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Finite density QCD sum rules for nucleons
Progress in Particle and Nuclear Physics PERGAMON
Progress in Particle and Nuclear Physics 50 (2003) 659-675 hup:Nwww.elsevier.com/locate/npe
Finite Density QCD Sum Rules for Nucleons E.G. DRUKAREV Petenhurg
Nuclear
Physics Institute,
Carchino.
St Petersburg
18R300, Russia
[email protected]
It is shown how the QCD sum rules can he applied for the investigation sity dcpcndcncc through
of the nucleon parameters.
the expectation
approximations
values of QCD operators
the expectation
also a new knowledge.
difficulties of the approach
in nuclear
of the dew
can be
exprc~~sed
matter.
In certain
values arc related Lo the observable%
First appli-
cations of the approach reproduced providing
These characteristics
some of the basic features of nuclear physics,
The propam
of the future work is presented.
The
are discussed.
Contents 1. Motivation 2. QCD sum rules in vacuum 2.1. Dispersion relations 2.2. Sum rules 3. QCD
sum
rules in nuclear
matter
3.1. The problems 3.2. Lowest order OPE terms 3.3. Gas approximation.
The role of aN sigma-term
3.4. Beyond the gas approximation.
in nuclear physics
A possible saturation
mechanism
3.5. Higher order OPE terms 3.6. New knowledge 3.7. Self-consistent 3.8. A sub-plot:
scenario
Goldstone pions never condense
4. Summary The lect,ure is not, addressed of t,hc rcscarchcrs obtained
who just started
in collaboration
to t,hc experts. to study
The aim of the talk is rather
the subject.
The third section
with M. G. R.yskin, V. A. Sadovnikova
0146.6410/03/$ see front matter 0 2003 Published PII: SO146-6410(03)00060-7
by Elsevier
Science
to attract
is based on the results
and E. M. Lcvin. BV.
attention
‘._
660
1
E.G. Drukarev / Prog. Part. Nucl. Phys. SO (2003) 6S9475
Motivation
i’he theory of nuclear matter
leaves some room for the improvement.
i.e. those close to the saturation Since the pioneering was succeeded
mesons.
of Wale&a
by still more successful
meson-exchange
picture
of nucleon
interactions
complete.
hadrodynamics
in both nuclear matter
of O- and w-mesons
place at the distances
where the nucleons cannot
to develop
however, the “meson-exchange”
Another
description
are much discussed containing
MeV. In the
from the exchange
by CJ and LJ in describing
by Negele [4] and by Sliv et
the scalar
the scalar interactions.
and vector
as the point, particles
interactions
t,ake
any more. Finally,
of QHD. They are chosen usually to fit the
the approach
and the disintegration The possibility
the admixture
of heavier
in a straightforward
point.
the observablcs
study the high density
whose density
invented
through
the vacuum
(&CD operators
the concept,ion
are several matter
dependence
interesting
phenomena.
to the quark-gluon
of the other phase states
or of the “pion condensate”
for the astrophysics.
to express
plasma
of nuclear
have been studied
However the QHD parameters
cannot
be expanded
the interactions
gS+
to t,he higher densitics
in the scalar and vector channels
can be found separately.
can be realized.
et al. [8] succeeded
expectation
This would enable
to
the in-medium
the in-medium
If we succeed
modification
Recall t,hat the QCD sum rules (SR)
in expressing
values of several simplest
of the lowest dimensions).
through
of QIID,
nuclear physics.
by Shifman
case of finite densities,
of the existence
baryons
There are chances that both requirements method
There
of the nuclear
Hence, this approach
way. It is desirable
features
This would enable us to avoid the controversy
wit,h the high densities.
nowadays.
are defined at the saturat,ion
which has the attractive
above.
long ago [S, 71. These effects can be important
expressed
is treated
and finite nuclei [2, 3). However the model
be treated
conception.
mentioned
point is connected
The chiral phase tra,nsition
through
matter
properties.
of the small distance
matter
phenomenology
(QHD). It is quite successful
are so large that
go and gU are the free parameters
Thus it would be desirable avoiding,
in nuclear
IS rather an effective way of describing
Also the masses m,,,
saturation
A nucleon
The weak points of QHD were reviewed
that “0-meson”
constants
SchrGdingcr
these fields originate
al [5]. Here I mention
the coupling
successful
of the scalar and vector fields which are of several hundreds
of nuclcons
is not fundamentally
[l] the partially
Dirac phenomenology.
This model is known as quantum
most of the properties
the low densities,
value as well as the higher densities.
paper
as moving in superposition
This concerns
operators
properties
the Sit approach
of the values of the hadron
The calculation
of the hadrons
of the quark and gluon fields
in expanding
values of the QCD condensates.
of heavy meson (O and w) exchange.
the static
parameters
Such approach
for the would be
would not require
of the density dependence
of QCD
661
E.G. Drukarev / Prog. Part. Nucl. Phys. SO (2003) 659-675
condensates
would enable to use the approach
in the broad interval
Now I give a brief review of the SR method need at the finite values of density.
2
QCD
2.1
There are several detailed
of the system
is the dispersion
with the quantum G(q2)
freedom
mechanics
are convenient
relation
numbers =
; /
G(q2) is just, the particle
singu1aritir.s
of a hadron.
becorncs
increasingly
propagator.
In the field theory
constant
constructed presentation
cr,). The coefficients
of the expansion
known as operator
Technically
power expansion
with 17being the local operator ~(2) is the composition
= i/d4xc.
There
to the systems
are the other
“protonfpions”,
interacting freedom
quarks.
Such
of &CD. This
as the power series of q--2 (and of QCD values of local operators
values arc called “condensates”. [Ill, 1,rovides
the perturbative
physics is contained
Thus such expansion
of
in the condensates.
~i’““‘(Op-{7,(2)75(0)})0)
with the quantum
of the quark fields $(z).
numbers
(2)
of the considered
system.
The operator
For each quark field one can write in the lowest order
for a while about colours)
(0I~(l&)qd0)10)
=
& QJ:““’- aXI-,x, (01: (I(o)rXq(z) A
wit,h f = x,,~+‘. This is the direct consequence mq stands
for the quark mass.
set of the basic Dirac 4 x 4 matrices structures.
with
this means that one should start with the general presentation G(q2)
of os (forgetting
degrees of
for the system
mass.
are the expectation
(OPE)
effects, while the nonperturbative
different
part Im G(k2) = 0 at li2 < m2
due to the asymptotic
of quark and gluon fields. These expectation
the short-distance
indices,
-cm
G(q2) can be presented
the function
form it is (1)
the system as that of three strongly
simple at, q2 +
the
Im G(k2)dk2 k2 _ q’
at larger values of k2. There arc the cuts corresponding
means that, at q2 + -co coupling
[9, lo].
which describes
In the simplest
of the lowest lying pole, i.e. m is the proton
etc. On the other hand, one can consider description
G(q’)
for the function
regions of the values of q2. In particular,
in the different
being the position
71~
which we shall
reviews of the SR approach
the baryon and electric charges equal to unity Q = B = 1, the imaginary with
on the points
relations
The basic point, of SR method
In quantum
focusing
values.
sum rules in vacuum
Dispersion
propagation
in vacuum,
of the density
of Wick theorem.
In the second with the scalar,
: IO)
(3)
In Eq. (3) cr and fi are the Lorentz
term of the rhs of Eq. (3) lYx is the complete vector,
pseudoscalar,
The first tcrrn in the rhs of Eq. (3) is just the froc propagator.
pseudovector
and tensor
662
E.G. Drukarev / Prog. Part. Nucl. Phys. 50 (2003) 659-675 III the theories
of Eq.d (3) vadshes
with the empty vacuum, e.g. in quantum due to the normal ordering.
one vanish due to the Lorentz breaking
invariance.
of the chiral invariance.
quark can propagate with the vacuum presented
*no +(;y2x2P
&
the space-time
in a gauge-invariant
due to the spontaneous
- ;
(Ol~(0)qa(Lc)JO)
points “0” and “z” as a free particle pairs.
Note, however,
q(z) depends
that
Eqs.
The
or by the exchange (3) and (4) are not
on the gauge of the gluon fields). The
form of Eq. (4) is
faia +2p)‘2x2
$
(OITq;(s)$(O)lO) =
cY(& - ;
in powers of x2 corresponding
To demonst,rate
except the scalar
Note that Eq.(4) h as a simple physical meaning.
way (the operator
rigorous and gauge-invariant
the well-known
In QCD the scalar term survives
sea of the quark-antiquark
with the expansion
In any field theory all the structures
for the colour indicts.
between
the second t,erm in rhs
Thus
(012-q: (2)c$(O)IO)= with “0” and “6” standing
electrodynamics
to the expansion
the power of the dispersion
Gell-Mann,
Oakes-Reuncr
(0~4”(O)q(O)(O)+
relations
relation
O(2)
of G(q*) in powers of q-“.
I present,
following
(lo]: the dcrivat,ion of
(GMOR.) 1121
z
(O(2L.u + ddl0) = -
(‘3)
u
with fir and nz, being the decay constant, and the mass of the pion. Recall that Eq. (6) is true in the chiral limit, ml + 0, m,, + 0. The quantum
numbers
as by the pscudoscalar
of pion can bc carried by the axial current
current
P(s)
= 1:ti(~)-y&(~). Consider
G(q*)
= i
A,,(s)
the dispersion
= UNIT& relation
for the function
q@J d4sei(qz)(OITA~(2)P(0)10)
(7)
4* using Eq. (5) for t,he quark propagators. a
wtof
x2
2
L*
one
3 + md) -In &i*
= i(m,
the rhs of Eq. (1) recall about
by t,he equation
PA,(a)
= &f,map(z)
Im G(k’) with the term R(k’) contain
The corresponding
integral
diverges at small 2. Introducing
finds in the limit oy = 0 G(q*)
To obtain
as well
describing
=
the partial
L* (Oltiu + &IO) + i -q* 4* conservation
with ‘p standing
one more factor mz compared
lying states.
by axial current
b(k’ - ma) + R(k*) Using PCAC
expressed
(9)
one finds the term R(li*) to
to the first term in rhs of Eq. (9). H crm, in the chiral limit we
can neglect R(k*) as well as the first in rhs of Eq. (8). Thus the dispersion
i (01~
(PCAC)
for the pion field. Present
(OlA,k”l~(k))(~(k)lPlO)
the higher
(8)
+ ddl0) 4*
=
(OlA,k+r)(nIPlO) m2kq2 x
'
relation
is
(10)
663
E.G. Drukarev / Prog. Part. Nucl. Phys. 50 (2003) 659475
with I? = m$. Calculat,ing
the matrix
elements
in the rhs by using PCAC and assuming
-q*
> rnz
we come to Eq. (6).
