1. Spectral sum rules and duality violations Maarten Golterman (SFSU) work with Oscar Catà and Santi Peris (BNL workshop Domain-wall fermions at 10 years)

2. Physics from (the OPE of)  LR : 1) In the chiral and large-N c limits and is proportional to the 1/Q 6 coefficient, while is an integral over  LR (Q 2 ). 2) The OPE part of  V (Q 2 )+  A (Q 2 ) “contaminate” the determination of  s from  decays. (Braaten, Narison and Pich)

3. Outline: 1)Relating the OPE to data: what is the problem? 2)Our model for the LR two-point function (N c =  ) 3)Finite N c : including finite widths 4)Testing proposed methods (duality points, pinched weights) 5)Can we do better?

4. Getting OPE coefficients from data: The OPE for  (Q 2 ) =  LR (q 2 = -Q 2 ) is an asymp. expansion for large Q 2  (t) = Im  (t) known from data up to a scale s 0 = m  2 Cauchy’s theorem: (P any polynomial) Re q 2 Idea: substitute  (Q 2 )   OPE (Q 2 ) on the right-hand side (“duality”) Assumption: s 0 already in the asymptotic regime Problem: not valid even for large s 0 near positive real axis! Im q 2

5. Comments: - expectation: duality violations decrease with increasing s 0 (not necessarily true at N c =  !) but: what is the size of the effect at some given s 0 ? - not much known in QCD!  resort to models - our model is not QCD (certainly not at N c =    but gives an idea how large effects can be: don’t ignore, but take as indication of uncertainties! - work in chiral limit

6. Our model at N c =  Infinite Regge-like sum over zero-width resonances: with  (z) = d log  (z) /dz, and setting  = 1 We can calculate everything in terms of F 0 = 0.086, F  = 0.134, F = 0.144, M  = 0.767, M V = 1.49, M A = 1.18,  = 1.28, all in GeV

7. D [0] (s 0 ) D [1] (s 0 ) D [2] (s 0 ) with the duality violations D [n] (s 0 ) defined through There are “duality points” at N c =  (in QCD!), but they are useless: Introduce widths: duality points move differently for different moments; slopes are finite, but very steep.

8. Our model at finite N c (Blok, Shifman and Zhang) Replace -q 2 - i  by z = (-q 2 - i  ) ,  = 1 - a/(  N c ) and  (q 2 ) by Expand in 1/N c  width  n) = aM(n)/N c (Breit-Wigners near poles)  (q 2 ) analytic for all q 2 except cut along the positive real axis (note: no multi-particle continuum)

9. data: Aleph and Opal (pion removed) blue line: model for a = 0.72 (total 7 parameters)

10. This leads to the following estimates for the spectral function: - large N c and large t limits do not commute (at N c = , Im  (t) is sum over Dirac  -functions) - duality violating part Im  (t) missed by OPE; it is exponentially suppressed, but (in model) by exp(-0.9s 0 )  numerically large effects at s 0 = m 2  ?

11. Equations for OPE coefficients: with D [n] (s 0 ) again representing the duality violations, we get (Note: cannot ignore b’s! Come with positive powers of s 0 !) - duality violations (RHS) are exponentially small -- but numerically?  test methods in use on model

12. dashed: OPE, solid: moments of spectral function

13. Tests: 1)Finite-energy sum rules (Peris et al., Bijnens et al.) determine duality point s 0 * from M 0,1 (s 0 )  0, and predict s 0 * = 1.472 GeV 2 : A 6 = -4.9 * 10 -3 GeV 6, A 8 = 9.3 * 10 -3 GeV 8 s 0 * = 2.363 GeV 2 : A 6 = -2.0 * 10 -3 GeV 6, A 8 = -1.6 * 10 -3 GeV 8 exact: A 6 = -2.8 * 10 -3 GeV 6, A 8 = 3.4 * 10 -3 GeV 8 Note: 2nd duality point only sets M 0 (s 0 ) = 0, not M 1 (s 0 ) b 6 s 0 * = -1.4 * 10 -3 GeV 8 at 2nd duality point! (Smaller in QCD?)

14. 2)Pinched weights (e.g. Cirigliano et al., ‘05) fit OPE coefficients to moments obtained with P 1 = (1 - 3t/s 0 ) (1 - t/s 0 ) 2, P 2 = (t/s 0 ) (1 - t/s 0 ) 2 and fit over range 1.5 GeV 2 < s 0 < 3.5 GeV 2 find: A 6 = -3.8 * 10 -3 GeV 6, A 8 = 6.5 * 10 -3 GeV 8 exact: A 6 = -2.8 * 10 -3 GeV 6, A 8 = 3.4 * 10 -3 GeV 8 3)Minimal hadronic ansatz (MHA) (de Rafael et al.) (with one vector and one axial vector) find: A 6 = -3.6 * 10 -3 GeV 6, A 8 = 5.4 * 10 -3 GeV 8 -orders of magnitude ok -quantitavely poor -- e.g. ~100% errors in Q 8 WME

15. Can we do better? try model the duality violations: fit to (range 1.5 < s 0 < 3.5 GeV 2 ) find  = 0.026,  = 0.591 GeV -2,  = 3.323,  = 3.112 GeV -2 with this, predict duality points for higher moments, find s 0 * = 2.350 GeV 2 for n = 2, s 0 * = 2.307 GeV 2 for n = 3, etc. and A 6 = -2.5 * 10 -3 GeV 6, A 8 = 3.3 * 10 -3 GeV 8 (exact: A 6 = -2.8 * 10 -3 GeV 6, A 8 = 3.4 * 10 -3 GeV 8 ) order 10% errors up to A 16  worth trying in QCD?

16. Conclusions Semi-realistic model suggests that duality violations cannot be ignored. (large effect also with higher duality points, pinched weights, etc.) Over a range duality violations can be successfully modeled  try to do the same thing in QCD! (take result as systematic error coming from duality violations) Need to assume 1) data below s = m   in asymptotic regime; 2) reasonable model in this regime It would be interesting to compute  V,A (Q 2 ) on the lattice, for instance with staggered sea and valence DWF. (test OPE effects in determination of  s from  decay?)