Localization Transition in QCD at the Roberge-Weiss Point

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Dipartimento di Fisica E. Fermi

Corso di Laurea Magistrale in Fisica

Localization Transition in QCD at the

Roberge-Weiss Point

Candidate:

Francesco Garosi

Prof. Massimo D'Elia

Supervisor:

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Introduction

Quantum Chromodynamics (QCD) is the theory of strong interactions: it describes Nf avours of quarks,

which exist in Ncdierent colours, coupled to a non abelian SU(Nc)gauge eld which represents the gluons

(in the physical case Nf = 6 and Nc = 3). Its formulation is very similar to Quantum Electrodynamics

(QED) but it shows peculiar properties, such as connement and asymptotic freedom. Thanks to the latter, the theory is weakly coupled at high energies and can be studied perturbatively. On the other hand at low energy the theory is strongly coupled and we need non perturbative methods to study all the interesting phenomena that occur at such energy scales, for instance connement, chiral symmetry restoration and the phase diagram of the theory in general.

One of these non perturbative techniques is lattice QCD: we perform numerical simulations based on Monte-Carlo methods, using a discretized version of the action as probability distribution for the eld congurations and on these congurations we compute quantities of interest. Numerical simulations have been intensively used to study the phase diagram of the theory, rst of all in absence of baryon chemical potential. The limits of zero and innite quark masses are particularly interesting, because in both cases the theory shows a spontaneously broken symmetry and there is a phase transition between the phases with broken/restored symmetry. In the rst case there is the chiral SU(Nf)V × SU (Nf)A symmetry, while in the opposite limit

the theory has the Z3 center symmetry. In both cases we can dene an order parameter which is zero when

the symmetry is exact and nonzero in the broken phase: simulations showed that the chiral symmetry is broken below a temperature Tχ≈ 140MeV, while the center symmetry has the opposite behaviour, i.e. it is

broken above a temperature TD≈ 270MeV. The interesting thing about center symmetry is that it can be

used to dene connement: its order parameter, the Polyakov loop L, which is the trace of a straight Wilson line in the time direction, is linked with the free energy of a pair of quarks with innite spatial separation and nonzero values of hLi mean nite free energy. It follows that when the center symmetry is broken we need a nite energy to separate the pair, i.e. above TD the theory deconnes and quarks and gluons

become the eective degrees of freedom. Dynamical fermions (quarks with nite masses) break explicitly both symmetries, so in the physical case there is not a real phase transition. However there is a cross-over region in temperature where the thermodynamic quantities quickly change their behaviour (Ref. [5]): this region separates a conned phase with broken chiral symmetry and a chirally symmetric quark-gluon plasma. In the cross-over region we can identify a pseudo-critical temperature Tc, but of course its determination is

dicult due to the absence of a real transition.

When we study the phase diagram at nite density, i.e. we introduce a nonzero baryon chemical potential µB,

we have to face the so called sign problem: the action becomes complex and we can not perform numerical simulations because we do not have a probability distribution for the elds.

The sign problem disappears when the chemical potential is purely imaginary, µB = iµB,I: in this case the

eect is just to shift the time component of the gauge eld, the action is real and positive denite and we can perform numerical simulations. The aim is to obtain the properties at real µB with analytic continuation.

The study of QCD with imaginary chemical potential is also important because in correspondence of the Roberge-Weiss (RW) points θB ≡ µB,I/T = kπ, with k an odd integer, the theory has a Z2symmetry, which

is the only one in presence of nite quark masses. Moreover this symmetry is spontaneously broken and it is a remnant of the Z3 center symmetry of the pure gauge theory, so it can be used to dene connement:

above a transition temperature TRW the symmetry is broken and the theory deconnes.

Thanks to the presence of this transition, it is interesting to study the properties of QCD at the RW point and to compare them with the results obtained at µB= 0. In this work we study the Anderson localization

transition of the eigenmodes of the QCD Dirac spectrum at the RW point with Nf = 2 + 1fermions, i.e. two

degenerate light quarks and a heavier one, on the basis of the results in Refs. [28] and [29], which proved that such a transition occurs at µB = 0 both in QCD with physical quark masses and in the pure gauge theory.

Those works showed that at low temperatures the Dirac spectrum is well described by the Gaussian Unitary Ensemble (GUE) in Random Matrix Theory (RMT) and this situation corresponds to completely delocalized modes, because the gauge eld uctuations mix the eigenmodes nearby in the spectrum. In the opposite case uctuations cannot mix the eigenmodes, which become localized (they spread uniformly in a fraction of the

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total four-volume and are zero everywhere else): in this limit the eigenvalues are all independent and they obey the Poisson statistics. Numerical simulations showed that above a critical temperature the lowest part of the Dirac spectrum becomes localized and higher up in the spectrum there is a crossover between Poisson and GUE statistics. The interesting thing is that the result for the critical temperature of the localization transition is compatible with TD in the pure gauge theory and with the cross-over region in the physical

case. The name "Anderson localization" comes from an analogous phenomenon discovered by Anderson about the diusion in disordered materials (Ref. [3]): for a number of dimensions d ≥ 3 and in presence of strong enough disorder (for instance, impurities in the material) diusion does not take place because all the eigenfunctions of the Hamiltonian are localized, while for weak disorder some of them become delocalized. In this work we repeated the analysis at the RW point in order to nd the critical temperature Tl at which

localized modes appear in the Dirac spectrum and to compare it with TRW: do they coincide or not? Previous

studies, for instance Ref. [22], showed that the Anderson localization may be the mechanism that drives the chiral phase transition and the evidences that TRW = Tχ in the chiral limit (Ref. [7]) and that the RW

endpoint is compatible with the analytic continuation of the pseudo-critical transition line present at zero density (Ref. [9]) gave us a hint that they may coincide. The relation TRW = Tl = Tχ, if true, may be

important for the zero density physics too: in the large Nc limit, which gives quite good answers to open

questions in QCD, for instance the mass of the η0 _{meson, the eect of the chemical potential is suppressed.}

Then, if the limit is quite well realized, we can expect to have at zero density the same separation of phases present at the RW point, i.e. a chirally symmetric quark-gluon plasma with some localized Dirac modes and a conned, non-chiral phase with all delocalized modes: within this picture, in the latter case the two phases are exactly separated by the critical temperature common to the three transitions, while in the former there is just a cross-over.

We performed our simulations with the RHMC algorithm on Nt×Ns3isotropic lattices with lattice spacing a,

determined tuning the temperature of the system T = 1/(aNt), and we explored three dierent values of the

time extension: Nt= 4, 6, 8. At xed Nt and for many values of the temperature above the corresponding

TRW we computed the lowest eigenvalues of the Dirac operator, which have the form m + iλ, and for each

temperature we calculated the mobility edge λc(T, Nt), which is the boundary value of λ between localized

and delocalized modes of the Dirac operator. Finally with a t of the data T − λc(T, Nt) we computed

the critical temperature Tl(Nt) for which λc(Tl) = 0, i.e. at which localized modes disappear, and we

extrapolated the continuum limit Nt→ ∞with the three data obtained.

The thesis is structured in ve chapters, followed by the conclusions and the appendices:

• In the rst chapter we introduce the continuum formulation of QCD, starting with the construction of the Lagrangian and its quantization, and then we focus on some properties, such as the asymptotic freedom and the symmetries.

• The second chapter is a review of the lattice formulation of QCD and of the Monte-Carlo methods used in the simulations.

• In the third chapter we introduce the imaginary baryon chemical potential and we show some liter-ature results about the RW point.

• In the fourth chapter we describe the Anderson localization both in condensed matter and in QCD, showing some literature results and explaining the reasons to study it at the RW point too.

• The fth chapter collects all our numerical results about the localization transition.

• In the conclusions we summarize the results obtained and we discuss some possibilities for further studies.

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Chapter 1

Introduction to QCD

In this chapter we introduce the continuum formulation of QCD, that is the non abelian SU(Nc) gauge

theory with Nf fermions describing the quarks. We start writing its Lagrangian and then we recall the

main results about the path integral formulation of a Quantum Field Theory (QFT), both in Minkowski and Euclidean spaces, which is the basis of Monte-Carlo simulations, as we will explain in chapter 2. Finally we show QCD's main properties, like asymptotic freedom and its symmetries, which will be useful in the next chapters.