This derivation textbooks
of GMOR relation
to the standard
(see, e.g. [13]) which is based on physics of small momenta
the way to determine
the cxpcctation
one presented
q. Usually
the value of (Oltiiu + (-7rlO). However in the framework
it can be viewed as the relation
2.2
is complementary
which expresses
the combination
in the QCD
GMOR is treated
of the developed
as
approach
rnz f,’through
of the pion parameters
value (Oltiu + &IO).
Sum rules
Unfortunately, a simple form. the low&
the example Usually
considered
above is the only case when the dispersion
there is no reason to neglect the higher lying physical
one. If the second lowest singularity Im G(lc’)
f (k*) is
with X2 being the residue while Eq. (1) can be expanded
the spectral
f (k*)~(k’-
function.
states
takes such
with respect
to
at the point W,$ WCcan present
is the cut starting
= X26(!? - m’) +
relation
W$,)
(11)
Following previous discussion
the Ihs of
in powers of q-* and thus Eq. (1) takes the form Go,&*)
=
---$--+
+ i
J
f (kZ)dk2 ~ k* - q*
(12)
W,‘h
with the unknown
parameters
to obtain the parameters approximately.
m, X’, and the unknown
of the lowest lying state.
The approximation
is prompted f(k*)
=
spectral
of k’, replacing
The standard
by the asymptotic ;
behaviour
AGow(k2)
(13)
ansatz consists in extrapolation
also the physical threshold
-la7 7l /
f (k2) -dk2 k* - q2
W$, by the unknown = &
relation
-q”)
exceeding
the contribution
of Eq. (12) to the lower values
effective threshold
W2, i.e.
O”A Gm(k*) dkz
/
k2
_
q2
(14)
’
W2
W&s
and thus the dispersion
The aim of the SR is
Hence, the second term of rhs of Eq. (12) is treated
at k* > 1q2(. The integral over k2 >> lq’l provides the terms N ln(L’/ of the pole by this factor.
f (k*).
function
(1) takes the form
Gm(q2) =
mz
x*
+
& /O”A Gx#) k2 - q2
dk2
(15)
WZ
Such approximation
of the spectrum
The lhs of Eq. (15) contains m, X2 and W2. However
is known as the “pole + continuum”
the QCD condensates.
both Ihs and rhs depend
The rhs contains
on q’. The OPE becomes
model. three unknown
parameters
increasing$
true ;4t la&
664
~
values of -q*. treated
E.G. Dmkarev / frog. Part. Nucl. Phys. 50 (2003) 659475
The “pole + continuum”
approx_imately
decomcs
increasingly
model has sense only if the contribution
does not exceed the contribution
true at small values of [q*[. The problem
OPE and “pole + continuum”
descriptions
subtractions.
Bore1 transform
terms
To improve
the overlap
happens,
useful features
(15) which are caused
since the Bore1 transform
subtractions.
between
relations
presented
by Eq. (15)
the QCD and phenomcnological
defined as
are several
in the Ihs of Eq.
Thus the model
is to find the region of /q*l where both
of the dispersion
Q2 =
was used in (81. There
exactly.
model are valid.
Such region is unlikely to exist in any channel with the necessary
of the polcj treated
of the continuum,
Also it emphasizes
eliminates
-q2
;
hf2
=
of the Bore1 transform.
Q’ln It removes
by the free quark loops -
all the polynomials
the contribution
the divergent
see, e.g., Eq.
(8).
This
in q*. Thus WCdo not need to make
of the lowest lying states
in rhs of Eq. (15) due to
the relation 1 = e-m=/.W D--Q2 + m2 The Bore1 transformed
dispersion
GopE(M2)
=
(17)
relations AGo&k’)
dk2e-k2/M2
A2e-m2’M2
(18)
+kJ W’
have been analyzed
first for the vector mesons [8] and for the nuclcons
of the values of the Bore1 mass M were found, achieved.
In particular,
condensates.
Ioffc [14] calculated
that Eq. (19) has a simple physical
and nucleons
meaning.
go = (OI$G,,G,,IO) of p and ‘p mesons
resembles
new knowledge
was extracted
of rhs and lhs of
by Vainshtein
and from QCD analysis
(19)
and of the four-quark propagator,
condensate
(O[qqqqlO).
similar to Eq. (3). One can see
The nucleon mass is caused by the quark exchange
was applied successfully
[14, 151. It provided
of QCD
C(O~tiuIO)
=
of the two-quark
the vacuum sea of qq pairs. The mechanism The QCD SR method
mass as the function
to be needed to provide the matching
of the gluon condensate
The latter can be viewed as the expansion
(18) was
as
7rL
with C < 0 being the function
of rhs and lhs of Eq.
the value of the meson
Several terms of OPE appeared
Eq. (18). The result of [14] can be treated
where the matching
[14]. In bot,h casts the regions
that of the Nambu and .Jona-Lasinio
to calculation
of the static properties
as well. For example, et al.
of charmonium
model.
of mesons [8]
the value of gluon condensate
[16] from the analysis spectrum.
with
of leptonic
This condensate
decays
is a very
665
E.G. Drukarev / Prog. Part. Nucl. Phys. 50 (2003) 659475
important
characteristics
The investigations preprint
of QCD vacuum,
based on the vacuum
since it is directly
related
to the vacuum
energy density.
QCD sum rules are going on until now.
[17] which the value of ~0 has been cdlculated
more accurately
The latest
was published
HEP
several months
ago.
3
QCD
sum rules in nuclear matter
The problems
3.1
Now we discuss characteristics
the possibility
of the nucleons in nuclear matter.
of the nucleon will be expressed some of the qualitative
through
the expectation
will be considered
invariance
on two variables,
connected
in the investigations
of the
of the characteristics in medium.
Although
of the finite nuclei, only
below. In other words, the density
/d4seicqz) (MIT
but not on q’ only.
than that of the vacuum correlator with the nucleon
the problems
the modification
values of QCD operators
is lost. the correlation
G”‘(q) = i
complicated,
If we succeed,
to the investigation
of the distribution
of
is the same in all t,hc space points.
Since the Lorentz
depends
of QCD SR approach
results may find the applications
the infinite nuclear matter the nucleons
of the extension
(proton)
is to find the proper
function
{~(x)fj(O)} IM)
The spectrum
of the function
G”(q) is much more
G(q*) defined by Eq. (2). The singularities
placed into the matter, variables,
in medium
as well as with the matter
can be
itself. One of
which would enable us to focus on the properties
of our
probe proton. In the papers
[X3, 191 it was suggested
to USCq2 as one of the variables.
of the nucleon pole m,,, - m would bc the urlknown parameter
to be determined
The shift of the position from the SR equations.
On the other hand 711,, -
with U being the single-particle the equation variable
of state.
is needed.
u(1+o(X))
energy of the nucleon.
To separate Considering
711 =
this singularity the nuclear
matter
This is the very characteristics
which enters
from the other ones a proper.choice
of the second
as the system
of A nucleons
with momenta
p,,
introduce
with
p = 0 in the rest frame of the matter.
singularities
connected
with the excitation
be fixed by the condition
that
Under the choice of s = (p + q)* =const of two nuclcons
the probe proton
[18]--[20]. The constant
is put on the Ferrni surface
WC avoid the
value of s should
of thematter.,
In the
E.G. Drukarev / Prog. Part. Nucl. Phys. 50 (2003) 659475
666
simplified s = 4m2.
case when the Fermi motion of the nuclcons of the matter
is neglected,
WCcan just assume
__
Thus we shall write the dispersion
relations
for the function
Gm(q2, s) - G(q2) with the functions
G and G’” defined by Eqs. (2) and (20). It was shown in [21] that the nucleon pole is still the lowest lying singularity
of the function
the other singularities
G”‘(q2, s) until we do not, include the three-nucleon
are lying at larger values of q2 being quenched
Eq. (17). Thus WC use the “pole + continuum”
to use the SR equations “a priori”
3.2
one should find the density
if the OPE series converges
expectation
dependence
All due to
Gm(q2, s).
of t,he function
values of QCD operators.
of these condensates.
Thus
It is not clear
indeed.
Lowest order OPE terms
In the lowest order of OPE particular,
the expectation
there are t.he scalar expectation
values of the lowest dimension
with qi standing
for “2~” or “8’ quark.
the form v:(p)
values in vacuum, the conservation
= v’(p)&
=
=
standing
The correlator standing unknowns
for the number
for the unit 4 x 4 matrix.
more unknown
position
i.e. ~~(0) = 0. Due to
n@’+ ,@)
2 * p
*
three structures
from SR equations.
(25)
being proportional
(neutron).
Tpqp: and I with I
to y,$‘,
Thrse are the three parameters
through
is the shift of the position
in the same interval
of the nucleon which
C, and X,7 and also the shift of the value of the residue A; - A’.
