Topological Properties of SU(3) Yang-Mills Theory With Double Trace Deformation

Topological Properties of SU (3) Yang-Mills

Theory with Double Trace Deformation

Candidato:

Marco Cardinali

Relatore :

Prof. Massimo D’Elia

Introduction i

1 Yang-Mills Theory and Non Perturbative Properties 1

1.1 Abelian and Non-Abelian Gauge Theories . . . 1

1.2 The Path Integral Formulation . . . 7

1.3 Asymptotic Freedom . . . 13

1.4 Confinement. . . 15

1.5 Finite Temperature Yang-Mills Theory. . . 18

1.6 The Large N Limit . . . 20

1.7 θ Dependence and Topology . . . 21

2 Lattice Yang-Mills Theory 33 2.1 Lattice Formulation of YM Theory . . . 33

2.2 The Continuum Limit . . . 35

2.3 Monte Carlo Methods . . . 37

2.4 Algorithms for Gauge Theories . . . 43

2.5 Finite Temperature on the Lattice . . . 46

2.6 Topology on the Lattice . . . 50

3 Double Trace Deformation 55 3.1 Why a Deformed Theory? . . . 55

3.2 Double Trace Deformation . . . 57

3.3 Double Trace Deformation and MC Simulations. . . 58

4 Numerical Results 63 4.1 Adjoint Polyakov Loop . . . 63

4.2 Topological Properties of The Reconfined Phase . . . 64

Yang-Mills theory (YM) is a non abelian gauge theory with SU (N ) as the gauge group. This the-ory was proposed for the first time in 1954 by C. N. Yang and R. L. Mills, when they tried to write a Lagrangian that was locally invariant under the gauge group SU (2) (isospin transformations). Nowadays QCD, the theory that describes strong interactions, is based on a SU (3) Yang-Mills theory. Yang-Mills theory has a very peculiar property: asymptotic freedom. When this theory is studied at high energies (ultraviolet limit) it is weakly coupled and it is possible to perform a perturbative expansion. On the contrary, when it is studied at low energies (infrared limit) it becomes strongly coupled and perturbation theory is not reliable any more. Intrinsically non perturbative phenomena, such as confinement, chiral symmetry breaking and topology, can be studied on the lattice using Monte Carlo simulations.

In this work we consider SU (3) Yang-Mills theory discretised onR3× S1, i.e. with a compacti-fied direction and periodic boundary conditions imposed on that direction. This is equivalent to consider the system at finite temperature. In particular, if the length of the compactified di-rection decreases, the temperature increases, and vice versa. We are interested in considering the theory in the limit of small compactification radius (high temperatures). In such a limit we have a high energy scale (the inverse of the compactification length) which, due to asymp-totic freedom, allows us to perform semiclassical methods to study the theory. However, if we squeeze the compactification radius too much the system undergoes a phase transition. In this phase transition centre symmetry is spontaneously broken and the Polyakov loop, the order pa-rameter, acquires a non-zero value. A centre symmetry transformation consists in multiplying all the temporal links at a given time slice by an element of the centre of the gauge group, the centre of SU (3) being Z3. Polyakov loop is the ordered product of the links in the time direc-tion and its mean value is different from zero if centre symmetry is spontaneously broken. This phase transition is known in the literature as the deconfinement phase transition, since the cor-relator of the Polyakov loop is related to the free energy of a quark-antiquark pair and thus is related to the confining mechanism. Because of this phase transition the theory is composed

of two different phases that are not analytically connected and so it is not possible to study the high temperature regime and obtain, from that, information about properties that are typical of the low temperature one (confinement, topology). However, this phase transition is not sur-prising because when we consider the theory in the small compactification limit we are in the weakly coupled regime. In this regime it is not possible to study with semiclassical methods properties that are relevant in the strong coupling regime.

In this framework M. Unsal and L. Yaffe in Ref. [1] proposed a deformed action for SU (3) Yang-Mills theory with the purpose of restoring centre symmetry even for small length of the com-pactified direction. The deformation is proportional to the square of the trace of the Polyakov loop and explicitly suppresses gauge configurations with a different from zero value of the mean Polyakov loop, thus it is possible to restore centre symmetry even for small values of the compactification radius. We may now ask if this new theory in which centre symmetry is preserved possesses the same non perturbative properties of the undeformed one. In this work we studied, using Monte Carlo simulations, some of the properties of the deformed theory try-ing to give an answer to the question: are the deformed theory and the undeformed SU (3) YM theory equivalent?

First of all we implemented on the lattice the deformed theory and then we checked that cen-tre symmetry was recovered even at high temperature. In doing so we reproduced the results of the only work in the literature, Ref. [2], in which numerical simulations with double trace deformation were performed. Then we computed the adjoint Polyakov loop, which is an ob-servable that gives information on how centre symmetry is recovered, in particular it is sensible to the local eigenvalues of the Polyakov loop. Next we explored the topological properties of the deformed theory. We considered the Lagrangian of SU (3) YM theory with a non-zeroθ param-eter. Theθ term breaks explicitly parity and time reversal. Moreover there are quite stringent experimental bounds on it:θ/10−10. Nevertheless it enters in different aspects of the hadron phenomenology, e.g. the solution of the U (1)Aproblem. Performing numerical simulations us-ingθ 6= 0 in the SU(3) Lagrangian is not possible because we will have an imaginary action, but it is possible to expand the free energy aroundθ = 0. The first two coefficients of such expan-sion areχ, the topological susceptibility, and b2. We can relate bothχ and b2to the momenta of the topological charge distribution computed atθ = 0. From the numerical point of view we have to make a remark on how we computed the coefficient b2. This is a rather noisy ob-servable because it involves the computation of the fourth order momentum of the topological charge distribution. If we want to measure b2atθ = 0, we need approximately 106independent gauge configurations to get a value with an error of ≈ 20% and this is very expensive from the computational point of view. In order to have a good result with less numerical effort we used the imaginaryθ method. We added to the Lagrangian an imaginary θ term and we performed

simulations using different values of this imaginaryθ term. The idea is to determine the cumu-lants of the topological charge as a function of the imaginaryθ and then perform a combined fit of them to extract b2. We measured bothχ and b2in the deformed theory. The topological susceptibility is a dimensional observable, thus we had to make an assumption on the relation between the lattice spacing and the deformation in order to compare it with its value in the undeformed theory: we assumed that the deformation does not modify the lattice spacing. On the contrary, b2is a dimensionless observable and no assumptions on the lattice spacing are required. We explored different values of the deformation and we performed simulations on lattices with different lengths of the compactified direction and different bare couplings.

This thesis is organised in four chapters.

Ï Chapter 1. We make a summary of the main properties of Yang-Mills theory in the con-tinuum, focusing especially on the non-perturbative aspects. We introduce asymptotic freedom, confinement, finite temperature, topology andθ dependence.

Ï Chapter 2. We discuss the lattice formulation of Yang-Mills theory and Monte Carlo methods.

Ï Chapter 3. We introduce the double trace deformation at first from a theoretical point of view and than we explain how we implemented it on the lattice. Next we show the litera-ture results about numerical simulations of SU (3) YM with double trace deformation. Ï Chapter 4. We show our original results about the adjoint Polyakov loop and the

topo-logical properties.

Ï Conclusions. We summarise all the results obtained and we give some possible ideas for further studies.

bative Properties

In this chapter we give a brief introduction on Yang-Mills theory. We start with its formulation in the continuum and we introduce also the path integral formalism. Then we show some of the key properties of this theory, such as confinement, asymptotic freedom and the limit of large number of colours. The last part of the chapter is dedicated to theθ dependence and topology.

