On the eta-eta' complex in the SD-BS approach

Abstract

The bound-state Schwinger-Dyson and Bethe-Salpeter (SD-BS) approach is chirally well-behaved and provides a reliable treatment of the eta-eta' complex although a ladder approximation is employed. Allowing for the effects of the SU(3) flavor symmetry breaking in the quark-antiquark annihilation, leads to the improved eta-eta' mass matrix.

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arXiv:hep-ph/0012267v1 20 Dec 2000
On the η–η′complex in the SD–BS approach
D. Klabuˇ cara, D. Kekezb, M. D. Scadronc
aPhysics Department, Faculty of Science, University of Zagreb,
Bijeniˇ cka c. 32, Zagreb 10000, Croatia
bRudjer Boˇ skovi´ c Institute, P.O.B. 180, 10002 Zagreb, Croatia
cPhysics Department, University of Arizona, Tucson, AZ 85721, USA
Abstract
The bound-state Schwinger-Dyson and Bethe-Salpeter (SD–BS) approach is
chirally well-behaved and provides a reliable treatment of the η–η′complex
although a ladder approximation is employed. Allowing for the effects of the
SU(3) flavor symmetry breaking in the quark–antiquark annihilation, leads
to the improved η–η′mass matrix.
I. η–η′PHENOMENOLOGY AND GOLDSTONE STRUCTURE
The physical isoscalar pseudoscalars η and η′are usually given as
|η? = cosθ|η8? − sinθ|η0? ,|η′? = sinθ|η8? + cosθ|η0? ,(1)
i.e., as the orthogonal mixture of the respective octet and singlet isospin zero states, η8and
η0. In the flavor SU(3) quark model, they are defined through quark–antiquark (q¯ q) basis
states |f¯f? (f = u,d,s) as
1
√6(|u¯ u? + |d¯d? − 2|s¯ s?) ,
1
√3(|u¯ u? + |d¯d? + |s¯ s?) .
|η8? =
(2a)
|η0? =(2b)
The exact SU(3) flavor symmetry (mu = md = ms) is nevertheless badly broken. It is
an excellent approximation to assume the exact isospin symmetry (mu= md), and a good
approximation to take even the chiral symmetry limit for u and d-quark (where current
quark masses mu= md= 0), but for a realistic description, the strange quark mass msmust
be significantly heavier than muand md. The same holds for the constituent quark masses,
denoted by ˆ m for both u and d quarks since we rely on the isosymmetric limit, and by ˆ ms
for the s-quark. They are nonvanishing in the chiral limit (CL). In the strange sector, CL
is useful only qualitatively, as a theoretical limit. (CL would reduce ˆ msto ˆ m, on which CL
has almost negligible influence.)
Thus, with |u¯ u? and |d¯d? being practically chiral states as opposed to a significantly
heavier |s¯ s?, Eqs. (2) do not define the octet and singlet states of the exact SU(3) flavor
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symmetry, but the effective octet and singlet states. Hence, as in Ref. [1] for example, only in
the sense that the same q¯ q states |f¯f? (f = u,d,s) appear in both Eq. (2a) and Eq. (2b) do
these equations implicitly assume nonet symmetry (as pointed out by Gilman and Kauffman
[2], following Chanowitz, their Ref. [8]). However, in order to avoid the UA(1) problem, this
symmetry must ultimately be broken at least at the level of the masses. In particular, it
must be broken in such a way that η → η8becomes massless but η′→ η0remains massive
(as in Ref. [1]) when CL is taken for all three flavors, mu,md,ms→ 0. Nevertheless, the
CL-vanishing octet eta mass mη8is rather heavy for the realistically broken SU(3) flavor
symmetry; for the empirical pion and kaon masses mπand mK, the Gell-Mann-Okubo mass
formula m2
of η(547) and η′(958), the singlet η0mass mη0(nonvanishing even in CL) can be found from
the mass–matrix trace
π+3m2
η8= 4m2
Kyields mη8≈ 567 MeV. In that case, and for the empirical masses
m2
η8+ m2
η0= m2
η+ m2
η′ ≈ 1.22 GeV2, giving mη0≈ 947 MeV. (3)
Alternatively, one can work in a nonstrange (NS)–strange (S) basis:
|ηNS? =
1
√2(|u¯ u? + |d¯d?) =
?
