Nuclear Physics B237 (1984) 59-76 © North-Holland Pubhshmg Company

A M O D E L FOR C H I R A L S Y M M E T R Y B R E A K I N G IN Q C D

J GOVAERTS 1

Instttut de Phystque Th~ortque, Umversltd Catholtque de Louvam, 1348 Louvam-la-Neuve, Belgtum

J E MANDULA

Phystcs Department, Washington Untverstty, St Louts, MO 63 130, USA

J WEYERS

lnstltut de Phystque Th~orzque, Umversttd Cathohque de Louvam, 1348 Louvam-la-Neuve, Belgium

Recewed

A recently proposed model for dynamical breaking of chlral symmetry in QCD is extended and developed for the calculation of plon and choral symmetry breaking parameters The plon is exphc~tly reahzed as a massless Goldstone boson and as a bound state of the constituent quarks We compute, m the hmlt of exact chlral symmetry, M o, the consntuent quark mass, f~, the plon decay couphng, (t~u), the constituent quark loop density, bt2/mq, the ratio of the Goldstone boson mass squared to the bare quark mass, and (r2)~, the plon electromagnetic charge radms squared

1. Introduction

In this paper, we extend the analysis of the phenomenologica l consequences of

a model of dynamical chiral symmetry breaking in qua n t um chromodynamlcs that

we have recent ly proposed [1].

The model is based on the N a m b u - J o n a - L a s i n i o picture [2], in which the vacuum

ts a coheren t state of f e r m i o n - a n t i f e r m i o n pairs which causes the breaking of chiral

symmetry and gives an effective mass to the fermlons. The model approximates the

Q C D vacuum by a condensa te of flavor and color slnglet q u a r k - a n t i q u a r k pairs.

The at tract ive s ingle-gluon exchange be tween quarks and an t iquarks is responsible

for all the dynamical effects considered.

A variat ional calculat ion [3] shows that for a value of the color coupling constant c~ = g2/47r larger than 9 ~, this exchange causes the per turba t ive vacuum to decay by

spon taneous creat ion of qq pairs, which indicates that the t rue vacuum must be a

condensa te of such pairs.

In Q C D , because of d imens iona l t r ansmuta t ion and asymptotic f reedom [4], we

expect the ground state to have propert ies which vary w~th distance, with an abrup t

cross-over at the scale where the effective coupl ing constant is 9. A t small distances

it will have essentially the propert ies of the per turba t ive vacuum, while at large

distances it will have those of a condensed quark pair phase.

59

60 J. Govaerts et al / Chtral symmetry breakmg ,n QCD

For large coupling, one also expects gluon condensation effects [5]. Although they surely influence the detailed structure of the quark condensate, they are not directly responsible for the breaking of chiral symmetry. The model does not incorporate them.

A detailed analysis of the quark pair structure of the vacuum in the model has been given previously [1]:

The minimization of the QCD hamiltonian over a set of coherent states of zero momentum, angular momentum, flavor, and color quark-antlquark pairs gives the condensate wave function as the solution to a non-linear integral equation which is solved numerically.

The pseudoparticle excitations of that condensate, which have the same quantum numbers as the bare quarks and which are identified with the constituent quarks, are created and annihilated by Bogoliubov-Valatin transforms of the quark creation and annihilation operators. It is precisely the fact that these pseudoparticles have a finite mass, the constituent quark mass M o, which shows that chlral symmetry is broken.

The model is equivalent to a self-consistent Har t ree-Fock approximation to the Schwinger-Dyson equation for the quark self-energy where only instantaneous one gluon exchange is taken into account.

In this paper, we extend that analysis to other phenomenological consequences of chiral symmetry breaking:

By an infinitesimal chiral rotation of the quark propagator, we obtain a homogeneous Bethe-Salpeter equation for massless pseudoscalar bound states of constituent quarks at zero-energy, whose solution is explicitly related to the con- densate wave function. This shows that the model contams the zero mass pion Goldstone bosons [6] associated with the dynamical breaking of chiral symmetry as bound states of the constituent quarks.

From the pion wave function at zero momentum we compute the static properties of the condensate and the pxon, namely f~, the plon decay coupling, ( t iu)= (dd), the constituent quark loop density, and i z 2 / m q , the ratio of the Goldstone boson mass squared to the quark bare mass. These quantities are calculated from matrix elements of operators between vacuum states or massive pion states at rest, with the hmit mq~0 (or equwalently 2 ~ 0 ) taken at the end of the calculation. The relation tzz~f 2 =-2mq(t2u), which follows from PCAC, is explicitly satisfied.

In addition, we outline the calculation of the pion form factor and explicitly compute (r2)~, the pion electromagnetic charge radius squared. These calculations reqmre the wave function of the moving pion. We obtain it in this non-covariant model, by Galilei transforming the wave function of a (massive) pion at rest.

