hep-ph/9411327 17 Nov 1994
August 1994
CHIRAL SYMMETRY BREAKING
FROM DYSON-SCHWINGER EQUATIONS
y
J.R.CUDELL
PhysicsDepartment,McGillUniversity,3600 University Street
Montreal, Quebec H3A2T8,Canada
cudell@hep.physics.mcgill.ca
ABSTRACT
I rep ort on recent progress inthe study of mass generation in non-ab elian gauge
theories,andconcentrate ontheoriginofconstituentmassesinQCD.Iarguethat
aconsistentformalismhasb eendevelop edtodescrib echiralsymmetrybreakingvia
multiplicativelyrenormalisableDyson-Schwinger equations. Its mainconsequence
istheexistence ofacriticalvalueof thecouplingb eyondwhichmassless solutions
donotexist.
It iswell-known that quarksare describ ed by twodi erent kindsofmasses. First
of all,there are the masses which enter as parameters in the lagrangian, or in
sum-rules, the current masses. For light quarks, these are small parameters (m
u
0:005
GeV, m
d
0:01 GeV). On the other hand, the kinematics of quarks (e.g. in B
or D decays) and the masses of hadrons imply much larger masses for quarks, the
constituent masses, of the order of 0.3 GeV for u and d quarks. These masses come
from the interactions of quarks propagating in a non-trivial vacuum, similarto the
interactions of electrons with the background eld of a crystal. Hence 97% of the
observedhadronicmass comesfromtheinteraction ofquarkswiththe QCDvacuum.
In the limitwhereone neglectscurrentmasses, the SU(3) symmetryof the QCD
lagrangian gets doubled into aleft- and aright-SU(3), as the left-handed and
right-handed chiralities do not mixwithout a mass term. Chirality then b ecomes a
con-served quantity, and the QCD lagrangian acquires a chiral symmetry. One then
exp ects the dynamics of the theory to break this symmetry into and SU(3) with
mass terms,app earing as p oles in the quark propagators. We shall write the quark
propagator as S(q 2 ) 1 a(q 2 ) 1q+6(q 2 ) F(q 2 ) 1q+G(q 2 ) (1)
Note that we have Wick-rotatedto Euclidean space, where we shall work hereafter.
Chiral symmetrybreaking o ccurs if 6(k
2
), or equivalentlyG(k
2
), are nonzero. Note
y
the formalism we are going to develop can b e applied to any gauge theory (e.g. to
technicolour).
Let us rst explain why Dyson-Schwinger equations are relevant in this context.
First of all, p erturbation theory, at least in lowest order, fails to pro duce a mass
for massless objects: the one-lo op corrections to the propagator merely multiply
the propagator 1= 1 p by a function of p
2
, and do not shift the p osition of the
p ole. In fact, a renormalisation group argument probably due to Callan, Dashen
and Gross [1], indicatesthat dynamical mass generationlies in the non-p erturbative
regime of QCD. Indeed,the quark mass m =
q
6(m 2
) can only b e a function of the
coupling constant and of the renormalisation p oint : m(g;) = M(g), with M a
dimensionless function. If the mass is to b e physical, it cannot dep end on , hence
dm=d = m+ (g)@m=@g = 0, with = @g=@. In QCD with two avours, the
one-lo op -function is (g)0 29 48 2 g 3
, which leadsto:
m(;g)exp (0 24 2 29g 2 ) exp(0 0:6 S ) (2)
Such a dep endence on
S
cannot b e pro duced via p erturbation theory, even if
re-summed,as mdo es nothavean asymptoticexpansionas
S
!0. Hencethe problem
is intrinsicallynon-p erturbative. A caveat is here in order. It was mentioned ab ove
that p erturbation theory fails to pro duce a mass. This is true only if one is careful
ab outgaugeinvariance. Anyadditionalterm,regularizationmetho d,orresummation
ansatz that breaks gauge invariancewill in generalpro duce a mass term,which will
come,not fromthe physics,butratherfromthe clumsinessofthe metho d. Hencethe
preservation of gauge invarianceisof utmost imp ortancein this problem.
