Universality in few-body physics around the unitary limit

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Corso di Laurea Magistrale in Fisica Curriculum Fisica Teorica

Universality in few-body physics around

the unitary limit

Candidate: Supervisor:

Simone Salvatore Li Muli Prof. Alejandro Kievsky

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Introduction

Universality in physics generally refers to situations in which systems that are very different at short-distances show identical behavior at long-distances. These kind of situations occur whenever the wavefunction of the system is so much delocalized to not being able to capture the short-distance structure of the interaction. When a two-body system is loosely bound, namely its interaction is strong enough to just support a low-energy state very close to the zero-low-energy threshold, the wavefunction of the system shows a long tail that makes that almost completely localized out of the interacting region. When this condition occurs the scattering length of the two-body system be-comes large compared to the natural length-scale of the interaction.

The s-wave scattering length a is the most important parameter governing the in-teraction of low-energy particles. Its absolute value can be determined by measuring the low-energy limit of the elastic cross-section σ(E), if the particles are indistinguish-able (distinguishindistinguish-able) the asymptotic low-energy expression of the scattering length, as E → 0, is

σ(E) → 8π(4π)a2. (1.1)

The scattering behavior of the system can be analyzed by using a partial-wave expansion of the scattering amplitudes. The most important quantities of this expansion are the phase-shifts δL(k), where k is the wave number of the system and L the angular

momenta. If particles interact with short-range interactions, then the phase-shift δL(k)

approaches zero like k2L+1. Thus in the low-energy domain the scattering behavior is dominated by the s-wave phase-shift δ0(k). At sufficiently low energies, δ0(k) can be

expanded in powers of k2, this expansions is known as the effective-range expansion (ERE): k cot δ0(k) = − 1 a+ 1 2rsk 2_{+ ....} _(1.2)

In this expression the scattering length is defined positive if the system supports a bound state while being negative otherwise, whereas rs is well-known as the effective

range of the interaction.

In most cases, the scattering length a is comparable in magnitude to the range of the interaction, but in exceptional cases it can be much larger. This condition requires necessarily a fine-tuning of the parameters that structure the interaction. Under these

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circumstances low-energy observables show universal behavior, independently of the short-distance structure of the two-body interaction. For example, the large value of the scattering length governs the energy of the shallow dimer, Ed≈ ~2/(ma2).

The first evidence of universality in three-body systems arose in 1971 when Vitaly Efimov found a remarkable effect in the quantum spectrum of three-particles. He con-sidered particles interacting through short-range, almost resonant, interactions. Under these conditions he found that an effective long-range three-body attraction arises, this attraction is able to support infinite many three-body bound states in which the three particles are bound at larger and larger distances. This effect has some striking features: • Although the two-body interactions are short-range, the particles feel a long-range three-body attraction. This situation can be explained by the fact that an effective interaction is mediated between two particles by the third particle moving back and forth. The three particles thus bind at distances far greater than the range of the two-body interaction, typically at distances of the order of the scattering length.

• At the unitary limit, namely when the scattering length becomes infinite, the ef-fective interaction extends to infinite distances. The system becomes scale invari-ant and the quinvari-antization of this three-body effective interaction gives an infinite series of bound states. This situation is known as the Efimov effect and the states show a discrete scale invariance.

• Even though the two-body interaction is not able to support bound states, the effective long-range three-body interaction can still bind particles. This possibility of binding N particles while the N − 1 subsystems are unbound is known as ”Borromean binding”.

In recent years, it has been realized that the Efimov effect gives rise to a broad class of phenomena that have been referred as Efimov physics. Its consequences are not con-fined to three-body systems and a huge amount of theoretical, as well as experimental, effort has been deployed in studying this effect in general few-body and many-body systems.

Due to the universal nature of the Efimov effect, even though its origin comes from nuclear physics, it can be applied to virtually any field of quantum physics, be it atomic and molecular physics, nuclear physics, condensed matter or high-energy physics. How-ever the universality in the Efimov effect does not mean that it occurs in any system. It means that any system meeting the conditions of its appearance can shows Efimov physics. These conditions turn out to be quite restrictive, which is why it has been so difficult to find a clear evidence of Efimov physics in nature.

The most strikingly results of Efimov physics are shown in (Fig. 1.1), starting from the first prediction of the Efimov effect in 1971 to the final observation in 2016 of the excited state of the 4He trimer which, up to date, is known as the clearest evidence of a Efimov state in nature.

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Figure 1.1: History of Efimov physics from the original theoretical prediction by Vi-taly Efimov to the latest experimental observations, along with a number of related publications (source: [4])

In this thesis we study the universality in few-boson systems with emphasis to nuclear and atomic physics. Looking to loosely bound systems as the atomic He dimer or the Deuteron, due to the appearance of universal behavior and disregarding specific forms of the interaction, we propose that the gross features of the universal behavior can be captured by using Gaussian effective interactions. The Gaussian interaction, being a regularized zero-range interaction, is able to reproduce the universal behavior while introducing range corrections to the theory. We argue that these range corrections, introduced by the Gaussian interaction, are good representations of the first order corrections obtained in a perturbative expansion in terms of rs/a.

Chapter 2. We start the study of the universality in two-body systems and we show that for these systems the universal behavior arises in the simplest and clearest way. Accordingly we take the Deuteron and the Helium dimer as our references for nuclear and atomic physics with large scattering length and we show that it is possible to describe with good precision both systems using a Gaussian two-body interaction.

Chapter 3. Here we study the properties of three-body universal systems. We deal with the three-body problem using a zero-range theory and the adiabatic hyperspherical (AH) expansion of the wavefunction, we take the approximation of cutting the expan-sion to the lowest adiabatic channel. The system exhibits a problematic feature well known in the literature as the Thomas Collapse, namely the lowest adiabatic potential diverges at short-distances and the system have not a well defined ground state. In order to overcome this problematic feature, a regularization scheme is needed. I studied different schemes and compared them in order to explain advantages and disadvantages between different methods. Finally the Efimov spectrum is computed with particular emphasis on reproducing the behavior of the4He3 system.

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Chapter 4. We leave out the zero-range theory in favor of two-body Gaussian inter-actions. We employ the hyperspherical harmonic (HH) expansion of the wavefunction, with the approximation on retaining only the lowest hyperspherical channel. We solved again the three-body problem within this formalism in order to compare with the result obtained in the previous chapter, then we generalized the formalism in order to deal with general A-bosons systems. Accordingly the universal behavior has been analyzed in order to study how the universality propagates by increasing the number of particles. Chapter 5. It is devoted to the conclusions and final thoughts.

