Some developments inthe physics of the light quarks

H. Leutwyler

University of Bern

Symposium in honour of Jan Stern, Orsay, June 3, 2008

– p. 1/51

Part I

Reminiscences

– p. 2/51

Once upon a time, when Jan and I studied physics . . .

in 1962, when I completed my studies at the University of Bern,I had heard about QED, the Fermi theory of β decay andnonrelativistic potential models for the nuclear forces, but as far asparticle physics is concerned, that was about itthere was nothing to miss. The theory of the strong interaction was a genuine mess: elementary fields forbaryons and mesons, Yukawa interaction for the strong forces, perturbation theory with couplingconstants of order 1, nuclear democracy, bootstrap . . . absolutely nothing worked even half ways, beyondgeneral principles like Lorentz invariance, causality, unitarity, crossing, dispersion relations

Jan was more lucky in this regard: started a little later and studied atthe Charles University in Prague, where there was an active groupin particle physics: Votruba, Vancura, Havlı́cek, Formanék, . . .

⇒ during his studies, he must have encountered some of the newideas which paved the way to an understanding of the stronginteraction

– p. 3/51

The new ideas

the strong interaction has a hidden, approximate symmetry Nambu 1960

eightfold way Gell-Mann, Ne’eman 1961

pattern of symmetry breaking, Ω− Gell-Mann, Okubo 1961/1962

quark model Gell-Mann, Zweig 1962

puzzle: why are the symmetries not exact ?exact consequences of approximate properties ?charges & currents form an exact algebraeven if they do not commute with the Hamiltonian Gell-Mann 1964

test of current algebra: size of 〈N |Aµ|N〉 ∼ gAAdler 1965, Weisberger 1966

prediction from current algebra: ππ scattering lengths Weinberg 1966

– p. 4/51

Mass of the pion

formula for pion mass Gell-Mann, Oakes & Renner 1968

M2π = (mu +md) × |〈0| q̄ q |0〉| ×1

F 2π⇑ ⇑explicit spontaneous

at that time, the existence of quarks was still questionable

quarks were treated like the veal used to prepare a pheasant inthe royal French cuisine

⇒ formula does not appear like this in the paper

the GMOR relation explains why the energy gap of the stronginteraction is so small . . . in terms of a new mystery:why aremu andmd so small ?

– p. 5/51

Quark masses

even before the discovery of QCD, attempts at estimating themasses of the quarks were made

bound state models for mesons and baryonsmu +mu +md ≃ Mp mu ≃ md

⇒ mu ≃ md ≃ 300 MeV “constituent masses”

remarkably simple and successful pictureexplains the pattern of energy levels without QCD

model for spontaneous symmetry breakdownrequires much smaller fermion masses Nambu & Jona-Lasinio 1961

same conclusion from sum rules for currents Okubo 1969

⇒ conceptual basis of royal French cuisine ?

smart people considered Regge theory very promisingVeneziano model 1968

This was the state of our art when I met Jan

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Where I presumably met Jan for the first time

– p. 7/51

Abstract for those who prefer it in Russian

– p. 8/51

In natura

– p. 9/51

First paper we wrote together: February 1970

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Extended version: March 1970

– p. 11/51

QCD

QCD was discovered in 1973

many considered this a wild speculation

for all quantum field theories encountered in nature before, thespectrum of physical states was the same as the one of thekinetic part of the Lagrangian Pauli

also true of the electroweak theoryGlashow 1961, Weinberg 1967, Salam 1968

only gradually, particle physicists abandoned their outposts in noman’s and no woman’s land, returned to the quantum fields andresumed discussion in the good old Gasthaus zu Lagrange Jost

⇒ Standard Model, clarified the picture enormously

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Standard Model

Standard Model appeared like a miracle:

weak, e.m. and strong interactions are very differentnevertheless, they are all generated by gauge fields

IG Physik, Gesellschaft mit besonderer Haftung, advertisement ca. 1973

Im Falle eines Fallesklebt ein EICHFELD

wirklich alles !

Bezugsquellennachweis

H. Weyl, Z. Phys. 56 (1929) 330, C. N. Yang and R. Mills, Phys. Rev. 96 (1954) 191

gauge fields are renormalizable in d = 4

miracle leads to new puzzle: SM cannot be the full truthmust be an effective theory adequate at low energiesno reason for an effective theory to be renormalizable

⇒ why is the SM renormalizable ?

