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Instantons and chiral symmetry breaking
Nuclear Physics B245 (1984) 293-312 © North-Holland Pubhshmg Company
I N S T A N T O N S AND C H I R A L SYMMETRY BREAKING C E I CARNEIRO and N A McDOUGALL Department of Theoretwal Phystcs, Unwerszty of Oxford, 1 Keble Road, Oxford OXI 3NP, UK
Received 21 November 1983 (Revised 21 May 1984) A detaded investigation of chlral symmetrybreaking due to mstanton dynamicsis carried out, w~thln the framework of the ddute gas approximation, for quarks m both the fundamental and adjolnt representatmons of SU(2) The momentum dependence of the dynamical mass is found to be very s~mdar m each representation
I. Introduction For many years, a qualitative understanding of the phenomenology of light quark systems has been based on the idea of choral symmetry breaking certain symmetries of the lagranglan, associated with the near masslessness of the quarks, not being respected by the vacuum state The consequences of chiral symmetry breaking are manifested in the fact that most hadrons are very massive, when compared to the lagrangian masses of the quarks from which the hadron is formed, and the Goldstone nature of the plon octet of light pseudoscalar mesons An understanding of the mechanism which causes the spontaneous breakdown of the chlral symmetry is, therefore, crucial for an understanding of the dynamics of light quarks The experimental data are consistent with the strong Interaction possessing an apparent chlral SUv(3) × SUA(3) symmetry with the SUA(3) symmetry being reahsed In the N a m b u - G o l d s t o n e mode, the pion octet corresponding to the Goldstone bosons associated with the spontaneous symmetry breaking. However, the lagranglan of QCD, for three flavours of quark, formally possesses a Uv(3)×UA(3) global symmetry in the chlral limit, the extra Uv( 1) symmetry reflecting baryon conservation and the U(1) problem being associated with the extra UA(I) symmetry The spontaneous breakdown of the SUA(3) symmetry is indicated by the chiral order parameter (fqJ) being non-vamshlng in the physical vacuum. Since ( ~ ) ¢ 0 also violates the UA(1) symmetry, the experimental data imply that this symmetry is also spontaneously broken and there IS, therefore, a spontaneously broken UA(3 ) symmetry Thus, naively, nine light pseudoscalar states should be produced, not just the eight observed by experiment. The solution to this apparent paradox has provided deep insight into the mechanisms which may be responsible for the chiral symmetry breaking 293
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The resolution of the U(I) problem has indicated that the dynamics contained wlthan the non-trivial topological sectors of the QCD vacuum, with lnstanton dynamics possibly playing a dominant role, Is intimately connected with chlral symmetry breaking The relevance of mstantons to chlral symmetry breaking was first reahsed in 1976 by 't Hooft [1], who demonstrated that the Ua(l ) chlral symmetry was spontaneously broken by mstantons m a manner such that the assocmted Goldstone excitation coupled to a gauge-variant conserved current and hence there was no need for a ninth light pseudoscalar particle to appear xn the physical spectrum The suggestion that instanton dynamics may be responsible for breaking the S U ( N ) × S U ( N ) chiral symmetry, in addition to the UA( 1) symmetry, was first made by Callan, Dashen and Gross [2] Quanhtatlve calculations by Caldl [3] indicated that indeed the interactions were sufficiently strong to generate a dynamical quark mass A detailed investigation of the self-consistency equation, whose solution determines the dynamical mass, was carried out by Carhtz [4] and Carhtz and Creamer [5] for up to four quark flavours m the fundamental representation of colour SU(2) They found that for any number of quark flavours there was a non-trwial solution of the self-consistency equation, although it became increasingly difficult, as the number of quark flavours increased, to generate a self-consistent mass within the regime of validity of the dilute gas approximation, the required N 2 - 1 Goldstone bosons, formed as bound states of a quark and an anti-quark, also resulted from the same dynamics However, there resulted no massless flavour smglet pseudoscalar state since, in this channel, the quark-anti-quark force, generated by mstanton dynamics, was repulsive, this ~s, of course, in complete accord with a resolution of the U(1) problem being obtained from mstanton dynamics In this paper, we shall build upon these prewous mvestxgations and extend the calculation to include quarks m the adjomt representation of SU(2) In principle, the physical spectrum resulting from QCD may be calculated by computer, without any knowledge or understanding of the underlying dynamical mechanisms, if the theory if solved on a sufficiently large lattice However, to date, it has only been possible to perform calculations on rather small lattices and with a drastic approximation (the quenched approximation) imposed on the handling of the quark degrees of freedom Since chlral symmetry breaking is claimed to have occurred in these calculations, it ~s important to determine whether the chlral symmetry breaking m the lattice calculations is occurring m the same manner as m continuum QCD The fact that lattice calculations obtain (qTO) non-vanishing need not mean that chiral symmetry breaking ~s occurring m a manner consistent w~th a resolution of the U(1) problem In lattice calculations, the fermlon action chosen often introduces a large degree of explicit chlral symmetry breaking (the states with the quantum numbers of the pions need not therefore be Goldstone excitations), the validity of using the quenched approx~mation for simulating the dynamics of QCD is not proven and chiral symmetry breaking does not occur, m the chlral limit, on any lattice of finite s~ze, thus, an understanding of the dominant dynamical
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mechanisms in the lattice calculations is of great interest. The lattice calculations have investigated the temperature (energy scale) at which chlral symmetry breakdown occurs for quarks in both the fundamental and adjolnt representations of the gauge group SU(2) and, at first, found a change by a factor of about fifty [6], but this result has recently been revised and a more modest change is now quoted [7] In view of the preceding discussion, it seemed worthwhile to determine the change in energy scale expected from lnstanton dynamics and thus th~s investigation of xnstantons and chlral symmetry breaking, including quarks in representations other than the fundamental, was initiated Since we are lnvesUgatlng an example of spontaneous symmetry breaking, it is not possible to obtain the dynamical mass by perturbing about the chlrally symmetric vacuum, and thus we must assume that the chlrally asymmetric vacuum is favoured and determine the dynamical mass by a self-consistent calculation In sect 2 is described the condition which must be satisfied if the calculation of the dynamical mass IS to be self-consistent The self-consistency equation, for one quark flavour in the adjolnt representation of SU(2), is solved in sect 3 by separating the momentum dependence, contained in the zero elgenvalue modes, from the instanton scale size dependence In sect 4, the self-consistency equation is considered for an arbitrary number of quark flavours in both the fundamental and adjoint representations and the quenched approximation is simulated by setting the number of quark flavours equal to zero In sect 5, the dynamical mass for the adjoint representation is compared with that obtained previously for the fundamental representation The qualitative conclusions to be deduced from this investigation are briefly described in sect 6
2. C a l c u l a t i o n of quark m a s s e s generated by instantons
Consider one quark flavour in the adjolnt representation Interacting with SU(2) gauge fields (later we shall extend the analysis to include more than one quark flavour) The dynamics of this system is contained in the lagranglan 1 4g 2
We formulate our theory in euclidean space with metric g . . = 6.. and E1234 1 A convenient realization of the Dlrac y-matrices is =
=
(0
,
a ~ = ( - z { r , 1 ) = O/p,, -*
(2 lb)
where the %, are hermltian and satisfy the anticommutatlon relations {Y/x, Yv} = 2~/zv,
{Y/x, "Y5}~-'0 ,
")/5 ~ "YlY2Y3Y4
(2 l c )
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D . denotes the covarlant derivative in the a d j o m t representation
(D~)~h = O~6a~+e.~bA.,, F.~. = i~Av. - O.A.~ + e~b,A~.bA.,,
(2 ld) (2 le)
In order to c o m p u t e the quark mass generated by lnstantons we examine the quark p r o p a g a t o r which, in the dilute gas approximation, is given by [5] 1
s - So'-s,,'(s',+~+s ', ' ) s , , ' '
(2 2)
where So is the free p r o p a g a t o r and S]+'(S] -') is the connected one lnstanton (anti-instanton) contribution to the p r o p a g a t o r
x f DilDqq(x)gl(y)exp[- I glk~qd4x]
(23)
To obtain the above expression, we e x p a n d e d A . a r o u n d the lnstanton or antilnstanton
A.=A.+a.,
A ~ = -r/.~a0~ In
1+
.
(24)
where "q..~ ~ are the 't H o o f t antlsymmetrlc symbols (~k~ v = ekt~, "qk4. = qz6k~), and kept only terms which are O(a~,) in F ~ F , . a and O(0q) in the f e r m m n lagranglan (semi-classical approximation) N c and N o are n o r m a h s a t i o n factors and for simplicity we have not written down the gauge fixing term and ghost fields We see that the pseudo-particles generate a quark self-energy (in m o m e n t u m space)
.~(p)=-ggl(p)(g',+)(p)+g~, ' ( p ) ) g o ~ ( p ) lnvarlance X ( p ) = q~A(p2)+m(p 2) The terms
(25)
m(P 2) = ½Ys{Ys.S ( p ) }
(26)
Due to Lorentz proportional to p contribute to the r e n o r m a h s a t l o n of the quark wave function and are not related to chlral symmetry breaking effects Hence. the dynamical mass is
In order to determine this dynamical mass we must evaluate expression (23) The integration over the quark fields m a y be easily performed [8] and is p r o p o m o n a l to ~ . e-I±~'x'-I±~t( , t )t~, Y )" [I A, l
AK
,
(27)
l#l
where A, and 0~,± ~are respectively the eigenvalues and elgenfunctlons of the operator 1/~ ± (the covarlant derivative in the mstanton or anti-instanton b a c k g r o u n d field)
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297
and F is the number of quark flavours The expression (2.7), except for the case of one quark flavour with isospin ½, is zero in the chlral hmlt due to the existence of zero eigenvalues The number no of these zero elgenvalues, for each quark flavour, is given by the index theorem [9]
no=2T(T+I)(2T+I),
(2 8)
where T IS the quark lsospln For the fundamental representation T = ½and no = 1, and for the adjolnt representation T = 1 and no = 4 Since S~l±~(p) and consequently ~ ( p ) vanish there is apparently no dynamical mass generation This result, however, corresponds to perturbing about the chirally symmetric vacuum, a non-vanishing dynamical mass may be obtained if we perturb self-consistently about the physical chlrally asymmetric vacuum [10]. First we rewrite (2 la) as 1 = --~g2 F~.,.Ff.~,~- gt(.O + m)q + Ctmq, (2 9) where
glrnq IS a non-local mass term, fd4xglmq=fd4xd4y(l(x)m(x-y)q(y)=f&~(p)m(p)~(p),
(210)
and m is proportional to the unit matrix both in colour and euclidean space Next we consider the mass counterterm as belonging to the interaction lagranglan, that is we redefine the free and Interacting lagrangians and assume that m is a small quantity so that ~lmq can be treated perturbatlvely. Now, if m is the true dynamical mass the complete propagator must have the form S ( p ) = (zp+m(p)) -I However, we know that self-energy effects will change the free propagator. Self-consistency is achieved if the mass counterterm, treated perturbatively, cancels these self-energy effects, 1 e if
m(p)-½Ts(Ts, ~ ( P ) } = 0 -
(2 11)
(As mentioned previously, terms proportional to p renormalise the quark wave function and can be neglected). ~ ( p ) is still given by (2 5) but now It is a function of the unknown mass re(p) Thus, from (2.11) we obtain an integral equation (self-consistency equation) from which we may determine re(p)
3. The solution of the self-consistency equation The one-pseudo-particle contributions S~±~(p) to the quark propagator, for one quark flavour in the adjolnt representation of SU(2). are calculated in the a p p e n & x
37'(p)
7: Ts)arP ;
?[G2(pp)ml,(p)m~3(p) + K~(pp)m~l(p)m33(p)]L(Ixo, p) , o
(3 la)
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where
m,,(p)=~p 4 , a q q m q-)~K~(pq),
O(x)= (Ko(x)
and
Kl(x)
,'
L~-\l
.
