Abstract
As the quark masses and Higgs boson mass have been measured more precisely over the last few years at the LHC as summarize by the Particle Data Group through its 2018 data, and in lattice determinations using the quark mass measurements in the FLAG 19 report, it is worth re-examining two phenomenological relationships between them: the LC & P hypothesis and the extended Koide's formula for quarks.
The LC & P Hypothesis
The global LC & P relationship between the fundamental particle masses and the Higgs vev proposed in 2013 that the sum of the Yukawas of the fundamental particles of the Standard Model is exactly 1 still holds to within 1.3 sigma of the currently measured masses of the top quark and Higgs boson.
This hypothesis, if true, strongly constrains the parameter space of beyond the Standard Model massive particles that derive their mass from the Higgs mechanism to a mass range that has been thoroughly tested in a manner that rules out non-Standard Model particles of most kinds.
But, it is increasingly clear that the contributions of the fundamental fermions and the fundamental bosons are not equal, although this equality is still not entirely ruled out. The sum of the Yukawas of the fundamental fermions using the best available data is 0.4940 and the sum of the Yukawas of the fundamental bosons using the best available data is 0.5022. It takes a top quark value about 2.6 sigma from the measured value and a Higgs boson value about 3.3 sigma from the measured value to bring both of them to 0.5 exactly.
The approximate symmetry between fermions and bosons in this relationship echos supersymmetry and may help explain why supersymmetric models can serve as approximations of Standard Model phenomena. This also suggests a new unsolved question in physics (that really isn't truly a problem any more than the hierarchy problem or naturalness problem are), which is:
"Why are fundamental boson masses slightly heavier relative to fundamental fermion masses than we might naively expect?"
The Extended Koide Formula
Rivero's extended Koide's formula from 2011 for quarks can be adjusted in a manner that I suggested in 2013 based upon the ansatz that the relative values of the charged fermion masses arises dynamically from the W boson interactions between these fermions and the fermions that they can be transformed into via W boson interactions.
How does the adjusted extended Koide's formula for quarks compare to the experimental measurements?
Quark Type - Adjusted Extended Koide Mass - FLAG 19 Mass- PDG Mass (all in MeV)
FLAG 19 and Other Experimental Data Concerning Fundamental Constant Masses
The main competitor to the Particle Data Group is the Flavour Lattice Averaging Group (FLAG). FLAG's "aim is to provide an answer to the frequently posed question “What is currently the best lattice value for a particular quantity?” in a way that is readily accessible to those who are not expert in lattice methods.
As the quark masses and Higgs boson mass have been measured more precisely over the last few years at the LHC as summarize by the Particle Data Group through its 2018 data, and in lattice determinations using the quark mass measurements in the FLAG 19 report, it is worth re-examining two phenomenological relationships between them: the LC & P hypothesis and the extended Koide's formula for quarks.
The LC & P Hypothesis
The global LC & P relationship between the fundamental particle masses and the Higgs vev proposed in 2013 that the sum of the Yukawas of the fundamental particles of the Standard Model is exactly 1 still holds to within 1.3 sigma of the currently measured masses of the top quark and Higgs boson.
This hypothesis, if true, strongly constrains the parameter space of beyond the Standard Model massive particles that derive their mass from the Higgs mechanism to a mass range that has been thoroughly tested in a manner that rules out non-Standard Model particles of most kinds.
But, it is increasingly clear that the contributions of the fundamental fermions and the fundamental bosons are not equal, although this equality is still not entirely ruled out. The sum of the Yukawas of the fundamental fermions using the best available data is 0.4940 and the sum of the Yukawas of the fundamental bosons using the best available data is 0.5022. It takes a top quark value about 2.6 sigma from the measured value and a Higgs boson value about 3.3 sigma from the measured value to bring both of them to 0.5 exactly.
The approximate symmetry between fermions and bosons in this relationship echos supersymmetry and may help explain why supersymmetric models can serve as approximations of Standard Model phenomena. This also suggests a new unsolved question in physics (that really isn't truly a problem any more than the hierarchy problem or naturalness problem are), which is:
"Why are fundamental boson masses slightly heavier relative to fundamental fermion masses than we might naively expect?"
The Extended Koide Formula
Rivero's extended Koide's formula from 2011 for quarks can be adjusted in a manner that I suggested in 2013 based upon the ansatz that the relative values of the charged fermion masses arises dynamically from the W boson interactions between these fermions and the fermions that they can be transformed into via W boson interactions.
How does the adjusted extended Koide's formula for quarks compare to the experimental measurements?
