strong coupling Antonio Vairo INFN and University of Milano

Transcription

1 potential Non-Relativistic QCD strong coupling Antonio Vairo INFN and University of Milano

2 For most of the quarkonium states: 1/r mv Λ QCD (0) V (r) (GeV) 2 1 Υ Υ Υ Υ η ψ c χ ψ ψ r(fm) -1 weak coupled pnrqcd strong coupled pnrqcd

3 For most of the quarkonium states: 1/r mv Λ QCD (0) V (r) (GeV) 2 1 Υ Υ Υ Υ η ψ c χ ψ ψ r(fm) -1 weak coupled pnrqcd strong coupled pnrqcd In this talk we will deal with EFT s suited to describe these states.

4 Quarkonium Scales MeV Υ(4S) BB threshold The non-relativistic hierarchy 500 Υ(3S) ψ(3s) Χ (2P) b m mv p 1/r 0 Υ(2S) ψ(3770) ψ (2S) η (2S) c Χ (1P) b Χc(1P) DD threshold h (1P) c mv 2 E exists regardless of the fact that 500 Υ(1S) J/Ψ η c (1S) mv Λ QCD S states P states Normalized with respect to χ b (1P ) and χ c (1P )

5 Quarkonium Scales m mv QCD perturbative matching perturbative matching NRQCD µ µ The non-relativistic hierarchy m mv p 1/r mv 2 E exists regardless of the fact that mv 2 nonperturbative matching (long range quarkonium) pnrqcd perturbative matching (short range quarkonium) mv Λ QCD

6 Quarkonium Scales The non-relativistic hierarchy m mv p 1/r mv 2 E exists regardless of the fact that mv ΛQCD mv 2 µ perturbative matching (short range quarkonium) nonperturbative matching (long range quarkonium) mv m µ perturbative matching perturbative matching QCD NRQCD pnrqcd

7 NRQCD

8 ) ' 2 -/ 8 ; / , + Spectrum from lattice NRQCD (*) ' % % &! & " # % % $ $! " # % % $ $ : 3 47 : B o t t o m o n i u m ; < ; < 3 46 : HPQCD and UKQCD coll. 02, Davies 02

9 NRQCD power counting * Counting in α s (m): α s c F = π log m µ +... c S = 2c F 1 c D = α s 9 π log m µ +... f = O(α s ) Imf = O(α 2 s) Lattice matching coefficients are known at tree level.

10 NRQCD power counting * Counting in v: 1) d 3 x ψ ψ 1 ψ 2 1 ( x) 3 m3 v 3 2) K (d) (mv) d (e.g. ge, gb m 2 v 2, D mv) 3) D 0 mv 2 (virial theorem)

11 NRQCD power counting * Counting in v: 1) d 3 x ψ ψ 1 ψ 2 1 ( x) 3 m3 v 3 2) K (d) (mv) d (e.g. ge, gb m 2 v 2, D mv) 3) D 0 mv 2 (virial theorem) The power counting is not unique. E.g. in Lepage et al. 92 ( standard NRQCD power counting ): ga 0 mv 2, ga mv 3, ge m 2 v 3, gb m 2 v 4.

12 NRQCD power counting * Counting in v: 1) d 3 x ψ ψ 1 ψ 2 1 ( x) 3 m3 v 3 2) K (d) (mv) d (e.g. ge, gb m 2 v 2, D mv) 3) D 0 mv 2 (virial theorem) The power counting uses arguments rigorously suited only for bound states in a non-relativistic quantum-mechanical context.

13 Lattice NRQCD precision In the bottomonium case, the precision of the radial splittings is α s v % in the standard NRQCD power counting; α s v % in the most conservative one, while for the fine and hyperfine splittings is α s %. In the charmonium case, the precision of the radial splittings is not smaller than 10% in the standard counting (20% in the conservative counting). Order α s v 4 (α s v 3 in the conservative counting), 1/m 2 corrections to the Yang Mills sector of the NRQCD Lagrangian and 4-fermion operators also have to be taken into account.

