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Dipartimento di Fisica

Corso di Laurea Magistrale in Fisica

QCD Phase Diagram in the Presence

of a Magnetic Background Field

Candidato:

Relatore:

Floriano Manigrasso

Prof. Massimo D’Elia

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Contents iv

1 QCD and LQCD 6

1.1 The QCD Action in the Continuum Limit . . . 6

1.2 Chiral Symmetry of QCD . . . 8

1.3 The Running of αS . . . 11

1.4 Path Integral Formulation of QCD . . . 14

1.4.1 Path Integral Formulation of QM . . . 14

1.4.2 Path Integral and Statistical Mechanics . . . 16

1.4.3 Path Integral Formulation of QFTs . . . 18

1.5 QCD on the Lattice . . . 19

1.6 Dynamical Fermions . . . 20

1.6.1 “Na¨ıve” Discretization of Fermions and the Doubling Problem . . . 20

1.6.2 Nielsen-Ninomyia Theorem . . . 23

1.6.3 Wilson Fermions . . . 24

1.6.4 Staggered Fermions (Kogut-Susskind) . . . 25

1.6.4.1 Staggered Action . . . 25

1.6.4.2 Continuum limit . . . 26

1.6.5 “The Fourth-root Trick” and the Non-locality of the Staggered Action . . . 28

1.6.6 Fermionic Gauge-Invariant Action on the Lattice . . . 29

1.6.6.1 Stout smearing . . . 30

1.7 SU(3) Gauge Action on the Lattice . . . 32

1.7.1 Symanzik Improvement Procedure . . . 33

1.8 The Continuum Limit of Lattice QCD . . . 34

2 QCD in a Background Magnetic Field 37 2.1 Magnetic Background Field in Lattice QCD . . . 37

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2.2 QCD Thermodynamics on the Lattice at Finite B . . . 40

2.2.1 Thermodynamic Observables . . . 41

2.2.2 The QCD Phase Diagram for B = 0 . . . 42

2.3 Magnetic Catalysis . . . 45

2.3.1 Magnetic Catalysis from Dimensional Reduction D = 3 + 1 → 1 + 1 . . . 45

2.3.2 Magnetic Catalysis in LQCD . . . 47

2.4 Towards Inverse Magnetic Catalysis . . . 51

2.5 Inverse Magnetic Catalysis in LQCD . . . 54

2.5.1 Sea Effect . . . 55

2.6 Possible Causes of Discrepancy . . . 56

2.7 Summary and Motivations . . . 59

3 Lines of Constant Physics (LCP) 61 3.1 Scale Setting . . . 61

3.1.1 Scale Setting Methods Based on the Gradient Flow . . 62

3.2 Pion Mass Determination . . . 64

3.3 Lines of Constant Physics . . . 69

4 Numerical Results 72 4.1 Simulation Details . . . 73

4.2 Lines of Constant Magnetic Field . . . 73

4.3 Observables and their Renormalization . . . 75

4.4 Dependence of the Pseudocritical Temperature on the Pion Mass at B = 0 . . . 76

4.5 Pseudocritical Temperature . . . 77

4.5.1 Chiral Condensate . . . 78

4.5.2 Chiral Susceptibility . . . 81

4.5.3 Polyakov Loop . . . 83

4.6 Behaviour of the Chiral Condensate . . . 84

4.7 Curvature of the Pseudocritical Line . . . 88

5 Conclusions and Perspectives 91 Bibliography 100 A Useful Formulas 106 A.1 Gaussian Integrals over Grassmann Variables . . . 106

A.2 Gaussian Integral . . . 107

A.3 Computation of Kmnαβ−1 . . . 107

B Algorithms 109 B.1 Dynamical Monte Carlo . . . 109

B.1.1 Metropolis-Hastings Algorithm . . . 111

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B.3 Hybrid Monte Carlo . . . 112 B.4 HMC in QCD with Staggered Fermions . . . 113 C An Example of Scale Setting Procedure by Means of the

w0 Scale 115

D Simulation Parameters 117

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Quantum Chromodynamics (QCD) is the quantum theory that describes strong interactions. It is a non-Abelian gauge theory with gauge group SU (3). The quanta of the SU (3) gauge field are called gluons, while the matter content of the theory consists of quarks. The latter are fermions which differentiate into six distinct flavours, each of them assigned to the fundamental representation of the local gauge group.

One of the key features of QCD, which differentiates it from Quantum Electro-dynamics (QED), is asymptotic freedom: in the high energy regime the theory is weakly coupled, hence it is possible to apply perturbative techniques. How-ever, many interesting aspects of QCD emerge in the low energy limit, where non-perturbative effects are dominant. One of the most important property that Quantum Chromodynamics exhibits in this energy regime is color con-finement : at sufficiently low temperature and density, the asymptotic states of QCD are color singlets. Even though there is not yet an analytic proof of color confinement, there is strong evidence for this phenomenon in numerical simulations. Moreover, in the massless limit, the QCD action has the chiral symmetry which gets spontaneously broken by the formation of the chiral condensate. For light quarks an approximate symmetry is still present and gets spontaneously broken for temperatures below the pseudocritical point T ∼ 155 MeV, leading to the appearance of pseudo-Goldstone bosons in the hadronic spectrum (i.e. pions).

In addition to strong interactions, quarks are also subject to electroweak in-teractions (EW). In particular, if the strength of the electromagnetic field is comparable with the QCD scale ΛQCD ≈ 200 MeV, these contributions are of

the same order of pure strong dynamics effect.

Recently, great interest has been shown in the subject of QCD in the presence of strong magnetic background fields. These physical systems have import-ant phenomenological relevance: in non-central heavy ion collision experiments conducted at LHC and RHIC, charged ion beams produce strong magnetic im-pulses which can reach the value of √eB ∼ 0.5 GeV [1]. Furthermore, strong

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magnetic fields (√eB ∼ 2 GeV) may have been produced during the cosmo-logical electroweak phase transition [2–4] of the early universe. Finally, some neutron stars called magnetars posses strong magnetic dipole fields (√eB ∼ 1 MeV) [5, 6].

The study of these physical phenomena, which manifest many of their promin-ent features in the low energy regime, cannot be approached through perturb-ative techniques. In this framework, lattice simulations represent a research tool of primary importance: Monte Carlo simulations allow us to carry out an in-depth analysis of non-perturbative strong dynamics effects from first prin-ciples.

In Fig. 0.1 the phase diagram of QCD (temperature vs. baryon chemical potential vs. magnetic field) is shown [7]. Our study has been carried out at zero baryon chemical potential µB = 0. In this physical situation, a crossover

divides the phase diagram in two distinct regions, namely the Quark-Gluon Plasma (QGP) and the Hadron Gas phases. The QGP phase, which consists of deconfined strong-interacting quarks and gluons, occurs in extreme con-ditions of temperature and/or density. In absence of background magnetic fields, the pseudo-critical temperature at which deconfinement and chiral res-toration take place is Tc ∼ 155 MeV [8]. There are indications [9] that the

crossover persists up to eB ∼ 10 GeV2, while, according to the same study,

for magnetic fields which exceed this threshold, a first order phase transition might take place.

Figure 0.1: Phase diagram of QCD: temperature T vs Baryon chemical po-tential vs. Magnetic field. Figure taken from Ref. [7]

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In order to completely understand the properties of strong interactions in the presence of background magnetic fields, it is now crucial to characterize the dependence of the pseudo-critical crossover temperature (i.e. the temper-ature at which the chiral restoration and deconfinement take place) on the magnetic field itself.

The first study which attempted to find this dependence [10] made use of unim-proved staggered quarks, larger than physical quarks masses (Mπ0 ∼ 200 − 500

MeV) and lattice spacing of the order of 0.3 fm. It showed an increase of the pseudocritical temperature and of the strength of the transition as a function of the magnetic field. This result was expected on the basis of the well known phenomenon of magnetic catalysis, i.e. the enhancement of the magnitude of the chiral condensate if it is already present at zero magnetic field, or the induction of chiral symmetry breaking and the appearance of a condensate when the symmetry is intact at B = 0 (this is also referred to as dynamical symmetry breaking by a magnetic field). Indeed, the increasing trend of the chiral condensate may naively suggest that the pseudocritical point move to-wards higher temperatures with increasing B.

A second study [8], performed with improved gauge and smeared fermionic action, three flavours (two of them degenerate, Nf = 2 + 1), continuum

ex-trapolations and physical masses, showed that the pseudocritical temperature TC(B) is a decreasing function of the magnetic field strength. In Fig. 0.2

the pseudo-critical temperature is shown as a function of the magnetic field, as obtained in Ref. [8]; the regions of phenomenological relevance have been highlighted.

