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EFFECTIVE FIELD THEORY:
INTRODUCTION AND APPLICATIONS
Johan Bijnens
Lund University
[email protected]://www.thep.lu.se/∼bijnens
Various ChPT: http://www.thep.lu.se/∼bijnens/chpt.html
lavinet
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.1/103
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Overview
What is effective field theory?
Introduction to ChPT including possible problems
Results for two-flavour ChPT
Results for three flavour ChPT
Partial quenching and ChPT
η → 3πSome comments about Higgs sector and ChPT
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Wikipedia
http://en.wikipedia.org/wiki/Effective_field_theory
In physics, an effective field theory is an approximate theory(usually a quantum field theory) that contains theappropriate degrees of freedom to describe physicalphenomena occurring at a chosen length scale, but ignoresthe substructure and the degrees of freedom at shorterdistances (or, equivalently, higher energies).
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Wikipedia
http://en.wikipedia.org/wiki/Chiral_perturbation_theory
Chiral perturbation theory (ChPT) is an effective field theoryconstructed with a Lagrangian consistent with the(approximate) chiral symmetry of quantumchromodynamics (QCD), as well as the other symmetries ofparity and charge conjugation. ChPT is a theory whichallows one to study the low-energy dynamics of QCD. AsQCD becomes non-perturbative at low energy, it isimpossible to use perturbative methods to extractinformation from the partition function of QCD. Lattice QCDis one alternative method that has proved successful inextracting non-perturbative information.
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Effective Field Theory
Main Ideas:
Use right degrees of freedom : essence of (most)physics
If mass-gap in the excitation spectrum: neglect degreesof freedom above the gap.
Examples:
Solid state physics: conductors: neglectthe empty bands above the partially filledoneAtomic physics: Blue sky: neglect atomicstructure
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Power Counting
➠ gap in the spectrum =⇒ separation of scales➠ with the lower degrees of freedom, build the mostgeneral effective Lagrangian
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Power Counting
➠ gap in the spectrum =⇒ separation of scales➠ with the lower degrees of freedom, build the mostgeneral effective Lagrangian
➠ ∞# parameters➠ Where did my predictivity go ?
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.6/103
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Power Counting
➠ gap in the spectrum =⇒ separation of scales➠ with the lower degrees of freedom, build the mostgeneral effective Lagrangian
➠ ∞# parameters➠ Where did my predictivity go ?
=⇒ Need some ordering principle: power counting
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.6/103
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Power Counting
➠ gap in the spectrum =⇒ separation of scales➠ with the lower degrees of freedom, build the mostgeneral effective Lagrangian
➠ ∞# parameters➠ Where did my predictivity go ?
=⇒ Need some ordering principle: power counting➠ Taylor series expansion does not work (convergenceradius is zero)➠ Continuum of excitation states need to be taken intoaccount
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.6/103
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Why is the sky blue ?
System: Photons of visible light and neutral atomsLength scales: a few 1000 Å versus 1 ÅAtomic excitations suppressed by ≈ 10−3
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Why is the sky blue ?
System: Photons of visible light and neutral atomsLength scales: a few 1000 Å versus 1 ÅAtomic excitations suppressed by ≈ 10−3
LA = Φ†v∂tΦv + . . . LγA = GF 2µνΦ†vΦv + . . .Units with h/ = c = 1: G energy dimension −3:
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.7/103
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Why is the sky blue ?
System: Photons of visible light and neutral atomsLength scales: a few 1000 Å versus 1 ÅAtomic excitations suppressed by ≈ 10−3
LA = Φ†v∂tΦv + . . . LγA = GF 2µνΦ†vΦv + . . .Units with h/ = c = 1: G energy dimension −3:
σ ≈ G2E4γ
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.7/103
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Why is the sky blue ?
System: Photons of visible light and neutral atomsLength scales: a few 1000 Å versus 1 ÅAtomic excitations suppressed by ≈ 10−3
LA = Φ†v∂tΦv + . . . LγA = GF 2µνΦ†vΦv + . . .Units with h/ = c = 1: G energy dimension −3:
σ ≈ G2E4γ
blue light scatters a lot more than red{
=⇒ red sunsets=⇒ blue sky
Higher orders suppressed by 1 Å/λγ.
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.7/103
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Why Field Theory ?
➠ Only known way to combine QM and special relativity➠ Off-shell effects: there as new free parameters
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Why Field Theory ?
➠ Only known way to combine QM and special relativity➠ Off-shell effects: there as new free parameters
Drawbacks• Many parameters (but finite number at any order)any model has few parameters but model-space is large
• expansion: it might not converge or only badly
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.8/103
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Why Field Theory ?
➠ Only known way to combine QM and special relativity➠ Off-shell effects: there as new free parameters
Drawbacks• Many parameters (but finite number at any order)any model has few parameters but model-space is large
• expansion: it might not converge or only badly
Advantages• Calculations are (relatively) simple• It is general: model-independent• Theory =⇒ errors can be estimated• Systematic: ALL effects at a given order can be included• Even if no convergence: classification of models oftenuseful
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.8/103
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Examples of EFT
Fermi theory of the weak interaction
Chiral Perturbation Theory: hadronic physics
NRQCD
SCET
General relativity as an EFT
2,3,4 nucleon systems from EFT point of view
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references
A. Manohar, Effective Field Theories (Schladminglectures), hep-ph/9606222
I. Rothstein, Lectures on Effective Field Theories (TASIlectures), hep-ph/0308266
G. Ecker, Effective field theories, Encyclopedia ofMathematical Physics, hep-ph/0507056
D.B. Kaplan, Five lectures on effective field theory,nucl-th/0510023
A. Pich, Les Houches Lectures, hep-ph/9806303
S. Scherer, Introduction to chiral perturbation theory,hep-ph/0210398
J. Donoghue, Introduction to the Effective Field TheoryDescription of Gravity, gr-qc/9512024
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School
PSI Zuoz Summer School on Particle PhysicsEffective Theories in Particle Physics, July 16 - 22, 2006
http://ltpth.web.psi.ch/zuoz_school/previous_summerschools/zuoz2006/index.html
R. Barbieri: Effective Theories for Physics beyond the Standard Model
M. Beneke: Concept of Effective Theories, Heavy Quark Effective Theory andSoft-Collinear Effective Theory
G. Colangelo: Chiral Perturbation Theory
U. Langenegger: B Physics and Quarkonia
H. Leutwyler: Historical and Other Remarks on Effective Theories
A. Manohar: Nonrelativistic QCD
L. Simons: Pion-Nucleon Interaction
M. Sozzi: Kaon Physics
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Chiral Perturbation Theory
Exploring the consequences of the chiral symmetry of QCDand its spontaneous breaking using effective field theorytechniques
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.12/103
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Chiral Perturbation Theory
Exploring the consequences of the chiral symmetry of QCDand its spontaneous breaking using effective field theorytechniques
Derivation from QCD:H. Leutwyler, On The Foundations Of Chiral Perturbation Theory,Ann. Phys. 235 (1994) 165 [hep-ph/9311274]
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The mass gap: Goldstone Modes
UNBROKEN: V (φ)
Only massive modesaround lowest energystate (=vacuum)
BROKEN: V (φ)
Need to pick a vacuum〈φ〉 6= 0: Breaks symmetryNo parity doubletsMassless mode along bottom
For more complicated symmetries: need to describe thebottom mathematically: G → H =⇒ G/H (explain)
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The two symmetry modes compared
Wigner-Eckart mode Nambu-Goldstone mode
Symmetry group G G spontaneously broken to subgroup H
Vacuum state unique Vacuum state degenerate
Massive Excitations Existence of a massless mode
States fall in multiplets of G States fall in multiplets of H
Wigner Eckart theorem for G Wigner Eckart theorem for H
Broken part leads to low-energy theorems
Symmetry linearly realized Full Symmetry, G, nonlinearly realized
unbroken part, H, linearly realized
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Some clarifications
φ(x): orientation of vacuum in every space-time point
Examples: spin waves, phonons
Nonlinear: acting by a broken symmetry operatorchanges the vacuum, φ(x) → φ(x) + αThe precise form of φ is not important but it mustdescribe the space of vacua (field transformationspossible)
In gauge theories: the local symmetry allows the vacuato be different in every point, hence the GoldstoneBoson must not be observable as a massless degree offreedom.
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.15/103
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The power counting
Very important:
Low energy theorems: Goldstone bosons do notinteract at zero momentum
Heuristic proof:
Which vacuum does not matter, choices related bysymmetry
φ(x) → φ(x) + α should not matterEach term in L must contain at least one ∂µφ
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Chiral Perturbation Theory
Degrees of freedom: Goldstone Bosons from ChiralSymmetry Spontaneous Breakdown
Power counting: Dimensional countingExpected breakdown scale: Resonances, so Mρ or higher
depending on the channel
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.17/103
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Chiral Perturbation Theory
Degrees of freedom: Goldstone Bosons from ChiralSymmetry Spontaneous Breakdown
Power counting: Dimensional countingExpected breakdown scale: Resonances, so Mρ or higher
depending on the channel
Chiral Symmetry
QCD: 3 light quarks: equal mass: interchange: SU(3)V
But LQCD =∑
q=u,d,s
[iq̄LD/ qL + iq̄RD/ qR − mq (q̄RqL + q̄LqR)]
So if mq = 0 then SU(3)L × SU(3)R.
