1001.0045.txt

arXiv:1001.0045v1  [hep-ph]  31 Dec 2009EPJ manuscript No.
(will be inserted by the editor)
Axial Anomaly and Mixing Parameters of
Pseudoscalar Mesons
Yaroslav N. Klopot1,a, Armen G. Oganesian2,b, and Oleg V. Teryaev1,c
1Joint Institute for Nuclear Research, Bogoliubov Laborato ry of Theoretical Physics, Dubna 141980,
Russia
2Institute of Theoretical and Experimental Physics, B.Cher emushkinskaya 25, Moscow 117218, Russia
Abstract. In this work the analysis of mixing parameters of the system i nvolving
η,η′mesons and some third massive state Gis carried out. We use the generalized
mixing scheme with three angles. The framework of the disper sive approach to
Abelian axial anomaly of isoscalar non-singlet current and the analysis of exper-
imental data of charmonium radiative decays ratio allow us t o get a number of
quite strict constraints for the mixing parameters. The ana lysis shows that the
equal values of axial current coupling constants f8andf0are preferable which
may be considered as a manifestation of SU(3) and chiral symmetry.
1 Introduction
This work is developing the approach of the papers [1,2] and is devot ed to the significant
problem of mixing of pseudoscalar mesons. It is especially important w ith a number of current
and planned experiments.
The problem of η-η′mixing has been studied for many years. The usual approach with on e
mixingangledominatedfordecades,butintherecentyearsthemor eelaboratedschemesappear
to be unavoidable [3–8]. In particular, the theoretical ground of th is was based on the recent
progress in the ChPT [9–11]. On the other hand, it was shown, that t he current experimental
data cannot satisfactory describe the whole set of experiments w ithin the one-angle mixing
scheme.
The mixing schemes are usually enunciated either in terms of SU(3) or quark basis. In
our paper [2] we construct and use the generalization of SU(3) basis similar to the mixing of
massive neutrinos. This is because we use the dispersive approach t o axial anomaly ( [12], [13]
for a review) to find some model-independent and precise restrictio n on the mixing parameters.
It was shown that any scheme with more than one angle unavoidably d emands an additional
admixture of higher mass state. If we restrict ourselves to only on e additional state G(denoted
as a glueball without really specifying its nature) then the general m ixing scheme can be
described in terms of 3 angles. In particular cases the number of an gles can be reduced to two.
In the paper [2] the analysis of different conventional (and most ph ysically interesting)
particular cases was performed (including two-angle mixing schemes ) basing on the dispersive
representation of axial anomaly from one side and charmonium deca ys ratio from the other
side.
ae-mail:[email protected]
be-mail:[email protected]
ce-mail:[email protected] Will be inserted by the editor
The main conclusion of the paper [2] is that in all considered cases the only reasonable
solutions appear at f8=f0≃fπ. The main aim of this work is to check whether this relation
remains valid in the most general case with some specific constraints imposed.
This paper is organized as follows. In the Sec. 2 we introduce our not ation and the general
approach to the mixing. In Sec. 3 we derive the basic equations relyin g on the dispersive
approach to Abelian axial anomaly of isoscalar non-singlet current J8
µ5and the charmonium
radiativedecayratio RJ/Ψ, while in Sec. 4 we performthe numerical analysisofthese equations .
Finally, in Sec. 5 we present the conclusion.
2 Mixing scheme
We start with a ( N-component) vector of physical pseudoscalar fields consisting of the fields of
the lightest pseudoscalar mesons and other fields:
/tildewideΦ≡
π0
η
η′
G
...
. (1)
We are not able to specify the physical nature of the other compon ents with higher masses,
the lowest of which Gcan be either a glueball or some excited state1. Let us also introduce,
following [16,17], a set of SU(3) fields ϕ3,ϕ8,ϕ0(Φ1,Φ2,Φ3) and complement them with other
(sterile) fields gi(Φi,i= 4..N)
Φ=
ϕ3
ϕ8
ϕ0
g
...
. (2)
The three upper fields ϕ3,ϕ8,ϕ0are the only ones which define the generalized PCAC
relationforaxialcurrent Ja
µ5=qγµγ5λa
√
2q(nosummationover acontraryto jandkisassumed):
∂µJa
µ5=faδ∆L
δΦa=FajMjkΦk, a= 3,8,0, j,k= 1..N, (3)
where∆Lis the mass term in the effective Lagrangian with a non-diagonal mass matrixM(as
fieldsΦkare not orthogonal to each other):
∆L=1
2ΦTMΦ, (4)
andFis a matrix of decay constants2:
F≡
f30 0 0...0
0f80 0...0
0 0f00...0
. (5)
In order to proceed from initial SU(3) fields Φto physical mass fields /tildewideΦthe unitary (real,
as the CP-violating effects are negligible) matrix Uis introduced
1Note, that the mixing with the excited states is usually(e.g . [14,15]) supposed to be suppressed.
