from decays and (2+1)-flavor lattice QCD
We present a lattice-QCD calculation of the semileptonic form factors and a new determination of the CKM matrix element . We use the MILC asqtad 2+1-flavor lattice configurations at four lattice spacings and light-quark masses down to 1/20 of the physical strange-quark mass. We extrapolate the lattice form factors to the continuum using staggered chiral perturbation theory in the hard-pion and SU(2) limits. We employ a model-independent parameterization to extrapolate our lattice form factors from large-recoil momentum to the full kinematic range. We introduce a new functional method to propagate information from the chiral-continuum extrapolation to the expansion. We present our results together with a complete systematic error budget, including a covariance matrix to enable the combination of our form factors with other lattice-QCD and experimental results. To obtain , we simultaneously fit the experimental data for the differential decay rate obtained by the BaBar and Belle collaborations together with our lattice form-factor results. We find where the error is from the combined fit to lattice plus experiments and includes all sources of uncertainty. Our form-factor results bring the QCD error on to the same level as the experimental error. We also provide results for the vector and scalar form factors obtained from the combined lattice and experiment fit, which are more precisely-determined than from our lattice-QCD calculation alone. These results can be used in other phenomenological applications and to test other approaches to QCD.
pacs:13.20.He, 12.38.Gc, 12.15.Hh
Present address: ]Department of Physics and Astronomy, University of Iowa, Iowa City, IA, USA
Present address: ]Laboratory of Biological Modeling, NIDDK, NIH, Bethesda, Maryland, USA
Fermilab Lattice and MILC Collaborations
The Cabibbo-Kobayashi-Masakawa (CKM) matrix Cabibbo:1963yz (); Kobayashi:1973fv () element is one of the fundamental parameters of the Standard Model and is an important input to searches for CP violation beyond the Standard Model. Constraints on new physics in the flavor sector are commonly cast in terms of over-constraining the apex of the CKM unitarity triangle. In contrast to the well-determined angle of the unitarity triangle, the opposite side is poorly determined, and the uncertainty is currently dominated by . This is due to the fact that charmless decays of the meson have far smaller branching fractions than the charmed decays, as well as the fact that the theoretical calculations are less precise than for , , or . Currently the most precise determination of is obtained from charmless semileptonic decays, using exclusive or inclusive methods that rely on the measurements of the branching fractions and the corresponding theoretical inputs. Exclusive determinations require knowledge of the form factors, while inclusive determinations rely on the operator product expansion, perturbative QCD, and non-perturbative input from experiments. There is a long standing discrepancy between determined from inclusive and exclusive decays: the central values from these two approaches differ by about . It was argued in Ref. Crivellin:2014zpa () that this tension is unlikely to be due to new physics effects, and it is therefore important to examine the (theoretical and experimental) inputs to the determinations. With the result obtained in this paper, the tension is reduced to .
In the limit of vanishing lepton mass, the Standard Model prediction for the differential decay rate of the exclusive semileptonic decay is given by
where is the pion momentum in the -meson rest frame. To determine , the form factor must be calculated with nonperturbative methods. The first unquenched lattice calculations of with 2+1 dynamical sea quarks were performed by HPQCD hep-lat/0601021 () and the Fermilab/MILC collaborations 0811.3640 () several years ago. Here we extend and improve Ref. 0811.3640 () in several ways.
The most recent exclusive determination of from the Heavy Flavor Averaging Group (HFAG) 1207.1158 () is based on combined lattice plus experiment fits and yields , where the error includes both the experimental and theoretical uncertainties. The experimental data included in the average are the BaBar untagged six--bin data 1005.3288 (), the BaBar untagged twelve--bin data 1208.1253 (), the Belle untagged data 1012.0090 (), and the Belle hadronic tagged 1306.2781 () data. The theoretical errors on the form factors from lattice QCD 0811.3640 () are currently the dominant source of uncertainty in 1209.4674 (). Hence a new lattice calculation of with improved statistical and systematic errors is desirable 111Note that there are several other efforts with 2 1211.6327 () and 2+1 flavors of sea quarks Flynn:2015mha (); 1310.3207 ().. To compare, the value of from the inclusive method quoted by HFAG is about 1207.1158 () using the theory of Ref. Bosch:2004bt ().
