The taming of the red dragon

Transcription

1 The taming o the red dragon H. Leutwyler University o Bern Spontaneous topics in theoretical physics Bern, May 5, 2006 The taming o the red dragon p.1/26

2 There is the broad object seen in ππ scattering, oten called background, which extends rom about 400 MeV up to about 1700 MeV. This object we consider as a single broad resonance 2 which we identiy as the lightest glueball with quantum numbers J P C = we reer to it as red dragon P. Minkowski and W. Ochs, Eur. Phys. J. C9 (1999) 283 The taming o the red dragon p.2/26

3 Low energy pion physics Plays a crucial role whenever the strong interaction is involved at low energies Example: Standard model prediction or muon magnetic moment Main experiments on ππ scattering were done in the seventies. What s new? Signiicant theoretical progress, based on ChPT + dispersion theory New precision data: K ππlν E865 Brookhaven pionic atoms DIRAC CERN K 3π NA48/2 CERN Lattice results on M π, F π, a 2 0, r2 s The taming o the red dragon p.3/26

4 Analyticity and crossing ππ scattering is special: crossed channels are identical Re T (s, t) can be represented as a twice subtracted dispersion integral over Im T (s, t) in physical region S.M. Roy 1971 The 2 subtraction constants can be identiied with the S-wave scattering lengths: a 0 0, a2 0 isospin angular momentum Representation leads to dispersion relations or the individual partial waves: Roy equations The taming o the red dragon p.4/26

5 Roy equations Pioneering work on the physics o the Roy equations: Basdevant, Froggatt & Petersen 1974 Dispersion integrals converge rapidly (2 subtractions) Crude phenomenological inormation on Im T (s, t) or energies above 800 MeV suices Given a 0 0, a2 0, the scattering amplitude can be calculated to within small uncertainties Ananthanarayan, Colangelo, Gasser & L Descotes, Fuchs, Girlanda & Stern 2002 a 0 0, a2 0 are the essential parameters at low energy Main problem in early work: a 0 0, a2 0 poorly known Experimental inormation near threshold is meagre The taming o the red dragon p.5/26

6 Low energy theorems Chiral perturbation theory provides the missing piece: theoretical prediction or a 0 0, a2 0 Weinberg 1966, Gasser & L. 1983, Bijnens, Colangelo, Ecker, Gasser & Sainio 1996 Most accurate results or a 0 0, a2 0 are obtained by matching the chiral and dispersive representations near the center o the Mandelstam triangle Colangelo, Gasser & L In combination with the low energy theorems or a 0 0, a2 0, the dispersion relations or the partial waves ix the ππ scattering amplitude to an incredible degree o accuracy The taming o the red dragon p.6/26

7 ents Predictions or the S-wave ππ scattering lengths a Universal Band tree, one loop, two loops low energy theorem or scalar radius Colangelo, Gasser & Leutwyler 2001 a Sizeable corrections in a 0 0, while a2 0 nearly stays put The taming o the red dragon p.7/26

8 Tests o the predictions or a 0 0, a2 0 : experiment & lattice Universal band tree (1966), one loop (1983), two loops (2000) Prediction (ChPT + dispersion theory, 2001) E 865 (2003) DIRAC (2005) NA48 (2005) MILC (2004) NPLQCD (2005) a a 0 0 Theory is ahead o experiment... The taming o the red dragon p.8/26

9 The red dragon I. Caprini, G. Colangelo and H. Leutwyler, Phys. Rev. Lett. 96 (2006) Does QCD have a resonance near threshold? Why care? Concerns the nonperturbative domain o QCD Quark and gluon degrees o reedom useless there Understanding very poor, pattern o energy levels? Lowest resonance: σ? ρ? Resonance pole on second sheet Poles are universal Pole position is unambiguous, even i width is large Where is the pole closest to the origin? The taming o the red dragon p.9/26

