Charming Penguins in B Meson Decays

• Charming penguins are non-factorizable long-distance charm-loop contributions in weak B-meson decays that exhibit helicity- and q²-dependent behavior.
• They induce shifts in short-distance amplitudes, complicating the extraction of Wilson coefficients and the interpretation of lepton universality tests.
• Recent data-driven fits and lattice-QCD strategies aim to quantify these nonperturbative effects, thereby refining predictions for New Physics signals.

Searching arXiv for the specified papers to ground the article in current source records. Charming penguins are long-distance charm-loop contributions in weak $B$-meson decays. In exclusive $b\to s\ell^+\ell^-$ transitions they arise from the non-local matrix elements of the current–current operators $Q_{1,2}^{c}$, while in charmless two-body decays they are associated with the amplitude component $A_c$ induced by internal $c\bar c$ loops. Their common feature is that they generate non-factorizable hadronic effects that can be helicity-dependent, $q^2$-dependent, and, in some formulations, intrinsically nonperturbative. Because these effects enter precisely in channels used to test Lepton Universality Violation (LUV) and to extract short-distance Wilson coefficients, their treatment materially affects both Standard Model predictions and New Physics inference. Recent work has also formulated a lattice-QCD strategy to compute these amplitudes using spectral-density methods, including the charmonium region (Ciuchini et al., 2021, Zhu, 2010, Frezzotti et al., 5 Aug 2025).

1. Definition and conceptual scope

In charmless two-body $B$ decays, the decay amplitude is decomposed using CKM unitarity as

$$A(B\to M_1M_2)=V_{ub}V_{uq}^*\,T+V_{tb}V_{tq}^*\,P,$$

with

$T=A_u-A_c,\qquad P=A_t-A_c,$

where $q=d,s$. In this notation, $b\to s\ell^+\ell^-$0 denotes contributions from loops with internal charm quarks. The nonleptonic operators with a $b\to s\ell^+\ell^-$1 pair can generate long-distance effects of order $b\to s\ell^+\ell^-$2; these are dubbed “charming penguins” (Zhu, 2010).

In exclusive $b\to s\ell^+\ell^-$3 decays, the dominant short-distance amplitudes are generated by the local operators $b\to s\ell^+\ell^-$4, $b\to s\ell^+\ell^-$5, and the radiative dipole $b\to s\ell^+\ell^-$6. However, because the charm quark in the photon-penguin loop remains “light” at scale $b\to s\ell^+\ell^-$7, one must also account for the non-local matrix element of the current–current operators

$b\to s\ell^+\ell^-$8

the so-called “charming penguins.” These produce an a priori unknown, helicity- and $b\to s\ell^+\ell^-$9-dependent shift in the short-distance amplitude (Ciuchini et al., 2021).

The theoretical interpretation differs across factorization frameworks. In QCDF, the entire amplitude $Q_{1,2}^{c}$0, including the $Q_{1,2}^{c}$1 loop, is treated perturbatively in terms of factorizable form factors, LCDAs, and perturbative kernels, with modeled endpoint divergences. In SCET, by contrast, the charm-loop contribution is argued to induce nonfactorizable operators whose Wilson coefficients are dominated by the nearly on-shell $Q_{1,2}^{c}$2 region, so that the corresponding charming-penguin matrix elements are nonperturbative parameters to be fitted from data (Zhu, 2010).

2. Helicity amplitudes and general parameterization in $Q_{1,2}^{c}$3

For $Q_{1,2}^{c}$4, the full helicity amplitudes are decomposed as

$Q_{1,2}^{c}$5

$Q_{1,2}^{c}$6

$Q_{1,2}^{c}$7

The seven factorizable form factors $Q_{1,2}^{c}$8 are smooth functions of $Q_{1,2}^{c}$9, known from light-cone sum rules or lattice QCD, whereas all genuine non-factorizable QCD effects are encoded in the complex functions $A_c$0 (Ciuchini et al., 2021).

