| Summary: | Behind succeeding measurements of anomalies in semileptonic decays at LHCb and several collider experiments hinting at the possible violation of lepton flavor universality, we undertake a concise review of theoretical foundations of the tree- and loop-level <i>b</i>-hadron decays, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo stretchy="false">→</mo><mi>c</mi><mi>l</mi><msub><mi>ν</mi><mi>l</mi></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo stretchy="false">→</mo><mi>s</mi><msup><mi>l</mi><mo>+</mo></msup><msup><mi>l</mi><mo>−</mo></msup></mrow></semantics></math></inline-formula> along with experimental environments. We revisit the world averages for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><mrow><mi>D</mi><mo>(</mo><msup><mi>D</mi><mo>*</mo></msup><mo>)</mo></mrow></msub></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><mrow><mi>K</mi><mo>(</mo><msup><mi>K</mi><mo>*</mo></msup><mo>)</mo></mrow></msub></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></msub></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><msub><mi>η</mi><mi>c</mi></msub></msub></semantics></math></inline-formula>, for the semileptonic transitions and provide results within the framework of the relativistic independent quark model in addition to the results from model-independent studies. If the ongoing evaluation of the data of LHC Run 2 confirms the measurements of Run 1, then the statistical significance of the effect in each decay channel is likely to reach 5 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>. A confirmation of these measurements would soon turn out to be the first remarkable observation of physics beyond the Standard Model, providing a wider outlook on the understanding of new physics.
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