Chiral symmetry breaking in dimensionally regularized nonperturbative quenched QED

Gusynin, V. P.; Schreiber, Andreas Wolfgang; Sizer, Tom; Williams, Anthony Gordon Physical Review D, 1999; 60(6):065007

©1999 American Physical Society

PERMISSIONS

“The author(s), and in the case of a Work Made For Hire, as defined in the U.S.

Copyright Act, 17 U.S.C.

§101, the employer named [below], shall have the following rights (the “Author Rights”):

[...]

3. The right to use all or part of the Article, including the APS-prepared version without revision or modification, on the author(s)’ web home page or employer’s website and to make copies of all or part of the Article, including the APS-prepared version without revision or modification, for the author(s)’ and/or the employer’s use for educational or research purposes.”

17th April 2013

Chiral symmetry breaking in dimensionally regularized nonperturbative quenched QED

V. P. Gusynin,^1,2A. W. Schreiber,^2,3T. Sizer,^2,3and A. G. Williams^2,3

1Bogolyubov Institute for Theoretical Physics, Kiev, 252143 Ukraine

2Special Research Centre for the Subatomic Structure of Matter, University of Adelaide, 5005, Australia

3Department of Physics and Mathematical Physics, University of Adelaide, 5005, Australia 共Received 24 November 1998; published 18 August 1999兲

In this paper we study dynamical chiral symmetry breaking in dimensionally regularized quenched QED within the context of Dyson-Schwinger equations. In D⬍4 dimensions the theory has solutions which exhibit chiral symmetry breaking for all values of the coupling. To begin with, we study this phenomenon both numerically and, with some approximations, analytically within the rainbow approximation in the Landau gauge. In particular, we discuss how to extract the critical coupling␣c⫽␲/3 relevant in 4 dimensions from the D dimensional theory. We further present analytic results for the chirally symmetric solution obtained with the Curtis-Pennington vertex as well as numerical results for solutions exhibiting chiral symmetry breaking. For these we demonstrate that, using dimensional regularization, the extraction of the critical coupling relevant for this vertex is feasible. Initial results for this critical coupling are in agreement with cut-off based work within the currently achievable numerical precision.关S0556-2821共99兲06916-7兴

PACS number共s兲: 11.30.Qc, 11.15.Tk, 11.30.Rd, 12.20.Ds

I. INTRODUCTION

It is fairly well established that quantum electrodynamics 共QED兲, and in particular quenched QED, breaks chiral sym- metry for sufficiently large couplings. This phenomenon has been observed both in lattice simulations关1兴as well as vari- ous studies based on the use of Dyson-Schwinger equations 关2–4兴. These latter calculations have generally relied on the use of a cut-off in Euclidean momentum in order to regulate divergent integrals, a procedure which breaks the gauge in- variance of the theory.

On the other hand, continuation of gauge theories to D

⬍4 dimensions has long been used as an efficient way to regularize perturbation theory without violating gauge in- variance. In nonperturbative calculations, however, the use of this method of regularization is rarely used关5兴. Within the context of the Dyson-Schwinger equations 共DSEs兲 only a few publications 关6,7兴 have employed dimensional regular- ization instead of the usual momentum cut-off.

It is the purpose of the present paper to study dynamical chiral symmetry breaking and the chiral limit within dimen- sionally regularized quenched QED. We are motivated to do this by the wish to avoid some gauge ambiguities occurring in cut-off based work, which we discuss in Sec. II. In that section we also outline some general results which one ex- pects to be valid for D⬍4, independently of the particular vertex which one uses as an input to the DSEs. Having done this we proceed, in Sec. III, with a study of chiral symmetry breaking in the popular, but gauge non-covariant, rainbow approximation. Just as in cut-off regularized work, the rain- bow approximation provides a very good qualitative guide to what to expect for more realistic vertices and has the consid- erable advantage that, with certain additional approxima- tions, one may obtain analytical results. We check numeri- cally that the additional approximations made are in fact quite justified. Indeed, it is very fortunate that it is possible to obtain this analytic insight into the pattern of chiral symme- try breaking in D dimensions as it provides us with a well

defined procedure for extracting the critical coupling of the 4 dimensional theory with more complicated vertices. We pro- ceed to the Curtis-Pennington共CP兲vertex in Sec. IV. There we derive, for solutions which do not break chiral symmetry, an integral representation for the exact wave function renor- malization function Zin D dimensions. We also provide an approximate, but explicit, expression for this quantity. The latter is quite useful, in the ultraviolet region, even if dy- namical chiral symmetry breaking takes place as it provides a welcome check for the numerical investigation of chiral symmetry breaking with the CP vertex with which we con- clude that section. Finally, in Sec. V, we summarize our results and conclude.

II. MOTIVATION AND GENERAL CONSIDERATIONS Although chiral symmetry breaking appears to be univer- sally observed independently of the precise nature of the ver- tex used in DSE studies, it has also been recognized for a long time that the critical couplings with almost all¹of these vertices show a gauge dependence which should not be present for a physical quantity. With a bare vertex this is not surprising as this vertex Ansatz breaks the Ward-Takahashi identity. However, even with the Curtis-Pennington 共CP兲 vertex, which does not violate this identity and additionally is constrained by the requirement of perturbative multiplica- tive renormalizability, a residual gauge dependence remains 关11,12兴.

Apart from possible deficiencies of the vertex, which we do not investigate in this paper, the use of cut-off regulariza- tion explicitly breaks the gauge symmetry even as the cut-off is taken to infinity. This is well known in perturbation theory

1Some vertex Ansa¨tze exist which lead to critical couplings which are strictly gauge independent关8–10兴. However, these involve ei- ther vertices which have unphysical singularities or ensure gauge independence of the critical coupling by explicit construction.

共see, for example, the discussion of the axial anomaly in Sect. 19.2 of Ref.关13兴兲and was pointed out by Roberts and collaborators 关14兴 in the present context. The latter authors proposed a prescription for dealing with this ambiguity which ensures that the regularization does not violate the Ward-Takahashi identity.

As may be observed in Fig. 1, this ambiguity has a strong effect on the value of the critical coupling of the theory. The two curves in that figure correspond to the critical coupling

␣c of Ref. 关12兴 as well as the coupling ␣c⬘ one obtains by following the prescription of Roberts, et al. It is straightfor- ward to show, following the analysis of Ref.关12兴, that these couplings are related through

␣c⬘⫽ ␣c

1⫹␰␣c

8␲

. 共1兲

Also plotted in this figure are previously published numerical results关11,15兴obtained with both of the above prescriptions.

