Study of some U(1)
axial condensates
in QCD at finite temperature
Candidata: Nicoletta Carabba
Relatore: Prof. Enrico Meggiolaro
Corso di Laurea Magistrale in Fisica
Curriculum: Fisica Teorica
Università di Pisa
Dipartimento di Fisica E. Fermi
Giugno 2020
Contents
Introduction 5 1 Quantum Chromodynamics 9 1.1 QCD Lagrangian . . . 9 1.2 Properties of QCD . . . 10 1.3 Global symmetries . . . 121.4 Chiral symmetry breaking . . . 13
1.4.1 Phenomenological motivation . . . 14
1.4.2 Chiral condensate . . . 15
1.5 U (1) axial anomaly . . . 16
1.5.1 The ABJ anomaly . . . 17
1.5.2 The Atiyah-Singer theorem . . . 18
1.6 Instantons . . . 21
1.7 Lattice formulation . . . 22
2 Symmetry restoration 25 2.1 QCD phase transition . . . 25
2.2 Is the U (1) axial symmetry restored? . . . 26
2.3 Local U (1) axial condensate . . . 30
3 Spectral-density analysis 33 3.1 Euclidean Dirac spectral density . . . 33
3.2 Chiral condensate . . . 35
3.3 Chiral susceptibilities . . . 36
3.3.1 Symmetry restoration . . . 43
3.4 Hypotheses on the spectral density . . . 45
3.4.1 Zero-mode contribution . . . 46
3.4.2 Power-law behaviour . . . 49
3.4.3 Gap hypothesis . . . 53 3
4 CONTENTS
3.5 Symmetry restoration for Nf light flavours . . . 55
3.6 U (1) axial condensates . . . 58
3.6.1 Properties of the multilocal operator . . . 60
3.6.2 Spectral-density analysis . . . 61
3.7 Global U (1) axial condensate for Nf light flavours . . . 63
4 Instanton-background computation of the U (1)Acondensates 65 4.1 ’t Hooft’s effective vertex . . . 66
4.2 Direct computation of the U (1)Acondensates in the instanton background . . . 73
4.3 U (1)A condensates in the instanton background at high tem-perature T . . . 78
5 Conclusions 85
A Conventions and Wick rotation 89
B Transformations of quark bilinears 91
Introduction
Quantum Chromo-Dynamics (QCD) is the quantum field theory of strong interactions and describes hadrons as bound states of quarks, which inter-act by exchange of gluons. The vacuum state of QCD is charinter-acterized by certain non-vanishing condensates which cannot be understood in the frame-work of perturbation theory. In the so-called chiral limit, in which Nf quark masses are sent to zero (Nf = 2 and Nf = 3 being the physically relevant
cases), the QCD Lagrangian turns out to be symmetric under the chiral group U (1)V ⊗ U (1)A⊗ SU (Nf)V ⊗ SU (Nf)A. In the quantum theory, the
subgroup SU (Nf)V ⊗ SU (Nf)Ais spontaneously broken down to SU (Nf)V
because of the condensation of quark-antiquark pairs, which gives rise to the so-called chiral condensate Σ ≡ hqqi =PNf
i=1hqi(x)qi(x)i, where the brackets
h. . .i stand for the vacuum expectation value at zero temperature or, more generally, for the thermal average at finite temperature T . On the other hand, the U (1)A (axial) symmetry is broken by the quantum anomaly : at the quantum level, under U (1)A transformations the action acquires a con-tribution proportional to the so-called topological charge Q. Although Q is the integral of a total divergence, it contributes to the path integral because of the existence of topologically non-trivial gauge configurations known as instantons, which are Euclidean solutions of the classical equations of motion with finite action and integer topological charge.
Moreover, it is also known that, at a certain critical temperature Tc ≈ 150 MeV, QCD (in the chiral limit) undergoes a phase transition which restores the SU (Nf)V ⊗ SU (Nf)A symmetry: the chiral condensate Σ, which is just
an order parameter of this symmetry, vanishes above Tc. Instead, the fate of the U (1)Asymmetry above the transition remains unclear. Although the
quantum anomaly is present at any finite temperature, at some point its effects could become practically negligible: if so, the U (1)Asymmetry would be effectively restored. It has been argued that this effective restoration could occur simultaneously to the chiral one at Tc, affecting the order of the
transition: this issue is, however, highly controversial. 5
6 INTRODUCTION Traditionally, this question has been investigated, at least in the case Nf = 2,
by studying (mainly by lattice simulations) the so-called chiral ities. For each meson channel M (σ, δ, π and η), the chiral susceptibil-ity χM is defined as the integral over four-space of the two-point
corre-lation function hJM(x)JM† (0)i of the corresponding interpolating operator
JM(x) = q(x)ΓMq(x), for some proper matrix ΓM in Dirac and flavour space.
The importance of these objects lies in the fact that the various meson chan-nels are mixed under SU (2)A and U (1)A transformations: as a symmetry
gets restored, the susceptibilities of the meson channels that are mixed under that symmetry become degenerate. In particular, besides the chiral conden-sate Σ, also the differences χπ − χσ and χδ− χη can be regarded as order
parameters of the SU (2)Asymmetry and must vanish above the critical tem-perature Tc, as confirmed by lattice simulations. On the other hand, χπ− χδ
and χσ− χη behave as order parameters of the U (1)A symmetry. Several
lattice simulations, measuring these quantities, have been carried out, but the results achieved so far are not yet conclusive: most of the studies find that the U (1)A-breaking difference χπ− χδ is still non-zero above the chiral
transition, but some others report an effective U (1)A restoration already at Tc.
The aim of this work is to study some possible U (1) axial condensates in the high-temperature chirally-restored phase of QCD, by means of two non-perturbative analytical techniques: (i ) by expressing the functional averages h. . .i in terms of the spectral density of the Euclidean Dirac operator /D, and (ii ) through an approximate evaluation of the path integral in the instanton background.
The U (1)A condensates that we shall first consider are functional averages
of local 2Nf-fermion operators of the form OU (1)(x) ∼ detstqs(x)1+γ25qt(x),
where s, t are flavour indices (while the colour indices can be contracted in different possible ways, so to give a colour singlet), so that they are invari-ant under the whole chiral group except for the U (1)A transformations. In this work, we shall also consider global U (1) axial condensates, taking the functional average of multilocal operators of the form OU (1)(x1, . . . , xNf) ∼
εi1...iNf
q1(x1)1+γ2 5qi1(x1) . . . qNf(xNf)
1+γ5
2 qiNf(xNf) (i.e., performing a "point
splitting" of the Nf quark bilinears contained in the expression of the local operator OU (1)(x)), and then integrating over the four-space coordinates.
The thesis is organized as follows.
INTRODUCTION 7 we shall discuss the global symmetries of the theory and the spontaneous breaking of the chiral symmetry, the axial anomaly and the special role of the instantons, pointing out, in particular, the relation between the topolog-ical charge and the zero modes of the Dirac operator (i.e., the Atiyah-Singer theorem). Moreover, since in this thesis we shall often mention and com-ment several numerical lattice results, we shall also briefly resume the basic aspects of the lattice formulation of QCD.
In chapter 2 we shall present the theory at finite temperature: in par-ticular, we shall discuss the chiral phase transition and the current under-standing of the U (1)A symmetry’s fate above Tc, introducing the chiral
sus-ceptibilities and the local U (1) axial condensates.
In chapter 3, after a systematic and critical review of the results which can be obtained by analysing the chiral susceptibilities using the spectral-density technique (for different possible choices of the spectral spectral-density), we shall apply this same technique to study also the above-mentioned U (1) ax-ial condensates and their relations with the chiral susceptibilities, as well as with the so-called topological susceptibility χtop (which is, essentially, the zero-momentum two-point correlation function of the topological charge den-sity). In this context, we shall introduce also the new global U (1) axial con-densates mentioned above.
In chapter 4, finally, we shall explicitly compute the U (1) axial conden-sates in the high-temperature phase, using the instanton-background approx-imation of the functional integral. In this way, besides proving that these condensates are indeed different from zero in the high-temperature regime, we shall also derive their asymptotic temperature dependence and compare it with that of the topological susceptibility χtop.
In chapter 5 we shall summarize our results and draw our conclusions, discussing also some possible future developments of this work.