Field theory contains few nonperturbative statements. Some of these, particularly
suited for the present study, are the equations of motion of Green's functions, the
Dyson-Schwinger(DS) equations. WesketchinFig. 1the equationfor the two-p oint
function, S(p
2
) whichwe exp ect to have ap ole at the physical mass of the quark.
( ) ( ) = { q q k=q-p -1 -1 q p
Figure 1: A pictorialrepresentation of the Dyson-Schwinger equation for quarks.
The exact two-p ointfunctions are indicated by ahatched circle, the exact
three-p ointfunctionbyadoublyhatchedcircle.
Notethat weworkinthe axialgauge,hence no ghost diagrams arepresent. This DS
equation relatesthe exactquark propagator to two a prioriunknown quantities: the
D (q 2 )=q 2 D (q 2 )D 0 (q 2 ) (3) withD 0
thep erturbativegluonpropagator. Weshallhere ratherconcentrateon the
approximationsone canmaketothe three-p oint function. One has to guess itsform,
and three levelsof approximation are p ossible. First of all, the simplest assumption
is to take the p erturbative vertex and assume that this is go o d enough. This leads
to the \rainb ow approximation", and to the breakdown of gauge invariance. The
second level of approximation [3] is to use another non-p erturbative statement, the
Ward-Takahashi (WT) identities, and cho ose a vertex that will resp ect these, and
hence not explicitly break gauge invariance. As we shall see, this is not restrictive
enough, and one needs to go to the third levelin the imp ositionof gauge invariance,
and demandone further constraint [4], namelythe matching to p erturbation theory
at highq
2
, inorder to obtaina sensible equationfor chiral symmetrybreaking.
Let us rst concentrate on WT identities,as shown in Fig. 2. One can see that
the projection ofthe three-p ointfunctiononto the gluonmomentumisrelatedto the
di erenceof two inversepropagators. Hence the part of the vertexfunction whichis
parallel to the gluon momentum,i.e. the longitudinal part, is known. The choice of
vertex function is then reduced to guessing the transverse part 0
T
, which do es not
enterthe WTidentities.
.k = -1 { ( ) ( ) -1 q p k q p
Figure2:ApictorialrepresentationoftheWard-Takahashiidentitiesusingthesame
conventionasinFig.1.
The simplest assumption [3] is to take 0
T
= 0. One then obtains [5] two linear
decoupled equations for F and G:
1 = F(q 2 )+ S F(q 2 ) Z dk 2 K 1 (k 2 ;q 2 )+ S Z dk 2 K 2 (k 2 ;q 2 )F(k 2 ) (4) 0 = G(q 2 )+ S G(q 2 ) Z dk 2 K 1 (k 2 ;q 2 )+ S Z dk 2 K 2 (k 2 ;q 2 )G(k 2 ) (5) The kernels K 1 and K 2
are the samein each equation, and can b efound in Ref. [5].
They are b oth prop ortional to the exact gluon propagator, and hence to D (q
2
) of
Eq. (3). Their most imp ortant prop erty is that the integral of K
1
diverges
loga-rithmicallyfor large k
2
duces a renormalisation p oint such that F(q 2 ) = Z f ( 2 )F R (q 2
), and one cho oses
2 F R ( 2
)=1. Asthereisno quarkmassterminour lagrangian,themass squared6
of Eq.(1) do es not get renormalised. This amountsto sayingthat Gand F havethe
same renormalisation factor Z
f (
2
). One also has to renormalise the gluons, which
is done by similarly renormalisingthe gluon propagator (3): D (q
2 ) =Z( 2 )D R ( 2 ).
The (in nite) renormalisation constant Z(
2
) is absorb ed in the de nition of the
renormalisedcoupling. Onethensubtracts fromEqs.(4,5)theirvalueat q
2
=
2 and
this subtraction removesthe remainingrenormalisationconstant Z
f (
2 ).