10 ! " # $ &% '( ) * + +, * * / # ;: ; < ACB D E FCG H I J B I J J G K ML MON P Q RM S MT ML U VWYX Z [ =\ > P P ]P ^`_ a = b L X c ^ X d _ eg Bh I I G ikj ml i pon q rst u v ew Bh I I G ikj ml i pon q rst u v e Bh I I G i j ml i pon q rst u v e Bh I I G i j ml i pon q rst u v x y z{ z } z ~ ƒ y { ˆŠ Ž Œ Ž ƒ Yš ŵœ ƒž Yš ˆ ƒ Yš ŵœ ƒž Yš ˆ ƒ Yš ŵœ ƒž Yš ˆ ƒ Yš ŵœ ƒž Yš ˆ { ª y«y z { z { y y «ª } z} y± }² {± } { ³ µz ³ z ³ { z ƒ ž ŵœ ƒ ˆ º¹» ¼ ¹ ¾ ½ Á à œ ƒ Ä ž ŵœ ƒ Å ž ˆ ÇÆ º º Æ É y } z y«ƒ Ä ŵœ ƒ ˆ ž`ê º Ë ¼ ƒ ž ž ŵœ ƒ Å ˆ ¹ Æ Æ ž ž œ Å º¹ É» ƒ Å ž ŵœ ƒ ˆ ɺ x Æ º ƒ ž Ã Ä œ Ìž ŵœ ƒ Ã Å Ì œ ž ˆ Ì Í Î Ï ƒž Ð ˆ Ï Ì œ Ì Ì Å È È ³ Ñ Ñ ƒ Ä Ã ž œ Å ŵœ ƒ Ì Ã ž œ ˆ Í Î $ Ò Ò œ ž Í x x Í } }± y± } zy± ƒ ž ŵœ ƒ Å ˆ» x» ³ Ñ Ñ ³ Ó Ó œ Å º º ³ Ñ Ñ Ì œ ž º º $ ³ Ñ Ñ œ º º ³ Ñ Ñ ³ Ô Ô ƒ ž à œ ˆ œ ƒ Ì ˆ x È ÉÍ Ä $ É y } z y«œ º ÉÍ Ä ³ Ñ Ñ ƒ Ì Ì ŵœ ƒ Ì Å ˆ Ä Í Î Å ƒ žå ŵœ ƒ Å ˆ» x» Å $ ³ Ñ Ñ ³ Å œ Õ x Ö È Ì ³ Ñ Ñ ³ Ñ ³ Ô ƒ œ ž ˆ º $ ɹ Ò Ò ž œ Õ º ɹ Ò Ò ž œ žå Õ ž º ɹ Ò Ò ž ž ÄÅ œ ž Õ Ø x x ³ Ñ Ñ œ Ì ÉÍ È ƒ Ì ŵœ ƒ Å ž ˆ» x» ³ Ñ Ñ žå œ ž Ì Ì Ë $ ³ Ñ Ñ Ä œ ž Õ Ì Ë ž ³ Ñ Ñ ÄÅ œ žå Õ Å ÄÅ ³ Ñ Ñ Å œ Ä Ä ¹ È x ÌÄ ³ Ñ Ñ ³ Ô Ô ³ ƒå ŵœ ƒ ˆ ¹» Å ³ Ñ Ñ ƒ ÌÌ ˆ œ ƒž Å ˆ Ê Ê Ê Å ž Í ¹ É ³ Ñ Ñ Ì œ žå ¹ Î È x Å The taming o the red dragon p.10/26

11 Model independent determination o the pole All o the results quoted by the PDG are obtained by (a) parametrizing the data or real values o s (b) continuing this parametrization analytically in s Result is sensitive to the parametrization used The taming o the red dragon p.11/26

12 Model independent determination o the pole All o the results quoted by the PDG are obtained by (a) parametrizing the data or real values o s (b) continuing this parametrization analytically in s Result is sensitive to the parametrization used We ound a model independent method: 1. Poles on second sheet are zeros on irst sheet 2. The Roy equations are valid or complex values o s, in a limited region o the irst sheet Exact representation o the partial waves in terms o observable quantities, valid or complex values o s The taming o the red dragon p.11/26

