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H. Kleinert, COLLECTIVE FIELDS
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Science is facts; just as houses are made of stones,
so is science made of facts; but a pile of stones is not a house
and a collection of facts is not necessarily science
Henri Poincaré (1854–1912)
5
Hadronization of Quark Theories
In this chapter we shall study a simple model of quantum field theory which shows
how quark theories can be converted into bilocal field theories via functional techniques. The new basic field quanta of the converted theory are approximate the
quark-antiquark meson bound states. They are obtained by solving the BetheSalpeter bound-state equation in the ladder approximation. These will be called
bare mesons and we shall introduce mesonic Feynman graphs describing their propagation. In the limit of infinitely heavy gluon mass, the bilocal fields become local
and describe π, ρ, A1 , and σ-mesons. Their fluctuations are governed by a collective chirally invariant Lagrange density, which has been known for a long time as a
so-called SU(3)-symmetric linear σ-model.
Many interesting relations are found between meson and quark properties such as
m2ρ ≈ 6M 2 , where M is the non-strange quark mass after spontaneous breakdown of
chiral symmetry, the so-called constituent quark mass. There is a simple formula
linking these quark masses with the small bare masses appearing as parameters in
the Lagrange density. The quark masses also determine the vacuum expectations of
scalar densities.
5.1
Introduction
In attempting to understand the physics of strongly interacting particles, the
hadrons, two fundamentally different theoretical approaches have been developed.
One of them, the dual approach, is based on complete democracy among all strongly
interacting particles. Within this approach, an elaborate set of rules assures the construction of certain lowest order vertex functions for any number of mesons [3]. The
other approach assumes the existence of a local field equation involving fundamental quarks bound together by vector gluons [4]. Here strong interaction effects on
electromagnetic and weak currents of hadrons can be analyzed in a straight-forward
fashion without detailed dynamical computations [5]. Either approach has its weakness where the other is powerful. Dual models have, until now, given no access
to currents while quark theories have left the problem of mesonic vertex function
316
5.1 Introduction
317
intractable. Not even an approximate bound state calculation is available (except
in 1 + 1 dimensions [6] or by substituting the field couplings by simple ad-hoc forces
[7]).
At present it appears that a Lagrangian field theory of the Yang-Mills type
is the correct fundamental theory of elementary particles. The gluon fields form
flux tubes between quarks and antiquarks which have a fixed diameter of the order
of the Compton wavelength of the pion. Since the field energy is to lowest order
proportional to the square of the field strength, the energy of the flux tube grows
linearly with the length and leads to quark confinement. For large distances, the
thickness of the flux tube can be ignored and the physics of the Yang-Mills theory
approaches the theory of a hadronic string. This approximation can, however, only
be used for long, i.e., highly excited mesons. Low-lying mesons are held together by
short fat flux tubes which become spherical for the ground state. The string picture
breaks down completely for these states, which look more like bags. In fact, the
low-lying states of a string have a negative norm. Only in spacetime of much higher
dimension than four does the norm become positive.
When theorists ran into these problems of the string approximation they abandoned the entire model. Others, who had spent a lot of effort in studying the
mathematics of strings decided to postulate the string picture to be physical representations of entirely new fundamental particles. Therefore they had to assume
that we live in a space with more than four dimensions, and proposed many possible
consequences of which there has been no evidence so far. They even derive from
this picture the existence of black mini-holes which could be produced in present-day
accelerators such as the Large Hadron Collider at the European laboratory CERN
in Geneva.
An important defect of the string model so far is that there exists no quantum
field theory for a grand-canonical ensemble of strings. Only the first-quantized
language of a single string is well developed. There are only rudimentary attempts
at a second-quantized field theory of strings [8].
In order to learn how a translation between the different languages might operate
we consider here the simplified field theory in which quarks are colorless, have N
flavors, and are held together by vector gluons of arbitrary mass µ. This theory
incorporates several realistic features of strong interactions, for example current
algebra and PCAC. Moreover, the case N = 1 and µ = 0 includes ordinary quantum
electrodynamics (QED). This will provide a good deal of intuition as well as the
possibility of a detailed test of our results.
First we demonstrate how functional methods can be employed to transform the
local quark gluon theory into a new completely equivalent field theory involving only
bilocal fields. The new free field quanta coincide with quark-antiquark bound states
when calculated by ladder exchanges only. They may be considered as bare mesons.
Accordingly, the transition from the local quark- to the bilocal meson-theory will
be named mesonization. In the special case of QED, bare mesons are positronium
atoms in ladder approximation.
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5 Hadronization of Quark Theories
The functional technique will ensure that bare mesons have exactly the correct
interactions among each so that mesonization preserves the equivalence to the original quark gluon theory. It is simple to establish the connection between classes of
Feynman graphs involving quarks and gluons with single graphs involving mesons.
The topology of meson graphs is the same as that of dual diagrams. It is interesting to observe the appearance of a current-meson field identity for photons just as
employed in phenomenological discussions of vector meson dominance. Moreover,
since the theory is bilocal, this identity can be extended to bilocal currents which
are measured in deeply inelastic electromagnetic and weak interactions.
The limit of a very heavy gluon mass can be mesonized most simply. Here
the bilocal fields become local and describe only a few mesons with the quantum
numbers of π, ρ, A1 , and σ-mesons. The Lagrange density coincides with that of the
standard chirally invariant σ-model which is known to account quite well for the lowenergy aspects of meson physics. Here mesonization renders additional connection
between quark and meson properties. It also makes transparent the relation between
the very small bare quark masses (which describe the explicit breakdown of chiral
symmetry) and the constituent quark masses (which include the dynamic effects due
to spontaneous symmetry violations).
5.2
Abelian Quark Gluon Theory
Consider now a system of N quarks ψ(x) held together by an abelian gluon field
Gν (x) of mass µ via a Lagrangian
µ2
1 2
(x) + G2ν .
L(x) = ψ̄(x) (i/
D − M) ψ(x) − Fµν
4
2
(5.1)
Here D
/ is the Dirac covariant derivative D
/ ≡ γ µ Dµ , in which the ordinary covariant
derivative are Dµ ≡ ∂µ − igGµ associated with the gluon fields are contracted with
the Dirac matrices, and Fµν ≡ ∂µ Gν − ∂ν Gµ is the usual spacetime curl of Gµ . In
the special case in which N = 1, µ = 0, and g 2 = 4πα, this Lagrangian describes
quantum electrodynamics. In other cases it may be considered as a model field
theory which carries many interesting properties of strong interactions, for example
approximate SU(3) symmetry, chiral SU(3)× SU(3) current algebra, PCAC, and
scaling up to small corrections. Certainly, this model will never be able to confine
quarks, give symmetric baryon wave functions, and explain infinitely rising meson
trajectories. For this it would have to contain an additional, exactly conserved,
color symmetry with Gν (x) being its non-abelian gauge mesons. Before attempting
to deal with this far more complicated situation we shall develop our tools for the
less realistic but much simpler model (5.1) without color.
The generating functional of all time ordered Green function is
Z [η, η̄, j ν ] = const ×
Z
DψD ψ̄DGei
R
dx(L+ψ̄η+η̄ψ+gj ν Gν )
.
(5.2)
H. Kleinert, COLLECTIVE FIELDS
319
5.2 Abelian Quark Gluon Theory
The exponent is quadratic in Gν (x), such that the functional integration over the
gluon field can be performed [9, 10] using formula (1.80). The result is
ν
Z [η, η̄, j ] = const ×
Z
DψD ψ̄eiA[ψ,ψ,η,η,j
ν]
(5.3)
with an action
h
A ψ, ψ̄, η, η̄, j
ν
i
=
Z
dxdy
L(x) + ψ̄(x)η(x) + η̄(x)ψ(x) δ(x − y)
i
− g 2 D(x − y) ψ̄(x)γ ν ψ(x) + j ν (x) ψ̄(y)γν ψ(y) + jν (y) . (5.4)
2
By employing the Fierz identity:
1 ν
1
ν
γαβ
⊗ γνγδ = 1αδ ⊗ 1αβ + (iγ5 )αδ ⊗ (iγ5 )γβ − γαβ
⊗ γν γβ − (γ ν γ5 )αδ ⊗ (γν γ5 )γβ ,
2
2
(5.5)
the four-Fermion quark interaction term can be written in a different fashion:
i 2
g D(x − y) ψ̄(x)ψ(y)ψ̄(y)ψ(x) + ψ̄(x)iγ5 ψ(y)ψ̄(y)iγ5ψ(x)
2
1
1
ν
ν
− ψ̄(x)γ ψ(y)ψ̄(y)γν ψ(x) − ψ̄(x)γ γ5 ψ(y)ψ̄(y)γν γ5 ψ(x) .
2
2
This will be written short as
i 2
g D(x − y)ψ̄α (x)ψδ (y)ξαδ,γβ ψ̄γ (y)ψβ (x),
2
(5.6)
(5.7)
where the matrix ξαδ,γβ denotes the right-hand side of Eq. (5.5). This is the point
where our elimination of quark fields in favor of new bilocal fields starts.
Let S(x, y), P (x, y), V ν (x, y), Aν (x, y) be a set of hermitian auxiliary fields, i.e.
S(x, y) = S(y, x), P (x, y) = P ∗ (y, x), etc.
(5.8)
With these field, we can construct the following functional identities [11]
Z
Z
Z
Z
i
DS(x, y)e− 2 |S(x,y)+ig
i
2 D(x−y)ψ̄(y)ψ(x)|2 /ig 2 D(x−y)
DP (x, y)e− 2 |P (x,y)+ig
DV (x, y)ei|V
DA(x, y)eiA
= const. ,
2 D(x−y)ψ̄(y)iγ ψ(x)|2 /ig 2 D(x−y)
5
ν (x,y)− i g 2 D(x−y)ψ̄(y)g γ ψ(x)|2 /ig 2 D(x−y)
2
= const. ,
= const. ,
ν |S(x,y)+ i g 2 D(x−y)ψ̄(y)γ ν γ ψ(x)|2 /ig 2 D(x−y)
5
2
= const. ,
(5.9)
which are independent of the fields ψ(x). If we now multiply Z [η, η̄, j ν ] in (5.3)
by these constants, and make use of (5.6), all four-Fermion quark terms are seen to
cancel. The generating functional becomes
Z [η, η̄, j ν ] = const ×
Z
DψD ψ̄DSDP DV DAeià ,
(5.10)
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5 Hadronization of Quark Theories
where the new action à is obtained as an integral
i
h
à ψ, ψ̄, S, P, V, A, η, η̄, j ν =
Z
dxdyL(x, y)
(5.11)
over a bilocal Lagrange density:
L(x, y) ≡
n
o
ψ̄(x) (i∂/ − M) ψ(x) + ψ̄(x)η(x) + η̄(x)ψ(x) δ (4) (x − y)
i
−ψ̄(x)m(x, y)ψ(y) − g 2 D(x − y)j ν (x)jν (y)
2
1 2 1 2
1
2
2
− |S| + |P | − |V | − |A|
.
(5.12)
2
2
2
ig D(x − y)
Here m(x, y) has been introduced as an abbreviation for the combined field:
m(x, y) ≡ S(x, y) + P (x, y)iγ5
+ V ν (x, y) + δ (4) (x − y)
Z
(5.13)
dzig 2 D(x − z)j ν (z) γν + Aν (x, y)γν γ5 .
Due to (5.8), the matrix m(x, y) is self-adjoint in the sense
m(x, y)
αβ
0
≡ γαα
m∗ (x, y)T
′
α′ β ′
γβ0′ β = mαβ (y, x).
(5.14)
At this place it is worth remarking that the Lagrangian (5.12) shows its equivalence
to the previous form (5.4) also quite directly. Extremizing the action we obtain the
Euler-Lagrange equations for the fields S, P, V, A, which are seen to be dependent
fields coinciding with the corresponding bilocal quark expressions
S(x, y) = −ig 2 D(x − y)ψ̄(y)ψ(x),
(5.15)
P (x, y) = −ig 2 D(x − y)ψ̄(y)iγ5ψ(x),
i 2
V ν (x, y) =
g D(x − y)ψ̄(y)γ ν ψ(x),
2
i 2
ν
g D(x − y)ψ̄(y)γ ν γ5 ψ(x),
A (x, y) =
2
(5.16)
(5.17)
(5.18)
if fluctuations are neglected. Inserting these relations back into (5.12), we reproduce
(5.4). In the action (5.11), quark fields enter only in a quadratic form such that they
can be integrated out according to formula (1.80). The functional matrix A is given
by [compare (1.250)]
A(x, y) = (−∂/ − M) δ (4) (x − y) − m(x, y).
(5.19)
Hence A−1 (x, y) ≡ −iG(x, y) is the Green function associated with the equation
Z
h
i
dy (i/
∂ − M) δ (4) (x − y) − m(x, y) G(y, z) = iδ (4) (x − z).
(5.20)
H. Kleinert, COLLECTIVE FIELDS
321
5.2 Abelian Quark Gluon Theory
With this notation, the quark integration brings the functional (5.10) to the form
Z [η, η̄, j ν ] = const ×
Z
ν
Dm(x, y)eiA[η,η̄,j ] ,
(5.21)
with
A [m, η, η̄, j ν ] =
Z
dxdy −itr ln iG−1 (x, y)δ (4) (x − y)
(5.22)
(4)
2
1 1
2 [δ (x − y)]
− Tr m(x, y)ξ −1m(y, x)
+
iη̄(x)G(x,
y)η(y)
−
ig
2
ig 2 D(x − y)
D(0)
)
Z
2
ν
′
′ ν
′
V (x, x)D(x − y)jν (y) + dzdz D(z − x)D(y − z )j (z)jν (z ) .
−
D(0)
Here we have introduced the notation
Dm(x, y) ≡ DSDP DV DV,
(5.23)
for brevity. Note that the effect of the matrix ξ −1 defined in Eq. (5.6) is simply
to divide the projections into S, P, V, A by 4, −4, −2, 2, respectively.1 The trace tr
refers only to Dirac indices. The new functional (5.21) is identical to the original
one in Eq. (5.2). As a consequence, a quantum theory based on the action (5.22)
must be completely equivalent to the original quantized quark gluon theory.
A word is in order concerning the internal symmetry SU(N) among the N quarks
(i = 1, . . . N) under consideration. Since the gluon is an SU(N) singlet, the interacP
ν
tion in Eq. (5.1) is g N
i=1 ψ̄i γ ψi Gν (x). In the Fierz transformed version (5.6) the
indices i and j appear separated
i 2
g D(x − y)ψ̄ j (x)ψi (y)ξ ψ̄ i (y)ψj (x).
2
Hence, in the presence of N quarks, the fields m(x, y) have to be thought of as
matrices in SU(N) space [m(x, y)]i j . This carries over to the action with the traces
including Dirac- as well as SU(N)-indices.
Let us now develop a quantum theory for the new action. In general, the field
m(x, y) may oscillate around some constant non-zero vacuum expectation value
m0 δ (4) (x − y). It is convenient to subtract such a value from m(x, y) and introduce
the field
m′ (x, y) ≡ m(x, y) − m0 δ (4) (x, y).
(5.24)
With this and the definition
Eq. (5.20) can be rewritten as
Z
1
Since
matrices.
1
41
h
M ≡ M + m0 ,
i
dy (i∂/ − M) δ (4) (x − y) − m′ (x, y) G(y, z) = iδ (4) (x − z).
(5.25)
(5.26)
⊗ 1, − 41 (iγ5 ) ⊗ (iγ5 ) , 14 γ ν ⊗ γν , − 41 γ ν γ5 ⊗ γν γ5 are the corresponding projection
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5 Hadronization of Quark Theories
