|
|
 |
|
 |
Doctoral
Program in Theoretical Physics |
 |
|
DOKTORATSKOLLEG GRAZ "Hadrons in Vacuum, Nuclei and Stars" in collaboration with the European Graduate School Jena |
|
|
|
|
|
|
Doctoral
Program in Theoretical Physics |
|
|
DOKTORATSKOLLEG GRAZ "Hadrons in Vacuum, Nuclei and Stars" in collaboration with the European Graduate School Jena |
Within this subproject we aim at a significantly
improved description of strongly-interacting matter at non-vanishing
densities and temperatures within functional approaches. As one main
objective the interplay of the chiral and (de-)confinement properties
at the crossover or phase transition will be investigated. This
especially includes the region of a possible critical
endpoint in
the phase diagram (see, e.g., [1]) as well as colour-superconducting
phases. We plan to apply non-perturbative functional approaches like
the functional renormalization group (FRG) method [2] and
Dyson-Schwinger equations (DSEs) [3]. The FRG method is best suited to
investigate such systems in particular near phase transitions where
fluctuations are important. On the other hand, the DSEs provide better
access to the Green functions in a given phase. These functional
techniques will complement QCD lattice calculations, cf. Subproject C3.
Besides
other results the self-consistent relation of the infrared structure of
the Landau gauge quark-gluon vertex to quark mass generation found in
functional approaches suggests that there is a direct connection
between confinement and dynamical breaking of chiral symmetry [4]. As
the quark-gluon vertex is the most basic Green's function providing the
link between the gauge and matter sector it seems to be a
straightforward idea to generalize existing investigations from
vanishing to non-vanishing densities and temperatures. To this end one
has to note, however, that due to the larger tensor structure for
Poincaré non-invariant systems the respective functional equations
increase dramatically in complexity. Therefore, we will continue
studies of the three-point vertex function of a gluon coupled with a
fundamentally charged scalar. These calculations will be extended to
non-vanishing densities and temperatures. Based on the experience
gained thereby a determination of the quark-gluon vertex in different
phases will be attempted.
In all calculations also recently suggested order parameters, like,
e.g., the dual condensate [5], will be determined.
[1] S. Leupold et al., Lect. Notes Phys. 814 (2011) 39.
[2] J. Berges, N. Tetradis and C. Wetterich, Phys. Rept. 363 (2002) 223.
[3] R. Alkofer and L. von Smekal, Phys. Rept. 353} (2001) 281.
[4] R. Alkofer et al., Annals Phys. 324 (2009) 106.
[5] C. Gattringer, Phys. Rev. Lett. 97 (2006) 032003.
This subproject part aims at a comprehensive
understanding of the properties and structure of mesons and baryons,
and hereby especially of nucleons, in an unified approach building on
QCD bound state equations,see, e.g., [1]. We plan to study:
(i) mixing of mesons with two-meson states as e.g. sigma -> pi pi -> sigma,
and mixing of baryons with baryon-meson states,
(ii) transition form factors, as e.g. Delta -> N pi,
(iii) time-like electromagnetic form factors and
(iv) Generalized Parton Distributions.
Based
on detailed studies of hadronic masses and space-like form factors with
different techniques (Lattice QCD, Functional Methods, Relativistic
Quark Models, etc.), see [2] and references therein, we want to employ
Dyson-Schwinger, Bethe-Salpeter and Poincaré-invariant Faddeev
equations to gain important insight into hadron structure. The
corresponding lattice calculations described in Subprojects C7 and C8
will be used to cross-check these investigations. With respect to
space-like form factors note that (while experimental access to
hadronic scattering processes involving two photons has been the
phenomenological foundation for the development of the Generalized
Parton Distribution picture) the importance of two-photon processes in
the context of electromagnetic form factors has been rediscovered only
recently: Two-photon corrections are believed to play a major role in
explaining the discrepancy in the two distinct measurements of the
proton's form factor ratio GE/GM [3], a result which certainly deserves
further clarification.
The hadronic properties we want to calculate explicitly from bound state
amplitudes (i.e. from the quark-gluon substructure) include the decay widths of hadrons,the longitudinal vs. transverse momentum fractions of quarks and gluonsin a hadron, the distribution of spin and orbital angular momentum among its constituents,and a three-dimensional tomography of hadrons, cf. [4] and references therein.
