QCD-2004

Lesson 3 :Non-perturbative

1) Simple quantities: masses and decay constants

2) Sistematic errors

3) Semileptonic decays

4) Non leptonic decays

Guido Martinelli Bejing 2004

COULD WE COMPUTE THIS PROCESS WITH SUFFICIENT COMPUTER POWER ?

THE ANSWER IS: NOIT IS NOT ONLY A QUESTION OF COMPUTER POWER BECAUSE THERE ARE COMPLICATED FIELD THEORETICAL PROBLEMS

LATTICE FIELD THEORY IN FEW SLIDES

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} eJ ij i j Ising Model

2N = 2L3 ≈ 10301 for L = 10 !!!

Wick RotationZ -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 = SSeveral important sampling methods can be used, for example the Metropolis technique, to extract the fields with weight e - S()

< > = Z -1 ∑{ (x)}n n(x1) n(x2) n(x3) n(x4)

Z = ∑{ (x)}n 1 = N

t -> i tE

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 invarianceThis problem is common to all regularizations like for example Pauli-Villars, dimensional regularization etc.

LOCAL GAUGE INVARIANCE

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 ∫x

y i g0 GA (x) tA dx ] q(x)

y

x

q(x + a ) q(x)

q(y) P [ exp ∫xy i g0 GA

(x) tA dx ] q(x)is Gauge invariant x y

exp [i g0 GA (x + a /2) tA ]

LINK U†(x)

Plaquette

W(x) = U(x) U(x + a )U†(x + a )U†

(x) ≈ 1 + i a2 g0 G(x) - a4 g0

2 /2 G (x) G(x) + ...

1 ∑x∑ < Re Tr [1-W(x)] ->g0

2

a4 /4 ∑x ∑ G (x) G(x) ->

1/4 ∫ G (x) G(x) + O(a2)

U(x) -> V(x) U(x) V†(x + a )

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 theseformulations and return to the calculation of physical quantities like masses, decay constants etc.

Determination of hadron masses and simple matrix elements

An example from the 4 theory

The field can excite one-particle, 3-particle etc. states

At large time distances the lightest (one particle) states dominate :

t/a

Log[G(t)]

m a

<> = 1/ m a is the dimensionless correlationlength (and the size of the physical excitations)

For a particle at rest we have

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

A0(x,t)A†

0(x,t)

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]

2f M ~ Z

Mass and decay constant in lattice units M = m a

In the chiral limit

u

d

u,d,s

gluonsA A A

A

anomalyabsent in the case of ,K and 8

Continuum limit

= 1/ m

a Formal lim a->0 SLattice() -> SContinuum()

a/ = m a ~1 The size of the object is comparable to the lattice spacing

a/ <<1 i.e. m a -> 0 The size of the object is much larger than the lattice spacing

Similar to a ∑n -> ∫ dx

Measured in thenumerical simulation

Calibration of the lattice spacing a

Let us start for simplicity with massless quarks mq = 0

Mproton = Mproton (g0 ,a ) = mproton a

Physical proton mass

a (g0 ) = Mproton

mproton

Then we predict m , m , m , f , ….we cannot predict m since m2

mq

Calibration of the lattice spacing a

Mproton = Mproton(g0 ,a , mup = mdown , mstrange ) = mproton a

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

Continuum limit a 0

Mproton = mproton a = mproton QCD-1 e-1/(2 0g0

2)

= Cproton e-1/(2 0g02) 0

Using asymptotic freedom a d g0 = 0g3

0 + 1g50+O (g7

0) d a

a (g0) ~ QCD-1 e-1/(2 0g0

2)

a 0 when g0 0

Mproton = mproton a = mproton QCD-1 e-1/(2 0g0

2)

= Cproton e-1/(2 0g02) + O (a) 0

a 0 when g0 0

Mproton / M Cproton / C =const.

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%

3-point functions

D†(t1)

D†(t1) = ∑x D†(x, t1) exp[-i pD x]K(t2) = ∑x K(x, t2) exp[+i pK x]

K(t2)

Jweak(0)

< K | Jweak(0) | D >

also electromagnetic form

factors, structure functions, dipole

moment of the neutron, ga/gv, etc.

e+

e

from the 2-point functions

1) Kl3 namely

2)

3)

General consideration on non-perturbative methods/approaches/models

Models a) bag-model b) quark modelnot 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.

