Quark distribution functions in the chiral quark-soliton model: cancellation of quantum anomalies

17 Pages·0.19 MB·English

by  K. Goeke

RUB/TPII-18/99
Quark distribution functions in the chiral quark-soliton
model: cancellation of quantum anomalies
K. Goekea, P.V. Pobylitsaa,b, M.V. Polyakova,b,
P. Schweitzera, D. Urbanoa,c
0
0 a Institute for Theoretical Physics II, Ruhr University Bochum, Germany
0
2 b Petersburg Nuclear Physics Institute, Gatchina, St. Petersburg 188350, Russia

n c Faculdade de Engenharia da Universidade do Porto, 4000 Porto, Portugal
a
J

6
2 Abstract

1
In the framework of the chiral quark-soliton model of the nucleon we investigate
v
2 the properties of the polarized quark distribution. In particular we analyse the so
7 called anomalous difference between the representations for the quark distribution
2
functions in terms of occupied and non-occupied quark states. By an explicit ana-
1
0 lytical calculation it is shown that this anomaly is absent in the polarized isoscalar
0 distribution ∆u+∆d, which is ultraviolet finite. In the case of the polarized isovec-
0
tor quark distribution ∆u − ∆d the anomaly can be cancelled by a Pauli-Villars
/
h
subtraction which is also needed for the regularization of the ultraviolet divergence.
p
-
p
e 1 Introduction
h
:
v
i Recently a rather successful program of computing the quark distribution functions in the
X
framework of the effective quark-soliton model was developed [1, 2, 3, 4, 5, 6]. The quark
r
a soliton model [7] includes the chiral pion field U = eiπaτa/Fπ and the quark field ψ whose
interaction is described by the Lagrangian
L = ψ¯(iγµ∂ −MUγ5)ψ. (1)
µ
In the mean field approximation (justified in the limit of the large number of quark colors
N [8]) the nucleon arises as a soliton of the chiral field U
c
xa
U(x) = exp[i(naτa)P(r)], na = , r = |x|. (2)
r
This effective theory allows a quantum field-theoretical approach to the calculation of
thequarkandantiquarkdistributionsinthenucleon. Incontrasttonaivequarkcomposite
models and to the bag model here we have a consistent approach reproducing the main
features of the QCD parton model like positivity of the quarkand antiquark distributions,
various sum rules etc.
1
In terms of the quark degrees of freedom this picture of the nucleon corresponds to
occupying with N = 3 quarks the negative continuum levels as well as the valence level
c
of the one-particle Dirac Hamiltonian H
H = −iγ0γk∂ +Mγ0Uγ5 , (3)
k
in the background soliton field U. For the pion field (2) one can find the spectrum of the
Hamiltonian (3)
H|ni = E |ni. (4)
n
Various nucleon observables can be naturally represented as sums over eigenstates |ni
of the Dirac Hamiltonian H. For example, the nucleon mass M is given by the sum over
N
all occupied states or alternatively by the minus sum over all non-occupied states
M = E = − E . (5)
N n n
n,occ n,non−occ
X X
In this expression the subtraction of similar sums is implied where the eigenstates |ni and
the eigenvalues E are replaced by those of the free Hamiltonian,
n
H |n(0)i = E(0)|n(0)i, H = −iγ0γk∂ +Mγ0. (6)
0 n 0 k
The physical reason for the existence of the two equivalent expressions in (5) is that
the polarized Dirac sea picture can be formulated either in terms of quark or in terms of
antiquark states (occupied antiquark states correspond to non-occupied quark states).
Formally the equivalence of two representations (5) for M follows from the identity
N
E + E = E = TrH = 0. (7)
n n n
n,occ n,non−occ n
X X X
At the last step we took into account that the trace of H over the spin indices vanishes.
Strictly speaking, this naive argument is not safe since the sums (5) over the occupied and
non-occupied states are ultraviolet divergent and must be regularized. In principle, the
ultraviolet regularization could lead to an anomalous difference between the summation
over occupied and non-occupied states but in the case of the nucleon mass (5) one can
check that in the regularizations like Pauli-Villars or proper-time ones the anomaly is
absent. E.g. the proper-time regularized version of (7) with the ultraviolet cutoff Λ is
lim E exp −E2/Λ2 = 0. (8)
n n
Λ→∞
Xn h i
We can reformulate this verbally as the “absence of the anomaly” in the nucleon mass
M (in the proper time regularization). The usage of the word “anomaly” is invoked by
N
the similarity with the axial anomaly which can be interpreted as nonvanishing trace of
γ in e.g. the proper-time regularization.
