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QCD corrections to B → X_1tnsl_1hn+l_1hn- in the standard model and beyond [Elektronische Ressource] / Christoph Bobeth

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+QCD Corrections to B!X l l in thesStandard Model and BeyondChristoph BobethSeptember 2003Physik Department T31Prof. A. J. BurasPhysik{DepartmentTechnische Universitat Munc henInstitut fur Theoretische PhysikLehrstuhl Univ.-Prof. Dr. Andrzej J. Buras+QCD Corrections to B!X l l insthe Standard Model and BeyondChristoph BobethVollst andiger Abdruck der Fakult at fur Physik der Technischen Universit at Munc hen zurErlangung des akademischen Grades einesDoktors der Naturwissenschaften (Dr. rer. nat.)genehmigten Dissertation.Vorsitzender: Univ.-Prof. Dr. Stephan PaulPrufer der Dissertation:1. Univ.-Prof. Dr. Andrzej J. Buras2. Dr. Manfred LindnerDie Dissertation wurde am 28. August 2003 bei der Technischen Universit at Munc hen ein-gereicht und durch die Fakult at fur Physik am 11. September 2003 angenommen.Abstract+The inclusive semileptonic B decay B!X l l will be discussed in this work. Here Xs sdenotes an arbitrary state of total strangeness 1 and the leptons are considered to bel =eor. The theoretical predictions of inclusive quantities are preferable with respect to exclu-sive ones since non-perturbative e ects are under control in the framework of heavy quarkexpansion (HQE). The non-perturbative corrections are small in certain regions of the dilep-ton invariant mass spectrum whereas the major contribution consists in the perturbatively+calculable parton decay b!sl l .

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Published 01 January 2003
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+QCD Corrections to B!X l l in thes
Standard Model and Beyond
Christoph Bobeth
September 2003
Physik Department T31
Prof. A. J. BurasPhysik{Department
Technische Universitat Munc hen
Institut fur Theoretische Physik
Lehrstuhl Univ.-Prof. Dr. Andrzej J. Buras
+QCD Corrections to B!X l l ins
the Standard Model and Beyond
Christoph Bobeth
Vollst andiger Abdruck der Fakult at fur Physik der Technischen Universit at Munc hen zur
Erlangung des akademischen Grades eines
Doktors der Naturwissenschaften (Dr. rer. nat.)
genehmigten Dissertation.
Vorsitzender: Univ.-Prof. Dr. Stephan Paul
Prufer der Dissertation:
1. Univ.-Prof. Dr. Andrzej J. Buras
2. Dr. Manfred Lindner
Die Dissertation wurde am 28. August 2003 bei der Technischen Universit at Munc hen ein-
gereicht und durch die Fakult at fur Physik am 11. September 2003 angenommen.Abstract
+The inclusive semileptonic B decay B!X l l will be discussed in this work. Here Xs s
denotes an arbitrary state of total strangeness 1 and the leptons are considered to bel =e
or. The theoretical predictions of inclusive quantities are preferable with respect to exclu-
sive ones since non-perturbative e ects are under control in the framework of heavy quark
expansion (HQE). The non-perturbative corrections are small in certain regions of the dilep-
ton invariant mass spectrum whereas the major contribution consists in the perturbatively
+calculable parton decay b!sl l .
We will extent the existing next-to leading order (NLO) QCD calculation of the parton
result to the next-to-next-to leading order (NNLO) in QCD within the Standard Model
(SM) of elementary particle physics. In particular, the NNLO matching contributions to the
+Wilson coe cien ts of the operators mediating b!sl l will be calculated.
These results are used in the evaluation of the dilepton invariant mass distribution at the
parton level taking into account the NNLO renormalization group evolution of the Wilson
coe cien ts and the NNLO matrix element calculation of the corresponding operators. As a
result we are able to reduce a large uncertainty of16% to 6% of the dilepton invariant
mass distribution which was mainly due to the renormalization scale of the top-quark mass.
2 2 3 3Furthermore, we will include all known =m and =m non-perturbative cor-QCD c QCD b
rections in the calculation of the dilepton invariant mass distribution at the hadronic level.
The analysis will be restricted to low values of the dilepton invariant mass in the region
s^2 [0:05; 0:25] and to high values in the regions^2 [0:64; 0:78]. The HQE is not invalidated
by intermediate hadron resonances in both regions and allows predictions without model-
dependencies. The poorly known matrix elements of heavy quark e ectiv e theory (HQET)
3 3operators of the =m corrections introduce further theoretical uncertainty.QCD b
The uncertainties of the branching ratio of the low- and high-s^ regions originating from
the residual renormalization scale dependence and the poorly known parameters of HQET
are analyzed. The residual renormalization scale dependencies are found to be of the order
of10% and 16% (compared to 20% and 22% at NLO order) in the low- and high-s^
3 3region, respectively. Additionally, the HQET parameters of the order =m induce anQCD b
uncertainty of 5% in the low-s^ region and a large uncertainty of 15% in the high-s^ region.
