Meson Chiral Perturbation Theory
Bastian Kubis
Helmholtz-Institut für Strahlen- und Kernphysik (Theorie)
Bethe Center for Theoretical Physics
Universität Bonn, Germany
Bosen, 1/9/2009
Meson ChPT, Bosen, 1/9/2009 – p. 1
Outline
Introduction
• Chiral perturbation theory on one page
The pion mass and ππ scattering
• What one can learn from ππ scattering lengths
• Experiments and how to interpret them:
Ke4 , pionium, and K → 3π
Chiral SU(3) and quark mass ratios
• Quark mass ratios, Leutwyler’s ellipse, and Dashen’s theorem
• η → 3π
What about the η ′ ?
Conclusions
Meson ChPT, Bosen, 1/9/2009 – p. 2
Chiral perturbation theory
Chiral perturbation theory (ChPT):
Weinberg 1966; Gasser, Leutwyler 1984
• chiral symmetry strongly constrains properties and interactions
of would-be Goldstone bosons, the pions
• systematic expansion of low-energy observables in terms of
(small) quark masses and (small) momenta
• ChPT is the effective field theory of QCD / the Standard Model
at low energies:
Meson ChPT, Bosen, 1/9/2009 – p. 3
Chiral perturbation theory
Chiral perturbation theory (ChPT):
Weinberg 1966; Gasser, Leutwyler 1984
• chiral symmetry strongly constrains properties and interactions
of would-be Goldstone bosons, the pions
• systematic expansion of low-energy observables in terms of
(small) quark masses and (small) momenta
• ChPT is the effective field theory of QCD / the Standard Model
at low energies:
• Weinberg’s conjecture on effective field theory:
Weinberg 1979
Quantum Field Theory has no content besides
unitarity, analyticity, cluster decomposition, and
symmetries.
Meson ChPT, Bosen, 1/9/2009 – p. 3
Chiral perturbation theory
Chiral perturbation theory (ChPT):
Weinberg 1966; Gasser, Leutwyler 1984
• chiral symmetry strongly constrains properties and interactions
of would-be Goldstone bosons, the pions
• systematic expansion of low-energy observables in terms of
(small) quark masses and (small) momenta
• ChPT is the effective field theory of QCD / the Standard Model
at low energies:
• Weinberg’s conjecture on effective field theory:
Weinberg 1979
To calculate the S-matrix for any theory below
some scale, simply use the most general effective Lagrangian consistent with these principles in
terms of the appropriate asymptotic states.
Meson ChPT, Bosen, 1/9/2009 – p. 3
Chiral perturbation theory
Chiral perturbation theory (ChPT):
Weinberg 1966; Gasser, Leutwyler 1984
• chiral symmetry strongly constrains properties and interactions
of would-be Goldstone bosons, the pions
• systematic expansion of low-energy observables in terms of
(small) quark masses and (small) momenta
• ChPT is the effective field theory of QCD / the Standard Model
at low energies:
• Weinberg’s conjecture on effective field theory:
Weinberg 1979
To calculate the S-matrix for any theory below
some scale, simply use the most general effective Lagrangian consistent with these principles in
terms of the appropriate asymptotic states.
• ChPT equivalent to QCD
Leutwyler 1994
Meson ChPT, Bosen, 1/9/2009 – p. 3
Quark mass expansion of the pion mass
Gell-Mann–Oakes–Renner relation for the pion mass M :
M 2 = B(mu + md )
B=−
h0|ūu|0i
F2
• B 6= 0 is sufficient (but not necessary) condition for
chiral symmetry breaking; order parameter
Meson ChPT, Bosen, 1/9/2009 – p. 4
Quark mass expansion of the pion mass
Gell-Mann–Oakes–Renner relation for the pion mass M :
M 2 = B(mu + md )
B=−
h0|ūu|0i
F2
• B 6= 0 is sufficient (but not necessary) condition for
chiral symmetry breaking; order parameter
• at next-to-leading order (O(p4 ), one-loop):
Mπ2
4
M
6
ℓ̄
= M2 −
+
O(M
)
3
32π 2 F 2
• correction to Gell-Mann–Oakes–Renner relation:
Mπ2 = B(mu + md ) + A(mu + md )2 + O(m3q )
• how do we know that the leading term dominates?
