Dynamical Mass from The Massless Quantum Field Theory
E. J. Jeong
Department of Physics, The University of Michigan, Ann Arbor,
Michigan 48109
Abstract
A method of dynamical mass generation from the massless quantum field theory is presented. The arbitrariness of subtracting the infinities in the self energy loop diagrams is utilized in the process. Even though the conventional renormalization group parameter

is to accommodate the arbitrariness of the renormalization, the finite shift of the self energy diagram seems still free to choose. The conventional renormalization is not necessary to every order so that the symmetry of the formal Lagrangian is not broken.
Still the mass appears due to the finite shift in the self energy diagram. The irregularity of the perturbation ordering and the masslessness of the tree diagrams are indicated to be the major drawback of the scheme.
1. Introduction
The standard Glashow-Weinberg-Salem (1) model of the electroweak interaction has been very successful in predicting the low energy phenomena. With the recent discovery
(2) of the W and Z gauge bosons, the expectations for the detection of the remaining
Higgs scalar have increased dramatically.
However, although quite a few methods of spontaneous symmetry breaking mechanism have been suggested, the mass of the Higgs boson has not been predicted yet. It has been considered for a long time that a better method of symmetry breaking mechanism would be desirable to avoid the proliferation of the Higgs fields and the corresponding coupling constants.
In this note, it is suggested that by the redefinition of the conventional infinities, it is possible to generate mass of the fields without the Higgs mechanism. The symmetry of the original Lagrangian is preserved yet the masses are inherent in the self energy loop diagrams. Even though the renormalization group arguments (3) are supposed to take care of the arbitrariness of the renomalization prescription, one is still free to choose the initial fixed constant after the regularization.
It can be written symbolically
   
C f

C a
, where C f
is a fixed constant parameter which determines the mass and C is an arbitrary running constant. a

In the following, examples are shown within the dimensional regularization scheme. It is straightforward to show that the prescription is independent of the regularization method employed.
2. Examples
(1)

4 theory
Let’s look at the structure of the one loop self energy diagram which is written
One loop= m
0
2
16


2

 n
1

4

1
2

( 2 )

1
2 ln m o
2
4
 2
 
( n

4 )

 , [1] where

( 2 ) is a constant given in general,

( n

1 )

1

1

.......

1
 
(
 
0 .
5772 ....)
2 n and

is an arbitrary constant with the dimension of mass. The reason for the renormalization is obvious for the nonzero bare mass M
0 in the n

4
, since the first term is divergent
limit. However, in the zero bare mass limit, the necessity of the renormalization is not very obvious. The problem can be stated precisely as following.
Which has to be done first? Renormalization or taking the M
0

0 limit?
If one proceeds the renormalization first, one is faced with an awkward situation in the
M
0

0 limit, since there is not a term left. One can define a theory with this prescription, in which the scalar field is strictly massless. However, if one proceeds taking M
0

0 limit first, which is thought to be the correct order, one has the expression, lim m
0 n


0
4

 m
0
2


16

2
C s
,

16
 2

 n
1

4

1
2

( 2 )

1
2 ln m o
2
4
 2
 
( n

4 )



  lim m
0 n


0
4

16
 2 n m
0
2

4
[2]
0 where the undetermined form is expressed in terms of the parameter C s which can be
0 identified in conjunction with the coupling constant

as the mass of the field.
As a result of this prescription, the conventional mass renormalization becomes unnecessary. However, the momentum dependent divergent term appears in the higher order loop diagram which needs to be renormalized as the usual manner. The mass correction terms appear in the form,

 m k
 lim m
0 n


4
0 m
0
2

 n


4

 k
  k
C k
[3] in the successive order.
Therefore, the massless

4 scalar field theory starts to have mass from the one loop self energy diagram. Recalling that the

4 massless scalar field theory is the simplest case of the supersymmetric theories, it also provides us a clue to a possible mass generation mechanism for the supersymmetric particles. What are the advantages of this procedure?
In the cases shown above, the advantage is not very conspicuous since an arbitrary small but finite parameter M
0
is needed to establish the diagram [1] at all, even if it is set to vanish afterwards. However, it becomes obvious in the case of the gauge theories that the advantage is conspicuous especially in QCD. The fact that the mass parameter in the
Lagrangian doesn’t have anything to do with the real mass of the field and its sole purpose is to provide a parameter which has to be determined by experiment has already suggested that they can be eliminated from the original Lagrangian but can be generated by a clever adjustment of the conventional renormalization. It is straightforward to apply this procedure for QED and QCD cases.
(2) QED
The self energy of diagram of the electron without the mass parameter in the Lagrangian is given by

