RMP_v065_p0001

7/24/2019 RMP_v065_p0001

1/46

Correlation functions in the QCD vacuumEdward V. S huryak

Department of Physics State U niversity of New York Stony Brook New York 11794

Correlation functions are one of the key tools used to study the structure of the QCD vacuu m. They areconstructed out of the fundamental fields and can be calculated using quantum-field-theory methods, suchas lattice gauge theory. One can obtain many of these functions using the rich phenomenology of hadronphysics. They are also the object of study in various quark mod els of hadr onic str uctu re. This review be

gins with available phenomenological information about the correlation functions, with their most important properties emphasized. These are then compared with predictions of various theoretical approaches,including lattice numerical simulations, the operator product expansion, and the interacting instanton approximation.

CONTENTS I. INTRODUCTION

I. Introduction 1A. Preface 1B. Why the correlation functions? 2C. Different types of correla tion functions 3D . General relations and inequalities 5

II . Phenomenology of Mesonic Correlation Functions 6A. Vector currents and correlators 6

B. Vector 1= 1 (or p ) channel 7C. o an d

7/24/2019 RMP_v065_p0001

2/46

2 Edward V. Shuryak: Correlation functions in the QCD vacuum

theo ret ic al predi ct ion s . As we shal l see, i t i s qui tes t ra igh t fo rward to " t rans la te back" the main resu l t s o fthe OPE and in s tan ton f rameworks to the coo rd ina terep resen ta t ion o f the co r re la to rs .

Unfo r tuna te ly, mos t l a t t i ce stud ies of co r re la t ion funct ions use ei ther sources in the form of three-dimensional" w a l l s " o r even more compl ica ted non loca l sou rces .However, the po in t - to -po in t co r re la to rs a re the bas ic obj e c t s ; t hey ca r ry more in fo rmat ion and can be measu redby s imp le modificat ion of the exis ting techn ique s . A pa rtf rom the qua l i t a t ive ly new sho r t - range phenomena to belea rned , one may ob ta in more accu ra te and re l i ab le compar i sons be tween l a t t i ce and empi r i ca l co r re la to rs . Th i scon t ras t s wi th the t rad i t iona l t echn ique o f on ly s tudy ingthe asympto t i c l a rge-d i s t ance behav io r o f the co r re la to rsrelated to the masses of the l ightes t hadrons .

I t is hoped that some part of th is review may be of interes t to experimental is ts working in d ifferent brancheso f s t rong - in te rac t ion ph ys ics . We com me n t o f ten on themutual consis tency of various sets of data , on most desirab le new exper imen ts , on the main sou rce o f exper imenta l uncer ta in t i es , e t c . A b r ie f summary o f the exper imental s ide of the problem can be found in Sec. VLB.

Th e pape r is organ ized as fo l lows. In Sec. II we d iscuss the ava i l ab le phenomeno log ica l in fo rmat ion abou tmesonic correlat ion funct ions , out l in ing a set of majorfacts to be explain ed by the theo ry. Sect ion III is devo ted to theo re t i ca l p red ic t ions , wh ich we cons ider on lybriefly on a matter-of-fact level , concentrat ing on the resul ts and th eir relat ion to experi me nt . In Sec. IV we discuss some o ther co r re la t ion func t ions , inc lud ing l igh t -and -heavy mesons and baryons , as wel l a s o rd inarybaryons , po in t ing ou t s evera l o ther impor tan t observat ions re la ted to quark p rop er t i es and in te rac t ions . Therea l so we cons ider exper imen ta l in fo rmat ion and theo re t ical ideas toge ther . Sect ion V is devo ted to correl at ionfunc t ions a t nonze ro t em pera tu re and /o r dens i ty. Thehad rons a re expec ted to "mel t " a t some c r i t i ca l t emperature in to free quarks and g luons, and one can s tudy th isphe nom enon us ing the co r re la t ion func t ions . Ou r conclusions and suggest ions are summarized in Sec. VI.

The reader who would l ike to see from the s tart whichopera to rs and co r re la t ion func t ions a re cons idered in th i sreview is invi ted to look at Table I .

B. Why the correlation functions?

Here we def ine what we mean by the co r re la t ion funct i ons , d i scuss the i r asympto t i c behav io r a t smal l andlarge d is tances , and then t ry to explain why they p laysuch an impor tan t ro le in s tud ies o f the QCD vacuum.

Below we dea l wi th two types of opera to rs : meson icones of the type

and baryon ic ones ,

0^x) = Mjfk Jk d.2)

Here the co lo r ind ices i>j,k are expl ici t ly shown; we shal lom it them below . Ot her indices like sp in and flavor areno t here specified , bu t shal l be later. As al l co lor indicesare properly contracted and al l quark f ields are taken atthe same po in t JC these operators are manifes t ly gauge invar ian t .

The correlat ion funct ion is defined as the vacuum expec ta t ion va lue (VEV) o f the p roduc t1 o f two opera to rstaken a t two po in t s x a n d y:

K(x-y) = (0\O(x)O(y)\0) . (1.3)

The f i rs t comm en t i s tha t th e vacuum i s hom ogene ous ;so one of the points can be the orig in of the coordinatesystem, say, y = 0 . A second com me n t is tha t we as sumeth roughou t th i s paper tha t the separa t ion be tween thep o i n t s (x-y) i s spacel ike. Th e reason is we prefer to dealwi th v i r tua l p ropaga t ion o f quarks o r had rons f rom onepoint to another, to have s imple decaying funct ions ins tead of funct ions having a complicated osci l latory behav io r.

There is an o ld quest ion that one inevi tably asks at th ispoin t : why is the re a corr elat io n between fields outs id ethe light cone? I t was essent ial ly answ ered by Fey nm an:par t i c l es can p ropaga te a long any pa th go ing f rom x t o y.Depend ing on the re fe rence f rame, an observer can consider the path t6 be a sequence of spontaneous pair-c rea t ion and ann ih i l a t ion even t s . Th i s co r re la t ion doesnot contradict causal i ty, because one cannot use i t fors ignal t ransfer. See textb ook s on qu an tu m field theor yfor more.

One can look at the pairs of points of the correlator intwo ways. Ei th er they are two poin ts in space, take n atthe same ins tant of t ime and separated by the spat ial d ist a n c e x, o r they are two events separated by some in terva l in imag inary o r Euc l idean t ime: ix0iy0 = r. Belowwe use bo th in te rp re ta t ions , depend ing on wh ich i s moreconven ien t a t the mo men t . We hope the reader wil l no tbe confused by our us ing the symbols x a n d r i n te rchangeab ly.

Let us now discuss the behavior of the correlat ionfunct ions at small and large d is tances . At small x( r e me mb e r, y =0) t he asympto t i c f reedom o f QC D te l l sus that quarks and g luons propagate freely, up to smalland ca lcu lab le rad ia t ive co r rec t ions . There fo re K(x) inthe mesonic (baryonic) case is essent ial ly the square (orcube) o f the f ree -quark p ropaga to r, S(x)= (q(x)q(O)),d epend ing on whether a meson ic o r baryon ic co r re la to ri s under cons idera t ion . F r om d imens iona l a rgum en ts ,t h e q u a r k p r o p a g a t o r S(x) is seen to scale as S(x)~x ~ 3 ,

Actually, it is the time-ordered product that is usually denoted by T: it is always implied below. We do not go into detailshere, but only mention that such ^-ordering just corresponds tousing the standard path integrals and Feynman propagators forparticles propagating from x to y.

Rev. Mod. Phys., Vol. 65, No. 1 , January 1993

7/24/2019 RMP_v065_p0001

3/46

Edward V. Shuryak: Correlation functions in the QCD vacuum 3

igno r ing smal l quark masses .2 So for the meson ic orbaryon ic co r re la to rs , the re fo l lows a smal l x l imitK(x)~ x ~6 (o r ~x ~9) f rom the s imp le d imens iona l a rgum e n t s3 a lone.

I f quark s a re a l lowed to p ro paga te to la rger d i s t ances ,they s tart in teract ing more s t rongly with vacuum fields .If correct ions are not too large, one can take these effects

in to accoun t u s ing the opera to r p roduc t expans ion (OPE)form alism (see Sec. III .B ). A t in te rm edia te d is tances ,descr ip t ion o f the co r re la t ion func t ions becomes , in genera l , ve ry compl ica ted , and one may on ly eva lua te themby us ing e i ther l a t t i ce numer ica l s imu la t ions o r some vacuum models (e .g . , the ins tanton model described in Sec.I I I .C) .

At l a rge d i s t ances one can aga in unders tand the behavior of the correlat ion funct ions , us ing now completelydifferent k ind s of arg um en ts . Ins te ad of th in kin g inter ms of fund am enta l f ields , one may jus t use the for malrelat ion for the t ime evolut ion of an operatorO(t) = eiH TO(0)e~iH t, where i f i s a Ham i l ton ian , andthen insert a complete set of physical in termediate s tatesbe tween the two opera to rs :

K(t) = J,\(0\O(0)\n)\2e~iEnt . (1.4)n

Now one can ana ly t i ca l ly con t inue the co r re la t ion funct ion in to the Euc l idean t ime r=it and get a sum over decreas ing exponen t s .4

Phys ica l ly, app l i ca t ion o f such re la t ions in QC D mean stha t one cons ider p ropaga t ion o f phys ica l exc i t a t ions o rhad rons be tween ou r two po in t s , l ead ing to the p red ict ion tha t K(x)~exp( mx) for large x, w h e r e m is themass o f the l igh tes t pa r t i c l e wi th the co r respond ing quantu m num ber s . Thi s is essent ial ly the idea of Yu kaw a, torelate the range of the nuclear forces with the p ion mass .

I t i s now easy to unders tand why the co r re la t ion funct i o n s a r e s o i mp o r t a n t i n n o n p e r t u r b a t i v e Q C D a n d h a -dro nic physics . Th e reason is tha t the same funct ion canbe consider ed on tw o different levels : (1) in term s of thefun dam enta l Q C D fields , quark a nd g luon s, or (2) inte rms o f the phys ica l in te rmed ia te s t a tes , u s ing the vas thad ron ic phenomeno logy o f masses , coup l ing cons tan t s ,form factors , e tc .

Mor eover, the re i s a th i rd ap p ro ach to the co r re la t ionfunct ion s . Th ere are useful mo dels orig in at ing from the

2

The coefficient is also easy to find by solving the Dirac equation for free massless particles: S(x)=z(iytJjdti){ l/4ir2x2).3Of course, QCD does have a dimensional parameter AQC D,

which eventually fixes the scale of all dimensional quantities.However, in perturbation theory it only comes in via the radiative corrections. Therefore, at small x, those produce corrections to our estimates above containing as(x)~ l / ln(;cA).

4A reader who does not like Euclidean time can repeat this exercise for spatially separated points and sum over virtual momenta of the intermed iate states. The result is the same, due tothe four-dimensional symmetry of the Euclidean space-time.

o r ig ina l quark model o f the ' 60s based on "cons t i tuen t"qua rks and their effect ive in terac t ion s . I t is ins tr uct iv e toexp la in what we wan t to l ea rn f rom the co r re la t ion funct ions in th is languag e: i t is the in te rqu ark effective inte rac t ion .

