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Chiral Perturbation Theory, Non-leptonic Kaon Decays, and the Lattice

In this talk, I first motivate the use of Chiral Perturbation Theory in the context of Lattice QCD. In particular, I explain how partially quenched QCD, which has, in general, unequal valence- and sea-quark masses, can be used to obtain real-world (i.e. unquenched) results for low-energy constants. In the second part, I review how Chiral Perturbation Theory may be used to overcome theoretical difficulties which afflict the computation of non-leptonic kaon decay rates from Lattice QCD. I argue that it should be possible to determine at least the O(p^2) weak low-energy constants reliably from numerical computations of the K to pi and K to vacuum matrix elements of the corresponding weak operators.


Introduction
Ideally, Lattice QCD should lead to the unambiguous, first-principle determination of hadronic quantities. However, it does not work that way in practice, basically because of limits on available computational power. Because of these limits, approximations are needed which lead to systematic errors, and we need a theoretical understanding of these in order to obtain estimates of hadronic quantities with known and controlled errors. This talk addresses some of the limitations of Lattice QCD with respect to the physics of Goldstone Bosons (GBs), and explains how Chiral Perturbation Theory (ChPT) can be used to extract physical quantities from the often unphysical results of lattice computations.
The basic idea is that ChPT parametrizes QCD correlation functions involving GBs in terms of a number of parameters, the low-energy constants (LECs). To a given order in the chiral expansion, physical quantities are determined completely in terms of a finite number of LECs, and these LECs can then be obtained from (possibly other) correlation functions at often unphysical choices of masses and momenta which are more easily accessible to numerical computation. Such a program only works if the chiral expansion converges in the regions of interest (both physical and on the lattice), but this can in principle be checked by considering successive terms in the chiral expansion, once the LECs have been determined from fits to lattice results. Obviously, for such checks at least two orders in the chiral expansion need to be considered, and one-loop ChPT is essential.
A simple example is the pion mass, which is extracted from the appropriate euclidean two-point function. The computations are done in a finite volume L 3 , and, to avoid finite-size distortions, one has to require that m π L ≫ 1. Since, in practice, L ≈ 3 fm is (currently) the largest attainable volume, demanding (say) m π L > 5 leads to the requirement that m π > 330 MeV on the lattice; in other words, the light quark masses have to be chosen unphysically large. But if ChPT converges in the range 300 MeV< m π < 700 MeV or so, it can be used to determine the relevant LECs and extrapolate to the physical pion mass (e.g. to determine light quark masses). Once this works, there is actually no need to perform numerical computations in volumes large enough to contain a physical pion. Instead, additional computational power could for instance be used to go to smaller lattice spacings at the same physical volume, in order to reduce scaling violations.
In this talk, I will discuss two topics from this point of view. The first topic will be the use of "partially quenched" QCD (PQQCD), which extends the idea that, for a number of quantities of interest, simpler unphysical lattice computations may be all one needs. The second topic will be that of weak matrix elements for non-leptonic kaon decays. In this case, accessible euclidean correlation functions are "not physical" for a number of reasons, but ChPT can be used to extract relevant information.

Why partially quenched?
In a lattice calculation, the quark fields are integrated over analytically, leading to a gluonic expectation value of the product of the fermion determinant and various quark propagators. For example, (1) It is clear that, in a computation, one could choose the quark mass in the determinant (the "sea" quark mass) unequal to those in the propagators (the "valence" quark masses), as indicated in eq. (1). This situation is referred to as "partial quenching." The (fully) quenched theory is obtained by omitting the sea quarks altogether, or, equivalently, by sending m = m sea → ∞. A simple reason to do this is that it is much cheaper to compute a propagator than a determinant, so that one can vary the valence masses almost "for free," unlike the sea-quark mass. This then raises two questions: is this useful for anything, and, can this be understood with field-theoretic methods?