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Two body problem with large

scattering length

2.1 The two body problem in Q.M.

At non relativistic energies the dynamic of a two body system could be described using the Schrodinger equation. Instead of working with the six degrees of freedom relative to the position vectors of the two particles we could as well work in the center of mass frame. The interesting dynamic of the system arise from the relative motion of the two particles, it is possible to factorize out the center of mass motion from the whole dynamic and deal with the following form of the Schrodinger equation:

−~

2

2µ∇

2_{+ V (r)}_{Ψ(r) = EΨ(r),} _(2.1)

where µ is the reduced mass of the system. For reasons that will be clear later on, we would like to expand this wave function in eigenstates of the angular momentum operator. In order to do so we use the spherical coordinates expression for the Laplace operator ∇2 = 1 r2 ∂ ∂r r2 ∂ ∂r −Lˆ 2_{(θ, ϕ)} r2 , (2.2)

and we express the wave function as a combination of the spherical harmonics functions, which are the eigenfunctions of the angular momentum operator ˆL2_{(θ, φ) that we were}

looking for. It is possible to take into account also for the spin degrees of freedom of the two particles, even though at some point in this thesis work we will start to consider just spinless boson systems, for the sake of completeness in this part of the work we start by describing a system with general total spin quantum number S and we add to the wave function a pure spin part χS,Sz, where the second index represent

the projection of the spin onto the ˆz axis. Thus, the wavefunction of the system can be expanded in the following way:

Ψ(r) = X

L,M,S,Sz

RL(r)YL,M(θ, ϕ)χS,Sz, (2.3)

where L is the total angular momentum quantum number and M is its projection onto the ˆz axis. Both L and S are angular momentum quantum numbers, so they could be

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added up giving a new quantum number J representing the total angular momentum (of both spin and orbital motion) of the system. It is possible, and useful, to change base using the Clebsch-Gordan coefficients, and instead of working in eigenfunctions of ( ˆL, ˆS, ˆM , ˆSz) we could as well work in eigenfunction of ( ˆJ , ˆJz, ˆL, ˆS), for this reason we

define: h YLχS i J,Jz = X M,Sz CL,M,S,Sz,J,JzYL,MχS,Sz, (2.4)

and use this to express the wave function as a combination of eigenfunctions of the total angular momentum operator ˆJ

Ψ(r) =X L,S RL(r) h YLχS i J,Jz . (2.5)

The Schrodinger equation simplifies greatly if we define a reduced radial wave function uL(r) satisfying uL(r) = rRL(r). Making this substitution we arrive at the following

form of the Schrodinger equation X L,S h −~ 2 2µ ∂2 ∂r2 − L(L + 1) r2 + V (r)iuL(r) h YLχS i J,Jz = EuL(r) h YLχS i J,Jz . (2.6)

Let us stop for a moment in order to make some considerations about the structure of the potential V (r) we are considering. Our intention is to apply this formalism for finding a low energy description of the nuclear force. Realistic nuclear potentials are really complicated, they are composed of various parameters in order to take into account for the spin and isospin dependence of the nucleon-nucleon interaction, they have some spin-orbit coupling terms and by no means they are centrals. The reason for not considering a central interaction in nuclear systems comes from the experimental evidence that the Deuteron is a mixtures of a 3S1 and a 3D1 state. In fact if the

mixture with the D-wave were not present the electric quadrupole moment would have been zero which is in contrast with experimental results. The fraction of the 3D1

state is, however, around the 4% which could be considered quite low, so we make the drastic assumption of considering, as a low energy description of nuclear systems, a central interaction. Everywhere in our work we should remember that this is just an approximation of the real structure of the interaction, but it is an approximation that simplifies greatly our job. For central potentials it is possible to project the Schrodinger equation into eigenstates of fixed total angular momentum, obtaining the following differential equation for the reduced radial wave function

h −~ 2 2µ ∂2 ∂r2 − L(L + 1) r2 + V_L,SJ (r)iuL(r) = EuL(r). (2.7)

It is possible to consider different kind of solutions of this Schrodinger equation. We will first deal with scattering processes, where E > 0, we should warn the reader that this would not be a complete description of the scattering theory in quantum mechanics, for example we will not deal at all with anelastic processes, but we will describe at length the general principles of the scattering theory and we will go further in detail in those aspects that could be more useful for the scope of this work. After a lengthy walkthrough into the scattering theory we will introduce bound state solutions

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of the Schrodinger equation, where E < 0 and, in particular, our intention will be to analyze how bound states that are near the threshold effect the behavior of the whole system. We will see that the mere presence of these low energy states is the starting point of our study of universality highlighted by a large and unnatural value of the scattering length.

2.1.1 Scattering theory

The nuclear interaction is short-range with a range (∼ 1.4fm) given essentially by a one pion exchange interaction, we should exploit right away this property by considering the potential V (r) that appears in the Schrodinger equation a short range potential with range r0 (it does not take a precise value for the moment). Considering a short

range interaction will simplify even more our job and, in particular, it will make possible for us to neglect all the phase shifts related to angular momentum quantum number L > 0 when dealing with low energy scattering processes. Under this assumption the asymptotic form of the Schrodinger equation takes the form:

h − ∂ 2 ∂r2 + L(L + 1) r2 i uL(r) = k2uL(r) , r > r0 (2.8)

where we defined the wave number k for the relative motion of the two particles as k2= 2µE

~2 , E > 0. (2.9)

It is possible to arrange this differential equation in a more natural form by defining a new quantity z = kr. Having done this we came up with a well known differential equation h ∂2 ∂z2 − L(L + 1) z2 + 1 i uL= 0, , r > r0 (2.10)

its solutions are the Bessel functions FL = krJL(kr), GL = −krYL(kr), where JL

is regular at the origin while YL is not. The whole radial part of the wave function

RL(r) = r−1uL(r) has to be regular at the origin, but we can not make any gain from

this condition because the origin is explicitly left out from the range of the previous differential equation. For this reason the general solution for the reduced radial part of the wave function is a combination of FLand GL

uL= AFL+ BGL , r > r0. (2.11)

The constants A and B should take appropriate values that satisfy the boundary conditions, namely that makes the solution and its derivative at r > r0 matching

per-fectly with the solution at r < r0. Usually there are three possible type of combinations

of A and B that are used in the literature: • u(K)_L = FL+ kLGL (K-wave solution) • u(T )_L = FL+ TL GL+ iFL (T-wave solution) • u(S)_L = − GL− iFL + SL GL+ iFL (S-wave solution).

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It is useful to consider T-wave and S-wave solutions because the combination r−1(GL± iFL) asymptotically at large distances has the form of an incoming or an

outgoing spherical wave GL± iFL −→ exph±ikr − Lπ 2 i . (2.12)

It is possible to demonstrate that the parameters SL, TL and kLare strictly correlated

with the partial wave phase shift δL.

• k_L= tan(δL),

• T_L= eiδL_sin(δ L),

• S_L= ei2δL_.

If we were considering free states, the boundary conditions for the wave function would have set δL= 0 so, for example, it would have left us with only the FL solution of the

Schrodinger equation in the T-wave and K-wave, or with a combination of an incoming radial wave and an outgoing radial wave if we would have chosen an S-wave solution. The interaction manifests its appearance in the asymptotic part of the wave function by the presence of the phase-shifts. Let us try to elate the phase-shifts of the wavefunction in the asymptotic region with measurable quantities like the scattering amplitude f (θ) and the scattering cross section σ(θ).

The scattering cross section is probably the most important quantity to measure in a scattering experiment, it could be related with the form of the wave function at large distances from the scattering point, where the potential could be considered zero and the wave function could be considered a superposition of an incident plane wave and an outgoing spherical wave

Ψ(r) −→ Aeikz+ f (θ)e

ikr

r

(2.13) where A is an overall normalization constant. The incident wave is considered traveling along the ˆz axis and θ is the angle subtended by the versors ˆz and ˆr. The quantity f (θ) is known as the scattering amplitude and represent the fraction of wave function scattered in the direction identified by the angle θ. For a general problem f (θ) is also a function of the angle ϕ which identify the direction of the scattered wave along the x−y plane, but for our particular problem, where the potential is central and we are not considering polarized initial or final states there is a completely symmetry for rotation of the system around the ˆz axis, so the scattering amplitude can depend only on the angle θ.

