C O N T E N T S
INTRODUCTION iii
I Topological solitons 1
1 topological solitons 3
1.1 Introduction to topology . . . 3
1.2 Topological solitons in Field Theory . . . 8
1.3 Abelian Vortex . . . 12
1.4 Moduli space and soliton dynamics . . . 18
2 low-energy theories 21 2.1 A particle in Rn . . . 21
2.2 Higher-derivative corrections in Field Theory . . . 28
II Non-Abelian Vortex 39 3 non-abelian vortex theory 41 3.1 General theory and static solution . . . 41
3.2 Low-energy dynamics . . . 48
3.2.1 Derivative expansion . . . 49
3.3 Effective theory . . . 51
3.3.1 The zeroth-order Lagrangian . . . 51
3.3.2 The second-order Lagrangian . . . 51
3.4 Zero modes . . . 54
4 higher-derivative corrections to the effective lagrangian 63 4.1 The fourth-order Lagrangian . . . 63
4.1.1 General strategy . . . 63
4.1.2 Fourth-order Lagrangian solutions . . . 70
4.2 Analysis in 0+1 dimensions . . . 77
4.3 Analysis in 1+1 dimensions . . . 80
4.4 Analysis in 2+1 dimensions . . . 83
III Monopole in the Higgs phase 85
5 monopole as deformed non-abelian vortex 87
5.1 General theory . . . 87
5.2 Low-energy dynamics . . . 91
5.2.1 The zeroth-order Lagrangian . . . 91
5.2.2 The second-order Lagrangian . . . 92
5.3 Zero modes . . . 97
6 higher-derivative corrections to the effective lagrangian 101 6.1 The fourth-order Lagrangian . . . 101
6.1.1 General strategy . . . 101
6.1.2 Fourth-order Lagrangian solutions . . . 105
6.2 Analysis in 0+1 dimensions . . . 107
CONCLUSIONS 111
I N T R O D U C T I O N
In the 1960s a new approach to quantum field theory developed becoming a new fron-tier of research. Physicists and mathematicians began to study the classical field equa-tions in their fully non-linear form trying to interpret their soluequa-tions as new particle-like objects. Differently from the elementary particles that arise from the quantization of the wave-like excitations, these new particles, called solitons, appear as smooth, clas-sical solutions, whose energy-density is concentrated in a localized area. Their stability is ensured by their topological structure that depends on the particular boundary con-dition of the field. Indeed, to such boundary map can be associated a topological classification that, once fixed, cannot be changed without breaking the finiteness of the energy or the continuity of the field. In this way, it represents an invariant of the solution during the time-evolution and prevents the soliton from decaying into the vac-uum. In many cases, the topological information of the field is captured by an integer N, called topological charge, that is directly connected with the choice of the boundary conditions [1] [2] [3] [4] [11].
Solitons arise in many different examples of field theories and their physical inter-pretation varies depending on the specific model considered. Prominent examples of solitons are istantons that arise in an Euclidean Yang-Mills theory. Being classical so-lutions of an Euclidean Action, their contribution becomes significant in the study of quantum processes in which the Euclidean functional integral is involved [4] [5].
An-other example of soliton is given by the t’Hooft-Polyakov monopole that emerges in a non-Abelian gauge theory spontaneously broken. This theory allows for the existence of smooth soliton solutions that asymptotically behave as magnetic monopoles [6] [7].
Other important examples of solitons are the vortices, which are the main focus of this thesis, sigma model Lumps, Kinks, domain walls, skyrmions and so on [9] [10] [14] [43] [44].
Vortices are one of the first examples of solitons that have been considered in Field Theory and their discovery was connected with the study of condensed matter phe-nomena, such as superconductors. Within the Ginzburg-Landau (GL) approach in de-scribing superconductors, a complex field, that can be interpreted as the wave-function of the Cooper electron-pair, is coupled to the electromagnetic gauge potential. The su-perconducting phase is characterized by the spontaneous symmetry breaking of the Abelian gauge group U(1)that introduces a length scale to the electromagnetic field. Due to the existence of such a scale, the magnetic field cannot propagate freely into the superconducting medium but it decays exponentially on its surface. This physical phenomenon, which consists in the expulsion of the magnetic field from the supercon-ductor, is known as Meissner effect. It was discovered by Abrikosov [12] in 1957 that
the GL energy function allows for the existence of topological vortices. Arising from
the breaking of an Abelian gauge symmetry, these solutions are commonly known as Abelian vortices. In three dimensions a single Abelian vortex appears as an infinite mag-netic flux tube, squeezed into the superconductor by the Meissner effect, whose total energy is proportional to its length [13]. The magnetic flux of such tube is proportional
to the topological charge N, that classifies the different solutions. Abelian vortices are experimentally observed in Type II superconductors under appropriate conditions of magnetic field.
These features of the Abelian vortices remind us of the semi-classical picture of the QCD-string generated by a quark-antiquark pair. In such a picture, the chromo-electric field, generated by the quarks charges, is squeezed into a flux tube whose energy-density per unit length is constant. Following this idea, in the mid-1970s ’t Hooft, Mandelstam and Nambu proposed a soliton model for the QCD confinement, known as dual-Meissner effect [26] [27] [28]. In such a conjecture, the QCD vacuum
might be a kind of dual-superconductor, in which a chromo-magnetic charge con-denses and the chromo-electric field is expelled from the bulk. In such a medium, a quark-antiquark pair would be confined by a dual vortex. In spite of many investiga-tions, the Nambu-’t Hooft-Mandelstam conjecture has not found convincing evidences. Anyway, a proposal for the realization of a non-Abelian superconductor, in which a chromo-magnetic charge pair would be confined, has been recently made in [16] .
As in the Abelian case, the non-Abelian superconductor allows for the existence of topological vortices. A non-Abelian vortex consists in a magnetic flux tube, as in the Abrikosov theory, but with a further internal degree of freedom. Indeed, due to the presence of an exact non-Abelian symmetry, the static vortex solution can be rotated with respect to this group. Every solution is therefore characterized by a set of pa-rameters, the orientation moduli, that represent the possible non-Abelian orientations of the vortex [11] [17] [23]. Since the fluctuations of such moduli represent the zero
modes of the system, in the limit of low-energy, they constitute the relevant degrees of freedom to describe the non-Abelian vortex dynamics. Therefore, as it happens in many examples of effective theories, a low-energy model for the non-Abelian vortex can be constructed by a derivative expansion of the Lagrangian with respect to these "light" degrees of freedom. The reason for using a derivative expansion depends on the fact that the value of a derivative is related to the scale of the kinetic energy and then, in the limit of low-energy, it can be used as parameter of a Taylor series. Such a procedure for the construction of an effective Lagrangian for the non-Abelian vor-tex has been proposed in [29]. In that work, the lower orders of the effective theory
describe the free motion of the orientation moduli or, equivalently, the rigid evolution of the vortex along its zero modes [11] [16] [19] [31]. At this order of approximation,
known as the moduli approximation [15], the soliton behaves like a rigid body that, as
long as the internal excitations can be neglected, evolves without modifying its struc-ture. In order to improve such an approximation, we need to take into account the small deformations of the soliton shape. To this end, in [29] a small fluctuation is
introduction v
added to the rigidly evolving non-Abelian vortex obtaining, in this way, a next order of the effective theory. The new effective Lagrangian contains now the higher deriva-tive corrections for the orientation moduli that give a non-vanishing contribution to the physical observables. The presence of higher derivative corrections is a common feature of different effective Quantum Field Theories, such as the Chiral Perturbation Theory [42]. In that model, the pions represent the zero-modes fluctuations, taking the
role of the orientation moduli, and the low-energy Lagrangian is performed, in the same way, with a derivative expansion of the field. We thus deduce that the soliton effective theory constitutes an example of a more general framework of low-energy theories.
