Spin Dynamics Of qqq Wave Function
On Light Front In High Momentum
Limit Of QCD : Role Of qqq Force
A.N.Mitra ∗
244 Tagore Park, Delhi-110009, India
Abstract
The contribution of a spin-rich qqq force ( in conjunction with
pairwise qq forces) to the analytical structure of the qqq wave func-
tion is worked out in the high momentum regime of QCD where the
confining interaction may be ignored, so that the dominant effect is
Coulombic. A distinctive feature of this study is that the spin-rich
qqq force is generated by a ggg vertex ( a genuine part of the QCD
Lagrangian ) wherein the 3 radiating gluon lines end on as many quark
lines, giving rise to a (Mercedes-Benz type) Y -shaped diagram. The
dynamics is that of a Salpeter-like equation ( 3D support for the ker-
nel) formulated covariantly on the light front, a la Markov-Yukawa
Transversality Principle (MYTP) which warrants a 2-way intercon-
nection between the 3D and 4D Bethe-Salpeter (BSE) forms for 2
as well as 3 fermion quarks. With these ingredients, the differential
equation for the 3D wave function φ receives well-defined contribu-
tions from the qq and qqq forces. In particular a negative eigenvalue
of the spin operator iσ1.σ2 × σ3 which is an integral part of the qqq
force, causes a characteristic singularity in the differential equation,
signalling the dynamical effect of a spin-rich qqq force not yet consid-
ered in the literature. The potentially crucial role of this interesting
effect vis-a-vis the so-called ‘spin anomaly’ of the proton, is a subject
of considerable physical interest.
∗Email: (1)ganmitra@nde.vsnl.net.in; (2)anmitra@physics.du.ac.in
http://arxiv.org/abs/0704.1103v1
1 Introduction
The concept of a fundamental 3-body force (on par with a 2-body force)
is hard to realize in physics, leaving aside certain ad hoc representations of
higher order effects, for example those of ∆, N∗ resonances in hadron physics.
At the deeper quark-gluon level on the other hand, a truly 3-body qqq force
shows up as a folding of a ggg vertex ( a genuine part of the gluon Lagrangian
in QCD) with 3 distinct q̄gq vertices, so as to form a Y -shaped diagram (see
fig 1 below). Indeed a 3-body qqq force of this type, albeit for ‘scalar’ gluons,
was first suggested by Ernest Ma [1], when QCD was still in its infancy. [ A
similar representation is also possible for NNN interaction via ρρρ or σσσ
vertices, but was never in fashion in the literature [2]]. We note in passing
that a Y -shaped (Mercedes-Benz type) picture [3] was once considered in the
context of a preon model for quarks and leptons.
In the context of QCD as a Yang-Mills field, a ggg vertex has a momentum
representation of the form [4]
Wggg = −igsfabc[(k1 − k2)λδµν + (k2 − k3)µδνλ + (k3 − k1)νδλµ] (1.1)
where the 4-momenta emanating from the ggg vertex satisfy k1+k2+k3 = 0,
and fabc is the color factor. When this vertex is folded into 3 q̄gq vertices
of the respective forms gsū(p
1)iγµ{λa1/2}u(p1), and two similar terms, the
resultant qqq interaction matrix (suppressing the Dirac spinors for the 3
quarks) becomes
Vqqq =
[iγ(1).(k2 − k3)γ(2).γ(3) + {2}+ {3}]{λ1λ2λ3}/{k21k22k23} (1.2)
where ki = pi − p′i; λi are the color matrices which get contracted into the
corresponding scalar triple products in an obvious notation. [Note that the
flavour indices are absent here since the quark gluon interaction is flavour
blind].
This interaction will be considered in conjunction with 3 pairs of qq forces
within the framework of a Bethe-Salpeter type dynamics to be specified be-
low. Before proceeding in this direction, it is in order to explain a possi-
ble motivation behind the use of a direct qqq force with such a rich spin
dependence. Apart from the intrinsic beauty of this term, the immediate
provocation for its use comes from the issue of ”proton spin” which, after
making headlines about two decades ago, has come to the fore once again,
thanks to the progress of experimental techniques in polarized deep inelastic
scattering off polarized protons, and their variations thereof, which allow for
an experimental determination of certain key QCD parameters by relating
them to certain observable quantities emanating from external probes ; ( see
a recent review [5] for references and other details). On the other hand it
is also of considerable theoretical interest to determine these very quantities
directly from the intrinsic premises of QCD provided one has a ” good ”
qqq wave function to play with. Such a plea would have sounded rather
utopian in the early days of QCD when phenomenology was the order of
the day. Today however many aspects of QCD are understood well enough
to make such studies worthwhile by hindsight, with possible ramifications
beyond their educational value. The role of the direct 3-body force may be
seen in this light, while working for simplicity in the experimentally accessi-
ble regime of valence quarks. To that end this paper is specifically concerned
with the effect of the 3-body force (1.2) on the analytical structure of the
qqq wave function, while the formalism dealing with the contributions of the
various operators ( iγµγ5, 2-gluon effects, etc) to the proton spin is reserved
for a subsequent paper.
1.1 Theoretical Ingredients
In the valence quark regime, we need to consider a qqq system governed by
pairwise qq forces as well as a direct 3-quark force of the type (1.2). A further
simplification occurs in the high momentum regime where the effect of confin-
ing forces may be neglected, so that only coulombic forces need be kept track
of. We shall take the dynamics of a qqq hadron in the high − momentum
regime to be governed by a Bethe-Salpeter Equation (BSE) whose kernel
is a sum of three pairs of (coulombic) qq forces plus a single 3-body term
(1.2). Unfortunately the 4D form of the BSE is too general to be of prac-
tical value for qqq dynamics, so a better option is the Salpeter Equation [6]
representing its instantaneous form. And, except for its lack of covariance,
the Salpeter equation has the remarkable property of 3D-4D interlinkage, a
feature that had been present all along in the original formulation itself [6],
but had somehow remained hidden from view in the literature, until clarified
[7] in the context of a comprehensive two − tier BSE formalism developed
independently in a covariant manner [8]. (A practical significance of the
‘two-tier’ formalism is that the reduced 3D form is ideal for the determi-
nation of hadron mass spectra of both qq̄ and qqq types [9, 10], while the
reconstructed 4D form is convenient for the evaluation of transition ampli-
tudes [9] via Feynman techniques for loop diagrams; see also Munz et al
ref.(9) ). A covariant formulation of the BSE is centered around the hadron
4-momentum Pµ in accordance with the Markov-Yukawa Transversality Prin-
ciple (MYTP) [11, 12], which was shown to be a ‘gauge principle’ in disguise
[13]. It ensures that the interactions among the constituents be transverse
to the direction of Pµ. In the high momentum regime to be considered here,
the confining interaction has been ignored for simplicity, which leaves the
3D form of the BS dynamics inadequate for mass spectral determination, yet
its dynamical on the spin-structure of the wave function should be realistic
enough for dealing with the hadron spin in the high momentum limit.
Now to another vital element of the theory: Although the Salpeter Equa-
tion, as the instant form of BSE, admits of a covariant formulation in the
rest frame of the total hadron 4-momentum Pµ, it suffers from certain ill-
defined 4D loop integrals due to a ‘Lorentz-mismatch’ among the rest-frames
of the participating hadronic composites, resulting in time-like momentum
components in the exponential/gaussian factors associated with their vertex
functions. This is especially true of triangle loops, such as applicable to the
pion form factor and ρ − ππ coupling [14] where this disease causes unwar-
ranted ”complexities” in the amplitudes, while two-quark loops just escape
this pathology. For a possible remedy against this disease, without losing the
benefit of an MYTP - based 3D-4D interconnection, a promising candidate
is the light-front approach of Dirac [15] by virtue of its bigger (7) stabil-
ity group compared with 6 for the instant form theory. Its basic simplicity
was first noticed by Weinberg [16] who formulated the infinite momentum
frame towards the same end. A covariant generalization of MYTP on the
light front, requiring the use of two null 4-vectors nµ and ñµ that satisfy
n2 = ñ2 = 0 and n.ñ = 1, cures the ‘Lorentz mismatch’ disease noted above
[17]. The remaining problem of n- dependence of transition amplitudes is
solved through a simple ansatz of ‘Lorentz completion’ [17, 18], so as to yield
a Lorentz-invariant pion e.m. form factor in accord with experiment . A sim-
ilar approach had been considered by Carbonell et al [19] in the context of
the Kadychevsky-Karmanov formalism [20, 21], except for their missing out
on the second (dual) null-vector ñ which happens to be crucial for recovering
a Lorentz - invariant structure in a more natural way.
1.2 Plan of the Paper
The plan of the paper is based on an interlinked 3D-4D BSE formalism char-
acterized by a Lorentz-covariant 3D support for its kernel a la MYTP [11, 12],
adapted to the light front (LF) [17]. The full structure of the kernel is a sum
of 3 pairs of coulombic qq forces plus a qqq force, Eq.(1.2), whose 3D support
is implied by the fact that all internal momenta q be transverse to Pµ ,
viz., q̂µ = qµ − q.PPµ/P 2. However the propagators are left untouched in
their standard 4D forms. [The light front formulation requires a collinear
frame [17] which is further specified in Section 2 ]. The strategy now lies
in a step-wise reduction of the (Salpeter-like) 4D Master Equation involving
the actual (4D) fermionic wave function Ψ. Step A consists in expressing
Ψ in terms of an auxiliary (bosonic) 4D quantity Φ satisfying an equivalent
(bosonic) 4D (Salpeter-like) equation. Step B involves a ” Gordon reduction
” to eliminate the γµ matrices in favour of σµν matrices. Finally Step C con-
sists in a 3D reduction of this ‘bosonized’ Master Equation, and a subsequent
reconstruction of the original Ψ in 4D form by a suitable reversal of steps.
In the process a 3D scalar wave function φ is introduced which not only fa-
cilitates an explicit solution of the 3D (albeit fully covariant) Salpeter-like
equation but is also a key component of the (reconstructed) 4D fermionic
wave function Ψ. To facilitate the process of 3D-4D interlinkage, a Green’s
function approach is employed a la [22], from which it is straightforward to
derive the corresponding wave functions via the appropriate ‘pole’ limits.
The entire exercise involves a close correspondence between the (earlier) in-
stantaneous form [22] and the (later) LF form [17], so as to project only the
latter by making free use of the results of the former. A new element will be
a generalization of the earlier formalism [22] so as to include the effect of the
3-body force, Eq (1.2), on the structure of the qqq Green’s function which
will require a more elaborate strategy, keeping in view the relative strengths
of qq and qqq forces.
After a short account of the correspondence between the instant and LF
forms of the dynamics, Section 2 is devoted to Steps A and B for converting
the Master Equation for the actual 4D qqq wave function Ψ (fermionic) to
an equivalent 4D (bosonic) form involving Φ, with all γ-matrices eliminated
in favour of Pauli matrices, by exploiting the effective 3D support for the
qq and qqq kernels on the light-front, as described above. Section 3 de-
scribes the first part of Step C, viz., the use of Green′s- function method
for the 3D reduction of the Master Equation for the 4D Green’s function
G corresponding to Φ, resulting in an integral equation for the 3D Green’s
function Ĝ corresponding to phi, specialized to the large momentum regime
on the light-front. The second part of Step C, namely its reconstruction
back to the 4D quantity G a la [22], leading to a formal connection between
Φ and φ, is the subject of Section 4, now with the added complexity behind
the inclusion of the 3-body qqq force along with the (dominant) pairwise qq
forces. [ The subsequent reconstruction of the 4D fermionic form Ψ, follows
from the results of Section 2]. As a further aid to the understanding of the
3D-4D interconnection, Appendix A describes the complete procedure for a
prototype qq subsystem, whose results are freely used for justifying several
steps in Sections 3 and 4. In Section 5, the full 3D BSE for φ is set up in
a simplified form designed to check for the dynamical effect of spin present
in both the qq and qqq forces. on the structure of the differential equation
for φ (the dominant effect being of the latter ! ). Section 6 is devoted to
a short critique of the role of the Vqqq term on the analytic srtucture of the
qqq wave function, while the derivation of the requisite QCD parameters for
the baryon from the forward scattering amplitude off the iγµγ5 operator,
via the quark constituents, as well as a more elaborate 2-gluon contribution
to the proton spin, is reserved for a subsequent paper.
2 Salpeter Eq on LF : From Ψ To Φ
2.1 Master Eq : Instant Vs LF Forms of Dynamics
Since the first step in the formulation of a covariant Salpeter-like equation
on the light front (LF) is to establish a correspondence between the instant
and LF forms of the dynamics, we first recall some definitions [17] for the LF
quantities p± = p0±p3 defined covariantly as p+ = n.p
2 and p− = −ñ.p
while the perpendicular components continue to be denoted by p⊥ in both
notations. Now for a typical internal momentum qµ, the parallel component
P.qPµ/P
2 of the instant form translates in the LF form as q3µ = zPnnµ,
where Pn = P.ñ, and z = n.q/n.P . As a check, q̂
2 = q2
+ z2M2 which shows
that zM plays the role of the third component of q̂ on LF. Next, we collect
some of the more important definitions / results of the LF formalism [17]
q⊥ = q − qnn; q̂ = q⊥ + zPnn; z = q.n/P.n; qn = q.ñ; (2.1)
Pn = P.ñ;P.q = Pnq.n + P.nqn; q̂.ñ = P⊥.q⊥ = 0;
P.q̂ = Pnq.n; q̂
2 = q2
+M2z2;P 2 = −M2
For a qqq baryon, there are two internal momenta, each separately satisfying
the relations (2.1). Note that for any 4-vector A, A.n and −An correspond
to 1/
2 times the usual light front quantities A± = A0 ± Az respectively.
