Effective Chiral Theory for Pseudoscalar and Vector Mesons

12 Pages·0.21 MB·English

by  E. Gedalin

Effective Chiral Theory for Pseudoscalar and Vector Mesons
E.Gedalin , A.Moalem and L.Razdolskaya
∗ † ‡
Department of Physics, Ben Gurion University, 84105,Beer Sheva, Israel
We consider the vector meson mixing scheme and mass splitting within the framework of an
extendedU(3)L U(3)R chiraleffectivefieldtheorybasedonthehiddenlocalsymmetryapproach
where, the pseudoscalar and vector meson nonets play the role of dynamic variables. Unlike other
variants of thisNmodel, we show that the diagonalization of the vector meson mass matrix and
the assumption that its eigenvalues are identical with the physical meson masses, determines the
mixing scheme as well as the model free parameters. We show that VMD can be derived from our
lagrangian and that for electromagnetic processes at low momenta the VMD model is a good first
approximation. The model reproduces nicely theradii of the charged pions and kaons.
2
0
I. INTRODUCTION
0
2
n In a recent contribution [1] it was indicated that, for any effective field theory (EFT) of colorless meson fields,
a the mixing schemes of particle states and decay constants are determined in a unique way by the kinetic and mass
J lagrangiandensities. Inageneralcase,thesedensitiesarebilinearintermsoftheintrinsicfields,involvingnondiagonal

2 kinetic and mass matrices. However, they can be reduced into a standard quadratic form by transforming the
intrinsic fields into the physical ones in three consecutive steps. These steps include : (i) the diagonalization of the

1 kinetic matrix, (ii) rescalling of intrinsic fields to restore the standard normalizationof the kinetic term, and (iii) the
v
diagonalizationof the resulting mass matrix. In such case where the dimensions of the nondiagonalkinetic and mass
3
submatricesarerespectively,k k andn n,thisprocedureleadstoschemeswhichinvolve[k(k 1)/2]+[n(n 1)/2]
0 × × − −
0 mixing angles and k field rescalling parameters. The commonly used mixing schemes correspond to specific choice
1 of the kinetic and mass matrices. In particular, η η′ mixing requires one mixing angle scheme, if and only if, the
−
0 kinetic term for the intrinsic fields has a quadratic form. Such are the traditional scheme [2] and the one proposed
2 by Feldmann et al. [3–5], the so called quark flavor basis (QFB) scheme. For bilinear kinetic and mass lagrangian
0
densities (i.e. with nondiagonal kinetic and mass matrices) mixing schemes involve two mixing angles similar to the
/
h ones proposed by Escribano and Fr`ere [6].
p There have been several attempts to incorporate within a single effective lagrangian a wide range of different
- electroweak and strong processes [7–10]. Particularly interesting are the Bando, Kugo and Yamawaki (BKY) model
p
[7], based on the hidden local symmetry (HLS) approach, and variants of this model [8,9], where the vector mesons
e
h (ρ,ω,K∗,K¯∗,φ) are identified with the dynamical gauge bosons of the HLS nonlinear chiral lagrangian. It was
: indicated by Benayoun et al. [9], that the original BKY model neither reproduces the observed vector mass splitting
v
nor preserves current conservation for the pseudoscalar octet-singlet sector. A variant of this model proposed by
i
X Bramon et al. [8] also does not predict the masses correctly, although it guarantees current conservation, explicitly.
r Yetanothervariantofthismodel[9]includestheη′ mesonalso,andtosomeextentcuresthesedeficiencies. However,
a
once the HLS lagrangian is extended to include the η meson, the trace of the pseudoscalar (and vector) nonet
′
field matrix no longer vanishes, and therefore, a general expression for the lagrangian must include additional terms
which are quadratic in the traces of the nonet field matrices. More recently, a most general lagrangian based on
localU(3) U(3) symmetry wasconstructed[11,12],by combiningthe HLS approachwitha generalprocedureof
L R
includingtheη mesonintoachiraltheory[13–16]. Boththesymmetricpartsofthelagrangian,aswellasthesymmetry
′
N
breakingcompanions,include terms quadraticinthe tracesofthe fieldmatrices. Likewise,the lagrangianincludesan
explicit U(3) U(3) symmetry violating mass term for the pseudoscalar mesons [14,16]. These additional terms
L R
influence the meson mass spectra and more importantly, by means of the diagonalization procedure of Ref. [1], not
N
only the mixing scheme is defined uniquely but also, taking the eigenvalues of the resulting mass matrix to be equal
to the physical masses, it was possible to fix a number of the model free parameters. All observables are calculated
fromthelagrangianstraightforwardlywithoutadditionalassumptions. Albeit, thelagrangianconstructedreflectsthe
∗[email protected]
†[email protected]
‡[email protected]
1
fundamentalsymmetriesoftheQCDlagrangianandprovidesacoherenttheoreticalframeworkwherethepseudoscalar
and vector meson nonets are treated on equal footing.
Our main interest in the present work is to further develop and explore the U(3) U(3) chiral theory of Ref.