As was discussed previously [1], to incorporate asymptotic freedom in the model without losing the advantages of its simple formulation, we replace the coupling constant a by an effective running coupling constant a (q2) which falls with distance.

z Govaerts et al / Chzral symmetry breaking m QCD 61,

This introduces a scale which regulates the integral equations and allows for the minimization of the Q C D hamiltonian over the coherent trial states.

To investigate the sensitivity of the results to the asymptotic behavior of the effectwe coupling function, we use two different analytical forms for it. The model is solved numerically, and we find results that are almost identical for both coupling functions. This confirms our intuition: the only proper ty which is relevant is that, at small distances, the effective coupling becomes smaller than a critical value for which the per turbatwe vacuum becomes unstable to qq condensation. In fact, the scale at which this occurs is found to be essentially the same for both coupling functions.

Some of the results presented here have already been given in succinct form elsewhere [7].

The paper is organized as follows. In sect. 2 we review the assumptions underlying the model and summarize the main results discussed m previous papers. In sect. 3 we show explicitly how the pion emerges simultaneously as a Goldstone boson and as a constituent quark bound state. We extend the model to the case of massive quarks and derive the Bethe-Salpeter wave function for a massive pion at rest. In sect. 4 we list the analytical formulae for f~, Iz2/mq and (~iu) in the limit of exact chiral symmetry. In sect. 5 we construct the wave function for a moving pion and derive the formula for the pion electromagnetic charge radius. Finally, in sect. 7 we solve the equations numerically and compute the values of the physical para- meters of the condensate and of the pion.

2. A short review of the model

To describe chiral symmetry breaking, the Q C D ground state is approximated by a coherent state of color and flavor singlet quark-ant iquark excitations on the perturbat ive vacuum w~th zero total momentum and angular momentum. The relative wave function O(IPl) of the quarks and antiquarks in the condensate is determined through mimmizatlon of the Q C D hamdtonian over the set of trial states

1 4 , ) - - - ~ e x p • Z d 3 P 4 ~ ( l p l ) s b + ( p , s ) d + ( - p , - s ) 10), (2.1) I ~ n = l t = l s = ± l

where b,+~ (p, s), d,~ (p, s) are the creation operators for massless quarks and anti- quarks of momentum p, helicity s, flavor n and color i (the case of interest is of course Nf = 2 and Nc = 3).

These operators are considered m the Coulomb gauge; they create physical quark and antiquark states along with the longitudinal color fields required by Gauss ' law.

The [if) states do not contain any transverse gluon fields: this allows the replace- ment, m the computation of expectation values (~0I:HQcD:I+), of the full Q C D hamiltonian by an effectwe hamiltonian m which these physical color fields are zero.

62 J Govaerts et al / Chlral symmetry breaking m Q C D

Note that this formulation is not covariant because of the fixed time description of the condensate and the choice of the Coulomb gauge.

An examination of the expectation values (q,l:HocD: 10) shows that the minimizing wave function q'([Pl) must be real and that it may be chosen to be everywhere positive.

Performing the minimization of this matrix element with respect to

6 ~ ( I p l ) (4 ' l :HocD: l~ ' )=0 , (2.2)

yields a non-linear integral equation for o(Ipl)

Iplsin2°(lpl)=4g2 C 3 d3q 1 1 (2rr)3 (p_q)2 ~[cos 20(Ipl) sin 2o(ql) +if- q sin 2o(Ipl)

where

x (1 - cos 20(Iql))], (2.3)

2~o(Ipl) ( 2 . 4 ) sin 2 0 (Ipl) - 1 + O2(Ipl) '

c o s 2 0 ( I p f ) -- 1 - ~'2(Ipl) 1 + ~o:(Ipl ) ' ( 2 . 5 )

From eq. (2.3) we immediately deduce:

0(0) = 1, (2.6)

4'(IPl) ~ 0. (2.7) Ipj~o~

The density of bare quark pairs in the condensate is given by

f d3p (1 -cos 20(Ipl)) (2.8) p = u f s c

These results are obtained through a Bogoliubov-Valatin transformation of the creatmn and annihilation operators of the bare quarks. Acting on the condensate 10), these Bogoliubov operators create and annihilate states with the same quantum numbers as the bare quarks and with the energy-momentum relation

o,(ipl)=lplcosZO(lpl)+~g=f d3q 1 1 . . (27r)3 (p_q)2 2 [sin 20 (Ipl) sin2o(Iql)

- t ~ 4 cos 20(Ipl)(1 - c o s 20([ql))]. (2.9)

We identify these pseudoparticles as constituent quarks and thus interpret w(0) as the constituent quark mass M o [8]:

Mo=w(O)=~g2 f d3q 1 1 sin 20(Iql) (2.10) J (27r)3 q2 2

J Govaerts et aL / Chzral symmetry breakmg m OCD 63

The same results may be derived from the Schwinger-Dyson equation for the quark self-energy in the approximation of instantaneous one gluon exchange.

