Erratum to: The new physics reach of null tests with $D \rightarrow \pi \ell \ell$ and $D_s \rightarrow K \ell \ell$ decays

1 Erratum to: Eur. Phys. J. C (2020) 80:65 https://doi.org/10.1140/epjc/s10052-020-7621-7

In this Erratum we correct terms involving finite $m_\ell$ and $C_T$ in $D \rightarrow \pi \ell \ell$ distributions. We further adapt the $q^2$ integration limits corresponding to the LHCb analysis [1]. We give the modifications regarding these two points, as well as further corrections. Note, further numerics of the article are unaffected and the conclusion remains unchanged.

$D \rightarrow \pi \ell \ell$ distributions Errors in Eqs. (7), (17), (18) and (21) involving $C_T$ and finite $m_\ell$-terms have been fixed. The corrected distributions are in agreement with Ref. [2], and therefore footnote 1 has been removed. The correct expressions read

$$\begin{aligned} {\text {d}\varGamma \over \text {d}q^2}&= {G_F^2 \alpha _e^2 \over 1024\pi ^5 m_D^3} \sqrt{\lambda _{DP} \biggl ( 1 - {4m_\ell ^2 \over q^2} \biggr )}\nonumber \\&\quad \times \biggl \{{2\over 3} \biggl | C_9 + C_9^R + C_7 {2m_c \over m_D + m_P} {f_T \over f_+} \biggr |^2 \nonumber \\&\quad \times \left( 1 + {2m_\ell ^2 \over q^2} \right) \lambda _{DP} f_+^2 \biggr . \nonumber \\&\quad + |C_{10}|^2 \biggl [ {2\over 3} \left( 1 - {4m_\ell ^2 \over q^2} \right) \lambda _{DP} f_+^2 \biggr .\nonumber \\&\quad + {4m_\ell ^2 \over q^2} (m_D^2-m_P^2)^2 f_0^2 \biggl .\biggr ]\nonumber \\&\quad + \left[ |C_S|^2 \left( 1 - {4m_\ell ^2 \over q^2} \right) + |C_P + C_P^R|^2 \right] \nonumber \\&\quad \times {q^2 \over m_c^2} (m_D^2-m_P^2)^2 f_0^2 \nonumber \\&\quad + {4\over 3} \left[ |C_T|^2 + |C_{T5}|^2 \right] \left( 1 - {4m_\ell ^2 \over q^2} \right) {q^2 \,\lambda _{DP} f_T^2 \over (m_D + m_P)^2} \nonumber \\&\quad + 8 \,\text {Re}\left[ \left( C_9 + C_9^R + C_7 {2m_c \over m_D + m_P} {f_T \over f_+} \right) C_T^*\right] \nonumber \\&\quad \times {m_\ell \over m_D + m_P} \lambda _{DP} f_+ f_T\nonumber \\&\quad + 4\,\text {Re}\left[ C_{10} \left( C_P + C_P^R \right) ^* \right] {m_\ell \over m_c} (m_D^2-m_P^2)^2 f_0^2 \nonumber \\&\quad \biggr .+16\,|C_T|^2 \frac{m_\ell ^2}{(m_D+m_P)^2}\lambda _{DP}f_T^2 \biggr \}, \end{aligned}$$

(7)

$$\begin{aligned} A_\mathrm{FB}(q^2)&= {1 \over \varGamma } \left[ \int _0^1 - \int _{-1}^0 \right] {\text {d}^2\varGamma \over \text {d}q^2 \text {d}\cos \theta } \text {d}\cos \theta = {b(q^2) \over \varGamma } \nonumber \\&= {1\over \varGamma } {G_F^2 \alpha _e^2 \over 512\pi ^5 m_D^3} \lambda _{DP} \biggl ( 1 - {4m_\ell ^2 \over q^2} \biggr ) \nonumber \\&\quad \times \biggl \{\biggr . \text {Re}\left[ \left( C_9 + C_9^R + C_7 {2m_c \over m_D + m_P} {f_T \over f_+} \right) C_S^* \right] \nonumber \\&\quad \times {m_\ell \over m_c} f_+ \nonumber \\&\quad + 2\text {Re}\left[ C_{10} C_{T5}^* \right] {m_\ell \over m_D + m_P} f_T \nonumber \\&\quad + \text {Re}\left[ C_S C_T^* + \left( C_P + C_P^R \right) C_{T5}^* \right] \nonumber \\&\quad \times {q^2 \over m_c (m_D + m_P)} f_T \biggl .\biggr \} (m_D^2-m_P^2) f_0 , \end{aligned}$$

