Topologyand θ -dependenceinQCDandQCD-liketheories PhdThesis

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Corso di Dottorato in Fisica

Phd Thesis

Topology and θ-dependence in QCD

and QCD-like theories

PhD candidate:

Supervisor:

Claudio Bonanno

Prof. Massimo D'Elia

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Introduction

Quantum Chromo-Dynamics (QCD) is the quantum eld theory that de-scribes strong interactions inside the Standard Model of Particle Physics. This model is formulated as a non-Abelian SU(Nc) gauge theory, with Nc = 3 the

number of colors, and describes the strong nuclear force as the result of the quan-tum interaction among quarks and gluons. Since the discovery of asymptotic freedom [1, 2], QCD has proven to be a solid and predictive theory, being able to accurately describe much experimental evidence about high-energy hadron physics in the perturbative regime. On the other hand, extensive studies of the more elusive non-perturbative properties of QCD have been and are still car-ried on as well, having many aspects of this regime still to be fully unveiled and being them tightly connected with interesting theoretical and phenomenological features, such as connement or chiral symmetry breaking.

In this context, the study of the topological properties of QCD is a very active eld of research because of their relation with intriguing theoretical features of gauge theories and with interesting phenomenological aspects of Standard Model and Beyond Standard Model physics. A hot topic which has been thoroughly investigated in the literature, both in QCD and in QCD-like theories, is the dependence of physical observables on the parameter θ, coupling the topological charge Q to the action. In particular, an observable whose dependence on θ emerges in several dierent contexts is the free energy (density)

f (T, θ) = 1 2χ(T )θ 2 " 1 + ∞ X n=1 b2n(T )θ2n # ,

whose Taylor expansion around θ = 0 can be parametrized in terms of the topo-logical susceptibility χ and of the coecients b2n, related to the cumulants of the

topological charge distribution at θ = 0. Among the many physical frameworks in which this quantity plays an important role, it is worth quoting:

The strong CP problem and the PecceiQuinn axion

A precise knowledge of f(T, θ) at high temperatures T is necessary to de-scribe the physics of the PecceiQuinn axion, the hypothetical particle in-troduced to explain the absence of CP-violating eects in strong interac-tions [36], at early times of the Universe evolution. For example, it is a fundamental input to compute the axion mass (related to the QCD topolog-ical susceptibility) today, which is a relevant quantity for current and future

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experimental researches of this particle, or the axion relic abundance, which could explain part of the observed Dark Matter [79].

The U(1)A anomaly, the large-Nc limit and the physics of the η0 meson

The study of the θ-dependence of the vacuum energy (i.e., the T = 0 free energy) of large-NcSU (Nc)pure-gauge theories is extremely interesting for

its relation to the U(1)A puzzle and to its solution [1015], which has been

found in the peculiar framework of the 1/Nc expansion around Nc = ∞.

Indeed, the anomalous breaking of the U(1) axial symmetry due to the topological charge, apart from being interesting from a theoretical point of view per se, is also relevant for hadron phenomenology, as the physical parameters of the η0 _{meson can be related to pure YangMills topological}

observables in the large-Nc limit.

Topology, large-N limit and θ-dependence in lower dimensional models Apart from four-dimensional gauge theories, topics related to topological properties have been also extensively studied in lower-dimensional simpler theories, such as the 2d CPN−1 _{models, very well-known toy models which}

share many fundamental properties with QCD [11, 1618], such as con-nement, the existence of a topological θ parameter and the possibility of performing a 1/N expansion of the theory in the large-N limit (analogous to the large-Nc limit of pure-gauge theories). For these reasons, and for

their moderate numerical cost compared to 4d non-Abelian gauge theories, these theories are ideal theoretical laboratory where non-perturbative prop-erties can be investigated both analytically and numerically, thus providing a useful environment for the numerical validation of analytic results, as well as for the testing of new algorithms or numerical techniques.

Generally speaking, the θ-dependence of the free energy can be computed analytically only in some specic regimes using suitable approximations, for in-stance for QCD close to the chiral limit using the Chiral Eective Lagrangian approach, or for asymptotically high temperatures using the Dilute Instanton Gas Approximation (DIGA), which leads to the well-known prediction [19]

fDIGA(T, θ) = χDIGA(T ) (1− cos θ) ,

where leading order perturbation theory predicts χDIGA(T ) ∼ T−8 for 3 light

quark avors [19].

For these reasons, in order to precisely assess θ-dependence of the free energy and how it changes with the temperature, lattice numerical computations are a natural rst-principle and fully non-perturbative approach that is well suitable to this end.

The study of topology by means of lattice Monte Carlo simulations, however, is far from being a simple task, as it is plagued by a series of highly non-trivial numerical challenges which need to be properly tackled, such as:

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(i) Rare uctuations of the topological charge

The topological susceptibility is usually determined from the variance hQ2_{i =}

V χof the probability distribution P (Q). Since χ drops very rapidly at high temperatures above Tc, on typically aordable lattice volumes the variance

of P (Q) results much smaller than 1. Being Q integer valued, this ultimately makes tunneling between two dierent topological sectors very unlikely to happen, so that charge uctuations occur very rarely, requiring unreason-ably large statistics to obtain a meaningful sampling of P (Q).

(ii) Non-chiral lattice fermions and large lattice artifacts

The topological susceptibility suers from somewhat large nite lattice spac-ing corrections when a fermionic discretization that explicitly breaks the chiral symmetry is adopted, such as the Wilson or the staggered ones. Such a choice prevents exact fermion zero-modes from appearing in the Dirac spectrum in the background of a non-zero charge conguration, unlike in the continuum theory. Since lowest-lying eigenvalues are shifted from zero, would-be zero-modes make the determinant of the Dirac operator larger than it would be in the continuum, ultimately leading to a larger weight of non-zero charge congurations in the path-integral, being them weighted with a factor ∝ detD + m/ quark

. Therefore, being non-zero charge con-gurations not properly suppressed, large lattice artifacts plague the topo-logical susceptibility at nite lattice spacing, strongly impacting the scaling towards the continuum of this observable. This is particularly problematic at high temperatures, where, being χ strongly suppressed compared to the zero-temperature case, discretization errors are so large that the continuum extrapolation is out of control in the lattice spacing ranges employed in typical QCD simulations. As an example, in Ref. [20] it is shown that, at high T , the value of the topological susceptibility can even drop by almost a factor 20 by just halving the value of the lattice spacing.

(iii) Critical slowing down of topological modes and topological freezing

To mitigate the impact of lattice artifacts on the continuum extrapolation of χone could, in principle, suitably reduce the value of the lattice spacing and approach the continuum limit, where chiral symmetry is recovered and the magnitude of nite lattice spacing corrections should be reduced. However, this is in practice prevented by the severe critical slowing down (CSD) of topological modes [2127]. This computational problem consists in the dramatic growth of the auto-correlation time of the topological charge as the continuum limit is approached and is due to the fact that, when a → 0, the nite free-energy barriers separating dierent topological sectors at nite lattice spacing rapidly diverge, so that a proper separation between them is recovered in the continuum. This means that standard algorithms need more and more updating steps to decorrelate the value of the topological charge, causing the Monte Carlo Markov chain to remain trapped in a xed sector for a long time. Practically, this means the Monte Carlo evolution

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of the charge is frozen when a is small, making simulations at very small lattice spacings practically unfeasible unless unreasonably large statistics are collected.

(iv) Worsening of the topological freezing in the large-Nc limit

The local nature of standard Monte Carlo algorithms employed in pure-gauge simulations cause the freezing of the time evolution of the topological charge as the continuum limit is approached, analogously to what happens in full QCD simulations. More precisely, an exponential growth of the auto-correlation time of Q as a → 0 is observed from Monte Carlo data [23,25]. However, this problem becomes more and more severe as Nc is increased,

causing the auto-correlation time of Q to grow exponentially also as a func-tion of Nc at xed lattice spacing [23, 25], causing topological freezing to

become overwhelming already at moderate sizes of the lattice spacing when Nc is large. This makes more and more dicult to perform reliable

con-tinuum extrapolations of topological observables as the large-Nc limit is

approached, especially for the b2n coecients, which are expected to vanish

in the large-Nclimit and thus require a stronger control over the continuum

extrapolation to obtain reliable continuum limits.