In [29] it has been proved that the explicit form of the internal vortex fluctuations
is not necessary in order to construct the effective Lagrangian with higher derivative corrections. This approach, however, does not give the opportunity of an appropriate analysis of such fluctuations, that can be useful for a future description of the exact vortex dynamics. With this aim, in this thesis, we solve the full equations for the vortex fluctuations and then we provide an appropriate discussion on the effects of the higher orders on the vortex dynamics. Afterwards, we repeat the same analysis for the case of a monopole in Higgs phase, a new soliton configuration that arises adding a new term to the vortex Lagrangian. The main original contribution of this thesis work is presented in Chapter4and Chapter6below.
The present thesis is organized into six Chapters with the following structure.
• In Chapter1we review the general theory of solitons, giving the basic concepts
of Topology. Here we analyse the case of the Abelian vortex.
• Chapter 2 is dedicated to reviewing different examples of low-energy effective
theories. In particular, we analyse in detail an example of low-energy approx-imation in Classical Mechanics and we provide different examples of higher derivative corrections in field theories. This discussion shows how the derivative-expansion is a common feature of effective theories.
• In Chapter3we discuss the general theory of non-Abelian Vortex and the general
procedure to construct a low-energy model for it. In particular, we analyse the lower orders of such effective theory that describe the rigid evolution of the vortex, i.e. the evolution along its zero modes. To this end, we reproduce an explicit calculation of the non-Abelian vortex zero modes using different gauge choices.
• In Chapter 4 we deal with a higher order of the low-energy approximation in
which the internal fluctuations of the vortex are taken into account. Here, we discuss a strategy for the resolution of such fluctuations giving, at the end, an appropriate solution. Then, we calculate the higher derivative corrections to the non-Abelian vortex effective theory, proving how this result reproduces correctly
the effective Lagrangian in [29].
Afterwards, we analyse the effective low-energy theory showing how the field-fluctuations deform the vortex profile. This discussion is proposed for different space-time d+1 dimensions with d=0, 1, 2.
• The Chapter 5 is dedicated to the case of a monopole in the Higgs phase,
con-sisting in a deformed configuration of the non-Abelian vortex. This new config-uration, obtained adding a new field to the Lagrangian, can be interpreted as a single Abelian monopole trapped inside the superconductor. For this model, we discuss the general theory and the low-energy dynamics. Here, we point out the similarities and the differences with respect to the vortex case.
• In Chapter6we take into consideration the higher corrections of the low-energy
theory of the monopole and we resolve the problem of the soliton internal fluc-tuations. Then, we analyse the effective monopole theory with the higher deriva-tive corrections, for which we provide a discussion in 0+1 dimensions.
Part I
Topological solitons
1
T O P O L O G I C A L S O L I T O N S
In this Chapter, we discuss the theory of topological solitons, a class of stable classical solutions characterized by non-trivial topological boundary conditions. In order to give a more complete analysis of such solutions, we previously review the main ideas of Topology, focusing our attention on the different classifications of the continuous maps.
In Section 1.3, we analyse an explicit example of soliton known as Abelian vortex.
This solution appears as an infinite magnetic flux-tube, extended along the space, whose energy for unit of length is constant. In this Section, we discuss the main characteristics of the vortices such as their stability and their topological classification.
In Section1.4, we deal with the idea of soliton dynamics, for which we review the
first example of low-energy approximation known as moduli approximation.
1.1
introduction to topology
Homotopy theory studies the relations and the equivalences among different maps de-fined on manifolds. This theory results very important in Physics and in particular in Field Theory since any field solution consists in a map, continually deformed by time evolution, between the space of spatial coordinates and the space of fields. Therefore, in order to classify these solutions, we need some basic tools of topology.
Let X and Y be two manifolds without boundary and let x0 ∈ X and y0 ∈Y be two points on the manifolds: we define the set of continuous based maps F : X→Y such that f(x0) = y0 for all f ∈ F. Now, a based map f0 : X → Y is said to be homotopic to another map f1 if f0 can be continuously deformed into f1. More rigorously, f0 is homotopic to f1 if exists a continuous map
˜f(x, τ): X× [0, 1] →Y (1.1.1) with τ ∈ [0, 1]such that ˜f(x, 0) = f0, ˜f(x, 1) = f1 and ˜f(x0, τ) = y0 for all τ. In this case, we say that ˜f deforms f0 into f1. The Homotopy-relation between two maps f and g represents an equivalence relation and then we write f ∼ g. In fact, we verify that
• It is symmetric since the “τ” flow can be inverted: if ˜f(x, τ) deforms f into g, then ˜f(x, 1−τ)deforms g into f .
• it is reflexive being f already “deformed” into itself.
• it is transitive because the “τ” interval can be split: if ˜f deforms f into g and ˜g deforms g into h we can define
˜s(x, τ) =( ˜f(x, 2τ) 0<τ<
1 2 ˜g(x, 2τ−1) 12 < τ<1
(1.1.2) that deform f into h.
Using this equivalence relation, we can divided the space F into different equivalence classes, called homotopy classes. In this sense, it is possible to classify different maps depending on their topological classes. Moreover, from this classification we can also obtain information about the target manifold Y.
An important classification of the continuous maps concerns the case in which the manifold X is a n-sphere, Sn. The set of homotopy classes of the continuous based maps F : Sn →Y is called the n-homotopy group, denoted by the symbol πn(Y). In fact, defining a composition operation between different classes of maps, we can demon-strate that πn(Y), with n ≥ 1, is closed with respect to this operation and that it possesses all the propriety of a group. Let us show an example of composition for n = 1: taking two continuous based maps f(θ) and g(θ), as representative of two
homotopy classes, from the set of F : S1 → Y with base points 0 ∈ [0, 2π] ∈ S1 and y0 ∈Y, we define h(θ) = ( f(2θ) 0<θ < π2 g(2θ−2π) π 2 <θ <2π (1.1.3) as the composition of the two maps. Note that h is still a continuous based map with h(0) = f(0) = y0 and then it can be a representative of an homotopy class, i.e. still an element of π1(Y). As we can trivially demonstrate the constant map, for which f(θ) = y0 for all θ, represent the identity, i.e. the neutral element. Furthermore, for every map f(θ)it exists the inverse f−1 = f(2π−θ)that, after the composition with
f , gives the constant map. All these properties can be extended for any generic n-homotopy group with n ≥ 1. Note that, if the manifold Y is connected, any different choice of the base points does not change the group, meaning that the result of our operations depends only on the particular topology of the target manifold.
As simple example of homotopy theory, we can study the continuous maps F such that
F : S1→R2/{0} (1.1.4) where we consider the real plane without a point in the origin. We arbitrarily choose as base point θ = 0 in the interval [0, 2π] that parametrizes the manifold S1, and a point (x0, y0)inR2 different from the origin.
Any map that we can construct is basically a circumference in the 2D plane. If we have a map that does not encircle the origin we can continuously deform it till it coincides with the base point, i.e. the constant map. Generalizing, all the maps that does not
1.1 introduction to topology 5
encircle the zero point are homotopically equivalent to the constant map and then they constitute the trivial element of π1(R2/{0}). Anyway if a map encircles one time the origin it cannot be deformed into the constant map since the point zero cannot be crossed. Therefore, all this kind of maps represents another equivalent class. If the map goes around the origin, passing from θ = 0 to θ = 2π, clockwise then the inverse goes anticlockwise and their composition is equivalent to the constant map. We intuitively understand that all the homotopy classes are characterized by an integer number that counts how many times the map encircle the origin, with the sign plus or minus, depending on the clockwise or anticlockwise direction. Calculating rigorously the structure of the first homotopy group we obtain
π1(R2/{0}) =Z (1.1.5) where the symbol equal means “being isomorphic”.