In order to accomplish our goal to relate the scattering cross section to the phase-shifts we would need to evaluate the probability that some part of the wave function is scattered on the direction identified by a particular value of θ, this could be done by defining a probability current density jp. Given a generic wave function we define the

probability current density jp as:

jp = ~

2iµ

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We just need to apply the above definition for the incoming and the outgoing wave function separately. For the outgoing wave function a direct calculation shows that

jout = ~k

µA

2|f (θ)|2

r2 , (2.15)

while for the incoming wave function we have jin= ~k

µ A

2_. _(2.16)

If the scattered part of the wave function is observed by a detector placed at distance r from the scattering point in the direction identified by θ we have that the infinitesimal cross section identified by the solid angle dΩ subtended by the area of the detector is

dσ = r2dΩjout jin = |f (θ)|2dΩ, (2.17) as a consequence dσ dΩ = |f (θ)| 2_. _(2.18)

Our next task is to express the quantity f (θ) as a function of the phase-shifts carried by the asymptotic part of the wave function. We consider a T-wave solution of the Schrodinger equation outside the range of interaction and then take the limit for r r0: uT_L(r) = FL(r) + TL GL+ iFL −→ sinkr − Lπ 2 + TLexp h ikr −Lπ 2 i . (2.19) We also make an expansion of the incident wave function in terms of eigenfunction of the orbital angular momentum operator ˆL2, namely in terms of spherical harmonics functions

Ψinc= Aeikr = 4πA

X

L,M

iLY_L,M∗ (ˆr)YL,M(ˆr)

FL(kr)

kr . (2.20)

This one represent the free wave function, the one we would have if it were not for the potential. The presence of the potential change this form of the wave function into:

Ψ = AX L,M 4πiLY_L,M∗ (ˆr)YL,M(ˆr) FL(kr) kr + TL (GL+ iFL) kr = = Aeikr+ 4πAX L,M iLY_L,M∗ YL,M exp h i(kr − Lπ₂ ) i kr TL= = Aeikr+ 4πAX L,M Y_L,M∗ YL,M eikr kr TL (2.21)

where we used the fact that exp(−Lπ₂) = i−L. We have just found the value of the relative coefficient between the incoming wave function and the outgoing one, but this value is no other than the scattering amplitude f (θ) defined above. For this reason we have that f (θ) =X L,M 4πY_L,M∗ (ˆr)YL,M(ˆr) TL k . (2.22)

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The spherical harmonic functions are strictly related to the Legende polynomials, so we can use

X

M

4πY_L,M∗ (ˆr)YL,M(ˆr) = (2L + 1)PL(cos θ) (2.23)

and express the scattering amplitude in terms of the Legendre polynomials, which will come at hand later on

f (θ) =X

L

(2L + 1)PL(cos θ)

TL

k . (2.24)

So we have been successful in finding a relation between the scattering amplitude and the phase shifts of the asymptotic wave function (TLis defined as a function of the

phase shift δL). Equivalently we could have defined the scattering amplitude in terms

of the SL matrix instead of the TL one, and it would have been in the following form:

f (θ) =X L (2L + 1)PL(cos θ) SL− 1 2ik . (2.25)

An usefull quantity to define is the partial scattering amplitude fL, namely the

scattering amplitude relative only to the L orbital angular momentum quantum number fL= SL− 1 2ik = 1 k cot δL− ik , (2.26)

using this definition the total scattering amplitude becomes f (θ) =X

L

(2L + 1)PL(cos θ)fL. (2.27)

Let us now try to obtain, from the scattering amplitude f (θ), the total scattering cross section. The Legendre polynomials satisfy the following integral identity

Z 1

−1

P_L∗(µ)PL0(µ)dµ = 2 δL,L 0

2L + 1, (2.28)

so we just need to use the identity above to integrate the modulus square of the scat-tering amplitude over the whole solid angle, the calculation is straightforward and gives the following value for the total scattering cross section

σ = 4πX L (2L + 1)sin 2_(δ L) k2 . (2.29)

2.1.2 Effective range analysis

Let us focus more on our main objective, namely trying to find a valuable low energy description of the nuclear interaction. It could be shown that, if the potential decreases fast enough at large distances, when we deal with low-energies scattering processes, the phase-shifts scale as δL → k2L+1 and the partial scattering amplitudes scale as

fL→ k2L[1]. Having considered a short range potential we are sure that the interaction

goes fast enough to zero at long distances and the low energy behavior of the phase shifts will be the one we stated above. For this reason when we deal with slow scattering

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systems only the S-wave phase shift makes considerable contributions to the scattering cross section. Furthermore the scattering cross-section should remain finite in the limit where the wave number k goes to zero, so we define the scattering length a as:

lim

k→0

dσ dΩ = a

2_. _(2.30)

It will be shown that the scattering length is the most evident indicator of univer-sality showed by some particular systems, interesting phenomena happen when it is unnaturally large. The scattering length and the s-wave phase shift are strictly related, for an overall sign factor their relation could be expressed as

a = lim k→0R h −1 ke iδ0_sin(δ 0) i , (2.31)

where the above definition takes into account that for a positive there is a bound state while for a negative the bound state is not present. On the other hand the above equation implies that for low energies the following relation is valid at first order in k 1.

k cot(δ0) = −

1

a. (2.32)

The first correction to the above formula, comes from a parameter known as the effective range rs of the interaction. The expression for this parameter and for the

second order terms in the above expansion comes from the following reasoning. Let us consider the Schrodinger equation for a two-body system in the L = 0 channel, for two different energies we have that the solution of the reduced radial wave function must satisfy. ∂2 ∂r2 − V (r) + k 2 1 u0(k1, r) = 0, (2.33) ∂2 ∂r2 − V (r) + k 2 2 u0(k2, r) = 0. (2.34)

Multiplying the first equation with u0(k2, r) and the second one with u0(k1, r) and

then integrating the difference in dr we obtain Z ∞ 0 h u0(k2, r) ∂2 ∂r2u0(k1, r)−u0(k1, r) ∂2 ∂r2u0(k2, r) i dr = (k2₂−k2 1) Z ∞ 0 u0(k1, r)u0(k2, r)dr, (2.35) we could easily evaluate the first integral by part, doing so we come up with:

h u0(k2, r) ∂u0(k1, r) ∂r − u0(k1, r) ∂u0(k2, r) ∂r i∞ 0 = (k 2 2 − k12) Z ∞ 0 u0(k1, r)u0(k2, r)dr. (2.36) Everything so far is valid for every kind of potential that appears in the Schrodinger equation. We will consider now solutions of another type of Schrodinger equation, this time it will be a free system with V (r) = 0. With an equivalent reasoning we obtain the following equation.

h v0(k2, r) ∂v0(k1, r) ∂r −v0(k1, r) ∂v0(k2, r) ∂r i∞ 0 = (k 2 2−k12) Z ∞ 0 v0(k1, r)v0(k2, r)dr. (2.37)

The potential is short-range and for this reason the Schrodinger equation for u0(k, r)

