Hadronic tau decays and the strong coupling

Transcription

1 Hadronic tau decays and the strong coupling Diogo Boito, Maarten Golterman, Kim Maltman, San;ago Peris (also: Oscar Catà, MaBhias Jamin, Andy Mahdavi, James Osborne) Mini Workshop on Tau Physics, Cinvestav, Mexico City May 22-23,

2 Values for the strong coupling from tau decays (ALEPH data): Pich, Rodríguez-Sánchez (PRD94 ( 16) ): Davier et al. (EPJC74 ( 14) 2803): Boito et al. (PRD91 ( 15) , 95 ( 17) ): (including OPAL data: s (m 2 )=0.309(9) ) s (m 2 )=0.328(12) s (m 2 )=0.332(12) s (m 2 )=0.301(10) PDG (2016): At the Z mass (from tau): laace: world average w/o tau: s (m 2 Z)=0.1192(18) s (m 2 )=0.325(15) s (m 2 Z)=0.1184(11) s (m 2 )=0.315(9) s (m 2 Z)=0.1179(11) s (m 2 )=0.314(9) Can we do beber? Note disagreements well outside errors! Technical note: these are averages between CIPT and FOPT values. 2

3 Experimental data vector non-strange axial non-strange ALEPH, Davier et al. (EPJC74 (2014) 2803) 3

4 Experimental data vector non-strange axial non-strange OPAL, Ackerstaff et al. (EPJC7 (1999) 571) 4

5 decays leptonic: hadronic: + R ud =! apple hadrons ud =3S EW V ud 2 1+ s! e e +5.2 s use this to determine s (m ) from non-strange decays see!! pions, not! jets = (770), (1450), (1700) (and others: incl. axial, kaons, ) rela;on with perturba;ve regime? 5

6 (! hadrons ud ) Z s0 R ud ( ) = 12 S EW V ud 2 0 ds 1 s (op;cal theorem) s Im (s) } (s) = = (s) s =(q W ) 2 0 apple s apple = m 2 depending on how much momentum carries away 6

7 (s) = 1 Im (s) inclusive non-strange spectral func;on, measured in tau decays (ALEPH, OPAL) (z = q 2 = s) current two point func;on, gives access to strong coupling Cauchy: Z s0 0 z = s = q 2 ds w(s) (s) = plane 1 2 i polynomial weight I dz w(z) (z) z = (z) = pert (z)+ nonpert OPE (z)+ nonpert DV (z) s resonances OPE (q 2 )= C 4 q 4 C 6 q 6 + C 8 q

8 (s) = 1 Im (s) OPE inclusive non-strange not valid!! spectral func;on, measured in tau decays (ALEPH, OPAL) DeVil of Duality Viola;ons: (z = q 2 = s) current two point func;on, gives access to strong coupling resonances Cauchy: Z s0 0 z = s = q 2 ds w(s) (s) = plane 1 2 i polynomial weight I dz w(z) (z) z = (z) = pert (z)+ nonpert OPE (z)+ nonpert DV (z) s resonances OPE (q 2 )= C 4 q 4 C 6 q 6 + C 8 q

9 V+A non-strange spectral func;on (Davier et al., 2014, ALEPH) V+A (s) perturba;on theory + OPE s 9

10 Blow up of large s region: 2 2 V+A (s) pert. part dependent on s s[gev 2 ] Duality viola;ons resonance effects are not small! 10

11 Two approaches to non-perturba;ve contamina;on ALEPH (Davier et al.), OPAL, Pich & Rodriguez ( Truncated-OPE approach ): Ignore Duality Viola;ons, but suppress dangerous region by pinching : choose only higher-order polynomials with mul;ple zeroes at s = = m 2 Fit s (m 2 ) and C 4, C 6, C 8, set higher orders and DVs to zero Problem: inconsistent treatment of the OPE Boito et al. ( DV-model approach ): Treat OPE consistently, keep only low orders: choose simple polynomials s Model DVs with ansatz DV (s) =e sin ( + s) Vary between s min 1.55 GeV 2 and m 2 Fit s (m 2 ) and C 6, C 8,,,, Problem: need to model DVs 11

12 Truncated-OPE approach: take = m 2 and weights ( x z/ ) w 00 (x) =(1 x) 2 (1 + 2x) w 10 (x) =(1 x) 3 (1 + 2x) w 11 (x) =(1 x) 3 (1 + 2x)x w 12 (x) =(1 x) 3 (1 + 2x)x 2 w 13 (x) =(1 x) 3 (1 + 2x)x 3 (ALEPH) Assume: C 10 = C 12 = C 14 = C 16 =0 and Duality Viola;ons negligible fit four parameters ( s, C 4, C 6, C 8 ) to five data spectral integrals I 1 However: dz z n C 2k 2 i z k = C 2(n+1) k,n+1 C 16 need OPE coefficients up to resonance oscilla;ons around OPE clearly visible in V+A spectral func;on! 12

13 This approach can be tested: fake data Start from model with, by construc;on, lower s (m 2 )=0.312 (CIPT) and non-negligible DVs, compa;ble with the experimental spectral func;on. Test Truncated-OPE approach V (s) A (s) s s Generate fake data from this model (using real-data covariances). Perform Truncated-OPE type fits on these fake data. Compare the parameter values ( s (m 2 ) ) obtained from these fits to the input value, s (m 2 )=

