Letters in Mathematical Physics 15 (1988) 165-170. 9 1988 by Kluwer Academic Publishers.
165
On the Mass Spectrum of Elementary Particles J. K. K O W A L C Z Y / q S K I Institute of Physics, Polish Academy of Sciences, Al. Lotnikbw 32/46, 02-668 Warsaw, Poland (Received: 20 November 1987) Abstract. A simple theory of the elementary particle mass spectrum is proposed. It originates from the Dtrac idea of the free electron motion and from the transformed Klein-Gordon equation. The theory is based on an equation that includes the squared mass operator having an infinite sequence of orthogonal eigenfunctions and a discrete spectrum ofeigenvalues. A discrete mass formula is derived. It yields values of mass that are in agreement with present-day empiric data for elementaly particles.
1. Introduction The mass spectrum of elementary particles is a puzzle. This Letter attempts to approach this problem. A simple theory is given in Section 3. It is preceded by its heuristic premises presented in Section 2. The theory is verified and discussed in Section 4, where it is shown that it is successful within the framework of present-day empiric data.
2. Heuristic Premises All our considerations concern free particles. One of the known peculiarities of the Dirac electron is that its velocity operator has only two eigenvalues + c [ 1]. Together with other specific properties of the spinorial wave, this makes a classical description of the Dirac particle extremely difficult. If one insists, however, on giving such a description, then one may imagine a rapidly oscillating particle [ 1] having, e.g., a periodic spiral-like trajectory determined by a classical motion equation including terms periodic in time. This kind of trajectory could only represent the classical rectilinear free motion as an average (cf. [1]). Such a picture, though somewhat vague, is automatically contained in the Dirac equation and, consequently, in the Klein-Gordon equation, which, as yet, is obligatory for every free particle. As a free extension of the Dirac idea, it is therefore tempting to introduce an equation including terms periodic in time and being a modification of the Klein-Gordon equation. The slower-than-light free particle has a straight time-like world line in terms of relativity, and is represented by a plane harmonic de Broglie wave in terms of quantum mechanics. Thus, after applying the appropriate Lorentz transformation, the particle can be described by expressions with only one variable. The particle proper time s (affine parameter of the world line) is the most natural choice. Then the Klein-Gordon
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equation reduces to h2 d2
C4 ds 2 ~r
= m 2 O(s).
(2.1)
If we observe the particle at rest, then s is, of course, the laboratory time. Thus, by Equation (2.1), operator - c - 4 h 2 d 2 / d s 2 could be considered as a squared rest mass operator. However, its spectrum of eigenvalues is continuous, while the elementary particle masses form a discrete set of values. Basing upon our previous considerations, we can assume that this operator is only a zero approximation, being too coarse to perceive the frequent oscillations of the particle. Figuratively, we can compare this operator with a vehicle running on big wheels that do not feel the small ruggednesses of the road surface. The ruggednesses will be felt if the vehicle is provided with sufficiently small wheels. We assume that to obtain a discrete spectrum of the mass eigenvalues, the operator should be generalized so as to make it adequately sensitive to the periodic oscillations of the particle. In mathematical terms, we assume that such a sensitivity can be realized as the operator singularities, which can be understood as a mathematical exaggeration frequent in descriptions of nature. Thus, we shall generalize the operator - c - 4 h 2 d2/ds 2 by adding some terms that are periodically singular with period T' such that the ratio TIT' is a natural number, and where T is the period (in s) of the hypothetical particle oscillations. In Section 3 we shall assume TIT' = 2. Though our considerations consist of free associations, the equation postulated in Section 3 gives the mass values of the elementary particles in unexpected agreement with present-day empiric data. Thus, it is not excluded that this equation is the first step to solving the puzzle of elementary particle masses by periodic oscillations of the free particle in vacuum.
3. Squared Mass Operator and the Mass Formula Let us consider the equation
h2 [
C-4
d2
d
g 2 + COtan(coS) - - +
c020~2 2(1 - sinco
c02/~2 )
+
]
2(1- mco )_l ~k= m=@,
(3.1)
where - oc < s < + 0% ~ and/~ are dimensionless real constants, and co is a constant frequency such that
co = 2~/r.
