Hadronic wave-functions and physical processes

320

Nuclear Physics B (Proc. Suppl.) 7B (1989)320-331 North-Holland, Amsterdam

Hadronic Wave-functions and Physical Processes ~

Tao HUANG Center of Theoretical Physics, CCAST (World Lab.) Physics, Beijing, China

and Institute of High Energy

The connections between hadronic wave-functions and physical processes are discussed. We compare several models for the pionic non-perturbative wave-functlon to analyze the experimental data in the framework of QCD theory. They can provide phenomenological constraints on the Fock state wave-functions to develop a deeper understanding of the hadronic structure in QCD.

1

Introduction

One of the most serious complications in testing quantum chromodynamics is that we have little information on the hadronic wave-function. In general, the physics of the bound state wave-function exhibits the full complexity of non-perturbative QCD and the hadronic wave-functions are the underlying links between hadronic phenomena in quantum chromodynamics at large and small distances. A convenient description of hadronic wave-functions is given by a set of n-parton momentum space amplitudes [1],

ql,(xi,]e±;,Ai),

i = 1,2,....n

(1)

defined on the free quark and gluon Fock basis at equal "light-cone time" T = t + z in /,,~-+

the physical "light-cone" gauge A + = A ° + A 3 = 0. (Here x i = pZ~_+ ( ~ i x i = 1) is the

~:±i,

light-cone m o m e m t u m fraction of i-th quark (or gluon) in the n-parton Fock state ; with ~ i le±i = 0, is its transverse momentum relative to the total momentum pU ; and hi is its helicity). A hadron state can be expanded in terms of this complete set of Fock states at equal r:

rH > =

> n,,\t

with the normalization condition,

nr/~ I

where the sum is over all Fock states and helicities, and 1The project supported by National Science Foundation of China.

0920-5632/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

(2)

T. tluang / Hadronic wave-functions

,

j

i

321

j

The wave-function kon(xi, kJ.i, Ai) is the amplitude for finding n bare partons with momenta zi and Icii, which is independent of the hadron's momentum. In addition we have employed the physical light-cone gauge, ~7 " A = A + = 0 , for the gluon field so unphysical degrees of freedom do not appear. The amplitudes {~I'~} are infrared finite for color-singlet bound states. There are many advantages obtained by quantizing a renormalizable local theory at equal r = t + z. In light-cone quantization, the Fock state vacuum is an eigenstate of the full light-cone Hamiltonian, conjugate to r. Consequently, all of the bare quanta in a hadronic Fock state are associated with the hadron ; disconnected elements of the vacuum are excluded. Thus, perturbative calculations are enormously simplified by the absence of vacuum to pair production amplitudes because of the fact that all particles must have k + > 0. It is also convenient to use ~'-ordered light-cone perturbation theory for the analysis of light-cone dominated processes such as deep inelastic scattering or large p± exclusive reactions. The structure functions G~/H,(x,Q), and the distribution amplitudes q)H(x~,Q) are specific, basic measures of ffl :

G~/H(z,,Q) = ~ f

Q

~

2

(4)

t~ ]e±).

(5)

~

[dz][dk±]l~)(z,,k±,)l ~ ( z - za)

and

=jfQd2k'l-m(Q)

In the case of inclusive reactions all of the hadron Fock states generally are involved. As for the high momentum transfer Q exclusive reaction, perturbative QCD predicts that only the lowest particle number (valence) Fock state contributes to leading order in 1/Q [2]. The essential gauge-invariant input is the distribution amplitude ~H(X,, Q). One of the most interesting examples of higher twist phenomena is the set of "direct" processes. For instance, the hadron is directly produced at large PT in the subprocess among partons. Recently, an experiment has been especially designed to single out such an effect and its results seem to answer positively the question for productions of p0 meson at medium to high P r [3]. The general scheme for calculating the rates of prompt resonance production has been widely discussed in the literature [4,5]. The main point of the theoretical calculation is the factorization of the short-distance physics from the long-distance one. Thus the required wave-function for calculating a direct hadron subprocess amplitude is the distribution amplitude, ~M(xi, Q), which is originally defined for exclusive processes. Therefore if these wave-functions were accurately known, then, an extraordinary number of phenomena, including decay amplitudes, exclusive processes, higher twist contributions, structure functions, and low transverse momentum phenomena could be interrelated. Conversely, these processes can provide phenomenological constraints on the Fock state wave-functions which are important for understanding the dynamics of hadrons in QCD. The study of phenomenology of hadron wave-functions in QCD has shown that even without explicit solutions for q~n, it is still possible to systematically incorporate the effects of the hadronic wave-function in large momentum transfer

