Quantum chromodynamics (QCD) possesses a rich phase structure. Hereby the transition at non-vanishing temperatures (and small chemical potentials) has been on the focus of many recent investigations thereby providing some substantial insight into the related part of the QCD phase diagram. On the other hand, for many planned experiments (and hereby most prominently the ones at FAIR) as well as for some astrophysical applications the medium to high-density, small-temperature region is of central interest.
In this subproject we plan to continue our investigations employing non-perturbative functional approaches like the functional renormalization group (FRG) method [1], Dyson-Schwinger equations (DSEs) [2], and n-Particle-irreducible (nPI) actions [3]. To calculate observables all these methods as well as suitably chosen combinations thereof require appropriate truncations of the respective infinite sets of equations. To this end it is important to note that the program of determining the properties of the primitively divergent Green functions (PDGFs) of Landau gauge QCD in the vacuum made good progress in the last years. In addition, exploratory studies of the phase transition into the conformal window by increasing the number of charged matter flavors (cf. subproject D2) provided further insight into potential changes the PDGFs experience when crossing to a chirally symmetric phase.
In this project truncations schemes verified in the vacuum will be extended to the case of non-vanishing matter density. In order to assess the validity of these truncations also calculations for QCD-like theories with the QCD gauge group SU(3) replaced by SU(2) or G2 will be performed. For these theories lattice results at non-vanishing chemical potential are available (cf. also subprojects D7) and direct comparisons and verifications are thus possible. In the planned calculations several different order parameters which can be relatively easily computed once the elementary Green functions are known (like,
e.g., the dual condensate), will be determined and compared to the respective lattice results.
Within this project it is mandatory to include all quark back-coupling effects in all investigated phases (confining and chiral-symmetry breaking, deconfined and chirally symmetric, color-superconducting, etc.). It should be noted that the required high level of self-consistency can only be achieved by employing tools which treat several aspects as, e.g., derivation of the equations or decomposition into and projection onto tensor components, in an automated way. Correspondingly the existing tools like, e.g., DoFun, will be generalized to settings without full Lorentz invariance.
To summarize, the main objective of this subproject is to gain information on the elementary Green functions of QCD and QCD-like theories at non-vanishing densities and to subsequently calculate order parameters for the different identified phases.
[1] see e.g. J. Berges, N. Tetradis and C. Wetterich, Phys. Rept. 363 (2002) 223 and refs. therein
[2] see e.g. R. Alkofer and L. von Smekal, Phys. Rept. 353} (2001) 281 and refs. therein
[3] see e.g. M.E. Carrington, Phys.Rev. D87 (2013) 045011 and refs. therein
In the last decade a description of mesons and baryons as bound states within the rainbow-ladder approximation of the Bethe-Salpeter or covariant Faddeev equation has been fully developed. As it is known that this approximation fails in determining several properties of hadrons substantial effort has gone into treating the Bethe-Salpeter equation in a consistent symmetry-preserving manner beyond the ubiquitous rainbow-ladder approximation. Although these calculations show some improvements in the description of hadronic properties progress is hindered by substantial technical challenges, especially if one wants to include non-trivial aspects of the quark-gluon vertex function (which has recently been calculated in Graz).
In this subproject we plan to follow another approach based on Functional Renormalization Group (FRG). Within the different formulations of exact FRG equations the one derived by Wetterich [1] has proven to be very well suited for numerical calculations. Optimized regulator functions have increased this applicability as well as the flexibility in the necessary truncation of the system. Employing an RG-scale dependent Hubbard-Stratonovich bosonisation bound state equations can be derived [2] which describe the dynamical and scale-dependent changes in the relevant degree of freedom (``dynamical hadronisation''). It has been shown already quite some time ago in which limits such equations reduce to the rainbow-ladder Bethe-Salpeter equation. Also, in the non-relativistic case more elaborate truncation schemes have proven successful, e.g., in the field of ultra-cold atom gases. However, for relativistic bound states the use of optimized regulators spoils the analytic properties of the equations, and the analytic continuation necessary to implement physical masses (which require time-like momenta in the equations) becomes very cumbersome. The same is true for exponential regulators whereas algebraic regulators introduce additional singularities and substantially decrease the convergence of the numerical methods.
