chiral perturbation theory for decay amplitudes
We use one-loop chiral perturbation theory ( ChPT) to study the behaviour of the form-factors for semileptonic decays with the pion mass at and at , where is the momentum transfer. At , the final-state pion has an energy of approximately (for ) and so is not soft, nevertheless it is possible to compute the chiral logarithms, i.e. the corrections of . We envisage that our results at will be useful in extrapolating lattice QCD results to physical masses. A consequence of the Callan-Treiman relation is that in the chiral limit (), the scalar form factor at is equal to , the ratio of the kaon and pion leptonic decay constants in the chiral limit. Lattice results for the scalar form factor at are obtained with excellent precision, but at the masses at which the simulations are performed the results are about 25% below and are increasing only very slowly. We investigate the chiral behaviour of and find large corrections which provide a semi-quantitative explanation of the difference between the lattice results and . We stress the generality of the relation in the chiral limit, where or and briefly comment on the potential value of using this theorem in obtaining physical results from lattice simulations.
One of the most precise methods to extract the element of the CKM-matrix is to use semileptonic decays ( decays), where is an electron or a muon. The combination can be determined from the experimental rate
where is a phase space integral which can be evaluated from the experimentally determined shape of the form-factors and , and contain the calculable corrections due to isospin breaking, electromagnetic and short-distance electroweak effects respectively. is the Clebsch-Gordan coefficient for the neutral (charged) kaon decay and is the form factor defined from
where is the momentum transfer . At we have . The Particle Data Group(2008)  quotes
so that in order to obtain we need to determine . In the last four years, following ref. , lattice QCD calculations of have been undertaken using dynamical simulations with or flavours of sea quarks[3, 4, 5, 6, 7], thus enabling the extraction of .
In this paper we investigate the behaviour of decay amplitudes with the masses of the and quarks. This is an interesting problem in itself, but the immediate motivation is the need to extrapolate the lattice results for , obtained with and quarks heavier than the physical ones (typically with pions with masses in the range ), to their physical values.
The mass dependence of the form factors is studied below using chiral perturbation theory (ChPT) at next-to-leading order (NLO) 111For compactness of notation in the remainder of this paper we refer to ChPT (for or 3) as ChPT.. Conventionally, following the seminal paper of Gasser and Leutwyler , it has been ChPT which has been applied to the study of decay amplitudes. However, following the study of the quark mass dependence of physical quantities computed in a lattice simulation using Domain Wall Fermions , together with our colleagues from the RBC and UKQCD collaborations we concluded that it may be better to use ChPT, at least for some quantities. This conclusion is primarily based on the large one-loop effects in ChPT found for the leptonic decay constant of ‘pions’ with masses in the range in which the simulations were performed. Note also that the strange quark mass () in lattice simulations can be chosen to be at its physical value and so ChPT is not needed to perform the corresponding extrapolation (although, since the bare strange quark mass is chosen before the simulation is undertaken, in practice there may have to be a small extrapolation to correct for the difference between the used in the simulation and its physical value; ChPT may provide useful guidance for this). Of course, in using rather than ChPT we sacrifice some symmetry and therefore some information.
In ref. , together with the RBC and UKQCD collaborations, we developed and used ChPT for kaon physics; in particular we studied the dependence on the pion mass of the mass of the kaon , the leptonic decay constant and the -parameter which contains the non-perturbative QCD effects in – mixing. We recall the main features of the formalism in section 2. An important difference between and ChPT is that with ChPT, powers of , where is the mass of the kaon in the limit and is the scale of chiral symmetry breaking, are absorbed into the low-energy constants (LECs). In ChPT at -loop order on the other hand, there remain errors of . The corresponding uncertainties in ChPT are of and .
In this paper we study two aspects of the chiral behaviour of the form factors:
The behaviour of with and . In order to determine from eq. (3) we need the form factor at for physical values of the quark masses. The result presented in eq. (31) below represents the behaviour of the form factor with the pion mass at NLO (one-loop order) and can be used to extrapolate the lattice results obtained at larger values of to the physical point.
