Framework for non-perturbative analysis of a Z(3)-symmetric effective
  theory of finite temperature QCD
HIP-2007-18/TH
Framework for non-perturbative analysis of a Z(3)-symmetric
effective theory of finite temperature QCD
A. Kurkela1
Theoretical Physics Division, Department of Physical Sciences,
P.O.Box 64, FI-00014 University of Helsinki, Finland
Abstract
We study a three dimensional Z(3)-symmetric effective theory of high temperature QCD.
The exact lattice-continuum relations, needed in order to perform lattice simulations with
physical parameters, are computed to order O(a0) in lattice perturbation theory. Lattice
simulations are performed to determine the phase structure of a subset of the parameter
space.
1aleksi.kurkela@helsinki.fi
http://arxiv.org/abs/0704.1416v3
1 Introduction
At high temperature, QCD matter undergoes a deconfinement transition, where ordinary had-
ronic matter transforms into strongly interacting quark-gluon plasma [1, 2, 3]. In the absence
of quarks, Nf = 0, the transition is a symmetry-breaking first order transition, where the order
parameter is the thermal Wilson line [4, 5]. The non-zero expectation value of the Wilson line
signals the breaking of the Z(3) center symmetry of quarkless QCD at high temperatures.
The transition has been studied extensively using lattice simulations [6, 7, 8]. Thermody-
namical quantities, condensates and various correlators can be measured on the lattice and the
equation of state can be estimated. This approach, however, becomes computationally exceed-
ingly expensive at high temperatures, and thus cannot be applied to temperatures T above
∼ 5Tc. The complementary approach has been to construct perturbatively effective theories,
such as electrostatic QCD or EQCD, using the method of dimensional reduction [9, 10, 11] to
quantitatively describe high temperature regime of QCD [12, 13, 14, 15]. In the dimensional
reduction procedure, however, one expands the temporal gauge fields around one of the Z(3)
vacua and thus explicitly violates the center symmetry. The range of validity of these theories
therefore ends for T below ∼ 5Tc, where the fluctuations between different vacua become im-
portant. There have also been several attempts to the build models for Wilson line, respecting
the center symmetry [16, 17, 18, 19, 20, 21]. These models give a qualitative handle on the
transition but cannot be perturbatively connected to QCD.
As a unification of these strategies, an effective field theory of high temperature QCD respect-
ing the Z(3) center symmetry has been constructed in [22]. At high temperatures, the effective
theory reduces to EQCD guaranteeing the correct behavior there, but the model still preserves
the center symmetry. The effective theory is further connected to full QCD by matching the
domain wall profile separating two different Z(3) minima.
Being a three dimensional model, the new theory relies on the scale separation between the
inverse correlation length and the lowest non-zero Matsubara mode, which is still modest at Tc
[23, 24]. Thus, one hopes that the range of validity of this theory would extend down to Tc.
The effective theory is a confining one, so perturbative analysis breaks down. Non-perturbative
methods, i.e. lattice simulations, are thus needed to find out the physical properties, such as
correlation lengths, condensates, and most importantly the phase structure of the theory, to test
its regime of validity.
The effective theory is super-renormalizable, and thus the connection between the continuum
MS and lattice regulated theories can be obtained exactly to the desired order in the lattice
spacing a. The matching of the parameters of the Lagrangian to order O(a0), which is needed
in order to perform simulations with MS scheme parameters and to obtain physical results,
requires a two-loop lattice perturbation theory calculation. The one-loop terms remove any
linear 1/a divergences, while two-loop terms remove the logarithmic log(1/a) divergences and
the constant differences in the mass terms of the theory. In addition, the condensates have lattice
spacing dependence and constant differences between the two schemes and can be calculated to
order O(a0) by performing a two-loop calculation for operators up to cubic order and a four-
loop calculation for the quartic condensates. In this paper we perform the needed two-loop
calculations.
This paper is organized as follows. In Sections 2 and 3 we define the theory in continuum
MS regularization and on the lattice, respectively. In Section 4 we study the phase diagram of
a subset of the parameter space of the theory. Details of the matching between the continuum
and lattice theories are given in the appendices.
2 Theory
The theory we are studying is defined by a three dimensional continuum action, which we
renormalize in the MS scheme
d3−2ǫx
TrF 2ij +Tr
+ V0(Z) + V1(Z)
, (1)
where
Fij = ∂iAj − ∂jAi + ig3[Ai, Aj ] (2)
Di = ∂i − ig3[Ai, ] (3)
and Z is a 3 × 3 complex matrix, which in the limit ǫ → 0 has dimension dimZ =
The gauge fields Ai are Hermitean traceless 3 × 3 matrices and can be expressed using gener-
ators of SU(3), Ai = A
a, with TrT aT b = 1
δab . The covariant derivative is in the adjoint
representation. The potentials V0, the “hard” potential, and V1, the “soft” potential, are
V0(Z) = c1Tr[Z
†Z] + 2c2Re(Det[Z]) + c3Tr[(Z
†Z)2], (4)
V1(Z) = d1Tr[M
†M ] + 2d2Re(Tr[M
3]) + d3Tr[(M
†M)2], (5)
where M = Z − 1
Tr[Z]1 is the traceless part of Z. Here, the gauge coupling g3 has a positive
mass dimension dim[g23 ] =GeV, making the theory super-renormalizible. Because of the super-
renormalizibility, the coefficients c2, c3, d2, and d3 are renormalization scale independent and
only the mass terms c1 and d1 acquire a scale dependence in the MS renormalization scheme.
The scale dependence in the mass terms arises from a two-loop calculation and has the form:
c1(µ̄) =
64c3g
d1(µ̄) =
c23 − 64d3g23 +
2d3c3 + d
, (7)
where Λ is a constant specifying the theory and µ̄ is the MS scale parameter. The coefficients
ci, di, and g3 are matched to the parameters of full thermal QCD by imposing the condition
that the theory reduces to EQCD at the high temperature limit, and that the theory reproduces
the domain wall profile of full QCD [22]. This defines a subset of parameter values (with a
limited accuracy due to perturbative matching), for which the theory describes thermal QCD3.
