CNR-IEI Report
B4-19 1998


New Polynomial Law of Hadron Mass

Sergio Bottini


Consiglio Nazionale delle Ricerche, Istituto di Elaborazione della Informazione,

I-56100Pisa, Italy



Abstract

A simple and general law of mass, intrinsically with zero freely-adaptable parameters, is

shown to be valid for all the hadrons with one or more flavored (s,c ,or
b )

quarks/antiquarks, both baryons and mesons. It establishes correspondencesH{Pi}

between these hadrons with at least one flavored constituent, H, and specific sets of lighter

particles, {P }, in which the total baryon and lepton numbers are conserved. The law is
i

polynomial with a varying degreethat univocally depends only on the particles involved.

Its statement may be given straightforwardly in terms of the binomial series converging to

-1
(1-
) 2. It asserts that, for each hadronH, there exists a certain set of lighter particles

{P }, together with a specific reference hadronh that fixes the mass scale, such that, for a
i 0

value of the sum of this power series equal to the mass of H, a definite partial sum of the

series equals the total mass of particlesPi. The starting, independent data in the rule are the

masses of the hadrons with exclusively unflavored (uand
d) constituents (among which,

our hadronsh , such as the proton or pion) and the masses of leptons. The consequence
0

of the law is a pre-discretization of the mass spectrum of the hadrons containing at least one

flavored constituent, on account of which the mass of any of them may only assume one of

the values aprioristically determined by the total masses of lighter particles. By its

simplicity, great accuracy, complete extensiveness and exhaustiveness, this rule may be

regarded as a potential new physical law of mass.



(Submitted to Physical Review D 1991; revised 1997)


I. INTRODUCTION


Despite the increasing complexity of phenomena emerging in the various areas of science at

deeper levels, nature sometimes appears to obey unexpectedly simple and elegant laws.

Generally, a new law of nature seems to follow no logical necessity. Its statement may be

completely unforeseen: initially guessed, it is then checked by the level of agreement with

experimental data. Usually, only the pragmatic assumption that nature can be simple is the

leading motivation behind any search for such laws.

On this basis, we have found that the masses of hadrons obey a simple

relationship[1, 2, 3]. Our rule differs from a typical mass formula in that it consists of a

condition on the masses which is only necessary. Thus, it appears to describe an

unsuspected new property of hadron mass, fortuitously representable in the form of a law.

The law is polynomial, essentially with no freely-adaptable parameter: only observed

masses are involved. It concerns all hadrons, which are partitioned into two different roles

depending on their quark contents: flavored or unflavored. According to this law, the

masses of the hadrons with one or more flavored constituents, namely strange (s), charm

(c), or bottom (b) quarks, and/or their respective antiquarks, may only take on their values

within a certain set of permitted values, which are fixed by the masses of lighter particles.

Such a pre-discretization of mass then represents the way in which our rule would take part

in the generation of the mass spectrum of these hadrons. The masses of the remaining

hadrons, i.e. those with exclusively unflavored constituents, namely up(u) and down(d)

quarks/antiquarks, such as the proton and pion, are, together with the masses of leptons,

the independent values in the rule, i.e. the starting data. The wide and complete validity of

the rule, for both mesons and baryons, suggests that it may originate from a physical

cause.





2


II. NEW LAW OF MASS


A. Definition

For each observed hadron (baryon or meson) containing one or more flavored (I=0)

quarks/antiquarks (namely, s, c or b), indicated as hadronH, there exists a certain set of

lighter particles, {P }
i , together with a specific hadron containing only unflavored (I=
1 )
2

constituents (namely, u and d) which fixes the mass scale, , called reference hadronh ,
0

i.e. there exists the correspondence


H 
h 0 {P }i, (1)

such that the value =
for which
H



(H) = S( ) (2)
H

also satisfies

S ( ) =  (P
n+1 H i), (3)
i
where
1  -1
S() = = 2 (-)k (4)
 k
1 -  k  =  0


(defined for -1<1);
S is a definite partial sum of this binomial series, i.e. the
n+1

sum of its first n+1 terms, with
n univocally depending only on the particles involved;

and
m ()
()= , (5)
m(h )
0

i.e. the masses m() of H and P are re-expressed adimensionally as () in units of the
i

mass of h . This h is a baryon or meson according to H, with S=
C=
B=0,
I0,
0 0

and L(h )=
L(H).
0

In Eq.(1), the total baryon and lepton numbers and the total electric charge,

between H and {P },
i are conserved. The types considered of these correspondences

