Instanton vacuum beyond chiral limit

0.17 MB·

by  M. Musakhanov

INSTANTON VACUUM BEYOND CHIRAL LIMIT1
M. Musakhanov
Department of Physics, Pusan National University,
609-735 Pusan, Republic of Korea
& Theoretical Physics Dept, Uzbekistan National University,
Tashkent 700174, Uzbekistan
E-mail: [email protected]
5
0 Abstract
0
2 Inthistalkitisdiscussedthederivationoflow–frequenciespartofquarkdeterminant
and partitionfunction. As a first application, quark condensate is calculated beyond
n
a chiral limit with the account ofO(m), O( 1 ), O( 1 m) andO( 1 mlnm) corrections.
J Nc Nc Nc
It was demonstrated complete correspondence of the results to chiral perturbation
5
1 theory.

1
v
3
3 Introduction
1
1
0 Instanton vacuum model assume that QCD vacuum is filled not only by perturbative
5
but also very strong non-perturbative fluctuations – instantons. This model provides a
0
/ natural mechanism for the spontaneous breaking of chiral symmetry (SBCS) due to the
h
p delocalization of single-instanton quark zero modes in the instanton medium. The model
-
is described by two main parameters – the average instanton size ρ ∼ 0.3fm and average
p
e inter-instanton distance R ∼ 1fm. These values was found phenomenologically [1] and
h
theoretically [2] and was confirmed by lattice measurements [3, 4, 5, 6, 7]. On the base of
:
v
this model was developed effective action approach [8, 9, 10], providing reliable method
i
X
of the calculations of the observables in hadron physics at least in chiral limit.
r
a On the other hand, chiral perturbation theory makes a theoretical framework incorpo-
rating the constraints on low-energy behavior of various observables based on the general
principles of chiral symmetry and quantum field theory [11].
It is natural expect, that instanton vacuum model leads to the results compatible with
chiral perturbation theory.
One of the most important quantities related with SBCS is the vacuum quark con-
densate < q¯q >, playing also important phenomenological role in various applications of
QCD sum rule approach. Previous investigations [12] shows that beyond chiral limit and
at small current quark mass m ∼ fewMeV these quantity receive large so called chiral
log contribution ∼ 1 m ln m with fixed model independent coefficient. On the typical
Nc
scale 1GeV it become leading correction since | 1 ln m| ≥ 1. It was shown, that this
Nc
correction is due to pion loop contribution [12, 11].
So, to be consistent we have to calculate simultaneously all of the corrections of order
m, 1 , 1 ln m in order to find quark condensate beyond chiral limit.
Nc Nc
1
Talk given at the XVII International Baldin Seminar ”Relativistic Nuclear Physics and Quantum
Chromodynamics”, Sept.27-Oct.2, 2004 (JINR, Dubna, Russia).
1
In our previous papers [13, 14] on the base of low–frequencies part of light quark
determinant Det , obtained in [15, 8, 16], was derived effective action. In this framework
low
was investigated current quark mass m dependence of the quark condensate, but without
meson loop contribution [14].
In the present work we refine the derivation of the low–frequencies part of light quark
determinant Det . The following averaging of Det over instanton collective coor-
low low
dinates is done independently over each instanton thanks to small packing parameter
π(ρ)4 ∼ 0.1 and also by introducing constituent quarks degree of freedoms ψ. This proce-
R
dure leads to the light quarks partition function Z[m]. We apply bosonisation procedure
to Z[m], which is exact one for our case N = 2 and calculate partition function Z[m]
f
with account of meson loops. This one provide us the quark condensate with desired
O(m), O( 1 ), O( 1 m ln m) corrections.
Nc Nc
Low–frequencies part of light quark determinant
The main assumption of previous works [8, 9, 10] (see also review [16]) was that at very
small m the quark propagator in the single instanton field A can be approximated as:
i
1 |Φ >< Φ |
0I 0I
S (m ∼ 0) ≈ + (1)
I i∂ˆ im
It gives proper value for the < Φ |S (m ∼ 0)|Φ >= 1 , but in S (m ∼ 0)|Φ >=
0I I 0I im I 0I
Φ0I>+ 1|Φ > second extra term has a wrong chiral properties. We may neglect by this
| im i∂ˆ 0I
one only for the m ∼ 0.