In addition, the authors of [8] found a complex dependence of the condensate on B and T: even though the magnetic field causes an enhancement of the chiral condensate in the low temperature regime, around Tc it decreases as a

function of B, i.e. we have the inverse magnetic catalysis phenomenon (IMC). Recent studies [11–13] indicate that the origin of IMC lies on the confining properties of QCD rather than on the chiral ones. In particular, the authors of [12] found that even at low temperatures, at which the chiral condensate shows no sign of inverse magnetic catalysis, the string tension is suppressed in the presence of a magnetic background field. This result supports the idea that the modification of the confining properties may play a dominant role in the mechanism that determines the trend of TC(B), which may be described

in terms of deconfinement catalysis. This point of view is also shared by recent calculations in holographic models [13].

The purpose of our work is to clarify the incompatibility of the results men-tioned above in order to deeply understand the nature of the inverse magnetic catalysis phenomenon. In particular, we can ascribe the origin of this discrep-ancy to three possible causes:

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Figure 0.2: Pseudocritical temperature T (MeV) as a function of the magnetic field eB (GeV). The two lines, red and blue, correspond to different definition of the pseudocritical temperature which are respectively the inflection points of the renormalized chiral condensate and the strange quark number suscept-ibility. The region of the phase diagram relevant for RHIC and LHC have been underlined. Figure taken from Ref. [8]

◦ presence of lattice artefacts in the study [10], which consisted of coarse lattice spacing (a ∼ 0.3 fm), an unimproved gauge action and the absence of smearing of the fermion action;

◦ different number of flavours;

◦ different masses: contrary to the first study, the work [8] which analyses the QCD phase diagram in presence of background magnetic fields made use of physical quarks masses;

In order to deeply understand the reason behind this discrepancy, an inter-esting direction to investigate is deconfinement catalysis. If the effect of the magnetic field on the confining properties is the leading mechanism which de-termines the trend of TC(B), than a consequence would be that, in contrast

with the conjecture made in Ref. [8], the origin of the discrepancy between the results mentioned above lies on the presence of lattice artefacts due to a coarse lattice spacing. If this is the case, we expect that the decreasing be-haviour of the pseudocritical temperature with respect to the magnetic field strength is preserved for larger than physical quark masses. More specifically, the following questions are still open:

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1. what is the fate of the decreasing behaviour of the pseudocritical temper-ature with improved gauge action for larger than physical quark masses? 2. Is the presence of the inverse magnetic catalysis phenomenon dependent

on the quark masses?

3. How does the variation in the pseudocritical temperature as a function of the magnetic field strength depend on the pion mass?

In order to address these issues, we have proceeded as follows: adopting the same numerical setup of [8] (i.e. staggered fermions, tree-level Symazik im-proved gauge action, smeared fermionic action and Nf = 2 + 1 flavours),

we have performed numerical simulations on a lattice with a lattice spacing a ∼ 0.15 − 0.2 MeV near the transition, at different values of the pion mass Mπ0, ranging from ∼ 340 MeV up to 650 MeV.

As a first step we have determined the lines of constant physics (LCP): for ∼ 10 values of the bare gauge coupling β = 2Nc/g2 (which correspond to

different lattice spacings, and thus temperatures T = 1/(Nta)) we tuned the

bare quark masses mu = md and ms in a such a way to constrain the pion

mass Mπ0 to a given value.

Later, we performed simulations at three different values of the pion mass along the LCPs previously determined, in a temperature range around the crossover, for different values of the magnetic field. The measurement of dif-ferent observables that describe the crossover allow us to have independent determination of the pseudocritical temperature Tc. In particular, we

meas-ured the renormalized chiral condensate of the two light flavours h¯uui and h ¯ddi, that represents an approximate order parameter for the chiral transition. The measure of its inflection point allows us to have a first determination of Tc.

Then we also measured the Polyakov loop, that is a pure gauge quantity that represents an approximate order parameter for the confinement/deconfinement crossover. The comparison between the measure of Tcdetermined through the

Polyakov loop with the former one, allowed us to analyse the possibility of a splitting of deconfimenent and chiral transition, which has been predicted by some low energy effective models [14].

This work represents a significant step forward towards the comprehension of the origin of the IMC phenomenon, which may have important consequences on the formulation of low energy effective models and holographic models, valid for the description of QCD in presence of magnetic background fields in the crossover region. The continuum extrapolation of the results obtained in this work is now an issue of particular interest, which should be further investig-ated in future studies.

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QCD and LQCD

1.1

The QCD Action in the Continuum Limit

The matter content of QCD is made up of quarks, which are massive fermions described by Dirac 4-spinors. Let ψ be a vector composed by Nf fermion fields

(flavours) ψ ≡      ψ1 ψ2 .. . ψNf      ,

each of the fields ψi has also Dirac and colour indices. The free lagrangian for

the vector of fermionic fields ψ reads LF = ¯ψ (iγµ∂µ− m) ψ = N X i=1 ¯ ψi(iγµ∂µ− m) ψi

and it is invariant under global SU (3) transformations in colour space ( ψ 7→ ψ0 = U ψ ¯ ψ 7→ ¯ψ0 = ¯ψU†, with U = expiPN a=1θaTa 

, where Ta are the group generators Ta = λa/2,

and λa are the Gell-Mann matrices. The group SU (N ) has N2− 1 generators

that in the fundamental representation are traceless hermitian N ×N matrices. They obey the relation [Ta, Tb] = ifabcTc, with fabc structure constants of the

group, and the conventional normalization condition is Tr [TaTb] = δab/2.

The global SU (3) global symmetry has to be promoted to a local one, requiring the lagrangian to be invariant under

(

ψ 7→ ψ0 = U (x)ψ ¯

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with U = expiPNC

a=1θa(x)Ta



. In order to write down a lagrangian possess-ing the local SU (3)C invariance, we define NC2 − 1 gauge fields Aaµ(x) (being

NC = 3 the number of colours) and the parallel transport

W (Cy←x) ≡Pe igR

Cy←xAµ(z)dzµ

,

where Aµ(x) ≡ Aaµ(x)Ta, Cy←x is a curve joining the space-time points xµ to

yµ and P is the path ordering operator.

We notice thatW (Cy←x) ∈ SU (3) and it is a map W (Cy←x) : Vx 7→ Vy , where

Vx is the vectorial space defined on xµ. It obeys the following properties

1. W (∅) = 1;

2. W (C2◦ C1) =W (C2)W (C1);

3. W (−C) = W−1(C);

4. ψ(x) → ψ0(x) = U (x)ψ(x) ⇒W (Cy←x) →W0(Cy←x) =

= U (y)W (Cy←x)U−1(x),

where ∅ is the trivial path of zero length, C2◦ C1 represents the conjunction of

two paths and −C is the path C covered in the opposite direction. According to the last property we find that

˜

ψ(y) ≡W (Cy←x)ψ(x) ∈ Vy ⇒ ˜ψ0(y) = U (y) ˜ψ(y).

Moreover, from the transformation law for W (Cy←x) it is simple to show that

the gauge fields transform under the gauge group SU (3)C in the following way

Aµ(x) → A0µ(x) = U (x)Aµ(x)U†(x) +

i

g(∂µU (x))U

(x).

We are now able to define the covariant differential, i.e. the infinitesimal variation of the field ψ(x) which transforms under local SU (3)C like

Dψ(x) ≡W (Cx←x+dx)ψ(x + dx) − ψ(x) =W (Cx+dx←x)−1ψ(x + dx) − ψ(x) ≈ (1 + igAµ(x)dxµ)ψ(x + dx) − ψ(x) ≈ (∂µψ(x) + igAµ(x)ψ(x)) | {z } ≡Dµψ(x) dxµ,

where Dµψ(x) = (∂µ+ igAµ(x))ψ(x) is the covariant derivative. By definition

we have that the transformation law for the covariant derivative is D0µ = U (x)DµU (x)−1. At this point we are able to construct a gauge invariant action

for the fermion fields, following the so-called minimal prescription LF = ¯ψ(iγµ∂µ− m)ψ 7→ ¯ψ(i /D − m)ψ.

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It consists of the free lagrangian for the fermion fields plus a coupling between two quarks and a gauge field, i.e. a gluon. We have now to find the kinetic terms for the gauge fields Aaµ. Having introduced the covariant derivative, it is a straightforward task finding a gauge invariant dimension four operator, starting from the gauge fields and their derivatives up to the 2nd order. Indeed, we notice that

[Dµ, Dν] ψ(x) = ig{∂µAν(x) − ∂νAµ(x) + ig[Aµ, Aν]}ψ ≡ igFµνψ, (1.1)

where the tensor Fµν(x) = Fµνa (x)Ta has been introduced. Its transformation

law under local SU (3)C is

Fµν → Fµν0 = 1 ig D 0 µ, D 0 ν = U (x)Fµν(x)U−1(x),

so we can build a gauge invariant dimension four term by taking the trace of FµνFµν. In conclusion, the QCD lagrangian reads

LQCD = 1 2Tr[FµνF µν] + ¯ψ(i /D − m)ψ = −1 4F a µνFaµν+ Nf X f =1 ¯ ψf(i /D − m)ψf. (1.2)

At this point we would like to add some final remarks. In the first place we could have been tempted to add a mass term for the gauge field ∝ m2Aa

µAµa, but

this would break the gauge symmetry. Furthermore, we can simply build gauge invariant terms with higher powers of the gauge fields, like Tr[(FµνFµν)2].