– p. 13/51

Quantum theory on null planes

for three years, Jan was engaged as a guest professor at theUniversity of Bern: 1977/78 and 1982/84

I had a wonderful time discussing physics with himmy prejudices were not always identical with his, but this onlyanimated the discussions. . .

he, Jiři and I vigorously pursued the idea that formulating thequantum theory on null planes might give a coherent description ofmesons and baryons in terms of quarks

our efforts led to a very thorough paper in Annals of Physics, whichyou might take pleasure reading, even now, 30 years later . . .

– p. 14/51

– p. 15/51

Suivez le guide !

– p. 16/51

Making bets in the coffee room, on a Friday evening

I thank Ottilia Haenni for the picture and Hans Bebiefor managing to make the scribbling legible

– p. 17/51

Exerpts

Ottilia’s offer: 5 bottles if Jan goes on hungerstrike for ≥ 24 hours

Jan’s answer: 2 bottles if Ottilia joins him on the 21st hour

Vorwarnung: Demnächst∗ Institutsausflug → Paris∗ 15. Sept. 78 Bitte innerlich sammeln

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Part II

Recent developments in thephysics of the light quarks

– p. 19/51

How well do we understand the strong interaction today ?

QCD with massless quarks is the ideal of a theory:no dimensionless free parameters

high energy side looks like what we are used to: relevant degrees offreedom are visible in the Lagrangian, can treat the interaction as aperturbation

gives rise to a rich structure at low energies

low energies are out of reach of perturbation theory⇒ progress in understanding is slow

∃ many models that resemble QCD: instantons, monopoles, bags,superconductivity, gluonic strings, linear σ model, hidden gauge,NJL, AdS/CFT, but . . .

– p. 20/51

Lattice results forMπ

Gell-Mann-Oakes-Renner formula can now be checked:can determineMπ as a function of mu=md=m

0 0.01 0.02 0.03am0

0.02

0.04

0.06

0.08

(amπ)2

mπ∼676 MeV

484

381

294

fit to 4 pointsfit to 5 points

(amPS)2

(aµ)

0.0160.0120.0080.0040

0.08

0.06

0.04

0.02

0

Lüscher, Lattice conference 2005 ETM collaboration, hep-lat/0701012

no quenching, quark masses are sufficiently light⇒ legitimate to use χPT for the extrapolation to the physical values ofmu,md

– p. 21/51

Lattice

quality of data is impressive

proportionality ofM2π to the quark mass appears to hold out tovalues ofmu,md that are an order of magnitude larger than innature

main limitation: systematic uncertaintiesin particular: Nf = 2 → Nf = 3

– p. 22/51

Chiral perturbation theory

consequences of hidden, approximate symmetry can be worked outby means of an effective field theory Weinberg 1979

hidden symmetry controls energy gap of QCD

⇒ can calculate how gap grows as the symmetry breaking parametersmu,md are turned on

hidden symmetry also determines the interaction of the Goldstonebosons at low energies, among themselves, as well as with otherhadrons

– p. 23/51

Expansion ofM2π in powers of the quark mass

Gell-Mann-Oakes-Renner formula represents leading term of χPT

disregard from isospin breaking, setmu = md = m

at NLO, the expansion contains a logarithm Langacker & Pagels 1973

M2π = M2

{

1 +M2

32π2F 2πlnM2

Λ 23+O(M4)

}

M2 ≡ 2Bm

coefficient is determined by pion decay constantsymmetry does not determine the scale Λ3

crude result, based on SU(3)×SU(3):

0.2 GeV

Lattice allows more accurate determination of Λ3

0

0

2

2

4

4

6

6

Gasser & L. 1984

MILC 2007

Del Debbio et al. 2006, Nf = 2

ETM 2007, Nf = 2

RBC/UKQCD 2008

JLQCD 2007, Nf = 2

PACS-CS (preliminary)

horizontal axis shows the value of ℓ̄3 ≡ lnΛ

2

3

M2π

range for Λ3 obtained in 1984 corresponds to ℓ̄3 = 2.9 ± 2.4

result of RBC/UKQCD 2008: ℓ̄3 = 3.13 ± 0.33stat

± 0.24syst

– p. 25/51

Expansion of Fπ in powers of the quark mass

also contains a logarithm at NLO:

Fπ = F

{

1−M2

16π2F 2lnM2

Λ 24+O(M4)

}

M2π = M2

{

1+M2

32π2F 2lnM2

Λ 23+O(M4)

}

F is value of pion decay constant in limitmu,md → 0

structure is the same, coefficients and scale of logarithm aredifferent

quark mass dependence of Fπ can also be measured on the lattice⇒ measurement of Λ4

alternative method: determine the scalar form factor of the pion,

radius 〈r2〉s ⇔ ℓ̄4

– p. 26/51

Lattice results for Λ4

3

3

4

4

5

5

Gasser & L. 1984

Colangelo, Gasser & L. 2001

MILC 2007

ETM 2007, Nf = 2

JLQCD 2007, Nf = 2

RBC/UKQCD 2008

PACS-CS (preliminary)

systsyst

ℓ̄4 = lnΛ 24M2π

lattice results beautifully confirm the prediction for the sensitivity ofFπ tomu,md:

Fπ

F= 1.072(7) Colangelo and Dürr 2004

– p. 27/51

Predictions for the S-wave ππ scattering lengths

0.16 0.18 0.2 0.22 0.24 0.26

-0.06

-0.05

-0.04

-0.03

19661983 1996

Universal Bandtree, one loop, two loopslow energy theorem for scalar radiusColangelo, Gasser & L. 2001

a00

a20

sizable corrections in a00, while a20 nearly stays put

– p. 28/51

Lattice result for a20

lattice allows direct measurement of a20 via volume dependence ofenergy levels

0.16 0.18 0.2 0.22 0.24

a00

-0.05 -0.05

-0.04 -0.04

-0.03 -0.03

-0.02 -0.02

a20

Universal bandtree (1966), one loop (1983), two loops (1996)Prediction (χPT + dispersion theory, 2001)l4 from low energy theorem for scalar radius (2001)

NPLQCD (2005, 2007)

– p. 29/51

Consequence of lattice results for ℓ3, ℓ4

uncertainty in prediction for a00, a20 is dominated by the uncertainty

in the effective coupling constants ℓ3, ℓ4

can make use of the lattice results for these

0.16 0.18 0.2 0.22 0.24

a00

-0.05 -0.05

-0.04 -0.04

-0.03 -0.03

-0.02 -0.02

-0.01 -0.01

a20

Universal bandtree (1966), one loop (1983), two loops (1996)Prediction (χPT + dispersion theory, 2001)l4 from low energy theorem for scalar radius (2001)

NPLQCD (2005, 2007)l3 and l4 from MILC (2004, 2006)

l3 from Del Debbio et al. (2006)

l3 and l4 from ETM (2007)

l3 and l4 from RBC/UKQCD (2007)

l3 and l4 from PACS-CS (preliminary)

– p. 30/51

Experiments on light flavours at low energy

production experiments πN → ππN , ψ → ππω . . .

problem: pions are not produced in vacuo

⇒ extraction of ππ scattering amplitude not simple

accuracy rather limited

K± → π+π−e±ν data: CERN-Saclay, E865, NA48/2

K± → π0π0π± cusp near threshold: NA48/2

π+π− atoms, DIRAC

– p. 31/51

Ke4 decay

K → ππeν allows clean measurement of δ00 − δ11

theory predicts δ00 − δ11 as function of energy

0.28 0.3 0.32 0.34 0.36 0.38 0.4GeV

-5

0

5

10

15

20

δ00− δ11

theoretical prediction (2001)Geneva-Saclay (1977)E865 (2003) isospin correctedNA48/2 (2008) isospin corrected

preliminary

there was a discrepancy here, because a pronounced isospin

breaking effect fromK→π0π0eν→π+π−eν had not beenaccounted for in the data analysis

Colangelo, Gasser, Rusetsky 2007, Bloch-Devaux 2007

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a00, a2

0: prediction, lattice & experiment

0.16 0.18 0.2 0.22 0.24

a00

-0.05 -0.05

-0.04 -0.04

-0.03 -0.03

-0.02 -0.02

-0.01 -0.01

a20

Universal bandtree (1966), one loop (1983), two loops (1996)Prediction (χPT + dispersion theory, 2001)l4 from low energy theorem for scalar radius (2001)

