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Introduction to Perturbative QCD

Introduction to Perturbative QCD. Chaehyun Yu Yonsei University. Outline. Beauty in B physics Basic Concepts The Standard Model RG and RGE OPE and Effective theory QCD Factorization Naïve Factorization QCD Factorization Perturbative QCD Basic Concepts of PQCD Ingredients of PQCD

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Introduction to Perturbative QCD

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  1. Introduction to Perturbative QCD Chaehyun Yu Yonsei University Introduction to PQCD

  2. Outline • Beauty in B physics • Basic Concepts • The Standard Model • RG and RGE • OPE and Effective theory • QCD Factorization • Naïve Factorization • QCD Factorization • Perturbative QCD • Basic Concepts of PQCD • Ingredients of PQCD • Scales and Penguin Enhancements • Strong Phases and CP asymmetries • Remarks Introduction to PQCD

  3. Beauty in B Physics Introduction to PQCD

  4. We have observed in K meson system: • indirect CPV: 1964 in KLπ+π– • direct CPV: 1999 in CERN-NA48 and FNAL-KTeV exp’ts • B factories have produced a lot of interesting results, particularly in measuring indirect CPV in the B system • sin2β=0.725 0.037 from ΨKS data (ICHEP)indicating that B physics isentering a precision era • Finally, we have recently observed direct CPV in B system • ( • Most branching ratios of charmless B P P and P V are currently measured with errors about 10%20% (mostly based upon 100M B anti-B pairs) 5% 10% errors on amplitudes • One hopes to have <5% errors on the amplitudes of most modes more precise information on α and γ (<10±) Motivations for Studying B Decays Introduction to PQCD

  5. Beauty in the B Meson • The beauty of B mesons lies in its large mass or the mass hierarchy: • In the heavy quark limit, mQ 1, we discover: • Flavor symmetry:dynamics unchanged under heavy flavor exchange (b  c), corrections incorporated in powers of 1/mb–1/mc; • Spin symmetry:dynamics unchanged under heavy quark spin flips, corrections incorporated in powers of 1/mb. • B mesons provide an ideal system for studying heavy-to-heavy transitions. • Much progress has been made in understanding heavy-to-light transitions in recent years: • Perturbative approach:naïve factorization, generalized factorization, QCD-improved factorization, pQCD, and SCET; • Nonperturbative approach:flavor SU(3) symmetry. Introduction to PQCD

  6. Basic Concepts Introduction to PQCD

  7. The Standard Model • Based on the gauge theory – SU(3)C ⅹSU(2)LⅹU(1)Y. • SU(3)C : asymptotic freedom, confinement. • At short distance : αs is small → perturbative calculation. • At long distance : αs is large → confined into colorless hadrons. • Higgs mechanism : SU(2)LⅹU(1)Y → U(1)QED. • Current Higgs mass bound : mH ≥ 114.1GeV. • Hierarchy problem : mH<< Λ 1018GeV. • Fermion Mass : Yukawa interaction . • CKM matrix : Unitary → no FCNC. • CP violation Introduction to PQCD

  8. Within SM, CPV in the quark sector is explained using the CKM matrix, which is unitary and complex: 3 mixing angles 1 weak phase (Kobayashi and Maskawa, 1973) small in magnitude and least known but contain largest CPV phases Both these elements can be explored using various B meson mixing and/or decays: Vub from tree-level decays; Vtd from loop-induced processes. The Matrix Introduction to PQCD

  9. The CKM matrix written in terms of Wolfenstein parameters (l, A, r, and h) becomes [to O(l3)][Wolfenstein, PRL 51, 1945 (1983)] l 0.2264 A 0.801 / e– ib / e– ig • The ultimate goal of studying B physics is not only to achieve precision measurements of the above parameters, but also to discover evidences of new physics and possibly its type (e.g. GUT, SUSY, XD). • One way to detect new physics is to perform consistency checks for the sizes and phases of the CKM elements. • Even if no deviation is seen from SM in these studies, we can still obtain useful and stringent bounds on new physics scales. The Matrix Reloaded Introduction to PQCD

