NON PERTURBATIVE EFFECTS AND QCD SUM RULES

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NON PERTURBATIVE EFFECTS AND QCD SUM RULES

H. Rubinstein

To cite this version:

H. Rubinstein. NON PERTURBATIVE EFFECTS AND QCD SUM RULES. Journal de Physique

Colloques, 1982, 43 (C3), pp.C3-249-C3-253. �10.1051/jphyscol:1982349�. �jpa-00221903�

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CoZZoque suppZ6ment au n o 12, Tome 43, d6cembre 1982 page

NON PERTURBATIVE EFFECTS AND QCD SUM RULES H.R. Rubinstein

Weizmann I n s t i t u t e of Science, Rehovot, Israel ChaZmers University of TeehnoZogy, Gijteborg, Sweden

- I n t r o d u c t i o n . Consider a p o l a r i z a t i o n operator generated by some c u r r e n t , p h y s i c a l o r unphysical

,

t h a t has t h e general form j ( r ) = q r q . The c o r r e l a t i o n f u n c t i o n o f t h i s c u r r e n t generates, t o z e r o t h order a quark l o o p i n a given p a r t i a l wave w i t h quan- tum numbers determined by

T.

A t v e r y s h o r t distances because o f asymptotic freedom t h e amplitude i s determined by t h i s term. Separating t h e quarks (by t a k i n g d e r i v a - t i v e s i n t h e conjugate v a r i a b l e ) one must i n c l u d e c o r r e c t i o n s . Gluon exchanges cannot keep quarks from escaping. Therefore, a f t e r normal o r d e r i n g o t h e r operators besides t h e u n i t operator survive. T h e i r Wilson c o e f f i c i e n t s a r e c a l c u l a t e d p e r t u r - b a t i v e l y w h i l e t h e m a t r i x element, which cannot be evaluated, parametrizes our i g - norance o f what goes on a l o n g d i s t a n c e

.

These operators l i k e <GG> and <qq> a r e gauge and r e n o r m a l i z a t i o n group i n v a r i a n t and t h e r e f o r e u n i v e r s a l . The expression obtained i s now matched w i t h a sum o f resonances and e v e n t u a l l y a continuum. One t h e r e f o r e o b t a i n s a d e t e r m i n a t i o n o f masses and couplings i n terms o f fundamental Lagrange parameters.

Since e a r l i e r work has been described i n e a r l i e r conferences ( I ) I w i l l make a c r i - t i c a l e v a l u a t i o n o f these r e s u l t s and discuss new c a l c u l a t i o n s (2,3,4,5,7,8,14).

Heavy quark systems. I n t h i s case t h e mass o f t h e quark f i x e s t h e scale. The d e r i - v a t i v e s o f t h e p o l a r i z a t i o n f u n c t i o n become:

where Q O i s a spacelike reference p o i n t upon which t h e r e s u l t s should n o t depend. 2

J J 2 2

I n t h e QCD s i d e we have Mn(c)=An(l+an(J ,e)as+bn(J ,c)@), where c=Qo/4mh and

4x2 2

4 9

< $

G a G a >(4mc ) - I i s t h e famous gluon condensate. On t h e resonance s i d e

vv I.lV 2

1 m n ( s ) = 9 m ~ / ~ ~ 2 6 ( s - m ~ ) + continuum. S a t u r a t i n g o n l y w i t h one resonance one o b t a i n s t h e formula: r n ( F ) = ( ~ i + ~ i ) - l . I n f i g u r e 3 , , I one sees t h e r e s u l t f o r a charmonium s t a t e (". The breakdown

" = = , " - . - - -

f o r h i g h n s i g n a l s as i t can be checked t h a t t h e t h e o r y i s no l o n g e r v a l i d . For small n t h e disagreement has t o do w i t h t h e breakdown o f t h e one resonance approximation. The smooth matching works t o o w e l l .

b, For t h e v e c t o r c u r r e n t and n=4 t h e q(3100) i s s u f f i - c i e n t t o s a t u r a t e as i t can be checked d i r e c t l y from experiment. One may ask which a r e t h e parameters i n - 3.6 volved. For every f l a v o u r t h e r e i s t h e mass o f t h e

*

-

3 S O quark (mu=md=D, e t c ) . For a l l p a r t i a l waves and f l a -

3.aLl-

3.4 ⁴ ⁶ ⁸ ¹⁰ ¹² ¹⁴

^,,

vours one has hgCD and t h e values o f t h e m a t r i x e l e - ments l i k e <qq> and <G:~G:~>. AOCD agrees w i t h pre-

