#### Probing the two-scale-factor universality hypothesis by exact rotation symmetry-breaking mechanism

Eur. Phys. J. C
Probing the two-scale-factor universality hypothesis by exact rotation symmetry-breaking mechanism
J. F. S. Neto 0
K. A. L. Lima 0
P. R. S. Carvalho 0
M. I. Sena-Junior 1 2
0 Departamento de Fi?sica, Universidade Federal do Piaui? , Teresina, PI 64049-550 , Brazil
1 Escola Polite?cnica de Pernambuco, Universidade de Pernambuco , Recife, PE 50720-001 , Brazil
2 Instituto de Fi?sica, Universidade Federal de Alagoas , Maceio?, AL 57072-900 , Brazil
We probe the two-scale-factor universality hypothesis by evaluating, firstly explicitly and analytically at the one-loop order, the loop quantum corrections to the amplitude ratios for O(N ) ??4 scalar field theories with rotation symmetry breaking in three distinct and independent methods in which the rotation symmetry-breaking mechanism is treated exactly. We show that the rotation symmetrybreaking amplitude ratios turn out to be identical in the three methods and equal to their respective rotation symmetrybreaking ones, although the amplitudes themselves, in general, depend on the method employed and on the rotation symmetry-breaking parameter. At the end, we show that all these results can be generalized, through an inductive process based on a general theorem emerging from the exact calculation, to any loop level and physically interpreted based on symmetry ideas.
1 Introduction
The identical critical behavior displayed by different
physical systems, as a fluid and a ferromagnet, near a continuous
phase transition, had lead to the genesis of the universality
concept [
1?6
] related to the scaling hypothesis [
7?10
]. The
critical behavior of such systems is characterized by an
identical set of critical exponents. When the critical behavior of
two or more systems is described by equal critical exponents,
we say that they belong to the same universality class [
11?
13
]. This occurs when they share the same dimension d, N
and symmetry of some N -component order parameter and if
the interactions present are of short- or long-range type. We
will deal with the general O(N ) universality class which is a
generalization of the specific models with short-range
interactions: Ising (N = 1), XY (N = 2), Heisenberg (N = 3),
self-avoiding random walk (N = 0), spherical (N ? ?)
etc. [
14
]. Furthermore, different systems can be represented
by a single universal equation of state, once one has fixed
two independent thermodynamic scales, as the order
parameter and its conjugate field scales. Then the equation of state
and amplitude ratios for the thermodynamic functions are
universal and thus satisfy the thermodynamic universality
hypothesis [
15,16
]. Stauffer, Ferer and Wortis [17]
generalized the thermodynamic universality concept to the
twoscale-factor universality hypothesis for correlation functions,
where, before that work, it was suggested that universality
for correlation functions would be inferred after the choice
of three scales, with the additional scale to the
thermodynamic ones being the length scale. This hypothesis asserts
that, near the critical point, the length scale is not
independent and it is related to the thermodynamic scales. Thus the
universal correlation function can be fully determined after
the choice of just two independent scales. Unlike the critical
exponents themselves, the critical amplitudes of the
thermodynamic and correlation functions, near the critical point, are
not universal quantities. The universal quantities in this case
are some amplitude ratios of these. The aim of this work is
to evaluate these amplitude ratios.