\,,-
,
'
xV
(3 l b )
]
13 lc)
are m o d i f i e d Bessel f u n c t i o n s ) ,
L(l~o,p)=21°rr6p4go8exp
877"2 22 8
-~-+(~-~)ln(/xop)+a(l)
]
(3 l d )
R e c a l h n g that, to l o w e s t o r d e r in m, So I = t/~, we o b t a i n
-et p) = -go'( g',++ g, ')go' =47r2p 2 f dP[G2(pp)ml,(p)m3~(p)+K~(pp)rn~(p)rn~s(p)]L(txo, p) P
(3 2)
It is o b s e r v e d t h a t -Y(p), to l e a d i n g o r d e r in m, d o e s n o t c o n t a m t e r m s p r o p o r t i o n a l to p a n d t h e r e f o r e - ~ ( p ) c o m m u t e s w t t h 3'5 a n d c l e a r l y p l a y s t h e role o f a m a s s t e r m H e n c e f r o m t h e e x p r e s s t o n (2 11 ) t h e s e l f - c o n s i s t e n c y c o n d i t i o n for the d y n a m i cally g e n e r a t e d q u a r k m a s s
m(p)
is g i v e n by t h e i n t e g r a l e q u a t i o n
m(p 2) = 4 ~ r 2 p 2 f dP[G2(pp)m,,(p)m~3(p) +K~(pp)m~(p)m~(p)]L(Ixo, p),
(3 3)
3 P a n d t h e p o s s i b i l i t y o f l n s t a n t o n d y n a m i c s , t r e a t e d w i t h i n the d i l u t e gas a p p r o x i m a tion, c a u s i n g choral s y m m e t r y b r e a k i n g f o r a q u a r k m t h e a d j o l n t r e p r e s e n t a t i o n o f S U ( 2 ) rests o n f i n d i n g a n o n t r t v m l s o l u t t o n to (3 3)* B e f o r e s o l v i n g (3 3), let us c o n s i d e r t h e f u n c t t o n L(/xo, p) g t v e n m (3 l d ) c o e f f i c i e n t m u l t i p l y i n g In
(/xop)
for t h e S U ( 2 ) r u n n i n g c o u p h n g representation
-
The
is t h a t w h i c h a p p e a r s m the o n e - l o o p e x p r e s s i o n c o n s t a n t w i t h o n e q u a r k f l a v o u r In the a d j o i n t
T h u s we m a y w i n e
gO J L
L gZ(p)j,
g2(p)
,
* For one quark flavour m the fundamental representation of SUI 2 ~,the right-hand side ofthe expre~slon analogous to {3 3) has no powers of m(p), the Dlrac operator in this case having but a single zero elgen~alue mode Therefore, for one quark flavour m the fundamental representation, dynamical mass generation does occur, the magmtude of the dynamical mass being propomonal to an integration over mstanton scale s~ze With respect to the powers of re(p) on the right-hand s~de of 13 3), one quark flavour m the adjomt representation of SU(2I behaves as four quark flavours m the fundamental representation Thus, it ~s seen that the structure of the self consistency equation for the fundamental and adjomt representations is very d~fferent, pamcularly m the case of one quark flavour
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299
where A Is the scale parameter (In the Pauh-Vdlars regulanzatlon scheme) and we see that the effect of the one-loop ra&atlve correcUons is simply to replace go by the running couphng constant g(p) On physical grounds, g(p) should be substituted wherever go appears (the final result clearly should not depend on the renormahzatlon point/xo) The major obstacle to finding the exact soluuon of (3 3) is lack of knowledge regarding the correct manner of performing the mtegraUon over mstanton scale size Any method of solving (3 3) requires making assumptions about a regime where non-perturbaUve physics wdl dominate, 1 e at values of g(p) where the application of perturbatlve results is no longer vahd However, the quahtatlve features of the one-loop expression may be expected to extend into the non-perturbatlve regime If g(p) continues to increase monotomcally as p increases, we note that (87r2/g2(p)) 4 exp (--8~2/g2(p)) approaches zero at both large and small p, and It may be possible to isolate a well-defined region where the mtegrand m (3 3) is significant In fact, using the one-loop expression, (81r2/g2(p)) 4 exp (-8~2/g2(p)) peaks at g2/87r2 = 0 25 or pA = 0 42, l e where the couphng xs still relatively small A graph of (8"IrZ/g2(fl))4 exp (-8~2/gZ(p)) versus pA is shown m fig 1 Thus, the magnitude of lnstanton effects may be estimated by integrating over lnstanton scale size (m the range 0 to A -1) using the one-loop expression (Non-perturbatlve dynamics cause the couphng to grow at an even faster rate [2] than that given by the one-loop expression and hence the peak m the mtegrand may be very sharp. The one-loop expression will m&cate the scale at which mstanton effects start to become important and therefore the scale where the mtegrand peaks should be close to that given by the one-loop expression, namely pA = 0 42) Since we are most interested m the momentum behavlour of m ( p 2) and we wish to isolate the uncertainty assocmted with the integration over mstanton scale s~ze, we shall assume that the mstanton density peaks sharply at a scale p = ~ ~ (an estimate would be
5 1 3 2 1
02
0/~
06
08
1~0
pA Fig 1 A graph showing (1)
(8"n'2/g2(p))4exp(--8"n'2/g2(p)) and 87r2/g2(p)= 14 In (p 1)
(2)
8~2/g2(p) versus pA with
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C E I Carnelro, N A McDougall/ Instantons
~ 2 4A ) and approximate (3 3) by
P \g2(p)# exp
g2(p)#,
(3.5)
l e we consider the functions O:(pp), K~(pp), mlt(p), and m33(P ) tO be slowly varying in the region where the lnstanton density is significant The uncertainty associated with the integration over lnstanton scale size is therefore wholly contained in a dimensionless constant c 2,
~c2=_e<.,>f ~--l,,g----~p).] dp[ 8~r2 ~4 e x p \[_ g~(p)), 87./.2
(36)
o ~ ( l ) ~--~0 4 4 3
Eq (3 3) has now been reduced to a form where it may be solved analytically with no further assumptions Recalling that m~
(1) fo dss3G2 s 3
=2/z~
4
m33(1) =2-~4fo
X/t/
oO
m(s2 )
(3 7a)
'
dss3Kl(S)m(s2)'\l~l
(37b)
we see that if we multiply (3 5) by (3/2iz4)p3G2(p/tz) and (3/2tx4)p3K2(p/tx), integrate in both cases over p from 0 to oo and then make a change of variable P/t* -+ x, we obtain the two equations ml,(~) = (~)2mlt(l)m23(1)+(~)2m121(~)m33(]~), m33(1) = ( ~ ) 2 m l l ( 1 ) m 2 3 ( 1 ) + ( ~ ) 2 m ~ l ( 1 ) m 3 3 ( 1 )
(3 8a) ,
(3 8b)
where
oz2=fl 2~-
dxxSG4(x)~-O 0292,
£7
~2=_
(3 8c)
dxxSG2(x)K~(x)~-O 0358,
(3 8d)
dxxSK~(~) ~-0 236
(3 8e)
Neglecting the trivial solution m(p 2) = 0 and using the fact that m,,(p)~> 0, we obtain
-"x/ ~ ( a y ' +3-')'
m33 = c
(3 9a)
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C 3'
roll
301
McDougall / Instantons
a(C~3`+fl 2)
(3.9b)
Comparing the expressions (3 7) and (3 8c, d, e), we see that a convenient ansatz for is
m(p 2)
p2
2
m(p2)=A--~ G2(p) +BP K~(P)
(3.10)
Inserting (3 10) into (3.7) we obtain m,l(l) =3tzot2A+atzfl2B ,
(3 l l a )
m33(~ ) ----3/J,/~2A+3/z3`2B
(3 lib)
Since ml~(1//z) and ma3(1//x) are also given by (3 9a) and (3.9b) we have therefore reduced the system of integral equations to a linear system in A and B which has the following soluUon: A --
(fl 2 +a3`) -3/2 ,
(3 12a)
B=
(f12 +o~3`)-3/2 ,
(3 12b)
and thus (3.13) For
p/tx ~-0 G2
=
(3 14)
(3 15) Hence, 2~,
/ ~ \ I/2
/3 The non-trivial solution of(3 5) for the dynamically generated quark mass is therefore gwen by m(p 2) = m(0)
Ga
+K 2
,
(3.16b)
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302 10 08 ~06 E04
',? \
02
I
2
3
z~
5
6
7
Fig 2 A graph showing (1) the dynamical mass function for the adjolnt representation (2) the contribution from (p/l~)2(y/a)G2(p/#), (3) the contribution from (p/~)ZK~(p/#)
where the uncertainty associated with the Integration over mstanton scale size is contained in the value of ~t and the dimensionless constant c in the expression for m(0) Since G2(p/i.z) and K~(p/l~) are known functions (see (3 lc)), we may plot m(p)/m(O) Fig 2 shows the graph of m(p)/m(O) and the behavlour expected of a dynamical mass is observed, the mass falling rapidly (asymptotically as p-6) as the momentum of the quark increases
4. The self-consistency equation for an arbitrary number of quark flavours Since the lattice calculations of Kogut et al [6, 7] were performed using the quenched approximation, it is of Interest to investigate how the results obtained in sect 3 are modified if the quenched approximation is simulated m the calculation of chlral symmetry breaking due to instanton dynamics In the quenched approximation, Det E ~ is taken to be unity (internal fermlon loops are neglected during the evaluation of the path integral) Since Det D is the product of the elgenvalues of and when the dynamical mass (contributions to the quark propagator arising from a single interaction with either an mstanton or anti-lnstanton) is calculated, one elgenvalue is removed from the product of all elgenvalues (and replaced by the square of the elgenfunctlon), the self-consistency equation for the dynamical mass becomes, in the quenched approximation, m A ( p ) = e ~l,p2
d p[G2(pp) L ~ +
K~(pp)]l' 87r2"~4
m---~-) j k u - ~ p ) } exp
-g~P)
,
(4 1)
with 8rr2/g2(p)=-3 2 In (pA), there being no influence on the ¢3 function arising from quark loops (Note that the trlwal solution of the self-consistency equation, corresponding to the chlrally symmetric vacuum with re(p)= 0, is not a possible
C E I Carne~ro,NA McDougall/ lnstantons
303
solution of (4 1)). For NA quark flavours in the adjolnt representation, the selfconsistency equation for the dynamical mass is a simple extension of (3.3)
rnN~,(p) =
e(2NA--')'~(l)p
2
f dP [G2(po)ml,(p)mZ3(p) + K~(po)m21(o)m33(P)] P
X(mlI(p)m33(p))2(NA-I~p 4NA
g-tP)
exp
(
-
,
g P
(4 2)
with 8rr2/g:(p)= _~22_~ ~3 3 NA)In (pA) to a one-loop level It IS observed that the self-consistency equation for the quenched approximation (4 1) simply corresponds to the NA~ 0 limit of (4 2) So let us solve (4 2) for an arbitrary NA and the solution resulting from the simulation of the quenched approximation will be obtained by setting NA = 0 Separating the momentum dependence from the instanton scale size dependence in the usual manner we obtain