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.17/103
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Chiral Perturbation Theory
Degrees of freedom: Goldstone Bosons from ChiralSymmetry Spontaneous Breakdown
Power counting: Dimensional countingExpected breakdown scale: Resonances, so Mρ or higher
depending on the channel
Chiral Symmetry
QCD: 3 light quarks: equal mass: interchange: SU(3)V
But LQCD =∑
q=u,d,s
[iq̄LD/ qL + iq̄RD/ qR − mq (q̄RqL + q̄LqR)]
So if mq = 0 then SU(3)L × SU(3)R.Can also see that via v < c, mq 6= 0 =⇒
v = c, mq = 0 =⇒/Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.17/103
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Chiral Perturbation Theory
〈q̄q〉 = 〈q̄LqR + q̄RqL〉 6= 0SU(3)L × SU(3)R broken spontaneously to SU(3)V8 generators broken =⇒ 8 massless degrees of freedomand interaction vanishes at zero momentum
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.18/103
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Chiral Perturbation Theory
〈q̄q〉 = 〈q̄LqR + q̄RqL〉 6= 0SU(3)L × SU(3)R broken spontaneously to SU(3)V8 generators broken =⇒ 8 massless degrees of freedomand interaction vanishes at zero momentum
Power counting in momenta: (explain)
p2
1/p2
∫
d4p p4
(p2)2 (1/p2)2 p4 = p4
(p2) (1/p2) p4 = p4
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.18/103
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Chiral Perturbation Theory
Large subject:
Steven Weinberg, Physica A96:327,1979: 1789citations
Juerg Gasser and Heiri Leutwyler,Nucl.Phys.B250:465,1985: 2290 citations
Juerg Gasser and Heiri Leutwyler, AnnalsPhys.158:142,1984: 2254 citations
Sum: 3777
Checked on 18/2/2007 in SPIRES
For lectures, review articles: seehttp://www.thep.lu.se/∼bijnens/chpt.html
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.19/103
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Chiral Perturbation Theor ies
Baryons
Heavy Quarks
Vector Mesons (and other resonances)
Structure Functions and Related Quantities
Light Pseudoscalar MesonsTwo or Three (or even more) FlavoursStrong interaction and couplings to externalcurrents/densitiesIncluding electromagnetismIncluding weak nonleptonic interactionsTreating kaon as heavy
Many similarities with strongly interacting HiggsOdense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.20/103
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Lagrangians
U(φ) = exp(i√
2Φ/F0) parametrizes Goldstone Bosons
Φ(x) =
0
B
B
B
B
B
B
@
π0√2
+η8√6
π+ K+
π− − π0
√2
+η8√6
K0
K− K̄0 −2 η8√6
1
C
C
C
C
C
C
A
.
LO Lagrangian: L2 = F20
4 {〈DµU †DµU〉 + 〈χ†U + χU †〉} ,
DµU = ∂µU − irµU + iUlµ ,left and right external currents: r(l)µ = vµ + (−)aµ
Scalar and pseudoscalar external densities: χ = 2B0(s + ip)quark masses via scalar density: s = M + · · ·〈A〉 = TrF (A)
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External currents?
in QCD Green functions derived as functionalderivatives w.r.t. external fields
Green functions are the objects that satisfy Wardidentities
By introducing a local Chiral Symmetry, Ward identitiesare automatically satisfied (great improvement overcurrent algebra)
QCD Green functions form a connection QCD⇔ChPT∫
[dGdqdq]eiR
d4xLQCD(q,q,G,l,r,s,p) ≈∫
[dU ]eiR
d4xLChP T (U,l,r,s,p)
so also functional derivatives are equal
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.22/103
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Lagrangians
L4 = L1〈DµU †DµU〉2 + L2〈DµU †DνU〉〈DµU †DνU〉+L3〈DµU †DµUDνU †DνU〉 + L4〈DµU †DµU〉〈χ†U + χU †〉+L5〈DµU †DµU(χ†U + U †χ)〉 + L6〈χ†U + χU †〉2
+L7〈χ†U − χU †〉2 + L8〈χ†Uχ†U + χU †χU †〉−iL9〈FRµνDµUDνU † + FLµνDµU †DνU〉
+L10〈U †FRµνUFLµν〉 + H1〈FRµνFRµν + FLµνFLµν〉 + H2〈χ†χ〉
Li: Low-energy-constants (LECs)Hi: Values depend on definition of currents/densities
These absorb the divergences of loop diagrams: Li → LriRenormalization: order by order in the powercounting
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.23/103
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Lagrangians
Lagrangian Structure:2 flavour 3 flavour 3+3 PQChPT
p2 F,B 2 F0, B0 2 F0, B0 2p4 lri , h
ri 7+3 L
ri , H
ri 10+2 L̂
ri , Ĥ
ri 11+2
p6 cri 52+4 Cri 90+4 K
ri 112+3
p2: Weinberg 1966p4: Gasser, Leutwyler 84,85p6: JB, Colangelo, Ecker 99,00
➠ replica method =⇒ PQ obtained from NF flavour➠ All infinities known➠ 3 flavour special case of 3+3 PQ: L̂ri ,K
ri → Lri , Cri
➠ 53 → 52 arXiv:0705.0576 [hep-ph]Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.24/103
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Chiral Logarithms
The main predictions of ChPT:
Relates processes with different numbers ofpseudoscalars
Chiral logarithms
m2π = 2Bm̂ +
(
2Bm̂
F
)2 [1
32π2log
(2Bm̂)
µ2+ 2lr3(µ)
]
+ · · ·
M2 = 2Bm̂B 6= B0, F 6= F0 (two versus three-flavour)
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.25/103
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LECs and µ
lr3(µ)
l̄i =32π2
γilri (µ) − log
M2πµ2
.
Independent of the scale µ.
For 3 and more flavours, some of the γi = 0: Lri (µ)
µ :
mπ, mK : chiral logs vanish
pick larger scale
1 GeV then Lr5(µ) ≈ 0 large Nc arguments????compromise: µ = mρ = 0.77 GeV
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.26/103
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Expand in what quantities?
Expansion is in momenta and masses
But is not unique: relations between masses(Gell-Mann–Okubo) exists
Express orders in terms of physical masses andquantities (Fπ, FK)?
Express orders in terms of lowest order masses?
E.g. s + t + u = 2m2π + 2m2K in πK scattering
See e.g. Descotes-Genon talk
I prefer physical masses
Thresholds correct
Chiral logs are from physical particles propagatingOdense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.27/103
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An example
mπ =m0
1 + am0f0fπ =
f0
1 + bm0f0
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.28/103
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An example
mπ =m0
1 + am0f0fπ =
f0
1 + bm0f0
mπ = m0 − am20f0
+ a2m30f20
+ · · ·
fπ = f0
(
1 − bm0f0
+ b2m20f20
+ · · ·)
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.28/103
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An example
mπ =m0
1 + am0f0fπ =
f0
1 + bm0f0
mπ = m0 − am20f0
+ a2m30f20
+ · · ·
fπ = f0
(
1 − bm0f0
+ b2m20f20
+ · · ·)
mπ = m0 − am2πfπ
+ a(b − a)m3π
f2π+ · · ·
mπ = m0
(
1 − amπfπ
+ abm2πf2π
+ · · ·)
fπ = f0
(
1 − bmπfπ
+ b(2b − a)m2π
f2π+ · · ·
)
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.28/103
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An example
mπ =m0
1 + am0f0fπ =
f0
1 + bm0f0
mπ = m0 − am20f0
+ a2m30f20
+ · · ·
fπ = f0
(
1 − bm0f0
+ b2m20f20
+ · · ·)
mπ = m0 − am2πfπ
+ a(b − a)m3π
f2π+ · · ·
mπ = m0
(
1 − amπfπ
+ abm2πf2π
+ · · ·)
fπ = f0
(
1 − bmπfπ
+ b(2b − a)m2π
f2π+ · · ·
)
a = 1 b = 0.5 f0 = 1Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.28/103
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An example: m0/f0
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5
mπ
m0
mπLO
NLONNLO
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.29/103
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An example: mπ/fπ
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5
mπ
m0
mπLO
NLOpNNLOp
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.30/103
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Two-loop Two-flavour
Review paper on Two-Loops: JB, hep-ph/0604043 Prog. Part.Nucl. Phys. 58 (2007) 521
Dispersive Calculation of the nonpolynomial part in q2, s, t, u
Gasser-Meißner: FV , FS: 1991 numerical
Knecht-Moussallam-Stern-Fuchs: ππ: 1995 analytical
Colangelo-Finkemeier-Urech: FV , FS: 1996 analytical
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.31/103
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Two-Loop Two-flavour
Bellucci-Gasser-Sainio: γγ → π0π0: 1994Bürgi: γγ → π+π−, Fπ, mπ: 1996JB-Colangelo-Ecker-Gasser-Sainio: ππ, Fπ, mπ:1996-97
JB-Colangelo-Talavera: FV π(t), FSπ: 1998
JB-Talavera: π → ℓνγ: 1997Gasser-Ivanov-Sainio: γγ → π0π0, γγ → π+π−:2005-2006
mπ, Fπ, FV , FS, ππ: simple analytical forms
Colangelo-(Dürr-)Haefeli: Finite volume Fπ,mπ2005-2006
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.32/103
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LECs
l̄1 to l̄4: ChPT at order p6 and the Roy equation analysis inππ and FS Colangelo, Gasser and Leutwyler, Nucl. Phys. B 603 (2001)125 [hep-ph/0103088]
l̄5 and l̄6 : from FV and π → ℓνγ JB,(Colangelo,)Talavera
l̄1 = −0.4 ± 0.6 ,l̄2 = 4.3 ± 0.1 ,l̄3 = 2.9 ± 2.4 ,l̄4 = 4.4 ± 0.2 ,
l̄6 − l̄5 = 3.0 ± 0.3 ,l̄6 = 16.0 ± 0.5 ± 0.7 .