(v) Degrading of the signal-to-noise ratio of higher-order cumulants in the ther-modynamic limit

The b2n coecients, being related to higher-order cumulants of the

topo-logical charge distribution, encode the deviations of P (Q) from a Gaussian. These quantities have a well-dened thermodynamic limit V → ∞ which becomes harder and harder to detect on large volumes because of the cen-tral limit theorem. In particular, their signal-to-noise ratio decays with a power of 1/V , which grows increasing the order of the cumulant [28]. For this reason, on the lattice volumes needed to keep nite-size eects under control, unreasonably large statistics are needed to obtain reliable measures of these coecients. Moreover, this problem further worsens as the large-Nc limit is approached, being the physical signal that has to be detected

suppressed by powers of 1/Nc.

Since the goal of this thesis is to investigate topological properties and θ-dependence in QCD and in QCD-like theories in the physical contexts introduced earlier and by means of lattice Monte Carlo simulations, it is therefore unavoidable to con-front all the computational problems described so far, which constitute the main obstacles in making substantial progresses in the state of the art of these elds.

Let us start from the discussion of θ-dependence in hot QCD. Our interest relies in the determination of the high-temperature behavior of the topological susceptibility for axion cosmology purposes, whose investigation by means of lat-tice numerical simulations has been carried on in several works [20, 2935]. To this end, the relevant issues to consider are essentially the one numerated as (i), (ii) and (iii) in the above list.

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To deal with the problem of rare topological uctuations at high temperatures, a possible solution is the one adopted in Refs. [20,36], where this issue was circum-vented using a multicanonical approach. The idea is to add a Q-dependent bias potential to the action, so that the probability of visiting suppressed topological sectors is enhanced. Then, the mean over the original probability distribution is exactly obtained through a standard reweighting procedure. However, although this strategy allows to solve the problem of rare topological uctuations at high T, the issue of large lattice artifacts remains still overwhelming and prevents from obtaining reliable continuum extrapolations of χ in this regime. Therefore, a way to reduce lattice artifacts or to access smaller lattice spacing is still necessary.

A possible heuristic strategy to reduce discretization errors coming from non-chiral fermion dynamics, explored in Ref. [32] with staggered fermions, is to reweight congurations with non-zero topological charge by hand using the corre-sponding continuum lowest-lying eigenvalues of the Dirac operator, which are just equal to the fermion mass. This way, the contribution of would-be zero-modes, which is not properly depressed by the fermion determinant, is suppressed by hand.

Another possible way to deal with this problem, which is better posed from a eld-theoretical point of view, is to adopt a dierent and more suitable lattice discretization of the topological charge operator, giving up the standard gluonic denition. In particular, being the problem of large lattice artifacts coming from the contributions of dynamical fermions, one can speculate that a fermionic def-inition of the topological charge, based on the index theorem [37], could improve the convergence of the topological susceptibility.

In this respect, a very promising discretization is the one based on spectral pro-jectors over eigenspaces of the Dirac operator, which up to now has been dened for Wilson fermions [38, 39] and tested both in the quenched case [40, 41] and in zero-temperature simulations of full QCD with Wilson fermions [42]. In par-ticular, the results of Ref. [42] support the aforementioned conjecture; indeed, there it is shown that the topological susceptibility measured with spectral pro-jectors exhibits much smaller lattice artifacts compared to the standard gluonic denition.

In light of these results, it is interesting to apply spectral projectors also to the case of high temperatures, where they might allow to achieve a major improve-ment, since a strong reduction of lattice artifacts would permit to have much more control over the continuum extrapolation already in the typical lattice spacing ranges currently aordable in typical QCD simulations, so that the necessity of pushing towards smaller lattice spacing, where the CSD becomes overwhelming, can be avoided.

Our goal is to extend the denition of spectral projectors from the Wilson to the staggered case in view of an application to our ongoing multicanonical high-temperature full QCD simulations with dynamical staggered fermions, in order to improve the continuum scaling of the topological susceptibility. In this thesis we will present the results originally appeared in Ref. [43], where a detailed

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deriva-tion of the extension of the spectral projectors method to the case of staggered fermions starting from the Wilson case was shown. We will also review numeri-cal results obtained in that work using staggered fermions spectral projectors in the pure SU(3) gauge theory, where this discretization can be easily tested and compared with the standard gluonic denition, which can be used as a bench-mark being the problem of the fermion determinant completely absent in pure gluodynamics. Moreover, we will also present preliminary results obtained with staggered spectral projectors in full QCD with dynamical staggered fermions, which is an ongoing study that will be the subject of a forthcoming work.

Let us now pass to the discussion of θ-dependence in the case of large-Nc

YangMills theories. In this case, our main interest resides in the lattice numerical validation of large-Nc predictions about the behavior of the vacuum energy as a

function of θ. Analytic large-Nc calculations predict that the coecients of the

free energy scale at large Nc as [1214,44]:

χ = ¯χ + O 1/N_c2, b2n = ¯b2n/Nc2n+ O 1/Nc2n+2

,

where the large-Nc limit of the topological susceptibility can be obtained from

the WittenVeneziano equation, ¯χ ' (180 MeV)4_{, while no analytic prediction is}

known for ¯b2n from large-Nc calculations in standard QCD. Therefore, given the

intrinsic non-perturbative nature of topological properties, the lattice oers a nat-ural environment to obtain more quantitative results about large-Ncθ-dependence

from rst principles, as well as a solid ground to non-perturbatively test large-Nc predictions and the hypotheses underlying them. Up to now, large-Nc

an-alytic results have been veried on the lattice for the topological susceptibil-ity [23,28, 4548], where ¯χ is known within a few percent, and also for b2, where

¯b2 is known at the level of ∼ 10% [23, 28, 49, 50]. Concerning b4, instead, it is

known only for Nc = 2 [51], while only upper bounds have been reported for

higher values of Nc [28, 50]. The larger uncertainties aecting b2, as well as the

absence of reliable results for the higher-order coecients, are essentially related to the computational problems (iv) and (v) listed earlier in this introduction.

A possible solution to the problem of the poor signal-to-noise ratio of higher-order cumulants relies on analytic continuation towards imaginary values of θ, a largely-adopted expedient [5162] which closely resembles the imaginary chemical potential method widely employed in QCD simulations [6370]. The addition of an imaginary-θ term avoids the sign problem and at the same times enhances the signal-to-noise-ratio of the lower-order cumulants since it acts as a Q-source in the action. Besides, information on higher-order cumulants is now contained in the dependence on the imaginary θ parameter of the lower ones, from which one can extract χ and the b2n coecients in an improved way.

Concerning the problem of the exponential CSD suered from topological modes at large-Nc, although an exact solution that completely avoids topological

freezing is known only for simpler models [27], several numerical strategy have been devised to mitigate its eects [7174]. A popular solution is to give up peri-odic boundary conditions in the time direction in favor of open boundaries [72].

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This way, the topological charge can ow in and out from the open side, causing a substantial decrease of the auto-correlation time of Q, which now diverges only as ∼ 1/a2 _{in the continuum limit. However, a drawback of this method is that now}

nite-volume eects are more prominent since the actual physical information is contained only in the bulk of the lattice.

Another possible strategy to alleviate topological freezing, which has been pro-posed in Ref. [75] by Martin Hasenbusch, is to smartly combine simulations with periodic and open boundaries in a parallel tempering framework in order to have the best of both worlds: on one hand parallel tempering transfers congurations from the open to the periodic replica, allowing to obtain a fast decorrelation of the topological charge even for the copy with standard boundaries; on the other hand, measures are taken on the periodic lattice, where complications related to the adoption of open boundaries are completely avoided.

The Hasenbusch algorithm has been originally proposed and checked for 2d CPN−1 models, whose lattice simulations performed with standard algorithms suer from an exponential CSD when studying topological quantities at large-N, analogously to what happens for large-Nc YangMills theories.

