Nuclear Physics B (Proc . Suppl .) 21 (1991) 317-322North-Holland

QCD GENERATES THE p-RESONANCE

Nikolaos PAPADOPOULOS

Institut für Physik, Johannes Gutenberg-Universität, Staudinger Weg 7, Postfach 3980, D-6500 Mainz,West Germany

ABSTRACT

The question whether the asymptotic QCD amplitude contains potentially hadronic resonances isexamined by a mathematically rigorous method, based on the theory of maximally converging sequencesof polynomials and conformal mappings . It is shown that the extrapolated amplitude to the physicalcut exhibits indeed a bump structure which corresponds to the p-resonance .

It is by now well known that the inclusionof nonperturbative terms to the perturbativeQCD amplitude allows the determination of ha-dronic properties . The connection between theso obtained asymptotic QCD amplitude and ha-drons can be made by methods usually calledQCD-sum rulesl ' 2 ' 3 . One of the most strikingand impressive results was the determinationof the p-resonance parameters by Shifman-Vain-shtein-Zakharov l .

In order to appreciate properly this re-

sult, one has to realize the following points .The determination of low energy parametersfrom the asymptotic QCD amplitude is by farnot a straightforward task since it is accom-

panied by the presence of very serious insta-bilities . It is generally an "ill posed" prob-lem and has to be considered within the frame-

work of inverse problems4 ' S . That means everyQCD sum rule method has to be tested to seewhether it is stable in the sense of the in-

verse problems . For the above determination ofthe p-mass, for instance, the inverse La-place-kind sum rule was used, of which it isnot proven that it is stable . Of course, this

does not mean that the inverse Laplace-likesum rules are not at all useful . It means thatone has to be careful . The "stability" we

usually test in most phenomenological applica-

tions is what we may call a relative stabilitysince it concerns only the chosen set of para-

Elsvvicr Sci( nce Publishers I1N. (North-Holland)

317

meters . A second different choice of parame-ters may lead to completely different results .The determination of the p -resonance parame-ters was also achieved by more rigorous me-thods which are proven to be stable in the ap-propriate sense . 6 So the question of the sta-bility in the determination of hadronic reso-nance parameters may be considered as appro-priately solved .

There still remains another problem which

is more fundamental than just the determina-

tion of a resonance parameter : Is the asympto

tic QCD amplitude at all able to generate aresonance in the cut region? This question

which lies in the air since the pioneering

work of the SVZ towards the end of the seven-

ties, has not been answered till now , even if

we can determine the p-resonance parameters in

agreement with experiment . The reason is that

in all the above determinations we have as-

sumed the existence of a p -resonance within

the QCD amplitude and with a given parametri-

zation we were able to determine some of its

parameters . In principle this could also be

possible even if no sign of a resonance in the

QCD amplitude existed . We may state our ques-

tion also differently . Can the asymptotic QCD

amplitude produce by itself a resonance bump

in the cut region? It is understood that this

is the first problem (the existence of a reso-

318

nance in the QCD amplitude) and what has been

solved till now (the correct and precise de-

termination of its parameters), is the second

problem .

Here we shall deal with the first problem

and give its solution . The answer to the above

question is affirmative. QCD is able to gene-

rate hadronic resonances . This strengthens

enormously our belief in QCD . It is true, even

given the fact that till now in general even

the best concrete predictions on physical ob-

servables are mostly calculable within 30-50% .

The purpose of what follows is not to de-

termine in high precision resonance parameters

but to prove or disprove the existence of such

resonances . In order to achieve our goal, we

use rigorous mathematical methods which were

developed by ï. They are based on the theory

of "maximally" converging polynoms"8 and on

the ideas of conformal mappings .9 To our know-

ledge it is the first time that such methods

are applied in connection with QCD.

Our starting point is the isospin 1

two-point function 11(g2 ) corresponding to the

P-particle :

ijdxelgx <OJT JP (x) JV(0)IO> =

(gpqV - gpvg2

) il(g2) -

(1)

For the QCD amplitude HQCD(g2) we take the

perturbative and nonperturbative parts given

by SVZ1 (see below) . The domain of analyticity

of the physical amplitude II (s=q2) is the com-

plex

cut plane,

and the QCD amplitude 11 QCD (s)

is valid for asymptotic values of s (s---) . We

assume (for technical reasons at this moment)

that we

know 11QCD

only

on

an interval

[a,b],

a, b E ~

in

the

negative

real

axis

in

the

s-plane .

We

may

call

the

QCD amplitude ITQCD

given in this interval our data and the inter-

val [a,b] the data domain . In this case the

data domain is an interval, but in general itcan be considered as a two-dimensional domain .

N. Papadopoulos/QCD generates the p-resonance

Our task now is, starting from the data do-

main with our data (the QCD amplitude II QCD) to

extrapolate and- to obtain hopefully (and of

course approximatively) the physical amplitude

in the resonance region on the physical cut .

