Bled Workshops in Physics Vol. 9, No. 1 p. 50

Testing the usage of a generalization of the Witten-Veneziano relation in a bound-state approach to n and n' mesons*
D. Horvatica, D. Blaschkeb>c>d, Yu. Kalinovskye, D. Kekezf, D. Klabucar9>h
a Physics Department, Faculty of Science, University of Zagreb, Bijenicka c. 32,10000 Zagreb, Croatia
b Institute for Theoretical Physics, University of Wroclaw, Max Born pl. 9, 50-204 Wroclaw, Poland
c Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia
d Institute of Physics, University of Rostock, D-18051 Rostock, Germany e Laboratory for Information Technologies, Joint Institute for Nuclear Research, 141980 Dubna, Russia
f Rudjer Boskovic Institute, P.O.B. 180,10002 Zagreb, Croatia
9 Physics Department, Faculty of Science, University of Zagreb, Bijenicka c. 32,10000 Zagreb, Croatia
h Senior Associate of Abdus Salam ICTP, Trieste, Italy
Abstract. The results of the Dyson-Schwinger approach utilizing the Witten-Veneziano relation to obtain a description of the n and n' mesons, are compared with the results obtained when Shore's generalization of the Witten-Veneziano relation is used instead. On the examples of three different model interactions, we find that irrespective of the concrete model dynamics, our Dyson-Schwinger approach is phenomenologically more successful in conjunction with the standard Witten-Veneziano relation than with the generalization valid, at least in principle, in all orders in the 1 /N c expansion.
1 Introduction
The complex of the n and n' pseudoscalar mesons is an intriguing problem in the light-quark sector of the nonperturbative Quantum Chromodynamics (QCD). The mixing of the pertinent isospin-zero states should be such that the physical n meson is one of the (almost-)Goldstone bosons of the dynamical chiral symmetry breaking (DChSB) of QCD, whereas its partner n' must be very massive (~ 1 GeV) and remain such even in the chiral limit. For the correct n' mass behavior, the non-abelian (gluon) axial anomaly of QCD is essential, and a way to extract its contribution is through the Witten-Veneziano (WV) relation [1,2].
We are particularly interested in the Dyson-Schwinger (DS) approach [3-8] to QCD and its modeling. In some variants of the DS approach (e.g., in Ref. [9]), the
* Talk delivered by D. Klabucar
WV relation has been used to obtain the description of the n-n' complex. In the present paper, for three different DS models, we compare the usage of the WV relation with the usage of its recent generalization recently proposed by Shore [10,11]. The present paper in the Bled 2008 proceedings, is a shortened version of Ref. [12].
The DS approach [3-8] is the chirally well-behaved bound-state approach and thus the most suitable one to treat the light pseudoscalar mesons (those composed of the u, d and s quarks), for which DChSB is essential. One solves the DS equations (DSEs) for dressed quark propagators, which are then employed in Bethe-Salpeter equations (BSEs). Their solving yields quark-antiquark (qq') bound state amplitudes and corresponding masses Mqq'.