L R
[12] by considering the vector meson sector. It is demonstrated that in this case the relevant model parameters can
N
allbefixedthroughthediagonalizationprocedureofRef.[1]. Ithasbeenillustratedalready[7]thatthevectormeson
dominance(VMD)modelemergesasanaturalconsequenceofthebasicassumptionoftheHLSapproachwherevector
mesons play the role of gauge bosons. Particularly, the VMD model becomes a dynamical consequence for a specific
choice of the HLS model parameter a=2. Detailed analyses of data argue for departures from this value [9,17], and
that in addition to the vector meson poles in the charged meson form factors, a direct photon- pseudoscalar meson
coupling may also contribute significantly. More recently, Harada and Yamawaki[18] have shown that in an effective
field theory based on HLS, VMD may be badly violated. We demonstrate in this work that VMD derived from the
lagrangianconstructed in Ref. [12] is a good first approximationfor electromagnetic processes at low momenta. Our
paper is organized as follows. In Section II we present briefly the lagrangian. In section III we discuss vector mixing
scheme and mass splitting corresponding to the lagrangian presumed in the simple case of exact isospin symmetry
and in the more evolved case of isospin symmetry breaking. In Section IV we consider the photon interaction with
hadronic matter and discuss the relation between the lagrangian constructed and the traditional VMD model. We
summarize and conclude in Section V.
II. THE EFFECTIVE LAGRANGIAN
As in Ref. [12], the lagrangianis written in the form,
1
L=L +L¯ +L +a(L +L¯ ) Tr(V Vµν)+L +... , (1)
A A m V V µν WZW
− 4
whereL (L¯ )andL (L¯ )arethesymmetricparts(symmetrybreakingcompanions)forthepseudoscalarandvector
A A V V
mesons,L isamostgeneralU(3) U(3) symmetryviolatingmassterm[14,16],L standsforthewellknown
m L R WZW
Wess-Zumino-Wittenterm[19,20],andtheellipsis”...”representstermsaccountingfortheregularizationofoneloop
N
contributions [22,23,13]. All terms in the expression above are constructed from boson gauge fields V , and vector
µ
and axial-vector covariants defined as [21,24,25],
i 1
Γ = ξ ,∂ ξ + ξ r ξ+ξl ξ , (2)
µ † µ † µ µ †
2 2
i(cid:2) (cid:3) 1(cid:0) (cid:1)
∆ = ξ ,∂ ξ + ξ r ξ ξl ξ . (3)
µ † µ † µ µ †
2 2 −
(cid:8) (cid:9) (cid:0) (cid:1)
Here r and l represent the standard model external gauge fields; r = v +a and l = v a , with v and
µ µ µ µ µ µ µ µ µ
−
a being the vector and axial vector external electroweak fields, respectively. The non-linear representation of the
µ
pseudoscalar nonet fields is taken as [13,14,16],
√2 2 1
U(P,η0+F0ϑ) ξ2(P,η0+F0ϑ) exp i P +i η01 , (4)
≡ ≡ ( F8 r3F0 )
with η0(x) being the pseudoscalar singlet and P(x) the pseudoscalar Goldstone octet matrix,
1 π0+ 1 η8 π+ K+
√2 √6
P = π− −√12π0+ √16η8 K0  , (5)
 K− K¯0 −√26η8 
 
withobviousnotations. Thevacuumangleϑ(x)isanauxiliaryfieldwhichrendersthe variableX(x) √6η0(x)/F0+
≡
ϑ(x) to be invariant under U(3) U(3) transformations [21]. In this representation, the unimodular part of U
L R
contains the octet degrees of freedom while the phase detU = exp(i√6η0/F0) involves the singlet only. To lowest
order (i.e., smallest number of deriNvatives)the lagrangianis built from traces of the covariants∆ , Γ gV , ∆ ∆µ,
µ µ µ µ
(Γ gV )(Γµ gVµ), Dµϑ, and arbitrary functions of the variable X(x), all being invariant under U−(3) U(3)
µ µ L R
− −
transformations. Explicit expressions of various lagrangian parts are listed for convenience in the Appendix. Their
N
derivation can be found in Ref. [12]. Although the lagrangian, Eqn.1 appears similar to that of Bando et al. [7] and
2
other variants of this model [8,9], the terms L and their symmetry breaking companions L¯ (i = A,V) are different,
i i
including additional terms proportional to Tr(∆ )Tr(∆µ), Tr(∆ )Dµϑ and Tr(Γµ gVµ)Tr(Γ gV ). Clearly,
µ µ µ µ
− −
suchtermsmustbeincludedinanyU(3) U(3) symmetrybasedHLStheory,wherethetracesof∆ andΓ gV
L R µ µ µ
−
do not vanish. It is demonstrated below that these terms are needed in order to reproduce the meson masses from
N
the lagrangianwithout imposing additional constraints. We add also that the matrix used to generate the symmetry
breaking companions is a universal Hermitian matrix defined as,
B =ξbξ+ξ bξ ; b=diag(b ,b ,b ) , (6)
† † u d s
where b = m /(m + m + m ). This form guarantees that the ratio of SU(3) symmetry breaking scales to
i i u d s F
isospinsymmetry breakingscalesbe identicalto the onceofthe underlyingQCD.Note that, the lagrangian1involves
coefficientfunctions W (X)whichareabsentin theSU(3)symmetry limit. Againsuchfunctions mustbe included in
i
the processofextending the SU(3) SU(3) basedmodel intoa generalU(3) U(3) theory. As a finalremark
L R L R
we note that the symmetry violating mass term in Eqn. 1 has the exact form derived by Herera-Sikl´ody et al. [16].
