Testing QCD with τ Decays. Antonio Pich IFIC, Univ. Valencia - CSIC

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1 Testing QCD with τ Decays Antonio Pich IFIC, Univ. Valencia - CSIC Mainz Institute for Theoretical Physics, Johannes Gutenberg University Determination of the Fundamental Parameters in QCD, 7-1 March 016

2 Theoretical Framework R τ = Γ[τ ν τ hadrons] Γ[τ ν τ e ν e ] m τ = 1π 0 ds m τ = R τ,v +R τ,a +R τ,s ( 1 s ) [( mτ 1+ s ) ] mτ ImΠ (1) (s)+imπ (0) (s) ( ) Π (J) (s) V ud Π (J) ud,v (s) + Π(J) ud,a (s) + V us Π (J) us,v+a (s) Davier et al, (v 1 +a 1 )(s) 3.5 ALEPH Perturbative QCD (massless) Parton model prediction (v 1 -a 1 )(s) 3.5 ALEPH Perturbative QCD/parton model v 1 = πimπ (1) ud,v (s) a 1 = πimπ (1) ud,a (s) s (GeV ) s (GeV ) A. Pich τ and QCD

3 i [ d 4 x e iqx 0 T J µ ij ] ν (x)jij (0) 0 = ( g µν q +q µ q ν ) Π (1) ij,j (q )+q µ q ν Π (0) ij,j (q ) A ω J (s 0) s0 s th ds s 0 ω(s) ImΠ (J) J (s) = 1 i s =s 0 ds s 0 ω(s) Π (J) J (s) Imq sth Req s 0 Π (J) J (s) Π(J) J (s)ope = D O (J) D,J ( s) D/ δ DV 1 { ds ω(s) Π (J) J (s) Π(J) J πi (s)ope} = 1 [ ds ω(s) Im Π (J) J (s) Π(J) J s =s 0 π (s)ope] s 0 Add residues whenever there are poles within the integration contour A. Pich τ and QCD 3

4 Left-Right Correlator Π(s) Π (0+1) ud,lr (s) Π(0+1) ud,v (s) Π(0+1) ud,a (s) = f π s mπ +Π(s) Pure non-perturbative quantity: Known short-distance constraints: Ideal to test the OPE Π(s) OPE m q=0 = D 6 O D ( s) D/ 1 1st WSR: nd WSR: 3 πsr: ds 1 s th π ImΠ(s) = f π ds s 1 s th π ImΠ(s) = f πm π ( s ) 1 ds s log s th Λ π ImΠ(s) = ( m π 0 m ) π + em m q=0 8π 3α f π mq=0 χpt: Π(s) = 8L 10 +χlogs+s [16C 87 +χlogs]+o(s ) A. Pich τ and QCD 4

5 Non-Pinched & Pinched Weights González-Alonso, Pich, Rodríguez-Sánchez L eff 10 = 1 s0 ds s s th π ImΠ(s) Ceff 87 = 1 16 s0 s th ds s 1 π ImΠ(s) ω 1,0 = s 1 (1 s/s 0 ), ω 1 = s 1 (1 s/s 0 ), ω,0 = s (1 s /s 0 ), ω = s (1 s/s 0 ) (1 + s/s 0 ) A. Pich τ and QCD 5

6 Playing with Duality Violations Gonza lez-alonso, Pich, Rodrı guez-sa nchez Ad-hoc Ansatz: ρ(s > sz ) = κ e γs sin{β(s sz )} (Cata et al) randomly generated tuples (κ, γ, β, sz ) Select Acceptable Spectral Functions, satisfying: 1 Consistent with ALEPH at 90% C.L., above s = 1.7 GeV Fulfill 1st and nd WSRs, and πsr (s) tuples selected 0.10 (red band) A. Pich τ and QCD s (GeV ) 6

