Nucleon-nucleon interaction: Central potential and pion production

Nucleon-nucleon interaction: Central potential and pion production

C. M. Maekawa,1J. C. Pupin,3,2and M. R. Robilotta2

1_{Kellogg Radiation Laboratory, 106-38 California Institute of Technology, Pasadena, California 91125} 2_{Nuclear Theory and Elementary Particle Phenomenology Group, Instituto de Fı´sica, Universidade de Sa˜o Paulo,}

C.P. 66 318, 05315-970, Sa˜o Paulo, SP, Brazil

3_{Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, 01405-900, Rua Pamplona 145, Sa˜o Paulo, SP, Brazil}

共Received 25 October 1999; published 27 April 2000兲

We show that the tail of the chiral two-pion exchange nucleon-nucleon potential is proportional to the pion-nucleon (␲N) scalar form factor and discuss how it can be translated into effective scalar meson inter-actions. We then construct a kernel for the process NN→␲NN, due to the exchange of two pions, which may be used in either three-body forces or pion production in NN scattering. Our final expression involves a partial cancellation among three terms, due to chiral symmetry, but the net result is still important. We also find that, at large internucleon distances, the kernel has the same spatial dependence as the central NN potential and we produce expressions relating these processes directly.

PACS number共s兲: 13.75.Cs, 13.75.Gx, 11.30.Rd

I. INTRODUCTION

The description of pion production in nucleon-nucleon (NN) interactions near threshold is a traditional problem in hadron physics. In recent times interest in it was renewed, due to a wealth of precise experimental data: n p→d␲0 关1兴, and p p→pp␲0 _{关2,3兴, pp→d␲}⫹_{关4,5兴, and pp→pn␲}⫹_关6兴. On the theoretical side, the availability of chiral perturbation theory 共ChPT兲 allowed the problem to be tackled in a sys-tematic manner. However, in spite of all the effort made, a satisfactory picture is still not available.

There are two classes of interactions involved in this pro-cess, associated with either nucleon correlations or the emis-sion of the external pion. In the procedure developed by Koltun and Reitan关7兴, these interactions are encompassed in wave functions and interaction kernels. The former corre-spond to solutions of the Schro¨dinger equation with realistic potentials, whereas the latter are described by models based on Feynman diagrams.

In the discussion of the production kernel, one usually distinguishes between long and short range contributions. The former are shown in Fig. 1, where the first diagram represents the impulse approximation and the second the pion rescattering term. These two processes were considered by Koltun and Reitan关7兴 in their description of the␲0 chan-nel, but the corresponding cross section proved to underesti-mate recent data关1–3兴 by a factor of 5 关8兴. The rescattering term used in that work came from on-shell␲N amplitudes, whereas the pion exchanged in diagram 1共b兲 is off shell. Models which take pion virtuality into account enhance the cross section and tend to reduce the underprediction关9–12兴. Heavy baryon ChPT calculations 关13–15兴 also stressed the importance of this rescattering term at leading order. How-ever, in these works the rescattering and impulse terms came out with opposite signs and the net result was again smaller than in phenomenological calculations 关16兴. Hence other mechanisms are needed to improve the description.

The next natural step concerns shorter range interactions, especially those involving two pions. The treatment of un-correlated two-pion exchange is rather complex and, in the

case of pion production, this part of the interaction has been described by effective heavy meson exchanges. Their contri-butions correspond to the last diagram of Fig. 1, known as z graph, since positive frequency nucleon propagation, already included in the wave function, is subtracted. The inclusion of ␴, ␻, and ␳ mesons, either explicitly 关12兴 or in a general axial current density 关17兴, gave rise to good fits to the pp

→pp␲0 _{cross section at threshold. However, the study of} relativistic effects in ␲0 production 关18兴 demonstrated that z-graph effects are small and hence heavy mesons in ChPT 关13,14兴 do not give rise to the required increase in cross sections. Extension to other channels (␲⫹ and ␲⫺), with inclusion of nucleon resonances关15,19,20兴, also did not im-prove the situation.

Some time ago, Coon, Pen˜a, and Riska 关21兴 produced a three-body potential based on the exchanges of a pion and a scalar meson, which proved to be able to reduce the gap between theory and experiment for the binding energy of trinuclei. Later on, we derived an equivalent result, using a nonlinear Lagrangian, which included an effective chiral sca-lar meson coupled to nucleons关22兴. In that work, the effec-tive field was designed to simulate the two-pion exchange potential. We stress that the exchange of two uncorrelated pions, formulated in the framework of chiral symmetry and including delta degrees of freedom, explains quite well the tail of the scalar-isoscalar nucleon-nucleon potential关23–25兴 and there is no need at all for a true scalar meson to describe that channel. On the other hand, the treatment of uncorre-lated two-pion exchange requires the calculation of many Feynman diagrams and, in problems where one is more