Now let us assume that the oscillations m′ (x, y) are sufficiently small as to permit
a perturbation expansion for G(x, y):
G(x, y) = GM (x, y) − i (GM m′ Gm ) (x, y) − (GM m′ GM m′ GM ) (x, y) + . . . (5.27)
where GM (x, y) are the usual propagators of a free fermion of mass M:
GM (x, y) ≡ GM (x − y) ≡
Z
d4 p −ipx i
e
.
(2π)4
p/ − M
Using this expansion, the action (5.22) takes the form2
A [m′ , η, η̄, j ν ] = A1 [m′ ] + A2 [m′ ] + Aint [m′ ] + Aext [m′ , η, η̄, j ν ] ,
#
"
Z
(5.28)
δ(y, x)
.
A1 [m′ ]≡ dxdy trSU(2) GM (x−y)m′ (x, y)−ξ −1m′ (x, y)m0 2
ig D(x−y)
(5.29)
and A2 being quadratic in m′
′
A2 [m ] ≡
Z
dxdy trSU(2)
"
#
i
1
1
.
GM mGM m′ (x, y) − ξ −1 m′ (x, y)m′ (y, x) 2
2
2
ig D(x−y)
(5.30)
The term Aint [m′ ] collects all remaining powers in m′
′
Aint [m ] ≡
Z
"
#
∞
X
(−i)n+1
n
dxTr −
(GM m′ ) (x, x) .
n
n=3
(5.31)
The last piece Aext , finally, contains all interactions with the external sources
′
ν
Aext [m , η, η, j ] =
Z
dxdy {iη̄(x)G(x, y)η(y)
i
2
− g 2 D(x − y)j ν (x)jν (y) −
V ν (x, x)D(x − y)jν (y)
2
D(0)
)
Z
(4)
(4)
′
′ ν
′
2 δ (0)δ (x − y)
dzdz D(z − x)D(y − z )j (z)jν (z ) .
−ig
D(0)
For the quantization we shall adopt an interaction picture. As usual, the quadratic
part of the action, A2 [m′ ], serves for the construction of free-particle Hilbert space.
According to the least action principle, the free equations of motion are obtained
from δA2 [m′ ]/δm′ (x, y) = 0, rendering
m′ (x, y) = g 2 ξD(x − y) (GM m′ GM ) (x, y).
2
(5.32)
A trivial additive constant has been dropped.
H. Kleinert, COLLECTIVE FIELDS
323
5.2 Abelian Quark Gluon Theory
Going to momentum space
Z
′
m (p2 , p1 ) ≡
dx2 dx1 ei(x2 p2 −x1 p1 ) m′ (x2 , x1 ) ,
and introducing relative and total momenta
P ≡ (p2 + p1 ) /2, q ≡ (p2 − p1 ) /2,
together with the notation
m′ (P |q) ≡ m′ (p2 , p1 ),
the field equation becomes
′
m (P |q) = ξg
2
Z
q
q
d4 P ′
m′ (P ′ |q) GM P ′ −
. (5.33)
D (P ′ − P ) GM P ′ +
4
(2π)
2
2
In this form we easily recognize the Bethe Salpeter equation [13] for the vertex
function of quark-antiquark bound states in the ladder approximation:
ΓH (P |q) ≡ NH P +
q
2
Z
dzeiP z h0|T ψ
q
z
z
,
ψ̄ − |0iGM P ′ −
2
2
2
(5.34)
where NH is some normalization factor. As a consequence our free field m′ (x, y)
can be expanded in a complete set of ladder bound state solutions. These are the
bare quanta spanning the Hilbert space of the interaction picture. Because of their
bound quark-antiquark nature, they are called bare mesons. In the special case of
QED, the “quarks” are electrons and the bare mesons are positronium atoms.
For mathematical reasons it is convenient to solve (5.33) for fixed q 2 ∈ (0, 4M 2 )
2
and all possible coupling constants g 2 , to be called gH
(q 2 ), i. e.
H
Γ (P |q) =
2
ξgH
q
2
Z
d4 P ′
q
q
ΓH P ′ −
.
D (P − P ′) GM P ′ +
4
(2π)
2
2
(5.35)
A useful normalization condition is
−i
Z
q
d4 P
′
ΓH (P |q)Γ¯H ′ (P |q) = ǫH δ HH .
Tr GM P +
4
(2π)
2
(5.36)
Here we have allowed for a sign factor ǫH (q) which cannot be absorbed in the normalization NH of (5.34). It may take the values +1, − 1 or zero. Then the expansion of the free field m′ (x, y) in terms of meson creation and annihilation operators
a†H (q), aH (q) can be written as
m′αβ (x, y)
n
=
Z
Z d4 P
d44 q X (+) 2 2 2
gH q − g
δ
(2π)4 H
(2π)4
(5.37)
o
× e−i(q(x+y)/2+P (x−y)) ΓH (P |q)nH aH (q)+ e−i(q(x+y)/2−P (x−y)) Γ̄H (P | − q)n∗H a†H (q) ,
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5 Hadronization of Quark Theories
where nH are appropriate factors giving aH (q) the standard normalization
h
i
aH (q), a†H ′ (q′ ) = (2π)3 δ (3) (q − q′ ) 2ω H (q)ǫH (q).
(5.38)
Here the sign factor ǫH (q) controls the norm of the mesonic state a+
H |0i ≡ |Hi.
In general there will be many states with unphysical norms since the bare mesons
are produced by ladder diagrams only and may not be directly related to physical
particles. This situation presents no fundamental difficulty. There are many interactions among bare mesons which are capable of excluding unphysical states from the
S-matrix. In fact, the equivalence of the mesonized theory to the healthy original
quark gluon version is a guarantee for physical results (on shell).
The propagator of the free field m′ (x, y) can be found most directly by adding
an external disturbance to the free action
A2 [m′ ] → A2 [m′ ] −
Z
dxdy Tr [m′ (x, y)J(y, x)] .
(5.39)
This current enters the equation of motion as
m′ (x, y) = −ξig 2 D(x − y)J(x, y) + ξg 2D(x − y) (GM m′ GM ) (x, y). (5.40)
The propagator
Gαβ,α′ β ′ (x, y; x′ , y ′) ≡ m′ (x, y)m′ (x′ , y ′)
(5.41)
is then defined as the solution of (5.40) for the δ-function disturbance
Jαβ (x, y) = iδ(x − x′ )δ(y − y ′)δαα′ δββ ′ .
(5.42)
It satisfies the inhomogeneous Bethe-Salpeter equation
Gαβ,α′ β ′ (x, y; x′ , y ′) = ξαβ,β ′ α′ g 2D(x − y)δ(x − x′ )δ(y − y ′)
+ξ
αβ,α′ β ′
Z
D(x − y) dx̄dȳ GM (x − x̄)αᾱ Gᾱβ̄,α′ β ′ (x̄, ȳ; x′ y ′) GM (ȳ − y)β̄β .
This expression is immediately recognized as the equation for the two-quark transition matrix in the ladder approximation [see Eq. (5A.20) in Appendix 5A].
We can now give an explicit representation of the Green function in terms of the
solutions ΓH (P |q) of the homogeneous equation (5.35). If Gαβ,α′ β ′ (P, P ′|q) denotes
the Fourier transform
(2π)4 δ (4) (q − q ′ ) Gαβ,α′ β ′ (P, P ′|q) ≡
Z
dxdydx′dy ′ei[P (x−y)+q(x+y)/2−P
(5.43)
′ (x′ −y ′ )−q ′ )x
+y
′ )/2]
Gαβ,α′ β ′ (x, y; x′ y ′) ,
it can be written as the sum over all meson solutions:
Gαβ,α′ β ′ (P, P ′|q) = −ig
X
H
ǫH (q)
′
H
ΓH
αβ (P |q)Γ̄β ′ α′ (P | − q)
,
2
gH
(q 2 ) − g 2
(5.44)
H. Kleinert, COLLECTIVE FIELDS
325
5.2 Abelian Quark Gluon Theory
where the sum comprises possible integrals over a continuous set of solutions. If
quarks and gluons were scalars, the sum would be discrete for q 2 ∈ (0, 4M 2 ) since
the kernel of the integral equation (5.35) would be of the Fredholm type. A more
detailed discussion is given in Appendix A. Here we only note that a power series
expansion of the denominator
′
Gαβ,α′ β ′ (P, P |q) = −i
∞ X
X
n=1 H
g2
2
gH
(q 2 )
!n
′
H
ǫH (q)ΓH
αβ (P |q)Γ̄β ′ α′ (P | − q)
(5.45)
renders explicit the exchange of one, two, three, etc. gluons. Hence one additional
gluon can be inserted (or removed) by multiplying (or dividing) (5.44) by a factor
2
g 2 /gH
(q 2 ). This fact will be of use later on.
Seen microscopically in terms of quarks and gluons, the free meson propagator
(5.44) is given by the sum of ladders (see Fig. 5.1). Graphically, this will be repre-
Figure 5.1 Ladder diagrams summed by solution of the Bethe-Salpeter equation in
Eq. (5.44).
sented by a wide band. In the last term of Fig. 5.1 we have also given a visualization
of the expansion (5.44). Here, the fat line denotes the propagator
∆H (q) = −iǫH (q)
g2
,
2
gH
(q 2 ) − g 2
(5.46)
where upper and lower bubbles stand for the Bethe Salpeter vertices ΓH (P |q) and
ΓH (P ′ | − q), respectively. This picture suggests another way of representing the new
bilocal theory in terms of an infinite component meson field depending only on the
average position X = (x + y)/2. For this we simply expand the interacting field
m′ (P |q) in terms of the complete set of vertex functions
m′ (P |q) =
X
H
ΓH (P |q)mH (q).
(5.47)
Inserting this expansion into (5.28), the free action becomes directly
A2 [m′ ] =
1Z
2
q 2 /g 2 mH (X),
dXmH (X) 1 − gH
2
(5.48)
implying the free propagator (5.46) for the field mH (X). With this understanding
of the free part of the action we are now prepared to interpret the remaining pieces.
326
5 Hadronization of Quark Theories
Figure 5.2 Ladder diagrams summed in tadpole term A1 [m′ ] of Eq. (5.44).
Consider first the linear part A1 [m′ ]. The first term in it can graphically be
presented as shown in Fig. 5.2. When attached to other mesons it produces a
tadpole correction. When interpreted within the underlying quark gluon picture,
such a correction sums up all rainbow contributions to the quark propagator (see
Fig. 5.3). Also the second term in A1 [m′ ] has a straight-forward interpretation. First
Figure 5.3 Rainbow diagrams in tadpole term of previous figure.
of all, the division by ξig 2D(x − y) has the effect of removing one rung from the
ladder sum (such that the ladder starts with no rung, one rung, etc. ) and creating
two open quark legs. This can be seen directly from (5.35) and (5.44): Suppose a
meson line ends at
the interaction
Z
−
h
i
dxdy Tr m′ (x, y)ξ −1m0 /ig 2D(x − y)δ (4) (x − y).
−1
Then the factor [ξig 2D(x − y)]
indices)
h
2
ξg D
i−1
applied to G(P, P ′|q) gives (leaving out irrelevant
G = −ig
2
X
H
Due to (5.35), this is equal to
h
ξg 2 D
i−1
G = −ig 2
X
H
ǫH
−1
[ξg 2D] ΓH Γ̄H
ǫH 2 2
.
gH (q ) − g 2
H
H
(q ) GM Γ GM Γ̄
.
2
g2
gH
(q 2 ) − g 2
2
gH
2
(5.49)
(5.50)
2
As discussed before, the factor gH
(q 2 ) /g 2 amounts to the removal of one rung.
R
Multiplication by −m0 and integration over dP/(2π)4 yields the total contribution
of this meson graph
i m0
X
H
×
2
gH
(q 2 )
2
gH
(q 2 ) − g 2
Z
q
q
d4 P
ΓH (P |q)GM P −
Tr GM P +
4
(2π)
2
2
ǫH (q)Γ̄H (P ′ | − q). (5.51)
H. Kleinert, COLLECTIVE FIELDS
327
5.2 Abelian Quark Gluon Theory
As far as quarks and gluons are concerned, this amounts to the insertion of a mass
term m0 on top of a ladder graph with one rung removed (this being indicated by
a slash in Fig. 5.4). The quark gluon picture leads us to expect that m0 must be a
Figure 5.4 Ladder of gluon exchanges summed in a meson tadpole diagram marked by
a slash. Compare Fig. 5.1.
cutoff dependent quantity cancelling the logarithmic divergence in every upper loop
of the ladder sum of Fig. 5.2. Numerically, m0 is most easily calculated by cancelling
the infinity contributed by A1 [m′ ] to the equation of motion (5.33). If we include
A1 [m′ ], this equation reads
m0 (2π)4 δ (4) (q) + m′ (P |q) =
#
"
Z
d4 P ′
′
′
2
D(P − P )G(P ) (2π)4 δ (4) (q)
ξig
(2π)4
Z
q
q
d4 P ′
′
′
′
′
′
2
m (P |q)GM P −
.
D(P − P )GM P +
+ξg
(2π)4
2
2
(5.52)
The first term on the right-hand side is exactly the usual self-energy Σ(P )(2π)4 δ (4) (q)
in second order
Σαβ (P ) ≡ −ξαβ,γδ i
Z
d4 P ′
1
1
.
(2π)4 (P − P ′ )2 − µ2 P/ − M
(5.53)
Normalizing Σ(P ) on the mass shell one finds the usual expression
Σ(P ) = Σ0 + Σ1 × (/
P − M) + ΣR (P )
(5.54)
where ΣR is the regularized self-energy. The cutoff dependent term
1
3 g2
M log Λ2 /M 2 +
Σ0 =
4π 4π
2
(5.55)
must be balanced by choosing m0 = −Σ0 on the left-hand side of (5.52). Also, the
second term Σ1 is cutoff dependent:
Σ1 =
9
1 g2
log Λ2 /M 2 + + 2 log µ2 /M 2 ,
4π 4π
2
(5.56)
328
5 Hadronization of Quark Theories
and a renormalization is necessary to cancel this infinity. Most economic is the
introduction of an appropriate wave function counter term (Z2 − 1) ψ̄(i/
∂ − M) in
the original Lagrangian (5.1). Such a term would enter Eq. (5.20) as
Z
n
dy (i∂ − M) δ (4) (x − y)
(5.57)
o
+ (Z2 − 1) (i/
∂ − M) δ (4) (x − y) − m(x, y) G(y, x) = iδ (4) (x − y).
Instead
m(x,iy) should now be assumed to fluctuate around
h
of (5.24),
−1
m0 + Z2 − 1 (i/
∂ − M) δ (4) (x − y). By defining a new m′ (x, y) via
h
i
m′ (x, y) ≡ m(x, y) − m0 + Z2−1 − 1 (i/
∂ − M) δ (4) (x − y),
(5.58)
the full action (5.28) is obtained exactly as before, except for the linear part A in
which the new wave function renormalization term enters together with m0 :
Z
′
A1 [m ] =
−1
dxdy trSU(2) {GM (x − y)m′(x, y)
h
′
−ξ m (x, y) m0 +
Z2−1
)
i
δ (4) (x − y)
− 1 (i/
∂ − M)
.
ig 2 D(x − y)
By choosing
Z2−1 − 1 = −Σ1
(5.59)
the cutoff dependent term Σ1 is exactly compensated in the equation of motion
(5.52). After this renormalization procedure, only the finite term ΣR (P ) is left. the
regularized action is
′
A1 [m ]R =
Z
dxdyΣR (x − y)m′ (x, y)
1
ig 2D(x
− y)
.
(5.60)
Using the expansion (5.47), this can be rewritten as
′
A1 [m ]R =
XZ
dX fH (− )mH (X),
(5.61)
H
with
fH q
2
=i
Z
q
q
d4 P
ΓH (P |q)GM P −
Tr ΣR (/
P )GM P +
4
(2π)
2
2
2
gH
(q 2 )
. (5.62)
g2
By momentum conservation, the tadpole momentum always vanishes such that only
fH (0) is needed eventually.
Let us now proceed to the discussion of the interaction part Aint [m′ ] of Eq. (5.31).
Take as an example the term of the third order in m′ . If a meson line ends at every
m′ , it can be represented graphically as shown in Fig. 5.5. Employing the expansion
H. Kleinert, COLLECTIVE FIELDS
329
5.2 Abelian Quark Gluon Theory
Figure 5.5 Gluon diagrams contained in a three-meson vertex.
(5.47), this interaction term can be rewritten as
′
A3hadr
int [m ]
Z
1 X
=−
3 H1 H2 H3
"
Z
d4 q3 d4 q2 d4 q1
(2π)4 δ (4) (q1 + q2 + q3 )
(2π)4 (2π)4 (2π)4
!
!
q − 2 d4 P
q3 H2
H
×
P
+
q
+
q
G
(P
+
q
q
)
Γ
q2
tr
P
−
Γ
1
3
M
1
2
SU(2)
(2π)4
2
2 q1
H1
P + |q1 GM (P ) mH3 (q3 )mH2 (q2 )mH1 (q1 )
GM (P + q1 ) Γ
2
Z
1 X
H3
H2
H1
, i∂X
, i∂X
mH3 (X)mH2 (X)mH1 (X), (5.63)
dXυH3 H2 H1 i∂X
×
3 H1 H2 H3
Hi
H3
H2
H1
, i∂X
, i∂X
whose derivatives ∂X
are to be
with a vertex function υH3 H2 H1 i∂X
applied only to the argument of the corresponding field mHi (X). A similar formula
holds for every power of m′ .
Notice that the flow of the quark lines in every interaction is anticlockwise. When
drawing up mesonic Feynman graphs it may sometimes be more convenient to draw
a clockwise flow. A simple identity helps to write down directly the corresponding
Feynman rules. Consider a graph for a three meson interaction and cross the upper
band downwards (see Fig. 5.6). The interaction appears now with the mesonic bands
in anticyclic order, and the fermion lines in the meson vertex flowing clockwise.
This is topologically compensated by twisting every band once. Mathematically,
this deformation displays the following identity of the vertex functions
υH3 H2 H1 (q3 , q2 , q1 ) = ηH3 ηH2 ηH1 υH1 H2 H3 (q1 q2 q3 ) ,
(5.64)
where the phase ηH denotes the charge parity of the meson H. This phase may be
absorbed in the propagator characterizing the twisted band.
The proof of this identity (5.64) is quite simple. Let C be the charge conjugation
matrix
C=
c
0
0 −c
!
,
(5.65)
330
5 Hadronization of Quark Theories
Figure 5.6 Three-meson vertex drawn in two alternative ways.
where c is the 2 × 2-matrix
2
c = −iσ =
0 −1
1
0
!
.
(5.66)
This matrix is the two-dimensional representation of rotation around the 2-axis by
an angle π:
2
c = e−iπσ /2 ,
(5.67)
as can easily be verified using
R'ˆ (ϕ) = e−i'·/2 = cos
ϕ
ϕ
ˆ sin .
− i · '
2
2
(5.68)
2
and the property σ i = 1. From the rotation property of σ i under C it follows
directly that

