Robert Kirchner
580 California St., Suite 400
San Francisco, CA, 94104
Topology in SU(2) Yang-Mills theory
1998, Nuclear physics
https://doi.org/10.1016/S0920-5632(97)00817-7visibility
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Abstract
New results on the topology of the SU (2) Yang-Mills theory are presented. At zero temperature we obtain the value of the topological susceptibility by using the recently introduced smeared operators as well as a properly renormalized geometric definition. Both determinations are in agreement. At non-zero temperature we study the behaviour of the topological susceptibility across the confinement transition pointing out some qualitative differences with respect to the analogous result for the SU (3) gauge theory.
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arXiv:hep-lat/9709074v1 20 Sep 1997
1
Topology in SU(2) Yang-Mills theory∗
B. Allésa† , M. D’Eliab , A. Di Giacomob and R. Kirchnerb
a
Dipartimento di Fisica, Sezione Teorica, Università degli Studi di Milano and INFN,
Via Celoria 16, 20133 Milano, Italy.
b
Dipartimento di Fisica, Università di Pisa and INFN,
Piazza Torricelli 2, 56126 Pisa, Italy.
New results on the topology of the SU (2) Yang-Mills theory are presented. At zero temperature we obtain the
value of the topological susceptibility by using the recently introduced smeared operators as well as a properly
renormalized geometric definition. Both determinations are in agreement. At non-zero temperature we study
the behaviour of the topological susceptibility across the confinement transition pointing out some qualitative
differences with respect to the analogous result for the SU (3) gauge theory.
1. INTRODUCTION
A relevant quantity to understand the breaking
of the UA (1) symmetry in QCD is the topological
susceptibility χ in pure Yang-Mills theory [1,2]
Z
χ ≡ d4 xh0|T (Q(x)Q(0))|0iquenched,
(1)
where
Q(x) =
g 2 µνρσ a
a
ǫ
Fµν (x)Fρσ
(x)
64π 2
(2)
is the topological charge density.
The prediction is [1,2]
χ=
fπ2
m2η + m2η′ − 2m2K ≈ (180 MeV)4 (3)
2Nf
where Nf is the relevant number of flavours. Eq.
(3) implies a well defined prescription [1,2] to deal
with the x → 0 singularity in eq. (1).
In [3] χ was evaluated at zero and finite temperature for SU (3) Yang-Mills theory. The value obtained at zero temperature was in agreement with
the prediction of Eq. (3). The value of χ at finite
temperature displayed a sharp drop beyond the
deconfinement transition. Here we give a short review of a similar calculation for the SU (2) gauge
∗ Partially
supported by EC Contract CHEX-CT92-0051
and by MURST.
† Speaker at the conference.
group [4]. In addition we discuss the comparison
with the geometric method, and we show that,
after a proper renormalization of the latter, the
two procedures give consistent results [5].
2. RENORMALIZATIONS
Let QL (x) be any definition of the topological
charge density on the lattice. The lattice topological susceptibility χL is defined as
X
χL ≡ h
QL (x)QL (0)i.
(4)
x
The lattice regulated QL (x) is related to the continuum M S Q(x) by a finite renormalization [6]
QL (x) = Z(β)Q(x)a4 + O(a6 )
(5)
where β ≡ 2Nc /g 2 in the usual notation.
Since χL does not obey in general the prescription [1,2] leading to eq. (3), besides the multiplicative renormalization of eq. (5) there is also
an additive renormalization
χL = Z(β)2 χa4 + M (β) + O(a6 )
(6)
where M (β) contains mixings with operators of
dimension ≤ 4.
In order to extract the physical signal χ from
eq. (6), we need a determination of the renormalization constants M and Z. We will determine
them using the non–perturbative method of ref.
[7,8,3].
2
3. THE MONTE CARLO SIMULATION
We have used Wilson action and the usual heatbath updating algorithm. The scale a(β) was
fixed by using the results of ref. [9,10].