con-FIG. 1. Contributions to the process NN→␲NN: 共a兲 impulse,

共b兲 rescattering, and 共c兲 z graphs; nucleons, pions, and heavier

cerned with simplicity than with fine details, it may be useful to replace all processes associated with the scalar-isoscalar channel by a single effective field. In this conceptual frame-work, our Lagrangian gave rise to a strong pion-scalar-nucleon contact interaction, which corresponds to the kernel for the reaction NN→␲NN due to the exchange of two pions. This kernel was then applied to the ␲0 and␲⫹ pro-duction channels and theoretical results were found to be comparable to the experimental ones关26兴. Recently, calcula-tions based on both relativistic关27兴 and heavy baryon ChPT 关28兴 dealt with such a transition operator and large contribu-tions were again found, involving several cancellacontribu-tions.

The purpose of the present work is twofold. In Sec. II, we discuss the relationship between the actual tail of the two-pion exchange potential and that provided by an effective scalar meson. The pion production kernel is considered in Sec. III and, in order to determine the role of chiral cancel-lations, we study the leading term, constructed by means of the ␲N→␲N and ␲N→␲␲N subamplitudes. Our results are based on a partial cancellation, involving three large fac-tors. We also find that, at large internucleon distances, the kernel has the same spatial dependence as the central NN potential and hence, in Sec. IV, we produce expressions re-lating these interactions directly. Finally, in Sec. V we present a summary and conclusions.

II. CENTRAL POTENTIAL

The two-pion exchange potential 共TPEP兲 is closely re-lated to ␲N scattering. The isoscalar amplitude for the pro-cess NN→NN is represented in Fig. 2 and given by

TS_⫽⫺ i 2!

冕

d4Q 共2␲兲4 3关T⫹兴(1)关T⫹兴(2) 共k2_⫺␮2_兲共k

_⬘

2_⫺␮2_兲, 共1兲

where T⫹ is the isospin symmetric part of amplitude for the process␲aN→␲bN and Q⫽(k

⬘

⫹k)/2.

In recent times, chiral symmetry has been systematically applied to this problem and one has learned关23–25兴 that it is convenient to separate T⫹ into a contribution T_N⫹, due only to pion-nucleon interactions, and a remainder T_R⫹, involving other degrees of freedom. One then writes symbolically 关T⫹_兴⫽关T

N

⫹_兴⫹关T

R

⫹_{兴 for each nucleon and the potential is}

then proportional to 关T⫹兴(1)关T⫹兴(2)⫽关T_N⫹兴(1)关T_N⫹兴(2) ⫹兵关TN⫹兴 (1)_关T R ⫹_兴(2)_⫹关T R ⫹_兴(1)_关T N ⫹_兴(2)_其⫹关T R ⫹_兴(1)_关T R ⫹_兴(2)_{. The} numerical study of these contributions has shown that the term within curly brackets is largely dominant关24兴 and due to the subamplitudes T_N⫹⫽g 2 m ¯u

再

1⫺

冋

m 共p⫹k兲2_⫺m2⫺ m 共p⫺k

⬘

兲2_⫺m2

册

Q”

冎

u, 共2兲 T_R⫹⫽A¯⫹共␯⫽0,t⫽4␮2兲u¯u⫽␣00 ⫹ ␮¯ u,u 共3兲 where g is the␲N coupling constant, m is the nucleon mass, and the bar over the isospin even ␲N subamplitude A⫹ in-dicates the subtraction of the pseudoscalar Born term. The constant␣₀₀⫹ may be expressed as a combination关24兴 of␲N subthreshold coefficients 关29兴. The leading contribution to

TS _{is written as} TS_⬵3␣00 ⫹ ␮ 关u¯u兴(1)关⌫N⫹兴 (2)_{⫹共1↔2兲, 共4兲} where 关⌫N⫹兴 (2)_⫽⫺i 2

冕

d4Q 共2␲兲4 ⫻ 关TN⫹兴 (2) 关共Q⫺⌬/2兲2_⫺␮2_{兴关共Q⫹⌬/2兲}2_⫺␮2_兴, 共5兲 with⌬⫽(k

⬘

⫺k). Using Eq. 共2兲, we have

关⌫N⫹兴 (2)_⫽⫺i 2 g2 m

再

关u¯u兴 (2)

冕

d 4_Q 共2␲兲4 1 关共Q⫺⌬/2兲2_⫺␮2_{兴关共Q⫹⌬/2兲}2_⫺␮2_兴 ⫺关u¯␥␮u兴(2)