One runs howeverinto several problems:
In the usual multiplicative renormalisation (MR), one de nes the renormalised
coupling as R S () = Z() bar e S
, and this leads to the renormalisation group. Here
however, this de nition is not sucientto remove Z() from the DS equation. One
has to use a de nition that breaks MR:
R S ()= Z() bar e S 10Z() bar e S R dk 2 K 1 (k 2 ; 2 ) (6) As the kernel K 1
issp eci c to the quark equation, one lo oses the universalityof the
renormalisedQCDcoupling: gluonscoupledi erentlyfromquarks! Furthermore,one
also obtains the wrong asymptoticb ehaviourfor
S (q 2 ) at large q 2 .
TherenormalisedequationforG(q
2
)atlargeq
2
isdominatedbythe termsresulting
from K
1
, and one can show that these b ehave like log (log (q
2
)). Hence the leading
termsof Eq. (5) lo ok likeG(q
2 ) = S ()G(q 2 )log (log (q 2
)), and the equation has no
consistent nonzero solution.
If one pro ceedsand solves theequation forF,one obtains asolution for any
S ()
and these solutions have a p ole at the origin. Hence the quarks are massless and
uncon ned.
All these problems come from the large k
2
region, which is where we exp ect
p erturbation theory to hold. In the region where q
2
k
2
>>>p
2
, one can calculate
the one-lo op p erturbative corrections to the vertex, and hence, by subtracting the
longitudinalvertexderivedfrom the WTidentities,get an asymptoticexpressionfor
0 T : lim q >>>p 0 T (p;q)= S log (q 2 =3 2 ) 4q 2 (0q 1p+ 1q 1p) (7)
where isthe quarkanomalousdimension. Furthermore,onehas the constraintthat
by de nition, for any p and q, 0
T
1(q0p) =0. One can then show that the general
formof 0 T , for 6=0, is: S 01 0 T S =[q 2 F(q 2 )0p 2 )F(p 2 )] (q 2 0p 2 )+(q +p )( 1p0 1q) D (8)
whereDb ehaveslikeq
4
atlargeq
2
,issymmetricinpandqand mustb enonsingular.
Wecho ose in the following D=(q
2 +p 2 ) 2 .
Eq. 4, with new kernels ~ K 1 and ~ K 2 replacing K 1 and K 2
. These kernels are again
giveninRef.[5]. Theiressentialprop ertyisthatthedivergencethatusedtob einthe
K 1
term is now shifted to
~ K 2
. As a consequence, the de nition of the renormalised
S
, Eq. (6), nowcontains a nite integral over
~ K 1
, and hence di ers from the usual
MR prescription by a nite calculable factor, hence the usual rede nition
R S () = Z() bar e S
nowworksand pro duces an equation free of singularities.
The renormalisedequation lo oks as follows:
(q 2 )q 2 F R (q 2 ) = ( 2 )+ Z dk 2 [ ~ K 2 (k 2 ;q 2 )0 ~ K 2 (k 2 ; 2 )]F R (k 2 ) (9) with (q 2 ) = 10 S ( 2 ) Z dk 2 ~ K 1 (k 2 ;q 2 )
This equationhas tworegimes. If (q
2
)>0forall q
2
, thenthere isasolution, which
meansthatF 6=0,G=0isallowedandthat chiralsymmetryremainsunbroken. On
the other hand, if (q
2
)= 0for some value of q
2
then there is no solution for F and
G=0 is not allowed.
