Status report:
Moments of Pion Distribution Amplitudes
M. Donellan, J.M. Flynn, A. Jüttner,
J-I. Noaki, and C.T. Sachrajda
plus contributions from Edinburgh...
Edinburgh, UKQCD-meeting
School of Physics & Astronomy
University of Southampton
23 January 2006
Moments of the PDA, Andreas Jüttner
1/12
Exclusive processes at large Q 2
e−
e−
γ
π(p ′ )
hπ(p ′ )|Vµ |π(p)i = Fπ (Q 2 )(p + p ′ )µ
Vµ = 23 e ūγµ u − 31 e d̄γµ d
π(p)
N
Brodsky and Lepage: factorise F (Q 2 ) at Q 2 → ∞
→ perturbative hard scattering amplitude (process
dependent)
→ non-perturbative wave fn.’s describing the hadron’s overlap
with lowest Fock state (universal)
Impact on exclusive non-leptonic decays like B → ππ, KK
currently studied at BarBar and Belle
Moments of the PDA, Andreas Jüttner
1/12
Factorisation and OPE
γ
v̄ p
π(p ′ )
(ū + u = 1)
′
ūp
φin
φout
π(p)
up
vp ′
hπ(p ′ )|ūαa (x)U(x, y)dβb (y)|0i × [T (x, y, z, 0)]abcd
× h0|ūγc (z)U(z, 0)dδd (0)|π(p)i
αβγδ
Ry
Wilson line : U(y, x) = P exp ig x dωµ Aµ (ω)
R1
hπ(p)|ū(z)γµ γ5 U(z, −z)d(−z)|0i = ifπ pµ due i(2u−1)pz φ(u, Q 2 ) (ξ ≡ ū − u,)
0
φ(u, Q 2 ) - probability of finding lowest fock state with momentum fractions u
and ū
Moments of the PDA, Andreas Jüttner
2/12
Factorisation and OPE
γ
v̄ p
π(p ′ )
(ū + u = 1)
′
ūp
φin
φout
π(p)
up
vp ′
hπ(p ′ )|ūαa (x)U(x, y)dβb (y)|0i × [T (x, y, z, 0)]abcd
× h0|ūγc (z)U(z, 0)dδd (0)|π(p)i
αβγδ
Ry
Wilson line : U(y, x) = P exp ig x dωµ Aµ (ω)
R1
hπ(p)|ū(z)γµ γ5 U(z, −z)d(−z)|0i = ifπ pµ due i(2u−1)pz φ(u, Q 2 ) (ξ ≡ ū − u,)
0
φ(u, Q 2 ) - probability of finding lowest fock state with momentum fractions u
and ū
(Gegenbauer) moments of the distribution amplitude
!
P
φ(u) = 6u ū 1 +
an Cn3/2 (2u − 1)
n≥1
Moments of the PDA, Andreas Jüttner
2/12
Moments of a distribution
2
' (x; 20)
1.75
an = hξn i =
Z1
1.5
1.25
d ξξn φ(ξ, Q 2 )
−1
1
0.75
0.5
0.25
0
0.2
0.4
0.6
0.8
1
x =u
where ξ = ū − u difference in parton’s (longit.) momentum
hξ0 i = 1
hξ1 i - average (Iso spin → hξ1π i = 0, hξ1K i =?)
hξ2 i - shape of the distribution (hξ2π,K i =?)
Moments of the PDA, Andreas Jüttner
3/12
Other calculations
hξ1K i
1st moment
Ball&Zwicky
SR
0.06(3)
2nd moment
a
−1
asymptotic
Bakulev et al.
