mid-range nucleon-nucleon interaction in the linear sigma model

Nuclear Physics A512 (1990) 637-668 North-Holland

M I D - R A N G E N U C L E O N - N U C L E O N I N T E R A C T I O N I N T H E L I N E A R

S I G M A M O D E L *

Wei LIN and Brian D. SEROT

Physics Department and Nuclear Theory Center, Indiana University, Bloomington, Indiana 47405, USA

Received 25 August 1989 (Revised 22 November 1989)

Abstract: The mid-range attraction between two nucleons is described by correlated two-pion exchange in the linear o- model. In contrast to previous work, chiral symmetry is incorporated dynamically, rather than by enforcing chiral constraints by hand. The input chiral o- mass is a free parameter, but the range and strength of the nucleon-nucleon (NN) attraction are insensitive to its value, which can be taken to be very large. A large o- mass reduces the strength of the nonlinear o--meson interactions. The resulting strong NN attraction and small many-nucleon forces qualitatively reproduce the scalar dynamics in the Walecka model.

To describe correlated two-pion exchange, the s-wave ~-~- phase shift is computed by summing the tree-level amplitude using the lowest-order Pad6 approximant. Unitarity and dispersion relations then determine the s-wave NI~ ~ ~-~ amplitude in the pseudophysical and physical regions. This amplitude contains important ~'~- rescattering effects and respects chiral symmetry. Unitarity and dispersion relations are then used again to calculate the spectral function for the scalar-isoscalar part of the NN interaction. The result can be approximated by a light scalar meson with a broadly distributed mass, which is consistent with the earlier results of Durso, Jackson and Verwest; moreover, the computed mass distribution is insensitive to the input o- mass, as long as it is large (~>1 GeV). Finally, nuclear matter properties are calculated in a Hartree approximation with the model scalar-isoscalar interaction and an elementary w meson.

1. Introduct ion

E v e r s i n c e Y u k a w a ' s p i o n e e r i n g w o r k o n t h e m e s o n t h e o r y o f n u c l e a r fo rces , o n e

g o a l o f n u c l e a r t h e o r y h a s b e e n to d e s c r i b e n u c l e a r s y s t e m s in t e r m s o f t h e i r h a d r o n i c

c o n s t i t u e n t s . T h i s g o a l is e s p e c i a l l y r e l e v a n t n o w , in v i ew o f t h e n e w a c c e l e r a t o r s

t h a t wil l p r o b e n u c l e i w i t h h i g h e n e r g i e s a n d h i g h p r e c i s i o n u s i n g e l e c t r o n s , h a d r o n s ,

a n d h e a v y ions . T h e s e a c c e l e r a t o r s a re e x p e c t e d to r e v e a l n e w p h y s i c s i n v o l v i n g t h e

p r o p e r t i e s o f m e s o n s i n s i d e n u c l e i , t h e n u c l e a r m a t t e r p h a s e d i a g r a m , t h e ro le o f

r e l a t i v i t y , a n d t h e d y n a m i c s o f t h e q u a n t u m v a c u u m . I n d e e d , we h o p e to l e a r n n o t

o n l y a b o u t t h e d y n a m i c s o f h a d r o n s b u t a l so a b o u t t h e ro le o f e x p l i c i t q u a r k a n d

g l u o n d e g r e e s o f f r e e d o m .

O n e f r a m e w o r k fo r d e s c r i b i n g n u c l e i in t e r m s o f h a d r o n s is k n o w n as q u a n t u m

h a d r o d y n a m i c s ( Q H D ) , w h i c h t r e a t s t h e h a d r o n s as r e l a t i v i s t i c q u a n t u m f ie lds

* Supported in part by DOE contract DE-FG02-87ER40365.

0375-9474/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)

638 IV.. Lin, B.D. Serot / Mid-range N N interaction

interacting through a local lagrangian density 1). QHD is consistent in the sense that the dynamical assumptions (such as the relevant degrees of freedom, the form of the lagrangian, and the normalization conditions) are made at the outset, and one then attempts to extract concrete results from the implied formalism. One goal of QHD is a unified description of nuclear matter, nuclear structure, the nucleon- nucleon (NN) interaction, and pion dynamics. Moreover, by searching for break- downs in a consistent approach based on hadrons, one hopes to discover signals of essential quark and gluon degrees of freedom in nuclei. However, in spite of much work over a long period, a consistent, unified hadronic description of these phenomena still eludes us.

The Walecka model 2) (called QHD-I in ref. 1)) has been used to study the properties of nuclear matter and finite nuclei. The NN dynamics arises in this model from the exchange of a scalar meson, which provides mid-range attraction, and a vector meson, which provides short-range repulsion. With a small number of parameters, QHD-I reproduces nuclear matter saturation in various approximations (mean-field, relativistic Hartree, Dirac-Hartree-Fock, and Dirac-Brueckner), pre- dicts sensible electromagnetic response functions in the relativistic random-phase approximation, and describes the bulk and single-particle properties of nuclei reasonably well 3-17). The key feature of QHD-I is the presence of strong scalar and vector fields at typical nuclear densities; these fields shift the nucleon mass and spectrum by several hundred MeV and introduce a new energy scale into the nuclear structure problem. These strong fields have been shown to survive throughout all of the approximations mentioned above ~2). They also provide a natural connection to the successful studies of proton-nucleus scattering using both Dirac phenomenology and the relativistic impulse approximation 18-22).

On the other hand, ~rN scattering has been studied by extending QHD-I to include other mesons (~-, p , . . . ) 1.23.24). Although it is possible, by a suitable choice of

parameters, to achieve reasonable low-energy pion dynamics in free space, these extended models cannot successfully describe the same dynamics when extrapolated into the nuclear medium. Some form of (approximate) chirai symmetry must be imposed to guarantee sensible pion dynamics at finite density.

One is therefore led to construct hadronic models with chiral symmetry. For example, the linear cr model with an additional vector (60) meson to provide short-range repulsion has been extensively applied to the nuclear matter problem in recent years 25 37). Unfortunately, the chiral cr - to model cannot reproduce nuclear matter saturation in the mean-field approximation 25), and this result persists even when exchange terms are included 37). The problem is that chiral models invariably contain strong many-body forces that destroy the successful phenomenology obtained in QHD-I. Although several methods based on corrections to the mean-field theory (MFT) have been proposed for taming these many-body forces 24,29-37), most require a d hoc cancellations between large contributions and cannot be regarded as reliable. Even when nuclear matter saturation can be reproduced, the resulting

W. Lin, B.D. Serot / Mid-range N N interaction 639

parameters imply an unrealistic N N interaction in free space. (In other words, the familiar picture of short-range repulsion and mid-range attraction is lost.)

Nevertheless, chiral symmetry is crucial for producing the correct pion dynamics, so it cannot be discarded. The most serious problems would be cured by precise cancellations among large terms, which might be achieved by imposing a larger

symmetry to constrain the model more tightly. For example, consider a chiral lagrangian that is U(1)®SU(2)L®SU(2)R symmetric, in which baryon number conservation is elevated to a local U(1 ) gauge symmetry. With spontaneous symmetry breaking and the Higgs mechanism, the pions become Goldstone bosons, the Higgs boson is the o-, and the U(1) gauge boson becomes the massive w meson. This larger symmetry provides additional constraints on the parameters and requires new couplings involving o-oJw, o-o-ww, and 7rTr~ow vertices. The resulting model is similar to that proposed by Boguta 3t). This model avoids problems associated with the Lee-Wick state 25,26), and nuclear matter saturation can be reproduced in the MFT, if one is willing to abandon a universal U(1) coupling. The resulting chiral o- mass is between 650 MeV and 1 GeV. Unfortunately, in spite of the successful nuclear matter results, calculations of finite nuclear ground states bear no resemblance to ordinary nuclei 38,39).

Thus, while chiral theories produce the correct pion dynamics, additional symmetry does not eliminate the undesirable (and unobserved) nonlinearities. The only known way to reduce the strength of these terms without enforcing ad hoc cancellations is to choose a large chiral o- mass; since the nonlinearities scale as inverse powers of this mass, this procedure reduces the strength uniformly. We remark that although these difficulties with chiral models have sometimes been referred to as the "pion problem" 4o), it is clearly the scalar dynamics, not the pion dynamics,

that needs to be further explained. The di lemma arising from these arguments is obvious. The successful

phenomenology of nuclei in Q H D and in modern models of the N N interaction 41-43) implies that there is a strong, medium-range (~1 fm) attraction in the spin-zero, isospin-zero channel that has the character of a Lorentz scalar. I f one ascribes this attraction to the exchange of a light scalar particle (the "Walecka o-"), the small mass produces large nonlinearities when chiral invariance is imposed on the theory. In contrast, if one chooses a large mass to reduce chiral nonlinearities, one-o- exchange will not produce the observed mid-range N N attraction, and one must rely on some other mechanism.

Since two pions in the s-wave, isospin-zero state have the same quantum numbers as a scalar-isoscalar particle, one may naturally suppose that the observed mid-range attraction is produced by correlated two-pion exchange between the nucleons. An intermediate state of mass 2m, =280 MeV clearly has a sufficiently long range; indeed, these contributions must be included in the N N interaction kernel if one includes one-boson exchange of heavier mesons. This conjecture is supported by the work of Durso, Jackson and Verwest 44), who constructed a phenomenological

640 W. Lin, B.D. Serot / Mid-range N N interaction

model to describe the mutual interaction of two pions in the J = 0 state. This two-pion

interaction produces a strong mid-range scalar-isoscalar attraction in the NN interaction. In their approach, chiral symmetry was enforced by hand by imposing

the proper soft-pion limits on the scattering amplitudes, and no explicit chiral ~r meson was introduced. Moreover, although the separable model yields an excellent fit to the observed s-wave 7rTr phase shift, it is impossible to relate the model directly to an underlying lagrangian.

