Chiral-scale Perturbation Theory About an Infrared Fixed Point

Chiral-scale Perturbation TheoryAbout an Infrared Fixed Point

Lewis C. Tunstallwith R.J. Crewther

Albert Einstein Centre for Fundamental PhysicsUniversitat Bern

13th International Conference on Meson-Nucleon Physics andthe Structure of the Nucleon, Rome, Italy, 30 September 2013

1 | QCD at low energies

Two powerful tools to analyse non-perturbative QCD

chiral symmetry

effective field theory [Weinberg 79]

Symmetry is hidden for 3 light quark flavours u, d , s

SU(3)L × SU(3)R −→ SU(3)V

〈qq〉vac 6= 0 most likely origin of chiral symmetry breaking

physical spectrum contains 8 pseudo-NG bosons {π,K , η}Model-independent marriage = chiral perturbation theory χPT3



chiral symmetry



SU(3)L × SU(3)R −→ SU(3)V





chiral symmetry



SU(3)L × SU(3)R −→ SU(3)V


physical spectrum contains 8 pseudo-NG bosons {π,K , η}

Model-independent marriage = chiral perturbation theory χPT3



chiral symmetry



SU(3)L × SU(3)R −→ SU(3)V




trace anomaly associated with dilatations x → eξx

∂µDµ = θµµ =β(αs)

4αsG aµνG

aµν +(1 + γm(αs)

) ∑q=u,d ,s

mqqq

Gluonic anomaly −→ generates most of nucleon mass

MN = 〈N|θµµ|N〉 =β(αs)

4αs〈N|G a

µνGaµν |N〉+ O

(m2

K

)This talk

phenomenological implications for β → 0 in IR limit

extension of NG sector in 3-flavour theory

replacement of χPT3 with new effective field theory




4αsG aµνG


) ∑q=u,d ,s

mqqq



4αs〈N|G a

µνGaµν |N〉+ O

(m2

K

)

This talk







4αsG aµνG


) ∑q=u,d ,s

mqqq



4αs〈N|G a

µνGaµν |N〉+ O

(m2

K

)This talk




3 | Chiral SU(3)L × SU(3)R perturbation theory

χPT3

effective field theory for low-energy π,K , η interactions

well established tool to study non-perturbative QCD

Scattering amplitudes calculated via asymptotic series

A ={ALO +ANLO +ANNLO + . . .

}χPT3

powers of O(mK ) momentum and mu,d ,s = O(m2K )

Key premise —NG sector {π,K , η} dominates the non-NG sector {ρ, ω, . . .}PCAC = pole-dominance


χPT3





}χPT3




χPT3





}χPT3



4 | Problems with SU(3)L × SU(3)R?

Old observation — lowest order χPT3 fails for amplitudes with

a 0++ channel and O(mK ) extrapolations in momenta

Notable examplesFinal state ππ interactions important [Truong 84, 90]

K`4 decays [Truong 81]Non-leptonic K decays [Neveu & Scherk 70, Truong 88]Non-leptonic η decays [Roisnel 81, G & L 86, 96]

Γ(KL → π0γγ) only 1/3 measured value [Ecker et al. 87]

LO prediction for σ(γγ → π0π0) [Donoghue/Bijnens et al. 88]

February 1, 2008 8:4 WSPC/INSTRUCTION FILE review-mpla-rev

8 M.R. Pennington

Fig. 6. Integrated cross-section for γγ → ππ as a function of c.m. energy M(ππ) from Mark II 46, CrystalBall 47,48 and CLEO 49. The π0π0 results have been scaled to the same angular range as the charged data and byan isospin factor. Below are graphs describing the dominant dynamics in each kinematic region, as discussed inthe text.

Fig. 7. Cross-section for γγ → π0π0 integrated over | cos θ∗| ≤ 0.8 as a function of the ππ invariant mass M(ππ).The data are from Crystal Ball 47,48. The line is the prediction of χPT at one loop (1%) 53. The shaded bandshows the dispersive prediction 59,61,62 described later — its width reflects the uncertainties in experimentalknowledge of both ππ scattering and vector exchanges.

Figure: Pennington 07









8 M.R. Pennington












8 M.R. Pennington












8 M.R. Pennington





Consider KL → π0γγ in lowest order of χPT3

KL

π0

γ

γπ±, K±

Include dispersive NLO corrections and truncate series for A:

AKL→π0γγ '{ALO +ANLO

}χPT3

Fit to data achieved only for∣∣ANLO

∣∣χPT3

&√

2∣∣ALO

∣∣χPT3

Reconcile with success of χPT3 elsewhere?

