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Three-Body Interactions Among Nucleons and Pions

arXiv:hep-ph/9209257v1 20 Sep 1992UTTG-11-92Three-Body Interactions Among Nucleons and PionsSteven Weinberg∗Theory GroupDepartment of PhysicsUniversity of TexasAustin, Texas 78712AbstractA chiral invariant effective Lagrangian may be used to calculatethe three-body interactions among low-energy pions and nucleonsin terms of known parameters. This method is illustrated by thecalculation of the pion-nucleus scattering length.∗Research supported in part by the Robert A. Welch Foundation and NSF Grant PHY9009850.

Recent articles1,2 have described a systematic effective Lagrangian frame-work for the calculation of reactions involving arbitrary numbers of nucleonsas well as pions of low 3-momentum. To leading order in small momenta,the ‘potential’ for such reactions is given entirely by the tree graphs in whichonly two of the pions and/or nucleons interact; further, their interaction iscalculated using the original effective chiral Lagrangian3 , which consists ofterms with only the minimum numbers of derivatives or pion mass factors,supplemented by contact interaction terms among nucleons.

The correctionsto these two-body interactions of second order in small momenta involve notonly one-loop graphs, but also a large number of new terms4in the La-grangian with additional derivatives, so many that not much can be learnedabout pion-nucleon or nucleon-nucleon interactions in this way. Fortunately,these two-body interactions can instead be taken from phenomenologicalmodels that incorporate experimental information on nucleon-nucleon, pion-nucleon, and pion-pion scattering.

The only remaining contributions to thepotential of the same order in small momenta consist of graphs in which threeparticles (or two pairs of particles) interact, their interactions given by treegraphs calculated from the original effective chiral Lagrangian. Thus we canuse the three-body interactions calculated in terms of known parameters fromthe original effective chiral Lagrangians together with experimental data ontwo-body scattering to calculate all corrections to the potential of first andsecond order in small momenta.1

This method will be illustrated here in the calculation of the amplitudesfor pion scattering on complex nuclei. But first, a reminder of some general-ities.Consider the amplitude for a process with Nn nucleons and Nπ pions inthe initial state and the same numbers of nucleons and pions in the finalstate, all with 3-momenta no larger than of order mπ.

We wish to developa perturbation theory for this amplitude, based on an expansion in powersof the ratio of these small momenta (and the pion mass) to some momen-tum scale that is characteristic of quantum chromodynamics, such as mρ.In counting the number of powers of small momenta in any given “old fash-ioned” (time-ordered) diagram for this process, we must distinguish betweenenergy denominators of two types.Those of the first type arise from in-termediate states that differ from the initial and final states in the numberof pions and/or in the pion energies, and are therefore of the order of thesmall momenta or the pion mass. The energy denominators of the secondtype arise from intermediate states that differ from the initial and final statesonly in the nucleon momenta, and are therefore much smaller, of the order ofthe nucleon kinetic energies.

A given graph is called irreducible if it containsonly energy denominators of the first type. These are graphs for which theinitial particle lines cannot all be disconnected from the final particle linesby cutting through any intermediate state containing Nn nucleons and eitherall the initial or all the final pions.

We shall consider disconnected as well2

as connected irreducible graphs, because a general connected graph is builtup from a sequence of both disconnected and connected irreducible graphsinterleaved with small energy denominators of the second type. (Howeverin all graphs considered here, each of the initial particle lines must be con-nected to one or more of the final particle lines, and vice versa.) This sum ofdisconnected and connected irreducible graphs is what was referred to aboveas the potential.Because irreducible graphs do not contain anomalously small energy de-nominators of the order of nucleon kinetic energies, it is easy to count thenumber ν of powers of small momenta or pion masses in these graphs.

For anirreducible graph with Vi vertices of type i, L loops, and C separate connectedpieces, the number of powers of small momenta or pion masses is1,2ν = 4 −Nn −2C + 2L +XiVi ∆i ,(1)where ∆i is an index for an interaction of type i, given in terms of the numberni of nucleon field factors and the number di of derivatives (or powers of pionmass) in the interaction, by∆i = di + 12ni −2 . (2)(In deriving this result, we count −3 powers of small momenta for eachline passing without interaction through the diagram, because the associatedmomentum-space delta function reduces the number of momentum factorsin the total connected amplitude by that amount.