14 Decay in NRQCD Γ(H LH) = 2 Im H H H = n 2 Im f (n) m d n 4 H ψ K (n) χχ K (n) ψ H Γ(H EM) = n (n) 2 Im f em m d n 4 H ψ K (n) χ vac vac χ K (n) ψ H Bodwin et al. 95 H f H

15 P-wave decays at O(mv 5 ) Γ(χ J LH) = 9 Im f 1 Γ(χ J γγ) = 9 Im f γγ R (0) Im f 8 πm 4 m 2 R (0) 2 πm 4 J = 0, 2 χ O 8 ( 1 S 0 ) χ * Octet and singlet contribute to the same order. The IR divergences of Im f 1 are absorbed into the non-perturbative operator χ O 8 ( 1 S 0 ) χ. Bodwin Braaten Lepage 92

16 P-wave decays at O(mv 5 ) Γ(χ J LH) = 9 Im f 1 Γ(χ J γγ) = 9 Im f γγ R (0) Im f 8 πm 4 m 2 R (0) 2 πm 4 J = 0, 2 χ O 8 ( 1 S 0 ) χ * Bottomonium and charmonium (below threshold) P-wave decays depend on 6 non-perturbative parameters [3 w.f. + 3 octet ].

17 S-wave decays at O(mv 5 ) Γ(V (ns) LH) = 2 m 2 Im f 1( 3 S 1 ) V O 1 ( 3 S 1 ) V +Im f 8 ( 3 S 1 ) P O 8 ( 1 S 0 ) P + Im f 8( 1 S 0 ) P O 8 ( 3 S 1 ) P 3 +Im g 1 ( 3 S 1 ) P P 1( 1 P S 0 ) P J m 2 + (2J + 1)Im f! 8( 3 P J ) P O 8 ( 1 P 1 ) P 3 m 2 Γ(P (ns) LH) = 2 m 2 Im f 1( 1 S 0 ) P O 1 ( 1 S 0 ) P +Im f 8 ( 1 S 0 ) P O 8 ( 1 S 0 ) P + Im f 8( 3 S 1 ) P O 8 ( 3 S 1 ) P 3! +Im g 1 ( 1 S 0 ) P P 1( 1 S 0 ) P m 2 + Im f 8 ( 1 P 1 ) P O 8( 1 P 1 ) P m 2 * Bottomonium and charmonium (below threshold) S-wave decays depend on 30 non-perturbative parameters.

18 S-wave e.m. decays at O(mv 5 ) Γ(V (ns) e + e ) = 2 m 2 Im f ee( 3 S 1 ) V O EM ( 3 S 1 ) V! +Im g ee ( 3 S 1 ) P P 1( 1 S 0 ) P m 2 Γ(P (ns) γγ) = 2 m 2 Im f γγ( 1 S 0 ) P O EM ( 1 S 0 ) P! +Im g γγ ( 1 S 0 ) P P 1( 1 S 0 ) P m 2 * Bottomonium and charmonium (below threshold) S-wave e.m. decays depend on 10 extra non-perturbative parameters.

19 NRQCD matrix elements By fitting charmonium P -wave decay data O 1 ( 1 P 1 ) hc (1P ) GeV 5 and O 8 ( 1 S 0 ) hc (1P ) GeV 3 in MS and at the factorization scale of 1.5 GeV. Maltoni 00 In quenched lattice simulations O 1 ( 1 P 1 ) hc (1P ) GeV 5, O 8 ( 1 S 0 ) hc (1P ) GeV 3 and O 1 ( 1 S 0 ) ηc (1S) 0.33 GeV 3 in MS and at the factorization scale of 1.3 GeV. Bodwin Sinclair Kim 96 In lattice simulations with three light-quark flavors (extrapolation) O 1 ( 1 S 0 ) ηb (1S) 4.1 GeV 3, O 1 ( 1 P 1 ) hb (1P ) 3.3 GeV 5 and O 8 ( 1 S 0 ) hb (1P ) GeV 3 in MS and at the factorization scale of 4.3 GeV. Bodwin Sinclair Kim 01 Some further recent (quenched) determinations are in Bodwin Lee Sinclair 05

20 E.m. widths on the lattice Gray et al 02

21 pnrqcd mv Λ QCD

22 pnrqcd for mv Λ QCD All quarks with energy mv 2 and momentum mv are integrated out.

23 pnrqcd for mv Λ QCD All quarks with energy mv 2 and momentum mv are integrated out. All gluonic excitations between heavy quarks are integrated out since they develop a gap of order Λ QCD with the static Q Q energy.

