WHAT IS RESONANCE SATURATION ?
SANTI PERIS
(IFAE - UA Barcelona)
EuroFlavor07, Orsay, Nov. 2007
Based on work done in collaboration with
Pere Masjuan
arXiv:0704.1247 [hep-ph] pub. in JHEP
WHAT IS RESONANCE SATURATION ? – p.1/15
Introduction
fπ2
LEC
z}|{
µ
†
µ
†
• Lχ =
Tr Dµ U D U + L1 Tr Dµ U D U Tr Dµ U D U + . . .
{z
}
|4
{z
} |
µ
†
O(p4 )+O(p6 )+...
O(p2 )

O(p4 ) →



 O(p6 ) →
• Proliferation of LECs:
8
O(p
)→



..

.
∼ 101
∼ 102
∼ 103 ?
• LECs are indispensable to make predictions.
• PHYSICS (LECs) = PHYSICS (short distances).
HOW ?
WHAT IS RESONANCE SATURATION ? – p.2/15
Resonance Saturation (aka VMD + extensions)
< π(p′ )|Vµ |π(p) >= F (Q2 ) (p′ − p)µ
2
F (Q )
=
1−Q
2
∞
X
R
2
CR
2
Q 2 + MR
αs (Q2 )
+ ...
Q2
≈
16πFπ2
≈
Q2
1 − 2L9 2 + ...
Fπ
≃
MV2
Q2 + MV2
,
(Nc → ∞,
−Q2 = (p′ − p)2
one subtr.,
meromorphic)
(Q2 large, PQCD)
(Q2 small)
(M HA)
,
1 Fπ2
−3
L9 ≡
∼
7
×
10
= L9 (Mρ )
2 MV2
0.9
Spacelike
π
0.8
2
Q |Fπ(Q )| (GeV )
P. Zweber
Sakurai ’69
Ecker et al. ’89
Donoghue et al. ’89
Moussallam ’97
------------Knecht, de Rafael ’98
Perrottet, de Rafael, S.P. ’98
0.7
0.6
VDM
2
0.5
0.4
2
0.3
0.2
PQCD(asy)
0.1
0
0
2
4
6
2
8
10
12
14
2
Q (GeV )
WHAT IS RESONANCE SATURATION ? – p.3/15
2
F (Q )
=
1−Q
∞
X
2
R
(⋆)
≈
2
CR
2
Q 2 + MR
MV2
Q2 + MV2
• What is this approximation (⋆) ?
(1,2,....∞)
• Where in the complex Q2 plane does (⋆) converge ?
• How are the poles/residues of the approx. (⋆) related to the physical
counterparts ?
WHAT IS RESONANCE SATURATION ? – p.4/15
High Energy: WSRs
< V V − AA > with regulator cutoff Λ:
2
2
Q ΠLR (Q ) = lim


Λ→∞ 
NV (Λ)
X
V
NA (Λ)
−F02
−Q
2
X
A
NV (Λ)


X
FA2
FV2
2
+Q
2
2
2 + M2 
Q + MA
Q
V
V
NA (Λ)
X
FV2
FA2
Λ2
1
∼
∼
log
+
O(
)
2 + M2
2
2
Q2 + MV2
Q
Q
Λ
A
A
Universality ⇒ ΠV,A (Q2 ) invariant under Λ → Λ + a ,
,
(Λ2 ≫ Q2 → ∞)
a≪Λ
M
+a
V
A
spectra
NV (Λ)
→
NV (Λ + a) = NV (Λ)+1
NA (Λ)
→
NA (Λ + a) = NA (Λ)
WHAT IS RESONANCE SATURATION ? – p.5/15
High Energy: WSRs
(and II)


NA (Λ)
NV (Λ)
X
X

2
2
−F02 −
F
+
F
A
V


A
V
|
{z
}
Q2 ΠLR (Q2 )|Q2 →∞ ∼ lim
Λ→∞
W SR1
−

1 
Q2
|
NA (Λ)
X

NV (Λ)
2
FA2 MA
−
A
X
FV2 MV2  +
V
{z
}
W SR2



1

O( 4 ) 

