Infrared fixed point and approximate chiral-scale symmetry ... · Infrared xed point and...

Infrared fixed point and approximate chiral-scalesymmetry in non-perturbative QCD

Lewis C. Tunstallwith R.J. Crewther

arXiv:1203.1321 & 1312.3319

Albert Einstein Centre for Fundamental PhysicsUniversitat Bern

TU Dresden, 10 July 2014

Chiral-Scale Perturbation Theory χPTσ

Assume Nf = 3 IR fixed point

O

β

αsαIR

Nf = 3 proposal

UV IR

What is the EFT for IR expansions about αIR?

Prompted by

poor χPT convergence in 0++ channels [Meissner 1991]

f0(441− i 272) 0++ pole on second sheet [Caprini et al 2006]

old puzzle in K -decays: ∆I = 1/2 rule [Gell-Mann 1954]

Low energy QCD

strong interactions described by SU(3)c gauge theory

LQCD = −1

4G aµνG

aµν +∑Nf

q(i /D −mq)q

for µ . 1 GeV, running coupling αs ∼ O(1)⇒ breakdown inperturbation theory

key non-perturbativephenomena:

confinement

limR→∞

VQQ(R) = σ · R +const

R

Figure: Leinweber 2003

Low energy QCD

strong interactions described by SU(3)c gauge theory

LQCD = −1

4G aµνG

aµν +∑Nf

q(i /D −mq)q

for µ . 1 GeV, running coupling αs ∼ O(1)⇒ breakdown inperturbation theorykey non-perturbativephenomena:

dynamical chiralsymmetry breaking:〈qq〉vac 6= 0

‘mass from nothing’ ∼2% of nucleon mass fromcurrent-quark masses

Figure: Roberts 2011

Broken scale invariance

decouple heavy quarks t, b, c and consider mu,d ,s → 0

⇒ LQCD inv. under global

{x → eξx

SU(3)L × SU(3)R

quantum corrections give rise to trace anomaly

∂µDµ = θµµ =β(αs)

4αsG aµνG

aµν +[1 + γm(αs)

] ∑q=u,d ,s

mqqq

UV perturbative expansion about αs ' 0:

β(αs) = µ2∂αs

∂µ2= −b1

α2s

16π2− b2

α3s

64π3+ O(α4

s )

hadrons remain massive in mq → 0 limit:

MN = 〈N|θµµ|N〉 =β(αs)

4αs〈N|G 2|N〉+(1+γm)

∑u,d ,s

〈N|mqqq|N〉︸︷︷︸σ terms

Spontaneously broken chiral symmetry

Two powerful tools to analyse non-perturbative QCD

chiral symmetry

effective field theory [Weinberg 1979]

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}masses from small explicit breaking

m2φ ∝ mq [Gell-Mann et al 58]

Model-independent marriage = chiral perturbation theory χPT3

3-flavour chiral perturbation theory χPT3

encode π,K , η in SU(3) matrix U = U(π,K , η)

treat quark masses M ∝ diag(mu,md ,ms) small perturbation

construct most general Lagrangian consistent with underlyingsymmetries

LQCD[q†, q,Aµ] −→ Leff [U, ∂U, . . . ,M] = L2 + L4 + . . .

low energy amplitudes calculated via asymptotic series

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

}χPT3

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

L2 =Fπ

2

4Tr(∂µU∂

µU†) + Tr(MU† + UM†)

3-flavour chiral perturbation theory χPT3

χPT3 provides a systematic framework to compute low energyQCD Green’s functions respecting:

unitarity / analyticity / symmetry


}χPT3

method is rigourously established and yields a set of rules:

LO tree graphs with L2

NLO tree graphs with L4

1-loop graphs with L2

NNLO tree graphs with L6

2-loop graphs with L2

1-loop graphs with one vertex from L4

main obstacle to precision: proliferation of LECs

The χPT3 dilemma

Old observation [Meissner 1991]

— lowest order χPT3 typically fails for amplitudes with

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

Final state ππ interactions important [Truong 1984, 1990]

π

π

π

π

= . . .

K`4 decays [Truong 1981]

Non-leptonic K decays [Neveu & Scherk 1970, Truong 1988]

Non-leptonic η decays [Roisnel 81, Gasser & Leutwyler 86, 96]

The χPT3 dilemma

Notable examples

Γ(KL → π0γγ) only 1/3 measured value [Ecker et al. 1987]

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

The χPT3 dilemma

Lowest order prediction for σ(γγ → π0π0) [Donoghue et al. &

Bijnens, Cornet 1988]

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

8 M.R. Pennington

Fig. 6. Integrated cross-section for !! ! "" as a function of c.m. energy M("") from Mark II 46, CrystalBall 47,48 and CLEO 49. The "0"0 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 !! ! "0"0 integrated over | cos #"| # 0.8 as a function of the "" invariant mass M("").The data are from Crystal Ball 47,48. The line is the prediction of $PT 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 "" scattering and vector exchanges.

Figure: (left) from Donoghue et al. 1988, (right) from Pennington 2007

The χPT3 dilemma

Extreme case KS → ππ

Experimental data exhibit the striking ratios

Γ(K 0S → π+π−)

Γ(K+ → π+π0)' 463

Γ(K 0S → π0π0)

Γ(K+ → π+π0)' 205

|A0/A2| ' 22 (∆I = 1/2 rule)

KS

π

π

g8,27 theoretical estimates: 1 ∼ g8,27 ∼ 5

data requires: |g8/g27| ∼ 22 (octet dominance)

|ANLO/ALO| ' 70× 0.3 (!)