The purpose of the present work is to examine 7rvr rescattering in an explicit lagrangian model, namely, the linear ~r model. Our primary aim is to study the implied N N scalar attraction as a function of the chiral ~r mass to see if the observed strength of this attraction constrains the allowed values of this parameter. I f a strong mid-range attraction arises from ~rTr rescattering even with a very massive chiral o,, this could provide a resolution of the so-called "pion problem" discussed above. It is certainly reasonable that if the bulk of the scalar attraction is determined by two-pion intermediate states, the result should be insensitive to the mass of an elementary scalar particle in this channel. However, to our knowledge, there have been no investigations of this question in a dynamical model.

The description of the NN interaction in terms of one-pion and multi-pion exchanges has a long history. In the 1950's and 1960's, S-matrix theory 45) was the

dominant technology. The basic idea is that by knowing certain singularities of the S-matrix elements, one can determine all the other singularities; unitarity relates different matrix elements and gives the residues at poles and the discontinuities along branch cuts. The hope was that the required S-matrix elements could be calculated directly from a set of integral equations, so that the introduction of fields

to describe the strongly interacting particles would be unnecessary. With the discovery of several new hadrons in the early 1960's (as explained by

the original quark model of Gell-Mann and Ne 'eman) , the application of S-matrix theory became more complex, and variations on this approach became fashionable. A dispersion-theoretic approach to the NN problem was formulated in the early 1960's 45-47), using double-dispersion relations developed by Mandelstam 48). This

formalism connects the NIK!-* 7r~" amplitude to the two-pion-exchange contribution in the NN amplitude. Rather than carry out the ambitious goal of S-matrix theory by solving the integral equations, dispersion theory was used to relate the required NN-~ ~-~- amplitudes to observed 7rN scattering. This technique still requires some analytic continuation, because the important Nlq-+ ~-~- amplitudes occur in the pseudophysical region below the physical NI~ threshold.

Calculations based on this formalism were continued in the 1970's in an effort to achieve a quantitative description the NN interaction 49 54). However, ambiguities remained in the determination of the s-wave isoscalar NI~ ~ 7rTr amplitude 52), which

is crucial for computing the NN scalar-isoscalar attraction. These ambiguities were due in part to the continuation into the pseudophysical region and in part to the paucity of relevant data. In particular, one needs ~-Tr scattering data in addition to

IV. Lin, B.D. Serot / Mid-range N N interaction 641

7rN data, and various extrapolations based on these data 49,52-56) gave conflicting

results. A significant breakthrough on this problem was made by Durso, Jackson and

Verwest 44), who constructed a phenomenological model for s-wave 7r~r scattering

that involved a coupling to the K K channel. This coupling is necessary to reproduce the structure in the ~rTr phase shifts due to the S*(993) resonance, and the seven- parameter model gave an excellent fit to the existing 7rTr data. The N/c,/-> 7r~r transition matrix elements were computed using both N and A pole terms. Chiral symmetry was imposed by forcing all input amplitudes to obey the chiral soft-pion limits, and unitarity was ensured by solving the Blankenbecler-Sugar equation to determine the scattering amplitudes. The results for the N N interaction in the scalar-isoscalar channel imply that ~-Tr rescattering produces a strong attraction with a broad peak centered at a scalar "mass" of about 650 MeV. These authors also found significant attraction from the NA and AA intermediate states ("box diagrams"), but this is about one-third as strong as the pion rescattering contribution and shows no resonant structure. Moreover, the strength of the NA and AA pieces are sensitive to vertex cutoffs, and there are unresolved questions concerning the consistency of crossed diagrams and the counting of amplitudes when an elementary 7rNA vertex is included in the model 57.58).

These results play an important role in the construction of modern boson-exchange models of the NN interaction 41-43). These calculations include NN, NA, and AA

boxes in addition to resonant two-pion exchange and achieve accurate quantitative fits to N N scattering data. It is verified that the bulk ( ~ ) of the scalar attraction comes from the 7rTr rescattering contribution, with the remainder supplied by the box diagrams. However, these calculations show that the low-energy N N observables are insensitive to the details of the rescattering, and equally successful fits are obtained by replacing this contribution with an elementary scalar meson [which is called ~r' in ref. 41)]. Thus, modern boson-exchange potentials do not have a dynami-

cal model of the ~-~- interaction and, as a result, do not connect the N N problem to the 7rN or ~'~- scattering observables.

The present work is motivated by these successful investigations. We use the linear o~-model lagrangian to construct a dynamical model for the s-wave ~'Tr scattering amplitude. Our basic aim is to study the sensitivity of the resulting scalar-isoscalar N N attraction to the input chiral cr mass, in view of the relevance of this parameter to the Q H D dilemma discussed above. This calculation provides another link between hadronic theories of the relativistic nuclear many-body problem and the N N interaction. Because of the simplicity of our model, we ake no attempt at a detailed description of the NN scattering data; in fact, we are concerned only with the scalar-isoscalar channel of the interaction.

There are several important reasons for building a dynamical model. First, we can directly relate boson-exchange descriptions of the N N interaction to other processes, such as 7rN and ~-~- scattering. Second, since we generate explicit


analytical expressions for the required amplitudes, extrapolations into the pseudophysical region are unambiguous. Finally, a lagrangian field theory provides a f ramework for calculating modifications to the basic amplitudes at finite density.

Our approach is based on dispersion relations, which allow us to connect the NN, Niq ~ 7rTr, and ~ r scattering amplitudes. The dispersion relations are of course

satisfied by the Feynman diagrams in the model, and in principle, one could perform similar calculations using the diagrams alone. The advantage to dispersion theory is that we can selectively include the most important intermediate states, thus simplifying the calculation. In particular, we can focus on the contributions with the longest range and avoid contamination from short-distance behavior.

For the ~-~ scattering amplitude, we consider only the two-pion channel, whose dynamics is determined by the exchange of an elementary (chiral) o- and the 7r 4 coupling present in the o- model. Although, as pointed out in ref. 44), it is necessary

to include the KI ( channel to reproduce the high-energy phase shifts, we find that the most important contributions for our purposes come from lower energies, where the ~-rr scattering is determined primarily by chiral symmetry and unitarity. [This is consistent with the conclusion of ref. 44).] With only one free parameter, we

achieve satisfactory agreement with the low-energy, J = 0, 7rTr phase shift (fig. 6). The NN--> ~-~- transition amplitudes are given by the N and ~ pole terms, and the soft-pion limits are respected automatically because of the chiral symmetry. Although A pole terms may also contribute significantly, these contributions affect only the overall strength of the scalar-isoscalar attraction and not its resonant behavior; in particular, these terms do not change the sensitivity of this behavior to the ~r mass.

Our simple model produces a strong, resonant attraction in the scalar-isoscalar channel, with a broadly distributed mass centered at about 600 MeV. Most impor- tantly, this resonant behavior is essentially independent of the chiral tr mass, as long

as this mass is greater than about 1 GeV. Thus it is possible to use a very massive chirai tr, which reduces the implied many-body forces in nuclear matter, and rely on correlated two-pion exchange to produce the necessary attraction that is represen- ted by the Walecka-model cr meson.

To see if the strength of the attraction is sufficient, we perform a nuclear matter calculation using the scalar-isoscalar NN amplitude at zero momentum transfer in a Hartree approximation. (An elementary to is added to supply repulsion.) We find that there is more than enough attraction, and our simple nuclear matter calculation produces saturation at too high a density and binding energy. We discuss these results in relation to Q H D models of the nuclear many-body problem and comment on some corrections that may modify the scalar-isoscalar strength found here.

The outline of this work is as follows. In the next section, we review the formalism for calculating the N N scattering kernel due to two-pion exchange and show how to relate this to the N l q ~ ~r~r scattering amplitude s9 ~). (Our conventions for the relevant amplitudes are defined in the appendix.) In sect. 3, we develop a model of the s-wave 7r~ and N l q ~ 7rTr amplitudes based on the linear ~ model and


partial-wave dispersion relations. Our results for the weight function for two-pion

exchange in the scalar-isoscalar channel are discussed in sect. 4, together with our mean-field nuclear matter calculation based on the model NN amplitude. Sect. 5 contains some remarks and our conclusions.

2. Dispersion relations for the NN amplitude

The analytic properties of the scattering amplitudes play the most important role in our discussion. We first briefly review the Landau-Bjorken-Cutkosky rules 62-64). Then we use them to motivate a dispersion relation for the NN interaction that connects the scalar-isoscalar part of the two-pion-exchange interaction to the

pseudophysical N I ~ ~-~- amplitude.