13 Model independent determination o the pole All o the results quoted by the PDG are obtained by (a) parametrizing the data or real values o s (b) continuing this parametrization analytically in s Result is sensitive to the parametrization used We ound a model independent method: 1. Poles on second sheet are zeros on irst sheet 2. The Roy equations are valid or complex values o s, in a limited region o the irst sheet Exact representation o the partial waves in terms o observable quantities, valid or complex values o s 3. Can evaluate this representation to good precision and determine the zeros numerically The taming o the red dragon p.11/26

14 Pole on second sheet zero on irst sheet S0 0(s) = η0 0 (s) exp 2iδ0 0 (s) s-plane S 0 0 (s) is analytic in the cut plane 0 4 M π 2 physical region The taming o the red dragon p.12/26

15 Pole on second sheet zero on irst sheet S 0 0 (s) = η0 0 (s) exp 2iδ0 0 (s) s-plane S 0 0 (s) is analytic in the cut plane 0 4 M π 2 physical region For 0 < s < 4Mπ 2, S0 0 (s) is real S 0 0 (s ) = S 0 0 (s) x in elastic interval: S 0 0 (x ± iɛ) = exp ±2iδ0 0 (x) The taming o the red dragon p.12/26

16 Pole on second sheet zero on irst sheet S 0 0 (s) = η0 0 (s) exp 2iδ0 0 (s) s-plane S 0 0 (s) is analytic in the cut plane 0 4 M π 2 physical region For 0 < s < 4Mπ 2, S0 0 (s) is real S 0 0 (s ) = S 0 0 (s) x in elastic interval: S 0 0 (x ± iɛ) = exp ±2iδ0 0 (x) Second sheet is reached by continuation across the elastic interval o the right hand cut S0 0(x iɛ)ii = S0 0(x + iɛ)i = 1/S0 0 (x iɛ)i Analyticity S 0 0 (s)ii = 1/S 0 0 (s)i valid s Pole in S 0 0 (s)ii zero in S 0 0 (s)i The taming o the red dragon p.12/26

17 Roy equation or the isoscalar S-wave S 0 0 (s)=1 + 2 iρ t0 0 (s) ρ = 1 4M 2 π /s The taming o the red dragon p.13/26

18 Roy equation or the isoscalar S-wave S0 0 (s)=1 + 2 iρ t0 0 1 (s) ρ = 4Mπ 2/s t 0 0 (s)=a + (s 4M 2 π ) b + 4M 2 π ds { K 0 (s, s ) Im t 0 0 (s ) + K 1 (s, s ) Im t 1 1 (s ) + K 2 (s, s ) Im t 2 0 (s ) } + higher partial waves The taming o the red dragon p.13/26

19 Roy equation or the isoscalar S-wave S 0 0 (s)=1 + 2 iρ t0 0 (s) ρ = 1 4M 2 π /s t 0 0 (s)=a + (s 4M 2 π ) b + 4M 2 π ds { K 0 (s, s ) Im t 0 0 (s ) + K 1 (s, s ) Im t 1 1 (s ) + K 2 (s, s ) Im t 2 0 (s ) } + higher partial waves The subtraction constants are determined by a 0 0, a2 0 : a = a 0 0, b = (2a0 0 5a2 0 )/(12M 2 π ) The taming o the red dragon p.13/26

20 Roy equation or the isoscalar S-wave S 0 0 (s)=1 + 2 iρ t0 0 (s) ρ = 1 4M 2 π /s t 0 0 (s)=a + (s 4M 2 π ) b + 4M 2 π ds { K 0 (s, s ) Im t 0 0 (s ) + K 1 (s, s ) Im t 1 1 (s ) + K 2 (s, s ) Im t 2 0 (s ) } + higher partial waves The subtraction constants are determined by a 0 0, a2 0 : a = a 0 0, b = (2a0 0 5a2 0 )/(12M 2 π ) The kernels are elementary unctions, e.g. K 0 (s, s ) = 1 π(s s) }{{} r.h.cut + 2 ln{(s+s 4Mπ 2 )/s } 3π(s 4Mπ 2 ) 5s +2s 16M 2 π 3πs (s 4M 2 π ) } {{ } l.h.cut The taming o the red dragon p.13/26