1
√3|η8? +
1
√3|η0? .
?
2
3|η0? ,(4a)
|ηS? = |s¯ s? = −
2
3|η8? +
(4b)
In analogy with Eq. (3), in this basis one finds
m2
ηNS+ m2
ηS= m2
η+ m2
η′ ≈ 1.22 GeV2, (5)
whereas the NS–S mixing relations, diagonalizing the mass matrix, are
|η? = cosφP|ηNS? − sinφP|ηS? ,
The singlet-octet mixing angle θ, defined by Eqs. (1), is related to the NS–S mixing angle φ
above as [3] θ = φ − arctan√2 = φ − 54.74◦.
Although mathematically equivalent to the η8–η0basis, the NS–S mixing basis is more
suitable for most quark model considerations, being more natural in practice when the
symmetry between the NS and S sectors is broken as described in the preceding passage.
There is also another important reason to keep in mind the |ηNS?-|ηS? state mixing angle
φ. This is because it offers the quickest way to show the consistency of our procedures
and the corresponding results obtained using just one (θ or φ) state mixing angle, with the
two-mixing-angle scheme considered in Refs. [4–10], which is defined with respect to the
mixing of the decay constants. For clarification of the relationship with, and our results in
the two-mixing-angle scheme, we refer to Ref. [11], particularly to its Appendix. Here, we
simply note that our considerations will ultimately lead us to φ ≈ 42◦, practically the same
as the result of Refs. [6–8,10] and in agreement with data (e.g., see Table 2 of Feldmann’s
review [10]).
|η′? = sinφP|ηNS? + cosφP|ηS? . (6)
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FIGURES
P
f
f
–
f´
f´
–
P´
FIG. 1.
diagram. It shows the transition of the f¯f pseudoscalar P into the pseudoscalar P′having the
flavor content f′¯f′. The dashed lines and full circles depict the q¯ q bound-state pseudoscalars and
vertices, respectively.
Nonperturbative QCD quark annihilation illustrated by the two-gluon exchange
As for a theoretical determination of the η–η′mixing angle φ or θ, we follow the path of
Refs. [3]. The contribution of the gluon axial anomaly to the singlet η0mass is essentially just
parameterized and not really calculated, but some useful information can be obtained from
the isoscalar q¯ q annihilation graphs of which the “diamond” one in Fig. 1 is just the simplest
example. That is, we can take Fig. 1 in the nonperturbative sense, where the two-gluon
intermediate “states” represent any even number of gluons when forming a C+pseudoscalar
qq meson [12], and where quarks, gluons and vertices can be dressed nonperturbatively,
and possibly include gluon configurations such as instantons. Factorization of the quark
propagators in Fig. 1 characterized by the ratio X ≈ ˆ m/ˆ msleads to the η–η′mass matrix
in the NS–S basis [3]
√2βX
√2βX2m2
?m2
π+ 2β
K− m2
π+ βX2
?
−→
φ
?m2
0
η0
m2
η′
?
,(7)
where β denotes the total annihilation strength of the pseudoscalar q¯ q for the light flavors
f = u,d, whereas it is assumed attenuated by a factor X when a s¯ s pseudoscalar appears.
(The mass matrix in the η8-η0basis reveals that in the X → 1 limit, the CL-nonvanishing
singlet η0mass is given by 3β.) The two parameters on the left-hand-side (LHS) of (7), β
and X, are determined by the two diagonalized η and η′masses on the RHS of (7). The
trace and determinant of the matrices in (7) then fix β and X to be [3]
β =(m2
η′ − m2
4(m2
π)(m2
K− m2
η− m2
π)
π)
≈ 0.28 GeV2,X ≈ 0.78 ,(8)
with the latter value suggesting a S/NS constituent quark mass ratio X−1∼ ˆ ms/ˆ m ∼ 1.3 ,
near the values in Refs. [12–16], ˆ ms/ˆ m ≈ 1.45.
This fitted nonperturbative scale of β in (8) depends only on the gross features of QCD.