The fundamental group, as we call the first homotopy group π1(Y), gives us some information about the manifold Y. In particular if Y is connected and π1(Y) = I, where I denotes the trivial group with just the identity element, the space Y is said to be simply connected. In the last example, we then say that the manifoldR2/{0}is connected but not simply connected.
Another important application of the homotopy theory concerns the Lie groups. The studies of the homotopy group of some Lie groups play an important role in Field Theory and in particular in the classification of topological solitons since, as we will see later, the symmetry groups determine the vacuum manifold of the theory. Let us consider, for example, the first homotopy group of the U(1) group that consists in the set of phase rotation of angle θ. Any maps f from the S1 space, parametrized by ˜θ ∈ [0, 2π], into U(1) can be written as f(˜θ) = eiθ(˜θ). For simplicity we choose as base points ˜θ = 0 and θ = 0 such that θ(0) = 0. Any maps must be continuous and then we need to impose the condition θ(2π) = θ(0) +2πm = 2πm, where m is
an integer number also called the winding number. In fact essentially this number counts how many times the map turns around the circumference in the complex plane of U(1). Any maps is characterized by m and it is easy to verify that two different maps, f1(˜θ)and f2(˜θ)or equivalently θ1(˜θ)and θ2(˜θ), with the same number m can be continuously deformed into each other. The continuous map
h(˜θ, τ) =ei (1−τ)θ1(˜θ)+τθ2(˜θ)
(1.1.6) deforms f1 into f2 with τ ∈ [0, 1]. Otherwise it is impossible to deform maps with different m into each other without changing the condition of continuity. From these considerations we deduce that every homotopy class is characterized by m and then
π1(U(1)) =Z (1.1.7) Obviously any element of the U(1)group can be seen as a point in a unitary circumfer-ence defined on the complex plane. It is therefore possible to define an isomorphism
between the Lie group U(1)and the manifold S1, meaning that U(1) is topologically equivalent to S1. We deduce that
π1(S1) =Z (1.1.8)
This intuitively relation can be extended to any n-homotopy group of an n-sphere Sn obtaining a general relation
πn(Sn) =Z ∀n≥1 (1.1.9) For the S2 sphere the winding number m counts the map cover the surface of the sphere and so on for all the higher dimensions.
Other important groups in Physics are for example SU(2), that is topologically equivalent to the sphere S3, and the quotient group SU(2)/U(1), topologically equiv-alent to S2. With the above relation (1.1.9) is easy to compute
π2(SU(2)/U(1)) =Z (1.1.10)
π3(SU(2)) =Z (1.1.11) A special consideration must be given for the 0-homotopy classes π0(Y). The equiva-lence classes of π0(Y)are maps from a single point to Y, up to homotopy equivalence. In particular, we consider maps, with image-point in the same connected component of Y, as equivalent and then π0(Y)is the set of distinct, connected, component of the manifold. As we have already specified π0(Y)is generally not a group.
Topological degree
In the above Section, we calculated some simple examples of homotopy groups even if, in the general case, this calculation is not easy. There are different techniques for classifying continuous maps and one of the most efficient is topological degree theory. Beside identifying the different homotopy classes in a set of continuous maps, we need a way to understand, given a certain map, what class it belongs to. Topological degree gives a solution to this problem.
Let X and Y be two oriented manifold with the same dimension n and let f be a continuous based map from X to Y. In this case, we need f to be differentiable every-where with continuous derivatives. LetΩ be a volume form on Y such that
Z
YΩ
=1 (1.1.12)
We define the topological degree of f as the integral on X of the pull-back ofΩ through f
deg f=
Z
X f
1.1 introduction to topology 7
In details, introducing the coordinates x over X and y over Y, the map f is represented by the function y(x)and then, writing explicitly the n-form over Y as
Ω= β(y)dy1∧dy2∧...∧dyn (1.1.14)
the pull-back ofΩ is given by
f∗(Ω) =β y(x)∂y 1 ∂xjdx j∧ ∂y2 ∂xkdx k∧...∧ ∂yn ∂xidx i =β y(x)det ∂yi ∂xjdx 1∧dx2∧...∧dxn (1.1.15) where J(x) =∂yi ∂xj
is the Jacobian of the map at x. This is form is now defined over X (remember that X and Y have the same dimension) and, integrating over ˜X= y−1(Y), we would obtain the same result of (1.1.12). On the contrary, we integrate over all
the manifold X and then, in some sense, we integrate the form (1.1.12) many times
as the number of time that the map y(x) “cover” the manifold Y. It is possible to demonstrate that topological degree is always an integer and then it constitutes an homotopy invariant of the map.
Considering a continuous map f : S1 → S1, we must verify that the degree of the map is equal to the winding number. We choose as normalized form over S1 the volume form Ω = 2π1 dθ and we want to calculate the degree of a generic map θ(˜θ), where ˜θ parametrizes the first manifold S1. Using our definition we obtain
deg θ= 1 2π Z 2π 0 dθ d ˜θd ˜θ= 1 2π θ(2π) −θ(0) = m (1.1.16) where we use the continuity of the map.
In general it exists a second, apparently independent, way to compute the topolog-ical degree of a map. Given a continuous map f : X → Y, let choose a point y on Y, such that the set of preimages of y, the points on X mapped to y, is a set of isolated points x1, ..., xm(the set can be empty) where the Jacobian of the map is non-zero. This condition for such a point y occurs almost everywhere on Y. We compute the degree of the map ˜deg f as
g deg f= M
∑
m=1 sign J(xm) (1.1.17) where sign J(xm) is the sign of the Jacobian at every xm. In other words we can say that gdeg f counts the preimages of y with their multiplicity 1 or -1 depending on the locally orientation of the maps f . gdeg f is therefore an integer and its value is independent of the choice of the point y. To prove it, we demonstrate that gdeg f =deg f . Let deform the volume form Ω on Y so that it is concentrated on a small neighbourhood of the point y and still normalized. Now the pull-back ofΩ, i.e. f∗(Ω),
is concentrated on small neighbourhoods of each preimages x1, ..., xm. Let us calculate the integral of the pull-back form in one of this small domain. If the neighbourhood is small enough then the continuous function f =y(x)is locally invertible in this domain and then, in order to compute the integral, we can transform back the variable x into y so that we introduce a factor|J|−1. Now the integral is the same as (1.1.12) up to a
sign, 1 or -1, depending on the sign of the Jacobian at that point. Therefore, integrating the form f∗(Ω)in all the manifold X, the formula for deg f reduces to the expression for gdeg f in (1.1.17).
1.2
topological solitons in field theory
In this Section, we show how Topology plays an important role in Classical Field Theory. Let us write a generic classical action S defined on the flat Minkowski space M. Considering a multiplet of n scalar fields, φ = (φ1, ..., φn), with a continuous potential V(φ1, ..., φn)such that
S= Z dd+1x h 1 2∂µφl∂ µ φl−V(φ1, ..., φn) i (1.2.1) where l = 1, ..., n and d is the number of spatial dimensions. Any field configuration of this theory is a continuous map
φ(t,~x): M→Y (1.2.2)
from the Minkowski space coordinates to Y = Rn, that represents the manifold in which the fields live, i.e. the target manifold. (All the theories in which the target manifold is a flat space asRn are called linear theories).
