Journal of Modern Physics, 2013, 4, 551-554
http://dx.doi.org/10.4236/jmp.2013.44076 Published Online April 2013 (http://www.scirp.org/journal/jmp)
On Approximating Fermion Masses in Terms of
Stationary Super-String States
Joseph Towe
Department of Physics, The Antelope Valley College, Lancaster, USA
Received January 31, 2013; revised March 3, 2013; accepted March 10, 2013
Copyright © 2013 Joseph Towe. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
AdS/CFT correspondence is adopted and fermion masses are modeled as analogues of Weyl curvature states, which
occur by hypothesis when closed spin-2 strings sweep out closed world tubes. Admissible curvature states are estab-
lished by gauge invariance and fundamental mass is attributed to admissible curvature. A consequent spectrum of
masses forms an SU(3) symmetry that is invariant under appropriate realizations of the SUGRA GUT interaction. Fi-
nally the spin-ħ/2 nature of the masses that are attributed to curvature emerges as a necessary condition for the relevant
SUGRA GUT realizations. Calibration of the proposed model reveals a spectrum of fermion masses that corresponds
approximately to observation. Moreover, the proposed model predicts a new quark that is characterized by I3 = −1/2 and
by a mass of about 30 GeV/c2.
Keywords: String Theory; Conformal Field Theory; Quarks; Super-Gravity
1. Introduction
The proposed model is partially motivated by the geome-
try of Hermann Weyl in which the parallel displacement
of a vector around a closed curve admits an increment of
vector magnitude. The theory of Weyl associates incre-
ments of vector magnitude with curvature states and
gauge transformations [1]:
d
exp lexp d
x
l





 (1)
The model that is proposed here replaces the parallel
transports of vectors around closed curves with the dis-
placements of closed, spin-2 strings that sweep out closed
world tubes; i.e. the model that is proposed here replaces
Weyl curvature with curvature states that associate with
increments of string scale:
dS
WS 

, (2)
this analogue of Weyl curvature will be known as W-cur-
vature. Based upon a hypothesis introduced by F. Lon-
don, a curvature state (2) will be regarded as admissible
if and only if gauge is preserved in the sense that



expexp 2πiWn: (3)
n = ±1 [2]. An admissible curvature state will be regard-
ed, by hypothesis, as a curvature state that can occur in
the physical world; i.e. as an element of a proposed phy-
sical geometry.
Secondly, the proposed model considers the compound
world tube that is generated when the outer circumfer-
ence of a closed world tube itself sweeps out a closed
world tube:
2
00
dd 2!
WW W
WW
 . (4)
The composite (4) is regarded as a second order ele-
ment of the proposed physical geometry if and only if
each of the two world tubes that together constitute the
composite (4) preserves gauge in the sense that


exp!exp 2πi
n
Wn n


 : (5)
n = ±1, ±2. Finally it is argued that generalizations of the
proposed second order process generate composites:
00
dd !
WW n
W
WW
n


 (6)
or
curvature !
n
WWn



1, 2, 3,,nm
: (7)

 . Once again, each curvature state
(7) is regarded as an element of the proposed physical
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J. TOWE 552
geometry if and only if


exp !
n
Wn


exp 2πin
1, 2,,nm 
: (8)
W > 0, [3]. 
AdS/CFT correspondence and the string background
AdS7XS4 are now adopted by hypothesis [4,5].
2. Theoretical Lepto-Quark States
The W-curvature classes (7) (established as admissible by
(8)) will now be associated with specific masses, which
are regarded as equivalent to the W-curvature states that
are described by (7) (the proposed model parallels Whee-
ler’s ideal, which attributes mass to curvature [6]). These
masses are initially regarded as significant because they
are attributed to curvature, but they assume additional
importance in the context of the proposed model because
they establish a context in which spin-(ħ/2) is intrinsi-
cally associated with postulated orientation states AL and
aL as a necessary condition for a relevant realization of
SUGRA GUT interactions. Specifically an admissible,
closed world tube (or a closed, compound world tube) is
interpreted, by hypothesis, as a transformation from the
mass-less, spin-2ħ field, which will be designated
L

