Bound state mass in quenched QED

Physics Letters 285 (1992) 277-284 North-Holland

PHYSICS LETTERS B

Bound state mass in quenched QED M. Carena and C.E.M. Wagner Max Planck InstitutJ~r Physik, Werner Heisenberg Institut, F6hringer Ring 6, W-8000 Munich 40, FRG 1 and Department of Physics, Purdue University, West Lafayette, IN 47907, USA

Received 19 March 1992

The dynamical properties of the composite scalar bound states in quenched planar QED with chirally invariant four Fermi operators are studied. The composite scalar mass, m~, is computed combining analytical and numerical methods, for any value of the gauge coupling constant, 0 ~
The dynamical breakdown o f symmetries [ 1 ] plays an important role in various physical scenarios. In particular, quenched planar quantum electrodynamics ( Q E D ) exhibits a dynamical behavior [ 2,3 ], which includes a nontrivial chiral symmetry phase structure. Due to the nonperturbative nature of the chiral symmetry breaking, the methods o f investigation are quite limited. The Schwinger-Dyson equations provide the possibility o f an analytical study o f the quenched theory by considering a self-consistent summation o f diagrams, contributing to all orders in the gauge coupling constant. An interesting issue is, of course, to determine which features o f the analytical study are preserved beyond this truncation o f QED, for which lattice simulations appear to be the most reliable tool [4]. However, the most promising application o f quenched planar Q E D is through its relation to more complex gauge theories with slowly running coupling constants, such as walking technicolor models [5,61. In view o f the chiral phase structure o f quenched QED, it is natural to include chirally invariant four fermion operators in the theory [ 1-3,7,8 ]. For any value o f the gauge coupling ot below its critical end point a c = Ire, the four Fermi coupling acquires a critical value, G¢, above which the chiral symmetry of the theory is dynamically broken. When G is tuned to Go, dynamical composite scalar bound states appear in the model. It is our intention in this letter to focus on the dynamical properties o f such scalar fields in the broken phase of the theory. In order to do that, we shall consider the gauged U ( 1 ) N a m b u - J o n a - L a s i n i o ( N J L ) model, whose lagrangian is given by £P=i~ ~-

a o C V + ½G[ (gT~')2 - (g77s~/) 2 ] ,

(1)

where ~ is the covariant derivative,/2o is an explicit soft chiral symmetry breaking mass for the fermion and the last term above is a chiral invariant four fermion interaction with G the corresponding four Fermi coupling constant. The Schwinger-Dyson equations for the fermion self-energy, 27(p), in the ladder approximation and in the Landau gauge [ 9 ] lead to a second order differential equation with given infrared and ultraviolet boundary conditions [ 31, Address after 1 September 1991. 0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

277

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PHYSICS LETTERSB

d(p4+Z(p))=-a dp 2

9 July 1992

PEZ(P) 40to p2+ v'2(p) "

(2)

The ultraviolet boundary condition determines the expression for the fermion bare mass, d mo=X(A)+ A~-~pX(p)

v=A ,

(3)

where A is the ultraviolet cutoff. The finiteness of 27(p=O) is required as the infrared boundary condition. Furthermore, from the explicit expression for the fermion condensate (qT~u), it follows that (~u)-2n

A 3 ac d S(p) p=A"

(4)

2 a dp

In the ultraviolet asymptotic regime (p2 >> 272 ), the solution to eq. (2) reads [ 10 ] A 272 [(pe~]W_(pe~]-~]_(A)

S(p)=~71_\So /

\So /

d

\w/

3e3`w

w+l So~,2_W,p3~_, 8(w-2) '

