UNIVERSIT `
A DEGLI STUDI DI PISA
Facolt`
a di Scienze Matematiche Fisiche e Naturali
Corso di Laurea Magistrale in Fisica
Anno Accademico 2016-2017
Tesi di Laurea Magistrale
Curvature of the pseudocritical line in QCD
Relatore Candidato
Contents
Introduction 9
1 Lattice QCD 13
1.1 SU (N ) gauge theories . . . 13
1.2 Numerical simulation of field theories . . . 15
1.3 QCD on the lattice . . . 17
1.3.1 Example: the static q ¯q potential . . . 18
1.4 Dynamical fermions . . . 19
1.4.1 Wilson fermions . . . 20
1.4.2 Kogut-Susskind fermions . . . 20
1.5 Improved actions . . . 21
1.6 Continuum limit, lattice spacing and lines of constant physics . . . . 22
2 QCD thermodynamics 25 2.1 QCD at finite temperature and density . . . 25
2.2 The phase diagram of QCD . . . 26
2.3 Chiral symmetry . . . 27
2.4 Z3 symmetry . . . 28
2.5 Finite quark masses . . . 29
2.6 The sign problem . . . 30
2.7 Previous research on the pseudocritical line . . . 31
2.8 Phenomenology of heavy-ion collisions . . . 32
3 Curvature of the pseudocritical line 35 3.1 Calculation of the curvature coefficient by Taylor expansion . . . 35
3.2 Calculation of derivatives with respect to µB . . . 37
3.3 Observable used to locate the transition . . . 38
3.4 Numerical set-up . . . 38
4 Numerical results 41 4.1 Results on the 163× 6 lattice . . . 41
4.2 Results on the 243× 6 lattice . . . 47
4.3 Results on the 323× 8 lattice . . . 53
4.4 Extrapolation to the continuum limit . . . 58
A Appendix 65
A.1 Monte Carlo integration . . . 65
A.2 The Metropolis-Hastings algorithm . . . 66
A.3 The HMC algorithm . . . 67
A.4 Noisy estimators . . . 68
A.5 Space-filling curves . . . 69
List of Figures
1 QCD phase diagram (Source: [1]) . . . 10
2.1 Coupling constant αS as a function of the energy scale (Source: [33]) 26 2.2 Columbia plot (Source: [32]) . . . 30
2.3 Curvature coefficient kB: previous determinations . . . 32
4.1 Renormalized chiral condensate h ¯ψψr1i (163× 6 lattice) . . . 42
4.2 Renormalized chiral condensate h ¯ψψr2i (163× 6 lattice) . . . 42
4.3 Renormalized chiral condensate h ¯ψψr1i (163× 6 lattice, local fit) . . 43
4.4 Renormalized chiral condensate h ¯ψψr2i (163× 6 lattice, local fit) . . 44
4.5 First derivative with respect to µ2B of h ¯ψψr1i (163× 6 lattice) . . . . 45
4.6 First derivative with respect to µ2B of h ¯ψψr2i (163× 6 lattice) . . . . 46
4.7 Renormalized chiral condensate h ¯ψψr1i (243× 6 lattice) . . . 47
4.8 Renormalized chiral condensate h ¯ψψr2i (243× 6 lattice) . . . 48
4.9 Renormalized chiral condensate h ¯ψψr1i (243× 6 lattice, local fit) . . 49
4.10 Renormalized chiral condensate h ¯ψψr2i (243× 6 lattice, local fit) . . 49
4.11 First derivative with respect to µ2B of h ¯ψψr1i: comparison between the results obtained on the 163 and the 243 lattices . . . 50
4.12 First derivative with respect to µ2B of h ¯ψψr2i: comparison between the results obtained on the 163 and the 243 lattices . . . 51
4.13 First derivative with respect to µ2B of h ¯ψψr1i (243× 6 lattice) . . . . 52
4.14 First derivative with respect to µ2B of h ¯ψψr2i (243× 6 lattice) . . . . 52
4.15 Renormalized chiral condensate h ¯ψψr1i (323× 8 lattice) . . . 54
4.16 Renormalized chiral condensate h ¯ψψr2i (323× 8 lattice) . . . 54
4.17 Renormalized chiral condensate h ¯ψψr1i (323× 8 lattice, local fit) . . 55
4.18 Renormalized chiral condensate h ¯ψψr2i (323× 8 lattice, local fit) . . 56
4.19 First derivative with respect to µ2B of h ¯ψψr1i (323× 8 lattice) . . . . 57
4.20 First derivative with respect to µ2B of h ¯ψψr2i (323× 8 lattice) . . . . 57
4.21 Curvature coefficient kB: extrapolation to the continuum limit . . . 59
5.1 Curvature coefficient kB: comparison with previous determinations . 64 5.2 Comparison of the pseudocritical line with the chemical freeze-out curve 64 A.1 Lebesgue-like curve filling a 48 × 12 lattice . . . 70
List of Tables
3.1 List of the bare quark masses and lattice spacings used in the simulations 39 4.1 Values obtained for the critical temperature (163× 6 lattice) . . . . 43 4.2 Values obtained for A(Tc0) and A000(Tc0) (163× 6 lattice) . . . 44 4.3 Values obtained for A(Tc0) and A000(Tc0) (163×6 lattice, fit with quintic
polynomial) . . . 44 4.4 Values obtained for B(Tc0) and B00(Tc0) (163× 6 lattice) . . . 46 4.5 Curvature coefficient kB (163× 6 lattice) . . . 46
4.6 Values obtained for the critical temperature (243× 6 lattice) . . . . 48 4.7 Values obtained for A(Tc0) and A000(Tc0) (243× 6 lattice) . . . 50 4.8 Values obtained for A(Tc0) and A000(Tc0) (243×6 lattice, fit with quintic
polynomial) . . . 50 4.9 Values obtained for B(Tc0) and B00(Tc0) (243× 6 lattice) . . . 53 4.10 Curvature coefficient kB (243× 6 lattice) . . . 53
4.11 Values obtained for the critical temperature (323× 8 lattice) . . . . 55
4.12 Values obtained for A(Tc0) and A000(Tc0) (323× 8 lattice) . . . 56 4.13 Values obtained for A(Tc0) and A000(Tc0) (323×8 lattice, fit with quintic
polynomial) . . . 56 4.14 Values obtained for B(Tc0) and B00(Tc0) (323× 8 lattice) . . . 58 4.15 Curvature coefficient kB (323× 8 lattice) . . . 58
