Lattice theory

A lattice scale theory is a gauge theory that is defined on a discrete spacetime . Lattice-scale theories are among the few possibilities to make non -perturbative calculations in quantum field theories .

The method became particularly important in the context of quantum chromodynamics  (QCD). Because the lattice regularization is a non-perturbation-theoretical regularization , one can also perform calculations for low energies in lattice-scale theories, which are not accessible for perturbation theory. This can u. a. the masses of hadrons , d. H. Investigate bound quark states, thermodynamic quantities or important topological excitations ( monopoles , instantons and solitons ).

In addition to QCD, other gauge theories and spin systems on the grid are also examined, in particular those with a non-Abelian gauge group (general Yang-Mills theories analogous to QCD).

idea

The basic idea is to regularize the theory by introducing a minimum distance in space and time (see Minkowski space ) so that divergences no longer occur at high energies. This minimum distance corresponds to a cut-off energy in the momentum space . A steady reduction in the minimum grid spacing corresponds to the transition to the original theory in continuous space by removing the highest energies in momentum space.

In order to enable simulations of lattice scale theories on computers, a wick rotation is usually also performed, which leads to Euclidean space . Then there is a relationship to statistical physics , and the powerful tool of the Monte Carlo simulation can be used.

formulation

The lattice theory of quantum chromodynamics introduced by Kenneth Wilson in 1974 discretizes the effect of QCD on a four-dimensional cubic lattice with a lattice spacing . An important principle in Wilson's construction of this lattice theory is that its effect is explicitly gauge invariant even with finite lattice spacing. Furthermore, the Wilson effect is chosen in such a way that the continuum effect results in the limit value . Usually the formulation for the calibration sector is considered separately from that for fermions, since the transfer of the chiral symmetry of the fermion fields to the lattice is a problem of its own. ${\displaystyle \Lambda (a)}$ ${\displaystyle a}$ ${\displaystyle a\rightarrow 0}$

Pure gauge theory

For the discretization of the Yang-Mills effect that the dynamics of the bosons describes one defines the link variables , which neighboring grid points connect (English link ): ${\displaystyle U_{\mu }}$

${\displaystyle U_{\mu }=\exp \left(iaA_{\mu }\left(n+{\frac {\hat {\mu }}{2}}\right)\right).}$

Here are

• the gauge fields elements of the adjoint representation of the algebra of the gauge group of the QCD, SU (3) ${\displaystyle A_{\mu }}$
• the link variable elements of the calibration group , i.e. SU (3) matrices, which each connect two neighboring grid points; In terms of differential geometry , they can be understood as finite parallel transport. ${\displaystyle U_{\mu }}$

The calibration field portion of the effect can now be represented as the trace over closed loops of link variables. Every track through such Wilson loops is gauge invariant. A simple calibration effect can therefore be written as: ${\displaystyle S}$

${\displaystyle S_{G}[U]={\frac {2}{g^{2}}}\sum _{n\in \Lambda }\sum _{\mu <\nu }\operatorname {Re} \;\operatorname {tr} \left(1-U_{\mu \nu }^{(4)}(n)\right).}$

The (the badge variables) are defined as the sizes belonging to the smallest closed rectangular loops: ${\displaystyle U_{\mu \nu }^{(4)}}$

${\displaystyle U_{\mu \nu }^{(4)}(n)=U_{+\mu }(n)U_{+\nu }(n+{\hat {\mu }})U_{-\mu }(n+{\hat {\mu }}+{\hat {\nu }})U_{-\nu }(n+{\hat {\nu }})\,,}$

analogous to the geometry of a square that z. B. is indicated by the four numbers 1, 2, 3 and 4 with a positive direction of rotation.