Q
| {z } 
log Q2 ?
W SRs not invariant under NA → NA and NV → NV +1 (Golterman, S.P. ’03)
Regge =⇒
MV2n ,An ∼ n
W SR1
∼
FV2n ,An ∼ F 2
,

lim 
Λ→∞
NA (Λ)
X
A
(parton model O.K.)
NV (Λ)
1−
X
V

1 =
??
◮ Physical masses and decay constants do not obey WSRs.
WHAT IS RESONANCE SATURATION ? – p.6/15
Low Energy: L8
Z
dt
< SS − P P > : Π’S−P (Q ) = lim
ρS (t) − ρ’P (t)
Λ→∞ 0
t + Q2
Z Λ2
Z Λ2
dt dt
∼
lim
ρS (t) − ρ’P (t) = lim
lim ρS (t) − ρ’P (t)
Λ,Nc →∞ 0
Λ→∞
N
t
t c →∞
0


NS (Λ)
N
(Λ)
P
2
X F 2 (n)
X
F
(n) 
S
P
−
∼ lim 
Λ→∞
MS2 (n)
MP2 (n)
n
n
2
L8
Λ2
2
Regge ⇒ MS,P
(n) ∼ n and
2
2
FS,P
(n) ∼ MS,P
(n) (parton model O.K.)

∴ L8 ∼ lim 
Λ→∞
NS (Λ)
X
n
NP (Λ)
1−
X
n

1 =
??
◮ no decoupling at large resonance masses.
◮ physical masses and decay constants do not obey L8 sum rule.
L8 ∼lim Q
Q2 →∞



2
lim
N
,N

 2S P →∞2
M
S
(NS )/M
P
NS
X
(NP )→1
n



FS2
′
−
(S
→
P
)
(Golterman,
Cata,
S.P.
06)

MS2 (MS2 + Q2 )