)3

Eq. (1) is useful because chiral invariance rules out any terms in theLagrangian with ∆i < 0.

It follows that for any given number of externallines, the leading irreducible graphs (those with smallest ν) are the treegraphs (i.e., L = 0) with the maximum number C of connected parts,constructed solely from vertices with ∆i = 0.The contribution of thesevertices can be read offfrom the effective interaction Hamiltonian (in theinteraction picture):Hint,∆=0=12(D2 −1) ˙π2 + 12(D−2 −1)⃗∇π · ⃗∇π + 12m2π(D−1 −1)π2+2F −4π (N(t × π)N)2+Nh2F −1π gAD−1t · (⃗σ · ⃗∇π) + 2F −2π Dt · (π × ˙π)iN+12CS(NN)(NN) + 12CT(N⃗σN)(N⃗σN) . (3)which is derived from the most general chiral-invariant Lagrangian with ∆i =0 :L∆=0=−12D−2∂µπ · ∂µπ −12D−1m2ππ2+Nhi∂0 −2D−1F −2π t · (π × ∂0π) −mN −2D−1F −1π gAt · (⃗σ · ⃗∇)πiN−12CS(NN)(NN) −12CT(N⃗σN) · (N⃗σN)(4)where gA ≃1.25 and Fπ ≃190 MeV are the usual axial coupling constantand pion decay amplitude; t is the nucleon isospin matrix; CS and CT areconstants whose values can be inferred from the singlet and triplet neutron-proton scattering lengths; and D ≡1+π2/F 2π.

(As discussed in ref. 2, terms4

involving time-derivatives of the nucleon field are eliminated by a suitableredefinition of that field, while corrections to the non-relativistic treatmentof the nucleon in (3) and (4) appear as terms in the effective Hamiltonianand Lagrangian with ∆i > 0.) The number C of connected parts is givenits maximum value C = Nn + Nπ −1 by including only graphs (with onequalification to be discussed later) for a single πN, NN, or ππ scattering,with all other lines passing without interaction through the diagram.The corrections to these leading terms with only one extra factor of smallmomenta (or mπ) arise from (a) tree graphs, with the maximum numberC = Nn+Nπ−1 of connected parts, that involve a single vertex (such as thosearising from non-zero u and d quark masses) with ∆i = 1, plus any numberof vertices with ∆i = 0.

The next corrections, with two extra factors of smallmomenta (or mπ), arise from (b) one-loop graphs with C = Nn + Nπ −1involving only vertices with ∆i = 0; (c) tree graphs with C = Nn + Nπ −1involving either two vertices with ∆i = 1 or one vertex with ∆i = 2 (whichserve as counterterms for the infinities encountered in one-loop graphs), aswell as any number of vertices with ∆i = 0 ; (d) tree graphs, constructedentirely from vertices with ∆i = 0, that have one less than the maximumnumber of connected parts, i. e., with C = Nn + Nπ −2.As already mentioned, the vertices with ∆i = 2 that contribute to cor-rections of type (c) contain so many free parameters4 that little of value canbe learned by using the effective Lagrangian to calculate these corrections.5

On the other hand, these corrections as well as the leading terms and thecorrections of types (a) and (b) all only contribute to the maximally discon-nected irreducible graphs, that consist of a connected piece involving just twoof the incoming nucleons and/or pions, plus disconnected lines passing with-out interaction through the diagram for all of the other incoming nucleonsand pions. But instead of trying to use the effective Lagrangian to calculatesuch two-body interactions, we can draw on various phenomenological mod-els that incorporate not only chiral symmetry but the whole body of presentexperimental information about low energy nucleon-nucleon, nucleon-pion,and pion-pion scattering.There remain only the corrections of type (d), with C = Nn + Nπ −2.These are to be calculated from tree graphs involving only the ∆i = 0 Hamil-tonian (3), which involves no unknown parameters.

