Z. Phys. C 74, 131–143 (1997) ZEITSCHRIFTFUR PHYSIK Cc© Springer-Verlag 1997

Instantons and meson correlation functions in QCDMarcus Hutter

Sektion Physik der Universitat Munchen, Theoretische Physik, Theresienstrasse 37, D-80333 Munchen, Germany(e-mail: hutter@hep.physik.uni–muenchen.de)

Received: 28 April 1996

Abstract. Various QCD correlators are calculated in the in-stanton liquid model in zeromode approximation and 1/Nc

expansion. Previous works are extended by including dy-namical quark loops. In contrast to the original “pertur-bative” 1/Nc approximation, not all quark loops are sup-pressed. Renormalization of the instanton density allows theidentification of the density with the gluon condensate evenin presence of dynamical quark loops. In the flavor singletmeson correlators a chain of quark bubbles survives theNc → ∞ limit causing a massiveη′ in the pseudoscalarcorrelator while keeping massless pions in the triplet cor-relator. The correlators are plotted and meson masses andcouplings are obtained from a spectral fit. They are com-pared to the values obtained from numerical studies of theinstanton liquid and to experimental results.

1 Introduction

QCD successfully describes the strong interaction for morethan 20 years. Despite its formal similarity to QED, there areimportant differences. Most important are asymptotic free-dom and instantons. Non-trivial solutions of the euclidianYang-Mills equations have been found in the year 1975 [1].They significantly influence the structure of QCD at low en-ergies. The one instanton1 solution has the well-known form

AaIµ(x) = Oab

I ηQI

bµν

(x− zI )ν(x− zI )2

2ρ2

(x− zI )2 + ρ2

γI = (zI , OI , ρI , QI ) = (location, orientation, radius,

topological charge) (1)

The instanton parametersγI reflect the symmetries of theQCD action (translation, rotation, scale, parity) invariance.Many exact results are known for the one instanton vac-uum [2, 3]. An introduction to the theory of instantons can

This work is supported by DFG and GIF1 In this work instantons and anti-instantons will both be called instan-

tons and are distinguished by their topological chargeQI = ±1

be found in [2, 4, 5]. The most interesting result for phe-nomenology is the explicit breaking of the axialU (1) sym-metry [6]. On the other hand, an one instanton approxima-tion, like a tree level calculation in perturbation theory, isnot able to describe bound states or spontaneous symmetrybreaking.

The next step was the analysis of exact [7] and approxi-mate multi instanton solutions [8]. The sum of widely sepa-rated instantons (QI = +1) and anti-instantons (QI = −1) isan approximate solution of the YM equations:

Aaµ(x) =

N∑I=1

AaIµ(x) (2)

There are two useful pictures. In one picture instantons areviewed as tunneling processes between topologically distinctvacua. In the other picture they describe ensembles of ex-tended (pseudo-)particles in a 4 dimensional Euclidian space.The partition functionZ is dominated by these ensembles.In the simplest caseZ describes a dilute gas of independentinstantons at temperatureg2 in a large 4-dimensional volumeV4:

Z =∞∑N=0

ZN , ZN ≈ 1N !

(V4n)N (3)

The sum is dominated by the total instanton densityn

n =∫ ∞

0dρD(ρ) , D(ρ) ∼ ρb−5 , b =

113Nc .

D(ρ) is the density of instantons of sizeρ determined inone loop approximation in [6]. Unfortunatelyn is infiniteand the assumption of a dilute instanton gas is inconsistent.The probability of small size instantons is low becauseD(ρ)vanishes rapidly for small distances. On the other hand, forlarge distancesD(ρ) blows up and soon gets large. Thisis the origin of the infrared problem which made a lot ofpeople no longer believing in instanton physics. Those whowere not deterred by that have thought of the following out-come [9]. For larger and larger distances, the vacuum getsmore and more filled with instantons of increasing size. Atsome scale the instanton gas approximation breaks downand one has to consider the interaction between instantons

132

which might be repulsive to stabilize the medium. The sta-bilization might occur at distances at which a semiclassicaltreatment is still possible and at densities at which the var-ious instantons are still well separated objects – say – notmuch deformed through their interaction. So there is a nar-row region of allowed values for the instanton radius. Thispicture of the vacuum is called the instanton liquid model.The idea has been refined and confirmed in the course ofyears.

The first suggestion was to introduce a cut-offρc and toignore large instantons [8]:

nρc =∫ ρc

0dρD(ρ) . (4)

The cut-off is chosen small enough to make the spacetimefraction f filled with instantons less than one so that thedilute gas model is justified

f =2Nc

∫ ρc

0dρ

12π2ρ4D(ρ) � 1 (5)

This simple cut-off procedure can be improved by introduc-ing a scale invariant hardcore repulsion between instantons,which effectively suppresses large instantons [10]. This pro-cedure has the advantage of respecting the scaling Wardidentities which are otherwise violated by the simple cut-offansatz. In [11] such an repulsion has been found leading toa phenomenologically welcomed packing fraction. Unfortu-nately this repulsion is an artefact of the sum-ansatz as hasbeen shown by [13], but see also [14] for a contrary discus-sion. Therefore the infrared problem is still unsolved.

Nevertheless it is possible to make successful predictionsby simply assuming a certain instanton density and some av-erage radius. It seems that the vacuum can be described byeffectively independent instantons of size ¯ρ = (600 MeV)−1

and mean distanceL0 = (200 MeV)−1. The total instantondensity is bounded by the experimentally known gluon con-densate [12]:

n = N/V4 = 1/L40 ≤ 〈Ga

µνGµνa 〉/32π2 = (200 MeV)4exp. (6)

The ratioL0/ρ is estimated in different works [11, 9] to be(L0/ρ)theor. ≈ 3. If one assumes, that instantons saturate thegluon condensate, then the instanton liquid model isdefinedby the following density:

D(ρ) = nδ(ρ− ρ), n = (200MeV)4, ρ = (600MeV)−1(7)

Numerical studies of this model have allowed the determina-tion of various hadronic parameters. Meson masses, baryonmasses, hadron wave functions and much more has beensuccesssfully calculated in the last 13 years [9, 15–18]. Thesuccess of this model and the comparison with lattice QCD[19, 20] suggests that instantons indeed describe correctlythe vacuum structure of QCD.

