Lattice analysis of semi-leptonic form factors

TL;DR

This paper reports lattice QCD simulations analyzing semi-leptonic form factors, comparing results with experimental data and HQET, focusing on $Q^2$ dependence, quark mass extrapolation, and renormalization effects.

Contribution

It provides preliminary lattice results for semi-leptonic form factors, including the Isgur-Wise function, and discusses their consistency with experimental data and theoretical models.

Findings

Results are consistent with experimental data for D decays.

Estimated Isgur-Wise function slope: $\xi'(w=1) = 0.97(6).

Analysis of $Q^2$ behavior and renormalization dependence.

Abstract

We present preliminary results from simulations done on 170 $32^3 \times 64$ lattices at $\beta = 6.0$ using quenched Wilson fermions. This talk focuses on the $Q^2$ behavior of the form-factors, extrapolation in quark masses, dependence on renormalization scheme, and comparison with heavy-quark effective theory (HQET). Even though we cannot estimate errors due to quenching and discretization, our results are consistent with experimental results for $D$ decays. We present results for the Isgur-Wise function and estimate $\xi'(w=1) = 0.97(6)$.

Tables

Table 1. Table 1: Estimates of form factors in 3 commonly used renormalization schemes defined in [ 5 ] .

	TAD1	TAD$\pi$	TADU0
$f_{+}$	0.71(4)	0.75(4)	0.66(4)
$f_0$	0.73(3)	0.77(3)	0.68(2)
$f_V$	1.28(7)	1.36(7)	1.19(6)
$f_{A_0}$	0.84(3)	0.85(3)	0.79(3)
$f_{A_1}$	0.72(3)	0.74(3)	0.68(3)
$f_{A_2}$	0.49(9)	0.50(9)	0.46(8)
$f_{A_3}$	0.85(3)	0.87(3)	0.80(3)
$f_V/f_{A_1}$	1.78(7)	1.84(8)	1.76(7)

Table 2. Table 2: Estimates in T ​ A ​ D ​ 1 𝑇 𝐴 𝐷 1 TAD1 scheme at Q 2 = 0 superscript 𝑄 2 0 Q^{2}=0 . The variations give estimates of systematic errors.

		Pole		Best
		$m_s\left(M_K\right)$	$m_s\left(M_{\varphi }\right)$	$m_s\left(M_K\right)$	$m_s\left(M_{\varphi }\right)$
$f_{+}$	$M_1$	$0.78\left(2\right)$	$0.80\left(2\right)$	$0.70\left(4\right)$	$0.71\left(4\right)$
	$M_2$	$0.79\left(2\right)$	$0.80\left(2\right)$	$0.67\left(5\right)$	$0.68\left(4\right)$
$f_0$	$M_1$	$0.70\left(2\right)$	$0.72\left(2\right)$	$0.71\left(3\right)$	$0.73\left(3\right)$
	$M_2$	$0.72\left(2\right)$	$0.73\left(2\right)$	$0.68\left(3\right)$	$0.70\left(2\right)$
$f_V$	$M_1$	$1.27\left(7\right)$	$1.28\left(6\right)$	$1.27\left(7\right)$	$1.28\left(7\right)$
	$M_2$	$1.22\left(6\right)$	$1.23\left(6\right)$	$1.22\left(9\right)$	$1.23\left(9\right)$
$f_{A_0}$	$M_1$	$0.85\left(3\right)$	$0.85\left(3\right)$	$0.83\left(3\right)$	$0.84\left(3\right)$
	$M_2$	$0.84\left(3\right)$	$0.84\left(3\right)$	$0.80\left(3\right)$	$0.81\left(3\right)$
$f_{A_1}$	$M_1$	$0.67\left(2\right)$	$0.68\left(2\right)$	$0.71\left(3\right)$	$0.72\left(3\right)$
	$M_2$	$0.65\left(2\right)$	$0.66\left(2\right)$	$0.67\left(4\right)$	$0.69\left(4\right)$
$f_{A_2}$	$M_1$	$0.46\left(9\right)$	$0.47\left(8\right)$	$0.49\left(9\right)$	$0.49\left(9\right)$
	$M_2$	$0.42\left(9\right)$	$0.44\left(9\right)$	$0.50\left(14\right)$	$0.52\left(12\right)$
$f_{A_3}$	$M_1$	$0.84\left(3\right)$	$0.85\left(3\right)$	$0.84\left(3\right)$	$0.85\left(3\right)$
	$M_2$	$0.81\left(3\right)$	$0.82\left(3\right)$	$0.81\left(4\right)$	$0.82\left(4\right)$
$\frac{f_0}{f_{+}}$	$M_1$	$0.93\left(1\right)$	$0.94\left(1\right)$	$1.02\left(4\right)$	$1.02\left(3\right)$
	$M_2$	$0.95\left(1\right)$	$0.95\left(1\right)$	$1.03\left(4\right)$	$1.02\left(4\right)$
$\frac{f_V}{f_{A_1}}$	$M_1$	$1.83\left(7\right)$	$1.81\left(6\right)$	$1.79\left(8\right)$	$1.78\left(7\right)$
	$M_2$	$1.85\left(7\right)$	$1.83\left(6\right)$	$1.81\left(11\right)$	$1.79\left(10\right)$
$\frac{f_{A_2}}{f_{A_1}}$	$M_1$	$0.70\left(10\right)$	$0.70\left(9\right)$	$0.69\left(12\right)$	$0.68\left(11\right)$
	$M_2$	$0.65\left(11\right)$	$0.67\left(11\right)$	$0.74\left(18\right)$	$0.75\left(15\right)$