21 Roy equation or the isoscalar S-wave S 0 0 (s)=1 + 2 iρ t0 0 (s) ρ = 1 4M 2 π /s t 0 0 (s)=a + (s 4M 2 π ) b + 4M 2 π ds { K 0 (s, s ) Im t 0 0 (s ) + K 1 (s, s ) Im t 1 1 (s ) + K 2 (s, s ) Im t 2 0 (s ) } + higher partial waves The subtraction constants are determined by a 0 0, a2 0 : a = a 0 0, b = (2a0 0 5a2 0 )/(12M 2 π ) The kernels are elementary unctions, e.g. K 0 (s, s ) = 1 π(s s) }{{} r.h.cut + 2 ln{(s+s 4Mπ 2 )/s } 3π(s 4Mπ 2 ) 5s +2s 16M 2 π 3πs (s 4M 2 π ) } {{ } l.h.cut Let hand cut is essential or convergence: K 0 (s, s ) 1/s 3 or large s The taming o the red dragon p.13/26

22 Domain o validity o the Roy equations Roy derived his equations or real energies in the interval 4M 2 π < s < 60M 2 π The taming o the red dragon p.14/26

23 40 Domain o validity o the Roy equations Roy derived his equations or real energies in the interval 4Mπ 2 < s < 60M π 2 Equations are valid or complex s in a limited region o the irst sheet I. Caprini, G. Colangelo and H. Leutwyler Phys. Rev. Lett. 96 (2006) Ims Res s in units o M 2 π The taming o the red dragon p.14/26

24 Ims Domain o validity o the Roy equations Roy derived his equations or real energies in the interval 4Mπ 2 < s < 60M π 2 Equations are valid or complex s in a limited region o the irst sheet 0 Axiomatic ield theory Mandelstam analyticity I. Caprini, G. Colangelo and H. Leutwyler Phys. Rev. Lett. 96 (2006) Proo is based on irst principles, general quantum ield theory A. Martin, Scattering Theory: Unitarity, Analyticity and Crossing Lecture Notes in Physics, vol. 3, Res G. Mahoux, S. M. Roy and G. Wanders Nucl. Phys. B 70 (1974) 297 s in units o M 2 π Exact representation or S0 0 (s) in this region Do not need to parametrize the amplitude The taming o the red dragon p.14/26

25 Evaluation o the pole position Insert our solutions o the Roy equations For the central solution, S0 0 (s) has two pairs o zeros in the region o validity o the representation: s = (6.2 ± i 12.3) Mπ 2 σ s = (51.4 ± i 1.4) Mπ 2 0 (980) The taming o the red dragon p.15/26

26 Evaluation o the pole position Insert our solutions o the Roy equations For the central solution, S0 0 (s) has two pairs o zeros in the region o validity o the representation: s = (6.2 ± i 12.3) Mπ 2 σ s = (51.4 ± i 1.4) Mπ 2 0 (980) Ims 0-20 σ σ ρ ρ 0 (980) 0 (980) Res s in units o M 2 π The taming o the red dragon p.15/26

27 Evaluation o the pole position Insert our solutions o the Roy equations For the central solution, S0 0 (s) has two pairs o zeros in the region o validity o the representation: s = (6.2 ± i 12.3) Mπ 2 σ s = (51.4 ± i 1.4) Mπ 2 0 (980) Ims 0-20 σ σ ρ ρ 0 (980) 0 (980) The eyes o the red dragon Tail at 1.7 GeV: s 150 M 2 π Res s in units o M 2 π The taming o the red dragon p.15/26