As in Refs. [7–9], the sum a(L +L¯N) contains, amongst other contributions, aNvector meson mass term V Vµ,
V V µ
a vector-photon conversion factor V Aµ and the coupling of pseudoscalar pairs to both vectors and photo∼ns. The
µ
∼
latter coupling can be eliminated by choosing a = 2 allowing to incorporate the strict VMD lagrangian [7,26]. We
shall attempt to determine the value of this parameter from data in order to assess the validity of the traditional
VMD model. Altogether, the lagrangianEqn. 1 (see Appendix), involves symmetry breaking scales c ,d (i=A,V), a
i i
generalcoupling constantg and a free parametera. The symmetry breaking scalesc ,d andr are determined from
A A
global fit analysis of radiative decay widths of pseudoscalar and vector mesons [12]. The other symmetry breaking
scales c ,d and ag2 are determined through the diagonalization of the vector mass matrix.
V V
III. VECTOR MESON MASSES AND MIXING
We now turn to consider mixing and masses of vector mesons. The vector part a(L +L¯ ) of the lagrangian 62
V V
(See Appendix) involvesa vector meson mass term V Vµ which determines as already indicated the vector mixing
µ
∼
scheme, uniquely. Generally speaking, the vector nonet matrix V is the sum of an octet matrix V8, involving the
ρ0,ρ±,K∗0,K¯∗0,K∗± and ω8 meson fields, and an SU(3)F singlet ω0 matrix, e.g.,
1
V =V8+rV ω0 1 . (7)
√3 ×
Here a departure of the parameter r from one accounts for possible nonet symmetry breaking. However, unlike the
V
case of pseudoscalar mesons, the light vector meson spectrum shows no clear evidence for such a departure. In the
discussion to follow we take r = 1, assuming exact nonet symmetry. Then, the vector meson nonet matrix has the
V
form (for brevity the Lorentz indices of the vector fields are omitted),
ρ0 + ω8 + ω0 ρ+ K +
√2 √6 √3 ∗
V = ρ− −√ρ02 + √ω86 + √ω03 K∗0  . (8)
K K¯ 0 2ω8 + ω0
 ∗− ∗ −√6 √3 
 
As was already indicated in the introduction, the kinetic and mass terms of the lagrangian can be reduced to a
standard quadratic form by first diagonalizing the kinetic matrix via a unitary transformation Υ, rescale the fields
through a transformation R to restore the standard normalization of the kinetic term, and finally diagonalizing the
resulting mass matrix through another unitary transformation Ω. These three steps are sufficient to define the state
mixing scheme [1].
From Eqns. 8, 56 and 62 the vector meson kinetic and mass terms amounts to,
1 1
L = (∂ V ∂ V )(∂µVν ∂νVµ)+ (V 2Vµ) , (9)
km −4 µ ν − ν µ − 2 µMV
with,
1
2VµM2VVµ =aF82g2[Tr(VµVµ)+w˜4Tr(Vµ)Tr(Vµ) +
cV (Tr(BVµVµ)+w˜4Tr(BVµ)Tr(Vµ))+dV (Tr(BVµBVµ)+w˜4Tr(BVµ)Tr(BVµ))] . (10)
3
Thusthekinetic termhasthe standardnormalizedquadraticform,andfollowing[1]the mixingschemeisdetermined
through the diagonalization of the mass matrix only and involves one mixing angle. The nonvanishing elements of
the vector meson mass matrix read,
c d
µ2(ρ0ρ0)=µ2V 1+ 2V (bu+bd)+ 2V [b2u+b2d+w˜4(bu−bd)2] , (11a)
(cid:18) (cid:19)
c
µ2(ρ+ρ+)=µ2 1+ V (b +b )+d b b , (11b)
V 2 u d V u d
(cid:16) c d(cid:17)
µ2(ω8ω8)=µ2V 1+ 6V (bu+bd+4bs)+ 6V [(b2u+b2d+4b2s)+w˜4(bu+bd−2bs)2] , (11c)
(cid:18) (cid:19)
c d
µ2(ω0ω0)=µ2V 1+3w˜4+ 3V (bu+bd+bs)(1+3w˜4)+ 3V [b2u+b2d+b2s+w˜4(bu+bd+bs)2] , (11d)
(cid:18) (cid:19)
1
µ2(ρ0ω8)=µ2V 2√3(bu−bd)(cV +dV[bu+bd+w˜4(bu+bd−2bs)]) , (11e)
1 3
µ2(ρ0ω0)=µ2V 6(bu−bd) cV(1+ 2w˜4)+dV[bu+bd+w˜4(bu+bd+bs)] , (11f)
r (cid:18) (cid:19)
1 3
µ2(ω8ω0)=µ2V 3√2 cV(bu+bd−2bs)(1+ 2w˜4)+dV[b2u+b2d−2b2s+w˜4(bu+bd−2bs)(bu+bd+bs)] , (11g)
(cid:18) (cid:19)
c d
µ2(K K )=µ2 1+ V (b +b )+ V b b , (11h)
∗± ± V 2 u s 2 u s
(cid:18) (cid:19)
c d
µ2(K∗0K∗0)=µ2(K¯0K¯0)=µ2V 1+ 2V (bd+bs)+ 2V bdbs , (11i)
(cid:18) (cid:19)
with,
µ2 =2ag2F2 , (12)
V 8
We stress here that even in the limits of exact chiral symmetry and nonet symmetry, i.e. c = d = 0, r = 1,
V V V
the singlet vector meson mass, mω0 =µV√1+3w˜4, differs from that of the octet mω8 =µV. This distinct property
occurs due to the term W˜4(X)Tr(Γµ gVµ)Tr(Γµ gVµ) in the lagrangianEqn. 1. Only for w˜4 =0 (corresponding
to lagrangians previously used) all no−net meson ma−sses are equal. 1. Clearly, the mass matrix is nondiagonal due to
the ω8 ω0, ρ0 ω8 andρ0 ω0 mixing. Notice though that the ρ0 ω8 and ρ0 ω0 admixtures areextremely small
− − − − −
and and vanish in the limit of exact isospin symmetry (b =b ).
