements. For a (not completely faithful) real life analogy you can think about you counting a stack of 1cent coins. You count it twice so that you are more sure about the total.

Now, not all measurements are made the same, some are more precise some are less. If you want to get the best result, that is the most accurate measurement, you want to combine your two most precise measurements - these two "channels" you mentioned happen to be the two most precise ones for the Higgs boson.

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There exists a whole paper on mass resolution!

From the abstract:

Although mass resolution is conventionally defined as the closest distinguishable separation between two peaks of equal height and width, the required mass resolving power can be ∼10× higher for equal width peaks whose peak height ratio is 100 : 1.

and it goes on from there to:

Thus, the answer to the question of “how much is enough mass resolving power” requires that one first specify S/N ratio, dynamic range, digital resolution, mass-to-charge ratio, and (if available) isotopic fine structure.

Those of us working for years in particle physics have the simplest definition in mind for mass resolution, and the truism : the "cleanest" the decay the better. Electromagnetic and weak interactions are "cleaner" they are not accompanied, at the level of statistics of the experiment, with higher order strong interaction radiation which increase the uncertainty in the measured value. The two channels used involve a limited number of leptons, and the decays can be calculated more accurately then when the strong interactions are involved.

It is the reason why a lepton collider is proposed, so that "cleaner" data can be gathered for an accurate determination of the parameters of the standard model, or statistically significant deviations found.

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