F i g u r e 1

sent estimates and t h e ' o t h e r s have been e s t a b l i s h e d by o t h e r meth d 1 i ke c u r r e n t algebra and l a t t i c e simulations. 965

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982349

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C3-250 JOURNAL DE PHYSIQUE

Though t h e r e i s no systematic study o f h i g h e r dimensional operators and many times t h e i r Wilson c o e f f i c i e n t s a r e n o t known.Inallcases when estimates a r e a v a i l a b l e t h e i r e f f e c t i s small. Other hidden parameters are: 1) t h r e s h o l d values f o r t h e continuum. I n charmonium these e f f e c t s a r e v e r y small, i n baryons too. For massless quark systems these e f f e c t s a r e e s s e n t i a l b u t p h y s i c a l assumptions o r experimental i n f o r m a t i o n unables t h e p r a c t i t i o n e r t o make accurate p r e d i c t i o n s . 2) I n t h e case o f baryons t h e r e i s ambiguity i n t h e choice o f c u r r e n t (798).

I So = 3.01 .02 Gev 3 ~ 1 = 3.10 .O1 Gev 3 ~ 0 = 3.40 .O1 Gev 3 ~ 1 = 3.50 .O1 Gev 3 ~ 2 = 3.56 .O1 Gev 'PI = 3.51 .01 Gev m -m = 60 Mev

Y

QB 2 2

m, = 1.25 Gev(p =-m )

As an example we see t h e masses p r e d i c t e d f o r t h e charmonium s t a t e s and some r e s u l t s on t h e bottonium spectrum. I n t h e case o f t h e bottom quark t h e r e a r e prob- lems t h a t r e q u i r e some f u r t h e r assumptions ( 9 ) b u t d i f f e r e n t techniques y i e l d v e r y s i m i l a r r e s u l t s and one i s t h e r e f o r e c o n f i d e n t on t h e r e s u l t s . These c a l c u l a - t i o n s a r e t h e b e s t k own method t o e s t a b l i s h t h e mass o f t h e heavy quarks 110). The l p l s t a t e has n o t been observed b u t i t s p o s i t i o n seems t o be p r e d i c t e d a t 3.51 by a l l models (11).

L i g h t mesons w i t h 1=1. The l o w e s t l y i n g mesons were s t u d i e d i n Ref. 3 . Here I de- s c r i b e t h e c a l c u l a t i o n s of Ref. 3 which have completed t h e knowledge o f t h e spectrum up t o s p i n 2. I n t h e case o f massless quarks one must i n t r o d u c e a s c a l e i n t o t h e problem. This i s accomplished by c a l c u l a t i n g a t q l a r g e and t a k i n g a Borel 2 transform.

J 2

iMn (Q )=1 i m

$:

and Q ~ / ~ = M ~ o f (n-1 ) ! - ' ~ ~ ~ ( - d / d ~ ) ' ~ n ~ ( Q ~ ) g i v i n g

2 3

I exp(-s/M ) Imn (s)ds

I n t h e case of t h e A, meson which we discuss as an example t h e p o l a r i z a t i o n o p e r a t o r

where t h e operators discussed e a r l i e r a r e shown t o appear. The o t h e r s can be expres- sed i n terms o f t h e s e o r estimated s i n c e t h e i r c o n t r i b u t i o n i s small. A f t e r a Borel t r a n s f o r m t h e sum r u l e ( i n c l u d i n g t h e continuum which i s important) reads

A s i m i l a r sum r u l e can be obtained using t h e a x i a l c u r r e n t . The r e s u l t s seen i n Fig.

2 a r e q u i t e s a t i s f a c t o r y . I n these cases t h e continuum i s important b u t t h e value t h a t g i v e s best r e s u l t s i s very reasonable. Large number o f s t a t e s can be c a l c u l a t e d except f o r t h e B meson where a c c i d e n t a l l y t h e presence o f s u b t r a c t i o n s makes t h e c a l c u l a t i o n impossible. Couplings as shown can a l s o be calculated. For completeness

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F i g u r e 2

750k30 776 750+30 780

920 892

1070 1020

1270 1270

1500 1516

1320 1317

1420 1434

1000 981

1000 980

1350 1300

1150 1100-1300

1270 1285

1460 1418

n o t

available1231

A3 2 - I 1630 1660

Theory Exp Table 1

2

f T =

2

=I25 Mev (133 Mev),

%

^=2.3*.1 (2.36?.18), ff=0.037r.003 (0.04) ( 5

Couplings f o r R=O and 1 l i g h t quark mesons

-

we i n c l u d e t h e r e s u l t s o f Shifman e t a1 w i t h o u t discussing them.(12)

N o t i c e t h a t t h e parameters a r e mn=md=O, ms=150 Mev, <iq> ( i t s breaking i s un- important) and

4.