1.1 Construction of the Lagrangian

First of all we write the Lagrangian of the theory. To do so, we start from the Lagrangian of a free fermion with mass m:

L = ¯ψ(i /∂ − m)ψ (1.1) where /∂ = γµ_∂

µ, γµ are the Dirac gamma matrices, which satisfy the anticommutation relation γµ; γν =

2ηµν_{1, ψ is the spinor which describes the fermion and ¯}_{ψ = ψ}†_γ0_{. If the quark exists in N}

c colours, the

Lagrangian is just the sum of Nc copies of the one in eq. (1.1). In a compact form we can write (f stands

for fermions) Lf = Nc X j=1 ¯ ψj(i /∂ − m)ψj= ¯Ψ(i /∂ − m)Ψ (1.2) Ψ = (ψ1, ψ2, . . . , ψNc) T _(1.3)

where T stands for transposed, i.e. Ψ is a column vector. Now we perform a global SU(Nc)transformation:

Ψ → Ψ0= ΩΨ; Ψ → ¯¯ Ψ0 = ¯ΨΩ† (1.4) with Ω ∈ SU(Nc)and independent on the space-time coordinates x. Since Ω†Ω =1 by denition and the

operator i(/∂ − m) does not act on Ω, the Lagrangian (1.2) is invariant under this transformation:

Lf → ¯ΨΩ†(i /∂ − m)ΩΨ = ¯ΨΩ†Ω(i /∂ − m)Ψ = Lf (1.5)

In the gauge theory, the Lagrangian in invariant under a local transformation, that is the matrix Ω now is x-dependent. In this case the Lagrangian (1.2) is not invariant, because the derivative acts on Ω too. The problem is simply solved by the substitution of the derivative with an operator /D = γµ_D

µ such that /DΨ

transforms as Ψ under SU(Nc):

/

DΨ → Ω /DΨ (1.6)

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In order to nd this operator, we introduce the parallel transport W (Cx→y)from x to y along the path C.

If we consider the local gauge transformation Ψ(x) → Ω(x)Ψ(x) and we call Vx the vector space generated

by the vectors Ψ(x), W (Cx→y)is a SU(Nc)matrix that maps Vxin Vy with the following properties:

• W is the identity for the path with zero length (∅):

W (∅) = 1 (1.7)

• if C is the composition of two paths C1 and C2then

W (C = C2◦ C1) = W (C2)W (C1) (1.8)

• for the inverse path −C (that is C with opposite direction)

W (−C) = W (C)−1 (1.9) • under a local SU(Nc)transformation, W changes as follows:

W (Cx→y) → W0(Cx→y) = Ω(y)W (Cx→y)Ω−1(x) (1.10)

Eq. (1.10) tells us that if we take a vector Ψ(x) ∈ Vx and we transport it with W along a path C towards

y, then W Ψ(x) ∈ Vy, as we can see performing a gauge transformation:

˜

Ψ(y) ≡ W (Cx→y)Ψ(x) → Ω(y)W (Cx→y)Ω−1(x)Ω(x)Ψ(x) = Ω(y)W (Cx→y)Ψ(x) = Ω(y) ˜Ψ(y) (1.11)

Since W is a SU(Nc)matrix acting on a Nc-vector, its generic form is

W = eiαaTa (1.12)

where a = 1, . . . , N2

c − 1. Ta are the SU(Nc) generators in the fundamental representation (which has

dimension Nc): they are traceless and hermitian and they satisfy the following relations:

[Ta, Tb] = ifabcTc

Tr[TaTb] = 1₂δab

(1.13) For an innitesimal path from x to x + dx, in order to recover eq. (1.7) in the limit dx → 0, we write the parallel transport as follows

W (Cx→x+dx) = e−igAµ(x)dx

µ

(1.14) where Aµ = AaµTa. The Nc2− 1 Aaµ are the gauge elds, which we will call gluons. With this denition the

properties from (1.7) to (1.9) are automatically satised, while eq. (1.10) holds only if the gauge eld Aµ

transforms in a proper way under SU(Nc). Since dx is innitesimal, we can expand everything at rst order:

W (Cx→x+dx) = e−igAµ(x)dx µ ≈1 − igAµ(x)dxµ W0_(C x→x+dx) = e−igA 0 µ(x)dx µ ≈1 − igA0 µ(x)dxµ Ω(x + dx) ≈ Ω(x) + ∂µΩ(x)dxµ (1.15)

Inserting these expansions in eq. (1.10) we nd, discarding the terms of the second or higher order in dx:

W0≈ Ω(x) + ∂µΩ(x)dxµ 1 − igAµ(x)dxµΩ−1(x) =1 − ig h Ω(x)Aµ(x)Ω−1(x) + i g∂µΩ(x)Ω −1_(x)i_dxµ (1.16)

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CHAPTER 1. INTRODUCTION TO QCD 3 Comparing this result with the expansion of W0 _{in eq. (1.15) we nd the transformation rule of the gauge}

eld under the local gauge transformation:

A0_µ= Ω(x)Aµ(x)Ω−1(x) +

i

g∂µΩ(x)Ω

−1_(x) _(1.17)

With the parallel transport and the transformation law (1.17) we can dene the operator Dµ: we generalize

the notion of dierential by writing

DΨ = W (Cx+dx→x)Ψ(x + dx) − Ψ(x) = W†(Cx→x+dx)Ψ(x + dx) − Ψ(x) (1.18)

that at rst order in dx becomes

DΨ ≈ ∂µΨ(x) + igAµ(x)Ψ(x)dxµ≡ DµΨ(x)dxµ (1.19)

The operator Dµ is called covariant derivative and it has the following form:

Dµ= ∂µ+ igAµ (1.20)

By construction, DΨ transforms as Ψ under SU(Nc), so eq. (1.6) holds and we can nally write the fermionic

Lagrangian Lf,I: Lf,I = Nc X j=1 ¯ ψj(i /D − m)ψj= ¯Ψ(i /D − m)Ψ = Lf − g ¯ΨγµAµΨ (1.21)

We obtained the Lagrangian of a quark, which exists in Nc copies, interacting with the gauge eld Aµ. Now

we have to write the kinetic term of the gauge eld, that is the Lagrangian of the so called Yang-Mills (YM) or pure gauge theory. In analogy with Quantum Electrodynamics (QED), we write this term as

LY M = − 1 2Tr(FµνF µν_{) = −}1 4F a µνF µν a (1.22)

where Fµν is the eld tensor, which can be obtained from the commutator of two covariant derivatives:

Fµν= Fµνa Ta =

1

ig[Dµ, Dν] = ∂µAν− ∂νAµ+ ig[Aµ, Aν] = (∂µA

a

ν− ∂νAaµ− gfabcAbµAcν)Ta (1.23)

We can easily obtain the transformation law of the eld tensor: using eqs. (1.4) and (1.6) we get

D0_µΨ0 = ΩDµΨ = ΩDµΩ−1Ψ0 (1.24)

and so

D0_µ= ΩDµΩ−1 (1.25)

Inserting this result in eq. (1.23) we immediately nd

Fµν0 = ΩFµνΩ−1 (1.26)

Thanks to the transformation rule (1.26) and the cyclic property of the trace, the pure gauge Lagrangian is gauge invariant, as it has to be. Putting all the pieces together, we nally obtain the Lagrangian:

L = −1 4F

a

µνFaµν+ ¯Ψ(i /D − m)Ψ (1.27)

The Lagrangian in eq. (1.27) can be easily generalized to Nf avours of quarks: for each quark we build the

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LQCD = − 1 4F a µνF µν a + Nf X f =1 ¯ Ψf(i /D − mf)Ψf (1.28)

Before showing some properties of QCD, we derive a relation between a parallel transport along an innites-imal parallelogram and Fµν, which we will use in chapter 2 to build the lattice version of the pure gauge

Lagrangian. We start writing the formula for a generic parallel transport: let us consider a path C from x to y parameterized by the curve z(τ), with τ ∈ [0, 1], z(0) = x and z(1) = y. We take an intermediate point z(τ ) and let Cτ be the path from x to z(τ): then, according to the composition rule (1.8), we have

W (Cτ +dτ) = W (Cz(τ )→z(τ +dτ ))W (Cτ) (1.29)

Using the rst order form of W (Cz(τ )→z(τ +dτ )), we nd the dierential equation for W :

dW (Cτ)

dτ = −igAµ z(τ ) dz

µ

dτ W (Cτ) (1.30) The solution of this equation is well known:

W (Cx→y) =P exp ( − ig Z 1 0 Aµ z(τ ) dz µ dτ dτ ) = = +∞ X n=0 (−ig)n n! Z 1 0 dτ1· · · Z 1 0 dτnP Aµ1 z(τ1) dz µ1 dτ1 . . . Aµn z(τn) dz µn dτn (1.31)

where the symbol P stands for path-ordering: PAµ z(τ1)Aν z(τ2)

= θ(τ1− τ2)Aµ z(τ1)Aν z(τ2) + θ(τ2− τ1)Aν z(τ2)Aµ z(τ1) (1.32)

Using eq. (1.31) we can nd the second order expression for a parallel transport along a straight path Cx→x+dx, for which z(τ) = x + τdx and dzµ/dτ = dxµ:

W (Cx→x+dx) =1 − ig Z 1 0 dτ (Aµ(x) + ∂νAµ(x)τ dxν)dxµ+ −g 2 2 Z 1 0 dτ1 Z 1 0 dτ2Aµ(x)Aν(x)dxµdxν = exp n − igAµ(x + dx 2 )dx µo (1.33)

where the path-ordering has been omitted because Aµ and Aν are computed at the same τ.