(17)

$$\begin{aligned} F_H(q^2)&= {2\over \varGamma } [a(q^2) + c(q^2)] \nonumber \\&= {1 \over \varGamma } {G_F^2 \alpha _e^2 \over 1024\pi ^5 m_D^3} \sqrt{\lambda _{DP} \biggl ( 1 - {4m_\ell ^2 \over q^2} \biggr )}\, \nonumber \\&\quad \times \biggl \{\biggr . \biggl | C_9 + C_9^R + C_7 {2m_c \over m_D+m_P} {f_T \over f_+} \biggr |^2 {4m_\ell ^2 \over q^2} \lambda _{DP} f_+^2 \nonumber \\&\quad + |C_{10}|^2 {4m_\ell ^2 \over q^2} (m_D^2-m_P^2)^2 f_0^2 \nonumber \\&\quad + \left[ |C_S|^2 \left( 1 - {4m_\ell ^2 \over q^2} \right) + |C_P + C_P^R|^2 \right] \nonumber \\&\quad \times {q^2 \over m_c^2} (m_D^2-m_P^2)^2 f_0^2 \nonumber \\&\quad + 4\,\left[ |C_T|^2 + |C_{T5}|^2 \right] \left( 1 - {4m_\ell ^2 \over q^2} \right) \nonumber \\&\quad \times {q^2 \over (m_D + m_P)^2} \lambda _{DP} f_T^2 \nonumber \\&\quad + 8\,\text {Re}\left[ \left( C_9 + C_9^R + C_7 {2m_c \over m_D + m_P} {f_T \over f_+} \right) C_T^*\right] \nonumber \\&\quad {m_\ell \over m_D + m_P} \lambda _{DP} f_+ f_T\nonumber \\&\quad + 4\,\text {Re}\left[ C_{10} \left( C_P + C_P^R \right) ^* \right] {m_\ell \over m_c} (m_D^2-m_P^2)^2 f_0^2 \nonumber \\&\quad +16\,|C_T|^2 \frac{m_\ell ^2}{(m_D+m_P)^2}\lambda _{DP} f_T^2 \biggr .\biggr \}, \end{aligned}$$

(18)

$$\begin{aligned} {\text {d}\varGamma \over \text {d}q^2} - {\text {d}\overline{\varGamma }\over \text {d}q^2}&= {G_F^2 \alpha _e^2 \over 256\pi ^5 m_D^3} \sqrt{\lambda _{DP} \biggl ( 1 - {4m_\ell ^2 \over q^2} \biggr )}\, \nonumber \\&\quad \times \biggl \{\biggr . {2\over 3} \,\text {Im}\left[ C_9 + 2C_7 {m_c \over m_D + m_P} {f_T \over f_+} \right] \nonumber \\&\quad \times \text {Im}\left[ C_9^R \right] \left( 1 + {2m_\ell ^2 \over q^2} \right) \lambda _{DP} f_+^2 \nonumber \\&\quad + \text {Im}\left[ C_P \right] \text {Im} \left[ C_P^R \right] {q^2 \over m_c^2} (m_D^2-m_P^2)^2 f_0^2 \nonumber \\&\quad + 4\,\text {Im}\left[ C_T \right] \text {Im}\left[ C_9^R \right] \nonumber \\&\quad \times {m_\ell \over m_D + m_P} \lambda _{DP} f_+ f_T\nonumber \\&\quad + 2\,\text {Im}\left[ C_{10} \right] \text {Im}\left[ C_P^R \right] {m_\ell \over m_c} (m_D^2-m_P^2)^2 f_0^2 \biggr .\biggr \}. \end{aligned}$$