(which include cases formally analogous to those produced by hadronic, radiative or

semileptonic decays with the minimum number of final hadrons) are





3


P P 
generally,wherethe
P arelighterhadronswithanykindofquarks
1 2 i

H P ()
onlyforthesamequarkcontentof
H and
P (6)
1 1
P (/e)
onlyif
P isan
S=
C =
B =0,
I0ground-statehadron
0 0


with L(P)L(H) (note that P contains only unflavored constituents). The value of n for
0

the (n+1)th partial sum of our power series, in Eq.(3), is determined by H, Pi and h0

through


n +1ifHcontainstwoormoreflavored(
I=0)quarks/antiquarks
min
w i t h  t h e  s a m e  a b s o l u t e  e l e c t r i c  c h a r g e
n = ( 7 )  

n 
o t h e r w i s e ,
m i n


where n is the minimum integer for which
min



Sn  (P
min+1(1-
) > max{ i)}, (8)
i


with (=) 1-
being a test-value very close to one (where these partial sums approach

their upper bounds), let be =
aexp[-b(n +1) 2(H)] with a = 2.9710-3 and
min

b=3
10-5 (note that for < of interest).
H ~0.99 it will be ~a for all nmin

The precise values to be used for the masses of the hadrons H and P in
i

Eqs.(2)and(3) are the average values of the masses of those isospin states of them that

intervene in the various charge-conserving variants of the same instance of

correspondence(1). On the right-hand side of Eq.(8), we consistently have the largest

value of the total mass of each {P } (depending on the permitted isospin states of the
i

hadrons involved). We specify that for the value of m(h ) in Eq.(5) in the cases of the
0

nucleon and pion we take the mass of the proton (p) and the neutral pion (0), i.e. the

lowest-lying baryon and meson, respectively. Finally, for the hadronsH with excitations

of a radial type, the relative quantum numbersN follow: N (h ) N (H ) and
0

N (P )N (H ).

This law of mass can be summarized as:

Statement:

-1
{P }, h | S ( ) = 2
 (P )=(1
- ) (9)
i 0 n+1 H i) for (H) = S(H H
i





4


with ()=
m()/m(h ), where {P } is found to satisfy Eq.(6), and n to obey Eqs.(7)
0 i

and (8), i.e. n is univocally determinable a priori, independently of Eq.(3), as a function

of the content of flavored quarks/antiquarks of H and of the magnitude of the total mass of

particles P in units of m(h ).
i 0


B. The rule re-written as a polynomial law

Our law of mass can be expressed equivalently in the form of a polynomial law of a

varying degree, as follows: for each H there exist {P } and h such that
i 0


Q [ -2(H)] =
n  (Pi) (10)
i
is satisfied, where n
 -1
Q [ -2(H)] = 2 [ -2(H)-1]k (11)
n  k
k  =  0


is the polynomial function of degreen in the dimensionless variable -2(H), with

0< -2(H)2, obtained by substituting from Eq.(2), i.e.
H


=1
- -2(H), (12)
H



into Eq.(3). Degree
n is fixed apriori by Eqs.(7) and (8), with Qn () replacing the
min

left-hand side in Eq.(8). The mass scale is set by h through Eq.(5).
0

Conversely, property

l i m
[ -2(H)] = (H) (13)
n Qn

of polynomials(11) makes it possible to re-transform this alternative formulation of the

law, Eq.(10), into Eqs.(2)and(3).


C. A more detailed criterion for n

Although it is not essential for the complete validity of the rule, we wish to take into

account a minor variant of the criterion that establishes the value ofn in Eq.(7),

exclusively for values of very close to one, i.e. 0.99< <1. Specifically, for
H ~H
>^ (n )= 1- cexp[- d (n +1)], with c=1.71 0 -2 and d =4.81 0 -2, the
H

following





5


n=
n (with 1) (14)
min H

holds for any quark content of H, replacing the specialized Eq.(7). The few cases falling

under this additional condition will be identified.


D. Comments and specifications

In its definition, our rule has essentially zero freely-adaptable parameters. This is strictly

true for the whole sector of known baryons. Note that is a simple representation of the
H

experimental mass of H, according to Eq.(12). Only in the choice of the small forming

the test-value of =1-
(1) in Eq.(8), is there a certain, though low, level of

arbitrariness. However, this may generally have a just slight effect on the determination

ofn only for quite high values (i.e., for some mesons, when also 1).
H

In almost all cases, we will have0<
<1, and accordingly m(H)>m(h ).
H 0

The permitted negative values of (i.e.-1
 <0) make it possible to cover the rare
H H

cases of the hadronsH which are lighter than any possible reference hadronh with the
0

same L, for m(h )/
2 m(H)<m(h ), thus preserving our condition L(h )=L(H).
0 0 0

This happens, for example, for the L=1 baryon
(1405)(=H) which is lighter than the

lowest-lying L =1 baryon with only unflavored constituents,
N (1520) or

N(1535)(=
h ). However, although it is interesting that these cases too have a solution in
0

the rule still with L(h )=
L(H), their very low number may not rule out the possibility that
0

condition >0 should always be satisfied. As a matter of fact, alternative solutions with
H

>0, i.e. m(H)>m(h ), and by necessity here with L(h )<L(H), would also exist
H 0 0

for such few cases.


In defining Eq.(6), we referred in passing to our correspondences H{P } as
i

including the formal analogues of decays of hadronsH only because this representation

rapidly conveys some correct information about the kinds of correspondences permitted in

the rule. Therefore, the involvement of the massless as well as the neutrinos is purely

nominal in this context. Our rule would be consistent with the limitation to the

correspondences of type(6) which, apart from non-massive particles, may formally be

obtained through one or more steps of possible decays. However, to be specific, only the




6


conservation of the baryon number and electric charge is strictly imposed on Eq.(1). In

this following, we will accept any correspondence(1) which satisfies these conditions,

under the restriction that {P } represents two terms, P and P , of which at least one is
i 1 2

always a hadron and the other, when it is not a hadron itself, may be, in accordance with

the cases described by Eq.(6), either an essentially massless term or a term characterized

-
by the mass of the muon. For example, given 0, we may consider 0e+e


as substantially equivalent; and hence e .