Now let us consider an innitesimal parallelogram Cx→x with sides dx and dy, that is

Cx→x= Cx+dy→x◦ Cx+dx+dy→x+dy◦ Cx+dx→x+dx+dy◦ Cx→x+dx (1.34)

Using eqs. (1.8) and (1.9) we have:

W (Cx→x) = W−1(Cx→x+dy)W−1(Cx+dy→x+dy+dx)W (Cx+dx→x+dx+dy)W (Cx→x+dx) (1.35)

Now we use the second order form of each parallel transport and we apply the Baker-Campbell-Hausdor formula for the product of the exponentials of two matrices A and B:

eλAeλB= eλ(A+B)+λ22[A,B]+O(λ 3₎

(1.36) After some algebra, we nd:

W (Cx→x) = exp

− igFµνdxµdyν

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CHAPTER 1. INTRODUCTION TO QCD 5

1.2 Path integral formulation of a QFT

In this section we briey recall the path integral formulation of a Quantum Field Theory (QFT). We want to write a formula to compute the n-points Green functions (or correlation functions) of the theory:

Ga1...an(x1, . . . , xn) = h0|T [φa1(x1) . . . φan(xn)]|0i (1.38)

where |0i is the ground state (or vacuum), φa are the elds (the index a runs over all the dierent elds of

the theory, possible Lorentz indices are omitted) and T is the time ordered product:

T [φ(x1)φ(x2)] = θ(x01− x02)φ(x1)φ(x2) + θ(x02− x01)φ(x2)φ(x1) (1.39)

We start with a bosonic theory (elds with integer spin). It can be shown, starting from transition amplitudes in Quantum Mechanics (QM) and generalizing to QFT, that the Green functions can be computed as follows:

Ga1...an(x1, . . . , xn) = R P BC[dφ]e iS_φ a1(x1) . . . φan(xn) R P BC[dφ]e iS (1.40)

where the measure [dφ] means integration over all the elds in all the innite space-time points: [dφ] = Y

t,~x,a

dφa(t, ~x) (1.41)

P BC are periodic boundary conditions in time, φ(t1, ~x) = φ(t2, ~x)and S is the action of the theory:

S = Z d3~x Z t2 t1 dtL (1.42)

where t1= −T e−i and t2= T e−i, in the limits T → ∞ and → 0. In the following we will assume these

boundary conditions and limits, without writing them explicitly. Now we introduce the generating functional Z[J]:

Z[J ] = Z

[dφ]eiR d4x(L+Jaφa) (1.43)

where Ja are external sources, one for each type of eld. Using eq. (1.40), we see that the n-points Green

functions can be obtained with n functional derivatives of Z[J], computed at J = 0: Ga1...an(x1, . . . , xn) = (−i)n Z[0] δn_{Z[J ]} δJa1(x1) . . . δJan(xn) _{J =0} (1.44) For fermionic theories (elds with half-integer spin) the elds anticommute and we can not use the previous formulas. To dene a path integral for the fermions, we introduce the Grassmann variables ηi(i = 1, . . . , N):

ηi, ηj = ηiηj+ ηjηi= 0 ∀i, j (1.45)

with the following integration rules (Berezin rules): Z

dηi= 0;

Z

dηiηi= 1 (1.46)

In the same way, we dene two derivatives of a function f(η) with respect to a Grassmann variable, called left (L) and right (R) derivatives:

( _∂ L(ηiF ) ∂ηi = F ∂R(F ηi) ∂ηi = F (1.47)

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and both derivatives are zero if ηi does not appear in f(η). These rules dene completely the integral and

the derivative of a function of Grassmann variables, because each ηi can appear only once in a function,

otherwise it is zero thanks to eq. (1.45).

Now let us consider 2N Grassmann variables ψ1, . . . , ψN, ¯ψ1, . . . , ¯ψN and we dene the following generating

functional:

Z[ρ, ¯ρ] = Z

[d ¯ψdψ]e− ¯ψiAijψj+ ¯ψiρi+ ¯ρiψi _{= e}ρ¯iA−1ijρjdetA (1.48)

where [d ¯ψdψ] =QN

l=1d ¯ψldψl. In the computation of Z[ρ, ¯ρ] we used a change of variables and the following

integral:

Z

[d ¯ψdψ]e− ¯ψiAijψj ₌detA (1.49)

From this generating functional, we can obtain the Green functions by taking the proper left and right derivatives, for instance:

hψiψ¯ji = 1 Z[0, 0] ∂R ∂ρj ∂L ∂ ¯ρi Z[ρ, ¯ρ] _ρ=ρ=0_¯ = A−1_ij (1.50) For a fermionic QFT, we extend the previous results introducing a set of Grassmann variables ψα(x), ¯ψα(x)

for each one of the innite points of the space-time. α is the spinorial index, which in the rest of the thesis we will call Dirac index. In conclusion, our generating functional is

Z[ρ, ¯ρ] = Z

[d ¯ψdψ]eiR d4x L+ ¯ψα(x)ρα(x)+ ¯ρα(x)ψα(x)

(1.51)

1.2.1 Quantization of a gauge theory

From eqs. (1.43), (1.44), (1.50) and (1.51) we see that the two points Green function, or propagator, of a eld is given by the inverse of the operator that appears in the quadratic part of the action. For a gauge theory this procedure has a problem, because that operator is not invertible. Let us consider the QCD case, with the Lagrangian in eq. (1.22). Using eq. (1.23) we have:

F_µνa F_aµν = 2(∂νAaµ∂ ν_Aµ

a − ∂νAaµ∂ µ_Aν

a) + . . .

where we omitted terms with more than two gauge elds. Integrating by parts, we nd that the quadratic part of the action is

SY M,q= −

1 2

Z

d4xd4yAµ(x)aδabδ(4)(x − y)(∂yµ∂νy− ηµνy)Abν(y) (1.52)

where = ∂µ∂µ. We see that we have to invert the following operator:

K_µνab(x, y) = δabδ(4)(x − y)(∂yµ∂ ν y − η

µν

y)

Unfortunately, K is not invertible because it has zero modes of the form ∂νθ:

(∂µ∂ν− ηµν

)∂νθ = (∂µ − ∂µ)θ = 0

One way to solve this problem, due to the redundancy coming from the gauge invariance, is to follow the Faddeev-Popov procedure, adding to the Lagrangian a gauge xing term (GF) and new unphysical elds c and ¯c called ghosts and antighost (GH) (one pair for each gauge eld), which are fermions (they are expressed in terms of Grassmann variables) but with no spinorial indices and with a scalar propagator:

LGF = −_2λ1(Ga)2

LGH∝ ¯ca δG_δAab µ

δcAbµ

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CHAPTER 1. INTRODUCTION TO QCD 7 where G is the gauge xing operator, and δcAbµ is the variation of the gauge eld under an innitesimal

gauge transformation with the ghost as parameter. Explicitly in the Lorenz gauge Ga _{= ∂}µ_Aa

µ we nd that

the complete Lagrangian is

L = LQCD+ LGF+ LGH = − 1 4F a µνF µν a + Nf X f =1 ¯ Ψf(i /D − mf)Ψf− 1 2λ(∂ µ_Aa µ) 2_{+ ¯}_ca_(−∂µ_)D(agg) µ,ab cb (1.54) where D(agg)

µ,ab is the covariant derivative in the adjoint representation on SU(Nc).

1.3 The Euclidean formulation of a QFT and application to QCD

The construction we built so far lies in the four-dimensional Minkowski space, where the metric has signature (1, −1, −1, −1). Now we want to switch to the formulation in the Euclidean space, where the metric is positive denite. To do so we perform a Wick rotation, that is we consider imaginary time, while the space components are the same:

t = x0= −ixE,4; xj = xE,j for j = {1, 2, 3} (1.55)

We will call xE,4Euclidean time. With this rotation we can rewrite the exponent iS in the following way:

iS = i Z d3~x Z dtL ≡ −SE= − Z d3x~E Z dxE,4LE (1.56)

where SE and LE are the Euclidean action and Lagrangian respectively. In the following we will use the

notation d4_x_{and d}4_x

Efor the integration measure in Minkowski and Euclidean spaces, with xE= ( ~xE, xE,4).

