Blogg
E_Heims_Mass_Formula_1982
01.05.2014 22:04
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
1
Heim’s Mass Formula
(1982)
Original Text by Burkhard Heim
for the Programming of his Mass Formula
Reproduction by Research Group
Heim's Theory
IGW Innsbruck,2002
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
2
On the Description of Elementary Particles
(Selected Results)
by Burkhard Heim
Northeim, Schillerstraße 2,
2-25-1982
A) Invariants of Possible Basic Patterns (Multiplets)
Symbols:
k Configuration number, k = 0 : no ponderable particle (no rest mass). For
ponderable particles only k = 1 and k = 2 possible, not k > 2. k is a metrical
index number.
e so-called “time-helicity“. Refering to the R4 e = +1 or e = -1 decides whether it
concerns an R4 - structure or the mirror-symmetrical anti-structure (e = -1).
G the number of quasi-corpuscular internal sub-constituents of structural kind.
bi symbol for these 1 £ i £ G internal sub-constituents of an elementary particle.
B baryonnumber
P double isospin P = 2s .
P1,2 locations in P-interval, where multiplets appear multiplied (doubled).
I number of components x of an isospin-multiplet, i.e. 1 £ x £ I .
Q double space-spin Q = 2J .
Q value of Q at P1,2 .
k(l) “doublet-number“, which distinguishes between several doublets by
k(l) = 0 or k(l) = 1 .
L Upper limit of k-interval 1 £ l £ L .
C structure-distributor, identical with sign of charge of the strangeness quantum number.
qx electrical charge quantum number with sign of the component x of the isospinmultiplet.
q amount of charge quantum number q = ½qx ½.
Uniforme Description of Quantum Numbers by k und e
G = k + 1
B = k - 1
P1 = 2 - k
P2 = 2k - 1
I = P + 1 , 0 £ P £ G }(I)
Q(P) = k - 1
Q(P) = 2k - 1
k(l) = (1 - d1l ) d1P , 1 £ l £ L = 4 - k
C = 2(PeP + QeQ)(k - 1 + k)/(1 + k)
eP,Q = e cos aP,Q
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
3
aP = pQ(k + ( ) 2
P )
aQ = pQ[Q(k - 1)+ ( ) 2
P ] } (II)
2qx = (P - 2x)[1 - kQ(2 - k)] + e[k - 1 - (1 + k)Q(2 - k)] + C , 0 £ x £ P , q = ½qx½
Possible configurations k = 1, k = 2 with e = ± 1
Possible Multiplets of Basic States
Multiplet xn of serial number n for e = +1 and anti-multiplet xn with e = -1.
General Representation: xn (eB,eP,eQ,ek)eC(q0,...,qP)
Mesons: k = 1, G = 2 (quark?), B = 0, 0 £ P £ 2, i.e from singlet I = 1 to triplet I = 3.
Q = 0, Q = 1, L(k=1) = 3, k(1) = 0, k(2) = k(3) = 1
Baryons: k = 2, G = 3 (quark?), B = 1, 0 £ P £ 3 from singulett I = 1 to quartet I = 4,
Q = 1, P1 = 0, P2 = 3, Q = 3, L(k=2) = 2, k(1) = 0, k(2) = 1
____________________
Possible multipletts for e = +1:
k = 1: x1 (0000)0(0) º (h)
x2 (0110)0(0,-1) º (e0,e-), (is the existence of e0 possible ? )
x3 (0111)0(-1,-1) º x3 (0111)0(-1) º (m-) pseudo-singlet }(III)
x4 (0101)+1(+1,0) º (K+, K0)
x5 (0200)0(+1,0,-1) º x5 (0200)0(±1,0) º (p±, p0) anti-triplet to itself
k = 2: x6 (1010)-1(0) º (L)
x7 (1030)-3(-1) º ( W-)
x8 (1110)0(+1,0) º (p,n)
x9 (1111)-2(0,-1) º (X0,X-) }(IV)
x10 (1210)-1(+1,0,-1) º (S+,S0,S-)
x11 (1310)-2(+1,0,-1,-2) º (o+,o0,o-,o--), (existence possible ?)
x12 (1330)0(+2,+1,0,-1) º (D++, D+, D0, D-), (thinkable as a basic state ?)
______________________
Abbreviations:
h = p/(p4 + 4)1/4
hkq = p/[p4 + (4+k)q4]1/4
J = 5 h + 2 Öh + 1 }(V)
A1 = Öh11 (1 - Öh11)/ (1 + Öh11)
A2 = Öh12 (1 - Öh12)/ (1 + Öh12)
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
4
Planck’s constant: h = h/2p, light-velocity: c = (e0m0)-1/2 , wave-resistance of empty space
R3 (electro-magnetic): R - = cm0 , with e0 and m0 constants of influence and induction.
Elektrical elementary charge: e± = 3C± with
C± = ± 2Jh / R- /(4 p)2 (possibly electr. quark-charge ?)
Finestructure-constant: aÖ(1- a2) = 9J (1 - A1A2) / (2p)5 , a > 0 .
Solution: a(+) (positive branch) and a(-) (negative branch).
Numerical: a(+)
- 1 = 137,03596147
a(-)
- 1 = 1,00001363
[A better formula, 1992, yields a(+) = 1/137,0360085 and a(-)_ = 1/1,000026627]
What is the meaning of that strong coupling a(-) ?
Abbreviation: a(+) = a , a(-) = ß » 137 a .
B) Mass-Spectrum of Basic Patterns and its Resonances
Used constants of nature and pure numbers:
Planck’s constant: h = h/2p = 1,0545887 x 10-34 J s,
light-velocity: c = 2,99792458 x 108 m s-1,
Newton’s constant of gravitation: g = 6,6732 x 1011 N m2 kg-2
constant of influence e0 = 8,8542 x 10-12 A sV-1 m-1,
constant of induction m0 = 1,2566 x 10-6 A-1 s V m-1,
vacuum wave-resistance R- = (m0/e0)1/2 = 376,73037659 V A-1
derived constants of nature (mass-element):
m = 4 p pg g -
0
3
0
3 hs h / 3c s 1 , s0 = 1 [m] (gauge factor) (VI)
Basis of natural logarithms: e = 2,71828183
number p = 3,1415926535
geometrical constant: x = 1,61803399
[Limes of the “creation-selector“] limn®¥ an : an-1 = x by the series an = an-1 + an-2 .
(till the 8th decimal place, represented by x = (1 + Ö5)/2).
Auxiliary functions:
h = p/(p4 + 4)1/4 (VII)
t = 1 - 2/3 x h2 (1 - Öh)
a+ = t (h2 h1/3 )-1 - 1} (VIII)
a- = t (hh1/3 )-1 - 1
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
5
Quantum numbers by (A):
hqk = p/[p4 + (4+k)q4]1/4
N1 = a1
N2 = (2/3) a2 ,
N3 = 2 a3 ,
with
a1 = ½ (1 + Öhqk ) ,
a2 = 1/ hqk , }(IX)
a3 = e(k-1) /k - q {a/3 [( 1 + Öhqk ) (x/hqk
2)](2k +1) hqk
3 +
+ [h(1,1)/ e hqk] (2 Ö xhqk)k [(1 - Öhqk) /(1 + Öhqk)]2 }
Invariants of metrical steps-structure (abbreviation s = k² + 1):
Q1 = 3 × 2 s - 2 ,
Q2 = 2s - 1 , }(X)
Q3 = 2s + 2(-1)k ,
Q4 = 2s - 1 - 1 .
Fourfold R3-construct 1£ j £ 4 . Qj = const. with respect to time t. Parameter of occupation
nj = nj(t) caused radioactive decay. Mass elements of occupations of the configurations zones
j are ma+ .
