QCD-2004 Lesson 3 :Non-perturbative

1. QCD-2004Lesson 3 :Non-perturbative Simple quantities: masses and decay constants Sistematic errors 3) Semileptonic decays 4) Non leptonic decays Guido Martinelli Bejing 2004

2. COULD WE COMPUTE THIS PROCESS WITH SUFFICIENT COMPUTER POWER ? THE ANSWER IS: NO IT IS NOT ONLY A QUESTION OF COMPUTER POWER BECAUSE THERE ARE COMPLICATED FIELD THEORETICAL PROBLEMS LATTICE FIELD THEORY IN FEW SLIDES

3. Z -1 ∫ [d ] (x1) (x2) (x3) (x4) ei S() On a finite volume (L) and with a finite lattice spacing (a ) this is now an integral on L4 real variables which can be performed with Important sampling techniques Z = ∑{=1} eJij i j Ising Model 2N = 2L3 ≈ 10301 for L = 10 !!!

4. Wick Rotation Z -1 ∫ [d ] (x1) (x2) (x3) (x4) ei S() -> Z -1 ∫ [d ] (x1) (x2) (x3) (x4) e - S() This is like a statistical Boltzmann system with  H = S Several important sampling methods can be used, for example the Metropolis technique, to extract the fields with weight e - S() t -> i tE <  > = Z -1 ∑{ (x)}n n(x1) n(x2) n(x3) n(x4) Z = ∑{ (x)}n1 = N

5. Z -1 ∫ [d ] (x1) (x2) (x3) (x4) ei S() This integral is only a formal definition because of the infrared and ultraviolet divergences. These problems can be cured by introducing an infrared and an ultraviolet cutoff. 1) We introduce an ultraviolet cutoff by defining the fields on a (hypercubic) four dimensional lattice (x) ->(a n) where n=( nx , ny , nz , nt ) and a is the lattice spacing;   (x) ->   (x) = ((x+a n) - (x)) /a ; The momentum p is cutoff at the first Brioullin zone, |p|  π / a The cutoff can be in conflict with important symmetries of the theory, as for example Lorentz invariance or chiral invariance This problem is common to all regularizations like for example Pauli-Villars, dimensional regularization etc.

7. LOCAL GAUGE INVARIANCE x y q(x + a  ) q(x) q(y) P [ exp ∫xy i g0 GA (x) tA dx ] q(x) is Gauge invariant GA (x) tA -> V(x) [GA (x) tA] V†(x)+i/ g0 [V(x)] V†(x) q(x) -> V(x) q(x) q(x) -> q(x) V†(x) q(y) = P [ exp ∫xy i g0 GA (x) tA dx ] q(x) y x exp [i g0 GA (x + a  /2) tA ] LINK U†(x)

8. Plaquette U(x) -> V(x) U(x) V†(x + a  ) W(x) = U(x) U(x + a )U†(x + a )U†(x) ≈ 1 + i a2 g0 G(x)- a4 g02/2 G (x)G(x)+ ... 1 ∑x∑ < Re Tr [1-W(x)] -> g02 a4 /4 ∑x ∑  G (x)G(x)-> 1/4 ∫ G (x)G(x) +O(a2)

9. Fermion action(s) We may define many (an infinite number of) lattice actions which all formally converge to the same continuum QCD action: Naïve, Kogut-Susskind, Wilson, Clover, Domain Wall, Overlap. We postpone the discussion of these formulations and return to the calculation of physical quantities like masses, decay constants etc.

10. Determination of hadron masses and simple matrix elements An example from the  4 theory The field  can excite one-particle, 3-particle etc. states

11. Log[G(t)] At large time distances the lightest (one particle) states dominate : For a particle at rest we have  = 1/ m a is the dimensionless correlation length (and the size of the physical excitations) <> m a t/a

12. HADRON SPECTRUM AND DECAY CONSTANTS IN QCD Define a source with the correct quantum numbers : “”  A0(x,t) = ua (x,t) (0 5 ) da (x,t) a=colour =spin G(t) = ∑x <A0(x,t)A†0(x,t) > = ∑n |< 0 | A0|n >|2 exp[- En t] 2 En ->|< 0 | A0|  >|2 exp[- Mt] 2 M -> f2 M exp[- Mt] 2 A†0(x,t) A0(x,t) f M~Z Mass and decay constant in lattice units M = m a

