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Hard (semi) exclusive processes in QCD and properties of hadronic wave functions
Nuclear Physics B (Proc. Suppl.) 7B (1989) 297-319 North-Holland, Amsterdam
297
H a r d (semi) e x c l u s i v e p r o c e s s e s in Q C D a n d p r o p e r t i e s of h a d r o n i c wave f u n c t i o n s V. L. C H E R N Y A K 1 Coll~ge de France - Laboratoire de Physique Corpusculaire 11, Place Marcelin Berthelot - 75231 PARIS Cedex 05 - F R A N C E and Institute of Nuclear Physics, 630090 NOVOSIBIRSK 90, USSR
Abstract A short review of the properties of hard exclusive processes and hadronic wave functions is given. The connection with the higher twist contributions in inclusive processes is considered.
Introduction I shall try to give a short overview of our present understanding of hard exclusive processes. The main points touched upon are : 1) General properties of hard exclusive processes, 2) Properties of hadronic wave functions, 3) Where and how the wave function properties can be checked experimentally. But before, because this is a workshop on the Higher Twists, let me put some questions of relevance to both Higher Twists and Exclusive Processes: What is really interesting in the Higher Twist (HT) contributions ? W h a t can we learn about the hadronic interactions and about the internal structure of hadrons from HT ? In what way the HT contributions can be used to obtain a new useful information ? The specific property of HT contributions is that they are between the pure exclusive processes and the usual inclusive ones, i.e., they are "semi-exclusive", in some sense. It is this property which determines their role. The HT contribution looks as a convolution of the hard kernel describing the short distance interactions, with the hadron wave function describing a soft non-perturbative interaction responsible for hadron formation. 0920 5632/89/$03.50 (~) Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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V.L. Chernyak / Properties of hadronic wave functions
It is clear that a separation of HT contributions is not the best way to study the properties of the hard kernels (i.e., the properties of perturbative QCD). This can be done much more easily by means of the usual leading twist (LT) contributions in hard inclusive processes. Therefore, H T are of interest just for studying properties of the hadronic wave functions. Although this can be done, in principle, in pure exclusive hard processes also, the inclusive HT are frequently more suitable experimentally. Besides, because the hard production of each additional hadron leads to a further power suppression (an additional form factor), the 1-particle HT are minimally suppressed as they correspond to hard production of only one hadron. At large m o m e n t u m transfers they are preferred, in this respect, in comparison with pure exclusive processes. A short but informative and clear review of the experimental results in HT physics is given in [1]. Well, let us return now to the points 1-3 above and let us look at the characteristic properties of exclusive processes.
1
G e n e r a l p r o p e r t i e s of hard e x c l u s i v e p r o c e s s e s
The simplest hard exclusive amplitude, the meson form factor at large m o m e n t u m transfer, is presented on Fig. 1 (c.m.s.) :
"x I
'
'1
I
o
"J
"~" ~ h a r d
M 2 (
~2
kernel Th
Figure 1: The meson form factor : kl ~- Zlpl + k±, k2 ~- z2pl - k±, 11 ~- ylp2 + Ix, 12~-y2p2-1x. It is a convolution of the initial and final hadron wave functions, q/,(z, ~:±) and g2*(y, l±),
(l ki)
with a hard kernel Th x, y, -0-' q - ' q
which is the form factor of the two-quark system [4] :
".L. Chernyak / Properties of hadronic wave functions
F(q) = f dy dF± = (l/q)
A
(y, F±)
T'
299
]~±) dx d;.l ,
W'
T h e wave function gJ(z, re±) is the a m p l i t u d e for finding two quarks w i t h longitudinal m o m e n t u m fractions z a n d 1 - x (at p~ ) oo) and transverse m o m e n t a k± and ( - k ± ) within the m e s o n (for m o r e details, see p a r t 2 below). T h e h a r d kernel Th represents the small-distance interaction and so it can be calculated reliably within p e r t u r b a t i v e Q C D . I want to e m p h a s i z e t h a t there is nothing u n k n o w n in it. If we neglect the intrinsic quark transverse m o m e n t a k± and l± in Th (indeed, they give only power corrections ~ O ( k ~ / q ~ ) ) , then the m e s o n form factor takes the f o r m :
F(q) = Co/(q )-L where C O ~-
(9,)
1 d x d y ~o~(y) f h a l : d ( ~ , y ) 731(x) ,
~0
(3)
T h e power N in (2) follows from a simple dimensional counting [2] : N = nman -- 1, there nr~n is the m i n i m a l n u m b e r of constituents within a given h a d r o n (nmm = 2 for mesons, nmin ---- 3 for baryons). At the s a m e time, to predict the absolute n o r m a l i z a t i o n (the constant Co in (2)), you have to know the h a d r o n wave function 7~(z). However, the dimensional counting is working only if there are no infrared power singularities which increase the a s y m p t o t i c s (i.e., if the integrals over z a n d y in (3) are not divergent as a power), and if there are no selection rules which suppress the a s y m p t o t i c s . T h e r e are no infrared singularities for form factors, but this is not the case, in general, for the elastic scattering a m p l i t u d e s [3]. (Indeed, the I S R e x p e r i m e n t s on high energy p-p elastic scattering have found the b e h a v i o u r : d a / d t ,,~ t - s at s >> Itl >> ~0~, which violates the dimensional counting prediction and agrees with the Landshoff one [3]). Let us r e t u r n now to the wave function (w.f.) ~ z ) in (3), which d e t e r m i n e s the final answer. It is the projection of the total w.f. @(x,k±) onto the Lz -- 0 state. You see, therefore, t h a t there is a selection rule [4] : the leading contributions give only the L . = 0 c o m p o n e n t s of the total h a d r o n w.f. Clearly, in this w.f. c o m p o n e n t : A hadron = ~
A quark i
,
(4)
i
(A is the helicity). A n d consequently, only meson states with A = 0 and ]A[ = 1, and b a r y o n states with ])~[ = 1/2 a n d ])~[ = 3/2 give leading contributions in h a r d exclusive processes. (For glueballs [A[ = 2 is also allowed). Besides, because the q u a r k helicity is conserved at large m o m e n t a , 2-, initial
hadron
=
2_.., final
A hadron
•
V.L. Chernyak / Properties of hadronic wave functio1~s
300
If, for instance, you are looking for a production of mesons with higher ])~1, then L: # 0 in such meson states. To produce a final state with Lz = n, you should have to separate out of the kernel
Th z , y , - ~ ,
a t e r m which contains, at least, n powers of/'± :
n
But because /~± enters Th only as (l±/q), you will have ~ O ( 1 / q ~) suppression of the hard amplitude. Proceeding in such a way, you can obtain finally a general formula for the power behaviour of any form factor [4] :
< P2, J2, .X~IJxIPl, J1, At > ~ ( l / q ) I \ ' - ' \ ' l + ~ " - : ~ ,
(5)
(here Jx is the vector or axial vector current with helicity $, and $ = $1 + $2 in the c.m.s.).
To illustrate (5), let us consider a simple example which contains all the characteristic features responsible for the behaviour (5). The contribution to the < $2 = 2 ] J1 I $l = - 1 > form factor is shown on Fig. 2 (arrows show quark helicities). In comparison with a dimensional behaviour ~ ( l / q ) , there are additional suppressions : (a) ~,, (~±/q)2 due to L~2 = 1 and L~ 1 = - 1 , (b) ~-, (k± + m ) / q due to the quark helicity fllp.
q
~q
L I= - 1
I !
t I I ! ,
L.Z=I 2
Figure 2: The contribution to the form factor < )t2 = 2 [ J,\=l [ )~1 : - 1 > On the whole, you have just the behaviour given by (5). Now, if we use the Born (i.e., the lowest order) approximation for the h a r d kernel Th , then the factorization of the total amplitude into a product of hadronic wave functions and Th is self-evident (see Fig. 1). But when the higher-loop p e r t u r b a t i v e theory corrections are taken into account, it is then a difficult technical problem to show that factorization is still working, and to sum up loop logs.
V.L. Cher~D'ak / Properties of hadronic wave functions
301
Nevertheless, this work was p e r f o r m e d in [5 - 9], and it was shown t h a t the factorization survives and t h a t all loop corrections are accounted for by the usual a n o m a l o u s dimensions. All this leads finally to an additional slow evolution with q2 (like t h a t ix/the deep inelastic e-p scattering), and at available Iq21 it is not of great i m p o r t a n c e , really. Let me dwell shortly on the space-time picture of a h a r d exclusive process (Fig. 3).
x....Z 2~,hard kernet Figure 3: T h e s p a c e - t i m e r e p r e s e n t a t i o n of the m e s o n form factor, z~ ~ z~ ~ (zl - z~) ~ ~ 1/q ~ •
ra.--,1Kj.
hard kernet
Figure 4: T h e s p a c e - t i m e picture of the m e s o n form factor : $ is the p h o t o n wave length, r L is the m e s o n longitudinal size; r± is the m e s o n transverse size. T h e p h o t o n wave length is )~ --- O ( 1 / q ) in the c.m.s., the h a r d kernel Th is c o n c e n t r a t e d in a small s p a c e - t i m e region with a linear size ~-- O ( 1 / q ) also, and it seems t h a t only the value of the w.f. at the origin should enter the answer for F ( q ) : F ( q ) ,~ ¢ ~ ((q2)N 0)¢1(0)
'
¢(0) = f dz dfe± ~2(z,/~±) ,
(6)
(i.e., the function fhard in (1) should be independent of z and y). This is not the case really (see Figs. 3 a n d 4, c.m.s, is used). T h e reason is t h a t the h a d r o n with a large m o m e n t u m p~ ~ q is L o r e n t z - c o n t r a c t e d along his m o t i o n , so t h a t its longitudinal size is : r L ~ " 1/q . Its transverse size, r3_, r e m a i n s large, however : r± ~ l/[f¢±l -,~ 1 fro. So, the p h o t o n with wavelengh $ ~-, 1/q "feels" the distribution of quarks in longitudinal m o m e n t u m fractions z~ inside the h a d r o n , a n d it does not resolve their distribution in/~±.
V.L. Chernyak / Properties of hadronic wave functions
302
2
Properties of the hadronic wave functions
Let us come now to the properties of w.f. Ti(x) (just these determine the absolute values of leading exclusive amplitudes ). The space-time meson w.f. which we use is defined (schematically) as [4] : ¢ ( z , p ) = < 0 I q(z) I q ( - z ) I M(p) > = ¢(zp, z ~) , I = exp{i 9
// z
(7) d~.A.(o')} .