Corrections to LO χPT3 should be ∼ 30% at most∣∣ANLO

/ALO

∣∣χPT3

. 0.3 , acceptable fit



KL

π0

γ

γπ±, K±



}χPT3


∣∣χPT3

&√

2∣∣ALO

∣∣χPT3



/ALO

∣∣χPT3




KL

π0

γ

γπ±, K±



}χPT3


∣∣χPT3

&√

2∣∣ALO

∣∣χPT3



/ALO

∣∣χPT3



Origin of large dispersive effects?

dispersive analysis of ππ-scattering + precise Ke4 dataindicates complex pole

√sp = M − iΓ/2 on second Riemann

sheet:

Mf0 = 441+16−8 MeV , Γf0 = 544+18

−25 MeV [Caprini et al. 06]


4 M.R. Pennington

Fig. 2. An illustration of the sheets and cut structure of the complex energy plane in a world with just onethreshold and how these are connected. This represents the structure relevant to ππ scattering near its threshold.Experiment is performed on the top shaded sheet, just above the cut along the real energy axis. The cross on thelower, or second, sheet indicates where the σ-pole resides at E2 = s = sR. The ellipse above this on the top, orfirst, sheet indicates where the S -matrix is zero.

chiral constraints, allows the I = J = 0 ππ partial wave to be determined everywhere onthe first sheet of the energy plane 19, Fig. 2. As shown by Caprini et al. 17, this fixes azero of the S -matrix (symbolically depicted by the solid ellipse in Fig. 2) at E = 441 −i 227 MeV, which reflects a pole (denoted by the cross) on the second sheet at the sameposition. This not only confirms the σ as a state in the spectrum of hadrons but locatesthe position of its pole very precisely with errors of only tens of MeV. This is within theregion found by Zhou et al 20, who also took into account crossing and the left-hand cut.While the chiral expansion of amplitudes can have no poles at any finite order, particularsummations may. The inverse amplitude method 21 is one such procedure. Application ofthis by Pelaez et al. 22 did indeed find a pole in the same domain as Caprini et al. but manyyears earlier. However, without a proof that the Inverse Amplitude Method, rather thanany other, provided precision unitarisation of the low order chiral expansion, the presentauthor rather believed the analysis of a wide range of data of Ref. 10 that indicated nopole (or perhaps a very distant one). Now we know differently. There is a pole with awell-defined location. This is far from the position proposed by the treatment of Ishida etal. 23,24,25,26,27, the deficiencies of which were explained long ago in Ref. 21. (Bugg 28has added to these arguments in response to the discussion on the position of the κ by thesame group 29 .)

With a narrow resonance, there would naturally be a close correlation between the phasevariation of the underlying amplitude on the real axis and in the complex plane, as onepasses the pole. However, for the very short-lived σ, “deep” in the complex energy plane,this simple connection is lost. In Fig. 3 we show the phase of the I = J = 0 ππ → ππscattering amplitude along two lines in the complex energy plane. The phase along thereal axis is compared with relevant data from scattering 32 and Ke4 decays 6 in Fig. 3.One sees how different this phase is compared with that on the lower sheet of Fig. 2 atImE = −0.25 GeV. That deep in the complex plane shows the 180o phase change expectedof a resonance. It is the dramatic variation in the amplitude as one moves away from the


Result is model-independent — derived from general principlesof QFT (Roy equations)


PDG have recently updated their entry for f0

Mf0 = (400− 550) MeV , Γf0 = (400− 700) MeV .

Dispersive techniques provide (in principle) controlled errorcalculation

Figure: Albaladejo and Oller 12


Conclude?

the existence of f0(500) well established

dispersion theory for K`4, nonleptonic K and η decays farbetter understood

But . . .

does not alter fact that LO of χPT3 fits these data so poorly


}χPT3

if first term poor fit −→ any truncation unsatisfactory

low energy expansion diverges


Conclude?



But . . .



}χPT3




Conclude?



But . . .



}χPT3




Conclude?



But . . .



}χPT3




Conclude?



But . . .



