Flavour breaking effects in the pseudoscalar meson decay constants

The SU(3) flavour symmetry breaking expansion in up, down and strange quark masses is extended from hadron masses to meson decay constants. This allows a determination of the ratio of kaon to pion decay constants in QCD. Furthermore when using partially quenched valence quarks the expansion is such that SU(2) isospin breaking effects can also be determined. It is found that the lowest order SU(3) flavour symmetry breaking expansion (or Gell-Mann-Okubo expansion) works very well. Simulations are performed for 2+1 flavours of clover fermions at four lattice spacings.


Introduction
One approach to determine the ratio |V us /V ud | of Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, as suggested in [1], is by using the ratio of the experimentally determined pion and kaon leptonic decay rates (where M K + , M π + and m μ are the particle masses, and δ em is an electromagnetic correction factor). This in turn requires the determination of the ratio of kaon to pion decays constants, f K + / f π + , E-mail address: rhorsley@ph.ed.ac.uk (R. Horsley). a non-perturbative task, where the lattice approach to QCD may be of help. For some recent work see, for example, [2][3][4][5][6][7][8][9][10].
The QCD interaction is flavour-blind and so when neglecting electromagnetic and weak interactions, the only difference between the quark flavours comes from the mass matrix. In this article we want to examine how this constrains meson decay matrix elements once full SU (3) flavour symmetry is broken, using the same methods as we used in [11,12] for hadron masses. In particular we shall consider pseudoscalar decay matrix elements and give an estimation for f K / f π and f K + / f π + (ignoring electromagnetic contributions).

Approach
In lattice simulations with three dynamical quarks there are many paths to approach the physical point where the quark masses take their physical values. The choice adopted here is to ex- (where numerically m = m 0 ). From this definition we have the trivial constraint The path to the physical quark masses is called the 'unitary line' as we expand in the same masses for the sea and valence quarks. Note also that the expansion coefficients are functions of m only, which provided we keep m = const. reduces the number of allowed expansion coefficients considerably.
As an example of an SU (3) flavour symmetry breaking expansion, [12], we consider the pseudoscalar masses, and find to next-to-leading-order, NLO, (i.e. O ((δm q ) 2 )) where m a , m b are quark masses with a, b = u, d, s. This describes the physical outer ring of the pseudoscalar meson octet (the right panel of Fig. 1). Numerically we can also in addition consider a fictitious particle, where a = b = s, which we call η s . We have further extended the expansion to the next-to-next-to-leading or NNLO case, [13]. As the expressions start to become unwieldy, they have been relegated to Appendix A. (Octet baryons also have equivalent expansions, [13].) The vacuum is a flavour singlet, so meson to vacuum matrix el- The SU (3) flavour symmetric breaking expansion has the simple property that for any flavour singlet quantity, which we generically This is already encoded in the above pseudoscalar SU (3) flavour symmetric breaking expansions, or more generally it can be shown, [11,12], that X S has a stationary point about the SU (3) flavour symmetric line.
As a further check, it can be shown that this property also holds using chiral perturbation theory. For example for mass degenerate u and d quark masses and assuming χ PT is valid in the region of the SU (3) flavour symmetric quark mass we find where the expansion parameter is given by δχ l = χ − χ l with χ = 1 3 (2χ l + χ s ), χ l = B 0 m l , χ s = B 0 m s , f 0 is the pion decay constant in the chiral limit, L i are chiral constants and L(χ ) = χ /(4π f 0 ) 2 × ln(χ / 2 χ ) is the chiral logarithm. In eq. (8), as expected, there is an absence of a linear term ∝ δχ l .
The unitary range is rather small so we introduce PQ or partially quenching (i.e. the valence quark masses can be different to the sea quark masses). This does not increase the number of expansion coefficients. Let us denote the valence quark masses by μ q and the expansion parameter as δμ q = μ q − m. Then we havẽ (9) and  where in addition to the PQ generalisation we have also formed the ratios . . . (see Appendix A for the NNLO expressions). This will later prove useful for the numerical results. We see that there are mixed sea/valence mass terms at NLO (and higher orders). The unitary limit is recovered by simply replacing δμ q → δm q .