Chapter 1
Quantum Chromodynamics
1.1
QCD Lagrangian
Quantum Chromodynamics (QCD) is the quantum field theory of strong interactions. It is a non-Abelian gauge theory with gauge group SU (3)c,
where c stands for colour, the associated quantum number. The classical QCD Lagrangian density in Minkowski space is
L = q(i /D − M )q − 1
2Tr(FµνF
µν). (1.1)
The matter field q is a vector in flavour space: it consists of six quark fields qf, corresponding to different flavour states f (up, down, strange, charm,
bottom and top). Furthermore, each field qf is a vector in colour space,
namely it has three different components qfα (we will use greek letters for colour indices) and transforms according to the fundamental representation of SU (3)c:
SU (3)c: qf → Ω(x)qf, (1.2)
with
Ω(x) = eiθa(x)Ta, (1.3)
where θa(x) are the parameters of the local gauge transformation, while Ta are the SU (3)c generators and thus satisfy the commutation rule
[Ta, Tb] = ifabcTc, (1.4)
fabc being the SU (3) structure constants. The generators also obey to the
following normalization condition:
Tr[TaTb] = 1
2δab. (1.5)
10 CHAPTER 1. QUANTUM CHROMODYNAMICS Moreover, M = diag(mu, md, ms...) is the quark mass matrix and
/
D ≡ γµDµ (1.6)
is the Dirac operator in Minkowski space. The covariant derivative Dµ is defined as
Dµ= ∂µ+ igAµ, (1.7)
where g is the QCD coupling constant and Aµ is the matrix gluon gauge field : Aµ= 8 X a=1 AaµTa, Aaµ being the gluon gauge fields.
As concerns the pure gauge term, Fµν is the field density tensor :
Fµν = DµAν− DνAµ= ∂µAν− ∂νAµ+ ig[Aµ, Aν] = 8 X a=1 Fµνa Ta, where Fµνa = ∂µAaν− ∂νAaµ− gfabcAbµAcν.
In order for the Lagrangian (1.1) to be gauge invariant, we must require, besides Eq. (1.2), the following transformation laws under the gauge group:
Aµ→ ΩAµΩ†+
i
g(∂µΩ)Ω
†,
Fµν → ΩFµνΩ†,
where Ω is given by Eq. (1.3).
1.2
Properties of QCD
The quark model was proposed by Gell-Mann and Ne’eman in the early 60’s [1] and succeeded in explaining the hadronic spectrum as the multiplet struc-ture of an approximate flavour symmetry: the Gell-Mann SU (3). However, beside other problems, the model could not explain why we do not observe in Nature neither quarks nor bound states belonging to the sextet of SU (3). Therefore, a new quantum number, named colour, was introduced and the confinement postulate was proposed: that any hadronic state must be in-variant under the colour SU (3)c group, i.e. it must be a colour singlet. As a
1.2. PROPERTIES OF QCD 11 consequence, the quarks are confined into hadrons and cannot be observed as asymptotic states.
Another important property of QCD is the so-called asymptotic freedom: strong interactions become weaker as energy increases (i.e., as distance de-creases). This property can be understood in the framework of perturbation theory. Indeed, the β-function
β ≡ µ∂gR
∂µ , (1.8)
where gRis the renormalized coupling constant and µ is the renormalization
scale, has the following expansion in powers of gR: β = −β0g3R− β1gR5 + O(gR7),
where the first coefficient β0 is given by
β0= 1 (4π)2 11 3 Nc− 2 3Nf, (1.9)
Nc and Nf being the number of colours and flavours, respectively. Given
that Nc= 3, β0 is positive as long as Nf < 17. Substituting this expression
into Eq. (1.8) and solving for gR(µ), one finds that:
gR2(µ) = 1 2β0ln(ΛQCDµ )
, (1.10)
where the constant of integration ΛQCD ' 250 MeV (in the M S
renormal-ization scheme) is the QCD scale parameter [2]. Therefore, identifying the renormalization scale µ with the energy scale E of a given physical process, one finds that the so-called running coupling constant gR(E)
asymptoti-cally vanishes at high energies. On the contrary, as energy is decreased, i.e. at larger distances, gR becomes O(1) and perturbation theory is no longer valid (infrared slavery). This behaviour provides an heuristic explanation of confinement: as one tries to move the quarks away from each other, the interaction between them becomes stronger and stronger.
If there were no colour symmetry, as in QED, the coefficient β0 would be
negative, being Nc= 0, and the theory would behave in the opposite way:
then one could use perturbation theory to investigate the low-energy regime. Instead, the fact that QCD is strong interacting at small energies results in the necessity of a nonperturbative approach, such as the lattice formulation, which will be briefly described in section 1.7.
12 CHAPTER 1. QUANTUM CHROMODYNAMICS
1.3
Global symmetries
The Lagrangian (1.1) is invariant under global transformations of the group ˆ
G = U (1)u⊗ U (1)d⊗ U (1)s⊗ ... ⊗ U (1)t, (1.11)
under which the various quark flavours transform as ˆ
G : qf → eiαfqf,
the associated Noether currents being Jfµ = qfγµqf. Now, in the limit
in which Nf1 quark masses are sent to zero (Nf = 2 and Nf = 3 being
the physically relevant cases), the symmetry group enlarges. In order to understand this special limit, it is convenient to rewrite the Lagrangian (1.1) by separating the left and right chiral components of the quark fields:
qL≡
1 + γ5
2 q, qR≡
1 − γ5
2 q, (1.12)
where we define γ5 ≡ −iγ0γ1γ2γ3 = diag(I2, −I2). Indeed, the Lagrangian
can be expressed as:
L = iqLDq/ L+ iqRDq/ R− qLM qR− qRM qL−
1
2Tr[FµνF
µν]. (1.13)
Without mass terms, the two chiral components are completely decoupled and the Lagrangian (1.13) is invariant under independent rotations of qLand
qR. Therefore, in this so-called chiral limit, the new symmetry group is
G = U (Nf)L⊗ U (Nf)R, (1.14)
known as chiral group. The action of G on the quark fields is the following: G : qL→ ˜VLqL, qR→ ˜VRqR, (1.15)
where ˜VL, ˜VR∈ U (Nf) = U (1) ⊗ SU (Nf) and they can be written in terms
of the generators Ta of SU (Nf) as ˜ VL= eiαLeiθ a LT a ≡ eiαLV L, V˜R= eiαReiθ a RT a ≡ eiαRV R.
Any SU (Nf)L⊗ SU (Nf)R transformation can be expressed as the
compo-sition of a vector and an axial one, the first having VL = VR ≡ V , while
1
In this work we shall concentrate on the light flavours only, as the others are not relevant for our discussion.
1.4. CHIRAL SYMMETRY BREAKING 13 the second has VL = VR† ≡ A. The same is true also for the subgroup
U (1)L⊗ U (1)R, and in this case the parameters αV and αA of the vector
and axial transformations, respectively, are related to αR and αLas follows: αV =
αL+ αR
2 , αA=
αL− αR
2 .
Therefore, the chiral group G can be expressed as
G = U (1)V ⊗ U (1)A⊗ SU (Nf)V ⊗ SU (Nf)A, (1.16)
and its action on the quark fields is the following: U (1)V : q → eiαVq, U (1)A: q → eiαAγ5q, SU (Nf)V : q → eiθ a VTaq, SU (Nf)A: q → eiθ a AT aγ 5q. (1.17)
Finally, to each generator of G we can associate a Noether current: U (1)V : Jµ= qγµq, U (1)A: J5µ= qγ µγ 5q, SU (Nf)V : Vaµ= qγµTaq, SU (Nf)A: Aµa = qγµγ5Taq, (1.18)
and the corresponding conserved charge is defined as Q = R d3x J0(x). At the classical level, these currents are all conserved in the chiral limit. If we turn on the quark masses, almost all the symmetries (1.17) are explicitly broken. Only the subgroup U (1)V is an exact symmetry of the complete
Lagrangian (1.13), being U (1)V ⊂ ˆG, and its conserved charge QU (1)V is related to the barionic number. Furthermore, looking at Eq. (1.13), it is easy to realize that if the mass matrix M is a multiple of the identity, which means that the light quark masses are set all equal, then also SU (Nf)V is
still a symmetry: this is just the isospin symmetry for Nf = 2 or Gell-Mann
SU (3) for Nf = 3. Conversely, the axial part is always explicitly broken
when the quark masses are different from zero.
1.4
Chiral symmetry breaking
A continuous global symmetry of the classical Lagrangian can be imple-mented, in the corresponding quantum theory, in two different ways, which lead to different properties of the particle spectrum.
14 CHAPTER 1. QUANTUM CHROMODYNAMICS • A symmetry is exact, or realized à la Wigner-Weyl, if it is implemented, as in quantum mechanics, by a unitary operator U = eiθaQa acting on the Hilbert space H, Qa being the quantum Noether charges. The vacuum is invariant:
U |Ωi = |Ωi, (1.19)
and therefore annihilated by the Noether charges:
Qa|Ωi = 0 ∀a. (1.20)
As a consequence, the particle spectrum is organized in multiplets cor-responding to the irreducible representations of the symmetry group. • Symmetries can also be spontaneously broken à la Nambu-Goldstone.
