Correlations between the strange quark condensate, strange quark mass, and kaon PCAC relation

Correlations between the strange quark mass, strange quark condensate $\langle \bar s s\rangle$, and the kaon partially conserved axial current (PCAC) relation are developed. The key dimensionless and renormalization-group invariant quantities in these correlations are the ratio of the strange to non-strange quark mass $r_m=m_s/m_q$, the condensate ratio $r_c=\langle \bar s s\rangle/\langle \bar q q\rangle$, and the kaon PCAC deviation parameter $r_p=-m_s\langle \bar s s+\bar q q\rangle/2f_K^2m_K^2$. The correlations define a self-consistent trajectory in the $\{r_m,r_c,r_p\}$ parameter space constraining strange quark parameters that can be used to assess the compatibility of different predictions of these parameters. Combining the constraint with Particle Data Group (PDG) values of $r_m$ results in $\{r_c,r_p\}$ constraint trajectories that are used to asses the self-consistency of various theoretical determinations of $\{r_c,r_p\}$. The most precise determinations of $r_c$ and $r_p$ are shown to be mutually consistent with the constraint trajectories and provide improved bounds on $r_p$. In general, the constraint trajectories combined with $r_c$ determinations tend to provide more accurate bounds on $r_p$ than direct determinations. The $\{r_c,r_p\}$ correlations provide a natural identification of a self-consistent set of strange quark mass and strange quark condensate parameters.

In QCD sum-rules analyses of hadronic systems containing strange quarks, the strange quark mass m s and the strange quark condensate ss are important parameters. Depending upon the system, the condensate may emerge as ss or be accompanied with quark mass factors (e.g., m s ss , m c ss ). However, ss is also used within determinations of higher-dimension condensates, including the vacuum-saturation approximation for dimension-six quark condensates and the dimension-five mixed condensate [6,7] 1 2 gsσ µν λ a G a µν s = gsσGs = (0.8 ± 0.1) GeV 2 ss .
Because of these multiple roles in determining different QCD condensates, the strange quark condensate is an essential parameter in QCD sum-rules and provides insight into SU(3) flavour symmetry of the QCD vacuum. QCD sum-rules studies of the strange quark condensate are based upon pseudoscalar and scalar correlation functions combined with low-energy theorems (see e.g., Refs. [8][9][10]) where SU(2) isospin symmetry of the vacuum implies qq = ūu = d d .
Furthermore, as qq < 0, if r c < 1 then Ψ(0) < 0, so the sign of the low-energy constant Ψ(0) provides valuable information on qualitative aspects of SU(3) flavour symmetry breaking by the vacuum.
There is a wide range of r c theoretical determinations. QCD sum-rules analyses of light scalar and pseudoscalar meson correlation functions have been used to determine r c [9,[11][12][13], and a combined estimate from these results is r c = 0.57 ± 0.12 [4]. QCD sum-rules for baryon mass splittings tend to yield larger values ranging from the earliest determinations r c = 0.8 ± 0.1 [14,15] to the updated values r c = 0.75 ± 0.08 [16] and r c = 0.74 ± 0.03 [17]. Because the QCD sum-rules for heavy baryon mass splittings have better perturbative convergence compared to the light meson analyses, the most precise sum-rule value is r c = 0.74 ± 0.03 [17]. A more conservative sum-rule value obtained from an average of light meson and baryon systems is r c = 0.66±0.10 [4]. The lattice QCD determination r c = 1.08 ± 0.16 [18] (multiple sources of uncertainty have been combined) is considerably larger than the sum-rules determinations. An analysis combining aspects of QCD  Table 1 for key to references.
sum-rules and lattice QCD results yields r c = 0.8 ± 0.3 [19]. The various theoretical determinations of r c are shown in Figure 1 and Table 1.
