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Vector-meson mass generation in the chiral Schwinger model
Volume 224, number 3
PHYSICS LETTERS B
29 June 1989
V E C T O R - M E S O N M A S S G E N E R A T I O N IN T H E C H I R A L S C H W I N G E R M O D E L She-Sheng X U E
Dipartimentodi Fisica, Universit~di Milano and 1NFN- Sezionedi Milano, Via Celoria 16, 1-20133Milan, Italy Received 17 October 1988
It is shown that an arbitrary mass is generated for the vector meson in the chiral Schwingermodel, a model which has caused some controversy. Our arguments are based on ambiguities in the dimensional regularization of quantum field theory with 75.
1. O f all the two-dimensional field-theoretical models, the Schwinger model [ 1 ] has probably been studied most extensively. It is an exactly solved model and is equivalent to a free massive gauge theory in which the mass of the gauge boson caused by quantum effects is eZ/Yg (here e is a vector-like coupling with the gauge field in this model ). On the contrary, in the chiral Schwinger model, the fermions have a ( V+A )-like coupling with gauge bosons, i.e. only right- or left-handed chiral fermions couple to the gauge fields. This chiral Schwinger model has recently aroused some controvercy [2,3 ]. In ref. [3 ], Jackiw and Rajaraman exploited the fact that the fermionic determinant has some arbitariness associated with the regularization of fermionic radiative connections, and therefore an arbitrary parameter a depending on the regularization scheme exists in the effective action on the chiral Schwinger model:
e 2 (ag~'~- (g~'~+ ~'~) ~~,~ l[A ] = f d2x [ - ~Fu,F ~ + -~nA~, - (g~._c~))A. 1
(1)
with gauge boson mass
mZ=eZa2/4n(a- 1 ).
(2)
For a > 1, they pointed out that eqs. ( 1 ), (2) can be defined as an unitary, renormalizable and consistent theory although it suffers from anomalous non-conservation of the gauge current. This result sparked interest and some authors studied the problem further in different ways [4]. Most of them arrived at the same conclusion as suggested by ref. [ 3 ], eqs. ( 1 ), (2). In this letter we shall show that the arbitrariness in eqs. ( 1 ), (2) may arise from ambiguities in the definition of the matrix Y5 in the dimensional regularization scheme. 2. In what follows, we shall work in euclidean space, in which the analytical continuation to d-dimensional spacetime has been performed, and, hence, where the antihermitian y-matrices satisfy the rules {7,,,Y,}=-26,,, tr
Y*~=-Yu,
72 = - 1
for/~,u=l,2,3,...,d,
I=f(d) = f ( a ) + f , ( a ) ( a - d) + ....
(3)
Here f ( d ) is an arbitrary function satisfying the condition f ( d ) = d (d denotes physical dimensions). The problem we face now is how to define the matrix Ys. Here we adopt the approach suggested by 't Hooft and Veltman [ 5 ]. The matrix 75 is defined as usual for the physical dimensions/2 = 0, 1 as Ys =iyoYl,
{Ys, Y~}= 0 ,
y~- = -Ys,
y2 _- _ 1.
0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
(4) 309
Volume 224, number 3
PHYSICS LETTERS B
29 June 1989
For unphysical dimensions p__,the Yu's and Y5 given above commutate:
[ys, ~,,] =0.
(5)
From the definition, the following relations are easily proved: 7l~7~ = i ~ 7 5 ,
~_,75= 75 7~,
(6)
where e~ is a 2D antisymmetric tensor. These definitions are not Lorentz invariant on the full space, but only on the first two physical dimensions. The lack of full Lorentz invariance is a nuisance, but it does give the correct result later on. With the above necessary preliminaries, we are in a position to consider the radiative corrections of the chiral Schwinger model in euclidean space. The action reads
Iz = f dax {~[i~-ey¢,(1
+ i75)A~,] ~'+
~Fu,,F~,,,+ ½).(0.A.) 2}
(7)
and the corresponding F e y n m a n n rules are
-~
~
p2 ,
.u
: k2
u
,
:e[7~,(1+i75)].~.
