Properties of strong interactions in background fields: the Chiral Magnetic Effect

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Dipartimento di Fisica E. Fermi

Corso di Laurea Magistrale in Fisica

Curriculum Fisica Teorica

Properties of strong interactions

in background fields: the Chiral

Magnetic Effect

Candidate:

Supervisor:

Lorenzo Dini

Prof. Massimo D’Elia

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Abstract

In a system composed by quasi-free fermions, initially chirally symmetric, it is possible to show that a strong magnetic field, in the presence of an imbalance of chirality, induces an electric dipole moment and then a current along the magnetic field. This is called the “Chiral Magnetic Effect” (CME).

The Quark Gluon Plasma (a phase of extremely hot matter consisting of quarks and gluons) is, indeed, a chirally symmetric phase, where chirality imbalance can be induced by the presence of gluonic configurations with non trivial topol-ogy. QGP can be created in heavy-ion collision, where the presence of a strong magnetic field in non central collision can generate the CME.

In order to investigate the consequences of the CME in QGP, we will measure on the lattice the local correlations between the topological charge density and electric dipole moment of quarks, induced by a constant external magnetic field. A similar analysis was done in Ref. [33], but comparing the continuum extrap-olation of topological susceptibility in Ref. [15] with the one in Ref. [33] we can see that the results of Ref. [33] are quite distant from the chiral limit, which is an important requirement for a correct investigation of this effect.

Accordingly, we will initially measure the local correlation between the topolog-ical charge density and electric dipole moment closer to the chiral limit. Then we will investigate the dependence of this correlation on the magnetic field and we will perform a simulation at higher temperature, extending the analysis done until now in a fully deconfined phase of QCD.

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Theoretical Introduction

1.1 Quantum Chromodynamics

1.1.1 An Introduction to QCD

Quantum Chromodynamics (QCD) is the theory of the strong interaction, one of the four fundamental forces in nature. It describes the interactions between quarks and gluons, and in particular how they bind together to form the hadrons. In this section I will try to introduce it, passing through the group of symmetry of quark flavour, that will be very important for the treatise of the chiral sym-metry.

In the late sixties, indeed, studies on the classification of hadrons strongly sug-gested that they were made of some fundamental and unknown building blocks: in fact, in that time, more and more strongly interacting ’elementary particles’ had been discovered. It was difficult to believe that all these hadrons were truly elementary.

So, one of the first theoretical evidences of the existence of these building blocks was provided by the so called Quark Model (we can find a more detailed expla-nation of this model in Ref. [3]): Gell-Mann and Ne’eman (1961) pointed out that we could group all these new mesons or baryons with the same spin and parity on the T3, Y (isospin and ipercharge, respectively) plot, and if we do so,

they look very much like representations of the SU(3) group.

One notable feature of the hadron spectrum in this scheme is that the fun-damental representation of SU(3) is not identified with any known particles. Against this background, the quark model was proposed, in which all hadrons are built out of spin 1/2 particles called quarks, which transform as members of the fundamental representation of SU(3). Thus, in this simple model, there are three types (flavours) of quarks, in the fundamental representation, 3. Their antiparticles, called antiquarks, are in the conjugate representation, 3∗.

Then, using tensor products of fundamental representations we can build all these new hadrons, and we can understand their valence quark content. Hence, the mesons are qq bound states, so from 3 × 3∗= 1 + 8 we have mesons in SU(3) singlets and octets. We can see a mesons octet in Fig.1.1.a. The baryons, on the other hand, are qqq bound states. Using the following products of representations:

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(a) Octet for 0−mesons _{(b) Decuplet for} 3 2 +

baryons

Figure 1.1: Higher dimensional representations of SU(3). Images taken from Ref. [3]

3 × 3 = 3∗+ 6 3 × 6 = 8 + 10

We get:

3 × 3 × 3 = 1 + 8 + 8 + 10 (1.1) And we can see an example of baryons decuplet in Fig. 1.1.b.

Of course, if this flavour symmetry was an exact one, all members of a multiplet would be strictly degenerate in mass, and this is not the case. But, as I stressed before, the importance of the quark model for this work is that this is the first example of a flavour symmetry, on which we will build the chiral group of symmetry, a fundamental part of this text.

Then, in 1974 the J/Ψ particle was discovered, and it was interpreted as a bound state of a new heavy quark, the charm quark c and its antiparticle c. So, with this new quantum number the flavour symmetry can be enlarged from SU(3) to SU(4). Of course SU(4) is badly broken because charm mass is much heavier than the other three quarks (mc∼ 1.3 GeV). Then in 1977 another set

of narrow-resonance Υs were discovered and they were successfully interpreted as bound states of another heavy quark, b (bottom, or beauty). Finally, only in 1995 the last heavy quark, t (top or truth) was observed. Hence, according to the Standard Model and experimental observations, there are six flavours of quarks, even if the flavour symmetry is a rather good one only if we consider the first three light quarks.

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1. Why we do not observe free stable quarks?

2. Why we do not observe qq and qqqq bound states? It is difficult to find a physical reason for the absence of such hadron states with masses compa-rable to the observed particles.

3. We stress that JP _{= 3/2}+ _{decuplet baryon wavefunctions seem to}

vi-olate the connection between spin and statistics (for example N∗++ in Fig. 1.1.b), because the overall wavefunction is totally symmetric with respect to the interchange of any pairs of constituent quarks.

The way out of all these difficulties is to postulate that each quark has a hidden degree of freedom, called color, and further postulate that only color singlet states are physically observable states. More specifically, each flavour of quark forms a triplet under a new, exact, color SU(3) symmetry group. Quantum Chromodynamics is the gauge theory built using this group.

1.1.2 QCD as a non-Abelian Gauge Theory

Now, let us start talking about the mathematical formulation of Quantum Chro-modynamics.

Introduction to Gauge Theories

In the framework of field theory, we can write a free Lagrangian of a model, simply imposing the symmetry of our theory. For example if we impose Lorentz invariance, renormalizability, and the presence of only one derivative (in order to get a bounded Hamiltonian), only these terms are allowed in a Lagrangian:

L = ψ(i /∂ − m)ψ (1.2) That is the free Lagrangian of fermions, with one flavour.

We notice that Lagrangian above enjoys U(1) global symmetry, i.e. is invariant under this transformation of the fields:

(

ψ → eiθQ_ψ

ψ → e−iθQ_ψ (1.3)

Where:

U = eiθQ (1.4)

is an element of U (1) group, Q its generator, and θ the parameter of the transformation. Then, a simple but deep way to obtain an interactive theory, is to gauge the theory, using the so-called Gauge Principle. We promote trasfor-mations above from global to local, thus:

U = eiθ(x)Q (1.5)

Hence, the transformation parameter depends on x, and then it is called a local transformation. If we use this transformation, we can see that our Lagrangian is no more invariant under local gauge simmetry. So, following the

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idea of Gauge Principle, we introduce a new gauge field, Aµ, and change the

standard derivative with covariant derivative, that is:

∂µ −→ Dµ= ∂µ+ igAµ (1.6)

With g is a coupling gauge constant. If our field transforms under local U (1) in this way:

Aµ−→ U (x)Aµ(x)U−1(x) +

i

g(∂µU (x)

−1_{)U (x)} _(1.7)

Our Lagrangian is still invariant under local gauge transformation.

The last thing we have to do is introduce a non trivial dynamic for the new gauge field, but if we impose gauge invariance, non violation of CP, and renor-malizability, the only term that remain in our Lagrangian is:

L = ψ(i/∂ − m)ψ − 1 4F

µν_F

µν (1.8)

At the end, we have the QED Lagrangian. If we generalize this method, fixing the group of symmetry of our model and promoting it to local, using the Gauge Principle, we obtain an interactive theory. We can use the same method even for non-Abelian gauge theories, thus theories that have a set of generators Ta, instead of a single generator, Q. Quantum Chromodynamics is, like we said

before, a non-Abelian gauge theories, based on SU (3) group. We can use the same passages, using Aµ = AaµTa, with N2− 1 gauge fields for a SU (N ) theory.

