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Chiral symmetry breaking and static gauge fields
Volume 139B, number 3
PHYSICS LETTERS
10 May 1984
CHIRAL SYMMETRY BREAKING AND STATIC GAUGE FIELDS ~ D. ESPRIU
Departmentof TheoreticalPhysics, 1 KebleRoad, Oxford OX1 3NP, UK Received 10 December 1983 Revised manuscript received 17 January 1984
We work out the spectral density of zero modes for a class of static SU(2) gauge fields recently proposed as being responsible for the spontaneous breakdown of chiral symmetry. Only for a zero measure set the spectral density is both nonvanishing and finite.
Not long ago a possible mechanism to account for the spontaneous breakdown ofchiral symmetry for light quarks in QCD was suggested by Floratos and Stern [ 1 ]. They proposed that the gauge field configurations responsible for a non-vanishing chiral order parameter ( ~ ) should be looked for in the p = 0 sector of QCD. More precisely, among static gauge configurations (A u = Au(x) ) with zero winding number. One considers the SU(2) eucfidean gauge theory with N f ( > 1) quarks o f a common bare mass m. The quantity
(Ol~i(Y)G (y, x) ~ki(x)lO)
lf[dA] [d~] I d a ] tr{~i(Y)G(y,x)~i(x)) × exp
[-S(A, q2, ~ ) ] ,
(1)
with
G(y'x)=Pexp( ig ra~) 2 dz#A~a(Z)) '
(2)
X
is the single flavor (no summation over i is assumed), gauge invariant, correlation function. If (1) does not vanish in the m ~ 0 , x ~ y limit one recognises the breakdown o f non-abelian chiral symmetry as this signals the appearance o f a non-zero value for (t~ff), ob¢' Work supported by Comissi6 Interdepartamental de Recerca i Innovaci6 Tecnol6gica de la Generafitat de Catalunya. 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
tained through the operator product expansion o f eq. (1). It has been argued that since any gauge configuration with p 4 : 0 should necessarily possess (due to the index theorem [2]) at least one normalisable, discrete fermionic zero mode (i.e. a normalisable eigenfunction o f i ~ ( A ) with zero eigenvalue, ~ ( A ) being the covariant derivative in the given external gauge field A~), the determinant o f gJ(A) is necessarily zero in the limit o f vanishing mass. Once the fermionic degrees o f freedom associated with the Nf - 1 flavors that do not appear on the LHS o f eq. (1) have been integrated out the measure o f the functional integral over the gauge fields contains a factor det I~(A)Nf -1. Thus any gauge configuration that leads to a zero value for det I~(A) would not be expected to contribute to the functional integral (1). The above discussion clearly holds at the lagrangian level, but quantum corrections may well lead to a different conclusion. In fact this problem is usually circumvented by arguing that due to the very breakdown o f chiral symmetry a dynamical mass for fermions can be generated through quantum corrections. Then, even for massless quarks det ~ ( A ) would be diferent from zero in the presence of fields with a non-zero winding number. Once the issue has been formulated in this way, what one can do is to look for a self-consistent relation for (~ ¢~). Nevertheless, to reach a definite answer one is forced to accept a number o f hypotheses (dilute gas approximation, semiclas183
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sical methods, ...) whose validity is far from being clear. On the contrary, as pointed out several times [3], the breaking of non-abelian chiral symmetry is very likely to be beyond these approximations. As a matter of fact, simulations on a lattice have provided numerical evidence, for both SU(2) and SU(3) gauge theories [ 4 - 6 ] , that the expected properties o f the vacuum - and, particularly, chiral symmetry breaking - indeed follow the expected patterns. Though the corresponding masses (in physical units) for the quarks involved in most simulations can hardly be described as "light" (typically they range in the scale of hundreds of MeV), the calculations shed some light on spontaneous breaking. The inclusion of the fermionic determinant in the measure shifts the statistical equilibrium towards more ordered configurations [5], thus lowering the value for (t~qJ). Clearly this effect becomes larger as N f increases. Lowering the mass should also reduce the value o f @~k), as it happens. Thus the predictions in the quenched approximation can only put an upper bound on @~k), as the contribution from dynamical quarks proves to be rather significant [5]. All this suggests that the fermionic determinant should also play a crucial role in the determination of (1) in the continuum. For massless quarks, chiral symmetry breaking can only show up in the infinite volume limit. Recent Monte Carlo simulations [7] show that this is indeed the case - for any finite size of the lattice (~qJ)= 0. What has to be done is to calculate for different lattice sizes and masses close to zero so as to obtain some indication of what the behaviour in the V = oo limit would be. The authors o f ref. [7] found a behaviour consistent with a breakdown of the chiral symmetry in the limit of an infinite lattice size. It is claimed in ref. [1] that a large set of SU(2) static fields in the v = 0 sector can actually be found for which a spontaneous breakdown of chiral symmetry occurs in the m ~ 0, infinite volume limit. A typical representative of this set - it is said - is a gas of m o n o p o l e - a n t i m o n o p o l e pairs. Furthermore, as this requires a non vanishing but regularly-behaved continous density of eigenstates of fib(A) in a neighbourhood o f the zero eigenvalue,it is claimed that for these configurations det ~ ( A ) admits a finite zero mass limit and, hence, for small masses, the functional measure may effectively be independent o f m and, eventually, the regularised fermion determinant may be close to one. 184
10 May 1984
The key point in this analysis is certainly the existence of a finite density of zero modes. The authors of ref. [ 1] managed to fred solutions to the equation iD(A)~ = 0 ,
(3)
for a wide class of SU(2) static gauge fields. Namely
AT(x) -- ~ j a j U(x) + ria(x) ,
A~(x) = OaV(x) ,
(4)
where U and V are entirely arbitrary functions and Tia is symmetric and traceless. However this fact, in itself, proves nothing as what one actually needs is a finite value for the spectral density of zero modes, lim O(X;x,x) = lim ~ b + ( X , x ) ~ ( X , x ) 4 : 0 , ~,~0 h ~0 where the q~'s are solutions to il~(A) $(a, x) = Xq~(X,x ) ,
(6)
and the sum in the RHS o f e q . (5) extends over all possible degeneracy. Indeed, after standard manipulations, eq. (1) can be written as
(0 [~i(Y)G(y, x) ~i(x)i0) oo m dX ~ t r { p O t ; y , x ) G ( y ,
-ftdAl f
x)}
0 × det ( ~ + m)~Rfexp [-S(A)] ,
(7)
where it has been taken into account that eigenfunctions to eq. (6) always occur in pairs of opposite chirality. In the m ~ 0 limit,
-f[dA] tr {p(0;y, x) G(y, x)} X exp ( - f d4x[L(A) +Lloop(A)l) .
