The chiral anomaly and the eta-prime in vacuum and at low temperatures

The chiral anomaly and the eta-prime in vacuum and at low temperatures Stefan Leupold, Carl Niblaeus, Bruno Strandberg Department of Physics and Astronomy Uppsala University St. Goar, March 2013 1

Table of Contents 1 Introduction 2 Consequences of chiral anomaly in vacuum 3 What changes in a medium? 4 Summary and outlook 2

Introduction Stefan Leupold neglecting quark masses smaller than Λ QCD chiral U R (3) U L (3) symmetry of QCD Lagrangian subgroups (U R (3) U L (3) =U V (3) U A (3)): U V (1): baryon number conservation SU V (3): flavor multiplets SU A (3): spontaneously broken 8 Goldstone bosons in vacuum and chiral restoration at some finite temperature U A (1): would also be spontaneously broken (chiral condensate not invariant w.r.t. U A (1)) would lead to 9th Goldstone boson (flavor singlet with quantum numbers of η, η ) 3

No symmetry of Quantum Chromodynamics chiral anomaly: U A (1) is not symmetry of quantized theory NB: actually one can decide whether to break Lorentz invariance or U A (1) by quantization path integral measure D ψ Dψ breaks U A (1), Dψ Dψ breaks Lorentz invariance experimental fact: nature seems to prefer Lorentz invariance consequences... P f f f f P 4

Consequences of U A (1) anomaly in vacuum parameter-free prediction of π 0 2γ in terms of pion decay constant f π agreement with experiment within error bars < 10% prediction strictly valid in the chiral limit ;-) coefficient of 1 f π ε µναβ π 0 F µν F αβ fixed by anomaly corrections suppressed m 2 π ε µναβ π 0 F µν F αβ in power counting of chiral perturbation theory: anomaly O(q 4 ) otherwise O(q 6 ) with generic momentum q m π 5

Consequences of U A (1) anomaly in vacuum II coupling constant of ε µναβ µ π 0 ν π + α π A β O(q 4 ) fixed by anomaly corrections (=dominant in absence of anomaly) O(q 6 ) predictive power for reactions e + e 3π and/or γ + π 2π close to threshold? How far is threshold away from idealized case of chiral limit? 6

e + e 3π 1000 Cross section nb, logarithmic scale 100 10 Bruno Strandberg, Master Thesis, Uppsala 2012 1 650 700 750 800 850 900 950 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 m π already quite sizable chiral limit 7

γ + π 2π Stefan Leupold 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 rescattering using dispersion theory solid line: prediction from anomaly dashed line: size of anomaly scaled up by about 30% can use whole range to pin down anomaly, not just threshold region 8

Consequences of U A (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 boson Veneziano formula for mass of eta singlet: mη 2 1 = 2N f τ 1 fπ 2 N c with topological susceptibility τ = i d 4 x 0 ω(x) ω(0) 0 and ω Gµν a µν G a note: nine light pseudoscalars in the combined chiral and large-n c limit starting point of large-n c chiral perturbation theory (χpt) (Kaiser/Leutwyler, EPJ C 17, 623 (2000)) 9

What changes in a medium? a lot of aspects already covered by Mario Mitter - thanks! π-γ-γ coupling in vacuum 1/f π f π is order parameter of SU(N f ) chiral symmetry breaking enhanced decay π 0 2γ? no! instead: decay decouples from anomaly, suppressed decay due to SU(N f ) chiral restoration (Pisarski/Trueman/Tytgat, PRD 56, 7077 (1997)) 10

Chiral multiplets Stefan Leupold at chiral restoration of SU(3) (no statement about U A (1) needed!): chiral multiplets (L, R) instead of just flavor multiplets in particular look at spin 0: ( 3, 1) (1, 3) = ( 3, 3) nonet parity commutes with QCD Hamiltonian and changes ( 3, 3) to (3, 3) q La q Rb (with flavor indices a, b) at and above transition: 18plet of degenerate states, with quantum numbers of π, K, η, η, a 0, κ, 2 f 0 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 11

A light η meson? might imply that η becomes 9th Goldstone boson, effective restoration of U A (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 transition and if states survive as quasi-particles P f f f f P 12

shopping list (not done yet!) use large-n c 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 and realistic 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 13

Why is this useful? Why a low-temperature calculation? Aren t we interested in extreme conditions? 14

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 for chiral condensate (and for pion decay constant) next slide chiral perturbation theory has something to say about not too low temperatures 14

Drop of quark condensate uses interacting pions and resonance gas produces realistic transition temperature Gerber/Leutwyler, Nucl.Phys.B 321, 387 (1989) 15

Mass shift? Stefan Leupold systematic calculation of thermal modifications of properties of η using large-n c 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 spectral functions instead of dropping masses 16

Mass shift vs. broadening Spectral Function "ρ" Dropping Masses? "a 1 " pert. QCD Spectral Function Melting Resonances? "ρ" pert. QCD "a 1 " Mass Mass chiral restoration demands degeneracy of spectra of chiral partners (btw: for spin 1: 16-plets, not 18-plets) melting of resonances might be hadronic precursor to deconfinement figures from Rapp, Wambach, van Hees, arxiv:0901.3289 [hep-ph] 17

Much celebrated example: ρ meson Im R 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 in-medium vacuum 0.00 0.2 0.4 0.6 0.8 1.0 1.2 s [GeV] M. Post, SL, U. Mosel, Nucl.Phys.A 741, 81 (2004) -Im D ρ (GeV -2 ) 9 8 7 6 5 4 3 2 1 ρ(770) 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 M (GeV) T=0 T=120 MeV T=150 MeV T=175 MeV R. Rapp, J. Wambach, H. van Hees, arxiv:0901.3289 [hep-ph] different groups (with different models) obtain broad ρ meson spectra, essentially no mass shift 18

Thermal width of η task: use large-n c chiral perturbation theory to determine in-medium life time of η in a gas of Goldstone bosons (Carl Niblaeus, Master thesis, Uppsala, ongoing) so far: leading-order calculation only 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 + K) next-to-leading-order calculation gas of all Goldstone bosons (study also chiral limit) 19

Thermal width of η preliminary! Width [kev] 25 20 15 10 Thermal width of η as function of temperature Vacuum decay width of η : 199±9 kev [PDG 2012] very small thermal width from pion gas, < 25 kev (vacuum width 200 kev) 5 0 0 50 100 150 Temperature [MeV] Carl Niblaeus, Master thesis, Uppsala, ongoing 20

Summary and outlook ongoing: experimental checks of consequences of chiral anomaly in vacuum (with support from theory) thermal modifications? to do: use large-n c 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 and realistic quark masses feasible: leading and next-to-leading order calculation 21

Outlook Stefan Leupold large-n c chiral perturbation theory allows for a systematic, model-independent low-temperature calculation; medium represented by gas of Goldstone bosons from experience with corresponding calculations for chiral condensate etc. results can be relevant for interesting temperature regime ( 100 MeV) 22

Outlook Stefan Leupold large-n c chiral perturbation theory allows for a systematic, model-independent low-temperature calculation; medium represented by gas of Goldstone bosons from experience with corresponding calculations for chiral condensate etc. results can be relevant for interesting temperature regime ( 100 MeV) stay tuned 22