Lattice 2003
The XXI International Symposium on Lattice Field Theory
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
Tsukuba, Ibaraki, Japan
July 18, 2003
Lattice 2003
The XXI International Symposium on Lattice Field Theory
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 g8.
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 g8 coefficient.
Four loops computations in 3d SU(3) (plus Higgs)
Tsukuba, Ibaraki, Japan
July 18, 2003
Lattice 2003
The XXI International Symposium on Lattice Field Theory
Outline
»
»
»
»
Sketch of Physical Framework
The Algorithm & The Code
Results in Pure Gauge Sector
Perspective in Complete Theory
Four loops computations in 3d SU(3) (plus Higgs)
Tsukuba, Ibaraki, Japan
July 18, 2003
Lattice 2003
The XXI International Symposium on Lattice Field Theory
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 Tc ), 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
Four loops computations in 3d SU(3) (plus Higgs)
Tsukuba, Ibaraki, Japan
July 18, 2003
Lattice 2003
The XXI International Symposium on Lattice Field Theory
Sketch of Physical Framework (II)
Dimensional Reduction & Effective Theory
»
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:

1 TrF 2 + Tr[D ,A ] + m2 TrA2 +  (TrA 2 )2
=

3
i
ij
3d
3
0
0
0
2
»
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 nonperturbative 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
Four loops computations in 3d SU(3) (plus Higgs)
Tsukuba, Ibaraki, Japan
July 18, 2003
Lattice 2003
The XXI International Symposium on Lattice Field Theory
The Algorithm (I)
Stochastic Quantization(*)
»
A stochastic dynamical system on the field configuration space dictated
by the general Langevin equation
x,t
S[]
———  ——— + (x,t)
x,t
t
such that averages on the noise h converge to averages on the Gibbs
measure:
 [
t 
 

 t 
1   [ [exp(S[
Z
(*) G.Parisi and Wu Yongshi, Sci. Sinica 24 (1981) 35
Four loops computations in 3d SU(3) (plus Higgs)
Tsukuba, Ibaraki, Japan
July 18, 2003
Lattice 2003
The XXI International Symposium on Lattice Field Theory
The Algorithm (II)
Numerical Stochastic Perturbation Theory(*)
»
In the Stochastic Quantization approach, perturbation theory is
performed through a formal substitution of the expansion
U(x,t) 

k
gkU(k)(x,t)
in the Langevin equation

i U
t U=[i S[U] 
obtaining a system of equation that can be solved numerically via
discretization of the stochastic time t = nt.
(*) F. Di Renzo, G. Marchesini and E. Onofri, P. Marenzoni Nucl. Phys. B426 (1994) 675
Four loops computations in 3d SU(3) (plus Higgs)
Tsukuba, Ibaraki, Japan
July 18, 2003
Lattice 2003
The XXI International Symposium on Lattice Field Theory
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.
Place for a
C++class:
complex,
matrix,
serie or
what you
want
»
Such environment is quite
general: we arrange a lattice
structure of classes and methods
which allowes, in principle, to do
both perturbative and nonperturbative simulations.
link
site
Four loops computations in 3d SU(3) (plus Higgs)
Tsukuba, Ibaraki, Japan
July 18, 2003
Lattice 2003
The XXI International Symposium on Lattice Field Theory
The Code (II)
»
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.
Rims
Bulk
»
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.
Four loops computations in 3d SU(3) (plus Higgs)
Tsukuba, Ibaraki, Japan
July 18, 2003
Lattice 2003
The XXI International Symposium on Lattice Field Theory
Extracting data I
Langevin Dynamic
0
-10
4
g
2
g
Langevin Dynamic
0
-10
g
1
-10
6
4
g
2
g
Langevin Dynamic
0
-10
g
1
8
g6
-10
4
g
2
g
2
-10
0
2
4
6
8
10
nt - Langevin steps (stochastic evolution)
12
14
x 10
4
8
g
1
6
g
-10
2
-10
0
2
4
6
8
10
nt - Langevin steps (stochastic time evolution)
12
14
x 10
4
g
8
2
-10
0
2
4
6
8
10
nt - Langevin steps (stochastic time evolution)
12
14
x 10
4
Four loops computations in 3d SU(3) (plus Higgs)
Tsukuba, Ibaraki, Japan
July 18, 2003
Lattice 2003
The XXI International Symposium on Lattice Field Theory
Extracting data II
-2.665
-1.92
-1.925
-2.67
g2 coefficient
g4 coefficient
-1.93
-2.675
-1.935
-2.68
-1.94
-2.685
-2.69
-1.945
0
5
10
t
15
20
25
-1.95
0
5
10
t
15
20
25
-32.6
-6.65
-32.7
-6.66
g8 coefficient
-32.8
-6.67
g6 coefficient
-32.9
-33
-6.68
-33.1
-33.2
-6.69
-33.3
-6.7
-33.4
-6.71
0
5
10
t
15
20
25
-33.5
0
5
10
t
15
20
25
Four loops computations in 3d SU(3) (plus Higgs)
Tsukuba, Ibaraki, Japan
July 18, 2003
Lattice 2003
The XXI International Symposium on Lattice Field Theory
Results
1.96
2.67
2
2.665
g
g
1.94
2.66
1.92
2.655
1.9
2.65
1.88
2.645
1.86
2.64
4
6
8
10
12
Lattice Size
14
16
18
1.84
4
6
8
10
12
Lattice Size
14
16
18
8
10
12
Lattice Size
14
16
18
34
6.8
6.7
4
g
6
33
6.6
g
8
32
6.5
31
6.4
30
6.3
29
6.2
28
6.1
27
6
26
5.9
5.8
4
6
8
10
12
Lattice Size
14
16
18
25
4
6
Four loops computations in 3d SU(3) (plus Higgs)
Tsukuba, Ibaraki, Japan
July 18, 2003
Lattice 2003
The XXI International Symposium on Lattice Field Theory
Checking
»
The perturbative expansion of the plaquette expectation value in
arbitrary dimension is known analytically up to b2 ~ g4 order(*):
 N   N 2  2