and we find
c−1
0
0 −c−1



−σ 1
σ1

−1 
2 
σ2 
c  σ c = 
,
3
3
−σ
σ
!
0 σµ
σ̃ µ 0
!
c
0
0 −c
!
=
(5.69)
−c−1 σ µ c
− c−1 σ̃ µ c
0
0
!
= (−γ 0 , γ 1 , −γ 2 , γ 3 ) = −γ µT ,
(5.70)
so that (5.65) satisfies:
C −1 γ µ C = −γ µT ,
(5.71)
thus ensuring a sign change of the electromagnetic interaction Aµ ψ̄γ µ ψ. As a consequence, the vertices satisfy:
CΓH (P |q) C −1 = ηH ΓH (−P |q)T .
(5.72)
Inserting now CC −1 between all factors in (5.63) and observing Cγ µ C −1 = −γ µT
[recall (5.71)], one has
υH3 H2 H1 (q3 , q2 , q1 ) =
H. Kleinert, COLLECTIVE FIELDS
331
5.2 Abelian Quark Gluon Theory
−ηH3 ηH2 ηH1
Γ H2
Z

!T

q3 dP
H3
−P
+
q3
tr
Γ
SU(2)

(2π)4
2
q2 −P − q1 − q2
2
!T
i
−/
P − q/ 1 − M
!T
Γ H1
i
−/
P − q/ 1 − q/ 2 − M
!
q1 −P − q1
2
!T
i
−/
P −M
!T 


