微扰QCD方法计算稀有衰变B +→D+sK0

单位代码10635学号112015315001012硕士学位论文微扰QCD 计算K K K B c c (3770),(2S),(2S)ψηψ→的衰变论文作者：段丰博指导教师：余先桥学科专业：理论物理研究方向：强子物理提交论文日期：2018年4月论文答辩日期：2018年6月学位授予单位：西南大学中国∙重庆2018年6月目录摘要....................................................................................................................................IV Abstract. (VI)第1章引言...........................................................................................................................1第2章B 物理与弱相互作用. (4)2.1B 物理简介 (4)2.2弱相互作用 (5)2.2.1弱相互作用简介 (5)2.2.2弱相互作用的V-A 理论 (6)第3章低能有效哈密顿方法和微扰QCD 因子化 (9)3.1基本理论框架 (9)3.2算符乘积展开 (9)3.3强子跃迁矩阵元 (9)3.4Sudakov 因子 (13)3.5阈值重求和 (16)第4章微扰计算 (18)4.1光锥坐标系 (18)4.2c B 介子衰变的有效哈密顿量 (19)4.3介子波函数 (19)4.3.1c B 介子波函数 (19)4.3.2K 介子波函数..............................................................................................204.3.3c c 粲偶态波函数 (21)4.4)1(13D ψ的分布振幅 (21)4.5)1(13D B c ψ→的跃迁形状因子 (23)4.6衰变分支比 (25)4.6.1计算K X B c c c →的衰变振幅 (27)4.6.2计算KBX的衰变分支比 (29)c cc4.7数值结果 (30)4.8小结 (33)第5章总结与展望 (35)附录A Wilson系数 (36)附录B计算中用到的公式 (37)参考文献 (38)致谢 (42)攻读硕士学位期间的工作情况 (43)微扰QCD 计算K K K B c c (3770),2S ,(2S)ψηψ→的衰变理论物理专业硕士研究生：段丰博指导教师：余先桥摘要粒子物理标准模型是关于微观物质最小单元即轻子和夸克，最基本的三种规范相互作用强相互作用、电磁相互作用和弱相互作用，产生粒子质量的黑格斯机制以及轻子、夸克、希格斯粒子和规范相互作用的量子相互作用和转化的基础理论。