冕

d4Q 共2␲兲4 2mQ␮ 关共Q⫺⌬/2兲2_⫺␮2_{兴关共Q⫹⌬/2兲}2_⫺␮2_兴关Q2_⫹2mV 2•Q⫺⌬2/4兴

冎

⫽1 2 g2 m 1 共4␲兲2关Jc,c共t兲⫺Jc,sN (1) _{共t兲兴关u¯u兴}(2)_{. 共6兲}

In deriving this result we used V₂⫽(p₂

⬘

⫹p₂)/2m and the symmetry of the integrand under Q→⫺Q. The functions J, defined in Ref.关30兴, are given by

Jc,c共t兲⫽C共d,⌳兲⫺␮2

冕

0 1 d␣

冕

0 1_d␤ ␤ ␭2 t⫺␭2_␮2, 共7兲 J_c,sN(1) 共t兲⫽⫺2m2

冕

0 1 d␣1⫺␣ ␣

冕

0 1 d␤1⫺␤ ␤ 1 t⫺␩2␮2, 共8兲 with t⫽⌬2 and ␭2_{⫽1/ 关␣共1⫺␣兲␤兴, 共9兲} ␩2_{⫽关共1⫺␣兲}2_{共1⫺␤兲}2_m2_/␮2_⫹1 ⫺共1⫺␣兲共1⫺␤兲兴/关␣共1⫺␣兲␤兴. 共10兲 The integral J_c,c contains a function C(d,⌳), where d is the number of space-time dimensions and ⌳ is the mass scale that arises in dimensional regularization. In the limit d→4 this function becomes divergent and needs to be re-moved by renormalization. We neglect this contribution be-cause it has zero range and overlaps with other short distance effects not considered here.

The function 关⌫_N⫹兴 is related to the scalar form factor ␴(t) by

具

p

⬘

兩Lsb兩p

典

⫽⫺␴(t)关u¯u兴, where Lsb is the

symme-try breaking Lagrangian. Its long range structure, as dis-cussed by Gasser, Sainio, and Sˇvarc关31兴, is associated with diagrams 2共a兲 and 2共b兲, and hence, in our notation, one has the equivalence

␴共t兲关u¯u兴⫽3␮2_关⌫

N

⫹_{兴, 共11兲}

which is valid for large distances. This allows the asymptotic scalar potential to be written as

TS_⬵2␣00 ⫹ ␮ ␴共t兲 ␮2 关u¯u兴 (1)_关u¯u兴(2)_{. 共12兲}

This result is interesting because it sheds light into the structure of the interaction. The picture that emerges is that of a nucleon, acting as a scalar source, disturbing the pion cloud of the other. The function ␴(t) is related to the ␲N ␴-term by ␴(0)⫽␴N and its value at the Cheng-Dashen

point t⫽2␮2 may be extracted from experiment.

In some situations, it may be useful to use an effective parametrized version of␴(t). In this case, the t dependence of Eqs.共7兲 and 共8兲 suggests that one should use the form

␴共t兲⬇⫺ c

t⫺m_s2, 共13兲

where the free parameters c and m_smay be written in terms of ␴(2␮2) and␴(0) as c⫽␴共0兲m_s2, 共14兲 m_s2⫽ 2␴共2␮ 2_兲 ␴共2␮2_{兲⫺␴共0兲}␮ 2_{. 共15兲}

The coupling constant of this effective scalar state to nucleons may be obtained by comparing Eq.共12兲 with

TS_⬇⫺ gs

2 t⫺m_s2关u¯u兴

(1)_关u¯u兴(2)_{, 共16兲}

and one has

g_s2⬇2␣₀₀⫹ms 2

␴共0兲

␮3 . 共17兲

In Table I we display the values of gs and ms obtained

from input factors found in the recent literature. In most cases, the scalar mass is close to that used in the Bonn po-tential 关35兴, but the coupling constant is smaller. We would like to stress, however, that the purpose of this exercise is not to predict these values. Instead, it is to show that the actual asymptotic exchange of two uncorrelated pions may be natu-rally simulated in terms of an effective scalar interaction. As a final comment, one notes that the coupling constant given by Eq. 共16兲 vanishes in the chiral limit and hence the effec-tive approach is not equivalent to the linear ␴-model, in which this does not happen.

The nonrelativistic potential in configuration space is

VS共x兲⫽⫺2␣₀₀⫹ ␮ 4␲

冋

4␲ ␮4

冕

d3⌬ 共2␲兲3 e ⫺i⌬"r_␴共⫺⌬2_兲

册

⫽⫺

冋

3␣₀₀⫹␮ m

冉

g 4␲

冊

2

册

␮ 4␲关Sc,c共x兲⫺Sc,sN (1) _共x兲兴, 共18兲 where x⫽␮r and Sc,c共x兲⫽

冕

0 1 d␣

冕

0 1 d␤␭ 2 ␤ e⫺␭x x , 共19兲 S_c,sN(1) 共x兲⫽2m 2 ␮2

冕

₀ 1 d␣1⫺␣ ␣

冕

0 1 d␤1⫺␤ ␤ e⫺␩x x . 共20兲 TABLE I. Predictions for gsand msfrom Eqs.共14兲 and 共15兲.