28 Evaluation o the pole position Insert our solutions o the Roy equations For the central solution, S0 0 (s) has two pairs o zeros in the region o validity o the representation: s = (6.2 ± i 12.3) Mπ 2 σ s = (51.4 ± i 1.4) Mπ 2 0 (980) Ims σ σ ρ ρ 0 (980) 0 (980) Res s in units o M 2 π 1. Lowest resonance o QCD has vacuum quantum numbers 2. Pole on lower hal o sheet II occurs in vicinity o m σ = 441 i 272 MeV = M σ i 2 Γ σ The taming o the red dragon p.15/26

29 Error analysis Results depend on phenomenological input used when solving the Roy equations, subject to uncertainties Can ollow error propagation explicitly Pole position o σ mainly depends on 3 input variables: a 0 0, a2 0, δ A δ0 0 (800 MeV) Substantial uncertainties in phenomenology o δ A Use the range: δ A = The taming o the red dragon p.16/26

30 Error analysis Noise rom remaining input variables is very small: m 0 σ = (441 ± 4) i(272 ± 6) MeV The taming o the red dragon p.17/26

31 Error analysis Noise rom remaining input variables is very small: m 0 σ = (441 ± 4) i(272 ± 6) MeV Values o a 0 0, a2 0, δ A are crucial: m σ = ( i 3.8) a (0.8 i 4.0) a (5.3 + i 3.3) δ A numbers in MeV The taming o the red dragon p.17/26

32 Error analysis Noise rom remaining input variables is very small: m 0 σ = (441 ± 4) i(272 ± 6) MeV Values o a 0 0, a2 0, δ A are crucial: m σ = ( i 3.8) a (0.8 i 4.0) a (5.3 + i 3.3) δ A numbers in MeV Final result: insert the predictions or a 0 0, a2 0, use the phenomenological range or δ A and add errors up: m σ = i MeV The taming o the red dragon p.17/26

33 Comparison with compilation o PDG Our result Im[m σ ] -400 PDG estimate: ( ) -i( ) MeV Re[m σ ] The taming o the red dragon p.18/26

34 Vicinity o the pole PDG estimate CCL (05) Zhou (05) Anisovich (05) Pelaez (04) CGL (01) Hannah (99) Oller (99) Locher (98) Tornqvist (96) Janssen (95) Kaminski (94) Zou (94) Zou (93) Beveren (86) -200 Im[m σ ] Focus on this region -200 Im[m σ ] -400 PDG estimate: ( ) -i( ) MeV Re[m σ ] Re[m σ ] Results or Re[m σ ] and Im[m σ ] are strongly correlated The taming o the red dragon p.19/26

35 Ignore the theoretical predictions or a 0 0, a2 0 Replace the low energy theorems or a 0 0, a2 0 by the experimental results rom E865, DIRAC and NA48 a 0 0, a2 0 universal band Universal band a 0 0, a0 2 rom E865 a 0 0, a0 2 rom DIRAC a 0 0, a0 2 rom NA48 PDG estimate CCL (05) Zhou (05) Anisovich (05) Pelaez (04) CGL (01) Hannah (99) Oller (99) Locher (98) Tornqvist (96) Janssen (95) Kaminski (94) Zou (94) Zou (93) Beveren (86) Re[m σ ] -200 Im[m σ ] The taming o the red dragon p.20/26

36 Why are our errors so incredibly small? The σ occurs at low energies At low energies, the subtraction term dominates t 0 0 (s) a0 0 + (2a0 0 5a2 0 )(s 4M 2 π ) 12M 2 π The taming o the red dragon p.21/26

37 Why are our errors so incredibly small? The σ occurs at low energies At low energies, the subtraction term dominates t 0 0 (s) a0 0 + (2a0 0 5a2 0 )(s 4M 2 π ) 12M 2 π Insert low energy theorem or a 0 0, a2 0 Roy equation reduces to Weinberg ormula t 0 0 (s) (2s M 2 π ) 32πF 2 π Dispersion integrals only represent a correction The taming o the red dragon p.21/26

