recoil corrections in the bag model applied to the photoproduction of the d13(1520) resonance

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Nuclear Physics A423 (1984) 419-428 @ North-Holland Publishing Company RECOIL CORRECTIONS IN THE BAG MODEL APPLIED TO THE PHOTOPRODUCTION OF THE D&1520) RESONANCE+ CH. HAJDUK* and B. SCHWESINGER** Physics Department, State University of New York, Stony Brook, NY 11794, USA Received 30 June 1983 (Revised 29 December 1983) Abstract: Proper treatment of recoil and c.m. motion in the bag eliminates large parts of the frame dependence of the electroproduction amplitude for the D,,(1520) resonance. Experimentally this amplitude vanishes in collinear pr -, sN reactions requiring a bag radius of 0.6 fm to fit the data. 1. Introduction The cleanest way to extract the size of a nucleon is to have it interact with an external probe. Pion photoproduction off nucleons shows a conspicuous behaviour which seems well suited for the purpose ‘): experimentally the total cross section for the reaction py+ r+n shows three clear peaks which are associated with the P,,(1235), Di3( 1520) and F& 1690) resonances, respectively. In the case of collinear production (8,+ = 0), however, only the first peak is seen, the others are absent. Generally the total cross section involves the helicity amplitudes for J, = i and .I, = 2. In the case of collinear production, however, only the amplitude for J, = $ can contribute, since the pion carries zero angular momentum and thus the final state has J, = $. Therefore the experimental data seem to indicate that the J, = $ photopro- duction amplitudes for the D,,(1520) and F,,(1690) resonances vanish. [See ref. ‘) for further details and references.] Theoretically this has been explained as a cancellation between convection-current and magnetization-current contributions to the transition amplitude ‘). Since such a cancellation requires specific values of the size parameter involved, it has been used to extract an oscillator constant for nonrelativistic quark models ‘). Unfortu- nately, however, the extraction of a size parameter is not unambiguous, but depends on the Lorentz frame in which the amplitude is calculated. This is exemplified by the case of the nonrelativistic quark model where the J, =f helicity amplitude for * Supported by the Deutsche Forschungsgemeinschaft (DFG). ** Supported in part by a fellowship from the Scientific Committee of the NATO via the German Academic Exchange Service (DAAD). ’ Work supported in part by US DOE contract DE-AC02-76ER13001. 419

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Page 1: Recoil corrections in the bag model applied to the photoproduction of the D13(1520) resonance

Nuclear Physics A423 (1984) 419-428 @ North-Holland Publishing Company

RECOIL CORRECTIONS IN THE BAG MODEL APPLIED TO THE PHOTOPRODUCTION

OF THE D&1520) RESONANCE+

CH. HAJDUK* and B. SCHWESINGER**

Physics Department, State University of New York, Stony Brook, NY 11794, USA

Received 30 June 1983 (Revised 29 December 1983)

Abstract: Proper treatment of recoil and c.m. motion in the bag eliminates large parts of the frame dependence of the electroproduction amplitude for the D,,(1520) resonance. Experimentally this amplitude vanishes in collinear pr -, sN reactions requiring a bag radius of 0.6 fm to fit the data.

1. Introduction

The cleanest way to extract the size of a nucleon is to have it interact with an external probe. Pion photoproduction off nucleons shows a conspicuous behaviour which seems well suited for the purpose ‘): experimentally the total cross section for the reaction py+ r+n shows three clear peaks which are associated with the P,,(1235), Di3( 1520) and F& 1690) resonances, respectively. In the case of collinear production (8,+ = 0), however, only the first peak is seen, the others are absent. Generally the total cross section involves the helicity amplitudes for J, = i and .I, = 2. In the case of collinear production, however, only the amplitude for J, = $ can contribute, since the pion carries zero angular momentum and thus the final state has J, = $. Therefore the experimental data seem to indicate that the J, = $ photopro- duction amplitudes for the D,,(1520) and F,,(1690) resonances vanish. [See ref. ‘) for further details and references.]

Theoretically this has been explained as a cancellation between convection-current and magnetization-current contributions to the transition amplitude ‘). Since such a cancellation requires specific values of the size parameter involved, it has been used to extract an oscillator constant for nonrelativistic quark models ‘). Unfortu- nately, however, the extraction of a size parameter is not unambiguous, but depends on the Lorentz frame in which the amplitude is calculated. This is exemplified by the case of the nonrelativistic quark model where the J, =f helicity amplitude for

* Supported by the Deutsche Forschungsgemeinschaft (DFG). ** Supported in part by a fellowship from the Scientific Committee of the NATO via the German

Academic Exchange Service (DAAD). ’ Work supported in part by US DOE contract DE-AC02-76ER13001.

