Stefan Leupold Chiral anomaly and eta-prime

The chiral anomaly and the eta-prime

in vacuum and at low temperatures

Stefan Leupold, Carl Niblaeus, Bruno Strandberg

Department of Physics and AstronomyUppsala University

St. Goar, March 2013

1

Stefan Leupold Chiral anomaly and eta-prime

Table of Contents

1 Introduction

2 Consequences of chiral anomaly in vacuum

3 What changes in a medium?

4 Summary and outlook

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Stefan Leupold Chiral anomaly and eta-prime

Introduction

neglecting quark masses smaller than ΛQCD

chiral UR(3)×UL(3) symmetry of QCD Lagrangian

↪→ subgroups (UR(3)×UL(3) =UV (3)×UA(3)):

UV (1): baryon number conservationSUV (3): flavor multipletsSUA(3): spontaneously broken

8 Goldstone bosons in vacuum andchiral restoration at some finite temperatureUA(1): would also be spontaneously broken(chiral condensate not invariant w.r.t. UA(1))

would lead to 9th Goldstone boson(flavor singlet with quantum numbers of η, η′)

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Stefan Leupold Chiral anomaly and eta-prime

No symmetry of Quantum Chromodynamics

chiral anomaly: UA(1) is not symmetry of quantized theory

NB: actually one can decide whether to breakLorentz invariance or UA(1) by quantization

↪→ path integral measure Dψ̄Dψ breaks UA(1),Dψ†Dψ breaks Lorentz invariance

↪→ experimental fact: nature seems to prefer Lorentz invariance

↪→ consequences ...

P

f

f –

f´

f´–

P´

4

Stefan Leupold Chiral anomaly and eta-prime

Consequences of UA(1) anomaly in vacuum

parameter-free prediction of π0 → 2γ in terms ofpion decay constant fπ

agreement with experiment within error bars < 10%

prediction strictly valid in the chiral limit ;-)

↪→ coefficient of 1fπεµναβ π0 Fµν Fαβ fixed by anomaly

↪→ corrections suppressed ∼ m2π ε

µναβ π0 Fµν Fαβ

in power counting of chiral perturbation theory:

anomaly ∼ O(q4)otherwise ∼ O(q6)with generic momentum q ∼ mπ

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Stefan Leupold Chiral anomaly and eta-prime

Consequences of UA(1) anomaly in vacuum II

coupling constant of εµναβ ∂µπ0 ∂νπ

+ ∂απ− Aβ ∼ O(q4)

fixed by anomaly

corrections (=dominant in absence of anomaly) ∼ O(q6)

↪→ predictive power for reactions e+e− → 3π and/or γ + π → 2πclose to threshold?

↪→ How far is threshold away from idealized case of chiral limit?

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Stefan Leupold Chiral anomaly and eta-prime

e+e− → 3π

650 700 750 800 850 900 950

1

10

100

1000

Total incoming energy s @MeVD

Cro

ss-

sect

ion

@nbD

,log

arith

mic

scal

e

Bruno Strandberg,Master Thesis,Uppsala 2012

data dominated by ω peak

dashed black line: without anomaly (just vector mesons),red full line: including anomaly

↪→ anomaly has some effect, but does not dominate a region

↪→ threshold at 3 mπ already quite sizable 6= chiral limit

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Stefan Leupold Chiral anomaly and eta-prime

γ + π → 2π

M. Hoferichter, B. Kubis, D. Sakkas,Phys.Rev.D86, 116009 (2012)

expect future data from COMPASS@CERN(pion beam, Primakov effect)calculation includes anomaly and pion-pion rescatteringusing dispersion theorysolid line: prediction from anomalydashed line: size of anomaly scaled up by about 30%

↪→ can use whole range to pin down anomaly,not just threshold region

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Stefan Leupold Chiral anomaly and eta-prime