Further auxiliary functions of zones occupations:
K = n1
2 (1+n1)²N1 + n2 (2n2²+3n2+1)N2 + n3 (1+n3)N3 + 4n4 ,
G = Q1²(1+Q1)²N1 + Q2(2Q2²+3Q2+1)N2 + Q3(1+Q3)N3 + 4Q4 ,
}(XI)
H = 2n1Q1[1+3(n1+Q1+n1Q1) + 2(n1²+Q1²)]N1 + 6n2Q2(1+n2+Q2)N2 + 2n3Q3N3
F = 3 P/(pÖhqk) (1 - a-/a+)(P+Q)(-1)P+Q[1-a/3+p/2 (k-1) 31-q/2 ]
*{1+2 k k/(3 h2) x[1 + x²(P-Q)(p 2-q)]} [1 +( 4 x( ) 2
P /k)(x /6)q] - 1
*[ 2 Öh11Öhqk + qh2 (k - 1)] (1+4pa/hÖh)(1+Q(1-k)(2-k)n1/Q1]
+ 4 (1 - a-/a+)a(P+Q)/x2 + 4 qa-/a+
Uniform Mass spectrum:
M = ma+ (K + G + H + F) (XII)
Not each quadruple nj yields a real mass! To the selection rule: in the fourfold R3-construct
1£j£4 configurations zones n(j=1), m(j=2), p(j=3), s(j=4). Increase of occupation with
metrical structure elements:
central zone n cubic,
internal zone m quadratic,
meso-zone p linear (continuation to the empty space R3),
external zone s selective.
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
6
Principle of increase of the configurations zones:
n4+Q4 £ (n3+Q3)a3 £ (n2+Q2)² a2 £ (n1+Q1)3a3 (XIII)
Selection rule for the Occupation of Configuration Zones
(n1+Q1)3a1 + (n2+Q2)² a2 + (n3+Q3)a3 + exp[1-2k(n4+Q4)/3Q4] + iF(G) = (XIV)
= Wnx{1 + [1-Q(2-k)(1-k)][anxN/(N+2) + bnx N(N - 2) ]}.
Wnx = g(qk) wnx ,
Basis rise: g(qk) = Q1
3a1 + Q2² a2 + Q3a3 + exp[(1-2k)/3] for nj = 0. (XV)
Structure power of the discussed state wnx = (kPQk)eC(qx) as component x of multiplets n
is:
wnx = {(1-Q)[A11-P(A12+A13qk/hqk) - ( ) 2
P (A14-A15q/hqk)] + kQhqkA16}2 - k +
+ {(q-1)A21 + (1-P)A22 + ( ) 2
P [A23-qxhqk(1+A24(+qx))- 1A25] + }(XVI)
+ k(A26+qhqk²A31) + ( ) 3
Q hqkA32 + ( ) 3
P [A33q3(qx - (-1)q)/(3-q) +
+
e( )h( ) /
( )
P Q
A
q q
q q
-
-
+
-
1 4
66
8 1
(1 - q(2-q)A34
1 - q
xA35/hqk) hqk/h² - A36]}k - 1 .
w(1) = (1-Q)[A11 - P(A12+ A13qk/hqk ) - ( ) 2
P (A14 - A15q/hqk)] + kQhqkA16 (XVII)
and
w(2) = (q-1)A21 + (1-P)A22 + ( ) 2
P [A23 - A25qxhqk(1 + A24(1+qx)) - 1] +
+ k(A26 + qhqk²A31) + ( ) 3
Q hqkA32 + ( ) 3
P {A33q3[qx - (-1)q)/(3-q)] + (XVIII)
+
e( )h( ) /
( )
P Q
A
q q
q q
-
-
+
-
1 4
66
8 1
[1 - q(2-q)A34
1 - q
xA35/hqk] hqk/h² - A36}
in wnx = [w(1)]2 - k + [w(2)]k - 1 (XIX)
can become w(2) = 0 for single sets of quantum numbers at k = 1 or w(1) = 0 at k = 2 ,
which leads to terms 00 , which but must have always have the value 1 as parts of structure
power. Therefore it is recommended for programming to complete w(1) and w(2) by the
numerical non-relevant summands k-1 and 2-k . Since always w(1) ¹ -1 and w(2) ¹ -1
remain, but only k=1 or k=2 is possible, the actually terms in the expression
wnx(k) = [k-1+w(1)]2 - k + [2-k+w(2)]k - 1
do no more appear. By this correction it is evident that for mesonical structures wnx (k=1) =
1 + w(1) and for barionical structures wnx (k=2) = 1 + w(2) holds.
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
7
As a basis of resonance holds anx = A41 (1 + anaq)/k (XX)
with an = PA42 [1 - kA43 (1 + A44 (-a)2 - k A45
k - 1)*
*(1 - kQA46(2-k)) - A51(k-1)(1-k)] (XXI)
and aq = 1 -qA52(1 - 2A53
k)[1 + qx(3-qx)(k-1)(1-k)/6] (XXII)
Resonance grid is
bnx = {A54A55
k - 1 [1 - PA56(1-kA61A62
1 - k)(1 + qA63(1 + kA64))] * (XXIII)
* (1-k- 1 (A65(q+k-1))2 - k ( ) 2
P (1 - ( ) 3
P )}/[kP(1+P+Q+kh2 - q)] .
The coefficients Ars can be seen as elements of the quadratic coefficient matrix $A = (Ars)6
with Ars ¹ Asr and ImArs = 0 .
Proposal for the determination of matrix elements (reduction to p, e and x):
A11 = (x² p e)² (1 - 4 p a² ) / 2 h² ,
A12 = 2 p x² (J/24 - e p h a² / 9)
A13 = 3 (4 + h a)[1 - (h²/5)((1 - Öh)² /(1 + Öh)² ]
A14 = [1 + 3 h (2 h a - e²x(1 - Öh)²/(1 + Öh)²)/4x]/ a
A15 = e²(1 - 2ea²/h)/3
A16 = (pe)²[1 + a(1+6a/p)/5h]
A21 = 2(ea/2h)²(1 - a/2x²)
A22 = x[1 - x(ax/h²)²]/12
A23 = (h² + 6xa²)/e
A24 = 2x²/3h
A25 = x(pe)²(1 - ß2)
A26 = 2{1 - [p(exa)²Öh]/2}/ex2
A31 = (pea)²[1 - (pe)²(1 - ß²)]
A32 = x²[1 + (2ea/h)²]/6
A33 = (pexa)²[1 - 2p(ex)²(1 - ß²)]
A34 = h 2ph
(XXIV)
A35 = 3a/ex²
A36 = [1 - pe(xe)²(1 - ß²)] - 1
A41 = {x[2 + (xa)²] - 2ß}/(2ß - a)
A42 = [px²h(ß - 3a)]/2
A43 = x/2
A44 = 2(h/x)²
A45 = (3ß - a)/6x
A46 = pe/xh - eh²a/2
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
8
A51 = (2a + 1)²
A52 = 6a/h²
A53 = (x/h)3
A54 = a(ß -a)Ö(3/2)
A55 = x²
A56 = (x/h)4
A61 = px(2ß - a)/12ß
A62 = p²(ß - 2a)/12
A63 = (Öh)/9
A64 = p/3h
A65 = p/3x
A66 = xh
The order of resonance N ³ 0 (positive integer) selects the admitted quadruple nj with
1 £ j £ 4 . With
f(N) = [1 - Q(2 - k)(1 - k)][anx N/(N+2) + bnx N(N - 2) ] (XXV)
follows that the unknown function F(G) remains 0 for all N ¹ 1 (right side is real).
In the case of N = 0 is f = 0 , so that
(n1 + Q1)3a1 + (n2 + Q2)² a2 + (n3 + Q3) a3 + exp[(1-2k)(n4+Q4)/3Q4] = Wnx (XXVI)
describes the nj of the state xnx and hence the mass M0(nx) of the component x of the
multiplet xn . The N ³ 2 assign xnx to a spectrum of occupation-parameter quadruples and
with that, according to the mass-formula, resonance-masses MN(nx) (for each component xnx
a spectrum of masses). In the case of N = 1 no spectral term. Here is not f(N ) ³ 0, f(1) is
complex.
Real part: (n1+Q1)3a1 + (n2+Q2)² a2 + (n3+Q3) a3 + exp[(1-2k)(n4+Q4)/3Q4] =
= Wnx{1+[1-Q(2-k)(1-k)]anx/3} (XXVII)
Imaginary part F(G) = Wnx[1-Q(2-k)(1-k)]bnx . (XXVIII)
The nj and F(G) are somehow related with N to the complete bandwidths G . Also there
must be a connection QN = Q(N) between doubled spin quantum-number Q and N . How
could this connection be like?
