The ABC effect and chiral-bag-model form factors

Nuclear Physics A426 (1984) 599-605 @ North-Holland Publishing Company

THE ABC EFFECT AND CHIRAL-BAG-MODEL G. KALBERMANN ~epar~ent

FORM FACTORS

and J.M. EISENBPRG

of Physics and Astronomy, Tel Auiv University, 49978 Tel Aviv, Israel

Received 29 November 1983 (Revised 16 February 1984) Abstract: We investigate the ABC effect in the reaction n +p+ d +(?r?r)’ using the chiral lagrangian of Weinberg in the tree approximation. Chin&bag-model form factors are used for the vertices. We obtain cross sections as a function of energy for deuteron scattering angles Bd= 0” and 10.59 The results are compared with the latest experimental measurements. The fit to the absolute value of the cross section is obtained taking the bag radius Iz as a free parameter; the fitted value of R turns out to be that expected in the light of previous studies in the framework of chiral bag models. The ABC effect is almost completely accounted for in terms of the lowest-order diagrams.

1. Introduction The ABC effect ‘) is an enhancement in the cross sections of the processes n+p+d+(vsr)c, n+d-tt+(rr)‘, and p+t+a +(rrr)‘that is not observed in the corresponding charged reactions p + p -+ d + (r=)+, etc. Despite theoretical efforts in the last decade to interpret the experimental data the effect remains largely unexplained. Two models 293)are available for the ABC effect in the reaction n +p + d + (7~7r)‘.The first 2, considers a AA resonant intermediate state which decays into two pions and a deuteron. The model reproduces qualitatively the features of the cross section for deuteron scattering angle t),,= 0”, 4.5” but the spectrum does not agree for Cfd> 4.5”. The second model 3, uses nucleon exchange graphs in the s-, t-, and u-channels and the missing mass spectra obtained are qualitatively good only for 19~= 0”; absolute cross sections are not given. The AA graph 2, is one of a large class of diagrams of fourth order in the pion-nucleon coupling constant g, which are the lowest-order diagrams for the process. The graphs involve pion-nucleon vertices of third order in the pionic field. One therefore needs a lagrangian model that is reliable to this order. Such a lagrangian is given in Weinberg’s chiral model “). It was constructed so as to incorporate current algebra for multipion processes in a straightforward manner. It was later shown 5, that this model yields the correct WN phenomenology up to third order in g for pions below 350 MeV/c momentum. The Iagrangian is to be used in the tree approximation, the effect of loops being accounted for in the renormalized coupling constant and the physical nucleon and pion masses. The extension of the model to quark-pion interactions has been carried out 6, in the framework of chiral 599

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G. Kdbermann,

J. M. Eisenberg / 7he ABC effect

bag models+. The main difference between Weinberg’s pion-nucleon lagrangian ‘) and the analogous quark pion lagrangian 6, is that the latter considers the nucleon as an extended object coupled to a pointlike pionic field. The coupling constant is therefore replaced by a form factor depending on the nucleon size parameter R. In the absence of a form factor, many of the diagrams of NN processes yield divergent results. The major shortcoming of the chiral-bag-model form factor 6, is its noncovariance, stemming from the use of static bag-model solutions 6). Both the noncovariant form factor and the soft-pion lagrangian ‘) call for restrictions on the momenta of the pions involved in the reactions to values ~350 MeV/c. In the ABC reaction n +p + d +(nrr)’ for an initial proton momentum ‘) of 1.88 GeV/c the pions are mostly produced at momenta around 200 MeV/ c, which is well below the estimated bound. The applications of this model 6, to various N, A( 1232) and Roper (1470) processes are quite successful for a bag model radius R = 0.8 fm. Its use in T- absorption on two nucleons 9, showed that third-order pionic effects are essential for a good description of NN interactions. The calculation of the ABC effect in the n +p+ d + (TT)’ reaction requires the deuteron wave function as input. In order to avoid possible complications introduced by dealing with the wave function in a relativistitally moving frame for deuteron momenta 1 < Pd < 2 GeV we choose to work in the deuteron rest frame. We include d-state admixtures for the deuteron that were omitted in previous works 2*3).We limit ourselves to nucleon intermediate states and to the lowest-order graphs. The use of particular form factols yields absolute values for the cross sections without any resealing. Among the various conjectures tested in this calculation is the original suggestion of the ABC discoverers ‘“) that there may exist an I = 0 TT resonance which produces the enhancement in the cross section; we find this assumption unnecessary. Our calculation also checks the validity of the bag-model form factor 6), whose parameter R as determined by the absolute normalization of the cross section will turn out to be in the expected range of values as found in previous studies 6). 2. Formalism and results The lagrangian density for the pion-nucleon U=i@$ly-

!FM,ly+(l

1 a,rl* ap?r +2 (1 +g%?)?