App l ica t ion o f these model s to had ron ic spec t ro scopyreminds one of nuclear physics in i ts early days , when

on ly l imi ted in fo rmat ion abou t the nuc lear fo rces wasknow n . Bes ides knowledg e o f the bound s ta tes , such asthe deu te ron , one had on ly qua l i t a t ive in fo rmat ion tha tthe po ten t i a l was a t t rac t ive and o f sho r t range .

Indeed , po ten t i a l - type quark model s a re success fu l lyapp l ied to the eva lua t ion o f had ron ic pa ram ete rs . Th i s i sd iscussed in detai l by Godfrey and Isgur (1985) formesons and by Capst ick and Isgur (1986) for baryons.One obtains the average characteris t ics of the few lowesthad ron ic s t a tes in each channe l , and the theo ry i s s ens it ive main ly to in te rquark in te rac t ion averaged over thes ize o f these s t a tes . The had ron ic phenomeno logydemonstrates the exis tence of f lavor- and spin-independen t con f in ing fo rces , complemen ted by some

shor t - range sp in -sp in in te rac t ion .However, we l ack de ta i l ed knowledge o f how the in te r

quark in te rac t ion depends on d i s t ance and momen ta .Re tu rn ing to the ana logy wi th nuc lear phys ics , we commen t tha t on ly the ex tens ive s tud ies o f nuc leon -nuc leonscat tering eventual ly showed al l the detai ls of nuclearfo rces wi th the i r compl ica ted sp in - i so sp in s t ruc tu re .

A l t h o u g h qq o r qq s ca t t e r ing i s exper ime n ta l ly imposs ib le to s tudy, due to confinement , a set of various mesonicco rre la t ion func t ions K(x) p lays essent ial ly the sam e ro leas tha t p layed in nuc lear phys ics by the sca t t e r ing phaseshi f t s . The se corr elat io n funct ion s are d iscussed below.Rough ly speak ing , we sha l l desc r ibe v i r tua l qq o r qqscat tering , us ing wave packets of variable s ize ins tead ofphys ica l had rons .

C. Different types of correlation functions

The co r re la t ion func t ions in Euc l idean space- t imeK{x) [o r K(T)] defined above are the objects of ou r d iscuss ion in wha t fo l lows . As the i r a rgum en t x i s the d ist ance be tween the two po in t s in Euc l idean space- t ime , weca l l them po in t - to -po in t co r re la t ion func t ions , o r co r re lators .

We use th is specific name because in various appl icat ions peop le have u sed o ther rep resen ta t ions o f co r re lators related to the above ones by some in tegral t ransfor

ma tion . W e co mp are her e their defin i tions and brieflycommen t on the i r advan tages and d i sadvan tages .5

I f one makes a Fou r ie r t rans fo rm o f K(x), t he resu l ting funct ion Kmo m(q2) d e p e n d s o n t h e mo me n t u mtrans fe r q flowing from one op era tor to ano the r. Fo r

5As we actually do not use any of them in what follows, thereader may well skip this section.


7/24/2019 RMP_v065_p0001

4/46


c la r i ty we use the fo l lowing no ta t ions , in t roduc ingmomen tum squared wi th a nega t ive s ign Q2= q2. W eare in te res ted in space l ike momen tum t rans fe rs , a s insca t t e r ing exper imen ts , fo r wh ich q2 0.

Due to causa l i ty, the Fou r ie r t rans fo rm o f the po in t -to-point correlat ion funct ion sat is f ies the usual d ispers ionrelat ion ,

0 r Im Kmo m{s)Kmom(q2) = (l/Tr)fds m m . (1.5)

is q z )

The numera to r on the r igh t -hand s ide , Im Kmo m(s), is thephysic al spectr al densi ty. I t describes the squa red m atri xe lemen ts o f the opera to r in ques t ion be tween the vacuumand a l l had ron ic s t a tes wi th the invar ian t mass sl / 2 , andis nonzero only for posi t ive s. Because we a re cons idering only negat ive q2, we never come across a vanishingdenomina to r and there fo re may igno re ie , which is usually put in the deno mi nat or. This s implif icat ion is poss ib lebecause our d iscuss ion is res tr icted to v ir tual processes ,a l though in the r igh t -hand s ide we sha l l u se in fo rmat ion

coming from the real processes of part icle creat ion andann ih i l a t ion .6

Equat ion (1 .5) is the basis of the so-cal led QCD sumru l e s . The ir general idea is as fo l lows. Supp ose onek n o w s Kmo m(q2) in som e region . Th is implies tha t someintegral of the spectral densi ty is known, which can beused to f ix a set of phys ical par am ete rs . Un fort una tely,such fin i te-energy sum rules are not very produ ct iv e, because the d ispers ion in tegrals are usual ly d ivergent , leading to useful sum rules only after some subtract ions .Th is in t roduces ex t ra paramete rs and s ign i f i can t ly un -de te rmines the i r p red ic t ive power.

Let us be more specific , tak ing the mesonic correlat ionfunct ions as an exam ple. As mentio ned earl ier, the

meson ic co r re la to rs a re g iven by a s imp le loop d iag ramfor small x, co r respond ing in the coo rd ina te rep resen tat ion to the free-quark pro pa ga tor squa red . I t is notd iff icul t to see that the imaginary part of th is d iagram,co rrespond ing to the p roduc t ion o f a qq pair, isIm Kmo m(s)~s. Thi s is also obvio us on d im ensio nalgro un ds. Th en from the d ispers ion in tegra l we see tha tat large s t he Fou r ie r- t rans fo rm ed c o r re la to r depend s ons a s Kmo m(s)~s ln( s). How ever, the d i spers ion integral is a lso d ivergent , which s ignals that something ismiss ing in the las t arg um ent . On e s imple way to getaround th is d iff icul ty is to consider the second derivat iveover Q2: then one deals with the funct ion Kf^om(s)fwhich is defined by a convergent d ispers ion relat ion .

However, while going back to the orig inal funct ionKmo m(s), one has to f ix two in teg ra t ion cons tan t s co r responding to the miss ing terms in Kmo m(s ) o f the type

6In principle, virtual processes contain all the information;but, of course, in practice, it is much more difficult to go in theopposite direction and reproduce the physical spectral densityfrom the point-to-point correlators.

cls-\-c2y whi ch have no ima gina ry par t . W e can safelyignore them below in our d iscuss ion of K(x), p r o v i d e d xi s never ze ro , because these co r respond in the coo rd ina tespace to contact terms, 8 funct ions , and their derivat ives .Ho we ve r, in finite-energy sum rule s, the se tw o undefin edcons tan t s need to be de te rm ined a l so f rom the d a ta .

Several o ther ideas have been suggested to improvethese sum rules . Firs t , after taking a sufficient number ofder iva t ives , one may t ake Q=0 and arrive at the so-cal led moments of the spectral densi ty,

Mn=(l/7r)fdsImKmo m(s)/sn + l . (1.6)

Fol lowing ideas presented in the orig inal paper of Shif-ma n , Vainsh te in , and Zakh arov (1979a) , th i s met hod i scommonly u sed in the d i scuss ion o f "charmon ium sumr u l e s , " wh ich a re re la ted to co r re la to rs o f Zc cu r ren t s .

Another idea, suggested in the same paper (Shifmanet al.y 1979a), is to in troduce the Borel t ransform of thefunct ion Kmo m(Q), defined as follows,

Kbor(m)= li m Q2

\ (-d/dQ2

)"Kmo m(Q2

) .n > oo \n 1 ;

s* oo

m2 = Q2/n 2

(1.7)

Applying th is to the d ispers ion relat ion (1 .5), we obtainthe sum ru les in the Bore l - t rans fo rmed rep resen ta t ion :

KboT(m) = (l/7r)fdsImKmo m(s)exp(-s/m2) . (1.8)

Now the in tegral is cut off at large s by the exponen t ia lfunct ion . This form ula also has ano the r useful feature:usual ly we know the contribut ion of the lowest s tates ( thefi rs t resonance) bet ter than the contribut ion of the mult i -

body high-energy part ; so the exponent ial cutoff h idesou r igno ranc e and is therefo re welcom ed. Such forms ofthe sum rules have been used in many papers based onthe O P E (see, e .g ., references in Shu ryak , 1984, 1988aand Shifman, 1992).

However, most of the resul ts obtained by th is technique can also be prese nted in a mu ch s imp ler way. Ins tead o f the Bore l t rans fo rmat ion , one can Four ie r t ransform back to coordinate space; then the d ispers ion relat ion has the t ransparent form (Shuryak, 1984, 1988a)

K(x) = (\/7r)fdsImKmo m(s)D(sW2yx) . (1.9)

Here func t ion Im Kmo m(s) d esc r ibes the amp l i tude o f p roduct ion of al l in termediate s tates of mass sl/2, while thefunct ion

D(m,x) = (m/47r2x)Kl(mx) (1.10)

i s no th ing more than the p ropaga to r o f these s t a tes to ap o i n t x. In prac t ice there is not muc h difference betw eenthis equat ion and Borel sum rules . At large x t he p ropagator goes as exp( mx); t he re fo re one has an exponent ial cutoff, b ut w ith the facto r exp( sl/2x) ins tead ofexp( s/m2). How ever, the space- t ime d i spers ion re la -

Rev. M o d . Ph y s . , Vo l . 6 5 , No . 1 , J an u ary 1 9 9 3

7/24/2019 RMP_v065_p0001

5/46


t ion has a much c lea re r phys ica l in te rp re ta t ion , and weshal l keep to i t in what fo l lows.

Fo r comple teness , l e t u s a l so men t ion one more type o fco r re la t ion func t ion , the one t rad i t iona l ly u sed in l a t t i cegauge theo ry. Th i s i s the so -cal l ed p lane- to -p lane c o r relat ion funct ion obtained from K(x) by an in teg ra t ionover a th ree-d imens iona l p lane :

^ p l a n e t o p l a n e ( ^ ) = < / ^ ^ ^ ( ^ ^ ^ ( 0 , 0 ) > . (1.11)

In o ther words , a spa t i a l in teg ra t ion se lec t s in te rmed ia tes ta tes o f momen tum zero ; so d i spers ion re la t ions a redon e in energy only. Th e above funct ion can be relate dto a physical spectral densi ty by

^pianetopiane(r) = ( 1 / ^ ) / r f ^ Im/sT m o m (m )exp ( -~rm ) .

(1.12)

The mass o f the lowes t had ron can be ob ta ined d i rec t lyf rom the logar i thmic der iva t ive o f th i s func t ion a t l a rger. Ho wev er, i ts essent ial d isad van tage is tha t it mixes

con t r ib u t ions o f smal l and l a rge d i s t ances . Th i s make s i td iff icul t to match with the OPE-derived funct ions atsmall d is tances , and also obscures the physics going on atin te rmed ia te d i s t ances .

However, in the impor tan t case o f heavy - l igh t mesons(see Sec. IV.A), a la t t ice evaluat ion of point-to-pointcorr elat i on funct ions ha s been mad e. In th is case ther e isno d i ffe rence be tween po in t - to -po in t and p lane- to -p laneo n e s , b ecause the super-heavy quark does no t p ropaga tein space; so the in tegral in (1 .11) has only a 8 funct ioncon t r ibu t ion .