Note that, curiously, the critical couplings obtained with the prescription of Ref. 关14兴 共i.e. the calculation which restores the Ward-Takahashi identity兲 exhibits a stronger gauge de- pendence, at least for the range of gauge parameters shown in Fig. 1.

Gauge ambiguities such as the one outlined above are absent if one does not break the gauge invariance of the theory through the regularization procedure. Hence, we now turn to dimensionally regularized 共quenched兲 QED. The Minkowski space fermion propagator S( p) is defined in the usual way through the dimensionless wave function renor- malization function Z( p²) and the dimensionful mass func- tion M ( p²), i.e.,

S共p兲⬅ Z共p²兲

p”⫺M共p²兲. 共2兲

The dependence of Z and M on the dimensionality of the space is not explicitly indicated here. Furthermore, note that to a large extent we shall be dealing only with the regular- ized theory without imposing a renormalization procedure, as renormalization 关15,16兴is inessential to our discussion.

In addition to the above, we shall consider the theory without explicit chiral symmetry breaking 共i.e. zero bare mass兲. This theory would not contain a mass scale were it not for the usual arbitrary scale 共which we denote by ␯⁾ introduced in D⫽4⫺2⑀dimensions which provides the con- nection between the dimensionful coupling␣Dand the usual dimensionless coupling constant ␣⫽e²/4␲^:

␣D⫽␣␯^2⑀. 共3兲 As␯ is the only mass scale in the problem, and as the cou- pling always appears in the above combination with this scale, on dimensional grounds alone the mass function must be of the form

M共p²兲⫽␯␣^1/2⑀M˜

冉

^{␯2p␣21/⑀,⑀}

冊

^共4兲

where M˜ is a dimensionless function and in particular M共0兲⫽␯␣^1/2⑀M˜共0,⑀兲. 共5兲 Moreover, as ⑀ goes to zero the ␯ dependence on the right hand side must disappear and hence the dynamical mass M (0) is either zero共i.e. no symmetry breaking兲 or goes to infinity in this limit. This situation is analogous to what hap- pens in cut-off regularized theory, where the scale parameter is the cut-off itself and the mass is proportional it.

Note that M˜ (0,⑀) is not dependent on ␣. This implies immediately that there can be no non-zero critical coupling in D⫽4 dimensions: if M (0) is non-zero for some coupling

␣ then it must be non-zero for all couplings.

Given these general considerations 共which are of course independent of the particular Ansatz for the vertex兲 it be- hooves one to ask how this situation can be reconciled with a critical coupling␣cof order 1 in four dimensions. In order to see how this might arise, we shall extract a convenient numerical factor out of M˜ and suggestively re-write the dy- namical mass as

M共0兲⫽␯

冉

^␣␣c

冊

^{1/2⑀M¯共0,⑀兲. 共6兲}

At present there is no difference in content between Eq.共5兲 and Eq. 共6兲. However, if we now define ␣c by demanding that the behavior of M (0) is dominated by the factor (␣^/␣c)^1/2⑀as⑀ goes to zero, which is equivalent to demand- ing that

关M¯共0,⑀兲兴^⑀——˜

⑀˜0

1, 共7兲

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

FIG. 1. The critical coupling for the CP vertex. The solid line is taken from the bifurcation analysis carried out in Ref.关12兴, which agrees with the numerical results共open squares兲of Ref.关11兴. The dashed line corresponds to the bifurcation analysis carried out with the ‘‘gauge violating term’’ removed 共as suggested in Ref. 关14兴兲 and agrees with the numerical results共open triangles兲of Ref.关15兴.

M共0兲 ——_⑀˜˜

0

0, ␣⬍␣c 共8兲

M共0兲 ——˜

⑀˜0 ⬁, ␣⬎␣c.

Note that in the above we have not addressed the issue of whether or not there actually is a M¯ (0,⑀) with the property of Eq. 共7兲. 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 sym- metry is indeed broken for D⬍4.² Notwithstanding this, as one knows from cut-off based work that there actually is a non-zero critical coupling for D⫽4, one can at this stage already come to the conclusion that M¯ (0,⑀) exists and hence that quenched QED in D⬍4 dimensions has a chiral sym- metry breaking solution for all couplings. Furthermore, even though the dimensionless coupling ␣ does not get modified by vacuum polarization effects in the quenched approxima- tion, the effective dimensionless coupling␣^{¯ (q2) in D}⬍4 di- mensions nevertheless runs as a function of q²:

␣

¯共q²兲⫽␣共␮²兲

冉

^␮q22

冊

^{⑀, 共9兲}

where ␮ is the renormalization scale 共for details see, for example, the discussion of the renormalization group equa- tions in Chapter 3.2 of Ref.关17兴兲. In the infrared the effective coupling increases without bound, suggesting not only that the theory is likely to break chiral symmetry for any cou- pling共for D⫽4) but also that it may actually be confining.

Although we will not pursue this further in this paper, this feature of quenched QED bears some similarity to QCD so that it may be of interest to investigate the theory without taking the limit⑀˜0. In this connection, note also that in the ultraviolet the running coupling ␣^{¯ (q2}) vanishes; i.e., in D

⫽4 dimensions quenched QED exhibits ‘‘asymptotic free- dom’’.

In summary, as the trivial solution M ( p²)⫽0 always ex- ists as well, we see that in D⬍4 dimensions the trivial and symmetry breaking solutions bifurcate at␣⫽0 while for D

⫽4 the point of bifurcation is at ␣⫽␣c; i.e., there is a discontinuous change in the point of bifurcation. As D ap- proaches four 共i.e. as ⑀ approaches 0兲 the generated mass M (0) decreases共grows兲 roughly like (␣^/␣c)^1/2⑀ for ␣ⱗ␣c

(ⲏ␣c), respectively, becoming an infinite step function at

␣⫽␣cwhen ⑀goes to zero.

