Krzysztof Cichy
The strong interaction of quarks, the elementary particles found in nucleons, via gluon exchange, is described by the theory of Quantum Chromodynamics (QCD), proposed and developed in the 1960s and 70s. The QCD coupling constant, which describes the strength of the quark-gluon and gluon-gluon interactions, depends on the energy scale or, equivalently, on the distance between particles. At high energies, or at small distances, of the order of 20 GeV (0.01 fm), the QCD coupling constant is small and the interactions can be described by QCD perturbation theory (PT), similarly to the prototype quantum field theory - the extremely successful quantum theory of electromagnetism, Quantum Electrodynamics (QED). On the other hand, at low energies (large distances), of the order of 200 MeV (1 fm), the QCD coupling becomes large and perturbative methods fail. Although it is possible to model this non-perturbative regime phenomenologically, the only known method to yield quantitative predictions from first principles, i.e. from the QCD Lagrangian, is the formulation of the theory on an Euclidean spacetime grid (the lattice), which serves as a non-perturbative regulator, and evaluation of the relevant path integrals numerically, using Monte Carlo algorithms implemented for the world's largest supercomputers. This approach is called Lattice QCD (LQCD).
Nucleons (protons and neutrons) are complex dynamical systems, consisting of quarks and gluons that interact with one another and these interactions are described by the theory of QCD. Our knowledge concerning this structure is summarized in the so-called distribution functions, in particular the parton distribution functions (PDFs). These functions can be obtained by fitting experimental data, but they are, in principle, computable theoretically. Given the non-perturbative nature of the PDFs, one can naturally expect this to be feasible using the lattice approach. However, for many years, lattice physicists have only been able to compute the low moments of the PDFs, which are accessible from matrix elements of local operators. The full computation of PDFs on the lattice remained elusive, as they are defined on the light-cone and hence require going to effectively zero spacetime intervals. In 2013, a new method was suggested by X. Ji to evaluate a different kind of object, the so-called quasi-PDF, which is defined from purely spatial correlations and hence avoids the above problem. The quasi-PDF can then be matched to the true PDF via a known matching procedure. In this project, we are exploring this approach and we have so far obtained very encouraging results. In our 2017 paper, published as a Frontiers Article in Nuclear Physics B, we introduced a non-perturbative renormalization procedure for quasi-PDFs. In 2018, we presented the first computation at the physical pion mass, with proper renormalization and matching procedures, which was published in Physical Review Letters and the Rapid Communications section of Physical Review D. At the end of 2018, together with my collaborator Martha Constantinou, we wrote a review of the field of x-dependent partonic functions, as a contribution to a special issue of Advances in High Energy Physics. In 2019, we continued our work with a thorough investigation of systematic uncertainties in quasi-PDF computations. Together with my collaborators from the NNPDF group, we also investigated reconstruction of PDFs from lattice data in the robust NNPDF neural network framework. In 2020, we investigated PDFs of the Delta baryon, PDFs from the pseudo-distribution approach and the continuum limit of quasi-PDFs at a non-physical pion mass. We also explored novel directions of generalized parton distributions (GPDs) - our exploratory study of unpolarized and helicity GPDs was published in Physical Review Letters. In 2021, we continued this thread by determining transversity GPDs. In 2020, we also started exploring twist-3 distributions from the quasi-PDF approach. We derived the matching relations and performed a lattice study of the twist-3 g_T(x) and h_L(x) functions. Also in 2020, we started a project of determining transverse-momentum dependent PDFs (TMDs) from the lattice and our first publication on this was accepted in Physical Review Letters. In 2021, the review of 2018 was updated with two proceedings contributions from my two plenary talks at LATTICE21 and vCONF21.
Selected publications:
21. M. Bhat, W. Chomicki, K. Cichy, M. Constantinou, J.R. Green and A. Scapellato, Continuum limit of parton distribution functions from the pseudo-distribution approach on the lattice, arXiv:2205.07585 [hep-lat].