For these theories, the state of the art about the large-N θ-dependence of the vac-uum energy E(θ) looks like the opposite compared to that of YangMills theories: while E(θ) is completely known analytically at leading order in 1/N [16,50,53,76] and even at next-to-leading order for the topological susceptibility [77], lattice re-sults about topology so far were quite limited, mainly being circumscribed to the leading large-N behavior of χ [25, 71, 75, 78]. In addition, latest numerical re-sults pointed out a discrepancy between analytic predictions and lattice rere-sults; in particular, the next-to-leading-order coecient in the 1/N expansion of the topological susceptibility appears to be of opposite sign compared to the analytic computation [25, 56, 75], and the large-N limit of b2 appears to be almost a

fac-tor of 2 larger than the analytic prediction [51]. A possible way to solve these apparent tensions, as observed in [51], is to assume that, unlike for pure Yang Mills theories, where the large-Nc predicted scaling is observed to hold already

for Nc≥ 3 [23, 48, 50], the 1/N series has a very slow convergence and large

cor-rections to the predicted leading behavior appear in the currently explored range of N. In Ref. [51] the value N ∼ 30 was reached, but it was shown that at least N ∼ 50 is required to check the consistency of this hypothesis.

Since simulations at large N using standard algorithms become rapidly not fea-sible due to the CSD of topological modes [25, 51, 56, 79] similarly to the Yang Mills case, in order to go beyond the present state of the art about large-N θ-dependence in the CPN−1 _{models, and also in view of an application to 4d gauge}

theories, in Ref. [80] we adopted the Hasenbusch algorithm, in combination with simulations at non-zero imaginary values of θ, to investigate θ-dependence of CPN−1 at large-N up to N = 51. Results obtained in Ref. [80] are presented and discussed in this thesis, where we will show that, thanks to the larger values of N reached with the new algorithm, it is possible to prove the existence of large corrections to the predicted large-N behavior, solving the apparent tensions with

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analytic predictions pointed out in previous studies.

These results motivate the application of the Hasenbusch algorithm to the case of 4d SU (Nc) YangMills theories, since, being them plagued by topological

freez-ing in the large-Nc limit like the CPN−1 models, it is natural to expect a major

improvement in the study of large-Nc θ-dependence with parallel tempering also

for these models. Our investigation of the large-Nc limit of YangMills theories

with this algorithm has been reported in Ref. [81], and will be reviewed in this thesis. In particular, we will compare the eciency of parallel tempering with that of standard algorithms and we will employ it to rene state-of-the-art results about the large-Ncθ-dependence of the vacuum energy beyond the leading O(θ2)

order.

Being the convergence of the large-N limit much slower for CPN−1

mod-els compared to 4d SU(Nc) gauge theories, it is interesting to understand why.

A possible explanation may reside in the fact that, unlike YangMills theories, CPN−1 models behave drastically dierent for small values of N. For example, considering the special case N = 2, where the CP1 _{model just reduces to the}

well-known O(3) σ-model, perturbation theory predicts a divergence of the topological susceptibility in the continuum limit [8286]. The origin of this pathological be-havior can be traced back to the divergence of the instanton density distribution dI(ρ, N )∝ ρN−3for small sizes ρ. This is an important dierence between CPN−1

models and YangMills theories, where the case Nc = 2 is observed to be

regu-lar [51,87].

Concerning numerical results about small-N θ-dependence, there are several aspects that deserve further investigations. Some works about CP1 _{found results}

compatible with a divergent topological susceptibility in the continuum limit, see, e.g., Refs. [8895]; however, since χ is expected to diverge as ∼ log a for N = 2, diculties in disentangling this behavior from a slowly convergent power-law may arise, thus requiring to span a very extended range of a to actually distinguish between the two.

The actual continuum behavior of χ is unclear even for N = 3, since disagreeing results are found in Refs [92, 96]: while the former reports a nite value of χ for N = 3, the latter points out a bad continuum scaling of the topological susceptibility, suggesting that also in this case the theory is ultraviolet dominated. This apparent discrepancy may be an indication that, as in the N = 2 case, also the continuum limit of N = 3 has to be handled with care because of the contribution of small instantons.

Finally, concerning the behavior of the higher-order coecients of E(θ), it is natural to ask whether they diverge in the N → 2 limit too or not. An intriguing hypothesis is that, being CPN−1 _{models asymptotically free, the small instantons}

characterizing their small-N behavior can be well described by the DIGA, which would lead to the prediction that only χ diverges for N = 2. However, this answer has not been addressed yet in the literature, apart from some nite lattice spacing results about b2 for N = 2 reported in Ref. [94].

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This scenario motivated the studies presented originally in Ref. [97], which will be reviewed and discussed in this thesis. Our aim is to study the small-N behavior of E(θ) up to O (θ4₎_{, and in particular to understand the behavior of χ and of}

the b2 coecient in the peculiar cases N = 2 and 3. We tackled the problem

from two independent fronts: on one hand, we made extensive simulations at N = 2and N = 3 on a lattice spacing range spanning over an order of magnitude to carefully study the continuum scaling of these topological observables; on the other hand, we performed simulations for 3 < N < 10 in order to deduce the behavior in N = 2 and N = 3 from a small-N extrapolation of results obtained in this interval.

The organization of this thesis essentially follows the discussion ow of this introduction: in Chap. 1 we make a brief overview of the lattice formulation of QCD and of the main known results about θ-dependence and its related the-oretical and phenomenological implications; in Chap. 2 we discuss the spectral projectors method and its extension to the case of staggered fermions, presenting numerical results obtained in the pure SU(3) gauge theory [43] as well as pre-liminary results obtained in full QCD; in Chap. 3 we present results obtained for the θ-dependence of the 2d CPN−1 _{models both in the large-N [80] and in the}

small-N limits [97] and in Chap. 4 we discuss numerical large-Ncresults obtained

with the Hasenbusch algorithm for SU(Nc) YangMills theories [81]. Finally, in

the last chapter we draw our conclusions and discuss possible future outlooks of our works.

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Chapter 1 Topological properties and lattice

formulation of QCD

In this chapter we make a brief overview of general aspects regarding topo-logical properties and θ-dependence of QCD and their connection to interesting phenomenological aspects of Standard Model and Beyond Standard Model physics. Then, we review the main results obtained for θ-dependence with analytic tech-niques such as the Dilute Instanton Gas Approximation, Chiral Perturbation The-ory or the 1/Nc expansion for large number of colors. Finally, we recall the main

aspects of the lattice approach, which is the non-perturbative method adopted in this thesis to study topology and θ-dependence.

1.1 The QCD partition function in the continuum

Quantum Chromo-Dynamics is formulated as a non-Abelian SU(Nc) gauge

theory, where Nc = 3 is the number of colors, minimally coupled to Nf = 6

quark avors. Gluons are described through N2

c − 1 gauge elds living in the

adjoint representation of the gauge group: Aµ(x) ≡ Aaµ(x)Ta, where Ta are the

N2

c − 1 traceless Hermitian generators of the algebra of SU(Nc) chosen so that

they satisfy 2 Tr{TaTb} = δab. The quark elds are instead described through a

collection of Nc spinor elds ψα = (ψ1α, . . . , ψαNc)that transform according to the

fundamental representation of SU(Nc). The theory, dened on a 4d Minkowski

space-time, is described by the well known Lagrangian LQCD =− 1 2Tr (G µν_G µν) + Nf X f =1 ψ_f i /D_{− m}f ψf ≡ LYM +LF. (1.1)

Here LYM is the SU(Nc) YangMills Lagrangian, describing pure gluodynamics

in terms of the eld strength

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with g the strong coupling, while LF is the Dirac Lagrangian for the quarks,

which are minimally coupled to the gluons through the covariant Dirac operator /

D_{≡ γ}µDµ= γµ(∂µ+ igAµ) = /∂ + ig /A, (1.3)

with γµ _{the Dirac matrices. With this construction, the Lagrangian is}

automati-cally invariant under the local gauge transformation Ω(x) ∈ SU(Nc):

ψf(x) → Ω(x)ψf(x), (1.4) ψf(x) → ψf(x)Ω†(x), (1.5) Aµ(x) → Ω(x)Aµ(x)Ω†(x) + i gΩ(x)∂µΩ †_(x), _(1.6)

The denition of the theory at the quantum level relies on the Feynman's path integral. The starting point is the construction of the generating functional, from which every correlation function of the theory can be derived:

Z(t) =Z dψdψdAeiSQCD[ψ,ψ,A] =Z dψdψdAexp i Z t 0 dx0 Z d3xLQCD(x) , (1.7) where we have introduced the action SQCD and the path-integral measures

dψdψ≡Y x Nc Y i=1 Nf Y f =1 dψi_f(x)dψ_fi(x), [dA]≡Y x Y µ N2 c−1 Y a=1 dAa_µ(x). (1.8)

This denition, however, is only formal because, since we are dealing with gauge theories, the path integral is carried over also unphysical degrees of freedom intro-duced by the gauge symmetry. In the continuum, this problem can be solved in-troducing the FaddeevPopov ghosts, unphysical elds whose contribution, xing the gauge, allows to eliminate the singularities emerging in the path integral due to the redundancy of gauge degrees of freedom. In the lattice formalism, instead, the path-integral is naturally regularized and gauge xing is not required [98].