If instead one treats the QCD graph of Fig. 1 in the perturbative sense of literally two gluons
exchanged, then one obtains [17] only β2g∼ 0.09 GeV2, which is about 1/3 of the needed
scale of β found in (8). (This indicates that just the perturbative “diamond” graph can
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hardly represent even the roughest approximation to the effect of the gluon axial anomaly
operator ǫαβµνGa
(8) can be converted to the NS–S η–η′mixing angle φ in (6) from the alternative mixing
relation tan2φ = 2√2βX(m2
αβGa
µν.) The above fitted quark annihilation (nonperturbative) scale β in
ηS− m2
?(m2
(2m2
ηNS)−1≈ 9.02 to [3]
φ = arctan
η′ − 2m2
K− m2
K+ m2
π− m2
π)(m2
η)(m2
η− m2
η′ − m2
π)
π)
?1/2
≈ 41.84◦. (9)
This kinematical QCD mixing angle (9) or θ = φ − 54.74◦≈ −12.9◦has dynamical analogs
[1,11], namely the coupled SD-BS approach discussed below, in Sec. II. Since this predicted
η–η′mixing angle in (9) is compatible with the values repeatedly extracted in various empir-
ical ways [13,14], and more recently from the FKS scheme and theory [6–10], we confidently
use the value (9) in the mixing angle relations (6) to infer the nonstrange and strange η
masses,
m2
m2
ηNS= cos2φ m2
ηS= sin2φ m2
η+ sin2φ m2
η+ cos2φ m2
η′ ≈ (757.9 MeV)2
η′ ≈ (801.5 MeV)2.
(10a)
(10b)
Thus it is clear that the true physical masses η(547) and η′(958) are respectively much
closer to the Nambu-Goldstone (NG) octet η8(567) and the non-NG singlet η0(947) configu-
rations than to the nonstrange ηNS(758) and strange ηS(801) configurations inferred in Eqs.
(10). However, the mean η–η′mass (548 + 958)/2 ≈ 753 MeV is quite near the nonstrange
ηNS(758). But since η8(567) appears far from the NG massless limit we must ask: how close
is η0(947) to the chiral-limiting nonvanishing singlet η mass?
To answer this latter question, return to Fig. 1 and the quark annihilation strength
β ≈ 0.28 GeV2in Eq. (8). These qq states presumably hadronize into the UA(1) singlet
state (2b), for effective squared mass in the SU(3) limit with β remaining unchanged [17]:
m2
η0= 3β ≈ (917 MeV)2.(11)
This latter CL η0mass in (11) is only 3% shy of the exact chiral-broken η0(947) mass found
in Eq. (3). (Such a 3% CL reduction also holds for the pion decay constant fπ≈ 93 MeV
→ 90 MeV [18] and for f+(0) = 1 → 0.97 [19], the K–π Kl3form factor.)
Our η–η′mixing analysis on the basis of phenomenological mass inputs thus tells us that
the physical η(547) is 97% of the chiral-broken NG boson η8(567). Also the mixing-induced
CL singlet mass of 917 MeV in (11) is 97% of the chiral-broken singlet η0(947) in (3), which
in turn is 99% of the physical η′mass η′(958). This can be viewed as the phenomenological
resolution of the UA(1) problem of the masses and (quasi-)Goldstone boson structure of the
observed η(547) and η′(958) mesons. Or rather, from a more microscopic standpoint, the
above represents phenomenological constraints that microscopic, more or less QCD–based
studies of the η–η′complex must respect.
II. BOUND-STATE SD–BS APPROACH TO η–η′
The coupled Schwinger-Dyson (SD) and Bethe-Salpeter (BS) approach [20] can be formu-
lated so that it has strong and clear connections with QCD, the fundamental theory of strong
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interactions. In this approach, by solving the SD equation for dressed quark propagators of
various flavors, one explicitly constructs constituent quarks. They in turn build q¯ q meson
bound states which are solutions of the BS equation employing the dressed quark propagator
obtained as the solution of the SD equation. If the SD and BS equations are so coupled in
a consistent approximation, the light pseudoscalar mesons are simultaneously the q¯ q bound
states and the (quasi) Goldstone bosons of dynamical chiral symmetry breaking (DχSB).
