Quasilocal Quark Models as Effective Theory of Non-perturbative QCD

0.14 MB·English

by  A. A. Andrianov

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International JournalofModernPhysicsA
(cid:13)c WorldScientificPublishingCompany
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0 Quasilocal Quark Models as Effective Theory of Non-perturbative QCD∗
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2

n
A.A.Andrianov
a
J Dipartimento di Fisica, Universita’ di Bologna and IstitutoNazionale di Fisica Nucleare,
2 Sezione di Bologna, 40126 Bologna, Italia, [email protected]

1 V.A.Andrianov
v
V.A.Fock Department of Theoretical Physics, St.Petersburg State University,
5
198504 St.Petersburg, Russia, [email protected]
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WeconsidertheQuasilocalQuarkModelofNJLtype(QNJLM)asaneffectivetheoryof
0
non-perturbativeQCDincludingscalar(S),pseudoscalar(P),vector(V)andaxial-vector
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(A) four-fermioninteraction with derivatives. In the presence of a strong attraction in
0
the scalar channel the chiral symmetry is spontaneously broken and as a consequence
/
h the composite meson states are generated in all channels. With the help of Operator
p Product Expansion the appropriate set of Chiral Symmetry Restoration (CSR) Sum
- Rules in these channels are imposed as matching conditions to QCD at intermediate
p energies. The mass spectrum and some decay constants for ground and excited meson
e statesarecalculated.
h
: Keywords: QCD;Effective Quasilocal QuarkModels; OPE; ChiralSymmetry Restora-
v
tionSumRules
i
X
r
a 1. Introduction
TheQCD-inspiredquarkmodelswithfour-fermioninteractionareoftenappliedfor
the effective descriptionof low-energyQCD in the hadronizationregime.The local
four-fermioninteractionisinvolvedtoinducethedynamicalchiralsymmetrybreak-
ing (DCSB) due to strong attraction in the scalar channel. As a consequence, the
dynamicalquarkmassm iscreated,aswellasanisospinmultipletofpions,mass-
dyn
less in the chiral limit, and a massive scalar meson with mass m = 2m arises.
σ dyn
1
However it is known from the experiment that there are series of meson states
with equal quantum numbers and heavier masses, in particular, 0 +(π,π ,π ,...);
− ′ ′′
0++(σ(f ),σ ,σ ,...);1 (ρ,ρ,ρ ,...).Duetoconfinement,oneexpectsaninfinite
0 ′ ′′ −− ′ ′′
numberofsuchexcitedstateswithincreasingmasses.Thereforeinordertodescribe
the physics of those resonances at intermediate energies one can extend the quark
2
model with local interaction of the Nambu-Jona-Lasinio (NJL) type taking into
account higher-dimensional quark operators with derivatives, i.e. quasilocal quark
interactions3,4,5,6,7,8,10. For sufficiently strong couplings the new operators pro-
∗TalkattheInternational Symposium”MENU2004”,29.August-4.September, Beijing,China.
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2 A. A. Andrianov, V. A. Andrianov
moteformationofadditionalnewmesonstates.Suchaquasilocalapproach(seealso
11,12,13)representsasystematicextensionoftheNJL-modeltowardsthecomplete
effective action of QCD where many-fermion vertices with derivatives possess the
manifest chiral symmetry of interaction, motivated by the soft momentum expan-
sion of the high-energy perturbative QCD effective action.
8
Another idea is to impose CSR Sum Rules at high energies . In particular,
at intermediate energies the correlators of QNJLM can be matched to the Oper-
9
ator Product Expansion (OPE) of QCD correlators . This matching realizes the
correspondence to QCD and improves the predicability of QNJLM. It is based on
the large-N approach which is equivalent to planar QCD. In this approximation
c
the correlators of color-singlet quark currents are saturated by infinite number of
narrow meson resonances.
9
Ontheotherhandthehigh-energyasymptoticisprovided bytheperturbation
theory of QCD and the OPE due to asymptotic freedom of QCD. Therefrom the