In the application of the criterion of assignment of n, Eq.(7), to the particular case
- - -
of the mixed states (uu ,d d ,s s ), we may consistently expect n= n or n +1
min min
-
depending on whether the percentage of the amount of ss content is less than or greater

than a certain threshold in between fifty and a hundred per cent. We will assign n=
nmin

to mesons (548) and '(958), as well as to f (980), and n=
n +1 to the almost pure
o min
-
ss states, such as (1020) and f '(1525). For certain I=0 mesons, the percentage content
2

of their strange constituents is a controversial matter. For the states with masses above

1GeV, we will take n=
n +1.
min

Finally, note that the (n+1)th partial sums in Eq.(3), S ( ), are represented, in
n+1 H

their respective domains implicitly given by Eqs.(7) and (8), by polynomialcurves

whose effective orders range from the first to, at most, the third, irrespective of the value,

even if high, ofn. In fact, when expressed in powers of 1- , they become
H

n n
 k -1

S ( ) = (1- )k' (-1)k-k'( 2
k ) , (15)
n+1 H  H ' k
k '  =  0 k  = 
k '

where any terms of a higher degree than the third can always be neglected for our purposes

(i.e., k'3) because, as is easy to see, the largest value that variable 1- may assume
H

sufficiently approaches zero with n increasing. By substituting 1- with -2(H),
H

Eq.(15) gives the corresponding expression of polynomial Q [ -2(H)] in powers of
n

-2(H).





7


III. AGREEMENT WITH THE DATA


All the known baryons and mesons with one or more flavored constituents, i.e.

hadronsH, are found to satisfy our law, Eqs.(2) and (3), or equivalently the single

Eq.(10), with a very good approximation. The respective sets of particles {P }
i are listed

in TableI for L(H)=0 and, for some significant cases, in TableII for
L(H)>0. The

rule is illustrated graphically in Fig.1. Acomprehensive view of its excellent agreement

with the data is cumulatively given in Fig.2 for all our reference hadrons
h .
0

To show the high degree of accuracy of the rule, we have derived the mass values

of hadronsH, m (H), from the experimental masses of the corresponding (lighter)
calc

particles P (see TablesI and II). Small variations in the mass values of the P correspond,
i i

in fact, to larger (and, with approaching 1, increasingly larger) changes in the calculated
H

masses of the H's, by a factor(>1) equal to the ratio between the derivatives of the sum

of series(4) and of its (n+1)
th partial sum for = , i.e. S'( )/S' ). Thus, in
H H n+1(H

comparing the rule with the data, the sum of the masses of the particles of a given set {P }
i

is taken as an input, and, accordingly, the mass of H, calculated through Eq.(10), is the

output of the rule. The experimental uncertainties on the masses which enter the

calculations produce the uncertainties associated with the values of m (H). The
calc

experimental masses are taken from[4] (more recent data for the [5] and * [6] are
b c
+
available). Recently, CLEO[7] reported the discovery of the (JP=
3 ) * whose mass
2 c

further confirms the validity of our rule.

Because of the experimental uncertainties, it should be realized that a reference

hadron h whose mass is not known with great accuracy may introduce a small margin of
0

choice. Clearly, in the important cases of the highly accurate reference masses of the proton

and 0 such an arbitrariness is reduced to zero. Moreover, for relatively low values of,

the exact determination of a reference mass inside its uncertainty interval is not crucial. To

form m(h ) in Eq.(5), we might, in general, establish to take the I=1/2 or
I=0 state of
0

h according to whether it is a baryon or meson. In the Tables, we have precisely used the
0

experimental mean/nominal values of mass for all the reference hadron states considered




8


except, to a very small (and justified) extent, in the case of h = (770) (see TableI), for
0

which, in order to show the best fit to data, we have expressed a certain, though restricted,

choice within its experimental uncertainty. The optimal value turns out to be 766.64 MeV,

that we assume to refer to the neutral state 0 (with the mass of  taken, say, 1.5MeV

higher). Consider, however, that just using the experimental mean values for the (770),

too, would still very well support the evidence for the complete validity of the rule (note

that the mean experimental data for , 766.91.2MeV, is very close to our value).

Therefore, the rule does not rely essentially on any adjustable parameter.

For the sake of completeness, consider that the (I=0)
(782) might, in principle,

be included as a possible additional L=0 reference meson since, like
(770), it contains

only unflavored constituents. In particular, for h =
, correspondences K*()
0
-
and  K K , found for h =
(TableI), would be even more accurately satisfied,
0

with 894 and 1018MeV calculated vs. 894 and 1019 MeV experimental for the masses of

K* and, respectively.


