Vacuum pseudoscalar susceptibility

Abstract

We derive a novel model-independent result for the pion susceptibility in QCD via the isovector-pseudoscalar vacuum polarisation. In the neighbourhood of the chiral-limit, the pion susceptibility can be expressed as a sum of two independent terms. The first expresses the pion-pole contribution. The second is identical to the vacuum chiral susceptibility, which describes the response of QCD's ground-state to a fluctuation in the current-quark mass. In this result one finds a straightforward explanation of a mismatch between extant estimates of the pion susceptibility. Comment: 4 pages

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arXiv:0912.2687v1 [nucl-th] 14 Dec 2009
Vacuum pseudoscalar susceptibility
Lei Chang,1Yu-Xin Liu,2,3,4Craig D. Roberts,2,5Yuan-Mei Shi,6Wei-Min Sun,7,8and Hong-Shi Zong7,8
1Institute of Applied Physics and Computational Mathematics, Beijing 100094, China
2Department of Physics, Peking University, Beijing 100871, China
3State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China
4Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou 730000, China
5Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
6Department of Physics, Nanjing Xiaozhuang College, Nanjing 211171, China
7Department of Physics, Nanjing University, Nanjing 210093, China
8Joint Center for Particle, Nuclear Physics and Cosmology, Nanjing 210093, China
(Dated: December 14, 2009)
We derive a novel model-independent result for the pion susceptibility in QCD via the isovector-
pseudoscalar vacuum polarisation. In the neighbourhood of the chiral-limit, the pion susceptibility
can be expressed as a sum of two independent terms. The first expresses the pion-pole contribution.
The second is identical to the vacuum chiral susceptibility, which describes the response of QCD’s
ground-state to a fluctuation in the current-quark mass. In this result one finds a straightforward
explanation of a mismatch between extant estimates of the pion susceptibility.
PACS numbers: 11.30.Rd, 12.38.Aw, 12.38.Lg, 24.85.+p
Colour-singlet current-current correlators or, equiva-
lently, the associated vacuum polarisations, play an im-
portant role in QCD because they are directly related
to observables. The vector vacuum polarisation, e.g.,
couples to real and virtual photons.
sic to the analysis and understanding of the process
e+e−→hadrons [1, 2]. In addition, analysis of the large
Euclidean-time behaviour of a carefully chosen correlator
can yield a hadron’s mass [3, 4]; and correlators are also
amenable to analysis via the operator product expansion
and are therefore fundamental in the application of QCD
sum rules [5].
In the latter connection, the vacuum pseudoscalar sus-
ceptibility (also called the pion susceptibility) plays a role
in the sum-rules estimate of numerous meson-hadroncou-
plings; e.g., the strong and parity-violating pion-nucleon
couplings, gπNN and fπNN, respectively [6, 7, 8]. Fur-
thermore, as will become plain herein, the pion suscepti-
bility is as intimate a probe of QCD’s vacuum structure
as the scalar susceptibility [9] but its veracious analysis
is more subtle, with conflicts and misconceptions being
common [7, 8, 10, 11, 12].
We approach the vacuum pseudoscalar susceptibil-
ity via the isovector-pseudoscalar vacuum polarization,
which can be written1
It is thus ba-
ωij
5(P;ζ) = NctrZ4
?Λ
q
i
2γ5τiS(q+)iΓj
5(q;P)S(q−), (1)
where the trace is over flavour and spinor indices; ζ is the
renormalisation scale; Z4(ζ,Λ) is the Lagrangian mass-
term renormalisation constant, which depends implicitly
1In our Euclidean metric: {γµ,γν} = 2δµν; γ†
γ4γ1γ2γ3; a · b =P4
µ = γµ; γ5 =
i=1aibi; and Pµ timelike ⇒ P2< 0.
on the gauge parameter;2and?Λ
resents a symmetry-preserving regularisation of the inte-
gral, with Λ the regularisation mass-scale which is taken
to infinity as the last step in a complete calculation.
Herein we will subsequently assume isospin symmetry;
viz., equal u- and d-quark current-masses, in considering
the isovector-channel. An extension to three flavours and
the flavour-singlet channel can be pursued following the
methods of Ref.[13].