7 Statistical Distributions of Selected Tuples González-Alonso, Pich, Rodríguez-Sánchez GeV units A. Pich τ and QCD 7

8 Statistical Distributions of L eff 10, Ceff 87, O 6 and O 8 González-Alonso, Pich, Rodríguez-Sánchez Non-pinched & pinched weights L eff 10 = ( ±0.05) 10 3 = ( 6.48 ±0.05) 10 3 C eff 87 = ( ±0.18) 10 3 GeV = (8.40 ±0.18) 10 3 GeV O 6 = ( ±0.5) 10 3 GeV 6 = ( ) 10 3 GeV 6 O 8 = ( 1.0 ±0.3±0.) 10 GeV 8 = ( 1.0 ±0.4) 10 GeV 8 A. Pich τ and QCD 8

9 O 6,8 with Pinched Weights, ignoring DV s0 s th ds 1 π ImΠ(s) (s s 0) {1, (s + s 0 )} González-Alonso, Pich, Rodríguez-Sánchez A. Pich τ and QCD 9

10 O 6,8 with Pinched Weights, ignoring DV s0 s th ds 1 π ImΠ(s) (s s 0) {1, (s + s 0 )} González-Alonso, Pich, Rodríguez-Sánchez Result with DV fully taken into account Pinched weights suppress very efficiently duality violation effects Big reduction of errors mainly due to short-distance constraints which overcome the large data uncertainties at large s 0 values A. Pich τ and QCD 10

11 Comparison with Previous Works González-Alonso, Pich, Rodríguez-Sánchez 10 3 L eff C87 eff 10 3 O 6 10 O 8 Reference Comments (GeV ) (GeV 6 ) (GeV 8 ) 6.45 ± ± ± 3.3 BPDS 06 ALEPH 05 + DV= ASS 08 ALEPH 05 + DV= ± ± 0.14 GPP 08 ALEPH 05 + DV=0 6.44± ± ± ±0.5 GPP 10 ALEPH 05+DV V A 6.45± ± ± ±0.5 Boito 1 OPAL 99+DV V/A 6.50 ± ± ± 0.5 DHSS 15 ALEPH 14 + DV=0 6.45± ± ± ±0.6 Boito 15 ALEPH 14+DV V/A 6.4± ± this work OPAL 99+DV V A 6.48± ± ±0.4 this work ALEPH 14+DV V A A. Pich τ and QCD 11

12 V +A Correlator: R τ (ud) R τ = Γ[τ ν τhadrons] Γ[τ ν τe ν e] = R τ,v +R τ,a +R τ,s m τ = 1π 0 ds m τ ( 1 s ) [( 1+ s ) ImΠ (0+1) (s) s ] ImΠ (0) (s) mτ mτ mτ ( ) Π (J) (s) V ud Π (J) ud,v (s) + Π(J) ud,a (s) + V us Π (J) us,v+a (s) Davier et al, (v 1 +a 1 )(s) 3.5 ALEPH Perturbative QCD (massless) Parton model prediction (v 1 -a 1 )(s) 3.5 ALEPH Perturbative QCD/parton model v 1 = πimπ (1) ud,v (s) a 1 = πimπ (1) ud,a (s) s (GeV ) s (GeV ) A. Pich τ and QCD 1

13 A R τ = N C S EW (1+δ P +δ NP ) Perturbative (m q =0) d (01) K0 K1 1, K , K , K (Baikov-Chetyrkin-Kühn 08 Le Diberder- Pich 9 P n1 K n ( n) A ( ) a a 7a s 1 ( ) ( ) K s s s s ds n 4 n0 3 4 a n ( n) 1 dx 3 4 s( s) n s x x x a a ) x 1 s m ( ) (1 ) ; ( / i x n A. Pich τ and QCD 13

14 A R τ = N C S EW (1+δ P +δ NP ) Perturbative (m q =0) d (01) K0 K1 1, K , K , K (Baikov-Chetyrkin-Kühn 08 Le Diberder- Pich 9 P n1 K n ( n) A ( ) a a 7a s 1 ( ) ( ) K s s s s ds n 4 n0 3 4 a n ( n) 1 dx 3 4 s( s) n s x x x a a ) x 1 s m ( ) (1 ) ; ( / i x n Power Corrections Braaten-Narison-Pich 9 C O (0+1) n n OPE ( s) n n ( s) C4 O4 0sG G 0 1 C O C O 3 n n NP dx (1 3x x ) x n i n ( xm ) m m 6 V A Suppressed by m [additional chiral suppression in C6 O 6 ] A. Pich τ and QCD 13 C O

