Dire Threads

Hadronic light-by-light scattering in the muon g − 2:
a dispersive approach
Martin Hoferichter
¨ Darmstadt
Institut fur
¨ Kernphysik, Technische Universitat
ExtreMe Matter Institute EMMI, GSI, Darmstadt
Dark Matter, Hadron Physics and Fusion Physics
Messina, September 26, 2014
G. Colangelo, MH, M. Procura, P. Stoffer, JHEP 09 (2014) 091, arXiv:1309.6877
G. Colangelo, MH, B. Kubis, M. Procura, P. Stoffer, PLB 738 (2014) 6
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
1
Outline
1
Hadronic contributions to the muon g − 2
2
Hadronic light-by-light scattering
One-pion intermediate states
Two-pion intermediate states
Preliminary numbers
3
Summary and outlook
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
2
Anomalous magnetic moment of the muon
Experimental precision 0.5 ppm BNL E821 2006
exp
aµ
= (116 592 089 ± 63) · 10−11
HMNT (06)
JN (09)
Davier et al, (10)
Theory error of similar size
+ –
Davier et al, e e (10)
Deviation from SM prediction around 3σ
New experiment at FNAL (E989) aiming at
JS (11)
HLMNT (10)
HLMNT (11)
experiment
0.14 ppm, beam in 2016/2017
BNL
BNL (new from shift in λ)
J-PARC aiming at 0.1 ppm, new approach
with ultra-cold muons, R&D in progress
170
aµ ×1010 – 11659000
180
190
200
210
Hagiwara et al. 2012
⇒ Need to improve theory by a factor of 4
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
3
Overview of SM prediction
experiment
aµ 10−11
116592089.
∆aµ 10−11
QED O(α )
116140973.21
0.03
QED O(α 2 )
413217.63
0.01
0.00
3
63.
QED O(α )
30141.90
QED O(α 4 )
381.01
0.02
QED O(α 5 )
5.09
0.01
116584718.85
0.04
QED total
electroweak, total
153.6
1.0
HVP (LO)
6949.
43.
HVP (NLO)
−98.
1.
HLbL (LO)
116.
40.
HVP (NNLO)
12.4
HLbL (NLO)
3.
2.
116591855.
59.
theory
M. Hoferichter (IKP & EMMI, TU Darmstadt)
Schwinger 1948
0.1
HLbL scattering: a dispersive approach
Messina, September 26, 2014
4
Overview of SM prediction
experiment
aµ 10−11
116592089.
∆aµ 10−11
QED O(α )
116140973.21
0.03
QED O(α 2 )
413217.63
0.01
0.00
3
63.
QED O(α )
30141.90
QED O(α 4 )
381.01
0.02
QED O(α 5 )
5.09
0.01
116584718.85
0.04
QED total
electroweak, total
153.6
1.0
HVP (LO)
6949.
43.
HVP (NLO)
−98.
1.
HLbL (LO)
116.
40.
HVP (NNLO)
12.4
HLbL (NLO)
3.
2.
116591855.
59.
theory
M. Hoferichter (IKP & EMMI, TU Darmstadt)
e, µ, τ
Sommerfield, Petermann 1957
0.1
HLbL scattering: a dispersive approach
Messina, September 26, 2014
4
Overview of SM prediction
experiment
aµ 10−11
116592089.
∆aµ 10−11
I(a)
I(b)
I(c)
I(d)
I(e)
I(f)
I(g)
I(h)
I(i)
I(j)
II(a)
II(b)
II(c)
II(d)
II(e)
II(f)
III(a)
III(b)
III(c)
IV
63.
QED O(α )
116140973.21
0.03
QED O(α 2 )
413217.63
0.01
QED O(α 3 )
30141.90
0.00
QED O(α 4 )
381.01
0.02
QED O(α 5 )
5.09
0.01
116584718.85
0.04
V
QED total
electroweak, total
153.6
1.0
HVP (LO)
6949.
HVP (NLO)
−98.
1.
HLbL (LO)
116.
40.
43.
HVP (NNLO)
12.4
HLbL (NLO)
3.
2.
116591855.
59.
theory
M. Hoferichter (IKP & EMMI, TU Darmstadt)
VI(f)
VI(a)
VI(b)
VI(c)
VI(d)
VI(e)
VI(g)
VI(h)
VI(i)
VI(j)
VI(k)
Kinoshita et al. 2012
0.1
HLbL scattering: a dispersive approach
Messina, September 26, 2014
4
Overview of SM prediction
experiment
aµ 10−11
116592089.
∆aµ 10−11
QED O(α )
116140973.21
0.03
QED O(α 2 )
413217.63
0.01
QED O(α )
30141.90
0.00
QED O(α 4 )
381.01
0.02
QED O(α 5 )
5.09
0.01
116584718.85
0.04
3
QED total
electroweak, total
153.6
63.
6949.
HVP (NLO)
−98.
1.
HLbL (LO)
116.
40.
12.4
HLbL (NLO)
W
Z
ν
t
h
1.0
HVP (LO)
HVP (NNLO)
W
γ, Z
43.
1-loop: Jackiw, Weinberg and others 1972
2-loop: Kukhto et al. 1992, Czarnecki, Krause, Marciano
0.1
1995, Degrassi, Giudice 1998, Knecht, Peris, Perrottet, de
3.
2.
¨
Rafael 2002, Vainshtein 2003, Heinemeyer, Stockinger,
116591855.
59.
Weiglein 2004, Gribouk, Czarnecki 2005
theory
M. Hoferichter (IKP & EMMI, TU Darmstadt)
Update after Higgs discovery: Gnendiger et al. 2013
