The XXI International Symposium on Lattice Field Theory Lattice 2003 Tsukuba, Ibaraki, Japan July 18, 2003 Four loop computations in 3d SU(3) (plus Higgs) Francesco Di Renzo (†) Andrea Mantovi (†) Vincenzo Miccio (†) York Schröder (‡) (†) Dipartimento di Fisica, Università di Parma and INFN, Gruppo Collegato di Parma, Italy (‡) Center for Theoretical Physics, MIT, Cambridge, MA 02139, USA

The XXI International Symposium on Lattice Field Theory Lattice 2003 Tsukuba, Ibaraki, Japan July 18, 2003 Four loops computations in 3d SU(3) (plus Higgs) The Matter of Our Computation » We use the methods of Numerical Stochastic Perturbation Theory to compute the plaquette in a 3d pure gauge SU(3) theory, up to g 8. We do our measurements for different lattice sizes in order to extrapolate the infinite-volume value of each coefficient of the serie from its lattice-size dependence. In particular we found the logarithmic divergence of the g 8 coefficient.

The XXI International Symposium on Lattice Field Theory Lattice 2003 Tsukuba, Ibaraki, Japan July 18, 2003 Four loops computations in 3d SU(3) (plus Higgs) Outline » » » » Sketch of Physical Framework The Algorithm & The Code Results in Pure Gauge Sector Perspective in Complete Theory

The XXI International Symposium on Lattice Field Theory Lattice 2003 Tsukuba, Ibaraki, Japan July 18, 2003 Four loops computations in 3d SU(3) (plus Higgs) Sketch of Physical Framework (I) High Temperature QCD » An observable to look after the confinement phase transition of QCD is the Free Energy Density, or the pressure of the quark-gluon plasma (*). » 4d finite temperature lattice simulations (**) cover (due to computational resource limits) the relatively low-temperature regions (till about 4÷5 times T c ), whereas » the analytic perturbative approach (***) (because of poor convergence of the series) cover the extremely high-temperature regions. (*) A. Papa, Nucl. Phys. B 478 (1996) 335; B. Beinlich, F. Karsch, E. Laermann and A. Peikert, Eur. Phys. J. C 6 (1999) 133 (**) G. Boyd et al, Nucl. Phys. B 469 (1996) 419; F. Karsch et al, Phys. Lett. B 478 (2000) 447 (***) C. Zhai and B. Kastening, Phys. Rev. D 52 (1995) 7232

The XXI International Symposium on Lattice Field Theory Lattice 2003 Tsukuba, Ibaraki, Japan July 18, 2003 Four loops computations in 3d SU(3) (plus Higgs) » A way to cover the intermediate regions is to construct, via dimensional reduction, an effective theory by integrating out the “hard” modes (*). The result is a 3d SU(3) + adjoint Higgs model for the “soft” modes: » This theory is confining, therefore non-perturbative; yet we can explore it by lattice methods, with less effort than the full 4d theory (**). » We need lattice perturbation theory to connect the non- perturbative 3d lattice results with the continuum 3d theory. (*) E. Braaten and A. Nieto, Phys. Rev. D 53 (1996) 3421 (**) K. Kajantie et al, Phys. Rev. Lett. 86 (2001) 10 Sketch of Physical Framework (II) Dimensional Reduction & Effective Theory

The XXI International Symposium on Lattice Field Theory Lattice 2003 Tsukuba, Ibaraki, Japan July 18, 2003 Four loops computations in 3d SU(3) (plus Higgs) » A stochastic dynamical system on the field configuration space dictated by the general Langevin equation such that averages on the noise  converge to averages on the Gibbs measure: The Algorithm (I) Stochastic Quantization (*) (*) G.Parisi and Wu Yongshi, Sci. Sinica 24 (1981) 35

The XXI International Symposium on Lattice Field Theory Lattice 2003 Tsukuba, Ibaraki, Japan July 18, 2003 Four loops computations in 3d SU(3) (plus Higgs) » In the Stochastic Quantization approach, perturbation theory is performed through a formal substitution of the expansion in the Langevin equation obtaining a system of equation that can be solved numerically via discretization of the stochastic time t = n . The Algorithm (II) Numerical Stochastic Perturbation Theory (*) (*) F. Di Renzo, G. Marchesini and E. Onofri, P. Marenzoni Nucl. Phys. B426 (1994) 675

The XXI International Symposium on Lattice Field Theory Lattice 2003 Tsukuba, Ibaraki, Japan July 18, 2003 Four loops computations in 3d SU(3) (plus Higgs) » Such environment is quite general: we arrange a lattice structure of classes and methods which allowes, in principle, to do both perturbative and non- perturbative simulations. The Code (I) » We set up a set of C++ classes in order to place our simulation on a PC-cluster, using the MPI language to handle communications between nodes. link site Place for a C++class: complex, matrix, serie or what you want

The XXI International Symposium on Lattice Field Theory Lattice 2003 Tsukuba, Ibaraki, Japan July 18, 2003 Four loops computations in 3d SU(3) (plus Higgs) The Code (II) BulkRims » The phisical allocation of memory is especially suited for not to overload the communications time respect to CPU time: to carry (between nodes) more data fewer times for each lattice sweep. » We also use some expedients of “template programming” to both to be able to quickly change the simulation parameters for a new run (lattice size, order of expansion, and so on...), and to optimize the code in such a way that a heavier compilation can make the code faster.

