Theta-dependence in QCD with dynamical fermions

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Università di Pisa

Tesi di Laurea Magistrale

Curriculum: Fisica Teorica

θ dependence in QCD

with dynamical fermions

Candidato: Tommaso Centrone

Relatore: Massimo D’Elia

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Introduction

Quantum Chromodynamics (QCD) is the theory which describes the strong interactions between hadrons in the framework of Quantum Field Theory. In this context, hadrons are seen as bound states of elementary massive fermionic particles (the quarks) held together by gluons, which are massless bosons.

The gluons constitute the gauge fields of QCD, and are the counterpart of photons in QED: as photons play the role of mediators of the electromagnetic force between electric charged particles, gluons mediate the strong interaction between particles which posses “color” charge. The main difference between the two theories, which makes QCD so different (and puzzling) with respect to QED, is the non-abelianess of the gauge group. The non-abelianess of SU(Nc) gives rise to the

most peculiar features which can be resumed in the two main properties: color confinement and asymptotic freedom.

The first one states that only color-singlet particles can constitute asymptotic states of the theory; it was introduced because there were no experimental evidences of free quarks (or other color-charged particles) and it is not yet proved analytically.

The second one states that the coupling constant vanishes at arbitrary high energy, thus the quarks at very small distance can be considered free; it was proved in 1973 by David Gross and Frank Wilczek and independently by David Politzer. This is the reason why the standard-perturbative approach is very limited into studying most of the properties of QCD, while the non-perturbative lattice formulation is more suitable, especially in the low energy regime.

Non-abelianess is also the cause of another important difference with respect to quantum electrodynamics: the non-trivial topology of the gauge fields configurations, which has relevant consequences on hadron phenomenology. The gauge fields, can be classified in homotopy classes (topological sectors), and cannot be transformed continuously from one sector to the other.

The study of the topological content of QCD turned out to be crucial to understand two of the major problems of the theory: the strong CP problem and the U(1)A problem. The first arises

because no violation of CP has ever been observed in strong interactions, but there is no theoretical reason to exclude in the QCD action a pure gauge term which breaks CP. It is possible to show that such a term has the form θQ, hence the CP problem is strictly connected to the topology of the gauge configurations. Experimentally we know that |θ| < 10−10 _{[1], but a non-trivial θ dependence}

allows to explain other puzzles of QCD: in the framework of the limit of large number of colors, 5

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6 CONTENTS Witten [2] found an explanation to the large mass of the η0 _{(with respect to the π meson), showing}

that its square mass is proportional to the topological susceptibility χ. The latter quantity is defined as the second derivative with respect to θ of the free energy density f, and can be interpreted as a measure of the fluctuations of the topological charge: χ ∼ hQ2_i

θ=0.

This is one of the results which led to extensive studies of the θ dependence of the free energy density, the argument of this work. Since θ is expected to be small, one is normally interested in studying only the first coefficients of the Taylor expansion of f, which are proportional to χ and χb2≡ −hQ4ic,θ=0/(12 V ), where hQnic is the nth cumulant of the probability distribution of the

topological charge. In addition to the mass of the η0_{, there are other physical quantities directly}

related to the topological susceptibility, e.g. the axion mass; also b2 has a physical interpretation,

since is related to the η0_-η0 _{elastic scattering amplitude [3]. For these reasons a reliable determination}

of the values of χ and b2 is of great interest.

Analytical studies have been tempted to investigate the topology of QCD, but ordinary perturba-tion theory is not suitable to study topological properties (which are non-perturbative): indeed the main results have been obtained by Chiral Perturbation Theory (ChPT) through the construction of effective Lagrangians which reproduce the symmetries of the original theory: In alternative to the the analytical approach, the lattice numerical simulations are one of the most useful tool to study the non-perturbative properties of QCD, and have the advantage of starting from the first principles of the theory. Many numerical works have been done in the last two decades to obtain a precise estimate of χ and b2 at T=0 and in the high temperature regime.

With the state of the art algorithms, this task can be quite expensive from a computational point of view, especially when we consider the full-QCD theory with dynamical fermions. One of the main problem is the so-called topological freezing, i.e. the difficulty which the sampling algorithms encounter in changing topological sector when the continuum limit is approached.

Other problems regard more specifically the choice of the strategy for the measure of χ and b2:

the traditional approach used both in pure gauge [4–7] and full-QCD theory [8,9] is to measure at θ= 0 the cumulants hQn_i

cof the probability distribution of the topological charge, which are related

to the coefficients of the Taylor expansion of f. This method turned out to be valid for the measure of χ ∼ hQ2_i

c, but very expensive for b2 ∼ hQ4ic. Indeed a valid estimate of the 4th cumulant

requires a statistics of the order of 106 _{measures; this value, in full-QCD simulations with the current}

computing power, would correspond to the prohibitive time of ∼ 10 years of uninterrupted runs. A different approach is the imaginary theta method proposed in Ref. [10] and refined in Ref. [11]: in order to study explicitly the θ dependence of the pure gauge theory, the topological term is introduced into the euclidean action by the analytic continuation of f from an imaginary θ. The coefficients of the Taylor expansion of f are then extracted as fit parameters. Considering imaginary values of θ is necessary, because the statistical weight in the euclidean partition function must be real and positive. This method turned out to be more accurate (with the same machine-time) than the traditional one, for what regard the determination of b2in pure gauge simulations.

For this reason this work aims at extending the imaginary theta approach, used in Yang-Mills simulations, to the full-QCD theory with dynamical fermions, to give an estimate of χ and in

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CONTENTS 7 particular b2 at T = 0. This target is challenging not only from the computational point of view

(full-QCD simulations are much slower than in Y.M. theory) but also from a theoretical perspective; the imaginary theta approach, in pure gauge theory, relies upon the fact that the lattice topological charge and the continuum one are related by a finite multiplicative constant [12]. When we add fermions to the theory, the situation gets more complicated and QL can mix with other operators

with the same quantum numbers [13].

The contents of this work are organized as follows. The first chapter resume briefly the main concepts of QCD, recalling particularly basic notions of the path integral formulation and of per-turbation theory. Chap. 2 is devoted to explain the topological content of the theory. We show how instantonic solutions emerge in the euclidean formulation, and then we review two of the main analytical tools to study the θ dependence of the theory: the large Nc limit and Chiral Perturbation

Theory. In Chap. 3 we introduce the lattice formulation, focusing specifically on the discretization adopted in this work; in the end of the chapter we treat in detail the smoothing methods used to reduce finite lattice spacing artifacts. Chap. 4 is dedicated to the Monte Carlo method: we discuss the main sampling algorithms, focusing on the ones used in QCD simulations, and then we explain in detail the features of the one used in this work: the Hybrid Monte Carlo. In Chap. 5 we resume how the imaginary theta method works to measure χ and b2 in the pure gauge theory, then we

explain how it can be extended when we add dynamical fermions: we propose some tests to verify if the renormalization mixing effects expected in the full-QCD theory are negligible.

In the final chapter we report the numerical results of this work. We discuss the stability of the combined fit when the regularization scheme is changed (the different number of stout smearing steps used to smooth QL) and then we try to extrapolate the continuum limit of χ and b2, in order

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Chapter 1

Quantum Chromodynamics

In this first chapter we recall quickly the theoretical tools necessary to understand the framework in which our work takes place. We present the QCD Lagrangian, its flavor symmetry and its gauge group. We then explain the path integral formalism, the perturbative expansion and some aspects of renormalization.

