QCD condensates with flavor SU(3) symmetry breaking from the instanton vacuum

0.2 MB·English

by  Seung-il Nam

PNU-NTG-01/2006
PNU-NURI-01/2006
QCD condensates with flavor SU(3) symmetry breaking from the
instanton vacuum
Seung-il Nam∗ and Hyun-Chul Kim†
Department of Physics and Nuclear Physics & Radiation Technology Institute (NuRI),
Pusan National University, Busan 609-735, Republic of Korea
(Dated: December 2006)
Abstract
7 We investigate the effects of flavor SU(3)-symmetry breaking on the quark, gluon, and mixed
0
quark-gluon condensates, based on the nonlocal effective chiral action from the instanton vacuum.
0
2 We take into account the effects of the flavor SU(3) symmetry breaking in the effective chiral
n action, so that the dynamical quark mass depends on the current quark mass (mf). We compare
a the results of the present approach with those without the current quark mass dependence of the
J
dynamical quark mass. Itis foundthat theresult of the quarkcondensate is decreased by about 30
5
2 % as m increases to 200 MeV, while that of the quark-gluon mixed condensates is diminished by
f

about 15%. We obtain the ratios of the quark and quark-gluon mixed condensates, respectively:
2
v [hs¯si/hu¯ui]1/3 = 0.75 and [hs¯σµνGµνsi/hu¯σµνGµνui]1/5 = 0.87. It turns out that the dimensional
1 parameter m2 = hq¯σ Gµνqi/hq¯qi =1.60 ∼ 1.92GeV2.
4 0 µν
0
5 PACS numbers: 11.15.Tk,14.40.Aq
0
Keywords: Gluon condensates, Quark condensates, Quark-gluon mixed condensates, Instanton vacuum,
6
0 Flavor SU(3)-symmetry breaking
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∗Electronic address: [email protected]
†Electronic address: [email protected]
1
1. Understanding the QCD vacuum is very complicated, since both perturbative and
non-perturbative fluctuations come into play. In particular, the quark and gluon conden-
sates, being the lowest dimensional ones, characterize the non-perturbative structure of the
QCD vacuum. The quark condensate is identified as the order parameter for spontaneous
chiral-symmetry breaking (SχSB) which plays an essential role in describing low-energy
phenomena of hadrons: In the QCD sum rule, these condensates come from the operator
product expansion and are related to hadronic observables [1], while in chiral perturbation
theory (χPT), the free parameter B is introduced in the mass term of the effective chiral
0
Lagrangian at the leading order [2] measuring the strength of the quark condensate [3]. On
the other hand, the gluon condensate is not the order parameter but measures the vacuum
energy density [1], which was first estimated by the charmonium sum rule [4].
While the quark and gluon condensates are well understood phenomenologically, higher
dimensional condensates suffer from large uncertainty. Though it is still possible to estimate
dimension-six four-quark condensates in terms of the quark condensate by using the factor-
ization scheme which is justified in the large N limit, the dimension-five mixed quark-gluon
c
condensateisnoteasilydetermined phenomenologically. Inparticular, themixedcondensate
is an essential parameter to calculate baryon masses [5], exotic hybrid mesons [6], higher-
twist meson distribution amplitudes [7] within the QCD sum rules. Moreover, the mixed
quark condensate plays a role of an additional order parameter for SχSB, since the quark
chirality flips via the quark-gluon operator. Thus, it is naturally expressed in terms of the
quark condensate:
hψ¯σµνG ψi = m2hψ¯ψi (1)
µν 0
with the dimensional parameter m2 which was estimated in various works [5, 8, 9, 10, 11, 12]
0
and with the definition G = Ga λa/2.
µν µν
The instanton picture allows us to study the QCD vacuum microscopically. Since the
instanton picture provides a naturalmechanism forSχSB due to the delocalization of single-
instantonquarkzeromodesintheinstantonmedium, thequarkcondensatecanbeevaluated.