1.1 Abelian and Non-Abelian Gauge Theories

A gauge theory is a theory that possesses a local symmetry induced by a gauge group. A first example of a gauge theory is Quantum Electrodynamics (QED), in which we have both a global and local U (1) symmetry. We start with a fermion fieldψ with spin 1₂ and charge q = Q|e| and the free lagrangianL = ψ¡i /∂− m¢ψ where /∂ = γ_µ∂µ andγµ ’s are the gamma matrices. The global U (1) is connected with the conservation of the electric charge and on theψ field it acts in the following way

     ψ(x) −→ ψ0_{(x) = e}iQχ_ψ ¯ ψ(x) −→ ¯ψ0_{(x) = e}−iQχ_ψ¯ (1.1)

whereχ is a constant. It is clear that L is invariant under such transformation. To such sym-metry is associated a conserved current Jµ= ¯ψγµψ. On the contrary, in order to implement the

U (1) gauge symmetry we have to makeL invariant under a local phase transformation of the ψ field, i.e.      ψ(x) −→ ψ0_{(x) = e}iQθ(x)_ψ ¯ ψ(x) −→ ¯ψ0_{(x) = e}−iQθ(x)_ψ¯ (1.2) 1

whereθ(x) is a function of the space-time point x. Since ∂_µacts also onθ(x) we have to intro-duce a new field A_µ(the photon, or gauge, field) to compensate the variation. If we substitute

∂µwith the covariant derivativeDµ

∂µ−→ Dµ= ∂µ+ iQ|e| Aµ (1.3)

and let the A_µfield transform as

A_µ−→ A0µ= Aµ− 1

|e|∂µθ(x), (1.4)

we have

L =ψ¡i /D − m¢ψ −→ L0_=ψ0³_{i¡ /∂ + iQ|e| /A}0_{¢ − m}´_ψ0₌ =ψe−iQθ(x)³i¡ /_{∂ + iQ|e| /A − iQ /∂θ(x)¢ − m}´eiQθ(x)ψ(x) =

=ψ(i /D − m)ψ −Qψ(/∂θ(x))ψ +Qψ(/∂θ(x))ψ =ψ(i /D − m)ψ = L .

In this wayL is invariant under the U(1) local symmetry.

The last thing left to be done is writing down the dynamic term for the photon field. To do so we introduce the tensor Fµν= ∂µAν− ∂νAµcalled the field strength. It is easy to derive Fµν computing the commutator of two covariant derivatives:

h Dµ,Dν i ψ = ·_³ ∂µ+ iQ|e| Aµ ´ ,¡ ∂ν+ iQ|e| Aν¢ ¸ ψ = iQ|e| µ_h ∂µ, Aν i +hA_µ,∂_νi ¶ ψ = = iQ|e| µ_³ ∂µ(Aνψ) − Aν(∂µψ) + Aµ(∂νψ) − ∂ν(Aµψ) ´¶ = = iQ|e|³(∂µAν)ψ + Aν(∂µψ) − Aν(∂µψ) + Aµ(∂νψ) − (∂νAµ)ψ − Aµ(∂νψ) ´ = = iQ|e|³∂µAν− ∂νAµ ´ ψ ≡ iQ|e|Fµνψ. Thus, adjusting the numerical factor, the QED lagrangian is

LQE D = −1

4FµνF

µν_{+ψ¡i /D − m¢ψ. (1.5)}

The interest in non abelian gauge theories arose from the study of strong interactions. The hadrons are the particles that interact under strong interactions and they can be divided in

two categories: the mesons, which possess a integer spin, and the baryons, which possess a half-integer spin. In 1932 Heisenberg, studying the neutron, noticed that proton and neutron possess roughly the same mass and they behave very similarly under strong interactions, in-deed they were also called nucleons. Therefore he proposed a global symmetry based on the group SU (2) called isospin and organised the known hadrons in multiplets of such group. Pro-ton and neutron formed a duplet and the three pions a triplet.

In 1954 C.N. Yang and R.L. Mills, Ref. [3], promoted the global SU (2) symmetry of isospin to a local symmetry. Their aim was to write a Lagrangian that was invariant under a local SU (2) transformation.

Ne’emann (1961) and Gell-Mann(1962) observed that hadrons fitted SU (3) representation. In particular they noticed that mesons fitted octets and singlets while baryons fitted the octets and decuplets. Since in 1961 only nine baryon resonances were known they predicted the existence of a tenth resonance: theΩ−particle with a mass approximately of 1680 MeV.

In 1964 this particle was found in Brookhaven National Laboratory and in the same year Gell-Mann, Ref. [4], and Zweig, Ref. [5], presented the quark model. They claimed that hadrons where bound states made of quarks, spin1₂particles, which fill the fundamental representation of SU (3). However this model left two open problems: why quarks have never been observed and how the wave function of the state∆++, which has spin J =3₂, can be made antisymmetric if it is composed by three identical quarks?

The solution to these problems arrived in 1965 by M.Y. Han and Y. Nambu with the colour hy-potheses, Ref. [6], and the postulate of confinement. They introduced an exact SU (3) gauge symmetry, and thus another quantum number, the colour. With this new symmetry the prob-lem of the∆++_{state is easily solved since now the three quarks possess another quantum} num-ber and the state can be made antisymmetric. On the other hand the postulate of confinement claims that only singlets of this SU (3) gauge symmetry can be observed and so quarks, which are not singlets, can not be observed as asymptotic states. In this way we can distinguish two different SU (3) symmetries: a flavour SU (3)Fsymmetry which is approximate and a local exact colour SU (3)Csymmetry.

The choice of SU (NC) with NC= 3 colours as the gauge group is strongly supported by two ex-perimental results. The first is the decay of the particleπ0in two photonsγ. If we consider the

π0_{composed of coloured quarks the decay width is proportional to N}2

Cand it is in agreement with the experimental results only if NC= 3. The second experimental result concerns the cross section

e+e− → hadrons,

electron and of the quarks we obtain σ =4πα 2 3s NC Nf(s) X f =1 Q2_f,

whereα =₄e_π1, s is the centre of mass energy, Nf(s) is the number of flavours with a mass mf such that 2mf <ps and Qf is the electric charge of the flavour f . We now normalise this cross section using the cross section of the process

e+e− → µ+µ−, obtaining the ratio

R =σ(e +_e− _{→ hadrons)} σ(e+_e− _{→ µ}+_µ−₎ = NC Nf(s) X f =1 Q2_f.

Also in this case the experiments are in agreement with the theory only if NC= 3.

The concepts exposed above are the foundation of Quantum Chromodynamics (QCD), the cur-rent theory of strong interactions.

Knowing that the theory has an SU (3) local, or gauge, symmetry we are able to build the La-grangian. We follow exactly the same steps as in the case of QED, remembering that now we have a non abelian gauge group with 8 generators2. A generic element U ∈ SU (NC) can be written as U = exp    i g N2 C−1 X a=1 θa_(x)Ta    ,

where Ta’s are the generators in the fundamental representation of SU (NC). We recall that in the fundamental representation we have

£Ta, Tb¤ = i fabcTc, (1.6) tr¡TaTb¢ = 1 2δab, (1.7) Ta=λ a 2 , (1.8)

1_{We are using natural units: ħ = c = 1.}

where theλa’s are the Gell-Mann matrices.

Consider the lagrangian

L0= NF X f =1 ψf ³ i /∂ − mf ´ ψf,

in which f is the flavour index andψf is the quark field. We wantL0to be invariant under the transformation

ψ −→ ψ0_{= U ψ U ∈ SU (N} C).