mN,,(p)~--e(2N,-"~"')p2tx4:~, [ G2( P~)mlt( l--~m~3(1-~-I \[~]~]A/
L
\I~NA /
\]~NA/
Jr-/ 2(-~P ~ g/'/21(~l ~ 1,r/33(--~1~ ] \~"LNA/
\['LN A /
\~JLNA/ A
x[rn,,(__~l/rn33( I )]2(N~-" I dp[ 87r 2 ~4 L \/xuA/ \/zu^/_l P \g-~p)] exp
(_
87T2 ~
g2(p)]
(4.3)
Defining
do(
3C~ _----e'-~A
J P \g2(p),l
4
(_ exp \
g2(o)],
2 22 8 g2(p)-- (T--sNA)
In (pA),
(4 4) the self-consistency equation (4.3) may be solved in the manner described in sect 3 with the results
mN'(p)=mNA(o)( P ~2[3`G2( La \/XNA/ P~I +K~(#-~)]
(45a)
where "~
[ O l \ (2NA--I)/f4N A 2)
- / cNA,"l 1/(2NA--|)~\T] mNA(0)=~..~A~
(/32+~3`)--~4NA--,,/,4N~ 2,. (4 5b)
32=00292, /32=00358, 3,2=0236 Thus, we see that the graphs of m(p)/m(O) obtained in this manner are independent of the number of quark flavours Chlral symmetry breaking due to lnstanton dynamics, treated within the dilute gas approximation, has previously been investigated by Carhtz [4] and Carhtz and
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304
Creamer
[5] for quarks
of SU(2)
transforming
For NF flavours,
accordmg
to the fundamental
the self-consistency
equation
representation
becomes
(4 6a)
where a(i)=0 level, f(5)= 1, ,(l),
146, a!(l)=0443
and 8~TT2/gZ(P)=-(~-_NNF)ln(p.~)
toaone-loop
-2~[~,(5)~"(5)-z"(5)K,(5)1-2~,(5)K,(5), f(O)=1,
K, ,(l) bemg modified
Bessel functions,
(4 6b)
and
X m(p)=$
Eq (4 6a) may be solved by the same method mN,(P)
=
(4 6c)
dppm&).#pp)
I0
as (4 2), yteldmg
mNF(0if2
the solutton (4 7a)
p (
2PW, > 3
wnh WIN&O)= /1.N,(cx~6)~“‘Nr-“6’ c&=$exp
(-a(
1) +2N+($))
s=4
)
N,#2,
[T(-j$)‘exp(-$j$),
’ dy_$(y)=O34, I0
(4 7b) (4 7c)
(4 7d)
and pJ,L IS the mstsnton scale size at whrch the mstanton density peaks For two quark flavours, terms of higher order m m(p) are required rn order to determme the magnitude of the dynamical mass
5. Comparison
of the dynamically
generated mass for the fundamental representations
and adjoint
As was mentioned m the mtroductton, this mvestrgatron of mstanton-generated chnal symmetry breaking was nntlated as a result of lattice calculations of chnal symmetry breaking for quarks m the adJoint representation of SU(2) Early in 1982, Kogut et al [6] published results which claimed that the choral symmetry restoration temperature for adjoint representation quarks was much higher than that for fundamental representation quarks VIZ TA/TF= 56 k 5 The mterpretatton of this result was that choral symmetry breakdown for quarks m the adJomt representatton occurred at a higher energy scale (shorter distance) than for fundamental representation quarks Since tt IS not clear that choral symmetry breakmg m SU(2) lattice
C E I Carnelro, N A McDougall / lnstantons
305
calculations, especially if the quenched approxlmatton ts used, occurs m the same manner as m the continuum QCD, an understanding of thts result was considered to be of importance Smce the peak m the mstanton density is largely determined by the behavlour of the coupling constant, which in the quenched approximation is independent of the quark representation, the lattice result &d not lmme&ately appear to be consistent with chlral symmetry breaking, for both representations, occurring due to mstanton dynamics However, when the quark representation ~s changed, there is a &fferent number of vanishing e~genvalue modes (of the massless Dxrac operator) with possibly very different momentum-space behavlour and thus the m o m e n t u m scale at which the dynamical mass begins to grow rapidly could be significantly modified From the results presented later in this section, it is clear that the chiral symmetry breaking scales for the fundamental and adjomt representations should be very similar if chlral symmetry breakdown, in both representations, occurs due to mstanton dynamics, the conclusion being unchanged if the quenched approx~matlon is simulated during the calculations This suggested either that the lattice result was unrehable or that chlral symmetry breaking m the lattice calculation did not occur due to mstanton dynamics for both quark representations Recently, however, a re-analysis of their data has caused Kogut et al to rewse their estimate of TA/Tv and they now quote [7] TA/TF= 8 6 + 4 5, which, m wew of the large uncertainty and the assumption of the approximate equivalence of the chlral symmetry restoration temperature and the chiral symmetry breaking m o m e n t u m scale (the analysis of finite temperature lnstanton dynamics [11] is rather more comphcated than for zero temperature), is not m conflict with chiral symmetry breaking on the lattice occurring due to instanton dynamics for quarks m both the fundamental and adjomt representations Using the results given in eqs (4 5a) and (4 7a) we may draw a graph of re(p)~ re(O) for both the fundamental and adjomt representations This is shown in fig 3, where 10 08 o
E 06
~
%
O2
0
1
2
3
t) pip
5
6
7
Fig 3 The dynamical mass functions for the fundamental representation (sohd curve) and the adjomt representation (dashed curve) obtained from (4 7a) and (4 5a) respectively
306
C E 1 Carnetro, N A McDougall / lnstantons
~t is observed that there is no radical &fference in the momentum dependence the shapes of the graphs are very similar and the momentum scales where there are sudden increases in the dynamical masses are the same to within a factor of two When the quark representation is changed, the expression for the coupling is modified and this will produce a shift m the peak of the instanton density For one flavour in the adjolnt representation, using the one-loop expression, th~s peak occurs at p~l = 0 42 (i e / z A - 2 4A) while for one flavour in the fundamental representation the peak occurs at p~| = 0 55 {1 e tZF--1 8A) Hence, for one quark flavour, ~z~- 1 3/zF and this difference should be borne in mind when looking in detail at fig 3 Thus, we may conclude that there is no evidence, from the zero temperature calculation, for any radical change in the momentum dependence of the dynamical mass between the fundamental and adjomt representations Furthermore, since the graphs of m(p)/m(O) are independent of the number of quark favours and the quenched approximation corresponds to the limit of zero quark flavours, the result shown in fig 3 should refect the behavlour expected from a lattice calculation using the quenched approximation
6. Concluding discussion We have presented a calculation of chlral symmetry breaking due to lnstanton dynamics, treated within the &lute gas approximation, for quarks in both the adjolnt and fundamental representations of SU(2) Several interesting features have emerged Firstly, within the regime of validity of the calculations, the behavlour of the momentum dependent mass is similar in each case (see fig 3) We believe that this result is non-trivial the zero elgenvalue modes can have very different momentum space behavlour (compare p2G2(p) and p2K~(p) In fig 2) and It Is not clear that the functions re(p) obtained for each representation need have been so similar (even for large p the dynamical mass m each case has the asymptotic behaviour p-6) To determine whether this result is a coincidence would require investigation of the zero elgenvalue modes for quarks m higher representations of the gauge group As expected from an mstanton calculation, there IS no evidence for a radical change in the length scale of the configurations responsible for the chlral symmetry breaking Thus, the revised result, using the quenched approximation, from Kogut et al [7] T~/TF= 8 6 ± 4 5. IS not In conflict with chlral symmetry breaking in lattice calculations arising due to lnstanton dynamics However, we believe that the crucial test of whether chlral symmetry breaking in lattice calculations occurs in the same manner as in continuum QCD lies in the resolution of the U(I) problem. The absence of the UA(1) Goldstone boson in the physical spectrum requires ( ~ ) to have a specific 0-dependence (depending on the number of hght quark flavours)
C E I Carnezro,N A McDougall/ Instantons
307
Thus, a lattice c a l c u l a t i o n to d e t e r m i n e the 0 - d e p e n d e n c e of (q~q/) or a n investigation of the massless p s e u d o s c a l a r states p r o d u c e d w o u l d be of great interest* We wish to express our t h a n k s to F. dos Aldos, N H. Parsons a n d J.F W h e a t e r for useful &scusslons. O n e of us (C.E I C ) wishes also to t h a n k Funda~fio de A m p a r o a P e s q m s a do Estado de Silo P a u l o for f i n a n o a l s u p p o r t while the other a c k n o w l e d g e s the receipt of a U K Science a n d E n g i n e e r i n g Research C o u n c d Postdoctoral F e l l o w s h i p
Appendix THE ONE-PSEUDO-PARTICLE CONTRIBUTION TO THE PROPAGATOR The o n e - p s e u d o - p a r t i c l e c o n t r i b u t i o n to the q u a r k p r o p a g a t o r in the self-consistent scheme for the case of one q u a r k flavour is given by
S(~±)(x-y)=N~tf Da~,exp[I~G(a~,,a~,)d4x]N~l
Iol +m qd']. (A 1) If we i n t r o d u c e source terms q~7, ¢1q for the f e r m l o n fields we can rewrite the integral over the f e r m i o n fields as hm ,,,~-o
~#(x)6rl(y)
exp
d4x '
d y 6-~
m(x -y') ~
x f DglDq e s° =- b°~±~,
N~ (A 2a)
J where (
So-- - - J
d4x
[(t(x)k~q(x) +fT(x)q(x)
+tT(x)r/(x)].
