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Damping of Quark-Hadron Form Factors and Diffraction Scattering.

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Annalen der Physik. 7. Folge, Band 39, Heft 1, 1982, S. 43-49
J. A. Barth, Leipzig
Damping of Quark-Hadron Form Factors
and Diffraction Scattering
By M. FRENZEL,
W. KALLIES,and K. LEWIN
Sektion Physik, Bereich Theorie der Teilchen und Felder, Humboldt-Universitat zu Berlin
Abstract. Damping of hadron bound state wave functions in relative quark momentum as
necessary to describe the beginning of power scaling at momentum transfer t = 2-3 GeV2 is obtained approximately from a corresponding behaviour of four- and six-quark Green’s functions in t
which is required by diffraction scattering.
Dampfung der Quark-Hadron-Formfaktorenund Diffraktionsstreuung
Inhaltsiibersicht. Das Abklingen gebundener Hadronzustinde in den relativen Quarkimpulsen, wie es erforderlich ist zur Erklarung der Quarkzahlregeln beginnend bei iibertragenen Impulsen
von t 2: 2-3 GeW, ergibt sich naherungsweise aus dem entsprechenden Verhalten von vier-Quarkund sechs-Quark-QreenschenBunktionen in t, das durch Diffraktionsstreuung vorgegeben ist.
1. Introduction
The understanding of power scaling [I, 21 as large momentum transfer behaviour
of hadron scattering amplitudes and electromagnetic form factors in field theory
presently requires phenomenological assumptions about the hadrons as bound states of
quark particles. These bound states are assumed t o exist, for instance, as solutions of
a corresponding Bethe-Salpeter equation [3]. If they are damped sufficiently in relative
quark momentum, power scaling is obtained by counting the number of far off-shell
quark propagators in tree-like diagrams. Up to now this large relative momentum
damping behaviour cannot, be deduced in vector gluon theories [4],so that the main argument for the damping is the assumed existence of the bound state wave function in
x space a t zero interquark space-time distance (see ref. [3]). But this argument is too
general to explain the beginning of power scaling a t non-asymptotic momentum transfers
of a few GeV2 which requires a sufficiently strong decrease of the bound states at relative
quark momentum in the region around 1GeV/c. A summary of “early” scaling and
corresponding experimental data is contained in ref. [ 5 ] .
The aim of this work is to find arguments for a sufficient damping of the bound states
a t such non-asymptotic relative quark momenta. Explicit information on a corresponding behaviour of the Kernel of the Bethe-Salpeter equation as twoparticle or threeparticle irreducible four-quark or six-quark Green’s function, respectively, is not
available because of the known difficulties in quantum chromodynamics a t lower momentum transfer. Therefore we go a more phenomenological way to obtain information
on the behaviour of the four-point Green’s function in momentum transfer using the
known t behaviour of diffraction scattering. As well-known, in the diffraction region the
differential cross section of elastic hadron-hadron scattering strongly falls off in t nearly
(see ref. [7]). The use of this behaviour for our
independently of the incident energy
vl
M. FRENZEL
u. a.
44
purpose becomes possible, if quark diagrams which describe hadron-hadron scattering
and contain no higher than four-quark Green’s functions contribute non-negligibly in
the intermediate region between diffraction scattering and niore localized quark-quark
interactions. Since interaction between quarks essentially is a matter of momentum
transfer, the transition from large relative longitudinal momentum in diffraction scattering to a small one in the bound state should not affect the t dependent factor which
governs the damping. After discussing the relation between the general four-quark
Green’s function and the irreducible Kernel in an iterative procedure to solve the bound
state equation in a restricted region, it is shown that the iterative solution exhibits the
expected damping of the bound state which is needed to guarantee “early” power scaling.
The structure of the paper is as follows: Section 2 gives a short review of the role of
hadron bound states in connection with power scaling. Section 3 deals with the information from diffraction scattering and in section 4 the iterative procedure in the twoparticle bound state equation is described. I n addition, it contains some aspects of
generalisation to three-particle bound states appearing in meson-baryon and baryonbaryon scattering.