The explicit form of the SR equations can be achieved
=
of the pole can bc expressed
parameter
vanish,
K have nonzero
Thus WCobtain three QCD sum rules. There are four independent
arc the vector and scalar self-energies ‘rho shift of the position
The condensates
of the valence quarks of the flavour “i” in the proton
G”(q2) contains
to be determined
(24)
we find immediately v’(p)
with @‘)
values
WlW,%W)
values of the vector condensates
of the vector current
In
(23)
in the rest frame of the matter.
while the vacuum
only.
Wlq67,lW
There are also the vector expectation 7$(P)
taking
are involved
values
n’(p)
OIW
by the Bore1 transform
model for the spectrum
of Gm(qZ, s) are the in-medium
The OPE coefficients
interactions.
is prcscnted,
the self-energies,
of the threshold
i.e. m,,, - rn = C, +X,.
W,f, - W2.
e.g. in [21]. The matching
of the values of M2 as in the case of vacuum.
of the nucleon pole was found to be a superposition m, m,--m
= CL+)
of the rhs and lhs The shift. of the
of the vector and scalar condensates
(201
+ &V(P)
= U(p)
(26)
667
E.G. Drukarev / Prog. Part. Nucl. Phys. SO (2003) 659475
with the last equality coming form Eq. (21), K = P + ICY.This provides the simple picture of formation of t,hc value of m,,,. Our probe proton that in vacuum.
exchanges
quarks
with the sea of Qq pairs which differs from
This forms the Dirac effective mass m’ = m + C,. This mechanism
the first term in rhs of Eq. (26). The exchange
3.3
We shall see that, numerically
Gas approximation.
is exactly
start, with the gas approximation Thus our probe proton
interacts
role of TN sigma-term
the condensates
linear in p, the condcnsatcs
$(p) = (MI$G,,IM)
should be taken
it is not, very important.
The
We saw t,hc lowest OPE to contain
by
with the valence quarks adds the second term.
In the next to leading order of OPE the gluon condensate into accourlt.
is described
in nuclear
v(p), x(p) and g(p). While the vector condensate
K(P) and g(p) arc more complicated
in which the matter wit01 the system
physics
is treated
functions
of density.
WC
as ideal Fermi gas of the nucleons.
of non-interacting
nucleons.
In this approximation
P81
The matrix elements
(27)
g(p)
(28)
=
point.
relation
In particular,
elastic scattering
T denotes
the amplitude
provide the data on the physical amplit,ude Tp,, =
of extrapolation
of observable
amplitude
to the unphysical
-C/f, point,
by Gasscr et al. [23]. They found (T =
Note that from the point, of chiral expansion,
MeV
(45f7)
the difference C-a
Not,e also the physical meaning of the expectation state.
to thr pion-nucleon
[22]. H owever in the latter
The experiments
with C = (60 f 7) McV. The method was developed
G,,,G,#‘)
sigma term which is connected
T by the relation T = -o/f:
in certain unphysical
J/(O) + P(N:
in rhs of Eqs. (27) and (28) can be related to the obscrvablc,s.
with (T being the pion-nucleon amplitude
K(P) = 4) + PGwrll~) >
Ansehnino
is of higher order, i.e. (X-a)/~
value of t,he operator
and Forte [24, 251 showed that under reasonable
as the total number
(30) N m,.
qq averaged over a hadron
model assumptions
it, can be treated
of quarks and antiquarks.
As to gluon condensate,
the expectation
value is [26]
(31) with j standing
for U, d and s quarks.
over the nucleon state
This equation
with the account
comes from the averaging
of the additional
relations
of the QCD Hamiltonian
found in [26]. In ‘tile chir+l limit
E.G. Dmkarev / Prog. Part. Nucl. Phys. 50 (2003) 659475
668 only
the strange quarks contribute.
In the chiral SU(3) limit the second term in brackets
in rhs of Eq.
(31) tHrns toJ_Fro. Solving the SR equations potential
in the leading
order of OPE
in the gas approximation
WC find the
energy to be [20]
U(p) = bigg[66v(p) - X$(p) with the diflercnce the potential
n(p) - n(0) being described
energy is presented
condensate
and a negative
po = 0.17 fme3 = 1.3. 10-“GeV3 MeV. The gluon condensate
K(O))] GeV-'
by the second
as the superposition
term proportional
-
term of the rhs of Eq.
of a positive
term proportional
to t,he scalar condensate.
(27).
Thus
to the vector
At, the saturation
point p =
we find the two terms in the rhs of Eq. (32) to be 200 McV and -330
adds about
OPE Similar results were obtained
10 McV to the vector term in the next to leading order of
in [27] in another
SR approach
based on the dispersion
relations
in
Qo. These are the common however,
that
The interactions
channel
are expressed
parameters
through
the observable
and the Walecka. model
[20]. Note,
like gw and gu of QHD. The exchanges
in the vector channel arc calculated
through
the exchanges
explicitly.
RN sigma-term.
by the uncorrelated
The interactions
Hence,
by the
in the scalar
in SR approach
the a-term
the linear part of the scalar interactions.
Beyond
the gas approximation.
?Jow we shall try to go beyond OPE.
the SR approach
quarks (i.e. by the mesons) are expressed
quarks.
3.4
between
we did not need the fitting
strongly correlated
determines
points
Account
of the interactions
the vector condensate. by averaging
A possible
the gas approximation, between
remaining,
the nucleons
However the scalar condensate
saturation however,
of the matter obtains
mechanism
in the lowest orders of
does not change Eq.
additional
contributions
(25) for
S(p) caused
of the operat,or @,Jover the meson cloud. Thus we have
44 = 44 + PWIBW) + S(P) with a nonlinear Assume
behavior
of S(p) at small values of p.
that, the meson cloud consists
of all kinds of the mesons
[18] that in the chiral limit ml -+ 0 (neglecting function
Even beyond the nonlinear
states contribute
the chiral limit we expect
term S(p).
This is because
etc.).
It was shown in
one can obtain the
pp - p’l”. The lowest order term - fly
Fock term (known also as the Pauli blocking
nucleons in the intermediate
(r,w,
also the finite size of the nucleons)
S(p) as a power series in Fermi momenta
the one-pion
(33)
term).
The two-pion
exchanges
as pp; [28]. The heavier mesons cont.ributc the pion cloud to provide
the contributions
comes from
the leading
with the
as pp:. - p2.
contribution
of various mesons X contains
to
the meson
669
E.G. Drukarev / Prog. Part. Nucl. Phys. 50 (2003) 659-675
expectation
= nx with nx being the total number of quarks and antiquarks.
values (X14X)
expect, nx z 2. However the pion expectation provides
(4+)
mechanism.
account of the nonlinear
to the saturation
which contains
value of density
the C-term and the pion-nucleon
coupling constant
u
(~7 = 47.8 McV) which is consistent, E = -9 MeV. The incompressibility
the Pauli blocking term only
coefficient
data.
energy
MeV ,
gr;NN as the only parameters.
After
at p = po if we put C = 62.8 McV The binding
energy appears
which defines the shape of the saturation
to be
curve also has
value K z 180 MeV.
to the exact value of the 23 term. rigorous treatment
should not be taken too seriously.
This is caused by the simplified
the saturation
value.
the saturation
propert,ics
of these effects [32, 331 still provides
Thus the nonlinear
as the sign that further Thus the nonlinear
of the matter. development, behaviour
behaviour
of the approach
of the scalar condensate
on the density Anyway,
dependence
0 = J‘$@(p, further
mechanism.
The
close to
may be responsible can be considered
for only
t6 bc fruitful.
is a possible source of the saturation.
Here
Recall that in Walccka model the saturation
of the “scalar density”
of the nucleons
ps = J &&?(p~
- p)
- p).
development
tcrrns is well as the analysis
S(p) < 0 at the densities
of the scalar condensate
may appear
effects.
effects in the propagat,ion
However the results of this subsection
we find a certain analog of the QHD saturation caused by a complicated
They are very sensitive
model of the nonlinear
of the pion cloud requires the account of the multinuclcon
of the pions [30, 311. Inclusion
of the approach
of the condcnsatcs
Higher order OPE
requires
beyond
investigation
of the higher order OPE
the gas approximation.
terms
As WC have seen, the vector and scalar condensates
only contribute
in the leading order of OPE. The
to t,he terms of the relative order 9-’ of the OPE of Gm(9’, s). We saw it
gluon condensate
contributes
to be nurncrically
small, cu well as some other contributions
[33]). However t,here are some problems condensates
[29]
signals the possible
the potential
obtains the minimum
wit.h the experimental
Of course, the results for the saturation
3.5
~0. This provides
g + 133
=
technique
[21] wi tl I p F. = 268 MeV/c being the Fermi momentum
adding the kinetic energy the energy functional
a reasonable
terms in the scalar condensate
(198 - 42. A)
U(p)
algebra
is enhanced.
the chiral limit m,’ = 0 and including
Assuming
we can present, K(P) = K(O) + *p-3.2gp corresponding
by the current
M 12. Thus the pion cloud contribution
= *
Note that the simplest sat,uration
value calculated
We can
hXY(p) = (A41&XqQrYqIM).
of this order which we did not discuss (see
with the terms of the order 9-4 which contain
the four-quark
E.G. Drukurev / Prog. Part. Nucl. Phys. SO (2003) 659-675
670
The expectation
values of the four-quark
The usual as$!lmption dominating
is the factori&ion
in all the channels.
the configuration
operators
arc not well established [8] with the intermediate
approximation
As to the in-medium
with one of the products
even in vacuum. vacuum
values of the scalar condensate,
qq acting on vacuum states
states
one can separate
while the other one act,s on the
nucleon st,ates [20]. Thus in the gas approximation
In the second term of the rhs all t,he quark operators enable us to find the magnitude all the four-quark
condensates
act inside the nucleon.
