arXiv:hep-th/0609172v4 20 Aug 2007
Potential for inert adjoint scalar field in SU(2) Yang-Mills
Francesco Giacosa†and Ralf Hofmann∗
†Institut f¨ ur Theoretische Physik
Universit¨ at Frankfurt
Johann Wolfgang Goethe - Universit¨ at
Max von Laue–Str. 1
60438 Frankfurt, Germany
∗Institut f¨ ur Theoretische Physik
Universit¨ at Heidelberg
69120 Heidelberg, Germany
A scalar adjoint field is introduced as a spatial average over (anti)calorons
in a thermalized SU(2) Yang-Mills theory. This field is associated with the
thermal ground state in the deconfining phase and acts as a background for
gauge fields of trivial topology. Without invoking detailed microscopic infor-
mation we study the properties of the corresponding potential, and we discuss
its thermodynamical implications. We also investigate the gluon condensate
at finite temperature relating it to the adjoint scalar field.
The potential importance of topological field configurations in generating a finite
correlation length in the dynamics of thermalized, nonabelian gauge fields  was
emphasized a long time ago . In particular, calorons, i.e. instantons at finite
temperature T, play an important role in the description of pure SU(2) and SU(3)
Yang-Mills theories in their deconfining phase. However, the inclusion of topologi-
cally nontrivial field configurations when evaluating thermodynamical quantities is
complicated because of the nonlinearity of the theory . Also, there is no a pri-
ori infrared-cutoff when integrating out residual quantum fluctuations about these
finite-action configurations .
In  the possibility that a scalar adjoint field φ can act as a thermal average
upon calorons and anticalorons was addressed. More precisely, BPS saturated and
stable (trivial holonomy) configurations of topological charge modulus |Q| = 1,
Harrington-Shepard solutions , were integrated into an adjoint scalar field φ in
the deconfining phase. This field φ, which is part of the thermal ground state,
affects thermodynamical quantities. The scalar field φ induces an adjoint Higgs
mechanism: some of the gauge modes acquire a (temperature-dependent) mass on
tree-level. Notice that rotational symmetry in the thermal system together with the
perturbative renormalizability of the fundamental Yang-Mills action  forbid the
emergence of any other composite field, induced by BPS saturated, nonpropagating,
fundamental field configurations, but an adjoint scalar field1.
To be more definite, let us denote by LY M= −1
Yang-Mills Lagrangian, with Fa
tal coupling). In the partition function all possible configurations of the field Aµ
have to be considered, including the topologically nontrivial ones. Upon the emer-
gence of the scalar adjoint field φ we are led to consider the effective Lagrangian
effective coupling constant. The effect of topology is embodied in the scalar ad-
joint field φ, which acts as a background for the dynamics of propagating gauge
fields. While the field φ acts as an infrared-cutoff in the spatial coarse-graining
performed in the fundamental theory  it represents an ultraviolet cutoff in the
coarse-grained theory, implying a rapidly converging loop expansion [5, 9]. The po-
tential V (|φ|2) = Λ6/|φ|2, where Λ is the Yang-Mills scale, was evaluated in  by
performing the spatial average over calorons explicitely.
In this work we would like to show that apart from general properties such
µν)2the fundamental SU(2)
ν(g is the fundamen-
E+ (Dµφ)2+ V (|φ|2)?where GE,a
µrefer to (coarse-grained) topologically trivial fluctuations and e denotes the
1For a more detailed discussion see .
renormalizability implies that φ transforms homogeneously under changes of the gauge. On the
fundamental level the only homogeneously transforming quantity is the field strength and (nonlo-
cal) products thereof which can always be reduced to spin-0 and spin-1 representations of SU(2).
While the former is irrelevant on BPS saturated configurations the latter plays an important role.
A rigorous version of the here-sketched argument will be presented in .
The argument involves the fact that perturbative
as gauge invariance and the periodicity of φ in the euclidean time the potential
V (|φ|2) follows by requiring BPS saturation. That is, no explicit calculation in-
volving calorons is needed to derive V (|φ|2). Over and above the existence of a
Yang-Mills scale Λ, which needed to be assumed in , turns out to be redundant
since Λ is shown to be a purely nonperturbative constant of integration.