13. In the chiral limit u u,d,s A A A A gluons d anomaly absent in the case of ,K and 8

14. Continuum limit a Formal lim a->0SLattice() -> SContinuum() a/  = m a ~1 The size of the object is comparable to the lattice spacing  = 1/ m a/  <<1 i.e. m a -> 0 The size of the object is much larger than the lattice spacing Similar to a ∑n -> ∫ dx

15. Calibration of the lattice spacing a Let us start for simplicity with massless quarks mq = 0 Mproton=Mproton(g0 ,a ) = mprotona Physical proton mass Measured in the numerical simulation a (g0 ) = Mproton mproton Then we predict m ,m  ,m  ,f , …. we cannot predict m since m2 mq

16. Calibration of the lattice spacing a Mproton=Mproton(g0 ,a ,mup=mdown , mstrange) = mprotona M=M (g0 ,a ,mup=mdown , mstrange) = ma MK=MK(g0 ,a ,mup=mdown , mstrange) = mKa …. a (g0 ,mup=mdown , mstrange) Then we predict m ,m  ,m  ,f , …. …. everything including the quark masses

17. Continuum limit a 0 Using asymptotic freedom a d g0 = 0g30 + 1g50+O (g70) d a a (g0) ~ QCD-1 e-1/(2 0g02) a 0 when g00 Mproton=mprotona =mprotonQCD-1 e-1/(2 0g02) =Cprotone-1/(2 0g02) 0

18. a 0 when g00 Mproton=mprotona =mprotonQCD-1 e-1/(2 0g02) =Cprotone-1/(2 0g02) + O(a)0 These are discretization errors due to the use of a finite lattice spacing; they vanish exponentially fast in g0 With inverse lattice spacings of order 2-4 GeV (+improvement/extrapolation) discretization errors range from O(10%)to less than 1% Mproton / M Cproton / C =const.

19. 3-point functions e+ D†(t1) = ∑xD†(x, t1) exp[-i pD x] K(t2) = ∑xK(x, t2) exp[+i pK x] Jweak(0) e D†(t1) K(t2) from the 2-point functions Kl3 namely <K | Jweak(0) | D > also electromagnetic form factors, structure functions, dipole moment of the neutron, ga/gv,etc.

20. General consideration on non-perturbative methods/approaches/models Models a) bag-model b) quark model not based on the fundamental theory; at most QCD “inspired”; cannot be systematically improved Effective theories c) chiral lagrangians d) Wilson Operator Product Expansion (OPE) e) Heavy quark effective theory (HQET) based on the fundamental theory; limited range of applicability; problems with power corrections (higher twists), power divergences & renormalons; need non perturbative inputs (f, < x >, 1,  ) Methods of effective theories used also by QCD sum rules and Lattice QCD f ) QCD sum rules based on the fundamental theory + “condensates” (non-perturbative matrix elements of higher twist operators, which must be determined phenomenologically; very difficult to improve; share with other approaches the problem of renormalons etc.

21. LATTICE QCD Started by Kenneth Wilson in 1974 Based on the fundamental theory [Minimum number of free parameters, namely QCD and mq ] Systematically improvable [errors can me measured and corrected, see below] Lattice QCD is not at all numerical simulations and computer programmes only. A real understanding of the underline Field Theory, Symmetries, Ward identities, Renormalization properties is needed. LATTICE QCD IS REALLY EXPERIMENTAL FIELD THEORY

22. Major fields of investigation • QCD thermodynamics • Hadron spectrum • Hadronic matrix elements ( K ->  , structure functions, etc. see below ) QCD { • Strong interacting Higgs Models • Strong interacting chiral models EW { • Surface dynamics • Quantum gravity

23. LATTICE QCD Leptonic decay constants : f,fK, fD, fDs, fB, fBs, f, .. Electromagnetic form factors : F(Q2),GM(Q2), ... Semileptonic form factors : f+,0(Q2), A0,..3(Q2), V(Q2)K ->, D -> K, K*, , , B -> D, D*, ,  B -> K*  The Isgur-Wise function B-parameters :  K0 | Q S=2| K0 and  B0 | Q B=2| B0   | Q S=1| K  and    | Q S=1| K  Weak decays : Matrix elements of leading twist operators :