2.1. A N A L Y T I C I T Y A N D G E N E R A L I Z E D U N I T A R I T Y

We define the on-shell, Lorentz-invariant scattering amplitude M in free space

through the S-matrix element for a generic reaction i - ~ f as

St, = a r , - i(27r)4t~(41(~ - Pi)~rl/2Clgl'/2J/[r,, (2.1)

where Pn is the total four-momentum in state n, and ~n is a phase-space factor:

Ntj

q',, =- II 2 f j a / ( 2 m j ) ' , . (2.2) j I

Here Nn is the number of particles in state n; each particle has mass mj, momentum __ _ 2 ~ .~2~ I /2 p~, energy E ~ = ( / , j T m i ) , and fermion number sj(0 for bosons, 1 for fermions);

and /2 is the quantization volume. -iJ/L1i is given by the Feynman rules 63).

In general, J// can be decomposed into invariant amplitudes that are functions of some invariant momentum variables zi, which can take on complex values. The analytic properties of these functions are given by the Landau-Bjorken rules 62.63),

which can be understood in terms of "reduced graphs." A reduced graph is a skeleton diagram of a Feynman graph obtained by shrinking subgraphs down to points. Each of the remaining lines has an on-shell four-momentum. Whenever these four-momenta satisfy certain kinematic conditions, there is a singularity of the original Feynman graph.

The singularities thus determined are simple poles or branch cuts corresponding to possible real intermediate states. The Cutkosky rules 64) give the residue Rg of a pole at z = zg or the discontinuity [J/ / (z)]c along a cut starting at the branch point (threshold) z¢7 from the associated reduced graph. If the singularities correspond to physically accessible intermediate states, the Cutkosky rules reproduce the unitar-


ity condition on the S matrix expressed in terms of ~ :

[ ] J , = - -

N,,[ d4pj (2"rr)~+(e~-mZ)(2mj)S' ] = - i ~ 5~. fjH t L(21r) 4

X (27r)4~(4) (P- j~ ' 1 pj)J/[fn~/[ni. (2.3)

Here P =-- Pr = Pi, 5f is the symmetry factor that produces the correct phase space for identical particles in an intermediate state:

species 5¢= H (Nk t ) - ' , (2.4)

k

3 + signifies that only the positive root for pO is to be taken, and a sum on fermion spin is implied. We define J~n-= ~ ; when fermions are involved, the bar signifies a Dirac adjoint. Cutkosky generalized the unitarity condition to apply to any Landau singularities, including cuts that correspond to anomalous thresholds or virtual intermediate states, i.e., states above an unphysical threshold and below the physical reaction threshold.

Once all the singularities are determined, the invariant amplitude can be expressed as a dispersion integral using Cauchy's theorem:

~(z)=2 Rg + 1 ~ fz ° [~/(z ' ) ]o dz' (2.5) g Z-Zg 2"n'i ~; z ' - z

This dispersion relation allows us to find the amplitude at any point knowing only its properties at the poles and along the cuts. It reveals the relationship between different physical (and sometimes pseudophysical) processes.

Dispersion theory is also useful when one makes a dynamical calculation starting from a lagrangian. Eq. (2.5) and Cutkosky's generalization of eq. (2.3) show that higher-order Feynman graphs can be constructed from lower-order ones. In general, there will be many branch cuts, so the calculation is still very complicated. However, depending on the problem, one can usually make an approximation by keeping only the most important branch cuts. Furthermore, a reasonable approximation for the discontinuities along the cuts may greatly simplify the calculation. In short, dispersion relations allow for approximation schemes that differ from those obtained by selecting a set of Feynman diagrams.

These ideas can be applied to study NN scattering. To make the different reactions easier to recognize, we use F to denote the scattering amplitude for NN ~ NN (and the crossed-channel reaction N1KI-> NI~), we use ~" for ~rN-> ~'N (and the crossed channels Nlq ~ 7rlr and ~Tr ~ Nlq), while for ~wr-~ 7rTr, we use T. The kinematic variables for the NN and crN scattering amplitudes are defined in figs. 1 and 2, and some of the properties of these amplitudes are given in the appendix.

W. Lin, B.D. Serot / Mid-range NN interaction

(a)

>--< (b) (c)

645

O O Q

Fig. 1. (a) The NN scattering amplitude; (b) one-boson exchange (OBE) process (the wavy line represents any boson); (c) two-pion exchange (TPE) process (a dashed line denotes a pion).

2.2. TWO-PION-EXCHANGE CONTRIBUTION FROM DISPERSION RELATIONS

The N N force is media ted by meson exchange, and the range of the force generated

by a single meson is de te rmined by its C o m p t o n wavelength. By looking at fig. lb ,

we find that the N N ampl i tude has a t -channel pole at the square of the meson

--, qt X k

t ~ m

Fig. 2. The 7rN~ 7rN and NIq-> 7rTr scattering amplitude.


mass with a residue equal to the square of the renormalized coupling. Thus the singularities of the amplitude in the complex t plane are directly related to the range and strength of the N N force. Consequently, choosing z-= t in eq. (2.5) is most suitable for studying the NN interaction.

To accurately describe the N N ~ NN amplitude at low and intermediate energies, one must go beyond the one-boson-exchange (OBE) contribution to the kernel shown in fig. lb. To incorporate intermediate states with ranges similar to those of heavy bosons, the two-pion-exchange (TPE) process illustrated in fig. lc must be included. This point will become clear from the following discussion, where we apply the Landau-Bjorken-Cutkosky rules.

The analytic properties of the full /6 amplitude are shown in the Mandelstam diagram of fig. 3. These properties can be incorporated into a double-dispersion relation, as discussed by Mandelstam 4s) and Chew 45). I f one of the variables s, t,

or u is held fixed, one can derive a one-dimensional dispersion relation involving integrals over the other two variables 45,65). In the present analysis, this single-

dispersion relation simplifies because we consider only the direct term in fig. 1, which contains only some of the singularities shown in fig. 3. The direct amplitude is suitable for constructing a kernel that can be inserted into a scattering equation, which then ensures unitarity by producing singularities in the s-channel; the result can be antisymmetrized at the end (by a Fierz transformation) to calculate observables. This strategy for constructing the kernel is similar to the one described in ref. 5o).

/ // "/ // //A 7r X" NN A N~ / f ~ , ~ s

\',V NN \ / / /

\ / t ~ /) ~

" NN--* NN // \ \ / \ \/NN--~NN

,.'./// \ 'A'/,? .','/ \ //\ 'A,-?,

NN u

Fig. 3. Mandelstam diagram for NN and NN scattering. The kinematic regions for physical scattering processes are shaded. The dashed lines represent the one-pion and deuteron poles, while the solid lines

show the thresholds for the various cuts.

W. Lin, B.D. Serot / Mid-range NN interaction 647

Thus, in analogy to eq. (2.5), we can write a fixed-s dispersion relation for the

direct ampli tude/~( t , s) of fig. 1 as

2 1 I ± ~ [ F ( t ' , s ) ] G d t , / ~ ( t , s ) g ' • ' + - - • . ( 2 . 6 ) = ~ - 7 - - - - ~ ( 7 , p~ t~ , ) t ' - t i t - m i 27ri G ,,~

Here the sum in the first term is over the species of contributing mesons, and (7i is the operator that appears at the meson-nucleon vertex. Note that /~(t , s) describes both N N ~ N N and NI~-~ NN, as discussed in the appendix. The branch cuts in the second term are on the real t axis and they fall into two categories: left-hand

cuts that extend from the threshold tG to t = - - ~ , and right-hand cuts that extend from the threshold tG to t = +O0. The left-hand t-channel cuts are generated by u-channel cuts on the positive real u axis, as can be seen from fig. 3. These left-hand

cuts correspond to intermediate states of the u-channel reaction. A right-hand t-cut contribution to the dispersion integral can be interpreted as

the t-channel exchange of a particle of distributed mass x/~ ranging from tG to oO with strength proport ional to [/~(t', s ) ]c . In our calculation, we will retain only the two-pion contributions to the discontinuity [fig. lc]: [ F ( t ' , S ) ] T p E . Our motivation is that other right-hand branch cuts have higher thresholds and are thus generally less important, as the equivalent exchanged particles are more massive and contribute to the short-range part of the NN interaction only.

Notice that there is no left-hand, t-channel, two-pion cut from the diagram in fig. lc because we consider only the direct term, which does not contain a ~rTr cut in the u channel. [After antisymmetrization, fig. lc will produce a 7rTr continuum

cut in the u channel with UG = 4 m ~ . ] There are, however, u-channel cuts from crossed-pion diagrams, but these occur with higher thresholds (UG ~ 4M2), so it is sensible to neglect them in the calculation of the direct term. With these approximations, the direct ampli tude contains no left-hand t-channel cuts at all.

With only the two-pion intermediate states for the t-channel reaction, the amplitude /~(t, s) has a branch cut on the real t axis starting from the pseudophysical threshold t = 4m~. This is the only branch cut we consider. Eq. (2.6) can then be written as

/~ ( t , s )= g~ l f4° I m F ( t " S ) d t ' , (2.7) ~ t - - ~ 2 t~(p) " ~ I ' ) + ,.~ t ' - t - i e

where the infinitesimal - i e reminds us that Im/~(t , s) is to be calculated near the real t axis in the upper half-plane. An analysis of the double-dispersion relations shows tha t /~ is hermitian on a finite segment of the real t axis 65). This allows us to use the Schwartz reflection principle to relate portions of /~ above and below the axis, so that the discontinuity can be rewritten as 2i Im/~. Note also that when one decomposes /~ into Dirac operators and invariant functions, the invariant functions must not contain kinematic singularities that modify the analytic structure.

