N ELSEVIER

UCLEAR PHYSIC~

Nuclear Physics B (Proc. Suppl.) 55A (1997) 40-43

P R O C E E D I N G S S U P P L E M E N T S

How strange a non-strange heavy baryon? Ariel R. Zhitnitsky ~

~Physics Department, University of British Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada

V~re give some general arguments in favor of the large magnitude of matrix elements of an operator associated with nonvalenee quarks in heavy hadrons. In particular, we estimate matrix element (AblgdAb) to be of order of 1 for Ab baryon whose valence content is b, u, d quarks. The arguments are based on the QCD sum rules and low energy theorems. The physical picture behind of the phenomenon is somewhat similar to the one associated with the large strange content of the nucleon, i.e. with the large magnitude of the matrix element (pigs]p) --- 1. We discuss some possible applications of the result.

1. I n t r o d u c t i o n a n d M o t i v a t i o n .

Nowadays it is almost accepted that a non- va lence component in a hadron could be very high, much higher than one could expect from the naive perturabative estimations. Experimentally, such a phenomenon was observed in a number of places. Let me mention only few of them.

First of all anomalies in charm hadroproduc- tion: As is known, the cross section for the pro- duction of ,l/c)ts a t high transverse momentum at the Tevatron is a factor ,-, 30 above the standard perturbative QCD predictions. The production cross sections for other heavy quarkoniuin states also show similar anomalies[i].

The second example of the same kind is the charm structure function of the proton. It is measured by EMC collaboration [2] to be some 30 times larger at x u j = 0.47, Q2 = 7 5 G e V 2 than that predicted on the standard calculation of photon-gluon fusion ~,~St g ~ cO,.

Next example is the matrix element (Xl~dX) which does not vanish, as naively one could ex- pect, but rather, has the same order of magnitude as valence matrix element (NlddlX}.

One can present man3' examples of such a kind, where the "intrinsic" nonvalenee component plays an important role. This is not the place to ana- lyze all these unexpected deviations from the s- tandard perturbative predictions. The only point we would like to make here is the following. Few examples mentioned above ( for more examples see recent review [3]) unambiguously suggest that a nonvalence component in a hadron in general is

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not small. In QCD-terms it means that the cor- responding matrix element has nonperturbative origin and has no as suppression which is naively expected from perturbative analysis (we use the term "intrinsic component" to describe this non- perturbative contribution in order to distinguish from the "extrinsic component" which is always present and is nothing but a perturbative ampli- tude of the ghon splitting g --* Q Q with nonva- lence quark flavor Q ).

The phenomenon we are going to discuss in this talk is somewhat similar to those effects men- tioned above. We shall argue that a nonvalence component in a heavy-light quark system could be very large. However, before presenting our ar- gumentation of why, let say, the matrix element (AblgSlAb) is not suppressed (i.e. has the same order of magnitude as valence one <ad~ulAb>), we would like to get some QCD-based explana- tion of the similar effects we mentioned earlier. Before doing so, remember that for a long time it was widely believed that the admixture of the pairs of nonvalence quarks in hadrons is smal- l. The main justification of this picture was the constituent quark model where there is no room, let say, for a strange quark in the nucleon. It has been known for a while that this picture is not quite true: In scalar and pseudoscalar channels one can expect a noticeable deviation from this naive prediction. This is because, these channels are very unique in a sense that they are tightly connected to the QCD-vacuum fluctuations with 0 + , 0 - singlet quantum numbers. Manifestation of the uniqueness can be seen, in particular, in

A.R. Zhitnitsky/Nuclear Physics B (Proc. Suppl.) 55,4 (1997) 40~43 41

the existence of the axial anomaly (0 - channel) and the t race anomaly (0 + channel).