Let us consider the four-dimensional quark propagator

i S (p ) - (2.11)

p - X ( p ) + ie

with a three-dimensional self-energy function X ( p ) parametrized in terms of two scalar functions A and B

~ ( p ) = IplA(Ipl) + p - ,tB(I t,I) • (2.12)

These are related to 0([p I) by

A(IPl) sin 20( Ip l ) , c(Ipl) 4A2(IPl) + c2(Ipl) 4A2(Ipl) + c2(Ipl)

C([p[) = 1 + B(Ipl).

=cos 20(Ipl) ,

(2.13)

(2.14)

The Schwinger-Dyson equation given in fig. 1 reads:

f d4q 1 X,(p) =~g2 (27r)4 ( p _ q ) 2 Y ° ( S ( q ) - S o ( q ) ) Y ° , (2.15)

where So(q) is S(q) with X(q )= 0. From eq. (2.15) it is possible to derive the condensate equation eq. (2.3). Taking

its trace gives:

f d3q 1 1 . IPlA(Ipl) __= _~g2 (2~,)3 ( t ,_q) 2 ~sm 20(Iql). (2.16) J

Finally, let us remark that the dressed quark propagator S (p ) has a physical pole at

Po = ]p][ a Z(Ipl) + C2(Ip]) ] 1/2. (2.17)

Multiplying eq. (2.15) by

sin 2o(Ipl) -t~" v cos 2o(]t,I), (2.18)

. _ _._.c22x

Fig 1 The approximate Schwmger-Dyson equahon for the quark self-energy The open orcle zs the proper mass, the hatched orcle ~s the full propagator and the wiggly hne is the instantaneous one-gluon

exchange

64 J Govaerts et al. / Chtral symmetry breakmg m O C D

and taking the trace gives

IPlU A ~(I Pl) + c=(Ipl) ] '/= = Ipl cos 2o(Ipl)

.]4 2 f d3q 1 1 [sin 20([/,I) sin 20(]ql) ~g (2,/7.)3 (p_q)2 2

-~. qcos[20(lp[)](1-cos20(lql))]. (2.19)

The right-hand side is the expression eq. (2.9) for w(lp[). This result justifies the interpretation of the pseudoparticle excitations of the

condensate as constituent quarks which acquire an effective mass through chiral symmetry breaking.

3. The p ion as a Golds tone boson

A chiral rotation on the quark fields

q,(x)--> e '~"~T°~v~(x), (3.1)

transforms the coherent states I g') into

l , t~,) t m= 1 f Nf Nc f } s)(e ),rod,,. ( - p , - s ) ]0). 14'~a)=~exp ~.,Z~ ,=1 ~ s=±l • d3pq'(IP[)sb'+(P' ,s,~o,o +

(3.2)

In both these equations, a a (a = 1, 2, 3) are the parameters of the chiral rotation and r a the Pauli matrices (i.e. the generators of SU(Nf= 2) in the fundamental representation).

For these chirally transformed coherent states, we have

(~lolal~tj~ a) = O, If e '~°~° ~s e,~ °T° ' (3.3)

(0a a I:HocD:]4'c~> ---- (q'I:HocD:I0), (3.4)

which shows that they are all degenerate and orthogonal to each other, and that the condensate [4') used as an approx~maUon to the true QCD ground state is not chiral invariant.

In such a s~tuatlon of symmetry breaking the Goldstone theorem [6] apphes, and therefore massless pseudoscalar Goldstone bosons, which are the chiral hmit approximation of the pions, should exist In the model.

The equation for these states is obtained through an infinitesimal chiral rotation of eq. (2.15) for the quark self-energy. Defining the pion vertex P(p) by

raP(p) = - t~-~a ~ {e"~°(~"):',.,Y(p) e'C'°(~'°)'~-~}l,~_ o (3.5a)

~, 5 . y (p ) } (3 5b) =~7" iT ,

J Govaerts et al / Chlral symmetry breakmg m Q C D

i I I I i I

Fig 2 T h e B e t h e - S a l p e t e r e q u a t i o n fo r t he p lon ve r t ex

65

an infinitesimal chiral rotation on eq. (2.15) gives

p(p) =_~g2 f d4q (-i)

(2~r)4 (p_q)2 Y°S(q)P(q)S(q)Y ° , (3.6)

which is represented in fig. 2. Eq. (3.6) is the Bethe-Salpeter equation, m the instantaneous one-gluon exchange

approximation, for a zero-energy massless pseudoscalar bound state of the con- stituent quarks. The fact that this Bethe-Salpeter equation follows d~rectly from the condensate equation (2.3) shows explicitly that the model contains zero-mass pion Goldstone bosons as bound state of the constituent quarks.

To compute pion static properties, such as f . o r l..t2/mq, one needs the pion Bethe-Salpeter wave function m other kinematical situations than for a massless pion at zero-energy.

Rather than using the wave function of a massless pion at non-zero momentum, we choose instead to use the wave function for a masstve pion at rest, m which case chiral symmetry is explicitly broken in the QCD lagrangian by non-zero current masses of the bare quarks. The Goldstone bosons then become massive.

