Jarzynski's theorem and free energy estimates for strongly interacting matter

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UNIVERSITÀ DEGLI STUDI DI PISA Corso di laurea magistrale in fisica

Gaspare Maria di Fede

JARZYNSKI’S THEOREM AND FREE ENERGY ESTIMATES FOR STRONGLY INTERACTING MATTER

Tesi di Laurea Magistrale

Relatore:

Prof. Massimo D’Elia

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Introduction

Quantum Chromodynamics (QCD), with gluons and quarks as degrees of free-dom, is the Quantum Field theory which describes Strong interactions. In 1964 Gell-Mann and Zweig proposed the "quark model" postulating the existence of thre different quarks with different "flavors": u(up), d(down), s(strange), and that hadrons are not fundamental particles but bound state of these quarks. However, the ∆++ baryon, being composed of three up quarks with parallel spins,

seem-ingly violated Pauli exclusion principle. To resolve this issue, a new quantum number, the color, was introduced. The existence of the color quantum number was later confirmed experimentally by measuring the ratio R = σ(e+_e−_→hadrons)

σ(e+_e−_→µ+_µ−₎ , whose value is consistent with a number of colors equal to 3. During the ’70, physicists predicted on theoretical bases the existence of other 3 quarks (charm, topand bottom), which were later discovered experimentally. Quarks seem to be confined in ordinary matter, due to the color confinement phenomenon which is not fully understood. On the other hand, in the high energy limit QCD reveals its asymptotic freedom property: the coupling costant decreses going asymptot-ically to 0. Thus, for high temperature a new state of the strongly interacting matter could exist. Lattice QCD simulations revealed the existence of two differ-ent phases: one of these, the deconfined one (known as Quark Gluon Plasma), is reached in extreme condition: high temperature and/or high baryon density. Several lattice QCD studies proved that at zero chemical baryon potential µB the

transition is a crossover around Tc∼ 150 − 170M eV [1] [2] [3]. For µB > 0 a first

order transition between the two phases could verify (see Fig.1), with a possible critical endpoint and then a crossover region.

The QGP (Quark Gluon Plasma) is also thought to have filled the Universe in its early stages. For this reason the study of the strongly interacting matter in a magnetic background has a relevant interest. Indeed cosmological models sug-gest that strong magnetic fields (√eB ∼ 2GeV) could be reproduced during the electroweak phase transition of the early universe, and consequently might have an impact on subsequent effects where strong interaction were involved. Another example of large magnetic fields (√eB ∼ 0.1 MeV) is present inside dense neu-tron stars called magnetars. Furthermore, in a noncentral heavy ion collision , the spectators could generate magnetic fields which reach up to √eB ∼ 0.5GeV [4]. Lattice QCD simulations are convenient tools for studying magnetic prop-erties of this "material", even if the usual lattice setup leads to some technical issues due to toroidal geometry, which imposes the quantization of the magnetic background. Consequently, evaluating free-energy derivatives with respect to the

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Figure 1: A possible phase diagram of QCD in the space of the state parameters: the temperature T and the baryon density.

magnetic field becomes a tricky task, furthermore, evaluating free energy differ-ences has always been an awkward goal in statistical mechanics.

Several lattice simulations [5] ,[6], [7] have been done to investigate magnetic properties of such "medium". In [6] the magnetic susceptibility of a SU(2 + 1) strongly interacting system was studied, using the thermodynamics integration

method, with an improved action and with physical quarks masses. The magnetic

background taken into consideration reaches up to√eB ∼ 0.1 GeV, varying the temperature T in a wide range (below and above the transition temperature) and keeping µB = 0. In [6] was found that, up to

√

eB ∼ 0.1GeV, the response of the material seems to be linear in B, within errors, with a paramagnetic behavior in almost the whole range of temperatures studied.

In [8], [7], using HRG (Hadron Resonance Gas) computations, a weak diamag-netism was suggested for low temperature (T ≤ 100 MeV) due to the dominant contribution from pions (see discussion in Appendix B of [7]). Such diamagnetic behavior is not proven by any numerical investigation yet. Only suggestions of weak diamagnetism have been detected [6], [7].

The proposal of this thesis is to adopt and study a new free energy measurement technique, which involves the usage of the Jarzynski’s relation [9] [10], reproduc-ing/improving measurements done in precedent studies [6]. The identity relates the average (over all possible realizations) of the exponentiated work W executed on a system during an evolution process, during which an Hamiltonian’s parame-ter λ is switched from an initial value λin to a final one λf in, to the exponentiated

free energy difference ∆F related to such process: e−βW_{= e}−β(F (λf in)−F (λin))_.

Denoting with forward process an evolution process during which λ is switched from λinto λf in, it is easy to convince oneself that the identity above is still valid

(with appropriate changes of sign) taking into account the reverse process, during which λf in → λin. Thus, one can obtain finite free energy estimates averaging

over forward processes or over reverse processes. Furthermore, analyzing the "physical" content of Jarzynski relation and proving its bond with the second law

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5 of Thermodynamics and with the Entropy production fluctuation theorem [11], we will show that more precise free energy estimates can be obtained, combining the data of both forward and reverse processes. Thus, the "pathological" issues and properties of an algorithm based on this relation will be discussed, proving that good free energy estimates are a consequence of a good sampling of a precise region of the work distributions. At the same time the efficiency of this new technique in comparision with the Thermodynamics integration method will be discussed.

The thesis is structured as follows: Chapter 1 will briefly introduce the reader to Non-Abelian gauge theories and then to Lattice implementation of such theories. Chapter 2 is dedicated to the QCD thermodynamics in the presence of a magnetic background. In Chapter 3 the numerical setup adopted for the implementation of the magnetic background is explained, and free energy measurements tech-niques developed so far in LQCD simulations involving magnetic backgrounds will be discussed. Chapter 4 deals with properties and issues regarding the algo-rithm based on Jarzynski relation. Finally, Chapter 5 and 6 are dedicated to the discussion of results, conclusions and perspectives.

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List of Figures

1 A possible phase diagram of QCD in the space of the state param-eters: the temperature T and the baryon density. . . 4 2.1 αs in function of the energy scale Q,image taken from [18] . . . . 28

2.2 order of the phase transition varying quark masses [20]. . . 30 2.3 Continuum extrapolated lattice results for the change of the

con-densate a s function of B, taking in consideration six different temperatures. . . 32 2.4 Magnetic susceptibility of QCD as a function of the temperature.

Results with dfferent lattice approaches are collected. A compar-ison with HRG model can be done in the low temperature region [25]. . . 33 2.5 From the last figure we can deduce the diamagnetic behavior for

T ∼ 90MeV . . . 36 4.1 ∆ ˆA is the free energy estimator. Image taken from [36]. . . 47 4.2 Comparison between the work (denoted with ∆U) distribution

P0(∆U ) and P0(∆U )e−β∆U [36] . . . 48

4.3 Graphic representation of trajectory space for enlightening the re-lation between "typical" and "dominant" regions [11]. . . 49 4.4 Distributions of work values when hWF

dissi > −hW R

dissi [11] . . . . 49

4.5 Different work distributions Pα (in this plot denoted with pα) for

different value of α [40]. . . 53 4.6 The estimator ∆ ˆf seems to reach its asymptotic value for N > 100. 55 4.7 Convergence of ∆ ˆf compared with the convergence measure

(de-noted with a instead of δ) . . . 56 4.8 A scatter plot of the deviation ∆ ˆf − ∆f versus the convergence

measure (indicated with a) for many individual estimates in de-pendence of the sample size N. . . 57 4.9 Convergence behavior of the covnergence measure and ∆F ,

sam-pling work values from gaussian distributions. . . 58 5.3 Behavior of δ (without error bars to better observe the bias,) in

function of the sample size N for nstep= 100 and nstep = 400. . . 61

5.5 Behavior of bias for each estimator, varying nstep. . . 63

5.6 Behavior of δ for different L for the nstep = 800 case, without

errorbars. . . 64 9

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5.7 ∆FB vs b. . . 65

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List of Tables

5.1 For such simulation we have adopted a protocol with nstep = 800

and L = 40. . . 65

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Chapter 1 Lattice QCD

1.1 Non-Abelian gauge fields theories

Quantum Chromodynamics, involving gluons (gauge fields) and quarks (fermions with three colored component) is a non-Abelian gauge theory and it is described by a Lagrangian density invariant under local SU(Nc= 3) transformation of the

fields involved in the theory. Then is useful to discuss SU(N) gauge theories before introducing the lattice formulation of QCD. A free-fermion Lagrangian invariant under global SU(N) transformation is of the type:

ψ =      ψ1 ψ2 ... ψN      L = ¯ψ(iγµ∂µ− m)ψ, (1.1)

where the fields transform as follows:

ψ → U ψ ¯

ψ → ¯ψU†, under U ∈ SU(N), which takes the form :

U = exp N2₋₁ X a=1 iTaωa ! ,

with Ta the generator of the SU(N) group complying the relations :

[Ta, Tb] = ifabcTc

tr[TaTb] =

δab

2 13

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To make the lagrangian in (1.1) locally invariant, a covariant differential is needed. For this reason we introduce the gauge fields Aa

µ (with a = 1, ..., N2− 1) and the

parallel transport along a path Cx→y between two points of the space-time defined

by :

W (Cx→y) = P exp −ig

Z

Cx→y

Aµ(z)dzµ

!