5.1 Curvature coefficient kB: comparison with the results reported by
Introduction
Since ancient times, man has wondered what is the structure of matter and what are its smallest constituents. Indeed, already in the fourth century BC, Greek philoso-pher Democritus theorized that matter is composed of indivisible units he called atoms. During the nineteenth century AC, Dalton, inspired by Democritus’ ideas, developed the first scientific theory of atoms. Dalton’s theory held that chemical elements are made of atoms, indivisible particles having the same physical proper-ties (mass, size, etc.), and that combinations of atoms of different elements form all chemical compounds found in Nature.
At the end of the nineteenth century, however, the discovery of the electron by Thomson cast some doubts on the fundamental nature of the atom and led to think that atoms were not indivisible particles, but they were rather composed of electrons either enclosed in a positively charged sphere (Thomson’s plum pudding model) or orbiting around a positive charge at the center (Nagaoka’s planetary model). In 1909, the Rutherford-Geiger-Marsden experiment offered experimental evidence that, contrary to Thomson’s model, atoms had to be composed of electrons orbiting around a positively charged nucleus. After the discovery of isotopes by Soddy in 1910 and the discovery of the neutron by Chadwick in 1932, atomic nuclei were also found to be composite objects, bound states of positively charged and neutral particles, protons and neutrons, collectively known as nucleons. To explain why many nuclei are stable despite the electrostatic repulsion between protons, physicists postulated the existence of the strong interaction and in 1935 Yukawa developed a quantum field theory for the strong force where the interaction is mediated by virtual bosons of mass ≈ 200 M eV , later identified as the pions.
During the 050, dozens of hadrons, many of them unstable, were discovered. It was then realized that hadrons can be grouped into isospin multiplets with similar mass: this symmetry was explained in 1964, when Gell-Mann proposed that hadrons are not fundamental particles, but bound states of three types or flavors (u, d and s) of elementary particles, which he called quarks. In the late 060, evidence of an internal structure within the proton was provided by the deep inelastic e-p scatter-ing experiments conducted at SLAC. However, the ∆++ baryon, being composed of three up quarks with parallel spins, seemingly violated Pauli’s exclusion principle. To resolve this issue, a new quantum number, the color, was introduced. While initially born as an ad hoc hypothesis to save Pauli’s exclusion principle, the exis-tence of the color quantum number was later confirmed experimentally by measuring the σ(e+e− → hadrons)/σ(e+e− → µ+µ−) ratio, whose value is consistent with a
number of colors equal to 3. During the 070, physicists predicted on theoretical grounds the existence of other 3 quarks (c, t and b), which were later discovered experimentally.
is a non-abelian gauge theory describing the strong interaction between quarks (Dirac spinors ψf a(x) having both a flavor f and a color a index) through a
La-grangian which is invariant under local SU (3) transformations in color space. Two of the most interesting properties of QCD are confinement and asymptotic freedom. Because of confinement, only color singlets exist as free particles, thus quarks are confined within hadrons and free quarks are not observed. On the other hand, ac-cording to the property of asymptotic freedom, discovered by Gross, Politzer and Wilczek in 1973, the strong coupling constant decreases at high energies. Due to asymptotic freedom, at high temperatures the formation of a new state of matter where quarks are deconfined, the quark-gluon plasma, is possible.
A property related to asymptotic freedom is the fact that, while it decreases at high energies, the strong coupling constant increases at low energies. Therefore, at low energy scales, it is impossible to apply the perturbative techniques routinely used in QED calculations. However, thanks to the improvement of algorithms and the exponential growth of computing power over the last five decades, non-perturbative numerical simulations can now be performed within lattice QCD, a formulation of QCD introduced by Wilson in the070.
In this work, lattice QCD is used to study the phase transition of hadronic matter from the confined state to the deconfined state, specifically the focus is on the construction of the phase diagram on the µB− T plane, where T is the temperature
and µB is the baryochemical potential, a thermodynamic parameter introduced to
study QCD thermodynamics at finite density. The phase diagram of QCD, whose conjectured form is shown in fig. 1, is a subject of outmost importance due to its relevance to various fields of physics: the region of high T and low µB is relevant
to the study of the early Universe and to heavy-ion collision experiments, while the region of low T and high µB is relevant to the physics of neutron stars.
