The Koide Formula and Geometry of Circles

Jerzy Kocik Department of Mathematics, Southern Illinois University, Carbondale, IL 62901∗

A remarkable formal similarity between Koide’s Lepton mass formula and a generalized Descartes circle formula is reported. Keywords: Lepton mass, , Descartes circle theorem, geometry.

I. INTRODUCTION formula. Descartes – in his 1654 letter to the princess of Bohemia, Elizabeth II – showed that the curvatures and are believed to be the fundamental of four mutually tangent circles (reciprocal of radii), say particles of matter yet their nature is still far from being a,b,c,d, satisfy the following “Descartes’s formula”: understood. One of the exciting puzzles is a formula Theorem II.1 (Descartes’s Circle Formula, 1654 [2]). involving the of the three leptons, discovered by The curvatures of four circles in Descartes’s configura- [6, 7]: tion satisfy this equation: 2 √ √ √ m + m + m = ( m + m + m )2. (I.1) 2 2 2 2 2 e µ τ 3 e µ τ (a + b + c + d) = 2(a + b + c + d ) (II.1)

See Table 1 for the corresponding numerical values (from where a = 1/r1, b = 1/r2, etc., denote reciprocals of [10]). 2. Fromradii Descartes called to bendsKoide , which are signed curvatures. In Fig. 1 in the middle, circle D is a boundary of an unbounded 2 In this note we want to call attention to a curious formal similarity of Koide’s formula to me = 0.510998910 MeV/c disk (region outside D) hence its bend is negative. Circle DescartesD circlein the formula. right figureDescartes has – in bend his letter equal to the zero. princess of Bohemia, Elizabeth = 1 me √ II, showed in 1645 that the curvatures of four mutually tangent circles (reciprocal of me = 1 radii), say a,b,c,d, satisfy the following “Descartes formula”:

2 mµ = 105.658367 MeV/c B A A D = 206.768282 me D √ 1/2 C C mµ = 14.37943957 me B A C 2 B mτ = 1776.84 MeV/c

= 3477.1894 me √ mτ = 58.97 FIG.Figure 1: Four 1: Four circle circles in various in various Descartes Descartes configurations.configurations

TABLE I: Lepton masses. Theorem 1 (Descartes Circle Formula, 1654 [Des]): The curvatures of four circles in Descartes Theconfiguration similarity satisfy of this Descartes’s equation: circle formula to Koide’s

Quite remarkably, Koide used his formula to predict lepton mass formula is striking:2 2 represent2 2 2 masses as re- (a + b + c + d) = 2 (a + b + c + d ) (2) the mass of the tau lepton with surprising accuracy: ciprocals of disk areas, equivalently, their curvatures as were asquare=1/r1, b=1/ rootsr2, etc., of denote masses, reciprocals and we of getradii(II.1) called bends except, which for are a signed old measured mτ = 1784 ± 4 [1970’s] curvaturesdifferent. In Figure coefficient 1 in the middle, for the circle sum D of is squares:a boundary 2 of instead an unbounded of disk predicted by Koide m = 1776.97 [1982] (region2/3. outside The D) hence fourth its bend term is maynegative. be Circle taken Das in the zero, rightd figure= 0, has cor- bend equal τ zero. new measured mτ = 1776.99 ± .3 [2002] responding geometrically to a line as a special circle. The similarityThe factor of Descartes of 2/3 circle invalidates formula to the Koide’s exact lepton correspondence mass formula is striking: newer measured mτ = 1776.84 ± .17 [2011] representwith masses Descartes’s as reciprocals formula. of disk areas, However, equivalently the their present curvatures author as square roots (all in MeV/c2). Some interesting formal associationsof masses,has and found we get a formula(2) except forfor a circles different in coefficient general for position the sum thatof squares: 2 insteadgeneralizes of 2/3. The fourth Descartes’s. term may be taken as zero, d =0, corresponding geometrically have been noticed since [1, 4, 9, 11], but after almost 30to a line as a special circle. arXiv:1201.2067v1 [physics.gen-ph] 5 Jan 2012 years the consensus is that the “mystery of the lepton mass formula” [8] remains unsolved. TheTheorem factor of 2/3 II.2 invalidates(General the exact Circle correspondence Formula [5]) with. TheDescartes’ radii formula. However,of nthe+ present 2(n − author1)-spheres has foundC1 ,...,Ca formulan+2 forin circles general in general position position that generalizesin R Descartes’.n satisfy a quadratic equation

