Breaking of chiral symmetry in quenched QED in dimension D < 4

ELSEVIER

UCLEAR PHYSIC.(

Nuclear Physics B (Prec. Suppl.) 102&103 (2001) 355-362

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Breaking of Chiral Symmetry in Quenched QED in Dimension D < 4

V. P. Gusynin a *

aBogolyubov Institute for Theoretical Physics, Kiev-143, 03143 Ukraine

It is shown that quenched QED in dimensions D < 4 has solutions which violate chiral symmetry for all values of the coupling. The equation for the fermion mass function in the rainbow approximation as well as with the Curtis-Pennington vertex is studied thoroughly, both analytically and numerically. In particular, we demonstrate haw to extract the critical coupling c~ relevant in four dimensions from the D dimensional theory.

1. I n t r o d u c t i o n 2. G e n e r a l C o n s i d e r a t i o n s

It is well know that quantum electrodynam- i,::s (QED), and in particular quenched QED, exhibits a new phase with broken chiral symmetry for sufficiently large couplings [1-3]. This was es- tablished in various studies based on the use of a t=uncated set of Dyson-Schwinger (DS) equations as well as in lattice simulations (for reviews see Refs.[3,4]).

Although chiral symmetry breaking appears to be universally observed independently on the pre- cise nature of the vertex used in DSE studies, ii, has been recognized that the critical couplings ~trith all ans~tze for vertices show a gauge dependence which should not be present for a physi- (al quantity. Though for a bare vertex this is r,ot surprising as this vertex breaks the Ward- Takahashi (WT) identity, the gauge dependence remains even for the Curtis-Pennington (CP) vertex which does not violate WT identity [5,6]. This residual gauge dependence can be traced to the use of a cut-off in Euclidean momentum in order t3 regulate divergent integrals, a procedure which }~reaks the gauge invariance of the theory. Thus ~ne naturally comes to studying the problem of dynamical chiral symmetry breaking within the quenched QED applying the gauge invariant dimensional regularization instead of the usual momentum cut-off.

We outline first some general results which one expects to be valid for dimensions D < 4 independently of the particular vertex used as an input to DSEs.

The Minkowski space fermion propagator S(p) is defined in the usual way through the dimensionless wavefunction renormalization function Z(p 2) and the dimensionful mass function M(p2) , i.e.

Z(p ~) S(p) - l~ _ M(p2 ) • (1)

We shall be dealing only with the regularized theory without imposing a renormalization procedure, as renormalization is inessential to our discussion. In addition, we shall consider the theory without explicit chiral symmetry breaking (i.e. zero bare mass). This theory would not contain a mass scale were it not for the usual arbitrary scale (which we denote by ~) introduced in D = 4 - 2e dimensions which provides the con- nection between the dimensionful coupling o~ D and the usual dimensionless coupling constant a = e2/4~:

O:D ----- o~v 2e. (2)

As v is the only mass scale in the problem, and as the coupling always appears in the above com- bination with this scale, on dimensional grounds alone the mass function must be of the form

(p2) "Department of Physics, Nagoya University, Nagoya 464- M (p 2) ---- ua ~ M ~ , e 8602, Japan \ p C~7

C920-5632/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. l>I1 S0920-5632(01)01579-1

(3)

356 V.P. Gusynin/Nuclear Physics B (Proc. Suppl.) 102&I03 (2001) 355-362

where M is a dimensionless function and in particular

M(0) = va~_llT/(0, e) . (4)

Moreover, as e goes to zero the v dependence on the right hand side must disappear and hence the dynamical mass M(0) is either zero (i.e. no symmetry breaking) or goes to infinity in this limit. This situation is analogous to what happens in cut-off regularized theory, where the scale parameter is the cut-off itself and the mass is propor- tional it.

Note that Jl~/(0, e) is not dependent on a. This implies immediately tha t there can be no non- zero critical coupling in D ~ 4 dimensions: if M(0) is non-zero for some coupling a then it must be non-zero for all couplings.