Let us restrict our interest in the static theory, neglecting the time derivative. For such a configuration the total Energy takes the form
E= Z ddxh 1 2∂iφl∂iφl+V(φ1, ..., φn) (1.2.3) where i=1, ..., d. The static configuration field is a continuous map from the flat space to Y
φ(~x):Rd →Y (1.2.4)
For any theory, we request that a physical field-configuration must have finite energy (1.2.3). Let us discuss how this condition can be respected.
Let us assume that the potential V(φ1, ..., φn)takes its minimal value Vmin =0 on a submanifold Γ ⊂ Y, i.e. the vacuum manifold of the theory. Then we must impose that, as long as the space-coordinates move to infinity, the field-map φ must take its value in Γ, in order to ensure a zero energy-density on the boundary of the space-coordinates (this condition is necessary but not sufficient to guarantee the finiteness
1.2 topological solitons in field theory 9
of the energy). Since in any direction at infinity, the field must approach the vacuum manifold, we are basically defining a map from the sphere at the boundary of Rd, i.e. Sd∞−1and the manifoldΓ as
φ∞ : Sd∞−1 →Γ (1.2.5)
Therefore, we can define a new equivalence-relation that says that two field configura-tions φ and ˜φwith distinct asymptotic data φ∞ and ˜φ∞ are homotopic if the maps φ∞
and ˜φ∞are homotopic. The topological classification of a finite-energy static field φ(~x)
is therefore determined by the homotopy class of the boundary map φ∞, i.e. by
πd−1(Γ) (1.2.6)
The same classification is still valid for time-depending field since the time evolution is a continuous deformation of the map that does not change the homotopy class of the field.
If we want to change the boundary condition, passing from one homotopy class to another one, we must necessarily pass through a set of field-configurations with infinite energy, since the boundary map φ∞ should live outside the vacuum manifold. The homotopy class of a field-map is therefore an invariant of the time evolution.
Minimizing the static energy in (1.2.3), we find the static solutions of our theory.
One of them is the vacuum solution that correspond to the constant map φ(~x) = φ0 such that V(φ0) = 0. This configuration corresponds to the trivial element of the group πd−1(Γ). Any static field solution with a non trivial topological classification is known as topological soliton or soliton. The name and the characteristics of every soliton depend on which particular homotopy group that is involved in the topological classi-fication. Let us list some examples.
d = 1: here Sd∞−1 consists of two points ±∞ in R. Therefore the topological classes of
this theory are elements of π0(Γ) ×π0(Γ). If Γ consists of two points in R, i.e. two distinct vacua, then the fields are classified byZ2×Z2. This is the case of topological solitons called Kinks.
d = 2: here Sd−1
∞ is a circle. The fields are topologically characterized by the element of π1(Γ), the fundamental group of Γ. If the vacuum manifold is a circle, as it hap-pens with the symmetry breaking of a group U(1), then the solutions are classified by
π1(S1) =Z and they are generically called Vortices.
d = 3: here Sd∞−1 is the surface S2. If the theory is characterized by the symmetry breaking of a group SU(2)down to U(1), then the vacuum manifold is a S2 surface too (SU(2)/U(1) =S2). The solutions classified by π
2(S2) =Z are called Monopoles. All these considerations about the topological classification of a Field Theory are valid for linear theories, in which the target manifold is the flat spaceRn. We turn now to nonlinear theories with field φ
where Y is a closed manifold and d≥2. Let considerΓ a non-trivial vacuum manifold Γ⊂ Y. It can happen that, even if a field approaches the vacuum manifold at infinity, the gradient term of the static energy results divergent at the boundary. To avoid such a divergence, the field must tend to a constant value at infinity, independently of the direction. Thus φ∞: Sd∞−1 →Y must be a constant map with value ,for example, y0. (In the absence of the potential y0is arbitrary, if a potential is present then it must be inΓ). Since all the map must respect this condition, we take y0 as the base point of Y. This boundary condition allows for a topological compactification of the coordinate-space Rd into Sd. In fact, if all the points at infinity are mapped with y
0, we do not lose any information if we consider them as a single point at infinity. This picture is the same of the stereographic projection of a Sn sphere on the planeR where a single point at the north pole of the sphere correspond to all the points at infinity onRn. Therefore, we now identify every map with finite energy φ :Rd →Y with the set of based maps
φ : Sd → Y with x0 ∈ Sd, the north pole, and y0 as based points. The topological classes of this theory are given by the element of the group πd(Y). Let us give some example.
d=2: here the field configurations are labeled by the element of π2(Y). A non trivial classification occurs for example when Y=S2that is the case of the O(3)sigma model: a model consisting in a triplet of real scalar fields φ= (φ1, φ2, φ3)with the constraint
~φ2=1.
d=3: here the relevant homotopy group is π3(Y). Topological solitons labeled by this group occur in Skyrme model and they are called Skyrmions. In this model the fields are organized in the matrix U with U∈ SU(2)and therefore the target space consists in SU(2) 'S3.
Derrick’s theorem
Let us consider only Field Theory defined in flat space as Rd: a spatial rescaling is a map x → µx, with µ > 0. Let ψ(x) be a finite energy configuration and let ψ(x)(µ)be the 1-parameter family of field obtaining from ψ(x)after the transformation
x →µx. Then we define e(µ) = E(ψ(µ)(x))as the static energy value for a given filed
configuration ψ(µ)(x)that depends on µ.
The Derrick’s theorem says: suppose that for an arbitrary, finite energy field con-figuration ψ(x), different from the vacuum, the function e(µ)has no stationary point.
Then the theory has no static solutions of the field equation with finite energy other than the vacuum.
Let us explicitly analyse the function e(µ)for a simple theory as (1.2.3). For a simple
scalar field φ(x)one defines
φ(µ)(x) =φ(µx) (1.2.8)
and the gradient is therefore
1.2 topological solitons in field theory 11
where we have simply defined ˜x = µx. Note that at infinity, on S∞d−1, the field φ(µ)
still tends to φ∈ Γ and then, as µ varies, the energy remains finite and the topological
class of the field does not change.
We now decompose the static finite energy of a generic field configuration φ in two parts such as E(φ) = Z ddx 1 2∇φ· ∇φ+V(φ) = E2+E0 (1.2.10)
where E2comes from the gradient part and E0 from the potential term. Now we want to find the static energy of the rescaled field configuration φ(µ)that gives
e(µ) =E(φ(µ)) = Z ddx 1 2∇φ (µ)· ∇ φ(µ)+V(φ(µ)) = Z ddx µ 2 2 ∇˜ φ(˜x) ·∇˜φ(˜x) +V(φ(˜x)) = µ2−dE2+µ−dE0 (1.2.11)
where the last step follows by a change of variable from x to ˜x. Being E2 and E0 positive, the function e(µ)crucially depends on the spatial dimensions d as
e(µ) = µE2+µ1E0 d=1 E2+µ12E0 d=2 1 µE2+ 1 µ3 d=3 (1.2.12)
In the case of d=2 or d= 3 the function e(µ)decrease monotonically and then there
are not stationary points. Following the Derrick’s theorem these theories do not admit non-trivial static solution different from the vacuum. Different is the case of d = 1 where a stationary point exists at µ =√E0/E2. Therefore in one dimension the possi-ble non-trivial static solutions are not ruled out but their existence is not guaranteed. Kink solitons are example of solution in d=1.