Quark Type - Adjusted Extended Koide Mass - FLAG 19 Mass- PDG Mass (all in MeV)
top 172,743 v. 173,000 +/- 400 (-0.64 sigma) v. 173,000 +/- 400 (-0.64 sigma)
bottom 4192.98 v. 4,198 +/- 12 (-0.49 sigma) v. 4,180 +40-/-30 (+ 0.32 sigma)
charm 1293.21 v. 1,282 +/- 17 (+0.66 sigma) v. 1,275 +25/-35 (+ 0.73 sigma)
strange 92.274758 v. 93.12 +/- 0.69 (-1.22 sigma) v. 95+9/-3 (- 0.91 sigma)
down 5.32 v. 4.88 +/- 0.2 (+2.2 sigma) v. 4.7 + 0.5/-0.4 (+1.24 sigma)
up 0.0356 v. 2.5 +/- 0.17 (-14.5 sigma) v. 2.2 +0.5/-0.4 (- 5.4 sigma)
These values are consistent at or near the two sigma level, depending upon the measurements used for comparison purposes, with all of the quark masses except the up quark mass which is grossly at odds with the experimentally measured value.
So, while Rivero's extended Koide's formula is a good first order approximation for quarks based solely on the electron and muon masses, and my adjustments are a good second order approximation for quarks that incorporates CKM matrix data, at least one more layer of adjustment is needed to actually capture physical reality.
This also suggests a new unsolved question in physics (that really isn't truly a problem any more than the hierarchy problem or naturalness problem are), which is:
"Why does the up quark have more than a negligible mass?"
Note that even though these are merely phenomenological relationships based on some hunches about a possible underlying physical mechanism (and Koide and others have also proposed different underlying physical mechanisms), even if they are not accurate for the reasons proposed, they do give us a deeper understanding of the manner in which they quantities are related that is more than merely random.
Reductions in the uncertainty in the measurements of the top quark mass and Higgs boson mass that are likely to occur in the foreseeable future will greatly increase the extent to which these hypotheses are tested empirically in a convincing manner.
Note that even though these are merely phenomenological relationships based on some hunches about a possible underlying physical mechanism (and Koide and others have also proposed different underlying physical mechanisms), even if they are not accurate for the reasons proposed, they do give us a deeper understanding of the manner in which they quantities are related that is more than merely random.
Reductions in the uncertainty in the measurements of the top quark mass and Higgs boson mass that are likely to occur in the foreseeable future will greatly increase the extent to which these hypotheses are tested empirically in a convincing manner.
FLAG 19 and Other Experimental Data Concerning Fundamental Constant Masses
The main competitor to the Particle Data Group is the Flavour Lattice Averaging Group (FLAG). FLAG's "aim is to provide an answer to the frequently posed question “What is currently the best lattice value for a particular quantity?” in a way that is readily accessible to those who are not expert in lattice methods.
FLAG's most recent review is for 2019 and for the most part, they are willing to agree on most precise measurements of Standard Model fundamental constants as state of the art and correct than the Particle Data Group.
The current (2018) Particle Data Group values for the quarks charged lepton masses, and mass fundamental bosons (in MeV) are:
top quark 173,000 +/- 400.
bottom quark 4,180 +40-/-30
charm quark 1,275 +25/-35
strange quark 95+9/-3
down quark 4.7 + 0.5/-0.4
up quark 2.2 +0.5/-0.4
tau lepton 1,776.86 +/- 0.12
muon 105.658375 +/- 0.000024
electron 0.51099895 +/- 0.00000031
W boson 80,379 +/- 12
Z boson 91,187.6 +/- 2.1
Higgs boson 125,180 +/- 160
The PDG also provides a value for Fermi's Constant (GF) which is proportional to the weak force coupling constant. The Particle Data Group reports its value as 0.00002663787 +/- 0.00000000006 GeV-2. The Higgs vacuum expectation value (vev) is a function of Fermi's constant. It is equal to one divided by the square root of the product of the square root of two and Fermi's Constant. On this basis we have one more physical constant whose value can be expressed in MeV units:
Higgs vev 246,219.65 +/- 0.60
FLAG 19 doesn't have separate values for the charged leptons or fundamental bosons, but does have its own set of values for the five quarks that are confined, determined with the best available lattice QCD methods.
bottom quark 4,198 +/- 12
charm quark 1,282 +/- 17
strange quark 93.12 +/- 0.69
down quark 4.88 +/- 0.2
up quark 2.5 +/- 0.17
It is also possible to consider a W boson mass based upon a global fit of the W boson mass, the Higgs boson mass, and the top quark mass considering relationships between them:
This would suggest a value (in MeV) of the W boson of about 80,362 +/- 2 rather than the current PDG value of 80,379 +/- 12.