Finally, it is of interest to speculate as to how the above analysis might be carried over to other field theories, in par- ticular full QED and QCD. Indeed, it is clear that Eq.共5兲 关as well as Eq. 共6兲 in theories which have a non-zero critical coupling兴, which is based purely on a dimensional argument, will remain valid in any theory which does not contain an explicit mass scale. However, it should be kept in mind that

the coupling occuring on the right hand side of these equa- tions is the unrenormalized coupling␣. In full QED or QCD, as opposed to quenched QED, this is not a convenient quan- tity as the bare coupling will itself be a function of ⑀ ^{if the} renormalized coupling is kept fixed as the regulator is re- moved共indeed, in lattice studies of QCD the unrenormalized coupling goes to zero as the lattice spacing is taken to zero;

see Section 9.2 of Ref. 关18兴兲. Hence Eqs. 共5兲 and 共6兲, al- though valid, loose their utility.

Nevertheless, it is possible to write down an equivalent expression in terms of the dimensionless renormalized cou- pling ␣^R(␮²). We may use the relationship between

␣^R(␮^2), ␣ ^and␯ in order to eliminate the scale␯^{. Further-} more, as in D dimensions the dependence on the renormal- ized coupling enters through the dimensionful coupling ␣D R

⬅␣^R␮^2⑀, the appropriate equation for the unrenormalized mass M (0) becomes

M共0兲⫽␮„␣^R共␮²兲…^1/2⑀M˜共0,⑀兲. 共10兲 In addition, should the theory under consideration have a non-zero critical coupling 共as has been observed in some Dyson-Schwinger calculations of full QED, with a small number of fermion flavors N_f; see Ref.关4兴兲, then Eq.共6兲and the subsequent discussion remains applicable as long as the coupling ␣ is replaced by the renormalized coupling

␣^R(␮²). In particular, it would be possible to explore the dependence of the critical coupling on N_f by investigating the small ⑀ behavior of the relevant M˜ (0,⑀). On the other hand, if the four-dimensional theory breaks chiral symmetry for all couplings共e.g. QCD兲only Eq.共5兲 remains valid and there is no essential difference between D⫽4 and D⬍4. In this case the limit⑀˜0 could be smooth.

III. THE RAINBOW APPROXIMATION

Let us now consider an explicit vertex. To begin with, we consider the rainbow approximation to the Euclidean mass function of quenched QED with zero bare mass in Landau gauge. It is given by

M共p²兲⫽共e␯^⑀兲²共3⫺2⑀兲

冕

_共2^d␲^Dk兲^D

M共k²兲 k²⫹M²共k²兲

1 共p⫺k兲².

共11兲 Note that the Dirac part of the self-energy is equal to zero in the Landau gauge in rainbow approximation even in D⬍4 dimensions and hence that Z( p²)⫽1 for all p².

It is of course possible to find the solution to Eq. 共11兲 numerically—indeed we shall do so—however it is far more instructive to first try to make some reasonable approxima- tions in order to be able to analyze it analytically. First, as the angular integrals involved in D-dimensional integration are standard 共see, for example, Refs.关7兴 and关17兴兲 we may reduce Eq. 共11兲to a one-dimensional integral, namely

2The reader will note that as neither M˜ (0,⑀) nor M¯ (0,⑀) are func- tions of the coupling␣, the value of␣ccan be determined indepen- dently of the strength␣of the self-interactions in D⬍4 dimensions.

M共p²兲⫽␣␯^2⑀c_⑀

冕

^{0⬁ dk2}k^共2k⫹^2兲M^1⫺⑀2共^Mk^2共兲^k2兲

⫻

冋

^p12F

冉

^{1,⑀;2⫺⑀;kp22}

冊

^{␪共p2⫺k2兲}

⫹1

k²F

冉

^{1,⑀;2⫺⑀;pk22}

冊

^{␪共k2⫺p2兲}

册

^{, 共12兲}

where

c_⑀⫽ 3⫺2⑀

共4␲兲^1⫺⑀⌫共2⫺⑀兲

冉

^c0⫽43␲

冊

^{. 共13兲}

Note that for D⫽4 the mass function in Eq.共12兲reduces to the standard one in QED₄.

In D⫽4 dimensions the hypergeometric functions in Eq.

共12兲 preclude a solution in closed form. However, note that these hypergeometric functions have a power expansion in⑀ so that for small⑀ one is not likely to go too far wrong by just replacing these by their⑀⫽0 共i.e. D⫽4) limit. After all, the reason for choosing dimensional regularization in the first place is in order to regulate the integral, and this is achieved by the factor of k^⫺2⑀, not the hypergeometric func- tions. 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⑀^{that the} approximation is not too bad in these regions.³

Making this replacement, i.e.

M共p²兲⫽␣␯^2⑀c_⑀

冋

^p12

冕

^{0p2 dk2k共2k⫹2兲M1⫺⑀2共Mk2共兲k2兲}

⫹

冕

^{p⬁2 dk}k^2共2k⫹^2兲M^⫺⑀2共^Mk^2共兲^k2兲

册

^{, 共14兲}

allows us to convert Eq. 共12兲 into a differential equation, namely

关p⁴M⬘共p²兲兴⬘⫹␣␯^2⑀c_⑀ ^共p

2兲^1⫺⑀

p²⫹M²共p²兲M共p²兲⫽0, 共15兲 with the boundary conditions

p⁴M⬘共p²兲兩p²⫽0⫽0, 关p²M共p²兲兴⬘兩p²⫽⬁⫽0. 共16兲 Unfortunately, the differential equation 共15兲 still has no so- lutions in terms of known special functions. Since the mass function in the denominator of Eq. 共14兲serves primarily as an infrared regulator we shall make one last approximation and replace it by an infrared cut-off for the integral, which

can be taken as a fixed value of M²(k²) in the infrared re- gion共for convenience we shall call this value the ‘‘dynami- cal mass’’ m). This simplifies the problem sufficiently to allow the derivation of an analytical solution.

