Direct measurement of the top quark mass
ATLAS: lepton+jets events at 8 TeV (20.2 fb−1 ). ... 172.69 ± 0.48 GeV, with a relative uncertainty of 0.28%. ...
CMS: dilepton events at 13 TeV (35.9 fb−1 ). ... 172.33 ± 0.14 (stat) +0.66 −0.72 (syst) GeV, with a total relative uncertainty of approximately 0.42%. ...
CMS: all-jets events at 13 TeV (35.9 fb−1 ). ... 172.26 ± 0.61 GeV, with a relative uncertainty of 0.36%. ...
ATLAS: lepton+jets with an additional soft µ at 13 TeV (36.1 fb−1 ). ... 174.48 ± 0.40 (stat) ± 0.67 (syst) GeV, with a total relative uncertainty of 0.45%. ...
Indirect determination of the top quark mass
ATLAS: inclusive tt cross section in the eµ channel at 13 TeV (36.1 fb−1 ). ... 173.1 + 2.0 − 2.1 GeV. ...
ATLAS: differential cross section for lepton+jets tt+1jet events at 8 TeV (20.2 fb−1 ). ... 171.1 ± 0.4 (stat) ± 0.9 (syst) +0.7 −0.3 (theo) GeV [pole mass]. ...
CMS: triple-differential cross section in dilepton events at 13 TeV (35.9 fb−1 ). ... 170.5 ± 0.8 GeV [pole mass]. ...
CMS: invariant jet mass distribution for boosted jets in lepton+jets events at 13 TeV (35.9 fb−1 ). ... 172.56 ± 2.47 GeV. ...
CMS: running of the top quark mass in eµ events at 13 TeV (35.9 fb−1 ). ... The observed evolution of the mt(µk) values agrees with the prediction from renormalization group equations at 1- loop precision within 1.1 standard deviations. The no-running hypothesis is excluded at 95% confidence level.
Abstract
The ATLAS and CMS Collaborations have performed a variety of measurements of the top quark mass, taking advantage of the abundant production of top quarks at the LHC. The most recent measurements are reported here, based on data collected at 8 and 13 TeV, with particular emphasis on the distinction between the so-called "direct" measurements and the "indirect" evaluations obtained from cross sections and differential cross sections.Andrea Castro (on behalf of the ATLAS and CMS Collaborations), "Top Quark Mass Measurements in ATLAS and CMS" (November 21, 2019).
The final combined value from the two Tevatron experiments which were the first to measure the top quark mass was 174.30 ± 0.35 ± 0.54 GeV.
The Particle Data Group entity for this is here, but not as up to date. The latest direct measurement average from the PDG is 172.9 ± 0.4 GeV. The latest indirect measurement average from PDG is 173.1 ± 0.9 GeV. The weighted average of those two measurements was 172.96 GeV.
Combining The Results For A New World Average
The Particle Data Group entity for this is here, but not as up to date. The latest direct measurement average from the PDG is 172.9 ± 0.4 GeV. The latest indirect measurement average from PDG is 173.1 ± 0.9 GeV. The weighted average of those two measurements was 172.96 GeV.
Combining The Results For A New World Average
To recap, the nine independent measurements at Tevatron and the LHC combined to date are:
172.69 ± 0.48 GeV
172.33 ± 0.14 (stat) +0.66 −0.72 (syst) GeV
172.26 ± 0.61 GeV
174.48 ± 0.40 (stat) ± 0.67 (syst) GeV
173.1 + 2.0 − 2.1 GeV
171.1 ± 0.4 (stat) ± 0.9 (syst) +0.7 −0.3 (theo) GeV
170.5 ± 0.8 GeV
172.56 ± 2.47 GeV
174.30 ± 0.35 ± 0.54 GeV
Feel free to calculate an error weighted world average at your leisure (which I would if I had time).
Step One
First combine the errors for each measurement by taking the square root of the sum of the squares of the errors (and the arithmetic average where upper and lower bound errors differ before combining different types of error). This gives you this (with extreme values in each category noted).
LHC Results (likely to have some correlated errors)
Thus the latest experimental result from the LHC have pulled down the global average top quark mass by about 0.31 GeV from the value currently listed by the Particle Data Group.
Step Three
Then, calculate the margin of error of the weighted average from the inputs. A prior post somewhere at this blog explains how to do with simplifying assumptions.
But, to really do it rigorously, you have to consider the fact that the systemic and theoretical errors are not fully independent of each other, particularly for results that are all from LHC (which tends to make the total combined margin of error greater), and that historically, in this kind of experiment, actual errors have had fatter tails than a Gaussian (i.e. "normal") distribution and are distributed in something closer to a student t-test distribution with different parameters established by the historical data, which also tends to increase the total combined margin of error. The combined error should be a little lower than, but fairly close in magnitude to, the lowest margin of error of any of the individual entries, unless there is very wide scatter of precise measurements, in which case the combined error should be higher than the lowest margin of error of the individual entries, because this pattern implies that one or more of the margins of error is underestimated.