(8)
The vacuum polarization in this case therefore is k
;
~
k
~ o
~ --e 2
~ dap
tr[gbyo(l+i75)(~b+~)Tp(l+i75)]
p2(p'4"k)2
j (27¢)d
= - - e 2 tr[7~Tp(1 +i75)7~7~(1 +i75)] ×
1 ~B(½d,
1(
d~2 _e2tr[7,,Tp(lq_i~s)?v?~(l+i?5) ] ~
k2 +
(k2)d/2-'( k~,k, ½d)) ½d) \-~Sj - --~5-F(2-½d)+½6~,F( 1,
(9)
where e=d-2. We have to be very careful to treat the trace of the y-matrices in eq. (9), which involves the matrix 75, because of the indices p, a associated with the external gauge fields Ap and Ao, which cannot be unphysical components at the end of the calculations. On the contrary, the indices/x, v associated with internal m o m e n t a are unphysical dimensions, in addition to the physical dimensions. Therefore, eq. (9) can be rewritten as follows: P
;
- ~
= tr [ 7¢y/~( 1 + iy,) 7; yo( 1 + i7,) ]F~o
+ tr [ 7~_W¢(1 +iYs)7~7~( 1 + i 7 , ) ]F~,~ + tr [~,_,7¢( 1 + its) y._)~o(1 + i t s ) IF~_.,
(10)
where F.~ is
F,~/j=_e21( 2g
k~ka k2
+
~_)
.
(11)
The second term of eq. (10) vanishes for I ~ = 0, which can be worked out by eq. ( 11 ). The first term of eq. (10) is straightforwardly calculated:
- 2e---~2~r(fip,~-ep.) 310
k2
+6.~
(6~+e/j.).
(12)
Volume 224, number 3
PHYSICS LETTERS B
29 June 1989
Here, the indices p, c~, fl, a are integral numbers 0, 1 and we drop ^ for simplicity. Obviously, the third term of eq. (10), in which there are two sorts of algebraical relations between the matrices ?'3, the y~_,'sand the ?'~'s is then ill-defined. It has to be separated into two parts to avoid this ambiguity; the parts correspond to the algebra (4) or ( 5 ), respectively. Thus, tr [?'~_?'i~(1 +i?'s) ?',?'o( 1 +iys) ] F~_,~_ = ( 1 - b 2) tr{yu[?'~( 1 +iy5 ) ]L~ [?'e( 1 + i?'5) ]} Fu_~+ b 2 tr{7¢[ ( 1 + iy5) ?',] ?'el ( 1 + i?'5)?', ] } F_~_,
(13)
where the arbitrary positive constant b 2 is due to the ambiguity mentioned above. It is easy to show that the second term ofeq. (13) is zero because ofeq. (5) and ( 1 - i?'s) ( 1 +i?'s) =0. The first term ofeq. (13) becomes (1 - b 2) tr{T,_[?'~(1 +i?'s) ] 7,_[?'e(1 +i75)]} Fa__e = ( 1 - b 2 )~_wtr (7,_,?'a ?'._?'~) (6a~ - cab) (6~e - eda) = - ( 1 - b 2 ) 4F~,~,a_~2 ( 1 - b 2)2e2/n.
( 14 )
Taking eqs. ( 11 ) - (14) into account, we get the one-loop vacuum polarization of the chiral Schwinger model:
k
(~
P
2e2(
k =--
(~,.-E,~)
k.kp
)
which gives then the following expression for the effective action of the theory: I~=
fl
d2x
~FZu.+½a(OuAu)2+ T A p ( x )
(
(l+b2)6p~-(6p~-ep.)
0~8, (6~,, + t~,~) ]Ao(x)
)1
.
(16)
This expression is the same as that suggested by Jackiw and Rajaraman [ 3 ]. Looking at eq. (16), one finds that the arbitrary parameter a = ( 1 + b 2) > 1 is always valid [ b 2 = 0 is a trivial case in the separation ( 13 ) ] and the theory is thus consistent. All subsequences and interpretations of the model for arbitrary a, as envisaged in refs. [ 3,4 ] therefore remain intact. This obviously refutes the assertion in ref. [2] that the effective action in this model is unique. 3. Let us examine where the arbitrary constant comes from and how it gets in this approach. The separation in eq. (13) seems to be a trivial identity at first glance. For the Schwinger model without an existing ?'5, the separation in eq. ( 13 ) is indeed trivial, and produces nothing. On the contrary, for the chiral Schwinger model with an existing ?'5, eq. (13) is no longer trivial since one has to divide the trace of the ?'-matrices so as to give definite algebraical relations for the matrices ?'3, the ?'u's and the ?'~'s. It seems that the arbitrary constant b 2 is introduced manually for the freedom of such a separation. However, the fact that the second term of eq. ( 13 ) vanishes by virtue of the algebra ( 5 ) leads to the explicit existence of arbitrariness in the theory. In this sense, obviously, chiral anomalies arise from the one-loop quantum correction, force the spontaneous breaking of gauge symmetry, which is valid at the classical level, through the generation of a mass for the gauge boson. The arbitariness is caused by the regularization freedom of the chiral Schwinger model. The author is grateful to Dr. T.D. Kien for discussions when he was at the University of Edinburgh. The initiate part of this work was supported by SERC Grant No. NG17094. The author is now supported by a postdoctoral fellowship of the Istituto Nazionele di Fisica Nucleare, Italy.