HLbL scattering: a dispersive approach
Messina, September 26, 2014
4
Overview of SM prediction
experiment
aµ 10−11
116592089.
∆aµ 10−11
QED O(α )
116140973.21
0.03
QED O(α 2 )
413217.63
0.01
0.00
3
63.
QED O(α )
30141.90
QED O(α 4 )
381.01
0.02
QED O(α 5 )
5.09
0.01
116584718.85
0.04
QED total
electroweak, total
153.6
1.0
HVP (LO)
6949.
43.
HVP (NLO)
−98.
1.
HLbL (LO)
116.
40.
HVP (NNLO)
12.4
HLbL (NLO)
3.
2.
116591855.
59.
theory
M. Hoferichter (IKP & EMMI, TU Darmstadt)
Hagiwara et al. 2011
0.1
HLbL scattering: a dispersive approach
Messina, September 26, 2014
4
Overview of SM prediction
experiment
aµ 10−11
116592089.
∆aµ 10−11
QED O(α )
116140973.21
0.03
QED O(α 2 )
413217.63
0.01
0.00
3
63.
QED O(α )
30141.90
QED O(α 4 )
381.01
0.02
QED O(α 5 )
5.09
0.01
116584718.85
0.04
QED total
electroweak, total
153.6
1.0
HVP (LO)
6949.
43.
HVP (NLO)
−98.
1.
HLbL (LO)
116.
40.
HVP (NNLO)
12.4
HLbL (NLO)
3.
2.
116591855.
59.
theory
M. Hoferichter (IKP & EMMI, TU Darmstadt)
Calmet et al. 1976, Hagiwara et al. 2011
0.1
HLbL scattering: a dispersive approach
Messina, September 26, 2014
4
Overview of SM prediction
experiment
aµ 10−11
116592089.
∆aµ 10−11
QED O(α )
116140973.21
0.03
QED O(α 2 )
413217.63
0.01
0.00
3
63.
QED O(α )
30141.90
QED O(α 4 )
381.01
0.02
QED O(α 5 )
5.09
0.01
116584718.85
0.04
QED total
electroweak, total
153.6
1.0
HVP (LO)
6949.
43.
HVP (NLO)
−98.
1.
HLbL (LO)
116.
40.
HVP (NNLO)
12.4
HLbL (NLO)
3.
2.
116591855.
59.
theory
M. Hoferichter (IKP & EMMI, TU Darmstadt)
Hayakawa, Kinoshita, Sanda 1995
Bijnens, Pallante, Prades 1995
Knecht, Nyffeler 2001
Jegerlehner, Nyffeler 2009
0.1
HLbL scattering: a dispersive approach
Messina, September 26, 2014
4
Overview of SM prediction
experiment
aµ 10−11
116592089.
∆aµ 10−11
QED O(α )
116140973.21
0.03
QED O(α 2 )
413217.63
0.01
0.00
3
63.
QED O(α )
30141.90
QED O(α 4 )
381.01
0.02
QED O(α 5 )
5.09
0.01
116584718.85
0.04
QED total
electroweak, total
153.6
1.0
HVP (LO)
6949.
HVP (NLO)
−98.
1.
HLbL (LO)
116.
40.
43.
HVP (NNLO)
12.4
HLbL (NLO)
3.
2.
116591855.
59.
theory
M. Hoferichter (IKP & EMMI, TU Darmstadt)
e
Kurz, Liu, Marquard, Steinhauser 2014
0.1
HLbL scattering: a dispersive approach
Messina, September 26, 2014
4
Overview of SM prediction
experiment
aµ 10−11
116592089.
∆aµ 10−11
QED O(α )
116140973.21
0.03
QED O(α 2 )
413217.63
0.01
0.00
3
63.
QED O(α )
30141.90
QED O(α 4 )
381.01
0.02
QED O(α 5 )
5.09
0.01
116584718.85
0.04
QED total
electroweak, total
153.6
HVP (LO)
6949.
HVP (NLO)
−98.
1.
HLbL (LO)
116.
40.
43.
HVP (NNLO)
12.4
HLbL (NLO)
3.
2.
116591855.
59.
theory
M. Hoferichter (IKP & EMMI, TU Darmstadt)
e
1.0
Colangelo, MH, Nyffeler, Passera, Stoffer 2014
0.1
HLbL scattering: a dispersive approach
Messina, September 26, 2014
4
Overview of SM prediction
experiment
aµ 10−11
116592089.
∆aµ 10−11
QED O(α )
116140973.21
0.03
QED O(α 2 )
413217.63
0.01
0.00
3
63.
QED O(α )
30141.90
QED O(α 4 )
381.01
0.02
QED O(α 5 )
5.09
0.01
116584718.85
0.04
QED total
electroweak, total
153.6
1.0
HVP (LO)
6949.
43.
HVP (NLO)
−98.
1.
HLbL (LO)
116.
40.
HVP (NNLO)
12.4
HLbL (NLO)
3.
2.
116591855.
59.
theory
M. Hoferichter (IKP & EMMI, TU Darmstadt)
exp
−11
aµ − aSM
[2.7σ ]
µ = (234 ± 86) · 10
⇒Theory error comes almost
exclusively from hadronic part
0.1
HLbL scattering: a dispersive approach
Messina, September 26, 2014
4
Hadronic vacuum polarization
General principles yield direct connection with experiment
Gauge invariance
Analyticity
= −i k 2 g µν − k µ k ν Π k 2
Z∞
k2
Im Π(s)
Πren = Π k 2 − Π(0) =
ds
π
s s − k2
4Mπ2
Unitarity
Im Π(s) =
α
s
σtot e+ e− → hadrons = R(s)
4πα
3
1 Lorentz structure, 1 kinematic variable, parameter-free
Dedicated e + e − program under way: BaBar, Belle, BESIII, CMD3, KLOE2, SND
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
5