Tostudy the p ositivity of forageneralgluon propagator, one canuse a K
allen-Lehmann representation for the gluon propagator, D (k
2 ) = R d()=(k 2 +). The
factthat the kernel
~ K 1 is linearinD (k 2 ), ~ K 1 (q 2 ;k 2 )~ 1 (k 2 ;q 2 )D (k 2 ),enablesus to rewrite as: (q 2 ) = Z d()H(;q 2 ) (10) with:H(;q 2 ) = 1 2 + 0 S ( 2 ) Z dk 2 ~ 1 (k 2 ;q 2 ) k 2 + H(;q 2
)>0impliesthatone hasamasslesssolution. WeshowinFig.3the regionin
( S
;)spaceforwhichthefunctionH isp ositivede nite,andthe upp ercurveshows
the values of S ( 2 ) for which H(; 2
)=0, i.e. for which the e ectivecoupling (6)
b ecomes in nite,and where one exp ects the equation not to have asolution.
Figure3: Theshadedregionis wherethefunctionH isp ositivede niteforallq
2 ;
thethickcurve showsthevaluesof
S
forwhichH(;
2
)=0.
Hence we see that massless solutions exist only for small values of
S
(), and that
invariantsuchas S
() D
R
( ))b eyondwhichoneceasestohavemasslesssolutions.
Futhermore, the chiral symmetry of QCD can also b e destroyed by the presence of
massive,high- mo des inthe gluon propagator. Although the exact value of
cr itical
S
willdep endon thegluonKallen-Lehmanndensity(),itisclearthatacriticalvalue
of the coupling willalways exist.
To show what happ ens to the massless solution in an explicit case, we can as a
toy-mo deluse the gluon propagator derivedin Ref.[2]:
2 D R (k 2 )= 1 0:88 k 2 2 0:22 00:95 k 2 2 0:86 +0:59 log h 2:1 k 2 2 +4:1 i (11)
The equation leading to this solution followed the old BBZ approach [3] and hence
violatedMR,and didnotgiveriseto therenormalisationgroup. Asaresult,wewere
able to derivea solution only for a given
S
(),
S
()=1:4, as the renormalisation
groupwhichcomp ensatesavariationof
S
byavariation ofD (k
2
)didnotwork. The
solution has a cut b ehaviour: near k
2 =0,D (k 2 )(k 2 ) 00:22
,and hence corresp onds
tocon nedgluons,withnop oleneartheorigin. Asoneexp ectsthep oletob etap ered
by nonp erturbative e ects,and as a theorem[3]requires the axialgauge propagator
to b e in nite at the origin, one exp ects an improvedsolution satisfying MR to lo ok
roughly the same. Hence we use the solution found in [2] and the b est we can do
at present is to change the value of
S
() only in the quark equation. One then
obtains the curves of Fig. 4. They show that as the value of
S
() increases, the
quark propagator exp eriencesoscillations, until eventuallyno solution can b e found,
foracriticalvalueof
S
of the orderof1.5. It isinterestingthatthe value of
S that
we obtained fromthe gluonequation is very closeto the critical value.
Figure 4: The solution for massless quark propagators that come from the
non-p erturbative gluonpropagatorof Ref.1,for
S
() =0.2(plain),0.6(dashed), 1.0
(dot-dashed)and1.4(dashed).
To conclude,we now havea formalismto handle the Dyson-Schwinger equations
whichtakesinto account everythingwe know ab out QCD:
The renormalisation is multiplicative, and hence gives rise to the renormalisation
The quark propagator matches smo othly with the p erturbative ansatzat large
mo-mentum.
This formalismpredicts that one keeps massless solutions onlyfor small
S
, and
hence there is a critical value of
S
b eyond which a mass must b e generated for the
equation to havea solution. One can also show that the equationfor the mass term
has a consistentb ehaviour at large momentum.
One ofcourse nowneedsto ndthe solutions inthe massivephaseand to extend
thepresentformalismtothegluoncase. Onewillthenhaveaconsistent eldtheoretic
description of chiral symmetrybreaking at the quarklevel.
Acknowledgements
This work was done in collab oration with A.J. Gentles and D.A. Ross and was
supp orted in part by by NSERC (Canada), les fonds FCAR (Queb ec) and PPARC
(United Kingdom).
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