[1] G. Eichmann, PhD Thesis (Advisor: R. Alkofer), arXiv:0909.0703 [hep-ph].
[2] G. Eichmann, arXiv:1104.4505 [hep-ph].
[3] C. E. Carlson and M. Vanderhaeghen, Ann. Rev. Nucl. Part. Sci. 57 (2007) 171.
[4] M. Burkardt, A. Miller, and W. D. Nowak, Rept. Prog. Phys. 73 (2010) 016201
With the running and upcoming experiments at CERN,
RHIC, GSI and Dubna, our knowledge of experimental facts about the QCD
phase diagram is expected to increase considerably. Also the
theoretical side is challenged to contribute to this development.
Besides the identification of the phase boundaries and the order of the
transitions, one of course would also like to understand the physical
mechanisms that drive the various transitions, which are often related
to symmetries of the underlying theory and their spontaneous breaking.
In
some situations we have a clear idea which symmetries drive the
transitions: The deconfinement transition of pure gauge theory is
related to the spontaneous breaking of the center symmetry [1], along
with a percolation of suitably defined center clusters of local
Polyakov loops [2]. The center properties of local Polyakov loops
display a strong response also for full QCD [3], and recently we were
able to show that an effective theory for the center degrees of freedom
of QCD reproduces many features expected for the QCD phase diagram [4].
In
this project we want to explore the role of center symmetry further. In
particular we will focus on analyzing canonical fermion determinants
projected to a fixed net quark quark number q. These transform in a
simple way under center transformations according to q mod 3, and it
was shown, that they implement the center properties very strongly [5].
The behavior of the full fermion determinant is then governed by the
interplay of the canonical determinants with interesting physical
consequences such as center selection rules for observables in the
center symmetric phase or regions with only mild breaking of center
symmetry. Based on newly developed numerical methods for canonical
determinants these features will be studied.
[1] L. McLerran, B. Svetitsky, Phys. Rev. D 24 (1981) 450.
L.G. Yaffe, B. Svetitsky, Phys. Rev. D 26 (1982) 963; Nucl. Phys. B 210 (1982) 423.
[2] C. Gattringer, Phys. Lett. B 690 (2010) 179.
C. Gattringer, A. Schmidt, JHEP 1101 (2010), 051.
[3] Y.D. Mercado, H.G. Evertz and C. Gattringer, Phys. Rev. Lett. 106 (2011) 222001.
[4] J. Danzer, C. Gattringer, S. Borsanyi, Z. Fodor, PoS LATTICE 2010, 176, and work in preparation.
[5] E. Bilgici, J. Danzer, C. Gattringer, C.B. Lang, L. Liptak, Phys. Lett. B697, 85 (2011).
In recent years the canonical approach to
finite-density lattice QCD has seen quite some attention. The canonical
determinants are obtained by a projection to a fixed quark number,
which may, e.g., be implemented by a Fourier transformation with
respect to a generalized temporal boundary condition for the quark
fields. An important technical development are new exact factorization
results for the fermion determinant that give rise to a dimensional
reduction, which in turn leads to a considerable speed-up in the
numerical evaluation of the canonical determinants [1-4]. Interesting
physical properties such as the role of the center sectors (compare
also Subproject C3) were analyzed. In this more technically oriented
Subproject C4 we plan to further explore the potential of the canonical
approach.
The dimensional reduction formulas [1-4] not only allow
for an efficient evaluation of canonical determinants via Fourier
transformation, but also may be used for their perturbative
calculation. While [1] and [2] are suitable for an expansion around net
quark number q ~ 0, the results [3,4] give rise to elegant expressions
for very high net quark numbers. The plan of Subproject C4 is to
explore the possibilities opened up by the new results: Firstly we plan
to use the expressions for canonical determinants at very high q based
on [3, 4] in numerical simulations of QCD at very high densities. At
least on sufficiently small volumes we expect that the complex phase
problem can be brought under control allowing for simulations in a
density range which so far is completely uncharted territory on the
lattice. A second aspect will be to develop the dimensional reduction
techniques further, with the goal of developing expressions that allow
to access net quark numbers in an intermediate range.
[1] Julia Danzer, Christof Gattringer, Phys. Rev. D78 (2008) 114506.