20. K. Cichy, Overview of lattice calculations of the x-dependence of PDFs, GPDs and TMDs, EPJ Web Conf. 258 (2022) 01005, arXiv:2111.04552 [hep-lat], proceedings of plenary talk in Virtual Tribute to Quark Confinement and the Hadron Spectrum
19. K. Cichy, Progress in x-dependent partonic distributions from lattice QCD, arXiv:2110.07440 [hep-lat], proceedings of plenary talk in LATTICE 2021
18. Y. Li, S-C. Xia, C. Alexandrou, K. Cichy, M. Constantinou, X. Feng, K. Hadjiyiannakou, K. Jansen, Ch. Liu, A. Scapellato, F. Steffens and J. Tarello, Lattice QCD Study of Transverse-Momentum Dependent Soft Function, Phys. Rev. Lett. 128 (2022) no.6, 062002, arXiv:2106.13027 [hep-lat].
17. C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou, K. Jansen, A. Scapellato and F. Steffens [ETM Collaboration], Transversity GPDs of the proton from lattice QCD, Phys. Rev. D105 (2022) no.3, 034501, arXiv:2108.10789 [hep-lat].
16. S. Bhattacharya, K. Cichy, M. Constantinou, A. Metz, A. Scapellato and F. Steffens, Parton distribution functions beyond leading twist from lattice QCD: The hL(x) case, Phys. Rev. D104 (2021) no.11, 114510, arXiv:2107.02574 [hep-lat].
15. C. Alexandrou, K. Cichy, M. Constantinou, J. Green, K. Hadjiyiannakou, K. Jansen, F. Manigrasso, A. Scapellato and F. Steffens [ETM Collaboration], Lattice continuum-limit study of nucleon quasi-PDFs, Phys. Rev. D103 (2021) no.9, 094512, arXiv:2011.00964 [hep-lat].
14. C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou, K. Jansen, A. Scapellato and F. Steffens [ETM Collaboration], Unpolarized and helicity generalized parton distributions of the proton within lattice QCD, Phys. Rev. Lett. 125 (2020) no.26, 262001, arXiv:2008.10573 [hep-lat].
13. S. Bhattacharya, K. Cichy, M. Constantinou, A. Metz, A. Scapellato and F. Steffens, The role of zero-mode contributions in the matching for the twist-3 PDFs e(x) and h_L(x), Phys. Rev. D102 (2020) no.11, 114025, arXiv:2006.12347 [hep-ph].
12. S. Bhattacharya, K. Cichy, M. Constantinou, A. Metz, A. Scapellato and F. Steffens, One-loop matching for the twist-3 parton distribution g_T(x), Phys. Rev. D102 (2020) no.3, 034005, arXiv:2005.10939 [hep-ph].
11. M. Bhat, K. Cichy, M. Constantinou and A. Scapellato, Parton distribution functions from lattice QCD at physical quark masses via the pseudo-distribution approach, Phys. Rev. D103 (2021) no.3, 034510, arXiv:2005.02102 [hep-lat].
10. S. Bhattacharya, K. Cichy, M. Constantinou, A. Metz, A. Scapellato and F. Steffens, Insights on proton structure from lattice QCD: the twist-3 parton distribution function g_T(x), Phys. Rev. D102 (2020) no.11, 111501 (Rapid Communications), arXiv:2004.04130 [hep-lat].
9. Y. Chai, Y. Li, S. Xia, C. Alexandrou, K. Cichy, M. Constantinou, X. Feng, K. Hadjiyiannakou, K. Jansen, G. Koutsou, C. Liu, A. Scapellato and F. Steffens, Parton distribution functions of Δ^+ on the lattice, Phys. Rev. D102 (2020) no.1, 014508, arXiv:2002.12044 [hep-lat].
8. K. Cichy, L. Del Debbio, T. Giani, Parton distributions from lattice data: the nonsinglet case, JHEP 10 (2019) 137, arXiv:1907.06037 [hep-ph].
7. C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou, K. Jansen, A. Scapellato and F. Steffens [ETM Collaboration], Systematic uncertainties in parton distribution functions from lattice QCD simulations at the physical point, Phys. Rev. D99 (2019) no.11, 114504, arXiv:1902.00587 [hep-lat].