l7 ∼ 5 · 10−3 from π0-η mixing Gasser, Leutwyler 1984Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.33/103
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LECs
Some combinations of order p6 LECs are known as well:curvature of the scalar and vector formfactor, two morecombinations from ππ scattering (implicit in b5 and b6)
Note: cri for mπ, fπ, ππ: small effect
cri (770MeV ) = 0 for plots shown
expansion in m2π/F2π shown
General observation:
Obtainable from kinematical dependences: known
Only via quark-mass dependence: poorely knownOdense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.34/103
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m2π
0
0.05
0.1
0.15
0.2
0.25
0 0.05 0.1 0.15 0.2 0.25
mπ2
M2 [GeV2]
LONLO
NNLO
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.35/103
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m2π (l̄3 = 0)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.05 0.1 0.15 0.2 0.25
mπ2
M2 [GeV2]
LONLO
NNLO
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.36/103
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Fπ
0
0.02
0.04
0.06
0.08
0.1
0.12
0 0.05 0.1 0.15 0.2 0.25
Fπ
[GeV
]
M2 [GeV2]
LONLO
NNLO
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.37/103
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Two-loop Three-flavour,≤2001
ΠV V π, ΠV V η, ΠV V K Kambor, Golowich; Kambor, Dürr; Amorós, JB, Talavera
ΠV V ρω Maltman
ΠAAπ, ΠAAη, Fπ, Fη, mπ, mη Kambor, Golowich; Amorós, JB, Talavera
ΠSS Moussallam Lr4, L
r6
ΠV V K , ΠAAK , FK , mK Amorós, JB, Talavera
Kℓ4, 〈qq〉 Amorós, JB, Talavera Lr1, Lr2, Lr3
FM , mM , 〈qq〉 (mu 6= md) Amorós, JB, Talavera Lr5,7,8,mu/md
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.38/103
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Two-loop Three-flavour,≥2001
FV π, FV K+, FV K0 Post, Schilcher; JB, Talavera Lr9
Kℓ3 Post, Schilcher; JB, Talavera Vus
FSπ, FSK (includes σ-terms) JB, Dhonte Lr4, Lr6
K,π → ℓνγ Geng, Ho, Wu Lr10ππ JB,Dhonte,Talavera
πK JB,Dhonte,Talavera
relation lri and Lri , C
ri Gasser,Haefeli,Ivanov,Schmid
Finite volume 〈qq〉 JB,Ghorbani
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.39/103
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Two-loop Three-flavour
Known to be in progress
η → 3π: being written up JB,GhorbaniKℓ3 iso: preliminary results available JB,Ghorbani
Finite Volume: sunsetintegrals being written up JB,Lähde
relation cri and Lri , C
ri Gasser,Haefeli,Ivanov,Schmid
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.40/103
-
Cri
Most analysis use:Cri from (single) resonance approximation
π
πρ, S
→ q2π
π|q2|
-
Cri
LV = −1
4〈VµνV µν〉 +
1
2m2V 〈VµV µ〉 −
fV
2√
2〈Vµνfµν+ 〉
− igV2√
2〈Vµν [uµ, uν ]〉 + fχ〈Vµ[uµ, χ−]〉
LA = −1
4〈AµνAµν〉 +
1
2m2A〈AµAµ〉 −
fA
2√
2〈Aµνfµν− 〉
LS =1
2〈∇µS∇µS − M2SS2〉 + cd〈Suµuµ〉 + cm〈Sχ+〉
Lη′ =1
2∂µP1∂
µP1 −1
2M2η′P
21 + id̃mP1〈χ−〉 .
fV = 0.20, fχ = −0.025, gV = 0.09, cm = 42 MeV, cd = 32 MeV, d̃m = 20 MeV,
mV = mρ = 0.77 GeV, mA = ma1 = 1.23 GeV, mS = 0.98 GeV, mP1 = 0.958 GeV
fV , gV , fχ, fA: experiment
cm and cd from resonance saturation at O(p4)
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.42/103
-
Cri
Problems:
Weakest point in the numerics
However not all results presented depend on this
Unknown so far: Cri in the masses/decay constants andhow these effects correlate into the rest
No µ dependence: obviously only estimate
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.43/103
-
Cri
Problems:
Weakest point in the numerics
However not all results presented depend on this
Unknown so far: Cri in the masses/decay constants andhow these effects correlate into the rest
No µ dependence: obviously only estimate
What we did about it:
Vary resonance estimate by factor of two
Vary the scale µ at which it applies: 600-900 MeV
Check the estimates for the measured ones
Again: kinematic can be had, quark-mass dependencedifficult
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.43/103
-
Inputs
Kℓ4: F (0), G(0), λ E865 BNL
m2π0, m2η, m
2K+, m
2K0 em with Dashen violation
Fπ+
FK+/Fπ+
ms/m̂ 24 (26) m̂ = (mu + md)/2
Lr4, Lr6
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.44/103
-
Outputs: Ifit 10 same p4 fit B fit D
103Lr1 0.43 ± 0.12 0.38 0.44 0.44
103Lr2 0.73 ± 0.12 1.59 0.60 0.69
103Lr3 −2.35 ± 0.37 −2.91 −2.31 −2.33
103Lr4 ≡ 0 ≡ 0 ≡ 0.5 ≡ 0.2
103Lr5 0.97 ± 0.11 1.46 0.82 0.88
103Lr6 ≡ 0 ≡ 0 ≡ 0.1 ≡ 0
103Lr7 −0.31 ± 0.14 −0.49 −0.26 −0.28
103Lr8 0.60 ± 0.18 1.00 0.50 0.54
➠ errors are very correlated➠ µ = 770 MeV; 550 or 1000 within errors➠ varying Cri factor 2 about errors➠ Lr4, L
r6 ≈ −0.3, . . . , 0.6 10−3 OK
➠ fit B: small corrections to pion “sigma” term, fit scalar radius
➠ fit D: fit ππ and πK thresholdsOdense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.45/103
-
Correlations
-5-4
-3-2
-10
L3r
00.2
0.40.6
0.81
L1r
0
0.2
0.4
0.6
0.8
1
1.2
1.4
L2r
(older fit)103 Lr1 = 0.52 ± 0.23103 Lr2 = 0.72 ± 0.24103 Lr3 = −2.70 ± 0.99
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.46/103
-
Outputs: II
fit 10 same p4 fit B fit D
2B0m̂/m2π 0.736 0.991 1.129 0.958
m2π: p4, p6 0.006,0.258 0.009,≡ 0 −0.138,0.009 −0.091,0.133
m2K : p4, p6 0.007,0.306 0.075,≡ 0 −0.149,0.094 −0.096,0.201
m2η: p4, p6 −0.052,0.318 0.013,≡ 0 −0.197,0.073 −0.151,0.197
mu/md 0.45±0.05 0.52 0.52 0.50F0 [MeV] 87.7 81.1 70.4 80.4FKFπ
: p4, p6 0.169,0.051 0.22,≡ 0 0.153,0.067 0.159,0.061
➠ mu = 0 always very far from the fits
➠ F0: pion decay constant in the chiral limit
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.47/103
-
ππ
p4
p6
-0.4 -0.2 0 0.2 0.4 0.6 103 L4
r -0.3-0.2
-0.10
0.10.2
0.30.4
0.50.6
103 L6r
0.195
0.2
0.205
0.21
0.215
0.22
0.225
a00
p4
p6
-0.4 -0.2 0 0.2 0.4 0.6 103 L4
r -0.3-0.2
-0.10
0.10.2
0.30.4
0.50.6
103 L6r
-0.048
-0.047
-0.046
-0.045
-0.044
-0.043
-0.042
-0.041
-0.04
-0.039
a20
a00 = 0.220 ± 0.005, a20 = −0.0444 ± 0.0010Colangelo, Gasser, Leutwyler