␣00⫹ ␴(2␮ 2

) 共MeV兲 关␴(2␮2)⫺␴(0)兴 共MeV兲 gs ms共MeV兲

3.68a 60c 7.3b 7.22 564 3.68a 60c 15c 4.36 393 6.74d 88d 15c 9.09 478 4.61e 90e 15c 7.71 483 a_{Reference关29兴.} b_{Reference关31兴.} c_{Reference关32兴.} d_{Reference关33兴.} e_{Reference关34兴.}

It is important to note that these functions S are not of the Yukawa type and hence cannot be represented over their full range by terms proportional to e⫺msr_{/r, irrespectively of the} value chosen for the parameter ms. In Fig. 3 we display this

potential together with that due to the exchange of an effec-tive scalar meson.

III. KERNEL

In this section we construct a kernel for pion production in NN scattering and due to the exchange of two pions. It is represented in Fig. 4, denoted byT, and based on Tcba and

Tba, the amplitudes for the processes ␲N→␲␲N and␲N

→␲N, respectively. The kernelT for an outgoing pion with momentum q and isospin index c is

Tc⫽⫺i 1 2!

冕

d4Q 共2␲兲4 TcbaTba 共k2_⫺␮2_兲共k

_⬘

2_⫺␮2_兲. 共21兲 The basic subamplitudes have the isospin structures

T_ba⫽␦_abT⫹⫹i⑀_bac␶_cT⫺, 共22兲 Tcba⫽⫺i兵␦bc␶aTA⫹␦ac␶bTB⫹␦ab␶cTC⫹i⑀cbaTE其,

共23兲 and hence Tc⫽␶c (1)_T 1⫹i共␶(1)⫻␶(2)兲cT12⫹␶c (2)_T 2, 共24兲 where T1⫽⫺ 1 2

冕

d4Q 共2␲兲4 关TA⫹TB⫹3TC兴 (1)_关T⫹兴(2) 共k2_⫺␮2_兲共k

_⬘

2_⫺␮2_兲 , 共25兲 T12⫽⫺ 1 2

冕

d4Q 共2␲兲4 关TA⫺TB兴(1)关T⫺兴(2) 共k2_⫺␮2_兲共k

_⬘

2_⫺␮2_兲, 共26兲 T2⫽⫺ 1 2

冕

d4Q 共2␲兲4 2关TE兴(1)关T⫺兴(2) 共k2_⫺␮2_兲共k

_⬘

2_⫺␮2_兲. 共27兲 We begin by discussing the process ␲b(k

⬘

)N( p)

→␲a_(k)␲c_{(q)N( p}

_⬘

_{). The amplitude T}

cba is given by the

sum of T_cba␲ , a t-channel pion-pole contribution, and a re-mainder, denoted by T¯cba. The explicit forms of these terms,

for a system containing just pions and nucleons, was pre-sented in Ref. 关36兴 and here we just quote the main results.

The pion-pole amplitude for on-shell nucleons is

iT_cba␲ ⫽⫺mgA

f_␲ 关u¯␶d␥5u兴

T_dcba␲␲

共p

⬘

⫺p兲2_⫺␮2, 共28兲 where f_␲ and g_A are the pion and axial decay constants, whereas T_dcba␲␲ is the pion scattering amplitude. At tree level, it is given by T_dcba␲␲ ⫽1 f_␲2兵␦ad␦bc关共q⫺k

⬘

兲 2_⫺␮2_兴⫹␦ bd␦ac关共q⫹k兲2⫺␮2兴 ⫹␦cd␦ab关共k

⬘

⫺k兲2⫺␮2兴其 共29兲

and one has

TA␲⫽⫺

mgA f_␲3

共p

⬘

⫺p⫹k兲2_⫺␮2

共p

⬘

⫺p兲2_⫺␮2 . 共30兲 The evaluation of T¯cbarequires the calculation of a large

number of diagrams. However, long ago Olsson and Turner 关37兴 showed that its leading contribution comes from the effective Lagrangian

FIG. 3. Scalar-isoscalar potential: asymptotic two-pion ex-change共solid line兲, Eq. 共18兲, and effective scalar exchange 共dashed line兲, Eq. 共16兲, with␣00⫹⫽4.61, gs⫽7.71, and ms⫽483 MeV taken from the last line of Table I.

