SPIN AND QUARK MASS EFFECTS IN QCD SPECTROSCOPY

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SPIN AND QUARK MASS EFFECTS IN QCD SPECTROSCOPY

H. Rubinstein

To cite this version:

H. Rubinstein. SPIN AND QUARK MASS EFFECTS IN QCD SPECTROSCOPY. Journal de Physique Colloques, 1985, 46 (C2), pp.C2-71-C2-75. �10.1051/jphyscol:1985207�. �jpa-00224518�

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SPIN AND QUARK MASS EFFECTS IN QCD SPECTROSCOPY

H.R. Rubinstein + *

Physics Department, University of Stockholm, Sweden

Résumé - Les forces de spin dépendent de la saveur. ELl.es induisent ainsi des effets importants à travers la brisure chirale. La brisure d'isospin peut aussi être importante. En utilisant des méthodes basées sur la QCD, nous discutons un certain nombre d'effets qui sont non négligeables et qui peuvent être mesurés à LEAR.

Abstract - Spin forces are flavour dependent. They also induce large effects through chiral breaking. Isospin breaking also can be Large. We discuss using QCD methods a variety of effects that are large and can be measured at LEAR.

The purpose of this Meeting is to discuss spin effects and learn about strong forces from their spin dependence. Spin dependent forces are obviously important in spectroscopy. In fact the pattern of spin dependent splittings has shown unexpected structure. The large separation between the vector and pseudoscaJar states of charmonium were at the origin of the discovery of the gluon condensate.

In this lecture I will concentrate on the theory of spin splittings and related problems due to quark mass dependence of the interactions. As a somewhat related subject 1 will discuss isospin violations dictated by quark mass splittings and the possibility of observing all. these phenomena at LEAR, the antiproton facility at CERN.

Many theorists in this Meeting are interested in polarization effects at high ener- gies. It is disappointing that our methods have very little to say about these problems except for some general, remarks. These observations are not very optimistic.

Our only incursion into that field will, be a discussion of electromagnetic form factors where one can see that the perturbative QCD calculations alone cannot des- cribe the form factor behaviour for a wide range of Q^. Introduction of non-perturbative effects has been shown to explain the data very nicely.

The success of the potential, model, is based on the original idea of Appelquist and Politzer that very heavy quarks will behave nonrel.ativistical ly and exchanging gluons in the weak coupling regime. As it turns out many qualitatively aspects of quark bound state problems are well described by this approximation. This is mainly due to spontaneous chiral breaking. The main effect of this condensate is to add to the mass of light quarks so that the resulting binding energy is relatively small.

However, the other condensates also contribute to the binding energy. This is con- ceptually necessary since we know that confinement forbids quark-quark physical, dis- continuities. In the potential, model this is faked by a linear portential. with some complications to insure asymptotic freedom as well. However there are two things wrong with this approximation, a) there are effects beyond the potential both local and nonlocal, b) it is assumed that the potential is spin independent in so far as the confining part goes just linear. Making a Breit hamiltonian approximation one hopes to reproduce all spin effects. Since it does not work one then makes "QCD like" corrections. Moreover, the strength of the gluon exchange is adjusted arbitra- rily. This criticism is not intended to show that the potential model is useless.

It simply remarks that it is limited and it is not expected to work well for more On leave of absence from the Weizmann Institute, Rehovot, Israel.

This work was partially supported by the Israel Academy of Sciences and the Minerva Foundation, F.R.G.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985207

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C2-72 JOURNAL DE PHYSIQUE

d e t a i l e d m a t t e r s a s f i n e s t r u c t u r e .

T h i s i s why an i n c r e a s i n g number of people f i n d t h e QCD sum r u l e approach of

Shifman, V a i n s h t e i n and Zacharov a more a p p e a l i n g method f o r s p e c t r o s c o p i c problems.