In this paper, we present a new lattice-QCD calculation of the semileptonic form factors and a determination of . Our calculation shares some features with the previous Fermilab/MILC calculation 0811.3640 () but makes several improvements. We quadruple the statistics on the previously used ensembles and improve our strategy for extracting the form factors by including excited states in our three-point correlator analysis. In addition, we include twice as many ensembles in this analysis. The new ensembles have smaller lattice spacings, with the smallest lattice spacing decreased by half. This analysis also includes ensembles with light sea-quark masses that are much closer to their physical values ( versus ). The smaller lattice spacings and light-quark masses provide much better control over the dominant systematic error due to the chiral-continuum extrapolation. We find that heavy-meson rooted staggered chiral perturbation theory (HMrSPT) in the SU(2) and hard-pion limits provides a satisfactory description of our data. All together, these improvements reduce the error on the form factors by a factor of about 3. Finally, we introduce a new functional method for the extrapolation over the full kinematic range.
The determination of from a combined fit to our lattice form factors together with experimental measurements also yields a very precise determination of the vector and scalar form factors over the entire kinematic range. These form factors will be valuable input to other phenomenological applications in the Standard Model and beyond. An example is the rare decay , which we will discuss in a separate paper.
Because our primary goal in this work was a reliable and precise determination of , we employed a blinding procedure to minimize subjective bias. At the stage of matching between the lattice and continuum vector currents, a slight multiplicative offset was applied to the data that was only known to two of the authors. The numerical value of the blinding factor was only disclosed after the analysis and error-estimation procedure, including the determination of , were essentially finalized.
This paper is organized as follows. In Sec. II, we present our calculation of the form factors. We describe the lattice actions, currents, simulation parameters, correlation functions and fits to extract the matrix elements, renormalization of the currents, and adjustment of the form factors to correct for quark-mass mistunings. In Sec. III, we present the combined chiral-continuum extrapolation, followed by an itemized presentation of our complete error budget in Sec. IV. We then extrapolate the form factors to the full range through the functional expansion method in Sec. V. We also perform fits to lattice and experimental data simultaneously, to obtain . We conclude with a comparison to other results and discussion of the future outlook in Sec. VI. Preliminary reports of this work can be found in Refs. Du:2013kea (); Bailey:2014fpx ().
Ii Lattice-QCD simulation
In this section, we describe the details of the lattice simulation. We briefly describe the calculation of the form factors in Sec. II.1. We also calculate the tensor form factor, which follows a analysis similar to that of the vector and scalar form factors. The tensor form factor enters the Standard-Model rate for decay, and our final result for will be presented in a forthcoming paper. In Sec. II.2, we introduce the actions and simulation parameters used in this analysis. This is followed, in Sec. II.3, by a brief discussion of the currents and lattice correlation functions. The correlator fits to extract the lattice form factors are provided in Sec. II.4. In Sec. II.5, we discuss the renormalization of the lattice currents. In Sec. II.6, we correct the form factors a posteriori to account for the mistuning of the simulated heavy -quark mass.
ii.1 Form-factor definitions
The vector and tensor hadronic matrix elements relevant for semileptonic decays can be parameterized by the following three form factors:
where , and . In lattice gauge theory and in chiral perturbation theory, it is convenient to parameterize the vector-current matrix elements by hep-ph/0101023 ()
where is the four velocity of the meson and is the projection of the pion momentum in the direction perpendicular to . The pion energy is related to the lepton momentum transfer by . With this setup, we have
where no summation is implied by the repeated indices here. The form factors and are
ii.2 Actions and parameters
The lattice gauge-field configurations we use have been generated by the MILC Collaboration 0903.3598 (); Bernard:2001av (); Aubin:2004wf (), and some of their properties are listed in Table 1. These twelve ensembles have four different lattice spacings ranging from fm to fm with several light sea-quark masses at most lattice spacings in the range . The parameter range is shown in Fig. 1. We use the Symanzik-improved gauge action Weisz:1982zw (); Weisz:1983bn (); Luscher:1984xn () for the gluons and the tadpole-improved (asqtad) staggered action hep-lat/9609036 (); Lepage:1998vj (); hep-lat/9806014 (); hep-lat/9805009 (); hep-lat/9903032 (); hep-lat/9912018 () for the 2+1 flavors of dynamical sea quarks and for the light valence quarks. Both Table 1 and Fig. 1 also indicate the ensembles used in the previous Fermilab/MILC calculation 0811.3640 (). The current analysis benefits from an almost quadrupled increase in the statistics over that of Ref. 0811.3640 (), as well as finer lattice spacings and lighter sea-quark masses. All ensembles have large enough spatial volume, , such that the systematic error due to finite-size effects is negligible compared to other uncertainties.