For instance, let us consider a standard Lagrangian of a scalar eld φ, such that after the Wick rotation φ(x) = φE(xE):

L = 1

2∂µφ(x)∂

µ_{φ(x) − V (φ(x))} _(1.57)

Applying eq. (1.55) we nd for the derivatives ∂0=

∂ ∂x0

= i ∂ ∂xE,4

≡ i∂E,4; ∂j = ∂E,j (1.58)

and so we obtain iS = i Z d4x 1 2∂0φ(x)∂0φ(x) − 1 2∂iφ(x)∂iφ(x) − V (φ(x)) = = i Z (−i)d4xE − 1 2∂E,4φE(xE)∂E,4φE(xE) − 1 2∂E,iφE(xE)∂E,iφE(xE) − V (φE(xE))

Using the denition in eq. (1.56) we conclude that the Euclidean Lagrangian is LE=

1

2∂E,µφE(xE)∂E,µφE(xE) + V (φE(xE)) (1.59) where ∂E,µφE∂E,µφE= ∂E,iφE∂E,iφE+ ∂E,4φE∂E,4φE because of the signature of the metric.

Now we want to nd the Euclidean action of QCD, starting with the Yang-Mills action. First of all we notice from eq. (1.20) that the gauge eld must transform as a derivative under a Wick rotation, so that the covariant derivative has the proper transformation rule:

A0(x) = iAE,4(xE); Aj(x) = AE,j(xE) (1.60)

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F0i(x) = iFE,4i(xE); Fij(x) = FE,ij(xE) (1.61)

where we dened

FE,µν = ∂E,µAE,ν− ∂E,νAE,µ+ ig[AE,µ, AE,ν] (1.62)

Now we are ready to write the euclidean Lagrangian/action: LY M = − 1 2Tr(FµνF µν_{) = −}_Tr(F 0iF0i) − 1 2Tr(FijF ij_{) = −}_Tr(F E,4iFE,4i) − 1 2Tr(FE,ijFE,ij) where we used that F is antisymmetric and that in Minkowski space F0i= −F0i and Fij = Fij.

Using eq. (1.56) we nally obtain the Euclidean action: SE,Y M = Z d4xELE,Y M = Z d4xE 1 2Tr(FE,µνFE,µν) (1.63) Now we consider the fermionic action. The Euclidean action is obtained introducing the Euclidean gamma matrices, linked with the usual Dirac matrices by the following relations:

γE,4= γ0; γE,j= −iγj (1.64)

These matrices satisfy the same anticommutation relation as in Minkowski space but with the new signature, i.e.:

γE,µ; γE,ν = 2δµν1 (1.65)

With these matrices, the Euclidean action is

SE,f = Z d4xELE,f = Z d4xE Nf X f =1 ¯ Ψf( /DE+ mf)Ψf (1.66)

where /DE = γE,µDE,µ= γE,µ(∂E,µ+ igAE,µ)is the covariant derivative in the Euclidean space.

Putting all the pieces together, the complete Euclidean action of QCD is SE,QCD = Z d4xE ( 1 2Tr(FE,µνFE,µν) + Nf X f =1 ¯ Ψf( /DE+ mf)Ψf ) (1.67)

1.3.1 Euclidean path integral

All the results obtained for the path integral in Minkowski space can be immediately extended to the Euclidean space performing the Wick rotation. For instance, the Green functions can be computed as follows:

Ga1...an(xE,1, . . . , xE,n) =

R P BC[dφE]e −SE_φ Ea1(xE,1) . . . φEan(xE,n) R P BC[dφE]e−SE (1.68) with SE= Z d3x~E Z ∞ −∞ dxE,4LE (1.69)

The Euclidean formulation is important because we can use the path integral to compute the partition function of the system at temperature T :

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CHAPTER 1. INTRODUCTION TO QCD 9 Z(T ) =Tr e−βH = Z P BC [dφE]e−SE = Z P BC [dφE]e−R d 3_x_~ ER0βdxE,4LE (1.70)

where β is the inverse of the temperature, H is the Hamiltonian of the system and φ stands for a generic set of bosonic elds. For fermionic elds the result is the same, but with antiperiodic boundary conditions. For simplicity we sketch the proof in the one dimensional QM case and then we generalize to QFT: labelling with |x, tithe state which describes a particle in position x at time t, the transition amplitude K(xf, tf, x0, t0)

from |x0, t0ito |xf, tfiis given, in the path integral formulation, by

K(xf, tf, x0, t0) = hxf|e−iH(tf−t0)|x0i =

Z

x(t0;f)=x0;f

[dx]eiS[x]

which in the Euclidean formulation becomes (from now on we omit the subscript E for the position variable and the elds)

K(xf, τf, x0, τ0) = hxf|e−H(τf−τ0)|x0i =

Z

x(τ0;f)=x0;f

[dx]e−SE[x]

where we set τ = xE,4. Computing the trace in equation (1.70) over the position eigenstates we obtain

Z(T ) =Tr e−βH = Z dx hx|e−βH|xi = Z dxK(x, β, x, 0) = Z x(β)=x(0) [dx]e−SE[x]

This ends the proof in the QM case. The formula can be extended to a QFT discretizing the space, considering the eld as a set of variables, one for each space point, and nally taking the continuum limit. The periodic (antiperiodic) boundary conditions for bosons (fermions) are the generalization of x(β) = x(0).

We see that if the Euclidean action is real and positive denite we have a probability distribution for the eld congurations:

P [φ][dφ] = e

−SE_[dφ]

R [dφ]e−SE (1.71)

Let us see what happens in QCD: using eqs. (1.49), (1.67) and (1.70) we can integrate over the fermions and write the partition function as (we drop all the subscripts E, they are understood for the rest of this section) Z = Z [dAµ] Y f [d ¯ΨfdΨf]e−SQCD = Z [dAµ] Y f DetMf[A]e−SY M (1.72)

where Mf = /D + mf is the Dirac operator for a single avour of quark. SY M is real, so the exponential is

positive denite and to see that we can dene a probability distribution for the gauge elds it is enough to prove that DetMf is real and positive. Since the operator /D is antihermitian, an eigenvalue of the Dirac

operator is α = mf+ iλ, with λ ∈ R. Let ϕ be the eigenvector of M with eigenvalue α, that is /Dϕ = iλϕ:

then γ5ϕ1 is an eigenvector with eigenvalue α?. The proof is immediate, because γ5 anticommutes with all

the gamma matrices:

(mf+ /D)γ5ϕ = mfγ5ϕ − γ5Dϕ = (m/ f− iλ)γ5ϕ = α?γ5ϕ

We conclude that each eigenvalue comes in pair with its complex conjugate, so the determinant of the Dirac operator is real and positive:

DetMf = Y α αα?=Y λ (m2_f+ λ2) > 0

This result is really important, because we can dene a probability distribution for the eld congurations, which can be used in numerical simulations based on Monte-Carlo methods, as we will see in chapter 2.

1_γ₅ _{is the fth Dirac matrix. In Minkowski space its expression is γ}₅ _{= iγ}₀_γ₁_γ₂_γ₃_{, which in Euclidean space becomes}

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1.4 Renormalization and asymptotic freedom

Now we briey recall the asymptotic freedom of QCD, which is the key item to dene a proper continuum limit on the lattice. This property comes from the renormalization of the theory, necessary to remove the divergences that arise once we compute the quantum corrections to the Green functions.

We substitute the elds and the parameters in the Lagrangian with the bare ones, which will be divergent, in the following way (the quantities without subscripts are the renormalized ones):

Aµ_B = Z_A1/2Aµ; ΨB= Z 1/2

ψ Ψ; caB= Zc1/2ca; gB= Zgg; mf B= Zmmf; λB = Zλλ

(1.73) The constants Z must be chosen in order to cancel the divergences. The full calculation can be found in Ref. [33], here we are just interested in the following result, obtained at one loop order in dimensional regularization and in the MS scheme:

Zg= 1 − g2 (4π)2 11Nc− 2Nf 6 1 + O(g 4₎ _(1.74)

where is the dimension of the coupling gB and it is linked with the dimension d of the space-time used to

make the computations in dimensional regularization: = 4 − d

2 (1.75)

In the limit → 0 the divergences appear in the poles 1/. In dimensional regularization we introduce a renormalization scale µ such that

gB= µZgg (1.76)

In this way the renormalized coupling g is dimensionless in d dimensions.