To summarize th is sect ion , we have noted f ive d ifferentcorr elat io n funct ions in use: (1) the orig inal poin t -to-po in t func t ion K(x) i n coo rd ina te space ; (2 ) the Fo u r ie r

t r a n s f o r m Kmo m(q2

); (3) the mo me n ts o f the spec t ra density Mn; (4) the Bo rel- t ra nsfor m funct ion Kmo m(m); and(5) the p lan e-to- plan e corr elat io n funct ion funct ion** p l a n e t o p l a n e v ^ ' *

Although each co r re la t ion func t ion has i t s advan tages ,we sugges t tha t fo r the unders tand ing o f the under ly ingphysics i t is bet ter to use the orig inal poin t-to-point funct ion K(x)9 and we shal l do so in wh at fo l lows.

D. General relations and inequalities

One can class ify correlat ion funct ions according toquark pa th s , recogn iz ing two d i ffe ren t types o f d iag rams :(1 ) the one- loop d iag rams , in wh ich qu arks p rod uced byone ope ra to r g o to ano the r one , and (2) the two- loop d iag r a m s , where quark l ines a re c lo sed on the same operator.

Fo r exam ple, in the isospin 7 = 1 chan nels l ike t r+ , onehas opera to rs l ike u Td, w h e r e T i s any Di rac mat r ix . Inth i s case , obv ious ly, on ly the one- loop d iag rams con t r ibu t e . On the o ther hand , cons ider ing the nond iagona lcorrelators in f lavor, say, (u(x)Tu(x)d(0)d(0)), on e isres t r i c t ed to the second type . Fo r mo s t 7 = 0 cases , one

has bo th types o f con t r ib u t ions . La t t i ce ca lcu la t ions dea lmain ly wi th one- loop d iag rams , and there fo re wi th the1 = 1 channe l s , fo r t echn ica l reasons . Some genera l s t a temen ts can be made abou t the one- loop d iag rams , wh ichwe wou ld l ike to ou t l ine here , fo l lowing Weingar ten(1983).7

To derive the relat ions , we fi rs t note the fo l lowing formu la fo r the p ropaga to r in the backward d i rec t ion ,8

S(x9y)=-r5S+(y,x)y5 . (1.13)

One nex t decom poses i t in to Di r ac mat r i c es S = 2 a/ r / ,where r f = 1 , 7 5 , 7 ^ / 7 5 7 ^ 1 7 ^ 7 , , (f*=v). F ina l ly, onecons iders a l l d iagona l one- loop co r re la to rs o f the typen = T r [ 5 ( x , ^ ) r/ S ( j ; , x )F t] and eva lua tes the t races .

The mos t in te res t ing resu l t appears fo r the p seudosca-lar (p ion) cor rela tor: in th is case one has a sum of al lcoefficients squared,

n P s / n ie = ( | a 1 |

2 + | a 5 |2 + | a M |

2 + | a M 5 l2 + | a M V |

2 ) / | a 0 |2 ,

(1.14)

while , for example, the scalar one is

n s/n |ee

= ( ~ | a 1 |2 ~ | f l 5 |

2 + | f l M |2 + | a ^ |2 - | i i M V |

2 ) / | a o l 2 .

(1.15)

Here we have no rmal ized the co r re la to r to i t s asympto t ical ly free vers ion , contain ing free propagators of masslessquarks . Assuming tha t the p ropaga t ion t akes p lace inthe t ime d i rec t ion , the p ro pag a to r i s Sf r e e = 70/(27r2X())and the only nonzero coefficient is a0 = 1 / 2 W2X Q ) . C o mpar ing the above two equa t ions , we ob ta in the Weingarten inequa l i ty. Th i s s t a tes tha t the p seudosca la r co r re la

tor exceeds the scalar one at a l l d is tances , | FIPS > | I IS .The nontriv ial th ing is that the physical p ion is very

l ight , while scalars are heavy; therefore for x > 0 .5 fm th esca la r co r re la to r i s p rac t i ca l ly ze ro , wh i le the p seudoscalar rat io is very large. This requ ires a very del icate c ancel lat ion between the d ifferent a{ i n the p ropaga to r.

Add i t iona l in fo rmat ion i s p rov ided by s imi la r re la t ionsfo r vec to r (p) and axial (Ax) channe l s ,

n F / nf r = ( 2 | a 1 |

2 - 2 | a 5 |2 + | a / i |

2 - | a i t i 5 |2 ) / | a 0 |

2 ,

(1.16)

n / , / nf r = ( 2 | a 1 |

2 + 2 | a 5 |2 4 - | a / x |

2 - | a / i 5 |2 ) / | a 0 |

2 ,

(1.17)

7In preparation of this section J. Verbaarschot has helped a lottoward th e understanding of the meaning of these relations. Healso found a few new ones.

8Readers who wonder why 75 is needed should take as an example a free massive propagator and notice that the terms proportional to {xy)^Y^ and to m behave differently under thetransformation xy.

Rev. Mod. Phys., Vol. 65, No. 1 January 1993

7/24/2019 RMP_v065_p0001

6/46

Edward V. Shuryak: Correlation functions in the QC D vacuum

TAB LE I. Set of the opera tors and correlation functions discussed in this paper.

Channel Current Section Info

PCO

K*

A x77

K

VV '

v'scalarsTJ5-type mesonsheavy baryonsNA

{uytlu~-dy^d)/2ul

(u7( 1u^dy^d)/2l/2

IYVLS

uy5u-dy5d)(i/2l/2)uiy5s(uy5u~dy1d-2Jy5s)/(i/6W2)(uYsYvU -^-dysy^d + sy5Ylts)/(1/31/2)GGqqbymubQTqq{uTCd)u-(uTCy5d)y5u(u TCytlu )u

TCTnQ

I I .BI I .CI I .CI I . D

I I .EI I . FI I . FI I . FI I . GI I . GI I . HI I I .AI V. AIV.CI V. DI V. D

e + e -^Nir,N evene+e~-+N7r,N odde+e--+KK + Nv T-\-K*

decay r>vT+i V17., N o d dp ion decayK d e c a y

J/xfj-+y-\-r] e tc .J/ip-^y-\-7] e tc .masses , gene ra l i t i e se+e~>BB+ p ionsma sses of hea vy f lavored m eso nsmassesO PE p r e d i c t i o n sO PE p r e d i c t i o n s

and the fo l lowing inequal i t ies may be proven

nPS /n |e > i( n v m f

e+nA /nAee)

free_ rfree\n P S /nt

Pr

se e> i ( n F / n r - n ^ / n

t r)

(1.18)

(1.19)

Wit ten (1983) has found another in teres t ing inequal i tybetween vector and axial correlators , but i t appl ies onlyto the momen tum rep resen ta t ion , and we do no t d i scussi t here.

As these inequal i t ies are ident i t ies , they are sat is f iedfor any configurat ion of the gauge field , and they therefore are not very res tr ict ive from a theoret ical poin t ofview. How ever , they can be used to chec k consis tency ofexperimental data , as d iscussed below.

On the o ther hand , the d iagona l co r re la to rs themse lvesare posi t ive monotonously decreasing funct ions , as isclear from the spectral decomposi t ion d iscussed in theprev ious sect ion . Th is cond i t ion is t r iv ial to sat is fy fromthe empi r i ca l de te rmina t ion o f co r re la to rs ; bu t f rom thetheo re t i ca l po in t o f v iew, i t p roduces non t r iv ia l l imi tat ions for the ensemb le of vac uum fields . S omeconf igu ra t ions do p rod uce nega t ive co r re la to rs , e spec ia llyin the scalar chann el . If their weig ht in the ensem ble ofvacuum fields is too large, the posi t iv i ty and monotonici-ty may be v io la ted . These cond i t ions may p ro v ide inte res t ing new cond i t ions on the model s o f the vacu um .

currents real ly exis t in nature, evidenced by their coupl ing to weak and electromagnet ic f ields , in contras t toma ny othe r opera tors to be d iscussed. In several casesthe comple te spec t ra l dens i ty o f the co r respond ing co r relat ion funct ions is experimental ly known, subject , ofcou rse , to some exper imen ta l uncer ta in ty, f rom e^e~ann ih i l a t ion in to had rons .

The vec to r cu r ren t s and the i r co r re la t ion func t ions tobe d iscussed below wil l be denoted by the name of thel igh tes t meson in the co r respond ing channe l ; in par t i cular, we define the p, t h e co , and the (f> currents as the fo llowing quark cu r ren t s ,

y = ( l / 21 / 2 ) [ i 7 y M - J y ^ ] o r uy^d

j^(\/2x/2

)[uY^u^dy^d} ,

(2.1)

(2.2)

(2.3)

Further defin i t ions may be found in Table I . The elect romagne t ic cu r ren t i s the fo l lowing combina t ion o f thequark cu r ren t s :

j^=\uyfXu-\dy^d +

= ( l / 2 1 / 2 ) y - ( l / 2 1 / 2 3 ) y + .

The vector correlat ion funct ions are defined as

(2.4)

II . PHENOMENOLOGY

OF MESONIC CORRELATION FUNCTIONS

A. Vector currents and correlators

We start the d iscuss ion of the correlat ion funct ionswith vector curre nts for an obvio us reason : these

9These were shown to me by J. Verbaarschot (private communication).

n ifAiVU)=iU)7I>(0)|o> , (2.5)

and the Four ie r t rans fo rm ( in Minkowsk i space- t ime) i st rad i t iona l ly wr i t t en as

ifd4x e ^ n / > v ( x ) = n / ( ^ 2 ) ( ^ ^ v - ^ 2 g / x v ) . (2.6)

The right-hand s ide is expl ici t ly t ransverse, i .e . , i t vani shes when mu l t ip l i ed by momen tum q. This is necessaryfo r conserva t ion o f the vec to r cu r re n t .

The d ispers ion relat ions for the scalar funct ions I I ; ( # 2)are

Rev. Mo d. Phys. , Vol. 65 , No. 1 , Janu ary 1993

7/24/2019 RMP_v065_p0001

7/46


Ili(Q2^- q2) = (l/7r)fds ' , (2.7)

where the phys ica l spec t ra l dens i ty Iml l / t e ) i s d i rec t ly related to the cross sect ion of e+e~ ann ih i l a t ion in to had -r o n s . As th i s quan t i ty i s d imens ion les s , i t i s p ropo r t iona lto the no rmal ized c ross sec t ion

*/ = < V . - ^ > / < Ve - ^ + / 1 - < * > > ( 2.8 )

wher e the c ross sec t ion o f mu on pa i r p ro duc t i on (neg lec ting the m uo n mass) is jus t a + - + - = (47ra2/3s) and

J e e >/i ft

a is th e fine-structure co ns tan t. If bo th qu ar ks in th ecur ren t consider ed have the same flavor, as , for exam ple,th e si i n the (f> cu r ren t , one ob ta in s

lmlli(s)=Ri(s)/(l27re2) , (2.9)

w h e r e eq i s qua rk e lec t r ic charg e . Gen era l i za t ion to p>channe l s i s s t ra igh t fo rw ard : in s tead of the charge ther es tands a corresponding coefficient in the equat ion for the

e lec t romagne t ic cu r ren t , e .g . ,

Jmnp{s) = -^RAs) . (2.10)

The reader may wonder how the d i ffe ren t vec to r co r relato rs are d is t ing uished expe rime ntal ly . I t is c learenough for the charge and beauty heavy flavors : i f thefinal s tate has a pair of such quarks , i t i s much more l ikely tha t they were d i rec t ly p roduced in the e lec t romagne tic current than that they were produced by final-s tate intera ct io ns . W e shal l a lso use th is arg um ent later for thes tra nge qu ark , al th ou gh i t is less jus t i f ied in tha t case. Tosepara te the l igh t quark p,co channe l s , we make use o fthe i r i so sp in and G p a r i ty. Th e two channe l s have ad i ffe ren t i so sp in 7 = 1 , 0 wh ich i s conserved by any s t rongfinal-state in te ract i on . As i t is wel l kn ow n, C par i ty p lusisotopic invariance leads to the so-cal led G-pari ty conserva t ion , and p ions have nega t ive G p a r i ty. There fo res t rong in te rac t ions do no t mix s t a tes wi th even and oddnum bers of p ions . The cu r ren t s jp,jm ha ve fixed G p a r i tyas wel l , and therefore p ionic s tates created by them canhave only even or odd numbers of p ions , respect ively.