The quark propagator and the meson correlators can alsobe determined by analytical methods. The most importantpredictions are the breakdown of chiral symmetry (SBCS)[11] and the absence of an axial singlet Goldstone boson.

In this article, I want to extend the analytical methodsand to evaluate the results numerically.

In Chap. 2, the propagator of a light quark is calcu-lated. The approximations which have to be made are stated

and discussed. I have extended [11] by including dynamcialquark loops, which renormalize the instanton density. Thisrenormalized instanton density can be identified with thegluon condensate despite the presence of dynamical quarks.This is the most important new result in this chapter. Furtherit is shown that for one quark flavor the 1/Nc approxima-tion is exact. The constituent quark masses and the quarkcondensates are calculated foru,d ands quarks.

The same approximations are used in Chap. 3 to calculatethe 4 point functions. Special attention is payed to the singletcorrelator where a chain of quark loops contributes and isnotsuppressed in the large-Nc limit. Within one and the sameapproximation we get Goldstone bosons in the pseudoscalartriplet correlator but no massless singlet boson.

In Chap. 4 meson correlators are discussed and plotted.Employing a spectral ansatz, it is possible to extract var-ious meson masses and couplings. They are compared tothe values obtained from extensive numerical studies of theinstanton liquid [15] and to the experimental values.

Conclusions are given in Chap. 5. Useful formulas forthe zeromode of the Dirac operator in the background of oneinstanton can be found in appendix A. All factors 2π in thefollowing chapters are absorbed in the convenient definitions

δ−d(· · ·) := (2π)dδ(· · ·) ,

∫d−dp :=

∫ddp

(2π)d.

2 Light quark propagator

In this chapter, the average quark propagator in the multi-instanton background will be calculated. The work ofDiakonov [11] is extended by including dynamical quarkloops. We will sum upall feynman graphs corresponding toscattering processes of quarks at the instanton background.This is possible in the case of one quark flavour within theso-called zeromode approximation due to an exact cancela-tion of certain graphs und by a renormalization of the in-stanton density. As a byproduct, the question, whether itis allowed to identify the instanton density with the gluoncondensate or not, when including dynamical quarks, is pos-itively answered. This is the most interesting new result. Thequark condensate and a constituent quark mass are extractedfrom the quark propagator. In the last section it is shownthat the case of two or more quark flavours can be reducedto the one flavour case in the limit ofNc → ∞. The re-sults obtained in the one flavour case are therefore still validwhen making this further approximation. A more detaileddescription of the following resummation procedure can befound in [21, 22].

Selfconsistency equation for the quark propagator

It is well known how to calculate correlators in the presenceof an external classical gauge field at least as perturbationseries in powers of the external fieldAa

µ(x). In the caseof QCD (or more accurately in classical chromodynamics)within the instanton liquid model, the external field is a sumof well separated scatterersA =

∑I AI called instantons

with a fixed radiusρ and distributed randomly and indepen-dently in Euclidian space.

133

For a while we will restrict ourselves to the case ofonequark flavorand ignore gluon loops. In the last section wewill show, that the case of many flavors reduces to this casein the limit Nc →∞. The Euclidian feynman rules have thefollowing form:

� =1

p/ + im= S0(p) ,

��

AI (x)

= A/I (x) .

x

(8)

To average a graph over the instanton parametersγI one hasto perform the following integration for each instanton:

〈. . .〉I = n

∫dγI . . . =

N

2

∑QI=±1

1V4

∫d4zI

∫dOI . . . (9)

Perturbation theory is suitable to study scattering processes.To achieve chiral symmetry breaking or bound states onehas to sum up infinite series of a subclass of graphs or solveSchwinger Dyson or Bethe Selpeter equations.

The first thing one can do is to sum up successive scat-terings at one instanton

�� ��

�VI := ��AI

+ ���AIx x

+ ����AI

+ . . . ≈ 1im

/υI (p)/υ†I (q)

x

VI is the exact 1PI scattering amplitude of a quark at thebackground of one instanton. In the last expression, the exactscattering amplitude has been approximated by the scatteringat the zeromode of the Dirac operatoriD//υI = 0 only. Thisapproximation is good for large as well as for small momenta[11]. /υI is given in appendix A in singular gauge. This gaugehas to be used when studying QCD at low energies as inour case [23]. For more details on the spectrum of the Diracoperator and its zeromodes, see [24, 25].

The next step is to derive an expression for the scatter-ing amplitudeMI at the instantonI but now in the presenceof further instantons. Let us consider a quark line with twoscatterings atVI and insert in between a number of instan-tons which differ fromI and from all other instantons oc-curring elsewhere in the graph. This enables us to averageover these enclosed instantons independently from the restof the graph. Summation over all possible insertions withat least one instanton just yields the exact quark propagatorS(p) minus the free propagatorS0(p). Remember that directrepeated scattering atAI (x) has already been included inVI .

�� ��VI�:=��

��MI� + �

��VI� �� ��VI

.<

+ � ��VI� � ��VI

. �� ��VI

. + . . . ≈ 1iµ

/υI (p)/υ†I (q)< <

. := − � = S − S0< <

(10)

Due to the product structure ofVI the geometrical seriescan simply be summed up. The only effect of the instanton

bath is the replacement of the current quark massm by aneffective massµ, with

µ = m + i∫d−4p /υ†I (p)S−1

0 (p)

×[S(p)− S0(p)]S−10 (p)/υI (p) (11)

This is similar to an electron propagating in a solid with aneffective massm∗

e instead ofme.We still have to construct the averaged propagatorS

of a quark in the multi instanton background. Although wehave already resummed a lot of graphs, there are still manygraphs containing dynamical quark loops with scatterings atMI . However due to the following identity

- � ��MI-

� ��MI

��

-� ��MI

� �

� ��MI

-+ = 0· · ·...

(12)

it happens that whenever anMI occurs twice or more thantwice in a graph there exists another graph with oppositesign. Both contributions cancel each other and can be ig-nored. So a quark can effectively scatter only once at everyinstanton. The origin of (12) is again the product structureof MI and the usual change of sign when interchanging twoquark lines. This can also be seen in another way: Becauseof Fermi statistics, every state can be occupied only once,and there is only one state for each quark in the zero modeapproximation, namely the zeromode.