1.1 Quantum Chromodynamics

Quantum Chromodynamics is the theory which describes the strong interactions. The main idea behind this theory was to extend the SU(2) isospin symmetry (which was known to be conserved in strong processes) to an approximate SU(3) flavor symmetry. Although many new hadrons discovered in the ’60s were found to obey this new symmetry, there was no particle that could be placed in the fundamental representation of the group. This induced Gell-Mann and Zweig in 1964 to postulate the existence of three new particles with spin 1/2: the quarks. The theory, however, led to two paradoxes: no one had ever observed free quarks, and the state |∆++_{, J} _{= 3/2i = |u}↑_u↑_u↑_i

l=0 had

the wrong statistics (it is completely symmetric, while it should be completely anti-symmetric). Both the problems were solved by the introduction of a new charge called color, and the related symmetry group SU(Nc). Passing from this global symmetry to a local one, as in QED for the

group U(1)e.m., is the last step to obtain a gauge theory of strong interaction.

The QCD Lagrangian is: L= −1 4Fµνa F µν a + Nf X f =1 ¯ ψf(iγµDµ− mf)ψf, (1.1)

where ψfand ¯ψf are respectively the fermionic field of the quark and the antiquark of flavor f, each of

which is a vector of Nccomponents. The flavor index f runs from 1 to 6, since three new quarks (top,

bottom, charge) were discovered in addition to the original three (up, down, strange). Dµ = ∂µ+igAµ

is the covariant derivative, where the gluon field Aµ= P N_c2−1

a=1 A a µT

a _{is written in a basis of su(N} c),

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10 CHAPTER 1. QUANTUM CHROMODYNAMICS since it belongs to the algebra of the gauge group SU(Nc); Fµν ≡_gi[Dµ, Dν] = P

N_c2−1

a=1 F a

µνTa is the

gluon field strength tensor. As in QED the gauge field Aµ is introduced in the theory to make the

QCD Lagrangian invariant under gauge transformation of the quarks fields: ψf → ψ0f= Ω(x)ψf,

¯

ψf → ¯ψ0f= ¯ψfΩ(x)†,

(1.2) where Ω(x) is an SU(Nc) matrix. It is easy to show that to ensure the invariance of the Lagrangian

of Eq. (1.1) the gauge field should transform as: Aµ→ A0µ= ΩAµΩ†−

i gΩ∂µΩ

†_. _(1.3)

The QCD Lagrangian exhibits a global (approximated) flavor symmetry which acts separately on the two chiral components of the Dirac field (ψL and ψR). For this reason the flavor symmetry

group can be written as G = U(1)L× U(1)R× SU(Nf)L× SU(Nf)R. Let us consider a general

transformation of the group:

ψL,f→ ψL,f0 = U L f gψL,g, ψR,f → ψR,f0 = U R f gψR,g. (1.4) The transformations with UL _{= U}R _{are called vectorial transformations, while the ones with}

UL = UR† are called axial transformations. It is possible to show that the gauge group can be rewritten as G = U(1)V × U(1)A× SU(Nf)V × SU(Nf)A, where with the common abuse of notation

we have called the axial subset SU(Nf)A, even though it is not a subgroup. U(1)V is always

conserved, while the other flavor symmetries are approximated because of the quark masses, in particular: SUV(Nf) is explicitly broken by the quark mass differences (it would be exact if mf = m

for all the flavors) and SUA(Nf) is broken by any non-zero value of the quark masses, i.e. it would

be exact only if all the flavors have mf = 0 (chiral limit). SU(Nf)Ais also spontaneously broken in

the chiral limit, thus the Nambu-Goldstone theorem states the presence of one massless Goldstone boson for each broken generator; these bosons have been identified with the mesonic octet. The non-zero masses are then explained by the explicitly symmetry breaking (pseudo-Goldstone bosons). U(1)A is broken by the quantum anomaly, and as we will see in Chap. 2, its breaking is strictly

related to the non-trivial topology of the gauge group. We conclude this topic noting that even though the symmetry group G is explicitly broken, the approximation remains good because the quark masses are small compared to the typical energy scale of the theory ΛQCD (see Sec. 1.3).

1.2 Feynman’s Path Integral

Path integral is one of the most useful tool in QFT, in particular in lattice field theory, where the Euclidean formulation of the path integral is the only way to study quantum mechanical system as statistical system through Monte Carlo simulations. In the following, we give the definition of path integral and review its main properties.

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1.2. FEYNMAN’S PATH INTEGRAL 11 Let us consider in Quantum Mechanics a system described by the Hamiltonian H(Q; P ) = T(P ) + V (Q), where P and Q are respectively the momentum and the position operators. Let |q; ti = eiHt_|qi_{be the eigenvector of the position operator at the time t in the Heisenberg picture.}

We want to compute the transition amplitude between two of these states at time which differs of an infinitesimal quantity t0_{= t + . We have:}

hq0_{; t}0_|q_{; ti = hq}0_|e−iH(Q,P )_|qi₌Z _dpe−iH(q0,p)_hq0_|pihp|qi

=Z _2πdpexp[−iH(q0_{, p}_{) + i(q}0_{− q}_)p],

(1.5) where we have used the completeness relation for the momentum eigenstates. To compute the same quantity when t0_{− t ≡ τ} _{is finite, we divide the time interval in N + 1 infinitesimal intervals of}

length and write e−iH(t0−t)_{= (e}−iH₎N +1_{. Then, we insert between each factor a complete set of}

position eigenstates, which differ from each other of an infinitesimal time , and exploit the above result. We find:

hq0; t0|q; ti = Z

dq1. . . dqNhq0|e−iH(Q,P )|qNi . . . hq1|e−iH(Q,P )|qi

=Z N Y k=1 dqk N Y k=1 dpk (2π)exp " i N X k=1 (−H(q(tk), p(tk)) + ˙q(tk)p(tk) + O()) # , (1.6)

where we have introduced the interpolating function q(tk) = qk and p(tk) = pk−1. Now we can

integrate out the momentum variable, exploiting the Gaussian integral in p1 _{an then take the limit}

N → ∞: hq0_{; t}0_|q_{; ti = N}Z q(t0)=q0 q(t)=q Y τ dq(τ) exp " i Z t0 t dτ L(q(τ), ˙q(τ))dτ # (1.7) ≡ N Z q(t0)=q0 q(t)=q Dq eiS[q]_, _(1.8)

where N is a divergent normalization factor (in general may depend upon q) and Dq is the measure of the functional integral over all the possible paths from q to q0_.

If in the previous formula we pass to the Euclidean formulation by means of the Wick rotation t → −itE we obtain that the factor which ‘weights’ the paths becomes e−SE= e−

R

d4xLE, where

the euclidean Lagrangian is formally equivalent to the Hamiltonian of the system. Now to find the relation with the usual partition function of thermodynamics Z we have to put q = q0 _(periodic

boundary condition) in Eq. (1.7): Z(β) = Tre−βH₌ Z dqhq|e−βH|qi= Z q(0)=q(β) Dqexp " − Z β 0 dtELE # , (1.9)

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12 CHAPTER 1. QUANTUM CHROMODYNAMICS where we have identified the euclidean time extent τE with β. This tells us that we can always

describe a quantum mechanical system in D spatial dimensions as a statistical system in D + 1 spatial dimensions with temperature T = 1/β = 1/τE. From this equivalence we can write the

expectation value of an observable as: hO(Q)iβ =

1 Z(β)

Z

DqO(q)e−SE[q]_, (1.10)

which we underline, is suitable to extract the mean value of a generic observable with Monte Carlo method.