what if ℓ̄3 is anomalously large? Knecht, Moussallam, Stern, Fuchs 1995
Meson ChPT, Bosen, 1/9/2009 – p. 4
ππ scattering at next-to-leading order
• I = 0 ππ scattering length at O(p4 ):
o
2 n
7M
π
4
1
+
a00 =
ǫ
+
O(M
)
π
2
32πFπ
2
5Mπ
3
21
21
ℓ̄
ℓ̄
ℓ̄
ℓ̄
ǫ =
+
2
+
+
+
1
2
3
4
84π 2 Fπ2
8
10
8
Meson ChPT, Bosen, 1/9/2009 – p. 5
ππ scattering at next-to-leading order
• I = 0 ππ scattering length at O(p4 ):
o
2 n
7M
π
4
1
+
a00 =
ǫ
+
O(M
)
π
2
32πFπ
2
3
21
5Mπ
21
+
+
ℓ̄
ℓ̄
ℓ̄
ℓ̄
ǫ =
+
2
+
1
2
3
4
84π 2 Fπ2 | {z } 8
10 |{z} 8
d-waves
hr 2 iS
π
Meson ChPT, Bosen, 1/9/2009 – p. 5
ππ scattering at next-to-leading order
• I = 0 ππ scattering length at O(p4 ):
o
2 n
7M
π
4
1
+
a00 =
ǫ
+
O(M
)
π
2
32πFπ
2
3
21
5Mπ
21
+
+
ℓ̄
ℓ̄
ℓ̄
ℓ̄
ǫ =
+
2
+
1
2
3
4
84π 2 Fπ2 | {z } 8
10 |{z} 8
d-waves
hr 2 iS
π
Mπ2 2 S 200π 2 2 0
Mπ2
2
Fπ Mπ a2 + 2a2 −
15ℓ̄3 − 353
=
hr iπ +
3
7
672π 2 Fπ2
• therefore: all we need to do is to measure a00 ⇒ extract ℓ̄3
know how much of Mπ2 is due to linear term in quark masses!
• . . . but how do you actually measure ππ scattering lengths?
Meson ChPT, Bosen, 1/9/2009 – p. 5
Experiments vs. theory on ππ scattering
Experiments:
• reactions on nucleons, e.g. πN → ππN
• K + → π + π − e+ νe (Ke4 )
BNL-865, NA48/2, KLOE
• pionium (= π + π − atom) lifetime
DIRAC
• cusp in K + → π 0 π 0 π +
NA48/2
Meson ChPT, Bosen, 1/9/2009 – p. 6
Experiments vs. theory on ππ scattering
Experiments:
• reactions on nucleons, e.g. πN → ππN
• K + → π + π − e+ νe (Ke4 )
BNL-865, NA48/2, KLOE
• pionium (= π + π − atom) lifetime
DIRAC
• cusp in K + → π 0 π 0 π +
NA48/2
Theory prediction:
• 2-loop ChPT + Roy equations (dispersion theory):
a0
a2
a0 − a2
=
=
=
0.220 ± 0.005
−0.0444 ± 0.0010
0.265 ± 0.004
(for QCD in the isospin limit) ⇒ ≃ 1.5% theoretical precision
Colangelo, Gasser, Leutwyler, PLB 488 (2000) 261, NPB 603 (2001) 125
Meson ChPT, Bosen, 1/9/2009 – p. 6
Ke4 decays and ππ scattering
e+
νe
π
π+
π
π−
K+
Why can one measure ππ scattering in such a process?
Meson ChPT, Bosen, 1/9/2009 – p. 7
Ke4 decays and ππ scattering
e+
νe
π
π+
π
π−
K+
Why can one measure ππ scattering in such a process?
• Answer:
⊲ decay Ke4 described by form factors
⊲ (partial waves of) form factors have phases of ππ interaction
(Watson’s final state theorem)
⊲ measure interference between partial waves δ00 − δ11
Meson ChPT, Bosen, 1/9/2009 – p. 7
Ke4 decays and ππ scattering
e+
νe
π
π+
π
π−
K+
Why can one measure ππ scattering in such a process?