But since a physical amplitude must not depend on the orientation n, a
simple device termed Lorentz−completion via the collinear trick [17] yields
a Lorentz-invariant amplitude for a transition process with three external
lines P, P ′, P ′′(= P+P ′) as follows. Since collinearity implies P⊥.P
= 0, the
4-scalar product P.P ′ = P.nP ′n+PnP
′.n+P⊥.P
simplifies to P.nP ′n+PnP
Then ‘ Lorentz completion ’ simply amounts to reversing the last step via the
‘zero’ quantity P⊥.P
, so as to recover the Lorentz invariant quantity P.P ′ at
the end ! And as a practical simplification, one does not even have to use the
n, ñ symbols; it suffices to use the more familiar light-front components A±
for the covariant quantities
2[A.n,−An] respectively. For ready reference,
the precise correspondence between the instant and LF definitions of the
‘parallel (z)’ and ‘time-like (0) ’ components of the various 4- momenta for
a qqq baryon ( i = 1,2,3) [22, 23] :
piz; pi0 =
; p̂i ≡ {pi⊥, piz} (2.2)
The last part of Eq.(2.2) defines a covariant 3-vector on the LF that will
frequently appear in the reduced 3D BSE for the qqq proton. Our goal is
to write down ( and solve) the Master equation for three fermion quarks
complete with all internal d.o.f.’s, in the presence of both qq and direct qqq
forces which reads [8] :
Ψ(p1p2p3) =
SF (p1)SF (p2)g
d4q′12
(2π)4
γ(1)µ γ
ν Dµν(k12)Ψ(p
2, p3)
+SF (p1)SF (p2)SF (p3)
d4q′12d
(2π)8
VqqqΨ(p
3) (2.3)
where the definitions for the various momenta, and the phase conventions for
the quark propagators are those of [8], while the direct 3-quark interaction
Vqqq in the last term is new, and given by (1.2). The central problem is now
the reduction of this Master equation (2.3) through the three steps (A, B,
C) vide Sect (1.2), as outlined below.
2.2 Reduction of Master Eq from Ψ to Φ
Since permutation symmetry plays a crucial role for a qqq hadron for fermion
quarks, we define at the outset a pair of internal variables (ξ; η) with, say,
the index #3 as basis, as [22]
2ξ3 = p1 − p2;
6η3 = −2p3 + p1 + p2; P = p1 + p2 + p3 (2.4)
where the time-like and space- like parts of each are given by (2.2), and
the corresponding 3-vector defined as p̂i ≡ {pi⊥, piz}. Two identical sets of
momentum pairs ξ1, η1 and ξ2, η2 are similarly defined, but can be expressed
in terms of the set (2.4) via permutation symmetry. Now Step A, which is
designed to dispose of the fermion d.o.f.’s for a qqq system, consists in defining
an auxiliary scalar function Φ related to the actual BS wave function Ψ by
Ψ = Π123S
F i (−pi)Φ(pip2p3)W (P ) (2.5)
Here the quantity W (P ) is independent of the internal momenta but includes
the spin-cum-flavour wave functions χ, φ of the 3 quarks involved (see [?] :
W (P ) = [χ′φ′ + χ′′φ′′]/
2 (2.6)
where φ′, φ′′ are the standard flavour functions of mixed symmetry [25] [not to
be confused with the 3D wave function φ !], and χ′ ,χ′′ are the corresponding
relativistic spin functions. The latter may be defined either in terms of the
quark # indices as in Eqs (1.2) or (2.3), or sometimes more conveniently in
a common Dirac matrix space as [24, 26]
|χ′ >; |χ′′ >= [
M − iγ.P
[iγ5; iγ̂µ/
2]βγ ⊗ [[1; γ5γ̂µ]u(P )]α (2.7)
where the first factor is the βγ-element of a 4 x 4 matrix in the joint spin
space of the quark #s 1, 2 [26], and the second factor the α element of a 4
x 1 spinor in the spin space of quark # 3; C is a charge conjugation matrix
with the properties [27]
−γ̃µ = C−1γµC; γ̃5 = C−1γ5C;
and γ̂µ is the component of γµ orthogonal to Pµ. Finally, the representations
of the flavour functions φ′, φ′′ satisfy the following relations in the ”3” basis
< φ′′|1;~τ (3)|φ′′ >=< φ′|1;−
~τ (3)|φ′ >
2.3 Gordon Reduction on Vqq3 & Vqqq
At this stage we indicate the effect of Gordon reduction on the pairwise
kernels V (ξ̂iη̂i) and the 3-body kernel Vqqq, following the original treatment of
[29], (also quoted in [8] ). The strategy lies in a close scrutiny of Eqs.(2.3-2.5)
with a view to eliminate the Dirac matrices in favour of the Pauli matrices
σµν . To that end, we express the Dirac propagators SF (pi) as
iSF (pi) = (mq − iγ(i).pi)/∆i; ∆i = m2q + p2i
Now employing the notation
V (i)µ = (mq − iγ(i).pi)iγ(i)µ (2.8)
the result of Gordon reduction is expressed by the formula [29, 8]
V (1).V (2) = −M212−(q̂+q̂′)2+i(q̂+q̂′).(σ̂1+σ̂2)×(q̂−q̂′)−σ
ik (q−q′)j(q−q′)k
(2.9)
where we have used a mixed vector-tensor notation for the various 3-vectors
on the light-front in the sense of (2.2) and a short-hand notation q for q12. It
is now possible to identify the pairwise kernel arising from the first group of
terms on the rhs of Eq.(2. 3) after expressing it in LF notation of Eq.(2.2)
as follows:
Vqq3 ≡ V (ξ̂3, ξ̂′′3 ) = g2sV (1).V (2) ×
(q̂12 − q̂′′12)2
(2.10)
where we have included a net color factor of (−4/3) arising from a folding
of the factor F12 for each qq pair in a 3
∗ state, into λ3/2 for the spectator
quark # 3 in a 3 state; the corresponding gluon propagator has been taken
on the light-front, in the notation of (2.2). Similarly the 3-body kernel Vqqq
in Eq.(2.3) may be identified as a sum of 3 terms arising from a folding of
(1.2) with the 3 Dirac factors (mq − iγ(i).pi) in the second term of (2.3).
Vqqq =
(−4/3)g4s [V (1).(k̂2 − k̂3)× V (2).V (3) + {2}+ {3}]/{k̂21k̂22k̂23} (2.11)
using the notation V (i)µ of (2.8). The net effect of the (antisymmetric) com-
bination of the color matrices λ is represented by the factor (−4/3) sitting
in front; and ki = pi − p′i. The ‘bosonized’ form of Eq. (2.3) is now
Φ(p1p2p3) =
d4q′12
(2π)4
V (ξ̂3, ξ̂
3 )Φ(p
2, p3)
∆1∆2∆3
d4q′12d
(2π)8
VqqqΦ(p
3) (2.12)
Figure 1: (a) Pictorial view of 2- & 3- body interactions; (b) ‘Mercedez-Benz’
diagram for qqq-force
in terms of the structures (2.10) and (2.11) for the 2- and 3-body kernels
respectively. And the Gordon reduction formula (2.9 ) may be repeatedly
used to used to further simplify the structures of Vqq[Eq (2.10)] and Vqqq[ Eq.
(2.12)], which are collected as follows.
2.3.1 Further Reduction of Vqq3 & Vqqq
The 2-body and 3-body interactions are pictorially represented by fig 1(a)
where the vertices of the triangleABC stand for the position vectors (r1, r2, r3)
of the 3 quarks in configuration space, while the centroid O has the position
vector R = ((r1 + r2 + r3)/3. The corresponding momenta of the gluon
lines connecting them for pairwise qq interactions are precisely the quanti-
ties qij − q′ij that appear as arguments in Eq. (2.10), while the momenta
associated with the gluon lines connecting all the 3 quarks together, a la the
Y -shaped fig.(1b), are the quantities k̂i corresponding to the vectors (ri −R).
In the notation of Eq.(2.4), the former momenta are precisely ξ̂i, while the
latter are the dual variables η̂i. [This interpretation will prove useful in the
subsequent analysis].
The pairwise terms Vqqi were earlier considered in detail [8]. The ‘con-
vective’ terms are of the form (p1 + p2).(p
1 + p
2) which simplifies to −M212 ≈
−4M2/9, plus smaller terms. The spin-orbit terms are unimportant for the
ground state of the qqq proton, but the spin-spin terms survive for its ground
state S = 1/2; their angular average works out as :
σ̂i.σ̂j =
[Σ2 − 9]
where we have introduced the total spin operator Σ for the proton state, with
Σ2 = 3. Collecting these results the sum of the 3 pairwise interactions (2.10)
work out as
Vqq3 = g
∑ 4M2p3z
(q̂12 − q̂′′12)2
− 3p3z(Σ2 − 9)] (2.13)
We now consider the more interesting term Vqqq, Eq.(2.12). The numerator
can be broken up into 3 distinct groups : a) convective; b) spin-orbit; c) spin-
sp[in-spin. Using the notation 2p̄i = pi+p
i, the convective part vanishes (due
to antisymmetry with too few d.o.f.’s) :
4[p̄1.p̄2][2p̄3.(k1 − k2)] ⇒ 0
The spin-orbit part comes from two groups of terms, one of which vanishes:
2p̄1.(k1 − k2)[σ̂3.(¯̂p3 × k̂3) ⇒ 0
where a mixed notation is used for the Pauli matrices, light-faced tensor for
the 4D and hatted vector for the 3D forms. The other group gives rise to a
non-zero contribution as
4p̄1.p̄2[iσ
µν k3ν(k1µ − k2µ)] ≈ (−4iM2/9)
σ̂i.ξ̂3 × η̂3 (2.14)
in the notation of Eq.(2.4). Note the overall antisymmetry of the last factor
(independent of the index 3).
Finally the (totally antisymmetric) spin-spin-spin contribution SSS is
defined as :
SSS =
[iσ(1)µν k̂1ν(k̂2µ − k̂3µ)iσ
λρ k̂2ρiσ
λσ k̂3σ (2.15)
The simplification of this expression (which is elementary) is helped by the
identity σ(i)µν = ǫµνaσ̂
a . The next step is angular averaging over each of the
independent 3-vectors involved in each of the three terms above, namely,
< k̂ik̂j >= k̂
2δij/3. The resulting expression for SSS is
SSS =
{k̂21k̂22 k̂23}σ̂1.σ̂2 × σ̂3
Substituting for SSS in Eq (2.11) and ignoring the spin-orbit terms (for the
ground state of the proton) the net 3-body force is
Vqqq =
−8ig4s
(σ̂1.σ̂2 × σ̂3)
(2.16)
3 3D Reduction By Green’s FunctionMethod
We are now in a position to implement Step C of our program ( establishing a
3D-4D interconnection between the corresponding wave functions) , for which
it is convenient to employ the Green’s function approach [22]. Calling the
4D Green’s functions associated with Ψ and Φ by GF and GS respectively,
the connection between them analogously to Eq.(2.5) may be written as
GF (ξη; ξ
′η′) = W (P )⊗Π123S−1F i (−pi)GS(ξη; ξ′η′)Π123S−1F i (−p′i)W̄ (P ′) (3.1)
where we have indicated the 4-momentum arguments of the Green’s functions
involved, in a common S3 basis (ξ, η), and expressed the spin-flavour depen-
dence of GF as a matrix product implied by the notation W (P )⊗ W̄ (P ′).
Now to show the 3D-4D interconnection it is enough to work at the level
of the ‘scalar’ Green’s function GS ( relabelled as G for simplicity), since it is
trivial to include the spin-flavour d.o.f.’s in a matrix notation via (2.5) later.
The main steps for the scalar qqq Green’s function are shown next, while
Appendix A sketches the corresponding process for a typical qq subsystem,
from 3D reduction to 4D reconstruction, so as to serve as a simpler prototype
for the actual qqq case.
3.1 Reduction from 4D G(ξη; ξ′η′) to 3D Ĝ(ξ̂η̂; ξ̂′η̂′)
We shall now outline a Green’s function method to establish a 3D-4D inter-
linkage between the 4D Φ and the 3D φ, following a generalization of the
procedure developed some time ago [22], so as to include the effect of the
direct 3-body force Vqqq. To that end we shall rederive the qq denominator
functions Dij of [22] arising from single integrations over dξi0, to show that
they are exactly proportional to the 3D denominator function D123 arising
from a double integration over dξ0dη0 in the direct 3-body term to be consid-
ered here. [This is a big improvement over the earlier derivation [22] where
this property was missing, and helps pave the way for putting together the
effects of both 2- and 3-body forces within a common dynamical framework
The full 4D Green’s function for ‘scalar’ quarks is G(ξη; ξ′η′) (taking out
the c.m δ-fn), while its 3D counterpart Ĝ is [22]
Ĝ(ξ̂η̂; ξ̂′η̂′) =
dξ0dη0dξ
0G(ξη; ξ
′η′) (3.2)
where the time-like subscript ‘0’ should be read as the corresponding LF
component in the sense of Eq.(2.2). Note that both G and Ĝ are S3 sym-
metric, since the combination dξ0dη0 has this property. But the two hybrid
Green’s functions defined as [22]
G̃3ξ(ξ̂3η3; ξ̂
dξ30dξ
30G(ξη; ξ
′η′);
G̃3η(ξ3η̂3; ξ
dη30dη
30G(ξη; ξ
′η′) (3.3)
are not S3 symmetric (hence they are indexed), since the integration now
involves only one of the two ξ,η variables. Next, the 4D G(ξη; ξ′η′) satisfies
a BSE of the form
i(2π)4G(ξη; ξ′η′) =
d4ξ′′
4∆1∆2
V (ξ̂3, ξ̂
3 )G(ξ
3η3; ξ
d4ξ′′d4η′′
9(2π)4∆1∆2∆3
VqqqG(ξ
3η3; ξ
3) (3.4)
which is analogous to Eq. (2.12) for the corresponding wave functions, except
for a change of variables wherein the factors 1/4 and 1/9 in the first and
second groups of terms stem from the relations (2.4) among the corresponding
variables. It is understood that both these types of interaction have covariant
3D support, and they incorporate the effect of ‘Gordon reduction’ as outlined
in Sect 2.3 above. The 3D reduction of (3.4) is now achieved by integrating
it a la (3.2), leading to a structure of the form :
(2π)3Ĝ(ξ̂η̂; ξ̂′η̂′) =
d3ξ̂′′3V (ξ̂3, ξ̂
3 )Ĝ(ξ̂
3 ; ξ̂
3(2π)3D123
d3ξ′′d3η′′VqqqĜ(ξ
3 ; ξ
3) (3.5)
To explain the structure of certain factors, Eq (3.5) shows that we have two
types of 3D denominator functions Dij and D123, associated with pairwise
2-body and direct 3-body forces respectively. It is easily shown that they are
simply related to each other. To that end, Appendix A already shows the
structure of Dij :
D12 =
MD12+
; D12+ = 2M [ω
1⊥p2+ + ω
2⊥p1+ − P12−p1+p2+] (3.6)
where ω2i⊥ equals m
q + p
i⊥. Using the on-shellness (∆3 = 0) of the spectator
(#3) then gives P12− = P−− p3− which reduces further to P−−ω23⊥/p3+. Its
substitution back in (3.6) leads to the S3 symmetric result:
p3+D12+ ≡
D123 = 2
{p2+p3+ω21⊥} − 2p1+p2+p3+P− (3.7)
Here we have anticipated the structure of D123 associated with the Vqqq term
whose formal definition is
P 2+dq12−dp3−
4M2(2iπ)2∆1∆2∆3
(3.8)
where the (double) contour integrations in the indicated variables leads to
the desired result. The resultant 3D BSE for Ĝ is now
(2π)3D123Ĝ(ξ̂η̂; ξ̂
′η̂′) =
d3ξ̂′′3V (ξ̂3, ξ̂
3)Ĝ(ξ̂
3 ; ξ̂
3(2π)3
d3ξ′′d3η′′VqqqĜ(ξ
3 ; ξ
3) (3.9)
The structure of this equation reveals some interesting symmetries, when
(2.13) and (2.16) are substituted for Vqq3 and Vqqq respectively. Namely, in
the first group of terms the η variable is the spectator since the integration is
only over the ξ variable, while in the second group of terms, their relative roles
are interchanged with the ξ variable effectively a spectator ! This fact is not
immediately apparent because of the double integration involved, but a little
reflection shows that the absence of the ξ variable in (2.16) effectively ensures
its spectator status in the argument of the corresponding Green’s function.