T. Huang/ Hadronic wave-functions

322

exclusive, inclusive reactions and higher twist processes, and to develop a deeper understanding of the detailed structure of hadrons. Section 2 gives some constraints on the meson distribution amplitudes which have been obtained by using perturbative QCD theory, QCD sum rule and lattice gauge theory. The Bethe-Salpeter equation provides a description for the bound state system; since the kernel is very comphcated in the QCD theory, one can not solve the Bethe-Salpeter equation exactly at present. However some information on the hadronic wave-function can be obtained from the approximate solution of the Bethe-Salpeter equation in the rest frame. We shall discuss these related topics in Section 3. Some conclusions are summarized in Section 4.

2

Meson

distribution

amplitude

The meson distribution amphtude q~(z, Q) in Eq.(5) is the probabiltiy amphtude for finding the qq Fock state in the meson. ¢(z, Q) is only weakly dependent on Q, which is controlled by evolution equations derivable from perturbation theory [2] or the operator product expansion [6]. Its solution depends on the initial form ff(zi, Q0), and ~(zi, Q0) is determined by the non-perturbative theory. For very large Q, only the leading term remains, so that we have

, +a,(*) = V~Y'MX(1 - .), ~2M(X'Q) Q__oo

(6)

where fM is the decay constant of the meson and is related to the meson wave-function at the origin

fM

=

2v"-3fdz~(z, Q) = 2~/3J(2-~-#-~z ' - " d4k ss(k).

(7)

It may be seen that the asymptotic behavior of the amplitude, q)~8(x), is a universal one

(except the constant fM) and does not depend on the initial form ~M(X, Q0). Actually the initial form ~M(Z~,Qo) can differ greatly from its asymptotic form ~a,(X) [7, 8] and the true distribution amplitude q~(x, Q) will become much like q~a,(x) only at extremely large Q, because the evolution with Q is extremely slow at large Q. The most detailed information on the structure of hadron can come from the analysis of exclusive processes at large momentum transfer, because each exclusive process amphtude depends in detail on quark and gluon scattering amplitude at short distances and the hadronic wave-function. GeneraUy one finds that such an amphtude factorizes into a convolution of quark distribution amplitudes ~(zi, Q), one for each hadron, with a hard scattering amplitude TH, where TH is the irreducible subprocess amplitude with the hadron replaced by the valence Fock state of the hadron. The pionic electromagnetic form factor, for example, can be written as

F~(Q2) =

/ol/? d~ dyc~*(x,O)Tu(x, y, Q)~(y, Q)

in leading order, where

T H - 16zrCFcts(Q) Q2

1 (1 - z)(1 - y)"

(8)

T. Huang / Hadronic wave-functions

323

In fact, all of theoretical predictions are smaller than the data if the asymptotic form ~as(x) is used in all physical processes. For example, the theoretical prediction for the pion form factor is about 50 % smaller than the data at intermediate Q, dependent on A, and for X ~ 2 r processes which have more singular TH (as x ---+0, 1) the theoretical predictions are smaller than experimental results by two orders of magnitude. Therefore this means that ~ ( z i , Q) should be much wider than its asymptotic form. It is clear that one has to know what is the behavior of the true distribution amplitude at the available energies of the present experiments since it will control the predictions of all exclusive processes. In order to fit the pionic data an empirical way to account for the non-perturbative effects is to modify the power form,

X/'3fMZ,Z2 ~

VIM

6B(~ + 1,~

+

1) (xlx2)t~'(;gl -]- x2

=

1)