However, such operators would render the theory non-renormalizable. In ad-dition, we can add for instance a term ∝ θµνρσF

ρσFµν that is a dimension

four gauge invariant operator, known as θ-term. Even though QCD

lag-rangian would be renormalizable in the presence of such a term, it breaks CP symmetry. Since experiments do not indicate any CP violation in the QCD sector of the Standard Model, the coupling θ is severely constrained from above, |θ| < 10−10 [15]. For further details about the strong CP problem see Ref. [16]. Finally, the pure gauge term in the QCD lagrangian include also cubic and quartic vertices as shown if Fig. 1.1, which characterize non abelian Yang-Mills theories.

1.2

Chiral Symmetry of QCD

Let us consider the QCD fermionic lagrangian for L flavours of quarks L = L X f =0 ¯ ψf(i /D − mf)ψf. (1.3)

The field ψ can be expressed in the chirality eigenstates basis ψR,L=

(1 ∓ γ5)

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Figure 1.1: Cubic and quartic gluon self-interaction vertices. Curled lines represent gluonic fields.

with γ5 ≡ −iγ0γ1γ2γ3. In the Dirac basis γ5 reads

γ5 =  0 −1 −1 0  ,

where we have introduced the right PR = (1 − γ5)/2 and left PL = (1 + γ5)/2

projectors. The matrix γ5 satisfies the relations

γ52 = 1 and {γ5, γµ} = 0,

while the projection operators obey the following properties

PR,L† = PR,L, PR,L2 = PR,L, PRPL = PLPR = 0, (PR+ PL) = 1.

We notice that, while ψ can be straightforwardly decomposed in its chiral components ψ = ψR + ψL, for ¯ψ = ψ†γ0 we have to take into account the

presence of the extra γ matrix: ( ¯ψ R= ψ † Rγ0 = ψ†P † Rγ 0 = ψγ0P L = ¯ψPL ⇒ ¯ψL= ¯ψPR.

We are now ready to rewrite the lagrangian (1.3) in terms of the chiral com-ponents of the fermion field

◦ ¯ψψ = ψ¯R+ ¯ψL (ψR+ ψL) = ¯ψ(PL + PR)(PR + PL)ψ = ¯ψPLPLψ +

¯

ψPRPRψ = ¯ψRψL+ ¯ψLψR ⇒ the fermionic mass term couples

compon-ents with different chirality of the field ψ;

◦ ¯ψγµψ = ¯ψ(PL+ PR)γµ(PL+ PR)ψ = ¯ψ[PLγµPL+ PLγµPR+ PRγµPL+

+PRγµPR]ψ = ¯ψRγµψR+ ¯ψLγµψL ⇒ the covariant derivative term in

the fermionic action does not couple different chiralities. In conclusion, we can rewrite the lagrangian (1.3) as

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where M =    m1 · · · 0 . .. 0 · · · mL    and ψ =    ψ1 .. . ψL   .

In the limit M → 0 we observe that the lagrangian (1.4) is invariant under the transformations of the group U (L)L× U (L)R ≈ U (1)L× U (1)R× SU (L)L×

SU (L)R, known as the chiral group.

We want now to analyse more in detail the symmetry groups composing the chiral group. The action of U (1)L× U (1)R on the fermion field is

(

ψL → ψL0 = eiαLψL

ψR → ψR0 = eiαRψR.

(1.5)

We can define now the U (1) vectorial and axial groups:

◦ U (1)V (vectorial) if αL = αR = α ⇒ ψ = ψL+ ψR→ ψ0 = eiαψ;

◦ U (1)A (axial) if αL= −αR= α ⇒ ψ = ψL+ ψR → ψ0 = eiβγ5ψ.

It is straightforward to show that U (1)L× U (1)R = U (1)V × U (1)A, i.e. any

transformation of U (1)L× U (1)Rcan be written as a composition of axial and

vectorial transformations.

We perform now the same analysis for the subgroup SU (L)L× SU (L)Rof the

chiral group. The action of this subgroup on the fermion field is (

ψL → ψL0 = VLψL

ψR → ψR0 = VRψR,

with VL,R ∈ SU (L). It is now possible to introduce vectorial and axial

trans-formations, defined as follows

◦ SU (L)A (axial) transformations do not constitute a group. They are

defined by VL= VR† = A, A ∈ SU (L) ( ψL→ ψL0 = AψL ψR→ ψR0 = A †ψ R;

◦ SU (L)V (vectorial) transformations are defined by VR = VL = V , V ∈

SU (L).

Even though SU (L)Ais not a group, we rewrite the chiral groupG improperly

as

G = SU(L)V × SU (L)A× U (1)A× U (1)V,

meaning that any chiral transformation can be interpreted as a composition of axial and vectorial transformations.

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The group SU (L)V × SU (L)A with 2(L2 − 1) generators, is spontaneously

broken down to SU (L)V. As a consequence of the Goldstone theorem, there

are (L2−1) massless bosons (Goldstone bosons), which for L = 2 are the three pions π0, π+, π−, while for L = 3 they constitute the pseudoscalar mesons

octet (Fig. 1.2). Taking into account the quark mass term M one obtains K0 K+ π+ η π0 K− K¯0 π−

Figure 1.2: Pseudoscalar mesons octet, pseudo Goldstone bosons associated with the spontaneous symmetry breaking ¯G = SU(3)V × SU (3)A → H =¯

SU (3)V.

L2 − 1 pseudo Goldstone bosons, i.e. they acquire mass. For L = 2, chiral

perturbation theory χPT gives a prediction for the pion mass in terms of quark masses:

Mπ2 = B(mu+ md). (1.6)

The mass formula (1.6) tells us that pion mass squared is linear in the quark masses. The chiral condensate is an approximate order parameter for the spontaneous symmetry breaking of the chiral symmetry

hΩ| ¯ψfψf|Ωi = − hΩ|

∂LF

mf

|Ωi .

For the sake of completeness we must highlight that at the quantum level the chiral group reduces to G = SU(L)V × SU (L)A× U (1)V, since U (1)A is

anomalous. For further details about the QCD chiral symmetry see Ref. [17].

1.3

The Running of α

S

Up to this point, we have analysed the QCD action at the classical level. In order to study strong interactions at the quantum level one has to face the problem that, because of gauge invariance, the canonical quantization approach fails with Yang-Mills theories. The quantization of QCD can be performed by means of the so-called Faddeev-Popov procedure that is based on the path integral formulation of a quantum field theory. The latter will be introduced in Sec. 1.4.

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We want now to analyse some of the fundamental aspects related to quantum chromodynamics renormalization. The QCD lagrangian (1.2)

LQCD =

1

2Tr[FµνF

µν] + ¯ψ(i /D − m)ψ,

where Fµν = ∂µAν(x) − ∂νAµ(x) + ig[Aµ, Aν] and Dµ= ∂µ+ igAµ(x) contains

only a dimensionless coupling g. Together with gauge invariance, this assures that the theory is renormalizable. We report here some of the main results of

QCD αs(Mz) = 0.1181 ± 0.0011

pp –> jets

e.w. precision fits (N3LO)

0.1 0.2 0.3

α

s

(Q

2

)

1 10 100

Q [GeV]

Heavy Quarkonia (NLO)

e+e– jets & shapes (res. NNLO)

DIS jets (NLO)

April 2016 τ decays (N3LO) 1000 (NLO pp –> tt(NNLO) ) (–)

Figure 1.3: The experimental determination of the running coupling of αs =

g2

s/(4π)2. Figure taken from [18].

perturbative renormalization performed using the MS renormalization scheme together with dimensional regularization. For further details about this topic see Ref. [19]. The explicit non perturbative renormalization of observables of interest in this work is going to be analysed in Sec. 4.3, where the insertion of an additional static magnetic field in the QCD lattice lagrangian will be studied.

Given the relation g = ZggR between renormalized and bare couplings, the

renormalization constant Zg reads

Zg = 1 − g2 R (4π)2 NC 6 (11 − 2Nf) 1 ε + O(g 4 R), (1.7)

with ε = (4 − D)/2 and D is the number of space-time dimensions. We define now the Gell-Mann Low beta function

β ≡ µd˜gR(µ)

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where ˜gR ≡ gRµε. Using the independence of the bare parameter g on the

renormalization scale µ, µ∂g/∂µ = 0, we obtain β(gR) = −

εZggR ∂

∂gR (ZggR)

. (1.8)

Substituting now (1.7) into (1.8) and sending ε to zero, we finally get β(gR) = − g3R (4π)2  11NC − 2Nf 3  + O(gR5) = −g3Rβ0+ O(g5R). (1.9)

In particular, for NC = 3 and Nf = 6 one obtains β0 = 21/(48π2). In a

per-turbative regime, for gR  1, Eq. (1.9) leads to a fundamental consequence:

given the fact that β0 > 0 we have obtained that

dgR(µ)

dµ < 0.