In order to derive eq. (31) one has to overcome a subtlety. At ,
so that , the energy of the pion in the rest frame of the kaon, is approximately equal to and is not small (i.e. it is not of ) for . Since ChPT is an expansion in powers of masses and momenta of the pions, the fact that the external pion in semileptonic decays is hard complicates this power counting. Nevertheless, by integrating by parts, we show in section 3 that an expansion in small masses and momenta of is possible and results in eq. (31). This is possible because the chiral logarithms arise from soft regions of phase-space for the internal pions.
[MeV] [ GeV] 671(11) 0.00235(4) 1.00029(6) 556(9) 0.01252(20) 1.00192(34) 416(7) 0.03524(62) 1.00887(89) 329(5) 0.06070(107) 1.02143(132) Table 1: Results for from ref.  at four values of the quark masses, corresponding to the pion masses given in the first column.
The behaviour of with and . The maximum physical value of is , corresponding to the pion and kaon both at rest. Using the double ratio techniques proposed in ref. , is evaluated with remarkable precision in lattice simulations. For illustration we reproduce in table 1 the results from the RBC and UKQCD collaborations’ simulation on a spatial lattice . The point which we particularly wish to underline here is that in the chiral limit (), the Callan-Treiman relation  implies that
where and are the kaon and pion leptonic decay constants in the chiral limit. The Callan-Treiman relation was derived for the unphysical value of , nevertheless, in the chiral limit it also holds for . We shall show however, that the corrections to the relation are of and not the standard ChPT corrections of . The ratio of physical decay constants is and in the chiral limit it is a little larger, e.g. the lattice study of ref.  finds a ratio 222Enno Scholz private communication. This particular result is not quoted directly in .. In ref.  the entries in table 1 were obtained with a strange quark mass which is a little larger than the physical one and the corresponding value of is about 1.28. This is the value which we use in the numerical estimates below, together with , which is the central value found in . We restrict the comparison of ChPT with table 1 to the entries with and , since our experience from ref.  is that one-loop ChPT is less reliable at the heavier masses. The values of in table 1 are equal to 1 within 2% or so, and although they are increasing as the quark masses are reduced, the observed increase is very slow indeed. As decreases from to , increases only from 1.00 to 1.02 which is still a long way from the expected value of about 1.28 in the chiral limit. We investigate the chiral behaviour of up to one-loop order in section 4 and find that the chiral logarithm has a large coefficient but the wrong sign to account for the extrapolation to . The coefficient of the linear term in , which is not calculable in ChPT, can be estimated by converting the results of ref.  to the theory. We find that it is large with the correct sign but predicts too large a ratio between in the chiral limit and at the masses where lattice simulations are performed. We also study the full prediction from ChPT, which reproduces qualitatively (and semi-quantitatively) the observed behaviour.
The plan for the remainder of this paper is as follows. In the following section we briefly recall some of the main features of ChPT for kaon physics. Within this context, we also derive eq. (4). Sections 3 and 4 contain the studies of the chiral behaviour of the form factors at and respectively. Our calculations have some overlap with those of semileptonic decays of -mesons [11, 12, 13, 14, 15, 16, 17] and we discuss the similarities and differences in section 5.
2 Chiral Perturbation Theory for Kaons
We start by briefly summarising the formalism introduced in section II.B of ref.  which we apply in the following sections to decays. We write the pion matrix, the quark mass matrix and the kaon fields in the form:
We work in the isospin limit so that represents . The pion matrices and are defined in the standard way:
where is the pion decay constant in the chiral limit, . As with all LECs in ChPT, depends on , the mass of the strange quark. Throughout this paper we define the pion and kaon decay constants using a normalization in which the physical value for the pion is .
We need to construct the chiral Lagrangian and operators which transform in a specified way under chiral transformations out of the fields in eqs. (5) and (6). Under global left and right handed transformations, and respectively, these fields transform as follows:
where is a function of and the meson fields which reduces to a global vector transformation when . From the transformations in eq. (7) we construct operators with the required flavour and chiral quantum numbers.