FIG. 4. Contributions to the NN→␲NN kernel: 共a兲 pion pole,

L¯_⫽ gA 8 f_␲3␺

¯_␥

␮␥5␶␺•␾⳵␮␾2, 共31兲

which gives the following contribution to T¯A:

T ¯_A⫽2gA

8 f_␲3共2m⫹k” 兲. 共32兲

The corresponding expressions for TB and TC are

ob-tained by making k→⫺k

⬘

and k→q, respectively.

The main implication of this structure of the ␲N

→␲␲N interaction for our study is that the leading contri-bution toT comes from diagrams 4共a兲 and 4共b兲. As the NN interaction due to the exchange of two pions is dominated by the scalar-isoscalar channel, in this work we consider only the amplitude T₁, Eq. 共25兲, and postpone the discussion of the remaining components to another occasion.

Diagram 4共a兲 yields

T1␲⫽

冋

mgA f_␲3

册

1 共p1

⬘

⫺p1兲2⫺␮2 关u¯␥5u兴(1) 1 2

冕

d4Q 共2␲兲4 ⫻关2共p1

⬘

⫺p1兲2⫹2q•⌬⫹3⌬2/2⫺5␮2⫹2Q2兴关T⫹兴(2) 关共Q⫺⌬/2兲2_⫺␮2_{兴关共Q⫹⌬/2兲}2_⫺␮2_兴 . 共33兲 We note that the last two terms in the result Q2⫽(␮2 ⫺⌬2_{/4)⫹关(Q⫺⌬/2)}2_⫺␮2_{兴/2⫹关(Q⫹⌬/2)}2_⫺␮2_{兴/2 allow} the cancellation of pion propagators and therefore corre-spond to short range effects that will be neglected here. We then obtain T1␲⫽i

冋

mgA f_␲3

册

再

3⫹ 3q2⫹4q•共p₁

⬘

⫺p1兲 共p1

⬘

⫺p1兲2⫺␮2

冎

关u¯␥5u兴(1)关⌫N⫹兴 (2)_, 共34兲 where关⌫_N⫹兴(2) _{is given by Eq.共5兲. This result may be} asso-ciated with the scattering of a pion emitted in one of the nucleons by the pion cloud of the other, indicating that the kernel one is considering here is not fully disentangled from that usually called pion rescattering. Indeed, the description of the rescattering process is based on an intermediate ␲N amplitude for off-shell pions, which satisfies a Ward-Takahashi identity 关39兴. In the isospin symmetric channel, this identity may be expressed as

T⫹共q

⬘

2,q2兲⫽T_N⫹⫹q

⬘

2_⫹q2_⫺␮2

f_␲2␮2 ␴共t兲关u¯u兴⫹r

⫹_{, 共35兲}

where T_N⫹ is the nucleon pole 共Born兲 term evaluated with pseudovector coupling, q and q

⬘

are the momenta of the pions, ␴(t) is the scalar form factor, and r⫹ is a remainder that does not include leading order contributions.

The only term that depends strongly on off-shell effects is that proportional to the scalar form factor and hence one writes T⫹共q

⬘

2,q2兲⫽T⫹共␮2,␮2兲⫹␦T⫹, 共36兲 with ␦T⫹⫽共q

⬘

2_⫺␮2_兲⫹共q2_⫺␮2_兲 f_␲2␮2 ␴共t兲关u¯u兴. 共37兲 The contribution of this factor to the pion rescattering amplitude on nucleon 2 reads

T1␦⫽i

冋

mgA f_␲3

册再

3⫹ 3共q2⫺␮2兲 共p1

⬘

⫺p1兲2⫺␮2

冎

关u¯␥5u兴(1)关⌫N⫹兴(2), 共38兲 using Eq.共11兲. Adding this result to the on-shell␲N ampli-tude derived by Gasser, Sainio, and Sˇvarc 关Ref. 关31兴, Eq. 共A.35兲兴, one recovers Eq. 共34兲. Therefore, in the sequence, we no longer consider the term proportional to the pion pole in that expression, with the understanding that it should be included in the on-shell rescattering amplitude.

The evaluation of diagram 4共b兲 is more straightforward and produces T¯1⫽⫺i

冋

mgA f_␲3

册

冋

u ¯

再

_2⫹ q” 2m

冎

␥5u

册

(1) 关⌫N⫹兴(2). 共39兲

Using the Goldberger-Treiman relation and Eq. 共11兲, we have T1␲⫹T¯1⫽i

冋

g 3␮2f_␲2

册

␴共t兲

冋

u ¯

再

_1⫺ q” 2m

冎

␥5u

册

(1) 关u¯u兴(2)_. 共40兲 One notes that when T₁␲ and T¯₁ are added together, a cancellation occurs, which springs from the same mechanism and is very similar to that noticed long ago in the study of exchange currents in pion-deuteron scattering 关38兴.