38 At low energies, the subtraction term dominates 0.4 Ret Imt 0 0 Subtraction term Weinberg Adler zero σ s in units o M 2 π s = (0.41 ± 0.06) M 2 π s = (6.2 i 12.3) M 2 π Adler zero pole rom σ The taming o the red dragon p.22/26

39 Estimate pole position on back o an envelope Approximate t 0 0 (s) with the Weinberg ormula t 0 0 (s) = (2s M π 2) 32πFπ 2 Where are the zeros o S0 0 (s) in this approximation? The taming o the red dragon p.23/26

40 Estimate pole position on back o an envelope Approximate t 0 0 (s) with the Weinberg ormula t 0 0 (s) = (2s M π 2) 32πFπ 2 Where are the zeros o S0 0 (s) in this approximation? i 1 4Mπ 2/s t0 0 (s) = 0 The taming o the red dragon p.23/26

41 Estimate pole position on back o an envelope Approximate t 0 0 (s) with the Weinberg ormula t 0 0 (s) = (2s M π 2) 32πFπ 2 Where are the zeros o S0 0 (s) in this approximation? i 1 4Mπ 2/s t0 0 (s) = 0 Cubic equation or s Pair o complex zeros, m σ = 365 i 291 MeV The taming o the red dragon p.23/26

42 Estimate pole position on back o an envelope Approximate t 0 0 (s) with the Weinberg ormula t 0 0 (s) = (2s M π 2) 32πFπ 2 Where are the zeros o S0 0 (s) in this approximation? i 1 4Mπ 2/s t0 0 (s) = 0 Cubic equation or s Pair o complex zeros, m σ = 365 i 291 MeV Correction rom higher orders amounts to m σ = i MeV For the quantity that counts, the accuracy is modest The taming o the red dragon p.23/26

43 Estimate pole position on back o an envelope Approximate t 0 0 (s) with the Weinberg ormula t 0 0 (s) = (2s M π 2) 32πFπ 2 Where are the zeros o S0 0 (s) in this approximation? i 1 4Mπ 2/s t0 0 (s) = 0 Cubic equation or s Pair o complex zeros, m σ = 365 i 291 MeV Correction rom higher orders amounts to m σ = i MeV For the quantity that counts, the accuracy is modest Real zero on sheet II, near s = 0 (ull amplitude has kinematic singularity: vanishes on sheet II at s = 0) The taming o the red dragon p.23/26

44 Conclusion Low energy pion physics: theory ahead o experiment Precision experiments carried out and under way Lattice makes slow, but steady progress So ar, all tests conirm the theory The taming o the red dragon p.24/26

45 Conclusion Low energy pion physics: theory ahead o experiment Precision experiments carried out and under way Lattice makes slow, but steady progress So ar, all tests conirm the theory Limitations o our approach: Calculations cannot be done on back o an envelope Method only covers low energies Only a ew applications have been worked out: ππ scattering, pion orm actors, hadronic vacuum polarization in SM prediction or muon g 2 γγ π 0 π 0 M. Pennington, hep-ph/ Much is yet to be done: J/ψ ωππ, D 3π,... πk, κ,... The taming o the red dragon p.24/26

46 Conclusion Model independent method or analytic continuation The lowest resonance o QCD occurs at M σ = MeV Γ σ = MeV and carries vacuum quantum numbers Crossing symmetry plays an essential role: Fixes contributions rom let hand cut Ensures ast convergence, low energy dominance Pole occurs at low value o s, closer to let hand cut than to singularities rom K K, 0 (980) Result or Γ σ relies on theory or a 2 0 Experiments concerning a 2 0 would be most welcome The taming o the red dragon p.25/26

47 VISIT THE TAMED RED DRAGON GENTLE ANIMAL LOOK IN HIS EYES FROM CLOSE SMELL HIS GOOD BREATH BRING YOUR PIONS ALONG AND FEED HIM YOURSELF The management denies responsibility or incidents involving the dragon s tail The taming o the red dragon p.26/26