The article is organized as follows: In Sec.2 we discuss the implications of BPS
saturation for the dynamics of the field φ. The properties of (coarse-grained) topo-
logically trivial fluctuations and the combined effect of the ground state and the
excitations on basic thermodynamical quantities are investigated in Sec.3. In Sec.4
we show that a gluon condensate emerges in the framework of the effective theory
and that its T-dependence is in agreement with lattice results [10, 11, 12].
2 Gauge invariance, BPS saturation, and inert-
In the deconfining phase of thermalized SU(2) Yang-Mills theory we study the pos-
sibility that an adjoint scalar field φ describes (part of) the thermal ground state.
Namely, we postulate that φ = φ(τ), where 0 ≤ τ ≤ β ≡ 1/T, emerges from
by virtue of a spatial average over (anti)selfdual fundamental field configurations
in euclidean spacetime. Intuitively, we assume that the nontrivial nature of the
Yang-Mills ground state can be described by a scalar-adjoint field φ.
Without the need to perform the average explicitely the field φ enjoys the follow-
ing general properties as a consequence: (i) Since φ is obtained by a spatial coarse-
graining over noninteracting, stable, BPS saturated field configurations (topology
changing energy and pressure free fluctuations) it is itself BPS saturated, thus the
associated energy density vanishes. (ii) Originating from periodic-in-τ field configu-
rations (in a given gauge) it is itself periodic. (iii) The gauge invariant modulus |φ|
does not depend on spacetime (trivial expansion into Matsubara frequencies due to
coarse-graining over energy and pressure free configurations).
We will now show that conditions (i)-(iii) uniquely fix the potential V for the
field φ ≡ φa(τ)λa(trλaλb= 2δab, a = 1,2,3) when working with a canonical kinetic
term in its euclidean Lagrangian density Lφ
Lφ= tr?(∂τφ)2+ V (φ2)?
Since the coarse-graining is over exact solutions to the Yang-Mills equations the
emerging field φ must minimize the effective action. Thus φ satisfies the Euler-
2As long as this term contains two powers of time derivatives this is not a constraint on gener-
ality due to (iii). Moreover, although we ignore the connection to the microscopic physics in the
present work we surely can make an appropriate choice of gauge such that the coarse-graining over
noninteracting topological defects generates Aµ= 0 on the macroscopic level.
Lagrange equations subject to Eq.(1):
∂ |φ|2φa↔ ∂2
τφ =∂V (φ2)
where |φ| ≡
potential) in Eq.(2) implies that the solution has to describe motion in a plane of
the three-dimensional vector space spanned by the Lie-algebra valued generators of
SU(2) Yang-Mills theory. (The angular momentum is a constant of motion in a
Without restriction of generality (a global gauge choice) we choose the plane
(φ1,φ2,0). Thus the solution takes the following form:
2trφ2. The gauge invariance of the potential V = V (|φ|2) (central
φ = |φ|λ1exp(iλ3θ(τ)) = |φ|(λ1cos(θ(τ)) + λ2sin(θ(τ))) ,
or in components
(φ1,φ2,φ3) = |φ|(cos(θ(τ)), sin(θ(τ)), 0). (4)
According to (ii) the function θ(τ) needs to satisfy the following condition
θ(τ + β) = θ(τ) + 2πn,(5)
where n is an integer. Finally, condition (i) implies the vanishing of the (euclidean)
energy density HE(φ):
Substituting Eq.(3) into Eq.(6) we have
HE(φ) = tr(∂τφ)2− V (φ2)
= 2?(∂τφa)2− V (|φ|2)?= 0 ,∀β .(6)
|φ|2(∂τθ(τ))2− V (|φ|2) = 0. (7)
According to (iii) the potential V (|φ|2) does not depend on τ. As a consequence of
Eq.(7), we then have ∂τθ(τ) = const. Together with Eq.(5) this yields:
up to an inessential constant phase (global gauge choice). Notice that the case n = 0
is excluded if we impose that3V ?= 0. Now Eq.(8) implies that ∂2
thus we obtain from Eqs. (3) and (2) that
τθ(τ) = 0, and
(∂τθ(τ))2= −∂V (|φ|2)
3In a similar way, the case Q = 0 is excluded for BPS saturated, microscopic field configurations
if we insist on a nonvanishing action.