Given these general considerations (which are of course independent of the particular ansatz for the vertex) one could ask how this situation can be reconciled with a critical coupling a~ of order 1 in four dimensions. In order to see how this might arise, we shall extract a convenient numerical factor out of -~/ and rewrite the dynamical m a s s as

1

M(0) = v M (0, e) . (5)

At present there is no difference in content between Eq. (4) and Eq. (5). However, if we now define ac by demanding tha t the behaviour of M(0)

( ) 1 / ~ is dominated by the factor ~ as e goes to

zero, which is equivalent to demanding that

[M (0, 1, (6)

then the intent becomes clear: even though M(0) may be nonzero for all couplings in D < 4 dimensions, in the limit tha t e goes to zero we obtain

M(0) ~o- ~ 0, a < a~, (7)

M ( 0 ) , ~ c~, a > a¢.

Note that in the above we have not addressed the issue of whether or not there actually is a M (0, e) with the property of Eq. (6). In fact, the numerical and analytical work in the following sections is largely concerned with finding this function and

hence determining whether or not chiral symme- t ry is indeed broken for D < 4. Notwithstand- ing this, as one knows from cut-off based work tha t there actually is a non-zero critical coupling for D = 4, one can at this stage already come to the conclusion that M (0, e) exists and hence tha t quenched QED in D < 4 dimensions has a chiral symmetry breaking solution for all couplings. Furthermore, even though the dimensionless coupling a does not get modified by vacuum polarization effects in the quenched approximation, the effective dimensionless coupling ~(q2) in D < 4 dimensions runs as a function of q2:

\ q 2 ] , (s)

where # is the renormalization scale. In the infrared the effective coupling increases without bound, suggesting the theory is likely to break chiral symmetry for any coupling (for D ~ 4). This feature of quenched QED bears some simi- larity to QCD. Note also tha t in the ultraviolet the running coupling ~(q2) vanishes, i.e. in D 4 dimensions quenched QED exhibits "asymptotic freedom" (for similarities between QED3 and QCD see, for example, Ref.[7] and references therein).

In summary, as the trivial solution M ( p 2) = 0 always exists as well, we see tha t in D < 4 dimensions the trivial and symmet ry breaking solutions bifurcate at a = 0 while for D = 4 the point of bifurcation is at a = ac; i.e., there is a discon- tinuous change in the point of bifurcation. As D approaches four (i.e. as e approaches 0) the generated mass M(0) decreases (grows) roughly like (Ot/O~c) 1/2e for a <~ ac (>~ ac), respectively, be- coming an infinite step function at a = a~ when e goes to zero.

3. T h e rainbow approximation

We consider first the rainbow approximation to the Euclidean mass function of quenched QED with zero bare mass in Landau gauge. I t is given by

dDk M ( k 2) 1 M ( p 2) = ~ (27r)V k2 + M2(k2) (p _ k) 2 , (9)

Et? Gusynin/Nuclear Physics B (Proc. Suppl.) 102&103 (2001) 355-362 357

where a = (ev~)s(3 - 20 . Note that the Dirac par t of the self-energy is equal to zero in the Landau gauge in rainbow approximation even in D < 4 dimensions, hence Z ( p 2) = 1 for all p2. Eq.(9) being the simplest nonper turbat ive approximation to the DS equation for the fermion propagator , can serve as start ing point in some regular i teration scheme [8].