We shall see this feature more clearly in the coordinate space representation
of the φ equation (corresponding to Eq (3.9) for Ĝ, to be considered in Section
5. But first we turn our attention to the reconstruction of the 4D Green’s
function from the 3D form Ĝ satisfying (3.9). To that end we note that
the qq and qqq group of terms in (3.9) need different strategies and are best
handled separately, one at a time, (temporarily) ignoring the presence of the
other.
4 Reconstruction of 4D Wave Function
4.1 Reconstruction of G with qq Forces Only
Considering first the qq group, we proceed exactly as in (A.10) of Appendix
A for the qq system. Namely, first express G̃3η, eq.(3.3), in terms of the 3D
Ĝ, using the notation of (2.2) and the result of (3.7):
G̃3η(ξ3η̂3; ξ
2iπp3z∆1∆2
Ĝ(ξ̂η̂; ξ̂′η̂′)
D′123
2iπp′3z∆
(4.1)
In a similar way the fully 4D G function is expressible in terms of the hybrid
function G̃3ξ as
G(ξη; ξ′η′) =
2iπp3z∆1∆2
G̃3ξ(ξ̂3η3; ξ̂
D′123
2iπp′3z∆
(4.2)
Now since the G̃3ξ function is not determined from qqq dynamics alone, we
invoke an ansatz similar to, but more symmetrical than, [22] :
G̃3ξ(ξ̂3η3; ξ̂
3) = Ĝ(ξ̂η̂; ξ̂
′η̂′)F (p3, p
3) (4.3)
where the balance of the p3 (spectator) dependence is in the (as yet unknown)
F function, subject to an explicit self-consistency check. To that end, try
the ( symmetrical) Lorentz-invariant form
F (p3, p
δ(∆3 −∆′3) (4.4)
and integrate both sides of (4.4) w.r.t. dp30dp
30, in the notation of (2.2),
to show that the consistency check is met with the mass shell value of the
spectator momentum :
4p3zp
Substitution from A3 in (4.4) then gives the symmetrical form
F (p3, p
3) = 4p3zp
δ(∆3 −∆′3)
2πi∆3
whence the 4D G-fn in terms of Ĝ via the sequence (4.3) and (4.2), works
out as
G(ξη; ξ′η′) =
Ĝ(ξ̂η̂; ξ̂′η̂′)
D′123
δ(∆3 −∆′3)
(2πi)5
(4.5)
4.2 Reconstruction of G with qqq Forces Only
Next we consider only the qqq group of terms for a reconstruction from Ĝ (
Eq (3.5)) to G (Eq (3.4)). This case is more akin to the qq case of Appendix
A in the sense that if we use the collective indices (ξ, η) ≡ ρ and (ξ′, η′) ≡ ρ′
for the initial and final state arguments of the G-function we can define two
kinds of G̃ as follows.
G̃(ρ̂; ρ′) =
dρ0G(ρ; ρ
′); G̃(ρ; ρ̂′) =
dρ′0G(ρ; ρ
′) (4.6)
where the integrals on the RHS are each of the double integral type, so that
the definitions of the denominator functions involved are given by Eq.(3.8).
Thus, following (A. 9), G̃ and Ĝ are connected as follows
G̃(ρ, ρ̂′) =
D123(ρ̂)
(2iπ)2∆1∆2∆3
Ĝ(ρ̂; ρ̂′) (4.7)
together with a second one with the roles of ρ, ρ′ interchanged. Continuing
exactly as in Appendix A, G of Eq (3.4) gets expressed in terms of G̃ on
the RHS, since the interaction Vqqq does not involve the variables ρ
0. Thence
another application of Eq.(4.7) for expressing G̃(ρ; ρ̂′) in terms of Ĝ finally
yields the desired connection:
G(ρ; ρ′) =
D123(ρ̂)
(2iπ)2∆1∆2∆3
Ĝ(ρ̂; ρ̂′)
D123(ρ̂
(2iπ)2∆′1∆
(4.8)
where the symbol ρ stands collectively for (ξ, η), as does ρ′. Note that in this
case the reconstruction of G in terms of of (the fully reduced) Ĝ is unique,
and does not require any extra ansatz like (4.4).
4.3 Putting Both Vqq & Vqqq Together
We have now two distinct types of 3D-4D interconnections valid for pure qq
and pure qqq forces respectively. But when both types of forces are present,
an exact interconnection is very difficult to derive. We shall therefore strive
for an approximate but sufficiently realistic solution based on the relative
strengths of the two forces, namely, an overwhelming preponderance of qq
over qqq forces. To give effect to this strategy, we first note that the 3D
level does not involve any approximation since Eq. (3.9) treats both types
of interaction on par. It is only at the 4D level of reconstruction that an
approximation is necessary for putting the two forces together. Now taking
account of the dominance of qq over qqq forces, the simplest choice is to
prefer the structure of Eq (4.5) over that of Eq. (4.8). And although the
ratio of the two 3D-4D interconversion jackets in (4.5) over (4.8) is a singular
quantity
δ(∆3 −∆′3)× (2πi)
it causes no harm for physical amplitudes since such singular quantities get
ironed out on integration over the various momenta [23]. With this un-
derstanding, Eq.(4.5) represents the reconstruction of the full 4D Green’s
function G(ξη; ξ′η′) in terms of the 3D quantity Ĝ(ξ̂η̂; ξ̂′η̂′) where the latter
now satisfies Eq (3.9) which includes the effect of both qq and qqq forces.
4.4 3D-4D Interconnection for Wave Functions
Finally the spectral representations of G(ξη; ξ′η′) and Ĝ(ξ̂η̂; ξ̂′η̂′) for the qqq
system, exactly on the lines of (A.12-A.13) for a qq subsystem near a bound
state pole P 2 = −M2 , are
G(ξη; ξ′η′) =
Φn(ξ, η)Φ
′, η′)/(P 2 +M2); (4.9)
and a similar equation for Ĝ vis-a-vis φ. These give the connection between
the 4D wave function Φ which satisfies Eq (2.12), and the 3D wave function
φ corresponding to the Green’s function Ĝ which satisfies (3.9). For purposes
of evaluating transition amplitudes via Feynman diagrams, it is convenient
to index Φ as Φ1 +Φ2 +Φ3, and a corresponding indexing for the associated
vertices as V = V1 + V2 + V3 , as in ref [22], so as to keep track of which
quark is involved in which vertex.
Φ3(ξ, η) ≡
∆1∆2∆3
φ(ξ̂, η̂)
δ(∆3)
(2πi)5/2
(4.10)
The δ-function in eq.(4.7) has nothing to do with any connectedness problem;
see ref.[22] for detailed reasons. Finally, the use of Eq.(2.5) with (4.10) yields
an explicit structure for the actual (fermionic) wave function Ψ as
Ψ(ξ, η) = Π123SF (pi)D123
[φ(ξ̂, η̂)
δ(∆3)∆3
(2πi)5/2
]×W (P ) (4.11)
5 Complete φ Equation In Coordinate Space
Our final task is to set up (and solve) the 3D BSE for φ which may be inferred
from (3.9) by making use of a spectral representation similar to (4.9), and
going to the pole P 2 = −M2 :
(2π)3D123φ(ξ̂, η̂) =
d3ξ̂′′3Vqq3φ(ξ̂
3 , η̂3)
3(2π)3
d3ξ′′d3η′′Vqqqφ(ξ̂
3 , η̂
3) (5.1)
where Vqq3 is given by (2.13); Vqqq by (2.16); and D123 by (3.7). To transform
this equation in coordinate space, define the combinations analogous to (2.4)
2s3 = r1 − r2;
6t3 = −2r3 + r1 + r2 (5.2)
Then a Fourier transform to the coordinate− space representation gives for
the pairwise terms Vqq3 the coulombic structure
V (2)(s) =
d3ξ̂′3
(2π)3
exp [iξ̂′3.s3]× [
∑ 8M2g2sp3z
27(ξ̂3 − ξ̂′3)2
(Σ2 − 9)p3z] (5.3)
which multiplies the coordinate space wave function φ(s, t) ( S3 symmetric
in its arguments). Similarly, the (S3 symmetric) double Fourier transform of
Vqqq is defined as
V (3)(s, t) =
−4ig4s
d3ξ̂′d3η̂′
3(2π)6
exp [iξ̂′.s+ iη̂′.t]×
(η̂3 − η̂′3)2
[σ̂1.(σ̂2× σ̂3)]
(5.4)
The integration in (5.3) involves a 3D δ-function δ3(ŝ3) ; and the double
integration in (5.4) additionally involves the factot 1/|t3| due to the η in-
tegration. The δ function being singular may be ‘regularized’ by using a
differential representation (acting on the scalar function φ(s, t)), with |ŝ3| =
|s3| :
δ3(ŝ3)
(2π)3)
φ(s, t) =⇒
4π|s3|
[∂2s3]φ(s, t) (5.5)
The result of integration in (5.3) is then expressed as
V (2)(s) =
8M2αsp3z
27|s3|
2 − 9)
27|s3|
] (5.6)
Next the 3-body term term (5.4) integrates, with the help of (5.5), to
[iσ̂1.(σ̂2 × σ̂3)]
Now the product 1/s3t3 can be approximately replaced by the S3 symmetric
expression 2/[s23 + t
3] which can be taken out of the summation sign (with
index ‘3’ dropped) to give
∂2s3 which in turn equals (3/2)(∂
s + ∂
t ), an S3
symmetric sum with index ‘3’ dropped again. This finally leads from (5.4)
to the S3 symmetric form of the 3-body term in coordinate space:
V (3)(s, t) = 4α2s
[iσ̂1.(σ̂2 × σ̂3)]
3(s2 + t2)
[∂2s + ∂
t ] (5.7)
For further manipulations it is convenient to replace the antisymmetric spin
operator A = [iσ̂1.(σ̂2 × σ̂3)] by one of its possible eigenvalues as follows:
Squaring this quantity yields in a simple way
A2 = −A− 15 + Σ2; Σ ≡ σ̂1 + σ̂2 + σ̂3
from which one obtains the successive results
A3 + A2 + 15A = Σ2A; A2(A+ 1)2 = [Σ2 − 15]2
The last equation yields the four solutions
A = −1
15− Σ2 − 1/4; A = −1
Σ2 − 15− 1/4 (5.8)
whose substitution in Eqs (5.4) and (5.7) summarises the full content of the
total spin effect of the Vqqq term. Finally, the denominator term D123 is a
differential operator in coordinate space :
D123/4 =
[(−m2q + ∂23⊥)∂1z∂2z]− iM∂1z∂2z∂3z (5.9)
Thus φ satisfies the following equation in coordinate space :
D123φ(s, t) = [V
(2)(s) + V (3)(s, t)]φ(s, t) (5.10)
where the various operators are given by (5.6) - (5.9).