(1.2) Where Aµ= AaµTa and P is the path-ordering operator.

Aµ(x) under logal gauge transformation becomes:

A0_µ= U (x)Aµ(x)U†(x) −

i

g(∂µU (x))U

†

(x)

Thus under a local gauge transformation : W (Cx→y) → W0(Cx→y) = U (y)W (Cx→y)U†(x),

and ˜ψ(y) = W (Cx→y)ψ(x)will transform to ˜ψ0(y) = W0(Cx→y)ψ0(x) = U (y) ˜ψ(y).

Therefore, using an infinitesimal parallel transport its possible to define a covari-ant differential :

Dψ(x) = W (Cx+dx→x)ψ(x + dx) − ψ(x) (1.3)

Since : W (Cx+dx→x) = exp(−igAµ(x)dxµ) ' 1 − igAµdxµ, eq. (1.3) becomes:

Dψ(x) ' [∂µψ(x) + igAµ(x)ψ(x)]dxµ

Thus, defining the covariant derivative : Dµ = (∂µ+ igAµ), the fermionic locally

invariant lagrangian is:

LF = ¯ψ(iγµDµ− m)ψ (1.4)

Defining Fµν = −_gi[Dµ, Dν] = ∂µAν − ∂νAµ + ig[Aµ, Aν], the pure gauge term

lagrangian is : LG = − 1 4F a µνF a µν (1.5)

Thus an SU(N) gauge theory with fermions is described by : L = LF + LG

Quantum Chromodynamics, as said above, is a non-abelian gauge theory which describes the interactions with Nf flavors of quarks, thus the QCD lagrangian

density is the following :

L = −1 4F a µνF a µν + Nf X f =1 ¯ ψf(iγµDµ− mf)ψf

This Lagrangian, taking in to consideration only the three lightes flavors, is also approximately invariant under global SU(Nf = 3)transformations in flavor space.

It also enjoys, in the massless limit, the so called Chiral symmetry which will be discussed briefely in the following section.

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1.2. CHIRAL SYMMETRY 15

1.2 Chiral symmetry

Suppose we have a QCD Lagrangian with L flavors :

L = L X f =1 ¯ ψf(iγµDµ− mf)ψf, (1.6)

if the masses are all the same, i.e. mf = m ∀f, it is invariant under global U(L)

transformations. Considering that each Dirac spinor has two chiral components: ψR,L =

1 ∓ γ5

2 ψ,

in the massless limit the Lagrangian (1.6) is invariant under the chiral group U (L)L⊗ U (L)R:

ψ_L→ ULψL

ψR → URψR

, (1.7)

where UL ∈ U (L) and UR ∈ U (L) acts only in the left/right handed component

of ψ respectively.

Defining the vectorial and the axial chiral transformations as follows:

U (1)V = ψL → eiαψL ψR → eiαψR U (1)A= ψL→ eiβψL ψR→ e−iβψR SU (L)V = ψ_L→ V ψL ψR→ V ψR SU (L)A= ψ_L→ AψL ψR→ A†ψR ,

where α, β are real parameters, and V , A ∈ SU(L), we can rewrite the chiral symmetry group as :

U (L)L⊗ U (L)R= U (1)L⊗ U (1)R⊗ SU (L)L⊗ SU (L)R=

U (1)V ⊗ U (1)A⊗ SU (L)V ⊗ SU (L)A .

(1.8) However, it should be noted that SUA(L) is not a subgroup of SU(L), since the

axial transformations are not a closed set. Even if the Lagrangian above is invari-ant under axial and vectorial chiral transformations (in the massless limit), the SU (L)A axial symmetry is spontaneously broken. Indeed, the expectation value

of the chiral condensate h ¯ψψi(which is not invariant under axial chiral transfor-mations) on the vacuum is not zero. This fact should lead to the appearance of massless Goldstone bosons (one for each generator of the spontaneously broken symmetry group), instead, since this symmetry is only approximately realized in the real world, we can see in the hadronic spectrum the "pseudo" Goldstone bosons which are not massless, like pions, kaons etc.

We also notice that, considering a single flavor f calling λ a given eigenvalue of the Dirac operator 6 Df = γµDµ,f and ρ(λ)1 the spectral density of the same

1_{ρ(λ) is defined in such a way that N}

λ= ρ(λ)∆λ is the number ov eigenvalues in [λ, λ + ∆λ]

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operator in the massless limit, the Banks-Casher relation [12] holds:

ρ(0) = −h ¯ψfψfi π .

Thus, when there is a spontaneous symmetry breaking the Dirac spectrum has non-zero density at λ = 0.

1.3 Path integral formalism and the Monte-Carlo

approach

Lattice QCD is based on the path integral formulation of quantum field theories. Thus, it is worth discussing such formalism.

In quantum mechanics the states of the system are described by vectors in a Hilbert space, and observables are represented by hermitian operators acting on this space.

Let us consider a system, with qα (α = 0, .., n) coordinate degrees of freedom and

respective conjugate momenta pα, described by the following Hamiltonian:

H(q, p) = 1 2 n X α=1 p2_α+ V (q) (1.9)

The matrix element between two coordinate states in different times (in the Heisenberg picture) will be (setting ~ = 1) :

hq0_{, t}0_{|q, ti =< q}0_|e−iH(t0_−t)

|q > (1.10)

Inserting a complete set of energy eigenstates to the right and left of the expo-nential in (1.10) we get:

hq0, t0|q, ti =X

n

e−iEn(t0−t)_ψ

n(q0)ψn∗(q),

where ψn(q) =< q|En >is the eigenfunction of H with energy En. The sum over

n includes the discrete as well as the continuous spectrum of the Hamiltonian. The above expression can now be continued to imaginary time, making the re-placements t → −iτ , t0 _{→ −iτ}0 _:

hq0, τ0|q, τ i =X n e−En(τ0−τ )_ψ n(q0)ψ∗n(q) =< q 0_|e−H(τ0_{−τ )} |q >, (1.11) which is dominated by the ground state in the limit τ0 _{− τ → ∞}_{. The path −}

integral representation of the right-hand side of (1.11) is : hq0, τ0|q, τ i =

Z q0

q

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1.3. PATH INTEGRAL FORMALISM AND THE MONTE-CARLO APPROACH17 where SE = Z τ0 τ dτ00LE(q(τ00), ˙q(τ00)) LE(q(τ00), ˙q(τ00)) = X α 1 2q˙ (l)2 α + V (q (l)_).

Thus, taking the limit τ0 _{→ ∞}_{, τ → −∞, given an observable O(q(τ)), its}

expectation value on the ground state is :

hO(q(τ ))i = lim

τ0_{→∞,τ →−∞} Rq0 q DqO(q(τ ))e −SE[q] Rq0 q Dqe −SE[q] (1.13)

The path-integral formalism allows to use these formulae also for a general Quan-tum Field theory in a 4D euclidean space.

