Numerical challenges in the study of topology in high T QCD: a Multicanonical approach

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Università di Pisa

Tesi di Laurea Magistrale

Curriculum di Fisica Teorica

Numerical challenges in the study of

topology in high T QCD: a

Multicanonical approach

Author:

Antonino Todaro

Supervisor:

Prof. Massimo D’Elia

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iii

Introduction

Quantum Chromodynamics (QCD) is the theory that describes the strong interactions be-tween quarks and gluons, the constituents of hadronic matter, and is part of the Standard Model of elementary particles. For a long time, it has been thought that a predictive theory of strong interactions was impossible to be formulated within the framework of quantum field theories. The reason was that QFT can be treated analytically only in a perturbative regime, namely by expanding in powers of the coupling constant, thus it seems doomed to fail in approaching a set of phenomena that, by the same name, are characterized by the presence of a strong force. For this reason, in the early 60’s, physicists started to search for alternative theories, most of which based on symmetries, more suited for dealing with this challenge; in this regard, Lev Landau said about the fate of QFT [1]

"It is well known that theoretical physics is at present almost helpless in deal-ing with the problem of strong interactions. We are driven to the conclusion that the Hamiltonian method for strong interactions is dead and must be buried, although of course with deserved honour."

Obviously, things have gone differently from his prevision. The turning point was represented by the experiment of deep inelastic scattering carried out at the Stanford Linear Accelerator Center (SLAC) in 1968. It consisted in firing high accelerated electrons against protons in order to investigate their internal structure; results obtained at high transfered momentum, were compatible with the picture of a proton made up of quasi-free constituent (called partons). This suggested the picture of a coupling related to the strong force that becomes small at high energies. The energy dependence of the strength of an interaction was a well known phenomenon in QED, where the effective charge decreases with distance (thus increases with the energy scale) because of the screening due to vacuum polarization. Here the desired behavior was the opposite, and a theory was needed showing this property, the so-called asymptotic freedom. Theoretical physicists in those years were quite skeptical about the existence of such a theory in the framework of QFT; in 1973, Coleman and Gross worked on a general proof for the non existence of asymptotically free quantum field theories in four dimensions [2]; the proof had a ’loophole’, it was not valid for non-abelian gauge theories (and in particular Yang-Mills theories, which were the generalization of the pure gauge theory of QED to SU (N ) gauge groups). They asked their doctoral students, Politzer and Wilczek, to explicitly calculate the running coupling for these theories in order to complete the proof.

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They won the Nobel Prize in 2004. What they found is that the derivative of the coupling respect to the energy scale, the so-called β-function, assumes for these theories a negative value in accordance with what expected by asymptotic freedom. From that, the formulation of Quantum Chromodynamics as nowadays known was only a small step. Gross himself, referring to the newborn theory, will say

“Like an atheist who has just received a message from a burning bush, I became an immediate true believer.”

Nowadays, we know that most of the nontrivial properties of Quantum Chromodynamics comes from the non-abelianess of its gauge group SU (3); as already cited, asymptotic freedom is one of them. Another one, although analytically unproven, is color confinement, namely the fact that all the asymptotic states which can be observed in nature are neutral under color charge; from this it follows, in particular, that quarks cannot be found as free isolated particles. This fact has been largely confirmed both from experiments and from lattice simulations.

Another feature coming from non-abelianess is the nontrivial topological structure of

the gauge fields space. This allows to sort gauge configurations into homotopy classes,

characterized by an integer number that represents the amount of windings in the gauge group, when you turn once in the 4 dimensional space-time. This integer is also known as topological charge and there are some relevant implications related to its existence. One of them is related to the strong CP problem, i.e. the question of why there is no experimental evidence of CP violation in the strong interactions without any theoretical argumentation. Indeed, it is possible, in principle, to add a term to the QCD Lagrangian proportional to the product of the chromoelectric and chromomagnetic fields without violating any fundamental theory of the SM; the term actually violates CP, but this symmetry is already spoiled by weak interactions. The proportionality constant is the so-called θ angle, and the additional term is often referred to as the θ-term. Nontrivial topology comes into play by noting that this term can actually be rewritten as a total four-divergence; its integration in the action, brings a surface term which is zero, thus has not physical effects, only if the field has trivial topology. The existence of homotopy classes prevent this from happen and de facto poses the problem. A physical implication of such a term would be an electric dipole moment for the neutron; it is the measure of this quantity which provides the nowadays more stringent

bound on the θ value (|θ| < 10−9). A possible solution of the puzzling question was proposed

in the late 70’s by Roberto Peccei and Helen Quinn. They proposed the introduction of a

new particle, the axion1, coming from a symmetry breaking mechanism, which couples to the

topological charge, thus experiences the free energy density of QCD as an effective potential,

1_{Actually, it is usually referred to as "QCD axion" for distinguishing it from other pseudoscalar particles}

not included in the Standard Model, that are usually called axions too, but are originated by different mechanisms.

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Contents 3

that has in θ = 0 its minimum; the null value is thus reached dynamically. The mass of this new particle is proportional to the second term of the potential expansion around θ = 0, a quantity known as topological susceptibility χ.

Nowadays axion is also a viable candidate for explaining the dark matter observed in the Universe; for a reliable estimation of the axion abundance, however, a quantitative knowledge of the parameters entering the evolution equations is required, in particular the mass dependence from the temperature, which is directly related to χ(T ). The knowledge of the axion abundance in terms of the axion decay constant allows to compare this quantity with the amount of observed dark matter present in the Universe, in order to obtain a window for the value of the axion mass at the present day.

A full analytic calculation is impracticable; perturbation theory can not provide any information about topology because of its non-perturbative nature, whereas alternative ap-proaches such as semi-classical expansion and chiral perturbation theory, provide good esti-mations only in the high and in the low temperature regime respectively. Therefore the only alternative for obtaining a solid computation of χ(T ) is constituted by lattice simulations.

Lattice QCD introduces a discretization of the spacetime into a four dimensional lattice on which matter fields (on sites) and gauge fields (on links between sites) are defined; then by mean of Monte Carlo algorithms, a configuration is extracted in accordance with a

sta-tistical weight ∝ e−S. It is the most powerful theoretical tool at our disposal for obtaining

physical information starting from first principles in a totally non-perturbative way, thus it is particularly suited for investigating topological quantities like χ(T ). Recently, several works have attempted to accomplished this task because of the cosmological relevance of the topic; results present in the literature, however, show discrepancies each other, in particular for what concerns the drop rate of χ(T ) at high T . These disagreements can be ascribed to the difficulties generally related to such a computation; there are some peculiar issues that affect this measurement, making necessary an huge computational effort for a reliably estimation. A recent work [3] seems to succeed in overcoming these problems, providing values for χ(T ) for a wide range of temperatures; however, the new algorithms used for accomplishing this goal rest their bases on assumptions and approximations which, even if reasonable, should be carefully checked.

An independent verification of these results would be desirable, and it is the main aim of this work. In particular, we check the feasibility of an already existing algorithm, the Multi-canonical algorithm [4], usually employed in other contexts (statistical simulation in presence of strong first order transitions), to the QCD case, in order to resolve the major difficulty which affects computation of χ(T ) at high T. The contents of this work are structured in the following way. In Chapter 1, we recall the main concepts of Quantum Chromodynamics, from its construction as a gauge theory to its lattice formulation. We also briefly cite the most used Monte Carlo algorithms for simulating lattice QCD, with particular attention to

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the ones used in this work. In Chapter 2 we provide a deeper insight into the topological structure of Yang-Mills theories, analyzing also the analytic approximations available and the relation between the topological susceptibility and the axion physics. In Chapter 3 we focused our attention on the state-of-the-art for what regards the computation of χ by means of lattice simulations. We also provide in the first part, the basis techniques for dealing with topology on the lattice. In Chapter 4 we introduce the new algorithm employed; we briefly explain the main ideas of the Berg original implementation and how we translate it to our problem. Also results from a simple toy model are discussed, such as comparison with other similar methods present in literature. Finally in Chapter 5 we summarize results of our simulative work. The chapter is divided into two parts and follows, in a certain sense, the chronological flow of our simulations; in the first part we illustrate several attempts made for finding a viable potential, whereas in the second part we show our continuum extrapolations at two different temperatures. Conclusions and future perspectives follow.