Therefore in a first step the experience of the Principal Investigator with the analytic continuation of Green functions determined from Dyson-Schwinger equations will be exploited to derive suitable regulator functions for the task at hand. The recently gained knowledge on the quark-gluon vertex in Landau gauge QCD will serve as valuable input in developing suitable truncation schemes for the resulting bound state equations. It is planned to study the respective ground state mesons for different spin and parity first (pseudoscalar, scalar, vector and axialvector mesons). Besides the respective masses also properties like the electromagnetic form factors and decay widths are hereby of interest.
Benchmarks for this subproject will be provided when extending to QCD-like theories. On the one hand, this is a natural continuation of the studies of the conformal window of gauge theories performed so far. On the other hand, due to the interest in technicolor theories, lattice results for ``mesons'' within different gauge groups and several types of matter content (as, e.g., adjoint fermions) are available. Reproducing spectra for such theories provides a highly non-trivial cross-check on the suitability of the employed methods. Besides validating the QCD calculation they are of interest in their own right, as an example we mention the determination of branching ratios of a potential techni-rho-meson.
[1] C. Wetterich, Phys.Lett. B301 (1993) 90
[2] H. Gies and C. Wetterich, Phys.Rev. D65 (2002) 065001
In the last two decades lattice QCD has matured into a powerful quantitative tool. However, one instance where lattice QCD has remained behind is the analysis of lattice field theories at finite densities. The problem is that at finite chemical potential the action S develops an imaginary part which leads to a Boltzmann factor exp(-S) with a complex phase. This phase prevents one from using the Boltzmann factor as a probability in a stochastic process and this so-called "complex action problem" or "sign problem" has slowed down the analysis of the QCD phase diagram considerably.
In the last three years several lattice field theories at finite density were rewritten exactly into a form where the partition sum has only real and positive contributions. In this so-called "dual formulation" the new degrees of freedom are loops of conserved matter flux and gauge fields correspond to surfaces that are either closed or bounded by matter flux. This implies that the new variables have to obey constraints that need to be taken into account in a Monte Carlo simulation, which is possible with generalizations of the Prokofev-Svistunov Worm algorithm. Our group has considerably contributed to this development (see e.g. [1]-[5] for examples).
In the new funding period we plan to further develop the dual approach, in particular aiming at the generalization to non-abelian gauge theories, where we currently explore new ideas that go beyond the unsuccessful character expansion techniques that can be found in the literature. A second challenge will be a more complete inclusion of fermion contributions, which we currently can take into account only with the leading terms of their hopping expansion. Aspects of finite density field theories will be studied - in particular various condensation phenomena.
[1] Dual lattice simulation of the U(1) gauge-Higgs model at finite density - an exploratory proof-of-concept study, Ydalia Delgado Mercado, Christof Gattringer, Alexander Schmidt, Phys. Rev. Lett. 111 (2013) 141601.
[2] Spectroscopy in finite density lattice field theory: An exploratory study in the relativistic Bose gas, Christof Gattringer, Thomas Kloiber, Phys. Lett. B720 (2013) 210-214.
[3] Surface worm algorithm for abelian Gauge-Higgs systems on the lattice, Ydalia Delgado Mercado, Christof Gattringer, Alexander Schmidt, Comput. Phys. Commun. 184 (2013) 1535-1546.
[4] Lattice study of the Silver Blaze phenomenon for a charged scalar phi^4 field, Christof Gattringer, Thomas Kloiber, Nucl. Phys. B869 (2013) 56-73.
[5] The QCD phase diagram according to the center group, Ydalia Delgado Mercado, Hans Gerd Evertz, Christof Gattringer, Phys. Rev. Lett. 106 (2011) 222001.