For a gauge field Aν, coupled to a scalar fieldΦ, we define
A(νµ)(x) =µAν(˜x) (1.2.13)
in order to have the same transformation of the derivative in (1.2.9). For the scalar
field we have the same transformation as before with Φ(µ)(x) =Φ(
µx) (1.2.14)
The field strength involves one derivative more then the gauge potential and then we have
In general for a gauge theory the finiteness of the static energy is guaranteed when at spatial infinity, on Sd∞−1, we have Fνλ = 0, DνΦ = 0 and Φ ∈ Γ. Also in this case the
rescaled fields respect these boundary conditions and therefore the energy remains finite and the topological class unchanged for every µ>0.
Now let us write a generic form of the energy functional E=
Z
ddx |Fνλ|2+ |DνΦ|2+V(Φ)
= E4+E2+E0
(1.2.16)
assuming that E4, E2 and E0 are all positive. Replacing the fields with the rescaled ones, the functional e(µ)depends on µ as
e(µ) =µ4−dE4+µ2−dE2+µ−dE0 (1.2.17) Now for d= 2 and d = 3 e(µ)has a minimum for some µ and solitons solutions are
not ruled out. This is respectively the case of Vortices and Monopoles. On the contrary for d =4 the theory has not solitons solutions. The analysis of the Derrick’s theorem can change if we study theories where for example E2 and E0are zero like the case of a pure Yang-Mills theory. The procedure can be reproduced for any specific case of Classical Field Theory.
1.3
abelian vortex
Vortex-solitons can emerge in many examples of Field Theory with a non-trivial vac-uum manifold. In 1957, the existence of topological vortices was discovered by Abrikosov [12] studying the superconductivity under particular condition of magnetic field. Later,
in 1973 Nielsen and Olesen considered the relativistic version of the vortex, after the advent of the Higgs model in high-energy physics [8] . The simplest example of
vor-tex, that we analyse in this Section, emerge in a U(1)gauge model with spontaneus symmetry breaking an it is known as Abelian vortex. A single vortex, defined in 3+1 dimensions, appears as an infinite tube-flux (string), characterized by constant energy per unit length, i.e. the string tension. In the following, we briefly discuss the general theory of Abelian vortices, providing an analysis of their main characteristics.
Let us discuss a theory consisting in a single complex scalar field, defined in 2+1 dimensions, with a U(1)global symmetry. The Lagrangian of such a theory results
L = |∂µφ|2−U(φ) (1.3.1)
where
1.3 abelian vortex 13
The U(1)group acts as a phase rotation
φ→φ 0 = eiθφ φ∗ →φ∗ 0 =e−iθφ∗ (1.3.3) A potential of the type (1.3.2) breaks the global symmetry, due to the
vacuum-expectation-value choice
φvac =veiθ (1.3.4)
with θ arbitrary phase. The vacuum manifold of the theory, i.e. the space of minimal energy configuration, is therefore a S1 circle, defined on the complex plane by all the possible values of θ.
Now, we take into consideration only the static energy of the model written as E=
Z
d2xh∂iφ∗∂iφ+U(φ)
i
. (1.3.5)
To ensure the finiteness of the static energy, we need that, at the boundary of the space coordinates|x|→∞, the field-map approaches the vacuum manifold, defining in this
way the boundary condition:
φ∞ : S1→S1. (1.3.6)
Therefore, every field-configuration, characterized by finite static-energy, can be classi-fied by the homotopy group of such boundary map, i.e.
π1(S1) =Z. (1.3.7)
As general rule, we can immediately deduce the form of the vacuum manifold and, thus, the relevant homotopy group of the model looking at the symmetry-breaking pattern. Being a vacuum-solution a point of minimum of the potential in the field-space, we can act on it with the symmetry group G in order to find all the other points of minimum, i.e. the whole vacuum manifold. If it exists a subgroup H that leaves the vacuum unchanged, i.e. the exact subgroup, the relevant group-elements that define all the points of the vacuum manifold belong to the quotient group G/H. Therefore, the topological classification of the theory can be equivalently made by πd−1(G/H), with d space dimensions.
The symmetry breaking pattern of this theory simply reads
U(1) →I (1.3.8)
and then we simply obtain again π1(U(1)) =Z.
Before to construct the static solutions of the model, we check the Derrick’s no-go theorem. As we have proved in (1.2.12) for a non-gauge theory, any stable static
solution, different from the vacuum, is ruled out in d=2 spatial dimensions. A global U(1)theory, therefore, does not allows for any stable soliton-vortices. Moreover, trying
to construct a field configuration with non-trivial topology, we find a further problem with the energy-finiteness. Let suppose to have a field map that “wind” at infinity, with boundary consisting in
φ(r, α) →veinα at r→∞. (1.3.9)
Here, we use the polar coordinates r = px2+y2 and α as angle, while n represents the winding number. The boundary behavior of the gradient of field tends to
∂iφ∼in∂iα= −ineij xj
r2 as r→∞ i, j=1, 2. (1.3.10) Looking at the static energy (1.3.5), even if we guarantee the convergence of the
po-tential part with the condition (1.3.9), the winding at infinity implies a logarithmic
divergence of the derivative-component Z
d2x ∂iφ∗∂iφ→
Z dr
r →∞ (1.3.11)
The origin of this divergence is clear in polar coordinates, where the α-derivative does not vanish at infinity due to the winding of the field.
All these problems will be resolved passing to a gauge theory, in which the simple derivative is replaced by the covariant derivative.
A model that allows for the existence of vortex-solitons, with finite energy, is the U(1)
gauge theory in Higgs phase. The model is described by the Lagrangian
L = − 1 4e2FµνFµν+ |Dµφ|2−λ |φ|2−v2 2 (1.3.12) where Fµν is the photon field strength tensor
Fµν= ∂µAν−∂νAµ (1.3.13)
and the covariant derivative is defined by
Dµφ=∂µ+iAµφ. (1.3.14)
Note that the complex field has charge Q = 1. The gauge transformations act on the fields as
φ→eiβ(x)φ, Aµ → Aµ−∂µβ(x). (1.3.15)
The potential energy is chosen to guarantee the Higgs mechanism, i.e. the spontaneous symmetry breaking of the gauge group, in which the phase of the complex scalar field is eaten out by the gauge field. This mechanism is equivalent to impose a gauge condition (unitary gauge) that eliminates the phase of φ, leaving the complex field as a real scalar field. Using this choice, the vacuum expectation value results
1.3 abelian vortex 15
Obviously any other gauge choice represents the same physical vacuum configura-tion. Let consider the static energy of the model. To this end, let firstly expand the Lagrangian in this way
L = 1
2e2EiEi+D0φD0φ− 1
4e2FijFij−DiφDiφ−U(φ) (1.3.17) with the “kinetic” fields
Ei = F0i =∂0Ai−∂iA0 D0φ=∂0φ+iA0φ
(1.3.18) If we set the gauge choice A0 = 0 and we request that the fields do not depend on time then the Kinetic energy vanishes and the static energy is written as
E=
Z d2x 1
4e2FijFij+DiφDiφ+U(φ). (1.3.19) A necessary condition to ensure the convergence of E implies that U(φ) →0 at|x| →
∞, i.e.
|φ| →v for|x| →∞ (1.3.20)
This condition, as for the global theory, defines the classification of the static solution with respect to the homotopy group π1(S1) =Z.