In this case the vacuum is no longer invariant, that is, there exist charges which do not annihilate the vacuum:
∃Qa t.c. Qa|Ωi 6= 0,
which are referred to as broken generators.2 The unbroken generators, which continue to annihilate the vacuum, generate a subgroup H, re-alized à la Wigner-Weyl. The original symmetry group G is said to be broken down to the subgroup H.
The Goldstone’s theorem asserts that for each broken generator a mass-less spin zero particle, called Goldstone boson, appears in the spectrum, with the same quantum numbers as those of the broken generator. The hypothesis is that the corresponding Noether current is conserved not only classically, but also at the quantum level.
1.4.1 Phenomenological motivation
It is easy to see, by using a phenomenological reasoning, that the SU (Nf)V⊗
SU (Nf)A chiral symmetry must be broken. If this were not the case, then
for each hadron |hi there would exist another state |h0i ≡ Qa
A|hi belonging
to the same multiplet, i.e. having the same mass (since QaA commutes with the Hamiltonian), but with opposite parity. Indeed, if
P |hi = ηh|hi,
2Actually the definition of Qa as an integral over space requires carefulness. Indeed,
the norm of the state Qa|Ωi is not well-defined and it is necessary to introduce a regulator. We will not go into detail in this regard.
1.4. CHIRAL SYMMETRY BREAKING 15 then, being QaAan axial charge, it follows that
P |h0i = P QaAP†P |hi = −ηh|hi.
Given that these states are not observed in Nature, the symmetry group must be broken. But since hadrons do organize in approximate SU (Nf)V
multiplets, this subgroup must be realized à la Wigner-Weyl. Therefore, the symmetry-breaking pattern is
SU (Nf)R⊗ SU (Nf)L→ SU (Nf)V. (1.21)
We recall that the chiral group G (1.16) is a true symmetry only in the chiral limit, but since the quark masses are not strictly zero, the hadronic SU (Nf)V multiplets are not exactly degenerate and the pseudo-Goldstone
bosons related to the axial generators are not exactly massless. These last ones are identified with the Nf2 − 1 pseudoscalar mesons, which are much lighter than the others and have the correct quantum numbers.
In Nature mu, md, ms ΛQCD, which means that there is an approximate
SU (3)V ⊗ SU (3)A symmetry, resulting in the pseudoscalar mesonic octet of
pseudo-Goldstone light particles (π±,π0,K±,K0,K0,η). Finally, since it also holds that mu, md ms, the approximate SU (2)V ⊗ SU (2)A symmetry is
more precise and indeed the three pions are much lighter than the other particles of the octet.
1.4.2 Chiral condensate
The phenomenon of symmetry breaking is ruled by an order parameter that acquires a non-zero value when the breaking occurs. The natural order parameter of the chiral symmetry, known as chiral condensate, is defined as:
Σ ≡ hqqi, (1.22)
where the brackets stand for the vacuum expectation value. Assuming that SU (Nf)V is realized à la Wigner-Weyl, one finds that:
h[QAa, iqγ5Tbq]i =
1 Nf
δabΣ,
where the commutator is taken at equal times. Evidently, if the SU (Nf)A
charges annihilated the vacuum, then the chiral condensate would be zero. Chiral symmetry breaking is a fully nonperturbative phenomenon, since the
16 CHAPTER 1. QUANTUM CHROMODYNAMICS first order contribution to Eq. (1.22) from ordinary perturbation theory van-ishes in the chiral limit.
A non-zero chiral condensate not only indicates the SU (Nf)A
break-ing, but also implies that neither the U (1)A symmetry can be realized à la Wigner-Weyl. Indeed, let us assume that the U (1)A symmetry is
im-plemented in the Hilbert space by a unitary operator U = eiαQ5, where
Q5 =R d3x qγ0γ5q is the U (1) axial charge operator. Then the fields would
transform as follows:
U (1)A: q → q0 = U†qU = eiαγ5q, q → q0= U†qU = qeiαγ5. (1.23)
Using the vacuum invariance U |Ωi = |Ωi and the unitarity of U , one con-cludes that the chiral condensate would be invariant:
Σ = hqqi = hU†qqU i = hq0q0i. (1.24) But, from Eq. (1.23) it also follows that:
hq0q0i = e−2iαhqRqLi + e2iαhqLqRi,
which, for α = π2, implies Σ → Σ0= −Σ. Therefore, if the chiral condensate differs from zero, it cannot be invariant under the U (1)A symmetry, which
means that this last must be broken. Nevertheless, the breaking mechanism is not that of Nambu-Goldstone and will be illustrated in the next section.
1.5
U (1) axial anomaly
Consistently with the barionic number conservation, the U (1)V symmetry is realized à la Wigner-Weyl. On the other hand, the U (1)A symmetry must be broken. This follows from a phenomenological reasoning similar to that of section 1.4.1, with the only difference that |h0i ≡ Q5|hi, Q5 being the
U (1) axial charge. Moreover, as discussed above, the U (1)A-breaking follows
also from the fact that Σ is different from zero, being the chiral symmetry SU (Nf) ⊗ SU (Nf) broken.
Furthermore, the U (1)Asymmetry cannot be broken à la Nambu-Goldstone neither. Indeed, this would require the existence of another pseudoscalar light particle which should be massless in the chiral limit. In real QCD its mass mswould not be exactly zero and Weinberg [3], using chiral
perturba-1.5. U (1) AXIAL ANOMALY 17 tion theory, determined an upper limit for it:3
ms<
√
3mπ0, (1.25)
which is not satisfied by neither of the two candidates found in Nature, i.e. the η and η0 mesons.
1.5.1 The ABJ anomaly
The crucial point is that the fermionic measure is not invariant under the U (1)A symmetry, but is affected by the so-called Adler-Bell-Jackiw (ABJ)
anomaly [4, 5]. Under an infinitesimal axial transformation q → q0 = eiαγ5t
generated by t, where t is either one of the SU (Nf) generators or the identity
If in flavour space, the fermionic measure transforms as [2]:
Dq Dq → eiαR d4xAtDq Dq, (1.26)
where At is the anomaly function:
At(x) = −2 Tr[γ5t]δ4(x − x). (1.27)
In the path integral formalism, this exponent can be incorporated into the Lagrangian, and thus the non invariance of the measure can be understood as an explicit symmetry breaking term, driven by quantum effects. Its con-sequences in terms of conservation laws are explained below.
Since the delta-function diverges while the trace over Dirac indices is null, it is necessary to regularize the expression (1.27). This can be done, in a gauge-invariant manner, by inserting a differential operator f ( /D2x/M2) acting on the delta-function before taking its argument to zero [2, 6]:
At(x) = lim
M →∞y→xlim−2 Tr[γ5f ( /D 2
x/M2)t]δ4(x − y), (1.28)
where /Dxis the Dirac operator defined in Eqs. (1.6) and (1.7), acting on the
variable x, and f (s) is a smooth function with the following properties: f (0) = 1, f (∞) = 0;
sf0(s) = 0 for s = 0, ∞. (1.29)
3Actually for the smallest eigenvalue of the mixing mass matrix between the singlet
pseudo-Goldstone boson S, generated by U (1)A spontaneous breaking, and the eighth
18 CHAPTER 1. QUANTUM CHROMODYNAMICS With this regularization, the anomaly function can be expressed as [2]:
At(x) = −
g2
32π2µνρσF µν
a (x)Faρσ(x) Trft, (1.30)
where Trf indicates the trace over flavour indices. It is clear that the result is non-zero only for the singlet transformations U (1)A, generated by t = If,
since in the other cases the trace vanishes.
The anomaly results in the non conservation, at quantum level, of the current J5µdefined in Eq. (1.18), even in the chiral limit, i.e. in the absence of mass terms that explicitly break the symmetry. Indeed, it is possible to demonstrate that, in this limit:
∂µJ5µ= −A, (1.31) where A = −Nf g 2 32π2µνρσF µν a (x)Faρσ(x). (1.32)
Since the Noether current is not conserved at the quantum level, the Gold-stone theorem does not apply and therefore one does not expect in the spec-trum a massless particle resulting from the U (1)A breaking. Indeed, the singlet (η0 for Nf = 3) has a non-zero mass, even in the chiral limit, and is known as would be Goldstone, since it would be a Goldstone boson if the anomaly were suppressed. This is what happens in the limit of large Nc,
and the singlet mass can be explicitly computed to the first order in 1/Nc, through the Witten-Veneziano mechanism [7, 8].