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5

Chapter 1

Quantum Chromodynamics

In this introductory chapter we briefly provide the theoretical framework of our work. We introduce some general but fundamental concepts taking a rapid look at the main ideas that characterize them. In particular in the first part we refresh the standard arguments for introducing a quantum gauge theory in order to build the QCD Lagrangian. Then, the path-integral formulation of QFTs is rapidly shown, where the parallelism with statistical systems becomes manifest. In the second part we illustrate the standard arguments regarding lattice QCD, with particular attention to the construction of the lattice action. Finally, we devote the last section to discuss Monte Carlo methods and algorithm used with dynamical fermions

1.1 Quantum Chromodynamics

In the early’60s, an American physicist, Murray Gell-Mann found that the large number of new particles discovered in those years could be arranged in the irreducible representations of an SU (3) group. That drove him and, independently, George Zweig to postulate the

existence of charged spin-1₂ particles, named quarks [5], which combine under the effect of

the strong interaction to form the hadronic spectrum. Quarks were thought to stay in the fundamental representation of this SU (3) group, and thus they were initially distinguished in three different types (or flavors): up, down and strange. Later on, other quarks with heavier mass have been discovered, the so-called charm, bottom and top quarks, increasing to 6 the total number of flavors.

Because of an apparent violation of the Pauli-principle for the ∆++particle, an additional

SU (3) internal symmetry was proposed, the color charge; unlike the flavor, this is an exact symmetry, and it constitutes the basis for the gauge theory of Quantum Chromodynamics.

As for every gauge theory, the main ingredient is the promotion of a global symmetry, usually a transformation under elements of a symmetry group, to a local one, by the insertion of a gauge field that transforms under the action of the group in such a way to preserve the symmetry of the whole Lagrangian. As an example, the Lagrangian of Quantum Electrody-namics (QED) describing the electromagnetic interaction, is obtained starting from the free fermionic theory, that is invariant under a global U (1) group, by upgrading this symmetry

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to a local one; this procedure gives rise to a new term of interaction between the fermionic field and the photon.

In the QCD case, the Lagrangian that describes the free quark field has the form:

Lf ree = Nf X i Ψi(iγµ∂µ− mi)Ψi(x) with Ψi =       ψi1 ψi2 .. . ψiNc       (1.1)

where i is the flavor index, Nf the number of flavors, Ψi is a vector of Nc color fields,

and the Dirac index is implicit. Since the mass term mi depends on the flavor index, the

Lagrangian is not invariant with respect to SU (Nf) transformations in the flavor space;

restricted to the lightest quarks, they are an approximated symmetry.

Besides that, the Lagrangian enjoys an exact symmetry under SU (Nc) transformations

in color space, explicitly:    Ψi(x) 7→ Ψi(x)0 = U Ψi(x) Ψi(x) 7→ Ψi(x)0 = Ψi(x)U† ∀i, where U ∈ SU (Nc) (1.2)

The transformation act for every flavor, and we will omit index i in the following. U is

an element of the fundamental representation of SU (Nc) and can be written as

e−iPN 2ac −1θaTa _(1.3)

where Ta are the generators of the group, i.e. a basis of the traceless hermitian Nc× Nc

matrix with [Ta, Tb] = ifabcTc and Tr[TaTb] = 1/2 δab.

If we change U → U (x) in (1.2), the derivative term in (1.1) is not invariant anymore; it produces an extra term due to the variation of U (x). To obtain gauge invariance, we have to define a parallel transport, namely a rule that allows the comparison between Ψ at different points.

Formally, it is an application W that associates to every path Cy←x that links x to y,

a map from Vx to Vy, where Vx is the vector space of Ψ defined in x, with the following

properties

• W (Cy←x) ∈ SU (Nc).

• W (∅) = 1 with ∅ a zero-length curve.

• W (Cz←y) ◦ W (Cy←x) = W (Cz←y)W (Cy←x)

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1.1. Quantum Chromodynamics 7

• W (Cy←x) 7→ U (y)W (Cy←x)U†(x) under a local gauge transformation.

It can be proven that, starting from this assumptions, W (Cy←x) can be written in the

general form

W (Cy←x) = Pe

−igR

Cy←xAµ(z)dzµ y=x+dx_≈ _e−igAµ(x)dxµ _(1.4)

where P is the path-ordering operator, and Aµ ≡

P

aA a

µ(x)Ta with Aaµ(x) the gauge fields.

From the transformation properties of Wy←x, and the infinitesimal form of (1.4) we can

derive the behavior of Aµ(x) under gauge transformations.

Aµ(x) 7→ U (x)Aµ(x)U (x)†+

i

g(∂µU (x))U

†

(x) (1.5)

Now we can define a covariant differential

DΨ(x) ≡ Ψ(x + dx) − Wx+dx←xΨ(x) ≈ (∂µ+ igAµ(x))Ψ(x)

| {z }

DµΨ(x)

dxµ (1.6)

where Dµ is the covariant derivative, that transforms like

DµΨ(x) 7→ U (x)DµU†(x)U (x)Ψ(x) = U (x)DµΨ(x) (1.7)

From (1.7) we conclude that by replacing the derivative term in (1.1) with the covariant

derivative, we restore the gauge invariance of the Lagrangian. Since the fields Aa

µ(x) we have

introduced are dynamical variables, we have to add their kinetic term, namely

LG= −

1

2Tr[FµνF

µν_] _(1.8)

This is the so-called Yang-Mills Lagrangian, where Fµν, the field strength tensor, is defined

as Fµν ≡ ∂µAν − ∂νAµ+ ig[Aµ, Aν] = (∂µAaν − ∂νAaµ− gfabcAbµA c ν | {z } Fa µν )Ta (1.9)

Finally we can write down the Lagrangian for QCD (Nc = 3)

LQCD = Nf X i Ψi(i /Dµ− mi)Ψi(x) − 1 2Tr[FµνF µν_] _(1.10)

The eight gauge fields Aa_µ are the gluonic fields; gluons, like photons for electrodynamics,

are the mediators of the strong interaction. Unlike them, however, gluons possess color charge, that means they "feel" the strong force, apart from mediating it. This feature, that comes from the non abelianity of the gauge group, makes QCD significantly harder to analyze

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than QED. For better understanding, consider the Lagrangian (1.8). We can expand the

Fµν tensor by exploiting (1.9) in order to obtain

LG= − 1 2(∂µA a ν∂ µ_Aν a− ∂νAaµ∂ µ_Aν a) | {z } f ree gluons + gfabc(∂µAaν)A µ bA ν c | {z } 3−gluons interaction −g 2 4fabcfcdeA a µA b νA µ cA ν d | {z } 4−gluons interaction (1.11)

This means that also the pure gauge theory itself, i.e. the theory with only LG, describes

highly non trivial phenomena due to the presence of the 3-gluons and 4-gluons interaction terms.

1.1.1 Chiral symmetry

As previously seen, quarks with different flavors have different masses, thus the Gell-Mann

SU(3) group of flavor transformations cannot be an exact symmetry of LQCD. Nevertheless,

in nature, it seems that such a symmetry still provides a good description of the particles spectrum, or equivalently, that the violation effects are quite small. This because the lightest three quarks have masses that can be considered small with respect to the typical scale of the

theory (i.e. ΛQCD, see next section); we can safely neglect heavier quarks in the following.

Thus we can put mu = md = ms = 0, respectively the mass of the quark up, down and

strange, and study this limit of the theory. Now the Lagrangian (1.10) remain unchanged under an independent rotation of the left and the right part of the Dirac fields in the flavor space, or more explicitly:

ΨL_i (x) 7→ Ψ0L_i (x) = U_ijLΨL_j(x) ΨR_i (x) 7→ Ψ0R_i (x) = U_ijRΨR_j(x)

UL∈ U (3)L

UR∈ U (3)R

(1.12)

where i, j = 1, 2, 3 are flavor indices, and UL/R = U_fL/R⊗ 1c, i.e. does not act over the color

space. Therefore, the symmetry group becomes

G = U (3)L× U (3)R = SU (3)L× SU (3)R× U (1)L× U (1)R (1.13)

Consider a transformation of the type (1.12) with UL _{= U}R _{or U}L _{= U}†R_{; they are named}

respectively vectorial and axial transformations.