a00 = 0.159 a20 = −0.0454 at order p2
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.48/103
-
ππ and πK
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
-0.4 -0.2 0 0.2 0.4 0.6
103
L6r
103 L4r
ππ constraints
a20C1
a03/2
C+10
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
-0.4 -0.2 0 0.2 0.4 0.6
103
L6r
103 L4r
πK constraints
C+10a0
3/2
C1a20
preferred region: fit D: 103Lr4 ≈ 0.2, 103Lr6 ≈ 0.0
General fitting needs more work and systematic studies
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.49/103
-
Quark mass dependences
Updates of plots inAmorós, JB and Talavera, hep-ph/0003258, Nucl. Phys. B585 (2000) 293
Some new onesProcedure: calculate a consistent set of mπ,mK ,mη, fπ withthe given input values (done iteratively)
vary ms/(ms)phys, keep ms/m̂ = 24m2π, m
2K , Fπ, FK
vary ms/(ms)phys keep m̂ fixedm2π, Fπvary mπ, keep mK fixedf+(0): the formfactor in Kℓ3 decays
f+(0),f+(0)/(
m2K − m2π)2
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.50/103
-
m2π fit 10
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
0 0.2 0.4 0.6 0.8 1
mπ2
[GeV
2 ]
ms/(ms)phys
LONLO
NNLO
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.51/103
-
m2π fit D
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
0 0.2 0.4 0.6 0.8 1
mπ2
[GeV
2 ]
ms/(ms)phys
LONLO
NNLO
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.52/103
-
m2K fit 10
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8 1
mK
2 [G
eV2 ]
ms/(ms)phys
LONLO
NNLO
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.53/103
-
m2K fit D
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8 1
mK
2 [G
eV2 ]
ms/(ms)phys
LONLO
NNLO
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.54/103
-
Fπ fit 10
0.086
0.088
0.09
0.092
0.094
0.096
0.098
0.1
0 0.2 0.4 0.6 0.8 1
Fπ
[GeV
]
ms/(ms)phys
LONLO
NNLO
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.55/103
-
FK fit 10
0.085
0.09
0.095
0.1
0.105
0.11
0.115
0 0.2 0.4 0.6 0.8 1
FK [G
eV]
ms/(ms)phys
LONLO
NNLO
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.56/103
-
FK/Fπ fit 10
1
1.05
1.1
1.15
1.2
1.25
0 0.2 0.4 0.6 0.8 1
FK/F
π
ms/(ms)phys
NLONNLO
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.57/103
-
m2π fit 10, fixed m̂
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
0 0.2 0.4 0.6 0.8 1
mπ2
[GeV
2 ]
ms/(ms)phys
LONLO
NNLO
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.58/103
-
Fπ fit 10, fixed m̂
0.086
0.088
0.09
0.092
0.094
0.096
0.098
0.1
0 0.2 0.4 0.6 0.8 1
Fπ
[GeV
]
ms/(ms)phys
LO
NLO
NNLO
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.59/103
-
f+(0) fit 10, fixed mK
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
corr
ectio
n to
f +(0
)
mπ2/(mK
2)phys
sump4
p6 2-loopsp6 Li
r
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.60/103
-
(f+(0))/(
m2K − m2π)2
fit 10, fixed mK
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 0.2 0.4 0.6 0.8 1
(cor
rect
ion
to f +
(0))
/(m
K2 −
mπ2 )
2 [G
eV−4
]
mπ2/(mK
2)phys
sump4
p6 2-loopsp6 Li
r
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.61/103
-
f0(t) in Kℓ3
Main Result: JB,Talavera
f0(t) = 1 −8
F 4π(Cr12 + C
r34)
(
m2K − m2π)2
+8t
F 4π(2Cr12 + C
r34)
(
m2K + m2π
)
+t
m2K − m2π(FK/Fπ − 1)
− 8F 4π
t2Cr12 + ∆(t) + ∆(0) .
∆(t) and ∆(0) contain NO Cri and only depend on the Lri at
order p6
=⇒All needed parameters can be determined experimentally
∆(0) = −0.0080 ± 0.0057[loops] ± 0.0028[Lri ] .Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.62/103
-
≥ 3-flavour: PQChPT
Essentially all manipulations from ChPT go through toPQChPT when changing trace to supertrace and addingfermionic variables
Exceptions: baryons and Cayley-Hamilton relations
So Luckily: can use the n flavour work in ChPT at two looporder to obtain for PQChPT: Lagrangians and infinities
Very important note: ChPT is a limit of PQChPT=⇒ LECs from ChPT are linear combinations of LECs
of PQChPT with the same number of sea quarks.
E.g. Lr1 = Lr(3pq)0 /2 + L
r(3pq)1
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.63/103
-
PQChPT
One-loop: Bernard, Golterman, Sharpe, Shoresh, Pallante,. . .with electromagnetism: JB,Danielsson, hep-lat/0610127
Two loops:
m2π+ simplest mass case: JB,Danielsson,Lähde, hep-lat/0406017Fπ+: JB,Lähde, hep-lat/0501014Fπ+, m2π+ two sea quarks: JB,Lähde, hep-lat/0506004
m2π+: JB,Danielsson,Lähde, hep-lat/0602003
Neutral masses: JB,Danielsson, hep-lat/0606017
Lattice data: a and L extrapolations needed
Programs available from me (Fortran)Formulas: http://www.thep.lu.se/∼bijnens/chpt.html
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.64/103
-
Partial Quenching and ChPT
Mesons
=
Quark FlowValence
+
Quark FlowSea
+ · · ·
Lattice QCD: Valence is easy to deal with, Sea very difficult
They can be treated separately: i.e. different quark massesPartially Quenched QCD and ChPT (PQChPT)
One Loop or p4: Bernard, Golterman, Pallante, Sharpe,Shoresh,. . .
Two Loops or p6: This talk JB, Niclas Danielsson, Timo Lähde
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.65/103
-
PQChPT at Two Loops: General
Add ghost quarks: remove the unwanted free valence loops
Mesons
=
Quark FlowValence
+
Quark FlowValence
+
Quark FlowSea
+
Quark FlowGhost
Possible problem: QCD =⇒ ChPT relies heavily on unitarity
Partially quenched: at least one dynamical sea quark=⇒ Φ0 is heavy: remove from PQChPT
Symmetry group becomes SU(nv + ns|nv) × SU(nv + ns|nv)(approximately)
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.66/103
-
PQChPT at Two Loops: General
Essentially all manipulations from ChPT go through toPQChPT when changing trace to supertrace and addingfermionic variables
Exceptions: baryons and Cayley-Hamilton relations
So Luckily: can use the n flavour work in ChPT at two looporder to obtain for PQChPT: Lagrangians and infinities
Very important note: ChPT is a limit of PQChPT=⇒ LECs from ChPT are linear combinations of LECs
of PQChPT with the same number of sea quarks.
