Work Report about B to VT Run-Hui Li

content  About tensor mesons  Comparison of tensor and vector mesons  Calculation in pQCD, wave functions and the counting rules  About our programs  The preliminary results

Momentum definition In the rest frame of B meson, the lightcone momentum of B, tensor and vector mesons are defined as r 2 2 and r 3 2 are kept for the convenience of polarization definitions. However, little momentum violation, so the terms which proportional to the r 2 should be neglect to keep the momentum conservasion.

About tensor  J P =2 +  5 polarization states with J z =2,1,0,-1,-2  Nonet: a 2 (1320),K 2 * (1430),f 2 (1270),f 2 ’ (1525) Isovector

Mixing between f 2 and f 2 ’ D.M. Li, H. Yu, and Q.X. Shen, JPG 27,807(2001) PDG, PLB 667,1(2008).

Polarization tensor  The spin-2 polarization tensor can be constructed by the spin-1 polarization vector:

By defining a new polarization vector the tensor wave function can be expressed as The wave function of the vector meson is v is the light cone direction opposite to the one along which the tensor meson moves.

Calculation in pQCD -----Start from the weak effective Hamiltonian

10 Calculation in pQCD

Topological Diagrams in pQCD  Emission diagrams  Annihilation diagrams

Wave functions B meson wave functionAsymptotic form of vector meson wave function

Counting rules for B to VV  Emission diagrams  Annihilation diagrams

Wave function of tensor Hard to establish the similar counting rules

Functions for all the figures  By inserting different operators in the diagrams, one can obtain all the possible functions for the figures. ex. When the tensor and vector meson is longitudinally polarised, the function for the (V-A)(V-A) operator of the factorizable emission diagram is  For one polarization case, there are 20 nozero such functions. 3 polarization cases, 60 nonzero functions.  The decay amplitude can be expressed by these 60 functions.

The expressions of the amplitude  For each F function, different operators may contribute. EX. Tree QCD penguin Electroweak penguin

 By collecting the contributed figures and judging the Wilson coefficients for each figure, one can obtain the expressions of the amplitude. EX.

About the programs  The next step is to obtain the numerical values for each F function on the above page and get the numerical results for the amplitude.  Purpose of the designing  Not only for B to VT, but also for B to AT,VA decays.  For the B to VT decays, 62 channels. The number of the B to AT decays are also large. In the purpose of making it easier from the B to VT to B to AT (or even repeat the B to VA calculations), I design a program to generate the decomposition of the amplitude expressions automatically.  For the same purpose, we make many modules, trying to separate the process-independent functions.

About the programs  13 files  About 6000 lines, except the Vegas module.  After correcting all the bugs, the B to AT can be calculated by substituting the 2 amplitude files and setting some parameters in Wilsoninitial.for file.

Some preliminary results Large deviations from the experimental data. Perhaps there are still some bugs in the programs.

Plans for solving the problems  Making use of the similarities of the tensor and vector mesons, check our formulas for the 60 F functions again to make sure that they are correct. EX.

 For the bugs in the programs, maybe we can figure it out by comparing the data of similar channels, or by analyzing the data of each figure.  Doing some form factor calculation, compare the results with the pQCD results for form factors.