Semileptonic decays of $D_{(s)}$ mesons

A symmetry-preserving continuum approach to meson bound states in quantum field theory, employed elsewhere to describe numerous $\pi$- and $K$-meson electroweak processes, is used to analyze leptonic and semileptonic decays of $D_{(s)}$ mesons. Each semileptonic transition is conventionally characterized by the value of the dominant form factor at $t=0$ and the following results are obtained herein: $f_{+}^{D_s\to K}(0)=0.673(40)$; $f_{+}^{D\to \pi }(0)=0.618(31)$ and $f_{+}^{D\to K}(0)=0.756(36)$. Working with the computed $t$-dependence of these form factors and standard averaged values for $|V_{cd}|$, $|V_{cs}|$, one arrives at the following predictions for the associated branching fractions: ${\mathcal{B}}_{D_s^{+}\to K^0{e}^{+}{\nu }_e}=3.31(33)\times {10}^{-3}$, ${\mathcal{B}}_{D^0\to {\pi }^{-}{e}^{+}{\nu }_e}=2.73(22)\times {10}^{-3}$, and ${\mathcal{B}}_{D^0\to K^{-}{e}^{+}{\nu }_e}=3.83(28)\%.$ Alternatively, using the calculated $t$-dependence, agreement with contemporary empirical results for these branching fractions requires $|V_{cd}|=0.221(9)$, $|V_{us}|=0.953(34)$. With all $D_{(s)}$ transition form factors in hand, the nature of SU(3)-flavor symmetry breaking in this array of processes can be analysed; and just as in the $\pi \text{−}K$ sector, the magnitude of such effects is found to be determined by the scales associated with emergent mass generation in the Standard Model, not those originating with the Higgs mechanism.

Figure 1

Pictorial representation of Table 3; namely, maximum recoil ($t=0$) value of $D_{(s)}$ semileptonic transition form factors computed herein (blue circles) compared with inferences from experiment (cyan squares) [1, 2] and lQCD (purple diamonds) [11] (where the latter are available).

Figure 2

$D_s\to K$ semileptonic transition form factors, defined by Eqs. (3) and (7) with the associated coefficients listed in Table 2. Legend: $f_{+}^{D_s^d}$, solid blue curve; $f_0^{D_s^d}$, dashed green curve; and $f_{-}^{D_s^d}$, dot-dashed red curve. The shaded band around each of these curves indicates the $1\sigma$ confidence level for the associated SPM extrapolation. (See Sec. 3.) Empirical datum-cyan square [2].

Figure 3

$D\to \pi$ semileptonic transition form factors, defined by Eqs. (3) and (7) with the associated coefficients listed in Table 2. Legend: $f_{+}^{D_u^d}$, solid blue curve; $f_0^{D_u^d}$, dashed green curve; and $f_{-}^{D_u^d}$, dot-dashed red curve. The shaded band around each of these curves indicates the $1\sigma$ confidence level for the associated SPM extrapolation. (See Sec. 3.) Empirical data-cyan squares [1]; and lQCD results, purple diamonds [11].

Figure 4

$D\to K$ semileptonic transition form factors, defined by Eqs. (3) and (7) with the associated coefficients listed in Table 2. Legend: $f_{+}^{D_d^s}$, solid blue curve; $f_0^{D_d^s}$, dashed green curve; and $f_{-}^{D_d^s}$, dot-dashed red curve. The shaded band around each of these curves indicates the $1\sigma$ confidence level for the associated SPM extrapolation. (See Sec. 3.) Empirical data-cyan squares [1]; and lQCD results, purple diamonds [11]. Long-dashed purple curve: least-squares fit to data $f_{+}^{\mathrm{fit}}(t)=(0.70+0.27t)/(1-0.089t)$.

Figure 5

Computed $t$-dependence of the ratio in Eq. (21): the “U-spin symmetry” hypothesis becomes quantitatively unreliable as $t$ increases away from the maximum recoil point. The shaded band around the curve indicates the $1\sigma$ confidence level derived from the associated SPM extrapolations. (See Sec. 3.)