• Answer:
⊲ decay Ke4 described by form factors
⊲ (partial waves of) form factors have phases of ππ interaction
(Watson’s final state theorem)
⊲ measure interference between partial waves δ00 − δ11
• Problem:
⊲ enormous precision of the data
⊲ what about Watson’s theorem and isospin breaking?
NA48/2
Meson ChPT, Bosen, 1/9/2009 – p. 7
Isospin breaking in Ke4
Colangelo, Gasser, Rusetsky, EPJC 59 (2009) 777
• two isospin-breaking effects:
e 6= 0 ⇒ Mπ2+ 6= Mπ20 and mu − md (e.g. in π 0 η mixing)
K+
π+
π+
π0
η
s̄γµ γ5 u
π−
π−
π0
π0
π0
Meson ChPT, Bosen, 1/9/2009 – p. 8
Isospin breaking in Ke4
Colangelo, Gasser, Rusetsky, EPJC 59 (2009) 777
• two isospin-breaking effects:
e 6= 0 ⇒ Mπ2+ 6= Mπ20 and mu − md (e.g. in π 0 η mixing)
K+
π+
π0
π+
π0
π0
η
s̄γµ γ5 u
π−
π0
π−
• generates isospin- breaking
correction phase
• result:
Bloch-Devaux, KAON 09
a0 = 0.2209 ± 0.0049(stat)
±0.0018(syst) ± 0.0064(th)
ψ 0 - δ0 [millirad]
• effect on the scattering length
a0 ≈ 10%!
30
25
Isospin correction to be applied to the phase δ0
20
15
10
280
300
320
340
360
380
400
Mππ [MeV]
Meson ChPT, Bosen, 1/9/2009 – p. 8
Pionium lifetime
• π + π − system, bound by electromagnetism
calculate energy levels as in quantum mechanics for the
hydrogen atom!
• energy levels perturbed by strong interactions:
ground state not stable, decays: Aπ+ π− → π 0 π 0 , γγ, . . .
• (improved) Deser formula for the width:
Γ
δ
Deser 1954, Gasser et al. 2001
2 3
=
α p|A(π + π − → π 0 π 0 )thr |2 (1 + ǫ)
9
2 3
=
α p|a0 − a2 |2 (1 + δ)
9
= 0.058 ± 0.012
• taking scattering length information from theory, predict lifetime:
τ = (2.9 ± 0.1) × 10−15 s
• turn argument around: measure τ , extract a0 − a2
DIRAC 2005
Meson ChPT, Bosen, 1/9/2009 – p. 9
The cusp effect in K ±
→ π±π0π0
x 10 2
50000
1600
45000
1400
40000
35000
1200
30000
25000
0.076
1000
0.077
0.078
0.079
0.08
800
600
400
200
0
0.08
0.09
0.1
0.11
0.12
0.13
• cusp at Mπ0 π0 = 2Mπ+
Batley et al., PLB 633 (2006) 173
Meson ChPT, Bosen, 1/9/2009 – p. 10
The cusp effect in K ±
→ π±π0π0
π+
π+
π+
x 10
K+
2
π
50000
1600
π
45000
1400
0
35000
1200
π+
30000
25000
0.076
0.077
0.078
0.079
0.08
s
π−
800
600
400
200
0
0.08
0.09
0.1
0.11
+
K+
π−
π0
π +π − → π 0π 0
40000
1000
0
π0
0.12
0.13
• cusp at Mπ0 π0 = 2Mπ+
Batley et al., PLB 633 (2006) 173
i
= ...+
v± (s)
16π
 q 2
4M +
2

π
 i
−
1
,
s
<
4M
+
π
s
v± (s) =
q

4M 2+

1 − sπ ,
s > 4Mπ2+
• interference tree + 1-loop
below π + π − threshold
• square-root behaviour = cusp
Cabibbo, PRL 93 (2004) 121801
Meson ChPT, Bosen, 1/9/2009 – p. 10
Non-relativistic effective field theory
• theoretical tool for scattering length extraction from cusp:
non-relativistic effective field theory
Colangelo, Gasser, Kubis, Rusetsky, PLB 638 (2006) 187
⊲
⊲
⊲
⊲
parametrise T directly in terms of scattering lengths etc.