Eliminating (∂τθ(τ))2from Eqs.(7) and (9) we have:
= −∂V (|φ|2)
Notice that Eq.(10) is valid for all values of β. The unique solution to the first-order
differential equation (10) is
V (|φ|2) =
where the mass scale Λ enters as a constant of integration. On dimensional grounds
Λ has to appear with the sixth power. We interprete Λ as the Yang-Mills scale which,
however, is not operational on the level of BPS saturated dynamics, see below. (On
this level the energy-momentum tensor vanishes.)
Let us now determine the modulus |φ|. By inserting the potential V (|φ|2) =
Λ6/|φ|2and Eq.(8) into Eq.(7) we have
This implies that the field φ vanishes in a power-like way with increasing temper-
ature. The value of the integer n can not be determined within the macroscopic
approach we have applied to deduce φ’s potential. Microscopically, one observes
that the definition of φ’s phase does only allow for the contribution of Harrington-
Shepard solutions  of topological charge modulus |Q| = 1 which implies that
n = ±1 .
Finally, we point out the inertness of the field φ. According to Eqs.(11) and (1)
the square of the mass Mφof potential (radial) fluctuations δφ is given as (setting
|n| = 1)
T2 = 48π2≫ 1, and no thermal excitations exist. On the other hand, we
λ > λc= 13.87, see . Since |φ| is the maximal resolving power allowed in the
effective theory we conclude that quantum fluctuations of the field φ do not exist.
= 48π2T2. (13)
|φ|2 = 12λ3where λ ≡
Λ. For λ ≫ 1 one has that
|φ|2 ≫ 1. In practice,
3Topologically trivial fluctuations
3.1Effective Lagrangian and ground state
For the reader’s convenience we briefly repeat the derivation of  leading to the
complete ground-state description of SU(2) Yang-Mills thermodynamics in its de-
If topological fluctuations were absent then renormalizability  would assure
that the action of the fundamental theory is form-invariant under the applied spatial
coarse-graining. Since the topological part is integrated into an inert field φ this still
holds true for the part of the effective action induced by Q = 0-fluctuations aµ. We
thus are confronted with the following, gauge invariant effective Lagrangian for the
dynamics of coarse-grained Q = 0 fluctuations aµsubject to the background φ:
Leff= L[aµ] = tr
= ∂µaν− ∂νaµ− ie[aµ,aν] = Ga,µν
aµ = aa
Dµφ = ∂µφ − ie[aµ,φ], (15)
and e denotes an effective gauge coupling. According to Eq.(14) the equation of
motion for the field aµis:
E= ie[φ,Dνφ] .(16)
This is solved by the pure-gauge configuration aµ= ags
e(∂µΩ)Ω†with Ω = e±i2π
2 ⇒ Dνφ = 0 (17)
The entire ground state thus is described by the ags
pressure Pgsand energy density ρgsgiven as Pgs= −4πΛ3T = −ρgs: The inclusion
of gluon fluctuations, which contribute to the dynamics of the ground-state, by
virtue of aµ= ags
BPS saturated configurations alone, to finite values. This makes the Yang-Mills
scale Λ (gravitationally) visible. Turning to propagating fluctuations δaµ in the
effective theory it is advantageous to work in unitary gauge.
µ,φ implying a ground-state
µafter coarse-graining shifts the vanishing results, obtained from
3.2 Unitary gauge and Higgs mechanism
By performing a gauge rotation4U = e−iπ
e(∂µU)U†= 0 and φ = λ3|φ|. This is the unitary gauge. The field strength
4λ2Ω we have that ags
Eand the covariant derivative Dµφ are functionals of the fluctuations δaµonly.
eff = L[δaµ] =1
4Notice that U is smooth and antiperiodic. One can introduce a center jump to make it periodic
by sacrificing its smoothness . However, the associated electric center flux does not carry any
energy or pressure and the periodicity of effective gluon fluctuations is maintained. These are the
physical reasons why the transformation to unitary gauge is admissible.
remains massless representing the fact that SU(2) is broken to its subgroup
U(1) by the field φ. One has
are massive in a temperature dependent way while the mode
3= 0. (19)
3.3Energy density, pressure and running coupling
From the effective Lagrangian (18) we derive the energy density ρ and the pressure
p on the one-loop level. This is accurate on the 0.1%-level as shown in [5, 15].
