four loop computations in 3d su ( 3 ) (plus higgs)

The XXI International Symposium on Lattice Field Theory The XXI International Symposium on Lattice Field Theory Lattice 2003 Lattice 2003

Tsukuba, Ibaraki, Japan Tsukuba, Ibaraki, Japan July 18, 2003July 18, 2003

Four loop computations in Four loop computations in 3d3d SUSU((33) ) (plus Higgs)(plus Higgs)

Francesco Francesco Di Renzo Di Renzo (†)(†)

Andrea Andrea Mantovi Mantovi (†)(†)

Vincenzo Vincenzo Miccio Miccio (†)(†)

York York Schröder Schröder (‡)(‡)

(†) (†) Dipartimento di Fisica, Università di Parma Dipartimento di Fisica, Università di Parma andand INFN, Gruppo Collegato di Parma, Italy INFN, Gruppo Collegato di Parma, Italy(‡)(‡) Center for Theoretical Physics, MIT, Cambridge, MA 02139, USA Center for Theoretical Physics, MIT, Cambridge, MA 02139, USA



Four loops computations in Four loops computations in 3d SU3d SU((33)) (plus Higgs)(plus Higgs)

The Matter of Our ComputationThe Matter of Our Computation

»» We use the methods of We use the methods of

Numerical Stochastic Perturbation Theory Numerical Stochastic Perturbation Theory

to compute the plaquette in a 3d pure gauge SU(3) to compute the plaquette in a 3d pure gauge SU(3) theory, up to gtheory, up to g88..

We do our measurements for different lattice sizes in We do our measurements for different lattice sizes in order to extrapolate the infinite-volume value of each order to extrapolate the infinite-volume value of each coefficient of the serie from its lattice-size coefficient of the serie from its lattice-size dependence.dependence.

In particular we found the logarithmic divergence of In particular we found the logarithmic divergence of the gthe g88 coefficient. coefficient.




OutlineOutline

»»»»»»»» Sketch of Physical FrameworkSketch of Physical Framework

The Algorithm & The CodeThe Algorithm & The Code

Results in Pure Gauge SectorResults in Pure Gauge Sector

Perspective in Complete TheoryPerspective in Complete Theory




Sketch of Physical Framework (I)Sketch of Physical Framework (I)

High Temperature QCDHigh Temperature QCD

»» An observable to look after the confinement phase transition of QCD An observable to look after the confinement phase transition of QCD is the Free Energy Density, or the pressure of the quark-gluon plasmais the Free Energy Density, or the pressure of the quark-gluon plasma (*)(*)..

»» 4d finite temperature lattice simulations 4d finite temperature lattice simulations(**)(**) cover (due to cover (due to computational resource limits) the relatively low-temperature regions computational resource limits) the relatively low-temperature regions (till about 4÷5 times T(till about 4÷5 times Tc c ), whereas ), whereas

»» the analytic perturbative approach the analytic perturbative approach(***)(***) (because of poor convergence (because of poor convergence of the series) cover the extremely high-temperature regions.of the series) cover the extremely high-temperature regions.

(*) A. Papa, Nucl. Phys. B 478 (1996) 335; (*) A. Papa, Nucl. Phys. B 478 (1996) 335; B. Beinlich, F. Karsch, E. Laermann B. Beinlich, F. Karsch, E. Laermann and A. Peikert, Eur. Phys. J. C and A. Peikert, Eur. Phys. J. C 6 (1999) 1336 (1999) 133(**)(**) G. Boyd et al, Nucl. Phys. B 469 (1996) 419; G. Boyd et al, Nucl. Phys. B 469 (1996) 419; F. Karsch et al, Phys. Lett. B 478 (2000) 447F. Karsch et al, Phys. Lett. B 478 (2000) 447(***)(***) C. Zhai and B. Kastening, Phys. Rev. D 52 (1995) 7232C. Zhai and B. Kastening, Phys. Rev. D 52 (1995) 7232




»» A way to cover the intermediate regions is to construct, A way to cover the intermediate regions is to construct, viavia dimensional dimensional reduction, an effective theory by integrating out the “hard” modesreduction, an effective theory by integrating out the “hard” modes (*)(*)..The result is a 3d SU(3) + adjoint Higgs model for the “soft” 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 This theory is confining, therefore non-perturbative; yet we can explore it by lattice methods, with less effort than the full 4d theoryit by lattice methods, with less effort than the full 4d theory (**)(**)..

»» We need lattice perturbation theory to connect the non- We need lattice perturbation theory to connect the non-perturbative 3d lattice results with the continuum 3d theory.perturbative 3d lattice results with the continuum 3d theory.