no quark-mass expansion of these parameters (↔ ChPT)
do not predict these, extract as parameters from data
retain recoil corrections ⇔ correct relativistic kinematics
Meson ChPT, Bosen, 1/9/2009 – p. 11
Non-relativistic effective field theory
• theoretical tool for scattering length extraction from cusp:
non-relativistic effective field theory
Colangelo, Gasser, Kubis, Rusetsky, PLB 638 (2006) 187
⊲
⊲
⊲
⊲
parametrise T directly in terms of scattering lengths etc.
no quark-mass expansion of these parameters (↔ ChPT)
do not predict these, extract as parameters from data
retain recoil corrections ⇔ correct relativistic kinematics
• combined expansion in a (ππ scattering lengths) and
non-relativistic parameter ǫ ∝ |pπ |/Mπ
Meson ChPT, Bosen, 1/9/2009 – p. 11
Non-relativistic effective field theory
• theoretical tool for scattering length extraction from cusp:
non-relativistic effective field theory
Colangelo, Gasser, Kubis, Rusetsky, PLB 638 (2006) 187
⊲
⊲
⊲
⊲
parametrise T directly in terms of scattering lengths etc.
no quark-mass expansion of these parameters (↔ ChPT)
do not predict these, extract as parameters from data
retain recoil corrections ⇔ correct relativistic kinematics
• combined expansion in a (ππ scattering lengths) and
non-relativistic parameter ǫ ∝ |pπ |/Mπ
• complete to O(ǫ4 , aǫ5 , a2 ǫ4 ); valid to all orders in quark masses
π+
π
π+
π+
+
K+
π0
π0
π+
π0
π0
K+
π+
π0
π−
K+
π−
π0
K+
π0
π−
π0
π−
π0
π+
π+
π0
• subleading sensitivity to a2 via two-loop effects
Meson ChPT, Bosen, 1/9/2009 – p. 11
Radiative corrections; result for scattering lengths
• radiative corrections change analytic structure near threshold:
Bissegger et al., NPB 806 (2009) 178
G(s) =
i
v±
16π
Meson ChPT, Bosen, 1/9/2009 – p. 12
Radiative corrections; result for scattering lengths
• radiative corrections change analytic structure near threshold:
Bissegger et al., NPB 806 (2009) 178
i
α
2
v± −
log −v± + C
G(s) =
16π
32π
⇒ large effect on scattering length extraction
shifts a0 − a2 by almost 10%!
Meson ChPT, Bosen, 1/9/2009 – p. 12
Radiative corrections; result for scattering lengths
• radiative corrections change analytic structure near threshold:
Bissegger et al., NPB 806 (2009) 178
i
α
2
v± −
log −v± + C
G(s) =
16π
32π
⇒ large effect on scattering length extraction
shifts a0 − a2 by almost 10%!