T h i s method s t a r t s f r o m t h e QCD L a g r a n g i a n and works w i t h t h e f u n d a m e n t a l p a r a m e t e r s l i k e as and t h e q u a r k masses. A few o t h e r u n i v e r s a l p a r a m e t e r s come i n : t h e con- d e n s a t e s and a p h y s i c a l a s s u m p t i o n a b o u t r a p i d convergence of t h e s p e c t r a l f u n c t - i o n s . T h i s i s t h e c a s e i n d u a l models and a l s o can b e v e r i f i e d i n t h e d a t a when t h e c u r r e n t s a r e p h y s i c a l a s i t i s t h e c a s e i n t h e v e c t o r c h a n n e l s .

One p r o c e e d s a s f o l l o w s :

C o n s i d e r t h e vacuum p o l a r i z a t i o n f u n c t i o n g e n e r a t e d by two c o m p o s i t e o p e r a t o r s (sometime c a l l e d c u r r e n t s ) and g i v e n by t h e e x p r e s s i o n

w h e r e ~ ( Q ' ) i s t h e p o l a r i z a t i o n o p e r a t o r w i t h t h e s t a n d a r d a n a l y t i c p r o p e r t i e s . The f o r m u l a s h o u l d c a r r y t h e a p p r o p r i a t e i n d i c e s which we h a v e o m i t t e d . A d e t a i l e d d i s c u s s i o n of t h e method can be s e e n e l s e w h e r e ( 1 ) . The c o m p o s i t e o p e r a t o r s h a v e t h e g e n e r a l form

where t h e q u a r k s c a n be l i g h t o r heavy and t h e o p e r a t o r i n between i s a c o m b i n a t i o n of D i r a c m a t r i c e s and d e r i v a t i v e s t o i n s u r e t h e d e s i r e d quantum numbers. F o r mesons t h e c h o i c e i s unique a s l o n g a s one u s e s t h e l o w e s t p o s s i b l e dimension. For baryons t h e r e a r e a m b i g u i t i e s though t h e s e t e c h n i c a l c o m p l i c a t i o n s a r e n o t c r u c i a l .

Formula (1) i s e v a l u a t e d by a Wilson e x p a n s i o n i n t h e p e r t u r b a t i v e vacuum. A s poin- t e d o u t by SVZ t h i s c a n be done even i f t h e vacuum i s non t r i v i a l provided one a l l o w s f o r o t h e r o p e r a t o r s b e s i d e 2 . The s t a t u s o f t h e o p e r a t o r e x p a n s i o n , t h a t h a s been c h a l l e n g e d , i s n o t t o be d i s c u s s e d h e r e . S u f f i c e s t o s a y t h a t t h e r e i s no c o n t r a d i c t i o n a t p r e s e n t . The p h y s i c a l i n f o r m a t i o n i s t h e n d i s t r i b u t e d i n two p a r t s . The vacuum s t r u c t u r e d e s c r i b e d by l o n g r a n g e e f f e c t s b u r i e d i n t h e e x p e c t a t i o n v a l u e o f t h e c h i r a l o p e r a t o r and t h e g l u o n c o n d e n s a t e , and t h e s h o r t d i s t a n c e e f f e c t s c o n t r o l l e d by t h e Wilson c o e f f i c i e n t . While t h e f o r m e r i s a g e n e r a l p r o p e r t y o f QCD and i t i s p r o c e s s i n d e p e n d e n t , t h e l a t t e r i s d e p e n d e n t on t h e s p i n and c h a r g e con- j u g a t i o n o f t h e s t a t e . A l l s t a t e s c a n now b e c a l c u l a t e d a s a f u n c t i o n of a few p a r a - m e t e r s : q u a r k m a s s e s , c o u p l i n g c o n s t a n t and two o r , sometimes, ( b a r y o n s ) t h r e e c o n d e n s a t e s . T h i s way, around one hundred numbers have been c a l c u l a t e d . Many d e s - c r i b e known s t a t e s and some a r e p r e d i c t i o n s .

