Kinks and Particles in Non-integrable
Quantum Field Theories
G. Mussardoa,b
aInternational School for Advanced Studies
Via Beirut 1, 34013 Trieste, Italy
bIstituto Nazionale di Fisica Nucleare, Sezione di Trieste
Abstract
In this talk we discuss an elementary derivation of the semi-classical spec-
trum of neutral particles in two field theories with kink excitations. We also
show that, in the non-integrable cases, each vacuum state cannot generically
support more than two stable particles, since all other neutral exitations are
resonances, which will eventually decay.
http://arxiv.org/abs/0704.0549v1
1 Introduction
Two–dimensional massive Integrable Quantum Field Theories (IQFTs) have proven
to be one of the most successful topics of relativistic field theory, with a large
variety of applications to statistical mechanical models. The main reason for this
success consists of their simplified on–shell dynamics which is encoded into a set of
elastic and factorized scattering amplitudes of their massive particles [1, 2]. The two-
particle S-matrix has a very simple analytic structure, with only poles in the physical
strip, and it can be computed combining the standard requirements of unitarity,
crossing and factorization together with specific symmetry properties of the theory.
The complete mass spectrum is obtained looking at the pole singularities of the
S–matrix elements. Off–mass shell quantities, such as the correlation functions, can
be also determined once the elastic S–matrix and the mass spectrum are known.
In fact, one can compute the exact matrix elements of the (semi)local fields on the
asymptotic states with the Form Factor (FF) approach [3], and use them to write
down the spectral representation of the correlators. By following this approach, it
has been possible, for instance, to tackle successfully the long-standing problem of
spelling out the mass spectrum and the correlation functions of the two dimensional
Ising model in a magnetic field [2, 4], as well as many other interesting problems of
statistical physics (for a partial list of them see, for instance, [5]).
The S-matrix approach can be also constructed for massless IQFTs [6, 7, 8, 9],
despite the subtleties in defining a scattering theory between massless particles in
(1 + 1) dimensions, and turns out to be useful mainly when conformal symmetry
is not present. In this case, massless IQFTs generically describe the Renormaliza-
tion Group trajectories connecting two different Conformal Field Theories, which
respectively rule the ultraviolet and infrared limits of all physical quantities along
the flows.
Given the large number of remarkable results obtained by the study of IQFTs,
one of the most interesting challenges is to extend the analysis to the non–integrable
field theories, at least to those obtained as deformations of the integrable ones and
to develop the corresponding perturbation theory. The breaking of integrability
is expected to considerably increase the difficulties of the mathematical analysis,
since scattering processes are no longer elastic. Non–integrable field theories are
in fact generally characterized by particle production amplitudes, resonance states
and, correspondingly, decay events. All these features strongly effect the analytic
structure of the scattering amplitudes, introducing a rich pattern of branch cut
singularities, in addition to the pole structure associated to bound and resonance
states. For massive non–integrable field theories, a convenient perturbative scheme
was originally proposed in [10] and called Form Factor Perturbation Theory (FFPT),
since it is based on the knowledge of the exact Form Factors (FFs) of the original
integrable theory. It was shown that, even using just the first order correction of
the FFPT, a great deal of information can be obtained, such as the evolution of
their particle content, the variation of their masses and the change of the ground
state energy. Whenever possible, universal ratios were computed and successfully
compared with their value obtained by other means. Recently, for instance, it has
been obtained the universal ratios relative to the decay of the particles with higher
masses in the Ising model in a magnetic field, once the temperature is displayed away
from the critical value [11] (see also the contibution by G. Delfino in this proceedings
[12]). For other and important aspects of the Ising model along non-integrable lines
see the references [13, 14, 15, 16]. Applied to the double Sine–Gordon model [17], the
FFPT has been useful in clarifying the rich dynamics of this non–integrable model.
In particular, in relating the confinement of the kinks in the deformed theory to the
non–locality properties of the perturbed operator and predicting the existence of a
Ising–like phase transition for particular ratios of the two frequencies – results which
were later confirmed by a numerical study [18]. The FFPT has been also used to
study the spectrum of the O(3) non-linear sigma model with a topological θ term,
by varying θ [19, 20].
In this talk I would like to focus the attention on a different approach to tackle
some interesting non-integrable models, i.e. those two dimensional field theories
with kink topological excitations. Such theories are described by a scalar real field
ϕ(x), with a Lagrangian density
(∂µϕ)
2 − U(ϕ) , (1.1)
where the potential U(ϕ) possesses several degenerate minima at ϕ
a (a = 1, 2, . . . , n),
as the one shown in Figure 1. These minima correspond to the different vacua | a 〉
of the associate quantum field theory.
The basic excitations of this kind of models are kinks and anti-kinks, i.e. topological
configurations which interpolate between two neighbouring vacua. Semiclassically
they correspond to the static solutions of the equation of motion, i.e.
∂2x ϕ(x) = U
′[ϕ(x)] , (1.2)
with boundary conditions ϕ(−∞) = ϕ(0)a and ϕ(+∞) = ϕ(0)b , where b = a ± 1.
Denoting by ϕab(x) the solutions of this equation, their classical energy density is
(A) (B)
Figure 1: Potential U(ϕ) of a quantum field theory with kink excitations (A) and
istogram of the masses of the kinks (B).
given by
ǫab(x) =
+ U(ϕab(x)) , (1.3)
and its integral provides the classical expression of the kink masses
Mab =
ǫab(x) . (1.4)
It is easy to show that the classical masses of the kinks ϕab(x) are simply proportional
to the heights of the potential between the two minima ϕ
a and ϕ
: their istogram
provides a caricature of the original ptential (see Figura 1).
The classical solutions can be set in motion by a Lorentz transformation, i.e.
ϕab(x) → ϕab
(x± vt)/
1− v2
. In the quantum theory, these configurations de-
scribe the kink states | Kab(θ) 〉, where a and b are the indices of the initial and final
vacuum, respectively. The quantity θ is the rapidity variable which parameterises
the relativistic dispersion relation of these excitations, i.e.
E = Mab cosh θ , P = Mab sinh θ . (1.5)
Conventionally | Ka,a+1(θ) 〉 denotes the kink between the pair of vacua {| a 〉, | a+ 1 〉}
while | Ka+1,a 〉 is the corresponding anti-kink. For the kink configurations it may
be useful to adopt the simplified graphical form shown in Figure 2.
The multi-particle states are given by a string of these excitations, with the adja-
cency condition of the consecutive indices for the continuity of the field configuration
| Ka1,a2(θ1)Ka2,a3(θ2)Ka3,a4(θ3) . . .〉 , (ai+1 = ai ± 1) (1.6)
In addition to the kinks, in the quantum theory there may exist other excitations in
the guise of ordinary scalar particles (breathers). These are the neutral excitations
a,a+1K K a+1,a
| a+1>
| 0 >
| a  >
| n  >
Figure 2: Kink and antikink configurations.
| Bc(θ) 〉a (c = 1, 2, . . .) around each of the vacua | a 〉. For a theory based on a
Lagrangian of a single real field, these states are all non-degenerate: in fact, there
are no extra quantities which commute with the Hamiltonian and that can give
rise to a multiplicity of them. The only exact (alias, unbroken) symmetries for a
Lagrangian as (1.1) may be the discrete ones, like the parity transformation P , for
instance, or the charge conjugation C. However, since they are neutral excitations,
they will be either even or odd eigenvectors of C.