Finite density QCD is plagued by the so-called complex action problem, the fact that at non-zero chemical potential the action S has an imaginary part and the Boltzmann factor exp(-S) cannot be used as a probability weight in a Monte Carlo simulation. A rewriting to a dual representation (see Subproject D3) might not be possible for all theories of interest and also alternative (although not as powerful) strategies should be explored. In Subproject D4 we will address two such approaches, the density of states method [1] and a newly developed improved Taylor expansion [2]-[4].
A recently developed iterative technique [1] allows the computation of the coefficients of a suitably parameterized density of states with exponential error suppression. In a preliminary study [2] together with K. Langfeld we are currently exploring the new approach in the Z3 spin model [3] that has a dual representation, such that the results can be cross-checked with the outcome of a dual simulation. The findings are very encouraging and part of Subproject D4 will be devoted to the further development of the density of states approach, in particular to theories with fermions.
Another technique that we have already assessed in the Z3 spin model is an improved Taylor series for finite chemical potential. The idea is to use a double Taylor expansion in the parameter (exp(+/-mu) - 1), which captures some features of the fugacity expansion and can also be viewed as a partly resumed conventional Taylor series. First tests show that for some parameter sets the improved Taylor series is considerably better that the conventional one and the goal will be to implement the improved Talyor series in full QCD.
[1] The density of states in gauge theories, Kurt Langfeld, Biagio Lucini, Antonio Rago, Phys. Rev. Lett. 109 (2012) 111601.
[2] In preparation.
[3] Taylor- and fugacity expansion for the effective Z(3) spin model of QCD at finite density, Eva GrĂĽnwald, Ydalia Delgado Mercado, Christof Gattringer, submitted to JHEP [arXiv:1403.2086 [hep-lat]].
[4] A test of Taylor- and modified Taylor-expansion, Max Wilfling, Christof Gattringer, PoS (Lattice 2013) 452, [arXiv:1311.7436 [hep-lat]].
[5] Improved Taylor expansion, Ydalia Delgado Mercado, Christof Gattringer, Hans-Peter Schadler, Max Wilfling, in preparation.
A consistent and systematic picture of hadrons made of light quarks is missing. There is a general understanding that both confinement and spontaneous breaking of chiral symmetry are important for hadronic mass generation.
In ref. [1,2] it has been noticed that upon removal of the chiral symmetry breaking dynamics from the valence quarks (by means of subtraction of the lowest-lying Dirac eigenmodes from the full quark propagators) a large degeneracy of mesons develops.
All possible chiral multiplets of the J=1 mesons get degenerate. They exhaust all possible chiralities of quarks and antiquarks, i.e., their spin orientations, as well as possible spatial and charge parities for non-exotic mesons.
The corresponding radial energy levels could be interpreted as radial energy levels of the dynamical QCD string that connects the ultra-relativistic quark and antiquark with the total spin $J=1$. If this string interpretation of the observed degeneracy is correct, then one expects even a higher degeneracy, namely a degeneracy of the radial excitations with the rotational and transverse excitations of the string.
Given the existing overlap gauge configurations it would be interesting and important to measure on the lattice masses of the lowest hybrid mesons with $J=1$ and their evolution upon truncation of the lowest Dirac modes. Within the string picture the hybrid mesons are transverse excitations of the string.
Consequently, if the string interpretation of the observed degeneracy is correct, then one should see upon truncation of the lowest Dirac eigenmodes a degeneracy of the observed radial levels with the hybrid states.
[1] L. Ya. Glozman, C.B. Lang, M. Schroeck, Phys. Rev. D 86 (2012) 014507
[2] M. Denissenya, L. Ya. Glozman, C. B. Lang, arXiv:1402.1887
Upon artificial restoration of chiral symmetry by means of subtraction of the lowest-lying Dirac eigenmodes from the valence quark propagators [1,2] the meson spectrum indicates a large symmetry of hadrons, larger than U(2)_L x U(2)_R.
Namely, all possible J=1 meson chiral multiplets get degenerate. A possible interpretation of this symmetry is that we observe the quantum levels of a dynamical QCD string that connects the ultrarelativistic quark and antiquark.