I t is of course possible to find the solution to F,q. (9) numerically, however it is more instructive to first t ry to make some reasonable approximations in order to be able to analyze it analytically. [ irst, let us consider the linearized version of Eq. (9) replacing M2(k s) in the denominator by the constant Ms(0) - m 2 (for convenience we shall call this value the "dynamical mass" m). We can convert the linearized equation to a Schr6dinger- like equation by introducing the function

dDk e ! k r M ( k 2)

~'(r) : (27r)D kS + mS . (10)

With this definition the equation becomes

H e ( r ) = - m 2 ¢ ( r ) , (11)

where H = - D + V ( r ) is the Hamiltonian, E = .... m s plays the role of an energy and the potential V(r) given by

l ' (r) ~ r (1 - e) - - r 0 - 2 ' ~? -- -4~Y-7 e2v2~(3-2e) . (12)

l?or D = 3 the coefficient r/ is 2va while near 11) = 4 it is 3av2¢ It is well known from any sr~andard course of quantum mechanics tha t po- tentials behaving as 1 / r 8 at infinity, with s < 2, always support bound states (actually, an infinite number of them). This can be seen by consider- ing the SchrSdinger equation (11) for zero energy, i e. E = 0. We can est imate the lowest energy eigenvalue variationally by using the trial wave- flmction

(2~) D ~'(r) = Ce -~r, IC[2 - f ]o[ ' (D) " (13)

Here C is determined by demanding tha t ¢ is nor- realized and ~ ' ~ D is the volume of a D-dimensional sphere. Calculating the expectation value of the

"Hamiltonian" H on the trial wave function in Eq. (13) we find

20-2 3 Eo(~ s) = (¢1H1¢) = ~2 1 - _ _ _ ~ 0 - - 4 ~ J . (14)

The minimum of the "ground state energy" in Eq. (14), Eo(~), is reached at

2 0 - 3 ~ 4 - n = (D - 2) F---(~ ~ (15)

(for D = 3 the parameter a is v a while near D = 1

4 it is v ~77/2 and is given by the expression

2 ) 2 D - 4 E 0 = - m s - - ~ s 1 D - 2 = ~ D -2' (16)

where the 1 is the contribution from the kinetic energy while the 2 / ( D - 2) corresponds to the potential energy. For D > 2 the potential is at trac- tive and for 2 < D < 4 it is always larger than the kinetic energy, so for this case we get dynamical symmetry breaking for any value of a. For example, for D = 3, one obtains E0 = _~2 = _v2as which coincides precisely with the ground-state energy of the hydrogen a tom (not surprisingly, as we have used the ground-state hydrogen wave function as our trial function). In this case the dynamical mass is m = va.

For D near 4, on the other hand, we obtain from Eq. (16) that

m -~ v(e)l/2 ( - ~ ) ~ (17)

This is of the general form anticipated in Sec- tion 2, with ac = ~. Indeed, for D = 4, the SchrSdinger equation (11) becomes an equation with the singular potential

V(r)=- , r l = r r / 3 . (18)

Again, it is known from standard quantum mechanics that the spectrum of bound states for such a potential depends on the strength r I of the potential: it has an infinite number of bound states with E < 0 if rl > 1 and bound states are absent if rl < 1. Thus, the true critical value for the coupling is expected to be ac = ~r/3 instead

358 V.P Gusynin/Nuclear Physics B (Proc. SuppL) 102&103 (2001) 355-362

of the c~c = ~r/2 obtained with the help of the variational method (which made use of the exponential Ansatz for the wavefunction and thus only gave an upper bound for the energy eigenvalue).

Now we turn to solving Eq. (9) to prove that indeed the critical coupling is ac = 7r/3. As the angular integrals involved in D-dimensional integration are s tandard we may reduce Eq. (9) to a one-dimensional integral, namely

oo / dk2(k2)l-~M(k 2)

M(p2) = av2"c~ k 2 -4- M2(k 2) 0

X [--~F(e,e;2-e;~2)O(P 2 -k2 )

1( ] + ~ F 1 , e ; 2 - e ; ~ O(k e -p2) , (19)

where

3 - 2~ . 3 _ c, = ( 4 ~ ) 1 - ~ r ( 2 - e)' (co = --)4~ (20)

