Chapter 2

More on the Nucleon-Nucleon Force

2.1 Meson Exchange in the Era of QCD

It should be of no surprise that the study of nuclear forces has attracted the interest ofmany theoretical as well as experimental physicists and has been the topic of vigorousresearch for more than fifty years. It not only represents the force between two nucleonsand the basis for an understanding of nuclei, but it is also a prominent example of a stronginteraction between particles.

Nowadays, almost everybody believes that quantum chromodynamics (QCD)is the theoryof strong interactions. Therefore, the nucleon-nucleon (NN) interaction is completelydetermined by the underlying dynamics of the fundamental constituents of hadrons, i.e.,quarks and gluons. Nevertheless, the meson exchange picture keeps its validity as asuitable effective description of the NN interaction in the low-energy region relevant innuclear physics for the following reasons: Due to asymptotic freedom, QCD in terms ofquarks and gluons can be treated perturbatively only for large momentum transfers, i.e., atdistances smaller than 0.2 fm or so, which typically occur in high-energy physics processes.At larger distances relevant in nuclear physics, a description of strong interaction processesin terms of the fundamental constituents, because of the highly non-perturbative characterof QCD in this region, becomes extremely complicated and is not possible, neither atpresent no in the foreseeable future.

Fortunately, at distances larger than the nucleon extension, which dominate nuclearphysics phenomena, color confinement dictates that nucleons can only interact by ex-changing colorless objects, i.e., just mesons. Only at smaller distances, at which the twonucleons overlap, genuinely new processes may occur involving explicit quark-gluon ex-

32

change. Due to the repulsive core of the NN interaction, however, both nucleons do notcome very close to each other unless the scattering energy is rather high. Thus, there isgood reason to believe these processes not to dominate the NN interaction for energiesrelevant in nuclear physics. Consequently, meson exchange (to be considered as a conve-nient, effective description of complicated quark-gluon processes) should remain a validconcept for deriving a realistic NN interaction, representing a reliable starting point fornuclear structure calculations.

2.2 What Do We Know Empirically About the Nu-

clear Force?

The basic qualitative features of the nuclear force are the following [1]:

a) Nuclear forces have a finite range, in contract to the Coulomb force. This can be easilydeduced from the saturation properties of heavy nuclei: Here, the binding energy pernucleon as well as the density are nearly constant. If the nuclear force were of long range,both quantities would increase with the nucleon number.

b) The nuclear force is attractive at intermediate ranges. The attractive character of thenuclear force is clearly established in nuclear binding. The range of this attraction canbe obtained from the central density of heavy nuclei, which is about 0.17 fm−3, giving toeach nucleon a volume of about 6 fm3. Therefore, in the interior of a heavy nucleus, theaverage distance between two nucleons is roughly 2 fm.

c) The nuclear force is repulsive at short distances. This is most easily seen in the empirical1S0 and

1D2 NN partial wave phase shifts (in the conventional notation2S+1LJ , where

L(J) denotes the orbital (total) angular momentum and S the total spin), which arededuced from NN scattering data (cross sections, polarization observables) by means ofa phase shift analysis. For small lab energies (up to about 250 MeV), the 1S0 phaseshift is positive, which corresponds to attraction. For high energies, it becomes negative(equivalent to repulsion), whereas the 1D2 phase shift stays positive up to about 800 MeV.This is consistent with a repulsion of short range since an S-state is sensitive to the innerpart of the force, whereas in a D-state the nucleons are kept apart by the centrifugalbarrier.

d) The nuclear force contains a tensor part. This is most clearly established in thepresence of a deuteron quadrupole moment and the so-called D/S ratio of the deuteronwave function [2].

33

e) The nuclear force contains a spin-orbit part. This is clearly seen in nuclear spectra.Furthermore, a quantitative description of triplet-P waves require a strong spin-orbitforce.

There are additional spin-dependent terms in the NN force, namely a spin-spin and aquadratic spin-orbit term. They are, however, of minor importance.

In fact, from general invariance principles (translation, Galilei, rotation, parity, time-reversal), the most general form of a nonrelativistic potential contains just these fiveterms, i.e., central (c), spin-spin(s), tensor (t), spin-orbit (LS) and quadratic spin-orbit(LL):

V =∑

i

Vi Oi (2.1)

with

Oc = 1

Os = ~σ1 · ~σ2Ot ≡ S12 ≡

3~σ · ~r ~σ2 · ~rr2

− ~σ1 · ~σ2OLS = ~L · ~SOLL = (~L · ~S)2 (2.2)

where ~S = 12(~σ1 + ~σ2).

The coefficients Vi can in general depend on the distance r, the relative momentum ~p2

and ~L2; they are completely undetermined. Phenomenological potentials like the Hamada-Johnson [3], the Reid [4] potential or its update by the Nijmegen group make an ansatzfor Vi(r), with a total of about 50 parameters, and fix these by adjusting them to the NNscattering data. However, such potentials cannot provide any basic understanding of theinteraction mechanism, and the parameters have no physical meaning.

2.3 Historical Background

The development of a microscopic theory of nuclear forces started around 1935 withYukawa’s fundamental hypothesis [5] that the nuclear force is generated by massive-particle exchange, leading to an interaction of the type e

−mαr

rwhere mα is the mass of the

34

exchange particle and r is the distance between two nucleons. This is quite analogous tothe electromagnetic case in which the interaction is known to be generated by (massless)photon exchange yielding the well-known Coulomb-potential being proportional to 1

r.

The original Yukawa idea of a scalar field interacting with nucleons was soon extendedto vectors (Proca [6]) and to pseudoscalar and pseudovector fields (Kemmer [7]). Theconsideration of a pseudoscalar field was dictated by the discovery of the quadrupole mo-ment of the deuteron [8], whose sign was correctly given by the exchange of an (isovector)pseudoscalar meson. Almost ten years later, in 1947, a pseudoscalar meson, the pion, wasindeed found [9].