We r e f e r t h e i n t e r e s t e d r e a d e r t o o u r P h y s i c s R e p o r t where t h e s e c a l c u l a t i o n s a r e d e s c r i b e d i n d e t a i l . I t i s perhaps worth m e n t i o n i n g t h a t t h e method i s q u i t e cumber- some and some o f t h e a t t r a c t i v e s t r u c t u r e o f t h e d u a l model i s l o s t i n t h e compli- c a t e d c a l c u l a t i o n a l scheme. However, i n o u r o p i n i o n t h e s u c c e s s o f t h e c a l c u l a t i o n s and t h e absence of o t h e r a n a l y t i c methods t o c a l c u l a t e bound s t a t e s j u s t i f i e s t h e e f f o r t . Most i m p o r t a n t , some of t h e p r e d i c t e d p a t t e r n s a r e y e t t o be s e e n . T h i s i s where we want t o put some emphasis s i n c e t h e s e e x p e r i m e n t s s h o u l d be performed.

I r r e s p e c t i v e of t h i s t h e o r y , i t i s worth r e m a r k i n g t o a n e x p e r i m e n t a l a u d i e n c e t h a t , e x c e p t f o r s t r a n g e mesons, no i n f o r m a t i o n e x i s t s f o r mesons w i t h s p i n g r e a t e r t h a n 1 and u n e q u a l mass q u a r k s . S i n c e t h e sti-ange q u a r k c a n be t h o u g h t a s m a s s l e s s t h i s i s a s e r i o u s gap i n o u r knowledge n o t t o know i f mass s p l i t t i n g s f o r s p i n two open b e a u t y o r charm mesons i s v e r y d i f f e r e n t from what we o b s e r v e f o r c h i r a l q u a r k s . We w i l l i l l u s t r a t e t h i s s i t u a t i o n by l o o k i n g a t t h e s t r u c t u r e o f t h e sum r u l . e s once more.

E v a l u a t i o n o f t h e f o r m u l a (1) l e a d s t o a n e x p r e s s i o n o f t h e form :

where

i s c a l l e d t h e moment o f t h e p o l a r i z a t i o n o p e r a t o r . An i s t h e b a r e l o o p , a n i s t h e p e r t u r b a t i v e c o e f f i c i e n t a s i n QED, d i s t h e s t r o n g c o u p l i n g c o n s t a n t a s g i v e n

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renorma!ization g r o u p i n v a r i a n t , c m u l t i p l i e s t h e c h i r a l c o n d e n s a t e and dn m u l t i - p l i e s t h e n e x t non v a n i s h i n g h i g h e ? o r d e r o p e r a t o r s .

The v a l u e o f t h e s e vacuum a v e r a g e s i s known from c u r r e n t a l g e b r a and l a t t i c e simu1.- a t i o n s ( t h e c h i r a l c o n d e n s a t e ) and by c a 1 . c u l a t i o n s i n t h e 1 . a t t i c e o f t h e g l u o n c o n d e n s a t e . The h i g h e r d i m e n s i o n a l o n e s a r e assumed t o b e o f reasonab1.e magnitude.

S i n c e t h e c a l c u 1 . a t i o n s s u c c e e d n i c e l y and t h e Wilson c o e f f i c i e n t s a r e s m a l l t h i s i s b e l i e v e d t o b e s o . T h e r e a r e c l a i m s t o t h e c o n t r a r y .

The 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 i n p e r t u r b a t i o n t h e o r y . T h e r e f o r e t h e r e a r e r e s t r i c - t i o n s a b o u t t h e i r s i z e . I n t h e c a s e o f open charm t h e r e i s no regime i n which a l l . t e r m s a r e s i m u l t a n e o u s l y s m a l l enough a s t o be a b l e t o n e g l e c t h i g h e r o r d e r c o r r e c t - i o n s . The c u l - p r i t i s a n i n t e r e s t i n g one. There i s a term t h a t i s v e r y l a r g e :

m heavy 9q

'

l i g h t ( 5 )

S i n c e we know from l i g h t q u a r k s t h a t mlilhr&q blight ^CCG> G;,,), o b v i o u s l y t h e t e r m i n ( 5 ) i s v e r y 1.arge. T h i s i s a non-local. erm t h a t c a n n o t b e a b s o r b e d a s a P o l i t z e r mass term. I n t h e c a s e o f open bottom a s i m i l a r t e r m i s b i g b u t smal.1 enough t o i n s u r e v a l . i d i t y o f t h e e x p r e s s i o n s and p r e d i c t a v e r y l a r g e s p l i t t i n g f o r d-wave mesons. No p o t e n t i a l model h a s s u c h a s t r a n g e consequence o f c h i r a l b r e a k i n g coming n a t u r a l l y .