Asymptotic freedom is linked with the behaviour of g with the renormalization scale, which is contained in the β function: β(g) = µdg dµ _B (1.77)

where the subscript B stands for constant bare parameters. This function can be computed noticing that the same derivative applied to gB is zero:

0 = µdgB dµ _B = µd(µ _Z gg) dµ _B = µZgg+µµ d(Zgg) dµ _B = µZgg+µ d(Zgg) dg µ dg dµ _B = µZgg+µ d(Zgg) dg β(g)

Keeping the lowest order in g in the limit → 0 we nd β(g) = − d log(Zgg) dg −1 ≡ −β0g3+ O(g5) (1.78) with β0= 1 (4π)2 11Nc− 2Nf 3 (1.79)

which is positive for Nf < 11Nc/2, including the physical case Nc= 3, Nf = 6.

Integrating eq. (1.77) and using the one loop result in eq. (1.78) we nd g2(µ) = 1 β0log µ 2 Λ2 QCD (1.80)

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CHAPTER 1. INTRODUCTION TO QCD 11 where ΛQCD is an integration constant.

Figure 1.1: Behaviour of the coupling g2 _{with the renormalization scale. If β}

0 > 0 g2(µ) is dened for

µ > ΛQCD and the theory shows asymptotic freedom (g2(∞) = 0) and infrared slavery (g2(ΛQCD) = ∞).

On the other hand, if β0< 0 g2(µ)is dened for µ < ΛQCD and µ = ΛQCD, in analogy to QED, is a Landau

pole of the theory (g2_(Λ

QCD) = ∞).

As shown in g. (1.1), in the physical case the coupling goes to zero for large µ. Once we insert this result in the solution of the renormalization group equation, which says that the Green functions computed at momenta sk are related to the same Green functions computed at momenta k with coupling g(s), we nd that the theory is asymptotically free, that is for high energy (s → ∞) the coupling vanishes. It can be shown that this result is independent on the renormalization scheme used, because the rst two coecients of the β function are scheme-independent. It is important to say that asymptotic freedom holds at each loop order, since when the coupling is small higher order terms are suppressed, so that asymptotically the behaviour of the coupling is the one reported in eq. (1.80). In the opposite regime (low energy), we can not trust the divergence at µ = ΛQCD, known as infrared slavery, because the coupling grows and the one loop

order calculation is no more valid. For the same reasons, we can not trust the divergence at µ = ΛQCD in

the β0< 0case (Landau pole).

1.5 Symmetries of QCD

To conclude this chapter, we show that there are two limits in which QCD has a spontaneously broken symmetry. An internal symmetry is a transformation of the elds that leaves the action unchanged, that is the Lagrangian is invariant up to a total derivative. Explicitly: if the eld φ (a generic eld, we omit possible Lorentz or spinorial indices) under an innitesimal transformation changes by αaδaφ, then

L(φ, ∂φ) → L(φ, ∂φ) + αa∂µJaµ (1.81)

where the variation is given by

∂µJaµ= δL δφδaφ + δL δ∂µφ ∂µδaφ (1.82)

At the classical level, Noether's theorem states that for each symmetry there exists an on-shell (i.e. when the equation of motion are satised) conserved current:

jaµ= −

δL δ∂µφ

δaφ + Jaµ (1.83)

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δL δφ − ∂µ

δL δ∂µφ

= 0

From the current we dene the conserved charge as Qa(x0) =

Z

d3~xj_a0(x0, ~x) (1.84)

Using the canonical equal time commutation/anticommutation relations for bosons/fermions, it can be shown that the following relation holds:

[Qa(x0), φ(y)]x0=y0 = iδaφ(y) (1.85)

for a generic eld φ.

Starting from a classical symmetry of the theory, at quantum level there are three possible scenarios: • The charges annihilate the vacuum state, Qa|0i = 0, and they are constant: in this case the symmetry

is exact and we can dene a unitary operator U that implements the transformation:

U = exp(iαaQa); φ → φ0= U†φU ; U |0i = |0i (1.86)

• There are some charges that do not annihilate the vacuum, Qa|0i 6= 0. These charges are not well

dened and we refer to them as broken generators. In the typical situation we have a symmetry group Gwith n generators and only the m generators of a subgroup H ⊂ G are unbroken: we say that there is spontaneous symmetry breaking G → H.

• The measure of integration over the elds is not invariant under the transformation and the current is no more conserved at quantum level: the symmetry is anomalously broken.

We are interested in the second scenario.

1.5.1 Chiral symmetry

Let us consider the fermionic Lagrangian in eq. (1.28). In the general case where the quark masses mf are

all dierent, the only symmetry is a global U(1)Nf, which gives a dierent phase to each avour:

Ψf → eiαfΨf (1.87)

This symmetry implies that strong interactions preserve the avour. Now let us suppose that the rst L avours have the same mass m and we insert the L spinors Ψf in a vector Ψ (we change notation with

respect to sec. 1.1): Ψ =    Ψ1 ... ΨL    (1.88)

The Lagrangian of these L avours can be written as L = ¯Ψ(i /D − M )Ψ = L X f =1 Nc X i=1 ¯ ψf,i(i /D − m)ψf,i (1.89)

where M is the L × L mass matrix, M = diag(m, . . . , m). Now the symmetry group is larger than before, because L is invariant under U(L) rotations:

Ψ → V Ψ (1.90)

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CHAPTER 1. INTRODUCTION TO QCD 13

G = U (1) × SU (L) × U (1)L+1× · · · × U (1)Nf (1.91)

The SU(L) symmetry is a good approximation for L = 2 (isospin) and L = 3 (Gell-Mann SU(3) avour). The next step is to consider the limit in which the L quarks are massless. Writing the Lagrangian in terms of the left (L) and right (R) components (eigenstates of γ5, which we will call chiral components) it can be

shown that only the mass term couples components with dierent chirality. So when the masses are zero the Lagrangian becomes

L = ¯ΨLi /DΨL+ ¯ΨRi /DΨR (1.92)

Now we can perform two dierent U(L) rotations, one for the left and one for the right components, without changing the Lagrangian:

ΨL→ VLΨL

ΨR→ VRΨR (1.93)

with VL, VR ∈ U (L). In conclusion, the symmetry group that acts on the L massless avours is the chiral

group:

G = U (1)L× U (1)R× SU (L)L× SU (L)R (1.94)

The subgroup with VL = VR is the same U(L) that appears when the L masses are equal but nonzero.

Now we have another subgroup, which rotates left and right components in opposite ways: VL= V_R†. These

two subgroups are called vectorial (V) and axial (A) respectively. It can be shown that a generic chiral transformation can be obtained by the composition of a vectorial and an axial transformations, so that we can write the chiral group as2

G = U (1)V × U (1)A× SU (L)V × SU (L)A (1.95)

It can be shown that U(1)A is anomalous and that SU(L)V × SU (L)A is spontaneously broken to SU(L)V.

In the following we focus on the chiral symmetry SU(L)V × SU (L)A. Using eq. (1.83) we nd for the axial

current

Aµ_a = ¯Ψγµγ5TaΨ (1.96)

where Ta are the generators of SU(L) in the fundamental representation.

Now let us consider the following quantity:

[QA_a(0), ¯Ψγ5TbΨ(0, ~0)] = − ¯Ψ{Ta, Tb}Ψ(0,~0) = −

1 Lδab

¯

ΨΨ − dabcΨT¯ cΨ (1.97)

The rst equality can be proven by explicit computation starting from the expression of the axial charge QA_{or using eq. (1.85), the second one comes from a property of the SU(L) generators in the fundamental}

representation:

{Ta, Tb} =

1

Lδab+ dabcTc (1.98) where dabc is the completely symmetric tensor of SU(L). Taking the vacuum expectation value of the

commutator in eq. (1.97), we obtain an order parameter for the chiral symmetry, that is a quantity which is zero if the symmetry is exact: nonzero values of the order parameter imply broken symmetry. Indeed, if the axial symmetry were exact, QA

a |Ωi = 0and we would nd

2_{It is usual to write the chiral group as the direct product of the vectorial and axial subgroups, but actually SU(L)} Ais not

a subgroup. Indeed the composition of two axial transformations with matrices V1, V2 ∈ SU (L)is not, in general, an axial

transformation: ΨL→ V2V1ΨL, ΨR→ V2†V †

1ΨR = (V1V2)†ΨR6= (V2V1)†ΨR. The inequality follows from the fact that two

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χ ≡ h0|[QA_a(0), ¯Ψγ5TbΨ(0, ~0)]|0i = 0 (1.99)

Of course, if we nd that χ 6= 0 necessarily the symmetry is broken.