Evidently, it is Lorentz and gauge invariant. To describe the leading contributions to hard exclusive amplitudes, we need only the w.f. taken on the light cone : ¢(zp)=¢(zp,
z2=0),
(pz"~Q~oo,
z~,~ l / Q , z 2 ~ l / Q 2,
z p ~ l) .
It can be represented in the form : ¢(zp)
~01 dzldz2 6(1 - zl - z2)
=
f'd¢
e ~Z(Zlp)
--
iZ(X2P) ~(X)
=
(8)
ei~(zp) qa(~)
1
=
xt-z2,
xl-t-z2= 1 ,
and ;E1 and x2 are the hadron m o m e n t u m fractions of the two quarks. Simple examples of T(~) are shown on Fig. 5. The non-relativistic w.f. looks like a 6-function, its width is ,-~ (binding e n e r g y / h e a v y quark mass) << 1. The Tpert(~) "" (1 - (2) corresponds to a perturbative (or, equivalently, asymptotic) distribution of two massless quarks. Expanding (7) in z you will obtain :
< 0l 9(0)
z
q(O) [ U(p)
> = (zp)"
(" ~(()
, z2 ~
o .
(9)
Therefore, if you know these m a t r i x elements, then you know the w.f. moments. The method of the QCD sum rules [10] allows us to find out the values of such m a t r i x elements. The idea of the approach is as follows. The correlator
I(q) =
fdz e 'q~ < 0 I T J r ( z ) J2(0) l0 > ,
J = qDnq , D = iO + gA
(10)
is considered, and its behaviour at Iq2l >> ~02 is calculated, (Fig. 6). The leading contribution at large [q2[ gives, of course, the perturbative theory (and the usual loop corrections to it). In these contributions the large m o m e n t u m q is divided more or less equally between all the lines (these lines with a virtuafity ,-- q2 are the boldface ones). Besides, there are contributions where the large m o m e n t u m q goes only through some lines, while other ones are soft. For such contributions we calculate explicitly the hard (small distance) part of diagrams using the p e r t u r b a t i o n theory, while the soft fines represent now the nonperturbative quark and gluon fields which are averaged over their fluctuations in the physical vacuum.
. L. Chern 3 ak / Properties of hadronic wave functions
/
303
b
)IL -1
Figure 5: a / T h e
p e r t u r b a t i v e (or, equivalently, asymptotic) wave function
b/The
non-relativistic wave f u n c t i o n : V~non_~el(Z) ~'- 6(zl - ½) ~ 6(¢) .
Physically, all this is a kind of multipole expansion : we calculate a radiation of quarks and gluons with a characteristic wave length ,-- 1/#0 (the scale of the v a c u u m fields) by a small system of the size.--- 1/q. T h e sum rules obtained in this way can be represented (qualitatively) in the form : rl
fl
r 1
Ave(C,/~F).
(11)
Here ~ ( ~ ) is the true m e s o ~ w.f. we are looking for, ~pe,t(~) -- ~(1 - ~ ) is the perturbative w.f. for two massless quarks and A~o is the n o n - p e r t u r b a t i v e contribution to the w.f. :
A%o(C, ATI) = Ci
11714
6(i - C) + 6(1 + ~) + (12)
< 0],~ql0 >2 {6'(1 - (~) - 5'(1 + (~)} + . . . +C2 /~-6
V.L. Chernyak / Properties of hadronic wave functions
304
pert. theory
~xpert. loop corrections
[d)
.~non-pert. vacuum quark field
-~"
•
.
gluon field
m
c)
(b )
(c)
t;uum
/'quark fields
Figure 6: Various perturbative and non-perturbative contributions to the two-point correlator.
".L. Chernyak / Properties of hadronic wave fun ctions
305
(C 1 and C2 are known coefficients and /1~ ~_ 1 GeV 2 is a characteristic scale for a given correlator, its precise value is determined by a special fitting procedure). You see that while ~Vpert(~) corresponds to a smooth symmetrical distribution of the meson m o m e n t u m between the two quarks, the non-perturbative contributions are highly assymmetric. T h e y correspond to distributions such that nearly all the meson m o m e n t u m is carried by one quark. The sign and the absolute value of A ~ depend strongly on the hadronic q u a n t u m numbers, while ¢Ppert is common to all mesons. As a result, the moment values of the true meson wave function differ strongly, in general, from those of ~pert , and this means that the form of the w.f. also differs strongly from ~pert(~) • The non-perturbative effects (i.e., A~) are very large and positive for the pion w.f. [11]. Therefore, the true m o m e n t values are much larger t h a n the perturbative ones, and the true pion w.f. is much wider t h a n ~pert(~) • The results for the lowest moments are shown in the table. The moment values of the simple model w.f. : epsumrule(<)_-- !~(1 _ <2)~2 (13)
are also given therein.
< <~4>
Vpert(¢)
sum rule
~sum rule(~)
1
1
1
0.20
0.44
0.43
0.086
0.27
0.24
T a b l e : ~Ppert(~)-- 43-(1 - ~ 2 ) ,
~psumrule(~) = ~ ( 1 -~2)~2
There are general arguments originating from the quark-hadron duality, showing that (in the chiral limit) the true w.f. has the same parametical behaviour at xi ~ 0 and xi ~ 1 as ~ p e r t ( X ) , i.e, ~ xlx2 • This excludes models like those proposed by Brodsky and Lepage : ~ r ( z ) ~ (ZlZ2) 1/4,
(14)
We see therefore (Fig. 7), that (at Pz -* oo) the distribution of the pion m o m e n t u m between two quarks is.highly assymmetric. There is a dip at ~ = 0 (i.e., x~ = x2 = 1/2), and this means t h a t the pion m o m e n t u m is rarely divided equally between two quarks. Usually~ the largest part of the pion m o m e n t u m is carried by one quark. For for the [25]. For is very
the pr - px=0 meson, the non-perturbative effects are also positive, but smaller t h a n pion. As a result, the pn-meson w.f. is still wider than ~ p e r t ~ but narrower than ~,~ p± - PI~\I=I meson the non-pert, effects are negative. As a result the p x - m e s o n w.f. narrow (see Fig. 8) [251
<~2
>pert
<~2>pL
----0.20 ' ~- 0.30,
<~2>~-----0.44, ~2 >p± -~ 0.14.
<
}
, < (2 > i = /_ll d~ ( 2
The nucleon state at pz ---+oo can be represented (in the leading twist) as:
¢Pi(~)
•
(15)
306
V.L. Chernyak / Properties of hadronic wa~v functions
[0 Figure 7: a / T h e
1
pion wave function T~(~) o b t a i n e d from s u m rules :
~,~um rule(C) = ~(1 - ~2)¢2 b/The
-1
p e r t u r b a t i v e pion wave f u n c t i o n : ~Opert(() = 3(1 - (2)
0
1
Figure 8: a / T h e non-relativistic wave f u n c t i o n : ~0non.rel(~) = ~(~) , b / T h e wave function of the p± = Pl~l=x m e s o n o b t a i n e d from the s u m rules ~ , ± ( ~ ) = ~(115 _ ~ 2 ) 2 , c/ T h e p e r t u r b a t i v e p ± - m e s o n wave f u n c t i o n : Tpert(~) = 3(1 - ~2).
".L. Chernyak / Properties of hadronic wave functions
3{17
The nucleon w.f. ~,~,'(=1, Z2, =3) was also investigated [12], and its profile is shown on Fig. 9. It is seen that it is highly assymetric : the largest part (-~ 60%) of the proton m o m e n t u m is carried by one u-quark with its spin parallel to the proton spin, while each of remaining two quarks carries _~ 20% of the total m o m e n t u m .
=.1
XI=
X]=l Figure 9: The profile of the proton wave function obtained from the sum rules. Let me summarize now (qualitatively) what we have learned about the hadronic wave functions on the light cone :
~(z) = ]all
'~,(=, k_~) •
1. The distribution of quarks in longitudinal m o m e n t u m fractions is wide, in general. This means that quarks are highly relativistic inside the hadrons, and we can't use the models with non-relativistic constituents. 2. The form of the true w.f. differs greatly, in general, from the perturbative (asymptotic) w.f., Tpert(Z) ~ and this means that complicated non-perturbative interactions influence strongly the w.f. form. 3. The w.f. form depends strongly on the hadron q u a n t u m numbers, in particular on its helicity (while 7~pert(Z) is universal and does not depend on the hadron considered). This is because the non pert. effects differ strongly in channels with different q u a n t u m numbers.