I t would be v e r y i n t e r e s t i n g t o measure t h e p o s i t i o n o f t h e open bottom 2' s t a t e . We r e f e r once more f o r d e t a i l s t o o u r P h y s i c s R e p o r t . Here we want t o b r i e f l y d i s c u s s form f a c t o r s . T h i s i s n o t b e c a u s e i t i s a s u c c e s s f o r QCD sum r u l e s b u t a l s o b e c a u s e i t shows t h a t c a l . c u l . a t i o n s w i t h f o u r p o i n t f u n c t i o n s u s i n g s i m p l e minded p e r t u r b a t i v e QCD c a n b e total.:l.y m i s l e a d i n g .

I o f f e and Smilga amongst o t h e r s have p o i n t e d o u t t h a t t h e r e a r e l a r g e n o n - p e r t u r b a t - i v e e f f e c t s i n form f a c t o r c a l c u l a t i o n s a t known e n e r g i e s . The p o s s i b i l i t y of c a l . c u l a t i n g t h r e e p o i n t f u n c t i o n s f i r s t s u g g e s t e d w i t h t h i s method by R e i n d e r s , Yazaki and m y s e l f . We c a l c u l a t e d some r a d i a t i v e d e c a y s and a l s o t h e c o u p l i n g cons- t a n t s t o b a r y o n s and some G o l d s t o n e mesons w i t h v e r y good r e s u l t s . The r e s u l t s on t h e form f a c t o r s g i v e f u r t h e r s t r e n g t h t o t h e i d e a t h a t n o n - p e r t u r t m t i v e e f f e c t s c a n n o t be d i s c a r d e d and t h a t c a l c u l a t i o n s c o n c e r n i n g q u a r k s wi1.1. n o t be s i m p l e b e f o r e v e r y h i g h e n e r g i e s a r e r e a c h e d .

The n e x t t o p i c we want t o d i s c u s s i s t h e q u e s t i o n of s p e c i f i c QCD s t a t e s . S i n c e t h e r e a l . i z a t i o n t h a t mesons a r e made of qtj and baryons of qqq, t h e r e h a s been s t r i k i n g c o n f i r m a t i o n o f t h e c l a s s i c a l scheme. Only 35 o f SU(6) f o r mesons and 56 and 70 1=1 f o r b a r y o n s . I n g e n e r a l . one e x p e c t s t h a t gbuons w i l l b i n d and t h a t s t a t e s w i t h quantum numbers o t h e r t h a n 35, 56 o r 70 shou1.d a p p e a r . The c l . a s s i c a 1 . examp1.e is a g l u e b a l . 1 , a s s t a t e s w i t h o u t q u a r k s a r e o f t e n c a l l e d . The dynamical. c a l c u l a t i o n s c o n c e r n i n g g 1 u e b a l l . s a r e n o t a l l . i n a g r e e m e n t . L a t t i c e s and phenomenological b a g mode1.s p r e d i c t them t o b e low and t h e ground s t a t e t o b e 0++ . QCD sum r u l e s seem t o p r e d i c t a 2++ ground s t a t e . The problem i s i n d e e d v e r y d i f f i c u l t b e c a u s e g l u e - bal1.s have vacuum quantum numbers a n d m i x i n g w i t h i n s t a n t o n s and o t h e r e f f e c t s a r e l a r g e . Moreover, t h e c a l . c u l . a t i o n s a r e t e c h n i c a l l y d i f f i c u l t and n o t c o m p l e t e . T h e r e i s a f u r t h e r d i f f i c u l - t y t h a t we d i s c u s s i n some d e t a i l . s i n c e i t a p p l . i e s t o t h e n e x t t o p i c a s w e l l . I n normal. q u a r k s p e c t r o s c o p y t h e o r y and e x p e r i m e n t a g r e e a b o u t t h e d u a l i t y i n t e r v a l . s a t u r a t e d by 1.ow l y i n g r e s o n a n c e s . More p r e c i s e l y , t h e d e n s i t y o f s t a t e s is r e l a t i v e l y low, t h e s e p a r a t i o n of s t a t e s i s g i v e n by t h e formula embo- d i e s i n t h e Regge l i n e a r t r a j e c t o r y s p e c t r u m which p r e d i c t s