The lattice calculations that have led to this result had been performed with dynamical overlap gauge configurations on a lattice of size 1.9 fm. This rather small volume could affect the excited radial level and shift it up with respect to the actual level. This may modify the actual quantization law of the string. One needs larger lattice volumes to clarify the situation and to discriminate between different possibilities.
A large-volume dynamical simulation with overlap fermions cannot be done at present due to the numerical demands. However, if the observed radial energy levels are levels of a dynamical QCD string there is no need to use unquenched gauge configurations, because the string (color- electric flux tube) is a purely gluonic object. Consequently, this physics can be addressed with the quenched gauge configurations.
We will generate quenched gauge configurations on physically large volume and then calculate the hadron propagators with truncated overlap fermion propagators. Upon unbreaking of the chiral symmetry one can study both radial, rotational and transverse excitations of a possible string with the large lattice volume and extract its actual quantization law.
[1] L. Ya. Glozman, C.B. Lang, M. Schroeck, Phys. Rev. D 86 (2012) 014507
[2] M. Denissenya, L. Ya. Glozman, C. B. Lang, arXiv:1402.1887
Neutron stars are highly interesting, as their structure is determined by particle physics, especially the strong interactions, while at the same time they are macroscopically observable objects. However, their description using first-principle approaches is a serious challenge, especially due to the so-called sign problem of numerical lattice simulations. At the same time, the importance of complex bound states, especially baryons, also hampers continuum approaches, like functional methods.
Thus, no single method seems so far able to describe neutron stars alone. Hence, a synergistic approach of several methods seems to be more suitable to address this problem. An approach already thoroughly tested for Yang-Mills theory [1] is a combination of lattice calculations and functional methods.
However, this requires a starting point accessible to both methods. But because of the sign problem, this cannot be QCD. In fact, currently, even the static phase diagram of cold, dense matter in generic gauge theories is largely unknown, with various effective models predicting an enormous spread of possible phases. A first principle determination of the static phase diagram of at least some gauge theories is therefore necessary. Since it is expected that neutron star properties are dominated by fermionic baryons, theories including them are the most desirable candidates.
To this end, we have designed G2-QCD, QCD with the gauge group SU(3) replaced by the exceptional Lie group G2. This theory has no sign problem and can thus be simulated on the lattice, and it has fermionic baryons [2]. In this subproject, the lattice investigation will be performed, while the investigation using functional methods is the focus of subproject D1.
The aim of this project is twofold. One is to better understand the phase diagram of this theory, to identify those phases which appear to be most closely related to the ones expected in QCD. This is necessary, as in G2-QCD additional bound states beyond the well-known mesons and baryons are possible, especially diquarks and hybrids. This will also require to improve the spectroscopy of bound states in this theory. That is a topic in which the previous stages of the DK have already provided substantial methodological progress [3].
The second step, in analogy to the less realistic case of 2-color QCD [4], is to determine the basic correlation functions of gluons and quarks as a function of chemical potential. The purpose of determining these gauge-dependent quantities is twofold [1]. One is that the results can serve as inputs to effective calculations using functional methods, here in subproject D1. Though such calculations are not self-consistent, such calculations have provided in the past conceptual insights as well as helped in the design of self-consistent truncation schemes. The second purpose is to serve as benchmarks for truncation schemes: If the conceptual construction of a truncation scheme is sound, it should be able to describe both G2-QCD and QCD. If it successfully describes vacuum and finite-temperature QCD as well as G2-QCD throughout the phase diagram, its application to finite-density QCD is likely justified.