correlators under discussion increase at large p2: ΠC(p2) |p2→∞∼ p2lnµp22. When
comparing the two approaches one concludes that the infinite series of resonances
withthesamequantumnumbersshouldexistinordertoreproducetheperturbative
asymptotic.
Meantime the differences of correlators of opposite-parity currents rapidly de-
crease at large momenta 8,12: (ΠP(p2)−ΠS(p2))p2→∞ ≡ ∆pS4P +O(p16),
∆∆SP ≃24π16απsαhq¯qiq¯2qa2n,dw1h0,e1r1e: (q¯ΠqV(isp2a)−quΠarAk(pc2o)n)dp2e→ns∞at≡e, w∆pVe6Aha+veOd(pe18fi)n,ed for V,A
VA s
fieldsΠ≃V−,A(p2) h( iδ p2+p hp )iΠV,A(p2),andthevacuumdominancehypothesis9
µν ≡ − µν µ ν
in the large-N limit is adopted.
c
Therefore the chiral symmetry is restored at high energies and the two above
differences represent genuine order parameters of CSB in QCD. As they decrease
rapidly at large momenta one can perform the matching of QCD asymptotic by
means of few lowest lying resonances that gives a number of constraints from the
CSR. They may be used both for obtaining some additional bounds on the model
parametersandforcalculatingofsomedecayconstants(seein13,14 andreferences
therein). In the presenttalk the QNJLMis presentedwith two channels where two
pairs of SPVA-mesons are generated. Respectively it is expected to reproduce the
lower part of QCD meson spectrum.
2. Quasilocal Quark Model of NJL-type
The minimal n-channel lagrangianof the QNJLM has 4,5,6 the following form,
2
1
L=q¯i∂ˆq+ 4N Λ2 X {akl[q¯fkq·q¯flq+q¯fkiγ5q·q¯fliγ5q]
c
k,l=1
+b [q¯f iγ q q¯f iγ q+q¯f iγ γ q q¯f iγ γ q] , (1)
kl k µ l µ k µ 5 l µ 5
· · }
where a ,b represent symmetric matrices of real coupling constants and f are
kl kl k
formfactors. We will restrict ourselves by the case n = 2 and describe the ground
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Quasilocal Quark Models as Effective Theory 3
meson states and their first excitations only.
The observables should not depend on the cutoff Λ. The scale invariance is
achievedwiththehelpanappropriateprescriptionofcutoffdependenceforeffective
coupling constants a ,b . Namely, we require the cancelation of quadratic diver-
kl kl
gences and parameterize the matrices of coupling constants in the vicinity of poly-
critical point as follows: 8π2a−kl1 =δkl− ∆Λk2l;16π2b−kl1 =δkl− 43∆¯Λk2l;∆kl,∆¯kl ≪Λ2.
The last inequalities provide the masses to be essentially less than the cutoff.
The parameters∆ just describe the deviation froma criticalpoint anddeter-
kl
mine the physical masses of scalar mesons. The CSB is generated by the dynamic
quark mass function corresponding to nontrivial v.e.v.’s of scalar fields σ ,σ :
1 2
M(τ)=σ f (τ)+σ f (τ); M M(0)=2σ ; τ ∂2.
1 1 2 1 0 ≡ 1 ≡−Λ2
The physical mass spectrum can be found from solutions of the corresponding
secular equation, det(Aˆp2+Bˆ)=0; m2 = p2, where Aˆ and Bˆ represent the
phys − 0
kinetic term and the momentum independent part correspondingly. Let us display
themass-spectrumforgroundmesonstatesandtheirfirstexcitations.Weintroduce
the notations, σ2 σ2 + 2√3σ σ +3σ2 > 0, d 3∆¯ +2√3∆¯ +∆¯ , and
≡ 1 3 1 2 2 ≡ 11 12 22
take into account the consistency inequalities ∆ < 0,∆¯ < 0. The spectra for
22 22
scalar and pseudoscalar mesons are: m = 4σ = 2M ; m = 0; m2 4∆ +
σ 1 0 π π′ ≃ −3 22
σ2; m2 m2 2σ2 > 0. The spectra for vector and axial-vector mesons are:
σ′ − π′ ≃
m2 det∆¯kl ; m2 m2 +6M2; m2 4∆¯ d m2; m2
ρ ≃ −2∆¯22lnMΛ202 a1 ≃ ρ 0 ρ′ ≃ −3 22− 6lnMΛ202 − ρ a′1 −
m2 3(m2 m2 ) 3σ2 > 0. The prime labels everywhere an excited meson
ρ′ ≃ 2 σ′ − π′ ≃
state.
We identify σ with f (400 1200), σ with f (1370), π with π(1300), ρ with
0 ′ 0 ′
− 1
ρ(770), ρ with ρ(1450) and a with a (1230). The experimental data give us:
′ 1 1
mσ = 400 1200Mev; mσ′ = 1200 1500MeV; mπ′ =1300 100 MeV; mρ =
÷ ÷ ±
770±0.8 MeV; mρ′ = 1465±25 MeV; ma1 = 1230±40 MeV. The prediction
for the mass of σ-meson is then m 800 MeV, which is close to the averaged
σ
≃