}χPT3




Two philosophies

limits to applicability of χPT3 — expect failures in a few cases

or

consistent trend of failure in 0++ channels which can andshould be corrected

Proposal: Modify lowest order of χPT3

Must preserve LO successes of χPT3 for reactions which donot involve f0(500) and O(mK ) extrapolations

spontaneously broken scale invariance identified as relevantsymmetry


Two philosophies


or






Two philosophies


or






Two philosophies


or






Two philosophies


or






Two philosophies


or





10 | QCD Infrared fixed point

Scale invariance a generic feature at fixed points


4αsG aµνG


) ∑q=u,d ,s

mqqq

O

β

αsαIR

Nf = 3 proposal

UV IR

Literature extensive, yet inconclusive on existence of αIR

lattice — no αIR for Nf < 8 [Appelquist et al. 09]

Dyson-Schwinger — αIR for Nf = 0, 3 [Alkofer, Fischer 97-03]

effective charges — ‘freezing’ [Mattingly & Stevenson 94, 13]

light-front holography — [Brodsky & de Teramond 10]




4αsG aµνG


) ∑q=u,d ,s

mqqq

O

β

αsαIR

Nf = 3 proposal

UV IR









4αsG aµνG


) ∑q=u,d ,s

mqqq

O

β

αsαIR

Nf = 3 proposal

UV IR






11 | f0(500) as a QCD dilaton

Gluonic anomaly absent at αIR −→ scale invariance

θµµ∣∣αs=αIR

=(1 + γm(αIR)

)(muuu + md dd + ms ss)

→ 0 , SU(3)L × SU(3)R limit

Conclude?

〈qq〉vac acts as both a chiral and scale condensate

NG sector extended to include 0++ dilaton σ = f0

{π,K , η} −→ {π,K , η, σ/f0}

replace χPT3 −→ chiral-scale perturbation theory χPTσ

in χPTσ, ms sets scale of m2f0

as well as m2K and m2

η

PCDC = σ-pole dominance in 0+ channels




=(1 + γm(αIR)



Conclude?



{π,K , η} −→ {π,K , η, σ/f0}




η





=(1 + γm(αIR)



Conclude?



{π,K , η} −→ {π,K , η, σ/f0}




η





=(1 + γm(αIR)



Conclude?



{π,K , η} −→ {π,K , η, σ/f0}




η





=(1 + γm(αIR)



Conclude?



{π,K , η} −→ {π,K , η, σ/f0}




η


12 | Separation of scales

χPT2

χPT3

NG bosons

p·p′=O(m2π) Not NG bosons

π f0 K η(mass)2

0

0

χPTσ

(mass)2

(mass)2

π

f0

K η

ρ

π f0 K η ρ

Not NG bosons0

0

NG bosons p·p′=O(m2K)

NG bosons p·p′=O(m2K)

(mass)2

Not NGbosons

scaleseparation

scaleseparation

rules for counting powers of mK changed in χPTσ— f0 pole amplitudes promoted to LO

13 | Infrared expansions

both χPT3 and χPTσ involve the limit

mi ∼ 0 , mi/mj fixed, i , j = u, d , s

amplitudes expanded in powers and logs of

{momenta}/χch � 1 , χch ≈ 1 GeV

χch ' 4πFπ in χPT3 (‘dim. analysis’) [Manohar & Georgi 84]

chiral scale χch also sets mass scale for non-NG particles






















14 | Nucleon mass

Contrast different pictures for nucleon mass

χPT3 ⇒ no sense in which gluonic anomaly is small

MN = 〈N|θµµ|N〉 =χPT3

β(αs)

4αs〈N|G a

µνGaµν |N〉+ O

(m2

K

)assumes f0(500) pole terms can be neglected

or equivalently

f0 couples weakly to G 2 and qq

small f0 mass ⇒ no scale separation in χPT3

14 | Nucleon mass




β(αs)

4αs〈N|G a

µνGaµν |N〉+ O

(m2

K

)

assumes f0(500) pole terms can be neglected

or equivalently



14 | Nucleon mass




β(αs)

4αs〈N|G a

µνGaµν |N〉+ O

(m2

K


or equivalently



14 | Nucleon mass




β(αs)

4αs〈N|G a

µνGaµν |N〉+ O

(m2

K


or equivalently



15 | Nucleon mass

In χPTσ

infrared regime emphasizes values of αs close to αIR

O

β

αsαIR

Nf = 3 proposal

UV IR

combined limit must be considered

mu,d ,s ∼ 0 and αs . αIR

β(αs) small ⇒ gluonic anomaly small as an operator

but can produce large amplitudes when coupled to dilatons

15 | Nucleon mass

In χPTσ


O

β

αsαIR

Nf = 3 proposal

UV IR





15 | Nucleon mass

In χPTσ


O

β

αsαIR

Nf = 3 proposal

UV IR





15 | Nucleon mass

In χPTσ


O

β

αsαIR

Nf = 3 proposal

UV IR





16 | Nucleon mass

θµµ

N N

σ

gσNN

σ couples to vacuum via divergence of symmetry current

〈σ|θµµ|vac〉 = −m2σFσ = O(m2

σ) , mσ → 0

nucleon mass remains massive in scaling limit

analogue of Goldberger-Treiman relation

FσgσNN ' MN

analysis of NN scattering: gσNN ' 9 [Calle-Cordon et al. 08]