1.5.2 The Atiyah-Singer theorem
In order to discuss further results, it is convenient to perform the analytic continuation to Euclidean space: x0 → −ixE0. We use Hermitian Euclidean
gamma matrices, which satisfy the condition {γµE, γEν}= 2δµν and are related to the ones in Minkowski space in the following way:
γ0E = γ0M,
γiE = −iγiM. (1.33)
Therefore, the Euclidean Dirac operator /
1.5. U (1) AXIAL ANOMALY 19 is anti-Hermitian. The Euclidean fifth γ matrix is just equal to the one in Minkowski space: γ5E ≡ −γE
0 γ1Eγ2Eγ3E = γ5M. Furthermore, we use the
convention E0123 = 1. From now on, we will always work in Euclidean space and omit the subscript E. With our conventions (we refer to the appendix A for more details), the left member of Eq. (1.31) acquires a factor i, when expressed in terms of the Euclidean variables, while the right one acquires a factor −i. Therefore, if we define
A = Nf g2 32π2µνρσF a µν(x)Fρσa(x), (1.35) it holds that: ∂µJµ5= −A. (1.36)
Finally, Eqs. (1.26)-(1.29) are unchanged in Euclidean space.
The anomaly function (1.35) is related to the topological charge density q, defined as: q ≡ g 2 16π2 Trc[FµνF˜µν], (1.37) with ˜ Fµν ≡ 1 2µνρσFρσ. Indeed, it is easy to see that:
A = 2Nfq. (1.38)
We note that this relation would have an extra minus sign in Minkowski space. Integrating q(x) over the four-space we obtain the topological charge:4
Q ≡ Z
d4x q.
It can be shown that Q is the gauge field winding number [9], which is an integer in Euclidean space. Furthermore, it is related to the zero modes of the Dirac operator through the Atiyah-Singer index theorem [10].
The eigenvalue equation for the Euclidean Dirac operator (1.34) can be expressed as:5
i /Duk= λkuk, (1.39)
4
With our conventions, qM → −iqE and QM → −QE
20 CHAPTER 1. QUANTUM CHROMODYNAMICS with real λk, being i /D Hermitian. Since Eq. (1.39) implicitly contains the
gauge field Aµ, the spinor eigenfunctions uk are vectors in colour space too.
They satisfy the following orthonormality and completeness relations: Z
d4x u†iαk (x)uiαk0(x) = δkk0, (1.40)
X
k
ui,αk (x)u†j,βk (y) = δ4(x − y)δijδαβ, (1.41)
where i, j and α, β are Dirac and colour indices respectively.
Another important property of the Dirac operator is that it anticommutes with the γ5 matrix:
{ /D, γ5} = 0. (1.42)
As a consequence, the non-zero eigenvalues come in pairs: for every eigen-function uk with eigenvalue λk 6= 0 there exists an eigenfunction u−k with
eigenvalue −λk:
u−k ≡ γ5uk,
i /Du−k = −λku−k.
(1.43) If k is such that λk6= 0, it is clear that uk and u−k are orthogonal, that is:
λk6= 0 ⇒
Z
d4x ukγ5uk= 0. (1.44)
The space integral of A, from Eq. (1.28) with t = If, assumes the following form [2]: 2NfQ = Z d4x A = −2Nf Z d4xX k u†kγ5uk, (1.45)
where the factor Nf in the right member comes from the trace of If. Now, from Eq. (1.44) it is clear that only the zero modes give a non-trivial contribution. Eq. (1.42) implies that the Euclidean Dirac operator commutes with γ5 on its kernel, where the two can be simultaneously diagonalized.
We can choose as simultaneous eigenfunctions the spinors φ+i , φ−i with γ5 eigenvalues +1 and −1 respectively:
i /Dφ+i = 0, γ5φ+i = φ + i ; i /Dφ−i = 0, γ5φ−i = −φ − i . (1.46) The first ones are referred to as left-handed zero modes, while second ones as right-handed. Then Eq. (1.45) becomes:
1.6. INSTANTONS 21 being n± the number of left/right-handed zero modes. This result is known
as Atiyah-Singer index theorem.
1.6
Instantons
It is known that the topological charge density q turns out to be a four-divergence, rendering the anomaly a boundary term. In fact, it is easy to demonstrate that: q = ∂µKµ, (1.48) where Kµ= g 2 8π µνρσTr c[Aν(∂ρAσ+ 2 3igAρAσ)]
is known as the Chern-Simons current. Therefore, one would expect the anomaly to have no physical effects. However, in a non-Abelian gauge the-ory, the anomaly does have physical effects, due to the existence of topologi-cally non trivial gauge configurations with finite action but non-zero winding number Q [11]. The most important among these are called instantons.
An instanton is a self dual configuration Fµν = ˜Fµν,
with topological charge Q = 1 and minimal action, which takes the value SI =
8π2
g2 . (1.49)
An anti-instanton has Fµν = − ˜Fµν and Q = −1, but the action is always
given by (1.49). ’t Hooft found [12, 13] the explicit form of the solution for a SU (2) gauge group, in terms of two parameters: the instanton centre x0 and its characteristic size ρ. The zero-centred solution is:
Aaµ(ρ; x) = 2η
a µνxν
x2+ ρ2, (1.50)
where ηµνa are the so-called ’t Hooft symbols
ηa00= 0, ηaij = aij, η0ia = −ηai0= −δai,
with i, j = 1, 2, 3. Since the problem is translationally invariant, by shifting x → x − x0 one can obtain a solution centred at x = x0. Moreover, it
22 CHAPTER 1. QUANTUM CHROMODYNAMICS is possible to demonstrate that all the gauge configurations with Q = 1 and minimal action can be represented in the form (1.50), up to rescalings, translations and gauge transformations [9]. Finally, in the case of three colours, one has to specify, besides the instanton size and its centre, also its orientation in SU (3).
We will discuss these issues in more detail in chapter 4.
1.7
Lattice formulation
A nonperturbative, gauge-invariant regularization of QCD can be imple-mented through the discretization of the Euclidean space; that is, introducing a finite lattice spacing a, which results in a finite ultraviolet cutoff ΛU V ∼ 1a
on the impulses and therefore removes all the divergences generated in per-turbation theory. This regularization was proposed by Wilson in 1974 [14], as a tool to study quark confinement in the strong-coupling regime. The physical, renormalized theory is then obtained through a continuum limit, where the lattice spacing a vanishes and all the correlation lengths ξ defined on the lattice diverge, in such a way that the physical quantities are kept finite. In other words, the statistical system defined on the lattice undergoes a second order phase transition as a → 0. The existence of this particular limit is guaranteed by the asymptotic freedom of QCD.
The lattice formulation of QCD [15] allows us to make Monte Carlo simulations, sampling discrete field configurations weighted with e−SQCD.
Obviously, this is possible only in Euclidean space. Although the work of this thesis is analytic and developed in the continuum, we recall a few aspects of the discretized theory, for we shall compare our results with those found in lattice simulations.
The QCD Lagrangian in the continuum, Euclidean space is L = q( /D + M )q + 1
2Tr(FµνFµν), (1.51)
and the QCD partition function Z is formally defined as: Z =
Z
DqDqDAµe−SQCD.
Since it is not possible to sample the Grassmann variables q and q, one has to perform the integration over the quark fields. Being /D + M the gauge-dependent fermionic quadratic form, Z becomes:
Z = Z
1.7. LATTICE FORMULATION 23 where SG = 12Tr(FµνFµν). At this point, one can sample discretized gauge
configurations, weighted with a probability det( /D + M )e−SG≥ 0. The
posi-tivity of the fermionic determinant is guaranteed, in the continuum,6 by the anti-hermiticity (1.39) of /D and by the relation (1.42) of anti-commutativity with γ5. From these properties it follows that non-zero eigenvalues come in
pairs, and therefore:
det /D =Y k iλk= 0 Y k λ2k, (1.53) whereQ0
k is restricted to the positive λk.
In order to implement the gauge symmetry on the lattice one has to in-troduce parallel transports from one site to another. These so-called links are the fundamental gauge variables of the discretized theory and are indi-cated as Uµ(n), where the n stands for the lattice site while µ indicates the
direction of the parallel transports. Then, one can define the elementary closed loop, the plaquette Πµν:
Πµν(n) = Uµ(n)Uν(n + ˆµ)Uµ†(n + ˆµ)U † ν(n).