(5)

with 27o---S ( p = 0) being the dynamically generated fermion mass scale, w= x/1 -ot/otc and where we have assumed that ot < ore. The solution depends on the adimensional parameters Oand A, which are functions of a and will be computed numerically after considering also the infrared contributions to Z(p). The last term on the right-hand side of eq. ( 5 ) is subdominant in the ultraviolet regime, except in the limit a--, 0 in which it becomes of the same order, S3/p 2, as the second term inside the square brackets and it is crucial in obtaining the correct critical behavior in the NJL limit, a = 0. For ot--,ac, there are higher order terms which cancel the apparently divergent contribution of the last term above, and the asymptotic solution is well described by the two terms between brackets in eq. (5). Considering the asymptotic solution for the dynamically generated fermion mass, the explicit expressions for mo and (V~v) are determined from eqs. (3) and (4). The requirement mo = / t o - G ( ~ v ) yields the gap equation, from which the fermion mass scale Zo is determined,

I.to_ Ae'~W('o'~2-w A z ( (A) 3

e 3aw

nz 2

/Zo\3~z-~'/

+

(3

)+ A e - a~~ ( S ,_)

AEaw-5 ) oz + n z _w-1 _ G

A2

(

1)

AZ

) (6)

From the gap equation, in the chiral limit, IZo--,0, the requirement of a nontrivial solution for Zo determines the critical value of the four fermion coupling G¢ = (nE/A 2) ( 1 + w)2, above which the chiral symmetry is dynamically broken [ 8 ]. For values of ot above its strong coupling critical end point ot~, the gauge interactions alone are sufficient to induce the chiral symmetry breaking. For ot < 0~¢, instead, attractive four Fermi interactions are necessary to drive the system into the broken phase. Introducing the composite scalar and pseudoscalar degrees of freedom, a = # o - G ~ and n = - G~y5 ~, the above lagrangian, eq. ( l ), can be rewritten as follows: L~=i~ ~ - ~ ( a + i y s n ) ~ -

1

~

[ ( a - ~ o ) 2 + n 2] ,

(7)

and the full fermion bare mass mo is identified with ( a ) . Integrating out the fermionic short distance components, the effective potential [ 7,1 1,1 0 ] 1

v(a, n)= w(a, n)+ 2-G [(a-~°)~+zr2] 278

(8)

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9 July 1992

is obtained, where the form of W(a, n) arises from the fermionic determinant, 0moW(mo, 0 ) = ( ~ ¥ > , and is derived to be 1 m g + A4 °tcFA2 (A o 2 w ~4/(2--w) W(mo, 0 ) = - - 2Go ~5n2--~-L--~- ( 2 - w ) w+lAeaW ]

2 {mo~ 4] (w+i-~\--A-] ._1"

(9)

In the above expression, we have substituted So by its dominant power behavior,

~0d°m=A(A° w-I-12A-e 6ww) ./(2-w, Hence, it is valid for any value of ot away from ac. Replacing m ~ effective potential in the/~o~ 0 limit then reads

V(O', ~) = ~

~Z2( 1

A 2 2-wore + 4n ~ Tot

W)~ (0.2..~_~2 )

Ol 1 +l) ac 2 n Z ( w

( a - # o ) z + n 2 we get W(a, n). The full

3 (az+n2)z

,~4/(2--w)

2w I ( w + l ) A e '~wJ

A4(~-w)l(2-')(a2+rc2)21(2-w) .

(10)

Away from the scaling region in which G is tuned to its critical value, the effective potential includes a quadratically divergent effective mass term, which renders the composite fields static. When G is tuned to G¢, instead, a and zt appear as dynamical fields. The effective mass term becomes negative for values of G > Go, signaling the breakdown of the chiral symmetry. Moreover, in describing the critical behavior of the theory, it is useful to define various critical exponents which determine the approach to criticality of the chiral symmetry breaking order parameter, the vacuum potential and the scalar mass. Evaluating these critical exponents, which fulfill the usual scaling relations [7], it is possible to compute the anomalous dimension of the scalar field a, ½r/= 1 - w . The physical dimension of a follows to be do = 1 + ½q--- 2 - w. Thus, for/~o = 0 and on the critical line, the dominant last term in the potential is scale invariant. The subdominant second term in the above expression becomes relevant only in the limit a = 0. In order to compute the effective potential at the critical point a~a~, we consider the expression for the asymptotic solution