A very general parameterization models these functions as low-order Taylor expansions in $A_c$1 around zero: $A_c$2

$A_c$3

$A_c$4

In this language, $A_c$5 acts like a universal shift $A_c$6, and $A_c$7 like a lepton-universal $A_c$8, while higher-order $A_c$9 are genuine long-distance effects that cannot be absorbed into local Wilson coefficients. This structure is central to separating hadronic dynamics from short-distance semileptonic physics (Ciuchini et al., 2021).

3. Conservative hadronic treatment and fitted charm-loop structure

A conservative treatment of hadronic uncertainties adopts a data-driven fit in which fourteen real parameters,

$c\bar c$0

are treated as free. To account for flavor-$c\bar c$1 breaking in $c\bar c$2 relative to $c\bar c$3, one writes

$c\bar c$4

with Gaussian priors $c\bar c$5. All $c\bar c$6-parameters are assigned broad, flat priors, and the fit is driven by differential rates and angular observables. Two more model-dependent alternatives are also considered: full extrapolation of LCSR results via dispersion relations over the entire $c\bar c$7 range, and LCSR constraints only for $c\bar c$8, letting the data determine the $c\bar c$9 at higher $q^2$0 (Ciuchini et al., 2021).

When fitted to the latest LHCb and Belle measurements, the extracted helicity combinations $q^2$1, $q^2$2, and $q^2$3 exhibit a non-trivial $q^2$4 shape, comparable in magnitude to the short-distance $q^2$5 rather than to the $q^2$6 QCDF corrections. In the fully data-driven fit, the 68% highest-posterior-density intervals include

$q^2$7

so that $q^2$8, $q^2$9, and $B$0 deviate from zero at the $B$1 level. The reported 95% contours for $B$2 lie above the purely perturbative expectation throughout the low-$B$3 region. This suggests that long-distance charm effects are neither small nor reducible to a universal local shift (Ciuchini et al., 2021).

4. Consequences for LUV observables and New Physics interpretation

In the pure Standard Model, where $B$4 and $B$5 matrix elements are lepton-universal, non-local charm effects cancel out almost exactly in the ratios

$B$6

leading to the predictions

$B$7

Once New Physics enters through a shift in $B$8 or $B$9, the interference with the hadronic term $$A(B\to M_1M_2)=V_{ub}V_{uq}^*\,T+V_{tb}V_{tq}^*\,P,$$0 reintroduces a residual uncertainty (Ciuchini et al., 2021).

The numerical impact depends strongly on how charming penguins are modeled. In a scenario with only $$A(B\to M_1M_2)=V_{ub}V_{uq}^*\,T+V_{tb}V_{tq}^*\,P,$$1 (“scenario A”), the prediction for $$A(B\to M_1M_2)=V_{ub}V_{uq}^*\,T+V_{tb}V_{tq}^*\,P,$$2 is

$$A(B\to M_1M_2)=V_{ub}V_{uq}^*\,T+V_{tb}V_{tq}^*\,P,$$3

in the data-driven hadronic fit,

$$A(B\to M_1M_2)=V_{ub}V_{uq}^*\,T+V_{tb}V_{tq}^*\,P,$$4

in the intermediate LCSR fit, and

$$A(B\to M_1M_2)=V_{ub}V_{uq}^*\,T+V_{tb}V_{tq}^*\,P,$$5

in the full Khodjamirian LCSR fit. The more optimistic hadronic model therefore enhances the NP-induced deviation in $$A(B\to M_1M_2)=V_{ub}V_{uq}^*\,T+V_{tb}V_{tq}^*\,P,$$6 by approximately $$A(B\to M_1M_2)=V_{ub}V_{uq}^*\,T+V_{tb}V_{tq}^*\,P,$$7. Similarly, $$A(B\to M_1M_2)=V_{ub}V_{uq}^*\,T+V_{tb}V_{tq}^*\,P,$$8 and $$A(B\to M_1M_2)=V_{ub}V_{uq}^*\,T+V_{tb}V_{tq}^*\,P,$$9 undergo shifts of $T=A_u-A_c,\qquad P=A_t-A_c,$0 in their central values and of order $T=A_u-A_c,\qquad P=A_t-A_c,$1 in their uncertainties when moving from the conservative to the optimistic hadronic treatment (Ciuchini et al., 2021).