( p )
 n

2 i

4 e
2
16

2
  i e
2
8

2


1
2

( 1
 
)
  1
0 dx

( 1
 x ) ln p
2 x ( 1

4

2 x )

 [4]
The first term is divergent for a nonzero p so that it has to be subtracted by adding a counter term in the Lagrangian.
Now, it is desirable for our purpose to write
  
| p

0
 
| p

0
[5] by which the rest state is separated from the moving state. The self energy diagram can be written with this provision,


( p )

 i e
2
8

2

2 i n

4 e
2
16

2


1
2

( 1
 


| p

0
 det

| p

0
)
  1
0 dx

( 1
 x ) ln
 p
2 x ( 1

4

2 x )



O ( n

4 )
[6]
After subtracting the divergent term, one is left with an arbitrary parameter proportional to e
2 p lim det

0 n


4
, which is defined as the mass of the electron, plus the original finite n

4 mass term. It can be seen that the scale parameter
 can not simply incorporate the finite shift chosen above. As can be seen from the vertex correction in the

4 scalar field theory, the

parameter allows an arbitrary finite shift of the vertex diagram. If that is the case, why not the finite shift for the self energy diagram? Since there are three basic primitive divergences, three arbitrary parameters are necessary to incorporate all the arbitrariness of the renormalization. If

is for the vertex, the other two are for the electron self energy and the vacuum polarization diagram respectively. However, it is not likely that all three parameters can vary arbitrarily (due to the Ward identities). It seems that the other parameters are determined by the initial condition, for example, depending on whether the particle described is an electron or a muon so on.
The successive mass correction terms appear in the form proportional to
 m k

1
 p lim

0 e
2 k
( n
 det

4 ) k k

2 n

4
[7]
The vacuum polarization diagram of the photon can be written,


( p )
 e
2
2

2





 

 2
 

3 ( n
1

4 )

1
6
   1
0 dxx ( 1
 x ) ln p
2 x ( 1

2

2 x )



O ( n

4 ) [8]
Again, the first term is divergent for a nonzero p so that it has to be subtracted by adding a counter term in the Lagrangian. By writing for our purpose,




 

 2 





 

 2 p
 

0
 



 

 2 p


0
[9]
The rest state is separated from the moving state. The vacuum polarization diagram can be written with this provision,



( p )




3 ( n
1

4 )
 e
2
2

2
1
6

 




 

 2

| p

0
  1
0 dxx ( 1
 x ) ln
 det(




 

 2
) | p

0 p
2 x ( 1

2

2 x )



O ( n

4 )

[10] p=0 is not necessary for a pure gauge as written above. However, it ensures the proper limit for the fixed gauge. After the subtraction, one is left with a completely undetermined form proportional to p lim

0

 det(



 n


4


 2
)

 which can be identified n

4 as the photon mass constant C
 and the usual finite term. As is well known, the photon mass is a must for the treatment of the infrared divergence problem.
It can be seen that the effect of the finite shift by the amount of the photon mass constant is equivalent to the rescale of the scale parameter
   e
3 C

[11]
Since the photon mass is considered to be extremely small, the change given by [11] is not significant, however, it is obvious that the effect of C

can not simply be incorporated without affecting the parameter
 in other parts of the diagram, which means that C

has to be an independent contribution which can not be accounted for by
 alone.
The generalization of this prescription to QCD is obvious and straightforward. Thereby the quarks and gluons acquire mass exactly the same way as above. Fortunately, it can be applied to any kind of renormalizable field theories as a mass generation mechanism. The presence of infinity in the diagram is a must for the application of this procedure.
3. Advantages and Drawbacks
An obvious reason to avoid the mass term in the Lagrangian is to preserve the symmetry.
Now, after removing the mass term, the tree diagrams are all massless! How do these two facts have to be reconciled? In fact, it must be noted that this is a generic problem for any kind of dynamical mass generation mechanism since one has to start with a massless theory. A crude and convenient argument is that any kind of the real external current is on shell and dressed, which is a physical requirement rather than a formal one.
A subtle and elaborate treatment on this topic would be necessary to resolve the conflict between the physical and formal requirements, which is left as an open question.
Regardless of these questions, the treatment of QED and QCD in the same footing, generation of the quark mass parameter, compactness of the scheme, consistency,