We have to introduce eight gauge fields, one for each generator of the group, called the gluons

Aµ(x) = Aa_µ(x)Ta. The transformation law of A_µis

A_µ−→ A0_µ= U (x)A_µ(x)U−1(x)

| {z }

this vanishes if the gauge group is abelian

+ i g ³ ∂µU (x) ´ U−1(x), (1.9)

thus we introduce the covariant derivative

Dµ= ∂µ+ i g Aµ= ∂µ+ i g A_µaTa

and we compute, as in QED,

h Dµ,Dν i ψ = ·_³ ∂µ+ i g Aµ ´ ,¡ ∂ν+ i g Aν¢ ¸ ψ = = i g µ ∂µAν− ∂νAµ+ i g h A_µ, A_νi ¶ ψ ≡ i gFµ νψ,

with F_µν= ∂µAν− ∂νAµ+ i ghA_µ, A_νi. Now we are able to write the full Lagrangian of QCD

LQC D = −1 4F a µνFaµν | {z }

pure gauge sector + Nf X f =1 ψf ³ i /D − mf ´ ψf | {z } quark sector . (1.10)

An alternative way to deduce the transformation law of A_µ(x) and the form of the field strength

F_µνis using the parallel transport. Consider two fermion fields defined in two different space-time points:ψ(x) and ψ(y). Under a gauge transformation they transform with two different elements of the gauge group

ψ(x) −→ ψ0_{(x) = U (x)ψ(x),}

ψ(y) −→ ψ0_{(y) = U (y)ψ(y).}

The parallel transport maps a vector that transform with U (x) into another one that transforms under U (y). Consider a path C_y←xconnecting x and y. We define parallel transport a matrix

W (C_y←x) ∈ SU (N ) such that:

• W (;) = 1, with ; a path with zero length. • W (C2◦C1) = W (C2)W (C1).

• W (−C ) =¡W (C )¢−1, where −C is the path C with the opposite direction. • W (C_y←x) −→ W0(C_y←x) = U (y)W (Cy←x)U†(x).

Using the last property we have immediately

˜

ψ(y) = W (Cy←x)ψ(x) −→ ˜ψ0(y) = U (y)W (Cy←x)U†(x)U (x)ψ(x) = = U (y)W (Cy←x)ψ(x) = U(y) ˜ψ(y).

Consider now an infinitesimal rectilinear path x −→ x +d x, since W (C_{x+d x←x}) ∈ SU (N ) we can write

W (C_{x+d x←x}) ' exph−i g Aµ(x)d xµi' 1 − i g Aµ(x)d xµ,

where A_µ(x) is the gauge field and A_µ(x) = N2

C−1

X a=1

Aa_µ(x)Tain which the Ta are the generators of

SU (N ) in the fundamental representation. Using again the last property itemised above, we

W0(C_{x+d x←x}) ' 1 − i g A0_µ(x)d xµ= U (x + d x)W (Cx+d x←x)U†(x) = = U (x + d x)³1 − i g Aµ(x)d xµ

´

U†(x) =

= U (x + d x)U†(x) − i gU (x + d x)AµU†(x)d xµ= =³1 + (∂µU (x))U†(x)d xµ´− i gU (x)Aµ(x)U†(x)d xµ= = 1 − i g " U (x)A_µ(x)U†(x) +i g(∂µU (x))U †_(x) # d xµ.

So we get exactly Eq. (1.9). For what concerns the field strength, we use the general form of the parallel transport along a generic path C_τparametrised by the function zµ(s)

W (Cτ) = P  exp ( −i g Z τ 0 Aµ¡z(s)¢ dzµ(s) ds ds ) , (1.11)

whereP is the path-ordering operator. Now we consider a closed path C_x←x, we expand Eq. (1.11) up to the second order and get

W (C_x←x) ' exph−i g Fµν(x)d xµd xν i

, (1.12)

with F_µν≡ ∂µAν− ∂νAµ+ i g [Aµ, Aν].

1.2 The Path Integral Formulation

In the rest of the work the Path Integral (PI) formulation will be largely used, so we will re-call now some of the main properties. Within the PI formulation, it is straightforward to write n-point correlation functions of fields, indeed consider a scalar fieldϕ(x) whose dynamic is governed by the Lagrangian

L =1 2ϕ(x)

³

2− m2´ϕ(x) −V (ϕ(x)) = L0− V (ϕ(x)),

where V (ϕ(x)) is a potential and it is a polynomial in the field ϕ(x). The n-point correlation function can be written using the well known formula

Gn(x1, ··· , xn) = 〈Ω|T©ϕ(x1)ϕ(x2) ···ϕ(xn)ª |Ω〉 =

R [dϕ]ϕ(x1)ϕ(x2) ···ϕ(xn)ei S

in which |Ω〉 is the ground state, R [dϕ] = R Πdϕ(xi) is the integration over all the possible field configurations and S =R d4_{xL (x) is the action of the system. Eq. (1.13) is very similar to an} ensemble average as commonly encountered in statistical mechanics. The main difference is that now we have an imaginary exponential instead of a real one, however it is useful to define the partition function Z as

Z = Z [dϕ]exp µ i Z d4xL (x) ¶ .

In general the integral of Eq. (1.13) can not be computed directly unless V (ϕ(x)) = 0 in which case we are dealing with a Gaussian integral. Correlation functions such as the one above are computed perturbatively, we first define a functional generator

Z [J ] = Z [dϕ]exp µ i Z d4x© L (x) + J(x)ϕ(x)ª ¶ , (1.14)

where J (x) is an external source coupled to the fieldϕ. All the correlation functions can now be computed just deriving the functional generator and then putting the external source equal to zero Gn(x1, ··· , xn) =(−i ) n Z [0] Ã δn_{Z [J ]} δJ(x1) ··· J(xn) ! J =0 . (1.15)

Now we consider the potential V (ϕ(x)). If, as we required at the beginning, it is a polynomial in the fields, we can write

Z [J ] = Z [dϕ]e−iR d4xV (ϕ(x))eiR d4x{L0+J (x)ϕ(x)} = = ∞ X n=0 (−i )n n! Z [dϕ] µZ d4xV (ϕ(x)) ¶n eiR d4x{L0+J(x)ϕ(x)} = = ∞ X n=0 (−i )n n! Z [dϕ]   Z d4xV Ã −i δ δJ(x) !  n Z0[J ],

where Z0[J ] = eiR d4x{L0+J(x)ϕ(x)}. Using the above formula it is possible to perform perturbative analysis.

This was the case of a bosonic field, if we are dealing with fermionic fields we can derive the same equations but we have to pay attention that now our fields are anti-commuting variables.

To treat them we introduce the Grassmann variablesη1· · · ηn, that are numbers that obey the anti-commutation rule n ηi,ηj o = ηiηj+ ηjηi= 0 ∀i , j.

Then we define the Berezin integration rules

Z

dηi= 0, Z

dηiηi= 1.

With this new variables we can write the functional integral for a fermionic fieldψ(x) in the following way

Z [ρ, ¯ρ] =Z [dψd ¯ψ]eiR ©d4xL (x)+ ¯ψρ+ ¯ρψª, (1.16)

whereρ and ¯ρ are Grassmann variables and L = ¯ψ¡i /∂− m¢ψ. All the correlation functions can be obtained from Eq. (1.16) deriving with respect toρ or ¯ρ. One last comment is on the form of the Gaussian integral in the case of fermionic variables. Consider the integral

I [A] = Z N Y i =1 d ¯ψidψiexp    − N X i , j =1 ¯ ψiAi jψj    , (1.17)

where A is a non singular matrix andψiare Grassmann variables. The result of such integral is

I [A] = det(A). (1.18)

1.2.1 The Yang-Mills Case

When we want to write the path integral formulation for a gauge theory, some problems arise due to the gauge invariance. Consider QED action

SQE D[A] = Z d4xLG[A] = − 1 4 Z d4xFµνFµν= = −1 2 Z d4x · ¡ ∂νAµ¢ ¡∂νAµ¢ − ³ ∂νAµ ´_¡ ∂µAν¢ ¸ = =1 2 Z d4x A_µ(x)hgµν∂2− ∂µ∂νiA_ν(x),

where gµν is the Minkowski metric. If we want to perform perturbation theory, we need to invert the operatorhgµν∂2− ∂µ∂νi, but this operator possesses some zero modes!