Our first step will be to expand the data

in a set of polynomials { Pn (s), nE:LNI and to

hope that such an expansion will be valid also

outside of the data domain . This means, we

essentially try a certain generalization of

the Taylor expansion. The point so which car-

ries the information about a given function,

say 11(s), and the Taylor polynomials tn (s),

n E9V which approximate the given function H(s)

11 (s)

:Z: tn(s)

with

n(k) k

(2)t (s) = 1 11

(s ) sn k=O o

are replaced by the data domain and a set of

maximal

converging

polynomials

i Pn(s)}

or

Isn(z)I

(see

below) .

As

is

well

known,

the

most important parameter in the Taylor expan-

sion is the convergence radius P which corres-

ponds to the maximal convergence domain KP

(disk) and its boundary @KP , the maximal con-

vergence curve (the circle with radius P, S P

aKP ) .

We also have of course for every R <P

a

convergence domain (KR e KP ) and a convergence

curve SR = aKR(the circle with radius R < P) .

In our more general case it can be proven that

the so-called maximally converging polynomials

{Pn (s)I exist together with the corresponding

convergence domains BR , in general bigger than

the

data

domain . 8 In particular,

the maximum

convergence domain BP is given when the con-

vergence curve CP = 3BPhits the first singu-

larity of n which in our case would be the

cut .

This result brings us a step further . The

situation in the generic case is given in

fig .1 . This result, although mathematically

not trivial and quite interesting, is physi-

cally useless for our purpose . What we want is

not only to approach the cut, but we have to

hit the whole cut .

Fig . 1 :

H =(a,b] : data domainBR

: convergence domain (R < p)

CR = B8R : convergence curve

Cp = N o : maximum convergence curve

At this point, in order to achieve our

goal, we need two things : first a new idea and

second, some luck . The idea is to try to make

an expansion not in polynomials in s but in

analytic functions Ifn (s) n E DV} in s which

have a similar analytic domain as the

two-point function II (s) in such a way that the

new maximal convergence curve coincides with

the cut! To be more precise, the functions

Ifn (s)} will correspond to maximal converging

polynomials in a new complex variable z

IsnM,

n E OV}

obtained from s by a conformal

mapping : z = 4'(s) . This means that fn(s) :=

Sn (T(s))We can immediately understand that such a

particular conformal mapping which makes the

relevant convergence curves hit the cut, is

something very special . In the general situa-

tion this is even not possible . That it is

possible in our physical problem, is a highly

nontrivial result . This means that we are in-

deed in the lucky situation which is necessary

to obtain a physically useful result . In addi-

tion to this we have to point out that it is

furthermore proven that the whole rathod is

stable10 (in the sense of inverse problems)

and that by the above conformal transformation

we have an improvement of the convergence

velocity (a further nontrivial result is rela-

ted to the principle of Groetzsch) .

N. Papadopoulos/QCD generates the p-resonance

With the above preparations we are now co-ming to the construction of the appropriate

"maximal converging analytic functions" forthe extrapolation of the asymptotic ampli-

tude 11QCD(s) on the cut in the s-plane. The

QCD amplitude we start with, for the isovector

current Jp = 1/2(ûyuu - dyed) which is domina-

ted by the p -resonance, is given by

flQCD(s) =

]'per(s) + IInp(s)

.

(3)

This perturbative term is the well-known ex-

pression

where

41r2 11per(s) =

1/2 { - k - a1 lnk - a2 ln)k/ k -

(a2 - a12F) /)L + const .}

,

al = 12/ (33-2nf) ,

a2 = 72(153-19nf)/(33-3nf) 3

F = 1 .986-0.115nf,

k = ln(-s/A2 ) .

given as

(mu+md )

<qq> _

a<qq>2 = 1 .8 * 10-4 (GeV) 6 .

+ (1r2/6s2 ) <a/1GG> + (448n3/81s3 ) a<qq>2

- 2f2 m,22 _ - 1.8 * 10-4 (GeV) 4

<a/1GG> = 1.2 * 10-3 (GeV)4

319

We shall use a value of 150 NteV for the QCD

scale parameter A and of = 3 . For the nonper-

turbative term we take the expression o¬ SVZ1

412 11np (t) = (n2/s

2) 2(mu+md) <qq>

(5)

with the corresponding numerical values for

the condensates,

We now proceed in three steps . First we

make a conformal (biholomorphic) transforma-

tion . Second, we determine the expansion coef-

ficients for the maximal converging polyno-

mials and third we return to the s-plane and

to the expansion on the physical cut .

320

As already pointed out, it is not trivial

that there exists a conformal mapping Y with

essentially the following properties :

C - cut E

(7)

with E the interior of an ellipsis .