419

Page 2: Recoil corrections in the bag model applied to the photoproduction of the D13(1520) resonance

420 Ch. Hajduk, B. Schwesinger / Recoil corrections

0 N’ J,=l/Z

f

P &=-l/2

Y Y 1,-l

Fig. 1. Collinear photoproduction of nucleon resonances in the J, =f channel.

protons Ad2 is ‘) AF;,* - 1 -(x&)*, independent of the frame, and

Km = M$ - M;

2M, ’ in the laboratory system

K=

K = M;.- M;

r.r.s. 2M,* ’

in the resonance rest frame.

For the Di3( 1520) resonance &b/K,.,.,. = 1.6 giving rise to a variation of the

extracted oscillator length x0 by the same factor. In the present paper we want to analyse the photoproduction amplitudes for the

D,,(1520) in the bag model. The analysis should give some clues on the bag size, to be contrasted with numbers determined from ground-state properties. However, we are already warned by the above example that a proper treatment of the recoiling nucleon is essential for such an analysis. For this purpose we propose to calculate the photoproduction amplitudes in the lab system and in the resonance rest frame. The discrepancies between the results will then tell us to what extent our treatment of the recoil is justified and what uncertainties are still associated with the resulting bag radius.

2. Approximate boost of bag states

The relevant matrix elements we have to investigate are the transition matrix elements of the quark-photon coupling i* A between a nucleon at rest N(0) and the recoiling resonance N*(K),

(N*(K)Ij* AIN( 9 in the lab system , (24

or those between a nucleon moving with -K and the resonance at rest,

(N*(O)lj* A/N(+)) > in the resonance rest frame. (2b)

To our knowledge no satisfactory prescription has been obtained yet of how to calculate above matrix elements for tightly bound systems. Once one has left the regime of perturbation theory the problem that arises is: How can one account for retardation effects from the fields mediating the interactions or for pair creations by an integration over wave functions with a finite number of particles?

Page 3: Recoil corrections in the bag model applied to the photoproduction of the D13(1520) resonance

Ch. ~~j~~~, B. Schwesinger / Recoil co~ec~io~s 421

It has been shown i3) under the assumption that the kernel of the Bethe-Salpeter

equation is instantaneous that the matrix elements of the electromagnetic current can be calculated by taking integrals over the Bethe-Salpeter amplitudes at a universal time T. The notion of instantaneous and universal time here refers to the c.m. system of the composite particle. This assumption seems reasonable for weakly bound systems. It should therefore apply to perturbative QCD at high momentum transfer 14).

Whether a similar situation pertains in the bag model is unclear: bag states are deeply bound and they are not obtained as solutions of a Bethe-Salpeter equation.

On the other hand, the bag boundary represents something like an instantaneous action on the quarks.

We choose the prescription

(N*(K)lj- AIN( = I

d3rr d3rz d3r3 (N*(K) [x1x2x3)

x if, Rx,) * A(xd(x,xzx~ IN(O)) , (3)

fixing the individual times to a universal time T in either the c.m. or the lab system and integrating over the spatial coordinates only. In both cases either the initial or the final bag state is at rest. This prescription allows us to make connection to nonrelativistic physics and is the one used in calculations of magnetic moments and charge radii in bag models *“,*1V15).

If the three quarks inside the bag were free particles then

&x2x31N*(K)) = S-‘(a)(x:x;x;~N*(O)) (4)

would correctly describe the Lorentz boost of the system. The Lorentz transforma- tion is specified by

X:=aXi, (5)

and the spinor transformation reads

qa> = i e-w2)~wi ) tanho=u. i=l

We will assume that eq. (4) holds approximately for the bag states too. The im~rtance of the spinor rotation (6) has been pointed out before*5);

however, in refs. 3,4) th e coordinate transformation (5) is neglected. We do not think this is justified, since the coordinate transformation gives rise to the exp (iK- It,.,.) of the familiar nonrelativistic c.m. motion. The bag model does not provide a rigorous derivation of an analogous result, but there are several plausible approaches. They differ because the baryon masses are not just the sum of quark kinetic energies, i.e. they are not assembled from free quarks. We will present one possibility and only comment upon the others.