The neutral particles must be identified as the bound states of the kink-antikink
configurations that start and end at the same vacuum | a 〉, i.e. | Kab(θ1)Kba(θ2) 〉,
with the “tooth” shapes shown in Figure 3.
| 0 >
| 0 >
| 0 >
Figure 3: Kink-antikink configurations which may give rise to a bound state nearby
the vacuum | 0 〉a.
If such two-kink states have a pole at an imaginary value i ucab within the physical
strip 0 < Im θ < π of their rapidity difference θ = θ1 − θ2, then their bound states
are defined through the factorization formula which holds in the vicinity of this
singularity
| Kab(θ1)Kba(θ2) 〉 ≃ i
θ − iuc
| Bc 〉a . (1.7)
In this expression gcab is the on-shell 3-particle coupling between the kinks and the
neutral particle. Moreover, the mass of the bound states is simply obtained by sub-
stituing the resonance value i ucab within the expression of the Mandelstam variable
s of the two-kink channel
s = 4M2ab cosh
−→ mc = 2Mab cos
. (1.8)
Concerning the vacua themselves, as well known, in the infinite volume their
classical degeneracy is removed by selecting one of them, say | k 〉, out of the n
available. This happens through the usual spontaneously symmetry breaking mech-
anism, even though – stricly speaking – there may be no internal symmetry to break
at all. This is the case, for instance, of the potential shown in Figure 1, which does
not have any particular invariance. In the absence of a symmetry which connects the
various vacua, the world – as seen by each of them – may appear very different: they
can have, indeed, different particle contents. The problem we would like to examine
in this talk concerns the neutral excitations around each vacuum, in particular the
question of the existence of such particles and of the value of their masses. To this
aim, let’s make use of a semiclassical approach.
2 A semiclassical formula
The starting point of our analysis is a remarkably simple formula due to Goldstone-
Jackiw [23], which is valid in the semiclassical approximation, i.e. when the coupling
constant goes to zero and the mass of the kinks becomes correspondingly very large
with respect to any other mass scale. In its refined version, given in [24] and redis-
covered in [25], it reads as follows1 (Figure 4)
(θ) = 〈Kab(θ1) | ϕ(0) | Kab(θ2)〉 ≃
dx eiMab θ x ϕab(x) , (2.9)
where θ = θ1 − θ2.
1The matrix element of the field ϕ(y) is easily obtained by using ϕ(y) = e−iPµy
ϕ(0) eiPµy
and by acting with the conserved energy-momentum operator Pµ on the kink state. Moreover, for
the semiclassical matrix element FG
(θ) of the operator G[ϕ(0)], one should employ G[ϕab(x)]. For
instance, the matrix element of ϕ2(0) are given by the Fourier transform of ϕ2
Figure 4: Matrix element between kink states.
Notice that, if we substitute in the above formula θ → iπ − θ, the corresponding
expression may be interpreted as the following Form Factor
(θ) = f(iπ − θ) = 〈a | ϕ(0) | Kab(θ1)Kba(θ2)〉 . (2.10)
In this matrix element, it appears the neutral kink states around the vacuum | a〉
we are interested in.
Eq. (2.9) deserves several comments.
1. The appealing aspect of the formula (2.9) stays in the relation between the
Fourier transform of the classical configuration of the kink, – i.e. the solu-
tion ϕab(x) of the differential equation (1.2) – to the quantum matrix ele-
ment of the field ϕ(0) between the vacuum | a 〉 and the 2-particle kink state
| Kab(θ1)Kba(θ2) 〉.
Once the solution of eq. (1.2) has been found and its Fourier transform has
been taken, the poles of Fab(θ) within the physical strip of θ identify the
neutral bound states which couple to ϕ. The mass of the neutral particles can
be extracted by using eq. (1.8), while the on-shell 3-particle coupling gcab can
be obtained from the residue at these poles (Figura 5)
θ→i uc
(θ − iucab)Fab(θ) = i gcab 〈a | ϕ(0) | Bc 〉 . (2.11)
2. It is important to stress that, for a generic theory, the classical kink config-
uration ϕab(x) is not related in a simple way to the anti-kink configuration
ϕba(x). It is precisely for this reason that neighbouring vacua may have a
different spectrum of neutral excitations, as shown in the examples discussed
in the following sections.
ab ba
ab ba
Figure 5: Residue equation for the matrix element on the kink states.
3. It is also worth noting that this procedure for extracting the bound states
masses permits in many cases to avoid the semiclassical quantization of the
breather solutions [22], making their derivation much simpler. The reason
is that, the classical breather configurations depend also on time and have,
in general, a more complicated structure than the kink ones. Yet, it can be
shown that in non–integrable theories these configurations do not exist as
exact solutions of the partial differential equations of the field theory. On
the contrary, in order to apply eq. (2.9), one simply needs the solution of the
ordinary differential equation (1.2). It is worth notice that, to locate the poles
(θ), one only needs to looking at the exponential behavior of the classical
solutions at x → ±∞, as discussed below.
In the next two sections we will present the analyse a class of theories with only
two vacua, which can be either symmetric or asymmetric ones. A complete analysis
of other potentials can be found in the original paper [27].
3 Symmetric wells
A prototype example of a potential with two symmetric wells is the ϕ4 theory in its
broken phase. The potential is given in this case by
U(ϕ) =
ϕ2 − m
. (3.12)
Let us denote with | ±1 〉 the vacua corresponding to the classical minima ϕ(0)± =
. By expanding around them, ϕ = ϕ
± + η, we have
± + η) = m
2 η2 ±m
λ η3 +
η4 . (3.13)
Hence, perturbation theory predicts the existence of a neutral particle for each of
the two vacua, with a bare mass given by mb =
2m, irrespectively of the value of
the coupling λ. Let’s see, instead, what is the result of the semiclassical analysis.
The kink solutions are given in this case by
ϕ−a,a(x) = a
, a = ±1 (3.14)
and their classical mass is
ǫ(x) dx =
. (3.15)
The value of the potential at the origin, which gives the height of the barrier between
the two vacua, can be expressed as
U(0) =
M0 , (3.16)
and, as noticed in the introduction, is proportional to the classical mass of the kink.
If we take into account the contribution of the small oscillations around the
classical static configurations, the kink mass gets corrected as [22]
+O(λ) . (3.17)
It is convenient to define
> 0 ,
and also the adimensional quantities
; ξ =
1− πcg
. (3.18)
In terms of them, the mass of the kink can be expressed as
. (3.19)
Since the kink and the anti-kink solutions are equal functions (up to a sign), their
Fourier transforms have the same poles. Hence, the spectrum of the neutral particles
will be the same on both vacua, in agreement with the Z2 symmetry of the model.
We have
f−a,a(θ) =
dx eiMθ xϕ−a,a(x) = i a
By making now the analitical continuation θ → iπ−θ and using the above definitions
(3.18), we arrive to
F−a,a(θ) = 〈a | ϕ(0) | K−a,a(θ1)Ka,−a(θ2)〉 ∝
(iπ−θ)
) . (3.20)
The poles of the above expression are located at
θn = iπ (1− ξ n) , n = 0,±1,±2, . . . (3.21)
and, if
ξ ≥ 1 , (3.22)
none of them is in the physical strip 0 < Im θ < π. Consequently, in the range of
the coupling constant
1 + πc
= 1.02338... (3.23)
the theory does not have any neutral bound states, neither on the vacuum to the
right nor on the one to the left. Viceversa, if ξ < 1, there are n =
neutral
bound states, where [x] denote the integer part of the number x. Their semiclassical
masses are given by
= 2M sin
= n mb
n2 + ...