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 FieldTheory, Symmetries, Ward identities, Renormalization properties is needed.

LATTICE QCD IS REALLY EXPERIMENTAL FIELD THEORY

Major fields of investigation

QCD {• QCD thermodynamics• Hadron spectrum • Hadronic matrix elements ( K -> , structure functions, etc. see below )

EW

{• Strong interacting Higgs Models• Strong interacting chiral models

• Surface dynamics• Quantum gravity

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

Weak decays : | Q S=1 | K and | Q S=1 | K

Matrix elements of leading twist operators :

Lattice QCD is really a powerful approach

BUT … FOR SYSTEMATIC ERRORS

SYSTEMATIC ERRORS

Lattice QCD is really a powerful approach

QUENCHEDUNQUENCHED

• 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

Quenching errors

Almost all groups are now moving to unquenched calculations

(partially,two-flavours, three?, etc.)

THE ULTRAVIOLET PROBLEM

1/MH >> a

For a good approximationof the continuum

O(a) errors {mq a << 1

p a << 1

Typically a-1 ~ 2 ÷ 5 GeV

mcharm ~ 1.3 GeV mcharm a ~ 0.3

mbottom ~ 4.5 GeV mbottom a ~ 1See talk by Sommer

SYSTEMATIC ERRORSP

a

Naïve solution: extrapolate measuresperformed at different values of thelattice spacing. Price: the error increases

1/MH

fH M1/2H

Physical behaviour

effect of lattice artefacts IMPROVEMENT

SYSTEMATIC ERRORS

BOX SIZE

THE INFRARED PROBLEM

L >> = 1/MH >> a

To avoid finite size effectsFor a good approximationof the continuum

Finite size effects are not really a problemfor quenched calculations; potentially more problematic for the unquenched case Is L 4 ÷ 5

sufficient ? O(exp[- /L])

Particularly in the unquenched case, because ofthe limitations in computer resourcesVOLUMES CANNOT BE LARGE ENOUGH TO WORK AT THE PHYSICAL LIGHT QUARK MASSES(min. pseudoscalar mass is ~ MK, needed ~M )

an extrapolation in mlight to the physical point isnecessary

Test if the quark mass dependence is described by Chiral perturbation Theory (PT),Then the extrapolation with the functional formsuggested by PT is justified

Typical quark mass ms /2 < mq < ms

3-point functions

D†(t1)

D†(t1) = ∑x D†(x, t1) exp[-i pD x]K(t2) = ∑x K(x, t2) exp[+i pK x]

K(t2)

Jweak(0)

< K | Jweak(0) | D >

also electromagnetic form

factors, structure functions, dipole

moment of the neutron, ga/gv, etc.

e+

e

from the 2-point functions

1) Kl3 namely

2)

3)

Vud Vus Vub

Vcd Vcs Vcb

Vtb Vts Vtb

Quark masses &Generation Mixing

NeutronProton

e

e-

downup

W

| Vud |

| 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)

-decays

< K(pK) | Jweak(0) | D(pD) > = [(pD + pK) -q (M2

D - M2K)/q2] f + (q2) +

q (M2D - M2

K)/q2 f 0 (q2 )

< K*(pK*,) | Jweak(0) | D(pD) > = * T

IN THE ELICITY BASIS:

Vector meson polarization

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+

1-

0-

1-

1+

= t-channel

quantum numbers

1+

0-

Pole Dominance

D†(t1) K(t2)

Jweak(0)

e+

e

f + (q2) =f + (0)

1 - q2 / Mt2

Mt

works well for the pion electromagnetic form factor , dipole in the case of the proton f (0)

(1 - q2 / Mt2) 2

Scaling behavior for the Form Factors

at q2 ≈ (q2 )max

Form Factor t-channel mQ dependence

B ->

f+ 1- mQ

1/2

f0 0+ mQ

-1/2B ->

V 1- mQ

1/2

A1 1+ mQ

-1/2

A2 1+ mQ

1/2

A3 1+ mQ

3/2

A0 0 - mQ

1/2

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 ACCOUNTTHE SCALING AT LARGE AND SMALL MOMENTUM TRANSFER

THE POLE CONTRIBUTION, THE KINEMATICAL CONSTRAINT AND THEANALITICITY PROPERTY OF THE FORM FACTORS

IS THE BK PARAMETRIZATION

f+ (q2) =

(1 - q2 / Mt2 ) (1 - + q2 / Mt

2 )