  10. Renormalization : eliminate infinities. • Redefinition of the fields and parameters in the Lagrangian. • Renormalizable : all divergences are reabsorbed in Z . • Counter term method : • : proportional to (Z-1) and treated as new interaction. • Renormalization constants are determined such that the contributions from these new interactions cancel the divergences in the Green function. → They are fixed up to an arbitrary subtraction of finite parts. • MS scheme, MS scheme→ different finite parts define different renormalization scheme. Renormalization and Renomalization Group • Ultraviolet divergences in internal loops. • Regularization – well defined to all orders in petrubation theory. • Dimensional regularization : preserve all symmeries (D=4-2ε). →NDR scheme : gμμ=D, same anticommuting rules in γ mat. HV scheme : complexity of calculation. Introduction to PQCD

  11. Every renormalization procedure necessitates to introduce a dimensionful parameter μ into the theory. • Even after renormalization the theoretical predictions depend on the renormalization scaleμ. • One defines the theory at the scale μ. • To determine the renormalized parameters from experiments, a specific choice of μis necessary. • Renormalization group : parameter sets with different μ. • Renormalization group equation. Introduction to PQCD

  12. Breakdown of Peturbation Theory at μ=ΛMS • A particular useful application of the RG is the summation of large logarithms. Introduction to PQCD

  13. Operator Product Expansion • Effective low energy theory describing the weak interactions. OPE For small separation, the product of two field operators can be expanded in local operators Qi. • For b → csu decay, the tree-level W-exchange amplitude is Introduction to PQCD

  14. Effective Theories The scale enters in αs(μ) or the running quark mass, mt(μ), mb(μ), and mc(μ). In Principle, a physical quantity (decay rate, etc.)cannot depend on the renormalization scale. However, as we have to truncate the perturbative series at some fixed order , this property is broken. • How to obtain the Wilson coefficients Ci (μ)? Matching the full theory onto the effective one. • Operator renormalization : mix under renormalization. • Factorization of SD and LD : WCs and ME. • Typical scale of ME : mB→ large logarithm ln(MW2/μ2) at WCs. Renormalization Group Equations for WCs. • Renormalization scale dependence? Introduction to PQCD

  15. Effective Hamiltonian Effective Hamiltonian for nonleptonic weak decays was first put forward by Gaillard, Lee (74), and developed further by Shifman, Vainshtein, Zakharov (75,77); Gilman, Wise (79). At scale , integrate out fermions & bosons heavier than   Heff=c()O() O(): 4-quark operator renormalized at scale  operators with dim > 6 are suppressed by (mh/MW)d-6 Why effective theory ? When computing radiative corrections to 4-quark operators, the result will depend on infrared cutoff and choice of gluon’s propagator, etc. The merit of effective theory allows factorization: WCs c() do not depend on the external states, while gauge & infrared dep. are lumped into hadronic m.e. Radiative correction to O1=(db)V-A(ub)V-A will induce O2=(db)V-A(uu)V-A - - - - Introduction to PQCD

  16. Penguin Diagram Penguin diagram [dubbed by John Ellis (77)] was first discussed by SVZ (75) motivated by solving I=1/2 puzzle in kaon decay It is a local 4-quark operator since gluon propagator 1/k2 is cancelled by (k k-gk2) arising from quark loop as required by gauge invariance Responsible for direct CPV in K & B decays as dynamical phase can be generated when k2>4m2 (time-like) Bander,Silverman,Soni (79) Fierz transformation of (V-A)(V+A)  -2(S-P)(S+P)  chiral enhancement of scalar penguin matrix elements  dominant contributions in many S=1 rare B decays Introduction to PQCD

  17. QCD penguins Gilman, Wise (79) • EW penguins induce four more EW penguin operators • Effective Hamiltonian Buras et al (92) Introduction to PQCD

  18. WC c()’s at NLO depend on the treatment of 5 in n dimensions: • NDR (naïve dim. regularization) {5, }=0 • HVBM (‘t Hooft, Veltman; Breitenlohner, Maison) =mb LO NDR HV c1 1.144 1.082 1.105 c2 -0.308 -0.185 -0.228 c3 0.014 0.014 0.013 c4 -0.030 -0.035 -0.029 c5 0.009 0.009 0.009 c6 -0.038 -0.041 -0.033 c7/ 0.045 -0.002 0.005 c8/ 0.048 0.054 0.060 c9/ -1.280 -1.292 -1.283 c10/ 0.328 0.263 0.266 • Results of WCs ci(i=1,…,10) were first obtained by Buras et al (92) • For details about WCs, see Buras et al. RMP, 68, 1125 (96) • In s 0 limit, c1=1, ci=0 for i1 • c3  c5  –c4/3  –c6/3 • c9 is the biggest among EW penguin WCs Introduction to PQCD