Parallel Transport

In this section, I will speak briefly about an alternative way to the Gauge Prin-ciple to obtain a local gauge theory, a more geometrical one, that it will be more useful for this work.

If we promote a global symmetry to a local one we see that some difficulty arises when we try to write terms including derivatives (see e.g. Ref. [2]): the derivative of ψ(x) in the direction of the vector nµ _{is defined, indeed, by the}

limiting procedure:

nµ∂µψ(x) = lim →0

1

[ψ(x + n) − ψ(x)] (1.9) However, in a theory with local phase invariance, this definition is not very sensible, since the two fields that are subtracted in equation above, have com-pletely different transformations under a local gauge symmetry. In order to subtract the values of ψ(x) at neighboring points in a meaningful way, we must introduce a factor that compensates for the difference in phase transformations from one point to the next.

To do that, let us introduce a parallel transport operator, a member of SU (3) group, in the case of QCD, that has the property:

ψ(y) = W (Cy←x)ψ(x) (1.10)

Hence, W send ψ(x), i.e. a vector who lives in a vectorial space valuated in x (we can call it Vx), in ψ(y), a vector who lives in another vectorial space

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Figure 1.2: Parallel transport among a close, infinitesimal path ([8])

transformation with a parameter θ(x), while ψ(y) transforms with a parameter θ(y).

Now, let us think about a possible form of the infinitesimal parallel transport, i.e W (Cx+dx←x). We know that is a member of SU(3), so we can write it using

the exponential notation. We know that it has to depend from a connection that we will call Aa_µ (we know from the Gauge Principle that it will be used to modify the derivative considering local symmetry, that is our goal), that has to be a Lorentz scalar and obviously it has to depend from dxµ. So, at the first

order in dxµ:

W (Cx+dx←x) ' e−igAµ(x)dx

µ

(1.11) Now, we just say that using the geometrical properties of parallel transport and the definition of covariant differential (the new correct way to differentiate our field, i.e Dψ = W (Cx+dx←x)ψ(x + dx) − ψ(x)) we can found the definition

(1.6) for covariant derivative, and the (1.7) law of transformation of connection Aµ. With the same approach, we can define Fµν, linking it at a topological

quantity. It is easy to demonstrate that, at second order in dxµ:

W (Cx←x) ' e−igFµνdx

µ_dyν

(1.12) Where C is the curve in Fig. 1.2, i.e. a close, infinitesimal path in µν plane. To demonstrate Eq. (1.12), we write W (Cx+dx←x) like a product of the single

parallel transport in Fig. (1.95) expanded at second order in dxµ, than using

the Campbell-Baker-Haussdorf formula we easily find our thesis, recalling that Fµν for a non-Abelian theory is:

Fµν = ∂µAν− ∂νAµ+ ig[Aµ, Aν] (1.13)

In conclusion, we have described every gauge quantity in function of topo-logical ones, and using geometrical argument we have deduced all their most important properties.

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Lagrangian of QCD

Following this strategy we can find the classical Lagrangian of QCD, considering a color SU (3) symmetry and a Nf different spinorial fields, describing quarks.

L = Nf X f =0 ψf(i /D − m)ψf− 1 4F µν a F a µν (1.14)

A full treatment of quantizing this theory is not the goal of this work, so I limit myself to expose the results of the quantization, hinting at the ideas behind the mathematical demonstrations. First of all, we recall that Gauge symmetry becomes a problem when I try to quantize a field theory, even an Abelian one. One of the simplest way to see it, see e.g. Ref. [1], is to look at what happens when I try to use canonical quantization of a pure-gauge Lagrangian, i.e. only the second term of Eq. (1.14).

Following the canonical formalism we have to define a conjugated field of Aa µ,

and then promote them to operators, setting their commutation relation. The momentum conjugated to our gauge field is:

Πa_µ= ∂L ∂ ˙Aa

µ

= −F_0µa (1.15)

And the canonical commutation relation: [Abν(x), Π

a

µ(y)] = iδabgµνδ3(~x − ~y) (1.16)

So, setting b = a and µ = ν = 0 we get:

[Aa₀(x), Πa₀(y)] = ig00δ3(~x − ~y) (1.17)

When, if we look at Eq. 1.15), we obtain:

Πa₀ = 0 (1.18)

And, of course, if we look at Eq. (1.17) we see that is inconsistent with Eq. 1.18). This problem can be solved modifying the Lagrangian, adding a con-straint of the gauge field, choosing a particular form for Aµ, using, for example,

Lagrange multiplier.

However, it is simpler to quantize the theory using functional formalism. Let us recall the generating functional, for the Yang-Mills theory:

Z = Z

DAeiSG[A] _(1.19)

Where SG is the pure-gauge action.

In functional formalism, fixing the gauge is equivalent to integrating over DA choosing a representative field for each gauge equivalence class. For this purpose we make the following restriction:

GµAa_µ= Ba (1.20) Where Gµ and Bashould be chosen in an appropriate way, depending of the gauge choice. For example, for Gµ:

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     Coulomb Gauge : Gµ_{= (0, ~}_∇) Lorentz Gauge : Gµ_{= ∂}µ Temporal Gauge : Gµ_{= (1, 0, 0, 0)}

Again, I will omit the details of the calculation, but using the famous Fadden-Popov trick is possible to gauge fix the Lagrangian using the functional formal-ism.

The final result is, introducing two new anticommuting fields Ca(x) and Ca(x),

the so-called Ghost fields, the complete Lagrangian of QCD, thus:

LQCD = Nf X f =0 ψ_f(i /D−m)ψf− 1 4F µν a F a µν− 1 2α(G µ_A µ)2−Ca(x)∂µ(Dµ(agg))abCb(x) (1.21) Where D(agg)µ is the covariant derivative in the adjoint representation (for

more details see Ref. [1]).

Functional Formalism and Analytical Continuation

In the previous paragraph I hint at the important role of the path integral in the formulation of a quantum field theory. Before the end of this subsection I want to properly introduce the analytic continuation and the Euclidean space, because of their importance in the functional formalism.

First of all let us recall the definition of this change of variables. The analytic continuation of a physical quantity to imaginary times, is obtained by replacing:

x0→ −ix4 (1.22)

One of the most important property of the analytic continuation is that this technique reveals a connection between quantum mechanics (and field theory) and statistical mechanics, especially if it is used in the functional formalism. In order to highlight this connection, we recall the functional approach to quan-tum mechanics: it can be shown (see e.g. [2],[6]) that a quanquan-tum transition between two states can be written as a path integral in the following way:

hqf| ei ˆHt/~|qii = Z q(0)=qi q(t)=qf Dq e i ~ Z t 0 dt0L(q, ˙q) (1.23)

It is possible to write the partition function of a statistical system, i.e. Z = Tr[e−βH], in the functional formalism too:

Z = Z q(0)=q(β) Dq e −β Z β 0 dτ H(q, q0) (1.24) Where τ is an integration variable and q0= ∂q/∂τ . Finally, it easily can be shown that the partition function of the classical system above coincides with the quantum mechanical amplitude:

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Z = Z

dq hq| eit ˆH/~|qi |_~=1/β

t=−iτ

(1.25) Where we can interpret τ as an “imaginary time” τ = it and Planck’s con-stant is identified with the “temperature”, ~ = 1/β. To see this explicitly, we have to substitute τ and β in Eq. 1.23) and notice that there would be a change of the sign of the kinetic energy term in the Lagrangian.

So, we showed that thanks to the functional formalism, using analytic continu-ation, it is possible to emphasize a parallelism between statistical and quantum mechanics. This connection turned out to be one of the major driving forces behind the success of path integral techniques in modern field theory and statis-tical mechanics. It offered, for the first time, a possibility to draw connections between systems that had seemed unrelated. This connection can be general-ized to higher dimensions, and hence in quantum field theory.

In addition, we must say that the concept of imaginary times not only provides a bridge between quantum and classical statistical mechanics, but also plays a role within a purely quantum mechanical context. Consider, indeed, the partition function of a single-particle quantum mechanical system:

Z = Tr[e−β ˆH] = Z

dq hq| e−β ˆH|qi (1.26) Hence, using analytic continuation, the partition function can be interpreted as a trace over the transition amplitude hq| e−β ˆH_{|qi evaluated at an imaginary}

time t = −i~β. So, another important conclusion is that real time dynamics and quantum statistical mechanics can be treated on the same footing.