C+ (1 - + )

f0(q2) = C+ (1 - + )

(1 - q2 / (+ Mt2 ) )

C+ ≈ mQ-1/2

(1 - + ) ≈ mQ-1

(1 - + ) ≈ mQ-1

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

Becirevic et al.SPQR-APE COLLABORATION

LARGE q2 DEPENDENCE COULD BE DIRECTLY MEASURED BY EXPERIMENTS AND COMPARED WITH LATTICE RESULTSPARTICULARLY FOR B ->

Typical lattice results QCD Sum (LC)Rules

1) No unquenched calculations

2) No extrapolation to zero lattice spacing

3) No serious study of the chiral extrapolation

compare directly to experiments the large

q2 region

Unquenched Calculation of D ->K,Semileptonic Decays

1) Improved Staggered light Fermions + Clover Action (with Fermilab interpretation) for the charm quark

2) 3-unquenched flavours

3) Value of the lattice spacing is a-1 = 1.6 GeV (discretization ?)4) Minimal value of the light quark mass about ms/8 (finite volume ?)

C. Aubin et al. - Fermilab Lattice, MILC & HPQCD Collaborations 26/8/04hep-ph/0408306

Advantages: unquenched, no extrapolation in the charm and strange quarkmasses, rather small light-quark masses

HQET

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

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

experimenthep-ex/0406028

or provide and independent determination of the CKM matrix elements

TESTING THE UNITARITY OF THE CKM MATRIX:

PDG 2002 QUOTES A 2.2 DEVIATION FROM UNITARITY

From F. Mescia ICHEP 2004

TYPICAL LATTICE CALCULATIONS OF FORM FACTORS HAVE ERRORSOF ORDER 5-10%, HERE AN ERROR BELOW 1% IS NEEDED

A feasibility study by Becirevic et al. hep-ph/0403217, see also for hyperonsGuadagnoli, Simula, Papinutto, GM 3 MAIN POINTS:

1. Evaluation of f0 at maximal recoil using the Fermilab double ratio

method

2. Extrapolation of f0 from q2max to q2=0 using suitable ratios of correlation

functions 3. Extrapolation to the physical point after the subtraction of the chiral logs

(see later)

1) 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

2) Extrapolation of f0(q2max ) to q2=0 and extraction

of f+(q2 =0 ): errors on f0(q2 ) much larger than for f+(q2) New ratio:

1)improved at O(a2) using the O(a)-improvement program in fermionic bilinear operators;

2) exactly normalized to 1 in the SU(3)-symmetric limit;

3) statistical fluctuations almost cancel out.

COMPARISON OF THE SLOPES WITH EXPERIMENTS

linear fit: f0(q2) = f (0) (1 + 0 q2)

quadratic fit: f0(q2) = f (0) (1 + 0 q2 + c0 q4)

polar fit: f0(q2) = f (0) / (1 - 0 q2)

€

+ =0.025 ± 0.002 λ 0 = 0.012 ± 0.002

€

+ =0.02411± 0.00036 λ 0 = 0.01362 ± 0.00073

LATTICE (EXTRAPOLATED TO THE PHYSICAL POINT)

K TeV (2004) hep-ex/0406003

KTeV results are 3/5 times more precise than previous determinations (PDG)

3) Extrapolation of f+(0) to the physical point upon subtraction of the quenched chiral logs

The extrapolation in the quark masses is one of the largest sources of systematic error

(twisted mass QCD)

EXPERIMENTAL NOVELTIES

1%)

Vus from hyperon semileptonic decays

new data analysis from Cabibbo, Swallow and Winston (‘03):

1) the ratio g1(0) / f1(0) is extracted from data,

2) SU(3) symmetry is assumed for f1(0) [as well as for g2(0)].

