Vol.2, No.1, 398-401 (2010) Natural Science
http://dx.doi.org/10.4236/ns.2010.24048
Gauge boson mass generation–without Higgs–in the
scalar strong interaction hadron theory
F. C. Hoh
Dragarbrunnsg, 55C, Uppsala, Sweden;
Received 6 December 2009; revised 23 December 2009; accepted 5 January 2010.
ABSTRACT
It is shown that the gauge boson mass is natu-
rally generated–without Higgs–in the pion beta
decay using the scalar strong interaction had-
ron theory. This mass generation is made pos-
sible by the presence of relative time between
quarks in the pion in a fully Lorentz covariant
formalism.
Keywords: No Higgs; Gauge Boson Mass; Scalar
Strong Interaction
The nonobservation of Higgs, needed in the standard
model [1], has led to various supersymmetry models that
have no experimental support. This gauge boson mass
generation problem is resolved here in the scalar strong
interaction hadron theory (SSI) [2,3], an alternative to
low energy QCD. The equations of motion of mesons,
not yet quantized, read [2,3].

 0,
),(),(
)(
2
)(


III
ea
pr
IIImmprIII
fbpr
ef
II
ba
I
xx
xxMxxx






(1)





rpmprIII
dbpr
IIImmprIII
ec
prdeII
cbI
mmMxxx
xxMxx


2
1
,0,
),(,
)(
2
)(





(2)

 

III
ab
III
ba
IIIbaIII
ab s
IIImIII
xxxxxxxx
g
xx
,,,,
4
,
4






 

(3)
Here, xI and xII are the quark and antiquark coordi-
nates, 
I = /xI,

II = /xII,

and  are the meson wave
functions with the spinor indices a, b,..., undotted and
dotted, running from 1 to 2, 
m the scalar interquark
potential, gs the strong quark charge, mp and mr the quark
masses, and p, r the quark flavors (1 for u and 2 for d
quark). An epistemological background of this theory
has been published earlier this year [4]. Eqs.1-3 have
been rather successfully applied to confinement and
meson spectra [5] and some basic meson decays [6-9].
In these references, the transformation

 2/1,1,  mIImImIII axaxaXxxx

(4)
has been made. The relative space time x
 = (x0,x) are
hidden variables [4] reflecting the fact that no free
quarks exists. Generation of gauge boson mass without
Higgs is shown here by the example of pion beta decay
  0e 
e. Formally, this requires a field theoretical
treatment but here attempt is made to describe such de-
cays on the quantum mechanical level, analogous to
some semiclassical treatments of radiation in quantum
mechanics. The justification is that the energies involved
are low so that field-theoretical effects such as vacuum
polarization are small, just like that analogous effects are
small in QED at low energies.
The starting point is the total action [10]
mLmLrLlGBT SSSSSS 



(5)
3
4
1
1
4
GBl l
l
SdXGG




 
 (6)




kjjkllXlXl WgWWWXG (7)
4
1..
4
ab
LrLa XLb
SidX cc   

 (8)

3 4
3
2
,..
42
2
b
L aa
LlL LXabb
L
ab
WW
ii
SdX gcc
WW



 



 


  









(9)

4
1..
2
a
LmLL La
SdXm cc   

 (10)
where SGB is the action for the gauge boson fields W, W¯
and W3 and SL the SU(2) part of the lepton action in the
standard model. Sm is the SSI meson action generalized
to include SU(2) gauge fields

F. C. Hoh / Natural Science 2 (2010) 398-401
399













 















..
22
44
2
1
ccMM
DDDD
xXddS
aeprmmpr
ea
rp
ea
prmmpraerp
fb
qr
efsqII
ae
rpbapsI
fbqr
ef
sqIIaerp
ba
psI
m






 





(11)
 
 
111
222
1
4
abab ababababab
I
XIpsXpsll ps
ps
ab ab
Ipsl l
ps
DigWX
igW X







 


(12)
 
3
12
3
2,2
2
ab
ab
ll
ps
WW
WX WWiW
WW











 (13)
The superscript  in Eq.11 denotes hermitian conjuga-
tion. Lorentz and gauge invariance of Eqs.4-10 has been
established in [2,3]. Here,

 
 
1
0
2
1
0
2
,
pr





 






(14)
Variation of Eq.11 with respect to

 and

, with
boundary conditions specified in [12], yields



20 ab feae
Ips IIsqmprmpr
qr bf
DD M   

 (15)