We can repeat these passages to generalize the formalism to the framework of QFT; we have to divide the 4-dimensional space into infinitesimal hyper-cubes, and then proceed as before. However there are some differences, because in QFT the path integral formalism is not mathematically rigorous. A considerable difference is that the statistics of the field changes the boundary condition in the partition function: we will have periodic boundary condition for the bosons and anti-periodic boundary condition for the fermions. Furthermore in the functional integral we have to use anti-commuting variables, called Grassmann numbers. This require a new definition of derivative and integral (Berezin integral) over these variables. We report the result of the Gaussian integrals in the the two cases:

• Bosonic field φ: Z DφDφ∗e−φ∗Aφ= 1 det[A/(2π)] (1.11) • Fermionic field ψ: Z D ¯ψDψe− ¯ψM ψ= det(M) (1.12)

Another complication arises in Quantum Chromodynamics, due to the redundancy of the gauge invariance, which render the quantization of the theory not straightforward.2 _{The F addeev − P opov}

method remedies this problem, by integrating out the superfluous degree of freedom and fixing the gauge; the result is that we must add to the Lagrangian of Eq. (1.1) two pieces:

Lf ix+ Lghost = − 1 2α(∂ µ Aaµ)2+ ∂ µ_¯c a∂µca− gfabc¯ca∂µ(Acµc b_), _(1.13)

where ca _{and ¯c}a_{are the so-called ghost fields, bosonic field which obey to the Fermi-Dirac statistics}3_.

2_{The Hamiltonian formulation we used to derive the path integral is not possible. In the pure gauge theory, the} zero component of the conjugate momentum Πµ= −F0µof the gauge field is zero, thus the canonical commutation relations are not valid anymore.

3_{A bosonic anti-commuting field would violate the spin-statistics theorem; however, since these ghost are an} artifact of the Faddeev-Popov method, they cannot appear in asymptotic states but only as loops in the Feynman’s diagram.

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1.3. PERTURBATION THEORY: THE GELL-MANN-LOW β-FUNCTION 13

1.3 Perturbation theory: the Gell-Mann-Low β-function

Assuming that the coupling constant g is small, we can do perturbation theory using the above described Lagrangian. QCD is renormalizable, thus we can reabsorb the divergences which arise in the diagrams adding a finite number of counterterms. We are interested in particular in studying the dependence of the renormalized coupling constant gR upon the arbitrary energy scale µ, so we

introduce the Gell-Mann-Low β-function: β(gR) ≡ µ ∂gR(µ) ∂µ = −β0g 3 R− β1gR5 + O(g 7 R). (1.14)

where in the last equality, we have expanded in powers of the renormalization constant gR. The

coefficients depend in general upon the renormalization scheme used, however, the first two are scheme independent. From the 1-loop computation one can find the first one:

β0= 1 4π2 11 3 Nc−2 3NF , (1.15)

and using this value, resolve at leading order the differential equation in Eq. (1.14), obtaining: g(µ)−2= 2β0log µ ΛQCD , (1.16)

where ΛQCD '200 MeV is a constant of integration, which sets the physical energy scale of QCD.

From Eq. (1.16) we can notice that when the scale of energy µ approaches infinity the renormalized coupling constant goes to zero. This behavior which is a direct consequence of the sign of β0 is

called asymptotic freedom, because at arbitrarily high energy µ (short distances) the theory become decoupled.

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Chapter 2

Topology in QCD

In this chapter we present the link between topology and Quantum Chromodynamics; we define the topological charge, show how it enters the action and discuss the physical consequences of a non-zero θ. We provide the mathematical framework to understand the role of the topological charge as winding number and then we present the connection between the strong-CP problem and the U(1)A

problem in the framework of the large Nc limit.

In the second part of the chapter, we deal with the main subject of this work: the θ dependence of the vacuum energy. We explain how the most recent numerical simulations approach this issue, and on the other hand, which are the analytical predictions of Chiral Perturbation Theory.

2.1 Importance of the topological term

We now briefly resume why topology has such a great interest in the framework of Quantum Chromodynamics, and the consequences related to the insertion of the θ term. We will work from now on in the Euclidean formulation of the theory, unless otherwise specified.

In the euclidean version of the QCD Lagrangian of Eq. (1.1) one can add a term Lθ= −iθq(x) that

explicitly breaks CP symmetry, where θ is a dimensionless parameter and q is the topological charge density defined as:

q(x) = g 2 64π2µνρσF a µν(x)F a ρσ(x). (2.1)

Although this term can be written as a divergence of a current (the so-called Chern-Simons current) q(x) = ∂µKµ, its contribution to the theory is not zero: there exist euclidean solutions with finite

action and non topologically trivial content (instantons) [14]. Hence Q = R d4_{x q}_{(x) is nonzero and}

it is possible to show that it can assume only integer values on any classical field configurations with finite action.

The introduction of the θ-term has a direct repercussion on various aspects of QCD phenomenology (e.g. the electric dipole moment of the neutron, the energy of the ground state, etc.). For this reason,

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16 CHAPTER 2. TOPOLOGY IN QCD the study of the θ dependence is of great interest, even though θ is expected to be very small. In fact, there are no experimental evidences of violation of the CP symmetry in strong interactions, and a very stringent upper bound can be set from the measure of the electric dipole moment of the neutron: dN(θ) '

M2

π

M3

N

|θ| ≤10−16_{θ e ·}_{cm. From the experiments we know that d}

N <10−26e ·cm

and hence we have for the vacuum angle the bound |θ| < 10−10 _{[1]. This fact goes under the name}

of strong-CP problem and is an example of problem of fine tuning in QCD, because there are no theoretical reasons to explain why the the vacuum-angle should have such a small value.

The last reason to study the vacuum angle term is that it is strictly related to the Adler-Bell-Jackiw axial anomaly: the non trivial dependence on θ of the theory furnishes an explanation of the U(1)A

problem in the framework of the limit of large number of colors [15] as proposed by Witten and Veneziano.

2.2 Instantons

We introduced instantons as fields configuration with finite action and non-zero topological charge; in this section we will show that such solutions exist and we will explain the geometrical meaning of the topological charge. We mainly present the arguments as in Ref. [16].

Let us consider the Yang-Mills theory, i.e. we are dealing with the Lagrangian of Eq. (1.1) without adding fermions to the theory. In order to have solutions with finite action, the Lagrangian should drop to zero as the Euclidean infinity is approached. This means that the gauge field Aµ

should tends to a generic pure gauge term: Aµ(r)

krk→∞

−−−−−−→ i

gΩ(n)∂µΩ(n)

†_, _(2.2)

where Ω is a generic element of the gauge group G and we have explicited the dependence of the pure gauge term on the direction n of the 4-vector r = krkn, from which infinity approached. The condition n · n = 1, clearly defines the 3-dimensionial sphere S3 _{in the euclidean space. Thus,}

Eq. (2.2) can be read as a mapping S3_{→ G}_{. In the space of all this mappings X , we can introduce}

an equivalence relation ∼, given by the homotopy group, thus we will say that two of these mappings are equivalent if it is possible to “continuously deform” one into the other, without passing through configurations with infinite action. These equivalence classes are called homotopy classes or more significantly topological sectors, and each sector is characterized by an integer Q ∈ Z which is the topological charge. More formally, we can say that the topological charge is isomorphic the quotient set X / ∼. We will not prove the equivalence with the definition we gave in Eq. (2.1), nevertheless we want to point out further the meaning of Q in a simple example.