€

′ B ( ′ p ) V μ B( p) = u ′ p ( ) f1 q2( )γ

μ − f2 q2( )

iσ μν qν

′ M + M+ f3 q2

( )qμ

′ M + M

⎧ ⎨ ⎩

⎫ ⎬ ⎭u p( )

′ B ( ′ p ) Aμ B( p) = u ′ p ( ) g1 q2( )γ

μ − g2 q2( )

iσ μν qν

′ M + M+ g3 q2

( )qμ

′ M + M

⎧ ⎨ ⎩

⎫ ⎬ ⎭γ5 u p( )

€

rate ∝ Vus2

f1

2 1+ 3g1

2

f12

+ 4′ M − M

M '+Mg1

f1

g2

f1

⎧ ⎨ ⎩

⎫ ⎬ ⎭

Only few form factors are relevantto the decay rate

SU(3)-symmetric limit

f1(0) = 1

g2(0) = 0

€

R0 ≡CΣn

0 tx , ty( ) ⋅CnΣ0 tx , ty( )

CΣΣ0 tx , ty( ) ⋅Cnn

0 tx , ty( ) tx >>1, tx −ty >>1 ⏐ → ⏐ ⏐ ⏐ ⏐ n V 0 Σ ⋅ Σ V 0 n

Σ V 0 Σ ⋅ n V 0 n

FNAL double ratio

Guadagnoli,Simula SPQR

€

f0 q2( ) ≡ f1 q2

( ) +q2

M 2 − ′ M 2f3 q2

( ) at qmax2 = ′ M − M( )

2

scalar form factor

120 confs. @ =6.20

V = (24)3 · 56

1.5 < MB(GeV) < 1.8

very high-precision !!!

0.995

1.000

1.005

-0.06 -0.03 0.00 0.03 0.06

f 0(q

2max

)

a2 (M

2 - Mn

2 )

€

MΣ ≈ A + B ms + 2ml( )

MΣ2 − Mn

2 ∝ ms − ml( )

Momentum dependence of the form factors

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-0.2 -0.1 0.0

f 0(q

2)

a2 q

2

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-0.2 -0.1 0.0

f 1(q

2)

a2 q

2

- - - - monopole fits

f0(1)(q2) = f(0) / (1 - 0(1) q2)

______ dipole fits

f0(1)(q2) = f(0) / (1 - 0(1) q2)2

Error slighlty larger than in the case of Kl3 still very promisingMore work and computer effort is needed

SUMMARY

NON LEPTONIC DECAYS

• Theoretical framework• Present situation e outlook

PROTOTYPE

The Effective Hamiltonian

W(q)s

u

u

d

u

u

d

s

New local four-fermion operators are generated

Q1 = (sLA uL

B) (uLB dL

A) Current-Current Q2 = (sL

A uLA) (uL

B dLB)

Q3,5 = (sR

A dLA)∑q (qL,R

B qL,R

B) Gluon Q4,6 = (sR

A dLB)∑q (qL,R

B qL,R

A) Penguins

Q7,9 = 3/2(sRA dL

A)∑q eq (qR,LB

qR,LB) Electroweak

Q8,10 = 3/2(sRA dL

B)∑q eq (qR,LB

qR,LA) Penguins+ Chromomagnetic end electromagnetic operators

Theoretical Methods for the Matrix Elements (ME)

• Lattice QCD Rome Group, M. Ciuchini & al.

• NLO Accuracy and consistent matching PT (now at the next to leading order) and quenching • no realistic calculation of <Q6> • Fenomenological Approach Munich A.Buras & al.

• NLO Accuracy and consistent matching • no results for <Q6,8> which are taken elsewhere • Chiral quark model Trieste S.Bertolini & al. • all ME computed with the same method • model dependence, quadratic divergencies,matching

I=1/2 and ′/• K π π from K π and K 0

• Direct K π π calculation

Multiparticle states(DIS, inclusive decays etc.)

