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Philipp Böer

Publications and source records attributed to Philipp Böer.

At least 19 recordsLinked to original sources
arXiv2026

Three-particle di-light-cone distribution amplitudes of the $B$-meson in heavy-quark effective theory

We present a systematic study of the three-particle di-light-cone distribution amplitudes (DLCDAs) of the $B$-meson. They are defined through $B$-meson--to--vacuum matrix elements of trilocal HQET operators, in which the light antiquark and the gluon field-strength tensor are located on two back-to-back light rays. In this sense, the DLCDAs generalise the conventional $B$-meson light-cone distribution amplitudes to configurations where soft fields couple to collinear degrees of freedom in two distinct directions. As such, they parametrise the non-perturbative dynamics associated with non-factorisable soft-gluon contributions in rare and non-leptonic exclusive $B$-meson decays. We derive the complete Lorentz decomposition of the matrix elements of generic trilocal operators, identify eight independent DLCDAs, and organise them in a basis of definite twist. Using local operator identities and equations-of-motion constraints, we obtain tree-level relations for their normalisation integrals and first moments in terms of a minimal set of hadronic parameters. These relations allow us to construct simple momentum-space models for all independent DLCDAs. For the leading-twist distribution, we further incorporate the perturbative radiative tail at order $α_s$ and discuss its impact on the resulting parametrisation.

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arXiv2026

$B_c \to η_c$ form factors at large recoil: SCET analysis and a three-loop consistency check

The double-logarithmic series of non-relativistic $B_c \to η_c$ form factors at large recoil is governed by a coupled set of integral equations, reflecting an intricate interplay between arbitrarily many soft-quark and soft-gluon exchanges. Whereas we previously derived these integral equations with diagrammatic resummation techniques, we analyze the form factors in the limit $m_b \gg m_c \gg Λ_{\rm QCD}$ with methods from Soft-Collinear Effective Theory (SCET) in this work. Although the resulting factorization theorem for the so-called soft-overlap contribution is known to be spoilt by endpoint divergences, it can still be used at the level of bare (regularized) quantities at any fixed order in perturbation theory. By calculating the required ingredients, we show that the SCET analysis confirms the predictions of the integral equations up to three-loop order. We also argue that the iterative structure and the intertwined soft-quark and soft-gluon effects can be derived from standard renormalization-group equations of the $B_c$-meson light-cone distribution amplitudes, provided their inverse moments are regularized with an appropriate cutoff.

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arXiv2025

NLO analysis of the subleading-power $Q_1-Q_{7γ}$ interference in $\bar{B}\to X_sγ$ at large photon energies

We report on progress towards including next-to-leading order (NLO) radiative corrections to the subleading-power factorization formula for the $Q_1^{q}-Q_{7γ}$ interference contribution in $\bar{B}\to X_sγ$ at large photon energies, $m_b - 2E_γ= \mathcal{O}(Λ_{\rm QCD})$. The novel ingredients for a NLO analysis are the one-loop anomalous dimension of the subleading shape function $g_{17}(ω,ω_1;μ)$, which was recently computed by us, and the two-loop corrections to the jet function with an internal charm loop. Here we summarize the renormalization of $g_{17}(ω,ω_1;μ)$, and further discuss that our insights allow for a simplified renormalization of a specific generalization of a three-particle $B$-meson light-cone distribution amplitude, but with non-aligned light-like separations of the fields.

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arXiv2025

$B_c \to η_c$ form factors at large recoil: Interplay of soft-quark and soft-gluon dynamics

We perform an all-order analysis of double-logarithmic corrections to the so-called soft-overlap contribution to heavy-to-light transition form factors at large hadronic recoil. Specifically, we study $B_c \to η_c$ transitions within a perturbative non-relativistic framework, treating both the bottom and charm quarks as heavy with the hierarchy $m_b \gg m_c \ggΛ_{\rm QCD}$. Our diagrammatic analysis shows that double-logarithmic corrections arise from two distinct sources: Exponentiated soft-gluon effects described by standard Sudakov factors, and rapidity-ordered soft-quark configurations, leading to implicit integral equations, which so far have only been studied in the context of energetic muon-electron backward scattering. We find that the all-order structure of the double logarithms is governed by a novel type of coupled integral equations, which encode the non-trivial interplay between these two effects. Whereas a closed-form solution to these equations is currently unknown, we present useful iteration formulas, and derive the asymptotic behaviour of the soft-overlap form factor for infinitely large recoil energies, showing that the Sudakov suppression is somewhat weakened by the intertwined soft-quark and soft-gluon corrections. In a broader context, our findings shed light onto the physical origin and mathematical structure of endpoint divergences arising from soft-collinear factorization and the related Feynman mechanism for power-suppressed hard exclusive processes.