An example of zero mode is the gauge transformation of the the gauge field Aµ= 0. Indeed the gauge transformation is 0 −→ 0 + ∂µθ(x) and thus

³

gµν∂2− ∂µ∂ν´∂νθ(x) = ∂2∂µθ(x) − ∂2∂µθ(x) = 0.

A way to overcome this problem is to fix the gauge field A_µand consider all its possible gauge transformations. Doing so we obtain a set of gauge orbits and when we define the path integral we integrate only over one field per gauge orbit. This method is known as the Faddeev-Popov method (Ref. [7]). Due to this gauge fixing in the partition function it is necessary to add the determinant of the Faddeev-Popov matrix and we can introduce two auxiliary fields called ghost and use Eq. (1.18) to write that determinant as a Gaussian integral over Grassmann variables. We underline the fact that these auxiliary fields are unphysical because they do not obey the spin-statistic theorem, since they are anti-commuting fields with spin 0.

1.2.2 The Euclidean Formalism

As we mentioned in the section above there is a similarity between the correlation functions such as Eq. (1.13) and the ensemble average in statistical mechanics. The difference relies on the fact that in the field theory we are dealing with an imaginary exponential while the Boltz-mann factor e−βHof the statistical system is real. Nevertheless we can recover a real exponential even in the field theory performing the so-called Wick Rotation. We pass from a "real" time to an "imaginary" time imposing

t −→ −i T.

More explicitly we perform the change in the coordinates space:

x =³x0,~x´−→¡−i xE 4,~xE¢ ≡ xE,

with x0= −i xE 4and~x = ~xE. In the momentum space the change is k0= i kE 4and ~k = −~kE. In this way we have

x2= xµxµ= (x0)2−¯¯~x¯¯ 2

= −xE 4−¯¯~xE¯¯ 2

= −xE2.

Also the derivative changes its form

∂0= ∂ ∂x0−→ i ∂ ∂xE 4⇒ ∂µ= ¡ ∂0, ∇¢ −→ ∂E=¡i∂0, ∇¢ .

The subscript E stands for Euclidean since the space in which we are writing our field theory has now the metric ofR4and not the Minkowski metric. Using this formalism and setting the notationϕ(xE) = ϕE(xE) the correlation function of Eq. (1.13) becomes

〈Ω|T ©ϕE(xE ,1)ϕE(xE ,2) ···ϕE(xE ,n)ª |Ω〉 =

R [dϕE]ϕE(xE ,1)ϕE(xE ,2) ···ϕE(xE ,2)e−SE R [dϕE]e−SE

, (1.19)

where SEis the Euclidean action defined as SE=R d4xELE(xE) with

LE(xE) = 1 2(∂EϕE) 2 +1 2m 2_ϕ2 E+ V (ϕE).

Now we will see how to write non-abelian gauge theories in the Euclidean formalism. The gauge field Aµ(x) acts as the derivative∂Ein order to keep the covariantDEas the common derivative. The field strength is then:

F0i= ∂0Ai− ∂iA0+ i g£ A0, Ai¤ = i ³ ∂E 4AEi− ∂EiAE 4+ i g£ AE 4, AEi¤ ´ ≡ i FE ,4i, Fi j≡ FE ,i j.

Inside the Euclidean framework the Lagrangian for a YM theory can be written as:

LY M(xE) = − 1 2tr h F_µν(xE)Fµν(xE) i = = −1 2 ½ 2trhF0iF0i i − trhFi jFi j i¾ = =1 2 ½ 2tr£FE ,0iFE ,0i¤ + tr h FE ,i jFE ,i j i¾ = = −1 2tr h FE ,µνFE ,µν i ≡ −LE ,Y M(xE).

In the Euclidean formalism the relation between the field theory and the corresponding sta-tistical mechanic system is more explicit and we could interpret e−SE _{as the Boltzmann factor}

with which we weight the field configurations. This interpretation will be used in Chapter 2 to introduce Monte Carlo methods. Moreover there is also a clear relation between the statistical mechanic system at finite temperature T and the corresponding field theory. This relation can be understood starting from the partition function of a classical statistical system in an heat bath with temperature T

Z (T ) = tr³e−βH´ (1.20)

in whichβ = _T1 is the inverse of the temperature3and H is the Hamiltonian of our system. If we consider a scalar fieldΦ we can rewrite Eq. (1.20) using the path integral formulation. Due to the trace we obtain the field theory of a scalar field with periodic boundary conditions along the time direction.

Z (T ) = Z [dΦ]e−S[Φ], (1.21) with S[Φ] = Z β 0 d t Z R3d 3_xL³_Φ(t,~x),∂ µΦ(t,~x) ´ Φ(0,~x) = Φ(β,~x). (1.22)

If we had the gauge field A_µ(t ,~x) instead of the scalar field Φ we would have the same equations with the Euclidean Yang-Mills action and periodic boundary conditions A_µ(0,~x) = A_µ(β,~x). Now the space in which we study our theory is no more anR4but aR3× S1where the temporal coordinate can assume the values [0,β]. Our system now is in a space with infinite spatial vol-ume but at a fixed temperature T =1_βand the limit of zero temperature correspond toβ → ∞. We want underline the fact that in pure gauge theory we have only a bosonic field A_µand no fermions, so if we consider the theory inR4and we compactify a generic direction imposing periodic boundary conditions on the fields, this is equal to study the theory at finite tempera-ture.

3_{Here we use the convention in which k}

1.3 Asymptotic Freedom

One of the key properties of QCD is asymptotic freedom. We now recall here some of the main features. We start with the QCD Lagrangian of Eq. (1.10)

LQC D = −1 4F a µνFaµν+ Nf X f =1 ψf ³ i /D − mf ´ ψf,

the gauge coupling g is written inside the covariant derivativeD_µ = ∂µ+ i g Aµ(x) and it is a dimensionless coupling, thus the theory is renormalizable. We consider the relation between the bare gauge coupling and the renormalized one, g = ZggR and we take Zg at 1-loop order in the MS scheme using the dimensional regularisation. The details of the computation can be found in Ref. [8]. The result obtained is

Zg= 1 − g_R2 (4π)2 1 6¡11NC− 2NF ¢ 1 ε+ O ³ g_R4´, (1.23)

where NC is the number of colours, NF is the number of flavours andε = 4−D₂ is a parameter due to the dimensional regularisation used here to perform the divergent integrals. The dimen-sional regularisation introduces an arbitrary energy scaleµ

gr → ˜gR= gRµε.

Now we define the Gell-Mann and Lowβ-function of QCD, defined as follows:

β¡gR(µ)¢ ≡ µ d gR(µ)

dµ . (1.24)

We can rewrite this expression as

µd gR(µ) dµ = −εgR− µ Zg d Zg dµ gR. (1.25)

If we substitute the 1-loop expression for Zg and we take the limitε → 0 we obtain the 1-loop β function4

β¡gR(µ)¢ = −β0g_R3+ O ³

g_R5´, (1.26)

with β0= 1 (4π)2 · 11NC− 2NF 3 ¸ . (1.27)

Asymptotic freedom is a consequence of the sign ofβ0. First of all we should solve the differen-tial equation (1.24) dµ µ = d gr(µ) β¡gr(µ)¢ ⇒ µ2= µ1exp ( Z gR(µ2) gR(µ1) d gr(µ) β¡gR(µ)¢ ) .