In conclusion, as we will see in section (1.2.1), analytic continuation in the Euclidean space can be a very useful technique for the numerical computation of some observables.

1.1.3 Asymptotic Freedom

In this section I will briefly face one of the most important property of QCD: asymptotic freedom.

By the late 1950’s high energy physicists had a good comprehension of weak and electromagnetic interaction, described by a quantum field theory. On the other hand, it was generally believed that the strong interactions may not be described in any sense by the perturbative method of quantum field theory. Experiments of deep inelastic scattering, indeed, in the late 1960’s provided a confirmation of the Feynmann parton model and Bjorken scaling: a simple way to theoretically obtain the Bjorken scaling is, indeed, to assume the ex-istence of free independent point-like particles (quarks) inside the proton. So, Bjorken scaling suggests that the quark dynamics must have the property of asymptotic freedom, i.e., the property that the quark interaction gets weaker at short distances. All the known quantum field theories at that time were sur-veyed as possible candidates for quark dynamics and were shown not to enjoy the asymptotic freedom property. Nevertheless, now we know that non-Abelian gauge theories are in fact asymptotically free: the coupling constant becomes small at large momenta (or at short distances). This result, hence, indicates the applicability of non-Abelian gauge theory, in particular SU(3), to model the

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strong interactions.

In this section I will briefly introduce some features of this crucial property. Beta Function

The most simple way to demonstrate the asymptotic freedom is to find the run-ning coupling constant. In this paragraph I will just hint at the basic ideas of the method to calculate the renormalization group function β(g), performed in the MS renormalization scheme. A more detailed calculation can be found, e.g. in Ref. [1].

We recall that in the dimensional regularization scheme, the renormalized cou-pling constant ˜gr is related with the adimensional one gr in the following way:

˜

gr= grµ (1.27)

Where µ is the so-called renormalization scale, that has the dimension of an energy. In the dimensional regularization scheme, indeed, the dimension in energy of the renormalized coupling is:

= 4 − D (1.28)

With D the generic number of dimensions. Now, I recall that β is defined as:

β = µdgr(µ)

dµ (1.29)

It is easy demonstrate, using equations above and the definition of renor-malization constant Zg (i.e. ˜g = Zg˜gr), that:

β = −gr−

µ Zg

dZg

dµ gr (1.30)

Then, if we calculate the renormalization constant Zgin one-loop order and

we substitute it in equation above, we find: β = −β0gr3+ o(g 5 r) (1.31) Where: β0= 1 (4π)2 11N_c− 2N_f 3 (1.32) Now, let us pass from the beta function to the running coupling constant. Running Coupling Constant

First of all, let us define the scale parameter of QCD, i.e. ΛQCD.

Starting from the definition (1.29) of the beta function, if we solve the equation using the separation of variables, integrating from µ1to µ2(two arbitrary points)

we obtain the so-called Gell-Mann - Low equation, thus:

µ2= µ1e

Z gr(µ2)

gr(µ1)

dgr

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If now we substitute Eq. 1.31) at the leading-order in the Gell-Mann - Low equation above and we solve the integral, we find:

µ2e−1/(2β0g 2 r(µ2))_{= µ} 1e−1/(2β0g 2 r(µ1))

We stress that we found a physical quantity that is independent from the renormalization scale µ, so we call it scale parameter of QCD, i.e.:

ΛQCD= µe

− 1

2β0gr2(µ) _(1.34)

We recall that g2r = αS, thus the strong coupling constant, and µ is an

arbitrary parameter with the dimension of an energy, so we can put µ = √s, i.e. the Mandelstam variable. If we do that, and isolate g2

r from the equation

above, we get: αS( √ s) = 1 β0ln(s/Λ2QCD) (1.35) We found a coupling constant that depends from the center-of-mass energy, so a running coupling constant. In this equation αShas a pole for µ = ΛQCD, the

so-called Landau Pole, but only because we used the leading-order beta function in our calculation. Nevertheless, this is a very simple way to qualitatively see the trend of αS at high energy: we can see that asymptotic freedom occurs if

and only if β0> 0, and if we look at Eq. 1.32), the condition for the asymptotic

freedom reads:

Nf < 33/2 (1.36)

Hence Quantum Chromodynamics enjoys the property of asymptotic free-dom in so far as the number of quark flavors is less than 16. So, for six quark flavours we found that QCD is asymptotic free.

1.1.4 Chiral Symmetry

Until now we introduced the QCD and one of its most important properties, the asymptotic freedom. Now, we will begin to analyze the symmetries of this theory. In particular, in this section I will discuss the chiral symmetries of QCD, one of the most important ingredients of the Chiral Magnetic Effect (CME). I will summarize the quantum implementation of this symmetry and briefly talk about spontaneous chiral symmetry breaking and chiral anomaly, particularly interesting for the study of CME in the Quark Gluon Plasma.

Introduction to Chiral Symmetry

In section 1.1.1 I qualitatively introduced the flavour symmetries. Let us analyze some mathematical details: the most general flavour group is defined:

U(l) = U(1) ⊗ SU(l) (1.37) Where ψf are defined, indeed, as vectors under the fundamental

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Starting from these fermionic fields in the flavour space, ψf(x), we can define

the so-called chiral components of the fields, using projectors:

ψL(x) = 1 − γ₅ 2 ψ(x) ψR(x) = 1 + γ₅ 2 ψ(x)

Where I did not write explicitly the flavour index anymore. Here R means “right-handed” while L is for “left-handed”.

Now, one can think to ‘rotate’ these two components in an independent way, each one with a different U(l) transformation. If we do that, the group of this new kind of flavour transformations is the so-called chiral group.

G = U(l)R⊗ U(l)L (1.38)

Now let us look closer at the flavour and chiral symmetries of the QCD Lagrangian (1.21), focusing on the fermionic parts, i.e. Lf = ψf(i /D − Mf)ψf.

First of all we note that U(1) is obviously a symmetry for Lf.

Concerning SU(l), it is a symmetry group only if M ∝ 1: covariant derivative is 1 in the flavour space, indeed. In conclusion, the flavour group U(l) is a symmetry group if all quarks have the same mass. As we said before, indeed, flavour symmetry is not an exact symmetry.

Now let us look at the chiral symmetry of QCD. It is easy to rewrite the fermionic Lagrangian of QCD in chiral components, using the properties of projectors:

Lf = ψLi /DψL+ ψRi /DψR− ψLM ψR− ψRiM ψL (1.39)

As I said for the flavour group, the kinetic part is invariant under chiral transformations. For the mass terms, if we apply transformations (1.38) to Lagrangian above we get:

L0 f = ψLi /DψL+ ψRi /DψR− ψLV † LM VRψR− ψRV † RM VLψL

Where VL and VR are elements of U(l)L,R. Because the two chiral

compo-nents are independently transformed, V_L†VR 6= 1f. So, even if M ∝ 1, then

L0₆₌_{L . Hence, the chiral group (1.38) is a symmetry only if M → 0. For this}

reason, this is called chiral limit.

It is easy to show that chiral group is equivalent to:

G = U(l)V ⊗ U(l)A (1.40)

Where the vector-axial (V-A) transformations are defined as:

U(1)V : ψ0 = eiαψ U(1)A: ψ0= eiαγ5ψ SU(l)V : ψ0= V ψ SU(l)A: ( ψ0 L= AψL ψ0_R= A†ψR

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With α a real phase parameter, and V, A ∈ SU(l). We stress that, by defini-tion, SU(l)A is not a formal group, because it is not closed under composition.

So, neither does U(l)A.

Spontaneous Breaking of Chiral Symmetry

Now let us discuss the quantum implementation of the chiral symmetry. First, we will see some mathematical properties of these symmetries. We define classical Noether currents for SU(l)V and SU(l)A, thus:

V_aµ= ψγµTaψ (1.41)

Aµ_a = ψγµTaγ5ψ (1.42)

Where Ta are generators of SU(l). Using definition below, where Jµa is a

generic Noether current:

Qa(t) = Z

d3xJ0a(~x, t) (1.43)

We definecharge operators Qa _{and Q}a

5, respectively for Vµa and Aa5.