(3.2)
Thus, on the left-hand side of Equation (3.I) we have a new squared mass operator with eigenfunctions ~ and eigenvalues m 2. Equation(3.1) is a generalization of Equation (2.i) in agreement with what has been said in Section 2, sincethe terms added to the operator -h2c-4de/ds2 are periodically singular. Of course, its concrete form is arbitrarilypostulated. The spectrum of the elementary particle rest masses m is discrete. Our operator too
THE MASS SPECTRUM OF ELEMENTARY PARTICLES
has a discrete spectrum of eigenvalues following eigenfunctions:
m 2
(for given ct and fl) corresponding to the
= ~(,~'~)(s)'= (1 - x ) :'/2 (1 + x ) t ~ / 2 P ~ " t ~ ) ( x ) , x : - sin cos,
167
(3.3a) (Y3b)
where n = 0, 1, 2 , . . . and P ~ ' a ) ( x ) are the n-degree Jacobi polynomials. Definition (3.3b) means that - 1 ~ x ~< 1. F r o m Equations (3.1)-(3.3) we get rn = M [ ( n
+ 7) (n + 7 + 1 ) ] 1/2 ,
(3.4a)
M : = hco/c 2 = h / c 2 T ,
(3.4b)
7"= 89 + fl),
(3.4c)
which is our mass formula for the elementary particles. Functions ~b given by Equation (3.3) are periodic in s, with the period T given by Equation (3.2), since they are algebraic in terms of the auxiliary variable x. They form a sequence of orthogonal functions (of x) numbered by n, for every given :r and fl such that e > - 1 and fl > - 1 [2]. (Here we confine ourselves to the usual Jacobi polynomials for which c~> - 1 and fl > - 1.) O f course, we can use their normalized version, since Equation (3.1) is homogeneous. (The periodicity tempts to assume an alternative (but compatible) hypothesis that the rest m a s s of the elementary particles is a quantity periodic in s. The constant rest mass measured in experiment, and the constant m from Equation (3.1), would then be an average of this quantity.) We assume that 7 < 1.
(3.5)
This restriction seems to be in agreement with c o m m o n sense, since the integer part of 7/> 1 can be absorbed by n in formula (3.4a). (Note that 7 > - 1 because ~ > - 1 and fl > - 1, and additionally, by Equations (3.4), we should have 7 > 0 in the case n = 0.) It seems necessary to assume that the squares of our functions qJ have everywhere the continuous derivative with respect to s, i.e. to assume that functions qj2(s) belong to class C ~ at every s ~ ( - oo, + oo). This assumption is equivalent to the following alternation of conjunctions: c~ = fl = 0 (then our functions ~O(x) are the Legendre 1 1 1 1 polynomials) or c~ = 0 and fl>/~ or ~>~i and ]3 = 0 or ~ > ~ and fl>/~. Then the following two relations hold: either7=0orT>/~; if~<~ 7 < ~ (which is the case considered in Section 4) then either c~= 0 or fl = 0.