UKQCD
QCDSF/
SR
Nf = 0
2.67GeV
Nf = 2
1.6 − 2.7GeV
UKQCD
µ2 /GeV 2
1
hξ2π i
2
µ /GeV 2
0.2
∞
0.19
1.35
0.280(49)+0.030
−0.013
7MS
0.281(28)
5MS
flat continuum extrapolation
flat chiral extrapolation
Moments of the PDA, Andreas Jüttner
4/12
Our goal
calculate hξ1K ,K ∗ i
’... an independent calculation on the lattice would be both
timely and useful ...’ [Ball & Zwicky, hep-ph/0601086]
calculate hξ2π,ρ,K ,K ∗ i
DWF wt. Ls = 8, 16 and L = 16, 24
two independent analyses
NPR (Rome-Southampton)
Chiral extrapolation
dream: continuum extrapolation
Moments of the PDA, Andreas Jüttner
5/12
Moments of the PDA
hπ(p)|ū(z)γµ γ5 U(z, −z)d(−z)|0i = ifπ pµ
R1
due i(2u−1)pz φ(u, Q 2 )
0
1st moment:
∂ν hπ(p)|ū(z)γµ γ5 U(z, −z)d (−z)|0i|z 2 =0
←
→
= hπ(p)|ūγµ γ5 (D ν − D ν )d |0i
= fπ (ipµ )(ipν )
R1
0
du(2u − 1)φ(u, Q 2 )
= fπ (ipµ )(ipν )hξ1 i
→
→
D µ = ∂ µ + igAµ ,
Moments of the PDA, Andreas Jüttner
←
←
D µ = ∂ µ − igAµ
6/12
Moments of the PDA
hπ(p)|ū(z)γµ γ5 U(z, −z)d(−z)|0i = ifπ pµ
R1
due i(2u−1)pz φ(u, Q 2 )
0
2nd moment:
∂ν ∂ρ hπ(p)|ū(z)γµ γ5 U(z, −z)d (−z)|0i|z 2 =0
← ←
→
←
→ → = hπ(p)|ūγµ γ5 D ν (D ρ − D ρ ) − (D ρ − D ρ )D ν d |0i
= fπ (ipµ )(ipν )(ipρ )
R1
0
du(2u − 1)2 φ(u, Q 2 )
= fπ (ipµ )(ipν )(ipρ )hξ2 i
Moments of the PDA, Andreas Jüttner
7/12
Bare lattice correlation functions
1st moment hξ1 i:
P
~x
←
→
ei ~p~x hū(x)γρ γ5 (D µ −D µ )d(x)P † (0)i
P
~x
ei ~p~x hAρ (x)P † (0)i
=
P
~x
5 (x)P † (0)i
ei ~p~x hOρµ
P
~x
t→∞
=
ei ~p~x hAρ (x)P † (0)i
ipµ hξ1 i
2nd moment hξ2 i:
P
~
x
← ← →
← → → ei ~p~x hū(x)γρ γ5 D µ (D ν −D ν )−(D ν −D ν )D µ d(x)P † (0)i
P
~x
ei ~p~x hAρ (x)P † (0)i
=
~x
5 (x)P † (0)i
ei ~p~x hOρµν
P
~x
t→∞
=
Moments of the PDA, Andreas Jüttner
P
ei ~p~x hAρ (x)P † (0)i
(ipµ )(ipν )< ξ2 >
8/12
Bare results - hξ1i
163 × 32 × 8,
45 independent configs, 4 pos. of the source
0.2
0.15
0.1
= 2π/L)
DEGENERATE
amu,d = 0.04
ams = 0.04
0.05
hξ1 i
hO{41} P † i
(p
hA4 P † i
0
−0.05
−0.1
−0.15
−0.2
0
5
10
15
20
25
30
t
hξ1 i = 0(0)
Moments of the PDA, Andreas Jüttner
9/12
Bare results - hξ1i
163 × 32 × 16,
85 configs (every 10th traj.), 1 pos. of the source
0.2
0.15
0.1
= 2π/L)
non-degenerate
amu,d = 0.02
ams = 0.04
0.05
hξ1 i
hO{41} P † i
(p
hA4 P † i
0
−0.05
−0.1
−0.15
−0.2
0
5
10
15
20
25
30
t
hξ1 i = 0.0373(34)
Moments of the PDA, Andreas Jüttner
9/12
Bare results - hξ1i
163 × 32 × 16,
85 configs (every 10th traj.), 1 pos. of the source
0.2
0.15
0.1
= 2π/L)
non-degenerate
amu,d = 0.01
ams = 0.04
0.05
hξ1 i
hO{41} P † i
(p
hA4 P † i
0
−0.05
−0.1
−0.15
−0.2
0
5
10
15
t
20
25
30
hξ1 i = 0.?(?)
Poor statistics ?
Moments of the PDA, Andreas Jüttner
9/12
Bare results - hξ2i
hO{421} P † i
(p
hA4 P † i
=
√ 2π
2L)
163 × 32 × 16, ams = 0.04, 85 cfgs. sep. by 10 traj.
amu,d = 0.02
amu,d = 0.01
0.4
0.3
0.3
0.2
0.2
hξ2 i
hξ2 i
0.4
0.1
0.1
0
0
−0.1
−0.1
−0.2
0
5
10
15
t
20
hξ2 i = 0.148(8)
Moments of the PDA, Andreas Jüttner
25
30
−0.2
0
5
10
15
20
25
30
t
hξ2 i = 0.155(10)
10/12
Bare results - hξ2i - multiple sources
hO{421} P † i
(p
hA4 P † i
=
√ 2π
2L)
163 × 32 × 8, amu,d = 0.04, ams = 0.04
#35, 40 traj. sep., 4 src.
#140, 10 traj. sep., 1 src.
0.4
0.3
0.3
0.2
0.2
hξ2 i
hξ2 i
0.4
0.1
0.1
0
0
−0.1
−0.1
−0.2
0
5
10
15
t
20
25
30
−0.2
0
5
10
15
t
20
25
30
Prefer multiple sources over increased stats!
Moments of the PDA, Andreas Jüttner
11/12
Twisted boundary conditions
Todo:
renormalisation (non-pt. and pt. under way)
chiral extrapolation
impact of twisted boundary conditions? - under way
extend analysis to ρ and K ∗
What we need:
ms = 0.04/mud = 0.03 run has to catch up: at least
2 × 256er
Strong preference for multiple sources over increased
statistics
Moments of the PDA, Andreas Jüttner
12/12