(8)
Lloop(A ) is defined by
-Ne f In X2[p(X; x, x) - .fr=(X)l,
(9)
0
Pfree stands for the spectral density calculated in the a
perturbative vacuum A u = 0. This regularisation ensures the convergence of the determinant [8].
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Unfortunately, since in the infinite volume limit, we are dealing with a continuum spectrum, in order to guarantee p (0; x, x) =~ 0 it is not enough to know the zero mode solutions. We will show that for most o f the configurations in (4), p ( 0 ; x , x) is either zero or divergent and certainly the set of gauge configurations where this does not occur is not likely to survive the functional integral at all. Moreover, at finite temperature, we shall see, the suggested mechanism is clearly ruled out, since only those configurations in (4) with U ~ -ar% a 2> 1 for r-+ ~ have any chance o f producing the desired spectral density, and these configurations are exponentially suppressed. Certainly one does not expect any sharp transition to occur at T = 0 for (t~k), but rather at some finite temperature T c [6,9]. Let us consider eq. (6) for the class o f gauge fields
(4) 1 a
a
T~(ib~ - 5 r A~)~b(X, x) = XqS(X,x ) ,
(10)
in the chiral representation for 7 matrices 3'4
=
-I
'
-toy
, '
a and r are Pauli matrices acting on spinor and isospinor indices, respectively. Writing [ exp [V(x) - ½ V(x)] ~b+ '~ ~b= exp ( - i P 4 x 4 ) ~exp [U(x) + ½V(x)] $_ ] '
gP+
("l)
(u3)
U2
U4,
(12)
u2
03-01 )
2 \ 04 -- 02 '
( i t 3 ) = 1 ( 01+03 / u4 2 \ O2 + 041 "
(13,
The set of equations now reads
g(r) ~ 2t + 1 ! -g(r) ( ~ )
rtm-1/2(°'¢~)
l/2 ylm+l/2(O, ~)
g/m =
[ l - m + 3 ] 1/2
- i f ( r ) \ 2-277~]
YI+Im-I/2( 0, ~)
(l+m +3 ]1/2 YI+I rn+l/2(O, ~9) -if(r) ~2 l + 3 ]
(P4 -- ~')Vl -- ~v4/aXl +iav4/ax2 - Ov3/ax3 = 0 ,
l= 0, 1,2,...,
(/04 - ~)v 2 - Ov3/ax 1 - i~v3/ax 2 + av4/~x 3 = o, (P4 + )k)O3 -- ~O2/~Xl + iav2/~x2 - aOl/aX3 = 0 ,
One might wonder whether the only solutions to eq. (10) are those satisfying oaq)+_= raq~_+ - that is, those where q~v+nis diagonal, v(n) being the Dirac (isospin) index. For )t = 0, the answer to that question is yes - these are the only solutions, as can be easily seen from the results of ref. [10]. Indeed, Jackiw and Rebbi have studied a similar equation for the triplet Higgs model coupled to fermions in the presence of a monopole - a particular case of which is eq. (3) and found that for k = 0 all eigenfunctions fulfil this relation between internal and external indices. Then, though other families o f solutions for )t@0 may eventually exist, they cannot contribute to p(0; x, x). (Obviously, the presence o f some solutions for X = 0 does not imply p(0; x, x) =/=0, but their absence does imply p(0; x, x) = 0.) Clearly this discussion disregards the possibility that the limit 2,-+ 0 is not continuous in P, but if this were the case also eq. (8) would become invalid and, as noted in ref. [1 ], also the limit m -+ 0 need not be interchangable with the functional integral - nothing could be concluded. Though system (14) can easily be solved e.g. in a plane wave basis, in order to construct the spectral density O(2t; x, x) one actually needs a set o f orthonormal functions. So, to proceed further, the form of the functions U(x), V(x) needsto be specified at this point. We will assume for simplicity V = 0, U(x) = U(r); r = Ixl. Eq. (14) can then be solved in a spherical wave basis. There are two linearly independent families of solutions [ 11 ].