pqchpt: submitted to World Scientific on November 3, 2000 To the first question, the answer is yes! For example, a partially quenched simulation with N = 3 degenerate light sea quarks (and arbitrary valence masses) gives information about the real world. 1 Consider again the pion mass. With N sea quarks of mass m s , and two valence quarks of mass m v , partially quenched ChPT (PQChPT; see next subsection) gives 2,3 where we follow the notation of Gasser and Leutwyler 4 (but the pion decay constant f (N ) in the chiral limit is normalized such that it equals 132 MeV at m π ). We explicitly indicate the dependence on N . (The QCD dynamics (i.e. the determinant) and hence the LECs do not depend on the valence quarks.) Since the LECs depend only on N , it is clear that if we take N = 3, the values obtained from fitting this expression to lattice results are the realworld values. Note that only degenerate sea quarks are needed to get 2L 8 −L 5 and 2L 6 − L 4 separately. If we only use unquenched QCD, non-degenerate sea quarks would be needed to get the same information, or one would have to consider matrix elements like π|ss|π , which involve -numerically much harder -disconnected diagrams. It is clear that the N = 3 LECs cannot be obtained from quenched QCD, which has N = 0. If we can simulate PQQCD with three dynamical degenerate quarks at roughly the strange quark mass to a high enough accuracy to fit eq. (2), one could thus, for example, decide the fate of the Kaplan-Manohar ambiguity 5 from first principles.

Partially quenched field theory
PQQCD can be defined as a path integral, by considering the lagrangian 6 where q i v , i = 1 . . . K are the valence quarks and q α s , α = 1 . . . N the sea quarks. The valence quarks are made into such by adding K "ghost" quarksq j , j = 1 . . . K, which are identical to the valence quarks q i v (with the same masses m i v ), except that they are given opposite statistics. 7 Therefore, the valence and ghost determinants cancel, leaving only the sea-quark determinant, as in eq. (1). Obviously, the theory defined in this way is not a "healthy" field theory (it violates spin-statistics), but the interesting point is that one can pqchpt: submitted to World Scientific on November 3, 2000 study its unphysical, euclidean correlation functions, and extract information about the real world, as in the example of the previous subsection.
For this, we need to adapt ChPT to the partially quenched situation. This can be done systematically by observing that L has a large chiral symmetry group, and vice versa. Based on this, PQChPT can be developed in much the same way as in the usual case, using supertraces and superdeterminants to build graded-group invariants. 8,6 Quenched QCD corresponds to N = 0. Note that partially quenched QCD, unlike quenched QCD, contains QCD (for which N = 3, K = 0) as a special case. 6 I have not enough space to review (P)QChPT, 9 and will limit myself to a brief discussion of neutral mesons (i.e. those containing quarks and antiquarks of the same flavor). The tree-level two-point function for mesons q i γ 5 q i (no sum over i) is, 6,2,1,10 in the limit of large η ′ mass, equal to where M V V (M SS ) is the mass of a meson made from (degenerate) valence (sea) quarks. In the quenched theory (N = 0), the η ′ cannot be decoupled, leading to a two-point function 8,11 where m 0 is the parameter corresponding to the singlet part of the η ′ mass in the unquenched theory. The remarkable new feature in these two-point functions is the double-pole term. (Note that it disappears for M V V = M SS , N = 0, as it should.) This double pole leads to several infrared diseases, which are a consequence of (partial) quenching, and not present in full QCD: • "Enhanced chiral logarithms." 8,6,11 Instead of having the usual chiral logs like M 2 log M 2 , one finds logs such as ( • "Enhanced finite-volume effects." 12,13 See subsection 3.2.