Then, first we focus on the vectorial part of the chiral symmetry. We can see that U(l)V = SU(l)V ⊗ U(1)V is an exact symmetry in the chiral limit: Lagrangian

(1.39) with M = 0 is, indeed, invariant under these transformations, so we can define a conserved current and a charge operator that is the generator of these groups, i.e. Qa _{for SU(l)}

V and Q for U(1)V. These operators annihilate the

ground state (see e.g. Ref. [4] for demonstrations), thus, we can state that U(l)V

is an exact symmetry of this theory.

From this point we will look at the axial part of chiral symmetry, focusing first on the analysis of SU(l)A.

Using phenomenological reasoning we can easily conclude that SU(l)A can not

be implemented as the previous vector symmetry: if we had a Wigner realization of this symmetry, indeed, each hadron state should be accompanied by another hadron state with the same mass and the opposite parity, because of the pseudo-vectorial properties of Qa

5.

If the symmetry is not exact we have two other possibilities: if we are able to construct a conserved current, but the corresponding charge operator does not annihilate the vacuum, thus the symmetry is spontaneously broken. Instead, if the Lagrangian is still invariant under a certain transformation, but we can not implement a quantum conserved Noether current, we have a so-called anomaly of the symmetry. We will discuss that later, because anomaly can be a very important ingredient of the CME.

We state that if the symmetry is spontaneously broken then Q is not well-defined. We can convince ourselves of this fact computing the norm of the state Q |0i.

Using definition (1.43):

h0|Q2_{|0i =}Z _d3_{x h0| J}

0(~x, 0)Q |0i

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Z

d3x h0| J0(~0, 0)Q |0i

And this is clearly infinite if the symmetry is spontaneously broken, thus Q |0i 6= 0. So we can not find an operator U = eiαaQa _{that implements the}

symmetry group in the quantum framework. Nevertheless there are many in-teresting consequences of spontaneous symmetry breaking, like the Goldstone Theorem and the Higgs Mechanism. We recall that Goldstone Theorem says that when a continuous symmetry is spontaneously broken, then there is nec-essarily one scalar, massless particle for each generator of the symmetry that is broken.

If we apply Goldstone Theorem to the axial SU(l)Asymmetry we find, for l = 2

three massless Goldstone bosons composed by the first two light quarks, i.e. the pions. Obviously, as we said before, chiral symmetry is not an exact symmetry, so when we insert the mass term in the chiral Lagrangian, pions acquire mass. An useful way to demonstrate chiral spontaneous symmetry breaking using lat-tice simulations, and so very interesting for this work, is through the chiral condensate.

First of all, let us define the chiral condensate:

χ = h0| ψψ |0i (1.44)

This physical quantity can be measured on the lattice. In addition, chiral condensate enjoys a very important property: using Leibniz rule and the well known charge algebra, it can be shown that

h0| [Q5

a(0), ψ(0)γ5Tbψ(0)] |0i = −

1

2δabχ (1.45) Hence, we can measure χ on the lattice and if χ 6= 0 looking at equation above we immediately see that Q does not annihilate the vacuum, and SU(l)A

is spontaneously broken. We can use the same method to demonstrate that if χ 6= 0 also U(1)A symmetry does not have a Wigner-Weyl realization.

Now, let us discuss the symmetry properties of the strong interaction at high energy: at low temperatures, the dependence of the running coupling constant from energy, predicts that quarks enjoy the property of the confinement, as I as explained in section (1.1.3). At high temperatures, though, asymptotic freedom predicts that the coupling between quarks and gluons will approach zero: so we switch from a confined phase to an extremely hot deconfined plasma of weakly coupled quarks and gluons. This phase is called Quark Gluon Plasma (QGP). Until now it is not clear if there is a particular temperature, called the critical temperature TC, at which the transition between this two phases

occurs, or if it is a smoother process. In literature (see e.g. Ref. [23]), it is said that the temperature scale for the formation of QGP is of the order 1012K, i.e. ∼ 170 MeV, which is achieved between 10−5s and 10−4s after the Big Bang: so, the study of QGP has also an interesting analogy in the physics of the Early Universe, considering that just after the Big-Bang, the universe itself was in the quark-gluon plasma phase. If we analyze the dependence of the chiral condensate on temperature using lattice simulations, in the deconfined phase we see that χ → 0 so, from Eq. 1.45), we conclude that we have a restoration of the chiral symmetry. Thus, chiral condensate can be used as an

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order parameter for the chiral restoration. In analogy with the more familiar ferromagnetic Ising model, chiral condensate plays the role of the magnetization for the breaking/restoration of the Z2 symmetry. In conclusion, we can state

that QGP is a deconfined and chirally symmetric phase.

Chiral properties of QGP are very important for this work, because they are strictly connected to the Chiral Magnetic Effect. We will examine in depth this topic in the next chapter.

Chiral Anomaly

Until now we focused on U(l)V and SU(l)A. We still did not analyze U(1)A

group. Using the chiral condensate we already concluded that U(1)A

sym-metry does not have a Wigner-Weyl realization. Then, one may assume that U(1)A too, is spontaneously broken. In this case, however, there should be a

corresponding isosinglet pseudoscalar Goldstone boson. But, like it is said in Ref. [11], using chiral perturbation theory, it is possible to estimate that such a particle, away from the chiral limit, should have a mass less than √3mπ. The

closest candidates, though, are the η mesons, whose masses are larger then 500 MeV, then clearly beyond that bound.

This is the so-called “U(1) problem”. A first solution to this problem was pro-posed by ’t Hooft who showed that U(1)Ais not a true symmetry of the theory,

and found the effect called anomaly. The final solution to the U(1) problem, instead, was originally proposed by Witten ([12]) and then refined by Veneziano ([13]).

In this paragraph I will show the idea of a simple derivation of the anomaly. We said that even if the Lagrangian is still invariant under a symmetry group and you can define a conserved Noether current operator, but you can not define a charge operator Q, then you have a Goldstone realization of the symmetry (or a spontaneous symmetry breaking). However, if you can not even define a quantum conserved current, then this is the case of the anomaly of the symme-try.

To discuss the anomaly I will use the functional formalism. During the quan-tum implementation of a generic symmetry using functional formalism, anomaly occurs if a measure is not invariant under the transformation, i.e:

Dφ0= eiA[α]Dφ

Where I exponentiated the variation of the measure. Then I parametrize the variation in this way:

A[α] = Z

d4xαa(x)Aa(x) (1.46)

Where αa(x) is the local parameter of the symmetry we are considering,

and Aa(x), the anomalous term. Thus, the anomaly adds a new term to the

Lagrangian (L ).

Now, it is possible to demonstrate that because of this new term, the derivative of the Noether current associated to the transformation becomes:

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Hence, even if the transformation was a symmetry of the classical Lagrangian (i.e. δaL = 0), when I quantize the symmetry I can not define a conserved

cur-rent operator. This is the effect of the anomaly.

Now that we have discussed the generic case of an anomalous symmetry let us focus on the chiral symmetry.

If ψ0(y) = R d4_xU(x)δ(4)_{(x − y)ψ(y), where U is a generic element of a}

trans-formation group, then using the properties of the Grassman variables it is easy to demonstrate the so-called Barazin relation, i.e.:

(DψDψ)0= (detM detM )−1DψDψ (1.48) Where:

M (x, y) = U(x)δ(4)_{(x − y)} _(1.49)

If we use U ∈ SU(l)V, i.e. U(x) = eiαaTa, we immediately see that U = U†,

and using the definition above and Binet theorem we get: detM detM = detM M = 1

Hence, looking at Eq. 1.48), we see that fermionic measure is invariant under vectorial symmetries, and then they are non-amolaous symmetries.

If we consider now an axial SU(l) symmetry, U(x) = eiαaTaγ5_{, then it is easy}

to verify that this time U = U, because of the anti-commuting properties of γ5.