Under the most conservative data-driven treatment, two NP patterns are singled out as most economical. A pure axial coupling, $T=A_u-A_c,\qquad P=A_t-A_c,$2 (“scenario C”), gives

$T=A_u-A_c,\qquad P=A_t-A_c,$3

with $T=A_u-A_c,\qquad P=A_t-A_c,$4. A pure left-handed coupling through the SMEFT operator $T=A_u-A_c,\qquad P=A_t-A_c,$5, corresponding at low energy to $T=A_u-A_c,\qquad P=A_t-A_c,$6 (“scenario B”), gives

$T=A_u-A_c,\qquad P=A_t-A_c,$7

with $T=A_u-A_c,\qquad P=A_t-A_c,$8. By contrast, a pure vector shift $T=A_u-A_c,\qquad P=A_t-A_c,$9 is less favored once the $q=d,s$0 are allowed to float, and can admit two disconnected solutions, one with $q=d,s$1 and $q=d,s$2, the other with $q=d,s$3. The reported conclusion is that long-distance effects reduce the need for a vector-like NP contribution $q=d,s$4, while the most robust emerging NP patterns feature either $q=d,s$5 or $q=d,s$6, with best-fit Wilson coefficients of order unity at $q=d,s$7 (Ciuchini et al., 2021).

5. Charming penguins in charmless $q=d,s$8 decays: QCDF, SCET, and the ratio $q=d,s$9

For $b\to s\ell^+\ell^-$00, the amplitude is written as

$b\to s\ell^+\ell^-$01

with

$b\to s\ell^+\ell^-$02

In QCDF the tree coefficient is

$b\to s\ell^+\ell^-$03

where the brackets are free of endpoint singularities except in $b\to s\ell^+\ell^-$04, whose hard-spectator uncertainty is parameterized by

$b\to s\ell^+\ell^-$05

Using Beneke–Neubert inputs and $b\to s\ell^+\ell^-$06, the numerical result is

$b\to s\ell^+\ell^-$07

with the quoted error dominated by $b\to s\ell^+\ell^-$08 and $b\to s\ell^+\ell^-$09. On this basis, $b\to s\ell^+\ell^-$10 is described as reliably predicted at the few-percent level (Zhu, 2010).

Using this QCDF value together with flavor-$b\to s\ell^+\ell^-$11, one predicts

$b\to s\ell^+\ell^-$12

to be compared with the CDF measurement $b\to s\ell^+\ell^-$13. Two explanations are identified. One is that the form-factor ratio is smaller than in light-cone sum rules, $b\to s\ell^+\ell^-$14. The other is that the QCDF estimate of $b\to s\ell^+\ell^-$15 is too large because the charming penguin $b\to s\ell^+\ell^-$16 is nonperturbative and gives a sizable shift in $b\to s\ell^+\ell^-$17; in SCET, this can reduce $b\to s\ell^+\ell^-$18 by $b\to s\ell^+\ell^-$19–$b\to s\ell^+\ell^-$20 (Zhu, 2010).

A particularly clean discriminator is

$b\to s\ell^+\ell^-$21

Because the penguin-over-tree ratios are below $b\to s\ell^+\ell^-$22, their squared effects largely cancel, leading to

$b\to s\ell^+\ell^-$23

With $b\to s\ell^+\ell^-$24, $b\to s\ell^+\ell^-$25, and $b\to s\ell^+\ell^-$26, QCDF predicts

$b\to s\ell^+\ell^-$27

independent of the form factor. If charming penguins are important, a global SCET fit gives, for example,

$b\to s\ell^+\ell^-$28

so that $b\to s\ell^+\ell^-$29, or in another solution $b\to s\ell^+\ell^-$30, both well below the QCDF value. A measured value near $b\to s\ell^+\ell^-$31 would support small, perturbative charming penguins as in QCDF, whereas $b\to s\ell^+\ell^-$32 would signal sizable nonperturbative charming-penguin contributions beyond QCDF expectations. The proposed experimental sensitivity is at the $b\to s\ell^+\ell^-$33 level with LHCb and Belle II (Zhu, 2010).