These corrections arisefrom graphs that either consist of (d1) a connected piece involving just threeof the incoming nucleons and/or pions, or (d2) two connected pieces eachinvolving just two of the incoming nucleons and/or pions, plus in both casesdisconnected lines passing without interaction through the diagram for all ofthe other incoming nucleons and pions. Graphs of type (d2) may, like thegraphs of types (a), (b), and (c), be taken from suitable phenomenologicalmodels based on experimental information about two-body scattering pro-cesses.

This leaves only the three-body graphs of type (d1), which can becalculated from first principles in terms of known constants.6

Let’s first see how this applies to processes involving only nucleons. Mult-inucleon scattering amplitudes and bound-state wave functions are found bysolving an inhomogeneous Lippman-Schwinger or homogeneous Schr¨odingerequation with the effective potential taken as the sum of irreducible graphs.The graphs for the three-nucleon terms in the effective potential are shown inFigures (1) and (2).

A cancellation (to leading order in small momenta) wasnoted in reference 2 among the graphs of Figure (1), the only graphs thatinvolve the part of the effective Hamiltonian (3) that is non-linear in thepion field. It is instructive to look at the reason for this cancellation.

Thesegraphs all involve a single quadratic interaction 2F −1π gAN t · (π × ˙π)N pluslinear interactions of the two pion fields in this interaction with the othertwo nucleons. In each individual time-ordered graph, the time derivative inthe quadratic interaction makes a contribution of the order of a pion energy.However, by summing up the old-fashioned graphs for all the time-orderingsof these three vertices, we obtain a Feynman diagram in which energy isconserved at each vertex, so that the time-derivative yields a difference ofnucleon kinetic energies, smaller by a factor at most of order mπ/mN.This leaves the 3-nucleon graphs of Figure (2).

These are genuine contri-butions to what we have defined as the 3-nucleon potential, but they involveonly the contact and pion-exchange nucleon-nucleon interactions, and theireffect is actually cancelled by terms in the expansion of the reducible three-nucleon graphs of Figure (3) in powers of the ratio of nucleon to pion kinetic7

energies.Again, the reason for this cancellation is not hard to find. Al-though in Figure (2) we are not summing over all time orderings, so thatthese graphs do not make up a complete Feynman diagram, the sum of allthe time-ordered graphs of Figures (2) and (3) makes up several completeFeynman diagrams, in which energy denominators are replaced with pionand nucleon propagators, and energy is conserved at each vertex.

Since thevirtual pion energies in these Feynman diagrams are equal to differences ofnucleon kinetic energies, and hence negligible compared with the virtual pion3-momenta ⃗q, the pion propagators in these diagrams are just (⃗q2 + m2π)−1.But these Feynman diagrams with such pion propagators are just what wewould get from the old-fashioned diagrams of Figure (3) if we were to neglectnucleon kinetic energies in energy denominators for states containing a pion.Thus we may calculate the multi-nucleon potential to second order in smallmomenta by ignoring nucleon kinetic energies in the energy denominators ofthe leading pion-exchange contributions to the potential, and ignoring thethree (or more) - nucleon contributions altogether. This is more or less whatnuclear physicists have always done anyway.∗The three-body forces are more interesting in processes involving a pion.For definiteness, consider the low-energy elastic scattering of a pion from anucleus of nucleon number A.

General considerations of scattering theory tell∗I am grateful to J. Friar for pointing out that in some treatments of the nuclear three-body problem the pion exchange forces are calculated neglecting nucleon kinetic energiesin energy denominators, and that the corrections to this approximation are of the sameorder as the other corrections considered in this work.8

us that the S-matrix element for this process is simply given by the matrixelement between nuclear wave functions of the sum of all irreducible graphswith Nn = A and Nπ = 1. (In applying the effective chiral Lagrangianto such processes we are making use of the fact that typical 3-momenta ofnucleons in nuclei are of order mπ or less.) The leading irreducible graphsare those in which the pion scatters offa single nucleon, evaluated usingthe ∆i = 0 vertices in the tree approximation.∗∗To second order in smallmomenta, the corrections to these leading terms arise from corrections tothe pion-nucleon scattering amplitude (from loop graphs and from verticeswith ∆i = 1, 2) which can be taken from phenomonological models of pion-nucleon scattering, together with connected three-body interactions amongtwo nucleons and the pion, calculated from tree graphs evaluated with the∆i = 0 vertices in Eq.