Now we use the 1-loop approximation of Eq. (1.26) and we get:

µ2= µ1exp ( − Z gR(µ2) gR(µ1) d gr(µ) β0g_R3 ) = µ1exp ( 1 2β0g_R2(µ2) − 1 2β0g_R2(µ1) ) , so µ2e −₂ 1 β0g2R(µ2)= µ1e −₂ 1 β0g2R(µ1) | {z }

same expression computed at two different energyµ2andµ1

≡ ΛQCD. (1.28)

ΛQCDis a constant of integration and it is called the mass parameter of QCD. Now it is easy to get gR(µ) µe− 1 2β0g2_R(µ) = ΛQC D ⇒ g_R2(µ) = 1 β0ln µ µ2 Λ2 QC D ¶ . (1.29)

Eq. (1.29) is valid both forβ0> 0 and β0< 0. In both of the cases g_R2(µ) has a divergence for

µ = ΛQC D, the difference is ifµ approaches ΛQC Dfrom the right or from the left. If we consider QCD we have NC= 3 and NF= 6 ⇒ β0> 0 ⇒ µ > ΛQC D. This means that if we consider the the-ory at high energy (µ → ∞ ) the coupling goes to zero and the theory becomes weakly coupled. Instead, forµ ≈ ΛQC D there is a divergence of gR(µ) and the theory became strongly coupled. An important remark is that it can be proved thatβ0andβ1, the first and the second term of the loop expansion, are independent of the renormalization scheme used.

As we mentioned before,µ is an arbitrary energy scale, but we can fix it. The most convenient way is to fix it at the same energy of the process that we are studying with perturbation theory,

however we can fix it independently of the energy of the process in exam, but we will need to compute more terms of the perturbative expansion.

In Fig.(1.1) we show the experimental results concerning asymptotic freedom and the diver-gence in the low energy region.

Figure 1.1:Experimental results for the coupling of QCD. Figure taken from Ref. [10].αS= g 2

(4π)2.

1.4 Confinement

One of the key features of QCD is confinement. In nature there is no evidence of isolated quarks and it is believed that only singlets of the gauge group SU (3)C can be observed. So far there is no analytic proof of quark confinement but only strong evidences and most of them come from numerical simulations. However there are reasons to believe that confinement relies on the non-abelian properties of the gauge group. If we consider a pure SU (3) gauge theory, we should compute the potential of a q ¯q or q q q system and see if it raises with distance. We can

not use the perturbative expansion to study such a potential because we are interested in the long distance regime, so at low energies (infrared limit), where the theory is characterised by strong coupling.

On the contrary when we consider full QCD with dynamical quarks there is a difference in the form of the potential. In this case the potential starts raising with the distance, but then it

stabilises. This is due to the fact that at very large distance there is enough energy for the pro-duction of two mesons.

1.4.1 The Static q ¯q Potential and the Wilson Loop

The technique used to extract the static q ¯q potential is the same technique used to extract the

ground state energy of a quantum mechanical system. First of all we have to choose an operator

O(t ) with approximately the same quantum numbers of the state we want to study, in this case

the bound state q ¯q. Then we write such operator in the Heisenberg picture

O(t ) = ei H tO(0)e−i H t,

where H is the Hamiltonian that describes the dynamics of our system. Next we compute the time evolution and we insert a complete set of eigenstates of H

〈Ω|O†_{(t )O(0) |Ω〉 = X} n 〈Ω|O †_{(t ) |n〉〈n|O(0)|Ω〉 =} =X n e−i (En−EΩ)t¯ ¯〈n|O(0) |Ω〉¯¯ 2 .

If we pass to the Euclidean time t → −i T and we perform the limit T → ∞ we select only the en-ergy of the ground state. We can observe from the above expression that it is crucial to choose wisely the operator O(t ) because it must have a non vanishing overlap with the state we want to study.

Consider now an heavy quark Q of mass MQ in the position y and an heavy antiquark ¯Q with the same mass MQin the position x. The fields of this two quarks are respectively ¯ψ( ¯Q)(~x,t) and ψ(Q)_{(~y,t). The operator chosen to extract the potential of the heavy Q ¯Q system is}

O(~x,~y;t) = ¯ψ( ¯Q)(~x,t)W (C_x←−y)ψ(Q)(~y,t), (1.30)

where W (C_x←−y) is the parallel transport from the point y to the point x along the path C . It is possible to study the time evolution5of the operator of Eq. (1.30) in the static limit, MQ→ ∞. Doing so we obtain the relation

〈WC[A]〉A≡ W (R, T ) = C (R)e−V (R)T (1.31)

where R is the spatial distance between the two quarks, C (R) is the overlap between our trial op-erator and the ground state of the Q ¯Q system, V (R) is the potential of the pair Q ¯Q and 〈WC[A]〉A is the expectation value over the gauge fields of the Wilson loop, defined as the ordered product of the gauge variables around the closed path C , as represented in Fig. (1.2)

(~x,0)

T

(~x,T ) (~y,T )

(~y,0)

T

Figure 1.2:The Wilson loop.

From Eq. (1.31), in the limit T → ∞, we can obtain

V (R) = − lim T →∞

1

Tln¡W (R,T )¢. (1.32)

The potential of Eq. (1.32) can be computed on the lattice by numerical simulations. Assuming

R, T → ∞ it has been observed in pure Yang-Mills, see also Fig. (1.3), that

W (R, T ) ' C (R)e−σRT, (1.33)

whereσ is called the string tension and represents the linearly raising part of the potential. For large distances V (R) → σR, thus it grows indefinitely with distance and we can say that this is a confining potential. Although this result derives from numerical simulations and is not an analytic proof of confinement, this is a quite strong hint that we are dealing with a confining theory.

integration of the force-3 loops bosonic string

Figure 1.3:The static Q ¯Q potential obtained with numerical simulations in the case of pure gauge theory. Figure taken from

Ref. [12]. r0is approximately 0.5 fm.

A more precise form of the static potential used in numerical simulations is the Cornell

Poten-tial

V (R) =α

r + σR, (1.34)

whereσ is the string tension and α is a parameter that takes into account the Coulomb part of the potential arising from the weak coupling regime. As we said in the previous section this scenario is valid only in the case of pure gauge theory. In full QCD with dynamical quarks the potential between the pair Q ¯Q does not grows indefinitely with distance, but it starts raising

and then it stabilises. This happens because for large distances there is enough energy for the production of two mesons and so the string connecting the two quarks breaks.

1.5 Finite Temperature Yang-Mills Theory

So far we have discussed Yang-Mills theory in the limit of zero temperature, in particular we focused our attention on the lowest excitations above the vacuum structure of the theory, i.e. the hadrons. However there are several reasons for which it is worth studying Yang-Mills theory even in a regime of high temperature.

The first reason of interest is that in extreme conditions (high temperature) hadronic matter is expected to behave differently with respect to the usual atoms, in particular there is a phase transition from a low temperature confined phase to a high temperature deconfined phase to which we usually refer as the quark gluon plasma.

Another reason is that the extreme conditions of temperature are possible in nature, for exam-ple in the early universe shortly after the Big-Bang.