Let us consider for simplicity the gauge group SU(2); a generic matrix belonging to this group can be parameterized by Ω = a01 + i~a~σ, where the parameters must satisfy the normalization

condition a2

0+ ~a2= 1. Thus, we are dealing with the mappings of the 3-dimensional sphere into

itself; to distinguish the sphere in the 4-dimensional physical space and the one in the space of the parameters of the gauge group, we will denote the two spheres n2_{= 1 and a}2_{= 1 respectively with}

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2.2. INSTANTONS 17 S3

phys and S3gauge. Some simple examples of mappings could be:

Ω0(n) = σ1 (2.3)

Ω1(n) = n01 + i~n~σ (2.4)

Ωm(n) ≡ Ω1(n)m (2.5)

The first one does not depends on the direction n: all the points of S3

phys are mapped into the

same point of the sphere S3

gauge, thus the topological charge is equal to 0. In the second example

each point of S3

phys is mapped into only one point of S3gauge(the mapping is bijective), thus the

topological charge is equal to 1. Regarding the mapping Ωm(n), there are m different points on

S3

phys which correspond to the same point on Sgauge3 ; in this case we have Q = m. In other words, Q

represents the number of time that the sphere in the space of the parameters of the group is covered by the mapping. If we consider, instead, Ω†

m(n), the gauge group is covered m times as in Eq. (2.5),

but in the opposite direction, thus we will have Q = −m.

We now return to the integral form (called Pontryagin index) of Eq. (2.1) to show the connection between this expression and Q as “winding number”; it is possible to verify that the topological charge density q(x) can be written as a total derivative:

q(x) = ∂µKµ(x) = ∂µ _g2 16π2µνρσTr Aν(∂ρAσ+ 2 3igAρAσ) , (2.6)

where Kµ is called Chern-Simons current. By Gauss theorem, the integral over all the Euclidean

space becomes the flux of Kµ through the sphere at infinity. Finally, one can express the

Chern-Simons current at infinity exploiting Eq. (2.2) for some particular mappings and check that the values found for Q are the ones discussed above.

We call instanton the gauge configuration that minimizes the action, i.e. it is a solution of the equations of motion. When Q = 1 the solution takes the name of BPST instanton after the name of the discovers A. Belavin, A. Polyakov, A. S. Schwarz and Yu. S. Tyupkin. From the Yang-Mills action we can derive the Bogomol’nyi inequality:

1 4 Z d4xF_µνa F_µνa = 1 8 Z d4x(F_µνa ∓ ˜F_µνa )(F_µνa ∓ ˜F_µνa ) ±1 4 Z d4xF_µνa F˜_µνa ≥ ±8π 2_Q g2 , (2.7)

which suggest that minimal configurations should saturate the inequality and thus should have:

F_µνa = ˜F_µνa (self-dual) if Q >0, (2.8)

F_µνa = − ˜F_µνa (anti-self-dual) if Q <0. (2.9) The solution to the self-duality (anti-self-duality) equation exists, for example for Q = 1 it has the form: Aa_µ= 2 g ηa µνxν x2+ ρ2, F a µν = − 4 g ρ2_ηa µν (x2+ ρ2)2, (2.10)

where we have introduced the t’Hooft symbols:

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18 CHAPTER 2. TOPOLOGY IN QCD

2.3 The Large N

c

limit

In 1974 t’Hooft proposed to consider Nc as a free parameter and study the limit of large number of

colors [15]. In taking the limit we have to impose that the physical quantities remain fixed to their values, thus we have to fix the physical energy scale of Eq. (1.16), which we remind is:

ΛQCD = µ exp−1/(2β0g2R(µ)) . (2.12)

Since in this limit β0∼ NC, we have gR2 = O(1/Nc). Now we can reorganize the Feynman’s diagram

according to their power of 1/Nc, considering the number of fermionic and gluonic lines and the

global power of gR. Given a generic diagram with L quarks loops and topological number H 1, we

can find a rule (from the well-known Euler’s identity), which tells us the power r in number of colors of the diagram. This relation, called t’Hooft rule, states:

r= N_c2−2H−L. (2.13) Witten assumed in his work [2] that the θ dependence at leading order was of order 1/Nc, and

found out an apparent paradox. In Sec. 2.5 we will see that is possible to “rotate” the vacuum angle term into the mass matrix, therefore if even only one massless quark exists, the theta dependence can be completely eliminated and the topological susceptibility is zero. Now let us consider the pure gauge theory, assuming that the the topological susceptibility in the limit of infinite colors is non-zero; in other words we assume (we use the Minkowsy formulation in this section)

χ(k) = −i Z

d4xeikxhT {q(x)q(0)}i = U0(k) + U1(k) + . . . , (2.14)

where we have expanded in powers of Nc the topological susceptibility in Fourier Transformation

with Um(k) = O(Nc−m). Since q(x) ∼ gR2, the t’Hooft rule for the two point correlation function of

Eq. (2.14) becomes:

r= (g2_R)2N_c2−2H−L= N_c−2H−L, where the lowest order is given by O(N0

c) when H = 0 (planar diagram) and L = 0 ( U0 is a pure

gauge contribute). U1 is given by all the planar diagrams (H = 0) with one quark loop (L = 1),

which come from the contribution of intermediate meson states: U1(k) = X nmesons |h0|q(0)|ni|2 k2_{− m}2 n (2.15) When we add massless fermions the vacuum angle dependence must vanish and we should have zero topological susceptibility. However this is not possible because the t’Hooft rule states that quarks loops brings a non-leading contribute O(1/Nc) , and thus the theta dependence should not be

affected. Indeed the cancellation between U0and U1cannot happen at any values of the momentum

k, but it can happen at k = 0.

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2.4. THE FREE ENERGY 19 If we assume the existence of a meson flavor singlet (the η0_{) with the same quantum number of}

the topological charge density q and squared mass m2

η0 = O(Nc−1), then, its propagator becomes

of order Nc at k = 0, and the one quark loop term can eliminate the pure gauge contribution in

Eq. (2.14). Imposing this cancellation, leads to the prediction of the η0 _{mass. This result refined by}

Veneziano [3] leads to the famous formula: 4Nf

f2

π

χ= m2_η0 + m2_η−2m2_K (2.16)

which, we remark, is very appealing from a computational point of view, since the Yang-Mills topological susceptibility is quite easily measurable with Monte Carlo simulation.

2.4 The free energy

The aim of this work is to study numerically the θ dependence of the free energy density of QCD with dynamical fermions at zero temperature. Similar studies have been done over the last two decades, in order to determine the vacuum angle dependence in Yang-Mills theory [7,10, 11] and in full-QCD [8, 9], using different approaches. Furthermore chiral perturbation theory [17, 18] at zero temperature provides a reliable result with which numerical simulations must be compared.