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the two solution are the same for r that goes to infinity. Namely v0(k, r) = u0(k, r)

as r → ∞. Taking also into account that u0(k, 0) = 0, because the whole radial wave

function for the interacting system should be regular in zero, we have: h u0(k2, r) ∂u0(k1, r) ∂r −u0(k1, r) ∂u0(k2, r) ∂r i∞ 0 = limr→∞ h v0(k2, r) ∂v0(k1, r) ∂r −v0(k1, r) ∂v0(k2, r) ∂r i , (2.38) and everything so fare makes the following equation satisfied

v0(k1, 0) ∂v0(k2, 0) ∂r −v0(k2, 0) ∂v0(k1, 0) ∂r = (k 2 2−k21) Z ∞ 0 h v0(k1, r)v0(k2, r)−u0(k1, r)u0(k2, r) i dr. (2.39)

Having previously imposed that v0(k, r) should be equal to u0(k, r) in the

asymp-totic region we cannot have that v0(k, r) is zero at the origin, but we can normalize this

wave function in order to impose v0(k, 0) = 1, doing so the previous equation simplify

greatly ∂v0(k2, 0) ∂r − ∂v0(k1, 0) ∂r = (k 2 2−k12) Z ∞ 0 h v0(k1, r)v0(k2, r)−u0(k1, r)u0(k2, r) i dr. (2.40) We need now to specify the form of v0(k, r), it should be equal to u0(k, r) in the

asymptotic region and also it must be normalized to one at the origin, the only possible form it could have is v0(k, r) = sin−1(δ0) sin(kr + δ0). We should use this definition of

v0(k, r) and considering two energies with wave number k1 and k2 very close to each

other we arrive at the following differential equation ∂ ∂k2 k cot(δ0) = Z ∞ 0 h v₀2(k, r) − u2₀(k, r)idr. (2.41) We can now define the effective range rsas two times the integral on the right part

calculated in k = 0 rs = 2 Z ∞ 0 h v₀2(k, r) − u2₀(k, r)i k=0dr. (2.42)

Making a Taylor expansion of k cot(δ0) around k = 0 and neglecting contributions

of order k4 or higher, we have: k cot(δ0) ' k cot(δ0) k=0+ h ∂ ∂k2 k cot(δ0) i k=0k 2 ₌ = −1 a+ 1 2rsk 2_. (2.43)

So we have defined the scattering length and the effective range as the first two parameters in the expansion around the zero energy that define the S-wave phase shift and, consequently the whole scattering process at low energy.

2.1.3 Bound states

We found previously that the asymptotic form of the scattered wave function could be expressed as a superposition of an incoming and an outgoing radial wave, their relative amplitude is given by the S-matrix which for central potentials and elastic scattering takes the simple form SL= e2iδLnamely it is a diagonal matrix with phase factor on all

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its terms. We want to consider only low-energy scattering processes where it is possible to take into account only for the L = 0 contribution to the scattering amplitude. Furthermore we would like to compare the asymptotic form of the scattered wave far away from the scattered point:

RL=0= u(S)_L=0 r = − GL=0− iFL=0 r +SL=0(k) GL=0+ iFL=0 r −→ SL=0(k) eikr r − e−ikr r (2.44) to the bound state wave function outside the range of interaction which, leaving out an overall normalization constant, has the following form

Rbound,L=0 −→

e−kdr

r = eikr

r , (2.45)

with k = ikd. There are two difference between the previous two equations, the first one

is that for the bound state k has a purely imaginary value k = ikd while for scattering

states it is real value, the second one is that for the bound state there is no incoming radial wave, so for bound states it is possible the existence of an outgoing radial wave without the existence of the incoming one. The relative amplitude of the outgoing and the incoming radial wave in the wave function is governed by the S-matrix, we conclude that, by extending the range of validity of the wave number k to the whole complex plane, we could describe bound states as poles of the S-matrix along the imaginary axis. The S-matrix could be expressed as:

SL=0=

k cot(δ0) + ik

k cot(δ0) − ik

, (2.46)

the poles of the S-matrix comes for k satisfying: k cot(δ0) − ik = − 1 a+ 1 2rsk 2_{− ik = 0,} _(2.47)

where we used the effective range expantion for k cot(δ0) valid only for low value of

|k|. By using the expression k = ik_d where kd is real and represent the point on the

imaginary axis where the S-matrix has a pole, we see that kd must satisfy:

kd= 1 a+ 1 2rsk 2 d. (2.48)

This is a remarkable result, it says that if the potential is strong just enough to have a shallow bound state positioned nearly at the threshold then we could obtain its energy at first approximation by just experimentally measuring the scattering amplitude for a low energy scattering. This statement could be improved with the introduction of more information about the potential structure, the first correction comes from the introduction of the effective range of the interaction.

As an example we could take the deuteron, the3S1 state of the np nuclear system.

It is a weakly bound state with binding energy ED = 2.22MeV. The scattering length of

the system is aD = 5.42fm which is almost four time larger than the natural low-energy

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`Youkawa ∼ 1.4fm. The big value of the scattering length and the small value of the

binding energy seem to suggest that the deuteron could be described with the tools developed in the effective range analysis. At first approximation we could just say that kD = a−1 → ED = ~2(ma2)−1 where m is the average mass of proton and neutron,

this approximation gives a binding energy of ED ∼ 1.4MeV which is reasonably close

to the real value. Precision could be improved including range corrections, the main discordance between our approximation and experimental results comes from the fact that a is just four times larger than the effective range of the interaction, in the limit case where rsa−1 → 0 we should have that kD would be exactly equals to a−1.

2.1.4 Resonant scattering

It is interesting to study a scattering process where the interaction is strong enough to have a shallow bound state with energy ( < 0) near the threshold ( ∼ 0). In this case if the scattering is performed at low energies, the energy of the scattered particle is almost in resonance with the energy of the bound state, as a consequence the scattering amplitude is considerably greater than it could have been if the potential would have not support a shallow state. Let us consider a low-energy scattering process where the interaction is short-range and where the relative wave number of the incident particles k satisfy kr0 1, here r0 stands for the range of the interaction. For natural

scattering processes, namely processes where the potential does not have a two-body bound state near the threshold, we should expect that the scattering amplitude in the S-wave channel (the only one considerably different from zero at low-energy) should be of the order of the range of interaction fL=0∼ r0. We could deal with the resonance of

a shallow bound state with the following reasoning [1]. Let us consider the Schrodinger equation for a two body system with a short-range interaction that is strong enough to allow for a low-energy bound state (with low energy we mean energy much lower than the typical strength of the interaction). We do not want to complicate the system by considering the spin degrees of freedom, as well as we do not want to consider interaction that are not central. With this assumption the Schrodinger equation for the reduced S-wave radial part of the wave function is:

∂2 ∂r2u(r) + 2µ ~2 h E − V (r) i u(r) = 0. (2.49)

We do not know the exact structure of the potential V (r) but the interesting thing is that, for our purpose, we do not need to, it is necessary to know only some of its general properties, in particular that it is short-range, central and strong enough to support a shallow bound state. Inside the range of interaction (r < r0) we can neglet

the energy of the scattered state compared with the potential: ∂2

∂r2u(r) −

2µ

~2V (r)u(r) = 0, (2.50) while outside the range of interaction r > r0the potential goes to zero so the Schrodinger

equation becomes:

∂2

∂r2u(r) +

2µE

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The solution of the previous differential equation is straightforward, it is just a sinusoidal of the form u(r) = B sin(kr + δ0), where the parameter B and δ0 should be

chosen in order to match the solution inside the potential and make the wave function continuous and differentiable everywhere. Instead off apply this kind of boundary conditions we could equally well specify a value of the logarithmic derivative of the asymptotic solution at r = r0. We see that just because the wave number is low

compared to the range of the interaction (kr0 1) the argument of the sine function

varies very slowly inside the interacting region, instead of evaluating the boundary condition at r = r0 we could equally well evaluate that at r = 0

u0(r)

u(r) = −c, (2.52)

where choosing the quantity c appropriately is equivalent to mach the outside and the inside solution and make the wave function differentiable everywhere. We highlight that, just because we have considered low-energy scattering processes, the Schrodinger equation inside the potential gap does not depend on the energy of the scattered wave, this equation gives the boundary condition that we should use to match with the solution outside the range of interaction, so the conditions do not depend on the energy of the system. In other words c, that takes the place of the boundary conditions, is a constant with respect to the energy and for this reason its value should also be the same for both a scattered and a bound state solution of the Schrodinger equation. We previously required that the potential is able to hold a shallow bound state with wave number kd 1, the corresponding wave function outside the range of the interaction

should be

uB(r) = Ae−kdr, (2.53)

and, just because the constant c which carries the boundary conditions has the same value for scattering and bound states, we can calculate the logarithmic derivative in the case of a bound state, evaluate the value of the constant c and then use this value in the scattered state, we obtain

hu0

B(r)

uB(r)

i

r=0 = −kd= −c. (2.54)

We need just to apply the boundary condition to the scattered state, that again in the asymptotic region outside the range of the interaction should have the form of a sinusoidal, the calculation is quite straightforward and leaves us with a constraint condition in the values of the S-wave phase-shift δ0, scattered wave number k and wave

number of the bound state kd

k cot(δ0) = −kd. (2.55)

Let us now apply the results obtained so far in the calculation of the scattering am-plitude. The phase-shifts for the wave functions with angular momentum wave number L > 0 remain small compared with the S-wave solution because we are considering low-energy scattering and a short-range potential. So the scattering amplitude has the following structure: f (θ) ∼ f0 = 1 2ik e2iδ0 _{− 1}₌ 1 k cot δ0− ik = − 1 kd+ ik , (2.56)

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and the total scattering cross-section is σ = 4π

k2 d+ k2

. (2.57)

We see that the scattering cross-section is isotropic as we should have expected for a low-energy scattering process, but it also depends on the energy of the bound state. In the resonant region where k ∼ kd, it has a much greater value that its natural

counterpart which is of the order σnatural ∼ r20, because by definition k −1

d r0. The

remarkable property we want to stress resides in the fact that the results obtained so far for the scattering cross section and the scattering amplitude do not depend on the interaction mechanisms of the particles at short-distances, the only information we needed to known about the interaction are: it is short-range (in order to being sure that we could neglect all the partial scattering amplitudes except for the one with L = 0), it is central and it is able to support a loosely bound state just below the threshold. We saw that, as a consequence for the presence of this loosely bound state, the scattering amplitude do not depend on the particular form of the interaction, but it does depend on the energy of the shallow state. We introduce that when the system allows for the existence of this kind of shallow states (it seems to not being relevant if they are real bound states, or virtual ones), the scattering cross section is not the only observable that shows this strange behavior. Also observables like the binding energy, the mean squared radius and the asymptotic normalization constant of the system show independence of the particular form of the interaction, in fact we will show that at first approximation their behavior is governed only by the scattering length and every kind of potential that gives the same scattering length is able to describe the low-energy behavior of the system.

2.1.5 Identical particles

In quantum mechanics the identity of particles comes with a permutation interaction that has no analogous in classical mechanics, this interaction is a main character in scattering processes of identical particles and can not be neglected. For the spin and statistic theorem we have that the total wave function of identical bosons must be symmetric under the exchange of the two particles, while for identical fermions it must be anti-symmetric. This has great consequences for the scattering process but before we deal with the general result let us consider two easy examples.

Let us suppose that we have two identical bosons with spin zero. Their wave function is composed only of the radial part which is symmetric or anti-symmetric corresponding respectively of even and odd values of the relative angular momentum quantum number L. For the spin and statistic theorem we must have that not every solution of the Schrodinger equation is realized on the real world, but we need to select only those solution with an even angular momentum quantum number.

We now consider a system of two identical fermions with spin 1/2. Again for the spin and statistic theorem the total wave function must be anti-symmetric. This time

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the wave function is composed of a radial part, which as before is symmetric or anti-symmetric corresponding to even and odd values of L, and a purely spin part, which is symmetric if the total spin of the two particles is 1(odd) and it is anti-symmetric if the total spin of the system is 0(even). We arrive at the result that also this time not all the solution of the Schrodinger equation are realized in the real world, but this time we need to select wave function with L = even values if the total spin of the system is 0 and L = odd if the total spin of the system is 1.

The results so far could be generalized [1] [2] by stating that if the total spin of the system is even, then by the spin and statistic theorem we should have that the angular momentum quantum number of the wave function is even; if the total spin of the system is odd, then the angular momentum quantum number of the wave function must be odd. For this reason the asymptotic form of the radial wave function, which is a solution of the Schrodinger equation, must be symmetrized or antisymmetrized corresponding to even and odd values of the total spin of the system.

In the center of mass frame the exchange of the positions of the two particles correspond to te following transformations:

• r −→ r, • θ −→ π − θ, • z −→ −z.

For this reason, leaving out an overall normalization constant which is not relevant for our purpose, we could construct a symmetrized or antisymmetrized radial wave function by making the following combination:

Ψ(r, θ) = eikz± e−ikz+hf (θ) ± f (π − θ)i1 re

ikr_. _(2.58)

The procedure for determining the value of the scattering cross section is completely the same as before, it depends only on the relative constant between the incident wave and the outgoing wave, so it is determined by the coefficient in front of the outgoing radial wave

dσ = |f (θ) ± f (π − θ)|2dΩ0, (2.59) where the factor dΩ0 comes instead of dΩ as indicating that, right now we are not integrating over the overall solid angle but the integration on θ is performed only from θ = 0 to θ = π/2.

Indicating with S the total spin of the two-body system, and considering the relation we have found between S and the parity of Ψ(r, θ), we should have:

S = Even dσ = |f (θ) + f (π − θ)|2dΩ0 S = Odd dσ = |f (θ) − f (π − θ)|2dΩ0

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We should remember the form of the scattering amplitude f (θ) =P

L(2L+1)fLPL(cos θ),

and considering that for L = Even the Legendre polynomial satisfy PL(cos θ) = PL(− cos θ)

while for L = Odd they satisfy PL(cos θ) = −PL(− cos θ) we must have:

σ = 8πP L(2L + 1) sin2_(δ L) k2 S = Even L = Even σ = 0 S = Even L = Odd σ = 0 S = Odd L = Even σ = 8πP L(2L + 1) sin2_(δ L) k2 S = Odd L = Odd

The solution of the Schrodinger equation should be symmetrized or antisymmetrized accordingly to the statistic of the system, as a consequence not all the values of the angular quantum number are possible for the system, but when there is a reliable combination of S and L the scattering cross section is twice with respect to the case of distinguishable particles.