There is one special pair of graphs which has to betreated with more care:

� ��MI

��� ��MI

not allowed

� ��MI

��+ = 0

...� ��MI

�

��

>

<

most general tadpole

·

·

The inclusion of the first graph would lead to a double count-ing of graphs, because repeated scattering at one instantonhas already been included inMI . Therefore it is not allowedand the second graph cannot cancel against the first graph. Itmust be included explicitly in the calculation of the propaga-tor. Fortunately the second graph is the most general tadpole.EveryMI can be surrounded by tadpoles which contributewith a universal multiplicative factor which can be absorbedin a redefinition of the instanton densityn. Using this renor-malized density2 nRNc, the pairing of graphs is now perfectand the statement “everyMI occurs only once” becomestrue.

In the presence of dynamical quarks this renormalizeddensity has to be identified with the gluon condensate insteadof the “bare” density since the same tadpoles contribute tothe gluon condensate too. A typical subgraph contributingto the gluon condensate is [26]

2 nR is defined as the renormalized and byNc divided density!

134

� ��MI

...� ��MI

�

��

>·

Therefore, the identification of the instanton density withthe gluon condensate is still valid, when including dynam-ical quarks. This discussion leaves untouched the questionwhether instantons indeed dominate the gluon condensate orwether there are other nonpertubative gluon configurationscontributing to〈GG〉.

Now we are able to derive a selfconsistency equation forthe quark propagator. Quark loops are no longer possible be-cause they cannot be connected to another part of the graphvia a common instanton. All graphs which can contributeto the propagator are chains of differentM ′

Is. Therefore thepropagator is

�nMI�=

⟨�< +

+ nMI� �nMJ + nMI

� nMJ�nMK + . . .

⟩I/=J/=...

� ��

The interested reader may check, that indeedall possiblefeynman graphs, which can be constructed out of (8) con-tribute exactly once to the above expression including dy-namical quark loops. The cancelation (12) and the renormal-ization of the instanton density have to be taken into account,of course.

TheM ′Is can be averaged independently

Mp ≡M (p) := i〈MI〉 =nR2µ

p2ϕ′2(p) , (13)

whereϕ′ is defined in appendix A. The propagator is nowa simple geometrical series which can be summed up andleads to

S(p) =1

p/ + i(m +M (p))(14)

There is just one thing to do: We have to solve the circulardependence

nR Mµ

S���?@@I

(14)

(11)

(13)

-

but µ is just a number, which makes the solution very sim-ple. Inserting (13) and (14) into (11) one gets the followingequation forµ

µ = m +∫d−4p

2ϕ′2(p)Mp

p2 + (m +Mp)2(p2 +m(m +Mp)) (15)

which may be solved numerically for different current massesm.

Fig. 1. Constituent quark mass M(p)

Some phenomenological results

In the chiral limit (15) reduces to

µ20 = nR

∫d−4p

p4ϕ′4(p)p2 +M2

p

= αnRρ2 +O(n2

R) (16)

µ20 is proportional tonR and thusM is proportional to

√nR

in contrast to a linear dependence onnR obtained from anaive density expansion.

In the last expression the denominator has been expandedin the density and

α = ρ−2∫d−4p p2ϕ′4(p) = 6.6 (17)

is a universal number. For the standard values ofnR andρ,one gets [11]

µ20 = 6.6nRρ

2 = (100MeV)2 ,

M (p = 0) = 7.7ρ√nR = 300MeV . (18)

The exact solution of (16) which has been obtained numer-ically by iteration, differs from the leading density value by15%:

M (0) = 345MeV . (19)

The momentum dependence of the quark mass is shown inFig. 1. The massm+M (p) may be interpreted as the mass ofa constituent quark. At high energies it tends to the currentmass, at low momentum chiral symmetry breaking occurs,and the quark gets its constituent mass M(0). Note that thisis not a pole mass but a virtual mass at zero momentumsquared.

Let us now take into account a small current massmformally of the order

√nR. The selfconsistency equation

now reads

1 =m

µ+µ2

0

µ2+O(nR) . (20)

135

Solving it for µ leads to

µ0 ≤ µ =12m +

√14m2 + µ2

0 ≤ m + µ0 (21)

For the strange quarkµ is increased by a factor of 2:

µ(ms = 150MeV) = 200MeV (22)

It is interesting thatms +M (0) remains to be 300MeV. Forzero momentum the increase of the current mass is just com-pensated by an equal decrease of the dynamical massM (0).This seems to be to small when interpreted as a constituentmass. Probably the neglection of non-zeromodes is respon-sible for the small value ofM (0).

From the propagator, one can obtain the quark conden-sate

〈/υ/υ〉 := limx→0

trCD(S(x)− S0(x))

= Nc

∫d−4ptrD(S(p)− S0(p)) . (23)

In leading order in the density one gets

i〈/υ/υ〉 =nRNc

µ= 〈Ga

µνGµνa 〉/32π2µ . (24)

This leads to the following condensates foru,d ands quarks:

i〈uu〉 = i〈dd〉 = (250MeV)3 , 〈ss〉 = 0.5〈uu〉 . (25)

For heavy quarks, there exists a similar relation

i〈/υ/υ〉 = 〈GaµνG

µνa 〉/48π2m +O(m−3) , (26)

which leads within 10% to the same value for the strangequark condensate. This nicely confirms the hypothesis thatthe strange quark can be treated as a light quark as well as aheavy quark. This hypothesis is used in heavy to light quarkmatching formulas.

LargeNc approximation

Consider now the case ofNf light quark flavorsu, d, s, . . ..The discussion of the one flavor case in the previous sectionscan be copied up to the pairing and cancelation of graphswhich contain more than oneMI (12). This is still true in themultiple flavor case but now both quark lines in (12) musthave the same flavor becauseMI always connects quarksof the same flavor. So we have the theorem: “everyMI

occurs only once for each flavor”. From this point on, thediscussion of the one flavor case breaks down because thereare now graphs contributing to the propagator containingquark loops. The simplest new contribution has the form

� ��MJ

��� ��MI

�

...� ��MI

...� ��MJ

u u u

�-

Is this contribution small in some sense? Yes it is! Quarkloops are suppressed by a factor 1/Nc. In perturbative con-text, this is extensively discussed in [27], in instanton physicsit was first used by [11]. Although 1/3 is not a very small

number, the largeNc approximation seems to be a goodapproximation in various cases.

