Electromagnetic splittings
and gµ − 2 light-by-light contribution.
Taku Izubuchi
RIKEN BNL Reserch Center
Kanazawa University
based on a collaboration with
Tom Blum, Masashi Hayakawa, and Norikazu Yamada
helpful discussions from the RBC collaboration.
Taku Izubuchi, Edinburgh, 10/Mar/2005
1
Motivations
• The first principle calculations of isospin breaking effects due to electromagnetic
(EM) and the up, down quark mass difference are necessary for accurate hadron
spectrum, quark mass determination.
• EM splittings are measured very accurately :
mπ± − mπ0 = 4.5936(5)MeV,
mN − mP = 1.2933317(5)MeV
• From Γ(π + → µ+νµ, µ+νµγ) + Vud(exp)
fπ+ = 130.7 ± 0.1 ± 0.36MeV
PDG 2004
the last error is due to the uncertainty in the part of O(α) radiative corrections
that depends on the hadronic structure of the π meson.
Γ(P S
+
+
+
→ µ νµ, µ νµγ) ∝ [1 + CPS α]had. struc.
Cπ ∼ 0 ± 0.24, Cπ − CK = 3.0 ± 1.5
c.f. Marciano 2004 : Vus from fπ /fK (MILC) + Γ(πl2)/Γ(Kl2).
Taku Izubuchi, Edinburgh, 10/Mar/2005
2
Motivations...
• A practice for gµ − 2 light-by-light calculation.
gµ − 2
exp
−11
aµ =
= 116, 592, 080(60) × 10
2
SM
QED
had
EW
new
2
aµ = aµ
+ aµ + aµ ,
aµ
∼ O((mµ/Mnew ) )
had
had,LO
had,HO
had,LBL
aµ = aµ
+ aµ
+ aµ
exp
aµ =
116, 592,
080(60) × 10
QED
aµ
=
116, 584,
706(3) × 10
had,LO
aµ
=
6,
963(62)(36) × 10
had,HO
aµ
=
−
100(6) × 10
had,LBL
aµ
=
EW
aµ =
(1)
−11
−11
4
+ O(α )
−11
−11
134(25) × 10
−11
154(1)(2) × 10
(before : 86(35) × 10
−11
)
−11
−11
• aexp
− ath
(e+e−, A. Vainshtein et. al. 2004)
µ
µ = (220 ± 100) × 10
Taku Izubuchi, Edinburgh, 10/Mar/2005
3
EM splittings
• Axial WT identity with EM
LEM
(NF = 2, mu = md),
e
= −iAµ(x)q̄Qγµq(x), Q =
3
a
„
«
1
3
T +
,
2
b
a
∂µAµ(x) = 2mJ5 (x) − ieAµ(x)f3abAµ
neutral current, A3µ(x), is conserved: π 0 is still a NG boson.
• ChPT with EM
”
1 2“
1 2
†
†
†
†
Lχ = FEM + f0 DµU DµU + χU + χ U + C(QR U QLU ),
4
4
iΦ
U =e ,
DµU = ∂µU − iQR AµU + iU QLAµ .
symmetric under
†
†
U → gR U gL, χ → gR χgL,
†
QI → gI QI gI , (I = L, R)
C is a new low energy constant.
Taku Izubuchi, Edinburgh, 10/Mar/2005
4
• at O(p4, p2e2) :
2
Mπ± = 2mB0 + 2e
2C
f02
2
2
2
Mπ0 = 2mB0
2
2
2
+O(m log m, m ) + I0e m log m + K0e m
2
2
2
+O(m log m, m ) + I±e m log m + K±e m
Dashen's theorem : The difference of squared pion mass is independent of quark
mass upto O(e2m),
2
2
2
∆Mπ ≡ Mπ± − Mπ0 = 2e
2C
f02
2
2
+ (I± − I0)e m log m + (K± − K0)e m
I±, I0 is known in terms of C .
Taku Izubuchi, Edinburgh, 10/Mar/2005
5
EM splittings on lattice
• The correlator for neutral meson is calculated using the interpolation field of the
third component of isospin:
E
D
E
i
1 hD uu
uu†
dd
dd†
JX (t)JX (0)
+ JX (t)JX (0)
CX 0 (t) =
conn
conn
2
• For degenerate quark mass, mu = md, contributions from disconnected diagrams
to meson masses are small, O(e4) , which we will neglect:
D
D
=⇒
3
3
J5 (t)J5 (0)
3
s
J5 (t)J5 (0)
E
2
e2
E
4
= Gconn(e ; t) + e Gdisc(t)
2
e2
= O(e )
2
D
3
EM EM
3
|Vµ Vµ |π
Eπ0 = Eπ3 + e π
˛ E
˛ E
˛ 0
˛ 3
2˛
0¸
˛
˛π = ˛π + e η
T3
Taku Izubuchi, Edinburgh, 10/Mar/2005
E
˛D
E˛
˛ 3 EM EM 0 ˛2
+ e ˛ π |Vµ Vµ |π ˛ ,
4
T3
6
EM splittings...