The free energy density is related to the partition function Z of the full QCD theory by f(θ, T ) = −_V1 log(Z), where V is the four-dimensional volume. f can be expressed in Taylor expansion of even powers of θ around zero, as:

f(θ, T ) = f(0, T ) +1

2χ(T )θ2s(θ, T ) (2.17)

where the dimensionless function s is equal to:

s(θ, T ) = 1 + b2θ2+ b4θ4+ O(θ6). (2.18)

This parameterization is useful because s describes the deviations from the quadratic behavior of the free energy density, i.e. the non-Gaussianities in the probability distribution of Q, as pointed out below. Only even terms appear in the expansion because of the CP invariance of the partition function of the theory. This is due to the fact that the coefficients of the expansion are proportional to the cumulants of the distribution of the topological charge. Performing the derivative with respect to θ in zero of the free energy density, one obtains :

∂n_f_{(θ, T )} ∂θn _θ=0= − in V hQ n_i c,θ=0, (2.19)

where we have used the expression of the partition function by means of the path integral Z=

Z

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20 CHAPTER 2. TOPOLOGY IN QCD Since under CP transformation the global topological charge Q is odd, the moments of odd order, and thus the cumulants k2n+1≡ hQn2+1ic,θvanish, when computed in θ = 0. For instance the first

coefficients which appear in the Taylor expansion of f are: χ=hQ 2_i θ=0 V , (2.21) b2= − hQ4_i θ=0−3hQ2i2θ=0 12hQ2_i θ=0 , (2.22) and b4= [hQ6_{i −}_15hQ4_ihQ2_i_{+ 30hQ}2_i3_] θ=0 360hQ2_i θ=0 . (2.23)

In order to determine numerically the coefficients b2n of the Taylor expansion of f (Eq. (2.18))

two different approaches have been adopted in pure gauge theory:

• the first one relies on Eq. (2.19) and consists of evaluating directly the cumulants of the global topological charge by means of Monte Carlo simulation at θ = 0;

• the second, proposed in Ref. [10] and refined in Ref. [11], consists of studying explicitly the dependence on θ of the cumulants of the topological charge, by means of MC simulations also at nonzero θ. The Taylor expansion of the cumulants around θ = 0 is clearly governed by the same coefficients which appear in the expansion of the free energy density, as one could see substituting Eq. (2.17) in Eq. (2.19).

For the sake of shortness, we will refer hereafter to the two methods respectively as the Taylor expansion method and the imaginary theta method. However, regarding the latter, the form of the euclidean action is not suitable for Monte Carlo simulation, due to the topological term which is purely imaginary2_{. A possible solution is to consider the analytic}

continuation from a purely imaginary θ = −iθI and to study the derivative with respect to θI

of ˜f(θI, T) ≡ f(−iθI, T); in this way Eq. (2.19) becomes:

∂n_f˜_(θ I) ∂θn I = −1 VhQ n_i c,θI. (2.24)

and the first cumulants of the topological charge read: hQic,θI V = χθI(1 − 2b2θ 2 I + 3b4θI4) + O(θ 7 I), hQ2_i c,θI V = χ(1 − 6b2θ 2 I+ 15b4θI4+ O(θ 6 I)), hQ3_i c,θI V = χ(−12b2θI + 60b4θ 3 I) + O(θ 5 I), hQ4_i c,θI V = χ(−12b2+ 180b4θ 2 I) + O(θ 4 I). (2.25)

2_{The problem of a not real positive distribution, which can not be directly sampled with standard MC simulation,} is known as the numerical sign problem. A way to overcome this problem is the procedure of reweighting, as explained in section 4.2.

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2.5. CHIRAL PERTURBATION THEORY 21 From this set of equation is possible to extract the values of the b2n coefficients and the

topological susceptibility, knowing the cumulants at different values of θI.

A not negligible drawback of the first method (evaluating the cumulants at zero topological parameter) is that an increasing amount of statistics is needed to determine with sufficient accuracy the cumulant of higher order. In fact, determining the coefficients b2n is equivalent to find the deviations from the

purely Gaussian behavior of the distribution of the topological charge p(Q) ∼ exp−1 2V χθ

2_{+ ....}

Due to the Central Limit Theorem this is a difficult task, because the deviations from the Gaussian behavior become more and more hardly detectable as the 4-volume increases.

The imaginary theta method allows to avoid this problem, but faces a different issue: the discretized version on the lattice QL of the topological charge is related to the continuum one by a finite

renormalization. In Chap. 5 we will discuss the different ways in which Q renormalizes, respectively in the pure gauge and in the full-QCD theory, in order to understand if the approach of Ref. [11] is generalizable to simulations with dynamical fermions.

2.5 Chiral Perturbation Theory

We already mention ChPT as a different approach to study the topological content of Quantum Chromodynamics from an effective Lagrangian; this method is useful to study analytically the zero temperature behavior of the theory, while DIGA (Dilute Instanton Gas Approximation) gives a reliable method to investigate the high temperature properties. In this section we present the main ideas and results of this approach, and finally we report an analytic prediction for χ and b2.

The starting point is to construct a new Lagrangian which reproduces the same flavors symmetries and the same breaking patterns of the original theory in the low energy limit (p-expansion). This effective theory will describe only the slow degree of freedom, i.e. the Goldstone bosons generated by the SSB. The full QCD Lagrangian in the chiral limit is invariant under the flavor group G = SU(Nf)V × SU(Nf)A × UV(1), which spontaneously breaks to SU(Nf)V × UV(1). The

chiral effective Lagrangian which reproduces at the leading order (LO) in the p-expansion the abovementioned breaking pattern is (we now use the Minkowsky formulation):

LCh = Fπ2 4 Tr∂µU ∂µU† + 2B TrMU + M†_U†_, _(2.26) where B = M2

π/(mu+ md), M is the mass matrix of the quarks and U(x) = exp{2iφa(x)Ta/Fπ}

is a SU(Nf) matrix which contains the physical mesons fields (pions) φa(x). Under a chiral

transformation the field U(x) transforms according to:

U(x) → U0(x) = VLU(x)VR†, (2.27)

where VLand VR are special unitary matrices which belong respectively to SU(Nf)Land SU(Nf)R.

Expanding U in terms of φa in Eq. (2.27), it is easy to show that under a vectorial transformation

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22 CHAPTER 2. TOPOLOGY IN QCD (VL= V_R†) they transform non-linearly. The mass term is chosen such that, if we consider a complex

mass matrix M that transforms under the chiral group as:

M → M0= V_L†MVR, (2.28)

the chiral symmetry is restored. In this way, when we go back to the real quark mass matrix M= M = diag(mu, md, ms), the symmetry is explicitly broken in the same way as in the original

theory.

Now the simplest way to add the theta term into the effective Lagrangian, is to exploit the anomalous breaking of the U(1)A. Since, under a transformation of the form exp{iαγ5}, the fermionic

measure of integration in the partition function changes as:

D ¯ψDψ → D ¯ψDψe−2iNfαQ_, (2.29)

it is possible to eliminate the theta term, by choosing α = θ

2Nf. However the transformation changes

the mass term which breaks explicitly the symmetry, thus we have:

det M → det M0_{= det Me}2iα_{= det Me}iθ/Nf_. (2.30)

Due to this arbitrariness, it is reasonable to define a new vacuum angle θphys= θ + arg(det M),

which is invariant under transformations of the group U(1)Aand thus is the only meaningful quantity.