References
[ l ] J. Schwinger, Phys. Rev. 128 (1962) 2425. [2] C.R. Hagen, Ann. Phys. (NY) 81 ( 1973 ) 67; Nuovo Cimento 5 IA (1967) 1033; 5 IB (1967) 169; A. Das, Phys. Rev. Lett. 55 (1985) 2126. 311
Volume 224, number 3
PHYSICS LETTERS B
[3] R. Jackiw and R. Rajaraman, Phys. Rev. Lett. 54 (1985) 1219, 2060 (E); 55 (1985) 2224 (C). [4] R. Banerjee, Phys. Rev. Lett. 56 (1986) 1889; K. Harada, H. Kubota and I. Tsutsui, Phys. Lett. B 173 ( 1986 ) 77; A.J. Niemi and G.W. Semenoff, Phys. Lett. B 175 ( 1986 ) 439; D. Boyanovsky, preprint UCD-PUB-87-1; I.G. Halliday, E. Rabinovici and A. Schwimmer, Nucl. Phys. B 268 (1986) 413; M.S. Chanowitz, Phys. Lett. B 171 (1986) 280. [5] G. 't Hooft and M. Veltman, Nucl. Phys. B 44 (1972) 189; P. Breitenlohner and D. Maison, Commun. Math. Phys. 52 (1977) 11.
312
29 June 1989
PHYSICS LETTERS B
29 June 1989
V E C T O R - M E S O N M A S S G E N E R A T I O N IN T H E C H I R A L S C H W I N G E R M O D E L She-Sheng X U E
Dipartimentodi Fisica, Universit~di Milano and 1NFN- Sezionedi Milano, Via Celoria 16, 1-20133Milan, Italy Received 17 October 1988
It is shown that an arbitrary mass is generated for the vector meson in the chiral Schwingermodel, a model which has caused some controversy. Our arguments are based on ambiguities in the dimensional regularization of quantum field theory with 75.
1. O f all the two-dimensional field-theoretical models, the Schwinger model [ 1 ] has probably been studied most extensively. It is an exactly solved model and is equivalent to a free massive gauge theory in which the mass of the gauge boson caused by quantum effects is eZ/Yg (here e is a vector-like coupling with the gauge field in this model ). On the contrary, in the chiral Schwinger model, the fermions have a ( V+A )-like coupling with gauge bosons, i.e. only right- or left-handed chiral fermions couple to the gauge fields. This chiral Schwinger model has recently aroused some controvercy [2,3 ]. In ref. [3 ], Jackiw and Rajaraman exploited the fact that the fermionic determinant has some arbitariness associated with the regularization of fermionic radiative connections, and therefore an arbitrary parameter a depending on the regularization scheme exists in the effective action on the chiral Schwinger model:
e 2 (ag~'~- (g~'~+ ~'~) ~~,~ l[A ] = f d2x [ - ~Fu,F ~ + -~nA~, - (g~._c~))A. 1
(1)
with gauge boson mass
mZ=eZa2/4n(a- 1 ).
(2)
For a > 1, they pointed out that eqs. ( 1 ), (2) can be defined as an unitary, renormalizable and consistent theory although it suffers from anomalous non-conservation of the gauge current. This result sparked interest and some authors studied the problem further in different ways [4]. Most of them arrived at the same conclusion as suggested by ref. [ 3 ], eqs. ( 1 ), (2). In this letter we shall show that the arbitrariness in eqs. ( 1 ), (2) may arise from ambiguities in the definition of the matrix Y5 in the dimensional regularization scheme. 2. In what follows, we shall work in euclidean space, in which the analytical continuation to d-dimensional spacetime has been performed, and, hence, where the antihermitian y-matrices satisfy the rules {7,,,Y,}=-26,,, tr
Y*~=-Yu,
72 = - 1
for/~,u=l,2,3,...,d,
I=f(d) = f ( a ) + f , ( a ) ( a - d) + ....