However, in this paper we consider the model in general, and do not restrict ourselves only to
the physical region.
The action is defined only for the number of colors Nc = 3, but for generality, we give some of
the perturbative results for any Nc. For analytic calculations the scalar field Z can be expanded
around the vacuum:
(φ+ iχ)1+ (H + iA), (8)
2Our notation is obtained from that in [22] by scaling with g3: Ai → g3Ai, Z → g3Z, c1 → c1, c2 → g
c3 → g
c3, c̃1 → g
d1, c̃2 → g
d2, and c̃3 → g
3In the matching, the hard potential is parametrically larger than the soft potential, explaining the terminology.
where φ and χ are real scalars and H and A are Hermitean traceless matrices. Fields H and A
can be written with the generators of the SU(3) group, H = HaT a and A = AaT a, where Ha
and Aa are real scalars.
The action is invariant under local gauge transformations, with Z transforming in the adjoint
representation:
Ai(x) −→ G(x)
Ai(x)−
G−1(x), (9)
Z(x) −→ G(x)Z(x)G−1(x), (10)
where G(x) ∈SU(3). In addition to this, there are further global symmetries in the potentials.
The potential V0 is invariant in global SU(3)×SU(3) transformations
Z(x) −→ LZ(x)R, (11)
where L and R are SU(3) matrices. The potential V1 is invariant under Z(3) transformations
M → zM, (12)
where z = ei2πn/3, which generalizes into a U(1) symmetry if d2 = 0. This implies that in the
presence of the both potentials (with non-zero coefficients), the overall global symmetry of the
Lagrangian is Z → zZ.
3 Lattice action
In order to perform non-perturbative simulations, the theory has to be formulated on the lattice.
On the lattice, the scalar field Z lives on the sites of the lattice, and the gauge fields Ai are traded
for link variables Ui, which are elements of SU(Nc) and live on the links connecting adjacent
sites. The lattice action corresponding to the continuum theory can be written as S = SW +SZ ,
where
SW = β
x,i<j
ReTr[Uµν ]
is the standard the Wilson action with the lattice coupling constant
corresponding to a lattice spacing a. The continuum limit is taken by β → ∞, and there the
Wilson action reduces to the ordinary pure gauge action.
The kinetic term, Tr
, is discretized by replacing the covariant derivatives by
covariant lattice differences. Then the scalar sector of the action reads:
SZ = 2
Ẑ†Ẑ − Ẑ†(x)Ui(x)Ẑ(x+ î)U †i (x)
ĉ1Tr[Ẑ
†Ẑ] + 2ĉ2ReDetẐ + ĉ3Tr[(Ẑ
†Ẑ)2]
d̂1Tr[M̂
†M̂ ] + 2d̂2ReTrM̂
3 + d̂3Tr[(M̂
†M̂)2]
. (14)
where ĉi, d̂i are dimensionless numbers, and M̂ and Ẑ are dimensionless Nc×Nc complex matri-
ces. Only the mass terms ĉ1 and d̂1 require non-trivial renormalization and all the other terms
can be matched to order O(a0) on tree-level by simply scaling with g3:
Z = g3Ẑ, M = g3M̂ (15)
c2 = g
3 ĉ2, d2 = g
3 d̂2 (16)
c3 = g
3 ĉ3, d3 = g
3 d̂3. (17)
For the mass terms, renormalization has to be carried out, so that the physical masses of the
fields are the same in both regularization schemes. A two-loop calculation gives (the details of
the calculation and the definitions of the numerical constants are given in the appendix):
ĉ1 =
2ĉ3β −
64ĉ3 +
(log β + ζ) +
16Σ2 − 64δ
+O(β−1)
6.3518228ĉ3β
64ĉ3 +
(log β + 0.08849) + 37.0863ĉ3
+O(β−1) (18)
d̂1 =
β(1 +
16Σ2 − 64δ
ĉ23 − 64d̂3 +
2d̂3ĉ3 + d̂
[log β + ζ]
πΣ− 4δ − 6ρ+ 2κ1 − κ4
+O(β−1)
3.17591 + 5.64606d̂3
41.780852 + 37.0863d̂3
ĉ23 − 64d̂3 +
d̂3ĉ3 +
d̂23 +
[log β + 0.08849]
+O(β−1). (19)
Here, we have set the renormalization scale to be µ̄ = g23 in Eqs. (6) and (7), and denote
c1 = c1(g
3) and d1 = d1(g
3). By making this choice, we get the logarithmic term to be a
function of the lattice coupling constant β. There are also higher order corrections (corrections
of order O(β−1) corresponding to order O(a) in lattice spacing), but their effect vanishes in the
continuum limit.
Various operators also need to be renormalized on the lattice in order to convert their expec-
tation values to continuum regularization. The Z3-symmetry protects the lowest dimensional
condensate 〈g−13 TrZ〉 from acquiring any additive renormalization, while a two-loop calculation
gives for the quadratic condensates:
〈g−23 TrZ
†Z〉MS = 〈TrẐ
†Ẑ〉a −
2Nc(N
c − 1)
log β + ζ +
+O(β−1)
= 〈TrẐ†Ẑ〉a − [0.3791β + 0.3040(log β + 0.66796)] +O(β−1), (20)
〈g−23 TrM
†M〉MS = 〈TrM̂
†M̂〉a −
N2c − 1
2Nc(N
c − 1)
log β + ζ +
+O(β−1), (21)
〈g−23 TrZ
†TrZ〉MS = 〈TrẐ
†TrẐ〉a −
β +O(β−1), (22)
where the subscript a denotes the lattice regularization. For the cubic condensates we get:
〈g−33 2ReDetZ〉MS = 〈2ReDetẐ〉a
N2c − 1
ĉ2 +
8/Nc − 10Nc + 2N3c
ĉ2 + d̂2)
(log β + ζ) +O(β−1)
〈g−33 2ReTrM
3〉MS = 〈2ReTrM̂
3〉a −
− 30Nc + 6N3c
ĉ2 + d̂2) (log β + ζ) +O(β−1)
The effect of subtraction of the divergences can be seen in Fig.1. The renormalization of the
quartic operators to order O(β0) would require a four-loop calculation, which we do not perform
here since they are not measured at this stage.