All analytic calculations are done with massive quarks and pions, and in the final expressions the limit of exact chiral symmetry (Le. mq-* 0 and / 2 _~ 0), for which the condensate equation is solved numertcally, is taken.

It is straightforward to reformulate the model with non-zero current quark masses (which we take to be equal for u and d quarks; mu = m d = mq)*. We give no details here since the analysis is exactly the same as in the massless case.

From the definition eq. (2.1) of the coherent states I~h), where b +, d + are now the creation operators for massive quarks, the matrix elements (6]:HOCD:I6) are easily evaluated by using a Bogohubov-Valat in transformation, and the minimization with respect to 6(Ip]) gives a non-linear integral equation which becomes eq. (2.3) m the hmtt m q ~ 0.

The pseudoparticle excitations of the condensate are identified with the constituent quarks having a constituent mass different from mq [ 8 ] .

The same results can again be expressed as the instantaneous one-gluon exchange approximation to the Schwinger-Dyson equation for the quark self-energy 2(p)

* If m u a n d m d a re u n e q u a l , all resul t s ho ld wi th mq be ing the a v e r a g e of the u p a n d d o w n c u r r e n t q u a r k masse s

66 J. Govaerts et al / Chzral symmetry breaking m QCD

(see eq. (2.15)), with the four-dimensional quark propagator

l

S(p) p - 2 ( p ) - m q + i e ' (3.7)

2(p) --IpIA(Ipl) ÷ p vB(lpl). (3.8) (Although these functions are different from those defined in eqs. (2.11) and (2.12), we will keep the same notation since they reduce to them in the limit mq-> 0. The precise meaning of the notation should be clear from the context).

A chiral rotation on eq. (2.15) gives an Integral equation for a pseudoscalar vertex at zero four-momentum which is not on the pion mass shell and which is not in the form of a Bethe-Salpeter equation. The Bethe-Salpeter equation for the pion is not a consequence of the Schwinger-Dyson equation for 2 ( p ) , since when mq # 0, chiral rotations do not leave the QCD lagrangian invarlant.

We define P(p, tz~) as the pion vertex for a zero-momentum pion, with relative three-momentum p between the quark fields.

The Bethe-Salpeter equation for P(p, /z~), given in fig. 3, imposes the general form

P(p, tz~)=Pp(IPl)ys+IZ~PA(IPl)YoYs+tX~PT([pl)p''YYo'Ys. (3.9)

In the chiral symmetry limit, the component functions are given by

P.(Ipl) --IplA(Ipl), (3.10a)

f d3q 1 [qlA(lql)P([q[) PA(IPl) __4g2 (27/.)3 (p_ q)2 4,o3(1q[) , (3.10b)

IplP~(Ipl)=~g=f d3q ff'q Iqlf(lql)P(lql) (3.10c) (277)3 ( p _ q ) 2 4to3(lql) ,

where P([Pl) is defined as

P(Ipl) =Po(IPl)÷21PlEA(IPl)PA(IPl)÷c(IPl)IPlPT(IPl)] (3.11) It is determined by the integral equation

p([pl)=pp([pj)+4g2[ J d3q 1 1 (2~.)3 (p_q)2 2 3(iql)

x r l P l A ( I P l ) l q l A ( q l ) + p " qC( IP l )c ( l# l ) ]P ( Iq l~ . (3.12)

I

Fig 3 The Bethe-Salpeter equauon for a masswe plon at rest.

J Govaerts et al / Chtral symmetry breaking m Q C D 67

These relations specify completely the pion vertex for a massive pion at rest in the limit of exact chiral symmetry.

4. Static pion properties

The pion decay coupling f= is defined by the matrix element

( n IT(4;(0) y,%4~1- 3) ~ (o))l o ( p ) ) = if=p,, (4.1)

where 0 is the quark isospin doublet (~) and I ~ ) the QCD ground state. Taking the divergence of the anomaly free axial current in eq. (4.1), we have also

2imq(alT(~(O)75(½1-3)O(o))lrrO(p)) _ 2. - L/x ~ (4.2)

By expressing these matrix elements in our model, the coupling f~ and the ratio 1~2/2mq can be directly evaluated, in the hmit of exact chiral symmetry, from the Bethe-Salpeter wave function for a pion at rest. For ~r °, ~r ± states, the wave functions are

1 1-3p(p, tx=), (4.3a) N

~/~ 1-±P(p, Ix=), (4.3b) N

where

1-± = 1(1-14- i1-2) , (4 .4 )

and N is the Bethe-Salpeter normahzatlon factor which has still to be computed. From eq. (4.1) and fig. 4, we have

f d4q { 1 1 1 } #=)S(qo-~l~=, q)

_ ~#~, N if=/x==(-1) ( -~ )4Tr yoys(11-3)S(qo+ q) "rap(q,

(4.5)

where the trace is understood to be on flavor, color and spinor indices. The q0 integral is easily performed, and eq. (4.5) reduces in the chiral symmetry

limit to

L = _ N c [" d3q [qlA(lql)P(lql) (4.6)