(A.2b)
We s h o w e d m sect. 2 that the o p e r a t o r , D ±, m the a d j o m t r e p r e s e n t a t i o n , has four zero e l g e n v a l u e modes, ~0(±'') ~= 1, 2, 3, 4, a n d for this reason it does n o t have an inverse However, if we remove the zero elgenvalue m o d e s we m a y calculate the * In the quenched approximation, it is not possible to obtain self-consistency m the 0-dependence of the dynamical mass [12] (1 e the same 0-dependence on both sides of (4 6a) with NF = 0), except for the trivial value 0 = 0, although self-consistency is possible for any other number of quark flavours Th~s result most probably in&cates that a resolution of the U(1) problem ~s not to be found wRhm the quenched approximation
C E I Carnesro, N A M~Dougall/ ln~tantons
308
reverse and perform the gaussmn m t e g r a t m n m the usual way [8] Using the completeness p r o p e r t y of the set {t~ ~± "~}, 4/~ "~ is the o g e n f u n c t m n of tD ± with exgenvalue )t., it is easdy shown that
S'<±'(x, y) = 5~'
0,± .,i x)q/~ .,l(y)
n
(A 3a)
hn
has the property ,1
l,O±S'~±'(x,Y)=64(x-Y) - 2
t~l+"'(x)t 91~' *(~),
(A3b)
where a prime in either }~ or l] denotes the removal of the zero elgenvalues C h a n g i n g the variables of lntegranon
q ( x ) ~ q ( x ) - z f d4y S'(x, y)rl(y) , 3 q ( x ) ~ q ( x ) -- t f d4y
fT(y)S'(y, x),
(A4a)
(A 4b)
we obtain, to lowest order m m, ~pt±) __
a~(x) 6rl(y) ~
6-~rn~
x - Y )~lorlty )/ ~'~o . A.
x ,=,[I (0 '±''~*, n)(rl, tk'±"')] ,
(A 5a)
where (A 5b)
(f" g) ~ I d4xf~(x)ga(x)
This calculation simplifies considerably if we choose a hnear c o m b i n a t i o n of the ~b(±"~t = l, 2, 3, 4 whtch dlagonallzes m, 1 e
f d4p ~?"'t(p)m(p)g~'~(p)=~,jrn,,
(A6)
in which case we obtain ~(±1
4
~b = NQ 1 E
I=l
4
6oI±,t) (x)Ob(± 1)~(v) II mjj[I',~ l=l
(A7)
k
The integrations over the gauge and ghost fields yield slmdar products of non-zero elgenvalues of the c o v a n a n t d e n v a n v e in the p s e u d o - p a m c l e b a c k g r o u n d field
C E I Carnetro, N A
309
McDougall / Instantons
These mstanton determinants were first calculated by 't Hooft [1] for fermlons in the fundamental representation We shall use the results of Chadha et al. [13] who have extended 't Hooft's computations to include fermlons in any representation of the gauge group Inserting (A 7) into (A 1) and using Chadha's expression for the determinants we obtain
S~b(x--y)=
ff
4 ~(± t)t
\--1± t)+t x 4
dct 2 t~.' tX)t~b" tY) I] mjj(p)L(txo, p), l=l
3=1
(A8a)
J~t
where a stands for the collective coordinates dp 4 da = ~- d z dO,
(A 8b)
(p being the mstanton scale size, z the lnstanton position and 0 the gauge orientation),
r 8~.2
L(/xo, p) = 21°~-6p4go8 exp 1_---~-o2+ (~-38-) In ( - / z 0 p ) + a ( 1 )
]
(A 8c)
To proceed further we reqmre the zero elgenvalue functions $(±")l = 1, 2, 3, 4 These modes have been constructed by Jacklw and Rebbl [14] for fermlons in the adjomt representation interacting with any self-dual (or anti-self-dual) SU(2) gauge field configuration We choose the following linear combination: ~---1,2
i
tO
~
0 t --- 3, 4 ,
o
),
(A.9)
~=1,2
,---3,4,
( a 10)
where u(lJ = U(3) = (10) ,
1-~(2)= I-4(4)~- (01) .
(A.1 1)
Using the properties of the a~, matrices [14], expressions (2 le), (2 4), normalislng the elgenfunctlons to unity and Inserting a gauge orientation factor U = exp (-IO~T), where U is an element of SU(2) m the adjomt representation and ~a
( E I Carnetro,NA McDougall/ lnstantons
310
are the x - i n d e p e n d e n t parameters which characterize that element, we obtain
(2: -' ~'o+"=
_d
(
0
(),-z).(x-z). u~,> U~,bO..O'bO~.(X_ Z)~[(X_ Z)2 +p~,]2
oi x - z ) ,
-~
,,_-1,2
1
t =3,4,
ul,i '
U~b'~.O-b [(x - z) 2 + p 2 y
(A 12)
m
(x - z)2[(x - z) -~+p2]~
1=1.2
0
= 3, 4
0
(A 13)
The F o u r i e r t r a n s f o r m s of these solutions for z = 0 are given by
t 2"n'pz U.~duo'ba,oP"P' 7 g'* "(p) =
1=1,2
a(pipl)u<"
0
),
t =3, 4,
(A 14)
P. -t2"n'p'- U,,hduo-b~[ K,(plPl)u ''~
(27rp2 U,~bOl~Crh~ G(plpl)u ~'')
g'o ,"tp~=
0
1=1,2
-12 rrp2Uob%,crh~p~ Kj(Plpl)u<" ) , 0
s=3.4,
(A 15)
(A 16) where Ko(x), Kl(X) are modified Bessel f u n c n o n s It is easy to check that (A 6) is satisfied In fact, u p o n e x a m i n a t i o n we notice that ~ ' ± ' ~ ( p ) is even m p for l = 1, 2 a n d odd for 1 = 3 , 4 As m e n n o n e d previously, the d y n a m i c a l mass is even m p , . due to Lorentz m v a n a n c e F u r t h e r m o r e , using the properties of the a . matrices [14], it is simple to show that m,, oc u~'rru ~J' (T stands for the t r a n s p o s e d ) Thus, the n o n - d i a g o n a l elements vanish either because u" ~Tu~,7 = 0 or because we are integrating two even a n d one odd f u n c t i o n Also, we see that
C E I Carnetro, N A McDougall / Instantons
311
the m,, are the same for both lnstantons and antHnstantons and roll
-
m=~o4
i0
. 2,[ (pp), opp ~ mtP)~K~(pp),
,t = 3 , 4 J '
.
=
m22 , m33 =
p=(p•p•),/2
m44
(A.17)
The solutions of the Dlrac equation have the form
U.h(0)6b
(X-- Z, p)
(A18)
We calculate the gauge average m (A 8a) using
f d~2 . U*c(g2)Uba(g2)=1 . . ~SabScd ,
(A 19)
and because our solutions depend only on z m the combination x - z go to m o m e n t u m space O m m m g the 6ah, we obtain
S~±)(P) =
~Idp 4 " 4 ~ ,=12 ~(a±'t)(P, P)~)(±")~f(P, P) J=l[l m j j ( P ) t ( ~ o ,
xt is easy to
P)
(A.20)
Using (A 17) we see that &~± ,,q~±,,),(q~±.3)~±,3,,) and ~'2>~±'2)*(4~±'4>4;(±'4)* are multlphed by the same factor m~m233(m~m33) and for this reason we can sum them
~±.2)~±.2,, = ½(1:7 ys)127r2p4GZ(pp), 4~±.3)q~±.3), + ~,4)q~:~,4), = ½(1:7 ys)12~2p4K~(pp), 4~7.,)4~7,,,, +
(A 21a) (A 21b)
(where the properties of the a . matrices l{ave again been used to combine the terms). Substituting these expressions into (A 20), the result for the one-pseudopartmle c o n t n b u t m n to the quark propagator ~s
g,l±,(p) = ½(1 :F Ts)4,n-2 fd
dp P
[O2(pp)m ' ,(p)m33(P)
+ K~(pp)m~,(p)m33(p)]L(I-~o,p),
(A 22)
the expression which is gwen m eq (3.1a)
References [1] [2] [3] [4] [5] [6] [7]
G 't Hooft, Phys Rev Lett 37 (1976) 8, Phys Re,, D14 (1976) 3432 C G Callan, R F Dashen and D J Gross, Phys Rev D16 (1977) 2526 D G Caldl, Phys Rev Lett 39 (1977) 121 R D Carhtz, Phys Rev DI7 (1978) 3225 R D Carhtz and D B Creamer, Ann of Phys 118 (1979) 429 J Kogut et al, Phys Rev Lett 48 (1982) 1140 J Kogut et al, Studies of chlral symmetry breaking m SU(2) lamce gauge theory, Ilhnols preprmt (1983) [8] S Coleman, The whys ofsubnuclear physics, 1977 Int school ofsubnuclear phys, Ence, ed Zlctnchl (Plenum, 1979)
312
C E I Carnelro, N A McDougall / Instantons
[9] M Atlyah, V Patodl and I Singer, Math Proc Camb Phdos Soc 77 (1975) 43, 78 (1975) 405 79 (1976) 71 [10] Y N a m b u and G Jona-Lasmlo, Phys Rev 122 (1961) 345, C G Callan, R F Dashen and D J Gross, Semsclasslcal methods m q u a n t u m chromodynam:cs toward a theory of hadron structure, lectures at La Jolla lnshtute Workshop on Particle Theory, August 1978 [11] D J Gross, R D P l s a r s k l a n d L G Yaffe, Rev Mod Phys 53 (1981) 43 [12] N A McDougall, Nucl Phys B211 (1983~ 139, Chlral symmetry breaking and the U(1) problem, Oxford preprmt (July 1984) [13] S C h a d h a , P Di Vecchla, A D'Adda and F Nlcodeml, Phys Lett 72B (1977~ 103 [14] R Jacklw and C Rebbl, Phys Re~ D16 (1977) 1052
I N S T A N T O N S AND C H I R A L SYMMETRY BREAKING C E I CARNEIRO and N A McDOUGALL Department of Theoretwal Phystcs, Unwerszty of Oxford, 1 Keble Road, Oxford OXI 3NP, UK
Received 21 November 1983 (Revised 21 May 1984) A detaded investigation of chlral symmetrybreaking due to mstanton dynamicsis carried out, w~thln the framework of the ddute gas approximation, for quarks m both the fundamental and adjolnt representatmons of SU(2) The momentum dependence of the dynamical mass is found to be very s~mdar m each representation
I. Introduction For many years, a qualitative understanding of the phenomenology of light quark systems has been based on the idea of choral symmetry breaking certain symmetries of the lagranglan, associated with the near masslessness of the quarks, not being respected by the vacuum state The consequences of chiral symmetry breaking are manifested in the fact that most hadrons are very massive, when compared to the lagrangian masses of the quarks from which the hadron is formed, and the Goldstone nature of the plon octet of light pseudoscalar mesons An understanding of the mechanism which causes the spontaneous breakdown of the chlral symmetry is, therefore, crucial for an understanding of the dynamics of light quarks The experimental data are consistent with the strong Interaction possessing an apparent chlral SUv(3) × SUA(3) symmetry with the SUA(3) symmetry being reahsed In the N a m b u - G o l d s t o n e mode, the pion octet corresponding to the Goldstone bosons associated with the spontaneous symmetry breaking. However, the lagranglan of QCD, for three flavours of quark, formally possesses a Uv(3)×UA(3) global symmetry in the chlral limit, the extra Uv( 1) symmetry reflecting baryon conservation and the U(1) problem being associated with the extra UA(I) symmetry The spontaneous breakdown of the SUA(3) symmetry is indicated by the chiral order parameter (fqJ) being non-vamshlng in the physical vacuum. Since ( ~ ) ¢ 0 also violates the UA(1) symmetry, the experimental data imply that this symmetry is also spontaneously broken and there IS, therefore, a spontaneously broken UA(3 ) symmetry Thus, naively, nine light pseudoscalar states should be produced, not just the eight observed by experiment. The solution to this apparent paradox has provided deep insight into the mechanisms which may be responsible for the chiral symmetry breaking 293
294
C E I Carnetro, N A l~l~Dougall / lnstantons