HLbL scattering
Large uncertainty and model dependence
q1 , µ
−q3 , λ
q2 , ν
k, σ
5 kinematic variables, (at least) 29 Lorentz structures
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
6
HLbL scattering
Large uncertainty and model dependence
q1 , µ
−q3 , λ
q2 , ν
k, σ
5 kinematic variables, (at least) 29 Lorentz structures
Dispersive point of view
Analytic structure: poles and cuts
֒→ residues and imaginary parts ⇒ by definition on-shell quantities
֒→ form factors and scattering amplitudes from experiment
Expansion: mass of intermediate states, partial waves
Pseudoscalar poles most important, next ππ cuts
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
6
HLbL scattering
Large uncertainty and model dependence
q1 , µ
−q3 , λ
q2 , ν
k, σ
5 kinematic variables, (at least) 29 Lorentz structures
Dispersive point of view
Analytic structure: poles and cuts
֒→ residues and imaginary parts ⇒ by definition on-shell quantities
֒→ form factors and scattering amplitudes from experiment
Expansion: mass of intermediate states, partial waves
Pseudoscalar poles most important, next ππ cuts
Decompose the tensor according to
0
π -pole
¯
Πµνλ σ = Πµνλ σ + ΠFsQED
µνλ σ + Πµνλ σ + · · ·
֒→ accounts for one- and two-pion intermediate states
¯ , but e.g. 3π more difficult
Generalizes immediately to η , η ′ , K K
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
6
Pion pole: pion transition form factor
Knecht, Nyffeler 2001
Master formula for pion-pole contribution
aπµ
0 -pole
= −e6
×
(
Z
d4 q1
(2π )4
Z
1
d4 q2
(2π )4 q12 q22 s (p + q1 )2 − m2 (p − q2 )2 − m2
Fπ 0 γ ∗ γ ∗ (q12 , q22 )Fπ 0 γ ∗ γ ∗ (s, 0)
s − Mπ2
T1 (q1 , q2 ; p) +
Fπ 0 γ ∗ γ ∗ (s, q12 )Fπ 0 γ ∗ γ ∗ (q22 , 0)
q22 − Mπ2
)
T2 (q1 , q2 ; p)
Crucial ingredient: pion transition form factor Fπ 0 γ ∗ γ ∗ (q12 , q22 )
Wick rotation: only space-like s, q12 , q22 contribute
Dispersive approach
π 0 , η, η ′
On-shell form factor
Fix parameters wherever data are available
Use analyticity to go to the space-like region
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
7
Pion transition form factor: physical regions
qs2
isoscalar
e−
π0
e−
e+
e+
π 0 → e+ e− e+ e−
qv2
−
e
π0
isovector
π 0 → γγ
e+
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
8
Pion transition form factor: physical regions
qs2
isoscalar
π0
e−
e−
e+
e+
π 0 → e+ e− e+ e−
qv2
−
e
π0
π 0 → γγ
isovector
related to γπ → ππ
e+
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
8
Pion transition form factor: physical regions
qs2
isoscalar
related to φ → 3π
Mφ2
related to ω → 3π
Mω2
π0
e−
e−
e+
e+
π 0 → e+ e− e+ e−
qv2
−
e
π0
π 0 → γγ
isovector
related to γπ → ππ
e+
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
8
Pion transition form factor: unitarity relations
process
unitarity relations
γv∗
γv∗
SC 1
SC 2
Fπ 0 γγ
P
γs
γπ → ππ
γs
F3π
P
γs∗
ω, φ
γv∗
ω, φ
Γπ 0 γ
γv∗
ω, φ
P
γs∗
γv∗
γs∗
Γ3π
γv∗
γs∗
P
σ (e+ e− → 3π )
F3π
ω → 3π , φ → 3π
d2 Γ (ω , φ → 3π )
dsdt
σ (e+ e− → π 0 γ )
γs∗
M. Hoferichter (IKP & EMMI, TU Darmstadt)
σ (γπ → ππ )
σ (γπ → ππ )
d2 Γ
dsdt (ω , φ → 3π )
γ ∗ → 3π
common theme:
resum ππ rescattering
σ (e+ e− → 3π )
HLbL scattering: a dispersive approach
Messina, September 26, 2014
9
Predicting σ (e+ e− → π 0 γ ) from σ (e+ e− → 3π )
fit isoscalar q 2 dependence to
σ (e + e − → 3π ):
a(q 2 ) =
F3π
3
+ β q2 +
q4
π
R∞
ds
q2
thr
Im a(s)
s 2 (s−q 2 )
e + e − → π 0 γ : both isoscalar and isovector
σ (e + e − → π 0 γ ) reproduced X
Next steps:
analytic continuation into space-like
region
generalize to doubly-virtual case using
e+ e− → π 0 γ as input
MH, Kubis, Leupold, Niecknig, Schneider (in preparation)
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
10
ππ intermediate states: FsQED contribution
0
-pole
FsQED
¯
Πµνλ σ = Ππµνλ
σ + Πµνλ σ + Πµνλ σ + ···
Separate terms with simultaneous cuts