[2] E. Bilgici, J. Danzer, C. Gattringer, C.B. Lang, L. Liptak, Phys. Lett. B697 (2011) 85.
[3] A. Alexandru, U. Wenger, Phys. Rev. D 83 (2011) 034502.
[4] K. Nagata, A. Nakamura, Phys. Rev. D 82 (2010) 094027.
At low resolution scales both confinement and chiral
symmetry are crucial phenomena. They influence mass and angular
momentum generation of hadrons. There is a way how to define and
measure a degree of chiral symmetry breaking in hadrons: One measures
in dynamical lattice simulations a ratio of couplings of different
operators (that form a complete basis with respect to the chiral group)
to a given hadron at different resolution scales [1]. A small
(infrared) resolution of a probe (interpolator) is achieved by a gauge
invariant Gaussian smearing of quarks fields in the interpolator in
spatial directions. Using this technique it is possible to extract the
chiral content of hadrons up to the infrared resolution of 1 fm.
Another interesting question is the spin content of hadrons.
Traditionally the spin content of hadrons is extracted from the parton
distributions in the infinite momentum frame. Then only about 30% of
the nucleon (meson) spin is carried by its valence quarks. The chiral
basis in the quark-antiquark system is a complete one and can be
connected to a complete angular momentum basis in the rest frame via a
unitary transformation [2]. Consequently, if we know a mixture of the
allowed chiral representations in a physical state, one is also able to
define the angular momentum content of this state in the rest frame.
This technique allows one to study the chiral and angular momentum
contents of both the ground and excited states of hadrons, which is an
interesting and important task. In principle it would be possible to
study systematics of chiral symmetry breaking in the low-lying states
and address a smooth transition to symmetries seen experimentally in
highly excited states. Our dynamical lattice simulations have shown
that the rho-meson in the rest frame is practically a 3S1 state at
scales 0.15 fm – 1.0 fm [3]. This definition of the spin content is
different from the traditional one that uses parton distributions in
the infinite momentum frame. It would be interesting to measure on the
lattice the spin content of mesons according to the traditional
definition [4] and to compare the results with those obtained in the
rest frame according to our definition. Then it would be important to
understand the physical reason for the difference between both results.
[1] L.Ya. Glozman, C. B. Lang, and M. Limmer, Phys. Rev. Lett. 103 (2009) 121601.
[2] L.Ya. Glozman and A.V. Nefediev, Phys. Rev. D 76 (2007) 096004; ibid., 80 (2009) 057901.
[3] L.Ya. Glozman, C. B. Lang, and M. Limmer, arXiv: 1106.1010.
[4] Ph. Hägler, Phys. Rept. 490 (2010) 49.
One of the most intriguing questions of the strong
interaction physics is the possibility for existence of the chirally
symmetric matter with confinement at low temperatures and large
densities. In the large NC limit confinement survives in a matter at
low temperatures up to arbitrary large density and it is possible to
define quarkyonic matter as dense nuclear matter with confinement but
where some properties are determined by the quark Fermi surface [1].
Assuming that confinement persists in dense matter up to a very large
density at NC = 3 (which is supported by lattice data with NC = 2), one
can ask the question whether a chiral restoration phase transition is
possible or not in such matter. If possible, then by what mechanism? We
cannot answer this question from first principles. What can be done,
however, is to construct a model. If demonstrated within such a model,
this scenario could also be realized in QCD and further theoretical
efforts to clarify this interesting question would be called for.
Assuming that a dense and a superdense baryonic (quarkyonic) matter
with confinement at NC = 3 is a liquid (as it is the case for the
standard nuclear matter) and neglecting quark back-reactions we have
constructed a confining model that admits at low temperatures a
confining but chirally symmetric phase at high density [2-4]. In this
conjectured phase a mass generation mechanism of the matter would not
be related to chiral symmetry breaking in a vacuum and the elementary
excitations would be chirally symmetric hadrons. What are properties of
such matter, if it exists? How would it be possible to observe it? Is
the chiral restoration phase transition of first or second order?
Answering these questions are the main objective of the subproject.
[1] L. McLerran and R. Pisarski, Nucl. Phys. A 796 (2007) 83.