A well-known example where this uniqueness shows up is a large magni tude of the strange con- tent of the nucleon. In formal terms one can show tha t the mat r ix element (:Vl~slN) has the same order of magn i tude as the valence matr ix element (NlddlX). w e shall give a QCD-based explanat ion of why a naively expected suppres- sion is not present there. After that , using an intuit ion gained from this analysis, we turn into our main subject : nonvahmce matr ix elements in heavy hadrons. We shall see tha t essential non- per turba t ive informat ion represents one and the same vacuum correlat ion fimctions which enters into the different physical-characteristics. So, we could ext rac t the unknown correlat ion function, let say, from ( X l ~ s l X ) and use this information in evaluat ion of the mat r ix element we are inter- ested in: (Ab[gS[Ab).

2. S t r a n g e n e s s in t he nucleon.

Let us s tar t f rom the s t andard a rguments (see e.g. the text book [4]) showing a large magni tude of of (Xl~slX) . Argument s are based on the re- sults of the fit to the da t a on 7rN scat ter ing which lead to the following est imates for the so-called a term

,nu + m d (plftu + ddlp) = 45MeV. (1) 2

(here and in wha t follows we omit kinematical s t ruc ture like pp in expressions for matr ix ele- ments.) Taking the values of quark masses to be m~ = 5 . 1 + 0 . 9 M e V ,rod = 9 . 3 + 1 . 4 M e V ,m~ = 175 4- 25MleV [5], f rom (1) we have

(pl~,,, + ddlp) "~ 6.2. (2)

Further , assuming oc te t - type SU(3) breaking to be responsible for the mass split t ing in the baryon octet , we find

(pl~ulp) _~ 3.5, (pl&lp) ~_ 2.8, (pl~slp) -~ 1.4 (3)

These very simple and very rough calculations ex- plicitly demons t r a t e t ha t the s t range matr ix ele- ment by no means is small. We would like to in- te rpre t the relations (3) as a combinat ion of two

very different (in sense of their origin) contr ibu- tions to the nucleon matr ix element:

(PlqqlP) - {PlqqlP)o + {plqqlp)l, (4)

where index 0 labels a (sea) vacuum contr ibut ion and index 1 a valence contr ibut ion for a quark q. In what follows we assume tha t tile vacuum con- t r ibut ion which is related to the sea quarks is the same for all light quarks u, d, s. Thus, the nonzero magni tude for the s t range mat r ix elements comes exclusively from the vacuum fluctuations. At the same time, the mat r ix elements related to the va- lence contr ibut ions are equal to

(plftu[p)l " 2.1, (PlddlP}l ~- 1.4. (5)

These values are in remarkable agreement with the numbers 2 and 1, which one could expect from the naive picture of the non-relat ivist ic con- st i tuent quark model. In spite of the very rough est imat ions presented above, we believe tha t t hey are sufficient to conclude tha t : a) a magn i tude of the nucleon matr ix element for as is not small; b ) the large value for this mat r ix element is due to the nontrivial QCD vacuum s t ruc ture where vacuum expecta t ion values of u, d , s quarks are developed and they have the same order in mag- nitude: (0[ddt0) ~ (0lieu[0) ~ (0[~sl0).

Oitce we realized tha t the phenomenon under discussion is related to the nontrivial vacuum structure , it is clear tha t the best way to under- s tand such a phenomenon is to use some me thod where QCD vacuum fluctuat ions and hadronic propert ies are s t rongly interrelated. We believe, tha t the most powerful analyt ical nonper tu rba - t ire me thod which exhibits these features is the QCD sum rules approach [6].

2.1. S t r a n g e n e s s in the nuc l eon a n d Q C D v a c u u m s truc ture ,

To calculate (NI~s[_Y > using the QCD -sum rules approach, we consider the following vacu- um correlat ion funct ion [7]:

/ T(q 2) = eiq~dxdy(O]Tlq(x) , ~s(y), q(0)}10 ) (6)

at _q2 ~ oc. Here q is an a rb i t r a ry current with nucleon q u a n t u m numbers. In part icular , this current may be chosen in the s t anda rd form

42 A.R. Zhitnitsky/Nuclear Physics B (Proc. Suppl.) 55A (1997) 40-43

q = eabCT, da(ubC?puC ). For tile future conve-

nience we consider the unit mat r ix kinematical s t ruc ture in (6).