+4]t/2
and assuming that oa¢_+ = ra¢_+, eq. (10) reduces to the free Dirac equation for ¢+, ~_. It is advantageous however, to rewrite it in the euclidean Dirac representation, rather than in the chiral one. The change is readily performed.
(Ul)=l(
10 May 1984
m=-l-],...,l+
,
(14)
(P4 + X)°4 - OVl/3Xl -iOvl/3X2 + 302/3x3 = 0 . 185
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10 May 1984
with
g rt, r[ 1)- ~m +½]1/2 l Ytm - I/2( 0, ¢)
~1 = -pIKIM(I~[ + 1,21Kl + 1, p),
{l+m+ l~ 1/2 g(r) ~ 2-~'-~1 Ylm+l/2(O'(~) (15)
Olin =
-if(r)(l~_T1)l/2yl_lm_l/2(O,¢)
if(r) (/~7/m 1
Yl-lm+l/2(O,¢) 1
l=0,1,...,
m = - l + ½ .... l - ~ .
g, f should satisfy the differential equations -i(p 4 + k)f-
(20)
~2 = plKIM(I~ h 21~1 + l , p ) ,
[dg/dr + (1 + ~)g/r] = O, (16)
- i ( p 4 - ?`)g + [df/dr + (1 - ~)f/r] = 0 ,
and p = 2r(P42 - X2) 1/2. As before K = - l - 1 for Olin and K = I for V~m. These radial functions require normalisation. Here comes the first surprise. Bothf(r) andg(r) for IP4I > k behave as eM2 when p -+ ~, so for these modes to contribute to the spectral density, one actually needs U(r) -ar% a > 1 as r - + ~ t o ensure the convergence of (17). For such configurations, however, the action diverges badly and, hence, they are exponentially suppressed. On the other hand, for [P41 ~< X the radial functions can be easily normalised. The only condition to demand of U(r) is that the integral
/dr exp [2U(r)]
(2 1)
,
where ~ = - l - 1 for Vim and K = I for V~m. Once f, g have been determined, they have to be normalised (the spherical harmonics being already normalised) according to
should converge. The spectral density of eigenrfiodes is thus given by
f e x p [2U(r)] r2(lfl 2 + [gl 2) dr = 1,
p(X; X, X) = (21) 2eXprr [2U(r)] r.u ~ f dp4 ((/+1) -k
(17)
k
X {l"2(pr) + [ ( k - p4)/(X +p4)]j2+l (Pr))N2(1, P)
When solving eq. (16) one has to distinguish between IP4~ > X and [ p 4 [ < X. For I p 4 1 <~ X, the regular normalisable solutions are the spherical Bessel functions [ 12]
g(r) = yK(pr), f(r) = i [(?` - P4)/(?` +P4)] 1/2y~_l(Pr),
K = - ( l + 1),
g(r) = j ~(pr) ,
(18)
f(r)=i[(?`-pa)/(?` +pa)]l}2]~_l(Pr),
+
?`) [(t94 -- ?`)/X] l[2p-le-p/2(~o 1 +~P2),
f(r) = 2i(p 4 -- ?`) [(P4 + k)/?`] l[2p-le-P[2(~o 1 -- ~P2), (19) 186
(22) We include, obviously, the sum/integral over all degener ate eigenstates. N and N ' are the respective normalisation constants. The sum over rn has been performed quite trivially using the relation +l
K = l.
where p = (),2 _ p2)1/2. Note that for non-singular gauge fields the exponentials in (17) cannot cancel out a singular behaviour o f f , g when r -+ 0. For IP41 >/X the solutions can be read directly from ref. [11], where the bound states in an electrostatic potential are discussed, in the limit of vanishing charge. They are given in terms of degenerate hypergeometric functions g(r) = -2(/9 4
+ l(1"~(pr) + [(X - P4)/(?` +P4)]/2-1 ( pr)}N'2(I' P)}"
E
m =-1
Y?mYlm - 2;+1 4n
(23)
'
and the identity
~n(Z) = ( - 1 ) n + l / _ n _ l ( z ) ,
n = 0, +1, ...,
(24)
has also been taken into account. Now, we are interested in the h ~ 0 limit. Unless the integrand in (22) contains a singularity in the X + 0 limit, we shall obtain p(0; x, x) = 0. In fact, the only way of getting a finite limit is if the integrand behaves asp -1 as X approaches zero. Using the expressions for the Bessel functions near the origin, we realise at once that they behave at most asp 0. The higher l is, the
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more rapidly/'/approaches zero as p ~ 0. Thus, in the X ~ 0 limit, we only need to k e e p j 0 and eq. (22) becomes h
o(X;x,x) x-~o ~ i21r)2 1 exp [2U(r)] fdP4/~(Pr) -k
× (N2(O,p)+
[(X-p4)/(X+p4)]N'2(1,p)).