V 2 V 2 
2
V

ΠFsQED
µνλ σ = Fπ q1 Fπ q2 Fπ q3 × 



Multiplication of sQED diagrams with FπV gives correct q 2 -dependence
֒→ not an approximation
¯ µνλ σ has cuts only in one channel
Remaining ππ contribution included in Π
֒→ partial-wave expansion, dispersion relations for this part
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
11
ππ intermediate states: reconstruction theorem
¯ µνλ σ = ∑ Aµνλ σ Πi (s) + Aµνλ σ Πi (t) + Aµνλ σ Πi (u)
Π
i,s
i,t
i,u
i
Πi functions with a right-hand cut only
֒→ similar to reconstruction theorem of ππ scattering Stern, Sazdjian, Fuchs 1993
Keep discontinuity of lowest partial waves
Dispersion integrals for Πi required to have correct soft-photon zeros
֒→ forces subtraction constants to zero
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
12
ππ intermediate states: reconstruction theorem
¯ µνλ σ = ∑ Aµνλ σ Πi (s) + Aµνλ σ Πi (t) + Aµνλ σ Πi (u)
Π
i,s
i,t
i,u
i
Πi functions with a right-hand cut only
֒→ similar to reconstruction theorem of ππ scattering Stern, Sazdjian, Fuchs 1993
Keep discontinuity of lowest partial waves
Dispersion integrals for Πi required to have correct soft-photon zeros
֒→ forces subtraction constants to zero
Gives relations such as
Π1 (s) =
s − q32
π
Z∞
ds′
s′ − q32
4Mπ2
′ 2 2 2
¯0
K1 (s′ ,s)Im h
++,++ s ; q1 ,q2 ; q3 ,0
0
Imh++,++
s; q12 , q22 ; q32 , 0 =
s
4Mπ2
σ (s) = 1 −
K1 (s′ , s) =
s
M. Hoferichter (IKP & EMMI, TU Darmstadt)
h0,++
h0,++
σ (s) ∗
h
s; q12 , q22 h0,++ s; q32 , 0
16π 0,++
s′ − q12 − q22
1
−
s′ − s λ (s′ , q 2 , q 2 )
1 2
λ (x , y , z) = x 2 + y 2 + z 2 − 2(xy + xz + yz)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
12
ππ intermediate states: non-diagonal terms
¯ µνλ σ = ∑ Aµνλ σ Πi (s) + Aµνλ σ Πi (t) + Aµνλ σ Πi (u)
Π
i,s
i,t
i,u
i
µνλ σ
Need to choose Ai
so that Πi are free of kinematic singularities
General procedure for finding such a basis Bardeen, Tung 1968, Tarrach 1975
Results in non-diagonal terms
Π1 (s) =
s − q32
π
Z∞
4Mπ2
ds′
2ξ 1 ξ 2
′
′
′
¯0
¯0
s
h
s
+
Im
h
K
(s
,s)Im
1
++,++
00,++
s′ − q32
λ (s′ ,q12 ,q22 )
Solved for S-wave, D-wave calculation in progress
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
13
ππ intermediate states: master formula
Colangelo, MH, Procura, Stoffer 2014
Master formula for ππ intermediate states
6
aππ
µ =e
Z
4
d q1
(2π )4
Z
∑ Ti (q1 , q2 ; p)Ii s, q12 , q22
d q2
i
(2π )4 q12 q22 s (p + q1 )2 − m2 (p − q2 )2 − m2
q1 , µ
−q3 , λ
q2 , ν
k, σ
4
Ii s, q12 , q22 : dispersive integrals over γ ∗ γ ∗ → ππ helicity partial waves
What is included? How?
֒→ sorted by analytic structure in the crossed channel
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
14
Preliminary numbers: FsQED






V 2 V 2 

2
V
ΠFsQED
µνλ σ = Fπ q1 Fπ q2 Fπ q3 × 

` factor FπV (s) = exp
Input for FπV q 2 : Omnes
Results for aFsQED
in units of 10−11
µ
∞
s R
π
4Mπ2
δ 1 (s ′ )
1
ds′ s′ (s
′ −s)
phase shift δ11
loop integrals
DR 1
DR 2
CCL
−13.77 ± 0.01
−15.87 ± 0.01
−14.57 ± 0.01
−14.65 ± 0.01
−16.90 ± 0.02
−15.53 ± 0.01
CCL +
ρ ′ , ρ ′′
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
15
Preliminary numbers: FsQED






V 2 V 2 

2
V
ΠFsQED
µνλ σ = Fπ q1 Fπ q2 Fπ q3 × 

` factor FπV (s) = exp
Input for FπV q 2 : Omnes
Results for aFsQED
in units of 10−11
µ
∞
s R
π
4Mπ2
δ 1 (s ′ )
1
ds′ s′ (s
′ −s)
phase shift δ11
loop integrals
DR 1
DR 2
CCL
−13.77 ± 0.01
−15.87 ± 0.01
−14.57 ± 0.01
−14.65 ± 0.01
−16.90 ± 0.02
−15.53 ± 0.01
CCL +
ρ ′ , ρ ′′
Dependence on FπV (s): analytic continuation can be stabilized using space-like data
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
15
Preliminary numbers: FsQED






V 2 V 2 

2
V
ΠFsQED
µνλ σ = Fπ q1 Fπ q2 Fπ q3 × 

` factor FπV (s) = exp
Input for FπV q 2 : Omnes
Results for aFsQED
in units of 10−11
µ
∞
s R
π
4Mπ2
δ 1 (s ′ )
1
ds′ s′ (s
′ −s)
phase shift δ11
loop integrals
DR 1
DR 2
CCL
−13.77 ± 0.01
−15.87 ± 0.01
−14.57 ± 0.01
−14.65 ± 0.01
−16.90 ± 0.02
−15.53 ± 0.01
CCL +
ρ ′ , ρ ′′
Dependence on FπV (s): analytic continuation can be stabilized using space-like data
µνλ σ
Basis Ai
not unique (but: Πi need to be free of kinematic singularities)
֒→ DR 1/2 equivalent for suitable high-energy behavior ⇒ theoretical uncertainty
Why does this work so well? it shouldn’t: double-spectral regions, only S-waves!
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
15
γ ∗ γ ∗ → ππ partial waves: unitarity relations
process
γv∗
building blocks and SC
left-hand cut
γv∗
π
γs∗
2π
γv∗
γs∗
γs∗
γv∗
ω, φ
ω, φ
γv∗
3π (∼ ω , φ )
γv∗
unitarity relations
γ
S,D
α1 ± β1 , α2 ± β2
γ
γ∗
S,D
γ
α1 q 2 ± β1 q 2 , ChPT
e+ e− → ππγ
e+ e− → e+ e− ππ
on-shell
singly-virtual
ChPT
γ∗
S,D
γ∗
M. Hoferichter (IKP & EMMI, TU Darmstadt)
(e+ e− → ππγ )
e+ e− → e+ e− ππ
HLbL scattering: a dispersive approach
doubly-virtual
Messina, September 26, 2014
16
Simplified input for γ ∗ γ ∗ → ππ : pion pole
` representation for S-wave
Omnes
I
I
h0,++
(s) = N0,++
(s) +
ΩI0 (s)
π
Zsm
4Mπ2
ds ′
sin δ0I (s ′ )
|ΩI0 (s ′ )|
1
s′ − s
−
s ′ − q12 − q22
2ξ1 ξ2
I
′
I
′
N0,++
N0,00
(s
)
+
(s
)
2
2
2
2
λ s ′ , q1 , q2
λ s ′ , q1 , q2
∗ → ππ
Starting point: Roy–Steinerequations for γ ∗ γ ` factors ΩI0 (s) = exp
Omnes
sm
s R
π
4Mπ2
δ I (s ′ )
ds′ s′ (s0 ′ −s)
I
LHC approximated by pion pole N0,
λ1 λ2 only
FπV