To obtain the chiral behavior as in QCD, DS and BS equations must be solved in a consistent approximation. The rainbow-ladder approximation (RLA), where DChSB is well-understood, is still the most usual approximation in phenomeno-logical applications. This also entails that in both DSE and BSE we employ the same effective interaction. Concretely, in the present paper we recall and utilize the results obtained i) in Refs. [13,14] by using the renormalization-group improved (RGI) interaction of Jain and Munczek [15], ii) in Ref. [9] by using the RGI gluon condensate-induced interaction [16], and iii) in Refs. [17,18] by using the separable interaction [19]. Such effective interactions must be modeled at least in the low-energy, nonperturbative regime in order to be phenomenologically successful - which above all means to be sufficiently strong in the low-momentum domain to yield DChSB. In the chiral limit (and close to it), light pseudoscalar (P) meson qq bound states (P = n0,±,K0,±,n) then simultaneously manifest themselves also as (quasi-)Goldstone bosons of DChSB. This enables one to work with the mesons as explicit qq bound states, while reproducing the results of the Abelian axial anomaly for the light pseudoscalars, i.e., the amplitudes for P —> yy and y* —> P0P+P-. This is unique among the bound state approaches - e.g., see Refs. [5,20,22,21] and references therein. Nevertheless, one keeps the advantage of bound-state approaches that from the qq substructure one can calculate many important quantities (such as the pion, kaon and sS pseudoscalar decay constants: fn, fK and fs5) which are just parameters in most of other chiral approaches to the light-quark sector. The treatment [13,14,23,9] of the n-n' complex is remarkable in that it is very successful in spite of the limitations of RLA. (Very recently, during the work on the present paper, the first and still simplified DS treatments of n and n' beyond RLA appeared [24,25]. However, RLA treatments will probably long retain their usefulness in applications where simple modeling is desirable, as in the calculationally demanding finite-temperature calculations [18].) The RLA treatments [13,14,23,9,18] of the n-n' complex relied on the Witten-Veneziano (WV) relation [1,2]. Nevertheless, Shore achieved [10,11] what can be considered as a generalization of the WV relation, and the purpose of the present paper is exploring the usage of this generalization in the DS context.
The paper is organized as follows: in the next section, we recapitulate the procedures and results of our previous treatments [14,9,18] relying on the WV relation (11), and present in Table I also their extension to the scheme of the four decay constants (and two mixing angles) of n and n'. In Section 3, we expose the
usage of the pertinent Shore's equations [10,11] in the context of DS approach. The last section concludes after giving the results of solving the pertinent equations.
2 ' mass matrix from Witten-Veneziano relation
All qq' model masses Mqq' (q, q' = u, d,s) used in the present paper, and corresponding qq' bound-state amplitudes, were obtained in Refs. [13,14,9,26,17,18] in RLA, i.e., with an interaction kernel which (irrespective of how one models the dynamics) cannot possibly capture the effects of the non-Abelian, gluon axial anomaly. Thus, when we form the n-n' mass matrix
M 2
M8s m8o NA" M2s M0o
(1)
in this case in the octet-singlet basis n8-n0 of the (broken) flavor-SU(3) states of isospin zero,
r|s = -^=(uu + dd — 2ss),	r|o = —L= (uu + dd + ss),	(2)
v 6	v 3
this matrix (1), consisting of our calculated qqq masses,
2 1
M28 = (nsl^Alns) = 3 (M25 + -M2un) ,	(3)
21
M20 = <r,o|KX2A|no) = - (^M-ss + M2U) ,	(4)
Mi0 = (nsl^Ablo) = M28 = MzuU - M25) < 0,	(5)
is purely non-anomalous (NA), vanishing in the chiral limit. In the isospin limit, to which we adhere throughout, the pion is strictly decoupled from the gluon anomaly and Muu = Mdd is exactly our model pion mass Mn. Also the unphys-ical ss quasi-Goldstone's mass Msq results from RLA BSE and does not include the contribution from the gluon anomaly. This is consistent with the fact that due to the Dashen-Gell-Mann-Oakes-Renner (DGMOR) relation, it is in a good approximation [13,14,9,18] given by M2, = 2MK — M^, i.e., by the kaon and pion masses protected from the anomaly by strangeness and/or isospin.
In our previous DS studies [13,14,9,26,17,18], to which we refer for all model details, the phenomenology of the non-anomalous sector was successfully reproduced, e.g., fn, fK, as well as the empirical masses Mn and MK (see the upper part of Table 1), yielding a strongly non-diagonal MNa (1). Its diagonalization leads to the eigenstates known as the nonstrange-strange (NS-S) basis,
riNs =-j=(uu + dd) ,	r|s = ss,	(6)
and to MNa = diagM^M2,]. In contrast to these mass-squared eigenvalues, the experimental masses are such that (Mn)expA2(M?)exp, and n' is too heavy,
(Mn') exp = 958 MeV, to be considered even as the ss quasi-Goldstone boson. This is the well-known UA (1) problem, resolved by the fact that the complete n-n' mass matrix M2 must contain the anomalous (A) part MA. That is, M2 = KXNa + MA.