If N = 1 is excluded, then F = 0 , and the real relationship
(n1 + Q1)3a1 + (n2 + Q2)² a2 + (n3 + Q3) a3 + exp[(1-2k)(n4+Q4)/3Q4] = Wnx (1+f) (XXIX)
has to be discussed. Generally f > 0 for N ³ 2 and f = 0 for N = 0. But in the case of the
multiplets x2 f = 0 for all N ³ 0, since only here is Q(2-k)(1-k) = 1 . Electrons according
to this image can not be stimulated !
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
9
For a numerical evaluation of Wnx , anx , bnx and Fnx (quantum number function in mass
spectrum M) not QN = Q(N) , but use Q = Q(0) of xn . For the evaluation of nj the
principle of increase of the occupations of configuration zones is considered. First determine
the right side Wnx (1+f(N)) = W1 numerically for an order of resonance N = 0 or N ³ 2 .
Determine according to the selection rule the maximal cubic number K1
3 whose product
with a1 is contained in W1 . Then insert W1 - a1K1
3 = W2 ³ 0 into
(n2 + Q2)² a2 + (n3 + Q3) a3 + exp[(1-2k)(n4+Q4)/3Q4] = W2 . (XXX)
Now maximal quadratic number K2² such, that a2K2
2 is still a factor of W2 , i.e.
W2 - a2K2
2 = W3 ³ 0 . Accordingly in
(n3 + Q3) a3 + exp[(1-2k)(n4+Q4)/3Q4] = W3 (XXXI)
Determine maximal number K3 in the way W3 - a3K3 = W4 ³ 0 .
Three possibilities for W4 : (a): W4 = 0 ,
(b): 0 < W4 £ 1 ,
(c): W4 > 1 .
General case (b): lnW4 £ 0 and K4(2k-1) = -3Q4lnW4 .
In case of (c) it is lnW4 > 0 and K < 0 . This is impossible, since always nj+Qj ³ 0 has to
be.
According to n4+Q4 £ (n3+Q3)a3 of the principle of rise K3 will be lowered by 1 and
a3K3 is added to K4 < 0 , so that a new value K4 ³ 0 will be generated., which requires
K3 > 0, since in that case K3 = 0. This dilatation can not happen because of the quadratic
rise of j = 2 , so that this order of resonance N does not exist for xnx (forbidden term).
In the case (a) W4 ® 0 would have as a consequence the divergence K4 ® ¥ , but this is
impossible according to K4 £ a3K3 (particularly there are no diverging self-potentials). For
that reason will be calculated in case of (a) the maximal value K4 = a3K3 . From the
computed Kj it follows nj = Kj - Qj . Beside nj ³ 0 also nj < 0 is possible , but it holds
always Kj ³ 0 , i.e. nj ³ -Qj . The quadruple nj determined in that way will be inserted with
Fnx in the spectrum of masses, which numerically yields MN(nx) as a spectral-term of
mass-spectrum at xnx .
Note: The Kj are always integers. But in the case of the evaluation of K4 generally decimal
figures will occur. In case of the decimal places ,99... 99 one has to use the identity
,99... 99 = 1 . But if the series of decimal places is different from this value, then one has not
to round up. The decimal places are to cut off , since the Kj are the numbers of structure
entities.
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
10
Limits of Resonance Spectra
General construction-principle of configuration-zones
n4+Q4 £ (n3+Q3)a3 ,
a3 (n3+Q3)(1+n3+Q3) £ 2a2(n2+Q2)² , (XXXII)
a2 (n2+Q2)[2(n2+Q2)² + 3(n2+Q2) + 1] £ 6a1 (n1+Q1)³ .
If by the increase of N between two zones equality is reached, then nj+Qj ® 0 in j, while j-1
will be raised by 1 to nj-1 + Qj-1 + 1 . The stimulation takes place “from outside to the
interior“. Always nj+Qj ³ 0 is an integer, since they are the numbers of structure entities.
Empty-space-condition: nj = -Qj , but (nj)max = Lj < ¥ (no diverging self-energy potentials).
Intervals -Qj £ nj £ Lj < ¥ cause 0 £ N £ L < ¥ of resonance-order. With M0(nx) = M0
holds
4ma+a1 (L1+Q1)³ = [2(P+1)]2 - kM0G (XXXIII)
with G = k+1 and from that by the construction-principle
a2 (L2+Q2)[2(L2+Q2)² + 3(L2+Q2) + 1] £ 6a1 (L1+Q1)³ ,
a3 (L3+Q3)(1+L3+Q3) £ 2a2(L2+Q2)² , (XXXIV)
L4+Q4 £ (L3+Q3)a3 .
For L implicitly the resonance-order is
(L1 + Q1)3a1 + (L2 + Q2)² a2 + (L3 + Q3) a3 + exp[(1-2k)(L4+Q4)/3Q4] =
=Wnx [1+f(L)] (XXXV)
Also in the evaluation of Lj and L do not round up, but cut off decimal digits! The Lj which
are obtained by the construction-principle, yield the absolute maximal masses Mmax , and the
quadruples, which are obtained from the L, yield the real limit-terms ML < Mmax , which are
to stimulate secondaryly with (Mmax - ML)c² and then reach Mmax .
Northeim,
Schillerstraße 2 gez. (Heim)
2-25-1982
Distributed to: Deutsches Elektronen-Synchrotron (DESY) Hamburg,
Eidgenössische Technische Hochschule (ETH) Zürich,
Max-Planck-Institut für Theoretische Physik, München,
Messerschmitt-Bölkow-Blohm GmbH (MBB), Ottobrunn bei München:
Dr. G. Emde, Dr. W. Kroy, Dipl.-Phys. I. v. Ludwiger.
Staatsanwalt G. Sefkow, Berlin, und H. Trosiner, Hamburg.
F_Heims_Mass_Formula_1989
01.05.2014 21:53
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
10
Heim’s Mass Formula
(1989)
According to a Report by Burkhard Heim
Prepared by the Research Group
Heim's Theory
IGW Innsbruck,2002
Content
· Introduction
· Mass of Basic States and of the Excited States of Elementary Particles
· The Average Life Times of the Basic States
· The Sommerfeld Finestructure Constant
· The Masses of Neutrino States
· Concluding Remarks
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
11
Introduction
After DESY physicists in 1982 had programmed and calculated the mass formula which was
published in the book Elementarstrukturen der Materie (Heim 1984), the mentioned formula by
B. Heim was extended and in 1989 a 57 pages report with a new formula and the results of the
calculations were sent to the company MBB/DASA. Unfortunately this later code could no more
be recovered today.
Parts of these formulae have now been programmed again by the research group „Heim
Theory“ (by Dr. A. Mueller). It was found that in the manuscript some brackets in very long
equations were lost during the process of writing; this had to be corrected at best estimate. The
code covers the masses of basic states only and no lifetimes.
Other than the program written in 1982, Heim’s 1989 computation also includes the life times
of the basic states, the neutrino masses, and the finestructure constant. Therefore, these equations
shall be given here, as far as they deviate from those given in the manuscript in 1982.
The structure distributor C (i.e. strangeness) given in eq. (I) of chapt. E has to be divided by k.
One of the angles by which the time helicity e is defined must read
aQ = p Q [Q + ( ) 2
P ] (B1)
The expression for the quantum number of charge other than in (II) now reads:
qx = ½ [ (P - 2x + 2) [1 - kQ(2 - k)] + e[k - 1 - (1 + k)Q(2 - k)] + C ] (B2)
All other constants are defined by eq.(I).
1. Mass of Basic States and of the Excited States of Elementary Particles
The modified mass formula of elementary particles is built up - other that in eq.(XII) - by the
following parts:
M = ma+ [(G + S + F + F) + 4 q a- ] (B3)
The parts G and S are the same as G and K in eq.(XII) (now using n, m, p instead of n1, n2, n3);
m is the mass element as in eq.(VI). The constants a± have the form:
a+ =
h
h
J
h
h h
h
6
2
1
2 1
1
2
²
( )
( )
-
-
+
é
ë êê
ù
û úú
æ
è
çç
ö
ø
÷÷
- 1 , a- = (a+ + 1)h - 1 (B4)
The calculated results for a+ and a- in (B4) are shown in a table VI/chapterG.