In Eq.(1), S is the dressed-quark propagator and Γ5
is the fully-dressed pseudoscalar vertex, both of which
depend on the renormalisation point. The propagator is
obtained from QCD’s gap equation; namely,
q:=?Λd4q/(2π)4rep-
S(p)−1= Z2(iγ · p + mbm) + Σ(p),
?Λ
q
Σ(p) = Z1
g2Dµν(p − q)λa
2γµS(q)λa
2Γν(q,p),
(2)
where Dµν(k) is the dressed-gluon propagator, Γν(q,p)
is the dressed-quark-gluon vertex, and mbmis the Λ-
dependent u- and d-quark current-quark bare mass. The
quark-gluon-vertex and quark wave-function renormali-
sation constants, Z1,2(ζ,Λ), also depend on the gauge
parameter.
The gap equation’s solution has the form
S(p)−1= iγ · pA(p2;ζ2) + B(p2;ζ2)(3)
and the mass function M(p2) = B(p2,ζ2)/A(p2,ζ2) is
renormalisation point independent. The propagator is
obtained from Eq.(2) augmented by a renormalisation
condition.
2Physical quantities obtained from Eq.(1) are manifestly gauge-
invariant.

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Since QCD is asymptotically free, the chiral limit is
defined by
Z2(ζ,Λ)mbm(Λ) ≡ 0, ∀Λ ≫ ζ ,(4)
which is equivalent to requiring that the renormalisation
point invariant current-quark mass is zero; i.e., ˆ m = 0.
A mass-independent renormalisation scheme can then be
implemented by fixing all renormalisation constants in
the chiral limit [14]; namely, one solves the chiral limit
gap equation subject to the requirement
S−1
ˆ m=0(p)??
p2=ζ2= iγ · p.(5)
This is implicit in the subsequent analysis. We note that
Z2(ζ,Λ)mbm(Λ) = Z4(ζ,Λ)m(ζ), (6)
where m(ζ) is the familiar running current-quark mass.
The pseudoscalar vertex is determined from an inho-
mogeneous Bethe-Salpeter equation; viz.,
[Γj
5(k;P)]tu= Z4[1
2γ5τj]tu+
?Λ
q
[χj
5(q;P)]srKrs
tu(q,k;P),
(7)
where k is the relative- and P the total-momentum of the
quark-antiquark pair; r, s, t, u represent colour, flavour
and spinor indices;
χj
5(k;P) = S(k+)Γj
5(k;P)S(k−), (8)
k± = k ± P/2, without loss of generality owing to
the symmetry-preserving nature of the regularisation
scheme; and K(q,k;P) is the fully-amputated two-
particle-irreducible quark-antiquark scattering kernel.
Much of the preceding material recapitulates results
familiar from QCD’s Dyson-Schwinger equations (DSEs)
[15, 16], which also provide the foundation for our sub-
sequent analysis. Consider, then, that in the presence
of a spacetime-independent pseudoscalar source, ? s5?= 0,
associated with a term
?
d4x ¯ q(x)i
2γ5? τ ·? s5q(x) (9)
in the action, one can define a vacuum pseudoscalar con-
densate, whose gauge-invariant, properly renormalised
form in QCD is
σj
5(? s5,m;ζ,Λ) = Z4Nctr
?Λ
q
i
2γ5τjS(q;? s5,m;ζ).(10)
This is analogous to the vacuum quark condensate [17]
σ(m;ζ,Λ) = Z4Nctr
?Λ
q
1
2τ0S(q;m;ζ),(11)
τ0= diag[1,1], whose source-term in the QCD action is
that associated with the current-quark mass.
It should be emphasised that when ˆ m = 0, it is only
foreknowledge of nonzero current-quark masses via the
Higgs mechanism which leads one to express dynamical
chiral symmetry breaking (DCSB) as
− ?¯ qq?0
ζ= lim
m→0σ(m;ζ,Λ) ?= 0, (12)
instead of choosing a different vacuum vector; e.g.,
(σ;? s5) ∝ (0;1,−i,0). Moreover, Eq.(12) defines what
we mean by an isoscalar-scalar configuration: isovector-
pseudoscalar correlations are by convention measured
with respect to this configuration.
These observations highlight the importance of the
pseudoscalar susceptibility
Xij
5(ζ) :=
∂
∂si
5
σj
5(? s5,m;ζ,Λ)
????
? s5=0
.(13)
Following the method of Ref.[9], it is straightforward to
show that
Xij
5(ζ) = −2ωij
5(P = 0;? s5= 0, ˆ m;ζ).(14)
NB. Whilst hitherto we have not specified a regularisa-
tion procedure for the susceptibility, it can rigorously be
defined via a Pauli-Villars procedure [9].