Consequences of UA(1) anomaly in vacuum III

P

f

f –

f´

f´–

P´figure from Klabucar/Kekez/Scadron,hep-ph/0012267

eta singlet is not a Goldstone bosonVeneziano formula for mass of eta singlet:

m2η1

=2Nf τ

f 2π

∼ 1

Nc

with topological susceptibility τ = −i∫

d4x 〈0|ω(x)ω(0)|0〉and ω ∼ G a

µνG̃µνa

note: nine light pseudoscalars in the combined chiral andlarge-Nc limit

↪→ starting point of large-Nc chiral perturbation theory (χPT)(Kaiser/Leutwyler, EPJ C 17, 623 (2000))

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Stefan Leupold Chiral anomaly and eta-prime

What changes in a medium?

a lot of aspects already covered by Mario Mitter - thanks!

π-γ-γ coupling in vacuum ∼ 1/fπ

↪→ fπ is order parameter of SU(Nf ) chiral symmetry breaking

enhanced decay π0 → 2γ?

↪→ no! instead: decay decouples from anomaly,suppressed decay due to SU(Nf ) chiral restoration(Pisarski/Trueman/Tytgat, PRD 56, 7077 (1997))

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Stefan Leupold Chiral anomaly and eta-prime

Chiral multiplets

at chiral restoration of SU(3) (no statement about UA(1) needed!):

chiral multiplets (L,R) instead of just flavor multiplets

in particular look at spin 0: q̄La qRb (with flavor indices a, b)

↪→ (3̄, 1)× (1, 3) = (3̄, 3) nonet

parity commutes with QCD Hamiltonianand changes (3̄, 3) to (3, 3̄)

at and above transition: 18plet of degenerate states,with quantum numbers of π, K , η, η′, a0, κ, 2 f0’s

at and below transition: Goldstone bosons stay light

η′ should become light at transition

thanks to D. Jido for pointing this out to me

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Stefan Leupold Chiral anomaly and eta-prime

A light η′ meson?

might imply that η′ becomes 9th Goldstone boson,effective restoration of UA(1)?

or might imply that η′ decouples from anomaly,i.e. change of decay constant of η′

caveats: only good argument if not strong first-order transitionand if states survive as quasi-particles

P

f

f –

f´

f´–

P´

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Stefan Leupold Chiral anomaly and eta-prime

shopping list (not done yet!)

use large-Nc χPT to determine temperature dependence of– topological susceptibility (maybe trivial)– mass of η′

– decay constant of η′ = coupling to singlet axial-vector current– mixing of η and η′

study systematically two- and three-flavor chiral limit andrealistic quark masses

feasible: leading and next-to-leading order calculations

this is a low-temperature calculation;medium represented by gas of Goldstone bosons

in part already performed by Tytgat et al. for pion gas

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Stefan Leupold Chiral anomaly and eta-prime

Why is this useful?

Why a low-temperature calculation?

Aren’t we interested in extreme conditions?

↪→ chiral perturbation theory allows for systematic,model independent calculations at low temperatures

provides excellent description of onset of chiral restoration forchiral condensate (and for pion decay constant) next slide

↪→ chiral perturbation theory has something to say about not toolow temperatures

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Stefan Leupold Chiral anomaly and eta-prime

Why is this useful?

Why a low-temperature calculation?

Aren’t we interested in extreme conditions?

↪→ chiral perturbation theory allows for systematic,model independent calculations at low temperatures

provides excellent description of onset of chiral restoration forchiral condensate (and for pion decay constant) next slide

↪→ chiral perturbation theory has something to say about not toolow temperatures

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Stefan Leupold Chiral anomaly and eta-prime

Drop of quark condensate

uses interacting pions and resonance gas

produces realistic transition temperature

Gerber/Leutwyler, Nucl.Phys.B 321, 387 (1989)15

Stefan Leupold Chiral anomaly and eta-prime

Mass shift?

systematic calculation of thermal modifications of properties ofη′ using large-Nc chiral perturbation theory could be interesting

↪→ mass shift?

but check first: does η′ survive at all in a thermal medium?

personal history in scepticism against mass shifts ;-)

hadronic many-body framework often produces broad spectralfunctions instead of dropping masses

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Stefan Leupold Chiral anomaly and eta-prime

Mass shift vs. broadening

Mass

Spec

tral

F

unct

ion

"a1"

"ρ"

pert. QCD

Dropping Masses ?

