Model independent constraints from vacuum and in-medium QCD Sum Rules

6 Pages·0.11 MB·English

by  F. Klingl

EPJ manuscript No.
(will be inserted by the editor)
Model independent constraints from vacuum and in-medium
QCD Sum Rules
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F. Klingl and W. Weise a
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n Physik-Department,Theoretische Physik,Technische Universit¨at Mu¨nchen, D-85747 Garching, Germany
a
J

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Received: date/ Revised version: date
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v Abstract. WediscussQCDsumruleconstraintsbasedonmomentsofvectormesonspectraldistributionsin
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5 thevacuumandinanuclearmedium.Sumrulesforthetwolowestmomentsofthesespectraldistributions
0
1 donotsufferfrom uncertaintiesrelated toQCDcondensatesofdimension higherthanfour.Weexemplify
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9 these relations for the case of the ω meson and discuss the issue of in-medium mass shifts from this
9
h/ viewpoint.
t
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l
c
u PACS. 12.40.Vv Vector-meson dominance – 24.85.+p Quarks, gluons, and QCD in nuclei and nuclear
n
:
v processes
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r QCD sum rules have repeatedly been used in recent four-quarkcondensatesintermsof q¯q 2,thesquareofthe
a h i
times to arrive at estimates for possible in-medium mass standard chiral condensate. The first objection is far less
shiftsofvectormesons[1,7].Thevalidityofsuchestimates serious for the ω meson which may well have a chance to
has been questioned, however, for several reasons. First, survive as a reasonably narrow quasi-particle in nuclear
for broadstructuressuchas theρ mesonwhose largevac- matter [3,4]. The second objection, however, is difficult
uum decay width is further magnified by in-medium re- to overcome:factorization of four-quark condensates may
actions, the QCD sum rule analysis does not provide a indeed be questionable.
reliableframeworkto extractanythinglike a“massshift”
[3,5].Secondly,notoriousuncertaintiesexistatthelevelof In the present note we focus on the two lowest mo-
factorizationassumptionscommonlyusedtoapproximate ments ( dssnR(s) with n = 0,1) of vector meson spec-
R
traldistributions,in vacuumas wellas in nuclear matter,
Send offprint requests to: [email protected]
a Work supported in part by BMBF and GSI andpointoutthatthesearesubjecttosumruleswhichdo
2 F. Klingl and W. Weise: Model independentconstraints from vacuumand in-medium QCD Sum Rules
notsufferfromtheuncertaintiesintroducedbyfour-quark tor product (Wilson) expansion gives
condensates.Thesesumrulesareshowntoprovideuseful, Q2 c c
12π2Π(q2 = Q2)= c Q2ln +c + 2 + 3 +...
− − 0 (cid:18)µ2(cid:19) 1 Q2 Q4
modelindependentconstraintswhichweexemplifyforthe
(3)
case of the ω meson spectral distribution and its change
Wespecifythecoefficientsc fortheisoscalarcurrentjµ =
i
in the nuclear medium. The sum rule for the second mo-
1(u¯γµu+d¯γµd), the case of the ω meson that we wish to
6
ment, dss2R(s),doesinvolvethefour-quarkcondensate.
use here for explicit evaluations. In vacuum we have:
R
In fact it can be used in principle to determine this par-
1 α 1
c = 1+ S , c = (m2 +m2), (4)
ticular condensate and test the factorization assumption. 0 6(cid:16) π (cid:17) 1 −2 u d
π2 α 2π2
c = S µν + m uu¯+m dd¯, (5)
The detailed analysis of this question will be defered to 2 µν u d
18h π G G i 3 h i
a longer paper. In this short note we confine ourselves to
while c involves combinations of four-quark condensates
3
conclusions that can be drawn without reference to four-
of (mass) dimension 6. The quark mass term c is small
1
quark condensates.
and can be dropped in the actual calculations. For the
The starting point is the current-current correlation gluon condensate we use αS 2 = (0.36GeV)4 [9], and
h π G i
function the(chiral)quarkcondensateisgivenby m u¯u+m d¯d
u d
h i≃
m u¯u+d¯d = m2f2 (0.11GeV)4 throughthe Gell-
Π (q)=i d4xeiq·x j (x)j (0) (1) qh i − π π ≃−
µν µ ν
Z hT i
Mann, Oakes, Renner relation.
where denotes the time-ordered product and the ex- In the nuclear medium with baryondensity ρ we have
T
pectation value is taken either in the vacuum or in the c (ρ) = c (ρ = 0)+δc (ρ) with c (0) given by eqs.(4,5),
i i i i
groundstate ofnuclear matter at rest.In vacuumthe po- and
larizationtensor(1)canbe reducedto a singlescalarcor- π2 4
δc (ρ)= M(0)+2σ +A M ρ (6)
2 3 (cid:20)−27 N N 1 N(cid:21)
relation function, Π(q2) = 1gµνΠ (q). In nuclear mat-
3 µν
to linear order in ρ. The first term in brackets is the
tertherearetwo(longitudinalandtransverse)correlation
leading density dependent correction to the gluon con-
functions which coincide for a meson at rest with respect
densate and involves the nucleon mass in the chiral limit,
to the medium (i.e. with qµ =(ω,q=0)).