The abbreviations for F and F, which depend on the quantum numbers, read:
F = 2 n Qn [1 + 3(n + Qn + n Qn) + 2(n² + Qn²)] + (B5)
+ 6 m Qm (1 + m + Qm)N2 + 2 p Qp N3 + j (p,s)*d(N)
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
12
F = P(-1)P+Q (P + Q) N5 + Q(P + 1) N6 (B6)
j = j (p,s) , d(0) = 1 (0 for N ¹ 0) (B7)
with
j =
N p
p
4
1
²
+ ²
s
s
s +
+
- =
- Q
BUWN 1
4 2 4
0
1
²
( ) +P(P - 2)²(1 + k(1 - q)/2aJ )(p/e)²Öh12(Qm-Qn) -
- (P + 1) ( ) 3
Q /a , (comp. with eq. B49)
U = 2Z [P² + 3/2 (P - Q) + P(1 -q) + 4kB (1 -Q)/(3 - 2q) +
+ (k - 1){P + 2Q - 4p(P - Q)(1 - q)/ 4 2 }] hqk
- ² (comp. with eq. B50)
and Z = k + P + Q + k (comp. with eq. B51)
j is a term of self-couplings, which depends on p and s and essentially determines the life time
of a basic state. j appears only in the basic states; therefore the symbol d(N) as a unit element is
used. The functions Qi from eq. (X) remain unchanged. For n1, n2, n3, n4 in eq. (B5) here n, m,
p, s will be written. The constants hq,k, J and h (with h10 = h, and J 1,0 =J ), as well as the
functions N1 and N2 read as in eq.(IX). The remaining Ni with i > 2 are:
ln (N3 k/2) = (k -1) [1 - p
1
1
1 1 1
1
1
1
-
+
- - - - +
h
h
h
J
a a h q k
q
q
q
u ,
,
,
,
{ ( / )( )²}] -
- 2/(3p e) (1 - h )² (6 p²e²/J
1
1
1 +
-
h
h
q, - 1) (B8)
N4 = (4/k) [1 + q(k - 1)] (B9)
N5 = A[1 + k(k - 1) 2k²+3 N(k) A
1
1
2 -
+
æ
è
çç
ö
ø
÷÷
h
h
q k
q k
,
,
] (B10)
A = (8/h)(1 - a-/a+)(1 - 3h/4) (B11)
N(k) = Qn + Qm + Qp + Qs + k(-1)k 2k²-1 (B12)
N6 = 2k/(p eJ ) [ k (k² - 1)
N k
k
( )
, h1
{q - (1 - q)
N k
Qn k
'( )
, h1
} +
+ (-1)k+1 ] h(1 - a a - + / ) 4
1
1
2 -
+
æ
è
çç
ö
ø
÷÷
h
h
Qs (B13)
N’(k) = Qn + Qm + Qp + Qs - 2k -1 (B14)
The calculated results for B8, B9, B10 and B13 can be found in a table VII/chapter G.
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
13
Let L be the upper barrier such that as soon as it is reached the filling of the zone x disappears
and the foregoing zone filling of next higher order is raised by 1. With the symbols L(x) (x - 1)
for this barrier and M0 = M(N=0) the limits of the fillings of structure zones corresponding to
eq.(XXXIII) are given by:
- Qn £ n £ L(n) =
(P )M
N
Qn
+
-
+
1
2
0
3
1 ma
(B15)
since in the case of the central region there are no further fillings.
For the series of numbers m the limitation holds:
- Qm £ m £ L(m)(n) (B16)
with 2(Qm + L(m)(n))³ + 3 (Qm + L(m)(n))² + Qm + Lm(n) = 4 N1(n + Qn)³/N2 (B17)
Correspondingly, we have
- Qp £ p £ L(p)(m) (B18)
with 2 L(p)(m) = 24 2 1
3
N
N
m Qm ( + )² + - 2Qp - 1 (B19)
and - Qs £ s £ L(s)(p) (B20)
with 2 L(s)(p) = N3(p + Qp) - 2 Qs (B21)
The calculated results for B15 can be found in a table IX/chapter G.
The selection rule which expresses the n, m, p, s by the quantum numbers
k, P, Q, k, q and N, is described by eq.(XXIX).
In that f(N) is the excitation function for N > 0. For the factor Wnx º WN=0 , which is
independent of the exciting state, holds:
WN=0 = A ex (1 - h)L + (P - Q)(1 - ( ) 2
P )(1 - ( ) 3
Q )(1 - h )² Ö2 (B22)
with
A = 8 g H[2 - k + 8H (k - 1)] - 1 (B23)
H = Qn + Qm + Qp + Qs (B24)
g = Qn² + Qm² + (Qp²/k) ek-1 + exp[(1- 2k)/3] - H(k - 1) (B25)
L = (1 - k) Q (2 - k) (B26)
x = [1 - Q - ( ) 2
P ](2 - k) + 1/4B [a1 + k³/(4H)(a2 + a3/(4B))] (B27)
B = 3 H [k² (2k - 1)] - 1 (B28)
The calculated results for B23, B24, and B28 can be found in a table VI/chapter G.
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
14
For the three parameters a1, a2, and a3 the following combinatorical relations hold:
a1 = 1 + B+ k(Q² + 1) ( ) 3
Q - k[(B - 1)(2 - k) - 3{H - 2(1 + q)}(P - Q) + 1] -
- (1 - k) [(3(2 - q) ( ) 2
P - Q{3(P + Q) + q})(2 - k) + [k(P + 1) ( ) 2
P +
+ {1 + B/k (k + P - Q)}(1 - ( ) 2
P )(1 - ( ) 3
Q ) - q(1 - q) ( ) 3
Q ] (k - 1)] (B30)
a2 = B [1 - ( ) 3
Q (1 - ( ) 3
P )] + 6/k - k[Q/2 (B - 7k) - (3q -1)(k - 1) +
+ ½ (P - Q){4 + (B + 1)(1 - q)}] - (1 - k) [(P(B/2 + 2 + q) -
- Q{B/2 + 1 - 4(1 + 4q)}) (2 - k) + ( ¼ (B - 2){1 + 3/2(P - Q)} - (B29)
- B/2 (1 - q) - ( ) 2
P [{ ½ (B + q - eqx) + 3 eqx}(2 - eqx) -
- ¼ (B + 2)(1 -q)]) (1 - ( ) 3
Q )(k - 1) - ( ) 3
P [2 (1 + eqx) +
+ ½ (2 - q){3(1 - q) + eqx - q } - q/4 (1 - q)(B - 4) - ¼ (B - 2) +
+ B/2 (1 - q)]]
a3 = 4 B y’/(y’+1) - (B + 4) - 1 (B31)
with
y’ 2 B = k[ h /k {4 (2 - Öh) - p e (1 - h) h }{k + e h (k - 1)} +
+
5 1
2 1
( )
( )
-
+ -
q
k k (4B + P + Q)] + (1 - k)[(P - 1)(P - 2){2/k² (H + 2) +
+ (2-k)/(2p)} + ( ) 2
P (1 - ( ) 3
Q )(q B/2 {B + 2(P - Q)} + {P (P + 2)B +
+ (P + 1)² - q(1 + eqx) [k(P² + 1)(B + 2) + ¼ (P² + P + 1)] -
- q (1 - eqx)(B + P² + 1)} (k - 1) + {(P - Q)(H + 2) +
+ P[5 B (1 + q) Q + k (k - 1) {k(P + Q)²(H + 3k + 1)(1 - q) -
- ½ (B + 6k)}]}(1 - ( ) 2
P )(1 - ( ) 3
Q ) + ( ) 3
P (2 - q) Q {eqx(B + 2Q + 1) +
+ q/(2k)(1 - eqx)(2k + 1) + (1 - q)(Q² + 1 + 2B)}]
The calculated results for B29, B30, B31 and B22 can be found in a table VIII/chapter G.