• Quark mass renormalization due to EM has dependence to renormalization prescriptoin. The counter term for quark mass due to EM:
3e2 2
2 2
−3
Q
m
log(µ
a
)
∼
10
mq
q
16π 2
for
µ→2×µ ,
thus the ambiguity of quark mass renormalization is tiny. ∼ 0.01 MeV for light
quarks and ∼ 0.1 MeV for strange quark.
• Isospin breaking due to `quark mass, mu − md, is ´higher order effect in psuedoscalar meson mass, O (mu − md)2, (mu − md)e2 .
This is not the case for Nucleons.
Taku Izubuchi, Edinburgh, 10/Mar/2005
7
EM splittings on lattice
• In 1996, Duncan, Eichten, Thacker carried out SU(3)×U(1) simulation to do the
EM splittings for the hadron spectroscopy using quenched Wilson fermion on
a−1 ∼ 1.15 GeV, 123 × 24 lattice.
• Using NF = 2 Dynamical DWF ensemble (RBC) would have advantages such as
better scaling and smaller quenching errors.
• Especially smaller systematic errors due to the the quark massless limits, mf →
−mres(Qi), has smaller Qi dependence than that of Wilson fermion, K →
Kc(Qi).
• Generate Coulomb gauge fixed (quenched) non-compact U(1) gauge action with
βQED = 1. UµEM = exp[−iAµ(x)].
• Quark propagator, Sqi (x) with EM charge Qi = qie
wall source
ˆ EM Q
SU (3) ˜
D (Uµ ) i × Uµ
Sqi (x) = bsrc,
with Coulomb gauge fixed
(i = up,down)
qup = 2/3, qdown = −1/3
Taku Izubuchi, Edinburgh, 10/Mar/2005
8
non-compact U(1) gauge
• The generation of configuration is very efficient using FFT with zero autocorrelation.
• Photon self coupling in compact QED, O(anα), which is dangerous for light-bylight calculation, vanishes.
• static lepton potential on 163 × 32 lattice (βQED = 100, 4,000 confs) vs lattice
Coulomb potential.
Coulomb potential V(r)-V(1)
ncU(1) simulation vs FFT prediction at beta=100
0
wilson_vs_r.dat_shift
V_t13.dat
V_t14.dat
V_t15.dat
-0.2
-0.4
-0.6
-0.8
0
1
2
Taku Izubuchi, Edinburgh, 10/Mar/2005
3
4
5
9
non-compact U(1) gauge...
• L=16 has significant finite volume effect for ra > 6 ∼ 1.5r0 ∼ 0.75 fm. We
hope it's less problematic for the calculation EM splitting of hadron due to
confinement. It would be worth considering for generation of U(1) on a larger
lattice and cutting it off.
Finite size effect
L=16
L=32
L=64
L=128
0
-0.2
-0.4
-0.6
-0.8
-1
0
5
Taku Izubuchi, Edinburgh, 10/Mar/2005
10
15
20
10
simulation parameters
• NF = 2 Dynamical DWF configuration
• a−1 = 1.691(53) GeV.
• degenerate quark mass at dynamical quark mass points,
mval = msea = (0.02), 0.03, 0.04 ∼ 50%, 75%, 100% of mstrange.
• 163 × 32 or (1.9 fm)3.
• Ls = 12, mresa = 0.0013 or a few MeV.
p
• EM charge: e = 1.0, 0.6, 0.3028 = 4π/137
• ∼ 94 configurations for each m
• one or two QED configuration per a QCD configuration.
• All 16 mesonic connected correlators + Neutron, Proton.
Taku Izubuchi, Edinburgh, 10/Mar/2005
11
Analysis methods
• Analysis method I :
Fit correlator for each charge combination separately,
then calculate the mass splittings under jackknife.
2
π :∆Mπ = (Mπ± − Mπ0 )(Mπ± + Mπ0 )
X = ρ, N :∆MX = MX ± − MX 0 ,
• Analysis method II :
Subtract charged correlator by neutral correlator,
and fit it by a linear function in t:
2
CX (t) = A(e )e
−MX (e2 )t
CX ± (t) − CX 0 (t)
= ∆MX × t + Const
CX 0 (t)
Taku Izubuchi, Edinburgh, 10/Mar/2005
12
results
• hJ5(0)J5(t)i at m = 0.04 and 0.03.