However, MA is inaccessible to RLA which yields our Goldstone pseudo-scalars. In Refs. [13,14,9,17,18], MA was extracted from lattice data through the WV relation [the second equality in Eq. (11)]. The main purpose of the present paper, instead, is to approach n and n' through Shore's [10,11] recent generalization of that relation.
Before that, nevertheless, we review the usage of the WV relation in Refs. [13,14,9,17,18]. The expansion in the large number of colors, Nc, indicates that the leading approximation in that expansion describes the bulk of main features of QCD. The gluon anomaly is suppressed as 1/Nc and can be viewed as a perturbation in the large Nc expansion. In the SU(3) limit, it is coupled only to the singlet combination no (2); only the no mass receives, from the gluon anomaly, a contribution which, unlike quasi-Goldstone masses Mqq''s comprising MNA, does not vanish in the chiral limit. As discussed in Refs. [13,9], in the present bound-state context it is thus meaningful to include the effect of the gluon anomaly just on the level of a mass shift for the no as the lowest-order effect, and retain the q q bound-state amplitudes and the corresponding mass eigenvalues Mqq as calculated by solving DSEs and BSEs with kernels in RLA.
References [13,14,9,17,18] thus break the UA(1) symmetry, and avoid the UA(1) problem, by shifting the no (squared) mass by an amount denoted by 3p (in the notation of Refs. [14,9]). The complete mass matrix M2 = MNa + MA then contains the anomalous part MA = diag[0,3p], where the anomalous no mass shift 3p is related to the topological susceptibility of the vacuum, but in the present approach must be treated as a parameter to be determined outside of our RLA model, i.e., fixed by phenomenology or taken from the lattice calculations [27]. (The possibility of employing an additional microscopic model for the gluon anomaly contribution, such as the one of Ref. [28], is presently not considered.)
The SU(3) flavor symmetry breaking and its interplay with the gluon anomaly [9] modifies MA = diag[0,3p] to
M a = P
|(1-X)2 ¿2(1_X)(2 + X) ■f(1-X)(2+X) }(2 + X)2
(7)
where X is the flavor symmetry breaking parameter. It is most often estimated as X = fn/fss - 0.7 - 0.8 (see, e.g., Refs. [30,29,14,9], although there are some other [14], of course related, estimates of X). Presently we also adopt X = fn/f ss, which means that X is a calculated quantity in our approach. The employed models achieved good agreement with phenomenology [13,14,9,18], e.g., fitted the experimental value of M_n + M^ ' for p around 0.26 - 0.28 GeV2. The anomaly contribution MA then brings the complete M2 rather close to a diagonal form for all considered models [13,14,9,18]; that is, to diagonalize M2, only a relatively small rotation (|6| - 13° ± 2°) of the ns-no basis states,
n = cos0ns — sin9n0 ,	n' = sin9ns + cos0no ,	(8)
is needed to align them with the mass eigenstates, i.e., with the physical n and n'. In contrast to this, the n-n' mass matrix in the NS-S basis (6),
M2 + 2(3 \/2|3X %/2(3X M25 + (3X2
M 2
M2 M2
nœ nsnNs M2 M2
MnNsns Mns
Mn o o Mn '
(9)
$
is then strongly off-diagonal. The indicated diagonalization, given by
n = cos ^nNs - sin ^ns,	n' = sin ^nNs + cos ^ns, (10)
is thus achieved for a large NS-S state-mixing angle ^ ~ 42° ± 2°. Of course, this is again in agreement with phenomenological requirements [14,9], since c[> is fixed to the angle 0 by the relation § = 0 + arctan 1/2 = 9+ 54.74°.