. (3.24)
Note that the leading term is given by multiples of the mass of the elementary boson
| B1〉. Therefore the n-th breather may be considered as a loosely bound state of n of
it, with the binding energy provided by the remaining terms of the above expansion.
But, for the non-integrability of the theory, all particles with mass mn > 2m1 will
eventually decay. It is easy to see that, if there are at most two particles in the
spectrum, it is always valid the inequality m2 < 2m1. However, if ξ <
, for the
higher particles one always has
mk > 2m1 , for k = 3, 4, . . . n . (3.25)
According to the semiclassical analysis, the spectrum of neutral particles of ϕ4 theory
is then as follows: (i) if ξ > 1, there are no neutral particles; (ii) if 1
< ξ < 1, there
Figure 6: Neutral bound states of ϕ4 theory for g < 1. The lowest two lines are the
stable particles whereas the higher lines are the resonances.
is one particle; (iii) if 1
< ξ < 1
there are two particles; (iv) if ξ < 1
there are
particles, although only the first two are stable, because the others are resonances.
Let us now briefly mention some general features of the semiclassical methods,
starting from an equivalent way to derive the Fourier transform of the kink solution.
To simplify the notation, let’s get rid of all possible constants and consider the
Fourier transform of the derivative of the kink solution, expressed as
G(k) =
dx eikx
cosh2 x
. (3.26)
We split the integral in two terms
G(k) =
dx eikx
cosh2 x
dx eikx
cosh2 x
, (3.27)
and we use the following series expansion of the integrand, valid on the entire real
axis (except the origin)
cosh2 x
(−1)n+1n e−2n|x| . (3.28)
Substituting this expression into (3.27) and computing each integral, we have
G(k) = 4i
(−1)n+1n
ik + 2n
−ik + 2n
. (3.29)
Obviously it coincides with the exact result, G(k) = πk/ sinh π
k, but this derivation
permits to easily interpret the physical origin of each pole. In fact, changing k to
the original variable in the crossed channel, k → (iπ − θ)/ξ, we see that the poles
which determine the bound states at the vacuum | a〉 are only those relative to
the exponential behaviour of the kink solution at x → −∞. This is precisely the
point where the classical kink solution takes values on the vacuum | a〉. In the case
of ϕ4, the kink and the antikink are the same function (up to a minus sign) and
therefore they have the same exponential approach at x = −∞ at both vacua | ±1〉.
Mathematically speaking, this is the reason for the coincidence of the bound state
spectrum on each of them: this does not necessarily happens in other cases, as we
will see in the next section, for instance.
The second comment concerns the behavior of the kink solution near the minima
of the potential. In the case of ϕ4, expressing the kink solution as
ϕ(x) =
2x − 1
2x + 1
, (3.30)
and expanding around x = −∞, we have
ϕ(t) = − m√
1− 2t+ 2t2 − 2t3 + · · · 2 (−1)ntn · · ·
, (3.31)
where t = exp[
2x]. Hence, all the sub-leading terms are exponential factors, with
exponents which are multiple of the first one. Is this a general feature of the kink
solutions of any theory? It can be proved that the answer is indeed positive [27].
The fact that the approach to the minimum of the kink solutions is always
through multiples of the same exponential (when the curvature ω at the minimum
is different from zero) implies that the Fourier transform of the kink solution has
poles regularly spaced by ξa ≡ ωπMab in the variable θ. If the first of them is within
the physical strip, the semiclassical mass spectrum derived from the formula (2.9)
near the vacuum | a 〉 has therefore the universal form
mn = 2Mab sin
. (3.32)
As we have previously discussed, this means that, according to the value of ξa,
we can have only the following situations at the vacuum | a 〉: (a) no bound state
if ξa > 1; (b) one particle if
< ξa < 1; (c) two particles if
< ξa <
; (d)
particles if ξa <
, although only the first two are stable, the others being
resonances. So, semiclassically, each vacuum of the theory cannot have more than
two stable particles above it. Viceversa, if ω = 0, there are no poles in the Fourier
transform of the kink and therefore there are no neutral particles near the vacuum
| a 〉.
4 Asymmetric wells
In order to have a polynomial potential with two asymmetric wells, one must nec-
essarily employ higher powers than ϕ4. The simplest example of such a potential is
obtained with a polynomial of maximum power ϕ6, and this is the example discussed
here. Apart from its simplicity, the ϕ6 theory is relevant for the class of universality
of the Tricritical Ising Model [28]. As we can see, the information available on this
model will turn out to be a nice confirmation of the semiclassical scenario. .
A class of potentials which may present two asymmetric wells is given by
U(ϕ) =
ϕ− b m√
ϕ2 + c
, (4.33)
with a, b, c all positive numbers. To simplify the notation, it is convenient to use the
dimensionless quantities obtained by rescaling the coordinate as xµ → mxµ and the
field as ϕ(x) →
λ/mϕ(x). In this way the lagrangian of the model becomes
L = m
(∂ϕ)2 − 1
(ϕ+ a)2(ϕ− b)2(ϕ2 + c)
. (4.34)
The minima of this potential are localised at ϕ
0 = −a and ϕ
1 = b and the
corresponding ground states will be denoted by | 0 〉 and | 1 〉. The curvature of the
potential at these points is given by
U ′′(−a) ≡ ω20 = (a+ b)2(a2 + c) ;
U ′′(b) ≡ ω21 = (a+ b)2(b2 + c) .
(4.35)
For a 6= b, we have two asymmetric wells, as shown in Figure 7. To be definite, let’s
assume that the curvature at the vacuum | 0 〉 is higher than the one at the vacuum
| 1 〉, i.e. a > b.
The problem we would like to examine is whether the spectrum of the neutral
particles | B 〉s (s = 0, 1) may be different at the two vacua, in particular, whether it
would be possible that one of them (say | 0〉) has no neutral excitations, whereas the
other has just one neutral particle. The ordinary perturbation theory shows that
both vacua has neutral excitations, although with different value of their mass:
m(0) = (a+ b)
2 (a2 + c) , m(1) = (a+ b)
2 (b2 + c) . (4.36)
Let’s see, instead, what is the semiclassical scenario. The kink equation is given
in this case by
= ±(ϕ + a)(ϕ− b)
ϕ2 + c . (4.37)
Figure 7: Example of ϕ6 potential with two asymmetric wells and a bound state only
on one of them.
We will not attempt to solve exactly this equation but we can present nevertheless its
main features. The kink solution interpolates between the values −a (at x = −∞)
and b (at x = +∞). The anti-kink solution does viceversa, but with an important
difference: its behaviour at x = −∞ is different from the one of the kink. As a matter
of fact, the behaviour at x = −∞ of the kink is always equal to the behaviour at
x = +∞ of the anti-kink (and viceversa), but the two vacua are approached, in this
theory, differently. This is explicitly shown in Figure 8 and proved in the following.
-4 -2 0 2 4
0.005
0.015
0.025
0.035
Figure 8: Typical shape of
, obtained by a numerical solution of eq. (4.37).