5.1 Simplified Form of the φ Equation (5.10)
Since this paper is intended as a preliminary mathematical framework for
the dynamical effect of the spin terms, especially the spin-rich Vqqq term
on the spatial structure of the proton wave function, pending a full-fledged
study of the proton spin anomaly, we shall at this stage merely outline a
qualitative procedure to determine the nature of the solution of Eq. (5.10),
with particular reference to the possible role of the eigenvalues (5.8) of A, on
the solution of this differential equation. To that end, we shall make some
drastic simplifications, starting with the differential operator (5.9) whose
principal terms (up to second order ), in momentum space, work out as
D123 ≈
) + [
− 2m2q](ξ2z + η2z) +
(m2q −M2/9)
Further simplification arises with the ‘special’ value M = 3mq which then
yields a simple yet transparent expression in coordinate space :
D123 ≈
[−∂2s − ∂2t ] (5.11)
an operator with full rotational invariance and S3 symmetry in the 3D space
( on light-front) defined by Eq. (2.2). In a similar vein, the 2-body terms
(5.6) can be simplified by the replacements piz ≈ M/3, so as to yield
V (2)(s, t) =
2M3αs
s2 + t2
Mαs(Σ
2 − 9)
s2 + t2
[∂2s + ∂
t ] (5.12)
where we have made a further simplification based on certain standard in-
equalities [31]
s2 + t2
And the 3-body term (5.7) may be written more compactly in terms of of
the operator A with eigenvalues (5.8) as
V (3)(s, t) = α2s
3(s2 + t2)
[∂2s + ∂
t ] (5.13)
One has now a differential equation for (5.10), with the simplified operators
(5.11-13) which exhibit a 6D symmetry. Taking R2 = s2 + t2, the 6D Lapla-
cian for the ground state of the proton (with all angular d.o.f.’s dropped)
takes the simpler form
[4M2/9 +
Mαs(Σ
2 − 9)
4Aα2s
](∂2R +
∂R)φ(R) +
8M3αs
φ(R) = 0
which on rescaling with the dimensionless variable X = MR leads to the
simpler form
2 − 9)
Aα2s√
](∂2X +
∂X)φ+
φ = 0 (5.14)
5.2 Spin Effects of Vqq & Vqqq Terms on φ Singularity
The first thing to notice from the φ Equation, (5.14), is that the spin-
dependent parts of both Vqq and Vqqq appear in the multiplying factor with
the 6D Laplacian acting on φ, and are proportional to αs and α
s respectively,
with the Vqq term having a milder singularity (∼ X−1) than the Vqqq term
(∼ X−2). Further information on the singularity of the differential equa-
tion (at points other than R = 0) hinges on the nature of the eigenvalues of
the spin operator A which appears in above multiplying factor. Now these
eigenvalues are given by (5.8), two of which are complex for the state of the
proton (Σ2 = 3), but the other two are real. Of the real solutions the positive
eigenvalue does not yield a zero in this multiplying factor but the negative
eigenvalue (≈ −4) does give a zero for a real value of R at a point X20 between
X2 = 0 and X2 = ∞ :
X20 −
Aα2s√
= 0;X0 = +
[α2s/32−
Aα2s√
] (5.15)
where the value of Σ2 = 3 for the proton state has been substituted. Indeed
this zero ( for A ≈ −4) is a key element of the dynamical effect of the spin
- rich 3-body force term, although the spin effect of the 2-body Vqq term is
marginal.
To study this effect more quantitatively, we seek an approximate solution
of Eq. (5.14) in the neighbourhood of the zero at (5.15) for which a crude nu-
merical estimate suggests the following location. Taking a 3-flavour structure
αs = 2π/[9 lnM/Λ], with Λ ≈ 150MeV , one finds αs ≈ 0.39, whence
X0 ≈ 0.0689±X1; X1 ≡ 0.5841
This shows that the qq term gives ∼ 10% shift around the central value of
X1, which corresponds to R ≈ 0.12fm, and is almost entirely due to the spin
effect of the Vqqq term. Therefore in the spirit of this qualitative investigation,
it makes sense to drop this 10% effect, in which case Eq (5.14) simplifies to
](∂2X +
∂X)φ+
φ = 0; X1 = 0.584 (5.16)
To bring it nearer to a standard form, transform to the independent variable
z according to zX20 = X
0 −X2, which yields
z(1 − z)∂2zφ− 3z∂zφ− β
1− zφ = 0; β ≡ αs
X1 ≈ 0.058 (5.17)
This equation is almost (not quite) of the hypergeometric form but it can be
reduced to a standard one (with singularities located at z = 0, 1,∞) [32] by
exploiting the smallness ( β = 0.058 ) of the last term to replace it with a
constant ( with value corresponding to z = 1/2) :
z(1 − z)∂2zφ− 3z∂zφ−
φ = 0 (5.18)
An equivalent equation may be obtained with the transformation z = 1−x :
x(1 − x)∂2xφ+ 3(1− x)∂xφ−
φ = 0 (5.19)
where x = 1 corresponds to the location X = X1 of the singularity, as
indicated in Eq. (5.16). [ Note that this singularity corresponds to the
negative eigenvalue of A, thus reflecting the dynamical effect of the spin-rich
Vqqq term, while a positive or complex eigenvalue of A would result in the
disappearance of this singularity ]. The solution of Eq.(5.19) is then given
by [32]
φ = F (a, b|3|x); a + b = 2; ab =
(5.20)
with the entire machinery of hypergeometric functions available [32] for ex-
ploring its properties according to need. The scaled variable x is related to
the 6D distance R by MR = X1x where X1 ≈ 1.5αs is directly related to
the location of the singularity induced by the spin structure of the Y-shaped
3-body force . And the reconstruction of the full Bethe Salpeter wave func-
tion Ψ as a sum of 3 pieces Ψi is now only a matter of substituting (5.20) in
(4.11), whence the corresponding vertex functions Vi are immediately iden-
tified [22, 23] for specific transition amplitudes a la Feynman diagrams.
6 Retrospect : Critique Of The 3-Body Force
In retrospect, we have considered, in conjunction with pairwise qq forces, the
effect of a new type of 3-body force Vqqq, rich in spin content, on the ana-
lytical structure of the qqq wave function in the high momentum regime of
QCD where the confining interaction is unimportant, rendering the dominant
force Coulombic. As to the anatomy of this (spin-rich) Vqqq , we have taken
it to be generated by a ggg vertex ( a genuine part of the QCD Lagrangian
) wherein the 3 radiating gluon lines end on as many quark lines, giving rise
to a (Mercedes-Benz type) Y -shaped diagram, a la fig 1. From a physical
point of view it is natural to expect that such a spin-rich structure should
play a potentially crucial role in the so-called ‘spin anomaly’ of the proton,
a subject that seems once again to have raised its head in the context of new
polarized beam techniques now available [5] for resolving the issue experi-
mentally. With that end in view, our strategy has been to determine the
dynamical effect of the spin dependence of Vqq and Vqqq forces on the ana-
lytical structure of the internal 3D wave function φ. [It is emphasized that
this effect must be carefully distinguished from the ” kinematical ” effect of
spin, which manifests in several other ways, namely through the presence of
various γ matrices that appear in equations like (4.11) connecting φ to the
full 4D wave function Ψ]. Indeed we found in Section 5 that while the spin-
dynamical effect of 2-body forces is marginal, that of the spin-rich 3-body
force is quite pronounced, and shows up through the possible ” eigenvalues
”, Eq (5.7), of the spin operator (iσ1.σ2 × σ3) which is a part of Vqqq : only
a negative eigenvalue (there is only one !), induces a singularity in the dif-
ferential equation for φ, but not others. The resulting dynamical effect is
expressed by a hypergeometric function, Eq.(5.20), which gets folded into
the 4D BS amplitude Ψ via Eq.(4.11), thus fulfilling the original (limited)
motivation behind this study.
Unfortunately, the dynamical framework underlying the contents of this
paper has had a long history born out of the author’s long involvement with
the so-called Bethe-Salpeter Equation (BSE), often with extended periods of
stalemate (and frustration !), but mostly centred around the quest for a sat-
isfactory definition of probability within the BSE framework. Eventually it
became possible to settle for a toned down version of BSE, that of a Salpeter-
like equation ( 3D support for the kernel), which is amenable to a probability
interpretation at the 3D level. And its 4D features, although present in the
original formulation itself [6], had for decades remained hidden from view, but
were finally dug out [7] in the context of an independent approach designed to
explore a 3D-4D interconnection between the corresponding BS amplitudes
when the kernel has a 3D support [8]. Further, the lack of covariance in the
original formulation [6] was subsequently remedied via a special (instanta-
neous) frame of reference in which the composite hadron of 4-momentum Pµ
is at rest [33]. This result turned out to be in conformity with the Markov-
Yukawa Transversality Principle (MYTP) [11, 12], which happens to be a
‘gauge principle’ in disguise [13] : It ensures that the interactions among the
constituents be transverse to the direction of Pµ. Subsequent refinements
have been mostly technical , especially the use of Dirac’s light-front formula-
tion so as to extend the dynamical range of validity of the BS framework by
overcoming certain practical problems like ‘Lorentz mismatch disease’ [17]
associated with different vertices of a given Feynman diagram.
It therefore looked worthwhile to employ this old-fashioned formalism
in the present context of a new kind of 3-body force Vqqq, with enough de-
tails put in for a reasonably self-contained description without having to dig
frequently into the original sources. This has also given an opportunity to
make several refinements, especially the derivation of a common denomina-
tor function associated with both the qq and qqq forces and extending the
earlier formalism to accommodate the Vqqq force, for which this theoretical
framework seems to be well suited.
The next part of this programme involves the derivation of the requi-
site QCD parameters for the baryon spin, for which some key ingredients
are i) the forward scattering amplitude off the iγµγ5 operator, inserted at
the individual quark lines, and ii) a more elaborate 2-gluon contribution to
the proton spin, which is reserved for a subsequent paper within the same
formalism.
7 Acknowledgements
Several colleagues have contributed to the evolution of the BS formalism,
with and without active authorships, and the present paper also shares the
general acknowledgement. Yet two items stand out in the specific context
of this paper : First, the analysis in Section 5 owes its origin to a simple-
minded methodology due to his late Father, Jatindranath Mitra. Second,
the pedagogical techniques of qqq symmetry with several d.o.f.,s, which have
played a key role in this paper, are due to the late Mario Verde as described
in Hand Buch der Physik Vol 24, 1957. The author also acknowledges the
comments of Aalok Mishra which led to the use of the eigenvalues of the spin
operator A in Section 5. He is also grateful to Vineet Ghildyal for help with
figure 1.
Appendix A : qq Subsystem Formalism
Using the correspondence (2.2) of the text, most of the covariant instant form
results of [22] may be taken over to the present light-front situation [7]. In this
Appendix, we outline the structure of the 4D / 3D interconnection between
the corresponding Green’s functions for the qq sub-system as a prototype for
the actual qqq system considered in Section 2 of the text. The qq Green’s
functions satisfy the respective equations [22]:
(2π)4iG(q, q′;P ) =
d4q′′V (q̂, q̂′′)G(q′′, q′;P ); (A.1)
Ĝ(q̂, q̂′) =
dq0dq
0G(q, q
′;P ) (A.2)
Integrating both sides of (A.1) gives via (A.2), the 3D BSE for a bound state
(no inhomogeneous term !):
(2π)3D(q̂)Ĝ(q̂, q̂′) =
d3q̂′′V (q̂, q̂′′)Ĝ(q̂′′, q̂′) (A.3)
where the 3D denominator function D(q̂) is defined as
D(q̂)
(A.4)
leading (for general unequal mass kinematics) to [8]
D(q̂) =
D+(q̂); D+(q̂) = 2P+[q̂
λ(M2, m21, m
] (A.5)
where λ is the triangle function of its arguments. Now define the hybrid
Green’s functions [22]:
G̃(q̂, q′) =
dq0G(q, q
′;P ); G̃(q, q̂′) =
dq′0G(q, q
′;P ) (A.6)
Using (A.6) on the RHS of (A.1) gives
(2π)4iG(q, q′;P ) =
d3q̂′′V (q̂, q̂′′)G̃(q̂′′, q′) (A.7)
Integrating (A.1) w.r.t. dq′0 only, and using (2.7) again, gives
(2π)4iG̃(q, q̂′) =
d3q̂′′V (q̂, q̂′′)Ĝ(q̂′′, q̂′) (A.8)
From (A.8) and (A.3), G̃ and Ĝ are connected as:
G̃(q, q̂′) =
D(q̂)
2iπ∆1∆2
Ĝ(q̂, q̂′) (A.9)
Interchanging q and q′ in (A.9) gives the dual result
G̃(q̂, q′) =
D(q̂′)
2iπ∆′1∆
Ĝ(q̂, q̂′) (A.10)
Substitution in (A.7) gives the desired 3D-4D interconnection
G(q, q′;P ) =
D(q̂)
2iπ∆1∆2
Ĝ(q̂, q̂′;P )
D(q̂′)
2iπ∆′1∆
(A.11)
Next, spectral representations for the 4D and 3D G-fns in (A.11) give [22]
G(q, q′;P ) =
Φn(q;P )Φ
′;P )/(P 2 +M2); (A.12)
Ĝ(q̂, q̂′) =
φn(q̂)φ
′)/(P 2 +M2) (A.13)
where Φn and φn are 4D and 3D wave functions, so that their interconnection
(valid for any bound state n), is expressed by:
Γ(q̂) ≡ ∆1∆2Φ(q;P ) =
D(q̂)φ(q̂)
(A.14)
which tells us that the covariant vertex function Γ on the light-front is a
function of q̂ only [22] via its definition (2.2). This derivation is a prototype
for the qqq case of text.
References
[1] Ernest Ma, Phys. Rev.D12, 2105 (1975).
[2] B. McKellar, private communication ; 1994.
[3] A. N. Mitra, Phys Rev. D28, 1745 (1983).
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Press, 1978.
[5] S.D. Bass Rev. Mod. Phys. 77, 1257-1302 (2005).
[6] E.E. Salpeter, Phys.Rev. 87 (1952) 328.
[7] A. N. Mitra and B. M. Sodermark, Nucl. Phys. A 695, 328 (2001).
[8] S. Chakrabarty et al, Prog. Part. Nucl. Phys. 22, 43-180 (1989).
[9] N. N. Singh and A.N. Mitra, Phys. Rev.D38, 1454 (1988);
C.R. Munz et al, Nucl. Phys. A578, 397 (1994).
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[11] M.A. Markov, Sov. J. Phys. 3 (1940) 452.
[12] H. Yukawa, Phys.Rev. 771950) 219.
[13] J. Lukierski and M. Oziwicz, Phys. Lett.B69, 339 (1977).
[14] I. Santhanam et al, Intl J Mod Phys E2, 219 (1993)
[15] P.A.M. Dirac, Rev. Mod. Phys. 21 (1949) 392.
[16] S. Weinberg, Phys. Rev. 150 (1966) 1313
[17] A.N. Mitra, Phys. Lett.B463 (1999) 293.
[18] A.N. Mitra, Nucl. Phys. A684, 313c (2000)
[19] J. Carbonell et al, Phys. Rep.400C, 215 (1998)
[20] V. Kadychevsky Nucl. Phys. B6, 125 (1968)
[21] V.A. Karmanov, Nucl.Phys. B166, 378 (1980).
[22] A.N. Mitra Int J Mod Phys.A14, 4781 (1999).
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[24] A.N. Mitra and A. Mittal, Phys.Rev.D29, 1399 (1984).
[25] R.P. Feynman et al, Phys.Rev. D3, 2706 (1971)
[26] R. Blankenbecler et al, Nucl.Phys.12, 629 (1959)
[27] A.S. Davidov Quantum Mechanics, Pergamon Press Oxford, 1965.