第26卷 增刊 高能物理与核物理V o1.26，Supp. 2002年12月HIGH ENERGY PHYSICS AND NUCLEAR PHYSICS Dec.，2002 Two-Body B Decays to Pseudoscalar and Vector Mesonsin QCD Factorization ApproachYANG Mao-Zhi1 YANG Ya-Dong21 (Institute of High Energy Physics, CAS, Beijing 100039, China)2 (Department of Physics, Technion, Haifa 32000, Israel)Abstract Motivated by recent CELO measurements and the progress of the theory of B decays,B→PV(P=π, K; V= K*, ρ, ω) decay modes are studied in the framework of QCD factorization.All the measured branching ratios are well accommodated in the reasonable parameter space andpredictions for other decay modes are well below the experimental upper limits.Key words factorization, weak decay, mesonB physics is one of the most important fields nowadays because it is of great help for testing the quark flavor mixing theory of the standard model and exploring the source of CP violation. Most of the theoretical studies of B decays to pseudocalar and vector final states are based on the popular Naive Factorization approach[1]. As it was ponited out years ago in Ref. [2], the dominant contribution in B decays comes from the so-called Feynman mechanism, where the energetic quark created in the weak decay picks up the soft spectator softly and carries nearly all of the final-state meson's momentum. It is also shown that Pion form factor in QCD at intermediate engery scale is dominated by Feynman mechanism[3—5]. From this point, we can understand why the naive factorization approach have worked well for B and D decays, and the many existing predictions for B decays based on naive factorization and spectator ansatz do have taken in the dominant physics effects although there are shortcommings. However, with the many new data available from CLEO and an abundance of data to arrive within few years from the B factories BaBar and Belle, it is demanded highly to go beyond the naive factorization approach.Recently, Beneke et al., have formed an interesting QCD factorization formula for B exclusive nonleptonic decays[6,7]. The factorization formula incorporates elements of the naive factorization approach (as leading contribution) and the hard-scattering approach (as subleading corrections), which allows us to calculate systematically radiative(subleading nonfactorizable) corrections to naive factorization for B exclusive nonleptonic decays. An important product of the formula is that the strong final-state interaction phases are calculable, which arise from the2 高能物理与核物理(HEP &NP) 第26卷hard-scattering kernel and hence process dependent. The strong phases are very important for studying CP violation in B decays.The amplitude of B decays to two light mesons, say M 1 and M 2, is obtained through the hadronic matrix element <M 1(p 1) M 2(p 2)⏐O i ⏐B (p )>, here M 1 denotes the final meson that picks up the light spectator quark in the B meson, and M 2 is the another meson which is composed of the quarks produced from the weak decay point of b quark. Since the quark pair, forming M 2, is ejected from the decay point of b quark carrying the large energy of order of m b , soft gluons with the momentum of order of ΛQCD decouple from it at leading order of ΛQCD /m b in the heavy quark limit. As a consequence any interaction between the quarks of M 2 and the quarks out of M 2 is hard at leading power in the heavy quark expansion. On the other hand, the light spectator quark carries the momentum of the order of ΛQCD , and is softly transferred into M 1 unless it undergoes a hard interaction.Any soft interaction between the spectator quark and other constituents in B and M 1 can be absorbed into the transition form factor of B →M 1. The non-factorizable contribution to B →M 1 M 2 can be calculated through the diagrams in Fig.1.Fig. 1. Order αs non-factorizable contributions in B →M 1M 2 decays.The O i 's incorporated in Fig.1 are the operators in the effective Hamiltonian for B decays [8], ⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛+++⎟⎟⎠⎞⎜⎜⎝⎛++=∑∑∑∑====21103g g ccqcb 21103g g uuq