Let us consider the limit x → −∞ of the kink solution. For these large values
of x, we can approximate eq. (4.37) by substituting, in the second and in the third
term of the right-hand side, ϕ ≃ −a, with the result
≃ (ϕ+ a)(a+ b)
a2 + c , x → −∞ (4.38)
This gives rise to the following exponential approach to the vacuum | 0〉
ϕ0,1(x) ≃ −a+ A exp(ω0x) , x → −∞ (4.39)
where A > 0 is a arbitrary costant (its actual value can be fixed by properly solving
the non-linear differential equation). To extract the behavior at x → −∞ of the
anti-kink, we substitute this time ϕ ≃ b into the first and third term of the right
hand side of (4.37), so that
≃ (ϕ− b)(a+ b)
b2 + c , x → −∞ (4.40)
This ends up in the following exponential approach to the vacuum | 1〉
ϕ1,0(x) ≃ b− B exp(ω1x) , x → −∞ (4.41)
where B > 0 is another constant. Since ω0 6= ω1, the asymptotic behaviour of the
two solutions gives rise to the following poles in their Fourier transform
F(ϕ0,1) →
ω0 + ik
(4.42)
F(ϕ1,0) →
ω1 + ik
In order to locate the pole in θ, we shall reintroduce the correct units. Assuming to
have solved the differential equation (4.37), the integral of its energy density gives
the common mass of the kink and the anti-kink. In terms of the constants in front
of the Lagrangian (4.34), its value is given by
α , (4.43)
where α is a number (typically of order 1), coming from the integral of the adimen-
sional energy density (1.4). Hence, the first pole2 of the Fourier transform of the
kink and the antikink solution are localised at
θ(0) ≃ iπ
1− ω0
1− ω0
(4.44)
θ(1) ≃ iπ
1− ω1
1− ω1
2In order to determine the others, one should look for the subleading exponential terms of the
solutions.
If we now choose the coupling constant in the range
, (4.45)
the first pole will be out of the physical sheet whereas the second will still remain
inside it! Hence, the theory will have only one neutral bound state, localised at
the vacuum | 1 〉. This result may be expressed by saying that the appearance of a
bound state depends on the order in which the topological excitations are arranged:
an antikink-kink configuration gives rise to a bound state whereas a kink-antikink
does not.
Finally, notice that the value of the adimensional coupling constant can be chosen
so that the mass of the bound state around the vacuum | 1 〉 becomes equal to mass
of the kink. This happens when
. (4.46)
Strange as it may appear, the semiclassical scenario is well confirmed by an
explicit example. This is provided by the exact scattering theory of the Tricritical
Ising Model perturbed by its sub-leading magnetization. Firstly discovered through
a numerical analysis of the spectrum of this model [29], its exact scattering theory
has been discussed later in [30].
5 Conclusions
In this paper we have used simple arguments of the semi-classical analysis to in-
vestigate the spectrum of neutral particles in quantum field theories with kink ex-
citations. We have concentrated our analysis on two cases: the first relative to a
potential with symmetric wells, the second concerning with a potential with asym-
metric wells. Leaving apart the exact values of the quantities extracted by the
semiclassical methods, it is perhaps more important to underline some general fea-
tures which have emerged through this analysis. One of them concerns, for instance,
the existence of a critical value of the coupling constant, beyond which there are
no neutral bound states. Another result is about the maximum number n ≤ 2 of
neutral particles living on a generica vacuum of a non-integrable theory. An addi-
tional aspect is the role played by the asymmetric vacua, which may have a different
number of neutral excitations above them.
Acknowledgements
I would like to thank G. Delfino and V. Riva for interesting discussions. I am
particularly grateful to M. Peyrard for very useful and enjoyable discussions on
solitons. This work was done under partial support of the ESF grant INSTANS.
References
[1] A.B. Zamolodchikov and Al.B. Zamolodchikov, Ann. Phys. 120 (1979) 253.
[2] A.B. Zamolodchikov, Adv. Stud. Pure Math. 19 (1989), 641.
[3] F. A. Smirnov, Form Factors in Completely Integrable Models of Quantum Field
Theory, (World Scientific, Singapore, 1992); M. Karowski and P. Weisz, Nucl.
Phys. D 139, (1978), 455.
[4] G. Delfino and G. Mussardo, Nucl. Phys. B 455, (1995), 724; G. Delfino and P.
Simonetti, Phys. Lett. B 383, (1996), 450.
[5] G. Mussardo, Phys. Rept. 218 (1992), 215.
[6] Al.B.Zamolodchikov, Nucl.Phys. B 358, (1991), 524.
[7] A.B.Zamolodchikov and Al.B.Zamolodchikov, Nucl.Phys. B 379, (1992), 602.
[8] P. Fendley, H. Saleur and N.P. Werner, Nucl.Phys. B 430, (1994), 577.
[9] G.Delfino, G.Mussardo and P.Simonetti, Phys. Rev. D 51, (1995), 6622.
[10] G.Delfino, G.Mussardo and P.Simonetti,Nucl.Phys. B 473, (1996), 469.
[11] P. Grinza, G. Delfino and G. Mussardo, hep/th 0507133, Nucl. Phys. B in press.
[12] G. Delfino, Particle decay in Ising field theory with magnetic field, Proceedings
ICMP 2006.
[13] B.M. McCoy and T.T. Wu, Phys. Rev. D 18 (1978), 1259.
[14] P. Fonseca and A.B. Zamolodchikov, J.Stat.Phys.110 (2003), 527.
[15] S.B. Rutkevich, Phys. Rev. Lett. 95 (2005), 250601.
[16] P. Fonseca and A.B. Zamolodchikov, Ising Spectoscopy I: Mesons at T < Tc,
hep-th/0612304.
http://arxiv.org/abs/hep-th/0612304
[17] G. Delfino and G. Mussardo, Nucl. Phys. B 516, (1998), 675.
[18] Z. Bajnok, L. Palla, G. Takacs, F. Wagner, Nucl.Phys. B 601, (2001), 503.
[19] D. Controzzi and G. Mussardo, Phys. Rev. Lett. 92, (2004), 021601.
[20] D. Controzzi and G. Mussardo, Phys. Lett. B 617, (2005), 133.
[21] G. Delfino, P. Grinza and G. Mussardo, Nucl. Phys. B 737 (2006), 291.
[22] R.F.Dashen, B.Hasslacher and A.Neveu, Phys. Rev. D 10 (1974) 4130;
R.F.Dashen, B.Hasslacher and A.Neveu, Phys. Rev. D 11 (1975) 3424.
[23] J. Goldstone and R. Jackiw, Phys.Rev. D 11 (1975) 1486.
[24] R. Jackiw and G. Woo, Phys. Rev. D 12 (1975), 1643.
[25] G. Mussardo, V. Riva and G. Sotkov, Nucl. Phys. B 670 (2003), 464.
[26] G. Mussardo, V. Riva and G. Sotkov, Nucl. Phys. B 699 (2004), 545.
G. Mussardo, V. Riva and G. Sotkov, Nucl. Phys. B 705 (2005), 548
[27] G. Mussardo, Neutral bound states in kink-like theories, hep-th/0607025, to
appear on Nucl. Phys. B.
[28] A.B. Zamolodchikov, Sov.J.Nucl.Phys. 44 (1986), 529.
[29] M. Lassig, G. Mussardo and J.L. Cardy, Nucl. Phys. B 348 (1991), 591.