(9)

with fl = ¼, which was first suggested by Brodsky and Lepage [9]. It can give the corresponding form factor of the pion compatible with experiment and a bit bigger than the experimental values for X --+ 2~r [9]. This fact tells us that it is possible to construct a broad distribution amplitude of the pion to fit the data, although the subprocess amplitude TH has a very different end-point behaviour. The data of hadron-hadron collisions and 7-hadron collisions support the picture that large PT phenomena are dominated by hard scattering subprocesses. It is believed that the interactions of partons in these subprocesses are controlled by the perturbative QCD. In the framework of hard scattering subprocesses one would also expect to find a prompt meson production. The theoretical results for inclusive meson production in the -),-hadron scattering has been investigated since several years. The observed meson directly comes from these subprocesses, such aS

7q ~ Mq, q(t ~ Mg, qg ~ Mq, gg , Mg, etc ... Obviously the meson wave-function controls the leading high-twist contribution to the inclusive meson production at high PT. Recently the experimental evidence of such processes has been obtained [3, 10]. In order to get a good agreement in prompt p°-meson production and T/'-meson production higher twist contributions are included. They use the meson wave-function ¢ _ ArM(mix2) I/4 and reproduce the observed dependence on ~E"

Benayoun et al. [5] have done the detailed calculations of the direct meson production. They show that all higher twist production amplitudes only depend on distribution amplitudes ¢~M(~1, x2) through moments like

C('71, 72) -where

=

/o1d z , d z ~ z ? ' z ~ 2 ~ M ( z ~ m 2 ) 6 ( 1

-

z~ -

z2),

(I0)

higher twist production cross-section for any

s-wave quarkonium meson is proportional to Of , C, = C ( - 1 , 0 ) -- /oldx~M-(x x)- .

(11)

T. Huang / Hadronic wave-functions

324

If using the wave-function (9), one can get C,

6fl

~ 1

as

1

as f l =

1/4.

(12)

Therefore the experimental information about a prompt meson production of the higher twist process will give constraints on the meson wave-functions. Although one cannot solve the bound-state problem from the non-perturbative QCD theory at present, an interesting progress has been made by the QCD sum-rule method [7, 8]. They give reliable evaluations of the first three moments of< ~2~ >, which.are defined as

for ~r- and p-meson. Their results have been confirmed by lattice gauge theory calculation recently [11]. These values of the distribution amplitude moments impose a strong restriction on the behavior of the meson distribution amplitude. They have shown that the distribution amplitude behaviors are quite distinct for different mesons. In the case of the light quark system, the non-perturbative influences are very different from each other due to the different interactions (depending on their helicity) with vacuum condensates. In particular, the pionic distribution amplitude is very strongly affected by the non-perturbative contributions and is extremely relativistic. In the case of the heavy-quark system the gluon condensates with the heavy quark make the distribution amplitude ~(z~, Q) go over to the non-relativistic form, g-like function [8]. Chernyak and Zhitnitsky [7] suggest, based on the moment values, a distribution amplitude • (*,,-2)

=

-

(14)

for the pion (see Fig.l) and

for any meson M. The parameters A and B are suitably optimized for each meson. It seems that the functions (14) (9) and are consistent with the experimental values of the pionic form factor and the processes X -----* 27r. A careful analysis, however, reveals that the reason why the function (9) and (14) can fit the data of the pionic form factor and the branching ratios Br (X --+ 27r) is the following : an important contribution to the pion form factor and the decay width comes from the end-point regions [10]. In the end-point regions, the transverse momenta of the ~r-meson in the gluon propagators cannot be neglected, and the process amplitude cannot be factorized.

T. Huang / tfadronic wave-functions

325

sSSS~''~

',,

,'-

%%%

\

/ SS

0.5

0

1.0

X

FIGURE 1 Meson distributions amplitude. The curves correspond to a) the pionic asymptotic amplitude x(1 - z), and b) the CZ amplitude for the pion 5z(1 - x)(2z - 1) 2 .