The instanton vacuum is validated by the two parameters: The average instanton size
ρ¯ ∼ 1/3fm and average inter-instanton distance R ∼ 1fm. These essential numbers were
suggested by Shuryak [13] within the instanton liquid model and were derived from Λ by
MS
Diakonov and Petrov [14]. These values were recently confirmed by lattice QCD simulations
[15, 16, 17, 18, 19].
In the present work, we want to investigate the effect of flavor SU(3)-symmetry breaking
on the above-mentioned three QCD condensates, i.e. quark, gluon, and mixed quark-gluon
condensates, based on the instanton liquid model for the QCD vacuum [20, 21, 22]. The
model was later extended by introducing the current quark masses [23, 24, 25, 26, 27, 28].
Since we are interested in the effect of explicit flavor SU(3)-symmetry breaking, we follow
the formalism of Ref. [23] in which the dependence of the dynamical quark mass on the
current quark mass has been studied in detail. Though the mixed quark-gluon condensate
was already studied in the instanton vacuum [11], explicit SU(3)-symmetry breaking was
not considered. Hence, we want now to extend the work of Ref. [11], emphasizing the effect
of flavor SU(3)-symmetry breaking on the QCD vacuum condensates. We will show that
with a proper choice of the m dependence of the dynamical quark mass [29] the gluon
f
condensate is independent of m . The corresponding results are summarized as follows:
f
The ratio [hs¯si/hu¯ui]1/3 = 0.75, [hs¯σ Gµνsi/hu¯σ Gµνui]1/5 = 0.87, m2 = 1.60GeV2, and
µν µν 0,u
m2 = 1.84GeV2, with isospin symmetry assumed.
0,s
2
2. We start with the effective low-energy QCD partition function from the instanton
vacuum with SU(3)-symmetry breaking taken into account [23, 24, 25]:
YNf N+ YNf N−
Z = Dψ†Dψexp d4x ψ†(i/∂ +im )ψ + − , (2)
 f f fVMNf VMNf 
Z Z f
X
    
where ψ and ψ† denote the quark fields and m stands for the current quark mass for a
f f f
given flavor. We consider here the strict large N expansion 1. V and MNf stand for the
c
space-time volume and for the dimensional parameter, respectively. Note that, by dropping
the current quark mass m in the first bracket in the r.h.s., Eq. (2) turns out to be the
f
same as that derived in the chiral limit [20, 21]. Y in Eq. (2) represents a ’t Hooft-type
±
2N -quark interaction generated by instantons:
f
d4k d4p
YNf = dρd(ρ) dU d4x f f [2πρF (k )][2πρF (p )]
± (2π)4(2π)4 f f f f
Z Z Z f Z
Y
× exp−x· kf − pf Uiα′(Uβj′)†ǫii′ǫjj′ f iψf†(kf)αi1±2γ5ψf(pf)βj , (3)
Xf Xf h i (cid:20) (cid:21)
  
where ρ denotes the instanton size and U represents the color orientation matrix. Since we
are interested in the vacuum condensates, it is enough to consider the case of N = 1 in
f
which the integration over U becomes trivial. We employ the instanton distribution of a
delta-function type, so that we get a simple quark-quark interaction for a given flavor f:
i d4k 1±γ
Y = [2πρ¯F (kρ¯)]2 ψ†(k) 5ψ (k) . (4)
± N (2π)4 f f 2 f
c Z (cid:20) (cid:21)
F (k) is the Fourier transformation of the fermionic zero mode solution Φ . We implicitly
f II¯
assumed that F (k) is a function of the current-quark mass. This assumption will be verified
f
in what follows. Φ satisfies the following Dirac equation under the (anti)instanton effects
II¯
A :
II¯
[i/∂ +A/ +im ]Φ = 0. (5)
II¯ f II¯
Instead of computing Φ directly from Eq. (5), being equivalently, we follows the course
II¯
suggested in Ref. [29] in which Pobylitsa used a elaborated and systematic expansion of
the quark propagator hx|(i/∂ +A/ +im )−1|xi. By doing this, one can immediately obtain
I f
analytical form of F (k). Here, we make a brief explanation on this method. Diakonov et
f
al. made a zero-mode approximation for the quark propagator in the instantons [21]:
1 1 Φ (x)Φ†(y)
x y ≃ x y + I I . (6)
* (cid:12)i/∂ +A/I +im(cid:12) + * (cid:12)i/∂ +im(cid:12) + im
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
We(cid:12)note that this z(cid:12)ero-mode(cid:12)approxim(cid:12)ation causes one difficult problem of breaking the con-
(cid:12) (cid:12) (cid:12) (cid:12)
servation of vector and axial-vector currents. To amend this current conservation problem,
1
Recently, it is shown that the effects of mesonic loops on the quark condensate, which is the next-to-
leading order in the Nc expansion, is not small [30]. However, a proper adjustment of the parameters ρ¯
and R may yield approximately the same results as those without the meson-loop corrections.