Thus, for µ → ∞ ⇒ gR → 0. This is a peculiar property of QCD called

asymptotic freedom since for asymptotically large energies (i.e. small dis-tances) the theory becomes decoupled. This phenomenon was discovered by David Gross and Frank Wilczek [20], and independently by David Politzer [21] in 1973: for this work all three were awarded the Nobel Prize in Physics in 2004 [22].

Finally, it is now possible to solve the differential equation (1.8) defining the beta function Z µ2 µ1 dµ µ = Z gR(µ2) gR(µ1) dgR β(gR) µ2 = µ1exp Z gR(µ2) gR(µ1) dgR β(gR) ! . Substituting now the 1-loop beta function (1.9)

µ exp  − 1 2β0g2R(µ)  ≡ ΛQCD.

ΛQCD is a scheme-dependent quantity with the dimension of a mass (the

ap-pearance of a dimensional parameter such as ΛQCD is sometimes referred to

as dimensional transmutation). We have now the one-loop expression for the renormalized coupling constant as a function of the renormalization scale µ

g2R(µ) = 1

2β0ln (µ/ΛQCD)

. (1.10)

Notice that, for µ = ΛQCD the 1-loop running coupling diverges. However,

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is weakly coupled.

For µ ∼ ΛQCD the coupling is gR(µ) ∼ 1 and the theory becomes strongly

coupled. In this energy regime QCD exhibits one of most important prop-erty, namely colour confinement : only colour singlet states are present in the hadronic spectrum, i.e. singlets of SU (3)C. Even though there is not yet

an analytic proof of color confinement, there is strong evidence for this phe-nomenon in numerical simulations.

The experimental determination of αR = gR2/(4π)2 has been reported in Fig.

1.3.

1.4

Path Integral Formulation of QCD

1.4.1

Path Integral Formulation of QM

In this section we will briefly introduce the main ideas behind the construction of the Feynman path integral and its applications in quantum field theories. Consider a non relativistic particle whose hamiltonian operator is

ˆ

H = ˆT + ˆV = pˆ

2

2m + ˆV.

Suppose now that the particle is observed at the position qi at time ti and qf

at time qf and that it is not observed in the time interval in between ti and

tf. From a quantum mechanically point of view it does not make sense saying

that the particle goes from qi to qf along a given path q(t), but we have to

take into account all the possible trajectories it could follow. In particular we are interested in calculating the amplitude

hqftf|e−

i ˆHt

~ |qitii , (1.11)

where e−i ˆHt~ is the time evolution operator.

The idea behind Feynman’s path integral approach is easy to formulate: we describe the time evolution for infinitesimal small times ∆t = (tf−ti)/N , with

N  1. We will use the identity e−i ˆHt~ =

h e−i ˆH∆t~

iN

and in the end we will take the limit ∆t → 0 whilst keeping t = ∆tN fixed. Thus, we can use the Baker-Campbell-Hausdorff formula (1.42) that allows us to write e−i ˆH∆t~ = e− i ˆT ∆t ~ e− i ˆV∆t ~ + O(∆t2).

Using this expansion in powers of ∆t we can rewrite the amplitude (1.11) in as hqftf| h e−i ˆH∆t~ iN |qitii ≈ hqftf|1e− i ˆT ∆t ~ e− i ˆV∆t ~ 1...1e− i ˆT ∆t ~ e− i ˆV∆t ~ |qitii ,

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where the identity operator is 1 = Z dqn Z dpn|qni hqn|pni hpn| .

The states hqn| and hpn| represent a complete set of position and momentum

eigenstates respectively. Moreover the index n = 1, . . . , N tracks the position at which the unit operator is inserted. Now, making use of the momentum eigenfunctions hq|pi = eiqp~ /

2π~ it is straightforward to show that hqftf|e−i ˆHt/~|qitii ≈ Z N −1 Y n=1 qn=qf q0=qi dqn N Y n=1 dpn 2π~e −i∆t ~ PN −1 n=0  V (qn)+T (pn+1)−pn+1qn+1−qn∆t  .

The next step consists in performing the limit N → ∞ wit N ∆t = t fixed. Thus, the sum over t and the time derivative read

∆t N −1 X n=0 7→ Z t 0 dt0, qn+1− qn ∆t 7→ ∂t0q|t0=tn ≡ ˙q|t0=tn.

Finally, we define the integration measure as

lim N →∞ Z N −1 Y n=1 qn=qf,q0=qi dqn N Y n=1 dpn 2π~ ≡ Z q(t)=qf q(0)=qi DpDq. In conclusion, we obtain hqf|e−i ˆHt/~|qii = = Z q(t)=qf q(0)=qi DpDq exp i ~ Z t 0 dt0(p ˙q − H(p, q))  = Z q(t)=qf q(0)=qi Dq exp  −i ~ Z t 0 dt0V (q)  Z Dp exp  −i ~ Z t 0 dt0 p 2 2m − p ˙q  = Z q(t)=qf q(0)=qi Dq exp i ~ Z t 0 dt0L(q, ˙q)  = Z q(t)=qf q(0)=qi Dq exp i ~ S  , (1.12) where we have integrated out the impulses using Eq. (A.5) so, in the last line

Dq = lim N →∞  N m it2π~ N/2 N −1 Y n=1 dqn.

Notice that the integrand includes the classical action of the particle in a given potential V(q) and we take care of quantum mechanics principles by

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integrating over all paths and not just on the subset of solutions of the clas-sical equations of motion. Furthermore, we made use of the definition of the lagrangian L(q, ˙q) = [p ˙q − H(p, q)], with ˙q = ∂H(p, q)/∂p. The previous dis-cussion, where we have exposed the example of a particle which moves in one dimension under the influence of a given potential V, is valid in general for any quantum mechanical system whose hamiltonian is quadratic in the momenta.

1.4.2

Path Integral and Statistical Mechanics

The partition function of a quantum system reads

Z ≡ Trhe−β ˆHi, (1.13)

where the trace is carried out over all possible configurations of the system and β = 1/T . For a 1d quantum system, such as the particle in a given potential V(q) described in the previous section, we can express the state of the system in terms of position eigenvalues. As a consequence

Z = Z

dq hq|e−β ˆH|qi .

We notice that the integrand in the last expression is very similar to the amplitude in Eq. (1.11) where qf = qi and t 7→ −iβ~. In order to recover the

partition function Z from the path integral in Eq. (1.12), we can follow the steps:

◦ carry out a Wick rotation, τ ≡ it;

◦ restrict the integration interval to τ ∈ [0, β~]; ◦ require the periodicity over τ .

We now perform these steps, obtaining hq|e−β ˆH|qi = Z q(t)=q q(0)=q Dq exp i ~ Z t 0 dt0L(q, ˙q) 

Wick rotation τ ≡it

−−−−−−−−−−→ Z q(t)=q q(0)=q Dq exp  −1 ~ Z τ 0 dτ0LE(q, ∂q/∂τ )  τ ∈[0,β~] −−−−→ Z q(t)=q q(0)=q Dq exp  −1 ~ Z β~ 0 dτ0LE(q, ∂q/∂τ )  ,

where LE is the Wick rotated (i.e. euclidean) lagrangian, that is obtained

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we are now ready to write down the partition function in the path integral formalism Z = Z dq Z q(t)=q q(0)=q Dq exp  −1 ~ Z β~ 0 dτ0LE(q, ∂q/∂τ )  . (1.14)

The expectation value for an observable O may be written as

hOi ≡ 1 ZTr h Oe−β ˆHi= 1 Z Z dq Z q(t)=q q(0)=q DqO exp  −1 ~ Z β~ 0 dτ0LE(q, ∂q/∂τ )  . (1.15) Moreover, if we rewrite hOi using a complete set of hamiltonian eigenstates |ni, we get hOi = P ne −Enβhn|O|ni e−Enβ .

Thus, in the limit β → ∞ (i.e. T → 0), one obtains

hOi−−−→ hΩ|O|Ωi ,β→∞ (1.16)

where Ω is the ground state of the system. The meaning of Eq. (1.16) is that, in the limit of infinite Euclidean time, the expectation value of an observable O reduces to the expected value of O on the ground state.

The path integral formulation of the partition function allows us to perform numerical simulations of quantum theories. To compute the expectation value hOi one may indeed replace the time interval [0, β]1 with a 1-dimensional

lattice of N + 1 points and spacing a = β/N . So, the discretized path integral reads:

hOi ≈ 1

Z Z

dx0dx1...dxN −1Oe−SE({xi})/~, (1.17)

where we do not integrate over the variable xN, which is constrained by the

periodic boundary condition x0 = xN. The Euclidean discretized action SE

reads SE = Z dτ0LE ≈ a N −1 X i=0 1 2m  xi+1− xi a 2 + V (xi).

Now the problem has been reduced to the numerical evaluation of the expect-ation value of the observable O with respect to the probability distribution

P ({xi}) =

1

Z exp (−SE({xi})/~), which in the continuum limit gives

P [q]Dq = exp (−SE[q]/~)Dq R Dq exp (−SE[q]/~)

.

1

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The discretized path integral in Eq. (1.17) allows us to perform numerical evaluation of the expectation value of an observable O using Monte Carlo methods.