(25)
The reader will realise that only i f N o r N ' behave as p - i / 2 will a non zero and finite spectral density be obtained. For reasonably behaved potentials, the normalisation constants can be calculated and will essentially depend on how rapidly exp [2U(r)] decreases as r goes to infinity. For instance, if exp [2U(r)] ~ (r 2 + a2) - ~ , N,N' ~pl--a/2. This implies that only ifc~ = 3 [that is, for vector potentials such that U(r) ~ l n ( r 2 + a2) -3/2] a non-vanishing spectral density for zero modes is achieved. This tiny set can certainly not be expected to survive the integral over all gauge configurations. Including V(r) does not change our conclusions, but makes the analysis more cumbersome. On normalisability grounds V(r) should decrease more rapidly than U(r). It could be objected that one does not necessarily have to limit oneself to U and V depending only on r = Ixl. This is certainly true, but it is hard to imagine how allowing for less symmetric configurations could raise dramatically the density o f eigenstates. We mentioned that new problems arose as one departs from the zero temperature theory. For/3 = lIT < ~ one has to demand antiperiodic boundary conditions for the solutions to eq. (6). This implies that only solutions with P4 = (2K + i)7r//3, K = 0, -+1, +2, ... are acceptable. Then as X approaches zero, only the solutions corresponding to IP41 ~> X contribute to the spectral density. We have already seen that the normalisability o f these solutions puts a stringent limit on the behaviour o f U(r). Though the gauge fields described by (4) cover essentially all the static configurations which satisfy 3 a _ Y.a=lAa(x)O,it does not seem possible to obtain a non-zero chiral order parameter from them. At least, without taking into account the fermionic determinant. The configurations leading to a finite, non-zero density o f zero modes seem to be a zero measure set among the static gauge fields. On the other hand, the
10 May 1984
divergent contributions to p(0; x, x ) can only be compensated b y the fermionic determinant * a. This indicates that the quenched approximation is not suited at all for studying the breakdown o f c h i r a l symmetry. Of course, other static fields and time-dependent gauge configurations should have something to say as far as chiral symmetry breaking is concerned. In any o f these categories, one expects to Fred some gauge fields leading to a non-vanishing density o f zero modes, which could be found in the same way as we have worked out the spectral density in one particular case. Unfortunately, for most gauge configurations, the associated Dirac equation cannot be solved and, hence, it appears extremely difficult to find any general pattern (similar to the one we have found for a class o f static, traceless vector potentials) in which the gauge configurations are to be blamed for the breakdown o f chiral symmetry. Thus, any attempt to get a non-zero chiral order parameter out o f a particular set (more or less arbitrarily chosen) or sector ,of gauge fields seems to be definitely hopeless. At present, only numerical methods can be used to deal with this problem. Let us summarize, however, some o f the results we have obtained. We have seen that traceless, static fields cannot generate a non-vanishing (t~ff) for non-zero even small temperature. On the other hand, for these configurations to contribute to @~), the fermionic determinant turns out to be essential. For many static configurations, this determinant is strongly dependent on the smallest eigenvalues and, hence, very far from the quenched assumption. As we mentioned, this fact becomes apparent in actual Monte Carlo calculations o f @ ~ ) and thus it is very likely to remain true for most gauge field configurations. Quite recently, it has been conjectured that the spectrum o f QCD does not distinguish between 0 = 0 and 0 = 7r [ 13] (0 being the chirally invariant vacuum angle). A necessary condition for this to happen is that (t~ ~) should flip its sign if instead o f taking the ,1 In fact, all seems to indicate that a divergent contribution to O(0; x, x) is not only compensated but actually suppressed by exp[ - j'd4x Lloop(A)] . If p diverges at X = 0 the limit m ~ 0 is not legitimated in (7). Introducing an infrared cut-off in the integrals over the eigenvalue X and, finally, letting m go to zero it is not difficult to convince oneself that the fermionic determinant overwhelms the spectral density. 187
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limit m ~ 0 +, one takes rn ~ 0 - . This claim has been supported by recalling the mechanism suggested in ref. [ 1]. Without aiming to rule out the above conjecture, it is clear that this mechanism does not fulfil the required features of spontaneous chiral symmetry breaking. The author is indebted to M. Gross and N.A. McDougall for several comments. Thanks are also due to I J . R . Aitchison for reading the manuscript.
References [ 1] E. Florato s and J. Stern, Plays. Lett. 119B (1982) 419. [2] A.S. Schwarz, Phys. Lett. 67B (1977) 172.
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[3] See, e.g., S. Coleman, in: The whys of subnuclear physics, ed. A. Zichichi (Plenum, New York, 1979). [4] J. Kogut et al., Phys. Rev. Lett. 48 (1982) 1140. [5] H.W. Hamber, E. Marinari, G. Parisi and C. Rebbi, Phys. Lett. 124B (1983) 99. [6] J. Engels and F. Karsch, Phys. Lett. 125B (1983) 483. [7] I.M. Barbour, J.P. Gilchrist, H. Schneider, G. Schierholz and M. Teper, Phys. Lett. 127B (1983) 433. [8] G. 't Hooft, Phys. Rev. D14 (1976) 3432. [9] H. Satz, Bielefeld preprint, BI-TP 83]20; J. Kogut et al., Nucl. Phys. B225 (1983) 326. [10] R. Jackiw and C. Rebbi, Phys. Rev. D13 (1976) 3398. [11] H.A. Bethe and E.E. Salpeter, Quantum mechanics of one- and two-electron atoms (Plenum, New York, 1957). [ 12] M. Abramowitz and I. Stegun, eds., Handbook of mathematical functions (Dover, New York, 1965). [13] H. Leutwyler, CERN preprint TH 3739 (1983).