  19. QCD Factorization Introduction to PQCD

  20. Naive Factorization For a given effective Hamiltonian, how to evaluate the nonleptonic decay B M1M2 ? In mb limit, M2 produced in point-like interactions carries away energies O(mb) and will decouple from soft gluon effect M2 B M1 M2 is disconnected from (BM1) system  factorization amplitude  creation of M2 BM1 transition  decay constant  form factor Naïve factorization = vacuum insertion approximation Introduction to PQCD

  21. - - - - Consider B--0 and H=c1O1+c2O2=c1(du)(ub)+c2(db)(uu) - d u u b B- 0 u from O1 color allowed 0 u u d b - B- u from O2 color suppressed Neglect nonfactorizable contributions from O1,2 ~ Introduction to PQCD

  22. Two serious problems with naïve factorization: • Empirically, it fails to describe color-suppressed modes for c1(mc)=1.26 and c2(mc)=-0.51, while Rexpt=0.55 • Theoretically, scheme and scale dependence of ci() doesn’t get compensation from Of as V and A are renor. scale & scheme independent  unphysical amplitude from naïve factorization Introduction to PQCD

  23. How to overcome aforementioned difficulties ? • Bauer, Stech, Wirbel (87) proposed to treat ai’s as effective parameters and extract them from experiment. (Of course, they should be renor. scale & scheme indep.) • If ai’s are universal (i.e. channel indep)  generalized factorization • Test of factorization means a test of universality of a1,2 • Problems: • Penguin ai’s are difficult to determine • Cannot predict CPV • How to predict ai from a given effective Hamiltonian ? Introduction to PQCD

  24. For problem with color-suppressed modes, consider nonfactorizable contributions • To accommodate DK data   -0.35 • In late 70’s & early 80’s, it was found empirically by several groups that discrepancy is greatly improved if Fierz-transformed 1/Nc terms are dropped so that a1  c1, a2  c2. Note that c2+c1/Nc=-0.09vs. c2=-0.51 • [Fukugita et al (77); Tadic & Trampetic (82); Bauer & Stech (85)] • This is understandable as 1/Nc+  0 ! • Buras, Gerard, Ruckl  large-Nc (or 1/Nc) approach (86) •  for charm decays has been estimated by Shifman & Blok (87) using QCD sum rules Nowadays, it is known that one needs sizable nonfactorizable effects and/or FSIs to describe hadronic D decays Introduction to PQCD

  25. If large-Nc approach is applied to B decays  a1eff=c1(mb)  1.10, a2eff=c2(mb)  -0.25  destructive interference in B-D0- just like D+K0+ A(B-D0-)= a1O1+a2O2, while A(B0D-+) = a1O1 supported by sum-rule calculations (Blok, Shifman; Khodjamirian, Ruckl; Halperin) Bigsurprise from CLEO (93): constructive interference as B-D0- > B0D-+ Generalized factorization(I)[HYC (94), Kamal (96)] with 1/Nceff=1/Nc+  determined from experiment For BD decays, Nceff 2 rather than  ,  is positive ! _ Introduction to PQCD

  26. For problem with scheme and scale dependence, consider vertex and penguin corrections to four-quark matrix elements penguin corrections Apply factorization to Otree rather than to O() Introduction to PQCD

  27. Z, Compute corrections to 4-quark matrix elements in the same 5 scheme as ci() : NDR or ‘t Hooft-Veltman Then, in general Ali, Greub (98) Chen,HYC,Tseng,Yang (99) V: anomalous dim., rV: scheme-dep constant, Pi: penguin Gauge & infrared problems with effective WCs [Buras, Silvestrini (99)] are resolved using on-shell external quarks [HYC,Li,Yang (99)] Introduction to PQCD

  28. It is more convenient to define ai=ci+ci1/Nc for odd (even) i CF=(Nc2-1)/(2Nc) • Scale independence of ai or cieff • Scheme independence can be proved analytically for a1,2 and • checked numerically for other ai’s (Vertex & penguin corrections have not been considered in pQCD approach) A major progress before 1999! Introduction to PQCD