The next period started around 1950, and again Japanese physicists initiated it. Taketani,Nakamura and Sasaki [10] (TNS) proposed to subdivide the range of the nuclear forceinto three regions: a ”classical” (long range, r > 2 fm), a ”dynamical” (intermediaterange, 1 fm < r < 2 fm) and a ”core” (short range, r ≤ 1 fm) region. The classicalregion is dominated by one-pion exchange (OPE). In the intermediate range, the two-pion exchange (TPE) is supposed to dominate, although heavier-meson exchange (tobe introduced later) become relevant, too. Finally, in the core region, many differentprocesses should play a role: multi-pion, heavy-meson and (according to our currentunderstanding) genuine quark-gluon exchange. Thus, in view ofQCD-inspired approachesto the nuclear force, this division is still most meaningful.

In the 1950’s, the one-pion exchange became well established as the long-range partof the nuclear force. Tremendous problems occurred, however, when the 2π exchangecontribution to the NN interaction was attacked. Apart from various uncertainties in theresults (the best known being those of Taketani, Machida and Onuma [11], and Bruecknerand Watson [12]), it was impossible to derive a sufficient spin-orbit force from the 2πexchange [13]. For that reason, Breit [14] in 1960 suggested to look for heavy vectorbosons in order to account for the empirically well-established, short-ranged spin-orbitforce. In fact, such mesons (ρ, ω) with a mass of nearly 800 MeV were soon discovered[15].

This led to the next step, namely the development of one-boson exchange (OBE) models.Their basic assumption is that multi-pion exchange can be well accounted for by theexchange of multi-pion resonances, i.e., that uncorrelated multi-pion exchange (apartfrom iterative contributions which are generated by the unitarizing equation) can beneglected. Such OBE models [16, 17, 18] (the contribution of the Bonn group is reviewedin Ref.. [17]), provide a relatively simple expression for the nuclear force; indeed, theycan account quantitatively for the empirical NN data using only very few parametersand thus convincingly demonstrate the importance of correlated interactions with two (ormore) pions.

35

In all OBE models, the intermediate-range attraction is generated by the exchange of ascalar-isoscalar boson with a mass around 600 MeV (representing a 2πS-wave resonance),which, although appearing in Particle Data Tables of the sixties, has not been confirmedempirically. This has to be considered as a serious drawback of OBE models. Therefore,the program of a realistic 2π-exchange calculation was taken up again; however, in contractto the fifties, with the goal to include not only the uncorrelated 2π-exchange contributioninvolving nucleon intermediate states, but also those involving nucleon excitations likethe ∆ isobar. Furthermore, from the experience with OBE models, it was clear from thebeginning that correlated 2π exchanges should be included, too.

In the dispersion-theoretic approach to the 2π exchange, empirical πN - (and ππ−) dataare used to derive the corresponding NN amplitude, with the help of causality, unitarityand crossing. Correlated as well as uncorrelated 2π exchange is automatically included.

Corresponding NN potentials are developed in the 1970’s, in particular by the StonyBrook [19] and the Paris [20] group, adding to the dispersion-theoretic 2π-exchange con-tribution OPE- and ω-exchange as well as some arbitrary phenomenological potential ofessentially short-range nature. In case of the Paris-potential, the final result is parame-terized by means of static Yukawa terms [21].

However, such a simplified representation of the nuclear force is probably insufficientin many areas of nuclear physics. For example, a consistent evaluation of three-bodyforces and meson-exchange current corrections to the electromagnetic properties of nucleirequires an explicit and consistent description of the NN interaction in terms of field-theoretic vertices. Also, a well-defined off-shell behavior and modifications of the nuclearforce when inserted into the many-body problem (e.g., Pauli-blocking of the 2π-exchangecontribution) are natural consequences of meson exchange. Only a field-theoretical ap-proach can account for these.

Work along the field theoretical line was taken up in the late 1960’s by Lomon and collab-orators [22, 23]. They evaluated the 2π-exchange Feynman diagrams with nucleons andrepresented their result in the framework of the relativistic three-dimensional reductionof the (four-dimensional) Bethe-Salpeter [24] equation, suggested by Blankenbecler andSugar [25]. In subsequent work [23], they also studied the correlated 2πS-wave contribu-tion. However, they did not include processes involving the ∆-isobar (an excited state ofthe nucleon with a mass of 1232 MeV and spin-isospin 3/2) in intermediate states, whichare known to contribute substantially to the nuclear force. Further, nonresonant 3π- and4π-exchange has to be considered since their range is about that of ω-exchange, which isincluded in all models.

Since the 1970’s the Bonn group has pursued a program that includes all relevant dia-grams in a field-theoretical model. In the early period [26], a relativistic three-dimensional

36

equation was used together with the principle of minimal relativity [27]; later the treat-ment has been based on relativistic, time-ordered perturbation theory [28]. A final statusof the Bonn model is described in detail in Ref. [29].

Let me finally mention attempts to derive the nucleon-nucleon interaction in the con-stituent quark model, based on one-gluon exchange. Corresponding calculations startedabout 15 years ago and many groups have been involved [30]. Indeed, certain qualitativefeatures of the short-range part of the NN interaction emerged, namely some inner repul-sion and spin-orbit force. However, all models of this kind create either too little or nointermediate-range attraction (which is sometimes artificially cured by adding a suitableattraction arising from scalar boson exchange). Note that pion exchange has to be addedin any case. Thus, the meson exchange concept for constructing the NN interactionclearly keeps its validity at the low and intermediate energies relevant to nuclear physics.