$2 [-A e'~W- A e -'~" w

27(P) = --p-

I

+

A e'~W+ A e -'~w [ P ,~2.-] 2 logtro) j,

(ll)

where the parameters in each term are computed numerically while including the infrared contributions to S ( p ) (see below). We obtain that (A e a ' - A e-a"~)/w--,kl =0.7467 and (A eaW+A e-~W)/2--*k2 =0.5207 as w tends to zero. Considering the expressions for mo and < ¢7~u)asa function of the asymptotic solution, eq. ( 11 ), for A >> 27o 1

V(a,g)-~

1

~-~-7

1

(a2+~2)+4~-7(~2+g2)log[A2/(a2+~2)]

(12)

is obtained. The existence of a flat direction in the potential at oz= o~ is also reflected in the fact that, unlike for any other value of o~, only a very mild fine-tuning of the four fermion coupling is required to induce a huge hierarchy of mass scales (Z'o<< A) at the strong gauge coupling end critical point. This is due to the exponential dependence of the fermion mass scale on the critical four fermion coupling [ 10], So~A--, exp ( - 2 G / A G ) with On the other hand, the NJL results can be obtained by considering the limit of the gauge coupling constant close to zero in the above general expressions. However, in the small gauge coupling regime, the renormalization group techniques prove to be a better alternative. Studying the gauged NJL model in the chiral limit, at the scale 279

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9 July 1992

A, and computing the radiative corrections which at scales/~ << A give rise to kinetic terms for the scalar and pseudoscalar fields, the modified lagrangian reads [ 12 ] ~l M o (2a

A=i~_~(a+i7sn)llz+½ZH[(Oua)2+(OuTQ2]

2 -I- ]IS2 ) - - 1/I, 0 (0"2 "l- ~ 2 ) 2 •

(13)

In the one loop approximation, the running of the parameters ZH, M~ and 20 are obtained by integrating the renormalization group equations subject to the boundary conditions, M~ ~ 1/G, 20--+0 and Z H - 0 , at the scale A. Thus,

,[

ZH= 1 - ~ ~

l

1-

a-o , -

1

,lo<

4/t 2

M2 = G

+ 4r~z(1-3ot/2n)

1 [1 2°= 1-~-

(~)6,~/~]

~o

,_

[ jr2

'

A2

21~z2l ° g ~

1 clio, - G

4~"2 '

.

(14)

Observe that, although in the continuum limit ZH remains finite for any value of oz> 0, it diverges in the NJL limit (or = 0 ), signaling the decoupling of the composite fields in the absence of gauge interactions. Furthermore, the unrenormalized quartic coupling, 20, diverges with ZH at A/#--,oo. Rescaling the scalar and pseudoscalar fields so that they have a canonically renormalized kinetic term, a ~ a/ ~-Hn, rt--,n/x/~n, and provided the four fermion coupling G is fine-tuned so that the mass parameter M2/ZH is of order/t 2, the scalar and pseudoscalar masses become [ 12 ]

Jm ,

m2B~O, m2~2

(15)

with the fermion mass mr= ( t r ) / x / ~ - The mass of the scalar field is of the order of the fermion mass and in the limit ct = 0 the NJL result, m~= 2mr, is recovered. The mass of the pseudoscalar vanishes and it becomes the Nambu-Goldstone bosch of the spontaneous chiral symmetry breaking. The results for the masses obtained above are only valid for moderate values of the gauge coupling constant a. It is now our intention to compute the masses of the composite states for any value of or, including the strong gauge coupling critical point. For this purpose we consider the second derivative of the effective potential, eq.