Fσ ≈ 100 MeV cf. Fπ ' 93 MeV

16 | Nucleon mass

θµµ

N N

σ

gσNN



σ) , mσ → 0



FσgσNN ' MN



16 | Nucleon mass

θµµ

N N

σ

gσNN



σ) , mσ → 0



FσgσNN ' MN



17 | Nucleon mass

In χPTσ

gluonic anomaly and the quark mass term in for θµµ cancontribute to MN in the chiral-scale limit

NG boson requirement

m2σ = O(m2

K ) = O(mu,d ,s)

Allows the constants FG2 and Fqq given by

β(αs)/

(4αs)〈σ|G 2|vac〉 = −m2σFG2 ,

{1 + γm(αs)}∑

q=u,d ,s

mq〈σ|qq|vac〉 = −m2σFqq

to remain finite in chiral-scale limit θµµ → 0

MN ' FG2gσNN + FqqgσNN

17 | Nucleon mass

In χPTσ



m2σ = O(m2

K ) = O(mu,d ,s)


β(αs)/

(4αs)〈σ|G 2|vac〉 = −m2σFG2 ,

{1 + γm(αs)}∑

q=u,d ,s




17 | Nucleon mass

In χPTσ



m2σ = O(m2

K ) = O(mu,d ,s)


β(αs)/

(4αs)〈σ|G 2|vac〉 = −m2σFG2 ,

{1 + γm(αs)}∑

q=u,d ,s




17 | Nucleon mass

In χPTσ



m2σ = O(m2

K ) = O(mu,d ,s)


β(αs)/

(4αs)〈σ|G 2|vac〉 = −m2σFG2 ,

{1 + γm(αs)}∑

q=u,d ,s




18 | Chiral-Scale Lagrangian

Consider physical region 0 < αs < αIR

L[σ,U,U†

]= Ld=4

inv + Ld>4anom + Ld<4

mass

Building blocks

chiral invariant QCD dilaton σ

SU(3) field U = U(π,K , η)

Operator dimensions satisfy

dinv = 4 and 1 6 dmass < 4 [Wilson 69]

Callan-Symanzik equation for QCD amplitudes

dmass = 3− γm(αIR

)and danom = 4 + β′

(αIR

)> 4

consistency requires Lanom = O(∂2,M)



L[σ,U,U†

]= Ld=4


mass

Building blocks








(αIR

)> 4




L[σ,U,U†

]= Ld=4


mass

Building blocks








(αIR

)> 4




L[σ,U,U†

]= Ld=4


mass

Building blocks








(αIR

)> 4




L[σ,U,U†

]= Ld=4


mass

Building blocks








(αIR

)> 4




L[σ,U,U†

]= Ld=4


mass

Building blocks








(αIR

)> 4




L[σ,U,U†

]= Ld=4


mass

Building blocks








(αIR

)> 4




L[σ,U,U†

]= Ld=4


mass

Building blocks








(αIR

)> 4



Want formula for L[σ,U,U†

]= Ld=4


mass

dilaton transforms non-linearly

σ → σ − 14Fσ log

∣∣ det(∂x ′/∂x)∣∣

d = 1 building block for scale symmetry: eσ/Fσ

K[U,U†

]= 1

4F2πTr(∂µU∂

µU†) Kσ =1

2∂µσ∂

µσ

at lowest order in χPTσ

Ld=4inv =

{c1K + c2Kσ + c3e

2σ/Fσ}e2σ/Fσ

Ld>4anom =

{(1− c1)K + (1− c2)Kσ + c4e

2σ/Fσ}e(2+β′)σ/Fσ

Ld<4mass = Tr(MU† + UM†)e(3−γm)σ/Fσ

vacuum stability −→ c3,4 ∼ O(M)