37

Bibliography

[1] For more details, see, e.g.: R. Wilson, The Nucleon-Nucleon Interaction (Inter-science, New York, 1963); M.J. Moravcsik, The Two-Nucleon Interaction (Claren-don Press, Oxford, 1963).

[2] T.E.O. Ericson and M. Rosa-Clot, Nucl. Phys. A 405 (1983) 497.

[3] T. Hamada and I.D. Johnson, Nucl. Phys. 34 (1962) 382.

[4] R.V. Reid, Ann. Phys. 50 (1968) 411.

[5] H. Yukawa, Proc. Phys. Math. Soc. Japan 17 (1935) 48.

[6] A. Proca, J. Phys. Radium 7 (1936) 347.

[7] N. Kemmer, Proc. Roy. Soc. (London) A166 (1938) 127.

[8] J. Kellog et al., Phys. Rev. 56 (1939) 728; 57 (1940) 677.

[9] C.M.G. Lattes, G.P.S. Occhiadini and C.F. Powell, Natura 160 (1947) 453, 486.

[10] M. Taketani, S. Nakamura and M. Sasaki, Prog. Theor. Phys. 6 (1951) 581.

[11] M. Taketani, S. Machida and S. Onuma, Prog. Theor. Phys. 6 (1951) 581.

[12] K.A. Brueckner and K.M. Watson, Phys. Rev. 90 (1953) 699; 92 (1953) 1023.

[13] N. Hoshizaki and S. Machida, Prog. Theor. Phys. 27 (1962) 288.

[14] G. Breit, Proc. Nat. Acad. Sci. (U.S.) 46 (1960) 746; Phys. Rev. 120 (1960) 287.

[15] B. Maglich et al., Phys. Rev. Lett. 7 (1961) 178; C. Alff et al., Phys. Rev. Lett. 9(1962) 322.

[16] R.A. Bryan and B.L. Scott, Phys. Rev. 135 (1964) B434.

[17] K. Erkelenz, Phys. Reports 13C (1974) 191.

38

[18] M.M. Nagels, T.A. Rijken and J.J. deSwart, Phys. Rev. D17 (1978) 768.

[19] A.D. Jackson, D.O. Riska and B. Verwest, Nucl. Phys. A 249 (1975) 397; G.E. Brownand A.D. Jackson, The Nucleon-Nucleon Interaction, (North-Holland, Amsterdam,1976).

[20] M. Lacombe et al., Phys. Rev. D12 (1975) 1495; R. Vinh Mau, in Mesons in Nuclei,Vol. I, eds. M. Rho and D. Wilkinson (North-Holland, Amsterdam, 1979), p. 151.

[21] M. Lacombe et al., Phys. Rev. C 21 (1980) 861.

[22] M.H. Partovi and E.L. Lomon, Phys. Rev. D 2 (1970) 1999.

[23] F. Partovi and E.L. Lomon, Phys. Rev. D 5 (1972) 1192; E.L. Lomon, Phys. Rev. C14 (1976) 2402; D22 (1980) 229.

[24] E.E. Salpeter and H.A. Bethe, Phys. Rev. 84 (1951) 1232.

[25] R. Blankenbecler and R. Sugar, Phys. Rev. 142 (1966) 2051.

[26] K. Holinde, K. Erkelenz and R. Alzetta, Nucl. Phys. A 194 (1972) 161; K. Holindeand R. Machleidt, Nucl. Phys. A 247 (1975) 495; A256 (1976) 479; A280 (1977)429.

[27] G.E. Brown, A.D. Jackson and T.T.S. Kuo, Nucl. Phys. A 133 (1969) 481.

[28] K. Kotthoff, K. Holinde, R. Machleidt and D. Schütte, Nucl. Phys. A 242 (1975)429; K. Holinde, R. Machleidt, M.R. Anastasio, A. Faessler and H. Müther, Phys.Rev. C 18 (1978) 870; C19 (1979) 948; C24 (1981) 1159; K. Holinde, Phys. Reports68C (1981) 121; K. Holinde and R. Machleidt, Nucl. Phys. A 372 (1981) 349; X.Bagnoud, K. Holinde and R. Machleidt, Phys. Rev. C 24 (1981) 1143; C29 (1984)1792.

[29] R. Machleidt, K. Holinde and Ch. Elster, Phys. Reports 149(1987) 1.

[30] see, e.g.: A. Faessler et al., Nucl. Phys. A 402 (1983) 555; M. Harvey, J. Le Tourneuxand B. Lorazo, Nucl. Phys. A 424 (1984) 428; K. Maltman and N. Isgur, Phys. Rev.D 29 (1984) 952; J. Suzuki and K.T. Hecht, Nucl. Phys. A 420 (1984) 525; F. Wangand C.W. Wong, Nucl. Phys. A 438 (1985) 620; S. Takeuchi, K. Shimizu and K.Yazaki, Nucl. Phys. A 504 (1989) 777.

[31] J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics, (McGraw-Hill, NewYork, 1964).

[32] J.W. Durso, A.D. Jackson and B.J. Verwest, Nucl. Phys. A 345 (1980) 471.

[33] K. Holinde, Nucl. Phys. A 415 (1984) 477.

39

[34] Ch. Elster et al., Phys. Rev. C 37 (1988) 1647; C38 (1988) 1828.

[35] A.W. Thomas, S. Théberge and G.A. Miller, Phys. Rev. D 24 (1981) 1216; A.W.Thomas, Adv. in Nucl. Phys. 13 (1983) 1.

[36] R. Büttgen, K. Holinde. A. Müller-Groeling, J. Speth and P. Wyborny, Nucl. Phys.A 506 (1990) 586; A. Müller-Groeling, K. Holinde and J. Speth, Nucl. Phys. A, inprint.