2.2 Universality for large scattering length in two body

systems

We saw in the previous section that whenever the two-body potential is able to support a loosely bound state just below the threshold, the two-body system exhibits unnatural properties that do not depend on the shape of the interaction. This was the case for the total cross section that we calculated explicitly and showed that, indeed, the mere presence of the shallow state (we will refer to shallow states as both states that are just loosely bound and virtual states that would be bound if the interaction strength would be increased by a tiny amount) makes the total scattering cross section large compared to its natural values. The particular value of the scattering cross section was completely independent on the shape of the interaction at short distances and it was determined only by the binding energy of the shallow state. This is just a particular case of a more general situation, in fact every time the scattering length of the system is large compared to the natural low-energy length scale, which for a short range potential is just the range r0 of the interaction, we see that systems behave unnaturally and their

low energy observables become independent on the structure of the interaction at short distances. In this work we study the behavior of three low energy two-body observables, namely the binding energy of the shallow dimer ED its mean quadratic radius r2 and

its asymptotic normalization constant CA. We saw already how the binding energy

behave when the scattering length is large compared to the range of the interaction, at first approximation we can neglect all the information we have about the potential, like the effective range, and we see that the binding energy scales as

k2_d= 1 a2 =

2µED

~2 . (2.60)

Another example comes from the mean quadratic radius of the shallow dimer that scales as [3]:

r2 = a

2

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or from the normalization coupling constant CAthat scales as:

CA=

r 2

a. (2.62)

By analyzing the dependence of these observables to the scattering length we can give a more general result. Whenever the scattering length is so large that we can neglect all the information we have on the interaction at short distances, for example we neglect the effective range and stop on the first term in the effective range expansion, the two-body sector shows a continuous scaling symmetry that consists of rescaling the scattering length a, the coordinate r, and the time t by appropriate powers of a positive number λ:

a → λa , r → λr , t → λ2t. (2.63) Under the continuous scaling symmetry, two body observables, such as binding ener-gies and cross sections, scale with the powers of λ implied by dimensional analysis. We have for example that the dimer binding energy scale as ED(λa) = λ−2ED(a), or the

scattering cross section scale as σ(λ−2E; λa) = λ2σ(E; a). This situation is known in the literature as the scaling limit, namely the limit where we take the range of interac-tion to be zero r0 → 0 holding the scattering length a fixed. Trivially in this situation

the scattering length is large compared to the natural low energy length scale and we recover an exact universality in physical systems. Within the scaling limit different systems, such as atomic and nuclear systems, behave similarly.

However, although the behaviors of all the different physical systems with large scattering length are similar, it is still possible to find differences. Furthermore we saw previously that even for the Deuteron, which holds a scattering length four time larger than its natural effective range, the scaling limit was not enough to describe its binding energy accurately, it was close but a better result could have been obtained using more information about the structure of the potential, namely its effective range rs. Let us

analyze the dependence of the binding energy of the shallow dimer on the scattering length and the effective range. We saw previously that for the dimer binding energy the following relation holds

kd= 1 a+ 1 2rsk 2 d, (2.64)

by substituting the value of kd in terms of the binding energy we obtain the following

equation _√ 2µED ~ = 1 a + 2µ ~2 rs 2ED, (2.65)

we can take the square on both sides and after rearranging the terms appropriately and neglecting the small term proportional to E2

D we obtain the following relation

2µ ~2ED 1 −rs a = 1 a. (2.66)

We can exploit now the fact that the scattering length is large compared to the range of the interaction |a| r0 ∼ rs, obtaining the following identity

a2ED

2µ ~2 = 1 +

rs

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where the other terms stand for higher therms in power of rs/a. So we see that, at least

for the case of the binding energy, at first order we recover the scaling limit, where the range of the interaction is set to be zero. But by leaving the range of interaction to not being zero, and considering the scattering length |a| rs we are able to calculate

scaling violations to the universal law. This kind of limit, namely where a goes to infinity |a| → ∞ with r0 held fixed is known as the resonant, or unitary, limit. In this

chapter we try to move numerically the system around the resonant limit, in particular we did so by changing a parameter on the potential used to implement the numerical analysis, in order to calculate correction to the universal law for the binding energy, mean squared radius and asymptotic normalization constant.

2.2.1 Effective theories in Q.M.

Our objective is to demonstrate that a good description of nuclear and atomic physics with large scattering length could be obtained using Gaussian potentials. The general idea comes from effective theories, although this is a subject that has been carried out almost only within the context of quantum field theory, Lepage [5] showed that their principles could be applied equally well to problems in quantum mechanics. Modern renormalization theory tells us that the low-energy behavior of a theory is independent on the details of the short distance dynamic. The fact that long and short distance behaviors are unrelated means that there are infinitely many theories that have the same low-energy behavior while being completely different at high energies. For this reason it is possible to replace the short-distance dynamic with something different, and perhaps simpler, without changing the low-energy behavior. On the same article mentioned before, Lepage showed that, by following three simple steps, one has the possibility to redesign the short distance dynamic and create effective theories that model arbitrary low energy data sets with arbitrary precision.

1. Incorporate in the effective theory the correct long-range behavior, in particular the dynamic of the interaction must be known at long distances.

2. Introduce an ultraviolet cutoff to exclude high momentum states. This cutoff has also the effect of making the interaction regular at the origin r = 0.

3. Add local correction terms to the effective Hamiltonian: these take the place of the high momentum state excluded by step 2. Each correction term consist of a coupling constant multiplied by a theory-independent local operator.

The coupling constants introduced in step 3 should be functions of the cutoff since they must account for quantum fluctuations excluded by the cutoff. More or less is excluded as the cutoff increases or decreases and therefore the coupling constants must be adjusted to compensate. For this reason these couplings are known as ”running coupling constants”.

Let us suppose that we want to describe the low energy behavior of a particular system, we should know the long range dynamic of the interaction but let us imagine that we do not know the short range structure of the potential. As an example, we

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could consider a nuclear force. It is short range so the long range behavior is simply V (r) = 0 for r > r0 where r0 is the range of the nuclear force. Another example comes

from atomic physics, the interaction between atoms could be divided in a short-range interaction and a long range one; the short range dynamics is quite complicated but the long range one is very simple, it is just a Van der Walls interaction V (r) ∼ −C6/r6 for

r > r0 where r0is the range of the short-range part of the atomic potential. In order to

describe these kind of systems we just replace the real potential with an effective one Vreal(r) → Veff(r; a1, a2, ..., an) with n adjustable parameters. The effective potential

should be the same as the real one in the asymptotic region, but it need not to bear any similarity with the real one at short-distances as long as its structure is obtained by some adjustable parameters, adequately tuned in order to describe the low-energy observables of the system. We can use this effective potential to make calculations, but first we need to fine-tune the n adjustable coupling constants in order to fit n low-energy observables, such as the scattering length, the effective range or the bind-ing energy. To achieve higher accuracy it is always possible to add others adjustable parameters. As an example if one tunes n parameters and uses these parameters to describe with the best precision possible the scattering amplitude of the system, the errors in the S-wave scattering amplitude can be reduced to order (E/E0)n, and the

errors in other low-energy S-wave observables will scale like (E/E0)n for |E| |E0|,

where E0 is the typical energy scale at which the errors become roughly 100%[3].