648 W. Lin, B.D. Serot / Mid-range NN interaction

The discont inui ty [/~(t, S)]Tp E along the two-p ion cut is given by a general izat ion of the N l q ~ ~ r ~ N N uni tary discussed in sect. 2.1. A physical ~rTr in termedia te state must be above the Nlq threshold t ~ 4 M 2. In the " p s e u d o p h y s i c a l " region 4m~ ~ t < 4 M 2, the two nucleons are on the mass shell (k 2 = M 2) but have imaginary

th ree -momenta . Fig. 4 shows schemat ica l ly how to evaluate [/~(t, s)]zpE and defines the four-

m o m e n t a of the particles. With the helicity and par t ia l -wave ampl i tudes defined in the append ix , the unitari ty condi t ion (2.3) for NI~I-> TrTr ~ Nlq can be writ ten as 66)

"W (t)l doo(O) i m F ] ± ) ( t , s ) = K ( t ) ~ 2 J + l ~±)s 2 s 4,17"

Im r(?)( t, s)= -K( t) ~ ~ l l~.(+~( t)l=dgo( O) ,

2 J + 1 I ~±)s 2 J , lm F~3~)(t, s) = K ( t ) ~ ~ r+_ (t) I d, l (O)

~ 2 J + l Im r]~)(t, s) = - K ( t ) s - ~ Ir~+~s(t)12d~_,(O),

i m F ~ ± ) ( t , s ) = K ( t ) ~ 2 J + l ~±)s , ~±)J s s 4 ~ ---r++ (t) r+_ ( t )do l (O) . (2.8)

Here we have defined

K ( t ) - 1 t - 4 m ~ (2.9) 1287r 2 t

Since our interest is u l t imately in the sca la r - i sosca la r channel , we keep only the J = 0 terms. From the re la t ionship be tween the N I q ~ ~-~r par t ia l -wave helicity ampl i tudes and the invar iant ampl i tudes [eq. (A.18)] and the crossing proper t ies of A and B [ref. 67)], we find

Im F] ) -- Im F~- ' = Im F~3 ±' = Im F~4 ±) = Im F~5 ±)= 0 . (2.10)

Fig. 4. A schematic of the two-pion-exchange contribution to the discontinuity in NN scattering.


Eq. (A.9) then leads to

2 1 M ( + ) s = o , . , 2

I m S F ( + ) ( t , s ) = ~ K ( t ) - - ~ - z++ tt) , (2.11)

Im SF~-) = Im vF~±) = Im ~F ~±~ = Im AFI±~ = Im PF ~±) = 0. (2.12)

Only the scalar-isoscalar NN amplitude SF(+) gets a contribution from the J = 0

helicity amplitude of the N/q o 0r~- process, which follows because angular momentum and parity are conserved, and the pions are identical bosons.

In summary, the two-pion-exchange contribution to the scalar-isoscalar part of

the NN invariant amplitude is given by the dispersion integral

SF(T+p)E(t, S) =--1 f ~ Im SF~+)(t', s ) d t ' , (2.13) J . 2 t ' - t - i e • r m~

where the spectral function Im sF~+) is approximated by eq. (2.11). Note that with sr<+) t , s) is independent of s. Thus, the two intermediate our approximations, --VpE~-,

pions in the J = 0 channel behave precisely like a scalar-isoscalar meson with a

distributed mass; they do not contribute to any other Fermi invariant. Since K (t) < 0 S t ' (+) [ t S ) for physical t, the two-pion-exchange contribution is attractive. However, ~VPE~',

contains a piece arising from iterated one-pion exchange, which must be removed

to identify part of a kernel for a scattering equation. Our procedure for removing

the iterated one-pion contribution is discussed in sect. 4. We turn first to the calculation of 'r(+++)J=°(t).

3. A ch ira l m o d e l o f the N l q ~ ~1r a m p l i t u d e

A dynamical model of the Nlq-> ~mr amplitude should respect (approximate)

chiral symmetry and reproduce the correct soft-pion limit. The NiKI--> 7rrr amplitude must also contain rrTr rescattering effects 44), because the outgoing pions interact strongly. The phenomenological study of the 7r~r interaction in ref. 44) found that

7rTr correlations affect both the phase of the amplitude and its magnitude. [It was previously believed that only the phase was modified 68).] Thus we must study both

NiKI -> ~-Tr and ~rTr scattering within the same dynamical model. We base our analysis on the chirally symmetric linear o- model 69-72). After

spontaneously breaking the symmetry, we can write 24)

~ = t~[ iy~O ~ - M + g=s - ig=~r. "~ys] ~b

+½(a~,sa~s_ 2 2 , "- -m~-2"2) m s s )+~(O.zr O"~

2 2 2 2 + g= ms - m ~ s(s2 + 7~.2 ) _ g2 ms - m ~ (s 2 + ~2) 2. (3.1)

2M 8M 2

Here g= is the ~ N N and crNN coupling constant (gZ /4~r = 14.4), and M = 939 MeV

and m= = 140 MeV are the physical nucleon and pion masses. The chiral o, field is

650 W. Lin, B.D. Serot I Mid-range N N interaction

denoted by s, and its mass rn~ is the only unknown parameter in the model. Note

that since the vacuum o--field scales like m~ -1, the scalar self-interactions become

weak in the limit ms-->~, whereas interactions involving pions remain strong 7~). The Feynman rules for the linear o--model lagrangian (3.1) are given in ref. 72).

By virtue of the chiral symmetry, calculations of processes involving pions are

guaranteed to have the correct low-energy behavior. Moreover, by using an explicit lagrangian, we can study consequences of chiral symmetry beyond the low-energy regime, and we have a framework for calculating modifications at finite nuclear

density. We will compute the ¢r7r scattering amplitude in a simple approximation and

apply partial-wave dispersion relations to determine the s-wave N/q --> ~-~r amplitude.

We are primarily interested in the behavior of the resulting N/q--> 7rrr amplitude as

a function of the chiral ~r mass ms.

3.1. THE Ir~'+Trrr AMPLITUDE AND PHASE SHIFT

For 7rTr scattering [see fig. 5], elastic unitarity holds below particle-production threshold. We make the same approximation as we did for N/q --> N/r,/and Nlq --> ~-~-,

namely, that only two-pion intermediate states are kept. Then, just as in eq. (2.8), eq. (2.3) implies a unitarity condition for the partial-wave ~-Tr--> ~-rr amplitudes 66) :

Im T~,(t) = K ( t ) I T ~ , , ( t ) I ~ , (3.2)

k,,y,

, 4 , a

(~)

\ / \ /

- - + / \

/ \

(b)

\ / -< / \ / \ /

× + + -f-- + , , , / / \\ / \

. , \ \ / \

O O Q

(c) (d) (~) Fig. 5. Tree-level diagrams for ~-¢r scattering in the linear ~ model. The dashed lines denote pions and

the wavy line is the chiral ~ meson.


where K ( t ) is defined in eq. (2.9), and the partial-wave amplitudes T~, are defined in the appendix. This relation allows us to write

K ( t) T~,( t) =-sin 6~,(t) exp [ i6Jt,( t) ] , (3.3)

and the elastic ~r~" phase shift ~ , ( t ) is real. We are concerned only with the s-wave, /t = 0 amplitude, and we henceforth denote s:o T~+~ T~,=o as and the phase shift 6J=°~,=o a s t ~ .

In the linear tr model, the lowest-order contributions to the ~'rr scattering amplitude come from the tree-level graphs in fig. 5b-e. From a straightforward application of the Feynman rules 72), we find

l-t-m m s - m = t . . . . L ~ Tab, dc( t , S==) = g 2 M 2 t~ab6d c _~ - - - - ~ - ~ 6ac6bd

1ArrTr -- m s

+ s ,~ - m= 2 6adObe " (3.4)

STrTr -- m s

This expression is explicitly crossing symmetric and agrees in form with eq. (A.19). Moreover, in the m~-~ oo limit, this amplitude reproduces the Weinberg result 7~)

2 tree ----> - g ~ 2 2 + ( u ~ - rn~)~ ,~bd Tob.d~(t, S~.) M 2 [ ( t -

From eqs. (A.21), (A.22) and ampli tude

+ ( S ~ -- m~)6ad6b,.]. (3.5)

(3.4), we obtain the tree-level, s-wave, isoscalar

2 2 [ 2 2 2 (+4q2 1 (+) 2 m s - - m = t - - m = m s - - m ' l n 1 --5- ,

Ttree(/) = 4¢rg~, M2 2 + 3 - - (3.6) t - m~ + ie 2q z rn~,l J

where q2 1 2 = z t - r n ~ . We emphasize that these tree-level graphs give the correct low-energy behavior because of cancellations between tr exchange and the 7r 4 contact interaction that are guaranteed by chiral symmetry. Nevertheless, to compute the 7rTr phase shift for t ~>4m~, we need a unitarized amplitude. One way to unitarize the ampli tude is to solve a scattering equation, such as the Blankenbecler-Sugar equation, with Tl+)~(t) as the kernel. A simpler alternative, which we adopt here, is based on Pad~ approximants 72,73).

Pad6 approximants have the property that unitarity is ensured by construction; thus, we can compute elastic 7rTr phase shifts at all energies. Although higher-order calculations can be made 72_75), for simplicity we construct the lowest-order Pad6 approximant of T ~+) by

(+) T~+)(t) = Ttreo(t) (3.7)

1 - i K ( t ) --tree,T~+)(t) "


This contains the most impor tant features o f ~-rr dynamics at low and intermediate

energies, namely, chiral symmetry and unitarity. Note that whereas T~+)(t) may

give meaningful results for physical t, this funct ion does not have the same analytic

structure as the exact scattering amplitude, so extrapolat ion to other regions o f the complex t plane is unwise.