1

system is ‘)

+g2~2)-‘~y”(gg,/g”y5T.

$&IT-g27*(axQr))*

M27r2 (1 +g%2)

(1)

where Cy is the nucleon field, IT is the pion field, MN and m are the nucleon and pion masses, g = G/2 MN is the hadronic coupling constant and gv, g, are the weak ’ These models are reviewed thoroughly in ref. ‘). Efforts to derive realistic dynamics systems from quark models are discussed in ref. ‘).

of two-baryon

601

G, Kiifbennann, J. M. Eisenberg / The ABC eflect

vector and axial vector couplings. We use this lagrangian to order T’, involving the interactions * a,‘llqY,

%.Jn = g(g*/g,)%‘Y,r 2 NNnn= -g2@y+

Y NNnna

=

(2)

* (‘II xc3,lT)P,

-g3wgd~Y’Ys~

Y Tr?mT = fg2m2rr4- g2m2a,71

(3)

* $T2F

(4)

. arm,

(5)

depicted in fig. 1. Fig. 2 shows all the fourth-order (g”) diagrams used in the calculation of n +p+ d +(TT)’ in the tree approximation 3. The initial state consists of a proton and a neutron, and the final state contains a deuteron and two pions. We attach to each TNN vertex a form factor f(s) = 3A(qR)lqR,

(6)

q being the momentum transfer on the nucleon line and R the bag radius parameter. The rest-frame deuteron wave function is taken from the analytic approximation of McGee ‘I):

@d(r)

=

L NNrrr

LNN~

/

/

/

/ // p_

/

,’ 0

I /-

/-

---

+

____

A

LNNmrr

/‘L

lr7rTnr

/ Fig.

1. Vertices used in the calculation,

eqs. (2)-(S). The solid line represents a nucleon and the dashed line a pion.

G. Kdbermann,

602

J.M. Eisenberg / The ABC e$ect

,’

/

/

1’

\L \

+-\

/ //

‘\

I

Fig. 2. Fourth-order diagrams. The initial state consists of a proton and a neutron and the final state of a deuteron (symbolically represented by a D) and two pions (dashed lines).

G. Kdbermann,

where nucleon

(Y, p, rzj, Cj, Cl, E;, N are parameters relative

distance

603

.I.M. Eisenberg / 7he ABC eficf

and Xd is a spin-l

given in that paper, deuteron

spin

function is easily translated to momentum space. The initial state wave function is a plane wave, the distortions

r is the nucleon-

function.

This

wave

being provided

by

the interactions depicted in fig. 2. If one chooses to use distorted waves in the initial state, care must be taken to avoid the possibility of double-counting. Anyhow, the impinging nucleon is very high in energy (about 2 GeV) so a plane wave is expected to be a reasonable approach to its wave function. A total of 80 different diagrams were included in the calculation. The matrix element obtained is very cumbersome. Its structure contains terms of the form

where f is the form factor in eq. (6), k,, k2 are the outgoing pion momenta, the indices I, 2 refer to the spin spaces of nucleons I and 2, E, is the projectile energy (here 2.1 GeV/c in the lab frame) and wI =m. The integration can only be done numerically using a very dense grid, which necessitates about four hours on a CYBER 170/855 computer for each point in the spectrum. The accuracy achieved by this procedure is about l&15%. The only free parameter is the bag radius R of eq. (6). We fitted it to the absolute value of the cross section for 6d = 0” and Pd = 1500 MeV/c, obtaining R = 0.75 fm. Raising this value of the radius to R = 0.8 fm decreases the cross section by a factor of 1.5 and decreasing R to 0.7 increases the cross section by a factor of 2. Previous calculations suggest 6, R = 0.8 fm. The agreement is surprisingly good. The main shortcoming of the present approach is the lack of intermediate nucleon excited states (A, Roper, etc.). The introduction of these higher resonances is somewhat complicated for two reasons. Firstly, Weinberg’s lagrangian uses effective constants and masses which could probably account for part of the strength states. Secondly, one needs a coherent relativistic treatment of both the and its excited states. Chiral bag models provide such a treatment for static

coupling of those nucleon nucleons

only. We thus choose to omit the higher excitations. It is plausible that the introduction of A degrees of freedom will change the cross section by roughly a factor of 1.5, thus suggesting a bag radius of R = 0.75 f 0.05 fm. We note that in the work of Bar-Nir et al. 3, the form factor for the TNN vertex had an equivalent bag radius of about R = 0.4 fm. This low value can be understood in light of the fact that only one diagram of a whole class of graphs was included in that calculation. Comparison of the theoretical results with the experimental data points ‘) and previous calculations **‘) is depicted in fig. 3 for ed = 0” and ed= 10.5”. It is seen that the spectra fit the experimental data points very well and represent a significant improvement over previous calculations, particularly at 10.5”. The suppression of the deuteron d-state wave function in eq. (8) from the calculation yields a change of about 20% at ed = 0” and of lO-30% at 10.5“. This is because

604

G. Kiilbermann, J.M. Eisenberg / The ABC effect

0.6

1.0

1.2

1

1.Q

I. 2

1.6

1.8

r

P”’ 188 QoV/c

0.8

I.4

e, 8lob .’ 10.5.