These transformations can be written in a more compact form as

UV = eiPaθaTa_, _UA_{= e}i

P

aγ5θaTa _(1.14)

where a labels the generators of the group plus the identity T0 _{≡ 1, and the γ}5 _{matrix in}

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the group G can be re-arranged into

G = SU (3)V × SU (3)A× U (1)V × U (1)A (1.15)

where an abuse of notation has been committed in writing SU (3)A, because axial

transfor-mations do not form a group. The SU (3)V is the already cited Gell Mann SU (3) group;

the Lagrangian stays invariant under this symmetry also at non zero quark masses, as long

as the mass matrix is proportional to the identity (mu = md = ms). The SU (3)A is

in-stead explicitly broken by non zero quark masses; what was found by Yoichiro Nambu in the early ’60s, is that this symmetry is also spontaneously broken in the chiral limit. The Nambu-Goldstone theorem thus, assures the presence of 8 massless spin-less particles (the

Goldstone bosons) with the same quantum numbers of the broken generators γ5Ta; they can

be naturally identified with the mesonic octect. The non zero mass of the mesons is then explained by the introduction of the quark mass term of explicit symmetry breaking.

This breakthrough marked the beginning of a broad development of effective field theories aiming at describing the low energy behavior of QCD, hardly derivable by perturbative approaches.

The order parameter of the spontaneous chiral symmetry breaking is the chiral condensate

Σ = h0|ΨiΨi|0i. It is known from lattice calculations, that its value changes abruptly at the

critical temperature Tc ≈ 170 MeV; for higher T the chiral symmetry is restored (Σ = 0),

quarks are no more bound into mesons but live deconfined leading to a new state of matter, the quark gluon plasma.

In order to complete our digression, we note that the U (1)V, that consist in multiplying

a constant phase to all the flavors, is always conserved, and is related to the conservation

of the baryonic number. The case of U (1)A is more tricky; as for the SU (3)A, spontaneous

symmetry breaking occurs, but in this case, there is also another term that spoils the con-servation of the Noether current, the anomaly. We will see more in details how this issue is related to the non trivial topology of the gauge fields in chapter 2.

1.1.2 Path integral formalism

The usual mechanism of quantization of classical fields (individuate the conjugate momenta and then promote all to quantum operators and impose the canonical commutation rules) is an unpractical point of view if we are interested in lattice simulations. To this aim the path integral formulation, is the most suitable approach since it has the advantage of dealing with averages over configurations with a given probability distribution, thus there is a parallelism with statistical systems which are usually studied by means of Monte-Carlo simulations.

We briefly illustrate the main ideas for the quantum mechanical case and then we gener-alize to field theory. Suppose to have a system described by a time independent Hamiltonian

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H = T (ˆp) + V (ˆx), where T (ˆp) is the kinetic part, ˆp the momentum operator and V (ˆx) a co-ordinate depending potential, and look at the probability amplitude between two eigenstates of position at different times. We can divide the time interval into N portions of length and take the limit N → ∞ and → 0 keeping the product N fixed. In this limit it can be proved that we can safely write:

hxf, tf|xi, tii = hxf|e−iH(tf−ti)|xii

= hxf|e−iN (T (ˆp)+V (ˆx))|xii

N →∞ →0

≈ hxf|(e−iT (ˆp)e−iV (ˆx))N|xii

(1.16)

where the last equality comes from the Lie-Trotter formula.

The last ingredient is to add complete sets of position and momentum eigenstates between each of the N factors, obtaining

hxf, tf|xi, tii = N −1

Y

j=0

Z

dxjdk hxj+1|e−iT (ˆp)|ki hk|e−iV (ˆx)|xji

= N −1 Y j=0 Z dxjdk e−i k2 2me−iV (xj)hx j+1|ki hk|xji =r m 2π N N −1 Y j=0 Z dxje−iV (xj)ei m 2 (xj+1−xj )2 =r m 2π NZ N −1 Y j=1 dxje iPN −1 j=0 m 2 _xj+1−xj 2 −V (xj) ≡ N Z x0≡x(ti)=xi xN≡x(tf)=xf Y τ dx(τ )ei Rtf ti [ m 2x˙ 2_{(τ )−V (x(τ ))}_]_dτ = N Z Dx eiS[x] (1.17)

where N is a divergent normalization factor, Dx is the functional integral over the paths x(τ ) with fixed endpoints and S is the action of the classical system.

Often one prefers to turn the Minkowski space-time into an Euclidean space-time, by Wick rotating the time coordinate t → −iτ . One of the reasons is that the partition function of a quantum system Z at temperature T can be written as:

Z = Tr[e−βH] = Z dx hx|e−βH|xi = N Z x(0)=x(β) Dxe−SE _(1.18)

where β = 1/T and the last passage follows from Eq. (1.17) at imaginary times. Notice that the path integral now is performed over all the paths periodic in the euclidean time direction, with a time extent equal to β. From this it follows that the expectation value of

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an observable can be put into the form

h ˆOi_T = Tr[e −βH_O]_ˆ Z = R Dx O[x]e−SE R Dx e−SE ≡ Z Dx O[x]P[x] (1.19)

The quantity in (1.19) is suited for being computed by Monte Carlo methods. We can

consider again a discretized time interval, and for every time step an xi position value; thus

we extract a chain {xi} with probability

P[{xi}] =

1

Ze

−SE[{xi}] _(1.20)

and take the average of O over the chain.

The situation can be naively generalized to quantum field theory; the main differences being:

• Q

τdx(τ ) →

Q

x,τdφ(x, τ ), because now the path integral is done over the field

vari-ables, labeled by the coordinates of a discretized space-time.

• As for the quantum mechanical case, the euclidean theory at finite time extent τf −

τi = β corresponds to the theory at finite temperature T = 1/β. The trace in

the partition function here leads to periodic/antiperiodic boundary conditions for the bosonic/fermionic fields.

What we obtain, for example, in the scalar field case is

h0|O[ ˆφ(y, t)]|0i =

Z φ(x,−∞)=φ(x,+∞) Y x,τ dφ(x, τ ) O[φ(y, t)]e −SE[φ] Z (1.21) where Z is defined as Z(T ) = Z φ(x,−∞)=φ(x,+∞) Y x,τ dφ(x, τ ) e−SE[φ] _(1.22)

Notice that in the left term in Eq. (1.21) we have an operator, whereas within the integral in the right part with deals only with c-numbers. Things change when we consider fermions, because they have to satisfy anti-commuting relations thus they cannot be expressed by simple c-numbers; hence fermionic fields are anti-commuting variables, also called Grassmann variables. Integrals over Grassman variables have some specific properties (Berezin integrals) and rules that differ from usual integration; we briefly list the two main results which are

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useful in calculation, the gaussian integrals for the bosonic and fermionic case Z Y j dφ∗_jY i dφie−φ ∗ iMijφj _{= det} M 2π −1 Z Y j dχ_jY i dχie−χiMijχj = det(M ) (1.23)

If we want to compute (1.21) by numerical simulations we need a four dimensional N3

s×Nt

lattice, with Ns = Nt in the T = 0 case, and then extract a set of φi for each i site according

to

P [{φi}] =

1

Ze

−SE[{φi}] _(1.24)

Obviously, the path integral formulation of QCD is a bit more involved. The first com-plication comes from the fact that we are in the presence of a gauge theory thus the above derivation of the path integral slightly changes. The reason is easily explainable; our starting

point was the Hamiltonian formulation of the theory, but the A0 component of the gauge

field has a zero conjugate momentum thus Hamilton equations cannot be defined on it. The way of proceeding is seen in more detail in Chapter 2, when we treat the topology of the Yang Mill fields; here we anticipate that, by imposing a constraint over the Hilbert space, we recovery the usual output

Z = Z

[DA][Dψ][Dψ]eiR dx4LQCD[Aµ,ψ,ψ] _(1.25)

where [DA] =Q

x,τ,µ,adAaµ(x, τ ) represent the functional measure, and (anti)periodic

bound-ary conditions are imposed over the (fermionic) gluonic fields. The second problem arise when we perform the perturbative expansion of (1.25). Before doing that we have to eliminate the spurious degrees of freedom coming from the gauge invariance; in other word, we have to select only one representative for each gauge orbit in order to correctly count the degrees

of freedom. This can be done in QED just by adding a gauge fixing term ∝ (∂µAµ)2 into

the Lagrangian. For QCD, however, such a simple procedure give rise to a non unitary the-ory; the right way of doing that was firstly discovered by Faddeev and Popov [6]. Without entering into details (available on quantum field theories standard textbooks Ref. [7]), the final result provides the addition of two terms into the Lagrangian; a gauge fixing term as for the abelian case plus a term involving a new unphysical field, which interact with the gauge field Aµ. L(perturb)_QCD = LG+ LF − 1 2α(∂ µ_Aa µ) 2_{+ ∂}µ_c a∂µca− gfabcca∂µ(Acµcb) (1.26)

where ca is the so-called ghost field. It is an anti-commuting spinless field, thus it would

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diagrams, which means that it is unphysical (cannot appear in asymptotic states).