For the excitation function f from eq.(XXXV) Heim got the expression
f (N) = a N/(N+1) + b N (B32)
with the substitutions (a is the finestructure constant):
a =
P
kX q k q k
²
, , h2 h (1 - k/4) + (k -1){p/4 ( ) 3
P - h1,1h1,2 ( ) 2
P } (B33)
X = k[4a
( )
( )
( )
( )
[ /
( )
( )
]
B k
q
+ + e
-
+
-
-
æè ç
öø ÷
- - -
+
-
1
1
1 5
1 5
2
3
4
2 1
1
2 2 1 6
2
2
2
2
a 2
a
a
a
p
p a
p h
J a
+ 1 (B34)
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
15
b =
1
2h2 h qk qk
[aJ /8 (P² + 1)[ ½ (1 + h )(1 + h1,1 h1,2 (3/4) ( ) 3
P (k - 1)] +
+ (k - 1) {J 1,2/J - 8 ( ) 2
P (P² + 1 ) - 1}] - C (B35)
C = p (1 - h )² [1 + p (k - 1) + P/k³ (3/e + q(8 + hqk) +
+ (4 pe/ h )(1 - k)[1 - q
3
5
ph
ehqk
] - 2(k - 1) ( ) 2
P (3 - P){2 e (h + hqk) }(B36)
+ eqx pe/(3 h ) } +
8p k 1
h h
e k e q
e
( - )
-
æ
è çç
ö
ø ÷÷
)] + (2 e k q/ h² )(2 - k)(1 - h)²
The excitations can lead to a change of angular momentum. Since Q is the double quantum
number of angular momentum, Q(N = 0) could change additive by an even number 2z with the
integer function z(N), such that:
Q (N) = Q (N = 0) + 2 z (N), (B37)
where z(N) is yet unknown.
One has to hold in mind, that the s-fillings of the external region of a term M(N) can get an
additional excitation because of their external character. If the zones nN, mN, pN, and sN are
occupied and if
L(s)(p) = ½ N3 (p + Qp) - 2 Qs with - Qs £ s £ L(s)(p) , (B38)
is the complete occupation of the external region related to pN , then
KB = L(s)(p) - sN (B39)
describes a real number, which as a bandwidth determines the number of the possible excitations
of the external field of an excitation state M(N). For KB £ 0 there is no possibility of an external
field excitation.
If L(N) describes the maximal occupation of all the four structure zones 0 £ N £ L(N) < ¥, then
the equation of the excitation limit is given by eq.(XXXV) and eq.(B32) with N = L(N).
If the quantum numbers k, P, Q, k, and qx , as well as the excitation N, are given for a basic
state, then the right-hand side of eq. (XXXV), i.e.
(n + Qn)³a1 + (m + Q m)² a2 + (p + Qp) a3 + exp[-(2k - 1) /3Qs(s + Qs)] =
= WN=0(1 + f(N)) (B40)
with a1 = N1 , a2 = 3/2 N2 , a3 = ½ N3, and eq.(B22) to eq.(B36) can be calculated numerically.
By an exhaustion process based on
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
16
w = WN=0(1 + f) (B41)
n, m, p, and s can be determined using eq.(B15) to eq.(B21) and (B40).
Let be K ³ 1 the series of natural numbers. Then, first of all, w - K³a1 ³ 0 will be formed. K
will be raised as long as K = Kn changes its sign. Then Kn will reduced by 1, which results in:
w - (Kn - 1)³ a1 = w1 (B42)
The process will be repeated with w1 in the form w1 - K² a2 ³ 0 . With K = Km
w1 - (Km - 1)² a2 = w2 (B43)
will be generated. In the same way w2 - K a3 ³ 0 yields the relation
w2 - (Kp - 1) a3 = w3 (B44)
and with the abbreviation ß = (2k-1)/3Qs
w3 - e -ßK £ 0 (B45)
is determined, which changes its sign for K = Ks . Next, Ks will be reduced by 1.With the limits
now known, Kn to Ks , the n, m, p, s can be calculated:
n = Kn - 1 - Qn m = Km - 1 - Qm
(B46)
p = Kp - 1 - Qp s = Ks - 1 - Qs
With these quantum numbers the mass formula (B3) with its parts eq.(B4) to eq.(B14) can be
calculated.
2. The Average Life Times of the Basic States
Let be T the average life time of the masses of elementary particles determined by eq. (B3). If
TN = T(N) << T is a function depending on N, so that T0 = 0 for N = 0, then according to Heim
the unified relation for the times of existence is:
(T - TN) =
=
192
1 1 1 2 2 1 1 1 2 0
hHy
Mc²[ ( )²( )²( )²](H n m p )(n m p ß ) , , , ( ) h - h - h - h + + + + s + +
d
(B47)
where d = d(N) is as in eq.(B7) .M is taken from eq.(B3), and H from eq.(B24). The substitution
y is given by:
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
17
y = F [j + (-1)s (1 + j)(b1 + b2/WN=0)] (B48)
with
j =
N p
p
4
1
²
+ ²
s
s
s +
+
- =
- Q
BUWN 1
4 2 4
0
1
²
( ) + P (P - 2)²(1 + k(1 - q)/(2aJ ))
(p/e)²Öh12(Qm-Qn) -- (P + 1) ( ) 3
Q /a , (B49)
U = 2Z [P² + 3/2 (P - Q) + P(1 -q) + 4kB (1 -Q)/(3 - 2q) + (k - 1){P + 2Q -
- 4p(P - Q)(1 - q)/ 4 2 }] /hqk² (B50)
and Z = k + P + Q + k (B51)
The calculated results for B48 and B49 can be found in a table IX.
B will be calculated from eq.(B28). It is(B52)
F= 1 - 1/3 (1 - q)(P - 1)²(3 - P)(1 + P - Q - e C P/2)(1 + ß(0)(-1)k) - ( ) 3
P (1 + D), (B52)
s= 2 - k + e C + (2kQ - kP) + ( ) 3
Q : 1/k (P-1)(P-2)(P-3) (B53)
b1= [P { 7 + 6(1 - q)(C - ( ) 2
P ) - 2q (1 - ( ) 2
P )} + k Q{(3 Z - 1) B + 1}] (2 - k) +
+ ½ (1 - k){(q - eqx - 2) Q + e C P + 2 (P + 1) - }(B54)
- (1 - q)
P P
P P
( )
( ² )
-
+ -
3
1 1
(4 B - 6 + P)}(k - 1) - ( ) 3
P (q - e qx)
b2= B(5B+3) +
2 3
1
H
P
-
+
+ Ck{B( 3B+2(H+1)) + H + ½ }(1 - q) - Q {B(2(B+H) - 1) +
+ H/2 + 3} + k q {B (3B + 1) - 5/2}(k-Q) - ( ) 2
P P²(P + Q)²[8B+1 -
-{5B - (2H+1)(1 + 2 ( ) 3
P - Q) + 2} q] - ( ) 2
P H(1-q) - (B -3/4)²(P-1)(P-2)(P-3)(-1)k-1 +
+ (Q-q)(1-q + Bq){3(H+B) + pe/h - q/4}(P+1)³(k-1) + k{(-1)1-q [7HB+3(H+B)-5/2 +
+ (1-q){H(3B-4) + B+7/2}](k-1) + Q ( ) 2
P {(2 -q)(1 + e qx)[B/2(H+2) + ¾ ] +5/2HB +
+ 3H -
B
P
+
+
5
1
} - 5/2 H² ( ) 3
P {q (1+p/3(2-q) h2,2) B - (2-q)(1 - q)} (B55)
with ß(0) =
2a
pe
1
1
2 -
+
æ
è çç
ö
ø ÷÷h
h
(B56)
and D = [1 + 4 q²(q - 1)(2q + 1)] - 1 h ß(0) (1-Öh)4 P2+eq (P - 1)(q-1)q/2/(3Ö2) (B57)
With the systems eq.(B3) to eq. (B14) and with the quantum numbers (Table I) the particular
masses M can be calculated, and from eq.(B47) to eq.(B57) the life times T of all the multiplet
components for N = 0 can be determined numerically and compared with empirical values (Table
II). The life times T are shown in multiples of 10 - 8 seconds.
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
18
3. The Sommerfeld Finestructure Constant:
In j and ß(0) the finestructure constant a is contained. The value in chapter D, section 8 is
calculated only approximately. Heim now also gives the exact formula for a:
According to eq.(8.21) we get:
a 1
9
2
- a = 5 1-
J
p
²
( )
( C') (B58)
with 1 - C’ = 1 -
1 1
1
2 2
1 1 1 2
2 + -
+
æ
è çç
ö
ø ÷÷
h
hh h
h
h
,
, ,
= Ka (B59)
With the abbreviation
D’ =
(2 )
9
p 5
J a K
(B60)
it follows for the reciprocal square of these solutions:
a(±)
-² = ½ D’²(1 ± 1- 4 / D'²) (B61)
With eq. (V/chapter E) the values for both branches are calculated:
a+ = 0.72973525 ´ 10 - 2 and a- = 0.99998589 (B62)
1/a(+) = 137,03601 1/a( ) = 1,0000142
which, compared with the empirical value (Nistler & Weirauch 2002) for the finestructure
constant,
1/a(+) = 137,0360114 ± 3.4 .10 - 8
yields a value which falls into the tolerance region of measurement. The negative branch shows
an extremely strong interaction, which probably is based on the inner connections of the four
zones in an elementary particle. But Heim did not investigate this further.