E.g. Lr1 = Lr(3pq)0 /2 + L
r(3pq)1
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.67/103
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PQChPT at Two Loop: Papers
valence equal mass, 3 sea equal mass:
m2π+: JB,Danielsson,Lähde, hep-lat/0406017
Other mass combinations:
F 2π+: JB,Lähde, hep-lat/0501014
F 2π+, m2π+ two sea quarks: JB,Lähde, hep-lat/0506004
In progress: the other charged masses
Actual Calculations:
➠ heavy use of FORM Vermaseren➠ Main problem: sheer size of theexpressions
No fits to lattice data (yet): a and L extrapolations needed
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.68/103
-
Long Expressions
=⇒
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.69/103
-
Long Expressions
=⇒
δ(6)22loops = π16 L
r0
[
4/9 χηχ4 − 1/2 χ1χ3 + χ213 − 13/3 χ̄1χ13 − 35/18 χ̄2]
− 2 π16 Lr1 χ213− π16 Lr2
[
11/3 χηχ4 + χ213 + 13/3 χ̄2
]
+ π16 Lr3
[
4/9 χηχ4 − 7/12 χ1χ3 + 11/6 χ213 − 17/6 χ̄1χ13 − 43/36 χ̄2]
+ π216[
−15/64 χηχ4 − 59/384 χ1χ3 + 65/384 χ213 − 1/2 χ̄1χ13 − 43/128 χ̄2]
− 48 Lr4Lr5 χ̄1χ13 − 72 Lr24 χ̄21− 8 Lr25 χ213 + Ā(χp)π16
[
−1/24 χp + 1/48 χ̄1 − 1/8 χ̄1 Rpqη + 1/16 χ̄1 Rcp − 1/48 Rpqη χp − 1/16 Rpqη χq+ 1/48 Rηpp χη + 1/16 R
cp χ13
]
+ Ā(χp)Lr0
[
8/3 Rpqη χp + 2/3 Rcp χp + 2/3 R
dp
]
+ Ā(χp)Lr3
[
2/3 Rpqη χp
+ 5/3 Rcp χp + 5/3 Rdp
]
+ Ā(χp)Lr4
[
−2 χ̄1χ̄ppηη0 − 2 χ̄1 Rpqη + 3 χ̄1 Rcp]
+ Ā(χp)Lr5
[
−2/3 χ̄ppηη1 − Rpqη χp+ 1/3 Rpqη χq + 1/2 R
cp χp − 1/6 Rcp χq
]
+ Ā(χp)2
[
1/16 + 1/72 (Rpqη)2 − 1/72 RpqηRcp + 1/288 (Rcp)2
]
+ Ā(χp)Ā(χps)[
−1/36 Rpqη − 5/72 Rpsη + 7/144 Rcp]
− Ā(χp)Ā(χqs)[
1/36 Rpqη + 1/24 Rpsη + 1/48 R
cp
]
+ Ā(χp)Ā(χη)[
−1/72 RpqηRvη13 + 1/144 RcpRvη13]
+ 1/8 Ā(χp)Ā(χ13) + 1/12 Ā(χp)Ā(χ46)Rηpp
+ Ā(χp)B̄(χp, χp; 0)[
1/4 χp − 1/18 RpqηRcp χp − 1/72 RpqηRdp + 1/18 (Rcp)2 χp + 1/144 RcpRdp]
+ Ā(χp)B̄(χp, χη; 0)[
1/18 RηppRcp χp − 1/18 Rη13Rcp χp
]
+ Ā(χp)B̄(χq, χq; 0)[
−1/72 RpqηRdq + 1/144 RcpRdq]
− 1/12 Ā(χp)B̄(χps, χps; 0)Rpsη χps − 1/18 Ā(χp)B̄(χ1, χ3; 0)RqpηRcp χp+ 1/18 Ā(χp)C̄(χp, χp, χp; 0)R
cpR
dp χp + Ā(χp; ε)π16
[
1/8 χ̄1Rpqη − 1/16χ̄1 Rcp − 1/16 Rcp χp − 1/16 Rdp
]
+ Ā(χps)π16 [1/16 χps − 3/16 χqs − 3/16 χ̄1] − 2 Ā(χps)Lr0 χps − 5 Ā(χps)Lr3 χps − 3 Ā(χps)Lr4 χ̄1+ Ā(χps)L
r5 χ13 + Ā(χps)Ā(χη)
[
7/144 Rηpp − 5/72 Rηps − 1/48 Rηqq + 5/72 Rηqs − 1/36 Rη13]
+ Ā(χps)B̄(χp, χp; 0)[
1/24 Rpsη χp − 5/24 Rpsη χps]
+ Ā(χps)B̄(χp, χη; 0)[
−1/18 RηpsRzqpη χp− 1/9 RηpsRzqpη χps
]
− 1/48 Ā(χps)B̄(χq, χq; 0)Rdq + 1/18 Ā(χps)B̄(χ1, χ3; 0)Rqsη χs+ 1/9 Ā(χps)B̄(χ1, χ3; 0, k)R
qsη + 3/16 Ā(χps; ε)π16 [χs + χ̄1] − 1/8 Ā(χp4)2 − 1/8 Ā(χp4)Ā(χp6)
+ 1/8 Ā(χp4)Ā(χq6) − 1/32 Ā(χp6)2 + Ā(χη)π16[
1/16 χ̄1 Rvη13 − 1/48 Rvη13 χη + 1/16 Rvη13 χ13
]
+ Ā(χη)Lr0
[
4Rη13 χη + 2/3 Rvη13 χη
]
− 8 Ā(χη)Lr1 χη − 2 Ā(χη)Lr2 χη + Ā(χη)Lr3[
4Rη13 χη + 5/3 Rvη13χη
]
+ Ā(χη)Lr4
[
4 χη + χ̄1 Rvη13
]
− Ā(χη)Lr5[
1/6 Rηpp χq + Rη13 χ13 + 1/6 R
vη13 χη
]
+ 1/288 Ā(χη)2 (Rvη13)
2
+ 1/12 Ā(χη)Ā(χ46)Rvη13 + Ā(χη)B̄(χp, χp; 0)
[
−1/36 χ̄ppηηη1 − 1/18 RpqηRηpp χp + 1/18 RηppRcp χp+ 1/144 RdpR
vη13
]
+ Ā(χη)B̄(χp, χη; 0)[
−1/18 χ̄ηpηpη1 + 1/18 χ̄ηpηqη1 + 1/18 (Rηpp)2Rzqpη χp]
− 1/12 Ā(χη)B̄(χps, χps; 0)Rηps χps − Ā(χη)B̄(χη, χη; 0)[
1/216 Rvη13 χ4 + 1/27 Rvη13 χ6
]
− 1/18 Ā(χη)B̄(χ1, χ3; 0)R1ηηR3ηη χη + 1/18 Ā(χη)C̄(χp, χp, χp; 0)RηppRdp χp + Ā(χη; ε)π16 [1/8 χη− 1/16 χ̄1 Rvη13 − 1/8 Rη13 χη − 1/16 Rvη13 χη
]
+ Ā(χ1)Ā(χ3)[
−1/72 RpqηRcq + 1/36 R13ηR31η + 1/144 Rc1Rc3]
− 4 Ā(χ13)Lr1 χ13 − 10 Ā(χ13)Lr2 χ13 + 1/8 Ā(χ13)2 − 1/2 Ā(χ13)B̄(χ1, χ3; 0, k)+ 1/4 Ā(χ13; ε)π16 χ13 + 1/4 Ā(χ14)Ā(χ34) + 1/16 Ā(χ16)Ā(χ36) − 24 Ā(χ4)Lr1 χ4 − 6 Ā(χ4)Lr2 χ4+ 12 Ā(χ4)L
r4 χ4 + 1/12 Ā(χ4)B̄(χp, χp; 0) (R
p4η)
2 χ4 + 1/6 Ā(χ4)B̄(χp, χη; 0)[
Rp4ηRηp4 χ4 − Rp4ηRηq4 χ4
]
− 1/24 Ā(χ4)B̄(χη, χη; 0)Rvη13 χ4 − 1/6 Ā(χ4)B̄(χ1, χ3; 0)R14ηR34η χ4 + 3/8 Ā(χ4; ε)π16 χ4− 32 Ā(χ46)Lr1 χ46 − 8 Ā(χ46)Lr2 χ46 + 16 Ā(χ46)Lr4 χ46 + Ā(χ46)B̄(χp, χp; 0)
[
1/9 χ46 + 1/12 Rηpp χp
+ 1/36 Rηpp χ4 + 1/9 Rηp4 χ6
]
+ Ā(χ46)B̄(χp, χη; 0)[
−1/18 Rηpp χ4 − 1/9 Rηp4 χ6 + 1/9 Rηq4 χ6 + 1/18 Rη13 χ4]
− 1/6 Ā(χ46)B̄(χp, χη; 0, k)[
Rηpp − Rη13]
+ 1/9 Ā(χ46)B̄(χη, χη; 0)Rvη13 χ46 − Ā(χ46)B̄(χ1, χ3; 0) [2/9 χ46
+ 1/9 Rηp4 χ6 + 1/18 Rη13 χ4] − 1/6 Ā(χ46)B̄(χ1, χ3; 0, k)Rη13 + 1/2 Ā(χ46; ε)π16 χ46
+ B̄(χp, χp; 0)π16[
1/16 χ̄1 Rdp + 1/96 R
dp χp + 1/32 R
dp χq
]
+ 2/3 B̄(χp, χp; 0)Lr0 R
dp χp
+ 5/3 B̄(χp, χp; 0)Lr3 R
dp χp + B̄(χp, χp; 0)L
r4
[
−2 χ̄1χ̄ppηη0χp − 4 χ̄1Rpqη χp + 4 χ̄1Rcp χp + 3 χ̄1Rdp]
+ B̄(χp, χp; 0)Lr5
[
−2/3χ̄ppηη1χp − 4/3 Rpqη χ2p + 4/3 Rcp χ2p + 1/2 Rdp χp − 1/6 Rdp χq]
+ B̄(χp, χp; 0)Lr6
[
4 χ̄1χ̄ppηη1 + 8 χ̄1R
pqη χp − 8 χ̄1Rcp χp
]
+ 4 B̄(χp, χp; 0)Lr7 (R
dp)
2
+ B̄(χp, χp; 0)Lr8
[
4/3 χ̄ppηη2 + 8/3 Rpqη χ
2p − 8/3 Rcp χ2p
]
+ B̄(χp, χp; 0)2
[
−1/18 RpqηRdp χp + 1/18 RcpRdp χp+ 1/288 (Rdp)
2]
+ 1/18 B̄(χp, χp; 0)B̄(χp, χη; 0)[
RηppRdp χp − Rη13Rdp χp
]
plus several more pages
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.69/103
-
Why so long expressions
Many different quark and meson masses (χij)
Charged propagators: −iGcij(k) =ǫj
k2−χij+iε (i 6= j)
Neutral propagators: Gnij(k) = Gcij(k) δij − 1nsea G
qij(k)
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.70/103
-
Why so long expressions
Many different quark and meson masses (χij)
Charged propagators: −iGcij(k) =ǫj
k2−χij+iε (i 6= j)
Neutral propagators: Gnij(k) = Gcij(k) δij − 1nsea G
qij(k)
−iGqii(k) =Rdi
(k2−χi+iε)2 +Rci
k2−χi+iε +Rπηii
k2−χπ+iε +R
ηπii
k2−χη+iε
Rijkl = Rzi456jkl, R
di = R
zi456πη,
Rci = Ri4πη + R
i5πη + R
i6πη − Riπηη − Riππη
Rzab = χa − χb, Rzabc =χa−χbχa−χc , R
zabcd =
(χa−χb)(χa−χc)χa−χd
Rzabcdefg =(χa−χb)(χa−χc)(χa−χd)(χa−χe)(χa−χf )(χa−χg)
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.70/103
-
Why so long expressions
Many different quark and meson masses (χij)
Charged propagators: −iGcij(k) =ǫj
k2−χij+iε (i 6= j)
Neutral propagators: Gnij(k) = Gcij(k) δij − 1nsea G
qij(k)
−iGqii(k) =Rdi
(k2−χi+iε)2 +Rci
k2−χi+iε +Rπηii
k2−χπ+iε +R
ηπii
k2−χη+iε
Rijkl = Rzi456jkl, R
di = R
zi456πη,
Rci = Ri4πη + R
i5πη + R
i6πη − Riπηη − Riππη
Rzab = χa − χb, Rzabc =χa−χbχa−χc , R
zabcd =
(χa−χb)(χa−χc)χa−χd
Rzabcdefg =(χa−χb)(χa−χc)(χa−χd)(χa−χe)(χa−χf )(χa−χg)
Relations =⇒ order of magnitude smallerOdense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.70/103
-
PQChPT at Two Loop
Problem: Plotting with many input parameters
Plot masses as a function of lowest order mass squared
I.e. of quark mass: χi = 2B0mi = m2(0)M
Remember: χi ≈ 0.3 GeV 2 ≈ (550 MeV )2 ∼ border ChPT
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.71/103
-
PQChPT at Two Loop
Problem: Plotting with many input parameters
Plot masses as a function of lowest order mass squared
I.e. of quark mass: χi = 2B0mi = m2(0)M