The resulting relativistically covariant bound-state model (such as the variant of Ref. [21]) is
consistent with current algebra because it incorporates the correct chiral symmetry behavior
thanks to DχSB obtained in an, essentially, Nambu–Jona-Lasinio fashion, but the SD–BS
model interaction is less schematic. In Refs. [1,21–25] for example, it is combined nonpertur-
bative and perturbative gluon exchange; the effective propagator function is the sum of the
known perturbative QCD contribution and the modeled nonperturbative component. For
details, we refer to Refs. [1,21–24], while here we just note that the momentum-dependent
dynamically generated quark mass functions Mf(q2) (i.e., the quark propagator SD solu-
tions for quark flavors f) illustrate well how the coupled SD-BS approach provides a modern
constituent model which is consistent with perturbative and nonperturbative QCD. For ex-
ample, the perturbative QCD part of the gluon propagator leads to the deep Euclidean
behaviors of quark propagators (for all flavors) consistent with the asymptotic freedom of
QCD [23]. However, what is important in the present paper, is the behavior of the mass
functions Mf(q2) for low momenta [q2= 0 to −q2≈ (400MeV)2], where Mf(q2) (due to
DχSB) has values consistent with typical values of the constituent mass parameter in con-
stituent quark models. For the (isosymmetric) u- and d-quarks, our concrete model choice
[21] gives us Mu,d(0) = 356 MeV in the chiral limit (i.e., with vanishing ? mu,d, the explicit
in vanishing pion mass eigenvalue, mπ= 0, in the BS equation), and Mu,d(0) = 375 MeV
[just 5% above Mu,d(0) in the chiral limit] with the bare mass ? mu,d= 3.1 MeV, leading to
The simple-minded constituent masses in both NS and S sectors, ˆ m and ˆ msemployed in
Sec. I, have thus close analogues in the coupled SD–BS approach which explicitly incor-
porates some crucial features of QCD, notably DχSB. Thanks to DχSB, this dynamical,
bound-state approach successfully incorporates the partially Goldstone boson structure of
the mixed η(547) and η′(958) mesons [1].
Before addressing its mass matrix, let us briefly recall what the SD–BS approach re-
vealed [1,11] about the mixing angle inferred from η,η′→ γγ decays. The SD–BS approach
incorporates the correct chiral symmetry behavior thanks to DχSB and is consistent with
current algebra. Therefore, and this gives particular weight to the constraints placed on the
mixing angle θ by the SD-BS results on γγ decays of pseudoscalars, this approach reproduces
(when care is taken to preserve the vector Ward-Takahashi identity of QED) analytically
and exactly the CL pseudoscalar → γγ decay amplitudes (e.g., π0→ γγ), which are fixed by
the Abelian axial anomaly. (Note that they are otherwise notoriously difficult to reproduce
in bound-state approaches, as discussed in Ref. [23].)
General and robust considerations in this chirally well-behaved approach showed [1]
that, unlike the pion case, η8,η0→ γγ (and therefore also their mixtures η,η′→ γγ) decay
amplitudes cannot be given through their respective axial-current decay constants fη8,fη0,
and also gave strong bounds on these amplitudes with respect to the pion decay constant
chiral symmetry breaking bare mass term in the quark propagator SD equation, resulting
a realistically light pion, mπ= 140.4 MeV. Similarly, for the s quark, Ms(0) = 610 MeV.
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fπ(i.e., w. r. to the π0→ γγ amplitude). All this says that in models relying on quark
degrees of freedom, reasonably accurate reproduction of the empirical η,η′→ γγ widths is
possible only for θ-values less negative than −15◦. For the concrete [22,21] model adopted
in Ref. [1], our calculated η,η′→ γγ widths fit the data best for θ = −12.0◦.
For the very predictive SD-BS approach to be consistent, the above mixing angle ex-
tracted from η,η′→ γγ widths, should be close to the angle θ predicted by diagonalizing
the η–η′mass matrix. In this section, it is given in the quark f¯f basis:
ˆ M2= diag(M2
u¯ u,M2
d¯d,M2
s¯ s) + β



1 1 1
1 1 1
1 1 1

 .