2Note, that matrix of decay constants Fis non-square expressing the fact that generally the number
ofSU(3) currents is less then the number of all possible states in volved in mixing. The similar situation
takes place (see e.g. [18]) in one of the extensions of the Sta ndard Model – neutrino mixing scenario
involving sterile neutrinos.Will be inserted by the editor 3
/tildewideΦ=UΦ (6)
that diagonalizes the mass matrix
UMUT=/tildewiderM≡diag(m2
π0,m2
η,m2
η′,m2
G,...), (7)
wheremπ,mη,mη′andmGare the masses of the π,η,η′mesons and glueball state G,
respectively.
Simple transformations of Eq.(3) read:
∂µJµ5=FUT/tildewiderM/tildewideΦ (8)
This formula is close to those obtained in [16,17] (in the limit of small mix ing). When the decay
constants are equal, it is reduced to formula (3.40) in [19].
The matrix elements of ∂µJµ5between vacuum state and physical states |/tildewiderΦk∝angbracketright
∝angbracketleft0|∂µJa
µ5|/tildewiderΦk∝angbracketright=Fa
i(UT/tildewiderM)i
k (9)
can be compared to the standard definition of the ”physical” couplin g constants of axial cur-
rents:
∝angbracketleft0|Ja
µ5|/tildewiderΦk∝angbracketright=ifa
kqµ. (10)
From (9) and (10) follows the relation
fa
k=Fa
i(UT)i
k=fa(UT)a
k. (11)
This expression (recall, that there is no summation over a) clearly shows that fa
kare obtained
bymultiplicationofeachlineof UTbyrespectivecoupling faandformanon-diagonal(contrary
toF) matrix.
Taking into account the well-known smallness of π0mixing with the η,η′sector [16,17,20]
and neglecting all higher contributions we restrict our consideratio n to three physical states
η,η′,Gand two currents J8
µ5,J0
µ5. Then the divergencies of the axial currents (recall, that Gis
a first mass state heavier than η′):
/parenleftbigg∂µJ8
µ5
∂µJ0
µ5/parenrightbigg
=/parenleftbigg
f80 0
0f00/parenrightbigg
UT
m2
η0 0
0m2
η′0
0 0m2
G

η
η′
G
. (12)
Exploring the mentioned similarity of the meson and lepton mixing, we us e the Euler
parametrization for the mixing matrix U(we use notation ci≡cosθi,si≡sinθi):
U=
c8c3−c0s3s8−c3s8−c8c0s3s3s0
s3c8+c3c0s8−s3s8+c3c8c0−c3s0
s8s0 c8s0 c0
. (13)
In the following consideration we will need the divergency of the octe t current ∂µJ8
µ5, so let
us write it out explicitly:
∂µJ8
µ5=f8(m2
ηη(c8c3−c0s3s8)+m2
η′η′(s3c8+c3c0s8)+m2
GG(s8s0)). (14)
As soon as in the chiral limit J8
µ5should be conserved, from Eq.(14) follows that coefficients
of the terms m2
η′,m2
Gmust decrease at least as ( mη/mη′,G)2. More specifically, we expect the
following limits for the terms of Eq.(14):
|s8s0|
|s3c8+c3c0s8|/lessorsimilar/parenleftbiggmη
mG/parenrightbigg2
. (15)4 Will be inserted by the editor
3 Abelian axial anomaly and charmonium decays ratio
In our paper the dispersive form of the anomaly sum rule will be exten sively used, so we remind
briefly the main points of this approach (see e.g. review [13] for det ails).