5
The main object of interest in this paper is the study of the quark distribution func-
tions. In the mean field approach (justified in the large N limit) the quark distributions
c
can be represented as single or double sums over occupied or non-occupied one-particle
eigenstates (4) of the Dirac Hamiltonian (3). We shall see that for the same parton distri-
bution one can write two naively equivalent representations but whether this equivalence
2
persists or not when one takes into account the ultraviolet regularization is a rather subtle
question and the situation is different for different distributions. Moreover, even in the
limit of the large cutoff, the cancellation of this anomalous difference between the naively
equivalent representations is sensitive to the regularization used.
Let us start from the unpolarized isosinglet quark distribution u(x) + d(x) which is
given by the following expressions ([1]) in the leading order of the 1/N expansion
c
d3p p3 +E
u(x)+d(x) = N δ n −x hn|pi(1+γ0γ3)hp|ni
c
n,occZ (2π)3 MN !
X
d3p p3 +E
= −N δ n −x hn|pi(1+γ0γ3)hp|ni. (9)
cn,non−occZ (2π)3 MN !
X
Also here the subtraction of similar sums with the eigenstates and eingenvalues of the
Hamiltonian (3) replaced by those of the free Hamiltonian (6) is implied. The result (9)
has a transparent physical meaning of the probability to find a quark with momentum
fraction x in the nucleon in the infinite momentum frame. In ref. [1] it was shown that
in the Pauli-Villars regularization the sums over occupied and non-occupied states in (9)
really give the same result.
We stress that the fact of the equivalence of the two representations for parton dis-
tributions is crucial for the positivity of unpolarized distributions and for the validity
of various sum rules inherited by the model from QCD [1]. Therefore the check of this
equivalence is an essential part of the calculation of parton distributions in the chiral
soliton model.
Now let us turn to the polarized quark distributions. In the leading order of the 1/N
c
expansion only the isovector polarized distribution survives
1 d3p p3 +E
n
∆u(x)−∆d(x) = − N δ −x
3 cn,occZ (2π)3 MN !
X
×hn|pi(1+γ0γ3)τ3γ5hp|ni. (10)
Compared to the expression (9) for u(x)+d(x) here we have an extra factor τ3γ5 which
reflects the fact that now we deal with the isovector polarized distribution. The factor of
1/3 comes from a careful treatment of the rotation of the soliton [1].
One can ask whether the summation over the occupied quark states in (10) can be
replaced by the summation over non-occupied states
1 d3p p3 +E
n
∆u(x)−∆d(x) = N δ −x
3 cn,non−occZ (2π)3 MN !
X
×hn|pi(1+γ0γ3)τ3γ5hp|ni. (11)
In this paper we shall show that in the case of the Pauli-Villars regularization (the sum
over states n in (10) is logarithmically divergent) the two representations (10) and (11)
are really equivalent.
We stress that the equivalence of the summation over the occupied and non-occupied
states is very sensitive to the choice of the regularization. For example, if instead of the
3
Pauli–Villars regularization we simply cut thesummation over quark states in(10)includ-
ing only states with |E | < ω then a nonzero difference between the two representations
n 0
(10) and (11) will remain even in the limit of the infinite cutoff ω → ∞. The mechanism
0
how this anomalous difference appears is similar in many respects to the famous axial
anomaly. In particular, such similarity manifests itself in the fact that the anomalous
difference between the two representations (10) and (11) can be computed analytically in
the limit ω → ∞. The calculation of the anomalous difference is presented in this paper.
0
Although the regularization including only states with |E | < ω is not acceptable as
n 0
a physical one and the Pauli–Villars regularization is more preferable in this respect, we
want to emphasize that in the practical calculations based on the numerical diagonaliza-
tion of the Dirac operator in the background soliton field, the |E | < ω regularization
n 0
appears naturally. Indeed, in the numerical calculation one can work only with a finite
amount of quark states so that one actually uses both Pauli-Villars subtraction (with the
regulatormassM )andthe|E | < ω regularization. ThepurePauli-Villarssubtraction
PV n 0
is simulated by working with ω ≫ M . The numerical calculation is rather involved
0 PV
and the analytical result for the anomaly in the |E | < ω regularization is very helpful
n 0
for the control of numerics even if the anomaly cancels after the Pauli-Villars subtraction.