The invariant trace of the mass matrix (9), together with M;?5 = 2MK — M^ (from the DGMOR relation), gives the first equality in
p(2 + X2) = M2+M2,-2M2 =-|xym.	(11)
The second equality is the Witten-Veneziano (WV) relation [1,2] between the n, n' and kaon masses and xym, the topological susceptibility of the pure gauge, Yang-Mills theory. Thus, p does not need to be a free parameter, but can be determined from lattice results on xym, so that no fitting parameters are introduced. For the three models [15,16,19] utilized in our treatments [13,14,9,18] of n and n', the bare quark mass parameters and the interaction parameters were fixed already in the non-anomalous sector, by requiring the good pion and kaon phenomenology. (See the n and K masses and decay constants in Table 1.) Then, following Refs. [9,18] in adopting the central value of the weighted average of the recent lattice results on Yang-Mills topological susceptibility [31-33],
Xym =(175.7 ± 1.5 MeV)4 ,	(12)
we have obtained the good descriptions of the n-n' phenomenology [13,14,9,18], exemplified by the first three columns (one for each DS models used) of the middle part of Table 1, giving the predictions for the n and n' masses and for the NS-S mixing angle
The lowest part of the table, below the second horizontal dividing line, contains the results on the quantities (0o, 08, etc.) defined in the scheme with four n and n' decay constants and two mixing angles, introduced and explained in the following Section 3. Table 1 also compares these results of ours (in the first three columns) with the corresponding results of Shore's approach [10,11], in which the experimental values of the meson masses Mn, MK, Mn, and Mn', as well as the decay constants fn and fK (in contrast to our qq bound-state model predictions for these quantities) are used as inputs enabling the calculation of various decay constants in the n-n' complex and the two mixing angles 0o and 08 (corresponding to ^ = 38.24° in our approach).
3 Usage of Shore's equations in DS approach
The WV relation was derived in the lowest-order approximation in the large Nc expansion. However, considerations by shore [10,11] contain what amounts to
from	[14]	[9]	[18]	Shore	
Ref.	& WV	& WV	& WV	[10,11]	Experiment
M„	137.3	135.0	140.0		M 38 Aiisospin ■ ' o • ^ J average
Mk	495.7	494.9	495.0		Î495 7)isosPta y^/u./ )average
Mss	700.7	722.1	684.8		
fn	93.1	92.9	92.0		92.4 ± 0.3
fK	113.4	111.5	110.1		113.0 ± 1.0
fss	135.0	132.9	119.1		
Mn	568.2	577.1	542.3		547.75 ±0.12
Mn,	920.4	932.0	932.6		957.78 ±0.14
*	41.42°	39.56°	40.75°	(38.24°)	
e0	-2.86	-5.12°	-6.80°	-12.3°	
e8	-22.59°	-24.14°	-20.58°	-20.1°	
fo	108.8	107.9	101.8	106.6	
fs	122.6	121.1	110.7	104.8	
f° n	5.4	9.6	12.1	22.8	
f°, ri '	108.7	107.5	101.1	104.2	
f* n	113.2	110.5	103.7	98.4	
f8, 'n'	-47.1	-49.5	-38.9	-37.6	
Table 1. The results of employing the WV relation (11) in our DS approach for the three dynamical models used in Refs. [14,9,18], compared with the results of Shore's analysis [10,11] and with the experimental results. The first column was obtained by the WV-recalculation of the results of Ref. [14], which in turn used the Jain-Munczek Ansatz for the gluon propagator [15]. Column 2: the results based on Ref. [9], which used the OPE-inspired, gluon-condensate-enhanced gluon propagator [16]. Column 3: the results based on Ref. [18], which utilized the separable Ansatz for the dressed gluon propagator [19]. Column 4: The results of Shore [10,11], who used the lattice result xym = (191 MeV)4 of Ref. [32], and not the weighted average (12), in contrast to us. Column 5: the experimental values. All masses and decay constants are in MeV, and angles are in degrees. For more details, see text.