24. Lattice QCD is really a powerful approach BUT… FOR SYSTEMATIC ERRORS

25. Lattice QCD is really a powerful approach SYSTEMATIC ERRORS

26. Quenching errors UNQUENCHED QUENCHED (partially,two-flavours, three?, etc.) • MH/M almost right • Kaon B-parameter essentially the same • effect on fD estimated at 10% level • nucleon -term and polarized structure functions wrong • problems with chiral logarithms • problems with unitarity for two-body decays Almost all groups are now moving to unquenched calculations

27. Extrapolation in the heavy quark mass DISCRETIZATION ERRORS THE ULTRAVIOLET PROBLEM 1/MH >> a mq a << 1 O(a) errors { pa << 1 Typically a-1~2÷ 5 GeV mcharm~ 1.3 GeV mcharm a~ 0.3 mbottom~ 4.5 GeV mbottom a~ 1 For a good approximation of the continuum See talk by Sommer

28. SYSTEMATIC ERRORS P DISCRETIZATION ERRORS Naïve solution: extrapolate measures performed at different values of the lattice spacing. Price: the error increases a fH M1/2H Physical behaviour IMPROVEMENT effect of lattice artefacts 1/MH

29. SYSTEMATIC ERRORS FINITE VOLUME EFFECTS THE INFRARED PROBLEM BOX SIZE L >>  = 1/MH >> a To avoid finite size effects For a good approximation of the continuum Finite size effects are not really a problem for quenched calculations; potentially more problematic for the unquenched case Is L  4 ÷ 5  sufficient ? O(exp[-  /L])

30. Particularly in the unquenched case, because of the limitations in computer resources VOLUMES CANNOT BE LARGE ENOUGH TO WORK AT THE PHYSICAL LIGHT QUARK MASSES (min. pseudoscalar mass is ~ MK, needed ~M ) Typical quark mass ms /2 < mq < ms an extrapolation in mlight to the physical point is necessary Test if the quark mass dependence is described by Chiral perturbation Theory (PT), Then the extrapolation with the functional form suggested by PT is justified

31. 3-point functions e+ D†(t1) = ∑xD†(x, t1) exp[-i pD x] K(t2) = ∑xK(x, t2) exp[+i pK x] Jweak(0) e D†(t1) K(t2) from the 2-point functions Kl3 namely <K | Jweak(0) | D > also electromagnetic form factors, structure functions, dipole moment of the neutron, ga/gv,etc.

32. Quark masses & Generation Mixing e- -decays | Vud | = 0.9735(8) | Vus | = 0.2196(23) | Vcd | = 0.224(16) | Vcs | = 0.970(9)(70) | Vcb | = 0.0406(8) | Vub | = 0.00363(32) | Vtb | = 0.99(29) (0.999) W e down up Neutron Proton | Vud |

33. IN THE ELICITY BASIS: 1- < K(pK) | Jweak(0) | D(pD) > = [(pD + pK) -q (M2D - M2K)/q2]  f + (q2) + q (M2D - M2K)/q2 f 0 (q2 ) < K*(pK*,) | Jweak(0) | D(pD) > = * T 0+ 0- 1- 1+ = t-channel quantum numbers Vector meson polarization 1- T = 2 V(q2) / (MD + MK*)  (pD)(pK*)  + - i (MD + MK*) A1 (q2) g + i A2 (q2) / (MD + MK*)  (pD + pK*)q + - i A(q2) 2 MK* / q2 (pD + pK*)q  A(q2)= A0 (q2) -A3 (q2) 1+ 0-

34. Pole Dominance e+ Jweak(0) e f + (0) Mt f + (q2) = D†(t1) K(t2) 1 - q2 / Mt2 works well for the pion electromagnetic form factor , dipole in the case of the proton f (0) (1 - q2 / Mt2) 2

35. Scaling behavior for the Form Factorsat q2 ≈ (q2 )max Form Factor t-channel mQ dependence B ->  f+ 1- mQ1/2 f0 0+ mQ-1/2 B ->  V1- mQ1/2 A1 1+ mQ-1/2 A2 1+ mQ1/2 A3 1+ mQ3/2 A0 0 - mQ1/2

