Effective chiral theory of mesons, coefficients of ChPT, axial-vector symmetry breaking

Abstract

A review of effective chiral theory of pseudoscalar, vector, and axial vector mesons is presented. 10 coefficients of the ChPT are predicted. A new symmetry breaking-axial-vector symmetry breaking is found and applied to EW theory. The masses of W and Z are dynamically generated and in agreement with data.

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arXiv:hep-ph/9808441v1 27 Aug 1998
Effective chiral theory of mesons, coefficients of ChPT,
axial-vector symmetry breaking
Bing An Li
Department of Physics and Astronomy, University of Kentucky
Lexington, KY 40506, USA
Abstract
A phenomenological successful effective chiral theory of pseudoscalar, vector, and
axial-vector mesons is introduced. Based on this theory all the 10 coefficients of ChPT
are predicted. It has been found that a new symmetry breaking-axial-vector symmetry
breaking is responsible for the mass difference between ρ and a1mesons. It is shown
that in an EW theory without Higgs mW and mZare dynamicallu generated by the
combination of the fermion masses and the axial-vector symmetry breaking: mW =
1
√2gmt, mZ = ρmW/cosθW with ρ ≈ 1, and GF = 1/(2√2m2
t). Two gauge fixing
terms of W and Z fields are dynamically generated too.
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1Effective chiral theory of mesons
The chiral perturbation theory is rigorous and phenomenologically successful in describing
the physics of the pseudoscalar mesons at low energies(E < mρ). Models attempt to deal
with the two main frustrations that the ChPT is limited to pseudoscalar mesons at low
energy and contains many coupling constants which must be measured.
I have proposed an effective chiral theory of pseudoscalar, vector, and axial-vector mesons[1].
It provides a unified study of pseudoscalar, vector, and axial-vector meson physics at low
energies.
The ansatz made in this theory is that the meson fields are simulated by quark operators.
For example,
ρi
µ= −
1
gρm2
ρ
¯ψτiγµψ.(1)
The ansatz can be tested. Applying PCAC, current algebra, and this expression to the decay
ρ → ππ, under the soft pion approximation it is derived
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2fρππgρ= 1 (2)
which is just the result of the VMD.
As a matter of fact, using fermion operator to simulate a meson field has a long history:
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1. More than six decades ago, Jordan et al observed
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√π∂µφ =¯ψγ5γµψ (3)
in 1 + 1 field theory.
2. A similar relation is proved in the bosonization of 1+1 field theory.
3. Use of quark operators to simulate meson fields has been already exploited in the
Nambu-Jona-Lasinio(NJL) model, a model of four quark interactions.
4. Quark operators have been taken as interpolating fields in lattice gauge calculations.
The simulations of the meson fields by quark operators are realized by a Lagrangian
which is constructed by chiral symmetry
L =¯ψ(x)(iγ · ∂ + γ · v + γ · aγ5− mu(x))ψ(x)
ψ(x)Mψ(x) +1
2m2
−¯
0(ρµ
iρµi+ K∗µK∗
µ+ ωµωµ+ φµφµ
+aµ
iaµi+ Kµ
1K1µ+ fµfµ+ fµ
1sf1sµ)(4)
u = exp{iγ5(τiπi+λaKa+λ8η8+η0)}. To avoide double couting, there are no kinetic terms of
mesons, which are generated dynamically. Using the least action principle, the relationships
between meson fields and quark operators are found from the Lagrangian. Taking the case
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of two flavors as an example, from the least action principle we obtain
Πi
σ
x = (i¯ψγ5ψ −Πi
µ= −1
ωµ= −1
= i(¯ψτiγ5ψ + ix¯ψτiψ)/(¯ψψ + ix¯ψγ5ψ),
σ
¯ψτiψ)/(¯ψψ + iΠi
σ
¯ψτiγ5ψ),
ρi
m2
0
¯ψτiγµψ,ai
µ= −1
fµ = −1
m2
0
¯ψτiγµγ5ψ,
m2
0
¯ψγµψ,
m2
0
¯ψγµγ5ψ,(5)
where σ + iγ5τ · Π = ue−iηγ5, σ =
√1 − Π2, and x = tanη. The pseudoscalar fields have
very complicated quark structures. Substituting these expressions into the Lagrangian of
two flavors, a Lagrangian of quarks is obtained. It is no longer a theory of four quarks. This
model is different from NJL model.
Using path interal to integrate out the quark fields, the effective Lagrangian of mesons
is derived.
LE= lndetD,
Lre=1
2lndet(D†D),
Lim=1
2lndet(D/D†).
The question is whether the masses, decay widths, and interactions of mesons can be
described by the quark operators correctly. Taking masses as an example,
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1. Pseudoscalar mesons
m2
π± =
4
f2
π±{−1
4
f2
3<¯ψψ > (mu+ md) −F2
3<¯ψψ > (mu+ md) −F2
3<¯ψψ > (mu+ ms) −F2
3<¯ψψ > (md+ ms) −F2
{−1
4(mu+ md)2},
m2
π0 =
π0{−1
4
f2
4
f2
4
f2
η8
2(m2
u+ m2
d)},
m2
K+ =
K+{−1
K0{−1
4(mu+ ms)2},
m2
K0 =
4(md+ ms)2},
m2
η8=
3<¯ψψ >1
3(mu+ md+ 4ms)
−F2
6(m2
u+ m2
d+ 4m2
s)},(6)
The formulas at the first order in quark masses are Gell-Mann, Oakes, and Renner
chiral perturbation theory.
2. Vector mesons
m2
ρ= m2
ω= 6m2[2].
3. Axial-vector mesons
(1 −
1
2π2g2)m2
a= 6m2+ m2
ρ,
(1 −
1
2π2g2)m2
f= m2
ρ+ m2
ω,
The masses of vector originate in dynamical chiral symmetry breaking. This is the reason
why they are much heavier than pseudoscalars. The masses of the axial-vector mesons are
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