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V.L. Chernyak / Properties of hadronic wave fimctions
These last two properties were the most surprising for m a n y physicists, because it was expected (from the experience with the old-fashioned SU(6)-constituent quark model) t h a t the w.fs. of hadrons entering the same SU(6) multiplet are like each other and do not differ greatly from ~pert (x) . To understand to what extent these expectations can be justified, we need to know the connection between different 2-particle w.fs. a) The old SU(6)-constituent quark model (CQM) originates from the old picture of nonrelativistic quarks inside the hadron and uses the so-called equal-time w.f., q r , and the "equal-time" basis (this formalism is a generalisation of the standard hamiltonian formalism of non-relativistic q u a n t u m mechanics). In the meson rest frame, the two-quarks state of this formalism describes two on-shell quarks w i t h : p~ -- r a ~ , p~ = my , ff = ff~ +/~2 = 0 , and p0 _ p0 _ p0 # 0 due to the interaction (p0 and/~ are the meson energy and m o m e n t u m ) . The w.f. q T ( k = ffl -- if2) is defined as a projection of the meson state onto this two-quarks state. One of the main assumptions of this approach to a description of hadrons is that we can neglect the non-valence states in this basis when describing the static properties and soft amplitudes (in the hadron rest frame). It is well known that a simple and successful SU(6) classification of low-lying hadrons then emerges, and simple phenomenological models can be developed, which are able to describe well enough (on the whole) m a n y soft hadronic processes. (It is known that the numerical results for the static hadron properties and soft amplitudes do not depend strongly on the precise form of the potential used, if the free parameters available in the model are adjusted properly). b) The light-front (or the light-cone) formalism is also widely used. In the infinitem o m e n t u m frame (p~ --+ oc , if± -- 0), the two-quarks state of this formalism describes two on-shell quarks with : p~ = m~ , p22 = m~ , p+ = p+ + p + , /Y± = /Y1±+/~2± , and P- - Pl - P~ # 0 due to the interaction (p+ = p0 ± P z , P1+2 = zl,2P + , zl + x2 = 1). The light-front w.f. q~Lr(Zl -- x 2 , /~± = ig1± -- ig2J_ ) is analogously defined as a projection of the meson state onto this 2-particle light-front state. c) Finally, the Lorentz-invariant w.f. q~(z,p) used everywhere above in this p a p e r is defined through the m a t r i x element of the 2-quark Heisenberg operator, see Eq. (7). If we introduce its Fourier transform : q~( z, p) = fd4pl d4p2 g~(p, + p2 - p)e 'z(p'-p~' ~(k = p, - P2, P) = fd~k e ~2k * ( k , p ) ,
(16)
then we can consider pl and P2 as the 4-momenta of two quarks, p = Pl + P2 , and these quarks are off mass shell due to the interaction, p~ # m ~ , p~ # m ~ . Really, in the operator formalism, in which q2(k, p) is defined, the name "two-particles" is simply a convention because the Heisenberg operator does not correspond to a definite number of on-shell particles. It is not difficult to obtain the asymptotic connection between k~(k,p) and # c r ( z , k ± ) . Let us return to the calculation of the meson form factor, Fig. 1, and substitute into this diagram the meson w.f. ~ ( z , p ) in the form (16), (d4k = d k + d k _ d k ± ). The four-momenta of the two quarks, k 1 and k2 (and analogously ll and 12), can be represented in the form : k, = (k +, £~±, k~-) = (z~ p+, F~±,ki- ) , k2 = ( z 2 p + , - k x , k ~ ) kT -
, k? + k f = p - = (p° - p~) ~_ M2/2p~ , 2kl~
'
k~ -
2k2:
".L. Chernyak / Properties of hadronic wave functions
309
Because ka~ ..~ k2, ". Pz ~ Q ~ o 0 , and < k]_ > , < kl2 > , < k~ > are all .-. O(/z02) inside the meson, we can neglect k~- and k~- in the hard kernel Th(kl, k2,11,12, q). Then, evidently, the w.f. ~(k, p), Eqs. (7,16), which was introduced in the operator formalism description of hard exclusive processes [4,5], and the w.f. ~(x,/~±) which was used in Eq. 1 are connected as follows :
On the other hand, the same expressions (2,3) were obtained in the light-front formalism [8], and Co is expressed here through the light-front w.f.
=f Therefore, we have the asymptotic connection between the w.fs. used in these two formalisms (in the c.m.s, at pz ~ oe) : ~'LF(Z) = T(x). Let us return now to our question : can we expect that the w.fs. ~(x,/~±) of different hadrons obey the same approximate SU(6) symmetry as the equM-time w.fs. ~r(~e) ? To answer this question, we would have to know the relations between these w.fs.. In general, the relations between ~T(/~) and ql(k) or ~LF(X, fez) are unknown, because their form is tightly connected with the properties of the interaction Hamiltonian, Hin t . Nevertheless, it was a common belief that, for instance, w.fs. ~ z ( x , le±) also obey approximate SU(6) summetry. Moreover, it was proposed (see for instance [14]) that, in a reasonable approximation, the light-front w.fs. ql~y(X , k±) can be obtained from the equaltime w.fs. ~ r ( k ) by neglecting completely the interaction and making only the kinematical substitutions like : ~LF(X, ]~±)= qtT (]~2
~Z i
/~I,i + r a ~ . ~i
(17)
/
This prescription leads to the w.fs. ~0i(~) which are much like ~0pert(X) and besides, SU(6)-like relations connect the w.fs. of different hadrons (and the w.fs. of the same hadron states with different helicities). Comparing with the results described above, obtained from QCD sum rules, you see that this prescription, and the picture it leads to, is not correct 2. What is the main reason? Let me give only one example. Let us suppose that you have chosen the standard SU(6) constituent quark model with an oscillator potential. So, you have a model for the 2-particle pion w.f. ~I'~-(k) (in the meson rest frame). Now, if you use the free-field prescription (17) to find out qlLF(X,/~±) (in the Pz --+ ~ frame) from q/~(k), then you will obtain the w.f. ~o(x) = f d k ± ~ L F ( Z , k±) which is much like ~Opert(Z). What is wrong ? What happened is that using the free-field prescription (17), you have used really the wrong time evolution operator (Hfree instead of H = /-/free + Hint)" As a result, you have obtained with this prescription not the true pion w.f., but some mixture of rr, a~=°(1270),...w.fs.. But it is known from QCD sum rules (see, for instance, [16]) that the true ~r and a) =° w.fs. are respectively much wider and much narrower than ~Opert(~ ). Because the "pion w.f." which you have obtained using (17) is really a mixture of the wide pion and narrow al~=° w.fs., it is not surprising that this "pion w.f." is much like ~0pert(~). 2It was shown in [13] using QCD sum rules that the w.fs. of the nucleon and the As3-resonance, ~aN(Z) and T,x(z), also differ greatly.
V.L. Chernyak / Properties of hadronic wave functio.s
310
It is clear that various non-pert, effects which are contained in Hin t are not small and non trivial in any sense. And neglecting t h e m even in part you can obtain here the wrong results. So, the problem is very complicated, in general. W h a t is interesting, however, is that taking account (phenomenologically) of neglected effects even in part, it is possible to reproduce some characteristic properties of the true w.fs. (as obtained from the QCD sum rules). This was shown by Z. Dziembowski and L. Mankiewicz [15]. They used the usual constituent quark model with the oscillator potential and accounted for a coupling between the quark relative m o m e n t a and their helicities, when going from the equal-time w.f. to the light-front one. And even this simple approach allowed t h e m to reproduce characteristic properties of the pion and the nucleon w.fs, qo~(x) and ~.~,(~:), which were obtained from QCD sum rules. 3 Therefore, it seems, their model is capable to reproduce the good description of hard exclusive processes with pions and nucleons. Moreover, it was shown [15] that it is capable to give a good description of the soft pion and nucleon amplitudes. Really, all this is very appealing. The main drawback of this model is that it seems impossible to have simultaneously a good description of soft and hard amplitudes without changing the main p a r a m e t e r of this model, /3, which determines the m e a n value of the quark transverse momentum. But this model is only a first step, and we can hope that it is possible to improve all this and to have soon good models for the hadronic w.fs. 4
Now, few words about the characteristic values of the quark transverse m o m e n t a in hadrons. Let us consider, as an example, the pion w.f. :
i/.P.Sd d'k,exp (i<(._.+)-,ir¢1} @:(<,¢±)+...,
< id~d'k±
g2,(~,/~±)=l,
/~=0,
,+=0.
Expanding the left- and the right-hand sides in Z± we obtain : < 0[ d(0) "/,75 0 2 u(0) [~r(p) > = if~p~, < ;~ > , + ... , < So, if we know the value of this matrix element, then we know the firstm o m e n t in k~_. The value of the above m a t r i x element was obtained from the QCD sum rules (see, i.e., [16]), and the result looks as :
< ;l
(325 MeV)'
This n u m b e r characterizes the values of the intrisic quark m o m e n t a inside hadrons. 3But their model fails to reproduce the narrow pi-meson w.f. 4The partial wave decomposition of the w.f. fit(k, p) is described in [17].
V.L. Chernyak / Properties of hadronic wave functions
3
311
How can the wave function properties be checked experimentally ?
Let us go now to the last question about the comparison of theoretical calculations with experiment. Many various applications have been considered (see, i.e., the review [16] and more recent papers) : charmonium decays into light mesons and baryons, hadronic form factors, 7V --~ two hadrons, etc... Let me give you only two typical examples.
Figure 10: The decay of the heavy quarkonium 3p0 into two pions.
q__.
! ! !
Figure 11: Typical diagram contributing to the asymptotic behaviour of the nucleon form factor.
a) The 3p0 , ~r+Tr- charmonium decay is described by the diagram on Fig. 10, and the branching ratio looks as [9, 11] :
Br
=
~s16~Mf-~ 27 f l d(1 ~o.(~1)
I0
~, /_'d{ 1 d(2 ~.((2)
~(() 1
= 1 [1 +
1(¢, - ¢.)']. 4 1---~2 J
(18)
V.L. Chernyak / Properties of hadronic wa~v functions
312
For : Br = 2 . 5 10 -3 ;
1.
T h e non-relativistic w.f.
2.
T h e pert. w.f. S0pert(() = 3(1 - ~2) :
Br = 3.5 10 -.2 ;
3.
T h e " s u m rule w.f." V,~(~) = ~ ( 1 - ~2)~2 :
Br = 1.1% .
The experiment So, you see, the that o b t a i n e d from b ) T h e nucleon obtains :
¢Pnon-rel(() = ~5(() :
gives [27] : Br = (0.8 -4- 0 . 2 ) % . answer is highly sensitive to a precise form of the pion w.f., and it is the Q C D s u m rules which is in a g r e e m e n t with experiment. form factor is described by d i a g r a m s like those shown on Fig. 11. One
- 3 10 -3 4
p
for ~0~9n-rel
2
Q GM(Q ) 1 GeV 4
Q4G~M 1 GeV 4
--~
0 0.95
_~
for
~pert
(19)
for ~sXum rule
+0.2 10 -3 for W_non-tel N pert 3.8 10 -2 for ~oN --0.47 for ¢psum rule
(20)
You see t h a t
. non-tel and ~opert ~v N N give not only too small predictions, but even wrong signs, while the " s u m rule w.f." agrees with the data. T h e s e two e x a m p l e s are typical. T h e s i m p l e - m i n d e d models of w.f. (like ~on°n-rel(x), e t c . . . ) give ordinarily too small predictions, and only the w.f. o b t a i n e d f r o m the s u m rules gives simultaneously a good enough description of a lot of the e x p e r i m e n t a l data. To illustrate the high sensitivity of the results to a precise form of the w.f., let me mention the nucleon m o d e l w.f. p r o p o s e d by M. Gari and N.G. Stefanis [18]. It was chosen so t h a t four of its lowest six m o m e n t values are t a k e n from the Q C D s u m rules, but two other ones are (arbitrarily) changed significantly (and are far away from the limits allowed by the s u m rules). As a result, this w.f. describes well the p r o t o n form factor, b u t fails (by a factor ~_ 50) to describe the J / ¢ --+ ~ p decay.