[2] L. Ya. Glozman and R. F. Wagenbrunn, Phys. Rev. D 77 (2008) 054027.
[3] L. Ya. Glozman, Phys. Rev. D 79 (2009) 037504.
[4] L. Ya. Glozman, Phys. Rev. D 80 (2009) 037701.
The scalar meson multiplet (σ, f0, a0, κ) is a
challenge in QCD, both in continuum calculations and even more in the
lattice approach. The heavier mesons seem to fit into a qbar-q picture,
but the light nonet may have a significant tetraquark (q-q-qbar-qbar)
component with diquarks coupled to antidiquarks [1] or, alternatively,
strongly bound hadronic molecules [2]. The problem in lattice
calculations is that the full meson propagators involve disconnected
contributions. In a recent study [3] these have been disregarded, as it
is customary due to the technical complications. Only connected
contributions of local two-meson interpolators are included and an
indication of the resonances σ and κ with strong tetraquark components
was found.
It is stringent, however, in a full calculation to
include also the disconnected contributions. This becomes possible with
new tools which extend the space of quark sources [4]. Enhancing the
space of coupled hadrons (including meson-meson interpolators) is
important for the extraction of the energy spectrum of the correlation
matrix. From these energies in finite volume one can derive values of
the scattering phase shift in continuum and infinite volume [5], which
then allows to derive the resonance parameters as usual. While in the
standard method [5] several lattice volumes have to be used, one may
also rely on interpolators with non-vanishing total momentum, which
allows to obtain results at further values of the center-of-momentum
energies.
The analysis will be done within a larger collaboration
(QCDSF) for ensembles of gauge configurations with 2+1 dynamical quarks
at pion masses near 250 MeV and on large lattcies. The CPU resources
are provided within the collaboration on a high performance computer at
LRZ Munich.
[1] G. 't Hooft et al., Phys. Lett. B 662, 424 (2008), arXiv:0801.2288.
[2] J.A. Oller, E. Oset and J.R. Pelaez, Phys. Rev. D 59, 074001 (1999), hep-ph/9804209.
[3] S. Prelovsek et al., Phys. Rev. D 82, 094507 (2010), arXiv:1005.0948.
[4] M. Peardon, J. Bulava, J. Foley et a. (Hadron Spectrum Collaboration), Phys. Rev. D 80, 054506 (2009), arXiv:0905.2160 [hep-lat].
[5] M. Lüscher, Nucl. Phys. B 354, 531 (1991); ibid. B 364, 237 (1991).
Except for the pion and the nucleon all hadrons are
resonances. This poses fundamental problems in the lattice formulation
of QCD. Whereas in experiments cross section and phase shift analyses
lead to the resonance properties, in lattice QCD one observes a
discrete energy spectrum of the hadronic correlation functions. It was
shown by Lüscher [1] that in the elastic regime these discrete spectra
on finite lattices may be related to values of the scattering phase
shift in the continuum. Only recently the lattice tools have become
strong enough to apply these considerations to realistic problems like
the rho-decay [2]. Crucial for the analysis of correlators is the
inclusion of several hadronic interpolators of different shape and
Dirac structure. The so-called variational method allows, via
diagonalization of the cross-correlation matrix, to identify the lowest
lying eigenstates and derive their energies level. These then give
information on ground states and excitations, or better, scattering
phase shifts. It has become clear that one should include the decay
channel interpolators (like pion-nucleon) explicitly in order to get
the correct spectrum. For this, however, one has to calculate also
disconnected (or partially disconnected) contributions, a challenge in
lattice simulations. Recently developed methods (like the distillation
method [3]) allow to deal with this task efficiently. In this project
we want to study the nucleon (both parities) sector and hope to shed
light on the notorious problem of the Roper state and the unusual level
ordering. This will be done first on gauge configurations with only two
dynamical quarks and pion masses around 270 MeV. If the method proves
successful, we plan to extend it to other ensembles of gauge
configurations on finer lattices and with 2+1 dynamical quark flavors.
[1] M. Lüscher, Nucl. Phys. B 354, 531 (1991); ibid. B 364 (1991) 237.
[2] C.B. Lang, D. Mohler, S. Prelovsek and M. Vidmar, arXiv:1105.5636 [hep-lat].