Feynman diagrams can be constructed from (1.26), and usual perturbative calculations can be performed. The theory is renormalizable, that means we can define a finite number of counterterms which, whatever the order of the loop expansion, are able to reabsorb the divergences arising from internal loops. In particular, we have

Aa_µ,B = z₃1/2Aa_µ,R ψB= z21/2ψR ca_B= ˜z₃1/2ca_R gB = zggR mB = zmmR αB = z3αR (1.27)

where B stands for the bare, cut-off dependent quantities whereas R labels the renormalized

ones. Focusing on the coupling constant gR, we know that this quantity depends on a energy

scale µ introduced due to the renormalization scheme; studying the behavior of gR(µ), the

so-called running coupling constant, give us informations about how the interaction changes at different energy scales. Defining the Gell-Mann-Low beta function as

β(gR) ≡ µ

dgR(µ)

dµ = −β0g

3

R− β1g5R− β2gR7 . . . (1.28)

we have for the 1-loop contribution (which is scheme independent)

β0 =

1 16π2

(11Nc− 2Nf)

3 (1.29)

that, for the physical case (Nc = 3, Nf = 6) implies β0 > 0. The negative sign of the first

perturbative term in (1.28) has an huge impact over physics. In order to see that, we can neglect higher order loops contributions, and resolve the differential equation in order to find

gR(µ), which reads

g_R2(µ) = 1

2β0log(_Λ_QCDµ )

if µ > ΛQCD (1.30)

where ΛQCD ' 200 MeV is a scheme-dependent constant that characterize the theory and

has the dimension of a mass. The appearance of a typical scale for a theory which has a dimensionless coupling constant can be at a first sight counterintuitive; this phenomenon is called dimensional transmutation.

Eq. (1.30) implies that gR(µ) → 0 for µ → ∞, or in other words, that the strong

inter-action becomes weaker at short distances (high energies) whereas it rises when we approach

to low energy scales.1 _{This behavior is also called asymptotic freedom and it is in principle}

compatible with confinement. Experimental evidences from deep inelastic scattering, where

1_{Actually, µ is a free parameter related to the renormalization scheme. It can be shown however that, in}

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Figure 1.1: αs = g2/(4π) measured at different energies by different experiments.

Figure taken from [8]

collisions at high energies allows to observe quarks behaving as free particles, and other hadronic collisions confirm the theoretical result (Figure 1.1).

1.2 Lattice formulation

As briefly seen above, perturbative methods applied to QCD are bound to fail in exploring low energy physics because of the growing of the coupling constant g that invalidates the weak coupling expansion. On the other hand, effective field theories, which recreates the right chiral properties of the theory, providing predictions about the hadrons spectrum in good accordance with experiments, do not derive from the first principles of the QCD Lagrangian and, moreover, becomes useless for high temperature estimations. In this framework, lattice QCD constitutes an inestimable tool for investigating the non perturbative aspects of the theory, such as confinement, topology and physics of the quark-gluon plasma.

The first step of this approach consists in using the Euclidean formulation of the theory;

as already anticipated, in this way the contribution of a fields configuration is ∝ e−SE _{and can}

be interpreted as a probability, thus it can be computed by means of numerical simulations like a standard statistical system. The QCD Lagrangian (1.10) at imaginary time (t → −iτ ) reads L(E)_QCD = 1 2Tr[F (E) µν (xE)Fµν(E)(xE)] + X i Ψi(γµ(E)D (E) µ + m)Ψi(xE) (1.31)

Indeed the Wick rotation implies A0(x) → iA

(E)

0 and ∂µ → i∂

(E)

µ ; if we redefine gamma

matrix as γ_i(E) ≡ −iγi_{, γ}(E)

0 = γ0, we obtain the expression above.

Then we have to discretize the euclidean space-time into a lattice of N3

s × Nt sites, at

distance a from each other. This introduces a non perturbative regularization for the theory, i.e. a momentum UV cut-off which is the inverse of the lattice spacing a and takes care of the divergences, allowing a well-defined path integral formulation.

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1.2. Lattice formulation 15

Now we have to put fields on the lattice. We will separately treat the gluonic and the fermionic field discretization. Usually it is preferred to work with dimensionless quantities, which will be indicated by theˆsymbol; rescaling all with respect to a, the replacements read

xµ= aˆnµ mf = a−1mˆf, ψij = a−3/2ψˆij Z d4x → a4X ˆ n ∂µ→ a−1∂ˆµ (1.32)

where ˆnµ = (n1, n2, n3, n4) is a 4-uple that labels the sites, and there are different possible

choice for ˆ∂µ as we will see in the following.

1.2.1 Gluons on the lattice

Although in principle a lattice theory has not to preserve all the symmetries of the continuous counterpart, as long as they are recovered when a → 0, in practice it is highly preferred to work with theories that maintain the symmetries when discretized. For this reason we want to construct a QCD lattice Lagrangian that shows gauge invariance.

When we introduced gauge fields we stressed the role of the parallel transport operator in ensuring the local symmetry, thus the natural choice for generalizing gluonic fields on lattice is to relate gauge degrees of freedom to the links of the lattice, i.e. the oriented segments between two close points.

Then we define the analog of W for connecting two contiguous sites along the shortest path

W (Cx+dx←x) 7→ Un+ˆˆ µ,ˆn≡ Uµ(ˆn) where Uµ(ˆn) ∈ SU (3) (1.33)

where ˆµ is a four vector with 1 in the µ position and zero elsewhere. Uµ(ˆn) is the link

variable; we can also consider the conjugated U_µ†(ˆn) ≡ Un,ˆˆn+ˆµ, i.e. the same path but with

opposite direction. As for the continuous case (we derived the relation in Minkowsky space, the passage to Euclidean four space, however, does not alter its shape), in the limit of infinitesimal path, we have:

Uµ(ˆn) ≈ eigaAµ(ˆn) for a → 0 (1.34)

where Aµ =

P

aAaµTa. This relation is useful for checking that quantities defined on lattice

have the right naive continuum limit.

Under a local gauge transformation, the link variable Uµ(ˆn) transforms according to:

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U

(3)

U

(1)

U

(2)

U

(5) (m−1)

U

(m−2)

U

(4)

U

(m) (1) (2) (3) (5) (4) (m−2) (m−1) (m)

Figure 1.2: An example of closed path. Arrows stands for link variables, (i) label the site. The dashed lines indicates several others links taken in such a way that the final length is m.

+

n

ν

n

+

_µ

U

µ(n) µ (n+ )

U

ν (n+ )ν

U

µ

U

ν(n)

n

ν µ Figure 1.3: A graphical representation of the plaque-tte Uµν(ˆn).

Therefore, if we want to construct a gauge invariant quantity, we have to consider the trace

of Uµ products along a closed path (see Figure 1.2); indeed

Tr[U(1)U(2)· · · U(m)_{] 7→ Tr[Ω}(1)_U(1)_Ω†(2)

Ω(2)U(2)Ω†(3)· · · Ω(m)_U(m)_Ω†(1)

]

= Tr[U(1)U(2)· · · U(m)_] (1.36)

The simplest closed path one can chose is the plaquette (Figure 1.3): Uµν(ˆn) ≡ Uµ(ˆn)Uν(ˆn + ˆµ)Uµ†(ˆn + ˆν)U

†

ν(ˆn) (1.37)

where the path ordering is important since the SU (Nc) elements do not commute.