IV. SIDE CONSIDERATIONS


A. Probability of fitting the data by chance alone

The combinations of particles P permitted by Eq.(6), when used as inputs in Eq.(10),
i

yield solutions which include the mass spectrum of the hadrons H as a subset. An

estimation of how likely a full agreement would be between our rule and the data if it were

to occur only by chance, to the same accuracy as our fit, for the well-established L=0

ground-state hadronsH alone, with h =
0/ for mesons and h =p for baryons, gives
0 0

a probability of less than10-16. Although this probability is ideal, in that it has been

calculated by replacing the outputs of the rule with as many random outcomes that have a

uniform distribution, nevertheless it is clearly indicative of our case. Such an extremely

small value for the probability of fitting the data accidentally in our rule indirectly supports

the conjecture that it may have a physical nature.





9


B. Counts of coincidences for fictitious reference masses

By a computer analysis, we directly checked that the masses of the proton, pion and

(770), in their role as L=0 reference hadrons
h , are really special values for the rule,
0

in that they produce sharp peaks in the number of the L=0 hadrons
H that satisfy

correspondences H{P }
i in Eq.(10), against the number of accidental coincidences

obtained by taking any other, arbitrary value as a (fictitious) trial reference mass (see

Fig.3). For the mesons, the test was made when the available experimental data were

those of[8], which however, for our purposes, are essentially equivalent to the present

data of[4] (used for baryons). The hadrons taken as Pi were ground-state hadrons.

Thus, the rule has the remarkable ability of finding out the masses of hadrons with

only unflavored constituents, on the basis of their property (when taken as reference

masses) of yielding good solutions in Eq.(10) for the hadrons
H concerned (as soon as a

sufficiently high number of these hadronsH is known).

This is also confirmed in the case of L=1, where the optimal agreement with the

data is reached for a reference mass in the region of N(1520)/N(1535) for baryons (namely

1526MeV), and b (1235)/a (1260) for mesons (namely 1228MeV, still within the
1 1

smaller experimental uncertainty of b ), i.e., respectively, the lightest L=1 baryon and
1

meson with only unflavored constituents.


C. Generative criterion of mass spectra of

L =0 ground-state hadrons
H

Our law consists of a necessary, but not sufficient, condition for a given value to be the

mass of a hadronH. Therefore, it cannot be seen as a typical mass formula which

reproduces a certain spectrum by itself. However, in order to provide a qualitative

evaluation of which part the rule appears to take in the formation of the mass spectrum of

the hadrons with at least one flavored constituent, we have found that, for example, in the

+
important case of the known ground-state JPC=0
-+, 1-- mesonsH and JP=1 2
baryonsH, by adding a few fair constraints of an essentially topological character to this

law of hadron mass, we can generate the masses correctly.





10


The generative criterion of these masses can be described through a process of

growth of directed graphs, the rooted trees shown in Fig.4, where roots, vertices, and

edges represent particles. Specifically, as roots: the same hadrons that may be our reference

hadrons(h ); as vertices: hadronsH; and as edges: any particles P . These graphs develop
0 i

upwards from their roots, producing verticesH, consistently with the requirement that

correspondences {P } are read downwards, i.e. in the inverse direction. Thus, for
i

each vertexH, the respective set {P } is constituted by the particles which represent the
i

adjacent lower vertex/root and their joining edge. With each of these verticesH we

associate a weight, w(H), defined as the total mass of the final particles, excluding the root,

that are obtained by iterating the application of {P }, sequentially, also to those P
i i

which still contain flavored constituents, if any, until only hadrons with exclusively

unflavored constituents, or leptons, or 's remain. For example, from the cascade

, N, we obtain N as final particles, and then, excludingrootN,

w()=2
m(). We see that the contributions to the weight assigned to a vertexH come

from the edges in the chain connecting it to the root.

The starting data are the masses of the L=0 ground-state hadrons with only

unflavored constituents (namely , , , N and ), and the masses of the leptons

(essentially, the ). The hadronsH generated as new vertices may be re-used as P , i.e.
i

lower vertices or edges, to keep on climbing the mass spectrum. The trees for the hadrons

with the lesserJ (i.e. 0 for mesons and 1/2 for baryons) are formed first.

For the L=0 hadrons
H considered, our correspondences H
h0 {P }
i are

correctly selected by taking the next higher outputs of the rule under the following

constraints:

-i)for a vertexH being the lowest-lying meson/baryon with one or two constituents of the

same given flavor: w=min, with a number of ending consecutive edges of
-type at

least equal to the number of its flavor unities;

-ii)J(H) = lesserJ only if
/N{h ,P} for mesons/baryonsH; and finally (with rare
0 i

occurrences)





11


-iii)maximizing the outputs within small ranges: if two sets of particles, {P } and {P '},
i i

with the same flavors and values of I involved, yield close outputs for the same h , say to
3 0

within100MeV, the higher one is preferred;

where w is the increase in weight w(H) relative to the case with the same number of

quarks/antiquarks of the right lighter flavor, and min stands for a minimum value of mass

equal to the total mass of one or more massive particles.