24 pnrqcd for mv Λ QCD All quarks with energy mv 2 and momentum mv are integrated out. All gluonic excitations between heavy quarks are integrated out since they develop a gap of order Λ QCD with the static Q Q energy. 3 2 E (0) 1 [E(r)-E(r 0 ]r E (0) 0 Bali et al. 98 (r fm) -2 quenched κ =

25 pnrqcd for mv Λ QCD All quarks with energy mv 2 and momentum mv are integrated out. All gluonic excitations between heavy quarks are integrated out since they develop a gap of order Λ QCD with the static Q Q energy. The singlet quarkonium field S of energy mv 2 and momentum mv is the only the degree of freedom of pnrqcd(up to ultrasoft light quarks, e.g. pions).

26 pnrqcd for mv Λ QCD L = Tr ) } {S (i 0 p2 m V s S

27 pnrqcd for mv Λ QCD L = Tr ) } {S (i 0 p2 m V s S The potential V s (Re V s + i Im V s ) is a mixture of perturbative and non-perturbative contributions to be determined by the matching.

28 Matching I The first step consists in integrating out quarks of momentum p m ΛQCD Λ QCD and energy Λ QCD (semihard). (Λ QCD, Λ QCD ) (Λ QCD, Λ QCD ) (mv 2, mv) (mv 2, mv) (E, p) (E, ξ) (E, q) Z d 3 ξ (2π) 3 V (p ξ) 1 E ξ 2 /m V (ξ q) α2 s α 2 s Z d 3 ξ (2π) 3 1 ξ 4 1 E ξ 2 /m 1 Λ QCD 1 p m ΛQCD

29 Matching I Semihard quarks induce the following vertices in NRQCD SH SH = O sh (R, r)r ges sh(r, r) SH P = χ(r) D 2 ψ (R) V (0) s (r) S sh (R, r)

30 Matching I (a) (b) P SH SH P P SH SH P V s V s (c) (d) P SH P P SH SH SH P V s V s

31 Matching I Contribution to the spectrum: Re δv = Γ(9/2) π α2 s E E 7/2 δ 3 (r) m 3/2 Contribution to the decay width: Im δv = Γ(7/2) Im f α s E E 5/2 δ 3 (r) m 5/2 E E n = 1 N c Z 0 dτ τ n vac ge(τ) ge(0) vac E

32 Matching I Contribution to the spectrum: δe nl = Γ(9/2) α2 s E E 7/2! R nl (0) 2 m 3/2 δ l0 O mv 3 mα s α s p mλqcd, Contribution to the decay width: δγ(v Q (ns) LH) = 1 π δγ(p Q (ns) LH) = 1 π δγ(v Q (ns) e + e ) = 1 π δγ(p Q (ns) γγ) = 1 π Rn0 V (0) 2 m 2 Im f 1 ( 3 68 S 1 ) 3Γ(7/2) Rn0 P (0) 2 m 2 Im f 1 ( 1 68 S 0 ) 3Γ(7/2) Rn0 V (0) 2 m 2 Im f ee ( 3 68 S 1 ) 3Γ(7/2) Rn0 P (0) 2 m 2 Im f γγ ( 1 68 S 0 ) 3Γ(7/2) α s E E 5/2 m 1/2 α s E E 5/2 m 1/2 α s E E 5/2 m 1/2 α s E E 5/2 m 1/2 Brambilla Pineda Soto Vairo 04

33 Matching II The second step consists in integrating out quarks of momentum mv and energy Λ QCD, and gluons of energy Λ QCD.

34 Matching II The matching condition is: H H H = nljs p2 m +X n V (n) s m n nljs

35 Matching II The matching condition is: H H H = nljs p2 m +X n V (n) s m n nljs H=H (0) + δh(1) m + δh(2) m 2 + δh(3) m 3 + δh(4) m Z H (0) = d 3 x 1 n Xf 2 (Πa Π a + B a B a ) q id γ q Z D δh (1) = d 3 x ψ 2 «2 + c F g S B ψ + antip

36 Matching II The matching condition is: H H H = nljs p2 m +X n V (n) s m n nljs In a QM language: H (0) n; x 1, x 2 (0) = E (0) n (x 1, x 2 ) n; x 1, x 2 (0) n; x 1, x 2 (0) = ψ (x 1 )χ(x 2 ) n; x 1, x 2 (0) x j are the quark positions n : CP,... 0 (0) = (Q Q) 1 Quarkonium Singlet n > 0 (0) = (Q Q)g (n) Higher Gluonic Excitations

37 Matching II The matching condition is: H H H = nljs p2 m +X n V (n) s m n nljs Expanding in 1/m: 0; x 1, x 2 = 0; x 1, x 2 (0) + X n 0 Z d 3 z 1 d 3 z 2 n; z 1, z 2 (0) (0) n; z 1, z 2 δh (1) 0; x 1, x 2 (0) E (0) 0 (z) E(0) n (x) H 0; x 1, x 2 nljs +...