(8),

I 0a--~ i=mo,<.>=o -

o~w 02{1 "~ 0( ~l//> 1 1 0a 2 mo,o+ 0 a 2 \ 2G (trE+n2),] too,0-- - - 0 t o o + ~ =Bs(0) + ~ =Ds(0) .

(16)

Bs (0) is the scalar bare bubble function at zero momentum transfer, while Ds (0) is the zero momentum denominator of the scalar propagator, which will signal a mass for the scalar. The zero momentum denominator of the pseudoscalar propagator, Dp(0) = 1/G+Bp(O) = 1/G+ ( ~ ) / m o , instead, vanishes in the/z0~0 limit for all values of or, implying the existence of the massless Nambu-Goldstone pseudoscalar of the spontaneous symmetry breaking. As done before in the renormalization group approach, a rescaling by the appropriate wavefunction renormalization factors is required. Considering the quadratic part of the tr scalar lagrangian at the minimum in the infrared regime, ~ 1

1 2 2 ~ Z H (OuO) 2 -- ~Znm~a

,

(17)

the scalar propagator is given by zJo(k 2 ) = [ Z H (m 2 - - k 2 ) ] -1. Therefore, the zero momentum transfer denominator of the scalar propagator is directly related to the renormalized scalar mass, as follows: 280

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m2=Ds(O)Z~= 0tr2 ,,ooZ~ l .

(18)

The expression for the scalar bare bubble function at zero momentum transfer can be readily computed. In fact, for any value of w ~ 0 it reads

[

Bs(0)=-

n2(w+l)2

1-4e-Zaw 2-w

--a

,27.2w 3A2e2W(27o4-w] ~ w-Z~ k~-I

_1"

(19)

Thus, the zero momentum transfer scalar denominator function has also a known analytical expression. To compute the wavefunction renormalization factor, ZH, we shall use an analog to the Pagels-Stokar formula for the pseudoscalar decay constant, f , [ 13 ]. The chiral invariance of the theory implies a common wavefunction renormalization constant for ~ and zr. We introduce an external pseudovector gauge field A 5u,which couples to the axial vector current, q77uYsq/, and to the pseudoscalar Goldstone field. Explicitly, the derivative coupling ofA u5 to the renormalized pseudoscalar field isf, = 2mo x/~n. Then, we obtain the expression for the scalar mass in terms of the zero m o m e n t u m transfer scalar denominator function, the scalar field vacuum expectation value and the pseudoscalar decay constant, 2 4Ds(0) m2 m~ = f~

(20)

Evaluating the A u-zt 5 two point function, at small external momentum, it is possible to derive the expression for f 2. In fact, considering the Ward identity for the axial vector vertex, q u/-5 (p + q, p ) = 5: - 1( p + q ) ~5 + 75 5: - 1 ( p ) , with 5~- l (p) = ~ _ 27(p) being the inverse of the fermion propagator, and the small momentum transfer expression, F~ (p + q, p) = YuYs-f,Y5 ~(P + q, P)qu/q2+ h.o., the pseudoscalar vertex involved in the above two point function computation is self-consistently evaluated to be f¢(p + q, p) = [27( (p + q)2) +27(p2) ]/f,~. It is then quite straightforward to derive the Pagels-Stokar formula for the squared pseudoscalar decay constant: f 2 = 212t2f dp2 p2 Z2(p2)- ½P2~(PZ) ( d/dp2)Z(P2) [p2+ 272(p2) ]z

(21)

In order to compute the above integral, eq. (21), we need to know the fermion self-energy function at all momentum scales. The ultraviolet asymptotic solution for S ( p ) is known, but to evaluate the infrared contributions to f~ numerical computations are required. The most efficient way to compute the pseudoscalar decay constant is to combine analytical calculations, for the contributions to eq. (21) coming from the ultraviolet regime, with numerical computations for the contributions coming from the low energy momentum regime. Using the second order differential equation, eq. (2) and expanding S ( p 2 ) at p 2= 0, the first and second derivatives of the fermion self-energy with respect to p2, 27' and 27", at zero momentum transfer are given by

° 27 1p~=o=271 =

8ac27o'

°

1)

lZa~

2~o3 .