Our aim is to use this effective formalism for finding a low energy description of nuclear systems by using only short-range Gaussian potentials, these potentials have in particular two adjustable parameter that could be use to describe accurately two energy observables (usually the dimer energy and the scattering length). At low-energies the dimer binding energy and the scattering length are related by kd = 1_a +

1

2rskd2 so by tuning these two parameters it is possible to obtain also the effective range

of the interaction for free.

2.2.2 Systems with large scattering length

We end this section by giving an overview of systems known in nature to have unnat-urally large scattering length compared to the natural low-energy length scale of their interaction. It can be shown [3] that the probability of systems to exhibit an unnatu-rally large value of the scattering length is usually quite low and requires an accidental fine tuning of some interaction parameters. This fine-tuning can be due to fortuitous value of the fundamental constants in nature.

The simplest example in atomic physics of an atom with a large positive scattering length is the helium dimer. The low energy length scale for this system comes from the Van Der Waals interaction and its value is `VdW∼ 10.2a0 where a0 is the Bohr radius

a0 = 5.29177 · 10−11m. A pair of 4He atoms has a single two body bound state that

is very weakly bound. From measurement of the size of the 4He dimer the scattering length has been determined to be a = 197+15₋₃₄a0 [6] that is much larger than the low

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Figure 2.1: Scattering length and length scales for alkali atoms in units of a0: the

equilibrium radius req, the van der Waals length `vdW, the spin-singlet scattering length

as, and the spin-triplet scattering length at. (Taken from: [3] pag.272)

The binding energy of the dimer, calculated with the realistic LM2M2 potential, is E2= 1.303mK which is much smaller than the natural low energy scale ~2(me`2VdW)−1∼

400mK. We have expressed these energies in terms of the temperature unit mK. The conversion factors to eV and to the natural atomic energy unit ~2(mea0)−1 are:

1mK = 8.61734 · 10−8eV = 3.16682 · 10−9 ~

2

mea20

. (2.68)

In (Fig. 2.1) we show some values of scattering lengths for various alkali atoms. The fortuitous fine-tuning is illustrated in the fact that7Li, whose mass is 17% larger than

6_{Li has a natural value for the scattering length in the triplet channel of the spin a} t.

Another example of the same type of fine tuning comes from the87Rb whose mass is 2.3% larger than that of85Rb and instead of its counterpart it has a natural value for as, namely the scattering length in the singlet state of the spin.

Systems with large scattering length arise also in nuclear and particle physics. As an example we could consider the system composed of two neutrons. Neutrons are spin 1/2 particles and their interaction is highly governed by the spin and statistic theorem, when we scatter two neutron one another and we polarize them in order to be prepared with opposite spin then an S-wave scattering channel is available. The scattering length and the effective range measured for this system are respectively a = −18.5fm and rs = 2.8fm, the natural low energy length scale is comparable with

the effective range measured (the interaction is in fact governed by a one pion exchange with range `π ∼ 1.4fm) and the absolute value of the scattering length is almost one

order of magnitude larger than the effective range. We note that the scattering length, although is large compared to the natural low-energy length scale, has a negative value, in fact the system does not support a bound state, we say then that it has a shallow virtual state. Physic does not care if there actually is a real bound state near the threshold or if the state is only a virtual one, it will show universal behavior anyway. Probably the best known example in nuclear physics of a system with large and positive

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scattering length is the Deuteron. Nuclear physics respects an approximate SU(2) isospin symmetry, in order to implement that in the quantum description of the system we will deal with the proton and the neutron as different state of the same particle known as the nucleon N. The Deuteron is the state of the NN system with total isospin quantum number T = 0 and total spin quantum number S = 1. It holds a L = 0 as well as a L = 2 wave function and its values of the s-wave scattering length and effective range are respectively a = 5.42fm and rs = 1.76fm. The scattering length is

more than three times larger than the effective range, so we should expect that this system should hold a state near the threshold. In fact the ground state of the Deuteron is shallow with a binding energy of Ed= 2.225MeV, that is far smaller than the natural

binding energy of nuclear systems of ∼ 8MeV per nucleon, furthermore the Deuteron has not any excited state. We just mention also the αα state, where α stands for the

4_{He nucleus, this state has a scattering length estimated to be a ∼ 5fm and an effective}

range approximately of rs ∼ 2.5fm, again the scattering length is almost four time

larger than the natural low energy length scale `π ∼ 1.4fm.

2.3 Theory and numerical calculations

Our aim is to find a valuable model for the description of low-energy nuclear systems with large scattering length. We will use only few information about the interaction for developing our theory, namely we will consider only central and short range interactions, in order to avoid complications carried by the spin degrees of freedom we will just try to describe systems of spin-less bosons. At first approximation we could try to describe the system with a zero range theory, under this assumption in the non relativistic quantum field theory formalism a system of spin-less bosons could be described by the following Lagrangian density

L = Ψ†i∂0+ ∇2 2m Ψ −g2 4 Ψ†Ψ2, (2.69) where Ψ is the boson field and g2 is a coupling constant. For simplicity we worked in

units of ~ = c = 1. Using this model the interaction is represented by a δ-like potential in coordinate space. The introduction of this singular interaction requires a regular-ization. It can be done by imposing an hard cutoff (Λ) in momentum space ass-well by using others type of regulator in order to suppress momenta above the cutoff (Λ). Observable will be independent of the arbitrary value of the cutoff Λ if the coupling constants are functions of Λ.

It can be shown [7] that by using a Gaussian regulator we obtain a Gaussian po-tential in coordinate space of the following form

V (r) = V0(g0, Λ) exp h − r r0(Λ) i , (2.70)

where V0(g0, Λ) is a function of both the coupling and the cutoff and it is the parameter

that will govern the strength of the interaction, while r0(Λ) is a function only of the

cutoff and it will take its place by smearing the δ function interaction over distances ∼ r0. Working with a Gaussian potential means in quantum mechanics to work with

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a Schrodinger equation that can not be solved analytically, for this reason we need to implement a numerical algorithm that will provide us with numerical solutions for the wave function of the system. We choose to work with the Numerov algorithm.

2.3.1 The Numerov Algorithm

The Numerov algorithm is a valuable tool for finding numerical solutions of a second order differential equation with no first order term, such as the Schrodinger equation. It is a three point algorithm which, given two point on a grid, is able to find a third point satisfying

∂2Yi

∂r2 i

= WiYi, (2.71)

where Wi is a known function that can be calculated point by point and implemented

on a grid. In order to solve this differential equation it is necessary to use an algorithm able to compute numerical differentiation of point-functions. In this thesis work we employed a five-point forward and a five-point backward numerical algorithm in order to provide the derivatives of the wavefunction. The forward and backward recurrence relations used to obtain the third point given the two nearest ones are respectively:

Zi+1= 12Yi− 10Zi− Zi−1, (2.72)

and

Zi−1= 12Yi− 10Zi− Zi+1, (2.73)

where Zi has to be considered a grid of points obtained from the grid of solutions Yi as

Zi = 1 −h 2 12Wi Yi, (2.74)

with h the separation value from two successive grid points. As a consequence of this procedure we came up with values of Zi and equivalently of Yi that have errors of the

order of h6. It is possible to think that we could try to reduce h as long as the numerical computation power permits us, but usually there is a typical value of h where it is not possible to gain anymore in precision because the reduction of the error on the single value of Zi is compensated by propagating errors among the grid points.

