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
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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´
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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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