[30] F. Colomo, A. Koubek and G. Mussardo, Int. Journ. Mod. Phys. A 7 (1992),
5281.
http://arxiv.org/abs/hep-th/0607025
	Introduction
	A semiclassical formula
	Symmetric wells
	Asymmetric wells
	Conclusions

  In this talk we discuss an elementary derivation of the semi-classical
spectrum of neutral particles in two field theories with kink excitations. We
also show that, in the non-integrable cases, each vacuum state cannot
generically support more than two stable particles, since all other neutral
exitations are resonances, which will eventually decay.

Introduction
Two–dimensional massive Integrable Quantum Field Theories (IQFTs) have proven
to be one of the most successful topics of relativistic field theory, with a large
variety of applications to statistical mechanical models. The main reason for this
success consists of their simplified on–shell dynamics which is encoded into a set of
elastic and factorized scattering amplitudes of their massive particles [1, 2]. The two-
particle S-matrix has a very simple analytic structure, with only poles in the physical
strip, and it can be computed combining the standard requirements of unitarity,
crossing and factorization together with specific symmetry properties of the theory.
The complete mass spectrum is obtained looking at the pole singularities of the
S–matrix elements. Off–mass shell quantities, such as the correlation functions, can
be also determined once the elastic S–matrix and the mass spectrum are known.
In fact, one can compute the exact matrix elements of the (semi)local fields on the
asymptotic states with the Form Factor (FF) approach [3], and use them to write
down the spectral representation of the correlators. By following this approach, it
has been possible, for instance, to tackle successfully the long-standing problem of
spelling out the mass spectrum and the correlation functions of the two dimensional
Ising model in a magnetic field [2, 4], as well as many other interesting problems of
statistical physics (for a partial list of them see, for instance, [5]).
The S-matrix approach can be also constructed for massless IQFTs [6, 7, 8, 9],
despite the subtleties in defining a scattering theory between massless particles in
(1 + 1) dimensions, and turns out to be useful mainly when conformal symmetry
is not present. In this case, massless IQFTs generically describe the Renormaliza-
tion Group trajectories connecting two different Conformal Field Theories, which
respectively rule the ultraviolet and infrared limits of all physical quantities along
the flows.
Given the large number of remarkable results obtained by the study of IQFTs,
one of the most interesting challenges is to extend the analysis to the non–integrable
field theories, at least to those obtained as deformations of the integrable ones and
to develop the corresponding perturbation theory. The breaking of integrability
is expected to considerably increase the difficulties of the mathematical analysis,
since scattering processes are no longer elastic. Non–integrable field theories are
in fact generally characterized by particle production amplitudes, resonance states
and, correspondingly, decay events. All these features strongly effect the analytic
structure of the scattering amplitudes, introducing a rich pattern of branch cut
singularities, in addition to the pole structure associated to bound and resonance
states. For massive non–integrable field theories, a convenient perturbative scheme
was originally proposed in [10] and called Form Factor Perturbation Theory (FFPT),
since it is based on the knowledge of the exact Form Factors (FFs) of the original
integrable theory. It was shown that, even using just the first order correction of
the FFPT, a great deal of information can be obtained, such as the evolution of
their particle content, the variation of their masses and the change of the ground
state energy. Whenever possible, universal ratios were computed and successfully
compared with their value obtained by other means. Recently, for instance, it has
been obtained the universal ratios relative to the decay of the particles with higher
masses in the Ising model in a magnetic field, once the temperature is displayed away
from the critical value [11] (see also the contibution by G. Delfino in this proceedings
[12]). For other and important aspects of the Ising model along non-integrable lines
see the references [13, 14, 15, 16]. Applied to the double Sine–Gordon model [17], the
FFPT has been useful in clarifying the rich dynamics of this non–integrable model.
In particular, in relating the confinement of the kinks in the deformed theory to the
non–locality properties of the perturbed operator and predicting the existence of a
Ising–like phase transition for particular ratios of the two frequencies – results which
were later confirmed by a numerical study [18]. The FFPT has been also used to
study the spectrum of the O(3) non-linear sigma model with a topological θ term,
by varying θ [19, 20].
In this talk I would like to focus the attention on a different approach to tackle
some interesting non-integrable models, i.e. those two dimensional field theories
with kink topological excitations. Such theories are described by a scalar real field
ϕ(x), with a Lagrangian density
(∂µϕ)
2 − U(ϕ) , (1.1)
where the potential U(ϕ) possesses several degenerate minima at ϕ
a (a = 1, 2, . . . , n),
as the one shown in Figure 1. These minima correspond to the different vacua | a 〉
of the associate quantum field theory.
The basic excitations of this kind of models are kinks and anti-kinks, i.e. topological
configurations which interpolate between two neighbouring vacua. Semiclassically
they correspond to the static solutions of the equation of motion, i.e.
∂2x ϕ(x) = U
′[ϕ(x)] , (1.2)
with boundary conditions ϕ(−∞) = ϕ(0)a and ϕ(+∞) = ϕ(0)b , where b = a ± 1.
Denoting by ϕab(x) the solutions of this equation, their classical energy density is
(A) (B)
Figure 1: Potential U(ϕ) of a quantum field theory with kink excitations (A) and
istogram of the masses of the kinks (B).
given by
ǫab(x) =
+ U(ϕab(x)) , (1.3)
and its integral provides the classical expression of the kink masses
Mab =
ǫab(x) . (1.4)
It is easy to show that the classical masses of the kinks ϕab(x) are simply proportional
to the heights of the potential between the two minima ϕ
a and ϕ
: their istogram
provides a caricature of the original ptential (see Figura 1).
The classical solutions can be set in motion by a Lorentz transformation, i.e.
ϕab(x) → ϕab
(x± vt)/
1− v2
. In the quantum theory, these configurations de-
scribe the kink states | Kab(θ) 〉, where a and b are the indices of the initial and final
vacuum, respectively. The quantity θ is the rapidity variable which parameterises
the relativistic dispersion relation of these excitations, i.e.
E = Mab cosh θ , P = Mab sinh θ . (1.5)
Conventionally | Ka,a+1(θ) 〉 denotes the kink between the pair of vacua {| a 〉, | a+ 1 〉}
while | Ka+1,a 〉 is the corresponding anti-kink. For the kink configurations it may
be useful to adopt the simplified graphical form shown in Figure 2.
The multi-particle states are given by a string of these excitations, with the adja-
cency condition of the consecutive indices for the continuity of the field configuration
| Ka1,a2(θ1)Ka2,a3(θ2)Ka3,a4(θ3) . . .〉 , (ai+1 = ai ± 1) (1.6)
In addition to the kinks, in the quantum theory there may exist other excitations in
the guise of ordinary scalar particles (breathers). These are the neutral excitations
a,a+1K K a+1,a
| a+1>
| 0 >
| a  >
| n  >
Figure 2: Kink and antikink configurations.
| Bc(θ) 〉a (c = 1, 2, . . .) around each of the vacua | a 〉. For a theory based on a
Lagrangian of a single real field, these states are all non-degenerate: in fact, there
are no extra quantities which commute with the Hamiltonian and that can give
rise to a multiplicity of them. The only exact (alias, unbroken) symmetries for a
Lagrangian as (1.1) may be the discrete ones, like the parity transformation P , for
instance, or the charge conjugation C. However, since they are neutral excitations,
they will be either even or odd eigenvectors of C.