At the present case of non-small m we assume:
|Φ >< Φ | 1
ˆ 0I 0I ˆ
S ≈ S +S i∂ i∂S , S = (2)
I 0 0 c 0 0 i∂ˆ+im
I
where
ˆ ˆ ˆ
c = − < Φ |i∂S i∂|Φ >= im < Φ |S i∂|Φ > (3)
I 0I 0 0I 0I 0 0I
The matrix element < Φ |S |Φ >= 1 , more over
0I I 0I im
1 1
S |Φ >= |Φ >, < Φ |S =< Φ | (4)
I 0I 0I 0I I 0I
im im
as it must be.
In the field of instanton ensemble, represented by A = A , full quark propagator,
I I
expanded with respect to a single instanton, and with account Eq. (2) is:
P
S = S + (S −S )+ (S −S )S 1(S −S )
0 I 0 I 0 0− J 0
I I=J
X X6
+ (S −S )S 1(S −S )S 1(S −S )+...
I 0 0− J 0 0− K 0
I=J,J=K
6 X6
1 1 1
ˆ ˆ
= S + S i∂|Φ > + T +... < Φ |i∂S
0 0 0I 0J 0
C C C
I,J (cid:18) (cid:19)IJ
X
1
ˆ ˆ
= S + S i∂|Φ > < Φ |i∂S (5)
0 0 0I 0J 0
C −T
I,J (cid:18) (cid:19)IJ
X
2
where
ˆ ˆ
C = δ c = −δ < Φ |i∂S i∂|Φ >,
IJ IJ I IJ 0I 0 0I
ˆ ˆ
(C −T) = − < Φ |i∂S i∂|Φ > (6)
IJ 0I 0 0J
We are calculating Det using the formula:
low
m
˜ ˜
lnDet = Tr idm(S(m)−S (m)) (7)
low ′ ′ 0 ′
ZM1
Within zero-mode assumption (Eq. (2)) the trace is restricted to the subspace of instan-
tons:
1
Tr(S −S ) = − < Φ |i∂ˆS2i∂ˆ|Φ >< Φ |( )|Φ > (8)
0 0,J 0 0,I 0,I ˆ ˆ 0,J
i∂S i∂
I,J 0
X
Introducing now the matrix
ˆ ˆ
B(m) =< Φ |i∂S i∂|Φ > (9)
IJ 0,I 0 0,J
it is easy to show that
m B(m) 1
lnDet = Tr idm(S(m)−S (m)) = (dB(m) )
low ′ ′ 0 ′ ′ II
B(m)
ZM1 I ZB(M1) ′
X
B(m)
= Trln = lndetB(m)−lndetB(M ) (10)
1
B(M )
1
which is desired answer. The determinant detB(m) from Eq. (10) is the extension of the
Lee-Bardeen result [15] for the non-small values of current quark mass m.
Light quark effective action beyond chiral limit
Averaged Det leads to the partition function Z[m], which for N = 2 has the form:
low f
2
Z[m] = dλ+dλ DψDψ†exp[ d4x ψf†(i∂ˆ + imf)ψf (11)
−
Z Z f=1
X
K K
+λ Y+ +λ Y +N ln +N ln ],
+ 2 − 2− + λ+ − λ
−
hereλ aredynamicalcouplings(K isunessentialconstant,whichprovideunder-logarithm
±
expression dimensionless) [9, 13, 14]. Values of them are defined by saddle-point calcula-
tions. Y are t’Hooft type interaction terms [10]:
2±
1 1 1
Y = d4x[(1− )det iJ (ρ,x)+ det iJ (x)] (12)
2± N2 −1 2N ± 8N µ±ν
c Z c c
d4k d4l 1±γ
J (x) = f g expi(k −l )xq+(k ) 5q (l )
f±g (2π)8 f g f f 2 g g
Z
d4k d4l 1±γ
J (x) = f g expi(k −l )xq+(k ) 5σ q (l )
µ±ν,fg (2π)8 f g f f 2 µν g g
Z
3
where q(k) = 2πρF(k)ψ(k). The form-factor F(k) is due to zero-modes and has explicit
formF(k) = −d[I (t)K (t)−I (t)K (t)] .Inthefollowing wewill neglect by J (x)
dt 0 0 1 1 t=|k|ρ µ±ν,fg
2
interaction term, since it give a O( 1 ) contribution to the quark condensate. Since
N2
c
q(x) = d4k exp(ikx) q(k), J (x) = q+(x)1 γ5q (x),
(2π)4 f±g f ±2 g
R
iJ+(x) iJ (x)
−
det +det (13)
g g
1
= (−(q+(x)q(x))2 −(q+(x)iγ ~τq(x))2 +(q+(x)~τq(x))2 +(q+(x)iγ q(x))2).