[37] B. Holzenkamp, K. Holinde and J. Speth, Nucl. Phys. A 500 (1989) 485.

[38] Th. Hippchen, K. Holinde and W. Plessas, Phys. Rev. C 39 (1989) 761; J. Haiden-bauer, Th. Hippchen, K. Holinde and J. Speth, Zeitschr. für Physik A 334 (1989)467.

2.4 Bethe-Salpeter Equation

In nonrelativistic scattering the equation describing the scattering process is the Lippmann-Schwinger equation. In a relativistic description, one starts from the Bethe-Salpeterequation (derived ≈ 1950), in which the relativistic scattering amplitude M is given byprocesses like

M

1’ 2’ 1’ 1’ 1’2’ 2’ 2’

1 2 1 1 12 2 2

1" 2" 1" 2"α

α

α α

α

’’

Figure 2.1.1 Relativistic scattering amplitude M

Analogously to the Lippmann-Schwinger equation, the Bethe-Salpeter equation can bewritten as integral equation

M = K + KGM

= K + KGK + KGKGK + · · · (2.3)

40

where K is the sum of all irreducible diagrams.

K = + + +

Figure 2.1.2: Irreducible kernel of the BS equation

K can be interpreted as a ”relativistic potential.” Take, e.g., the first term in Fig. 2.1.2as ”potential,” i.e.,

K(0)

=

Figure 2.1.3: Largest orderdiagram to K

Then the scattering amplitude M (0) takes the following form:

(0)=M + + +

Figure 2.1.4: Interaction of K(0)

M (0) = K(0) + K(0) GK(0) + K(0) GK(0) GK(0) + · · · (2.4)

41

Eq. (2.4) represents the so-called ”ladder approximation” of the BS equation. As it turnsout, it is not a good approximation to the BS equation, since the crossed diagrams areequally important as part of the irreducible kernel.

G is the two-nucleon propagator and describes the two nucleons in the intermediate states.Nucleons are spin-1

2particles, thus a starting point for obtaining the propagator is the

Dirac equation. Define Sα,β(x, y) with α, β = 1, 2, 3, 4 as propagator (4 × 4 matrix).Insertion into the Dirac equation leads to

(

iγµ∂

∂xµ− m1

)

αγ

Sγβ(x, y) = δ(4)(x− y) δαβ (2.5)

Ansatz:

Sγβ(x, y) =

∫

d4q

(2π)4e−iq(x−y) Sγβ(q) (2.6)

which gives with (2.5)

∫

d4q

(2π)4(γµqµ − m1)αγ Sγβ(q) e−iq(x−y) = δ(4)(x− y) δαβ

=

∫

d4q

(2pi)4e−iq(x−y) δαβ (2.7)

In the last relation, the definition of the δ-function was used. A comparison of thecoefficients gives

(γµqµ − m1)αγ Sγβ(q) = δαβ (2.8)

This leads to the Ansatz for the two-nucleon propagator in momentum space

Sγβ(q)(γµqµ + m1)γβ

q2 − m2 , (2.9)

42

where q2 = qµqµ. Proof by insertion into (2.8):

(γµqµ − m1)αγ (γνqν + m1)γβ = (γµγν qµqν − m21)αβ

=

(

1

2[γµγν + γνγµ] qµqν − m21

)

αβ

= (Gµν1 qµqν − m21)αβ= (qνqν − m2) (1)αβ= (q2 −m2) δαβ

which shows that the Ansatz for Sαβ was correct. Thus, the two nucleon propagation isgiven as

S(q) =γνqν + m1

q2 − m2 + iǫ =:1

γνqν − m1 + iǫ(2.10)

Since there are singularities at q2 = m2, S(q) has to be defined off-mass shell. Thus, oneobtains for the two-nucleon propagator:

G =

[

1

γµq(1)µ − m1 + iǫ

](1) [

1

γµq(2)µ − m1 + iǫ

](2)

=

[

γµq(1)µ + m1

q(1) 2 − m2 + iǫ

](1) [

γµq(2)µ + m1

q(2) 2 − m2 + iǫ

](2)

(2.11)

2.5 Relativistic Kinematics

2.5.1 Single Scattering

Consider the following diagram.

43

q q

q’q’

= q’-q∆

1 2

2

1 1

1

Figure 2.2.1: Singlescattering diagram

with

qµ1 = (q01, ~q1) ; q

µ′1 = (q

0′

1 , ~q′

1)

qµ2 = (q02, ~q2) ; q

µ′2 = (q

0′2 , ~q2

′) (2.12)

and let ∆ be an exchanged meson, e.g., at the vertices, one has 4-momentum conservation,i.e.,

∆ = q′1 − q1 = q2 − q′2 (2.13)and conservation of the total momentum P :

P ′ = q′1 + q′

2 = q1 + q2 = P (2.14)

We define center-of-mass momentum P µ and relative momentum qµ as

P µ = (p0, ~p) = qµ1 + qµ2

qµ = (q0, ~q) =1

2(qµ1 − qµ2 )

qµ′

= (q0′, ~q′) =1

2(q +µ′1 − qµ′2 ) (2.15)

44

From (2.14) and (2.15) follows

qµ1 =1

2P µ + qµ

qµ2 =1

2P µ − qµ

qµ′

1 =1

2P µ + qµ

′

qµ′

2 =1

2P µ − qµ′ (2.16)

and thus qµ′

1 − qµ′

1 = qµ′ − qµ. In the c.m. system, we have ~P = 0, thus

qµ1 =

(

1

2P 0 + q0, ~q

)

qµ2 =

(

1

2P 0 − q0, − ~q

)

qµ′

1 =

(

1

2P 0 + q0

′

, ~q ′)

qµ′

2 =

(

1

2P 0 − q′1, − ~q ′

)