The small d i f f e r e n c e s l i k e f, A2 can be understood ( d i f f e r e n t thresholds), b u t o v e r a l l t h e theory p r e d i c t s 1=0,1 degeneracy i n agreement w i t h experiment. There a r e s t i l l c o n f l i c t i n g models f o r CD sum r u l e s on s c a l a r s (14).

Our r e s u l t s seem confirmed by l a t t i c e s i m u l a t i o n s ?6

1.

L i h t heavy quark systems.(16)(open bottom) Here we o n l y mention t h a t f -200 Mev

mg1

which i s incompatible w i t h models t o e x p l a i n r D + / ~ ~ o and mo++

-

in0-!$800 Mev.

This r e s u l t i s due t o t h e term mb<uu> which s p l i t s s t r o n g l y o p p o s i t e p a r i t y states.

T h i s term seems " n o n - p o t e n t i a l " .

Baryons. For t h e baryon o c t e t t h e r e a r e two p o s s i b l e c u r r e n t s . The n a t u r a l choice i s t h e one t h a t has an SU(6) non r e l a t i v i s t i c l i m i t and couples t o t h e non p e r t u r b a t i v e operators. For t h e decuplet t h e choice i s unique.

The nucleon p o l a r i z a t i o n f u n c t i o n i s given by

4 2 2

I d x e i P ' X < ~ ~ ~ n n ( x ) ~ n ( ~ ) ~ ~ > = PFi(p )+l.F2(p

1.

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C3-252 JOURNAL DE PHYSIQUE

On dimensional counting i t follows t h a t F1 i s even and F2 i s odd. As a consequence the function F2 i s proportional t o <qq> without a mass f a c t o r and i s furthermore enhanced by the elimination of one loop i n t e g r a l . I t completely overwhelms the bare loop term. The other sum rule i s standard and several terms including the two con- densates and bare loop compete. After Borel transform the two sum rules become:

0 ,,

4 4 2 - M ~ / M ~

2 a M = ZIT) AN MN e where a= - ( ~ n ) ~ < q q > , b = 2

<G

a G a

>,

and MN i s the I.lv .I.lv

mass of t h e nucleon and A , the coupling t o three quarks, a quantlty t h a t appears in proton decay calculations"(15). M i s the Borel variable as usual.

Solving f o r the nucleon mass one obtains:

Figure 3

and a s l i g h t l y more complicated formula i f the continuum i s allow- ed. In Figure 3 we see the r e s u l t s as a function of M. The r e s u l t s show how f o r massless quarks the chiral condensate generates the nucleon mass. The continuum im- proves the r e s u l t s but i t i s a minor e f f e c t . Analogous formulae can be written f o r the o c t e t and decuplet and the agreement i s ex- c e l l e n t . In the decuplet there i s a-new operator contributing:

< q ~ ; ~ h ~ q > = m ~ < q q > and i t s value has been estimated elsewhere. I t i s i n t e r e s t i n g t h a t these calculations-depend c r u c i a l l y on y = ( < u ~ > - < q q > ) / < u u > . ( 17)

Allowing f o r d i f f e r e n t strange and u quark masses but s e t t i n g y=O drives the E below t h e C. The calculation also yields a correc- MN=900(940) MA=1070(1 115) Mz=l 170(1185) tion t o proton l i f e t i m e t h a t M1=1370(1320)

-

MN*;1240(1235) My*=1370(1 385)

seems

Out simp1est S U ( 5 ) ' M~*=1510(1520) Ma-=1650(1670) I t i s a l s o possible t o apply the

theory t o three point functions.

Table 2 though we can o t discuss these

r e s u l t s here 1181, one can compute the pion nucleon coupling constant and obtain

in remarkable agreement with experiment. Calculations of higher partial waves f o r baryons show the negative parity s t a t e s f a r above the 56 representation a s desired.