WHAT IS RESONANCE SATURATION ? – p.7/15
What is resonance saturation ? (Nc → ∞)
• It’s a Pade Approximant to a meromorphic function.
2
◮ F (Q ) ≈
2
MV
2
Q2 +MV
is the PA P10 (Q2 ) to F (Q2 ).
A
• NA,V resonances in Q2 ΠLR (Q2 ) = −F02 − Q2 PN
A
2
FA
2
Q2 +MA
+ (A → V )
N
=⇒ PN
(Q2 ) with N = NA + NV
N −2
⊕ 1/Q4 fall-off =⇒ PN
(Q2 )
◮ PA’s parameters do obey WSRs.
• N resonances in Π’S−P (Q2 ) =⇒ PNN −1 (Q2 ).
◮ PA’s parameters do obey L8 sum rule.
∴ Begin to see the tip of the iceberg:
Parameters (residues + poles)
of Pade Approx.
6=
Decay constants and masses
of Green′ s functions
WHAT IS RESONANCE SATURATION ? – p.8/15
Pade Approximants
(Physics ⇔ z ≡ Q2 )
Let
G(z)|z→0 ≈ G0 + G1 z + G2 z 2 + G3 z 3 + ....
M
Define rational function PN
(z) such that
M
PN
(z) ≡
QM (z)
≈ G0 + G1 z + G2 z 2 + ... + GM +N z M +N + O(z M +N +1 )
RN (z)
M
If G(z) ∼ 1/z K , choose PM
+K (z).
(Pommerenke ’73)
Convergence Theorem
Let G(z) be meromorphic and analytic at the origin. Then,
M
lim PM
+K (z) = G(z)
M →∞
for z ∈ compact set in C, except on isolated points.
WHAT IS RESONANCE SATURATION ? – p.9/15
Convergence Map
2
Im(q )
poles of
2
G(q )
Re(q 2)
WHAT IS RESONANCE SATURATION ? – p.10/15
Convergence Map
2
Im(q )
z
poles of
2
G(q )
Re(q 2)
z*
WHAT IS RESONANCE SATURATION ? – p.10/15
Convergence Map
2
Im(q )
z
poles of
2
G(q )
Re(q 2)
z*
WHAT IS RESONANCE SATURATION ? – p.10/15
Convergence Map
2
Im(q )
z
poles of
2
G(q )
Re(q 2)
z*
_ ``defect´´
= pole U zero =
WHAT IS RESONANCE SATURATION ? – p.10/15
Convergence Map
2
Im(q )
P
M
N
~
~
2
G(q )
z
poles of
2
G(q )
Re(q 2)
z*
Pommerenke ‘73
_ ``defect´´
= pole U zero =
WHAT IS RESONANCE SATURATION ? – p.10/15
Simple case: Stieltjes
G(Q2 ) =
Z
∞
M2
∞
X
dν(t)
2n
=
G
Q
n
t + Q2
n=0
with ν(t) a bounded non-decreasing function Gn = (−1)n
R∞
M2
dν(t)
.
tn
E.g.: Vac. Polarization ΠV (Q2 ) (with one subtraction).
∞
X
FR2
2
dν(t) = dt
δ(t−MR
)
t
R=0
,
M 0 = Mρ ,
ΠV (0) − ΠV (Q2 )
G(Q ) =
Q2
2
etc...
(t)
t
WHAT IS RESONANCE SATURATION ? – p.11/15
Simple case: Stieltjes (II)
E.g.: Fn = 1 ,
Mn2
2
= n , n = 1, 2, 3, . . . ⇒
G(Q2 )|Q2 →0 ≈
∞
X
G(Q ) =
ψ(Q2 +1)+γ
Q2
ζ(p + 2) (−Q2 )p
,
, ψ(z) =
d
dz
log Γ(z)
i.e. Rconv. = 1
p=0
∞
G(Q2 )|Q2 →∞
X
log Q2 + γ
1
B2p
≈
+
−
Q2
2Q4 p=1 2p(Q2 )2p+1
2
X G(Q )
OPE
1
2
0.9
P3
0.8
1
P2
0.7
Im(q 2 )
0
P1
0.6
0
5
10
15
20
Q
2
0
33
5
P6
52
10
20
Re(q 2 )
WHAT IS RESONANCE SATURATION ? – p.12/15
Main Properties of PAs
• In general, a PA may have complex poles (but it’s O.K. !).
◮ E.g. P20 to Q2 ΠLR has MV,A complex. (Masjuan,S.P., ’07)
• Pommerenke’s Thm ⇒ ∃ convergence in complex Q2 plane, away from
singularities.
◮ Residues and poles of PAs are not physical.
• PAs approximate original function at the expense of altering residues
and/or poles hierarchically (more the farther away from the origin).
50
• E.g.: P52
approximation to Regge-like model of Q2 ΠLR (Q2 )
2
Im (q )
15
P. Masjuan, S.P., JHEP05 (2007) 040.
10
5
5
-5
10
15
20
25
Re (q 2 )
-1 0
-1 5
WHAT IS RESONANCE SATURATION ? – p.13/15
• Pade-Type Approximants are PAs with poles predetermined at the
physical masses. Then, residues pay full price i.e. last residue
considered, completely off (recall previous •).
• E.g.: QCD, Pade-Type prediction for O(p6 ) contribution to < V V − AA >.
Physical masses from PDG.
T3 1
T4 2 a
T4 2 b
T5 3 a
T5 3 b
T6 4
0
-0.01
C4 H10-3 L
-0.02
-0.03
-0.04
-0.05
Masjuan, S.P., in preparation.
-0.06
T3 1
T4 2 a T4 2 b T5 3 a
Pade Type
T5 3 b
T6 4
WHAT IS RESONANCE SATURATION ? – p.14/15
Conclusions and Outlook
• Resonance saturation at large-Nc can be understood from the theory of
Pade Approximants to meromorphic functions.
(Rate of convergence ?)
• Chiral expansion allows to construct rational approx. at finite Q2 in
region free of poles.
• At the last poles, the approximation is unreliable:
◮ Last Residues/poles in rational approx. not physical.
E.g., form factors not to be extracted from rational approx. to 3-point
functions (Bijnens et al. ’03).
• Parameters in a Lagrangian of large-Nc QCD with a finite number of
resonances do not in general coincide with physical decay constants and
masses.
( Lagrangian of the rational approx.?)
WHAT IS RESONANCE SATURATION ? – p.15/15