PHYSICS LETTERS
10 May 1984
CHIRAL SYMMETRY BREAKING AND STATIC GAUGE FIELDS ~ D. ESPRIU
Departmentof TheoreticalPhysics, 1 KebleRoad, Oxford OX1 3NP, UK Received 10 December 1983 Revised manuscript received 17 January 1984
We work out the spectral density of zero modes for a class of static SU(2) gauge fields recently proposed as being responsible for the spontaneous breakdown of chiral symmetry. Only for a zero measure set the spectral density is both nonvanishing and finite.
Not long ago a possible mechanism to account for the spontaneous breakdown ofchiral symmetry for light quarks in QCD was suggested by Floratos and Stern [ 1 ]. They proposed that the gauge field configurations responsible for a non-vanishing chiral order parameter ( ~ ) should be looked for in the p = 0 sector of QCD. More precisely, among static gauge configurations (A u = Au(x) ) with zero winding number. One considers the SU(2) eucfidean gauge theory with N f ( > 1) quarks o f a common bare mass m. The quantity
(Ol~i(Y)G (y, x) ~ki(x)lO)
lf[dA] [d~] I d a ] tr{~i(Y)G(y,x)~i(x)) × exp
[-S(A, q2, ~ ) ] ,
(1)
with
G(y'x)=Pexp( ig ra~) 2 dz#A~a(Z)) '
(2)
X
is the single flavor (no summation over i is assumed), gauge invariant, correlation function. If (1) does not vanish in the m ~ 0 , x ~ y limit one recognises the breakdown o f non-abelian chiral symmetry as this signals the appearance o f a non-zero value for (t~ff), ob¢' Work supported by Comissi6 Interdepartamental de Recerca i Innovaci6 Tecnol6gica de la Generafitat de Catalunya. 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
tained through the operator product expansion o f eq. (1). It has been argued that since any gauge configuration with p 4 : 0 should necessarily possess (due to the index theorem [2]) at least one normalisable, discrete fermionic zero mode (i.e. a normalisable eigenfunction o f i ~ ( A ) with zero eigenvalue, ~ ( A ) being the covariant derivative in the given external gauge field A~), the determinant o f gJ(A) is necessarily zero in the limit o f vanishing mass. Once the fermionic degrees o f freedom associated with the Nf - 1 flavors that do not appear on the LHS o f eq. (1) have been integrated out the measure o f the functional integral over the gauge fields contains a factor det I~(A)Nf -1. Thus any gauge configuration that leads to a zero value for det I~(A) would not be expected to contribute to the functional integral (1). The above discussion clearly holds at the lagrangian level, but quantum corrections may well lead to a different conclusion. In fact this problem is usually circumvented by arguing that due to the very breakdown o f chiral symmetry a dynamical mass for fermions can be generated through quantum corrections. Then, even for massless quarks det ~ ( A ) would be diferent from zero in the presence of fields with a non-zero winding number. Once the issue has been formulated in this way, what one can do is to look for a self-consistent relation for (~ ¢~). Nevertheless, to reach a definite answer one is forced to accept a number o f hypotheses (dilute gas approximation, semiclas183
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sical methods, ...) whose validity is far from being clear. On the contrary, as pointed out several times [3], the breaking of non-abelian chiral symmetry is very likely to be beyond these approximations. As a matter of fact, simulations on a lattice have provided numerical evidence, for both SU(2) and SU(3) gauge theories [ 4 - 6 ] , that the expected properties o f the vacuum - and, particularly, chiral symmetry breaking - indeed follow the expected patterns. Though the corresponding masses (in physical units) for the quarks involved in most simulations can hardly be described as "light" (typically they range in the scale of hundreds of MeV), the calculations shed some light on spontaneous breaking. The inclusion of the fermionic determinant in the measure shifts the statistical equilibrium towards more ordered configurations [5], thus lowering the value for (t~qJ). Clearly this effect becomes larger as N f increases. Lowering the mass should also reduce the value o f @~k), as it happens. Thus the predictions in the quenched approximation can only put an upper bound on @~k), as the contribution from dynamical quarks proves to be rather significant [5]. All this suggests that the fermionic determinant should also play a crucial role in the determination of (1) in the continuum. For massless quarks, chiral symmetry breaking can only show up in the infinite volume limit. Recent Monte Carlo simulations [7] show that this is indeed the case - for any finite size of the lattice (~qJ)= 0. What has to be done is to calculate for different lattice sizes and masses close to zero so as to obtain some indication of what the behaviour in the V = oo limit would be. The authors o f ref. [7] found a behaviour consistent with a breakdown of the chiral symmetry in the limit of an infinite lattice size. It is claimed in ref. [1] that a large set of SU(2) static fields in the v = 0 sector can actually be found for which a spontaneous breakdown of chiral symmetry occurs in the m ~ 0, infinite volume limit. A typical representative of this set - it is said - is a gas of m o n o p o l e - a n t i m o n o p o l e pairs. Furthermore, as this requires a non vanishing but regularly-behaved continous density of eigenstates of fib(A) in a neighbourhood o f the zero eigenvalue,it is claimed that for these configurations det ~ ( A ) admits a finite zero mass limit and, hence, for small masses, the functional measure may effectively be independent o f m and, eventually, the regularised fermion determinant may be close to one. 184
10 May 1984
The key point in this analysis is certainly the existence of a finite density of zero modes. The authors of ref. [ 1] managed to fred solutions to the equation iD(A)~ = 0 ,
(3)
for a wide class of SU(2) static gauge fields. Namely
AT(x) -- ~ j a j U(x) + ria(x) ,
A~(x) = OaV(x) ,
(4)
where U and V are entirely arbitrary functions and Tia is symmetric and traceless. However this fact, in itself, proves nothing as what one actually needs is a finite value for the spectral density of zero modes, lim O(X;x,x) = lim ~ b + ( X , x ) ~ ( X , x ) 4 : 0 , ~,~0 h ~0 where the q~'s are solutions to il~(A) $(a, x) = Xq~(X,x ) ,
(6)
and the sum in the RHS o f e q . (5) extends over all possible degeneracy. Indeed, after standard manipulations, eq. (1) can be written as
(0 [~i(Y)G(y, x) ~i(x)i0) oo m dX ~ t r { p O t ; y , x ) G ( y ,
-ftdAl f
x)}
0 × det ( ~ + m)~Rfexp [-S(A)] ,
(7)
where it has been taken into account that eigenfunctions to eq. (6) always occur in pairs of opposite chirality. In the m ~ 0 limit,
-f[dA] tr {p(0;y, x) G(y, x)} X exp ( - f d4x[L(A) +Lloop(A)l) .