Consider a matrix element of a transition of the axial current to two photons with momenta
pandp′
Tµαβ(p,p′) =∝angbracketleftp,p′|Jµ5|0∝angbracketright. (16)
The general form of Tµαβfor a case p2=p′2can be represented in terms of structure
functions (form factors):
Tµαβ(p,p′) =F1(q2)qµǫαβρσpρp′
σ+
1
2F2(q2)[pα
p2ǫµβρσpρp′
σ−p′
β
p2ǫµαρσpρp′
σ−ǫµαβσ(p−p′)σ],(17)
whereq=p+p′. The functions F1(q2),F2(q2) can be described by dispersion relations with
no subtractions and anomaly condition in QCD results in the sum rule:
∞/integraldisplay
0Im F1(q2)dq2= 2αNc/summationdisplay
e2
q, (18)
whereeqare quark electric charges and Ncis the number of colors. This sum rule [21] was
developed by Jiˇ r´ ı Hoˇ rejˇ s´ ı [22], and later generalized [23]. Not ice that in QCD this equation
does not have any perturbative corrections [24], and it is expected that it does not have any
non-perturbative corrections as well due to the ’t Hooft’s consist ency principle [25]. It will be
important for us that as q2→ ∞the function ImF1(q2) decreases as 1 /q4(see discussion
in Ref. [2]). Note also that the relation (18) contains only mass-indep endent terms, which is
especially important for the 8th component of the axial current J8
µ5containing strange quarks:
J8
µ5=1√
6(¯uγµγ5u+¯dγµγ5d−2¯sγµγ5s). (19)
The general sum rule (18) takes the form:
∞/integraldisplay
0Im F1(q2)dq2=2√
6α(e2
u+e2
d−2e2
s)Nc=/radicalbigg
2
3α , (20)
whereeu= 2/3,ed=es=−1/3,Nc= 3.
In order to separate the form factor F1(q2), multiply Tµαβ(p,p′) byqµ/q2. Then, taking the
imaginary part of F1(q2), using the expression for ∂µJ8
µ5from Eq.(12) and unitarity we get:
ImF1(q2) =Im qµ1
q2∝angbracketleft2γ|J(8)
µ5|0∝angbracketright=
−f8
q2∝angbracketleft2γ|[m2
ηη(c8c3−c0s3s8)+m2
η′η′(s3c8+c3c0s8)+m2
GGs8s0]|0∝angbracketright=
πf8[Aηδ(q2−m2
η)(c8c3−c0s3s8)+Aη′δ(q2−m2
η′)(s3c8+c3c0s8)+AGδ(q2−m2
G)(s8s0)].
(21)
If we employ the sum rule (20), we obtain a simple equation:
(c8c3−c0s3s8)+β(s3c8+c3c0s8)+γ(s8s0) =ξ, (22)
whereWill be inserted by the editor 5
β≡Aη′
Aη=/radicaligg
Γη′→2γ
Γη→2γm3η
m3
η′, γ≡AG
Aη=/radicaligg
ΓG→2γ
Γη→2γm3η
m3
G, (23)
ξ≡/radicaligg
α2m3η
96π3Γη→2γ1
f2
8, Γη→2γ=m3
η
64πA2
η. (24)
Note that if we include higher resonancesin this equation, they will be suppressed as 1 /m2
res
by virtue of the mentioned above asymptotic behavior of F1(q2)∝1/q4. For the last two terms
in (22) we can specify this constraint as follows:
|s8s0|
|s3c8+c3c0s8|/lessorsimilarβ
γ/parenleftbiggmη′
mG/parenrightbigg2
. (25)
As an additional experimental constraint we use, following [26,27], t he data of the decay
ratioRJ/Ψ= (Γ(J/Ψ)→η′γ)/(Γ(J/Ψ)→ηγ).
As it was pointed out in [28], the radiative decays J/Ψ→η(η′)γare dominated by non-
perturbative gluonic matrix elements, and the ratio of the decay ra tesRJ/Ψ= (Γ(J/Ψ)→
η′γ)/(Γ(J/Ψ)→ηγ) can be expressed as follows:
RJ/Ψ=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∝angbracketleft0|G/tildewideG|η′∝angbracketright
∝angbracketleft0|G/tildewideG|η∝angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenleftbiggpη′
pη/parenrightbigg3
, (26)
wherepη(η′)=MJ/Ψ(1−m2
η(η′)/M2
J/Ψ)/2. The advantage of this ratio is expected smallness of
perturbative and non-perturbative corrections.