On the lattice, the fields Aµ are replaced with the link variables Uµν(n) and
the pure gauge action SG is replaced with the Wilson action SW [14]:
SW[U ] = β X n X µ<ν 1 Nc < Tr(1 − Πµν(n)), (1.54) where β ≡ 2Nc g2 0
and g0(a) is the bare gauge coupling, defined on the lattice. What is more relevant to this thesis is how fermions and chiral symme-try are implemented on the lattice. This is a delicate issue, indeed if the fermionic action is discretized "naively", requiring that the Dirac operator on the lattice K has the same properties of the one in the continuum (i.e. it is anti-Hermitian and it anticommutes with γ5), new unphysical degrees of freedom appear in the continuum limit: each quark field exists in 16 copies, known as doublers. In order to solve this so-called doubling problem, Wilson proposed to modify the action by adding a term proportional to qq, which vanishes as a → 0 but explicitly breaks the chiral symmetry on lattice, in a way that the additional degrees of freedom are removed. The explicit chiral symmetry breaking results in the fact that K is no longer anti-Hermitian and
24 CHAPTER 1. QUANTUM CHROMODYNAMICS that {K, γ5} 6= 0.
The Nielsen-Ninomiya’s no-go theorem asserts that one cannot eliminate the doublers while maintaining the full chiral symmetry on the lattice. The best option is to reduce these additional degrees of freedom and at the same time preserve some residue of the original chiral symmetry, which will be fully re-established only in the continuum limit. This can be achieved in many ways, and many different fermionic actions have been implemented in the lattice simulations.
Chapter 2
Symmetry restoration
From now on we shall focus on the finite-temperature theory and on the issue of symmetry restoration, investigating, in particular, the fate of the U (1)A symmetry at high temperature T . In this chapter, after resuming
some aspects of the QCD phase transition, we will briefly review the current understanding of the question and introduce the local U (1) axial condensates. In order to take into account the finite-temperature effects, it is sufficient to substitute the vacuum expectation values with the quantum thermal av-erages:
hΩ|O|Ωi → hOiT =
Tr[e−βHO]
Tr[e−βH] , (2.1)
where H is the Hamiltonian. This amounts to restrict the set of field con-figurations included in the path integral to those periodic or antiperiodic in the Euclidean time (in the case of bosonic or fermionic fields, respectively), with period β = kT1 . From now on we will omit the subscript T .
In the following, we shall not consider real QCD with six flavours and three colours but leave Nf (which is here the number of light flavours) and Nc as
free parameters, although in many cases it will be convenient to set Nf = 2.
2.1
QCD phase transition
From lattice studies [16] it is known that, at a certain critical temperature Tc' 150 MeV, QCD undergoes a phase transition which restores the chiral
symmetry SU (Nf)V ⊗ SU (Nf)A: in other words, for T ≥ Tcthe chiral
con-densate (1.22), which is an order parameter of this symmetry, is zero in the chiral limit. The critical temperature Tc practically coincides with that of
26 CHAPTER 2. SYMMETRY RESTORATION the deconfinement transition, which separates the confined hadronic phase from the deconfined one, described at high T as a quark-gluon plasma. Instead, the fate of the U (1)A symmetry is not well understood. Although the anomaly, due to the non-invariance of the fermionic measure, is always present, it could (practically) cease to have effects on the correlation func-tions at some temperature TU (1), at which U (1)A would then be effectively restored. Moreover, if this were the case, we might also ask ourselves whether the two symmetries, U (1)Aand SU (Nf)A, are restored separately or
simul-taneously. The nature of the QCD transition at T = Tcstrongly depends on which of these possibilities is realized [17, 18].
We note that it must necessarily be TU (1) ≥ Tc: indeed, at temperatures
below Tc the chiral condensate Σ is different from zero, implying not only the chiral symmetry breaking but also, as discussed in subsection 1.4.2, the one of the U (1)Asymmetry.
The question has been studied mostly in the case of two flavours and the following scenarios have been considered in literature:
• If TU (1) Tc, or if the U (1)A symmetry is not restored at all (i.e. TU (1)→ ∞), the chiral restoration at T = Tcoccurs through a second-order phase transition with the O(4) critical exponent. Indeed, the symmetry breaking pattern SU (2)L⊗ SU (2)R→ SU (2)V is equivalent to O(4) → O(3) [17, 18].
• If TU (1) ≈ Tc the symmetry breaking pattern is U (2)L⊗ U (2)R → U (2)V. The transition may be of first-order [17] or second-order but
belonging to a universality class different from O(4)[18].
Therefore, a possibility to investigate the fate of the U (1)A symmetry near the chiral transition is through the study of its critical exponents. This is a controversial matter: indeed, some studies [19] find evidence of a O(4) scaling, and thus support the scenario in which the U (1)Asymmetry remains broken at T ≈ Tc, while others do not [20].
2.2
Is the U (1) axial symmetry restored?
The fate of the U (1)A symmetry in the chirally-restored phase is a long de-bated question. In 1984 Pisarski and Wilczek [17] argued that «if instantons themselves are the primary chiral symmetry breaking mechanism, then it is very difficult to imagine how the anomaly effects could be unsuppressed at T = Tc», these being due precisely to the instantons. Indeed, it is known
2.2. IS THE U (1) AXIAL SYMMETRY RESTORED? 27 that at high T instantons are suppressed by the so-called Pisarski-Yaffe sup-pression factor [21]:
dn(T, ρ) ∼ dn(T = 0, ρ)e−(πρT )23 (2Nc+Nf), (2.2)
where dn(T, ρ) is the instanton density and ρ is the instanton size. However, this suppression factor is absent below Tc and Eq. (2.2) is really applicable only for T & 2Tc, therefore it is not clear how this suppression could be strong enough to eliminate the anomaly already at T = Tc [22]. Moreover,
the qualitative picture of instanton-driven chiral symmetry restoration that is today accepted has significantly changed since 1984. The chiral transition seems to be driven not by an instanton suppression but by a "rearrange-ment" of the instantons, from a disordered liquid phase at low temperature to a strongly correlated chirally symmetric phase of polarized instanton-anti-instanton molecules [23]. Within this picture, the "strength" of the anomaly seems to be essentially unchanged near Tc [24].
A possibility to investigate the fate of the U (1)Asymmetry in two-flavour
QCD is through the two-point correlation functions hJM(x)JM† (0)i for
vari-ous (scalar and pseudoscalar) meson channels M [22, 25, 26]: Jσ = qq,
Jδa= qτaq, Jη = iqγ5q,
Jπa= iqγ5τaq,
(2.3)
where τa, with a = 1, 2, 3, are the Pauli matrices, normalized so as Tr τaτb =
2δab. The bilinear Jπa interpolates the three pions πa, while η is the Nf = 2
version of the singlet η0. All the operators defined in Eqs. (2.3) are Hermitian and they have all quark indices (Dirac, flavour and colour) contracted. In addition, one can define the currents Jπ±= iqγ5τ1√±iτ22q, which are no longer
Hermitian and have the quantum numbers of the charged pions π± (analo-gously, Jδ± = qτ1√±iτ2
2 q interpolate δ ±).
The importance of these correlators lies in the fact that under SU (2)A and U (1)A transformations the meson channels are mixed as follows:
σ ←→ U (1)A η SU (2)Al l SU (2)A π ←→ U (1)A δ (2.4)
28 CHAPTER 2. SYMMETRY RESTORATION For example, recalling Eq. (1.17), one can see that:1
U (1)A: Jσ = qq → qeiθγ5q = θ=π/2i(q † RqL− q † LqR) = iqγ5q = Jη, (2.5) and similarly U (1)A: Jδa= qτaq → θ=π/2iqτ aγ 5q = Jπa. (2.6)
As regards the special group, we refer to the appendix B.
The restoration of a certain symmetry results in the degeneracy between the correlation functions of the channels that are mixed under that symmetry. Therefore, the SU (2) ⊗ SU (2) restoration implies the π-σ and δ-η degenera-cies, while an eventual U (1) ⊗ U (1) restoration would yield also the π-δ and σ-η degeneracies, meaning that all the correlators would be identical (in the chiral limit). Moreover, it is convenient to introduce, for each meson channel M , the so-called chiral susceptibility, defined as:
χM = Z d4xhJM(x)JM† (0)i = 1 V Z d4xd4yhJM(x)JM† (y)i, (2.7)
where translational invariance has been used. Now, from our discussion it follows that the differences χπ− χσ and χπ− χδ behave as order parameters for the SU (2) ⊗ SU (2) and U (1) ⊗ U (1) symmetries, respectively.