1.5.1 The Polyakov Loop and the Free Energy of a q ¯q Pair

In order to study the confining properties of pure SU (3) Yang-Mills theory at finite temperature we consider the free energy of a quark-antiquark pair Fq ¯q. This quantity can be related to an important observable, called the Polyakov loop or Wilson thermal line. The definition of such observable is the following

P (~x) = tr

·

P³eiR d x4A4(~x,x4)´ ¸

, (1.35)

whereP is the path-ordering operator. Now we want to relate this object to the free energy

Fq ¯q. Consider the operatorsψa(~x,t) and ψ†a(~x,t) which respectively annihilate and create a quark with colour index a in the position (~x,t). The charge conjugates of these operators create and annihilate an antiquark with the same colour charge in the same point instead. The quark and antiquark created with the operators above obey, in the infinite mass limit, the static time evolution equation ¡ ∂t− i A4(~x,t)¢ψ(~x,t) = 0, (1.36) which gives ψ(~x,t) = T µ ei Rt 0d t0A4(~x,t0) ¶ ψ(~0,t). (1.37)

In Eq. (1.37) we recognise the Polyakov loop in the time ordered product. Now we consider the free energy of the pair q ¯q

e−βFq ¯q

=X s

〈s| e−βH|s〉 , (1.38)

where the states |s〉 on which we perform the trace are of the form

so they represent a state with an heavy quark with colour index a in the position (~x,t). If we now consider an heavy quark in the position (~x,t) and an heavy antiquark in the position (~y,t) we can substitute the expression of Eq. (1.37) in Eq. (1.38) and after some calculation (the full computation can be found in Ref. [13]) we get

e−βFq ¯q_{= 〈P (~x)P}†(~y)〉. _(1.40)

If we write~y =~x + ~R we can rewrite Eq. (1.40) as a function of the distance R =¯¯~x −~y¯¯,

e−βFq ¯q(R)_{= 〈P (~x)P}†(~x + ~_{R)〉. (1.41)}

We have seen that the expectation value of the product of two Polyakov loop in opposite di-rection in two different spatial position gives the free energy of a q ¯q pair. It is now interesting

considering the limit of large spatial separation. If we assume cluster property for Polyakov loops we get e−βFq ¯q(R)_{= 〈P (~x)P}†(~x + ~_{R)〉 −→} R→∞ ¯ ¯〈P 〉 ¯ ¯ 2 , (1.42)

where we have used the translational invariance to write P =_V1R d3_{xP (~x). From Eq. (1.42) we} can see that if 〈P〉 = 0, then the free energy of the quark antiquark pair grows indefinitely with distance and this means that we are considering a confining system. If instead 〈P〉 6= 0 means that Fq ¯q(R) approaches a finite value as R → ∞ so the system is not confining any more.

1.6 The Large N Limit

In 1974 ’t Hooft introduced a new possible expansion in order to study particle physics: the _N1 expansion or, in other words, the large N limit (Ref. [9]). The main idea is to generalise the number of colours of the theory from the usual 3 of QCD to a generic N and organise the Feyn-man diagrams in an expansion of the parameter _N1. Consider now the renormalized coupling of Eq. (1.29) defined in Sec. (1.3), we notice that if we keepΛQC Dfixed, we have g_R2∼_N1 for large values of N. We define then the ’t Hooft coupling

λ ≡ g2

and then we perform the ’t Hooft large N limit

N → ∞ keeping λ fixed. (1.44)

In order to organise Feynman diagrams in the _N1 expansion we simply count the powers of N of such diagrams, what we find is that the planar diagrams are the most relevant ones. In general when we insert in a diagram a fermionic loop we are introducing an _N1 factor while when we insert a non planar crossing we are adding an_N12 term. A detailed description of the organisation of Feynman graphs in the _N1 expansion can be found in Ref. [14]. The large N expansion is a quite remarkable tool to study the phenomenology of hadrons and a lot of non trivial predictions can be done, for example the explanation of the U (1)A problem (Ref. [15] and Ref. [14]). A summary of other important results is in Ref. [16].

The large N expansion is interesting also if we want to study the topological properties of pure gauge theories at finite temperature. What happen is that there is a different large N behaviour of the theory if we consider the confined or the deconfined phase, a more complete discussion will be found in Sec. (1.7.4) and in Chapter 2, in which also numerical results will be presented.

1.7

θ Dependence and Topology

In this section we want to discuss the dependence of 4D SU (N ) gauge theories on the topolog-ical termθ at non zero temperature T . The first step in order to introduce topology is to discuss the topological charge. The interested reader can find a full and detailed discussion in Ref. [17].

1.7.1 Topological Charge

Consider the Euclidean Yang-Mills action

SE= β Z

d4xF_µνa F_µνa . (1.45)

We look for field configurations that possess a finite Euclidean action. It is necessary that the field strength F_µνa fall off fast enough at infinity in order for the integral of Eq. (1.45) to be convergent. Nevertheless this does not implies that the gauge field Aa_µmust be equal to zero at infinity, indeed it is sufficient that

A_µ= −i³∂µΩ ´

Ω†_{, (1.46)}

withΩ a gauge transformation. We notice that Eq. (1.46) is a gauge transformation of the 0 gauge field so in this case A_µis a pure gauge transformation. Ω will be in general a function of the direction n_µ (n2= 1) along which Euclidean infinity is approached, thus Ω(nµ) can be viewed as a map from the sphere S3onto the gauge group we are considering:

Ω(nµ) : S3 −→ SU (N ).

The space of such maps can be divided into an infinite number of equivalence classes such that a transformation belonging to a certain class cannot be deformed continuously into an-other transformation belonging to a different equivalence class. We refer to this equivalence classes as homotopy classes. We will see later in this section that all of this homotopy classes can be characterised by an integer number q, called the topological charge. When q 6= 0 we say that the transformation is topologically nontrivial. An intuitive way to understand what is the topological charge q is to consider the Schwinger Model, which is an abelian gauge theory in two Euclidean dimensions. In this theory we have

Ω(x) = eiχ(x)_{, (1.47)}

where x is a spacetime point. Topologically U (1), which is the gauge group of this theory, is equivalent to the circumference S1, thus if we rewrite the gauge transformation as a function

a(ϕ) of the direction along which we are approaching Euclidean infinity we have the map

Ω(ϕ) = ei a(ϕ): S1 −→ U (1) = S1. (1.48)

We define the topological charge as the number of winding so, for a map of charge q we have that

a(ϕ + 2π) = a(ϕ) + 2πq. (1.49)

We could say, in a quite suggestive way, that the topological charge represent how many times the Euclidean infinity wraps onto the gauge group. It is easy to notice that q can be write as an integral over the circumference

q = 1 2π Z 2π 0 A_ϕdϕ = 1 2π Z Cεµν A_νdσ_µ, (1.50)

where dσ_µ is the length element of a distance contour C multiplied by a unit vector pointing outside. Using Gauss theorem we can rewrite the integral in this way

q = 1 2π Z Z d2x∂µ ³ εµνAν ´ = 1 4π Z Z d2xεµνFµν, (1.51)

where the topological charge is now expressed as an integral over the whole space time. If we consider a non-abelian gauge group we have to follow the same steps to define the topological charge and we get

q = 1 16π2 Z d4xtrnF_µνF˜_µνo, (1.52) where ˜ F_µν=1 2εµνρσFρσ (1.53)

It is important to notice that the topological charge can be written as the integral of a total derivative of the so-called Chern Simons current, indeed

1 16π2tr n F_µνF˜_µνo= ∂µKµ= ∂µ   1 8π2εµναβtr ( A_ν · ∂α−2i₃ AαAβ ¸)  . (1.54) 1.7.2 θ Term

We consider now the Lagrangian of QCD in the presence of a non-zeroθ term

Lθ=1₄Fµνa (x)Fµνa (x) − i θ g 2

64π2εµνρσF a

µν(x)Fρσa (x), (1.55)

where q(x) =₆₄g_π22εµνρσFµνa (x)Fρσa (x) is the topological charge density and Q =R d4xq(x) is the topological charge. We remark the fact that theθ term can be written as a total derivative (the Chern-Simons current mentioned before) and that becomes important when topologically non trivial gauge transformations came into play. We now want to discuss briefly the importance of theθ term in QCD. The first thing that we notice is that this term breaks explicitly both parity and time reversal. An intuitive way to see this is to notice that F_µνa F˜_µνa ≈ ~EaB~a where Ea and

Ba are the chromoelectric and chromomagnetic fields which are respectively odd and even under time reversal so their product changes the sign after that transformation. However it is known from high energy experiments that neither parity or time reversal are broken in strong interactions, thusθ is zero or very small. The best experimental restriction can be found in Ref. [18] and it comes from the electric dipole moment of the neutron (dn)

dn.e−25e · cm ⇒ θ.10−9. (1.56)

Even if the experimental bounds are quite stringent theθ dependence is still relevant in strong interaction phenomenology, the most famous example being the solution of the U (1)A prob-lem.