Now let us turn to the polarized isoscalar quark distribution ∆u(x) + ∆d(x) which
gets the first nonzero contribution only in the subleading order of the 1/N expansion
c
N M 1
c N
∆u(x)+∆d(x) =
2I E −E
m,alln,occ n m
X X
×hn|τ3|mihm|(1+γ0γ3)γ5δ(E +P3 −xM )|ni
n N
N ∂
+ c hn|(1+γ0γ3)τ3γ5δ(E +P3 −xM )|ni. (12)
n N
4I ∂x
n,occ
X
Here P3 is the quark momentum projection on the third axis
∂
P3 = −i , (13)
∂x3
and I is the moment of inertia of the soliton [9].
Another representation for ∆u +∆d can be written in terms of the summation over
non-occupied states n
N M 1
c N
∆u(x)+∆d(x) = −
2I E −E
m,alln,non−occ n m
X X
×hn|τ3|mihm|(1+γ0γ3)γ5δ(E +P3 −xM )|ni
n N
N ∂
− c hn|(1+γ0γ3)τ3γ5δ(E +P3 −xM )|ni. (14)
n N
4I ∂x
n,non−occ
X
The numerical calculation of ∆u+∆d with the Pauli–Villars subtraction was presented
in paper [5]. Unfortunately there the question about the equivalence of the two repre-
sentation (12) and (14) was not investigated properly. Also the Pauli-Villars subtraction
was used in paper [5] without proper justification.
4
In this paper we show that if one cuts the sum over occupied (non-occupied) states n
allowing only |E | < ω in the eqs. (12), (14) then in the infinite cutoff limit ω → ∞
n 0 0
1) both representations (12) and (14) have a finite limit (i.e. ∆u(x) +∆d(x) has no
ultraviolet divergences),
2) the two representations (12), (14) give the same result.
Comparing the last terms in the rhs of representations (12) and (14) for ∆u + ∆d
with expressions (10) and (11) for ∆u−∆d we see that the total expression for ∆u+∆d
contains a contribution proportional to ∂ [∆u(x)−∆d(x)].
∂x
Therefore we start our analysis by investigating the anomaly of ∆u−∆d which we do
in section 2. In section 3 we show by explicit calculation that for the quark distribution
∆u + ∆d there is no anomalous difference between the summations over occupied and
non-occupied states. In section 4 we discuss the numerical results and compare them the
GRSV parametrization of experimental data.
2 Anomaly of ∆u(x) − ∆d(x)
As it was explained in the introduction one of our aims is to investigate whether the two
representations (10) and(11) for the polarized isovector quark distribution ∆u(x)−∆d(x)
are equivalent. The answer to this question is sensitive to the ultraviolet regularization.
Let us start from the regularization that allows only the quark states n with |E | < ω .
n 0
In this regularization eq. (10) can be rewritten as follows.
[∆u(x)−∆d(x)]ω0
occ
Elev+0
1
= − N M dωTr δ(H −ω)δ(ω+P3 −xM )τ3(1+γ0γ3)γ . (15)
c N N 5
3
−Zω0 h i
Here H is the Dirac Hamiltonian (3) and P3 is momentum operator (13). Similarly,
representation (11) becomes
[∆u(x)−∆d(x)]ω0
non−occ
ω0
1
= N M dωTr δ(H −ω)δ(ω +P3 −xM )τ3(1+γ0γ3)γ . (16)
c N N 5
3
EleZv+0 h i
The main results of this section can be formulated as follows
1) Both [∆u(x)−∆d(x)]ω0 and [∆u(x)−∆d(x)]ω0 are logarithmically divergent
occ non−occ
in the limit of large cutoff ω → ∞
0
N M M2 ω
[∆u(x)−∆d(x)]ω0 ∼ [∆u(x)−∆d(x)]ω0 = c N ln 0
occ non−occ 12π2 M
d3k
× Sp (U˜[k])+τ3U˜(k) θ k3 −|x|M +... . (17)
(2π)3 fl N
Z h i (cid:16) (cid:17)
5
2) In the difference [∆u(x)−∆d(x)]ω0 − [∆u(x)−∆d(x)]ω0 the ultraviolet di-
occ non−occ
vergences cancel and the ω → ∞ limit of this difference reduces to the following finite
0
expression
lim [∆u(x)−∆d(x)]ω0 −[∆u(x)−∆d(x)]ω0
ω0→∞ occ non−occ
n o
1 d3k |xM +k3|
= − N M M2 log N Sp τ3(U˜[k])+U˜(k) . (18)
12π2 c N (2π)3 |xM | fl
Z N h i
where U˜(k) is the Fourier transform of the chiral mean field U(r) entering the Dirac
Hamiltonian (3)
U˜(k) = d3re−i(kr)[U(r)−1] . (19)
Z
Note that [∆u(x)−∆d(x)]ω0 and [∆u(x)−∆d(x)]ω0 separately are given by compli-
occ non−occ
cated functional traces (15) and (16) which can be computed only numerically. The fact
that the anomalous difference between the representations in terms of the occupied and
non-occupied states reduces to a simple momentum integral (18) is highly nontrivial and
is similar to the well known fact that the famous axial anomaly gets its contribution only
from the simplest diagram.