3
Pobylitsa expand the quark propagator as follows. Having summed up the planar diagrams,
one arrives the following integral equation for a quark propagator:
1
S = x x
* (cid:12)i/∂ +A/I +im(cid:12) +
(cid:12) (cid:12)
(cid:12) (cid:12) −1
(cid:12) 1 (cid:12) N 1
= x(cid:12) x +(cid:12) tr d4Z hx|(i/∂ +A/ +im)|xi− +(I → I¯) , (7)
c I I
* (cid:12)(cid:12)i/∂ +im(cid:12)(cid:12) + 2VNc Z " /A# 
(cid:12) (cid:12)
(cid:12) (cid:12)  
where ZI(cid:12)I¯ indicat(cid:12)es the collective coordinate for the (anti)instanton. Then the inverse of
the quark propagator can be approximately written with an arbitrary scalar function to be
determined:
λ[2πρ¯]2
S−1(i∂) = i/∂ +im +iM F2(i∂), M = . (8)
f 0 f 0 N
c
Note that we choose the scalar function to be M F2(i∂) for later convenience. The value of
0 f
M will be determined by the self-consistent equation (so called “saddle-point equation”) of
0
the model. To obtain F (k), Inserting Eq.(8) into Eq.(7), we obtain an integral equation for
f
F (i∂):
f
iF2(i∂) ≃ M tr d4Z /A /A +i/∂ +im −iM F2(i∂) −1(i/∂ +im )+(I → I¯) . (9)
f 0 c I I I f 0 f f
(cid:20)Z (cid:16) (cid:17) (cid:21)
Having carried out a straightforward manipulation, finally, we arrive at:
m2 m 0.083858πρ¯
M (k) = M F2(k) 1+ f − f = M F2(k)f(m ), d =
f 0 f s d2 d  0 f f s 2N R2
c
  1
F (k) = 2t I (t)K (t)−I (t)K (t)− I (t)K (t) ≃ 0.198GeV, t = |k|ρ¯/2. (10)
f 0 1 1 0 1 1
t
(cid:20) (cid:21)
Thus, we have exact form of F (k) in terms of explicitly broken flavor SU(3) symmetry
f
(m 6= 0).