Apart from the Standard Model we will consider also a special scenario of the Minimal
Supersymmetric Model (MSSM) with a heavy decoupled gluino and a minimal
a vor violation inspired texture of soft supersymmetry breaking parameters. Analogously
to the SM within this scenario the complete set of NNLO matching contributions to the
+Wilson coe cien ts of the operators mediatingb!sl l will be evaluated. Such corrections
can become numerically important when approximate cancellations occur among the new
physics contributions and/or the SM one. The impact of the NNLO corrections on the
dilepton invariant mass distribution will be discussed in connection with the reduction of
renormalization scale uncertainties.Contents
1 Introduction . ... ..... .... ..... .... ..... .... ..... .... ..... ..... .... ..... ... 9
2 The Standard Model . .... ..... .... ..... .... ..... .... ..... ..... .... ..... .. 15
3 E ective Theories ... ..... .... ..... .... ..... .... ..... .... ..... ..... .... ... 23
3.1 E ectiv e Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 The Group Equation . . . . . . . . . . . . . . . . . . . . . . 30
4 The Minimal Supersymmetric Standard Model .. ..... .... ..... .... ..... .... 33
4.1 The MSSM Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 MSSM with MFV and Gluino Decoupling . . . . . . . . . . . . . . . . . . . 39
+5 B ! X l l . ..... .... ..... .... ..... ..... .... ..... .... ..... .... ..... .... 43
5.1 Matching Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Renormalization Group Evolution . . . . . . . . . . . . . . . . . . . . . . . 50
5.3 Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.4.1 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.4.2 The MSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.5 Non-perturbative Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6 Conclusions and Outlook .. ..... .... ..... .... ..... ..... .... ..... .... ..... . 83
Appendix AMSSM Lagrangian .. ..... .... ..... .... ..... .... ..... .... ..... .... 87
Appendix BNon-Physical Operators ... ..... ..... .... ..... .... ..... .... ..... .. 91
Appendix CWilson Coe cients ..... ..... .... ..... .... ..... .... ..... .... ..... . 95
C.1 i =W { \top quark { W boson" . . . . . . . . . . . . . . . . . . . . . . . . . 96
C.2 i =H { \top { charged Higgs" . . . . . . . . . . . . . . . . . . . . . . 98
C.3 i =~ { \chargino { up-squark" . . . . . . . . . . . . . . . . . . . . . . . . . 99
C.4 i = 4 { \c { up-squark (quartic)" . . . . . . . . . . . . . . . . . . . . 101
Appendix DAuxiliary Functions ..... .... ..... .... ..... ..... .... ..... .... ..... . 103
Bibliography . . .... ..... .... ..... ..... .... ..... .... ..... .... ..... .... ..... . 105
1 Introduction
The current description of particle physics { the so called Standard Model (SM) { has been
formed as a result of a combination of many theoretical concepts combined with experimental
observations. As a quantum eld theory being locally gauge invariant the ideas of quantum
theory, relativistic eld theory and group theory are needed. The SM successfully incorpo-
1rates almost all known properties of three out of the four known fundamental interactions
{ the strong, the weak and the electro-magnetic force { which is re ected by the invariance
under local transformations of the non-abelian gauge symmetrySU(3)
SU(2)
U(1) .C W Y
In principle local gauge invariance implies massless gauge bosons and consequently pre-
dicts long range forces. Therefore local gauge theories might not appear to be the proper de-
scription of the observed short range weak force mediated by massive gauge bosons. This ap-
parent shortcoming can be solved with the help of spontaneous symmetry breaking achieved
through the Higgs mechanism [1{4]. Furthermore, the proof of the renormalizability of local
non-abelian (Yang-Mills) gauge theories [5] in the early 70ties was extended to spontaneously
broken gauge theories [6]. Both, the Higgs mechanism and the proof of renormalizability
were the foundations of the application of local gauge theories to describe short range weak
interactions and the starting point of the formulation of the SM.
The spontaneous breaking of the gauge symmetry is necessary to properly model the
short range interactions and requires the introduction of at least one scalar particle { the
so called Higgs particle. Up to now the Higgs particle escaped the direct detection at high
energy colliders being the last missing part for the experimental con rmation of the SM.