1.4

Pd (GeV/c

I.6

I.8

21)

22

2.4

2.2

2.4

I

2.0

1

Fig. 3. Experimental data at 0” and 10.5” (dots) compared with the theoretical calculations: present work (solid line), AA model 2, (dash-dot line), and nucleon-exchange model ‘) (dashed line).

at a momentum transfer of q 3 300 MeV/ c the deuteron d-state form factor begins to dominate, and in our process momenta of mainly 200 MeV/c enter. We investigated the hypothesis of the I = 0 OTTresonance ‘) by enhancing the n4 interaction of eq. (5). The results obtained by this procedure are much worse than those displayed in fig. 3. The enhancement suppresses the backward and forward peaks in favor of the region around Pd= 1500 MeV/ c. It is easy to identify the graphs that dominate in each part of the spectrum. The backward and forward peaks are due to those graphs where both pions are produced either at the projectile or at the struck nucleon line, and the intermediate region, 1.3 < Pd< 1.5 GeV/ c, relates to graphs where the pions are produced at different nucleon lines. The charged reaction p+p+d +(~n)+ was calculated using the same diagrams of fig. 2. The results are a factor of 4 lower than those of fig. 3 and the central maximum disappears completely. At the same time the cross section grows smoothly from threshold without any backward or forward peaks. These results are in agreement with those shown in ref. I’) fqr incident proton momenta of 1.69 GeV/c. The reason for the different behavior of the charged case is twofold: Firstly, at the central peak the neutral ( TT)’ emission is dominated by operators proportional to

G. Kdbemann,

J.M. Eisenberg / 77ze ABC effect

605

k, + kz = 2k,, whereas the charged (err)+ emission depends on k, - k2 causing the

vanishing of the main contribution to the peak enhancement. Secondly, most of the dominant diagrams that determine the behavior at the phase space limits vanish due to isospin or antisymmetrization, thus yielding a smooth spectrum a factor of 4 lower than the neutral one. We conclude that the lagrangian of eq. (1) modified by the inclusion of the nucleon form factor as obtained from chiral bag models produces correct cross sections. The form factor of eq. (6) with R = 0.8kO.05 is appropriate at least for momenta k < 350 MeV/ c. The uncertainty in R estimates the possible contribution of nucleon resonances. We find no need for new mesonic resonant states. This research was supported in part by the US-Israel Binational Science Foundation and by funds from the Yuval Ne’eman Chair in Theoretical Nuclear Physics. References 1) 2) 3) 4) 5)

6) 7) 8)

9) IO) II) 12)

F. Plouin, J. Duflo and R. Goldzahl, Nucl. Phys. A302 (1978) 413, and references therein 1. Bar-Nir, T. Risser and M.D. Schuster, Nucl. Phys. B87 (1975) 109 J.C. Anjoos, D. Levy and A. Santoro, Nucl. Phys. B67 (1973) 37 S. Weinberg, Phys. Rev. Lett. 18 (1967) 188 C.W. Bjork et al, Phys. Rev. Lett. 44 (1980) 62; R. Aaron el al, Phys. Rev. Lett. 44 (1980) 66; M.G. Olsson and L. Turner, Phys. Rev. Lett. 20 (1968) I I27 G. KBlbermann and J.M. Eisenberg, Phys. Rev. DZS ( 1983) 66, 71: D29 (1984) 577 A.W. Thomas, Adv. Nucl. Phys. 13 (1983) I J.C.H. van Doremalen and H.J. Weber, Phys. Rev. D26 (1982) 275; B.L.G. Bakker, M. Bogoian, J.N. Maslow and HJ. Weber, Phys. Rev. CZS (1982) 1134; H.J. Weber. Phys. Rev. C26 (1982) 2333 G. KBlberrnann and J.M. Eisenberg, Phys. Rev. C28 (1983) 1318 A. Abashian et aL, Phys. Rev. 132 (1963) 2296 I. McGee, Phys. Rev. 151 (1966) 772 G.W. Barry, Nucl. Phys. B85 (1975) 239