.
Taking the transpose inside the trace and changing the dummy variable P to −P ,
the vertices appear in anticyclic order and the right-hand side coincides indeed with
ηH3 ηH2 ηH1 υH1 H2 H3 (q1 , q2 , q3 ). Twisted propagators are physically very important.
They describe the strong rearrangement collisions of quarks and certain classes of
cross-over gluon lines. Fig. 5.7 shows some twisted graphs together with their quark
gluon contents. Meson scattering rearrangement collisions shown in Fig. 5.7(a) have
Figure 5.7 Quark-gluon exchanges summed in meson exchange diagrams.
roughly the same coupling strength as direct (untwisted) exchanges. In QED they
are the source of the main binding forces in molecules.
The exchange of two twisted meson lines (Fig. 5.7b) seems to be an important
part of diffraction scattering (Pomeron).
Two more examples are shown in Fig. 5.8. Note that in the pseudoscalar channel
these graphs incorporate the effect of the Adler triangle anomaly.
In this connection it is worth pointing out that all fundamental meson vertices
are planar graphs as far as the quark lines are concerned. Non-planar graphs are
generated by building up loops involving twisted propagators. With propagator
bands, their twisted modifications and planar fundamental couplings meson graphs
are seen to possess exactly the same topology as the graphs used in dual models
[14] except for the stringent dynamical property of duality itself: In the present
mesonized theory one still must sum s and t channel exchanges and they are by no
means the same. Only after introduction of color and the ensuing linearly rising mass
spectra one can hope to account also for this particular aspect of strong interactions.
332
5 Hadronization of Quark Theories
Figure 5.8 Quark-gluon diagrams summed in a meson loop diagram.
The similarity in topology should be exploited for a model study of an important
phenomenon of strong interactions. the Okubo, Zweig, and Iizuka rule. Obviously
all meson couplings derived by mesonization exactly respect this rule. All violations
have to come from graphs of the so called cylinder type [15] (for example Fig. 5.7b).
If it is true that the topological expansion [14] is the correct basis for explaining this
rule3 , it may also provide the appropriate systematics for organizing the mesonized
perturbation expansion.
Let us finally discuss the external sources. From Aext in (5.42) we see that
external fermion lines are connected via the full propagator G which after expansion
in powers of m′ amounts to radiation of any number of mesons (see Fig. 5.9). These
Figure 5.9 Multi-meson emission from a quark line.
mesons interact further among each other as quantum fields. Diagrammatically,
every bubble carries again a factor ΓH (P |q).
It has to be watched out that mesons are always emitted to the right of each line.
For example, the lowest order quark-quark-scattering amplitude should initially be
drawn as shown in Fig. 5.10 in order to avoid phase errors due to twisted bands. Then
Figure 5.10 Twisted exchange of a meson between two quark lines.
the graphical rules yield directly the expression (5.44) as they should. Afterwards,
arbitrary deformations can be performed if all twisted factors ηH are respected.
3
See the fourth paper in Ref. 14).
H. Kleinert, COLLECTIVE FIELDS
333
5.2 Abelian Quark Gluon Theory
External vectors fields such as photons interact with mesons according to the
third term in Eq. (5.32)
2
− 2
g D(0)
Z
dxdyV ν (x, x)g 2 D(x − y)jν (y).
(5.73)
Hence every external vector field enters the mesonic world only via an intermediate vector particle and there is a current field identity as has been postulated in
phenomenological treatments of vector mesons (VMD). Here one finds a non-trivial
coupling between the gluon and the vector mesons: As discussed before, the division
by g 2 D amounts to a removal of one rung from the ladder of the incoming meson
propagator and takes care of the direct coupling of the gluon to the quarks without
2
the ladder corrections. This is accounted for a factor gH
(q 2 )/g 2 in the propagator
sum (5.44). Thus the direct coupling of the vector meson field mH (x) to an external
vector field Aext
ν (x) can be written as:
g
XZ
H
d4 q
(2π)4
Z
d4 P
(2π)4
×Tr γ ν GM P +
q
q
ΓH (P |q)GM P −
2
2
2
gH
(q 2 )
mH (q)Aext
ν (−q). (5.74)
g2
In a mesonic graph, the removal of one rung will be indicated by a slash. As an
example, the lowest order contribution to the quark gluon form factor is illustrated
in Fig. 5.11. The slash guarantees the presence of the direct coupling. The free
Figure 5.11 Vector meson dominance in coupling of external photon to a quark line.
propagator of an external gluon is given by the second term of Eq. (5.32). The
lowest radiative corrections consist in an intermediate slashed vector meson (see
Fig. 5.12). Here the slash is important to ensure the presence of one single quark
loop.
The divergent last term in the external action (5.32) has no physical significance
since it contributes only to the external gluon mass and can be cancelled by an
appropriate counter term.
A final remark concerns the bilocal currents as measured in deep inelastic electron
and neutrino scattering. These are vector currents of the type
j ν (x, y) ≡ ψ̄(x)γ ν ψ(y).
(5.75)
334
5 Hadronization of Quark Theories
Figure 5.12 Vector meson dominance in photon propagator.
It is obvious that also for bilocal currents there is a current-field identity with the
bilocal field V µ (x, y). In fact, if one would have added an external source term
Cν (x, y) in the quark action:
∆Aext ≡
Z
dxdy ψ̄(x)γ ν ψ(y)Cν .(x, y)
(5.76)
This would appear in the mesonized version in the form
∆Aext =
Z
dxdy
1
ig 2D(x
− y)
V ν (x, y)Cν (x, y),
(5.77)
which proves our statement. Again, a rung has to be removed in order to allow for
the pure quark contribution (see Fig. 5.13).
Figure 5.13 Gluon diagrams in dashed meson propagator.
Bilocal currents carry direct information on the properties of Regge trajectories
[18]. Therefore the present bilocal field theory seems to be the appropriate tool for
the construction of a complete field theory of Reggeons [19], which is again equivalent to the original quark gluon theory. Technically, such a construction would
proceed via analytic continuation of the propagators (5.34) in the angular momentum (and the principal quantum number) of the mesons H. The result would be a
“reggeonized” quark gluon theory. The corresponding Feynman graphs would guarantee unitary in all channels. Present attempts at such a theory enforces at channel
unitarity only [20].
5.3
Limit of Heavy Gluons
As an illustration of the mesonization procedure we now discuss in detail the limit
of very heavy gluons [1], [34]. Apart from its simplicity, this limit is quite attractive
H. Kleinert, COLLECTIVE FIELDS
335
5.3 Limit of Heavy Gluons
on physical grounds since it may yield a reasonable approximation to low energy
meson interactions. This is suggested by the following arguments:
Suppose hydrodynamics follows a colored quark gluon theory. In this theory
the color degree of freedom is very important for generating a potential between
quarks rising at long distances which can explain the observed great number of
high-mass resonances. However, as far as low-energy interactions among the lowest
lying mesons are concerned, color seems to be a rather superfluous luxury:
First, many fundamental aspects of strong interaction dynamics are independent
of color. Examples are chiral SU(3)×SU(3) current algebra relations with PCAC
(together with the low-energy theorems derived from both) and the approximate
light cone algebra.
Second, there is no statistics argument concerning the symmetry of the meson
wave function as there is for baryons [21].
Third, high-lying resonances are known to contribute very little in most dispersion relations of low-energy amplitudes. For example, the low-energy value of
the isospin odd ππ-scattering amplitude is given by a dispersion integral over the
mesons ρ and σ with ≈ 90% accuracy [22]. Similarly, πρ-scattering is saturated
by the intermediate mesons π and A1 . By looking at all scattering combinations
we can easily convince ourselves that the resonances π, ρ, A1 , σ form an approximately closed “subworld” of mesons as far as dispersion relations are concerned. As
a consequence, it would not at all be astonishing if the neglect of color in a quark
gluon theory would not change the dynamics when restricting the attention to this
mesonic “subworld”.4 The point is now that, in the limit of a large gluon mass
µ → ∞, exactly this restricted set of mesons appears as particles in the mesonized
quark gluon theory (5.1) without color. Thus it might be considered as some approximation to the low-energy aspects of the colored version. Indeed, we shall see
that the mesonized theory coincides exactly with the well-known chirally invariant
σ-model. In the past, this model has proven to be an appropriate tool for a rough
description of low-energy meson physics [23]. Our derivation of the σ model via
mesonization will render several new relations between meson and quark properties
[1]. We shall at first confine ourselves to SU(2) quarks only, such that symmetry breaking may be neglected. The extension to broken SU(3) will be performed
afterwards.
In order to start with the derivation observe that in the limit µ → ∞, the gluon
propagator approaches a δ-function:
iD(x − y) →
1 (4)
δ (x − y).
µ2
(5.78)
The equation of motion (5.32) forces m′ (x, y) to become a local field m′ (x):
m′ (x, y) → m′ (x)δ (4) (x − y),
(5.79)
4
There exists a simple estimate concerning the electromagnetic decay of π 0 → γγ based on
short-distance arguments and therefore depends on color [23]. However, the same decay can be
estimated also via intermediate distance arguments, namely by using the coupling ρωπ and vector
meson dominance. Then color does not enter the argument.
336
5 Hadronization of Quark Theories
which satisfies the free field equation
g2
m (x) = −i 2 ξ
µ
′
Z
dyGM (x − y)m′ (y)GM (y − x).
(5.80)
In the local limit, the action without external sources takes the form
A[m′ ] =
Z
1
(GM m′ GM m′ ) (x, x)
2
)
µ2 1 ′ 2 1 ′
′ n
(GM m ) (x, x) − 2 m (x) − m (x)m0 ,
g 2ξ
ξ
dx trSU(2) GM (x, x)m′ (x) −
+
X
n
(−i)n−1
n
(5.81)
R
where (GM m′ GM m′ ) (x, x) stands short for dyGM (x − y)m′ (y)GM (x − y) etc. As
in the earlier general discussion, the constant m0 is determined by the vanishing of
the tadpole parts in (5.81) which amounts to balancing the constant contributions
in the wave equation. Due to the singularity of GM (x − y) for x → y this condition
has a meaning only if a cutoff is introduced such that GM (0) is finite:
[GM (0)]αβ =
Z
= M
"
d4 P
i
(2π)4 P/ − M
#
=M
αβ
Z
0
Λ
d 4 PE 2
2 −1
P
+
M
δαβ
(2π)4 E
π2 2
2
2
2
δαβ ≡ MQδαβ .
Λ
−
M
log
Λ
/M
(2π)4
(5.82)
Here the dP 0 integration has been Wick-rotated by 900 such that the momentum
P µ = (P 0, P) becomes (iP 4 , P) with P 4 ∈ (−∞, ∞) along the integration path.
The new real momentum (P 4 , P) has been denoted by PEµ , and its euclidean scalar
product by PEµ = P 42 + P2 = −P 2 . The tadpoles can now be cancelled by setting
m0 = 4
g2
QM.
µ2
(5.83)
Remembering the relation to the bare quark mass m0 = M − M, this determines
the connection between the “true” quark mass M and the bare mass M contained
in the Lagrangian:
M =M+4
g2
QM.
µ2
(5.84)
Equation (5.82) is often called “gap equation” because of its analogous appearance
in the theory of superconductivity (see Chapter 3 and Ref. [24]).
Consider now the free part A2 [m′ ] of the action. Performing again a decomposition of type (5.13) but with the local field m′ (x), it can be written in the form
′
Z
A2 [m ] = dx trSU(2)
)
i
µ2 h
1 ′
mi (x)Jij (i∂)mj (x) − 2 S 2 (x)+P 2(x)−2V 2 (x)−2A2 (x) ,
2
2g
(5.85)
H. Kleinert, COLLECTIVE FIELDS
337
5.3 Limit of Heavy Gluons
where m′i (x)(i = 1, 2, 3, 4) stands short for the fields5 S(x), P (x), V (x), A(x) and
the trace runs only over internal SU(2) indices. The coefficients Jij (q) are given by
the integrals
Jij (q) ≡ −4
Z
1
1
d 4 PE
tij (P |q),
2
4
2
(2π) (P + q/2)E + M (P − q/2)2E + M 2
where tij (P |q) denotes the Dirac traces
(
!
q/
q/
1
tij (P |q) ≡ trDirac Γi P/ + + M Γj P/ − + M
4
2
2
!)
,
(5.86)
(5.87)
with Γi (i = 1, 2, 3, 4) abbreviating the standard Dirac covariants 1, iγ5 , γ ν , γ ν γ5 .
The traces are displayed in Appendix A Eq. (5A.33). Some of them grow quadratically in P . The corresponding integrals Jij (q) are quadratically divergent for large
cutoffs. The others diverge logarithmically. If one introduces the basic integral
L≡
Z
!
d 4 PE
π2
Λ2
1
=
log
−1 ,
(2π)4 (PE2 + M 2 )2
(2π)4
M2
(5.88)
the divergent parts of jij are (see Appendix 5B)
Jss (q) =
Jpp (q) =
JV µ V ν (q) =
JAµ Aν (q) =
!
q2
− 2M 2 ,
Q+L
2
q2
Q + L ; JP Aν = −iLMq ν ,
2
1 2 µν
− q g − q µ q ν L,
3
i
1 h 2 µν
q g − q µ q ν − 6M 2 g µν L,
−
3
JAµ P (q) = iLMq µ ,
(5.89)
(5.90)
(5.91)
(5.92)
(5.93)
with all other integrals vanishing.
If we neglect the finite contributions as compared with these divergent ones, the