8g2 5 5
Here color factor g2 = (Nc2−1)2Nc.
(2Nc 1)
In the following we will−take equal number of instantons and antiinstantons N =
+
N = N/2 and corresponding couplings λ = λ.
− ±
Now it is naturalto bosonize quark-quark interaction terms (13) by introducing meson
fields. For N = 2 case it is exact procedure. We have to take into account the changes
f
of q and q under the SU(2) chiral transformations:
†
δq = iγ ~τα~q, δq+ = q+iγ ~τα~
5 5
to introduce appropriate meson fields, changing under SU(2) chiral transformations as:
~ ~
δσ = 2α~φ, δφ = −2α~σ, δη = −2α~~σ, δ~σ = 2ηα~.
Then δq+(σ+iγ ~τφ~)q = 0, δq+(~τ~σ+iγ η)q = 0 means that these combinations of fields
5 5
are chiral invariant 2. So, the interaction term has an exact bosonized representation:
iJ+ iJ
d4xexp[λ(det +det −)] (14)
g g
Z
λ0.5 1
= DσDφ~DηD~σexp d4x[ q+i(σ +iγ ~τφ~ +i~τ~σ +γ η)q − (σ2 +φ~2 +~σ2 +η2)]
5 5
2g 2
Z Z
Then the partition function is
K
~
Z[m] = dλDσDφDηD~σexp[N ln −N (15)
λ
Z
−1 d4x(σ2 +φ~2 +~σ2 +η2)+Trln pˆ+im+iλ20g.5(2πρ)2F(σ +iγ5~τφ~ +i~τ~σ +γ5η)F]
2 pˆ+im
Z
(Tr(...) means here tr d4x < x|(...)|x >, where tr is the trace over Dirac, color,
γ,c,f γ,c,f
and flavor indexes.) In the following we assume m = m = m. Then common saddle
R u d
point on λ, σ (= const) (others = 0)is defined by Eqs. ∂V[m,λ,σ] = ∂V[m,λ,σ] = 0, where
∂λ ∂σ
the potential
K 1 pˆ+i(m+M(λ,σ)F2(p))
V[m,λ,σ] = −N ln +N + Vσ2 −Trln (16)
λ 2 pˆ+im
2Certainly, quark-quark interaction term Eq. (13) is non-invariant over U(1) axial transformations,
as it must be.
4
and we defined M(λ,σ) = λ0.5(2πρ)2σ. Then the common saddle-point on λ and σ is
2g
given by Eqs.:
1 iM(λ,σ)F2(p) 1
N = Tr = Vσ2. (17)
2 pˆ+i(m+M(λ,σ)F2(p)) 2
The solutions of this Eqs. are λ and σ = (2N)1/2 = 21/2R 2. It is clear that M =
0 0 V − 0
M(λ ,σ ) has a meaning of dynamical quark mass, which is defined by this Eqs.. At
0 0
typical values R 1 = 200MeV, ρ 1 = 600MeV we have σ2 = 2(200MeV)4, and in chiral
− − 0
limit m = 0 M → M = 358MeV, λ ≈ M2 . It is clear that due to saddle-point
0 00 00 00
equation (17) M (and λ ) become the function of the current mass m. This dependence
0 0
was investigated in [14].
Vacuum with account of quantum corrections
The account of the quantum fluctuations around saddle-points σ ,λ will change the
0 0
potential V[m,λ,σ] to V [m,λ,σ] (it is clear that the difference between these two
eff
potentials is order of 1/N ). Then, the partition function is given by Eq.
c
Z[m] = dλexp(−V [m,λ,σ]) (18)
eff
Z
There is important difference between this instanton generated partition function Z[m]
and traditional NJL-type models – we have to integrate over the coupling λ here. As was
mentioned before, this integration on λ by saddle-point method leads to exact answer.