Example K π π

THE ULTRAVIOLET PROBLEM

Although rather complicated in practice, because we have composite operators of dimension six, in the presence of a hard cutoff (the inverse lattice spacing 1/a)

the theoretical framework for the renormalization of the relevant operators has been completely developed

THE INFRARED PROBLEM

The IR problem arises from two sources:• The (unavoidable) continuation of the theory to Euclidean space-time (Maiani-Testa theorem) • The use of a finite volume in numerical simulations

An important step towards the solution of the IR problem has been achieved by L. Lellouch andM. Lüscher (LL), who derived a relation between

the K π π matrix elements in a finitevolume and the physical amplitudes

Commun.Math.Phys.219:31-44,2001 e-Print Archive: hep-lat/0003023

Here I discuss an alternative derivation based on the behaviour of correlators of local operator when V D. Lin, G.M., C. Sachrajda and M. Testa hep-lat/0104006 (LMST)

Consider the following Euclidean T-products (correlation functions) G(t,tK)= ‹ 0 | T [J(t) Q(0) K+ (tK)] |0 ›, G(t) = ‹ 0 | T [J(t) J(0) ] |0 ›, GK (t) = ‹ 0 | T [K(t) K+(0) ] |0 ›,

where J is a scalar operator which excites (annhilates) zero angular momentum () states from (to) the vacuum and K is a pseudoscalar source which excites a Kaon from the vacuum (t > 0 ; tK < 0)

G(t,tK) G(t) GK (t)

J(t)

Q

K+(tK) J+(0)

J(t) K+(0) K(tK)

K

At large time distances:G(t,tK) V n ‹ 0 |J | n›V ‹ n |Q(0) |K ›V ‹ K |K+ |0 ›V

exp[ -(Wn t +mK| tK |) ] G(t) = V n ‹ 0 |J | n›V ‹ n|J | 0›V exp[ -Wn t ]

From the study of the time dependence of G(t,tK), G(t) and GK (t) we extract • the mass of the Kaon mK • the two- energies Wn

• the relevant matrix elements in the finite volume‹ K |K+ |0 ›V , ‹ 0 |J | n›V , and ‹ n |Q(0) |K ›V

We may also match the kaon mass and the two pion energy, namely to work with mK = Wn* Necessary to obtain a finite I=1/2 matrix element

The fundamental point is that it is possible to relate the finite-volume Euclidean matrix element with the absolute value of

the Physical Amplitude |‹ E |Q(0) |K ›| by comparing, at large values of V, the finite volume correlators to the infinite volume ones

|‹ E |Q(0) |K ›| = √F ‹ n |Q(0) |K ›V

F = 32 2 V2 V(E) E mK/k(E) where k(E) = √ E2/4- m2

and

V(E) = (q ’(q) + k ’(k))/4 k 2 is the expression which one would

heuristically derive by interpreting V(E) as the density of states in a finite volume (D. Lin, G.M., C. Sachrajda and M. Testa)

On the other hand the phase shift can be extractedfrom the two-pion energy according to (Lüscher):

Wn = 2 √ m2 + k2 n - (k) = (q)

Wn is determined from the time dependence of the correlation functions

G(t,tK) = V ‹ K |K+ |0 ›V exp( -mK| tK | ) n ‹ 0 |J | n›V ‹ n |Q(0) |K ›V exp (- Wn t ) = n A n exp (- Wn t )

From Wn it is possible to extract the FSI phase (for a different method to obtain (E) =(k) see LMST)

Wn = 2 √ m2 + k2 n - (k) = (q)

1) IT IS VERY DIFFICULT TO ISOLATE Wn WHEN n IS LARGE !

2) THIS METHOD HAS BEEN USED FOR THE I=3/2 AND 1/2 TRANSITIONS

SERIOUS PROBLEM FOR QUENCHED CALCULATIONS DUE TO THE LACK OF UNITARITY (D. LIN, G.M., G.M. AND C. SACHRAJDA)

THE CHIRAL BEHAVIOUR OF ‹π π IHW I K ›I=2 by the SPQcdRCollaboration and a comparison with JLQCD Phys. Rev. D58 (1998) 054503

Aexp= 0.0104098 GeV3

This work 0.0097(10) GeV3

no chiral logs included yet, analysis under way

Lattice QCD finds BK = 0.86 and a value of ‹π π IHW I K ›I=2 compatible with exps

1) the final state interaction phase is not universal, since it depends on the operator used to create the two-pion state. This is not surprising, since the basis of Watson theorem is unitarity;