M(0) 0.75GeV[6].The secondpartproportionalto the
Thereducedcorrelationfunctioniswrittenasa(twice N ≃
nucleon sigma term σ 45MeV is the first order cor-
N
subtracted) dispersion relation, ≃
rectionofthe quarkcondensate,andthethirdtermintro-
q4 ImΠ(s)
Π(q2)=Π(0)+Π′(0)q2+ ds . (2) duces the first moment of the quark distribution function
π Z s2(s q2 iǫ)
− −
in the nucleon:
whereΠ(0)vanishesinvacuumbutcontributesinnuclear
matter. At largespacelike Q2 = q2 >0 the QCD opera- A1 =2 dxx u(x)+u¯(x)+d(x)+d¯(x) . (7)
Z
− (cid:2) (cid:3)
F. Klingl and W. Weise: Model independentconstraints from vacuumand in-medium QCD Sum Rules 3
It represents twice the fraction of momentum carried by s0dss2R(s)= s30c +c . (12)
0 3
Z 3
0
quarks in the proton. We take A 1 as determined by
1
≃
Note thatthe firsttwo sumrules arewelldetermined and
deep-inelastic lepton scattering at Q 2GeV. Note that
∼
δc (ρ ) 4 10−3GeV4atρ=ρ =0.17fm−3,thedensity represent useful constraints for the spectrum R(s). Only
2 0 0
≃ ·
the third sum rule (12) involves four-quark condensates
of nuclear matter, and almost all of this correctioncomes
which are uncertain. In this short paper we concentrate
from the term proportional to A .
1
on eqs. (10,11). A detailed analysis of eq.(12) will be pre-
Next we introduce the Borel transform of eq. (3):
sented in a forthcoming longer paper. It is instructive to
∞
12π2Π(0)+ dsR(s)e−s/M2 =c 2+c + c2 + c3 +...
Z 0M 1 2 2 4 illustratethesumrules(10,11)fortheω mesoninvacuum
0 M M
(8)
using Vector Meson Dominance (VMD) for the resonant
with R(s)= 12πImΠ(s) andΠ(0)= ρ/4M ,the vec-
− s − N part of R(s). In this model we have
tor meson analogue of the Thomson term in photon scat-
m2 1 α
R(s)=12π2 ω δ(s m2)+ 1+ S Θ(s s ). (13)
tering. gω2 − ω 6(cid:16) π (cid:17) − o
We separate the spectrum R(s) into a resonance part
with g = 3g 16.8 (using the vector coupling constant
ω
≃
with s s and a continuum R (s) which must approach
≤ 0 c g = 5.6). We can neglect the small quark mass term c1
the perturbative QCD limit for s>s :
0 and find from eq. (10):
1 α
R (s)= 1+ S Θ(s s ). (9) 8π2m2 α
c 6(cid:16) π (cid:17) − o ω =1+ S, (14)
g2 s π
0
The factor 1 is again specific for the isoscalar channel.
6 whichfixes√s =1.16GeVusingα (s ) 0.4andm =
0 S 0 ω
≃
The Borel mass parameter must be sufficiently large
M 0.78GeV. It is interesting to identify the spectral gap
so that eq.(8) converges rapidly, but otherwise it is ar-
∆=√s with the scale for spontaneous chiral symmetry
0
bitrary. We choose > √s0 so that e−s/M2 can be ex-
M breaking, ∆ = 4πf , where f = 92.4 MeV is the pion
π π
pandedinpowersofs/ 2fors<s .Theremaininginte-
0
M decayconstant.IntheVMDmodel,takingthezerowidth
gral ∞dsR (s)e−s/M2 isevaluatedinsertingtherunning
s0 c limit,eq.(14)holdsforboththeωandρmeson,withequal
R
couplingstrengthα (s )ineq.(9).Thentheterm-by-term
S o mass m = m = m . Inserting s = 16π2f2 in eq.(14)
V ρ ω 0 π
comparison in eq.(8) gives the following set of sum rules
one recovers the famous KSFR relation m =√2 gf up
V π
for the moments of the spectrum R(s) (see also refs. [7,
to a small QCD correction.