N J (21-r) 3 ,o3(Iql)

YoYb~

Fig 4 The matrix element eq (4 1)

68 J Govaerts et al / Chtral symmetry breakmg m QCD

3

Fig 5. The mamx element eq. (4.2)

Similarly, from eq. (4.2) and fig. 5 we obtain

2imq = J ~ - ~ T r ys(l~-3)S(qo+llx=,q) .r3p(q,l.L~.)S(qo-llz~.,q),

which reduces in the chiral symmetry limit to

2mq (2"/1") 3 2Pp(lq[)

~o(Iq])

(4.7)

(4.8)

The normalization factor N has to be calculated from a matrix element whose absolute normalization is fixed independently. We use for this the pion electromag- netic form factor [9] since at ze ro-momentum transfer its value is the pion electric charge.

The pion electromagnetic vertex has the kinematic form

( rc( p')lJ~ ( O )l~( p) ) = ieQ( p + p') ~F~( q2) , (4.9)

where Q is the pion charge, F,~(q 2) the electromagnetic form factor normalized to F~(0) = 1, q = p ' - p , and J , the electromagnetic current.

The vertex matrix element is related to the pion wave function as shown in fig. 6. By computing it for a massive ~r + at rest and in the kinematic situation of fig. 7, we have:

2ie l .~Q=(_l) ie(Q2_Q1) f d4q Tr{S(qo+½1~,q ) x/2 (2rr) [ -~- r+P(q, I~=)

X 1 42 } S(qo-~l*~, q)yoS(qo -~l*~,~ q) - ~ r-P(q, - I ~ ) , (4.10)

with Q the rr + charge and Q~, Q2 the u and d charges respectively (Q = Q1 - Q2 = 1). Here we have used the fact that the sum of the two diagrams of fig. 6 is proportional

Y F~g. 6 The plon electromagnetic vertex.

J Govaerts et al / Chiral symmetry breakmg m QCD 69

It= 0

y t t = 0

Fxg. 7 The pmn electromagnetic vertex at zero momentum

to ( O 1 - 02) and that the expression for the second diagram of fig. 7 is the same as that for the first one with the change/z~, ~ (-/x~).

Computing the q0 integral in the right-hand side of eq. (4.10) and taking the chiral symmetry limit/z~ ~ 0, we get

N 2 = N ~ I d3q Pp(Iql)P(lql) (4.11) (2rr) 3 ~o3(Iql) ,

which does not fix the sign of N. Since we have chosen the condensate wave function ~,(Ip[) to be positive

everywhere, from which it follows that all other functions are also positive, and because of the minus sign in eqs. (4.6) and (4.8), the normalization factor is most conveniently chosen negative, which gives

N = _ [ N c f d3q Pp(lql)P(lq])] 1/2 (4.12) k J (277-) 3 ,o3(Iql) I "

To verify that the relation

~z2~ f~ = -2mq( aU) , (4.13)

which follows from PCAC, is satisfied in our model, we compute the matrix element

(alT(C,(O)u(O))la) = (niT(d(0) d(0))ls2), (4.14)

which is given in fig. 8. In the chiral symmetry limit, this corresponds to

f d4q =-Ncf (277.) 3d3~q 21qlA(lql)w([q]) ' ( -1 ) ~ - s T r S(q) (4.15)

(no trace on flavor indices). Recalling that e0(Ipl)--IplA(Ipl) in that limit, one verifies from eqs. (4.6), (4.8), (4.12) and (4.15) that the PCAC relation eq. (4.13) is indeed satisfied.

u~d tt

Fig 8 The matrix element (a(0)u(0))

70 J Govaerts et al / Chtral symmetry breakmg m OCD

5. The pion torm factor and its charge radius

In contrast to the static physical pion parameters listed in sect. 4, the calculation of the form factor at non-zero momentum transfer reqmres the knowledge of the Bethe-Salpeter wave function for a mowng pion.

If the wave function for a pion at rest were known from a covariant approximation

to its Bethe-Salpeter equation, the wave function for a moving pion would be simply obtained from it by a Lorentz boost.

However , our model is not covariant, since it represents an instantaneous one- gluon exchange approximation. The calculation of such dynamic quantities as form factors requires an extension of the ideas developed up to now to describe plon states w~th non-zero momentum. Three possible ways of inferring the wave function of a moving plon are: to solve the instantaneous one-gluon exchange Bethe-Salpeter equation for pseudoscalar bound states with non-zero three-momentum; to Lorentz boost the wave function of a massive plon at rest (and then take the ~z~ ~ 0 limit); or to boost the wave function of a pion at rest with a galilean transformation. Here we will choose the third possibility, which has the effect that the relatwe three- momentum dependence for a moving pion is the same as for a pion at rest.