The divergencies of singlet and octet components of the axial curr ent in terms of quark
fields can be written as:
∂µJ8
µ5=1√
6(muuγ5u+mddγ5d−2mssγ5s), (27)
∂µJ0
µ5=1√
3(muuγ5u+mddγ5d+mssγ5s)+1
2√
33αs
4πG/tildewideG. (28)
Following [26], neglect the contribution of u- and d- quark masses, th en the matrix elements
of the anomaly term between the vacuum and η,η′states are:
√
3αs
8π∝angbracketleft0|G/tildewideG|η∝angbracketright=∝angbracketleft0|∂µJ(0)
µ5|η∝angbracketright+1√
2∝angbracketleft0|∂µJ(8)
µ5|η∝angbracketright, (29)
√
3αs
8π∝angbracketleft0|G/tildewideG|η′∝angbracketright=∝angbracketleft0|∂µJ(0)
µ5|η′∝angbracketright+1√
2∝angbracketleft0|∂µJ(8)
µ5|η′∝angbracketright. (30)
Using Eq. (12), (26), (29), (30) we deduce:
RJ/Ψ=/bracketleftiggf0(−s3s8+c3c8c0)+1√
2f8(s3c8+c3c0s8)
f0(−c3s8−c8c0s3)+1√
2f8(c8c3−c0s3s8)/bracketrightigg2
×/parenleftbiggmη′
mη/parenrightbigg4/parenleftbiggpη′
pη/parenrightbigg3
.(31)
4 Analysis
For further analysis it is convenient to rewrite the equations (22), (31) in terms of angles
θ1≡θ8+θ3,θ8andθ0:
1
2(c1+c2−c0(c2−c1))+β
2(s1−s2+c0(s1+s2))+γ(s8s0) =ξ. (32)6 Will be inserted by the editor
RJ/Ψ=/bracketleftiggf0(c1−c2+c0(c1+c2))+1√
2f8(s1−s2+c0(s1+s2))
f0(−s1−s2−c0(s1−s2))+1√
2f8(c1+c2−c0(c2−c1))/bracketrightigg2/parenleftbiggmη′
mη/parenrightbigg4/parenleftbiggpη′
pη/parenrightbigg3
,(33)
whereθ2≡2θ8−θ1.
The angles θ1,θ8,θ0have the explicit physical meaning. From the definition (13) of the
mixing matrix Uone can see that the angle θ1describes the overlap in the η−η′system with
an accuracy ∼θ2
0/2 and coincides with their mixing angle as θ0→0. At the same time θ0is
responsible for the glueball admixture to η−η′system, and s8s0describes the contribution of
the glueball state Gto the octet component of axial current ∂J8
µ5only.
In the further analysis we will use the following assumptions:
I) As we discussed in Sec. 2, the last term in (14) should be suppress ed as (mη/mG)2. So
we impose the following constraint:
|s8s0|
|s3c8+c3c0s8|/lessorsimilar/parenleftbiggmη
mG/parenrightbigg2
. (34)
II) In sec 3 we found another constraint, which follows from the as ymptotic behavior of
ImF1(see 25):
|s8s0|
|s3c8+c3c0s8|/lessorsimilarβ
γ/parenleftbiggmη′
mG/parenrightbigg2
. (35)
III) In our numerical analysis we suppose that γcannot exceed 1 (i.e. ΓG→2γ/m3
G/lessorsimilar
Γη→2γ/m3
η). This restriction corresponds to the assumption that 2-photon decay widths of
pseudoscalar mesons grow like the third power of their masses, or in other words, the glueball
coupling to quarks is of the same order as for the meson octet stat es.
IV) We accept that the decay constants obey the relation f8/greaterorsimilarf0/greaterorsimilarfπ(for various kinds
of justification see, e.g., [3,9]).
For the purposes of numerical analysis, the values of RJ/Ψ(RJ/Ψ= 4.8±0.6), masses
and two-photon decay widths of η,η′mesons are taken from PDG [29]. Using the values
mη,mη′,Γη→2γ,Γη′→2γ, we see that the relation for the constraint (34) is more strict tha n the
constraint (35). Supposing the minimal mass of the glueball to be of ordermG≃3mη≃1.5
GeV, we get the estimation:
|s8s0|/|s3c8+c3c0s8|/lessorsimilar0.1. (36)
On Fig. 1 the plots of the equations (32) and (33) in the parameter s pace (θ8,θ1) are shown
for different values of decay constants f8,f0and mixing angle θ0. The dashed curves denote
experimental uncertainties. The intersection points of the curve s represent the solutions of both
equations (32),(33). The filled area indicates the region, where the constraint (36) is valid. The
plotted range of angle θ1is limited to the physically interesting region, where the solution for
relatively small angles θ0exists. Let us note for completeness, that there is another solut ion for
θ1∼90◦,θ0/greaterorsimilar50◦which does not seem to have a physical sense.
