Theoretical framework for lattice QCD computations of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>B</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi>K</mml:mi> <mml:msup> <mml:mo>ℓ</mml:mo> <mml:mo>+</mml:mo

We develop a strategy for computing the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>B</a:mi> <a:mo stretchy="false">→</a:mo> <a:mi>K</a:mi> <a:msup> <a:mo>ℓ</a:mo> <a:mo>+</a:mo> </a:msup> <a:msup> <a:mo>ℓ</a:mo> <a:mo>−</a:mo> </a:msup> </a:math> and <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:msub> <d:mover accent="true"> <d:mi>B</d:mi> <d:mo stretchy="false">¯</d:mo> </d:mover> <d:mi>s</d:mi> </d:msub> <d:mo stretchy="false">→</d:mo> <d:mi>γ</d:mi> <d:msup> <d:mo>ℓ</d:mo> <d:mo>+</d:mo> </d:msup> <d:msup> <d:mo>ℓ</d:mo> <d:mo>−</d:mo> </d:msup> </d:math> decay amplitudes using lattice QCD (where <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msup> <i:mo>ℓ</i:mo> <i:mo>±</i:mo> </i:msup> </i:math> are charged leptons). We focus on those terms which contain complex contributions to the amplitude, due to on-shell intermediate states propagating between the weak operator and electromagnetic current(s). Such terms, which are generally estimated using model calculations and represent significant uncertainties in the phenomenological predictions for these decays, cannot be computed using standard lattice QCD techniques. It has recently been shown that such contributions can be computed using spectral density methods and our proposed strategy, which we discuss in detail, is built on this approach. The complex contributions include the “charming penguins” [matrix elements of the current-current operators <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msubsup> <k:mi>O</k:mi> <k:mn>1</k:mn> <k:mrow> <k:mo stretchy="false">(</k:mo> <k:mi>c</k:mi> <k:mo stretchy="false">)</k:mo> </k:mrow> </k:msubsup> </k:math> and <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:msubsup> <o:mi>O</o:mi> <o:mn>2</o:mn> <o:mrow> <o:mo stretchy="false">(</o:mo> <o:mi>c</o:mi> <o:mo stretchy="false">)</o:mo> </o:mrow> </o:msubsup> </o:math> defined in Eq. (6) below], in which the charm-quark loop can propagate long distances, particularly close to the region of charmonium resonances. They also include the contributions from the chromomagnetic operator [ <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:msub> <s:mi>O</s:mi> <s:mn>8</s:mn> </s:msub> </s:math> in standard notation, defined in Eq. (8) below]. We discuss the renormalization of the ultraviolet divergences, and in particular those which arise due to “contact” terms, and explain how those which appear as inverse powers of the lattice spacing can be subtracted nonperturbatively. We apply the spectral density methods in an instructive exploratory computation of the charming penguin diagram in <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mi>B</u:mi> <u:mo stretchy="false">→</u:mo> <u:mi>K</u:mi> <u:msup> <u:mo>ℓ</u:mo> <u:mo>+</u:mo> </u:msup> <u:msup> <u:mo>ℓ</u:mo> <u:mo>−</u:mo> </u:msup> </u:math> decays in which the virtual photon is emitted from the charm-quark loop [the diagram in Fig. 1(a) below] and discuss the prospects and strategies for the reliable determination of the amplitudes in future dedicated computations; computations which are, however, beyond the scope of the present paper.