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[29] A.N. Mitra, Zeits. f. Phys. Particles & Fields, C8, 25 (1981).
[30] A.N. Mitra and B.M. Sodermark, hep-ph/0104219
[31] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions,
Nat. Bur. of Standards 1965
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http://arxiv.org/abs/hep-ph/0104219
http://arxiv.org/abs/hep-ph/0104219
	 Introduction
	Theoretical Ingredients
	 Plan of the Paper
	 Salpeter Eq on LF : From  To 
	 Master Eq : Instant Vs LF Forms of Dynamics
	 Reduction of Master Eq from  to  
	Gordon Reduction on  Vqq3 & Vqqq
	 Further Reduction of Vqq3 & Vqqq 
	 3D Reduction By Green's Function Method
	 Reduction from 4D G(; ' ' ) to 3D "705EG("705E"705E;"705E'"705E') 
	 Reconstruction of 4D Wave Function
	Reconstruction of G with qq Forces Only
	 Reconstruction of G with qqq Forces Only
	 Putting Both Vqq & Vqqq Together
	3D-4D Interconnection for Wave Functions
	 Complete  Equation In Coordinate Space 
	 Simplified Form of the  Equation (5.10)
	 Spin Effects of Vqq & Vqqq  Terms on  Singularity 
	 Retrospect : Critique Of The 3-Body Force
	Acknowledgements

  The contribution of a spin-rich $qqq$ force (in conjunction with pairwise
$qq$ forces) to the analytical structure of the $qqq$ wave function is worked
out in the high momentum regime of QCD where the confining interaction may be
ignored, so that the dominant effect is $Coulombic$. A distinctive feature of
this study is that the spin-rich $qqq$ force is generated by a $ggg$ vertex (a
genuine part of the QCD Lagrangian) wherein the 3 radiating gluon lines end on
as many quark lines, giving rise to a (Mercedes-Benz type) $Y$-shaped diagram.
The dynamics is that of a Salpeter-like equation (3D support for the kernel)
formulated covariantly on the light front, a la Markov-Yukawa Transversality
Principle (MYTP) which warrants a 2-way interconnection between the 3D and 4D
Bethe-Salpeter (BSE) forms for 2 as well as 3 fermion quarks. With these
ingredients, the differential equation for the 3D wave function $\phi$ receives
well-defined contributions from the $qq$ and $qqq$ forces. In particular a
$negative$ eigenvalue of the spin operator $i \sigma_1.\sigma_2\times \sigma_3$
which is an integral part of the $qqq$ force, causes a characteristic
singularity in the differential equation, signalling the dynamical effect of a
spin-rich $qqq$ force not yet considered in the literature. The potentially
crucial role of this interesting effect vis-a-vis the so-called `spin anomaly'
of the proton, is a subject of considerable physical interest.

Introduction
The concept of a fundamental 3-body force (on par with a 2-body force)
is hard to realize in physics, leaving aside certain ad hoc representations of
higher order effects, for example those of ∆, N∗ resonances in hadron physics.
At the deeper quark-gluon level on the other hand, a truly 3-body qqq force
shows up as a folding of a ggg vertex ( a genuine part of the gluon Lagrangian
in QCD) with 3 distinct q̄gq vertices, so as to form a Y -shaped diagram (see
fig 1 below). Indeed a 3-body qqq force of this type, albeit for ‘scalar’ gluons,
was first suggested by Ernest Ma [1], when QCD was still in its infancy. [ A
similar representation is also possible for NNN interaction via ρρρ or σσσ
vertices, but was never in fashion in the literature [2]]. We note in passing
that a Y -shaped (Mercedes-Benz type) picture [3] was once considered in the
context of a preon model for quarks and leptons.
In the context of QCD as a Yang-Mills field, a ggg vertex has a momentum
representation of the form [4]
Wggg = −igsfabc[(k1 − k2)λδµν + (k2 − k3)µδνλ + (k3 − k1)νδλµ] (1.1)
where the 4-momenta emanating from the ggg vertex satisfy k1+k2+k3 = 0,
and fabc is the color factor. When this vertex is folded into 3 q̄gq vertices
of the respective forms gsū(p
1)iγµ{λa1/2}u(p1), and two similar terms, the
resultant qqq interaction matrix (suppressing the Dirac spinors for the 3
quarks) becomes
Vqqq =
[iγ(1).(k2 − k3)γ(2).γ(3) + {2}+ {3}]{λ1λ2λ3}/{k21k22k23} (1.2)
where ki = pi − p′i; λi are the color matrices which get contracted into the
corresponding scalar triple products in an obvious notation. [Note that the
flavour indices are absent here since the quark gluon interaction is flavour
blind].
This interaction will be considered in conjunction with 3 pairs of qq forces
within the framework of a Bethe-Salpeter type dynamics to be specified be-
low. Before proceeding in this direction, it is in order to explain a possi-
ble motivation behind the use of a direct qqq force with such a rich spin
dependence. Apart from the intrinsic beauty of this term, the immediate
provocation for its use comes from the issue of ”proton spin” which, after
making headlines about two decades ago, has come to the fore once again,
thanks to the progress of experimental techniques in polarized deep inelastic
scattering off polarized protons, and their variations thereof, which allow for
an experimental determination of certain key QCD parameters by relating
them to certain observable quantities emanating from external probes ; ( see
a recent review [5] for references and other details). On the other hand it
is also of considerable theoretical interest to determine these very quantities
directly from the intrinsic premises of QCD provided one has a ” good ”
qqq wave function to play with. Such a plea would have sounded rather
utopian in the early days of QCD when phenomenology was the order of
the day. Today however many aspects of QCD are understood well enough
to make such studies worthwhile by hindsight, with possible ramifications
beyond their educational value. The role of the direct 3-body force may be
seen in this light, while working for simplicity in the experimentally accessi-
ble regime of valence quarks. To that end this paper is specifically concerned
with the effect of the 3-body force (1.2) on the analytical structure of the
qqq wave function, while the formalism dealing with the contributions of the
various operators ( iγµγ5, 2-gluon effects, etc) to the proton spin is reserved
for a subsequent paper.
1.1 Theoretical Ingredients
In the valence quark regime, we need to consider a qqq system governed by
pairwise qq forces as well as a direct 3-quark force of the type (1.2). A further
simplification occurs in the high momentum regime where the effect of confin-
ing forces may be neglected, so that only coulombic forces need be kept track
of. We shall take the dynamics of a qqq hadron in the high − momentum
regime to be governed by a Bethe-Salpeter Equation (BSE) whose kernel
is a sum of three pairs of (coulombic) qq forces plus a single 3-body term
(1.2). Unfortunately the 4D form of the BSE is too general to be of prac-
tical value for qqq dynamics, so a better option is the Salpeter Equation [6]
representing its instantaneous form. And, except for its lack of covariance,
the Salpeter equation has the remarkable property of 3D-4D interlinkage, a
feature that had been present all along in the original formulation itself [6],
but had somehow remained hidden from view in the literature, until clarified
[7] in the context of a comprehensive two − tier BSE formalism developed
independently in a covariant manner [8]. (A practical significance of the
‘two-tier’ formalism is that the reduced 3D form is ideal for the determi-
nation of hadron mass spectra of both qq̄ and qqq types [9, 10], while the
reconstructed 4D form is convenient for the evaluation of transition ampli-
tudes [9] via Feynman techniques for loop diagrams; see also Munz et al
ref.(9) ). A covariant formulation of the BSE is centered around the hadron
4-momentum Pµ in accordance with the Markov-Yukawa Transversality Prin-
ciple (MYTP) [11, 12], which was shown to be a ‘gauge principle’ in disguise
[13]. It ensures that the interactions among the constituents be transverse
to the direction of Pµ. In the high momentum regime to be considered here,
the confining interaction has been ignored for simplicity, which leaves the
3D form of the BS dynamics inadequate for mass spectral determination, yet
its dynamical on the spin-structure of the wave function should be realistic
enough for dealing with the hadron spin in the high momentum limit.
Now to another vital element of the theory: Although the Salpeter Equa-
tion, as the instant form of BSE, admits of a covariant formulation in the
rest frame of the total hadron 4-momentum Pµ, it suffers from certain ill-
defined 4D loop integrals due to a ‘Lorentz-mismatch’ among the rest-frames
of the participating hadronic composites, resulting in time-like momentum
components in the exponential/gaussian factors associated with their vertex
functions. This is especially true of triangle loops, such as applicable to the
pion form factor and ρ − ππ coupling [14] where this disease causes unwar-
ranted ”complexities” in the amplitudes, while two-quark loops just escape
this pathology. For a possible remedy against this disease, without losing the
benefit of an MYTP - based 3D-4D interconnection, a promising candidate
is the light-front approach of Dirac [15] by virtue of its bigger (7) stabil-
ity group compared with 6 for the instant form theory. Its basic simplicity
was first noticed by Weinberg [16] who formulated the infinite momentum
frame towards the same end. A covariant generalization of MYTP on the
light front, requiring the use of two null 4-vectors nµ and ñµ that satisfy
n2 = ñ2 = 0 and n.ñ = 1, cures the ‘Lorentz mismatch’ disease noted above
[17]. The remaining problem of n- dependence of transition amplitudes is
solved through a simple ansatz of ‘Lorentz completion’ [17, 18], so as to yield
a Lorentz-invariant pion e.m. form factor in accord with experiment . A sim-
ilar approach had been considered by Carbonell et al [19] in the context of
the Kadychevsky-Karmanov formalism [20, 21], except for their missing out
on the second (dual) null-vector ñ which happens to be crucial for recovering
a Lorentz - invariant structure in a more natural way.
1.2 Plan of the Paper
The plan of the paper is based on an interlinked 3D-4D BSE formalism char-
acterized by a Lorentz-covariant 3D support for its kernel a la MYTP [11, 12],
adapted to the light front (LF) [17]. The full structure of the kernel is a sum
of 3 pairs of coulombic qq forces plus a qqq force, Eq.(1.2), whose 3D support
is implied by the fact that all internal momenta q be transverse to Pµ ,
viz., q̂µ = qµ − q.PPµ/P 2. However the propagators are left untouched in
their standard 4D forms. [The light front formulation requires a collinear
frame [17] which is further specified in Section 2 ]. The strategy now lies
in a step-wise reduction of the (Salpeter-like) 4D Master Equation involving
the actual (4D) fermionic wave function Ψ. Step A consists in expressing
Ψ in terms of an auxiliary (bosonic) 4D quantity Φ satisfying an equivalent
(bosonic) 4D (Salpeter-like) equation. Step B involves a ” Gordon reduction
” to eliminate the γµ matrices in favour of σµν matrices. Finally Step C con-
sists in a 3D reduction of this ‘bosonized’ Master Equation, and a subsequent
reconstruction of the original Ψ in 4D form by a suitable reversal of steps.
In the process a 3D scalar wave function φ is introduced which not only fa-
cilitates an explicit solution of the 3D (albeit fully covariant) Salpeter-like
equation but is also a key component of the (reconstructed) 4D fermionic
wave function Ψ. To facilitate the process of 3D-4D interlinkage, a Green’s
function approach is employed a la [22], from which it is straightforward to
derive the corresponding wave functions via the appropriate ‘pole’ limits.
The entire exercise involves a close correspondence between the (earlier) in-
stantaneous form [22] and the (later) LF form [17], so as to project only the
latter by making free use of the results of the former. A new element will be
a generalization of the earlier formalism [22] so as to include the effect of the
3-body force, Eq (1.2), on the structure of the qqq Green’s function which
will require a more elaborate strategy, keeping in view the relative strengths
of qq and qqq forces.
After a short account of the correspondence between the instant and LF
forms of the dynamics, Section 2 is devoted to Steps A and B for converting
the Master Equation for the actual 4D qqq wave function Ψ (fermionic) to
an equivalent 4D (bosonic) form involving Φ, with all γ-matrices eliminated
in favour of Pauli matrices, by exploiting the effective 3D support for the
qq and qqq kernels on the light-front, as described above. Section 3 de-
scribes the first part of Step C, viz., the use of Green′s- function method
for the 3D reduction of the Master Equation for the 4D Green’s function
G corresponding to Φ, resulting in an integral equation for the 3D Green’s
function Ĝ corresponding to phi, specialized to the large momentum regime
on the light-front. The second part of Step C, namely its reconstruction
back to the 4D quantity G a la [22], leading to a formal connection between
Φ and φ, is the subject of Section 4, now with the added complexity behind
the inclusion of the 3-body qqq force along with the (dominant) pairwise qq
forces. [ The subsequent reconstruction of the 4D fermionic form Ψ, follows
from the results of Section 2]. As a further aid to the understanding of the
3D-4D interconnection, Appendix A describes the complete procedure for a
prototype qq subsystem, whose results are freely used for justifying several
steps in Sections 3 and 4. In Section 5, the full 3D BSE for φ is set up in
a simplified form designed to check for the dynamical effect of spin present
in both the qq and qqq forces. on the structure of the differential equation
for φ (the dominant effect being of the latter ! ). Section 6 is devoted to
a short critique of the role of the Vqqq term on the analytic srtucture of the
qqq wave function, while the derivation of the requisite QCD parameters for
the baryon from the forward scattering amplitude off the iγµγ5 operator,
via the quark constituents, as well as a more elaborate 2-gluon contribution
to the proton spin, is reserved for a subsequent paper.
2 Salpeter Eq on LF : From Ψ To Φ
2.1 Master Eq : Instant Vs LF Forms of Dynamics
Since the first step in the formulation of a covariant Salpeter-like equation
on the light front (LF) is to establish a correspondence between the instant
and LF forms of the dynamics, we first recall some definitions [17] for the LF
quantities p± = p0±p3 defined covariantly as p+ = n.p
2 and p− = −ñ.p
while the perpendicular components continue to be denoted by p⊥ in both
notations. Now for a typical internal momentum qµ, the parallel component
P.qPµ/P
2 of the instant form translates in the LF form as q3µ = zPnnµ,
where Pn = P.ñ, and z = n.q/n.P . As a check, q̂
2 = q2
+ z2M2 which shows
that zM plays the role of the third component of q̂ on LF. Next, we collect
some of the more important definitions / results of the LF formalism [17]
q⊥ = q − qnn; q̂ = q⊥ + zPnn; z = q.n/P.n; qn = q.ñ; (2.1)
Pn = P.ñ;P.q = Pnq.n + P.nqn; q̂.ñ = P⊥.q⊥ = 0;
P.q̂ = Pnq.n; q̂
2 = q2
+M2z2;P 2 = −M2
For a qqq baryon, there are two internal momenta, each separately satisfying
the relations (2.1). Note that for any 4-vector A, A.n and −An correspond
to 1/
2 times the usual light front quantities A± = A0 ± Az respectively.