In terms of the dimensionless variables x⫽p²/␯^{2, y}

⫽k²/␯^{2 and a}⫽m²/␯² the linearized equation becomes

M共x兲⫽␣^c⑀

冋

^1x

冕

^{ax d y y1⫺⑀yM共y兲⫹}

冕

^{x⬁ dy y⫺⑀yM共y兲}

册

^;

共17兲 for simplicity, we do not explicitly differentiate between M (x) and M ( p²). This may be written in differential form as 关x²M⬘共x兲兴⬘⫹␣^c⑀x^⫺⑀M共x兲⫽0, 共18兲 with the boundary conditions

M⬘共x兲兩x⫽a⫽0, 关x M共x兲兴⬘兩x⫽⬁⫽0. 共19兲 This differential equation has solutions in terms of Bessel functions

M共x兲⫽x^⫺1/2

冋

^C1J␭

冉 冑

^{⑀4x␣⑀/2c⑀}

冊

^⫹C2J⫺␭

冉 冑

^{⑀4x␣⑀/2c⑀}

冊 册

^,

共20兲 where we have defined␭⫽1/⑀in order to avoid cumbersome indices on the Bessel functions. The ultraviolet boundary condition Eq.共19兲gives C₂⫽0 while the infrared boundary condition leads to

C₁

冋

^J␭

冉 冑

^{⑀4x␣⑀/2c⑀}

冊

^⫹

冑

^{4x␣⑀/2c⑀J␭⬘}

冉 冑

^{⑀4x␣⑀/2c⑀}

冊 册

x⫽a

⫽0. 共21兲

This equation may be simplified using the relation among Bessel functions

zJ_␭⬘共z兲⫹␭J_␭共z兲⫽zJ_␭⫺1共z兲, 共22兲 and becomes

C₁

冋 冑

^{⑀4x␣⑀/2c⑀J␭⫺1}

冉 冑

^{⑀4x␣⑀/2c⑀}

冊 册

x⫽a

⫽0. 共23兲

Clearly this equation is satisfied by C₁⫽0, which corre- sponds to the trivial chirally symmetric solution M (x)⫽0.

However, for values of a which are such that the argument of the Bessel function in Eq. 共23兲 corresponds to one of its zeroes, the equation is also satisfied for C₁⫽0, i.e. for these values of a there exist solutions with dynamically broken chiral symmetry. If we define j_␭⫺1,1⫽

冑

⁴␣^c⑀/⑀^{a⑀/2to be the} smallest positive zero of Eq. 共23兲, the dynamical mass for this solution becomes

m⫽␯^a1/2⫽␯␣^1/2⑀

冉

^⑀

冑

^{j␭⫺4c1,1⑀}

冊

^{1/⑀. 共24兲}

3It is however possible to show that a linearized version of Eq.

共11兲always has symmetry breaking solutions even without making this approximation of the angular integrals. We indicate how this may be done in Appendix A.

Note that for this solution the normalization C₁ is not fixed by Eq.共17兲 as this equation is linear in M (x). Later on we shall fix C₁ by demanding that M (a)⫽m, however there is no compelling reason to do this and one might alternatively fix the normalization in such a way as to approximate the true 共numerical兲 solutions of Eq. 共11兲 as well as possible.

Finally, note that, as expected, a dynamical symmetry break- ing solution exists for any value of the coupling and that the expression for the dynamical mass is in agreement with the general form expected from dimensional considerations关i.e.

Eq. 共5兲兴.

In order to extract␣c, we need to look at the behavior of m as⑀goes to zero共i.e.␭˜⬁). This may be done by noting that the positive roots of the Bessel function J_␭ have the following asymptotic behavior共see, for example, Eq. 9.5.22 in Ref.关19兴兲:

j_␭,s⬃␭z共␨兲⫹_k

兺

_⫽^⬁_{1 ␭}^fk2k共␨⫺^兲1, ␨⫽␭^⫺2/3a_s, 共25兲 where a_sis the sth negative zero of Airy function Ai(z), and z(␨) is determined 关z(␨⁾⬎1兴 from the equation

2

3共⫺␨兲^3/2⫽

冑

^z2⫺1⫺arccos 1

z. 共26兲 For large␭the variable␨is small and so it is valid to expand z around 1. Writing z⫽1⫹␦ ^{we obtain}

arccos 1

1⫹␦^⬃

冑

²␦⫺5

冑

²

12 ␦^3/2, 共27兲 and so␦⯝⫺␨^/21/3yielding, to leading order,

z⫽1⫺ a₁

2^1/3␭^2/3. 共28兲 If we define

␥⫽⫺ a₁

2^1/3⬃1.855757 共29兲 then the leading terms in the expansion of j_␭⫺1,1are

j_␭⫺1,1⬃␭⫹␥␭^1/3⫺1⫹O共␭^⫺1/3兲. 共30兲 Also, the coefficient c_⑀ appearing in Eq. 共24兲 may be ex- panded

c_⑀⬃

⑀˜0

3

4␲^共1⫹d⑀兲, d⫽ln共4␲兲⫹1

3⫹␺共1兲 共31兲 so that for small⑀ the dynamical mass becomes

m⬃␯␣^1/2⑀

冋

^{␲3共1⫹d⑀兲}

册

^1/2⑀

共1⫹␥⑀^2/3⫺⑀兲^1/⑀⬃␯

冉

^␲␣/3

冊

^{1/2⑀e1⫹d/2⫺␥⑀⫺1/3.}

共32兲

Note that the behavior of the first term 共for⑀ going to zero兲 dominates over the exponential function, as required in Eq.

共7兲. Hence the critical coupling in four dimensions is given by␲/3, as expected from cut-off based work关2,3兴.

Returning now to the mass function itself, we may substi- tute the expression for the dynamical mass, i.e., Eq. 共24兲, together with our choice of normalization condition

M共p²⫽m²兲⫽m, 共33兲 into Eq.共11兲in order to eliminate C₁. One obtains

M共p兲⫽m² 兩p兩

J_␭

冋

^{j␭⫺1,1•}

冉

^兩mp兩

冊

^⑀

册

J_␭关j_␭⫺1,1兴 . 共34兲 Note that the explicit dependence on ␯ 共and hence ␣^{) has} been completely replaced by m in this expression.

So far we have taken␣ independent of the regularization.

As we have seen this leads to a dynamically generated mass which becomes infinite as the regulator is removed. Fomin, et al. 关2兴 examined 共within cut-off regularized QED兲 a dif- ferent limit, namely one where the mass m is kept constant while the cut-off is removed. In our case this limit necessi- tates that the coupling␣ is dependent on ⑀ ^through

␣⯝␲

3共1⫹2␥⑀^2/3兲 共35兲 关see Eq. 共32兲; note that ␣c is approached from above兴. The limit may be taken analytically in Eq.共34兲by making use of the known asymptotic behavior of the Bessel functions 共see Eq. 9.3.23 of Ref.关19兴兲, i.e.