(8)
Lloop(A ) is defined by
-Ne f In X2[p(X; x, x) - .fr=(X)l,
(9)
0
Pfree stands for the spectral density calculated in the a
perturbative vacuum A u = 0. This regularisation ensures the convergence of the determinant [8].
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Unfortunately, since in the infinite volume limit, we are dealing with a continuum spectrum, in order to guarantee p (0; x, x) =~ 0 it is not enough to know the zero mode solutions. We will show that for most o f the configurations in (4), p ( 0 ; x , x) is either zero or divergent and certainly the set of gauge configurations where this does not occur is not likely to survive the functional integral at all. Moreover, at finite temperature, we shall see, the suggested mechanism is clearly ruled out, since only those configurations in (4) with U ~ -ar% a 2> 1 for r-+ ~ have any chance o f producing the desired spectral density, and these configurations are exponentially suppressed. Certainly one does not expect any sharp transition to occur at T = 0 for (t~k), but rather at some finite temperature T c [6,9]. Let us consider eq. (6) for the class o f gauge fields
(4) 1 a
a
T~(ib~ - 5 r A~)~b(X, x) = XqS(X,x ) ,
(10)
in the chiral representation for 7 matrices 3'4
=
-I
'
-toy
, '
a and r are Pauli matrices acting on spinor and isospinor indices, respectively. Writing [ exp [V(x) - ½ V(x)] ~b+ '~ ~b= exp ( - i P 4 x 4 ) ~exp [U(x) + ½V(x)] $_ ] '
gP+
("l)
(u3)
U2
U4,
(12)
u2
03-01 )
2 \ 04 -- 02 '
( i t 3 ) = 1 ( 01+03 / u4 2 \ O2 + 041 "
(13,
The set of equations now reads
g(r) ~ 2t + 1 ! -g(r) ( ~ )
rtm-1/2(°'¢~)
l/2 ylm+l/2(O, ~)
g/m =
[ l - m + 3 ] 1/2
- i f ( r ) \ 2-277~]
YI+Im-I/2( 0, ~)
(l+m +3 ]1/2 YI+I rn+l/2(O, ~9) -if(r) ~2 l + 3 ]
(P4 -- ~')Vl -- ~v4/aXl +iav4/ax2 - Ov3/ax3 = 0 ,
l= 0, 1,2,...,
(/04 - ~)v 2 - Ov3/ax 1 - i~v3/ax 2 + av4/~x 3 = o, (P4 + )k)O3 -- ~O2/~Xl + iav2/~x2 - aOl/aX3 = 0 ,
One might wonder whether the only solutions to eq. (10) are those satisfying oaq)+_= raq~_+ - that is, those where q~v+nis diagonal, v(n) being the Dirac (isospin) index. For )t = 0, the answer to that question is yes - these are the only solutions, as can be easily seen from the results of ref. [10]. Indeed, Jackiw and Rebbi have studied a similar equation for the triplet Higgs model coupled to fermions in the presence of a monopole - a particular case of which is eq. (3) and found that for k = 0 all eigenfunctions fulfil this relation between internal and external indices. Then, though other families o f solutions for )t@0 may eventually exist, they cannot contribute to p(0; x, x). (Obviously, the presence o f some solutions for X = 0 does not imply p(0; x, x) =/=0, but their absence does imply p(0; x, x) = 0.) Clearly this discussion disregards the possibility that the limit 2,-+ 0 is not continuous in P, but if this were the case also eq. (8) would become invalid and, as noted in ref. [1 ], also the limit m -+ 0 need not be interchangable with the functional integral - nothing could be concluded. Though system (14) can easily be solved e.g. in a plane wave basis, in order to construct the spectral density O(2t; x, x) one actually needs a set o f orthonormal functions. So, to proceed further, the form of the functions U(x), V(x) needsto be specified at this point. We will assume for simplicity V = 0, U(x) = U(r); r = Ixl. Eq. (14) can then be solved in a spherical wave basis. There are two linearly independent families of solutions [ 11 ].