For example, for only one flavored constituent (i.e. for K, D, B, , , ), it is
c b

w = w = m() and w =m() for mesons, and w = w = m(), w = m()
s b c s c b

for baryons. Still as an example, pointiii implies that K' (with a slightly higher output) is

preferred to K* for H=
D*, with h =
. It should be noted that the only constraints i
0

andii would suffice for all the L=0 ground-state mesons and baryons with the lesser
J

+
up to a content of two flavored constituents. The extension to the JP=
3 baryonsH
2

would require the addition of a few other general constraints (already satisfied

spontaneously by all the above L=0 hadrons
H).

As supplementary cases, Fig.4 also reports the selected correspondences for

mesons B , B*, and , and baryon , which predict plausible values for their masses
c c b cc

(see TableIII). The further predictions of the graphs would also be consistent with the

expected values for the masses of the hadrons with top quarks, although, in all these cases,

the outputs of the rule are, because of their very high values, very strongly dependent on

the variations of the masses in input. For example, from  +, with h =
p(938)
t b 0

and n=29, we would have
m( ) =181GeV for a value of the mass of
in input of
t b

5633.7MeV.

This adjunctive criterion simply shows how a small set of essentially not

quantitative constraints may be sufficient, in many significant cases, to correctly select the

masses of the hadrons with at least one flavored constituent from the set of the values

which are pre-determined by our law, Eq.(9), as their possible candidates. In other

words, it is as though these constraints were a "measure" of how much information our

law would need (by its nature of being an only necessary condition on the masses) in order

to reproduce the mass spectrum of these hadronsH properly.



12


V. SUMMARY AND CONCLUDING REMARKS


We have shown that it is possible to define a rule, essentially with zero freely-adaptable

parameters, which maps all the observed hadrons with at least one flavored constituent, H,

onto specific sets of lighter particles, i.e. H{P }, Eq.(6), under the condition that the
i

total baryon and lepton numbers and the total electric charge are conserved. With each of

these correspondences, a specific reference hadronh is associated containing only
0

unflavored constituents, such as the proton or pion, which fixes the mass scale ,

Eq.(5), with L(h )=
L(H). The rule asserts that if the mass value of a hadron H is set
0 - 1
equal to the sum of the binomial series S()= 
 k =(1-
) 2, then a definite
k
k  =  0
partial sum of this power series must equal the total mass of certain lighter particles P ; i.e.,
i

the rule consists of mapping H  {P }
i established by S ( )=
 (P
n+1 H i) for
i
(H )= S( ), Eqs.(3) and (2). Alternatively, by combining together these two
H

equations, which separately involve the partial and total sums of the series, the rule can be

expressed equivalently as a polynomial law of a varying degreen, between the

inverse-square of the mass of H and the total mass of theP , namely:
i

{P },h |Q [ -2(H)]=  (P ) ,
i 0 n i)), Eq.(10), with ()= m ()/m (h 0
i
Eq.(5); where the coefficients of polynomial function
Q , Eq.(11), are determined as
n

being the first n+1 coefficients of our binomial series. Degree
n (or, equivalently, the

point of truncation of the series) is a function of the triple(H, {P },
i h ): specifically, n is
0

jointly and univocally fixed by the content of flavored quarks/antiquarks of H, Eq.(7),

and by the magnitude of the total mass of particles P in units of m(h ), Eq.(8). We thus
i 0

have a family of polynomial functions of which only one is selected apriori in order to

satisfy each given correspondence H{P }
i . Although n can be large, our polynomials

are represented in their domains by curves whose effective orders only reach, at most, the

third one [see Eq.(15)].

While the latter formulation of the law, given in terms of a single polynomial

equation, is more concise and usual, the former formulation has the interesting property

that all the masses are treated on the same footing, i.e. also the mass of H, Eq.(2), like




13


those of the P , Eq.(3), is included in a linear term. According to this form of the law, the
i

hadrons with strange, charmed or bottom constituents appear to be endowed with an

unsuspected new property by which specific fractions of their masses would be constrained

to correspond in value to the total masses of certain lighter particles.

The rule expresses a necessary condition on mass, thus implying a prohibition: the

values that do not satisfy this condition, Eq.(10), are forbidden as mass values for the

hadrons with at least one flavored (I=0) constituent. It has predictive capabilities: for

example, as a partial recapitulation, the mass of meson K is determined by (the mass of) the

; the mass of by the and ; the mass of ' by the and ; the mass of baryon by

the N and ; the mass of by the ; the mass of by the and ; the mass of by the

and K; the mass of by the and ; and so on. In general, for any mass interval
c

expected to contain hadrons with one or more flavored constituents, the rule determines

certain values which would be the only possible candidates for those masses. Thus, the

whole mass spectrum of such hadrons would be "pre-discretized": their masses are a subset

of the values permitted a priori by the rule on the basis of the masses of lighter particles.

The law is in very good agreement with the experimental data, without exceptions. The

high number of its accurate solutions can be regarded as a very sound guarantee against

chance, i.e. in favor of the conjecture that it may actually reveal physical regularities of the

hadron mass spectrum.





14


References




[1] S. Bottini, Nuovo Cimento A 102, 1321 (1989).

[2] S. Bottini, Abstracts of the LXXIX Congresso Nazionale della Societ Italiana di

Fisica, Udine, 199 (1993).

[3] S. Bottini, Report CNR-IEI B 2/93-08 (1993).