The neutral particles must be identified as the bound states of the kink-antikink
configurations that start and end at the same vacuum | a 〉, i.e. | Kab(θ1)Kba(θ2) 〉,
with the “tooth” shapes shown in Figure 3.
| 0 >
| 0 >
| 0 >
Figure 3: Kink-antikink configurations which may give rise to a bound state nearby
the vacuum | 0 〉a.
If such two-kink states have a pole at an imaginary value i ucab within the physical
strip 0 < Im θ < π of their rapidity difference θ = θ1 − θ2, then their bound states
are defined through the factorization formula which holds in the vicinity of this
singularity
| Kab(θ1)Kba(θ2) 〉 ≃ i
θ − iuc
| Bc 〉a . (1.7)
In this expression gcab is the on-shell 3-particle coupling between the kinks and the
neutral particle. Moreover, the mass of the bound states is simply obtained by sub-
stituing the resonance value i ucab within the expression of the Mandelstam variable
s of the two-kink channel
s = 4M2ab cosh
−→ mc = 2Mab cos
. (1.8)
Concerning the vacua themselves, as well known, in the infinite volume their
classical degeneracy is removed by selecting one of them, say | k 〉, out of the n
available. This happens through the usual spontaneously symmetry breaking mech-
anism, even though – stricly speaking – there may be no internal symmetry to break
at all. This is the case, for instance, of the potential shown in Figure 1, which does
not have any particular invariance. In the absence of a symmetry which connects the
various vacua, the world – as seen by each of them – may appear very different: they
can have, indeed, different particle contents. The problem we would like to examine
in this talk concerns the neutral excitations around each vacuum, in particular the
question of the existence of such particles and of the value of their masses. To this
aim, let’s make use of a semiclassical approach.
2 A semiclassical formula
The starting point of our analysis is a remarkably simple formula due to Goldstone-
Jackiw [23], which is valid in the semiclassical approximation, i.e. when the coupling
constant goes to zero and the mass of the kinks becomes correspondingly very large
with respect to any other mass scale. In its refined version, given in [24] and redis-
covered in [25], it reads as follows1 (Figure 4)
(θ) = 〈Kab(θ1) | ϕ(0) | Kab(θ2)〉 ≃
dx eiMab θ x ϕab(x) , (2.9)
where θ = θ1 − θ2.
1The matrix element of the field ϕ(y) is easily obtained by using ϕ(y) = e−iPµy
ϕ(0) eiPµy
and by acting with the conserved energy-momentum operator Pµ on the kink state. Moreover, for
the semiclassical matrix element FG
(θ) of the operator G[ϕ(0)], one should employ G[ϕab(x)]. For
instance, the matrix element of ϕ2(0) are given by the Fourier transform of ϕ2
Figure 4: Matrix element between kink states.
Notice that, if we substitute in the above formula θ → iπ − θ, the corresponding
expression may be interpreted as the following Form Factor
(θ) = f(iπ − θ) = 〈a | ϕ(0) | Kab(θ1)Kba(θ2)〉 . (2.10)
In this matrix element, it appears the neutral kink states around the vacuum | a〉
we are interested in.
Eq. (2.9) deserves several comments.
1. The appealing aspect of the formula (2.9) stays in the relation between the
Fourier transform of the classical configuration of the kink, – i.e. the solu-
tion ϕab(x) of the differential equation (1.2) – to the quantum matrix ele-
ment of the field ϕ(0) between the vacuum | a 〉 and the 2-particle kink state
| Kab(θ1)Kba(θ2) 〉.
Once the solution of eq. (1.2) has been found and its Fourier transform has
been taken, the poles of Fab(θ) within the physical strip of θ identify the
neutral bound states which couple to ϕ. The mass of the neutral particles can
be extracted by using eq. (1.8), while the on-shell 3-particle coupling gcab can
be obtained from the residue at these poles (Figura 5)
θ→i uc
(θ − iucab)Fab(θ) = i gcab 〈a | ϕ(0) | Bc 〉 . (2.11)
2. It is important to stress that, for a generic theory, the classical kink config-
uration ϕab(x) is not related in a simple way to the anti-kink configuration
ϕba(x). It is precisely for this reason that neighbouring vacua may have a
different spectrum of neutral excitations, as shown in the examples discussed
in the following sections.
ab ba
ab ba
Figure 5: Residue equation for the matrix element on the kink states.
3. It is also worth noting that this procedure for extracting the bound states
masses permits in many cases to avoid the semiclassical quantization of the
breather solutions [22], making their derivation much simpler. The reason
is that, the classical breather configurations depend also on time and have,
in general, a more complicated structure than the kink ones. Yet, it can be
shown that in non–integrable theories these configurations do not exist as
exact solutions of the partial differential equations of the field theory. On
the contrary, in order to apply eq. (2.9), one simply needs the solution of the
ordinary differential equation (1.2). It is worth notice that, to locate the poles
(θ), one only needs to looking at the exponential behavior of the classical
solutions at x → ±∞, as discussed below.
In the next two sections we will present the analyse a class of theories with only
two vacua, which can be either symmetric or asymmetric ones. A complete analysis
of other potentials can be found in the original paper [27].
3 Symmetric wells
A prototype example of a potential with two symmetric wells is the ϕ4 theory in its
broken phase. The potential is given in this case by
U(ϕ) =
ϕ2 − m
. (3.12)
Let us denote with | ±1 〉 the vacua corresponding to the classical minima ϕ(0)± =
. By expanding around them, ϕ = ϕ
± + η, we have
± + η) = m
2 η2 ±m
λ η3 +
η4 . (3.13)
Hence, perturbation theory predicts the existence of a neutral particle for each of
the two vacua, with a bare mass given by mb =
2m, irrespectively of the value of
the coupling λ. Let’s see, instead, what is the result of the semiclassical analysis.
The kink solutions are given in this case by
ϕ−a,a(x) = a
, a = ±1 (3.14)
and their classical mass is
ǫ(x) dx =
. (3.15)
The value of the potential at the origin, which gives the height of the barrier between
the two vacua, can be expressed as
U(0) =
M0 , (3.16)
and, as noticed in the introduction, is proportional to the classical mass of the kink.
If we take into account the contribution of the small oscillations around the
classical static configurations, the kink mass gets corrected as [22]
+O(λ) . (3.17)
It is convenient to define
> 0 ,
and also the adimensional quantities
; ξ =
1− πcg
. (3.18)
In terms of them, the mass of the kink can be expressed as
. (3.19)
Since the kink and the anti-kink solutions are equal functions (up to a sign), their
Fourier transforms have the same poles. Hence, the spectrum of the neutral particles
will be the same on both vacua, in agreement with the Z2 symmetry of the model.
We have
f−a,a(θ) =
dx eiMθ xϕ−a,a(x) = i a
By making now the analitical continuation θ → iπ−θ and using the above definitions
(3.18), we arrive to
F−a,a(θ) = 〈a | ϕ(0) | K−a,a(θ1)Ka,−a(θ2)〉 ∝
(iπ−θ)
) . (3.20)
The poles of the above expression are located at
θn = iπ (1− ξ n) , n = 0,±1,±2, . . . (3.21)
and, if
ξ ≥ 1 , (3.22)
none of them is in the physical strip 0 < Im θ < π. Consequently, in the range of
the coupling constant
1 + πc
= 1.02338... (3.23)
the theory does not have any neutral bound states, neither on the vacuum to the
right nor on the one to the left. Viceversa, if ξ < 1, there are n =
neutral
bound states, where [x] denote the integer part of the number x. Their semiclassical
masses are given by
= 2M sin
= n mb
n2 + ...