The purpose of this paper is to employ field-theoretic
renormalization-group and -expansion techniques for
computing, firstly explicitly and analytically at the one-loop
order, the loop quantum corrections to the amplitude ratios
for rotation symmetry-breaking O(N ) ??4 scalar field
theories. This task plays a similar role in the description of a given
universality class, although the evaluation of amplitude ratios
is, in general, harder than those for critical exponents. There
is another field-theoretic renormalization-group approach for
evaluating critical quantities. It is called the field-theoretic
renormalization-group at fixed-dimension approach [
18,19
]
and is based on the computation of critical quantities directly
in three dimensions. One important application of a
rotation symmetry-breaking scalar field theory on the research
area of high energy physics is explaining the Higgs behavior
through the recently proposed rotation symmetry-breaking
Higgs sector of the extended standard model [
20?22
]. In
a conventional rotation-invariant theory, the critical
amplitudes are the amplitudes of the scaling thermodynamic
functions and correlation functions, defined above and below the
critical temperature. These functions, in turn, are a result of
some derivative operations, some of them being derivatives
of the magnetization M , of the reduced temperature, i.e.,
a parameter that is proportional to the difference between
some arbitrary temperature and the critical one, t ? T ? Tc,
etc. with respect to the free energy density or, in the present
language, the effective potential with spontaneous
symmetry breaking. The effective potential at the loop level
considered explicitly and analytically here, the one-loop order,
is composed of two terms. The first term is responsible for
the so-called Landau approximation values to the amplitude
ratios, valid for d ? 4. In the Landau regime, the
fluctuations of the scalar field ?, whose mean value is
identified to the magnetization of the system are discarded. The
second one, representing corrections to the Landau
approximation, which takes into account the fluctuations as loop
quantum corrections, valid for 2 < d < 4, is the
infinite sum of all the one-loop 1PI vertex parts with
amputated external legs. Initially, the effective potential is
written in its bare or nonrenormalized form, thus plagued by
infrared divergences, typical for massless theories, as the
treated in this work. These divergences must be removed
of the theory and are contained in just a few 1PI vertex
parts, the ?B(2), ? (4) and ? (2,1) functions commonly called
B B
primitively divergent 1PI vertex parts. All the others
divergent 1PI vertex parts, obtained through a skeleton expansion
[
23
] of the former, turn out to be automatically
renormalized once one has renormalized the primitive divergent ones.
For attaining the renormalized theory, we will apply three
independent renormalization schemes: normalization
conditions [
24
], the minimal subtraction scheme [
26
] and the
Bogoliubov?Parasyuk?Hepp?Zimmermann (BPHZ)
methods [
27?29
]. Universality is satisfied if the final results for
the amplitude ratios, in the three distinct methods, are
identical, although the amplitudes themselves, in general, depend
on the renormalization scheme employed and on the rotation
symmetry-breaking mechanism through the introduction of
an appropriate rotation symmetry-breaking parameter to be
defined below. As universality means that the critical
exponents do not depend on the system in a given universality
class, the corresponding critical exponents must be the same
when obtained through any renormalization scheme.
Universality then arises in any renormalization scheme through
the flow of the renormalized coupling constant to its fixed
point value in which scale invariance is manifest. The present
calculation of amplitude ratios in just one renormalization
scheme would be enough for showing that they are
independent of the symmetry-breaking tensor K?? at one-loop
order. But similar calculations in different renormalization
schemes besides providing a check of the final results that
must be identical, give more robustness on the
two-scalefactor universality hypothesis validity. Furthermore, the
minimal subtraction method for obtaining the amplitude ratios
presented here, as opposed to the normalization conditions
method, is not found in the literature. The probing of a
possible effect of the rotation symmetry-breaking mechanism on
the universality properties of the systems studied here, starts
with the introduction into the rotation-invariant standard
theory, the kinetic rotation symmetry-breaking O(N )
operator K?? ????? ?, as introduced for the pioneer evaluation
of rotation symmetry-breaking critical exponents by one of
the authors and co-workers [
30?32
], although non-exactly in
the rotation symmetry-breaking mechanism through tedious
calculations in powers of K?? . The dimensionless,
symmetric, constant rotation symmetry-breaking coefficients K?? =
K?? are equal for all the N components of the field and leave
intact the O(N ) symmetry of the N -component field.
Physically, they act as a constant background field. If the rotation
symmetry-breaking coefficients are kept at arbitrary values,
the rotation symmetry symmetry is violated if these
coefficients do not transform as a second order tensor under rotation
transformations. As for the earlier work on the computation
of rotation symmetry-breaking critical exponents, the
rotation symmetry-breaking theory can be used for studying the
symmetry aspects of the O(N ) two-scale-factor universality
class in the rotation symmetry-breaking scenario, now
treating the rotation symmetry-breaking mechanism exactly.