action A2 [m′ ] is seen to correspond to the local Lagrangian6
(
"
#
1 ′
µ2
L(x) = trSU(2)
+ 4M 2 L − 2 S ′ (x)
S (x) 4Q − 2
2
g
#
"
2
µ
1
P (x) 4Q − 2 L − 2 P (x)
+
2
g
"
#
2
2µ
1
4
+
Vµ (x) ( g µν − J µ ∂ ν ) L + 2 Vν (x)
2
3
g
The Lorentz indices of V ν and Aν fields are suppressed.
Since m(x) and m′ (x) differ only by a Dirac scalar constant m0 1α,β there is no difference
between primed and unprimed fields except for S ′ (x) = S(x) − m0 .
5
6
338
5 Hadronization of Quark Theories
+
"
#
4
1
2µ2
Aµ (x)
( g µν − ∂ µ ∂ ν ) L + 8M 2 g µν L + 2 Aν (x)
2
3
g
µ
)
µ
+ 2ML [∂µ P (x)A (x) + A (x)∂µ P (x)] .
(5.94)
If we insert the gap equation (5.84) in this Lagrangian, the quadratically divergent
terms Q can be eliminated. The mixed terms may be removed by introducing a new
field Aeµ (x) via
and fixing λ as
Aµ (x) = Aeµ (x) + λ∂ µ P,
(5.95)
λ = −3M/m2A ,
(5.96)
m2A = m2V + 6M 2 ,
(5.97)
m2V = 3µ2 /(2g 2L).
(5.98)
where m2A stands short for
with
This substitution produces additional kinetic terms for the pseudoscalar fields which
now appears with a factor
2
−trSU(2) (P (x) P (x)) 1+ m2A λ2 +4Mλ L = trSU(2) (∂µ P (x)∂ µ P (x)) ZP−1 L. (5.99)
3
Using (5.96), this renormalization factor becomes
ZP−1 = 1 − 6M 2 /m2A .
(5.100)
After this diagonalization, the quadratic part of the collective Lagrangian reads
L(x) = trSU(2)
1
∂µ S ∂ S − 4M + m2V M/M S ′2
3
1
−∂µ P ∂ µ P ZP−1 − m2V M/MP 2
3
1 µν V
2 2 2
− FV Fµν + mV Vµ
3
3
1 µν Ã 2 2 2
− Fà Fµν + mA õ × L
3
3
′ µ
′
2
(5.101)
where F µν V,Ã are the usual field tensors of vector and axial vector fields. The particle
content of this free Lagrangian is now obvious. There are vector mesons of mass
m2V , axial-vector mesons of mass m2A and scalar and pseudoscalar mesons of mass
1
M
,
m2S = 4M 2 + m2V
3
M
1 2M
m2P =
m
ZP .
3 VM
(5.102)
(5.103)
H. Kleinert, COLLECTIVE FIELDS
339
5.3 Limit of Heavy Gluons
With (5.98), the constant (5.100) can also be written as
ZP−1 =
m2V
.
m2A
(5.104)
As we have argued before, there is a good chance that the fields P, V, S, A describe
approximately the lowest lying mesons π, ρ , σ, A. Let us test this hypothesis as
far as the masses are concerned. Since experimentally m2A1 ≈ 2m2ρ the factor Zπ
becomes ≈ 2. Furthermore, Eq. (5.97) determines the quark mass as:
6M 2 = m2A1 − m2ρ ; M ≈ 310MeV,
(5.105)
in good agreement with other estimates [25]. The small pion mass yields via (5.103)
M ≈ 15 MeV.
(5.106)
Thus the bare quark mass has to be extremely small. Also this result has been
obtained by many authors [26]. It is common to all models in which the smallness
of the pion mass is related to the approximate conservation of the axial current
(PCAC).
The scalar meson finally is predicted from (5.102) to have a mass
mS ≈ 2M ∼ 620 MeV.
(5.107)
This agrees well with the observed broad resonance in ππ-scattering [27] [22].
One disagreement with experiment appears in connection with the SU(2)-singlet
pseudoscalar mass (the η-meson). According to (5.103) it should be degenerate with
the pion. The resolution of this problem will be discussed later when the theory has
been extended to SU(3).
After these first encouraging results we shall rename the fields P, S, V, S, A by
the corresponding particle symbols
√
LP ≡ π,
−
√
′
′
LS ≡ σ ,
−
s
2
LV µ ≡ ρµ ,
3
−
s
2 µ
LA ≡ Aµ1
3
(5.108)
where a normalization factor has been introduced in order to bring the kinetic terms
in the Lagrangian to a conventional form.
A comment is in order concerning the appearance of a quadratic divergence
in equations (5.84), (5.89). Such a strong divergence indicates, that the limiting
procedure µ → ∞ of Eq. (5.78) has been performed too carelessly. In fact, if one
2
inserts (5.78) into the action (5.4), the theory becomes of the ψ̄ψ type and thus
non-renormalizable. In order to keep the renormalizability while dealing with a
large gluon mass µ2 ≫ Λ2 . Then the quadratic divergence becomes actually of the
logarithmic type (compare (5.53)):
Z
1
1
d 4 PE
2
2
4
2
(2π) PE + M PE + µ2
"
!
!#
π2
Λ2
Λ2
2
2
=
µ log
+ 1 − M log
+1 .
µ2 − M 2
µ2
M2
Q =
(5.109)
340
5 Hadronization of Quark Theories
(which in the careless limit µ2 → ∞ reduces again to (5.82)). The logarithmic divergence (5.88) on the other hand becomes in this more careful treatment independent
of the cutoff which is replaced by the large gluon mass
L =
Z
1
1
d 4 PE
2
2
4
2
2
(2π) (PE + M ) (PE + µ2 )
"
#
π2
µ2
µ2
π2
µ2
2
2
2
×
µ
log
≈
+
M
−
µ
log
M
(2π)4
M2
(2π)4 (µ2 − M 2 )2
(5.110)
Hence all our results refer to a renormalizable theory if one reads both Q and L as
logarithmic expression once in the cutoff and once in the gluon mass, respectively.
Let us now proceed to study the interaction terms. The n’th order contribution
to the action is given by
(−i)n−1
An [m ] =
n
′
Z
dxTr (GM m′ )
n
(5.111)
In momentum space this can be written as the one loop integral
Z
(−1)n−1
d4 qn
d4 q1
.
.
.
(2π)4 δ (4) (qn + . . . + q1 )
n
(2π)4
(2π)4
Z 4
1
1
d PE
·
·
·
×
2
(2π)4 (P + qn + . . . + q1 )E + M 2
(P + q1 )2E + M 2
An [m′ ] = 4
h
i
× tin ...i1 (P |qn−1, . . . , q1 ) trSU(2) m′in (qn ) · . . . · m′i1 (q1 ) ,
where tin ...i1 (P |qn−1, . . . , q1 ) is the generalization of the tensor (5.87):
(5.112)
tin ...i1 (P |qn−1 , . . . , q1 ) ≡
(5.113)
i
h
1
P + q/ n−1 + . . . + q/ 1 + M) Γin−1 . . . Γi2 (/
P + q/ 1 + M) Γi1 (/
P + M) .
Tr Γin (/
4
The result is hard to evaluate in general (except in a 1 + 1 dimensional space).
With the approximation of a large cutoff one may however, neglect again all contributions which do not diverge. This considerable simplifies the results. Since
tin...i−1 (P |qn−1 , . . . , q1 ) are polynomials in P of order n, the integral is seen to converge for n > 4. For n = 4 there is a logarithmic divergence with only the leading
momentum behavior of tin . . . i1 contributing. For n ≤ 3 also lower powers in momentum P of tin . . . i1 (P |qn−1 , . . . , q1 ) diverge logarithmically. A simple but somewhat
tedious calculation of all the integrals (see Appendix 5B) yields the remaining terms
in the Lagrangian. They can be written down in a most symmetric fashion by employing the unshifted fields S(x) ≡ M ∗ S ′ (x) rather than S ′ , or in renormalized
form7
√
(5.114)
σ(x) = − LM + σ ′ (x).
7
Note that with this notation m(x) = m0 + m′ (x) = (M − M) + m′ (x) = −M + S + Pi γ5 +
V γµ + Aµ γµ γs
µ
H. Kleinert, COLLECTIVE FIELDS
341
5.3 Limit of Heavy Gluons
Then the Lagrangian reads
L(x) = Tr SU(2)
i
nh
(Dµ σ)2 + (Dµ π)2 + M02 σ 2 + π 2
i
1 V 2 1 A2
2 h
− γ 2 σ 4 + π 4 − 2σπσπ − Fµν
− Fµν
3
2 2
2 2√
2
2
2
+mV Vµ + Aµ − mV LM .
3
(5.115)
Here Dµ σ and Dµ π are the usual covariant derivatives:
Dµ σ = ∂µ σ − iγ [Vµ , σ] − γ {Aµ , π} ,
Dµ π = ∂µ π − iγ [Vµ , π] + γ {Aµ , σ} ,
(5.116)
V
A
and Fµν
, Fµν
are the covariant curls
V
Fµν
= ∂µ Vν − ∂ν Vµ − iγ [Vµ , Vν ] − iγ [Aµ , Aν ] ,
A
Fµν
= ∂µ Aν − ∂ν Aµ − iγ [Vµ , Aν ] − iγ [Aµ , Vν ] .
(5.117)
The constant γ is
γ=
s
3
.
2L
(5.118)
It describes the direct coupling of the vector mesons to the currents, i. e. it coincides with the coupling conventionally denoted by γρ . Here γ has its origin in the
renormalization of the fields. The mass term stands short for
1
2M02 ≡ 2M 2 − m2V M/M.
3
(5.119)
Actually, the so-defined mass quantity has an intrinsic significance. This can be
seen by deriving the Lagrangian in a different fashion from the beginning. Consider
the tadpole terms of the action
A1 [m1 ] =
Z
dx trSU(2)
(
)
1
GM (x, x)m (x) − m′ (x)m0 .
ξ
′
(5.120)
In the former treatment we have eliminated m0 completely by giving the quarks a
mass M satisfying the gap equation
g2
M − M ≡ m0 = 4 2 QM.
µ
(5.121)
Instead, we could have introduced an auxiliary mass M0 satisfying the equation
M0 = 4
g2
Q0 M0
µ2
(5.122)
342
5 Hadronization of Quark Theories
where Q0 is the same function of M0 as Q is of M. The connection between this M0
and the other masses is obtained by inserting M = M0 δM into Q:
Q = Q0 − 2M0 δM (1 + δM/2M0 ) L
(5.123)
which holds exactly in δM with only small corrections for large cutoffs (notice that
at this accuracy L0 = L). Inserting this into (5.121) we find
M=4
g2
L2M0 M (1 + δM/2M0 ) δM
µ2
(5.124)
M=
12
M0 M (1 + δM/2M0 ) δM.
m2V
(5.125)
and using m2V from
If now m(x) is split in a different fashion
m(x) = m̃0 + m′′ (x)
(5.126)
with a new m̃0 = M0 − M then the propagator G(x, y) would have an expansion
G(x, y) = GM0 (x − y) − i (GM0 m′′ GM0 ) (x, y) + .
(5.127)
For this reason, the derivation of all Lagrangian terms yields exactly the same results
as before only with m′′ , M0 , L0 , and Q0 occurring rather than m′ , M, L and Q,
respectively. There are only two differences: First, due to the gap equation (5.122),
the scalar and pseudoscalar mass terms become 4M02 and O rather than (5.102),
(5.103) second, the tadpole terms in this derivation do not cancel completely. Instead
one finds from (5.120)
Z
(
!
)
M0 − M
A1 [m ] =
dxtrSU(2)
4Q0 M0 −
m′′ (x)
g 2/µ2
Z
Z
2
µ2
=
dx 2 trSU(2) {Mm′′ (x)} = m2x dxtrSU(2) {Mm′′ (x)} . (5.128)
g
3
′
These tad pole terms provide exactly the necessary additional shifts in the fields
which are needed in order to bring the scalar and pseudoscalar masses from 4M02
and O to their correct values m2σ and m2π . The symmetric form (5.115) of the
Lagrangian is again reached by introducing the original unprimed fields
√
S(x) ≡ M0 + S ′′ (x), σ = − LM0 + σ ′′ .
(5.129)
Then the mass term appears as an SU(3) × SU(3) invariant
2M02 σ 2 + π 2 .
H. Kleinert, COLLECTIVE FIELDS
343
5.3 Limit of Heavy Gluons
8
Notice now this coincides exactly with the former calculation which rendered (see
4.38)
1 2
2M − mV M/M σ 2 + π 2 .
3
2
Inserting here M = M0 + δM and (5.125) gives
1
δM
2M − m2V M/M = 2M02 + 4M0 δM + 2(δM)2 − 4M0 1 +
3
M0
2
= 2M0 .
2
!
δM
(5.130)
Hence the SU(3) symmetric mass M0 defined by the gap equations (5.122) coincides
with the mass M0 introduced as an abbreviation to the mass combination (5.119).
The Lagrangian (5.115) is recognized as the standard chirally invariant σ model.
Its symmetry transformations are for isospin
δσ = i [α, σ] ;
δπ = i [α, π] ,
1
δV µ = i [α, V µ ] + ∂ µ α;
γ
δAµ = i [α, Aµ ] .
(5.131)
For axial transformations the fields change according to
δ̄σ = − {ᾱ, π} ;
δ̄V µ = i [ᾱ, Aµ ] ;
δ̄π = {ᾱ, σ} ,
1
δ̄Aµ = i [ᾱ, V µ ] + ∂ µ ᾱ.
γ
(5.132)
The only term in the Lagrangian which is not invariant is the last linear term. In
fact from
√
2
(5.133)
δ̄L = i m2V M L {ᾱπ} ≡ i {ᾱ, ∂A}
3
one finds
∂A(x) = fπ m2π Zπ−1/2 π(x).
(5.134)
Introducing the conventional pion decay constant via
∂A(x) ≡ fπ m2π Zπ−1/2 π(x)
8
(5.135)
With this substitution, the unprimed field S really coincides with the formerly introduced field
S since now
m(x)
=
(M0 − M) + S ′′ (x) + P (x)iγ5 + . . . = −M + S(x) + P (x)iγ5 + . . . ,
whereas before
m(x)
= (M − M) + S ′ (x) + P (x)iγ5 + . . . = −M + S(x) + P (x)iγ5 + . . . .
344
5 Hadronization of Quark Theories
one can read off
√
2
f πm2π = Zπ1/2 m2V LM.
3
(5.136)
Inserting m2π from (5.103) this gives
√
fπ = Zπ−1/2 2M L.
By squaring this and using γ =
f π2 =
(5.137)
q
3/2L, one obtains
m2ρ m2A − m2ρ
6M 2 1
=
,
Zπ γ 2
γ2
m2A
(5.138)
which for m2A ≈ 2m2ρ renders the well-known KSFR relation. Apart from that, the
model has the usual predictions
gρππ = γρ
m2 − m2
1− A 2 ρ
2mA
!
3
≈ γρ ,
4
(5.139)
and
mρ
1 2
γ Zπ fπ ≈
≈ 4, hA1 ρπ = 0,
2mρ
2fπ
mρ
≈ 8,
= γZπ1/2 ≈
fπ
2 1 2
mσ √
2 ≈ 9.
=
γ fπ Zπ3/2 ≈
mσ 3
fπ
gA1 ρπ =
(5.140)
gA1 σπ
(5.141)
gσππ
(5.142)
When compared with experiment, the only real discrepancy consists in the d-wave
A1 ρπ coupling, hA1 ρπ, being absent. Additional chirally invariant terms are needed
in the Lagrangian, for example, the so-called δ-term:
δTr
h
i
A
V
Fµν
+ Fµν
D µ (σ + iπ) D ν (σ − iπ) − (V → −V, π → −π) .
(5.143)
Such terms appear in our derivation if the approximation of large µ2 is improved by
terms which do not grow logarithmically in µ.
Let us now determine the couplings of π, σ, ρ, A1 , A1 to external quark fields.
The external propagation proceeds via
iη̄Gη = iη̄GM0 η + η̄GM0 m′′ GM0 m′′ GM0 µ − . . . .
(5.144)
If we define the couplings by
L ≡ gπQQ ψ̄iγ5 τa ψπ a + gσQQ ψ̄τa ψσ a + gV QQ Ψ̄γ µ
τa
τa
ΨVµa + gAQQ Ψ̄γ µ γ5 ψAaµ ,
2
2
(5.145)
H. Kleinert, COLLECTIVE FIELDS
345
5.3 Limit of Heavy Gluons
we can read off
1
M
M
1
, gρQQ = gA1 QQ = γ. (5.146)
; gσQQ = √ =
gπQQ = √ Zπ1/2 =
1/2
fπ
2 L
2 L
fπ Zπ
We see that the vector coupling to quark-antiquark pairs shows vector-meson dominance. Due to PCAC, also the Goldberger-Treiman relation is respected
gπQQ = gA
M
,
fπ
(5.147)
since the axial charge of the quark is gA = 1. With the quark mass being roughly
M ≈ mN /3, the pionic coupling to quarks is considerably smaller than to nucleons.
Numerically this implies that
2
2
gπQQ
1 gπN
N
≈
≈ 0.86.
4π
17 4π
(5.148)
The σ-meson couples like
2
gσQQ
≈ 0.43.
4π
Vector- and axial-vector mesons, on the other hand, couple as strongly to quarks as
to nucleons:
2
2
2
gρQQ
gAQQ
gρN
γ2
N
=
=
≈ 2.6 ≈
4π
4π
4π
4π
!
.
(5.149)
We are now ready to extend our consideration to SU(3) (and higher groups).
In this case the explicit symmetry breaking in the Lagrangian is too large to be
neglected. Thus the bare masses M of the quarks have to be considered as a matrix


