# Dalitz plot analysis of $B^0 \to \overline{D}^0 \pi^+\pi^-$ decays

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## Abstract

The resonant substructures of $B^0 \to \overline{D}^0 \pi^+\pi^-$ decays are studied with the Dalitz plot technique. In this study a data sample corresponding to an integrated luminosity of 3.0 fb$^{-1}$ of $pp$ collisions collected by the LHCb detector is used. The branching fraction of the $B^0 \to \overline{D}^0 \pi^+\pi^-$ decay in the region $m(\overline{D}^0\pi^{\pm})>2.1$ GeV$/c^2$ is measured to be $(8.46 \pm 0.14 \pm 0.29 \pm 0.40) \times 10^{-4}$, where the first uncertainty is statistical, the second is systematic and the last arises from the normalisation channel $B^0 \to D^*(2010)^-\pi^+$. The $\pi^+\pi^-$ S-wave components are modelled with the Isobar and K-matrix formalisms. Results of the Dalitz plot analyses using both models are presented. A resonant structure at $m(\overline{D}^0\pi^-) \approx 2.8$ GeV$/c^{2}$ is confirmed and its spin-parity is determined for the first time as $J^P = 3^-$. The branching fraction, mass and width of this structure are determined together with those of the $D^*_0(2400)^-$ and $D^*_2(2460)^-$ resonances. The branching fractions of other $B^0 \to \overline{D}^0 h^0$ decay components with $h^0 \to \pi^+\pi^-$ are also reported. Many of these branching fraction measurements are the most precise to date. The first observation of the decays $B^0 \to \overline{D}^0 f_0(500)$, $B^0 \to \overline{D}^0 f_0(980)$, $B^0 \to \overline{D}^0 \rho(1450)$, $B^0 \to D_3^*(2760)^- \pi^+$ and the first evidence of $B^0 \to \overline{D}^0 f_0(2020)$ are presented.

## Figures and captions

 Examples of tree diagrams via $\bar{b} \rightarrow \bar{c}u\bar{d}$ transition to produce (a) $\pi ^+ \pi ^-$ resonances, (b) nonresonant three-body decay and (c) $\overline{ D }{} {}^0 \pi ^-$ resonances. Fig1.pdf [187 KiB] HiDef png [318 KiB] Thumbnail [113 KiB] *.C file Invariant mass distribution of $B^0 \rightarrow \overline{ D }{} {}^0 \pi^+\pi^-$ candidates. Data points are shown in black. The fit is shown as a solid (red) line with the background component displayed as dashed (green) line. Fig2.pdf [21 KiB] HiDef png [175 KiB] Thumbnail [145 KiB] *.C file Density profile of the combinatorial background events in the Dalitz plane obtained from the upper $m(\overline{ D }{} {}^0 \pi^+\pi^-)$ sideband with a looser selection applied on the Fisher discriminant. Fig3.pdf [17 KiB] HiDef png [179 KiB] Thumbnail [181 KiB] *.C file Efficiency function for the Dalitz variables obtained in a fit to the LHCb simulated samples. Fig4.pdf [54 KiB] HiDef png [569 KiB] Thumbnail [176 KiB] *.C file Dalitz plot distribution of candidates in the signal region, including background contributions. The red line shows the Dalitz plot kinematic boundary. Fig5.pdf [26 KiB] HiDef png [429 KiB] Thumbnail [320 KiB] *.C file Projections of the data and Isobar fit onto (a) $m^2(\pi^+\pi^-)$ and (c) $m^2(\overline{ D }{} {}^0 \pi^-)$ with a linear scale. Same projections shown in (b) and (d) with a logarithmic scale. Components are described in the legend. The lines denoted $\overline{ D }{} {}^0 \pi^-$ and $\pi^+\pi^-$ include the coherent sums of all $\overline{ D }{} {}^0 \pi^-$ resonances, $\pi^+\pi^-$ resonances, and $\pi^+\pi^-$ S-wave resonances. The various contributions do not add linearly due to interference effects. Fig6a.pdf [74 KiB] HiDef png [200 KiB] Thumbnail [163 KiB] *.C file Fig6b.pdf [72 KiB] HiDef png [251 KiB] Thumbnail [194 KiB] *.C file Fig6c.pdf [35 KiB] HiDef png [200 KiB] Thumbnail [171 KiB] *.C file Fig6d.pdf [34 KiB] HiDef png [232 KiB] Thumbnail [174 KiB] *.C file Projections of the data and K-matrix fit onto (a) $m^2(\pi^+\pi^-)$ and (c) $m^2(\overline{ D }{} {}^0 \pi^-)$ with a linear scale. Same projections shown in (b) and (d) with a logarithmic scale. Components are described in the legend. The lines denoted $\overline{ D }{} {}^0 \pi^-$ and $\pi^+\pi^-$ include the coherent sums of all $\overline{ D }{} {}^0 \pi^-$ resonances, $\pi^+\pi^-$ resonances, and $\pi^+\pi^-$ S-wave resonances. The various contributions do not add linearly due to interference effects. Fig7a.pdf [76 KiB] HiDef png [200 KiB] Thumbnail [164 KiB] *.C file Fig7b.pdf [73 KiB] HiDef png [255 KiB] Thumbnail [198 KiB] *.C file Fig7c.pdf [35 KiB] HiDef png [202 KiB] Thumbnail [173 KiB] *.C file Fig7d.pdf [34 KiB] HiDef png [232 KiB] Thumbnail [174 KiB] *.C file Comparison of the $\pi^+\pi^-$ S-wave obtained from the Isobar model and the K-matrix model, for (a) amplitudes and (b) phases. The K-matrix model is shown by the red solid line, two scenarios for the Isobar model with (black long dashed line) and without (blue dashed line) $f_0(1370)$ and $f_0(1500)$ are shown. Fig8a.pdf [29 KiB] HiDef png [242 KiB] Thumbnail [195 KiB] *.C file Fig8b.pdf [186 KiB] HiDef png [191 KiB] Thumbnail [165 KiB] *.C file Distributions of $m^2(\pi^+\pi^-)$ in the $\rho(770)$ mass region. The different fit components are described in the legend. Results from (a) the Isobar model and (b) the K-matrix model are shown. Fig9a.pdf [64 KiB] HiDef png [312 KiB] Thumbnail [238 KiB] *.C file Fig9b.pdf [41 KiB] HiDef png [283 KiB] Thumbnail [230 KiB] *.C file Distributions of $m^2(\overline{ D }{} {}^0 \pi ^- )$ in the $D_J^*(2760)^-$ mass region. The different fit components are described in the legend. Both results from (a) the Isobar model and (b) the K-matrix model are shown. Fig10a.pdf [28 KiB] HiDef png [239 KiB] Thumbnail [198 KiB] *.C file Fig10b.pdf [24 KiB] HiDef png [244 KiB] Thumbnail [199 KiB] *.C file Invariant mass distributions of (a) $m(\overline{ D }{} {}^0 \pi^+\pi^-)$ and (b) $m(\overline{ D }{} {}^0 \pi^-)$ for $B^0 \rightarrow D^*(2010)^-\pi^+$ candidates. The data is shown as black points with the fit superimposed as red solid lines. Fig11a.pdf [21 KiB] HiDef png [282 KiB] Thumbnail [255 KiB] *.C file Cosine of the helicity angle distributions in the $m^2(\overline{ D }{} {}^0 \pi^-)$ range [7.4, 8.2] GeV$^2/c^4$ for (a) the Isobar model and (b) the K-matrix model. The data are shown as black points. The helicity angle distributions of the Dalitz plot fit results, without the $D^*_J(2760)^-$ and with the different spin hypotheses of $D^*_J(2760)^-$, are superimposed. Fig12a.pdf [25 KiB] HiDef png [314 KiB] Thumbnail [226 KiB] *.C file Fig12b.pdf [26 KiB] HiDef png [316 KiB] Thumbnail [225 KiB] *.C file Mixing angle as a function of form factor ratio for the (a) $q\bar{q}$ model and (b) $[qq'][\bar{q}\bar{q'}]$ tetraquark model. Green band gives 1$\sigma$ interval around central values (black solid line). Fig13a.pdf [26 KiB] HiDef png [115 KiB] Thumbnail [101 KiB] *.C file Fig13b.pdf [37 KiB] HiDef png [142 KiB] Thumbnail [117 KiB] *.C file The first eight unnormalised Legendre polynomial weighted moments (0 to 7 correspond to (a) to (h)) for background-subtracted and efficiency-corrected $B^0 \rightarrow \overline{ D }{} {}^0 \pi^+\pi^-$ data and the Dalitz plot fit results as a function of $m^2(\overline{ D }{} {}^0 \pi^-)$. Fig14.pdf [78 KiB] HiDef png [732 KiB] Thumbnail [517 KiB] *.C file The first eight unnormalised Legendre polynomial weighted moments (0 to 7 correspond to (a) to (h)) for background-subtracted and efficiency-corrected $B^0 \rightarrow \overline{ D }{} {}^0 \pi^+\pi^-$ data and the Dalitz plot fit results as a function of $m^2(\pi^+\pi^-)$. Fig15.pdf [71 KiB] HiDef png [651 KiB] Thumbnail [465 KiB] *.C file The first eight unnormalised Legendre polynomial weighted moments (0 to 7 correspond to (a) to (h)) for background-subtracted and efficiency-corrected $B^0 \rightarrow \overline{ D }{} {}^0 \pi^+\pi^-$ data and the Dalitz plot fit results as a function of $m^2(\overline{ D }{} {}^0 \pi^-)$. Only results in the region $m^2(\overline{ D }{} {}^0 \pi^-)< 10$ GeV$^2$/$c^4$ are shown. Fig16.pdf [198 KiB] HiDef png [767 KiB] Thumbnail [537 KiB] *.C file The first eight unnormalised Legendre polynomial weighted moments (0 to 7 correspond to (a) to (h)) for background-subtracted and efficiency-corrected $B^0 \rightarrow \overline{ D }{} {}^0 \pi^+\pi^-$ data and the Dalitz plot fit results as a function of $m^2(\pi^+\pi^-)$. Only results in the region $m^2(\pi^+\pi^-)< 3$ GeV$^2$/$c^4$ are shown. Fig17.pdf [202 KiB] HiDef png [760 KiB] Thumbnail [540 KiB] *.C file Animated gif made out of all figures. PAPER-2014-070.gif Thumbnail

## Tables and captions

 Results of the fit to the invariant mass distribution of $B^0 \rightarrow \overline{ D }{} {}^0 \pi^+\pi^-$ candidates. Uncertainties are statistical only. Table_1.pdf [59 KiB] HiDef png [87 KiB] Thumbnail [41 KiB] tex code The K-matrix parameters used in this paper are taken from a global analysis of $\pi^+\pi^-$ scattering data [22]. Masses and coupling constants are in units of ${\mathrm{ Ge V /}c^2}$ . Table_2.pdf [58 KiB] HiDef png [121 KiB] Thumbnail [57 KiB] tex code Resonant contributions to the nominal fit models and their properties. Parameters and uncertainties of $\rho(770)$, $\omega(782)$, $\rho(1450)$ and $\rho(1700)$ come from Ref. [92], and those of $f_2(1270)$ and $f_0(2020)$ come from Ref. [32]. Parameters of $f_0(500)$, $f_0(980)$ and K-matrix formalism are described in Sec. 4. Table_3.pdf [57 KiB] HiDef png [151 KiB] Thumbnail [75 KiB] tex code Systematic uncertainties on $\cal B ( B ^0 \rightarrow \overline{ D }{} {}^0 \pi ^+ \pi ^- )$. Table_4.pdf [45 KiB] HiDef png [75 KiB] Thumbnail [36 KiB] tex code Statistical significance ($\sigma$) of $\pi^+\pi^-$ resonances in the Dalitz plot analysis. For the statistically significant resonances, the effect of adding dominant systematic uncertainties is shown (see text). Table_5.pdf [36 KiB] HiDef png [33 KiB] Thumbnail [16 KiB] tex code Measured masses ($m$ in ${\mathrm{ Me V /}c^2}$ ) and widths ($\Gamma$ in $\mathrm{ Me V}$ ) of the $D_0^*(2400)^-$, $D_2^*(2460)^-$ and $D_3^*(2760)^-$ resonances, where