Equations
(6) and (29)
of the first; term of the rhs. If the first term estimates
the values of
the term of the order q -4 of OPE is numerically
indeed,
the leading term at the values of the Bore1 mass where t,he SR equations doubts
in the convergence
demonstrated
the situation
that there is a st.rong cancellation
lack of informat,ion the approach
3.6
of OPE. Fortunately
on the four-quark
are solved.
is not as bad as that.
larger than
This would cause et ol. [34]
Cclenza
between t,he two terms in rhs of Eq. (34). However the
condensate
have been an obstacle
during many years. The recent calculations
for the further development
[35] are expected
of
to improve the situation.
New knowledge
Independently
of the magnitude
certain
new features
3.6.1
Anomalous
correlated
earlier,
qq pairs
of uncorrclated
of the
@ is caused
between
V corresponds
nucleon-meson
our probe
pairs with the same quantum
nucleon
numbers
the SR predict
vertices
to exchange
to density
which is a more complicated
and the mat,ter
(the mesons).
function
and the exchange
(i-
by st,rongly
QIID picture this the vector
while the scalar interaction
In the mean field approxirnation
of the density
exchange
p.)$ = (rrL++)$
by the vector mesons with the matter
p_ while the scalar interaction
between
In the convent,ional
for the nucleon in the nuclear matter
by the scalar mesons exchange.
V is proportional
condcnsatcs,
the QCD sum rules can be viewed as a connection
means that in the Dirac equation interaction
of the four-quark
of the nuclear forces.
structure
As we have stated
of the contribution
is proportional
the vector interaction
to the “scalar density”
p. Thus V = V(p), while Q, = +(ps).
ps
We have
seen that QCD sum rules provide similar pict,ure in the lowest orders of OPE: vector and scalar parts G,,, of the correlator GT = GT(lc(p)). terms,
G”’ depend
However WCfind a somewhat
say, Gy = Gy(~(p),v(p)),
the four-vector contribution
on vector and scalar condensates
condensates.
depending
In particular,
(Ol~ulO)(MldyodlM).
more complicated
correspondingly: dependence
Gr = G~(TJ(~));
in the higher order OPE
on both scalar and vector condensates. the scala.r-vector
This t,erm is proportional
condcnsatc
(MIti~&~dlM)
to the vcct,or condensate,
This is due to contains
the
contributing,
671
E.G. Drukarev /Prog. Part. Nucl. Phys. 50 (2003) 6594575
however, to the SR. for the scalar structure. of the scalar interaction
In QHD picture this corresponds
4 on the density
mean field level. These contributions
p. Such dependence
correspond
is not included
to the anomalous
bet,wccn the nucleon and the scalar meson in the meson exchange (OI~~UIO)(MI~~UIM)contributing
for the condensate
Charge-symmetry
3.6.2
There is an old problem with the systems the difference masses
breaking
and on the isospin
by the SR approach exprcsscd
in a number
The SR analysis
of the charge-symmetry scalar channel. the elect.
takes place
of the proton
[36]. In framework
of the function
value y = “$$Y$$‘.
of works [37]. The proton-neutron
provided
breaking
Similar situation
interactions
and neutrons
depcndencc
cxpcctation
some qualitative
results
and neutron of SR method
G” on the current The problem
binding
rnd - rn, and the condensate
t,hrough the quark mass difference
assumpt,ions.
the strong
of protons
by the explicit
breaking
picture.
of vertex of the interaction
to the SR for the vcct,or structure.
between
equal numbers
can be caused
structure
in QHD, at least on the
forces
of the difference
containing
to the explicit dependence
energy
quark
was attacked difference
y under various additional
which may be useful in the building
nuclear forces (CSB). One of them is the importance
of CSB in the
Earlier there was a common belief that the CSR in the vector channel are responsible
Another
point is that the mixed structures
in the leading terms of OPE. Thus explicit
described
dependence
was
in Subset.
3.6.1, manifest
for
themselves
of vector and scalar forces on both p and ps
may become important.
3.7
A self-consistent
A rigorous
calculation
the consistent, presented
scenario
of the contribution
treatment
[32,33] of the pion cloud to the scalar condensate
of the pion dynamics
in nuclear
matter.
The nonlinear
K(P) requires
term S(p) can be
as (36)
with C, being the pion cont,ribution presentation
S = dV/drn,,
and taking into account propagator
t,o the nucleon self-energy.
(with V standing the pion contribution
D which satisfies
scatt,ering Finite
only. The self-energy
=
operator
Theory
by Cohen et al. [38]
C, contains
the in-medium
pion
II. The latter can be expressed
(FFST)
introduced
(37)
Do due to the particle-hole
(this provides the value II”). The short-range
Fermi System
obtained
Do + D,,IID
D differs from the free propagator
by the polarization
energy)
by using the
the Dyson equation D
The propagator
for the interaction
This can be obtained
through
correlations
excitations
the amplitude
described
of the forward TN
can be described
by rncans of the
by Migdal [30]. If only the nucleon-kale
exci,tatiois
E.G. Drukarev / Prog Part. Nucl. Phys. 50 (2003) 659475
672 are included,
the operator
from the expqimcntal
II0 turns to II = &,/(l
data.
The account
+gNno)
of long-range
with .9N being a PFST constant
determined
[39] modifies the effective value of
correlations
LlN.
The function coupling
constant
axial coupling
S(p)
thus depends
on the effective
mass of the nucleon
CJ:,,,~(~). The latter
can be prcscnted
through
constant
which can be expanded
Q,J by Goldbcrgcr-Treiman
relation
!kNN -=-
gA
2m
2f,
K(P). If we succeed in describing
the dependence
should originate
This would enable to describe
’
of the nucleon effective mass on the condensate
equations
from the sum rules. The last one is just the combination
the baryon parameters
up t,o the point of the chiral phase transition
The possibility at certain
Goldstone
value of density
appeared
an admixture to be sensitive
for the delta-isobar. in the hadron
pions never condense
of the “pion condensation”
was first discussed
p = pn the pion propagator
of the oscillations
of
phase of the baryon matter
where K(P) = 0.
energy E, = 0. This would signal the degeneracy contains
fr and
[40]
Eqs. (33), (36) and (38). One should add similar equations
A sub-plot:
parameters
of gA(P, K(P)) (the first, steps wcrc made in [20]) and
p), WCshall come to the set, of self-consistent
The first three equations
3.8
the fundamental
and on the TN
to the case of the finite density.
On the other hand, the SR provide the dependence
of f,(ri(p),
m*(p)
by Migdal [7]. The observation
presented
by Eq. (36) obtains
of the ground state of the system.
with the quantum
to the values of FFST
[32,33,41,42]
constants,
numbers
of the pions.
is that
the pole at the
The ground state The value of p*
being px 2 2p0 in most of the hadronic
models. The analysis shows that the singularity sation”
leads to the divergence
at P --f f+, while the density
of the function increases.
of the pion propagator S(p) expressed
the zero value and pT, i.e. 0 < PC,, < &.
restored
at p = &h. One cannot
value po to the region p > pd. of the existing
hadronic
as the Goldstone
models.
by Eq. (35). This provide
n(p) + +m
On the other hand, K(O) < 0. Thus K = 0 at, certain
between
expand
in the point of the “pion condcn-
However, since &h)
the hadronic
= 0, the chiral symmetry
physics of the densities
Thus the “pion condensation”
point cannot
Anyway, once the chiral symmetry
is
close to the saturation be reached
is restored,
bosons any more. Thus, there is no “pion condensation”
point ~~6
in framework
the pions do not exist
of the Goldstone
pions.
E.G. Dmkarev /Pros. Part. Nucl. Phys. 50 (2003) 659475
The subject SR directly,
is analyzed
in details in the papers
this result is the outcome
[41, 421. Although
of investigation
673
not being connected
of the scalar condensate
with the
stimulated
by studies
of the sum rules.
4
Summary
We saw that in-medium
the first steps
nucleon
paramctcrs
energy where expressed condensates
in the application are successful.
through
arc expressed
of QCD sum rules method The effective
QCD condensates
through
to investigation
mass and the single-particle
in the lowest OPE orders.
the obscrvables.
The pion-nucleon
a-term
appeared
The approach
nucleon moves in superposition
of scalar and vector fields which cancel to large extent.
and avoids the cont,roversial The approach lous structures Another
conception
of the meson-nucleon
Even the sirnplified
vertices.
model for the in-medium
It is caused by the nonlinear
the rigorous trcatrncnt, the expression
for the pion contribution
The complete
set of the equations
in medium equations
and for the in-medium is the subject
The development The calculation
The numerical parameters
usually are not included
scalar condensate
provides
breaking
in QHD.
forces.
a possible mechanism
of the pion cloud to this expectation
of
value. In
the sum rule for the effective mass of the nucleon m* and
to the condensate
should
the
by the point nucleons.
in the charge-symmetry
contribution
of the pion dynamics
of QHD picture:
about the nuclear forces. These are the anoma-
Such contributions
of the scalar channel
to determine
does not use the litting
of the heavy meson exchange
provides also some new knowledge
point is the importance
saturation.
one of the key points
with those of QHD. However the approach
potential
The quark and gluon
the value of the scalar forces.
values are consistent
reproduced
of the
include
forrn the set of self-consistent
equations.
the sum rules for the axial coupling
pion decay constant,
f;.
constant
Investigation
of the complete
of the troublesome
four-quark
9;
set of the
of the future work. of the method
of complete
requires calculation
set of these expectation
values in framework
condensates.
of the convincing
models is
in progress. I thank years.
M. G. Ryskin,
I am indebted
E.E. Saperstein Deutschc
V. A. Sadovnikova
to V.M. Braun,
and C.M. Shakin
Forschungsgemeinschaft
for Basic Research
(RFBR)
-
and E. M. Levin for fruitful cooperation
G.E. &own,
for mlmerous (DFG) -
during many
E.M. Henley, B.L. Ioffe, L. Kisslinger,
discussions.