NPLQCD (2005, 2007)l3 and l4 from MILC (2004, 2006)

l3 from Del Debbio et al. (2006)

l3 and l4 from ETM (2007)

l3 and l4 from RBC/UKQCD (2007)

l3 and l4 from PACS-CS (preliminary)

E865 Ke4 (2003) isospin correctedDIRAC (2005)NA48 K3π (2006)NA48 Ke4 (preliminary) isospin corrected

– p. 33/51

Conflict with ππ data on exotic S-wave

fit to the data on ππ scattering (Losty, Hoogland A) disagrees withthe theoretical prediction by about 2σ

Descotes-Genon, Fuchs, Girlanda & Stern 2002

-0.06

-0.055

-0.05

-0.045

-0.04

-0.035

-0.03

0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3

a00

a 02

0.3 0.4 0.5 0.6 0.7 0.8

E (GeV)

-20

-15

-10

-5

0

δ2 0

Colangelo, Gasser & L. 2001Descotes, Fuchs, Girlanda & Stern 2002central value of "global" fitLostyHoogland AHoogland B

if the DFGS analysis is applied to the new precision data fromNA48/2, the disagreement may well persist: K-decay and thelifetime of pionium give a strong constraint only on a00 − a

20

– p. 34/51

Conclusions for SU(2)×SU(2)

expansion in powers ofmu,md yields a very accurate low energyrepresentation of QCD

lattice results confirm the GMOR relation

⇒ Mπ is dominated by the contribution from the quark condensate

⇒ energy gap of QCD is understood very well

lattice approach allows an accurate measurement of the effectivecoupling constant ℓ3 already now

even for ℓ4, the lattice starts becoming competitive with dispersiontheory

expect significant results for effective coupling constants of NNLO inSU(2)×SU(2) very soon

despite the beautiful new experimental results in low energy pion

physics, theory is still ahead, particularly concerning a20 . . .

– p. 35/51

Expansion in powers ofms

theoretical reasoning:

the eightfold way is an approximate symmetry

the only coherent way to understand this within QCD:ms−md,md−mu can be treated as perturbations

sincemu,md ≪ ms

⇒ ms can be treated as a perturbation

⇒ expect expansion in powers ofms to work,but convergence to be comparatively slow

in principle, this can now also be checked on the lattice

those effective coupling constants that are relevant forMπ ,MK , Fπ , FK have been determined on the lattice,both for SU(2)×SU(2) and SU(3)×SU(3)

LO: B,F,B0, F0 NLO: ℓ3, ℓ4, L4, L5, L6, L8

– p. 36/51

Paramagnetic inequalities

consider the limitmu,md → 0,ms physical

F is value of Fπ in this limit

Σ is value of |〈0| ūu| |0〉 in this limit

B is value ofM2π/(mu +md) in this limit

exact relation: Σ = F 2B

inequalities set up by Jan and collaborators:both F and Σ should decrease ifms is taken smaller

F > F0 , Σ > Σ0for a recent discussion, see S. Descotes-Genon at Lattice 2007

ifNc is taken large, F,Σ andB become independent ofms

⇒ (F/F0−1), (Σ/Σ0−1), (B/B0−1) violate the OZI rule

the lattice results confirm the parametric inequalities, but do not yetallow to draw conclusions about the size of the OZI-violations

– p. 37/51

Condensate

1

1

1.2

1.2

1.4

1.4

1.6

1.6

1.8

1.8

2

2

RBC/UKQCD 2007statistical errors only

MILC 2007

PACS-CS preliminarydirectwith FSEwithout FSE

systsyst stat stat

Moussallam 2000one loop formula with

the range quoted for L6

Gasser & L. 1985

Σ/Σ0

central values of RBC/UKQCD and PACS-CS for Σ/Σ0 lead toqualitatively different conclusions concerning OZI-violations

⇒ discrepancy indicates large systematic errors

– p. 38/51

Results forB, F

1

1

1.1

1.1

1.2

1.2

1.3

1.3

syst syststat stat

B/B0

1

1

1.1

1.1

1.2

1.2

1.3

1.3

RBC/UKQCD 2007statistical errors only

MILC 2007

PACS-CS preliminarydirectwith FSEwithout FSE

systsyst statstat

Moussallam 2000one loop formula with the range quoted for L4

Gasser & L. 1985

F/F0

results forB are coherent, indicate small OZI-violations inB

⇒ F/F0 is the crucial factor in Σ = F2B

– p. 39/51

Expansion in powers ofms

B = B0(1 +ms rB1 +m

2s r

B2 + . . .)