J_␭共␭⫹␭^1/3z兲⬃

冉

^␭2

冊

^{1/3Ai共⫺21/3z兲, 共36兲}

as well as the asymptotic expansion of j_␭⫺1,1 in Eq. 共30兲. One obtains

M共p兲⫽m²

p

冉

^{lnmp ⫹1}

冊

^{, 共37兲}

which agrees with the result in Ref.关2兴.

To conclude this section, we analyze the validity of the approximations made by solving Eq. 共11兲 numerically and comparing it to the Bessel function solution in Eq. 共34兲. In Fig. 2a we have plotted the mass function共divided by␯^{) as} a function of the dimensionless momentum x for a moder- ately large coupling (␣⫽0.6) and⑀⫽0.03. The solid curve corresponds to the exact numerical result关Eq.共11兲兴while the dashed line is a plot of Eq.共34兲for these parameters. As can be seen, the approximation is not too bad and could actually be made significantly better by adopting a different normal- ization condition to that in Eq. 共33兲. However, no further insight is gained by doing this and we shall not pursue it further.

One might naively think that most of the difference be- tween the Bessel function and the exact numerical solution comes from the linearization of Eq. 共11兲—i.e., the approxi-

mation made by going from Eq.共14兲to Eq.共17兲, as the only approximation made prior to this is to replace the hypergeo- metric functions by unity, which is expected to be good to order ⑀ 共i.e. in this case, 3%兲. This turns out to be not the case; the dotted curve in Fig. 2a corresponds to the共numeri- cal兲 solution of Eq. 共14兲. Not only is the difference to the true solution essentially an order of magnitude larger than expected共about 30%—note that Fig. 2a is a log-log plot兲, it is actually of opposite sign to the equivalent difference for the Bessel function. In other words, the validity of the two approximations is roughly of the same order of magnitude and they tend to compensate.

Why are the quantitative differences rather larger than expected? On the level of the integrands the approximations are actually quite good. In Fig. 2b we show the integrands of Eqs. 共11兲,共17兲and Eq.共14兲 for a value of x in the infrared (x⬇7.1⫻10^⫺11). Clearly the replacement of the hypergeo- metric functions by unity is indeed an excellent approxima- tion, as is the linearization performed in Eq. 共17兲 共except in the infrared, as expected. Note that when estimating the con- tribution to the integral from different y one should take into account that the x-axis in Fig. 2b is logarithmic兲. The real source of the ‘‘relatively large’’ differences observed for the integrals in Fig. 2a is the fact that these are integral equations for the function M (x)—small differences in the integrands do not necessarily guarantee small differences in M (x). To illustrate this point, consider a hypothetical ‘‘approxima- tion’’ to Eq.共17兲in which we just scale the integrands by a constant factor 1⫹⑀ and ask the question how much this affects the solution M (x). For x⫽0 the answer is rather simple: the hypothetical approximation just corresponds to a rescaling of␣^{by 1}⫹⑀and as M (0) scales like␣^{1/2⑀we find} that the solution has increased by a factor (1⫹⑀^{)1/2⑀. In}

other words, even in the limit⑀˜0 there remains a remnant of the ‘‘approximation,’’ namely a rescaling of M (0) by a factor e^1/2⬇1.6!

IV. THE CURTIS-PENNINGTON VERTEX

We shall now leave the rainbow approximation and turn to the CP vertex. The expressions for the scalar and Dirac self-energies for this vertex, using dimensional regularization and in an arbitrary gauge, have already been given in Ref.

关7兴. Before we discuss chiral symmetry breaking for this ver- tex we shall first examine the chirally symmetric phase. We remind the reader that in this phase in four dimensions the wave function renormalization has a very simple form for this vertex关20兴, namely

Z共x,␮²兲兩M (x)⫽0⫽

冉

^␮x2

冊

^{␰␣/4␲, 共38兲}

where the renormalized Dirac propagator is given by S共p兲⫽Z共x,␮²兲

p” . 共39兲

Here␰is the gauge parameter and␮^{2 is the}共dimensionless兲 renormalization scale. This power behavior ofZ(x) is in fact demanded by multiplicative renormalizability关21兴as well as gauge covariance关14兴. 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)⫽0 as long as xⰇ关M (x)/␯兴².

A.Z„p²… in the chirally symmetric phase

In the chirally symmetric phase, the unrenormalizedZ(x) corresponding to the CP vertex in D dimensions is given by

Z共x兲⫽1⫹ ␣ 4␲

共4␲兲^⑀

⌫共2⫺⑀兲␰

冕

0

⬁

dy y^⫺⑀

x⫺yZ共y兲

⫻

冋

^{共1⫺⑀兲}

冠

^1⫺I1D

冉

^yx

冊 冡

^⫹y⫹yxI1D

冉

^yx

冊册

^{. 共40兲}

This equation may be obtained from Eq.共A6兲of Ref.关7兴by setting b( y ) equal to zero in that equation and by using Eq.

共A8兲 of the same reference in order to eliminate the terms with coefficient a²( y ). The angular integral I₁(w) is defined to be

I₁^D共w兲⫽共1⫹w兲2F₁共1,⑀^;2⫺⑀^;w兲, 0⭐w⭐1 共41兲 I₁^D共w兲⫽I₁^D共w^⫺1兲, w⭓1. 共42兲 In four dimensions the solution to Eq. 共40兲 is given by a Z(x) having a simple power behavior while for D⬍4 this is clearly no longer the case. Nevertheless, it is possible to derive an integral representation of the solution of Eq. 共40兲 by making use of the gauge covariance of this equation. We do so in Appendix B, with the result

FIG. 2. The mass function for rainbow QED for␣⫽0.6 and⑀

⫽0.03 as a function of x⬅p²/␯². The dynamical mass is specified by m/␯⫽2.24⫻10^⫺6, where ␯ is the scale introduced by dimen- sional regularization. The solid line corresponds to the exact nu- merical solution of Eq.共11兲, the dashed line is the Bessel function of Eq. 共34兲 and the dotted line is the solution of the Dyson- Schwinger equation with the hypergeometric function replaced by unity关Eq.共14兲兴. In共a兲the actual mass function is shown, while in 共b兲we show the integrand at a particular value of x.