0 0.01 0.02 0.03 0.04 0.05
25 <Tr(Z✝Z)>
<Tr(Z
 - κ1/β
<Tr(Z
-κ1/β-κ2(logβ+κ2’)
0 0.01 0.02 0.03 0.04
<Tr(M
M)> -div
<Tr(Z
Z)>-div 
Figure 1: The effect of subtraction of the divergences in 〈TrZ†Z〉 and 〈TrM †M〉 in a fixed physical
volume with d1 = 6.6 and d3 = 0.01. On the left panel, the effect of subtracting the divergent parts
of 〈TrZ†Z〉 is plotted. The constants κ1, κ2 and κ′2 are the coefficients of the linear, logarithmic, and
constant differences between lattice and MS regularizations form equation (20). On the right panel:
the continuum limit of the condensates. Notice the negative values of the quadratic condensates in the
symmetric phase.
4 Phase diagram of the soft potential
A simpler model is obtained from the original theory by setting ci = 0. In this model, the trace
of Z decouples and can be integrated over as a free scalar field. The relevant degree of freedom
is thus a traceless complex matrix M , or two traceless hermitian matrices H and A. This can
be viewed as a natural generalization4 of EQCD to complex values of the adjoint higgs field Aa0.
The simpler model is defined by the action:
TrF 2ij +TrDiM
†DiM + d1TrM
†M + 2d2Re(Tr[M
3]) + d3Tr(M
TrF 2ij +TrDiADiA+TrDiHDiH + d1TrA
2 + d1TrH
+ 2d2Tr[H
3 − 3HA2] + d3Tr[H4 +A4 + 4H2A2 − 2HAHA]
. (26)
If the cubic term d2 is zero, the Lagrangian is invariant under a U(1) global symmetry
M → gM , g ∈U(1). The breaking of the symmetry is signalled by a local order parameter:
〈TrA3〉2 + 〈TrH3〉2. (27)
This operator remains a valid order parameter after the renormalization since it has no additive
renormalization, if d2 = 0. In the symmetric phase A is strictly zero and in the broken phase the
order parameter obtains a non-zero vacuum expectation value, while the two phases are separated
by a first order transition. In the broken phase 〈TrM †M〉 is larger than in the symmetric phase.
After the inclusion of the cubic term, A is no longer strictly an order parameter, since the U(1)
symmetry is explicitly broken. However, the first order transition remains and is accompanied
with a significant discontinuity in A and 〈TrM †M〉.
4.1 Perturbation theory
In the limit of small d3/g
3 , the transition becomes very strong, and we expect a semiclassical
approximation to produce the correct behavior of the critical line [25, 26]. We parametrize a
constant diagonal hermitian background field in a fixed Landau gauge as follows:
〈M〉 = 2pT3 + 2
3qT8 =
q + p 0 0
0 q − p 0
0 0 −2q
 , (28)
where p and q are real scalars with dimensions of g3.
Lattice simulations suggest that the A → −A symmetry is not broken spontaneously at any
non-zero value of d2, so that it is sufficient to consider only hermitian background fields. Using
this parametrization, the 1-loop effective potential V1(d1, d2, d3; p, q) can be calculated:
4In EQCD with gauge group SU(3), there is only one linearly independent quartic gauge invariant opera-
tor namely TrA40. In the complex case, however, there are four different Z3-symmetric operators: Tr(M
†M)2,
(TrM†M)2, Tr[M†M†MM ] and Tr[M†
]Tr[M2]. In the case of unitary M , i.e. in the minimum of the hard
potential, these operators collapse into a single one. However, since there is no such restriction in our model, the
operators are linearly independent. From these operators, we choose to include only the one appearing in the
original theory, Tr(M†M)2.
V1(d1, d2, d3; q, p) = (2p
2 + 6q2)d1 + 36q(p
2 − q2)d2 +
(2p2 + 6q2)2d3
(8|p|3 + |p− 3q|3 + |p+ 3q|3)g33
d1 + 3(p − q)d2 + 2(p2 + 3q2)d3
d1 + 6qd2 + 2(p
2 + 3q2)d3
d1 − 3(p + q)d2 + 2(p2 + 3q2)d3
d1 + 4(p
2 + 3q2)d3 − 2
3(p2 + 3q2)d22 + 18q(p
2 − q2)d2d3 + (p2 + 3q2)2d23
d1 + 4(p
2 + 3q2)d3 + 2
3(p2 + 3q2)d22 + 18q(p
2 − q2)d2d3 + (p2 + 3q2)2d23
]3/2 }
d1 − 6qd2 + 2(3p2 + q2)d3
d1 + 3(p + q)d2 + 2(p
2 − 4pq + 7q2)d3
d1 − 3(p − q)d2 + 2(p2 + 4pq + 7q2)d3
(p2 + 3q2)d3 −
27(p2 + 3q2)d22 + 54q(q
2 − p2)d2d3 + (p2 + 3q2)2d23
(p2 + 3q2)d3 +
27(p2 + 3q2)d22 + 54q(q
2 − p2)d2d3 + (p2 + 3q2)2d23
]3/2 }
where the first term is the classical potential, the second one comes from one-loop vector dia-
grams and the fourth and the fifth from one-loop scalar diagrams of H and A, respectively5.
The effective potential has a symmetry arising from the permutations of the diagonal elements
of the background 〈M〉 and has the following invariance:
V1(d1, d2, d3; q, p = ±3q) = V1(d1, d2, d3;−2q, 0). (30)
More generally, the potential is invariant under rotations of 2π/3 in the (p, 3
q)-plane and in the
reflections of p:
q → p−q
p → p+3q
q → −p−q
p → p−3q
q → q
p → −p
Thus there is a fundamental region, which determines the potential over the whole plane. We
choose the fundamental region to be bounded by the two lines p = 0 and p = −3q together with
the condition p ≥ 0.