[1] A. Maas, Phy. Rept. 524 (2013) 203
[2] B. Wellegehausen et al.,PRD 89 (2014) 056007; A. Maas et al. PRD 86
(2012), 111901(R)
[3] G. Engel et al., PRD 85 (2012), 034508
[4] T. Boz et al., EPJ A49 (2013), 87
While the static properties of strongly-interacting matter at high densities and low temperatures are a necessary first step to understand neutron stars, most observational data on the internal structure of neutron stars are expected to be obtained indirectly from either neutron star formation or merging. In both cases, neutrinos, and thus weak interaction effects, play an important role [1]. Any equation of state which should be used to describe neutron stars must therefore capture the effect of neutrinos, and thus also beta-decay. However, weak interactions violate parity, which is a serious obstacle for lattice calculations [2]. Furthermore, the extremely different energy scales involved make numerical lattice simulations also unfeasible. Hence functional methods are the prime candidate for a unified description, especially at densities where the quark and gluon degrees of freedom play the most important role. Especially during mergers, it appears likely that this is the case.
This subproject therefore aims at a unified description of QCD, parity-violating beta-decays, and possibly electromagnetic interactions, using functional methods. However, even in the most violent neutron star mergers, it is not expected that a full resolution of weak interaction effects plays a significant role. Hence, an effective description of weak interactions in form of the Fermi theory appears adequate. To evade ultraviolet problems, such an interaction must be appropriately modeled.
An ideal framework for this is the functional renormalization group framework [3], which offers a natural way to implement effective low-energy operators using appropriate regulators. Furthermore, the effective inclusion of bound states in this formalism is possible using dynamical hadronization. Finally, thermodynamic consistency is imprinted in the equation from the beginning. Hence, it is a suitable framework for this topic. Furthermore, the same methods are used in subproject D2, supporting methodological cooperation between both.
However, this approach requires to solve an infinite set of coupled, non-linear integro-differential equations. Any solution thus requires approximations. To this end, comparison with results from lattice QCD, including ultimately the G2-QCD version at finite density of subprojects D1 and D7, will be very useful to constrain these approximations. Moreover, beta-decay is well studied, and experimental input in the vacuum can be used to cross-check the inclusion of weak interaction effects as effective operators.
However, this area is new, and this project will pioneer it. Hence, a full description at finite density, though the ultimate motivation, is not feasible in a single PhD thesis. Therefore, the scope of this project is to describe the beta-decay of a neutron in vacuum using the functional renormalization group. This will combine, for the first time, the existing description of QCD in this approach [3,4], dynamical hadronization at the baryon level [3], and weak interaction effects.
Electromagnetic effects will be included to the extent necessary to describe the difference between up quarks and down quarks, but not for a full inclusion of electromagnetic effects, e.g. formfactors of the baryons. The aim is to describe the pion and possibly neutron beta-decay at the 20% level quantitatively accurate.
[1] J. Faber et al. Living Rev.Rel. 15 (2012), 8; Y. Sekiguchi et al.
PRL 107 (2011), 051102
[2] P. Hasenfratz et al., JHEP 0802 (2008), 079
[3] H. Gies, Lect. Notes Phys. 852 (2012), 287
[4] L. Fister et al., arXiv:1112.5440 [hep-ph] (2011)
Several relativistic approaches (relativistic constituent quark models, PoincarĂ©-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 foregoing work on p-pbar->Lambda_c-Antilambda_c [1] and p-pbar->D-Dbar [2] 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 â€“ either a heavy quark mass or large momentum transfer -- various perturbative mechanisms (generalized parton picture [3], hard-scattering approach [4], Landshoff mechanism [5]) are available to describe such processes. These mechanisms differ by the way how perturbative and non-perturbative pieces 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 [6]. 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 ansĂ¤tze 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 Prof. Bernard Pire (Ecole Polytechnique, Palaiseau).
[1] Proton-antiproton annihilation into a Lambda_c-Antilambda_c pair, A.T. Goritschnig, P. Kroll, and W. Schweiger, Eur. Phys. J. A 42 (2009) 43.
[2] Double handbag description of proton-antiproton annihilation into a heavy meson pair, A.T. Goritschnig, B. Pire, and W. Schweiger, Phys. Rev. D87 (2013) 014017; Erratum, Phys. Rev. D88 (2013) 079903.
[3] Exclusive meson pair production in gamma* gamma scattering at small momentum transfer, J.P.Lansberg, B. Pire, L. Szymanowski, Phys. Rev. D73 (2006) 074014.