The explicit form of the order parameter can be simplied noticing that hΩ| ¯ΨTcΨ|Ωi = 0. To prove this, we

dene

CBA≡ h0| ¯ΨAΨB|0i

Using eq. (1.86) for the vectorial symmetry, which is exact, we nd:

h0| ¯ΨAΨB|0i = h0|U†Ψ¯AU U†ΨBU |0i = h0| ¯Ψ0AΨ0B|0i

Under a vectorial transformation Ψ0 _{= V Ψ}_{and ¯Ψ}0_{= ¯}_ΨV†_{, so we have}

CBA= h0| ¯ΨAΨB|0i = (V†)CAVBDh0| ¯ΨCΨD|0i = (V†)CAVBDCDC= (V CV†)BA

or in matrix form

C = V CV† =⇒ [C, V ] = 0 ∀V ∈ SU (L)

Using Schur's lemma we nd that C is proportional to the identity: h0| ¯ΨAΨB|0i = kδAB=⇒ h0| ¯ΨAΨB|0i =

δAB

L h0| ¯ΨΨ|0i

where the second relation is obtained taking the trace. Since the generators of SU(L) are traceless, we nd h0| ¯ΨTcΨ|0i = (TC)AB

δAB

L h0| ¯ΨΨ|0i = 1

Lh0| ¯ΨΨ|0iTr(Tc) = 0

In conclusion the order parameter of the chiral symmetry, which is called chiral condensate, is χ = −1

Lδabh0| ¯ΨΨ|0i (1.100) As we will see in the next chapter, the chiral condensate can be computed on the lattice and it is nonzero under the temperature Tχ ≈ 140MeV and zero above.

1.5.2 Center symmetry

Now we focus on the limit of innite quark masses. In this limit the quarks are frozen and the theory corresponds to the pure gauge one (quenched theory). Formally, we can see this from eq. (1.72): when mf → ∞the determinant of the fermionic matrix becomes a constant independent of the gauge elds and

it can be taken out of the integral. In this way, once we compute the Green functions as in eq. (1.68) this constant disappears and we obtain the same path integral of the Yang-Mills theory. In this limit the theory has the so called center symmetry. A center transformation looks like the gauge transformation in eq. (1.17) but it does not respect the boundary conditions (periodic for bosons and antiperiodic for fermions). In particular at temperature T the transformation matrix Ω(~x, x4)(in the Euclidean space) is such that

Ω(~x, x4+ β) = Ω(~x, x4) gauge transformation

Ω(~x, x4+ β) = zkΩ(~x, x4) center transformation (1.101)

where zk is an element of the center of the gauge group, that is the set of elements which commute with all

the elements of the group. For SU(N) the center is given by the Nth roots of the unit:

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CHAPTER 1. INTRODUCTION TO QCD 15 Using the transformation rule of the gauge eld in eq. (1.17) we see that the transformed eld is still periodic, because the phases zk cancel out:

A0_µ(~x, x4+ β) = Ω(~x, x4+ β)Aµ(~x, x4+ β)Ω−1(~x, x4+ β) + i g∂µΩ(~x, x4+ β)Ω −1_(~_{x, x} 4+ β) = = zkΩ(~x, x4)Aµ(~x, x4)zk∗Ω−1(~x, x4) + i gzk∂µΩ(~x, x4)z ∗ kΩ−1(~x, x4) = A0µ(~x, x4)

In conclusion, in the pure Yang-Mills theory the center transformation respects the boundary conditions, so it is a symmetry of the theory. The presence of dynamical quarks breaks explicitly the center symmetry, because it does not preserve the antiperiodic boundary conditions, even if the action is invariant (the proof is the same of the invariance under a gauge transformation, because the center phases always cancel out):

ψ0(~x, x4+ β) = Ω(~x, x4+ β)ψ(~x, x4+ β) = −zkΩ(~x, x4)ψ(~x, x4) 6= −ψ0(~x, x4) (1.103)

To nd an order parameter for the center symmetry we have to build a physical quantity, i.e. gauge invariant, which is not invariant under a center transformation: if the symmetry is exact this quantity must be zero, otherwise the symmetry is spontaneously broken. The order parameter is the Polyakov loop, that is the trace of a parallel transport along the time direction (which at temperature T is compactied from 0 to β). Using eq. (1.31), the Polyakov loop can be written as

L(~x) =Tr PeiRβ 0 dx4A4(~x,x4) (1.104) Using eqs. (1.10) and (1.101) and the cyclic property of the trace, we nd that L transforms as we want under a gauge/center transformation:

L0(~x) =Tr Ω(~x, β)PeiR0βdx4A4(~x,x4)_Ω−1_(~_{x, 0)} = L(~x) gauge transformation zkL(~x) center transformation (1.105)

In conclusion L is the order parameter of the center symmetry and in the next chapter we will see how to compute it on the lattice. Like for the chiral symmetry, there is a transition from a broken phase and an unbroken one, but in this case numerical simulations showed that center symmetry is broken above the temperature TD≈ 270MeV. The subscript D stands for deconnement, because center symmetry is linked

to one of the peculiar properties of QCD: connement. At high temperatures quarks and gluons are the eective degrees of freedom and they form the so called quark-gluon plasma, but at low temperatures they are conned in states which are colour singlets. If we consider a heavy quark-antiquark (q¯q) pair, this means that in the conning phase their free energy diverges when the distance between them goes to innity, that is we need innite energy to separate the pair. It can be shown that this free energy is linked with the correlator of two Polyakov loops computed at the positions of the two particles by the following relation:

e−βFq ¯q( ~R)_{= hL(~}_x)L†_(~_{x + ~}_R)i (1.106)

where R is the separation between q and ¯q. When R → ∞, assuming the clustering property for the Polyakov loops and using invariance under translations we nd

e−βFq ¯q( ~R) ₌

R→∞hL(~x)i hL

†_(~_{x + ~}_{R)i = | hLi |}2 _(1.107)

where L is the average over the volume of the Polyakov loop: L = 1 V R d

3_xL(~_x)_.

From eq. (1.107) it follows that when the center symmetry is exact L = 0 and Fq ¯q → ∞for R → ∞, so the

theory connes. In the other case, when the Polyakov loop is nonzero and the center symmetry is broken, the free energy of the heavy q¯q pair remains nite even when R is innite, so we can have a quark and an antiquark located at arbitrarily large distances: the theory deconnes.

In conclusion for T > TDthe center symmetry is spontaneously broken and the theory deconnes, while for

T < TD center symmetry is restored and quarks and gluons are conned in colour singlets. As we will see in

chapter 2 for nite values of the quark masses, when both the chiral and the center symmetries are explicitly broken, there is a region in temperature, the so called cross-over region, in which the thermodynamic properties of the system (for instance the pressure) rapidly become the typical ones of a quark-gluon gas.

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Chapter 2

Lattice QCD

In this chapter we introduce the lattice formulation of QCD, which is the basis for numerical simulations. Following standard textbooks as Refs. [23] and [36], we discretize the Euclidean space as a 4-dimensional isotropic Nt× Ns3lattice with sites separated by a distance a, which is called lattice spacing. Using equation

(1.70) and calling V = L3_{the spatial volume (which will be innite in the thermodynamic limit), this means}

L = aNs and β = 1/T = aNt. We label each site with four integers n = (n1, n2, n3, n4) = (~n, n4), such

that its coordinates are x = (an1, an2, an3, an4). We start with the discretization of the action, writing a

lattice version that in the continuum limit a → 0 reproduces the QCD one. Then, we show how to properly perform the continuum limit and we briey discuss the Monte-Carlo methods used for the simulations. In the end we show how to compute some physical quantities on the lattice, focusing in particular on the ones linked with the chiral and the center symmetries. In the following, everything is referred to the Euclidean formulation and we will drop the subscript E.

2.1 Discretization of the action

2.1.1 Pure gauge action

We start with the discretization of the pure gauge action, using as variables the parallel transports (or links) between adjacent sites. We call Uµ(n)the parallel transport from n + ˆµ to n, where ˆµ is a unit vector in one

of the four directions, for instance:

n ± ˆ2 = (n1, n2± 1, n3, n4)

Using eq. (1.14) we can write Uµ as follows:

Uµ(n) = eig0aAL;µ(n) (2.1)

where g0 is the bare gauge coupling and AL;µ(n)reproduces the gauge eld in the continuum limit:

AL;µ(n) = Aµ(x) + O(a) (2.2)

We begin the construction of the gauge action considering a plaquette Πµν(n), which is a closed parallel

transport along a square, starting and arriving in n with sides in directions µ and ν, as shown in g. (2.1). 17

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n n + ˆµ n + ˆµ + ˆν n + ˆν

Figure 2.1: The plaquette Πµν(n).

Using eqs. (1.8), (1.9) and (1.37) we have

Πµν(n) = Uµ(n)Uν(n + ˆµ)Uµ†(n + ˆν)U † ν(n) = e

ig0a2FL;µν(n) (2.3)

where FL;µν(n) is given by eq. (1.62), with the substitution Aµ(x) → AL;µ(n) and it reproduces the eld

tensor in the continuum limit. From now on, we will omit the subscript L and we will refer to the quantities on the lattice.