The pion Lagrangian at lowest order is well known:
where is the standard lowest order LEC and to this order . For the interactions of kaons, which are not considered soft in the ChPT formalism, with soft pions the chiral Lagrangian has been introduced by Roessl  and at lowest order is given by
where the covariant derivative is constructed using the vector field ,
and is defined by
In the following it will be necessary also to introduce the pion axial vector field defined by
When constructing Feynman diagrams from the Lagrangian and the effective theory local operators, we expand the vector and axial fields in terms of the pion fields,
so that the first term in the expansion of the vector field contains two pions and that for the axial field starts with a single pion.
Similar calculations for and semileptonic decays using heavy meson ChPT were undertaken in ref. [15, 16] and we will discuss the similarities and differences with kaon decays in more detail in section 5. Here we simply point out that in the heavy meson ChPT, the limit is taken before performing the chiral expansion. The resulting spin symmetry implies that the vector meson is degenerate with the pseudoscalar , and so the interactions (where the pion is soft), and hence diagrams containing propagators, must also be included. In our kaon case, the mass splitting is considered to be of and so the corresponding diagrams are absent.
2.1 Lowest Order vector and axial currents
We end this section by discussing the lowest order vector and axial currents in the effective theory. As already mentioned in the Introduction, it will not be enough to consider only the lowest order contributions and we will have to extend the present discussion in the following two sections.
The left handed QCD current is
where or . It is convenient to promote to be a 2-component vector with components and and to introduce a 2-component constant spurion vector in order to be able to project and as required; specifically we write the left-handed current as
The current in eq. (15) would be invariant under transformations if transformed as . We now construct the form of the left-handed current in the effective theory. This is a linear combination of all operators which are linear in and which would be invariant under transformations if transformed as above. At lowest order in the chiral expansion we identify two possible independent terms and, following the notation of ref. , we write the left-handed current as
where and are LECs and is the pion axial current defined in eq. (12). Note that since (and also ), no new independent operator is obtained by replacing by the covariant derivative in the second term on the right-hand side of eq. (16).
For the right-handed current, we take the transformation on to be and obtain two possible operators at lowest order,
which transform as . Noting that parity transformations, under which
transform into (where is the right-handed current), so that the same LECs, and appear in the right-handed current,
In some applications, the term can be considered to be sub-leading since the derivative is on the pion rather than the kaon field. The vector () and axial-vector () currents can now readily be determined:
The LEC appears in both the vector and axial-vector currents and we will see in section 4 that it is this feature which allows us (in the chiral limit) to relate the matrix element of the axial-vector current and the matrix element of the vector current and hence to derive eq. (4). Evaluating the matrix element in the chiral limit immediately shows us that
3 form factors at
As discussed in the Introduction, in order to study the chiral behaviour of the form factor at we have to deal with the fact that in this case and hence we cannot neglect operators with an arbitrary numbers of derivatives on the external pion field. This situation is reminiscent of the light-cone dominated process of deep-inelastic scattering. To illustrate the point consider the matrix element
at . In spite of the additional derivative acting on the pion field relative to the first term in in eq. (20), the matrix element in (23) does give a leading contribution in the chiral expansion since by inspection we see that there is a contribution of times the matrix element of . Nevertheless, as we now show, the leading contribution is simply proportional to the matrix element of (with a constant of proportionality which depends on but not on ) and so the chiral logarithms are the same and the number of LECs remains the same. To see this, note that at ,
Before discussing the chiral behaviour of the operators on the right-hand side of eq. (3) we clarify our power counting. The external pion is hard in the sense that and so we need to keep to any power. This is the reason the matrix element in eq. (23) is of leading order. We accept that the corrections of are multiplied by an unknown constant and so we do not attempt to calculate such terms. We do however, calculate the chiral logarithms, i.e. the corrections of and in order to evaluate these we can treat the internal pion momenta as being soft, i.e. of .