(2.17)

and for the exchange particle

∆ = (q0′ − q0, ~q ′ − ~q) = (qµ′ − qµ) (2.18)

In the initial state, particles 1 and 2 are on-mass-shell (real particles), thus

(qµ1 )2 = m2 = (q01)

2 − (~q1)2

q01 =√

q21 + m2 := Eq1 = Eq

q02 =√

q22 + m2 := Eq2 = Eq (2.19)

and

P 0 = q01 + q02 = 2Eq

q0 =1

2(q01 − q02) = 0 (2.20)

Thus (suppressing the 4-indices):

q1 = (Eq, ~q) ; q2 = (Eq, − ~q)q′1 = (Eq + q

0′ , ~q ′) ; q′2 = (Eq − q′,0, − ~q ′) (2.21)

45

For the single scattering diagram in Figure 2.2.1, particles 1 and 2 are also on-mass-shell,i.e., q0′1 = q

0′2 =

√

q2 + m2 := Eq′, from which follows

P 0 = 2Eq′

q0′

= 0

q′1 = (Eq′, ~q′) = (Eq, ~q

′)

q′2 = (Eq′ − ~q ′) = (Eq, − ~q′) (2.22)

from which follows | ~q ′ | = | ~q |.

Thus, for the physical scattering, the single scattering diagram looks as follows:

(0, q’-q)

(E , q) (E , -q)

(E , q’) (E , -q’)q q

qqFigure 2.2.2: Kinematics ofthe single scattering

46

2.5.2 Double Scattering

kk - q

q q

k

1

1 11

1

11

2

2

2

q’ q’

q’ - k

Figure 2.2.3: Double scat-tering diagram

Here K1, K2 describe intermediate states. Again one has 4-momentum conservation atall vertices, i.e.,

P = q1 + q2 = k1 + k2 = q′

1 + q′

2 (2.23)

and

K =1

2(k1 − k2) . (2.24)

Let us consider the c.m. system. Again, in the initial state particles 1 and 2 are on-massshell, and in the final state both particles are on-mass shell. However, in the intermediatestates, the particles can have all possible independent values of k0, ~k, i.e., both particlesare in general off-mass shell in the intermediate states, i.e., are virtual states.

Thus, one has the following kinematic diagram:

47

(E , -q)(E , q)

(k , k-q)(E +k , k) (E -k , -k)

(E -q ’ ,-q’)(E +q ’ , q’)

q q

qq

q q0

000

(q ’-k , q’-k)0 0

0

Figure 2.2.4: Kinematics of the double scattering

In the c.m. system, the Bethe-Salpeter equation can be written as (~P = 0)

M(q′, q | P 0) = K(q′, q | P 0) + 1(2π)4

∫

d4k K(q′, k | P 0) G(k, P 0) M(k, q | P 0) .(2.25)

Here P 0 = 2Eq, i.e., in the initial state both particles are on-mass shell. The integrationimplies all possible intermediate states. The propagator G(k, P 0) is given as

G(k, P 0) =

[

1

γν (12P + k)ν − m1 + iǫ

](1) [

1

γν (12P − k)ν − m1 + iǫ

](2)

(c.m.)=

[

1

γ0 (12P 0 + k0)− ~γ · ~k − m1 + iǫ

](1)

×[

1

γ0 (12P 0 − k0) + ~γ · ~k − m1 + iǫ

](2)

. (2.26)

Eq. (2.25) is a 4-dimensional integral equation. Even after partial wave decomposition itis still 2-dimensional, thus more complicated to solve. As a technical detail, one assumesfor the solution that the particles in the final state are not on-mass shell and picks thenthe physical solution. (Compare solution of Lippmann-Schwinger equation.)

48

2.6 Structure of the Two-Nucleon Propagator

One has according to (2.10)

S(p) =γνpν + m1

p2 − m2 + iǫ , (2.27)

where in general p0 6= Ep =√

p2 +m2. Introduce projection operators on particles 4× 4matrices):

Λ+(p) =

2∑

i=1

u(i)(p)ū(i)(p) =1

2m(γ0Ep − ~γ · ~p +m1) (2.28)

and on anti-particles

Λ−(p) = −2

∑

i=1

v(i)(p) ~v(i)(p) =1

2m(−γ0Ep + ~γ · ~p + m1) (2.29)

where v(1)(p) = u(3)(−p) and v(2)(p) = u(4)(−p). [For notation, see Bjorken-Drell.] Theprojection operators have the following properties:

(Λ±)2 = Λ±

Λ+ + Λ− = 1

Λ+ · Λ− = 0 (2.30)

With

Λ−(−p) = −2

∑

i=1

v(i)(−p)v̄(i)(−p) = 12m

(−γ0Ep − ~γ · ~p + m1)

follows

Λ+(p) − Λ−(−p) =Epm

γ0

Λ+(p) + Λ+(−p) =1

m(−~γ · ~p + m1) (2.31)

49

Consider the numerator of (2.10):

γνpν + m1 = mp0

Ep· Epm

γ0 + m−~γ · ~p + m1

m

=mp0

Ep

[

Λ+(p) − Λ−(−p)]

+ m

[

Λ+(p) + Λ−(−p)]

=m

Ep

[

(p0 + Ep)Λ+(p) − (p0 − Ep)Λ−(−p)]

Thus the relativistic single nucleon propagator is given by

S(p) =γνpν + m1

p02 − E2p + iǫ=

m

Ep

[

Λ+(p)

p0 − Ep + iǫ− Λ−(−p)

p0 + Ep − iǫ

]

(2.32)

The poles are at p0 = Ep (particles) and p0 = −Ep (anti-particles). The two-nucleon

propagator contains particles and anti-particles, thus there are four different possibilitiesfor the intermediate states.

