重庆大学物理系（吴兴刚）：Some new applications of the Principle of Maximum Conformality（PPT讲稿，英文）

Some new applications of the Principle of maximum Conformality 具兴刚 Xing-Gang Wu 重庆大学物理系 Department of Physics, Chongging University 全国第十六届重味物理和CP破坏研讨会河南工业大学

Some new applications of the Principle of Maximum Conformality 吴兴刚 Xing-Gang Wu 重庆大学物理系 Department of Physics, Chongqing University 全国第十六届重味物理和CP破坏研讨会 河南工业大学

OUTLINE 微扰论 The principle of Maximum Conformality(PMC) Some applications Summary and Outlook

OUTLINE Summary and Outlook The principle of Maximum Conformality (PMC) Some applications 微扰论

The perturbative theory: A physical observable p could be written as the perturbative form =En P(AR)+r1a.p+1 (μ)+x2aP+2(k)+. Q s Di yts Reliable High precision a。<1 High-order prediction; Introducing renormalization scheme/scale regularization, renormalization, /scale -setting =QCDa(M2)=01181±00011 ame import Q(GeVT Up to infinite order, there is no scheme-and scale- dependence, i.e., any choice of scheme/scale should result in the same prediction. Standard Renormalization group invariance(RGI)

The perturbative theory: A physical observable  could be written as the perturbative form Up to infinite order, there is no scheme- and scale￾dependence, i.e., any choice of scheme/scale should result in the same prediction. “Standard Renormalization group invariance (RGI)” High-order prediction; Introducing renormalization scheme/scale regularization、renormalization、scale-setting same importance 𝛼𝑠 < 1 Reliable High precision

Conventional scale-setting approach =>Choose the scale Q to be''seemingly"typical momentum transfer = Vary in a certain range, e. g [Q/2, 2Q],[Q/3, 3Q], to discuss its uncertainty Main problems of conventional scale-setting 1)Convergence depends on as-power suppression Once inconvergence appears, one cannot judge whether is its intrinsic property or caused by improper choice of scale 2) By finishing more loop-terms, the scale-dependence could be smaller, it is however caused by cancellation among different orders

Conventional scale-setting approach =>“Choose” the scale Q to be ``seemingly” typical momentum transfer =>“Vary” in a certain range, e.g. [Q/2, 2Q], [Q/3, 3Q], to discuss its uncertainty Main problems of conventional scale-setting 1) Convergence depends on s -power suppression； Once inconvergence appears, one cannot judge whether it is its intrinsic property or caused by improper choice of scale。 2) By finishing more loop-terms, the scale-dependence could be smaller, it is however caused by cancellation among different orders

Even if the prediction of quessing scale agrees with the data It, in fact, cannot answer why this is the case It is important to achieve reliable fixed-order prediction at low-order levels such as nlo and nnlo The reason for scale dependence at fixed-order? Mismatching under conventional treatment p=x0(1)aB(1)+x1(1)a3+1(1)+x2(1)aB2(1)+ p=x0(2)aB(2)+x1(2)a31(2)+x2(2)a32(42)+ Directly replacing H1->H2=> Mismatching of coefficients and alphas values = one reason for large scale uncertainty

Even if the prediction of guessing scale agrees with the data. It, in fact, cannot answer why this is the case It is important to achieve reliable fixed-order prediction at low-order levels, such as NLO and NNLO ? The reason for scale dependence at fixed-order ? Mismatching under conventional treatment Directly replacing 1 -> 2 => Mismatching of coefficients and alphas values => one reason for large scale uncertainty

寻找对方案和能标变化的稳定点 任意物理量可定义有效荷=>有效能标 ===============================================5H Any observable nf-term QED不存在重整化能标: e.g. USIng a5u2a)+… ED极限=GML方案 设定问题 n"=== ======== ======= (csR=>解决方案不确定性 1/137-本身数倒小

早期解决方案＝寻找最优能标 Optimized perturbation theory – minimize the higher-order contributions – PMS How about directly set it to satisfy the RG invariance 寻找对方案和能标变化的稳定点 BLM=> nf-term QED极限＝GM-L方案 CSR＝>解决方案不确定性 Any observable an effective coupling constant (idea useful) Fastest Apparent Convergence (FAC) How about directly cut off all higher-order-terms ? 任意物理量可定义有效荷=>有效能标 Using RGE Runs from 1 -> 2 e.g. using s (1 )=s (2 )+… 典型 四类

