| Summary: | Anomalies recently observed in semileptonic <inline-formula> <math display="inline"> <semantics> <mrow> <mi>b</mi> <mo>→</mo> <mi>c</mi> <msup> <mi>ℓ</mi> <mo>-</mo> </msup> <msub> <mover accent="true"> <mi>ν</mi> <mo>¯</mo> </mover> <mi>ℓ</mi> </msub> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>b</mi> <mo>→</mo> <mi>s</mi> <msup> <mi>ℓ</mi> <mo>+</mo> </msup> <msup> <mi>ℓ</mi> <mo>-</mo> </msup> </mrow> </semantics> </math> </inline-formula> transitions point to violation of Lepton Flavour Universality. Strategies for new analyses of different modes are required, in particular for the modes induced by the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>b</mi> <mo>→</mo> <mi>u</mi> </mrow> </semantics> </math> </inline-formula> transition. We describe the purely leptonic <i>B</i> decay, the <inline-formula> <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi>B</mi> <mo>¯</mo> </mover> <mo>→</mo> <mi>π</mi> <msup> <mi>ℓ</mi> <mo>-</mo> </msup> <msub> <mover accent="true"> <mi>ν</mi> <mo>¯</mo> </mover> <mi>ℓ</mi> </msub> </mrow> </semantics> </math> </inline-formula> channel and the <i>B</i> semileptonic modes to <inline-formula> <math display="inline"> <semantics> <mrow> <mi>ρ</mi> <mo>(</mo> <mn>770</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>a</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mn>1260</mn> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> in extensions of the Standard Model involving Lepton Flavour Universality violating <inline-formula> <math display="inline"> <semantics> <mrow> <mi>b</mi> <mo>→</mo> <mi>u</mi> </mrow> </semantics> </math> </inline-formula> operators. In particular, we review the observables in the four-dimensional angular <inline-formula> <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi>B</mi> <mo>¯</mo> </mover> <mo>→</mo> <mi>ρ</mi> <mrow> <mo>(</mo> <mi>π</mi> <mi>π</mi> <mo>)</mo> </mrow> <msup> <mi>ℓ</mi> <mo>-</mo> </msup> <msub> <mover accent="true"> <mi>ν</mi> <mo>¯</mo> </mover> <mi>ℓ</mi> </msub> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi>B</mi> <mo>¯</mo> </mover> <mo>→</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>ρ</mi> <mi>π</mi> <mo>)</mo> </mrow> <msup> <mi>ℓ</mi> <mo>-</mo> </msup> <msub> <mover accent="true"> <mi>ν</mi> <mo>¯</mo> </mover> <mi>ℓ</mi> </msub> </mrow> </semantics> </math> </inline-formula> distributions, suitable to pin down deviations from the Standard Model. We discuss the complementarity among the various modes for New Physics searches.
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