The derivative of the topological susceptibility at zero momentum and an estimate of $\eta'$ mass in the chiral limit

The anomaly-anomaly correlator is studied using QCD sum rules. Using the matrix elements of anomaly between vacuum and pseudoscalars $\pi, eta$ and $\eta'$, the derivative of correlator $chi'(0)$ is evaluated and found to be $\approx 1.82 \times 10^{-3}$ GeV$^2$. Assuming that $\chi'(0)$ has no significant dependence on quark masses, the mass of $\eta'$ in the chiral limit is found to be $\approx$723 MeV. The same calculation also yields for the singlet pseudoscalar decay constant in the chiral limit a value of $\approx 178$ MeV.

The topological susceptibility χ(q 2 ) defined by is of considerable theoretical interest and has been studied using a variety of theoretical tools like latice guage theory, QCD sum rules, chiral perturbation theory etc. In particular the derivative of the susceptibility at q 2 = 0 enters in the discussion of the proton-spin problem [1,2,3,4,5]. In the QCD sumrule approach, one can determine χ ′ (0) as follows. Using despersion relation one can write Defining the Borel transform of a fuction f (q 2 ) bŷ one gets from Eq.(4) According to Eq.(2), ℑ(χ(s)) receives contribution from all states |n such that 0|Q|n = 0. In particular we have [6] 0|Q|π The matrix elements, when |n is |η or |η ′ , can be determined as follows. It is known from both theoretical considerations based on chiral perturbation theory as well as phenomenological analysis that one needs two mixing angles θ 8 and θ 0 to describe the coupling of the octet and siglet axial vector currents to η and η ′ [7,8,9]. Introduce the definition where J 8,0 µ5 are the octet and singlet axial currents : The |P (p) represents either η or η ′ with momentum p µ . The couplings f a P can be equivalently represented by two couplings f 8 , f 0 and two mixing angles θ 8 and θ 0 by the matrix identity Phenomenological analysis of the various decays of η and η ′ to determine f a P has been carried out by a number of authors [7,8,9]. In a recent analysis [9] Escribano and Frere find, with the other three parameters to be The divergence of the axial currents are given by Since m u , m d << m s , one can neglect them [10] to obtain Using Eqs. (7), (16) and (17) we get the representation of χ(q 2 ) in terms of physical states as On the other hand, χ(q 2 ) has an operator product expansion [11,12,1,5] In Eq.(19), the first term arises from the perturbative gluon loop with radiative corrections [12], the second, third and fourth terms are from the vacuum expectation values of G 2 , G 3 and G 4 . The 0|G 4 |0 term has been expressed as 0|G 2 |0 2 using factorization [11]. The fifth term proportional to the quark mass has been computed by us and is indeed quite small compared to other terms numerically. Finally, the last two terms represent the contribution to χ(q 2 ) from the direct instantons [11]. n(ρ) is the density of instanton of size ρ, K 2 is the Mc Donald function and Q 2 = −q 2 . In a recent work [13], Forkel has emphasized the importance of screening correction which almost cancels the direct instanton contribution (cf. especially Fig.8 and Secs. V and VI of Ref. [13] ). For this reason we shall disregard the direct instanton term and screening correction for the present and return to it later.
From Eq. (6), we now obtain Here E 0 (x) = 1 − e −x and takes into account the contribution of higher mass states, which has been summed using duality to the perturbative term in χ OP E , and W is the effective continuum threshold. We take W 2 = 2.3 GeV 2 , and Writing as in Ref. [5], we take ǫ = 1 GeV 2 . We also have the PCAC relation, For f 0 , f 8 , θ 8 and θ 0 we use the central values given in Eqs. (12) and (13). Let us now examine how the various terms in the r.h.s. of Eq.(20) add up to remain a constant. The pion term is small and has little variation because of its low mass, η and η ′ are significantly larger and η is even larger that η ′ . In Fig.1 the upper line gives the combined contribution of π, η and η ′ which we denote as χ ′ poles and it is seen that it has a gentle increase with M 2 . The OPE term given by the last three lines in Eq.(20), which we denote by χ ′ OP E , so that is also plotted in Fig.1. It is seen that χ ′ OP E is roughly about 25% of χ ′ poles also increases with M 2 , with the result that χ ′ (0) is nearly a constant w.r.t M 2 .
We expect this trend of compensating variation in χ ′ poles and χ ′ OP E to be maintained when variation in χ ′ poles due to uncertainties in θ 8 , θ 0 , f 8 , f 0 [see Eqs. (12)] and (13) and the variations in χ ′ OP E due to uncertainties in the estimates of the vacuum condensates are taken into account. We can then obtain, from Fig.1, the value We note that the above determination, Eq.(24), is in agreement with an entirely different calculation by two of us [14] from the study of the correlator of isoscalar axial vector currents Π I=0 1 (q 2 = 0) can be computed from the spectrum of axial vector mesons. In Ref. [14] a value We now compare our result for χ ′ (0) with some earlier results. In Ref. [1], Narison et al obtained a value for χ ′ (0) ≈ 0.7 × 10 −3 GeV 2 , substantially different from the value derived here. Since the expression for χ OP E used by us is identical to theirs, albeit the estimate used for the gluon condensates are slighly different, we need to explain the difference in the end result for χ ′ (0). The most important difference is in the expression of χ(q 2 ) in terms of physical intermediate states. We have seen that both η and η ′ contribute, and in fact η makes a larger contribution than η ′ . In Ref. [1] only η ′ (958) state is taken into account. We have also seen that if we were to take the chiral limit then η and η ′ contribution to χ(q 2 ) is representable by η χ with mass m χ = 723 MeV, which is substantially different form the physical η ′ mass. This also explains why Narison et al. find stability in the sum rule only for rather larger W 2 (=6 GeV 2 ) instead of our W 2 =2.3 GeV 2 . We must also add that while our Eq.(6) involves only [χ ′ (q 2 )/q 2 ], Narison et al. use the linear combination of two sum rules (cf. Eq.(6.22) of Ref. [1]).
In Ref. [2], Ioffe and Khodzhamiryan's claim that the OPE for χ(q 2 ) does not converge is based on the following. They computed the correlators where, J q µ5 =qγ µ γ 5 q (q = u, d, s) with m u = m d = 0 but m s = 0 and j 0 µ5 is the flavor singlet axial current. Introducing the definition If SU (3) symmetry were to be exact, this ratio would be unity. Insisting that the ratio in Eq.(32) should be close to unity even when m s = 0, they concluded that their result signals a breakdown of OPE [2]. As discussed earlier, 0|J 8 µ5 |η ′ = 0. In fact using the values given in Eqs. (12) and (13), it is easy to obtain g s which is close enough to the estimate of Ref. [2]. As in the case of Narison et al [1], Ioffe and Samsonov [5] and Forkel [13] also do not take into account the π, η matrix element of the anomaly in their sum rules involving χ(q 2 ).
In conclusion we find a value of χ ′ (0) ≈ 1.82 × 10 −3 GeV 2 without incorporating direct instantons. Screening corrections to the latter appears to be significant. We also obtain an estimate m χ = 723 MeV and f ηχ = 178 MeV.