  29. Generalized Naive Factorization • Exp shows that the Wilson coefficients are not really universal. • Due to nonfactorizable correction? • Fine tune the mode-dependent parameters to data. • Equivalently, effective number of colors in . • Difficulties: • Gluon’s momentum k2 is unknown, often taken to be mB2/2. It is OK for BRs, but not for CPV as strong phase is not well determined • a6 & a8 are associated with matrix elements in the form mP2/[mb()mq()], which is not scale independent ! • a2,3,5,7,10 (especially a2, a10) are sensitive to Nceff. Expt’l data of charmless B decay  a2  0.20  Nceff 2 Introduction to PQCD

  30. QCD Factorization PRL, 83, 1914 (99) Beneke, Buchalla, Neubert, Sachrajda (BBNS) TI: TII: hard spectator interactions At O(s0) and mb, TI=1, TII=0, naïve factorization is recovered At O(s), TI involves vertex and penguin corrections, TII arises from hard spectator interactions M(x): light-cone distribution amplitude (LCDA) and x the momentum fraction of quark in meson M Introduction to PQCD

  31. twist-2 & twist-3 LCDAs: Twist-3 DAs p &  are suppressed by /mb with =m2/(mu+md) Cn: Gegenbauer poly. with 01 du (u)=1, 01 du p,(u)=1 Introduction to PQCD

  32. In mb limit, only leading-twist DAs contribute The parameters ai are given by ai are renor. scale & scheme indep except for a6 & a8 strong phase from vertex corrections Introduction to PQCD

  33. Penguin contributions Pi have similar expressions as before except that G(m) is replaced by Gluon’s virtual momentum in penguin graph is thus fixed, k2 xmb2 • Hard spectator interactions (non-factorizable) : not 1/mb2 power suppressed: i). B() is of order mb/ at =/mb   d/ B()=mB/B ii). fM  , fB  3/2/mb1/2, FBM  (/mb)3/2  H  O(mb0) [ While in pQCD, H  O(/mb) ] Introduction to PQCD

  34. Power corrections 1/mb power corrections: twist-3 DAs, annihilation, FSIs,… We encounter penguin matrix elements from O5,6 such as formally 1/mb suppressed from twist-3 DA, numerically very important due to chiral enhancement: m2/(mu+md)  2.6 GeV at =2 GeV Consider penguin-dominated mode B K A(BK)  a4+2a6/mb where 2/mb 1 & a6/a4 1.7 Phenomenologically, chirally enhanced power corrections should be taken into account  need to include twist-3 DAs p &  systematically OK for vertex & penguin corrections Introduction to PQCD

  35. Not OK for hard spectator interactions: • The twist-3 term is divergent as p(y) doesn’t vanish at y=1: Logarithmic divergence arises when the spectator quark in M1 becomes soft • Not a surprise ! Just as in HQET, power corrections are a priori nonperturbative in nature. Hence, their estimates are model dependent & can be studied only in a phenomenological way • BBNS model the endpoint divergenceby • with h being a typical hadron scale  500 MeV. • Relevant scale for hard spectator interactions • h=(h)1/2 (hard-collinear scale),s=s(h) • as the hard gluon is not hard enough • k2=(-pB+xp1)2xmB2 QCDmb 1 GeV2 Introduction to PQCD

  36. ai for B K at different scales black: vertex & penguin, blue: hard spectatorgreen: total Introduction to PQCD

  37. Annihilation topology Weak annihilation contributions are power suppressed • ann/tree  fBf/(mB2 F0B)/mB • Endpoint divergence exists even at twist-2 level. In general, ann. amplitude contains XA and XA2 with XA10 dy/y • Endpoint divergence always occurs in power corrections • While QCDF results in HQ limit (i.e. leading twist) are model independent, model dependence is unavoidable in power corrections Introduction to PQCD

  38. Classify into (i) (V-A)(V-A), (ii) (V-A)(V+A), (iii) (S-P)(S+P) • (V-A)(V-A) annihilation is subject to helicity suppression, in analog to the suppression of e relative to  • Helicity suppression is not applicable to (V-A)(V+A) & penguin- induced (S-P)(S+P) annihilation  dominant contributions • Since k2  xymB2 with x,y O(1), imaginary part can be induced from the quark loop bubble when k2> mq2/4 Gerard & Hou (91) Introduction to PQCD

