The relative branching fractions of $B^+ \to h^+h^{\prime +}h^{\prime }$ decays, where $h^{(\prime)}$ is a pion or kaon, are measured. The analysis is performed with a data sample, collected with the LHCb detector, corresponding to an integrated luminosity of $3.0 {\rm fb}^{1}$ of $pp$ collisions. The results obtained improve significantly on previous measurements of these quantities, and are important for the interpretation of Dalitz plot analyses of threebody charmless hadronic decays of $B^+$ mesons.
Example Feynman diagrams that contribute to $ B ^+ \rightarrow h^+h^{\prime +}h^{\prime }$ decays. (Top row) treelevel processes with external $W$ emission coupling to (left) pion and (right) kaon; (second row) (left) $\bar{b} \rightarrow \bar{s}$ and (right) $\bar{b} \rightarrow \bar{d}$ looplevel processes with $u\bar{u}$ production; (following rows) same, but with (third row) $d\bar{d}$ and (bottom row) $s\bar{s}$ production. Where finalstate particles other than $\pi ^+ $ and $ K ^+ $ are given, it should be understood that a range of resonances are possible, and where these are unflavoured in many cases decays to both $\pi ^+ \pi ^ $ and $ K ^+ K ^ $ are possible. Other types of Feynman diagrams that can also contribute, such as internal $W$ emission and annihilation amplitudes as well as rescattering processes, are not shown. 
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Total efficiency, $\epsilon^{\rm tot}$, as a function of SDP position for (top left) $ B ^+ \rightarrow K ^+ K ^+ K ^ $ , (top right) $ B ^+ \rightarrow \pi ^+ K ^+ K ^ $ , (bottom left) $ B ^+ \rightarrow K ^+ \pi ^+ \pi ^ $ , and (bottom right) $ B ^+ \rightarrow \pi ^+ \pi ^+ \pi ^ $ . 
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Invariantmass distributions of the (top) $ B ^+ \rightarrow K ^+ K ^+ K ^ $ and (bottom) $ B ^+ \rightarrow \pi ^+ K ^+ K ^ $ candidates compared to the results of the simultaneous fit with (left) linear and (right) logarithmic $y$axis scales. 
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Invariantmass distributions of the (top) $ B ^+ \rightarrow K ^+ \pi ^+ \pi ^ $ and (bottom) $ B ^+ \rightarrow \pi ^+ \pi ^+ \pi ^ $ candidates compared to the results of the simultaneous fit with (left) linear and (right) logarithmic $y$axis scales. 
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Comparisons of the measured branching fraction ratios, with $\mathcal{B} ( B ^+ \rightarrow K ^+ K ^+ K ^ )$ as denominator, with the current world averages \cite{PDG2020}. Light (dark) bands associated with the branching fraction ratio correspond to the $\pm1\sigma$ total (statistical) uncertainty intervals. For horizontal and vertical bands taken from the PDG only the total uncertainty is shown. 
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Animated gif made out of all figures. 
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Current knowledge of the branching fractions of $ B ^+ \rightarrow h^+h^{\prime +}h^{\prime }$ decays \cite{PDG2020}. 
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Probability (%), due to the particle identification requirements, for each of the four signal modes to be correctly identified, or to form a crossfeed background to one of the other final states. Empty entries correspond to values below $0.05 \%$. The decays $ B ^+ \rightarrow \pi ^+ K ^+ K ^ $ and $ B ^+ \rightarrow K ^+ \pi ^+ \pi ^ $ can, through both $ K ^+ \rightarrow \pi ^+ $ and $\pi ^+ \rightarrow K ^+ $ misidentification, appear as a crossfeed background in the correct final state with probabilities of below $0.05 \%$ and $0.4 \%$, respectively. 
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Ordering of finalstate particles used in definitions of the SDP variables. 
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Charm veto efficiencies as determined from samples of the four signal modes generated according to the most recent Dalitzplot models. 
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Fitted signal yields and associated statistical uncertainties. 
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Absolute systematic uncertainties on the branching fraction ratios. All values are given multiplied by 100. Uncertainties are presented for all ratios of one mode to another, even though not all are independent. 
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Measured relative branching fractions of $ B ^+ \rightarrow h^+h^{\prime +}h^{\prime }$ decays, where the first uncertainty is statistical and the second is systematic. Results are presented for all ratios of one mode to another, even though not all are independent. 
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Statistical correlations between the measured branching fraction ratios. 
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Systematic correlations between the measured branching fraction ratios. 
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Created on 27 February 2021.