+4]t/2
and assuming that oa¢_+ = ra¢_+, eq. (10) reduces to the free Dirac equation for ¢+, ~_. It is advantageous however, to rewrite it in the euclidean Dirac representation, rather than in the chiral one. The change is readily performed.
(Ul)=l(
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m=-l-],...,l+
,
(14)
(P4 + X)°4 - OVl/3Xl -iOvl/3X2 + 302/3x3 = 0 . 185
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with
g rt, r[ 1)- ~m +½]1/2 l Ytm - I/2( 0, ¢)
~1 = -pIKIM(I~[ + 1,21Kl + 1, p),
{l+m+ l~ 1/2 g(r) ~ 2-~'-~1 Ylm+l/2(O'(~) (15)
Olin =
-if(r)(l~_T1)l/2yl_lm_l/2(O,¢)
if(r) (/~7/m 1
Yl-lm+l/2(O,¢) 1
l=0,1,...,
m = - l + ½ .... l - ~ .
g, f should satisfy the differential equations -i(p 4 + k)f-
(20)
~2 = plKIM(I~ h 21~1 + l , p ) ,
[dg/dr + (1 + ~)g/r] = O, (16)
- i ( p 4 - ?`)g + [df/dr + (1 - ~)f/r] = 0 ,
and p = 2r(P42 - X2) 1/2. As before K = - l - 1 for Olin and K = I for V~m. These radial functions require normalisation. Here comes the first surprise. Bothf(r) andg(r) for IP4I > k behave as eM2 when p -+ ~, so for these modes to contribute to the spectral density, one actually needs U(r) -ar% a > 1 as r - + ~ t o ensure the convergence of (17). For such configurations, however, the action diverges badly and, hence, they are exponentially suppressed. On the other hand, for [P41 ~< X the radial functions can be easily normalised. The only condition to demand of U(r) is that the integral
/dr exp [2U(r)]
(2 1)
,
where ~ = - l - 1 for Vim and K = I for V~m. Once f, g have been determined, they have to be normalised (the spherical harmonics being already normalised) according to
should converge. The spectral density of eigenrfiodes is thus given by
f e x p [2U(r)] r2(lfl 2 + [gl 2) dr = 1,
p(X; X, X) = (21) 2eXprr [2U(r)] r.u ~ f dp4 ((/+1) -k
(17)
k
X {l"2(pr) + [ ( k - p4)/(X +p4)]j2+l (Pr))N2(1, P)
When solving eq. (16) one has to distinguish between IP4~ > X and [ p 4 [ < X. For I p 4 1 <~ X, the regular normalisable solutions are the spherical Bessel functions [ 12]
g(r) = yK(pr), f(r) = i [(?` - P4)/(?` +P4)] 1/2y~_l(Pr),
K = - ( l + 1),
g(r) = j ~(pr) ,
(18)
f(r)=i[(?`-pa)/(?` +pa)]l}2]~_l(Pr),
+
?`) [(t94 -- ?`)/X] l[2p-le-p/2(~o 1 +~P2),
f(r) = 2i(p 4 -- ?`) [(P4 + k)/?`] l[2p-le-P[2(~o 1 -- ~P2), (19) 186
(22) We include, obviously, the sum/integral over all degener ate eigenstates. N and N ' are the respective normalisation constants. The sum over rn has been performed quite trivially using the relation +l
K = l.
where p = (),2 _ p2)1/2. Note that for non-singular gauge fields the exponentials in (17) cannot cancel out a singular behaviour o f f , g when r -+ 0. For IP41 >/X the solutions can be read directly from ref. [11], where the bound states in an electrostatic potential are discussed, in the limit of vanishing charge. They are given in terms of degenerate hypergeometric functions g(r) = -2(/9 4
+ l(1"~(pr) + [(X - P4)/(?` +P4)]/2-1 ( pr)}N'2(I' P)}"
E
m =-1
Y?mYlm - 2;+1 4n
(23)
'
and the identity
~n(Z) = ( - 1 ) n + l / _ n _ l ( z ) ,
n = 0, +1, ...,
(24)
has also been taken into account. Now, we are interested in the h ~ 0 limit. Unless the integrand in (22) contains a singularity in the X + 0 limit, we shall obtain p(0; x, x) = 0. In fact, the only way of getting a finite limit is if the integrand behaves asp -1 as X approaches zero. Using the expressions for the Bessel functions near the origin, we realise at once that they behave at most asp 0. The higher l is, the
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more rapidly/'/approaches zero as p ~ 0. Thus, in the X ~ 0 limit, we only need to k e e p j 0 and eq. (22) becomes h
o(X;x,x) x-~o ~ i21r)2 1 exp [2U(r)] fdP4/~(Pr) -k
× (N2(O,p)+
[(X-p4)/(X+p4)]N'2(1,p)).