Questions for previous solutions PMS一破坏标准重整化群不变性,只能近似有效 FAC-由实验反定,破坏理论预言能力 BLM-单圈成功,但如何拓展到高圈,有不少失败尝试 如 seBUM等等一-基于大βo近似,目的变为提高微扰收敛性 RGE一降低依赖,未解决问题,一般用于估算未知高阶(RG- improved) 新一轮尝试:真正解决重整化 能标及重整化方案不确定性 最大共形原理一PMc 可以选择任意初始重整化能标、重整化方案完成微扰论计算;但经过PMC能标设定 步骤之后,最终获得的微扰表达式与初始重整化能标和重整化方案的选择无关 优点: 1)可确定正确“物理”动量流动值一与初始能标选择无关,但不同方案下得到的动 量流动值不一样,与共形系数相匹配=>CSR=>总预言与重整化方案无关 )可自然改善微扰收敛性、可更好估箅未知高阶贡献

Questions for previous solutions PMS － 破坏标准重整化群不变性，只能近似有效 FAC － 由实验反定，破坏理论预言能力 BLM － 单圈成功，但如何拓展到高圈，有不少失败尝试： 如seBLM等等－－基于大0近似，目的变为提高微扰收敛性 RGE －降低依赖，未解决问题，一般用于估算未知高阶（RG－improved） 新一轮尝试：真正解决重整化 能标及重整化方案不确定性 最大共形原理－PMC 可以选择任意初始重整化能标、重整化方案完成微扰论计算；但经过PMC能标设定 步骤之后，最终获得的微扰表达式与初始重整化能标和重整化方案的选择无关 优点： I）可确定正确“物理”动量流动值－与初始能标选择无关，但不同方案下得到的动 量流动值不一样，与共形系数相匹配＝> CSR ＝> 总预言与重整化方案无关 II）可自然改善微扰收敛性、可更好估算未知高阶贡献

Two equivalent ways to achieve the goal of PMC Multi-scale scale-sefting approach Single-scale scale-setting approach Key: Based on the RGe, achieving the correct beta-series at each order p(Q)=n10a(pP+20+m2l()+1+0+mm2x+(+1)1+pp+1) B32|a()2 +{0+p2+(p+1)31+24132+(p+2)1+ (p+1)(p+2) 6r4,2 p(p+1)(p+2) 3! n3a()+3 利用已知β项一类似于重求和一可准确确定耦合常数值 ∑ Man+i-1 (QM.D+ p(Q)=∑n0a(Q)y+n- i.O s Residual scale dependence Relatively large

Two equivalent ways to achieve the goal of PMC Multi-scale scale-setting approach Single-scale scale-setting approach Key: Based on the RGE, achieving the correct beta-series at each order 利用已知项－类似于重求和－可准确确定耦合常数值 Residual scale dependence Relatively large

Recent PMC applications e-J/+ne Zhan Sun, Yang Ma, XGW,SJB PRD2018 mass collision energy vs= 10.58 GeV [1 L○ NRQCD prediction Belle lete→J+×B24=36+9 2.3-5.5fb oete→J/p+nlxB2=25.6±28±3.4 BaBr176±281tb3 NLO-prediction can explain the data 14 Y J. Zhang, Y j. Gao and K. T Chao, Phys. Rev. Lett But strongly scale dependent 96.092001(2006) 15 B. Gong and J. X. Wang, Phys. Rev. D 77, 05-4028 Guessing: Hr 2-3 GeV Bell PMC scale Why such guessing Q1=Q2=2.30GeV scale is correct Babar 0+o +) PMC works

Recent PMC applications NLO-prediction can explain the data But strongly scale dependent Guessing: Belle BaBar LO-NRQCD prediction Why such guessing scale is correct？ PMC scale Zhan Sun, Yang Ma, XGW, SJB PRD2018 PMC works

Y→n orm factor puzzle Sheng-Quan Wang, Wen-Long Sang, XGW, SJB PRD2018 (n (P)EMIr(k, e))=ie-ewpEq k, F(2-) LO-roughly agrees with data NLO-agrees better with data PMC scale NNLO-highly inconsistent with data Conventional scale 100 Special: NNLO >NLO onv. uA=mc) .o.. NLO NNLO(PMC this workI NNLO(PRL 115.222001) A, and a, uncertainties BABAR measurements F(Q2)/F(O) ◆ BABAR data 平。 NRQCD wrong? PMC works

NRQCD wrong ? LO-roughly agrees with data NLO-agrees better with data NNLO-highly inconsistent with data Special: |NNLO|>|NLO| PMC works Sheng-Quan Wang, Wen-Long Sang, XGW, SJB PRD2018