Page 4: Recoil corrections in the bag model applied to the photoproduction of the D13(1520) resonance

422 Ch. Hajduk, B. Schwesinger / Recoil corrections

We start with the wave function at rest. It is an antisymmetrized product of independent quark wave functions (see next section):

(x;x;x;lN*(O))= d I+) i,bp,(ri) e-‘“sfi , i=l

(7)

where wP, is the energy of the ith quark orbital and d is the antisymmetrizer. The Lorentz transformation (5) connects primed and unprimed coordinates,

K. ri , ri + K. ri

MN*(E*+MN*) K ,

>

and therefore the t’ dependence in eq. (7) translates into

exp (iK.,R,) , RIM = i L wp,ri . i=i MN’

(9)

In the nonrelativistic limit, wei = mi and

1 R,,,, = R,.,, = -

M ( mlrl + mzr2+ m3r3) , N*

(10)

we recover the familiar c.m. motion as advertized above. Depending on how one distributes the total energy of the baryon in the exponent of eq. (7), one ends up with different answers for RM ; e.g. assuming each quark shares +M one finds RM =

$( rl + r, + r3). However, all these descriptions coincide in the nonrelativistic limit. In order to keep the calculations tractable we neglect the Lorentz contraction in

eq. (8). This higher-order relativistic correction should be of a minor importance compared to the effects of the spinor rotation and the mixing of space and time which are accounted for. Furthermore, since we are going to calculate the helicity amplitudes in two different frames, we maintain some control over the quality of our approximations. Eq. (11) gives our final approximation for a boosted-bag wave function:

(rlr2r31N*(K)) = & h e(“‘/2)“ui$pi(ri) eiK’(wO,x,wO,)ri i=l

(11)

with the additional replacement MN* = c j cop,.

3. The helicity amplitude

The structure of the odd-parity states has been explained very successfully with nonrelativistic quark models 6), and also with a bag model ‘). In either calculation the Dr3( 1520) resonance appears to be an excitation of the nucleon where the quark

spins are, to a large extent, in a pure S = t(p) mixed symmetry state. One quark

is lifted from an s-orbit to a p-orbit and the overall symmetry of the orbital states

Page 5: Recoil corrections in the bag model applied to the photoproduction of the D13(1520) resonance

Ch. Hajduk, B. Schwesinger / Recoil corrections 423

is mixed symmetric again - in order to represent a proper intrinsic excitation relative to the ground state.

The elimination of spurious nucleon excitations can be accomplished after decoup- ling the upper from the lower Dirac components by the Foidy-Wouthuysen transfor- mation exp (-pa - @@) [ref. *)I. With

j.A=i iK ri i&'&e , (12)

i=l

with the boosted wave functions (1 l), and after performing the Foldy-Wouthuysen transformation, the matrix elements in (2) and (3) are approximated by

(13) in the lab system and

x +-Ej (0/2)R.pi)(e-~,P,“j~,e ) 5TPSaL ) (14)

in the resonance rest frame. The states, denoted by their SU(3) structure, in the bra and kets of (13) and (14) are the upper-com~nent spinor states defined in ref. ‘).

The decom~sition of the operators appearing in (13) and (14) is straightforward. The result contains one-body, two-body and three-body operators and is given by eqs, (Ha-d) for the lab system:

(N*(~)i~*A/N(O))=(~T$S~Ll~~Q(i)+~jQ(~,j)/~=~s~’),

(15a) 3/2

;(++ 7$ eiKQ,-RMM)

(139

Q’“b(l,*)=-1 2 g&352- ir2) 1 K+ia2x(ir2+%2) * KjQ(1) , (1%)

Page 6: Recoil corrections in the bag model applied to the photoproduction of the D13(1520) resonance

424 Ch. Hajduk, B. Schwesinger / Recoil corrections

;;=jj/Jjj2. (1W

The resulting operator in the resonance rest system is again given by (15a-d) after replacing

Q’.‘.“(l, 2) = -@b(l, 2) . (16)

The reader is asked to consult ref. “) for more details on the velocity operators ir, ir which act on bra and ket states, respectively. It should be stressed that the c.m.

operator RM = f( ri + r2 + r3) has to be inserted into (Mb) making Q( 1) effectively a three-body operator.