From eqs. (3.3) and (3.7), the rrrr phase shift is given by

~,~(t) = a r c t a n [ K ( t ) (+) T t r e e ( / ) ] • ( 3 . 8 )

This is shown in fig. 6 for various values o f ms. Also shown are experimental data f rom refs. 7 6 - 8 1 ) , which are obtained primarily f rom p(~' ,2rr) experiments. The

analyses are somewhat model dependent , as a one-p ion-exchange reaction mechan-

ism is assumed, and various treatments are used for the initial- and final-state

interactions. This model dependence may explain the discrepancies between the

experimental results in the low-energy region. In any case, our simple, one-parameter model gives satisfactory results for 4m 2 ~< t <~ 40rn 2, as long as ms ~> 1 GeV. This is a consequence of chiral symmetry, which constrains the threshold behavior, and unitarity, which in our scheme implies that 6,,~ passes through 90 ° at t = ms 2.

Without performing a precise fit, it is clear f rom fig. 6 that ms ~ 1.5 GeV reproduces

the low- and medium- t data reasonably well. In particular, the s-wave rrTr scattering

length follows from eqs. (2.9), (3.6) and (3.8) as

1 g 2 ~ 2 2 , [ 9m2 2 r n ~ m~ao 327r M 2tmS-m'~)l\rns~---7-~+m~2 m2 ] . (3.9)

O) q)

v

s - w a v e It=O 7rTr P h a s e Shif t

15o . . . . I . . . . I . . . . I . . . . I . . . . I~

125

100

75

50

25 -

0 ' 0

S t , , , I . . . . I . . . . I . . . . I

250 500 750 1000 1250

t 1/z (MeV)

Fig. 6. The s-wave, isoscalar ~-Tr phase shift as a function of the total pion c.m. energy. The chiral o- masses used here are (a) m~=950 MeV, (b) 1400 MeV, and (c) 14GeV. The data are f rom ref. 76) (~);

ref. 77) ( ~ ) ; ref. 79) ([7); ref. 8°) (x ) ; and ref. 81) ( + ) .


For ms = 1.4 GeV, we find m=ao = 0.29, which agrees with the experimental result 82)

m,~ao = 0.26+0.05. Larger values of ms change the preceding value only slightly, and the ms~oo limit produces m=ao=0.28 . In contrast, small values of ms are

unacceptable; for example, ms= 500 MeV yields m~a0=0.41. This is additional

evidence that one should not attribute the mid-range NN attraction to the exchange

of a single chiral o- meson. Higher-order Pad4 approximants 73-75) give improved results for 6~= near ~ - -

1 GeV, but as pointed out in ref. 44), it is necessary to include a coupling to the KK

channel to produce accurate results in this regime. [The fit of ref. 44) uses only the data from refs. 76,78),] The most important region for our purposes is at lower t,

where our model is adequate, given the spread in the existing data. Note also that we need only the on-shell 7rTr amplitude in the subsequent analysis.

3.2. DISPERSION RELATIONS FOR THE NIKl~TrTr AMPLITUDE

The o--model tree-level contributions to the Nlq--> ~-Tr reaction, which reproduce

the pseudovector Born terms when ms--> oo, are shown in fig. 7b. If these terms are

used to construct a kernel for an NN scattering equation, the nucleon box diagram

must be omitted, as it is generated automatically through iterated one-pion exchange.

Even when the pseudovector crossed box is included in the kernel, the resulting mid-range attraction is much smaller than the empirical one 41,44). The strong Try-

correlations must be considered 44), and we include 7rTr rescattering in our model

/ / /

/

\ x

(~) (b)

4-

"X~ k' - +

QOO

(c) Fig. 7. (a) The Nlq ~ 7r~- amplitude; (b) tree-level graphs in the linear or model; (c) the ~-~- rescattering

contribution.

654 w. Lin, B.D. Serot / Mid-range NN interaction

of the NN-~ ~-~- amplitude. This is shown schematically in fig. 7c. For the ~rTr scattering amplitude T, we use eq. (3.7), where the intermediate pions are in the isoscalar s-wave and are on shell.

As we are considering only the s-wave N l q ~ 7r~r amplitude, we can use the partial-wave dispersion relations studied in refs. 67,83). The partial-wave amplitudes

are computed by integrating over the invariant amplitudes A and B. The analytic properties of the partial-wave amplitudes can be determined from the Landau- Bjorken rules and the integral representations.

Unfortunately, the partial-wave helicity amplitudes in eqs. (A.15) and (A.18) contain kinematic singularities. To avoid these complications and to write a dispersion relation for the s-wave N I ~ 7rTr amplitude, we define a new function F( t )=- M2 ~+~s=o~ "r++ t t ) / ln] , which is finite at t = 4M 2. (One can define similar amplitudes for higher partial waves.) The spectral function in eq. (2.11) for the NN-> NN two-pion dispersion integral (2.13) can then be rewritten as

1 K(t)IF(t)l~ Im SF~+l(t, s) - - 4 ~ M2 , (3.10)

which is clearly independent of s. As in sect. 2.2, we find that the partial-wave form of the unitarity relation for one

of the isoscalar Nlq ~ ~rTr helicity amplitudes is

Im ~ ' ~ s ( t ) = K ( t ) ( T ~ , = o ( t ) ) * ' r ~ s ( t ) , (3.11)

where T~,_o is the partial-wave ¢r~'-~ 1r~r amplitude defined in eq. (A.22). Eq. (3.11) is valid for t I> 4M 2, and for the s-wave, it leads to

Im F ( t) = K ( t ) F ( t) Tt+~( t) * . (3.12)

This can be generalized to the pseudophysical region using the Cutkosky rules, so it remains valid for 4 m 2 ~ t < ~ . Eq. (3.12) implies that F ( t ) and Tt+~(t) have the same phase, which follows since we have included only two-pion intermediate states.

By following the arguments in ref. 67) and working in analogy to eq. (2.7), we can write a dispersion relation for the s-wave amplitude F ( t ) as

F ( t ) = F B ( t ) + 1 f ~ K ( t ' ) [ T ~ + ) ( t ' ) * ] F ( t ' ) d t ' 7r d4m~ t ' - t - ie

F B ( t ) + I f4 "~ exp [- iS==(t ' )] sin 8==(t ' )F( t ' ) . , ~r _ _ m~ t ' - t - ie tit . (3.13)

In principle, the first term on the right-hand side should include the t-channel Born poles and all left-hand t-channel cuts. In empirical extrapolations 56), fixed-t dispersion relations are used to compute this term; here we approximate it by including only the Born terms FB(t ) of the linear 6r model [fig. 7b]. (Recall that to compute


SF~+p~E, we only need to know F(t) for t>~4m2.) From the Feynman rules and the

definition of F(t), we can use eq. (A.18) to obtain 66) M 2 [ t n 2 t : m ~ .1

Fa( t )=-4~rg~-~ zQo( z ) - -~ -+ M2 t - m ~ + i e J ' (3.14)

where Qo(z) is a Legendre function of the second kind and

2q 2 + m~

z ~ 21qllnl (3.15)

Since zQo(z) = 1 + Q~(z) is an even function of z, there are no kinematic singularities in Fa(t) .

The second term on the right-hand side of eq. (3.13) represents the important ~rTr

rescattering contribution. I f ~ is known, eq. (3.13) is an integral equation for F(t). Equations of this form were solved by Omn~s 84), with the result

FB( t) + l .J4 [~,n] FB( t') sint,_t~==(t')t - ie e-P{") dt' F(t)

= e i ~ " ) I F B ( t ) c o s , ~ ( t ) + l e p " ) P f a ° F~( t ' ) s in '~( t ' ) e p''')dt'] 7r ~ t'-- t '

(3.16)

where

I4 ~.~(t') P( t ) -=IP ~ t ' - t at' (3.17)

I f the principal-value integral for p(t) diverges, we define instead

p(t)=_ ¢__p f4 ° ~,,(t ') ~ m: c ( c - t------) dC, (3.18)

which corresponds to the replacement

ep( t ) -p( t ' ) . . .>ep( t ) -p(o) ep(O)-p(t ' ) ,

in the original version of eq. (3.16). In our model, ~,~(t) becomes constant as t~oo , so that p(t) defined in eq. (3.18)

is finite. Moreover, as only one subtraction is required to define p(t), this function behaves logarithmically at large t, and l im,_ ,~eO~' l - t -1. This implies that the

convergence of the integral in eq. (3.16) is determined by the behavior of FB(t) as t ~ oo. From eq. (3.14), we find lim,~oo Fa(t) -- O(t ~ In t), so that only one subtraction is needed to define the solution F(t). We discuss this subtraction in the next section. Note that the damping of Fa(t) at large t arises because the final two terms in eq. (3.14) cancel to leading order in t. This is a consequence of the finite value of ms in the renormalizable o" model; if the limit rns~ oo is taken first (as in Weinberg's nonlinear, nonrenormalizable model), FB(t)--O(t °) at large t, and an additional subtraction would be needed to define F(t).


