Investigation On The Relativistic Correction Of X_bJ?J/?+y Decay Width

In this paper we mainly studies the decay of bottom quarkonium to charm quarkonium J/? plus a photon.According to feynman's rule,the contribution of tree diagram of this process comes from quantum electrodynamics(QED),namely: J/? is obtained by a virtual photon fragmentation.However,due to the small electromagnetic coupling constant,the paper also considers the contribution of quantum chromodynamics(QCD).Since J/? is a color singlet,the contribution of the leading order comes from one loop diagram.The thesis will study these two kinds of processes separately.In this paper,the nonrelativistic quantum chromodynamics(NRQCD)factorization theorem will be used to study.The effective field theory of NRQCD can be obtained from the first principle of strong interaction and is an important tool for the study of heavy quarkonium.The NRQCD factorization theorem can expand the generation cross section and decay width of heavy quarkonium into a series of products of short range coefficients and long range matrix elements.The short range coefficient contains the perturbation scale contribution of the process,while the long range matrix element contains all the non-perturbation scale contribution of the process.The short-range coefficients can be expanded according to the strong coupling constant,and the different long-range matrix elementsare expanded according to the characteristic velocity(v)of the heavy quarks in the heavy quarkonium system.So NRQCD is a double expansion of the coupling constant and the velocity.The factorization theorem guarantees that the short-range coefficients are all infrared finite.In this paper,we will study and calculate the short-range coefficients in front of the long-range matrix elements of the leading order and the next-leading order.For the short-range coefficients,we will reserve the leading order of the perturbation expansion.Some recent literatures have pointed out that the existing NRQCD factorization theorem cannot guarantee that all short range coefficients are infrared finite for the generation process of double P charm quarkonium,that is,the short-range coefficients contain the non-perturbation contribution of the long range,so the NRQCD factorization theorem needs to be modified and expanded.Our calculations show that the leading order matrix elements corresponding to the drag coefficient is the infrared limited,however next leading order matrix elements in front of the short-range coefficient is remaining infrared divergence,which means that the NRQCD factor of theorem in the treatment of the relativistic correction of higher order when there are some problems,so the NRQCD need to introduce new additional operator to absorb the contribution of the species.The research of the paper is very valuable.The focus of this paper is to calculate the short-range coefficients.For effective field theory,the matching method is an important and common method to calculate the short-range coefficients.We first calculate the decay width of the process by replacing the heavy quarkonium with a pair of positive and negative heavy quark pairs,which have the same quantum number as the heavy quark quarkonium.Since there is no bound state in the process,the perturbation can be calculated strictly,and we will reserve to the next-leading orderofvelocity expansion(relativistic correction).Then we replace the heavy quarkonium in the matrix element in the NRQCD factorization theorem expansion formula with a pair of positive and negative heavyquark pairs with the same quantum number,so that the matrix element can be perturb calculated.Finally,if the perturbation decay width is the same as the width of the factor expansion,the short-range coefficients in front of different matrix elements can be obtained systematically.It should be emphasized here that replacing heavy quarkonium with free heavy quark does not change the perturbation contribution of the process,so if the factor theorem is true,the short-range coefficients obtained by our perturbation matching are exactly the short-range coefficients of the decay process of the real heavy quarkonium.In the calculation of this paper,we firstly use the method of spin projection operator to extract the corresponding spin states,and then synthesize the spin and orbital angular momentum into total angular momentum.Since the calculation is complicated to obtain the relativistic correction of the loop diagram,we expand the amplitude according to Lorentz invariance(expansion into formal factor).The relativistic correction mainly includes two sources:1.The relative momentum q of heavy quark pairs in heavy quarkonium is projected into orbital singlet(S wave);2.Correction for the difference between the mass of heavy quarkonium and the mass of heavy quarks.For convenience,we use the method of covariant projection operator to calculate the first contribution.And the second contribution,we're going to rescale the momentum so that the newly obtained momentum doesn't contain any relativistic modifications at all.After the formal factor is obtained,the decay width can be obtained by substituting the amplitude into the formula of the decay width.By calculation,the short-range coefficients in front of the leading order matrix element and the next leading order matrix element ofprocess are obtained.The results show that the short-range coefficients in front of the leading order matrix elements are exactly the same as those in the literature.However,the short-range coefficients corresponding to the next-to-leading order matrix elements have some infrared divergence residues,which means that the NRQCDfactorization theorem is not complete when studying the relativistic correction of this kind of process,so it is necessary to introduce additional operators to absorb the long-range contributions.In fact,recent literature has pointed out that NRQCD factorization theorem needs to be extended for double P wave heavy quarkonium production process.In addition,the results of the paper also show that for the decays of,and the residual infrared divergence in the short-range coefficients is different.We will do further research in the following work.