Clearly the first term of Q(l) is just the bag equivalent to the nonrelativistic current operator consisting of a convection and a magnetization part. The second term of Q(1) and Q( 1,2) originate from the spinor transformation (6) and have no nonrelativistic counterpart. The replacement of K. fi + K* (ri -Z&M) finally con- stitutes the difference between recoil corrections in refs. 3*4) and the present approach. The difference originates from the transformation of times from the rest system to the moving system (5).

The calculation of the matrix elements of the operators is straightforward but rather tedious. Eqs. (17), (18) give the results for both helicity amplitudes AYj2 and A$2 where the angular momentum of the resonance is J, = $ and J, =i, respectively:

-;1231(K1~ (I- rS, K~)-31m(K~, (I- YX, Kc,)) 1

Page 7: Recoil corrections in the bag model applied to the photoproduction of the D13(1520) resonance

Ch. ~ajduk, B. Schwesinger / Recoil co~~cfjons 425

where

R

fimn(% h c) = I

~~(~r)~~(~r)~~(cr)r* dr, 0

KO = o. = 2.041 R , K1 = w1 = 3.40/R,

R NO* c

I

R

r*ji( k,r) dr , NY’= 0 I

r2i:tk1r) dr, (17c) o

and R is the bag radius. In the lab system we have

K = KM,, M=MN, E = (M;*+K;b)1'2,

‘yo=wo/@o+4, y1= w1/(2wo+ 4 9 S=l, (184

while in the resonance rest frame

K = K.,, 3 M=M,., E = tMk+K:.+.s.)1’2 ,

yo=yyt=+, s=o. OW

4. Results

The size parameter for which the helicity amplitude vanishes is determined under the assumption that the bag radii of the ground and excited states are equal. This is in accordance with the findings of DeGrand and Jaffe r6) that the energy minimum of the bag for odd-parity states occurs at slightly (10%) larger bag radii due to the increased pressure exerted by the free moving quarks. We will argue later that an increased bag radius for the excited states will not help correct the discrepancies between the size parameter obtained here as compared to the one fitted to the ground-state properties. We can see no reason why one should assume a smaller bag radius for the excited states in the MIT bag model.

Table 1 summarizes our results for the bag radius at which the helicity amplitude for photoproduction of the D,,(1520) vanishes.

If no recoil correction is included (i.e. the effects of the spinor rotation are ignored) the ratio R’ab/R’.‘.S. - l/1.6 is the same as for oscillator lengths in the nonrelativistic translation-invariant case (see eq. (1) and following remarks). This ratio is fairly independent of whether or not we include the cm. correction --eiSRM (9). The

Page 8: Recoil corrections in the bag model applied to the photoproduction of the D13(1520) resonance

426 Ch. Hajduk, B. Schwesinger / Recoil corrections

TABLE 1

Bag radius R, at which Ay12(K, R) vanishes

Without cm. correction With cm. correction

no

recoil

spinor rotation incl.

l-body (l-+2)-body

no

recoil

spinor

rotation

R’.‘.“. ( fm) 0.82 0.66 0.59 (8.6) 0.98 0.67 (8.4) Rlab (fm) 0.51 0.41 0.46 (9.3) 0.64 0.53 (8.4)

Numbers in parentheses give the value of A:,,(K, R) X lo3 at the corresponding radius R.

c.m. correction only effects the size of the radius but does not reduce the frame dependence. The frame dependence is diminished by taking into account relativistic effects. The one-body term from the spinor rotation decreases the radii in both frames reducing also their difference. The two-body terms finally work in different directions closing the gap further though not completely. Again inclusion of the c.m. correction pushes the radius of cancellation up to higher values.

Eventually we are left with a 20% difference between the answers obtained in the two frames. That seems to be a fair agreement considering all the approximations involved. We have to remember that the bag is not translationally invariant and that the boost and Foldy-Wouthuysen transformations are only approximately valid in the presence of the bag boundary. We think that at least the c.m. problem has been partially removed. The bag wave function does not describe purely intrinsic motion. It contains a rather sizeable fraction of c.m. motion ‘). But the transforma- tion of time coordinates (5) leads to an operator -e iK’(q-R~) depending on intrinsic coordinates only. So upon integration the c.m. wave function just gives a factor to the matrix elements, provided it factorizes the total wave function. The latter is true for a product of s-state nonrelativistic oscillator wave functions. We believe it to hold approximately for the ground-state bag wave functions too. By construc- tion ‘) the c.m. wave functions of the nucleon and the resonance are identical, so the amplitudes calculated here should not depend strongly on the wrong c.m. wave function of the bag.