[3] M. Peardon, J. Bulava, J. Foley et al. (Hadron Spectrum Collaboration), Phys. Rev. D 80 (2009) 054506, arXiv:0905.2160 [hep-lat].
The description of low-energy hadrons in terms of
constituent quarks has been much refined over the recent years.
Especially the constituent-quark model, if formulated in a relativistic
framework and solved in a microscopically rigorous manner, has been
quite successful at least with regard to the ground-state properties of
hadrons with light and strange flavors. The relativistically invariant
predictions for their spectroscopy as well as electroweak structures
have been found in very reasonable agreement with both phenomenology
and available lattice QCD results [1]. On the other hand, shortcomings
have appeared with respect to a realistic description of hadron
resonance states [2].
The relativistic constituent-quark model
should now be extended to cover in a universal manner also the
heavy-flavor hadrons by modeling the heavy-heavy and light-heavy
quark-quark interactions, which are not yet so well known. With regard
to the resonance states, the relativistic constituent-quark model,
which has so far essentially relied on quark-antiquark configurations
for mesons as well as three-quark configurations for baryons, will be
extended to include additional degrees of freedom by explicit couplings
to the decay channels. The construction of a relativistic
coupled-channel model should then allow for a description better
adjusted to the nature of resonant states than achieved hitherto.
Promising investigations so far made for meson resonances [3,4] should
be further improved and extended to baryons too. The influences of
higher Fock components will be investigated also in comparison to
lattice studies for excited mesons and baryons (compare Subproject C8)
A better description of hadron reactions involving resonant states,
such as strong and weak decays as well as inelastic photon-, electron-,
and meson-induced processes, should thus be achieved.
[1] W. Plessas, PoS (LC2010) 017.
[2] T. Melde, W. Plessas, and B. Sengl, Phys. Rev. D 77 (2008) 114002.
[3] A. Krassnigg, W. Schweiger, and W.H. Klink, Phys. Rev. C 67 (2003) 064003.
[4] R. Kleinhappel, Resonances and decay widths within a relativistic coupled-channel approach, Master Thesis, Univ. of Graz, 2010.
So far the investigations of the electroweak hadron
structures, primarily of the nucleon and the pion, have been in the
focus of various approaches, ranging from relativistic
constituent-quark models, via functional methods to lattice-QCD
calculations. Recently, studies of electromagnetic as well as axial
form factors have been extended to the hyperon ground states and their
resonances. In particular, the covariant predictions of the
relativistic constituent-quark model obtained in the point form have
been found in remarkably good agreement with existing phenomenology
and/or available lattice-QCD results [1-4].
In this subproject the
investigation of elastic electromagnetic and weak form factors of
hadrons in the range of low and moderate momentum transfers will be
continued along several approaches. The calculations of the
relativistic constituent-quark model should be performed in parallel
also in the instant as well as front forms and compared to the results
of the Dyson-Schwinger formalism as well as lattice-QCD calculations
(cf. C2, C8 and C12). In this context also the question of the proper
current operators to be used will be further pursued in a broader
context.
The studies of the electroweak form factors should be
extended in the first instance to heavy-flavor baryons in the framework
of the relativistic constituent-quark model. Subsequently, also
electromagnetic and weak transition reactions between baryon ground and
resonant states, such as the N-N* and the N- transition form factors,
should be investigated along covariant approaches. In this context the
advances of Subproject C9 will be made use of as far as the description
of resonance states is concerned. Regarding weak processes special
attention will be devoted also to the axial charges of baryon
resonances, given their role in reflecting the possible chiral-symmetry
restoration in higher-lying resonance spectra (cf. also C5).
[1] T. Melde, K. Berger, L. Canton, W. Plessas, R. Wagenbrunn, Phys. Rev. D76 (2007) 074020
[2] Ki-Seok Choi, W. Plessas, and R.F. Wagenbrunn, Phys. Rev. C 81 (2010) 028201
[3] Ki-Seok Choi, W. Plessas, and R.F. Wagenbrunn, Phys. Rev. D 82 (2010) 014007
[4] Ki-Seok Choi, Electromagnetic and axial structures of baryon ground and resonant states, PhD Thesis, Univ. of Graz, 2011
Several relativistic approaches (relativistic
constituent-quark models, Poincaré-invariant Faddeev formalism,
perturbative QCD) will be followed to study exclusive hadron reactions
like, e.g., hadron decays, meson photo- and electroproduction, and
hadron-hadron scattering over a broad range of energy and momentum
transfers. In this way the various degrees of freedom dominating the
hadron structure at different excitation energies will be investigated.