(3)
Here f ( d ) is an arbitrary function satisfying the condition f ( d ) = d (d denotes physical dimensions). The problem we face now is how to define the matrix Ys. Here we adopt the approach suggested by 't Hooft and Veltman [ 5 ]. The matrix 75 is defined as usual for the physical dimensions/2 = 0, 1 as Ys =iyoYl,
{Ys, Y~}= 0 ,
y~- = -Ys,
y2 _- _ 1.
0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
(4) 309
Volume 224, number 3
PHYSICS LETTERS B
29 June 1989
For unphysical dimensions p__,the Yu's and Y5 given above commutate:
[ys, ~,,] =0.
(5)
From the definition, the following relations are easily proved: 7l~7~ = i ~ 7 5 ,
~_,75= 75 7~,
(6)
where e~ is a 2D antisymmetric tensor. These definitions are not Lorentz invariant on the full space, but only on the first two physical dimensions. The lack of full Lorentz invariance is a nuisance, but it does give the correct result later on. With the above necessary preliminaries, we are in a position to consider the radiative corrections of the chiral Schwinger model in euclidean space. The action reads
Iz = f dax {~[i~-ey¢,(1
+ i75)A~,] ~'+
~Fu,,F~,,,+ ½).(0.A.) 2}
(7)
and the corresponding F e y n m a n n rules are
-~
~
p2 ,
.u
: k2
u
,
:e[7~,(1+i75)].~.
(8)
The vacuum polarization in this case therefore is k
;
~
k
~ o
~ --e 2
~ dap
tr[gbyo(l+i75)(~b+~)Tp(l+i75)]
p2(p'4"k)2
j (27¢)d
= - - e 2 tr[7~Tp(1 +i75)7~7~(1 +i75)] ×
1 ~B(½d,
1(
d~2 _e2tr[7,,Tp(lq_i~s)?v?~(l+i?5) ] ~
k2 +
(k2)d/2-'( k~,k, ½d)) ½d) \-~Sj - --~5-F(2-½d)+½6~,F( 1,
(9)
where e=d-2. We have to be very careful to treat the trace of the y-matrices in eq. (9), which involves the matrix 75, because of the indices p, a associated with the external gauge fields Ap and Ao, which cannot be unphysical components at the end of the calculations. On the contrary, the indices/x, v associated with internal m o m e n t a are unphysical dimensions, in addition to the physical dimensions. Therefore, eq. (9) can be rewritten as follows: P
;
- ~
= tr [ 7¢y/~( 1 + iy,) 7; yo( 1 + i7,) ]F~o
+ tr [ 7~_W¢(1 +iYs)7~7~( 1 + i 7 , ) ]F~,~ + tr [~,_,7¢( 1 + its) y._)~o(1 + i t s ) IF~_.,
(10)
where F.~ is
F,~/j=_e21( 2g
k~ka k2
+
~_)
.
(11)
The second term of eq. (10) vanishes for I ~ = 0, which can be worked out by eq. ( 11 ). The first term of eq. (10) is straightforwardly calculated:
- 2e---~2~r(fip,~-ep.) 310
k2
+6.~
(6~+e/j.).