15 Perturbative Uncertainty in α (Ñτ ) d 1 K Kn n rn Kngn (01) n ( n) n s ( s) ( ) n a s P A ( s) r a ds 4 n0 n1 n0 CIPT FOPT ( n) 1 dx 3 4 n n A ( s) (1 x x x ) a ( xm ) a ; a ( ) / x 1 s m i x n K n g n r n The dominant corrections come from the contour integration Le Diberder- Pich 199 Large running of s along the circle s = m e i, [0, ] A. Pich τ and QCD 14

16 R τ suitable for a precise α s determination Im(s) mτ Re(s) [ ] R τ = 6πi (1 x) (1+x)Π (0+1) (xmτ) x Π (0) (xmτ) x =1 Π (J) J (s) Π(J) J (s)ope = D O (J) D,J ( s) D/ R τ large enough to safely use the OPE OPE only valid away from real axis: (1 x) pinched at s = m τ m u,d = 0 sπ (0) = 0 R τ,v+a = 6πi (1 3x +x 3 ) Π (0+1) ud,v+a (xm τ) x =1 δ NP 1/m τ 6 Strong suppression of non-perturbative effects D = 6 OPE contributions have opposite sign for V & A. Cancellation δ NP can be determined from data A. Pich τ and QCD 15

17 α s determination with ALEPH-like fit Rodríguez-Sánchez, A.P. ω kl (s) = ( 1 s ) +k ( ) l ( s 1+ s ) mτ mτ mτ (k,l) = (0,0) α s,o 6V/A,O 8 V/A (k,l) = (1,0) α s, a sgg,o 6 V/A,O 8V/A,O 10 V/A (k,l) = (1,1) α s, a sgg,o 6 V/A,O 8V/A,O 10 V/A,O 1V/A (k,l) = (1,) α s,o 6V/A,O 8 V/A,O 10V/A,O 1 V/A,O 14V/A (k,l) = (1,3) α s,o 8V/A,O 10 V/A,O 1V/A,O 14 V/A,O 16 V/A Channel α s(m τ)/π αs π GG O 6 O 8 (10 3 GeV 4 ) (10 3 GeV 6 ) 10 3 GeV 8 ) V (FOPT) V (CIPT) A (FOPT) A (CIPT) V+A (FOPT) V+A (CIPT) Good agreement with ALEPH A. Pich τ and QCD 16

18 1 Fit one more condensate to test stability/uncertainties Channel α s(mτ )/π αs π GG O 6 O 8 O 10 (10 3 GeV 4 ) (10 3 GeV 6 ) 10 3 GeV 8 ) (10 3 GeV 10 ) V (FOPT) V (CIPT) A (FOPT) A (CIPT) V+A (FOPT) V+A (CIPT) ± Good stability of α s with respect to previous fit Larger variation in condensates values and increased errors Take central values from first fit, adding differences as errors A. Pich τ and QCD 17

19 α s determination with ALEPH-like fit Rodríguez-Sánchez, A.P. ω kl (s) = ( 1 s ) +k ( ) l ( s 1+ s ) mτ mτ mτ Channel α s(m τ)/π αs π GG O 6 O 8 (10 3 GeV 4 ) (10 3 GeV 6 ) 10 3 GeV 8 ) V (FOPT) V (CIPT) A (FOPT) A (CIPT) V+A (FOPT) V+A (CIPT) α s(m τ) FOPT = α s(m τ) CIPT = α s (m τ ) = A. Pich τ and QCD 18

20 Experiment vs. (pinched) Perturbation Theory (only) ω (n0) (s = s 0 x) = (1 x) n, α s(m τ ) = Rodríguez-Sánchez, A.P. A. Pich τ and QCD 19