We thus use as the wave function for a moving pion ~ts wave function at rest given by eqs. (3.9) to (3.12). The results of such an approximation are of course not reliable at large momentum transfers. Since our model should reasonably well describe the low-energy phenomena associated with the breaking of chiral symmetry, we will concentrate on the form factor at small momentum transfer.

For the electromagnetic form factor F=(q 2) defined in eq. (4.9) the second term in an expansion in q2 is the pion charge radius squared (r2)= defined by

aF~(q 2) (r2)= = 6 0q2]q2=o . (5.1)

The remamder of this secnon is a calculation of (r2)~.

From the definition of the pion electromagnetic vertex in eq. (4.9) and the particular kinematic situation depicted in fig. 9, we have

1 I d4k f ~ ( - q 2) = N ~ ~- ~ T ~ Tr {S(ko+½E, k)P(k +~q, E)S(ko-½E, k +½q)'yo

× S(ko-½E, k -½q)P(k-~q , - E ) } , (5.2)

The right-hand side is understood as being evaluated in the chiral symmetry limit. It is most convement to compute eq. (5.2) as a ser,es in E and ]ql, since at ]ql 2 = 0

the chiral limit corresponds to E = 0. The coefficient of Iq] 2 gives directly the pion charge radius in that limit, the coefficient of ]q] vanishes, and the coefficient of [q]O

is (N2/Nc). The algebraic manipulations involved in that expansion are tedious but straightforward, and the calculation was done using the M A C S Y M A system at MIT.

J Govaerts et al / Chtral symmetry breakmg m Q C D

E * (ko+ ~,k)

n _ l i _ _ r~- = rt _ - ~ _ ~ ' - ' ) ( ' - ' )_ _'~Z"Z_ r t -

( E , ) ( E , ) E + E ÷ -" 2 2 ( k o - ~ , k

y ~ = 0

71

-> ¥ , ~ t = O+

, I[ ( t ' o 2 ' ~ 2 ~ -+

- r _ ~ ~=r&--

E -+ (ko-~,k)

Fig 9. The p~on electromagnetic vertex at non-zero momentum

The final result is expressed in terms of three functions Rp(k), RA(k), RT(k), defined below, and reads:

F.(-q2) = 2 Nc f o° dk 17 N 2 (27r) e 24 (k) {Rp(k)+RA(k)+RT(k)}' (5.3)

from which follows

N 2 ( '~ dk 1 --=2No Jo (2~') 2 24oS(k){Rp(k)+RA(k)+Ry(k)}l '~ '=° ' (5.4)

Nc {co dk 1 ( r 2 ) ~ = - 1 2 - ~ -o (2~r) 2 24wY(k) {Rp(k)+ RA(k )+ RT(k)}I~r2. (5 .5 )

The Iql-independent parts of these functions are

{Rp(k) + RA(k) + Ri(k)}l,ql=o = 24~o4(k) k2pp(k)

[Pp(k) + 2 k ( A ( k ) P A ( k ) + C(k)kPT(k))] , (5.6)

so that eq. (5.4) is the same as eq, (4.11) after angular integrations. The coefficients of Iql 2 are

Rp(k)frq? = w4[2pp( k4n'~ ) - 2(kZP'p)2 + 3Pp(kzP~)]

+ P2{~o2[-6( kC)( k 5 C") + 4( k 3 C') 2 - 6( kA)( kSA '') - 16(k3A') 2 ]

- 20( kA )2( k3C')2 + 40( kA )( k3A')( kC)( k3C ') + 20( kA )a( k3A') 2

- l lw2[(kC)(k3C' )+(kA)(k3A ' ) ]+o9215(kA)2-7oj2]} , (5.7a)

RA(k) i,,,z = 2w 4pA( kA)( k4n~ ) + ( k 2n,p ){w4[_ 12PA( k3A ') - 4( kA)( kep'A )]

+ 12w 2pA( kA)[( kC)( k3C ') + ( kA)(k3A')] + 3 ~o4PA( kA)}

+ Pp{~O4[4PA( kSA ") + 12( k2p'A )( k3A ' ) + 2( k4p'~ )( kA)]

+ w2[4PA(kA)[-3( k C) ( k sC ") - 3 ( kA) (kSA '') + (k3C') 2 _ 9(k 3A') 2 ]

- 12(k2pk ) ( kA) (kC) (k3C ') + (kA)(k3A')]]

72 J Govaerts et al / Chtral symmetry breakmg m QCD

-40PA( kA)2[ ( kA )( k3C' )2- 2( k3A')( kC)( k3C ') - ( kA)( k3A') 2]

+ w4[ IOPA( k3A') + 3( k2P'A)( kA)]

- 26~oZPA(kA)[(ke)(k3C ') + (kA)(kA' )]

-- ¢o2pa( kA)[15o92 - 12(kA)2]}, (5.7b)

RT(k)llql 2 = 2w4( kPr)( kC )( k 4pp ) + ( k2p'p){-4w4( kC)( k3p~- )

- 12~o2(kA)(kpv)[(kA)(k3C ') - (k3A')(kC)] + w4(kC)(kPT)}

+ Pp{W4[-8( kPT)( ks C") + 2( ksP~ )( kC) ]