]= Ld=4


mass



∣∣ det(∂x ′/∂x)∣∣


K[U,U†

]= 1

4F2πTr(∂µU∂

µU†) Kσ =1

2∂µσ∂

µσ


Ld=4inv =

{c1K + c2Kσ + c3e

2σ/Fσ}e2σ/Fσ

Ld>4anom =

{(1− c1)K + (1− c2)Kσ + c4e






]= Ld=4


mass



∣∣ det(∂x ′/∂x)∣∣


K[U,U†

]= 1

4F2πTr(∂µU∂

µU†) Kσ =1

2∂µσ∂

µσ


Ld=4inv =

{c1K + c2Kσ + c3e

2σ/Fσ}e2σ/Fσ

Ld>4anom =

{(1− c1)K + (1− c2)Kσ + c4e






]= Ld=4


mass



∣∣ det(∂x ′/∂x)∣∣


K[U,U†

]= 1

4F2πTr(∂µU∂

µU†) Kσ =1

2∂µσ∂

µσ


Ld=4inv =

{c1K + c2Kσ + c3e

2σ/Fσ}e2σ/Fσ

Ld>4anom =

{(1− c1)K + (1− c2)Kσ + c4e






]= Ld=4


mass



∣∣ det(∂x ′/∂x)∣∣


K[U,U†

]= 1

4F2πTr(∂µU∂

µU†) Kσ =1

2∂µσ∂

µσ


Ld=4inv =

{c1K + c2Kσ + c3e

2σ/Fσ}e2σ/Fσ

Ld>4anom =

{(1− c1)K + (1− c2)Kσ + c4e




20 | Strong interactions

From L obtain dilaton mass

m2σF

2σ = F 2

π (m2K + 1

2m2π)(3− γm)(1 + γm)− β′(4 + β′)c4] ,

and the effective σππ coupling

Lσππ ={(

2 + (1− c1)β′)|∂π|2 − (3− γm)m2

π|π|2}σ/(2Fσ)

Key feature: mostly derivative

small effect on ππ-scattering in SU(2)L × SU(2)R limit∂ = O(mπ)

vertex for on-shell amplitude for σ → ππ:

gσππ = −(2 + (1− c1)β′

)m2σ/(2Fσ) + O(m2

π)



m2σF

2σ = F 2

π (m2K + 1

2m2π)(3− γm)(1 + γm)− β′(4 + β′)c4] ,


Lσππ ={(

2 + (1− c1)β′)|∂π|2 − (3− γm)m2

π|π|2}σ/(2Fσ)




gσππ = −(2 + (1− c1)β′

)m2σ/(2Fσ) + O(m2

π)



m2σF

2σ = F 2

π (m2K + 1

2m2π)(3− γm)(1 + γm)− β′(4 + β′)c4] ,


Lσππ ={(

2 + (1− c1)β′)|∂π|2 − (3− γm)m2

π|π|2}σ/(2Fσ)




gσππ = −(2 + (1− c1)β′

)m2σ/(2Fσ) + O(m2

π)



m2σF

2σ = F 2

π (m2K + 1

2m2π)(3− γm)(1 + γm)− β′(4 + β′)c4] ,


Lσππ ={(

2 + (1− c1)β′)|∂π|2 − (3− γm)m2

π|π|2}σ/(2Fσ)




gσππ = −(2 + (1− c1)β′

)m2σ/(2Fσ) + O(m2

π)

21 | χPTσ at next to leading order

Add σ-loops to χPT3 analysis to test convergence of χPTσ

involves unknown LECs β′, γ, c1...4 in σσσ and σσππ vertices

π

ππ

π

π, Kσ

Apply Manohar & Georgi to first diagram

Aloop

/Atree ∼

1

16π2F 2π

× logarithms

in χPTσ must add ∼{

1

16π2F 2σ

andF 2π

16π2F 4σ

}× logarithms

in principle two χPTσ scales

χπ = 4πFπ and χσ = 4πFσ ; χπ ≈ χσ ≈ 1 GeV




π

ππ

π

π, Kσ


Aloop

/Atree ∼

1

16π2F 2π

× logarithms


1

16π2F 2σ

andF 2π

16π2F 4σ

}× logarithms






π

ππ

π

π, Kσ


Aloop

/Atree ∼

1

16π2F 2π

× logarithms


1

16π2F 2σ

andF 2π

16π2F 4σ

}× logarithms






π

ππ

π

π, Kσ


Aloop

/Atree ∼

1

16π2F 2π

× logarithms


1

16π2F 2σ

andF 2π

16π2F 4σ

}× logarithms






π

ππ

π

π, Kσ


Aloop

/Atree ∼

1

16π2F 2π

× logarithms


1

16π2F 2σ

andF 2π

16π2F 4σ

}× logarithms



22 | The f0 width in χPTσ

The f0(500) is almost as broad as it is heavy

is Γσ a LO effect?

invalidates PCDC?