Taking, for example, a scalar field theory, we can evaluate , given a general La-grangian density L[φ] :

hO(φ(τ, ~x))i = limτ0_{→∞,τ →−∞} Rφ0 φ DφO(q(τ, ~x))e −SE[φ] Rφ0 φ Dφe −SE[φ] and the thermal partition function, replacing τ0 _→ 1

T = β, τ = 0 and fixing boundary condition φ(β, ~x) = φ(0, ~x): Z = T r[e−βH] = Z φ Dφ0e−R0βdτ R∞ −∞d3xLE[φ0]

Consequently the task of a Monte Carlo method is to generate configurations of the system (took in consideration ) with a statistical weight e−SE,then estimating O(q(τ ))for each configuration m, and finally, mediating over an appropriate large number N of configurations, get an estimate of the expectation value:

hOi → 1 N N X m=1 Om

In a 4D Euclidean space, the QCD Lagrangian in the continuum limit is :

LE = 1 4F a µνF a µν+ Nf X f =1 ¯ ψf(γµE∂µ+ mf)ψf, (1.14) where γE

4 = γ0, γiE = −iγi accomplish the following anticommutation relation :

γE µ, γ

E

ν = 2δµν,

and the action :

SE = Z β 0 dτ Z V d3xLE, (1.15)

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then, the vacuum expectation value of an observable O, in the path integral formalism, is given by:

hOi = 1 Z

Z

DAD ¯ψDψO[A, ¯ψ, ψ]e−SE[A, ¯ψ,ψ]_, (1.16) where the following periodic and antiperiodic boundary condition are imposed, for gauge fields and fermions respectively :

Aµ(~x, β) = Aµ(~x, 0)

ψ(~x, β) = −ψ(~x, 0), (1.17)

and ¯ψ, ψ, are Grassmann variables.

It is well known, anyway, that the above functional integral is ill-defined. This problem can be solved fixing the gauge via the Fadeev-Popov procedure. Such problem , while mantaining the gauge invariance of the path integral formalism, will not be present in the lattice QCD formulation, as it will be explained in the following section.

1.4 Lattice regularization

1.4.1 Free fermion field on the lattice : the doubling

prob-lem

To implement fermion fields on the lattice we have to regularize the euclidean version of (1.1) :

LE = ¯ψ(γµE∂µ+ M )ψ, (1.18)

The lattice is a 4D hypercube (with Ns sites in three direction and Nτ in one)

space where the coordinate of each point are labeled by four integer numbers : xµ = nµa,

where a is the lattice spacing (the distance between two next-neighbors).

Firstly the discretization of integral operations and differential operators is needed: Z d4x → a4X n ∂µf (na) = f ((n + 1)a) − f ((n − 1)a) 2a f (na) = ˆ f (na) a2 = P

µ(f (na + ˆµa) + f (na − ˆµa) − 2f (na))

a2

Denoting now the fermion field with ψα (α ,β etc. will denote the 4 Dirac

com-ponents of ψ), we can, after some replacements, discretize SE =R d4xLE[ψ, ¯ψ]in

a "naive" way :

M → ˆ M a

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1.4. LATTICE REGULARIZATION 19 ψα(x) → 1 a3/2ψˆα(n) ∂µψα(x) → 1 a5/2∂ˆµψˆα(n) SE = X n,m,α,β ¯ ˆ ψα(n)Kα,β(n, m) ˆψβ(m), (1.19) where Kα,β(n, m) = X µ 1 2(γµ)α,β[δm,n+ˆµ− δm,n−ˆµ] + ˆM δm,nδα,β. The free fermion propagator will be :

ψα(x) ¯ψβ(y) = Z π/a −π/a d4_p (2π)4 [−iP γµp˜µ+ M ]α,β P µp˜2µ+ M2 eip·(x−y) (1.20) ˜ pµ= 1 asin(pµa) (1.21)

Taking the continuum limit of (1.20) we should get back the well known contin-uum expression of the fermion propagator. Instead, since, for a → 0 there are more than one zero of the sine-function in (1.14)(16 zeros, which are nothing else that the 16 corners of the Brillouin Zone ) :

pµ = (0, · · · , 0), (π/a, 0, · · · , 0), · · · , (π/a, π/a, · · · , π/a)

we won’t get the right formula.

This issue, which is usually called the doubling problem, rises in the lattice to keep the theory invariant under axial chiral transformation in the massless limit (see [13]).

An axial chiral transformation of the fermion field is defined by the following mapping:

ψ → eiγ5θ_ψ ¯

ψ → ¯ψeiγ5θ where θ is a parameter and γ5 = γ1γ2γ3γ4.

A massless interacting fermion theory, like QED or QCD, (in the continuum limit) is symmetric under these transformations. This implies the existence of a conserved axial vector current, which, by the way, has an anomalous divergence due to quantum fluctiations. On the other hand, in a lattice regularization, this symmetry implies that this current is strictly conserved for any lattice spacing. Thus, the lattice cancels out the anomaly introducing extra excitations (the

dou-blers), that have no analog in the continuum.

To solve such doubling problem many discretized fermion action have been pro-posed. However, solving this issue leads to the loss of some property of the continuum theory, indeed, the so called "no-go" theorem, due to Nielsen and Ni-nomiya, states that the doubling problem cannot be solved without breaking the chiral symmetry or losing the locality of the fermion action.

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1.4.2 Wilson fermions

A way to remove the extra zeros of (1.20) is to lift them by an amount proportional to the inverse lattice spacing. One action which accomplishes this task is the following: S_F(W ) = X n,m,α,β (ψ¯ˆα(n)Kα,β(n, m) ˆψβ(m)) − r 2 X n ¯ ˆ ψ(n) ˆ_ˆψ(n), (1.22)

where r is the Wilson parameter. Setting ˆψ = a3/2ψ and ˆ = a2_{it is easy to see} that the additional term in (1.22) vanishes linearly with a in the naive continuum limit. This action leads to the following propagator in the continuum theory :

ψα(x) ¯ψβ(y) = lima→0

Z π/a −π/a d4_p (2π)4 [−iγµp˜µ+ M (p)]α,β P µp˜2µ+ M (p)2 eip·(x−y), whit ˜pµ is given in (1.21) and

M (p) = M + 2r a

X

µ

sin2(pµa/2). (1.23)

From (1.23) we see that for any fixed value of pµ, M(p) approaches to M for

a → 0. Near the of the Brillouin Zone (except for pµ = 0 ), however, M(p)

diverges as we let a → 0. This eliminates the fermion doubling problem, but the chiral symmetry of the original action (1.19) in the massless limit has been broken, as predicted by the "no-go" theorem: preserving the locality we have lost the symmetry.

1.4.3 Staggered fermions

It is also possible, for removing the unwanted zeros, to reduce the Brillouin Zone, i.e. by doubling the effective lattice spacing.

Indeed, if we take a 4-dimensional space-time lattice and subdivide it into ele-mentary 4-dimensional hypercubes of unit length, and then we place a different degree of freedom at each site within a given hypercube, repeating this structure periodically through the lattice, we will get an effective doubled lattice spacing for each degree of freedom.

Since there are 4 sites within an hypercube, to fill each of this site with a different degree of freedom we need 4 different Dirac fields with different "taste" denoted by ψf

α, where f denotes the taste and α the spinor index.

Let us start from the naive action (1.19) and let us make a local change of variables: ˆ ψ = T (n)χ(n) ˆ ¯ ψ = ¯χ(n)T†(n), (1.24)

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1.4. LATTICE REGULARIZATION 21 where T (n) are unitary 4 × 4 matrices. It is possible to diagonalize in the "spin-sector" this expression by choosing the matrices T (n) in this way:

T†(n)γµT (n + ˆµ) = ηµ(n)1. (1.25)

Indeed, the matrices:

T (n) = γn1 1 γ n2 2 · · · γ n4 4 , (1.26)

satisfy eq. (1.25) with:

ηµ(n) = (−1)n1+n2+···+nµ−1, η1(n) = 1.

Thus, we can write the so called staggered action, in function of the new field χ: S_Fstag = 1 2 X µ,n ηµ(n)[ ¯χ(n)χ(n + ˆµ) − ¯χ(n)χ(n − ˆµ)] + ˆM X n ¯ χ(n)χ(n) (1.27) The above action involves only one degree of freedom per lattice site: the remnant of the Dirac structure are the phases ηµ(n). Labeling the center of each hypercube

inside the lattice with an integer 4-component vector 2Nµ,its corners position will

be given by:

ˆ

rµ = 2Nµ+ ρµ,

where ρµ= 0 or 1 . In this way χ(n) are relabeled :

χρ(N ) = χ(2N + ρ). (1.28)

Finally the correlation function hχρ(N ) ¯χρ(N0)i :

hχρ(N ) ¯χρ(N0)i = Z π −π d4_p (2π)4 −iP µΓ µ ρ,ρ0(p)sin( ˆ pµ 2 ) + ˆM δρ,ρ0 P µsin2(ˆpµ/2) + ˆM2 ei ˆp·(N −N0), (1.29) where Γµ_ρ,ρ0(ˆp) = ei ˆp·(ρ−ρ 0_)/2 [δρ+ˆµ,ρ0+ δ_ρ−ˆ_µ,ρ0]η_µ(ρ).