Among the many interesting properties of the generating functional, one of the most important is that it can be related, after a Wick rotation to imaginary times, to the thermal partition function of the theory. The prescriptions of the Wick rotation, which can be succinctly summarized in the following rules,

(x0, xi)_{→ (ix}0 _{≡ x}4, xi), (∂0, ∂i)→ (−i∂0 ≡ ∂4, ∂i), (γ0, γi)→ (γ0 _{≡ γ} 4,−iγi ≡ γi), (A0, Ai)→ (−iA0 ≡ A4, Ai), (ψ, ψ)→ (ψ, ψ), xµxµ= ηµνxµxν → −δµνxµxν =−xµxµ,

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lead to the Euclidean generating functional ZE(τ ) = Z dψdψdAe−SQCDE [ψ,ψ,A] =Z dψdψdAexp − Z τ 0 dx4 Z d3xLE QCD(x) , (1.9) where τ ≡ it and SE

QCD is the Euclidean QCD action dened in terms of the

Euclidean eld strength and of the Euclidean Dirac operator (whose denitions in terms of Euclidean elds remain invariant in form after the Wick rotation):

SQCDE = SYME + SFE = Z τ 0 dx4 Z d3xLE QCD(x) (1.10) = Z τ 0 dx4 Z d3_x    1 2Tr (GµνGµν) + Nf X f =1 ψ_f D + m/ f ψf   .(1.11) The relation between ZE(τ ) and the partition function at inverse temperature

β ≡ 1/T

Z(T )_{≡ Tr}e−βHQCD _, (1.12)

is immediately obtained setting τ = β and performing the path-integral over a compactied time direction where periodic and anti-periodic boundary conditions are taken respectively for the gluon and quark elds.

The thermal mean value of a generic observable O can be expressed through the path integral as well:

hOi (T ) = 1 Z

Z

dψdψdAe−SQCDE [ψ,ψ,A]Oψ, ψ, A. (1.13)

Expression (1.13) can be interpreted as a mean value over an ensemble of eld congurations extracted according to the probability distribution

P ψ, ψ, A= 1 Ze

−SE

QCD[ψ,ψ,A]. (1.14)

This interpretation is meaningful only if SE

QCD is real and SQCDE ≥ 0, so that P is

guaranteed to be real and P ≤ 1 for every eld conguration. To show this, it is useful to observe that, being the quark action quadratic in the fermion elds, one can perform the Gaussian integral over the quark sector in Eq. (1.9) and obtain the equivalent expression

ZE(τ ) = Z [dA] e−SYME [A] Nf Y f =1 detD [A] + m/ f . (1.15)

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Since SE

YM ≥ 0 by denition, the gluon contribution is trivially real and positive.

Concerning the quark contribution, the anti-hermiticity of the massless Dirac operator /D, which implies

/

Duλ = iλuλ, λ∈ R, (1.16)

guarantees that the fermion determinant is real and positive. Indeed, since /

Dγ5 =−γ5D, each eigenvector u/ λ has a corresponding twin γ5uλ with

complex-conjugated eigenvalue: /

Dγ5uλ =−γ5Du/ λ =−iλγ5uλ. (1.17)

As a consequence, the Dirac determinant can be rewritten in a manifestly positive way: detD + m/ = Y λ∈R (iλ + m) = mn0Y λ>0 [(iλ + m)(_{−iλ + m)]} = mn0Y λ>0 (λ2_{+ m}2₎_{≥ 0,} (1.18)

where n0 is the number of zero-modes of /D.

1.2 Gauge elds topology and index theorem

The space of gauge eld congurations with nite Euclidean YangMills ac-tion can be decomposed into separated sectors characterized by the value of the topological charge, dened in the continuum Euclidean theory as:

Q = Z d4x q(x) = g 2 32π2εµνρσ Z d4x Tr_{Gµν(x)Gρσ(x)} = g 2 16π2 Z d4x TrnGµν(x) ˜Gµν(x) o , (1.19) where q(x) is the topological charge density and ˜Gµν ≡ 1₂εµνρσGρσis the dual eld

strength tensor. This quantity yields always an integer number Q ∈ Z, which can be interpreted as the number of windings of the eld conguration around the gauge group at innity.

To make this interpretation manifest, it is useful to write the topological charge density q(x) as the total divergence of the ChernSimons current Kµ(x):

q(x) = ∂µKµ(x) = ∂µ g2 8π2εµνρσTr Aν(x)∂ρAσ(x) + 2

3igAν(x)Aρ(x)Aσ(x)

. (1.20) Substituting expression (1.20) in (1.19) leads to the following expression of Q as a surface integral at innity:

Q = Z d4x q(x) = Z d4x ∂µKµ(x) = lim r→∞ Z S3_(r) dSµKµ, (1.21)

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where S3_(r) _{is the 3-sphere with radius r.}

At large distances, it suces that Gµν ∼ 1/r2, with r ≡

√

x2, for SE

YM to be

nite. However, this does not imply Aµ = 0 at r = ∞, since any gauge copy of

zero would yield the same eld strength. Thus, Aµat large distances assumes the

following general form, cf. Eq. (1.6): Aµ(x) ∼ r→∞ i gΩ(ˆx)∂µΩ †_(ˆ_x), _Ω_{∈ SU(N} c), xˆ≡ x/r ∈ S3(1), (1.22)

where Ω depends only on the direction ˆx along which innity is approached, which can be described by 3 angles (α1, α2, α3) = ~α. Substituting (1.22) in (1.21), one

nally obtains Q = g 2 24π2εijk Z S3₍₁₎ d3α Tr Ω∂Ω † ∂αi Ω∂Ω † ∂αj Ω∂Ω † ∂αk = n∈ Z. (1.23) The integral on the left-hand side of Eq. (1.23) is the so-called ChernSimons characteristic of Ω, which can be shown to be equal to the winding number n of Ω [99]. This integer represents how many times Ω(ˆx) covers the gauge group when ˆx covers the 3-sphere once, i.e., it counts how many times the 3-sphere is mapped in the gauge group by Ω(ˆx), making manifest the topological nature of Q. A simple example of Ω(ˆx) with n = 1 for SU(2) is

Ω(ˆx) = ˆxµσµ, (1.24)

where σµ= (1, i~σ), with ~σ the Pauli matrices.