If we use the limit in eq. (1.34) and the Baker-Campbell-Hausdorff formula

eAeB= eA+B+12[A,B]+... (1.38)

we can rewrite (1.37) to the lowest orders in a as Uµν(ˆn) ≈ e−iga 2_F µν _{≈ 1 − iga}2_F µν− 1 2g 2_a4_F µνFµν (1.39)

From which it follows Re Tr[Uµν(ˆn)] = 1 2Tr[Uµν(ˆn) + U † µν(ˆn)] = Tr 1 − 1 2g 2_a4_F µνFµν + o(a6) = Nc− 1 2g 2_a4_Tr[F µνFµν] + o(a6) (1.40)

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where Re is the real part and repeated indices are not summed. Therefore we can conclude that, when a → 0, the lattice action

S_G(W )= X ˆ n,ν,µ>ν 2Nc g2 1 − 1 Nc Re Tr[Uµν(ˆn)] (1.41)

reproduces the standard euclidean pure gauge action L(E)_G up to o(a6_{); the pre-factor is}

usually referred to as β = 2Nc/g2

S_G(W ) is also called Wilson action, from K.G. Wilson who firstly developed a way to

formulate gauge theories on lattice. Obviously this is not the unique choice we can make since we can add any further term to the action as long as it vanishes in the a → 0 limit.

Often this is done in order to cancel o(an_{) corrections of the lattice action with the aim of}

mitigating the continuum convergence; this procedure can be repeated at every order n and is named Symanzik improvement program [9] [10]. As an example, suppose we want to cancel

o(a6_{) corrections arising in (1.40); this is the so-called tree level Symanzik improvement and}

is usually performed by adding the rectangle loop defined as U_µν1×2(ˆn) = Uµ(ˆn)Uµ(ˆn + ˆµ)Uν(ˆn + ˆµ)Uµ†(ˆn + ˆµ + ˆν)U

†

µ(ˆn + ˆν)U †

ν(ˆn) (1.42)

which is only one of the possible 6-links closed paths (they are listed in Figure 1.4). The resulting action is SSym = β Nc X ˆ n,µ6=ν 5 6Re Tr[1 − Uµν(ˆn)] − 1 12Re Tr[1 − U 1×2 µν (ˆn)] (1.43)

The coefficients between the plaquette and the rectangular contributions are determined by means of a perturbative expansion [11].

Figure 1.4: Three possible types of 6-links closed paths,from the left to the right: the "rectangle", the "parallelogram" and the "chair".

1.2.2 Fermions on the lattice: the doubling problem

Now we have to introduce the lattice version of the fermionic free fields. As anticipated, they are defined on sites, and a naive discretization follows from (1.31) by applying conversions

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(1.32). S_F(L) =X ˆ n Nf X i ˆ Ψi(γµ(E)∂ˆµ+ ˆmi) ˆΨi(ˆn) (1.44)

where the derivative discretization is not yet specified. In the following some examples of possible definitions of ˆ∂µ on lattice are reported

ˆ ∂_µFf (ˆn) ≡ f (ˆn + ˆµ) − f (ˆn) ˆ ∂_µBf (ˆn) ≡ f (ˆn) − f (ˆn − ˆµ) ˆ ∂_µSf (ˆn) ≡ (f (ˆn + ˆµ) − f (ˆn − ˆµ)) /2 forward backward symmetric (1.45)

In the continuum, the derivative is an anti-hermitian operator; if we desire to preserve this property we must choice the symmetric definition. Indeed

X ˆ n f (ˆn) ˆ∂_µFg(ˆn) =X ˆ n f (ˆn)[g(ˆn + ˆµ) − g(ˆn)] =X ˆ n0 [f (ˆn0− ˆµ) − f (ˆn0)]g(ˆn0) = −X ˆ n0 ˆ ∂_µBf (ˆn0)g(ˆn0) (1.46)

from which it follows

ˆ

∂_µF † = − ˆ∂_µB (1.47)

thus it is not anti-hermitian, whereas, by noting that ˆ ∂_µS = 1 2[ ˆ∂ F µ + ˆ∂ B µ] (1.48)

and using (1.47) we can see that the symmetric choice satisfies our request. Notice that the problem of anti-hermiticity arises because of the presence of a first derivative in the action; for the scalar case, where the action has a Laplacian term, thus a second derivative, the action stays hermitian whatever the choice of the discretization.

For fermions, instead, the choice is in this sense obligated, and brings nevertheless a more subtle issue also known as doubling problem, that we briefly show. Our aim is to write down the lattice version for the fermionic propagator. Exploiting the properties of integrals over Grassmann variables we have

h0| ˆΨαi(ˆn) ˆΨβj( ˆm)|0i = Z Y ˆ x,λ [dΨˆλ(ˆx) d ˆΨλ0(ˆx0)] Ψˆ_αi(ˆn) ˆΨ_βj( ˆm) e ˆ ΨM ˆΨ = (M−1)αβ,ij(ˆn, ˆm) (1.49)

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where α, β are Dirac index, i, j flavor ones, ˆx, λ stand for all types of index (spacial, Dirac,

flavour and color) and the M matrix is the action matrix, which reads

Mαβ,ij(ˆn, ˆm) = " δˆn ˆmδαβmˆi+ 1 2 X µ (γµ)αβ(δm,ˆˆ n+ˆµ− δm,ˆˆ n−ˆµ) # δij (1.50)

(one can multiply fields recovering (1.44)). In order to find M−1 one can Fourier transform

the matrix M and then anti-transform the inverse. For a function of discrete variables f (ˆn),

the Fourier transformation and anti-transformation are defined as ˜ f (k) ≡ +∞ X ˆ n=−∞ f (ˆn) e−ikµnˆµ f (ˆn) ≡ Z +π −π Y i dki 2π ˜ f (k) eikµnˆµ (1.51)

By exploiting the formula for the delta function

δ4(k − k0) = +∞ X ˆ n=−∞ ei(kµ−k0µ)ˆnµ _(1.52)

we obtain for the case of interest

( ˜M )−1_αβ,ij(k) = δij " ˆ miδαβ+ i X µ (γµ)αβsin(kµ) #−1 = δij ˆ miδαβ − i P µ(γµ)αβsin(kµ) ˆ m2 i + P µsin 2_(k µ) (1.53)

Thus, if we approach the continuum limit, we obtain h Ψαi(y)Ψβj(x) i = lim

a→0 a −3_{h ˆ} Ψαi(aˆn) ˆΨβj(a ˆm) i = lim a→0 a −3 Z +π/a −π/a a4 d 4_p (2π)4 " ami+ i X µ γµsin(apµ) #−1 eipµ(xµ−yµ) = lim a→0 Z +π/a −π/a d4_p (2π)4 mi− iγµp˜µ m2 i + |˜p|2 eipµ(xµ−yµ) (1.54)

where ˜pµ≡ sin(apµ)/a. The integral in (1.5) is dominated by small values of ˜pµ; we can see

from the plot of ˜p respect to p (Figure 1.5) that this happens in two different regions. It

appears clearly, indeed, that when a → 0 both points around pµ = 0 and pµ = ±π/a give

a finite contribute to the integral and this occurs for each space-time direction. Thus we

actually have 2d _{= 16 times the expected result for the propagator, and we have to cut off}

in some way the spurious degrees of freedom if we want to recover our original theory. It can be seen that this issue can be avoided if one picks a different discretization of

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Figure 1.5: Plot of ˜p respect to p in the integration interval. The straight line is for ˜

p = p. When a → 0 the dashed line rise, and finite contributes comes only from the origin and the corners of the Brillouin zone. Image taken from [12]

the derivative, however in that case the theory on the lattice loses hermiticity. As we will see later, this statement follows from a more general theorem that put some limits over the properties a fermionic discretized theory can have.