2 Amplitude ratios in normalization conditions scheme
The normalization conditions renormalization scheme is
characterized by fixing the external momenta of the
primitively divergent 1PI vertex parts at a nonzero value scaled
by some arbitrary momentum scale ?, at the symmetry points
S P and S P
? (2)( P2 + K?? P? P? = 0, g) = 0,
?? (2)( P2 + K?? P? P? , g)
?( P2 + K?? P? P? )
P2+K?? P? P? =?2
= 1,
? (4)( P2 + K?? P? P? , g)|S P = g,
? (2,1)( P1, P2, Q3, g)|S P = 1,
where for SP: Pi ? Pj = (?2/4)(4?i j ? 1), implying that
( Pi + Pj )2 ? P2 = ?2 for i = j . For S P: Pi2 = 3?2/4 and
P1 ? P2 = ??2/4. This implies ( P1 + P2)2 ? P2 = ?2,
of the multiplicatively renormalized primitively 1PI
ver(1)
(2)
(3)
(4)
tex parts ? (n,l)( Pi , Q j , g, ?) = Z n/2 Z ?l2 ? (n,l)( Pi , Q j , ?0)
? B
(i = 1, . . . , n, j = 1, . . . , l, where for (n, l) = (0, 2),
the function ? (0,2) is renormalized additively). These are
B
generated by the initially bare rotation symmetry-breaking
Lagrangian density
LB = 21 ???B ???B + K?? ???B ?? ?B + ?4B ?4B + 21 tB ?2B ,
!
where the conditions (1)?(4) permit us to renormalize the
bare field ?B , the coupling constant ?B and composite field
coupling constant tB parameters. Thus after the
renormalization of these parameters, we can write down the renormalized
rotation symmetry-breaking free energy density at the fixed
point with spontaneous symmetry breaking at one-loop level:
[
24
]
(5)
where g, t and M are the renormalized coupling constant,
composite field coupling constant and magnetization (as
being the renormalized nonzero field mean value M = ?
in the spontaneously broken direction), respectively. The
coupling constant g? is the fixed point of the theory, the
value for which the renormalized coupling constant flows
naturally when the renormalized theory is attained and, in
general, is obtained as the nontrivial root of the ?-function
of the respective theory in the respective renormalization
scheme. The ?fish? diagram SP , whose internal line is
given by the massless propagator ?1 ? q2 + K??q?q?,
is evaluated at the symmetry point S P after we set ?2 = 1,
because we can redefine all momenta of the diagrams in
units of ?. Thus the redefined external momenta turn out
to be dimensionless and the symmetry point now is given by
P2 + K?? P? P? = ?2 ? 1. Then we absorb the dependence
on ? of the diagrams into the coupling constant. The ?fish?
diagram was evaluated in an expansion in the dimensional
regularization parameter = 4 ? d and exactly in K [
33
];
see Sect. 5. In the analytical computation of the momentum
integral, we apply the well-known definition of Ref. [
26
]
in which the d-dimensional surface area factor is absorbed
into a redefinition of the coupling constant, since each loop
momentum integration is accompanied of this factor. Thus,
(7)
we can write
SP
=
1
1 +
where = 1/?det (I + K) is a rotation symmetry-breaking
full factor emerging from the exact calculation where we
change the variables [
33
] through coordinates redefinition
applied in momentum space directly in Feynman diagrams
q = ?I + K q resulting in the fact that each loop
integration is accompanied by a rotation symmetry-breaking full
factor and according to [
33
],
Now, we are in a position to evaluate the critical
amplitudes of the thermodynamic and correlation functions. The
existence of two independent scales, leads naturally to the
existence of ten relations among the 12 critical exponents
?, ? , ? , ? , ?, ? , ?, ?, ?, ?c, ?c, ?c as well as for the critical
amplitudes, with ten universal relations among the 12 critical
amplitudes
Fortunately, not all critical exponents must be evaluated,
because not all of them are independent. Some of them are
related, as the ones defined above and below the critical
temperature and through some scaling relations among them,
namely ? = ? , ? = ? , ? = ? , ? = ?(? ? 1), ? =
2 ? 2? ? ? , 2 ? ? = d?, ? = (2 ? ?)?, ?c = ?/??, ?c =
1 ? 1/?, ?c = ?/??, thus two independent ones remaining.