ub F eff2i i i i i i *i i i i i i *O C O C O C V V O C O C O C V V G H , (1)Where()()A V A V O －－ββααu 1b u u q ⋅=, ()()A V A V O －－αββαu 2b u u q ⋅=, ()()A V A V O －－ββααc 1b c c q ⋅=, ()()A V A V O －－αββαc 2b c c q ⋅=,()()AV A V O －－ββq αα3q q b q ′′⋅=∑′,()()AV A V O －－αβq βα4q q b q ′′⋅=∑′,()()AV A V O +′′′⋅=∑ββq αα5q q b q －, ()()A V A V O +′′′⋅=∑αβq βα6q q b q －, ()()A V A V e O +′′′′⋅=∑ββq q αα7q q b q 23－, ()()A V A V e O +′′′′⋅=∑αq q b q 23βq q βα8－,增 刊 杨茂志等：用QCD 因子化方法研究B →PV 两体弱衰变过程 3()()A V A V e O －－ββq q αα9q q b q 23′′⋅=∑′′, ()()A V A V e O －－αβq q βα10q q b q 23′′⋅=∑′′, ()()AA a g G b R m g O µνβαβµνλσ2/d π8/b 2s =. (2)Here q=d, s and (q'ε {u, d, s, c, b}), α and β are the SU (3) color indices and , A =1,...,8 are the Gell-Mann matrices, and denotes the gluonic field strength tensor. The Wilson coefficients evaluated at µ=m AαβλAG µνb scale are[8]C 1= 1.082, C 2=−0.185, C 3= 0.014, C 4=−0.035, C 5= 0.009, C 6=−0.041,C 7=−0.002/137, C 8=0.054/137, C 9=−1.292/137, C 10=0.262/137, C g =−0.143. (3) The non-factorizable contributions to B →M 1M 2 can be calculated through the diagrams in Fig.1. The details of the calculations can be found in Ref. [9]. In the numerical calculations we use[10]τ (B +) = 1.65×10－12s, τ (B 0) = 1.56×10－12s,M B = 5.2792GeV , m b = 4.8GeV , m c = 1.4GeV , f B= 0.180GeV , f π = 0.133GeV , f K = 0.158GeV , f K * = 0.214GeV , f ρ = 0.21GeV , f ω = 0.195GeV .For the chiral enhancement factors for the pseudoscalar mesons, we takeR π ±π= R K ±, 0 = －1.2 ,which are consistent with the values used in [6, 11, 12]. We should take care for R π0. As pointedout in Ref. [7], R π0 for π0 should be －2M /(m 2b (m u + m d )) and equal to R π± due to inclusion ofisospin breaking effects correctly.For the form factors, we take the results of light-cone sum rule[13,14]F B →π(0)=0.3, F B →K (0)=1.13F B →π(0), A =0.372, A =0.470,ρB 0→*K B 0→and assume (0)=1.2(0) since we find larger (0) is preferred by experimental data.ωB 0→A ρB 0→A ωB 0→A We take the leading-twist distribution amplitude (DA) φ(x ) and the twist-3 DA φ0(x ) of light pseudoscalar and vector mesons as the asymptotic form[15]φP,V (x ) =6x (1－x ), (x ) =1. (4) 0P φFor the B meson, the wave function is chosen as[16,17]()(),xM x x N x ⎥⎥⎦⎤⎢⎢⎣⎡=2B 22B 22B B 2exp 1ωφ－－ (5)with ωB =0.4GeV , and N B is the normalization constant to make(x ) =1. φ∫1Bd φx B (x ) is stronglypeaked around x =0.1, which is consistent with the observation of Heavy Quark Effective Theory that the wave function should be peaked around ΛQCD /M B .We have used the unitarity of the CKM matrix V *uq V ub +V *V cq cb +V *tq V tb =0 to decompose the4 高能物理与核物理(HEP &NP)第26卷amplitudes into terms containing , V *uq V ub and V *V cq cb , and⏐V us ⏐=λ=0.2196, ⏐V ub /V cb ⏐=0.085±0.02, ⏐V cb ⏐=0.0395±0.0017, ⏐V ud ⏐=1－λ2/2 . (6) We leave the CKM angle γ as a free parameter.The numerical results of the branching ratios B →PV are shown in Fig.2 as the function of CKM angle γ. We can see from Fig. 2(a), (b) and (c) that for the three detected channels the predicted branching ratios agree well with the CLEO experiment data [18]. Our predictions for other decay modes are well below their 90% C.L. upper limits.There are several works available with detailed analysis of the CLEO new data of the decays of B to charmless PV states[11,12,19]. It is worth to note that the shortcomings in the “generalizedfactorization” are resolved in the framework of QCD Factorization. Nonfactorizable effects are calculated in a rigorous way here instead of being parameterized by effective color number. Since the hard scattering kernals are convoluted with the light cone DAs of the mesons, gluon virtualityk 2=2b m x in the penguin diagram Fig. 1(e) has well defined meaning and leaves no ambiguity as tothe value of k 2, which has usually been treated as a free phenomenological parameter in the estimations of the strong phase generated though the BSS mechanism [20]. So that CP asymmetries are predicted soundly in this approach. We present the numerical result