F = F0(1 +ms rF1 +m

2s r

F2 + . . .) modulo logarithms

Σ = Σ0(1 +ms rΣ1 +m

2s r

Σ2 + . . .)

expansion only useful if 1 ≫ msr1 ≫ m2s r2

values of RBC/UKQCD for L4 and L6 imply

msrB1 = 0.043 ± 0.033

msrF1 = 0.20 ± 0.04 purely statistical errors

msrΣ1 = 0.43 ± 0.09

⇒ for the central values of RBC/UKQCD, the expansion looks like

B = B0(1 + 0.043 − 0.013)F = F0(1 + 0.20 + 0.03)Σ = Σ0(1 + 0.43 + 0.12)

⇒ large OZI-violations, but expansion in powers ofms looks OK

– p. 40/51

Conclusions for SU(3)×SU(3)

the available lattice data allow for very juicy OZI-violations, but arealso consistent withB/B0 ≃ F/F0 ≃ Σ/Σ0 ≃ 1

if the central value of RBC/UKQCD for F/F0 were confirmed withinsmall uncertainties, we would be faced with a qualitative puzzle:

Fπ is the pion wave function at the origin

FK is a little larger because one of the two valence quarksmoves more slowly → wave function more narrow→ higher at the origin: FK/Fπ ≃ 1.19

F/F0 = 1.23 indicates that the wave function is moresensitive to the mass of the sea quarks than to the mass of thevalence quarks . . . very strange → extraordinarily interesting

note: most of the numbers quoted are preliminary, errors purelystatistical, continuum limit, finite size effects, . . .

– p. 41/51

Puzzling results onKL → πµν

hadronic matrix element of weak current:

〈K0|ūγµs|π−〉 = (pK +pπ)µf+(t)+ (pK −pπ)

µf−(t)

scalar form factor ∼ 〈K0|∂µ(ūγµs)|π−〉

f0(t) = f+(t) +t

M2K −M2π

f−(t)

low energy theorem of Callan and Treiman (1966):

f0(M2K −M

2π) =

FK

Fπ

{

1 + O(mu,md)

}

≃ 1.19

f0(0) = f+(0) ≃ 0.96 relevant for determination of Vus

– p. 42/51

Comparison with experiment

0 0.05 0.1 0.15 0.2 0.25

t (GeV2)

1.0

1.05

1.1

1.15

1.2

1.25

1.3

f0(t)

physical region

MK2 - M

2π

Callan-Treiman

NA48

NA48, Phys. Lett. B647 (2007) 341

141 authors, 2.3 × 106 events

plot shows normalizedscalar form factor

f̄0(t) =f0(t)

f0(0)

Callan-Treiman relation in this normalization:

f̄0(M2K −M

2π) =

FK

Fπf+(0)

experimental value:FK

Fπf+(0)= 1.2446 ± 0.0041

Bernard and Passemar 2008

– p. 43/51

Corrections, extrapolation

Callan-Treiman-relation is exact only formu,md → 0

corrections of NLO were worked out long ago, are tiny Gasser & L. 1985

form factor now known to NNLO Post & Schilcher 2002,Bijnens & Talavera 2003, Cirigliano, Ecker, Eidemüller, Kaiser, Pich & Portoles 2005

including the uncertainties frommu,md 6= 0:

f̄0(M2K −M

2π) = 1.240 ± 0.009 Bernard & Passemar 2008

⇒ cannot blame the discrepancy on the prediction

CT–point is not in physical region, extrapolation needed

curvature can be calculated with dispersion theoryJamin, Oller & Pich 2004, Bernard, Oertel, Passemar & Stern 2006