q12 FπV q22 × 
I
Finite matching point: h0,++
(s) = 0 above sm
√
Take sm = 0.98 GeV
¯ ⇒ “σ -contribution”
֒→ no f0 (980) or coupling to K K
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach



Messina, September 26, 2014
17
Preliminary numbers: ππ rescattering for S-waves
−11
aππ
µ in units of 10
phase shift δ11
CCL
CCL + ρ ′ , ρ ′′
M. Hoferichter (IKP & EMMI, TU Darmstadt)
I = 0 DR 1
I = 0 DR 2
I = 2 DR 1
I = 2 DR 2
−7.13 ± 0.03
−6.75 ± 0.06
1.82 ± 0.01
1.68 ± 0.01
−7.79 ± 0.03
−7.38 ± 0.06
2.00 ± 0.01
1.84 ± 0.01
HLbL scattering: a dispersive approach
Messina, September 26, 2014
18
Preliminary numbers: ππ rescattering for S-waves
−11
aππ
µ in units of 10
phase shift δ11
CCL
CCL + ρ ′ , ρ ′′
I = 0 DR 1
I = 0 DR 2
I = 2 DR 1
I = 2 DR 2
−7.13 ± 0.03
−6.75 ± 0.06
1.82 ± 0.01
1.68 ± 0.01
−7.79 ± 0.03
−7.38 ± 0.06
2.00 ± 0.01
1.84 ± 0.01
Adding the FsQED contribution
phase shift δ11
CCL
CCL +
ρ ′ , ρ ′′
M. Hoferichter (IKP & EMMI, TU Darmstadt)
FsQED
sum DR 1
sum DR 2
−13.77 ± 0.01
−19.08 ± 0.03
−18.84 ± 0.06
−14.65 ± 0.01
−20.44 ± 0.03
−20.19 ± 0.06
HLbL scattering: a dispersive approach
Messina, September 26, 2014
18
Preliminary numbers: ππ rescattering for S-waves
−11
aππ
µ in units of 10
phase shift δ11
CCL
CCL + ρ ′ , ρ ′′
I = 0 DR 1
I = 0 DR 2
I = 2 DR 1
I = 2 DR 2
−7.13 ± 0.03
−6.75 ± 0.06
1.82 ± 0.01
1.68 ± 0.01
−7.79 ± 0.03
−7.38 ± 0.06
2.00 ± 0.01
1.84 ± 0.01
Adding the FsQED contribution
phase shift δ11
CCL
CCL +
ρ ′ , ρ ′′
FsQED
sum DR 1
sum DR 2
−13.77 ± 0.01
−19.08 ± 0.03
−18.84 ± 0.06
−14.65 ± 0.01
−20.44 ± 0.03
−20.19 ± 0.06
Comparing to the literature Jegerlehner, Nyffeler 2009
Contribution
BPP
HKS
KN
MV
BP
PdRV
N/JN
π 0 , η, η′
π , K loops
π , K loops + other subleading in Nc
85 ± 13
−19 ± 13
–
2.5 ± 1.0
−6.8 ± 2.0
21 ± 3
82.7 ± 6.4
−4.5 ± 8.1
–
1.7 ± 1.7
–
9.7 ± 11.1
83 ± 12
–
–
–
–
–
114 ± 10
–
0 ± 10
22 ± 5
–
–
–
–
–
–
–
–
114 ± 13
−19 ± 19
–
15 ± 10
−7 ± 7
2.3±
99 ± 16
−19 ± 13
83 ± 32
89.6 ± 15.4
80 ± 40
136 ± 25
110 ± 40
105 ± 26
116 ± 39
Axial vectors
Scalars
Quark loops
Total
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
–
22 ± 5
−7 ± 2
21 ± 3
Messina, September 26, 2014
18
Summary
Dispersive framework for the calculation of the HLbL contribution to aµ
Includes one- and two-pion intermediate states
Master formula for ππ in terms of γ ∗ γ ∗ → ππ partial waves
Next steps
Pion transition form factor
Refinement of γ ∗ γ ∗ → ππ input
Comprehensive treatment of D-waves
Error analysis: which input quantity has the biggest impact on aµ ?
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
19
Outlook: towards a data-driven analysis of HLbL
γπ → ππ
e+ e− → e+ e− π 0
e+ e− → π 0 γ
e+ e− → ππγ
ω, φ → ππγ
ππ → ππ
Pion transition formfactor
Fπ0 γ ∗γ ∗ q12 , q22
Partial waves for
γ ∗ γ ∗ → ππ
pion polarizabilities
e+ e− → 3π
ω, φ → 3π
e+ e− → e+ e− ππ
Pion vector
form factor FVπ
γπ → γπ
ω, φ → π 0 γ ∗
Reconstruction of γ ∗ γ ∗ → ππ , π 0 : combine experiment and theory constraints
֒→ simplified version for ππ in this talk, some first results for π 0
¯ , multi-pion channels (resonances), pQCD constraints, . . .
Beyond: η , η ′ , K K
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
20
Pion vector form factor
Dispersive approach: resum ππ rescattering FVπ as example
Unitarity for pion vector form factor
Im FVπ (s) = θ s − 4Mπ2 FVπ (s)e −i δ1 (s) sin δ1 (s)
FVπ
t1
֒→ final-state theorem: phase of FVπ equals ππ P-wave phase δ1
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Watson 1954
Messina, September 26, 2014
21
Pion vector form factor
Dispersive approach: resum ππ rescattering FVπ as example
Unitarity for pion vector form factor
FVπ