Let us consider finally some e x a m p l e s which are directly connected with the H T contributions and which were not considered here up to now. 1. High-p± b a r y o n p r o d u c t i o n In the process : p+p ---+p± + X , where the final p r o t o n has a large transverse m o m e n t u m , there is a H T c o n t r i b u t i o n described by the d i a g r a m on Fig. 12a. T h e a m p l i t u d e of this contribution is p r o p o r t i o n a l to the integral [19] : IN = r / d 3 z ~pN(Z) xl J X l X 2 X 3 1 - X2
(21)
V.L. Chernyak / Properties of hadronic wave functions
313
If you can separate out only those events in which this high p± proton is not surrounded by other high-p± hadrons (i.e., it is not a m e m b e r of a jet), then the above is the only contribution. It was estimated that then (for the nucleon w.f. obtained from the sum rules) o" ~ 4.10-35cm 2 at v'~ ~-- 40 GeV , p± > 2 GeV .
(22)
In the usual one-particle inclusive experiments (see [20] and the references therein) there is also the usual leading twist contribution, which corresponds to the large p± scattering
°
Figure 13: The H T contribution to e + e- ~ M± + X .
of two incoming quarks and subsequent fragmentation of the scattered quarks. Besides, there are, for instance, the contributions like those shown of Figs. 12b and 12c. (~P is the Pomeranchuk trajectory or the one gluon exchange). Initial protons scatter here quasi-diffractively on each other and produce then two large p± quark jets (Fig. 12b) or large p± diquark (with a small invariant mass go << p±) and ant±quark jets (Fig. 12c). The large p± proton originates from the fragmentation of these large p . jets. At large p± the H T contributions like those shown on Figs. 12b,c are the main power corrections to the leading twist contribution. The reason is that a diquark acts like a meson, and at large p± a hard process with a meson is parametrically preferred to those with a baryon, because the meson form factor falls off more slowly. The cross section has the form (qualitatively, E >> Pz) :
dp
-
LT
+
+
.
.
.
.
Here : (do'/dp2,)LT is the leading twist contribution, and the second and third terms in brackets describe the diquark (Figs. 12b,c) and the baryon (Fig. 12a) contributions respectively ; #d and /~B are some characteristic masses, which can be expressed through the integrals over the diquark and the baryon wave functions, see (21). In any case, if you are able to measure experimentally these H T contributions, you will have the numerical values for ~a and # s , i.e., for definite integrals over the diquark and the nucleon w.fs. You can check then what model w.fs. reproduce correctly the experimental values of/td and # s • The baryon production in deep inelastic scattering was considered in [21].
314
V.L. Chernyak / Properties of hadronic wave functions
q _,
q
=.
Fig. 12 a
q
G C
I !
q
;11 ~
i
~q I,q ~,q
G C Fig. 12 b
q q
./_/'
@ TP
@
/~ C
~q *q
t,q
Fig. 12 c
F i g u r e 12: T h e H T c o n t r i b u t i o n s to p + p --+ p± + X .
V.L. Chernyak / Properties of hadronic wave functions
2.
315
Isolated meson production in e+e - [22].
In e* e - -~ M± + X , where the meson has a large transverse m o m e n t u m but is not a m e m b e r of a jet, the process is described by the diagram on Fig. 13. This is the H T contribution corresponding to : e + e- --+ M± + q + ~ , and the meson has a large transverse m o m e n t u m p± with respect to its parent quark. If p± is large but much smaller than the parent quark energy, then there is a universal probability function for a quark to emit such a meson :
4 ,~2 zF2(z) 27 p~_
w(z,p±)-
Emeson
;
z = - - ;
Equar k
LI(I+z--z') ~(') dx. FA=o(Z) =
(1 - x) (1 - zx)
L1 ;
(1 --z-)-(i=
FI,\I=,(z ) = z
(23)
~(~.) d~ zz)
You see that by measuring the z-dependence you can, in principle, measure the form of the w.f. ~(z). It was estimated that for pion production (using the pion w.f. from the sum
rules): o.(e+e - .-~ lr+qq)
~r(e+e- --, g+g-)
3.
--~10 -3 at E b e a m > 4 G e V .
(24)
P r o m p t meson production in meson-nucleon collisions.
These are processes like : 7r + p --. M± + X , where the final meson has a large p±. The detailed description of the HT contributions for such (and similar) processes have been given by M. Benayoun, P. Leruste, J.L. Narjoux and R. Petronzio in [23]. For the above reaction the H T contributions considered in [23] correspond to hard subprocesses like : q + q ~ M± + g (Fig. 14 and other similar ones). The experiment was performed by the WA77 group and, for example, they see clearly the H T p0 signal and don't see the H T f0 signal [24]. The reason m a y be that the H T Pt:q=l production is highly enhanced kinematically in comparison with the H T P~=o production [23]. For instance (for simplicity, consider d+u ~p++g):
do- "~,t
~p2~JHT -
-
256 ~.~,~ t ~ + ~,~ £ 729 t2u 2 s -
\~JHT = 8 ~ ' ~ 1
S
i-',d~: +,~(~;) 1 ---(-~
J-1
1
_-~
' ,
(25) (26)
1
At the same time, the flAl=2 o o and flAt=l are not H T - p r o d u c e d in subprocesses like those shown in Fig. 14. Further, it seems [23, 24], that the WA77 d a t a point to a wide p±-meson w.f. (approximately as wide as the pion one obtained from sum rules, see above).
316
V.L. Chernyak / Properties of hadronic wave functions
/A) v
q
~-
(M:P,f,n' .... )
"'-g.
Figure 14: The HT contribution to lr N ~
q
-~
M± + X .
~.q
Figure 15: The additional HT contribution to 7r N --+ M ± + X , 79 is the Pomeranchuk trajectory (the analogous Pomeranchuk contributions are present in many others high-p± processes).
From my viewpoint, a more careful analysis of the WA77 data is needed, before making definite conclusions about the properties of the p±-meson w.L. One reason is that there are calculations of the charmonium decay 3P2 ~ pp [25,26] :
V.L. Chernyak / Properties of hadronic wave functions P±P±) = 6 B r ( 3 P 2 --+ r + r )
\f~]
I,~
~ 30
-
I,~
"
3l 7 (27)
You see that the p± production is strongly enhanced kinematically in comparison with that of 7r (or pL), similarly to (25, 26). Here, I~ and I,~ are definite integrals (similar to (18)) over the p±-meson and the pion w.fs., and they are such that if the p± and 7r w.fs. are much like each other, then [I~/I~ I > 1. Therefore, (using the experimental value Br (3P2 ---* 7r+ 7r-) = (0.19+0.10)% and neglecting the 3P2 ~ p~ p~, contribution) we should obtain in this case : Br(3P2~p°p°)
~ 3 0 B r ( 3 p 2 ~ r r + 7r-) _~ 6 % ,
(28) Br (3p2 --~/~0. KO*) ~ 12% . This contradicts strongly the available experimental data [27] : Br(3P2~p°p°)
<
Br (3P2--~p°Tr +Tr-)
= (0.67±0.40)%,
(29)
Br (3P2 ---,/~o* g 0 * ) < Br (3P2 ---* g °* K + 7r=F) = (0.47 5= 0.28)% . At the same time, if we use (see above) the wide pion w.f. and the narrow p±-meson w.f. obtained from the s u m rules, then I ~ / I , << 1 , and [25] : Br (zP2 -~ p0 p0) _~ Br (3P2 -~ 7r+ 7r-) = 0.2% ,
(30)
and this looks much more realistic. The second reason is that there are additional HT contributions, for instance those shown on Fig. 15 (79 is the Pomeranchuk or the one-gluon exchange). Here initial quarks scatter quasi-diffractively, producing in the (near) forward direction one light and one heavy (i.e., virtual, -.~ O(p~)) quark, the latter decaying into a meson and a q- It may be that such contributions are not small at high energies 5 (/~ >> p±). In any case, the H T effects are seen in the experiment but, it seem to me, a careful analysis of all experimental and theoretical uncertainties (i.e., K-factors, additional contributions, e t c . . . ) should be carried out before concluding definitely about the properties of the p i - m e s o n w.f. from the WA77-experiment.
You see from all the above examples and from other ones considered at this Workshop, that an investigation of the H T contributions has good perspectives of providing us with useful information about the properties of the hadronic wave functions. A hard work is required, however, before we will have sufficiently accurate and conclusive results. Acknowledgments : I a m deeply grateful to M. Benayoun for introducing me into the HT physics and to M. groissart and M. Benayoun for the very fruitful discussions on the properties of the wave functions. The kind hospitality of the Laboratoire de Physique Corpusculaire, Coll~ge de France, where this paper was prepared, is greatly acknowledged. 5Besides, such contributions can be important in many other processes at high energies, E >> p±.
318
V.L. Chernyak / Properties of hadronic wave functions
References 1. M. Benayoun, contribution to the XXIII Rencontres de Moriond, March 1988. 2. V.A. Matreev et al., Lett. Nuovo Cim.7 (1973) 719 ; S.J. Brodsky and G.R. Farrar, Phys. Rev. Lett. 31 (1973) 1153, Phys. Rev. D 11 (1975) 1309. 3. P.V. Landshoff, Phys. Rev. D l 0 (1974) 1027. 4. V.L. Chernyak and A.R. Zhitnitsky, Pisma ZhETP 25 (1977) 544 (Engl. Transl. JETP Left. 25 (1977) 510). . V.L. Chernyak, V. G. Serbo and A.R. Zhitnitsky, Pisma ZhETP 26 (1977) 760 (Engl. Transl. JETP Lett. 26 (1977) 594) ; Yad. Fiz. 31 (1980) 1053, 1068. 6. G.R. Farrar and D.R. Jackson, Phys. Rev. Lett. 43 (1979) 246. 7. A.V. Efremov and A.V. Radyushkin, Phys. Lett. B 94 (1980) 245 ; Teor. i Mathem. Fiz. 42 (1980) 147. 8. P. Lepage and S.J. Brodsky, Phys. Left. B 87 (1979) 359 ; Phys. Rev. Lett. 43 (1979) 545 ; Phys. P~ev. D 22 (1980) 2157. 9. A. Ducan and A. H. Mueller, Phys. Rev. D 21 (1980) 1636, Phys. Lett. B 93 (1980) 119. 10. M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B 147 (1979) 385, 448. 11. V.L. Chernyak and A.R. Zhitnitsky, Nucl. Phys. B 201 (1982) 492. 12. V.L. Chernyak and I.R. Zhitnitsky, Nucl. Phys. B 246 (1984) 52. I.D. King and C.T. Sachrajda, Nucl. Phys. B 279 (1987) 785. V.L. Chernyak, A.A. Ogloblin and I.R. Zhitnitsky, Preprint INP-87-135, Novosibirsk, 1987. 13. G.R. Farrar, A.A. Ogloblin, H. Zhang and I.R. Zhitnitsky, Preprint RU-88-14, Rutgers, 1988, Nucl. Phys. B, in press. 14. S. J. Brodsky, T. Huang and G.P. Lepage, in Quarks and Nuclear Forces, edited by D. Fries and B. Zeitnitz, Springer Tracts in Modern Physics, v. 100 (Springer-Verlag, New York, 1982). 15. Z. Dziembowski and L. Mankiewicz, Phys. Rev. Lett. 58 (1987) 2175, Z. Dziembowski, Phys. Rev. D 37 (1988) 778 ; Phys. Rev. D 3"/(1988) 2030. 16. V.L. Chernyak and A.R. Zhitnitsky, Phys. Rep. 112 (1984) 173. 17. M. Benayoun and M. Froissart, Preprint LPC-88-16, CollSge de France, Paris, 1988. 18. M. Gari and N.G. Stefanis, Phys. Lett. B 175 (1986) 462 ; Phys. Rev. D 35 (1987) 1074.