Equations

$$f(Q^2)=f(0)/(1-Q^2/M^2),$$

$$\xi (w)$$

$$w=v_i\cdot v_f=\frac{M_i^2+M_f^2-q^2}{2M_i{M}_f};R=\frac{2\sqrt{M_i{M}_f}}{M_i+M_f}.$$

$$h_{+}(w)$$

Full text

1 TECHNICAL DETAILS

We briefly mention some details of our analysis to extract the form-factors at $Q^{2}=0$ relevant to phenomenology. A full analysis will be presented elsewhere [2]. The details of the lattices are contained in the papers on the hadron spectrum [3] [4], and decay constants [5]. Preliminary results on a sub-set of lattices have been presented at LATTICE 94 [6] and DPF 94 [7]. We do not have data to extrapolate to $m_{b}$ or to $a=0$, thus our results are relevant for $D$ decays, $i.e.$ $D\to Kl\nu$, $D\to\pi l\nu$, $D\to K^{*}l\nu$, and $D\to\rho l\nu$, calculated at $a^{-1}=2.33(4)\ \mathord{\rm\;GeV}$.

The decaying $D$ meson is created at rest by using a $\vec{p}=0$ source. On each lattice we make two measurements by creating the $D$ meson at $t=7$ and $57$, for a total statistical sample size of 340. The sink for the final state meson is at $t=32$ in both cases. The time-slice of the weak operator is taken to vary between $10\leq t\leq 30$ and $35\leq t\leq 55$ respectively, and the insertion is at 5 lowest values of lattice momenta. The quark propagators are created using a Wuppertal smeared source as described in [3].

In order to isolate the desired matrix element ($ME$) we construct a ratio of 3-point to 2-point correlation functions. We have a choice of using either smeared-smeared ($SS$) or smeared-local ($SL$) 2-point correlation functions. We calculate the $ME$ both ways and take the average as our best estimate. Figure 1 shows a typical example of signal for the ratio of correlators in the $SL$ case: the quality in the $SS$ case is very similar. We have seen a steady improvement in the consistency between these two estimates of $ME$ with statistics. With the current sample they are within $1\sigma$ in all cases.

Pole dominance hypothesis (PDH): It states that all form-factors, $f(Q^{2})$, have the structure

[TABLE]

where $M$ is the mass of the nearest resonance with the right quantum numbers. To test PDH we make two kinds of fits: (i) single parameter “pole” fit where $M$ is the lattice measured value of the resonance mass, (ii) two parameter “best” fit where $M$ and $f(0)$ are free parameters. An example of these fits is shown in figure 2. Overall we find that only $f_{0}$ and $f_{V}$ are well described by the “pole” form. $f_{+}$ and $f_{A_{0}}$ are consistent with “pole” form with $M<M_{pole}$, while for $f_{A_{3}}$ we find $M>M_{pole}$. $f_{A_{1}}$ and $f_{A_{2}}$ show a much smaller $Q^{2}$ dependence than expected from pole dominance, however the data are too noisy to make a definite statement. We use results from “best” fit for our final estimates.