, to
a massive composite
L
LR
, where the pairs AL, aA 
R
A
and
L
a,
R
a consist of CPT conjugates. By hypothesis,
these four fields share a common spin magnitude. Each
composite
L
LR
is therefore characterized by a net
spin magnitude of 2ħ. Finally it is observed that AL,
aA 
R
A
,
L
a and
R
a can interact super-gravitationally: provided
that composite triplets
L
LL
, aaa
L
LL
, aaA
L
LL
A
Aa and
L
LL
A
AA are each of spin-(3ħ/2); i.e. provided that
L
a,
R
a, AL and
R
A
are individually characterized by a spin
of ħ/2; and provided that the above described triplets are
characterized by a second quantum number that ranges
over three values and that the consequent triplets
  
123
L
LL
, aa a
 
12
 
3
L
LL
etc., each represent sufficient
diversity in this new quantum number to avoid violation
of the Pauli exclusion principle. Subsequently the quan-
tum numbers “(1)”, “(2)” and “(3)” will be regarded as
referring to “generation”. It should be noted that if the
SUGRA GUT interaction depicted by Figure 1 is to oc-
cur (in the relevant forms which preserve the Figure 2
symmetry), then it is necessary that the constituents aL
and
aa A
R
A
of the generic composite
L
LR
associate
with a single generation. In the context of this model, the
constituents aL and
a  A
R
A
are respectively interpreted as a
lepton and an anti-quark of the same generation.
3. Calibration
The postulated model is calibrated by identification of
the W-curvature class W6 with the most massive lepto-
quark state that is exclusively characterized by I3 = +1/2.
Specifically, W6/6! is assigned the value [180 GeV/c2]/6!
L
A
LR
a
 
123
L
A
LL
Aa
L
LR
aA
  
123
L
LL
aaa
L
LR
A
a
   123
L
LL
A
Aa
L
LR
aA
t
Figure 1. Pure SUGRA interaction involving postu-lated ge-
neric composites.
= 0.25 GeV/c2, so that


622
6! 0.25GeVc180 GeVcW
6
W

, (9)
where is assigned to the quark-lepton state

VI VI
L
LRL LR
AaT






and where 180 GeV/c2 is belive-
ed to be the approximate mass of the top quark (Since the
mass of the tauon’s neutrino is relatively negligible, the
mass (9) is also regarded as approximating that of the
heaviest lepto-quark state that is characterized by I3 =
+1/2). The Roman numerals I through VI in Equation (9)
through (14) refer to six states that will subsequently be
identified as “six generations of I3 conjugates”.
The following equations are equivalent to (9) and col-
lectively indicate a spectrum of masses:


522
5!0.25 GeV c30GeV c,
VV
LLR
WA a 

(10)


422
4! 0.25GeVc6GeVc,
IV IV
LLR
WA a 

(11)


322
3!0.25GeV c1.5GeV c,
III III
LLR
WA a 

(12)


222
2!0.25 GeVc0.5GeV c
II II
LLR
WA a 

, (13)
and
 2
0.25 GeVc
II
LLR
WWA a , (14)
Interpretation of the mass (10) will be deferred until
the massive states described by expressions (11) through
(14) have been interpreted. The theoretical mass repre-
sented by (11) motivates the association of (11) with the
spin-2ħ composite
L
LR
B



or
L
LR
, where B 

L
B is
an LH bottom quark, where
R
B is an RH anti-bottom
quark (a mass of about 4.3 GeV/c2), where
L


is an LH
tauon (a mass of about 1.7 GeV/c2) and where
R


is an
RH anti-tauon. The bottom quark and the tauon are re-
garded as I3 = −1/2 partners in the heavy generation.
The mass of expression (12) motivates the association
of (12) with the composite spin-2ħ field
L
LR
C




or
L
LR
, where CL and C



R
C respectively represent the
LH charmed quark and the RH anti-charmed; and where
L



and
R



respectively represent the LH muon’s
neutrino and the RH anti-muon’s neutrino. The charmed

J. TOWE 553
quark and the muon’s neutrino are regarded as I3 = +1/2
partners in the moderately heavy generation.
3
1
2
I
The mass of expression (13) motivates the association
of (13) with the composite, spin-2ħ field
L
LR
or eS 

L
LR
, where SR and Se 

L
S respectively represent the RH
strange quark and the LH anti-strange, and where
R
e

and
L
e respectively represent the RH electron and the
LH anti-electron. The strange quark and the right-hand-
ed electron are (in the proposed model) regarded as I3
partners in the light generation.
3
1
2
I
30I
The mass of expression (14) motivates the association
of (14) with the average mass of the two spin-2ħ states
e
L
LR
U


 and
L
LR
, where UL and DL respectively
represent the LH up quark and LH down quark, where
De 

