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Helmholtz International Center for FAIR

Effective Theories for Hadrons

Stefan Leupold

Institut für Theoretische Physik, Justus-Liebig-Universität Giessen

March 6, 2008 Stefan Leupold 2

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r Understanding the spectrum of hadrons

Selection of key questions:

How can we understand the masses of hadrons and their decay pattern?

Are there hadrons which solely/dominantly consist out of gluons (glueballs)?

Do some/many hadrons have a hadronic substructure (“hadron molecules”)?

e.g. experimental Ds spectrum qualitative input from QCD: quarks and gluons form hadrons

quantitative: challenging

experiment: PANDA

complementary approaches (relevant for PANDA):

Lattice QCD ↔ Dyson-Schwinger Equations ↔ Effective Field Theory

March 6, 2008 Stefan Leupold 3

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r HIC for FAIR opportunity: glueball spectrum

consider resonance decay R → A + B problem of lattice QCD for light enough quarks,

i.e. for mA + mB < mR :

correlator yields lightest state in spectrum two-particle state A + B instead of resonance state R

way out: Lüscher’s phase shift analysis in finite box with variable box size numerically expensive need for effective field theory in HIC for FAIR

prediction from lattice QCD: glueball masses but: so far only gluons, no quarks in calculation necessary improvements: include quarks, i.e.

mixing with mesons (next slide)

states become resonances deal with finite width

March 6, 2008 Stefan Leupold 4

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r Mixing of glueballs with mesons

complementary to lattice QCD: Dyson-Schwinger (DS) equations

for quark-gluon n-point functions (quantum correlations)

input for Bethe-Salpeter equation for glueballs and mesons

mixing of glueballs and mesons microscopic understanding

(relevant degrees of freedom,...)

gluon propagator from lattice and from DS:

difficult for lattice QCD and Dyson-Schwinger: treatment of intermediate hadron-hadron states Effective Field Theory

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r Structure of resonances

Resonances decay into other “final-state” hadrons

Influence of hadrons and their interactions on resonance properties?

Examples for extreme cases: Resonance is dominantly quark-antiquark Resonance is formed by attractive interactions between hadrons

hadron molecule → fig.

HIC for FAIR opportunity and challenge: develop sophisticated approach for description of final-state hadrons and their interactions effective field theory = systematic approach unknown coupling constants from fit to data or from lattice QCD

March 6, 2008 Stefan Leupold 6

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r Axial-vector states as hadron molecules

axial-vectors decay into vectors + pseudoscalars, attractive interaction! strong enough to generate axial-vectors dynamically

(see also poster on PANDA theory)

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r Structure of resonances

Resonances decay into other “final-state” hadrons

Influence of hadrons and their interactions on resonance properties?

Examples for extreme cases: Resonance is dominantly quark-antiquark Resonance is formed by attractive interactions between hadrons

hadron molecule → poster

HIC for FAIR opportunity and challenge: develop sophisticated approach for description of final-state hadrons and their interactions effective field theory = systematic approach unknown coupling constants from fit to data or from lattice QCD

March 6, 2008 Stefan Leupold 8

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r Cross relation to CBM

effective theories can yield input for unknown cross sections necessary for transport in particular important for dedicated probes, e.g. dileptons, charm, ...

(e.g. N + → resonance → N + dilepton)

in addition: systematic treatment of in-medium modifications changes induce changes:

↔

selfconsistent approaches coupled integral equations

under development by several Hessian groups, but incoherent efforts: Weinhold/Friman (GSI): dynamics of pion-nucleon-Delta Post/Mosel/Leupold (JLU): rho meson in nuclear medium Riek/Knoll (GSI): omega meson in nuclear medium Röder/Ruppert/Rischke (FFM): mesons at finite temperature Leupold (JLU): how to satisfy conservation laws

March 6, 2008 Stefan Leupold 9

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r HIC for FAIR: Opportunities and challenges

Synergy for numerically and conceptually challenging developments

understanding the spectrum of hadrons(PANDA)

Effective Field Theory(final state interactions, hadron molecules,...)

lattice QCD(spectrum, coupling constants,...)

Dyson-Schwinger(microscopic understanding, mixing,...)

coherent starting point for

in-medium modifications

(selfconsistency,...)

March 6, 2008 Stefan Leupold 10

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r Backup slide: Effective field theories for hadrons

Systematic approach (instead of arbitrary model building) principles of scattering theory and effective field theory:

exact unitarity and analyticity, i.e. use of Bethe-Salpeter equation

coupled-channel dynamics (Lutz, Kolomeitsev, ...) systematic power counting

extension of chiral perturbation theory to include (at least) vector mesons and Delta decuplet

currently developed (e.g. Lutz/Leupold, arXiv:0801.3821 [nucl-th])

Goal: disentangle hadronic rescattering effects from “elementary” resonances (quark-antiquark, glueball,…)

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full analyticity (dispersion relations) requires serious treatment of left-hand cuts from t- and u-channels

coupled integral equations

get unknown coupling constants from lattice QCD e.g. three-point functions for vector mesons (V-V-V) numerically challenging

Backup slide: Shopping list