J = a + a ( ' m 2 ; d f u l GeV ² ( 6

a s a f u n c t i o n o f mass. I t i s n o t cl.ear t h a t s i m i l . a r c o n s i d e r a t i o n s a p p l y t o gluehal.1 s t a t e s and t h a t t h e s a t u r a t i o n o f t h e i m a g i n a r y p a r t o f a g i v e n p a r t i a l . wave c a n be a c h i e v e d by w i d e l y spaced s t a t e s . As you probabl-y know t h e r e i s v e r y l i t t l e s o l i d e v i d e n c e c o n c e r n i n g g l u e b a l l s t a t e s . S i n c e t h e i n t r i n s i c p r o p e r t i e s of t h e g l u e b a l l s a r e n o t v e r y d i f f e r e n t from normal 1=0 mesons, t h e o n l y hope i s a v e r y d e t a i l e d knowledge o f t h e s p e c t r u m . I f one t h e n f i n d s t h a t t h e r e i s a n e x c e s s of 1=0 s t a t e s o v e r t h e normal n o n e t number e x p e c t e d , one cou1.d i n f e r t h a t some of t h e s e

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C2-74 JOURNAL DE PHYSIQUE

states are glueballs. It seems that the controversy will last for a I.ong time before a given state is accepted as a glueball. The difficu1.t~ is compounded by the fact that mixing with normal mesons is expected and that normal. radial excitations are expected and known to exist.

In our opinion, a more promising and decisive test comes from hybrid states. These are defined to be states that cannot be formed by quark and antiquark.

Sometimes states that contain gl.ue but can also be quark antiquark states is used hut we prefer the restricted definition. A case in question is the state I-+, where the notation is J ~ C . A negative parity state requires even 1 since the intrinsic parity is negative. To reach spin 1 the quark spins must be in triplet and the charge conjugation follows to be negative. However if there is a gluon present there is enough freedom to construct any state. In the bag model. one constructs a wave function with three partic1.e~ and in the case of light quarks one finds a state at about 1.3 GeV. In the case of QCD sum rules the calculation is on better footing since the composite operator preserves gauge invariance and excites the state without prejudice. The calculation must assume a duality structure of pole plus continuum. As it turns out the results are not very convincing. We revert to the remarks above. The present methods to solve QCD are very primitive. We do not know the QCD scales in different problems and therefore it is not'easy to predict the ground state in a given partial wave. This is not as desperate as it might sound since for normal spectroscopy knowing one state per flavour gives quite good predictive power. Nevertheless, the situation is not as good as it might be. In the case of light flavours there has been a large number of cal.culations with con- flicting results. Slowly these calculations are converging to a common answer. For heavy quarks there is a recent calculation ( 3 ) that shows simi1.ar problems. The most interesting state does not come naturally and the threshold parameter (a tech- nical detail) which measures the isolation of the ground state is too important to feel at ease. It is rather unfortunate that in this case the mass of the interesting states are all. around 4.2 GeV, or higher. The search for these states becomes very difficult. Remember that they cannot be seen in formation experiments, even in experiments I.ike the ISR ones. This is because, though in this beautiful experiment a11 partial waves are excited, these states cannot communicate with fermion and antifermion !.