⇒ cannot blame the discrepancy on the extrapolation

– p. 44/51

Slope of the scalar form factor

definition of the slope f̄0(t) = 1 +λ0 t

M2π++ O(t2)

Callan-Treiman-relation implies sharp prediction:

λ0 = (16.0 ± 1.0) × 10−3

Jamin, Oller & Pich 2004

Update with current experimental information

λ0 = (15.0 ± 0.7) × 10−3

Bernard, Oertel, Passemar & Stern, preliminary

To be compared with the result of NA48:

λ0 = (8.9 ± 1.2) × 10−3

Fit with dispersive representation of BOPS

– p. 45/51

Implications

NA48 data onKL → πµν disagree with SM

if confirmed, the implications are dramatic:

⇒ right-handed currents ? Bernard, Oertel, Passemar & Stern 2006

there are not many places where the SM disagrees with observation

need to investigate these carefully

at low energies, high precision is required

– p. 46/51

New data from KLOE

0 0.05 0.1 0.15 0.2 0.25

t (GeV2)

1.0

1.05

1.1

1.15

1.2

1.25

1.3

f0(t)

physical region

MK2 - M

2π

NA48

KLOE

Prediction (χPT + dispersion theory)

I thank Emilie Passemar for some of the material shown in this figure

– p. 47/51

Comparison of results for the slope

10

10

15

15

20

20

λ0 in units of 10-3

ISTRA 2004 (K+)

KTeV 2004 (KL)

Jamin, Oller & Pich 2004

PDG 2007 (K+)

PDG 2007 (KL)

KLOE 2007 (KL)

NA48 2007 (KL)

fit2

fit1

Birulev 1981 (K

Bernard, Oertel, Passemar & SternStandard Model prediction of

– p. 48/51

Older measurements

10

10

15

15

20

20

30

30

40

40

50

50

60

60

5

5

25

25

λ0 in units of 10-3

Gasser & L. 1985

ISTRA 2004 (K+)

KTeV 2004 (KL)

Jamin, Oller & Pich 2004

PDG 2007 (K+)

PDG 2007 (KL)

KLOE 2007 (KL)

NA48 2007 (KL)

Bernard, Oertel, Passemar & Stern

fit2fit1

PDG 2004 (KL)

Birulev 1981 (KL)

Cho 1980 (KL)

Hill 1979 (KL)

Clark 1977 (KL)

Buchanan 1975 (KL)

Donaldson et al. 1974 (KL)

Standard Model prediction of

– p. 49/51

Conclusions forK → µνπ

experiment is difficult, discrepancies need to be resolved

Donaldson 1974: 1.6 × 106 events

ISTRA 2004: 0.54 × 106 events

KTeV 2004: 1.5 × 106 events

NA48 2007: 2.3 × 106 events

dispersion theory fixes the shape of the form factorspublishing linear fits is nonsensical

Jan and collaborators worked hard to improve the analysis of theKTeV collaboration, paper to appear soon

NA48 should improve their data analysis as well . . . and extend it tocharged kaons (isospin breaking)

⇒ plenty of work ahead for all of us !

– p. 50/51

na zdravı́ !

– p. 51/51

SPARES

– p. 51/51

Effective coupling constants of SU(3)×SU(3)

-2 -2

-1 -1

0 0

1 1

2 2

3 3

103L

L4 L5 L6 L8 2L6-L4 2L8-L5

Rosell, Sanz-Cillero & Pich 2007PACS-CS 2007 without FSEPACS 2007 with FSEMILC 2007RBC/UKQCD 2007MILC 2006Kaiser 2005MILC 2004Bijnens & Dhonte 2003Meissner & Oller 2001Moussallam 2000Gasser & L. 1985

– p. 51/51

Quark mass ratios

0

0

0.2

0.2

0.4

0.4

0.6

0.6

0.8

0.8

1

1

mumd

0

5

10

15

20

25

msmd

χPT fails RχPT

RBC 2007Bijnens & Ghorbani 2007Namekawa & Kikukawa 2006MILC 2006MILC 2004Nelson, Fleming & Kilcup 2003

Gao, Yan & Li 1997Leutwyler 1996

Schechter et al. 1993Donoghue, Holstein & Wyler 1992

Gerard 1990Cline 1989Gasser & Leutwyler 1982Langacker & Pagels 1979Weinberg 1977Gasser & Leutwyler 1975