a0 − a2
a2
= 0.2571 ± 0.0048(stat) ± 0.0025(syst) ± 0.0014(ext)
= −0.024 ± 0.013(stat) ± 0.009(syst) ± 0.002(ext)
Giudici for NA48/2, Chiral Dynamics (2009)
Meson ChPT, Bosen, 1/9/2009 – p. 12
Combined information on ππ scattering lengths
0.01
NA48/2 combined Ke4 + Cusp
(stat. + syst.) errors
68% CL contour
0
-0.01
-0.02
Cusp
-0.03
ChPT
-0.04
-0.05
-0.06
Ke4
DIRAC
0.24
0.25
0.26
0.27
0.28
0.29
0.3
figure courtesy of B. Bloch-Devaux
Meson ChPT, Bosen, 1/9/2009 – p. 13
Chiral SU(3)
• expand around chiral SU(2) limit mu = md = 0
SSB
SU(2)L × SU(2)R −→ SU(2)V , 3 Goldstone bosons π ± , π 0
Meson ChPT, Bosen, 1/9/2009 – p. 14
Chiral SU(3)
• expand around chiral SU(3) limit mu = md = ms = 0
SSB
SU(3)L × SU(3)R −→ SU(3)V , 8 Goldstone bosons 3π , 4K , η
⊲ interrelates more processes (kaons, η )
⊲ ms ≫ mu , md , perturbation series converges much worse
Meson ChPT, Bosen, 1/9/2009 – p. 14
Chiral SU(3)
• expand around chiral SU(3) limit mu = md = ms = 0
SSB
SU(3)L × SU(3)R −→ SU(3)V , 8 Goldstone bosons 3π , 4K , η
⊲ interrelates more processes (kaons, η )
⊲ ms ≫ mu , md , perturbation series converges much worse
• Gell-Mann–Oakes–Renner relation for the pseudoscalar octet:

Mπ2± = B(mu + md ) + 2Ze2 F 2 
 Dashen′ s
Mπ20 ≈ B(mu + md )

 theorem
2
2 2
MK ± = B(mu + ms ) + 2Ze F
2
MK
= B(md + ms )
0
Mη2 ≈ B3 (mu + md + 4ms )
Meson ChPT, Bosen, 1/9/2009 – p. 14
Chiral SU(3)
• expand around chiral SU(3) limit mu = md = ms = 0
SSB
SU(3)L × SU(3)R −→ SU(3)V , 8 Goldstone bosons 3π , 4K , η
⊲ interrelates more processes (kaons, η )
⊲ ms ≫ mu , md , perturbation series converges much worse
• Gell-Mann–Oakes–Renner relation for the pseudoscalar octet:

Mπ2± = B(mu + md ) + 2Ze2 F 2 
 Dashen′ s
Mπ20 ≈ B(mu + md )

 theorem
2
2 2
MK ± = B(mu + ms ) + 2Ze F
2
MK
= B(md + ms )
0
Mη2 ≈ B3 (mu + md + 4ms )
• quark mass ratios:
2
2
2
2
MK
mu
+ − MK 0 + 2Mπ 0 − Mπ +
=
≈ 0.55
2
2 − M2
+
M
md
MK
0
K+
π+
ms
md
=
2
2
2
MK
0 + MK + − Mπ +
≈ 22
2 − M2 + M2
MK
0
K+
π+
Meson ChPT, Bosen, 1/9/2009 – p. 14
Quark mass ratios beyond leading order
• calculate pion/kaon/eta masses beyond leading order:
π, K, η
Li
π, K, η
+
+
Meson ChPT, Bosen, 1/9/2009 – p. 15
Quark mass ratios beyond leading order
• calculate pion/kaon/eta masses beyond leading order:
n
o
Mπ2+ = B(mu + md ) 1 + O(mu , md , ms )
n
o
2
MK
= B(mu + ms ) 1 + O(mu , md , ms )
+
Meson ChPT, Bosen, 1/9/2009 – p. 15
Quark mass ratios beyond leading order
• calculate pion/kaon/eta masses beyond leading order:
n
o
Mπ2+ = B(mu + md ) 1 + O(mu , md , ms )
n
o
2
MK
= B(mu + ms ) 1 + O(mu , md , ms )
+
• form dimensionless ratios: (m̂ = 21 (mu + md ))
o
2
MK
ms + m̂ n
2
1 + ∆M + O(mq )
=
2
Mπ
mu + md
o
2
2
(MK
md − mu n
0 − MK + )strong
2
∆
+
O(m
1
+
=
M
q)
2
2
MK − Mπ
ms − m̂
∆M
=
2
8(MK
− Mπ2 )
(2L8 − L5 ) + chiral logs
Fπ2
• double ratio Q2 particularly stable:
n
o
2
2
2
2
2
M
M
−
M
m
−
m̂
π
2
K
K
=
1
+
O(m
Q2 = 2s
q)
2
2
2
2
md − mu
Mπ (MK 0 − MK + )strong
Meson ChPT, Bosen, 1/9/2009 – p. 15
Quark mass ratios beyond leading order
• Leutwyler’s ellipse:
mu
md
2
+
1
Q2
ms
md
2
=1
Meson ChPT, Bosen, 1/9/2009 – p. 16
Quark mass ratios beyond leading order
• Leutwyler’s ellipse:
mu
md
2
+
1
Q2
ms
md
2
=1
• Dashen’s theorem:
2
2
2
2
2
(MK
+ − MK 0 )em = (Mπ + − Mπ 0 )em + O(e mq )
2
2
⇒
⇒ determine (MK
+ − MK 0 )strong
QDashen = 24.2
Meson ChPT, Bosen, 1/9/2009 – p. 16
Quark mass ratios beyond leading order
• Leutwyler’s ellipse:
mu
md
2
+
1
Q2
ms
md
2
=1
• Dashen’s theorem:
2
2
2
2
2
(MK
+ − MK 0 )em = (Mπ + − Mπ 0 )em + O(e mq )
2
2
⇒
⇒ determine (MK
+ − MK 0 )strong
QDashen = 24.2
• are O(e2 mq ) corrections big?