4. The Masses of Neutrino States
Supposing that in the central region of an elementary particle an euclidian metric rules, i.e. that
there is no structure element, than that means: L(n) = - Qn .
According to eq.(B15) it means that there also is no ponderable mass M0 . According to
eq.(B16) to eq.(B21) it follows, that also the remaining structure zones are governed by an
euclidian metric. In eq.(B3) then we must substitute
n = - Qn , m = - Qm , p = - Qp und s = - Qs , (B63)
from which follows:
G + F + S = j (B64)
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
19
According to eq.(B49) generally j ¹ 0 holds, in spite of s + Qs = 0, and also F ¹ 0 is not
affected by the lower barrier of the n. m, p, s . If F + j ¹ 0, since P > 0 or Q > 0, then
eq.(B49) yields a field mass unequal zero, in spite of eq.(B63). This field mass is not
interpretable as a ponderable particle, but is - according to Heim - a kind of „spin-potence“
which as a „field catalyst“ permits transmutations of elementary particles or enforces the validity
of certain conservation principles (angular momentum). This behaviour is equivalent to those
properties which made the definition of neutrinos necessary by empirical reasons.
If according to eq.(B3) one substitutes for the mass of neutrinos in whole generality
Mn = ma+ (F + j0) (B65)
where j0 relates eq.(B49) to the lower bounds of n, m, p, s, than it follows, that Mn is
determined only by the quantum numbers k, k, P, and Q .
For Mn(kPQk) > 0 the following possibilities result:
Mn (1110) = Mn (1111) and Mn (1200) in the mesonic region, and
Mn (2110) and Mn (2111) in the baryonic region.
In addition there is another neutrino, which only transfers the angular momentum Q = 1 and
which is required by the ß-transfer. For this neutrino only the two possibilities exist:
Mn (2010) or Mn (1010).
Since in the case (2010) Mn < 0 would be, only Mn ( 1010) remains as a possibility for the ßneutrino.
With i = 1,...,5 the possible neutrino states ni are:
for k = 1: n1 (1010) , n2 (1110), n3(1200)
for k = 2: n4 (2110) , n5 (2111).
For each ni there exists the mirror-symmetrical anti-structure ni . From eq.(B3) with the
possibly non-zero quantum numbers the neutrino-masses may be determined.
The calculated results are collected in table II. The masses are given in electron volt.
The empirical ß-neutrino can be interpreted by n1 and the empirical m-neutrino by n2.
For the time being it cannot be decided whether the rest of the neutrinos also are implemented in
nature or whether it concerns merely logical possibilities.
5. Concluding Remarks
For the numerical investigation of the states N > 0 the system (B32) must be used, which is
uncertain because of the uncertain relations eq.(B33) to eq.(B36). The function z(N) in eq.(B37)
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
20
must still be determined. Since z is not given, also Q(N) for N > 0 remains
unknown. The mass values of the spectra N > 0 which belong to the basic states therefore still
have an approximate character. Also the life times TN of such states cannot be described yet. In
eq.(B49) the free eligible parameters for the expression j with eq.(B50) were fitted by empirical
facts [i.e. 4 2,(p / e)2 and 4p 4 1/ 2 ] .
The error Q(N) = Q(0) = Q based on the approximation z = 0 for all of the N only causes an
approximation error less than 0.1 MeV.
In spite of the mentioned uncertainties the numerical calculation of the relations eq.(B22) to
eq.(B36) and eq.(B3) yields a spectrum of excitations for each basic state, whose limits are given
by eq.(XXXV) with eq.(B32), and whose finestructure is described by eq.(B39).
In these spectra of excitation all empirical masses of short living resonances fit which were
available to Heim at that time (CERN - Particle Properties - 1973). But there are much more
theoretical excitation terms than were found empirically. That could be caused either by the
existence of a yet unknown selection rule for N, or the selection rule is only pretended since the
terms are not yet recordable by measurements.
In the tables IV and V Heim listed only such states N > 0 which seem to be identical with
empirical resonances. The N-description in the third column differs between N and N , where the
underlining means that a term is addressed which does not fit the selection rule for N of the
masses M(NB) - M(NA) > 0 with NB > NA . The values put in brackets in the 3rd and 4th column
(with KB from eq.(B39) ) are related to possible electrically charged components. For the D -
states, q = 2 was used. In the 5th column, the theoretical masses in MeV are indicated.
Here also the brackets are related to electrically charged components. The resonance states in
general are represented very well, in spite of the approximate character (because ofz(N) = 0), but
the uncertainty appears for k = 1 in the particles w(783) and h’(958), as well as for k = 2 in the
particle N(1688).
While the functions z(N) and TN yet have been searched for by Heim, he already possessed an
ansatz for a unified description of magnetic spin moments of particles with Q ¹ 0, which was not
yet published.
After discovering z and TN, Heim wanted to calculate the cross sections of interaction, which
regrettably could not more be done.
Apart from the above-mentioned incompleteness, it can be stated that on the basis of the farreaching
correspondence with the empirical data Heim’s structure theory meets all requirements
to be fulfilled by a mathematical scheme for a unified theory, and there is no other unified
structure theory which allows for more exact or much better confirmed descriptions of the
geometro-dynamical processes within the microregion.
G_Selected_Results
01.05.2014 21:46
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
1
Selected Results
Research Group Heim's Theory, IGW Innsbruck, 2003
Pay attention also to Heim,B. 1979/89/98, 1984
Ó IGW Innsbruck
Content
· Quantum Numbers of Basic States (N=0) 2
· Theoretical Data of Elementary Particles 3
with Mean Lives > 10-16 sec Calculated by B. Heim 1989