Thus, the factorization of the process gives the conditions zY >

< k . >2 4M~ '

(16)

< k . >~ (1-x)(1-y)>

4M~

'

(17)

where < k± > " 3 0 0 MeV. Therefore, from Eqs.(16) and (17) one can get the constraints on the integration region, zy>a

and

(1-a)(1-y)>a,

(18)

< k . >2 where a For example, the cut-off parameter a is taken 0.01 if ( k± > ~ 300 MeV and Mc~ ~3 GeV ~. The contribution from the end-point regions gives at least 50 % of the predictions if using the wave-function (9) and (14). It is thus hard to believe that this last perturbative QCD prediction is reliable, since in these regions, the processes will fail to be factorized and the reliability of the perturbative QCD theory is questionable. Therefore the reliable predictions should not be sensitive to the choice of the cut-off parameter a. This implies that the true distribution amplitude @(z,Q) should have such end-point behavior to render this region harmless.

326

3

T. Huang / Hadronic wave-functions

Quark model approach

To deal with the bound state problems, the Bethe-Salpeter equation in the relativistic quantum field theory is a useful tool. However one cannot exactly solve it at present and in practical calculations it can be solved only in the ladder approximation. Moreover, noticing that the QCD theory is a strong coupling one and the light quarks in a hadron are highly relativistic, one can not use the perturbative calculation to get the hadronic wave-functions by solving the Bethe-Salpeter equation. The kernel in the equation should be determined mainly by the non-perturbative contribution in QCD [12]. Fortunately, one has obtained the approximate bound-state solution of a hadron in terms of the quark model by solving Bethe-Salpeter equation in the rest frame. The rest frame wave-function ~CM(~) controls binding and hadronic spectroscopy. As we know, the harmonic oscillator wave-function describes the ground states of hadrons very weU for the static properties [13]. Therefore there is a possibility to get the infinite momentum frame wave-function if one can establish a connection between the equaltime wave-function in the rest frame and the light-cone wave-function in the infinite momentum frame. Brodsky-Huang-Lepage suggested [14] an (approximate) connection between the equal-time wave-function in the rest frame and the wave-function in the infinite momentum frame by equating the energy propagator M 2- e = M 2-

k

(19)

in the two frames. If we kinematically identify (qO + qS) 7%

+

q~

= _

+

(20)

L.

P+'

J

then the rest frame wave-function ¢2CM(~,), which controls binding and hadronic spectroscopy will indicate a corresponding form for the IMF wave-function, t~MF(Zi, k±i). In the case of two-particle state there is a possible connection, ( ~li 2-rn2 ~ C M ( ~ ) ~-+ ~[MF\ 4zlz2

m 2)

(21)

As an example, the wave-function of the harmonic oscillator model in the rest frame was obtained from the Bethe-Salpeter equation in an instantaneous approximation for mesons [13],

*cM(¢) =

(22)

By using the connection (20) we can get the wave-function in the infinite m o m e n t u m frame,

Xl

~2

t.

~12~2

a

Substituting Eq.(23) into Eq. (5), one can obtain the distribution amplitude ~M(Z~),

T. tluang/ Ha.dronic u'a~v-functions

:I'27

Ah (16,~2b~) xl~2 exp [k- m2b~], ~J_

~M(Xi)

(24)

which goes to the asymptotic form as m is set to zero. The end point contributions are

strongly suppressed by the wave-function (24) due to the existence of the exponential factor. Thus the reliability of the perturbative QCD theory to the subprocess can be guaranteed by this kind of wave-function in the infinite m o m e n t u m s frame. However, it is close to the asymptotic form ~,~,(z) = V'~fMz(1 - z), and it is ruled out by the present data for the pion. In order to fit the experimental data and to suppress the end-point contributions. A model for the pionic non-perturbative valence wave-function has been proposed in ref. [15],

• ~(~,~) = A(~l- ~2)2exp[-b 2ki-X +--~-], lX2 l

(25)

which is even under charge conjugation, and symmetric under the exchange between Zl and z2. It leads a distribution amplitude

'~-(xi)-

A

~ m2b 2"

16rr2b2(Z, -

~ ) % ~ 2 e x p /k- - -X/ l,X 2

(26)

J

which is almost the same as the Chernyak-Zhitnisky model distribution amplitudes except for the end-point regions (see Fig.2).