2
2
2
2
1 . (MK
+ − MK 0 )em /(Mπ + − Mπ 0 )em . 2.5
20.6 . Q . 24.2
Meson ChPT, Bosen, 1/9/2009 – p. 16
Where else to get information on Q2 ? —
η → 3π
• η → 3π isospin violating; two sources in the standard model:
HQCD (x)
HQED (x)
md − mu ¯
(dd − ūu)(x)
2
2 Z
e
dy D µν (x − y)T (jµ (x)jν (y))
= −
2
=
• electromagnetic contribution zero at leading order
ALO
c
B(mu − md )
√
=
3 3Fπ2
Sutherland 1967
3(s − s0 )
1+ 2
Mη − Mπ2
2
LO (s, t, u)
1 MK
3s − 4Mπ2
2
LO
2 M
√
=
Mπ − MK
, M (s, t, u) = 2
2
2
2
Q Mπ
Mη − Mπ2
3 3Fπ
⇒ (potentially) clean access to mu − md or Q2
• ChPT up to two loops
• strong experimental activities
Bijnens, Ghorbani, JHEP 0711 (2007) 030
WASA-at-COSY, PLB677 (2009) 24
MAMI-B, EPJA 39 (2009) 169
MAMI-C, PRC79 (2009) 035204
Meson ChPT, Bosen, 1/9/2009 – p. 17
Electromagnetic contributions in η
→ 3π
• NLO electromagnetic contributions O(e2 mq ), mq = m̂, ms small
Baur, Kambor, Wyler NPB 460 (1996) 127
• somewhat unnatural: excludes most obvious effects
⊲ photon loops ⇒ Coulomb pole in the charged amplitude
⊲ pion-mass difference Mπ2+ − Mπ20 = O(e2 ) ⇒
cusps in the neutral amplitude
both O(e2 (mu − md ))
• O(e2 (mu − md )) effects violate ∆I = 1 rule:
An (s, t, u) 6= Ac (s, t, u) + Ac (t, u, s) + Ac (u, s, t)
• however: EM corrections in general very small, although
corrections of order e2 (mu − md ) as large as e2 mq
Ditsche, Kubis, Meißner, EPJC 60 (2009) 83
Meson ChPT, Bosen, 1/9/2009 – p. 18
η → 3π : final-state interactions
• strong final-state interactions among pions (3 of them . . . )
Cronin 1967
⊲ tree level: Γ(η → π + π − π 0 ) = 66 eV
⊲ one-loop: Γ(η → π + π − π 0 ) = 160 ± 50 eV
Gasser, Leutwyler 1985
⊲ experimental: Γ(η → π + π − π 0 ) = 295 ± 17 eV
PDG 2006
• major source: large S-wave final-state rescattering ⇒
use dispersion relations to resum those beyond loop expansion
match subtraction constants to ChPT and/or to data
Kambor, Wiesendanger, Wyler, NPB 465 (1995) 215
Anisovich, Leutwyler, PLB 375 (1996) 335
Colangelo, Lanz, Passemar, in progress
Meson ChPT, Bosen, 1/9/2009 – p. 19
η → 3π : final-state interactions
• strong final-state interactions among pions (3 of them . . . )
Cronin 1967
⊲ tree level: Γ(η → π + π − π 0 ) = 66 eV
⊲ one-loop: Γ(η → π + π − π 0 ) = 160 ± 50 eV
Gasser, Leutwyler 1985
⊲ experimental: Γ(η → π + π − π 0 ) = 295 ± 17 eV
PDG 2006
• major source: large S-wave final-state rescattering ⇒
use dispersion relations to resum those beyond loop expansion
match subtraction constants to ChPT and/or to data
Kambor, Wiesendanger, Wyler, NPB 465 (1995) 215
Anisovich, Leutwyler, PLB 375 (1996) 335
Colangelo, Lanz, Passemar, in progress
Γ(η → 3π 0 )
• on the other hand: consider r =
Γ(η → π + π − π 0 )
ChPT: rtree = 1.54 , r1-loop = 1.46 , r2-loop = 1.47
PDG: r = 1.432 ± 0.026 (fit) , r = 1.48 ± 0.05 (average)
⇒ agrees rather well
Bijnens, Ghorbani, JHEP 0711 (2007) 030
Meson ChPT, Bosen, 1/9/2009 – p. 19
Combined result on quark mass ratio Q
Q
Lanz et al. (prel.)