In order to simplify the notation we will call from now on θphys as θ. To conclude, the effect of the

theta term can be taken into account in Eq. (2.26), by adding an imaginary phase to the physical mass matrix. The potential V (U, U†_{) in the Lagrangian with the “rotated mass matrix” becomes}

then:

V(U, U†) = 2B TrhM eiθ/Nf_U+ Me−iθ/Nf_U†i_. (2.31)

It is possible to show that the minimum of the potential with respect to U coincides with the free energy density, up to an irrelevant constant C which does not depend on theta:

f(θ) = min

U V(U, U

†_{, θ}_{) + C,} _(2.32)

thus, by minimizing V we obtain the free energy at zero temperature, as a function of theta. For two flavors (up and down quark) an exact analytical solution to the minimization exists, while for three flavors only an approximate one for θ 1. However the solution for Nf = 3 reproduces

correctly the one for Nf = 2 in the limit ms→ ∞, as expected when the strange quark decouples

from the theory; in the real world case ms mu, md, thus the solution for two flavors is a good

approximation. We report the results at the leading order for two flavors: F(θ) = −Mπ2Fπ2 s 1 −_{(1 + z)}4z ₂sin2 θ 2 , (2.33)

where z = mu/md. Expanding around θ = 0 one obtains:

χLO = z (1 + z)2M 2 πfπ2, (2.34) bLO₂ = − 1 12 1 + z3 (1 + z)3. (2.35)

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2.5. CHIRAL PERTURBATION THEORY 23 The next leading order corrections in the p-expansion to Eq. (2.33) have been computed in Ref. [19] and are of order of percent for physical values of the quark masses.

We conclude this topic reminding that there exist other effective Lagrangian models, which take into account also the flavor singlet meson field and implement the U(1)A anomaly. These

models make different assumptions upon the nature of the dominant fluctuations, we recall the most important two: the model proposed by Witten, Di Vecchia, Veneziano et al. assumes quantum fluctuation and is based upon the large Nc limit, while the so-called extended non linear sigma

model considers that the main contribute to the fluctuations is given by semi-classical instantons. In Chap. 6 we report the full account of values of χ and b2 computed Ref. [20] using these different

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Chapter 3

Lattice formulation

The lattice formulation in particle physics has become more and more important, since it provides a non-perturbative method to study Quantum Chromodynamics. Perturbation theory is useful only for small coupling constant g, while chiral perturbation theory, although it reproduces the right symmetries, doesn’t deduce its results from the first principles of the theory. For these reasons, the lattice formulation is a powerful instrument for studying QCD, in particular the non-perturbative aspects, as the topological properties we are interested in. In the following chapter we will show a possible way of discretizing the fermionic and the gluonic fields of QCD; first we’ll treat only the fermionic part of the action, then we’ll consider the pure gauge term. We will introduce the problem of doubling, and some of the possible answers to it, and finally we will treat the discretization of the topological charge. At the end of the chapter, we will discuss some of the most used “smoothing methods” to reduce lattice artifacts, focusing on the one used in this work.

We divide the 4D euclidean space in a lattice made of Nt× Ns3sites; the distance between two

adjacent sites is called lattice spacing.1 _{In this way, we are able to identify the position in space-time}

with the vector xµ = a nµ = a (n1, n2, n3, n4) . Since the formulation on the lattice is naturally

suitable for numerical simulation, which deals with no physical quantity but only pure numbers, it is common to work with dimensionless quantities. Hence, from now on, we will use the convention of indicate with the ’hat’ symbol ˆ, the lattice quantities, and define them as functions of only other lattice dimensionless quantities. For instance, if we are considering a field φ(xµ) with mass

dimension D, we will define the corresponding lattice quantity by ˆφ(nµ) = aDφ(nµa).

Firstly we will treat only the fermionic action, then we will introduce the gluonic field to define the parallel transport, and eventually we will discuss pure gauge action.

1_{We recall that in general there is no problem in choosing different lattice spacings for different directions; for} example in finite temperature simulation, sometimes is convenient to choose the temporal lattice spacing smaller than the space one (at< as). However this has the complication of having an anisotropic coupling which depends upon the direction.

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26 CHAPTER 3. LATTICE FORMULATION

3.1 The naive discretization of the free fermionic field

We introduce the so called naive discretization of fermions. Later in section 3.6 we will discuss the problem known as fermion doubling, i.e. the fact that using the naive discretization leads in the continuum limit to an action, which corresponds to 2d _{fermions instead of just one fermion. Now,}

following the previous prescription, we can place spinors at each vertex of the lattice: ˆ

ψ(n), ˆ¯ψ(n)

where we have suppressed the Dirac and color indices, and we consider only one flavor for simplicity. The integral over space-time in the continuum action, becomes a sum over all the vertices of the lattice, while the derivative can be discretized in different ways as follow:

Z d4x −→ a4X n , ∂µψ(x) −→ a5/2        ˆ∂S µψˆ(n) = 1 2( ˆψ(n + ˆµ) − ˆψ(n − ˆµ)) ˆ∂F µψˆ(n) = ( ˆψ(n + ˆµ) − ˆψ(n)) ˆ∂B µψˆ(n) = ( ˆψ(n) − ˆψ(n − ˆµ)) , (3.1)

where we have introduced the Symmetric, Forward and Backward derivative operators on the lattice, respectively ˆ∂S

µ, ˆ∂Fµ, ˆ∂µB. It is easy to show that ˆ∂Sµ =

1 2

ˆ∂F

µ + ˆ∂Bµ and that the symmetric

derivative operator is the only anti-hermitian. Let us consider for instance the forward derivative operator in the following expectation value:

h ˆψ ˆ¯∂_µFψiˆ =X n ˆ¯ψ(n)ˆ∂F µψˆ(n) = X n ˆ¯ψ(n) ˆψ(n + ˆµ) − ˆ¯ψ(n) ˆψ(n) = X n ˆ¯ψ(n − ˆµ) ˆψ(n) − X n ˆ¯ψ(n) ˆψ(n) = −X n ˆ∂B µ ˆ¯ψ(n) ˆψ(n) = = −hˆ∂B µ ˆ¯ψ ˆψi. (3.2)

Eqs. (3.2) proves that ˆ∂F †

µ = −ˆ∂Bµ, i.e. the Forward and the Backward derivative operators are

not anti-hermitian. On the other hand, expressing the symmetric derivative operator in terms of the other two, one finds immediately that ˆ∂S†

µ = −ˆ∂µS. Although all the three possible choices are

equally valid2_{, if we want to preserve the same symmetry of the continuum action we have to choose}

the symmetric derivative which is the only one anti-hermitian as the continuum one.

2_{By ’equally valid’ we mean that the three discrete derivatives possess the same naive continuum limit. We stress} out that the limit we are talking about is naive, because the use of the symmetric derivative instead of the right nor the left one, leads to different results, as we point out in Sec. 3.6 .

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3.2. DISCRETIZATION OF THE GLUONIC FIELD 27 The free fermionic action then reads:

SF,0[ ˆψ, ˆψ¯] = X n ˆ¯ψ(n) X4 µ=1 γµˆ∂µSψˆ(n) + ˆm ˆψ(n) ! =X n,m α,β ˆ¯ψα(n)ˆD0α,β(n, m) ˆψβ(m), (3.3)

where, in the last equality we have written explicitly the Dirac indices and we have defined the lattice Dirac operator as:

ˆ D0(n, m) = 4 X µ=1 γµ 1 2(δm,n+ ˆµ− δm,n− ˆµ) + ˆmδm,n1 ! . (3.4)

Having discretized the fermionic field we can deal with the gluon field.