(22)

With the above initial conditions we compute the value of the fermion self-energy and its first and second derivatives by taking steps in the squared momentum. We then compute the contribution t o f 2 at each step. The values of the parameters A and ~, which appear in the asymptotic solution, can be computed in terms of the fermion self-energy and its first derivative, A e~=

( 1 + w)~(AZl) + 2A 1~27'(A2) - (ot/Zac) (A eaW/w)3S6o-3WA3W-5/(w-2) 227o(So~A,) 1- ~

A e - a ~ = - ( w - 1 )S(Al2) - 2 A lZ27'(A~) + ( a / 4 a ~ ) (A ea'/w)3276-3":A3t~-5/(w- 1) 2Xo (27o/A 1) ~+1

(23) 281

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In the above expressions, A l has to be sufficiently large to secure that it is in the ultraviolet asymptotic regime and, thus, that eq. (5) is the solution o f S ( p ) . The last term in each of the numerators above come from the last term in eq. (5) and can hence be neglected for small values of w. The values of ( ~ ) , ( ~ u ) and Ds (0), which depend only on the asymptotic solution of X(p), have been computed numerically as well as analytically in order to compare the results, which turn to be in very good agreement. The expression for the squared pseudoscalar decay constant reads "-~"

- 2n 2 w 2 [

w-

L\~]

1

\ZoJ

_e_2a I+I(w+I)F(A'~ -2(w+l) (A,) -2(w+')] w+l

-

Too

3

5[(A) +2

Too

-2

(A')-2]} -

,

(24)

where f ~""m is the infrared contribution computed numerically between p 2 = 0 and p2 =A 2, while the other terms are the contribution obtained by considering the asymptotic solution of 27(p), eq. (5), between p2 =A 12 and p2 =A 2. In the above, we have neglected the subdominant term in p 3w-5 of 2;(p), since it is only relevant in the limit w ~ 1, in which only the dominant term gives an appreciable contribution t o f ~ . To obtain the continuum limit, we consider A/2Jo--, oe in eq. (24). For co--, 1, f ~ will be given by its dominant part in 1/o6 which reads f ~ _ oz¢ 272A 2 e 26w ot rt 2

(25)

Furthermore, in the continuum limit and for any value of ct ¢ 0 we have

m2DdO) = otc

2A 2

ot ~ z ( 2 - w ) 27o4.

(26)

Thus, for A/27o~OO and taking into account that for c~--,0, A--, ½ and e~--,2, we obtain m ~2 2 2 7 o2. Since f . is twice the vacuum expectation value of the renormalized scalar field, in the weak gauge coupling regime we can also compute the renormalized Yukawa and scalar quartic self-couplings through the relations h ~ = 4.r,2/f 2 and 2 2 2 = 2rndf,~, respectively. In the continuum limit we obtain h 2 = 12root,

2 = 12zwt.

(27)

The above expressions for the renormalized couplings coincide with those which can be obtained, also in the continuum limit, by using the renormalization group approach considered above. In fig. 1, we plot the ratio of the squared scalar mass to the fermion mass scale as a function of or, for small to moderate values of the gauge coupling constant and for different values of the cutoffA. We compare our results for the mass ratio with those obtained using renormalization group techniques. For values of c~<< 1 both results coincide. As the gauge coupling becomes larger the renormalization group approach becomes less reliable and the values computed within this method are slightly larger than those coming from our combined Schwinger Dyson equations and Pagels-Stokar method. In the continuum limit, we observe that mo/27o--.2 2 z as the gauge coupling tends to zero, as expected from the analytical calculations. For any finite value of the cutoff, instead, the mass ratio tends to the NJL result, mg/X g = 4, as or--,0. In fig. 2 we present the values of the ratio of the squared scalar mass to 272, evaluated while using our combined analytical and numerical technique, for all values of ot ~
Volume 285, number 3

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9 July 1992

'''1'''1'''1'''1'''1' 4

4

Ix ~',,\,

o

~-

¢a

3

I , 0

I I I , .05

i ,

f I , .i

, J j

+.