Consider a graph and add to it a new quark loop consist-ing of N new instantons. LetS be the total number of scat-terings at all instantons occuring in the loop, which might belarger thanN due to multiple occurancies of one instanton(S ≥ N ). E.g. for the loop aboveS = 2 andN = 0.

This multiplies the graph (see Table 1) by a factor ofO(N1+N−S

c ). The following cases are possible:

S = N : All instantons are new, therefore theloop is disconnected.

S = N + 1 : There is one ‘old’ instanton, the loopis a tadpole.

S > N + 1 : The loop is suppressed by at least onefactor 1/Nc.

Disconnected graphs are not allowed and tadpoles have beenabsorbed innR. So quark loops are indeed suppressed in thelimit Nc → ∞. The same is true for gluon loops. This canbe seen by the same argument using theNc dependencesfrom Table 1. More details on the gluon propagator in theinstanton background and an effective gluon mass analogousto the quark propagator can be found in [26].

3 Four point functions

In the last section, we have derived effective Feynman ruleswithin the zero mode approximation:

� ��MI

��p =1iµ

/υI (p)/υ†I (q)

� =1

p/ + im= S0(p)

q

/υI is the zeromode of the Dirac operator in the backgroundof one instanton defined in appendix A. A quark of a givenflavor can scatter only once at an instantonI via MI . Tad-pole graphs are not allowed, they are absorbed in the renor-malized instanton densitynR which has to be used whenaveraging over the instantons. In the largeNc limit, dynam-ical quark loops are suppressed andµ can be determined by(15). In the chiral limit,

µ2 = 6.6nRρ2 +O(n2

R) . (27)

In the following sections, we will calculate the correlatorsof four quark fields within the same approximations as usedfor the quark propagator. It is interesting, that not all quarkloops are suppressed in leading order in 1/Nc. Summationof all graphs leads to Bethe-Salpeter equations, which canbe solved.

Derivation of Bethe Salpeter equations

In this section selfconsistency equations for the 4 point func-tions will be derived in the case of two massless quark fla-vors in the limitNc → ∞ within the zero mode approx-imation. The only significant effect of including the smallup and down current quark masses would be the generation

136

Table 1. Dependence of various quantities on the parameters of the instanton liquid modelnR, ρ,Nc

Parameters nR, ρ,Nc

Instanton density nR ≈ 13(200MeV)4

Instanton radius ρ ≈ (600MeV)−1

Number of colors Nc = 3Gluon condensate 〈GG〉 ∼ nRNc

Quark condensate 〈/υ/υ〉 ∼ Ncρ−1√nRConstituent mass M (p) ∼ ρ

√nR

Meson correlator 〈/υΓ /υ(x)/υΓ /υ(0)〉conn. ∼ Nc

Quark loop j� ∼ Nc

Instanton scattering jI ∼ N−1c ρ/

√nR

Gluon loop ∼ N2c − 1

Instanton scattering jI ∼ (N2c − 1)−1ρ/

√nR

Instanton occurance ∼ NcnR

of the pion mass, which has already been discussed in [11],and an increase of theσ mass. The 4 point functions aredefined by

δ−(p− s + r − q)ΠΓΓ ′ (p, s, q, r) =< <

>>Γ

p

s

q

rΓ ′

= −∫dxdydzdw ei(py−qz+rw−sx)

× 〈0|T /υ(x)Γ /υ(y)/υ(z)Γ ′/υ(w)|0〉 . (28)

The /υ fields areu or d quark fields arbitrarily mixed. Theextension to an arbitrary number of flavors can be found in[22]. Without restriction to generality, we take the correlatorto be a color singlet. These 4 point functions can be used tostudy meson correlators (see Chap. 4) or form factors, e.g.the spin of the proton [28].

The most general graph for the quark propagator is asequence of different instantonsMI , according to the rulesstated above. Similarly the most general graph for the tripletcorrelator〈(ud)(du)〉 consists of two quark propagators eachcontaining every instanton only once. However, theu anddpropagator may contain common instantons, e.g.

s - mMI-

mMI��p q

r

... mMN-

mMJ�

mMP-

mMK�

mMK-

mML�

mMM-

mMM�

...

u u

dd

...

It is always assumed that the left and right hand sides of thegraphs form color singlets. Non-common instantons can beaveraged like in the propagator case to yield graphs of theform

� ��MI

� ��MI

...

<

>

<

> � ��MN

� ��MN

...

<

>� ��MK

� ��MK

...

<

>

u

d

u

d

(29)

where the thick line represents the full propagator

�nMI�+=

⟨�<

+ nMI� �nMJ

� + nMI� nMJ

�nMK +. . .

⟩I/=J/=...

��

It can be shown that non planar diagrams like

� ��MN

� ��MI<

>

<

> � ��MI

� ��MN <

>

u

d

u

d

· · ···

are suppressed by 1/Nc, where again the left and right handsides of the graphs have to form color singlets. So onlythe ladder diagrams shown in (29) contribute to the tripletcorrelator.

One might think that the mixed correlator〈(uu)(dd)〉 iszero because the graphs necessarily include quark loops, orthat only the two loop graphs contribute — but this is notthe case! Let us first state the result and then discuss it.Graphs contributing to the mixed correlator are chains ofquark bubbles

� ��MI � ��MI· · ·

<

>

<

>� ��MJ � ��MJ· · ·

<

>� ��MK � ��MK· · ·

<

>

u d u d

u d u d (30)

Application of theNc counting rules shows that this chainis of order 1/Nc. Taking the color trace at the left and righthand side of the chain we see that the mixed correlator is oforderNc. Using (12), it is clear that the triplet correlator isof the same order.

What is wrong with the derivation of quark loop sup-pression in the last chapter? The main assumption was thatevery graph containing a loop can be constructed from agraph not possessing this loop by simply adding the loop.Eliminating a loop from the bubble chain (30) yields a dis-connected graph, but we only consider connected 4 pointfunctions. So the quark loop chain cannot be constructed ina way needed to prove quark loop suppression.

In the case of meson correlators, one can take anotherpoint of view. The disconnected two loop contribution is oforderN2

c but (except for the scalar case) the contributionis zero. So the bubble chain is a subleading graph of orderNc and nothing has been said about the form of subleadinggraphs.