26
Kambor et al.
Bijnens, Ghorbani
η → 3π vs. various corrections to Dashen’s theorem:
25
Dashen
24
23
Lattice
Bijnens&Prades
Donoghue&Perez
22
21
Anant.&Moussallam
0.5
η −> 3π
1.0
1.5
2.0
2.5
3.0
3.5
(MK+ - MK0)em [MeV]
Meson ChPT, Bosen, 1/9/2009 – p. 20
η → π + π − π 0 : Dalitz plot parameter
• |Ac (x, y)|2 = |Nc |
2
2
2
1 + ay + by + dx + . . .
a
KLOE
Crystal Barrel
ChPT O(p6 )
dispersive
−1.090 +
−
y=
b
0.009
0.020
−1.22 ± 0.07
−1.271 ± 0.075
−1.33
x=
√
3
2Mη Qη ( u
3
2Mη Qη (s0
− t)
− s)
d
0.124 ± 0.012
0.057 +
−
0.394 ± 0.102
0.055 ± 0.057
0.22 ± 0.11
0.26
0.009
0.017
0.06 ± 0.04
0.10
KLOE, JHEP 0805 (2008) 006
Crystal Barrel, PLB 417 (1998) 197
Bijnens, Ghorbani, JHEP 0711 (2007) 030
Kambor, Wiesendanger, Wyler, NPB 465 (1995) 215
• note significant discrepancy theory vs. experiment for b
Meson ChPT, Bosen, 1/9/2009 – p. 21
η → 3π 0 : Dalitz plot parameter
• |An (x, y)| = |Nn | 1 + 2αz + . . . , z = x2 + y 2
2
2
α
KLOE
WASA@COSY
arXiv:0707.4137
PLB677 (2009) 24
MAMI-B
EPJA 39 (2009) 169
MAMI-C
PRC79 (2009) 035204
ChPT O(p6 )
JHEP 0711 (2007) 030
dispersive
NPB 465 (1995) 215
−0.027
+ 0.006
− 0.007
−0.027 ± 0.009
−0.032 ± 0.003
−0.032 ± 0.003
+0.013 ± 0.032
−(0.007. . .0.014)
• puzzle: why doesn’t two-loop ChPT come out closer to the
dispersive result "naturally"?
Meson ChPT, Bosen, 1/9/2009 – p. 22
And what about the cusp effect in η
→ 3π 0 ?
• similar cusps in KL → 3π 0 , η → 3π 0 , η ′ → ηπ 0 π 0 . . . however:
π
π0
π
π0
+
+
π−
0
G
H
π0
∝ a0 − a2
⇒ cusp strength ∝ H/G × (a0 − a2 )
⇒ sensitivity on a0 − a2 depends strongly on relative strength
"charged" to "neutral" amplitudes
• H/G is strongly different for different channels:
H
(K + → 3π) ≈ 2
G
H
1
(η → 3π) ≈
G
3
H
1
(KL → 3π) ≈
G
3
√
H ′
(η → η 2π) ≈ 2
G
Meson ChPT, Bosen, 1/9/2009 – p. 23
And what about the cusp effect in η
→ 3π 0 ?