.15

Fig. I. Comparison between the results coming from the combined Schwinger-Dyson equations and Pagels-Stokar method (dashed lines) and the renormalization group predictions for /t = L'o (solid lines ) for the ratio of the squared scalar mass to the squared fermion mass scale as a function of the gauge coupling c~, in the small ~ regime and for different values of the cutoffA. From top to bottom, A 2/4Z~ = l 0z, I 04 109, l 0 TM and oc.

t,, 0

I,,,I .2

.4

i i i I t i, .6

,,, .8

,

1

Fig. 2. Ratio of the squared scalar mass to I~ as a function of the gauge coupling a in the interval [0, ac= In], for the same values of the cutoffA as in fig. 1.

t e c h n i q u e s . F o r c~ = a¢, the v a l u e o f m , was r e c e n t l y c o m p u t e d by B a r d e e n a n d L o v e b a s e d on a c o n j e c t u r e d d i s p e r s i o n r e l a t i o n [ 1 0 ] . T h e i r final result, m ~ = 2 x / 1 - exp ( - 1 / k ) ,So, d e p e n d s o n an u n k n o w n c o n s t a n t k, w h i c h is e x p e c t e d to be o f o r d e r one. We agree w i t h ref. [ 10 ] on the p o i n t t h a t the scalar m a s s is f o u n d to be o f the o r d e r o f the f e r m i o n mass scale, So. Thus, in spite o f the a p p r o x i m a t e scale i n v a r i a n c e o f the theory, no a b n o r m a l l y light p a r t i c l e a p p e a r s in the s p e c t r u m at a = c~c. We h a v e also c o m p u t e d the ratio o f t h e scalar m a s s to 270 for i n t e r m e d i a t e v a l u e s o f the gauge c o u p l i n g c o n s t a n t a n d p r o v e d that it is o f o r d e r o n e in the w h o l e c~ regime. As has b e e n r e m a r k e d in ref. [ 10 ], it r e m a i n s an o p e n q u e s t i o n h o w the i n c l u s i o n o f the effects o f the C o u l o m b b o u n d states in the scalar c h a n n e l , w h i c h are negligible in the p e r t u r b a t i v e regime, w o u l d m o d i f y the q u a n t i t a t i v e p r e d i c t i o n s for the scalar m a s s in the p r o x i m i t y o f the critical point. We are grateful to W.A. B a r d e e n a n d S.T. L o v e for useful c o m m e n t s a n d suggestions. We w o u l d also like to t h a n k T. A p p e l q u i s t a n d T.E. C l a r k for i n t e r e s t i n g discussions.