The fact that the divergence (17) is proportional to M2 means that this divergence
can be removed by the Pauli–Villars subtraction so that the following combinations are
finite
[∆u(x)−∆d(x)]PV
occ
M2
= lim [∆u(x)−∆d(x)]ω0,M − [∆u(x)−∆d(x)]ω0,MPV (20)
ω0→∞( occ M2 occ )
PV
[∆u(x)−∆d(x)]PV
non−occ
M2
= lim [∆u(x)−∆d(x)]ω0,M − [∆u(x)−∆d(x)]ω0,MPV . (21)
ω0→∞( non−occ M2 non−occ)
PV
Next since the anomaly (18) is proportional to M2 we see that in the Pauli-Villars regu-
larization the summation over occupied and non-occupied states gives the same results:
[∆u(x)−∆d(x)]PV = [∆u(x)−∆d(x)]PV . (22)
occ non−occ
Now let us turn to the derivation of the result (18) for the anomalous difference
between the summation over occupied and non-occupied states. Subtracting (16) from
(15) we obtain
[∆u(x)−∆d(x)]ω0 −[∆u(x)−∆d(x)]ω0
occ non−occ
ω0
1
= − N M dωTr δ(H −ω)δ(ω +P3 −xM )τ3(1+γ0γ3)γ . (23)
c N N 5
3
−Zω0 h i
We use the following representation for the operator delta function δ(H −ω)
signω 1 1
δ(H −ω) = − (H +ω). (24)
2πi H2 −ω2 −i0 H2 −ω2 +i0
(cid:20) (cid:21)
6
The squared Dirac Hamiltonian (3) is
H2 = −∂2 +M2 +iM(γk∂ Uγ5). (25)
k
Now (23) takes the form
ω0
2 dω
[∆u(x)−∆d(x)]ω0 −[∆u(x)−∆d(x)]ω0 = − N M Im signω
occ non−occ 3 c N 2π
−Zω0
1
×Tr (ω −iγ0γk∂ +γ0MUγ5)
(−∂2 +M2 −ω2 −i0+iM(γk∂kUγ5) k
×δ(ω +P3 −xM )τ3(1+γ0γ3)γ . (26)
N 5
o
Next we expand the “propagator” in the rhs in powers of iM(γk∂ Uγ5)
k
1 1
=
−∂2 +M2 −ω2 −i0+iM(γk∂ Uγ5) −∂2 +M2 −ω2 −i0
k
1 1
− iM(γk∂ Uγ5) +... . (27)
−∂2 +M2 −ω2 −i0 k −∂2 +M2 −ω2 −i0
The first nonvanishing contribution to (26) comes from the term linear in iM(γk∂ Uγ5)
k
ω0
2 dω
[∆u(x)−∆d(x)]ω0 −[∆u(x)−∆d(x)]ω0 = − N M Im
occ non−occ 3 c N 2π
−Zω0
1 1
×Tr [−iM(∂ Uγ5)]
−∂2 +M2 −ω2 −i0 k −∂2 +M2 −ω2 −i0
(cid:26)
×(ω −iγ0γk∂ +γ0MUγ5)(1+γ0γ3)γ γkδ(ω +P3 −xM )τ3 . (28)
k 5 N
Computing the trace over the spin indices and turning to the momentumorepresentation
according to (19) we arrive at
[∆u(x)−∆d(x)]ω0 −[∆u(x)−∆d(x)]ω0
occ non−occ
8 d3k
= N M M2Im k3Sp (U˜[k])+U˜(k)τ3
3 c N (2π)3 fl
Z n o
ω0 dω d3p signω δ(ω +p3 −xM )
N
× . (29)
2π (2π)3|k+p|2 +M2 −ω2 −i0|p|2 +M2 −ω2 −i0
−Zω0 Z
The calculation of the integral over ω and p is straightforward and in the limit ω → ∞
0
one arrives at final result (18).