Once the subtraction is specified, the required integrals can be done straightfor- wardly with the Gauss-Legendre method. Notice that all the dynamics is specified by the nucleon Born terms and the 7rTr phase shift ~.~. The only adjustable parameter in the model is ms. Since the solution (3.16) involves ~ in the physical region only, one could certainly insert empirical phase shifts directly into the integral. (This necessitates a cutoff rather than a subtraction, since the phases are known only up to some finite energy.) The advantage of our model is that we can study the sensitivity of F( t ) to rn~, as the value of this parameter is important for hadronic theories of the relativistic nuclear many-body problem. We now proceed to discuss our numerical results.

4. Model results

The s-wave isoscalar Nlq ~ 7r~- amplitude F( t ) in the pseudophysical region plays an essential role in determining the N N scalar-isoscalar amplitude. In the literature, one usually considers instead an amplitude fo (+) that is defined by

/I 2

f(o+)(t) =- -F(t) (4~.)2M, (4.1)

and we shall present our results for N l q ~ 7rTr in terms o f f 0 (+). As described at the end of sect. 3, the insertion of our model 6== into the Omnrs

solution (3.16) necessitates one subtraction. We fix the subtraction by demanding

that Refo(+)(0) reproduce the value given by the Born terms Ref~÷)(0), which is -0.43 m~ for rn~ = 1400 MeV and +0.54rn= in the limit ms ~ oo. With this subtraction, the finite Omn~s solution can be written as

F(t )=ei~("IFB( t )c°s~( t )+tep( ' )Pf4 ~FB(t')sin~(t ')e-p(' ') r~ t'(t'--t) dt' ,

(4.2)

with p(t) from (3.18). The motivation is that chiral symmetry implies that Re F(0) -- O(m2/M2), and this is also true of the Born amplitude Re FB(0). The threshold

contribution from ~Tr rescattering must therefore be a small term of O(m~/M2), and for simplicity, we take it to be zero. We find numerically that eP(')~<2 for all t, so the error caused by the uncertainty of our subtraction is O(eP(')m~/M 2) O(m~/M 2) ~-5% for all t. This is of the same size as effects from chiral-symmetry breaking, which are essentially unknown.

The resulting N / q ~ ~Tr amplitude with ms = 1400 MeV is shown in fig. 8, where we plot Re fo (+) and I m f o (÷) separately. Evidently, the real part of the amplitude at t = 0 is relatively small, and our subtraction scheme yields an fo(+)(0) that differs only slightly from the analytic continuation of the empirical scattering amplitude 56)

[Re fo(+'(O) = -2.45m~].


O

80

60

40 // "'" "', / / / "'... "~.

/ , ' ' . . k . 20

0 . . . . . . . . . ~ \

-20

0 10 20 30 40 50 t (m~ -z)

Fig. 8. Our model NIq ~ 7r~- amplitude (solid curve - Ref(o +~, dashed - Imf(o +~) compared to the extrapolated empirical amplitude (dotted curve - Ref(o +~, dash-dotted curve - Imf(o+)). The model results are computed with rn s = 1400 MeV. The imaginary parts for the model and for ref. 56) coincide for 0 < t < 4rn~,

as they are determined essentially by the Born terms.

Fig. 8 also compares our result to the empirical ampl i tude in the pseudophys ica l

region. The agreement is quite good, consider ing the simplicity o f our model and the uncert~tinties in different analytic cont inuat ions 55,56) and in previous models 52.44)

o f f o C+). In fig. 9, we compare Ifo<+~l with the empirical extrapolat ion o f ref. 56) and with the result obta ined f rom the pseudovector Born terms alone 49). Our model is

close to the empirical extrapolat ion (as is the result in ref. 44)) and shows the

impor tance o f the ~rTr rescattering in the range 4m~<~ t ~ < 30m~, which is the most

relevant region.

Al though our model differs qualitatively f rom the empirical result at large t, this regime 0 f fo {+ ) (t) p roduces th e short-range part o f the two -pion-exchange interaction, which will be masked by the exchange of heavy vector mesons in any reasonable N N model. Thus the large-t behavior is un impor tan t for determining the mid-range

N N interaction. The results o f relevance here are determined primarily by chiral

symmetry and elastic unitarity.

The spectral weight funct ion Im SF(+)(t, s) for the N N interaction follows directly

f rom the N l q o ~-Tr ampli tudes [see eq. (3.10)]. To compute the scalar part o f the two-p ion-exchange kernel, however, we must remove the iterated one-pion (OPI) exchange f rom eq. (2.13). The simplest way to do this is to subtract the s-wave

contr ibut ion from the box diagram computed with pseudovector (PV) 7rN coupling, which is given by

1 21~Pv(t) l 2 (4.3) Im s_F(oT,(t , s) =~-~ K ( t ) M 2 ,

658 w. Lin, B.D. Serot / Mid-range NN interaction

+ v

o

80

60

40

0 , I . . . . I 20 30

t (mzr z)

20

' ' I ' '~ ' - '~1 . . . . I . . . . I . . . . / \

/ \ / \

-- / \ - -

/ / •

\ - - \

. . . . . . k . . . . . . . . . . . . . . . . . . . . . . . . . . \

" - - ~ " \

\

\ / \ \ / -

, , ] . . . . . . ] } / / , , ,

10 40 50

Fig. 9. The magnitude of our model N/q~ 7rrr amplitude (solid curve) compared to ref. 56) (dashed curve) and the PV Born terms (dash-dotted curve).

in which the direct PV ampli tude is

~pv(t) --= ½FB( t ) lm~ • (4.4)

To motivate this subtraction, we make the following argument. First, as pointed out in ref. 52), the definition o f the two-pion-exchange kernel depends on the

scattering equat ion into which it will be inserted, as this equat ion defines the form o f the OPI term. It is well known 41) that to achieve reasonable results for the N N

interaction, one should compute one-pion exchange with a PV coupling, or alterna-

tively, use a pseudoscalar (PS) coupl ing and enforce "pa i r suppress ion" by hand.

Thus it is clear that we do not want to subtract the iterated one-pion exchange using a simple PS coupling. In principle, we could carry out the preceding analysis by t ransforming the lagrangian (3.1) to PV representat ion 71.24) and then subtracting

the box diagram with PV coupling. However , since all o f our input ampli tudes are

evaluated at tree level and are on shell, we would find identical two-pion-exchange

results f rom the t ransformed lagrangian, even though the contributions f rom various

diagrams would differ. After subtracting the PV box diagram, the remaining contributions to the isoscalar s-wave ampli tude come f rom the crossed-box diagram, the impor tant ~r~r rescattering, and a term involving the t-channel exchange of a chiral o- meson; the last term vanishes in the m s~ ~ limit. None of these contributions resemble iterated one-p ion exchange, and thus our subtract ion procedure is

acceptable. In fig. 10, we show the weight functions ImSF(+) and ImSF~R+)~-

(Im SF(+)- Im SF~o~) for different chiral G masses ms. The results are insensitive

to the value o f ms. The b u m p at ~ ~ 600 MeV in Im SF~R+) suggests that two-pion

exchange can be approximated by the exchange o f a single scalar- isoscalar meson


I

v

÷

-1

- 2

- 3

-4

~ i I . . . . I . . . . I . . . . [ . . . . I ' - - . . . . . . - . , . . . . , . . .

. . . . . . . . . . . . ,

k " . . . . - /

\% .- s .>-o \ / / / / ~

\ / ? ~ i / - \ / / / . ~ - -

"% i 1 # •

,,I .... I .... I .... I .... I,, 400 600 800 1000 1200

t l lz (MeV)

Fig. 10. The scalar-isoscalar weight function before (solid curve - lm SF(--)) and after (dashed curve - Im SF~+)) the subtraction of the iterated PV box diagram. The dotted curve gives the contribution of the PV crossed box only. The scalar mass is m~ = 950 MeV, 1400 MeV and 14 GeV for the curves labeled A

(a), B (b), and C (c).

with dis t r ibuted mass. Using ms = 1400 MeV, we find roughly M~ ~ 600 MeV, F~ ~-

650 MeV, and G~/4~r ~ 15 for the mass, width, and t rNN coupl ing of the dis tr ibuted-

mass scalar meson. Note that in the m s - c o limit, the peak posi t ion moves to

approx imate ly 650 MeV, in agreement with the phenomeno log ica l model of ref. 44). The dot ted curve shows only the con t r ibu t ion from the s-wave part of the "PV

crossed box." [For the s-wave, this is ident ical to the con t r ibu t ion from the "direct

box" and is given by eq. (4.3).] Al though the crossed-box con t r ibu t ion is attractive,

it is significantly weaker than the rescattering term and has no resonant structure.

This conc lus ion agrees with that of ref. a4), but disagrees with the results of refs. 85,86).

However, the latter work used a model for the 7r~" rescattering that does not respect

chiral symmetry, and this could be responsible for the qual i tat ively different con-

clusions.

To examine the strength of the mid- range scalar a t t ract ion in our model , we

perform a simple nuc lear matter ca lcula t ion using the Hartree approximat ion . To

supply the short-range N N repuls ion, we add a vector- isoscalar meson (to) to the

chiral l agrangian (3.1). If we consider our two-p ion-exchange in teract ion together

with the exchange of a single to meson*, the N N scattering ampl i tude has only

scalar and vector terms and can be writ ten as

V ~ t /~ = s f l (p) l tn)+ -F"y~(p)T(n). (4.5)

Here s F ~ 3sF~+p~E, which is computed from eqs. (2.13), (3.10) and (4.2); we neglect

* Although there are now also two-omega-exchange diagrams in the model, these can be neglected since the corresponding contributions have very short range.


medium modifications to this amplitude. (We also neglect all contributions from

the chiral a, since they can be made as small as we wish by choosing a large value

of m,.) Recall that “r&L includes the iterated one-pion exchange.