Conclusions. A theory based on QCD and some dynamical assumptions about the conver- gence of dfspersion i n t e g r a l s can reproduce remarkably well t h e spectrum of hadrons with very few parameters. Moreover i t predicts the appearance of i n t e r e s t i n g terms in the spectrum of these s t a t e s . In particular: establishes the spin dependence of

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t h e c o n f i n i n g f o r c e s a r e s h o r t distances and unables t o demonstrate how c h i r a l breaking endows t h e p r o t o n w i t h mass. There a r e a l s o non l o c a l terms t h a t a r e neces- sary and t h e i r presence would be a d e c i s i v e element i n p r o v i n g i t s v a l i d i t y . L a t t i c e s i m u l a t i o n s o f these systems a r e o f g r e a t i n t r e t. There i s some work on l a t t i c e s t h a t might e x p l a i n t h e success o f t h e theory 7193. C h i r a l breaking seems t o occur a t much s h o r t e r distances than confinement dressing t h e quarks and e s t a b l i s h i n g t h e p r o p e r t i e s o f t h e bound s t a t e s a t s h o r t distances.

Because o f time I cannot discuss o t h e r i n t e r e s t i n g issues l i k e t h e Schwinger Smilga Cronstrom gauge (201, f u r t h e r c a l c u l a t i o n s on Wilson c o e f f i c i e n t s ( 2 1 ) , form f a c t o r s and many o t h e r t o p i c s . References a r e f o r guidance o n l y and c e r t a i n l y incomplete.

I would l i k e t o thank my f r i e n d s a t Weizmann f o r discussions and my c o l l a b o r a t o r s L.H. Reinders and S. Yazaki t h a t played a c r u c i a l r o l e i n a l l our r e s u l t s .

References

V. Zakharov, High Energy Physics Conference, Wisconsin 1980, p. 1235.

M.A. Shifman, I n t e r n a t i o n a l Symposium on Electromagnetic and Weak I n t e r a c t i o n s , Bonn 1981.

L.J. Reinders, H.R. Rubinstein and S. Yazaki, Nucl. Phys.

8186

(1981) 109.

L.J. Reinders, H.R. Rubinstein and S. Yazaki, Nucl. Phys. (1982) 125.

S. Narisson and E. de Rafael, Phys. L e t t . (1981) 57.

M.A. Shifman e t a l , Phys. L e t t . t o appear.

H. Hamber and G. P a r i s i , Phys. Rev. L e t t .

2

(1982) 1792, and p r e p r i n t s .

T. Banks, R. Horsley, H.R. Rubinstein and U. Wolf, Nucl. Phys. ( F S ~ ) (1981) 692.

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B188

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8191

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Pascual and Tarrach, p r e p r i n t , Madrid 1982.

For a discussion, see M.A. Shifman, Ref. 1.

See Ref. 2. For a l l quarks masses: P. Leutwyler and S. M a l l i k , Physics Reports t o be published.

This s t a t e should be observed by i s o s p i n v i o l a t i o n decay. See N. I s g u r , H.J. L i p k i n , H.R. Rubinstein and A. Schwimmer, Phys. L e t t . @J (1979) 79.

M.A. Shifman, A.I. Vainshtein and V . I . Zakharov, Nucl. Phys.

8147

(1979) 385.

V.A. Novikov. M.A. Shifman, A.I. Vainshtein, M.B. Voloshin and V . I . Zakharov,

~ h y s . Rep. 4 i (1978) I .

E.V. S h u r y a r a n d c o l l a b o r a t o r s , f o r example, discuss i n s t a n t o n e f f e c t s i n these channels

.

V.S. Berezinsky, B.L. I o f f e , Ya I. Kogan, Phys. L e t t . (1981) 33.

L.J. Reinders, H.R. Rubinstein and S. Yazaki, Phys. L e t t . (1981) 305.

E.V. Shuryak, Novosibirsk p r e p r i n t . See Shifman, Ref. 1.

L.J. Reinders, H.R. Rubinstein and S. Yazaki, t o be published.

L.J. Reinders, H.R. Rubinstein and S. Yazaki, Weizmann I n s t i t u t e p r e p r i n t 1982.

J. Kogut, M. Stone, H.W. Wyld, J. Shigemitsu, S.H. Shenker and D.K. S i n c l a i r , Phys. Rev. L e t t . 4 8 (1982) 735.

J. Schwinger, P a r E c l e s , Sources and F i e l d s , 1970, p. 271;

M.J.

Dulobikov, A.V.

Smilga, Nucl. Phys. 8185 (1981) 109; C. Cronstrom, Phys. L e t t . (1980) 267.

W. Hubschmid and S. m i k , p r e p r i n t butp 10/1982.