2) the Lüscher quantization condition for the two-pion energy levels does not hold. Consequently it is not possible to take the infinite volume limit at constant physics, namely

with a fixed value of W ;3) a related consequence is that the LL relation between the absolute value of the physical amplitudes and the finite volume matrix elements is no more valid;

4) whereas it is usually possible to extract the lattice amplitudes by constructing suitable time-independent ratios of correlation functions, this procedure fails in the quenched theory because the time-dependence of correlation functions corresponding to the same external states is not the same

I=0 States in the QuenchedTheory (Lack of Unitarity)

D. Lin, G.M., E. Pallante, C. Sachrajda and G. Villadoro

There could be a way-out …..

A = ∑i Ci() ‹ (π π) IQi () I K › ‹ (π π) IQi () I K › = ‹ (π π) IQi I K ›VIA B ()

VACUUM SATURATION & B-PARAMETERS

-dependence of VIA matrix elements is not consistentWith that of the Wilson coefficientse.g. ‹ (π π) IQ9 I K › I=2,VIA = 2/3 fπ (M2

K - M2π )

In order to explain the I=1/2 enhancement the B-parameters of Q1 and Q2 should be of order 4 !!!

Relative contribution of the OPS

-6-4-202468

1012

B2charm

B5 B8 B9 e'/e

PositiveNegative Total

B6

B83/2

'/ = 13 Im t [ 110 MeV]2 [B6 (1- IB) - 0.4 B8 ]

The Buras Formula that should NOT be used but is presented by everyone

t = Vtd Vts* = ( 1.31 1.0 ) 10-4

ms ()

a value of B6 MUCH LARGER than 1 (2 ÷ 3 ) is needed to explain the experiments

The situation worsen if also B8 is larger than 1

('/ )EXP = ( 17.2 1.8 ) 10-4

I=1/2 and ′/

• I=1/2 decays (Q1 and Q2)

• ′/ electropenguins (Q7 and Q8)• ′/ strong penguins (Q6)

• K π π from K π and K 0

• Direct K π π calculation

Physics Results from RBC and CP-PACSno lattice details here

Re(A0) Re(A2) Re(A0)/Re(A2)

′/

RBC 29÷3110 -8

1.1 ÷1.210 -8

24÷27 -4 ÷ -810 -4

CPPACS

16÷2110 -8

1.3÷1.510 -8

9÷12 -2 ÷ -710 -4

EXP 33.310 -8

1.5 10 -8 22.2 17.2 ±1.810 -4

Total Disagrement with experiments ! (and other th. determinations)

Opposite sign !

New Physics?

0.5

1.0

1.5

2.0

2.5

3.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5

B6

B8

'/ ~ 13 (QCD/340 MeV)Im t (110 MeV/ms )[B6 (1-IB ) -0.4 B8 ]'/ =0

Artistic representation of present situation

DonoghueDe Rafael

Physics Results from RBC and CP-PACS

Re(A0) Re(A2) Re(A0)/Re(A2)

′/

RBC 29÷3110 -8

1.1 ÷1.210 -8

24÷27 -4 ÷ -810 -4

CPPACS

16÷2110 -8

1.3÷1.510 -8

9÷12 -2 ÷ -710 -4

EXP 33.310 -8

1.5 10 -8 22.2 17.2 ±1.810 -4

• Chirality• Subtraction• Low Ren.Scale• Quenching • FSI• New Physics• A combination ?

Even by doubling O6 one cannot agree with the dataK π π and Staggered Fermions (Poster by W.Lee) will certainly help to clarify the situation I am not allowed to quote any number

Lattice B6 = 1 Lattice from K-

QM Trieste

From S. Bertolini

Typical Prediction5-8 10-4

CONCLUSIONS I

For many quantities (quark masses, decay constants, form factors, moments of structure functions, etc.) Lattice QCD is entering the stage of precision calculations, with errors at the level of a few percent and full control of unquenching, discretization, chiral extrapolation and finite volume effects.

CONCLUSIONS II

For non-leptonic decays (particle widths) theoretical progresses have been made but, with the exception of some imprecise calculation of DI=3/2 amplitudes,

NO QUANTITATIVE RESULTS EXIST and it remains open the problem of the decays above the elastic threshold

e.g. B