From (1.29) we can see that, because of the appearance of the factor 1/2 in the argument of the sine function in the denominator, the integral will be dominated (for a → 0 ) by the momenta in the immediate neighbourhood of ˆp = 0, avoiding the doubling problem.

Thus, the action in function of the "tasted" Dirac field can be reconstructed. Indeed making the following transformations (it should be noticed that now N = (N1, · · · , N4)labels the space-time points of a lattice with lattice spacing 2a, and

that the index ρ labels the 4 components of the new field χ): ˆ ψα,β(N ) = C X ρ (Tρ)αβ 2 χρ(N ) (1.30) ¯ ˆ ψαβ(N ) = C X ρ ¯ χρ(N ) (T_ρ†)αβ 2 ,

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and doing the substitutions ψ = b−3/2_ψ_ˆ_{, ∂} µ =

ˆ ∂µ

b (where b = 2a), we get :

S(stag). = X ρ,x,µ b4ψ(x)[(γ¯ µ⊗ 1)∂µ+ 1 2b(γ5⊗ γ ∗ µγ5)]ψ(x) + 2M X x ¯ ψ(x)1 ⊗ 1ψ(x), (1.31) where the former and the latter matrices in the product tensor act on the Dirac and "taste" components of ψ(x) respectively.

As we can see, in the massless limit, S(stag) _{preserves a continuous U(1) ⊗}

U (1) symmetry. This is the reason why the staggered fermions formulation is preferred: they can used to study the spontaneous breakdown of the remaining lattice symmetry and the associated Goldstone phenomenon. By the way, there are still 3 extra doublers in this formulation. In the next subsection the "rooting method" will be explained for getting rid of them.

1.4.4 Gauge fields on the lattice

If we introduce a "3-color" component χ(n) variables, the action (1.27) ,written in function of them , is invariant under global SU(3)ctransformations. Under local

transformations is not invariant anymore, due to the presence of bilinear terms like ¯χ(n)χ(n+ ˆµ). In the continuum formulation it’s well known how such bilinear terms should be modified in order to arrive at a gauge-invariant expression :

¯

χ(x)χ(y) → ¯χ(x)W (Cx→y)χ(y),

where W (Cy→x) is the parallel transport defined in (1.2). In the same way, the

gauge-invariant version of ¯χ(n)χ(n + ˆµ) is : ¯

χ(n)U (n, n + ˆµ)χ(n + ˆµ), where :

U (n, n + ˆµ) = Uµ(n) = eig ˆAµ(n), (1.32)

is an infinitesimal parallel transport along the ˆµ direction which lives between the sites n and n + ˆµ. For this reason it is usually called link variable. Then, the gauge-invariant version of (1.27) is :

S_Fstag = 1 2 X µ,n ηµ(n)[ ¯χ(n)Uµ(n)χ(n+ ˆµ)− ¯χ(n)Uµ†(n− ˆµ)χ(n− ˆµ)]+ ˆM X n ¯ χ(n)χ(n). (1.33) The pure gauge action term on the lattice is obtained from the so called plaquette variables : Uµν(n) = 1 3Uµ(n)Uν(n + ˆµ)U † µ(n + ˆν)U † ν(n), (1.34)

which for a → 0 become :

Uµν(n) ' eiga

2_F µν(n)_.

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1.5. IMPROVED ACTION 23 Thus : SG = 6 g2T r X n,µ<ν [1 −1 2(Uµν(n) + U † µν(n)], (1.35)

will converge to S(cont.) G =

6

g2T rR d4xFµνFµν for a → 0.

1.4.5 Rooting trick

As suggested in subsection (1.2), taking into consideration the action:

S[ ¯χ, χ, U ] = S_F(stag.)[ ¯χ, χ, U ] + SG[U ], (1.36)

(where χ = (χ1, χ2, χ3)has 3 color degrees of freedom and one flavor degree),we

could express the expectation value of an observable O[¯χ, χ U] as :

< O >= R DU D ¯χDχOe

−S[ ¯χ,χ,U ]

R DU D ¯χDχe−S[ ¯χ,χ,U ] ,

Since SF(stag.)can be expressed as SF =

P

n,m,c,c0χ¯n,cK[U ]n,m,c,c0χ_m,c0, we obtain :

< O >= R DU D ¯χDχO det(K[U ])e

−SG[U ]

R DU D ¯χDχ det(K[U ])e−SG[U ] (1.37) but we still have 4 doublers for each color. Thus, we could think that, if det(K[U])e−SG[U ] is a statistical weight for a theory which involves 4 different "tasted" fermion , the corresponding statistical weight for only one "taste" is det(K[U])1/4_e−SG[U ] . Then, for a Lattice QCD setup with 3 flavors (up, down, strange) the partition function is:

Z = Z

DU det(Ku[U ])1/4det(Kd[U ])1/4det(Ks[U ])1/4e−SG[U ]. (1.38)

This "rooting trick" is not theoretically well established, since the existence of a local theory with the previous Z can not be proven. Nevertheless, numerical results seem to enforce its validity (see [14] for more details).

We finally notice that, even though the gauge invariance is preserved, the path integral formulation in LQCD doesn’t require any gauge fixing, since the integra-tion is done over the link variables which live in the SU(3) group, the latter is compact and, consequently, the integral is finite.

1.5 Improved action

The action in eq. (2.23) reproduces the correct continuum limit, but still presents some regularization artifacts which can be mitigated. The pure gauge term has a discretization error ∼ O(a2₎ _{, since it differs from the continuum action by}

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convergence we can add other terms, which cancel out the O(a2₎ contributions

in SG using other Wilson loop operators, like the rectangle operator :

Rµν(x) =

1

3T r(Uµ(x)Uµ(x+aˆµ)Uν(x+2aˆµ)U

†

µ(x+2aˆµ+aˆν)U †

µ(x+aˆµ+aˆν)U †

ν(x+aˆν)).

(1.39) The latter is used to construct the tree level improved Symanzik action (see [15] for a detailed discussion) :

SG = −β X x,µ>ν 5 6ReT rUµν(x) − 1 12ReRµν(x) , (1.40)

which is accurate up to O(a4₎_{. Another issue of (1.33) is the taste symmetry}

violation, due to the fact that two fermions with different taste can interact with each other (see [16]). In fact, taking in mind Eq. (1.30), we observe that different tastes of the quark field, ψt_(x)_{, are different combinations of the staggered field,}

χ, on a hypercube, then, if gluon fields are present, the different tastes are not influenced at the same way by the different Uµ(x). Thus, we expect that, cutting

off the rapidly fluctuating components of the gauge fields, taste breaking would be decreased. Gauge-link smearing is an a algorithm to suppress UV fluctuations in the gauge field: Uµ(x) is replaced with a gauge covariant average over paths

connecting x to x + ˆµ.

Stout smearing is an analytic method of smearing, hence differentiable every-where, which is useful since it avoids the problem of projection over SU(3). De-noting with Cµ(x)the weighted sum of the perpendicular staples which begin at

lattice site x and terminate at neighboring site x + ˆµ. Cµ(x) = X ν6=µ ρµν(Uν(x)Uµ(x + ˆν)U † ν(x + ˆµ) + U † ν(x − ˆν)Uµ(x − ˆν)Uν(x − ˆν + ˆµ)), (1.41) where ρµν are tunable real parameters. Then,defining the Hermitian and traceless

matrix Qµ(x): Qµ(x) = i 2(Ω † µ(x)−Ωµ(x))− i 6T r(Ω † µ(x)−Ωµ(x)), Ωµ(x) = Cµ(x)Uµ†(x), (1.42)

eiQµ(x) is an element of SU(3). Thanks to this fact it is straightforward to define an iterative analytic link smearing algorithm in which the links U(n)

µ (x)at step n

are mapped into links U(n+1)

µ (x) using:

U_µ(n+1)(x) = exp(iQ(n)_µ (x))U_µ(n)(x). (1.43) The analyticity property allows to adopt RHMC (Rational Hybrid Montecarlo) algorithm, which adopts a molecular dynamics updating (see [17] and [13] for an introduction to Molecular dynamics based algorithms ).