3.2 Discretization of the gluonic field

To preserve the gauge invariance of the lattice fermion action Eq. (3.3), as in the continuum, we introduce the gauge fields. Gauge transformation of the Dirac spinors on the lattice is implemented by an element ˆΩ(n) ∈ SU(Nc) according to:

ˆ

ψ(n) → ˆψ0(n) = ˆΩ(n) ˆψ(n),

ˆ¯ψ(n) → ˆ¯ψ0_{(n) = ˆ¯}_ψ_(n)ˆΩ†_(n). (3.5)

The mass term of the action in Eq. (3.3) is invariant under local rotation of the Dirac field, while the kinetic term is not, because we have to define a covariant derivative, i.e. a parallel transport. Instead of considering the gauge field as elements of su(Nc) (the algebra of the group SU(Nc) ) is

more natural, on the lattice, to define the gauge field as link variable ∈ SU(Nc). So we define for

each lattice site the link variable Uµ(n) as the SU(Nc) operator which transports parallel ψ from

the site n + ˆµ to the site n3_{. The relation with the well-known form of the parallel transport in the}

continuum is given by:

Uµ(n) = exp

i ˆAµ(n)a + O(a2)

(3.6) The introduction of the link variables allows us to define a covariant derivative on the lattice4_:

Dµ(x)ψ(x) −→ ˆDµ(n) ˆψ(n) = 1 2a Uµ(n) ˆψ(n + ˆµ) − U−µ(n) ˆψ(n − ˆµ) . (3.7)

As in the continuum, after the substitution in Eq. (3.3) of the simple derivative ∂µ with the covariant

derivative Dµ, we obtain the gauge invariant fermion action. One can easily verify that the result is

gauge invariant, exploiting the action of a local rotation over the link variables, i.e.:

Uµ(n) −→ Uµ0(n) = Ω(n)Uµ0(n)Ω(n + ˆµ)† (3.8)

3_{There is a bit of ambiguity in literature regarding the definition of the parallel transport. The meaning of our} definition is that ¯ψ(n)Uµ(n) lives in the dual vector space of ψ(n + ˆµ), and therefore transforms in in the same way under gauge transformation.

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Figure 3.1: The 4 links variable that contributes to the plaquette Πµν(n) in the µν-plane. The inner arrow denotes the orientation of the parallel transport.

3.3 The Wilson gauge action

Now we can face the discretization of the pure gauge term of the QCD action, the first term of Eq. (1.1). We need to construct a gauge invariant term with only the link variables, which reproduces the continuous pure gauge Lagrangian as the lattice spacing goes to zero. First of all we can notice how a string of link variables transforms under the action of the gauge group. Let us call S[U]P the

string with k link variables connecting two points ni and nf along a path P. Under local rotation

the string transforms as:

S[U]P = Uµ1(ni)Uµ2(ni+ ˆµ1) . . . Uµk(nf−ˆµk) −→ S[U]

0

P = Ω(ni)S[U]PΩ(nf)†. (3.9)

This equation is consistent with the gauge transformation of the parallel transport in the continuum, and tell us how we can build a gauge invariant object: if we consider a closed path L (a loop), the operators which appear on the side of Eq. (3.9) are Ω(ni) and Ω(ni)†; thus if we take the trace of

S[U]L we obtain a gauge-invariant object. From this result it is clear that we can build a gauge

invariant action considering any closed path, nevertheless the easiest way is to consider the shortest non-trivial closed path, the so called plaquette shown in Fig. 3.1. We define the plaquette as:

Πµν(n) = Uµ(n)Uν(n + ˆµ)Uµ(n + ˆν)†Uν(n)†. (3.10)

Taking the real part of the sum over all the possible plaquettes, each with only one orientation, we obtain a quantity which must be related to the pure gauge action in the continuum limit. To see this we we substitute Eq. (3.6) in Eq. (3.10) and expand the plaquette in powers of the lattice spacing a, exploiting the Baker-Campbell-Haussdorf formula for the product of exponentials. We do not report all the calculations which give as results:

Πµν(n) = 1 + i g a2Fµν(n) −

1 2 g

2_a4_F

µν(n)Fµν(n) + O(a5). (3.11)

The Wilson gauge action is then defined as [21]: SG[U] = 2 g2 X n X µ<ν Re Tr[1 − Πµν(n)] = a4 2 X n X µ,ν Tr[Fµν(n)Fµν(n)] + O(a6) . (3.12)

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3.4. SYMANZIK IMPROVEMENT 29 In the literature is common to introduce a different coupling constant defined as β = 2Nc/g2. In

the last equality, we have used the expansion of the plaquette of Eq. (3.11); the resulting expression is correct to O(a2_{), because P}

na

4_{= R d}4_x_{. This proves that the Wilson gauge action reproduces,}

in the limit a → 0, the continuum pure gauge action.

3.4 Symanzik improvement

It is possible to choose different lattice gauge actions, considering different loops of links in addition the plaquette, e.g. all the closed path of perimeter 6a (Fig. 3.2); this leads to the so-called Symanzik improvement program [22], that is, an improvement in the corrections of the lattice action. This is sometimes preferable, even though it requires a higher computational cost, because the improved action has better convergence in the continuum limit. We can summarize how the Symanzik improvement program works in general, as follows:

• We discretize the quantity of interest, obtaining an expression with corrections of order k: ˆ

O= O + O(ak).

• In the continuum formulation, we find an analytic expression for the correction term: ˆ

O= O + C(k)_ak_.

• We add to the discretized quantity, a discretized version of the correction term with appropriate coefficient b, so that the correction of order k vanishes:

ˆ

O+ b ˆC(k)_ak_{= O + O(a}k+1_).

We can repeat the procedure to achieve the desired order of correction of the discretized quantity. Resuming the previous example, in the lattice gauge action we can consider in addition to the plaquette, the rectangle closed path, which is:

Π1×2

µν = Uµ(n)Uµ(n + ˆµ)Uν(n + 2ˆµ)Uµ†(n + ˆµ + ˆν)Uµ†(n + ˆν)Uν†(n). (3.13)

In principle, we should consider also the other possible nontrivial closed loops (Fig. 3.2), but as shown in Ref. [23], it is sufficient to consider only the rectangle. The result of the procedure to improve the precision of the Yang-Mills Lagrangian to order O(a4_{) is:}

SSym= 2 g2 X n X µ,ν 5 6Re Tr[1 − Πµν(n)] − 1 12Re Tr1 − Π1×2µν (n) (3.14) where the coefficients are obtained by means of perturbative expansion.

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Figure 3.2: Some possible contributions to the Symanzik improvement program to eliminate corrections of order O(a2) for the pure gauge action. We remark that, as demonstrated in Ref. [23], we can consider only the planar rectangular loops with an appropriate coefficient, and neglect the non-planar ones.

3.5 The fermionic determinant

In the continuum formulation, the fermionic contribute in the partition function can be rewritten as a determinant, after integrating the fermionic degrees of freedom. Thanks to the property of the gaussian integral with Grassmann variables, we have:

Z D ¯ψDψexp{−SF}= Z D ¯ψDψexp − Z dx4ψ¯(x)Dψ(x) = det D. (3.15) This form is very useful in the lattice formulation, since it allows to sample the fermionic degrees of freedom, which cannot be sampled directly in terms of the anticommutating field ψ (Grassmann variable). To bypass this problem, one has to rewrite the determinant of the Dirac operator in Eq. (3.15) with bosonic fields (pseudofermions), which are manageable in numerical simulations. Before explaining in detail how this method works, we’ll see some other features of the determinant of the Dirac operator.