38 The non-perturbative Potentials V s = V s (0) + V s (1) m + V s (2) m

39 The non-perturbative Potentials V (0) s = lim T i T ln W (r T ) = lim T β=6.0 β=6.2 fit i T ln 1 GeV r/fm Bali Schilling Wachter 97

40 The non-perturbative Potentials V s (1) = (0) 0 D 2 0 (0) = 2 + k 0 Since kinetic energy (0) k ge 0 (0) E (0) 0 E (0) k 2 E(t) E(0) = k (0) 0 ge k (0) 2 e ie(0) 0 T i(e(0) k E(0) 0 )t V (1) s = dt t E Brambilla Pineda Soto Vairo 00

41 The non-perturbative Potentials V (2) SD = 1 r kij 2rk (c F ɛ r i 0 E i j dt t 1 2 V (0) B s ) (S 1 + S 2 ) L c F 2ˆr iˆr j i ( i j dt δ ) ij ( 3 ) S 1 S 2 3(S 1 ˆr)(S 2 ˆr) 0 ( c F 2 i 0 dt 4 ( d ) ) 3 d 4 δ (3) (r) S 1 S 2 Eichten Feinberg 81, Gromes 84, Chen et al. 95 Brambilla Vairo 99

42 The non-perturbative Potentials V (2) SI = p i (i c2 F i 2 i 0 ) i j i j dt t 2 + p j B dt + (d d ) πα sc D δ (3) (r) 0 t1 t2 ( dt 1 dt 2 dt 3 (t 2 t 3 ) t1 + dt 1 dt 2 (t 1 t 2 ) 2 i 0 ( 0 i i i b 3 f abc d 3 x g G a µν(x)g b µα(x)g c να(x) c ) ) Brambilla et al , Pineda Vairo 00

43 Imaginary parts of the Potential Im V s P wave = Ω SJ ij i δ 3 (r) j [ 3 Im f 1( 2S+1 P J ) m 4 + E 27 Im f 8 ( 2S+1 S S ) m 4 ] where E = 18 n 0 = ge n n ge 0 (E (0) n E (0) 0 ) 4 dt t 3 ge a (t, 0)Φ ab (t, 0; 0)gE b (0, 0)

44 P-wave decays at O(mv 5 ) Γ(χ J LH) = R (0) πm 4 2 Γ(χ J γγ) = 9 Im f γγ [ 9 Im f 1 + Im f 8 9 R (0) 2 πm 4 J = 0, 2 E ] Brambilla et al. 01, 02, 03

45 P-wave decays at O(mv 5 ) Γ(χ J LH) = R (0) πm 4 2 Γ(χ J γγ) = 9 Im f γγ [ 9 Im f 1 + Im f 8 9 R (0) 2 πm 4 J = 0, 2 E ] * χ O 8 ( 1 S 0 ) χ = R (0) 2 18πm 2 E; E Brambilla et al. 01, 02, 03 0 dt t 3 Tr(gE(t) ge(0))

46 P-wave decays at O(mv 5 ) Γ(χ J LH) = R (0) πm 4 2 Γ(χ J γγ) = 9 Im f γγ [ 9 Im f 1 + Im f 8 9 R (0) 2 πm 4 J = 0, 2 E ] * χ O 8 ( 1 S 0 ) χ = R (0) 2 18πm 2 E; E Brambilla et al. 01, 02, 03 0 dt t 3 Tr(gE(t) ge(0)) * The quarkonium state dependence factorizes.

47 P-wave decays at O(mv 5 ) Γ(χ J LH) = R (0) πm 4 2 Γ(χ J γγ) = 9 Im f γγ [ 9 Im f 1 + Im f 8 9 R (0) 2 πm 4 J = 0, 2 E ] * χ O 8 ( 1 S 0 ) χ = R (0) 2 18πm 2 E; E Brambilla et al. 01, 02, 03 0 dt t 3 Tr(gE(t) ge(0)) * Bottomonium and charmonium (below threshold) P-wave decays depend on 4 non-perturbative parameters [3 w.f. + 1 corr. ].