In the fundamental region, there can be four different minima at the critical parameter
values d1, d2 and d3. The one at the origin (denoted by 1 in fig.2) is the symmetric minimum,
5By dropping the last term, i.e., the five last lines and scaling d1 → y, d2 → iγ3, and d3 → 2x, one obtains the
effective potential for EQCD in the presence of a finite (imaginary) chemical potential using the notation of [27]
−20 −15 −10 −5 0 5 10 15 20
−25 −20 −15 −10 −5 0 5 10 15 20 25
q/g 1
Figure 2: 1-loop effective potential in the (q,p)-plane at the critical point for d3 = 0.01 and d2 = 0 (left
panel) and d2 = 0.05 (right panel). Light areas represent the minima of the potential. Solid lines separate
the three identical sectors which are related by the permutation symmetry of the diagonal elements of the
background field 〈M〉. In the absence of d2 there is an additional U(1) symmetry making the directions
marked with dashed lines identical to the p = 0 direction. This symmetry explicitly broken by finite d2
as seen on the right panel.
-20 -10 0 10 20
Figure 3: 1-loop effective potential with p = 0 as a function of q at the critical point for d3 = 0.01 and
d2 = 0(line), 0.005(dotted),and 0.05(dashed).
the minima 2 and 3 are connected by the permutation symmetry and correspond to the same
physical broken minimum, with TrH3 < 0 and TrA3 = 0. The minimum 4 corresponds to a
phase with TrH3 > 0 and TrA3 = 0 and is connected continuously to the minimum 2 by a global
U(1) symmetry if d2 = 0. If d2 6= 0, the U(1) symmetry is lost and the minima 2 and 4 are no
longer equivalent. If d2 > 0 the minimum at 2 is favored over 4 and vice versa.
Setting d2 to zero and expanding in d3 up to order O(d23) the potential reads (for p = 0):
V1(d1, 0, d3; q, 0) = 18q
|q| −
4π2d3
+O(d23),
so that in the limit d3 → 0 the potential has two coexisting minima and a first order transition
d1 = d
4π2d3
≈ 0.0759909
, (33)
This sets the scaling of the critical line as a function of d3 at small d3. Corrections to this, and
d2 dependence, are obtained by minimizing the real part of Eq.(29) numerically. The results are
shown in Figs.4 and 5. The phase transition is accompanied with a discontinuity in q:
∆q = |qbroken − qsymmetric| =
, (34)
We see that the transition gets stronger as the coupling d3 gets smaller justifying a posteriori
the semiclassical approximation.
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
=0.00
=0.05
=0.10
=0.15
Symmetric
Broken
Figure 4: 1-loop perturbative phase diagram of the soft potential, V1, as function of d1, d2 and d3. A
first order critical line separates the two phases. The symmetric phase refers to the phase where with
d2 = 0 the order parameter vanishes and with d2 6= 0 is smaller than in the broken phase. The data points
represent non-perturbative lattice measurements with d2 = 0 on a N
3 = 123 lattice. The perturbative
result approaches the lattice data points for small values of d3 where the transition is very strong. Small
discrepancy between the perturbative result and lattice data points at small d3/g
is mostly due to finite
volume effects.
4.2 Lattice analysis
The perturbative calculation is valid only for small d3 and a non-perturbative lattice analysis
has to be performed to obtain the full phase structure of the model. For the simulations we
used a hybrid Monte-Carlo algorithm for the scalar fields and Kennedy-Pendleton quasi heat
bath and full group overrelaxation for the link variables [28, 29, 30].
-0.4 -0.2 0 0.2 0.4
=0.01
Broken
Symmetric
Figure 5: 1-loop perturbative phase diagram of the soft potential, V1, as function of d1 and d2 with
d3 = 2. A first order critical line separates the two phases. The non-analyticity at d2 = 0 is due to the
change of global minimum between minima 3 and 4.
The transition was found to be of the first order for all parameter values used in the simula-
tions (d3 ≤ 4 and d2 ≤ 0.15) accompanied with a large latent heat and surface tension; hysteresis
curves showing discontinuity around critical point in 〈TrM †M〉MS can be seen in Fig.8. The
probability distributions of TrM †M along the critical curve are very strongly separated (see
Fig.7). This makes the system change its phase very infrequently during a simulation, and mul-
ticanonical algorithm is needed to accommodate a phase flip in reasonable times for any system
of a modest size [31]. Even with the multicanonical algorithm, the critical slowing restricts us
to physical volumes up to V . 50/g63 .
The pseudo-critical point was determined requiring equal probability weight for TrM †M in
both phases. The simulations were performed with β = 12 and a lattice size N3 = 123, which
precludes the continuum extrapolation as well as the thermodynamical limit. However, these
limits were studied for one set of parameter values and the dependence of the critical point on
both lattice spacing and volume were found to be of order of five per cent for the lattice spacings
and volumes used (see Fig.6 and Table 1).
The phase diagram can be seen in Fig.9 and Fig.10. The non-perturbative critical line follows
the perturbative one for small values of d3, but for larger d3 fluctuations make the system prefer
the symmetric phase. The discontinuity in 〈TrM †M〉 along the critical line diminishes, as d3 gets
larger (see Fig.11), but it seems that the discontinuity persists, even if its magnitude diminishes
in the limit d3 → ∞ suggesting that there is a first order phase transition for any (positive)
value of d3.
5 Conclusions
In this paper, exact relations between the lattice and continuum MS regulated formulations of
the Z(3)-symmetric super-renormalizable effective theory of hot QCD, defined by Eqs.(1),(4),
0 0.005 0.01 0.015 0.02
1/(Vg
Figure 6: Volume dependence of the pseudo-critical point with d3 = 2 and d2 = 0.1. The pseudo-
critical point was determined by requiring equal probability weight for TrM †M in both phases. The line
represents a linear fit. The dependence on lattice spacing and volume seem to be within 5% for the lattice
spacings and volumes used.
β Lattice volumes
12 83, 103, 123, 163
16 123, 163, 203
20 163, 203, 243
Table 1: Lattices used in the continuum and thermodynamical extrapolation of the critical point, seen
in Fig.6.
and (5), have been calculated. The Lagrangians and the operators up to cubic ones have been
matched to O(a0). These results make the non-perturbative lattice study of the theory possible.
