Spectral sum rules and duality violations

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-Nc limits and is proportional to the 1/Q6 coefficient, while is an integral over LR(Q2) . 2) The OPE part of V(Q2)+A(Q2) “contaminate” the determination of s from  decays. (Braaten, Narison and Pich)

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

4. Im q2 Getting OPE coefficients from data: The OPE for (Q2) = LR(q2 = -Q2) is an asymp. expansion for large Q2 (t) = Im (t) known from data up to a scale s0 = m2 Cauchy’s theorem: (P any polynomial) Re q2 Idea: substitute (Q2) OPE(Q2) on the right-hand side (“duality”) Assumption: s0 already in the asymptotic regime Problem: not valid even for large s0 near positive real axis!

5. Comments: • expectation: duality violations decrease with increasing s0 • (not necessarily true at Nc = !) • but: what is the size of the effect at some given s0? • not much known in QCD!  resort to models • our model is not QCD (certainly not at Nc =  •  • 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 Nc=  Infinite Regge-like sum over zero-width resonances: with (z) = d log (z) /dz , and setting  = 1 We can calculate everything in terms of F0 = 0.086, F= 0.134, F = 0.144, M = 0.767, MV= 1.49, MA = 1.18,  = 1.28, all in GeV

7. D[0](s0) D[1](s0) D[2](s0) with the duality violations D[n](s0) defined through There are “duality points” at Nc = (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 Nc (Blok, Shifman and Zhang) Replace -q2 - i by z = (-q2 - i) ,  = 1 - a/(Nc) and (q2) by Expand in 1/Nc width n) = aM(n)/Nc (Breit-Wigners near poles) (q2) analytic for all q2 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 Nc and large t limits do not commute • (at Nc = , Im (t) is sum over Dirac -functions) • duality violating part Im (t) missed by OPE; • it is exponentially suppressed, but (in model) by exp(-0.9s0) •  numerically large effects at s0 = m2 ?

11. Equations for OPE coefficients: with D[n](s0) again representing the duality violations, we get • (Note: cannot ignore b’s! Come with positive powers of s0!) • duality violations (RHS) are exponentially small -- but numerically? • test methods in use on model

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

13. Tests: Finite-energy sum rules (Peris et al., Bijnens et al.) determine duality point s0* from M0,1(s0)  0, and predict s0* = 1.472 GeV2 : A6 = -4.9 * 10-3 GeV6, A8 = 9.3 * 10-3 GeV8 s0* = 2.363 GeV2 : A6 = -2.0 * 10-3 GeV6, A8 = -1.6 * 10-3 GeV8 exact: A6 = -2.8 * 10-3 GeV6, A8 = 3.4 * 10-3 GeV8 Note: 2nd duality point only sets M0(s0) = 0, not M1(s0) b6s0* = -1.4 * 10-3 GeV8 at 2nd duality point! (Smaller in QCD?)

14. Pinched weights (e.g. Cirigliano et al., ‘05) fit OPE coefficients to moments obtained with P1 = (1 - 3t/s0) (1 - t/s0)2, P2 = (t/s0) (1 - t/s0)2 and fit over range 1.5 GeV2 < s0 < 3.5 GeV2 find: A6 = -3.8 * 10-3 GeV6, A8 = 6.5 * 10-3 GeV8 exact: A6 = -2.8 * 10-3 GeV6, A8 = 3.4 * 10-3 GeV8 • Minimal hadronic ansatz (MHA) (de Rafael et al.) (with one vector and one axial vector) find: A6 = -3.6 * 10-3 GeV6, A8 = 5.4 * 10-3 GeV8 • orders of magnitude ok • quantitavely poor -- e.g. ~100% errors in Q8 WME

15. Can we do better? try model the duality violations: fit to (range 1.5 < s0 < 3.5 GeV2) find  = 0.026,  = 0.591 GeV-2,  = 3.323,  = 3.112 GeV-2 with this, predict duality points for higher moments, find s0* = 2.350 GeV2 for n = 2 , s0* = 2.307 GeV2 for n = 3 , etc. and A6 = -2.5 * 10-3 GeV6, A8 = 3.3 * 10-3 GeV8 (exact: A6 = -2.8 * 10-3 GeV6, A8 = 3.4 * 10-3 GeV8) order 10% errors up to A16 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(Q2)on the lattice, for instance with staggered sea and valence DWF. (test OPE effects in determination of s from  decay?)