Also, as H and ? are related on the critical isotherm, the
universal relation ?C c D1/? follows, and nine universal relations
among the critical amplitudes remain. More than nine
universal relations can be derived, some of them being dependent
of a minimal set of nine ones. This shows that we can choose
a given minimal set. The minimal set chosen in this paper
will be that whose the -expansion results are displayed in
Ref. [
34
], originally evaluated in the references therein.
Equation of state Before computing the amplitude ratios
themselves, it is important to begin by computing the
equation of state and its universal form. The equation of state is
obtained as a first derivative of the free energy with respect
to the magnetization, namely H = ?F /? M [
24
], whose
diagrammatic expression is given by [
35
]
The indices (1) and (N ?1) indicate that in the respective
diagram, the internal propagators are
evaluation of the Feynman diagrams
in d = 4 ? results in
and
, respectively. The
and
(9)
(10)
(11)
(12)
(13)
and we have a similar expression for with
g? M 2/2 ? g? M 2/6. The equation of state is clearly a
rotation symmetry-breaking one. But if H (x ) is normalized at
the values x = 0 and x = ?1 such that H = M ? and
H = 0, respectively, where x = t (g? M 2)?1/2? , the
rotation symmetry-breaking and f (2) factors disappear and it
assumes its well-known universal form [
25
]. Now we
proceed by evaluating the minimal set of amplitude ratios.
A+/ A?. The critical amplitudes for the specific heat can
be obtained as the second derivative of the free energy
F (t, M, g?) with respect to t . For describing the regions
above and below the transition point, the values of the
magnetization in the respective regions can be calculated by
minimizing the effective potential (equivalently the roots of
H ), giving the magnetization values of the O(N ) symmetric
phase and the spontaneously broken one. The results for the
referred amplitudes are
(N + 2)(N 2 + 30N + 56) .
2(4 ? N )(N + 8)2
C +/C ?. We can obtain the amplitude ratio C +/C ? by
computing the susceptibility in terms of the effective
potential or, equivalently, of the equation of state, through ? ?1 =
?2F /? M 2 = ? H/? M and evaluating the amplitudes above
and below the transition. We have to mention a
peculiarity here: Below the critical temperature, the susceptibility is
defined only for Ising systems due to the presence of
Goldstone modes. Thus we have
C + = 1 ? 2(NN++28) ,
.
Q1. The amplitude ratio Q1 is related to the R? one
through Q1 = R??1/?, where R? is defined as R? =
C + D B??1. Thus the universality of Q1 is ensured if R?
is universal. The amplitude C + is displayed in Eq. (14). The
amplitude D is obtained by normalizing H (x ) only at the
value x = 0 such that H = M ?, where x = t (g? M 2)?1/2? ,
implying that
1 1
D = 6 g?(??1)/2 1 + 2
1 ? ln 2 ? NN ?+ 18 ln 3
. (16)
The amplitude B can be calculated from the nonzero root of
H , namely
B =
N + 8
3
1 ? N + 8 (1 + ln 2)
9N + 42 1
? (N + 8)2 ? 2
1/2
.