|(fm)||volume||(Ref. 0811.3640 ()))|
In this calculation, we work in the full-QCD limit, so that the light valence-quark masses are the same as the light sea-quark masses , which are degenerate. For the bottom quarks, we use the Fermilab interpretation hep-lat/9604004 () of the Sheikholeslami-Wohlert clover action Sheikholeslami:1985ij (). In Table 2, we list parameters for the valence quarks.
Table 3 lists the values of on each ensemble, along with other derived parameters, where is the characteristic distance between two static quarks such that the force between them satisfies hep-lat/0002028 (); hep-lat/9310022 (). The absolute lattice scale is obtained by comparing the Particle Data Group value of with lattice calculations of from MILC 0910.2966 () and HPQCD 0910.1229 (), yielding the absolute scale fm 1112.3051 (). This value is consistent with the independent, but less precise, determination from RBC/UKQCD using domain-wall fermions Arthur:2012opa ().
ii.3 Currents and correlation functions
We calculate the two-point and three-point functions
where , labels the pseudoscalar meson, the operators () annihilate (create) the states with the quantum numbers of the pseudoscalar meson on the lattice, and are the lattice currents.
For the meson, we use a mixed-action interpolating operator which is a combination of a Wilson clover bottom quark and a staggered light quark 0811.3640 ():
where , , and is a smearing function. For the pion, we use the operator
which is constructed from two 1-component staggered quarks.
The current operators are constructed in a similar way:
where the heavy quark field spinor is rotated to remove tree-level discretization effects, via hep-lat/9604004 ()
Figure 2 illustrates the three-point correlation function used to obtain the lattice form factors. The current operator is inserted between the - and -quark lines. The three-point functions depend on both the current insertion time and the temporal separation between the and mesons. The signal to noise ratio is largely determined by . A convenient approach is to fix the source-sink separation in the simulations and then insert the current operators at every time slice in between. The source-sink separations at different lattice spacings, sea-quark masses, and recoil momenta are chosen to be approximately the same in physical units. To minimize statistical uncertainties and reduce excited-state contamination, we tested data with different source-sink separations before choosing those shown in Table 2. The meson is at rest in our simulation, while the daughter pion is either at rest or has a small three-momentum. The light-quark propagator is computed from a point source so that one inversion of the Dirac operator can be used to obtain multiple momenta. The spatial source location is varied randomly from one configuration to the next to minimize autocorrelations. The -quark source is always implemented with smearing based on a Richardson 1S wave function menscher2005 () after fixing to Coulomb gauge. We compute both the two-point function and three-point function at several of the lowest possible pion momenta in a finite box: , , , and , where contributions from each momentum are averaged over permutations of components. We find the correlation functions with momentum too noisy to be useful, so we exclude these data from our analysis.
ii.4 Two-point and three-point correlator fits
In this subsection, we describe how to extract the desired matrix element from two- and three-point correlation functions. With our choice for the valence-quark actions and for the interpolating operators, the two- and three-point functions take the form hep-lat/0211014 ()
where is the temporal length of the lattice and
Note that due to the staggered action used for the light quarks, the meson interpolating operators also couple to the positive parity (scalar) states which oscillate in Euclidean times and with the factors and .
Our goal is to extract , the ground state matrix element from these correlation functions. To suppress the contributions from the positive parity states to the ratio, we follow the averaging procedure of Ref. 0811.3640 (), which exploits the oscillating sign in their Euclidean time dependence. The time averages can be thought of as a smearing over neighboring time slices to significantly reduce the overlap with opposite-parity states. Denoting the averaged correlators by and , we then use the ratio 0811.3640 ()
where and are the ground-state pion energy and -meson rest mass, respectively. The uncertainty in the -meson rest mass has significant impact on the ratio , so we follow a two-step procedure. We first determine the pion and -meson ground-state energy as precisely as possible using the corresponding two-point functions. We then feed these ground-state energies into the ratio , preserving the correlations with jackknife resampling.