For the implementation of the Numerov algorithm for bound states we do not know the energies of the system right from the start, as a consequence we need to make a guess for them. Because we are not implementing the right values for the energy of the system(which carries the right boundary conditions for the differential equation), the wave function at some point will start to diverge, either positively or negatively. In order to account for this problem one usually implement both a forward and a backward Numerov algorithm starting from the first point and the end point of the grid, then it is possible to choose a matching point M (making sure that it stands outside the range of the short range potential) and evaluate at the matching point the left D_ML and right D_MR logarithmic derivative. If the energy implemented in the algorithm is the correct one, then these two derivative should be equal, but if the energy is a wrong one, then the two logarithmic derivative will be different from each other. A possible solution

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comes from assuming that the difference of the energy implemented in the algorithm and the real energy of the system is linear in δM, with δM = DLM− DMR. Imposing that

ED is the real energy of the system one has that

E = ED+ BδM. (2.75)

One then makes two guess E1 and E2 for the energy of the system, obtains with

the above procedure two values of δM and find a new value of the energy by making

use of the previous equation

E3= hE₁− E₂ δ1 M − δ2M i δ1_M, (2.76)

by iterating the procedure it is possible to obtain the right value of the energy with as much precision as needed.

2.3.2 Evaluation of integrals, scattering length and effective range In order to calculate the numerous observables of the system it is necessary to evaluate numerically various integrals. For the calculation of these integrals we decided to implement a Simpson algorithm, this is also a three point algorithm which calculates the integral of a point function by splitting it in groups of three elements and evaluating the integrals of each group. The algorithm for evaluating the integral of a single group of three elements is Z xi+2 xi f (x)dx = h 3 fi+ 4fi+1+ fi+2 + , (2.77)

where h is the distance between two adjacent point on the grid and is the error asso-ciated to the evaluation. It can be shown that this error scales as ∝ h5.

The scattering length has been calculated by solving the Schrodinger equation for the reduced radial wave function at zero energy with the Numerov algorithm. It is easy to see that outside the range of the potential the reduced radial wave function can be normalize to have the following form

u(r) = r − a , r > r0. (2.78)

Of course just after the implementation of the Numerov algorithm the wave function does not come out with the following normalization, but in general it would be of the form:

u(r) = A + Br , r > r0. (2.79)

So by evaluating the coefficient A and B we can transform the wave function in the way we need by a multiplication of an appropriate normalization constant and the value of the scattering length should be

a = −A

B. (2.80)

The effective range could be calculated by making use of its definition stated some section before. If v0(0, r) is the reduced radial wave function for the free solution of the

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Schrodinger equation at zero energy and u0(0, r) is the reduced radial wave function for

the solution of the Schrodinger equation with a short range potential at zero energy, then the effective range of interaction can be calculate as

rs= 2 Z ∞ 0 v₀2(0, r) − u2₀(0, r) dr. (2.81)

The integral can be evaluated by using the Simpson algorithm and, for practical purpose, its extension is not from 0 to ∞ but we stop the integral at some large value of r when the effective range does not change anymore by increasing the range of integration.

2.3.3 Results

We solved the Schrodinger equation for a Gaussian potential by using the Numerov algorithm. Our intention was first to calculate the first scaling violations of universal laws for low energy observables, secondly to show that a Gaussian potential could be considered a valuable effective potential for the description of low energy nuclear and atomic systems.

We first show the results we obtained for the scaling correction parameters of three observable: the binding energy of the shallowest dimer, the mean square radius and the asymptotic normalization constants. Then we look for universality in nuclear and atomic physics by comparing experimental data for the Deuteron and the Helium dimer with the results obtained numerically with a Gaussian potential.

Binding energy

The first observable we calculated with a Gaussian potential was the binding energy of the shallowest dimer. Our intention was to show that at first approximation the system shows the kind of universality that makes the binding energy depending only on the value of the scattering length, but we wanted also to calculate range corrections to the universal law. For this reason we made various calculation of the binding energy of the system, we hold the cutoff r0 of the Gaussian fixed at r0 = 1.0, but we varied

the strength of the interaction V0 from a value of Vmax= −112.5 to a value of Vmin=

−142.5 after every calculation of the binding energy we incremented the strength of the interaction by 0.5. In all the calculations the value of ~2/m was fixed to the value needed to describe the Deuteron system ~2_{/m = 41.471 MeV (Fm)}2_{. We then fitted}

the results obtained with a fourth degree polynomial ma2 ~2 E = a + b r0 a + c r₀ a 2 + dr0 a 3 + er0 a 4 . (2.82)

Below we report only the results obtained for the first three coefficient of the ex-pansion because coefficients d and e take contribution also from wave function with higher angular momentum quantum numbers [3]. We specify that the range of V0 has

not been chosen randomly, in particular we performed the calculation in regions where rs a. The goodness of the Gaussian interaction in capturing the range corrections of

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0.00

0.05

0.10

0.15

0.20 r

0 /a

1.0

1.1

1.2

1.3 E/

(

2 /m

a

2 )

Dimer_Energy

fit: a=1.00, b=1.46, c=1.25

Figure 2.2: Dimensionless binding energies (blue dots) for the shallowest dimer obtained with Gaussian potentials with different values of the strength V0 holding r0 fixed. The

values of the energies are graphed as function of r0/a where r0 is the fixed parameter

of the Gaussian and a is the scattering length. The red line is the best fit with a fourth order polynomial where the first three coefficients (a,b,c) are listed in the figure.

the analytical expansion in rs/a of the binding energy. This analytical expansion can

be obtained by squaring (Eq: 2.48) and then expanding for low values of rs/a:

ma2 ~2 E = 1 + rs a + 5 4 r_s a 2 + .... (2.83)

In (Fig. 2.2) and (Fig. 2.3) we have shown the results obtained with our fitting procedure. The precise values obtained for the first three parameter of the expansions in powers of r0/a and rs/a with the corresponding (1σ) errors relative only to the fitting

procedure are listed in the following tables.

rs/a Parameters Values Errors

a 0.99948 ±0.00004 b 1.007 ±0.002

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0.05

0.10

0.15

0.20

0.25 r

s

/a

1.0

1.1

1.2

1.3 E/

(

2 /m

a

2 )

Dimer_Energy

fit: a=1.00, b=1.01, c=1.27

Figure 2.3: Dimensionless binding energies (blue dots) for the shallowest dimer obtained with Gaussian potentials with different values of the strength V0 holding r0 fixed. The

values of the energies are graphed as function of rs/a where rsis the effective range and

a is the scattering length. The red line is the best fit with a fourth-order polynomial where the first three coefficients (a,b,c) are listed in the figure.

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r0/a Parameters Values Errors

a 0.99939 ±0.00002 b 1.456 ±0.002

c 1.25 ±0.03

Mean square radius

The second observable we calculated numerically for a system interacting through a Gaussian potential is the mean square radius of the dimer. Again we held fixed the value of the cutoff r0 and we varied the strength of the interaction V0. We used the

same value of Vmin and Vmax used for the binding energy and again we fitted the data

with a fourth order polynomial 2 a2r 2 _{= a + b}r0 a + c r₀ a 2 + d r₀ a 3 + e r₀ a 4 . (2.84)

As before we reported only the first three terms in the expansions, just because the other terms take contribution also from higher angular momentum value wave function. The results of the numerical computation and the fitting procedure are shown in (Fig. 2.4) and (Fig. 2.5).

Universality in few-body physics around the unitary limit