Let us s tart with a s imple example to show how theserelat ions lead to defin i te predict ions . The rat ios Rt(s)have a very s imple l imit at h igh energies s , because in th isl imi t quarks and an t iquarks a re p roduced as f ree par t ic l e s . Fo r cu r ren t s con ta in ing on ly one quark f l avo r q, t heonly d ifference w ith the m uo n is a different electr iccharge and a color factor:

limRqis) = e 2 N c , (2.11)

wh ich fo r the case g ives lims_+ O0R(p(s) = j . F o r t h e pa n d co cases , we expand the e lec t rom agne t ic cu r re n tequat ion (2 .4) in terms of (2 .1) and (2 .2) and obtain fromtha t rep resen ta t ion

lim Rp(s) = , l i m i ^ ( s ) = j . ( 2. 12 )S> 00 S> 00

As we shal l see short ly, these relat ions are wel l sat is f iedexp erim ental l y. In fact, th is was h is torica l ly one of thefirst and s im ples t jus t i f icat ions for Q C D .

Coming back to coo rd ina te rep resen ta t ion o f thed i spers ion re la t ion , one ob ta in s

9 1

X C^dsRAsWis1'2^) , (2.13)J o

whe re , we reca l l, D{m,x) from Eq. (1 .10) is jus t the propaga to r o f a sca la r mass -m par t i c l e to po in t x. C o n t r a c ting indices and using the equat ion d2D(m,x)= m2D(m>x)-\- con tac t t e rm , wh ich we d i s regard , thedispers ion relat ion f inal ly becomes

ni^(x) = (l/4ir2)fCOdssRi(s)D(sl/2,x) . (2.14)

This is our experimental defin i t ion of the vector correlat ion funct ions .

A f inal comm en t re la ted to ou r no ta t io n : as co r re la

tors are very s t rongly decreasing funct ions of x,

i t i s moreconven ien t to p lo t them no rmal ized to the f ree p ropagat o r s , n amely, as Tl (x)/Hf^(x) w h e r e n j ^e ( x ) c o r r esponds to the s imp le loop d iag ram descr ib ing f ree -quarkp rop aga t io n . In such ra t io s a l l un in te res t ing no rm al izat ion fac to rs , such as the quark e lec t romagne t ic charges ,dr op out . A t small d is tan ces these rat ios are al l c lose to ldue to asympto t i c f reedom.

B. Vector / = 1 (o r p) channel

Figu r e 1 shows a sample o f exper im en ta l da t a on Rp(s)at low energies . On e can see th at th is funct ion co nsis ts oftwo qu i t e d ifferen t pa r t s : (1 ) the p rom inen t p -m eson resona nce , seen in the 2-rr cha nne l , and (2) a mix tur e of m ul-t ip ion s t a tes , wh ich s t a r t s wi th (a t l eas t ) two "p r imed"resonances , p' (1450) and p' (1700), seen mainly in thefou r-p ion channe l . However, t aken toge ther wi th thes ix -p ion channe l , they add up to a ra ther smoo th non res -onance "con t inuum," and a l ready a t energ ies o f abou t1 .5 GeV th is spectral densi ty fo l lows the predict ionRp = - | made above.

We have para met r i zed the da ta in F ig . 1 by the fo l lowing funct ion , shown as the so l id l ine:

9 l+4(E~mp)2/T2p

+ (l+a s(E)/ir) - ^ r , ( 2.1 5)2 l + e x p [ ( i s0 E)/o]

w h e r e E0 = l.3 G e V, 8 = 0 . 2 G e V, a n d as{E) = 0JAnE/0.2 GeV ) . Th i s param et r i za t ion inc ludes a l l e s sen t ia ling red ien t s o f the da ta : the resonance peak an d(smoo thened ) t rans i t ion to the asympto t i c behav io r, co rrespond ing to the famous c ross sec t ion o f f ree -quark p roduc t ion . Fo r the h igh -energy con t r ib u t ion , we used asmoo thed func t ion in s tead o f the 6 funct ion , as i t is t radit iona l ly done in QCD sum ru les . The phys ica l mean ing


7/24/2019 RMP_v065_p0001

8/46


2.0

FIG. 1. Ratio of a(e+e~-+mr)/o-(e+e~-+p,+fj,~) with neven, as a function of the total invariant mass of the hadronicsystem. The data points correspond to the following states: theerror bars without points to 2TT (Barkov et al., 1985); stars t o 4-rr(Cosme et al, 1979; Cordier et al, 1982a; Bar kov et al, 1988;

Kurdadze et al, 1988); and triangl es to 677" states (Cosme et al,1979; Dolinsky et al, 1989). The solid line is our fit to the sumof all contributions with n even for th e total cross section in theI \ channel. We have not shown all data points available nearthe top of the p peak, nor the region 2E>2 GeV, where theagreement between data points and our fitted curve is verygood.

o f the paramete r E0 i s the same : i t is the energy a bovewhich the asym pto t i c f reedom i s res to red and s imp lequark model es t imates for cross sect ion become val id .

We now calculate the correlat ion funct ion using th isparamet r i za t ion and the d i spers ion re la t ion (2 .14 ) , t ak ingthe in tegral over al l energies . Th e resul t ing curv e isshown in F ig . 2 , where the con t r ibu t ions o f two componen ts o f the spec t ra l dens i ty men t ioned above a re a l so

show n separately. Th e fi rs t s t r ik ing observ at ion is tha t ,s t a r t in g w i th t h e r a t h e r c o mp l i c a ti o n f un c t io n I m l l ^ U ) ,we arrived at a very smooth funct ion of the separat ion x.Clear ly, the way back f rom the coo rd ina te rep resen ta t ionto phys ica l spec t ra l dens i ty wou ld be much moredifficult.

The second s tr ik ing observat ion (Shuryak, 1989a) istha t the con t r ibu t ions o f the lowes t meson and con t inuum complemen t each o ther in such a way tha t the ra t ioU(x)/IlfTee(x ) rem ains close to 1 up to d is tance s as largeas 1.5 fm. W e call this fine tu ni ng of all pa ra m ete rs su-perd ual i ty . As we shal l show , i t pers is ts in al l vectorchan nels . Fo r small d is tanc es i t is noth ing mor e tha nasymp to t i c f reedom. At ; c ~ 0 .3 fm, i t i s a consequenceof the so-cal led dual i ty between hadronic and quarkdescrip t ion . How ever , from 0.3 to 1 .5 fm, whe re theco rre la to r d rops by more than fou r o rders o f magn i tude ,i t i s an unexpec ted and remarkab le phenomenon

Complet ing th is sect ion , le t us examine the errors inthe de te rm ina t ion o f the co r re la to rs . Of cou rse , the exper imen ta l uncer ta in t i es a re the re , and the i r magn i tudesare seen in Fig. 1. In the p region (due essent ial ly toVEPP-2M da ta f rom Novos ib i rsk ) , the resu l t ing e r ro r a tl a rge x > 0 . 6 fm in the co r re la to r i s abou t 5% . Theh igh -energy dom ain i s covered by SP EA R da ta f romSLA C, wh ich f ix the no rm al iza t ion o f smal l -* reg ion a l soto with in a few percen t . Ho wev er, in the most in ter es ting med ium d i s tances , we have con t r ibu t ions f rom the p'energy region, and here the s i tuat ion is actual ly evenmo re uncer ta in than ou r F ig . 1 ind ica tes : the F rasca t t iand Orsay data do not agree, and the problem is not s tat is t ical . I t is qui te probab le that in th is region our parametrizat ion of the cross sect ion is off by as much as3 0 % , which may lead to error bars for the rat ion U ) / I I

f r e e( ; c ) p lo t t ed in F ig . 2 o f abou t 15% a t x - 0 . 6

fm. In v iew of the app are nt system atic deviat ion of thetwo sets of data , i t would not be useful to d isplay s tat is t ical error bars on the p lo ts of the correlators .

1 .0

1

0 . 5

n

1 1 1 | 1

- * \

U'

-. - I . 1 1 1

1 1 1 | 1 1 1 1 | 1 1 1

P X ,

^ . c o n t i n u u m1 . 1 1 1 . ~ Y - - t - - J - - . - - - . - - . i _

-

-

-

- L . -

0.5 1.5x ( f m )

FIG . 2. R atio of the 7= 1 vector correlation function to thatcorresponding to free-quark propagation vs the distance x. Thedot-dashed curve is the p-meson contribution calculated as thecontrib ution to the integral of the region below a total energy of1 GeV. The dashed curve labeled "co ntin uum " is the complementa ry contribu tion of all hadro nic states above 1 GeV, andthe solid curve is their sum .

C. o) an d t f> channels

The next channel we d iscuss is the isoscalar channelhav ing the quan tum numbers o f the co meson . The co r responding data for the cross sect ion of e+e~ ann ih i l a t ionin to an odd number o f p ions , now summed over a l l channe l s , are shown in Fig . 3 . The top of the co peak is no tshown because here the Bre i t -Wigner cu rve (wi th thewid th va lue t aken f rom Review of Particle Properties,H e r n a n d e z et al., 1990) is very accura te: the peak valu eof R i s abo ut 12 . O ne can also see a t rac e of the (f> p eakdue to the co-(f> mixing, which wil l be d isregarded in ourparam et r i za t ion . No te the change o f sca le and the es sent i a l ly l a rger e r ro r bars compared to the 1 = 1 channe l .Wi t h i n u n c e r t a i n t i e s t h e c o n t i n u u m ma g n i t u d e a pp roaches i t s asympto t i c va lue Rco = j a t about the sameenergies as in the 1= 1 channe l .

Fo r nar row resonances , we inc lude the resonance contribut ion in the s imple form

http://localhost/var/www/apps/conversion/tmp/scratch_7/~Y--t--J--.---.--.i_http://localhost/var/www/apps/conversion/tmp/scratch_7/~Y--t--J--.---.--.i_

7/24/2019 RMP_v065_p0001

9/46

Edward V. Shuryak: Correlation functions in the GCD vacuum

1.5

0.8 I2E(GeV)

FIG. 3. As in Fig. 1, the isospin 1= 0 final states, defined asthose having an odd number of pions. Data points are summedover all channels, compiled in Dolinsky et al. (1989), while thecurve is our fit discussed in the text.