Nevertheless, all connected graphs can be obtained start-ing from (30) by adding further instantons and bubbles —but nowNc counting rules tell us that every attempt resultsin a 1/Nc suppression. Therefore the bubble chain is themost general leading order graph.

137

Nothing has to be changed for the correlator〈(dd)(dd)〉except that chain (30) must start withd.

To calculate the connected 4 point functions one mustnow average and sum up the chains. Alternatively this can berepresented in recursive from usually called Bethe Salpeterequations:

>

<

G

u

d>

<u

d

=⟨iMI

iMI...

<

>

<

>

u

d

u

d

+ iMI

iMI...

<

>

<

>

u

d

u

d

G>

<u

d

⟩I

>

<

H

u

u>

<d

d

=⟨ iMIiMI· · ·

<

>

u

u

<

>

d

d

+ iMIiMI· · ·

<

>

u

u

<

>

d

d

K>

<d

d

⟩I

(31)

>

<

K

d

d>

<d

d

= iMIiMI· · ·

<

>

d

d

<

>

u

u

H>

<d

d

⟩I

⟨

Solution of the Bethe Salpeter equations

Before solving the BS equations we have to construct thekernel. The l.h.s. of the kernel always forms a color singletbecause of the restriction to color singlet correlators.

Contracting the color and Dirac indices on the l.h.s. andusing the formulas of appendix A, one gets

� ��MI

� ��MI

...

′

′

′

′

⟨Γ

p q

rs

⟩I

=1

(iµ)2〈r//υI (r)/υ†I (s)s/Γp//υI (p)/υ†I (q)q/〉I =

= −nRµ2

pϕ′(p)qϕ′(q)rϕ′(r)sϕ′(s)δ−(p− s + r − q)

×⟨

trD

(Γ

1± γ5

2

)1± γ5

2

⟩±

. (32)

The kernel can now be determined to be

′

′1

′

′p q

rs

:=

� ��MI

� ��MI

...

′

′

′

′

⟨ p q

rs

⟩I

= −⟨� ��MI � ��MI· · ·

′

′

′

′

⟩I

p q

rs

= − 1nRNc

√MpMqMrMsδ

−(p− s + r − q)

×(δipisδiriq + γip5 isγir5 iq

)δαpαsδ

αrαq . (33)

The result is just proportional to the non-local version ofthe ’t Hooft vertex between color singlet states.

The solutions of the BS equations have a very similarstructure:

′

′A

′

′p q

rs

:= − 1nRNc

√MpMqMrMsδ

−(p− s + r − q)

× (A0(t)δipisδiriq

+A5(t)γip5 isγir5 iq

)δαpαsδ

αrαq . (34)

A0 andA5 are scalar functions depending only ont = p−s =q− r. The proof is simple: The kernel has the structure (34)with A0 = A5 = 1. The product of two vertices yields thesame structure:

′

′A

p q

rs=B

′

′<

> ′

′AFB

′

′p q

rs=

= − 1nRNc

√MpMqMrMsδ

−(p− s + r − q) (35)

×(A0F0B0(t)δipisδiriq

+A5F5B5(t)γip5 isγir5 iq

)δαpαsδ

αrαq ,

F0(t) = −∫

(dpds)1nR

MpMstrD(S(p)S(s)) ,

F5(t) = −∫

(dpds)1nR

MpMstrD(S(p)γ5S(s)γ5) , (36)∫(dpds) =

∫d−pd−s δ−(p− s− t) , t = p− s = q − r .

In other words, the vertices of structure (34) build a closedalgebra. The reason for this simple result is that the kernel isa simple product function up to the momentum conservingδ.

Using (33), (34) and (35), the BS equations (31) re-duce to primitive algebraic equations forG0/5(t), H0/5(t)andK0/5(t):

G0/5(t) = 1 +F0/5(t)G0/5(t) ,

H0/5(t) = −1− F0/5(t)K0/5(t) ,

K0/5(t) = −F0/5(t)H0/5(t) ,

with the solution

G =1

1− F, H = − 1

1− F 2, K =

F

1− F 2(37)

where we have suppressed the index 0/5 and the argumentt.

It is now easy to construct the physically more interestingtriplet and singlet correlators, because of isospin symmetrySU (2)f , mesons form triplets and singlets. Replacing/υ/υ in(28) by the triplet and singlet combinations (borrowing thenotation from the pseudoscalar correlator)

π0 =1√2

(uu− dd), π+ = ud,

π− = du, η =1√2

(uu + dd), (38)

one gets

138

′′Ct

′′

= 12

(π0 π0

′′K ′′u u

uu− ′′H ′′u d

du− ′′H ′′d u

ud+ ′′K ′′d d

dd

)(39)

ThereforeCt = K−H = 11−F . This coincides withG = 1

1−Ffor the charged triplet correlator〈(π±)(π±)〉 as it should be.In the singlet case, we getCs = K +H = − 1

1+F .When adding propagators in (34) to the external legs,

the final result for the connected 4 point function is

ΠconnΓΓ ′ (p, s, q, r) = −Nc

nR

√MpMqMrMs

×[C0(t)trD(S(s)ΓS(p))trD(S(q)ΓS(r))

+C5(t)trD(S(s)ΓS(p)γ5)trD(S(q)ΓS(r)γ5)]

Cs/t0/5(t) = − 1

F0/5(t)± 1+ for the singlet correlator− for the triplet correlator

(40)

F0/5(t) are defined in (36). The correlators of a singlet witha triplet current are zero as expected.

The following graphs may contribute to the disconnectedpart:

= −N2c trD(ΓS(p))trD

×(Γ ′S(q))δ−(p− s)δ−(q − r)Γ Γ ′

p q

rs

= NctrD(ΓS(p)Γ ′S(s))×δ−(p− q)δ−(r − s)Γ Γ ′

p q

rs >

∧ ∨

<

(41)

depending on the flavor structure of the correlator. Note thatthe second two loop term is of the orderN2

c . However, asdiscussed above, in most applications it drops out or yieldsan uninteresting constant or only the connected part is con-sidered anyway.

For the triplet and singlet case, one gets

δ−(p− s + q − r)ΠdiscΓΓ ′ (psqr) = NctrD(ΓS(p)Γ ′S(s))

×δ−(p− q)δ−(r − s) +

{0 for triplet

2 · (41) for singlet (42)

The 4 point functions obtained in this section will be dis-cussed in the following chapter.