One would like to determine the nature and the exact location of the phase transition. It is well known that for µB= 0 a transition takes place at Tc0≈ 150 ÷ 170 M eV and
that this is not a strict phase transition, but a crossover [2, 3], where there is a rapid change in thermodynamic quantities but no discontinuities in the derivatives of the free energy. It is also thought that for some critical value µB = µcB the transition is
of second order and that for µB > µcB the transition is of first order. The small µB
region of the QCD phase diagram is currently being probed by heavy-ion collision experiments conducted at the LHC near Geneve and at the RHIC in Brookhaven and a region with higher µB will also be accessible to future experiments conducted at
the FAIR in Darmstadt, hence experimental data are available for a comparison with results obtained from lattice QCD and even more data will be available in the future. Investigating the case of nonzero µBby lattice simulations is a difficult task, because
the weight1 ∝ detM e−SG that at zero µ
B is interpreted as a probability measure in
order to evaluate observables by importance sampling becomes a complex number: this issue is the infamous sign problem [4]. Several methods have been proposed to tackle the sign problem for small µB, like reweighting [5–7], analytic continuation
from imaginary µB [8–15] and the Taylor expansion method [16–18].
The dashed line located in the small µB region of Fig. 1, known as the
pseudo-critical line, can be parametrized as Tc(µB) = Tc0 1 − kB µB T0 c 2 + O(µ4B) ! . (1)
Recently, many investigations have been carried out to calculate the curvature co-efficient kB, using both analytic continuation [11–15] and Taylor expansion [17, 18].
The results obtained by analytic continuation are somewhat larger than those ob-tained by Taylor expansion. Of particular relevance are the results reported by refs. [11] and [18], as they have found respectively kB ≈ 0.013 by analytic continuation
and kB ≈ 0.0066 by Taylor expansion, with a discrepancy of a factor ≈ 2, despite
having used the same discretization and the same observable, the chiral condensate, to locate the transition. Further investigation is thus of interest, as it might help to better understand possible sources of systematic uncertainty.
The purpose of this study is to determine an independent estimate of kB and to
investigate the systematics, specifically: • finite size effects
• effects due to the renormalization chosen for the observable • effects due to the prescription adopted to define Tc(µB)
The curvature coefficient of the pseudocritical line is calculated by Taylor expansion, using the inflection point of the chiral condensate h ¯ψψri to locate the transition. The
discretization used in this work is the same adopted in refs. [11, 12, 18] and both the renormalizations for the chiral condensate used respectively in refs. [11, 12] and [18] are adopted. Moreover, after computing kB under the hypothesis of constant h ¯ψψri
at Tc used in ref. [18], hence defining Tc0 as the inflection point of h ¯ψψri(T, µB =
0) and Tc(µB) by the relation h ¯ψψri(Tc(µB), µB) = h ¯ψψri(Tc0, 0), the analysis is
repeated using the prescription adopted in refs. [11, 12], that is defining Tc(µB)
as the actual inflection point of h ¯ψψri. This set-up allows, besides pursuing the 1
aforementioned objectives, to make a proper comparison of the results with those previously obtained by Taylor expansion and by analytic continuation.
Numerical simulations have been carried out on the GPU cluster at INFN -Sezione di Pisa and on the Galileo and Marconi A2 clusters at Cineca, using the code developed by the Pisa and Ferrara groups [44] and NISSA [45]. Both codes have been appropriately modified so as to include the missing operators needed by the simulations and, in NISSA, an improved site ordering algorithm also has been implemented.
The dissertation is organized as follows. Chapter 1 contains a review of lat-tice QCD, chapter 2 briefly discusses QCD thermodynamics and the sign problem, chapter 3 explains how to determine the curvature coefficient by Taylor expansion, chapter 4 reports the results of the numerical simulations performed in this work. Finally, the Appendix describes some of the numerical techniques needed to perform the numerical simulations.
Chapter 1
Lattice QCD
1.1
SU (N ) gauge theories
Quantum chromodynamics is an example of a Yang-Mills theory, that is a gauge theory whose symmetry group is SU (N ). Thus, before delving into the lattice formulation of QCD, it is useful to discuss SU (N ) gauge theories. Let ψ be a vector comprising N fermion fields
ψ ≡ ψ1 ψ2 .. . ψN (1.1)
and let LF be the following Lagrangian for N free fermion fields:
LF = ¯ψ(iγµ∂µ− m)ψ = N X i=1 ¯ ψi(iγµ∂µ− m)ψi . (1.2)
This Lagrangian is invariant under global SU (N ) transformations
ψ 7→ ψ0 = U ψ
¯
ψ 7→ ¯ψ0 = ¯ψU† , U ∈ SU (N ) . (1.3) Any element U of SU (N ) can be obtained by exponentiating the generators Ta of
the group,
U = eiPaθaTa . (1.4)
SU (N ) has N2− 1 generators, these are traceless Hermitian N × N matrices which are conventionally normalized so that T r[TaTb] = 12δab. The generators satisfy the
commutation relations [Ta, Tb] = ifabcTc, where fabc are the structure constants of
the group. For instance, SU (3) has 8 generators given by Ta= λ2a, where λadenotes
the Gell-Mann matrices.