V.L. Chernyak / Properties of hadronic wave functions
19. A.V. Grozin. Z. Phys. C 34 (1987) 531. 20. D. Drijard, contribution to this Workshop. 21. A.G. Grozin, Yad. Fiz. 37 (1983) 424 M. Fontannaz and H.F. Jones, Z. Phys. C 28 (1985) 371. 22. V.N. Baier and A.G. Grozin, Phys. Lett. B 96 (1980) 181. 23. M. Benayoun, P. Leruste, 3.L. Narjoux, R. Petronzio, Nucl. Phys. B 282 (1987) 653. 24. M. Benayoun et al. Phys. Lett. B 183 (1987) 412 ; M.T. Trainor (WA77), contribution to this Workshop. 25. V.L. Chernyak, A.R. Zhitnitsky and I.R. Zhitnitsky, Nucl. Phys. B 214 (1982) 477 ; Yad. Fiz. 38 (1983) 1074. 26. V.M. Baier and A.G. Grozin, Yad. Fiz. 35 (1982) 1021. 27. Particle Data (1986).
319
297
H a r d (semi) e x c l u s i v e p r o c e s s e s in Q C D a n d p r o p e r t i e s of h a d r o n i c wave f u n c t i o n s V. L. C H E R N Y A K 1 Coll~ge de France - Laboratoire de Physique Corpusculaire 11, Place Marcelin Berthelot - 75231 PARIS Cedex 05 - F R A N C E and Institute of Nuclear Physics, 630090 NOVOSIBIRSK 90, USSR
Abstract A short review of the properties of hard exclusive processes and hadronic wave functions is given. The connection with the higher twist contributions in inclusive processes is considered.
Introduction I shall try to give a short overview of our present understanding of hard exclusive processes. The main points touched upon are : 1) General properties of hard exclusive processes, 2) Properties of hadronic wave functions, 3) Where and how the wave function properties can be checked experimentally. But before, because this is a workshop on the Higher Twists, let me put some questions of relevance to both Higher Twists and Exclusive Processes: What is really interesting in the Higher Twist (HT) contributions ? W h a t can we learn about the hadronic interactions and about the internal structure of hadrons from HT ? In what way the HT contributions can be used to obtain a new useful information ? The specific property of HT contributions is that they are between the pure exclusive processes and the usual inclusive ones, i.e., they are "semi-exclusive", in some sense. It is this property which determines their role. The HT contribution looks as a convolution of the hard kernel describing the short distance interactions, with the hadron wave function describing a soft non-perturbative interaction responsible for hadron formation. 0920 5632/89/$03.50 (~) Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
298
V.L. Chernyak / Properties of hadronic wave functions
It is clear that a separation of HT contributions is not the best way to study the properties of the hard kernels (i.e., the properties of perturbative QCD). This can be done much more easily by means of the usual leading twist (LT) contributions in hard inclusive processes. Therefore, H T are of interest just for studying properties of the hadronic wave functions. Although this can be done, in principle, in pure exclusive hard processes also, the inclusive HT are frequently more suitable experimentally. Besides, because the hard production of each additional hadron leads to a further power suppression (an additional form factor), the 1-particle HT are minimally suppressed as they correspond to hard production of only one hadron. At large m o m e n t u m transfers they are preferred, in this respect, in comparison with pure exclusive processes. A short but informative and clear review of the experimental results in HT physics is given in [1]. Well, let us return now to the points 1-3 above and let us look at the characteristic properties of exclusive processes.
1
G e n e r a l p r o p e r t i e s of hard e x c l u s i v e p r o c e s s e s
The simplest hard exclusive amplitude, the meson form factor at large m o m e n t u m transfer, is presented on Fig. 1 (c.m.s.) :
"x I
'
'1
I
o
"J
"~" ~ h a r d
M 2 (
~2
kernel Th
Figure 1: The meson form factor : kl ~- Zlpl + k±, k2 ~- z2pl - k±, 11 ~- ylp2 + Ix, 12~-y2p2-1x. It is a convolution of the initial and final hadron wave functions, q/,(z, ~:±) and g2*(y, l±),
(l ki)
with a hard kernel Th x, y, -0-' q - ' q
which is the form factor of the two-quark system [4] :
".L. Chernyak / Properties of hadronic wave functions
F(q) = f dy dF± = (l/q)
A
(y, F±)
T'
299
]~±) dx d;.l ,
W'
T h e wave function gJ(z, re±) is the a m p l i t u d e for finding two quarks w i t h longitudinal m o m e n t u m fractions z a n d 1 - x (at p~ ) oo) and transverse m o m e n t a k± and ( - k ± ) within the m e s o n (for m o r e details, see p a r t 2 below). T h e h a r d kernel Th represents the small-distance interaction and so it can be calculated reliably within p e r t u r b a t i v e Q C D . I want to e m p h a s i z e t h a t there is nothing u n k n o w n in it. If we neglect the intrinsic quark transverse m o m e n t a k± and l± in Th (indeed, they give only power corrections ~ O ( k ~ / q ~ ) ) , then the m e s o n form factor takes the f o r m :
F(q) = Co/(q )-L where C O ~-
(9,)
1 d x d y ~o~(y) f h a l : d ( ~ , y ) 731(x) ,
~0
(3)
T h e power N in (2) follows from a simple dimensional counting [2] : N = nman -- 1, there nr~n is the m i n i m a l n u m b e r of constituents within a given h a d r o n (nmm = 2 for mesons, nmin ---- 3 for baryons). At the s a m e time, to predict the absolute n o r m a l i z a t i o n (the constant Co in (2)), you have to know the h a d r o n wave function 7~(z). However, the dimensional counting is working only if there are no infrared power singularities which increase the a s y m p t o t i c s (i.e., if the integrals over z a n d y in (3) are not divergent as a power), and if there are no selection rules which suppress the a s y m p t o t i c s . T h e r e are no infrared singularities for form factors, but this is not the case, in general, for the elastic scattering a m p l i t u d e s [3]. (Indeed, the I S R e x p e r i m e n t s on high energy p-p elastic scattering have found the b e h a v i o u r : d a / d t ,,~ t - s at s >> Itl >> ~0~, which violates the dimensional counting prediction and agrees with the Landshoff one [3]). Let us r e t u r n now to the wave function (w.f.) ~ z ) in (3), which d e t e r m i n e s the final answer. It is the projection of the total w.f. @(x,k±) onto the Lz -- 0 state. You see, therefore, t h a t there is a selection rule [4] : the leading contributions give only the L . = 0 c o m p o n e n t s of the total h a d r o n w.f. Clearly, in this w.f. c o m p o n e n t : A hadron = ~
A quark i
,
(4)
i
(A is the helicity). A n d consequently, only meson states with A = 0 and ]A[ = 1, and b a r y o n states with ])~[ = 1/2 a n d ])~[ = 3/2 give leading contributions in h a r d exclusive processes. (For glueballs [A[ = 2 is also allowed). Besides, because the q u a r k helicity is conserved at large m o m e n t a , 2-, initial
hadron
=
2_.., final
A hadron
•
V.L. Chernyak / Properties of hadronic wave functio1~s
300
If, for instance, you are looking for a production of mesons with higher ])~1, then L: # 0 in such meson states. To produce a final state with Lz = n, you should have to separate out of the kernel
Th z , y , - ~ ,
a t e r m which contains, at least, n powers of/'± :
n
But because /~± enters Th only as (l±/q), you will have ~ O ( 1 / q ~) suppression of the hard amplitude. Proceeding in such a way, you can obtain finally a general formula for the power behaviour of any form factor [4] :
< P2, J2, .X~IJxIPl, J1, At > ~ ( l / q ) I \ ' - ' \ ' l + ~ " - : ~ ,
(5)
(here Jx is the vector or axial vector current with helicity $, and $ = $1 + $2 in the c.m.s.).
To illustrate (5), let us consider a simple example which contains all the characteristic features responsible for the behaviour (5). The contribution to the < $2 = 2 ] J1 I $l = - 1 > form factor is shown on Fig. 2 (arrows show quark helicities). In comparison with a dimensional behaviour ~ ( l / q ) , there are additional suppressions : (a) ~,, (~±/q)2 due to L~2 = 1 and L~ 1 = - 1 , (b) ~-, (k± + m ) / q due to the quark helicity fllp.
q
~q
L I= - 1
I !
t I I ! ,
L.Z=I 2
Figure 2: The contribution to the form factor < )t2 = 2 [ J,\=l [ )~1 : - 1 > On the whole, you have just the behaviour given by (5). Now, if we use the Born (i.e., the lowest order) approximation for the h a r d kernel Th , then the factorization of the total amplitude into a product of hadronic wave functions and Th is self-evident (see Fig. 1). But when the higher-loop p e r t u r b a t i v e theory corrections are taken into account, it is then a difficult technical problem to show that factorization is still working, and to sum up loop logs.