M≈
Mn
Md
M
s


.
(5.150)
The derivation of the Lagrangian presented above [via the gap equation (5.122)]
has shown complete SU(3) symmetry of M02 . Hence when extending from SU(2) to
SU(3), no change occurs except in the last symmetry breaking term of (5.115). As a
consequence, the mass expressions for m2p and m2s remain as they are, only that the
renormalization constants Zp become more complicated SU(3)-dependent quantities
due to the involved mixing of pseudoscalar and axialvector mesons. For a complete
discussion of this SU(3) × SU(3) invariant chiral Lagrangian the reader is referred
to the review articles [23]. Here we only give a few results:
A best fit to π K meson masses requires9


9
M≈
15
15
435
For other determinations of Mu,d,s see Ref. [26].


 MeV.
(5.151)
346
5 Hadronization of Quark Theories
Thus the explicit symmetry breakdown of SU(3) caused by the bare masses is quite
large. The standard parameter [28] C characterizes this:
√1
3
C≡
Mn + Md − 2Ms
√
M
q
≈
−1.28(≈
−
=
2).
2
M0
(Mn + Md + Ms )
3
8
(5.152)
Inserting into (5.125) we find the shifts in the quark masses caused by dynamics


7
δM = 
7
127


 MeV,
(5.153)
and hence for the “physical” quark masses

3/2

3/2
M ≈
432


 MeV.
(5.154)
Thus contrary to the large explicit SU(3) violation the bare masses M, the physical
quark masses M show only the moderate violation:
C′ ≡
√1
3
8
M n + M d − 2M s
M
≈ −16%.
= q
2
n + M d + M s)
M0
(M
3
(5.155)
Since the quark masses M are produced almost completely by dynamical effects
we expect some symmetry breakdown to appear also in the vacuum. A measure of
this is provided by the expectation values of the scalar quark densities
h0|Ũ a |0i ≡ h0|ψ̄(x)
λa
ψ(x)|0i.
2
(5.156)
In the mesonized theory, the scalar densities are identical with the scalar fields, up
to a factor:
Sji (x) = −
g2 i
ψ̄ (x)ψj (x),
µ2
(5.157)
as can be seen most easily by considering the equations of constraint (5.15) following
from the Lagrangian (5.12) in the large −µ limit. Hence
h0|ũa |0i = −
µ2 X a j
µ2
µ2 a
i
a
λ
h0|S
|0i
=
−
tr
(Mλ
)
=
−
M .
SU(2)
i
j
g 2 i,j
2g 2
2g 2
(5.158)
Inserting (5.98) and (5.137), the factor becomes simply
1
µ2
=
2
2g
3
fπ2
1 2
≈ fπ2 ,
L = mV Zπ
2
2 g2
3
4M
L
3 µ2
1
H. Kleinert, COLLECTIVE FIELDS
347
5.3 Limit of Heavy Gluons
so that
s
2 n
M + M d + M s ≈ −8 × 10−3GeV3 ,
3
8
2
8
′
h0|ũ |0i ≈ −fπ M = c h0|ũ0 |0i.
0
h0|ũ |0i ≈
fπ2 M 0
=
−fπ2
(5.159)
(5.160)
This shows that the SU(3) violation in the vacuum equals that in the quark masses
≈ −16%.10 Notice that the three results (5.103), (5.125) and (5.157) are in complete
agreement with what one obtains by very general considerations using only chiral
symmetry and PCAC (see Appendix 5C).
The extension of the Lagrangian to SU(3) produces additional defects which are
well known from general discussions of chiral SU(3) × SU(3) symmetry [23]. For
example the vector mesons ω1 , ϕ are not mixed (almost) ideally as they should but
ϕ remains close to an SU(3) singlet. In general discussions, additional terms have
been added to the chiral Lagrangian in order to account for this. There are the so
called “current mixing terms”:
Tr
V
Fµν
+
A 2
Fµν
(σ + iπ)(σ − iπ) + (A → −A, π → −π) ,
(5.161)
as well as “mass mixing terms”
n
o
Tr (V µ + Aµ )2 (σ + iπ) (σ − iπ) + (V µ − Aµ )2 (σ − iπ) (σ + iπ) .
(5.162)
In our derivation these arise as a next correction to the µ2 → ∞ limit. Another
problem is the degeneracy of the ideally mixed isosinglet pseudoscalar meson µ′ideal
with the pion. In order to account for the fact that the η ′ (Ξ0 ) meson is almost a
pure SU(3) singlet and much heavier than the other pseudoscalar mesons one needs
some chirally symmetric term
det (σ + iπ) + det (σ − iπ) .
(5.163)
Such a term breaks PCAC for the ninth axial current. It is well known [30] that
the quark gluon triangle anomaly operates in the singlet channel. It might be
capable of producing such a PCAC violation. In fact, if this was not true, quantum
electrodynamics would possess an exactly massless Goldstone boson [31] with η
quantum numbers. The term (5.163) will also appear in the effective Lagrangian
when µ is no longer very large.
It is obvious that corrections to the µ2 → ∞ approximation will become even
more important if one tries to extend the consideration to SU(4) since then vector
and pseudoscalar masses are quite heavy. In addition, the narrow width of the SU(4)
vector meson ψ/J seems to indicate that short-distance parts of the gluon propagator
are probed. Thus the colorless quark gluon theory itself cannot be considered any
more a realistic approximation to the colored theory.
10
In Ref. [30], the breaking of SU(3)-symmetry in the vacuum was neglected. For a more general
discussion and earlier references see Ref. [29].
348
5 Hadronization of Quark Theories
At this place we should remark that present explanations of electromagnetic
mass differences require also a breakdown of SU(2)-symmetry in M [32]. This is
conventionally parametrized by
d≡
M3
Mn − Md
q
.
=
2
n + Md + Ms )
M0
(M
3
(5.164)
From meson masses (as well as from the electromagnetic η → 3π decay) one finds
[33]
d ≈ −3%.
(5.165)
This amounts to the bare quark masses