On the large circle S1∞, at the boundary of the 2D plane, we must choose
φ=vei f(α) for|x| →∞ (1.3.21)
with again α the polar angle. The simplest example of f(α) is given by f(α) = nα
that produces n winding at infinity (n∈Z). A positive value for n means a clockwise
winding. Formally, we can calculate the topological degree of the map, called also the topological charge, proceeding as in (1.1.16):
n= 1 2π Z 2π 0 dα d f(α) dα = 1 2π f(2π) − f(0). (1.3.22) n is a proper characteristic of every map and represents a gauge invariant for the theory. To prove it, let consider a smooth gauge transformation depending only of the spatial coordinates
U(x) =eiβ(x) (1.3.23) with β(x) a continuous function. Passing to polar coordinates, we consider that, for every value of r ≥ 0, U(x) is a map from the circle or radius r to the U(1) space with winding number supposing nr(U). This gauge transformation would change n to n+n∞(U). However, by continuity, nris independent of r and, proceeding to r→0, again the continuity imposes n0 = 0. If not, we would obtain different phase-values
for each directions to the origin. In this way, we demonstrate that n∞(U) =0 for every continuous U(x).
Differently from the global case, here the Derrick’s theorem allows for the existence of possible stable static solutions, as we proved in (1.2.17). Moreover, the
conver-gence of the space-derivative components of the static energy regards now the covari-ant derivative Diφ → ∂iφ. To ensure that Diφ → 0 fastly enough for the
energy-convergence we need that, at large r, the gauge potential tends to Ai = −n∂iα=neij
xj
r2 i, j=1, 2 (1.3.24) keeping Fij → 0 at r → ∞ and with Diφ → 0 faster than r12. Here, eij is the two
dimensional Levi-Civita tensor.
The boundary form of the gauge potential is in one-to-one correspondence with the form of the phase in the asymptotic behaviour of φ. Therefore, the topological information characterizing the soliton must be contained also in the gauge potential. Indeed, calculating the magnetic flux on the plane, we have
ΦB = Z d2x B= Z d2x F21 = I S1 ∞ ~ dx· ~A=2πn (1.3.25) where n is the winding number. Here, we used the Stokes theorem that permits us to integrate the boundary of the gauge potential. Since ΦB is proportional to the topological charge then it constitutes a topological invariant of the theory and classifies the different static solution for the vortex.
After the topological analysis, regarding the asymptotic field-conditions, we finally search for a vortex solution. In the general case, we must minimize the static functional (1.3.19) and resolve the second order differential equations, imposing the right
bound-ary conditions. In this analysis, anyway, we focus on a particular family of solutions, called BPS solutions, that energe from a system of first order differential equations. In particular, for a special value of the coupling λ
λ= e
2
2. (1.3.26)
the static energy, namely T as tension, can be re-written as T = Z d2x h 1 4e2F 2 ij+ |Diφ|2−e 2 2 |φ| 2−v22i = Z d2x ( 1 2 h1 eF21+e |φ| 2−v2i2 + |(D1+iD2)φ|2+v2F21 ) . (1.3.27)
Such a completion, called Bogomol’nyi completion, implies the following inequality
1.3 abelian vortex 17
for every map with positive topological charge n (here we use (1.3.25) to integrate the
last term of (1.3.27)). The tension energy is bounded from below and its minimum is
obtained for the field configurations that resolve F21+e2 |φ|2−v2=0
(D1+iD2)φ=0.
(1.3.29) We have so obtained a system of first order differential equations, called Bogomol’nyi-Prasad-Sommerfield (BPS) equations, for which the static energy results
T =2πn. (1.3.30)
Obviously any solutions of these equations resolve also the usual second order differ-ential equations of motion since they represent a stationary point of the energy. To solve the system (1.3.29), we must find an appropriate ansatz: for the elementary n=1
vortex it is convenient to introduce two profile functions ϕ(r)and f(r)as
φ(x) =vϕ(r)eiα, Ai = eij xj
r2[1− f(r)] (1.3.31) where r = px2+y2 and α is the polar angle. In this case, we set the center of the vortex in the origin of the plane. To obtain a different choice is sufficient to translate x→x−x0and y→y−y0. Note that the static-energy does not depend on this choice and then the two collective coordinates x0and y0 represent the moduli parameters of the Abelian string. Using the ansatz (1.3.31), we reduce the field equations to a system
for the only profile functions −1 r d f dr +e 2v2( ϕ2−1) =0 rdϕ dr − f ϕ=0. (1.3.32)
From the first equation, we immediately recognize the length scale of the vortex core ev∼1/L.
The boundary conditions for these functions derives directly from the finiteness energy as we have already discusses and result
ϕ(∞) =1, f(∞) =0. (1.3.33)
At the same time, at the origin, the smoothness of the fields requires
ϕ(0) =0, f(0) =1. (1.3.34)
Figure 1: Profile function f(r)
Figure 2: Profile function ϕ(r)
Looking at these solutions, we note that the magnetic field B decreases rapidly from the center of the vortex and, on the contrary, the scalar field vanishes in the core of the vortex. This fact reminds us of the Landau-Ginzburg theory of superconductivity, in which the condensate wave-function, represented by the complex scalar field, vanishes in presence of a magnetic field.
1.4
moduli space and soliton dynamics
Besides the static configurations, we are also interested in the study of the soliton dynamics. In general, the exact behavior of a time-evolving soliton can be provided only by solving the complete Euler-Lagrange equations of motion. However, these kinds of solutions are often unknown due to the complexity of the highly non-linear differential equations. The study of the soliton dynamics can be carried on working in the limit of low energies.
In this case, the idea of Manton [15] consists in treating the soliton like a rigid body
1.4 moduli space and soliton dynamics 19
fact that, as a whatever solid body, any static soliton-solution depends on a series of constant parameters, known as moduli, that describe its position or its orientation in the space. The space in which these parameters live is known as the Moduli space. If the excitations of the internal structure can be neglected, a slow-moving solid body or a slow-moving soliton can be approximated by the slow-variation of its moduli-parameters. This low-energy approximation, commonly known as the moduli approx-imation, is then realized promoting the constant parameters ai of the static solution to time-depending variable ai(t). Therefore, as long as the time-evolution is slow, a time-evolving soliton can be approximated by φ(x, ai(t)), being φ(x, ai) the family of static solutions.
If we substitute this ansatz inside the soliton-Lagrangian we have L=T−V= Z ddx 1 2∂0φ∂0φ−U(∂iφ, φ) = Z ddx 1 2 ∂φ ∂ai ∂φ ∂aj ˙ai˙aj−U(∂iφ, φ) = 1 2gij(a)˙ai˙aj−E0 (1.4.1)
in which we obtain a new Lagrangian for the only dynamic variables ai(t). Here, the potential energy E0is a constant since the static energy, for definition, does not depend on the specific value of the moduli parameters ai. The matrix
gij(a) ≡ Z ∂φ ∂ai ∂φ ∂aj ddx (1.4.2)
defines a metric over the moduli space and then the motion of a soliton is descibed by the geodesic equations of its parameters.
However, studying the simple case of a rigid body, we know that, during a contin-uous rotation, the internal structure of the body is deformed by the centrifugal force. Therefore, to improve the moduli approximation, we need to take into account the massive-mode excitations of the soliton and give a further order of approximation to the effective Lagrangian (1.4.1). This analysis represents the main focus of this thesis
2
L O W - E N E R G Y T H E O R I E S
In this Chapter, we review various examples of effective theories, constructed to de-scribe the low-energy limit of a given model. This discussion points out how, even if they emerge in different areas of Physics, the effective theories share a number of common features such as the use of the derivative expansion and the choice of the light degrees of freedom as variables of the model.