V.L. Cher~D'ak / Properties of hadronic wave functions
301
Nevertheless, this work was p e r f o r m e d in [5 - 9], and it was shown t h a t the factorization survives and t h a t all loop corrections are accounted for by the usual a n o m a l o u s dimensions. All this leads finally to an additional slow evolution with q2 (like t h a t ix/the deep inelastic e-p scattering), and at available Iq21 it is not of great i m p o r t a n c e , really. Let me dwell shortly on the space-time picture of a h a r d exclusive process (Fig. 3).
x....Z 2~,hard kernet Figure 3: T h e s p a c e - t i m e r e p r e s e n t a t i o n of the m e s o n form factor, z~ ~ z~ ~ (zl - z~) ~ ~ 1/q ~ •
ra.--,1Kj.
hard kernet
Figure 4: T h e s p a c e - t i m e picture of the m e s o n form factor : $ is the p h o t o n wave length, r L is the m e s o n longitudinal size; r± is the m e s o n transverse size. T h e p h o t o n wave length is )~ --- O ( 1 / q ) in the c.m.s., the h a r d kernel Th is c o n c e n t r a t e d in a small s p a c e - t i m e region with a linear size ~-- O ( 1 / q ) also, and it seems t h a t only the value of the w.f. at the origin should enter the answer for F ( q ) : F ( q ) ,~ ¢ ~ ((q2)N 0)¢1(0)
'
¢(0) = f dz dfe± ~2(z,/~±) ,
(6)
(i.e., the function fhard in (1) should be independent of z and y). This is not the case really (see Figs. 3 a n d 4, c.m.s, is used). T h e reason is t h a t the h a d r o n with a large m o m e n t u m p~ ~ q is L o r e n t z - c o n t r a c t e d along his m o t i o n , so t h a t its longitudinal size is : r L ~ " 1/q . Its transverse size, r3_, r e m a i n s large, however : r± ~ l/[f¢±l -,~ 1 fro. So, the p h o t o n with wavelengh $ ~-, 1/q "feels" the distribution of quarks in longitudinal m o m e n t u m fractions z~ inside the h a d r o n , a n d it does not resolve their distribution in/~±.
V.L. Chernyak / Properties of hadronic wave functions
302
2
Properties of the hadronic wave functions
Let us come now to the properties of w.f. Ti(x) (just these determine the absolute values of leading exclusive amplitudes ). The space-time meson w.f. which we use is defined (schematically) as [4] : ¢ ( z , p ) = < 0 I q(z) I q ( - z ) I M(p) > = ¢(zp, z ~) , I = exp{i 9
// z
(7) d~.A.(o')} .
Evidently, it is Lorentz and gauge invariant. To describe the leading contributions to hard exclusive amplitudes, we need only the w.f. taken on the light cone : ¢(zp)=¢(zp,
z2=0),
(pz"~Q~oo,
z~,~ l / Q , z 2 ~ l / Q 2,
z p ~ l) .
It can be represented in the form : ¢(zp)
~01 dzldz2 6(1 - zl - z2)
=
f'd¢
e ~Z(Zlp)
--
iZ(X2P) ~(X)
=
(8)
ei~(zp) qa(~)
1
=
xt-z2,
xl-t-z2= 1 ,
and ;E1 and x2 are the hadron m o m e n t u m fractions of the two quarks. Simple examples of T(~) are shown on Fig. 5. The non-relativistic w.f. looks like a 6-function, its width is ,-~ (binding e n e r g y / h e a v y quark mass) << 1. The Tpert(~) "" (1 - (2) corresponds to a perturbative (or, equivalently, asymptotic) distribution of two massless quarks. Expanding (7) in z you will obtain :
< 0l 9(0)
z
q(O) [ U(p)
> = (zp)"
(" ~(()
, z2 ~
o .
(9)
Therefore, if you know these m a t r i x elements, then you know the w.f. moments. The method of the QCD sum rules [10] allows us to find out the values of such m a t r i x elements. The idea of the approach is as follows. The correlator
I(q) =
fdz e 'q~ < 0 I T J r ( z ) J2(0) l0 > ,
J = qDnq , D = iO + gA
(10)
is considered, and its behaviour at Iq2l >> ~02 is calculated, (Fig. 6). The leading contribution at large [q2[ gives, of course, the perturbative theory (and the usual loop corrections to it). In these contributions the large m o m e n t u m q is divided more or less equally between all the lines (these lines with a virtuafity ,-- q2 are the boldface ones). Besides, there are contributions where the large m o m e n t u m q goes only through some lines, while other ones are soft. For such contributions we calculate explicitly the hard (small distance) part of diagrams using the p e r t u r b a t i o n theory, while the soft fines represent now the nonperturbative quark and gluon fields which are averaged over their fluctuations in the physical vacuum.
. L. Chern 3 ak / Properties of hadronic wave functions
/
303
b
)IL -1
Figure 5: a / T h e
p e r t u r b a t i v e (or, equivalently, asymptotic) wave function
b/The
non-relativistic wave f u n c t i o n : V~non_~el(Z) ~'- 6(zl - ½) ~ 6(¢) .
Physically, all this is a kind of multipole expansion : we calculate a radiation of quarks and gluons with a characteristic wave length ,-- 1/#0 (the scale of the v a c u u m fields) by a small system of the size.--- 1/q. T h e sum rules obtained in this way can be represented (qualitatively) in the form : rl
fl
r 1
Ave(C,/~F).
(11)
Here ~ ( ~ ) is the true m e s o ~ w.f. we are looking for, ~pe,t(~) -- ~(1 - ~ ) is the perturbative w.f. for two massless quarks and A~o is the n o n - p e r t u r b a t i v e contribution to the w.f. :
A%o(C, ATI) = Ci
11714
6(i - C) + 6(1 + ~) + (12)
< 0],~ql0 >2 {6'(1 - (~) - 5'(1 + (~)} + . . . +C2 /~-6
V.L. Chernyak / Properties of hadronic wave functions
304
pert. theory
~xpert. loop corrections
[d)
.~non-pert. vacuum quark field
-~"
•
.
gluon field
m
c)
(b )
(c)
t;uum
/'quark fields
Figure 6: Various perturbative and non-perturbative contributions to the two-point correlator.
".L. Chernyak / Properties of hadronic wave fun ctions
305
(C 1 and C2 are known coefficients and /1~ ~_ 1 GeV 2 is a characteristic scale for a given correlator, its precise value is determined by a special fitting procedure). You see that while ~Vpert(~) corresponds to a smooth symmetrical distribution of the meson m o m e n t u m between the two quarks, the non-perturbative contributions are highly assymmetric. T h e y correspond to distributions such that nearly all the meson m o m e n t u m is carried by one quark. The sign and the absolute value of A ~ depend strongly on the hadronic q u a n t u m numbers, while ¢Ppert is common to all mesons. As a result, the moment values of the true meson wave function differ strongly, in general, from those of ~pert , and this means that the form of the w.f. also differs strongly from ~pert(~) • The non-perturbative effects (i.e., A~) are very large and positive for the pion w.f. [11]. Therefore, the true m o m e n t values are much larger t h a n the perturbative ones, and the true pion w.f. is much wider t h a n ~pert(~) • The results for the lowest moments are shown in the table. The moment values of the simple model w.f. : epsumrule(<)_-- !~(1 _ <2)~2 (13)
are also given therein.
<
Vpert(¢)
sum rule
~sum rule(~)
1
1
1
0.20
0.44
0.43
0.086
0.27
0.24
T a b l e : ~Ppert(~)-- 43-(1 - ~ 2 ) ,
~psumrule(~) = ~ ( 1 -~2)~2
There are general arguments originating from the quark-hadron duality, showing that (in the chiral limit) the true w.f. has the same parametical behaviour at xi ~ 0 and xi ~ 1 as ~ p e r t ( X ) , i.e, ~ xlx2 • This excludes models like those proposed by Brodsky and Lepage : ~ r ( z ) ~ (ZlZ2) 1/4,
(14)
We see therefore (Fig. 7), that (at Pz -* oo) the distribution of the pion m o m e n t u m between two quarks is.highly assymmetric. There is a dip at ~ = 0 (i.e., x~ = x2 = 1/2), and this means t h a t the pion m o m e n t u m is rarely divided equally between two quarks. Usually~ the largest part of the pion m o m e n t u m is carried by one quark. For for the [25]. For is very
the pr - px=0 meson, the non-perturbative effects are also positive, but smaller t h a n pion. As a result, the pn-meson w.f. is still wider than ~ p e r t ~ but narrower than ~,~ p± - PI~\I=I meson the non-pert, effects are negative. As a result the p x - m e s o n w.f. narrow (see Fig. 8) [251
<~2
>pert
<~2>pL
----0.20 ' ~- 0.30,
<~2>~-----0.44, ~2 >p± -~ 0.14.
<
}
, < (2 > i = /_ll d~ ( 2
The nucleon state at pz ---+oo can be represented (in the leading twist) as:
¢Pi(~)
•
(15)
306
V.L. Chernyak / Properties of hadronic wa~v functions
[0 Figure 7: a / T h e
1
pion wave function T~(~) o b t a i n e d from s u m rules :
~,~um rule(C) = ~(1 - ~2)¢2 b/The
-1
p e r t u r b a t i v e pion wave f u n c t i o n : ~Opert(() = 3(1 - (2)
0
1
Figure 8: a / T h e non-relativistic wave f u n c t i o n : ~0non.rel(~) = ~(~) , b / T h e wave function of the p± = Pl~l=x m e s o n o b t a i n e d from the s u m rules ~ , ± ( ~ ) = ~(115 _ ~ 2 ) 2 , c/ T h e p e r t u r b a t i v e p ± - m e s o n wave f u n c t i o n : Tpert(~) = 3(1 - ~2).
".L. Chernyak / Properties of hadronic wave functions
3{17
The nucleon w.f. ~,~,'(=1, Z2, =3) was also investigated [12], and its profile is shown on Fig. 9. It is seen that it is highly assymetric : the largest part (-~ 60%) of the proton m o m e n t u m is carried by one u-quark with its spin parallel to the proton spin, while each of remaining two quarks carries _~ 20% of the total m o m e n t u m .
=.1
XI=
X]=l Figure 9: The profile of the proton wave function obtained from the sum rules. Let me summarize now (qualitatively) what we have learned about the hadronic wave functions on the light cone :
~(z) = ]all
'~,(=, k_~) •
1. The distribution of quarks in longitudinal m o m e n t u m fractions is wide, in general. This means that quarks are highly relativistic inside the hadrons, and we can't use the models with non-relativistic constituents. 2. The form of the true w.f. differs greatly, in general, from the perturbative (asymptotic) w.f., Tpert(Z) ~ and this means that complicated non-perturbative interactions influence strongly the w.f. form. 3. The w.f. form depends strongly on the hadron q u a n t u m numbers, in particular on its helicity (while 7~pert(Z) is universal and does not depend on the hadron considered). This is because the non pert. effects differ strongly in channels with different q u a n t u m numbers.