(12)
As in Sec. I, 3β (called λη in Ref. [1]) is the contribution of the gluon axial anomaly to
m2
BS equations for q¯ q pseudoscalars with the flavor content f¯f′(f,f′= u,d,s). However,
since Ref. [1] had to employ a rainbow-ladder approximation (albeit the improved one of
Ref. [21]), it could not calculate the gluon axial anomaly contribution 3β. It could only
avoid the UA(1)-problem in the η–η′complex by parameterizing 3β, namely that part of
the η0mass squared which remains nonvanishing in the CL. Because of the rainbow-ladder
approximation (which does not contain even the simplest annihilation graph – Fig. 1), the q¯ q
pseudoscalar masses Mf¯f′ do not contain any contribution from 3β, unlike the nonstrange
and strange η masses mηNS[in Eq. (10a)] and mηS[in Eq. (10b)], which do, and which
must not be confused with Mu¯ u = Md¯dand Ms¯ s. Since the flavor singlet gluon anomaly
contribution 3β does not influence the masses mπand mKof the non-singlet pion and kaon,
the realistic rainbow-ladder modeling aims directly at reproducing the empirical values of
these masses: Mu¯ u= Md¯d= mπand Ms¯d= mK. In contrast, the masses of the physical
etas, mηand mη′, must be obtained by diagonalizing the η8-η0sub-matrix containing both
Mf¯fand the gluon anomaly contribution to m2
Since the gluon anomaly contribution 3β vanishes in the large Nclimit as 1/Nc, while
all Mf¯f′ vanish in CL, our q¯ q bound-state pseudoscalar mesons behave in the Nc→ ∞ and
chiral limits in agreement with QCD and χPT (e.g., see [26]): as the strict CL is approached
for all three flavors, the SU(3) octet pseudoscalars including η become massless Goldstone
bosons, whereas the chiral-limit-nonvanishing η′-mass 3β is of order 1/Ncsince it is purely
due to the gluon anomaly. If one lets 3β → 0 (as the gluon anomaly contribution behaves
for Nc → ∞), then for any quark masses and resulting Mf¯fmasses, the “ideal” mixing
(θ = −54.74◦) takes place so that η consists of u,d quarks only and becomes degenerate
with π, whereas η′is the pure s¯ s pseudoscalar bound state with the mass Ms¯ s.
In Ref. [1], numerical calculations of the mass matrix were performed for the realistic
chiral and SU(3) symmetry breaking, with the finite quark masses (and thus also the finite
BS q¯ q bound-state pseudoscalar masses Mf¯f) fixed by the fit [21] to static properties of many
mesons but excluding the η–η′complex. The mixing angle which diagonalizes the η8-η0mass
matrix thus depended in Ref. [1] only on the value of the additionally introduced “gluon
anomaly parameter” 3β. Its preferred value turned out to be 3β = 1.165 GeV2=(1079
MeV)2, leading to the mixing angle θ = −12.7◦[compatible with φ = 41.84◦in Eq. (9)]
and acceptable η → γγ and η′→ γγ decay amplitudes. Also, the η mass was then fitted
to its experimental value, but such a high value of 3β inevitably resulted in a too high η′
η0, the squared mass of η0. We denote by Mf¯f′ the masses obtained as eigenvalues of the
η0.
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mass, above 1 GeV. (Conversely, lowering 3β aimed to reduce mη′, would push θ close to
−20◦, making predictions for η,η′→ γγ intolerably bad.) However, unlike Eq. (7) in the
present paper, it should be noted that Ref. [1] did not introduce into the mass matrix the
“strangeness attenuation parameter” X which should suppress the nonperturbative quark
f¯f → f′¯f′annihilation amplitude (illustrated by the “diamond” graph in Fig. 1) when
f or f′are strange. Ref. [11] concluded that it was precisely the lack of the strangeness
attenuation factor X that prevented Ref. [1] from satisfactorily reproducing the η′mass
when it successfully did so with the η mass and γγ widths.
One can expect that the influence of this suppression should be substantial, since X ≈
ˆ m/ˆ msshould be a reasonable estimate of it, and this nonstrange-to-strange constituent mass
ratio in the considered variant of the SD-BS approach [1] is not far from X in Eq. (8) and
from the mass ratios in Refs. [12,15,16], and is even closer to the mass ratios in the Refs. [14].
Namely, two of us found [1] it to be around Mu(0)/Ms(0) = 0.615 if the constituent mass
was defined at the vanishing argument q2of the momentum-dependent SD mass function
Mf(q2).