Remember: χi ≈ 0.3 GeV 2 ≈ (550 MeV )2 ∼ border ChPT
1+1 case: Valence: χ1 = χ2 = χ3Sea: χ4 = χ5 = χ6
Plot along curves: χ4 = tan θ χ1 or χ4 constantOdense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.71/103
-
PQChPT: 1+1 case, 3 sea quarksS
ea
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5
χ 4 [G
eV2 ]
χ1 [GeV2]
15o
30o
45o
60o
75o
A
Valence
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.72/103
-
PQChPT: 1+1 case, 3 sea quarksS
ea
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5
χ 4 [G
eV2 ]
χ1 [GeV2]
15o
30o
45o
60o
75o
A
-0.04
-0.02
0
0.02
0.04
0.06
0 0.1 0.2 0.3 0.4 0.5
δ(4)
/F02
χ1 [GeV2]
15o
30o
45o
60o
75o
A
Valence p4 mass
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.72/103
-
PQChPT: 1+1 case, 3 sea quarksS
ea
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5
χ 4 [G
eV2 ]
χ1 [GeV2]
15o
30o
45o
60o
75o
A
-0.04
-0.02
0
0.02
0.04
0.06
0 0.1 0.2 0.3 0.4 0.5
δ(4)
/F02
χ1 [GeV2]
15o
30o
45o
60o
75o
A
Valence p4 mass
Notice the Quenched Chiral Logs:m2πχ1
= 1+α
F 2χ4 log χ1+· · ·
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.72/103
-
PQChPT: 1+1 case, 3 sea quarks
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.1 0.2 0.3 0.4 0.5
δ(4)
/F02 +
δ(6)
/F04
χ1 [GeV2]
15o
30o
45o
60o
75o
A
p4 + p6 relative correction mass
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.73/103
-
PQChPT: 1+1 case, 3 sea quarks
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.1 0.2 0.3 0.4 0.5
δ(4)
/F02 +
δ(6)
/F04
χ1 [GeV2]
15o
30o
45o
60o
75o
A
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5δ(
4)/F
02 +δ(
6)/F
04
χ1 [GeV2]
15o
30o
45o
60o
75o A
p4 + p6 relative correction mass decay constant
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.73/103
-
PQChPT: 1+1 case, 2 sea quarks
-0.2
-0.1
0
0.1
0.2
0.3
0.4
χ1 [GeV2]
χ 3 [G
eV2 ]
∆M
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Relative Correction: Mass
χ1: valence mass, χ3: sea massOdense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.74/103
-
PQChPT: 1+1 case, 2 sea quarks
-0.2
-0.1
0
0.1
0.2
0.3
0.4
χ1 [GeV2]
χ 3 [G
eV2 ]
∆M
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Relative Correction: Mass
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
χ1 [GeV2]
χ 3 [G
eV2 ]
∆F
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Decay Constant
χ1: valence mass, χ3: sea massOdense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.74/103
-
PQChPT: 2 sea quarks
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 0.05 0.1 0.15 0.2
∆ M
χ1 [GeV2]
z = 1.4z = 1.2z = 1.0z = 0.8
z = 0.6z = 0.4
nf = 2dval = 1dsea = 2
θ = 70o
χ3 = χ1 tan(θ)
χ4 = z χ3
Relative Correction: Mass1+2 caseValence: χ1 6= χ2Sea: χ3 = χ4
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.75/103
-
PQChPT: 2 sea quarks
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 0.05 0.1 0.15 0.2
∆ M
χ1 [GeV2]
z = 1.4z = 1.2z = 1.0z = 0.8
z = 0.6z = 0.4
nf = 2dval = 1dsea = 2
θ = 70o
χ3 = χ1 tan(θ)
χ4 = z χ3
Relative Correction: Mass1+2 caseValence: χ1 6= χ2Sea: χ3 = χ4
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 0.05 0.1 0.15 0.2
∆ Mχ1 [GeV
2]
x = 0.8x = 1.0x = 1.2x = 1.4
x = 1.6x = 1.8
nf = 2dval = 2dsea = 1
θ = 70o
χ3 = χ1 tan(θ)
χ2 = x χ1
Mass2+1 caseValence: χ1 = χ2Sea: χ3 6= χ4
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.75/103
-
PQChPT: Fitting Strategy
For masses and Decay constants
At order p4: Lr4, Lr5, L
r6, L
r8
At order p6: all allowed quadratic quark masscombinations show up
Problem: Need Lr0, Lr1, L
r2, L
r3 from ππ, πK scattering but
only to order p4
Nonanalytic structure at p6 given (only one included innumerics shown)
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.76/103
-
Conclusions PQChPT Part
All relevant mass combinations for masses and decayconstants for charged pseudoscalar mesons nowknown to two loops
Decay constants and masses for two sea quarksconverge nicely
Masses for three sea quarks convergence slower
Looking forward to getting lattice data to fit
Expressions available fromhttp://www.thep.lu.se/∼bijnens/chpt.htmlFor the numerical programs contact the authors
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.77/103
-
Wishing list
General:
Quark-mass dependences everywhere
Not only fits but also continuum infinite volume resultsat a given quark-mass (so we can also fit ourselves forstudying other inputs)
More use of the existing two-loop calculations
Analytical NNLO only: not really fewer parameters
Only more in LECs: p4 scattering LECs also inmasses/decay constants
I.e. lr1, lr2 (nF = 2), L
r1, L
r2, L
r3 (nF = 3),
L̂ri , i = 0, 1, 2, 3 (PQChPT)
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.78/103
-
Wishing list: Two-flavour
(with input from Gasser et al.)
l̄3 and errors
l̄4: from Fπ(mq) and scalar radius: can lattice check thisrelation
a20 accurately predicted in terms of scalar radius: canlattice check this
Isospin breaking in ππ scattering (important for CPviolation in K → ππ)l̄5 − l̄6 Needed for π → ℓνγ, can be had from ΠAA
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.79/103
-
Wishing list: ≥ 3-flavour
Ideas on how to make all those calculations usable foryou
Three and more flavour: typically slow numerically
large Nc suppressed couplings: i.e. ms dependence ofmπ, Fπ
Lr4, Lr6
sigma terms and scalar radii
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.80/103
-
Conclusions and final comments
Lots of analytical work done in ChPT
Use the correct ChPT2-flavour for varying m̂ and possible for Nf = 2 andNf = 2 + 1 at fixed ms (but have different LECs)
otherwise 3-flavourthe various partially quenched versions
Remember at which order in ChPT you compare things
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.81/103
-
Conclusions and final comments
Lots of analytical work done in ChPT
Use the correct ChPT2-flavour for varying m̂ and possible for Nf = 2 andNf = 2 + 1 at fixed ms (but have different LECs)
otherwise 3-flavourthe various partially quenched versions
Remember at which order in ChPT you compare things
Seen a lot of lattice work, looking forward to seeingmore
LECs in many talks: Boyle, Matsufuru, Urbach,Kuramashi and many parallel session talks
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.81/103
-
η → 3π
Reviews: JB, Gasser, Phys.Scripta T99(2002)34 [hep-ph/0202242]JB, Acta Phys. Slov. 56(2005)305 [hep-ph/0511076]
ηpη
π+pπ+
π−pπ−
π0 pπ0
s = (pπ+ + pπ−)2 = (pη − pπ0)2
t = (pπ− + pπ0)2 = (pη − pπ+)2
u = (pπ+ + pπ0)2 = (pη − pπ−)2
s + t + u = m2η + 2m2π+ + m
2π0 ≡ 3s0 .
〈π0π+π−out|η〉 = i (2π)4 δ4 (pη − pπ+ − pπ− − pπ0) A(s, t, u) .