the generalization of the WV relation, which is valid to all orders in 1/Nc. Among the relations he derived through the inclusion of the gluon anomaly in DGMOR relations, the following are pertinent for the present paper:
(f°,)2M2, + (f0)2M2 = (f2M2 + 2f2M2) +6A
(13)
tl V 1 tl T) T) ^
2v/I(f^M2
f2M2)
(14)
(f«,)2M2, + (f«)2M2=--(f2M2-4f2M2)
(15)
where A is the full QCD topological charge parameter, which is presently unknown, but in the large Nc limit, it reduces to YM topological susceptibility: A = xYM + O(1/Nc). Besides fn, they contain fK and the four decay constants [34-36], f', fn, fn, and f', associated with the two pseudoscalars n and n'.
Adding Eqs. (13) and (15), one gets the relation
(fn' )2Mn' + (fn)2Mn + (fn)2Mn + (fn' )2m"' -
2f2M2
6A
(16)
which is the analogue of the standard WV formula (11), to which it reduces in the large Nc limit where A —> xYM, the ^', f^fx —> fn limit, and the limit of vanishing subdominant decay constants (since n and n' are dominantly ns and no, respectively), i.e., f], f' —> 0. Nevertheless, we will need to use not just this single equation, but the three equations (13)-(15) from Shore's generalization.
These four n and n' decay constants are often parameterized in terms of two decay constants, fs and f0, and two mixing angles, 0s and 0o:
fn = cos 0s fs
f! = -sin 0o fo , fn' = sin 0s fs , fn' = cos 0o fo . (17)
This is the so-called two-angle mixing scheme, which shows explicitly that it is inconsistent to assume that the mixing of the decay constants follows the pattern (8) of the mixing of the states ns and no [34-36,30,37,29].
The advantage of our model is that, as we shall see, we are able to calculate the fs and fo parts of the physical decay constants (17) from the qq substructure. However, we cannot keep the full generality of Shore's approach, which allows for the mixing with the gluonic pseudoscalar operators, and therefore employs the definition [10,11] of the decay constants which, in general, due to the gluonic contribution, differs from the following standard definition through the matrix elements of the axial currents Aa q(x):
(0|Aaq(x)|P(p)) = ifa pqe-ip'
a = 8,0;
p = n,n'
(18)
Nevertheless, Shore's definition [10,11] coincides with the above standard one in the non-singlet channel, where there cannot be any admixture of the pseudoscalar gluonic component. Similarly, since our BS solutions (from Refs. [13,14,9,18]) are the pure qq states, without any gluonic components, using Shore's definition would not help us calculate the gluon anomaly influence on the decay constants. We thus employ the standard definitions (18), also used by, e.g., Gasser, Leutwyler, and Kaiser [34-36], as well as by Feldmann, Kroll, and Stech (FKS) [30,37,29].
Equivalently to fn', fn, fn, and fn', defined by Eq. (18), one has four related but different constants fNS, fNS, fn, and fn', if instead of octet and singlet axial currents (a = 8,0) in Eq. (18) one uses the nonstrange-strange axial currents (a = NS, S)
1	[J	1
A£s(x) = —^=A8M(x) +	= - [u(x)y^y5u(x) + a(x)y^y5d(x)] , (19)
Aq (x)
1
-A8^(x) +	= ^s(x)y^y5s(x) .
V 3'v ^ ' v/T	v
The relation between the two equivalent sets is thus
(20)
fNS fS
fn fn
fNS fS
V VJ
f8 fO
fn fn
V n'J
i
73
rj
i
V3
(21)
Of course, this other quartet of n and n' decay constants can also be parameterized in terms of other two constants and two other mixing angles:
fN = cos ^NS fNS , = - sin ^S fs , f where fNS and fs are given by the matrix elements
n - cos ^ns fNS , f = - sin fo fS , fNS = sin ^ns fNS , fO'
cos ^s fs , (22)
(0|ANsMlnNs(p)) = ifNS pse-
apx
(0|A^(x)|ns(p)) = ifs PHe-
ap-x
(23)
while (0|ANS(x)|ns(p)) = 0 and (0|AsHMlnNs(p)} = 0 .
