B - physics

Sung Hyun Chang

11/04/2003

I summarize lecture of "B physics and CP violation" at '1995 EUROPEAN SCHOOL OF HIGH-ENERGY PHYSICS', written by Matthias Neubert. That lecture's sub-topics are heavy-quark symmetry, exclusive weak decays of B mesons, inclusive decay rate and lifetimes of b hadrons, and CP violation in B-meson decays. Among the these sub-topics, I summarize only two sub-topics, exclusive weak decays of B mesons and CP violation in B mesons decays

Introduction

The rich phenomenology of weak decays has always been a source of information about the nature of elementary particle interactions. A long time ago [eqn: beta]- and [eqn: mu```]-decay experiments revealed the structure of the effective flavor-changing interactions at low momentum transfer. Today, weak decays of hadrons containing heavy quarks are employed for tests of the Standard Model and measurements of its parameters. In particular they offer the most direct way to determine the weak mixing angles, to test the unitarity of the Cabibbo-Kobayashi-Maskawa(CKM) matrix, and to explore the physics of CP violation

weak decays of hadrons can be divided into three classes

- leptonic decays : only leptons appear in the final state.

- semi-leptonic decays : both leptons and hadrons appear in the final state.

- non-leptonic decays : final state consists of hadrons only.

In the real world, quarks are confined inside hadrons, bound by the exchange of soft gluons. The simplicity of the weak interactions is overshadowed by the complexity of the strong interactions. A complicated interplay between the weak and strong forces characterizes the phenomenology of hadronic weak decays More realistic representation of  a non-leptonic decay

Exclusive Semileptonic decays

Semileptonic decays of B mesons have received a lot of attention in recent years. The decay channel [eqn: bold {{bar B}`` -> ``D^* l` {bar nu}``}] has the largest branching fraction of all B-meson decay modes. The analysis of these decays provides much information about the strong forces that bind the quarks and gluons into hadrons.

The strength of the b -> c transition vertex is governed by the element [eqn: bold V_{rm cb}``]of the CKM matrix. The parameters of this matrix are fundamental parameters of the Standard Model. A primary goal of the study of semileptonic decays of B mesons is to extract with high precision the values of [eqn: left|bold V_{rm cb} right|] and

Semileptonic Branching Ratio

The semileptonic branching ratio of B mesons is defined as

[eqn: B_SL ={GAMMA`({barB}`->`X`e`barnu)}over{{SUM}_l ``GAMMA`(`bar B ``->`` X`` l`` {barnu}`)``+``GAMMA_{rm NL} ``+``GAMMA_{rm rare} }]

where [eqn: GAMMA_NL ```]and [eqn: GAMMA_{rm rare} ```] are the inclusive rates for non-leptonic and rare decays, respectively.

The average value at low energies is [eqn: B_{rm SL}=(10.37+-0.30)%```]. High energy measurements give [eqn: {B_{rm SL}}^{(b`)}=(11.11 +- 0.23)%] . If we take the average, we obtain

[eqn: B_{rm SL}`=`(10.90``+-``0.46`)`%``]

CP VIOLATION

The violation of CP symmetry is one of the most intriguing aspects of high-energy physics. Experimentally, it is one of the least tested properties of the Standard Model . All CP violating effects are related to the phase [eqn: delta``] of the CKM matrix.

This description has the major difficulties as follow.

: CP violation in the Standard Model is believed to be too small to account for the observed baryon asymmetry in the Universe.

P, C, and CP Transformations

The parity transformation is a space-time transformation, under which [eqn: t````->``t```],[eqn: vec x````->```-``vec x```]. It changes the sign of momenta, [eqn: vec p ````->``-``vec p````], leaving spins unchanged.

Charge conjugation is a transformation that relates particles and antiparticles.

The combined transformation, CP, acts on the pseudoscalar meson states as follows:

For neutral mesons, [eqn: {P^{0}```}]and [eqn: {{bar P}^{0}}```], one can construct the CP eigenstates.

which obey

Direct CP Violation in Weak Decays

Let [eqn: P```]and [eqn: bar P ```]be CP-conjugated pseudoscalar meson states, and [eqn: f```]and [eqn: bar f ```] some CP-conjugated final states.

The phases [eqn: varphi_P ```] and [eqn: varphi_f ```]are arbitrary. The CP-conjugated decay amplitudes, [eqn: A```]and [eqn: bar A ```], can be written as

where [eqn: H```]is the effective Hamiltonian for weak decays, and [eqn: A_i ```]are real partial amplitudes.

Indirect CP Violation in the Mixing of Neutral Mesons

The neutral mesons [eqn: {P^{0}```}] and [eqn: {{bar P}^{0}}```] can mix via common decay channels

An arbitrary neutral meson state can thus be written as a superposition of the flavour eigenstates [eqn: a``|{P}^{0}>``+``b``|{barP}^0 >```],which obeys the time-dependent Schrodinger equation.

where [eqn: rm bold M```]and [eqn: rm bold GAMMA```]are Hermitian 2×2 matrices, which are called the mass and decay matrices, respectively. Since the Hamilton operator,[eqn: rm bold H```], is not Hermitian, its eigenvectors.

are net orthogonal, and the eigenvalues

This reflects that the states [eqn: P_1 ```]and [eqn: P_2 ```]are resonances, net elementary particles.[eqn: M_i ```]are the masses of these resonances, and [eqn: GAMMA_i ```]are their decay widths.

CP Asymmetries in Neutral B-meson Decays

As mentioned above, decays of neutral B mesons into CP eigenstates provide for largely model-independent determinations of the angles of the unitarity triangle. In the B-meson system, up to corrections of order , we have

Most channels receive contributions from "tree" and "penguin" diagrams, which for a generic  decay contribute in the ratio

It follows that there are three possibilities to obtain the dominance of a single diagram

- If the CKM parameters of the penguin diagram are not enhanced with respect to

the tree diagram i.e if

the tree diagram dominates over the penguin diagram. Examples of such decays

are  ,   ,    ,   and   .

-If tree diagrams are forbidden, the penguin diagram dominates. Examples of such

decays are  ,    ,    ,   and

-If

both the tree and the penguin diagram have the same weak phase. In that case one

still has , i.e. no hadronic uncertainties. Examples of such decays are

,   and    .

Tree-dominant decays :

The decays  proceeds through the quark decay , for which both the tree and the penguin diagram have CKM parameters of order . Thus, the tree diagram is dominant, and to a good approximation.

Hence,

Tree-forbidden decays :

The decay  proceeds through the quark transition  , which is forbidden at the tree level in the Standard Model. Thus, the relevant diagram is the penguin transition

This is essential for decays with a single  , since only  and  transitions are allowed. It follows that

and therefore

Decays with a single weak phase :

The decay  is based on the quark transition , for which the tree diagram is dominant.

and therefore

The above-mentioned examples are only meant to illustrate the range of possibilities for performing model-independent measurements of CP-violating CKM parameters in neutral B-meson decays into CP eigenstates.

Reference

Matthias Neubert,"'B PHYSICS AND CP VIOLATION'-in EUROPEAN SCHOOL OF HIGH-ENERGY PHYSICS", 1996, GENEVA.