The work was supported
grant 438/RUS113/595/0-1
grants 0015096610 and 0002-16853.
and by Russian
M. Rho, in part
by
Foundation
674
E.G. Drukarev / Prog. Part. Nucl. Phys. 50 (2003) 659475
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Progress in Particle and Nuclear Physics 50 (2003) 659-675 hup:Nwww.elsevier.com/locate/npe
Finite Density QCD Sum Rules for Nucleons E.G. DRUKAREV Petenhurg
Nuclear
Physics Institute,
Carchino.
St Petersburg
18R300, Russia
[email protected]
It is shown how the QCD sum rules can he applied for the investigation sity dcpcndcncc through
of the nucleon parameters.
the expectation
approximations
values of QCD operators
the expectation
also a new knowledge.
difficulties of the approach
in nuclear
of the dew
can be
exprc~~sed
matter.
In certain
values arc related Lo the observable%
First appli-
cations of the approach reproduced providing
These characteristics
some of the basic features of nuclear physics,
The propam
of the future work is presented.
The
are discussed.
Contents 1. Motivation 2. QCD sum rules in vacuum 2.1. Dispersion relations 2.2. Sum rules 3. QCD
sum
rules in nuclear
matter
3.1. The problems 3.2. Lowest order OPE terms 3.3. Gas approximation.
The role of aN sigma-term
3.4. Beyond the gas approximation.
in nuclear physics
A possible saturation
mechanism
3.5. Higher order OPE terms 3.6. New knowledge 3.7. Self-consistent 3.8. A sub-plot:
scenario
Goldstone pions never condense
4. Summary The lect,ure is not, addressed of t,hc rcscarchcrs obtained
who just started
in collaboration
to t,hc experts. to study
The aim of the talk is rather
the subject.
The third section
with M. G. R.yskin, V. A. Sadovnikova
0146.6410/03/$ see front matter 0 2003 Published PII: SO146-6410(03)00060-7
by Elsevier
Science
to attract
is based on the results
and E. M. Lcvin. BV.
attention
‘._
660
1
E.G. Drukarev / Prog. Part. Nucl. Phys. SO (2003) 6S9475
Motivation
i’he theory of nuclear matter
leaves some room for the improvement.
i.e. those close to the saturation Since the pioneering was succeeded
mesons.
of Wale&a
by still more successful
meson-exchange
picture
of nucleon
interactions
complete.
hadrodynamics
in both nuclear matter
of O- and w-mesons
place at the distances
where the nucleons cannot
to develop
however, the “meson-exchange”
Another
description
are much discussed containing
MeV. In the
from the exchange
by CJ and LJ in describing
by Negele [4] and by Sliv et
the scalar
the scalar interactions.
and vector
as the point, particles
interactions
t,ake
any more. Finally,
of QHD. They are chosen usually to fit the
the approach
and the disintegration The possibility
the admixture
of heavier
in a straightforward
point.
the observablcs
study the high density
whose density
invented
through
the vacuum
(&CD operators
the concept,ion
are several matter
dependence
interesting
phenomena.
to the quark-gluon
of the other phase states
or of the “pion condensate”
for the astrophysics.
to express
plasma
of nuclear
have been studied
However the QHD parameters
cannot
be expanded
the interactions
gS+
to t,he higher densitics
in the scalar and vector channels
can be found separately.
can be realized.
et al. [8] succeeded
expectation
This would enable
to
the in-medium
the in-medium
If we succeed
modification
Recall t,hat the QCD sum rules (SR)
in expressing
values of several simplest
of the lowest dimensions).
through
of QIID,
nuclear physics.
by Shifman
case of finite densities,
of the existence
baryons
There are chances that both requirements method
There
of the nuclear
Hence, this approach
way. It is desirable
features
This would enable us to avoid the controversy
wit,h the high densities.
nowadays.
are defined at the saturat,ion
which has the attractive
above.
long ago [S, 71. These effects can be important
expressed
is treated
and finite nuclei [2, 3). However the model
be treated
conception.
mentioned
point is connected
The chiral phase tra,nsition
through
matter
properties.
of the small distance
matter
phenomenology
(QHD). It is quite successful
are so large that
go and gU are the free parameters
Thus it would be desirable avoiding,
in nuclear
IS rather an effective way of describing
Also the masses m,,,
saturation
A nucleon
The weak points of QHD were reviewed
that “0-meson”
constants
SchrGdingcr
these fields originate
al [5]. Here I mention
the coupling
successful
of the scalar and vector fields which are of several hundreds
of nuclcons
is not fundamentally
[l] the partially
Dirac phenomenology.
This model is known as quantum
most of the properties
the low densities,
value as well as the higher densities.
paper
as moving in superposition
This concerns
operators
properties
the Sit approach
of the values of the hadron
The calculation
of the hadrons
of the quark and gluon fields
in expanding
values of the QCD condensates.
of heavy meson (O and w) exchange.
the static
parameters
Such approach
for the would be
would not require
of the density dependence
of QCD
661
E.G. Drukarev / Prog. Part. Nucl. Phys. SO (2003) 659-675
condensates
would enable to use the approach
in the broad interval
Now I give a brief review of the SR method need at the finite values of density.
2
QCD
2.1
There are several detailed
of the system
is the dispersion
with the quantum G(q2)
freedom
mechanics
are convenient
relation
numbers =
; /
G(q2) is just, the particle
singu1aritir.s
of a hadron.
becorncs
increasingly
propagator.
In the field theory
constant
constructed presentation
cr,). The coefficients
of the expansion
known as operator
Technically
power expansion
with 17being the local operator ~(2) is the composition
= i/d4xc.
There
to the systems
are the other
“protonfpions”,
interacting freedom
quarks.
Such
of &CD. This
as the power series of q--2 (and of QCD values of local operators
values arc called “condensates”. [Ill, 1,rovides
the perturbative
physics is contained
Thus such expansion
of
in the condensates.
~i’““‘(Op-{7,(2)75(0)})0)
with the quantum
of the quark fields $(z).
numbers
(2)
of the considered
system.
The operator
For each quark field one can write in the lowest order
for a while about colours)
(0I~(l&)qd0)10)
=
& QJ:““’- aXI-,x, (01: (I(o)rXq(z) A
wit,h f = x,,~+‘. This is the direct consequence mq stands
for the quark mass.
set of the basic Dirac 4 x 4 matrices structures.
with
this means that one should start with the general presentation G(q2)
of os (forgetting
degrees of
for the system
mass.
are the expectation
(OPE)
effects, while the nonperturbative
different
part Im G(k2) = 0 at li2 < m2
due to the asymptotic
of quark and gluon fields. These expectation
the short-distance
indices,
-cm
G(q2) can be presented
the function
form it is (1)
the system as that of three strongly
simple at, q2 +
the
Im G(k2)dk2 k2 _ q’
at larger values of k2. There arc the cuts corresponding
means that, at q2 + -co coupling
[9, lo].
which describes
In the simplest
of the lowest lying pole, i.e. m is the proton
etc. On the other hand, one can consider description
G(q’)
for the function
regions of the values of q2. In particular,
in the different
being the position
71~
which we shall
reviews of the SR approach
the baryon and electric charges equal to unity Q = B = 1, the imaginary with
on the points
relations
The basic point, of SR method
In quantum
focusing
values.
sum rules in vacuum
Dispersion
propagation
in vacuum,
of the density
of Wick theorem.
In the second with the scalar,
: IO)
(3)
In Eq. (3) cr and fi are the Lorentz
term of the rhs of Eq. (3) lYx is the complete vector,
pseudoscalar,
The first tcrrn in the rhs of Eq. (3) is just the froc propagator.
pseudovector
and tensor
662
E.G. Drukarev / Prog. Part. Nucl. Phys. 50 (2003) 659-675 III the theories
of Eq.d (3) vadshes
with the empty vacuum, e.g. in quantum due to the normal ordering.
one vanish due to the Lorentz breaking
invariance.
of the chiral invariance.
quark can propagate with the vacuum presented
*no +(;y2x2P
&
the space-time
in a gauge-invariant
due to the spontaneous
- ;
(Ol~(0)qa(Lc)JO)
points “0” and “z” as a free particle pairs.
Note, however,
q(z) depends
that
Eqs.
The
or by the exchange (3) and (4) are not
on the gauge of the gluon fields). The
form of Eq. (4) is
faia +2p)‘2x2
$
(OITq;(s)$(O)lO) =
cY(& - ;
in powers of x2 corresponding
To demonst,rate
except the scalar
Note that Eq.(4) h as a simple physical meaning.
way (the operator
rigorous and gauge-invariant
the well-known
In QCD the scalar term survives
sea of the quark-antiquark
with the expansion
In any field theory all the structures
for the colour indicts.
between
the second t,erm in rhs
Thus
(012-q: (2)c$(O)IO)= with “0” and “6” standing
electrodynamics
to the expansion
the power of the dispersion
Gell-Mann,
Oakes-Reuncr
(0~4”(O)q(O)(O)+
relations
relation
O(2)
of G(q*) in powers of q-“.
I present,
following
(lo]: the dcrivat,ion of
(GMOR.) 1121
z
(O(2L.u + ddl0) = -
(‘3)
u
with fir and nz, being the decay constant, and the mass of the pion. Recall that Eq. (6) is true in the chiral limit, ml + 0, m,, + 0. The quantum
numbers
as by the pscudoscalar
of pion can bc carried by the axial current
current
P(s)
= 1:ti(~)-y&(~). Consider
G(q*)
= i
A,,(s)
the dispersion
= UNIT& relation
for the function
q@J d4sei(qz)(OITA~(2)P(0)10)
(7)
4* using Eq. (5) for t,he quark propagators. a
wtof
x2
2
L*
one
3 + md) -In &i*
= i(m,
the rhs of Eq. (1) recall about
by t,he equation
PA,(a)
= &f,map(z)
Im G(k’) with the term R(k’) contain
The corresponding
integral
diverges at small 2. Introducing
finds in the limit oy = 0 G(q*)
To obtain
as well
describing
=
the partial
L* (Oltiu + &IO) + i -q* 4* conservation
with ‘p standing
one more factor mz compared
lying states.
by axial current
b(k’ - ma) + R(k*) Using PCAC
expressed
(9)
one finds the term R(li*) to
to the first term in rhs of Eq. (9). H crm, in the chiral limit we
can neglect R(k*) as well as the first in rhs of Eq. (8). Thus the dispersion
i (01~
(PCAC)
for the pion field. Present
(OlA,k”l~(k))(~(k)lPlO)
the higher
(8)
+ ddl0) 4*
=
(OlA,k+r)(nIPlO) m2kq2 x
'
relation
is
(10)
663
E.G. Drukarev / Prog. Part. Nucl. Phys. 50 (2003) 659475
with I? = m$. Calculat,ing
the matrix
elements
in the rhs by using PCAC and assuming
-q*
> rnz
we come to Eq. (6).