Mass

Sp

ectr

al

Fu

nct

ion

"a1"

"ρ"pert. QCD

Melting Resonances ?

chiral restoration demands degeneracy of spectra of chiralpartners(btw: for spin 1: 16-plets, not 18-plets)

melting of resonances might be hadronic precursor todeconfinement

figures from Rapp, Wambach, van Hees, arXiv:0901.3289 [hep-ph]

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Stefan Leupold Chiral anomaly and eta-prime

Much celebrated example: ρ meson

0.2 0.4 0.6 0.8 1.0 1.20.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Im R

s [GeV]

in-medium vacuum

M. Post, SL, U. Mosel,

Nucl.Phys.A 741, 81 (2004)

0

1

2

3

4

5

6

7

8

9

0 0.2 0.4 0.6 0.8 1 1.2 1.4

-Im

Dρ (

GeV

-2)

M (GeV)

ρ(770)T=0

T=120 MeVT=150 MeVT=175 MeV

R. Rapp, J. Wambach, H. van Hees,

arXiv:0901.3289 [hep-ph]

different groups (with different models) obtainbroad ρ meson spectra, essentially no mass shift

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Stefan Leupold Chiral anomaly and eta-prime

Thermal width of η′

task:

use large-Nc chiral perturbation theory to determinein-medium life time of η′ in a gas of Goldstone bosons(Carl Niblaeus, Master thesis, Uppsala, ongoing)

so far:

leading-order calculationonly pion gas (most abundant medium constituent)only reaction with largest open phase space, i.e. η′ + π → η + π

figure on next slide

to do:

all channels (e.g. η′ + π → K + K̄ )next-to-leading-order calculationgas of all Goldstone bosons (study also chiral limit)

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Stefan Leupold Chiral anomaly and eta-prime

Thermal width of η′ — preliminary!

0 50 100 1500

5

10

15

20

25Thermal width of η′ as function of temperature

Wid

th [

keV

]

Temperature [MeV]

Vacuum decay width of η′ : 199±9 keV [PDG 2012]

Carl Niblaeus, Master thesis, Uppsala, ongoing

very small thermal widthfrom pion gas, < 25 keV(vacuum width ≈ 200 keV)

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Stefan Leupold Chiral anomaly and eta-prime

Summary and outlook

ongoing: experimental checks of consequences of chiral anomalyin vacuum (with support from theory)

thermal modifications?

to do:

use large-Nc χPT to determine temperature dependence of– topological susceptibility– mass and life time of η′

– decay constant of η′ = coupling to singlet axial-vector current– mixing of η and η′

study systematically two- and three-flavor chiral limit andrealistic quark massesfeasible: leading and next-to-leading order calculation

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Stefan Leupold Chiral anomaly and eta-prime

Outlook

large-Nc chiral perturbation theory

allows for a systematic, model-independent low-temperaturecalculation; medium represented by gas of Goldstone bosons

from experience with corresponding calculations for chiralcondensate etc.

↪→ results can be relevant for interesting temperature regime(≈ 100 MeV)

↪→ stay tuned

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Stefan Leupold Chiral anomaly and eta-prime

Outlook

large-Nc chiral perturbation theory

allows for a systematic, model-independent low-temperaturecalculation; medium represented by gas of Goldstone bosons

from experience with corresponding calculations for chiralcondensate etc.

↪→ results can be relevant for interesting temperature regime(≈ 100 MeV)

↪→ stay tuned

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