" / u^ ^ ' l re s" res> ' ' (2.16)

where the coup l ing cons tan t s o f the cu r ren t s to mesonsare defined as follows10 :

=fresmresefl . (2.17)

Here the eM i s the pol ariza t ion v ector of the vecto rmeson . These coup l ings and the par t i a l wid ths to the

e e channel are related as fo l lows:

/ .2 :res3 m r e s r ( r e s - > e

+ e )

4w a2(2.18)

For reference, the accepted values of the coupl ing cons tan t s o f the p , , a n d co mesons a re /fi > = 46 Me V,/ ^ 7 9 M e V , /p 152 MeV.

Nex t F ig . 4 shows the co r re la t ion func t ionTLa)(x)/Ilfree(x), aga in wi th con t r ib u t ions f rom the co re sonance and the con t inuum s ta te shown separa te ly and insum . Th e cu rve co r respond s to the fo l lowing para met r i z -at ion of the cross sect ion:

RJE) = 12

\+4(E-mJ2/ri

+ Ul+aAE)/ir)1

l+exp[(E0-E)/8] '

(2.19)

w h e r e n o w E0 = l.l G e V a n d S = 0 . 2 G e V . T h e e x p e r imen ta l e r ro r on the c o n t r i b u t i o n i s a b o u t 3 % , b u ta b o u t 2 0 % f o r t h e c o n t i n u u m .

In spi te of completely d ifferent f inal hadronic s tates

10There will, of course, be some am biguity in these definitions,if the resonanc e is broad.

3

0.5

~iiI r-

^ . c o n t i n u u mi I i i i i I F" " i > - - -

0.5 1.5x ( f m)

FI G. 4. Same as in Fig. 2, but for the 1 = 0 vector correlator,th e c o channel.

and a much smal le r c ro ss sec t ion , the co r re la to r in the cochannel is s imilar to the p co r re la to r. F igu re 2 fo r the pchannel and Fig . 4 for co ag ree to w i th in uncer ta in t i es ,and the only d ifference between them appears at d istances as large as about 2 fm

To unders tand what th i s phenomenon means , l e t u slook at the d ifference between the p a n d co co r re la to rs .As the fo rmer cu r ren t has the uudd f lavor s t r uct ure ,and the l a t t e r uu -\-dd, th is d ifference is the vectorflavor-changing correlator

K^(x)=(urflu(x)dyfld(0))- ' ^^COylJLfJL A l p ^ ^ ; (2.20)

Thus the da ta p resen ted above t e l l u s tha t th i s amp l itude i s fo r some reason ex t remely smal l . Unfo r tuna te ly,we do not real ly know how small i t i s a t in termediate d istanc es , up to 1 fm or so , beca use i t is with in th e exp erime nta l unc ertai n t ies . Only at d is tances as large as 2 fm

does the d ifference between the co a n d p co r re la to rs becom e clearly observa ble. I t me ans tha t the f lavor-chang ing co r re la t ion func t ion (2.20 ) becom es co mp arab leto th e flavor-diagonal on es1 1 on ly when the l a t t e r d ropsby many o rders o f magn i tude .

There a re two more s t r ik ing exper imen ta l observa t ionstha t suggest that the famous Zweig ru le , forbid ding theflavor-changing tra nsi tio ns, is inde ed surp risin gly strictin the vector chan nels : (1) the p-co ma ss difference is onl y12 MeV; (2) the co-(f> mixing angle is only l-3.

No general reasons for such s t rong suppress ion offlavor-changing t ran si t io ns in vect or cha nne ls are kno wn ,a l thoug h some in te res ting h in t s have been sugges ted . Inpar t i cu la r, a pe r tu rba t ive ana lys i s l eads to the idea tha t

in the vector case one needs at leas t three g luons in thein te rmed ia te s t a te , no t two as in the p seudosca la r case .However, th i s a rgumen t shou ld no t be app l i cab le to d i stanc es of the ord er of 1 fm and beyo nd. In th is respe ct ,

11 We show below that for pseud oscalar corre lators such deviation happens at much smaller distances, where the correlationfunction is about four orders of magnitude larger.


7/24/2019 RMP_v065_p0001

10/46


0.5

0.4 h-

peak , because i t i s nea r ly pe r fec t ; t he max ima lvalue i s R^ ~ 50 . Ins t ead we show how our f i t r ep roducesthe sum of a l l o the r con t r ibu t ions , shown by the so l idl i ne . Th e pa ra me t r i za t i on used he re was

0.5

FI G . 6. Same as in Fig. 2, but for the correlator.

RÊ) = 52.4

l + 4 ( - m ^ )2 / r 2

+ }{l+as(E)/ir) - ~ = ^ , (2 .2 1)3

l + e x p [ ( 2 ?0 2 ? ) / 8 ]1 .5 G e V a n d 5 = 0 . 4 G e V .we present in Fig. 6

w h e r e E0 ~-Fina l ly, we p re sen t i n F ig . 6 t he cor re l a to r

I I ^ ( x ) / nf r e e(x ) , w hich i s a lso su rp r i s ing ly s imi l a r1 2 to the

pyco cor re l a to r s shown above .

D. Strange vector (or K* ) channel

For comple t eness , l e t us a l so cons ide r t he s t r ange vecto r channe l . He re t he cur r en t is j uy^s and thelowest meson is the K*(%92). Pheno men olog ica l ana lys i sin this case i s not based on e lec t romagnet ic processes, butra the r on the vec to r pa r t of weak cur ren t s . Th e da t acome in t h i s ca se f rom the weak decay p rocessr > vr +h ad ron s . S ince t he had rons a re p rodu ced f rom av i r tua l W i n s t ead o f v i r t ua l pho ton , we ob ta in an admixtu re o f t he s t r ange cur ren t f rom the Cab ibbo mix ing o fthe weak c ur ren t .

Compar ing the Cab ibbo suppre ssed p roduc t ion o f K *to t he Cab ibbo a l l owed p roduc t ion o f p in this decay, onecan obta in the fol lowing re la t ion,

B[T-VT+K*) _ 2

B(r-+vT+p)ta n2(< 9c)

AfP

( l - m * / m 2 ) 2 ( l + 2 m * / m2 )

( l - m 2 / m 2 ) 2 ( l + 2 m 2 / m 2 )(2.22)

w h e r e 0C i s t he Cab ibbo ang le . Inse r t i ng on the l e f t -hands ide t he expe r im enta l r a t i o 0 .0143+ 0 .003 1 ( f rom Reviewof Particle Properties, H e r n a n d e z et al., 1990) , one obta ins the fol lowing ra t io of the coupl ing constants:

= 1.1+0.1 (2.23)

We do not have suff ic ient informat ion about the cross

12 He re our prese ntation is somewhat illogical, because we still measu re the correla tor in units of n ê , corresponding to the freepropagation of massless quarks. The decrease of Rîx) with distance is partly kinematical, due to nonze ro strange quark mass . Wehave not included this correction, in order to make comparison with nonstrange correlators in the same figure.

Rev. Mod . Phys., Vol. 65, No. 1 , January 1993

7/24/2019 RMP_v065_p0001

11/46


I . O

I

0.5

n

i i i |

- ~ X

-

/- < ' * . I

. . . . I . . i I . i

_ X.

K*V

X .

X c o n t i n u u m

i L. i i 1 i *T~'r--i---J-ni--i.,..

t t

-

-_

t i

0.5 1.5x ( fm)

FI G . 7. Same as in Fig. 2, but for the K * correlator.

sec t i on o f weak p roduc t ion o f nonre sonance s t a t e s wi ths u c h q u a n t u m n u m b e r s , K an d p ions . In add i t i on , t he rl ep ton mass se t s a r a the r r e s t r i c t i ve l im i t on t he ava i l ab l eene rgy. How eve r , one can s ti l l ma ke some a rgum ent sbased on spec t ros cop ic da t a . Indeed , i n t he s t r ange vecto r channe l t he re a re two p r imed re sonances , a t 1415 and1715 MeV, ve ry much s imi l a r t o two p' r e sonances nea r1600 Me V. W e sha l l t hus a ssume tha t t he nonre sona ncepa r t o f bo th K * a n d p c ross sec t i ons a re s imi l a r. The refo re t he same nonre sonan t con t r ibu t ion a s t ha t fo r t he pchanne l wi l l be t aken in t he pa rame t r i za t i on , sca l ed , o fcourse , to a di fferent l imi t a t inf ini te energies.

T h e c o r r e s p o n d i n g c u r v e s f o r n ^ * / I If r e e ( j c ) a re g ivenin Fig. 7 . The resul t ing curve f i t s perfec t ly be tween the pa n d curves d i scussed above , sugges t ing tha t a l l t he secomplete ly di fferent se ts of da ta are , in fac t , deeply connec t ed to each o the r.

E. Ax ial / = (o r A^) channel

No w we tu r n f rom vec to r t o ax i a l -vec to r chan ne l s ,con cen t ra t i ng on the 1 = 1 c h a n n e l . T h i s h a s t h e q u a nt u m n u m b e r s o f t h e A i m eson an d i s r e l a t ed t o t he fo ll owing cur ren t :

(2.24)

Data cor re spond ing to t h i s channe l a re a l so ob t a inab lef rom the r l ep ton decay in to t he cor re spond ing neu t r inoa n d h a d r o n s , b e c a u s e t h e w e a k c u r r e n t h a s b o t h v e c t o rand ax i a l com pon en t s . S ince we dea l wi th t he cha rg ecur ren t a ssoc i a t ed wi th t he W exchange , we do no t havea n 1 = 0 componen t ; so p roduc t ion o f odd numbers o fp ions i s now en t i r e ly due t o t he ax i a l pa r t o f t hec u r r e n t .1 3

s(GeV^

FIG. 8. Contribution of the 3rr channels to the spectral densityof the axial current, measured in the r lepton decay by theARGUS Collaboration (Albrecht et a/., 1986; the five-pion oneis small and rather uncertain). The curve is just a param etrization used in the theoretical paper by Peccei and Sola (1987),from which we took this figure, and it is not used here.

The expe r imenta l l y measured d i s t r i bu t ion in to t h reepions as a funct ion of the i r invar iant mass i s shown inFig . 8. Th e a symm et r i c peak a rou nd 1.2 GeV i s t he cont r ibu t ion o f t he Ax m e s o n .1 4 I t s dom inan ce is a l soconf i rmed by the obse rva t ion tha t two channe l s wi thth ree p ions , ir~Tr~Tr+ a n d T T W - , h a v e b r a n c h i n g r a t i o s(of all T>vT-f h a d r o n s d e c a y s) e q u a l t o ( 6 . 8 0 . 6 ) % a n d( 7 . 5 + 0 . 9 ) % , r e s p e c ti v e ly. T h e y a r e e q u a l w i t h i n u n c e rta int ies, and this i s prec ise ly what should be the case i ft hey a re domina t ed by A x decays . Fo r t hose rea sons , wetrea t the peak seen in the r decays as an "effec t ive" A xm e s o n .