This saddle-point is defined by Eq.:
dV [m,λ,σ]
eff
= 0 (19)
dλ
which leads to the λ as a function of σ, i.e. λ = λ(σ).
Then, the vacuum is the minimum of the effective potential V [m,σ], which is given
eff
by a solution of the equation
dV [m,σ,λ(σ)] ∂V [m,σ,λ(σ)]
eff eff
= = 0. (20)
dσ ∂σ
where it was used Eq. (19).
We denote a fluctuations as a primed fields Φ . The action and corresponding V
′i eff
now has a form:
S[m,λ,σ,Φ] = S [m,λ,σ]+S [m,λ,σ,Φ], (21)
′ 0 V ′
1 pˆ+i(m+M(λ,σ)F2) K
S [m,λ,σ] = V[m,λ,σ] = Vσ2 −Trln −N ln +N
0
2 pˆ+im λ
1
S [m,λ,σ,Φ] = d4x (σ′2 +φ~′2 +~σ′2 +η′2)
V ′
2
Z
1 iM(λ,σ)F2 2
~
− Tr (σ +iγ ~τφ +i~τ~σ +γ η ) ,
2σ2 "pˆ+i(m+M(λ,σ)F2) ′ 5 ′ ′ 5 ′ #
and
V [m,λ,σ] = S [m,λ,σ]+Vmes[m,λ,σ] (22)
eff 0 eff
5
Here second term in Eq. (22) is explicitly represented by
1 δ2S [m,λ,σ,Φ] V d4q 1 d4p
Vmes[m,λ,σ] = Trln V ′ = ln[1−tr
eff 2 δΦ (x)δΦ (y) 2 (2π)4 σ2 (2π)4
′i ′j i Z Z
X
M(λ,σ)F2(p) M(λ,σ)F2(p+q)
× Γ Γ ], (23)
pˆ+i(m+M(λ,σ)F2(p)) ipˆ+qˆ+i(m+M(λ,σ)F2(p+q)) i
wherethefactorsΓ = (1,iγ ~τ,i~τ,γ )andthesumoniiscounted allcorresponding meson
i 5 5
~
fluctuations σ ,φ,~σ ,η . tr here means the trace over flavor, color and Dirac indexes.
′ ′ ′ ′
Integrals in Eq. (23) are completely convergent one due to the presence of the form-
factors F.
Certainly the quantum fluctuations contribution will move the the coupling λ from λ
0
to λ +λ and σ as σ → σ +σ , where λ1 and σ1 are of order 1/N .
0 1 0 0 1 λ0 σ0 c
First, consider Eq. (19):
dV [m,λ,σ] 1 iM(λ,σ)F2 V d4q
eff
λ = N − Tr + (24)
dλ 2 pˆ+i(m+M(λ,σ)F2) 2 (2π)4
i Z
X
d4p M(λ,σ)F2(p) M(λ,σ)F2(p+q)
×[σ2 −tr Γ Γ ] 1
(2π)4pˆ+i(m+M(λ,σ)F2(p)) ipˆ+qˆ+i(m+M(λ,σ)F2(p+q)) i −
Z
d4p M(λ,σ)F2(p) M(λ,σ)F2(p+q)
×[−tr Γ Γ
(2π)4pˆ+i(m+M(λ,σ)F2(p)) ipˆ+qˆ+i(m+M(λ,σ)F2(p+q)) i
Z
d4p M(λ,σ)F2(p) 2 M(λ,σ)F2(p+q)
+itr Γ Γ ] = 0
(2π)4 pˆ+i(m+M(λ,σ)F2(p))! ipˆ+qˆ+i(m+M(λ,σ)F2(p+q)) i
Z
From this saddle-point Eq. we get λ = λ(σ).
From vacuum Eq. (20) we in similar manner arrive to:
∂V [m,σ,λ(σ)] iM(λ(σ),σ)F2 V d4q
σ eff = Vσ2 −Tr + (25)
∂σ pˆ+i(m+M(λ(σ),σ)F2) 2 (2π)4
i Z
X
d4p M(λ(σ),σ)F2(p) M(λ(σ),σ)F2(p+q)
×[σ2 −tr Γ Γ ] 1
(2π)4pˆ+i(m+M(λ(σ),σ)F2(p)) ipˆ+qˆ+i(m+M(λ(σ),σ)F2(p+q)) i −
Z
d4p M(λ(σ),σ)F2(p) 2 M(λ(σ),σ)F2(p+q)
×[2itr Γ Γ ] = 0
(2π)4 pˆ+i(m+M(λ(σ),σ)F2(p))! ipˆ+qˆ+i(m+M(λ(σ),σ)F2(p+q)) i
Z
Since we are believing to 1 expansion, it is natural inside quantum fluctuations contri-
Nc
bution (under the integrals over q) to take σ = σ , M(λ(σ),σ) = M .