Estimate width a la Manohar & Georgi

Γσππ ≈|gσππ|216πmσ

∼ m3σ

16πF 2σ

∼ 250 MeV

justified use of tree approximation to generate LO of χPTσ

methods like complex mass scheme may be necessary beyondLO and in degenerate cases KL − KS oscillations




invalidates PCDC?



∼ m3σ

16πF 2σ

∼ 250 MeV






invalidates PCDC?



∼ m3σ

16πF 2σ

∼ 250 MeV






invalidates PCDC?



∼ m3σ

16πF 2σ

∼ 250 MeV



23 | Weak interactions

In the leading order of χPT3

Lweak|χPT3= g8Q8 + g27Q27 + Qmw + h.c.

vacuum stability? [Crewther 86]

Qmw = Tr(λ6 − iλ7)(gMMU† + gMUM†

)absorbed by field rotation −→ cannot be used to explain∆I = 1/2 rule

in χPTσ, anomalous mass dimensions unrelated

3− γm(αIR) 6= 3− γmw (αIR)

Qmweσ(3−γmw )/Fσ cannot be eliminated by chiral rotation




















scale weak operators and align vacuum

Lalignweak =Q8

∑n

g8ne(2−γ8n)σ/Fσ + g27Q27e

(2−γ27)σ/Fσ

+ Qmw

{e(3−γmw )σ/Fσ − e(3−γm)σ/Fσ

}+ h.c.

residual interaction mixes KS and σ in leading order

gKσ = (γm − γmw )<e[(2m2K −m2

π)gM −m2πgM ]Fπ/Fσ

produces a pure ∆I = 1/2 amplitude

+K0S

π

π

g8,27

σ

gσππgKSσ



Lalignweak =Q8

∑n


(2−γ27)σ/Fσ

+ Qmw


}+ h.c.





+K0S

π

π

g8,27

σ

gσππgKSσ



Lalignweak =Q8

∑n


(2−γ27)σ/Fσ

+ Qmw


}+ h.c.





+K0S

π

π

g8,27

σ

gσππgKSσ


key processes: KS → γγ and γγ → ππ

|gKσ| ≈ 4.4× 103 keV2, 30% precision

to the extent that gσNN and hence Fσ can be determined

|Aσ-pole| ≈ 0.34 keV cf. data |A0|expt. = 0.33 keV

conclude g8 =∑

n g8n and g27 can have similar magnitudes

leading order of χPTσ explains the ∆I = 1/2 puzzle






conclude g8 =∑








conclude g8 =∑








conclude g8 =∑



Electromagnetic interactions

electromagnetic anomaly [Crewther 72; Chanowitz & Ellis 72]

θµµ∣∣strong + e′mag

= θµµ + (Rα/6π)FµνFµν ,

R =σ(e+e− → hadrons)

σ(e+e− → µ+µ−)

∣∣∣∣high−energy

anomaly realised in χPTσ through effective coupling

Lσγγ = 12gσγγσFµνF

µν , gσγγ =(RIR − 1

2 )α

3πFσ

dispersive methods determine f0 → γγ width [Pennington 06]

Γf0γγ = 1.98± 0.3 keV [Oller & Roca 08]

non-perturbative χPTσ prediction RIR ≈ 5






σ(e+e− → µ+µ−)





2 )α

3πFσ









σ(e+e− → µ+µ−)





2 )α

3πFσ









σ(e+e− → µ+µ−)





2 )α

3πFσ




Concluding remarks

The assumption of aninfrared fixed point in3-flavour QCD leads to anextended Goldstone sector{π,K , η, σ}, where σ isidentified with f0(500) as theNG-boson of spontaneouslybroken scale symmetry.

O

β

αsαIR

Nf = 3 proposal

UV IR

Model independent theory χPTσ promotes f0 pole terms toleading order ⇒ χPT3’s problems in the 0++ channelovercome.

Retains χPT3’s successful predictions elsewhere: NLOanalysis of strong sector in progress.

The ∆I = 1/2 rule for K -decays emerges as a consequence ofχPTσ.

χPTσ has implications for CP violation, rare kaon decays, andpossibly η → 3π.

Documents

Chiral-scale Perturbation Theory About an Infrared Fixed Point