In this section we use this me thod to demon- s t ra te a reasonable agreement with other ap- proaclles regarding the mat r ix element (plgslp) (3). In section 3 we use the same me thod to es- t imate some new characteristics of heavy hadron system. Let us note tha t due to the absence of the s -quark field in the nucleon current q, any substant ia l contr ibut ion to T(q 2) is connected on- ly with non-per turba t ive , so-called induced vacu- unl condensates . Such a contr ibut ion arises from the region, when some distances are large. Thus, this contr ibut ion can not be directly calculated in per turba t ive theory, but ra ther should be coded (parameter ized) in terms of a bilocal opera tor K:

- - I g t

<plasl;> ,--, (7)

f K = i ] dy<OlT{as(y), ~u(0)}10>. (8)

!

One can es t imate the value of K by expressing this in terms of sortie vacuum condensates [7]:

/ ( ,,, 18 (qq>2 ~ 0 .04GeV 2 (9) - b t-eaGZ ~ -

where b = ~ Nc 2 . . - 5 N f = 9 and we use the stan- dard values for the condensates[6]:

<c%G2 ~ ~,, 1 .2 .10-2GeV 4 (qq) "., - ( 2 5 0 ~ l e V ) 3. 7"( - - p u ~ - -

It is very impor t an t tha t our following formulas for tlle nonvalence content in heavy quark system ( next section ) will be expressed in terms of the s a m e correla tor K. Therefore, we could use the formula (7) to ext rac t the corresponding value for K from exper imental d a t a instead of using our es t imat ion (9).

Adopt ing our es t imate (9) for K, formula (7) gives the following expression for the nucleon ex- pec ta t ion value for as

lS (@> 2.4, (10) b

which is close to tile naive es t imat ion (3). Let us stress: we are not pre tending to have made a

reliable calculation of the mat r ix element <PFSlP> here. Rather , we wanted to use the qual i tat ive picture to demons t ra t e the close relation between nonvalence matr ix elements and QCD vacuum structure.

2 . 2 . L e s s o n s .

First of all, the result (10) means tha t s quark contr ibut ion into the nucleon mass is not small. Indeed, by definition

m : (N[ Z m q q q K V ) - ~<: \ ; ] -~G2~]N) , (11) q

where the sum is over all light quarks u ,d , 8. Adopt ing the vahles for (plsslp> - 2 and 175MeV [5], one can conclude tha t a noticeable par t of the nucleon mass is due to the s t range quark. We have ment ioned this, well known re- sult, in order to emphasize tha t the same phe- nomenon takes place in heavy quark sys tem al- so. Namely, we shall see tha t the s quark con- t r ibut ion to X = muQ - m Q I , , Q _ ~ for a heavy hadron HQ is not small. Our second remark is the observation tha t a variat ion of the s t range quark mass may considerably change some vac- uum characteristics, nucleon mat r ix elements as well as JL Therefore, the s t andard latt ice calcula- tions of those characteris t ics using a quenched ap- proximat ion is questionable simply because such a calculation clearly is not account ing for the fluc- tuat ions of the s t range (non-valence)quark.

3. H e a v y h a d r o n s

In this section we shall calculate the mat r ix el- ement {Ab]aS]Ab) as an explicit demons t r a t ion of general idea (formulated in the in t roduct ion) t ha t a nonvalence componen t (as) in a heavy quark sys tem (Ab ~ bud)could be large and comparab le with valence matr ix elements like (AblftulAb).