4. Verification The theory presented in Section 3 gives formula (3.4) but it does not determine the exact values of M and 7. Therefore, to verify the theory, we have to check whether such M
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J. K. KOWALCZYIqSKI
and 7 exist for which formula (3.4a) gives values o f m within the empiric error intervals (for appropriate n's, see comments at the end of this section). Thus, in our case the verification is a phenomenological step. Let the mass rn calculated from Equation (4.3a) be m ca~, that obtained from experiments m emp, and the + empiric error o f m emp be a positive quantity Am, i.e. the length of the empiric error interval is equal to 2Am. We shall mostly be using the empiric data from [3]. We shall use the Am values from pp. 11-20 (repeated on pp. 95-177; all page numbers in this paragraph refer to [3]), where they are said to correspond to a 9 0 ~ confidence level. (Note that there is no universal convention for the term 'confidence level', pp. 52 and 53.) The term 'error' is introduced on p. 6 in the meaning of one standard deviation (see also footnote on p. 36). However, the uncertainties of the electron and proton masses have the same values on pp. 11 and 16 (90~o confidence level) and on p. 36 (one standard deviation); cf. also p. 53. If we assume M comparable with the smallest intervals 2Am, then we obtain of course some 'verification' of formula (3.4a), but this would be a worthless arithmetical trick. We can speak of verification i f M is considerably greater than these intervals. Such an M was found with the help of a computer, M _ 0.025866642(51) MeV/c 2, where the figures in parentheses are the conventional presentation of the + uncertainty in the last two digits. This uncertainty is due to Am's of the particles listed in Table I. Our M gives T --- 1.5988551(52) • 10- 19 sec by Equations (3.4b) (the uncertainty of h (MeV sec, p. 36 of Ref. [3]) is also taken into account). Would this M (or T) be the fundamental constant having the dimension of mass (or of time)? Table I. Comparisonbetween the calculated and empiricmasses (the latter from [3]) of the particles having the smallest empiric error intervals. The n~ and n• cases are commented on in the text. Particle
n
mCal[MeV/c2]
memp(Am)[MeV/c2] 0.5110034(14) 105.65916(30) 134.9642(38) [134.9659(38)] 139.5685(10) [139.5702(11)] 938.2796(27) 939.5731(27)
e # no
19 4084 5217
0.511003471 105.659066 134.965972
n•
5395
139.570234
36273 36323
938.280406 939.573738
p n
Our M corresponds to V _~ 0.261631(39) (149 ppm). In this case, therefore, functions to class C 1 and there is either ~ = 0 or fl = 0 (see the last paragraph of Section 3). A strong verification of our M is the use of the difference between the neutron and proton masses, since this difference has been empirically determined with precision (dimensional) by two orders of magnitude greater than that of its components m nemp and
~bZ(s) belong
THE MASS SPECTRUM OF ELEMENTARYPARTICLES
169
mpmp. Namely, our M gives (m n - mp)cal ~ 1.2933321(26) MeV/c 2 (2 ppm) as compared with (ran - r a p ) e m p "~ 1.293323(16) MeV/c z (12 ppm) [3]. For calculating m~ we have assumed the middle value of our M, i.e. M ~ 0.025866642 M e V / c 2 ,
(4.1)
which gives ( m n - m p ) ca1 ~ 1.2933321 MeV/c 2. To obtain a more precise value of our ~ we can use the proton to the electron mass ratio. This ratio has been empirically determined with a precision much greater than that of its components /T/p m p and -"etqqemp" Unfortunately, there are several values of (mp/me) emp and some of them do not overlap (see p. 36 of Ref. [3], p. 23 of Ref. [4], p. $31 of Ref. [5], and footnotes therein). The older value, formally recommended, is 1836.15152(70) [ 3-5] and corresponds to 7 ~ 0.2616468(75) (29 ppm). However, we have chosen for three reasons one of the newer values (given in the just-mentioned footnotes in Refs. [3-5] where they are said to be better than that formally recommended), namely 1836.152701(100) [3] corresponding to 7 --- 0.2616341(11) (4 ppm). First, it is recent. Second, it overlaps that given in [4]. Third, its middle value is almost equal to the average of those newer values. Let us note that the differences between the values of 7, that are due to the differences between the values of (rap/me) emp under consideration, do not practically affect the values of m ~ for all the particles listed in Table I, except for the electron. In fact, because of the n magnitudes (see Table I), the differences between the values of a given m ca~, produced in this way are, by several orders of magnitude, smaller than the appropriate Am. Even in the electron case, the differences are smaller than ~1A m e . Since ? is a dimensionless quantity, it would be welcome from the quantum physics point of view if 7 were simply expressed by relatively small natural numbers. It appears that this is possible. We assume that 7 = 88
1/2 "~ 0.261634059,
(4.2)
which gives ( m p / m e ) cal ~_ 1836.1527068. Using M and 7 from Equations (4.1) and (4.2) in formula (3.4a) we obtain the m~a~'s presented in Table I. Excluding the electron case, we can use the simpler formula m ~ M ( n + ), + 1) instead of Equation (3.4a), since the resulting differences are by, at least two orders of magnitude, smaller than Am, because of the magnitudes of n. It is seen that mca~'s are in agreement with the recent fits [3] of memP'S within Am's, except for the rt + case. However, the recent fit of m emp and Am~+ given in Table I is burdened with the scale factor S (see pp. 6 - 9 and 20 of Ref. [3]) equal to 1.5 [3], which means that those values are unreliable due to the divergence of the empiric data ([3], p. 114). Such a situation is frequent among empiric data. The (rap~me) emp values considered in this section can serve as another example. The proton mass case is very striking (and shocking, since it deals with a stable and widespread object) because the difference between the older recommended value ofm/~ rnp [6] and the recent one [3-5] is four times greater than the larger Amp. This obliges us to be cautious when using empiric data, and to remember that the empiric error interval is only a statistical concept.