Rc. The ratio Rc is defined as Rc = A+C +/B2. All the
amplitudes necessary to the computation of Rc were
evaluated already in Eqs. (11), (14) and (17)
?0+/?0?. For calculating the amplitude ratio between the
correlation length above and below the transition, we have to
consider the momentum-dependent longitudinal correlation
function [
24
] with the diagrammatic expansion [
35
]
(14)
(15)
(17)
(18)
and
diagrams
After the -expansion of the
exactly in K , we have
where
L( P2 + K?? P? P? )
1
=
0
dx ln x (1 ? x )( P2 + K?? P? P? ) + t +
with an analog expression for with g? M 2/2 ?
g? M 2/6, now defining the correlation length as the
second moment of the spin?spin correlation function as ? 2 =
(d?L /d( P2+K?? P? P? ))|P2+K?? P? P? =0/? ( P2+K?? P? P?
= 0),
?0+ = 1 ? 4(NN ++28) ,
1 7
?0? = 2?1/2 1 ? 12 2 + ln 2
.
R?+ = ?0+( A+)1/d .
As we can see in Eq. (22), ?0? is not defined for all N =
1. The reason is the same as for the critical amplitude C ?:
the existence of divergences generated by the presence of
Goldstone modes.
R?+. The amplitudes needed for the evaluation of the ratio
R?+ are displayed in Eqs. (11) and (21) through the definition
?0+/?0T . The momentum-dependent transverse correlation
function [
24
] is given by its diagrammatic expansion [
35
]
(19)
g? M 2
2
,
(20)
(21)
(22)
(23)
where the index (1, (N ? 1)) in the referred diagram means
that its internal propagators are the and the
ones, respectively. The -expansion of this diagram results in
similar expressions to the (19)?(20) ones with the substitution
t + g? M 2/2 ? x (t + g? M 2/2) + (1 ? x )(t + g? M 2/6).
Thus defining the transverse correlation length by ?T ( P2 +
?0T =
Q2. For the evaluation of the ratio Q2 defined by Q2 =
(C +/C c)(?0c/?0+)2?? (the critical exponent ? can be set to
zero at the loop level here), we have computed the amplitudes
C + and ?0+ already in Eqs. (14) and (21), respectively. Now,
from the susceptibility and correlation length at the critical
point, we get
D = 1.
As can be seen, the rotation symmetry-breaking full
factor disappear in the final expressions to all the amplitude
ratios above and we obtain their rotation-invariant
counterparts [
34
].
3 Amplitude ratios in minimal subtraction scheme
In the minimal subtraction renormalization scheme, the
external momenta of the 1PI vertex parts to be
renormalized, by minimally subtracting the dimensional poles, are
held at arbitrary values, showing that this method is more
general and elegant than the earlier. Thus, Eqs. (1)?(4) must
not hold necessarily. Then, minimally renormalizing the bare
field ?B , coupling constant ?B and composite field coupling
constant tB , the renormalized effective potential F (t, M, g?)
of Eq. (6) assumes a similar form, but with the change
, where
is, for example, the
diagram or, similarly, the one with their external
momenta held at arbitrary values and [ ]S means that what
is to be considered inside the brackets are the singular terms
N
A+ = 4
A? =
1 +
where
1 +
1/(d?2)
,
,
1/2
,
1 1
D = 6 g?(??1)/2 1 ? 2
ln 2 + NN ?+ 18 ln 3
N + 8
3 ln 2
1 ? N + 8
9N + 42
? (N + 8)2
,
1/2
,
1 5
?0? = 2?1/2 1 ? 12 2 + ln 2
,
of the diagram, not the regular ones, as opposed to the
normalization conditions renormalization method in which the
regular terms are also taken into account. The nontrivial fixed
point, i.e., the nontrivial root of the ?-function in this scheme
is given by [
31
]
Thus, performing the same steps of the earlier section, the
expressions for the amplitudes are
esting feature of this method is that the amplitudes
themselves do not depend explicitly on the rotation
symmetrybreaking full factor. The absence of an explicit dependence
on non-universal features is not an exception of the
rotation symmetry-breaking properties in this method. It is also
observed in treating critical properties of rotation-invariant
finite-size systems subject to periodic and anti-periodic [
36
]
as well as Dirichlet and Newmann [
37
] boundary conditions.