Quenched ChPT and Lattice QCD
An important question is whether numerical computations are precise enough to discern one-loop effects in ChPT. The most obvious place to look is the quenched numerical results for GB masses and decay constants, for which the pqchpt: submitted to World Scientific on November 3, 2000 most extensive and statistically significant numerical results are available. In quenched ChPT, one finds, 8,11 for degenerate quarks f π /f = 1 + 4λ 5 y , (There are similar expressions for non-degenerate quark masses.) The parameter α parametrizes momentum dependence of the singlet GB two-point vertex, and appears in the renormalization of the η ′ field. 8,4 There are really two questions: 1) Given certain numerical results, can we see the chiral logs? and 2) Can we determine λ 5,8 ? CP-PACS 14 has high statistics results at quark masses ranging from m π ∼ 300 to 750 MeV, extrapolated to the continuum limit, and shows that this is sufficient to fit δ from eq. (6), setting α = 0. With this restriction, they see the chiral logs, and find δ ≈ 0.1. This answers the first question. A full analysis of their results has not appeared yet, but estimates for λ 5,8 can also be extracted. From their fits, I find, for example, 15 With these values, eq. (6) appears to converge well. These numbers were recently confirmed by Duncan et al., 16 who find δ = 0.065(13) (at one value of the lattice spacing, with improved Wilson fermions), 10 3 L 5 = 2.5(5) (cf. the value of the Alpha collaboration below!) and 10 3 (2L 8 − L 5 ) = 0.4(2) (again, taking α = 0). It remains to be seen what can be said when α is not constrained to vanish.
The Alpha collaboration 17 has recently presented high-statistics, continuum-extrapolated results in the range 590 − 670 MeV, which, as they point out, is not sufficient to demonstrate the presence of chiral logs. However, using eq. (6) with the CP-PACS value of δ = 0.12(2) (assuming that the one-loop chiral expansion converges for this mass range), they find, for α = 0 10 3 L 5 = 0.78 ± 0.05 ± 0.20 , 10 3 (2L 8 − L 5 ) = 0.28 ± 0.04 ± 0.10 ± 0.06 , (8) where the 1st (2nd, 3rd) error is statistical (higher-order uncertainty, error on δ). Arguing that, if α = 0.5, δ = 0.05(2) would be reasonable, they find with similar errors. It is clear that this value is very sensitive to α. All these estimates are quenched, and should not be directly compared to the real world. 3 Non-leptonic two-body kaon decays The simplest non-leptonic kaon decay is K + → π + π 0 , which is pure ∆I = 3/2. In order to determine this decay rate, one needs the weak matrix element where u L denotes the left-handed part of u, etc. On the lattice, one computes euclidean correlation functions, and this matrix element is contained in where π 0 (t 2 ) is a pion field with arbitrary momentum q at time t 2 , etc. (we take the kaon to be at rest). Inserting complete sets of states on both sides of O weak , and keeping only the leading exponential in t 1 on the kaon side, we find (with t 2 > t 1 > 0) The leading exponential corresponds to the state |n with the lowest energy, and this is the state with both pions at rest (i.e. q = 0), with E = 2m π ! This is a manifestation of the so-called Maiani-Testa theorem, 19 which states that it is impossible to obtain the physical matrix element from the large-time behavior of the euclidean correlation function when there are more particles in the final state. All we get is the unphysical matrix element with all mesons at rest. Note that this conclusion does not depend on what we pick q to be: only the total momentum in the decay is conserved. The desired matrix element, for which | q| = 1 2 m 2 K − 4m 2 π , is buried in the excited states. The insertion of O weak at a fixed time does not conserve energy.
At present, there are two approaches to this problem. One approach is to make use of the fact that the desired physical matrix element can be obtained from C(t 2 , t 1 ) in a finite spatial volume L 3 , in which the excited energy levels are discrete. 20 With periodic boundary conditions, the energies are quantized pqchpt: submitted to World Scientific on November 3, 2000 roughly as E n = 2 m 2 π + (2π n) 2 /L 2 (plus O(1/L 3 ) corrections due to finalstate interactions (FSI)), and one finds that the first excited states | n| = 1 have energy equal to the incoming energy m K if m π L ≈ 4 for realistic meson masses. So, if it is possible to determine the first excited-state exponential in C(t 2 , t 1 ) reliably on the lattice, one obtains the physical matrix element in a finite (but rather large) volume. This matrix element can be then converted to infinite volume, with a correction factor rather close to one. 20 This method, while theoretically "clean," may turn out to be hard to implement in practice. First, one needs numerical computations with a high enough precision to extract the first excited state. Then, one needs 2m π < m K < 4m π (the latter inequality such that the rescattering of the pions is elastic), and this implies rather small up and down quark masses (of order m strange /8 or less) if m K ≈ 500 MeV. For light quark masses at ∼ m strange /8 this means L ≈ 5 fermi, which is large by present standards. With a larger lattice kaon mass, L can be smaller. ChPT can be used to extrapolate to physical masses. 21 Because of the difficulty of the required numerical computations, one would like to test these ideas in a (partially) quenched setting, but it is not clear at the moment what the effects of quenching would be in this approach. (For instance, "enhanced" finite volume effects may occur, 12,13 see below.) It would be interesting to investigate this in (P)QChPT.