The operator in the second term on the right hand side of eq. (3) contains the insertion . This leads to a contribution which is suppressed by a factor of , with no chiral logarithm proportional to . Thus, up to the order to which we are working, we only need to consider the first term on the right-hand side of eq. (3), where we can replace by up to terms which are suppressed by . Note that the commutator contains two derivatives acting on two different pion fields, at least one of which must be Wick contracted to give a soft internal propagator. This leads to a suppression of without chiral logarithms and we arrive at the useful result that, up to corrections of (without chiral logarithms):
Thus in order to include the contribution of the matrix element (23) to the form factor , including the one-loop chiral logarithms, it is sufficient to replace the LEC in the definition of the vector current in eq. (20) by an unknown coefficient which depends on but not on the light-quark masses.
The discussion of the matrix element in (23) presented explicitly above can be generalised to other operators. Leading-order operators can have any number of covariant derivatives on the external pion field. If the Lorentz index of a covariant derivative acting on the external pion field is contracted with another derivative on the external pion then we obtain a non-leading correction of . Similarly if it is contracted with a derivative on a pion in an internal loop we also obtain a similar suppression. Finally if it is contracted with a derivative on the kaon field then we can reduce it to an operator which is proportional to the leading operator by integrating by parts as above. Note also that the kaon mass-squared, , has no chiral logarithms of the form and so no chiral logarithms are introduced by using the equations of motion.
From this discussion we see that to leading order at we have
where we recall that and are unknown constants which cannot be obtained from and alone. They depend on but not on the light-quark masses and hence we treat them as LECs, noting however that they are only relevant for the case . As a result of the fact that the matrix element at is written in terms of and rather than and , we lose the connection to in this case.
3.1 The Chiral Logarithms
The tree level contribution to the matrix element in eq. (3) is
This contribution is represented diagrammatically in fig. 1(a).
In order to obtain the chiral logarithms at one-loop order we need to evaluate the diagrams in fig. 1(b), (c) and (d), where fig. 1(d) represents the contribution to the pion’s wave-function renormalization. There is no one-loop chiral logarithm contributing to the kaon’s wave function renormalization which can be deduced from the structure of the term in the strong Lagrangian in (9). The vertex arises when one keeps a partial derivative from one of the factors and the current from the other. From eq. (13) we see that the expansion of starts with two pion fields, on one of which there is a single derivative. This derivative corresponds to a single momentum in the numerator of the tadpole loop and hence the momentum integration is odd and gives zero. The chiral logarithms from each of the diagrams in fig. 1 are presented in table 2 together with the total.
Gasser and Leutwyler have calculated the chiral logarithms in the theory as a function of . In this case the power counting is different from that here, in that is also considered to be small. It is instructive to check our calculation by converting the results to , using eq. (2.6) of ref.  and the expression for in eq. (A.7) of ref. . Expanding the Gasser-Leutwyler results in powers of , we confirm that the total one-loop chiral logarithms in table 2 are indeed correct.
From table 2 we now have all the ingredients to write down the NLO expression for the matrix element at . The expression is
where and and are LECs. ( and should not be confused with the leptonic decay constants for which we use the notation and .) The chiral logarithm is defined by
and the dependence of on is cancelled in expressions for physical quantities by the -dependence of the LECs (e.g. in eq. (29) the dependence of is cancelled by that of and ). Eq. (29) implies that the chiral behaviour of the form factors is given by
where again and are LECs, given in terms of the parameters present in eq. (29) (for example, ).
Eq. (31) is the NLO ChPT formula for extrapolating the lattice results for which are obtained at unphysical values of the up and down quark masses to the physical point. The two LECs and need to be determined by fitting the mass dependence of the measured values of to (31); the physical result of is then readily obtained. Of course, using ChPT the Ademollo-Gatto theorem  ensures that there are no LECs at one-loop order so that and are known and we can rewrite eq. (31) as
It is conventional for experimental results to be presented in terms of and so we have concentrated above on the chiral behaviour of the form factors at . We can perform a similar analysis for any value of with , but the effective LECs, i.e. the and depend on .