4 Amplitude ratios in the BPHZ method
In the BPHZ method, the most general and elegant one, the
divergences of the bare theory are eliminated by adding terms
to the initially bare Lagrangian density. The terms introduced
are called counterterms. This process can be applied order
by order in perturbation theory. At the loop level considered
in this work, we can once again renormalize the field ?B ,
the coupling constant ?B and the composite field coupling
constant tB by adding counterterms to turn them finite [
32
].
The resulting renormalized effective potential F (t, M, g?),
with g? being the same as in Eq. (28), will display once
again the form in Eq. (6), now with .
The symbol K( ) indicates that only the singular part of the
diagram is to be considered. Although this renormalization
process is distinct of that of the earlier section, at least at the
loop level treated here the effective potential is the same as the
one of the earlier section, thus leading to the same amplitudes
as computed in that section. This leads to identical amplitude
ratios found in the last two sections. Now we proceed to a
computation of the amplitude ratios valid for all-loop order.
5 All-loop order amplitude ratios
In this section, we show the universality of the amplitude
ratios for O(N ) rotation symmetry-breaking self-interacting
scalar field theories valid for any loop level in the BPHZ
method, the most general and elegant one (without loss of
generality, similar arguments can be used in the other
methods). We follow the same steps as Ref. [
35
] used for
rotationinvariant amplitude ratios. A given general amplitude AG is
of the form AG = X a Y b F (g?), where a and b are integer
numbers, critical exponents or a combination of them and
F a general function of g? only. The X and Y factors are
non-universal and are responsible for the scale-dependence
of the critical amplitudes. They are related, for example, to
the order parameter and conjugate field scales, which are
the two fixed independent scales needed for the
establishment of the two-scale-factor universality hypothesis. Another
source of non-universality is contained in the general
function F (g?), through the fixed point, whose value depends on
?0T =
?0c =
The amplitude ratios obtained in Eqs. (29)?(41) are the same
as the respective rotation-invariant ones [
34
], as well as
the equation of state in its universal form [
24
]. One
interthe renormalization scheme employed. It was shown, in a
general proof and therefore valid for any loop level [
35
], that
the amplitude ratios are independent of the fixed independent
scale factors. Thus, we have, additionally, to show that the
amplitude ratios are independent of the rotation
symmetrybreaking parameters K?? , thus being identical to the
corresponding rotation-invariant ones. This task will be achieved
if we show that the rotation symmetry-breaking general
function F (g?) is the same as its rotation-invariant counterpart,
i.e. is does not depend on K?? . We apply the general theorem
[
33
]
Theorem 1 Consider a given Feynman diagram in
momentum space of any loop order in a theory represented by the
Lagrangian density of Eq. (5). Its evaluated expression in
dimensional regularization in d = 4 ? can be written
as a general functional L G(g, P2 + K?? P? P? , , ?) if
its rotation-invariant counterpart is given by G(g, P2, , ?),
where L is the number of loops of the corresponding diagram.
Proof A general Feynman diagram of loop level L is a
multidimensional integral in L distinct and independent
momentum integration variables q1, q2, . . . , qL , each one with
volume element given by dd qi (i = 1, 2, . . . , L). As showed in
the last section, the substitution q = ?I + K q transforms
edadcqh =vodlud mqe/?eldeemt(eIn+taKs )d?dq d=d q?, det=(I +1/?K)ddedt(qI.+TKhu)s.
Then the integration in L variables results in a rotation
symmetry-breaking overall factor of L . Now putting q ?
P in the substitution above, where P is the transformed
external momentum, we have P 2 = P2 + K?? P? P? . So a
given Feynman diagram, evaluated in dimensional
regularization in d = 4 ? , assumes the expression L G(g, P2 +
K?? P? P? , , ?), where G is associated to the corresponding
diagram if the rotation-invariant Feynman diagram
counterpart evaluation results in G(g, P2, , ?).