21 Playing with the s 0 dependence Rodríguez-Sánchez, A.P. ω (0) (s = s 0 x) = (1 x) O 4, O 6 ω (1) (s = s 0 x) = (1 x) (1 +x) O 6, O 8 ω () (s = s 0 x) = (1 x) (1 +x +3x ) O 8, O 10 Fit to the (9) s 0 > GeV points. One moment only (avoid correlations) V+A channel Moment Method α s(mτ )/π Lower-D Condensate Higher-D Condensate (10 3 GeV D ) (10 3 GeV D ) A ω(0) (s 0 ) FOPT A ω(0) (s 0 ) CIPT ± ±1 A ω(1) (s 0 ) FOPT 0.10 ± ± A ω(1) (s 0 ) CIPT ± ±1 1± A ω() (s 0 ) FOPT ±0.003 ±3 0± A ω() (s 0 ) CIPT ± ± +3 A. Pich τ and QCD 0

22 Rodríguez-Sánchez, A.P. A ω(0) (s 0 ) CIPT Fit to last n = 4,,8 s 0 points vs. starting s 0 fitted value A. Pich τ and QCD 1

23 α s (m τ) = from V +A BUT... Bad quality fit (χ min /d.o.f.) Much worse behaviour in separate V & A channels Fitting the s 0 dependence removes pinching: Fitting m points of A (n0) (s 0 ) is equivalent to a fit of { A (n0) (s 0 ),A (n 10) (s 0 ),...,A (00) (s 0 ),ImΠ(s 0 ), d ImΠ(s } 0),..., dm n 1 ImΠ(s 0 ) ds 0 ds 0 Local duality assumed! Violations of Duality A. Pich τ and QCD

24 Boito et al. approach Ansatz: ρ DV V/A (s) = e (δ V/A +γ V/A s) sin(α V/A +β V/A s) s > s th 1.5GeV ω(x) = 1: Fit s 0 dependence of A (00) (s 0 ) = s 0 ds s 0 ImΠ(s) n-point fit {A (00) (s 0),ImΠ(s 0 + s 0),,ImΠ(s 0 +(n 1) s 0)} n 1 points dedicated just to fit the spectral function Uncertainties too large in A channel. Only V channel fitted FOPT Rodríguez-Sánchez, A.P. A. Pich τ and QCD 3

25 Boito et al. approach (cont) Fit s 0 dependence of A (00) (s 0 ) = s 0 ds s 0 ImΠ V (s) Rodríguez-Sánchez, A.P. FOPT Problematic fit. Model dependence. Instabilities. Very low p-value α s (m τ) =? A. Pich τ and QCD 4

26 Many more fits, tests, plots... Consistent value of α s (m τ) obtained (no surprises) Detailed analysis will be released soon A. Pich τ and QCD 5

27 Summary Many interesting test of QCD with spectral information Current data has limited accuracy & large correlations Better experimental data (high statistics & precision) needed Π V A allows clean tests of OPE and χpt Precise determination of α from τ decays: Π V+A (Π V,A ) Main limitation: perturbation theory Better understanding of higher orders needed Improved control of δ NP possible with better data A. Pich τ and QCD 6

28 BACKUP A. Pich τ and QCD 7

29 ( n) 1 dx 3 4 n n A ( a ) (1 x x x ) a ( xm ) a ; a ( ) / x 1 s m i x Im (s) m Re (s) (1) A ( a ) a 3 1a 1 a a a 1 a( s) a i a a log s / m 1 i a n n ; 0, FOPT expansion only convergent if < 0.14 (0.11) [at 1 (3) loops] Experimentally FOPT should not be used (divergent series) FOPT suffers a large renormalization-scale dependence (Le Diberder- Pich, Menke) The difference between FOPT and CIPT grows at higher orders A. Pich τ and QCD 8

30 Renormalon Hypothesis: Asymptotics already reached Modelling a better behaved FOPT (Beneke Jamin) o Large higher-order K n corrections could cancel the g n ones Happens in the large- 0 approximation (UV renormalon chain) o D = 4 corrections very suppressed in R n = IR renormalons can do the job (K n - g n ) o No sign of renormalon behaviour in known coefficients n = -1,,3 renormalons + linear polynomial 5 unknown constants fitted to K n (. K 5 = 83 assumed o Borel summation: large renormalon contributions. Smaller Nice model of higher orders. But too many different possibilities (Descotes-Genon Malaescu) A. Pich τ and QCD 9