LC & P
The LC & P Hypothesis
A May 17, 2013 paper by Lopez Castro and Pestieau (LC & P) observes that the sum of the square of the mass of each of the fundamental bosons in the Standard Model is close to the sum of the square of each of the fundamental fermions in the Standard Model, and that combined that they are equal to the square of the Higgs vacuum expectation value.
The LP & C Relationship With Updated Values
Six years later, how is this hypothesis looking?
The current value of the Higgs boson mass 125.18 +/ 0.16 GeV is lighter and significantly more accurate than it was in 2013. The value of the Higgs boson mass that makes the sum of the square of the masses of the fundamental bosons (W, Z and the Higgs boson, and if you like also the photon and gluon and even graviton, all of which would have zero masses) equal to one-half of the Higgs boson is 124.65 GeV. If you use the PDG value of the W boson mass this is about 3.29 sigma (i.e. standard deviations) below the current global average measured value of the Higgs boson mass. If you use the global fit value of the W boson, the gap is about 3.35 sigma.
The current value of the top quark mass 173.0 +/- 0.4 GeV is also lighter and significantly more accurate than it was in 2013. The value of the top quark mass that makes the sum of the square of the masses of the fundamental fermions (the quarks, the charged leptons and the neutrinos) equal to one-half of the Higgs boson is 174.04 GeV. This is about 2.6 sigma higher than the PDG value of the top quark mass.
Put another way, the hypothesis that the fundamental fermions separately have a sum of Yukawas equal to 0.5 and that the fundamental bosons have a sum of Yukawas equal to 0.5 is strongly disfavored. Instead, the data strongly favor a scenario in which the sum of the Yukawas of the fundamental bosons is a little bit greater than the sum of the Yukawas for the fundamental fermions.
Specifically, the sum of the Yukawas of the fundamental fermions using the best available data is 0.4940 and the sum of the Yukawas of the fundamental bosons using the best available data is 0.5022.
The sum of the Yukawas is a little bit under 1.0 but is very close. So, the overall hypothesis that the Yukawas of all of the fundamental particles of the Standard Model are equal to 1.0 is still very healthy. Using the best available values the sum is 0.9962 using the PDG value for the W boson and 0.9963 using the global fit value for the W boson.
How close is that?
If you adjusted up both the top quark mass and Higgs boson mass by 1.27 sigma, an amount entirely consistent with the current data, the Yukawas would equal exactly 1. This would imply a top quark mass of 173508 MeV and a Higgs boson mass of 125383 MeV.
If you adjusted up the top quark mass only by 1.64 sigma to 173654 MeV, which is also consistent with the current data, the Yukawas would equal exactly one. If you adjusted up the Higgs boson mass only, however, you would need to do so by 5.64 sigma to 126083 MeV which is ruled out by the current data.
Definitional Issues
The LP & C relationship evaluates the relevant masses at their pole mass values (i.e. at the entire scale where where the mass of the particle equals the energy scale at which the mass is evaluated).
Would evaluating the masses at a higher energy scale, the obvious one being the Higgs vev, solve the problem?
Probably not. Both the fundamental fermion masses and the fundamental boson masses fall at higher energy scales. So, evaluated at the Higgs vev, the sum of the square of the fundamental fermion masses divide by one half of the Higgs vev squared would be significantly less than 0.4940 and the sum of the square of the fundamental boson masses divided by the Higgs vev squared would be significantly less than 0.5022, so the total would be significantly less than 1.
If the boson masses were falling somewhat more quickly than the fermion masses (something I don't know one way or the other since I don't have data evaluating all of the Standard Model masses at that energy scale) they could end up equal in value at some energy scale, and perhaps that works for Yukawas of the Higgs boson which might be rescaled at that energy scale as well. But, this doesn't look promising.
Uncertainty Analysis
If you use the FLAG 19 values for the quark masses and the global fit value of the W boson, 86.00% of the uncertainty in the total sum of squares values (including the uncertainty in the Higgs vev) comes from uncertainty in the top quark mass, 13.76% comes from uncertainty in the Higgs boson mass, and 0.24% comes from uncertainty in other masses (mostly the bottom and charm quarks). So, improvements in the precision of mass measurements other than the top quark and Higgs boson are almost irrelevant to testing the LP & C hypothesis.