1.2.3 Fermions on the lattice: Wilson fermions

A possible solution to the doubling problem is the addition of an irrelevant term into the

lattice action (i.e. a term that vanishes when a → 0) which modifies the expression of ˜pµ in

order to suppress multiple contributions. This procedure, proposed by Wilson, consists of adding a second derivative term of the type

− r 2 X ˆ n ˆ Ψi(ˆn) ˆ ˆΨi(ˆn) (1.55)

where r is a free constant and ˆ_{is the lattice counterpart of the Laplacian operator}

ˆ ij,αβ(ˆn, ˆm) = X µ ˆ µ = X µ [δm,ˆˆ n+ˆµ+ δm,ˆˆ n−ˆµ− 2δn ˆˆm]δijδαβ ˆ −−→ aa→0 2X µ ∂µ∂µ (1.56)

From the last line we can see that the term (1.55) goes as a5; if we consider thatP

ˆ n7→ a

−4R the additional term goes to zero like a in the continuum limit as anticipated. This addition modifies the matrix in (1.50) as follows

M_αβ,ij(W )(ˆn, ˆm) = " δn ˆˆmδαβ( ˆmi+ 4r) − 1 2 X µ [(r − γµ)αβδm,ˆˆ n+ˆµ+ (r + γµ)αβδm,ˆˆ n−ˆµ] # δij (1.57)

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If we calculate the lattice propagator, we find a new expression that can be obtained from (1.54) by performing the substitution

mi 7→ Mi(pµ) ≡ mi+ 2r a X µ sin2(pµa/2) (1.58)

We can see that this term reproduces the correct continuum limit for each fixed value of

pµ, Mi(pµ) → mi. Moreover, at the corner of the Brillouin zone it gives a contribution

proportional to 2r/a which diverges thus it eliminates the spurious doublers.

The payback of such a method is the loss of chiral symmetry in the mi = 0 case because

of the presence of a r 6= 0 term, hence Wilson fermions are not suited for studying questions such as spontaneous symmetry breaking in QCD and other related topics. For this reason, other methods have been developed which provide a fermion discretization without losing chiral symmetry. They however manifest other unwanted features: the attempt of solving a problem inevitably generate another issue. This situation is unbeatable, as follows from the Nielsen and Ninomiya no-go theorem [13], which states that it is impossible to make a discretization that at the same time does not create doublers, shows chiral symmetry in the vanishing mass limit, is local and hermitian. Now we provide some detail about another scheme for putting fermions on lattice, the one we use in this work.

1.2.4 Fermions on the lattice: staggered fermions

The main idea of staggered fermions consists in distributing the fermionic degrees of freedom over the sites of the lattice in such a way that the effective length of the lattice spacing corresponds to 2a; in this way, the Brillouin zone is halved and spurious contributions are cut away. One has to recovery in the continuum limit the exact number of degrees of freedom, and in particular the naive limit of the staggered action has to reproduce the correct continuum action. To this end, the first step requires to decouple the Dirac component of the spinor fields; for simplicity we work with a single flavor: this is a convenient way of proceeding, though in the original formulation 4 different flavors were required (the reason will appear clear in the following). We define a transformation of the type

ˆ Ψα(ˆn) 7→ Tαβ(ˆn) ˆΨβ(ˆn) = χα(ˆn) ˆ Ψα(ˆn) 7→Ψˆβ(ˆn)T † βα(ˆn) = χα(ˆn) (1.59)

where Greek letters stand for Dirac index (the transformation does not act on color space), and T (ˆn) ≡ γnˆ1 1 γ ˆ n2 2 γ ˆ n3 3 γ ˆ n4 4 (1.60)

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The key point is that such a transformation possesses the following property

T†(ˆn)γµT (ˆn + ˆµ) = ηµ(ˆn)1 with ηµ≡ (−1)nˆ1+ˆn2+...+ˆnµ−1, η1(ˆn) = 1 (1.61)

thus (using also T†T = 1) we can rewrite (1.44) as

S_F(L) = ˆmX ˆ n,α χ_α(ˆn)χα(ˆn) + 1 2 X ˆ n,µ,α ηµ(ˆn)χα(ˆn) [χα(ˆn + ˆµ) − χα(ˆn − ˆµ)] (1.62)

Therefore, we reduce the degeneracy by picking only one of the four Dirac components (it can be done because now the action is diagonal).

S_F(stag.) = ˆmX ˆ n χ(ˆn)χ(ˆn) +1 2 X ˆ n,µ ηµ(ˆn)χ(ˆn) [χ(ˆn + ˆµ) − χ(ˆn − ˆµ)] (1.63)

Notice that the above action, in the ˆm = 0 limit, is invariant under a transformation of the

type

χ 7→ eiΓ5(n)θ0_χ, _{χ 7→ χe}iΓ5(n)θ0 _where _Γ

5(n) =    1 even n −1 odd n (1.64)

because the kinetic part connects only odd with even sites. As we will see better later, this is a remnant of the chiral symmetry and constitutes the practical utility of staggered fermions. Note also that we have not completely eliminated all the unwanted doublers; we still have a fourfold degeneracy, which is called taste degeneracy. In the original formulation of staggered fermions the taste coincided with the flavor degree of freedom, thus the staggered action reproduced in the continuum limit a four flavor theory. This analogy turned out to be quite unfruitful, because, for example, it is not possible to couple different masses to different tastes in order to explicitly break chiral symmetry, as it happens in the physical case. Thus what is currently done is to add a fourfold degenerate staggered fermion for each flavor and then takes care of the degeneracy in other ways (which we will discuss later on). In order to better investigate the connection with the continuum degrees of freedom,

often a different basis is used. Considering the χ(ˆn) field, where the α index drops because

we have picked only one component of the spinor, we can relabel the sites of the lattice as

2ˆn0 + ˆρ where ρ is a 4-uple composed by 0 and 1 only, that explores all the sites of the

hypercube with vertex in 2ˆn0. We also redefine

χ(2ˆn0+ ˆρ) ≡ χρ(ˆn0) (1.65)

In this way the variable ˆn0 lives on a lattice with double lattice spacing respect to the original one.

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Now we perform the change of basis ˆ ψαa(ˆn) = 1 8 X ρ (T ( ˆρ))αaχρ(ˆn) (1.66)

It can be proven that the staggered action in this basis (often called spin-taste basis) reads

S_Fstagg. =X ˆ n,µ ˆ ψ(ˆn) (γµ⊗ 1)∂µ(S)+ 1 2(γ5⊗ γ ∗ µγ5) ˆµ ˆ ψ(ˆn) + ˆmX ˆ n ˆ ψ(ˆn)(1 ⊗ 1) ˆψ(ˆn) (1.67)

where A ⊗ B means that the first operator acts on the α indices of ˆψ (which can be identified

with the Dirac indices) and the second on a indices (i.e. it acts on the taste space). In the continuum limit, namely when b = 2a → 0, the second term in square brakets goes to zero, while the remaining action becomes a continuum action for a degenerate four-taste fermion. When the lattice spacing is not zero the taste degeneracy is broken by this term; however a symmetry in the spin-taste space survives. This symmetry is just the one under the action

of (1.64)2_{, which in the new basis reads}

ˆ

ψ 7→ eiθγ5⊗γ5_ψˆ ˆ

ψ 7→ψeˆ iθγ5⊗γ5

(1.68)

The presence of such a symmetry is the great advantage of the staggered formulation; such a theory shows already on lattice some of the features of the continuum counterpart (for example, additive mass renormalization is suppressed in the presence of chiral symmetry and moreover Goldstone bosons are naturally produced).

1.2.5 Full QCD on the lattice

Finally, we can put all together and build the lattice version of the entire theory. As for the continuum case, the interaction is introduced by requiring the gauge invariance of the derivative term of the fermion action; because on lattice this term puts in contact neighbors sites, the natural procedure consist in adding a link variable to these terms. Thus the full action, (1.44) plus (1.41), becomes

S_QCD(L) = S_G(L)+X ˆ n ˆ Ψ(ˆn) ˆM ˆΨ(ˆn) +X ˆ n X µ h ˆ_Ψ(ˆ_n)γ µUµ†(ˆn) ˆΨ(ˆn + ˆµ) −Ψ(ˆˆ n + ˆµ)γµUµ(ˆn) ˆΨ(ˆn) i (1.69)

2_{It is actually a generator of the SU (3)}

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where ˆM = diag(mi) and in the last term within square brakets we have redefined ˆn.

