A New Field Theoretical Definition of the Mass of the Elementary Particles
E. J. Jeong
Department of Physics, The University of Michigan, Ann Arbor,
Michigan 48109
The mass of the elementary particles are defined within the field theoretical boundaries in
terms of the conventional renormalization using their property of becoming a completely

undetermined form in the limit m0  0 , n  4 , p  0 .
At the present, there is no convincing theory that explains the origin of the mass in the
gauge theories of the elementary particles(1). Especially QCD(quantum chromo
dynamics) is not clearly understood in the sense that the Lagrangian does not allow a bare
mass term and yet the quarks are supposed to have mass(2). Any information regarding
the mass of the fields obtained directly from the known calculational tool of the
Lagrangian will surely help understanding the mass of the fields in this regard.
With the confidence that QED and QCD should have such information that may be
hidden, we reexamine the self energy diagrams and their infinities. Surprisingly enough,
it can be shown that all the self energy diagrams of the renormalizable field theory have
0
the same property of becoming a completely undetermined form of , or 0   , in the
0

limit m0  0 , n  4 , p  0 .
It may be the nature’s way of the theories telling us that the mass of the fields can not be
precisely determined by the gauge theories alone so that the exact value of the mass has
to be determined by experiments. In fact, we have seen the similar situation in the
spontaneous symmetry breaking mechanism due to the presence of the Higgs(3) boson.
With these limitations in mind, we define the masses of the field in the following.
(1) Scalar field in the  4 theory
M s2 
lim  2 
mo
mo  0  16 2
n4
 1

mo2
1
1



(
2
)

ln
 0(n  4)  
Cs ,

2
2
2 4
n  4 2
  16
[1]
where  is an arbitrary scale parameter with the dimension of mass and  (2) is the
Euler-Mascheroni constant, and C s 
lim  mo2 

.
mo  0  n  4 
n4
(2) Electron mass from QED
Me 
e2
Ce
8 2
Ce 
lim  det  
p  0  n  4 
n4
[2]
(3) Photon(QED)
M 2 
e2
C
6 2
C 
2
lim  det(       ) 


p  0 
n4

[3]
n4

where p  0 is redundant for a pure gauge, however it ensures a proper limit for the
fixed gauge.
(4) Quark(QCD)
g2
M q  C3 2 C q
8
Cq 
4
lim  det  
, C3 


3
p  0 n  4
n4
[4]
(5) Gluon (Yang-Mill boson) from QCD
M
2
Y .M
4  g2
5
  C1  C2  2 CY .M
3  8
3
CY .M
2
lim  det(       ) 


 , [5]
p  0 
n4

n4
where C1 , C2 , C 3 are constants determined by the group structure of the nonabelian
gauge and the flavor number dependent constant.
Therefore, one can define the mass of the gauge fields as following. Suppose one loop
self energy diagram of a renormalizable field theory can be expressed by
1


f ( g ) h1 ( p, mo )
 h2 ( p, mo )
[6]
n4


in the dimensional regularization scheme, where g is a coupling constant and h2 ( p, mo ) is
a finite term.
In general, the mass of the fields can be defined by,
1


M N  lim  f ( g )
det h1 ( p, mo )
n4
n4

mo 0 
[7]
p 0
where N=1 for a fermion and N=2 for a boson.
The advantages of the above definition of the mass are;
1. It respects the gauge invariance of the Lagrangian ( m0  0 ).
2. It is natural in the sense that they are derived from the self energy diagrams.
3. It does not require an arbitrary constant put in by hand other than the one that is
generated by the limit taken by the definition.
4. It provides a heuristic mass ratio of the electron vs. quark and the photon vs. gluon.
5. It fulfills the paradigm of the 4-dimensional space.

6. The limit p  0 makes the masses to be of purely non kinetic origin so that one can
identify them easily as the rest mass or the potential mass of the fields.
Also, these masses do not depend on the regularization scheme employed. Since the
coupling constants are known to develop the scale dependences(4), we are confronted
with a nontrivial fact that the mass defined here also depends on the momentum scale.
We will discuss the implications of these results elsewhere.
References
[1] K. Huang, “Quark, Leptons, and Gauge Fields”, World Scientific Pub. Co. (1982).
[2] W. Marciano and H. Pagels, Phys. Reports 36C (1978) 141, see also the references
within.
[3] S. Weinberg, Phys. Rev. D13 (1976) 974; Phys. Rev. D19 (1978) 1277; L. Susskind,
Phys. Rev. D20 (1979) 2619; G. ‘tHooft, Lectures at Cargese Summer Institute (1979);
P. Langacker, Phys. Reports 72(1981) 185.
[4] P. Carruthers, Phys. Reports 1C (1971) 3; D. Gross and F. Wilczek, Phys. Rev. Lett
26 (1973) 1343; H. D. Politzer Phys. Rev. Lett 26 (1973) 1346