L e t u s i n t r o d u c e c o u p l i n g c o n s t a n t s fA s imi la r to

those o f vec to r r e sonances :

VT- AX)

B(r->vT+p)

/A

fp

(l-m2A{/m2

T)2 (l + 2m2A{/m2T)

(l-m2p/m2

T)2 (l+2m2p/m2T)

(2.26)

one deduces a va lue fo r t he coup l ing cons t an t ,

/ ^ / / p 1.0+0.07 . (2.27)

Having f i xed the r e sonance con t r ibu t ion , we nex t

13Decays into ne utrino and an even number of pions are, as inth e e*e~~ annihilation, related to the vector p-type current.The corresponding data are consistent with the e+e~ annihilation data, although they are much less accurate.

14The A i shape observed in the r decay and hadronic reactions is somew hat different. This point is discussed in Isgur(1989), which also contains further references. The data shownin Fig. 8 seem to suggest an admixture of some nonresonancebackground at the largest energies, but the errors are sti l l toolarge to allow any definite conclusions.

Rev. Mod. Phys., Vol. 65, No. 1, January 1993

7/24/2019 RMP_v065_p0001

12/46


proceed to the con t inuum s ta tes a t l a rger invar ian tmasses . Unfo r tuna te ly, th e r l ep ton is no t heavy enoug hto produce f inal s tates in the asymptot ic region; d irectobservat ion of the axial spectral densi ty is l imited by i tsm a s s , 1784 Me V. Mo reov er, as one can see from F ig . 8 ,the s tat is t ics of the exis t ing experiments are only good upt o s 1 / 2 1 . 4 - 1 . 5 G e V. T h e r e f o r e w e d o n o t s ee t h e mo s tin te res t ing reg ion in wh ich spec t ra l dens i ty app roachesi ts asymptot ic l imit .

However, we have some genera l a rgumen ts tha t a l lowone to f ix the con t inuum con t r ibu t ion wi th reasonab lysmall uncert ain t ie s . Firs t , in the chiral l imit , the fo l lowing inequa l i ty has been p rov en1 5 (Wit ten , 1983):

n^(g2)-n^;(^2)> o (for a iu2 < o ) . (2.28)This cond i t ion shou ld become an equa l i ty a t l a rge \q2\because the Od/q2) t e rm s , co r respo nd ing to theO (1 /x2) t e rms in the coo rd ina te rep resen ta t ion o f theco rre la to rs , shou ld be the same fo r vec to r and ax ia lco r re la to rs in the ch i ra l limi t . Th i s s t a temen t i s know n

as the second Weinberg sum ru le1 6

(Weinberg , 1967):

/ i f e ( I m nw 41 - I m n j ^ ) = 0 (2.29)

As the resonance con t r ibu t ion i s p ropo r t iona l tom2A f2A ~~m2pf1p>^y t he con t r ib u t ion o f the non reso -nance con t inu um shou ld be nega t ive . Assum ing ou r p rev ious paramet r i za t ion o f the con t inuum, we may therefo re conc lude tha t asympto t i c f reedom in the ax ia l channel should be recovered at larger energies , E0 l >E $9wh ich is ind eed th e case. Th e sum rul e (2.29) is satisfied

A

a t E0 l 1 . 5 G eV, if we use the same shape as tha t u sedfor the p case, with 8 = 0 .2 Ge V. Thi s is qui te a f i rm p red ic t ion , p rov ided the shape o f the con t inuum spec t rum i sthe same. How ever, to show the sens i tiv i ty o f the co r r elator to th is uncertain ty, we shal l d isp lay two curves forthe ax ia l co r re la to r, wi th E0 l = 1 .5 and 1.7 GeV . Fo r amore detai led d iscuss ion of the axial spectral densi ty, inc lud ing , in par t i cu la r, i t s re la t ion to rn +m 0 , see Pec-cei and Sola (1987).

Before p lo t t ing the correlat ion funct ion , le t us alsoclari fy a theoret ical poin t related to a general form of theax ia l co r re la to rs . I f ch i ra l sym met r y were exac t , wi th a l lquark masses ze ro , the nons ing le t ax ia l cu r ren t s wou ld beconserved . Because o f tha t , one mig h t th ink tha t theFour ie r t rans fo rm o f the i r co r re la to rs wou ld have on ly a

15The author is indebted to S. Nussinov for bringing thistheorem to his attention.

16This statement can be derived from the fact that, in thechiral limit, the only dimension-4 scalar operator is a gluonicfield strength squared, which contributes the same amount tovector and axial correlato rs (see Sec. III.B). However, fornonzero quark masses, there appear contributions of the typemqqq , different for vector and axial correlators.

t ransverse par t , namely,

V L v(a)-{q lqv-gyLVq2) . (2.30)

This is not the case. The exis tence of a Goldstone mode,the mass les s p ion , coup led to the ax ia l cu r ren t , p ro duces ,

in add i t ion , a long i tud ina l con t r ibu t ion :

n v{q)=iit(q )^q^v-g^q )+flq^v/r (2.31)

In the coo rd ina te rep resen ta t ion , the second t e rm ju s tg ives a s ingulari ty at x =0 , wh ich does no t spo i l cu r ren tconserva t ion .

Now we proceed further, d iscuss ing the real world inwh ich quar k masses are non zer o . W e s t i ll have a longi tud ina l pa r t due to a p ion con t r ibu t ion , wh ich now dependson x as 3/i 3vZ >( m7 r,x ) . Tak ing the d ivergence 3^(o r con t rac t ing ind ices fiv), we ob ta in d2D(m1T >x)= m 2 r D ( m 7 r , x ) +c o n t a c t t e r m . N o w w e h a v e a l o n g i

tud ina l con t r ibu t ion , non t r iv ia l ly depend ing on d i s t ance ,bu t i t i s p ropo r t iona l to m 2 . Thi s resul t is not u nexp ected: al th oug h in the real wor ld the axial cur ren t is notconserved, i ts d ivergence is O (mq ) = O (m %).

The conc lus ion f rom these theo re t i ca l cons idera t ions i sthat one can part ial ly get r id of the p ion s ignal in the A xcorrelat ion funct ion by s imply contract ing the indices onthe correl ator. Thi s wil l a lso ma ke bet t er contac t to theco rre la to rs fo r the vec to r channe l s . The con t rac t ionleads to the fo l lowing approximate relat ion for the axialco r re la to r :

3 r ,_ ._ . l+ax(E)/ir+ -f- fdEE3D(E,r)-4lT J 1 + exp[(E0-E)/8]

(2.32)

The fi rs t two terms are the contribut ions of the A { andthe 7r, and the th i rd t e r m i s the non reso nan t con t inuu m.The l a t t e r i s exp ressed in ou r u sua l way, wi th a per tu rba-t ive con t r ibu t ion s t a r t ing a t some E0, tak en to be 1.5 or1.7 GeV . As befo re , we took 8 = 0 .2 GeV .

Now everyth ing is f ixed, and the resul t ing correlator isshown in F ig . 9 . Com par in g i t wi th the p co r re la to r inF i g . 2 , one observes th at i t has a completely d ifferentshape . The A { con t r ibu t ion can be s l igh t ly l a rger thanth e p one at small x (again , beca use fAmA > / p m p )> ^ u tat larger d is tances i t drops due to larger A x mass . Eventual ly, a t large JC the axial correlator grows again , due tothe long -range p ion con t r ibu t ion .

Final ly let us emphasize that the d ifference betweenthe vector and axial correlators is ent i rely due to thech i ra l asym met ry o f the QC D vacuum . By s tudy ing howthis d ifference develops as a funct ion of d is tance, one canhope to l ea rn someth ing abou t the mechan i sms c rea t ingth i s asymmet ry.


7/24/2019 RMP_v065_p0001

13/46

Edward V. Shuryak: Correlation functions in the QC D vacuum 13

1 . 5 1 i i i | i i i i | i i i i i i i i r

0.5 I I .5 2x(fm)

FI G . 9. Same as in Fig. 2, but for the axial curre nt. The do t-dashed line is the contribution of the Al meson, while theshort-da shed one is that of the pion. Two long-dashed linesshow the contribution of the nonresonance continuum, if itsthreshold is E0 1 .5 or 1.7 GeV . Two solid lines show thesums of all contributions in these two cases; the true correlatoris somewhere between them.

F. Pseudoscalar correlation functions forthe SU(3) octet (the ir,K,r} channels)

Here we cons ide r co r re l a t i ons o f t he oc t e t pseudoscal a r q u a r k - a n t i q u a r k o p e r a t o r s

j = (i/2U2)(uy5u-dy5d) , (2.33)

j K = iuy5s , (2.34)

j v = (i/6W2)(uy5u+dy5d-2sy5s) . (2.35)

These cor re l a to r s a re ve ry impor t an t fo r t he unde rs t a n d i n g of Q C D v a c u u m s t r u c t u r e . O n e m i g h t n a i v e lyth ink tha t because t he pseudosca l a r s a re t he l owes t exc it a t i ons o f t he QCD vacuum, t hey t e l l u s p r imar i l y abou ti t s l ong- range s t ruc tu re . How eve r, a s we sha ll see shor tl y, t hey a l so p rov ide much puzz l ing in fo rma t ion a bou t i t sshor t - r ange s t ruc tu re a s we l l .

Gene ra l l y speak ing , t he pseudosca l a r and sca l a rmesons a re r a the r excep t iona l members o f t he f ami ly o fhad ron s . The re a re some su rp r i s ing ly l a rge num bers a tt a ched to t hem; i n pa r t i cu l a r, t he coup l ing cons t an t s t othe cor re spon d ing cur re n t s a re ve ry l a rge . The re fo re t hecon t r ibu t ions o f t he se pa r t i c l e s t o t he cor re l a to r s a re a l soi m p o r t a n t a t s m a l l x.

Before we come to co r re l a t i on func t ions , some gene ra lc o m m e n t s a b o u t p s e u d o s c a l a r s a r e i n o r d e r. T h r o u g h o u tthe h i s to ry o f had ron i c phys i c s , f rom na ive nonre l a t i v i s -t i c qua rk mode l s t o mode rn l a t t i c e ca l cu l a t i ons , somepuzz l e s r e l a t ed t o t he se pa r t i c l e s have p re sen t eddiff icul t ies, and they are in many cases st i l l unexpla ined.New, su rp r i s ing fac t s a re r evea l ed i f one cons ide r s t hecor re l a t i on func t ions .

The we l l -known obse rva t ion tha t t he p ion i s ex t raord inar i ly l ight was, in fac t , expla ined in c lassica l works ofthe ' 60s , even be fo re Q C D w as d i scove red : it is a G old-s tone mo de a ssoc i a t ed wi th ch i ra l sym met ry . In QC D

terms, i t s mass i s smal l due to the near vanishing of thel ight quar k masse s. Th is subjec t i s review ed in de ta i l byGa sse r and L eu twyle r (1987) , which a l so has o r ig ina lre fe rences .

