“Koide formula”的意思、由来-开放百科全书

The Koide formula is an unexplained empirical equation discovered by Yoshio Koide in 1981. In its original form, it relates the masses of the three charged leptons; later authors have extended the relation to neutrinos, quarks, and other families of particles.

Formula

where the masses of the electron, muon, and tau are measured respectively as m_e = {{val|0.510998946|(3)|ul=MeV/c2}}, m_μ = {{val|105.6583745|(24)|u=MeV/c2}}, and m_τ = {{val|1776.86|(12)|u=MeV/c2}}, and the digits in parentheses are the uncertainties in the last figures.^[1] This gives Q = {{val|0.666661|(7)}}.^[2]

It is clear that {{nowrap|{{frac|1|3}} < Q < 1}}. The superior bound follows if we assume that the square roots cannot be negative. By Cauchy–Schwarz inequality, the value

can be interpreted as the squared cosine of the angle between the vector

and the vector

(see dot product).

The mystery is in the physical value. Not only is this result odd in that three apparently random numbers should give a simple fraction, but also that Q is exactly halfway between the two extremes of {{frac|1|3}} (should the three masses be equal) and 1 (should one mass dominate).

While the original formula appeared in the context of preon models, other ways have been found to produce it (both by Sumino and by Koide, see references below). As a whole, however, understanding remains incomplete. Similar matches have been found for triplets of quarks depending on running masses.^[3]^[4]^[5] With alternating quarks, chaining Koide equations for consecutive triplets, it is possible to reach a result of 173.263947(6) GeV for the mass of the top quark.^[6]

Similar formulae

Quark masses depend on the energy scale used to measure them, which makes an analysis more complicated.

Taking the heaviest three quarks, charm (1.275 ± 0.03 GeV), bottom (4.180 ± 0.04 GeV) and top (173.0 ± 0.40 GeV), and without using their uncertainties gives the value cited by Cao (2012),^[7]

This was noticed by Rodejohann and Zhang in the first version of their 2011 article^[7] but the observation was removed in the published version,^[8] so the first published mention is in 2012 from F. G. Cao.^[9]

The experimental uncertainties of the top quark's mass, known to less decimal places than the other two, make the value of Q uncertain.

Similarly, the masses of the lightest quarks, up (2.2 ± 0.4 MeV), down (4.7 ± 0.3 MeV), and strange (95.0 ± 4.0 MeV), without using their experimental uncertainties yield,

a value also cited by Cao in the same paper.^[9] Note that {{frac|5|9}} is exactly midway between {{frac|1|9}} and 1. However, definite conclusions cannot be made unless experimental measurements are improved to more decimal places.

Running of particle masses

In quantum field theory, quantities like coupling constant and mass "run" with the energy scale. That is, their value depends on the energy scale at which the observation occurs, in a way described by a renormalization group equation (RGE).^[10] One usually expects relationships between such quantities to be simple at high energies (where some symmetry is unbroken) but not at low energies, where the RG flow will have produced complicated deviations from the high-energy relation. The Koide relation is exact (within experimental error) for the pole masses, which are low-energy quantities defined at different energy scales. For this reason, many physicists regard the relation as "numerology" (e.g.^[11]). However, the Japanese physicist Yukinari Sumino has constructed an effective field theory in which a new gauge symmetry causes the pole masses to exactly satisfy the relation.^[12] Goffinet's doctoral thesis gives a discussion on pole masses and how the Koide formula can be reformulated without taking the square roots of masses.^[13]

See also

References

1. ^{{cite journal |last1=Amsler|first1=C. |display-authors=etal |collaboration=Particle Data Group |title=Review of Particle Physics|journal=Physics Letters B |volume=667 |issue=1–5 |year=2008 |pages=1–6 |doi=10.1016/j.physletb.2008.07.018 |bibcode=2008PhLB..667....1A}}
2. ^Since the uncertainties in m_e and m_μ are much smaller than that in m_τ, the uncertainty in Q was calculated as

.
3. ^{{cite journal |last1=Rodejohann |first1=W. |last2=Zhang |first2=H. |year=2011 |title=Extension of an empirical charged lepton mass relation to the neutrino sector |journal=Physics Letters B |volume=698 |issue=2 |pages=152–156 |arxiv=1101.5525 |bibcode=2011PhLB..698..152R |doi=10.1016/j.physletb.2011.03.007}}
4. ^{{cite journal |last=Rosen |first=G. |year=2007 |title=Heuristic development of a Dirac-Goldhaber model for lepton and quark structure |url=http://home.comcast.net/~gerald-rosen/heuristicmpla.pdf |journal=Modern Physics Letters A |volume=22 |issue=4 |pages=283–288 |bibcode=2007MPLA...22..283R |doi=10.1142/S0217732307022621}}
5. ^{{cite arXiv |last=Kartavtsev |first=A. |year=2011 |title=A remark on the Koide relation for quarks |eprint=1111.0480 |class=hep-ph}}
6. ^{{cite arXiv |last1=Rivero |first1=A. |year=2011 |title=A new Koide tuple: Strange-charm-bottom |eprint=1111.7232 |class=hep-ph}}
7. ^{{cite arXiv |last1=Rodejohann |first1=W. |last2=Zhang |first2=H. |year=2011 |title=Extension of an empirical charged lepton mass relation to the neutrino sector |arxiv=1101.5525 |class=hep-ph}}
8. ^{{cite journal |last1=Rodejohann |first1=W. |last2=Zhang |first2=H. |year=2011 |title=Extension of an empirical charged lepton mass relation to the neutrino sector |journal=Physics Letters B |volume=698 |issue=2 |pages=152–156 |arxiv=1101.5525 |bibcode=2011PhLB..698..152R |doi=10.1016/j.physletb.2011.03.007}}
9. ^¹²{{cite journal |last1=Cao |first1=F. G. |year=2012 |title=Neutrino masses from lepton and quark mass relations and neutrino oscillations |journal=Physical Review D |volume=85 |issue=11 |page=113003 |arxiv=1205.4068 |bibcode=2012PhRvD..85k3003C|doi=10.1103/PhysRevD.85.113003}}
10. ^Green, D., Cosmology with MATLAB (Singapore: World Scientific, 2016), [https://books.google.com/books?id=7mhIDQAAQBAJ&pg=PA197 p. 197].
11. ^{{cite web |last=Motl |first=L. |date=16 January 2012 |title=Could the Koide formula be real? |url=http://motls.blogspot.com/2012/01/could-koide-formula-be-real.html |work=The Reference Frame |accessdate=2014-07-10}}
12. ^{{cite journal |last=Sumino |first=Y. |year=2009 |title=Family Gauge Symmetry as an Origin of Koide's Mass Formula and Charged Lepton Spectrum |journal=Journal of High Energy Physics |volume=2009 |issue=5 |pages=75 |arxiv=0812.2103 |bibcode=2009JHEP...05..075S |doi=10.1088/1126-6708/2009/05/075}}
13. ^{{cite thesis |last=Goffinet |first=F. |year=2008 |title=A bottom-up approach to fermion masses |url=https://dial.uclouvain.be/pr/boreal/object/boreal:20873 |type=PhD Thesis |publisher=Université catholique de Louvain}}

Formula

Similar formulae

Running of particle masses

See also

References

Further reading

External links