To construct a gauge theory, one promotes the global symmetry to a local sym-metry by requiring the Lagrangian to be invariant under the following space-time dependent transformation:
ψ 7→ ψ0 = U (x)ψ
¯
ψ 7→ ¯ψ0 = ¯ψU†(x) , U (x) ∈ SU (N ) . (1.5) In order to do so, one defines N2− 1 gauge fields Aa
µ(x) and the parallel transport
W (Cy←x) ≡ Peig R
Cy←xAµ(z)dz µ
where Aµ(x) ≡ Aaµ(x)Ta, Cy←x is a curve joining x to y and P is the path-ordering
operator. By requiring that under a gauge transformation the parallel transport transforms according to
W (Cy←x) 7→ W0(Cy←x) = U (y)W (Cy←x)U†(x) , (1.7)
one finds the following transformation law for the parallel transported spinor: W (Cy←x)ψ(x) 7→ W0(Cy←x)ψ0(x) = U (y)W (Cy←x)ψ(x) . (1.8)
It can also be shown that the gauge fields transform according to Aµ(x) 7→ A0µ(x) = U (x)Aµ(x)U†(x) −
i
g(∂µU (x))U
†(x) . (1.9)
Indeed, considering the infinitesimal parallel transport Wx+dx←x= eigAµ(x)dx µ , one finds that 1 + igA0µ(x)dxµ ≈ W0 x+dx←x= U (x + dx)Wx+dx←xU†(x) ≈ ≈ U (x + dx)(1 + igAµ(x)dxµ)U†(x) ≈ ≈ (U (x) + ∂µU (x)dxµ)(1 + igAµ(x)dxµ)U†(x) ≈
≈ 1 + (∂µU (x))U†(x)dxµ+ igU (x)Aµ(x)U†(x)dxµ .
(1.10)
By making use of the infinitesimal parallel transport, it is possible to define the covariant differential and the covariant derivative:
Dψ(x) ≡ Wx←x+dxψ(x + dx) − ψ(x) = Wx+dx←x−1 ψ(x + dx) − ψ(x) ≈ ≈ (1 − igAµ(x)dxµ)ψ(x + dx) − ψ(x) ≈ ≈ (1 − igAµ(x)dxµ)(ψ(x) + ∂µψ(x)dxµ) − ψ(x) ≈ ≈ ∂µψ(x)dxµ− igAµ(x)ψ(x)dxµ Dµψ(x) ≡ (∂µ− igAµ)ψ(x) . (1.11)
Since the covariant derivative transforms like
Dµψ(x) 7→ U (x)Dµψ(x) = (U (x)DµU†(x))(U (x)ψ(x)) , (1.12)
after replacing the partial derivative by the covariant derivative one obtains a locally symmetric Lagrangian
LF = ¯ψ(iγµDµ− m)ψ . (1.13)
Having introduced the gauge fields Aaµ(x), which in general can be dynamical vari-ables, it’s desirable to add a kinetic term and this may be accomplished using the field strength tensor Fµν:
Fµν ≡ [Dµ, Dν] = (∂µAν − ∂νAµ+ ig[Aµ, Aν]) ≡ Fµνa Ta LG = −1 2T r[FµνF µν] = −1 4F a µνFaµν . (1.14) Note that one could also be tempted to add a mass term ∝ mAaµAaµ for the gauge fields, but this term, unlike Fµνa Faµν, would break gauge invariance. Furthermore, adding higher order gauge invariant terms, for instance a term ∝ T r[(FµνFµν)2],
Quantum chromodynamics is a gauge theory having the color group SU (Nc)cas
local symmetry. This theory describes the interaction between Nf flavors of quarks
through the following Lagrangian:
LQCD = − 1 4F a µνFaµν+ Nf X f =1 ¯ ψf(iγµDµ− mf)ψf , where ψf ≡ ψf 1 ψf 2 .. . ψf Nc . (1.15)
This Lagrangian is invariant with respect to local SU (Nc)ctransformations in color
space and, as a consequence of the low masses of the 3 lightest quarks, it is approx-imately invariant with respect to global SU (3)f transformations in flavor space. It
also enjoys chiral symmetry in the massless limit, as will be discussed more in detail in chap. 2.3.
1.2
Numerical simulation of field theories
Consider a 1-dimensional quantum mechanical system governed by the Hamiltonian H. In the path integral formulation of quantum mechanics, the amplitude from the eigenstate |x1i at t = t1 to the eigenstate |x2i at t = t2 is given by the following
functional integral [19]: K(x2, t2; x1, t1) = hx2|e− i ~H(t2−t1)|x1i = Z [Dx(t)]e~iS[x(t)] = Z [Dx(t)]e~i Rt2 t1dt 0L(x, ˙x) L = 1 2m ˙x 2− V (x) , (1.16)
where S and L are respectively the action and the Lagrangian of the system. After a Wick rotation (t0 7→ −iτ0), which turns the Minkowski space-time into an Euclidean space-time, one obtains for the case x1 = x2 = x, t1= 0:
KE(x, τ2; x, 0) = hx|e− 1 ~Hτ2|xi = Z [Dx(τ )]e−1~SE[x(τ )] = Z [Dx(τ )]e−1~ Rτ2 0 dτ 0L E(x, ˙x) LE = 1 2m ˙x 2+ V (x) . (1.17)
The statistical properties of a system in thermodynamic equilibrium can be derived from the partition function Z ≡ P
ne−βEn = T r[e−βH]; letting β ≡ T1 ↔ τ2
~ and
comparing the definition of Z with eq. 1.17, it is easily seen that the partition function can be rewritten, like the propagator, as a path integral
Z ≡ T r[e−βH] = Z dxhx|e−1~Hτ2|xi = Z dx Z [Dx(τ )]e−1~SE[x(τ )] . (1.18)
Moreover, the expected value of an observable O may be written as hOi ≡ 1 ZT r[Oe −βH] = 1 Z Z dxhx|e−1~Hτ2O|xi = 1 Z Z dx Z [Dx(τ )]Oe−1~SE[x(τ )] (1.19) and by using a complete set of energy eigenstates {|ni}, one finds
hOi = P ne −Enτ2 ~ hn|O|ni P ne −Enτ2 ~ . (1.20)
It follows that, in the τ2 → ∞ limit, one has
hOi = h0|O|0i . (1.21) This relation shows that, in the limit of infinite Euclidean time, one recovers the expected value of O on the ground state.