Howeve r, t ak ing a c lose r l ook a t t h i s p rob lem, one a rr i ve s a t t he oppos i t e puzz l ing conc lus ion : t he p ion i ssu rp r i s ing ly heavy, g iven the l igh t qua rk masses . Indee d ,the p ion mass can be wr i t t en a s

ml = (mu+m d)K , (2.36)

w h e r e t h e c o n s t a n t K i s non ze ro i n t he ch i ra l lim i t . Th i scons t an t i s r e l a t ed t o t he qua rk co nden sa t e and the p ion-decay cons t an t /7 r = 2

l / 2 F 7 r = I3 l MeV by the f amous rel a t i on (Ge l l -Ma nn , Oakes , and Renne r, 1968)

K=2\{uu)\/fl . (2.37)

We do no t p re sen t i t s de r iva t ion he re and on ly no t e t ha tt he s t anda rd va lues o f t he qua rk masses1 7 a re (Gasse r andLeutwyler, 1987)

m d 7 M e V , m * 4 M e V . (2.38)

One then f inds a very large value of this constant associa t e d w i t h t h e q u a r k c o n d e n s a t e : K1700 MeV.1 8

Masses a re ex t e rna l t o QCD, bu t t he va lue o f K is anin t e rna l p rob lem, which shou ld be exp la ined by QCD.We fo rmula t e t h i s ques t i on in a s l i gh t ly more gene ra lwa y as th e first puzz le: (1) Why are the masses of thepseudoscalar octet mesons so sensitive to small quarkmasses?

The second we l l -known puzz l e r e l a t ed t o t he pseudosca l a r chan ne l s i s t he f amous W einbe rg (1975) "UA{\)prob lem ," whic h i s r e l a t ed t o t he SU(3) s ing l e t chan ne land th e 77' me son. Igno ring the u,d qua rk masses andconsider ing only the effec t of ra5, one can easi ly see tha tch i ra l pe r tu rba t io n the ory p re d i c t s 77' t o be li gh t e r t ha nth e 7] meson : t he fo rmer ha s ~ o f t h e " s t r a n g e " c o mponent , whi le the la t te r has f of i t .1 9 E x p e r i m e n t a l l y,m ^ 9 5 8 M e V, w h i c h is m u c h la rg e r t h a n t h e s e n a i v ees t ima te s . Le t us now fo rmula t e t h i s p rob le m so me wha tmo re genera l ly: (2) Why is the singlet chan nel so much

17 Quark masses are not physical, but are instead a kind oftheoretical parameter; so their values depend on their exactdefinition. In particular, they have perturbativ e anomalous dimensions; so the numbers depend on "resolution" (normaliza

tion point /x0) used. For example, speaking about bare qua rkmasses in the lattice Lagrangian, one has resolution on the scaleof lattice spacing fiQ a~x. The numbers mentioned correspondto the scale fi0= 1 GeV.

18 Accuracy of these "standard" numbers depends on whetherextrapolation of chiral perturbation theory is good for thestrange quark; see details in Gasser and Leutwyler (1987).

19We simplify discussion of this point for pedagogical reasons.The reader may consult the original paper (Weinberg, 1975) forhis estimates of the up per limit of the 77', with an d witho utOims) effects.


7/24/2019 RMP_v065_p0001

14/46


different from the octet ones? What is the mechanism respon sible for this splitting?

The th ird problem we address is also an o ld one, related to the fact that in pseudoscalar channel we do not seeeven a t rac e of the Zweig ru le . Na me ly, f lavor ch ang ingi s no t supp ressed in th i s chann e l , bu t ra ther enhan ced :(3) Why isn't the strange sector in the pseudoscalar multi-plet separated from the non strange one, as in other multi-plets? What is the mechanism of these mixings?

We now p roceed to d i scuss ion o f the p seudosca la rco r re la t ion func t ions . The main po in t i s tha t the coup l ing cons tan t s o f the mesons to the p seudosca la rcu r ren t s a l so can be exp ressed in t e rms o f known parame t e r s . For example, s tart ing with the defin i t ion of thep ion -decay cons tan t

7/24/2019 RMP_v065_p0001

15/46


Several es t imates of f h ave been pu t fo rward by No-v ikov et aL (1980), a l l suggest ing i t to be smaller than

/ V = ( 0 . 5 - 0 . 7 ) / , (2.48)

The s imples t es t imate is related to the J /tfj rad ia t ive

decay, wh ich i s a l so impor tan t because i t p rov ides somed i rec t in fo rmat ion abou t the mat r ix e lemen ts o f theg luon ic opera to r en te r ing the anomaly (Nov ikov et aL,1980). Inde ed, if the char me d quar ks are sufficientlyheavy, one can descr ibe the cc ann ih i l a t ion in t e rm s o f local ope rat ors . W e do not need to go in to detai l her e, buton ly commen t tha t , fo r the decays J/xp-^y-\-pseudo-sca la r meson , one has to dea l wi th the lowes t -d imens ion2 2

pseudosca la r g luon ic opera to r GG. Th e exa ct coefficientof th is operator in the effect ive Lagrangian is i rrelevant ,because we shal l only consider rat ios of the decay probabilities:

= 2 . 4 6 + 0 . 1 (2.50)

S ince tha t work was pub l i shed , ano ther l a rge con t r ibutio n in radia tiv e deca y of t/> ha s been foun d, th at of th edecay in to pho ton and T / (1 4 3 0 ) (orig inal ly cal led t ) . Repea t ing the same a rgumen t , one ob ta in s an even s l igh t lyla rger2 4 matrix element for th is part icle:

< 0 l G g l ^ ( 1 4 3 0 ) )= 1 1 2 0 2 ^0.74-f^.

Armed wi th th i s in fo rmat ion , l e t u s re tu rn to theco rre la t ion func t ions . U nfo r tu na te ly, the coup l ing cons tan t o f the p seudosca la r SU(3 ) s ing le t cu r ren t remainsunk now n . Never the les s , ju s t for the sake o f comp ar i sonwi th o ther p seudosca la r co r re la to rs , we have a l so p lo t t edin Fig . 10 the 77' con trib ut io n , mak ing an "ed uca tedg u e s s " based on the rat io of f^/f jus t deriv ed,^ 0 . 7 4 X i r

Whatever a re the uncer ta in t i es in th i s coup l ing , a qua li tat ive d ifference between the SU(3) octet and the s ingletco r re la to rs i s obv ious . Even i f the s ing le t I I (x ) / nf r e e ( x )is flat up to x ~ 0 .5 fm, the sp l i t t ing between them seemsto begin at *spii tt ing ~ 0 . 2 fm. In the whole in terval of inte rmed ia te d i s t ances x = 0 . 3 - 1 . 5 f m , t h e s i ng l et c o r r e lat ion func t ion i s abou t on e o rder o f ma gn i tu de smal le rth an th e 77 cor rela tor.

No w we swi tch to ano t her in te res t ing sub jec t : the

co rre la t ion func t ions o f the p seudosca la r g luon ic operat o r s . Genera l ly , we know very li t t le abou t them ; we dono t even have re l i ab le exper imen ta l in fo rmat ion abou tg lueba l l masses . Hea ted d i scuss ions on whether par t i cula r had ron ic resonances a re g lueba l l s t ake p lace a t spec ia l i zed con fe rences on had ron ic spec t ro scopy, and wecanno t go in to th i s ques t ion here .

Le t u s make on ly a genera l commen t tha t a l l g lueba l lcand ida tes a re ra ther heavy, wi th masses in the reg ion1 .5 -2 GeV . Th is i s qua l i t a t ive ly cons i s t en t wi th L G T

100

10

I tr

0.1 b-

0.01

z 1

_

-

I - - ^

-

1

' 1

71' /

1

1 \y

K

V

7)'

"

1 1

1 _

_-

:

-

%

i0.5 1.5

x ( f m )

FI G . 10. Norm alized pseudoscalar correlation functions vs distance x (in fm). The th ree solid lines show the 7r,K,rj channels,while the dashed line corresponds to the contribution of the 77'meson into the SU(3) singlet co rrelator.

Rev. Mod . Phys., Vol. 65, No. 1, January 1993

7/24/2019 RMP_v065_p0001

16/46


( lat t ice gauge theory) calculat ions [see rev iews in La t t i ce88 (1989), Lat t ice 89 (1990), Lat t ice 90 (1991), andTeraflop (1992)]. Quenched calculat ions also suggest thatthe l ightes t g luebal l is the scalar, with a ma s s of abou t1.3-1.5 GeV, wh i le p seudosca la rs are abou t twice ash e a v y. Of cou rse , resu l t s may be modified in ca lcu la t ionsgo ing beyond the q u e n c h e d a p p r o x i ma t i o n .

The genera l ques t ion of why all g lueba l l s have completely d ifferent mass scale , d is t inct from those typical ofh a d r o n s ma d e of quark s , remains es sen ti a l lyu n a n s w e r e d .2 6

The rea l ly re levan t ques t ion is the c o n t r i b u t i o n of v a r ious had ron ic s t a tes to g luon ic co r re la t ion func t ions , ind e p e n d e n t of w h e t h e r or not we cal l them gluebal ls .F r o m ou r d iscuss ion above we ob ta ined severa l m at r ixe lemen ts of the p seudosca la r g luon ic opera to r. Ou r es t imates d iscussed above lead to

< 0 | G G | - 7 > 0 . 9 G e V3 , iO\GG\^)^2.2 G e V3 ,(2.53)

< 0 | G G | T ? ( 1 4 4 0 ) > 2 . 9 GeV 3 ,

w h e r e we h ave a l so t aken a to be " f r o z e n " at a s 0 . 3 . Iti s t emp t ing to e x a mi n e the c o n t r i b u t i o n of t hese th rees ta tes to the p seudosc a la r g luon ic co r re la t ion func t ion .As befo re , in o r d e r to get an i dea of w h e t h e r the ma t r i xe lemen ts ob ta ined are l a rge or smal l , it is i n s t ruc t ive tonormal ize th i s con t r ibu t ion to the asym ptot ic al ly freeg luon ic con t r ibu t ion , wh ich is equa l to

9(N 2 1)K0(x)=(0\GG(x)GG(0)\0) = %-j . (2.54)

IT X

This equa t ion is d e r ived by p r o p a g a t i n g tw o g luons f romp o i n t 0 to x. The x d e p e n d e n c e is obvious , s ince theg luon opera to r GG has mass d imens ion 4. Apar t f rom

the color factor, the f o r mu l a is the s a me as in q u a n t u melec t rodynamics .

T h e e s t i ma t e d c o n t r i b u t i o n s of the i f s to the g luonc o r r e l a t o r are s h o w n in Fig. 11. We see t h a t the 7]r and7/(1440) matrix elements found above are i ndeed com parab le to the p e r tu rba t ive ones a l ready at d i s t ances assmal l as \ fm. M o r e o v e r, t h e y b e c o me a b o u t an o r d e r ofma g n i t u d e l a rg er at only s l ight ly larger d is tances .

I t is a mu s i n g to n o t e t h a t the 77' and 7?(1440) togetherc o n t r i b u t e to the g luon ic p seudosca la r co r re la to r in away very s imilar to the ^r,K,rj c o n t r i b u t i o n s to quarkpseudosca la r cu r ren t . The g enera l t endency of a rap idrise suggests a s t r o n g a t t r a c t i o n in t h i s channe l , s t a r t inga t abou t the s ame d i s t ances .

O n e may fu r ther sp ecu la te tha t the " t rue g luon ic

2 6In fact, in the interacting instanton approximation thedifference in mass scales is quite natural. In the IIA the quarkan d the gluon fields have completely different roles and differentdistribution in space-time. The former are distributed m ore orless homogen eously, while glue is concentrated in small spots ofthe stron g field, the instantons.