The XXI International Symposium on Lattice Field Theory Lattice 2003 Tsukuba, Ibaraki, Japan July 18, 2003 Four loops computations in 3d SU(3) (plus Higgs) Extracting data I

The XXI International Symposium on Lattice Field Theory Lattice 2003 Tsukuba, Ibaraki, Japan July 18, 2003 Four loops computations in 3d SU(3) (plus Higgs) Extracting data II

The XXI International Symposium on Lattice Field Theory Lattice 2003 Tsukuba, Ibaraki, Japan July 18, 2003 Four loops computations in 3d SU(3) (plus Higgs) Results

The XXI International Symposium on Lattice Field Theory Lattice 2003 Tsukuba, Ibaraki, Japan July 18, 2003 Four loops computations in 3d SU(3) (plus Higgs) in which represent the finite-volume correction to the leading term. Checking » The perturbative expansion of the plaquette expectation value in arbitrary dimension is known analytically up to b 2 ~ g 4 order (*) : » So we can check our code comparing our g 2 results at each lattice size with the analytic results, and comparing the infinite-volume value of the g 4 term with the extrapolation of our measurements. (*) U. Heller and F. Karsch, Nucl. Phys. B251 [FS13] (1985) 254

The XXI International Symposium on Lattice Field Theory Lattice 2003 Tsukuba, Ibaraki, Japan July 18, 2003 Four loops computations in 3d SU(3) (plus Higgs) Results: checking with order g 2 » Data suggest well clearly the inverse- volume form of the finite-volume correction » Both the coefficient of the volume- dependence and the infinite-volume extrapolation values are in good agreement with the analytic results á P ñ  -1 ( V ) = 2.667(1) - 2.8(3) V -1 á P ñ  -1 ( V ) = V - 1

The XXI International Symposium on Lattice Field Theory Lattice 2003 Tsukuba, Ibaraki, Japan July 18, 2003 Four loops computations in 3d SU(3) (plus Higgs) Results: checking with order g 4 á P ñ  -2 ( V= ¥) = » Different power laws for size-dependence are tried in the interpolation (as “effective” finite- size correction), leading to quite the same infinite-volume extrapolation » The value found is again in good agreement with the analytic result á P ñ  -2 ( V= ¥) = 1.95(1)

The XXI International Symposium on Lattice Field Theory Lattice 2003 Tsukuba, Ibaraki, Japan July 18, 2003 Four loops computations in 3d SU(3) (plus Higgs) Results: order g 6 á P ñ  -3 ( V= ¥) = 6.7(2) » Again we fit different power-law for the finite- size dependence in order to extract the asintotic value » In (*), they estimate this coefficient rescaling the analogous 4d value which is known (**). The number they use á P ñ  - 3 » 7.02 á P ñ  - 3 » 7.02 is in quite well agreement with the value we found (*) F. Karsch, M. Lütgemeier, A. Patkòs and J. Rank, Phys. Lett. B390, 275 (1997) (**) B. Allés, M. Campostrini, A. Feo and H. Panagopoulos, Phys. Lett. B324 (1994) 433

The XXI International Symposium on Lattice Field Theory Lattice 2003 Tsukuba, Ibaraki, Japan July 18, 2003 Four loops computations in 3d SU(3) (plus Higgs) Results: order g 8 » For this coefficient we expect a logarithmic divergence in the lattice volume (*). Indeed data fit better if we add a log-term in the interpolation-law » The coefficient of the lnV divergence must be the same as the continuum log-divergence in the cut-off of the dimensional-reduced theory » So we can do another indirect check comparing the expected value (**) (0.9765) with our result » If we use this analytical value in our fit, the estimate for the constat coefficient improves, leading to the result value of 25(2) (*) F. Karsch, M. Lütgemeier, A. Patkòs and J. Rank, Phys. Lett. B390, 275 (1997) (**) K. Kajantie, M. Laine, K. Rummukainen, Y. Schroder, Phys. Rev. D67, (2003) á P ñ  -4 ( V ) = 0.9(4) ln(L 3 ) + 23(5) + S k c k ( L - k )

The XXI International Symposium on Lattice Field Theory Lattice 2003 Tsukuba, Ibaraki, Japan July 18, 2003 Four loops computations in 3d SU(3) (plus Higgs) Conclusion & Perspective » We computed the plaquette up to g 8 » Now the code is ready to play Numerical Stochastic Perturbation Theory also with the Higgs field that the dimensional reduced theory couples to the gauge field » We are doing some prelimiary simulation and the signal is there for measuring the quadratic and the quartic condensate of the scalar field