20 ae
IpscaII sqedmprm qrpr cd
DD M   

  (16)
In the limit of g
 0, Eq.15 and Eq.16 reproduce
Eqs.1 and 2 together with subsidiary conditions, arising
from the c.c term in Eq.11, that are satisfied at least for
plane wave W, which refers to W here.
Following [8], let the meson wave functions be per-
turbed:



















10
0
0
10
01
0
,
exp
,,1
,,
ab
pr pr
pr
ab
ab
pr prK
pr Kpr K
pr
ab ab
pr pr
pr
ab
pr
Xxaa X
x
xiEXiKX
aX
Xx Xxa
Xx

 


 








 


(17)
Here, the index 1 denotes a first order quantity, apr is
unity here but is in a quantized case to be elevated to an
annihilation operator annihilating a initial meson with
flavor pr. Its complex conjugate *
rs
a is also unity and is
elevated to a creation operator creating a final state with
flavor rs. (1) 0 ()
ps
aX is a small amplitude that varies
slowly with time and, in the quantized case, becomes an
operator that “slowly” transforms the same initial state
meson to some virtual intermediate vacuum state. It is
zero at X0 =
 . (1)* 0 ()
rs
aX is the complex conjugate of
(1) 0
()
ps
aX
and, in the quantized case, becomes an opera-
tor that “slowly” creates the same final state as that cre-
ated by *
rs
a. The subscripts pr have also been attached
to EK and K of the meson. It has been shown that the last
of Eq.4, required by decay applications [8], leads to that
Eq.17 is independent of the relative time x0.
The terms in the actions can now be grouped in pow-
ers of the small parameter g. Only the lowest order and
independent quantities are listed in the two alternatives
below [3]:




bL
La
b
L
a
LGBXpr SXag 

 ,,in , , :orderFirst 01 
(18)
LLGBX SWg


,,in , , :orderFirst





01
:order SecondXapr (19)
Insert Eq.17 into Eq.15 and Eq.16, multiply Eq.15 by

*
0ea rp

 and Eq.16

*
0
dc
rp

, add them together, and inte-
grate over X and x. The first order quantities read
''
msmdSS  (20)
Here, '
md
S is linear in the first order quantity (1)0
()
ps
aX,
'44
md
SdXdx





 

 
 















ea
prdeII
cd
rpprpracI
fbpr
ef
IIaerpprpr
ba
I
aXa
aXa








00
01
0
0
01


(21)
The source part '
ms
S of the first order terms contains
the gW terms in Eq.11, ignoring the c.c. term there,
'44
ms
SdXdx







 






 






























s
ea
srdel
ps
l
cd
rpacI
ea
srdeIIacl ps l
cd
rp
fbsr
ef
l
ps
laerp
ba
I
fbsr
ef
II
ba
l ps laerp
RW
W
W
W
ig















00
00
0
0
0
0
4




(22)
where Rs is a surface term [3] which vanishes upon inte-
gration in Eq.29 below. As a rudimentary quantization
procedure, let



 

WKPfEPi ffi ,, 221122111212


 (23)

F. C. Hoh / Natural Science 2 (2010) 398-401
400
where Pi(12) represents the initial , Pf(1122) the final
state 0 and W the intermediary boson. The subscript K
in Eq.17 is zero for the initial  and is suppressed.
apr in Eq.17 is now elevated to become an annihila-
tion operator according to

 0
prpripr EPa (24)
Similarly, *
rs
a is interpreted as a equivalent creation
operator acting on |0> or an annihilation operator acting
on <f|. Along these lines, the decay amplitude has been
defined as [3]


iXaafS prrpfi.
0)1(   (25)
The zeroth order wave functions for a pseudoscalar
meson at rest are obtained by solving Eqs.1-3 using
Eq.17 and are [3,7]





  ,exp, 0
0XiErxX prpro
baba
pr

(26)



 
Gev864.0
,2exp
8
3
0


m
m
m
propro
d
rd
d
rrr


(27)
Inserting these into Eq.17 and Eq.2 1 and place Eq.20
between <f| and |i> yields:
0
0
2,
fims o
pr
Si fSidx
E




 (28)
where Epr is the mass of the initial . With Eq.22 and
Eqs.26 and 27, Eq.28 becomes


  
40
0 0
π
0
π0 ππ0
3
exp
4
2,
fi
ig
SdXiEEXiKX
E
EEWX KWX
dX
 




 


 

 