Reducing the uncertainty in the Higgs boson mass from 160 MeV to 80 MeV would reduce the total uncertainty by 3%. Reducing the uncertainty in the top quark mass from 400 MeV to 300 MeV would reduce the total uncertainty by 33.3%. Reducing the uncertainty in the top quark mass from 400 MeV to 200 MeV would reduce the total uncertainty by 64%. Reducing the uncertainty in the top quark mass from 400 MeV to 100 MeV would reduce the total uncertainty by 80%. If the uncertainty in the Higgs boson mass were reduced from 160 MeV to 80 MeV, and the uncertainty in the top quark mass were reduced from 400 MeV to 100 MeV, both of which are not unrealistic possibilities in the foreseeable future, the total uncertainty could be reduced by about 83%.
So, the prospects of testing the LP & C relationship more rigorously in the foreseeable future is good. Already, since 2013, the uncertainty in the Higgs boson mass has fallen from 400 MeV to 160 MeV and the uncertainty in the top quark mass has fallen from 900 MeV to 400 MeV.
Koide's Formula
In 1981, Yoshio Koide proposed a relationship between the masses of the electron, muon and tau leptons which is called Koide's formula (or Koide's law or Koide's rule). This rule wasn't a great fit to the data when the tau lepton mass was first measured, but has become a better fit with subsequent measurements of the three quantities and is now consistent with the measured data at the 0.83 sigma level.
Koide's formula implies a tau lepton mass of 1776.96894 +/- 0.00007 given the current ultra-precise values of the electron mass (known to 6 parts per billion) and the muon mass (known to 23 parts per billion).
In contrast, the experimentally measured value of the tau lepton mass is currently known to one part per 14,807.
The formula is independent of scale or units. It is a function only of the ratio of the lightest mass to the second lightest mass and the ratio of the second lightest mass to the heaviest mass.
The two-thirds value is exactly midway between the extremes of the values that Q can take for any three real valued constants. The lowest possible value of Q, in which all three have the same value is 1/3. The highest possible value of Q, in which two of the values are 0, is 1.
Thus, by this particular measure, the ratios of the charged leptons to each other is exactly equidistant between maximum equality and maximum inequality.
Note also, that Koide's formula, like LP & C is evaluated at the pole masses of the charged leptons, not at their running value at some other energy scale.
The Extended Koide Relationship For Quarks
It is also possible to consider a W boson mass based upon a global fit of the W boson mass, the Higgs boson mass, and the top quark mass considering relationships between them:
This would suggest a value (in MeV) of the W boson of about 80,362 +/- 2 rather than the current PDG value of 80,379 +/- 12.
LC & P
The LC & P Hypothesis
Using mH = (125.6±0.4) GeV and mt = (173.5±0.9) GeV and other masses from the PDG compilation, the left-hand side of Eq. (1) is in full agreement with the right-hand side because v^2 = 1/(√2GF ) = (246.2 GeV)^2.Another way of stating this hypothesis is that the sum of the Yukawas (basically, the Higgs boson coupling strengths) of the Standard Model fermions is 0.5, the sum of the Yukawas of the Standard Model bosons is 0.5, and the sum of all of the Standard Model Yukawas is 1.0. (It is a bit more complicated than that because the Yukawas for the Higgs boson and other Standard Model bosons have different names).
The LP & C Relationship With Updated Values
Six years later, how is this hypothesis looking?
The current value of the Higgs boson mass 125.18 +/ 0.16 GeV is lighter and significantly more accurate than it was in 2013. The value of the Higgs boson mass that makes the sum of the square of the masses of the fundamental bosons (W, Z and the Higgs boson, and if you like also the photon and gluon and even graviton, all of which would have zero masses) equal to one-half of the Higgs boson is 124.65 GeV. If you use the PDG value of the W boson mass this is about 3.29 sigma (i.e. standard deviations) below the current global average measured value of the Higgs boson mass. If you use the global fit value of the W boson, the gap is about 3.35 sigma.
The current value of the top quark mass 173.0 +/- 0.4 GeV is also lighter and significantly more accurate than it was in 2013. The value of the top quark mass that makes the sum of the square of the masses of the fundamental fermions (the quarks, the charged leptons and the neutrinos) equal to one-half of the Higgs boson is 174.04 GeV. This is about 2.6 sigma higher than the PDG value of the top quark mass.
Put another way, the hypothesis that the fundamental fermions separately have a sum of Yukawas equal to 0.5 and that the fundamental bosons have a sum of Yukawas equal to 0.5 is strongly disfavored. Instead, the data strongly favor a scenario in which the sum of the Yukawas of the fundamental bosons is a little bit greater than the sum of the Yukawas for the fundamental fermions.
Specifically, the sum of the Yukawas of the fundamental fermions using the best available data is 0.4940 and the sum of the Yukawas of the fundamental bosons using the best available data is 0.5022.