14 Fake data: (fake above s =1.55 GeV 2 ) Real data: s (GeV 2 ) (V+A spectral func;ons)

15 Results of this test: ALEPH s (m 2 ) C 4,V +A (GeV 4 ) C 6,V +A (GeV 6 ) C 8,V +A (GeV 8 ) 2 /dof true value fake data fit 0.334(3) (4) (3) (4) 0.95/1 optimal s (m 2 ) C 6,V +A (GeV 6 ) C 8,V +A (GeV 8 ) C 10,V +A (GeV 10 ) 2 /dof true value fake data fit 0.334(4) (4) (5) (3) 0.92/1 (CIPT, sta;s;cal errors only) Truncated-OPE approach gets it wrong (similar conclusion for FOPT), 2 despite good : systema;c overes;ma;on of big difference in behavior of OPE s 15

16 DV-model approach (Boito et al.): Use simple weights: w 0 (x) =1, w 2 (x) =1 x 2, w 3 (x) =(1 x) 2 (1 + 2x) hence C 6, C 8 only OPE coefficients needed No abempt to suppress DVs, hence use ansatz DV (s) =e s sin ( + s) for the DV part of the spectral func;on, and fit,,, To do this, vary s min apple apple m 2, make use of the data! Fit determines s min 1.55 GeV 2 16

17 Example: simplest fit, vector channel w 0 (x) = HGeV 2 L HGeV 2 L Blue: FOPT s (m 2 )=0.296(11) Red: CIPT s (m 2 )=0.310(14) Black: OPE contribu;on only 17

18 Example: 3-weight Blue: FOPT HGeV 2 L HGeV 2 L w 0,2,3 (x), vector channel fit s (m 2 )=0.296(10) HGeV 2 L Red: CIPT HGeV 2 L s (m 2 )=0.310(14) Black: OPE contribu;on only Vector as good as Vector+Axial! 18

19 This approach passes many tests (here V+A): Check dependence, should work above 1.5 GeV 2 ALEPH moments for Boito et al. (only w 00 used in fits; = m 2 minus diffs.): OPE, spectral integrals τ 0 spectral integrals OPE+DV integrals OPE, spectral integrals [GeV 2 ] τ 0 spectral integrals OPE+DV integrals w 5e w w 11 OPE, spectral integrals [GeV 2 ] spectral integrals OPE+DV integrals [GeV 2 ] OPE, spectral integrals 0-5e τ 0 spectral integrals OPE+DV integrals [GeV 2 ] 0-2e-05-4e-05 w 12 w 13-6e-05-8e-05 spectral integrals OPE+DV integrals [GeV 2 ] 19

20 Same tests for truncated-ope approach (again, V+A): Check dependence, should work above 2 GeV 2 ALEPH moments for P&R (all used in fit): OPE, spectral integrals τ 0 spectral integrals OPE integrals OPE, spectral integrals [GeV 2 ] τ 0 spectral integrals OPE integrals w 5e w w [GeV 2 ] OPE, spectral integrals spectral integrals OPE integrals [GeV 2 ] OPE, spectral integrals 0-5e τ 0 spectral integrals OPE integrals [GeV 2 ] 0-2e-05-4e-05 w 12 w 13-6e-05-8e-05 spectral integrals OPE integrals [GeV 2 ] 20

21 Model dependence of DV-model approach: Depends on a model for the effect of Duality Viola;ons = resonance effects Ingredients: Regge behavior of spectrum for s : s min M 2 (n) =M 2 (0) + n, n=0, 1, 2,..., 2 QCD 1 GeV 2 Large : N c (n) / M(n)/N c Model sa;sfying these constraints and analy;city (Blok, Shifman & Zhang 98, Bigi, Shifman, Uraltsev & Vainshtein 99 Catà, Golterman & Peris 05, 08) More general arguments: in prepara;on (Boito et al.) Important to test this with data! 21

22 FOPT and CIPT FOPT = fixed order perturba;on theory CIPT = contour improved perturba;on theory (Pivovarov 92, Le Diberder & Pich 92) Different ways of par;ally resumming perturba;on theory on the theory side of the sum rules. Differences: (5.1%) (Davier et al.) (4.9%) (Pich & Rodriguez) (4.7%) (Boito et al.) Renormalon analysis somewhat favors FOPT (Beneke & Jamin 08, Beneke, Boito & Jamin 12) 22

23 Outlook Recent theory advances may help extract s (m 2 ) from non-perturba;ve hadronic -decay data to higher precision, but need to understand the physics of resonances beber precision tests of the DV ansatz? Important! Belle and Belle-II will have thousands ;mes more pairs than ALEPH/OPAL! Focus on vector channel less clear that s apple m 2 is already asympto;c in axial channel; reduce errors especially for s>2.5 GeV 2 Will improving the! 4 non-strange vector decay already help? (Large errors in the region s>1.5 GeV 2 dominated by this decay) Open theory ques;on: CIPT vs. FOPT? (cf. Boito, Beneke & Jamin, 12) Comparison with value from other high-precision value, at common scale? 23

24 BACKUP SLIDES 24

25 Why does the Truncated-OPE approach get it wrong? Rely on uncontrolled assump;on about the OPE in higher orders. Assume that duality viola;ons (resonance effects) can be neglected, at least in V+A, without tes2ng this A V+A(s) PT 0.9 V V+A s 25

26 Why does the Truncated-OPE approach get it wrong? Rely on uncontrolled assump;on about the OPE in higher orders. Assume that duality viola;ons (resonance effects) can be neglected, at least in V+A, without tes2ng this. Poten;ally large effect at = m 2! Not excluded by data. V+A(s) PT A V V+A s 26