An interesting model with non-trivial dynamics is obtained by setting ci = 0 in Eq.(4). The
model amounts to a natural generalization of EQCD to complex variables. The phase diagram
of the model has been determined using lattice simulations. Two distinct phases were found, a
symmetric phase with small 〈TrM †M〉MS, and a broken phase with large 〈TrM
†M〉MS. The two
phases were found to be separated by a strong first order transition with a large surface tension
and discontinuities in the operators. In contrary to EQCD, where the first order line terminates
at a tricritical point, the model seems to have a first order transition with all values of d2 and
In the future, it is our goal to map out the phase diagram in the full parameter space of the
model, rather than in a restricted region as in the present exploratory study, in order to search
for regions in which the phase diagram would resemble that expected for the finite-temperature
SU(3) pure Yang-Mills theory.
0 1 2 3 4
1e-10
1e-09
1e-08
1e-07
1e-06
1e-05
0.0001
0.001
Figure 7: Histograms of TrM †M in logarithmic scale with d2 = 0 along the critical curve. Transition
channel between the peaks weakens and the transition gets stronger for decreasing d3. For d3/g
= 0.5,
the relative probability density in the tunneling channel is suppressed by a factor ∼ 10−10.
Acknowledgments
The author thanks K. Kajantie for suggesting this topic and for numerous comments concerning
the text. The author also thanks M. Laine, K. Rummukainen, Y. Schröder and A. Vuorinen for
invaluable advice. This research has been supported by Academy of Finland, contract number
109720 and the EU I3 Activity RII3-CT-2004-506078 HadronPhysics. Simulations were carried
out at CSC - Scientific Computing Ltd., Finland; the total amount of computing power used
was ∼ 1× 1016 flops.
A Details of renormalization
In this appendix, we give details of the calculation of the renormalization of the mass parameters
ĉ1 and d̂1 and the condensates 〈g−23 TrZ†Z〉, 〈g
3 TrM
†M〉, 〈g−23 TrZTrZ†〉,〈g
3 2DetZ〉, and
〈g−33 2ReTrM3〉.
The renormalization calculation compares ultraviolet properties of the two regularizations
and thus it is irrelevant, in which phase we carry out the computation. We chose to work
around the symmetric vacuum, since the Feynman rules are the simplest this way. However,
in this vacuum, all components of the gluon are massless and one therefore has to deal with
infrared divergences. The infrared divergences in the two regularizations are the same, and
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Figure 8: Discontinuity in the quadratic condensate in continuum regularization 〈TrM †M〉
for d3 =
0.1, 1, 3. The phase transition gets weaker as the coupling d3 grows. The metastable regions shrink and
the discontinuity diminishes.
cancel exactly in the final results.
Using the expansion (8), the potential V2 is a function of H and A only. The unit matrix
commutes with any SU(3) matrix, and the interaction with the gluon field arises from commu-
tator in the covariant derivative in adjoint representation. Thus the gluons couple, on tree-level,
only to H and A
Using this expansion, there are four tree-level mass terms in the Lagrangian that require
renormalization:
(ĉ1 + d̂1)H
(ĉ1 + d̂1)A
The coefficients ĉ1 and d̂1 have to be adjusted such that both regularization schemes give the
same physical masses for all fields φ, χ, H and A. The theory is super-renormalizible, and there
are divergent contributions up to two-loop level only. The masses are obtained from the low
momentum properties of the two-point correlator:
k,p→0
〈〈φ(k)φ(p)〉〉 = δ(3)(k + p) 1
k2 +m2
The difference between the correlators in the two schemes is in the mass and the wave function
renormalization. However, the effect of the wave function renormalization is of order O(a) and
can be neglected. To get the same masses in the different schemes, we enforce the condition that
the two-point correlators give identical values in the low momentum limit. The zero-momentum
lattice correlator can be written in a weak-coupling expansion
〈〈φ(0)φ(0)〉〉a =
〈φ(0)φ(0)〉a − 〈φ(0)φ(0)〉MS
〈φ(0)φ(0)(−SI )〉a − 〈φ(0)φ(0)(−SI )〉MS
〈φ(0)φ(0)S2I 〉a − 〈φ(0)φ(0)S2I 〉MS
+ 〈〈φ(0)φ(0)〉〉MS +O(a), (37)
where double and single brackets represent exact and Gaussian expectation values, respectively,
and subscripts give the regularization scheme. From here, we can read the condition for the
lattice mass term ĉ1 by requiring that the exact correlators give the same value up to order
0 1 2 3 4
=0.15
=0.10
=0.05
=0.00
Symmetric
Broken
Figure 9: The phase diagram of the soft potential as a function of d1, d2 and d3. First order critical
line separates two phases. Solid lines represent polynomial fits to the lattice data points and dashed lines
are the perturbative predictions. The symmetric phase refers to the phase where with d2 = 0 the order
parameter A vanishes and with d2 6= 0 is smaller than in the broken phase. Also 〈TrM †M〉 is significantly
smaller in the symmetric phase and the critical line was determined requiring equal propability weight
for 〈TrM †M〉 in both phases.
O(β−1):
ĉ1 =
〈φφ(−SI)〉a,1PI − 〈φφ(−SI)〉MS,1PI
〈φφS2I 〉a,1PI − 〈φφS2I 〉MS,1PI
+O(a) (38)
Similarly, we get for the mass term of adjoint fields:
d̂1 =
− ĉ1
〈HH(−SI)〉a,1PI − 〈HH(−SI)〉MS,1PI
〈HHS2I 〉a,1PI − 〈HHS2I 〉MS,1PI
+O(a) (39)
-0.2 -0.1 0 0.1 0.2
Broken
Symmetric
Figure 10: The the phase diagram as a function of d2, with d3 = 2. The symmetric phase refers to the
phase where with d2 = 0 the order parameter vanishes and with d2 6= 0 is smaller than in the broken
phase.
The correlators in both regularizations are in two-loop weak-coupling expansion infrared-divergent
quantities. However, since the infrared properties of the two regularization schemes are the same,
the infrared divergences cancel exactly in the difference.