References

[ 1 ] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 ( 1961 ) 345. [ 2 ] K. Johnson, M. Baker and R. Willey, Phys. Rev. 136 (1964) B 1111; 163 ( 1967 ) 1699; S.L. Adler and W.A. Bardeen, Phys. Rev. D 4 ( 1971 ) 3045; T. Maskawa and H. Nakajima, Prog. Theor. Phys. 52 (1974) 1326; 54 ( 1975 ) 860; R. Fukuda and T. Kugo, Nucl. Phys. B 117 (1976) 250; V.A. Miransky, Nuovo Cimento 90 A (1985) 149; Sov. Phys. JETP 61 (1985) 905; P.I. Fomin, V.P, Gusynin, V.A. Miransky and Yu.A. Sitenko, Nuovo Cimento 6 (1983) 1. [3] C.N. Leung, S.T. Love and W.A. Bardeen, Nucl. Phys. B 273 (1986) 649; B 323 (1989) 493; W.A. Bardeen, S.T. Love and V.A. Miransky, Phys. Rev. D 42 (1990) 3514; W.A. Bardeen, in: Proc. 1988 Nagoya Workshop on New trends in strong coupling gauge theories, eds. M. Bando et al. (World Scientific, Singapore, 1989) p. 110. [4] E. Dagotto, J. Kogut and A. Kocic, Phys. Rev. Lett. 60 (1988) 772; Nucl. Phys. B 317 (1989) 253, 271; B 333 (1990) 551; Phys. Rev. D43 (1991) R1763; A. Kocic, S. Hands, J.B. Kogut and E. Dagono, Nucl. Phys. B 334 (1990) 279; B 347 (1990) 217; M. G6ckeler, R. Horsley, E. L~irmann, P. Rakow, G. Schierholz, R. Sommer and U.J. Wiese, Nucl. Phys. B 334 (1990) 527; 283

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M. G~Sckeler, R. Horsley, P. Rakow, G. Schierholz and R, Sommer, Scaling laws, renormalization group flow and the continuum limit in noncompact lattice QED, preprint DESY-91-098. [ 5 ] B. Holdom, Phys. Lett. B 150 ( 1985 ) 30 l; T. Appelquist, D. Karabali and L. Wijewardhana, Phys. Rev. Lett. 57 (1986) 957; T. Appelquist and L. Wijewardhana, Phys. Rev. D 35 (1987) 774; D 36 (1987) 568; T. Appelquist, T. Takeuchi, M. Einhorn and L.C.R. Wijewardhana, Phys. Lett. B 220 (1989) 223; T. Takeuchi, Phys. Rev. D 40 (1989) 2697; V.A. Miransky, M. Tanabashi and K. Yamawaki, Phys. Lett. B 221 ( 1989 ) 177; Mod. Phys. Lett. A 4 ( 1989 ) 1043; R.S. Chivukula, A.G. Cohen and K. Lane, Nucl. Phys. B 343 (1990) 554. [ 6 ] T. Appelquist, J. Terning and L. Wijewardhana, Phys. Rev. D 44 ( 1991 ) 87 I. [7] W.A. Bardeen, S.T. Love and V.A. Miransky, Phys. Rev. D 42 (1990) 3514. [8] K.I. Kondo, H. Mino and K. Yamawaki, Phys. Rev. D 39 (1989) 2430; T. Appelquist, M. Soldate, T. Takeuchi and L. Wijewardhana, in: Proc. 12th Johns Hopkins Workshop on Current problems in particle theory, eds. G. Domokos and S. Kovesi-Domokos (World Scientific, Singapore, 1988 ); M. Inoue, T. Muta and T. Ochiumi, Mod. Phys. Lett. A 4 (1989) 605; K. Aoki, Kyoto preprint RIFP-758 ( 1989 ). [9] T. Appelquist, U. Mahanta, D. Nash and L.C.R. Wijewardhana, Phys. Rev. D 43 ( 1991 ) 646; B. Haeri, Phys. Rev. D 43 ( 1991 ) 2701. [ 10 ] W.A. Bardeen and S,T. Love, Effective lagrangian for scale invariant quantum electrodynamics, preprint PURD-TH- 91-08. [ 11 ] V.P. Gusynin and V.A. Miransky, Mod. Phys. Lett. A 6 ( 1991 ) 2443; Effective action in the gauged Nambu-Jona-Lasinio model, preprint UWO-DAM 16/91. [ 12 ] S.T. Love, in: Proc. 1990 Nagoya Workshop on Strong coupling gauge theories and beyond, eds. T. Muta and K. Yamawaki (World Scientific, Singapore, 1991 ) p. 309. [ 13] H. Pagels and S. Stokar, Phys. Rev. D 20 (1979) 2947.

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