In the NS-S basis, it is possible to recover a scheme with a single mixing angle ^ through the application of the Okubo-Zweig-Iizuka (OZI) rule [30,37,29]. For example, fNSfSsin(^NS — ^s) differs from zero just by an OZI-suppressed term [29]. Neglecting this term thus implies ^NS = (Refs. [30,37,29] denote fNS, fs, ^NS, by, respectively, fq, fs, , .) In general, neglecting the OZI-suppressed terms, i.e., application of the OZI rule, leads to the so-called FKS scheme [30,37,29], which exploits a big practical difference between the (in principle equivalent) parameterizations (17) and (22): while 08 and 0o differ a lot from each other and from the octet-singlet state mixing angle 0 « (08 + 0o)/2, the NS-S decay-constant mixing angles are very close to each other and both can be approximated by the state mixing angle: ^NS « « Therefore one can deal with only this one angle, and express the physical n-n' decay constants as
f8 fO
'o 'n
fO fO
V n'J
fNScos ^ —fssin ^ fNS sin ^ fs cos ^
i

2
2	j_
3	^3
(24)
This relation is valid also in our approach, where n and n' are the simple nNS-nS mixtures (10). In our present DS approach, mesons are pure qq BS solutions, without any gluonium admixtures, which are prominent possible sources of OZI violations. Therefore, our decay constants are calculated quantities, fNS = fuu = fdd = fn and fs = fss, in agreement with the OZI rule. Our DS approach is thus naturally compatible with the FKS scheme, and we can use the n and n' decay constants (24) with our calculated fNS = fn and fs = fss in Shore's equations (13)-(15).
4 Results and conclusions
All quantities appearing on the right-hand side of Eqs. (13)-(15), namely Mn, MK, fn, and fK, are calculated in our DS approach [14,9,18] (for the three dynamical models [15,16,19]), except the full QCD topological charge parameter A. Since it is at present unfortunately not yet known, we follow Shore and approximate it by the Yang-Mills topological susceptibility xym.
On the left-hand side of Eqs. (13)-(15), the model results for fNS = fn and fs = fss and Eq. (24) reduce the unknown part of the four n and n' decay constants fO', fO, and fO', down to the mixing angle The three Shore's equations (13)-(15) can then be solved for MO and MO', providing us with the upper three lines
Inputs:	from Ref. [14]		from Ref. [9]		from Ref. [18]	
1 /4 XYM	175.7	191	175.7	191	175.7	191
Mn	485.7	499.8	482.8	496.7	507.0	526.2
Mn,	815.8	931.4	818.4	934.9	868.7	983.2
*	46.11°	52.01°	46.07°	51.85°	40.86°	47.23°
e0	1.84°	7.74°	1.39°	7.17°	-6.69°	-0.33°
08	-17.90°	-12.00°	-17.6°	-11.85°	-20.47°	-14.11°
fo	108.8	108.8	107.9	107.9	101.8	101.8
f8	122.6	122.6	121.1	121.1	110.7	110.7
f° n	-3.5	-14.7	-2.6	-13.5	11.9	0.6
f°,	108.8	107.9	107.9	107.1	101.1	101.8
? 'n	116.7	119.9	115.4	118.5	103.7	107.4
f8, V	-37.7	-25.5	-37.6	-24.9	-38.7	-27.0
Table 2. The results of the three DS models obtained through Shore's equations (13)-(15) for the two values of xYM approximating A: (175.7MeV)4 and (191 MeV)4. Columns 1 and 2: The results when the non-anomalous inputs for Eqs. (13)-(15), namely Mn, MK ,fn = fNs, fss = fS and fK, are taken from Ref. [14], which uses Jain-Munczek Ansatz interaction [15]. Columns 3 and 4: The results for the non-anomalous inputs from Ref. [9] using OPE-inspired interaction nonperturbatively dressed by gluon condensates [16]. Columns 5 and 6: The results for the inputs from Ref. [18] using the separable Ansatz interaction [19]. All masses and decay constants, as well as xYm , are in MeV, and angles are in degrees.