But since a physical amplitude must not depend on the orientation n, a
simple device termed Lorentz−completion via the collinear trick [17] yields
a Lorentz-invariant amplitude for a transition process with three external
lines P, P ′, P ′′(= P+P ′) as follows. Since collinearity implies P⊥.P
= 0, the
4-scalar product P.P ′ = P.nP ′n+PnP
′.n+P⊥.P
simplifies to P.nP ′n+PnP
Then ‘ Lorentz completion ’ simply amounts to reversing the last step via the
‘zero’ quantity P⊥.P
, so as to recover the Lorentz invariant quantity P.P ′ at
the end ! And as a practical simplification, one does not even have to use the
n, ñ symbols; it suffices to use the more familiar light-front components A±
for the covariant quantities
2[A.n,−An] respectively. For ready reference,
the precise correspondence between the instant and LF definitions of the
‘parallel (z)’ and ‘time-like (0) ’ components of the various 4- momenta for
a qqq baryon ( i = 1,2,3) [22, 23] :
piz; pi0 =
; p̂i ≡ {pi⊥, piz} (2.2)
The last part of Eq.(2.2) defines a covariant 3-vector on the LF that will
frequently appear in the reduced 3D BSE for the qqq proton. Our goal is
to write down ( and solve) the Master equation for three fermion quarks
complete with all internal d.o.f.’s, in the presence of both qq and direct qqq
forces which reads [8] :
Ψ(p1p2p3) =
SF (p1)SF (p2)g
d4q′12
(2π)4
γ(1)µ γ
ν Dµν(k12)Ψ(p
2, p3)
+SF (p1)SF (p2)SF (p3)
d4q′12d
(2π)8
VqqqΨ(p
3) (2.3)
where the definitions for the various momenta, and the phase conventions for
the quark propagators are those of [8], while the direct 3-quark interaction
Vqqq in the last term is new, and given by (1.2). The central problem is now
the reduction of this Master equation (2.3) through the three steps (A, B,
C) vide Sect (1.2), as outlined below.
2.2 Reduction of Master Eq from Ψ to Φ
Since permutation symmetry plays a crucial role for a qqq hadron for fermion
quarks, we define at the outset a pair of internal variables (ξ; η) with, say,
the index #3 as basis, as [22]
2ξ3 = p1 − p2;
6η3 = −2p3 + p1 + p2; P = p1 + p2 + p3 (2.4)
where the time-like and space- like parts of each are given by (2.2), and
the corresponding 3-vector defined as p̂i ≡ {pi⊥, piz}. Two identical sets of
momentum pairs ξ1, η1 and ξ2, η2 are similarly defined, but can be expressed
in terms of the set (2.4) via permutation symmetry. Now Step A, which is
designed to dispose of the fermion d.o.f.’s for a qqq system, consists in defining
an auxiliary scalar function Φ related to the actual BS wave function Ψ by
Ψ = Π123S
F i (−pi)Φ(pip2p3)W (P ) (2.5)
Here the quantity W (P ) is independent of the internal momenta but includes
the spin-cum-flavour wave functions χ, φ of the 3 quarks involved (see [?] :
W (P ) = [χ′φ′ + χ′′φ′′]/
2 (2.6)
where φ′, φ′′ are the standard flavour functions of mixed symmetry [25] [not to
be confused with the 3D wave function φ !], and χ′ ,χ′′ are the corresponding
relativistic spin functions. The latter may be defined either in terms of the
quark # indices as in Eqs (1.2) or (2.3), or sometimes more conveniently in
a common Dirac matrix space as [24, 26]
|χ′ >; |χ′′ >= [
M − iγ.P
[iγ5; iγ̂µ/
2]βγ ⊗ [[1; γ5γ̂µ]u(P )]α (2.7)
where the first factor is the βγ-element of a 4 x 4 matrix in the joint spin
space of the quark #s 1, 2 [26], and the second factor the α element of a 4
x 1 spinor in the spin space of quark # 3; C is a charge conjugation matrix
with the properties [27]
−γ̃µ = C−1γµC; γ̃5 = C−1γ5C;
and γ̂µ is the component of γµ orthogonal to Pµ. Finally, the representations
of the flavour functions φ′, φ′′ satisfy the following relations in the ”3” basis
< φ′′|1;~τ (3)|φ′′ >=< φ′|1;−
~τ (3)|φ′ >
2.3 Gordon Reduction on Vqq3 & Vqqq
At this stage we indicate the effect of Gordon reduction on the pairwise
kernels V (ξ̂iη̂i) and the 3-body kernel Vqqq, following the original treatment of
[29], (also quoted in [8] ). The strategy lies in a close scrutiny of Eqs.(2.3-2.5)
with a view to eliminate the Dirac matrices in favour of the Pauli matrices
σµν . To that end, we express the Dirac propagators SF (pi) as
iSF (pi) = (mq − iγ(i).pi)/∆i; ∆i = m2q + p2i
Now employing the notation
V (i)µ = (mq − iγ(i).pi)iγ(i)µ (2.8)
the result of Gordon reduction is expressed by the formula [29, 8]
V (1).V (2) = −M212−(q̂+q̂′)2+i(q̂+q̂′).(σ̂1+σ̂2)×(q̂−q̂′)−σ
ik (q−q′)j(q−q′)k
(2.9)
where we have used a mixed vector-tensor notation for the various 3-vectors
on the light-front in the sense of (2.2) and a short-hand notation q for q12. It
is now possible to identify the pairwise kernel arising from the first group of
terms on the rhs of Eq.(2. 3) after expressing it in LF notation of Eq.(2.2)
as follows:
Vqq3 ≡ V (ξ̂3, ξ̂′′3 ) = g2sV (1).V (2) ×
(q̂12 − q̂′′12)2
(2.10)
where we have included a net color factor of (−4/3) arising from a folding
of the factor F12 for each qq pair in a 3
∗ state, into λ3/2 for the spectator
quark # 3 in a 3 state; the corresponding gluon propagator has been taken
on the light-front, in the notation of (2.2). Similarly the 3-body kernel Vqqq
in Eq.(2.3) may be identified as a sum of 3 terms arising from a folding of
(1.2) with the 3 Dirac factors (mq − iγ(i).pi) in the second term of (2.3).
Vqqq =
(−4/3)g4s [V (1).(k̂2 − k̂3)× V (2).V (3) + {2}+ {3}]/{k̂21k̂22k̂23} (2.11)
using the notation V (i)µ of (2.8). The net effect of the (antisymmetric) com-
bination of the color matrices λ is represented by the factor (−4/3) sitting
in front; and ki = pi − p′i. The ‘bosonized’ form of Eq. (2.3) is now
Φ(p1p2p3) =
d4q′12
(2π)4
V (ξ̂3, ξ̂
3 )Φ(p
2, p3)
∆1∆2∆3
d4q′12d
(2π)8
VqqqΦ(p
3) (2.12)
Figure 1: (a) Pictorial view of 2- & 3- body interactions; (b) ‘Mercedez-Benz’
diagram for qqq-force
in terms of the structures (2.10) and (2.11) for the 2- and 3-body kernels
respectively. And the Gordon reduction formula (2.9 ) may be repeatedly
used to used to further simplify the structures of Vqq[Eq (2.10)] and Vqqq[ Eq.
(2.12)], which are collected as follows.
2.3.1 Further Reduction of Vqq3 & Vqqq
The 2-body and 3-body interactions are pictorially represented by fig 1(a)
where the vertices of the triangleABC stand for the position vectors (r1, r2, r3)
of the 3 quarks in configuration space, while the centroid O has the position
vector R = ((r1 + r2 + r3)/3. The corresponding momenta of the gluon
lines connecting them for pairwise qq interactions are precisely the quanti-
ties qij − q′ij that appear as arguments in Eq. (2.10), while the momenta
associated with the gluon lines connecting all the 3 quarks together, a la the
Y -shaped fig.(1b), are the quantities k̂i corresponding to the vectors (ri −R).
In the notation of Eq.(2.4), the former momenta are precisely ξ̂i, while the
latter are the dual variables η̂i. [This interpretation will prove useful in the
subsequent analysis].
The pairwise terms Vqqi were earlier considered in detail [8]. The ‘con-
vective’ terms are of the form (p1 + p2).(p
1 + p
2) which simplifies to −M212 ≈
−4M2/9, plus smaller terms. The spin-orbit terms are unimportant for the
ground state of the qqq proton, but the spin-spin terms survive for its ground
state S = 1/2; their angular average works out as :
σ̂i.σ̂j =
[Σ2 − 9]
where we have introduced the total spin operator Σ for the proton state, with
Σ2 = 3. Collecting these results the sum of the 3 pairwise interactions (2.10)
work out as
Vqq3 = g
∑ 4M2p3z
(q̂12 − q̂′′12)2
− 3p3z(Σ2 − 9)] (2.13)
We now consider the more interesting term Vqqq, Eq.(2.12). The numerator
can be broken up into 3 distinct groups : a) convective; b) spin-orbit; c) spin-
sp[in-spin. Using the notation 2p̄i = pi+p
i, the convective part vanishes (due
to antisymmetry with too few d.o.f.’s) :
4[p̄1.p̄2][2p̄3.(k1 − k2)] ⇒ 0
The spin-orbit part comes from two groups of terms, one of which vanishes:
2p̄1.(k1 − k2)[σ̂3.(¯̂p3 × k̂3) ⇒ 0
where a mixed notation is used for the Pauli matrices, light-faced tensor for
the 4D and hatted vector for the 3D forms. The other group gives rise to a
non-zero contribution as
4p̄1.p̄2[iσ
µν k3ν(k1µ − k2µ)] ≈ (−4iM2/9)
σ̂i.ξ̂3 × η̂3 (2.14)
in the notation of Eq.(2.4). Note the overall antisymmetry of the last factor
(independent of the index 3).
Finally the (totally antisymmetric) spin-spin-spin contribution SSS is
defined as :
SSS =
[iσ(1)µν k̂1ν(k̂2µ − k̂3µ)iσ
λρ k̂2ρiσ
λσ k̂3σ (2.15)
The simplification of this expression (which is elementary) is helped by the
identity σ(i)µν = ǫµνaσ̂
a . The next step is angular averaging over each of the
independent 3-vectors involved in each of the three terms above, namely,
< k̂ik̂j >= k̂
2δij/3. The resulting expression for SSS is
SSS =
{k̂21k̂22 k̂23}σ̂1.σ̂2 × σ̂3
Substituting for SSS in Eq (2.11) and ignoring the spin-orbit terms (for the
ground state of the proton) the net 3-body force is
Vqqq =
−8ig4s
(σ̂1.σ̂2 × σ̂3)
(2.16)
3 3D Reduction By Green’s FunctionMethod
We are now in a position to implement Step C of our program ( establishing a
3D-4D interconnection between the corresponding wave functions) , for which
it is convenient to employ the Green’s function approach [22]. Calling the
4D Green’s functions associated with Ψ and Φ by GF and GS respectively,
the connection between them analogously to Eq.(2.5) may be written as
GF (ξη; ξ
′η′) = W (P )⊗Π123S−1F i (−pi)GS(ξη; ξ′η′)Π123S−1F i (−p′i)W̄ (P ′) (3.1)
where we have indicated the 4-momentum arguments of the Green’s functions
involved, in a common S3 basis (ξ, η), and expressed the spin-flavour depen-
dence of GF as a matrix product implied by the notation W (P )⊗ W̄ (P ′).
Now to show the 3D-4D interconnection it is enough to work at the level
of the ‘scalar’ Green’s function GS ( relabelled as G for simplicity), since it is
trivial to include the spin-flavour d.o.f.’s in a matrix notation via (2.5) later.
The main steps for the scalar qqq Green’s function are shown next, while
Appendix A sketches the corresponding process for a typical qq subsystem,
from 3D reduction to 4D reconstruction, so as to serve as a simpler prototype
for the actual qqq case.