u d
First we diagonalize the mass matrix in the limit of exact isospin symmetry. In this limit, taking b = 1 and
s
neglecting terms of the order of c b ,d b (i = u,d), the matrix mass assumes a particularly simple form with
V i V i
elements,
µ2(ρρ)=µ2 , (13a)
V
2c 2d
µ2(ω8ω8)=µ2V 1+ 3V + 3V (1+w˜4) , (13b)
(cid:18) (cid:19)
c d
µ2(ω0ω0)=µ2V (1+ 3V )(1+3w˜4)+ 3V (1+w˜4) , (13c)
(cid:18) (cid:19)
√2 3
µ2(ω8ω0)=−µ2V 3 cV(1+ 2w˜4)+dV(1+w˜4) , (13d)
(cid:18) (cid:19)
c
µ2(K )=µ2 1+ V , (13e)
∗ V 2
(cid:16) (cid:17)
which we diagonalize using the unitary transformation,
ω8 = cosθV sinθV ω . (14)
ω0 sinθV cosθV φ
(cid:18) (cid:19) (cid:18)− (cid:19)(cid:18) (cid:19)
1Incaseofbroken nonetsymmetryitisstillpossibletomaintainmω06=µV butwiththenonetsymmetrybreakingparameter
fixed arbitrary tobe rV2 =1/(1+3w˜4)
4
It is straightforwardto show that the resulting physical meson masses and mixing angle are,
1
m2ω = 2 µ2(ω8ω8)+µ2(ω0ω0)− (µ2(ω0ω0)−µ2(ω8ω8))2+4(µ2(ω0ω8))2 , (15)
1 (cid:16) p (cid:17)
m2φ = 2 µ2(ω8ω8)+µ2(ω0ω0)+ (µ2(ω0ω0)−µ2(ω8ω8))2+4(µ2(ω0ω8))2 , (16)
tan2θ =(cid:16) 2µ2(ω0ω8) p= (m2φ−m2ω)2 1 . (cid:17) (17)
V −(µ2(ω0ω0)−µ2(ω8ω8) −s(m2φ+m2ω−2µ2(ω8ω8)2 −
From Eqns. 13a-13e, with w˜4 = 0 the mixing angle reduces to the ideal mixing angle value θVideal = 54.7o, and
−
more importantly does not depend on the symmetry breaking scales c and d . Correspondingly, the ω, K and φ
V V
meson masses become,
m2 =m2 , m2 =m2(1+ 1c ) , m2 =m2(1+c +d ) . (18)
ω ρ K ρ 2 V φ ρ V V
Clearly, assuming now d =c2/4 leads to the BKY [7] mass relations,
V V
m2 m2 1
K = φ =1+ c . (19)
m2 m2 2 V
ρ K
Otherwise, taking d =0 leads to the mass relations of Bramon et al. [8]
V
m2 1 m2
K =1+ c , φ =1+c . (20)
m2 2 V m2 V
ρ ρ
It should be stress though that in all of these private cases, the ρ and ω masses are equal and θ = θ . To
V Videal
describe the actual vector meson masses we should keep dV as a free parameter and w˜4 =0. Based on the proximity
6
of the ρ and ω masses, the parameter w˜4 should be very small, and to first order the ω and φ meson masses are,
m2ω =m2ρ(1+2w˜4) , m2φ =m2ρ(1+cV +dV)(1+w˜4) . (21)
Thus, the parameter w˜4 determines the ρ−ω mass splitting while (cV +dV) = 2(m2φ −m2ρ)/(m2ρ +m2ω). At this
order,usingthe expressionsaboveandEq.18for m2 , alongwiththe experimentalmesonmassvalues[2]oneobtains,
K
cV = 0.69 0.01, dV 0.03 and w˜4 0.01. Indeed, w˜4 is very small but it plays an important role in reproducing
the ρ ω m±ass splittin≈g. Note alsotha≈tthe actualvalue ofthe parameterd is far smallerthanc2/4=0.12favoring
− V V
the Bramon et al. [8] relations Eqn.20. More accurate values of these parameters can be deduced from the exact
expressions for the vector meson masses. These are, w˜4 =0.009 0.0001 and dV =0.025 0.01. Correspondingly,
the mixing angle for the vector meson states is θ = (52 1)±o, rather close to ideal mix±ing. In turn, this value
V
− ±
constrains the field admixtures due to nonideal mixing to be very small. With nonideal mixing taken into account,
the matrix of the physical vector fields reads,
ρ0+ω+ǫφ ρ+ K +
√2 ∗
V = ρ− −ρ0+√ω2+ǫφ K∗0  . (22)
K K¯ 0 φ ǫω
 ∗− ∗ − 
 
where ǫ = 0.047 0.001, being a measure of a non-strange (strange) admixture in the physical φ and ω) fields. We
±
stress again that this parameter like the symmetry breaking scales cV,dV and ω˜4 are all determined through the
diagonalization of the mass matrix and the requirement that the eigenvalues of this matrix should reproduce the
physical masses.