of the branching ratios of B →PV decays in Table 1 with the relevant strong phases shown explicitly. It shows that the strong phases are generally mode dependent.Table 1. Strong phases in the branching ratios (in units of 10－6) for thecharmless decays modes studied by CLEO. (γ =Arg V *u b )B (B －→π－ρ0)=6.65⏐0.11e －i86.5°+e －i γ⏐2B (0B →π+ρ－)=19.79⏐0.11e i9.02°+e －i γ⏐2B (0B →π－ρ+)=13.43⏐0.03e i172°+e －i γ⏐2B (B －→π－ω)=10.59⏐0.065e i26.01°+e －i γ⏐2B (0B →π0ρ0)=0.11⏐0.21e 2.90°+e －i γ⏐2B (B －→π0ρ－)=10.81⏐0.176e i7.20°+e －i γ⏐2B (0B →π－ω)=1.49×10－3⏐1.64e i148°+e －i γ⏐2B (B －→K －ρ0)=0.55⏐0.24e －i162°+e －i γ⏐2B (B －→π－⎯K *0)=0.0012⏐56.4e －i15.7°+e －i γ⏐2B (B －→K －K *0)=0.030⏐2.86e i164°+e －i γ⏐2B (B －→π0K *－)=0.59⏐2.80e －i169°+e －i γ⏐2B (B －→K －ω)=0.80⏐0.48e －i9.23°+e －i γ⏐2B (0B →K 0ω)=0.72⏐0.81e －i 11.8°+e －i γ⏐2B (⎯B 0→K －ρ+)=0.96⏐0.63e －i7.20°+e －i γ⏐2B (0B →π0⎯K *0)=0.004⏐12.89e i67.61°+e －i γ⏐2Hou, Smith and W ürthwein have performed a model dependent fit using the recent CLEOdata and found γ =114degree. Using SU (3) flavor symmetry, Gronau and Rosner have analyzedthe decays of B to charmless PV final states extensively and found several processes are consistent with cos γ < 0. In this paper we find cos γ < 0 is favored by the B 2521+－－→π－ρ0 and ⎯B 0→π－ρ++π+ρ－ if their experimental center values are taken seriously. To meet its center value with cos γ < 0 , B －→π增 刊 杨茂志等：用QCD 因子化方法研究B →PV 两体弱衰变过程 5－ω would indicate larger form factor i.e. A (0) > A (0). In our numerical calculation, wehave taken A (0) = 0.446 which is still consistent with the LCSR results 0.372 ± 0.074ω→B 0ρ→B 0ω→B 0[13]. It isalso interesting to note that ⎯B 0→π+ρ－ is suppressed by cos γ < 0 while ⎯B 0→π－ρ+ is enchanced. The defference between Br (⎯B 0→π+ρ－) and Br (⎯B 0→π－ρ+) is much more sensitive to γ than their sum.6 高能物理与核物理(HEP &NP) 第26卷Summarywe have calculated the branching ratios and CP asymmetries of the charmless decays B →PV(P = (π, K), V= (ρ,ω, K *)) in QCD factorization approach. We have used LCSR formfactors F B →π,K (0) and A (0) as inputs. The results of Br (B *K ,0ρ－→π－ρ0) and Br (⎯B 0→π±ρ) agree with CLEO m [18]very well and favor cos γ < 0 if their experimental center values are taken seriously. To meet its experimental center value and cos γ < 0, the decay B －→π－ω will prefer larger form factor (0). For the other decay modes, the branching ratios are predicted well below their 90% C.L. upper limits given in Ref. [18].ωB 0→A References1 Bauer M, Stech B, Wirbel M. Z. Phys., 1985, C29: 637; Z. Phys., 1987, C34: 1032 Chernyak V L, Zhitnitsky L R. Nucl. Phys., 1990, B345: 1373 Isgur N, Llewelyn-Smith C H. Phys. Rev. Lett., 1984, 52: 1080; Nucl. Phys., 1989, B317: 5264 Radyushkin A V . Acta Phys., 1984, Pol. 15: 4035 Stefanis N G . hep-ph/99113756 Beneke M, Buchalla G , Neubert M. Phys. Rev. Lett., 1999, 83: 19147 Beneke M, Buchalla G , Neubert M et al. hep-ph/00061248 Buchalla G , Buras A J, Lautenbacher M E. Rev. Mod. Phys., 1996, 68: 1125 9 YANG M Z, YANG Y Y . Phys. Rev., 2000, D62: 114019 10 Particle Data Group. Eur. Phys. J., 1998, C3: 1 11 CHENG H Y , YANG K C. hep-ph/991029112 HOU W S, Smith J G , W ürthwein F W. hep-ex/9910014 13 Ball P, Braun V M. Phys. Rev., 1998, D58: 094016 14 Ball P. JHEP09, 005(1998)15Lepage G P, Brodsky S J. Phys. Lett., 1979, B87: 359; Chernyak V L, Zhitinissky A R. Phys. Rep., 1983, 112: 173; Braun V M, Filyanov I E. Z. Phys., 1990, C48: 239 16 Keum Y Y , LI H -N, Sanda A I. Phys. Lett., 2001, B504: 2; Phys. Rev., 2001, D63: 054008 17 LÜ C D, Ukai D, YANG M Z. Phys. Rev., 2001, D63: 07400918 CLEO Collaboration. CLEO CONF 99-13; CLEO Collaboration. CLNS 99/1652 and CLEO 99-19 19 Gronau M, Rosner J L. Phys. Rev., 2000, D61: 073008 20Bander M, Silverman D, Soni A. Phys. Rev. Lett., 1979, 43: 242增刊杨茂志等：用QCD因子化方法研究B→PV两体弱衰变过程7 用QCD因子化方法研究B→PV两体弱衰变过程杨茂志1 杨亚东21 (中国科学院高能物理研究所北京 100039)2 (Department of Physics, Technion, Haifa 32000, Israel)摘要基于最近CLEO实验和B介子物理中理论研究的进展, 在QCD因子化方案下研究了B介子到一个赝标π, K和一个矢量介子ρ, ω的两体弱衰变过程.在合理的参数范围内, 理论计算与实验相符得很好.关键词因子化弱衰变介子。