In order to construct a global solution with non-trivial topological charge, it is useful to start from the Bogomol'nyi inequality [100]:

SYME = 1 2 Z d4x Tr{Gµν(x)Gµν(x)} = 1 2 Z d4x Tr 1 2 h Gµν(x)∓ ˜Gµν(x) i2 ± Gµν(x) ˜Gµν(x) ≥ 1 2 Z d4x TrnGµν(x) ˜Gµν(x) o = 8π2_|Q| g2 , (1.25)

where it was used the fact that ˜GµνG˜µν = GµνGµν. Inequality (1.25) is saturated

by congurations called instantons, which minimize the action in their respective topological sector, i.e., they are solutions with non-trivial topology of the classical equation of motion of SE

YM. The minimum of the action SI = 8π2/g2 is reached

when Gµν =± ˜Gµν, i.e., for selfdual/anti-selfdual solutions. Solving this equation

is a simpler task compared to solving the equation of motion, being it rst-order in the derivatives of the gauge elds, and a solution with Q = 1 for Nc = 2 was

found by Belavin, Polyakov, Schwartz, and Tyupkin (BPST instanton) [101]: Aµ(x) = ηµνa

(x_{− z)}ν

(x− z)2_{+ ρ}2σ

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where ηa

µν ≡ ε0aµν + δaµδν0 − δaνδµ0 is the 't Hooft symbol, z is the center of

the instanton and ρ is the size of the instanton. A BPST anti-istanton with Q = _{−1 can be obtained replacing in (1.26) the 't Hooft symbol with the dual} 't Hooft symbol ¯ηa

µν ≡ ε0aµν− δaµδν0 + δaνδµ0. An interesting general

character-istic of instantons is that their contribution to the path integral, exhibiting the non-analytic dependence e−SI ∼ e−1/g2 on the coupling g, vanishes faster than

any power of g, thus being invisible in perturbation theory. This aspect makes manifest why topological properties are essentially non-perturbative.

So far, we have discussed topology in relation to gauge elds. However, the value of the topological charge can also be related to the properties of the spec-trum of the Dirac operator through the index theorem [37]. In particular, this theorem relates the value of the topological charge to the presence of zero-modes in the Dirac spectrum:

Q = IndexD/ = Tr_{γ5} = n+− n−, (1.27)

where n± stands for the number of left/right zero-modes of the Dirac

opera-tor /D. The last equality can be obtained from the properties of the eigenvec-tors uλ of the Dirac operator: indeed, since uλ and u0λ = γ5uλ have

complex-conjugated eigenvalues, if λ 6= 0, then uλ and u0λ are orthogonal to each other,

i.e., u†

λu0λ = u†λγ5uλ = 0. Zero-modes, instead, are chiral, i.e., they either have

u0L = 0 or u0R = 0, thus, left/right zero-modes satisfy u†0γ5u0 = ±1 assuming

normalized eigenvectors. Evaluating the trace of γ5 on a basis of eigenvectors of

/ Dimmediately yields Tr_{γ5} = X λ u†_λγ5uλ = X λ=0 u†_λγ5uλ = n+− n−. (1.28)

1.3 Topological term and θ-dependence of the free

energy

Being the topological charge density a renormalizable and gauge invariant quantity with dimension 4 in energy, the topological charge Q can be coupled to the ordinary QCD action via the dimensionless parameter θ:

S_QCDE _{→ S}_QCDE (θ) = S_QCDE _{− iθQ.} (1.29) Note that, in the Euclidean formulation, the topological therm is purely imaginary because, being Q ∝ εµνρσGµνGρσ, this quantity contains only one time derivative

of the gauge eld Aµ(x), therefore, the imaginary unit in the path integral weight

eiSQCD is not reabsorbed by the Wick rotation:

exp{iSQCD(θ)} = exp

iSQCD+ i Z d4x g 2 16π2 Tr n Gµν(x) ˜Gµν(x) o → exp−SQCDE + iθQ = exp_−S_QCDE (θ) .

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The addition of a θ-term to the action modies the partition function of the theory, introducing a non-trivial dependence on the parameter θ:

Z(T, θ)≡ Z [dψdψdA]e−SQCDE (θ) = Z [dA] e−SEYM+iθQ Nf Y f =1 detD + m/ f = ∞ X n=−∞ eiθn Z Q=n [dA] e−SYME Nf Y f =1 detD + m/ f = ∞ X n=−∞ eiθnZn(T ), (1.30)

where Zn(T ) is the partition function restricted to the nth topological sector.

Since θ appears only through phases in the partition function, it can be treated as an angular variable with periodicity 2π, being Z(T, θ) invariant under θ → θ+2πk for any integer k.

Because of its phenomenological implications, in this work we are interested in particular in the study of the θ-dependence of the free energy density F (T, θ), which is related to the partition function by

F (T, θ) = ₋ 1 βVs

log Z(T, θ) =₋1

V log Z(T, θ), (1.31) where Vsis the 3-dimensional spacial volume and where we used the fact that the

inverse temperature β is also equal to the space-time temporal extent, so that βVs = V, the 4-dimensional Euclidean space-time volume.

Assuming analyticity in θ = 0, it is possible to expand the free energy in Taylor series around this point. In this work, we will use the following parametrization:

f (T, θ)_{≡ F (T, θ) − F (T, 0) =} 1 2χ(T )θ 2 " 1 + ∞ X n=1 b2n(T )θ2n # , (1.32) where only even powers of θ appear because F (T, θ) is an even function in θ. This property immediately follows from the fact that the topological charge is odd under a CP transformation (i.e., a charge conjugation C followed by a parity transformation P), unlike SE

QCD. This implies that:

CP S_QCDE (θ) = S_QCDE _{− iθ(−Q) = S}_QCDE (_−θ), (1.33) thus, performing the change of variable Aµ(x)→ CP Aµ(x)inside Eq. (1.30), one

straightforwardly obtains Z(T, θ) = Z(T, −θ), i.e., f(T, θ) = f(T, −θ).

Another important property that can be derived directly from the denition of the partition function is the relation between the derivatives of f(θ) computed in

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θ = 0 and the cumulants kn of P (Q), the path-integral probability distribution

of the topological charge at θ = 0: dn_{f (θ)} dθn θ=0 =₋i n V kn θ=0 =− in V hQ n ic θ=0, (1.34) where hQn_i

cdenotes the connected mean value of Qn. Using parametrization (1.32),

the quadratic coecient χ, the topological susceptibility, can thus be related to the second cumulant k2 of P (Q)

χ≡ lim V→∞ k2 V θ=0 = lim V→∞ hQ2_i c V θ=0 = lim V→∞ hQ2_i V θ=0 , (1.35)

where we have used that hQi |θ=0= 0, being f(T, θ) even in θ. The b2ncoecients,

parametrizing the non-quadratic dependence on θ of the free energy, are instead related to the higher-order cumulants k2n+2 of P (Q):

b2n ≡ lim V→∞(−1) n 2 (2n + 2)! k2n+2 k2 θ=0 = lim V→∞(−1) n 2 (2n + 2)! hQ2n+2_i c hQ2_i θ=0 , (1.36)

where the rst two non-quadratic even cumulants read, using that hQ2k+1_{i |} θ=0= 0, hQ4ic θ=0 = hQ4i − hQ2i2 θ=0, (1.37) hQ6ic θ=0 = hQ6i − 15 hQ4i hQ2i + 30 hQ2i3 θ=0. (1.38)

Note that in the thermodynamic limit V → ∞ the cumulants kn are extensive

quantities, being them related to derivatives of the free energy (which is extensive as well), thus χ and the b2ncoecients are intensive quantities and possess a

well-dened innite-volume limit.

1.4 Phenomenological aspects of topology

In the following subsections we will briey review some interesting aspects of strong interactions where topological properties play an important role: the U (1)A anomaly and its connection with η0 physics, the strong CP problem and

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1.4.1 Axial anomaly, large-N

c

limit and WittenVeneziano

equation

The classical QCD Lagrangian enjoys, in the chiral limit mf → 0, f =

1, . . . , Nf, a global avor symmetry under the chiral group

G = U (Nf)L⊗ U(Nf)R = U (1)L⊗ U(1)R⊗ SU(Nf)L⊗ SU(Nf)R. (1.39)

Collecting the dierent avored quark elds in a single spinor ψ = (ψ1, . . . , ψNf),

the Dirac term can be rewritten as LEF =

Nf

X

f =1

ψ_f( /D + mf)ψf = ψLDψ/ L+ ψRDψ/ R+ ψLM ψR+ ψRM ψL, (1.40)

where ψL,R = (1± γ5) ψ/2 are the left and right component of ψ and M =

diag(m1, . . . , mNf) is the quark mass matrix. Now, the massless Dirac term is

manifestly invariant under the action of G:    ψL→ G ULψL= e iαLV LψL ψR→ G URψR= e iαRV RψR VL,R ∈ SU(Nf), UL,R∈ U(Nf). (1.41)

A convenient decomposition of the avor group is

G = U (1)V ⊗ U(1)A⊗ SU(Nf)V ⊗ SU(Nf)A, (1.42)

where the subscripts V and A refer to vectorial/axial transformations, i.e., trans-formations where the left and the right matrices are related by, respectively, UR = UL and UR = UL†. In real-world QCD, the only symmetry that remains

exact is U(1)V, however, avor symmetry G is approximately true when

consid-ering QCD at low energy scales. Indeed, in that regime, one expects that the leading contribution to QCD dynamics is given by the lightest quarks, and since the u, d and s masses mu ' 2.2 MeV, md ' 4.7 MeV, ms ' 93 MeV [102] are

smaller than the dynamically-generated scale ΛQCD ∼ 300 MeV [103], one can

consider the chiral-symmetric theory with Nf = 2 or even 3 an approximation of

the physical one.