We are interested in the value of Xij
bourhood of the chiral limit. Therein one may write [18]
5(ζ) in the neigh-
iΓj
5(k;0) =1
2iγ5τjER
5(k;0) +rπ
m2
π
Γj
π(k;0),(15)
where Γj
normalised Bethe-Salpeter amplitude; rπ(ζ), determined
by
π(k;P) is the pion bound-state’s canonically-
iδjkrπ(ζ) = ?0|¯ q1
2γ5τkq|πj?
?Λ
q
(16)
= NctrZ4
1
2γ5τkχj
π(q;P),(17)
is the residue of this bound-state in the inhomogeneous
pseudoscalar vertex; and ER
5(k;P) is a part of the in-
homogeneous pseudoscalar vertex which is regular as
P2+ m2
π→ 0.
These statements will not be surprising once one re-
calls that the solution of a linear, inhomogeneous integral
equation is a sum; viz., the regular solution of the inho-
mogeneous equation plus a solution of the homogeneous
equation, here, naturally, the canonically normalised so-
lution. In terms of this solution, the pion’s leptonic decay
constant is expressed through
δjkfπPµ = ?0|¯ q1
2γ5γµτkq|πj?
?Λ
q
(18)
= NctrZ2
1
2γ5γµτkχj
π(q;P).(19)
One can turn to the axial-vector Ward-Takahashi iden-
tity in order to determine ER
5(k;P); viz.,
PµΓj
5µ(k;P) + 2m(ζ)iΓj
= S−1(k+)1
5(k;P)
2iγ5τj+1
2iγ5τjS−1(k−),(20)

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where Γj
tex. At P = 0 with ˆ m ?= 0 there is no pole contribution
on the left-hand-side and hence Eq.(20) states
5µ(k;P) is the inhomogeneous axial-vector ver-
m(ζ)ER
5(k;P = 0) = B(k2;m;ζ2), (21)
namely, this regular piece of the pseudoscalar vertex is
completely determined by the scalar part of the ˆ m ?= 0
quark self-energy.NB. It is straightforward to ver-
ify Eq.(21), order-by-order, via the gap and Bethe-
Salpeter equations using the systematic, nonperturba-
tive, symmetry-preserving DSE truncation scheme intro-
duced in Refs.[19, 20].
We insert Eq.(15) into Eq.(14) to obtain
Xij
5(ζ)
X5(ζ)
ˆ m 0
=δijX5(ζ),
Xπ
(22)
=
5(ζ) + XR
5(ζ) + O(ˆ m);(23)
namely, in the neighbourhood of ˆ m = 0 the susceptibil-
ity splits into a sum of two terms. The first of these
expresses the contribution of the pion pole, O(ˆ m−1), and
can readily be expressed in a closed form
Xπ
5(ζ) =2rπ(ζ)2
m2
π
ˆ m=0
= −?¯ qq?0
ζ
m(ζ), (24)
where the last equality is proved in Ref.[18]. The second
term in Eq.(23), O(ˆ m0), is implicitly determined via
m(ζ)XR
5(ζ)δjk
ˆ m∼0
=−NctrZ4
?Λ
q
iγ5τkS(q)i
2γ5τjB(q2;m)S(q)
=δjkσ(m;ζ,Λ), (25)
where the last line is obtained using {γ5,γµ} = 0.
We can now proceed to our desired result.
tion(25) entails
Equa-
XR
5(ζ;m) = X(ζ) + O(ˆ m), (26)
where the vacuum chiral susceptibility is [9]
X(ζ) =
∂
∂m(ζ)σ(m;ζ,Λ)
????
ˆ m=0
.(27)
Hence we arrive at a model-independent consequence of
chiral symmetry and the pattern by which its broken in
QCD; namely,
X5(ζ)
ˆ m 0
= −?¯ qq?0
ζ
m(ζ)+ X(ζ) + O(ˆ m).(28)
For illustration, in TableI we report numerical values
computed from two models for the gap equation’s kernel.
Namely, we simplify the renormalisation-group-improved
effective interaction in Ref.[22]
Z1g2Dρσ(p − q)Γa
σ(q,p)
ρσ(p − q)λa
= G((p − q)2)Dfree
2Γσ(q,p),(29)
TABLE I: Vacuum pseudoscalar susceptibility and related
quantities, computed using the two kernels of the Bethe-
Salpeter equation described in connection with Eqs.(30),
(31 and (32).
Dimensioned quantities are listed in GeV,
κ := −(?¯ qq?0
cay constant. The entries were compiled from Refs.[9, 21].