[4] Exclusive processes in perturbative Quantum Chromodynamics, G.P. Lepage, S.J. Brodsky, Phys. Rev. D22 (1980) 2157.
[5] Model for elastic scattering at wide angle, P.V. Landshoff, Phys. Rev. D10 (1974) 1024.
[6] Exclusive production of heavy flavors in proton-antiproton annihilation, P. Kroll, B. Quadder, and W. Schweiger, Nucl. Phys. B316 (1989) 373.
The physical picture that is often associated with a hadron is a quark core that is surrounded by a meson cloud. It emerges quite naturally within the chiral constituent-quark model in which pions (and also the other pseudoscalar meson ground states) 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]. Further support for this kind of picture comes from recent calculations of electromagnetic nucleon form factors within the Bethe-Salpeter-Dyson-Schwinger framework [2] where "clear signals of missing pion-cloud effects" have been detected. In lepton-hadron scattering and the corresponding time-like processes one thus will not only probe the electroweak structure of the quark core but will also get contributions from the meson cloud. In this project we are interested in two types of contributions coming from the meson cloud:
i) Exchange-current contributions, where the photon (W or Z) couples either to the meson in the cloud, or the quark core when the meson is â€śin-flightâ€ť.
ii) Z-graph contributions, where the photon (W or Z) fluctuates into a (correlated) quark-antiquark pair which then couples to the quark core (or vice versa).
In order to investigate the resulting effects on the electroweak current and form factors we will use a hybrid constituent-quark model with instantaneous confinement and additional mesonic degrees-of-freedom that account for the meson cloud. Lepton-hadron scattering and electroweak hadron decays are then treated by means of a PoincarĂ© invariant coupled-channel framework that is based on relativistic point-form dynamics. This kind of relativistic formalism has already been tested for the calculation of electromagnetic pi- and rho-meson form factors [3,4] as well as for weak decay form factors of heavy-light mesons [5] employing simple constituent-quark models. With instantaneous binding forces it gave results that are equivalent with corresponding front-form calculations. The approach is, however, general enough to accommodate also for (dynamical) mesons coupling directly to the quarks. The necessary extensions of the formalism are summarized in Refs. [6,7], where exchange-current effects on the electromagnetic-nucleon form factors (for space-like momentum transfers) and Z-graph contributions for weak meson-decay form factors (involving time-like momentum transfers) are addressed. It is the main goal of this project to quantify exchange-current and Z-graph effects on the electroweak hadron structure within such a hybrid constituent-quark model in both, the space- and the time-like momentum-transfer regions. In addition we will try to substantiate these predictions by comparison with corresponding results coming from Bethe-Salpeter-Dyson-Schwinger calculations.
[1] Relativistic coupled-channel quark model for meson resonances, R. Kleinhappel, W. Plessas and W.Schweiger, Few Body Syst. 54 (2013) 339.
[2] Nucleon electromagnetic form factors from the covariant Faddeev equation, G. Eichmann, Phys. Rev. D84 (2011) 014014.
[3] Electromagnetic meson form factor from a relativistic coupled-channel approach, E.P. Biernat, W. Schweiger, K. Fuchsberger, W.H. Klink, Phys. Rev. C79 (2009) 055203.
[4] Electromagnetic rho-meson form factors in point-form relativistic quantum mechanics, E.P. Biernat and W. Schweiger, arXiv:1404.2440 [hep-ph].
[5] Electroweak form factors of heavy-light mesons: A relativistic point-form approach, M. Gomez-Rocha and W. Schweiger, Phys. Rev. D86 (2012) 053010.
[6] Meson-cloud effects in the electromagnetic nucleon structure, D. Kupelwieser and W. Schweiger, arXiv:1312.0863 [nucl-th].
[7] Hadron structure within the point form of relativistic quantum mechanics, M. Gomez-Rocha, W. Schweiger and O. Senekowitsch, arXiv:1311.1936 [hep-ph].