(*)(*) E. Braaten and A. Nieto, Phys. Rev. D 53 (1996) 3421 E. Braaten and A. Nieto, Phys. Rev. D 53 (1996) 3421(**) (**) K. Kajantie et al, Phys. Rev. Lett. 86 (2001) 10K. Kajantie et al, Phys. Rev. Lett. 86 (2001) 10

Sketch of Physical Framework (II)Sketch of Physical Framework (II)

Dimensional Reduction & Effective TheoryDimensional Reduction & Effective Theory

= F + Tr Tr[ ] + T r (T r )D ,A m A + A 123 d ij i 0 3 30 0

222 2 2




»» A stochastic dynamical system on the field configuration space dictated A stochastic dynamical system on the field configuration space dictated by the general Langevin equationby the general Langevin equation

such that averages on the noise such that averages on the noise converge to averages on the Gibbs converge to averages on the Gibbs measure:measure:

The Algorithm (I)The Algorithm (I)

——— ——— + ( )x ,t x ,t

x ,t t S [ ]

[ [ ex p ( [ S [ t 1Z

t

Stochastic QuantizationStochastic Quantization(*)(*)

(*) (*) G.Parisi and Wu Yongshi, Sci. Sinica 24 (1981) 35G.Parisi and Wu Yongshi, Sci. Sinica 24 (1981) 35




»» In the Stochastic Quantization approach, perturbation theory is In the Stochastic Quantization approach, perturbation theory is performed through a formal substitution of the expansionperformed through a formal substitution of the expansion

in the Langevin equationin the Langevin equation

obtaining a system of equation that can be solved numerically via obtaining a system of equation that can be solved numerically via discretization of the stochastic time t = ndiscretization of the stochastic time t = n..

The Algorithm (II)The Algorithm (II)

t U = i S[ [U i U]

Numerical Stochastic Perturbation TheoryNumerical Stochastic Perturbation Theory (*)(*)

g ( )U x,tU x ,t( ) k

k ( )k

(*) F. Di Renzo, G. Marchesini and E. Onofri, P. (*) F. Di Renzo, G. Marchesini and E. Onofri, P. Marenzoni Nucl. Phys. B426 (1994) 675Marenzoni Nucl. Phys. B426 (1994) 675




»» Such environment is quite Such environment is quite general: we arrange a lattice general: we arrange a lattice structure of classes and methods structure of classes and methods which allowes, in principle, to do which allowes, in principle, to do both perturbative and non-both perturbative and non-perturbative simulations.perturbative simulations.

The Code (I)The Code (I)

»» We set up a set of C++ We set up a set of C++ classes in order to place our classes in order to place our simulation on a PC-cluster, simulation on a PC-cluster, using the MPI language to using the MPI language to handle communications between handle communications between nodes.nodes.

linklink

sitesite

Place for a Place for a C++class: C++class: complexcomplex, , matrixmatrix, , serieserie or or

what you what you wantwant




The Code (II)The Code (II)

BulkBulk

RimsRims

»» The phisical allocation of memory is The phisical allocation of memory is especially suited for not to overload the especially suited for not to overload the communications time respect to CPU communications time respect to CPU time: to carry (between nodes) more time: to carry (between nodes) more data fewer times for each lattice sweep.data fewer times for each lattice sweep.

»» We also use some expedients of We also use some expedients of “template programming” to both to be “template programming” to both to be able to quickly change the simulation able to quickly change the simulation parameters for a new run (lattice size, parameters for a new run (lattice size, order of expansion, and so on...), and to order of expansion, and so on...), and to optimize the code in such a way that a optimize the code in such a way that a heavier compilation can make the code heavier compilation can make the code faster.faster.




Extracting data IExtracting data I

0 2 4 6 8 10 12 14

x 104

-102

-101

-100

Langevin Dynamic

n - Langevin steps (stochastic evolution)

g2

g4

g6

g8

0 2 4 6 8 10 12 14

x 104

-102

-101

-100

Langevin Dynamic

n - Langevin steps (stochastic time evolution)

g2

g4

g6

g8

0 2 4 6 8 10 12 14

x 104

-102

-101

-100

Langevin Dynamic

n - Langevin steps (stochastic time evolution)

g2

g4

g6

g8




Extracting data IIExtracting data II

0 5 10 15 20 25-2.69

-2.685

-2.68

-2.675

-2.67

-2.665

g2 coefficient

0 5 10 15 20 25-1.95

-1.945

-1.94

-1.935

-1.93

-1.925

-1.92

g4 coefficient

0 5 10 15 20 25-6.71

-6.7

-6.69

-6.68

-6.67

-6.66

-6.65

g6 coefficient

0 5 10 15 20 25-33.5

-33.4

-33.3

-33.2

-33.1

-33

-32.9

-32.8

-32.7

-32.6

g8 coefficient




ResultsResults

4 6 8 10 12 14 16 182.64

2.645

2.65

2.655

2.66

2.665

2.67

g2

Lattice Size4 6 8 10 12 14 16 18

1.84

1.86

1.88

1.9

1.92

1.94

1.96

g4

Lattice Size

4 6 8 10 12 14 16 185.8

5.9

6

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

g6

Lattice Size4 6 8 10 12 14 16 18

25

26

27

28

29

30

31

32

33

34

g8

Lattice Size




in which in which

represent the finite-volume correction to the leading term.represent the finite-volume correction to the leading term.