4 Correlators of light mesons

The meson correlators, often called polarisation functions,contain information about the meson spectrum. Poles atp2 =m2 show the existance of mesons with massm. With the 4point functions calculated in the last section, we also possessthe meson correlators analytically, besides integration.

Due to chiral symmetry breaking there are masslessGoldstone bosons in the pseudoscalar triplet channel, butnone in the singlet channel, because instantons explicitlybreak theU (1)A symmetry.

In order to determine the meson masses, we have to makean Ansatz for the meson spectrum. Comparison of the theo-retical curve with this Ansatz allows a determination of the

masses of the lightest mesons in various channels. Calcula-tion of the integrals and the fit are performed numerically.

Analytical expressions and results

In the last chapter we have calculated various quark 4 pointfunctions (28). The meson correlators or polarisation func-tions are just local versions of these vertices and can beobtained by simply settingx = y andz = w. In momentumspace the meson correlators have the form

ΠΓΓ ′ (t) = Πdisc(t) +Πconn(t) =� ��>< <

>∼′∼′Γ Γ ′

(43)

=∫

(dpds)∫

(dqdr)Π(p, s, q, r)

= −∫dx eitx〈0|T jΓ (x)jΓ ′ (0)|0〉 ,

js/tΓ (x) =

1√2

(uΓu(x)± dΓd(x)) ,

js/tΓ ′ (0) =

1√2

(uΓ ′u(0)± dΓ ′d(0)) ,

∫(dpds) =

∫d−pd−s δ−(p− s− t) ,∫

(dqdr) =∫d−qd−r δ−(q − r − t) ,

t = p− s = q − r .

From the explicit expressions of the 4 point functions(40) and (42) obtained in the last chapter, one can get, up tointegration, analytical expressions for the meson correlators:

ΠdiscΓΓ ′ (t) = Nc

∫(dpds) trD(ΓS(p)Γ ′S(s)) , (44)

ΠconnΓΓ ′ (t) = −Nc(C0(t)Γ 0

Γ (t)Γ 0Γ ′ (t) +C5(t)Γ 5

Γ (t)Γ 5Γ ′ (t)) ,

C0/5(t) = − 1F0/5(t)± 1

+ for singlet correlator− for triplet correlator

,

Γ 0Γ (t) =

1√nR

∫(dpds)

√MpMstrD(S(p)ΓS(s)) ,

Γ 5Γ (t) =

1√nR

∫(dpds)

√MpMstrD(S(p)ΓS(s)γ5) .

ΓΓ (t) may be interpreted as effective vertices.F0/5(t) aredefined in (36).S(p) = (p/ + iMp)−1 is the full propagatorof a ’constituent’ quark of dynamical massMp, defined in(13).

Performing the Dirac traces leads to the following ex-pressions forF andΓ :

F0/5(t) = − 4nR

∫(dpds)

MpMs(±(ps)−MpMs)(p2 +M2

p)(s2 +M2s )

,

Γ0/51/5 (t) =

4√nR

∫(dpds)

√MpMs(±(ps)−MpMs)

(p2 +M2p)(s2 +M2

s ), (45)

Γ 5µ5(t) =

4i√nR

∫(dpds)

√MpMs(Mpsµ −Mspµ)

(p2 +M2p)(s2 +M2

s ).

139

Table 2. Analogy between chiral symmetry breaking in QCD and the BCS theory of superconductors [30]

Instantons in QCD BCS-theoryPhenomenon Chiral symm.-breaking SuperconductivityBoundstates. . . qq-pairs=π-mesons e−e−-pairs=Cooper-pairsbounded by attractive. . . instanton-induced phonon-electron-

’t Hooft interaction interaction

� ��I

� ��I

...

<

>

<

>

...

...

< <

> >

in the . . . pseudo-scalar channel scalar channelOrderparameter Quark condensate Density of superconducting

〈/υ/υ〉 electrons (Cooper-pairs)〈ρ〉Spontaneous breaking of chiral symmetry gauge symmetryleads to. . . Goldstone bosons = Excitons =

massless pions massless density fluct.

Vacuum polarisationdiagrams make the �

��I· · ·

<

>

� ��J · · ·

"!

# · · · · · ·

<

>massless excitationsmassive in the flavor singlet case for the Coulomb interaction

=⇒ massiveη′ =⇒ massive plasma osc.

All other verticesΓ are zero. Consider the one instantonvertex (33) (the kernel). It contributes only to the scalarand pseudoscalar correlator. From this observation, one mayhave predicted that the connected part of all other channelsis small because a contribution has to be a multi-instantoneffect. Indeed, they are all zero as seen above except for theaxial correlator. Due to an extra factorM ∼ √

nR in thenumerator ofΓ 5

µ5 the connected part of the axial correlatoris suppressed byO(nR). Therefore it is small as expectedand will be neglected in the following.

Using the selfconsistency equation (16), one can see thatF5(t = 0) = 1 which leads to a pole att = 0 in the pseu-doscalar triplet correlator due to theF5(t) − 1 denominatorin (44). This is the massless Goldstone pion one expects inthe chiral limit. A more extensive discussion can be foundin [11]. On the other side, in the singlet correlator the de-nominator isF5 + 1 /= 0 and there is no Goldstone bosonin this case. Thus, the (two flavor)η′ meson is massive !Unfortunately we cannot make any reliable prediction of theη′ mass along the line discussed in the next section becausethe kernel is very repulsive in this channel and no bound-state is formed. There have to be other attractive forces, e.g.confinement forces, to built anη′ boundstate. Similar thingshappen in the scalar triplet channel (compare Fig. 3 and 4).But the most important point is that there is no masslesspseudoscalar singlet meson which is an important step to-wards discussing theU (1)A-problem. By combining instan-ton results with other methods, it is nevertheless possible todetermine the mass of theη′ [29].