The inverse calibration coupling is often used instead of the coupling constant ${\displaystyle g}$ ${\displaystyle \beta ={\frac {6}{g^{2}}}\,.}$

Since the form of the action is only determined by the continuum limit, the calibration action given above, the Wilson or Plakett action, is not unambiguous, but can be modified by terms that vanish in the continuum limit. This observation is used to construct improved effects with a faster continuum approximation. ${\displaystyle a\rightarrow 0}$

Fermions on the grid

While the link variables each connect two grid points, the fermion fields are defined on these points. This allows calibration-invariant combinations of the shape to be formed, which can be used as building blocks of the discretized covariant derivation . ${\displaystyle \psi (x)U_{\mu }(x)\psi (x+{\hat {\mu }})}$ ${\displaystyle x/a\in \Lambda }$

If one now replaces the derivatives in the Dirac effect with finite differences, one obtains a naive discretization of the theory, which describes not only a single fermion, but sixteen ( including for the number of dimensions). This phenomenon is known as the Doppler problem and is related to the realization of the chiral symmetry on the lattice. Indeed, the Nielsen-Ninomiya theorem says that on the lattice no Dirac operator with the correct continuum limit can be Doppler-free, local , translation-invariant and chiral-symmetric at the same time . ${\displaystyle 2^{d}}$ ${\displaystyle d=4}$

In order to consider the Doppler problem in a physically correct manner, various types of fermion discretization are used, which are described below.

Wilson fermions

To eliminate the Doppler, further terms can be included in the effect, which give the unphysical fermion modes additional mass. When the continuum limit is formed, the resulting Doppler modes decouple from the theory because their mass diverges. This is the approach of the Wilson fermion action: ${\displaystyle \sim 1/a}$

${\displaystyle S_{F}[\psi ,{\bar {\psi }},U]=a^{4}\sum _{f=1}^{N_{f}}\sum _{n\in \Lambda }\left({\bar {\psi }}^{f}(n)\sum _{\mu =\pm 1}^{\pm 4}(1-\gamma _{\mu }){\frac {U_{\mu }(n)\psi ^{f}(n+{\hat {\mu }})}{2a}}+\left(m^{f}+{\frac {4}{a}}r\right){\bar {\psi }}^{f}(n)\psi ^{f}(n)\right)\,,}$

in which

• f denotes   the flavor degree of freedom of the fermions
• r can be freely chosen as a prefactor before the newly introduced Wilson term.

For r = 0 the original naively discretized fermions with Dopplers are obtained, while for the usual choice r = 1 the Dopplers are eliminated as described above.

At finite  a , however, the chiral symmetry is explicitly broken by the Wilson term and is only restored in the continuum limit. A practical consequence is that, unlike for other effects, the grid artifacts occur in a linear order of the grid spacing.

To solve this problem, improved effects are used almost exclusively in numerical simulations . The most widespread are the clover fermions , for which a further term is added to the effect, the free parameter of which can be selected so that the leading lattice artifacts are eliminated. In addition, Wilson fermions with a modified mass term under the name twisted mass fermions are also used.

Staggered fermions

In addition to the Wilson fermions, the staggered fermions in particular are used. These use a spin diagonalization to reduce the number of Dopplers by a factor of 4.

In order to describe a theory with exactly one species of fermion, a theoretically controversial process known as rooting must be used.

Chiral fermions

In continuum theory, the Dirac operator  D of a chiral symmetric theory fulfills the relationship where γ ₅ , like  D , should be assumed to be known from Dirac theory . The Wilson term explicitly breaks this symmetry. However, this can be avoided by a weakened definition of chiral symmetry on the lattice: ${\displaystyle \{D,\gamma _{5}\}=0\,,}$

${\displaystyle \{D,\gamma _{5}\}aR\to a{\frac {1}{2}}(D\gamma _{5}R+R\gamma _{5}D)}$,

where R is a local lattice operator.

The behavior ~ a leads to an effective smoothing of the perturbing term ~ 4 / a of the Wilson functional .

This replacement gives the overlap operator , and the Wilson equation results in the Ginzparg-Wilson equation . In addition to exact solutions, there are also a number of common fermions whose Dirac operator only approximately satisfies the Ginsparg-Wilson equation. The best known are the domain wall fermions , which (in the case of infinite extension of a fifth dimension) correspond to the overlap operator. In practical simulations, however, this dimension always remains finite.

Alternative formulations

The aforementioned discretizations are the most commonly used methods of treating fermions on the lattice. There are also others, such as minimally doubled fermions , which can minimize the Doppler problem by modifying the lattice geometry .

Another variant is to break the translation invariance by introducing an additional dimension, as happens with the domain wall fermions .