Im FVπ (s) = θ s − 4Mπ2 FVπ (s)e −i δ1 (s) sin δ1 (s)
t1
֒→ final-state theorem: phase of FVπ equals ππ P-wave phase δ1
Watson 1954
` function Omnes
` 1958
Solution in terms of Omnes
FVπ (s) = P(s)Ω1 (s)
Ω1 (s) = exp
(
s
π
Z∞
4Mπ2
ds′
δ1 (s′ )
′
s (s′ − s)
)
Asymptotics + normalization ⇒ P(s) = 1
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
21
γ ∗ γ ∗ → ππ partial waves
Roy(–Steiner) equations = Dispersion relations + partial-wave expansion
+ crossing symmetry + unitarity + gauge invariance
On-shell case γγ → ππ
2011
Moussallam 2010, MH, Phillips, Schat
֒→ precision determination of σ → γγ coupling
Singly-virtual γ ∗ γ → ππ
Moussallam 2013
Doubly-virtual γ ∗ γ ∗ → ππ : anomalous thresholds
Colangelo, MH, Procura, Stoffer arXiv:1309.6877
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
22
γ ∗ γ ∗ → ππ partial waves
Roy(–Steiner) equations = Dispersion relations + partial-wave expansion
+ crossing symmetry + unitarity + gauge invariance
On-shell case γγ → ππ
2011
Moussallam 2010, MH, Phillips, Schat
֒→ precision determination of σ → γγ coupling
Singly-virtual γ ∗ γ → ππ
Moussallam 2013
Doubly-virtual γ ∗ γ ∗ → ππ : anomalous thresholds
Colangelo, MH, Procura, Stoffer arXiv:1309.6877
Constraints
Low energies: pion polarizabilities, ChPT
e−
π
π
π−
Primakoff: γπ → γπ (COMPASS), γγ → ππ (JLab)
π
e+
−
Scattering: e+ e− → e+ e− ππ , e+ e− → ππγ
(Transition) Form factors: FVπ , ω , φ → π 0 γ ∗
֒→ discuss these constraints in the following
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
e−
π
Z
π
e+
Messina, September 26, 2014
22
Left-hand cut
ω, φ
Pion pole: coupling determined by FVπ as before
Multi-pion intermediate states: approximate in terms of resonances
2π ∼ ρ : can even be done exactly using γ ∗ → 3π amplitude
֒→ see pion transition form factor
3π ∼ ω , φ : narrow-width approximation
֒→ transition form factors for ω , φ → π 0 γ ∗
Higher intermediate states also potentially relevant: axials, tensors
֒→ sum rules to constrain their transition form factors Pauk, Vanderhaeghen 2014
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
23
ω , φ → π 0 γ ∗ transition form factor
100
100
VMD
f1 (s) = a Ω(s)
once subtracted f1 (s)
twice subtracted f1 (s)
|Fωπ0 (s)|2
|Fφπ0 (s)|2
NA60 ’09
NA60 ’11
Lepton-G
VMD
Terschl¨
usen et al.
f1 (s) = a Ω(s)
full dispersive
10
10
1
1
0
0.1
0.2 √0.3
0.4
s [GeV]
0.5
0.6
0
0.1 0.2
0.3 √0.4
0.5 0.6 0.7 0.8
s [GeV]
Schneider, Kubis, Niecknig 2012
Puzzle of steep rise in Fωπ 0
֒→ measurement of Fφ π 0 would be extremely valuable
Clarification important for pion transition form factor, but also γ ∗ γ ∗ → ππ
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
24
Subtraction functions
` representation for S-wave
Omnes
"
h0,++ (s) = ∆0,++ (s) + Ω0 (s)
+
+
s(s − s+ )
2π
2q12 q22 s
π
Z∞
ds ′
4Mπ2
Z∞
4Mπ2
ds ′
1
1
(s − s+ )a+ q12 , q22 + (s − s− )a− q12 , q22 + q12 q22 b q12 , q22
2
2
sin δ0 (s ′ )∆0,++ (s ′ )
s(s − s− )
+
s ′ (s ′ − s+ )(s ′ − s)|Ω0 (s ′ )|
2π
′
′
sin δ0 (s )∆0,00 (s )
s ′ (s ′ − s+ )(s ′ − s− )|Ω0 (s ′ )|
#
Z∞
ds ′
4Mπ2
sin δ0 (s ′ )∆0,++ (s ′ )
s ′ (s ′ − s− )(s ′ − s)|Ω0 (s ′ )|
q
s± = q12 + q22 ± 2 q12 q22
Inhomogeneities ∆0,++ (s), ∆0,00 (s) include left-hand cut
Subtraction functions
q
b q12 ,q22 and a+ q12 ,q22 − a− q12 ,q22 multiply q12 q22 and q12 q22
֒→ inherently doubly-virtual observables ⇒ need ChPT (or lattice)
However: a(q12 ,q22 ) = (a+ (q12 ,q22 ) + a− (q12 ,q22 ))/2 fixed by singly-virtual measurements
֒→ compare with chiral prediction, uncertainty estimates for the other functions
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
25
Subtraction functions: chiral constraints
1-loop result for arbitrary qi2 , e.g.
0
aπ q12 ,q22 = −
