Spectral sum rules and duality violations. Maarten Golterman (SFSU) work with Oscar Catà and Santi Peris (BNL workshop Domain-wall fermions at 10 years). Physics from (the OPE of) LR :. 1) In the chiral and large- N c limits.
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Spectral sum rules and duality violations
Maarten Golterman (SFSU)
work with Oscar Catà and Santi Peris
(BNL workshop Domain-wall fermions at 10 years)
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)
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?
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 = m2
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!
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
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.
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)
data: Aleph and Opal (pion removed)
blue line: model for a = 0.72 (total 7 parameters)
This leads to the following estimates for the spectral function:
with D[n](s0) again representing the duality violations, we get
dashed: OPE , solid: moments of spectral function
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?)
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
(with one vector and one axial vector)
find: A6 = -3.6 * 10-3 GeV6, A8 = 5.4 * 10-3 GeV8
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?
(large effect also with higher duality points, pinched weights, etc.)
try to do the same thing in QCD!
(take result as systematic error coming from duality violations)
2) reasonable model in this regime
for instance with staggered sea and valence DWF.
(test OPE effects in determination of s from decay?)