preservation of the gauge symmetry and the unnecessity of the mass renormalization are the major advantages which can not simply be traded off with the drawbacks of the masslessness of the tree diagrams.
4. Mass and Coupling Constant Relations
It is a well known fact that people have tried to relate the electron mass with the electrostatic potential energy in classical electrodynamics, in which one has to define a fundamental length or the finite radius of the electron, where the electron mass is given by m e e
2

, r
0
[12] where r
0
is the classical radius of the electron. The relationship between the mass and the corresponding charge seems to be a universal feature even apart from the above classical argument.
From a totally different approach as presented in the previous section, the mass of the fields are directly related to the corresponding coupling constants by the following relations.
M s
2 

16

2
C s
C s
 lim m o n


4
0

 n m o
2

4


M e
 e
2
8

2
C e
M

2  e
2
6

2
C

M f
M
2
Y .
M

C
3 g
8

2
2
C f

5
3
C
1

4
3
C
2 g
8

2
2
C
Y .
M
C e
 p lim

0

 det n


4

 n

4
C

C f

C
Y .
M p lim

0

 det(



 n


4


 2
)

 n

4
 n p lim

0

 det n


4



4
 p lim

0

 det(



 n


4


 2
)

 , [13] n

4 where C
1
, C
2
, C
3
are constants determined by the group structure of the nonabelian gauge theory and the subindesis s, e,

, Y.M, f, indicate scalar, electron, photon, Yang-
Mills field and fermion respectively. These mass and the coupling constant relations suggest an interesting variation of mass due to the running coupling constant, more about which will be discussed in a separate note.

5. Internal Braking of The Symmetry
In the case of the free field theories, the mass tern in the Lagrangian seems mandatory for the massive fields. However, in case of interacting field theories, it is not obvious whether the mass term should exist since the coupled equation of motion may well provide mass as a result of the interactions. Since the procedure of renormalization has to be distinguished from the interaction itself, non-breaking of the explicit symmetry as a result of renormalization is not unnatural. Instead, it can be said that the symmetry is broken internally due to the presence of finite shifts which correspond to the mass of the fields. If one tries to build an effective Lagrangian after the stage of the first loop correction, one will automatically have an explicit mass term in the Lagrangian,
Therefore, the fundamental idea of the symmetry breaking is preserved internally, which is in a certain sense truly dynamical.
In this study, one may wonder whether the fields have mass after switching off the interaction, thereby the interacting theory transforms into the corresponding free field theory. According to the argument above, the fields are massless after turning off the interaction, which is correct if one agrees, for example, that the mass of the electron comes from the electromagnetic interaction and so forth.
Now, the more serious question is concerning with the well known analogy with the spontaneous mechanism in the condensed matter physics. It is hoped by many people that the same mechanism holds in the standard electroweak theory. However, it must be noted that the success of the scheme depends seriously upon the existence of the Higgs boson which does not affect the finite shift method presented above.
One of the obvious reasons that one is obliged to break the explicit symmetry is to have the explicit mass terms so that the equation of motion returns into the free field equation after the interaction term is removed, which is a strictly formal procedure. Since the explicit generation of the mass has necessarily introduced the breaking of symmetry, both have been used synonymously as representing the same effect. Now, when it is claimed that the mass can be generated implicitly or dynamically, it is not obvious whether the symmetry has to be broken explicitly as in the usual manner.
6. Conclusions
A method of the dynamical mass generation in the general quantum field theory is presented. The basic scheme is to utilize the arbitrariness subtracting infinites in the fundamental regularization procedure. The conventional mass renormalization is not necessary to every order of the diagram within the scheme. The masslessness of the tree diagrams and the mixed up of the perturbation ordering are observed to be the major drawback of the present scheme. A subtle and elaborate treatment is suggested to be required for the resolution of this problem. Or a certain systematic change of the calculation rule would be desirable.

With the present scheme, the standard SU(2)x U(1) model is a complete theory even without the Higgs field provided the above mentioned calculation rule is fixed. A set of the suggestive mass relations for the fields is also presented as a result of the scheme.
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