Wenowconsidertheeffectsofisospinsymmetrybreakingonthemasssplittingandmixingangle. Thenondiagonal
part of the mass matrix now reads
µ2(ρ0ρ0) µ2(ρ0ω8) µ2(ρ0ω0)
2(ρ0,ω8,ω0)= µ2(ρ0ω8) µ2(ω8ω8) µ2(ω8ω0) (23)
M µ2(ρ0ω0) µ2(ω8ω0) µ2(ω0ω0)
 
It is convenient to write the transformation Ω which diagonalize this submatrix in the form
5
Ω=Ω(ω8,ω0)Ω(ρ0,ω,φ) (24)
where the transformation Ω(ω8,ω0) diagonalize the ω8,ω0 submatrix
1 0 0
Ω(ω8,ω0)= 0 cosθV sinθV (25)
 
0 sinθ cosθ
V V
−
 
and the transformation Ω(ρ0,ω,φ) diagonalize the reduced mass matrix
m2 m2 m2
ρ ρω ρφ
Ω−1(ω8,ω0)M2(ρ0,ω8,ω0)Ω(ω8,ω0)= mm2ρ2ω m02ω m02  , (26)
ρφ φ
 
where,
1 1
m2ρω = 2m2ρ(bu−bd)cV , m2ρφ = 2√2m2ρ(bu−bd)cVw˜4 . (27)
Fromthe Eq.26itisclearthateffectsofisospinsymmetrybreakingontheρandω massesarenegligiblysmall 0.1%
Takingw˜4 =0.009asabove,itisrathereasytoshowthatthematrixelementm2ρφ isnegligiblysmallandconseq≈uently
the ρ0 φ mixing angle is practicallyzero. Therefore we may write the Ω(ρ0,ω,φ) transformationin the simple form
−
cosθ sinθ 0
ρω ρω
Ω(ρ0,ω,φ)= sinθ cosθ 0 (28)
ρω ρω
− 
0 0 1
 
where the ρ ω mixing angle is given by the expression,
−
m2
ρω
tan2θ = 2 . (29)
ρω − m2 m2
ρ− ω
With this in mind, we may now write the vector meson matrix in the form,
X1ρ0+X2ω+ǫφ ρ+ K +
√2 ∗
V = ρ− −X2ρ0+√X21ω+ǫφ K∗0  , (30)
K K¯ 0 φ+ǫ(ρ0sinθ ωcosθ )
 ∗− ∗ ρω − ρω 
 
with,
X1 =cosθρω sinθρω , (31)
−
X2 =cosθρω +sinθρω. (32)
Toevaluatethe ρ0 ω mixingangle,weneedanestimateofthe quarkmassesratiosb ,(i=u,d,s). Werecallthat,
i
−
1 1
m2 = m2(b +b )(1+ c ) , (33)
K+ 2(b +b ) π u s 2 A
u d
1 1
m2 = m2(b +b )(1+ c ) . (34)
K0 2(b +b ) π d s 2 A
u d
Then, we may estimate (b b ) from the kaon masses [2] according to,
u d
−
b b m2 m2
u− d =2 K+ − K0 , (35)
1 (b +b )/2 m2 +m2
− u d K+ K0
which yields,
b b = 0.0160 0.0005 . (36)
u d
− − ±
6
With this value of b b the calculated ρ0 ω mixing angle reads,
u d
− −
θ =( 8.9 0.3)o . (37)
ρω
− ±
For the parameter b we have [12],
s
2
m 2m 2b 1 m
s s s K
= = =2 1+ c 1 , (38)
A
m m +m 1 b 2 m −
u d − s (cid:18) (cid:19)(cid:18) π (cid:19)
which yields for c =0.64 0.06 (Alternative 1 of Ref. [12])
A
±
b =0.940 0.01 , (39)
s
±
and consequently b = 0.022 0.005, b = 0.038 0.005. For the Alternative II (c = 0.2 0.05) an QFB mixing
u d A
± ± ±
scheme (c = 1.4 0.1) we obtain b = 0.928, b = 0.028, b = 0.044 and b = 0.953, b = 0.0155, b = 0.0315
A s u d s u d
±
respectively. This complete defining the mixing scheme in the case of isospin symmetry breaking. Here as in the
analysis of the pseudoscalar meson sector [12], the exact form of the matrix B used to generate flavor symmetry
breaking(seeAppendix), guaranteesthattheratiosofSU(3)symmetrybreakingscalestoisospinsymmetrybreaking
scales are identical to the ones of the underlying QCD.