In Section 2.1, we review a general procedure, proposed in [29] , to construct a
low-energy model for a classical particle subjected to a potential. In particular, we focus our attention on the higher-derivative corrections that emerge in the effective theory when the massive-mode excitations are taken into account. The importance of this Section will result evident in the following of this thesis, when we deal with the higher derivative corrections of a soliton-effective theory. In particular, a series of issues, such as the double-counting of the zero-modes, arise in both the models and then we have here a first possibility of an appropriate discussion.
In Section2.2, we provide a series of examples of effective field theories in which
the higher-derivative corrections are involved. The purpose of this Section is not to give a complete review of these models, for which we provide a series of references, but to show how the effective soliton theories, that we discuss in this thesis, constitute further examples of a more generic framework of low-energy models.
2.1
a particle in r
nLet us now consider a simple mechanical system in which a given potential acts on a particle. The analysis of the low-energy dynamics for this case will be used as general idea for the constuction of an effective theory for solitons.
A generic Lagrangian for a particle inRnis written as L= m
2 ˙x·˙x−V(x) (2.1.1) with x(t) = (x1, x2, ..., xn). In this case, the static solution, i.e. the minimal stable energy configuration, is represented by the vector x(0)such that
∇V|x(0) =0. (2.1.2)
Now, we assume that the minimum of the potential V is not unique but it exists a set of continuous points for which the potential has the same minimum value. This
condition is well realized if, for example, the potential V has a rotation symmetry. Let us parametrize all these points with the parameters ϕi in such a way that the vector x(0)(ϕi) is always a minimum of V, for every value of ϕi. In this way, we define the
space of the minimal energy configurations, that we call “the moduli space” M: M = {x(0)(ϕi) ∈Rn |∇V =0}. (2.1.3)
If this system is in a state of equilibrium, in a stable static configuration, and we perturb it with a sufficiently small excitation, we can assume that the particle starts to move approximately along the flat directions, i.e. in the moduli space, in which the potential energy does not increase. In other words, if the kinetic energy is smaller than the scale of the potential-excitations, the trajectory of the particle can be approximately described by the vector x(0)(
ϕi). The possible ansatz for a slow-moving particle is
therefore
x(t) =x(0)(ϕi(t)) (2.1.4)
in which we promote the moduli parameters to dynamical variables that depend on time ϕi → ϕi(t). Beside this approximation, we are basically assuming that the
vari-ation on time of the moduli parameters ϕi is smaller then the scale of the massive modes, i.e the excitations of the potential energy. This scale depends on the type of potential V and is determined by the value of the non-zero eigenvalue of the matrix H, obtained by the Taylor expansion of V near the minimum
[H(x(0))]ab = ∂
2V
∂xa∂xb
|x=x(0). (2.1.5)
The non-zero eigenvalue of this matrix defines the frequency ω of the small oscillation around the minimum of the potential. Therefore, we can assume the validity of the approximated solution (2.1.4) only in the limit
∂
∂t ω. (2.1.6)
Substituting the ansatz (2.1.4) in the original Lagrangian, we obtain an effective
La-grangian in which the dynamical variable in not the generic Rn vector x(t) but it is now the moduli parameters ϕi(t):
Le f f =L (0) e f f +L (2) e f f = −V(x (0)) + m 2 ˙x (0)· ˙x(0) (2.1.7) where L(e f f0) = −V(x(0)) is constant, being the minimum of V, and does not depend on ϕi(t). The second part of the Lagrangian can be rewritten using the new variables
ϕi(t)as
L(e f f2) = m
2gij ˙ϕ
2.1 a particle in rn 23 where gij = ∂x(0) ∂ϕi ·∂x (0) ∂ϕj (2.1.9)
defines the metric of the moduli space M. The effective equation of motion for the moduli parameters takes the form of the geodesic equation
¨
ϕi+Γijk ˙ϕj ˙ϕk =0 (2.1.10)
where the connection is given by Γi jk= 1 2g il g lj,k+glk,j−gjk,l. (2.1.11) This approximation is known as geodesic-approximation, or moduli approximation, and can be applied to the theory of solitons as we have discussed in1.4.
Before to improve such an approximation, it is useful to clarify the idea of “zero modes” and “massive modes” in Classical Mechanics. For every point x(0)(
ϕi)of the
moduli space M we can expand the potential V obtaining V'V(x(0)) + 1
2δxaHabδxb+... (2.1.12) where the matrix H is defined in (2.1.5). If the minimum of V is not unique but it can
be parameterized by continuous parameters as in (2.1.3), then it will exist one or more
directions in which, after a fluctuation δx, the potential does not increase. The vectors oriented in such directions are called “zero modes” and all the other directions define the “massive modes”. Looking at (2.1.12) and defining for simplicity the normalized
matrixΠ as
H=mΠ (2.1.13)
then the i−zero modesΦ0
i correspond to the eigenvectors ofΠ with zero eigenvalue: Π·Φ0
i =0. (2.1.14)
On the contrary all the other j−massive modesΦMj are defined by non-zero eigenval-ues
Π·ΦMj =ω2jΦMj . (2.1.15)
Remember that the set of zero and massive modes form a basis and then i+j = n, where n is the number of total dimensions.
The number of i−zero modes and i−moduli parameters is the same. This intuitively relation is confirmed by fact that is always possible to define a set of i−zero modes as
Φ0 i =
∂x(0)
where ϕi is a set of moduli parameters. To prove it, we write the zero modes as the fluctuations along two static configurations
x(0)(ϕi+ei) =x(0)(ϕi) +aiΦ0i. (2.1.17)
and then we just need to expand the first point around ϕi and obtain therefore the form of the zero modes (2.1.16):
x(0)(ϕi+ei) =x(0)(ϕi) +ei∂x
(0)
∂ϕi +... (2.1.18)
The definition (2.1.16) permits us to write the moduli-space metric (2.1.9) as a scalar
product of zero modes (2.1.9) as
gij =Φ0i ·Φ0j. (2.1.19) Higher orders corrections
In order to improve the moduli approximation, we have to take into account, for the particle trajectory, the fluctuations near the flat directions, i.e. the fluctuations along the massive-modes of the system. To this end, we introduce a correction to the ansatz (2.1.4) adding a small fluctuation as
x(t) =x(0)(ϕi(t)) +δx(t). (2.1.20)
where δx(t)must be oriented along the massive modes.
The new effective Lagrangian, written substituting the new ansatz (2.1.20) in the
fundamental theory, results
L=L(e f f0) +L(e f f2) +δL δL= m˙x(0)·δ ˙x+m 2δ ˙x·δ ˙x− 1 2δxHδx+... (2.1.21)
In the limit of low-energy, the massive-mode fluctuation δx can be expanded with respect to the small parameter ∂t/ω 1, obtaining a low-energy derivative-expansion of the type δx=x(2)+x(4)+... x(n) ∼O ∂ t ω n . (2.1.22)
For simplicity, we assume the presence of a unique eigenvalue ω2 in (2.1.15), i.e. the
presence of a unique massive mode for the potential V. Note that, in the expansion (2.1.22), we have only the even powers of the time derivative, due to the reflection
symmetry t→ −t of the theory.
The Lagrangian-correction δL can now be expanded with respect to the derivative of the moduli parameters as
2.1 a particle in rn 25
Resolving order by order the Euler-Lagrange equations for the fluctuations x(n), we get a continuous approximation of the time-evolving solution. Moreover, substituting such solutions into the expanded Lagrangian, we obtain an effective theory consisting in a derivative-expansion of the moduli parameters. The new Lagrangian-correction (2.1.23) gives rise to the higher derivative terms of the theory.