8])
The sum rule (11) for the first moment gives
s0
Zs00 dsR(s)= ss02c0+c1−12π2Π(0), (10) 8gπ22m4ω = s220 (cid:16)1+ απS(cid:17)−π32 hhαπSG2i+12hmuu¯u+mdd¯dii.
dssR(s)= 0c c , (11)
Z 2 0− 2 (15)
0
4 F. Klingl and W. Weise: Model independentconstraints from vacuumand in-medium QCD Sum Rules
Inserting the values for the gluon and quark condensates – the in-medium case:
wefindindeedperfectconsistency.Givenamodelfortheω now we have to match the moments of the density
mesonspectralfunction inthe vacuumandin the nuclear dependent spectral distributions,
medium, the sum rules (10,11) therefore provide useful s0 s α (s ) 3π2
0 S 0
dsR(s,ρ)= 1+ + ρ, (17)
Z 6 (cid:18) π (cid:19) M
constraints to test the calculated spectra. 0 N
We nowcontinueonfromVMDtoamorerealisticap- together with
proach.Inrefs.[3,4]wehaveusedaneffectiveLagrangian s0 s2 α (s )
dssR(s,ρ)= 0 1+ S 0 c (0) δc (ρ).
2 2
Z 12(cid:18) π (cid:19)− −
based on chiral SU(3) SU(3) symmetry with inclusion 0
⊗ (18)
of vector mesons as well as anomalous couplings from
Using our calculated spectrum [4] shown in Fig. 1b,
the Wess-Zumino action in order to calculate the ω me-
we find √s0 = 1.08GeV at ρ = ρ0 = 0.17fm−3, with
son spectrum both in the vacuum and in nuclear mat-
s0dsR(s,ρ ) = 0.26GeV2. Then s0dssR(s,ρ ) =
0 0 0 0
ter. The resulting vacuum spectrum reproduces the ob- R R
0.11GeV4 is to be compared with the right hand side
served e+e− hadrons(I = 0) data very well [3] (see
→ of eq. (18) which gives 0.12GeV4, so there is again
Fig. 1a). The predicted in-medium mass spectrum (for
excellent consistency.
ω excitations with q = 0) shows a pronounced down-
Note again that these tests do not involve uncertain
wardshiftoftheω-mesonpeakandasubstantial,butnot
four-quarkcondensates.Furthermore,ifthein-medium
overwhelmingincreaseof its width from reactionssuch as
spectrumshowsareasonablynarrowquasiparticleex-
ωN πN, ππN etc. (see Fig. 1b). At large s>s , both
→ 0 citation, the quantity m¯2 = s0dssR(s)/ s0dsR(s)
0 0
spectra should approach the QCD limit (9). The consis- R R
canindeedbeinterpretedasthesquareofanin-medium
tency test of these calculated spectral distributions with
“mass”ofthisexcitation.Forourωmesoncasewefind
the sum rules (10) and (11) goes as follows:
m¯ =0.65GeVatρ=ρ ,asubstantialdownwardmass
0
– the vacuum case: shift as discussed in refs. [3,4]. (For the broad ρ me-
the two sides of eq. (10), sonspectrum,ontheotherhand,theinterpretationof
s0 s α (s ) m¯ as an in-medium mass is not meaningful as demon-
0 S 0
dsR(s)= 1+ , (16)
Z 6 (cid:18) π (cid:19)
0 strated in ref. [3]).
now match at √s0 = 1.25GeV, with 0s0dsR(s) = Amusingly, the spectral gap ∆ = √s0 decreases by
R
0.29GeV2. The sum rule for the first moment gives about 15 percent when replacing the vacuum by nu-
s0dssR(s)=0.19GeV4,tobecomparedwith 1s2(1+ clear matter. This is in line with the proposition that
0 2 0
R
α /π) c =0.22GeV4, so there is consistency at the this gap reflects the order parameter for spontaneous
s 2
−
10% level. chiral symmetry breaking and scales like the pion de-
F. Klingl and W. Weise: Model independentconstraints from vacuumand in-medium QCD Sum Rules 5
cay constant f (or, equivalently, like the square root
π
of the chiral condensate q¯q ).
h i
In summary, we have shown that the combination of
sum rules (10) and (11)for the lowestmoments of the
spectraldistributionsdoesserveasamodel-independent
consistency test for calculated spectral functions.
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6 F. Klingl and W. Weise: Model independentconstraints from vacuumand in-medium QCD Sum Rules
101 101 (b)
Barkov (3π)
Dolinsky (3π)
100 Dolinsky (I=0) ρ=0
ρ=ρ
100 0
s)
R( 10-1
10-1
10-2
((aa))
10-3 10-2
0.6 0.8 1.0 1.2 1.4 0.4 0.6 0.8 1.0 1.2
√s [GeV] √s [GeV]
Fig. 1. a) SpectrumR(s)in theω meson channelas calculated in ref. [3] (solid line). Thedatapointsrefer toe+e− →3π and
e+e− → hadrons (I =0) [10]
b) In-medium spectrum of ω meson excitations in nuclear matter at density ρ0 = 0.17fm−3 as calculated in refs.[3,4] (solid
line) in comparison with the vacuumspectrum (dashed line).