Note that the limit ω → ∞ should be taken after computing the ω and p integrals.
0
Otherwise if one first computes integrals ω and p3 at fixed p⊥ and with ω = ∞ then one
0
gets zero
∞ ∞
dω dp3 signω δ(ω +p3 −xM )
N
= 0. (30)
2π 2π |k+p|2 +M2 −ω2 −i0|p|2 +M2 −ω2 −i0
−Z∞ −Z∞
7
Actually after integrations over ω (in the interval −ω < ω < ω ) and p3 the integration
0 0
over |p⊥| is restricted at large ω to the interval
0
2ω min |xM |,|xM +k3| < |p⊥|2 < 2ω max |xM |,|xM +k3| . (31)
0 N N 0 N N
h i h i
In the limit of large cutoff ω → ∞ this interval of p⊥ is shifted to infinity which explains
0
why the nonzero result (18) is compatible with the vanishing integral (30).
A similar phenomenon occurs with the p⊥ integration region in the contributions
coming from the higher terms of the expansion (27). However, since the integrands of
these higher order terms decay faster at large |p⊥| these higher order terms give vanishing
contribution to the anomalous difference (18) in the limit of the large cutoff ω → ∞.
0
Restricting the integration over ω in eq. (26) to the interval −ω < ω < 0 or to
0
0 < ω < ω we can investigate separate distribution functions [∆u(x)−∆d(x)]ω0 or
0 occ
[∆u(x)−∆d(x)]ω0 . In this case one gets nonzero contributions from all terms of the
non−occ
infinite series (27). However, it is not difficult to check that only the first nonvanishing
term of this expansion is logarithmically divergent in the limit of large cutoff ω → ∞ and
0
this logarithmic divergence is given by (17). This logarithmic divergence is proportional
to M2 and therefore in our previous calculation of ∆u−∆d we could regularize it by the
Pauli-Villars subtraction. Moreover, since the anomaly (18) is also proportional to M2 it
is cancelled by the same Pauli-Villars subtraction [2].
3 Cancellation of the anomaly of ∆u(x) + ∆d(x)
Now we turn to the investigation of ∆u(x)+∆d(x). The ω cutoff version of (12) is
0
N M 1
[∆u(x)+∆d(x)]ω0 = c N
occ 2I E −E
Xm −ω0<XEn≤Elev n m
×hn|τ3|mihm|(1+γ0γ3)γ5δ(E +P3 −xM )|ni
n N
N ∂
+ c hn|(1+γ0γ3)τ3γ5δ(E +P3 −xM )|ni. (32)
n N
4I ∂x
−ω0<XEn≤Elev
Although we use notations corresponding to the discrete spectrum actually most of the
spectrum is continuous. The singularities corresponding appearing to E = E are as-
m n
sumed to be regularized according to the principal value prescription.
Making use of (15) we find
3 ∂
[∆u(x)+∆d(x)]ω0 = [∆u(x)+∆d(x)](1)ω0 − [∆u(x)−∆d(x)]ω0 , (33)
occ occ 4IM ∂x occ
N
where
N M 1
[∆u(x)+∆d(x)](1)ω0 = c N
occ 2I E −E
Xm −ω0<XEn≤Elev n m
×hn|τ3|mihm|(1+γ0γ3)γ5δ(E +P3 −xM )|ni. (34)
