Spectral sum rules and duality violations
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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). 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

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Spectral sum rules and duality violations

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

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)


Outline

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?


Spectral sum rules and duality violations

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!


Spectral sum rules and duality violations

  • 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


Our model at n c

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


Spectral sum rules and duality violations

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.


Our model at finite n c blok shifman and zhang

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)


Spectral sum rules and duality violations

data: Aleph and Opal (pion removed)

blue line: model for a = 0.72 (total 7 parameters)


Spectral sum rules and duality violations

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 ?


Equations for ope coefficients

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


Spectral sum rules and duality violations

dashed: OPE , solid: moments of spectral function


Tests

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?)


Spectral sum rules and duality violations

  • 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


Can we do better

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?


Conclusions

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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