G(k, p0) =

[

1

γ0(12P 0 + k0)− ~γ · ~k −m1+ iǫ

](1) [

1

γ0(12P 0 + k0) + ~γ · ~k − m1 + iǫ

](2)

=m2

E2k

[

Λ+(k)12P 0 + k0 − Ek + iǫ

− Λ−(−k)12P 0 + k0 + Ek − iǫ

](1)

×[

Λ+(−k)12P 0 − k0 − Ek + iǫ

− Λ−(k)12P 0 − k0 + Ek − iǫ

](2)

=m2

E2k

2∑

i,j=1

[

u(i)(~k) ū(i)(~k)12P 0 + k0 − Ek + iǫ

+v(i)(−~k) v̄(i)(−~k)

12P 0 + k0 + Ek − iǫ

](1)

×[

u(j)(−~k) ū(j)(−~k)12P 0 − k0 − Ek + iǫ

+v(j)(~k) v̄(j)(~k)

12P 0 − k0 + Ek − iǫ

](2)

(2.33)

2.7 One-Pion Exchange

Consider the Born term:

50

(1/2 P +p , p) (1/2 P -p , p)

(1/2 P +p ’, p’) (1/2 P -p ’, p’)

(p’ - p)ΓΓ (1) (2)

0 0 0 0

0000

Figure 2.2.5: One meson exchange

with the vertices in pseudoscalar coupling Γ(i) =√4π gps iγ

5 τ(i)j . In this case, one obtains

for K(p′, p | P 0):

K̂(p′, p | P 0) = 1(2π)3

Γ(1) Γ(2)

(p′ − p)2 −m2ps + iǫ

= − 4π(2π)3

g2psγ5γ5

[

(p0′ − p0)2 − (~p ′ − ~p)2 −m2ps + iǫ

]−1

τ (1) · τ (2) .

(2.34)

Both G(k, P 0) and K(p′, p | P 0) are 16× 16 matrices. Thus, the Bethe-Salpeter equationis in its original form a (16×16) matrix equation. Consider matrix elements. For positiveenergy, one obtains

K++,++s1′s2′s1s2

(p′, p | P 0) = ū(p′, s′1) ū(−p′s′1) K̂(p′, p | P 0) u(p1s1)u(−p, s2)

=−4π(2π)3

g2psū(p′s′1) γ

5 ū(−p′1s′2) γ5 u(−p1s2)(p′0 − p0)2 − (~p ′ − ~p)2 −m2ps + iǫ

(2.35)

which is valid for free particles. The indices si indicates the spin degrees of freedom.Similarly, one obtains

K+−,++s′2s′2s1s2

(p′p | P 0) = ū(p′s′1) v̄(p′s′2)K̂(p′p | P 0) u(p, s1)u(p, s2) (2.36)

51

as well as all other matrix elements.

One can now write down the Bethe-Salpeter equation for matrix elements. If one isinterested in NN-scattering, one needs only the positive energy solutions in the initial andfinal state. However, in the intermediate data all solutions can appear. Consider

M++,++s′1s′2s1s2

(q′q | P 0) = K++,++s′1s′2s1s2

(q′1q | P 0)

+∑

s′′1s′′2

1

(2π)4

∫

d4km2

E2k

[

K++,++s1′s2′s

′′

1s′′2

(q′1k | P 0)G++(k;P 0)M++,++s′′1s′′2,s1s2

(k, q | P 0)

+ K++,+− G+− M+−,++ + K++,−+ G−+ M−+,++

+ K++,−− G−− M−−,++

]

(2.37)

with

Gαβ(k, P 0) =1

k0 − ωα11

k0 − ωβ2(2.38)

and

ω+1 = Ek −1

2P 0 − iǫ

ω−1 = −Ek −1

2P 0 + iǫ

ω+2 = −Ek +1

2P 0 + iǫ

ω−2 = Ek +1

2P 0 − iǫ (2.39)

Location of the poles:

52

k

2.8 Reductions of the Bethe-Salpeter Equation

The starting point is the original Bethe-Salpeter equation

M = K + KGM (2.41)

Introduce a simpler propagator g

M = W + WgM (2.42)

W = K + K(G− g)W (2.43)Here we only rewrote (2.41) into a system of 2 coupled equations. Proof by insertion

M = W + WgM

= K + KGW − KgW + (K + K(G− g)W ) gM= K + KGW + KG Wg M + KgM − KgW − Kg Wg M= K + KGM + KgM − Kg M= K + KGM

The two systems of equations (2.41) and (2.42), (2.49) are equivalent, and due to (2.43),the second system is as complicated as the first. The requirement for g should be that itis simple, and the difference (G− g) should be small, so that (2.43) can be simplified as

W ≈ K + K(G− g)K (2.44)

which is no longer an integral equation. K represented the sum of infinitely many diagrams

K = K(2) + K(4) + · · · (2.45)

same isW = W (2) + W (4) + · · · (2.46)

The first terms are identical, i.e., K(2) = W (2) ≡ one-pion exchange (e.g.). Then

W (4) = K(4) + K(2) GK(2) − K(2) gK(2) . (2.47)

For K(4) being the cross diagram, we obtain diagrammatically

54

W (4)

= + -G G G G g g

Figure 2.5.1 Diagrammatic representation of W (4)

IfK(4) is included (very important with γ5-coupling), thenW (4) is small if g is a reasonablechoice of propagator. Under these circumstances, one would expect a good convergenceof the expansion.

Determination of g:

The requirement for g should be:

(a) g should be simple.