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arXiv2025

Renormalisation group evolution of the shape function $g_{17}$ in $\bar{B}\to X_s γ$ and $\bar{B} \to X_s \ell^+\ell^-$ at subleading power

We derive and solve the renormalisation-group (RG) equation of the shape function $g_{17}(ω,ω_1;μ)$, which appears at subleading power in the factorization of the inclusive decays $\bar{B} \to X_s γ$ and $\bar B \to X_s \ell^+\ell^-$. Our results provide the first ingredient for a next-to-leading order analysis of the respective resolved-photon ${Q}^{c}_{1} - {Q}_{7γ}$ interference contribution, whose current uncertainties are among the largest ones in both inclusive penguin modes. As a by-product of our study, we find that the analytic properties of the soft anomalous dimension as well as the jet functions in a factorization theorem allow for a simplified renormalisation of operators in Heavy-Quark Effective Theory that are composed of fields smeared along distinct light-cone directions. Their hadronic matrix elements become relevant in various inclusive and exclusive $B$-decays beyond leading power in the heavy-quark and large-energy expansion. Using these insights, we derive and solve a simpler ``reduced'' RG equation of an amplitude-level soft function that describes the long-distance QCD dynamics for penguin contributions to exclusive $\bar{B}_{d,s} \to γγ$ decays, and which has recently been discussed in the literature.

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arXiv2024

Renormalization-Group Improved Resummation of Super-Leading Logarithms

A new strategy is presented for systematically treating super-leading logarithmic contributions including higher-order Glauber exchanges for non-global LHC observables in renormalization-group (RG) improved perturbation theory. This represents an important improvement over previous approaches, as it allows for the consistent inclusion of the scale dependence of the strong coupling, thereby providing more reliable estimates of the scale uncertainties in theoretical predictions. The key idea is to rearrange the relevant RG evolution operator in such a way that all double-logarithmic corrections are exponentiated from the outset. This forms the starting point for the first resummation of super-leading logarithms at leading order in RG-improved perturbation theory for arbitrary $2\to M$ scattering processes. Moreover, the asymptotic scaling of subleading logarithmic corrections from higher-order Glauber exchanges is determined, demonstrating their parametric suppression.

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arXiv2024

Resummation of Glauber Phases in Non-Global LHC Observables for Large $N_c$

The Glauber series for non-global jet observables at hadron colliders simultaneously includes the super-leading logarithms alongside an arbitrary number of Glauber phases. Building on the formalism of arXiv:2405.05305, it is shown that the leading terms in this series for large $N_c$ can be resummed in closed form in renormalization-group improved perturbation theory. This remarkable observation suggests that large-$N_c$ methods might also be helpful to study other aspects of non-global logarithms at hadron colliders, and to combine our analytic results with amplitude-level parton showers.

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arXiv2023

Glauber Phases in Non-Global LHC Observables: Resummation for Gluon-Initiated Processes

The resummation of the ''Glauber series'' in non-global LHC observables is extended to processes with gluons in the initial state. This series simultaneously incorporates large double-logarithmic corrections, the so-called ''super-leading logarithms'', together with higher-order exchanges of pairs of Glauber gluons associated with the large numerical factor $(iπ)^2$. On a technical level, the main part of this work is devoted to the systematic reduction of the appearing color traces and construction of basis structures, which consist of thirteen elements for $gg$ and eleven elements for $qg$ scattering. Numerical estimates for wide-angle gap-between-jet cross sections at the parton level show that, in particular for $gg$ scattering at relatively small vetoes $Q_0$, the contribution involving four Glauber exchanges gives a sizeable correction and should not be neglected.