The expression just obtained for h0|O|0i is useful because path integrals are well suited for numerical simulations. To compute h0|O|0i, one may indeed replace the time interval [0, τ2] with a 1-dimensional lattice of N + 1 points and spacing a = τN2
and then discretize the path integral: hOi ≈ 1 Z Z dx0dx1. . . dxN −1Oe− SE ({xi}) ~ SE = Z dτ0LE ≈ a N −1 X i=0 1 2m xi+1− xi a 2 + V (xi) , (1.22)
where x0, x1, x2, . . ., xN have been defined as the values x(0), x(a), x(2a), . . ., x(τ2)
assumed by the path at times 0, a, 2a, . . ., T and the integrand is not integrated over dxN because of the boundary condition xN = x0. Now the problem has been
reduced to the numerical evaluation of the expected value of O with respect to the probability distribution P ({xi}) = 1 Ze −SE ({xi}) ~ . (1.23)
and may be solved by Monte Carlo methods, i.e. using the Metropolis algorithm. The path integral formulation of quantum mechanics can be generalized to quan-tum field theories by the following substitutions:
x(t) 7→ φ(t, ~x) Z dtL 7→ Z dtd3x L Z [Dx(t)] 7→ Z [Dφ(t, ~x)]
h0|O[x(t)]|0i 7→ h0|O[φ(t, ~x)]|0i . (1.24) In quantum field theories one is often concerned with the calculation of vacuum expected values (v.e.v.)
hOi = h0|O[φ(t, ~x)]|0i = R [Dφ(t, ~x)]Oe
−SE
R [Dφ(t, ~x)]e−SE . (1.25)
These can be evaluated numerically by replacing the Euclidean space-time with a 4-dimensional lattice of N3
s × Nt points and spacing a and then discretizing:
hOi ≈ 1 Z Z Y i∈sites dφi ! Oe−SE({φi}) . (1.26)
As in the case of 1-dimensional quantum mechanics, this calculation can be carried out by Monte Carlo methods.
1.3
QCD on the lattice
The basis of lattice QCD is the path integral formulation of quantum field theories. Starting from the Euclidean Lagrangian of QCD1,
LE QCD= 1 4F a µνFµνa + Nf X f =1 ¯ ψf(γµEDµ+ mf)ψf , (1.27)
the vacuum expected value of an observable may be expressed as a path integral. For instance, considering only one flavor for the sake of simplicity, the vacuum expected value of the observable O can be written as
hOi = 1 Z
Z
[DA][D ¯ψ][Dψ]O[A, ¯ψ, ψ]e−SQCD , (1.28)
where periodic and antiperiodic boundary conditions are imposed respectively for the gluon and fermion fields:
Aµ(x, T ) = Aµ(x, 0) (1.29)
ψ(x, T ) = −ψ(x, 0) . (1.30) After discretization, hOi may be evaluated by numerical simulations.
The gauge part of the integral is discretized using parallel transports as gauge variables,
Uµ(x) ≡ e−igAµ(x)a . (1.31)
These variables live on the links (x, µ) connecting neighboring sites of the lattice. It is worth mentioning that, as one can see by recalling that Aµ(x) ≡ Aaµ(x)Ta, in
continuum QCD the gluon fields lie in the algebra of SU (3), while in the lattice for-mulation of QCD they lie in the SU (3) group itself; since SU (3) is a compact group, in lattice QCD the functional integral is finite and gauge fixing is not necessary, on the other hand in continuum QCD the functional integral is ill-defined and gauge fixing via the Fadeev-Popov procedure is required. Using parallel transports it is possible to build the Wilson plaquette
Uµν(x) ≡ Uµ(x)Uν(x+µ)U−µ(x+µ+ν)U−ν(x+ν) = Uµ(x)Uν(x+µ)Uµ†(x+ν)U † ν(x) .
(1.32) Wilson plaquettes are parallel transports along squares of length a and are in turn used to construct the Wilson action
SW ≡ β X x,µ>ν 1 −1 3ReT rUµν(x) . (1.33)
From the transformation law for the parallel transport and from the cyclic property of the trace, it follows that the Wilson action is gauge invariant. Moreover, for β ≡ g62, it reproduces the continuum QCD gauge action in the a → 0 limit:
SW = Z d4x1 4F a µνFµνa + O(a2) . (1.34)
1The Euclidean gamma matrices are defined by γE
4 = γ0, γ1,2,3E = −iγ1,2,3 and they satisfy the
For the fermionic part of the action, since one cannot directly simulate anticommut-ing fields on a computer, the fermionic fields are integrated out, obtainanticommut-ing
hOi = 1 Z
Z
[DU ][D ¯ψ][Dψ]O[U, ¯ψ, ψ]e−SG− ¯ψM ψ= 1
Z Z
[DU ]O[U, M−1]detM e−SG .