We start f rom the definition of the fundamenta l pa ramete r of H Q E T (heavy quark effective theo- ry), see e.g. nice review paper [8]:

£ -mHQ -- m~l.~Q--~ (12)

All hadronic characteristics in HQET should be expressed in terms of ,~ which is defined as the

A.R. Zhitnitsky/Nuclear Physics B (Proc. Suppl,) 55A (1997) 40~43 43

following matr ix element:

2mH~(HQl~-~n, qqq+ G~IHQ> (13) q

Now we can use the same techniques that we have been using in the previous section to est imate the strange quark contribution to the mass of a heavy hadron:

1 A ( s ) - - - ( HQ[rn~gsI H Q ). (14)

2rIt HQ

Lessons we learned from the similar calculations teach us tha t this matr ix element might be large enough. Indeed, the calculation leads to the fol- lowing formula 1

1 3 K 2, , ,~ (AblgslAb) ~ (~S0 + E ~ ) - - _~ 1 - 2. (15)

- <qq)

where So and Er are the duality interval and the binding energy for the lowest state with given quantum numbers 2. Wha t is important for us is the fact tha t the nonper turbat ive correlation function K which enters into the expression (15) is the s a m e correlator we have been using for the calculation (p[gs[p). It sets the scale of the phe- nomenon. Moral: If we accept the large value for (p[gsIp) we should also accept the large value for

1 2mMQ (HQIm~sIHQ) ~ (200 + 300)MeI~\ (16)

as a consequence of absence of any suppression for nondiagonal correlator K.

4, C o n c l u s i o n

W'e have argued that the matr ix element (16) could be numerically large. The arguments are very similar to the case of the strange matr ix ele- ment of the nucleon and based on the fundamen- tal proper ty of nonperturbat ive QCD that there is

1As in the case of the nucleon, we use the local dua l i ty a r g u m e n t s (so-called, f inite energy s u m rules) to e s t i m a t e th is m a t r i x e lement . Besides t ha t , we use the s t anda rd technica l trick[9] which sugges t s us ing the combina t ion (q2 _ m 2 ) T ( q 2 ) in s u m rules (6) ra ther t h a n T(q 2) itself. Th i s trick allows exponen t i a l ly suppress ing an unknown con t r ibu t ion f rom t he nond iagona l t r ans i t i ons which in- c lude h igher resonances . 2~,Ve use So ~ Er ",~ (0.5 + 0 . 7 ) G e V for numer ica l e s t ima t ion .

no suppression for flavor changing ampli tudes in the vacuum channels 0 + (the Zweig rule in these channels is badly broken). Few consequences of the result (16) are:

1. The value of ,-~ continues to be controver- sial, because the QCD sum rules indicate that 7~ ,~ 0 . 5 G e V which does not contradict the low- er bound s temming from Voloshin's sum rules [10],[8]. At the same t ime the lattice calculations give a much smaller number: A ,-~ 0.2 - 0 . 3 G e V , see [8] for more details. A possible interpretat ion is: the lattice definition of/~, does not correspond to the continuum theory because the s-quark con- tribution (16) was not accounted properly.

2.Scalar and pseudoscalar light mesons (~, f0...) strongly interact with Ab; the ¢ meson does not interact with Ab.

3. We expect a similar situation for all heavy hadrons. Therefore, for the inclusive product ion of strangeless heavy hadrons we expect some ex- cess of strangeness in comparison with naive cal- culation. However, we do not know how to esti- mate this effect in an appropr ia te way.

R E F E R E N C E S

1 F.Abe, et al., CDF Collaboration, Phys. Rev. Lett. 69 (1992) 3704

2 J .J .Auber t , et al., Phys.Lett . B l l 0 (1982) 73. 3 S.Brodsky, hep-ph/9609415, SLAC-PUB

7306. 4 J.F.Donoghue, E.Golowich and B.R.Holstein,

Dynamics of the Standard Model, Cambridge University Press, 1992.

5 H.Leutwyler, hep-ph/9602255, 1996. 6 M.A.Shifman, Vacuum Structure and QCD

Sum Rules, North-Holland, 1992. 7 V.M.Khatsimovsky, I.B.Khriplovich,

A.R.Zhitnitsky, Z.Phys. C- Particle and Fields, 36, (1987), 455; A.R.Zhitnitsky, hep- ph/9604314.

8 M.A.Shifman, hep-ph/9510377, lectures giv- en at YASI, Boulder, Colorado, 1995.

9 V.M. Braun and A.Kolesnichenko, Nu- el.Phys. B283, (1987) 723.

10 M.Voloshin, Phys. Rev.D 46, (1992), 3062.