We therefore introduce the suppression parameter X the same way as in the NS–S mass
matrix (7), whereby the mass matrix in the f¯f basis becomes
ˆ M2= diag(M2
u¯ u,M2
d¯d,M2
s¯ s) + β



1
1
X X X2
1 X
1 X


 . (13)
The very accurate isospin symmetry makes the mixing of the isovector π0and the isoscalar
etas negligible for all our practical purposes. Going to a meson basis of π0and etas enables
us therefore to separate the π0and restrictˆ M2to the 2 × 2 subspace of the etas. In the
NS–S basis,
?m2
m2
ηNS
ηNSηSm2
m2
ηSηNS
ηS
?
=
?M2
u¯ u+ 2β
√2βX
√2βX
M2
s¯ s+ βX2
?
.(14)
To a very good approximation, Eq. (14) recovers Eq. (7). This is because not only mπ=
Mu¯ u= Md¯d, but also because M2
to the good chiral behavior of the masses Mf¯f′ calculated in SD-BS approach. (These M2
and the CL model values of fπ and quark condensate, satisfy Gell-Mann-Oakes-Renner
relation to first order in the explicit chiral symmetry breaking [22].) The SD-BS–predicted
octet (quasi-)Goldstone masses Mf¯f′ are known to be empirically successful in our concrete
model choice [21], but the question is whether the SD-BS approach can also give some
information on the X-parameter. If we treat both 3β and X as free parameters, we can
of course fit both the η mass and the η′mass to their experimental values. For the model
parameters as in Ref. [21] (for these parameters our independent calculation gives mπ =
Mu¯ u= 140.4 MeV and Ms¯ s= 721.4 MeV), this happens at 3β = 0.753 GeV2=(868 MeV)2
and X = 0.835. However, the mixing angle then comes out as θ = −17.9◦, which is too
negative to allow consistency of the empirically found two-photon decay amplitudes of η and
η′, with predictions of our SD-BS approach for the two-photon decay amplitudes of η8and
η0[1].
s¯ sdiffers from 2m2
K−m2
πonly by a couple of percent, thanks
f¯f′
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Therefore, and also to avoid introducing another free parameter in addition to 3β, we
take the path where the dynamical information from our SD-BS approach is used to estimate
X. Namely, our γγ decay amplitudes Tf¯fcan be taken as a serious guide for estimating the
X-parameter instead of allowing it to be free. We did point out in Sec. I that the attempted
treatment [17] of the gluon anomaly contribution through just the “diamond diagram” con-
tribution to 3β, indicated that just this partial contribution is quite insufficient. This limits
us to keeping 3β as a free parameter, but we can still suppose that this diagram can help
us get the prediction of the strange-nonstrange ratio of the complete pertinent amplitudes
f¯f → f′¯f′as follows. Our SD-BS modeling in Ref. [1] employs an infrared-enhanced gluon
propagator [21,23] weighting the integrand strongly for low gluon momenta squared. There-
fore, in analogy with Eq. (4.12) of Kogut and Susskind [27] (see also Refs. [28,29]), we can
approximate the Fig. 1 amplitudes f¯f → 2gluons → f′¯f′, i.e., the contribution of the
quark-gluon diamond graph to the element ff′of the 3 × 3 mass matrix, by the factorized
form
?Tf¯f(0,0)C?Tf′ ¯f′(0,0).(15)
In Eq. (15), the quantity C is given by the integral over two gluon propagators remaining
after factoring out?Tf¯f(0,0) and?Tf′ ¯f′(0,0), the respective amplitudes for the transition of
case into two gluons. The contribution of Fig. 1 is thereby expressed with the help of the
(reduced) amplitudes?Tf¯f(0,0) calculated in Ref. [1] for the transition of q¯ q pseudoscalars
“reduced” two-photon amplitudes obtained by removing the squared charge factors Q2
Tf¯f, the γγ amplitude of the pseudoscalar q¯ q bound state of the hidden flavor f¯f. Although
C is in principle computable, all this unfortunately does not amount to determining β,βX
and βX2in Eq. (13) since the higher (four-gluon, six-gluon, ... , etc.) contributions are
clearly lacking. We therefore must keep the total (light-)quark annihilation strength β as a
free parameter. However, if we assume that the suppression of the diagrams with the strange
quark in a loop is similar for all of them, Eq. (15) and the “diamond” diagram in Fig. 1 help
us to at least estimate the parameter X as X ≈?Ts¯ s(0,0)/?Tu¯ u(0,0). This is a natural way
(Recall that?Ts¯ s(0,0)/?Tu¯ u(0,0) ≈ ˆ m/ˆ msto a good approximation [11].)