The fate of each subgroup in the quantum theory is very dierent. The U(1)V

invariance remains a true symmetry (i.e., realized à la WignerWeyl) at the quantum level too and is associated to the conservation of the baryon number. The sub-group SU(Nf)V ⊗ SU(Nf)Ais spontaneously broken to SU(Nf)V by the

chiral condensate hψψi ≡ PNf

f =1hψfψfi. Chiral symmetry breaking is

respon-sible for the low mass of the lightest mesons, which are its associated (pseudo) NambuGoldstone bosons. Finally, U(1)A is not a quantum symmetry at all,

being explicitly broken by the axial anomaly.

Indeed, unlike what happens at the classical level, the divergence of the singlet axial current Jµ

5(x) = ψγµγ5ψ computed in the quantum theory is non vanishing

and is, in the chiral limit, proportional to the topological charge density [10]: ∂µJ5µ(x) = 2Nfq(x). (1.43)

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The anomalous term on the right hand side of Eq. (1.43) arises from the non-invariance of the fermion path-integral measure under a U(1)A transformation:

   ψL → U (1)A eiα_ψ L ψR → U (1)A e−iα_ψ R =⇒ [dψdψ] → U (1)A e−2iαNfQ_[dψdψ]. (1.44)

If this change of variable is performed inside the path-integral denition of the par-tition function, the presence of the anomaly leads ultimately to a non-invariance of the QCD action under U(1)A even in the chiral limit:

SQCD M =0 _{U (1)}→ A SQCD M =0− 2iαNfQ. (1.45)

This non-invariance has physical consequences on the low-energy spectrum of the theory. For instance, it is the origin of the large mass of the η0 _{meson, which,}

because of the anomalous breaking of the U(1)A symmetry, cannot be described

as a pseudo NambuGoldstone boson like the pions or the η meson [10]. More precisely, the squared mass of the η0 _{can be related to the topological}

suscepti-bility of the pure-gauge theory, which does not vanish in the chiral limit. This relation, as well as other relations between η0 _{parameters and pure-gluodynamics}

topological observables, can be derived in the framework of the large-Nc limit,

where the axial anomaly is expected to vanish and the η0 _{is expected to become}

lighter and lighter and behave as an actual NambuGoldstone boson.

First introduced by 't Hooft [104], the large-Nc limit consists in extending

QCD to an arbitrary number of colors and considering the limit Nc → ∞ while

keeping the 't Hooft coupling λ ≡ g2_N

c xed, so that the dynamically

gener-ated scale ΛQCD is kept xed too. This fact can be justied by the one-loop

perturbative expression of the dynamically-generated scale: Λ1 loop_QCD = µ exp −_2β 1 0g2(µ) , β0 = 1 16π2 11Nc− 2Nf 3 Nc∼→∞ Nc, (1.46) i.e., Λ1 loop

QCD at large Nc is a function of g2Nc = λ. This implies that the axial

anomaly (1.45) should vanish as 1/Nc since q ∝ g2 ∝ λ/Nc, justifying the

as-sumption that, in this limit, the η0 _{behaves as a proper NambuGoldstone boson.}

The large-Nc regime provides QCD with the small parameter 1/Nc, thus the

idea is that any observable can be non-perturbatively (with respect to the strong coupling) expanded in this parameter around 1/Nc = 0. Using large-Nc

argu-ments and the 1/Nc expansion, it is possible to establish a relation, at leading

order in 1/Nc and in the chiral limit, between m2η0 and the large-N_c limit of

the topological susceptibility of the pure YangMills theories. This relation was rst found by Witten (and later rened by Veneziano) and is known as Witten Veneziano equation [12,13]: m2η0 = 2Nf f2 π ¯ χ, (1.47)

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where fπ ' 92 MeV [102] is the pion decay constant and

¯

χ = lim

Nc→∞

χ, (1.48)

where χ denotes, in this subsection, the topological susceptibility of the pure-gauge theory (not to be confused with χQCD, the topological susceptibility of the

theory with fermions). This important result allowed to reconcile two facts that apparently seemed in contrast with each other:

In the chiral limit, QCD θ-dependence must vanish, i.e., fQCD(θ) = 0, thus,

in particular, χQCD = 0. More precisely, if at least one avor ¯f has vanishing

mass m_f¯ = 0, any non-zero θ-term can be canceled by applying a U(1)A

transformation to that avor with α = θ/2, cf. Eq. (1.68), resulting in a θ-independent theory.

The large-Nc analysis of vacuum diagrams shows that the fermion

contri-bution is suppressed by a factor 1/Nc compared to that of the gluons [104].

Since in the chiral limit, and for every value of Nc, the fermion contribution

has to cancel θ-dependence, the question is how it is possible if, at large Nc,

this contribution is sub-leading in 1/Nc compared to the that of the gluons.

Witten hypothesized that there is a avor-singlet state (identied with the η0

meson) with a squared mass of order 1/Nc and not vanishing in the chiral limit

whose contribution to χQCD is O (Nc0), so that it can cancel out the large-Nc

leading order contribution to χQCD of the gluons [11]. Using these arguments

and Eq. (1.43), it is possible to derive Eq. (1.47).

This equation requires a nite large-Nc limit for the topological susceptibility of

the pure-gauge theory in order to yield a square η0 _{mass of order 1/N}

c, being

fπ ∼ O

√ Nc

. In particular, using the experimental values of mη0 ' 957.8

MeV [102] and fπ, one nds [13]:

¯

χ_{' (180 MeV)}4.

The WittenVeneziano equation can be generalized to any self-coupling of the η0 _{[13]. Each of them can be related, at large N}

c, to topological observables of the

pure-gauge theory. More precisely, the functional dependence of the low-energy eective potential of the η0 _{is equal to the θ-dependence of the vacuum energy}

density (i.e., zero-temperature free energy density) of the pure YangMills theory at large Nc: Veff (η0) = EYM θ = s 2Nf f2 π η0 ! , (1.49)

where EYM(θ) = fYM(T = 0, θ). The b2 coecient, for example, is related to the

coecient of the η04 _{term, i.e., to the value of the η}0_η0 _{scattering amplitude.}

Given its connection with hadron phenomenology and with intriguing theoret-ical aspects of gauge theories, the study of the θ-dependence of the vacuum energy

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of the pure-gauge theory in the large-Nc limit is of great interest. Since, apart

from the phenomenological prediction for ¯χ, no other analytic result about large-Nc θ-dependence is known from QCD, lattice simulations are a natural choice

to obtain rst principle and fully non-perturbative results about EYM(θ, Nc).

Nevertheless, large-Nc arguments still allow to derive some interesting general

properties that can be tested on the lattice.

Generally speaking, the requirement that a non-trivial θ-dependence is present at leading order in 1/Nc implies that the most general form of EYM(θ, Nc) is [14,44]

EYM(θ, Nc) = Nc2E¯YM θ Nc + O 1 N2 c = χθ¯ 2 2 1 + ∞ X n=1 ¯b2n θ2n N2n c ! + O 1 N2 c . (1.50) The N2

c pre-factor is deduced from the fact that the YangMills Lagrangian has

O (N2

c) degrees of freedom, while the scaling variable θ/Nc can be understood

from the form of the θ-term, since θQ ∝ λ(θ/Nc).