f
Now, we are in position to discuss the saddle-point equation of the model derived from
the effective QCD partition function. Introducing the Lagrange multipliers λ [22], we are
±
able to exponentiate the partition function of Eq. (2):
dλ
Z = ± Dψ†Dψexp d4xψ¯ (i/∂ +im )ψ
f f f
2π
Z Z (cid:20)Z (cid:21)
N N
+ −
× exp N ln −1 +λ Y exp N ln −1 +λ Y . (11)
" + λ+VMNf ! + +# " − λ−VMNf ! − −#
In the following, we assume that N = N = N/2 and λ = λ = λ so that the partition
+ − + −
function can be simplified as follows:
dλ N
Z = Dψ†Dψexp d4xψ†(i/∂ +im )ψ +N ln −1 +λ(Y +Y ) .(12)
2π f f f 2λVMNf + −
Z Z (cid:20)Z (cid:18) (cid:19) (cid:21)
ItisalsointerestingtocompareEq.(12)withthepartitionfunctionproposedbyDiakonov
et al.:
dλ N
Z = Dψ†Dψexp d4xψ†i/∂ψ +N ln −1 +λ(Y +Y +2m ) .(13)
D 2π f f 2λVMNf + − f
Z Z (cid:20)Z (cid:18) (cid:19) (cid:21)
4
One can easily find the difference between the effective actions of Eq. (12) and of Eq. (13); In
Eq. (12), the current quark mass m appears as a mass term of the quark fields accounting
f
for the explicit breaking of chiral symmetry, while the m is placed in the quark-instanton
f
vertex in Eq. (13). We note that this configuration of the current quark mass is deeply
related to the parity-violating quark mass term (∝ ψ†γ ψ) in terms of the instanton number
5
fluctuation [21].
Concentrating on the first and third brackets in Eq. (12), we get the fermionic trace log
which relates to the quark propagator as follows:
d4x d4k tr ln −/k +imf + Niλc[2πρ¯Ff(kρ¯)]2
(2π)4 cγ h [−/k +im ] i
Z Z f
d4k [−/k +im +iM (k)]
f f
= N V tr ln , (14)
c (2π)4 γ [−/k +im ]
Z f
where the subscripts c and γ denote the color and Dirac-spin spaces. Thus, we obtain the
partition function in a compact form as follows:
dλ N d4k [−/k +im +iM (k)]
f f
Z = exp N ln −1 +N V tr ln . (15)
Z 2π " (cid:18) 2λVMNf (cid:19) c Z (2π)4 d [−/k +imf] #
Now, we perform a functional variation of the partition function with respect to λ:
δlnZ N d4k iM (k)/N
f c
= − +N V tr = 0,
δλ λ c (2π)4 γ−/k +im +iM (k)
Z f f
N d4k iM (k)[−/k −im −iM (k)]
f f f
= N V tr ,
λ c (2π)4 γ k2 +[m +M (k])2
Z f f
N d4k M (k)[m +M (k)]
f f f
= 4N . (16)
V c (2π)4 k2 +[m +M (k)]2
Z f f
Final formula in Eq. (16) is called the saddle-point equation of the model. We employ the
standard values of the instanton ensemble, N/V = 2004 MeV4 and ρ¯ ∼ 1/3 fm ≃ 1/600
MeV−1 for the numerical calculations. From these values, we obtain M = 0.350 GeV,
0
which is determined self-consistently by the saddle-point equation. Using this value, we
obtain ihu†ui ≃ 2503 MeV3 and pion weak decay constant F ≃ 93 MeV. We note that this
π
saddle-point equationiscloselyrelatedtothegluon-condensate[31]; hG Gµνi = 32π2N/V.
µν f
As for an arbitrary N , it becomes rather difficult to obtain the saddle-point equation,
f
since we need to perform a complicated integration over the color orientation. However,
keeping only the leading order in the large N limit and additional variation over the Her-
c
mitian N ×N flavor matrix (M in Ref. [21]), we have the same form of the saddle-point
f f
equation for each flavor as shown in Eq. (16). The similar argument is also possible for the
quark and mixed condensates [11, 21, 24].
Being similar to the saddle-point equation (the gluon condensate), the quark condensate
for each flavor can be obtained by the following functional derivative with respect to the
current quark mass m :
f
1 δlnZ d4k /k +i[m +M (k)] /k +im
f f f
hq¯qi = = −iN tr −
f V δmf cZ (2π)4 γ"k2 +[mf +Mf(k)]2 k2 +m2f#
d4k m +M (k) m
f f f
= 4N − . (17)
cZ (2π)4 "k2 +[mf +Mf(k)]2 k2 +m2f#
5
Onecanimmediatelyseethatwhenm → 0,Eq.(17)getsequaltothewell-knownexpression
f
for the quark condensate in the chiral limit.