Clearly for D = 4 the mass function in Eq.(19) reduces to the s tandard one in QED4. In D # 4 dimensions the hypergeometric functions in Eq. (19) preclude a solution in closed form. How- ever, since these hypergeometric functions have a power expansion in e for small e, we can replace these by their e = 0 (i.e. D = 4) limit. After all, the reason for choosing dimensional regularization in the first place is to regulate the integral, and this is achieved by the factor of k -2E, not the hypergeometric functions. In addition, this approximation also corresponds to just replacing the hypergeometric functions by their IR and UV limits, so that one might expect that even for larger e the approximation is not too bad in these regions. Since the mass function in the denominator of Eq. (19) serves primarily as an infrared regulator we shall make one more approximation and replace it by an infrared cut-off m 2 for the integral. This simplifies the problem sufficiently to allow the derivation of an analytical solution since the equation becomes a linear one

M (p2) = ac~ k2

[ ~dk2k-2"M(k2)] + J ~ . (21)

k2

This integral equation can be explicitly solved re- ducing it to the second order differential equation with two (infrared and ultraviolet) boundary conditions. The solution is given in terms of the Bessel function

M(p) = x-1/2C J~ \ ex~ /

where we have defined A = l /e , x = p2/u2 and the normalization constant C will be determined later. The dynamical mass m satisfies the tran- scendental equation

: 0 (23

If we define j:~-1,1 = ~ / e a ~/2 to be the small- est positive zero of Eq. (23), the dynamical mass for this solution becomes

!

m = va I/2 = v o ~ (24) \ e jx-l ,1

In order to extract ac, we need to look at the behaviour of m as e goes to zero (i.e. A ~ oo). This may be done by noting that the positive roots of the Bessel function Jx have the following asymptotic behaviour (see, for example, Eq. 9.5.22 in Ref. [9]):

j~,,8 ~ ),z(~) + ~_, ) ,2k-l , ~ = ) ,-2/3a8, (25) k = l

where as is the sth negative zero of Airy function Ai(z), and z(() is determined (z(¢) > 1) from the equation

(_~)s/2 = ~ z 2 _ 1 - a r c c o s - . (26) Z

For large A the variable ~ is small and so it is valid to expand z around 1. In our case

j A - I , 1 ~ A -4- ~'A 1 /3 -- 1 + O(A-1/3), 7 ~ 1.86.(27)

Then we find that for small e the dynamical mass becomes

m ~ v e ~ - , (28)

V.P Gusynin/Nuclear Physics B (Proc. Suppl.) 102&103 (2001) 355-362 359

where d = ln(41r) + 7/3 + ¢(1). Note that the behaviour of the first term (for e going to zero) dominates over the exponential function, as re- quired in Eq. (6). Hence the critical coupling in four dimensions is given by zr/3, as expected from cut-off based work [2-4].

Returning now to the mass function itself, we may substitute the expression for the dynamical mass, Eq. (24), together with our choice of nor- realization condition M ( m 2) = m into Eq. (9) in order to eliminate C. One obtains

m

M(p) = IP--/ &[jx- l ,1] (29)

Note that the explicit dependence on v (and hence a) has been completely replaced by m in this expression.

So far we have taken a independent of the regularization. As we have seen this leads to a dy- namically generated mass which becomes infinite as the regulator is removed. Fomin et al. [2,3] examined (within cut-off regularized QED) a different limit, namely one where the mass m is kept cc~nstant while the cut-off is removed. In our case this limit necessitates that the coupling a is dependent on e through

a - g 1 + 23 ~e~ (30)

[see Eq. (28); note that ac is approached from above]. The limit may be taken analytically in Eq. (29) by making use of the known asymptotic behaviour of the Bessel functions [9]

J:*" ()~'4-)tl/3z) (_~)1/3 ~- Ai ( -21 /3z) , (31)

m~ well as the asymptotic expansion of jx-l ,1 in Eq. (27). One obtains

m2 (la p + 1) (32) M ( p ) =

which agrees with the result in Refs. [2,3]. To conclude this section, we analyzed the valid-

ity of the approximations made by solving Eq. (9) numerically and compared it to the Bessel function solution in Eq. (29). It was shown that the approximation is not bad and could actually be

made significantly better by adopting a normalization condition different from M(m) = m (for details see Ref.[10]).