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1.6. CONTINUUM LIMIT 25

1.6 Continuum limit

In the sections above we have constructed a discrete action which for a → 0 converges to the correct continuum limit. However this action can be written in function of dimensionless variables, as has to be, since they are the only kind of variables we can handle in a Monte Carlo simulation. Anyway, when sending the lattice cutoff to infinity, i.e. sending a → 0, physical observables should agree with the experimental value and become indipendent of a. This, in general, will imply that the bare parameters have a nontrivial dependence on the lattic spacing a. Let us consider an observable O with a given mass dimension dO, and let be

ˆ

O the corresponding lattice quantity which can be determined numerically. ˆO will depend on the bare parameters of the theory, like the bare coulping or the fermions masses. In a pure gauge theory or in a massles full QCD theory the only dimensionless parameter is the bare coupling g. The existence of a continuum limit naively implies that

O(g, a)lat = (

1 a)

dO_{O(g) →}ˆ

a→0Ophys, (1.44)

where Ophys is the physical value which has to be reached in the continuum limit.

Thus, in order that Olat converges to Ophys the bare coupling must depend on the

lattice spacing in such a way that it is possible to take the limit for a → 0, and this dependence is encoded in the Beta function:

βlat(g) = −a

dg

da (1.45)

We can derive its analytic expression up to O(g7₎_{starting from the renormalized}

coupling costant gR(g(a), aµ) (where µ is an energy scale) as follows:

0 = adgR da = a ∂gR ∂g aµ dg da+a ∂gR ∂aµ g daµ da = − ∂gR ∂g aµ βLAT+µ dgR dµ g = −∂gR ∂g aµ βLAT+β, thus we obtain: βLAT = β ∂gR ∂g aµ . (1.46)

Knowing the perturbative expression of β up to O(g7 R) : β = −β0gR3 − β1g5R+ O(g 7 R), (1.47) where : β0 = 1 (4π)2 11Nc− 2Nf 3 (1.48) β1 = 1 (4π)4 34 3N 2 c − 13 3 Nc− 1 Nc Nf , (1.49)

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and knowing that gR(µ) = g + Ag3+ O(g5), we get :

βLAT(g) = −β0g3− β1g5 + O(g7). (1.50)

By solving the last equation we obtain:

a = 1 Λlat exp − 1 2β0g2(a) (β0g2) −β1 2β2₀_. (1.51)

This solution suggests that the continuum limit can be reached sending g → 0, thanks to the positivity of β0. As mentioned above, a numerical simulation can

produce only dimensionless results, therefore we need to determine a to get a physical meaningful measurement. In a pure gauge simulation (which means no fermion involved in the theory) one can fix some observables with their phe-nomenological value, like the string tension √σ = 440M eV, evaluate the adi-mensional ˆσ from lattice simulations and then obtain a. By the way, doing full QCD simulations (thus with light (up and down) and strange quarks involved), we have to deal with 3 bare parameters: g, mlight, ms. These parameters need to

be tuned among them to reproduce the "correct physics" through the continuum limit. As suggested, explained and made in refs. [1],[2],[3], a Line of Constant Physics (LCP) needs to be built up. A LCP can be constructed fixing a ratio between two well estabilished experimental values of some quantity as mK

mπ, then tuning (for a given g) mlight and ms to obtain from numerical simulations the

same fixed value of the ratio mentioned above. In such a way, lowering g and following a LCP we can lead the system to the correct physical continuum limit.

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Chapter 2 QCD Thermodynamics in a

magnetic background: overview

In this chapter a brief review regarding the QCD thermodynamics properties in a magnetic background will be exposed, introducing at first QCD Thermodynamics in absence of any external field.

2.1 QCD Thermodynamics : lattice results at

zero chemical potential

For studying QCD Thermodynamics the partition function at finite temperature T has to be introduced:

Z = Z

DAD ¯ψDψe−SE[A, ¯ψ,ψ]_{= T r}e−βH , (2.1) thus, the expectation value of a given observable O will be :

hOi = T rOe−βH₌ 1

Z Z

DAD ¯ψDψOe−βSE[A, ¯ψ,ψ]_. (2.2) The free energy density and the pressure density are defined as:

f = F/V = −1 β lnZ V , p = P V = −f, = − 1 V ∂lnZ ∂β (V → ∞). (2.3) Studying f and its derivatives one can determine eventually the presence and the order of a phase transition, and the equation of state of a thermodynamic system. Another fundamental observable in QCD is the chiral condesate defined by :

h ¯ψψi =X f T V ∂lnZ ∂mf =X f h ¯ψfψfi. (2.4) The temperature T = 1

β (choosing natural units) on the lattice can be varied

varying the bare coupling g since β = Nτa and a changes varying g as explained

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in chapter 2. In turn, in presence of fermions we need to impose antiperiodic boundaries in the "temporal" direction for fermions and, on the other hand, pe-riodic boundaries for bosons.

It’s well known the property of asymptotic freedom of QCD : the strong coupling constant αs goes to 0 in high energy processes as showed in fig.2.1 :

Figure 2.1: αs in function of the energy scale Q,image taken from [18]

This property is revealed when the system,being in extreme condition (high tem-perature, or density), goes from the hadronic phase to a "quark-gluon" plasma phase. Several lattice studies have been done to determine the critical tempera-ture Tc and the order of this phase-transition: it is well estabilished that,

consid-ering physical quark masses, there is not a real phase transition but a crossover [1][2][3], and that Tc = 150 − 170 MeV. To evaluate the critical region,usually

order parameters, which are related to some global symmetry restoration (or spontaneous breaking) are analyzed. For lattice QCD relevant order parameters are the Polyakov loop P and the chiral condensate h ¯ψψi, even if QCD with finite mass flavors doesn’t preserve neither Z3 symmetry (which will be discussed right

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2.2. Z3 SYMMETRY 29

2.2 Z

3

symmetry

The pure gauge action SG defined in Eq. 1.35 is invariant under the following

transformation : U4(n4, ~n) → zU4(n4, ~n) z = exp 2iπl 3 ∈ Z3, l ∈ 0, 1, 2, (2.5)

which is executed over every temporal link U4(n4, ~n) in a given time slice n4.

Instead the Polyakov loop, defined as :

L(~n) = T r

β

Y

n4=1

U4(n4, ~n), (2.6)

is not invariant under such transformation : L(~n) → zL(~n).

Furthermore, it can be proven that the following identity holds :

e−βFq ¯q _{= | hLi |}2_, (2.7) where Fq ¯q is the free energy of a static quark-antiquark pair infinitely apart from

each other, measured relative to that in the absence of the q¯q pair. Therefore we can conclude that if hLi = 0, then the free energy increases for large distances between the quarks, implying confinement. On the other hand, if hLi 6= 0, then the free energy of the quark-antiquark pair approaches a constant for large separations. Thus, that will be intepreted as signal of deconfinement. These facts suggest that, if there is the appearance of a deconfined phase for T ≥ Tc,

then hLi 6= 0 and the Z3 symmetry is necessary broken. Lattice pure gauge

simulations (for example [19]) have been proven that the transition is of the first order. Furthermore, this is valid for each SU(N) pure gauge theory with N ≥ 3, since a pure SU(N) gauge theory belongs to the same universality class of a Z(N) spin model.

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2.3 Nature of QCD transition as a function of

the quark masses

The Nature of the QCD transition strictly depends on the values of the quark masses. A possible scenario is shown in Fig. 2.2, which will be described below. In the case of Nf = 3 the Full QCD Lagrangian preserves chiral symmetry which

Figure 2.2: order of the phase transition varying quark masses [20]. is spontaneously broken since, for T = 0, ¯ψψ is non zero. It is known that in such case a chiral first order transition is expected [21] and thus the chiral symmetry is restored, i.e. ¯ψψ = 0. The extension of this first order region for higher values of the quark masses is not well estabilished yet[22], [23]. Any-way, this region is separated from the crossover region by a second order phase transition belonging to the 3-d Z(2) universality class. In the case of Nf = 2 the

transition is expected to belong to the universality class of a 3-d O(4) spin model, then, it is a second order phase transition. The second order Z(2) line separating the Nf = 2 + 1 first order and the crossover regions, the second order O(4) line

for Nf = 2 and the first order region for the Nf = 2 + 1 case are supposed to

meet at a tri-critical point characterized by a certain value of ms = mtrics .