1.7.3 Instantons

We want now write about solutions to the equation of motion with finite Euclidean action, but with a non zero value of the topological charge. In particular we look for solutions with topo-logical charge equal to one or minus one. These solutions are commonly known respectively as instantons and anti-instantons. A full description of instantons and their properties can be found in Ref. [19]. First of all we write the Euclidean action for SU (3) gauge theory

S =1

4 Z

d4xF_µνa F_µνa Ê 0. (1.57)

Remembering the definition ˜F_µν=1₂εµνρσFρσwe can rewrite Eq. (1.57) as

S =1 4 Z d4xtr ·_³ F_µν∓ ˜F_µν´ ³F_µν∓ ˜F_µν´ ¸ ±1 2 Z d4xtr³F_µνF˜_µν´, (1.58) thus, using Q ≡ Z d4xq(x) = g 2 16π2 Z d4xtr³FµνF˜µν ´ , (1.59) we obtain S =1 4 Z d4xtr · ³ F_µν∓ ˜F_µν´ ³F_µν∓ ˜F_µν´ ¸ ±8π 2 g2 Q Ê ± 8π2 g2 Q. (1.60)

From the above expression we get two possible solutions with finite action and topological charge equal to plus or minus one

Ï Fµν= ˜Fµν ⇒ S =8π 2 g2 Q, Ï Fµν= − ˜F_µν ⇒ S =8_gπ22 ¯ ¯Q ¯ ¯.

The first are instantons while the second are anti-instantons. Looking at the dependence on the coupling g it is clear that instantons (and anti-instantons) are, by their nature, non perturbative solutions, indeed we have

Z

[d A]e−S∼ e− 8π2

g 2Q _{Q ∈ N, (1.61)}

the coupling g is in the denominator and for high energies, where the perturbative regime is valid, it is small due to asymptotic freedom and so such configurations are largely suppressed. Nevertheless in the non-perturbative regime g is high and these configurations become more important.

1.7.4 θ Dependence of the Ground State Energy

We are now interested in studying theθ dependence of the ground state energy of 4D SU(N) gauge theories. Most of the topic discussed below can be found with more detail in the review Ref. [20]. We start with the definition of the ground state energy F (θ), which obviously depend onθ e−V F (θ)= Z [d A] exp µ − Z d4xL_θ ¶ , (1.62)

where V is the spatial volume of the system. It is useful working with dimensionless quantities, thus we introduce the standard scaling function

f (θ) =F (θ) − F (0)

σ2 , (1.63)

whereσ is the string tension computed at θ = 0. We remark that, due to the quantization of the topological charge, F (θ) and f (θ) will be periodic in θ with periodicity 2π. In order to study the

θ dependence we expand the function f (θ) around θ = 0. The usual parametrisation used to

f (θ) =1

2Cθ

2_{s(θ), (1.64)}

where s(θ) is a dimensionless even function of θ such that s(0) = 1 and C is the ratio_σχ2 in which

σ is again the string tension computed at θ = 0 and χ is the topological susceptibility defined as χ =〈Q2〉

V , Q =

Z

d4xq(x). (1.65)

Now we can expand s(θ) as a function of θ obtaining

s(θ) = 1 + b2θ2+ b4θ4+ · · · . (1.66)

The coefficients b2 j are related to∂j_∂θF (θ)j computed atθ = 0, the n-point correlation function of

the topological charge, and so they are related to the moments of the probability distribution of the topological charge. The coefficientsχ, b2and b4will be object of numerical simulations in the following chapters so we want to write them explicitly. The computation is very simple since it consists only in computing derivatives of the ground state energy with respect toθ, the coefficients are              χ =〈Q2_〉 V b2= −_12V1_χ h 〈Q4〉 − 3〈Q2〉2i θ=0 b4=_360V1 _χ h 〈Q6〉 − 15〈Q2〉〈Q4〉 + 30〈Q2〉3i θ=0

1.7.5 θ Dependence in the Large N Limit

We want now to discuss how theθ dependence changes when we are considering the large N limit. We remind that such limit is obtained considering N → ∞ while the quantity λ = g2N is

kept fixed. One could naively consider as a first attempt the dilute instanton gas approximation, in which the form of theθ dependence is

e− 8π2 g 2_eiθ₌ Ã e− 8π2 g 2 N_eiNθ !N . (1.67)

In this case the dependence is exponentially suppressed in N . However there are reasons to believe that theθ dependence is not suppressed in the large N limit. One of that reasons is

that the solution of the U (1) problem in the framework of the _N1 expansion implies that theθ dependence in the ground state energy is present at the leading order in the_N1 expansion. The dilute instanton gas approximation is also affected by bad infrared divergences, therefore we consider now a new possibleθ dependence originally proposed by Witten in Ref. [21] and Ref. [22].

Using large N scaling argument we can say that the relevant variable is

¯

θ = θ

N. (1.68)

Moreover, since the number of degrees of freedom of the system is N2the ground state energy

F (θ) should be proportional to it and so we could write

F (θ) = N2h( ¯θ), (1.69)

where h( ¯θ) is a function of ¯θ = _Nθ which possess a well defined limit for N → ∞. In addition to that we have the periodicity condition

F (θ) = F (θ + 2π). (1.70)

Conditions (1.69) and (1.70) are incompatible since it cannot exists a smooth function of _Nθ with periodicity 2π unless it is the constant function. The possible way out proposed by Witten is to consider F (θ) as a multibranched function since it possess many vacua which all become stable in the N → ∞ limit, but not degenerate. The form of this function should be

F (θ) = N2minkH

à θ +2πk

N

!

, (1.71)

where H is an unspecified function. For each value ofθ we have to minimise the function H with respect to k. It is clear that with this definition the function F is periodic with periodicity 2π as required, but it is not smooth since for some values of θ there is a jump between different branches. Whenθ = 0 the function F (θ) has its absolute minimum since the Euclidean path integral atθ = 0 is real and positive while for θ 6= 0 is not because of the factor eiθ. If the vacuum is unique atθ = 0, then the minimum must occur for k = 0. If we take into account arguments involving the U (1) problem in the theory with quarks we expect also that

d2h

dθ2 6= 0 at θ = 0. (1.72)

With this information we can parametrise the ground state energy at the leading order in the expansion_N1 as F (θ) = Amink ¡ θ + 2πk¢2+ Oµ 1 N ¶ , (1.73)

where A is positive and do not get higher contributions at the leading order in the_N1 expansion. For what concerns the b2 j coefficients of the expansion (1.66) it is easy to derive their large N behaviour. We start noticing that, if F (θ) = N2H ( ¯θ) we will have

f (θ) =F (θ) − F (0) σ2 = N 2_{f ( ¯¯ θ), (1.74)} where ¯ f ( ¯θ) =1 2C∞ ¯ θ2³ 1 + ¯b2θ¯2+ ¯b4θ¯4+ · · · ´ , (1.75)

in which C_∞is the leading term in the_N1 expansion of the ratio C =_σχ2. In general we will have

C = C∞+

c2

N2+ · · · . (1.76)

If we compare Eq. (1.75) with Eq. (1.64) we get the relation between the coefficients

b2 j= ¯

b2 j

N2 j + · · · . (1.77)

We thus notice that in the large N limit all the coefficients are suppressed.