· Experimental Data of Elementary Particles 4
with Mean Lives > 10-16 sec
· Approximated Meson Resonances 5
· Approximated Baryon Resonances 6-8
· Numerical Evaluations of Different Equations 9-11
· Relative Deviations of the Theoretically Determined 12
Particle Masses from the Experimental Mean Values for
Different Values of the Gravity Constant
· Relative Deviations of the Theoretically Determined 13
Particle Lifetimes from the Corresponding Experimental
Mean Values
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
2
Table I
Quantum Numbers of Basic States (N=0)
Partikel k n m p s P Q eqx eC Â
e- ,e + 1 0 0 0 0 1 1 -1 0 0
0 0 e ,e 1 0 0 0 1 1 1 0 0 0
m - ,m + 1 11 6 11 6 1 1 -1 0 1
h,h 1 18 22 17 14 0 0 0 0 0
K + ,K - 1 17 26 30 28 1 0 1 1 1
K 0 ,K 0 1 18 5 5 2 1 0 0 1 1
p ± ,p m 1 12 9 2 3 2 0 ±1 0 0
p 0 ,p 0 1 12 3 6 4 2 0 0 0 0
L,L 2 1 3 0 -11 0 1 0 -1 0
W- ,W+ 2 4 4 -1 -15 0 3 -1 -3 0
p, p 2 0 0 0 0 1 1 1 0 0
n, n 2 0 0 -2 17 1 1 0 0 0
X- ,X+ 2 2 7 -17 2 1 1 -1 -2 1
X0 ,X0 2 2 6 -1 6 1 1 0 -2 1
S+ ,S- 2 2 -7 -12 10 2 1 1 -1 0
S0 ,S0 2 2 -7 -14 -2 2 1 0 -1 0
S- ,S+ 2 2 -6 -5 -8 2 1 -1 -1 0
o ++ ,o -- 2 2 1 9 4 3 3 2 0 0
o + ,o - 2 2 -1 -1 -6 3 3 1 0 0
o 0 ,o 0 2 2 -1 -10 2 3 3 0 0 0
o - ,o + 2 2 -1 -16 -15 3 3 -1 0 0
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
3
Tabelle II
Theoretical Data of Elementary Particles
with Mean Lives > 10-16 sec Calculated by B. Heim 1989
(J = spin, P = parity, I = isospin, S = strangeness, B = baryon number)
Type Symbol Mass
MeV
J P I S B Mean Life
10-8 sec
Photons g 0 1 -1 - - 0 ¥
ne 0.00381 ´ 10-6 1/2 - - - 0 ¥
nm 0.00537 1/2 - - - 0 ¥
nt 0.010752 1/2 - - - 0 ¥
n4 0.021059 1/2 - - - 0 ¥
n5 0.207001 1/2 - - - 0 ¥
e 0.51100343 1/2 ±1 - - 0 ¥
e0 0.51617049 1/2 1 - - 0 ¥
Leptons
m 105.65948493 1/2 ±1 - - 0 219.94237553
p± 139.56837088 0 -1 1 0 0 2.60282911
p0 134.96004114 0 -1 1 0 0 0.84016427 ´ 10-8
h 548.80002432 0 -1 0 0 0 0.00233820 ´ 10-8
K± 493.71425074 0 -1 1/2 ±1 0 1.23709835
K0 497.72299959 0 -1 1/2 1 0 5.17900027
Mesons
K0 497.72299959 0 -1 1/2 -1 0 0.00887666
p 938.27959246 1/2 1 1/2 0 1 ¥
n 939.57336128 1/2 1 1/2 0 1 917.33526856 ´ 10 8
L 1115.59979064 1/2 1 0 0 1 0.02578198
S+ 1189.37409717 1/2 1 1 1 1 0.00800714
S- 1197.30443002 1/2 1 1 1 1 0.01481729
S0 1192.47794854 1/2 1 1 1 1 0.42908026 ´ 10-10
X- 1321.29326013 1/2 1 1/2 -2 1 0.01653050
X0 1314.90206200 1/2 1 1/2 -2 1 0.02961947
W- 1672.17518902 3/2 1 0 -3 1 0.01317650
o++,o--- 1232.91663788 3/2 1 3/2 0 1 5.99071759 ´ 10-16
o+,o-- 1234.60981181 3/2 1 3/2 0 1 5.72954997 ´ 10-16
o-,o + 1229.99529979 3/2 1 3/2 0 1 6.74230244 ´ 10-16
Baryons
o0,o 0 1237.06132359 3/2 1 3/2 0 1 5.08526841 ´ 10-16
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
4
Tabelle III
Experimental Data of Elementary Particles
with Mean Lives > 10-16 sec
(J = spin, P = parity, I = isospin, S = strangeness, B = baryon number)
Type Symbol Mass/MeV
(PDG,CERN 2002)
J P I S B Mean Life
10-8 sec
Photons g 0 1 -1 - - 0 ¥
ne £ 5 ´ 10-8 1/2 - - - 0 ¥
nm < 0.17 1/2 - - - 0 ¥
nt <18.2 1/2 - - - 0 ¥
e 0.51099907±0.00000015 1/2 ±1 - - 0 ¥
Leptons
m 105.658389±0.000034 1/2 ±1 - - 0 219.703±0.004
p± 139.57018±0.000351 0 -1 1 0 0 2.6033±0.0005
p0 134.9766±0.0006 0 -1 1 0 0 (0.84±0.06)´10-8
h 547.30±0.12 0 -1 0 0 0
K± 493.677±0.016 0 -1 1/2 ±1 0 1.2384±0.0024
K0 497.672±0.031 0 -1 1/2 1 0 5.2±0.5(Rohlf1994)
Mesons
K0 497.672±0.031 0 -1 1/2 -1 0 0.0089±0.0002 ( " )
p 938.27231±0.00026 1/2 1 1/2 0 1 ¥
n 939.56563±0.00028 1/2 1 1/2 0 1 (886.7±1.9)´10 8
L 1115.683±0.006 1/2 1 0 0 1 0.02632±0.0002
S+ 1189.37±0.07 1/2 1 1 1 1 0.00799±0.00004
S- 1197.449±0.03 1/2 1 1 1 1 0.01479±0.00011
S0 1192.642±0.024 1/2 1 1 1 1 (7.4±0.7)´10-12
X- 1321.32±0.13 1/2 1 1/2 -2 1 0.01639±0.00015
X0 1314.9±0.6 1/2 1 1/2 -2 1 0.029±0.0009
W- 1672.45±0.29 3/2 1 0 -3 1 0.00822±0.00012
D++ »1232 3/2 1 3/2 0 1
D+ »1232 3/2 1 3/2 0 1
D0 »1232 3/2 1 3/2 0 1
Baryons
D- »1232 3/2 1 3/2 0 1
The data are taken from the Particle Data Group homepage https://pdg.lbl.gov , CERN, (2002),
except for the life times of K0 and K0 , which are taken from J.W. Rohlf 1994: Modern Physics
from a to Z0, New York: John Wiley & Sons.
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
5
Table IV
Approximated Meson Resonances
(k=1)
Partikel P N(N)± KB(KB)± Theoretische Masse in
MeV
e 0 49 10 691,7094
w(783) 0 64 51 783,9033
h´(958) 0 144 28 956,8400
S * (993) 0 170 -1 992,6142
F(1019) 0 153 63 1019,6306
f (1270) 0 253 26 1274,5452
D(1285) 0 255 27 1286,1728
E(1420) 0 272 82 1414,1873
f ' (1514) 0 323 2 1517,8602
w(1675) 0 342 71 1664,0125
* (892)
- K 1 23(11)- 29(3) 891,1955(892,2211)
(1240) A K 1 83(69) 6(15) 1241,1180(1239,9767)
K* (1420) 1 98(101) 25(23) 1420,2213(1414,4956)
L(1770) 1 161(164) 65(11) 1775,2145(1764,9862)
p(770) 2 8(5) 30(34) 769,9833(769,3101)
d (970) 2 39(21) 19(5) 976,4931(973,6704)
(1100) 1 A 2 76(48) 41(5) 1106,9780(1106,7462)
B(1235) 2 93(79) 27(10) 1239,5340(1239,1994)
(1310) 2 A 2 127(86) 22(59) 1310,4695(1309,6730)
(1540) 1 F 2 182(145) 37(4) 1539,5100(1537,9095)
p' (1600) 2 215(156) 43(29) 1604,8640(1605,1008)
(1640) 3 A 2 221(160) 4(7) 1637,2669(1634,2138)
g(1680) 2 228(165) 28(5) 1686,0154(1678,6425)
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
6
Table Va
Approximated Baryon Resonances
(k=2)
Partikel P N(N)± KB(K)± theoretische Masse in MeV
N(1470) 1 13(12) 10(38) 1470,4888(1480,1770)
N(1520) 1 14(13) 29(8) 1509,6087(1515,7293)
N(1535) 1 18(17) -2(8) 1533,9788(1535,3254)
N(1670) 1 23(22) 8(0) 1657,9536(1679,5754)
N(1688) 1 24(23) -23(11) 1694,3687(1719,4898)
N(1700) 1 25(27) 63(-12) 1734,6717(1751,2494)
N(1770) 1 26(24) 14(65) 1771,8218(1769,0721)
N(1780) 1 31(29) -9(0) 1784,3644(1782,2884)
N(1810) 1 32(30) 38(40) 1808,3795(1808,5253)
N(1990) 1 37(35) 60(50) 1974,9129(1989,7028)
N(2000) 1 42(39) -3(-37) 2011,0552(2001,9706)
N(2040) 1 44(41) 7(30) 2044,8079(2034,6322)
N(2100) 1 40(44) 78(25) 2107,8085(2120,5890)
N(2190) 1 49(46) -14(21) 2200,5168(2195,5259)
N(2220) 1 50(47) 66(43) 2244,1911(2245,4563)
N(2650) 1 73(69) 2(-9) 2653,5304(2652,4071)
N(3030) 1 90(85) 41(54) 3036,2404(3033,5279)