2021.03科学技术创新P Q C D 因子化方法计算B S→ψ(3770)K π刘文(西南大学物理科学与技术学院,重庆400700)在B 介子三体非轻衰变过程中,由于存在两个硬胶子的交换,使得直接计算三体衰变十分困难。

在PQ CD 因子化框架下,将末态中的两个轻介子引入强子对的分布振幅中,因此就将三体衰变简化为类两体衰变。

因此,B 介子三体衰变振幅可以被普遍地表示为:其中的硬核H 表示在领头阶只存在一个胶子被交换的费曼图贡献,用微扰理论计算这部分。

波函数ϕβ,ϕh3,ϕh 1h 2是非微扰部分,波函数的具体表达式通常在实验中抽取或者通过非微扰方法计算获得。

对于类两体衰变B S 0→ψ(3770)K π,弱有效哈密顿量表示为:我们采用在光锥坐标系下描述衰变中有关的物理量。

假设初态B S 0介子处于静止态,K π介子对沿着类光矢量n=(1,0,0T )方向运动,粲偶素ψ(3770)介子沿着类光矢量v=(0,1,0T )方向运动。

衰变的费曼图如下所示。

图1在P Q C D 方法中B S 0→ψ(3770)K π衰变的最低阶费曼图、和的动量如下所示:与此同时,各个介子相对应的轻夸克动量可以表示为:式中,M B 0S是B 0S 介子质量,M ψ是ψ(1430)介子质量,r =M ψ/M B 0S。

变量η定义为η=ω2/(M 2B 0S-M 2ψ)。

x 1,z 和x 3分别表示介子中不参与衰变夸克的动量分数,它们的取值范围都是0~1。

B 介子被认为是重-轻系统,其波函数可参考文献[1]:分布振幅表示为N B 为归一化常数,其值可以通过关系式10∫dx ϕBs (x B ,b B =0)=f B 0S/22N c √与色因子量N c =3求出。

在整个计算过程中取对于粲偶素介子,采用谐振子基形式构造其波函数,这种形式的波函数在其它粲偶素的波函数构造中已经取得了很大成功,采用这种方法构造的波函数的计算结果也与实验观测结果符合的很好,粲偶素介子的波函数可参考文献[2]:作者简介:刘文(1993-),男,汉族,重庆万州人,硕士研究生,单位:西南大学物理科学与技术学院理论物理专业,研究方向:强子物理。

QCD改善基本原理方法应用实例汇总Quantum Chromodynamics (QCD)是目前最成功的强相互作用理论，用于描述夸克和胶子之间的相互作用。

在高能物理实验中，研究QCD改善基本原理方法的应用实例是非常重要的。

本文将汇总QCD改善的基本原理、方法和实例，并探讨其在粒子物理领域的应用。

一、QCD基本原理QCD是基于量子场论的理论框架，描述了强相互作用的基本原理。

它是一种基于SU(3)规范群的理论，其中夸克和胶子是强相互作用的基本粒子。

QCD的基本原理包括以下内容：1. 夸克的色荷：夸克有三种颜色荷，即红、绿和蓝。

颜色对应了SU(3)规范群的三维表示。

2. 胶子的色荷：胶子携带颜色荷，是传递夸克之间相互作用的粒子。

它们是自旋为1的矢量玻色子。

3. 强相互作用：夸克和胶子之间通过交换胶子相互作用，形成强相互作用。

二、QCD改善的基本方法QCD改善的方法旨在解决强相互作用的复杂性，并改善在高能物理实验中对粒子物理过程的预测精度。

以下是几种常用的QCD改善方法：1. 扰动展开：扰动展开方法是应用拉格朗日量计算物理过程的常用方法。

它将物理过程分解为层级相互作用，并计算各个层级的贡献。

这种方法在高能物理实验中具有广泛的应用。

2. 重整化：重整化是QCD中用于处理无穷大发散问题的方法。

通过引入耦合常数、质量等参数的修正，将计算结果与实验结果相一致。

3. 衰变和散射过程：衰变和散射过程是研究强相互作用的重要手段。

通过计算衰变和散射过程中的强相互作用，可以研究强子物理的性质以及粒子物理的基本参数。

三、QCD改善的应用实例汇总QCD的改善方法在粒子物理实验中有广泛的应用。

以下是一些常见的实例：1. 强子光子耦合的测量：粒子物理实验中，通过测量粒子与光子的相互作用，可以研究夸克和胶子的性质。

QCD改善方法用于提高强子光子耦合的理论预测，从而与实验数据相比较。

2. 强子形状因子的研究：强子形状因子是描述强子内部结构的重要物理量。