II. FROM DESCARTES TO KOIDE Theorem 2 (General Circle FormulaBFB [jk]):T The= 0 radii of n+2 (n–1)-spheres (II.2) C1,…, Cn+2 in general position in Rn satisfy a quadratic equation

In this note we want to call attention to a curious for- or simply T (3) mal similarity of Koide’s formula to Descartes’s circle BFB = 0 or simply FijFbiji bbijbj = 0 0

where Bwhere = [b1,…,Bb=n] is [b the1, .row . . , matrix bn] is (covector) the row of matrix the curvatures (covector) of the ofspheres the and F = –1 ∗Electronic address: [email protected] f is thecurvatures inverse of the of “configuration the spheres matrix” and Ff with= entriesf −1 is defined theinverse by of

(dij – ri – rj)/2rirj in general (4) fij = cos , if Ci and Cj intersect

th th th Here dij = distance between the centers of i and j spheres, ri = radius of the i sphere. For spheres that intersect,  is the angle of their intersection.

2

Consider the special case where the product for every pair is the same, say p. Then one calculates the inverse of the corresponding matrix f:

1 p p p  1  2 p p p p      p 1 p p p 1  2 p p p f =    F = 1    p p 1 p  3p2  2 p 1  p p 1  2 p p     p p p 1      p p p 1  2 p

Solving (3) gives the following geometric result:

Proposition: Four circles of curvatures a, b, c, d, respectively, intersecting pairwise at the angle  = arcos p satisfy the quadratic equation

p a 2  b 2  c 2  d 2  (a  b  c  d) 2 (5) 3p 1

Consider the special case where the product for every pair is the same, say p. Then one calculates the inverse of the corresponding matrix f: For circles mutually tangent we have p = 1 and the above formula reduces to Descartes formula. We are however interested in the value 2/3 for the coefficient on the 1 p p p  1  2 p p p p      p 1 p p right side. p 1  2 p p p f =    F = 1    p p 1 p  3p2  2 p 1  p p 1  2 p p     p p p 1 Corollary: Koide’s lepton mass formula may be interpreted as a Descartes-like circle    p p p 1  2 p formula (3) for p = 2/3, where the squared curvatures correspond to the masses of the Solving (3) gives the following geometricthree result: leptons. One mass is assumed zero and corresponds to a straight line. Each pair of circles intersects under the same angle: Proposition: Four circles of curvatures a, b, c, d, respectively, intersecting pairwise at the angle  = arcos p satisfy the quadratic equation  = arccos(2/3)  0.841 rad  15/56   48.2

p a 2  b 2  c 2  dInterestingly,2  ( athe b radii, c  d reciprocals) 2 of square(5) roots of masses, are close to integer values, 3p 1 namely, r = 1, r  4.10, re  58.97, but not close enough to warrant further attention.

For circles mutually tangent we have p = 1 and the above formula reduces to 2 Descartes formula. We are however interested in the value 2/3 for the coefficient on the electron the “configuration matrix” f with entriesright side. defined by electron ( Corollary: Koide’s lepton mass formula may be interpreted as a Descartes-like circle (dij − ri − rj)/2rirj in generalformula (3) for p = 2/3, where the squared curvatures correspond to the masses of the fij = three leptons. One mass is assumed zero and corresponds to a straight line. Each pair of cos ϕ, if C and C intersect. muon i circles intersectsj under the same angle: (II.3)  = arccos(2/3)  0.841 rad  15/56   48.2 th Here dij = distance between the centers of i and th th Interestingly, the radii, reciprocals of square roots of masses, are close to integermuon values, j spheres, ri = radius of the i sphere. For spheres namely, r = 1, r  4.10, re  58.97, but not close enough to warrant further attention. that intersect, ϕ is the angle of their intersection. (Quite tau tau interestingly, the above theorem can be derived from the fact that circles in the Euclidean plane may be regarded electron electron as vectors in the Minkowski space [5].) Consider the special case where the product for every Figure 3: Lepton masses via circle configuration (on the left a detail) pair is the same, say p. Then one calculates the inverse muon of the corresponding matrix f:

  −1 p p p muon  p −1 p p    f =   tau tau  p p −1 p  p p p −1 3

  FIG. 2: Lepton masses via circle configuration (detail on bot- 1 − 2 p pFigure p 3: Lepton massestom). via circle configuration (on the left a detail)