The lowest QCD resonance

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 [CCL 2006]

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

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 !! 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 "-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 !! 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 " 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 di!erently. 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 # 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 ", “deep” in the complex energy plane,this simple connection is lost. In Fig. 3 we show the phase of the I = J = 0 !! " !!scattering 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 di!erent 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

Figure: from Pennington 2007

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

The lowest QCD resonance

PDG have recently updated their entry for f0(500)

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

Dispersive techniques provide (in principle) controlled errorcalculation

Figure: from Albaladejo and Oller 2012

Conclude?

the existence of f0(500) well established

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

But . . .

poor fits to data in 0++ channel from lowest order χPT3


}χPT3

low energy expansion diverges

=⇒ either

{accept failure of χPT3 in a few caseslook for a systematic way to fix 0++ channel

Extend scope of pQCD?

pQCD region

Q � ΛQCD

{make UV series as convergent as possible

many schemes, e.g. FAC, PMS, BLM, PMC

β ∼ − b1α2s − b2α

3s −

∞∑k=3

bkαks

QQk................. ................. ................. ...................

............................................

arbitrary (’t Hooft 1977)

. . . generally too arbitrary.

Truncate only if ∃ truly perturbative IR fixed point (nocondensates)

b2 < 0, αIR = −b2

/b1 = small enough? [Banks & Zaks 82]

@@I9 6 Nf 6 16 = perturbative conformal window

Green’s functions explicitly conformal covariant:

powers ∼ (Q2/µ2)γ , no mass gap

Non-perturbative conformal window?

Literature extensive, yet inconclusive on existence of αIR

e.g. Walking TC: β ∼ 0 in IR

search on lattice for (Q/µ)γ dependence

define α(L) = g2(L)/4π as function of box length L

fails for Nf < 6: chiral condensate forms

2-loop univ.3-loop SF

0 2 4 6 8

5

10

15

20

25

Log!L"L0#

g2$L%

2-loop univ.3-loop SF

0 10 20 30 40

2

4

6

8

10

Log!L"L0#

g2$L%

Figure: from Appelquist et al. 2009: Nf = 8 (left) and Nf = 12 (right)

Non-perturbative running couplings

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

αgh(p2) =g2

4πG 2(p2)Z (p2) p2→0

∼ constant

−1

=

−1

-1

2

-1

2-

1

6

-1

2+

−1

=

−1

- 10-4

10-3

10-2

10-1

100

101

102

p2

[GeV2]

0

1

2

3

4

α(p

2) Sternbeck et al. (2005)

DSEDSE on torus

Figure: from Fischer 2006

QCD infrared fixed point

What if αIR exists for Nf = 3?

Nf = 0 lattice

O

βNf = 3 proposal

αsαIR

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

f0(500) as a QCD dilaton

Consequences?

qq non-singlet under scale and chiral SU(3)L × SU(3)Rtransformations

〈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

two-fold expansion about

αs . αIR and mu,d ,s ∼ 0

!PT2

!PT3

NG bosons

p·p!=O(m2!) Not NG bosons

" f0 K #(mass)2

0

0

!PT$

(mass)2

(mass)2

"

f0

K #

%

" f0 K # %

Not NG bosons0

0

NG bosons p·p!=O(m2K)

NG bosons p·p!=O(m2K)

(mass)2

Not NGbosons

scaleseparation

scaleseparation

Nucleon mass

scale invariant theories characterised by θµµ = 0

scale symmetry can be

{manifest⇒ no mass gapspontaneously broken⇒ mass gap

In 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

Nucleon mass

O

β

αsαIR

Nf = 3 proposal

UV IR

In χPTσ

physical region 0 < αs < αIR

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

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

MN = 〈N|θµµ|N〉 ' FσgσNN

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

Fσ ≈ 100 MeV cf. Fπ ' 93 MeV

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)

Chiral-Scale Lagrangian

Want formula for L[σ,U,U†

]= Ld=4

inv + Ld>4anom + 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)

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)

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 ‘naive dimensional analysis’ to first diagram

Aloop

/Atree ∼

1

16π2F 2π

×logarithms [Manohar & Georgi 84]

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

The f0 width in χPTσ

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

Mf0 = 441+16−8 MeV , Γf0 = 544+18

−25 MeV

is Γσ a LO effect?

= + + + . . .σ

π

π

Estimate width a la Manohar & Georgi

Γσππ ≈|gσππ|216πmσ

∼ m3σ

16πF 2σ

∼ 250 MeV

}NLO

justified use of tree approximation to generate LO of χPTσ

Objection?

Q: Aren’t you ruining everything we know about χPT?

A: No!

0++ channels absent:

limσ→0

χPTσ = χPT3

0++ channels present:

χPT3 expansion: LO + NLO{ large f0 pole︸︷︷︸ + small corrections︸︷︷︸} + . . .

χPTσ expansion:

QQQQQs

��

��

@@R

��

LO + NLO{small, including σ loops}

Objection?

Q: Aren’t you ruining everything we know about χPT?A: No!

0++ channels absent:

limσ→0

χPTσ = χPT3

0++ channels present:

χPT3 expansion: LO + NLO{ large f0 pole︸︷︷︸ + small corrections︸︷︷︸} + . . .

χPTσ expansion:

QQQQQs

��

��

@@R

��

LO + NLO{small, including σ loops}

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†

)Qmw absorbed by field rotation −→ cannot be used toexplain ∆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

Weak interactions

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

Weak interactions

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

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 = 2.0± 0.2 keV [Hofericther et al. 2011]

non-perturbative χPTσ prediction RIR ≈ 5 for large-Nc

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