Let us discuss the first example of higher derivative corrections. The fourth-order Lagrangian is written as
L(4)=m˙x(0)·˙x(2)−1
2x
(2)
Hx(2) (2.1.24) where the term x(4) vanishes since it is proportional to∇V(x(0)) =0.
This order of Lagrangian contains the second order fluctuation x(2) but it does not contain a kinetic term for such a fluctuation. For this reason, the massive mode x(2)is
not an independent variable of the system but it depends on the derivative of the zero order x(0).
Before to deal with the resolution of the fourth order equation, we need to discuss a crucial point regarding all the higher orders of the effective theory. As previously men-tioned, the fluctuation x(2) must be oriented along the massive modes of the system since the evolution of the particle along its zero modes has been already considered in lower order. To avoid the double-counting of such degrees of freedom, we must im-pose to the fluctuations x(2)to be orthogonal to the zero modes. This constraint can be fixed with the help of a Lagrangian multiplier, added to the fourth order Lagrangian
L(4) → L(4)+
λiφ0i ·x(2) (2.1.25) where i counts the number of zero modes.
In order to understand in which way the double-counting of the zero modes emerges in the higher orders of approximation, in what follows we do not impose any condi-tion to the variables x(2). During the calculation, we then recover by hands the right
conditions for the fluctuations x(2). The equation of motion for x(2)results
m¨x(0) = −Hx(2). (2.1.26) Let us decompose this equation with respect to a complete vector-basis, such as the all set of zero modes and the massive modes of the theory{Φ0
i,ΦM}as
aiΦ0i +bΦM =a0iΦ0i +b0ΦM (2.1.27) where ai and a0i are the coefficients of the i−zero modes and b and b0 for the massive ones. To obtain these coefficient, we just need to project the equation for every mode using the proprieties
Φ0 j·Φ0
i =gjkΦ0k·Φi0 =gjkgki =δij Φ0 j·ΦM =gjkΦ0
k·ΦM =0.
Let us project the right side of (2.1.26) along the zero modes:
a0i = −Φ0i·Hx(2)= −gijΦ0j·Hx(2) =0 (2.1.29) where we have used (2.1.14) and the hermiticity of H. We deduce that this side of the
equation contains only massive-mode components.
On the left side of (2.1.26), along the zero modes, we have
ai =mΦ0i·¨x(0)=mΦ0i ∂x (0) ∂ϕk∂ϕj ˙ϕ k ˙ϕj+ ∂x(0) ∂ϕk ϕ¨ k =m ϕ¨i+Γikj ˙ϕk ˙ϕj. (2.1.30)
The coefficient b for the massive mode depends on the particular system that we are studying. An explict example of such a coefficient can be provided adding a general hypothesis on the form of the potential V.
Let us suppose that the minimum of the potential is characterized by the condition x(0)·x(0)= R2 (2.1.31) with R2 constant. The moduli space for such a system consists in a (n−1)−Sphere parameterized by n−1 moduli, to which corresond n−1 zero modes. Differentiating the condition (2.1.31), we obtain
∂x(0) ∂φi ·x
(0)=
0 (2.1.32)
that means that the vector x(0) is orthogonal to all the zero modes and then it
consti-tutes the only massive mode of the system.
With this hypothesis on the form the potentiav V, we can determine the coefficient b of eq. (2.1.27) projecting along x(0) as
b= mx(0)·¨x(0)=mx(0) ∂x (0) ∂ϕk∂ϕj ˙ϕ k ˙ϕj+ ∂x(0) ∂ϕk ϕ¨ k =mx(0) ∂x (0) ∂ϕk∂ϕj ˙ϕ k ˙ϕj = −mΦk·Φj ˙ϕk ˙ϕj = −mgkj ˙ϕk ˙ϕj (2.1.33)
where we have used the relation x(0) ∂x (0) ∂ϕk∂ϕj + ∂x (0) ∂ϕk ∂x(0) ∂ϕj =0 (2.1.34)
2.1 a particle in rn 27
Summarizing, the equation (2.1.26) can be decomposed into the system
(
−m gkj ˙ϕk ˙ϕjx(0)= −Hx(2) ¨
ϕi+Γijk ˙ϕj ˙ϕk =0
(2.1.35) where the first equation regards the massive mode and the other i−equations represent the zero-modes components. The physical origin of these different equations depends on the term ¨x(0)in (2.1.26). This term is nothing but the acceleration of a point moving
along the minimum of the potential V and then it contains the “centrifugal force”, orthogonal to the motion, and the acceleration along the zero modes. The second line of the system (2.1.35) corresponds, therefore to the same equations of motion for the
moduli parameters obtained in the second-order Lagrangian. The presence of these equations represents one of the effects of the zero-modes double-counting, that we discussed above. At this order, we are interested in the fluctuations of the particle-trajectory along the massive modes of the system and then the only relevant equation is −m gkj ˙ϕk ˙ϕjx(0) = −Hx(2). (2.1.36) that is solved by x(2)= cx(0) with c = 1 ω2 gkj ˙ϕ k ˙ϕj. (2.1.37) Note that c contains a quadratic power in the time derivative as we expected in the expansion x(2) and the vector is correctly oriented along the massive mode.
However, adding a whatever combination of zero modes to such a solution, the final result
x(2) =cx(0)+diΦ0i (2.1.38) still represents a solution of the equation, due to the propriety of the matrix H. To avoid the presence of these redundant degrees of freedom, we must project the solu-tion along the massive mode and discard all the other components.
At the end of this discussion, let us verify how the same result can be obtained imposing an appropriate constraint to the Lagrangian as
L(R4) = L(4)+λiΦ0i ·x(2). (2.1.39)
The system of equations for the fluctuation x(2)reads (
m¨x(0) = −Hx(2)+λiΦ0i
x(2)·Φ0
i =0 ∀i
(2.1.40) Projecting along the zero modes, in the first equation, we resolve the Lagrangian mul-tiplier λi obtaining
Using such a solution for λi, the system reduces to ( −m gkj ˙ϕk ˙ϕjx(0)= −Hx(2) x(2)·Φ0 i =0 ∀i. (2.1.42) Note that this operation is equivalent to discard the equations of motion for the moduli variables from the system (2.1.35). The new system (2.1.42) is uniquely resolved by
(2.1.37).
Substituting this solution inside the fourth order Lagrangian, we finally obtain the higher derivative corrections for the effective theory (2.1.7)
L(e f f0)+(2)+(4)= −V(x(0)) + 1
2gij ˙ϕ
i ˙ϕj+ (order of(˙ϕi)4) (2.1.43)
2.2
higher-derivative corrections in field theory
Euler-Heisenberg effective theoryA first example of an effective model in QFT was given in QED, studying the scatter-ing light-by-light at very low energies. The Lagrangian of Quantum Electrodynamics (QED) is written as
L = L0+ Lint (2.2.1)
where the free part reads
L0=ψ(i∂µγµ−m)ψ−
1 4FµνF
µν (2.2.2)
and the interaction
Lint= −eψγµ
ψAµ. (2.2.3)
Here, ψ represents the fermionic field and Aµthe photon field with
Fµν = ∂µAν−∂νAµ. (2.2.4)
In case of light-by-light scattering, at very low photon energy ω m with m mass of electron, the electrons (and positrons) cannot be produced in the final state, due to the energy conservation, but their contribution emerges in the amplitute calculation via virtual processes. At the lowest order, the relevant Feynaman-diagram consists in a single fermionic-loop with four external photon “legs”, describing the initial and final photon-states. Since in the diagram every fermionic propagator contributes with a term of the type
s(x−y) ∼ 1
