# Charmless B-decays in Soft-Collinear Effective Theory

**Principal Investigator:** Guido Bell (Siegen)

**Participating Researchers:** Thorsten Feldmann (Siegen),
Björn O. Lange
(Siegen), Jan Piclum
(Siegen)

*B*-meson decays into light (charmless) hadrons play
a central role in testing the Cabibbo-Kobayashi-Maskawa (CKM)
mechanism of quark flavour mixing and CP violation. Within the
Standard Model (SM), charmless *B*-decays are mediated
by rare *b→ u* quark transitions or by loop-induced
*b→ d/s* decays that are sensitive to effects from new
heavy particles beyond the SM. The phenomenology of charmless
*B*-decays is extremely rich, ranging from semileptonic
*B→ X _{u}lν* decays to nonleptonic

*B→Kπ*, radiative

*B→ X*and electroweak penguin

_{s}γ*B→K*decays.

^{(∗)}llMost of the charmless *B*-meson decays will be
scrutinized at the Large Hadron Collider (LHC) and the
super-flavour factory Belle~II in the next few years. A
meaningful inter\-pretation of the experimental data requires
to control the underlying hadronic matrix elements in Quantum
Chromodynamics (QCD). The heavy quark expansion as well as its
field-theoretical incarnation in terms of effective field
theories (EFTs) provide a systematic framework to compute the
hadronic matrix elements in a twofold expansion in
Λ_{QCD}/ m_{b} and
α_{s}(m_{b}).

At leading power in the heavy quark expansion the hadronic
matrix elements factorize into perturbatively calculable
coefficient functions and process-independent hadronic
parameters. The factorization is particularly involved for
decays with energetic, massless particles in the final state
(with respect to the *B*-meson rest frame). The
energetic (*collinear*) degrees of freedom prevent a
local operator product expansion, and one is left with
convolutions of hard-scattering kernels with nonlocal hadronic
matrix elements. The relevant effective theory is called
Soft-Collinear Effective Theory (SCET) and one distinguishes
between two different versions, SCET_{I} and
SCET_{II}, depending on the physical context.

SCET_{I} is the appropriate effective theory for
inclusive *B*-meson decays as *B→
X _{u}lν* and

*B→ X*. Experimental cuts constrain the measurements of the decay spectra to the endpoint region in which the invariant mass of the hadronic system is small. The factorization of the differential decay rates in the endpoint region is well established. SCET

_{s}γ_{I}also provides the means to resum parametrically large logarithms to all orders in perturbation theory based on renormalization group (RG) techniques.

The theoretical understanding of exclusive
*B*-decays, as *B→Kπ* or
*B→K ^{(∗)}ll* at large hadronic recoil, within
SCET

_{II}is currently incomplete. The problem is related to the separation of the various (soft and collinear) long-distance modes which result in endpoint-divergent convolution integrals. At present, one typically tries to circumvent this problem by absorbing the endpoint-sensitive contributions into hadronic parameters which are not factorized but treated as nonperturbative inputs. While such a procedure limits the application of the EFT techniques, a better understanding of the endpoint dynamics is desirable. In particular, it would open the path for studying power corrections to exclusive

*B*-decays — a problem of primary importance for nonleptonic

*B*-decays as well as for the precision angular analysis of

*B→K*decays.

^{(∗)}llThe current project aims at improving the SCET techniques in
various respects. First and foremost, the factorization of
short- and long-distance effects in SCET_{II}
applications to exclusive *B*-meson decays will be
revisited. In the last few years, there has been substantial
progress in understanding SCET_{II} factorization
theorems for collider physics observables. The new techniques
go under the names *collinear anomaly* and *rapidity
renormalization group*, and the current project aims at
transferring these techniques to *B*-physics
applications. In addition, RG properties of nonperturbative
input parameters — the shape function for inclusive
*B*-decays and light-cone distribution amplitudes
(LCDAs) for exclusive *B*-decays — will be
investigated. The project also intends to derive new
factorization theorems as well as to refine their perturbative
input by computing the relevant hard and jet functions to
higher orders.