Recalling that the gauge group acts on fermionic fields as ˆ

Ψ(ˆn) 7→ Ω(ˆn) ˆΨ(ˆn) ˆ

Ψ(ˆn) 7→Ψ(ˆˆ n)Ω†(ˆn)

(1.70)

and that link variables change in accordance to Eq. (1.35), it can be verified that the above action is invariant under gauge transformations. Obviously, action (1.69) suffers from the doubling problem; the solutions previously seen, Wilson and staggered actions, have to been rearranged taking into account the presence of the interaction term. In particular, for what concerns Wilson fermions, now the added term has to be gauge invariant, thus the M matrix in Eq. (1.57) becomes M_αβ,ij,ab(W ) (ˆn, ˆm) = δn ˆˆmδαβδab( ˆmiδij + 4r)− 1 2 X µ (r − γµ)αβ(Uµ†)ab(ˆn)δijδm,ˆˆ n+ˆµ −1 2 X µ (r + γµ)αβ(Uµ)ab( ˆm)δijδm+ˆˆ µ,ˆn (1.71)

where Dirac (αβ), flavor (ij) and color (ab) indices are explicitly indicated. Considerations made in the previous sections hold also in this case; the mechanism that avoids doubling remains unaltered. Also for staggered fermions, the modification is quite direct; the M matrix acting on the fields χ reads

M(stag.)(ˆn, ˆn0) = ˆmδn,ˆˆn0 + 1 2 X µ ηµ(ˆn)[Uµ†(ˆn)δn+ˆˆ µ,ˆn0− U_µ(ˆn − ˆµ)δ_n−ˆ_ˆ _µ,ˆ_n0] (1.72)

After the basis change, the inclusion of link variables into the action (1.67) adds an irrelevant operator which explicitly breaks taste symmetry. The reason of that can be easily understood if we think that taste degrees of freedom are distributed on different lattice sites, thus different tastes coupled to different links. It is important to stress that this terms maintain the invariance of the action under transformations in Eq. 1.68, thus the remnant chiral symmetry is preserved. Often, effects of taste breaking are suppressed by replacing the link

variables Uµ with the respective smeared version. We will talk about smearing in Chapter

3; for now we simply anticipate that it is a method that proposes to change a link with a sum over longer paths that connect the same points. In this way a gauge configuration gets smoother, and short range fluctuations are eliminated. It appears clear, at least at the naive level, how this can help with taste breaking; if fields are averaged at short distances, different tastes get coupled now with similar link variables and additional terms in the action get suppressed. In this work we will use a 3 flavor staggered action with stout smearing improvement, as we will explicitly show in the following.

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We can now write down the path integral for SQCD using the lattice regularization

hOi_T = 1 Z Z Y ˆ n,µ dUµ(ˆn) Y ˆ m,λ dΨλ( ˆm) Y ˆ m0_,λ0 dΨλ0( ˆm0) O[U_µ, Ψ, Ψ]e−S (L) G e−ΨMFΨ _(1.73)

where the λ index abbreviates all the Dirac,flavor,color indeces related to fermions fields. Several considerations are mandatory. First of all, giving a look at the bosonic measure, one sees that, differently from the continuum formulation, the integration variable is the

link variable Uµ instead of the gauge field Aµ. Secondly, dU has to be a gauge invariant

measure of the gauge group, namely it must holds d(W U ) = d(U W ) = dU ∀W ∈ SU (3); for a compact Lie group such a measure exists (and is unique) and is called Haar measure. Also in this case the gauge invariance introduces a wrong count of the physical degree of freedom that can be cured by a gauge fixing; however this time the incorrect count gives rise only to a finite multiplicative factor because of the compactness of the group, which is immaterial if we are interested in expectation values because it cancels in the ratio with Z. Thus we can avoid the appearance of the Fadden-Popov matrix and the ghost fields on lattice.

For what concerns the fermionic part, we cannot directly perform simulations with non-commutative variables, thus the expression Eq. (1.73) is usually put in a more convenient form. Indeed, by exploiting the properties of the Grassmann integrals, we can integrate out fermionic fields, and Eq. (1.73) becomes

hOi_T = R [D U ] ˜O[Uµ] e −S_G(L)_det(M F[U ]) R [D U ] e−S_G(L)_det(M F[U ]) (1.74) where ˜O is defined as ˜ O ≡ R [D Ψ][D Ψ] O[U, Ψ, Ψ] e −ΨMFΨ R [D Ψ][D Ψ] e−ΨMFΨ (1.75)

Now one can extract a set of gauge configurations {U }(i) in accordance with the probability

distribution

P [U ]D[U ] = e

−SG[U ]_det(M

F[U ])

Z (1.76)

and then evaluate the average of the values O[{U }(i)]. This argument implicitly assumes that

Eq (1.76) can be interpreted as a statistical weight, but this is the case only if det(MF[U ]) ≥

0. One would have to check that this request is fulfilled for each fermion discretization; if the discretized Dirac operator is anti-hermitian (thus has only imaginary or null eigenvalues) and

anti-commutes with γ5 (thus non-zero eigenvalues occurs in conjugate pairs) then the proof

is easily shown. Indeed, the form of the Dirac spectrum allows to write the determinant as det(MF) = det( ˆDdirac+ ˆm) =

Y

i

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This holds for naive fermions as well as for staggered fermions, but it is in general not valid for Wilson fermions; the positivity of the determinant is fulfilled only for a range of the r parameter.

It can be seen that det(M [U ]) constitutes a non local function of the gauge links U by transforming

det(MF[U ]) = exp (log(det MF[U ]))

= exp (Tr[log(MF[U ])]) = exp

Tr log 1 ˆ m ˆ Ddirac+ 1 + log( ˆm) (1.78)

and noting that the expansion in powers of ˆm−1 reads

log 1 + 1 ˆ m ˆ Ddirac ≈ 1 ˆ m ˆ Ddirac+ 1 ˆ m2Dˆ 2 dirac+ . . . (1.79)

Now each Dirac operator connects neighboring sites, while the trace in Eq. (1.78) selects only closed paths; thus the expansion in Eq. (1.79) has only even powers, and includes arbitrarily long paths. Thus a local updating algorithm is extremely time-consuming in simulations with dynamical fermions, because each link enters a lot of paths in the expansion, thus

det(MF[U ]) has to be computed at each step. A first step fort overcoming the problem is the

"bosonization" of the fermions determinant. Exploiting the properties of gaussian integrals one can in principle write

det(MF) =

1

det(M_F−1) = Z

[D φ][D φ∗] e−φ∗MF−1φ (1.80)

where φ are called pseudofermion fields. They are boson fields (thus c-numbers, that means they can be included without problems in simulations), which have the same indices of the corresponding fermion fields; thus the problem is reduced to perform a certain number of matrix inversions. This is however still a partial step towards the non-local HMC algorithm which provides a net improvement and is illustrated at the end of this chapter.

Finally we concentrate on staggered fermions; in particular, we return to the problem of taste degeneracy in the continuum limit; the method usually employed in modern calculations to remove the residual degeneracy is the so called rooting trick. It consists in replacing, for each flavor, det(M_i(stag.)) 7→ det(M_i(stag.))1/4_{, that reads}

Z = Z [D U ] e−SG Nf Y i det((M_i(stag.)) = Z [D U ][D φ∗][D φ] e−SG_e−φ∗M−Nf /4φ _(1.81)

Taking a fractional power of the determinant in this way is not rigorously justified in field theory; naively, at zero lattice spacing, where taste degeneracy is restored, it is easy to

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see that the degenerate eigenvalues are properly counted, providing the correct result. The unanswered question is whether artifacts introduced on the the lattice disappear in the con-tinuum limit (the naive concon-tinuum limit is not enough); although the question is not totally solved, nowadays rooted staggered fermions are largely accepted as a standard practice in dynamical fermions simulations.

1.2.6 Continuum extrapolation

The last missing step for completing our rapid overview on lattice QCD is how to extract from the lattice, physical quantities related to the continuum theory.

What we expect is that, when the continuum limit is approached, the following relation holds for every physical observable

Ophys = lim a→0 1 a dO ˆ O(g0) (1.82)

where ˆO(g0, ˆm0) is the dimensionless lattice operator, dO is the dimension in mass units and

for simplicity we are considering only the pure gauge theory, thus the only bare parameter

is g0. This statement is equivalent to say that a lattice theory, in order to have a

contin-uum limit, has to show a region of parameters where correlation lengths diverge; therefore the continuum limit is equivalent to a second order phase transition for the corresponding statistical system.