Then Eq. 1.48) becomes:

(DψDψ)0 = (detM )−2DψDψ

Now it is possible to calculate the determinant of equation above, and then, using parametrization (1.46), to find the anomalous term. We will focus on a non-Abelian gauge theory with a set of generator {Ta}, and an axial symmetry

generated by a generic t. The result is the Adler-Bell-Jackiw (ABJ) anomaly, and the general anomalous term of an axial symmetry is:

A(x) = − g 2 16π2εµνρσF µν a F ρσ b Tr(TaTbt) (1.50)

All details of calculation can be found in Ref. [10] or in Ref. [5]. Here “Tr” denotes a trace only over indexes labelling the various fermion species and g is the gauge coupling constant.

We already said that SU(l)A of the QCD is spontaneously broken, now let us

look at its anomalous term. If we consider SU(l)A then t = Tc, so:

Tr(TaTbt) = Tr(TaTbTc)

= TrC(TaTb)Trf(Tc) = 0

Where I separated trace over color and flavour structures, and used Tr(Tc) =

0 because SU(l)A is a special group and thus has traceless generators. So, we

can state that SU(l)A is not an anomalous symmetry.

Now let us look closer at U(1)_A. This time the anomalous term is different from zero because U(1)_A is not a special group. In fact, if we use t = 1f, and

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A(x) = − g 2 32π2NfεµνρσF µν a F ρσ a (1.51)

With Nfthe number of flavours. If we substitute equation above in Eq. 1.47),

in the case of U(1)_A transformations we get:

∂µJ5µ= −δ5L − 2Nfq(x) (1.52)

Where q(x) is the so-called topological charge density, i.e.: q(x) = g 2 64π2εµνρσF µν a F ρσ a (1.53)

We stress that it is possible to write the topological charge density above as a total derivative of the so-called Chern-Simons current, Kµ. In the Euclidean

space we get:

q(x) ∝ ∂µKµ (1.54)

Where:

Kµ= 4εµνρσTr[Aν∂ρAσ+2₃AνAρAσ] (1.55)

Because of that, one can think that the contribution of the anomaly (1.51) in the Lagrangian can be neglected. In the following paragraph I will show that this is not the case.

In conclusion, we implemented in the quantum framework all symmetries of the chiral group G. We saw that U(l)V has a Wigner-Weyl realization, SU(l)A is

spontaneously broken and with U(1)Awe found an anomaly.

Treating the anomaly of U(1)A we found a very important physical quantity,

related to the topological properties of the gauge configurations, the topological charge density, indeed. Another important quantity related to the these prop-erties is the topological charge, i.e. the analytic continuation of the new term of the action added by the anomaly:

QE=

Z

d4xEqE(x) (1.56)

In next paragraph we will consider the topological properties of the gauge configurations with more details.

Instanton Solutions

Let me begin this paragraph discussing topology of gauge transformation in general, in order to better understand the following parts.

For simplicity I will focus on the SU(2) case, then we can easily extend these reasonings to the case of SU(3) (and thus to QCD).

We can easily convince ourselves that the manifold of SU(2) group elements is topologically equivalent to a sphere in four-dimensional Euclidean space. In-deed, if U is a generic element of SU(2), it is known that U can be parametrized as U = u01 + ~u · ~τ , where ~τ are the three Pauli matrices and u0 and ~u are real

coefficients, satisfying:

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Which follows from U†U = 1. This is clearly the equation for the sphere in four-dimensional space, that I will call S3_.

I define now a function f from the space of parameters {x0, ~x} :

f (x0, ~x) = x01 + ~x · ~τ (1.58)

These continuous functions are, indeed, mappings from a S3_{sphere to SU(2)}

space, and they can be divided in homotopy classes. Each class is made up of functions that can be deformed continuously into each other, and is charac-terized by a winding number, i.e. n. Thus, mapping functions belonging to different homotopy classes have a different winding number. It can be shown (see e.g. Ref. [3]) that winding number can be expressed as:

n = − 1 16π2

Z

V

d3xTr(εijkAiAjAk) (1.59)

Where, at this level, Ai is a function of the mapping, i.e.:

Ai= f−1(x)∂if (x) (1.60)

Now let us come back to a classical Yang-Mills SU(2) gauge theory, in the Euclidean space (i.e. where x2 _{= x}2

0+ ~x2). We define the gauge fields like we

did in section (1.1.2), but if, for notational convenience, we scale them as: Aa_µ−→ i gA a µ, F a µν −→ i gF a µν (1.61)

Then Eq. 1.7) becomes:

Aµ−→ U−1AµU + U−1∂µU (1.62)

If we want to answer the question asked at the end of the previous paragraph, thus why we can not neglect the anomalous term in the Lagrangian, we have to fix the correct boundary conditions. In fact, if the naive boundary conditions Aµ = 0, at spatial infinity are used, the anomalous term would not contribute

and U(1)_Awould appear to be a symmetry again. But ’t Hooft showed that the correct choice of the boundary conditions is that Aµ is a “pure gauge field”, i.e.

FµνFµν = 0, at infinity so that the Euclidean Yang-Mills action is finite (see

references therein Ref. [11]). This means that, in Euclidean space:

F_µνa −→ 0, for |x| → ∞ (1.63) From the viewpoint of gauge transformations, this is much less restrictive. Condition above, indeed, only requires Aµ to approach the configuration:

Aµ−→ U−1(∂µU ) (1.64)

which is obtained from Aµ = 0 by a gauge transformation.

Now, in order to identify the relation between the charge (1.56) and topological properties of gluon configurations, we will express the winding number in terms of gauge fields.

As we said before, it is possible to write the topological charge density (1.53) as a total derivative of the so-called Chern-Simons current (1.55). Using the

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definition of the dual of F , thus ˜Fµν = 1₂εµναβFαβ, and using notation (1.61) in

Euclidean space, we can write the following volume integral in this way: Z d4xTr(FµνF˜µν) = 1 2 Z d4x∂µKµ (1.65)

Now, using Gauss theorem, the volume integral (1.65) becomes: Z

d4xTr(FµνF˜µν) = 1₂

Z

S3

dsµKµ (1.66)

Where the hypersurface integral is over the S3 at infinity. At this point boundary conditions become very important: as we already said, with the naive conditions Kµ is zero. With t’ Hooft conditions, instead, Aµ is given by (1.64);

then substituting (1.64) in Eq. 1.55) we get: Kµ=

4

3εµνρσTr[(U

†_∂

νU )(U†∂ρU )(U†∂σU )] (1.67)

Finally we substitute equation above into (1.66). We stress that, according to Eq. 1.58), mappings functions are de facto group elements of SU(2), i.e. f (x) = U (x). Then, comparing the volume integral (1.66) after this substitution to the expression of the winding number in (1.59) we obtain:

n = 1 16π2

Z

d4xTr(FµνF˜µν) (1.68)

And comparing this result with Eq. 1.56) we find that Q = n, i.e. topological charge plays the role of the winding number.

Thus, this means that we can divide mapping functions of the gauge configura-tions in homotopy classes, identified by their topological charge. In this way we understood the topological meaning of Q and his importance in characterizing properties of gauge configurations.

With the discussion above we have demonstrated that despite the anomalous term can be written as a total derivative, with the correct boundary condition we can not neglect it.

Now it can be shown that it is possible to find some particular gauge configu-rations with non trivial topology (i.e. with Q 6= 0) that they are finite-action solutions to the classical Euclidean Yang-Mills theory. These solutions exist and they are called instanton.

Now, let us convince ourselves that these solutions effectively minimize the Eu-clidean action. We can rewrite it in the following way:

SE= 1 2 Z d4xETr(FµνFµν) =1 4 Z d4xETr[(Fµν∓ ˜Fµν)(Fµν∓ ˜Fµν)] ± 1 2 Z d4xETr(FµνF˜µν) (1.69)

It is easy to verify that the second term of the last equation is a constant. Looking to Eq. 1.56) and (1.53), indeed, we find:

1 2 Z d4xETr(FµνF˜µν) = 8π2 g2 Q (1.70)

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For the first term of equation (1.69), making use of the important positivity condition in Euclidean space, we stress:

1 4

Z

d4xETr[(Fµν∓ ˜Fµν)(Fµν∓ ˜Fµν)] ≥ 0 (1.71)

Thus, the Euclidean action satisfies the inequality: SE[A] ≥

8π2

g2 Q (1.72)

In fact the action is minimized when the first term of equation (1.69) is zero, thus when:

Fµν= ± ˜Fµν (1.73)

Then if there are any nontrivial self-dual gauge-field solutions, these instan-ton solutions are indeed minima of the action.

