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Analyzing Duality Violations in Spectral Functions and Their Implications for QCD

This work explores duality violations in the context of spectral functions, particularly using the Operator Product Expansion (OPE). By examining the behavior of the longitudinal and transverse correlators in both chiral and large-Nc limits, we derive relationships that highlight potential issues in determining αs from τ decays. Our model employs asymptotic expansions and finite Nc effects, offering insight into duality points and width influences. We emphasize the significance of modeling duality violations, suggesting their essential consideration in QCD analyses and highlighting the substantial uncertainties they introduce.

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Analyzing Duality Violations in Spectral Functions and Their Implications for QCD

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  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?)

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