Our whole discussion was based upon neglecting the inertia of the bag itself; if the kinetic energies of the quarks in the bag do not provide a major portion of the baryon mass our treatment of the boost becomes increasingly meaningless. However, there seems to be experimental evidence to support our assumptions. The number of odd-parity baryons observed as well as their quantum numbers and masses leave no room for excitations of the three-quark c.m. relative to an “empty” bag 7V6).

The bag radius at which A&,(K, R) vanishes - R = 0.6 fm - is less than half of the radius needed to fit the charge radius of the proton [R = 1.3 fm with recoil

Page 9: Recoil corrections in the bag model applied to the photoproduction of the D13(1520) resonance

Ch. Hajduk, B. Schwesinger / Recoil corrections 427

corrections r”‘“)] or the magnetic moment of the nucleon [R = 1.5 fm with recoil corrections roY”)]. A bag radius of R = 0.6 fm gives (r2)1’2 = 0.44 fm together with recoil corrections. Such a small size parameter is, however, in accordance with the one necessary to fit the position in energy of the odd-parity states - R = 0.7 fm [ref. ‘)I. It also agrees with the size parameter deduced from nonrelativistic oscillator models by fitting properties of excited states: (r2)1’2= x,, with x,, =0.46 fm [ref. “)I. This is reassuring since the relation between baryon mass and bag radius R used in the form factor here (9)~( 11) is consistent with the transferred momenta (1) for R - 0.6-0.7 fm. As we have argued, the model provides no reason to assume a decrease in radius for the excited states relative to the one for the ground state. Higher kinetic energy of the quarks will rather increase the pressure on the bag boundary. We have tried to enlarge the size of the excited baryon with the result that the ground-state radius required hardly moves from 0.6 fm here to 0.7 fm at an excited-state radius of 1.8 fm. So, apart from being inconsistent with (9) and giving unphysically large radii and far too low energies for the excited state, this modification does not improve on the discrepancy between the radii needed to fit the charge radius and the one for vanishing A$,(& R).

A possible explanation for this apparent discrepancy might come from chiral-bag models. Recent work on the skyrmion ‘*,19) - a model which describes baryons in a large NC limit as solitons of nonlinear meson fields - reproduces proton and neutron charge radii and magnetic moments with the same accuracy as the quark- model counterpart 19). The connection between skyrmion and chiral-bag models - where the center of the skyrmion is replaced by a 3-quark bag - has now also been established 20). It is conceivable that the meson fields surrounding a small quark bag account for large parts of the charge radius and the magnetic moments of the ground state (as they do in the skyrmion, when no bag is present at all). We conjecture the odd-parity states to be excitations of the quark core alone, leaving the meson cloud unchanged relative to the ground state; then the small radius needed to fit properties of excited states actually represents a measure for the size of the quark core. The radii needed to fit the ground-state charge dist~butions and magnetic moments have to simulate contributions from the neglected meson cloud and are therefore larger than the actual size of the quark core.

5. Conclusion

We have presented a reasonable procedure for treating recoil corrections for the MIT bag. In contrast to usual procedures we take proper care of the c.m. motion and ensure that in the nonrelativistic limit the boost prescription is a galilean boost. These cm. effects are not negligible. For instance the total recoil corrections to the baryon magnetic moments are approximately zero lo). A large correction from the spinor rotation is cancelled by the cm. correction essentially leaving the MIT value for the proton magnetic moment fir, = 1.93R unaltered. This has been noted indepen-

Page 10: Recoil corrections in the bag model applied to the photoproduction of the D13(1520) resonance

428 Ch. Hajduk, B. Schwesinger / Recoil corrections

dently in ref. 11) where the authors study the boosting of the wave function in the Friedberg-Lee model.

The bag radius at which the photoproduction amplitude for the D13( 1520) vanishes in collinear p y + TN reactions turns out to be R = 0.6 fm. Such a radius is consistent with radii needed to fit the spectrum of odd-parity baryons. A radius of 0.6 fm is too small to fit baryon magnetic moments and charge radii. Mesonic contributions along the lines of the chiral-bag models 12*18S20) must be considered and might resolve

the apparent discrepancy in bag radii needed to fit ground- and excited-state properties.

We would like to thank G.E. Brown for interesting us in the problem and acknowledge useful discussions with C. Carlson.

References

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1983, and Stony Brook preprint

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A.D. Jackson, private communication