(12)
Volume 224, number 3
PHYSICS LETTERS B
29 June 1989
Here, the indices p, c~, fl, a are integral numbers 0, 1 and we drop ^ for simplicity. Obviously, the third term of eq. (10), in which there are two sorts of algebraical relations between the matrices ?'3, the y~_,'sand the ?'~'s is then ill-defined. It has to be separated into two parts to avoid this ambiguity; the parts correspond to the algebra (4) or ( 5 ), respectively. Thus, tr [?'~_?'i~(1 +i?'s) ?',?'o( 1 +iys) ] F~_,~_ = ( 1 - b 2) tr{yu[?'~( 1 +iy5 ) ]L~ [?'e( 1 + i?'5) ]} Fu_~+ b 2 tr{7¢[ ( 1 + iy5) ?',] ?'el ( 1 + i?'5)?', ] } F_~_,
(13)
where the arbitrary positive constant b 2 is due to the ambiguity mentioned above. It is easy to show that the second term ofeq. (13) is zero because ofeq. (5) and ( 1 - i?'s) ( 1 +i?'s) =0. The first term ofeq. (13) becomes (1 - b 2) tr{T,_[?'~(1 +i?'s) ] 7,_[?'e(1 +i75)]} Fa__e = ( 1 - b 2 )~_wtr (7,_,?'a ?'._?'~) (6a~ - cab) (6~e - eda) = - ( 1 - b 2 ) 4F~,~,a_~2 ( 1 - b 2)2e2/n.
( 14 )
Taking eqs. ( 11 ) - (14) into account, we get the one-loop vacuum polarization of the chiral Schwinger model:
k
(~
P
2e2(
k =--
(~,.-E,~)
k.kp
)
which gives then the following expression for the effective action of the theory: I~=
fl
d2x
~FZu.+½a(OuAu)2+ T A p ( x )
(
(l+b2)6p~-(6p~-ep.)
0~8, (6~,, + t~,~) ]Ao(x)
)1
.
(16)
This expression is the same as that suggested by Jackiw and Rajaraman [ 3 ]. Looking at eq. (16), one finds that the arbitrary parameter a = ( 1 + b 2) > 1 is always valid [ b 2 = 0 is a trivial case in the separation ( 13 ) ] and the theory is thus consistent. All subsequences and interpretations of the model for arbitrary a, as envisaged in refs. [ 3,4 ] therefore remain intact. This obviously refutes the assertion in ref. [2] that the effective action in this model is unique. 3. Let us examine where the arbitrary constant comes from and how it gets in this approach. The separation in eq. (13) seems to be a trivial identity at first glance. For the Schwinger model without an existing ?'5, the separation in eq. ( 13 ) is indeed trivial, and produces nothing. On the contrary, for the chiral Schwinger model with an existing ?'5, eq. (13) is no longer trivial since one has to divide the trace of the ?'-matrices so as to give definite algebraical relations for the matrices ?'3, the ?'u's and the ?'~'s. It seems that the arbitrary constant b 2 is introduced manually for the freedom of such a separation. However, the fact that the second term of eq. ( 13 ) vanishes by virtue of the algebra ( 5 ) leads to the explicit existence of arbitrariness in the theory. In this sense, obviously, chiral anomalies arise from the one-loop quantum correction, force the spontaneous breaking of gauge symmetry, which is valid at the classical level, through the generation of a mass for the gauge boson. The arbitariness is caused by the regularization freedom of the chiral Schwinger model. The author is grateful to Dr. T.D. Kien for discussions when he was at the University of Edinburgh. The initiate part of this work was supported by SERC Grant No. NG17094. The author is now supported by a postdoctoral fellowship of the Istituto Nazionele di Fisica Nucleare, Italy.
References
[ l ] J. Schwinger, Phys. Rev. 128 (1962) 2425. [2] C.R. Hagen, Ann. Phys. (NY) 81 ( 1973 ) 67; Nuovo Cimento 5 IA (1967) 1033; 5 IB (1967) 169; A. Das, Phys. Rev. Lett. 55 (1985) 2126. 311
Volume 224, number 3
PHYSICS LETTERS B
[3] R. Jackiw and R. Rajaraman, Phys. Rev. Lett. 54 (1985) 1219, 2060 (E); 55 (1985) 2224 (C). [4] R. Banerjee, Phys. Rev. Lett. 56 (1986) 1889; K. Harada, H. Kubota and I. Tsutsui, Phys. Lett. B 173 ( 1986 ) 77; A.J. Niemi and G.W. Semenoff, Phys. Lett. B 175 ( 1986 ) 439; D. Boyanovsky, preprint UCD-PUB-87-1; I.G. Halliday, E. Rabinovici and A. Schwimmer, Nucl. Phys. B 268 (1986) 413; M.S. Chanowitz, Phys. Lett. B 171 (1986) 280. [5] G. 't Hooft and M. Veltman, Nucl. Phys. B 44 (1972) 189; P. Breitenlohner and D. Maison, Commun. Math. Phys. 52 (1977) 11.
312
29 June 1989