(3). The graphs for these three-body interactions areshown in Figure 4.This is a lot to calculate, but the problem becomes much simpler if werestrict our attention to the pion-nucleus scattering length, for which theincoming and outgoing pion have vanishing 3-momenta.

The leading termsas well as the corrections to pion-nucleon scattering give a scattering lengththat (apart from reduced-mass corrections) is just the sum of the scattering∗∗There are also nominally leading terms in which the incoming pion is absorbed by onenucleon and the outgoing pion is emitted by another, but when these are summed overdifferent time-orderings they cancel. Again, this is because summing over time-orderingsyields a Feynman diagram in which energy is conserved, but energy cannot be conservedin the emission or absorption of a single real pion by a single nucleon.9

lengths on the individual nucleons.This leaves only the three-body irre-ducible graphs, of which the only ones that survive in the limit of vanishingexternal pion 3-momenta are those shown in Figures 4(a) to 4(f).It is easiest to calculate the contributions of Figures 4(a)-4(c) and 4(f) bynoting that the sum over time orderings in graphs of each type [and lumpingtogether graph 4(f), produced by the interaction term 2F −4π (N(t × π)N)2in the Hamiltonian (3), with the other graphs] must give the same resultas the complete Feynman diagrams of type 4(a) - 4(c) calculated from theLagrangian (4) [which does not contain the interaction 2F −4π (N(t × π)N)2. ]The other graphs, 4(d) and 4(e), are not summed over all time-orderings(because the sum would include reducible as well as irreducible graphs) andso their contributions must be calculated using old-fashioned perturbationtheory.

These contributions to the pion-nucleon scattering length are:a[4(a)]ab=m2π2π4F 4π(1 + mπ/md)Xr

where subscripts a, b are pion isovector indices; r, s label individual nucleons;⃗qrs is the momentum transferred between nucleons r and s in their interactionwith the pion; ⃗σ(r) and t(r) are the Pauli spin vector and isospin vector ofnucleon r; and (t(π)c )ab = −iǫabc is the pion isospin vector.Note that asa result of a partial cancellation between (6) and (7), the integrand in thesum of these averages vanishes for ⃗qrs →∞, which makes the result lesssensitive to the behaviour of the nuclear wave function at small internucleonseparation.† To second order in small momenta, the pion-nuclear scatteringlength isaab = 1 + mπ/mN1 + mπ/AmNXra(r)ab + a[4(a)]ab+ a[4(b)]ab+ a[4(c)]ab+ a[4(d,e)]ab(9)where a(r)ab is the pion scattering length on the r’th nucleon.This all becomes much simpler in two special cases. One is double charge-exchange scattering, π+ + N →π−+ N′, where the scattering lengths a[r]abas well as the corrections (6) and (8) vanish.

The other, on which we shallconcentrate here, is pion scattering on an isoscalar nucleus. Here t(r)a t(s)b+t(s)a t(r)bmay be replaced with23δab t(r) · t(s), and Eq.

(8) vanishes.Moreimportant, the contributions of the nominally leading terms in the pion-nucleon scattering lengths vanish, because they involve an expectation valueof Pr t(r) · t(π), which vanishes in any isoscalar nucleus. The first term in (9)†This cancellation was noted by Robilotta and Wilkin5in the case of pion-deuteronscattering.

They used a different definition of the pion field, so their results for diagrams4(b) and 4(c) were different from (6) and (7), but the sum of their results agrees withwhat would be found for this process from the sum of (6) and (7).11

arises only from “σ-term” corrections to the pion-nucleon scattering lengths,and is therefore relatively small, making it feasible to compare calculationsof the corrections considered here with experimental measurements of thepion-nuclear scattering lengths.This may be illustrated in the paradigmatic case of pion-deuteron scat-tering. To evaluate the two-body terms here we need to use isotopic spininvariance to derive the pion-neutron scattering lengths from measured val-ues of the π+p and π−p scattering lengths.