Now we can write a general Feynman diagram in the
form L G(g, P2 + K?? P? P? , , ?) if its rotation-invariant
counterpart is given by G(g, P2, , ?), where L is the
number of loops of the referred diagram. Thus, as the
general function depends only on g and at a given of its
terms each term of one order of g is also a term of one
order of loop, it is given by F ( g?). The all-loop rotation
symmetry-breaking fixed point g?, taking into account the
rotation symmetry-breaking breaking mechanism exactly,
is related to its rotation-invariant counterpart g?(0) through
g? = g?(0)/ [
33
]. Now, substituting g? = g?(0)/ , we
have F ( g?) ? F (g?(0)). Then the rotation
symmetrybreaking general function is the same as the
corresponding rotation-invariant one, leading to the rotation
symmetrybreaking amplitude ratios identical to their rotation-invariant
counterparts. This completes our task.
6 Conclusions
In this paper, we have evaluated the all-loop quantum
corrections to the amplitude ratios for rotation symmetry-breaking
O(N ) ??4 scalar field theories, taking exactly the
rotation symmetry-breaking mechanism into account, through
a redefinition of the coordinates applied in momentum space
directly in Feynman diagrams, and thus avoiding the tedious
calculation in powers of K?? , by employing field-theoretic
renormalization group, dimensional regularization and
expansion techniques in three distinct methods. Firstly, we
have explicitly computed analytically the amplitude ratios
at the one-loop level and finally, in a proof by induction
through a general theorem emerging from the exact
calculation, computed the quantum corrections for any loop level.
We have showed that, although the same rotation
symmetrybreaking critical amplitude can be different in distinct
methods and thus can be dependent on the renormalization method
employed, the outcomes for the amplitude ratios are the same
and, furthermore, equal to their rotation-invariant
counterparts. This result reveals that the amplitude ratios do not
depend on the renormalization scheme employed and on
the exact rotation symmetry-breaking mechanism, thus being
universal quantities and ratifying the robustness of the O(N )
two-scale-factor universality hypothesis. We can interpret
physically this result by realizing that the symmetry-breaking
mechanism does not occur in the internal symmetry space of
the field, but in the one in which the field is defined [
38
]. Then,
if the amplitude ratios are really to be universal quantities,
the values of the amplitude ratios must be not affected by
this symmetry-breaking mechanism and they depend just, as
usual, on the d and N parameters and the symmetry of the N
component order parameter. Furthermore, this work can shed
light on the understanding of the exact rotation
symmetrybreaking properties of corrections to scaling and finite-size
scaling of rotation symmetry-breaking amplitude ratios as
well as critical exponents in geometries subjected to
different boundary conditions for systems undergoing a continuous
phase transition for the systems studied here as well as for
anisotropic ones [
39?42
].
Acknowledgements JFSN, KALL and PRSC would like to thank
FAPEPI-CAPES, CAPES (Brazilian funding agencies) and
Universidade Federal do Piau? for financial support, respectively. MISJ was
financially supported by FAPEAL (Alagoas State Research
Foundation) and CNPq (Brazilian Funding Agency).
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Funded by SCOAP3.