The situation with light states is more promising and LEAR should make an effort in this direction. Even if these states are not produced too strongly, it is possible to see their effects by interference to final states l i k e T Q that communicates with this and other partial waves. Asymmetries in angular distributions might show the effect. Remarkably enough all of these predictions agree with the concept of a constituent gluon with mass of 600-800 MeV. The idea of a constituent gluon is not easy to pin down but it has been used to explain anomalies in final state dis- tribution in photon-photon scattering and al.so has been extracted from lattice simulations as the mass necessary to trigger string breaking. Masses of states are then Zm a + mgluon and one gets the correct mass. I think that the predictions deserve a careful search. It is rather frustrating that al.1 of these predictions of states beyond the quark model cannot be confirmed. Though I do not believe that the situation is serious and that the difficulties in finding them catastrophic, it is a sokr- ing reminder that, though most peop1.e believe in the theory of quarks and gluons, some of the most striking consequences cannot be tested. It is also unfortunate that these consequences cannot be tested. It is also unfortunate that these consequences though very compelling cannot be calculated, so that if something or some state are not found, the theory would be in serious trouble.

It is also the occasion to refer to the recently revived question of the ground state of matter in QCD. Again, the question can appear in slight7.y different versions.

The generaL question is the following : if one had a large assembly of quarks and gluons, say a few hundred : which would be the ground state ? If the temperature was high enough it is even possible that the quarks are chirally symmetric, though the question can be asked about zero temperature as well. The point is that there is a competition between Fermi momentum and the penalty of creating strange quarks

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forms is energetically favorable. This peculiar form of matter, a collective state of quarks, a quark nugget, would have peculiar properties and spectacular signat- ures. Again, since the calculations are not exact and the error margins do not allow enough confidence, we must conclude that it is not possible to decide if these matter configurations are stable or not. There is speculation about their importance in cosmology. It has been suggested that these little balls might be around :

remnants of the early universe in which chiral invariance is preserved.

The last topic I want to briefly recall is the question of isospin invariance. It was pointed out sometime ago by several authors that isospin is an accidental sym- metry and that large violations of isospin can be expected. In particular in high spin states the violations of isospin could be as large as 30 per cent (4). Thls spin dependence of the isospin violations have to do with the alignment of high spin states with respect to the perturbation caused by the difference between U and d quarks. This is another area where the measurements of LEAR could be useful.

To conclude I will like to emphasize that there are very strong reasons to believe that the SU(3) colour piece of the standard model is doing fine. Calculations of the spectrum using the theory seem to agree we11 with the measurements. Though 1 have not discussed details,many subtle effects concerning current masses, running couplings and masses and universality of the relevant parameters are all in agree- ment with expectations. This does not mean that the calculations can be totally trusted since some assumptions and approximations schemes are necessary. Fortunate- ly, many of these can be checked against experiment. Ilnfortunately, when these assumptions are extended to other systems, one is not on firm ground. As a whole it is extremely interesting to notice that there 1s a great wealth of different phenomena depending on which quark family one studies. The systems with angular momentum and one heavy quark at least show effects that are not present in potential models. Experiments searching for these states are of great importance for QCD dynamics.

It is rather unfortunate that the ISR experiment with antiprotons on a fixed target was unable to establish the mass of the positive parity and charge conjugation charmonium state. This state is of some interest though most theories agree about its position. More important is the search for glueballs and true hybrids. We think that LEAR might play an important role in this area. The spectroscopy of heavy quarks is at its infancy and some peculiarities might prove useful. Two examples are toponium, that might determine the true value of the gluon condensate, and open bottom strange vector meson states that are predicted to be stable. The latter might be useful for CP physics.

Finally, confirmation of large isospin violations in decay of tensor or higher spin mesons will be of interest. If these effects are confirmed, they will strengthen the belief that non-perturbative effects are large and important.

It is a pleasure to thank the Physics Department at the University of Stockholm for their hospitality, and Mrs. Cohen-Solal for careful typing of this manuscript.

References

( 1 ) H. Reinders, H.R. Rubinstein and S. Yazaki, Physics Reports, to appear.

( 2 ) We refer again to our Report where the original references are given.

( 3 ) J. Govarts, H. Reinders, H.R. Rubinstein and J. Weyers, submitted to publication.

(4) See for example N. Isgur, H.J. Lipkin, H.R. Rubinstein and A. Schwimmer, Phys. Lett. 89, 78, 1979, and references therein.