4. Th e C u r t i s - P e n n i n g t o n vertex

We shall now turn to the CP vertex. Before we discuss chiral symmetry breaking for this vertex we shall first examine the chirally symmetric phase. We remind that in this phase in four dimensions the wavefunction renormalization has a very simple form for this vertex [11], namely

Here ~ is the gauge parameter and #2 is the (dimensionless) renormalization scale. This power behaviour of Z(x) is in fact demanded by mul- tiplicative renormalizability as well as gauge covariance. We shall derive the form of this self- energy in D < 4 dimensions, which will provide a very useful check on the numerical results even if M(x) # O.

In the chirally symmetric phase, the unrenor- malized Z(x) corresponding to the CP vertex in D dimensions satisfies the equation [12]

a (4r) ~ y-~ Z(x) = 1 + ~ F 7W--7~z-e) Jo dy Z(y) x - y

[ ( (34) x ( l - e ) 1 - I D x + y + x ' x "

The angular integral 11 (w) is defined to be

ID(w) = (l +w)2Fl (1 , e ;2 -e ;w) ,O<_w<_ 1

I1D(w) = I1D(w-I), W > 1. (35)

In four dimensions the solution to Eq. (34) is given by a Z(x) having a simple power behaviour while for D < 4 this is clearly no longer the case. Eq. (34) looks very complicated, nevertheless, it is possible to derive its explicit solution.

We seek the solution as an expansion in powers of x-e:

oo z ( x ) = cnx -n ' . (36)

rt----O For that consider first the RHS of Eq. (34) upon insertion of the power y~ in the place of Z(y).

360 V.P Gusynin/Nuclear Physics B (Proc. Suppl.) 102&I03 (2001) 355-362

Note that the integral converges only if e > 5 > e - 2. After some work the result is that the RHS of Eq. (34) becomes

_2 - e ~_~ [ r(2 - ~) 1 + c-----~-x L-(1 + 5 - e) F(e)

oo r ( e + n) 1 [

x ~ r ~ - - ~ + - - n ) n - - 5 + e " J (37)

For e < 1 this may simplified further by applying Dougall's formula (Eq. 1.4.1 in [13]) and Eq. (37) becomes

2 - ~ ~ C r ( 1 - e ) r ( 2 - c )

x r (1 - 5) (38)

Note that , as opposed to the integral represen- tation Eq. (34), this expression is defined for outside the range e > 5 > e - 2 and so we may use it as an analytical continuation of the integral. Furthermore, note that this last expression vanishes for integer 5 _ 1 hence we cannot obtain a simple power expansion around x -- 0 for Z(x) in this way. We may equate the coefficients of equal powers of x -~ after inserting the series (36) into both sides of Eq. (34). This way we obtain the recurrence relation for the coefficients cn ( c 0 = l ) as

Cn+ l c , - 2(n + 1) r ( - e ) r ( 3 - e)

r ( 1 + en + e ) r (2 - e - en) x r ( 2 - 2e - en ) r (1 + en) " (39)

This may be solved leading to

,,n F(2 - e)F(1 + ne) e. = [ r ( - ~ ) r ( 3 - ~)j K 2 E ~ - n T ) ~ . ' ' (40)

so that finally we obtain

r ( 1 + n~) Z ( z ) = r ( 2 - ~) r ( 2 E - e - : - - ~ ) n !