1.4.3

Path Integral Formulation of QFTs

In order to perform numerical evaluation of expectation values in QCD ther-modynamics by means of Monte Carlo techniques, the path integral formu-lation of quantum field theories is necessary. In particular, in QFTs one is often concerned with the calculation expectation values of operators that are functionals of the field variables. Thus, we are interested in expressing the partition function in Eq. (1.13) in the path integral formulation. The trace has to be performed over a complete set of Fock space states {|ni}. At this step we work with bosonic field variables φ(x, τ ). The procedure that leads to the construction of the path integral formulation of the quantum parti-tion funcparti-tion follows the same steps performed in Sec. 1.4.1 in the case of non-relativistic quantum mechanics. However, in QFT the equivalent of the identity operator (1.4.1) is made up by coherent states, i.e. eigenstates of the Fock state operators a and a† which obey the commutation relation

a, a† = 1.

We will report now the expression for the partition function in the path integral formulation in the euclidean time, which is analogous with Eq. (1.14):

Z = Z Dφ(x, τ ) exp  −1 ~ Z β~ 0 dτ Z d3x L (φ(x, τ ), ∂φ(x, τ ))  , where Dφ(x, τ ) ∼ Q

x,τdφ(x, τ ). Moreover, the field variable must obey the

periodical boundary condition φ(x, β) = φ(x, 0). For further details about the specific construction of the functional field integral see Ref. [23].

Performing the same steps we can express the partition function for a fermionic theory in the path integral formulation. It reads

Z = Z D ¯ψ(x, τ )Dψ(x, τ ) exp  − Z β~ 0 dτ Z d3x ¯ψ(x, τ ) /∂E + m ψ(x, τ )  , where the fields ¯ψ(x, τ ) and ψ(x, τ ) are Grassmann variables which obey an-tiperiodic boundary conditions

¯

ψ(x, ~β) = − ¯ψ(x, 0) ψ(x, ~β) = −ψ(x, 0) .

The expectation value of a field operator O in a bosonic theory reads hO(φ)i = R Dφ(x, τ )O(φ)e −1 ~ Rβ~ 0 dτR dx L(φ,∂φ) R Dφ(x, τ )e−1 ~ Rβ~ 0 dτR dx L(φ,∂φ) .

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The last expression can be evaluated numerically by replacing the Euclidean space-time with a four-dimensional lattice of size N3

s × Nt with lattice spacing

a hOi ≈ 1 Z Z Y i∈sites dφi ! Oe−SE({φi}).

As stated in Sec. 1.4.2, this kind of integrals can be carried out by means of Monte Carlo methods. Further details about the specific discretization of field variables will be analysed in the following sections.

1.5

QCD on the Lattice

Lattice gauge theories are based on the path integral formulation of QFTs. Let us consider the Euclidean QCD lagrangian

LE QCD = 1 4F a µνF a µν+ Nf X f =1 ¯ ψf(γµEDµ+ mf)ψf.

The vacuum expectation values of an observable O[A, ¯ψ, ψ] can be expressed in the path integral form

hOi = 1

Z Z

DU D ¯ψDψO[U, ¯ψ, ψ]e−SQCD, (1.18)

where periodic and antiperiodic boundary conditions are imposed respectively for the gluon and fermion fields

Aµ(x, T ) = Aµ(x, 0)

ψ(x, T ) = −ψ(x, 0).

Because of gauge invariance, the path integral of Eq. (1.18) is ill-defined in the continuum limit and gauge fixing by means of the Fadeev-Popov procedure is required. While in the continuum QCD the gluon fields lie in algebra of SU (3), in the lattice formulation of QCD the gauge dynamical variables lie in the SU (3) group itself, as we will see in the following section. Now, since SU (3) is a compact group, the lattice QCD functional integral is finite and gauge fixing is not necessary. For this reason, we have not included the gauge fixing term in the above discussion.

After discretization, hOi may be evaluated by numerical simulations. Before analysing the specific discretization procedure we followed in this work, we notice that, since we cannot directly simulate Grassman variables, the fermonic fields are integrated out in Eq. (1.18) using Eq. (A.2)

hOi = 1 Z Z DU D ¯ψDψO[U, ¯ψ, ψ]e−SG− Rβ 0 dτR d 3x ¯ψM ψ = 1 Z Z DU O[U, M−1] det M e−SG ,

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where M is the fermionic operator D + m/ f. The computation of det M is

a time expensive task: in the early days of lattice QCD, when computational resources were significantly slower than now, simulation were performed in the so-called quenched approximation which consists in setting det M = 1.

1.6

Dynamical Fermions

Lattice QCD simulations with dynamical fermions require to deal with the fermionic determinant det Mf, with Mf = ( /D + mf) an n × n complex matrix

with n = Nsites × NDirac × Ncolours. Even in the case of small lattices, with

Nx = Ny = Nz = 16 and Nt = 6, we would have to compute determinants of

square matrices of size ∼ 3 · 106. For this reason, it is preferred a stochastic approach rather than the direct computation of this quantity: we introduce an (or more than one) auxiliary bosonic field Φ, called pseudofermion, such that det Mf = det Mf−1 −1 = Z DΦ†DΦ e−Φ†Mf−1Φ.

Moreover, in order to correctly estimate the fermionic determinant by means of the pseudofermion method, it is necessary to work with an hermitian and positive definite matrix. In the case of two degenerate flavours, we can simply build a matrix with this properties as MuMd, being Mu and Md respectively

the Dirac euclidean matrix for the up and down quarks, which coincide in the degenerate case. In terms of pseudofermionic fields the determinant for the light quarks reads

detMuMd= detMl2 = detM =

Z

DΦ†DΦ e−Φ†M−1Φ.

For further details about the pseudofermionic method and for a general ana-lysis of Monte Carlo algorithms we made use of in the present work, see App. B. In the following sections we have reported some possible discretizations of the fermionic action, which is a fundamental aspect of unquenched simulations.

1.6.1

“Na¨ıve” Discretization of Fermions and the Doubling

Problem

While the lattice formulation of the free scalar theory poses no problems, regarding the free fermion field difficulties arise. In the following section the main issues about the lattice formulation of the free fermion field are going to be analyzed: for further details, see Ref. [24, 25].

The action for a free 1/2 spin field ψα in Minkovski space reads

SFψ, ¯ψ =

Z

(27)

where ¯ψ ≡ ψ†γ0. Here with Greek indices we refer to the four components of

the Dirac field ψ. Gamma matrices γµ satisfy the anticommutation relations

{γµ, γν} = 2ηµν, (1.20)

so they generate a matrix representation of the Clifford algebra. Here ηµν

is the flat space time metric with the “mostly minus” sign convention ηµν = diag(1, −1, −1, −1).

The classical equations of motion for the fields ψ and ¯ψ can be derived by means of an independent variation of the action (1.19):

(iγαβµ ∂µ− M δαβ)ψβ = (i /∂αβ− M δαβ)ψβ = 0. (1.21)

In order to write the free fermion action in the Euclidean space we must follow the prescriptions:

x = (x0, x) → ˜x = ˜x0, ˜x = (−ixE4, xE), (1.22)

k = (k0, k) → ˜k =˜k0, ˜k= (ikE4, −kE), (1.23)

where k is the four-momentum. From Eq. (1.22) follows that ∂0 → ˜∂0 = ∂ ˜x0 =

i∂x

E4. We are now ready to write down the euclidean action S

E F[ψ, ¯ψ]: iSF[ψ, ¯ψ] → i Z d4x ¯˜ψα(˜x)(iγµ∂˜µ− M )αβψβ(˜x) = Z d4xEψ¯α(xE)(−γ0∂E4− iγi∂Ei− M )αβψβ(xE) = − Z d4xEψ¯α(xE)(γEµ∂Eµ+ M )αβψβ(xE) ≡ − Z d4xELFE[ψ, ¯ψ, ∂ψ] = −S E F[ψ, ¯ψ], (1.24)

where we defined γµE(µ = 1, . . . , 4) as γ4E = γ0,γiE = −iγi. The anticommuta-tion relaanticommuta-tions (1.20) now become

µE, γνE} = 2δµν. (1.25)

The Lorentz invariance SO(3,1) of the lagrangian densityLF[ψ, ¯ψ, ∂ψ] has now

been replaced by the invariance under O(4) transformations ofLFE[ψ, ¯ψ, ∂ψ]. We want ow to write the euclidean action SE

F on a space-time isotropic lattice.

Given the lattice spacing a, the fields ψ, ¯ψ live on the lattice sites na. From now on, unless otherwise stated, we will drop the superscript E referring to fields defined in the Euclidean space.

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following replacements M → 1 a ˆ M , ψα(x) → 1 a(3/2)ψˆα(n), ¯ ψα(x) → 1 a(3/2) ¯ ˆ ψα(n), ∂µψα(x) → 1 a(5/2)∂ˆµψˆα(n),

where ˆ∂µ is the symmetric lattice derivative

ˆ ∂µψˆα(n) = 1 2  ˆψ α(n + ˆµ) − ˆψα(n − ˆµ)  .