I suspect that the combined margin of error is on the order of 0.35 GeV to 0.45 GeV. Using the higher figure to account for issues like correlated errors and non-Gaussian errors, the two sigma range for the top quark mass given current data is 171.75 GeV to 173.55 GeV.
Given comparison of the two sigma bands for each result and giving slightly more importance to the Tevatron value as it is more independent than the other values, I suspect that the true value of the top quark pole mass is probably between 172.92 GeV and 173.3 GeV, which favors the higher end of combined two sigma range.
Theoretical Comparison Points (Highly Speculative)
Step One
First combine the errors for each measurement by taking the square root of the sum of the squares of the errors (and the arithmetic average where upper and lower bound errors differ before combining different types of error). This gives you this (with extreme values in each category noted).
LHC Results (likely to have some correlated errors)
172.69 ± 0.48 GeV (171.73 - 173.65 GeV) ATLAS direct
172.33 ± 0.70 GeV (170.93 - 173.73 GeV) CMS direct
172.26 ± 0.61 GeV (171.04 - 173.48 GeV) CMS direct
174.48 ± 0.78 GeV (172.92 - 176.04 GeV) ATLAS direct
173.1 ± 2.05 GeV (169 - 177.2 GeV) ATLAS indirect
171.1 ± 1.10 GeV (168.9 - 173.3 GeV) ATLAS indirect
170.5 ± 0.80 GeV (168.9 - 172.1 GeV) CMS indirect
172.56 ± 2.47 GeV (167.62 -177.5 GeV) CMS indirect
Tevatron Results
Tevatron Results
174.30 ± 0.64 GeV (173.02 -175.58 GeV)
Step Two
Then, use the inverse of the margin of error for each measurement to construct of weight for each measurement and take the weighted average. This give you:
Step Two
Then, use the inverse of the margin of error for each measurement to construct of weight for each measurement and take the weighted average. This give you:
172.65 GeV
Thus the latest experimental result from the LHC have pulled down the global average top quark mass by about 0.31 GeV from the value currently listed by the Particle Data Group.
Step Three
Then, calculate the margin of error of the weighted average from the inputs. A prior post somewhere at this blog explains how to do with simplifying assumptions.
But, to really do it rigorously, you have to consider the fact that the systemic and theoretical errors are not fully independent of each other, particularly for results that are all from LHC (which tends to make the total combined margin of error greater), and that historically, in this kind of experiment, actual errors have had fatter tails than a Gaussian (i.e. "normal") distribution and are distributed in something closer to a student t-test distribution with different parameters established by the historical data, which also tends to increase the total combined margin of error. The combined error should be a little lower than, but fairly close in magnitude to, the lowest margin of error of any of the individual entries, unless there is very wide scatter of precise measurements, in which case the combined error should be higher than the lowest margin of error of the individual entries, because this pattern implies that one or more of the margins of error is underestimated.
I suspect that the combined margin of error is on the order of 0.35 GeV to 0.45 GeV. Using the higher figure to account for issues like correlated errors and non-Gaussian errors, the two sigma range for the top quark mass given current data is 171.75 GeV to 173.55 GeV.
Given comparison of the two sigma bands for each result and giving slightly more importance to the Tevatron value as it is more independent than the other values, I suspect that the true value of the top quark pole mass is probably between 172.92 GeV and 173.3 GeV, which favors the higher end of combined two sigma range.
Theoretical Comparison Points (Highly Speculative)
Other reference points from theory include the following conjectures, none of which is widely accepted among physicists, but is innocent enough to compare to the experimental results. The world average is at the low end of these predictions, but the first two are consistent to within two sigma with the current world average, while the third theoretical number is not quite consistent with the world average including all measurements at two sigma.
The Extended Koide's Rule Estimate
An extended Koide's rule estimate of the top quark mass using only the electron and muon masses as inputs, predicted a top quark mass of 173.263947 ± 0.000006 GeV. This would be 173.26 GeV to the greatest precision that would be non spurious to compare to current experimental results. This is probably within 1.74 sigma of the current world average and also within the preferred region I identify above.
If you read my prior posts about the extended Koide's rule, there is good reason to think that this value should receive a second order correction which is a small downward adjustment of this value. This is because it does not reflect top quark to down quark transition which are rare but not impossible. This rule as a whole should also be given something less than full confidence, because both the charm quark and up quark values estimated in this fashion are quite far from the experimentally measured values of these quarks, so it clearly needs some adjustment to accurately reflect reality and is only a first order approximation of the fermion mass matrix. This suggests that a corrected extended Koide's rule would need a roughly 151 MeV downward adjustment to the top quark mass.
The Higgs Vacuum Expectation Value Based (LP&C) Estimates
If you read my prior posts about the extended Koide's rule, there is good reason to think that this value should receive a second order correction which is a small downward adjustment of this value. This is because it does not reflect top quark to down quark transition which are rare but not impossible. This rule as a whole should also be given something less than full confidence, because both the charm quark and up quark values estimated in this fashion are quite far from the experimentally measured values of these quarks, so it clearly needs some adjustment to accurately reflect reality and is only a first order approximation of the fermion mass matrix. This suggests that a corrected extended Koide's rule would need a roughly 151 MeV downward adjustment to the top quark mass.