308
V.L. Chernyak / Properties of hadronic wave fimctions
These last two properties were the most surprising for m a n y physicists, because it was expected (from the experience with the old-fashioned SU(6)-constituent quark model) t h a t the w.fs. of hadrons entering the same SU(6) multiplet are like each other and do not differ greatly from ~pert (x) . To understand to what extent these expectations can be justified, we need to know the connection between different 2-particle w.fs. a) The old SU(6)-constituent quark model (CQM) originates from the old picture of nonrelativistic quarks inside the hadron and uses the so-called equal-time w.f., q r , and the "equal-time" basis (this formalism is a generalisation of the standard hamiltonian formalism of non-relativistic q u a n t u m mechanics). In the meson rest frame, the two-quarks state of this formalism describes two on-shell quarks w i t h : p~ -- r a ~ , p~ = my , ff = ff~ +/~2 = 0 , and p0 _ p0 _ p0 # 0 due to the interaction (p0 and/~ are the meson energy and m o m e n t u m ) . The w.f. q T ( k = ffl -- if2) is defined as a projection of the meson state onto this two-quarks state. One of the main assumptions of this approach to a description of hadrons is that we can neglect the non-valence states in this basis when describing the static properties and soft amplitudes (in the hadron rest frame). It is well known that a simple and successful SU(6) classification of low-lying hadrons then emerges, and simple phenomenological models can be developed, which are able to describe well enough (on the whole) m a n y soft hadronic processes. (It is known that the numerical results for the static hadron properties and soft amplitudes do not depend strongly on the precise form of the potential used, if the free parameters available in the model are adjusted properly). b) The light-front (or the light-cone) formalism is also widely used. In the infinitem o m e n t u m frame (p~ --+ oc , if± -- 0), the two-quarks state of this formalism describes two on-shell quarks with : p~ = m~ , p22 = m~ , p+ = p+ + p + , /Y± = /Y1±+/~2± , and P- - Pl - P~ # 0 due to the interaction (p+ = p0 ± P z , P1+2 = zl,2P + , zl + x2 = 1). The light-front w.f. q~Lr(Zl -- x 2 , /~± = ig1± -- ig2J_ ) is analogously defined as a projection of the meson state onto this 2-particle light-front state. c) Finally, the Lorentz-invariant w.f. q~(z,p) used everywhere above in this p a p e r is defined through the m a t r i x element of the 2-quark Heisenberg operator, see Eq. (7). If we introduce its Fourier transform : q~( z, p) = fd4pl d4p2 g~(p, + p2 - p)e 'z(p'-p~' ~(k = p, - P2, P) = fd~k e ~2k * ( k , p ) ,
(16)
then we can consider pl and P2 as the 4-momenta of two quarks, p = Pl + P2 , and these quarks are off mass shell due to the interaction, p~ # m ~ , p~ # m ~ . Really, in the operator formalism, in which q2(k, p) is defined, the name "two-particles" is simply a convention because the Heisenberg operator does not correspond to a definite number of on-shell particles. It is not difficult to obtain the asymptotic connection between k~(k,p) and # c r ( z , k ± ) . Let us return to the calculation of the meson form factor, Fig. 1, and substitute into this diagram the meson w.f. ~ ( z , p ) in the form (16), (d4k = d k + d k _ d k ± ). The four-momenta of the two quarks, k 1 and k2 (and analogously ll and 12), can be represented in the form : k, = (k +, £~±, k~-) = (z~ p+, F~±,ki- ) , k2 = ( z 2 p + , - k x , k ~ ) kT -
, k? + k f = p - = (p° - p~) ~_ M2/2p~ , 2kl~
'
k~ -
2k2:
".L. Chernyak / Properties of hadronic wave functions
309
Because ka~ ..~ k2, ". Pz ~ Q ~ o 0 , and < k]_ > , < kl2 > , < k~ > are all .-. O(/z02) inside the meson, we can neglect k~- and k~- in the hard kernel Th(kl, k2,11,12, q). Then, evidently, the w.f. ~(k, p), Eqs. (7,16), which was introduced in the operator formalism description of hard exclusive processes [4,5], and the w.f. ~(x,/~±) which was used in Eq. 1 are connected as follows :
On the other hand, the same expressions (2,3) were obtained in the light-front formalism [8], and Co is expressed here through the light-front w.f.
=f Therefore, we have the asymptotic connection between the w.fs. used in these two formalisms (in the c.m.s, at pz ~ oe) : ~'LF(Z) = T(x). Let us return now to our question : can we expect that the w.fs. ~(x,/~±) of different hadrons obey the same approximate SU(6) symmetry as the equM-time w.fs. ~r(~e) ? To answer this question, we would have to know the relations between these w.fs.. In general, the relations between ~T(/~) and ql(k) or ~LF(X, fez) are unknown, because their form is tightly connected with the properties of the interaction Hamiltonian, Hin t . Nevertheless, it was a common belief that, for instance, w.fs. ~ z ( x , le±) also obey approximate SU(6) summetry. Moreover, it was proposed (see for instance [14]) that, in a reasonable approximation, the light-front w.fs. ql~y(X , k±) can be obtained from the equaltime w.fs. ~ r ( k ) by neglecting completely the interaction and making only the kinematical substitutions like : ~LF(X, ]~±)= qtT (]~2
~Z i
/~I,i + r a ~ . ~i
(17)
/
This prescription leads to the w.fs. ~0i(~) which are much like ~0pert(X) and besides, SU(6)-like relations connect the w.fs. of different hadrons (and the w.fs. of the same hadron states with different helicities). Comparing with the results described above, obtained from QCD sum rules, you see that this prescription, and the picture it leads to, is not correct 2. What is the main reason? Let me give only one example. Let us suppose that you have chosen the standard SU(6) constituent quark model with an oscillator potential. So, you have a model for the 2-particle pion w.f. ~I'~-(k) (in the meson rest frame). Now, if you use the free-field prescription (17) to find out qlLF(X,/~±) (in the Pz --+ ~ frame) from q/~(k), then you will obtain the w.f. ~o(x) = f d k ± ~ L F ( Z , k±) which is much like ~Opert(Z). What is wrong ? What happened is that using the free-field prescription (17), you have used really the wrong time evolution operator (Hfree instead of H = /-/free + Hint)" As a result, you have obtained with this prescription not the true pion w.f., but some mixture of rr, a~=°(1270),...w.fs.. But it is known from QCD sum rules (see, for instance, [16]) that the true ~r and a) =° w.fs. are respectively much wider and much narrower than ~Opert(~ ). Because the "pion w.f." which you have obtained using (17) is really a mixture of the wide pion and narrow al~=° w.fs., it is not surprising that this "pion w.f." is much like ~0pert(~). 2It was shown in [13] using QCD sum rules that the w.fs. of the nucleon and the As3-resonance, ~aN(Z) and T,x(z), also differ greatly.
V.L. Chernyak / Properties of hadronic wave functio.s
310
It is clear that various non-pert, effects which are contained in Hin t are not small and non trivial in any sense. And neglecting t h e m even in part you can obtain here the wrong results. So, the problem is very complicated, in general. W h a t is interesting, however, is that taking account (phenomenologically) of neglected effects even in part, it is possible to reproduce some characteristic properties of the true w.fs. (as obtained from the QCD sum rules). This was shown by Z. Dziembowski and L. Mankiewicz [15]. They used the usual constituent quark model with the oscillator potential and accounted for a coupling between the quark relative m o m e n t a and their helicities, when going from the equal-time w.f. to the light-front one. And even this simple approach allowed t h e m to reproduce characteristic properties of the pion and the nucleon w.fs, qo~(x) and ~.~,(~:), which were obtained from QCD sum rules. 3 Therefore, it seems, their model is capable to reproduce the good description of hard exclusive processes with pions and nucleons. Moreover, it was shown [15] that it is capable to give a good description of the soft pion and nucleon amplitudes. Really, all this is very appealing. The main drawback of this model is that it seems impossible to have simultaneously a good description of soft and hard amplitudes without changing the main p a r a m e t e r of this model, /3, which determines the m e a n value of the quark transverse momentum. But this model is only a first step, and we can hope that it is possible to improve all this and to have soon good models for the hadronic w.fs. 4
Now, few words about the characteristic values of the quark transverse m o m e n t a in hadrons. Let us consider, as an example, the pion w.f. :
i/.P.Sd d'k,exp (i<(._.+)-,ir¢1} @:(<,¢±)+...,
< id~d'k±
g2,(~,/~±)=l,
/~=0,
,+=0.
Expanding the left- and the right-hand sides in Z± we obtain : < 0[ d(0) "/,75 0 2 u(0) [~r(p) > = if~p~, < ;~ > , + ... , < So, if we know the value of this matrix element, then we know the firstm o m e n t in k~_. The value of the above m a t r i x element was obtained from the QCD sum rules (see, i.e., [16]), and the result looks as :
< ;l
(325 MeV)'
This n u m b e r characterizes the values of the intrisic quark m o m e n t a inside hadrons. 3But their model fails to reproduce the narrow pi-meson w.f. 4The partial wave decomposition of the w.f. fit(k, p) is described in [17].
V.L. Chernyak / Properties of hadronic wave functions
3
311
How can the wave function properties be checked experimentally ?
Let us go now to the last question about the comparison of theoretical calculations with experiment. Many various applications have been considered (see, i.e., the review [16] and more recent papers) : charmonium decays into light mesons and baryons, hadronic form factors, 7V --~ two hadrons, etc... Let me give you only two typical examples.
Figure 10: The decay of the heavy quarkonium 3p0 into two pions.
q__.
! ! !
Figure 11: Typical diagram contributing to the asymptotic behaviour of the nucleon form factor.
a) The 3p0 , ~r+Tr- charmonium decay is described by the diagram on Fig. 10, and the branching ratio looks as [9, 11] :
Br
=
~s16~Mf-~ 27 f l d(1 ~o.(~1)
I0
~, /_'d{ 1 d(2 ~.((2)
~(() 1
= 1 [1 +
1(¢, - ¢.)']. 4 1---~2 J
(18)
V.L. Chernyak / Properties of hadronic wa~v functions
312
For : Br = 2 . 5 10 -3 ;
1.
T h e non-relativistic w.f.
2.
T h e pert. w.f. S0pert(() = 3(1 - ~2) :
Br = 3.5 10 -.2 ;
3.
T h e " s u m rule w.f." V,~(~) = ~ ( 1 - ~2)~2 :
Br = 1.1% .