The numerical analysis shows, that the solution of the equations (3 2) and (33) satisfying
the mentioned above constraint is possible only for rather small mixin g angleθ0and for decay
constants f8,f0closetoeachotherandcloseto fπ:forf8/fπ=f0/fπ= 1.0the possiblerangeof
mixing angle θ0isθ0= (0÷25)◦(see Fig. 1(a)-1(c) for demonstration), for f8/fπ=f0/fπ= 1.1
the possible range of mixing angle θ0isθ0= (0÷20)◦.
There is no solutions for decay constant values f8/fπ= 1.1,f0/fπ= 1.0 for any θ0(see Fig.
1(d)-1(f) for demonstration), and for any f0/lessorsimilarf8in case of f8/fπ≥1.2. The obtained results
are quite stable: even if we relax the constraint (36) making its r.h.s. several times larger, all
the conclusions are preserved.
Note finally, that this result is in contradiction with the prediction for the decay constant
f8/fπ= 1.34 [10] obtained in the Large NcChPT.Will be inserted by the editor 7
/Minus150/Minus100/Minus50050100150/Minus40/Minus30/Minus20/Minus100
Θ8/LBracket1degrees/RBracket1Θ1/LBracket1degrees/RBracket1
(a) (f8,f0) = (1.0,1.0)fπ,θ0=
0◦/Minus150/Minus100/Minus50050100150/Minus40/Minus30/Minus20/Minus100
Θ8/LBracket1degrees/RBracket1Θ1/LBracket1degrees/RBracket1
(b) (f8,f0) = (1.0,1.0)fπ,θ0=
5◦/Minus150/Minus100/Minus50050100150/Minus40/Minus30/Minus20/Minus100
Θ8/LBracket1degrees/RBracket1Θ1/LBracket1degrees/RBracket1
(c) (f8,f0) = (1.0,1.0)fπ,θ0=
30◦
/Minus150/Minus100/Minus50050100150/Minus40/Minus30/Minus20/Minus100
Θ8/LBracket1degrees/RBracket1Θ1/LBracket1degrees/RBracket1
(d) (f8,f0) = (1.1,1.0)fπ,θ0=
0◦/Minus150/Minus100/Minus50050100150/Minus40/Minus30/Minus20/Minus100
Θ8/LBracket1degrees/RBracket1Θ1/LBracket1degrees/RBracket1
(e) (f8,f0) = (1.1,1.0)fπ,θ0=
5◦/Minus150/Minus100/Minus50050100150/Minus40/Minus30/Minus20/Minus100
Θ8/LBracket1degrees/RBracket1Θ1/LBracket1degrees/RBracket1
(f) (f8,f0) = (1.1,1.0)fπ,θ0=
30◦
Fig. 1.The solutions of the Eq. (32) (thin curves, blue online) and ( 33)(thick curves, red online) with
the experimental uncertainties (dashed curves) for differe nt values of the parameters f8,f0andθ0. The
shaded area indicates the region, where the relation (36) is valid.
5 Conclusion
In this paper we studied what can be learnt about the mixing in the pse udoscalar sector from
the dispersive approach to axial anomaly.
Our analysis shows that the equal values of axial current coupling c onstants f8andf0are
favorablewhich may be considered as a manifestation of SU(3) and chiral symmetry. Moreover,
with a less definiteness the relation fπ≈f8≈f0[2] is also supported.
Theanalysisdemands f8<1.2fπwhich deviatesat 10%levelfrom theresultsofcalculations
within the chiral perturbation theory ( f8= 1.34fπ) [10].
Thevalueofthe mixingangle θ0,whichisresponsibleforthe glueballadmixturetothe η−η′,
is limited to θ0<25◦for (f8,f0) = (1.0,1.0)fπand toθ0<20◦for (f8,f0) = (1.0,1.0)fπ.
The improvement of the experimental data of RJ/Ψcan significantly limit the constraints
for the parameters θ0,θ8andf8,f0.
We thank J. Hoˇ rejˇ s´ ı, B. L. Ioffe and M. A. Ivanov for useful co mments and discussions.
Y. K. and O. T. gratefully acknowledge the organizers of the works hop for hospitality and
support. This work was supported in part by RFBR (Grants 09-02- 00732,09-02-01149),by the
funds from EC to the project ”Study of the Strong Interacting M atter” under contract N0.
R113-CT-2004-506078 and by CRDF Project RUP2-2961-MO-09.
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