This derivation textbooks
of GMOR relation
to the standard
(see, e.g. [13]) which is based on physics of small momenta
the way to determine
the cxpcctation
one presented
q. Usually
the value of (Oltiiu + (-7rlO). However in the framework
it can be viewed as the relation
2.2
is complementary
which expresses
the combination
in the QCD
GMOR is treated
of the developed
as
approach
rnz f,’through
of the pion parameters
value (Oltiu + &IO).
Sum rules
Unfortunately, a simple form. the low&
the example Usually
considered
above is the only case when the dispersion
there is no reason to neglect the higher lying physical
one. If the second lowest singularity Im G(lc’)
f (k*) is
with X2 being the residue while Eq. (1) can be expanded
the spectral
f (k*)~(k’-
function.
states
takes such
with respect
to
at the point W,$ WCcan present
is the cut starting
= X26(!? - m’) +
relation
W$,)
(11)
Following previous discussion
the Ihs of
in powers of q-* and thus Eq. (1) takes the form Go,&*)
=
---$--+
+ i
J
f (kZ)dk2 ~ k* - q*
(12)
W,‘h
with the unknown
parameters
to obtain the parameters approximately.
m, X’, and the unknown
of the lowest lying state.
The approximation
is prompted f(k*)
=
spectral
of k’, replacing
The standard
by the asymptotic ;
behaviour
AGow(k2)
(13)
ansatz consists in extrapolation
also the physical threshold
-la7 7l /
f (k2) -dk2 k* - q2
W$, by the unknown = &
relation
-q”)
exceeding
the contribution
of Eq. (12) to the lower values
effective threshold
W2, i.e.
O”A Gm(k*) dkz
/
k2
_
q2
(14)
’
W2
W&s
and thus the dispersion
The aim of the SR is
Hence, the second term of rhs of Eq. (12) is treated
at k* > 1q2(. The integral over k2 >> lq’l provides the terms N ln(L’/ of the pole by this factor.
f (k*).
function
(1) takes the form
Gm(q2) =
mz
x*
+
& /O”A Gx#) k2 - q2
dk2
(15)
WZ
Such approximation
of the spectrum
The lhs of Eq. (15) contains m, X2 and W2. However
is known as the “pole + continuum”
the QCD condensates.
both Ihs and rhs depend
The rhs contains
on q’. The OPE becomes
model. three unknown
parameters
increasing$
true ;4t la&
664
~
values of -q*. treated
E.G. Dmkarev / frog. Part. Nucl. Phys. 50 (2003) 659475
The “pole + continuum”
approx_imately
decomcs
increasingly
model has sense only if the contribution
does not exceed the contribution
true at small values of [q*[. The problem
OPE and “pole + continuum”
descriptions
subtractions.
Bore1 transform
terms
To improve
the overlap
happens,
useful features
(15) which are caused
since the Bore1 transform
subtractions.
between
relations
presented
by Eq. (15)
the QCD and phenomcnological
defined as
are several
in the Ihs of Eq.
Thus the model
is to find the region of /q*l where both
of the dispersion
Q2 =
was used in (81. There
exactly.
model are valid.
Such region is unlikely to exist in any channel with the necessary
of the polcj treated
of the continuum,
Also it emphasizes
eliminates
-q2
;
hf2
=
of the Bore1 transform.
Q’ln It removes
by the free quark loops -
all the polynomials
the contribution
the divergent
see, e.g., Eq.
(8).
This
in q*. Thus WCdo not need to make
of the lowest lying states
in rhs of Eq. (15) due to
the relation 1 = e-m=/.W D--Q2 + m2 The Bore1 transformed
dispersion
GopE(M2)
=
(17)
relations AGo&k’)
dk2e-k2/M2
A2e-m2’M2
(18)
+kJ W’
have been analyzed
first for the vector mesons [8] and for the nuclcons
of the values of the Bore1 mass M were found, achieved.
In particular,
condensates.
Ioffc [14] calculated
that Eq. (19) has a simple physical
and nucleons
meaning.
go = (OI$G,,G,,IO) of p and ‘p mesons
resembles
new knowledge
was extracted
of rhs and lhs of
by Vainshtein
and from QCD analysis
(19)
and of the four-quark propagator,
condensate
(O[qqqqlO).
similar to Eq. (3). One can see
The nucleon mass is caused by the quark exchange
was applied successfully
[14, 151. It provided
of QCD
C(O~tiuIO)
=
of the two-quark
the vacuum sea of qq pairs. The mechanism The QCD SR method
mass as the function
to be needed to provide the matching
of the gluon condensate
The latter can be viewed as the expansion
(18) was
as
7rL
with C < 0 being the function
of rhs and lhs of Eq.
the value of the meson
Several terms of OPE appeared
Eq. (18). The result of [14] can be treated
where the matching
[14]. In bot,h casts the regions
that of the Nambu and .Jona-Lasinio
to calculation
of the static properties
as well. For example, et al.
of charmonium
model.
of mesons [8]
the value of gluon condensate
[16] from the analysis spectrum.
with
of leptonic
This condensate
decays
is a very
665
E.G. Drukarev / Prog. Part. Nucl. Phys. 50 (2003) 659475
important
characteristics
The investigations preprint
of QCD vacuum,
based on the vacuum
since it is directly
related
to the vacuum
energy density.
QCD sum rules are going on until now.
[17] which the value of ~0 has been cdlculated
more accurately
The latest
was published
HEP
several months
ago.
3
QCD
sum rules in nuclear matter
The problems
3.1
Now we discuss characteristics
the possibility
of the nucleons in nuclear matter.
of the nucleon will be expressed some of the qualitative
through
the expectation
will be considered
invariance
on two variables,
connected
in the investigations
of the
of the characteristics in medium.
Although
of the finite nuclei, only
below. In other words, the density
/d4seicqz) (MIT
but not on q’ only.
than that of the vacuum correlator with the nucleon
the problems
the modification
values of QCD operators
is lost. the correlation
G”‘(q) = i
complicated,
If we succeed,
to the investigation
of the distribution
of
is the same in all t,hc space points.
Since the Lorentz
depends
of QCD SR approach
results may find the applications
the infinite nuclear matter the nucleons
of the extension
(proton)
is to find the proper
function
{~(x)fj(O)} IM)
The spectrum
of the function
G”(q) is much more
G(q*) defined by Eq. (2). The singularities
placed into the matter, variables,
in medium
as well as with the matter
can be
itself. One of
which would enable us to focus on the properties
of our
probe proton. In the papers
[X3, 191 it was suggested
to USCq2 as one of the variables.
of the nucleon pole m,,, - m would bc the urlknown parameter
to be determined
The shift of the position from the SR equations.
On the other hand 711,, -
with U being the single-particle the equation variable
of state.
is needed.
u(1+o(X))
energy of the nucleon.
To separate Considering
711 =
this singularity the nuclear
matter
This is the very characteristics
which enters
from the other ones a proper.choice
of the second
as the system
of A nucleons
with momenta
p,,
introduce
with
p = 0 in the rest frame of the matter.
singularities
connected
with the excitation
be fixed by the condition
that
Under the choice of s = (p + q)* =const of two nuclcons
the probe proton
[18]--[20]. The constant
is put on the Ferrni surface
WC avoid the
value of s should
of thematter.,
In the
E.G. Drukarev / Prog. Part. Nucl. Phys. 50 (2003) 659475
666
simplified s = 4m2.
case when the Fermi motion of the nuclcons of the matter
is neglected,
WCcan just assume
__
Thus we shall write the dispersion
relations
for the function
Gm(q2, s) - G(q2) with the functions
G and G’” defined by Eqs. (2) and (20). It was shown in [21] that the nucleon pole is still the lowest lying singularity
of the function
the other singularities
G”‘(q2, s) until we do not, include the three-nucleon
are lying at larger values of q2 being quenched
Eq. (17). Thus WC use the “pole + continuum”
to use the SR equations “a priori”
3.2
one should find the density
if the OPE series converges
expectation
dependence
All due to
Gm(q2, s).
of t,he function
values of QCD operators.
of these condensates.
Thus
It is not clear
indeed.
Lowest order OPE terms
In the lowest order of OPE particular,
the expectation
there are t.he scalar expectation
values of the lowest dimension
with qi standing
for “2~” or “8’ quark.
the form v:(p)
values in vacuum, the conservation
= v’(p)&
=
=
standing
The correlator standing unknowns
for the number
for the unit 4 x 4 matrix.
more unknown
position
i.e. ~~(0) = 0. Due to
n@’+ ,@)
2 * p
*
three structures
from SR equations.
(25)
being proportional
(neutron).
Tpqp: and I with I
to y,$‘,
Thrse are the three parameters
through
is the shift of the position
in the same interval
of the nucleon which
C, and X,7 and also the shift of the value of the residue A; - A’.
The explicit form of the SR equations can be achieved
=
of the pole can bc expressed
parameter
vanish,
K have nonzero
Thus WCobtain three QCD sum rules. There are four independent
arc the vector and scalar self-energies ‘rho shift of the position
The condensates
of the valence quarks of the flavour “i” in the proton
G”(q2) contains
to be determined
(24)
we find immediately v’(p)
with @‘)
values
WlW,%W)
values of the vector condensates
of the vector current
In
(23)
in the rest frame of the matter.
while the vacuum
only.