(21)

LHCb and constraints on Wilson coefficients We correct the integration limits of the full $q^2$–region according to [1] and add a footnote to Eq. (9). The bounds on Wilson coefficients are changed accordingly. The paragraph is changed to the following:

Table 2 Integrated branching fractions in the high $q^2$–bin ($\sqrt{q^2}\ge 1.25~\hbox {GeV}$) in the SM and in the NP benchmark scenarios as in Fig. 3. In the third to sixth column, upper entries correspond to NP-only branching ratios while for the lower entries the resonance contributions are taken into account

Using the experimental limits on the branching fraction of $D^+\rightarrow \pi ^+\mu ^+\mu ^-$ in high and full $q^2$–regions at 90% CL [1]¹,

$$\begin{aligned}&{\mathcal {B}}(D^+\rightarrow \pi ^+\mu ^+\mu ^-)\vert _{\mathrm{full}~q^2}< 7.3 \times 10^{-8} \nonumber \\&\quad \left( 250\,\mathrm{MeV}\le \sqrt{q^2}\le m_{D^+}-m_{\pi ^+}\right) ,\nonumber \\&{\mathcal {B}}(D^+\rightarrow \pi ^+\mu ^+\mu ^-)\vert _{\mathrm{high}~q^2} < 2.6 \times 10^{-8} \nonumber \\&\quad \left( \sqrt{q^2}\ge 1.25~\mathrm{GeV}\right) , \end{aligned}$$

(9)

and neglecting the SM contributions, we obtain the following constraints on the BSM Wilson coefficients in the full $q^2$-region,

$$\begin{aligned}&1.2|C_7|^2 + 1.2|C_9|^2 + 1.2|C_{10}|^2 + 2.4|C_S|^2 \nonumber \\&\quad + 2.5|C_P|^2 + 0.4|C_T|^2 + 0.3|C_{T5}|^2 \nonumber \\&\quad + 0.3\,{\mathrm {Re}}[C_9 C_T^*] + 1.0\,{\mathrm {Re}}[C_{10} C_P^*] + 2.4\,{\mathrm {Re}}[C_7 C_9^*] \nonumber \\&\quad + 0.6\,{\mathrm {Re}}[C_7 C_T^*] \lesssim 1. \end{aligned}$$

(10)

and in the high $q^2$–region,

$$\begin{aligned}&0.6|C_7|^2 + 0.7|C_9|^2 + 0.8|C_{10}|^2 + 4.4|C_S|^2\nonumber \\&\quad + 4.5|C_P|^2 + 0.4|C_T|^2 + 0.4|C_{T5}|^2 \nonumber \\&\quad + 0.3\,{\mathrm {Re}}[C_9 C_T^*] + 1.1\,{\mathrm {Re}}[C_{10} C_P^*] + 1.4\,{\mathrm {Re}}[C_7 C_9^*] \nonumber \\&\quad + 0.3\,{\mathrm {Re}}[C_7 C_T^*]\lesssim 1, \end{aligned}$$

(11)

Further corrections Updated values for the integrated branching fractions of $D^+ \rightarrow \pi ^+\mu ^+\mu ^-$ are given in Table 2.

The lifetime factor $\tau _D$ in Eq. (13) is added on the right hand side.

We correct Eq. (25).² It reads

$$\begin{aligned}&1.3\,\big (|K_9|^2 + |K_{10}|^2\big ) + 2.6\,\big ( |K_S|^2 + |K_P|^2 \big ) \nonumber \\&+ 0.4\,\big ( |K_T|^2 + \big |K_{T5}\big |^2 \big ) \nonumber \\&+ 0.5\,{\mathrm {Re}}\big [ K_{10} K_P^* \pm K_9 K_S^* \big ] \nonumber \\&+0.3\,{\mathrm {Re}}\big [ K_9 K_T^* \pm K_{10} K_{T5}^* \big ] \lesssim 50. \end{aligned}$$

(25)

The corrected version of the article is available at arXiv:1909.11108v3.