Taking the real part of the trace of the plaquette, we get: <TrΠµν(n) = Nc−

g2 0a4

2 TrFµνFµν + O(a

6₎ _(2.4)

where we expanded the exponential and we used that Fµν is traceless. We underline that in eq. (2.4) the

indices µ and ν are not summed. We then dene the plaquette action as follows: βGSpl= βG 1 − 1 Nc <TrΠµν(n) = βG g2 0a4 2NcTrF µνFµν + O(a6) (2.5)

where βGis a parameter that will be xed by the continuum limit. We nally obtain the lattice gauge action,

originally proposed by Wilson, taking the sum of Spl over all the plaquettes:

SW = βG X Spl= X n X µ6=ν βG g₀2a4 2Nc 1 2TrFµνFµν (2.6) where stands for sum over all the plaquettes and we inserted the additional factor 1/2 because the terms (µ,ν) and (ν,µ) are the same: switching µ ↔ ν we obtain the hermitian conjugate plaquette and once we take the real part and the trace we get the same contribution.

In the continuum limit, a → 0, the trace with the factor 1/2 becomes the YM Lagrangian and we have X

n

a4→ Z

d4x (2.7)

so SW reproduces the YM action if we set

βG=

2Nc

g2 0

(2.8) Because of eq. (2.8), βG is called inverse gauge coupling.

We notice that SW is not the only action with the good properties: for instance, we could have used rectangles

Π(2;1)µν . In this case there is an extra factor 2 at the exponent in eq. (2.3) and the 1/2 in eq. (2.6) disappears,

because now the exchange µ ↔ ν gives a dierent contribution, as shown in g. (2.2). The result is that the action with rectangles has to be multiplied by β0

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CHAPTER 2. LATTICE QCD 19 n n + 2ˆµ n + 2ˆµ + ˆν n + ˆν n n + ˆν n + ˆν + 2ˆµ n + 2ˆµ

Figure 2.2: The rectangles (µ,ν) and (ν,µ).

From eqs. (2.5) and (2.7) we see that building the action we have corrections O(a2₎_{, which disappear in the}

limit a → 0. To make the convergence more rapid, we want an action which reproduces the YM one up to O(a4). As shown by Symanzik, this can be obtained including in the action both the plaquettes and the rectangles with proper coecients:

SSY M = − βG 3 X n X µ6=ν 5 6Spl− 1 12Srec ! (2.9) where Srec is dened as in eq. (2.5) with the rectangle instead of the plaquette. This action is called tree

level improved Symanzik pure gauge action and it is the one we used in our numerical simulations.

2.1.2 Fermionic action

Now we want to nd a lattice version of the fermionic action. For simplicity, we will consider a single fermion (Nf = 1) with mass m. We start introducing in each site a variable ˆψ(n)which represents the fermion and

it is linked with the continuum eld by the following relation: ˆ

ψ(n) = a32ψ(x) (2.10)

In this way ˆψ(n)is dimensionless. Using eq. (2.7), we nd that the mass term of the action can be written as m Z d4x ¯ψψ → ˆmX n ¯ ˆ ψ(n) ˆψ(n) (2.11) where ˆm = am. The discretization of the derivative is less straightforward, because we want an operator ˆ∂ antihermitian, i.e. such that

Z dxgdf dx = − Z dxfdg dx (2.12)

where f and g are two generic functions and we assumed that the boundary terms are zero. On the lattice eq. (2.12) becomes: X n g(n) ˆ∂f (n) = −X n f (n) ˆ∂g(n) (2.13) A possible solution is the symmetric derivative:

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∂µψ → ˆ∂sψ =ˆ

ˆ

ψ(n + ˆµ) − ˆψ(n − ˆµ) 2a52

(2.14) With this discretization of the derivative, the free fermionic action becomes:

SF = X n n ˆ mψ(n) ˆ¯ˆ ψ(n) +1 2 ¯ ˆ ψ(n)X µ γµ[ ˆψ(n + ˆµ) − ˆψ(n − ˆµ)] o (2.15) With the gauge eld, we use the covariant derivative, obtained parallel transporting in n the elds in n + ˆµ and n − ˆµ: SF = X n n ˆ mψ(n) ˆ¯ˆ ψ(n) +1 2 ¯ ˆ ψ(n)X µ γµ[Uµ(n) ˆψ(n + ˆµ) − Uµ†(n − ˆµ) ˆψ(n − ˆµ)] o (2.16) Inserting explicitly all the indices, the action can be written as:

SF =ψ¯ˆα,i(n)M_αβij(n, m) ˆψβ,j(m) (2.17)

where the greek letters are Dirac indices and the latin ones are colour indices and M is given by

M_αβij(n, m) = ˆmδijδnmδαβ+ 1 2 X µ (γµ)αβ[Uµ(n)ijδm;n+ ˆµ− Uµ†(n − ˆµ)ijδm;n− ˆµ] ≡ ˆm1ij_αβ(n, m) + K_αβij(n, m) (2.18) M is known as Dirac operator of the theory and its inverse, according to eq. (1.50), is the fermionic propagator:

hψ¯ˆα,i(n) ˆψβ,j(m)i = (M−1)ijαβ(n, m) (2.19)

It is interesting to notice that with this discretization of the fermionic action the matrix K dened in eq. (2.18) is antihermitian, so the Dirac operator has the same properties discussed in sec. (1.3.1) for /D: in particular its determinant is real and positive.

2.1.3 Probability distribution for the links

With the discretized versions of the gauge and fermionic actions we are now able to write a distribution for the link variables using eqs. (1.71) and (1.72). The partition function is given by

Z = Z [dUµ]e−SW Y f DetMf[U ] (2.20)

where Mf is the Dirac operator of the avour f given in eq. (2.18) and [dUµ]is the Haar invariant measure

over the group G (in our case G = SU(Nc)), which has the following property:

[dU ] = [dU V ] = [dV U ] ∀V ∈ G (2.21) In this way the measure is invariant under a gauge transformation. The probability distribution of the links is

P [Uµ][dUµ] =

e−SWDetM

f[U ][dUµ]

Z (2.22)

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CHAPTER 2. LATTICE QCD 21

2.1.4 Fermion doubling and the staggered solution

The discretization we used so far for fermions has a problem, because it describes 16 avours of quarks. To see this phenomenon we write the fermion propagator in Fourier transform (for simplicity here we consider a free fermion): h ˆψα(n)ψ¯ˆβ(m)i = (M−1)αβ(n, m) = Z π −π d4_k (2π)4 M˜αβ(k) −1 eik(n−m) (2.23) where kµ is the dimensionless momentum dened on the lattice:

kµ= apµ (2.24) Using eq. (2.15) we nd: ˜ Mαβ(k) = X n−m Mαβ(n, m)e−ik(n−m)= δαβm +ˆ 1 2 X µ (γµ)αβ[eikµ− e−ikµ] = δαβm + iˆ X µ (γµ)αβsin kµ (2.25)

Inverting ˜Mαβ(k)and inserting the result in eq. (2.23) the propagator becomes

h ˆψα(n)ψ¯ˆβ(m)i = (M−1)αβ(n, m) = Z π −π d4k (2π)4e ik(n−m)δαβm − iˆ P µ(γµ)αβsin kµ ˆ m2₊P µsin 2_(k µ) (2.26) Now we want to check if this expression has the proper continuum limit, i.e. if the following relation is valid:

1 a3h ˆψ(n) ¯ ˆ ψ(m)i → hψ(x) ¯ψ(y)i = Z _d4_p (2π)4e ip(x−y) m − i/p p2_{+ m}2 with x = an; y = am (2.27)

Expressing all the quantities inside the integral in terms of the continuum ones we obtain 1 a3h ˆψα(n) ¯ ˆ ψβ(m)i = Z π_a −π a d4p (2π)4e ip(x−y)m − i a P µγµsin(apµ) m2₊ 1 a2 P µsin 2_(ap µ) (2.28) In the limit a → 0 the integrand goes to zero, unless sin(apµ) = sin(kµ) = 0. This happens when kµ= 0; ±π,

but ±π are the same point because the Fourier transform is 2π periodic. In other words, the integral receives contributions from the center and the corners of the rst Brillouin zone. In the rst case sin(kµ) ≈ apµ and

we recover eq. (2.27), but in total we have 24 _{= 16} _{contributions to the integral (two possible values for}

each of the four components of k), i.e. it is like having 16 fermions. This result can be easily generalized to ddimensions, where the degrees of freedom are 2d_{: there is a doubling of the degrees of freedom for each}

dimension and this is known as fermion doubling.