model [21]. This value of X agrees well with the other way of estimating X, namely the
nonstrange-to-strange constituent mass ratio of Refs. [12,15,16]. With X = 0.663, requiring
that the 2 × 2 matrix trace, m2
limiting nonvanishing singlet mass squared to 3β = 0.832 GeV2=(912 MeV)2, just 0.5%
below Eq. (11), while mηNS= 757.87 MeV and mηS= 801.45 MeV, practically the same as
Eqs. (10). The resulting mixing angle and η, η′masses are
the q¯ q pseudoscalar bound state for the quark flavor f and f′into two vector bosons, in this
to two real photons (k2= k′2= 0), while in general?Tf¯f(k2,k′2) ≡ Tf¯f(k2,k′2)/Q2
fare the
ffrom
to build in the effects of the SU(3) flavor symmetry breaking in the q¯ q annihilation graphs.
We get X = 0.663 from the two-photon amplitudes we obtained in the chosen SD-BS
η+ m2
η′, be fitted to its empirical value, fixes the chiral-
φ = 41.3◦or θ = −13.4◦;
These results are for the original parameters of Ref. [1]. Reference [11] also varied the
parameters to check the sensitivity on SD-BS modeling, but the results changed little.
mη= 588MeV,mη′ = 933MeV.(16)
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The above results of the SD-BS approach [1] are very satisfactory since they agree very
well with what was found in Sec. I by different methods. They also agree with the UKQCD
lattice results [30] on η–η′mixing. Their calculated mixing parameter xsscorresponds to our
βX2, and their mixing parameters xnnand xns(n = u,d), corresponding respectively to our
β and βX, are aimed to obey xnn≈ 2xssand x2
GeV2, xnn= 0.292 GeV2and xns= 0.218 GeV2. This, together with their preferred input
values Mn¯ n = 0.137 GeV and Ms¯ s = 0.695 GeV, give the NS–S mass matrix (14) with
elements reasonably close to ours, resulting in a rather close mixing angle, θ = −10.2◦.
ns≈ xnnxss. UKQCD prefers [30] xss= 0.13
III. CONCLUSION
We have shown that the treatment of the η–η′complex in the SD–BS approach [1] is
sensible in spite of employing the ladder approximation. This is confirmed especially by Ref.
[11] which showed its connection and robust agreement with the phenomenological studies
of the η–η′complex. It is therefore desirable to extend the SD–BS studies of the η–η′mass
matrix to finite temperatures. Usually, one has neglected all temperature dependences in
the mass matrix, except the one of the gluon anomaly contribution 3β which is assumed very
strong, which is appropriate if the UA(1) symmetry breaking is due to instantons [31–34].
However, rather strong topological arguments of Kogut et al. [35] that the UA(1) symmetry
is not restored at critical (but only at a higher, possibly infinite) T, motivates also the
scenario where 3β(T) ≈ const, while other entries in the mass matrix carry the temperature
dependence. The inclusion of their T–dependence is needed also because the scenario with
the instanton–induced, strongly T–dependent β should be carefully re-examined, since it
has lead to contradicting conclusions: the depletion of η′production in Ref. [32], but η′–
enhancement in Ref. [33].
The temperature dependence of mπ = Mu¯ u = Md¯d, Mu,d(q2), fπ and ?u¯ u?(= ?d¯d?),
was already studied in various SD–BS models [36,37], so that the extension [38] to the
T–dependence of the remaining needed ingredients, Ms¯ s, Ms(q2), fs¯ sand ?s¯ s?, should be
straightforward.
Acknowledgments: D. Kl. thanks thanks D. Blaschke, S. Schmidt and G. Burau, the or-
ganizers of the workshop “Quark Matter in Astro- and Particle Physics” (27.-29. November
2000, Rostock, Germany) for their hospitality and for the support which made his partici-
pation possible.
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