the first uncertainty is statistical, the second and the third are experimental and model-dependent systematic uncertainties, respectively. Table_6.pdf [43 KiB] HiDef png [68 KiB] Thumbnail [37 KiB] tex code The moduli of the complex coefficients of the resonant contributions for the Isobar model and the K-matrix model. The first uncertainty is statistical, the second and the third are experimental and model-dependent systematic uncertainties, respectively. Table_7.pdf [54 KiB] HiDef png [152 KiB] Thumbnail [75 KiB] tex code The phase of the complex coefficients of the resonant contributions for the Isobar model and the K-matrix model. The first uncertainty is statistical, the second and the third are experimental and model-dependent systematic uncertainties, respectively. Table_8.pdf [54 KiB] HiDef png [153 KiB] Thumbnail [81 KiB] tex code The fit fractions of the resonant contributions for the Isobar and K-matrix models with $m(\overline{ D }{} {}^0 \pi^{\pm})> 2.1$ ${\mathrm{ Ge V /}c^2}$ . The first uncertainty is statistical, the second and the third are experimental and model-dependent systematic uncertainties, respectively. Table_9.pdf [54 KiB] HiDef png [185 KiB] Thumbnail [96 KiB] tex code Correction factors due to the $D^*(2010)^-$ veto. Table_10.pdf [52 KiB] HiDef png [359 KiB] Thumbnail [138 KiB] tex code Measured branching fractions of $\cal B ( B ^0 \rightarrow r h_3) \times \cal B (r \rightarrow h_1 h_2)$ for the Isobar and K-matrix models. The first uncertainty is statistical, the second the experimental systematic, the third the model-dependent systematic, and the fourth the uncertainty from the normalisation $B^0 \rightarrow D^*(2010)^- \pi^+$ channel. Table_11.pdf [45 KiB] HiDef png [145 KiB] Thumbnail [76 KiB] tex code Systematic uncertainties on $r^f$. The sum in quadrature of the uncertainties is also reported. Table_12.pdf [47 KiB] HiDef png [279 KiB] Thumbnail [111 KiB] tex code Results of $R_{D\rho}$ and $\cos\delta_{D\rho}$. Table_13.pdf [51 KiB] HiDef png [67 KiB] Thumbnail [29 KiB] tex code Systematic uncertainties on the $\overline{ D }{} {}^0 \pi ^-$ resonant masses (MeV/$c^2$) and widths (MeV) for the Isobar model. Table_14.pdf [40 KiB] HiDef png [180 KiB] Thumbnail [90 KiB] tex code Systematic uncertainties on the moduli of the complex coefficients of the resonant contributions for the Isobar model. The moduli are normalised to that of $\rho(770)$. Table_15.pdf [40 KiB] HiDef png [228 KiB] Thumbnail [104 KiB] tex code Systematic uncertainties on the phases ($^{\circ}$) of the complex coefficients of the resonant contributions for the Isobar model. The phase of $\rho(700)$ is set to $0^{\circ}$ as the reference. Table_16.pdf [40 KiB] HiDef png [183 KiB] Thumbnail [87 KiB] tex code Systematic uncertainties on the fit fractions (%) of the resonant contributions for the Isobar model. Table_17.pdf [40 KiB] HiDef png [196 KiB] Thumbnail [87 KiB] tex code Systematic uncertainties on the $\overline{ D }{} {}^0 \pi ^-$ resonant masses (MeV/$c^2$) and widths (MeV) for the K-matrix model. Table_18.pdf [40 KiB] HiDef png [156 KiB] Thumbnail [78 KiB] tex code Systematic uncertainties on the moduli of the complex coefficients of the resonant contributions for the K-matrix model. The moduli are normalised to that of $\rho(770)$. Table_19.pdf [40 KiB] HiDef png [219 KiB] Thumbnail [105 KiB] tex code Systematic uncertainties on the phases ($^{\circ}$) of the complex coefficients of the resonant contributions for the K-matrix model. The phase of $\rho(700)$ is set to $0^{\circ}$ as reference. Table_20.pdf [40 KiB] HiDef png [182 KiB] Thumbnail [89 KiB] tex code Systematic uncertainties on the fit fractions (%) of the resonant contributions for the K-matrix model. Table_21.pdf [40 KiB] HiDef png [201 KiB] Thumbnail [95 KiB] tex code Interference fit fractions (%) of the resonant contributions for the Isobar model with $m(\overline{ D }{} {}^0 \pi^{\pm})>2.1$ ${\mathrm{ Ge V /}c^2}$ . The resonances are: ($A_0$) nonresonant S-wave, ($A_1$) $f_0(500)$, ($A_2$) $f_0(980)$, ($A_3$) $f_0(2020)$, ($A_4$) $\rho(770)$, ($A_5$) $\omega(782)$, ($A_6$) $\rho(1450)$, $(A_7)$ $\rho(1700)$, $(A_8)$ $f_2(1270)$, $(A_9)$ $\overline{ D }{} {}^0 \pi^-$ P-wave, $(A_{10})$ $D_0^*(2400)^-$, $(A_{11})$ $D_2^*(2460)^-$, $(A_{12})$ $D_3^*(2760)^-$. The diagonal elements correspond to the fit fractions given in Table 9. Table_22.pdf [33 KiB] HiDef png [58 KiB] Thumbnail [27 KiB] tex code Interference fit fractions (%) of the resonant contributions for the K-matrix model with $m(\overline{ D }{} {}^0 \pi^{\pm})>2.1$ ${\mathrm{ Ge V /}c^2}$ . The resonances are: ($A_0$) K-matrix S-wave, ($A_1$) $\rho(770)$, ($A_2$) $\omega(782)$, ($A_3$) $\rho(1450)$, $(A_4)$ $\rho(1700)$, $(A_5)$ $f_2(1270)$, $(A_6)$ $\overline{ D }{} {}^0 \pi^-$ P-wave, $(A_{7})$ $D_0^*(2400)^-$, $(A_{8})$ $D_2^*(2460)^-$, $(A_{9})$ $D_3^*(2760)^-$ The diagonal elements correspond to the fit fractions given in Table 9. Table_23.pdf [33 KiB] HiDef png [55 KiB] Thumbnail [28 KiB] tex code Statistical uncertainties on the interference fit fractions (%) of the resonant contributions for the Isobar model with $m(\overline{ D }{} {}^0 \pi^{\pm})>2.1$ ${\mathrm{ Ge V /}c^2}$ . The resonances are: ($A_0$) nonresonant S-wave, ($A_1$) $f_0(500)$, ($A_2$) $f_0(980)$, ($A_3$) $f_0(2020)$, ($A_4$) $\rho(770)$, ($A_5$) $\omega(782)$, ($A_6$) $\rho(1450)$, $(A_7)$ $\rho(1700)$, $(A_8)$ $f_2(1270)$, $(A_9)$ $\overline{ D }{} {}^0 \pi^-$ P-wave, $(A_{10})$ $D_0^*(2400)^-$, $(A_{11})$ $D_2^*(2460)^-$, $(A_{12})$ $D_3^*(2760)^-$. The diagonal elements correspond to the statistical uncertainties on the fit fractions given in Table 9. Table_24.pdf [47 KiB] HiDef png [88 KiB] Thumbnail [44 KiB] tex code Experimental systematic uncertainties on the interference fit fractions (%) of the resonant contributions for the Isobar model with $m(\overline{ D }{} {}^0 \pi^{\pm})>2.1$ ${\mathrm{ Ge V /}c^2}$ . The resonances are: ($A_0$) nonresonant S-wave, ($A_1$) $f_0(500)$, ($A_2$) $f_0(980)$, ($A_3$) $f_0(2020)$, ($A_4$) $\rho(770)$, ($A_5$) $\omega(782)$, ($A_6$) $\rho(1450)$, $(A_7)$ $\rho(1700)$, $(A_8)$ $f_2(1270)$, $(A_9)$ $\overline{ D }{} {}^0 \pi^-$ P-wave, $(A_{10})$ $D_0^*(2400)^-$, $(A_{11})$ $D_2^*(2460)^-$, $(A_{12})$ $D_3^*(2760)^-$. The diagonal elements correspond to the statistical uncertainties on the fit fractions given in Table 9. Table_25.pdf [33 KiB] HiDef png [88 KiB] Thumbnail [44 KiB] tex code Model-dependent systematic uncertainties on the interference fit fractions (%) of the resonant contributions for the Isobar model with $m(\overline{ D }{} {}^0 \pi^{\pm})>2.1$ ${\mathrm{ Ge V /}c^2}$ . The resonances are: ($A_0$) nonresonant S-wave, ($A_1$) $f_0(500)$, ($A_2$) $f_0(980)$, ($A_3$) $f_0(2020)$, ($A_4$) $\rho(770)$, ($A_5$) $\omega(782)$, ($A_6$) $\rho(1450)$, $(A_7)$ $\rho(1700)$, $(A_8)$ $f_2(1270)$, $(A_9)$ $\overline{ D }{} {}^0 \pi^-$ P-wave, $(A_{10})$ $D_0^*(2400)^-$, $(A_{11})$ $D_2^*(2460)^-$, $(A_{12})$ $D_3^*(2760)^-$. The diagonal elements correspond to the statistical uncertainties on the fit fractions given in Table 9. Table_26.pdf [33 KiB] HiDef png [92 KiB] Thumbnail [45 KiB] tex code Statistical uncertainties on the interference fit fractions (%) of the resonant contributions for the K-matrix model with $m(\overline{ D }{} {}^0 \pi^{\pm})>2.1$ ${\mathrm{ Ge V /}c^2}$ . The resonances are: ($A_0$)K-matrix S-wave, ($A_1$) $\rho(770)$, ($A_2$) $\omega(782)$, ($A_3$) $\rho(1450)$, $(A_4)$ $\rho(1700)$, $(A_5)$ $f_2(1270)$, $(A_6)$ $\overline{ D }{} {}^0 \pi^-$ P-wave, $(A_{7})$ $D_0^*(2400)^-$, $(A_{8})$ $D_2^*(2460)^-$, $(A_{9})$ $D_3^*(2760)^-$ The diagonal elements correspond to the statistical uncertainties on the fit fractions shown in Table 9. Table_27.pdf [33 KiB] HiDef png [79 KiB] Thumbnail [41 KiB] tex code Experimental systematic uncertainties on the interference fit fractions (%) of the resonant contributions for the K-matrix model with $m(\overline{ D }{} {}^0 \pi^{\pm})>2.1$ ${\mathrm{ Ge V /}c^2}$ . The resonances are: ($A_0$) K-matrix S-wave, ($A_1$) $\rho(770)$, ($A_2$) $\omega(782)$, ($A_3$) $\rho(1450)$, $(A_4)$ $\rho(1700)$, $(A_5)$ $f_2(1270)$, $(A_6)$ $\overline{ D }{} {}^0 \pi^-$ P-wave, $(A_{7})$ $D_0^*(2400)^-$, $(A_{8})$ $D_2^*(2460)^-$, $(A_{9})$ $D_3^*(2760)^-$ The diagonal elements correspond to the statistical uncertainties on the fit fractions shown in Table 9. Table_28.pdf [33 KiB] HiDef png [80 KiB] Thumbnail [41 KiB] tex code Model-dependent systematic uncertainties on the interference fit fractions (%) of the resonant contributions for the K-matrix model with $m(\overline{ D }{} {}^0 \pi^{\pm})>2.1$ ${\mathrm{ Ge V /}c^2}$ . The resonances are: ($A_0$) K-matrix S-wave, ($A_1$) $\rho(770)$, ($A_2$) $\omega(782)$, ($A_3$) $\rho(1450)$, $(A_4)$ $\rho(1700)$, $(A_5)$ $f_2(1270)$, $(A_6)$ $\overline{ D }{} {}^0 \pi^-$ P-wave, $(A_{7})$ $D_0^*(2400)^-$, $(A_{8})$ $D_2^*(2460)^-$, $(A_{9})$ $D_3^*(2760)^-$ The diagonal elements correspond to the statistical uncertainties on the fit fractions shown in Table 9. Table_29.pdf [33 KiB] HiDef png [78 KiB] Thumbnail [41 KiB] tex code The moduli and phases of the K-matrix parameters. The first uncertainty is statistical, the second the experimental systematic, and the third the model-dependent systematic. The moduli are normalised to that of the $\rho(770)$ contribution and the phase of $\rho(770)$ is set to 0$^{\circ}$. Table_30.pdf [43 KiB] HiDef png [117 KiB] Thumbnail [67 KiB] tex code Systematic uncertainties on the moduli of the K-matrix parameters. The moduli are normalised to that of $\rho(770)$. Table_31.pdf [40 KiB] HiDef png [115 KiB] Thumbnail [58 KiB] tex code Systematic uncertainties on the phases ($^{\circ}$) of the K-matrix parameters. The phase of $\rho(700)$ is set to 0$^{\circ}$ as the reference. Table_32.pdf [40 KiB] HiDef png [111 KiB] Thumbnail [58 KiB] tex code

Created on 05 November 2019.