  39. Comparison between QCDF & generalized factorization • QCDF is a natural extension of generalized factorization with the following improvements: • Hard spectator interaction, which is of the same 1/mb order as vertex & penguin corrections, is included  crucial for a2 & a10 • Include distribution of momentum fraction  1. a new strong phase from vertex corrections 2. fixed gluon virtual momentum in penguin diagram • For a6 & a8, V=6 without log(mb/) dependence !So unlike other ai’s, a6 & a8 must be scale & scheme dependent  Contrary to pQCD claim, chiral enhancement is scale indep. Introduction to PQCD

  40. Form factors • B D form factor due to hard gluon exchange is suppressed by wave function mismatch  dominated by soft process • For B , k2  h2  mb. Let FB=Fsoft+Fhard • It was naively argued by BBNS that Fhard=s(h)(/mB)3/2 & Fsoft=(/mB)3/2 • so that B to  form factor is dominated by soft process • In soft-collinear effective theory due to Bauer,Fleming,Pirjol,Stewart(01), • B light M form factor at large recoil obeys a factorization theorem • Writing FB(0)=+J, Bauer et al. determined  & J by fitting to B data • and found   J  (/mb)3/2 • In pQCD based on kT factorization theorem, <<J Beneke,Feldmann (01) Introduction to PQCD

  41. In short, for B M form factor QCDF: Fsoft>> Fhard, SCET: Fsoft Fhard, pQCD: Fsoft<< Fhard However, BBNS (hep-ph/0411171) argued that Fsoft>>Fhard even in SCET We compute form factors & their q2 dependence using covariant light-front model [HYC, Chua, Hwang, PR, D69, 074025 (04)] BSW=Bauer,Stech,Wirbel MS=Melikhov,Stech LCSR=light-cone sum rule B+ +0  F0B(0)  0.25 B0   A0B(0)  0.29 CLF BSW MS LCSR FB(0) 0.25 0.33 0.29 0.31 FBK(0) 0.35 0.38 0.36 0.35 A0B(0) 0.28 0.28 0.29 0.37 A0BK*(0) 0.31 0.32 0.45 0.47 Light meson in B M transition at large recoil (i.e. small q2) can be highly relativistic  importance of relativistic effects Introduction to PQCD

  42. Color-allowed Color-suppressed : universal Wilson coefficients Naive Factorization Naïve factorization (BSW) Introduction to PQCD

  43. Color Transparency Success due to “color transparency” D B Lorentz contraction Small color dipole Decoupling in space-time From the BD system To be quantitative, nonfactorizable correction? Large correction in color-suppressed modes due to heavy D, large color dipole Introduction to PQCD

  44. Shortcoming of Factorization Approach • The non-factorizable contributions are not predictable. • Form factors are not calculable, depend on experiments or other models. • There are difficulties in calculation of the annihilation type diagram (From factors unknown). • The strong phase can not be calculated well which is essential for CP violation prediction. Introduction to PQCD

  45. Perturbative QCD Introduction to PQCD

  46. High energy QCD processes can be factorized into convolution of hard parts with hadron wave function. • Hard parts : calculable in perturbative theory (scale Q). • Wave functions : non-calculable but universable (scale ΛQCD). • The leading order diagram is one gluon exchange diagram with a scale ofwhere . • With the final light meson, there must be a hard gluon to kick the light spectator quark in the B meson to form a fast moving meson. • Nonperturbative dynamics are absorbed into the wave function. • We cannot obtain the naïve factorization in the limit of neglection of gluon exchange. Basic Concepts of PQCD Introduction to PQCD

  47. Ingredients of PQCD A : Factorization in PQCD Introduction to PQCD

  48. All-order proof of factorization theorem • Factorization in momentum space Eikonal approximation • Factorization in spin space Fierz identity • Factorization in color space Ward identity for nonabelian gauge theory • For example, from Fierz identity i k j l Introduction to PQCD

  49. Collinear vs. k Factorization T • Collinear Factorization in B→π transition form factor (QCDF) Lowest order hard kernel is Asymptotic π wave function is End-point singularity → cannot calculate a heavy-to-light form factor FBπ should be treated as a soft object. Introduction to PQCD

  50. Soft Collinear Effective Theory (SCET) More sophisticated treatment of end-point singularity and scales. Factorizable : 2x2 term. Nonfactorizable : 1 term in the lowest-order hard kernel. End-point singularities arise only in the soft, nonperturbative form factors ζ. An end-point singularity in collinear factorization implies its breakdown. kT factorization theorem becomes a more appropriate. Introduction to PQCD

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