• sketches for one-loop cusps:
K + → π 0 π 0 π + (H/G ≈ 2):
0.276
0.278
0.280
1/2
s3
η → 3π 0 (H/G ≈ 1/3):
0.282
0.276
0.278
0.280
0.282
1/2
s3
• precision determinations of a0 − a2 much harder from η (or KL )
theoretical formalism exists
Bissegger et al., PLB 659 (2008) 576, NPB 806 (2009) 178
Gullström, Kupść, Rusetsky, PRC 79 (2009) 028201
• take cusp into account in precision determinations of α
Meson ChPT, Bosen, 1/9/2009 – p. 24
Some remarks on the η ′
• fermion part of the QCD Lagrangian in the chiral limit:
LQCD [qL , qR ] = iq̄L D
6 qL + iq̄R D
6 qR , q T = (u, d, s)
⇒ U(3)L × U(3)R = SU(3)L × SU(3)R × U(1)V × U(1)A symmetry
• U(1)A is anomalously broken, the flavour singlet η0 ≈ η ′ is no
Goldstone boson
• anomaly disappears in the limit Nc → ∞
⇒ different versions of U(3) ChPT,
with or without additional 1/Nc power counting
see talk by Escribano
• problems:
⊲ unclear whether derivative expansion for the η ′ works
⊲ large number of free parameters (low-energy constants)
Meson ChPT, Bosen, 1/9/2009 – p. 25
Hadronic decays of the η ′
• large final-state interactions expected in η ′ → ηππ and η ′ → 3π
e.g. Borasoy, Nißler, EPJA 26 (2005) 383
⇒ unitarised U(3) ChPT
• only includes two-particle unitarity:
Meson ChPT, Bosen, 1/9/2009 – p. 26
Hadronic decays of the η ′
• large final-state interactions expected in η ′ → ηππ and η ′ → 3π
e.g. Borasoy, Nißler, EPJA 26 (2005) 383
⇒ unitarised U(3) ChPT
• only includes two-particle unitarity:
• claim:
Gross, Treiman, Wilczek, PRD 19 (1979) 2188
′
0 + −
Γ(η → π π π )
∝
′
+
−
Γ(η → ηπ π )
md − mu
ms
2
assumptions: (a) A(η ′ → π 0 π + π − ) = ǫπ0 η A(η ′ → ηπ + π − )
(b) both amplitudes "essentially flat" in phase space
• refuted in unitarised U(3) ChPT: assumptions too simplistic
Borasoy, Meißner, Nißler, PLB 643 (2006) 41
Meson ChPT, Bosen, 1/9/2009 – p. 26
Prediction of the cusp in η ′
→ ηπ 0 π 0
BK, Schneider, EPJC 62 (2009) 511
• new ingredient: πη scattering in the final state
not know experimentally, very badly constrained from ChPT
• uncertainty can be absorbed in redefinition of polynomial terms:
• integrated event deficit ≈ 8% below the π + π − threshold
⇒ ELSA, MAMI-C, WASA@COSY, KLOE@DAφNE, BES-III
Meson ChPT, Bosen, 1/9/2009 – p. 27
Can one extract the πη scattering length?
• only possibility to distinguish πη rescattering from polynomial:
threshold effects at s1 = (Mπ + Mη )2
⇒ border of phase space, cannot go below the cusp
• square-root behaviour above threshold visible? no:
⇒ interference
∗
2-loop
×
tree
= −
1-loop
×
(1-loop)∗
• cusps at πη threshold cancel exactly
non-trivial 2-loop graph essential ingredient for this cancellation
would be missed in unitarised ChPT with 2-particle unitarity only
Meson ChPT, Bosen, 1/9/2009 – p. 28
Conclusions
• high-precision experiments require high-precision theory!
• isospin breaking & radiative corrections essential (Ke4 , K → 3π )
• combine ChPT with other rigorous, model-independent methods:
⊲ non-relativistic EFT (K → 3π , hadronic atoms)
⊲ dispersion relations (η → 3π )
• extension to the η ′ / into the resonance region a challenge
Meson ChPT, Bosen, 1/9/2009 – p. 29
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Meson ChPT, Bosen, 1/9/2009 – p. 30