NB. For quantitative comparison with some other studies
[7, 8, 10, 12], our results for X should be multiplied by (2π)2.
√D ωκ
1
2
1
√2
2
ζ)1/3and f0
πis the pion’s chiral-limit leptonic de-
Vertex
f0
π
m
pXπ
1.77
5
pXR
0.39
5
RL [Eq.(31)]10.25 0.091 0.0050
BC [Eq.(32)]
1
0.26 0.11 0.00641.660.28
wherein Dfree
propagator, through the choice
ρσ(p−q) is the Landau-gaugefree gauge-boson
G(s)
s
=4π2
ω6Dse−s/ω2,(30)
which is a finite width representation of the form intro-
duced in Ref.[23]. This interaction has been rendered as
an integrable regularisation of 1/k4[24]. Equation (30)
delivers an ultraviolet finite model gap equation. Hence,
the regularisation mass-scale can be removed to infinity
and the renormalisation constants set equal to one.
The kernel is completed by specifying the dressed-
quark-gluon vertex. At leading-order in the systematic
DSE truncation scheme [19, 20] the vertex is
Γσ(q,p) = γσ.(31)
This defines the rainbow-ladder (RL) truncation. One
can alternatively employ Ans¨ atze for the vertex whose
diagrammatic content is unknown. A class of such mod-
els, which has seen much use in diverse applications; e.g.,
Refs.[9, 21, 25, 26, 27, 28], can be characterised by [29]
iΓσ(k,ℓ) = iΣA(k2,ℓ2)γσ+ (k + ℓ)σ
?i
×
2γ · (k + ℓ)∆A(k2,ℓ2) + ∆B(k2,ℓ2)
?
,(32)
where
ΣF(k2,ℓ2) =
1
2[F(k2) + F(ℓ2)],
F(k2) − F(ℓ2)
k2− ℓ2
(33)
∆F(k2,ℓ2) =
,(34)
with F = A,B; viz., the scalar functions in Eq.(3). This
Ansatz satisfies the vector Ward-Takahashi identity and
is often referred to as the BC vertex.
Equation(28) is a remarkable result, which is nonethe-
less readily understood.Recall that in the absence
of a current-quark mass, the two-flavour action has a
SUL(2)⊗SUR(2) symmetry; and, moreover, that ascrib-
ing scalar-isoscalar quantum numbers to the QCD vac-
uum is a convention contingent upon the form of the
current-quark mass term.

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It follows that the massless action cannot distinguish
between the continuum of sources specified by
constant ×
?
d4x ¯ q(x)eiγ5? τ·?θq(x), |θ| ∈ [0,2π). (35)
Hence, the regular part of the vacuum susceptibility must
be identical when measured as the response to any one
of these sources, so that XR = X for all choices of?θ.
This is the content of the so-called “Mexican hat” po-
tential, which is used in building models for QCD. The
magnitude of X depends on whether the chiral symme-
try is dynamically broken, or not; and the strength of
the interaction as measured with respect to the critical
value required for DCSB [9]. When the symmetry is dy-
namically broken, then the Goldstone modes appear, by
convention, in the pseudoscalar-isovector channel, and
thus the pole contributions appear in X5but not in the
chiral susceptibility. It is valid to draw an analogy with
the Weinberg sum rule [25, 30].
With Eq.(28) we have, in addition, provided a novel,
model-independent perspective on a mismatch between
the evaluation of the pion susceptibility using either a
two-point or three-point sum rule.
point study [10] produces the pion pole contribution, Xπ
Namely, the two-
5,
which is also the piece emphasised in Ref.[11], whereas
a three-point method [8] isolates the regular piece, XR
because a vacuum saturation Ansatz is implemented in
the derivation. Thus, the analyses are not essentially
in conflict. Instead, they emphasise different, indepen-
dent pieces of the susceptibility, which, with care, can be
distinguished. However, in a sum-rules estimate of pion-
nucleon coupling constants, only the regular piece should
be retained [6].
We note in closing that the vacuum tensor susceptibili-
ties, which can be related to the nucleon’s tensor charges
[31], can similarly be analysed. Such a study is underway.
5,
Acknowledgments
This work was supported by:
ural Science Foundation of China, under Contract
Nos. 10425521, 10675007, 10705002, 10775069 and
10935001; the Major State Basic Research Development
Program under contract No. G2007CB815000; and the
United States Department of Energy, Office of Nuclear
Physics, contract no. DE-AC02-06CH11357.
the National Nat-
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