CheckingChecking

2

2 2 2 dd d d N I N N I N N P = ———————

ddI = — —d V

»» The perturbative expansion of the plaquette expectation value in The perturbative expansion of the plaquette expectation value in arbitrary dimension is known analytically up to arbitrary dimension is known analytically up to 22 ~ g ~ g44 order order(*)(*)::

»» So we can check our code comparing our g So we can check our code comparing our g22 results at each lattice size results at each lattice size with the analytic results, and comparing the infinite-volume value of the with the analytic results, and comparing the infinite-volume value of the gg44 term with the extrapolation of our measurements. term with the extrapolation of our measurements.

(*) U. Heller and F. Karsch, Nucl. Phys. B251 [FS13] (1985) 254(*) U. Heller and F. Karsch, Nucl. Phys. B251 [FS13] (1985) 254




Results: checking with order gResults: checking with order g22

»» Data suggest well Data suggest well clearly the inverse-clearly the inverse-volume form of the volume form of the finite-volume correctionfinite-volume correction

»» Both the coefficient Both the coefficient of the volume-of the volume-dependence and the dependence and the infinite-volume infinite-volume extrapolation values are extrapolation values are in good agreement with in good agreement with the analytic resultsthe analytic results

PP--

11VVVV

4 6 8 10 12 14 162.625

2.63

2.635

2.64

2.645

2.65

2.655

2.66

2.665

2.67

g2

Data interpolation Analytical result

Lattice size

PP--

11VVVV




Results: checking with order gResults: checking with order g44

PP-2-2V=V=

»» Different power laws Different power laws for size-dependence are for size-dependence are tried in the interpolation tried in the interpolation (as “effective” finite-(as “effective” finite-size correction), leading size correction), leading to quite the same to quite the same infinite-volume infinite-volume extrapolationextrapolation

»» The value found is The value found is again in good again in good agreement with the agreement with the analytic resultanalytic result

4 6 8 10 12 14 16 181.84

1.86

1.88

1.9

1.92

1.94

1.96

g4

Lattice size

PP-2-2V=V=




Results: order gResults: order g66

PP-3-3V=V=

4 6 8 10 12 14 16 185.8

5.9

6

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

g6

Lattice size

»» Again we fit different Again we fit different power-law for the finite-power-law for the finite-size dependence in order to size dependence in order to extract the asintotic value extract the asintotic value

»» In In (*)(*), they estimate , they estimate this coefficient rescaling this coefficient rescaling the analogous 4d value the analogous 4d value which is knownwhich is known(**)(**). The . The number they usenumber they use

PP--33is in quite well agreement is in quite well agreement with the value we foundwith the value we found

(*) F. Karsch, M. Lütgemeier, A. Patkòs and J. Rank, Phys. Lett. B390, 275 (1997) (*) 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(**) B. Allés, M. Campostrini, A. Feo and H. Panagopoulos, Phys. Lett. B324 (1994) 433




Results: order gResults: order g88

4 6 8 10 12 14 16 1825

26

27

28

29

30

31

32

33

34

g8

Lattice size

»» For this coefficient we expect a For this coefficient we expect a logarithmic divergence in the logarithmic divergence in the lattice volumelattice volume(*). (*). Indeed data fit Indeed data fit better if we add a log-term in the better if we add a log-term in the interpolation-lawinterpolation-law

»» The coefficient of the The coefficient of the lnlnV V divergence must be the same as the divergence must be the same as the continuum log-divergence in the continuum log-divergence in the cut-off of the dimensional-reduced cut-off of the dimensional-reduced theorytheory

»» So we can do another indirect So we can do another indirect check comparing the expected check comparing the expected valuevalue(**)(**) ( (0.97650.9765) with our result) with our result

»» If we use this analytical value in If we use this analytical value in our fit, the estimate for the constat our fit, the estimate for the constat coefficient improves, leading to the coefficient improves, leading to the result value of result value of 25(2)25(2)

(*) F. Karsch, M. Lütgemeier, A. Patkòs and J. Rank, Phys. Lett. B390, 275 (1997)(*) 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, 105008 (2003)K. Kajantie, M. Laine, K. Rummukainen, Y. Schroder, Phys. Rev. D67, 105008 (2003)

PP-4-4VVln(ln(LLk k cck k LLkk




Conclusion & PerspectiveConclusion & Perspective

»» We computed the plaquette up to gWe computed the plaquette up to g88

»» Now the code is ready to play Numerical Stochastic Now the code is ready to play Numerical Stochastic Perturbation Theory also with the Higgs field that the Perturbation Theory also with the Higgs field that the dimensional reduced theory couples to the gauge fielddimensional reduced theory couples to the gauge field

»» We are doing some prelimiary simulation and the We are doing some prelimiary simulation and the signal is there for measuring the quadratic and the signal is there for measuring the quadratic and the quartic condensate of the scalar fieldquartic condensate of the scalar field

four loop computations in 3d su ( 3 ) (plus higgs)

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