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arXiv2023

Glauber Phases in Non-Global LHC Observables: Resummation for Quark-Initiated Processes

It has been known for many years that jet cross sections at hadron colliders exhibit double-logarithmic corrections starting at four-loop order, arising from two soft Glauber-gluon interactions between the two colliding partons. The resummation of these "super-leading logarithms" has been achieved only recently by means of a renormalization-group treatment in soft-collinear effective theory. We generalize this result and, within the same framework and for quark-initiated processes, resum the double logarithms arising in the presence of an arbitrary number of Glauber-gluon exchanges. For typical choices of parameters, the higher-order Glauber terms give rise to corrections which are expected to be numerically of the same magnitude as the super-leading logarithms. However, we find that the Glauber series for jet cross sections is dominated by the two-Glauber contribution, if the colliding partons are quarks or anti-quarks.

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arXiv2023

Soft-overlap contribution to $B_c \to η_c$ form factors: diagrammatic resummation of double logarithms

Using diagrammatic resummation techniques, we investigate the double-logarithmic series of the "soft-overlap" contribution to $B_c \to η_c$ transition form factors at large hadronic recoil, assuming the scale hierarchy $m_b \gg m_c \gg Λ_{\rm QCD}$. In this case, the hadronic bound states can be treated in the non-relativistic approximation and the relevant hadronic matrix elements can be computed perturbatively. This setup defines one of the simplest examples to study the problem of endpoint singularities appearing in the factorization of exclusive $B$-decay amplitudes. We find that the leading double logarithms arise from a peculiar interplay of soft-quark "endpoint logarithms" from ladder diagrams with energy-ordered spectator-quark propagators, as well as standard Sudakov-type soft-gluon corrections. We elucidate the all-order systematics, and show that their resummation proceeds via a novel type of integral equations. The current status of the calculation, which includes all double logarithms in the Abelian limit, is reported.

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arXiv2023

On the gauge-invariance of SCET beyond leading power

We point out that the gauge-invariance of the subleading Lagrangian of soft-collinear effective theory is realised in an intricate way through momentum-conservation violating contributions. Although these terms are disregarded in diagrammatic calculations, the gauge invariance of any physical transition amplitude is preserved due to the soft equations of motion. When not working with gauge-invariant building blocks, individual manifestly gauge-invariant constituent terms in the Lagrangian may give rise to gauge-dependent matrix elements starting at $\mathcal{O}(λ^2)$. Implications for a gauge-invariant definition of radiative jet functions are discussed.

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arXiv2022

Muon-electron backward scattering: a prime example for endpoint singularities in SCET

We argue that energetic muon-electron scattering in the backward direction can be viewed as a template case to study the resummation of large logarithms related to endpoint divergences appearing in the effective-theory formulation of hard-exclusive processes. While it is known since the mid sixties that the leading double logarithms from QED corrections resum to a modified Bessel function on the amplitude level, the modern formulation in Soft-Collinear Effective Theory (SCET) shows a surprisingly complicated and iterative pattern of endpoint-divergent convolution integrals. In contrast to the bottom-quark induced $h \to γγ$ decay, for which a renormalized factorization theorem has been proposed recently, we find that rapidity logarithms generate an infinite tower of collinear-anomaly exponents. This can be understood as a generic consequence of the underlying $2\to 2$ kinematics. Using endpoint refactorization conditions for the collinear matrix elements, we show how the Bessel function is reproduced in the effective theory from consistency relations between quantities in a "bare" factorization theorem.

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arXiv2022

Light-cone distribution amplitudes of heavy mesons with QED effects

We discuss the QED-generalized leading-twist light-cone distribution amplitudes of heavy mesons, that appear in QCD$\times$QED factorization theorems for exclusive two-body $B$ decays. In the presence of electrically charged particles, these functions should be more appropriately regarded as soft functions for heavy-meson decays into two back-to-back particles. In this paper, we derive the one-loop anomalous dimension of these soft functions and study their behaviour under renormalization-scale evolution, obtaining an exact solution in Laplace space. In addition, we provide numerical solutions for the soft functions and analytical solutions to all orders in the strong and to first order in the electromagnetic coupling. For the inverse (and inverse-logarithmic) moments, we obtain an all-order solution in both couplings. We further provide numerical estimates for QED corrections to the inverse moments.