The amplitude f determines the NN interaction, and in the Hartree approxima-

tion, 1’ is needed only at t = 0. Thus only the integrated strength of the interactions

are important, and for the scalar part, this can be characterized by Cf -

-3’r:‘d,(O)M’. This quantity is extremely insensitive to the value of m,, and we

find Cf = 480 f 8 for 1s m, C 10 GeV. Each nucleon has a scalar-isoscalar interaction

with all the other nucleons through the exchange of strongly correlated virtual pion

pairs in the s-wave, and each nucleon sees a resulting “mean scalar field” in addition

to the mean vector field from one-omega exchange.

Since the amplitude (4.5) is analogous to that obtained in the one-boson-exchange

approximation to the Walecka model, the mean-field energy density of the system

follows immediately and is given by ‘)

(4.6)

where E*(k) = m, kF is the Fermi momentum, y = 4 is the spin-isospin

degeneracy, and m, and gv are the w mass and coupling constant. The minimization

of 8 with respect to M* yields the thermodynamic self-consistency condition

M*= M+ST(0)p,= M+ST(0) (2;)3 M*

d3kE*(k),

which determines M*. This looks exactly like the MFT of the Walecka model,

except that ‘T(O) takes the place of (-gf/mf). With C$= g$M’/rn$ = 273.8 from table V of ref. ‘) and Cf = 480 from the present

model, we obtain nuclear matter saturation at k, = 1.40 fm-’ with an energy/nucleon

8/pB- M = -87 MeV. This can be compared to the equilibrium values of kF= 1.30 fin-’ and 8/pB- M = -15.75 MeV obtained in ref. ‘) using C$= 273.8 and

C%=3.57.4. (These are the “finite Hartree” parameters.) By adjusting C$ to 381.1

(while keeping Cf = 480), we can achieve saturation at a binding energy of about

15.75 MeV and an equilibrium density corresponding to kF = 1.19 fm-‘; thus, our

model interaction is somewhat too attractive, but has qualitatively the correct

strength.

We emphasize, however, that the value of Cf =480 used above comes from a

direct evaluation of the dispersion integral (2.13) without any cutoff at large t’, and

fig. 9 shows that our model overestimates the large-t behavior of the two-pion-

exchange amplitude. While this would be unimportant in a nuclear matter calculation

that includes correlations (due to short-range anticorrelations produced by repulsive

w exchange), in the present approach it leads to an overestimation of the integrated

scalar strength. Any corrections that reduce our model NR+ rn amplitude in this

regime (which could result from an improved representation of the high-t a,, phase

shift or the presence of form factors at the rrNN vertices) would reduce the integrated


strength. For example, introducing an exponential damping factor (with a decay

constant of roughly 2ms) into the large-t behavior of the 7r~ phase shift left the peak position of the attractive strength essentially unchanged, but reduced the

integrated strength to C~ -~ 400 and verified that the important values of 6 ~ involve 4 m ~ < t<~ 50m~. Note also that our model C 2 is sensitive to small changes in the

pion coupling, since the two-pion-exchange contributions scale approximately as

g4. For example, with g~/4zr = 12.2, the finite Hartree value C~ = 357.4 is obtained. Evidently, the current model is too crude to make detailed comparisons with

models built specifically to reproduce NN scattering observables or the properties

of nuclear matter. The important message is that there is a sufficient scalar-isoscalar

attraction from correlated two-pion-exchange processes even when the chiral tr mass

is large.

5. Discussion

The purpose of this paper was to study scalar meson dynamics and the NN interaction in the linear tr model. We concentrated on three facets of this dynamics

that have been known for some time. First, the NN interaction contains a strong,

mid-range, scalar-isoscalar attraction. Second, if the range of this attraction is used to determine the chiral o- mass, the implied scalar nonlinearities are too strong to

allow for a reasonable description of nuclear matter. These strong nonlinearities

can be reduced by taking a large value for the o- mass. Third, the exchange of two correlated, s-wave pions between nucleons, when consistent with ~-N and ~-~-

scattering data, can produce a strong mid-range attraction. Our aims were to investigate these three aspects in an explicit dynamical model and to study the

sensitivity of the zrTr rescattering to the chiral o- mass.

Beginning with the linear o--model lagrangian, which has approximate chiral

symmetry, we calculated the s-wave ~-~- phase shift with a Pad6 approximant and

used unitarity and dispersion relations to compute the s-wave NN ~ ~'zr amplitude and NN ~ NN scalar-isoscalar amplitude. We found strong 7rzr correlations in the

s-wave that produce a scalar-isoscalar attraction in the NN interaction that is essentially independent of the chiral or mass, as long as the mass is greater than about 1 GeV. Our results are consistent with the empirical NI~ ~ zr~- and 7rTr data 56)

and the coupled-channel study using phenomenological ~'zr and KK separable potentials 44). Thus the important feature that the scalar-isoscalar NN strength

peaks at medium range is quite general. We also verified the important conclusion that the strong ~rzr correlations, and not the crossed box diagrams, are responsible for the bulk of the strength in the attractive scalar-isoscalar channel. In a simple

nuclear matter calculation, we found that the implied NN attraction was more than strong enough to explain the observed nuclear binding.

The difference between this work and previous studies is that instead of using a

phenomenological model of the zrzr interaction with chiral constraints enforced by


hand, we incorporated chiral symmetry dynamically. We worked directly with the degrees of freedom in the lagrangian and did not include A isobars or form factors*. Moreover, although we compared our calculations to existing ~-~ scattering data,

the only input parameters were the ~-NN coupling constant and the pion and nucleon masses. The only unknown parameter was the chiral o- mass, and our results were insensitive to its value, as long as the mass was large. The reason for this insensitivity is that the two-pion part of the N N interaction depends on the low- energy behavior of the 7rTr and N/q --> ~r~ amplitudes, and this behavior is determined primarily by chiral symmetry and unitarity.

There are several advantages to an explicit dynamical model. First, we can relate various scattering processes directly through the relevant Feynman diagrams and parameters in the lagrangian. Moreover, the lagrangian provides a framework for extrapolating amplitudes into pseudophysical regions and for computing the modifications at finite density and temperature. Finally, we gain insight into other lagrangian models of the relativistic nuclear many-body problem.

In particular, we found a qualitative understanding of the successful phenomenology contained in the Walecka model. The exchange of a light scalar meson can be viewed as an approximate description of the exchange of correlated pion pairs in the s-wave. Thus the claim that fashionable o--~o models do not include pion dynamics is somewhat overstated. Moreover, since the chiral o- mass can be taken to be very large, undesired scalar nonlinearities [which are absent in the Walecka- model lagrangian 2)] can be reduced. This approach has not been used in previous chiral studies of nuclear matter, as the chiral o- is still held responsible for the bulk of the N N attraction, and a rather low mass (m+ ~ 1 GeV) is chosen specifically to achieve delicate cancellations between large many-body forces 29-37). It is clearly

more advantageous to let the chiral ~ mass become very large and introduce another "Walecka ~r" to achieve the required mid-range N N attraction. This procedure may also reduce the size of higher-order vacuum corrections s7), since it is likely that the mass parameter in the scalar vacuum terms should be the chiral ~r mass.

This discussion brings into focus some of the difficulties in constructing consistent chiral theories of nuclear systems. Since it is important to include rrTr rescattering effects at an early stage, simple approximations based on the degrees of freedom in the lagrangian are inadequate. In addition, the construction of a consistent f ramework that contains both "elementary" hadrons and "effective" hadrons (which include both the light o- and the A) is an unsolved and difficult problem. This presents an important challenge for future work.

We are pleased to thank G.E. Brown, R.J. Furnstahl, C.J. Horowitz, J.T. Londer- gan, M.H. Macfarlane and J.D. Walecka for valuable discussions and comments.

* We also performed calculations including the A pole terms in the N/q-~ ~rTr Born amplitude using the formulas in ref. 44). These contributions increase the overall strength of the two-pion exchange but

do not change the resonant behavior, and the results are still insensitive to the chiral o- mass. We did not illustrate these calculations because there are still open questions about their consistency 57,58).

W. Lin, B.D. Serot / Mid-range N N interaction

Appendix

663

R E L A T I O N S I N V O L V I N G S C A T T E R I N G A M P L I T U D E S

A.1. The N N ~ NN and N]Q~ N]V amplitudes. The s-channel react ion for fig. la

is N N - ~ N N scattering. Only the direct d iagram will be discussed, in which the two

nucleons are treated as distinguishable, so that p~ and p~ (or n2 and n;) are the fou r -momenta o f the same nucleon in the initial and final states, respectively. The

on-shell s -channel ampl i tude for N N ~ N N scattering can be written as

Fs(p'~, n~; p~, n2) = a (p ' , , h'O~f'(p;)f~(n'2, A~)~*(p~)/~(t, s)

x u( pl , h~) ~( pl)u( n2, A2)~:(p2), (A.1)

where ~(p) is a (Pauli) isospinor, and our convent ion for Dirac spinors is

u ( k , h ) = N ( M + E ( k ) ~ ( ¢r.k ) \ cr.k ]Xa, v ( k , h )=N M+E(k) X ~, (A.2)

with N = [ 2 M ( M + E(k))] -~/2, so that t~u = -~Tv = 1 *. We define

t 2 s~(pl+n2) 2, t=--(pl-pl) , u=-(n~-p~) 2, (A.3)

as the Lorentz- invariant Mande l s tam variables. [The third variable u does not appear in eq. (A.1) and is related to the other two by s + t + u = 4M2.] The spin and isospin

dependence is suppressed on the left-hand side o f eq. (A.1).