Any-way, Several lattice simulations [24],[23] point out that the first order region may extend further in the ml= 0 region.

In the limit of infinite quark masses, fermions decouple and the thermodynamics of a pure SU(3) gauge theory is recovered, thus, as explained above, the decon-finement transition is of the first order. This first order region extends to lower quark masses and it ends at a second order critical line that belongs to the univer-sality class of the three dimensional Z(2) Ising model . Finally, between the two first order regions, there is, as mentioned above, a crossover region , i.e. there is no true phase transition between the hadronic and the quark-gluon plasma

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2.4. EFFECTS OF A MAGNETIC BACKGROUND 31 phases. Lattice simulations state that the physical point, i.e. ml = ml,phys and

ms = ms,phys, belongs to this region [1], [2], [3].

2.4 Effects of a magnetic background

The study of nonzero magnetic background is of great interest, since external magnetic fields play an important role in the evolution of the early universe, in strongly magnetized neutron stars and in non-central heavy-ion collisions. Mag-netic fields induce a variety of effects in the thermodynamics of QCD, for example they significantly affect the phase diagram [7][25], thus, it is also interesting to study the magnetic properties of the system below and above the pseudocritical temperature Tc, which is also influenced by the presence of the magnetic

back-ground [7][25].

Before exposing lattice simulation results obtained so far, we need to stress out that, in the presence of a magnetic background, the fundamental quantity of thermodynamics, i.e. the free energy density will have the following expression:

f = tot− T s − eB · M, (2.8)

where tot is the energy density of the medium plus the work needed to mantain

the constant external field, and M is the magnetization per unit volume. Thus, each thermodynamic observable is a derivative of the previous expression. Several lattice simulations brought to light magnetic properties of such "medium" and non trivial effects, due to the presence of the magnetic background, on the deconfinement transition and on the behavior of the chiral condensate. Indeed, lattice simulations have shown that h ¯ψψi depends on eB, and it increases as the strenght of the magnetic background is increased [7], [26], [27] (the so called

magnetic catalysis effect, this effect is quite visible as reported in [27] at low

temperature (T 6 130 MeV).

The magnetic catalysis is a phenomenon present also in a free-fermion system [28] [29] [30] and is essentially due to the dimensional reduction D → D − 2 in the dynamics of fermions in the presence of a magnetic constant field. Indeed, in this case, taking into consideration a non interacting fermion theory in a D = 3 + 1 spacetime , the energy spectrum is :

E = ±pm2_{+ 2|eB|n + p}2

z, n = 0, 1, 2, · · ·

where n = k + s + 1

2 is the Landau level, k is the quantum number associated

with the orbital motion on the orthogonal plane respect to the magnetic field, and s = ±1₂ is the spin projecton on the direction of the field. Furthermore, we notice that the lowest Landau level, i.e. n = 0, corresponds to the lowest orbital state k = 0and admits only s = −1₂, which means that it is a polarized state. It should also be noted that each Landau level has an additional infinite degeneracy. It can be shown that the number of this degenerate states per unit area in the x−y plane is|eB|

2π for n = 0 and |eB|

π for n > 0. At the lowest Landau level the energy spectrum

E =pm2_{+ p}2

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theory. This spectrum reflects the kinematic aspect of the dimensional reduction, which is the result (from the physiscs view point) of a partially restricted motion of fermions in the perpendicular plane to the magnetic field. When the fermion mass is much smaller than the magnetic energy scale, i.e. m << p|eB|, the low-energy sector is that with n = 0. Furthermore, if the spin contribution were absent (s = 0), the energy of the lowest Landau level would scale as√eB which is comparable with the next Landau level √3eB. Thus, the separation of a the low-energy sector with n = 0 would become unjustified. It can be proven [28] [30] that, the low-energy spectrum is directly related to the formation of a dynamical mass, which spontaneously breaks down chiral symmetry. For D = 3 + 1 the vacuum expectation value of the chiral condensate is (for m → 0)[30] :

h ¯ψψi ' −|eB|m 4πln Λ2 m2 ,

where Λ is an ultraviolet cutoff. The effect (and thus a generation of a fermion dynamical mass in function of the magnetic field) still remains even in the presence of the weakest attractive interaction between fermions, as can be shown taking into consideration low-energy effective model like the NJL model [28].

It was also observed that around the crossover region (T = 148 MeV, T = 153 MeV) the dependence of the chiral condensate on B is not monotonic and varies strongly with the temperature [27] (see Fig. 2.3). Furthermore, a lowering of the deconfining temperature Tc as eB is increased has been observed [31]. Denoting

by Σu,d(B, T )the renormalized lattice chiral condensate (with the renormalization

prescription used in ref. [27]) for the quark up and down respectively, we report (always from ref. [27]) a plot of the change of the condensate due to the magnetic field ∆Σu,d(B, T ) in function of the eB.

Figure 2.3: Continuum extrapolated lattice results for the change of the conden-sate a s function of B, taking in consideration six different temperatures.

Considering Eq. (2.8), the derivative of the magnetization with respect to B at vanishing magnetic field gives the magnetic susceptibility:

χB = ∂M ∂(eB) = − ∂2_f ∂(eB)2 B=0 (2.9)

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2.4. EFFECTS OF A MAGNETIC BACKGROUND 33 The thermal QCD medium is a strong paramagnet around and above the tran-sition region [7]. But, in turn, it seems to show a weakly diamagnetic behavior at low temperature T 6 100 MeV (seeing the results in [25]) as predicted by the HRG model. Indeed, the HRG model (see [20] and [32] for a brief introduc-tion of the model) predicts a diamagnetic behavior for temperature in the range T ∼ 100 − 150MeV,due to pions [8]. The model anyway has intrisic limitations and the results are reliable only at low temperature and low magnetic fields, which means T ' 130 − 150 MeV for B = 0, and a magnetic field such that eB < 0.6GeV2 _{is fullfilled (as explained in [8]). Furthermore in [7] was observed}

that its prediction breaks down already at T ' 120 MeV, since it over-estimates the pions diamagnetic contribution (see fig. 2.4) . In another study [6], it was pointed out, taking into consideration a wide temperature range (above and be-low Tc' 150 − 160MeV), that , within errors, the magnetic susceptibility seems

to vanish for T as low as 100 MeV. Thus, it is still not clear if the HRG prediction, about the diamagnetic behavior for low temperature, reflects the actual behavior of the medium.

Figure 2.4: Magnetic susceptibility of QCD as a function of the temperature. Results with dfferent lattice approaches are collected. A comparison with HRG model can be done in the low temperature region [25].

2.4.1 HRG model prediction

In the HRG model the free energy of the system is approximated in the thermody-namic limit, i.e. V → ∞, using the partition function of a gas of non-interacting free hadrons and resonances. Consequently, the free energy density of the model has the following expression :

f =X

h

dh· fh(eB, T, mh, qh/e, sh, gh), (2.10)

where fh is the contribution of the h-th hadron, which depends on the

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(mass, spin, gyromagnetic ratio and electic charge). Each hadron contributes in eq. (2.10) with some multiplicity dh . All the hadrons from pions up to Σ0baryon

are taken in account. The gyromagnetic factor is usually set to gh = 2for charged

particles, considering universal tree-level result. Thus, we are assuming that each hadron is a point-like particle.

Assuming that the interaction between hadrons is neglible, we can reconstruct the free energy, considering that each hadron is a relativistic free particle in the presence of a magnetic field B pointing in the positive z direction. Thus, labeling with k the Landau levels and taking in account that sh

z is a conserved quantity,

the energy levels of a charged hadron are given by :

E(pz, k, sz)h =

q p2

z+ m2h+ 2qhB(k + 1/2 − sz), (2.11)

and for neutral particle :

E0,h(p) =

q

|~p|2_{+ m}2

h. (2.12)

At arbitrary finite temperature, the free energy density can be written,for a given charged hadron h, as (see [8] and references cited in the paper):

fc,h = ∓ X sz ∞ X k=0 qB 2π Z _dp z 2π E(pz, k, sz)h 2 + T log(1 ± e −E(pz,k,sz)/T₎ , (2.13)

where the lower sign correpsonds to bosons and the upper to fermions (only fermions with s = 1/2 will be taken in account) and qB

2π is the degeneracy

multi-plicity of Landau levels. On the other hand, for neutral hadron :

fn,h= ∓ X sz Z d3p (2π)3 E_0,h(p) 2 + T log(1 ± e −E0(p)/T₎ . (2.14)

The total free energy density has a thermal and a vacuum component :

fvac = f |T =0 ftherm = f − fvac. (2.15)

Thus, to calculate the susceptibility it suffices to determine the thermal part of the free energy. Then, for a given hadron with spin s we have [7]:

fs(T ) − fs(0) = (−1)2s+1 qB 8π2 Z ∞ 0 dt t2e −m2_t 1 2 sinh(qBt)· h Θ3 φs, e−1/(4T 2_t) − 1i· s X sz=−s e−2qBszt_, (2.16)

where the ellptic Θ-function results from summing over Matsubara-frequencies and the factor 2 sinh(qBt) from summing over the angular momenta k. The first

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2.4. EFFECTS OF A MAGNETIC BACKGROUND 35 argument of the Θ-function is φs = 0 for s = 0, 1 and φs = π/2 for s = 1/2.