The general expression of Eq. (1.50) and the requirement χ = ¯χ + O 1 N2 c . (1.51)

needed for the WittenVeneziano mechanism, lead to the prediction: b2n = ¯b2n 1 N2n c + O 1 N2n+2 c , (1.52)

i.e., the b2n coecients are expected to vanish as Nc−2n. In all cases, corrections

to the leading large-Nc behavior are expected to be suppressed by a further 1/Nc2

factor since, as shown by 't Hooft [104], in the pure-gauge theory the gluon vacuum diagrams contributing to the vacuum energy all yield contributions which are even powers of 1/Nc, thus making 1/Nc2 the actual expansion variable in the

YangMills case.

Apparently, the fact that θ/Nc is the eective variable in the large-Nc limit

and that EYM(θ, Nc)must be 2π-periodic for every Ncseem in contrast with each

other. Actually, these two properties cannot be satised by an analytic function. To reconcile them, it was conjectured that EYM(θ), because of the presence of

multiple stable and degenerate vacua, is actually a multi-branched function which takes, at large-Nc, the general form [44]

EYM(θ, Nc) = Nc2min k K θ + 2πk Nc , (1.53)

with K being an unknown analytic function. This general functional form re-spects all the desired properties mentioned so far but is non-analytic in θ. This

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can be easily seen by taking the large-Nc limit of Eq. (1.53). Indeed, using

large-Nc predictions (1.51) and (1.52), one obtains:

EYM(θ, Nc) = ¯χ min k (θ + 2πk) 2 + O 1 N2 c , (1.54)

which has cusps in θ = π+2πk, where the curves (θ + 2πk)2 _{and (θ + 2π(k + 1))}2

cross; a similar behavior is expected also for large-N CPN−1 _{models, see Sec. 3.1.}

In any case, as long as θ is small, one can just assume analyticity and safely use the Taylor expansion around θ = 0.

1.4.2 The strong CP problem and axion phenomenology

As we already stated, coupling a non-zero θ parameter to the QCD action also introduces an explicit breaking of the CP symmetry, which is absent at θ = 0. This is at odds with experimental results, which so far have shown no evidence of parity violations from strong interactions. This fact points towards the possibility that θ = 0. The most precise experimental upper bound for θ can be obtained from the measure of the neutron electric dipole moment, which is expected, from Chiral Perturbation Theory, to be of order [105]

d_nChPT _{∼ θ · 10}−16e cm. (1.55) Since dexp

n is compatible with zero within ∼ 10−26e cm[106], then the

experimen-tal value of θ is bounded to be |θexp| . 10−10.

Since θ = 0 is the only value for which no CP violations is present in the strong sector, this result sets the necessity of a ne tuning on this parameter, which is an undesirable feature of the theory1_{. Instead, a fundamental and natural}

explanation of the absence of strong CP violations would be welcome.

A possible mechanism to solve this issue was proposed by Peccei, Quinn, Weinberg and Wilczek [36]. This solution relies on the introduction of a new, global U(1) axial symmetry, usually referred to as PecceiQuinn (PQ) symmetry or U(1)PQ, which is broken both spontaneously and by an axial anomaly (similarly

to the U(1)Asymmetry in QCD), so that the spectrum of the theory now contains

a new pseudo NambuGoldstone boson, the axion. Regardless of the ultraviolet (UV) details of the underlying fundamental theory, the axion eective action at low energies (i.e., below the scale of the breaking of the PQ symmetry) must take the general form

La(x) = 1 2∂µa(x)∂µa(x)− i a(x) fa q(x) +_Lint(∂µa/fa, . . . ), (1.56) 1_{Actually, S}E

QCD(θ)is naively CP-invariant also for θ = π, since a CP transformation changes π → −π, but −π ∼ π for the 2π-periodicity. However, for θ = π a spontaneous breaking of the CP symmetry can occur, for instance if f(θ) exhibits a cusp in this point. For more details about this topic we refer the reader to, e.g., Refs. [107,108].

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where a is the axion eld, fa is the scale at which the PecceiQuinn symmetry

spontaneously breaks down and Lint represents the interactions with other elds

of the Standard Model, in which the axion eld must enter through a derivative by virtue of being a pseudo NambuGoldstone boson. For the same reason, the axion-gluon term ∼ a(x)q(x) must appear in order to eectively reproduce the correct U(1)PQ axial anomaly (1.43) since, when applying the U(1)PQ

transfor-mation eiαPQ, the axion eld gets shifted a → a + α

PQfa by virtue of being a

pseudo NambuGoldstone boson.

This minimal set-up provides a simple solution to the strong CP problem; indeed, if we now add the axion contribution to the QCD Lagrangian at non-zero θ, we obtain the following modied θ-term:

La+LQCD(θ)⊃ −i a(x) fa + θ q(x). (1.57)

Now, the θ parameter can be dynamically relaxed to zero by acting with a U(1)PQ

transformation and choosing αPQ = θ, solving the strong CP problem exactly,

Besides, this choice also provides the axion eld with the correct vacuum value amin = 0, meaning that, when θ is shifted to zero by the action of the U(1)PQ

transformation, the axion eld a describes actual physical excitations above the vacuum. To show this fact, it is sucient to evaluate the minimum of the axion eective potential.

Being the axion eld coupled to the QCD action through the topological charge density, cf. Eq. (1.57), one can integrate out the QCD sector, treating the quantity a/fa as an external source eld, and relate the (temperature-dependent) axion

eective potential to the free energy density of QCD Z(T ) =

Z

[da]e−Sfreea [a]

Z [dψdψdA]e−SEQCDei R1/T 0 dx4R d 3_x₍a(x) fa +θ)q(x) = Z

[da]e−Sfreea [a]_e−V f(T,a/fa+θ) _≡

Z

[da]e−Seff[a]_,

(1.58) where the axion eective Lagrangian is

Leff(a, T ) = 1 2∂µa∂µa + Veff(T, a) = 1 2∂µa∂µa + f T, a fa + θ . (1.59) From the denition (1.30) of the QCD partition function, it can be easily inferred that f(T, θ) reaches its minimum for θ = 0; as a matter of fact, being the path-integral measure [dA]e−SYME [A] Nf Y f =1 detD[A] + m/ f

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positive-dened, the value of Z(θ) is always smaller than Z(θ = 0) because of the presence of the oscillating exponential factor eiθQ_{. Hence:}

Z(T, θ) _{≤ Z(T, θ = 0) =⇒ f(T, θ) ≥ f(T, θ = 0).} This relation, along with the fact that (cf. denition (1.32))

d2_f dθ2 θ=0 = χ≥ 0,

proves that θ = 0 is a global minimum of f(T, θ). This result implies that the minimum of Veff is reached for amin = −θfa, thus, the U(1)PQ transformation

with αPQ = θ, applied to cancel out the QCD θ angle, also shifts the minimum

of the axion potential to zero, so that now the axion eld a correctly describes above-vacuum excitations.

Using the expression of the eective potential (1.59), one can extract the expression of the eective axion mass from the θ2 _{term of f (θ), which is thus}

related to the topological susceptibility of QCD, cf. Eq. (1.32): m2_a(T ) = χ(T )

f2 a

. (1.60)

Higher-order eective self-interaction terms can be instead related to the b2n

coecients; indicating by λ2n+2 the coecient of the O(a2n+2) term, one gets:

λ2n+2(T ) = 1 2 χ(T ) f2 a b2n(T ) f2n a = m 2 a(T ) 2 b2n(T ) f2n a (1.61) Since current bounds set 108 _{GeV . f}

a . 1012 GeV [7, 109], a tiny mass for the

axion and even smaller higher-order self-interaction terms are expected, justifying the leading-order approximation Veff(a)' m2aa2/2. Besides, being all interaction

terms involving the axion and Standard Model particles suppressed as powers of 1/fa too, the PQ axion can also be a possible Dark Matter candidate. In this

context, it is interesting to compute the relic abundance of axion dark matter na. Indeed, this quantity is a fundamental input to compute the axion energy

density nowadays Ωa ≡ ρa/ρcrit = (mana)/ρcrit, which can account for a part of

the measured Dark Matter energy density ΩDM ≡ ρDM/ρcrit = 0.264(7) [110].