(1.35) Then one can deal with the fermionic determinant by rewriting it as a path inte-gral over auxiliary commuting fields and evaluating it by Monte Carlo methods, as described in chapters 1.4 and A.3. In the early days of lattice QCD, due to the limited computational resources available at that time, evaluating the fermionic de-terminant was prohibitively expensive and numerical simulations were performed in the so-called quenched approximation, where detM is manually set to 1. Before discussing how to simulate full QCD, given the historical relevance of the quenched approximation, it’s interesting to consider a pratical application of the quenched theory as a first example of a lattice simulation.
1.3.1 Example: the static q ¯q potential
In the quenched approximation, detM is set to 1 and the expected value of an observable O reduces to
hOi = 1 Z
Z
[DU ]O[U ]e−SG . (1.36)
The gauge action SG can be discretized using the Wilson action, which, up to an
additive constant that doesn’t affect the results, may be taken equal to SW =
−β3P
x,µ>νReT rUµν(x). Historically, one of the first applications of the quenched
theory has been the computation of the static q ¯q potential, which in turn can be used to determine the physical spacing a of the lattice. This can be accomplished by computing the expected value of the Wilson loop, defined as
W (T, R) ≡ T r Y (x,µ)∈L Uµ(x) , (1.37)
where L is a closed loop along a rectangle having temporal and spatial lengths respectively T and R. The physical meaning of the Wilson loop is determined by the relation [20]
limT →∞hW (T, R)i ∝ e−T aVq ¯q(R). (1.38)
From this relation it can be seen that, by computing the expected value of the Wilson loop for various values of T and R, it is possible to extract the static q ¯q potential. First one analyzes the results obtained for different values of T at fixed R looking for a plateau, in order to determine, for each R, what value T = TR∞ is large enough to consider valid the limit T → ∞. Then, being assured that the equation
aVq ¯q(R) = ln hW (T∞ R , R)i hW (TR∞+ 1, R)i (1.39) holds, one fits the values obtained for the rhs of eq. 1.39 with the Cornell potential −AR+ BR. Here R is a pure number and, to obtain the physical result, one makes the change of variable r = aR and finds
Vq ¯q(r) = . . . +
B
The term that multiplies r is the string tension σ. From the results of the fit and from the phenomenological value√σ = 440M eV [21], one can determine the lattice spacing a =
q
B σ.
1.4
Dynamical fermions
Unquenched simulations require to deal with the fermionic determinant detMf. Mf
is a n × n complex matrix, where n = Nsites· NDirac· Ncolors, and the computational
cost for a direct calculation of the determinant is ≈ O(n3). For Nsites = 163 × 6,
a direct calculation would be quite expensive, requiring ≈ O(1016) operations, and a stochastic approach provides a more efficient method to evaluate the fermionic determinant. Introducing an auxiliary bosonic field Φ, called pseudo-fermion, detMf
may be expressed as detMf = 1 detMf−1 = Z [DΦ†][DΦ]e−Φ†M −1 f Φ (1.41)
and the fermionic determinant can be estimated by Monte Carlo methods, using the conjugate-gradient method to calculate Mf−1Φ. Actually, to estimate the fermionic determinant by the pseudo-fermion method, one must work with an hermitian and positive definite matrix. It is usually possible to obtain such matrix from Mf; for
instance, in the case of two degenerate quarks u, d, the fermionic determinant is detMudetMd = detMl2 = detMlM
†
l ≡ detM and rewriting in terms of
pseudo-fermionic fields yields
detMl2 = detM = Z
[DΦ†][DΦ]e−Φ†M−1Φ , (1.42) where M is hermitian and positive definite.
In addition to a stochastic method to estimate detMf, unquenched simulations
also require a suitable discretization for the fermionic Lagrangian
LF = Nf X f =1 ¯ ψf(γµDµ+ mf)ψf . (1.43)
Approximating the derivative by finite differences ∂µψ ≈
ψ(x + aµ) − ψ(x − aµ)
2a (1.44)
and using parallel transports to discretize the covariant derivative Dµψ = Wx←x+aµψ(x+a µ)−W x←x−aµψ(x−aµ) 2a = = Uµ(x)ψ(x+µ)−Uµ†(x−µ)ψ(x−µ) 2a , (1.45)
one may construct the naive fermionic action SFnaive = a4P sitesψ(x)(γ¯ µDµ+ m)ψ(x) = = a4P sitesψ(x)¯ γµUµ(x)ψ(x+µ)−γµUµ†(x−µ)ψ(x−µ) 2a + mψ(x) . (1.46)
This action, however, is afflicted with the fermion doubling problem. Indeed, when computing the free fermion propagator
M−1(q) = i
aγµsin(qµa) + m −1
(1.47) one finds that in the massless case it has, besides the physical pole, 15 additional poles at the edges of the Brillouin zone (qµ = (πa, 0, 0, 0), . . ., (πa,πa,πa,πa)). It is
possible to eliminate these doublers, but not without drawbacks. This follows from the Nielsen-Ninomiya no-go theorem [22], according to which a lattice fermionic action SF =Px,yψ(x)M (x, y)ψ(y) =¯ Px,yψ(x)(D(x − y) + mδ¯ xy))ψ(y) can’t at the
same time: 1. be local
2. exhibit no doublers
3. have chiral symmetry in the massless limit ({D, γ5} = 0)
4. have the right continuum limit (D(q) → iγµqµ)
As examples of possible approaches to get rid of the doublers, let’s consider the Wilson and the Kogut-Susskind actions.