N(3245) 1 95(90) 61(28) 3234,0166(3231,8730)
N(3690) 1 119(113) 3(4) 3689,8085(3684,1957)
N(3755) 1 113(115) 37(31) 3751,7230(3728,0808)
L(1330) 0 25 10 1329,8831
L(1405) 0 22 79 1403,3999
L(1520) 0 37 36 1516,3419
L(1670) 0 54 4 1669,9762
L(1690) 0 55 61 1693,2832
L(1750) 0 58 25 1754,7613
L(1815) 0 70 -10 1815,4961
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
7
Table Vb
Approximated Baryon Resonances
(k=2 continuation)
Partikel P N(N)± KB(K)± Theoretische Masse in MeV
L(1830) 0 71 11 1830,4081
L(1860) 0 73 -5 1864,6313
L(1870) 0 74 1 1884,4529
L(2010) 0 87 17 2010,5372
L(2020) 0 88 18 2018,1998
L(2100) 0 94 0 2095,9533
L(2110) 0 84 34 2113,6593
L(2350) 0 116 30 2344,7465
L(2585) 0 136 5 2591,7184
X(1530) 1 4(2) 9(5) 1531,5487(1534,7628)
X(1630) 1 7(4) 30(20) 1621,5840(1661,1690)
X(1820) 1 16(10) 35(9) 1828,9065(1810,8367)
X(1940) 1 19(13) 59(27) 1944,8454((1945,2579)
X(2030) 1 25(19) -4(-3) 2027,8157(2037,5528)
X(2250) 1 31(24) 65(-4) 2247,4841(2241,9080)
X(2500) 1 42(35) 42(13) 2481,8202(2517,9008)
D(1650) 3 44 11 1651,0807
D(1670) 3 48 44 1678,6242
D(1690) 3 71 0 1690,0383
D(1890) 3 124 1 1887,9876
D(1900) 3 125 56 1900,8602
D(1910) 3 129 -27 1915,2764
D(1950) 3 134 59 1949,2695
D(1960) 3 137 38 1965,3571
D(2160) 3 211 33 2153,9221
D(2420) 3 302 12 2422,5186
D(2850) 3 419 63 2856,6694
D(3230) 3 572 34 3229,6911
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
8
Table Vc
Approximated Baryon Resonances
(k=2 continuation)
Partikel P (N)+N(N)- (KB)+KB(KB)- Theoretische Masse in MeV
S(1385) 2 (13)+6(13)- (11)+59(22)- (1383)+1382(1386)-
S(1440) 2 (16)8(16) (9)71(-5) (1441)1434(1441)
S(1480) 2 (18)20(18) (64)12(52) (1492)1490(1489)
S(1620) 2 (32)35(32) (18)10(20) (1624)1622(1616)
S(1670) 2 (34)27(35) (8)15(-23) (1664)1660 (1678)
S(1690) 2 (35)38(36) (-10)43(57) (1691)1683((1705)
S(1750) 2 (43)41(38) (-25)34(5) (1752)1747(1750)
S(1765) 2 (45)49(46) (9)10(-2) (1769)1766(1770)
S(1840) 2 (50)45(51) (19)11(47) (1847)1844(1848)
S(1880) 2 (42)57(43) (65)61(7) (1884)1887(1885)
S(1915) 2 (53)59(54) (28)16(24) (1909)1923(1908)
S(1940) 2 (54)60(55) (23)44(-10) (1932)1951(1931)
S(2000) 2 (63)70(64) (8)1(-45) (2003)2012(2002)
S(2030) 2 (66)72(59) (21)12(5) (2035)2031(2031)
S(2070) 2 (68)75(69) (2)38(40) (2066)2071(2064)
S(2080) 2 (69)76(70) (9)29(10) (2083)2089(2074)
S(2100) 2 (70)77(71) (31)52(6) (2103)2106(2093)
S(2250) 2 (76)84(78) (-12)33(35) (2243)2250(2252)
S(2455) 2 (94)104(85) (18)56(3) (2444)2458(2455)
S(2620) 2 (110)121(103) (27)-12(26) (2624)2625(2621)
S(3000) 2 (136)150(140) (-85)38(12) (2994)3001(3003)
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
9
Tables VI
Numerical Evaluations of the Equations V and VIII (chapter E)
symbol numerical value symbol numerical value
h 0,98998964 J 7,93991266
h1,1 0,98756399 J1,1 7,92534503
h1,2 0,98516776 J1,2 7,91095114
h2,2 0,84242385 J2,2 7,04779227
a+ 0,01832211 a- 0,00812835
Numerical Evaluations of the Equations
X and B23,B24,B28 (chapter E+F)
k Qn Qm Qp Qs B H A
1 3 3 2 1 27 9 2787,59025432
2 24 31 34 15 26 104 14727,57867072
Table VII
Numerical Evaluations of the Equations
IX and B8,B9,B10,B13 (chapter E+F)
Ni(k,q ) numerical value Ni(k,q ) numerical value
N1(1,1 ) 0,99688127 N4(1,1 ) 4
N1(1,0 ) 1 N4(1,0 ) 4
N1(2,1 ) 0,99627809 N4(2,1 ) 4
N1(2,0 ) 1 N4(2,0 ) 2
N1(2,2 ) 0,95891826 N4(2,2 ) 6
N2(1,1) 0,67506174 N5(1,1) 1,15773470
N2(1,0) 0,66666667 N5(1,0) 1,15773470
N2( 2,1) 0,67670370 N5( 2,1) 1,73247496
N2(2,0) 0,66666667 N5(2,0) 1,15773470
N2(2,2) 0,79136728 N5(2,2) 76,73214581
N3(1,1) 1,95731764 N6(1,1) 0.00000164
N3(1,0) 2 N6(1,0) 0,00000164
N3(2,1) 2,59881924 N6(2,1) 0,02518725
N3(2,0) 2,71828183 N6(2,0) -0.10493009
N3(2,2) 2,12190443 N6(2,2) 0,15580107
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
10
Table VIII
Numerical Evaluations of the Equations
B22,B29,B30,B31 (Chapter F)
Particle a1 a2 a3 WN=0
e- 35 11 89,96774158 38,70294226
e0 34 28 77,11059862 38,51308957
m 1 23 7,26891022 2830,2632345
p± 25 0 95,62488526 3514,46294316
K+ 16 31 7,26891022 8857,95769020
p0 22 2 -0,03225806 3419,16217346
K0 22 17 98,29474138 9332,35821820
h 28 33 48,65020426 9905,00599107
p 0 23 84,22944059 14792,56308050
S+ 21 30 26,15371691 18124,03136129
S- 21 47 94,49556347 18183,30294347
X- 26 25 15,61504747 18998, 73451193
W- 47 3 69,73881899 23157,61451004
o++ 23 27 82,92386515 18115,38391620
o+ 23 22 22,64335811 18467,56082305
o- 21 27 69,73881899 18448,51703290
n 0 36 101,15000035 14828,61089116
L 13 45 -0,033333333 16827,97671482
S0 21 46 83,86257747 18179,59733741
X0 26 22 71,62409771 18990,08927597
o0 23 39 93,76289283 18508,94119539
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
11
Table IX
Numerical Evaluations of the Equations
XXXV and B48,B49 (Chapter E+F)
Particle Y j L(N)
e- ,e + -408,54063248 0 1021
0 0 e ,e -53,97104336 0 1373
m - ,m + 1086,93016693 2,57120915 2340
h,h 0,26273140×10-8 5,06612007 3236
K + ,K - 184,84508008 -40,78574065 3258
K 0 147,94249859 -12,73395842 3166
K 0 0,25356917 -12,73395842 3166
p ± ,p m 17,08389288 -2,32863274 1485
p 0 ,p 0 3,70004027×10-8 -5,12094079 1833
L,L 0.06178705 0 1964
W- ,W+ 0,09369559 -137,03604095 2062
p, p 17,31698079 9,28034058 1841
n, n 1228,02191382 11,16885467 1932
X- ,X+ 0,10666692 23,44132266 2247
X0 ,X0 0,20184712 90,44612205 2382
S+ ,S- 0,04603481 -6,00947753 5785
S0 ,S0 211,63404729×10-8 11,78154008 6375
S- ,S+ 0.06836890 -2,01125294 5991
o ++ ,o -- 14,72282381×10-16 -1364,07751672 35510
o + ,o - 11,51525605×10-16 -623,74523006 5115
o 0 ,o 0 10,13617609×10-16 -985,00227539 5551
o - ,o + 10,19390807×10-16 -548,14408156 5102
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
12
Relative Deviations of the Theoretically Determined Particle
Masses from the Experimental Meanvalues for
Different Values of the Gravity Constant Gv
Introduction to Heim's Mass Formula
Ó IGW Innsbruck, 2003
13
Relative Deviations of the Theoretically Determined Particle Lifetimes
from the Corresponding Experimental Mean Values
Heim 1989, ? : an experimental value could not be found at the PDG data set,
Heim 1998, measuring uncertainty
Objekt: 31 - 33 av 85