6. K. Cichy, M. Constantinou, A guide to light-cone PDFs from Lattice QCD: an overview of approaches, techniques and results, accepted for publication in Special Issue of Advances in High Energy Physics Transverse Momentum Dependent Observables from Low to High Energy: Factorization, Evolution, and Global Analyses, arXiv:1811.07248 [hep-lat].
5. C. Alexandrou, K. Cichy, M. Constantinou, K. Jansen, A. Scapellato and F. Steffens [ETM Collaboration], Transversity parton distribution functions from lattice QCD, Phys. Rev. D98 (2018) no.9, 091503 (Rapid Communications), arXiv:1807.00232 [hep-lat].
4. C. Alexandrou, K. Cichy, M. Constantinou, K. Jansen, A. Scapellato and F. Steffens [ETM Collaboration], Light-Cone Parton Distribution Functions from Lattice QCD, Phys. Rev. Lett. 121 (2018) no.11, 112001, arXiv:1803.02685 [hep-lat].
3. C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou, K. Jansen, H. Panagopoulos and F. Steffens [ETM Collaboration], A complete non-perturbative renormalization prescription for quasi-PDFs, Nucl.Phys. B923 (2017) 394-415 (Frontiers Article), arXiv:1706.00265 [hep-lat].
2. C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou, K. Jansen, F. Steffens and C. Wiese [ETM Collaboration], Updated Lattice Results for Parton Distributions, Phys. Rev. D 96 (2017) 014513, arXiv:1610.03689 [hep-lat].
1. C. Alexandrou, K. Cichy, V. Drach, E. Garcia-Ramos, K. Hadjiyiannakou, K. Jansen, F. Steffens and C. Wiese [ETM Collaboration], Lattice calculation of parton distributions, Phys. Rev. D 92 (2015) 014502, arXiv:1504.0755 [hep-lat].
Despite the amazing successes of Lattice QCD, there are still areas where satisfactory description is elusive. These are in particular finite density and real-time simulations. Both of them have a sign problem in standard Euclidean Monte Carlo. Although there are ways of alleviating the problem, they have not led not a breakthrough. Therefore, alternative approaches are sought for. One of such alternatives are tensor network (TN) methods. They have proven to be a useful tool in analyzing the properties of condensed matter systems and in quantum information. In this branch of research, we have shown that they can also be effectively used to describe lattice gauge theories. We concentrated so far on a (1+1)-dimensional system, the Schwinger model, using the paradigmatic example of TN methods, the Matrix Product States (MPS) ansatz. We analyzed the spectrum, getting very precise estimates of the vector and scalar particle masses, and also the chiral symmetry breaking at zero and non-zero temperature. These served as proofs of principle that TN can handle quantum field theories, but these properties are also readily accessible with Monte Carlo simulations without a sign problem. In 2017, in a paper published in Physical Review Letters, we have analyzed a situation where the sign problem indeed appears - in the 2-flavour Schwinger model at non-zero chemical potential. We have shown that MPS can precisely describe such a setup, hence confirming that this is a prospective direction for sign problem plagued lattice gauge theories. In another 2017 paper, published in Physical Review X, we proposed an explicit formulation of the physical subspace for a non-Abelian SU(2) lattice gauge theory, where the gauge degrees of freedom are integrated out, which is potentially amenable for quantum simulations. We also analyzed the spectrum and entanglement properties of this theory. Recently, we are investigating the (1+1)-dimensional Thirring model, with the first paper devoted to its phase structure. In follow-up work, we are concentrating on the real-time dynamics and the possible existence of dynamical quantum phase transitions. The important next step is to go to higher dimensions, where the simple MPS ansatz has to be replaced by a more sophisticated one, e.g. the natural generalization of MPS to higher dimensions - the Projected Entangled Pair States (PEPS) ansatz. In 2019, together with my collaborator Mari Carmen Banuls, we wrote a review with a focus on the usage of tensor network methods for lattice field theories, published in Reports on Progress in Physics. In 2021, also with Mari Carmen, we wrote an essay on the perspectives of tensor networks for QCD, published as News&Views in Nature Physics.
Selected publications:
10. M.C. Banuls, K. Cichy, Tensors cast their nets on quarks, Nature Physics 17 (2021) 762.
9. M.C. Banuls, K. Cichy, Review on Novel Methods for Lattice Gauge Theories, Rept. Prog. Phys. 83 (2020) 024401, arXiv:1910.00257 [hep-lat].