L

represents a mass-less LH spin-2ħ field and where
e
L

 and
L
e respectively represent the LH electron’s
neutrino and the LH electron. The up quark and the LH
electron’s neutrino are regarded as I3 = +1/2 partners in
the light generation, and the down quark and LH electron
are regarded as I3 = −1/2 partners in the light generation.
Note that the masses described by expressions (11), (12),
(13) and (14) are approximately equal to those deter-
mined by observation [7,8].
To interpret the mass that is described by expression
(10), it is first observed that the LH muon
L


is not
included in the earlier discussion. Accordingly, the mass
that is described by (10) is interpreted as the spin-2ħ
composite 7
L
LR


 or 7
L
LR

. Paralleling the earlier
discussion, the composites 7
L
LR


 and 7
L
LR
are
interpreted as lepto-quark states (which consist of conju-
gates that form composites with spin-2ħ fields

L

), the
constituents of which share an I3 classification and a
generational classification. Thus the 7L is interpreted as
an unobserved LH quark that is characterized by I3 =
−1/2 and is regarded as an element of the moderately
heavy generation. Finally, since the mass of the
L


is
relatively negligible, the mass of the newly predicted
quark that is associated with expression (10) will be des-
ignated as approximately 30 GeV/c2.
The spin-2ħ composites (9) through (14) are associated
with three generations of I3 generators. These elements
are therefore interpreted as constituting an SU(3) sym-
metry, the preservation of which is regarded as equiva-
lent to the conservation of I3 and generation:
Referring to Figure 2, the counterclockwise sequence
beginning at 180 GeV/c2 indicates, in order of occurrence,
the masses that are described by expressions (9) through
(14). Again the mass 0.25 GeV/c2 and the associated I3
value of I3 = 0 represent the averages of the masses and I3
numbers of the up quark (together with its I3 lepton part-
ner) and the down quark (together with its I3 lepton part-
ner).
Note that the Figure 2 symmetry is partially broken
because the vertices represent a variety of masses. Note
however that an appropriate realization of the SUGRA
Figure 2. World symmetry.
e
L
LR
U 


RRR
TU



L
LR
T



e

R
RR






e
RRR
TU



L
LR
U 

t
L
LR
T




Figure 3. A pure SUGRA GUT interaction.
GUT interaction can preserve the partially broken sym-
metry that is depicted in Figure 2 by beginning and ter-
minating at a single vertex. Specifically, the Figure 2 sym-
metry is preserved by the SUGRA GUT interaction.
Note that preservation of the Figure 2 symmetry by
the Figure 3 interaction is indeed equivalent to the con-
servation of I3 and generation by the Figure 3 interac-
tion.
4. Conclusion
The proposed model is geometrically characterized by
Ricci flatness (all curvature is identified with the ana-
logues (7) of Weyl curvature) and by the SU(3) symme-
try that is described by Figure 2. This model appears to
represent an increment of progress because it is charac-
terized by a uniqueness that is established by gauge in-
variance and because significant physical consequences
and predictions emerge.
5. Acknowledgements
Figures designed by R. James Towe.

J. TOWE 554
REFERENCES
[1] H. Weyl, “Space, Time and Matter,” Metheun and Co.
Ltd., London, 1922.
[2] F. London, “Quantenmechanische Deutung der Theorie
von Weyl,” Zeitschrift fur Physik, Vol. 42, No. 5-6, 1927,
pp. 375-389. doi:10.1007/BF01397316
[3] J. Towe, “Stationary String States as Theoretical Fer-
mions,” Advanced Studies in Theoretical Physics, Vol. 6,
No. 8, 2012, pp. 369-377.
[4] O. Aharony, S. Gubser, J. Maldacina, H. Ooguri and Y.
Oz, “Large N-Theories, String Theory and Gravity,” 1999,
in press.
[5] E. D’Hoker and D. Freedman, “Super-Symmetric Gauge
Theories and the AdS/CFT Correspon-Dance,” 2002, in
press.
[6] J. A. Wheeler, “Curved, Empty Space as the Building
Material of the Physical World,” In: E. Nagel, Ed., Logic,
Methodology and the Philosophy of Science, Stanford
University Press, Palo Alto, 1962.
[7] R. Bock and A. Vasilesca, “Particle Detector Brief
Book,” 1999.
[8] SLAC Virtual Visitor Center, “SLAC National Accelera-
tor Laboratory,” Menlo Park, 2009.