The renormalization of the condensates is done very similarly. The condensates can be
expressed, in both regularization schemes, as derivatives with respect to mass parameters of the
free energy and thus they can be related. For the quadratic condensates we get:
〈TrZ†Z〉MS =
∂c1(µ)
= 〈TrZ†Z〉a +
∂(fMS − fa)
∂c1(µ)
〈TrM †M〉MS =
∂d1(µ)
= 〈TrM †M〉a +
∂(fMS − fa)
∂d1(µ)
, (41)
and for the cubic:
〈2ReDetZ〉MS =
= 〈2ReDetZ〉a +
∂(fMS − fa)
〈2ReTrM3〉MS =
= 〈2ReTrM3〉a +
∂(fMS − fa)
. (43)
0 0.2 0.4 0.6 0.8 1
=0.05
=0.10
=0.15
Figure 11: Discontinuity in 〈TrM †M〉 along the critical line dcrit
(d2, d3). Dotted lines represent second
order polynomial fits to the data, and the points on the y-axis represent extrapolations to infinite d3.
The large d3 extrapolation yields a finite value suggesting that the transition remains of first order even
at large d3.
Due to the super-renormalizability the difference in free energy is dimensionally of the form:
fMS − fa =
+D1,0
c1(µ̄)
+D1,1
d1(µ̄)
(4π)2
+ C2,1
+ C2,2
+E2,0c
2 + E2,1d
2 + E2,2c2d2
+D2,0g
3c1(µ̄) +D2,1c3c1(µ̄) +D2,2d3c1(µ̄)
+D2,3g
3d1(µ̄) +D2,4c3d1(µ̄) +D2,5d3d1(µ̄)
(4π)3
+ C3,1
g23c3
+ C3,2
g23d3
+ C3,3
+ C3,4
+ C3,5
(4π)4
B4,0g
3 + C4,1g
3c3 + C4,2g
3d3 + C4,3g
3 + C4,3g
3 + C4,4g
3c3d3
+C4,5c
3 + C4,6d
3 + C4,6c
3d3 + C4,7c3d
+O(a), (44)
where the dimensionless coefficients Ai,j, Bi,j, Ci,j, Di,j , and Ei,j are functions of a dimensionless
combination aµ̄ only. The coefficients Ci,j and Di,j follow from an i-loop computation. For the
quadratic and cubic condensates we need to know coefficients Di,j and Ei,j in order to obtain
the matching of the condensates to order O(a0), which follow from a two-loop calculation:
D1,0 = −ΣN2c
D1,1 = −Σ(N2c − 1)
D2,0 = −2Nc(N2c − 1)
+ ζ +
D2,1 = 0
D2,2 = 0
D2,3 = −2Nc(N2c − 1)
+ ζ +
D2,4 = 0
D2,5 = 0
E2,0 = −
(N2c − 1)
E2,1 = −12
E2,2 = −8
. (45)
For the quartic condensates, however, the coefficients Ci,j are needed and a four-loop lattice
perturbation theory calculation is required for the matching. For the gluon condensates, also the
Bi,j are needed. The coefficients B2,0 and B3,0 have been calculated in [32] and [33], respectively.
The coefficient B4,0 has been calculated for Nc = 3 using stochastic perturbation theory in [34].
B Feynman rules
Using the expanded fields, the potentials become:
V0(Z) =g
χ2 + ĉ1Tr[A ·A] + ĉ1Tr[H ·H]
2ĉ2√
2ĉ2√
φχ2 +
2ĉ2√
φTr[A ·A]−
2ĉ2√
φTr[H ·H]
4ĉ2Tr[H ·H ·H] +
2ĉ2√
χTr[A ·H]− 1
4ĉ2Tr[A · A ·H]
2ĉ3φ
2ĉ3χ
4ĉ3φ
2Tr[H ·H] + 1
2Tr[H ·H] + 1
2Tr[A ·A]
4ĉ3χ
2Tr[A ·A] +
ĉ3φχTr[A ·H]
6ĉ3φTr[H ·H ·H] +
6ĉ3χTr[A ·A · A]
6ĉ3φTr[A · A ·H] +
6ĉ3χTr[A ·H ·H]
24ĉ3Tr[A · A · A ·A] +
24ĉ3Tr[H ·H ·H ·H]
16ĉ3Tr[A ·A ·H ·H]−
8ĉ3Tr[A ·H ·A ·H]
V1(Z) =g
d̂1Tr[A ·A] + d̂1Tr[H ·H]
−6d̂2Tr[A · A ·H] + 2d̂2Tr[H ·H ·H]
d̂3Tr[A ·A ·A ·A] + 4d̂3Tr[A · A ·H ·H]
−2d̂3Tr[A ·H · A ·H] + d̂3Tr[H ·H ·H ·H]
. (47)
The gauge part of the scalar Lagrangian in Fourier space (momentum conservation, all
integrations over Brillouin zone, with measure
(2π)3
, and sums understood) becomes [35]:
SZ = i
fabc (p̃− q)iAa(p)Ab(q)Aci (r) +
2facef bde( p− q
)iδijA
a(p)Ab(q)Aci (r)A
j (s)
fabc (p̃− q)iHa(p)Hb(q)Aci (r) +
2facef bde( p− q
)iδijH
a(p)Hb(q)Aci (r)A
j (s),
where we use a compact notation:
= cos
, p̃i =
, and p̃2 =
In addition to these there is the pure gluon and gauge fixing sector [36]
p̃2Aai (−p)Aai (p) + p̃2c̄a(p)ca(p) + ig3fabcri
p̃ic̄
a(p)cc(r)Abi (q)
2(facef bde + fadef bce)s̃ip̃ic̄
a(p)Aci (q)A
i (r)c
b(s) +
Aai (−p)Aai (p) + S3 + S4.
Contributions of three and four gluon vertices S3 and S4 can be found in [37], Eqs. (15.39),(15.43)
and (15.53), where one needs to replace (2
)(δABδCD + . . .) with (
)(δABδCD + . . .) [36].
C Calculation of the diagrams
The perturbation theory calculations were done using symbolic manipulation language FORM[38].