An alternative, and in fact older, idea is to compute the unphysical matrix element with all mesons at rest from the dominant term in eq. (12), for which |n = |π + ( 0)π 0 ( 0) , 22 and use ChPT in order to convert the result to the physical matrix element. ChPT thus "corrects for" the systematic error from using unphysical (energy non-conserving) momenta, and unphysical quark masses. In addition, ChPT can also be used to estimate effects from finitevolume, excited states and (partial) quenching. Whether this works in practice depends on how well the chiral expansion converges for the situation at hand. For the physical amplitude, one finds, for ∆I = 3/2 (in infinite volume), 21,23 Here where c is a numerical constant (for a more extensive discussion, see Bernard and Golterman 12 and refs. therein). The combined one-loop corrections in eq. (14) are rather large for typical lattice values of the parameters, and tend to increase the size of the matrix element. 23,3 The most recent numerical computation of this matrix element was done by JLQCD, 25 in the quenched approximation (i.e. N = 0), at an inverse lattice spacing of 2 GeV. It was found that eq. (14) describes the numerical results reasonably well, in particular the volume dependence, which has no scale ambiguities. They obtained d ′′ (m ρ )/(4π) 2 ≈ 0.025 and d ′′ (1 GeV)/(4π) 2 ≈ 0.015, with the difference in agreement with eq. (14).
A variant of this method is to take, on the lattice, m K = 2m π . 26 In this case the matrix element with all mesons at rest conserves energy, and, in that sense, is physical. It is also closer to the real world than the case with m K = m π , the choice made by JLQCD and in earlier lattice simulations. However, chiral corrections are typically still large. The matrix element for ∆I = 1/2 non-leptonic kaon decay constitutes a more difficult case for many reasons. First, lattice computations are harder because of the so-called "eye" diagrams (containing quark propagators starting and ending at the same space-time point). Also, unlike the ∆I = 3/2 case, mixing with the lower-dimension operator (m s − m d )sγ 5 d occurs. This problem can be avoided by choosing degenerate (valence) quarks, 22 or, in the case that m K = 2m π , by using an improved action. 26 In practice, we are therefore limited to the computation of the matrix element with m K = m π or m K = 2m π and all mesons at rest, if we can only determine the leading large-time behavior of eq. (11).
This means that ChPT will be needed in order to extract the relevant LECs from the unphysical matrix element, in order to calculate the physical matrix element. The question arises again which order in ChPT will be needed in practice. At one loop a problem occurs (for both ∆I = 1/2, 3/2), as it turns out that the O(p 4 ) LECs needed for the physical matrix element cannot be determined from the unphysical matrix element. This can be seen as follows. With arbitrary external four-momenta p K = (im K , 0), p π1 and p π2 with p 2 π1 = p 2 π2 = −m 2 π , the contribution from O(p 4 ) operators is expected to be of the form Last but not least, there are new problems associated with (partial) quenching. Even in the mass-degenerate case, the ∆I = 1/2 operator couples to the η ′ , and therefore the matrix element is afflicted by the double pole in the η ′ two-point function (eqs. (4,5)). (If SU (3) is unbroken, the η ′ does not couple to the ∆I = 3/2 operator.) It turns out that enhanced chiral logs are pqchpt: submitted to World Scientific on November 3, 2000 either absent 13 (∆I = 1/2 at m K = m π ) or numerically small 27 (∆I = 3/2 at m K = 2m π ), at one loop in ChPT. However, in the ∆I = 1/2 case, enhanced finite volume effects occur at one loop, 13 giving rise to "corrections" of order L to the matrix element! The double pole can occur on the internal lines of the meson rescattering diagram, leading to terms proportional to (cf. eq. (15)) 1 In finite volume, the smallest momentum has | k| = 2π/L, leading to contributions of order (1/L 3 ) × L 4 = L. This is another manifestation of the infrared diseases afflicting the (partially) quenched theory. In the quenched theory (N = 0), enhanced finite-volume effects are proportional to m 2 0 (cf. eq. (5)), while in the partially quenched case (N = 0) they are proportional to M 2 SS − M 2 V V (cf. eq. (4)). (For a more detailed version of this argument, see Bernard and Golterman. 12 ) All this means that we may have to be more modest, at least at present. While we may not easily acquire sufficient information to determine nonleptonic decay rates to O(p 4 ), it would be already interesting to determine the O(p 2 ) weak LEC α 27 and the corresponding octet LEC α 8 1 from the lattice. This, however, can be accomplished using simpler weak matrix elements, which avoid the enhanced finite-volume complication described above.