. (3.24)
Note that the leading term is given by multiples of the mass of the elementary boson
| B1〉. Therefore the n-th breather may be considered as a loosely bound state of n of
it, with the binding energy provided by the remaining terms of the above expansion.
But, for the non-integrability of the theory, all particles with mass mn > 2m1 will
eventually decay. It is easy to see that, if there are at most two particles in the
spectrum, it is always valid the inequality m2 < 2m1. However, if ξ <
, for the
higher particles one always has
mk > 2m1 , for k = 3, 4, . . . n . (3.25)
According to the semiclassical analysis, the spectrum of neutral particles of ϕ4 theory
is then as follows: (i) if ξ > 1, there are no neutral particles; (ii) if 1
< ξ < 1, there
Figure 6: Neutral bound states of ϕ4 theory for g < 1. The lowest two lines are the
stable particles whereas the higher lines are the resonances.
is one particle; (iii) if 1
< ξ < 1
there are two particles; (iv) if ξ < 1
there are
particles, although only the first two are stable, because the others are resonances.
Let us now briefly mention some general features of the semiclassical methods,
starting from an equivalent way to derive the Fourier transform of the kink solution.
To simplify the notation, let’s get rid of all possible constants and consider the
Fourier transform of the derivative of the kink solution, expressed as
G(k) =
dx eikx
cosh2 x
. (3.26)
We split the integral in two terms
G(k) =
dx eikx
cosh2 x
dx eikx
cosh2 x
, (3.27)
and we use the following series expansion of the integrand, valid on the entire real
axis (except the origin)
cosh2 x
(−1)n+1n e−2n|x| . (3.28)
Substituting this expression into (3.27) and computing each integral, we have
G(k) = 4i
(−1)n+1n
ik + 2n
−ik + 2n
. (3.29)
Obviously it coincides with the exact result, G(k) = πk/ sinh π
k, but this derivation
permits to easily interpret the physical origin of each pole. In fact, changing k to
the original variable in the crossed channel, k → (iπ − θ)/ξ, we see that the poles
which determine the bound states at the vacuum | a〉 are only those relative to
the exponential behaviour of the kink solution at x → −∞. This is precisely the
point where the classical kink solution takes values on the vacuum | a〉. In the case
of ϕ4, the kink and the antikink are the same function (up to a minus sign) and
therefore they have the same exponential approach at x = −∞ at both vacua | ±1〉.
Mathematically speaking, this is the reason for the coincidence of the bound state
spectrum on each of them: this does not necessarily happens in other cases, as we
will see in the next section, for instance.
The second comment concerns the behavior of the kink solution near the minima
of the potential. In the case of ϕ4, expressing the kink solution as
ϕ(x) =
2x − 1
2x + 1
, (3.30)
and expanding around x = −∞, we have
ϕ(t) = − m√
1− 2t+ 2t2 − 2t3 + · · · 2 (−1)ntn · · ·
, (3.31)
where t = exp[
2x]. Hence, all the sub-leading terms are exponential factors, with
exponents which are multiple of the first one. Is this a general feature of the kink
solutions of any theory? It can be proved that the answer is indeed positive [27].
The fact that the approach to the minimum of the kink solutions is always
through multiples of the same exponential (when the curvature ω at the minimum
is different from zero) implies that the Fourier transform of the kink solution has
poles regularly spaced by ξa ≡ ωπMab in the variable θ. If the first of them is within
the physical strip, the semiclassical mass spectrum derived from the formula (2.9)
near the vacuum | a 〉 has therefore the universal form
mn = 2Mab sin
. (3.32)
As we have previously discussed, this means that, according to the value of ξa,
we can have only the following situations at the vacuum | a 〉: (a) no bound state
if ξa > 1; (b) one particle if
< ξa < 1; (c) two particles if
< ξa <
; (d)
particles if ξa <
, although only the first two are stable, the others being
resonances. So, semiclassically, each vacuum of the theory cannot have more than
two stable particles above it. Viceversa, if ω = 0, there are no poles in the Fourier
transform of the kink and therefore there are no neutral particles near the vacuum
| a 〉.
4 Asymmetric wells
In order to have a polynomial potential with two asymmetric wells, one must nec-
essarily employ higher powers than ϕ4. The simplest example of such a potential is
obtained with a polynomial of maximum power ϕ6, and this is the example discussed
here. Apart from its simplicity, the ϕ6 theory is relevant for the class of universality
of the Tricritical Ising Model [28]. As we can see, the information available on this
model will turn out to be a nice confirmation of the semiclassical scenario. .
A class of potentials which may present two asymmetric wells is given by
U(ϕ) =
ϕ− b m√
ϕ2 + c
, (4.33)
with a, b, c all positive numbers. To simplify the notation, it is convenient to use the
dimensionless quantities obtained by rescaling the coordinate as xµ → mxµ and the
field as ϕ(x) →
λ/mϕ(x). In this way the lagrangian of the model becomes
L = m
(∂ϕ)2 − 1
(ϕ+ a)2(ϕ− b)2(ϕ2 + c)
. (4.34)
The minima of this potential are localised at ϕ
0 = −a and ϕ
1 = b and the
corresponding ground states will be denoted by | 0 〉 and | 1 〉. The curvature of the
potential at these points is given by
U ′′(−a) ≡ ω20 = (a+ b)2(a2 + c) ;
U ′′(b) ≡ ω21 = (a+ b)2(b2 + c) .
(4.35)
For a 6= b, we have two asymmetric wells, as shown in Figure 7. To be definite, let’s
assume that the curvature at the vacuum | 0 〉 is higher than the one at the vacuum
| 1 〉, i.e. a > b.
The problem we would like to examine is whether the spectrum of the neutral
particles | B 〉s (s = 0, 1) may be different at the two vacua, in particular, whether it
would be possible that one of them (say | 0〉) has no neutral excitations, whereas the
other has just one neutral particle. The ordinary perturbation theory shows that
both vacua has neutral excitations, although with different value of their mass:
m(0) = (a+ b)
2 (a2 + c) , m(1) = (a+ b)
2 (b2 + c) . (4.36)
Let’s see, instead, what is the semiclassical scenario. The kink equation is given
in this case by
= ±(ϕ + a)(ϕ− b)
ϕ2 + c . (4.37)
Figure 7: Example of ϕ6 potential with two asymmetric wells and a bound state only
on one of them.
We will not attempt to solve exactly this equation but we can present nevertheless its
main features. The kink solution interpolates between the values −a (at x = −∞)
and b (at x = +∞). The anti-kink solution does viceversa, but with an important
difference: its behaviour at x = −∞ is different from the one of the kink. As a matter
of fact, the behaviour at x = −∞ of the kink is always equal to the behaviour at
x = +∞ of the anti-kink (and viceversa), but the two vacua are approached, in this
theory, differently. This is explicitly shown in Figure 8 and proved in the following.
-4 -2 0 2 4
0.005
0.015
0.025
0.035
Figure 8: Typical shape of
, obtained by a numerical solution of eq. (4.37).