The lattice version of the euclidean fermionic action (1.24) now reads SF = Z d4x ¯ψα(x)(γµ∂µ+ M )αβψβ(x) → X n " X µ ˆ ¯ ψα(n)γµαβ 1 2  ˆψ β(n + ˆµ) − ˆψβ(n − ˆµ)  +ψ¯ˆα(n)δαβM ˆˆψβ(n) # = X n,m " ¯ ˆ ψα(n) X µ γµαβ1 2(δm,n+ˆµ− δm,n−ˆµ) + δαβδmn ˆ M ! ˆ ψβ(m) # = X m,n ¯ ˆ ψα(n)Kmnαβψˆβ(m), (1.26) where we have defined Kαβ

mn=  P µγµαβ 1 2(δm,n+ˆµ− δm,n−ˆµ) + δ αβδ mnMˆ  . The action (1.26) is now explicitily written as quadratic form in the lattice fields, hence the generating functional Z[η, ¯η] has the form of a gaussian integral which we are able to perform analytically

Z[η, ¯η] = Z DψD ˆ¯ˆ ψe−SFh ˆψ,ψ¯ˆ i +P n[¯ηα(n) ˆψα(n)+ψ¯ˆα(n)ηα(n)] , (1.27)

where ηα(n) and ¯ηα(n) are the Grassman-valued source with respect to the

generating functional has to be differentiated in order to get the n-point cor-relation functions.

By using the formula (A.4) we obtain

Z[η, ¯η] = det K exp X m,n ¯ ηα(n) Kmnαβ −1 ηβ(m) ! . (1.28)

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The two-point function is defined as h ˆψα(n)ψ¯ˆβ(m)i = R DψD ¯¯ˆ ψ ˆψ α(n)ψ¯ˆβ(m)e −SE Fh ˆψ, ¯ ˆ ψ i R DψD ¯¯ˆ ψ e−SE Fh ˆψ, ¯ ˆ ψ i = 1 Z[0, 0] ∂Z[η, ¯η] ∂ ¯ηα(n) ∂ηβ(m) = Kmnαβ−1.

The explicit computation of Kmnαβ−1 has been carried out in App. A.3. Mak-ing use of Eq. (A.9) we obtain

hψα(n) ¯ψβ(m)i = lim a→0 1 a3 h ˆψα(n) ¯ ˆ ψβ(m)i = lim a→0 Z π/4 −π/4 d4k (2π)4 h M − iP µγ µ 1 asin (kµa) i αβ M2+P µ 1 a2 sin 2(k µa) eik·(x−y).

At this point we require that in the continuum limit this expression coincides with the propagator of a free fermion field

Z π/4 −π/4 d4k (2π)4 h M − iP µγ µ 1 asin (kµa) i αβ M2+P µ 1 a2 sin 2(k µa) eik·(x−y) a→0−−→ Z ∞ −∞ d4p (2π)4 eik·(x−y) M + i/p = Z ∞ −∞ d4p (2π)4 M − i/p M2 + /p2e ik·(x−y) (1.29)

and the condition is fulfilled ⇐⇒ 1asin pµa ∼ pµ. However, besides the correct

form of the propagator, there are other fifteen contribution (i.e. doublers) to the continuum limit of the free particle propagator, which are pure lattice artefacts. In fact, in the limit a → 0, given the fact that sin(π + ˜pµa) =

−˜pµa + O(a2), also the contribution coming from pµa → ±π give a finite

result for (1.29) . We can define the first Brillouin zone (BZ) that ranges from pµ = −π/a to pmu = π/a: the modes at the boundaries of the BZ give raise

to the doublers.

1.6.2

Nielsen-Ninomyia Theorem

The appearance of extra degrees of freedom in the continuum limit, whose propagator coincides (apart from sign) with the one of the free fermion, is a consequence of the particular type of discretization we performed. Specific-ally, this issue has been entitled fermion doubling problem and it arises as a byproduct of the symmetric lattice derivative employment. Indeed, if we used the forward discretization of the derivative ˆ∂µFψˆα(n) = ˆψα(n + ˆµ) − ˆψ(n), we

would obtain ˜ Kαβ(ˆk) = ˆM δαβ+ 2iX µ γµαβsinˆkµ/2  eiˆkµ/2,

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which leads to the correct continuum limit, but it is not hermitian. For this reason, the use of the forward lattice derivative would make the discretized Hamiltonian non-unitary.

As it has been shown, the simple solution to the doubling problem which we have analysed (i.e. the implementation of a new derivative discretization) introduces some drawbacks. This is a specific consequence of the more general Nielsen-Ninomiya no-go theorem [26], which states that it is not possible to formulate a lattice fermionic action that:

1. is local;

2. lacks of doublers;

3. exhibits chiral symmetry in the massless limit (i.e. {D, γ5} = 0);

4. has the right continuum limit.

Possible implementations of the fermions on the lattice lack of at least one of these properties.

1.6.3

Wilson Fermions

A possible solution to the doubling problem is the extension of the fermionic action with a new term

SF(W )= SF− r 2 X µ ¯ ˆ ψ(n) ˆ ˆψ(n) | {z } ∆S(W ) ,

where SF is the action (1.26) and the discretized d’Alambert operator is

ˆ

ψ(n) =X

µ

h ˆψ(n + ˆµ) + ˆψ(n − ˆµ) − 2 ˆψ(n)i ,

with ˆ = a2 = a2∂µ∂µ. Now, we can rewrite the new term in the action as

∆S(W )= −r 2 X µ,m,n ¯ ˆ ψ(n) [δm,n+ˆµ+ δm,n−ˆµ− 2δn,m] ˆψ(m).

Following the same procedure used in Sec. 1.6.1, we derive Kαβ(W )(n, m) = ( ˆM + 4r)δm,n− 1 2 X µ [(r − γµ)αβδm,n+ˆµ+ (r + γµ)αβδm,n−ˆµ] .

In particular, as expected on the basis of the Nielsen-Ninomiya theorem cited in the last section, this lattice formulation for fermion fields lacks a funda-mental property that is chiral symmetry in the massless limit. For further

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details on Wilson fermions see Refs. [24, 25].

For our purposes it is sufficient to report the expression for the two point function, showing its peculiar features:

hψα(x) ¯ψβ(y)i = lim a→0 Z π/a −π/a d4p (2π)4 −iγµ1asin (pµa) + M (p)  αβ P µ 1 a2 sin 2(p µa) + M (p)2 eip·(x−y), where M (p) = M +2ra P µsin 2(p

µa/2). We want this expression to recover the

correct continuum limit for the free spin 1/2 fermion propagator reported in the right hand side of Eq. (1.29).

We notice that, for pµa → 0 and a → 0, M (p) = M + 2ra Pµ ap2µ

2 + O ((pµa/2)4)

pµa→0, a→0

−−−−−−−→ M . However, if pµa → ±π with a → 0, the limit

for M (p) is M (p) = M + 2r a X µ 1 − ˜pµa 2 2 + O((˜pµa)4) ! pµa→±π, a→0 −−−−−−−−→ ∞.

Taken this result into account, we have verified that the fifteen extra contri-butions to the two point fermionic correlation function, which arise with the naive discretization shown in Sec. 1.6.1, have been removed. However, the doublers removal procedure introduced by Wilson leads to an action that does not preserve chiral symmetry for finite lattice spacing a in the massless limit, as stated before.

1.6.4

Staggered Fermions (Kogut-Susskind)

1.6.4.1 Staggered Action

As an alternative to Wilson fermions, one may introduce the Kogut-Susskind action [27], commonly known as staggered action. First consider the na¨ıve action (1.26) SF = 1 2 X µ,n h¯ ˆ ψ(n)γµψ(n + ˆˆ µ) −ψ(n)γ¯ˆ µψ(n − ˆˆ µ) i + ˆMX n ¯ ˆ ψ(n) ˆψ(n)

and define the unitary transformation ˆ ψ(n) = ζ(n)χ(n) ≡ γn1 1 γ n2 2 γ n3 3 γ n4 4 χ(n), (1.30) ¯ ˆ ψ(n) = ψˆ†(n)ζ(n)†= ¯χ(n)γn4 4 γ n3 3 γ n2 2 γ n1 1 , (1.31)

where n ≡ (n1, n2, n3, n4) and we used the anticommutation relations (1.25).

We notice that the transformation ζ(n) verifies the property ζ†(n)γµζ(n + ˆµ) = ηµ(n)1.

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Making use of Eq. (1.25), it is straightforward to verify that ηµ(n) = (−1)n1+n2+n3+...+nµ−1.

The transformation (1.30) allows us to “spin diagonalize” the action that now reads SFstag[ ¯χ, χ] = X n,α,µ ηµ(n) ¯χα(n) ˆ∂µχα(n) + ˆM X n,α ¯ χα(n)χα(n).

By means of this procedure we obtained that the action written in terms of the new field variable χα(n) is diagonal in the Dirac space. In principle we can

discard 3 of the 4 Dirac components in order to reduce the number of doublers by a factor of 4. This can be simply done by dropping the subscript α in Eq. (1.6.4.1).