For formalized computation, it is advantageous to write all the color tensors in the fundamental
representation, i.e. using the generators of the group:
Tr[T aT bT c] =
(dabc + ifabc) (51)
Tr[T aT b] =
δab. (52)
Then all the color contractions in loop calculations can be done systematically with repeated
use of the Fiertz identity:
T aijT
(δilδjk −
δijδkl). (53)
The following combinations are found in the action:
ifabc = 2Tr(T a[T b, T c]) (54)
dabc = 2Tr(T a{T b, T c}) (55)
fabcf bde = −2Tr[T a, T c]Tr[T b, T d] (56)
dabedcde = 2Tr{T a, T b}Tr{T c, T d} − 2
δabδcd (57)
In lattice perturbation theory, the numerators of the integrals contain complex trigonometric
objects. These can be systematically reduced to squares of sines, which also appear in the
denominator, by repeated use of the following formulae (no summation over repeated indices):
x̃+ yi = x̃iyi
+ ỹixi
(x+ y)i
x̃iỹi (59)
2 = δii −
x̃2i (60)
x̃iỹixi
(x̃+ y)2i − x̃2i xi
− ỹ2i xi
. (61)
This procedure generalizes trivially also to higher order loop calculations. The set of integrals
can be further reduced by applying a trigonometric identity (for j ≥ 2):
(2π)3
(x̃2 +m2)j
j − 1
(2π)3
x̃2 − 3
(x̃2 +m2)j−1
(2π)3
(x̃2 +m2)j
. (62)
D Diagrams for mass renormalization
In this section, we give the zero momentum diagrams that affect the mass renormalization.
The expressions are in lattice regularization and the symbol ”MS” refers to the result of the
corresponding diagram in the MS regularization. Solid and wiggly lines represent scalars and
gluons, respectively. Symbols in parentheses indicate fields running in the internal scalar lines.
The symmetry factors are included in the coefficients.
The following diagrams with zero incoming momenta contribute to the renormalization of
the mass term ĉ1 of φ-field (with external lines φ):
• One-loop: The mass mi refers to the mass of the field running in the loop. In the difference
between continuum and lattice regularization, the mass dependence cancels.
(φ) : −1
(χ) : −1
(A) : −1
(N2c − 1)
(H) : −(N2c − 1)
g23 ĉ3I(mi)
N2c g
3 ĉ3
+O(a) +MS (63)
• Two-loop:
– Terms proportional to ĉ23g
3 : Masses m1, m2 and m3 in the denominator refer to the
masses of the internal lines and m2d = g
3(ĉ1 + d̂1).


(φφφ) : 2
(χχφ) : 2
(AAφ) : 2
(N2c − 1)
(HHφ) : 2(N2c − 1)
(χAH) : 4
(N2c − 1)
(AAH) : −1
c − 1)
(HHH) : −Nc(N2c − 1)


g43 ĉ
×H(m1,m2,m3)
N2c −
g43 ĉ
+O(a) +MS (64)
– Terms proportional to ĉ3g
3 : The tadpoles are cancelled by the 1-loop counter terms.
(A) : −1
c − 1)
(H) : −Nc(N2c − 1)
g43 ĉ3
2H(md,md, 0) + (2I(0) − I(md))
I(md)
+ 4m2dH
′(md,md, 0) − a2G(md,md)
c − 1)
g43 ĉ3
+ ζ − δ
+O(a) + tadpoles. + MS (65)
(A) : 1
c − 1)
(H) : Nc(N
c − 1)
3I(0)(−∂m2
)I(md) + a
I(md)
I(0)I(md)
c − 1)
g43 ĉ3
+O(a) + tadpoles. + MS (66)
The following diagrams with zero incoming momenta contribute to the renormalization of
the mass term d̂1 of H-field (with external lines H):
• One-loop diagrams:
(φ) : −ĉ3
(χ) : −1
(A) : −(2Nc − 1/Nc)(ĉ3 + d̂3)
(H) : −(2Nc − 3/Nc)(ĉ3 + d̂3)
g23I(mi)
ĉ3 − 4(Nc − 1/Nc)(ĉ3 + d̂3)
+O(a) +MS (67)
= −3g23NcI(0) = −3g23Nc
+O(a) +MS (68)
= g23NcI(0) = g
+O(a) +MS (69)
• Two-loop diagrams:
– Terms proportional to g43 ĉ


(φφH) : 2ĉ23
(φχA) : 4
(AAφ) : −1
(HHφ) : −3Ncĉ23
(χχH) : 2
(χAH) : −2
(AAH) : −2(1− 3/N2c − 32N
c )(ĉ3 + d̂3)
(HHH) : −6(1− 3/N2c − 16N
c )(ĉ3 + d̂3)


g43H(m1,m2,m3)
− 4Nc)ĉ23 + (4N2c + 24/N2c − 8)(ĉ3 + d̂3)2
+O(a) +MS
– Terms proportional to g43(ĉ3 + d̂3):
(A) : −2(N2c − 12)
(H) : −2(N2c − 32)
g43(ĉ3 + d̂3)
2H(md,md, 0) + (2I(0) − I(md))
I(md)
+ 4m2dH
′(md,md, 0) − a2G(md,md)
= −8(N2c − 1)(ĉ3 + d̂3)
+ ζ − δ
+O(a) + tadpoles. + MS (71)
(A) : −2(N2c − 12)
(H) : −2(N2c − 32)
g43(ĉ3 + d̂
3I(0)(−∂m2
)I(md) + a
I(md)
I(0)I(md)
= −2(N2c − 1)g43(ĉ3 + d̂3)
+O(a) +MS (72)
– Terms proportional to g43 : The coupling of the adjoint fields A and H is exactly the
same as as in EQCD, so the term proportional to g43 can be taken from EQCD [36].