n = 0

as the series expansion of the solution to Eq. (34). We shall show now that Eq. (41) coincides pre-

cisely with the corresponding expansion of the

solution obtained via the Landau-Khalatnikov transformation. This transformation relates the coordinate space propagator Se(u) in one gauge to the propagator in a different gauge. Specifi- caily, with covariant gauge fixing, we have

S~(u) = e4~2"[~(°)-A(u)]S~=°(u), (42)

where A(u) is the Fourier transform of the gauge- dependent part of the photon propagator, i.e.

dO k e- ik .u A(u) = - ~ (27r)D k4 (43)

Hence we obtain

:~ (u) = e - r ( . ,,)2. ~ = o (u), (44)

where r = - (a /47r)F(-e)(Tr) ~. Substituting the coordinate-space propagator in Landau gauge (which is a free one according to Eq. (34))

uo

and carrying out the inverse Fourier transform of Eq. (44) one obtains the wavefunction renormalization function in an arbitrary gauge, namely

Z ( x ) - F(D/2) fdDueiP.uP_~e_r(vu)~, - 2i7r D/2 g u --

IC x-~21-~F(2 " e duu~-le-rU~'&-~(Vrxu). (46)

Note that for small x this function vanishes:

Z(x) - 4 ~ ) r - ~ x + o (x~). (47)

One can check explicitly that Eq. (46) reproduces the series in Eq. (41) by changing the variable of integration from u to u /x /~ and expanding the exponential in the integrand.

We shall now examine dynamical chiral symmetry breaking for the C P vertex. The compari- son to the analytic result in Sec.3 provides a very convenient check on the numerics. The check is provided by plotting the logarithm of M(0) against the logarithm of the coupling. Accord- ing to Eq. (4) this should be a straight line with gradient 1/2e. As can be seen in Fig. 1 not only does one observe chiral symmetry breaking down

V.P. Gusynin/Nuclear Physics B (Proc. Suppl.) 102&103 (2001) 355-362 361

10

-10

CD ~" -20

[ :

-50

I I I I

/ (:= 0.25 / / / ~ /1, 2= 10'

~ " F ~ O = -2.4513 + 20.001 In(a) -4O

-50 r I ~ i -2.0 -1.5 -1.0 -0.5 0.0 0.5

T~=o.15 In(a) a=1.51

4.0

5.5

5.0

2.5 o .

"6" 2.0

1.5

1.0

0.5 15

I I I

I r I

20 25 50 1/2~

35

Figure 1. The logari thm of the dynamical mass as a function of ln(a) for e = 0.025 (i.e. ~ = 20). The gauge paramete r is fixed at ~ -- 0.25 and the renormalization point is #2 __ 108. The open squares are the numerical values while the solid line is a linear fit to these points.

Figure 2. The logarithm of the dynamical mass as a function of 1/2e for a coupling of a = 1.20. All other parameters are as in Fig. 1. The open squares are the numerical values while the two lines are fits.

to couplings as small as a -- 0.15, the expected linear behaviour is confirmed to quite high preci- .,;ion. Although the numerics in D < 4 dimensions ~re clearly under control, the extraction of the (ritical coupling (appropriate in four dimensions) has proven to be numerically quite difficult. From the discussion in Sections 2 and 3, we anticipate tha t the logarithm of the dynamical mass has the general form

1 (o) In = In ~ + l n ( M ( 0 , e)) , (48)

where the last t e rm is subleading as compared to the first as e tends to zero. For sufficiently .,mall e, therefore, ac is related to the gradient of In (M(0)) plotted against e -1. In Fig. 2 we ~!~ttempt to extract ac in this way. The logarithm ,:)f M(0) was evaluated for e ranging from 0.03 down to e = 0.015 for a fixed gauge ~ = 0.25. The squares corresponds to a coupling constant ,:~ = 1.2, al though some of the points at lower e have actually been calculated at smaller a and Then rescaled according to Eq. (4). At present

we are unable, for these parameters , to decrease e significantly further without a significant loss of numerical precision.