The PCAC Gell-Mann-Oakes-Renner (GMOR) relation relates the RG-invariant combination of the non-strange quark condensate and mass parameter to pion properties [20] where in our conventions f π = 130/ √ 2 MeV [21]. Deviations from the pion PCAC relation are bounded by approximately 5% [10,18,19] and are thus a small numerical effect. The RG-invariant quark mass ratio [21] r m = m s m q = 27.3 ± 0.7 (9) can then be combined with r c to obtain the following strange quark condensate m s ss in terms of (8) A complementary approach to determinations of m s ss is through the deviation from the kaon PCAC relation as parameterized by r p where r p = 1 corresponds to the kaon PCAC result (in conventions where f K = 156/ √ 2 MeV [21]) and (9) implies that neglecting m u is a small numerical effect (see e.g., Eq. (6)). The PCAC deviation parameter r p can be determined in QCD sum-rules by using the low-energy theorem for the pseudoscalar correlation function (3) (see e.g., Ref. [8]). A sum-rule evaluation of Ψ 5 (0) thus allows determination of r p via Eqs. (11) and (3). Significant deviations from the kaon PCAC result have been found in this approach ranging from the earliest values r p = 0.63 ± 0.08 [8] and r p = 0.5 ± 0.17 [22], to later determinations from Laplace sum-rules r p = 0.57 ± 0.19 [12] 1 , QCD sum-rules r p = 0.66 +0. 23 −0.17 [11], lattice QCD r p = 0.74 ± 0.16 [18], and combined approaches (merging Laplace sum-rules, chiral perturbation theory, and lattice QCD input) r p = 0.39 ± 0.22 [19]. The most precise determination r p = 0.66 ± 0.05 emerges from finite-energy sum-rules [13]. The various determinations of r p are shown in Figure 2.     Table 1 for key to references.
The RG-invariant combination of the strange quark mass and condensate emerging from the kaon PCAC relation (11) is The two expressions (10) and (12) for m s ss are thus self-consistent if the following constraint is satisfied: providing a correlation in the {r m , r c , r p } parameter space. The {r c , r p } linear trajectories resulting from the PDG r m range (9) are shown in Fig. 3. Determinations of r c and r p that lie along these trajectories will thus be self-consistent for the PDG range of the strange quark mass. The correlation trajectories provide a relatively more stringent constraint on r p compared to r c . For example, the conservative range 0.6 < r c < 1.2 leads to a relatively narrow range 0.57 < r p < 0.86 corresponding to a significant deviation from the kaon PCAC relation. The most interesting analyses from the literature are those which simultaneously allow determination of both r c and r p because they map out a region in the {r c , r p } parameter space that can be compared with the constraint trajectories. In Fig. 4, the simultaneous determinations of {r c , r p } are compared with the linear constraint trajectories. The determinations from Refs. [4,[11][12][13]18] show good agreement with the trajectories, but Ref. [19], which has the smallest determination of r p , does not intersect the trajectories. However, Ref. [19] is somewhat different than the other simultaneous determinations in Fig. 4 because it combines different methodologies (the r p determination in Ref. [19] involves chiral perturbation theory whereas r c involves QCD sum-rules).  Table 1 for key to references.
As a final consideration, Fig. 5 assesses the most precise individual determinations r c = 0.74 ± 0.03 [17] and r p = 0.66 ± 0.05 [13] against the constraint trajectories. The two determinations delineate a compatible region of {r c , r p } parameter space, and as discussed above, the r c determination provides a tighter bound 0.62 < r p < 0.69 completely contained within the Ref. [13] determination. A similar feature for the Ref. [18] lattice determinations is shown in Fig. 6; the constraint trajectories combined with the r c determination again provides a tighter bound 0.69 < r p < 0.88 completely contained within the Ref. [18] determination. Thus the constraint trajectories combined with r c determinations tend to provide more accurate bounds on r p than direct determinations. Based on a small set of input parameters, we can generate a collection of self-consistent numerical results for some QCD condensates containing strange quarks. The strange quark condensate ss is often multiplied by a quark mass, i.e., M ss , an RG-invariant quantity. Using r c = 0.74 ± 0.03 and (8)-(10), we find m s ss = (−1.7 × 10 −3 ) GeV 4 .
In summary, we have developed the constraint (13) relating the strange quark parameters r m = m s /m q , r c = ss / qq and the kaon PCAC deviation parameter r p = −m s ss +qq /2f 2 K m 2 K . Using r m PDG [21] values, {r c , r p } theoretical determinations (see Table 1) are compared with the constraint trajectories (see Fig. 4). Theoretical predictions corresponding to the smallest value of r p show poor agreement with the constraint trajectories. However, Fig. 5 demonstrates that the most precise values r c = 0.74 ± 0.03 [17] and r p = 0.66 ± 0.05 [13] are mutually consistent with the {r c , r p } constraint trajectories and provide an improved determination 0.62 < r p < 0.69. The combination of {r c , r p } constraint trajectories with r c determinations to obtain improved r p bounds is also observed in Fig. 6 for the lattice values [18]. Thus the {r c , r p } constraint trajectories provide a valuable methodology for assessing self-consistency or improving accuracy of determinations of the condensate ratio r c = ss / qq and the kaon PCAC deviation parameter r p = −m s ss + qq /2f 2 K m 2 K .