1.4.1 Wilson fermions
A possible solution to the doubling problem consists in adding, to the naive action, the Wilson term
SFW = −a4X sites ra 2 ¯ ψ2ψ , (1.48)
where r is a constant, usually set to 1, the Laplacian is discretized by finite differences and gauge invariance is achieved by making use of parallel transports,
2ψ ≈ Pµ
ψ(x+aµ)+ψ(x−aµ)−2ψ(x) 2a2
7→ P
µ
Uµ(x)ψ(x+aµ)+Uµ†(x)ψ(x−aµ)−2ψ(x)
2a2 .
(1.49) The free fermion propagator becomes
M−1(q) = X µ i aγµsin(qµa) + X µ r a(1 − cos(qµa)) + m !−1 (1.50) and has the right continuum limit. The Wilson term removes the doublers, but the Nielsen-Ninomiya theorem still holds since at finite a the fermionic action has lost the chiral symmetry in the massless limit.
1.4.2 Kogut-Susskind fermions
Alternatively, one may construct the Kogut-Susskind action (or staggered action). Let’s redefine the fermionic fields as follows:
ψ(x) = A(x)χ(x) ≡ γx1 1 γ x2 2 γ x3 3 γ x4 4 χ(x) . (1.51)
From the property γµ2 = 1 it follows that the mass term of the fermionic La-grangian is invariant under such redefinition, ¯ψ(x)mψ(x) = ¯χ(x)mχ(x). On the
other hand, from the property γµ2 = 1 and from the anticommutation relations for the gamma matrices it follows that ¯ψ(x)γµψ(x ± µ) = ¯χ(x)ηµ(x)χ(x ± µ), where
ηµ(x) ≡ (−1) P
ν<µxν. Therefore, after replacing the derivatives by finite differences,
the kinetic term becomes ¯ ψγµ∂µψ ≈ ¯ ψ(x)γµψ(x+aµ)− ¯ψ(x)γµψ(x−aµ) 2a = = χ(x)η¯ µ(x)χ(x+aµ)− ¯χ(x)ηµ(x)χ(x−aµ) 2a (1.52) and one obtains the following action:
SFstagg = a4X sites ¯ χ(x) ηµ(x) Uµ(x)χ(x + µ) − Uµ†(x − µ)χ(x − µ) 2a + mχ(x) ! . (1.53) Since the gamma matrices have been replaced by the staggered phases ηµ(x), which are pure numbers, the fermionic action is now diagonal in Dirac space and in the above equation 3 of the 4 Dirac components can be discarded in order to reduce the number of doublers by a factor 4. This procedure is consistent because, after indexing the even sites of the lattice as nµ= 2Nµand a generic site as nµ= 2Nµ+ρµ
(ρµ being a vector whose 4 components are valued in {0, 1}), one may define the
fields
ψαf(N ) ≡ N0
X
ρ
A(ρ)αfφρ(N ) , (1.54)
where φρ(N ) ≡ χ(2N + ρ), α = 1 . . . 4, f = 1 . . . 4 and N0 is a normalization factor.
Then it can be shown [19] that by choosing a proper N0 the staggered action may
be rewritten as SFstagg = b4P f,Nψ¯f(N )(γµ∂µ+ M )ψf(N ) + . . . = = b4P Nψ(N )((γ¯ µ⊗ 1)∂µ+ M )ψ(N ) − b 2ψ(N )(γ¯ 5⊗ (γµγ5))2µψ(N ) . (1.55) Here b = 2a is the lattice spacing of the even sublattice, M = 2m, the derivatives are defined on the sublattice and the direct product acts on the Dirac ⊗ F lavor space. This action indeed describes 4 flavors (or tastes) of degenerate quarks in the con-tinuum limit. The second term breaks taste symmetry but, unlike the Wilson term, it leaves a remnant chiral symmetry (the action is invariant under ψ 7→ eiαγ5⊗γ5ψ).
Thus, when investigating the QCD phase transition by studying the spontaneous breaking of chiral symmetry, a sensible choice is to perform the numerical simula-tions using staggered fermions instead of Wilson fermions.
To get rid of the remaining 3 doublers, one usually employs the rooting procedure, that is one replaces the fermionic determinant with its fourth root:
detM 7→ (detM )14 . (1.56)
It can be shown that, in agreement with the Nielsen-Ninomiya theorem, the rooted action becomes non local [23].