Doublers in the na¨ıve formulation arise because sin(pµa) vanishes when its

ar-gument approaches the values ±π. Starting from this observation, the Kogut-Susskind formulation of fermion fields on the lattice gets rid of the doubling problem by reducing the BZ. In particular, the reduction of the effective Bril-louin zone is obtained by doubling the lattice spacing.

Our aim is to write down a discretized action for the spin 1/2 fermion field which respects the following properties:

1. reproduces the correct continuum limit;

2. the fermionic degrees of freedom are distributed on the lattice in such a way that the lattice spacing for each of them is twice the fundamental lattice spacing;

Consider now the four-dimensional space-time lattice: we subdivide its volume into four-dimensional hypercubes of unit length, which has 24 sites. Now it is

possible to assign a given degree of freedom to each hypercube site. A spinor has 4 components in d = 4: we can define 4 different flavoured fermions is such a way that each of them has an effective lattice spacing that is twice the original a. The procedure which leads to the construction of the staggered fermion action, which has been rapidly sketched here, will be analysed more in detail in the following section, where the continuum limit of this theory has been studied.

1.6.4.2 Continuum limit

We want now to show that by following the discretization procedure exposed in the last section, we recover the correct continuum limit

SFstag → Z d4xX α,β,f ¯ ψfα(x)(γµ∂µ+ M )αβψ f β(x).

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Our presentation will follow the guideline of Ref. [24]. Consider now an hypercube with origin at ˆxµ = 2hµ where hµ = 0, 1, ..., Nµ/2 − 12, and

Nµ = (Nt, Nx, Ny, Nz) are the dimensions of the lattice. We can indicate

the coordinates of the 2d sites within the selected hypercube as n

µ= 2hµ+ sµ,

with sµ= 0, 1.

We notice that with this parametrization the staggered phases are independent of hµ

ηµ(n) = ηµ(2h + s) = ηµ(s)

and this suggests that we can relabel the fields χ(n) in the following way χs(h) ≡ χ(2h + s).

At this stage, hµ labels the space-time points of a lattice with lattice spacing

2a and sµ labels the 2d components of the new field χ. We can now define

2d/2 spin 1/2 fermion fields (i.e. 2d/2 tastes) by taking appropriate linear

combinations of the new field components. ˆ

ψfα(n) = N0

X

s

(Ts)αfχs(n), (1.32)

where f = 1, ..., 2(d/2) is the taste index and α = 1, ..., 2(d/2) is the index which

labels the spinor components. Moreover, in Eq. (1.32) we have defined Ts = γ1s1γ

s2

2 ...γ sd

d .

If we choose the normalization constant N0 appropriately, the action reads

SFstag =X

n,f

¯ ˆ

ψf(n)(γµ∂ˆµ+ ˆM ) ˆψf(n) + . . . , (1.33)

where ˆ∂µ is the discretized (in unit of 2a) derivative and the dots represent

terms which vanish in the continuum limit.

It can be shown that the propagator in momentum space reads

S(p) = P

µ

h

−i(γµ⊗ 1)1b sin(pµb) + 2b(γ5⊗ tµt5) sin2 pµ b 2 i + M01 ⊗ 1 P µ 4 b2sin2 p µb 2 + M 2 0 ,

where b = 2a is the lattice spacing for the tastes, M0 = 2M and we defined

tµ = γµ∗, t5 = γ5. Moreover, the tensor product notation is Dirac ⊗ taste. It is

now straightforward to verify that the na¨ıve continuum limit b → 0 leads to S(p)−−→b→0 −i

P

µ(γµ⊗ 1)pµ+ M01 ⊗ 1

p2+ M2 0

2Here we take the lattice dimensions N

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that is the fermion propagator describing four degenerate flavoured Dirac particles.

While chiral symmetry is completely lost in the Wilson fermion action, the lagrangian density in Eq. (1.33) in the massless regime is invariant under an abelian subgroup generated by γ5⊗ t5 of the original U (Nf)R× U (Nf)L chiral

group. However, in contrast with the continuum QCD, the chiral symmetry group of the staggered action acts in taste space rather than flavour. Under the action of this group the fields transform as follows

ψ(n) → eiα(γ5⊗t5)ψ(n); (1.34)

¯

ψ(n) → ψ(n)e¯ iα(γ5⊗t5). (1.35)

As a consequence, given the fact that the staggered fermion action preserves a non-trivial piece of the full chiral symmetry, it is adequate for studying the approximate spontaneous symmetry breaking in QCD with physical quark masses.

1.6.5

“The Fourth-root Trick” and the Non-locality of the

Staggered Action

In Sec. 1.6.4.1 we have shown that it is possible to build up a fermion action which get rids of 12 out of 15 doublers and preserves a subgroup of the original chiral symmetry group in the massless limit. We want now to eliminate the three residual doublers by means of a procedure that is known as the “the fourth-root trick”.

Consider the lagrangian for continuum QCD with Nf flavours of quarks. We

want to compute the partition function which reads Z =

Z

DU D ¯ψDψ e−SF−SG,

where SF and SG are respectively the fermionic and pure gauge actions. We

integrate out the fermions obtaining Z =

Z

DU det M e−SG.

If we discretize the fermion action through the staggered prescription, we have now to face the taste multiplicity issue. In order to reduce the unwanted four-fold degeneracy it is a widespread practice to take the fourth root of the fermion determinant. Thus, the most general partition function for three flavours (e.g. up, down and strange) with the staggered fermion discretizaton reads Z = Z DU e−SG Y f =u,d,s detMstf 1/4 .

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In accordance with the Nielsen-Ninomyia theorem enunciated in Sec. 1.6.2, it can be shown that the rooted action is non local [28].

This procedure is controversial: in the continuum limit it is expected that the eigenvalues of the fermion matrix are degenerate leading to the conclusion that also the four tastes are degenerate, so the fourth root is well defined. With a finite lattice spacing a, the eigenvalues can be not-degenerate and this implies that the root procedure introduces artefacts. The question whether the continuum limit of the staggered theory is smooth enough that all artifacts vanish with the lattice spacing going to zero has not yet been successfully addressed. However, we have to highlight that some of the most successful and precise results in contemporary lattice QCD have been obtained making use of the rooted staggered action. For further details about the issue see Ref. [29].

In the present work we made use of the staggered fermion discretization and the fourth root trick. In particular we have generated configurations with Nf = 2+1 flavours, two degenerate light quarks with different electric charges,

and an heavier strange quark.

1.6.6

Fermionic Gauge-Invariant Action on the Lattice

The QCD lagrangian density in Minkowski space reads L(x) = −1 4Fµν(x)F µν (x) +X f ¯ ψf(x)(i /D − mf)ψf(x), (1.36)

where the minimal prescription (i.e. /∂ → /D ≡ γµ(∂µ+ igAµ)) has been

ap-plied in order to make the action gauge invariant. The matter fields transform under G(x) ∈ SU (3)C as follows

ψ(x) → ψ(x)0 = G(x)ψ(x) (1.37)

¯

ψ(x) → ψ¯0(x) = ¯ψ(x)G−1(x), (1.38)

while the gauge field transforms as

Aµ(x) → G(x)Aµ(x)G−1(x) −

i

gG(x) ∂µG

−1(x) .

We want now to show how to make the lattice QCD action gauge invariant. The na¨ıve discretization of the fermion action (1.26) includes bilinear terms such as ¯ψ(n)ψ(n + ˆµ). In the continuum limit ¯ψ(x)ψ(y) transform as

¯

ψ(x)ψ(y) → ¯ψ(x)G−1(x)G(y)ψ(y)

in agreement with Eq. (1.37). At this point we can introduce the parallel transport

U (x, y) =PeigRxydx 0 µAµ(x0),

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where P is the path ordering operator. By definition (see Sec. 1.1) U(x, y) transforms in the following way under SU (3)C

U (x, y) → G(x)U (x, y)G−1(y).

Starting from this definition, we can write the following gauge invariant bilin-ear expression

¯

ψ(x)U (x, y)ψ(y) = ¯ψ(x)PeigRxydx 0

µAµ(x0)ψ(y). (1.39)

The infinitesimal parallel transport U (x, x + dx) ≈ 1 + igAµ(x)dxµis useful to

build up a gauge-invariant expression for the fermionic action on the lattice. Indeed, we can make the following substitutions in eq. (1.26)

¯ ˆ ψ(n)γµψ(n + ˆˆ µ) → ¯ ˆ ψ(n)γµUµ(n) ˆψ(n + ˆµ); ¯ ˆ ψ(n + ˆµ)γµψ(n) →ˆ ψ(n + ˆ¯ˆ µ)γµUµ†(n) ˆψ(n),

where we have defined Uµ(n) = U (n, n+aˆµ) ≈ 1+iagAµand U †

n,n+ˆµ= Un+ˆµ,n.