In lattice simulations, the lattice spacing a is not an input parameter, and one has to

fine tune g0 in order to approach the correct continuum limit; the correct functional form

is provided by renormalization group theory. In principle one can invert the above relation to find the explicit dependence g0(a) = ˆO−1(adOOphys) but the explicit formula for g0(a) in

general depends on the observable; however, in the regime a 1 it is uniquely determined.

For obtaining it we can consider the lattice counterpart of the quantity gR(µ) defined in Eq.

(1.26). It will be g_R(L)(g0, ˆµ) → gR (it is dimensionless thus no a factors arise). By exploiting

Eq. (1.82) for this observable we have

0 = a d dagR(µ) = a d dag (L) R (g0, ˆµ) = a ∂g(L)_R ∂g0 dg0 da + a ∂g(L)_R ∂ ˆµ dˆµ da =⇒ adg0 da | {z } ≡βlat(g0) = − ∂gR ∂g0 −1 µ∂gR ∂µ | {z } =β(gR) (1.83)

where βlat gives the rate of change of g0 with respect to a, whereas β(gR) has been already

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relation gR= g0+ Cg03+ o(g05) we obtain

βlat(g0) = −β0g03− β1g05+ . . . (1.84)

where the coefficient are exactly the same of β(gR). This means that a relation of the type

(1.30) holds also for the lattice case, from which it follows

a = 1

Λlat

e− 1

2β0g20 (1.85)

meaning that, when a → 0, the coupling goes to the fixed point g0 → 0. It is worth noting

that the two β-functions are conceptually different, and moreover, higher order terms are in

general different.3 At that purpose, notice, for instance, that Λlat 6= ΛQCD. Finally we have

to define a procedure for finding the value of the lattice spacing in physical units, in order to give the correct dimension to results derived from the lattice, which are dimensionless. What is usually done is to pick an already known result of a physical quantity and then impose,

at fixed g0, the equality with what found on the lattice. From this you can find the lattice

spacing. If fermions are taken into account, we have also their masses as bare parameters, thus a quantity related to the spectrum is computed on the lattice and then masses are chosen in order to keep the desired physical result. Therefore we have a prescription for masses at fixed a; if we imagine the space of parameters, this constitutes a curve spanned

by a, mi(a) ∀i. This is generally called Line of Constant Physics.

For concluding this chapter, we furnish some basic ideas about Monte Carlo methods and illustrate the used algorithm for extracting configurations in accordance with Eq. (1.81).

1.3 Monte Carlo methods

Monte Carlo algorithms form the largest and most important class of numerical methods used for solving statistical physics problems. They are based on the fundamental concepts of importance sampling and detailed balance that we briefly illustrate in the following. Consider that usually, when we study a statistical system, physical quantities we are interested in are of the form hOi = P xOxe−βEx P xe−βEx (1.86) where x represents a state of the system and the sum goes over the entire number of possible states, which usually is an extremely huge quantity. The simpler way to compute such a quantity would be to extract N such states with flat probability and then calculate the O

estimator, namely the expression (1.86) over the xi states extracted. This choice is however

3_{The fact that their 2-loop expansion gives rise to the same function depends on the scheme-independence}

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1.3. Monte Carlo methods 29

quite impracticable because of the large number of states. Consider for instance that a

three-dimensional Ising model on a 10 × 10 × 10 lattice has 21000 ' 10300 _{possible configurations;}

putting together the best supercomputers available at the present days, each of which reaches

a computational power of about 100 petaFLOPS4_{, we could obtain a generous 10}19 _{of total}

FLOPS. Then, even if we suppose that a FLOP is enough for generating the value of a single spin, the time required for exploring all the possible states would exceed of several orders of magnitude the age of the universe. Notice that both of the sums in Eq. (1.86) are actually dominated by a small number of states, which contribute the most, whereas the vast majority give a total negligible contribution. From this the idea of extracting configurations accordingly with their statistical weight, i.e. with probability

P [x] = e

−βEx P

xe−βEx

(1.87)

and then estimate hOi as

O = 1

N X

i

Oxi (1.88)

This method works much better with respect to the direct approach because the small number of states which contribute the most to the sum are the ones mostly extracted. In order to obtain a list of states distributed according to a Boltzmannian distribution, we usually make use of Markovian chains. More generally a Markovian process is a stochastic process in which the probability of an event to happens depends on the current state of the system but not on previous events; Markovian chains are Markovian process with a discrete number of states.

Let us fix some formalism: i labels a possible state of the studied system, and Wji ≡ Pi→j

is the probability that starting from the state i we arrive to the state j, with the usual

properties Wji 6= 0 and P_jWji = 1 ∀ i, j. We can imagine an ensemble of Markov chains,

with Π(τ )_j the number of copies that stay in the j-th state after τ steps; we want to reach an

equilibrium density of states, where Π(τ +1)_j = Π(τ )_j or, using Wji

Π(eq)_j if WjiΠ(eq)i = Π (eq)

j (1.89)

or in other words, we are interested in eigenvectors of Wij with eigenvalue 1. Moreover, we

require that its multiplicity has to be equal to 1 and that other eigenvalues must be < 1; for such a W we have (W )N_jiΠ(0)_i = Π(eq)_j after a finite number of steps. It can be proven that these requirements are equivalent to the following ones:

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• Ergodicity: It means that

∀ i, j ∃k ∈ N

(Wk)ij 6= 0 (1.90)

Less formally we require that every state can be reached starting from every other state after a finite number of steps.

• Aperiodicity:

if we define the period of a Markov chain as the gcd of the integers d such that

(Wd)ii6= 0 (1.91)

a Markov chain is called aperiodic if di = 1, ∀ i. Aperiodicity ensures us that we deal

with an equilibrium process and not an oscillating one. Notice that from these requirements it follows that

X j WijΠj = Πi ⇒ WiiΠi+ X j6=i WijΠj = Πi( X j Wji) ⇒X j6=i WijΠj = X j6=i WjiΠi ∀ i (1.92)

or in other words that the probability to arrive in i starting from a different state is equal the probability to reach the j state starting from i; this is the equation of balance (a sort of conservation law of the probability flux).

The last condition can be replaced by the so-called detailed balance, which is a more strict constraint with respect to simple balance. Indeed, detailed balance requires

WijΠj = WjiΠi ∀ i, j (1.93)

Returning to our original problem, we want to create a Markov chain such that Π(eq)_i = Pi,

i.e. that the equilibrium configuration is the Boltzmann distribution we are interested in. Let’s see two of the most used algorithm that implement these ideas. We anticipate that the use of Markov chains cause resulting data to be correlated, a fact that has to be taken into account in correctly estimating errors.

1.3.1 Metropolis algorithm

The Metropolis-Hastings algorithm first proposed by Metropolis et al. in 1953 [14] and then generalized by Hasting for non symmetric selection matrices [15] is constituted by the following steps.

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1.3. Monte Carlo methods 31

• Starting from a state i, we select a state j(t) _{with probability A}

j(t)_i. The element A_ij has to be interpretable as a probability, thus only non negative values are allowed. • We construct the quantity

r_j(t)_i ≡ P_j(t) Pi A_ij(t) Aji (1.94)

• if r ≥ 1 we accept the change i → j = j(t)_{. If r < 1 we extract}

j =    j(t) with probability r i with probability 1 − r (1.95)

This entire procedure defines the matrix Wji.

Wji =    Aji if rji ≥ 1 Ajirji if rji < 1 (1.96)

We can explicitly check that such a procedure satisfies detailed balance. if rji ≥ 1 ⇒ rij < 1 WjiPi = AjiPi = Pi Pj Pj Aji Aij Aij = WijPj (1.97)

and it is specular for the case rji < 1. Usually one takes a symmetric selection matrix

Aji = Aij, thus the r term becomes rji = Pj/Pi.

1.3.2 Heat-bath algorithm

An alternative to the Metropolis method is represented by the heat-bath algorithm. Suppose we have a probability distribution depending on several variables

P (q1, q2, q3, . . .)

Y

i

dqi (1.98)

The idea is to update only a small number of variables, keeping the others fixed. If we divide

qi into two groups {qi} = (˜q, ˆq) and decide to change only ˆq, the conditioned probability

P (ˆq|˜q)) is the probability to have ˆq at given ˜q. The algorithm is made of the following steps

• Select a subgroup ˆq with probability Q;