8. M.C. Banuls, K. Cichy, Y.-J. Kao, C.-J. D. Lin, Y.-P. Lin, D. T.-L. Tan, Phase structure of the (1+1)-dimensional massive Thirring model from matrix product states, Phys. Rev. D 100 (2019) 094504, arXiv:1908.04536 [hep-lat].
7. M.C. Banuls, K. Cichy, J.I. Cirac, K. Jansen, S. Kühn, Tensor Networks and their use for Lattice Gauge Theories, review of the field, proceedings from a plenary talk in LATTICE 2018 given by M.C. Banuls, arXiv:1810.12838 [hep-lat].
6. M.C. Banuls, K. Cichy, J.I. Cirac, K. Jansen, S. Kühn, Efficient basis formulation for 1+1 dimensional SU(2) lattice gauge theory: Spectral calculations with matrix product states, Phys. Rev. X7 (2017) 041046 , arXiv:1707.06434 [hep-lat].
5. M.C. Banuls, K. Cichy, J.I. Cirac, K. Jansen, S. Kühn, Density induced phase transitions in the Schwinger Model - A study with matrix product states, Phys. Rev. Lett. 118 (2017) 071601, arXiv:1611.00705 [hep-lat].
4. M.C. Banuls, K. Cichy, K. Jansen, H. Saito, Chiral condensate in the Schwinger model with Matrix Product Operators, Phys. Rev. D 93 (2016) 094512, arXiv:1603.05002 [hep-lat].
3. M.C. Banuls, K. Cichy, J.I. Cirac, K. Jansen, H. Saito, Thermal evolution of the Schwinger model with Matrix Product Operators, Phys. Rev. D 92 (2015) 034519, arXiv:1505.00279 [hep-lat].
2. M.C. Banuls, K. Cichy, K. Jansen, J.I. Cirac, The mass spectrum of the Schwinger model with Matrix Product States, JHEP 1311 (2013) 158, arXiv:1305.3765 [hep-lat].
1. K. Cichy, A. Kujawa-Cichy, M. Szyniszewski, Lattice Hamiltonian approach to the massless Schwinger model: Precise extraction of the mass gap, Comput. Phys. Commun. 184 (2013) 1666, arXiv:1211.6393 [hep-lat].
One of the most important phenomena of QCD is the spontaneous breaking of chiral symmetry (SChSB). This mechanism is purely non-perturbative and hence invisible in perturbation theory, hence making LQCD the natural approach to address it. The order parameter of SChSB is called the chiral condensate. It is one of the low energy constants of chiral perturbation theory and can be extracted e.g. from fits of the quark-mass dependence of light pseudoscalar meson masses and decay constants. However, other methods also exist and in this thread of research, we have concentrated on two approaches that extract the chiral condensate from the spectrum of the Hermitian Dirac operator. The first one relies on the Banks-Casher relation between the chiral condensate and the spectral density of the Dirac operator. In 2008, an effective method (the spectral projector method) has been proposed by Giusti and Lüscher to effectively make use of this relation and explore the chiral properties of QCD on the lattice, in particular to compute the chiral condensate. The method consists in stochastically evaluating the number of eigenmodes (the mode number) of the Dirac operator below some threshold value M. The slope of the M-dependence of this mode number can be shown to be related to the chiral condensate, in the regime of renormalized M smaller than approx. 100 MeV. The spectral projector method can also be generalized to compute observables other than the mode number. As hinted by Giusti and Lüscher and then shown by Lüscher and Palombi, the spectrum of the Hermitian Dirac operator also contains information about the topological susceptibility.
Selected publications:
7. C. Alexandrou, A. Athenodrou, K. Cichy, M. Constantinou, D.P. Horkel, K. Jansen, G. Koutsou, C. Larkin, Topological susceptibility from twisted mass fermions using spectral projectors and the gradient flow, Phys. Rev. D 97 (2018) 074503, arXiv:1709.06596 [hep-lat].