On the lattice, neglecting an irrelevant multiplicative constant, we can always write the Dirac operator (independently of the discretization used) as:

ˆ

D = 1 + ˆK, (3.16)

where ˆK connects different sites of the lattice. However, even without specifying the exact form of the Dirac operator, we can say something about the properties of the fermion determinant in general. For example, if ˆK is antihermitian (i.e. it has only pure imaginary eigenvalues) and it anticommutes with γ5 _{(i.e. the eigenvalues appear in conjugate pair), then:}

det ˆD =Y

λ

(1 + iλ)(1 − iλ) ≥ 0. (3.17)

The condition of Eq. (3.17) is fundamental, because it ensures that the statistical weight of fermions in the partition function is real and positive. Some discretizations do not guarantee the positivity of

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3.6. THE DOUBLING PROBLEM 31 the determinant, thus we have the so-called numerical sign problem.

We can find other interesting properties if we expand the fermion determinant in powers of ˆK: det ˆD = expnTrhlog1 + ˆKio = expnTrhˆK + ˆK2_{+ . . .}io_. _(3.18)

From this equation we can notice that arbitrary high powers of ˆK can connect arbitrary distant sites, and so the determinant is non-local. On the other hand, the trace imposes that the starting site must be equal to the ending one, thus we can have only even powers in the expansion. The non-locality of the determinant has a big impact on the computational cost of the fermionic sampling.

In order to obtain a form exploitable for the sampling we rewrite the determinant as:

det D = _{det D}1 ₋₁ =Z DΦ†DΦ exp−Φ†_D−1Φ , (3.19) where we have used the property of the gaussian integral for a complex scalar field, and we have introduced the fictitious field Φ, which is called pseudo-fermion. Pseudo-fermions are commutating fields with the exact same quantum numbers of the fermions (we omit to write the indices of flavors, colors, etc..); this trick goes under the name of bosonization of the determinant.

We can get a more suitable action to sample if we substitute the Dirac operator D in the action of Eq. (3.19) with DD†_{. This leads to :}

det D → | det D|2_{= exp}

− Z dx4Φ†D†−1D−1Φ . (3.20)

In this form we can easily sample a new variable η = D−1_{Φ, which will be gaussian distributed. This}

substitution has the drawback of introducing a new degeneracy: we are describing twice the initial number of fermions. This issue, however, can be overcome with staggered fermions (see Sec. 3.8), by sampling only on even (or odd) sites; we can do so, because in the staggered formulation DD†

couples only even (odd) sites with even (odd) sites.

3.6 The doubling problem

As we anticipated the naive discretization of fermions leads, in the continuum limit, to lattice artifacts, the doublers. Although we haven’t yet given a lattice version of the path integral (Sec. 3.10), we will use implicitly some well-known results of the continuum formulation, which remain valid in the discrete theory. The doubling problem arises from the analyses of the fermionic propagator, that is, exploiting the well-known properties of the generating functional Z[η, ¯η], the inverse of the Dirac operator D−1_{(n, m).}

We can for simplicity consider the free fermion action S0

F Eq. (3.3), since the problem has to do

only with fermions and does not depend on the presence of the gauge field. Our starting point is the discretization of the Feynman path integral R D ˆ¯ψD ˆψ e−S0F[

ˆ ¯

ψ, ˆψ] _{(Sec. 3.10) from which we can}

extract the two points correlation function:

h ˆψ¯(n) ˆψ(m)i = R D ˆ¯ψD ˆψ ˆψ¯(n) ˆψ(m)e −S0 F[ψ, ˆˆ¯ψ] R D ˆ¯ψD ˆψ e−SF0[ ˆ ¯ ψ, ˆψ] = ˆD −1_{(n, m)} _(3.21)

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32 CHAPTER 3. LATTICE FORMULATION We need to compute the inverse of the Dirac operator, which is:

X

l

ˆ

Dαλ(n, l)ˆD−1λβ(l, m) = δnmδαβ. (3.22)

Using the discrete Fourier transform5_{, i.e. we apply the operator P}

n,me

i ˆp(n−m) _{to Eq. (3.22) , the}

above equation reads:

X

λ

˜

Dαλ(ˆp)˜D−1λβ(ˆp) = 1δαβ. (3.23)

Now, applying the Fourier transform to the Dirac operator Eq. (3.4), one obtain: ˜ Dαβ(ˆp) = ˆm δαβ+ i 4 X µ=1 γµ_αβsin(ˆpµ), (3.24)

and exploiting the anticommutation relation of the gamma matrices in the Euclidean formulation {γµ, γν}= 2δµν, one can easily guess that the inverse of the Dirac operator in momentum space is:

˜ D−1αβ(p) = ˆ m δαβ− iP 4 µ=1γ µ αβsin(ˆpµ) ˆ m2 + P4 µ=1sin 2_(ˆp µ) . (3.25)

The last step left is to come back from momentum to position space, applying the inverse Fourier transform, and performing the continuum limit (the Dirac indices are implied):

lim a→0 1 a3 Z π/a −π/a dp4_a4 2π4 e ip(x−y) " m a 1 − iP µγµsin(pµa) ma2 + P µsin 2_(p µa) # ≡lim a→0 Z π/a −π/a dp4 2π4e ip(x−y) " m 1 − iP µγµ˜pFµ m2 + P µ(˜pFµ)2 # (3.26)

where we have returned to physical dimensioned quantities expliciting the lattice spacing dependence, and we have multiplied the entire expression by a factor a−3_{, since the physical 2-point correlation}

function has mass dimension [ ¯ψ(x)ψ(y)] = 3. We also have defined ˜pF ≡sin(pµa)/a. Now in the

first Brillouin Zone (BZ), if p = 0, the result of the continuum limit is indeed ˜pF _→_{0. Nevertheless}

we obtain the same result, if one of the components of the momentum pµ is equal to ±π/a, that are

the edges of the BZ (Fig. 3.3).

5_{The discrete Fourier Transform is naturally defined from the continuous one: following the above discussed} prescription, we substitute the space integral with a sum, and defined the adimensional momentum ˆp = a p.

Theta-dependence in QCD with dynamical fermions

Università di Pisa

Tesi di Laurea Magistrale

Curriculum: Fisica Teorica

θ dependence in QCD

with dynamical fermions

Candidato: Tommaso Centrone

Relatore: Massimo D’Elia

Contents

Introduction

Chapter 1

Quantum Chromodynamics

1.1

Quantum Chromodynamics

1.2

Feynman’s Path Integral

1.3

Perturbation theory: the Gell-Mann-Low β-function

Chapter 2

Topology in QCD

2.1

Importance of the topological term

2.2

Instantons

2.3

The Large N

limit

2.4

The free energy

2.5

Chiral Perturbation Theory

Chapter 3

Lattice formulation

3.1

The naive discretization of the free fermionic field

3.2

Discretization of the gluonic field

3.3

The Wilson gauge action

3.4

Symanzik improvement

3.5

The fermionic determinant

3.6

The doubling problem