Z共x兲⫽x^⑀/22^1⫺⑀⌫共2⫺⑀兲

冕

0

⬁

du u^⑀⫺1e^⫺ru2⑀J_2⫺⑀共

冑

^xu兲, 共43兲 where r is defined in Eq.共B4兲. Although this result is exact it is somewhat cumbersome to evaluate numerically because, for ⑀˜0, the oscillations in the integrand become increas- ingly important. For this reason we shall approximate the integrand in Eq.共40兲by its IR and UV limits, as we did for the rainbow approximation共as before, this approximation is good to order⑀^{). Using}

I₁^D共w兲⫽1⫹ 2

2⫺⑀^{w⫹O关w2兴 共44兲} this approximation yields

Z共x兲⫽1⫹ ␣ 4␲

共4␲兲^⑀

⌫共2⫺⑀兲␰

冋

^2⫺⑀⑀

冕

^{0xd yyx1⫺⑀2 Z共y兲}

⫺

冕

x

⬁

d y y^⫺⑀⫺1Z共y兲

册

^{. 共45兲}

This may be converted to the differential equation Z⬙共x兲⫹3

xZ⬘共x兲⫽ ˜c

x^1⫹⑀

冋

^{Z⬘共x兲⫹21⫺x⑀Z共x兲}

册

^共46兲

where c˜ is defined to be

˜c⫽ ␣ 2␲

共4␲兲^⑀

⌫共3⫺⑀兲␰ 共47兲

and the appropriate boundary conditions are

x^2⫺⑀Z共x兲兩x⫽0⫽0, Z共x兲兩x⫽⬁⫽1. 共48兲 关The IR boundary condition arises from the requirement that the integral in Eq.共45兲needs to converge at its lower limit.兴 In order to solve Eq. 共46兲, it is convenient to change vari- ables to

z⫽ ˜c

2⫺⑀^{x⫺⑀, 共49兲}

and to define

a⫽2

⑀⫺1 共50兲

so that the differential equation becomes

zZ⬙⫺a共1⫺z兲Z⬘⫺a共a⫺1兲Z⫽0, 共51兲 while the boundary conditions now are

z^⫺aZ兩z⫽⬁⫽0, Z兩z⫽0⫽1. 共52兲 This equation is essentially Kummer’s equation 共see Eq.

13.1.1 of Ref.关19兴; we use the notation of that reference in

the following兲. Its general solution may be expressed in terms of confluent hypergeometric functions, i.e.

Z⫽z^a⫹1e^⫺az关C˜₁M共a,a⫹2;az兲⫹C˜₂U共a,a⫹2;az兲兴

⫽e^⫺az兵^C1关␥共a⫹1,⫺az兲⫹az␥共a,⫺az兲兴⫹C₂关1⫹z兴其^. 共53兲 The UV boundary condition is fulfilled if C₂⫽1 while C₁ is not fixed by the boundary conditions. Although Eq. 共45兲 is solved by Eq. 共53兲for arbitrary C₁ we shall concentrate on the solution with C₁⫽0. The reason for this is that the solu- tion to the unapproximated integral 关Eq.共43兲兴vanishes at x

⫽0 共see Appendix B兲while the term multiplying C₁ in Eq.

共53兲diverges like x^2⑀⫺2 and is therefore unlikely to provide a good approximation to Eq.共40兲. Hence we obtain

Z共x兲⫽

冋

^{1⫹2⫺˜c⑀x⫺⑀}

册

^exp

冉

^{⫺˜c⑀x⫺⑀}

冊

^{. 共54兲}

Finally, the renormalized functionZ(x,␮²) may be obtained from this by demanding thatZ(␮^2,␮²⁾⫽1 so that the renor- malized wave function renormalization becomes

Z共x,␮²兲⫽

1⫹ ˜c 2⫺⑀^x⫺⑀

1⫹ ˜c 2⫺⑀ ␮^⫺2⑀

exp

冉

^{⫺˜c⑀x⫺⑀}

冊

exp

冉

^{⫺˜c⑀ ␮⫺2⑀}

冊

^{. 共55兲}

Only in the limit D˜4 does this reduce to the usual power behaved function found in cut-off based work 关Eq. 共38兲兴 while for D⬍4 it vanishes non-analytically at x⫽0. On the other hand, note that the solution to Eq. 共40兲—for finite

⑀—only goes to zero linearly in x. For the purpose of this paper this difference in the analytic behavior in the infrared does not concern us as for solutions which break chiral sym- metry the infrared region is regulated by M²(x) so that we do not expect the chirally symmetricZto be a good approxi- mation in this region in any case.

B. Chiral symmetric breaking for the CP vertex We shall now examine dynamical chiral symmetry break- ing for the C P vertex in the absence of any explicit symme- try breaking by a nonzero bare mass, as before. Even for solutions exhibiting dynamical symmetry breaking, it is to be expected that the analytic result derived forZ(x)兩M (x)⫽0关Eq.

共55兲兴 remains valid as long as x is large compared to ( M (x)/␯^{)2 and}⑀is sufficiently small. That this is indeed the case is illustrated in Fig. 3, where we show a typical example of Z^⫺1(x) for a solution which breaks chiral symmetry. In this figure, as well as in the rest of this section, we shall be dealing with the renormalizedZ(x) and M (x) instead of the unrenormalized quantities in the previous sections. This makes no essential difference to the physics of chiral sym- metry breaking, although it of course effects the absolute scale ofZ(x). For a discussion of the renormalization of the dimensionally regularized theory we refer the reader to Ref.

关7兴.

The comparison to the analytic result in Fig. 3 provides a very convenient check on the numerics. Another check is provided by plotting the logarithm of M (0) against the loga- rithm of the coupling. According to Eq.共5兲this should be a straight line with gradient 1/2⑀. As can be seen in Fig. 4 not only does one observe chiral symmetry breaking down to couplings as small as␣⫽0.15, the expected linear behavior is confirmed to quite high precision.

Although the numerics in D⬍4 dimensions are clearly under control, the extraction of the critical coupling共appro- priate in four dimensions兲has proven to be numerically quite difficult. From the discussion in Secs. II and III, we antici- pate that the logarithm of the dynamical mass has the general form

ln

冉

^M共␯0兲

冊

^⫽21⑀ln

冉

^␣␣c

冊

^{⫹ln„M¯ 共0,⑀兲… 共56兲}

where the last term is subleading as compared to the first as

⑀ tends to zero. For sufficiently small ⑀, therefore, ␣c is related to the gradient of ln„M (0)… plotted against⑀^⫺1.