6. C. Alexandrou, A. Athenodrou, K. Cichy, A. Dromard, E. Garcia-Ramos, K. Jansen, U. Wenger, F. Zimmermann, Comparison of topological charge definitions in Lattice QCD , Eur.Phys.J. C 80 (2020) 424, arXiv:1708.00696 [hep-lat].
5. K. Cichy, S. Zafeiropoulos, Wilson chiral perturbation theory for dynamical twisted mass fermions vs lattice data - a case study, Comput. Phys. Commun. 237 (2019) 143, arXiv:1612.01289 [hep-lat].
4. K. Cichy, E. Garcia-Ramos, K. Jansen, K. Ottnad, C. Urbach [ETM Collaboration], Non-perturbative Test of the Witten-Veneziano Formula from Lattice QCD, JHEP 1509 (2015) 020, arXiv:1504.07954 [hep-lat].
3. K. Cichy, E. Garcia-Ramos, K. Jansen [ETM Collaboration], Short distance singularities and automatic O(a) improvement: the cases of the chiral condensate and the topological susceptibility, JHEP 1504 (2015) 048, arXiv:1412.0456 [hep-lat].
2. K. Cichy, E. Garcia-Ramos, K. Jansen [ETM Collaboration], Topological susceptibility from the twisted mass Dirac operator spectrum, JHEP 1402 (2014) 119, arXiv:1312.5161 [hep-lat].
1. K. Cichy, E. Garcia-Ramos, K. Jansen [ETM Collaboration], Chiral condensate from the twisted mass Dirac operator spectrum, JHEP 1310 (2013) 175, arXiv:1303.1954 [hep-lat].
For many physical quantities to be computed in Lattice QCD,
renormalization is an essential ingredient. Therefore, it is extremely important to have, ideally,
several different methods available for the determination of renormalization constants (RCs)
that are moreover non-perturbative.
Our main aim in this thread of research is to show that the extraction of RCs from the behaviour of
correlation functions in coordinate space (the X-space method) is
feasible and, besides providing the RCs in a gauge invariant manner,
has certain important advantages.
In addition to computing RCs for ETMC ensembles, we have also performed a benchmark quenched study of combining the X-space method with a step scaling approach, to investigate the running of RCs from scales of order 20 GeV down to 1.5 GeV.
Recently, we also extracted the running of the strong coupling from the X-space method and determined the Lambda-parameter of 3-flavor QCD.
A related study concerned the scale dependence of the quark mass anomalous dimension. The dependence of the mode number on the threshold value M, mentioned above in Chiral and topological properties of QCD, for moderate and large values of M (of order 1 GeV) is governed by the mass anomalous dimension. Hence, the spectral projector method can give access also to this quantity.
Selected publications:
4. S. Cali, K. Cichy, P. Korcyl, J. Simeth, Running coupling constant from position-space current-current correlation functions in three-flavor lattice QCD, Phys. Rev. Lett. 125 (2020) no.24, 242002, arXiv:2003.05781 [hep-lat].
3. K. Cichy, K. Jansen, P. Korcyl [ETM Collaboration], Non-perturbative running of renormalization constants from correlators in coordinate space using step scaling, Nucl.Phys. B913 (2016) 278, arXiv:1608.02481 [hep-lat].
2. K. Cichy, Quark mass anomalous dimension and Lambda_MSbar from the twisted mass Dirac operator spectrum, JHEP 1408 (2014) 127, arXiv:1311.3572 [hep-lat].
1. K. Cichy, K. Jansen, P. Korcyl [ETM Collaboration], Non-perturbative renormalization in coordinate space for Nf=2Nf=2 maximally twisted mass fermions with tree-level Symanzik improved gauge action, Nucl.Phys. B865 (2012) 268, arXiv:1207.0628 [hep-lat].
One of the classical aims of Lattice QCD is to compute the spectra of hadrons. In our work, we used twisted mass fermions to compute spectra of charm-containing mesons - D mesons, Ds mesons and charmonia. Another study concerned investigation of the possibility of existence of four-quark bound states, i.e. tetraquarks. Four-valence-quark systems are very difficult to investigate using lattice methods. However, if two quarks are very heavy (e.g. bottom quarks), one can treat them as static. One can then much more easily compute such systems, in particular the potential between two static-light mesons. Such potential can then be fitted into some phenomenologically inspired fitting ansatz and plugged into a Schrödinger equation to check whether binding is observed. If yes, the considered system is a tetraquark candidate.