The chiral symmetry breaking induced by instantons inQCD has a close analogon in the BCS theory of supercon-ductivity. The strong attractive ’t Hooft interaction in QCDin the pseudoscalar triplet channel leads to massless gold-stone bosons, the pions. In the BCS theory there is a strongattractive electron-phonon interaction in the scalar channelleading to the formation of Cooper pairs. The excitons are

the massless modes. Even the absence of a flavor singletgoldstone boson posses an analogon: inclusion of dynamicalelectron loops can lead to massive plasma oscillations. Fur-ther parallels can be read from Table 2. The developmentof the BCS theory along these lines can be found in [30].Further it is interesting to see that in leading order in theinstanton densityF0(t) = −F5(t), which leads to a mass-less pole in the scalar singlet correlator. Numerically theσ-meson indeed turns out to be very light. The experimentalsituation is rather unclear [31]

Spectral representation

To extract phenomenological information from the mesoncorrelators, we make use of the spectral representation

Π(p) =∫dx eipx〈0|T jΓ (x)jΓ ′ (0)|0〉

=∫ ∞

0dσ2D(σ, x)ρ(σ2) (46)

where

ρ(p2) = (2π)3∑n

δ(p− qn)〈0|jΓ (0)|n〉〈n|j′Γ (0)|0〉 (47)

is the spectral density and

D(m,x) =∫d−4p

e−ipx

p2 +m2=

14π2x2

(mx)K1(mx) (48)

is the free propagator of massm in coordinate representa-tion. We have chosen the coordinate representation ofΠ toallow a comparison with lattice calculations and with nu-merical studies of the instanton liquid [15].

The spectrum consists of mesonic resonances and thecontinuum contributions. If one is only interested in theproperties of the first resonance, one might approximate the

140

Table 3. Mesonic correlators

Correlator Γ = Γ ′ I=1 I=0Pseudoscalar Π5 = 〈j5j5〉 iγ5 π η′Scalar Π1 = 〈j1j1〉 11D δ σVector Πµµ = 〈jµjµ〉 γµ ρ ωAxialvector Π5

µµ = 〈j5µj

5µ〉 γµγ5 a1 f1

rest of the spectrum by the perturbatively calculated contin-uum.

One might think that the disconnected part only con-tributes to the continuum and the connected part will yieldthe boundstates. But this is not the case. On one hand, BetheSalpeter equations have bound as well as continuum solu-tions. On the other hand consider a theory with weak at-traction between particles of massm. It is clear that there isonly a cut above 2m and no boundstate pole in the free loop.However in the exact polarization function only a small por-tion of the continuum will be used to form a pole just belowthe threshold because the attraction is only weak. The Eu-clidian correlator will hardly be changed. Therefore, assum-ing weak attraction, we can already estimate the boundstatemass from the disconnected part. Of course in this exam-ple, we need not calculate anything because we know thatthe boundstate mass is approximately 2m with errors of theorder of the strength of the interaction.

Assuming that all other forces neglected in QCD so far,especially perturbative corrections, are small and attractivein the vector and axialvector channel, we can obtain bound-state masses although in these channels up to our approxi-mation there is no connected part. But things are less trivialthan in the example above because the quarks do not possesa definite mass and we have to inspect the correlator to ex-tract the meson masses.

Let us start with the scalar and pseudoscalar correlator.The lowest resonance of massm∗ is coupled to the currentwith strength

λ∗ = 〈0|j1/5(0)|p〉 . (49)

The rest of the spectrum is approximated by the continuumstarting at the thresholdE∗. * meansπ, η, δ or σ (see Ta-ble 3). E∗ is typically of the order 1.5 GeV and thereforethe continuum can be calculated perturbatively. The spec-trum thus has the form

ρ1/5(s) = λ2∗δ(s−m2

∗) +3s

8π2Θ(s− E2

∗) . (50)

Insertingρ into (46), one gets

Πfit1/5(x) = λ2

∗D(m∗, x) +E(E∗, x) , (51)

E(E∗, x) =3

π4x6

(E∗x)3

16(2K3(E∗x) + (E∗x)K2(E∗x))

x→0−→ 3π4x6

. (52)

In the next section,m∗, λ∗ and E∗ are obtained by fit-ting the phenomenological ansatzΠfit(x) to the theoreticalcurveΠsum(x) in the Euclidian region where the theoreticalcalculation is reliable.

Consider now the vector and axial vector correlator. Thevector current is conserved, thus the correlator is transverse

Fig. 2. Pseudoscalar triplet correlator normalized to the free massless quarkcorrelator. The pion coupling constantλπ and the continuum thresholdEπare fitted in order to match the spectral ansatz with the theoreticalsumofthe free and theconnected part

and only the vector meson can contribute. In the chiral limitthe same holds true for the axial current. In the singlet chan-nel, one has to be careful because there are two currents. Aconserved one and a gauge invariant one which contains ananomaly. Up to now we have only calculated the correlatorof the conserved current. Nevertheless to leading order in theinstanton density, the two correlators coincide and should beboth conserved.

For conserved vector and axial currents the spectral func-tion is transverse:

ρ(5)µν(p2) = (−δµν +

pµpνp2

)ρ(5)T (p2) . (53)

The coupling of the vector and axial meson to the current isgiven by

iλ∗εµ = 〈0|j(5)µ (0)|p〉 (54)

whereεµ is the meson polarization. The spectral and polar-ization functions have the form

−ρ(5)µµ(s) = 3λ2

∗δ(s−m2∗) +

3s4π2

Θ(s− E2∗) , (55)

−Πfit(5)µµ (x) = 3λ2

∗D(m∗, x) + 2E(E∗, x) .

Here * meansρ, ω, a1 or f1 (see Table 3).

Plot and fit of meson correlators

The meson correlators are shown in Figs. 2–7. The numericalevaluation of the integrals is discussed in the appendices of[21, 22]. The correlators are normalized to the free correlator

Π01/5(x) =

3π4x6

, Π0(5)µµ = − 6

π4x6. (56)

141

Table 4. Meson massm∗, coupling constantλ∗ and continuum thresholdE∗ obtained within the instanton liquid model in this work (1/Nc approximation),from numerical simulation and from experiment

Meson IG(JPC ) m∗[MeV]√λ∗[MeV] E∗[MeV] source

0 508±1 1276±33 1/Nc

π 1−(0−+) 142±14 510±20 1360±100 simulation138 480 — experiment/= 0 ? ? 1/Nc

η′ 0+(0−+) /= 0 ? ? simulation960 ? — experiment/= 0 ? ? 1/Nc

δ 1−(0++) /= 0 ? ? simulation970 ? — experiment433±3 506±3 1446±20 1/Nc

σ 0+(0++) 543 500 1160 simulation? ? — experiment930±5 408±4 1455±33 1/Nc

ρ 1+(1−−) 950±100 390±20 1500±100 simulation780 409±5 — experiment930±5 408±4 1455±33 1/Nc