Relationship between physical quantities and simulation parameters

A number of parameters can be set in a lattice QCD simulation: the number of lattice points in spatial and temporal direction, the lattice coupling and, if necessary, quark mass parameters and parameters that lead to the theoretical improvement of the continuum behavior. To translate such a "setup" from dimensionless numbers into physical units, i. H. In order to obtain dimensional values ​​such as the lattice spacing (in fm ) or hadron masses (in MeV / c ² ), selected physical objects (such as the mass or decay constant of the pion ) must be fixed to set the scale. All other calculated quantities are then predictions of the grid QCD for the given parameters. ${\displaystyle \beta }$

The required computer power increases with decreasing mass, so that reaching physical quark masses without further extrapolation still means an enormous effort and has not been achieved with all fermion discretizations. In addition, it is important to keep the systematic effects under control, which are caused by the extrapolation to vanishing grid spacing and infinite volume.

The asymptotic freedom of the QCD with its fixed point in the flow of the coupling constant ensures that the continuum limit for vanishing coupling ( or ) is reached. ${\displaystyle a\rightarrow 0}$ ${\displaystyle g\rightarrow 0}$ ${\displaystyle \beta \rightarrow \infty }$

development

The actual hour of birth of the lattice QCD is today considered to be the publication of the work of the physicist Kenneth Wilson in 1974, which very soon became a core area of ​​research at that time and triggered a rapid development of the method.

The lattice QCD method is analogous to special spin models that were set up in 1971 by Franz Wegner in a solid-state theoretical context . As in QCD, these lattice spin models are characterized by a local gauge invariance and a term analogous to the gauge field energy.

Although quantum chromodynamics is one of the main areas of application of the lattice theory, even in high-energy physics there are studies with lattice methods that go beyond QCD, e.g. B. to the Higgs mechanism .

For the QCD itself, u. a. a larger network centered at the University of Regensburg was formed, which bundles the activities of many leading research centers in Germany (and northern Italy) and can use a special high-performance computer, QPACE , to fall back on existing experience (see QCDOC in Wikipedia) and a future-oriented concept.

Selected results

Representation of a meson in a lattice QCD simulation. (From M. Cardoso et al.)

One advantage of simulations of lattice scale theories is that primarily gauge-invariant quantities are accessible. This led to a calculation of all meson - and baryon - ground states that contain up , down or strange quarks , so that the masses of nucleons, for example, were calculated with an accuracy of 1 to 2 percent (Budapest-Marseille-Wuppertal collaboration 2008). Three hadron masses (including Pion, Kaon) were used to determine the isospin-averaged masses of (u, d, s) quarks and the total energy scale. The error-controlled calculation was the result of decades of global development of the methods of QCD lattice theory, both from the theoretical side and from the side of algorithms and supercomputers. A more detailed analysis of how the nucleon masses are divided into individual QCD contributions was successful in 2018: only around 9 percent are due to quark contributions, one third to the kinetic energy of the trapped quarks, the rest to gluon contributions. A typical result (see the graphic) shows that not only the particles, quarks and antiquarks are important in a meson, but also the “flow tubes” of the gluon fields.

Such calculations can also be used to study collective effects that could be related to the confinement phenomenon. This can e.g. B. topological stimuli such as instantons, monopoles and solitons, or percolation effects of the center of the calibration group .

It is also possible to calculate the QCD at high temperatures to study the transition into the quark-gluon plasma , which was measured in experiments at particle accelerators above about 1.2 × 10 ¹²  Kelvin .

literature

• István Montvay and Gernot Münster : Quantum Fields on a Lattice Cambridge University Press, Cambridge 1994, ISBN 0-521-40432-0 .
• Heinz J. Rothe: Lattice Gauge Theories: An Introduction World Scientific Lecture Notes in Physics, 4th edition, 2012, ISBN 978-981-4365-85-7 .
• Thomas DeGrand and Carleton DeTar: Lattice Methods for Quantum Chromodynamics , World Scientific Publishing, Singapore 2006, ISBN 981-256-727-5 .
• Christof Gattringer and Christian B. Lang: Quantum chromodynamics on the lattice , Lect. Notes Phys. 788, 2010, ISBN 978-3-642-01849-7 .