3.1. Zero Temperature
The simulation was done on a 164 lattice.
We have used various definitions for QL . The
(i)
i-smeared field theoretical QL (x) is defined as
(i)
±4
X
−1
ǫ̃µνρσ ×
29 π 2 µνρσ=±1
(i)
Tr Π(i)
µν (x)Πρσ (x) .
QL (x) =
(7)
(i)
Πµν is the plaquette in the µ − ν plane con(i)
structed with i-times smeared links Uµ (x) [11].
We call M (i) and Z (i) the additive and multiplicative renormalization constants for the i-smeared
operators. Of course χ must be independent of
the choice of the operator QL .
is statistical and the second comes from the error
in ΛL [9,10].
In Fig. 1 we also report the geometric susceptibility χgL [12,13]. The usual determination
is done by identifying χgL = χa4 , and claiming
that the geometrical susceptibility has no additive renormalization. χgL is shown in Figure 1 by
the stars: it is one order of magnitude bigger than
the field theoretical determinations and no scaling is observed. By using the same method as
for the field theoretical determination, we have
measured the multiplicative renormalization Z g ,
finding Z g = 1 within errors. At the same time
we have determined and subtracted the additive
renormalization M g . This brings down the resulting χ by a factor of 10. The results are shown by
the circles in Figure 1 and are in agreement with
the field theoretical results. By renormalizing we
have eliminated the so–called dislocations.
3.
3.
4
1.
2.
0.
-5
10 χ/ΛL
4
5
(316 MeV)
10 GL(x)
2.
1.
−1.
(212 MeV)
0.
2.44
2.51
β
4
2.57
Figure 1. Topological susceptibility at zero temperature.
This is visible in Figure 1, for χ at zero temperature. Up and down-triangles indicate the
value of χ as obtained from the 0-smear and 2smear data respectively. There is good scaling
and (χ)1/4 = (198 ± 2 ± 6) MeV, the first error
0
2
4
6
8
x
Figure 2. Correlation function for the i-smeared
charges at β = 2.57. The lines are to guide the
eye. i=0,1,2 correspond to circles, squares and
triangles respectively.
To support the necessity of the subtraction of
M g , we have also computed the correlation function
GL (x) ≡ hQL (x)QL (0)i
(8)
By reflection positivity we expect that GL (x) ≤ 0
at x 6= 0. Since hQ2L i > 0, the susceptibility
3
8.
1.0
χ(T)/χ(T=0)
10 GL(x)
6.
4.
2.
SU(2) data
SU(3) data
0.5
0.
−2.
0
2
4
6
8
0.0
0.5
1.5
2.0
T/Tc
x
Figure 3. Correlation function for the geometrical
charge at β = 2.57. The line is to guide the eye.
1.0
Figure 4. Ratio χ(T )/χ(T = 0) for SU (3) and
SU (2).
REFERENCES
is mainly determined by the singularity at x =
0. This correlator, for x lying along a coordinate
axis, is shown in Figures 2 for the smeared charges
and in Figure 3 for the geometrical charge. The
peak at x = 0 for the geometric charge is 4 − 5
orders of magnitude larger than for the i-smeared
charges, indicating that M g is much bigger than
M i , i = 0, 1, 2, and is ∼ 80% of the observed χL .
3.2. Finite Temperature
The simulation was done on a 323 × 8 lattice. At this size the deconfining transition is located at βc = 2.5115(40) [10] which means that
Tc = 1/(Nt a(βc )) with Nt the temporal size of
the lattice.
The data show again a drop at the transition.
However this is less sharp than for the SU (3) case
[3,4]. In Figure 4 we show the behaviours for
SU (2) and SU (3) of the ratio χ(T )/χ(T = 0),
where χ(T ) indicates the physical susceptibility
at temperature T . The slope for the SU (3) data
is steeper. In both cases the data at T < Tc
show a constant value consistent with the value
at T = 0.
We thank Prof. Gerrit Schierholz for providing us with the fortran code for the geometrical
charge.
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