For the pion two-point functions at zero momentum, the oscillating states — the terms in Eq. (17) with odd powers of — do not appear. Thus, we fit the pion two-point functions using Eq. (17) with the lowest two non-oscillating states (). For the two-point functions with nonzero momentum, the contribution from oscillating states is small but noticeable. We find that we only need to include the lowest three states ( in the fits. Because the momenta we consider are typically small compared to , the continuum dispersion relation is satisfied within statistical errors, as shown in Fig. 3. In the main analysis, we therefore use the mass from the zero-momentum fit and the continuum dispersion relation to set for non-zero momentum. Because the zero-momentum energy has significantly smaller statistical error than that of nonzero momentum, using this choice and the dispersion relation for nonzero-momentum energy leads to a more stable and precise determination of . Table 4 lists the relevant fit ranges for the two-point fits. In the two-point correlators (except the zero-momentum pion two-point correlators), the noise grows rapidly with increasing , the distance away from the pion source in the temporal direction. The data points at large are not useful, and including them would lead to a larger covariance matrix which would be difficult to resolve given the limited number of configurations. We choose the upper end of the fit ranges such that the relative error does not exceed 20%. The lower end is chosen such that the excited state contamination is sufficiently small, i.e., the resulting central values of the ground state energy are stable against variations in as shown in Fig. 4 (left).
In our analysis, there are two places where quantities from the -meson two-point functions are needed. The first is for in Eq. (21). The second is for the -meson excited state energy in Eq. (22) below. For the determination of in Eq. (21) we use two-point functions constructed with a 1S-smearing function in the interpolating operators for the source and sink. The 1S-smeared operator has good overlap with the ground state and a much smaller overlap with the excited states than the local source operator, thus reducing excited-state contributions to the corresponding correlators. We fit the 1S-1S smeared -meson two-point correlators with relatively large to only two states ( in Eq. (17)). To choose , we again apply the 20%-rule on the relative error. The lower bound is chosen in a manner similar to the pion two-point fits and the stability plot is shown in Fig. 4 (right). The chosen fit ranges are shown in Table 4.
|0.12||0.01/0.05||[6, 30]||[9, 21]|
|0.007/0.05||[6, 30]||[9, 21]|
|0.005/0.05||[6, 30]||[9, 21]|
|0.09||0.0062/0.031||[9, 47]||[12, 32]|
|0.00465/0.031||[9, 47]||[12, 29]|
|0.0031/0.031||[9, 47]||[13, 29]|
|0.00155/0.031||[9, 47]||[14, 29]|
|0.06||0.0072/0.018||[13, 71]||[14, 41]|
|0.0036/0.018||[13, 71]||[14, 41]|
|0.0025/0.018||[13, 71]||[14, 41]|
|0.0018/0.018||[13, 71]||[15, 41]|
|0.045||0.0028/0.014||[17, 74]||[17, 61]|
We test for autocorrelations by blocking the configurations on each ensemble with different block sizes, and then using a single-elimination jackknife procedure to propagate the statistical error to the two-point correlator fits for and . We do not observe any noticeable autocorrelations in all the ensembles we use, as illustrated in Fig. 5 for the coarsest and finest ensembles, and choose not to block the data.
The ratios in Eq. (21) have the advantage that the wavefunction overlap factors cancel, but the trade-off is that we need an additional factor — the square root term on the right-hand side — to remove the leading dependence in the ratio. If the lowest lying states dominated the ratio , then it would be constant in and proportional to the lattice form factor . The subscript now runs over , , and , corresponding to the operators , , and , respectively. Our previous analysis employed a simple plateau fit constant in time. With our improved statistics, the small excited-state contributions to the ratio are significant and cannot be neglected. On the other hand, even with our improved statistics, we find that contributions to from wrong-parity states are still negligible. We use two different fit strategies to remove excited state contributions and use the consistency between them as an added check that any remaining excited state contamination is negligibly small.