Now, we discuss how to calculate the mixed condensate, hq¯σ Gµνqi. Actually, the local
µν
operatorinside thiscondensate corresponds to thequark-gluoninteractionof aYukawa type.
However, in the present work, the gluon field strength (G ) can be expressed in terms of
µν
the quark-instanton interaction [11, 21]. First, the one flavor quark and one instanton
interaction in Eq. (3) can be rewritten as a function of space-time coordinates x and color
orientation matrix U:
d4k d4k
Y (x,U) = 1 2 [2πρF (k )][2πρF (k )]e−ix·(k1−k2)
±,1 (2π)4(2π)4 f 1 f 2
Z
× Uiα′(Uβj′)†ǫii′ǫjj′ f iψf†(k1)αi1±2γ5ψf(k2)βj . (18)
h i (cid:20) (cid:21)
Here, we assume again the delta function-type instanton distribution. We define then the
field strength Ga in terms of the instanton configuration as a function of I(I¯) position and
µν
orientation matrix U:
1
Ga (x,x′,U) = λaUλbU† Gb (x′ −x). (19)
±µν 2 ±µν
h i
Gb (x′ − x) stands for the field strength consisting of a certain instanton configuration.
±µν
Using Eqs. (18) and (19), we define the field strength in terms of the quark-instanton inter-
action:
iN M
Gˆa = c d4x dUGa (x,x′,U)Y (x,U) (20)
±µν 4πρ¯2 ±µν ±,1
Z Z
Following the method in Ref. [11, 21], we finally obtain the quark-gluon mixed condensate
as follows:
hq¯σ Gµνqi = 2N ρ¯2 d4k1 d4k2 Mf(k1)Mf(k2)G(k1,k2)N(k1,k2) , (21)
µν f c (2π)4 (2π)4[k2 +q[m +M (k )]2][k2 +[m +M (k )]2]
Z Z 1 f f 1 2 f f 2
where G(k ,k ) and N(k ,k ) are defined as follows:
1 2 1 2
K (t) 4K (t) 2 8 8
G(k ,k ) = 32π2 0 + 0 + + K (t)− ,
1 2 " 2 " t2 (cid:18)t t3(cid:19) 1 t4##
1
N(k ,k ) = 8k2k2 −6(k2 +k2)k ·k +4(k ·k )2 (22)
1 2 (k −k )2 1 2 1 2 1 2 1 2
1 2
h i
with t = |k − k |ρ¯. The functions K and K stand for the modified Bessel functions of
1 2 0 1
the second kind of order 0 and 1, respectively. If we consider for arbitrary N the mixed
f
condensate, the situation may be somewhat different from the cases of the gluon and quark
condensates. We note that, though the mixed condensate has been calculated for the case
of N = 1, the same formula of Eq. (21) still holds for each flavor with arbitrary N as
f f
discussed previously [11].
3. We first discuss the numerical results for the gluon condensate, hG Gµνi/32π2 =
µν
N/V. As mentioned before, we use N/V = 2004 MeV4 [31]. In Fig. 1, we show the
6
260
14/2/π(32)[MeV]i221305.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................W...................................i..........t................h.....................o...............u.........................t..........................f..................(.................m...............................f....................)......................................................................................................................................................................
νi Withf(mf)
µ
G
185
ν
µ
G
h
h
160
0 50 100 150 200
m [MeV]
f
FIG. 1: Gluon condensate as a function of the current quark mass m . The solid curve draws the
f
gluon condensate with the m -correction factor f(m ) in Eq. (10) and the dashed one corresponds
f f
to that without it.