10
20
M≈
435


 MeV,
(5.166)
giving the “true” masses


M ≈
310
315
432


 MeV.
(5.167)
Thus the SU(2) breaking of the vacuum is very small:
d′ ≡
h0|ũ3|0i
M3
=
≈ −0.6%.
h0|ũ0|0i
M0
(5.168)
With all parameters fixed numerically we should finally check whether the approximation of a large gluon mass is self-consistent. From (5.137) we have
1 fπ2
≈ .046.
L≈
2 M2
(5.169)
Inserting this into (5.110) we calculate
log
µ2
∼ (4π)2 × 0.046 ≈ 7.3,
M2
(5.170)
and hence
µ2 ≈ 1500M 2 ≫ M 2 ,
(5.171)
µ ≈ 12BeV.
(5.172)
or
H. Kleinert, COLLECTIVE FIELDS
349
5.4 Summary
It is gratifying to note that this value is much larger than the mass of the vector
mesons. In this way it is assured that higher powers of q 2 / (PE2 + M 2 ) which were
neglected in the derivation of the Lagrangian remain really small as compared to
unity for all mesons of the theory (see Appendix 5B).
We should point out that the quark gluon theory in the limit µ2 → ∞ coincides
with the well-known Nambu-Jona Lasinio [24] model which has proven in the past
to be a convenient tool for studying the spontaneous breakdown of chiral symmetry
and the dynamical generation of PCAC. Those authors have demonstrated the close
analogy of the dynamic structure of this model with that of superconductivity. As
we have mentioned before the equation (5.121) removing the tadpoles in the action
is analogous to the gap equation for superconductors.
A similar analogy to superconductors exists also for the hadronized theory. The
classical version of it corresponds exactly to the classical Ginzburg-Landau equation
for type II superconductors in which the gap is allowed to be space time dependent.
In fact, the classical mesonized theory can be derived alternatively by assuming such
a dependence in the gap equation (5.82) [34, 3]. The advantage of our functional
derivation is that the mesonized theory is not merely some classical approximation
but becomes, upon quantization, completely equivalent to the original quark gluon
theory.
A final comment concerns the Okubo-Zweig-Iizuka rule. As argued in the general section, the meson Lagrangian exactly respects this rule. This can be checked
directly for all interaction terms in (5.115). Violations of this rule are all coming
from meson loops whose calculations allow to estimate their size.
5.4
Summary
We have shown that in the absence of color, quark gluon theories can successfully
be mesonized. The resulting quantum field theory incorporates correctly many features of strong interactions. Its basic fields are bilocal and the Feynman rules are
topologically similar to dual diagrams. Our considerations have take place at a
rather formal level. Certainly, there are many problems which have been left open.
For example, there is need for an understanding of the non-trivial gauge properties
of the bilocal theory. Also, a consistent renormalization procedure will have to be
developed in future investigations.
The inclusion of color is the challenging problem left open by this investigation.
If the color quark gluon theory is really equivalent to some kind of dual model the
corresponding mesonization program should not produce bilocal but multilocal fields
which are characterized by the position of a whole string rather that just its end
points. A field theory should be constructed for gauge invariant objects like
ψ̄(x) exp ig
Z
y
x
Gν (z)dzν ψ(y)
which depend on the whole path from x to y.
350
5 Hadronization of Quark Theories
The difficulty in a direct generalization of the previous procedure is the selfinteraction of the gluons. Only after the infrared behavior of gluon propagators will
be known, bare mesons can be constructed inside the corresponding potential well
and the “mesonization” methods can serve for the determination of the complete
residual interactions.
It is hoped that mesonic Feynman rules in the presence of color will follow a
pattern similar to that found here for the non-abelian theory.
Let us finally mention that an interesting field of applications of our methods
lies in solid-state physics. Semi-conductors in which conduction and valence bean
have only small separations may show a phase transition to what is called excitonic
insulator. The critical phenomena taking place inside such an exciton system will
find their most appropriate description by studying the scaling properties of the
bilocal field theory.
Appendix 5A
Remarks on the Bethe-Salpeter Equation
Consider the four Fermion Green function
(4)
Gαβ,α′ β ′ (x, y; x′y ′ ) ≡ h0|T ψα (x)ψ̄β (y)ψ̄α′ (x′ )ψβ ′ (y ′ ) |0i,
which becomes in the interaction picture
h
(4)
Gαβ,α′ β ′ (x, y; x′, y ′ ) ≡ h0|T eig
×h0|T eig
R
R
d4 z ψ̄(z)γ µ ψ(z)Gµ (z)
d4 z ψ̄(z)γ µA ψ(z)G
|0i
µ (z)
(5A.1)
i−1
(5A.2)
ψα (x)ψ̄β (y)ψα′ (x′ )ψβ ′ (y ′) |0i.
Expanding the exponential and keeping only the ladder exchanges corresponding to
the Feynman graph in Fig. 5.14, we obtain
(4)
Gαβ,α′ β ′ (xy, x′ y ′) = Gαα′ (x − x′ ) Gβ ′ β (y ′ − y)
+g 2
+g
4
Z
Z
(5A.3)
dx1 dy1Gαα1 (x − x1 ) γαµ1 α′ Gα′1 α′ (x1 − x′ ) Gβ ′ β1′ (y ′ − y1 ) γµβ ′ β Gβ1 β (y1 − y) D(x − y)
1 1
1
dx1 dy1dx2 dy2 Gαα1 (x −
x1 ) γαµ11α′ Gα′1 α2
1
(x1 −
x2 ) γαµ22α′ Gα′2 α2
2
(x2 − x′ )
D (x1 − y1 ) D (x2 − y2 ) Gβ ′ β2′ (y ′ − y2 ) γµ2 β ′ β Gβ2 β1′ (y2 − y1 ) γµ1 β1′ β1 Gβ ′ β (y1 − y)
2 2
+... .
(5A.4)
Figure 5.14 Diagrams in Bethe-Salpeter equation.
H. Kleinert, COLLECTIVE FIELDS
351
Appendix 5A Remarks on the Bethe-Salpeter Equation
The series can be summed by an integral equation
(4)
Gαβ,α′ β ′
′
′
′
′
(x, y; x , y ) =Gαα′ (x − x )Gβ ′ β (y − y) + g
2
(4)
Z
dx1 dy1
×Gαα1(x−x1 )γαµ1 α′ Gα′ β ′ ,α′ β ′ (x, y1 ; x′ y ′)γµ β1′ β1 Gβ1 β (y1 −y)D(x1 −y1 ) .
1
(5A.5)
1 1
With the abbreviation
ξα1 β1 ,β1′ α′1 = γαµ1 α′ γµβ1′ β1 = 1α1 β1 1β1′ α′1 + (iγ5 )α1 β1 (iγ5 )β ′ α′
1 1
1
1 µ
1 µ
γ
γµβ ′ α′ − (γ γ5 )α1 β1 (γµ γ5 )β ′ α′ , (5A.6)
−
1 1
2 α1 β1 1 1 2
this can be written as
(4)
Gαβ,α′ β ′ (x, y; x′, y ′) = Gαα′ (x − x′ ) Gβ ′ β (y ′ − y)
+g 2
Z
(4)
dx1 dy1 Gαα1 (x − x1 ) ξα1 β1 ,β1′ α1 D (x1 − y1 ) Gα′ β ′ ,α′ β ′ Gβ1 β (y1 − y), (5A.7)
1 1
or, symbolically, as
G(4) = GGT + GGT ξg 2DG(4) .
(5A.8)
The transition matrix T is defined by removing the external particle poles in the
connected part of G(4) :
(4)
Gαβ,α′ β ′
′
′
′
′
(x, y; x , y ) ≡ Gαα′ (x − x ) Gββ ′ (y − y) +
Z
dx1 dy1dx2 dy2
(5A.9)
× Gαα1 (x−x1 ) Gβ ′ β2 (y ′ −y2 ) Tα1 β1 ,α2 β2 (x1 , y1 ; x2 , y2 ) Gα1 α (x2 −x′ ) Gβ1 β (y1 −y) ,
which may be abbreviated by
G(4) = GGT + GGT T GGT .
(5A.10)
From (5A.7) and (5A.9), the transition matrix satisfies the integral equation
Tα1 β1 ,α2 β2 (x1 , y1; x2 , y2 ) = ξα1 β1 ,β2 α2 g 2 D (x1 − y1 ) δ (x1 − y1 ) δ (x2 − y2 )
Z
(5A.11)
+ξα1 β1 ,β̄1ᾱ1 D(x1 − y1 ) dx′1 dy1′ Gα1 α′1 (x1 −x′1 )Tα′1 β1 ,α2 β2 (x′1 , y1′ ; x2 , y2 ) Gβ ′ β1(y1′ −y2 ) ,
which is seen to coincide with the equation (5.43) for the propagator of the bilocal
field. In a short notation, this equation can be written as
T = ξg 2D + ξg 2DGGT T.
(5A.12)
The perturbation expansion
T = ξg 2D + ξg 2DGGT ξg 2D + . . .
(5A.13)
352
5 Hadronization of Quark Theories
Figure 5.15 Momenta in integral equation 5A.15.
reveals the one, two etc. photon exchanges of the ladder diagrams. In momentum
space, the four-particle Green function is defined by
(4)
(2π)4 δ 4 (q ′ − q) Gαβ,α′ β ′ (P, P ′|q)
=
Z
dxdydx′dy ′ei[(P + 2 )x+(P
q
′− q ′
2
)y′ −(P − 2q )y−(P ′ + 2q ′ )x] G(4)
αβα′ β ′
(xy, x′ y ′ ) . (5A.14)
The momenta are indicated in Fig. 5.15. The corresponding scattering matrix
T (P, P ′|q) satisfies the integral equation
T (P, P ′|q) = ξg 2D(P − P ′ )
(5A.15)
Z
′′
q
g
dP
T (P ′′ , P ′|q) G P ′′ −
.
D(P − P ′′ )G P ′′ +
+ ξg 2
4
(2π)
2
2
The ladder exchange is, in general, expected to produce quark-antiquark bound
states. Suppose |H(q)i is one of them. Inserting it into (5A.1) as an intermediate
state gives for x0 , y0 > x′0 , y0′ a contribution
(4)
Gαβ,α′ β ′ (x, y; x′y ′ ) =
Z
1
d3 q
′
Θ
R
−
R
−
(|z0 | + |zd′ )
0
0
2Eq (2π)3
2
× h0|T ψα (x)ψ̄β (y) |H(q)ihH(q)|T ψ̄α′ (x′ )ψβ ′ (y ′) |0i,
where R ≡ (x + y)/2 and z = x − y. The Heaviside function Θ is non-zero if
min (x0 , y0 ) > max (x′0 y0′ ) .
Using the standard integral representation of the Heaviside function,
Θ (x0 ) =
i
2π
Z
dae−q0 x0
1
,
q0 + iǫ
H. Kleinert, COLLECTIVE FIELDS
353
Appendix 5A Remarks on the Bethe-Salpeter Equation
we may write
Θ R0 −
1
i
− (|z0 | + |z0′ |) =
2
2π
R0′
Z
e−i(q0 −Eq )(R0 −R0 ) ei(q0 −Eq ) 2 (|z0 |+|z0|)
dq0
. (5A.16)
q0 − Eq + iη
1
′
′
Inserting now the Bethe-Salpeter wave functions
φαβ (x, y|q) ≡ h0|T ψα (x)ψ̄β (y) |H(q)i,
φ̄β ′ α′ (y ′x′ | − q) ≡ hH(q)|T ψβ ′ (y ′ )ψ̄α′ (x′ ) |0i,
and their momentum space versions
(5A.17)
Z
d4 P −iP (x−y)
e
φαβ (P |q),
(5A.18)
(2π)4
Z
′
′
d4 P ′ −iP ′ (x′ −y′ )
e
φα′ β ′ (P ′ | − q), (5A.19)
ψ̄α′ β ′ (x′ , y ′| − q) = ei(E1 R0 −qR )
(2π)4
ψαβ (x, y|q) = e−i(E1 R0 −qR)
the four-particle Green function in momentum space is seen to exhibit a pole at
q0 ≈ Eq :
(4)
Gαβα′ β ′ (P, P ′|q) ≈
−i
ψαβ (P |q) φ̄β ′ α′ (P ′ | − q) .
2Eq (q0 − Eq + iǫ)
(5A.20)
The opposite time ordering x0 , y0 < x′0 , y0′ contributes a pole at q0 = −Eq . Both
poles can be collected in a single expression by exchanging the pole term in (5A.20)
by
−
q2
i
.
− MH2 + iǫ
This factorization is consistent with the integral equation only for a specific normalization of the Bethe-Salpeter wave functions. In order to see this, we write (5A.8)
in the form
G(4) = GGT + GGT ξg 2DG(4) = 1 − GGT ξg 2D
= GGT 1 − ξg 2DGGT
−1
.
−1
GGT
(5A.21)
Suppose now that a solution is found for different values of the coupling constant
g 2 . Then the variation of G(4) for small changes of g 2 is
−1
−1
∂G(4)
T
2
2
T
T
=
1
−
GG
ξg
D
1
−
ξg
D
GG
ξD
×
GG
∂g 2
= G(4) ξDG(4) .
(5A.22)
If one remains in the vicinity of the pole at q 2 = MH2 (g 2), this becomes
−
i
i
∂
g
g
φ
(P
|q)
φ̄
(P
|
−
q)
=
φg (P |q)
(5A.23)
H
H
2
∂g 2 s − MH (g 2 )
s − MH2 (g 2 ) H
Z
i
dP̄ dP̄ ′ g
′
(
P̄
|
−
q)ξD
φgH (P̄ ′ |q)φ̄gH (P̄ ′ | − q)
φ̄
.
P̄
−
P̄
×
H
8
(2π)
s − MH2 (g 2)
354
5 Hadronization of Quark Theories
This can be true at q 2 = MH2 (g 2 ) only if
∂MH2 (g 2)
= −i
∂g 2
Z
dP dP ′
φH (P |q)ξD (P − P ′ ) φ̄H (P ′ | − q) .
(2π)4 (2π)4
(5A.24)
If we go over to the Bethe Salpeter vertex function (5.34):
q
q
P−
φH (P |q)G−1
M
2
2
q
φ̄H (P | − q)G−1
P+
ΓH (P | − q) = NH∗ G−1
P−
M
M
2
P+
ΓH (P |q) = NH G−1
M
,
q
,
2
(5A.25)
we obtain
gV2
Z
q
dP dP ′
q
ΓH (P |q)GM P −
Tr GM P +
4
4
(2π) (2π)
2
2
q
q
2
×gH
(q 2 )D(P − P ′ )GM P ′ −
Γ̄H (P ′ | − q) GM P ′ +
.
2
2
(5A.26)
∂MH2 (g 2 )
= i|NH |2
(q )
∂g 2
2
Using now the integral equation (5.35), we find the normalization condition
2
gH
(q 2 )
2 (q 2 )
∂gH
∂q 2
= −i|NH |2
Z
d4 P
(2π)4
× Tr GM P +
(5A.27)
q
q
ΓH (P |q)GM P −
Γ̄H (P | − q) ,
2
2
which determines |NH |2 to be
|NH |2 =
2
gH
(q 2 )
2 (q 2 )
∂gH
∂q 2
.
(5A.28)
Note that the normalization is defined for all q 2 with some NH (q 2 ). For real Γ(P |q),
we may choose a real NH (q 2 ), so that
Γ̄(P | − q) = Γ(P | − q).
Both satisfy the same integral equation.
′
The orthogonality of ΓH (P |q) and Γ̄H (P | − q) for different mesons is proved,
′
−1
−1
as usual, by considering (5.35), once for (ξg 2D) ΓH and once for (ξg 2D) Γ̄H ,
′
multiplying the first by Γ̄H and the second by ΓH , taking the trace and subtracting
the results from each other [assuming no degeneracy of gH (q 2 ) and gH ′ (q 2 ) ]. The
normalization (5A.28) is seen to be consistent with the expansion of the T -matrix
given in (5.44):
Tαβ,α′ β ′ (P, P ′|q) = −ig 2
X
H
H
′
ΓH
αβ (P |q)Γ̄β ′ α′ (P | − q)
.
2
gH
(q 2 ) − g 2
(5A.29)
H. Kleinert, COLLECTIVE FIELDS
355
Appendix 5A Remarks on the Bethe-Salpeter Equation
If q 2 runs into a pole MH2 this expression is singular as
Tαβ,α′ β ′ (P, P ′|q) ≈ −ig 2
1
(q 2
−
2 (q 2 )
∂gH
MH2 ) ∂q
2
′
H
ΓH
αβ (P |q)Γ̄β ′ α′ (P | − q) .
(5A.30)
According to (5A.10) this produces a singularity in G(4) (in short notation)
g2
G(4) ≈ −i ∂g2 (q2 )
H
∂q 2
2
g
= −i ∂g2 (q2 )
H
∂q 2
1
H
H
GΓ
G
G
Γ̄
G
q 2 − MH2
q2
1
|NJ |2 φH φ̄H ,
2
− MH
(5A.31)
which coincides with (5A.20) by virtue of (5A.28).
For completeness we now give the Bethe-Salpeter equation (5.32) the form projected into the different covariants:
m′ (P |q) = S(P |q) + P (P |q)iγ5 + V µ (P |q)γµ + Aµ (P |q)γµγ5 .
If mi (P |q)(i = 1, 2, 3, 4) abbreviates S, P, V, A, one has
mi (P |q) = −4ξi g 2
4
X
d4 P ′
1
1
2
2
4
′
2
j=1 (2π) (P − P )E + µ
P ′ + 2q + M 2
E
1
× tij (P ′ |q) mj (P ′ |q) ,
q 2
′
2
P − 2 +M
(5A.32)
E
with tij (P |q) being the traces defined in (5.87) and ξi = (4, + 4, − 2, − 2).
Explicitly, one finds
q2
+ M 4 , tSV (P |q) = tV S = 2MP µ ,
4
q2
− M 2 , tP Aµ (P |q) = −tAµ P = iMq µ ,
tP P (P |q) = P 2 −
4
!
q2
1
2
2
tV µ V ν (P |q) = − P −
− M g µν + 2P µ P ν − q µ q ν ,
4
2
!
q2
1
tAµ Aν (P |q) = − P 2 −
− M 2 g µν + 2P µ P ν − q µ q ν − 2M 2 g µν ,
4
2
tSS (P |q) = P 2 −
(5A.33)
(5A.34)
(5A.35)
(5A.36)
with all other traces vanishing. Notice that in the Bethe-Salpeter equation for
m′ (P |q), there is no tensor contribution due to the absence of such a term in the
Fierz transform of γ µ ⊗ γµ . The integrals in (5A.32) go directly over into (5.86) for
a large gluon mass µ.
356
5 Hadronization of Quark Theories
Appendix 5B
Vertices for Heavy Gluons
Here we present the calculation of the vertices A2 , A3, A4 for large µ. As discussed
in the text, all higher vertices remain finite when the cutoff and µ goes to infinity
like Λ2 ≫ µ2 ≫ M 2 , and will consequently be neglected.
Consider first A2 as described in (5.87) and (5A.33). The integrals Jij (q) are
evaluated by expanding
"
2
#−1 "
#−1 "
#−1
q 2
q 2
P−
P−
+M
+ M2
+ M2
=
(5B.1)
2 E
2 E
E
!)
(
i
h
(Pq )2
M4
q4
q 2 /2
2
2 −2
.
+
+O
1+ 2
PE + M
PE + M 2 (PE2 + M 2 )2
(PE2 + M 2 )2 (PE2 + M 2 )2 )
q
P+
2
2
Since tij (P |q) grows at most like PE2 [see (5A.33)], the terms O(M 4 /E 4 , q 4 /PE4 )
contribute finite amounts upon integration, and will be neglected. At this place we
have assumed q 2 to remain of the same order of M 2 . Actually, this is not true for
< µ2 /100, the neglected
vector- and axialvector meson fields11 but, since m2ρ , m2A1 ∼
terms are indeed very small.
The following integrals are needed, in addition to (5.82), (5.88) (neglecting finite
amounts):
Z
d 4 PE
1
P2
2
4
(2π) (PE + M 2 )2
Z 4
1
d PE
, Pµ Pν
2
4
(2π) (PE + M 2 )2
Z 4
d PE
1
Pµ Pν
2
4
(2π) (PE + M 2 )
Z 4
1
d PE
Pµ Pν Pλ Pκ
(2π)4 (PE2 + M 2 )4
= −(Q − M 2 L),
= −(Q − M 2 L)
= −L
=
(5B.2)
gµν
,
4
gµν
,
4
(5B.3)
(5B.4)
L
(gµν gλκ + gµλ gνκ + gµκ gνλ ) . (5B.5)
24
The results have been used in Eq. (5.89).
There is one subtlety connected with gauge invariance when evaluating the integrals JW (q) and JAA (q). In fact, the first of these integrals coincides with the
standard photon self-energy graph in quantum electro-dynamics. There the cutoff
procedure is known to produce a non-gauge invariant result. The cutoff calculation
yields:
1 2
1
q gµν − qµ qν L − (Q + M 2 L)gµν ,
3
2
i
1
1 h 2
q − 6M 2 gµν − qµ qν L − (Q + M 2 L)gµν .
= −
3
2
JV µ V ν (q) = −
JA µ A ν
11
(5B.6)
(5B.7)
m2ρ = 6M 2 ; m2A ≈ 12M 2 ; µ2 ≈ 144GeV2 .
H. Kleinert, COLLECTIVE FIELDS
357
Appendix 5B Vertices for Heavy Gluons
There are many equivalent ways to enforce gauge invariance. The simplest of these
proceeds via dimensional regularization. If one evaluates the integrals Q and L in
D = 4 − ǫ, dimensions with a small ǫ > 0, then
Z
D/2
D
π
1
1
d p
=
2
4
2
2
4
2
2−(D/2)
(2π) (PE + M )
(2π) (M )
2
π 2
=
+ O(ǫ),
(2π)4 ǫ
L =
Z
D
2
Γ 2−
Γ(2)
(5B.8)
Γ 1 − D2
π D/2
1
1
d D PE
=
Q =
(2π)4 PE2 + M 2
(2π)4 (M 2 )1−(D/2) Γ(2)
π2
2
=
+ O(ǫ).
M2 −
4
(2π)
ǫ
(5B.9)
Hence at the pole at ǫ = 0, Q and L become related such that Q + M 2 L = 0,
cancelling the last terms in (5B.6). Notice that when dealing with the renormalizable
theory with large gluon mass µ2 ≪ Λ2 , this cancellation is still present while the
other Q integrals in Jij (q) become unrelated with the L integrals, the first being
essentially µ2 log Λ2 /µ2 , the other log µ2 /M 2 .
Consider now the interaction terms A3 . Here the traces grow at most as PE2 .
Thus, as far as the divergent contributions are concerned, the denominators in the
integrals (5.108) can be approximated as
1
1
1
=
(5B.10)
2
2
2
2
2
(P + q2 + q1 )E + M (P + q1 )E + M PE + M 2
!)
(
q12
M2
1
P (2q1 + q2 )
+O
,
.
3 1+
PE2 + M 2
PE2 + M 2 PE2 + M 2
(PE2 + M 2 )
Since this expression decreases at least as 1/p6E , the traces have to be known only
with respect to their leading PE2 and PE2 behaviors. These are
tSSS (P |q2q1 )
tSP Aµ (P |q2q1 )
tSAµ P (P |q2q1 )
tSSV µ (P |q2q1 )
tSV µ S (P |q2q1 )
tSV µ V ν (P |q2q1 )
tSAµ Aν (P |q2q1 )
≃
≃
≃
≃
≃
≃
≃
3P 2M ≃ tSP P (P |q2 q1 ) ,
iP 2 P µ + 2iP µ [P (q1 + q2 )] − iP 2 q2µ ,
−iP 2 P µ − iP 2 (2q1 + q2 )µ ,
(5B.11)
2 µ
2 µ
µ
P P − P q2 + 2P [P (q1 + q2 )] ≃ tP P V µ (P |q2 q1 ) ,
P 2P µ + P 2 (2q1 + q2 )µ ≃ tP V µ P (P |q2 q1 ) ,
4MP µ P ν − MP 2 g µν ,
4MP µ P ν − 3MP 2 g µν ,
tV µ V λ V κ (P |q1q2 ) ≃ 4P µP λ P κ − P 2 P µ g λκ + P λg µκ + P κ g µλ
+2P µP λ q1κ + 2P λP κ (q1 + q2 )µ + 2P µ P κ (2q1 + q2 )λ
−2P κ P (q1 + q2 ) g µλ − 2P µ P q1 g λκ
h
i
−P 2 −g µλ q2 + g λκ q2 + g µκ (2q1 + q2 ) .
(5B.12)
358
5 Hadronization of Quark Theories
Using (5B.2) one obtains exactly the third order terms in the Lagrangian Eq. (5.110)
(if this is written in the σ ′ -form).
The fourth-order couplings in A4 are the simplest to evaluate. Here only the
leading P 4 behavior of tij (P |q3 q2 q1 ) contributes proportional to L and the propa−4
gator can directly be used in the form (PE2 + M 2 ) :
tSSSS (P |q3 q2 q1 ) ≃ tSSP P ≃ tP P P P ≃ P 4 ,
tSSV µ V ν (P |q3 q2 q1 ) ≃ tP P V µ V ν ≃ tSSAµ Aν ,
≃ tP P Aµ Aν ≃ −itSP Aµ V ν ≃ −itP SAµ V ν ,
≃ 2P 2 P µ P ν − P 4 g µν .
Appendix 5C
(5B.13)
(5B.14)
(5B.15)
(5B.16)
Some Algebra
Here we want to compare some of our results with traditional derivations [28], [29]
obtained by purely algebraic considerations together with PCAC.
The vector and axial-vector currents
Vµa (x) = ψ̄(x)γµa
λa
ψ(x),
2
Aaµ (x) = ψ̄(x)γµ γ5
λa
4(x)
2
(5C.1)
generate chiral SU(3)×SU(3), under which the quark gluon Lagrangian transforms
as
L = Lchirally invariant − u0 − cu8 − du3 ,
(5C.2)
where
u0 + cu8 + du3 = ψ̄Mψ ≡
s
X
a
Ma ψ̄
λa
ψ
2
λ0
2 u
M + Md + Ms ψ̄ ψ
(5C.3)
2
2
λ3
λ8
1 u
u
d
d
s
√
+
M + M − 2M ψ̄ ψ + M − M ψ̄ ψ.
2
2
3
=
By identifying
ua ≡ M0 ψ̄
λa
ψ,
2
(5C.4)
we see that the coefficients in (5C.2) are
c≡
M3
M8
,
d
=
.
M0
M0
(5C.5)
Afer defining also the pseudoscalar densities
υ a ≡ M0 ψ̄iγ5
λa
ψ
2
(5C.6)
H. Kleinert, COLLECTIVE FIELDS
359
Appendix 5C Some Algebra
we get ua and υ a A A (3̄, 3) ± (3, 3̄) -representation of SU(3) × SU(3):
h
i
[Qa5 , ua ] = idabc υ c , Qa5 , υ b = −idabc uc .
(5C.7)
From the equation of motion one finds the conservation law
∂
µ
Vµa (x)
=
=
∂ µ Aaµ (x) =
=
"
#
λa
λa
M ψ
∂ ψ̄γµ ψ = ψ̄
2
2
λc
if abc Mb ψ̄ ψ a = 0, 1, . . . , 8,
2
"
#
a
a
λ
λ
M γ5 ψ
∂ µ ψ̄γµ γ5 ψ = −ψ̄
2
2
λc
idabc Mb ψ̄iγ5 ψ a = 1, . . . , 8.
2
µ
(5C.8)
Let us neglect SU(2) breaking in M. By taking (5C.8) between vacuum and pseudoscalar meson states we find
m2 √
1 √
V
fπ m2π = √
2M0 + M8
LZπ1/2 ,
3
3
m2 √
1 √
1/2
V
fK m2K = √
2M0 + M8
LZK .
3
3
(5C.9)
Similar equations hold for the other members of the multiplet. We have used the
definitions [see the pseudoscalar version of (5.157)]:
h0|∂ µ Aaµ |πi ≡ fπ m2π ,
h0|ψ̄iγ5
λπ
µ2 1 1
µ2 Zπ1/2
m2V √ 1/2
√
√
ψ|πi =
h0|π|πi
=
=
LZπ .
2
2g 2 2g 2 L
2g 2 L
3
If we write the matrix M in the form