ω 0−(1−−) ? ? ? simulation780 390±5 — experiment1350±200 370±30 1050±80 1/Nc

a1 1−(1++) 1132±50 305±20 1100±50 simulation1260 400 — experiment1350±200 370±30 1050±80 1/Nc

f1 0+(1++) 1210±50 293±20 1200±50 simulation1285 ? — experiment

’

Fig. 3. Pseudoscalar singlet correlator normalized to the free massless quarkcorrelator. There is a strong repulsion in this channel and no boundstate isformed. The theoretical curve is compared to a curve obtained from a purecontinuum spectrum aboveEη′

The diagrams therefore show the deviation from the pertur-bative behaviour. The meson parameters obtained by fittingthe parameter ansatz to the theoretical curve are summa-rized in Table 4. Shuryak and Verbaarshot [15] have ob-tained the same parameters from a numerical investigationof the instanton liquid model. Their values are also shownin Table 4. The parameters of the vector channel coincideextremely well with ours. For theπ and σ meson there is

Fig. 4. Scalar triplet correlator normalized to the free massless quark cor-relator. There is a strong repulsion in this channel and no boundstate isformed. The theoretical curve is compared to a curve obtained from a purecontinuum spectrum aboveEδ

a large discrepancy in the masses but this is not surprising:We are working in the chiral limit thus the pion mass has tobe zero. A similar argument holds for theσ meson as dis-cussed above. The couplings fit very well. The discrepancyin the axial channels can have various origins, which areunder investigation. Alternatively one may directly comparethe graphs. They coincide very well even in cases wherea spectral fit does not work like in theδ and η′ channel.

142

Fig. 5. Scalar singlet correlator normalized to the free massless quark corre-lator. Theσ massmσ and couplingλσ and the thresholdEσ are obtainedfrom a spectral fit

Fig. 6. Axial vector correlator normalized to the free massless quark cor-relator. The triplet and singlet correlator are equal because the connectedpart has been neglected. Thea1 and f1 mass, coupling and threshold areobtained from a spectral fit

The conclusion is that the terms neglected in our analyticaltreatment but included in the numerical study [15] are smalland usually give an correction less than 10%. These are con-tributions from nonzero modes and higher order correctionsin 1/Nc. This is again an example for the surprisingly highaccuracy of the 1/Nc approximation. In the case of strangequarks the nonzero mode contributions will become moreimportant.

Fig. 7. Vector correlator normalized to the free massless quark correlator.The triplet and singlet correlator are equal because the connected part iszero. Theρ andω mass, coupling and threshold are obtained from a spectralfit

Finally, one should compare the numbers with exper-iment. As far as known, these numbers are also listed inTable 4. A general discussion of the meson correlators andcomparison with experimental results can be found in [15].

5 Conclusions

Various QCD correlators have been calculated in the instan-ton liquid model in zeromode approximation and 1/Nc ap-proximation. The 1/Nc approximation should be seen as asubstitute for the density expansion which fails in the pres-ence of light quarks. We have extended the work [11] byincluding dynamical quark loops. In contrast to the original“perturbative” 1/Nc expansion [27], not all quark loops aresuppressed. In the case of the propagator, all quark loopscan be absorbed in a renormalized instanton density. An in-teresting observation is that this renormalized density can beidentified with the gluon condensate. In this sense the per-turbative theorem, that all quark loops are suppressed in the1/Nc approximation, remains true in the instanton case forthe quark propagator. In the flavor singlet meson correlators,a chain of quark loops survives the 1/Nc limit causing theabsence of a Goldstone boson in the pseudoscalar singletchannel. The analytical expressions for the meson correla-tors have been evaluated numerically and meson masses andcouplings have been obtained from a spectral fit. Exceptfor the pseudoscalar singlet and the scalar triplet channel,the spectral ansatz matches very well the theoretical curve.The numbers obtained in this way are consistent with thoseobtained by numerical studies of the instanton liquid [15]within 10%. Comparison with experiment is also quite sat-isfactory.

143

Acknowledgements.I want to thank theDeutsche Forschungs-Gemeinschaftand theGerman Israeli Foundationfor supporting this work.

Appendix A: zeromode formulas

The covariant derivativeD/ in the background of an instantonhas one zeromode

(i∂/−A/I )/υI = 0 (57)

The solution for an instanton of topological chargeQI =±1 in singular gauge in coordinate and momentum space isgiven in terms of modified Bessel functionsI0/1 andK0/1

/υI (x + zI ) =√

2ϕ(x)x/χ±I , ϕ(x) =ρ

π|x|(x2 + ρ2)3/2

/υI (p) =

√2ϕ′(p)p/|p| eipzIχ±I , ϕ(p) =

∫d4x eipxϕ(x)

ϕ′(p) = πρ2 d

dz[I0(z)K0(z)− I1(z)K1(z)]z=|p|ρ/2

=

{− 2πρ

|p| : pρ� 1− 12πp4ρ2 : pρ� 1

(58)

ρ2∫d4xϕ2(x) =

∫d4xϕ2(x)x2 =

∫d−4pϕ′2(p) =

12

χ±I ∼ γµ1∓ γ5

2τ∓µ

∣∣∣∣some color & Dirac column

,

τ∓µ = (τ ,±i)As usual only the projectors can be given in a covariantway3:

χ±I χ±J =

116

(γµγν1± γ5

2)(UIτ

∓µ τ

±ν U

†J ) (59)

χ±I χ∓J = ∓ i

4(γµ

1∓ γ5

2)(UIτ

∓µ U

†J )

UI/J ∈ SU (Nc) is the orientation matrix of the instantonI/J . Although multi-instanton effects are studied in thiswork, formulas like (59) concerning the overlap of differ-ent instantons are not needed. (59) is only needed forI = Jin the following special cases:

NC〈χ±I χ±I 〉I = trCχ±I χ

±I =

12

(1± γ5

2) (60)

χχ = trCDχχ = 1 (61)

3 In Euclidian space the bar operation is the same as the hermitian con-jugate: noγ0 is introduced, thus ¯χ = χ†

Taking space, color and Dirac part together we get

/υ†I (p)/υI (p) = 2ϕ′2(p) =⇒∫d−4p /υ†I (p)/υI (p) = 1 (62)

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