Another contribution with the same two-pion range comes from diagrams 4共c兲 and 4共d兲, which yield

T1 z_⫽i

冋

g␣00 ⫹ m␮3

册

␴共t兲

冋

u ¯

再

_1⫺ mq” 共p

⬘

⫹q兲2_⫺m2 ⫺ mq” 共p⫺q兲2_⫺m2

冎

␥5u

册

(1) 关u¯u兴(2)_{. 共41兲}

This result includes the propagation of positive energy states that do not contribute to the kernel. Eliminating them and neglecting small noncovariant terms, we have

T1 z_⫽i

冋

g␣00⫹ m␮3

册

␴共t兲

冋

u ¯

再

_1⫹ q” 2m

冎

␥5u

册

(1) 关u¯u兴(2)_{. 共42兲}

Our final expression for the covariant kernel is obtained by adding Eqs.共40兲 and 共42兲 and reads

T1⫽i␴共t兲g

再

冋

1 3␮2f_␲2⫹ ␣00⫹ m␮3

册

u ¯_␥5u ⫺

冋

1 3␮2f_␲2⫺ ␣00⫹ m␮3

册

u ¯ q” 2m␥5u

冎

(1) 关u¯u兴(2)_{. 共43兲}

This covariant amplitude is our main result. IV. APPLICATION

In order to consider applications in low energy processes, we perform a nonrelativistic approximation in our results. In the case of the central potential, Eq.共11兲, one has

tS_⫽2␣

00

⫹_{␴共t兲/␮}3_{, 共44兲} where t⫽⫺⌬2 and we have discarded a normalization factor 4m2. On the other hand, the kernel, suited to be used with nuclear wave functions, is

t1⫽i␴共t兲 g 2m

再

⫺

冋

1 3␮2_f ␲ 2⫹ ␣00⫹ m␮3

册

␴•共p

⬘

⫺p兲 ⫹

冋

1 3␮2f_␲2⫺ ␣00⫹ m␮3

册

␴•q

冎

(1) . 共45兲

The term proportional to␴•(p

⬘

⫺p)/3␮2f_␲2 in this result coincides with that produced recently in Ref. 关27兴.

As discussed in Refs. 关21兴 and 关22兴, a pion production kernel such as t1 gives rise to three-body forces and again one has t⬵⫺⌬2. In the case of threshold pion production, on the other hand, t⬵␮2/4⫺⌬2. In order to test the influence of these different values of t, in Fig. 5 we plot the Fourier transform of the function␴(t), which dictates the space de-pendence of the kernel in the two cases. Inspecting it, one learns that the energy component of the four momentum transferred has little importance and hence the static result

also holds for the production kernel. This allows one to relate it directly to the central potential

t1⫽i gA f_␲ tS 2m

再

⫺

冋

␮m 6␣₀₀⫹f_␲2⫹ 1 2

册

␴•共p

⬘

⫺p兲 ⫹

冋

␮m 6␣₀₀⫹f_␲2⫺ 1 2

册

␴•q

冎

(1) . 共46兲

In order to use these results in actual calculations, in ei-ther momentum or configuration space, one has to evaluate the function␴(t) numerically and, then, the sandwich of the kernel between two-nucleon wave functions. Since the ker-nel and the central potential are closely related, consistency would require that the same dynamics should be used in the construction of both the operator t1 and the wave functions. However, at present, the potential due to the exchange of two pions is reliable at large distances only and hence it is not suited to determine wave functions by means of the Schro¨-dinger equation. Therefore, the possibility of using Eq. 共46兲 with one’s favorite scalar potential is an interesting one.

V. SUMMARY AND CONCLUSIONS

In this work we have shown that the central component of the NN potential at large distances, which is due to the change of two uncorrelated pions, may be naturally ex-pressed in terms of ␴(t), the scalar form factor. This func-tion is related to the␲N ␴-term and may, if one wishes, be parametrized as an effective scalar meson exchange. How-ever, the coupling of this state to nucleons vanishes in the chiral limit and hence this scalar meson does not correspond to that present in the linear ␴-model.

We have also obtained a two-pion exchange kernel for the process NN→␲NN, which can be applied in both three-body forces and pion production in NN scattering. The com-plete calculation of this kernel would require the evaluation of a large number of diagrams. Thus, in order to estimate the dominant term at large NN distances, we have used just the leading contributions to the subamplitudes ␲N→␲N and ␲N→␲␲N, in the framework of chiral symmetry. The sim-plified result so obtained involves a cancellation between contact-three-pion and pion-pole vertices. The latter may also be associated with an off-shell intermediate␲N ampli-tude and has been included into the Tucson-Melbourne 关40兴 two-pion exchange three-nucleon potential. This means that this force does include a term describing a two-pion ex-change between a pair of nucleons. Thus, the use of an on-shell␲N amplitude gives rise to a less ambiguous definition of the two-pion exchange three-body force关41兴.