Let us come back to the U(1) problem. We saw that a term like (1.51) can be added to the Lagrangian because, using the correct boundary conditions, we can not neglect it.

Hence, we can rewriteLQCD in the following way:

Lef f = LQCD+

θ

16π2Tr(FµνF˜

µν₎ _(1.74)

With the addition of the so-called theta term. This shows that U(1)A is

not a spontaneously broken symmetry and then, in this sense, solves the U(1) problem.

Nevertheless, this solution of the U(1) problem creates another one: the θ-term violates P and conserves C (hence violates CP), while experimentally CP is a good symmetry for QCD. In fact, the stringent experimental upper limit on neu-tron dipole moment can be translated into a bound on the QCD θ parameter, i.e. θ < 10−9. So, how to give a rationale for such a small value? One would think that the only plausible solution is that θ is effectively zero. Various ways to achieve this have been suggested. One of the most interesting solutions is the so-called Peccei-Quinn model, that introduce a new U(1) symmetry and a new scalar field (the axion field) in order to put θ = 0 using symmetry considerations (see [19], [20] for details).

In conclusion, axial anomaly is a very important concept for our work, because, as we will see in next chapter, Chiral Magnetic Effect can occurs, for example, when fermions interact via the axial anomaly with these instantonic configura-tions.

1.2 QCD on the Lattice

Now that I introduced the most important theoretical ingredients of the Chiral Magnetic Effect, in this section I will discuss basic ideas of Quantum Field Theory on the Lattice. These are in fact the two pillars of this work.

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1.2.1 Field Theory on the Lattice

First let me review some basic aspects of a general Quantum Field Theory on the lattice, starting with a free fermionic theory and ending with gauge theories. Discretization of a Free Field Theory

At the end of section (1.1.2) we have already examined some aspects of the euclidean formulation of Quantum Field Theory (QFT).

This formulation is, as we mentioned before, also very useful in numerical meth-ods. An easy way to convince ourselves of that is looking at the euclidean func-tional generator of a free bosonic theory. We start from the usual funcfunc-tional, i.e.:

Z = Z

DφeiS[φ] _(1.75)

Now, applying analytic continuation to equation above we get: Z = Z Dφe−SE[φ] _(1.76) Where: ( SE =R d4xELE LE = ∂µφ∂µφ +1₂mφ2 (1.77) We stress that in this way, in this simple case, we obtained a quantity suited for numerical calculations, because field configurations can be easily extracted with a real, exponential weight.

Now we want to write the euclidean form of the fermionic action. In order to do that it is convenient to define a new set of euclidean Dirac matrices γµ_E, satisfying the algebra:

{γ_Eµ, γ_Eν} = 2δµν _(1.78)

This euclidean matrices are related to the standard Dirac matrices by: ( γ0 E= γ 0 γi E= −iγi (1.79) With this choice, the euclidean action then takes the form:

S_FE= Z

d4xψ(x)(γ_µE∂µ+ M )ψ(x) (1.80)

Our purpose, although, is to arrive at a lattice formulation of QCD which describes the interaction of quarks and gluons. Hence, first of all, we have to discretize space-time and then we must learn how to deal with fermions on this lattice.

So, the transition from the continuum to the lattice is done by making:          xµ→ nµa Z d4x → a4X n (1.81)

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Where a is the lattice spacing.

Now, concerning fermions, we will rewrite functionals and Green functions in terms of dimensionless lattice variables, by scaling the following quantities with a according to their dimensions, in this way:

           M → 1 a ˆ M ψ(x) → 1 a3/2 ˆ ψ(n) ∂µψ(x) → 1 a5/2∂ˆµψ(n)ˆ (1.82)

And the analogous relations for the antifermionic fields. Here, ˆ∂µ is the

antihermitean lattice derivative, defined by: ˆ

∂µψ(n) =ˆ

1

2[ ˆψ(n + ˆµ) − ˆψ(n − ˆµ)] (1.83) Other possible choices for the lattice derivatives are the so-called backward and forward derivative, i.e.:

∂_µFψ(n) = ˆˆ ψ(n + ˆµ) − ˆψ(n) (1.84) ∂_µBψ(n) = ˆˆ ψ(n) − ˆψ(n − ˆµ) (1.85) Anyway, in this case the hermitean conjugate of these derivatives would be:

(∂_µF)†= −∂B_µ (1.86) (∂µB)†= −∂Fµ (1.87)

So, using derivatives that involve once the lattice spacing will lead to a non hermitian Hamiltonian, then we have to use the symmetric (and antihermitian) derivative (1.83). But in this way our derivative now involves twice the lattice spacing: a consequence of that is the so-called doubling problem. I will discuss this problem with more details in section (1.2.3), along with the solution we adopted in our simulation.

Proceeding in this naive way we can write the discretized form of fermionic ac-tion. Applying substitutions (1.82) to the euclidean action (1.80), and explicitly writing the quadratic form, we obtain:

S =X n,m α,β ˆ ψ_α(n)Kαβ(n, m) ˆψβ(m) (1.88) where: Kαβ(n, m) = X µ 1 2(γµ)αβ[δm,n+ ˆµ− δm,n− ˆµ] + ˆM δmnδαβ (1.89) In this way we have shown how to regularize free fermionic quantum field theory on the lattice, although ignoring until now the doubling problem. In next section I will discuss the discretization of a gauge theory, the next step to a lattice formulation of the QCD.

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Figure 1.3: geometrical meaning of link variables on the lattice. Image taken from [8].

Gauge Theory on the Lattice

In section (1.1.2) we already discussed the geometrical approach to gauge in-variance. To have a lattice analogue of that invariance we have to maintain the geometrical structure of gauge symmetries when we discretize the fields. This consideration suggest that to arrive at a gauge-invariant expression for the fermionic action on the lattice, we should use the following quantity:

Uµ(n) ≡ U (n, n + ˆµ) Uµ†(n) ≡ U (n + ˆµ, n) = U−1(n, n + ˆµ) (1.90)

Where the geometrical meaning of U (n, n + ˆµ) is well illustrated in Fig.(1.3): it plays the role of parallel transport in order to maintain gauge invariance. In fact, if G is a representation of a gauge symmetry group, under this transfor-mation:

U (n, n + ˆµ) → G(n)U (n, n + ˆµ)G−1(n + ˆµ) (1.91) U (n, n + ˆµ) transorm, indeed, exactly like parrallel transport (1.11). Notice that the group elements Uµ(n) live on the links connecting two

neigh-bouring lattice sites; hence we will refer to them as link variables and sometimes simply as links.

Because of this connection between these link variables and infinitesimal parallel transport (1.11), we can write:

Uµ(n) = eigaA

L

µ(n) _(1.92)

Where AL

µ(n) is the discretized version of the gauge field. At the continuum

limit a → dxµ and ALµ(n) → Aµ(x), then link variable becomes effectively the

parallel transport (1.11).

Then, it is easy to verify that the following bilinear: ˆ

ψ(n)Uµ(n) ˆψ(n + ˆµ) (1.93)

is now a gauge-invariant quantity, and hence we can substitute it in the fermionic action.

At this point we can also write covariant derivative in lattice regularization in this way: ψ(x)Dµψ(x) →ψ(n)ˆ 1 2[Uµ(n) ˆψ(n + µ) − U −1 µ (n − ˆµ) ˆψ(n − ˆµ)] (1.94)

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Finally, to complete our construction of the lattice action for QCD, we must obtain the lattice version of tensor (1.13). In section (1.1.2) we already saw that Fµν is related to the closed loops on space-time. Then we define an analogous

quantity on the lattice, and we call it plaquette:

Πµν(n) ≡ Uµ(n)Uν(n + ˆµ)Uµ†(n + ˆν)U †

ν(n) (1.95)

Then, inserting the expression (1.92) into (1.95) we find:

Π(n) = eiga2FµνL(n) _(1.96)

Where, again, FµνL(n) is the lattice version of (1.13). At the continuum limit

Πµν(n) becomes the parallel transport through the infinitesimal loop (1.12).