n N
8
Similarly (14) leads to
[∆u(x)+∆d(x)]ω0 = [∆u(x)+∆d(x)](1)ω0
non−occ non−occ
3 ∂
− [∆u(x)−∆d(x)]ω0 , (35)
4IM ∂x non−occ
N
where
N M 1
[∆u(x)+∆d(x)](1)ω0 = − c N
non−occ 2I E −E
Xm Elev<XEn<ω0 n m
×hn|τ3|mihm|(1+γ0γ3)γ5δ(E +P3 −xM )|ni. (36)
n N
We see that
[∆u(x)+∆d(x)]ω0 −[∆u(x)+∆d(x)]ω0
occ non−occ
= [∆u(x)+∆d(x)](1)ω0 −[∆u(x)+∆d(x)](1)ω0
occ non−occ
3 ∂
− [∆u(x)−∆d(x)]ω0 −[∆u(x)−∆d(x)]ω0 . (37)
4IM ∂x occ non−occ
N
n o
Here
[∆u(x)+∆d(x)](1)ω0 −[∆u(x)+∆d(x)](1)ω0
occ non−occ
N M 1
c N
=
2I E −E
Xm −ω0<XEn<ω0(cid:18) n m(cid:19)PV
×hn|τ3|mihm|(1+γ0γ3)γ5δ(E +P3 −xM )|ni. (38)
n N
We remind that here the principal value prescription for (E − E )−1 is implied. This
n m
can be rewritten in the form
ω0
M N
[∆u(x)+∆d(x)](1)ω0 −[∆u(x)+∆d(x)](1)ω0 = − N c dω
occ non−occ 4I
−Zω0
1 1
×Sp τ3δ(H −ω)+δ(H −ω)τ3
H −ω H −ω
(cid:26)(cid:20)(cid:18) (cid:19)P.V. (cid:18) (cid:19)P.V.(cid:21)
ω0
iM N dω
× δ(ω +P3 −xM )(1+γ0γ3)γ5 = − N c
N
4I 2π
o −Zω0
1 1
×Sp τ3 δ(ω +P3 −xM )(1+γ0γ3)γ5
N
H −ω +i0 H −ω +i0
(cid:26)(cid:20) (cid:21) (cid:27)
ω0
iM N dω
N c
+
4I 2π
−Zω0
1 1
×Sp τ3 δ(ω +P3 −xM )(1+γ0γ3)γ5 . (39)
N
H −ω −i0 H −ω −i0
(cid:26)(cid:20) (cid:21) (cid:27)
Hence
[∆u(x)+∆d(x)](1)ω0 −[∆u(x)+∆d(x)](1)ω0
occ non−occ
9
ω0
M N dω 1
= −Im N c Sp (H +ω)τ3(H +ω)
2I 2π (H2 −ω2 −i0signω
−Zω0
1
× δ(ω +P3 −xM )(1+γ0γ3)γ5
H2 −ω2 −i0signω N )
ω0
M N dω 1
= −Im N c signωSp (H +ω)τ3(H +ω)
2I 2π H2 −ω2 −i0
−Zω0 (cid:26)
1
× δ(ω +P3 −xM )(1+γ0γ3)γ5
H2 −ω2−i0 N
(cid:27)
ω0
M N dω 1
N c
= −Im signωSp
2I −Zω0 2π (−∂2 +M2 −ω2 −i0+iM(γk∂kUγ5)
×(ω −iγ0γk∂ +γ0MUγ5)τ3(ω −iγ0γk∂ +γ0MUγ5)
k k
1
× δ(ω −i∂ −xM )(1+γ0γ3)γ5 . (40)
−∂2 +M2 −ω2 −i0+iM(γk∂kUγ5) 3 N )
The rest of the calculation is similar to how we worked with expression (26) for the
anomaly of ∆u(x)−∆d(x).
Nonzero contributions to the anomaly come from the expansion of the propagators up
to terms linear and quadratic in iM(γk∂ Uγ5):
k
[∆u(x)+∆d(x)](1)ω0 −[∆u(x)+∆d(x)](1)ω0
occ non−occ
= A (x)+A (x). (41)
1 2
Here A (x) corresponds to terms linear in iM(γk∂ Uγ5)
1 k
ω0
M N dω 1
A (x) = Im N c signωSp iM(γl∂ Uγ5)
1 2I 2π l −∂2 +M2 −ω2 −i0
−Zω0 (cid:26)(cid:20)
×(ω −iγ0γk∂ +γ0MUγ5)τ3(ω −iγ0γk∂ +γ0MUγ5)
k k
+(ω −iγ0γk∂ +γ0MUγ5)τ3(ω −iγ0γk∂ +γ0MUγ5)
k k
1
× iM(γl∂ Uγ5)
−∂2 +M2 −ω2−i0 l
(cid:21)
1 2
× δ(ω −i∂ −xM )(1+γ0γ3)γ5 , (42)
(cid:20)−∂2 +M2 −ω2 −i0(cid:21) 3 N )
and A is quadratic in iM(γk∂ Uγ5)
2 k
ω0
M N dω 1
N c
A (x) = −Im signωTr
2 2I 2π −∂2 +M2 −ω2 −i0
−Zω0 (cid:26)
1 1
iM(γm∂ Uγ5) iM(γn∂ Uγ5)
m −∂2 +M2 −ω2 −i0 n −∂2 +M2 −ω2 −i0
(cid:20)
10