The sum of the Yukawas is a little bit under 1.0 but is very close. So, the overall hypothesis that the Yukawas of all of the fundamental particles of the Standard Model are equal to 1.0 is still very healthy. Using the best available values the sum is 0.9962 using the PDG value for the W boson and 0.9963 using the global fit value for the W boson.
How close is that?
If you adjusted up both the top quark mass and Higgs boson mass by 1.27 sigma, an amount entirely consistent with the current data, the Yukawas would equal exactly 1. This would imply a top quark mass of 173508 MeV and a Higgs boson mass of 125383 MeV.
If you adjusted up the top quark mass only by 1.64 sigma to 173654 MeV, which is also consistent with the current data, the Yukawas would equal exactly one. If you adjusted up the Higgs boson mass only, however, you would need to do so by 5.64 sigma to 126083 MeV which is ruled out by the current data.
Definitional Issues
The LP & C relationship evaluates the relevant masses at their pole mass values (i.e. at the entire scale where where the mass of the particle equals the energy scale at which the mass is evaluated).
Would evaluating the masses at a higher energy scale, the obvious one being the Higgs vev, solve the problem?
Probably not. Both the fundamental fermion masses and the fundamental boson masses fall at higher energy scales. So, evaluated at the Higgs vev, the sum of the square of the fundamental fermion masses divide by one half of the Higgs vev squared would be significantly less than 0.4940 and the sum of the square of the fundamental boson masses divided by the Higgs vev squared would be significantly less than 0.5022, so the total would be significantly less than 1.
If the boson masses were falling somewhat more quickly than the fermion masses (something I don't know one way or the other since I don't have data evaluating all of the Standard Model masses at that energy scale) they could end up equal in value at some energy scale, and perhaps that works for Yukawas of the Higgs boson which might be rescaled at that energy scale as well. But, this doesn't look promising.
Uncertainty Analysis
If you use the FLAG 19 values for the quark masses and the global fit value of the W boson, 86.00% of the uncertainty in the total sum of squares values (including the uncertainty in the Higgs vev) comes from uncertainty in the top quark mass, 13.76% comes from uncertainty in the Higgs boson mass, and 0.24% comes from uncertainty in other masses (mostly the bottom and charm quarks). So, improvements in the precision of mass measurements other than the top quark and Higgs boson are almost irrelevant to testing the LP & C hypothesis.
Reducing the uncertainty in the Higgs boson mass from 160 MeV to 80 MeV would reduce the total uncertainty by 3%. Reducing the uncertainty in the top quark mass from 400 MeV to 300 MeV would reduce the total uncertainty by 33.3%. Reducing the uncertainty in the top quark mass from 400 MeV to 200 MeV would reduce the total uncertainty by 64%. Reducing the uncertainty in the top quark mass from 400 MeV to 100 MeV would reduce the total uncertainty by 80%. If the uncertainty in the Higgs boson mass were reduced from 160 MeV to 80 MeV, and the uncertainty in the top quark mass were reduced from 400 MeV to 100 MeV, both of which are not unrealistic possibilities in the foreseeable future, the total uncertainty could be reduced by about 83%.
So, the prospects of testing the LP & C relationship more rigorously in the foreseeable future is good. Already, since 2013, the uncertainty in the Higgs boson mass has fallen from 400 MeV to 160 MeV and the uncertainty in the top quark mass has fallen from 900 MeV to 400 MeV.
Koide's Formula
In 1981, Yoshio Koide proposed a relationship between the masses of the electron, muon and tau leptons which is called Koide's formula (or Koide's law or Koide's rule). This rule wasn't a great fit to the data when the tau lepton mass was first measured, but has become a better fit with subsequent measurements of the three quantities and is now consistent with the measured data at the 0.83 sigma level.
Koide's formula implies a tau lepton mass of 1776.96894 +/- 0.00007 given the current ultra-precise values of the electron mass (known to 6 parts per billion) and the muon mass (known to 23 parts per billion).
In contrast, the experimentally measured value of the tau lepton mass is currently known to one part per 14,807.
The formula is independent of scale or units. It is a function only of the ratio of the lightest mass to the second lightest mass and the ratio of the second lightest mass to the heaviest mass.
The two-thirds value is exactly midway between the extremes of the values that Q can take for any three real valued constants. The lowest possible value of Q, in which all three have the same value is 1/3. The highest possible value of Q, in which two of the values are 0, is 1.
Thus, by this particular measure, the ratios of the charged leptons to each other is exactly equidistant between maximum equality and maximum inequality.
Note also, that Koide's formula, like LP & C is evaluated at the pole masses of the charged leptons, not at their running value at some other energy scale.