1.2.2 Focusing on QCD

Until now we use a naive lattice regularization for the fermionic action and we discuss the discretization of gauge fields. Now we have to put together this elements to obtain a complete formulation of QCD on the lattice.

Discretizing QCD

In this first part we will write the lattice version of the Yang-Mills action for a generic SU(N) theory.

First of all, we will write the euclidean formulation of the continuum classical action. Hence, we apply the analytical continuation that we discussed in section (1.1.2) to the gauge part of the QCD Lagrangian, remembering that we also have to transform gauge fields in the following way:

Aµ(x) = (iA0(xE), Ai(xE)) (1.97)

This is because the connection Aµtransform under a Wick rotation like ∂µ,

thus like pµ. In the same way, the field strength tensor (1.13) becomes:

Fµν →

(

F0i= iF0iE

Fij = FijE

(1.98) Thus, the Euclidean Yang-Mills action is:

SE = Z d4xE 1 2TrF E µνF E µν (1.99)

Now it can be shown (see e.g. Ref. [9]) that the regularized action takes the form of:

S = βW

X

Π

(1 −_N1<TrΠ) (1.100) And this is called the Wilson Action. It is easy to demonstrate the previous statement expanding the exponential in Eq. 1.96); then we notice that (dropping index E):

<Tr(Π) ∼ N −1 2g

2_a4_Tr(F

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Confronting equation above with (1.99) we conclude that the discretized action has to be proportional to 1 − _N1<TrΠ. Finally we take the continuum limit in order to calculate the constant βW, and we find:

βW =

2N

g2 (1.102)

Nevertheless, this is not the only way to regularize the pure gauge action of a non-Abelian gauge theory. For example it is possible to build other improved discretized action. Here the word “improvement” refers to a technique used to reduce the discretization errors ([36]): every discretization process, indeed, gives rise to lattice artifacts and systematic errors. These effects disappear only in the continuum limit. An elegant way of approaching the problem is a systematical reduction of the discretization errors via modification of the action itself. Because there is not an only one way to discretize an action, there are other choices that give rise to the same formal continuum limit. In particular one may combine different terms to obtain an improved lattice action with reduced discretization effects.

An example of improved action is the so-called Symanzik action, the one we used in our simulation (see e.g. [15] for more details). Symanzik action involves also 1 × 2 Wilson loop, and it can be write in the following way:

SY M = − β 3 X i,µ6=ν 5 6W 1×1 i;µν − 1 12W 1×2 i;µν (1.103)

Where β is a constant analogous to (1.102) and W_i;µν1×1 is a symbol for TrΠ. A generic W_i;µνn×m, instead, denotes the trace of the n × m Wilson loop built using the gauge links departing from the site in position i along the positive µ, ν directions.

Introduction to Monte Carlo Algorithms

Before I explain some details of the algorithm we used in ours simulation, I will outline the main idea of the Monte Carlo sampling.

Before passing to a QFT on the lattice, let us focus on a very simple discretized model: the four dimensional Ising model. Imagine a four dimensional lattice in which a single spin variables lives on each of its sites. Each of these spins may assume only two values: +1 or -1. This system can be, for example, a simple model for ferromagnetism and it has many other applications.

The expectation value of some observable O in this statistical ensamble is given by: hOi = 1 Z X {s} e−βH[s]O[s] (1.104) Where the sum runs over all possible spin configurations {s}. Now we want to measure this expectation value on the lattice, with a numerical simulation. Suppose we want to evaluate hOi following the definition above: a lattice with N4 _{sites has N}4 _{spin variables. Counting all possible combinations of spin}

val-ues we have 2N4 _{spin configurations. For a moderately large lattice we may}

have, for example, N = 16 and therefore 265536_{' 10}19728_{possible spin}

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all these configurations, clearly an impossible task. We have to find another way to evaluate that expectation values, providing an estimate for that sum. If we want to perform a simulation of QCD on the lattice, we have the same problem: for this reason we use the Monte Carlo method. In QFT or in statisti-cal physics, we already saw that all observables can be written in the functional formalism, i.e. as path integrals. The general problem, then, is how to numer-ically evaluate such integrals, that it is a problem analogous to evaluating the sum (1.104). In order to do that, let us start from an one dimensional system. In such a system the expectation value of some function f (x) with regard to a probability distribution with density ρ(x) is given by:

hf iρ= Z b a dx ρ(x)f (x) Z b a dx ρ(x) (1.105)

The main idea of the Monte Carlo integration is that the expectation value above can be approximated by an average over N values, {xn}. There are two

ways of sampling these variables: in principle it is possible to extract an uniform sample of variables {xn} between a and b, and then considering a new observable

Fρ(x) = [ρ(x)f (x)]/R dxρ(x). in this case the expectation value above can be

approximated by: hf iρ= lim N →∞ 1 N N X n=1 Fρ(xn) (1.106)

This technique is called simple sampling. Otherwise, each variables xn ∈

(a, b) can be randomly sampled with the normalized probability density: dP (x) = ρ(x)dx

Rb

adx ρ(x)

(1.107) Once that we have extracted the set of random variables {xn}, that we will

call a “configuration”, it is possible to perform the sum (1.106), using f instead of Fρ. This is the technique of ”importance sampling”. Using one of these

techniques we can approximate the integral (1.105). It is easy to show that, if the N measurements are statistically independent, then the error in the mean will be of order 1/√N . Anyway, if we compare these two techniques using two samples of data with the same N , we can see that the estimator calculated with the importance sampling MC converges more rapidly to the real value than the one estimated with the simple MC.

Now that we introduced the main idea of the MC, we can use the same strategy to perform simulation of QCD on the lattice. Using the discretization illustrated in the previous paragraphs, we usually want to compute the following generic ensemble average:

hOi =R DU O[U ]e

−S[U ]

R DU e−S[U ] (1.108)

Where S[U ] is a real functional of the link variables defined in the previous paragraph. Again, it is impossible to exactly perform the integral above, so we

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have to use a statistical method. We have to choose which technique of Monte Carlo is more efficient in this case: we stress that most of the link configurations will have an action which is very large, meanwhile only a small fraction of them will make a significant contribution to the integral (1.108). Hence in this case the most efficient way of computing the ensemble is the importance sampling technique: then, we have to generate a sequence of link variable configurations with a probability distribution given by the Boltzmann factor e−S[U ], and using them to evaluate the following sum:

hOi ' 1 N N X i=1 O({U }i) (1.109)

Again, the error will be of order 1/√N . I stress that in most of the simulation algorithms the configurations will not be statistically independent, so we have to find another way to estimate the errors. In this work we used the method of the Jackknife.

This is the main idea of the importance sampling Monte Carlo. In the next paragraph I will talk a little about how to sample the link variables.

Simulation of a non-Abelian Gauge Theory

In this paragraph we will still focus on the simulation of the “pure gauge” part of QCD Lagrangian. In particular, I will discuss some basic ideas of the numerical sampling of link variables.

In general, if we want to correctly sample a gauge theories we have to achieve the following objectives:

1. Define a regularized link variable that at the continuum limit corresponds to an element of the gauge group.

2. Because we usually work with local1 Monte Carlo algorithms we have to find the local form of the gauge action, that is the terms of the action which contain a particular link Uµ(x).

3. Finally, if we want to get a regularized form of the generating functional (1.19), we have to find a gauge invariant measure, sometimes called Haar measure.

In order to correctly simulate a SU(3) gauge theory, we will start discussing the simpler case of a SU(2) symmetry.

We already examined the discretization of the gauge fields in the link variables. We only have to find a suitable form for numerical simulations. In section (1.1.4), though, we mentioned that we can write the SU(2) link variables in the following way:

U = u01 + ~u · ~τ (1.110)

Where the four parameters are related by Eq. 1.57). Then, sampling three independent parameters of the four above is equivalent to sampling a link vari-able.