1. M.E. Fisher, Phys. Rev. Lett . 16 , 11 ( 1966 )
2. P.G. Watson , J. Phys. C 2 , 1883 ( 1969 )
3. P.G. Watson , J. Phys . C 2 , 2158 ( 1969 )
4. D. Jasnow , M. Wortis , Phys. Rev . 176 , 738 ( 1968 )
5. R.B. Griffiths , Phys. Rev. Lett . 24 , 1479 ( 1970 )
6. L.P. Kadanoff , in Phase Transitions and Critical Phenomena, ed. by M. S. Green , vol. 5A (Academic, New York, 1971 )
7. B. Widom , J. Chem . Phys. 43 , 3898 ( 1965 )
8. L.P. Kadanoff , Phys. 2 , 263 ( 1966 )
9. A.Z. Patashinskii , V.L. Pokrovskii , Zh Eksp, Teor. Fiz 50 , 439 ( 1966 )
10. J.W. Essam , M.E. Fisher, J. Chem . Phys. 39 , 842 ( 1963 )
11. K.G. Wilson, M.E. Fisher, Phys. Rev. Lett . 28 , 240 ( 1972 )
12. K.G. Wilson, Phys. Rev. Lett . 28 , 548 ( 1972 )
13. K.G. Wilson, J. Kogut, Phys. Rev. Lett . 12 , 75 ( 1974 )
14. A. Pelissetto , E. Vicari, Phys. Rep . 368 , 549 ( 2002 )
15. R.B. Griffiths , Phys. Rev . 158 , 176 ( 1967 )
16. M. Barmatz , P.C. Hohenberg , A. Kornblit , Phys. Rev. B 12 , 1947 ( 1975 )
17. D. Stauffer , M. Ferer , M. Wortis , Phys. Rev. Lett . 29 , 345 ( 1972 )
18. G. Parisi, J. Stat . Phys. 23 , 49 ( 1980 )
19. C. Bagnuls , C. Bervillier , D.I. Meiron , B.G. Nickel , Phys. Rev. B 35 , 3585 ( 1987 )
20. A. Ferrero , B. Altschul , Phys. Rev. D 84 , 065030 ( 2011 )
21. P.R.S . Carvalho, Phys. Lett. B 726 , 850 ( 2013 )
22. P.R.S . Carvalho, Phys. Lett. B 730 , 320 ( 2014 )
23. J. Zinn-Justin , Quantum Field Theory and Critical Phenomena (Oxford University Press, Oxford, 2002 )
24. E. Brezin, J.C. Le Guillou , J. Zinn-Justin, in Phase Transitions and Critical Phenomena, ed. by C. Domb , M.S.A. Green , Vol. 6 (Academic, New York, 1976 )
25. E. Brezin , D.J. Wallace , K.G. Wilson, Phys. Rev. B 7 , 232 ( 1973 )
26. D.J. Amit , V. Mart?n-Mayor , Field Theory , The Renormalization Group and Critical Phenomena (World Scientific Pub Co Inc , Singapore, 2005 )
27. N.N. Bogoliubov , O.S. Parasyuk , Acta Math. 97 , 227 ( 1957 )
28. K. Hepp , Commun. Math. Phys. 2 , 301 ( 1966 )
29. W. Zimmermann, Commun. Math. Phys. 15 , 208 ( 1969 )
30. W.C. Vieira, P.R.S. Carvalho , Europhys. Lett. 108 , 21001 ( 2014 )
31. W.C. Vieira, P.R.S. Carvalho , Int. J. Geom. Methods Mod. Phys . 13 , 1650049 ( 2016 )
32. P.R.S. Carvalho , Int. J. Mod. Phys. B 30 , 1550259 ( 2016 )
33. P.R.S. Carvalho , M.I. Sena-Junior, Eur . Phys. J. C. https://doi.org/ 10.1140/epjc/s10052-017-5304-9
34. V. Privman , P.C. Hohenberg , A . Aharony, in Phase Transitions and Critical Phenomena, ed. by C. Domb , J.L. Lebowitz , vol. 14 (Academic, New York, 1991 )
35. C. Bervillier, Phys. Rev. B 14 , 4964 ( 1976 )
36. J.B. Silva Jr., M.M. Leite , J. Math. Phys. 53 , 043303 ( 2012 )
37. M.V.S. Santos , J.B. Silva Jr. , M.M. Leite , ( 2015 ). arXiv: 1509 . 05793
38. A. Aharony, in Phase Transitions and Critical Phenomena, ed. by C. Domb , M.S.A. Green , vol. 6 (Academic, New York, 1976 )
39. M.M. Leite , Phys. Rev. B 67 , 104415 ( 2003 )
40. M.M. Leite , Phys. Rev. B 72 , 224432 ( 2005 )
41. P.R.S. Carvalho , M.M. Leite , Ann. Phys. 324 , 178 ( 2009 )
42. P.R.S. Carvalho , M.M. Leite , Ann. Phys. 325 , 151 ( 2010 )