The resolution of the U(I) problem has indicated that the dynamics contained wlthan the non-trivial topological sectors of the QCD vacuum, with lnstanton dynamics possibly playing a dominant role, Is intimately connected with chlral symmetry breaking The relevance of mstantons to chlral symmetry breaking was first reahsed in 1976 by 't Hooft [1], who demonstrated that the Ua(l ) chlral symmetry was spontaneously broken by mstantons m a manner such that the assocmted Goldstone excitation coupled to a gauge-variant conserved current and hence there was no need for a ninth light pseudoscalar particle to appear xn the physical spectrum The suggestion that instanton dynamics may be responsible for breaking the S U ( N ) × S U ( N ) chiral symmetry, in addition to the UA( 1) symmetry, was first made by Callan, Dashen and Gross [2] Quanhtatlve calculations by Caldl [3] indicated that indeed the interactions were sufficiently strong to generate a dynamical quark mass A detailed investigation of the self-consistency equation, whose solution determines the dynamical mass, was carried out by Carhtz [4] and Carhtz and Creamer [5] for up to four quark flavours m the fundamental representation of colour SU(2) They found that for any number of quark flavours there was a non-trwial solution of the self-consistency equation, although it became increasingly difficult, as the number of quark flavours increased, to generate a self-consistent mass within the regime of validity of the dilute gas approximation, the required N 2 - 1 Goldstone bosons, formed as bound states of a quark and an anti-quark, also resulted from the same dynamics However, there resulted no massless flavour smglet pseudoscalar state since, in this channel, the quark-anti-quark force, generated by mstanton dynamics, was repulsive, this ~s, of course, in complete accord with a resolution of the U(1) problem being obtained from mstanton dynamics In this paper, we shall build upon these prewous mvestxgations and extend the calculation to include quarks m the adjomt representation of SU(2) In principle, the physical spectrum resulting from QCD may be calculated by computer, without any knowledge or understanding of the underlying dynamical mechanisms, if the theory if solved on a sufficiently large lattice However, to date, it has only been possible to perform calculations on rather small lattices and with a drastic approximation (the quenched approximation) imposed on the handling of the quark degrees of freedom Since chlral symmetry breaking is claimed to have occurred in these calculations, it ~s important to determine whether the chlral symmetry breaking m the lattice calculations is occurring m the same manner as m continuum QCD The fact that lattice calculations obtain (qTO) non-vanishing need not mean that chiral symmetry breaking ~s occurring m a manner consistent w~th a resolution of the U(1) problem In lattice calculations, the fermlon action chosen often introduces a large degree of explicit chlral symmetry breaking (the states with the quantum numbers of the pions need not therefore be Goldstone excitations), the validity of using the quenched approx~mation for simulating the dynamics of QCD is not proven and chiral symmetry breaking does not occur, m the chlral limit, on any lattice of finite s~ze, thus, an understanding of the dominant dynamical
295
C E I Carnezro, N A McDougall / lnstantons
mechanisms in the lattice calculations is of great interest. The lattice calculations have investigated the temperature (energy scale) at which chlral symmetry breakdown occurs for quarks in both the fundamental and adjolnt representations of the gauge group SU(2) and, at first, found a change by a factor of about fifty [6], but this result has recently been revised and a more modest change is now quoted [7] In view of the preceding discussion, it seemed worthwhile to determine the change in energy scale expected from lnstanton dynamics and thus th~s investigation of xnstantons and chlral symmetry breaking, including quarks in representations other than the fundamental, was initiated Since we are lnvesUgatlng an example of spontaneous symmetry breaking, it is not possible to obtain the dynamical mass by perturbing about the chlrally symmetric vacuum, and thus we must assume that the chlrally asymmetric vacuum is favoured and determine the dynamical mass by a self-consistent calculation In sect 2 is described the condition which must be satisfied if the calculation of the dynamical mass IS to be self-consistent The self-consistency equation, for one quark flavour in the adjolnt representation of SU(2), is solved in sect 3 by separating the momentum dependence, contained in the zero elgenvalue modes, from the instanton scale size dependence In sect 4, the self-consistency equation is considered for an arbitrary number of quark flavours in both the fundamental and adjoint representations and the quenched approximation is simulated by setting the number of quark flavours equal to zero In sect 5, the dynamical mass for the adjoint representation is compared with that obtained previously for the fundamental representation The qualitative conclusions to be deduced from this investigation are briefly described in sect 6
2. C a l c u l a t i o n of quark m a s s e s generated by instantons
Consider one quark flavour in the adjolnt representation Interacting with SU(2) gauge fields (later we shall extend the analysis to include more than one quark flavour) The dynamics of this system is contained in the lagranglan 1 4g 2
We formulate our theory in euclidean space with metric g . . = 6.. and E1234 1 A convenient realization of the Dlrac y-matrices is =
=
(0
,
a ~ = ( - z { r , 1 ) = O/p,, -*
(2 lb)
where the %, are hermltian and satisfy the anticommutatlon relations {Y/x, Yv} = 2~/zv,
{Y/x, "Y5}~-'0 ,
")/5 ~ "YlY2Y3Y4
(2 l c )
296
C E I Carnetro, N A McDougaU / lnstanton~
D . denotes the covarlant derivative in the a d j o m t representation
(D~)~h = O~6a~+e.~bA.,, F.~. = i~Av. - O.A.~ + e~b,A~.bA.,,
(2 ld) (2 le)
In order to c o m p u t e the quark mass generated by lnstantons we examine the quark p r o p a g a t o r which, in the dilute gas approximation, is given by [5] 1
s - So'-s,,'(s',+~+s ', ' ) s , , ' '
(2 2)
where So is the free p r o p a g a t o r and S]+'(S] -') is the connected one lnstanton (anti-instanton) contribution to the p r o p a g a t o r
x f DilDqq(x)gl(y)exp[- I glk~qd4x]
(23)
To obtain the above expression, we e x p a n d e d A . a r o u n d the lnstanton or antilnstanton
A.=A.+a.,
A ~ = -r/.~a0~ In
1+
.
(24)
where "q..~ ~ are the 't H o o f t antlsymmetrlc symbols (~k~ v = ekt~, "qk4. = qz6k~), and kept only terms which are O(a~,) in F ~ F , . a and O(0q) in the f e r m m n lagranglan (semi-classical approximation) N c and N o are n o r m a h s a t i o n factors and for simplicity we have not written down the gauge fixing term and ghost fields We see that the pseudo-particles generate a quark self-energy (in m o m e n t u m space)
.~(p)=-ggl(p)(g',+)(p)+g~, ' ( p ) ) g o ~ ( p ) lnvarlance X ( p ) = q~A(p2)+m(p 2) The terms
(25)
m(P 2) = ½Ys{Ys.S ( p ) }
(26)
Due to Lorentz proportional to p contribute to the r e n o r m a h s a t l o n of the quark wave function and are not related to chlral symmetry breaking effects Hence. the dynamical mass is
In order to determine this dynamical mass we must evaluate expression (23) The integration over the quark fields m a y be easily performed [8] and is p r o p o m o n a l to ~ . e-I±~'x'-I±~t( , t )t~, Y )" [I A, l
AK
,
(27)
l#l
where A, and 0~,± ~are respectively the eigenvalues and elgenfunctlons of the operator 1/~ ± (the covarlant derivative in the mstanton or anti-instanton b a c k g r o u n d field)
C E I Carnelro, N A McDougall / lnstantons
297
and F is the number of quark flavours The expression (2.7), except for the case of one quark flavour with isospin ½, is zero in the chlral hmlt due to the existence of zero eigenvalues The number no of these zero elgenvalues, for each quark flavour, is given by the index theorem [9]
no=2T(T+I)(2T+I),
(2 8)
where T IS the quark lsospln For the fundamental representation T = ½and no = 1, and for the adjolnt representation T = 1 and no = 4 Since S~l±~(p) and consequently ~ ( p ) vanish there is apparently no dynamical mass generation This result, however, corresponds to perturbing about the chirally symmetric vacuum, a non-vanishing dynamical mass may be obtained if we perturb self-consistently about the physical chlrally asymmetric vacuum [10]. First we rewrite (2 la) as 1 = --~g2 F~.,.Ff.~,~- gt(.O + m)q + Ctmq, (2 9) where
glrnq IS a non-local mass term, fd4xglmq=fd4xd4y(l(x)m(x-y)q(y)=f&~(p)m(p)~(p),
(210)
and m is proportional to the unit matrix both in colour and euclidean space Next we consider the mass counterterm as belonging to the interaction lagranglan, that is we redefine the free and Interacting lagrangians and assume that m is a small quantity so that ~lmq can be treated perturbatlvely. Now, if m is the true dynamical mass the complete propagator must have the form S ( p ) = (zp+m(p)) -I However, we know that self-energy effects will change the free propagator. Self-consistency is achieved if the mass counterterm, treated perturbatively, cancels these self-energy effects, 1 e if
m(p)-½Ts(Ts, ~ ( P ) } = 0 -
(2 11)
(As mentioned previously, terms proportional to p renormalise the quark wave function and can be neglected). ~ ( p ) is still given by (2 5) but now It is a function of the unknown mass re(p) Thus, from (2.11) we obtain an integral equation (self-consistency equation) from which we may determine re(p)
3. The solution of the self-consistency equation The one-pseudo-particle contributions S~±~(p) to the quark propagator, for one quark flavour in the adjolnt representation of SU(2). are calculated in the a p p e n & x
37'(p)
7: Ts)arP ;
?[G2(pp)ml,(p)m~3(p) + K~(pp)m~l(p)m33(p)]L(Ixo, p) , o
(3 la)
298
( E I Carnetro, N A
M¢Dougall / In~tantons
where
m,,(p)=~p 4 , a q q m q-)~K~(pq),
O(x)= (Ko(x)
and
Kl(x)
,'
L~-\l
.