Thus, the susceptibility is :

χs(T ) = −∂ 2_[f s(T ) − fs(0)] ∂(eB)2 B=0 = (−1)2s 1 4π2(q/e) 2_· · Z ∞ 0 dt t e −m2_t/T2 Θ3 φs, e−1/(4t) − 1 ωs , (2.17) where : ω0 = −1/12, ω1/2 = 1/3, ω1 = 7/4

Due to the behavior of the Θ-function the susceptibility is negative for s = 0 and positive for s = 1/2, 1, then, charged pions contributes to diamagnetism, in turn, protons and charged ρ-mesons to paramagnetism. Such behavior can be understood qualitatively for boson channels, indeed,since the thermal part of the free energy contains exponential dependence on the effective mass m(B) of the hadron at non-zero magnetic field, for scalar hadrons the effective mass increases as growing B, instead, it decreases for vector hadrons. Once determined the susceptibility for each hadron channel, the total susceptibility will be :

χ(T ) =X

h

dh · χsh(T ) (2.18)

In the following some plot of the free energy density and susceptibility at low temperature T ∼ 90 MeV will be shown :

0 _{5 · 10}

−2

_0.1

_0.15

_0.2

0 · 10

0

2 · 10

−6

4 · 10

−6

6 · 10

−6

eB (GeV

2

)

∆

f

(GeV

4

)

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0 _{5 · 10}

−2

_0.1

_0.15

_0.2

−2 · 10

−3

−2 · 10

−3

−1 · 10

−3

−5 · 10

−4

0 · 10

0

eB (GeV

2

)

∂ 2 [f s (T )− fs (0)] ∂ (eB ) 2

Figure 2.5: From the last figure we can deduce the diamagnetic behavior for T ∼ 90MeV .

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Chapter 3 Magnetic background on the

lattice

In this chapter we discuss the numerical setup for the discretization of Nf = 2 + 1

quark flavors in a magnetic background at first, discussing issues and advantages of our setup.

3.1 Numerical setup

The external magnetic field is introduced inside the QCD Lagrangian through quark covariant derivatives Dµ = ∂µ+ igAaµTa+ iqfAµ, where Aµ is the abelian

gauge four potential and qf the electric charge of a given flavor. Since, as

ex-plained in chapter 2, on the lattice SU(3) covariant derivatives are written in terms of the link variables Uµ(n) (µ is the direction and n the position), in the

same way the introduction of an abelian gauge field amounts to add a U(1) phase: Uµ(n) → uµ(n)Uµ(n).

The euclidean action as usual will be of the type in Eq.(2.24). Then, SG[U ] is

equal to Eq.(2.28) (the tree level improved Symanzik action), instead, the Dirac operator is the following:

D_n,lf = amfδn,l+ 4 X ν=1 ηn,ν 2 uν;nU (2) ν (n)δn,l−ˆν − un−ˆν;νUν(2)†(n − ˆν)δn,l+ˆν , (3.1) where U(2)

µ (n) are the two-time stout-improved links defined in chapter 2.

Con-sidering the case of a constant external magnetic field ~B = (0, 0, B), that is pointing in the z direction, it is necessary to introduce a vector potential statis-fying ~B = ~∇ × ~A, for example:

Aµ= ( ~A, At) = (0, Bx, 0, 0). (3.2)

All the possible choises of the vector potential are connected by an abelian gauge transformation. Usually a lattice setup is characterized by a toroidal grid, thus

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periodic boundary conditions are imposed in the spatial directions for quantum

fields:

ψ(t, 1, j, k) = ψ(t, Ni, j, k)

Uµ(t, 1, j, k) = Uµ(t, Ni, j, k)

∀i = x, y, z.

Therefore the magnetic flux cannot be arbitrary. Indeed, taking in consideration the circulation of Aµalong any closed path, lying in the x−y plane and enclosing

an arbitrary region of area A, we have, thanks to Stoke’s theorem : I

Aµdxµ = AB.

In turn, due to the toroidal geometry, we can choose as enclosed surface also the complementary region of area LxLy− A (where Li = aNi is the spatial extention

of the lattice in the i-th direction) obtaining: I

Aµdxµ= (A − LxLy)B.

Such ambiguity can be solved admitting discontinuities in Aµ somewhere on the

lattice. Furthermore, one has to guarantee that the ambiguity is not visible by charged particles moving on the torus, thus, the phase factor taken by the charged particle moving along the closed path needs to be defined unambiguously:

exp(iqBA) = exp(iqB(A − LxLy)).

Then, the magnetic flux is quantized in terms of the area A of the system in the plane orthogonal to the external field [33]:

qB · A = 2πb b ∈ Z, (3.3)

where q is the elementary electric charge of the particles which populate the system (in our case q = |e|/3). In turn, A depends on the spatial lattice extensions orthogonal to the magnetic field and on the lattice spacing a as A = NxNya2.

The continuum vector potential (3.2) can be represented by the following complex phases uν ∈ U (1) :

u(q)_y = eia2qBnx_, _u(q)

x |nx=Nx = e

−ia2_qN

xBny (3.4)

and uν(n) = 1 otherwise, where nµ ∈ {0, · · · , Nµ}. In this way a constant

magnetic flux a2_B _{goes trhough all plaquettes in the xy plane, apart from a}

plaquette located at nx = Nx and ny = Ny where the magnetic flux is (1 −

NxNy)a2B, thus, the total flux through the torus is vanishing as it could be for a

closed surface. In the continuum limit, that corresponds to a uniform magnetic field plus a Dirac string piercing the lattice on a point. Anyway, if B is quantized, the string becomes invisible to all particles carrying electric charges multiple of q, then, the phase of the singular plaquette becomes equivalent modulo 2π, to that of all other plaquettes and the field is uniform. The quantization of the magnetic flux leads to an awkward issue: it is not possible to define differentiation with respect to eB or b.

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3.2. FREE-ENERGY MEASUREMENTS 39

3.2 Free-energy measurements

Since, due to the magnetic flux quantization, the magnetization is not accesible directly, several methods have been developed and used in literature to circum-vent this problem, some of them are based on the evaluation of finte free energy differences,for determining both thermodynamics and magnetic properties.

• Half-half method : instead of the uniform and quantized magnetic field, it is possible to work with an inhomogeneous field with zero flux, e.g. a field such that it is positive in one half and negative in the other half of the lattice. With this setup the field strenght is now a continuous variable, thus derivatives of lnZ with respect to eB are well defined. Even if this is quite advantageous, such setup has some drawbacks, for example the discontinuities in the magnetic field on the lattice may enhance finite volume effects.

• Thermodynamics integration method : due to Eq. (3.3), the derivatives of lnZ with respect to b are unphysical quantity, in turn they can still be measured for any real value of b, and their integral over b between integer values gives the change ∆f between these two fluxes. This method has been adopted in [6] to extrapolate magnetic susceptibility measurements. A detailed description of the procedure adopted will be given in the next section.

In this thesis we will prupose anew method based on the Jarzynski theorem [9] to evaluate finite free energy differences and thus evalueating magnetic properties of the medium. The method is new in the field of LQCD simulations, and it was adopted for the first time, for pure lattice gauge theory simulations, in ref. [34]. The Jarzynski’s relation will be largely discussed in the next chapter.

