# Presentation: 770495

## 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

- 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)

**Notes:**
J_mu: vector and axial-vector isospin 1 currents

## Outline:

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

**Notes:**
Approach: get OPE from hadronic tau decay data

## 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

**Notes:**
P any polynomial

## 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

## 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 *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

**Notes:**
These parameters give the correct parton model log for V,A, have no 1/Q^2 terms, and no 1/Q^4 in V-A.
Get “canonical” values for condensates, including gluon condensate.

##             D[0](s0)                            D[1](s0)                               D[2](s0)

-             *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.

**Notes:**
Note vertical scales!

## Our model at finite Nc                                 (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)

**Notes:**
Lambda=1

## data:   Aleph and Opal  (pion removed)

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

**Notes:**
With these parameters no 1/Q^2 and 1/Q^4 terms (Weinberg sum rules)

## 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(-0.9*s*0)
-     numerically large effects at *s*0 = *m*2 ?

**Notes:**
b_6 suppressed in large N, alpha_s suppressed in QCD

## 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

**Notes:**
In QCD b_6 small in practice, but not known about higher b’s

##           dashed:  OPE ,     solid:  moments of spectral function

-           dashed:  OPE ,     solid:  moments of spectral function

**Notes:**
Define A_6,8 from previous slide

## Tests:

- 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 GeV2 :  *A*6 = -4.9 * 10-3 GeV6 ,        *A*8 = 9.3 * 10-3 GeV8
*s*0* = 2.363 GeV2 :  *A*6 = -2.0 * 10-3 GeV6 ,        *A*8 = -1.6 * 10-3 GeV8
- exact:                     *A*6 = -2.8 * 10-3 GeV6 ,        *A*8 = 3.4 * 10-3 GeV8
- Note:    2nd duality point only sets *M*0(*s*0) = 0, not *M*1(*s*0)
-              *b*6*s*0* = -1.4 * 10-3 GeV8  at 2nd duality point!  (Smaller in QCD?)

## Pinched weights                                            (e.g. Cirigliano et al., ‘05)

-       fit OPE coefficients to moments obtained with
-       *P*1 = (1 - 3*t*/*s*0) (1 - *t*/*s*0)2  , *P*2 = (*t*/*s*0) (1 - *t*/*s*0)2
-       and fit over range 1.5 GeV2 < *s*0 < 3.5 GeV2
-          find:            *A*6 = -3.8 * 10-3 GeV6 ,        *A*8 = 6.5 * 10-3 GeV8
-         exact:         *A*6 = -2.8 * 10-3 GeV6 ,        *A*8 = 3.4 * 10-3 GeV8
- Minimal hadronic ansatz  (MHA)                               (de Rafael et al.)
-          (with one vector and one axial vector)
-       find:           *A*6 = -3.6 * 10-3 GeV6 ,        *A*8 = 5.4 * 10-3 GeV8
- orders of magnitude ok
- quantitavely poor  -- e.g. ~100% errors in *Q*8 WME

**Notes:**
Authors quote large errors on A_8, but suspicion that this is because of data -- 
they would claim smaller errors if better data were available!
MHA: parameters from absence of 1/Q^2,4, plus EM pion mass diff plus L_10.

## Can we do better?

- try model the duality violations:
- fit to (range 1.5 < *s*0 < 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  *s*0* = 2.350 GeV2  for  *n* = 2 ,  *s*0* = 2.307 GeV2  for  *n* = 3 , etc.
- and                  *A*6 = -2.5 * 10-3 GeV6 ,        *A*8 = 3.3 * 10-3 GeV8
- (exact:             *A*6 = -2.8 * 10-3 GeV6 ,        *A*8 = 3.4 * 10-3 GeV8)
- order  10% errors up to *A*16       worth trying in QCD?

## 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?)