\,,-
,
'
xV
(3 l b )
]
13 lc)
are m o d i f i e d Bessel f u n c t i o n s ) ,
L(l~o,p)=21°rr6p4go8exp
877"2 22 8
-~-+(~-~)ln(/xop)+a(l)
]
(3 l d )
R e c a l h n g that, to l o w e s t o r d e r in m, So I = t/~, we o b t a i n
-et p) = -go'( g',++ g, ')go' =47r2p 2 f dP[G2(pp)ml,(p)m3~(p)+K~(pp)rn~(p)rn~s(p)]L(txo, p) P
(3 2)
It is o b s e r v e d t h a t -Y(p), to l e a d i n g o r d e r in m, d o e s n o t c o n t a m t e r m s p r o p o r t i o n a l to p a n d t h e r e f o r e - ~ ( p ) c o m m u t e s w t t h 3'5 a n d c l e a r l y p l a y s t h e role o f a m a s s t e r m H e n c e f r o m t h e e x p r e s s t o n (2 11 ) t h e s e l f - c o n s i s t e n c y c o n d i t i o n for the d y n a m i cally g e n e r a t e d q u a r k m a s s
m(p)
is g i v e n by t h e i n t e g r a l e q u a t i o n
m(p 2) = 4 ~ r 2 p 2 f dP[G2(pp)m,,(p)m~3(p) +K~(pp)m~(p)m~(p)]L(Ixo, p),
(3 3)
3 P a n d t h e p o s s i b i l i t y o f l n s t a n t o n d y n a m i c s , t r e a t e d w i t h i n the d i l u t e gas a p p r o x i m a tion, c a u s i n g choral s y m m e t r y b r e a k i n g f o r a q u a r k m t h e a d j o l n t r e p r e s e n t a t i o n o f S U ( 2 ) rests o n f i n d i n g a n o n t r t v m l s o l u t t o n to (3 3)* B e f o r e s o l v i n g (3 3), let us c o n s i d e r t h e f u n c t t o n L(/xo, p) g t v e n m (3 l d ) c o e f f i c i e n t m u l t i p l y i n g In
(/xop)
for t h e S U ( 2 ) r u n n i n g c o u p h n g representation
-
The
is t h a t w h i c h a p p e a r s m the o n e - l o o p e x p r e s s i o n c o n s t a n t w i t h o n e q u a r k f l a v o u r In the a d j o i n t
T h u s we m a y w i n e
gO J L
L gZ(p)j,
g2(p)
,
* For one quark flavour m the fundamental representation of SUI 2 ~,the right-hand side ofthe expre~slon analogous to {3 3) has no powers of m(p), the Dlrac operator in this case having but a single zero elgen~alue mode Therefore, for one quark flavour m the fundamental representation, dynamical mass generation does occur, the magmtude of the dynamical mass being propomonal to an integration over mstanton scale s~ze With respect to the powers of re(p) on the right-hand s~de of 13 3), one quark flavour m the adjomt representation of SU(2I behaves as four quark flavours m the fundamental representation Thus, it ~s seen that the structure of the self consistency equation for the fundamental and adjomt representations is very d~fferent, pamcularly m the case of one quark flavour
C E I Carnetro, N A McDougall / Instantons
299
where A Is the scale parameter (In the Pauh-Vdlars regulanzatlon scheme) and we see that the effect of the one-loop ra&atlve correcUons is simply to replace go by the running couphng constant g(p) On physical grounds, g(p) should be substituted wherever go appears (the final result clearly should not depend on the renormahzatlon point/xo) The major obstacle to finding the exact soluuon of (3 3) is lack of knowledge regarding the correct manner of performing the mtegraUon over mstanton scale size Any method of solving (3 3) requires making assumptions about a regime where non-perturbaUve physics wdl dominate, 1 e at values of g(p) where the application of perturbatlve results is no longer vahd However, the quahtatlve features of the one-loop expression may be expected to extend into the non-perturbatlve regime If g(p) continues to increase monotomcally as p increases, we note that (87r2/g2(p)) 4 exp (--8~2/g2(p)) approaches zero at both large and small p, and It may be possible to isolate a well-defined region where the mtegrand m (3 3) is significant In fact, using the one-loop expression, (81r2/g2(p)) 4 exp (-8~2/g2(p)) peaks at g2/87r2 = 0 25 or pA = 0 42, l e where the couphng xs still relatively small A graph of (8"IrZ/g2(fl))4 exp (-8~2/gZ(p)) versus pA is shown m fig 1 Thus, the magnitude of lnstanton effects may be estimated by integrating over lnstanton scale size (m the range 0 to A -1) using the one-loop expression (Non-perturbatlve dynamics cause the couphng to grow at an even faster rate [2] than that given by the one-loop expression and hence the peak m the mtegrand may be very sharp. The one-loop expression will m&cate the scale at which mstanton effects start to become important and therefore the scale where the mtegrand peaks should be close to that given by the one-loop expression, namely pA = 0 42) Since we are most interested m the momentum behavlour of m ( p 2) and we wish to isolate the uncertainty assocmted with the integration over mstanton scale s~ze, we shall assume that the mstanton density peaks sharply at a scale p = ~ ~ (an estimate would be
5 1 3 2 1
02
0/~
06
08
1~0
pA Fig 1 A graph showing (1)
(8"n'2/g2(p))4exp(--8"n'2/g2(p)) and 87r2/g2(p)= 14 In (p 1)
(2)
8~2/g2(p) versus pA with
300
C E I Carnelro, N A McDougall/ Instantons
~ 2 4A ) and approximate (3 3) by
P \g2(p)# exp
g2(p)#,
(3.5)
l e we consider the functions O:(pp), K~(pp), mlt(p), and m33(P ) tO be slowly varying in the region where the lnstanton density is significant The uncertainty associated with the integration over lnstanton scale size is therefore wholly contained in a dimensionless constant c 2,
~c2=_e<.,>f ~--l,,g----~p).] dp[ 8~r2 ~4 e x p \[_ g~(p)), 87./.2
(36)
o ~ ( l ) ~--~0 4 4 3
Eq (3 3) has now been reduced to a form where it may be solved analytically with no further assumptions Recalling that m~
(1) fo dss3G2 s 3
=2/z~
4
m33(1) =2-~4fo
X/t/
oO
m(s2 )
(3 7a)
'
dss3Kl(S)m(s2)'\l~l
(37b)
we see that if we multiply (3 5) by (3/2iz4)p3G2(p/tz) and (3/2tx4)p3K2(p/tx), integrate in both cases over p from 0 to oo and then make a change of variable P/t* -+ x, we obtain the two equations ml,(~) = (~)2mlt(l)m23(1)+(~)2m121(~)m33(]~), m33(1) = ( ~ ) 2 m l l ( 1 ) m 2 3 ( 1 ) + ( ~ ) 2 m ~ l ( 1 ) m 3 3 ( 1 )
(3 8a) ,
(3 8b)
where
oz2=fl 2~-
dxxSG4(x)~-O 0292,
£7
~2=_
(3 8c)
dxxSG2(x)K~(x)~-O 0358,
(3 8d)
dxxSK~(~) ~-0 236
(3 8e)
Neglecting the trivial solution m(p 2) = 0 and using the fact that m,,(p)~> 0, we obtain
-"x/ ~ ( a y ' +3-')'
m33 = c
(3 9a)
C E I Carnetro, N A
C 3'
roll
301
McDougall / Instantons
a(C~3`+fl 2)
(3.9b)
Comparing the expressions (3 7) and (3 8c, d, e), we see that a convenient ansatz for is
m(p 2)
p2
2
m(p2)=A--~ G2(p) +BP K~(P)
(3.10)
Inserting (3 10) into (3.7) we obtain m,l(l) =3tzot2A+atzfl2B ,
(3 l l a )
m33(~ ) ----3/J,/~2A+3/z3`2B
(3 lib)
Since ml~(1//z) and ma3(1//x) are also given by (3 9a) and (3.9b) we have therefore reduced the system of integral equations to a linear system in A and B which has the following soluUon: A --
(fl 2 +a3`) -3/2 ,
(3 12a)
B=
(f12 +o~3`)-3/2 ,
(3 12b)
and thus (3.13) For
p/tx ~-0 G2
=
(3 14)
(3 15) Hence, 2~,
/ ~ \ I/2
/3 The non-trivial solution of(3 5) for the dynamically generated quark mass is therefore gwen by m(p 2) = m(0)
Ga
+K 2
,
(3.16b)
C E I Carnetro, N A McDougall/ lnstanton6
302 10 08 ~06 E04
',? \
02
I
2
3
z~
5
6
7
Fig 2 A graph showing (1) the dynamical mass function for the adjolnt representation (2) the contribution from (p/l~)2(y/a)G2(p/#), (3) the contribution from (p/~)ZK~(p/#)
where the uncertainty associated with the Integration over mstanton scale size is contained in the value of ~t and the dimensionless constant c in the expression for m(0) Since G2(p/i.z) and K~(p/l~) are known functions (see (3 lc)), we may plot m(p)/m(O) Fig 2 shows the graph of m(p)/m(O) and the behavlour expected of a dynamical mass is observed, the mass falling rapidly (asymptotically as p-6) as the momentum of the quark increases
4. The self-consistency equation for an arbitrary number of quark flavours Since the lattice calculations of Kogut et al [6, 7] were performed using the quenched approximation, it is of Interest to investigate how the results obtained in sect 3 are modified if the quenched approximation is simulated m the calculation of chlral symmetry breaking due to instanton dynamics In the quenched approximation, Det E ~ is taken to be unity (internal fermlon loops are neglected during the evaluation of the path integral) Since Det D is the product of the elgenvalues of and when the dynamical mass (contributions to the quark propagator arising from a single interaction with either an mstanton or anti-lnstanton) is calculated, one elgenvalue is removed from the product of all elgenvalues (and replaced by the square of the elgenfunctlon), the self-consistency equation for the dynamical mass becomes, in the quenched approximation, m A ( p ) = e ~l,p2
d p[G2(pp) L ~ +
K~(pp)]l' 87r2"~4
m---~-) j k u - ~ p ) } exp
-g~P)
,
(4 1)
with 8rr2/g2(p)=-3 2 In (pA), there being no influence on the ¢3 function arising from quark loops (Note that the trlwal solution of the self-consistency equation, corresponding to the chlrally symmetric vacuum with re(p)= 0, is not a possible
C E I Carne~ro,NA McDougall/ lnstantons
303
solution of (4 1)). For NA quark flavours in the adjolnt representation, the selfconsistency equation for the dynamical mass is a simple extension of (3.3)
rnN~,(p) =
e(2NA--')'~(l)p
2
f dP [G2(po)ml,(p)mZ3(p) + K~(po)m21(o)m33(P)] P
X(mlI(p)m33(p))2(NA-I~p 4NA
g-tP)
exp
(
-
,
g P
(4 2)
with 8rr2/g:(p)= _~22_~ ~3 3 NA)In (pA) to a one-loop level It IS observed that the self-consistency equation for the quenched approximation (4 1) simply corresponds to the NA~ 0 limit of (4 2) So let us solve (4 2) for an arbitrary NA and the solution resulting from the simulation of the quenched approximation will be obtained by setting NA = 0 Separating the momentum dependence from the instanton scale size dependence in the usual manner we obtain
