Bottom line: the mass of the top quark is 173,340 ± 760 MeV (combined, the error is ± 270 MeV statistical and ± 710 MeV systemic error). This is a precision of one part in 228 (i.e. ± 0.044%).

The top quark mass is the only quark mass that can be directly measured. All other quark masses must be inferred from the masses of hadrons believed to contain those quarks in a model dependent manner.

While the top quark mass is the most precisely known quark mass on a percentage basis, is known with only slight less precision than the Higgs boson mass, and is known more precisely than any of the neutrino masses (the masses of the W boson, Z boson and charged leptons, as well as the Higgs vacuum expectation value ("vev") are known more precisely).

But, as explained below, about 61% of the uncertainty in the sum of the absolute value of the Standard Model fundamental particle masses (including the Higgs vev), and about 72% of the uncertainty in the sum of the square square of the Standard Model fundamental particle masses (including the square of the Higgs vev) is due to uncertainty in the mass of the top quark. All but 6.3% and 0.85% of the balance of these uncertainties is due to uncertainty in the mass of the Higgs boson.

Thus, even a modest improvement in the precision of the top quark mass improves the overall precision of the measurements o the Standard Model fundamental particle masses considerably.

**Previous Estimates of the Top Quark Mass**

__Previous Direct Estimates of the Top Quark Mass__

The top quark mass is the only quark mass that can be directly measured. All other quark masses must be inferred from the masses of hadrons believed to contain those quarks in a model dependent manner.

The previous best estimate based upon direct measurements from the Particle Data Group had been 173,070 ± 888 MeV (combined, ± 520 MeV statistical, ± 720 MeV systemic).

The new result is 270 MeV higher than the previous best estimate (0.3 standard deviations from the previous best estimate, which is unsurprising since the new result uses most of the same data as the old result and merely analyzes it more rigorously and precisely) and has a 14% smaller margin of error (with almost all of the improvement coming from a much smaller reduced statistical error in a pooled data set).

The final estimate of the top quark mass from Tevatron alone (CDF and D0 combined) has been 173,200 ± 600 ± 800 MeV.

__Previous Top Quark Mass Global Fits__

A previous global electroweak observable based fit including early LHC data in 2012 had come up with a top quark mass of 173,520 ± 0.880 MeV which was about 450 MeV more than the previous best estimate and about 180 MeV more than the current combined analysis. These global fits are based upon Standard Model relationships between the Higgs boson mass, top quark mass and W boson mass. These fits are most sensitive to the W boson mass, then to the top quark mass, and are least sensitive to the precise value of the Higgs boson mass.

*The diagonal line shows the combinations of the top quark mass and W boson mass that are expected in the Standard Model at a Higgs boson mass of about 125,000 MeV, with the thickness of the line reflecting the range of uncertainty in that measurement. The latest estimate of the Higgs boson mass shift that line imperceptibly to the right, while the latest estimate of the top quark mass shift the center point of the 1 standard deviation confidence interval ellipse to the right by about a third of a hash mark. The greatest impact of a global fit is to favor a low end estimate of the W boson mass of about 80,362 MeV, rather than the current best estimate from the Particle Data Group of 80,385 ± 15 MeV.*

__The Extended Koide's Rule Fit To the Top Quark Masses (and Other Quark Masses)__

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 173,263.947 ± 0.006 MeV, which is about 80 MeV less than the latest direct measurement. This is within 0.1 standard deviations from the new directly measured value.

Prior to the new combined measurement, the extended Koide's rule estimate was 0.22 standard deviations from the measured value.

The fact that the extended Koide's rule estimate has become more precise than it was when it was devised, as the experimental value has been measured more precisely, is impressive. Indeed, the extended Koide's rule estimate was closer to the new measurement than it was to the old one.

On the other hand, new precision estimates of the bottom and charm quark masses, mentioned below, increase the number of standard deviations in the gap between the experimentally measured values of these masses and the extended Koide's rule estimates for them (from 0.58 sigma to 3.57 sigma for the bottom quark, and from 3.38 sigma to 14.4 sigma for the charm quark). In the case of the bottom quark, the extended Koide's rule estimate is about 0.7% too high. In the case of the charm quark, the extended Koide's rule estimate is about 6.8% too high).

To four significant digits the t-b-c triple's Koide ratio which is predicted to be 0.6667 to four significant digits is 0.6695 at the PDG values and is 0.6699 using the new combined value for the t quark mass, and the new precision values for the b quark and c quark masses. This is still a better fit than any of the other quark triples, although it is a less good fit than it was before the new precision measurements were reported.

The accuracy of the extended Koide's rule estimate for the strange, down and up quark masses is unchanged since there are no new estimates of these masses. The strange quark estimate is 2.9% low (0.55 standard deviations), the down quark estimate is 10.8% high (1.3 standard deviations), and the up quark estimate is 98.5% low (2.26 standard deviations).

The original Koide's rule predicts a mass of the tau lepton from the electron and muon masses that is within 0.93 standard deviations of the currently measured value.