IV. VECTOR MESON DOMINANCE MODEL AND PSEUDOSCALAR MESON FORM FACTORS
A particularly interesting aspect of the theory presented aboveis the interactionbetween the photon and hadronic
matter. It was already illustrated by Bando et al. [7] and O’Connell et al. [26] that the HLS lagrangian is closely
related to the well known VMD type 2 lagrangian and that by suitable transformation it can be reduced into the
VMD type 1 lagrangian. (See also Ref. [9] on this matter). The lagrangian, Eqn.1, contains terms proportional
to V Vµ and A Aµ and V Aµ, corresponding respectively to a vector meson and photon mass terms and
µ µ µ
∼ ∼
photon-vector meson interactions. likewise, it contains the coupling of both vector mesons and photons to pairs of
pseudoscalarmesons. ByfollowingargumentssimilartothoseofRef.[26]wederiveinthissectionexplicitexpressions
for the photon, vector, and pseudoscalar parts of the lagrangian, Eqn.1. We recall first that the covariants Γ and
µ
∆ , Eqns.2,3 can be expanded as,
µ
i ie
Γ = eA Q+ [ ,∂ ] A 2,Q Q +... , (40)
µ − µ 4F2 P µP − 2F2 µ {P }−P P
π π
1 e (cid:0) (cid:1)
∆ = ∂ +i A [Q, ]+... , (41)
µ µ µ
−F √2 P F √2 P
π π
whereA standsforthephotonfield,Q=diag(2/3, 1/3, 1/3)thequarkchargeoperator,and isthepseudoscalar
µ
− − P
nonetmatrixexpressedintermsofthe physicalfields(seeRef.[12]). The ellipsis”...”denotestermsinvolvinghigher
powers of the fields. The covariant Γ gV contains then terms proportional to (eA Q gV ). 2 By substituting
µ µ µ µ
the expression, Eqn.40 into the lagrang−ianpart L +L¯ we obtain the following photon a−nd vector meson terms :
V V
(i) a direct photon-vector meson coupling,
aL(V,γ)=aF2geAµ[Tr( V ,Q (1+c B))+
8 µ V
{ }
c Tr(V )Tr(BQ)+d Tr( BV ,BQ )+2d Tr(BQ)Tr(bV )] , (42)
V µ V µ V µ
{ }
(ii) a photon mass term,
F2e2 1
aL =a 8 A2 2+ (c +2d ) , (43)
mγ 3 µ 3 V V
(cid:18) (cid:19)
(iii) a photon-vector meson-pseudoscalar meson interactions,
2Strictly speaking, Γµ−gVµ involvestheterms −e(Aµ−tanθWZµ0)Q− 2sinϑWecosϑWTzZµ0− 2√2seinϑWWµ−gVµ which give
rise to direct V −A, V −Z0 and V −W couplings.
7
aie
aL(V,γ, )= z2 +A Tr( Q,[ ,∂µ ] )+c Tr(B Q,[ ,∂µ ] ) +
P −2 2 π µ{ { P P } V { P P }
d (Tr( BQ,B[ ,∂µ ] )+2Tr(BQ)Tr(B[ ,∂µ ])) +... , , (44)
V
{ P P } P P }
and (iv) a vector-pseudoscalarmeson interaction term,
agz2
aL(V, )= π Tr( V ,[ ,∂µ ] )+c [Tr(B Q,[ ,∂µ ] ) +
µ V
P −2 2 { { P P } { P P }
Tr(V )Tr(B[ ,∂µ ])]
µ
P P
d [Tr( BV ,B[ ,∂µcalP] )+2Tr(BV )Tr(B[ ,∂µ ])] +... . (45)
V µ µ
{ P } P P }
Similarly, substituting the expansion of ∆ , Eqn.41, into the pseudoscalar lagrangian part, L +L¯ , leads to the
µ A A
photon-pseudoscalar meson interactions,
ie
L(A,γ, )= z2A Tr( [Q, ],∂µ )+c Tr(B [Q, ],∂µ ) +... . . (46)
P −2 π µ{ { P P} A { P P} }
Here the ellipsis ... denote terms involving ∂ n, n 2. To summarize the photon-meson part corresponding to
PP ≥
the lagrangian,Eqn.1 is given by,
1
L = Tr(V Vµν)+aL(V,γ)+aL +aL(V,γ, )+aL(V, )+L(A,γ, ) . (47)
VMDII µν mγ
−4 P P P
This last expression, with its photon mass L term is rather similar to the popular VMDII model [26]. By suitable
mγ
transformation of the fields this term can be removed but prior to doing that some comments are in order. First, in
complete analogy with the discussion of Ref. [26], we state here that the dressed photon propagator has the proper
behavior of i/q2 at small photon momenta but significantly modified away from q2 = 0. This certainly may be
importantin−describingprocessesatsufficientlyhighmomenta. Secondly,itisclearthattheneutralρ0,ωandφmeson
mix with the photon, spontaneously breaking symmetry down to the U(1) . We may exploit the diagonalization
em
procedure described in section III to introduce physical photon and mesons. In terms of these physical fields, the
lagrangianhas no explicitcoupling betweenthe photon andneutralmesons, whilst there is a directcoupling between
the photon to hadronic currents in the limit of exact U(3) . Thirdly, so far the value of the parameter a is left free
F
and in order to assign its value one needs considering the dynamics underlying QCD. Taking a = 2 at the instance
of exact symmetry, i.e.c = d = c = d = 0, leads successfully to the phenomenological KSFR relations [27,28]
A A V V
m2 = 4F2g2, g = 2g F2 with g and g are the ργ and ρππ effective coupling constants, respectively.