The lattice
We use an O (a) non-perturbatively improved clover action with tree level Symanzik glue and mildly stout smeared 2 + 1 clover fermions, [14,15], for β ≡ 10/g 2 0 = 5.40, 5.50, 5.65, 5.80 (four lattice spacings). We set giving A κ value along the SU (3) symmetric line is denoted by κ 0 , while κ 0c is the value in the chiral limit. Note that practically we do not have to determine κ 0c , as it cancels in δμ q . (For simplicity we have set the lattice spacing to unity.) We first investigate the constancy of X S in the unitary region. In Based on this observation, we determine the path in the quark mass plane by considering M 2 Fig. 3 we show this for (β, κ 0 ) = (5.50, 0.120900), (5.50, 0.120950). We see that this is indeed the case. In addition κ 0 is adjusted so that the path goes through (or very close to) the physical value. For example we see that from the figure, β = 5.50, κ 0 = 0.120950 is very much closer to this path than κ 0 = 0.120900, [14].
The programme is thus first to determine κ 0 and then find the expansion coefficients. Then use 1 isospin symmetric 'physical' 1 Masses are taken from FLAG3, [16].  masses M * π , M * K to determine δm * l and δm * s . PQ results can help for the first task. As the range of PQ quark masses that can then be used is much larger than the unitary range, then the numerical determination of the relevant expansion coefficients is improved.
PQ results were generated about κ 0 , a single sea background, so γ 1 was not relevant. Also some coefficients (those ∝ (δμ a − δμ b ) 2 ) often just contributed to noise, so were then ignored. In Fig. 4  can also tune κ 0 using PQ results so that we get the physical values of (say) M * π , X * N and M * K correct. This gives κ 0 , δμ * l , δμ * s . The philosophy is that most change is due to a change in valence quark mass, rather than sea quark mass. Note that then 2δμ * l + δμ * s = 0 necessarily (while 2δm l + δm s always vanishes). For our κ 0 val- Table 1 Results for δm * l . Of course the unitary range is much smaller, as can be seen from the horizontal lines in Fig. 4. In the LH panel of Fig. 6 we show this range as a function of δm l for M 2 π , M 2 K and M 2 η s , together with the previously found fits. The expressions are given from eq. (9), setting δμ → δm q and then a → u, b → d with m u = m d ≡ m l for M 2 π etc. . Here we clearly observe the typical 'fan' behaviour seen in the mass of other hadron mass multiplets [12]. As we have mass degeneracy at the symmetric point, the masses radiate out from this point to their physical values. For both M 2 and f the LO completely dominates.
As can be seen from the LH panel of Fig. 6 when M π takes its physical value, M * π , this determines the physical value δm * l . These are given in Table 1. Note that due to the constraint given in eq.

Decay constants
The renormalised and O (a) improved axial current is given by [17] A ab;R μ = Z A A ab;IMP μ , (14) with A ab;IMP and A ab μ = q a γ μ γ 5 q b , P ab = q a γ 5 q b . (16) Using the axial current we first define matrix elements (1) , (17) giving for the renormalised pseudoscalar constants As indicated in Fig. 7 For b A (only defined up to terms of O (a)) we presently take the tree level value, b A = 1 + O (g 2 0 ).

f K / f π
As demonstrated in the RH panel of Fig. 6, we again expect LO behaviour for SU (3) flavour symmetry breaking for f to dominate in the unitary region. Using the coefficients for the SU (3) flavour breaking expansion for f as previously determined, and then extrapolating to the physical quark masses gives the results in Table 2. Finally using these results, we perform the final continuum extrapolation, using the lattice spacings given in [14], as shown in Fig. 8. (The fits have χ 2 /dof ∼ 3.3/2 ∼ 1.6.) For comparison, the FLAG3 values, [16], are shown as stars. (Note that although f η s helps in determining the expansion coefficients, there is no further information to be found from the various extrapolated values.) Continuum values are also given in Table 2   (for simplicity now dropping the superscripts). The first error is statistical; the second is an estimate of the combined systematic error due to b A , SU (3) flavour breaking expansion, finite volume and our chosen path to the physical point as discussed in Appendix C.

Isospin breaking effects
Finally we briefly discuss SU (2) isospin breaking effects. Provided m is kept constant, then the SU (3) flavour breaking expansion coefficients (α, G , . . .) remain unaltered whether we consider 1 + 1 + 1 or 2 + 1 flavours. So although our numerical results are for mass degenerate u and d quarks we can use them to discuss isospin breaking effects (ignoring electromagnetic corrections). We parameterise these 2 effects by (2) , and expanding in m = (δm d − δm u )/2 about the average light quark mass δm l = (δm u + δm d )/2 gives, using the LO expansions (which from Figs. 4, 5 or more particularly Fig. 6, have been shown to work well) 2 An alternative, but equivalent method is to first determine δm * u , δm * d directly.

Conclusions
We have extended our programme of tuning the strange and light quark masses to their physical values simultaneously by keeping the average quark mass constant from pseudoscalar meson masses to pseudoscalar decay constants. As for masses we find that the SU (3) flavour symmetry breaking expansion, or Gell-Mann-Okubo expansion, works well even at leading order.
Further developments to reduce error bars could include another finer lattice spacing, as the extrapolation lever arm in a 2 is rather large and presently contributes substantially to the errors, and PQ results with sea quark masses not just at the symmetric point (κ 0 ) but at other points on the m = const. line.

Appendix B. Correlation functions
On the lattice we extract the pseudoscalar decay constant from two-point correlation functions. For large times we expect that where A 4 and P are given in eq. (16). We have suppressed the quark indices, so the equations with appropriate modification are valid for both the pion and kaon. V S is the spatial volume and T is the temporal extent of the lattice. To increase the overlap of the operator with the state (where possible) the pseudoscalar operator has been smeared using Jacobi smearing, and denoted here with a superscript, S for Smeared. We now set where f , f (1) are real and positive. By computing C A 4 P S and C P S P S we find for the matrix element of A 4 , and for the matrix element of ∂ 4 P we obtain from the ratio of the C P P S and C A 4 P S correlation functions Some further details and formulae for other decay constants are given in [20,21].
(while 2δm l + δm s is always = 0). This gives for example for β = 5.80, δμ * ∼ −0.0001. Changing δm * l (or δm * s ) by this and making a continuum extrapolation (which is again most sensitive to this point) and comparing the result with that of eq. (20) results in a systematic error of ∼ 0.009.

Total systematic error
Including all these systematic errors in quadrature give a total systematic estimate in f K / f π of ∼ 0.013.