+ w2[(kP-r)[12(kA)2(ksC ") +4(kC)(k3C') 2 ]

+ (kA)(k3C')[80(kPT)(k3A') + 12(kA)(k3p~)]

- 12(kC)(kPr)[(kA)(kSA ,,) + 3(k3A') z] -

- 12(k3p~r)(kA)(k3A')(kC)]

- 4 0 ( kPy)( kA)2[( kC)( k3C')2 + 2( kA)( k3A')( k3C ')

- (k3A')2(kC)]

+ w4[-16( kPT)(k3C ') + 5(k3p!r)(kC)]

+ 32w2(kA)(kPT)[( kA) (k3C ') - (k3A')(kC)]

+ w2( kC)( kPT)[12( kA ) 2 - 11w2]},

where primes indicate differentiation with respect to ]k] 2.

(5.7c)

6. Numer ica l results

As was discussed previously [1] and pointed out in the introduction, the model as analysed so far does not contain any scale, and hence the Q C D hamiltonian cannot be minimized over the trial states ]~b). Physically, the source of this problem ~s that the dynamical effects included in the model are not those which cause the coupling and other parameters to effectively vary with distance [4]. This would reqmre both the introduction of bare gluon states into the vacuum trial states and renormalization of the variational calculation.

T o b y p a s s these difficulties and regulate the equations, we replace the couphng constant a = gZ/4rc by an effective running coupling constant a ( q 2) which falls off at small distance. This introduces a scale for the condensate wave function and bounds the expectation value of the hamiltonian in the trial states f rom below.

We used in ref. [1]

al (q2) = so l +q2 /A2 , (6.1)

J Govaerts et al / Chlral symmetry breakmg m QCD

TABLE 1

Best fit to plon and chlral symmetry breaking parameters for the two forms of the running couphng constant

73

O~l(q 2) Ot2(q 2) a o = 1 6 5 c~ o = 1 5 5

A = 1 6 G e V A = 0 6 G e V gc = 1 09 GeV qc = 1 07 GeV

Physical values

M o 300 MeV 300 MeV ~300 MeV

3 / p 147 MeV 149 MeV - -

f= 134 MeV 132 MeV 95 MeV

ixz=/2mq 2110 MeV 2800 MeV 1700 MeV (m u +mo = 11 MeV)

Iz2/2mqf~ 15 7 21 2 18

(uS) ( -340 MeV) 3 ( -360 MeV) 3 ( -250 MeV) 3

(r2)~ 0 64 fm 2 0.62 fm 2 0 46 fm 2 [10]

qc Is defined by a(qZ)_s_9

but in order to investigate the sensitivity of the results on the asymptotic behaviour of 0/1(q2), we use here, in addition, a second form for the running coupling constant whose asymptotic form agrees with the results of asymptotic freedom:

12Ir az(q2) ( 3 3 - 2Nf) In (q2/A2)' for Iql >i q0

= (6.2a) So, for Iql <~ qo,

where I6 / qo = A exp ( 3 3 - 2 N f ) a 0 (6.2b)

The two integral equations eqs. (2.3) and (3.12) for the functions o(Ipl) and P(Ipl) are solved numerically for both these couplings. From their solutions we compute all other functions necessary for the evaluation of physical parameters. The funcUons sin 2 0(Ipl), IpI/(lpl) and .,(Ipl) were described previously [11 for the effective couphng a l (q 2) with different values of c%.

The best fit to the physical values of all parameters for each form of a(q2) is shown in table 1. Although the analytic and especially the asymptotic forms of or1 and c~2 are quite different, the physical results differ only slightly. This confirms our expectation [1] that it IS only the fact that at some scale q~r the effective coupling c~ (q2) becomes less than the critical coupling C~cr, at which the perturbative vacuum becomes unstable to spontaneous creation of qCt pairs, which is relevant for the physical properties of the chiral symmetry breaking phase*. This scale is found to be essentially the same for both couplings, i.e qcr ~ 1.1 GeV (see table 1).

* For both forms of a(q2), the numerical solutions of the equations show that Ogcr hes between 1 10 and 1.20 This agrees with analytic results [3] derived from an analysis of small fluctuations (act = ~) and agrees with the corresponding result of Amer et al (gcr = 3/~r) [11]

74 J Govaerts et al / Chtral symmetry breaking m QCD

1.0

I

0.5

f~ l~

f~ ~

f~/MQ

• i i I I I

1.2 1.4 1 6 1 8 o

Fig 10. The dimensionless ratios f~/Mo, f~/3x/p, f~x/(r~ as functions of a o, for the coupling Otl(q 2)

Since massless QCD is a parameter free theory as a consequence of dimensional transmutation, dimensionless ratios of physical quantities computed m QCD are

pure numbers. The model of QCD described here, however, contains a dimensionless parameter a0. The degree to which dimensionless ratios computed in the model

depend on a0 measures how good an approximation to QCD this model can be.