3.1 Reduction from 4D G(ξη; ξ′η′) to 3D Ĝ(ξ̂η̂; ξ̂′η̂′)
We shall now outline a Green’s function method to establish a 3D-4D inter-
linkage between the 4D Φ and the 3D φ, following a generalization of the
procedure developed some time ago [22], so as to include the effect of the
direct 3-body force Vqqq. To that end we shall rederive the qq denominator
functions Dij of [22] arising from single integrations over dξi0, to show that
they are exactly proportional to the 3D denominator function D123 arising
from a double integration over dξ0dη0 in the direct 3-body term to be consid-
ered here. [This is a big improvement over the earlier derivation [22] where
this property was missing, and helps pave the way for putting together the
effects of both 2- and 3-body forces within a common dynamical framework
The full 4D Green’s function for ‘scalar’ quarks is G(ξη; ξ′η′) (taking out
the c.m δ-fn), while its 3D counterpart Ĝ is [22]
Ĝ(ξ̂η̂; ξ̂′η̂′) =
dξ0dη0dξ
0G(ξη; ξ
′η′) (3.2)
where the time-like subscript ‘0’ should be read as the corresponding LF
component in the sense of Eq.(2.2). Note that both G and Ĝ are S3 sym-
metric, since the combination dξ0dη0 has this property. But the two hybrid
Green’s functions defined as [22]
G̃3ξ(ξ̂3η3; ξ̂
dξ30dξ
30G(ξη; ξ
′η′);
G̃3η(ξ3η̂3; ξ
dη30dη
30G(ξη; ξ
′η′) (3.3)
are not S3 symmetric (hence they are indexed), since the integration now
involves only one of the two ξ,η variables. Next, the 4D G(ξη; ξ′η′) satisfies
a BSE of the form
i(2π)4G(ξη; ξ′η′) =
d4ξ′′
4∆1∆2
V (ξ̂3, ξ̂
3 )G(ξ
3η3; ξ
d4ξ′′d4η′′
9(2π)4∆1∆2∆3
VqqqG(ξ
3η3; ξ
3) (3.4)
which is analogous to Eq. (2.12) for the corresponding wave functions, except
for a change of variables wherein the factors 1/4 and 1/9 in the first and
second groups of terms stem from the relations (2.4) among the corresponding
variables. It is understood that both these types of interaction have covariant
3D support, and they incorporate the effect of ‘Gordon reduction’ as outlined
in Sect 2.3 above. The 3D reduction of (3.4) is now achieved by integrating
it a la (3.2), leading to a structure of the form :
(2π)3Ĝ(ξ̂η̂; ξ̂′η̂′) =
d3ξ̂′′3V (ξ̂3, ξ̂
3 )Ĝ(ξ̂
3 ; ξ̂
3(2π)3D123
d3ξ′′d3η′′VqqqĜ(ξ
3 ; ξ
3) (3.5)
To explain the structure of certain factors, Eq (3.5) shows that we have two
types of 3D denominator functions Dij and D123, associated with pairwise
2-body and direct 3-body forces respectively. It is easily shown that they are
simply related to each other. To that end, Appendix A already shows the
structure of Dij :
D12 =
MD12+
; D12+ = 2M [ω
1⊥p2+ + ω
2⊥p1+ − P12−p1+p2+] (3.6)
where ω2i⊥ equals m
q + p
i⊥. Using the on-shellness (∆3 = 0) of the spectator
(#3) then gives P12− = P−− p3− which reduces further to P−−ω23⊥/p3+. Its
substitution back in (3.6) leads to the S3 symmetric result:
p3+D12+ ≡
D123 = 2
{p2+p3+ω21⊥} − 2p1+p2+p3+P− (3.7)
Here we have anticipated the structure of D123 associated with the Vqqq term
whose formal definition is
P 2+dq12−dp3−
4M2(2iπ)2∆1∆2∆3
(3.8)
where the (double) contour integrations in the indicated variables leads to
the desired result. The resultant 3D BSE for Ĝ is now
(2π)3D123Ĝ(ξ̂η̂; ξ̂
′η̂′) =
d3ξ̂′′3V (ξ̂3, ξ̂
3)Ĝ(ξ̂
3 ; ξ̂
3(2π)3
d3ξ′′d3η′′VqqqĜ(ξ
3 ; ξ
3) (3.9)
The structure of this equation reveals some interesting symmetries, when
(2.13) and (2.16) are substituted for Vqq3 and Vqqq respectively. Namely, in
the first group of terms the η variable is the spectator since the integration is
only over the ξ variable, while in the second group of terms, their relative roles
are interchanged with the ξ variable effectively a spectator ! This fact is not
immediately apparent because of the double integration involved, but a little
reflection shows that the absence of the ξ variable in (2.16) effectively ensures
its spectator status in the argument of the corresponding Green’s function.
We shall see this feature more clearly in the coordinate space representation
of the φ equation (corresponding to Eq (3.9) for Ĝ, to be considered in Section
5. But first we turn our attention to the reconstruction of the 4D Green’s
function from the 3D form Ĝ satisfying (3.9). To that end we note that
the qq and qqq group of terms in (3.9) need different strategies and are best
handled separately, one at a time, (temporarily) ignoring the presence of the
other.
4 Reconstruction of 4D Wave Function
4.1 Reconstruction of G with qq Forces Only
Considering first the qq group, we proceed exactly as in (A.10) of Appendix
A for the qq system. Namely, first express G̃3η, eq.(3.3), in terms of the 3D
Ĝ, using the notation of (2.2) and the result of (3.7):
G̃3η(ξ3η̂3; ξ
2iπp3z∆1∆2
Ĝ(ξ̂η̂; ξ̂′η̂′)
D′123
2iπp′3z∆
(4.1)
In a similar way the fully 4D G function is expressible in terms of the hybrid
function G̃3ξ as
G(ξη; ξ′η′) =
2iπp3z∆1∆2
G̃3ξ(ξ̂3η3; ξ̂
D′123
2iπp′3z∆
(4.2)
Now since the G̃3ξ function is not determined from qqq dynamics alone, we
invoke an ansatz similar to, but more symmetrical than, [22] :
G̃3ξ(ξ̂3η3; ξ̂
3) = Ĝ(ξ̂η̂; ξ̂
′η̂′)F (p3, p
3) (4.3)
where the balance of the p3 (spectator) dependence is in the (as yet unknown)
F function, subject to an explicit self-consistency check. To that end, try
the ( symmetrical) Lorentz-invariant form
F (p3, p
δ(∆3 −∆′3) (4.4)
and integrate both sides of (4.4) w.r.t. dp30dp
30, in the notation of (2.2),
to show that the consistency check is met with the mass shell value of the
spectator momentum :
4p3zp
Substitution from A3 in (4.4) then gives the symmetrical form
F (p3, p
3) = 4p3zp
δ(∆3 −∆′3)
2πi∆3
whence the 4D G-fn in terms of Ĝ via the sequence (4.3) and (4.2), works
out as
G(ξη; ξ′η′) =
Ĝ(ξ̂η̂; ξ̂′η̂′)
D′123
δ(∆3 −∆′3)
(2πi)5
(4.5)
4.2 Reconstruction of G with qqq Forces Only
Next we consider only the qqq group of terms for a reconstruction from Ĝ (
Eq (3.5)) to G (Eq (3.4)). This case is more akin to the qq case of Appendix
A in the sense that if we use the collective indices (ξ, η) ≡ ρ and (ξ′, η′) ≡ ρ′
for the initial and final state arguments of the G-function we can define two
kinds of G̃ as follows.
G̃(ρ̂; ρ′) =
dρ0G(ρ; ρ
′); G̃(ρ; ρ̂′) =
dρ′0G(ρ; ρ
′) (4.6)
where the integrals on the RHS are each of the double integral type, so that
the definitions of the denominator functions involved are given by Eq.(3.8).
Thus, following (A. 9), G̃ and Ĝ are connected as follows
G̃(ρ, ρ̂′) =
D123(ρ̂)
(2iπ)2∆1∆2∆3
Ĝ(ρ̂; ρ̂′) (4.7)
together with a second one with the roles of ρ, ρ′ interchanged. Continuing
exactly as in Appendix A, G of Eq (3.4) gets expressed in terms of G̃ on
the RHS, since the interaction Vqqq does not involve the variables ρ
0. Thence
another application of Eq.(4.7) for expressing G̃(ρ; ρ̂′) in terms of Ĝ finally
yields the desired connection:
G(ρ; ρ′) =
D123(ρ̂)
(2iπ)2∆1∆2∆3
Ĝ(ρ̂; ρ̂′)
D123(ρ̂
(2iπ)2∆′1∆
(4.8)
where the symbol ρ stands collectively for (ξ, η), as does ρ′. Note that in this
case the reconstruction of G in terms of of (the fully reduced) Ĝ is unique,
and does not require any extra ansatz like (4.4).
4.3 Putting Both Vqq & Vqqq Together
We have now two distinct types of 3D-4D interconnections valid for pure qq
and pure qqq forces respectively. But when both types of forces are present,
an exact interconnection is very difficult to derive. We shall therefore strive
for an approximate but sufficiently realistic solution based on the relative
strengths of the two forces, namely, an overwhelming preponderance of qq
over qqq forces. To give effect to this strategy, we first note that the 3D
level does not involve any approximation since Eq. (3.9) treats both types
of interaction on par. It is only at the 4D level of reconstruction that an
approximation is necessary for putting the two forces together. Now taking
account of the dominance of qq over qqq forces, the simplest choice is to
prefer the structure of Eq (4.5) over that of Eq. (4.8). And although the
ratio of the two 3D-4D interconversion jackets in (4.5) over (4.8) is a singular
quantity
δ(∆3 −∆′3)× (2πi)
it causes no harm for physical amplitudes since such singular quantities get
ironed out on integration over the various momenta [23]. With this un-
derstanding, Eq.(4.5) represents the reconstruction of the full 4D Green’s
function G(ξη; ξ′η′) in terms of the 3D quantity Ĝ(ξ̂η̂; ξ̂′η̂′) where the latter
now satisfies Eq (3.9) which includes the effect of both qq and qqq forces.
4.4 3D-4D Interconnection for Wave Functions
Finally the spectral representations of G(ξη; ξ′η′) and Ĝ(ξ̂η̂; ξ̂′η̂′) for the qqq
system, exactly on the lines of (A.12-A.13) for a qq subsystem near a bound
state pole P 2 = −M2 , are
G(ξη; ξ′η′) =
Φn(ξ, η)Φ
′, η′)/(P 2 +M2); (4.9)
and a similar equation for Ĝ vis-a-vis φ. These give the connection between
the 4D wave function Φ which satisfies Eq (2.12), and the 3D wave function
φ corresponding to the Green’s function Ĝ which satisfies (3.9). For purposes
of evaluating transition amplitudes via Feynman diagrams, it is convenient
to index Φ as Φ1 +Φ2 +Φ3, and a corresponding indexing for the associated
vertices as V = V1 + V2 + V3 , as in ref [22], so as to keep track of which
quark is involved in which vertex.
Φ3(ξ, η) ≡
∆1∆2∆3
φ(ξ̂, η̂)
δ(∆3)
(2πi)5/2
(4.10)
The δ-function in eq.(4.7) has nothing to do with any connectedness problem;
see ref.[22] for detailed reasons. Finally, the use of Eq.(2.5) with (4.10) yields
an explicit structure for the actual (fermionic) wave function Ψ as
Ψ(ξ, η) = Π123SF (pi)D123
[φ(ξ̂, η̂)
δ(∆3)∆3
(2πi)5/2
]×W (P ) (4.11)
5 Complete φ Equation In Coordinate Space
Our final task is to set up (and solve) the 3D BSE for φ which may be inferred
from (3.9) by making use of a spectral representation similar to (4.9), and
going to the pole P 2 = −M2 :
(2π)3D123φ(ξ̂, η̂) =
d3ξ̂′′3Vqq3φ(ξ̂
3 , η̂3)
3(2π)3
d3ξ′′d3η′′Vqqqφ(ξ̂
3 , η̂
3) (5.1)
where Vqq3 is given by (2.13); Vqqq by (2.16); and D123 by (3.7). To transform
this equation in coordinate space, define the combinations analogous to (2.4)
2s3 = r1 − r2;
6t3 = −2r3 + r1 + r2 (5.2)
Then a Fourier transform to the coordinate− space representation gives for
the pairwise terms Vqq3 the coulombic structure
V (2)(s) =
d3ξ̂′3
(2π)3
exp [iξ̂′3.s3]× [
∑ 8M2g2sp3z
27(ξ̂3 − ξ̂′3)2
(Σ2 − 9)p3z] (5.3)
which multiplies the coordinate space wave function φ(s, t) ( S3 symmetric
in its arguments). Similarly, the (S3 symmetric) double Fourier transform of
Vqqq is defined as
V (3)(s, t) =
−4ig4s
d3ξ̂′d3η̂′
3(2π)6
exp [iξ̂′.s+ iη̂′.t]×
(η̂3 − η̂′3)2
[σ̂1.(σ̂2× σ̂3)]
(5.4)
The integration in (5.3) involves a 3D δ-function δ3(ŝ3) ; and the double
integration in (5.4) additionally involves the factot 1/|t3| due to the η in-
tegration. The δ function being singular may be ‘regularized’ by using a
differential representation (acting on the scalar function φ(s, t)), with |ŝ3| =
|s3| :
δ3(ŝ3)
(2π)3)
φ(s, t) =⇒
4π|s3|
[∂2s3]φ(s, t) (5.5)
The result of integration in (5.3) is then expressed as
V (2)(s) =
8M2αsp3z
27|s3|
2 − 9)
27|s3|
] (5.6)
Next the 3-body term term (5.4) integrates, with the help of (5.5), to
[iσ̂1.(σ̂2 × σ̂3)]
Now the product 1/s3t3 can be approximately replaced by the S3 symmetric
expression 2/[s23 + t
3] which can be taken out of the summation sign (with
index ‘3’ dropped) to give
∂2s3 which in turn equals (3/2)(∂
s + ∂
t ), an S3
symmetric sum with index ‘3’ dropped again. This finally leads from (5.4)
to the S3 symmetric form of the 3-body term in coordinate space:
V (3)(s, t) = 4α2s
[iσ̂1.(σ̂2 × σ̂3)]
3(s2 + t2)
[∂2s + ∂
t ] (5.7)
For further manipulations it is convenient to replace the antisymmetric spin
operator A = [iσ̂1.(σ̂2 × σ̂3)] by one of its possible eigenvalues as follows:
Squaring this quantity yields in a simple way
A2 = −A− 15 + Σ2; Σ ≡ σ̂1 + σ̂2 + σ̂3
from which one obtains the successive results
A3 + A2 + 15A = Σ2A; A2(A+ 1)2 = [Σ2 − 15]2
The last equation yields the four solutions
A = −1
15− Σ2 − 1/4; A = −1
Σ2 − 15− 1/4 (5.8)
whose substitution in Eqs (5.4) and (5.7) summarises the full content of the
total spin effect of the Vqqq term. Finally, the denominator term D123 is a
differential operator in coordinate space :
D123/4 =
[(−m2q + ∂23⊥)∂1z∂2z]− iM∂1z∂2z∂3z (5.9)
Thus φ satisfies the following equation in coordinate space :
D123φ(s, t) = [V
(2)(s) + V (3)(s, t)]φ(s, t) (5.10)
where the various operators are given by (5.6) - (5.9).