The formally as massless introduced gauge and \matter" elds (leptons and quarks) acquire
masses due to their couplings to the Higgs eld. Thus the Higgs mechanism represents the
mass generation mechanism of gauge and \matter" elds.
As a consequence of the spontaneous symmetry breaking the couplings of theW boson to
quarks are given in terms of the elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix
(V ) [7,8] that arises from the diagonalization of the quark mass matrices. In the SM itCKM ij
is this very matrix that is responsible for all weak decays of hadrons as well as for CP non-
2conservation . CP violation was rst observed in 1964 in kaon decay [11] and recently for
0the rst time in theB meson system in the decayB!J= K [12]. The CKM matrix playsS
a fundamental role in the description of weak decays and the origin of CP violation requiring
an exact knowledge and understanding of these parameters. Theoretical predictions of weak
hadron decays su er generally from uncertainties due to non-perturbative strong interaction
e ects preventing a straightforward determination of the CKM matrix elements. Their
1The experimental evidence for neutrino oscillations [9, 10] can be accommodated for example within
the SM by adding right-handed neutrinos, however there exist also other theoretically more favorable and
attractive alternatives.
2A di eren t source of CP violation appears in QCD when including non-trivial topological e ects due to
the \ -term" known as the strong CP problem.10 1 Introduction
improved understanding is therefore desirable in order to test the SM and perhaps to nd
new physics e ects.
The investigation ofB meson decays promises to provide these kind of insights. Especially
the high value of the b-quark mass in the range of 4 5 GeV leads to a special role of B
mesons for a vor physics studies. The theory of inclusive and also exclusive B decays has
been dramatically improved in the last decade. Especially the heavy quark expansion (HQE)
approach combined with heavy quark e e ctive theory (HQET) methods applied to inclusive
semileptonic and radiativeB decays o ers a quite systematic treatment of non-perturbative
e ects in a model independent way. Remarkably the perturbatively calculable parton de-
cay emerges as the leading contribution to physical observables whereas non-perturbative
2 2corrections are suppressed by =m with m being the heavy quark mass. It shouldQQCD Q
be emphasized that the validity of the HQE depends strongly on the kinematical con gura-
tion under consideration which can limit the reliability and applicability of this approach in
practice.
The class of avor changing neutral current (FCNC)B decays forms a special group among
B processes from a theoretical point of view. Within the SM their decay rate is naturally
suppressed as they proceed only at the loop level in perturbation theory. This mechanism
seems to be con rmed by the present experimental data through the fact that the branching
ratios of FCNC decays are tiny. So far the experimental results are in agreement with the
expected range of magnitude predicted by the SM, however still some decay channels are
unobserved. The \loop" suppression of FCNC processes leads to a strong dependence on
virtually exchanged particles, such as the top quark or the electroweak gauge bosons in the
SM. With respect to the consideration of physics beyond the SM the sensitivity of FCNC
processes to the exchange of unknown particles makes them attractive because they provide
tests of the SM and constrain parameter spaces of new physics models.
Among the rare FCNC processes the radiative and semileptonic channels mediated at
+the parton level by b! s and b! sl l , respectively, are experimentally observed. The
experimental and theoretical situations of the inclusive decay B!X have progressed tos
a very involved point. The present data from CLEO, BaBar, Belle and ALEPH give the
current world average [13] for the branching ratio B!X s
+0:41 4B[B!X (E > 1:6 GeV)] = (3:28 ) 10 (1.1)s exp 0:36
with an error of around 12%. The measurement of B!X requires the introduction of as
lower cut on the photon energy in theB meson rest-frame being typicallyE > 2:0 GeV [14]
to exclude the dominant charm background. The extrapolation of the branching ratio to
lower energy ranges down to 1:6 GeV with the help of theoretical models introduces besides
the statistical and systematical error also theoretical model dependencies. So far these model
dependencies do not dominate the total error of the experimental result but this might easily
happen in the near future. With the expected high luminosity of the B factories (BaBar
and Belle), an experimental accuracy below 10% appears to be achievable.
In the meantime the theoretical prediction of B ! X has reached an advanced levels
including a huge variety of corrections. As mentioned before within the HQE the leading
contribution is the perturbatively calculable parton decay b!s inclusive bremsstrahlung
corrections. It is known up to next-to leading order in QCD (NLO) as well as all enhanced
2logarithmic and 1= sin electroweak corrections. Further the leading non-perturbativeW
2 2 =m fQ =c;bg corrections are calculated. A careful reconsideration of this process [15]QCD Q