2. Hadron Bound States and Power Scaling
-
Considering, for simplicity, the meson-meson scattering amplitude T M M in the centerof-mass system with s = 2pg = 2p2 beyond the resonance region and s l t
O(l),
t=q2=q2 , one has the expression
where the convoluting bound states (4 x 4 spinor matrices)
are introduced as solutions of the homogeneous Bethe-Salpeter equation
+
+
a
b
d
e
C
Fig. 1. Decomposition of the amplitude TMMinto diagrams with n-quark Green’s functions M, of
different connectivity. Concerning diagrams of the types d and e see ref. [6]
45
Damping of Quark-Hadron Form Factors and Diffraction Scattering
and M as function of the four-vectors p , q, k,, . . ., k4 represents the amputated full
eight-quark Green's function decomposable into terms of different connectivity as is
illustrated in Fig. 1. As usual, yp(k)contains the two quark legs AF. The condition
J d4k y p ( k )= &(.z
= 0) < 00
(2-4)
guarantees the existence of the wave function a t zero interquark space-time distance.
The dominant contributions to the diagrams of Fig. 1 come from regions of the loop
variables belonging to small relative quark momenta a t the hadron vertices. Power scaling is contained in the diagrams a-c. The four-quark Green's functions of diagram d
have all quark legs near the mass shell and therefore should not be dominant if exponentiation of infrared logarithms in QCD perturbation theory works [el. I n the following
sections diagram c containing one four-point function will play a special role.
3. Information from Diffraction Scattering
The beginning of power scaling a t t 2 2 GeV2 (see Fig. 2) requires sufficient decrease of the hadron wave functions at relative quark momenta of order 1GeVJc. On
the other hand, hadron-hadron elastic scattering a t t ,- 0(1 GeV2) with t << s, i.e. diffraction scattering, shows damping of the differential cross section over- several orders of
magnitude for growing t nearly independently of the incident energy
As well-known,
experimental results are fitted in phenomenological work by exponentially decreasing
factors F(t) appearing in the differential cross section
1s.
do
- = F ( t ) . G(s, t),
at
F ( t ) = a . e-".
Fig. 2. Power scaling of pion and proton electromagnetic form factors (see ref. [ 5 ] )
(3.1)
46
M. FRENZEL
u. a.
G(s, t ) only weakly depends on t compared with F(t). What concerns meson-meson
diffraction scattering, this damping should be reflected by all non-negligibly contributing
diagrams, especially those of Fig. 1, if we assume that the strong damping in t is not
due t o coherent compensation effects among such diagrams of different connectivity.
For our purpose graph c containing one four-quark Green’s function is of special interest.
Analytically this double loop graph has the structure
where Mi,“,, denotes the spinor matrix element of M4 and the internal variables k , I,
- k , q/2
I appear as half relative momenta between the quark legs of corresponding
hadron vertices. The dependence on momentum transfer t, especially the behaviour of
the diffraction peak, is contained in
pi2
+
-
Fi(t) G:%(p, q, k , I)
(3.3)
and in the two bound states y ( q / 2 - k ) and y(q/2 1). The relation between F,(t)
and the full damping F ( t ) turns out in the next section.
Some remarks are in order concerning the four-point function (3.3). I n QCD perturbation theory the treatment of (3.3) is much more complicated in the diffraction region
than a t far asymptotic momentum transfers where the effective coupling constant is
small. But the following connection does not depend on the order of graphs: The function (3.3) appears in (3.2) at large longitudinal relative momentum and at t = q2 in the
diffraction region. Besides this situation (A) we consider a situation (B) with the same
momentum transfer t but with longitudinal relative quark momentum near zero. I n
both cases an arbitrary Feynman graph of higher order has the analytical structure
+
(3.4)
where the constant factors ai, bi, ci, di (i = 1, ..., n) are normalized by the conditions
N
2 ai = Zbi = ,Zci = Eli = 1
i=l
(3.5)
They describe a given distribution of the external momenta p and q over the internal
lines. The transition (A) + (B) only changes the arguments of f2 and thus cannot change
the factor Fl(t)in (3.3). This expresses the fact that interaction between quarks essentially is a matter of momentum transfer. The factor Fl(t) becomes important in the next
section where the case (B) is studied in detail.
4. Bethe-Salpetcr Equation in tho Damping Region
Returning to the meson bound states as introduced at the beginning wc look for
a n approximate iterative solution of the Bethe-Salpeter equation ( 2 . 3 ) in a restricted
region with not too small relative quark inoinentuni k. The iterative procedure starts
from a “zeroth” approximation
representing the exact solution with a cut off at some value x in the region 0.2 5 x
5 0.5 GeV/c corresponding t o an interquark distance near I fm. Of course, the existence
of the soliit ion yLo)(k)is an additional assumption, necessary in our phenomenological
47
Damping of Quark-Hadron Form Factors and Diffraction Scattering
approach. The exact solution beyond the cut off is assumed to be a regular function of
relative quark momentum and can be obtained by the following iterative procedure:
Insertion of (4.1) into the Bethe-Salpeter equation (2.3) gives the first approximation
(:
y p ( k ) =A, -- k
1
p 4 1
yp)(~)
~ (
(: + k 1
kI , ,p)l d F -
(4.2)
where ykl) = yio)for k, 5 x . We come to the intermediate region, i.e., relative quark
momenta of order 1GeV/c by adding a given momentum q. Then the substitution
p+p+q,
k+k+q/2
in eq. (4.2) leads to
where k,, I, 5 x . Improved approximations may be expressed by equations of the
type (4.2) or (4.3) with iterated Kernels
K(1)= K ,
K(')
= J d4k'
K ( k , k') d F K ( k ' ,I ) dF, ...