A possible way to evaluate the axion relic energy density is through the so-called misalignment mechanism [79]. Assuming that ination occurs before the breaking of the PQ symmetry, the dependence of a(t, ~x) on the spatial coordinates can be neglected and the axion eld time evolution in the FriedmannLemaître RobertsonWalker metric is described by the equation:

¨

a + 3H ˙a + V_e0 (a) = 0_{' ¨a + 3H ˙a + m}2_aa, (1.62) where H(t) ≡ ˙R(t)/R(t) is the Hubble parameter and R(t) is the Universe scale factor. At high temperatures (i.e., early times t), the PQ symmetry is eectively

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restored, therefore, the vacuum expectation value of the axion eld is in general non-zero, and has the form v = −θ0fa(where θ0 is the initial misalignment angle).

In this regime the time evolution of the axion is dominated by the friction term proportional to H, since it grows when approaching t → 0. As the Universe cools down, H(t) decreases, the eective potential term starts to dominate over the Hubble one and the axion eld starts oscillating. In this regime, the comoving number density of axions na = hmaa2i freezes and axions behave as cold Dark

Matter with mass

m2_a _' χ (T = 0) f2

a

. (1.63)

At the end of this process, the expected relic axion energy density is [111]: Ωmis a = 86 33 Ωγ Tγ n? a s?, (1.64)

where Ωγ and Tγ are the energy density and temperature of relic photons and n?a

and s? _{are the comoving axion number density and entropy, evaluated at a time}

t? _{suciently after the start of the oscillation, so that n}?

a/s? can be considered

constant. These results underline that a precise knowledge of the behavior of Veff(T, a) as a function of the temperature, i.e., of the temperature dependence

of f(T, θ), is needed in order to precisely estimate Ωmis

a and ma. In order to

obtain full non-perturbative results for f(T, θ) from rst principles, the lattice approach is a natural choice. However, it is also possible to obtain some analytic predictions about θ-dependence in the opposite limits T → 0 and T → ∞ using suitable approximations. These results will be briey reviewed in the following subsections.

1.4.3 Free energy θ-dependence at T = 0 from the Chiral

Eective Lagrangian

A possible non-perturbative way of investigating the low-energy sector of QCD (i.e., for energy scales µ ΛQCD) is to use the Chiral Eective Lagrangian

formulation [112]. The main idea behind this approach is to describe the theory at low energies in terms of an eective Lagrangian which reproduces the same symmetries of the fundamental theory but is expressed in terms of the lightest degrees of freedom of QCD, i.e., the N2

f − 1 (pseudo) NambuGoldstone bosons

arising from the spontaneous breaking of the chiral symmetry G0 _{= SU (N} f)L⊗

SU (Nf)R= SU (Nf)V ⊗ SU(Nf)A to SU(Nf)V. Usually, one considers the cases

Nf = 2 or even 3, since mu, md ΛQCD and ms . 4ΛQCD. For this reason,

the quark mass term is treated as a small perturbation of the chiral-symmetric massless case.

Focusing on the Nf = 2 case, the interesting degrees of freedom are the three

pions. Introducing the SU(Nf) eld

U _{≡ exp} i 1 fπ ~π_{· ~σ} ,

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where ~π collects the pions eld, at leading order O(p2₎ _{(i.e., with at most two}

derivatives in the eld U) the most general Lagrangian which is G0 _and

Lorentz-invariant is [113] L(2)eff(U ) = f2 π 4 Tr ∂µU†∂µU . (1.65)

The invariance under G0 _{can be immediately obtained recalling that the U eld}

transforms under G0 _as

U →

G0 U

0 _{= V}

LU VR†. (1.66)

Concerning the quark mass term, which instead explicitly breaks the chiral sym-metry (as in the full theory), the leading order term is linear in the quark mass matrix and has no U eld derivatives. Requiring that it matches the symmetry properties of the mass term of the full theory, one nds that the most general term that satises these constraints is

δ_L(M )_eff (U, θ) = Σ_{< Tr}e−iθ/Nf_{M U} _, (1.67)

where M = diag(mu, md) is the Nf = 2 quark mass matrix and Σ ≡ − hψψi

is the absolute value of the chiral condensate, which, being the order parameter of the spontaneous breaking of G0_{, acquires a non-vanishing value in the T → 0}

regime. The θ angle appears in the mass term as a phase e−iθ/Nf by, again,

analogy with the full theory. As a matter of fact, by performing a U(1)A axial

transformation in the QCD Lagrangian with non-vanishing quark masses, one obtains, cf. Eqs. (1.40), (1.44) and (1.45):

2<ψLM ψR → 2<e−2iαψLM ψR , θ→ θ − 2αNf, (1.68) therefore, the θ term can be traded for a complex phase e−iθ/Nf in the quark mass

matrix by choosing α = θ/(2Nf).

By minimizing the leading order massive Chiral Eective Lagrangian L(LO) eff =

L(2)eff + δL (M )

eff with respect to the eld U for a generic value of θ, one can compute

the θ-dependence of the zero-temperature free energy density in the Euclidean formulation up to an irrelevant constant as:

e−V F (θ) = Z(θ) = e−V L(LO )eff (Umin,θ) =⇒ F (θ) = L(LO)

eff (Umin, θ) (1.69)

Using expressions (1.65) and (1.67) one nds for the vacuum energy density [15, 111,114] EQCD(θ) ≡ F (θ) − F (0) = m2πfπ2 " 1− s 1− 4z (1 + z)2 sin 2 θ 2 # , (1.70)

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where m2 π = fΣ2

π(mu + md) =

Σ f2

π(1 + z)mu is the pion mass and z ≡ mu/md is

the up-down mass ratio. As a side note, we observe that expression (1.70) is 2π periodic, has a cusp in θ = π and that, if mu = 0, then EQCD(θ) = 0, i.e., there

is correctly no θ-dependence.

From Eq. (1.70), one obtains, by Taylor-expanding around θ = 0, the value of the topological susceptibility and of the b2n coecients (for brevity we only report

b2): χ = z 1 + zm 2 πfπ2, b2 =− 1 12 1 + z3 (1 + z)3, . . . , (1.71)

which, in this simple case, can be expressed using only the physical parameters fπ, mπ and z. Using the values mu = 2.2(2) MeV [102] and Σ = 0.0147(16)

GeV3 _{[115], one obtains [111]:}

( χ1/4_{= 75.5(5)} _{MeV, b} 2 =−0.029(2), z = 0.48(3), χ1/4_{= 77.8(4)} _{MeV, b} 2 =−0.022(1), z = 1, (1.72) where the rst prediction refers to the physical up/down mass ratio [111], while the second one refers to the case of two degenerate light quarks (isospin-symmetric case), which yields an acceptable approximation of the physical case, being the up-down mass dierence much smaller than ΛQCD. Using the former result and

Eq. (1.63), we nd the cold axion mass [111]: ma= χ1/2_{(T = 0)} fa = r z 1 + z mπfπ fa = 5.70(7) µeV 1012_GeV fa . (1.73)

1.4.4 Free energy θ-dependence at high T from the Dilute

Instanton Gas Approximation

Another possible approach to evaluate the partition function of QCD is to use semi-classical methods, i.e., by perturbatively expanding the quantum uctua-tions around a classical minimum of the QCD Euclidean action. While perturba-tive methods break down for in the conned phase, they are expected to become reliable in the high-T , deconned phase, since, due to asymptotic freedom, the strong coupling becomes small.

The simplest way to describe a classical QCD vacuum conguration is to model it as a dilute gas of non-interacting BPST instantons and anti-instantons, which is usually referred to as Dilute Instanton Gas Approximation (DIGA). In this ap-proximation scheme, instantons and anti-instantons are treated as non-interacting identical particles, and the partition function of the nth _{topological sector is}

Topologyand θ -dependenceinQCDandQCD-liketheories PhdThesis

Corso di Dottorato in Fisica

Phd Thesis