The first strategy starts with the ratio in Eq. (21) and minimally extends the plateau fitting scheme by including the first excited state of the meson in the following form:
where and are unconstrained fit parameters, is the lowest energy splitting of the pseudoscalar meson, and the prefactors are and . We choose the fit ranges for such that contributions from pion excited states to can be neglected. The fit parameter is determined by the -meson two-point correlators. In practice, we fit the ratio in Eq. (22) along with the -meson two-point correlation functions with as a common parameter. We find it beneficial in the combined fit to include both the local and smeared two-point correlation functions. We use 2+2 states for both correlators, but use a different set of fit ranges (listed in Table 5). The results of these two-point fits are shown in Fig. 6. The agreement in the -meson energies between the separate and combined fits is very good, but the combined fit leads to smaller errors.
|0.12||0.01/0.05||[9, 23]||[7, 21]||[6,12]||[6,12]||[6,12]|
|0.007/0.05||[9, 23]||[7, 21]||[6,12]||[6,12]||[6,12]|
|0.005/0.05||[9, 23]||[7, 21]||[6,12]||[6,12]||[6,12]|
|0.09||0.0062/0.031||[12, 32]||[9, 32]||[9,16]||[9,16]||[9,16]|
|0.00465/0.031||[12, 32]||[9, 29]||[9,16]||[9,16]||[9,16]|
|0.0031/0.031||[12, 32]||[9, 29]||[9,16]||[9,16]||[9,16]|
|0.00155/0.031||[12, 29]||[9, 29]||[9,15]||[9,15]||[9,15]|
|0.06||0.0072/0.018||[13, 41]||[9, 41]||[12,22]||[12,22]||[12,22]|
|0.0036/0.018||[13, 41]||[9, 41]||[12,22]||[12,22]||[12,22]|
|0.0025/0.018||[13, 41]||[9, 41]||[12,22]||[12,22]||[12,22]|
|0.0018/0.018||[13, 41]||[9, 41]||[12,21]||[12,21]||[12,21]|
|0.045||0.0028/0.014||[16, 61]||[10, 61]||[16,26]||[16,26]||[16,26]|
To summarize our strategy, for the case of zero momentum, we fit the ratio together with the local and smeared -meson two-point correlators , simultaneously. For non-zero momentum , is common to all three ratios, . Thus, we perform a combined fit to the five quantities: and . Figure 7 shows an example of these fits. Figure 8 shows the stability plots of against the variations in the fit ranges of the ratio fits, and the variations in the fit ranges of both two-point correlators. The preferred fit ranges are set to be in the stable region upon these variations.
Our second fit strategy includes excited-state contributions from both the pion and the meson. It starts with a different ratio, without time averages, which ensures that there are enough data points to constrain all the parameters:
where the two- and three-point correlators are defined in Eqs. (10) and (11). We fit with all the possible states with in Eqs. (17) and (LABEL:eq:C_3pt), combining the fits to the pion and -meson two-point correlators. We compare the fit results of the two different fit schemes in Fig. 9. The first (simple) fit model described in Eq. (22) gives, fitting either simultaneously or individually to the three lattice form factors , results that are consistent with the second fit model that includes the full set of first excited states in Eq. (23). In contrast, the plateau fits to defined in Eq. (21) yield results that are as much as one statistical smaller. In summary, we find that the first fit strategy described by Eq. (22) is sufficient to remove contributions from excited states, and we therefore adopt this method for the main analysis.
We match the lattice currents to continuum QCD with the relation,
where and denote the vector or tensor currents in the continuum and lattice theories, respectively, and “” means “has the same matrix elements” Kronfeld:2000ck (). We calculate the current renormalization with the mostly nonperturbative renormalization method hep-ph/0101023 (); Harada:2001fi (),
where and are the matching factors for the corresponding flavor-conserving vector currents. These factors capture most of the current renormalization. The remaining flavor off-diagonal contribution to the matching factor, , is close to unity.
We calculate the factors and nonperturbatively for each ensemble by computing the matrix elements of the flavor-conserving vector currents and using the relations
where the lattice current is a bilinear of light staggered quark fields and is a bilinear of clover heavy quark fields. The factors and are listed in Table 6. Because there is very little dependence in the factor , we use the same for ensembles with different light quark masses but the same lattice spacing. The factor depends crucially on the heavy quark mass, though it has negligible light quark mass dependence.