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Figure 1.4: examples of three staples around Uµ(x). This figure is taken from

Ref. [14].

Hence, in finding a way to sample a link variable, we accomplished the first of the objectives above. Now, we have to handle the second point of the previous list.

It is easy to find that the terms of the Wilson action in Eq. 1.100) which contain a particular link Uµ(x) are (see e.g. Ref. [9] for details):

Sloc[Uµ(x)] = −

β

N<Tr(Uµ(x)Σ

†

µ(x)) (1.111)

Where I dropped index W for the beta constant, and Σµ(x) is the sum of

the 2(d − 1) so-called “staples”, i.e.:

Σµ(x) =

X

ν6=µ

[Uν(x)Uµ(x+aˆν)Uν†(x+aˆµ)+Uν†(x−aˆν)Uµ(x−aˆν)Uν(x−aˆν +aˆµ)]

(1.112) A staple is the product of three link variables, graphically represented as Ui in Fig.1.4, with i = 1, 2, 3 in four dimension. Hence, the staple is a SU(2)

element. Fixing ν the first addendum of the Eq. 1.112) is one of the represented staples, while the second one is the staple which completes the other Wilson loop that involves Uµ(x). Then, in four dimension Σµ is the sum of six staples,

so it is not necessary a SU(2) element, but it is certainly proportional to a SU(2) matrix. Then, we can parametrize it in the following way:

Σµ(x) = F fµ(x) (1.113)

Where fµ is an element of SU(2) and F is a constant.

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Haar measure for SU(2).

We already stressed in section (1.1.4) that SU(2) is topologically equivalent to a sphere in four-dimensional Euclidean space. It is well known that the invariant measure of the four-dimensional sphere is:

dU = 1 π2d

4_uδ(|u|2_{− 1)} _(1.114)

Then, using spherical coordinates:

u0= |x|cosα u1= |x|sinαcosθ u2= |x|sinαsinθcosφ u3= |x|sinαsinθsinφ and: d4u = |u|3d|u|dΩ3 dΩ3= sin2(α)sin(θ)dαdθdφ

Integrating Eq. 1.114) in d|u| with the delta function and adding a normal-izing constant, we get:

dUµ=

1 2π2sin

2

(α)sin(θ)dαdθdφ (1.115) That is the Haar measure for SU(2).

Now we have all ingredients to sampling a SU(2) gauge theory. Combining these three elements and performing this change of variable:

˜

Uµ(x) = Σ†µ(x)Uµ(x) (1.116)

it is possible to find the following distribution of the link:

P ( ˜U ) = dcos˜θd ˜φp1 − ˜u0d˜u0eβF ˜u0 (1.117)

Then the problem is reduced to sampling the variable ˜u0, with the

distri-bution above, and it easily can be done, for example, with a Von Neumann algorithm.

Our goal, though, is to simulate a SU(3) theory. However, using the so-called Cabibbo–Marinari algorithm, SU(N) gauge links can be updated by applying sequential updates in SU(2) subgroups embedded in SU(N). In our case we consider three SU(2) subgroups of SU(3) parameterized by the matrices:

A1,2=   a11 a12 0 a21 a22 0 0 0 1   A2,3=   1 0 0 0 a11 a12 0 a21 a22   A2,3=   a11 0 a12 0 1 0 a21 0 a22   (1.118)

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Where: a = a11 a12 a21 a22 (1.119) With a ∈ SU(2). Then the SU(3) gauge link that we want to update is multiplied by a matrix A. Substituting U0 = AU in Eq(1.111) we can find a local action that depends from a. From here we follow the SU(2) algorithm described above to generate a matrix a, hence, from Eq(1.118), a matrix A. Thus, U0 is the new updated SU(3) link, and then the next SU(2) subgroup in the sequence is considered.

Hybrid Monte Carlo

In the last part of this section, I will say some words about the Hybrid Monte Carlo algorithm (HMC), that we used in our simulation (see Ref. [40] for more details).

The general problem is to generate a sequence of configurations {φi} distributed

according to e−S[φ]. This is can be done using the following ergodic algorithm. The main trick is to introduce a new set of random variables with a Gaussian probability density: P ({πi}) = Y i 1 √ 2π ! e X i 1 2π 2 i (1.120) Then, we do not try anymore to generate configurations {φi} but pairs of

{φi} and “momenta” {πi}, distributed according to e−H[π,φ], with:

H =X i 1 2π 2 i + S[φ] (1.121)

Hence, in the equation above we have an Hamiltonian H that governs the dynamics in a new ”time” variable (the simulation time). Then we can generate configurations using a discretized version of Hamilton’s equation of motion, starting from a pair {φi, πi} and obtaining a new pair {φ0i, πi0}. This mechanism

is called molecular dynamics. Therefore, we consider the equations of motion in their Hamiltonian form:

           ˙ φi = ∂H[φ, π] ∂πi ˙ πi = − ∂H[φ, π] ∂φi (1.122)

We want to discretize them in the time with time step . The result of this operation, integrating the Hamilton equations according to the so-called leapfrog scheme are the following equations, that tell us how to get φi(t + ) from φi(t),

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˜ πi(t) = πi(t) − 2 ∂S[φ] ∂φi(t) (1.123) φi(t + ) = φi(t) + ˜πi(t) (1.124) ˜ πi(t + ) = ˜πi(t) − ∂S[φ] ∂φi(t + ) (1.125) Where ˜π(t) is the momenta at time t + ₂ and this is the first step of the evolution.

Using a molecular dynamics allows one to update all φi variables at once, and

hence is better suited for studying systems with a non-local action (as is the case for QCD when fermions are included). But integrating Hamilton equations with a finite time step introduces a systematic error. One of the most commonly used algorithm that is free of systematic error is the Metropolis algorithm, but this algorithm becomes very slow when updating configurations with an action depending non-locally on the fields. This suggests that one should try to eliminate the systematic error in the hybrid algorithm by combining it with a Metropolis acceptance test. With this purpose in mind we notice that if the Hamiltonian equations of motion could be integrated exactly, then H would be constant along a molecular dynamics trajectory. But for finite then δH 6= 0. Therefore we perform a Metropolis test; thus, we define:

R = eδH (1.126)

Then we accept the new pair φi(t + ), πi(t + ) with a probability P =

min{1, R}.

Summarizing the algorithm:

1. Choose the starting set {φi} in some arbitrary way.

2. Extract the momenta {πi} from the Gaussian ensemble (1.120).

3. Perform the half step (1.123) and calculate ˜πi to start the integration

process.

4. Iterate Eqs (1.124, 1.125) for some number of time steps. Let {φ0_i, π0_i} be the last configuration generated in the molecular dynamics chain. 5. Perform Metropolis test: Accept {φ0_i, π0_i} as the new configuration with

probability:

P = min{1, R}

6. If the configuration {φ0_i, π0_i} is not accepted, start again with the old con-figuration {φi, πi}, and repeat the steps starting from (2). Otherwise use

the coordinates {φ0_i, π_i0} to generate a new configuration beginning with the step (2).

In conclusion the Hybrid Monte Carlo algorithm (HMC) is suited for study-ing systems with a non-local action and at the same time is free of systematic error caused by the finite time integration. In addition most of the configura-tions generated will be accepted by the Metropolis test, if the time step is not too large. This means that the system is moving fast through configuration space. For these reasons is one of the best algorithms for simulating QCD on the lattice.

Properties of strong interactions in background fields: the Chiral Magnetic Effect

Dipartimento di Fisica E. Fermi

Corso di Laurea Magistrale in Fisica

Curriculum Fisica Teorica

Properties of strong interactions

in background fields: the Chiral

Magnetic Effect

Candidate:

Supervisor:

Lorenzo Dini

Prof. Massimo D’Elia

Contents

Chapter 1

Theoretical Introduction

1.1

Quantum Chromodynamics

1.1.1

An Introduction to QCD

1.1.2

QCD as a non-Abelian Gauge Theory

1.1.3

Asymptotic Freedom

1.1.4

Chiral Symmetry

1.2

QCD on the Lattice

1.2.1

Field Theory on the Lattice

1.2.2

Focusing on QCD