[4] Particle Data Group, R. M. Barnett et al., Phys. Rev.D 54, 1 (1996).

[5] DELPHI Collaboration, P. Abreu et al., Phys. Lett. B 374, 351 (1996);

ALEPH Collaboration, D. Buskulic et al., Phys. Lett. B 380, 442 (1996);

CDF Collaboration, F. Abe et al., Phys. Rev.D 55, 1142 (1997).

[6] CLEO Collaboration, G. Brandenburg et al., CLEO CONF 96-13 (unpublished).

[7] CLEO Collaboration, P. Avery et al., Phys. Rev. Lett. 75, 4364 (1995);

CLEO Collaboration, L. Gibbons et al., Phys. Rev. Lett. 77, 810 (1996).

[8] Particle Data Group, J.J. Fernndez et al., Phys. Lett. B 239, 1 (1990).





15


Table Captions





TABLE I. Results of the fit to the masses of the L=0 hadrons
H, i.e. the L=0 baryons

and mesons with at least one flavored (I=0) quark/antiquark, including the radially

excited states. For each reported correspondence, H 
h 0 {P }i, the calculated mass value
of H, m (H), has been obtained from the experimental masses of the respective lighter
calc

particlesP , through Eqs.(2)and(3), or, equivalently, the single Eq.(10). The value
i

ofn, which indicates the point of truncation of binomial series(4) in our equations, is

determined by Eqs.(7) and (8). Hadron
h is the associated reference hadron which sets
0

the mass scale in the rule. Also reported are the calculated values of the dummy

parameter, =1-
m2(h )/m2 (H), which only has a formal role in the law. As
H,calc 0 calc

mass of (548) in input we used 547.93 MeV, taken within its experimental uncertainty

range; the outputs from its current mean value, 547.45MeV, are given in parentheses. A

few hadronsH are found to satisfy correspondences for both their possible reference

hadrons h . a(Extra) cases which follow Eq.(14) for
n.
0




TABLE II. Some examples of the fit to the masses of the hadronsH with L>0. The case

of the controversial L=1 meson
f (980) (=H) illustrates that imposing everywhere
o

condition >0 would sometimes necessarily require
L(h )=
L(H)-1; otherwise,
H 0

L(h )=
L(H) can always be obeyed. bUnderestimated value because we have taken zero
0

for the unknown mass difference inside the isospin doublet of N(1535).




TABLE III. Further values of mass generated, as predictions, by the adjunctive criterion

forming the hadron trees in Fig.4, through selection from the mass values permitted by

our rule. aSee footnote of TableI.





16


Figure Captions




FIG.1. Graphic representation of mapping H{P }
i established by the rule, in the case

of the L=0 ground-state baryons
H with h =
p(938). The mass ofH is reported
0

vertically on the left, the total mass of the corresponding particle set{P }
i on the right. The

scale of mass is set by the proton, being the current reference baryon(h ), i.e.
0

()=
m()/m(p). On the horizontal axis we have the values of which, as the left
H

vertical axis, again represent the experimental masses of hadronsH, expressed in

accordance with Eq.(12). The linear, quadratic, cubic,... curves (labeled with
n=1,

2,...) represent the consecutive partial sums of the binomial series with (total) sum

1/

1  - 
 (=  k), increasingly approaching their limit curve (in bold) as their
k
k= 0
degreen increases (here shown up to n=6). Our law of mass, expressed by Eqs.(2)

and (3), or equivalently by Eq.(10), asserts that for every hadron
H there must exist one

experimental point with coordinates [ =1
- -2(H),  (P
H i)] which belongs to the
i
one of these curves that is jointly and univocally determined a priori by the flavored quark

content ofH and the magnitude of the total mass of particlesPi, according to Eqs.(7) and

(8) respectively.




FIG.2. Cumulative view of the very good agreement of the rule with the data (reported in

TablesIandII), shown for
n7, in both sectors of baryons and mesons
H, for all our

reference hadronsh , in(a) for n=
n and in (b) for n=
n +1, depending on the
0 min min

flavored quark content ofH, Eq.(7). Masses are expressed in units of the mass of the

relativeh , and thus "normalized". For each H 
h 0 {P }, only one polynomial curve of a
0 i

certain degree n (=1,2,...) is pre-selected for the fit, within a specific domain, by

Eqs.(7) and (8). All the experimental points are well arranged along their appropriate

curves, i.e. they satisfy Eq.(10) with a very good approximation.





17


FIG.3. Number of coincidences of the L=0 hadrons
H satisfying the rule, Eq.(10), as

the trial reference mass varies. Sharp peaks are found exactly at the masses of reference

hadronsh (marked with ), namely: the proton(a)for the ground-state baryons
H, the
0

pion(b) and rho(c) for the ground-state mesons
H plus their radial excitations. The

required accuracy for the coincidences is fixed by two threshold values, T and T , which
1 2

are the permitted maximum distances, respectively, between the mean experimental and

calculated mass values of H, and between the relative uncertainty intervals. The cumulative

count of the coincidences for all these L=0 baryons and mesons, with the reciprocal

reinforcement of the peak, is shown in (d), where the results of (a)-(c) are summed

together by making the masses of the proton, pion, and rho coincide with the zero of the

axis.