1  p 1 − 2 p p  ⇒ F =   3p2 + 2p − 1  p p 1 − 2 p    that may be a candidate for geometry of generalized p p p 1 − 2 Koide formula: Solving (II.2) gives the following geometric result: √ √ √ 2 m1 + m2 + m3 + ... = κ · ( m1 + m2 + m3 3 + ...) . Proposition II.1. Four circles of curvatures a, b, c, The relation is thus d, respectively, intersecting pairwise at the angle ϕ = arccos p satisfy the quadratic equation 1 2 2 p p = and κ = ⇒ p = . (a2 + b2 + c2 + d2) = (a + b + c + d)2. (II.4) n + 1 − 1/κ 3 2n − 1 3p − 1 (II.6) For circles mutually tangent we have p = 1 and the above formula reduces to Descartes’s formula. We are, III. CONCLUSIONS however, interested in the value 2/3 for the coefficient on the right side. We conclude with some remarks: Corollary II.1. Koide’s lepton mass formula may be interpreted as a Descartes-like circle formula (II.2) for 1. Leptons and segments. If one considers three seg- p = 2/3, where the squared curvatures correspond to the ments on a line rather than circles in plane, formula masses of the three leptons. One mass is assumed zero (II.2) with n = 1, becomes and corresponds to a straight line. Each pair of circles p a2 + b2 + c2 = (a + b + c)2. intersects under the same angle: 2p − 1 ϕ = arccos(2/3) ≈ 0.841 rad ≈ 15/56 π ≈ 48.2◦. To make the coefficient in Koide’s formula equal κ = 2/3 we would need p = 2. The segments Interestingly, the radii, reciprocals of square roots of for e, µ, τ would be (−1.40, 1.40), (−1.49, 1.68), masses, are close to integer values, namely, rτ = 1, (−1.42, 1.47), respectively. rµ ≈ 4.10, re ≈ 58.97, but not close enough to warrant further attention. 2. . Recently, Brannen observed [1] that the same pattern is followed by the masses Since the Koide formula might be applicable for other under the condition that the square root of the first particle families, let us provide a solution to the configu- mass is taken to be negative. ration of n + 2 spheres in n-dimensional space: With his data, m1 = 0.000388eV , m2 = p 0.00895eV , m = 0.0507eV , the factor of 2/3 is ob- b2+b2+...+b2 = (b +...+b )2 (II.5) 3 1 2 n+2 (n + 1)p − 1 1 n+2 tained with the very good precision of 0.01% and Since the Koide’s formula might be applicable for other particle families, let us provide a solution to the configuration of n+2 spheres in n-dimensional space:

p b 2  b 2  ...  b 2  (b  ...  b ) 2 (5) 1 2 n2 (n 1) p 1 1 n2 that may be a candidate for geometry of generalized Koide formula:

m  m Since m the Koide’s...  formula ( m might bem applicable m  for...) other2 particle families, let us 1provide2 a solution3 to the configuration1 of n2+2 spheres3 in n-dimensional space:

The relation is thus 2 2 2 p 2 b1  b2  ...  bn2  (b1  ...  bn2 ) (5) 1 2 (n 1) p 1 2 p  and   3  p  (6) thatn may1  be1/ a candidate for geometry of generalized Koide2n  formula:1

2 3. Conclusions m1  m2  m3  ...   ( m1  m2  m3  ...)

We conclude with someThe relationremarks: is thus 1 2 p  and   2  p  (6) 1. Leptons and segments. If one considersn 1 three1/  segments on a line3 rather than2n  circles1 in plane, formula (3) with n=1, becomes 3. Conclusions 2 2 2 p 2 We concludea  b with somec remarks: (a  b  c) 2 p 1 1. Leptons and segments. If one considers three segments on a line rather than circles in To make the coefficientplane, informula Koide’s (3) with formula n=1, becomes equal  = 2/3 we would need p = 2. The p segments for e, ,  would be (–1.40, 1.40),a 2 (1.49, b 2  1.68),c 2  (1.42, (1.47),a  b  respectively.c) 2 2 p 1 2. Neutrinos. Recently, Brannen observed [Bra] that the same pattern is followed by the neutrino masses underTo themake condition the coefficient that thein Koide’s square formula root of equal the first = 2/3 mass we iswould taken need to pbe = 2. The segments for e, ,  would be (–1.40, 1.40), (1.49, 1.68), (1.42, 1.47), respectively. negative. 2. Neutrinos. Recently, Brannen observed [Bra] that the same pattern is followed by the

neutrino1 = 0.000388 masses MeV under =the 1 condition1 that the square 1 =root –1 of the first mass is taken to be negative. 2 = 0.00895 MeV = 23.06701 1 2 = 4.8028