The experiment So, you see, the that o b t a i n e d from b ) T h e nucleon obtains :
¢Pnon-rel(() = ~5(() :
gives [27] : Br = (0.8 -4- 0 . 2 ) % . answer is highly sensitive to a precise form of the pion w.f., and it is the Q C D s u m rules which is in a g r e e m e n t with experiment. form factor is described by d i a g r a m s like those shown on Fig. 11. One
- 3 10 -3 4
p
for ~0~9n-rel
2
Q GM(Q ) 1 GeV 4
Q4G~M 1 GeV 4
--~
0 0.95
_~
for
~pert
(19)
for ~sXum rule
+0.2 10 -3 for W_non-tel N pert 3.8 10 -2 for ~oN --0.47 for ¢psum rule
(20)
You see t h a t
. non-tel and ~opert ~v N N give not only too small predictions, but even wrong signs, while the " s u m rule w.f." agrees with the data. T h e s e two e x a m p l e s are typical. T h e s i m p l e - m i n d e d models of w.f. (like ~on°n-rel(x), e t c . . . ) give ordinarily too small predictions, and only the w.f. o b t a i n e d f r o m the s u m rules gives simultaneously a good enough description of a lot of the e x p e r i m e n t a l data. To illustrate the high sensitivity of the results to a precise form of the w.f., let me mention the nucleon m o d e l w.f. p r o p o s e d by M. Gari and N.G. Stefanis [18]. It was chosen so t h a t four of its lowest six m o m e n t values are t a k e n from the Q C D s u m rules, but two other ones are (arbitrarily) changed significantly (and are far away from the limits allowed by the s u m rules). As a result, this w.f. describes well the p r o t o n form factor, b u t fails (by a factor ~_ 50) to describe the J / ¢ --+ ~ p decay.
Let us consider finally some e x a m p l e s which are directly connected with the H T contributions and which were not considered here up to now. 1. High-p± b a r y o n p r o d u c t i o n In the process : p+p ---+p± + X , where the final p r o t o n has a large transverse m o m e n t u m , there is a H T c o n t r i b u t i o n described by the d i a g r a m on Fig. 12a. T h e a m p l i t u d e of this contribution is p r o p o r t i o n a l to the integral [19] : IN = r / d 3 z ~pN(Z) xl J X l X 2 X 3 1 - X2
(21)
V.L. Chernyak / Properties of hadronic wave functions
313
If you can separate out only those events in which this high p± proton is not surrounded by other high-p± hadrons (i.e., it is not a m e m b e r of a jet), then the above is the only contribution. It was estimated that then (for the nucleon w.f. obtained from the sum rules) o" ~ 4.10-35cm 2 at v'~ ~-- 40 GeV , p± > 2 GeV .
(22)
In the usual one-particle inclusive experiments (see [20] and the references therein) there is also the usual leading twist contribution, which corresponds to the large p± scattering
°
Figure 13: The H T contribution to e + e- ~ M± + X .
of two incoming quarks and subsequent fragmentation of the scattered quarks. Besides, there are, for instance, the contributions like those shown of Figs. 12b and 12c. (~P is the Pomeranchuk trajectory or the one gluon exchange). Initial protons scatter here quasi-diffractively on each other and produce then two large p± quark jets (Fig. 12b) or large p± diquark (with a small invariant mass go << p±) and ant±quark jets (Fig. 12c). The large p± proton originates from the fragmentation of these large p . jets. At large p± the H T contributions like those shown on Figs. 12b,c are the main power corrections to the leading twist contribution. The reason is that a diquark acts like a meson, and at large p± a hard process with a meson is parametrically preferred to those with a baryon, because the meson form factor falls off more slowly. The cross section has the form (qualitatively, E >> Pz) :
dp
-
LT
+
+
.
.
.
.
Here : (do'/dp2,)LT is the leading twist contribution, and the second and third terms in brackets describe the diquark (Figs. 12b,c) and the baryon (Fig. 12a) contributions respectively ; #d and /~B are some characteristic masses, which can be expressed through the integrals over the diquark and the baryon wave functions, see (21). In any case, if you are able to measure experimentally these H T contributions, you will have the numerical values for ~a and # s , i.e., for definite integrals over the diquark and the nucleon w.fs. You can check then what model w.fs. reproduce correctly the experimental values of/td and # s • The baryon production in deep inelastic scattering was considered in [21].
314
V.L. Chernyak / Properties of hadronic wave functions
q _,
q
=.
Fig. 12 a
q
G C
I !
q
;11 ~
i
~q I,q ~,q
G C Fig. 12 b
q q
./_/'
@ TP
@
/~ C
~q *q
t,q
Fig. 12 c
F i g u r e 12: T h e H T c o n t r i b u t i o n s to p + p --+ p± + X .
V.L. Chernyak / Properties of hadronic wave functions
2.
315
Isolated meson production in e+e - [22].
In e* e - -~ M± + X , where the meson has a large transverse m o m e n t u m but is not a m e m b e r of a jet, the process is described by the diagram on Fig. 13. This is the H T contribution corresponding to : e + e- --+ M± + q + ~ , and the meson has a large transverse m o m e n t u m p± with respect to its parent quark. If p± is large but much smaller than the parent quark energy, then there is a universal probability function for a quark to emit such a meson :
4 ,~2 zF2(z) 27 p~_
w(z,p±)-
Emeson
;
z = - - ;
Equar k
LI(I+z--z') ~(') dx. FA=o(Z) =
(1 - x) (1 - zx)
L1 ;
(1 --z-)-(i=
FI,\I=,(z ) = z
(23)
~(~.) d~ zz)
You see that by measuring the z-dependence you can, in principle, measure the form of the w.f. ~(z). It was estimated that for pion production (using the pion w.f. from the sum
rules): o.(e+e - .-~ lr+qq)
~r(e+e- --, g+g-)
3.
--~10 -3 at E b e a m > 4 G e V .
(24)
P r o m p t meson production in meson-nucleon collisions.
These are processes like : 7r + p --. M± + X , where the final meson has a large p±. The detailed description of the HT contributions for such (and similar) processes have been given by M. Benayoun, P. Leruste, J.L. Narjoux and R. Petronzio in [23]. For the above reaction the H T contributions considered in [23] correspond to hard subprocesses like : q + q ~ M± + g (Fig. 14 and other similar ones). The experiment was performed by the WA77 group and, for example, they see clearly the H T p0 signal and don't see the H T f0 signal [24]. The reason m a y be that the H T Pt:q=l production is highly enhanced kinematically in comparison with the H T P~=o production [23]. For instance (for simplicity, consider d+u ~p++g):
do- "~,t
~p2~JHT -
-
256 ~.~,~ t ~ + ~,~ £ 729 t2u 2 s -
\~JHT = 8 ~ ' ~ 1
S
i-',d~: +,~(~;) 1 ---(-~
J-1
1
_-~
' ,
(25) (26)
1
At the same time, the flAl=2 o o and flAt=l are not H T - p r o d u c e d in subprocesses like those shown in Fig. 14. Further, it seems [23, 24], that the WA77 d a t a point to a wide p±-meson w.f. (approximately as wide as the pion one obtained from sum rules, see above).
316
V.L. Chernyak / Properties of hadronic wave functions
/A) v
q
~-
(M:P,f,n' .... )
"'-g.
Figure 14: The HT contribution to lr N ~
q
-~
M± + X .
~.q
Figure 15: The additional HT contribution to 7r N --+ M ± + X , 79 is the Pomeranchuk trajectory (the analogous Pomeranchuk contributions are present in many others high-p± processes).
From my viewpoint, a more careful analysis of the WA77 data is needed, before making definite conclusions about the properties of the p±-meson w.L. One reason is that there are calculations of the charmonium decay 3P2 ~ pp [25,26] :
V.L. Chernyak / Properties of hadronic wave functions P±P±) = 6 B r ( 3 P 2 --+ r + r )
\f~]
I,~
~ 30
-
I,~
"
3l 7 (27)
You see that the p± production is strongly enhanced kinematically in comparison with that of 7r (or pL), similarly to (25, 26). Here, I~ and I,~ are definite integrals (similar to (18)) over the p±-meson and the pion w.fs., and they are such that if the p± and 7r w.fs. are much like each other, then [I~/I~ I > 1. Therefore, (using the experimental value Br (3P2 ---* 7r+ 7r-) = (0.19+0.10)% and neglecting the 3P2 ~ p~ p~, contribution) we should obtain in this case : Br(3P2~p°p°)
~ 3 0 B r ( 3 p 2 ~ r r + 7r-) _~ 6 % ,
(28) Br (3p2 --~/~0. KO*) ~ 12% . This contradicts strongly the available experimental data [27] : Br(3P2~p°p°)
<
Br (3P2--~p°Tr +Tr-)
= (0.67±0.40)%,
(29)
Br (3P2 ---,/~o* g 0 * ) < Br (3P2 ---* g °* K + 7r=F) = (0.47 5= 0.28)% . At the same time, if we use (see above) the wide pion w.f. and the narrow p±-meson w.f. obtained from the s u m rules, then I ~ / I , << 1 , and [25] : Br (zP2 -~ p0 p0) _~ Br (3P2 -~ 7r+ 7r-) = 0.2% ,
(30)
and this looks much more realistic. The second reason is that there are additional HT contributions, for instance those shown on Fig. 15 (79 is the Pomeranchuk or the one-gluon exchange). Here initial quarks scatter quasi-diffractively, producing in the (near) forward direction one light and one heavy (i.e., virtual, -.~ O(p~)) quark, the latter decaying into a meson and a q- It may be that such contributions are not small at high energies 5 (/~ >> p±). In any case, the H T effects are seen in the experiment but, it seem to me, a careful analysis of all experimental and theoretical uncertainties (i.e., K-factors, additional contributions, e t c . . . ) should be carried out before concluding definitely about the properties of the p i - m e s o n w.f. from the WA77-experiment.
You see from all the above examples and from other ones considered at this Workshop, that an investigation of the H T contributions has good perspectives of providing us with useful information about the properties of the hadronic wave functions. A hard work is required, however, before we will have sufficiently accurate and conclusive results. Acknowledgments : I a m deeply grateful to M. Benayoun for introducing me into the HT physics and to M. groissart and M. Benayoun for the very fruitful discussions on the properties of the wave functions. The kind hospitality of the Laboratoire de Physique Corpusculaire, Coll~ge de France, where this paper was prepared, is greatly acknowledged. 5Besides, such contributions can be important in many other processes at high energies, E >> p±.
318
V.L. Chernyak / Properties of hadronic wave functions
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319