0 0
To simplify the expressions introduce vertices V (q), V (q) and meson propagators
2i 3i
Π (q), which are defined as:
i
d4p M (p) M (p+q)
0 0
V (q) = tr Γ Γ (26)
2i (2π)4pˆ+iµ (p) ipˆ+qˆ+iµ (p+q) i
Z 0 0
d4p M (p) 2 M (p+q)
0 0
V (q) = tr Γ Γ (27)
3i Z (2π)4 pˆ+iµ0(p)! ipˆ+qˆ+iµ0(p+q) i
2
Π 1(q) = −V (q). (28)
−i R4 2i
6
Here M (p) = M F2(p), µ (p) = m+M (p) and was taken into account that σ2 = 2R 4.
0 0 0 0 0 −
From Eqs. (24) and (25) we have
M 2 1 M (p) 2 d4q
1 + Tr 0 = (iV3(q)−V (q))Π (q) (29)
M0 R4 V pˆ+iµ0(p)!  i Z (2π)4 i 2i i
X
σ  R4 d4q 
1
= − V (q)Π (q) (30)
σ 4 (2π)4 2i i
0 i Z
X
The vertices V (q), V (q) and the meson propagators Π (q) are well defined functions,
2i 3i i
providing well convergence of the integrals in Eqs. (29), (30).
It is of special attention to the contribution of pion fluctuations φ~ at small
′
pion momentum q. We shall demonstrate that this contribution leads to the famous
chiral log term with model independent coefficient in the correspondence with previous
calculations in NJL-model [18].
Pion inverse propagator of Π 1(q) at small q ∼ m is: Π 1(q) = f2 (m2 + q2). At
−φ~′ π −φ~′ kin π
lowest order on m, f ≈ f = 93MeV, m2 ∼ m.
kin,m=0 π π
The vertices in the right side of Eq. (29) at q = 0 and in chiral limit are:
2 d4p p2M2(p)
V (0) = , iV (0)−V (0) = 8N 0 (31)
2φ~′i,m=0 R4 3φ~′i,m=0 2φ~′i,m=0 c (2π)4(p2 +M2(p))2
Z 0
We see that the factor in the left side of Eq. (29) in the chiral limit is equal to:
d4p ipˆM (p)
0
tr = −2(iV (0)−V (0)) (32)
Z (2π)4(pˆ+iM0(p))2 3φ~′i,m=0 2φ~′i,m=0
~
Collecting all the factors we get small q ≤ κ contribution of pion fluctuations φ:
′
σ M 3 κ d4q 1
1 1
| = | = − (33)
σ φ~′,smallq M φ~′,smallq 2f2 (2π)4m2 +q2
0 0 π Z0 π
3 κ2 1 3 m2
= − q2dq2 = − (κ2 +m2 ln π )
32π2f2 f2(m2 +q2) 32π2f2 π κ2 +m2
π Z0 π π π π
Here we put m = 0 everywhere except m . We see that the coefficient in the front of of
π
m2 lnm2 is a model independent as it must be.
π π
Quick estimate, assuming κ = ρ 1, gives
−
σ M 3
1| ≈ 1| ≈ − (1+m2ρ2lnm2ρ2) ≈ −0.4(1+0.054ln0.054) (34)
σ φ~′ M φ~′ 32π2f2ρ2 π π
0 0 π
So, we expect that pion loops is provided not only non-analytical 1 mlnm term but also
Nc
very large contribution to 1 term.
Nc
This one dictate the strategy of the following calculations of σ and M :
1 1
1. we have to extract analytically 1 mlnm term from pion loops;
Nc
2. rest part of σ and M can be calculated numerically and expanded over m, paying
1 1
special attention to the pion loops and keeping 1 and 1 m terms.