Mπ2
2 2
8π 2 Fπ2 q12 − q2
+ q12 1 +
(
q12 + q22 + 2 Mπ2 q12 + q22 + q12 q22 C0 q12 ,q22
)
2 6q22
¯ q 2 + q 2 1 − 6q1
¯ q2
J
J
1
2
2
q12 − q22
q12 − q22
Special case: q12 = q22 = 0
¯l6 − ¯l5
π ±
Mπ
+ ··· =
α1 − β1
2α
48π 2 Fπ2
π 0
0
1
Mπ
aπ (0,0) = −
+ ··· =
α − β1
2α 1
96π 2 Fπ2
±
aπ (0,0) =
±
b π (0,0) = 0
0
b π (0,0) = −
1
+ ···
1440π 2 Fπ2 Mπ2
֒→ resum higher chiral orders into pion polarizabilities
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
26
Subtraction functions: dispersive representation
0.4
0.5
Akhmetshin (2003)
βρ , βω : fitted
βρ = βω = 0
Cross-section (nb)
Cross-section (nb)
0.5
0.3
0.2
0.1
0
0.4
Achasov (2002)
βρ , βω : fitted
βρ = βω = 0
0.3
0.2
0.1
0
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
0.6
0.65
0.7
q (GeV)
0.75
0.8
0.85
0.9
0.95
q (GeV)
Moussallam 2013
Singly-virtual case: phenomenological representation with chiral constraints
֒→ parameters fixed from e + e − → π 0 π 0 γ (CMD2 and SND) Moussallam 2013
Dispersive representation: imaginary part from 2π , 3π , . . .
֒→ analytic continuation from time-like to space-like kinematics
Example: I = 2 ⇒ isovector photons ⇒ 2π ∼ ρ
h
i
a2 q12 , q22 = α0 α 2 + α q12 F ρ q12 + q22 F ρ q22 + q12 q22 F ρ q12 F ρ q22
∗
∞
1 Z
q 3 (s) F V (s) Ω1 (s)
F ρ q2 =
ds ππ 3/2 π
π
s
s − q2
qππ (s) =
4Mπ2
r
s
− Mπ2
4
֒→ α0 and α can be determined from a2 (q 2 , 0) alone!
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
27
Quark-mass dependence: phase shifts
1-loop IAM with low-energy constants
´
from Hanhart, Pelaez,
R´ıos 2008
Quark-mass dependence of the
phase shifts
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
28
Quark-mass dependence: FsQED
Check accuracy of IAM
phase shift δ11
loop integrals
DR 1
DR 2
CCL
−13.77 ± 0.01
−15.87 ± 0.01
−14.57 ± 0.01
−14.65 ± 0.01
−16.90 ± 0.02
−15.53 ± 0.01
−13.72 ± 0.01
−15.82 ± 0.01
−14.51 ± 0.01
CCL +
ρ ′ , ρ ′′
IAM
Quark-mass dependence of FπV (s) via
` representation
phase shift in Omnes
Guo, Hanhart, Llanes-Estrada, Meißner 2009
֒→ relevant for large Mπ
Quark-mass dependence of aFsQED
µ
roughly ∝ Mπ−2
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
29
Quark-mass dependence: S-waves
DR 1
DR 1 + IAM
DR 2
DR 2 + IAM
I=0
−7.13 ± 0.03
−7.70 ± 0.04
−6.75 ± 0.06
−7.28 ± 0.06
I=2
1.82 ± 0.01
1.73 ± 0.01
1.68 ± 0.01
1.60 ± 0.01
Choose
√
sm = 2MK = 2
q
phys 2
MK
phys 2 + Mπ2 − Mπ
Beyond 3Mπ : σ and ρ become bound states
/2, compare to CCL δ11
֒→ highly non-trivial quark-mass dependence
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
30
Anomalous thresholds
q22 → −∞
q22 = 0
−
e
π
q22 = 4Mπ2 − q12
π
e+
q22 = 4Mπ2
q22 → ∞
Analytic continuation in qi2 in time-like region non-trivial in doubly-virtual case
Singularities from second sheet move onto first one
֒→ need to deform the integration contour
Problem already occurs for a simple triangle loop function C0 (s)
֒→ extra factor tℓ (s)/Ωℓ (s) is well defined in the whole complex plane
` solution
֒→ remedy in case of C0 (s) can be taken over to full Omnes
Becomes relevant for e + e + → e + e − ππ in time-like kinematics
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
31
Numerical check of anomalous thresholds
numerical
analytic
dispersive
numerical
analytic
dispersive
5
4
3
2
1
0
−Im C0 (s)
−Re C0 (s)
3
2
1
0
-1
-2
-10
-10
-5
-10
-5
0
q12
0
5
5
-5
q22
-10
-5
1010
0
q12
0
5
5
q22
1010
Comparison for s = 5, Mπ = 1
֒→ dispersive reconstruction of C0 (s) works!
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
32
Numerical check of anomalous thresholds
numerical
analytic
dispersive
numerical
analytic
dispersive
5
4
3
2
1
0
−Im C0 (s)
−Re C0 (s)
3
2
1
0
-1
-2
-10
-10
-5
-10
-5
0
q12
0
5
5
-5
q22
-10
-5
1010
0
q12