The two fits shown in Fig. 2 correspond to two different assumptions for the functional form of M(0, e) [10], which is a priori unknown. The curves do indeed appear to be well approximated by a straight line. The solid line corresponds to the assumption tha t the leading te rm in M(0, e) has the same form as what we found in the rainbow approximation, i.e.

1

In = ~ In + cl (49)

With this form the fit parameters ac and c 1 are found to be

ac ---- 0.966, cl = - 1 . 1 5 . (50)

The critical coupling is similar to what is found in cut-off based work (see Sec. 2; in Ref. [14] the value was 0.9208 for ~ = 0.25).

362 V.P. Gusynin/Nuclear Physics B (Proc. Suppl.) 102&I03 (2001) 355-362

5. C o n c l u s i o n s

In this paper we explored the phenomenon of dynamical chiral symmetry breaking through the use of Dyson-Schwinger equations with a regularization scheme which does not break the gauge covariance of the theory, namely dimensional regularization. It is necessary to do this as the cut- off based work leads to ambiguous results for the critical coupling of the theory precisely because of the lack of gauge covariance in those calculations.

We have shown on dimensional grounds alone and for an arbitrary vertex, that in D < 4 dimensions either a symmetry breaking solution does not exist at all or it exists for all non zero values of the coupling. For Dyson-Schwinger analyses employing the rainbow and CP vertices we have shown that it is the second of these possibilities which is realized. For these symmetry breaking solutions the limit to D = 4 is necessarily dis- continuous and so the extraction of the critical coupling of the theory (in 4 dimensions) is not as simple as in cut-off regularized work.

We have examined the symmetry breaking in the rainbow approximation in Landau gauge, both analytically and numerically. Converting the (linearized) Dyson-Schwinger equation to a SchrSdinger equation in 4 dimensions we explicitly showed that the theory always breaks chiral symmetry if D < 4. We also showed how the

7r usual critical coupling ~c = y may be extracted from the dimensionally regularized work.

We have derived the exact solution of the equation for the fermion wavefunction Z ( p 2) with the CP vertex in arbitrary gauge in the chirally symmetric phase and have shown that it coincided precisely with that one obtained by making use of the Landau-Khalatnikov transformation from the Landau gauge. Finally, we extracted the critical coupling corresponding to the CP vertex and found that, within errors, it agrees with the standard cut-off results.

This my talk is based on the results obtained together with A.W. Schreiber, T. Sizer and A.G. Williams whom I would like to thank for fruitful collaboration. I would like to thank also the or- ganizers of International conference "Supersym- metry and Quantum Field Theory" held on July

25-29, 2000 at Kharkov for invitation to take part and excellent atmosphere during the conference.

This research was supported in part by the Swiss National Science Foundation grant 7IP 062607 and Grant-in-Aid of Japan Society for the Promotion of Science (JSPS) No. 11695030. I wish to acknowledge the JSPS for financial support.

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3. P.I . Fomin et al., Riv. Nuovo Cim. 6 (1983) 1.

4. V.A. Miransky, Dynamical Symmetry Break- ing in Quantum Field Theories, World Sci- entific, Singapore, 1993; C.D. Roberts and A.G. Williams, Prog. Part. Nucl. Phys. 33 (1994) 477.

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6. D. Atkinson et al., Phys. Lett. B329 (1994) 117.

7. V. P. Gusynin, A. Hams and M. Reenders, Phys. Rev. D63 (2001) 045025.

8. V. E. Rochev, J. Phys. A: Math. Gen. 33 (2000) 7379.

9. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972.

10. V. P. Gusynin, A. W. Schreiber, T. Sizer, A. G. Williams, Phys. Rev. D60 (1999) 065007.

11. D.C. Curtis and M.R. Pennington, Phys. Rev. D44 (1991) 536.

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Breaking of chiral symmetry in quenched QED in dimension D < 4