In Fig. 5 we attempt to extract ␣c in this way. The loga- rithm of M (0) was evaluated for⑀ ranging from 0.03 down to ⑀⫽0.015 for a fixed gauge ␰⫽0.25. The squares corre- sponds to a coupling constant ␣⫽1.2, although some of the points at lower ⑀have actually been calculated at smaller␣ and then rescaled according to Eq. 共5兲. At present we are unable, for these parameters, to decrease ⑀significantly fur- ther without a significant loss of numerical precision. 共We also note in passing that it is quite difficult numerically to move away from small values of the gauge parameter; ␰

⫽20, which, judging from Fig. 1, would not require a very high numerical accuracy for␣c, is unfortunately not an op- tion.兲

The two fits shown in Fig. 5 correspond to two different assumptions for the functional form of M¯ (0,⑀), which is a priori unknown. The curves do indeed appear to be well approximated by a straight line, however we caution the reader that this does not allow an accurate determination of

␣cas the gradient is essentially determined by the ‘‘trivial’’

dependence on log(␣⁾ 共more on this below兲. The solid line corresponds to the assumption that the leading term in M¯ (0,⑀) has the same form as what we found in the rainbow approximation, i.e.

ln

冉

^M␯共0兲

冊

^⫽21⑀ln

冉

^␣␣c

冊

^⫹c1

冉

^21⑀

冊

^{1/3. 共57兲}

With this form the fit parameters␣cand c₁ are found to be

␣c⫽0.966, c₁⫽⫺1.15. 共58兲 Indeed, the critical coupling is similar to what is found in cut-off based work 共see Sec. II; in Ref. 关15兴 the value was 0.9208 for␰⫽0.25). At present it is difficult to make a more precise statement, let alone differentiate between the two curves plotted in Fig. 1, as␣cis quite strongly dependent on FIG. 3. A typical共inverse兲wave function renormalization func-

tionZ^⫺1(x,␮²) corresponding to a chiral symmetry breaking solu- tion. Note that the mass function M (x) is of the same order as Z^⫺1(x,␮²) itself. Nevertheless, the analytical chirally symmetric solution of Sec. IV A共thin line兲provides an excellent approxima- tion共better than one part in a thousand for x⬎␮²) for x⬎M (x).

FIG. 4. The logarithm of the dynamical mass as a function of ln(␣) for⑀⫽0.025共i.e. 1/2⑀⫽20). The gauge parameter 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. Note that the dependence on the coupling expected in Eq.共5兲is reproduced to high precision.

FIG. 5. The logarithm of the dynamical mass as a function of 1/2⑀for a coupling of␣⫽1.20. All other parameters are as in Fig.

4. The open squares are the numerical values while the two lines are fits共see main text兲.

the functional form assumed in Eq.共57兲. In fact, allowing an extra constant term on the right hand side of Eq.共57兲reduces the critical coupling to 0.920 and the addition of yet a further term proportional to⑀^1/3increases it again to 0.931. As these numbers appear to converge to something of the order of 0.92 or 0.93 one might think that␣c has been determined to this precision. However, it is not clear that the functional form suggested by the rainbow approximation should be taken quite this seriously. The dashed line in Fig. 5 corre- sponds to a fit where the power of ⑀of the subleading term has been left free, i.e.

ln

冉

^M␯共0兲

冊

^⫽21⑀ln

冉

^␣␣c

冊

^⫹c1

冉

^21⑀

冊

^{c2. 共59兲}

The optimum fit assuming this form for ln„M (0)… yields a power quite different to ¹₃ and a very much smaller␣c:

␣c⫽0.825, c₁⫽⫺0.801, c₂⫽0.688. 共60兲 To conclude this section, let us discuss why it is that the functional form of the subleading term M¯ (0,⑀) appears to be rather important even if⑀is already rather small. The reason for this is two-fold: most importantly, although the leading⑀ dependence of ln„M (0)…is indeed⑀^⫺1, the coefficient of this term 共leaving out the trivial␣ ^dependence兲is ln(␣c). As ␣c

is rather close to 1 one therefore obtains a strong suppression of this leading term, increasing the relative importance of the subleading terms. In addition, it appears as if the numerical results favor a subleading term which is not as strongly sup- pressed 共as a function of ⑀) as suggested by the rainbow approximation共i.e. the power of⑀^⫺1 of the subleading term appears to be closer to ²₃ rather than ¹₃). This again increases the importance of the subleading terms.

V. CONCLUSIONS AND OUTLOOK

The primary purpose of this paper was to explore 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. In particular, this should be kept in mind when using the expected gauge invariance of the critical coupling as a criterion for judging the suitability of a particular vertex.

To begin with, 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共in which case, however, it would also not exist in D⫽4 dimen- sions兲or it exists for all nonzero values of the coupling 共in which case a chiral symmetry breaking solution exists in D

⫽4 for ␣⬎␣c). 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 discontinuous and so the extraction of the critical coupling of

the theory 共in 4 dimensions兲 is not as simple as in cut-off regularized work.

We next turned to an examination of symmetry breaking in the rainbow approximation in Landau gauge, both analyti- cally and numerically. Indeed, for this vertex one could re- write the共linearized兲Dyson-Schwinger equation as a Schro¨- dinger equation in 4 dimensions and appeal to standard results from elementary quantum mechanics to explicitly show that the theory always breaks chiral symmetry if D

⬍4. We also showed how the usual critical coupling ␣c

⫽␲/3 may be extracted from the dimensionally regularized work.

We concluded this work with an examination of the CP vertex. By making use of the gauge covariance of the theory we derived an exact integral expression for the wave func- tion renormalization function Z( p²) of the chirally symmet- ric solution. Furthermore we obtained a compact expression for this quantity which is an excellent approximation to the trueZ( p²) even for solutions which break the chiral symme- try. Finally, we extracted the critical coupling corresponding to this vertex and found that, within errors, it agrees with the standard cut-off results.

In the future, we plan to increase the numerical precision with which we can extract this critical coupling for the CP vertex by an order of magnitude or so. The factor limiting the precision at present is that when solving the propagator’s Dyson-Schwinger equation with the CP vertex by iteration the rate of convergence decreases dramatically as ⑀ ^{is de-} creased below⑀⬇0.015. If this increase in precision can be attained it will enable one to make a meaningful comparison with the cut-off based results shown in Fig. 1.