 3 = 0. 0507 MeV = 130.6701 1 3 = 11.4311 1 = 0.000388 MeV = 1 1 1 = –1 3  = 0.00895 MeV = 23.06701   = 4.8028 Table2 2: Neutrino masses 1 2 3 = 0. 0507 MeV = 130.6701 1 3 = 11.4311 the resulting equation be that of six 4-spheres in a five-dimensional space The factor of 2/3 is obtained with the very goodTable precision2: Neutrino massesof 0.01%. The resulting with p = 2/9, giving an intersection angle of ϕ ≈ equation 2 √ √ √ 2 (3/7)π ≈ 77.2◦. We note that the precision grows The factorm of1 +2/3m is2 obtained+ m3 = with( −the verym1 +good mprecision2 + mof 30.01%.) The resulting 2 3 2 rapidly to 0.1% when only the last three quarks, c, equationm1 + m2 + m3 = (–m1 + m2 + m3) (7) 3 2 2 b and t, are considered. poses no problemm1 + m2 + to m3 our= (– geometricm1 + m2 + interpretation, m3) (7) 3 poses no problem to our geometricas negative interpretati curvatureon, as isnegative associated curvature to unboundedis associated to √ unbounded disk definedposes by no the diskproblem region defined to outsideour geometric by a the circle. regioninterpretati outsideon, as negative a circle. curvature is associated to up mu = 1mu mu = 1 unbounded disk defined by the region outside a circle. √ down md = 12mu md = 3.464 √ strange ms = 210mu ms = 14.491 √ charm mc = 2500mu mc = 50 √ bottom m = 9000mu m = 94.868 b √ b top mt = 348000mu mt = 589.915

TABLE II: masses

Figure 2: Neutrino masses via disk configuration (on√ left a detail) Figure 2: NeutrinoFIG. masses 3: Disk via disk configuration configuration for a(on negative left a detail)m. 4. Coda. Whether this intriguing connection with ge- 4 But estimations of neutrino masses change rapidly ometry will contribute to an understanding of the 4 with new experiments and only upper bounds are masses of leptons remains an interesting question measured. Most recent estimations are: electron and requires further investigation. The analogy de- neutrino: m < 2.2eV , muon neutrino: m < 170 scribed above may turn out to be merely superfi- keV, and tau neutrino: m < 15.5 MeV [3]. Thus, cial, but given the current state of understanding taking the highest value for the muon neutrino about the matter, any interesting structural paral- would suggest a rather low value of 2.4 MeV for lels are worthy of our consideration in the effort of the tau neutrino if Koide’s formula with ratio 2/3 reconstructing deeper patterns. were to be satisfied. 3. Quarks. Quark mass estimations taken from [11] Acknowledgements are presented in the table below. An extended ver- sion of Koide’s equation would suggest 2/3 with I am grateful to Philip Feinsilver for his encouraging a precision of 5% [11]. The geometry here would interest and priceless comments.

[1] C.A. Brannen, The Lepton Masses, http://brannenworks and Leptons and Cabibbo Mixing, Lett. Nuovo Cimento .com/MASSES2.pdf, (2006). 34 (1982) 201. [2] R. Descartes, Oeuvres de Descartes, Correspondence IV, [7] Y. Koide, Mod. Phys. Lett. A5 (1990) 2319. (C. Adam and P. Tannery, Eds.), Paris: Leopold Cerf [8] Y. Koide, Challenge to the Mystery of the Charged Lep- 1901. ton Mass Formula (2005), hep-ph/0506247. [3] CUPP, ”Laboratory measurements and limits for neu- [9] Nan Li and Bo-Qiang Ma, Estimate of neutrino masses trino properties” at http://cupp.oulu.fi/neutrino/ from Koides relation, Phys. Lett. B 609 (2005) 309, nd-mass.html a web page maintained by the Center for hepph/0505028. Underground Physics in Pyh¨asalmi. [10] K. Nakamura, The Review of Particle Physics, avail- [4] J. M. G´erard,F. Goffinet, and M. Herquet, A New Look able from at the Internet site at an Old Mass Relation, Phys. Lett. B 633 (2006) 563- http://pdg.lbl.gov/index.html as http://pdg.lbl. 566. hep-ph/0510289. gov/2010/tables/rpp2010-sum-leptons.pdf [5] J. Kocik, A matrix theorem on circle configuration [11] A. Rivero and A. Gsponer, The strange formula of Dr. (arXiv:0706.0372v2). Koide, arXiv:hep-ph/0505220v1 (2005). [6] Y. Koide, Fermion- Two Body Model of Quarks