numerical results of the gluon condensates with and without the m -correction factor f(m )
f f
in Eq. (10). As shown in Fig. 1, the result with f(m ) is noticeably different from that
f
without it. Since the gluon condensate is related to the vacuum energy density, it should
be independent of the current quark mass m . However, if we turn off the m -correction
f f
factor, the gluon condensate increases almost linearly. It indicates that it is essential to
have the m -correction factor f(m ) in order to consider SU(3) flavor symmetry breaking
f f
properly. In Table I we list the values of the gluon condensate when m = m = 5 MeV and
f u
m = m = 150 MeV with and without f(m ).
f s f
In Fig. 2, we depict the results of the quark condensate in a similar manner. In general,
the quark condensate decreases as the value of m increases, as already shown in Ref. [30].
f
The values of the quark condensate are listed in Table II. As for the u-quark condensate,
we have −2503 ∼ −2603 MeV3. Here, we assume isospin symmetry for the light quarks;
m = m = 5 MeV. We find that the nonstrange quark condensate does not depend much
u d
on f(m ). In contrast, the strange one is sensitive to the m correction factor as shown
f f
in Table II (and also in Fig. 2). We find that without the f(m ) the quark condensate is
f
decreased by about 10 % as m increases from 0 to 200 MeV, while it is diminished by
f
about 30 % with the f(m ). Being compared with the results of Refs. [9, 23, 32], the quark
f
condensate in the present work is obtained to be similar to them with the correction factor.
With f(m ) without f(m )
f f
hG Gµνi /32π2 2004 2024
µν u
hG Gµνi /32π2 1994 2434
µν s
TABLE I: Gluon condensates for m = 5 and m = 150 MeV [MeV4].
u s
7
260
13/−hqqi[¯][MeV] 122280240000..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................W............................W...............i.......t...................h..i..........t........o........h...............u......................tf....................(.......f............m...........(.................m........f.................)..........f...................).......................................................................................................................................................................................
160
0 50 100 150 200
m [MeV]
f
FIG. 2: Quark condensate as a function of the current quark mass m . The solid curve draws the
f
quark condensate with the m -correction factor f(m ) in Eq. (10) and the dashed one corresponds
f f
to that without it.
InFig.3, we draw theresults ofthe quark-gluonmixed condensate as functions ofthe m .
f
The correction factor f(m ) being taken into account, the mixed condensate is decreased
f
by about 15 %. However, it is almost constant when f(m ) is switched off. We also list the
f
values of the mixed condensate for the up and strange quarks in Table III.
We now consider the effect of flavor SU(3)-symmetry breaking by calculating the ratios
between the nonstrange condensates and the strange ones. As already discussed, the ratio
of the gluon condensates remains unity for the m -correction factor. As for the ratio of
f
the quark condensate, [hs¯si/hu¯ui]1/3, there is a great amount of theoretical calculations and
thoseresultslieintherange: 0.79 ∼ 1.03[9,23,32]. Ourresultsare0.75 ∼ 0.93forthequark
condensate, which is consistent with them. We note that [hs¯si/hu¯ui]1/3 = 0.75 with the m -
f
correction factor will be taken as our best result. Being compared to Refs. [34, 35, 36, 37],
our results are in good agreement with them.
We now study a dimensional quantity m2 defined as the ratio between the mixed and
0
quark condensates:
m2 = hq¯σ Gµνqi/hq¯qi, (23)
0 µν
which is an important input for general QCD sum rule calculations. We draw m2 in Fig. 4
0
With f(m ) without f(m )
f f
hu¯ui −2533 −2543
hs¯si −1913 −2353
TABLE II: Quark condensates for m = 5 and m = 150 MeV [MeV3].
u s
8
500
15/µν−hqσGqi[¯][MeV]µν 444440246800000............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................W........................................i...............t..............h.....................................f..........................(...........m.....................................f.........................).............................................................................W.........................................i..............t.............h...........................o...............u.........................t..............................f.....................(.................m..................................f......................)....................................................................................................................................................................................