It is instructive to express the U (1)A-breaking difference χπ− χδin terms of
the left-handed and right-handed quark fields [27, 28]. Let us consider the channels π+ and δ+:2 χδ+ = Z d4xhq(x)τ1√+ iτ2 2 q(x)q(0) τ1√− iτ2 2 q(0)i = 2 Z d4xhu(x)d(x)d(0)u(0)i = 2 Z d4xhu†R(x)dL(x)d†R(0)uL(0) + u†R(x)dL(x)d†L(0)uR(0) + R ↔ Li,
where u, d are the up and down quark fields. Similarly, we find χπ+ = Z d4xhq(x)iγ5 τ1√+ iτ2 2 q(x)q(0)iγ5 τ1√− iτ2 2 q(0)i = −2 Z d4xhu(x)γ5d(x)d(0)γ5u(0)i = −2 Z d4xhu†R(x)dL(x)d†R(0)uL(0) − u†R(x)dL(x)d†L(0)uR(0) + R ↔ Li.
1Here we adopt the notation q = (q
L, qR) in the chiral basis. 2
Actually, in chapter 3 we will demonstrate that the chiral susceptibilities in the isovec-tor channels are independent of a (i.e., the index of the flavour generaisovec-tors).
2.2. IS THE U (1) AXIAL SYMMETRY RESTORED? 29 Therefore, the following identity holds:
χπ+ − χδ+ = −4
Z
d4x hu†RdL(x)d†RuL(0) + u†LdR(x)d†LuR(0)i. (2.8)
Below Tc, the left-handed and right-handed components of a given light quark flavour can be connected through the chiral condensate Σ, rendering the integrand of Eq. (2.8) different from zero (even though it vanishes in perturbation theory). Above Tc, Σ vanishes in the chiral limit: therefore, in order for χπ − χδ to be non-zero, one should require the existence of a four-fermion local condensate which would be an order parameter for the U (1) axial symmetry and remain different from zero also above Tc [29–31].
The properties of this condensate are discussed in the next section.
Finally, let us briefly review the progress achieved both from a numerical and an analytical perspective.
• Lattice approach: In the last decades the correlators and suscepti-bilities defined above have been extensively studied on the lattice, but the results achieved so far are not yet conclusive. While the π-σ degen-eracy for T ≥ Tc has been always confirmed [25, 26, 32], the π-δ one is
controversial. Indeed, most of the groups find that the U (1)A-breaking
difference χπ− χδ is still non-zero above Tc[32–38], and, according to some of them [34, 37], it eventually goes to zero at a higher temper-ature TU (1) ' 1.3Tc ' 195 MeV; but other studies report an effective
U (1)A restoration [39–41] already at Tc.3
The use of different fermion actions (with different residue chiral sym-metries of the discretized theory, as discussed in section 1.7) may pro-vide an explanation for these conflicting outcomes, as pointed out, for example, in [36].
• Analytical approach: The problem has been studied also from an analytical point of view [42–47], mostly in the case Nf = 2. In 1996 Cohen [42] supported a U (1)Aeffective restoration at T = Tc, starting
from the same considerations as reference [17]: «since the same mecha-nism, namely instantons, is responsible both for SU (2) ⊗ SU (2) chiral symmetry breaking and for allowing the anomaly to have physical con-sequences»then in the chirally restored phase the U (1)Ahas to be
effec-tively restored, because «the instanton effects are turned off»[42]. As
3Most of the studies set N
f = 2. In some cases it has been investigated the Nf = 2 + 1
case, i.e. the up and down are massless (actually it is necessary an extrapolation to the chiral limit) while the strange is left massive.
30 CHAPTER 2. SYMMETRY RESTORATION argued above, this reasoning is misleading, since the chiral restoration does not imply that instantons are suppressed already at Tc [22, 23].
Moreover, Cohen’s considerations rely on an assumption, indeed he states that: «if the QCD functional integral does not get contributions from a set of zero measure, then the high-temperature chirally restored phase of QCD is effectively symmetric under U (2) ⊗ U (2) rather than SU (2) ⊗ SU (2)».
Conversely, in references [44, 45] it is claimed that, above the transi-tion, the anomaly does have physical effects, but only on the n-point correlators of quark bilinears JM with n ≥ Nf, which is confirmed also
by reference [46].
2.3
Local U (1) axial condensate
A non-zero chiral condensate Σ is a sufficient condition for the U (1)A -breaking: if Σ 6= 0 the U (1)A symmetry is broken. However, there is no
obvious reason for the reverse implication to be true. Therefore, it is natural to introduce a different condensate which is an order parameter only for the U (1)A symmetry. This was done by Meggiolaro in references [29–31], in the
context of a chiral effective Lagrangian model. The desired quantity must be invariant under all the chiral group except for the U (1)Asymmetry: the flavour determinant
OU (1) ∼ det
st qs
1 + γ5
2 qt + h. c. = detst [qsRqtL] + detst [qsLqtR], (2.9)
where s, t are flavour indices (while the colour indices can be contracted in different possible ways), satisfies this criterion. This quantity was initially proposed by ’t Hooft [12, 13] as an effective vertex of quark interaction due to an instanton. Recalling Eq. (1.15), we deduce that the operator (2.9) transforms under the chiral group U (Nf)L× U (Nf)R as
det st qs 1 + γ5 2 qt + h. c. −→ (det VR) ∗ det VLdet st qs 1 + γ5 2 qt + h. c., being VL, VR elements of U (Nf). Evidently, this quantity is invariant under
the SU (Nf)L⊗ SU (Nf)R⊗ U (1)V transformations, but not under the U (1)A
ones. Indeed, in this last case, from Eq. (1.17) one derives that: (det VR)∗det VL= e−2iNfα.
Furthermore, the operator OU (1) must be a colour singlet: that is, the 2Nf quark fields in Eq. (2.9) must have colour indices properly contracted. The
2.3. LOCAL U (1) AXIAL CONDENSATE 31 possible contractions are Nf, so that the most general operator, for an
ar-bitrary number of flavours, is extremely complicated. However, for Nf = 2 and arbitrary Nc, it is possible to demonstrate that OU (1) has the following general form [31]: OU (1)= Fβδαγstqα11 + γ5 2 q β sq γ 2 1 + γ5 2 q δ t + h. c., (2.10) with Fβδαγ ≡ Aδαβδδγ+ Bδδαδβγ, (2.11) where A and B are numerical constants and Greek letters corresponds to colour indices. The tensor Fβδαγ satisfies the relation
Fβδαγ = Fδβγα, (2.12)
which will be useful in the following. The local U (1) axial condensate4 is then defined as
CU (1)≡ hOU (1)i, (2.13)
and constitutes an order parameter of the U (1)A symmetry. We observe that, for Nf = 2, this object is precisely the four-fermion condensate needed
in order for the U (1)A-breaking difference (2.8) to be different from zero. Let us make a few comments on the structure of Eq. (2.10) in the space of Dirac and colour indices. Each quark bilinear has Dirac indices separately contracted, which we have omitted. Instead, the two terms in Eq. (2.11) represent the two possibilities of contracting colour indices: the first, with δαβδδγ, is such that the Dirac and colour indices are contracted in the same way; conversely, the one with δδαδβγ, is such that they are contracted in the opposite way. In the general case of Nf light flavours we have Nf possibil-ities of contracting the colour indices, and thus Nf corresponding terms in the tensor Fβα1...αNf
1...βNf .
Let us also observe that, just because of its more complicate structure in the space of Dirac and colour indices, the second contraction (i.e., the one with δδαδβγ) cannot be expressed in the simple form of Eq. (2.9), and there-fore we need to explicitly prove that the operator (2.10) has the desired transformation properties, i.e. it transforms as the determinant (2.9).
4
Actually, because of the different possibilities of colour contraction, one could talk about more condensates.
32 CHAPTER 2. SYMMETRY RESTORATION Chiral transformations properties
Under the chiral group, recalling Eqs. (1.15), one finds that OU (1)transforms as
OU (1)= Fβδαγstqα1RqsLβ qγ2RqtLδ + h. c. −→
Fβδαγst(VRf 1)∗VLsg(VRf02)∗VLtg0qαf RqgLβ qγf0Rqgδ0L+ h. c.,
which, by using that stVLsgVLtg0 = det VLgg0, can be rewritten as follows:
OU (1) −→ Fβδαγdet VLgg0(Vf 1 R ) ∗ (VRf02)∗qαf RqgLβ qγf0Rqδg0L+ h. c. Now, since f f0(Vf 1 R ) ∗(Vf02 R ) ∗ = (det V
R)∗, in order to prove that the operator
OU (1) actually transforms as the determinant (2.9), that is
OU (1)−→ det VL(det VR)∗OU (1), (2.14)
we just need to demonstrate that Fβδαγgg0qαf Rqβ gLq γ f0Rqgδ0L= Fβδαγf f0stqα1Rqβ sLq γ 2RqδtL.