(29)
This result can also be obtained starting from either
Eq.15 or Eq.16, without the addition operation men-
tioned below Eq.19.
Variation of Eqs.6 and 7 with respect to ba W
 de-
fined in Eq.13 yields




  WVgWWWS abCcd dc
X
abXab
ba
GB 


2
2
1
4
1
2
1

(30)
where VC is trilinear in W. Variation of the same order
part of Eq.11 yields


***
20(12)0(21) 00
0(12) 0(21)
4
***
0(21)0(12) 00
0(21) 0(12)
/
16
ab
m
ea ea
bf bf
fe
ea ea
bf bf
SW
gdxW

  
 





 










(31)
Inserting Eqs.26 and 27 into Eq.31 yields
abba
mW
g
WS 
 



0
2
2 (32)
The same variation applied to Eqs.8-10 yields
La
bL
ba
L
g
WS 


22

 (33)
where L on the right side refers to e+. With Eqs.30-33,
variation of Eq.5 with respect to ba W
 gives

 
La
bL
ab
W
abCcd
dc
X
abXab
g
WMWVgWW  





2
22
2
1 
(34)


0
22 gMW (35)
MW is the mass of the charged gauge boson [7] and its
square the ratio of an integral over the relative time x0
between the quarks of the pion and the normalization
volume  of the pion wave function. By the last of
Eq.28 and Eq.29, this ratio is     finite. The pions
here also play the role of the Higgs in the standard
model. That Higgs boson is not needed to generate MW
was first shown in [12].
Contracting Eq.34 by ba
ba 

 and yields

LaaLWC
g
WMWVgW
XX



22
0202
0 













(36)

La
bL
ba
W
C
g
WMWVg
W
X
W
XX
W




22
22
0
0






















(37)
Choose the gauge [3] to be the Coulomb type

 0  WX (38)
Further, the ordering Eqs.18 and 19 adopted relegates
the nonlinear g2VC terms in Eqs.36 and 37 to higher or-
der. In the absence of the lepton source terms on the
right of Eqs.36 and 37, it yields to lowest order
0
0 
W (39)
0
2 WMW W (40)
W is identified with the observed charged gauge
boson W [1] with the mass
Gev42.80

W
M (41)
The time component W associated with W in
Eqs.39 and 40 vanishes in agreement with the nonob-
servation of such a singlet charged gauge boson W
accompanying the observed triplet W. If Higgs boson
were used to generate the gauge boson mass, such a
singlet W
with same mass Eq.41 should also be seen,
contrary to observation.

F. C. Hoh / Natural Science 2 (2010) 398-401
401
If the Lorentz gauge
0 
cd
dc
XW
 (42)
were employed, Eq.40 remains unchanged and Eq.39
becomes
0
020  WMW W (43)
This implies that W
has an imaginary mass of Eq.41
and therefore must vanish and Eq.39 remains in effect
valid.
The energy and momentum of the virtual gauge boson
in Eqs.36 and 37 are determined by those of the lepton
pair and are small and can be dropped next to the mass
terms. Hence, Eqs.36 and 37 reduces to
   aLLaW
g
WM 


22
02 (44)
  
 LbaL ab
W
g
WM 



22
2 (45)
While the triplet W can exist freely and hence be seen,
as is shown in Eq.40, it can also be a virtual intermedi-
ate state in Eq.45 . On the other hand, the singlet W
cannot be observed by Eq.39, but can only be a charged,
virtual intermediate singlet as is seen in Eq.44 . These
results are due to that the signs of the 2
W
M
terms in
Eqs.36 and 37 are different, which in its turn stems from
that the meson wave functions Eqs.26 and 27 are not
scalar but the time component of a four vector in SSI. In
pseudoscalar meson decays, only the virtual W enters.
Because Eqs.26 and 27 are independent of flavor, any
pseudoscalar meson can generate the same MW. When
the above treatment is generalized to account for kaon
decay [3], MW is unaltered and the neutral gauge boson
mass becomes MZ=MW/cos (Weinberg angle)=91.02 Gev.
Decay of the W boson into a lepton pair is the same as
that in the standard model. Inserting Eqs.44 and 45 into
Eq.29 leads to a pion beta decay amplitude [3,6] that is
(E0/E)1/2  1 times that of the literature [11] assuming
conserved vector currents.
The value MW      finite cannot and should not be
determined in the present theory so far. If MW were
somehow obtained from some data, it implies a test of
the well-established Fermi constant with far reaching
consequences. This is due to that Fermi constant is
proportional to 2
W
M

and is hence also is a ratio    
finite.
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