0 50 100 150 200
m [MeV]
f
FIG. 3: Quark-gluon mixed condensate as a function of the current quark mass m . The solid
f
curve draws the mixed condensate with the m -correction factor f(m ) in Eq. (10) and the dashed
f f
one corresponds to that without it.
as a function of m and list the values of m2 in Table IV. The value of m2 increases as
f 0 0
m does, which implies that the mixed condensate is less sensitive to the m than the
f f
quark condensate. The values of m2 are in the range of 1.84GeV2 for the strange quark
0
and of 1.60GeV2 for the up quark. Thus, the strange m2 turns out to be larger than the
0,s
nonstrange m2 by about 15%.
0,u
We now consider the scale dependence of the condensates.
α (µ ) γ1/b α (µ ) γ2/b
s B s B
hO (µ )i = hO (µ )i, hO (µ )i = hO (µ )i, (24)
1 A 1 B 2 A 2 B
"αs(µA)# "αs(µA)#
where the subscripts 1 and 2 indicate different generic operators corresponding to the con-
densates in which we are interested. The µ and µ denote two different renormalization
A B
scales of the operators. γ is the corresponding anomalous dimension for the condensates. b
is defined as 11N /3−2N /3 and becomes 11−2 = 9 in the present work. Taking the ratio
c f
of the operators 1 and 2, we have:
hO (µ )i α (µ ) (γ1−γ2)/b hO (µ )i
1 A s B 1 B
= . (25)
hO2(µA)i "αs(µA)# hO2(µB)i
With f(m ) without f(m )
f f
hu¯σ Gµνui −4815 −4845
µν
hs¯σ Gµνsi −4185 −4835
µν
TABLE III: Quark-gluon mixed condensates for m = 5 and m =150 MeV [MeV5].
u s
9
2.2
22m[GeV]0112...680........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................W..............................W...........i..............t...........h..i................t.......o....h..................u....................tf.......................(.....f..............m..........(...................m.......f...................)...........f...................)............................................................................................................................................................................................
1.4
0 50 100 150 200
m [MeV]
f
FIG. 4: Ratio m2 as a function of the current quark mass m . The solid curve draws the gluon
0 f
condensate with the m -correction factor f(m ) in Eq. (10) and the dashed one corresponds to
f f
that without it.
If O and O stand for the mixed and quark condensates respectively, we have γ = −2/3
1 2 1
and γ = 4 resulting in (γ −γ )/b = −14/27 ≃ −0.52. On the contrary, we have γ /b =
2 1 2 1
−2/27 ≃ −0.07 and γ /b = 4/9 ≃ 0.44. Thus, m2 shows relatively strong dependence on
2 0
the scale, whereas the quark and mixed condensates are less influenced by the scaling. Thus,
m2 depends more strongly on the scale.
0
Finally, we would like to mention that the relation between the mixed condensate and the
expectation value of the transverse momentum for the leading-twist light-cone distribution
amplitude. This relation can be written as follows:
5 hu¯σ Gµνui 5m2
hk2i = µν = 0. (26)
T π 36 hu¯ui 36
Considering m2 for the light quarks in Table IV (m2 = 0.16GeV2), we get hk2i = 0.2GeV2.
0 0 T π
Note that the value estimated here is in qualitatively good agreement with that calculated
With f(m ) without f(m ) Other results
f f
[hG Gµνi /hG Gµνi ]1/4 1.00 1.20 −
µν s µν u
[hs¯si/hu¯ui]1/3 0.75 0.93 0.79 ∼ 1.03 [9, 23, 32]
[hs¯σ Gµνsi/hu¯σ Gµνui]1/5 0.87 1.00 0.75 ∼ 1.05 [34, 35, 36, 37, 38]
µν µν
m2 = hu¯σ Gµνui/hu¯ui 1.60 GeV2 1.60 GeV2 0.8±0.2 [8], 1.4 [11], 2.5 [12] [GeV2]
0,u µν
m2 = hs¯σ Gµνsi/hs¯si 1.84 GeV2 1.92 GeV2 −
0,s µν
TABLE IV: The ratios of the condensates for the different types of the m correction factors.
f
m = 5 MeV and m = 150 MeV are used.
u s
10