Let us consider the limit x → −∞ of the kink solution. For these large values
of x, we can approximate eq. (4.37) by substituting, in the second and in the third
term of the right-hand side, ϕ ≃ −a, with the result
≃ (ϕ+ a)(a+ b)
a2 + c , x → −∞ (4.38)
This gives rise to the following exponential approach to the vacuum | 0〉
ϕ0,1(x) ≃ −a+ A exp(ω0x) , x → −∞ (4.39)
where A > 0 is a arbitrary costant (its actual value can be fixed by properly solving
the non-linear differential equation). To extract the behavior at x → −∞ of the
anti-kink, we substitute this time ϕ ≃ b into the first and third term of the right
hand side of (4.37), so that
≃ (ϕ− b)(a+ b)
b2 + c , x → −∞ (4.40)
This ends up in the following exponential approach to the vacuum | 1〉
ϕ1,0(x) ≃ b− B exp(ω1x) , x → −∞ (4.41)
where B > 0 is another constant. Since ω0 6= ω1, the asymptotic behaviour of the
two solutions gives rise to the following poles in their Fourier transform
F(ϕ0,1) →
ω0 + ik
(4.42)
F(ϕ1,0) →
ω1 + ik
In order to locate the pole in θ, we shall reintroduce the correct units. Assuming to
have solved the differential equation (4.37), the integral of its energy density gives
the common mass of the kink and the anti-kink. In terms of the constants in front
of the Lagrangian (4.34), its value is given by
α , (4.43)
where α is a number (typically of order 1), coming from the integral of the adimen-
sional energy density (1.4). Hence, the first pole2 of the Fourier transform of the
kink and the antikink solution are localised at
θ(0) ≃ iπ
1− ω0
1− ω0
(4.44)
θ(1) ≃ iπ
1− ω1
1− ω1
2In order to determine the others, one should look for the subleading exponential terms of the
solutions.
If we now choose the coupling constant in the range
, (4.45)
the first pole will be out of the physical sheet whereas the second will still remain
inside it! Hence, the theory will have only one neutral bound state, localised at
the vacuum | 1 〉. This result may be expressed by saying that the appearance of a
bound state depends on the order in which the topological excitations are arranged:
an antikink-kink configuration gives rise to a bound state whereas a kink-antikink
does not.
Finally, notice that the value of the adimensional coupling constant can be chosen
so that the mass of the bound state around the vacuum | 1 〉 becomes equal to mass
of the kink. This happens when
. (4.46)
Strange as it may appear, the semiclassical scenario is well confirmed by an
explicit example. This is provided by the exact scattering theory of the Tricritical
Ising Model perturbed by its sub-leading magnetization. Firstly discovered through
a numerical analysis of the spectrum of this model [29], its exact scattering theory
has been discussed later in [30].
5 Conclusions
In this paper we have used simple arguments of the semi-classical analysis to in-
vestigate the spectrum of neutral particles in quantum field theories with kink ex-
citations. We have concentrated our analysis on two cases: the first relative to a
potential with symmetric wells, the second concerning with a potential with asym-
metric wells. Leaving apart the exact values of the quantities extracted by the
semiclassical methods, it is perhaps more important to underline some general fea-
tures which have emerged through this analysis. One of them concerns, for instance,
the existence of a critical value of the coupling constant, beyond which there are
no neutral bound states. Another result is about the maximum number n ≤ 2 of
neutral particles living on a generica vacuum of a non-integrable theory. An addi-
tional aspect is the role played by the asymmetric vacua, which may have a different
number of neutral excitations above them.
Acknowledgements
I would like to thank G. Delfino and V. Riva for interesting discussions. I am
particularly grateful to M. Peyrard for very useful and enjoyable discussions on
solitons. This work was done under partial support of the ESF grant INSTANS.
References
[1] A.B. Zamolodchikov and Al.B. Zamolodchikov, Ann. Phys. 120 (1979) 253.
[2] A.B. Zamolodchikov, Adv. Stud. Pure Math. 19 (1989), 641.
[3] F. A. Smirnov, Form Factors in Completely Integrable Models of Quantum Field
Theory, (World Scientific, Singapore, 1992); M. Karowski and P. Weisz, Nucl.
Phys. D 139, (1978), 455.
[4] G. Delfino and G. Mussardo, Nucl. Phys. B 455, (1995), 724; G. Delfino and P.
Simonetti, Phys. Lett. B 383, (1996), 450.
[5] G. Mussardo, Phys. Rept. 218 (1992), 215.
[6] Al.B.Zamolodchikov, Nucl.Phys. B 358, (1991), 524.
[7] A.B.Zamolodchikov and Al.B.Zamolodchikov, Nucl.Phys. B 379, (1992), 602.
[8] P. Fendley, H. Saleur and N.P. Werner, Nucl.Phys. B 430, (1994), 577.
[9] G.Delfino, G.Mussardo and P.Simonetti, Phys. Rev. D 51, (1995), 6622.
[10] G.Delfino, G.Mussardo and P.Simonetti,Nucl.Phys. B 473, (1996), 469.
[11] P. Grinza, G. Delfino and G. Mussardo, hep/th 0507133, Nucl. Phys. B in press.
[12] G. Delfino, Particle decay in Ising field theory with magnetic field, Proceedings
ICMP 2006.
[13] B.M. McCoy and T.T. Wu, Phys. Rev. D 18 (1978), 1259.
[14] P. Fonseca and A.B. Zamolodchikov, J.Stat.Phys.110 (2003), 527.
[15] S.B. Rutkevich, Phys. Rev. Lett. 95 (2005), 250601.
[16] P. Fonseca and A.B. Zamolodchikov, Ising Spectoscopy I: Mesons at T < Tc,
hep-th/0612304.
http://arxiv.org/abs/hep-th/0612304
[17] G. Delfino and G. Mussardo, Nucl. Phys. B 516, (1998), 675.
[18] Z. Bajnok, L. Palla, G. Takacs, F. Wagner, Nucl.Phys. B 601, (2001), 503.
[19] D. Controzzi and G. Mussardo, Phys. Rev. Lett. 92, (2004), 021601.
[20] D. Controzzi and G. Mussardo, Phys. Lett. B 617, (2005), 133.
[21] G. Delfino, P. Grinza and G. Mussardo, Nucl. Phys. B 737 (2006), 291.
[22] R.F.Dashen, B.Hasslacher and A.Neveu, Phys. Rev. D 10 (1974) 4130;
R.F.Dashen, B.Hasslacher and A.Neveu, Phys. Rev. D 11 (1975) 3424.
[23] J. Goldstone and R. Jackiw, Phys.Rev. D 11 (1975) 1486.
[24] R. Jackiw and G. Woo, Phys. Rev. D 12 (1975), 1643.
[25] G. Mussardo, V. Riva and G. Sotkov, Nucl. Phys. B 670 (2003), 464.
[26] G. Mussardo, V. Riva and G. Sotkov, Nucl. Phys. B 699 (2004), 545.
G. Mussardo, V. Riva and G. Sotkov, Nucl. Phys. B 705 (2005), 548
[27] G. Mussardo, Neutral bound states in kink-like theories, hep-th/0607025, to
appear on Nucl. Phys. B.
[28] A.B. Zamolodchikov, Sov.J.Nucl.Phys. 44 (1986), 529.
[29] M. Lassig, G. Mussardo and J.L. Cardy, Nucl. Phys. B 348 (1991), 591.
[30] F. Colomo, A. Koubek and G. Mussardo, Int. Journ. Mod. Phys. A 7 (1992),
5281.
http://arxiv.org/abs/hep-th/0607025
	Introduction
	A semiclassical formula
	Symmetric wells
	Asymmetric wells
	Conclusions