(i) g should contain only particle states, i.e.,

g(k, P 0) = g′(k, P 0) Λ+(k) Λ+(−k) , (2.48)

then

M++,++s′1,s′

2,s1s2

(q′q | P 0) = W++,++s′1,s′

2,s1s2

(q′, q | P 0)

+∑

s′′1s′′2

1

(2π)4

∫

d4k W++,++s′1s′2,s′′

1s′′2

(q′, k | P 0) G′(k, P 0)

M++,++s′′4s′′2s1s2

(k, q | P 0) (2.49)

Eq. (2.49) is still four-dimensional. One would like to have a three-dimensionalequation, so a possible definition of g is

(ii)

g′(k, P 0) = δ(k0 − F (~k, P 0)) ḡ(~k, P 0) (2.50)

With this the scattering amplitude reads:

55

Ms′′1s′′2s1s2(~q

′, ~q | P 0) = W++,++s′1s′2s1s2

(~q ′, ~q | P 0)

=∑

s′′1s′′2

1

(2π)4

∫

d3k W++,++s1′s2s

′′

1s′′2

(~q,~k | P 0) ~g(~k, P 0)

M++,++s′′1s′′2s1s2

(~k~q | P 0) (2.51)

Eq. (2.51) has the derived three-dimensional form. One has to determine

F (~k, P 0) andG(~k, P 0), which should be chosen such that the above requirementon G are fulfilled.

(b) g should closely represent G, i.e. (G− g) should be small:

(i) ḡ has to be chosen such that M fulfills the relativistic unitarity condition.

(ii) Ḡ is not uniquely determined by (i). A further choice has to be made forF (K,P 0) in G′(K,P 0). In principle, there are infinitely many possible choices.

Several suggestions are given in the literature:

Blankenbecler-Sugar (BbS, 1966):

g′(k, P 0) = δ(k0) 2πm2

Ek

114P 0 2 − E2k + iǫ

(2.52)

= δ(k0) 2πm2

Ek

1

q2 − k2 + iǫ (2.53)

For the last relation P 0 = 2Eq was used. The BbS choice looks as the non-relativisticLippmann-Schwinger equation.

Erkelenz-Holinde (EH, 1973):

g′(k, P 0) = δ(k0 − Ek +1

2P 0) 2π

m2

Ek

1

q2 − k2 + iǫ (2.54)

The scattering amplitude is given then as

MBbS,EHs′1s′2s1s2

(~q ′, ~q | P 0) = WBbS,EHs′1s′2s1s2

(~q′, ~q | P 0)

+∑

s1′′s2′′

m

(2π)3

∫

d3km

EkWBbS,EH

s′1s′2s1′′s2′′

(~q ′, ~k′ | P 0)

× 1q2 − k2 + iǫ M

BbS,EH

s′′1s′′2s1s2

(~k, ~q | P 0) (2.55)

56

Due to the factor m/Ek, the equation (2.55) is still covariant. For e.g., the one-pionexchange from Sect. 2.4 one has the following consequences:

KBSps (p′p | P 0) = − 4π

(2π)3g2ps

ū(p′s′1) γ5 u(p1s1) ū(−p′s′2)γ5 ū(−ps2)

(p0′ − p0)2 − (~p ′ − ~p)2 −m2ps + iǫ(2.56)

BbS-Choice: P 0′ = P 0 =⇒ P 01 = P 02 and P ′01 = P ′02 , i.e., both particles in theintermediate state are equally for off-mass shell. Or in other words, there is only three-momentum transfer by the exchanged particles, no energy transfer, and (P 0′ − P 0)2 = 0.

EH-Choice: P 0′ = E ′p − 12 ; P 0 = Ep − 12P 0 =⇒ (P 0′ − P 0)2 = (E ′p − Ep)2. Thus aretardation is included in the meson propagator. Particle 1 is on-mass shell, particle 2 ingeneral not.

(1/2 P , p) (1/2 P , -p)

(1/2 P , p’) (1/2 P , -p’)

(0, p’-p)

(E , p) (E , -p)0 0

0 0

p p

p

p p

BbS EH

(E’ , p’) (E’ , -p’)

(E’ -E , p’-p)p

Figure 2.5.2: Illustrations of different choices for G′(k, P 0)

For simplicity, spin degrees of freedom shell be omitted in the following consideration.After making the favorite choice for G′(K,P 0), one has the following three-dimensionalintegral equation for M :

M(~q ′, ~q) = W (~q ′, ~q) +m

(2π)3

∫

d3k W (~q ′, ~k)

√

m

Ek

1

q2 − k2 + iǫ

√

m

EkM(~k, ~q) (2.57)

57

multiplication from the left with√

mEq′

and from the right with√

mEq

gives

T (~q ′, ~q) = V (~q ′, ~q) +m

(2π)3

∫

d3k V (~q′, ~k)1

q2 − k2 + iǫ T (~k, ~q) , (2.58)

which has the form of a nonrelativistic Lippmann-Schwinger equation with the potential

V (~q ′, ~q) =

√

m

Eq ′W (~q′, ~q)

√

m

Eq. (2.59)

The factors√

mEq

are often called ”minimal relativity factors,” and they have their justifi-

cation through the derivation (2.57). From the t-matrix, one can in the usual way proceedto the K-matrix. If the potential is real, this is the logical choice. One has the Heitlerequation

T (E) = K(E) − iπ K(E) δ(E − H0) T (E) (2.60)and obtains

K(E) = V + V P1

E −H0K(E) . (2.61)

In partial waves: Tℓ ≈ eiδℓ sin δℓ and Kℓ ≈ tan δℓ.