However, at two-loop level diagrams with an adjoint scalar loop contribute two times
since there are two adjoint fields. The diagrams with adjoint loops are the following:
= N2c g
(2π)3
(2π)3
x̃2x̃2(ỹ2 +m2d)
=N2c g
(2π)3
(2π)3
x̃2x̃2(ỹ2 +m2
(2π)3
(2π)3
x̃2x̃2
a2m2d
(2π)3
(2π)3
x̃2x̃2(ỹ2 +m2d)
N2c g
˜(2x+ y)
(x̃2 +m2
)ỹ2ỹ2( ˜(x+ y)
= N2c g
H(md,md, 0)
(2π)3
(2π)3
x̃2x̃2
ỹ2 +md
+ 2m2d
(2π)3
(2π)3
x̃2x̃2
ỹ2 +md
˜(x+ y)
(2π)3
(2π)3
[ ˜(x+ y)2]2(x̃2 +m2d)(ỹ
2 +m2d)
The last line can be written in a more familiar form using the definition of ρ, Eq.(93),
and the trigonometric identity Eq.(62):
(2π)3
(2π)3
[ ˜(x+ y)2]2(x̃2 +m2
)(ỹ2 +m2
4π2a2
(2π)3
(2π)3
x̃2x̃2
(2π)3
(2π)3
ỹ2ỹ2
+O(a).
The infrared divergences in these two diagrams cancel and the sum of the diagrams
becomes:
+ = N2c g
+O(a) +MS
There are also two other diagrams:
= −N2c g43
(2π)3
(2π)3
i (2y)
x̃2x̃2(x̃2 +m2d)(ỹ
2 +m2d)
= N2c g
(2π)3
(2π)3
i x̃i
˜(x+ 2y)i
j x̃j
˜(x+ 2y)j
x̃2x̃2(x̃2 +m2d)(ỹ
2 +m2d)(
˜(x+ y)
+m2d)
After repeated use of Eqs.(61), the both diagrams can be written in the form
±N2c g43
(2π)3
(2π)3
x̃2 − 2a2
x̃2x̃2(x̃2 +m2d)(ỹ
2 +m2d)
, (79)
with the negative and the positive sign coming from the first and second diagram,
respectively, so that their sum cancels exactly.
The sum of diagrams proportional to g43 reads:
Σ2 + (
)πΣ − 4(δ + ρ) + 2κ1 − κ4
+ 2ρ+
+O(a) +MS (80)
It is noteworthy that the scale dependence from the diagrams containing only gauge
interactions with a single adjoint scalar field cancels exactly in the renormalization.
However, upon the inclusion of another adjoint scalar field this property is lost.
E Diagrams for operator renormalization
In this section, we give the results for the vacuum diagrams that affect the renormalization of
quadratic condensates present in the action. The diagrams needed for the quadratic condensates
〈TrZ†Z〉MS, 〈TrM
†M〉MS, and 〈TrZ
†TrZ〉MS:
• One-loop:
φ : 1
χ : 1
H : 1
(N2c − 1)J(md)
A : 1
(N2c − 1)J(md)
= (N2c − 1)J(md) + J(m) (81)
• Two-loop: the counter terms cancel the {c3, d3}-dependent linearly divergences terms, and
only the gauge diagrams contribute:
H : 1
A : 1
c − 1)
I(0)− 6I(0)I(md)− a2m2dI(0)I(md)
A : 1
H : 1
c − 1)
− I(md)I(md) + 4I(0)I(md)
− 4m2dH(md,md, 0)− a2G(md,md)
The diagrams needed for the cubic condensates: 〈2ReDetZ〉 and 〈2ReTrM3〉:
• Two-loop


φφφ : 1
φχχ : 1
φhh : 1
(N2c − 1)ĉ22
φaa : 1
(N2c − 1)ĉ22
χah : 1
(N2c − 1)ĉ22
hhh : 3(1/Nc − 54Nc +
N3c )(
ĉ2 + d̂2)
aah : 9(1/Nc − 54Nc +
N3c )(
ĉ2 + d̂2)


g63H(m1,m2,m3) (84)
N2c − 1
ĉ22 + 12
1/Nc −
ĉ2 + d̂2)
g63H(m1,m2,m3) (85)
F Basic lattice integrals and numerical constants
In this appendix, we list the basic lattice integrals and numerical constants defined and calculated
in [39, 35, 36].
Integrals:
J(m) ≡
(2π)3
ln(x̃2 +m2)
+O(am4)
I(m) ≡
(2π)3
x̃2 +m2
−m+O(am2)
H(m1,m2,m3) ≡
(2π)3
(2π)3
x̃2 +m21
ỹ2 +m22
x̃+ y
a(m1 +m2 +m3)
+ ζ +
+O(am)
G(m,m) ≡
(2π)3
(2π)3
(x̃2 +m2)(ỹ2 +m2)(x̃+ y
+O(m3a−1)
H ′(m1,m2,m3) = (∂m2
)H(m1,m2,m3) (90)
Numerical constants:
Σ ≡ 1
∫ π/2
i sin
2(xi)
≈ 3.175911535625 (91)
∫ π/2
d3xd3y
i sin
2(xi) sin
2(xi + yi)
i sin
2(xi))2
j sin
2(yj)
k sin(xk + yk)
≈ 1.942130(1) (92)
∫ π/2
d3xd3y
i sin
2(xi) sin
2(xi + yi)
i sin
2(xi))2
j sin(xj + yj)
i sin
i sin
2(xi))2
i sin
2(yi))2
≈ −0.313964(1) (93)
∫ π/2
d3xd3y
i sin
2(xi) sin
2(xi + yi)∑
i sin
2(xi)
j sin
2(yj)
k sin(xk + yk)
≈ 0.958382(1) (94)
∫ π/2
d3xd3y
i sin
2(xi) sin
2(xi + yi) sin
2(yi)
i sin
2(xi))2
j sin
2(yj)
k sin(xk + yk)
≈ 1.204295(1) (95)
ζ = lim
∫ π/2
d3xd3y
i sin
2(xi) + z)
j sin
2(yj)
k sin
2(xk + yk)
≈ 0.08849(1). (96)
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	Introduction
	Theory
	Lattice action
	Phase diagram of the soft potential
	Perturbation theory
	Lattice analysis
	Conclusions
	Details of renormalization
	Feynman rules
	Calculation of the diagrams
	Diagrams for mass renormalization
	Diagrams for operator renormalization
	Basic lattice integrals and numerical constants