The Higgs Vacuum Expectation Value Based (LP&C) Estimates
* The weak version of the LP & C hypothesis is probably true.
The sum of the squares of the pole masses of the Standard Model fundamental particles is almost precisely identical to the square of the vacuum expectation value of the Higgs field.
A fit comparing the two predicts a top quark mass of 173.1125 ± 0.0025 GeV. Most of the uncertainty in this value is due to the uncertainty in the Higgs boson mass. This is 173.11 GeV to the greatest precision that would be non spurious to compare to current experimental results. This is probably within 1.31 sigma of the current world average and also right in the middle of the preferred region I identify above based on harmonizing the two sigma ranges of the nine available mass measurements.
The boson side is also consistent with the weak version of this hypothesis at a roughly 1.3 sigma level.
I strongly suspect that this relationship is a true and accurate law of physics and that the top quark, as a result, has a true pole mass of 173.11 GeV.
* The strong version of the LP & C hypothesis is probably false.
The value of the top quark mass necessary to make the sum of the squares of the fermion masses equal to the sum of the square of the boson masses would with the combined amount equal to the square of the vacuum expectation value of the Higgs field is 174.974 GeV (with less than 0.0005 GeV of uncertainty). This is 174.97 GeV to the greatest precision that would be non spurious to compare to current experimental results. This is probably 6.6 sigma from the current world average and is at least 5.1 sigma from the current world average. It is also more than 5.1 sigma from the PDG direct measurement world average.
On the boson side, the strong version of the hypothesis would require a Higgs boson mass of 124.66 GeV, which is 5.3 sigma away from the current PDG value of 125.10 ± 0.14 GeV, which is also pretty much definitively ruled out by the experimental data.
Thus, the strong version of this hypothesis is ruled out by experimental data, at the more than five sigma level, for both fermions and for bosons.
This would most directly imply, without other new physics, a lack of perfect harmony between fundamental fermion pole masses and fundamental boson masses in the universe, even though they are very nearly balanced (in much the way that the pion is almost, but not quite, a Goldstone boson and is, instead, a pseudo-Goldstone), and there is a slight imbalance in favor of bosons in the universe, for reasons unknown.
To the extent that one thinks about this approximate balance of masses as being some sort of "supersymmetric" (in the less strict sense) balance in the universe between fermions and bosons, this broad sense supersymmetry is an approximate, rather than an exact, symmetry of the universe. This approximate symmetry may help explain why supersymmetry theories can provide informative and useful predictions despite the fact that there is no positive evidence for the existence of any of the new particles or other new physics predicted in supersymmetry theories.
The boson side is also consistent with the weak version of this hypothesis at a roughly 1.3 sigma level.
I strongly suspect that this relationship is a true and accurate law of physics and that the top quark, as a result, has a true pole mass of 173.11 GeV.
* The strong version of the LP & C hypothesis is probably false.
The value of the top quark mass necessary to make the sum of the squares of the fermion masses equal to the sum of the square of the boson masses would with the combined amount equal to the square of the vacuum expectation value of the Higgs field is 174.974 GeV (with less than 0.0005 GeV of uncertainty). This is 174.97 GeV to the greatest precision that would be non spurious to compare to current experimental results. This is probably 6.6 sigma from the current world average and is at least 5.1 sigma from the current world average. It is also more than 5.1 sigma from the PDG direct measurement world average.
On the boson side, the strong version of the hypothesis would require a Higgs boson mass of 124.66 GeV, which is 5.3 sigma away from the current PDG value of 125.10 ± 0.14 GeV, which is also pretty much definitively ruled out by the experimental data.
Thus, the strong version of this hypothesis is ruled out by experimental data, at the more than five sigma level, for both fermions and for bosons.
This would most directly imply, without other new physics, a lack of perfect harmony between fundamental fermion pole masses and fundamental boson masses in the universe, even though they are very nearly balanced (in much the way that the pion is almost, but not quite, a Goldstone boson and is, instead, a pseudo-Goldstone), and there is a slight imbalance in favor of bosons in the universe, for reasons unknown.
To the extent that one thinks about this approximate balance of masses as being some sort of "supersymmetric" (in the less strict sense) balance in the universe between fermions and bosons, this broad sense supersymmetry is an approximate, rather than an exact, symmetry of the universe. This approximate symmetry may help explain why supersymmetry theories can provide informative and useful predictions despite the fact that there is no positive evidence for the existence of any of the new particles or other new physics predicted in supersymmetry theories.
2 comments:
The 2021 PDG value for the top quark mass is 172.76 ± 0.3 GeV by direct measurements and 172.5 ± 0.7 GeV by indirect cross section measurements.
The 2021 PDG value for the Higgs boson mass is 125.25 ± 0.17 GeV which is about 3.5 sigma from the stronger LP & C prediction, but a decent weak to the weaker LP & C prediction.
The 2021 PDG value for the top quark mass is consistent with the weaker LP & C prediction.
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