More precisely : Y maps the complex s-plane to

the complex z-plane in such a way that the cut

is mapped on the ellipsis 3E with foci -1 and

}1 . the boundary of E (T(cut) = K), and the

data

domain H = [a,b] to the interval

p -1,+1] =

Y (H)

(the

new

data

domain

in

the

z-plane) . This is also represented in fig .2 .

Fà® .2 :

ï' (Cut)

= Co

Y R

; convergence domain (R < p)

-R= axR : convergence curve

Cp = Wo : maximum convergence curve

h(z) :_RQCD(T-1(z))

s-plane

Coming now to the second step, we consider

the extrapolation in the z-plane . Here we mayuse directly the method of maximal converging

polynomials .8 The most important fact is thatthe convergence domains kR which are associa-ted to the new data domain oy = T (H) are theinteriors of the ellipses, and the maximal

converging curve eP:= DKp coincides with the

picture of the cut T(cut) = fp ! For the set ofmaximal converging polynomials we may choosethe so-called Szegö-polynomials ISn (z)I

of de-gree

n,

n e W }

given

by

the

following

con-struction :

We first want to approximate our new datain the z-plane

(S)

within the new data domain

= [-1,+1] with

N. Papadopoulos/QCD generates the p-resonance

the polynomials Sn (z) . We use for that purpose

the scalar product in the interval [-1,1] gi-

ven by<fjg> := fdx f*(x) g(x)

(9)

and the corresponding orthonormal polynomials

ITn (z) i <TiITk>

= Sik,

i, k c

(N }

.

(10)

The Szegö-polynomials can be expressed with

the help of the above orthonormal polynomials,

and the coefficients ck are determined by thecondition

<Tkih - Sn> = 0

for

k = 0,1,2,

. . .

n

(12)

or equivalently by

nSn(z) = 1 ck Tk (z)

(11)k=0

ck = <Tk j h> .(13)

The so obtained set of maximal converging

polynomials ISn I does not only approximate the

new data in ?E= [-1,1], but by constructionalso the corresponding function in the full

domain, inside the ellipsis and on the ellipsis(cut) itself .

In the last step we obtain the set of maxi-

mal converging analytic functions ifn (s)} in

the complex s-plane by transforming back the

above set of the maximal converging polyno-

mials Sn (z) :

f n (s) := Sn (y(s)) -

(14)

The functions fn (s) for an appropriately cho-

sen n = no represent not only the asymptotic

QCD amplitude RQCD as it is given by egs .(3-6)in the interval [a,b], but also its extrapola-

tion which is, by the above construction, va-

lid in the full domain inside the cut, that

means in the E-cut and on the physical cut .So, all the useful theoretical information

we have in RQCD, eq .(3), is now contained in

the expression

QCDn (s) := fn (s) .(15)

This includes of course also the imaginary

part of R(s) we are interested in . In particu-

lar, we have for s real and positive the ima-

ginary part

There is only the n0 left to be determined .

It can be shown that there exists an optimal

value n0 which is a compromise between inter-

polation and extrapolation error. The discus-

sion of this point and further details are

presented in 10 and 11 . Here we shall deter-

mine no "minimally" just after we have reached

a reasonable approximation of the QCD data11QCD(s),

egs .(3-6) in the data domain . This

gives for a data domain (-100,-11 GeV no = 5.

So we expect the appropriate n0 to be between

6 and 8 . The result for the imaginary part for

the extrapolated QCD amplitude Ano (s) 9 , pro-

perly normalized to correspond to the quotient

RI=1 = 6(e+e - hadrons/ a(e+e - N+p ), is

given in fig .3 .

An (s)

:= lim Im flQCDn (s+iE)

.

(16)o E-"0 0

0

1

2

3

4

5

s [Gel/, 2]

FIGURE 3Extrapolation results corresponding to thequantity RI=1 .

This result goes without saying . The ques-

tion we stated at the beginning of this paper,

is answered affirmatively . The asymptotic QCD

N. Papadopoulos/QCD generates the p-resonance

amplitude is able to generate the p-resonance

in the cut region . This is a reliable result

since it was derived by rigorous mathematical

methods. There are only a few contents to be

added . In order to appreciate this result even

more, it should be noticed that even for the

positivity of the output An (s) there was no

guarantee by itself at the0 beginning . There

are in fact the nonzero values of the conden-

sates in eq . (6) which allow to obtain the re-

sonance structure at all .

The precise determination of resonance

parameters was not our aim. In spite of this

the p-mass came out quite stable near the ex-

perimental value :

mp QCD = 0.5 GeV2

.

(17)

321

For the other parameters and especially for

the height of the resonance, more refined me-

thods are necessary . It seems possible that

this can also be achieved ; it is left for ano-

ther publication .11

A question which naturally arises is whe-

ther the above results may be obtained also

for further hadronic resonances . Some prelimi-

nary encouraging results concerning the A re-

sonance were obtained, which we hope to in-

clude in a next publication .

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