6. First-principles lattice-QCD framework

A lattice-QCD framework has been developed for $b\to s\ell^+\ell^-$34 and $b\to s\ell^+\ell^-$35 that targets precisely the complex contributions generated by on-shell intermediate states between the weak operator and electromagnetic current(s), including charming penguins and the chromomagnetic operator $b\to s\ell^+\ell^-$36 (Frezzotti et al., 5 Aug 2025).

The relevant four-quark operators are

$b\to s\ell^+\ell^-$37

with $b\to s\ell^+\ell^-$38, together with

$b\to s\ell^+\ell^-$39

The time-ordered amplitude

$b\to s\ell^+\ell^-$40

is split into $b\to s\ell^+\ell^-$41 and $b\to s\ell^+\ell^-$42 pieces, where the latter admits a spectral representation

$b\to s\ell^+\ell^-$43

On the lattice one computes the Euclidean correlator

$b\to s\ell^+\ell^-$44

and reconstructs the Minkowski integral using the Hansen–Lupo–Tantalo expansion

$b\to s\ell^+\ell^-$45

so that

$b\to s\ell^+\ell^-$46

The connected charming-penguin diagram in which the virtual photon is emitted from the $b\to s\ell^+\ell^-$47 loop is isolated through a four-point correlator with smeared $b\to s\ell^+\ell^-$48- and $b\to s\ell^+\ell^-$49-interpolating fields. On-shell charmonium resonances such as $b\to s\ell^+\ell^-$50 and $b\to s\ell^+\ell^-$51 appear as plateaux in the effective mass

$b\to s\ell^+\ell^-$52

Renormalization is nontrivial. Each time ordering is individually UV-divergent as $b\to s\ell^+\ell^-$53, although the sum is only logarithmic; a three-term subtraction of the kernel separates a UV-finite part from a divergent part without an $b\to s\ell^+\ell^-$54 pole. In addition, $b\to s\ell^+\ell^-$55 mix with lower-dimension operators such as $b\to s\ell^+\ell^-$56 and $b\to s\ell^+\ell^-$57 with coefficients of order $b\to s\ell^+\ell^-$58 or $b\to s\ell^+\ell^-$59. With Wilson–Clover twisted-mass fermions at maximal twist, spurionic symmetries and nonperturbative subtraction conditions are used to remove these mixings (Frezzotti et al., 5 Aug 2025).

The full charm-loop amplitude is then

$b\to s\ell^+\ell^-$60

and contracting with the lepton current and photon propagator shows that charming penguins shift the effective Wilson coefficient, $b\to s\ell^+\ell^-$61. In an exploratory calculation on a single ETMC $b\to s\ell^+\ell^-$62 ensemble with $b\to s\ell^+\ell^-$63, $b\to s\ell^+\ell^-$64, $b\to s\ell^+\ell^-$65, and a lighter-than-physical heavy mass $b\to s\ell^+\ell^-$66, the $b\to s\ell^+\ell^-$67 signal is very clean and the $b\to s\ell^+\ell^-$68 signal is about an order of magnitude smaller. A stability analysis in the trade-off parameter $b\to s\ell^+\ell^-$69 yields a robust reconstruction for a range of $b\to s\ell^+\ell^-$70, and comparison with a “VSA + Breit–Wigner + perturbative continuum” model shows qualitative agreement in the shape and size of the real and imaginary parts of $b\to s\ell^+\ell^-$71 after a global $b\to s\ell^+\ell^-$72 renormalization. The stated long-term goal is a first-principles Standard Model prediction for $b\to s\ell^+\ell^-$73 and $b\to s\ell^+\ell^-$74 over the full $b\to s\ell^+\ell^-$75 range, including the charmonium region, once the charming-penguin and $b\to s\ell^+\ell^-$76 contributions are determined to $b\to s\ell^+\ell^-$77 accuracy (Frezzotti et al., 5 Aug 2025).