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arXiv2021

Light-cone distribution amplitudes of light mesons with QED effects

We discuss the generalization of the leading-twist light-cone distribution amplitude for light mesons including QED effects. This generalization was introduced to describe virtual collinear photon exchanges above the strong-interaction scale $Λ_{\rm QCD}$ in the factorization of QED effects in non-leptonic $B$-meson decays. In this paper we study the renormalization group evolution of this non-perturbative function. For charged mesons, in particular, this exhibits qualitative differences with respect to the well-known scale evolution in QCD only, especially regarding the endpoint-behaviour. We analytically solve the evolution equation to first order in the electromagnetic coupling $α_{\rm em}$, which resums large logarithms in QCD on top of a fixed-order expansion in $α_{\rm em}$. We further provide numerical estimates for QED corrections to Gegenbauer coefficients as well as inverse moments relevant to (QED-generalized) factorization theorems for hard exclusive processes.

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arXiv2021

QCD factorization of the four-lepton decay $B^-\rightarrow \ell \barν_\ell \ell^{(\prime)} \bar{\ell}^{(\prime)}$

Motivated by the first search for the rare charged-current $B$ decay to four leptons, $\ell \barν_\ell \ell^{(\prime)} \bar{\ell}^{(\prime)}$, we calculate the decay amplitude with factorization methods. We obtain the $B\to γ^*$ form factors, which depend on the invariant masses of the two lepton pairs, at leading power in an expansion in $Λ_{\rm QCD}/m_b$ to next-to-leading order in $α_s$, and at $\mathcal{O}(α_s^0)$ at next-to-leading power. Our calculations predict branching fractions of a few times $10^{-8}$ in the $\ell^{(\prime)} \bar{\ell}^{(\prime)}$ mass-squared bin up to $q^2=1~\text{GeV}^2$ with $n_+q>3~$GeV. The branching fraction rapidly drops with increasing $q^2$. An important further motivation for this investigation has been to explore the sensitivity of the decay rate to the inverse moment $λ_B$ of the leading-twist $B$ meson light-cone distribution amplitude. We find that in the small-$q^2$ bin, the sensitivity to $λ_B$ is almost comparable to $B^- \rightarrow \ell^- \barν_\ellγ$ when $λ_B$ is small, but with an added uncertainty from the light-meson intermediate resonance contribution. The sensitivity degrades with larger $q^2$.

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arXiv2021

QED factorization of two-body non-leptonic and semi-leptonic $B$ to charm decays

The QCD$\times$QED factorization is studied for two-body non-leptonic and semi-leptonic $B$ decays with heavy-light final states. These non-leptonic decays, like $\bar{B}^0_{(s)}\to D^+_{(s)} π^-$ and $\bar{B}_d^0 \to D^+ K^-$, are among the theoretically cleanest non-leptonic decays as penguin loops do not contribute and colour-suppressed tree amplitudes are suppressed in the heavy-quark limit or even completely absent. Advancing the theoretical calculations of such decays requires therefore also a careful analysis of QED effects. Including QED effects does not alter the general structure of factorization which is analogous for both semi-leptonic and non-leptonic decays. For the latter, we express our result as a correction of the tree amplitude coefficient $a_1$. At the amplitude level, we find QED effects at the sub-percent level, which is of the same order as the QCD uncertainty. We discuss the phenomenological implications of adding QED effects in light of discrepancies observed between theory and experimental data, for ratios of non-leptonic over semi-leptonic decay rates. At the level of the rate, ultrasoft photon effects can produce a correction up to a few percent, requiring a careful treatment of such effects in the experimental analyses.

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arXiv2020

QED factorization of non-leptonic $B$ decays

We show that the QCD factorization approach for $B$-meson decays to charmless hadronic two-body final states can be extended to include electromagnetic corrections. The presence of electrically charged final-state particles complicates the framework. Nevertheless, the factorization formula takes the same form as in QCD alone, with appropriate generalizations of the definitions of light-cone distribution amplitudes and form factors to include QED effects. More precisely, we factorize QED effects above the strong interaction scale $Λ_{\rm QCD}$ for the non-radiative matrix elements $\langle M_1 M_2|Q_i|\bar{B}\rangle$ of the current-current operators from the effective weak interactions. The rates of the branching fractions for the infrared-finite observables $\bar B\to M_1 M_2(γ)$ with photons of maximal energy $ΔE\llΛ_{\rm QCD}$ is then obtained by multiplying with the soft-photon exponentiation factors. We provide first estimates for the various electromagnetic corrections, and in particular quantify their impact on the $πK$ ratios and sum rules that are often used as diagnostics of New Physics.

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