3 1 , , , , . , , , , J , , , , . , j , r

x(fm)

FIG. 11. Normalized pseudoscalar gluonic correlation functionto that corresponding to the propagation of two free gluons.Three curves correspond to the contributions of 77,77', 7/(1440)mesons, respectively.

s t a t e s " c o n t r i b u t e r o u g h l y an amoun t tha t causes theK(x)/KfreQ(x ) ra t io to level off at 1 for x < \ fm. If so,the th resho ld E 0 is expec ted to be r a t h e r h i g h , of the ord er 2 GeV or so. In p r inc ip le , one can t e l l whe ther it ist r u e or not from s tudies of the rad ia t ive decayY y - h h a d r o n s ( s ) , in wh ich ha d ron ic sy s tems wi th co rrespond ing invar ian t mass are p r o d u c e d . M o r e o v e r, theloca l ann ih i l a t ion hypo thes i s is even bet ter fulfilled he ret h a n for c h a r me d q u a r k s .

H. General properties of the scalar co rrelators

We c o n c l u d e ou r su rvey of the p h e n o me n o l o g y of the

QCD co rre la t ion func t ions wi th some remarks abou t thescalar correlat ion funct ions .

F r o m the phenomeno log ica l s ide the s i tua t ion is farfrom clear. His to rical ly , the f i rs t candidate for scalarme s o n s was the famous enhan ceme n t seen in the i soscalarTTTT s ca t t e r ing near 500 MeV, k n o w n in l i t e ra tu re as the" s i g ma me s o n . " T h i s n a me was a lso used in the " s igmael" (Ge l l -Mann and Levi, 1960), in w h i c h the s ca la r pa r t icl e is essent ial ly the " r a d i a l " (1=0) osci l la t ion of theq u a r k c o n d e n s a t e . It is not recogn ized as re sonance , butstill can be s t rong ly coup led to the scalar 7 = 0 cu r ren t .

The nex t sca la r mesons are i sovec to r and i soscalarpa i rs of part icles , /0 ( 9 7 5 ) and a 0 (98 0 ) . The i r c lo semasses and p a r t i cu la r decay modes have led to the suspic ion tha t they are not regu la r qq me s o n s , bu t r a t h e rfou r-body qqqq mesons con ta in ing " in t r in s ic s t rangen e s s . "2 7 Th i s l a t t e r observa t ion makes it v e ry improbab letha t they p lay any ro le in the spec t ra l dens i ty of non-s t r a n g e q u a r k c u r r e n t s uu, dd.

27 The interested reader can consult the proceedings of anyconference on hadronic spectroscopy, where this topic is repeatedly discussed.

Rev. Mod . Phys., Vol. 65, No. 1, January 1993

7/24/2019 RMP_v065_p0001

17/46


T h e 1= 0 s ca la r chann e l has two more resonances l is ted in the Review of Particle Properties, the / 0 ( 1400) and/ 0 ( 1 5 9 0 ) . The fo rmer decays p redo mina n t ly in to twopions and can therefore be p lausib ly ass igned as a non-s t r a n g e qq me s o n .2 8

T h e f0( 1590) was pro du ced d iffract ively in one experimen t on ly, and i t has a very in te res t ing dominan t mode ,rj'rj, in sp i te of the fact that i t i s very much suppressed bysmal l phase space . Tak ing in to accoun t i t s p roduc t ionmechan i sm and spec i f i c decay pa t t e rn , wi th a "g luon ietouch," we see that i t is a good candidate for scalarg lu oniu m. I ts mass value also f its wel l with wh atquenched lat t ice data tel l us (see, e .g . , Teraflop , 1992).Unfo r tuna te ly, the re a re p rob lems wi th th i s in te rp re tat i o n o f /0 ( 1590) : in par t i cu la r, the re a re s t rong exper imen ta l l imi ta t ions on J/ip>y -i-rfr}. I f co r rec t , they imply that th is part ic le cann ot hav e a sufficient ly s t ron gc o u p l i n g t o t h e g l u o n i c o p e r a t o r ( 0 | G2 | / 0 ( 1590)) .

No i sovec to r sca la r resonances (o ther than aQ m e ntioned) are in the Review of Particle P roperties; so one has

to conc lude tha t such mesons p robab ly do no t ex i s t , o rthey are too heavy and wide.Let us now discuss the qual i tat ive behavior of

sca la r co r re la t ion func t ions . In the 7 = 1 ch anne l ,K(x)/Kfree(x ) shou ld s t ron gly fal l off with xy b ecause thelowest in ter me dia te s tate has a mas s of at leas t 1 Ge V ormo re . In the 1 = 0 ch anne l the s itua t ion i s d i ffe rent , because the co r re la t ion func t ion

Ksc&iaT fJ=oM=(qq(x)qq(0)) (2.55)

possesses the fac to r izab le con t r ibu t ion , (qq)1, wh ichdoes not fal l off at large d is tan ces . Lar ge meson ic masse sin th is case mean that t ransi t ion to th is region should bera ther sharp .

Our point now is that i t i s poss ib le to guess at whatd i s t ances th i s t rans i t ion t ake s p lace ju s t by com par i son o fper tu rb a t ive con t r ib u t ions . In t e rm s o f the ra t io we usual ly use, K(x)/Kfree, i t me ans a rap id increase s tart i ngf rom the po in t where

^ s c a l a r , / = o U ) / ^ f r e e ^ )= = ^ 4 < # > ^ 6 / 3 - l . ( 2 . 56 )

This es t imate t e l l s u s tha t th i s cu rve p robab ly tu rns upstart ing from rather small d is tances , about j fm (which isaga in re la ted to and f rom a ra ther l a rge magn i tude o f thequark condensa te ) .

Summar iz ing , we have two impor tan t observa t ions :one expects the curve for K(x)/Kfree (a) to cu rv e up in

th e 1 = 0 [or the SU(3) singlet] scalar case, but (b) tocu rve down in the 1 = 1 [o r the SU(3) oc te t ] case . In o ther t e rms , one expec t s the ex i s t ence o f some a t t rac t ion inthe s inglet and a repul s ion in the octet chann el . Let usnow compare these conc lus ions wi th the behav io r o f thepseudosca la r channe l s . No te tha t the i r behav io r i s exac t -

28However, the sigma meson of the sigma model (Gell-Mannand Levi, 1960) should be m uch w ider at this mass.

ly the opp osi te : a s imilar rat io (c) goes up for the oc tet(7 r9K9r]), but (d) goes down for the singlet (77') case.

Le t u s re fo rmu la te these fou r s t a temen ts in t e rms o fsome wha t d i ffe ren t co r re la t io n func t ions . Cons id er ingfor s implici ty u, d quarks only, we define ins tead of thefou r p rev ious co r re la t ion func t ions , s ca la r and p seudo-sca la r wi th 1 = 0 and 1 , the fo l lowing l inear c om binat ions:

K + +=u LuRuRuL~\-dLdRdRdL , (2.57)

K + ._=uLuRuLuR+d RdLdRdL , (2.58)

K-+=uLuRdRdL+dLdRuRuL , (2.59)

K =uLuRdLdRJtuRuLdRdL . (2.60)

H e r e L,R s tand for left and right chira l i ty. Th e notat ions are as fo l lows: the f i rs t index here corresponds toflavor, the second to ch iral i ty, ( + ) me ans th is qu arkp rop er ty rema ins unch ange d , ( ) mean s i t i s chang ed .At smal l d i s t ances the dominan t con t r ibu t ion comes

f rom f ree -quark p ropaga t ion , wh ich co r responds to dominant J + + .Based on the d iscuss ion above of both scalars and

pseudosca la r co r re la to rs , wi th 7 = 0 , 1 , one may reachtwo impo r tan t conc lus ions : (1 ) Th e qua l i t a tive behav io rof those correlat ion funct ions is consis tent with the assumpt ions tha t the dominan t t e rm p roduc ing sp l i t t ing inpari ty and isospin is K ; and (2) devia t ions fromasympto t i c f reedom are much more rad ica l than those invec to r and ax ia l channe l s , and they show up a t muchsmal le r d i s t ances , x \ - \ fm.

Consequences of these observat ions wil l be d iscussed inthe nex t sec t ion , and we no te here on ly tha t the Kampl i tude co r responds exac t ly to the quan tum numbers

o f the in s tan ton - induc ed ' t Hoof t in te rac t io n .

III. THEORY OF MESONICCORRELATION FUNCTIONS

A. Potential models and heavy quarkon ia

This sec t ion i s somewhat separa te f rom the o thers , because i t appl ies to the physics of heav y quar ks only. W ehave inc luded i t main ly fo r pedagog ica l reasons : he reone can u se s imp le non re la t iv i s t i c l anguage based on thein te rac t ion po ten t i a l be tween quarks , wh ich , we hope ,wil l make the d iscuss ion clear.

Our main goa l i s to show how s tud ies o f the co r re lat ion func t ions may he lp to revea l in fo rmat ion tha t i snearly impossib le to get from an analysis of s tat ionarys t a t e s . Th is d iscuss ion is based on a pap er (Shury ak a ndZh i rov , 1987) tha t a t t em p ted to f ind exper im en ta l ev idence fo r a s t rong Cou lomb law.

Very heavy quarks and an t iquarks fo rm nonre la t iv i s t i cbound s ta tes s imi la r to pos i t ron ium, wi th the in te rac t iondescr ibed by a Cou lomb- type po ten t i a l (Appe lqu i s t andPol i tzer, 1975). T he force is as fun dam enta l as a


7/24/2019 RMP_v065_p0001

18/46


C o u l o mb l a w of e l e c t r o d y n a mi c s or t he N ewt on l aw ofgravi ty ; so it i s ce r t a in ly wo r th t ry ing to me asu re it mo r eprecisely. In fac t, Q CD does not p red ic t exac t ly aCou lomb law, because o f the runn ing coup l ing cons tan t ,which effect ively depends on the d i s t ance be tweenquar ks . The equa t ion der ived in Appe lq u i s t and Po l i t ze r(1975) for the potential is

4 ^ Q C D 3 R

(llNc~2 Nf)ln[l/(RACoulomb)] R '

w h e r e N c a n d Nf a r e t h e n u mb e r of colo rs an d flavors.No te tha t th i s po ten t i a l con ta in s a p a r a me t e r AC o u l o m b.I t s me a s u r e me n t is c ruc ia l for set t ing the absolu te scalein QCD, which is also needed for lat t ice calculat ions (d iscussed below).2 9 H e a v y q u a r k o n i u m is in p r inc ip le anidea l p lace to measu re the QCD sca le , because the po tent ial is p e r t u r b a t i v e in n a tu re bu t s t i ll p roduce s l a rge observable effects.

Unfo r tuna te ly, ne i ther c no r even b quark s a re heavye n o u g h for th is s imple idea to be app l i cab le . How ever,these mesons are very wel l described by an effective poten t i a l V efr (r), a c o mb i n a t i o n of con f in ing and Cou lom bforces. To be specific , le t us consider tw o potent ia ls used:the Mart in potent i

RMP_v065_p0001

Documents

Transcript of RMP_v065_p0001