〈π0π0π0out|η〉 = i (2π)4 δ4 (pη − p1 − p2 − p3) A(s1, s2, s3)
A(s1, s2, s3) = A(s1, s2, s3) + A(s2, s3, s1) + A(s3, s1, s2) ,Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.82/103
-
η → 3π: Lowest order (LO)
Pions are in I = 1 state =⇒ A ∼ (mu − md) or αemαem effect is small (but large via mπ+ − mπ0)η → π+π−π0γ needs to be included directly
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.83/103
-
η → 3π: Lowest order (LO)
Pions are in I = 1 state =⇒ A ∼ (mu − md) or αem
ChPT:Cronin 67: A(s, t, u) =B0(mu − md)
3√
3F 2π
{
1 +3(s − s0)m2η − m2π
}
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.83/103
-
η → 3π: Lowest order (LO)
Pions are in I = 1 state =⇒ A ∼ (mu − md) or αem
ChPT:Cronin 67: A(s, t, u) =B0(mu − md)
3√
3F 2π
{
1 +3(s − s0)m2η − m2π
}
with Q2 ≡ m2s−m̂2
m2d−m2uor R ≡ ms−m̂md−mu m̂ =
12(mu + md)
A(s, t, u) =1
Q2m2Km2π
(m2π − m2K)M(s, t, u)3√
3F 2π,
A(s, t, u) =
√3
4RM(s, t, u)
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.83/103
-
η → 3π: Lowest order (LO)
Pions are in I = 1 state =⇒ A ∼ (mu − md) or αem
ChPT:Cronin 67: A(s, t, u) =B0(mu − md)
3√
3F 2π
{
1 +3(s − s0)m2η − m2π
}
with Q2 ≡ m2s−m̂2
m2d−m2uor R ≡ ms−m̂md−mu m̂ =
12(mu + md)
A(s, t, u) =1
Q2m2Km2π
(m2π − m2K)M(s, t, u)3√
3F 2π,
A(s, t, u) =
√3
4RM(s, t, u)
LO: M(s, t, u) = 3s − 4m2π
m2η − m2πM(s, t, u) =
1
F 2π
(
4
3m2π − s
)
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.83/103
-
η → 3π beyondp4: p2 and p4
Γ (η → 3π) ∝ |A|2 ∝ Q−4 allows a PRECISE measurement
Q ≈ 24 gives lowest order Γ(η → π+π−π0) ≈ 66 eV .
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.84/103
-
η → 3π beyondp4: p2 and p4
Γ (η → 3π) ∝ |A|2 ∝ Q−4 allows a PRECISE measurement
Q ≈ 24 gives lowest order Γ(η → π+π−π0) ≈ 66 eV .
Other Source from m2K+ − m2K0 ∼ Q−2 =⇒ Q = 20.0 ± 1.5Lowest order prediction Γ(η → π+π−π0) ≈ 140 eV .
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.84/103
-
η → 3π beyondp4: p2 and p4
Γ (η → 3π) ∝ |A|2 ∝ Q−4 allows a PRECISE measurement
Q ≈ 24 gives lowest order Γ(η → π+π−π0) ≈ 66 eV .
Other Source from m2K+ − m2K0 ∼ Q−2 =⇒ Q = 20.0 ± 1.5Lowest order prediction Γ(η → π+π−π0) ≈ 140 eV .
At order p4 Gasser-Leutwyler 1985:
∫
dLIPS|A2 + A4|2∫
dLIPS|A2|2= 2.4 ,
(LIPS=Lorentz invariant phase-space)
Major source: large S-wave final state rescattering
Experiment: 295 ± 17 eV (PDG 2006)Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.84/103
-
η → 3π beyondp4: DispersiveTry to resum the S-wave rescattering:
Anisovich-Leutwyler (AL), Kambor,Wiesendanger,Wyler (KWW)
Different method but similar approximationsHere: simplified version of AL
Up to p8: No absorptive parts from ℓ ≥ 2=⇒ M(s, t, u) =
M0(s)+(s−u)M1(t)+(s−t)M1(t)+M2(t)+M2(u)−2
3M2(s)
MI : “roughly” contributions with isospin 0,1,2
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.85/103
-
η → 3π beyondp4: Dispersive
3 body dispersive: difficult: keep only 2 body cuts
start from πη → ππ (m2η < 3m2π) standard dispersive analysisanalytically continue to physical m2η.
MI(s) =1
π
∫ ∞
4m2π
ds′ImMI(s
′)
s′ − s − iεImMI(s
′) −→ discMI(s) =1
2i(MI(s + iε) − MI(s − iε))
M0(s) = a0 + b0s + c0s2 +
s3
π
Z
ds′
s′3discM0(s′)
s′ − s − iε ,
M1(s) = a1 + b1s +s2
π
Z
ds′
s′2discM1(s′)
s′ − s − iε ,
M2(s) = a2 + b2s + c2s2 +
s3
π
Z
ds′
s′3discM2(s′)
s′ − s − iε .
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.86/103
-
η → 3π beyondp4
• Technical complications in solving• Only 4 relevant constants:M(s, t, u) = a + bs + cs2 − d(s2 + tu)
M0(s) +4
3M2(s) sM1(s) + M2(s) + s
24L3 − 1/(64π2)F 2π (m
2η − m2π)
converge better
c = c0 +4
3c2 =
1
π
Z
ds′
s′3
discM0(s′) +
4
3discM2(s
′)
ff
,
d = −4L3 − 1/(64π2)
F 2π(m2η − m2π)
+1
π
Z
ds′
s′3
˘
s′discM1(s′) + discM2(s
′)¯
Fix a, b by matching to tree level or p4 amplitudeOdense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.87/103
-
η → 3π beyondp4
-0.5
0
0.5
1
1.5
2
2.5
1 2 3 4 5 6 7 8
Re
M (
KW
W)
s/mπ2
tree
p4
dispersive
Along s = u KWW
-0.5
0
0.5
1
1.5
2
2.5
1 2 3 4 5 6 7 8
Re
M (
AL)
s/mπ2
Adler zeroAdler zero
tree
p4
dispersive
Along s = u AL
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.88/103
-
η → 3π beyondp4
-0.5
0
0.5
1
1.5
2
2.5
1 2 3 4 5 6 7 8
Re
M
s/mπ2
p2
dispersive (match p2)
dispersive (match (|M|)
dispersive (match Re M)
Along s = u BG
Very simplified analysisJB, Gasser 2002looks more like AL
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.89/103
-
Two Loop Calculation: why
In Kℓ4 dispersive gave about half of p6 in amplitude
Same order in ChPT as masses for consistency checkon mu/mdCheck size of 3 pion dispersive part
At order p4 unitarity about half of correction
Technology exists:Two-loops: Amorós,JB,Dhonte,Talavera,. . .Dealing with the mixing π0-η:Amorós,JB,Dhonte,Talavera 01
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.90/103
-
Two Loop Calculation: why
In Kℓ4 dispersive gave about half of p6 in amplitude
Same order in ChPT as masses for consistency checkon mu/mdCheck size of 3 pion dispersive part
At order p4 unitarity about half of correction
Technology exists:Two-loops: Amorós,JB,Dhonte,Talavera,. . .Dealing with the mixing π0-η:Amorós,JB,Dhonte,Talavera 01
Done: JB, Ghorbani, arXiv:0709.0230 [hep-ph]Dealing with the mixing π0-η: extended to η → 3π
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.90/103
-
Diagrams
(a) (b) (c) (d)
(a) (b)(c)
(d)
(e) (f) (g) (h)
(i) (j) (k) (l)
Include mixing, renormalize, pull out factor√
34R , . . .