mN,,(p)~--e(2N,-"~"')p2tx4:~, [ G2( P~)mlt( l--~m~3(1-~-I \[~]~]A/
L
\I~NA /
\]~NA/
Jr-/ 2(-~P ~ g/'/21(~l ~ 1,r/33(--~1~ ] \~"LNA/
\['LN A /
\~JLNA/ A
x[rn,,(__~l/rn33( I )]2(N~-" I dp[ 87r 2 ~4 L \/xuA/ \/zu^/_l P \g-~p)] exp
(_
87T2 ~
g2(p)]
(4.3)
Defining
do(
3C~ _----e'-~A
J P \g2(p),l
4
(_ exp \
g2(o)],
2 22 8 g2(p)-- (T--sNA)
In (pA),
(4 4) the self-consistency equation (4.3) may be solved in the manner described in sect 3 with the results
mN'(p)=mNA(o)( P ~2[3`G2( La \/XNA/ P~I +K~(#-~)]
(45a)
where "~
[ O l \ (2NA--I)/f4N A 2)
- / cNA,"l 1/(2NA--|)~\T] mNA(0)=~..~A~
(/32+~3`)--~4NA--,,/,4N~ 2,. (4 5b)
32=00292, /32=00358, 3,2=0236 Thus, we see that the graphs of m(p)/m(O) obtained in this manner are independent of the number of quark flavours Chlral symmetry breaking due to lnstanton dynamics, treated within the dilute gas approximation, has previously been investigated by Carhtz [4] and Carhtz and
C‘ E I Carne~ro, N A McDougall / Instanron~
304
Creamer
[5] for quarks
of SU(2)
transforming
For NF flavours,
accordmg
to the fundamental
the self-consistency
equation
representation
becomes
(4 6a)
where a(i)=0 level, f(5)= 1, ,(l),
146, a!(l)=0443
and 8~TT2/gZ(P)=-(~-_NNF)ln(p.~)
toaone-loop
-2~[~,(5)~"(5)-z"(5)K,(5)1-2~,(5)K,(5), f(O)=1,
K, ,(l) bemg modified
Bessel functions,
(4 6b)
and
X m(p)=$
Eq (4 6a) may be solved by the same method mN,(P)
=
(4 6c)
dppm&).#pp)
I0
as (4 2), yteldmg
mNF(0if2
the solutton (4 7a)
p (
2PW, > 3
wnh WIN&O)= /1.N,(cx~6)~“‘Nr-“6’ c&=$exp
(-a(
1) +2N+($))
s=4
)
N,#2,
[T(-j$)‘exp(-$j$),
’ dy_$(y)=O34, I0
(4 7b) (4 7c)
(4 7d)
and pJ,L IS the mstsnton scale size at whrch the mstanton density peaks For two quark flavours, terms of higher order m m(p) are required rn order to determme the magnitude of the dynamical mass
5. Comparison
of the dynamically
generated mass for the fundamental representations
and adjoint
As was mentioned m the mtroductton, this mvestrgatron of mstanton-generated chnal symmetry breaking was nntlated as a result of lattice calculations of chnal symmetry breaking for quarks m the adJoint representation of SU(2) Early in 1982, Kogut et al [6] published results which claimed that the choral symmetry restoration temperature for adjoint representation quarks was much higher than that for fundamental representation quarks VIZ TA/TF= 56 k 5 The mterpretatton of this result was that choral symmetry breakdown for quarks m the adJomt representatton occurred at a higher energy scale (shorter distance) than for fundamental representation quarks Since tt IS not clear that choral symmetry breakmg m SU(2) lattice
C E I Carnelro, N A McDougall / lnstantons
305
calculations, especially if the quenched approxlmatton ts used, occurs m the same manner as m the continuum QCD, an understanding of thts result was considered to be of importance Smce the peak m the mstanton density is largely determined by the behavlour of the coupling constant, which in the quenched approximation is independent of the quark representation, the lattice result &d not lmme&ately appear to be consistent with chlral symmetry breaking, for both representations, occurring due to mstanton dynamics However, when the quark representation ~s changed, there is a &fferent number of vanishing e~genvalue modes (of the massless Dxrac operator) with possibly very different momentum-space behavlour and thus the m o m e n t u m scale at which the dynamical mass begins to grow rapidly could be significantly modified From the results presented later in this section, it is clear that the chiral symmetry breaking scales for the fundamental and adjomt representations should be very similar if chlral symmetry breakdown, in both representations, occurs due to mstanton dynamics, the conclusion being unchanged if the quenched approx~matlon is simulated during the calculations This suggested either that the lattice result was unrehable or that chlral symmetry breaking m the lattice calculation did not occur due to mstanton dynamics for both quark representations Recently, however, a re-analysis of their data has caused Kogut et al to rewse their estimate of TA/Tv and they now quote [7] TA/TF= 8 6 + 4 5, which, m wew of the large uncertainty and the assumption of the approximate equivalence of the chlral symmetry restoration temperature and the chiral symmetry breaking m o m e n t u m scale (the analysis of finite temperature lnstanton dynamics [11] is rather more comphcated than for zero temperature), is not m conflict with chiral symmetry breaking on the lattice occurring due to instanton dynamics for quarks m both the fundamental and adjomt representations Using the results given in eqs (4 5a) and (4 7a) we may draw a graph of re(p)~ re(O) for both the fundamental and adjomt representations This is shown in fig 3, where 10 08 o
E 06
~
%
O2
0
1
2
3
t) pip
5
6
7
Fig 3 The dynamical mass functions for the fundamental representation (sohd curve) and the adjomt representation (dashed curve) obtained from (4 7a) and (4 5a) respectively
306
C E 1 Carnetro, N A McDougall / lnstantons
~t is observed that there is no radical &fference in the momentum dependence the shapes of the graphs are very similar and the momentum scales where there are sudden increases in the dynamical masses are the same to within a factor of two When the quark representation is changed, the expression for the coupling is modified and this will produce a shift m the peak of the instanton density For one flavour in the adjolnt representation, using the one-loop expression, th~s peak occurs at p~l = 0 42 (i e / z A - 2 4A) while for one flavour in the fundamental representation the peak occurs at p~| = 0 55 {1 e tZF--1 8A) Hence, for one quark flavour, ~z~- 1 3/zF and this difference should be borne in mind when looking in detail at fig 3 Thus, we may conclude that there is no evidence, from the zero temperature calculation, for any radical change in the momentum dependence of the dynamical mass between the fundamental and adjomt representations Furthermore, since the graphs of m(p)/m(O) are independent of the number of quark favours and the quenched approximation corresponds to the limit of zero quark flavours, the result shown in fig 3 should refect the behavlour expected from a lattice calculation using the quenched approximation
6. Concluding discussion We have presented a calculation of chlral symmetry breaking due to lnstanton dynamics, treated within the &lute gas approximation, for quarks in both the adjolnt and fundamental representations of SU(2) Several interesting features have emerged Firstly, within the regime of validity of the calculations, the behavlour of the momentum dependent mass is similar in each case (see fig 3) We believe that this result is non-trivial the zero elgenvalue modes can have very different momentum space behavlour (compare p2G2(p) and p2K~(p) In fig 2) and It Is not clear that the functions re(p) obtained for each representation need have been so similar (even for large p the dynamical mass m each case has the asymptotic behaviour p-6) To determine whether this result is a coincidence would require investigation of the zero elgenvalue modes for quarks m higher representations of the gauge group As expected from an mstanton calculation, there IS no evidence for a radical change in the length scale of the configurations responsible for the chlral symmetry breaking Thus, the revised result, using the quenched approximation, from Kogut et al [7] T~/TF= 8 6 ± 4 5. IS not In conflict with chlral symmetry breaking in lattice calculations arising due to lnstanton dynamics However, we believe that the crucial test of whether chlral symmetry breaking in lattice calculations occurs in the same manner as in continuum QCD lies in the resolution of the U(I) problem. The absence of the UA(1) Goldstone boson in the physical spectrum requires ( ~ ) to have a specific 0-dependence (depending on the number of hght quark flavours)
C E I Carnezro,N A McDougall/ Instantons
307
Thus, a lattice c a l c u l a t i o n to d e t e r m i n e the 0 - d e p e n d e n c e of (q~q/) or a n investigation of the massless p s e u d o s c a l a r states p r o d u c e d w o u l d be of great interest* We wish to express our t h a n k s to F. dos Aldos, N H. Parsons a n d J.F W h e a t e r for useful &scusslons. O n e of us (C.E I C ) wishes also to t h a n k Funda~fio de A m p a r o a P e s q m s a do Estado de Silo P a u l o for f i n a n o a l s u p p o r t while the other a c k n o w l e d g e s the receipt of a U K Science a n d E n g i n e e r i n g Research C o u n c d Postdoctoral F e l l o w s h i p
Appendix THE ONE-PSEUDO-PARTICLE CONTRIBUTION TO THE PROPAGATOR The o n e - p s e u d o - p a r t i c l e c o n t r i b u t i o n to the q u a r k p r o p a g a t o r in the self-consistent scheme for the case of one q u a r k flavour is given by
S(~±)(x-y)=N~tf Da~,exp[I~G(a~,,a~,)d4x]N~l
Iol +m qd']. (A 1) If we i n t r o d u c e source terms q~7, ¢1q for the f e r m l o n fields we can rewrite the integral over the f e r m i o n fields as hm ,,,~-o
~#(x)6rl(y)
exp
d4x '
d y 6-~
m(x -y') ~
x f DglDq e s° =- b°~±~,
N~ (A 2a)
J where (
So-- - - J
d4x
[(t(x)k~q(x) +fT(x)q(x)
+tT(x)r/(x)].