4 form factor at
We now turn our attention to the form factor , where . The tree-level diagram for the decay is drawn in fig. 1(a) and its contribution to the amplitude is given in the first row of table 3. By setting for example and recalling that (see eq. (22)) we see that in the chiral limit and hence establish (4).
In the remainder of this section we try to understand why the lattice results for in table 1 are significantly different from the value in the chiral limit, , and seem to be approaching this value very slowly, if at all. At the momentum of the external pion is small () and so the counting of contributions in terms of powers of is simpler than at . However, close to the chiral limit () the corrections to the matrix element of the vector current are linear in (and not quadratic). To see this, consider for example, the matrix element in eq. (23), , which is now manifestly linear in . The coefficient of the linear term is not calculable directly in ChPT.
Below we study the chiral behaviour of in three stages as follows:
We start in section 4.1 by calculating the one-loop chiral logarithms, i.e. the corrections of . This can be done within ChPT.
In order to estimate the remaining terms we use ChPT. In the second stage, in section 4.2 below we estimate the the coefficient of the linear term in by converting the one-loop expressions to .
Finally, in section 4.3 we also estimate the quadratic terms using ChPT .
We will find that at the chiral corrections are very large and provide only a qualitative or perhaps a semi-quantitative, explanation of the observed chiral behaviour. Nevertheless, the calculations confirm that the differences of the lattice results from are reasonable.
4.1 The chiral logarithms
The chiral logarithms from each of the diagrams in fig. 1 are presented in table 3. From the table, choosing the Lorentz index and neglecting terms of , with or without logarithms, we deduce that the chiral behaviour of the form factor is of the form:
where are low energy constants which depend on the strange quark mass but not on the light quark masses. Again one can readily verify that the coefficient of the chiral logarithm in (34) is indeed the result obtained by converting the general formulae of Gasser and Leutwyler [19, 8] to .
The coefficient is large (for example, at in eq. (29) the coefficient of is ) and the term with the chiral logarithm does give a sizeable contribution in the region of pion masses between the physical one and that where the lattice simulations of ref.  were performed. In fig. 2 we sketch with the physical mass of the -meson as the scale and with which is the central value found in  . The sign of the chiral logarithm however, is such as to make the form factor decrease as the mass of the pion is decreased towards the chiral limit, which is the opposite of what is required to account for the difference between the measured values of in table 1 and . Thus the chiral logarithms approximately double the size of the effect which should be explained.
4.2 Linear term in
We cannot evaluate using ChPT alone. To estimate whether the linear term in in (34) can account for the difference of the measured form factors from we convert the results of Gasser and Leutwyler [19, 8] to ChPT. In this way we can obtain an approximate value of , using which we rewrite (34) as:
where is the mass of the kaon in the chiral limit. Eq. (35) represents an approximation for since, within ChPT, contains higher powers of , whereas in (35) we have kept only those from one-loop ChPT. Setting and neglecting higher order terms, we plot the expression in square parentheses in (35) as a function of the pion mass as the solid curve in fig. 3, where we have set , and . We notice that the linear term in does indeed change the sign, the value of the form-factor does increase as we approach the chiral limit. The effect is too large however, and since the term is as large as 50-80% in the region where we have data, the stability of the chiral expansion is likely to be questionable.
We write the converted expression from ChPT as eq. (35) because this is the natural form for ChPT. To illustrate that the result may depend significantly on the higher order terms we also plot, as the dashed curve in fig. 3, the expression
which is equivalent to that in square parentheses (with ) in eq. (35) at one-loop order in ChPT but differs by terms which are powers of . We make this choice because it is the form obtained directly from one-loop ChPT. The difference in the curves in the region where we have lattice data is about 25-30%, confirming that the uncertainties due to higher order terms are indeed likely to be large.
Finally we use the Gasser-Leutwyler