FIG.4. Graphs illustrating the networks of the correspondences H{P } established
i

by the rule for the ground-state J P C  =  0
- + , 1- - mesonsH (a), and
+
J P =
1 baryons
H(b). The trees read top-down: for each hadron
H, represented by a
2

vertex, the corresponding particlesP are represented by the adjacent lower vertex/root and
i

their joining edge. A small set of adjunctive, essentially topological, constraints (given in

the text) can generate the graphs from their roots upwards, thus correctly reproducing the

mass spectrum of these L=0 hadrons
H. In (a) the edges drawn in bold lead to the

mesonsH with the lesserJ. The parentheses indicate hadrons whose masses are

unknown.





18


b
3.0 2814
3.0

c
(2645)
c
c
c n=6
[in units of m(p)]
c
i
5
c (P)

n=4 i
1
(H) (H) =
2.0 1- n=3 1876
2.0
H
n=2

(1530) (1385)
(1385)
n=1 





1.0 938
1.0 FIG. 1
0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0
-2
H (H)


3.0
n=7

n = n 0
n=6
min


n=5 2.5

baryons
mesons n=4
[in units of m(h )]

2.0
n=3 i
(P)

i
n=2
1.5

n=1



1.0
0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0
-2 (a)
(H) (= )
H





3.0


n = n + 1 n=7 0
min


2.5
n=6

baryons
mesons n=5
[in units of m(h )]

2.0
n=4 i
(P)

n=3 i
1.5
n=2




1.0 FIG. 2
0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0
-2 (b)
(H) (= )
H


L = 0 BARYONS


p ( 9 3 8 )
12
T = 7 MeV
1
T = 5 MeV
2

8



4
FIG. 3a

number of coincidences 0
5
5 0
0 0
0 7 0 0 9 0 0 1 1 0 0 1 3 0 0 1 5 0 0
(a) trial reference mass (MeV)





L = 0 MESONS


12 (135)


8 T = 20 MeV
1
T = 8 MeV
2

4
FIG. 3b

number of coincidences 0
0 5 0 1 0 0 1 5 0 2 0 0
(b) trial reference mass (MeV)


L = 0 MESONS

25 (770)

20 T = 20 MeV
1
T = 8 MeV
2

15


10


5 FIG. 3c
number of coincidences
0
2 0 0 6 0 0 1 0 0 0 1 4 0 0 1 8 0 0
( c ) trial reference mass (MeV)





L = 0 BARYONS + MESONS

50


40


30


20


10
number of coincidences FIG. 3d
0
- 2 0 0 - 1 0 0 0 1 0 0 2 0 0
(d) mass difference (MeV)


a meson trees
B ( )
b

0


(B ) B B D B D D (B )
c s s s c c

'
' D K 


J / D K K '
s

N 
c


N





b baryon trees

( ) ( )
cc
t

+


b c c




c c



N FIG. 4


H q u a r k {P
i} h n n m (H ) m (H )
0 H ,c a l c m i n calc expt
c o n t e n t (MeV) (MeV)

sqq N p(938) .2940 1 1 1117 1116

0 sqq () " .3782 1 1 1190 1193

ssq " .4904 1 2 1314 1314-1321

(1385) sqq () " .5432 1 1 1388 1382-1388

(1530) ssq (1232) " .6265 1 2 1535  2 1531-1536

sss (1385) " .6877 2 3 1679  2 1672

cqq " .83123 1 1 2284  1 2285
c
(2455) cqq (1232)  (/e) " .85400 1 1 2456  18 2451-2454
c
csq (1232) " .85562 3 3 2469 5 (24676) 2464-2472
c
(2645) csq K " .87455 3 3 2649  2 2644  2
c
or " + " .87354 3 3 2638  2 "

css 0 (782) " .87928 4 5 2700  3 2704  4
c
or " 0 (782) " .87888 3 4 2696  1 "

bqq " .97228 6 6 5635  33 5641  50
b c
sss K- (1232) .4527 1 2 1669 1672

(2530) cqq (782) " .7610 2 2 2526 2530  7
c
(2645) csq (1232) (770) " .7816 2 2 2642  5 2644  2
c
(1600) sqq N(1440) N(1440) .1907 1 1 1601 ~1600

(1660) sqq K " .2379 1 1 1650 ~1660

K sq- (135)0 .92667 3 3 498 494- 498

(ss-, qq-) (/e) " .94029 2 2 552 547

(2S) (ss-, qq-) " .98914 18 19 1295 1295  4
D cq- " .994789 21 21 1870 (1852  7) 1864-1870

D cs-  " .995279 21 21 1964 (1944  8) 1969
s
cc- " .997951 54 55 2982 (2947  14) 2980  2
c -
[J/(3097) cc- K K " .998114 44 44 a
3108  3 3097 ]

(2S) cc- (1020) 0 " .998588 59 60 3592 3594  5
c

B* bq- K*(892) " .9993521 46 46 5303  68 5325  2

B bs- K0 (782) " .99936765 71 72 5368  21 5369  2
s


K*(892) sq- (770)  (/e) (770) .2783 1 1 902 891 - 896

'(958) (ss-, qq-) (770) " .3601 1 1 958 958
-
(1020) ss- K K " .4406 1 2 1025 1019