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J. K. KOWALCZYlqSKI
The
cal is 9 in agreement with the recent me~ p -~ 139.5704(11) MeV/c 2 [7] ~r. Since m,~_+ this value has been based on the older fit [5] of m emp it should be reduced to 139.5702(11) MeV/c 2 if the recent fit [3] of m ~ mp w e r e taken into account. The latter value of me•mp is inserted in Table I in square brackets. Consequently, the recent value [3] of m ~ p should be increased to 134.9659(38)MeV/c2, the recent value [3] of (mr~ • -- m n o ) emp being retained, is inserted in Table I (in square brackets). As regards all the massive particles, having their Am's determined, other than those listed in Table I, the memP'S ' v e r i f y ' , of course, Equations (3.4a), (4.1), and (4.2), since the present-day values of intervals 2Am exceed the value ofM. A real verification will be possible after we get suitably smaller Am's. It seems, however, that real verification would be possible now if a rule for n's were found. The set of numbers n in Table I is a puzzle, and there should exist a (simple?) formula determining n for all or some elementary particles. Such a formula would be a supplement to Equations (3.4), (4.1), and (4.2), but has not yet been found.
Acknowledgements I wish to thank Professor B. Mielnik for critical remarks and helpful discussions, and Dr L. Dobaczewski, Dr J. A. Majewski, J. Pawlak, and Mgr T. Wiszowaty for their indispensable help in my struggles with the computers. I also thank Drs E. J. van der Wolk from North-Holland Physics Publishing Division for supplying Ref. [3].
References Dirac, P. A. M., The Principles of Quantum Mechanics, The Clarendon Press, Oxford, 1958, w 69. SzegS, G., Orthogonal Polynomials, American Mathematical Society, New York, 1959. Particle Data Group, Review of particle properties, Phys. Lett. 17011 (1986). Particle Data Group, Review of particle properties, Phys. Lett. l l l B (1982). Particle Data Group, Review of particle properties, Rev. Mod. Phys. 56, No. 2, Part II (1984). Taylor, B. N., Parker, W. H., and Langenberg, D. N., Rev. Mod. Phys. 41,375 (1969), Table XXXIII on p. 479. 7. Abela, R., et aL, Phys. Lett. 1468, 431 (1984). 8. Marushenko, V. I., et al., Pis'ma Zh. Eksp. Teor. Fiz. (Soviet JETP Lett.) 23, 80 (1976). 9. Carter, A. L., et al., Phys. Rev. Lett. 37, 1380 (1976).
1. 2. 3. 4. 5. 6.
* The m ~ l also lies within the empiric error intervals of some older mnemP'S,namely in that from [8] (where m~mp is the smallest) after correction made in [9] and in that from [9], though the middle values of these emp,s are closer to the recent fit [3] of ranr than to m cal m~• ~ (we take into account only the cases with the smallest Am.•