There are many solutions to this problem: we used the staggered fermions solution, for which the Dirac operator becomes (Mstf)(n, m) = ˆmfδnm+ 4 X µ=1 ηµ(n) 2 [Uµ(n)δm;n+ ˆµ− U † µ(n − ˆµ)δm;n− ˆµ] (2.29)

where we omitted the colour indices and ηµ(n)is the staggered sign function. The idea is to use variables

for which the rst Brillouin zone is halved, so that the contributions of the corners are no more present. The staggered action can be improved applying the smearing to the links, some details can be found in appendix A.

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2.2 Continuum limit for lattice QCD

Before we start the discussion on how to compute some quantities on the lattice, we need to dene a proper continuum limit. Let us consider an operator ˆO and its correlator, dened by

CO(τ ) = h0| Ô(τ ) Ô|0i − | h0| Ô|0i |2 (2.30)

where τ is the euclidean time, |0i is the fundamental state and ˆO is the operator computed at τ = 0. Inserting a complete set of eigenstates of the Hamiltonian

H |ni = En|ni

X

n

|ni hn| = 1 (2.31) and rewriting ˆO(τ )in terms of ˆO we have

CO(τ ) =

X

n

h0|eHτ_{O |ni hn| e}_ˆ −Hτ_{O|0i − | h0| ˆ}ˆ _{O|0i |}2₌X

n6=0

e−(En−E0)τ_{| h0| ˆ}_{O|ni |}2 _→

τ →∞e

−(E¯n−E0)_{| h0| ˆ}_O|¯_{ni |}2

(2.32) where ¯n is the state of minimum energy with h0| ˆO|ni 6= 0. In a QFT this minimum energy is linked with the mass at rest of the corresponding particle:

mO= En¯− E0≡

1 ξO

(2.33) where ξ is called correlation length. On the lattice we dene the corresponding dimensionless quantities:

mO= ˆ mO a ξO= a ˆξO mˆO= 1 ˆ ξO (2.34) where ˆmO and ˆξO are functions of the bare gauge coupling g0. It is clear that we can not simply take the

limit a → 0 with xed g0, because in this way the physical mass mO diverges. We need to change g0 with

a in order to have ˆmO → 0, or ˆξO → ∞: in conclusion we are looking for a second order critical point,

where the correlation lengths diverge and at the same time the ratios between dierent correlation lengths (or masses) are constant:

g0→ g0∗; ξˆi=constant

a → ∞ (2.35)

In this way when g0≈ g0∗ the mass m reaches its physical value and the ratios between dierent correlation

lengths are constant as we required. To nd the critical value g∗

0 we dene a renormalized coupling gR, function of g0 and of a certain scale of

energy µ. For instance we can consider a pair q¯q at distance r = aˆr and dene the energy scale on the lattice as ˆµ = aµ = 1/ˆr. We want this renormalized coupling to be a−independent, so we have

0 = adgR(g0, ˆµ) da = ∂gR ∂g0 _µ_ˆ adg0 da + ∂gR ∂ ˆµ _g 0 ad(aµ) da = ∂gR ∂g0 _µ_ˆ adg0 da + ∂µ ∂ ˆµ dgR dµ _g 0 aµ = ∂gR ∂g0 _µ_ˆ adg0 da + µ dgR dµ _g 0

In the last term we recognise the beta function dened in eq. (1.77) and we dene the beta function on the lattice as

βL(g0) = a

dg0

da (2.36)

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CHAPTER 2. LATTICE QCD 23 βL(g0) = − ∂gR ∂g0 _µ_ˆ −1 β(gR) (2.37)

It can be shown that the two couplings are linked by the following relation: gR= ˜Zgg0= g0+ Ag03+ O(g

5

0) (2.38)

After a few calculations, we obtain

βL(g0) = −β0g03− β1g05+ O(g 7

0) (2.39)

where the rst two coecients are the same that appear in β(gR), in particular β0 is given by eq. (1.79).

We can now write the relation a(g0)using eq. (1.80) with µ → 1/a and a new integration constant:

g2₀(a) = 1 β0log_a21_Λ2 L ; a(g0) = 1 ΛL exp − 1 2β0g20 (2.40) In conclusion, in the limit a → 0 the bare coupling goes to zero:

g0 → a→0g

∗

0 = 0 (2.41)

Summing up, to perform the continuum limit in the proper way we must send the lattice spacing to zero and at the same time the inverse gauge coupling βG to innity. When there are nontrivial quark masses,

i.e. ˆmf 6= (0; ∞), they enter in the ratios between hadronic physical masses. To keep these ratios xed

to their physical values, ˆmf must be tuned as g0 changes: numerical computations have established the so

called line of constant physics with the values of a and ˆmf corresponding to each value of g0. As said in the

introduction to this chapter, the temperature of the system is given by T = 1

aNt (2.42)

so we can x the lattice spacing choosing the temperature we want to simulate and then we set the values of the quark masses and of the coupling following the line of constant physics.

2.3 Monte-Carlo Methods

Now we are ready to see how to compute certain quantities on the lattice. We start the discussion of Monte-Carlo methods in general and in the next section we will focus on the QCD case. Monte-Monte-Carlo methods are useful to compute averaged values of an observable O function of a set of variables q ≡ (q1, . . . , qk)with a

certain probability distribution P (q)Dq:

hOi = Z

DqP (q)O(q) (2.43)

The procedure is as follows:

1. We generate a set of N congurations of the variables q. Each conguration qi is obtained from the

previous one with a proper algorithm.

2. On each conguration we measure the observable Oi≡ O(qi)and we compute the sample mean ¯O:

¯ O = 1 N N X i=1 Oi (2.44)

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3. For large N, according to the Central Limit Theorem, ¯Ois distributed as a Gaussian around hOi with variance ¯σ2_{= σ}2_/N_{, where σ}2_{is the variance of the observable and it can be estimated as}

σ = hO2i − hOi2≈ 1 N − 1 N X i=1 (Oi− ¯O)2 (2.45)

In conclusion our estimate will be

hOi = ¯O ± v u u t 1 N (N − 1) N X i=1 (Oi− ¯O)2 (2.46)

The estimate of the error in eq. (2.46) does not take into account the correlations between congurations and we need to correct it, for instance using the blocking technique or the bootstrap. The last solves another problem that arises when the observable to compute is not of the form (2.43), but a generic functional of the probability distribution F [P (q)], for instance a combination of sample averages.

This method can be applied to a QFT: if we want to compute the vacuum expectation value (vev) of a set of elds, for instance the Green functions given by eq. (1.68), we just need to use a proper algorithm to generate the eld congurations, with probability distribution e−SE;L, where S

E;Lis a discretized version of

the euclidean action.

Now we focus on the rst step. The congurations are generated as elements of a Markov chain, which is a stochastic path in the space Ω of all possible congurations with no memory: each step of the chain depends only on the previous one. Such a process is dened by a matrix W , whose elements Wbaare the probabilities

to reach the conguration b starting from the conguration a. Since we want to sample variables with a certain probability distribution, this path must visit a conguration b with frequency P (b), that is after an arbitrary number of steps the probability to reach b starting from a generic state a is P (b). Summing over the intermediate steps, we see that this condition is equivalent to

lim

k→∞(W k₎

ba= P (b) (2.47)

This is called equilibrium condition and it is reached only after some steps of the chain: in this thermalization phase the Markov process does not sample the right probability distribution, so congurations obtained in this phase must be discarded. At the equilibrium the probability distribution must remain unchanged, so P must be an eigenvector of the transition matrix W with eigenvalue 1 and it must be unique:

X

a

WbaP (a) = P (b) (2.48)

At the same time the other eigenvalues λ must satisfy |λ| < 1, so that other distributions (the other eigenvectors) are suppressed after a certain number of steps: all of these requests are guaranteed for regular chains. Here we recall the main results, without proofs:

• A chain is called ergodic if the probability of reaching a state b starting from a state a after k steps is nonzero, that is

∀a, b ∈ Ω ∃k|(Wk)ba6= 0 (2.49)

• The period of a state a is the greatest common divisor of all integers n such that the probability of returning in a after n steps is nonzero. For an ergodic chain all the states have the same period, so we can dene the period of the entire chain. If this period is 1 the chain is called aperiodic. An ergodic and aperiodic chain is also called regular.

• For a Markov chain, the eigenvalues of W satisfy |λ| ≤ 1 and there is always an eigenvector with eigenvalue 1.

• For an ergodic chain the eigenvector associated to the unit eigenvalue is unique and if the chain is regular the other eigenvalues satisfy |λ| < 1.

Now, given the distribution P we just need to nd the matrix W which satises eq. (2.48). It can be shown that a sucient condition is that W satises the detailed balance principle:

Localization Transition in QCD at the Roberge-Weiss Point

Dipartimento di Fisica E. Fermi

Corso di Laurea Magistrale in Fisica