Generalizing previous work on p-pbar->Lambda_c-Antilambda_c [1] we
will put particular emphasis on proton-antiproton annihilation into
meson pairs (p-pbar -> meson-antimeson). This kind of reactions will
be part of the forthcoming experimental program at the Facility for
Antiproton and Ion Research (FAIR) in Darmstadt. In the presence of a
hard scale – either a heavy quark mass or large momentum transfer -
various perturbative mechanisms (generalized parton picture,
hard-scattering approach, Landshoff mechanism) are available to
describe such processes. These mechanisms differ by the way how
perturbative and non-perturbative contributions are separated. By
considering the proton as a quark-diquark system the necessary
annihilation of 2 quarks and 2 antiquarks will become a simple
diquark-antidiquark annihilation [2]. One of the goals will be to
compare these different perturbative mechanisms. Thereby the
non-perturbative input will be treated in a consistent manner by
starting from appropriate models for the light-cone wave functions of
the hadrons involved. These will first be physically motivated ansätze
and in the sequel we intent to use wave functions derived from
Bethe-Salpeter amplitudes or relativistic constituent-quark models. The
studies on meson-pair production will be done in collaboration with
Bernard Pire (Palaiseau).
[1] A.T. Goritschnig, P. Kroll, and W. Schweiger, Eur. Phys. J. A 42 (2009) 43.
[2] P. Kroll, B. Quadder, and W. Schweiger, Nucl. Phys. B316 (1989) 373.
[3] G.P. Lepage, S.J. Brodsky, Phys. Rev. D22 (1980) 2157.
[4] P.V. Landshoff, Phys. Rev. D10 (1974) 1024.
[5] P. Kroll, B. Quadder, and W. Schweiger, Nucl. Phys. B316 (1989) 373.
One of the important tasks of the forthcoming
funding period of the DK will be a more realistic description of hadron
excitations as decaying resonances (cf. C2, C7 ,C8 ,C9). The physical
picture that is often associated with a resonance is a core of quarks
that is surrounded by a meson cloud. This kind of picture emerges quite
naturally within the chiral constitutent-quark model in which pions
(and also the other pseudoscalar mesons) couple directly to the
constituent quarks. If the dynamics of these pseudoscalar mesons is
explicitly taken into account within a coupled-channel framework, a
hadron state is not just a valence-quark state, but has also a
valence-quark-meson component [1]. The latter will also influence the
electromagnetic structure of the hadron and is responsible for
so-called “exchange currents” [2]. In addition to the spectator
current, that comes from the valence-quark state, dynamical
pseudoscalar meson exchange gives rise to exchange-current
contributions in which the photon couples either directly to the meson
or to one of the constituent (anti)quarks when the meson is “in-flight”.
In
order to derive the full electromagnetic current and calculate
electromagnetic form factors we will treat electron-hadron scattering
by means of a Poincaré-invariant coupled-channel framework that is
based on relativistic point-form dynamics. This kind of approach has
already been tested for the calculation of pi- and rho-meson form
factors with instantaneous binding forces, where it gave results that
are equivalent with corresponding front-form calculations [3,4]. The
approach is, however, general enough to accommodate also for interquark
forces that are caused by dynamical particle exchange. It is the main
goal of this project to quantify exchange-current effects on the
electromagnetic hadron structure within the chiral constituent-quark
model, first for mesons and then for baryons. In addition we will try
to substantiate these predictions by comparison with results coming
from Bethe-Salpeter and lattice methods.
[1] A. Krassnigg, W. Schweiger, W.H. Klink, Phys. Rev. C67 (2003) 064003.
[2] F. Gross, D.O. Riska, Phys. Rev. C36 (1987) 1928.
[3] E.P. Biernat, W. Schweiger, K. Fuchsberger, W.H. Klink, Phys. Rev. C79 (2009) 055203.
[4] E.P. Biernat, Electromagnetic properties of few-body systems within a point-form approach, PhD thesis, Graz University (2011).