-
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
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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
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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
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Bibliography
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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.
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[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).
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345 (1980) 471.
[33] K. Holinde, Nucl. Phys. A 415 (1984) 477.
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1828.
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(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.
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(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
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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
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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
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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