 √
2M0 + M8 √
Mu
1



Md
M=
2M0 + M8 √
= √ 
2
3
s
M


,
2M0 − 2M8
(5C.10)
then (5C.9) take the form
fπ2 m2π
fK2 m2K
which agrees with (5.136).
Mu + Md 1/2 2 2 √
=
Zπ mV L,
2
3
Mu + Ms 1/2 2 2 √
=
Zπ mV L,
2
3
(5C.11)
360
5 Hadronization of Quark Theories
By evaluating (5C.7) between vacuum states and saturating the commutator
with a pseudoscalar intermediate state, one finds
1 √
µ2 Zπ1/2
= √
fπ M0 2 √
2h0|u0|0i + h0|u8 |0i ,
2g
3
L
1/2
2
1 √
µ Z
1
= √
2h0|u0 |0i − h0|u8|0i ,
fK M0 2 √K
2g
2
3
L
(5C.12)
and similar for the other partners of the multiplet. Inserting the result of Eq.
(5.157),
h0|ua|0i = M0 h0|ũa|0i =
1 µ2 0 a
MM ,
2 g2
(5C.13)
and writing M in the same way as M in (5C.10), brings (5C.12) to the form
√ u
L M + Md ,
√
L (M u + M s ) ,
=
fπ Zπ1/2 =
1/2
fK ZK
(5C.14)
which agree exactly with (5.136) (written there in SU(3)-matrix form). Considerations of this type have led to the determination [28, 29, 33]
c ≈ −1.28,
(5C.15)
or
Mu + Md /2
Ms
≈
1
.
29
Including also SU(2)-violation in such a consideration gives [33]
d ≈ −.03,
(5C.16)
or
Mu − Md
1
≈− .
Mu + Md
4
There are numerous extensions to SU(4), but they have to be viewed with caution,
since it is hard to see how the large pseudoscalar and vector masses occuring there
can dominate the divergence of the axial current and the vector current, respectively.
Notes and References
H. Kleinert, COLLECTIVE FIELDS
Notes and References
361
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H. M. Chan, Phys. Letters B 28, 485 (1969);
Z. Koba and H. B. Nielsen, Nucl. Phys. B 12. 512 (1969);
Y. Nambu, Proc. Int. Conf. on Symmetries and Quark Models, Wayne State
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H. Nielsen, 15th Int. Conf. on High Energy Physics, Kiev 1970;
L. Susskind, Nuovo Cimento 69, 457 (1970).
[2] Y. Nambu, in Preludes in Theoretical Physics, North Holland (1966);
H. Fritzsch and M. Gell-Mann, Proc. XVI Intern Conf. on High Energy
Physics, Chicago 1972, Vol. 2, p. 135;
W. Barden, H. Fritzsch, M. Gell-Mann in Scale and Conformal Invariance in
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D.J. Gross and F. Wilczek, Phys. Rev. Letters 30, 1343 (1973), Phys. Rev. D
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See also:
R. A. Brandt and G. Preparata, Nucl. Phys. B 27, 541 (1971)
and the review by
R.A. Brandt, Erice Lectures 1972 in Highlights in Particle Physics, ed. by A.
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C.R. Hagen, Nucl. Phys. B 95, 477 (1975);
C.G. Callan, N. Coote, D.J. Gross, Phys. Rev. D 11, 1649 (1976);
T. Appelquist and H.D. Politzer, Phys. Rev. Letters 34 43 (1974).
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A. Zichichi;
R. H. Dalitz, Proc. XIIIth International Conference on High Energy Physics
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J.J.J. Kokkedee, The Quark Model, Benjamin, New York 1969,
H. Lipkin, Phys. Rep. C 8 173 (1973);
362
5 Hadronization of Quark Theories
M. Böhm, H. Joos and M. Krammer, Nucl. Phys. B 51 397 (1973) and Acta
Phys. Austriaca 38 (1973).
[6] See
H. Kleinert, Lettere Nuovo Cimento 4, 285 (1970) ;
M. Kaku and K. K. Kikkawa; Phys. Rev. D 10 1110 (1974); D 10, 1823 (1974);
D 10, 3943 (1974);
E. Gremmer and J. L. Sherk, Nucl. Phys. B 90, 410 (1975).
[7] For a detailed introduction see:
J. Rzewuski, Quantum Field Theory II, Hefner, New York, 1968;
S. Coleman, Erice Lectures 1974, in Laws of Hadronic Matter ed. by A.
Zichichi, p. 172.
[8] S. Hori, Nuclear Physics 30, 644 (1962).
[9] This is a generalization to bilocal auxiliary fields of an old method of P. T.
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[10] H. Kleinert, Phys. Letters B 62, 429 (1976).
[11] For a thorough review to
N. Nakanishi, Progr. Theor. Phys. Suppl. 43, 1 (1969).
The transition matrix T is discussed for scalar particles by
H. zur Linden, Nuovo Cimento A 65, 197 (1970), Phys. Rev. D 3, 1335 (1971),
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J. Schwinger, Journ. Math. Phys. 5, 1606 (1964).
Recent discussions on fermion-fermion Bethe-Salpeter equation:
W. Kummer, Nuovo Cimento 31, 219 (1964), 34, 1840 (1964);
K. Seto, Progr. Theor. Phys. 42, 1394, (1969);
H. Ito, Progr. Theor. Phys. 43, 1035 (1970);
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C. Lovelace, Phys. Lett. B 32 703 (1970);
V. A. Lessandrini, Nuovo Cimento A 2, 321 (1971);
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[13] See Ref. 2., the review articles by
H. D. Politzer, Phys. Reports C 14, 129 (1974) and the one dimensional colored
quark gluon model in Ref. 4.
[14] S. Okubo, Phys. Letters 5, 165 (1963);
G. Zweig, (unpublished);
J. Iizuka, Supplement to Progress of Theor. Phys. 37, 21 (1966);
H. Kleinert, COLLECTIVE FIELDS
Notes and References
363
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M.Y. Han and Y. Nambu, Phys. Rev. 139, B1006 (1965);
M. Gell-Mann, Acta Physica Austriaca Suppl. 9, 733 (1972).
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in Hadron Physics, ed. by R. Gatto (John Wiley and Sons, 1973), p. 139.
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G. S. Guralnik, Phys. Rev. B 136, 1404, 1417 (1963);
H. Umezahwa, Nuovo Cimento XL, 450 (1965);
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H. Matsumoto, H. Umezawa, N. J. Papastamation, Nucl. Phys. B 68, 236
364
5 Hadronization of Quark Theories
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L. Leplae, H. Umezawa, F. Mancini, Phys. Rev. C 10 (1974).
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G. Preparata, Erice Lectures 1974, in Lepton and Hadron Structure ed. by A.
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H. Sugawara, Phys. Rev. D 12, 3212 (1975);
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A. Chakrabarti and B. Hu, Phys. Rev. D 13, 2347 (1976);
G. Konisi, T. Saito, and K. Shigemoto, Phys. Rev. D 15 (1976).
These authors prove the operatorial validity of the “gap equation” by using
equations of motion rather than the functional method employed in Ref. 18
and extended to bilocal form in Ref. 10.
T. Eguchi, Chicago Preprint, 1976.
This author studies in detail the local version of mesonization via functional
methods which was outlined in Ref. 18 on p. 78.
H. Kleinert, COLLECTIVE FIELDS