ρ 8 ρ ρππ π ρ ρππ
Furthermore,witha=2thesymmetriclagrangianpartL(A,γ, )+aL(V,γ, )=0sothattheeffectiveγππcoupling
P P
constantvanishes. Thusinthelimitofexactsymmetrythepionformfactorissaturatedbytheρmesoncontribution.
All departures from the KSFR relations, nonvanishing effective γππ coupling g = 0, and the ρ-meson dominance
γππ
6
for the pion formfactoraredue to contributionsfromthe symmetry breakingcompanions. The valuea=2.13 0.06
±
deduced from the ρ meson mass and its decay width into two pions, indicates that VMD provides, a good first
approximationfortheelectromagneticpartofourlagrangian. Withthisvalueofaourlagrangianincludesinaddition
to the usual VMD corrections due to direct photon-pseudoscalarmeson coupling.
We now turn to derive a more elegant form of the lagrangian Eqn.47 with manifestly a massless photon. We do
that via transforming the vector meson fields into [26], 3
e
=V A Q , (48)
µ µ µ
V − g
and define Γ gV = Γˆ g where Γˆ does not involve the pure photon field contributions. In terms of
µ µ µ µ µ
these new field−s, the lagrang−ianVpart L + L¯ neither contains a photon mass term nor an explicit coupling of
V V
the photon to pseudoscalar currents, j [∂ , ]. Thus the terms L and aL(V,γ, ) are removed from L +
µ µ mγ V
L¯ . An explicit photon coupling to pse∼udoscPalaPr currents, is now confined to the L(AP,γ, ) term resulting from
V
P
the pseudoscalar L + L¯ part which is left unaffected by the transformation Eqn.48. Clearly, since the Γˆ as
A A µ
before involves terms like ieA ( Q 2,Q /2)/2F2 etc. the lagrangian L +L¯ still contains the photon
µ 8 V V
− P P −{P } −
vector meson pseudoscalar meson couplings. Correspondingly,since
−
3Note that the same way the momentum independent direct Vµ −Zµ0 and Vµ −Wµ couplings may be transformed into
momentum dependentcouplings
8
e
V = A Q , (49)
µν µν µν
V − g
the vector meson kinetic term transforms into,
1 1 e 2e2
Tr(V Vµν)= Tr( µν) 2 A Tr(Q µν)+ A Aµν , (50)
− 4 µν −4 VµνV − g µν V 3g2 µν
(cid:18) (cid:19)
where as usual A =∂ A ∂ A . The first term on rhs of Eqn. 50 is the common kinetic term of the vector fields
µν µ ν ν µ
−
while the second term describes the direct photon-vector meson coupling. The last term is associated with the
kVinetic term ofthe photon A Aµν/4 and canbe removedby rescallingthe chargee andthe photon field according
µν
−
to [26],
A = 1+2e2/3g2A ; e =e/ 1+2e2/3g2 , (51)
′µν µν ′
Inserting all of these modifications intpo the lagrangianEqn.47 leadps to,
1 e
L = Tr( µν) 2 A Tr(Q µν) +
VMDI µν µν
−4 V V − g V
(cid:18) (cid:19)
aF82g2[Tr( )+w˜4Tr( )Tr( )+cV (Tr(B )+w˜4Tr(B )Tr( ))+dV (Tr(B B )+w˜4Tr(B )Tr(B ))]
VV V V VV V V V V V V
agz2
π Tr( ,[ ,∂µ ] )+c [Tr(B Q,[ ,∂µ ] ) +Tr( )Tr(B[ ,∂µ ])]
µ V µ
−2 2 { {V P P } { P P } V P P
d [Tr( B ,B[ ,∂µ ] )+2Tr(B )Tr(B[ ,∂µ ])]
V µ µ
{ V P P } V P P }
ie
z2A Tr( [Q, ],∂µ )+c Tr(B [Q, ],∂µ ) , (52)
−2 π µ{ { P P} A { P P} }
whereforconveniencewehaveomittedthe fromA ande. ThelastexpressioniscloselyrelatedtotheVMDImodel
′ ′µ ′
with the characteristics of (i) a massless photon, (ii) a momentum dependent direct photon-vector meson coupling
Aµν, and (iii) a direct photon-pseudoscalar meson coupling as well as the coupling through direct photon-vector
µν
V
meson transitions.