This strong suppression of the effects of residual Q = 0 quantum fluctuations in
the effective theory takes place due to limited resolution, given by the modulus
|φ|, and due to emergent, temperature-dependent tree-level mass. Both phenomena
introduce nonperturbative aspects into the loop expansion based on the tree-level
action Eq.(18) which render the radiative corrections small.
On the one-loop level we have
ρ = ρ3+ ρ1,2+ ρgs,p = p3+ p1,2+ pgs, (20)
where the subscript 1,2 is understood as a sum over the two massive modes. Ex-
plicitely, we have:
ρ3 = 2π2
30T4, ρ1,2= 6
) − 1, ρgs= 2Λ6
1 − e−
|φ|2= 4πΛ3T .(21)
p3 = 2π2
90T4, p1,2= −6T
, pgs= −ρgs. (22)
The effective coupling constant e is a function of the temperature e = e(T), and so
is m. The function e = e(T) is deduced by requiring the validity of the Legendre
ρ = Tdp
in the effective theory.
By substituting the equations (20) into (23) we obtain:
, D(m) =
Solving the differential equation (24) and inverting the solution, the function e(T)
follows by virtue of Eq.(19).
Eq.(24) is of first order. Thus a boundary condition needs to be prescribed. It
was shown in  that the evolution at low temperature decouples from the boundary
physics at high temperature. That is, there exists a low-temperature attractor to the
evolution. This attractor is characterized by a logarithmic pole, e ∼ −log(T − Tc)
Figure 1: Scaled energy density ρ (black), pressure p (dark gray), and entropy density
s (light gray) in the deconfining phase of SU(2) Yang-Mills thermodynamics.
where Tc= 13.87Λ
e ≡√8π for T sufficiently larger than Tc, indicating magnetic charge conservation
for screened monopoles. In Fig.1 we indicate the (scaled) energy density ρ, pressure
p, and entropy density s as functions of temperature. A detailed comparison of
these results with those obtained on the lattice (for both differential and integral
method) is carried out in the first reference of . While there is good agreement for
the infrared safe quantity entropy density s the pressure p becomes negative close
to the phase boundary which is qualitatively in agreement with the result of the
differential method but not with that of the integral method.
2π, signalizing the presence of a phase transition, and by a plateau
4 Gluon condensate
We start with an intuitive discussion on the gluon condensate.
in , at T = 0 instantons are responsible for a nonzero gluon condensate .
In fact, the average
perturbation theory (Fµν is the fundamental stress-energy tensor). Instantons are
selfdual solutions in euclidean spacetime, which, in Minkowski space, are interpreted
as tunnelling events implying Ei,a= ±iBi,aand so generate a positive average?F2
in euclidean spacetime?F2
Mills dynamics we evaluate the action density by virtue of Eq.(14). Thus the average
where a proportionality between the average ?Leuc
= −4?LY M? = 2
is zero at any order in
An estimate of the gluon condensate is thus obtained by evaluating the action density
Y M? [13, 14].
In the framework of the effective theory for thermalized, deconfining SU(2) Yang-
Y M? ∝ 4?Lu.g.
eff? = 4ρgs= 16πΛ3T ,
Y M? over fundamental fields and the
We notice that the ground-state energy density ρgsis responsible for the emergence
of a nonvanishing thermal average?F2
group invariant object. This holds for
function for the fundamental coupling g, compare with . By virtue of the trace
anomaly we can evaluate this quantity as
eff ? evaluated in the effective theory with coarse-grained fields holds .
More precisely, the gluon condensate should be defined as a renormalization-
Twhere β(g) is the full beta-
= ρ − 3p.(26)
In  we have shown that within the effective theory a linear growth ρ − 3p =
6ρgs= 24πΛ3T for T ≫ Tcfollows. Such a linear growth has been found by lat-
tice simulations [10, 11, 12] and also in an analytical approach . In the latter a
momentum-dependent, universal modification of the dispersion relation for propa-
gating, fundamental gluon fields, motivated by the reduction of the physical state
space a la Gribov, is introduced. While this is interesting a direct comparison of
both approaches beyond the observation of a linear growth of the trace anomaly
would need a coarse-graining over the modified gluon propagation of .