The Extended Koide Relationship For Quarks
A relationship similar to that of Koide's formula was noted in a pre-print for a 2011 paper:
Rodejohann, W.; Zhang, H. (2011). "Extension of an empirical charged lepton mass relation to the neutrino sector". Physics Letters B. 698 (2): 152–156. arXiv:1101.5525.
But, this wasn't included in the published version. It was then mentioned in a published version of a 2012 paper:
Cao, F. G. (2012). "Neutrino masses from lepton and quark mass relations and neutrino oscillations". Physical Review D. 85 (11): 113003. arXiv:1205.4068.
The relationship holds most tightly for the top-bottom-charm quark triple. For the current PDG values of those masses it is:
Rivero took this further and proposed an extension of Koide's formula, ultimately deriving the masses of the tau lepton and all of the quarks solely from the masses on the electron and the muon (the two most precisely known fermion masses). See Rivero, A. (2011). "A new Koide tuple: Strange-charm-bottom". arXiv:1111.7232 (I should also credit Carl Brannen whose analysis of extensions of Koide's rule in areas including neutrinos including a 2006 paper is also important to this whole line of thought.)
Rivero's extension leads to the original Koide formula mass for the tau lepton and the following quark masses, which are, in MeV:
top 173,263.947
bottom 4,197.67589
charm 1,359.56428
strange 92.274758
down 5.32
up 0.0356
This relationship is surprisingly good for just random chance, although it also have some glaring misses. The discrepancies between Rivero's Koide extension values and the PDG values, measured in sigmas of PDG measurement uncertainty (there is only negligible uncertainty in the Rivero Koide Extension calculations relative to the PDG measurement uncertainties) is:
top +0.66 sigma
bottom -0.04 sigma
charm +4.46 sigma
strange -1.22 sigma
down +2.2 sigma
up -14.5 sigma
Thus, it is very close to the values for the top, bottom strange and down quark masses, but is significantly off (although of the right order of magnitude) for the charm quark and up quark.
Rivero's Koide extension also matches the global LP & C hypothesis quite well, producing a sum of Yukawas of 0.9962, mostly because it predicts a higher top quark mass than the currently measured value.
Given the major discrepancies in the cases of the charm quark and up quark, Rivero's Koide extension is clearly not a correct description of physical reality, but it is also a remarkably good first order approximation.
My Approach To Adjusting Rivero's Extended Koide's Formula
In a few posts in 2013 and 2014 (here and here and here), I considered what kind of physical reality might cause Koide's formula and its extension to work reasonably well, and used that heuristic understanding to propose an adjustment to the formula, which I have since then, further refined.
I considered a mechanism by which both Koide's formula and some variation on Rivero's Extended Koide's formula might make sense. In that mechanism the Higgs vev sets the overall mass scale for the fundamental particles of the Standard Model. But, in that mechanism, the distribution of the total mass available for the fundamental particles of the Standard Model is developed dynamically through W boson interactions that dynamically balance the masses of particles the decay into a given particle, with the masses of particles that arise from decays of a given particle.
In the case of the charged leptons, where the relationship works best, this relationship is particularly simple and clean. Lepton universality (which is either true or nearly true) means that the probabilities of one lepton becoming another kind lepton (mass-energy conservation permitting) is always exactly the same. Also, the masses of the neutrinos is so small relative to the charged lepton masses (on the order of meV, while the charged lepton masses are on the order of MeVs or GeVs), that any tweak to the charged lepton masses caused by their interactions with the neutrino masses in W boson interactions can be safely ignored.
In the case of the quarks, however, there are multiple possible W boson transitions. A Koide triple for a quark captures some, but not all of the possible W boson transitions.
For example, the bottom quark can transition via a W boson interaction from and into a top quark, a charm quark and an up quark. The t-b-c quark triple, captures the impact of the bottom quark transition into or from a top quark or a charm quark, but it ignores the possibility that a bottom quark could transition into an up quark. But, this deviation shouldn't be huge. We can look to the PDG values of the entries in the CKM matrix that determines the relatively likelihood of different types of W boson mediated quark flavor changes to see that.
The probability of a W boson interaction which includes a bottom quark leading to a transition (mass-energy permitting) into or from a top quark or a charm quark is 99.89%, while the probability of a W boson interaction which includes a bottom quark leading to a transition into or from an up quark is 0.11%. Unsurprisingly, the t-b-c quark triple is the best fitting of the quark triples of all, and the extended Koide's formula prediction for the value of the bottom quark is a mere 0.04 sigma off from the measured value.