By performing the aforementioned substitutions, the na¨ıvely discretized action becomes SF[ ˆψ,ψ, U ] =¯ˆ =X n ¯ ˆ ψ(n) " ˆ m ˆψ(n) +1 2 X µ γµ  Uµ(n) ˆψ(n + ˆµ) − Uµ−1(n − ˆµ) ˆψ(n − ˆµ)  # .

It is crucial now to underline that the procedure that allowed us to build up a gauge invariant fermion action on the lattice in this section can be performed also on the staggered action (1.6.4.1), leading to

SFstag[χ, ¯χ, U ] = X n,α,µ ηµ(n) ¯χα(n)Uµ(n)χα(n + ˆµ) − Uµ†(n − ˆµ)χα(n − ˆµ) + + ˆMX n,α ¯ χα(n)χα(n). 1.6.6.1 Stout smearing

In order to determine the lines of constant physics, as we will see in the next chapters, the calculation of the pseudoscalar pion (π0) mass is needed.

Spec-troscopy measurements are frequently performed by means of correlators. In particular, so as to extract meson masses we will need to analyse the long distance behaviour of correlation functions. However, gauge theories exhibit violent short distance fluctuations of the gauge field. As a consequence, in order to improve the correlation signal, one can perform the smearing of the gauge links. In addition, it is worth highlighting that this procedure consti-tutes an improvement of the fermionic action reducing the discretization error.

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Indeed, smeared variables substitute the original gauge links Uµ(n) in the

lat-tice QCD action.

Gauge fields can be smeared either only in time slices or even in space and time. There are different smearing techniques, but in general they consist of substituting the gauge link with an average over short paths connecting the link’s endpoint. The operators and correlators are than built up on the smeared gauge configurations. It is a gauge covariant procedure and, as a con-sequence, no gauge fixing is needed. As long as the averages are local enough, the long distance correlation signal should not be affected in the continuum limit. We notice that for a non-abelian gauge theory of gauge group SU (3)C,

the product of links is not proportional to a group element. As a consequence one has to project this average to an SU (3) matrix.

We will now briefly expose the idea behind the stout smearing procedure [33], which has been used in this study. In the first place, the original gauge link Uµ(n) is replaced by

Uµ(n) → Uµ0(n) = e

iQµ(n)U

µ(n), (1.40)

where Qµ(n) is a traceless hermitian matrix constructed starting from the

staples surrounding a given gauge link Qµ(n) = i 2  Ω†µ(n) − Ωµ(n) − 1 3trΩ † µ(n) − Ωµ(n)   ; Ωµ(n) = X µ6=ν ρµνCµν(n) ! Uµ†(n); Cµν(n) = Uν(n)Uµ(n + ˆν)Uν†(n + ˆµ) + U † ν(n − ˆν)Uµ(n − ˆν)Uν(n − ˆν + ˆµ).

The real weight factors ρµν are tunable parameters. Our numerical set-up

Uν(n) n n+ ^ν Uµ(n + ^ν) n+ ^µ+ ^ν Uy ν(n + ^µ) n+ ^µ Uy ν(n − ^ν) Uµ(n − ^ν) Uν(n − ^ν + ^µ) n− ^ν+ ^µ n− ^ν

Figure 1.4: The simple link Uµ(n) (the dotted arrow) is replaced by an average

performed over the staples whose endpoints correspond with the link extrema. includes two times isotropic stout-smeared gauge links ρµν = ρ. Indeed, such

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smearing steps can be iterated. In this case, the smearing affects not just the neighbouring gauge links, but extends over increasingly larger distances. One of the main pros of stout smearing is that the transformed gauge link is differentiable with respect to the link variable. This particular feature of the stout smeared gauge links is essential in order to sample full QCD configura-tions by means of the Hybrid Monte Carlo HMC algorithm since it requires to numerically solve the equation of motion derived from the action SQCD. The

HMC algorithm will be discussed in detail in App. B.

Summarizing, stout smearing consists in averaging the links in a local, differ-entiable and gauge covariant way. It significantly reduces UV-fluctuations of the gauge field. Finally it may improve the correlation signal and reduce the discretization errors.

1.7

SU(3) Gauge Action on the Lattice

We want to construct a lattice action for gauge fields which approaches the continuum action SY M as a goes to zero. A possible way of building up gauge

invariant product of link variables is to choose a closed loop L and to take the trace L[U ] = Tr   Y (µ,n)∈L Uµ(n)  .

In particular, the simplest non-trivial closed loop is the so-called plaquette Uµν(n): it is the product of four link variables

Uµν(n) ≡ Uµ(n)Uν(n + ˆµ)Uµ†(n + ˆν)U † ν(n). n n+ ^µ n+ ^µ+ ^ν n+ ^ν Uµ(n) Uν(n + ^µ) Uy µ(n + ^ν) Uy ν(n)

Figure 1.5: The simplest non-trivial closed loop on the lattice, the so-called plaquette.

We can now define the Wilson gauge action [34]

S(W ) X n,µ<ν β  1 − 1 2NC tr(Uµν(n) + Uµν† (n))  . (1.41)

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It is straightforward to verify that this action reduces to the correct continuum form in the limit a → 0. Recalling that Uµ(n) ≡ eigaAµ(n) and making use of

the Baker-Campbell-Hausdorff formula

eAeB = eA+B+12[A,B]+..., (1.42) one obtains Uµν(n) = exp(igaAµ(n) + igaAµ(n + ˆµ) − g2a2 2 [Aµ(n), Aν(n + ˆµ)] − igaAµ(n + ˆν) − iagAν(n) − g2a2 2 [Aµ(n + ˆν), Aν(n)] + g 2a2 2 [Aν(n + ˆµ), Aµ(n + ˆν)] + g2a2 2 [Aµ(n), Aν(n)] + g 2a2 2 [Aµ(n), Aµ(n + ˆν)] + g2a2 2 [Aν(n + ˆµ), Aν(n)] + O(a 3)). (1.43)

We now perform a Taylor expansion

Aν(n + ˆµ) = Aν(n) + a∂µAν(n) + O(a2)

and taking into account only contributions up to O(a2) we obtain

Uµν(n) = exp iga2(∂µAν(n) − ∂νAµ(n) + ig[Aµ(n), Aν(n)]) + O(a3)



= exp(ig2a2Fµν(n) + O(a3)).

If we substitute now this expression in Eq. (1.41), then the Wilson action reads S(W ) = X n,µ<ν β  1 − 1 2NC tr 21 − g2a4Fµν(n)Fµν(n) + O(a5)   = X n,µ<ν βg 2a4 2NC tr [Fµν(n)Fµν(n)] + O(a5) a→0 −−→ β Z d4x g 2 2NC tr [Fµν(n)Fµν(n)] + O(a2),

which coincides with the QCD continuum action (1.36) in the Euclidean space if we set β = (2Nc/g2) = (6/g2).

1.7.1

Symanzik Improvement Procedure

In order to extract quantitative predictions from Lattice QCD, the continuum limit a → 0 of physical quantities is necessary. The computational cost of lattice simulations increases rapidly as a is reduced: for this reason we need to construct the lattice action in such a way to weaken the dependence of phys-ical quantities on the cutoff, in order to rapidly approach the continuum limit.

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Then one may perform simulations at a larger lattice spacing. Furthermore, reducing the dependence of physical observables on the cutoff would eventually allow us to achieve enhanced symmetries already at non-zero lattice spacings. We have already introduced stout smearing, which reduces discretization er-rors. A further improvement procedure has been introduced by Symanzik [30, 31]: it consists in adding operators of dimension greater than four, which vanish in the continuum limit, to the Wilson action. These operators allow to achieve a better convergence to the continuum by cancelling the leading O(a2) error terms in the Wilson action. Moreover, they must not modify the

symmetries of the regularized action in order not to spoil the continuum limit, but can suppress lattice artefacts and remove the discretization effects. In conclusion, this improvement project led to the formulation of the tree-level Symanzik action [32] SY M = − β 3 X i,µ6=ν  5 6U 1×1 i;µν − 1 12U 1×2 i;µν  , where U1×2

µν is a rectangular loop 1 × 2 originating from the site i and oriented

along the µν plane, while Uµν1×1 is the plaquette.

1.8

The Continuum Limit of Lattice QCD

In this section will discuss the continuum limit of the lattice theory described above. For further details see Ref. [24].

In Sec. 1.6 and 1.7 we have described how to build lattice QCD action, re-quiring that in the limit a → 0 it reduces to the correct continuum limit. In principle there are infinite possible choices for the discretized version of the QCD continuum action satisfying this requirement. We notice that there is no a priori reason why a lattice action possessing this property should reproduce a continuum limit corresponding to QCD.

Consider now a SU (3) Yang-Mills theory whose partition function on the lat-tice reads

Z = Z

DU e−SG[U ], (1.44)

where the only bare parameter is the gauge coupling g.

On the lattice we can measure only dimensionless quantities, e.g. the ratios of two observables which have the same mass dimension. Given a particle contained in the spectrum of the theory, we can measure its mass in lattice units ˆm(g). The relation between the latter and the mass m expressed in physical units is

m(a, g) = m(g)ˆ

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