36. Kinematical constraints & scaling at q2 ≈ 0 f+ (0)= f0 (0) f+ (0)≈mQ-3/2 from Light cone behaviour A POPULAR PARAMETRIZATION WHICH TAKES INTO ACCOUNT THE SCALING AT LARGE AND SMALL MOMENTUM TRANSFER THE POLE CONTRIBUTION, THE KINEMATICAL CONSTRAINT AND THE ANALITICITY PROPERTY OF THE FORM FACTORS IS THE BK PARAMETRIZATION C+ (1 - + ) C+ ≈mQ-1/2 (1 - + ) ≈mQ-1 (1 - + ) ≈mQ-1 f+ (q2) = (1 - q2 / Mt2 ) (1 - + q2 / Mt2 ) C+ (1 - + ) f0(q2) = (1 - q2 / (+ Mt2 ))

37. MAIN LIMITATIONS FROM DISCRETIZATION The typical value of the lattice spacing is a-1 = 2-5 GeV i.e. |p|a ≈ 1 |mb|a ≈ 1 -> a large extrapolation in mQ and in the pion momentum is needed from mQ≈ mcharm to mb Moreover px,y,za =2 nx,y,z /L with L=24-48 (0.13-0.25) (also px,y,za =  /L possible with suitable boundary conditions - Bedaque, De Divitiis, Petronzio, Tantalo) We are thus confined in the region q2 ≈ (M2B - M2) ≈25 GeV2 It is possible to use the HQET for the heavy quark, this however does not solve the problem of the limited kinematical region covered in q2

38. Becirevic et al. SPQR-APE COLLABORATION

41. LARGE q2 DEPENDENCE COULD BE DIRECTLY MEASURED BY EXPERIMENTS AND COMPARED WITH LATTICE RESULTS PARTICULARLY FOR B -> 

42. Typical lattice results QCD Sum (LC)Rules compare directly to experiments the large q2 region No unquenched calculations No extrapolation to zero lattice spacing No serious study of the chiral extrapolation

43. Unquenched Calculation of D ->K,Semileptonic Decays C. Aubin et al. - Fermilab Lattice, MILC & HPQCD Collaborations 26/8/04 hep-ph/0408306 Improved Staggered light Fermions + Clover Action (with Fermilab interpretation) for the charm quark 3-unquenched flavours Value of the lattice spacing is a-1 = 1.6 GeV (discretization ?) Minimal value of the light quark mass about ms/8 (finite volume ?) Advantages: unquenched, no extrapolation in the charm and strange quark masses, rather small light-quark masses HQET

44. Total systematic error from chiral extr., BK parametrization, lattice calibration, estimated discretization uncertainty is about 10%

45. Heavy-Light Semileptonic Decays D -> K,K* DECAYS PROBE LATTICE (or model) RESULTS BY COMPARISON WITH EXPERIMENTAL DATA: (D -> K) = known constant |Vcs|2 |A|2 Also (D -> K*)L / (D -> K) T theory experiment hep-ex/0406028 or provide and independent determination of the CKM matrix elements

46. TESTING THE UNITARITY OF THE CKM MATRIX: PDG 2002 QUOTES A 2.2 DEVIATION FROM UNITARITY

47. From F. Mescia ICHEP 2004

49. TYPICAL LATTICE CALCULATIONS OF FORM FACTORS HAVE ERRORS OF ORDER 5-10%, HERE AN ERROR BELOW 1% IS NEEDED A feasibility study by Becirevic et al. hep-ph/0403217, see also for hyperons Guadagnoli, Simula, Papinutto, GM 3 MAIN POINTS: Evaluation of f0at maximal recoil using the Fermilab double ratio method Extrapolation of f0from q2max to q2=0 using suitable ratios of correlation functions Extrapolation to the physical point after the subtraction of the chiral logs (see later)

50. f0(q2max) using the Fermilab method: from ratio of suitable 3-point functions we get because of the Ademollo-Gatto theorem Note the accuracy !! Systematic errors (e.g. discretization errors) are also of order