Wlq67,lW
There are also the vector expectation 7$(P)
taking
are involved
values
n’(p)
OIW
by the Bore1 transform
model for the spectrum
of Gm(qZ, s) are the in-medium
The OPE coefficients
interactions.
is prcscnted,
the self-energies,
of the threshold
i.e. m,,, - rn = C, +X,.
W,f, - W2.
e.g. in [21]. The matching
of the values of M2 as in the case of vacuum.
of the nucleon pole was found to be a superposition m, m,--m
= CL+)
of the rhs and lhs The shift. of the
of the vector and scalar condensates
(201
+ &V(P)
= U(p)
(26)
667
E.G. Drukarev / Prog. Part. Nucl. Phys. SO (2003) 659475
with the last equality coming form Eq. (21), K = P + ICY.This provides the simple picture of formation of t,hc value of m,,,. Our probe proton that in vacuum.
exchanges
quarks
with the sea of Qq pairs which differs from
This forms the Dirac effective mass m’ = m + C,. This mechanism
the first term in rhs of Eq. (26). The exchange
3.3
We shall see that, numerically
Gas approximation.
is exactly
start, with the gas approximation Thus our probe proton
interacts
role of TN sigma-term
the condensates
linear in p, the condcnsatcs
$(p) = (MI$G,,IM)
should be taken
it is not, very important.
The
We saw t,hc lowest OPE to contain
by
with the valence quarks adds the second term.
In the next to leading order of OPE the gluon condensate into accourlt.
is described
in nuclear
v(p), x(p) and g(p). While the vector condensate
K(P) and g(p) arc more complicated
in which the matter wit01 the system
physics
is treated
functions
of density.
WC
as ideal Fermi gas of the nucleons.
of non-interacting
nucleons.
In this approximation
P81
The matrix elements
(27)
g(p)
(28)
=
point.
relation
In particular,
elastic scattering
T denotes
the amplitude
provide the data on the physical amplit,ude Tp,, =
of extrapolation
of observable
amplitude
to the unphysical
-C/f, point,
by Gasscr et al. [23]. They found (T =
Note that from the point, of chiral expansion,
MeV
(45f7)
the difference C-a
Not,e also the physical meaning of the expectation state.
to thr pion-nucleon
[22]. H owever in the latter
The experiments
with C = (60 f 7) McV. The method was developed
G,,,G,#‘)
sigma term which is connected
T by the relation T = -o/f:
in certain unphysical
J/(O) + P(N:
in rhs of Eqs. (27) and (28) can be related to the obscrvablc,s.
with (T being the pion-nucleon amplitude
K(P) = 4) + PGwrll~) >
Ansehnino
is of higher order, i.e. (X-a)/~
value of t,he operator
and Forte [24, 251 showed that under reasonable
as the total number
(30) N m,.
qq averaged over a hadron
model assumptions
it, can be treated
of quarks and antiquarks.
As to gluon condensate,
the expectation
value is [26]
(31) with j standing
for U, d and s quarks.
over the nucleon state
This equation
with the account
comes from the averaging
of the additional
relations
of the QCD Hamiltonian
found in [26]. In ‘tile chir+l limit
E.G. Dmkarev / Prog. Part. Nucl. Phys. 50 (2003) 659475
668 only
the strange quarks contribute.
In the chiral SU(3) limit the second term in brackets
in rhs of Eq.
(31) tHrns toJ_Fro. Solving the SR equations potential
in the leading
order of OPE
in the gas approximation
WC find the
energy to be [20]
U(p) = bigg[66v(p) - X$(p) with the diflercnce the potential
n(p) - n(0) being described
energy is presented
condensate
and a negative
po = 0.17 fme3 = 1.3. 10-“GeV3 MeV. The gluon condensate
K(O))] GeV-'
by the second
as the superposition
term proportional
-
term of the rhs of Eq.
of a positive
term proportional
to t,he scalar condensate.
(27).
Thus
to the vector
At, the saturation
point p =
we find the two terms in the rhs of Eq. (32) to be 200 McV and -330
adds about
OPE Similar results were obtained
10 McV to the vector term in the next to leading order of
in [27] in another
SR approach
based on the dispersion
relations
in
Qo. These are the common however,
that
The interactions
channel
are expressed
parameters
through
the observable
and the Walecka. model
[20]. Note,
like gw and gu of QHD. The exchanges
in the vector channel arc calculated
through
the exchanges
explicitly.
RN sigma-term.
by the uncorrelated
The interactions
Hence,
by the
in the scalar
in SR approach
the a-term
the linear part of the scalar interactions.
Beyond
the gas approximation.
?Jow we shall try to go beyond OPE.
the SR approach
quarks (i.e. by the mesons) are expressed
quarks.
3.4
between
we did not need the fitting
strongly correlated
determines
points
Account
of the interactions
the vector condensate. by averaging
A possible
the gas approximation, between
remaining,
the nucleons
However the scalar condensate
saturation however,
of the matter obtains
mechanism
in the lowest orders of
does not change Eq.
additional
contributions
(25) for
S(p) caused
of the operat,or @,Jover the meson cloud. Thus we have
44 = 44 + PWIBW) + S(P) with a nonlinear Assume
behavior
of S(p) at small values of p.
that, the meson cloud consists
of all kinds of the mesons
[18] that in the chiral limit ml -+ 0 (neglecting function
Even beyond the nonlinear
states contribute
the chiral limit we expect
term S(p).
This is because
etc.).
It was shown in
one can obtain the
pp - p’l”. The lowest order term - fly
Fock term (known also as the Pauli blocking
nucleons in the intermediate
(r,w,
also the finite size of the nucleons)
S(p) as a power series in Fermi momenta
the one-pion
(33)
term).
The two-pion
exchanges
as pp; [28]. The heavier mesons cont.ributc the pion cloud to provide
the contributions
comes from
the leading
with the
as pp:. - p2.
contribution
of various mesons X contains
to
the meson
669
E.G. Drukarev / Prog. Part. Nucl. Phys. 50 (2003) 659-675
expectation
= nx with nx being the total number of quarks and antiquarks.
values (X14X)
expect, nx z 2. However the pion expectation provides
(4+)
mechanism.
account of the nonlinear
to the saturation
which contains
value of density
the C-term and the pion-nucleon
coupling constant
u
(~7 = 47.8 McV) which is consistent, E = -9 MeV. The incompressibility
the Pauli blocking term only
coefficient
data.
energy
MeV ,
gr;NN as the only parameters.
After
at p = po if we put C = 62.8 McV The binding
energy appears
which defines the shape of the saturation
to be
curve also has
value K z 180 MeV.
to the exact value of the 23 term. rigorous treatment
should not be taken too seriously.
This is caused by the simplified
the saturation
value.
the saturation
propert,ics
of these effects [32, 331 still provides
Thus the nonlinear
as the sign that further Thus the nonlinear
of the matter. development, behaviour
behaviour
of the approach
of the scalar condensate
on the density Anyway,
dependence
0 = J‘$@(p, further
mechanism.
The
close to
may be responsible can be considered
for only
t6 bc fruitful.
is a possible source of the saturation.
Here
Recall that in Walccka model the saturation
of the “scalar density”
of the nucleons
ps = J &&?(p~
- p)
- p).
development
tcrrns is well as the analysis
S(p) < 0 at the densities
of the scalar condensate
may appear
effects.
effects in the propagat,ion
However the results of this subsection
we find a certain analog of the QHD saturation caused by a complicated
They are very sensitive
model of the nonlinear
of the pion cloud requires the account of the multinuclcon
of the pions [30, 311. Inclusion
of the approach
of the condcnsatcs
Higher order OPE
requires
beyond
investigation
of the higher order OPE
the gas approximation.
terms
As WC have seen, the vector and scalar condensates
only contribute
in the leading order of OPE. The
to t,he terms of the relative order 9-’ of the OPE of Gm(9’, s). We saw it
gluon condensate
contributes
to be nurncrically
small, cu well as some other contributions
[33]). However t,here are some problems condensates
[29]
signals the possible
the potential
obtains the minimum
wit.h the experimental
Of course, the results for the saturation
3.5
~0. This provides
g + 133
=
technique
[21] wi tl I p F. = 268 MeV/c being the Fermi momentum
adding the kinetic energy the energy functional
a reasonable
terms in the scalar condensate
(198 - 42. A)
U(p)
algebra
is enhanced.
the chiral limit m,’ = 0 and including
Assuming
we can present, K(P) = K(O) + *p-3.2gp corresponding
by the current
M 12. Thus the pion cloud contribution
= *
Note that the simplest sat,uration
value calculated
We can
hXY(p) = (A41&XqQrYqIM).
of this order which we did not discuss (see
with the terms of the order 9-4 which contain
the four-quark
E.G. Drukurev / Prog. Part. Nucl. Phys. SO (2003) 659-675
670
The expectation
values of the four-quark
The usual as$!lmption dominating
is the factori&ion
in all the channels.
the configuration
operators
arc not well established [8] with the intermediate
approximation
As to the in-medium
with one of the products
even in vacuum. vacuum
values of the scalar condensate,
qq acting on vacuum states
states
one can separate
while the other one act,s on the
nucleon st,ates [20]. Thus in the gas approximation
In the second term of the rhs all t,he quark operators enable us to find the magnitude all the four-quark
condensates
act inside the nucleon.
Equations
(6) and (29)
of the first; term of the rhs. If the first term estimates
the values of
the term of the order q -4 of OPE is numerically
indeed,
the leading term at the values of the Bore1 mass where t,he SR equations doubts
in the convergence
demonstrated
the situation
that there is a st.rong cancellation
lack of informat,ion the approach
3.6
of OPE. Fortunately
on the four-quark
are solved.
is not as bad as that.
larger than
This would cause et ol. [34]
Cclenza
between t,he two terms in rhs of Eq. (34). However the
condensate
have been an obstacle
during many years. The recent calculations
for the further development
[35] are expected
of
to improve the situation.