(4.4)
where, of course, iteration of K(") (n = 2, 3, . . .) leads to the same solution of the exact
integral equation as iteration of K(I).Even correspondingly normalized linear combinations of different iterated Kernels (4.4)
n
n
(4.5)
have the same exact solution. Note that in the iteration procedure the replacement
K ( l )+ KC.).,, gives a n approximate solution between ~ ( land
) y@).
I n other words: An iteration with mixed Kernels (4.5) converges more slowly towards the exact solution than the optimal iteration (4.4). For n + 00 and special coefficients ai + a? among the sums (4.5) is the s-channel two-quark reducible four-point
Green's function.
Kr$...a~?I!?(+)
M4
.
Prom eqs. (4.3), (4.5) and (4.6) we obtain
If K.
3
0.2 GeV/c and q2
-
0(1 GeV2) then roughly
Using eq. (3.3) with t + t/4 and G4 -+G; if (A) + (B), we have
(4.6)
48
M. FRENZEL
u. a.
and (4.7), (4.8) and (4.9) lead t o
where G; only weekly depends on q. Insertion of (4.10) into the integrand of T$& (see
eq. (3.2)) gives the structure
(4.11)
(4.12)
falls off at least by two orders of magnitude if t + 2t in the considered region, the
decrease of F1(t/4) in eq. (4.10) should be strong enough t o guarantee approximate
power scaling at t 2 2 GeV2.
Experimental information on the diffraction peak concerns meson-baryon and baryon-baryon scattering. Therefore we have t o extend the method described above to threequark bound states. This generalisation requires the additional approximation that
the Kernel X 6 of the three-particle Bethe-Salpeter equation as irreducible six-point
Green’s function can be decomposed into four-point functions as shown in Fig. 3. The
corresponding procedure leads t o the results
(4.13)
(4.14)
(4.15)
which can be considered as generalisations of eqs. (4.10) and (4.12) respectively. The
detailed calculations leading to eqs. (4.13)-(4.15) are given in ref. [8]. These results
indicate a damping behaviour of the baryon bound state in relative quark momenta
which coincides qualitatively with the meson case (see again ref. [8]).
Thus our result shows that in both meson and baryon bound states “early” damping
in relative quark momentum turns out by the same mechanism using the strong decrease
of the diffraction scattering cross section.
Fig. 3. Graphical structure of t’he three-particle Rethe-Salpeter equation and of the corresponding
kernel K6
Damping of Quark-Hadron Form Factors and Diffraction Scattering
49
References
[l]V. A. MATVEEV,R. MURADJAN,
and A. N. TAVKHELIDZE,
Nuovo Cimento Lett. 7 (1973) 719.
[2] S. J. BRODSKY
and G. R. FARRAR,
Phys. Rev. Lett. 31 (1973) 1153.
[3] S. J. BRODSKY
and G. R. FARRAR,
Phys. Rev. D 11 (1975) 1309.
[4] C. G. CUN and D. GROSS,Phys. Rev. D 11 (1975) 2905.
[5] S. J. BRODSKY
and B. S. CHERTOK,Preprint SLAC-PUB-1759(1976).
[6] J. M. CORNWALLand G. TIKTOPOULOS,
Phys. Rev. D 13 (1976) 3370.
[7] S. GASIOROWICZ,
Elementary Particle Physics, John Wiley Sons, Inc., New York-LondonSydney, Chapter 28.
[S] M. FRENZEL,
Diplomarbeit, Humboldt-Universitat zu Berlin (1980).
Bei der Redaktion eingegangen am 14. Mai 1981.
Anschr. d. Verf.: Dip].-Phys. M. FRENZEL,
Dr. W. KALLIES,and Dr. K. LEWIN
Sektion Physik der Humboldt-Universittt zu Berlin
DDR-1080 Berlin, Unter den Linden 6
4 Ann. Physik. 7. Folge, Bd. 39
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