3.3 K → ππ from K → π and K → 0 where in the K → π case we choose m K = m π = M . α 8 2 is the O(p 2 ) LEC associated with the weak mass term. 28,21,10 There are no FSI, and therefore none of the unphysical effects on the lattice associated with them. a Oneloop corrections to eq. (19) have been calculated in (P)QChPT including all contributions from O(p 4 ), 10,13 and are typically found to be large (assuming a natural cutoff and no sizable cancellations between chiral logs and O(p 4 ) contact terms). A number of lattice collaborations are presently pursuing this approach, and it should be interesting to see what can be learned by fitting numerical results to O(p 4 ) ChPT. If, for certain results from the lattice, the convergence of ChPT is reasonable, one obtains the O(p 2 ) LECs α 27 and α 8 1 . Even though that does only give kaon decay rates to lowest order in ChPT, lattice results for these LECs are interesting by themselves, and can be compared with phenomenological estimates. 24 Of course one would also obtain information about certain O(p 4 ) LECs from such fits, but, as in the case of unphysical K → ππ matrix elements, one does not get enough information to determine all O(p 4 ) LECs relevant for the physical matrix elements. 10 Results for α 27 and α 8 1 from K → π and K → 0 can of course be checked against those obtained from unphysical K → ππ matrix elements, in principle.

Conclusion
In this talk, I have argued that ChPT plays a crucial role in extracting physics from numerical computations using Lattice QCD. The generic setting of such computations is that of partially quenched QCD, and ChPT can be systematically adapted to this situation. In fact, PQQCD with three light quarks subsumes the real world, i.e. unquenched QCD with valence-quark masses equal to those of the sea quarks. It follows that LECs of the partially quenched theory are the same as those of the real world. This means that they can often be determined from computations using unequal (and thus unphysical) values of the sea-and valence-quark masses, allowing to pick choices which are accessible with presently available computer resources (cf. section 2.1).
In contrast, quenched QCD has no sea quarks, and is therefore different from QCD. Because it is relatively cheap to simulate, it is a useful tool for investigating the issues discussed in this talk, for instance the convergence of ChPT at realistic values of the parameters.
In the second part, I addressed non-leptonic kaon decays. These involve the computation of weak matrix elements with more than one strongly intera Another issue is whether ChPT at O(p 4 ) takes physical FSI reliably into account. See Pallante and Pich 29 and refs. therein. Their results for FSI corrections to α 8 1 appear to be in reasonable agreement with the one-loop corrections of ChPT. 24 acting particle in the final state, and this leads to several difficulties. The first difficulty is that, when final-state interactions are present, one does not easily obtain matrix elements for a physical choice of momenta. This difficulty originates in the fact that on the lattice we only have access to the large-time regime of euclidean correlation functions. The second difficulty is that (partial) quenching artifacts, due to the unphysical infrared behavior of (partially) quenched theories, are more severe in this case. However, ChPT provides an analytic framework for parametrizing all these unphysical effects in terms of the LECs of the physical theory (when three reasonably light sea quarks are used on the lattice), and therefore can be used to extract phenomenologically relevant information from unphysical matrix elements. In particular, I argued that it should be possible to determine the leading weak LECs α 27 and α 8 1 reliably from Lattice QCD.