In contrast, suppose that you are trying to determine the value of the charm quark mass using the analogous b-c-s quark triple (which is a poor fit to Koide's magic 2/3rd value). The b-c-s quark triple considers charm quark transitions into and from bottom quarks and strange quarks, but it omits the probability of a charm quark transition from a charm quark to a down quark.
What does the CKM matrix tell us about how often a charm quark to down quark transition happens?
It turns out that this happens in 4.9% of W boson transitions into or out of charm quarks (mass-energy permitting), which is about 45 times as often as a W boson transition of a bottom quark unaccounted for in the t-b-c triple. Unsurprisingly, the charm quark estimate from the extended Koide's formula is off by 4.46 sigma.
How then could be adjust the extended Koide's formula to reflect that some extended Koide triples are further off than others?
One approach is to take the CKM probability of a W boson transition that is not considered in that triple, by the difference in mass (using the first order extended Koide fomula values) between the quark whose mass you are trying to determine and the mass of the quark whose transition is omitted. But, since W boson transitions actually take place only when they are mass-energy permitted, only the omitted transitions to lower masses and not the transitions to higher masses, should be considered.
Calculating The Adjustments
Calculating The Adjustments
If you do this, in the case of the top quark, you consider (1) the probability of a transition from a top quark to a strange quark times the difference between the top quark mass and the strange quark mass, plus (2) the probability of a transition from a top quark to a down quark times the difference between the top quark mass and the down quark mass. Both of these adjustments should reduce the expected mass of the top quark since the omitted transitions are a downward tug on the top quark mass. In the case of the top quark, this leads to an adjustment down of the extended Koide's formula mass of 257 MeV to 172,743 MeV.
In the case of the bottom quark, as noted above, only one path is omitted, so you consider the probability of a transition from a bottom quark to a down quark times the difference between the bottom quark mass and the down quark mass. In the case of the bottom quark, this leads to an adjustment down of the extended Koide's formula mass of 4.6 MeV to 4192.98 MeV.
In the case of the charm quark, as noted above, only one path is omitted, so you consider the probability of a transition from a charm quark to a down quark times the difference between the charm quark mass and the down quark mass. In the case of the charm quark, this leads to an adjustment down of the extended Koide's formula mass of 66.36 MeV to 1293.21 MeV.
None of the omitted W boson paths for the strange quark, the down quark or the up quark are mass-energy permitted, so they are not adjusted.
How Close To The Measured Values Are The Adjusted Extended Koide Formula Masses?
How Close To The Measured Values Are The Adjusted Extended Koide Formula Masses?
What are the adjusted extended Koide formula quark masses (which are still solely a function of the electron mass and the muon mass) in MeV and how do they compare to the more precise FLAG 19 masses for those quarks (except the top quark for which the PDG value is used) and the less precise PDG masses for those quarks?
Quark Type - Adjusted Extended Koide Mass - FLAG 19 Mass- PDG Mass (all in MeV)
top 172,743 v. 173,000 +/- 400 (-0.64 sigma) v. 173,000 +/- 400 (-0.64 sigma)
bottom 4192.98 v. 4,198 +/- 12 (-0.49 sigma) v. 4,180 +40-/-30 (+ 0.32 sigma)
charm 1293.21 v. 1,282 +/- 17 (+0.66 sigma) v. 1,275 +25/-35 (+ 0.73 sigma)
strange 92.274758 v. 93.12 +/- 0.69 (-1.22 sigma) v. 95+9/-3 (- 0.91 sigma)
down 5.32 v. 4.88 +/- 0.2 (+2.2 sigma) v. 4.7 + 0.5/-0.4 (+1.24 sigma)
up 0.0356 v. 2.5 +/- 0.17 (-14.5 sigma) v. 2.2 +0.5/-0.4 (- 5.4 sigma)
Thus, the adjusted extended Koide's formula is less than 1 sigma from the true value for the three heavy quarks, less than two sigma for the strange quark and just 2.2 sigma from the down quark, with just the electron mass and the muon mass and the CKM matrix as inputs.
Moreover, the amount the experimental uncertainty is actually underestimated for the three heavy quarks, because it omits the not insignificant uncertainty in the CKM matrix values used in the adjustment, so the statistical significance of the differences between the adjusted extended Koide's formula values and the measured values of the heavy quark masses is actually somewhat level than the already under 0.7 sigma values shown.
The fact that the direction of the error is different between FLAG 19 and PDG for the bottom quark, and that of 9 distinct comparisons of quarks other than the up quark, four are underestimates and five are overestimates all suggest that the adjusted extended Koide masses for these five types of quarks are very close to the true values.