 

P= N Id  ——————— Id N2N 2 
d


in which



Id = —  —
V
d
represent the finite-volume correction to the leading term.
»
So we can check our code comparing our g2 results at each lattice size
with the analytic results, and comparing the infinite-volume value of the
g4 term with the extrapolation of our measurements.
(*) U. Heller and F. Karsch, Nucl. Phys. B251 [FS13] (1985) 254
Four loops computations in 3d SU(3) (plus Higgs)
Tsukuba, Ibaraki, Japan
July 18, 2003
Lattice 2003
The XXI International Symposium on Lattice Field Theory
Results: checking with order g2
»
2.67
2.665
g2
2.66
Data suggest well
clearly the inversevolume form of the
finite-volume correction
2.655
»
2.65
Both the coefficient
of the volumedependence and the
infinite-volume
extrapolation values are
in good agreement with
the analytic results
2.645
2.64
Data interpolation
Analytical result
2.635
2.63
2.625
4
6
8
10
12
14
16
Lattice size
Pb-1 (V) = 2.667(1) - 2.8(3) V-1
Pb-1 (V) = 2.66667 - 2.66667 V-1
Four loops computations in 3d SU(3) (plus Higgs)
Tsukuba, Ibaraki, Japan
July 18, 2003
Lattice 2003
The XXI International Symposium on Lattice Field Theory
Results: checking with order g4
1.96
»
1.88
Different power laws
for size-dependence are
tried in the interpolation
(as “effective” finitesize correction), leading
to quite the same
infinite-volume
extrapolation
1.86
»
1.94
g4
1.92
1.9
1.84
4
6
8
10
12
Lattice size
14
16
18
The value found is
again in good
agreement with the
analytic result
Pb-2(V=) = 1.(1)
Pb-2(V=) = 1.862
Four loops computations in 3d SU(3) (plus Higgs)
Tsukuba, Ibaraki, Japan
July 18, 2003
Lattice 2003
The XXI International Symposium on Lattice Field Theory
Results: order g6
»
Again we fit different
power-law for the finitesize dependence in order to
extract the asintotic value
6.8
6.7
6.6
g6
6.5
6.3
6.2
6.1
6
5.9
5.8
4
In (*), they estimate
this coefficient rescaling
the analogous 4d value
which is known(**). The
number they use
Pb-3  7.2
is in quite well agreement
with the value we found
»
6.4
6
8
10
12
14
16
18
Lattice size
Pb-3(V=) = 6.7(2)
(*) 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
Four loops computations in 3d SU(3) (plus Higgs)
Tsukuba, Ibaraki, Japan
July 18, 2003
Lattice 2003
The XXI International Symposium on Lattice Field Theory
Results: order g8
»
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
34
33
g
8
32
»
The coefficient of the lnV
divergence must be the same as the
continuum log-divergence in the
cut-off of the dimensional-reduced
theory
31
30
29
28
27
»
So we can do another indirect
check comparing the expected
value(**) (0.9765) with our result
26
25
4
6
8
10
12
14
16
18
Lattice size
»
Pb-4 (V) = .() ln(L3) + 23() +
S
k ck (L
-k)
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, 105008 (2003)
Four loops computations in 3d SU(3) (plus Higgs)
Tsukuba, Ibaraki, Japan
July 18, 2003
Lattice 2003
The XXI International Symposium on Lattice Field Theory
Conclusion & Perspective
»
We computed the plaquette up to g8
»
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
Four loops computations in 3d SU(3) (plus Higgs)
Tsukuba, Ibaraki, Japan
July 18, 2003