Q = 24.2∆M = 0

– p. 51/51

Value ofms

0

0

20

20

40

40

60

60

80

80

100

100

120

120

140

140

160

160

180

180

200

200

ms(2GeV)

Tomozawa 1969Okubo 1969

Gasser & Leutwyler 1982

Leutwyler 1973

pre QCD

early estimates

Chiu & Hsieh 2003Gamiz et al. 2003

Becirevic et al. 2003CP-PACS 2003MILC/HPQCD/UKQCD 2003

Gorbunov & Pivovarov 2005Narison 2005

Gamiz et al. 2005Baikov et al.. 2005Narison 2006Jamin et al. 2006QCDSF-UKQCD 2006Chetyrkin & Khodjamirian 2006

JLQCD 2007RBC 2007

since 2003

Weinberg 1977Vainshtein et al. 1978

PDG 2007

RBC/UKQCD 2007

Gamiz 05Baikov 05Narison 06Jamin 06Goeckeler 2006Chetyrkin 2006 – p. 51/51

ormalsize extsf {}ormalsize extsf {it Once upon a time, when Jan and I studied physics ldots }ormalsize extsf {it The new ideas}ormalsize extsf {it Mass of the pion}ormalsize extsf {it Quark masses}ormalsize extsf {it Where I presumably met Jan for the first time}ormalsize extsf {it Abstract for those who prefer it in Russian}ormalsize extsf {it In natura}ormalsize extsf {it First paper we wrote together: February 1970}ormalsize extsf {it Extended version: March 1970}ormalsize extsf {it QCD}ormalsize extsf {it Standard Model}ormalsize extsf {it Quantum theory on null planes}ormalsize extsf {}ormalsize extsf {it Suivez le guide !}ormalsize extsf {it Making bets in the coffee room, on a Friday evening}ormalsize extsf {it Exerpts}ormalsize extsf {}ormalsize extsf {it How well do we understand the strong interaction today ?}ormalsize extsf {it �oldmath Lattice results for $M_pi $}ormalsize extsf {it Lattice}ormalsize extsf {it Chiral perturbation theory}ormalsize extsf {it Expansion of $M_pi ^2$ in powers of the quark mass}ormalsize extsf {it Lattice allows more accurate determination of $Lambda _3$}ormalsize extsf {it Expansion of $F_pi $ in powers of the quark mass}ormalsize extsf {it Lattice results for $Lambda _4$}ormalsize extsf {it Predictions for the S-wave $pi pi $ scattering lengths}ormalsize extsf {it Lattice result for $a_0^2$}ormalsize extsf {it Consequence of lattice results for $ell _3$, $ell _4$}ormalsize extsf {it Experiments on light flavours at low energy}ormalsize extsf {it $K_{e4}$ decay}ormalsize extsf {it $a_0^0,a_0^2$ : prediction, lattice & experiment}ormalsize extsf {it Conflict with $pi pi $ data on exotic S-wave}ormalsize extsf {it Conclusions for SU(2)$imes $SU(2)}ormalsize extsf {it Expansion in powers of $m_s$}ormalsize extsf {it Paramagnetic inequalities}ormalsize extsf {it Condensate}ormalsize extsf {it Results for $B$, $F$}ormalsize extsf {it Expansion in powers of $m_s$}ormalsize extsf {it Conclusions for SU(3)$imes $SU(3)}ormalsize extsf {it Puzzling results on $K_Lightarrow pi mu u $}ormalsize extsf {it Comparison with experiment}ormalsize extsf {it Corrections, extrapolation}ormalsize extsf {it Slope of the scalar form factor}ormalsize extsf {it Implications}ormalsize extsf {it New data from KLOE}ormalsize extsf {it Comparison of results for the slope}ormalsize extsf {it Older measurements}ormalsize extsf {it Conclusions for $Kightarrow mu u pi $}ormalsize extsf {it Huge na zdrav'i ! }ormalsize extsf {}ormalsize extsf {it Effective coupling constants of SU(3)$imes $SU(3)}ormalsize extsf {it Quark mass ratios}ormalsize extsf {it Value of $m_s$}