On Approximating Fermion Masses in Terms of Stationary Super-String States

doi:10.4236/jmp.2013.44076

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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

Email: [email protected]

Received January 31, 2013; revised March 3, 2013; accepted March 10, 2013

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]:

exp lexp d









 (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:





, (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:

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

Wn n





 : (5)

n = ±1, ±2. Finally it is argued that generalizations of the

proposed second order process generate composites:

dd !

WW n





 (6)

curvature !

WWn







1, 2, 3,,nm

: (7)



 . Once again, each curvature state

(7) is regarded as an element of the proposed physical

J. TOWE

552

geometry if and only if





exp !





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



, to

a massive composite

, where the pairs AL, aA



and

a consist of CPT conjugates. By hypothesis,

these four fields share a common spin magnitude. Each

composite

is therefore characterized by a net

spin magnitude of 2ħ. Finally it is observed that AL,



a and

a can interact super-gravitationally: provided

that composite triplets

, aaa

, aaA

Aa and

AA are each of spin-(3ħ/2); i.e. provided that

a, AL and

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

, aa a

 

 

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

of the generic composite

associate

with a single generation. In the context of this model, the

constituents aL and



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!

a

 

123

aA

  

123

aaa

a

  

123

aA

Figure 1. Pure SUGRA interaction involving postu-lated ge-

neric composites.

= 0.25 GeV/c2, so that





622

6! 0.25GeVc180 GeVcW



, (9)

where is assigned to the quark-lepton state



VI VI

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,

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



0.25 GeVc

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







, where B





B is

an LH bottom quark, where

B is an RH anti-bottom

quark (a mass of about 4.3 GeV/c2), where





is an LH

tauon (a mass of about 1.7 GeV/c2) and where





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









, where CL and C







C respectively represent the

LH charmed quark and the RH anti-charmed; and where







and







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.

I

The mass of expression (13) motivates the association

of (13) with the composite, spin-2ħ field

or eS





, where SR and Se





S respectively represent the RH

strange quark and the LH anti-strange, and where



and

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.

I

30I

The mass of expression (14) motivates the association

of (14) with the average mass of the two spin-2ħ states





 and

, where UL and DL respectively

represent the LH up quark and LH down quark, where







represents a mass-less LH spin-2ħ field and where



 and

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





is not

included in the earlier discussion. Accordingly, the mass

that is described by (10) is interpreted as the spin-2ħ

composite 7





 or 7



. Paralleling the earlier

discussion, the composites 7





 and 7

are

interpreted as lepto-quark states (which consist of conju-

gates that form composites with spin-2ħ fields





), 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





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.







RRR



























RRR

















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

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