of Table 2. For each of the three different dynamical models which we used in our previous DS studies [13,14,9,26,17,18], these results are displayed for xYM = (175.7MeV)4 as in Refs. [9,18] and for xym = (191 MeV)4 [32] (adopted by Shore [10,11]). The lower part of the table, displaying various additional results, is then readily obtained through Eq. (24) and/or the following useful relations [29,14]:
fo =	I - 00 = ^ - arctan ^^^ j .	(26)
Note that fo and f8 do not result from solving of Eqs. (13)-(15), but are the calculated predictions of a concrete dynamical DS model, independently of Shore's equations.
For all three quite different (RGI [15,16] and non-RGI [19]) dynamical models which we used in our previous DS studies [13,14,9,26,17,18], the situation with the results turns out to be rather similar. The most conspicuous feature is that n and n' masses are both much too low when the weighted average xYM = (175.7 ± 1.5 MeV)4 of Refs. [31-33] is used, in contrast to the results from the standard WV relation, displayed in Table 1. If we single out just the highest of these values (191 MeV)4 [32]), the masses improve somewhat. However, other results are spoiled - e.g., the mixing angle ^ becomes too high to enable agreement with the experimental results on n,n' ->1 YY decays, which require ^ ~ 40° [9].
When we turn to the lower parts of Tables 1 and 2, where the results for the n and n' decay constants, and the corresponding two mixing angles 0o and 08, are given, we notice a feature common to all our results, as well as Shore's (also given in Table 1). The diagonal ones, fO' and fO, are all of the order of fn, being larger by some 10% to 30%. The off-diagonal ones, fO' and fO, are, on the other hand, in general strongly suppressed. This is expected, as n' is mostly singlet, and n is mostly octet. The feature that may be surprising is that Shore's results (which, to be sure, were obtained [10,11] in quite a different way from ours) are more similar to our results obtained through the standard WV relation, than to our results obtained through Shore's Eqs. (13)-(15).
All in all, inspection and comparison of the results in Table 2 with the results (in Table 1) from the analogous calculations but using the standard WV relation to construct the complete n-n' mass matrix, leads to the conclusion that the DS approach with the standard WV relation (11) is phenomenologically more successful, yielding the masses closer to the experimental ones. This may seem surprising, as Shore's generalization is in principle valid to all orders in 1 /Nc, while the standard WV relation is a lowest order 1 /Nc result. Nevertheless, one must be aware that we do not yet have at our disposal the full QCD topological charge parameter A, and that we (along with Shore) had to use its lowest 1 /Nc approximation, xYM. Also, we should recall from Sections 1 and 2 that the very usage of the RLA assumed that the anomaly is implemented on the level of the anomalous mass only, as a lowest order 1 /Nc correction [13,14,9,17,18]. Thus, with respect to the orders in 1 /Nc, the usage of the standard WV relation is consistent in the present formulation of our DS approach, whereas the usage of Shore's generalization is not, which is probably the cause of its lesser phenomenological success. However, the usage of Shore's generalization in the DS context as exposed here, will likely find its application at finite temperatures. Namely, there it may help alleviate the difficulties met due to the usage of the standard WV relation in the DS approach at T > 0, as discussed in Ref. [18].
Acknowledgments: D.H. and D.Kl. acknowledge the support of the project No. 119-0982930-1016 of MSES of Croatia. D.Kl. also acknowledges the hospitality and support through senior associateship of International Centre for Theoretical Physics at Trieste, Italy, where the presented research was started. D.Kl. also thanks the LIT of JINR for its hospitality in Dubna, Russia, in August 2007. D.Ke. acknowledges the support of the Croatian MSES project No. 098-0982887-2872. Yu.K. thanks for support from Deutsche Forschungsgemeinschaft (DFG) under grant No. BL 324/3-1, and the work of D.B. was supported by the Polish Ministry of Science and Higher Education under grant No. N N 202 0953 33.
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