For on-shell scattering, the opera tor /~ can be decomposed into invariants as follows:

/~(t, s) = }~ mF(t, s)Km(pl" K(~, (A.4) rrz - - S , V , T , P , A

where the set o f five invariant operators K,,(p~ • K("~ is not unique 88). We use the

so-called Fermi invariants defined by the Dirac matrices

Ks = 1 , Kv = ,)//x , K T = o "~" , Kp = 3/5, KA = 3/"75, (A.5)

i n and Krupp) • K(,) stands for Lorentz contract ion.

The only scalars available in isospin space are 1 and ~¢v)" 7(,)- Assuming isospin invariance, each invariant ampl i tude raP(t, s) can be written as

mF(t, s)=--3 ~F<+)(t, s ) + 2 ~F<-~(t, s)?(p)" 7~,), (A.6)

where ~ are the isospin Pauli matrices, and the numerical factors are convenient for the discussion in sect. 2.

• Note tha t this no rma l i za t i on conven t ion is not the one used in ref. ~).


We thus have a total o f ten invariant ampli tudes, a complicat ion arising from the

nucleon spin and isospin. A study of these ampli tudes in the whole s or t plane

will inevitably involve the physical t -channel ( N N o N I ~ ) ampli tude (see fig. 1),

which, by crossing symmetry, is described by the same ten functions in different regions o f the complex t and s planes 89). In particular, with/3 -= -p~ , p ~- p'~, n -= n2,

- = - n ~ , and the antiparticle helicities and isospins denoted with bars, we define

F(±)(p, ff; n, ~) = Y~ "r~±)( t, s)[~/(p, ap)Km(p)V(/~ , A,~)] m

• [~5(& a~)K("~)u(n, a . ) ] (A .7 )

as the Nlq-+ Niq helicity ampli tudes with definite Nlq total isospin I,. The super-

scripts ( + ) and ( - ) denote /, = 0 and /, = 1, respectively.

Due to the parity, time-reversal, and charge-conjugat ion invariance o f the strong interaction, there are five invariant ampli tudes for each isospin channel 89). Thus

there are ten independent Nlq--, Nlq helicity amplitudes, which we can define from

eq. (A.7) as I'~±)(t, s ) ( j = 1 , . . . , 5). The helicities are chosen in the same way as in ref. 12), except that one o f the nucleons is now an antinucleon. Our Nlq-+ Nlq

partial-wave helicity ampli tudes F~±)s ( t ) a r e also defined as in ref. 12), except that

the scattering angle 0 is given here in the Nlq ze ro -momen tum frame ( I /+ n = 0) as

s = ( - f f + n ) 2 = - ( p Z + n 2 + 2 [ p [ [ n l cos 0 ) = - 2 n 2 ( l + c o s 0) . (A.8)

We choose the azimuthal angle to be zero, so that p and n define the N/q scattering

plane• Note that we define ]n]---x/~4~t-M 2, which is imaginary when t is below

threshold. The relationships between the invariant ampli tudes " F (±~ and the helicity ampli-

tudes FI ~ can be deduced using the same procedures as in ref. 12). We obtain the

following matrix equat ion 66)

S F( ±

VF(±

rF~±

I PF (±

AF( ±

1 1 x x 2a 2 2a 2 2a 2 2a 2

1 1 0 0 2a 2 2a 2

1 1 0 0 4ol 2 4oL 2

1 1 a2X -- 1 1 + a2x 2/3 2 2/3 2 2a2/3 2 20e 2/32

1 1 0 0 2a 2 2a 2

f+ x t a2/3

1 a2/3

/3 2 a 2

X

0

(A.9)

where x-= cos O, y ~ sin O, a =- [nl/M, and 13 -= no/M.


A.2. The ~rN ~ 1rN and N N ~ 7rTr amplitudes. Fig. 2 illustrates both the rrN ~ ~rN and N/q-* 7r~- reactions. The s-channel reaction is ~-N~ rrN scattering, and its amplitude can be written as

"(~(q'b, k'A'p'; cla, kAp) = fi(k', A')~*(p')c~*(b)'~(tk, Sk)fb(a)u(k, A)sC(p), (A.10)

where t7 --- - q , and th(i) is the isospin wave function of the pion. With the assumption of isospin invariance, the decomposition into invariant amplitudes is performed by defining

"~(tk, Sk)--= ~(+)(tk, S k ) - - r ( )(tk, Sk)'~" "i', (A.11)

where t" is the isospin matrix for the pion, and

"~±~( tk, Sk) = AI~( tk, Sk) + %,Q" B~±)( tk, Sk) , (A.12)

with

Q=-l(gl+q'), sk=-(k+(l) 2, t k = - ( k ' - k ) : . (A.13)

Notice that there are two invariant amplitudes (A and B) in each isospin channel. For the t-channel reaction, the on-shell N / q ~ 7r~ amplitude is given by

r~±~(q,q';kA, kJt)=~(k,X)[A~±)(tk, Sk)+%,Q~B~±~(tk, Sk)]U(k,A) (A.14)

for each isospin channel, with h7--- -k ' . We can also define helicity and partial-wave amplitudes for N/q ~ 7rTr as

.C~+±+)(tk, Sk)=~.¢±)(q,q';k+,~C+)=~ 2 J + l J (±)J j 47r doo(0k)T++ (tk),

r~+~(tg, sk)------ ~-¢±l(q, q'; k+ , /~ - ) = ~ 2 J + 1 d~l(Ok) ei¢~r~+±w(tk), (A.15)

where the d~m,(O) are the rotation functions of Edmonds 90), which differ from those in refs. 59.91) by m-dependent phases. Here Ok is defined in the N/q zero-

momentum frame ( k + / ~ = 0 ) as the angle between q and k:

Sk = ( k + tT) 2= -(/¢2+ q=-21kllql cos Ok). (A.16)

Since we will ultimately combine two N l q ~ 7r~" amplitudes to compute N /q ~ Nlq, we define the scattering plane as in eq. (A.8). The intermediate-state pion momenta may thus be out of the scattering plane, and Ck is the azimuthal angle of q. The two other helicity amplitudes are not independent due to parity conservation:


"r(±)( q, q'; k - , £-) = --r(+~( tk, Sk) ,

r(±)(q, q'; k - , k + ) = e-2~*~r~Y( tk, Sk) . (A.17)

Note also that r (+) and r ~-) are proportional to the t-channel isoscalar (I, = 0) and

isovector (/, = 1) amplitudes, respectively. The readers are directed to refs. 66,67) for the relationship between the helicity

amplitudes and the invariant amplitudes, the partial-wave form of that relationship, and the crossing properties of A and B. The amplitudes used here differ only in some kinematic factors. For example, we obtain

{ ]klA(~)(t~+2J+llql r(+~+)J(tk)=2rr\- M , t k, [JB(J~-',(tk)+( J + l

~(+±)~(t~) = - 2 ~ IqlE(k------2) J(J,/YO-~ (B(j~)( tk)_ B ~ + ) ( t k ) ) " (A.18) M 2 J + l

A.3. The ¢rcr-~ 7rrr amplitude. The rrcr--> rr~- scattering process is depicted in fig. 5a, where we also define the four-momenta and isospin labels of the pions. The scattering amplitude has the isospin structure 63.72)

in which

Tob, d~( t=~, S=,~) = T( t=,~; u ~ , S~)6ob6dc + T(u==; s,~=, t,~,~)6o~6hd

+ T(s~ . ; t ~ , U~)t~od6b~ , (A.19)

s ~ , = - ( k ' - q ' ) 2 , t , ~ = - ( k + k ' ) 2 = - t , u ~ = - - ( k ' - q ) 2 . (A.20)

These Mandelstam variables satisfy s , ~ + t ~ + u ~ = 4 m ~ for on-shell scattering. Crossing symmetry in the cry---> rrrr reaction implies that the same function appears in all three terms on the right-hand side ofeq. (A.19) and that T(x; y, z) = T(x; z, y).

Amplitudes with definite t-channel total isospin It are given by

T,,=o(t, s ~ ) = 3T( t ; u ~ , s ~ ) + T (u ,~ ; s,,~, t )+ T ( s ~ ; t, u ~ ) ,

Tl,=,(t, s ~ ) = T ( u ~ ; s ~ , t ) - T ( s , , ; t, u~,) ,

Tt,_2(t, s ~ ) = T (u ,~ ; s ~ , t )+ T ( s ,~ ; t, u~ , ) , (A.21)

and each can be expanded in partial waves as

Tt,(t, s ~ ) = ~ 2 J + l Pj(cos O,~,,)T~,(t) (A.22) 4~

with 0 ~ defined as the angle between q and k in the t-channel zero-momentum frame:

s== = (q - k ) 2= - ( q 2 + k 2 - 2 l q l l k [ cos 0 ~ ) . (A.23)


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mid-range nucleon-nucleon interaction in the linear sigma model

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