At large distances, the kernel is closely related to the two-pion exchange scalar isoscalar NN potential. Indeed, in the case of three-body forces, we could show that the kernel and the potential have the same spatial dependence. For threshold pion production, this relationship is also approximately valid. These results led us to produce expressions that relate di-rectly the kernel to the potential. Using the extreme numeri-cal values for␣₀₀⫹ found in the literature, namely, 3.68关29兴

FIG. 5. Fourier transform of␴(t), which determines the space dependence of the three-body force and production kernels, as a function of distance r.

and 6.74关33兴, in Eq. 共46兲, one has t1⫽i gA f_␲ tS 2m兵⫺1.18␴•共p

⬘

⫺p兲⫹0.18␴•q其 (1) _共47兲 or t₁⫽igA f_␲ tS 2m兵⫺0.87␴•共p

⬘

⫺p兲⫺0.13␴•q其 (1)_{. 共48兲} These results are quite close to the kernel obtained by ourselves sometime ago 关22兴, given by

t1⫽i gA f_␲ tS 2m兵⫺␴•共p

⬘

⫺p兲其 (1)_{, 共49兲}

in the case of a model based on effective scalar-isoscalar mesons.1 This allows one to consider the relationship be-tween the kernel and the potential to be a rather general one. The reason for this generality springs from an old insight by Nambu and co-workers关42兴 and Weinberg 关43兴 that, for ge-neric states A and B, the leading contributions to the process A→␲B are obtained by inserting the pion, with gradient coupling, into the external lines of the process A→B.

Finally, we would like to point out that we may expect the contributions from the kernel t1 to be large. In order to see this, note that momentum conservation allows one to write

t1⫽i gA f_␲ tS 2m兵␴•共q⫺⌬兲其 (1) _共50兲

and, in the case of threshold pion production, in configura-tion space one has

t1⫽ g_A f_␲ ␮ 2m␴ (1)_•“ xVS共x兲. 共51兲

As the central potential contains Yukawa functions with effective masses which are not small, its gradient produces a large kernel, proportional to those masses.

ACKNOWLEDGMENTS

We thank Bira van Kolck and Carlos A. da Rocha for useful conversations. C.M.M. thanks the hospitality of the Instituto de Fı´sica Teo´rica 共Universidade Estadual Paulista兲 in the initial stage of this work and acknowledges the support of FAPESP 共Brazilian Agency, Grant No. 99/00080-5兲 and NSF 共Grant No. PHY_94-20470兲. J.C.P. acknowledges the support of FAPESP共Brazilian Agency, Grant No. 94/03469-7兲.

关1兴 D. A. Hutcheo et al., Nucl. Phys. A535, 618 共1991兲. 关2兴 H. O. Meyer et al., Phys. Rev. Lett. 65, 2846 共1990兲; Nucl.

Phys. A539, 633共1992兲.

关3兴 A. Bondar et al., Phys. Lett. B 356, 8 共1995兲. 关4兴 M. Drochner et al., Phys. Rev. Lett. 77, 454 共1996兲. 关5兴 P. Heimberg et al., Phys. Rev. Lett. 77, 1012 共1996兲. 关6兴 W. W. Daehnick et al., Phys. Rev. Lett. 74, 2913 共1995兲;

Phys. Lett. B 423, 213共1998兲; J. G. Hardie et al., Phys. Rev. C 56, 20共1997兲; R. W. Flammang et al., ibid. 58, 916 共1998兲.

关7兴 D. S. Koltun and A. Reitan, Phys. Rev. 141, 1413 共1966兲;

Nucl. Phys. B4, 629共1968兲.

关8兴 G. A. Miller and P. U. Sauer, Phys. Rev. C 44, 1725 共1991兲. 关9兴 F. Hachenberg and H. J. Pirner, Ann. Phys. 共N.Y.兲 112, 401

共1978兲.

关10兴 V. P. Efrosinin, D. A. Zaikin, and I. I. Osipchuk, Z. Phys. A

322, 573共1985兲; Phys. Lett. B 246, 10 共1990兲.

关11兴 E. Herna´ndez and E. Oset, Phys. Lett. B 350, 158 共1995兲. 关12兴 C. Hanhart, J. Haidenbauer, A. Reuber, C. Schu¨tz, and J.

Speth, Phys. Lett. B 358, 21共1995兲; J. Haidenbauer, C. Han-hart, and J. Speth, Acta Phys. Pol. B 27, 2893共1996兲.

关13兴 T. D. Cohen, J. L. Friar, G. A. Miller, and U. van Kolck, Phys.