In addition to the above, it would be interesting to extend the work described in this paper to unquenched QED. Chiral symmetry breaking in this theory has been much studied in four dimensions 共employing a cut-off as a regulator; for a review, and references, see Sec. 10.9 of Ref.关3兴兲as well as in three dimensions 共see Ref.关22兴as well as Sec. 3 of Ref.

关4兴兲. In particular, for D⫽3 it is known that共at least in the N_f˜⬁ limit, with fixed ␣^Nf) the theory has both an ultra- violet fixed point at ␣^¯⫽0 and an infrared fixed point at␣^¯

⬃1. These fixed points will survive in 4⫺2⑀ dimensions, however in this case 共within the same approximation兲 the value of the running coupling at the infrared fixed point will go to zero like ␣^¯⬃⑀. The issue of how chiral symmetry breaking is manifested in dimensionally regularized un- quenched QED, without a cutoff, will then become closely connected to the possible existence of additional fixed points in the theory when going beyond the large N_f approximation.

If there are none, full QED would be trivial and uninteresting without the introduction共even in D⫽4 dimensions兲of a new dimensionful scale signifying the onset of physics outside the theory.

ACKNOWLEDGMENTS

We would like to acknowledge illuminating discussions with D. Atkinson, A. Kızılersu¨, V. A. Miransky and M. Re- enders. V.P.G. is grateful to the members of the Institute for Theoretical Physics of the University of Groningen for hos-

pitality during his stay there. This work was supported by a Swiss National Science Foundation grant共Grant No. CEEC/

NIS/96-98/7 051219兲, by the Foundation of Fundamental Research of Ministry of Sciences of Ukraine 共Grant No 2.5.1/003兲and by the Australian Research Council.

APPENDIX A: CHIRAL SYMMETRY BREAKING IN RAINBOW APPROXIMATION

In this appendix we show that the linearized version of Eq. 共11兲, i.e.

M共p²兲⫽共e␯^⑀兲²共3⫺2⑀兲

冕

_共2^d␲^Dk兲^D M共k²兲 k²⫹m²

1 共p⫺k兲²,

共A1兲 has symmetry breaking solutions for all values of the cou- pling. Our aim here is to convert this equation to a Schro¨dinger-like equation, which we do by introducing the function

␺共r兲⫽

冕

_共2^d␲^Dk兲^D

e^ikrM共k²兲

k²⫹m² . 共A2兲 With this definition we have

共⫺䊐⫹m²兲␺共r兲⫽

冕

_共2^d␲^Dk兲^De^ikrM共k²兲 共A3兲 where䊐 is the D-dimensional Laplacian and so

共⫺䊐⫹m²兲␺共r兲⫽e²␯^2⑀共3⫺2⑀兲

冕

_共2^d␲^Dp兲^De^{i pr}

⫻

冕

_共2^d␲^Dk兲^D M共k²兲 k²⫹m²

1 共p⫺k兲².

共A4兲 After shifting the integration variable ( p˜p⫹k) the last equation can be written in the form of a Schro¨dinger-like equation

H␺共r兲⫽⫺m²␺共r兲, 共A5兲 where H⫽⫺䊐⫹V(r) is the Hamiltonian, E⫽⫺m² plays the role of an energy and the potential V(r) given by

V共r兲⫽⫺e²␯^2⑀共3⫺2⑀兲

冕

_共2^d␲^Dp兲^D e^{i pr}

p² ⫽⫺ ␩ r^D⫺2,

共A6兲 where

␩⫽⌫共1⫺⑀兲 4␲^{2⫺⑀ e}

2␯^2⑀共3⫺2⑀兲. 共A7兲 For D⫽3 the coefficient ␩ ^{is 2}␯␣ while near D⫽4 it is (3/␲⁾␣␯^2⑀. It is well known from any standard course of

quantum mechanics共see, for example, Ref.关23兴兲that poten- tials behaving as 1/r^s at infinity, with s⬍2, always support bound states 共actually, an infinite number of them兲. In the present case this can be seen by considering the Schro¨dinger equation 共A5兲 for zero energy, i.e. E⫽0. The s-symmetric wave function then satisfies the equation

␺⬙⫹D⫺1

r ␺⬘⫹ ␩

r^D⫺2␺⫽0. 共A8兲 The solution finite at the origin r⫽0 is

␺共r兲⫽const⫻r^⑀⫺1J_1/⑀⫺1

冉 冑

^⑀␩r⑀

冊

^{. 共A9兲}

The Bessel function in Eq. 共A9兲 has an infinite number of zeros, which means that there is an infinite number of states with E⬍0.

Returning now to Eq. 共A5兲, we can estimate the lowest energy eigenvalue variationally by using

␺共r兲⫽Ce^⫺␬r 共A10兲 as a trial wave function. Here C is related to␬by demanding that ␺ is normalized, i.e.

兩C兩²⫽ 共2␬兲^D

⍀D⌫共D兲, 共A11兲

where⍀D is the volume of a D-dimensional sphere. Calcu- lating the expectation value of the ‘‘Hamiltonian’’ H on the trial wave function in Eq.共A10兲we find

E₀共␬²兲⫽具␺兩H兩␺典⫽␬²

冋

^{1⫺⌫共2DD⫺2兲␬D⫺4␩}

册

^{. 共A12兲}

The minimum of the ‘‘ground state energy’’ in Eq. 共A12兲, E₀(␬), is reached at

␬^4⫺D⫽共D⫺2兲2^D⫺3

⌫共D兲␩ 共A13兲

„for D⫽3 the parameter ␬ ^is ␯␣ while near D⫽4 it is

␯关␣^/(␲^/2)兴^1/2⑀…and is given by the expression

共E₀兲var⫽⫺m²⫽␬²

冉

^1⫺D21⫺1

冊

^{⫽␬2DD⫺⫺42, 共A14兲}

where the 1 is the contribution from the kinetic energy while the (D/2⫺1)^⫺1 corresponds to the potential energy. For D

⬎2 the potential is attractive and for 2⬍D⬍4 it is always larger than the kinetic energy, so for this case we get dy- namical symmetry breaking for any value of␣. For example, for D⫽3, one obtains E₀⫽⫺␬²⫽⫺␯²␣² which coincides precisely with the ground-state energy of the hydrogen atom 共not surprisingly, as we have used the ground-state hydrogen wave function as our trial function兲. In this case the dynami- cal mass is m⫽␯␣