(~(x)

t

\

C

Y 0

\

0.5

1.0

FIGURE 2 The model distribution amplitude for the pion. The curves correspond to a) The harmonic oscillator model with m = 300 MeV, A~x(l-z)exp[

[_ z (m2b~ ], l-z)

b) The modified harmonic oscillator model with m = 300 M e V , [ _m2b ~ ] Avz(1 - z)(2z - 1)2 exp [-

X(I ~)J'

c) The pionic amplitude of CZ wave-functions, 5z(1 - z)(2z - 1) 2.

X

T. Huang / Hadronic wave-functions

328

Recently, a relativistic model of the hadron valence-quark structure has been presented [16]. This model amplitude is very close to the amplitude (26). They use the Brodsky-Huang-Lepage prescription [14] to get the exponential part of the function (25) from the harmonic oscillator wave-function and use a light-cone analog of the mockhadron method by Isgur together with the Mellosh transformation to get the factor (zl - x2) 2 from spin state. Similar to the method of ref.[1, 14], we can normalize the hadronic wave-function by two constraints for the pion, 1

-1

r d k±_~

f~

(27)

and

--Joldxruuuuwqq/,~(x,k'± ----0) -- ~

(28)

f~ ' which are determined by the processes 7r ~ #u and lr ° ~ 3'7 respectively [1]. The model wave-function (26) can be completely fixed by these two constraints, e.g. the parameters A and b in eq.(25) are approximately

A ~ as

f~

(29)

and

m2b 2 < < 1. It has been shown that the function (26) give the moment values [15], <(~>~0.4

and

<(4>~0.23,

(30)

which are consistent with the QCD sum-rule calculation. One can use it to fit the experimental d a t a for the pionic form factor, as well as for the processes X0.2 - - ~ 27r. It seems to be quite remarkable to fit both simultaneously with QCD formulae and this reasonable wave-functlon. In fact, the value Ct in Eq. (11) plays an important role in the pionic form factor and in the higher twist process of p r o m p t meson production. The predictions, which vary with the p a r a m e t e r m, are shown in Fig.3. The dashed curve is given by the harmonic oscillator model (23) and the value at m = 0 coincides with the asymptotic form. The solid curve is given by the modified harmonic oscillator model (25) and the value at m = 0 coincides with the C-Z wave-function [7]. In addition, m a n y observable processes can be analyzed in QCD to get more information about the hadronic wave-function, such as two-photon processes, heavy quarkonium decays, and p r o m p t meson production in hadron-hadron collisions. From Eqs.(25) and (29) we can get the probability of finding the valence qq state in the pion is thus

[da: d~ k j_ 9 P~q/¢ = J 16¢r3lq'q~/~(~'/~-)12 - 28

(31)

and the "radius" of the valence qq state 27

1

(Rqq/~) 2 - 407r2 f~ -~ (0.52fro) 2.

(32 /

T. Huang / Ifadronic wave-functions

329

1.0

O. 0.8 0.7 0.61

o.si 0.4 0.31 0.2 0.1 0

I SO

I 100

I 150

I 200

I 250

I ~ ITI 300 ( b l e V )

FIGURE 3 The values of C1 predicted by the model distribution amplitudes with the different p a r a m e t e r m. T h e dashed curve is given by the harmonic oscillator model (23) ; the solid curve is given by the modified harmonic oscillator model (25).

Therefore the valence wave-function of a hadron is more compact than the hadron radius [15]. In this way one hopes to develop a deeper understanding of the detailed structure of hadrons.