∆ G (t)/ G(t) ps
∆ G (t)/ G(t) ps
msea=mval=0.04
0.03
down-down
up-down
up-up
0.02
0.01
0
0
.5M
-0.01
∆M
5
eV
∆M=
2.5M
eV
-0.01
M
∆M
eV
-0.04
0
5
M
eV
V
Me
V
Me
-0.03
=
10
=
∆M
-0.02
10
=
∆M
-0.02
=
down-down
up-down
up-up
0.02
0.01
∆M=
2
msea=mval=0.03
0.03
-0.03
10
5
-0.04
15
0
5
10
15
• larger e2 doesn't help to reduce error for J5: relative error stays almost same.
• Fluctuations due to SU(3) are compareble to that from U(1): by double the QED
statistics: ∆Mπ reduces by ∼ 4, 10, (30) % for A4, J5, ρ, (N ) resp. at m = 0.04.
2
2
σQCD
+ 0.5σQED
2
2
σQCD
+ σQED
Taku Izubuchi, Edinburgh, 10/Mar/2005
2
= (0.9) =⇒ σQED /σQCD ∼ 0.85
13
∆Mπ2
• Method I, Method II are consistent within statistical error.
-4
8×10
-4
7×10
-4
∆M
2
ps
6×10
-4
5×10
A4
J5
-4
4×10
-4
3×10
-4
2×10
0
0.01
0.02
m+mres
0.03
0.04
0.05
• preliminary results
2
2
A4 : ∆Mπ |m=ml =1.20(24)GeV ,
2
J5 : ∆Mπ |m=ml =1.03(15) × 10
(experiment :1.26 × 10
−3
−3
2
GeV ,
2
GeV) .
2
2
( ∆Mps
∼ 1.6 × 10−3 GeV2 at m = mstrange/2, exp: ∆MK
=0.98 GeV2)
Taku Izubuchi, Edinburgh, 10/Mar/2005
14
∆Mρ2
•
-4
4×10
-4
∆Mρ
2×10
0
-4
-2×10
0
0.01
0.02
0.03
m+mres
0.04
0.05
2
∆Mρ |m=ml =0.08(45)GeV,
(experiment : ∼ 0).
• Nucleon mass splitting is also consistent with zero.
Taku Izubuchi, Edinburgh, 10/Mar/2005
15
Dashen's theorem
• Mπ,Q : pure QCD pion (e2 = 0)
2
2
2
2
Mπ0 − MπQ = Ie m log m + Ke m
0.02
m
2
2
−m
π0
πQ
0.015
0.01
0.005
0
0
Taku Izubuchi, Edinburgh, 10/Mar/2005
0.01
0.02
0.03
m+mres
0.04
0.05
16
Direct calculation of the each QED diagram
• Each of the perturbative pieces could be obtained by a direct calculation of QED
diagram treating QCD non-perturbatively. This method may have the following
advantages:
• Could obtain the contribution from each QED diagram separately, and/or
order by order in α.
• The charge, e and qi, could be input later than simulation.
• Would help or avoid the subtractions, for example in the light-by-light
calculation.
∂χ D
−1
(χ) = D
−1
∂χDD
−1
χ = m, (mu − md), e
• An example for meson EM splittings :
1
G(t; q1, q2) =
2
1
1
1
q12
e2 : q 1 q 2
2
Taku Izubuchi, Edinburgh, 10/Mar/2005
1
2 2
e4 : q 1 q 2
q22
2
q12 q22
2
2
1
2
17
Direct calculation ...
• Prepare a USU (3)µ, and an Aµ.
• 1. Solve
ˆ
˜
(1)
D USU (3) Sq (x) = b,
• 2. multiply −iAµ(x)Kµ or −ieqiAµ(x)Kµ to the first solution vector, Sq(1)(x),
where Kµ is the kernel of the conserved vector current: Vµcon(x) = ψ̄Kµψ .
• 3. Sequentially solve
ˆ
˜ (2)
(1)
D USU (3) Sq (x) = −iAµ(x)γµSq (x),
• ( 4. Repeat 2, 3. (n − 1) times for Sq(n) )
• Then Tr[Sq(2)Γγ5[Sq(2)]†γ5Γ] is the e2q1q2 diagram of the G(t; q1, q2) in the background fields,
1
1
1 ∂2
G(t; q1, q2) =
e2 ∂q1q2
e2 :
=⇒
q1 q2
2
Taku Izubuchi, Edinburgh, 10/Mar/2005
2
18
Future prospects
• EM splittings using non-perturbative QED.
• Nucleon mass splittings on non-degenerate quark mass.
• EM splittings using the direct calculation of the QED diagrams.
• O(α) contribution to gµ − 2 (pure QED).
=⇒
• O(α3) contribution (light-by-light) to gµ − 2.
(c)-1
(d)-1
Taku Izubuchi, Edinburgh, 10/Mar/2005
(c)-2
(d)-2
19