Nc Nc
7
For actual numerical calculations we are using simplified version of the form-factor
F(p) from [17] (with corrected high momentum dependence):
L2 1.414
F(p < 2GeV) = , F(p > 2GeV) = (35)
L2 +p2 p3
where L ≈ √2 = 848MeV.
ρ¯
At N = 3 semi-numerical calculations of M and σ lead to:
c 1 1
M
1
= −0.662−4.64m−4.01mlnm (36)
M
0
σ
1
= −0.523−4.26m−4.00mlnm (37)
σ
0
Here m is given in GeV. Certainly, in (36) the m lnm term is completely correspond to
Eq. (33). M1 is −66% in chiral limit and reach its maximum ∼ −20% at m ∼ 0.115GeV.
M0
The relative shift of the vacuum σ1 is −52% at the chiral limit and reach its maximum
σ0
∼ −2% at m ∼ 0.125GeV.
The main contribution to both quantities M1 and σ1 come from pion loops. Other
M0 σ0
mesons give the contribution ∼ 10% to O( 1 ) and O( 1 m) terms.
Nc Nc
Quark condensate
We have to calculate quark condensate beyond chiral limit taking into account O(m),
O( 1 ),O( 1 m)andO( 1 mlnm)terms. Quarkcondensateisextractedfromthepartition
Nc Nc Nc
function:
1 dV [m,λ,σ] 1 ∂(V[m,λ,σ]+Vmes[m,λ ,σ ])
< q¯q > = eff = eff 0 0
2V dm 2V ∂m
1 i i 1 ∂Vmes[m,λ ,σ ]
eff 0 0
= − Tr( − )+ (38)
2V pˆ+iµ(p) pˆ+im 2V ∂m
here λ = λ + λ , ,σ = σ +σ , M = M +M , µ(p) = m+MF2(p). First term of
0 1 0 1 0 1
Eq. (38) is
1 i i
− Tr( − ) (39)
2V pˆ+iµ(p) pˆ+im
d4p µ (p) m M M (p)(p2 −µ2(p))
= −4N 0 − + 1 0 0
cZ (2π)4 p2 +µ20(p) p2 +m2 M0 (p2 +µ20(p))2 !
Second term of Eq. (38) – meson loops contribution to the condensate is
1 ∂Vmes[m,λ ,σ ] i d4q d4p M (p) M (p+q)
eff 0 0 = tr 0 Γ 0 Γ
2V ∂m 2 i Z (2π)4 Z (2π)4(pˆ+iµ0(p))2 ipˆ+qˆ+iµ0(p+q) i!
X
2N d4p M0(p) M0(p+q) −1
× −tr Γ Γ (40)
V Z (2π)4pˆ+iµ0(p) ipˆ+qˆ+iµ0(p+q) i!
8
At m = 0 and without meson loops the condensate is
d4p M (p)
00
< q¯q > = −4N (41)
00 c (2π)4p2 +M2 (p)
Z 00
Here M ≡ M .
00 0,m=0
~
Let us to consider now the contribution of pion fluctuations φ to the quark condensate
′
at small q. First we consider:
1 ∂Vφ~′,smallq[m,λ ,σ ] d4p M2(p)µ (p) κ d4q
eff 0 0 = 12N 0 0 (42)
2V ∂m c (2π)4(p2 +µ2(p))2 (2π)4f2 (m2 +q2)
Z 0 Z0 kin π
We keep m only in m2. Then at m = 0 µ (p) ⇒ M (p) ⇒ M (p), f ⇒ f and we have
π 0 0 00 kin π
M d4p M (p)(p2 −M2 (p))
< q¯q > = < q¯q > − 14N 00 00 (43)
00 M c (2π)4 (p2 +M2 (p))2
0 Z 00
d4p M3 (p) κ d4q 1
+ 12N 00
c (2π)4(p2 +M2 (p))2 (2π)4f2(m2 +q2)
Z 00 Z0 π π
3 κ d4q 1
=< q¯q > 1− (44)
00 2 Z0 (2π)4fπ2(m2π +q2)!
Eq. (33) for M1 was applied here. We see that Eq. (44) is in the full correspondence with
M0
[11, 12].