This balancing of the direction of error also suggests that the overall scale of the Koide waterfall is almost exactly right, despite its rather mysterious source in setting a quark triple equal to three times the charge lepton triple in combined masses, which really had no obvious mechanism to explain it other than the fact that many quark decay calculations have to be tripled relative to their leptonic counterparts because there are three colors of quarks but leptons come in only one color.
Moreover, the amount the experimental uncertainty is actually underestimated for the three heavy quarks, because it omits the not insignificant uncertainty in the CKM matrix values used in the adjustment, so the statistical significance of the differences between the adjusted extended Koide's formula values and the measured values of the heavy quark masses is actually somewhat level than the already under 0.7 sigma values shown.
The fact that the direction of the error is different between FLAG 19 and PDG for the bottom quark, and that of 9 distinct comparisons of quarks other than the up quark, four are underestimates and five are overestimates all suggest that the adjusted extended Koide masses for these five types of quarks are very close to the true values.
This balancing of the direction of error also suggests that the overall scale of the Koide waterfall is almost exactly right, despite its rather mysterious source in setting a quark triple equal to three times the charge lepton triple in combined masses, which really had no obvious mechanism to explain it other than the fact that many quark decay calculations have to be tripled relative to their leptonic counterparts because there are three colors of quarks but leptons come in only one color.
So, the adjusted extended Koide formula for quarks is certainly close enough to reality to take seriously, although, as the gross inaccuracy in the up quark mass illustrates, this approach is still only a second order approximation and not the true physical reality, because the up quark value is clearly wrong and there needs to be, at least, a third order approximation to get the correct value.
Indeed, the performance of the LC &P relationship and the adjusted extended Koide formula compare favorable to the performance of orthodox and accepted QCD calculations, both in the precision of their fit to the data, and in the sense that there are a small number of unresolved anomalies where a straight forward application of the usual rules alone doesn't work. This is fitting because most of predicted quantities are used primarily in calculations that have a QCD component.
Indeed, the performance of the LC &P relationship and the adjusted extended Koide formula compare favorable to the performance of orthodox and accepted QCD calculations, both in the precision of their fit to the data, and in the sense that there are a small number of unresolved anomalies where a straight forward application of the usual rules alone doesn't work. This is fitting because most of predicted quantities are used primarily in calculations that have a QCD component.
Potential Improvements To The Formula
Simply using the adjusted extended Koide formula values to recalculate the light quark masses is probably not a sufficient adjustment, because although this would ease the tension in the down quark mass somewhat, it wouldn't resolve the gross disparity in the up quark mass.
My intuition is that the correct generalization of Koide's formula for quarks ought to relate four input quark masses and the corresponding CKM matrix entries relating each of those four quarks in it, and that doing this rather than using Koide triples adjusted by a missing element, would make all of the estimates slightly closer to the physical one and would fix the problem with the underestimated up quark mass.
For example, perhaps the up quark mass is being tugged up by the omitted up quark-bottom quark connection, but by less than the 4.6 MeV adjustment made to the bottom quark mass as a result of that omitted connection. Or, perhaps the up quark should borrow more of its mass from the down quark than it would in any other position because it is at the end of the chain of quarks from up to top, or something like that. But, improving the adjustments is a question for another day.
For example, perhaps the up quark mass is being tugged up by the omitted up quark-bottom quark connection, but by less than the 4.6 MeV adjustment made to the bottom quark mass as a result of that omitted connection. Or, perhaps the up quark should borrow more of its mass from the down quark than it would in any other position because it is at the end of the chain of quarks from up to top, or something like that. But, improving the adjustments is a question for another day.
2 comments:
There are five paths by which a top quark can decay to an up quark, in order of likelihood (in the absence of injected energy):
t-b-c-s-u 57.4% of the time
t-b-u 39.3% of the time
t-b-c-d-u 2.97%% of the time
t-s-u 0.16% of the time
t-d-u 0.00745% of the time.
(may not add to 100% due to rounding errors).
All decay paths for all quarks are on one or more of those five paths.
The extended Koide fit for the top, bottom and charm quarks is very good .
The strange is low by 0.855 MeV, the down is high by .44 MeV, and the up is low by 2.46 MeV.
The up is missing an up-bottom component, the down is missing a top-down component, the strange is missing a top-strange component. Could it be that these contributions are not absent but highly suppressed somehow? A suppressed adjustment could solve the strange and up disparities.
The suppressed top-down disparity goes the wrong way, however, and this is enhanced further by the effect of slightly higher adjusted up quark mass, but it gets tugged down by the more direct and stronger tug down on the charm quark mass. So, iteration may solve the down quark mass problem and suppressed but non-zero omitted anti-decay paths could solve the strange and up shortfalls.
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