experimental value. Furthermore we have the following prediction for the mass
of a′1-particle, ma′1 ∼= 1465÷ 1850 MeV. The large range for a possible mass of
a -meson is accounted for by a big experimental uncertainty for the mass of σ
′1 ′
and π mesons. If we accept the averaged values for them and use the CSR rules,
′
then ma′ mρ′ 30 MeV. One can confront this value with a phenomenological
1 − ≈
1
estimate, ma′ =1640 40 MeV .
1 ±
3. Chiral Symmetry Restoration Sum Rules
Let us exploit the constraints based on chiral symmetry restoration in QCD at
high energies. Expanding the meson correlators in powers of p2 one arrives to the
CSR Sum Rules. In the scalar-pseudoscalarcase they read: ZS ZP =0;
Pn n −Pn n
ZSm2 ZPm2 =∆ , and in the vector-axial-vector(1) one obtains:
Pn n S,n−Pn n P,n SP
ZV ZA =4f2; ZVm2 ZAm2 =0,
Pn n −Pn n π Pn n V,n−Pn n A,n
ZVm4 ZAm4 =∆ .ThefirsttworelationsarethefamousWeinberg
Pn n V,n−Pn n A,n VA
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4 A. A. Andrianov, V. A. Andrianov
Sum Rules, with f being the pion decay constant. The residues in resonance pole
π
contributionsinthevectorandaxial-vectorcorrelatorshavethestructure,Z(V,A) =
n
4f2 m2 , with f being defined as corresponding decay constants.
(V,A),n (V,A),n (V,A),n
In the scalar-pseudoscalar case it has been obtained 12,15 that the residues in
poles are of different order of magnitude;
the second CSR Sum Rule results in the estimation for splitting between the
σ - and π -meson masses: m2 m2 1m2; and the value L =(0.9 0.4) 10 3
′ ′ σ′ − π′ ≃ 6 σ 8 ± · −
16
from accepts m 800 MeV.
σ
≃
In the vector-axial-vector case all residues are found to be of the same order
18
of magnitude in contrast to the scalar-pseudoscalar channel . The first and the
second Sum Rule is fulfilled identically in the large-log approach. The third one
takes the form: Z1(m2a′1 −m2ρ′) ≃ 16παs <q¯q>2 . The structure of Zρ′ and Za′1
shows that if ma′ mρ′ then Za′ Zρ′ and thereforefa′ fρ′. As a consequence
1 ≃ 1 ≃ 1 ≃
these residues approximately cancel each other in Sum Rules and after evaluating
16
we get f 0.15 and f 0.06 to be compared with the experimental values
ρ a
≈ ≈
f = 0.20 0.01, f = 0.10 0.02. We have also a reasonable prediction for the
ρ a
± ±
chiral constant L for the ρ,a -mesons and their first excitations (n=2) one gets
10 1
L = 6.0 10 3, which is consistent with that one 16 from hadronic τ decays:
10 −
≈ − ·
L = (6.36 0.09) 0.16 ) 10 3.Itisworthtomentionalsothatwithin
10 expt theor −
− ± | ± | ·
thefour-resonanceansatz(n=2)andusingtwofirstWeinbergsumrulesoneobtains
the estimation of electromagnetic pion-mass difference ∆m4 (3.85 0.16)
17 π |em≃ ±
MeV, (see ) which improves the agreement between theoretical predictions and
the experimental value of ∆m expt (4.42 0.03) MeV.
π |em ≃ ±
4. Summary
Let us summarize the results presented in this talk.
(i) The mass of the second axial-vectorparticle with I =1 is predicted. It is com-
parable with the mass of the vector counter-partner: ma′ = 1465 1850MeV
1 ÷
and the most plausible value of the mass difference is ma′ mρ′ 30MeV;
1 − ≈
(ii) The estimation on the mass of the σ-meson does not contradict to existing
1
experimental data : m 800 MeV;
σ
≃
(iii) The couplings f ,f and the chiral constant L as well as the electromag-
ρ a 10
netic pion-mass difference ∆m4 17 are evaluated from CSR Sum Rules as
π |em
matching rules for QNJLM to QCD at intermediate energies.
FinallywewouldliketomentionthattheQNJLModelscanbeusedtodescribe
Higgs particles in extensions of the Standard Model, see 15,19.
Acknowledgements
WeexpressourgratitudetotheorganizersoftheInternationalSymposium”MENU
2004” in Beijing for hospitality and financial support. This work is supported by
February 2, 2008 1:54 WSPC/Guidelines-IJMPA menu50
Quasilocal Quark Models as Effective Theory 5
GrantRFBR 04-02-26939,by GrantINFN/IS-PI13andthe Program”Universities
of Russia”.
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