Fig. 10 shows three such ratios, f=/Mo, f=/3x/p and f=x/(--~=, plotted against s0 for So between 1.20 and 2.00 (in the case of the coupling eel(q2)). Their varianon in this range (which is an order of magnitude in ( s0 -ac r ) ) is reasonably small. The

raUo of purely static quantities, f,~/Mo, is nearly constant. By contrast, the ratio ix2~/2m, f~ shown in fig. 11 varies wildly with s0. This

non-QCD-like behaviour ~s a necessary artifact of the sort of model presented here. For a0 = ac,, ~ z2 will not vanish when mq does, causing the above ratio to diverge. This should occur in any model in which a fundamental parameter 's value chooses whether or not chlral symmetry is dynamically broken.

Let us remark that the identification of the pseudoparticle excitations of the condensate with constituent quarks is reasonable, since the density of bare quarks in the condensate, which is 20, is such that the average separation between these quanta is less than the size of a hadron. From table 1, we find an average separation of 185 MeV -~ or 1.04 fm which is smaller than a plon compton wavelength, the

characteristic size of a nucleon. Finally, the fact that the value of qc, is larger than the confinement scale

( - 7 0 0 MeV) shows that the model is consistent with its own approximations.

100

50

J Govaerts et al / Chtral symmetry breaktng m QCD

2

1.2

Fig l 1

I I I I I I I

i .4 i .6 1 .8

The dlmenslonless ratio 1~2~/2mqf= as a function of c%, for the couphng ~t(q 2)

75

Ct O

The model we have described here is a simple realization of the physical ideas underlying the Nambu Jona-Lasinio picture of chiral symmetry breaking. The main virtue of the model is undoubtedly that it allows, in the f ramework of QCD, defimte quantitative predictions to be made for the parameters related to chlral symmetry breaking.

To calculate these parameters several approximations had to be made: the only dynamical effect which IS taken into account is the attractive single-gluon exchange between quarks and antiquarks. All confinement effects are ignored and all calcula- tions are done m the limit of equal and vamshmg u and d quark masses.

Despite these approximations, the model leads to a fairly reasonable phenomenology for the physical parameters of chlral symmetry breaking as well as for the pion exphcitly reahzed as a Goldstone boson and as a constituent quark- antiquark bound state. The most important result is perhaps the self-consistency of the assumption that the scales of chiral symmetry breaking and of confinement are indeed different.

There is clearly much room for improving the model by taking various additional physical effects into account such as renormahzation, instantons [12], different quark masses, etc. This could be done for example in a covanant self-consistent approxima- tion to the Schwmger-Dyson equations for both the quark and gluon propagators.

The authors would like to thank Pro[ R. Brout, Prof. F. Englert and Dr. J. Finger for useful discussions. We thank the MIT Mathlab group for the use of MACSYMA.

76 J. Govaerts et al. / Chtral symmetry breakmg m Q C D

This w o r k was s u p p o r t e d in pa r t by t h e D e p a r t m e n t of E n e r g y u n d e r c o n t r a c t

n u m b e r E R - 7 8 - S - 0 2 - 4 9 1 5 .

References

[1] J Fmger and J E Mandula, Nucl. Phys. B199 (1982) 168 [2] Y Nambu and G. Jona-Lasimo, Phys Rev. 122 (1961) 345, 124 (1961) 246 [3] J. Finger, D Horn and J.E. Mandula, Phys. Rev D20 (1979) 3253 [4] G 't Hooft, Marseflle Conf 1971 (unpublished);

H. Pohtzer, Phys. Rev. Lett. 30 (1973) 1346, D. Gross and F. Wllczek, Phys Rev. Lett. 30 (1973) 1343

[5] R Fukuda, Phys Lett. 73B (1978) 33, V P. Gusynm and V.A Mlransky, Phys Lett 76B (1978) 585

[6] J Goldstone, Nuovo Clm 19 (1961) 154 [7] J. Finger, J E. Mandula and J Weyers, Phys. Lett 96B (1980) 367;

J. Govaerts, J.E. Mandula and J. Weyers, Pion properties in QCD, Preprlnt UCL-IPT 83/13 (1983) [8] H. Klelnert, Phys. Lett 62B (1976) 77 [9] S Mandelstam, Proc. Roy Soc. A233 (1955) 258

[10] S. Dubmcka, V A Meshcheryakov and J. Mllko, J Phys. G 7 (1981) 605 and references thereto [11] A. Amer, A. Le Yaouanc, L. Ohver, O. P~ne and J.-C. Raynal, Phys Rev. D28 (1983) 1530 and

private commumcation [12] G. 't Hooft, Phys. Rev. D14 (1976) 3432,

C G. Callan, R Dashen and D J. Gross, Phys. Rev. D17 (1978) 2717, R.D. Carhtz and D B Creamer, Ann of Phys 118 (1979) 429, D G Cal&, Phys Rev Lett 39 (1977)121