48 Determination of E from charmonium data Process Γ (MeV) Reference Γ(χ c0 LH) 9.7 ± 1.1 E835 Γ(χ c0 LH) 14.3 ± 3.6 BES Γ(χ c0 LH) 13.5 ± 5.4 CBALL Γ(χ c1 LH) 0.64 ± 0.12 E760 Γ(χ c2 LH) 1.71 ± 0.18 E760 From the charmonium decay-width ratios we get: 20 E E(1 GeV) = [exp] 15 E(µ) = E(m) + 96 ln α s(m) β 0 α s (µ) µ

49 Bottomonium P -wave decays Γ(χ b0 (1P ) LH) Γ(χ b1 (1P ) LH) = Γ(χ b0(2p ) LH) Γ(χ b1 (2P ) LH) = 8.0 ± 1.3 [E] (CleoIII 02) = 19.3 ± Γ χb0 /Γ χb NLO LO Brambilla Eiras Pineda Soto Vairo 01, Vairo 02, CLEO coll. 02 µ

50 S-wave octet matrix elements At leading order in the v and Λ QCD /m expansion: V O 8 ( 3 S 1 ) V = P O 8 ( 1 S 0 ) P = 3 R(0) 2 2π V O 8 ( 1 S 0 ) V = P O 8( 3 S 1 ) P 3 V O 8 ( 3 P J ) V 2J + 1 χ O 8 ( 1 S 0 ) χ = 1 6 = P O 8( 1 P 1 ) P 9 R (0) 2 πm 2 E 3 = 3 R(0) 2 2π = 3 R(0) 2 2π! E(2) 3 9m 2 c2 F B! 1 18m 2 «E 1 54 V P 1 ( 3 S 1 ) V = P P 1 ( 1 S 0 ) P = V P EM ( 3 S 1 ) V = P P EM ( 1 S 0 ) P = 3 R(0) 2 me (0) n0 2π E 1

51 S-wave decays R V n Γ(V (ns) LH) Γ(V (ns) e + e ) R P n Γ(P (ns) LH) Γ(P (ns) γγ) It is a prediction of pnrqcd that, for the states for which Λ QCD mv 2, the wave-function dependence drops out. [ Residual m dependence in 1/m, E (0) n0 and Imf ; residual n dependence in E (0) n0. ]

52 S-wave decays R V n R V m R P n R P m = 1 + = 1 + ( Im g1 ( 3 S 1 ) Im f 1 ( 3 S 1 ) Im g ) ee( 3 S 1 ) Mn M m, Im f ee ( 3 S 1 ) m ( Im g1 ( 1 S 0 ) Im f 1 ( 1 S 0 ) Im g ) γγ( 1 S 0 ) Mn M m. Im f γγ ( 1 S 0 ) m For m b = 5 GeV, R2 Υ /R3 Υ 1.3 [pdg 1.4] Im g 1 ( 1 S 0 )/Im f 1 ( 1 S 0 ) Im g γγ ( 1 S 0 )/Im f γγ ( 1 S 0 ) α s up to O(v 3 ), R P n is equal for all radial excitations.

53 NRQCD power counting & pnrqcd In pnrqcd: V P 1 ( 3 S 1 ) V = 3 R(0) n0 2 2π ( me (0) n0 E 1 ) Using the standard NRQCD power counting : Gremm Kapustin 97 V Q (ns) P 1 ( 3 S 1 ) V Q (ns) GK = 3 R(0) n0 2 2π me(0) n0

54 NRQCD power counting & pnrqcd In pnrqcd: V P 1 ( 3 S 1 ) V = 3 R(0) n0 2 2π ( me (0) n0 E 1 ) Using the standard NRQCD power counting : Gremm Kapustin 97 V Q (ns) P 1 ( 3 S 1 ) V Q (ns) GK = 3 R(0) n0 2 2π me(0) n0 The difference clarifies the range of validity of the standard NRQCD power counting : E (0) n0 mv 2, E 1 Λ 2 QCD : If Λ QCD mv, then E 1 me (0) n0 If Λ QCD mv 2, then E 1 me (0) n0

55 Conclusions Heavy quarkonium in the non-perturbative regime is accessible to a systematic study inside QCD. Wave-functions and potentials may be precisely defined in terms of QCD parameters. Accurate determinations interesting both for the phenomenology and for the structure of the QCD vacuum are possible. Future applications may include radiative and hadronic transitions.