New knowledge
Independently
of the magnitude
certain
new features
3.6.1
Anomalous
correlated
earlier,
qq pairs
of uncorrclated
of the
@ is caused
between
V corresponds
nucleon-meson
our probe
pairs with the same quantum
nucleon
numbers
the SR predict
vertices
to exchange
to density
which is a more complicated
and the mat,ter
(the mesons).
function
and the exchange
(i-
by st,rongly
QIID picture this the vector
while the scalar interaction
In the mean field approxirnation
of the density
exchange
p.)$ = (rrL++)$
by the vector mesons with the matter
p_ while the scalar interaction
between
In the convent,ional
for the nucleon in the nuclear matter
by the scalar mesons exchange.
V is proportional
condcnsatcs,
the QCD sum rules can be viewed as a connection
means that in the Dirac equation interaction
of the four-quark
of the nuclear forces.
structure
As we have stated
of the contribution
is proportional
the vector interaction
to the “scalar density”
p. Thus V = V(p), while Q, = +(ps).
ps
We have
seen that QCD sum rules provide similar pict,ure in the lowest orders of OPE: vector and scalar parts G,,, of the correlator GT = GT(lc(p)). terms,
G”’ depend
However WCfind a somewhat
say, Gy = Gy(~(p),v(p)),
the four-vector contribution
on vector and scalar condensates
condensates.
depending
In particular,
(Ol~ulO)(MldyodlM).
more complicated
correspondingly: dependence
Gr = G~(TJ(~));
in the higher order OPE
on both scalar and vector condensates. the scala.r-vector
This t,erm is proportional
condcnsatc
(MIti~&~dlM)
to the vcct,or condensate,
This is due to contains
the
contributing,
671
E.G. Drukarev /Prog. Part. Nucl. Phys. 50 (2003) 6594575
however, to the SR. for the scalar structure. of the scalar interaction
In QHD picture this corresponds
4 on the density
mean field level. These contributions
p. Such dependence
correspond
is not included
to the anomalous
bet,wccn the nucleon and the scalar meson in the meson exchange (OI~~UIO)(MI~~UIM)contributing
for the condensate
Charge-symmetry
3.6.2
There is an old problem with the systems the difference masses
breaking
and on the isospin
by the SR approach exprcsscd
in a number
The SR analysis
of the charge-symmetry scalar channel. the elect.
takes place
of the proton
[36]. In framework
of the function
value y = “$$Y$$‘.
of works [37]. The proton-neutron
provided
breaking
Similar situation
interactions
and neutrons
depcndencc
cxpcctation
some qualitative
results
and neutron of SR method
G” on the current The problem
binding
rnd - rn, and the condensate
t,hrough the quark mass difference
assumpt,ions.
the strong
of protons
by the explicit
breaking
picture.
of vertex of the interaction
to the SR for the vcct,or structure.
between
equal numbers
can be caused
structure
in QHD, at least on the
forces
of the difference
containing
to the explicit dependence
energy
quark
was attacked difference
y under various additional
which may be useful in the building
nuclear forces (CSB). One of them is the importance
of CSB in the
Earlier there was a common belief that the CSR in the vector channel are responsible
Another
point is that the mixed structures
in the leading terms of OPE. Thus explicit
described
dependence
was
in Subset.
3.6.1, manifest
for
themselves
of vector and scalar forces on both p and ps
may become important.
3.7
A self-consistent
A rigorous
calculation
the consistent, presented
scenario
of the contribution
treatment
[32,33] of the pion cloud to the scalar condensate
of the pion dynamics
in nuclear
matter.
The nonlinear
K(P) requires
term S(p) can be
as (36)
with C, being the pion cont,ribution presentation
S = dV/drn,,
and taking into account propagator
t,o the nucleon self-energy.
(with V standing the pion contribution
D which satisfies
scatt,ering Finite
only. The self-energy
=
operator
Theory
by Cohen et al. [38]
C, contains
the in-medium
pion
II. The latter can be expressed
(FFST)
introduced
(37)
Do due to the particle-hole
(this provides the value II”). The short-range
Fermi System
obtained
Do + D,,IID
D differs from the free propagator
by the polarization
energy)
by using the
the Dyson equation D
The propagator
for the interaction
This can be obtained
through
correlations
excitations
the amplitude
described
of the forward TN
can be described
by rncans of the
by Migdal [30]. If only the nucleon-kale
exci,tatiois
E.G. Drukarev / Prog Part. Nucl. Phys. 50 (2003) 659475
672 are included,
the operator
from the expqimcntal
II0 turns to II = &,/(l
data.
The account
+gNno)
of long-range
with .9N being a PFST constant
determined
[39] modifies the effective value of
correlations
LlN.
The function coupling
constant
axial coupling
S(p)
thus depends
on the effective
mass of the nucleon
CJ:,,,~(~). The latter
can be prcscnted
through
constant
which can be expanded
Q,J by Goldbcrgcr-Treiman
relation
!kNN -=-
gA
2m
2f,
K(P). If we succeed in describing
the dependence
should originate
This would enable to describe
’
of the nucleon effective mass on the condensate
equations
from the sum rules. The last one is just the combination
the baryon parameters
up t,o the point of the chiral phase transition
The possibility at certain
Goldstone
value of density
appeared
an admixture to be sensitive
for the delta-isobar. in the hadron
pions never condense
of the “pion condensation”
was first discussed
p = pn the pion propagator
of the oscillations
of
phase of the baryon matter
where K(P) = 0.
energy E, = 0. This would signal the degeneracy contains
fr and
[40]
Eqs. (33), (36) and (38). One should add similar equations
A sub-plot:
parameters
of gA(P, K(P)) (the first, steps wcrc made in [20]) and
p), WCshall come to the set, of self-consistent
The first three equations
3.8
the fundamental
and on the TN
to the case of the finite density.
On the other hand, the SR provide the dependence
of f,(ri(p),
m*(p)
by Migdal [7]. The observation
presented
by Eq. (36) obtains
of the ground state of the system.
with the quantum
to the values of FFST
[32,33,41,42]
constants,
numbers
of the pions.
is that
the pole at the
The ground state The value of p*
being px 2 2p0 in most of the hadronic
models. The analysis shows that the singularity sation”
leads to the divergence
at P --f f+, while the density
of the function increases.
of the pion propagator S(p) expressed
the zero value and pT, i.e. 0 < PC,, < &.
restored
at p = &h. One cannot
value po to the region p > pd. of the existing
hadronic
as the Goldstone
models.
by Eq. (35). This provide
n(p) + +m
On the other hand, K(O) < 0. Thus K = 0 at, certain
between
expand
in the point of the “pion condcn-
However, since &h)
the hadronic
= 0, the chiral symmetry
physics of the densities
Thus the “pion condensation”
point cannot
Anyway, once the chiral symmetry
is
close to the saturation be reached
is restored,
bosons any more. Thus, there is no “pion condensation”
point ~~6
in framework
the pions do not exist
of the Goldstone
pions.
E.G. Dmkarev /Pros. Part. Nucl. Phys. 50 (2003) 659475
The subject SR directly,
is analyzed
in details in the papers
this result is the outcome
[41, 421. Although
of investigation
673
not being connected
of the scalar condensate
with the
stimulated
by studies
of the sum rules.
4
Summary
We saw that in-medium
the first steps
nucleon
paramctcrs
energy where expressed condensates
in the application are successful.
through
arc expressed
of QCD sum rules method The effective
QCD condensates
through
to investigation
mass and the single-particle
in the lowest OPE orders.
the obscrvables.
The pion-nucleon
a-term
appeared
The approach
nucleon moves in superposition
of scalar and vector fields which cancel to large extent.
and avoids the cont,roversial The approach lous structures Another
conception
of the meson-nucleon
Even the sirnplified
vertices.
model for the in-medium
It is caused by the nonlinear
the rigorous trcatrncnt, the expression
for the pion contribution
The complete
set of the equations
in medium equations
and for the in-medium is the subject
The development The calculation
The numerical parameters
usually are not included
scalar condensate
provides
breaking
in QHD.
forces.
a possible mechanism
of the pion cloud to this expectation
of
value. In
the sum rule for the effective mass of the nucleon m* and
to the condensate
should
the
by the point nucleons.
in the charge-symmetry
contribution
of the pion dynamics
of QHD picture:
about the nuclear forces. These are the anoma-
Such contributions
of the scalar channel
to determine
does not use the litting
of the heavy meson exchange
provides also some new knowledge
point is the importance
saturation.
one of the key points
with those of QHD. However the approach
potential
The quark and gluon
the value of the scalar forces.
values are consistent
reproduced
of the
include
forrn the set of self-consistent
equations.
the sum rules for the axial coupling
pion decay constant,
f;.
constant
Investigation
of the complete
of the troublesome
four-quark
9;
set of the
of the future work. of the method
of complete
requires calculation
set of these expectation
values in framework
condensates.
of the convincing
models is
in progress. I thank years.
M. G. Ryskin,
I am indebted
E.E. Saperstein Deutschc
V. A. Sadovnikova
to V.M. Braun,
and C.M. Shakin
Forschungsgemeinschaft
for Basic Research
(RFBR)
-
and E. M. Levin for fruitful cooperation
G.E. &own,
for mlmerous (DFG) -
during many
E.M. Henley, B.L. Ioffe, L. Kisslinger,
discussions.
The work was supported
grant 438/RUS113/595/0-1
grants 0015096610 and 0002-16853.
and by Russian
M. Rho, in part
by
Foundation
674
E.G. Drukarev / Prog. Part. Nucl. Phys. 50 (2003) 659475
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