**Other Standard Model Fundamental Particle Mass Measurements.**

__Other Quark Mass Uncertainties__

*All of the other measured values of the Standard Model quark masses are model dependent estimates based on QCD and hadron masses, and definition issues arise because a quark's mass is a function of the energy scale at which it is measured. The pole mass of a quark is roughly speaking, the mass of a quark at an energy scale equal to its own rest mass and this is the number quotes for the bottom quark and charm quark as well as for the top quark. The pole masses of the strange quark, down quark and up quark are ill defined and instead their masses in what is known as the MS scheme at an energy scale of 2 GeV (i.e. slightly more than the mass of two protons, two neutrons or one proton and one neutron), is normally used instead.*

In the case of the bottom quark, the second heaviest of the quarks, the PDG estimate of the bottom quark mass has an uncertainty of 30 MeV (4,180 ± 30 MeV), but a new and consistent precision estimate using improved QCD approaches claims an uncertainty of just 8 MeV (4,169 ± 8 MeV). Similarly, in the case of the charm quark, the third heaviest of the quarks, the PDG estimate has an uncertainty of 25 MeV (1275 ± 25 MeV), but a new and consistent precision estimate using improved QCD approaches claims an uncertainty of just 6 MeV (1,273 ± 6 MeV). The best estimates of the up and down quark masses are about 2.3-0.5+0.7 MeV (about 25% precision) and 4.8-0.3+0.5 MeV (about 8% precision) respectively, but in each case the uncertainty in absolute terms is less than 1 MeV.

The strange quark mass (95 ± 5 MeV per the Particle Data Group) is known only to a roughly 5% precision. The QCD approaches used to reduce uncertainties in bottom quark and charm quark mass produce estimates of only about 10% precision in the case of the strange quark mass.

__Charged Lepton Mass Uncertainties__

The measured tau lepton mass is 1776.82 MeV with an uncertainty in the mass of the tau lepton is 0.16 MeV. The measured muon mass is 105.6583715 Mev with an uncertainty in the muon mass is 0.0000035 MeV. The measured electron mass is 0.510998928 MeV with an uncertainty in the electron mass is 0.000000011 MeV

__Standard Model Neutrino Mass Uncertainties__

The two differences in masses between the neutrino mass states are known to a precision of less than 10

^{-5}MeV and 10

^{-4}MeV respectively (the smaller one is about 0.007 eV, while the larger one is about 0.046 eV).

The absolute value of the lightest neutrino mass has been directly measured to be less than 2*10

^{-6}MeV (i.e. 2 eV), and is constrained in a model dependent way by cosmic background radiation and other astronomy measurements to be less than 10

^{-7}MeV (i.e. 0.1 eV). If the neutrinos have a "normal" rather than "inverted" mass hierarchy lightest electron neutrino mass is probably on the order of 0.001 eV.

__Gauge Boson Mass and Higgs vev Uncertainties__

The newly discovered Higgs boson has a global best fit measured mass of 125,900 MeV with an uncertainty about about ± 400 MeV, which is just over half of the absolute value of the uncertainty in the top quark mass. This is 79 MeV lower than the expectation if the Higgs boson mass is exactly equal to the W boson mass plus 1/2 of the Z boson mass (about 0.2 standard deviations from the measured value).

The accepted value of the Higgs vev is 246,227.9579 MeV with an uncertainty on the order of 0.001 MeV. This is measured via measurements of the lifetime of the mean lifetime of the muon (the Higgs vev is the inverse of the square root of two times the Fermi coupling constant). Both the Higgs vev and Fermi coupling constant are functions of the W boson mass and the weak force coupling constant, but the combined impact of these two factors is known much more precisely than the exact value of either of them.

As noted above, the uncertainty in the measured mass of the W boson is about 15 MeV (from a base number 80,385 MeV). The uncertainty in the Z boson mass is 2.1 MeV (from a base number of 91,187.6 MeV).

**Why Is The Accurate Determination Of The Top Quark Mass So Important?**

*The Absolute Value of the Top Quark Mass Uncertainty Is The Largest In The Standard Model*

In evaluating relationships between Standard Model fundamental particle masses, the uncertainty in the top quark mass is one of the dominant sources of uncertainty, because the top quark mass is the heaviest fundamental particle in the Standard Model and the absolute value of the uncertainty in this value is greater than that for any other Standard Model particle.

For example, the accuracy of the strange quark mass measurement is about 100 times less precise on a percentage basis than the top quark mass measurement, but the absolute value of the uncertainty in the top quark mass is still 760 MeV, compared to just 5 MeV for the strange quark mass.

The absolute value of the uncertainty in the top quark mass (760 MeV, which is about 61% of the total) is greater than the sum of the absolute values of the uncertainty in all of the other Standard Model fundamental particles combined (478.261 MeV, which is about 39% of the total), of which 400 MeV is the uncertainty in the Higgs boson mass and 78.261 MeV (which is about 6.3% of the total).

Thus, two fundamental particle mass measurements, one of which is just a couple of years old, account for 93.7% of all of the absolute value of the uncertainty in Standard Model particle mass measurements.