4

Conclusions

The fact that the Fock state representation of the hadronic wave-function provides a description to interrelate exclusive, inclusive and higher twist processes implies one has the opportunity of extending the QCD testing ground. We can systematically incorporate the effects of the hadronic wave-function in exclusive, inclusive and hadron-hadron processes where a meson or baryon is produced singly at large transverse m o m e n t u m . Their results strongly depend on the behavior of the hadronic wave-function. We have analysed the pionic form factor, the processes X0,2 ~ 27r and a direct meson production subprocess in the hadron-hadron collision. The d a t a tell us that the pionic wave-function with a broad ~ distribution behavior is needed. Although the functions (9) (with fl = 1/4) and (14) can fit the data, they do not satisfy the requirement that the end-point contribution has to be suppressed in order to make the QCD p e r t u r b a t i v e theory reliable.

330

T. Huang / tfadronic wave-futlctions

The hadronic wave-function depends essentially on the non-perturbative QCD theory, and one has little information on it at present. We have suggested the function (25), which is a reasonable form from the Bethe-Salpeter approximate solution in the rest frame, to fit the exclusive processes and the single meson production at large PT phenomena. The end-point contribution can be partly suppressed by the wave-function (25). Thus these physical processes will give phenomenological constraints on the valence Fock-state wave-functions which are essential for understanding the dynamics of hadrons in QCD. All phenomenological constraints allow us to construct a possible model which describes hadrons in QCD and is consistent with data at large and small distances. The function (25) is only an example of hadronic wave-functions and it will be improved further.

Acknowledgements : I wish to thank Professors M. Froissart and M. Benayoun and their colleagues for organizing an outstanding meeting and for their hospitality at College de France. I also wish to acknowledge the support and the hospitality of College de France, where this talk was prepared.

T. Huang / Hadronic wave-functions

331

References 1. S.J. Brodsky, T. ttuang and G.P. Lepage, in Particles and Fields 2, edited by A.Z. Capri, A.N. Kamal (Plenum, NY, 1983) p. 143 ; G.P. Lepage, S.J. Brodsky, T. Huang, and P. Mackenzie, Ibid, p. 83. 2. G.P. Lepage and S.J. Brodsky, Phys. Rev. D22 (1980) 2157. 3. M. Benayoun et al., Phys.Lett. 192 (1987) 447. 4. E.L. Berger, T. Gottschalk and S. Sivers, Phys.Rev. D23 (1981) 99 ; J.A. Bagger and J.F. Gunion, Phys.Rev. D25 (1982) 2287. 5. M. Benayoun, Ph. Leruste and J.L. Narjoux, Nucl. Phys. B282 (1987) 653. 6. S.J. Brodsky, Y. Frishman, G.P. Lepage and C. Sachrajda, Phys.Lett. 91B (1980) 239. 7. V.L. Chernyak and A. R. Zhitnisky, Nucl. Phys. B201, (1982) 492 ; Phys.Rep. 112 (1984) 173 ; Nucl. Phys. B246 (1984) 52. 8. T. Huang, X.D. Xiang and X.N. Wang, Chinese Phys. Lett. 2_ (1985) 76 ; Commun.Theor.Phys. 5 (1986) 117 ; Phys. Rev. DS5 (1987) 1013. 9. S.J. Brodsky and G.P. Lepage, Phys. Rev. D24 (1981) 1848. 10. X.N. Wang, X.D. Xiang and T. Huang, Commun.Theor.Phys. 5 (1986) 123. 11. G. Martinelli and C.T. Sachrajda, CERN-TH 4909 (1987). 12. T. Huang and Z. Huang, BIHEP-TH-88-26 (1988). 13. Elementary Particle Theory group, Peking University, Acta Physics Sinica 25, (1976) 415; N. Isgur, in the New Aspects of Subnuclear Physics, edited by A. Zichichi (Plenum, N.Y.) (1980), p. 107. 14. S.J. Brodsky, T. Huang and G.P. Lepage, SLAC-PUB-2540 ; T. Huang, Proceedings of XX-th International conference, (1980), Madison, Wisconsin. 15. T.Huang, Proceedings of the International Symposium on Particle and Nuclear Physics, edited by N. Hu and C.S. Wu, (1985), Beijing, China ; and Proceedings of second Asia Pacific Conference (1986), India. 16. Z. Dziembowski and L. Mankiewicz, Phys. Rev. Lett. , 58 (1988) 2175.