0
5
5
q22
1010
Ignore anomalous piece
֒→ substantial deviations for large virtualities!
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
33
γ ∗ γ ∗ → ππ
Left-hand cut approximated by pion pole + resonances
Unitarity for γ ∗ γ ∗ → ππ system: Watson’s theorem
disc f0 s; q12 , q22 = 2i σs f0 s; q12 , q22 t0∗ (s)
t0 (s) = σ1 ei δ0 (s) sin δ0 (s)
s
σs =
r
1−
f0
t0
4Mπ2
s
` function, e.g. for pion pole only
֒→ solution in terms of Omnes
∞
Ω (s) Z
N0 s′ ; q12 ,q22 sin δ0 (s′ )
f0 s; q12 ,q22 = 0
ds′
π
(s′ − s)|Ω0 (s′ )|
4Mπ2
q
λ (s,q12 ,q22 )
q
L = log
s − q12 − q22 − σs λ (s,q12 ,q22 )
s − q12 − q22 + σs
N0 s; q12 ,q22 =
σs
2L
q
λ s,q12 ,q22
λ (x,y ,z) = x 2 + y 2 + z 2 − 2(xy + xz + yz)
Analytic continuation in qi2 ?
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
34
γ ∗ γ ∗ → ππ : analytic continuation
L = log
√
2
s−q12 −q22 +σs λ (s,q12 ,q22 ) q2 →0
σs
√
−→ ± log 1+
2
2
2
2
1−σs
s−q1 −q2 −σs λ (s,q1 ,q2 )
Singularities of the log: anomalous thresholds
s± = q12 + q22 −
q12 q22
2Mπ2
±
1
2Mπ2
q
q12 q12 − 4Mπ2 q22 q22 − 4Mπ2
` derivation breaks down
֒→ usual Omnes
Idea: consider first the scalar loop function
d4 k
k 2 − Mπ2 (k + q1 )2 − Mπ2 (k − q2 )2 − Mπ2
2 2
L = −π i σs N0 s; q1 ,q2
Z
1
C0 (s) ≡ C0 (q1 + q2 )2 ; q12 ,q22 = 2
iπ
2π i
disc C0 (s) = − q
λ s,q12 ,q22
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
35
γ ∗ γ ∗ → ππ : anomalous thresholds
s+ = q12 + q22 −
q12 q22
2Mπ2
+
1
2Mπ2
q
q12 q12 − 4Mπ2 q22 q22 − 4Mπ2
Anomalous threshold usually on the
second sheet
Trajectory of s+ (q22 ) for 0 ≤ q12 ≤ 4Mπ2
֒→ moves through unitarity cut onto first sheet
q22 → −∞
q22 = 0
q22 = 4Mπ2 − q12
q22 = 4Mπ2
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
q22 → ∞
Messina, September 26, 2014
36
γ ∗ γ ∗ → ππ : anomalous thresholds
s+ = q12 + q22 −
q12 q22
2Mπ2
+
1
2Mπ2
q
q12 q12 − 4Mπ2 q22 q22 − 4Mπ2
Anomalous threshold usually on the
second sheet
Trajectory of s+ (q22 ) for 0 ≤ q12 ≤ 4Mπ2
֒→ moves through unitarity cut onto first sheet
q22 → −∞
q22 = 0
q22 = 4Mπ2 − q12
Need to deform the contour
C0 (s) =
1
2π i
Z∞
4Mπ2
ds′
disc C0 (s′ )
s′ − s
+ θ q12 + q22 − 4Mπ2
q22 = 4Mπ2
q22 → ∞
1
1 Z
∂ sx discan C0 (sx )
dx
2π i
∂x
sx − s
sx = x4Mπ2 + (1 − x)s+
M. Hoferichter (IKP & EMMI, TU Darmstadt)
0
4π 2
discan C0 (s) = q
λ s,q12 ,q22
HLbL scattering: a dispersive approach
Messina, September 26, 2014
36
` representation
γ ∗ γ ∗ → ππ : back to the Omnes
f0 s; q12 , q22 =
∞
Ω0 (s) R
π
4Mπ2
ds′
N0 (s ′ ;q12 ,q22 )sin δ0 (s ′ )
(s ′ −s)|Ω0 (s ′ )|
Integrand similar to the scalar-loop example
N0 s; q12 ,q22 sin δ0 (s)
disc C0 (s) sin δ0 (s)
disc C0 (s) t0 (s)
=−
=−
|Ω0 (s)|
π i σs
|Ω0 (s)|
πi
Ω0 (s)
֒→ additional factor independent of qi2 and well-defined in the whole s-plane
M. Hoferichter (IKP & EMMI, TU Darmstadt)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
37
` representation
γ ∗ γ ∗ → ππ : back to the Omnes
f0 s; q12 , q22 =
∞
Ω0 (s) R
π
4Mπ2
ds′
N0 (s ′ ;q12 ,q22 )sin δ0 (s ′ )
(s ′ −s)|Ω0 (s ′ )|
Integrand similar to the scalar-loop example
N0 s; q12 ,q22 sin δ0 (s)
disc C0 (s) sin δ0 (s)
disc C0 (s) t0 (s)
=−
=−
|Ω0 (s)|
π i σs
|Ω0 (s)|
πi
Ω0 (s)
֒→ additional factor independent of qi2 and well-defined in the whole s-plane
` representation for γ ∗ γ ∗ → ππ
Omnes
∞
Ω (s) Z
N0 s′ ; q12 ,q22 sin δ0 (s′ )
f0 s; q12 ,q22 = 0
ds′
π
(s′ − s)|Ω0 (s′ )|
4Mπ2
1
+ θ q12 + q22 − 4Mπ2
sx = x4Mπ2
M. Hoferichter (IKP & EMMI, TU Darmstadt)
Ω0 (s) Z
+ (1 − x)s+
2π i
0
dx
∂ sx discan f0 sx ; q12 ,q22
∂x
sx − s
8π
t0 (s)
discan f0 s; q12 ,q22 = − q
λ s,q12 ,q22 Ω0 (s)
HLbL scattering: a dispersive approach
Messina, September 26, 2014
37

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