Making the sum over flavour indices explicit, this expression becomes: Fβδαγ(qαf Rqβ1Lqγf0Rqδ2L− qαf Rq β 2Lq γ f0Rq1Lδ ) = Fβδαγf f0(qα1Rqβ 1Lq γ 2Rq δ 2L− qα1Rq β 2Lq γ 2Rq δ 1L). (2.15)
Now, for f = 1 and f0 = 2 the identity (2.15) is trivial. For f = f0 we can use the property (2.12) to prove that the first member is zero: indeed, the second term at the right side of the equation is just equal to
Fβδαγqαf Rqβ2Lqf Rγ q1Lδ = Fδβγαqγf Rq2Lδ qαf Rqβ1L= Fβδαγqαf Rq1Lβ qγf Rq2Lδ .
Finally, using again Eq. (2.12), it is easy to prove the identity (2.15) also for f = 2, f0 = 1.
In conclusion, the local operator OU (1)transforms just as Eq. (2.14): that is, it is invariant under all the chiral group except for the U (1)Atransformations.
Chapter 3
Spectral-density analysis
In this chapter we shall study the chiral susceptibilities and the U (1) axial condensates by means of the spectral density of the Euclidean Dirac oper-ator, adopting a fully nonperturbative approach. Besides the local U (1)A condensate introduced in the previous chapter, we shall also define a new "global" condensate that will be of great importance to this thesis.
3.1
Euclidean Dirac spectral density
The anti-Hermitian Euclidean Dirac operator is defined as /D[A] = γµEDµ[A],
through Eqs. (1.34) and (1.33). In section 1.5.2 some of its properties have been described: in particular, the anticommutative relation (1.42) with γ5 and its consequence on the Dirac spectrum, i.e. that its non-zero eigenvalues come in pairs. In a finite four-volume V the spectrum is discrete and the spectral density ρ(λ) is defined as [9]
ρ(λ) ≡ h1 V
X
k
δ(λ − λk[A])i, (3.1)
where λkis the set of background-dependent (that is, dependent on the gauge
field A, which is here considered as an external background) real eigenvalues of i /D and the variable λ is usually referred to as virtuality. Here the brackets stand for the following functional integral over the gauge configurations:
h. . .i = 1 Z
Z
DAe−SGdet( /D[A] + m)Nf(. . .),
where SGis the pure gauge action and m is the quark mass, which we assume
equal for all the quark flavours (the chiral limit will be taken later on). From 33
34 CHAPTER 3. SPECTRAL-DENSITY ANALYSIS the properties of the Dirac spectrum it follows that:
ρ(λ) = ρ(−λ). (3.2)
A useful relation, which allows us to relate various quantities to the spectral density, is the following decomposition of the fermionic propagator GA(x, y) in an external field A:
GAij,αβ(x, y) ≡ hqi,α(x)qj,β(y)iA=
X
k
ui,αk (x)u†j,βk (y) m − iλk
, (3.3)
where α, β and i, j are colour and Dirac indices respectively, while flavour indices are omitted and the identity in flavour space is implicit in the third member. The brackets h. . .iAstands for the path integral on fermionic fields only, A being an external source. We note that, since we are not integrating over the gauge fields, the propagator (3.3) is not translationally invariant. Let us verify that Eq. (3.3) is actually the quark propagator (in an external field): ( /D + m)ij,αβX k uj,βk (x)u†l,γk (y) m − iλk =X k
(m − iλk)ui,αk (x)u †l,γ
k (y)
m − iλk
=X
k
ui,αk (x)u†l,γk (y) = δ4(x − y)δilδαγ,
where we have used the completeness relation (1.41).
From the properties (1.42) and (1.43) of the Dirac operator one can derive the following relation for the propagator GA:
γ5GA(x, y)γ5= GA†(y, x). (3.4)
To prove this identity, it is sufficient to use the fact that the spinor function γ5uk has Dirac eigenvalue −λk: indeed,
γ5GA(x, y)γ5 = X k γ5uk(x)(γ5uk(y))† m − iλk =X k uk(x)uk(y)† m + iλk = GA†(y, x).
3.2. CHIRAL CONDENSATE 35
3.2
Chiral condensate
In this section we derive, following reference [9], the Banks-Casher relation [48], which relates the chiral condensate Σ (1.22) to the spectral density at zero virtuality (λ = 0), in the chiral limit. We recall that, in our conventions:
Σ = hqqi = Nf X f =1 hqfqfi, and therefore hqf(x)qg(y)i = δf g Σ Nf .
Using the translational invariance of the full quark propagator, we can ex-press the chiral condensate as
Σ = 1 V
Z
d4x hq(x)q(x)i. (3.5)
Let us perform the integration over the fermionic fields only, considering A as an external field, and substitute the background-dependent propagator (3.3):
Σ = h1 V
Z
d4x Tr GA(x, x)i,
where the brackets now stands for the integration over the gauge fields with the weight e−SGdet( /D[A]+m)Nf and the trace Tr is understood over flavour,
Dirac and colour indices. Since we are assuming equal quark masses, the first trace results in an overall factor Nf. Using the orthonormality relation (1.40) with k = k0 and the definition (3.1), we obtain
Σ = Nfh 1 V Z d4xu i,α k (x)u †i,α k (x) m − iλk i = Nfh 1 V X k 1 m − iλk i = Nfh 1 V Z ∞ −∞ dλX k δ(λ − λk) m − iλ i = Nf Z ∞ −∞ dλ ρ(λ) m − iλ.
Making use of the parity (3.2) of ρ(λ), we can express Σ as Σ = 2Nfm Z ∞ 0 dλ ρ(λ) m2+ λ2 = 2Nf Z ∞ 0 dz ρ(mz) 1 + z2, (3.6)
36 CHAPTER 3. SPECTRAL-DENSITY ANALYSIS where we have changed the integration variable to λ = mz. Now we under-stand how this nonperturbative condensate develops for T < Tc: the integral
in Eq. (3.6) acquires a contribution coming from the region of small λ which survives the chiral thermodynamic limit (i.e. first V → ∞ and then m → 0), yielding the famous Banks-Casher relation [48]:
Σ = Nfπρ(0). (3.7)
Therefore, ρ(λ = 0) differs from zero below Tc and vanishes in the
chirally-restored phase, in a sense that will be specified later on.
Finally, let us observe that, in a finite volume, the chiral condensate is always zero in the chiral limit, because, being the spectrum discrete, there cannot be an accumulation of eigenvalues near λ = 0.
3.3
Chiral susceptibilities
The chiral susceptibilities have been introduced in the previous chapter, through Eqs. (2.3) and (2.7). We have already pointed out their relevance to the symmetry restoration, as χπ−χσand χπ−χδbehave as order parameters
of SU (2) ⊗ SU (2) and U (1) ⊗ U (1), respectively. In this section we consider the case of Nf = 2 light flavours and express the chiral susceptibilities of all the various meson channels through the spectral density ρ, as we did for the chiral condensate. The relations that we shall derive can be also found, for example, in references [35, 36, 49].
Isoscalar (I = 0) scalar channel
The chiral susceptibility in the isoscalar scalar channel Jσ = qq,
correspond-ing the meson σ, is χσ = 1 V Z d4x Z d4yhJσ(x)Jσ†(y)i. (3.8)
After integrating over the quark fields, we obtain the two following Wick contractions:
hJσ(x)Jσ†(y)i = hqi,αf (x)qi,αf (x)qj,βg (y)qgj,β(y)i
= hTr GA(x, x) Tr GA(y, y)i − hTr[GA(x, y)GA(y, x)]i,
(3.9)
where we have explicitly written all the index: f, g are flavour, i, j Dirac and α, β colour indices. The brackets in the right member, as in the definition
3.3. CHIRAL SUSCEPTIBILITIES 37 (3.1) of the spectral density, stand for the integration over the gauge fields, and the trace is understood over all the indices. Let us define the connected and disconnected components of χσ, with respect to the fermionic lines in an external field (that is, before averaging over the gauge fields), as follows:
χσ,conn= − 1 V Z d4x Z
d4yhTr[GA(x, y)GA(y, x)]i,
χσ,disc= 1 V Z d4x Z
d4yhTr GA(x, x) Tr GA(y, y)i.
(3.10)
Evidently, the total chiral susceptibility χσ is given by
χσ = χσ,conn+ χσ,disc. (3.11)
Now, substituting the definition (3.3) of the background propagator, we can easily express the connected term in terms of the spectral density, in the same way as we did for the chiral condensate in the previous section.