Detailed numerical calculations lead to the semi-analytical formula for the quark con-
densate including all O(m), O( 1 ) and O( 1 mlnm)-corrections:
Nc Nc
< q¯q >=< q¯q > (1−18.53m−7.72mlnm) (45)
m=0
Here < q¯q > = 0.52 < q¯q > . Certainly, the mlnm term in Eq.(45) is in full
m=0 00
correspondence with Eq. (44), as it must be. < q¯q > / < q¯q > is a rising function of
m=0
m until m ∼ 0.04GeV and is a falling one in the region m > 0.04GeV.
The main contribution to O( 1 ), O( 1 m) and O( 1 mlnm) terms in <q¯q> is due to
Nc Nc Nc <q¯q>00
pion loops. Other mesons give the contribution ∼ few% to O( 1 ) and O( 1 m) terms.
Nc Nc
m − m effects in quark condensate
d u
Current quark mass become diagonal 2 × 2 matrix with m = m ,m = m , m =
1 u 2 d
m 1+τ3 + m 1 τ3 = m + δmτ3. Here m = m1+m2,δm = m − m . Let us introduce
1 2 2 −2 2 2 1 2
external field s . In our particular case it is s = iδm,s = s = 0. Our aim is to find the
i 3 2 1 2
asymmetry of the quark condensate <u¯u> <d¯d>, taking into account only O(δm) terms
−
<u¯u>
and neglecting by O( 1 δm), O( 1 δmlnm). It means that we neglect at all by meson
Nc Nc
loops contribution.
Inthepresence oftheexternalfield~sweexpect alsovacuumfield~σ. Effective potential
within requested accuracy is
V [σ,~σ,m] ≈ S [m,λ,σ,~σ] (46)
eff 0
V pˆ+i~τ~s+i(m+M(λ,σ,~σ)F2) K
= (σ2 +~σ2)−Trln −N ln +N.
2 pˆ+im+i~τ~s λ
9
λ,σ,~σ are defined by the vacuum equations:
∂V ∂V ∂V
eff eff eff
= 0, = 0, = 0. (47)
∂λ ∂σ ∂σ
i
They can be reduced to the following form:
1 F2(p)M (m +M F2(p))
i i i
Tr = N (48)
2 p2 +(m +M F2(p))2
i i
where M = λ0.5(2πρ)2(σ ± σ ). Solution of these equations leads to λ = λ[m,~s], σ =
i 2g 3
σ[m,~s] σ = σ [m,~s]. We have to put them into V and find V = V [m,~s]. Desired
i i eff eff eff
correlator is
∂V [m,~s]
eff
| (49)
∂s s3=δ2m,s1,2=0
3
We calculate this correlator within requested accuracy, taking into account only O(δm)
terms. So, the difference of the vacuum quark condensates of u and d quarks is
¯
< u¯u > − < dd > (50)
1 −i −i −i −i
= Tr( − )−Tr( − )
V " pˆ+i(mu +MuF2) pˆ+imu pˆ+i(md +MdF2) pˆ+imd #
¯
We expect that < dd ><< u¯u > if m > m .
d u
Typical values of light current quark masses [19] are m = 5.1MeV, m = 9.3MeV on
u d
the scale 1GeV (which is in fact close to our scale ρ 1 = 0.6GeV) leads to the asymmetry
−
¯
< u¯u > − < dd >
= 0.026 (51)
< u¯u >
From this asymmetry and using sum-rules [20] we estimate strange quark condensate at
m = 120MeV as:
s
< s¯s >
= 0.43, (52)
< u¯u >
which is rather small. The reason that the asymmetry (51) is rather large.
Conclusion
Inthe framework of instanton vacuum model it was calculated simplest possible correlator
– quark condensate with complete account of O(m), O( 1 ), O( 1 m) and O( 1 mlnm)
Nc Nc Nc
terms, demanding the calculation of meson loops contribution. Since initial instanton
generated quark-quark interactions are nonlocal and contain corresponding form-factor
induced by quark zero-mode, these loops correspond completely convergent integrals.
The main loop corrections come from the pions, as it was expected. We found that
O( 1 ) corrections are very large ∼ 50%, which request the ∼ 10% changing of the basic
Nc
parameters – average inter-instanton distance R and average instanton size ρ to restore
chiral limit value of the quark condensate < q¯q > and other important quantities as
m=0
f and m to their phenomenological values. This work in the progress.
π π
In general, it was demonstrated, that instanton vacuum model is well working tool
also beyond chiral limit and satisfy chiral perturbation theory.
10