5.1 Simplified Form of the φ Equation (5.10)
Since this paper is intended as a preliminary mathematical framework for
the dynamical effect of the spin terms, especially the spin-rich Vqqq term
on the spatial structure of the proton wave function, pending a full-fledged
study of the proton spin anomaly, we shall at this stage merely outline a
qualitative procedure to determine the nature of the solution of Eq. (5.10),
with particular reference to the possible role of the eigenvalues (5.8) of A, on
the solution of this differential equation. To that end, we shall make some
drastic simplifications, starting with the differential operator (5.9) whose
principal terms (up to second order ), in momentum space, work out as
D123 ≈
) + [
− 2m2q](ξ2z + η2z) +
(m2q −M2/9)
Further simplification arises with the ‘special’ value M = 3mq which then
yields a simple yet transparent expression in coordinate space :
D123 ≈
[−∂2s − ∂2t ] (5.11)
an operator with full rotational invariance and S3 symmetry in the 3D space
( on light-front) defined by Eq. (2.2). In a similar vein, the 2-body terms
(5.6) can be simplified by the replacements piz ≈ M/3, so as to yield
V (2)(s, t) =
2M3αs
s2 + t2
Mαs(Σ
2 − 9)
s2 + t2
[∂2s + ∂
t ] (5.12)
where we have made a further simplification based on certain standard in-
equalities [31]
s2 + t2
And the 3-body term (5.7) may be written more compactly in terms of of
the operator A with eigenvalues (5.8) as
V (3)(s, t) = α2s
3(s2 + t2)
[∂2s + ∂
t ] (5.13)
One has now a differential equation for (5.10), with the simplified operators
(5.11-13) which exhibit a 6D symmetry. Taking R2 = s2 + t2, the 6D Lapla-
cian for the ground state of the proton (with all angular d.o.f.’s dropped)
takes the simpler form
[4M2/9 +
Mαs(Σ
2 − 9)
4Aα2s
](∂2R +
∂R)φ(R) +
8M3αs
φ(R) = 0
which on rescaling with the dimensionless variable X = MR leads to the
simpler form
2 − 9)
Aα2s√
](∂2X +
∂X)φ+
φ = 0 (5.14)
5.2 Spin Effects of Vqq & Vqqq Terms on φ Singularity
The first thing to notice from the φ Equation, (5.14), is that the spin-
dependent parts of both Vqq and Vqqq appear in the multiplying factor with
the 6D Laplacian acting on φ, and are proportional to αs and α
s respectively,
with the Vqq term having a milder singularity (∼ X−1) than the Vqqq term
(∼ X−2). Further information on the singularity of the differential equa-
tion (at points other than R = 0) hinges on the nature of the eigenvalues of
the spin operator A which appears in above multiplying factor. Now these
eigenvalues are given by (5.8), two of which are complex for the state of the
proton (Σ2 = 3), but the other two are real. Of the real solutions the positive
eigenvalue does not yield a zero in this multiplying factor but the negative
eigenvalue (≈ −4) does give a zero for a real value of R at a point X20 between
X2 = 0 and X2 = ∞ :
X20 −
Aα2s√
= 0;X0 = +
[α2s/32−
Aα2s√
] (5.15)
where the value of Σ2 = 3 for the proton state has been substituted. Indeed
this zero ( for A ≈ −4) is a key element of the dynamical effect of the spin
- rich 3-body force term, although the spin effect of the 2-body Vqq term is
marginal.
To study this effect more quantitatively, we seek an approximate solution
of Eq. (5.14) in the neighbourhood of the zero at (5.15) for which a crude nu-
merical estimate suggests the following location. Taking a 3-flavour structure
αs = 2π/[9 lnM/Λ], with Λ ≈ 150MeV , one finds αs ≈ 0.39, whence
X0 ≈ 0.0689±X1; X1 ≡ 0.5841
This shows that the qq term gives ∼ 10% shift around the central value of
X1, which corresponds to R ≈ 0.12fm, and is almost entirely due to the spin
effect of the Vqqq term. Therefore in the spirit of this qualitative investigation,
it makes sense to drop this 10% effect, in which case Eq (5.14) simplifies to
](∂2X +
∂X)φ+
φ = 0; X1 = 0.584 (5.16)
To bring it nearer to a standard form, transform to the independent variable
z according to zX20 = X
0 −X2, which yields
z(1 − z)∂2zφ− 3z∂zφ− β
1− zφ = 0; β ≡ αs
X1 ≈ 0.058 (5.17)
This equation is almost (not quite) of the hypergeometric form but it can be
reduced to a standard one (with singularities located at z = 0, 1,∞) [32] by
exploiting the smallness ( β = 0.058 ) of the last term to replace it with a
constant ( with value corresponding to z = 1/2) :
z(1 − z)∂2zφ− 3z∂zφ−
φ = 0 (5.18)
An equivalent equation may be obtained with the transformation z = 1−x :
x(1 − x)∂2xφ+ 3(1− x)∂xφ−
φ = 0 (5.19)
where x = 1 corresponds to the location X = X1 of the singularity, as
indicated in Eq. (5.16). [ Note that this singularity corresponds to the
negative eigenvalue of A, thus reflecting the dynamical effect of the spin-rich
Vqqq term, while a positive or complex eigenvalue of A would result in the
disappearance of this singularity ]. The solution of Eq.(5.19) is then given
by [32]
φ = F (a, b|3|x); a + b = 2; ab =
(5.20)
with the entire machinery of hypergeometric functions available [32] for ex-
ploring its properties according to need. The scaled variable x is related to
the 6D distance R by MR = X1x where X1 ≈ 1.5αs is directly related to
the location of the singularity induced by the spin structure of the Y-shaped
3-body force . And the reconstruction of the full Bethe Salpeter wave func-
tion Ψ as a sum of 3 pieces Ψi is now only a matter of substituting (5.20) in
(4.11), whence the corresponding vertex functions Vi are immediately iden-
tified [22, 23] for specific transition amplitudes a la Feynman diagrams.
6 Retrospect : Critique Of The 3-Body Force
In retrospect, we have considered, in conjunction with pairwise qq forces, the
effect of a new type of 3-body force Vqqq, rich in spin content, on the ana-
lytical structure of the qqq wave function in the high momentum regime of
QCD where the confining interaction is unimportant, rendering the dominant
force Coulombic. As to the anatomy of this (spin-rich) Vqqq , we have taken
it to be generated by a ggg vertex ( a genuine part of the QCD Lagrangian
) wherein the 3 radiating gluon lines end on as many quark lines, giving rise
to a (Mercedes-Benz type) Y -shaped diagram, a la fig 1. From a physical
point of view it is natural to expect that such a spin-rich structure should
play a potentially crucial role in the so-called ‘spin anomaly’ of the proton,
a subject that seems once again to have raised its head in the context of new
polarized beam techniques now available [5] for resolving the issue experi-
mentally. With that end in view, our strategy has been to determine the
dynamical effect of the spin dependence of Vqq and Vqqq forces on the ana-
lytical structure of the internal 3D wave function φ. [It is emphasized that
this effect must be carefully distinguished from the ” kinematical ” effect of
spin, which manifests in several other ways, namely through the presence of
various γ matrices that appear in equations like (4.11) connecting φ to the
full 4D wave function Ψ]. Indeed we found in Section 5 that while the spin-
dynamical effect of 2-body forces is marginal, that of the spin-rich 3-body
force is quite pronounced, and shows up through the possible ” eigenvalues
”, Eq (5.7), of the spin operator (iσ1.σ2 × σ3) which is a part of Vqqq : only
a negative eigenvalue (there is only one !), induces a singularity in the dif-
ferential equation for φ, but not others. The resulting dynamical effect is
expressed by a hypergeometric function, Eq.(5.20), which gets folded into
the 4D BS amplitude Ψ via Eq.(4.11), thus fulfilling the original (limited)
motivation behind this study.
Unfortunately, the dynamical framework underlying the contents of this
paper has had a long history born out of the author’s long involvement with
the so-called Bethe-Salpeter Equation (BSE), often with extended periods of
stalemate (and frustration !), but mostly centred around the quest for a sat-
isfactory definition of probability within the BSE framework. Eventually it
became possible to settle for a toned down version of BSE, that of a Salpeter-
like equation ( 3D support for the kernel), which is amenable to a probability
interpretation at the 3D level. And its 4D features, although present in the
original formulation itself [6], had for decades remained hidden from view, but
were finally dug out [7] in the context of an independent approach designed to
explore a 3D-4D interconnection between the corresponding BS amplitudes
when the kernel has a 3D support [8]. Further, the lack of covariance in the
original formulation [6] was subsequently remedied via a special (instanta-
neous) frame of reference in which the composite hadron of 4-momentum Pµ
is at rest [33]. This result turned out to be in conformity with the Markov-
Yukawa Transversality Principle (MYTP) [11, 12], which happens to be a
‘gauge principle’ in disguise [13] : It ensures that the interactions among the
constituents be transverse to the direction of Pµ. Subsequent refinements
have been mostly technical , especially the use of Dirac’s light-front formula-
tion so as to extend the dynamical range of validity of the BS framework by
overcoming certain practical problems like ‘Lorentz mismatch disease’ [17]
associated with different vertices of a given Feynman diagram.
It therefore looked worthwhile to employ this old-fashioned formalism
in the present context of a new kind of 3-body force Vqqq, with enough de-
tails put in for a reasonably self-contained description without having to dig
frequently into the original sources. This has also given an opportunity to
make several refinements, especially the derivation of a common denomina-
tor function associated with both the qq and qqq forces and extending the
earlier formalism to accommodate the Vqqq force, for which this theoretical
framework seems to be well suited.
The next part of this programme involves the derivation of the requi-
site QCD parameters for the baryon spin, for which some key ingredients
are i) the forward scattering amplitude off the iγµγ5 operator, inserted at
the individual quark lines, and ii) a more elaborate 2-gluon contribution to
the proton spin, which is reserved for a subsequent paper within the same
formalism.
7 Acknowledgements
Several colleagues have contributed to the evolution of the BS formalism,
with and without active authorships, and the present paper also shares the
general acknowledgement. Yet two items stand out in the specific context
of this paper : First, the analysis in Section 5 owes its origin to a simple-
minded methodology due to his late Father, Jatindranath Mitra. Second,
the pedagogical techniques of qqq symmetry with several d.o.f.,s, which have
played a key role in this paper, are due to the late Mario Verde as described
in Hand Buch der Physik Vol 24, 1957. The author also acknowledges the
comments of Aalok Mishra which led to the use of the eigenvalues of the spin
operator A in Section 5. He is also grateful to Vineet Ghildyal for help with
figure 1.
Appendix A : qq Subsystem Formalism
Using the correspondence (2.2) of the text, most of the covariant instant form
results of [22] may be taken over to the present light-front situation [7]. In this
Appendix, we outline the structure of the 4D / 3D interconnection between
the corresponding Green’s functions for the qq sub-system as a prototype for
the actual qqq system considered in Section 2 of the text. The qq Green’s
functions satisfy the respective equations [22]:
(2π)4iG(q, q′;P ) =
d4q′′V (q̂, q̂′′)G(q′′, q′;P ); (A.1)
Ĝ(q̂, q̂′) =
dq0dq
0G(q, q
′;P ) (A.2)
Integrating both sides of (A.1) gives via (A.2), the 3D BSE for a bound state
(no inhomogeneous term !):
(2π)3D(q̂)Ĝ(q̂, q̂′) =
d3q̂′′V (q̂, q̂′′)Ĝ(q̂′′, q̂′) (A.3)
where the 3D denominator function D(q̂) is defined as
D(q̂)
(A.4)
leading (for general unequal mass kinematics) to [8]
D(q̂) =
D+(q̂); D+(q̂) = 2P+[q̂
λ(M2, m21, m
] (A.5)
where λ is the triangle function of its arguments. Now define the hybrid
Green’s functions [22]:
G̃(q̂, q′) =
dq0G(q, q
′;P ); G̃(q, q̂′) =
dq′0G(q, q
′;P ) (A.6)
Using (A.6) on the RHS of (A.1) gives
(2π)4iG(q, q′;P ) =
d3q̂′′V (q̂, q̂′′)G̃(q̂′′, q′) (A.7)
Integrating (A.1) w.r.t. dq′0 only, and using (2.7) again, gives
(2π)4iG̃(q, q̂′) =
d3q̂′′V (q̂, q̂′′)Ĝ(q̂′′, q̂′) (A.8)
From (A.8) and (A.3), G̃ and Ĝ are connected as:
G̃(q, q̂′) =
D(q̂)
2iπ∆1∆2
Ĝ(q̂, q̂′) (A.9)
Interchanging q and q′ in (A.9) gives the dual result
G̃(q̂, q′) =
D(q̂′)
2iπ∆′1∆
Ĝ(q̂, q̂′) (A.10)
Substitution in (A.7) gives the desired 3D-4D interconnection
G(q, q′;P ) =
D(q̂)
2iπ∆1∆2
Ĝ(q̂, q̂′;P )
D(q̂′)
2iπ∆′1∆
(A.11)
Next, spectral representations for the 4D and 3D G-fns in (A.11) give [22]
G(q, q′;P ) =
Φn(q;P )Φ
′;P )/(P 2 +M2); (A.12)
Ĝ(q̂, q̂′) =
φn(q̂)φ
′)/(P 2 +M2) (A.13)
where Φn and φn are 4D and 3D wave functions, so that their interconnection
(valid for any bound state n), is expressed by:
Γ(q̂) ≡ ∆1∆2Φ(q;P ) =
D(q̂)φ(q̂)
(A.14)
which tells us that the covariant vertex function Γ on the light-front is a
function of q̂ only [22] via its definition (2.2). This derivation is a prototype
for the qqq case of text.
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http://arxiv.org/abs/hep-ph/0104219
http://arxiv.org/abs/hep-ph/0104219
	 Introduction
	Theoretical Ingredients
	 Plan of the Paper
	 Salpeter Eq on LF : From  To 
	 Master Eq : Instant Vs LF Forms of Dynamics
	 Reduction of Master Eq from  to  
	Gordon Reduction on  Vqq3 & Vqqq
	 Further Reduction of Vqq3 & Vqqq 
	 3D Reduction By Green's Function Method
	 Reduction from 4D G(; ' ' ) to 3D "705EG("705E"705E;"705E'"705E') 
	 Reconstruction of 4D Wave Function
	Reconstruction of G with qq Forces Only
	 Reconstruction of G with qqq Forces Only
	 Putting Both Vqq & Vqqq Together
	3D-4D Interconnection for Wave Functions
	 Complete  Equation In Coordinate Space 
	 Simplified Form of the  Equation (5.10)
	 Spin Effects of Vqq & Vqqq  Terms on  Singularity 
	 Retrospect : Critique Of The 3-Body Force
	Acknowledgements