1.7.6 θ Dependence and Finite Temperature

The above discussion about the dependence of the ground state energy on the topological term and the behaviour in the large N limit is valid in the zero temperature regime. Now, however, we want to discuss the relation between the free energy of a 4D pure gauge SU (N ) theory at finite temperature and theθ term. As is shown in detail in Ref. [23], one expects a change in theθ dependence between the low temperature, before the deconfinement transition, and the

high temperature regime, after the deconfinement transition. In particular what is found is that theθ dependence obeys different large N scaling in the two regimes. At high temperature the system should be describable by a semiclassical instanton gas picture. As we said in the previous section, due to infrared divergences the instanton picture fails in the low temperature regime and the relevant variable turns out to be ¯θ =_Nθ. The finite-T behaviour is specified by the free energy

F (θ,T ) = − 1 V4 ln Z [d A] exp ( − Z _T1 0 d t Z d3xL_θ ) , (1.78)

in which V4= V_T is the Euclidean space-time volume and the gauge field A_µ obeys periodic boundary conditions along the temporal direction. We can parametrise the free energy F (θ,T ) in the same way we have done for the ground state energy6, however in this case we will use onlyχ(T ), the topological susceptibility, instead of the ratio_σχ2 used before

F (θ,T ) ≡ F (θ,T ) − F (0,T ) =1 2χ(T )θ

2_{s(θ,T ), (1.79)}

where, exactly as before

s(θ,T ) = 1 + b2(T )θ2+ b4(T )θ4+ · · · . (1.80)

In the low-T regime we expect the scenario described in the previous section, thus the param-eters will be

χ = χ∞+ O(N−2), (1.81)

b2 j= O(N−2 j), (1.82)

whereχ∞is the leading term in theN1 expansion. Numerical simulations performed at T = 0 are in agreement with predictions (1.81) and (1.82) and they remain valid for the low temperature regime before the critical temperature TC at which the deconfinement phase transition takes place. Below are reported numerical results for_σχ2 (Ref. [20]), b2(Ref. [24]) and b4(Ref. [24]) in the low temperature regime for the gauge group SU (3).

6_{In this case, since we are dealing with a system at finite temperature all the coefficients b}

2 jand the topological

χ

σ2= 0.028(2) ⇒ χ 1

4 _{= 180(3) MeV. (1.83)}

Here the physical value of the topological susceptibility has been obtained using the standard value for the string tensionpσ = 440 MeV.

b2= −0.0216(15),

¯ ¯b₄

¯

¯.4 × 10−4. (1.84)

Results concerning the large N limit can be found in Tab. (1.1).

Table 1.1: _σχ2, b2, b4for different values of N . We notice that the predictions (1.81) and (1.82) are well represented since χ σ2 seems to approach a finite value for large N , while b2and b4are suppressed as N increase.

N _σχ2 b2 b4

3 0.0289(13)[25] −0.0216(15)[24] . 4 ×10−4[24]

4 0.0248(8)[25] −0.0155(20)[25] −0.0003(3)[25]

6 0.0230(8)[25] −0.0045(15)[25] −0.0001(7)[25]

In the high temperature regime the dependence onθ changes. While in the low-T regime the dilute instanton gas approximation is not valid due to infrared divergences, in the

high-T regime the temperature acts as a sort of regulator and the overlap between instantons get

suppressed. We can then argue that after the deconfinement phase transition the theory can be well represented by an instanton gas. The free energy and the topological susceptibility in such approximation are given by

F (θ,T ) ≈ χ(T )³1 − cos¡

θ¢´, (1.85)

and

χ(T ) ≈ T4_e−_{g 2(T )}8π2

. (1.86)

From the above expressions we understand that the relevant variable in the high-T regime is justθ and not _Nθ. Moreover the topological susceptibility is exponentially suppressed and the values of the various coefficients b2 jget a well defined value independent ofθ. In particular we want to report the value of the first of those coefficients, b2, since it will be object of numerical simulations later on this work. To compute b2we expandF (θ,T ) around θ = 0

F (θ,T ) ≈ χ(T ) µ 1 − 1 +1 2θ 2 + · · · ¶ , (1.87)

comparing this equation with the one in terms of the b2 jcoefficients we get

b2= − 1

12. (1.88)

In Fig. (1.4) are shown numerical results about both the low-T and the high-T regime for b2. In particular it is shown that the change in theθ dependence happens around the critical temper-ature of the deconfinement phase transition. Agreement with the instanton gas approximation is reached closer and closer to TC as N increases.

-0.05 0 0.05 0.1 0.15

t

b

2

N=3, L

=10

N=6, L

=6

Figure 1.4:Numerical results for the coefficient b2as a function of t = (T − TC)/TC. The red points refers to SU (3) while the

black ones to SU (N ). Figure taken from Ref. [23].

The coefficient b2can give information also on the topological objects of our theory. Consider again the low-T limit in which we have a _Nθ dependence. We could argue that our system is composed of topological objects like the instantons, but with a topological charge equal to _N1. In this case we could try to use a dilute gas of this topological objects to describe our theory, exactly as in the case of the dilute instanton gas approximation, but with the difference that now the free energy will be

F (θ,T ) ≈ χ(T )  1 − cos à θ N ! . (1.89) This cos¡

θ/N¢ leads to a precise prediction for the coefficient b2, indeed F (θ,T ) ≈ χ(T ) Ã 1 − 1 +1 2 θ2 N2− 1 4! θ4 N4· · · ! =χ(T )θ 2 2N2 Ã 1 − 1 12 θ2 N2· · · ! , (1.90) thus b2= − 1 12N2 ⇒ b2= 0.0092 if N = 3. (1.91)

We can clearly see that from the values of Tab. (1.1) that this prediction is not in agreement with the numerical results. This could happen because topological objects with a fractional topological charge of _N1 does not exist, or because they are strong interacting objects and the dilute approximation is not valid.

In this chapter we introduce the lattice formulation of pure Yang-Mills theory and Monte Carlo methods. We will show the lattice discretisation of Yang-Mills theory and the continuum limit. Next we will give a brief overview of Monte Carlo methods focusing, in particular, on the algo-rithms used to simulate gauge theories. In the last part we will discuss finite temperature and topology from the lattice point of view.

2.1 Lattice Formulation of YM Theory

We discretise the space-time introducing a lattice spacing "a". We want to place the fields onto the lattice paying attention not to break gauge invariance. As field variable we use the parallel transport from one site to the other: the so called link. If we consider the site i and j = i + a ˆµ we have:

U_µ(i ) = ei g Aµ(i ) _{i −→ j = i + a ˆµ,}

U_µ†(i ) = e−i g Aµ(i )

i ←− j = i + a ˆµ,

With these variables we are able to write an action that, for a −→ 0, coincides with YM action

SY M. Let us consider first of all a plaquette U2(Fig.2.1): Then we build the quantity

S₂= β " 1 − 1 2NC tr³U₂+U₂†´ # = βh1 − Re¡tr(U₂)¢i , 33