Rev. C 53, 2661共1996兲.

关14兴 U. van Kolck, G. A. Miller, and D. O. Riska, Phys. Lett. B

388, 679共1996兲.

关15兴 C. A. da Rocha, G. A. Miller, and U. van Kolck, Phys. Rev. C

61, 034613共2000兲.

关16兴 C. Hanhart, J. Haidenbauer, M. Hoffmann, U. -G. Meißner,

and J. Speth, Phys. Lett. B 424, 8共1998兲.

关17兴 T.-S. H. Lee and D. O. Riska, Phys. Rev. Lett. 70, 2237 共1993兲.

关18兴 J. Adam, A. Stadler, M. T. Pen˜a, and F. Gross, Phys. Lett. B

407, 97共1997兲.

关19兴 C. Hanhart, J. Haidenbauer, O. Krehl, and J. Speth, Phys. Lett.

B 444, 25共1998兲.

关20兴 M. T. Pen˜a, D. O. Riska, and A. Stadler, Phys. Rev. C 60,

045201共1999兲.

关21兴 S. A. Coon, M. T. Pen˜a, and D. O. Riska, Phys. Rev. C 52,

2925共1995兲.

关22兴 C. M. Maekawa and M. R. Robilotta, Phys. Rev. C 57, 2839 共1998兲.

关23兴 C. Ordo´n˜ez, L. Ray, and U. van Kolck, Phys. Rev. Lett. 72,

1982共1994兲; Phys. Rev. C 53, 2086 共1996兲.

关24兴 M. R. Robilotta and C. A. da Rocha, Nucl. Phys. A615, 391 共1997兲; M. R. Robilotta, ibid. A595, 171 共1995兲.

关25兴 N. Kaiser, R. Brockman, and W. Weise, Nucl. Phys. A625,

758 共1997兲; N. Kaiser, S. Gertendo¨rfer, and W. Weise, ibid. A637, 395共1998兲.

关26兴 C. M. Maekawa and C. A. da Rocha, nucl-th/9905052. 关27兴 V. Bernard, N. Kaiser, and U.-G. Meißner, Eur. Phys. J. A 4,

259共1999兲.

关28兴 V. Dmitrasˇinovic´, K. Kubodera, F. Myhrer, and T. Sato, Phys.

Lett. B 465, 43共1999兲.

关29兴 G. Ho¨hler, in Numerical data and Functional Relationships in

Science and Technology, edited by H. Schopper, Lando¨lt-Bernstein, New Series, Group I, Vol. 9b, Pt. 2 共Springer-Verlag, Berlin, 1983兲.

1_{In that work we used g}

关30兴 M. R. Robilotta 共unpublished兲.

关31兴 J. Gasser, M. E. Sainio, and A. Sˇvarc, Nucl. Phys. B307, 779 共1988兲.

关32兴 J. Gasser, H. Leutwyler, and M. E. Sainio, Phys. Lett. B 253,

252共1991兲; 253, 260 共1991兲.

关33兴 W. B. Kaufmann and G. E. Hite, Phys. Rev. C 60, 055204 共1999兲.

关34兴 M. M. Pavan, R. A. Arndt, I. I. Strakovsky, and R. L.

Work-man,␲N Newsletter 15, 118 共1999兲.

关35兴 R. Machleidt, K. Holinde, and C. Elster, Phys. Rep. 149, 1 共1987兲.

关36兴 J. C. Pupin and M. R. Robilotta, Phys. Rev. C 60, 014003 共1999兲.

关37兴 M. G. Olsson and L. Turner, Phys. Rev. Lett. 20, 1127 共1968兲.

关38兴 M. R. Robilotta and C. Wilkin, J. Phys. G 4, L115 共1978兲. 关39兴 L. S. Brown, W. J. Pardee, and R. D. Peccei, Phys. Rev. D 4,

2801共1971兲.

关40兴 S. A. Coon, M. S. Scadron, and B. R. Barrett, Nucl. Phys.

A242, 467 共1975兲; S. A. Coon, M. S. Scadron, P. C. Mc-Namee, B. R. Barrett, D. W. E. Blatt, and B. H. J. McKellar, ibid. A317, 242 共1979兲; S. A. Coon and W. Glo¨ckle, Phys. Rev. C 23, 1790共1981兲.

关41兴 H. T. Coelho, T. K. Das, and M. R. Robilotta, Phys. Rev. C 28,

1812共1983兲; M. R. Robilotta, and H. T. Coelho, Nucl. Phys. A460, 645共1986兲.

关42兴 Y. Nambu and D. Lurie, Phys. Rev. 125, 1429 共1962兲; Y.

Nambu and E. Schrauner, ibid. 128, 862共1962兲.