(25)
The reader will realise that only i f N o r N ' behave as p - i / 2 will a non zero and finite spectral density be obtained. For reasonably behaved potentials, the normalisation constants can be calculated and will essentially depend on how rapidly exp [2U(r)] decreases as r goes to infinity. For instance, if exp [2U(r)] ~ (r 2 + a2) - ~ , N,N' ~pl--a/2. This implies that only ifc~ = 3 [that is, for vector potentials such that U(r) ~ l n ( r 2 + a2) -3/2] a non-vanishing spectral density for zero modes is achieved. This tiny set can certainly not be expected to survive the integral over all gauge configurations. Including V(r) does not change our conclusions, but makes the analysis more cumbersome. On normalisability grounds V(r) should decrease more rapidly than U(r). It could be objected that one does not necessarily have to limit oneself to U and V depending only on r = Ixl. This is certainly true, but it is hard to imagine how allowing for less symmetric configurations could raise dramatically the density o f eigenstates. We mentioned that new problems arose as one departs from the zero temperature theory. For/3 = lIT < ~ one has to demand antiperiodic boundary conditions for the solutions to eq. (6). This implies that only solutions with P4 = (2K + i)7r//3, K = 0, -+1, +2, ... are acceptable. Then as X approaches zero, only the solutions corresponding to IP41 ~> X contribute to the spectral density. We have already seen that the normalisability o f these solutions puts a stringent limit on the behaviour o f U(r). Though the gauge fields described by (4) cover essentially all the static configurations which satisfy 3 a _ Y.a=lAa(x)O,it does not seem possible to obtain a non-zero chiral order parameter from them. At least, without taking into account the fermionic determinant. The configurations leading to a finite, non-zero density o f zero modes seem to be a zero measure set among the static gauge fields. On the other hand, the
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divergent contributions to p(0; x, x ) can only be compensated b y the fermionic determinant * a. This indicates that the quenched approximation is not suited at all for studying the breakdown o f c h i r a l symmetry. Of course, other static fields and time-dependent gauge configurations should have something to say as far as chiral symmetry breaking is concerned. In any o f these categories, one expects to Fred some gauge fields leading to a non-vanishing density o f zero modes, which could be found in the same way as we have worked out the spectral density in one particular case. Unfortunately, for most gauge configurations, the associated Dirac equation cannot be solved and, hence, it appears extremely difficult to find any general pattern (similar to the one we have found for a class o f static, traceless vector potentials) in which the gauge configurations are to be blamed for the breakdown o f chiral symmetry. Thus, any attempt to get a non-zero chiral order parameter out o f a particular set (more or less arbitrarily chosen) or sector ,of gauge fields seems to be definitely hopeless. At present, only numerical methods can be used to deal with this problem. Let us summarize, however, some o f the results we have obtained. We have seen that traceless, static fields cannot generate a non-vanishing (t~ff) for non-zero even small temperature. On the other hand, for these configurations to contribute to @~), the fermionic determinant turns out to be essential. For many static configurations, this determinant is strongly dependent on the smallest eigenvalues and, hence, very far from the quenched assumption. As we mentioned, this fact becomes apparent in actual Monte Carlo calculations o f @ ~ ) and thus it is very likely to remain true for most gauge field configurations. Quite recently, it has been conjectured that the spectrum o f QCD does not distinguish between 0 = 0 and 0 = 7r [ 13] (0 being the chirally invariant vacuum angle). A necessary condition for this to happen is that (t~ ~) should flip its sign if instead o f taking the ,1 In fact, all seems to indicate that a divergent contribution to O(0; x, x) is not only compensated but actually suppressed by exp[ - j'd4x Lloop(A)] . If p diverges at X = 0 the limit m ~ 0 is not legitimated in (7). Introducing an infrared cut-off in the integrals over the eigenvalue X and, finally, letting m go to zero it is not difficult to convince oneself that the fermionic determinant overwhelms the spectral density. 187
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limit m ~ 0 +, one takes rn ~ 0 - . This claim has been supported by recalling the mechanism suggested in ref. [ 1]. Without aiming to rule out the above conjecture, it is clear that this mechanism does not fulfil the required features of spontaneous chiral symmetry breaking. The author is indebted to M. Gross and N.A. McDougall for several comments. Thanks are also due to I J . R . Aitchison for reading the manuscript.
References [ 1] E. Florato s and J. Stern, Plays. Lett. 119B (1982) 419. [2] A.S. Schwarz, Phys. Lett. 67B (1977) 172.
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[3] See, e.g., S. Coleman, in: The whys of subnuclear physics, ed. A. Zichichi (Plenum, New York, 1979). [4] J. Kogut et al., Phys. Rev. Lett. 48 (1982) 1140. [5] H.W. Hamber, E. Marinari, G. Parisi and C. Rebbi, Phys. Lett. 124B (1983) 99. [6] J. Engels and F. Karsch, Phys. Lett. 125B (1983) 483. [7] I.M. Barbour, J.P. Gilchrist, H. Schneider, G. Schierholz and M. Teper, Phys. Lett. 127B (1983) 433. [8] G. 't Hooft, Phys. Rev. D14 (1976) 3432. [9] H. Satz, Bielefeld preprint, BI-TP 83]20; J. Kogut et al., Nucl. Phys. B225 (1983) 326. [10] R. Jackiw and C. Rebbi, Phys. Rev. D13 (1976) 3398. [11] H.A. Bethe and E.E. Salpeter, Quantum mechanics of one- and two-electron atoms (Plenum, New York, 1957). [ 12] M. Abramowitz and I. Stegun, eds., Handbook of mathematical functions (Dover, New York, 1965). [13] H. Leutwyler, CERN preprint TH 3739 (1983).