We use lattice perturbation theory hep-lat/9209022 () to compute the remaining renormalization factors at one-loop. Due to the cancellation of the tadpole contributions in the radiative corrections to the left and right side of Eq. (25), the factors are very close to one. They have the perturbative expansion
where we take the strong coupling in the -scheme hep-lat/9209022 () at a scale that corresponds to the typical gluon loop momentum. In practice, we choose . The details of the calculation of the one-loop coefficients will be presented elsewhere. The values used in this work are shown in Table 6.
ii.6 Heavy-quark mass correction
In the clover action, the hopping parameter corresponds to the bare -quark mass. When we started generating data for this analysis, we had a good estimate for the bottom-quark on each ensemble, but not the final tuned values, which were obtained as described in Appendix C of Ref. Bailey:2014tva (). We therefore need to adjust the form factors a posteriori to account for the slightly mistuned values of .
The parameters are adjusted so that the corresponding kinetic masses match the experimentally-measured value Bailey:2014tva (). Table 7 shows both the simulation and final tuned values. For some ensembles, the difference between the two is as large as 7 of the statistical uncertainty associated with the tuning procedure. We study the -dependence of the lattice form factors by generating data on the fm, ensemble, with two additional values, and , and all other simulation parameters unchanged.
To generalize the dependence from this ensemble to others, we work with the quark kinetic mass instead of itself. We expand the form factor ( in about a reference point (which corresponds to the tuned ) as follows
where the masses and are all in units. To obtain at the reference point, we need to find the dimensionless normalized slope .
We use exactly the same procedure as described in Sec. II.4 for to obtain the form factors for the additional values and . We apply the matching factors given in Table 6. Finally, we take to be the kinetic mass corresponding to (the tuned kappa given in Table 7) and use it as the reference point. We fit each form factor at each momentum for the three data points to the linear form given in Eq. (29), taking and as fit parameters. The result is shown in Fig. 10 (left). As shown in the plot, the normalized slope has a very mild dependence. Therefore, for each form factor we perform a correlated fit to all momenta to obtain a single common normalized slope. The result is shown in Table 8. Fitting the data to a linear form in results in a slope statistically consistent with zero.
To examine the light-quark mass dependence of the normalized slopes, we repeat the same procedure for the semileptonic form factors with a heavier daughter valence quark , which is close to the physical strange-quark mass. The results are plotted in Fig. 10 (right). We fit the points of each form factor to a constant and tabulate the results in Table 8. Comparing the normalized slopes for and , taking into account statistical correlations, we observe a mild but statistically-significant light daughter-quark mass dependence. So we fit the slopes for and simultaneously to a linear form,
|Combined fit with dependence|
We use the parameters and in Table 8 to determine the normalized slope for each ensemble. Although the dependence of the normalized slopes on the light daughter-quark mass is resolvable, the effects are small for the ensembles we use in the analysis (with light daughter-quark masses ranging from to ). We expect similarly small effects from the spectator-quark masses. We also expect that the lattice-spacing dependence of the normalized slopes is small, because it is a dimensionless ratio. We therefore correct each lattice form factor in each ensemble by a factor
where and are the kinetic masses corresponding to the simulation and tuned , respectively. The resulting relative shift for each ensemble is shown in Table 7. Although the corrections to itself are significant for some ensembles, the corresponding corrections to the form factors are much smaller (), as a consequence of the small normalized slopes.
Iii Chiral-continuum extrapolation
Here we extrapolate the form factors at four lattice spacings with several unphysical light-quark masses to the continuum limit and physical light-quark mass. We use heavy-meson rooted staggered chiral perturbation theory (HMrSPT) 0704.0795 (); Aubin:2005aq (), in the hard-pion and SU(2) limits. We also incorporate heavy-quark discretization effects into the chiral-continuum extrapolation.
iii.1 SU(2) staggered chiral perturbation theory in the hard-pion limit
The full-QCD next-to leading order (NLO) HMrSPT expression for the semileptonic form factors can be written 0704.0795 ()
where . Note that the expressions are in units of the mass-independent scale and the coefficients have the dimension of . The leading-order terms are
with the coupling constant and the mass splitting. The terms and are the one-loop nonanalytic contributions in the chiral expansion, and depend upon the light pseudoscalar meson mass and energy 0704.0795 (). Note that in the heavy-quark expansion is proportional to up to . We therefore use the same pole location and nonanalytic corrections for as . The terms analytic in are introduced to cancel the scale dependence arising from the nonanalytic contribution in Eq. (32). The dimensionless variables are proportional to the quark mass, pion energy, and lattice spacing. We define