2.9 Time-Ordered Perturbation Theory

The technique that has historically been most useful in calculating the S-matrix is per-turbation theory, an expansion in powers of the interaction term V in a HamiltonianH = H0 + V , where H0 is the free Hamiltonian. The S-matrix can be written as

Sαβ = δ(β − α) − 2iπ δ(Eβ − Eα)T+µ (2.62)

withT+βα = 〈 φβ | V ψ+α 〉 . (2.63)

Here α, β characterize the initial, final states, | φβ〉 is a free state (solution to H0) and| ψ+α 〉 a scattering state satisfying a Lippmann-Schwinger equation

| ψ+α 〉 = φα +∫

dγTγαφγ

Eα − Eγ + iǫ. (2.64)

58

Multiplying with V and taking the scalar product with 〈φβ | gives the standard form ofthe operator L− S equation

T+βα = Vβα +

∫

dγVβγ T

+γα

Eα − Eγ + iǫ(2.65)

where Vβα ≡ 〈φβ | V | φα〉. The perturbation series for T+βα is obtained by iterating(2.65) as

T+βα = Vβα +

∫

dγVβα Vγα

Eα − Eγ + iǫ

+

∫

dγ dγ′VβγVγγ′ Vγ′α

(Eα − Eγ + iǫ) (Eα − E ′γ + iǫ)+ · · · (2.66)

This method of calculating the S-matrix is today called old-fashioned- perturbationtheory. Its obvious drawback is that the energy denominators obscure the underlyingLorentz invariance of the S-matrix. A rewritten version of (2.66) is known as time-dependent perturbation theory. This has the virtue of making the Lorentz structuremore obvious, while somewhat obscuring the contribution of the individual intermediatestates. The time-ordered perturbation expansion can be derived from the S-matrix in theform

S ≡ U(∞,−∞) ,where

U(τ, τ0) := eiH0τ e−iH(τ−τ0) e−iH0τ0 (2.67)

Differentiating (2.67) with respect to τ gives

id

dτU(τ, τ0) = V (τ) U(τ1τ0) (2.68)

whereV (t) = eiH0t V eoH0t , (2.69)

which corresponds to the definition of time dependence for an operator in the interactionpicture. Eq (2.68) together with the initial condition U(τ0, τ0) = 1 is satisfied by thesolution

U(τ, τ0) = 1 − i∫ τ

τ0

dt V (t) U(t, τ0) . (2.70)

By iterating this integral equation, one obtains an expansion for U(τ, τ0) in powers of V :

µ(τ, τ0) = 1 − i∫ τ

τ0

dt1 V (t1) + (−i)2∫ τ

τ0

dt1

∫ t1

τ0

dt2 V (t1) V (t2)

+ (−i)3∫ τ

τ0

dt1

∫ t1

τ0

dt2

∫ t2

τ0

dt3 V (t1) V (t2) V (t3) + · · · (2.71)

59

Setting τ ≡ ∞ and τ0 ≡ −∞ gives the perturbation expansion for the S-operator

S = 1 − i∫

∞

−∞

dt1 V (t1) + (−i)2∫

∞

∞

dt1

∫ t1

∞

dt2 V (T1) V (t2) + · · · (2.72)

This can also be derived directly from (2.66) by using the Fourier representation of theenergy factors in (2.66)

(Eα − Eγ + iǫ)−1 = −i∫

∞

0

dτ ei(Eα − Eγ)τ (2.73)

with the understanding that such integrals are to be evaluated by inserting a convergencefactor eǫτ in the integrand with ǫ→ 0+.

One can rewrite (2.72) in a way that proves very useful in carrying out manifestly Lorentz-covariant calculations. For this, define the time-ordered product of any time-dependentoperators as the product with factors arranged so that the one with the latest timeargument is placed leftmost, the next latest next to the leftmost, etc.

T{V (t)} = V (t) (2.74)T{V (t1) V (t2)} = θ(t1 − t2) V (t1) V (t2) + θ(t2 − t1) V (t2) V (t1) , (2.75)

where θ(τ) is the step function equal +1 for τ > 0, zero for τ < 0. Then the time-orderedproduct of n V ’s is the sum over all n! permutations of the V ’s, each of which gives thesame integral over all t1 · · · tn. Thus Eq. (2.72) may be written

S = 1 +

∞∑

n=1

(−i)n)n!

∫

∞

−∞

dt1 dt2 · · · dtn T{V (t1) · · · V (tn)} . (2.76)

This is sometimes known as Dyson series. If the V (t) at different times all commute,the series can be summed up as

S = exp

[

−i∫

∞

−∞

dt V (t)

]

. (2.77)

Of course, this is usually not the case – (2.76) does in general not converge.

One can now find a class of theories for which the S-matrix is manifestly Lorentz invariant.Since the elements of the S-matrix are the matrix elements of S taken between freestates, φα, φβ, the S-operator should commute with the operator U0(Λ, a), which producesLorentz transformations on free states. Equivalently, S must commute with the generatorsof µ0(Λ, a), namely H0, ~P0, ~J0, and ~K0.

60

To satisfy this requirement, assume that V (t) is given as

V (t) =

∫

d3x H(t, ~x) (2.78)

with H(x) being a scalar in the sense that

U0(Λ, a) H(x) U−10 (Λ, a) = H(Λx+ a) . (2.79)

Everything is now manifestly Lorentz invariant, except for the time ordering of the opera-tor product. The time ordering of two space-time points x1, x2 is Lorentz invariant unlessx1 − x2 is space-like, i.e., (x1 − x2)2 > 0, thus the time ordering in (2.79) introduces nospecial Lorentz frame if the H(x) commute at space-like or light-like separations

[H(x), H(x′)] = 0 for (x− x′)2 > 0 . (2.80)

61