3.2.1 Thermodynamics integration method

The procedure adopted in [6] permits to determine free energy differences f(b2) −

f (b1) (with b1 and b2 integrers) by integrating the derivative of an interpolating

function which is a suitable extension of f(b) to real values of b :

f (b2) − f (b1) =

Z b2

b1

∂f (b)

∂b db. (3.5)

The method is well defined and feasible. Thus, the problem of determining the ratio of two partition function is reduced to evalueate ∂f (b)

∂b . Anyway, it is

neces-sary to point out that this quantity has not direct relation with the magnetization of the medium, since it is just the derivative of an interpolating function of f(b). Since ∆f contains ultraviolet (UV) divergences, and they also have B-dependent terms, once ∆f has been determined, it is necessary to remove them. In refs. [7],[25] has been proven that these divergenses are all inside vacuum contributions, and that no other B-dependent divergences are present. Taking into account

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these facts, the renormalization prescription consists in subtracting the vacuum cntributions:

∆fR(B, T ) = ∆f (B, T ) − ∆f (B, 0), (3.6)

assuming that both terms on the right hand side are computed at the same lattice spacing a.

In turn, for small fields , the behavior of ∆fR gives direct acces to the magnetic

susceptibility :

∆fR = −

Z ~

M · d ~B, (3.7)

thus, assuming that the medium is linear,homogeneous and isotropic, one has ~

M = ˜χ ~B/µ0, where ˜χ is the magnetic susceptibility in SI units, and then it is

possible to write : ∆fR= − ˜ χ µ0 Z ~ B · d ~B = − χ˜ 2µ0 B2. (3.8)

The integrand in eq. (3.5) has to be evaluate following the expression (in function of the Dirac operator) :

M = a4∂f (b) ∂b = 1 4NtNs3 X f =u,d,s T r ∂D f ∂b D f −1 , (3.9)

with Ns and Nt are the spatial and temporal lattice extents in lattice spacing

units, and the average is done over a few hundred thermalized configurations for each value of b, adopting a noisy estimator and averaging over a given number of Z2 random vectors for each single measure (it is worth noting that adopting

noisy estimators does not lead to any bias, see Appendix B). To determine the integral in (3.5) between two given quanta one can determine M on a grid of a fixed number of points ∈ [b − 1, b[ (notice that,for integer values of b M vanishes since f(b) has a local minimum for those values, see [5] ), for constructing an interpolating function. Once this has been done for several quanta b , assuming that a4_{∆f (b) = c}

2b2+ O(b4) holds , a best fit with the following function can be

done:

a4(f (b) − f (b − 1)) ' c2(2b − 1). (3.10)

Then , after renormalization, once obtained c2,R , the magnetic susceptibility is :

˜ χ = −|e| 2_µ 0c 18~π2N 4 sc2,R, (3.11)

in SI units, instead, in natural units we have :

ˆ

χ = −N

4 sc2,R

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Chapter 4 Jarzynski theorem

This thesis is focused on an application in full QCD simulations, of a non-equilibrium method, used usually in numerical statistical mechanics or in Chemi-cal Physics simulations. Algorithms based on Jarzynski’s relation can be applied to lattice gauge theories formulated on an Euclidean lattice in a straightforward way, as it will be shown. This chapter will be structured as follows: at first a demonstration of Jarzynski’s relation for Markovian process will be given, after-wards several sections will be dedicated to a description of properties and issues of free energy estimators derived from Jarzynsk’s relation.

4.1 Non equilibrium work relation for

Marko-vian process

The Jarzynski’s relation was proven as a statistical mechanics relation for classical and quantum mechanical systems in ref. [9] and in ref. [35] where the following identity was stated:

e−βW_{= e}−β∆F

, (4.1)

where h· · · i is an average over all accessible microstates of the system in a given thermal equilibrium state, and β is the inverse temperature (in natural units : ~ = c = kB = 1). The identity relates fluctuations in the work W performed

during a thermodinamic process, in which a system is driven away from equilib-rium, to a free energy difference ∆F between two equilibrium states of the system realized respectively at an initial time tin and at a final time tf in. From now on,

we will schematize non-equilibrium evolutions with a Markovian non-equilibrium processes. Thus, since we are dealing with a system which follows a Markovian evolution, even during an evolution process which leads the system to the final state (usually not a thermal equilibrium state) through non-equilibrium interme-diate state, each updating of the system, from a starting configuration A to a final configuration B at a given n-th step of the Markovian chain, is driven by a transition matrix, which varies at each step. At each n-th step we will associate a discrete time tn, setting t0 = tin and (assuming that the Markovian chain is

constituted by N total steps )tN = tf in. Consequently, the average h· · · i is taken

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over a large number of realizations of a non-quilibrium evolution from the initial to the final ensemble.

Let us consider a system, with microscopic degrees of freedom denoted all to-gether with φ, whose dynamics is described by an Hamiltonian H, function of the degrees of freedom φ, which depends on a set fo parameters. In a thermal equilibrium with a large heatbath at temperature T = 1/β the partition function related to the system is :

Z =X

φ

e−βH, (4.2)

where Pφ denotes the multiple sum over all values that each microscopic degrees

can take. Then, the statistical distribution of φ configurations in thermodynamic equilibrium is given by :

π[φ] = 1 Ze

−βH

, (4.3)

obviuosly normalized to 1. Denoting the conditional probability that the system undergoes a transition from a configuration φ to a configuration φ0 _{as P [φ → φ}0_]_,

the sum of such probabilities over all possible distinct final configurations is one :

X

φ0

P [φ → φ0] = 1 (4.4)

Therefore, we will assume that the system satisfies the detailed-balance condition: π[φ]P [φ → φ0] = π[φ0]P [φ0 → φ] (4.5) Generally the Boltzmann distribution π will depend on the temperature and in the parameters appearing on the Hamiltonian, thus, denoting the latters collectively as λ, it is possible to emphasize such dependence by writing the configuration distribution as πλ and the related partition function and conditional probabilities

with Zλ and Pλ respectively. Let us introduce a time dependence for the

param-eters λ. Starting from an initial time t = tin = 0 in which the parameters of

the Hamiltonian take certain values, and the system is in a thermal equilibrium state at the temperature T , the parameters are modified as function of time, according to some specified protocol, λ(t) , and are driven to certain final values λ(tf in = 1) over an interval of time ∆t = tf in − tin = 1 (such time interval is a

matter of conventions, indeed, we can assume any finite value which simply esta-bilishes the "duration" of a single Monte Carlo evolution from the intial state to the final one, consequently the following discussion will be totally independent of such value). It’s important to notice (for discussing properties and consequences of the Jarzynski’s relation) that the system during such evolution will not be in thermal equilibrium. Now, we discretize ∆t in N sub-intervals of the same width τ = ∆t/N and define tn = tin+ nτ for integer values of n ranging from 0 to N;

correspondingly, the protocol λ(t) mentioned above can be discretized by a func-tion taking the value λ(tn) = λ(n) for tn 6 t < tn+1(the simplest protocol with

constant switching velocity is λ(n) = λ0+ (λf in− λ0)_Nn). We denote a possible

"trajectory" in the space of field configurations, or alternatively, a mapping be-tween the time interval ∆t and the configuration space of the system as φ(t). Thus

Jarzynski's theorem and free energy estimates for strongly interacting matter

Introduction

Contents

List of Figures

List of Tables

Chapter 1

Lattice QCD

1.1

Non-Abelian gauge fields theories

1.2

Chiral symmetry

1.3

Path integral formalism and the Monte-Carlo

approach

1.4

Lattice regularization

1.4.1

Free fermion field on the lattice : the doubling

prob-lem

1.4.2

Wilson fermions

1.4.3

Staggered fermions

1.4.4

Gauge fields on the lattice

1.4.5

Rooting trick

1.5

Improved action

1.6

Continuum limit

Chapter 2

QCD Thermodynamics in a

magnetic background: overview

2.1

QCD Thermodynamics : lattice results at

zero chemical potential

2.2

Z

symmetry

2.3

Nature of QCD transition as a function of

the quark masses

2.4

Effects of a magnetic background

2.4.1

HRG model prediction

0

5 · 10

0.1

0.15

0.2

0 · 10

2 · 10

4 · 10

6 · 10

eB (GeV

)

∆

f

(GeV

)

0

5 · 10

0.1

0.15

0.2

−2 · 10

−2 · 10

−1 · 10

−5 · 10

0 · 10

eB (GeV

)

Chapter 3

Magnetic background on the

lattice

3.1

Numerical setup

3.2

_{5 · 10}

_0.1

_0.15

_0.2

_{5 · 10}

_0.1

_0.15

_0.2