Selected publications:
3. K. Cichy, M.Kalinowski, M. Wagner [ETM Collaboration], Continuum limit of the D meson, Ds meson and charmonium spectrum from Nf=2+1+1 twisted mass lattice QCD, Phys. Rev. D 94 (2016) 094503, arXiv:1603.06467 [hep-lat].
2. P. Bicudo, K. Cichy, A. Peters, M.Wagner, BB interactions with static bottom quarks from Lattice QCD, Phys. Rev. D 93 (2016) 034501, arXiv:1510.03441 [hep-lat].
1. P. Bicudo, K. Cichy, A. Peters, B. Wagenbach, M.Wagner, Evidence for the existence of (u d bbar bbar) and the non-existence of ( s s bbar bbar ) and ( c c bbar bbar ) tetraquarks from lattice QCD, Phys. Rev. D 92 (2015) 014507, arXiv:1505.00613 [hep-lat].
The chirally symmetric overlap fermions provide a theoretically very attractive regularization of quarks. However, they are computationally very expensive, around two orders of magnitude more than e.g. Wilson-type fermions. A possibility that we investigated in this work (my Ph.D. thesis work) is to use overlap quarks only in the valence sector, using gauge field configurations generated with a cheaper discretization. Such setup leads to unitarity violations, which, however, disappear in the continuum limit. We have shown that certain conditions have to be met in order that the continuum limit is correct.
Another thread was to investigate cut-off effects at tree-level of perturbation theory. For this we used different fermion discretizations - twisted mass, overlap and Creutz fermions. We also analyzed situations that can be considered analogues of a mixed action setup in the free theory.
Selected publications:
4. K. Cichy, V. Drach, E. Garcia-Ramos, G. Herdoiza, K. Jansen [ETM Collaboration], Overlap valence quarks on a twisted mass sea: a case study for mixed action Lattice QCD, Nucl.Phys. B869 (2013) 131, arXiv:1211.1605 [hep-lat].
3. E.V. Luschevskaya, K. Cichy, Nf=2+1+1Nf=2+1+1 flavours of twisted mass quarks: cut-off effects at tree-level of perturbation theory, Nucl.Phys. B847 (2011) 17, arXiv:1012.5047 [hep-lat].
2. K. Cichy, G. Herdoiza, K. Jansen [ETM Collaboration], Continuum Limit of Overlap Valence Quarks on a Twisted Mass Sea, Nucl.Phys. B847 (2011) 179, arXiv:1012.4412 [hep-lat].
1. K. Cichy, J. Gonzalez Lopez, K. Jansen, A. Kujawa, A. Shindler [ETM Collaboration], Twisted Mass, Overlap and Creutz Fermions: Cut-off Effects at Tree-level of Perturbation Theory, Nucl.Phys. B800 (2008) 94, arXiv:0803.3637 [hep-lat].
I have also done some research in condensed matter theory. In my M.Sc. thesis research, I implemented the contractor renormalization group (CORE), invented by Morningstar and Weinstein, to investigate the Heisenberg and Hubbard models.
In another project, we analyzed the ground state of ultracold fermions in the presence of effects of orbital and Zeeman magnetic fields, finding interesting phases: unpolarized superconducting state, partially and fully polarized normal states and phase separated regions, partially or fully polarized. The system, in the presence of orbital synthetic magnetic field effects, shows non-monotonous changes of the phase boundaries when electron concentration is varied. We observed reentrant phenomena and also density dependent oscillations of different areas of the phase diagram.
Selected publications:
2. A. Cichy, K. Cichy, T.P. Polak, Competition between Abelian and Zeeman magnetic field effects in a two dimensional ultracold gas of fermions, Annals of Physics 354 (2015) 89.
1. K. Cichy, P. Tomczak, Application of Contractor Renormalization Group to the Heisenberg Zig-Zag and the Hubbard Chain, Acta Phys. Polon. A111 (2007) 773, arXiv:0805.2171 [cond-mat.str-el].