In this article we have derived the potential V (|φ|2) = Λ6/|φ|2for an inert, adjoint
scalar field φ by solely assuming its origin to be a spatial average over noninteract-
ing, BPS saturated topological field configurations in the underlying theory: SU(2)
Yang-Mills thermodynamics being in its deconfining phase. That is, no detailed
microscopic information on these configurations other than their stability and BPS
saturation is needed to derive the potential for the effective field φ. The conceptually
interesting implication of our present work is that the Yang-Mills scale Λ emerges as
a constant of integration: Λ’s existence needs not be assumed as in . For our pre-
sentation to be selfcontained we have repeated the derivation of the effective action,
involving the field φ as a background, for the coarse-grained, topologically trivial
fluctuations. We also have pointed out that the (linear) temperature dependence of
the gluon condensate agrees with that found in lattice simulations.
There is a host of applications of SU(2) Yang-Mills thermodynamics in particle
physics  and cosmology .
 A. D. Linde, Phys. Lett. B 96, 289 (1980).
 A. M. Polyakov, Phys.Lett.B 59, 82 (1975).
A. M. Polyakov, Phys.Lett.B 72, 477 (1978).
 F. Bruckmann, D. Nogradi, and P. van Baal, Nucl. Phys. B 666, 197 (2003).
P. Gerhold, E.-M. Ilgenfritz, M. Muller-Preussker, Nucl. Phys. B 760, 1 (2007).
E.-M. Ilgenfritz, B. V. Martemyanov, M. Muller-Preussker, A. I. Veselov, Phys.
Rev. D 73, 094509 (2006).
 D. Diakonov, N. Gromov, V. Petrov, S. Slizovskiy, Phys. Rev.D 70, 036003
 R. Hofmann, Int. J. Mod. Phys. A 20, 4123 (2005); Erratum-ibid. A 21 (2006)
R. Hofmann, Mod. Phys. Lett. A 21, 999 (2006); Erratum-ibid. A 21, 3049
R. Hofmann, hep-th/0609033.
U. Herbst and R. Hofmann, arXiv:hep-th/0411214.
 B. J. Harrington and H. K. Shepard, Phys. Rev. D 17, 105007 (1978).
 G. ’t Hooft, Nucl. Phys. B 33 (1971) 173. G. ’t Hooft and M. J. G. Veltman,
Nucl. Phys. B 44, 189 (1972). G. ’t Hooft, Int. J. Mod. Phys. A 20 (2005) 1336
 R. Hofmann, work in progress.
 R. Hofmann, arXiv:hep-th/0609033.
 D. E. Miller, Phys. Rept. 443, 55 (2007).
 D. E. Miller, Acta Phys. Polon. B 28 (1997) 2937. G. Boyd and D. E. Miller,
 K. Langfeld, E. M. Ilgenfritz, H. Reinhardt and G. Shin, “Gauge potential
singularities and the gluon condensate at finite Nucl. Phys. Proc. Suppl. 106
(2002) 501 [arXiv:hep-lat/0110024].
 D. Diakonov, arXiv:hep-ph/9602375.
 M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 147 (1979)
 D. Kaviani and R. Hofmann, arXiv:0704.3326, to appear in Mod. Phys. Lett.
 F. Giacosa and R. Hofmann, arXiv:0704.2526 [hep-th].
 F. Giacosa and R. Hofmann, arXiv:hep-th/0703127.
 D. Zwanziger, Phys. Rev. Lett. 94 (2005) 182301.
 M. Schwarz, R. Hofmann, and F. Giacosa, Int. J. Mod. Phys. A 22, 1213 (2007). Download full-text
M. Schwarz, R. Hofmann, and F. Giacosa, JHEP 0702, 091 (2007).
F. Giacosa, R. Hofmann, and M. Schwarz, Mod. Phys. Lett. A 21, 2709 (2006).
 F. Giacosa and R. Hofmann, Eur. Phys. J. C 50, 635 (2007).