(1440) (ss-, qq-) K (770) " .7044 2 3 1410 1415  10

K(2S) sq- K " .7225 1 1 1455 (1452) ~1460

(2S) ss- K0 '(958) " .7940 3 4 1689 1680  20
(3 -
S) (ss-, qq-) K K*(892) " .8073 2 3 1746  1 1760  11

K(3S) sq- K (782) " .8239 2 2 1827  1 ~1830

D*(2010) cq- K '(958) " .8544 3 3 2009  1 2006-2011

D*(2110) cs- (770) " .8681 2 2 2111 (2107) 2112
s
[ cc- (1020) '(958) " .93348 5 6 2972  1 2980  2 ]
c
-
J/(3097) cc- N N " .93857 4 5 3093 3097

[ (2S) cc- N
-(1232) " .95430 6 7 3586  5 3594  5 ]
c
(2S) cc- - " .95695 8 9 3695  1 3686

B bq- D " .978905 5 5 5278  20 5277-5281

B* bs- D K*(892) " .980011 10 11 5422  11 5416  3
s
-
(9460) bb p
- " .993424 14 14 a
9454  29 9460
c
or " (1385)0 - " .993438 23 24 9464  115 "
c
-
(2S) bb n
- " .994157 5 5 a
10029  2 10023
-  -
(3S) bb D*(2010) D*(2110) " .9945328 23 24 10368  36 10355
s
-
(4S) bb
- " .9947349 15 15 a
10565  36 10580  4


TABLE I


H L q u a r k {P
i} h n n m (H ) m (H )
0 H ,calc min calc expt

c o n t e n t (MeV) (MeV)

(1520) 1 sqq (1385) N(1520) .0024 1 1 1522 1520

(1670) 1 sqq (1520) " .1800 1 1 1679  2 1665-1685

(1690) 1 sqq " .1974 1 1 1687 1685-1695

(1820) 1 ssq (1670) " .3072 1 2 1826  12 1823  5

(2625) 1 cqq (1530)0 K+ " .6651 1 1 2627  1 2626
c
(1405) 1 sqq (1385)0 () (1535) -.1971 1 1 1403  1 1407  4

(1620) 1 sqq K " .0995 1 1 1618 ~1620

(1670) 1 sqq N(1520) " .1593 1 1 1674  13 1660-1680
(2593) 1 cqq N(1535) K " .6458 1 1 2579 b
2594
c
(1820) 2 sqq (1670) N(1680) .1592 1 1 1824  13 1815-1825

(1915) 2 sqq (1385) K " .2384 1 1 1925 1900-1935

(2030) 2 ssq (1232) (770) " .3099 1 2 2022  3 2025  5

(2030) 2 sqq (1530) K (1950) .0811 1 1 2034  1 2025-2040

f (980) 1 (ss-, qq-) (782) 0 (770) .3853 1 1 981 980  10
0
or " " (770)  (/e) b (1235) -.5804 1 1 979  1 "
1

h (1170) 1 (ss-, qq-) (1020) 0 " -.1243 1 1 1161 1170  20
1

K (1270) 1 sq- K (770) " .0529 1 1 1265  1 1273  7
1

or 1 " K (782) " .0759 1 1 1280 "

h (1380) 1 (ss-, qq-) b (1235) " .1946 1 2 1372 1380  20
1 1

K (1400) 1 sq- b (1235) " .2230 1 1 1396 1402  7
1 1

D (2420)0 1 cq- a (980) '(958) " .7417 2 2 2422  4 2422  2
1 0

D (2536) 1 cs- D () " .76284 2 2 2528  6 2535
s1 s

h (1P) 1 cc- a (1320)0 f (1420) " .87785 4 5 3522  11 3526
c 2 1

K*(1430) 1 sq- f (1270) a (1320) .1426 1 1 1423  2 1424-1434
2 2 2

or " " f (1285) " .1538 1 1 1433  1 "
1
f '(1525) 1 ss- (1020) K0 " .2537 1 2 1526 1525  5
2
-
D*(2460) 1 cq- K*(892) K*(892) " .7125 1 1 2458  4 2455-2463
2
(1P) 1 cc- a (1320) K (1400) " .86313 3 4 3563  27 3556
c2 2 1
-
(1P) 1 bb f (1270) (1P) " .9823286 10 11 9915  23 9913
b2 2 c1
K*(1780) 2 sq- a (980) (782) (1690) .0880 1 1 1771  1 1770  10
3 0 3
(1850) 2 ss- '(958) K*(892)0 " .1707 1 2 1857 1854  7
3

TABLE II


H q u a r k {P
i} h n n m (H )
0 H ,c a l c m i n calc
c o n t e n t (MeV)

B bc- '(958)  (135)0 .9995356 52 52 6263  50
c

B* bc- K*(892) '(958) (770) .984973 4 4 6254  33
c
-
bb 0 " .9931272 13 13 a 9248 108
b c
ccq p(938) .9340 5 6 3652  4
cc c



TABLE III