*The Top Mass Squared Uncertainty Is Even More Dominant*

The dominance of any imprecision in the top quark mass to overall model fits is further amplified in cases where the quantities compared are the square of the masses rather than the masses themselves (e.g. comparing the sum of squares of the Standard Model particle masses to the almost precisely identical square of the vacuum expectation value of the Higgs field).

About 72% of this imprecision is due to the top quark mass and about 99.15% of the imprecision is due to the top quark mass and Higgs boson masses combined.

The difference between the high end and low end of the square of the current combined estimate of various fundamental particle masses (i.e. their uncertainty) is as follows (in MeV^2, using PDG error bars for cases other than the top quark):

t quark mass squared: 526,953,600

All other Standard Model fundamental particles mass squared combined: 207,683,572 (about 28% of the total).

Higgs boson mass squared: 201,440,000

All other masses square except the t quark and Higgs boson masses: 6,243,572 (about 0.85% of the total).

W boson mass squared: 4,823,100

Z boson mass squared: 765,976

b quark mass squared: 501,600

c quark mass squared: 127,500

tau lepton mass squared: 22,497

s quark mass squared: 1,900

Higgs vev squared: 984.912

d quark mass squared: 7.84

u quark mass squared: 5.76

muon mass squared: 0.0015

electron mass squared: 0.0000000225

neutrino masses squared (combined): < 0.0000000003

**A Final Prediction Re Top Quark Mass**

**1. Assume that the W boson mass has its global fit value of 80,362 MeV, rather than its best fit measured value of 80,385 MeV (a 1.53 standard deviation shift).**

2. Assume that the Higgs boson mass is exactly equal to the W boson mass plus one half of the Z boson mass (i.e. that it is 125,955.8 MeV) (a 0.14 standard deviation shift).

3. Assume that the Tau lepton mass has its Koide's rule predicted value of 1776.97 MeV, rather than the 1776.82 MeV value that it is measured at today (a 0.93 standard deviation shift).

4. Assume that the bottom quark has the precision value of 4,169 MeV (a 0.37 standard deviation shift)

5. Assume that the charm quark has the precision value of 1,273 MeV (a 0.08 standard deviation shift)

6. Assume that the sum of the squares of the masses of the fundamental particles in the Standard Model equals the sum of the squares of the Higgs vev.

7. Assume that the electron, muon, up quark, down quark, and strange quark have their PDG values and that the neutrinos have masses of less than 2 eV each.

8. Assume that there are no fundamental particles beyond the Standard Model that contribute to the Higgs vev.

What is the best fit value for the top quark mass?

Answer:

**173,112.5 ± 2.5**

**MeV**(a 0.29 standard deviation shift of 227.5 MeV from the newly announced value).

N.B. 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 be about 174,974 MeV under the same set of assumptions, about 174,646 MeV with a Higgs boson mass at the 125,500 MeV low end of the current 68% confidence interval for the Higgs boson mass, and about 175,222 MeV with a Higgs boson mass at the 126,300 MeV high end of the 68% confidence interval for the Higgs boson mass. These are about 2.07, 1.65, and 2.38 standard deviations, respectively, from the measured value of the mass of the top quark, and thus is not grossly inconsistent with the evidence, despite being a less good fit the the Higgs vev contribution hypothesis (which I also find more compelling theoretically). But, in order to achieve this, the sum of the squares of all of the fundamental Standard Model particle masses must be about 0.7% or more in excess of the square of the Higgs vev.

While both relationships are possible within experimental error, they cannot be true simultaneously despite being quite similar, at least for the pole masses. It is possible to imagine some running mass scale where they might coincide, however, and if there is some energy scale at which this happens, this might be viewed as the energy scale at which fermion-boson symmetry (much like that of supersymmetry, but without the extra particles) breaks down.

The running of the charged lepton masses is almost 2% up to the top quark mass and 3.6% over fourteen orders of magnitude. In contrast, the Higgs boson self-coupling runs to zero at the GUT scale, and the W and Z boson masses at high energies appear to be functions of the running of the electromagnetic and weak force coupling constants. The electromagnetic force coupling constant gets stronger (from about 1/137 to 1/125 up to the electroweak scale) while the weak force coupling constant gets weaker. This appears to be more dramatic than the running of the fermion masses, so the equalization of masses between bosons and fermions shouldn't require too high an energy scale.

**Footnote**: The measured Higgs boson mass is very nearly the mass that minimizes the second loop corrections necessary to convert the mass of a gauge boson from an MS scheme to a pole mass scheme.

## 5 comments:

The latest combined result from CMS is 172,200 +/- 730 MeV.

This is a bit on the low side, although consistent at two sigma.

A final D0 result is problematically high.

A study based on ratios of quarkonia moments estimates the charm quark mass at 1261(18) MeV and the bottom quark mass at 4173(10) MeV, both of which are consistent with the precision estimates from the study cited in the original post.

A final Tevatron estimate of the top quark mass is 174.34+/-0.64 GeV/c^2.

D0 most precise ever single measurement:

mt=174.98±0.76 GeV.

http://arxiv.org/abs/1501.07912

This is still on the high end.

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