To conclude this sectionwe now use the lagrangianL , Eq.52,to calculate the pion and kaonform factors. In
VMDI
the three diagram approximation (see Fig. 1) one obtains,
a 1
F (q2)=1+q2 , (53a)
π 2m2 q2
ρ−
3a 1 1 c 1 2(1+c +d ) 1
F (q2)=1+q2 z2 + − V + V V , (53b)
K± 2 K"m2ρ−q2 3 m2ω −q2 3 m2φ−q2#
3a 1 1 c 1 2(1+c +d ) 1
FK0(q2)=q2 2 zK2 "m2ρ−q2 − −3 V m2ω−q2 − 3V V m2φ−q2# , (53c)
where z = 1/ 1+c /2 is a rescalling parameter defined from applying the diagonalization procedure to the
K A
pseudoscalar meson sector. It must be stressed that the three diagrams involving V A transition vertex contribute
to those terms pproportional to Q2, only. The meson charge radii are completely de−termined by diagrams involving
the conversionof ρ, ω, φ mesons into a photon (diagram 1b). We have
3a 1
<r2 >= , (54a)
π 2 m2
ρ
3a 1 1 c 1 2(1+c +d ) 1
<r2 >= z2 + − V + V V , (54b)
K± 2 K"m2ρ 3 m2ω 3 m2φ#
3a 1 1 c 1 2(1+c +d ) 1
<r2 >= z2 + − V + V V . (54c)
K0 2 K"−m2ρ 3 m2ω 3 m2φ#
Thecalculatedradiiofthe pionsandkaonsalongwithdataarelistedinTableI. Inthe firstthreecolumnswelistthe
results from our model for the Alternative I (c =0.64 0.06), Alternative II (c =0.20 0.05) and QFB scheme
A A
± ±
(c =1.4 0.1) [12]. The results from the Alternative I and Alternative II explain equally well the pion and kaon
A
radii altho±ugh the χ2 criteria is slightly better for the former. The results from the QFB scheme are by far inferior.
For comparison we show the radii calculated with a = 2.4 as determined by Benayoun and O’Connell [9]. Clearly,
this value can not explain the charged pion radius.
9
V. SUMMARY AND CONCLUSIONS
We have considered the vector meson sector within the framework of an effective field theory based on local
U(3) U(3) symmetry,wherethevectormesonsplaytheroleofthedynamicalgaugebosonsoftheHLSnonlinear
L R
chiral lagrangian. This supplement a previous study of the pseudoscalar sector [12] within this same framework.
N
The lagrangian is written in the most general way, including additional terms quadratic in the traces of the field
matrices, which no longer vanish once singlet contributions are included. With these terms included, particularly
the contribution from the small W˜4(X)Tr(Γµ gVµ)Tr(Γµ gVµ) term and its symmetry breaking companion, it is
− −
possibletoreproducetheω ρmasssplitting. Theexplicitformofthelagrangian,throughitskineticandmassterms
−
corresponding to vector mesons, determine the vector meson mixing scheme unambiguously. This observation which
has been overlooked in previous studies allows us to calculate the model symmetry breaking scales and the value
of ag2 through the diagonalization of the mass matrix. Once the value of g and a are disentangled, say from using
anotherdatapointlike the decaywidth ofρ ππ, the lagrangianis completely defined, andcanbe usedto calculate
→
all observables without any additional assumption. As an example we have calculated the form factors and radii of
the pions and kaons. Our calculations reproduce straightforwardly the proper normalization of the form factors and
explain nicely the charged radii. We finally indicate that VMD model emerges naturally as a first approximation of
our lagrangian. This stands in marked difference with Ref. [9,18] where VMD seems to be badly violated.
Acknowledgment This work was supported in part by the Israel Ministry of Absorption.
VI. APPENDIX
For convenience we quote here the axial and vector parts of the lagrangian1, explicitly. First the symmetric parts
are,
LA =W1(X)Tr(∆µ∆µ)+W4(X)Tr(∆µ)Tr(∆µ)+
W5(X)Tr(∆µ)Dµϑ+W6(X)DµϑDµϑ , (55)
LV =W˜1(X)Tr([Γµ gVµ][Γµ gVµ])+
− −
W˜4(X)Tr(Γµ gVµ)Tr(Γµ gVµ) , (56)
− −
where
D ϑ=∂ ϑ+Tr(r l ) , (57)
µ µ µ µ
−
V =∂ V ∂ V ig[V ,V ] . (58)
µν µ ν ν µ µ ν
− −
The symmetry breaking companions L¯ are constructed in two alternative ways. The first (referred to as
A
AlternativeI)breaksSU(3) symmetry,andcorrespondstoaquadraticformofthe Goldstonemesonkineticenergy
F
term,
L¯A =W1(X) cATr(B∆¯µ∆¯µ)+dATr(B∆¯µB∆¯µ) +
W4(X)dATr(B(cid:0)∆¯µ)Tr(B∆¯µ)+W5(X)cATr(B∆¯µ(cid:1))Dµϑ . (59)
where the axial vector covariant,
i 1
∆¯µ = 2 ξ8†,∂µξ8 + 2 ξ8†rµξ8−ξ8lµξ8† , (60)
n o (cid:16) (cid:17)
involves the pseudoscalar octet field matrix only. The second alternative (Alternative II) breaks U(3) symmetry
F
and corresponds to bilinear kinetic energy term. Namely,
L¯A =W1(X)(cATr(B∆µ∆µ)+dATr(B∆µB∆µ))
+W4(X)(cATr(B∆µ)Tr(∆µ)+dATr(B∆µ)Tr(B∆µ))+
W5(X)cATr(B∆µ)Dµϑ , (61)
with ∆ involves now the nonet pseudoscalar field matrix. Similarly the asymmetric companion of L is,
µ V
10