• Connected term −χσ,conn= 1 V Z d4x Z
d4yhTr[GA(x, y)GA(y, x)]i
= 2 Vh
Z
d4x d4yX
k
ui,αk (x)u†j,βk (y) m − iλk X k0 uj,βk0 (y)u†i,αk0 (x) m − iλk0 i = 2 Vh X k 1 (m − iλk)2 i = 2 V Z ∞ −∞ dλhX k δ(λ − λk) (m − iλ)2i = 2 Z ∞ −∞ dλ ρ(λ) (m − iλ)2 = 4 Z ∞ 0 dλρ(λ) m 2− λ2 (m2+ λ2)2 ⇒ χσ,conn= −4 Z ∞ 0 dλρ(λ) m 2− λ2 (m2+ λ2)2, (3.12)
where we have used the normalization (1.40) of uk and the parity of ρ(λ), i.e. Eq. (3.2). The factor 2 in the second passage comes from the trace over flavour indices, being Nf = 2.
• Disconnected term
38 CHAPTER 3. SPECTRAL-DENSITY ANALYSIS for the disconnected part, indeed:
χσ,disc= 1 V Z d4x Z
d4yhTr GA(x, x) Tr GA(y, y)i
= 4 Vh
Z
d4x d4yX
k
ui,αk (x)u†i,αk (x) m − iλk X k0 uj,βk0 (y)u†j,βk0 (y) m − iλk0 i = 4 Vh X k 1 m − iλk X k0 1 m − iλk0 i, (3.13)
where the factor 4 = Nf2 comes from the double trace over flavour indices. Using Eq. (1.40), we still remain with two independent sum-mations over k and k0 which cannot be directly written in terms of ρ(λ). Nevertheless, we are able to relate χσ,disc to a derivative of the
spectral density. Indeed, the total susceptibility χσ is related to the chiral condensate Σ as follows:
χσ =
∂Σ ∂m+ V Σ
2, (3.14)
where we have used that
Σ = 1 V
∂ ln Z
∂m . (3.15)
Since Σ = hJσ(0)i (see the first Eq. (2.3)), Eq. (3.14) can be rewritten as
Z
d4x[hJσ(x)Jσ(0)i − hJσ(x)ihJσ(0)i] =
∂Σ ∂m.
Now, it is convenient to redefine the chiral susceptibility as the left member of this expression, i.e. subtracting the disconnected product of the two expectation values R d4xhJσ(x)ihJσ(0)i:1
χσ ≡ Z
d4x[hJσ(x)Jσ(0)i − hJσ(x)ihJσ(0)i], (3.16)
so that
χσ = χσ− V Σ2=
∂Σ
∂m. (3.17)
1The chiral susceptibility defined this way is usually referred to as "connected", with
respect to the full theory. Here we shall use the term "connected" as in Eq. (3.10), i.e. with respect to the fermionic lines in an external field Aµ.
3.3. CHIRAL SUSCEPTIBILITIES 39 We also define the connected and disconnected components of χσ as
χσ,conn= χσ,conn,
χσ,disc= χσ,disc− V Σ2.
However, above Tc and in the chiral limit, the chiral condensate Σ vanishes and the two definitions become equivalent, i.e. χσ = χσ.
For convenience, in the following we shall continue to use the notation χσ. It is understood that, in order for the relations below to be true
in general, χσ should be defined subtracting V Σ2, as in Eq. (3.16).
Substituting Eq. (3.6) for Σ into Eq. (3.17) (and using the notation χσ
instead of χσ), we obtain χσ = ∂Σ ∂m = 4 Z ∞ 0 dλρ(λ, m) λ 2− m2 (m2+ λ2)2 + 4 Z ∞ 0 dλ∂ρ(λ, m) ∂m m m2+ λ2,
where the dependence of ρ on the quark mass m has been pointed out. This dependence comes from the weight det( /D + m)2 in the functional integral over the gauge fields. Since the first term is just equal to the connected susceptibility (3.12), it follows that:
χσ,disc= 4 Z ∞ 0 dλ∂ρ(λ, m) ∂m m m2+ λ2. (3.18)
Let us note that, in the following, we shall use these expressions to study the chiral restoration, which takes place in the high-temperature phase and only as m → 0: in this context, it actually makes no differ-ence which of the two definitions, (3.8) or (3.17), is used.
Isovector (I = 1) scalar channel
The chiral susceptibility in the isovector scalar channel Jδa = qτaq, corre-sponding to the meson δa, is
χδa = 1 V Z d4x Z d4yhJδa(x)(Jδa)†(y)i,
where there is no summation over a. In this case the disconnected term vanishes, because it is proportional to the trace of τa, which is null. This
40 CHAPTER 3. SPECTRAL-DENSITY ANALYSIS follows from assuming equal quark masses, so that GA is the identity in
flavour space. The integration over the quark fields yields: hJδa(x)Jb†δ(y)i = hqi,αf (x)τaf f0qi,α
f0 (x)qj,βg (y)τbgg0qj,β
g0 (y)i
= hTr[τaGA(x, x)] Tr[τbGA(y, y)]i − hTr[τaGA(x, y)τbGA(y, x)]i
= −hTr[τaGA(x, y)τbGA(y, x)]i = −2δabhTrDC[GA(x, y)GA(y, x)]i,
where a, b = 1, 2, 3 and for the other indices we use the same conventions as in Eq. (3.9). In the last passage the trace over flavour space has been performed (Tr τaτb = 2δab) and we are left with the trace TrDC over Dirac
and colour indices only. Therefore, the isospin symmetry implies that χδa
is the same for every flavour generator; moreover, from a comparison with Eq. (3.9), it is easy to see that:
χδ≡ χδa = χσ,conn. (3.19)
Evidently, this is true also for the channels δ+, δ− and δ0, corresponding to
τ1√+iτ2
2 , τ1√−iτ2
2 and τ3:
χδ = χδ+ = χδ− = χδ0.
Isoscalar (I = 0) pseudoscalar channel
The isoscalar pseudoscalar channel Jη = iqγ5q is associated to the singlet
η, which is the two-flavour version of the meson η0.2 As in the scalar chan-nel, the integration over the quark fields gives rise to a disconnected and a connected term: hJη(x)Jη†(y)i = hiq i,α f (x)γ ii0 5 q i0,α f (x)iq j,β g (y)γ jj0 5 qj 0,β g (y)
= −hTr[γ5GA(x, x)] Tr[γ5GA(y, y)]i + hTr[γ5GA(x, y)γ5GA(y, x)]i.
(3.20)
The chiral susceptibility χη is given by
χη = χη,conn+ χη,disc,
where the connected an disconnected terms (with respect to the fermionic lines in an external field) are defined analogously to Eq. (3.10). Again, the former can be straightforwardly written in terms of the spectral density ρ, while the latter does not.
2
Actually the physical state η0is a superposition of the singlet and the eighth Goldstone boson of the pseudoscalar octet.
3.3. CHIRAL SUSCEPTIBILITIES 41 • Connected term
Using Eq. (3.4), we can express the connected term as: χη,conn= 1 V Z d4x Z
d4yhTr[GA†(y, x)GA(y, x)]i
= 2 Vh
Z
d4x d4yX
k
ui,αk (x)u†j,βk (y) m + iλk X k0 uj,βk0 (y)u†i,αk0 (x) m − iλk0 i = 2 Vh X k 1 m2+ λ2 k i = 2 Z ∞ −∞ dλ ρ(λ) m2+ λ2 = 4 Z ∞ 0 dλ ρ(λ) m2+ λ2.
Comparing this result with Eq. (3.6) we obtain the following relation between χη,conn and the chiral condensate Σ:
χη,conn=
Σ
m. (3.21)
• Disconnected term
Instead, it is not possible to express the disconnected part in terms of the spectral density. Nevertheless, one can derive an important relation between χη,discand the topological susceptibility χtop, which is defined in Euclidean space as:
χtop= Z d4xhq(x)q(0)i = 1 V Z d4x Z d4yhq(x)q(y)i = 1 VhQ 2i, (3.22) where q is the topological charge density and Q is the topological charge. As in Eq. (2.7), the translational invariance has been used. Now, using Eqs. (1.40) and (1.44), it is easy to see that only the Dirac zero modes give a non-zero contribution to χη,disc:3
−χη,disc= 1 V
Z d4x
Z
d4yhTr[γ5GA(x, x)] Tr[γ5GA(y, y)]i
= 4 Vh Z d4x d4yX k γ5ijuj,αk (x)u†i,αk (x) m − iλk X k0 γ5i0j0ujk00,β(y)u†i 0,β k0 (y) m − iλk0 i = 4 Vh 1 m2(n+− n−) 2i = 4 m2 1 VhQ 2i = 4 m2χtop 3
The following calculation is very similar to the demonstration of the Atiyah-Singer theorem (1.47) Q = n−− n+ in section 1.5.2.