HQET: At leading order in $1/m_{c}$, HQET predicts that the Isgur-Wise function $\xi$ describes all form-factors relevant to $D\to Kl\nu$ and $D\to K^{*}l\nu$ decays. Neglecting $O(\alpha_{s})$ corrections, one gets [8]

[TABLE]

where $M_{i}$ and $M_{f}$ are the initial and final meson masses and

[TABLE]

The first HQET relation, $(1-q^{2}/(M_{i}+M_{f})^{2})^{-1}f_{0}=f_{+}$, is satisfied by our data as exemplified in figure 3 for $CU_{3}\rightarrow U_{1}U_{3}$ decay. Thus, both $f_{0}$ and $f_{+}$ cannot simultaneously satisfy the PDH. If leading order HQET holds, $i.e.$ $M_{i}+M_{f}\sim m_{c}+m_{s}\sim M_{pole}$, $f_{+}$ should obey a ‘dipole’ form as $f_{0}$ agrees with pole dominance. The data, as shown in figure 2, suggests that $f_{+}$ lies in between “pole” and “dipole” forms. Similar analysis for the vector form factors is under progress.

Dependence on quark mass: Figure 4 shows an example of the variation of $f_{+}(0)$ with quark masses. There is significant dependence on the mass of the quark $C$ decays into. This is a kinematic effect as shown in Section 4. Our data is good enough to expose slight dependence on $m_{spectator}$—the small decrease in slope between the transitions $CU_{i}\to U_{1}U_{i}$ and $CU_{i}\to SU_{i}$ is consistent with HQET.

Fixing $m_{s}$: We fix $m_{s}$ using $M^{2}_{K}/M^{2}_{\rho}$ and $M_{\phi}/M_{\rho}$. Our preferred way is $m_{s}(M_{\phi})$, and the variation of results with $m_{s}$ is shown in Table 2.

$O(ma)$** effects**: In Ref. [3] we discuss $O(ma)$ effects for heavy-light mesons. The kinetic mass $M_{2}\equiv d^{2}E/dp^{2}$ is given by $\sinh M$ as the data agree with the dispersion relation $\sinh^{2}{E/2}=\sin^{2}{p/2}+\sinh^{2}{M/2}$. We show variation of form-factors with $M$ in Table 2.

Renormalization Constants: To relate lattice results to experimental data we need the renormalization constants $Z_{A}$ and $Z_{V}$. We use three Lepage-Mackenzie tadpole improved schemes described in Ref. [5]. Our preferred scheme is $TAD1$, and the variation with the schemes is illustrated in Table 1.

2 RESULTS at $\beta=6.0$

Our final results for $D\rightarrow Kl\nu$ and $D\rightarrow K^{*}l\nu$ are shown in tables 1 and 2 along with variation with $m_{s}$, type of fit, heavy-light meson mass, and the renormalization scheme. Our preferred estimates are with $m_{s}(M_{\phi})$, “best” fit, $M_{1}$, and $TAD1$ scheme. We also get that $f_{+}^{\pi}/f_{+}^{K}=0.87(4)$.

3 $d\Gamma(Q^{2})$

As explained in section 1 there are considerable uncertainties involved in extrapolating the form factors to $Q^{2}=0$. Therefore, we also calculate $d\Gamma(Q^{2})/dQ^{2}$ by linearly extrapolating in $m_{q}$ the form-factors at fixed 3-momentum transfer. In figure 5 we show the results for $D\rightarrow Ke\bar{\nu}_{e}$. In figure 6 we show the longitudinal, transverse and the total decay widths for the process $D\rightarrow K^{*}e\bar{\nu}_{e}$.

4 Isgur-Wise function

As mentioned in section 1, at the leading order in the heavy quark mass, there is one universal ‘Isgur-Wise’ function $\xi$ which controls all the form factors. In figure 7 we present data for $\xi_{ren}(w)$,

[TABLE]

where $\hat{C}_{1}$, $\hat{C}_{2}$, $\hat{C}_{3}$ are HQET renormalization constants including the leading $O(\alpha_{s})$ corrections [8]. The data show a slight dependence on the spectator quark mass indicative of $m_{c}^{-1}$ corrections. Analysis from the vector form factors is in progress. From this data we estimate the slope to be $\xi^{\prime}(w=1)=0.97(6)$.