Two independent calculations (comparison majoramount of work)
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.91/103
-
Dalitzplot
0.08
0.1
0.12
0.14
0.16
0.08 0.1 0.12 0.14 0.16
t [G
eV
2]
s [GeV2]
mpiavmpidiff
u=ts=u
t-thresholdu thesholds threshold
x=y=0
x variation:vertical
y variation:parallel to t = u
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-
η → 3π: M(s, t = u)
-10
0
10
20
30
40
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
−M(s
,t,u
=t)
s [GeV2]
Li fit 10 and Ci
Re p2
Re p2+p
4
Re p2+p
4+p
6
Im p4
Im p4+p
6
Along t = u
-2
0
2
4
6
8
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
−M(s
,t,u
=t)
s [GeV2]
Li fit 10 and Ci
Re p6 pure loops
Re p6 Li
r
Re p6 Ci
r
sum p6
Along t = u parts
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.93/103
-
η → 3π: M(s, t = u)
-10
0
10
20
30
40
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
M(s
,t,u
=t)
s [GeV2]
Li = Ci = 0
Re p2
Re p2+p
4
Re p2+p
4+p
6
Im p4
Im p6
Along t = uLri = C
ri = 0
-10
0
10
20
30
40
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
−M(s
,t,u
=t)
s [GeV2]
Li fit 10 and Ci
Re p2+p
4
µ= 0.6 GeVµ= 0.9 GeV
Re p2+p
4+p
6
µ= 0.6 GeVµ= 0.9 GeV
Along t = u: µ dependenceI.e. where Cri (µ) estimated
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.94/103
-
η → 3π: M(s = u, t)
-10
0
10
20
30
40
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
−M(s
,t,u
=s)
s [GeV2]
Re p2
Re p2+p
4
Re p2+p
4+p
6
Im p4
Im p4+p
6
Along s = u
Shape agrees with AL
Correction larger:20-30% in amplitude
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-
Dalitz plot
x =√
3T+ − T−
Qη=
√3
2mηQη(u − t)
y =3T0Qη
− 1 = 3 ((mη − mπo)2 − s)
2mηQη− 1 iso= 3
2mηQη(s0 − s)
Qη = mη − 2mπ+ − mπ0
T i is the kinetic energy of pion πi
z =2
3
∑
i=1,3
(
3Ei − mηmη − 3m0π
)2
Ei is the energy of pion πi
|M |2 = A20(
1 + ay + by2 + dx2 + fy3 + gx2y + · · ·)
|M |2 = A20 (1 + 2αz + · · · )
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.96/103
-
Experiment: charged
Exp. a b d
KLOE −1.090 ± 0.005+0.008−0.019 0.124 ± 0.006 ± 0.010 0.057 ± 0.006
+0.007−0.016
Crystal Barrel −1.22 ± 0.07 0.22 ± 0.11 0.06 ± 0.04 (input)Layter et al. −1.08 ± 0.014 0.034 ± 0.027 0.046 ± 0.031
Gormley et al. −1.17 ± 0.02 0.21 ± 0.03 0.06 ± 0.04
KLOE has: f = 0.14 ± 0.01 ± 0.02.Crystal Barrel: d input, but a and b insensitive to d
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Theory: charged
A20
a b d f
LO 120 −1.039 0.270 0.000 0.000NLO 314 −1.371 0.452 0.053 0.027
NLO (Lri = 0) 235 −1.263 0.407 0.050 0.015NNLO 538 −1.271 0.394 0.055 0.025
NNLOp (y from T 0) 574 −1.229 0.366 0.052 0.023NNLOq (incl (x, y)4) 535 −1.257 0.397 0.076 0.004NNLO (µ = 0.6 GeV) 543 −1.300 0.415 0.055 0.024NNLO (µ = 0.9 GeV) 548 −1.241 0.374 0.054 0.025
NNLO (Cri = 0) 465 −1.297 0.404 0.058 0.032NNLO (Lri = C
ri = 0) 251 −1.241 0.424 0.050 0.007
dispersive (KWW) — −1.33 0.26 0.10 —tree dispersive — −1.10 0.33 0.001 —
absolute dispersive — −1.21 0.33 0.04 —error 18 0.075 0.102 0.057 0.160
NLO toNNLO:Littlechange
Error on|M(s, t, u)|2:
|M (6) M(s, t, u)|
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-
Experiment: neutral
Exp. α
KLOE 2007 −0.027 ± 0.004+0.004−0.006
KLOE (prel) −0.014 ± 0.005 ± 0.004Crystal Ball −0.031 ± 0.004
WASA/CELSIUS −0.026 ± 0.010 ± 0.010Crystal Barrel −0.052 ± 0.017 ± 0.010GAMS2000 −0.022 ± 0.023
SND −0.010 ± 0.021 ± 0.010
A2
0 α
LO 1090 0.000
NLO 2810 0.013
NLO (Lri = 0) 2100 0.016
NNLO 4790 0.013
NNLOq 4790 0.014
NNLO (Cri = 0) 4140 0.011
NNLO (Lri = Cri = 0) 2220 0.016
dispersive (KWW) — −(0.007—0.014)tree dispersive — −0.0065
absolute dispersive — −0.007Borasoy — −0.031
error 160 0.032
Note: NNLO ChPT gets a00 in ππ correct
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-
α is difficult
Expand amplitudes and isospin:
M(s, t, u) = A(
1 + ã(s − s0) + b̃(s − s0)2 + d̃(u − t)2 + · · ·)
M(s, t, u) = A(
3 +(
b̃ + 3d̃) (
(s − s0)2 + (t − s0)2 + (u − s0)2)
+ · ·
Gives relations (Rη = (2mηQη)/3)
a = −2Rη Re(ã) , b = R2η(
|ã|2 + 2Re(b̃))
, d = 6R2η Re(d̃) .
α =1
2R2η Re
(
b̃ + 3d̃)
=1
4
(
d + b − R2η|ã|2)
≤ 14
(
d + b − 14a2
)
equality if Im(ã) = 0
Large cancellation in α, overestimate of b likely the problemOdense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.100/103
-
r and decay rates
sin ǫ =√
34R + O(ǫ2)
Γ(η → π+π−π0) = sin2 ǫ · 0.572 MeV LO ,sin2 ǫ · 1.59 MeV NLO ,sin2 ǫ · 2.68 MeV NNLO ,sin2 ǫ · 2.33 MeV NNLOCri = 0 ,
Γ(η → π0π0π0) = sin2 ǫ · 0.884 MeV LO ,sin2 ǫ · 2.31 MeV NLO ,sin2 ǫ · 3.94 MeV NNLO ,sin2 ǫ · 3.40 MeV NNLOCri = 0 .
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-
r and decay rates
r ≡ Γ(η → π0π0π0)
Γ(η → π+π−π0)
rLO = 1.54
rNLO = 1.46
rNNLO = 1.47
rNNLO Cri =0 = 1.46
PDG 2006
r = 1.49 ± 0.06 our average .r = 1.43 ± 0.04 our fit ,Good agreement
Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.102/103
-
R and Q
LO NLO NNLO NNLO (Cri = 0)
R (η) 19.1 31.8 42.2 38.7
R (Dashen) 44 44 37 —
R (Dashen-violation) 36 37 32 —
Q (η) 15.6 20.1 23.2 22.2
Q (Dashen) 24 24 22 —
Q (Dashen-violation) 22 22 20 —
LO from R =m2K0 + m
2K+ − 2m2π0
2(
m2K0
− m2K+
) (QCD part only)
NLO and NNLO from masses: Amorós, JB, Talavera 2001
Q2 =ms + m̂
2m̂R = 12.7R (ms/m̂ = 24.4)
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OverviewWikipediaWikipediaEffective Field TheoryPower CountingWhy is the sky blue ?Why Field Theory ?Examples of EFTreferencesSchoolChiral Perturbation TheoryThe mass gap: Goldstone ModesThe two symmetry modes comparedSome clarificationsThe power countingChiral Perturbation TheoryChiral Perturbation TheoryChiral Perturbation TheoryChiral Perturbation Theor{�lue ies}LagrangiansExternal currents?LagrangiansLagrangiansChiral LogarithmsLECs and $mu $Expand in what quantities?An exampleAn example: $m_0/f_0$An example: $m_pi /f_pi $Two-loop Two-flavourTwo-Loop Two-flavourLECsLECs$m_pi ^2$$m_pi ^2$ ($�ar l_3 = 0$)$F_pi $Two-loop Three-flavour, $le $2001Two-loop Three-flavour, $ge $2001Two-loop Three-flavour$C_i^r$$C_i^r$$C_i^r$InputsOutputs: ICorrelationsOutputs: II$pi pi $$pi pi $ and $pi K$Quark mass dependences$m_pi ^2$ fit 10$m_pi ^2$ fit D$m_K^2$ fit 10$m_K^2$ fit D$F_pi $ fit 10$F_K$ fit 10$F_K/F_{pi }$ fit 10$m_pi ^2$ fit 10, fixed $hat m$$F_pi $ fit 10, fixed $hat m$$f_+(0)$fit 10, fixed $m_K$$(f_+(0))/left(m_K^2-m_pi ^2ight )^2$ fit 10, fixed $m_K$$f_0(t)$in $K_{ell 3}$$ge 3$-flavour: PQChPTPQChPTPartial Quenching and ChPTPQChPT at Two Loops: GeneralPQChPT at Two Loops: GeneralPQChPT at Two Loop: PapersLong ExpressionsWhy so long expressionsPQChPT at Two LoopPQChPT: 1+1 case, 3 sea quarksPQChPT: 1+1 case, 3 sea quarksPQChPT: 1+1 case, 2 sea quarksPQChPT: 2 sea quarksPQChPT: Fitting StrategyConclusions PQChPT PartWishing listWishing list: Two-flavourWishing list: $ge 3$-flavourConclusions and final comments$eta o 3pi $ $eta o 3pi $: Lowest order (LO)$eta o 3pi $ beyond $p^4$: $p^2$ and $p^4$$eta o 3pi $ beyond $p^4$: Dispersive$eta o 3pi $ beyond $p^4$: Dispersive$eta o 3pi $ beyond $p^4$$eta o 3pi $ beyond $p^4$$eta o 3pi $ beyond $p^4$Two Loop Calculation: whyDiagramsDalitzplot$eta o 3pi $: $M(s,t=u)$$eta o 3pi $: $M(s,t=u)$$eta o 3pi $: $M(s=u,t)$Dalitz plotExperiment: chargedTheory: chargedExperiment: neutral$alpha $ is difficult$r$ and decay rates$r$ and decay rates$R$ and $Q$