(A.2b)
We s h o w e d m sect. 2 that the o p e r a t o r , D ±, m the a d j o m t r e p r e s e n t a t i o n , has four zero e l g e n v a l u e modes, ~0(±'') ~= 1, 2, 3, 4, a n d for this reason it does n o t have an inverse However, if we remove the zero elgenvalue m o d e s we m a y calculate the * In the quenched approximation, it is not possible to obtain self-consistency m the 0-dependence of the dynamical mass [12] (1 e the same 0-dependence on both sides of (4 6a) with NF = 0), except for the trivial value 0 = 0, although self-consistency is possible for any other number of quark flavours Th~s result most probably in&cates that a resolution of the U(1) problem ~s not to be found wRhm the quenched approximation
C E I Carnesro, N A M~Dougall/ ln~tantons
308
reverse and perform the gaussmn m t e g r a t m n m the usual way [8] Using the completeness p r o p e r t y of the set {t~ ~± "~}, 4/~ "~ is the o g e n f u n c t m n of tD ± with exgenvalue )t., it is easdy shown that
S'<±'(x, y) = 5~'
0,± .,i x)q/~ .,l(y)
n
(A 3a)
hn
has the property ,1
l,O±S'~±'(x,Y)=64(x-Y) - 2
t~l+"'(x)t 91~' *(~),
(A3b)
where a prime in either }~ or l] denotes the removal of the zero elgenvalues C h a n g i n g the variables of lntegranon
q ( x ) ~ q ( x ) - z f d4y S'(x, y)rl(y) , 3 q ( x ) ~ q ( x ) -- t f d4y
fT(y)S'(y, x),
(A4a)
(A 4b)
we obtain, to lowest order m m, ~pt±) __
a~(x) 6rl(y) ~
6-~rn~
x - Y )~lorlty )/ ~'~o . A.
x ,=,[I (0 '±''~*, n)(rl, tk'±"')] ,
(A 5a)
where (A 5b)
(f" g) ~ I d4xf~(x)ga(x)
This calculation simplifies considerably if we choose a hnear c o m b i n a t i o n of the ~b(±"~t = l, 2, 3, 4 whtch dlagonallzes m, 1 e
f d4p ~?"'t(p)m(p)g~'~(p)=~,jrn,,
(A6)
in which case we obtain ~(±1
4
~b = NQ 1 E
I=l
4
6oI±,t) (x)Ob(± 1)~(v) II mjj[I',~ l=l
(A7)
k
The integrations over the gauge and ghost fields yield slmdar products of non-zero elgenvalues of the c o v a n a n t d e n v a n v e in the p s e u d o - p a m c l e b a c k g r o u n d field
C E I Carnetro, N A
309
McDougall / Instantons
These mstanton determinants were first calculated by 't Hooft [1] for fermlons in the fundamental representation We shall use the results of Chadha et al. [13] who have extended 't Hooft's computations to include fermlons in any representation of the gauge group Inserting (A 7) into (A 1) and using Chadha's expression for the determinants we obtain
S~b(x--y)=
ff
4 ~(± t)t
\--1± t)+t x 4
dct 2 t~.' tX)t~b" tY) I] mjj(p)L(txo, p), l=l
3=1
(A8a)
J~t
where a stands for the collective coordinates dp 4 da = ~- d z dO,
(A 8b)
(p being the mstanton scale size, z the lnstanton position and 0 the gauge orientation),
r 8~.2
L(/xo, p) = 21°~-6p4go8 exp 1_---~-o2+ (~-38-) In ( - / z 0 p ) + a ( 1 )
]
(A 8c)
To proceed further we reqmre the zero elgenvalue functions $(±")l = 1, 2, 3, 4 These modes have been constructed by Jacklw and Rebbl [14] for fermlons in the adjomt representation interacting with any self-dual (or anti-self-dual) SU(2) gauge field configuration We choose the following linear combination: ~---1,2
i
tO
~
0 t --- 3, 4 ,
o
),
(A.9)
~=1,2
,---3,4,
( a 10)
where u(lJ = U(3) = (10) ,
1-~(2)= I-4(4)~- (01) .
(A.1 1)
Using the properties of the a~, matrices [14], expressions (2 le), (2 4), normalislng the elgenfunctlons to unity and Inserting a gauge orientation factor U = exp (-IO~T), where U is an element of SU(2) m the adjomt representation and ~a
( E I Carnetro,NA McDougall/ lnstantons
310
are the x - i n d e p e n d e n t parameters which characterize that element, we obtain
(2: -' ~'o+"=
_d
(
0
(),-z).(x-z). u~,> U~,bO..O'bO~.(X_ Z)~[(X_ Z)2 +p~,]2
oi x - z ) ,
-~
,,_-1,2
1
t =3,4,
ul,i '
U~b'~.O-b [(x - z) 2 + p 2 y
(A 12)
m
(x - z)2[(x - z) -~+p2]~
1=1.2
0
= 3, 4
0
(A 13)
The F o u r i e r t r a n s f o r m s of these solutions for z = 0 are given by
t 2"n'pz U.~duo'ba,oP"P' 7 g'* "(p) =
1=1,2
a(pipl)u<"
0
),
t =3, 4,
(A 14)
P. -t2"n'p'- U,,hduo-b~[ K,(plPl)u ''~
(27rp2 U,~bOl~Crh~ G(plpl)u ~'')
g'o ,"tp~=
0
1=1,2
-12 rrp2Uob%,crh~p~ Kj(Plpl)u<" ) , 0
s=3.4,
(A 15)
(A 16) where Ko(x), Kl(X) are modified Bessel f u n c n o n s It is easy to check that (A 6) is satisfied In fact, u p o n e x a m i n a t i o n we notice that ~ ' ± ' ~ ( p ) is even m p for l = 1, 2 a n d odd for 1 = 3 , 4 As m e n n o n e d previously, the d y n a m i c a l mass is even m p , . due to Lorentz m v a n a n c e F u r t h e r m o r e , using the properties of the a . matrices [14], it is simple to show that m,, oc u~'rru ~J' (T stands for the t r a n s p o s e d ) Thus, the n o n - d i a g o n a l elements vanish either because u" ~Tu~,7 = 0 or because we are integrating two even a n d one odd f u n c t i o n Also, we see that
C E I Carnetro, N A McDougall / Instantons
311
the m,, are the same for both lnstantons and antHnstantons and roll
-
m=~o4
i0
. 2,[ (pp), opp ~ mtP)~K~(pp),
,t = 3 , 4 J '
.
=
m22 , m33 =
p=(p•p•),/2
m44
(A.17)
The solutions of the Dlrac equation have the form
U.h(0)6b
(X-- Z, p)
(A18)
We calculate the gauge average m (A 8a) using
f d~2 . U*c(g2)Uba(g2)=1 . . ~SabScd ,
(A 19)
and because our solutions depend only on z m the combination x - z go to m o m e n t u m space O m m m g the 6ah, we obtain
S~±)(P) =
~Idp 4 " 4 ~ ,=12 ~(a±'t)(P, P)~)(±")~f(P, P) J=l[l m j j ( P ) t ( ~ o ,
xt is easy to
P)
(A.20)
Using (A 17) we see that &~± ,,q~±,,),(q~±.3)~±,3,,) and ~'2>~±'2)*(4~±'4>4;(±'4)* are multlphed by the same factor m~m233(m~m33) and for this reason we can sum them
~±.2)~±.2,, = ½(1:7 ys)127r2p4GZ(pp), 4~±.3)q~±.3), + ~,4)q~:~,4), = ½(1:7 ys)12~2p4K~(pp), 4~7.,)4~7,,,, +
(A 21a) (A 21b)
(where the properties of the a . matrices l{ave again been used to combine the terms). Substituting these expressions into (A 20), the result for the one-pseudopartmle c o n t n b u t m n to the quark propagator ~s
g,l±,(p) = ½(1 :F Ts)4,n-2 fd
dp P
[O2(pp)m ' ,(p)m33(P)
+ K~(pp)m~,(p)m33(p)]L(I-~o,p),
(A 22)
the expression which is gwen m eq (3.1a)
References [1] [2] [3] [4] [5] [6] [7]
G 't Hooft, Phys Rev Lett 37 (1976) 8, Phys Re,, D14 (1976) 3432 C G Callan, R F Dashen and D J Gross, Phys Rev D16 (1977) 2526 D G Caldl, Phys Rev Lett 39 (1977) 121 R D Carhtz, Phys Rev DI7 (1978) 3225 R D Carhtz and D B Creamer, Ann of Phys 118 (1979) 429 J Kogut et al, Phys Rev Lett 48 (1982) 1140 J Kogut et al, Studies of chlral symmetry breaking m SU(2) lamce gauge theory, Ilhnols preprmt (1983) [8] S Coleman, The whys ofsubnuclear physics, 1977 Int school ofsubnuclear phys, Ence, ed Zlctnchl (Plenum, 1979)
312
C E I Carnelro, N A McDougall / Instantons
[9] M Atlyah, V Patodl and I Singer, Math Proc Camb Phdos Soc 77 (1975) 43, 78 (1975) 405 79 (1976) 71 [10] Y N a m b u and G Jona-Lasmlo, Phys Rev 122 (1961) 345, C G Callan, R F Dashen and D J Gross, Semsclasslcal methods m q u a n t u m chromodynam:cs toward a theory of hadron structure, lectures at La Jolla lnshtute Workshop on Particle Theory, August 1978 [11] D J Gross, R D P l s a r s k l a n d L G Yaffe, Rev Mod Phys 53 (1981) 43 [12] N A McDougall, Nucl Phys B211 (1983~ 139, Chlral symmetry breaking and the U(1) problem, Oxford preprmt (July 1984) [13] S C h a d h a , P Di Vecchla, A D'Adda and F Nlcodeml, Phys Lett 72B (1977~ 103 [14] R Jacklw and C Rebbl, Phys Re~ D16 (1977) 1052