Web links

• Gernot Münster, Lattice Quantum Field Theory, Scholarpedia

Individual evidence

1. ↑ The definition of the badge variables is analogous to that of the classical Stokes theorem .
2. ↑ J. Kogut, L. Susskind: Hamiltonian Formulation of Wilson's Lattice Gauge Theories , Phys. Rev. D11 (1975) 395.
3. ↑ cf. z. BM Creutz: "Why rooting fails", PoS (Lat2007) 007. arxiv : 0708.1295 .
4. ^ DB Kaplan, Phys. Lett. B288 (1992) 342.
5. ↑ Y. Shamir, Chiral fermions from lattice boundaries , Nucl. Phys. B406 (1993) 90.
6. ↑ L. Karsten: Lattice Fermions in Euclidian Space-Time , Phys. Lett. B104 (1981) 315.
7. ^ F. Wilczek: On Lattice Fermions , Phys. Rev. Lett. 59 (1987) 2397.
8. ↑ M. Creutz: Four-dimensional graphene and chiral fermions , JHEP 04 (2008) 017. arxiv : 0712.1201 .
9. ↑ Kenneth Wilson , Confinement of quarks . In: Physical Review D . Volume 10, 1974, pp. 2445-2459.
10. ^ ^{A b} John Kogut : Introduction to lattice gauge theory and spin systems , Reviews of Modern Physics, Vol. 51 (1979), pp. 659-713.
11. ↑ John Kogut: The lattice gauge theory approach to quantum chromodynamics , Rev. Mod. Phys., Vol. 55 (1983), pp. 775-836.
12. ↑ Kenneth Wilson : The Origins of lattice gauge theory , Nucl. Phys. Proc. Suppl. 140 (2005), pp. 3-19. arxiv : hep-lat / 0412043
13. ↑ It is interesting that Kenneth Wilson's work on the renormalization group , for which he received the Nobel Prize, was also made in a solid-state theoretical context (see the article on the person given).
14. ↑ F. Wegner, Duality in Generalized Ising Models and Phase Transitions without Local Order Parameter , J. Math. Phys. 12 (1971) 2259-2272. Reprinted in Claudio Rebbi (ed.), Lattice Gauge Theories and Monte Carlo Simulations , World Scientific, Singapore (1983), p. 60-73.
15. ^ E. Fradkin and SH Shenker: Phase diagrams of lattice gauge theories with Higgs fields , Phys. Rev. D 19, 3682-3697 (1979).
16. ↑ M. Cardoso et al., Lattice QCD computation of the color fields for the static hybrid quark-gluon-antiquark system, and microscopic study of the Casimir scaling , Phys. Rev. D 81, 034504 (2010), (abstract) , arxiv : 0912.3181
17. ↑ S. Dürr et al., Ab initio determination of light hadron masses , Science, Volume 322, 2008, pp. 1224-1227, Arxiv .
18. ^ Frank Wilczek, Mass by numbers, Nature, Volume 456, 2008, pp. 449-450, online
19. ^ André Walker-Loud: Viewpoint: Dissecting the Mass of the Proton , Physics, APS, November 19, 2018
20. ↑ Y.-B. Yang, J. Liang, Y.-J. Bi, Y. Chen, T. Draper, K.-F. Liu, Z. Liu, Proton mass decomposition from the QCD energy momentum tensor, Phys. Rev. Lett., Volume 121, 2018, p. 212001, Arxiv
21. ↑ C. Gattringer, M. Göckeler, PEL Rakow, S. Schaefer and A. Schäfer, A comprehensive picture of topological excitations in finite-temperature lattice QCD , Nucl. Phys. B 617, 101 and B 618, 205 (2001).
22. ↑ JM Carmonaa, M. D'Eliab, L. Del Debbioc, d, A. Di Giacomoc, B. Lucinie, G. Paffuti Color confinement and dual superconductivity in full QCD , Phys. Rev. D 66, 011503 (2002)
23. ↑ J. Danzer, C. Gattringer: Center clusters and their percolation properties in Lattice QCD , arxiv : 1010.5073v1 .
24. ↑ On the lattice calculations: see F. Karsch: The Phase Transition to the Quark-Gluon Plasma: Recent Results from Lattice-QCD , 1995, arxiv : hep-lat / 9503010v1
25. ↑ About the experiments: BNL press release about the generation of quark-gluon plasma, 2005