This is not entirely straightfor-ward, because we are interested here in the relatively small corrections tothe leading soft-pion results for which aπp + aπn = 0, and these correctionsarise in part from “sigma terms” proportional to u and d quark masses thatdo not even approximately conserve isospin. Fortunately to first order inquark masses the isospin violation in the sigma terms affects only processesinvolving at least one neutral pion,6 so that isospin relations can be used tocalculate aπn.

This gives the two-body terms in the π −d scattering lengthas71+mπ/mN1+mπ/md [aπp+aπn] = −(0.021±0.006)m−1π . Shifting to coordinate space,the remaining corrections are given by:a[4a)] = −m2ππ2F 4π(1 + mπ/md)Z ∞0(u2 + w2)rdr ,(10)anda[4(b,c)]=m2πg2A3π2F 4π(1 + mπ/md)14Z ∞0 (u2 + w2)1r −mπ2e−mπr dr−Z ∞0 uw√2 −w24!

1r + mπe−mπr dr#(11)12

where u and w are the s-wave and d-wave parts of the deuteron wave function,normalized so thatZ ∞0 (u2 + w2) dr = 1(12)The rescattering term (10) [but not (11)] has been previously consideredin the books of Eisenberg and Koltun and Ericson and Weise.7Becauseof the anomalously large radius of the deuteron, this term is considerablylarger than the remaining three-body term (11), so it should be calculatedincluding first-order corrections to the pion-nucleon scattering vertices inFigure 4(a). Fortunately these corrections can be taken from the measuredvalues of the scattering lengths.8In this way one finds that7a[4(a)] =−(0.026 ± 0.001)m−1π .

The remaining three-body terms (11) are calculated‡to be a[4(b,c)] = −0.0005m−1π(mostly arising from the interference between s-wave and d-wave parts of the wave function), in agreement with the numericalresult quoted in reference 5. This is small compared with the uncertainties inother terms, and so may be neglected here, though this may not be the casefor pion scattering on heavier nuclei.

This justifies the final theoretical resultof reference 7, aπd = −(0.050 ± 0.006)m−1π , which is in good agreement withthe experimental value −(0.056±0.009)m−1π . Although the use of chiral effec-tive Lagrangians has turned out here only to confirm previous calculations ofpion-deuteron scattering as well as nuclear binding, the systematic countingof momentum factors in chiral perturbation theory has proved its value in‡The calculation of the integrals in Eqs.

(10) and (11) was carried out by R. C.Mastroleo and U. van Kolck, using the Bonn wave function for the deuteron.13

explaining (as previous calculations did not explain) just why it is correct toconsider only certain graphs and certain terms in the effective Lagrangian.I am grateful for discussions with C. Dove, J. Friar, A. Gleeson, C.Ordo˜nez, U. van Kolck, and J. D. Walecka.14

References1. S. Weinberg, Physics Letters B 251 (1990) 288.2.

S. Weinberg, Nuclear Physics B 363 (1991) 3.3. S. Weinberg, Phys.

Rev. Lett.

18 (1967) 188 ; Phys. Rev.

166 (1968)1568 .4. C. Ordo˜nez and U. van Kolck, Texas preprint UTTG-01-92, to be pub-lished in Physics Letters B291 (1992).5.

M. R. Robilotta and C. Wilkin, J. Phys. G: Nucl.

Phys., 4 (1978)L115. Also see H. McManus and D. O. Riska, Phys.

Lett. 92B (1990)29.6.

S. Weinberg, in A Festschrift for I. I. Rabi, Transactions of the N. Y.Academy of Sciences 38 (1977) 185.7. J. M. Eisenberg and D. S. Koltun, Theory of Meson Interactions with Nuclei(Wiley-Interscience, New York, 1980); T. Ericson and W. Weise, Pions and Nuclei(Oxford University Press, Oxford, 1988).8.

V. M. Kolybasov and A. E. Kudryavtsev, Zh.Eksper.Teor.Fiz. (USSR), 63 (1972) 35; Sov.

Phys. JETP, 36 (1973) 18.15

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