1.5
Improved actions
The Wilson gauge action is constructed from parallel transports over 1 × 1 loops (plaquettes) and has a discretization error ≈ O(a2); this action is not unique, one may add further terms that vanish in the continuum limit and the resulting action
would still reproduce continuum QCD for a → 0. Adding larger loops in order to achieve a better convergence to the continuum by canceling the leading O(a2) error terms is the idea behind the Symanzik improvement project [24, 25]; it led to the formulation of the tree-level Symanzik action [26]
SG = − β 3 X i,µ6=ν 5 6ReT rU 1×1 µν (i) − 1 12ReT rU 1×2 µν (i) , (1.57)
where Uµν1×2(i) is a rectangular loop 1 × 2 starting from the site i and oriented along the µν plane. A further improvement can be obtained by replacing the gauge links in the fermionic action with weighted averages over neighboring links (fat links). A widely used method is stout-smearing [27], which consists in iteratively replacing the gauge links by
Uµ(n)(x) 7→ Uµ(n+1)(x) ≡ eiQ(n)µ (x)U(n)
µ (x) , (1.58)
where Qµ(x) ≡ 2i(Ω†µ(x) − Ωµ(x)) − 6iT r(Ω†µ(x) − Ωµ(x)) and Ωµ(x) is the product
of Uµ†(x) and the weighted sum of staples Σµν(x):
Ωµ(x) ≡ X ν6=µ ρµνΣµν(x) Uµ†(x) (1.59) Σµν(x) ≡ Uν(x)Uµ(x + ν)Uν†(x + µ) + U † ν(x − ν)Uµ(x − ν)Uν(x − ν + µ) .
Stout-smearing reduces the effects of taste symmetry breaking in simulations with staggered quarks [28].
1.6
Continuum limit, lattice spacing and lines of
con-stant physics
In lattice QCD the Euclidean space-time is replaced by an Ns3× Nt lattice having
spacing a. Considering a generic observable O, from the requirement that lattice QCD reproduces continuum QCD for a → 0 it follows that
Olat(a, g) =
1 a
dO
ˆ
Olat(g) →a→0Ophys , (1.60)
where ˆOlat(g) is the dimensionless estimate of O resulting from the simulations,
dO is the dimension of O and Ophys is the physical value to be recovered in the
continuum limit. One can see that for Olat to converge to the finite value Ophys, the
bare coupling must be a function of the lattice spacing, g = g(a). The functional dependence of g on a is determined by the lattice beta function
βlat ≡ −a
∂g(a)
∂a . (1.61)
Consider now the renormalized coupling gR ≡ gR(g(a), µa), which is a function of
the bare coupling g(a) and the energy scale µ. By taking the derivative of gR with
respect to a and then multiplying by a, one obtains 0 = a∂gR ∂a = ∂gR ∂g ∂g ∂a + ∂gR ∂aµ ∂aµ ∂a . (1.62)
This equation can be rewritten in terms of the QCD beta function βQCD≡ µ∂g∂µR as βlat= βQCD ∂gR ∂g . (1.63)
The QCD beta function has been calculated up to O(gR7) and is given by the expan-sion βQCD = −β0gR3 − β1g5R+ O(gR7) with β0 = 1 16π2 11 3 Nc− 2 3Nf β1 = 1 256π4 34 3 N 2 c − 10 3 NcNf− N2 c − 1 Nc Nf . (1.64) By inserting this expansion into eq. 1.63 and using gR= g + αg3+ O(g5), one finds
βlat= −β0g3− β1g5+ O(g7) (1.65)
and the solution of this equation is a = 1 Λlat (β0g2) −β1 2β2 0e − 1 2β0g2 . (1.66)
Therefore, as a consequence of the sign of β0 (and hence of asymptotic freedom),
the continuum limit a → 0 can be reached by sending g → 0.
Expectation values resulting from lattice simulations are pure numbers and to make a connection between the lattice theory and the physical theory one has to determine the lattice spacing a. Moreover, nothing has been said about how to choose the values of the bare quark masses. For Nf = 2 + 1 QCD, the bare
pa-rameters are g, ml and ms. At fixed g one may compute, for different values of
ml and ms, the ratios of some physical quantities, like hadron masses or decay
constants, and impose that the ratios resulting from lattice QCD equal the ratios known from experiments. This condition determines the lines of constant physics (or LCP) ml,s(g). The lattice spacing a can also be determined by comparing a known
quantity with the corresponding value obtained from numerical simulations. For instance, one may compute the static q ¯q potential Vq ¯q and compare the result with
the phenomenological value for the string tension √σ = 440M eV or with the Som-mer parameter r0 ≈ 0.5f m, defined as the distance such that r2 dVdrq ¯q|r=r0 = 1.65.
For this procedure to be consistent, the results should be independent from the quantities chosen to determine a and the bare parameters. This work adopts the tree-level Symanzik gauge action and the stout-smeared staggered fermion action: using this discretization, refs. [29, 30] determined the LCP and lattice spacing from
mK fK,
mK
mπ and fK with an accuracy of 2% and the consistency of the procedure has
been checked by redetermining the lattice spacing using different quantities and by measuring (and comparing with the respective experimental values) different ratios of physical quantities.
Chapter 2
QCD thermodynamics
2.1
QCD at finite temperature and density
The thermodynamic properties of QCD can be derived from its partition function, which may be written as a path integral over the gluon and fermion fields
Z ≡ T r[e−βH] ≡ T r[e−Hτ] = Z
[DA][D ¯ψ][Dψ]e−SE[A, ¯ψ,ψ] , (2.1)
with periodic b.c. for the gluon fields and antiperiodic b.c. for the fermion fields. For example, Z can be used to derive the expected values of the energy density hi = −1
V ∂lnZ
∂β or the pressure hpi = 1 β
∂lnZ
∂V . In the path integral formalism, the
expected value of an observable O is given by hOi = R [DA][D ¯ψ][Dψ]Oe
−SE
R [DA][D ¯ψ][Dψ]e−SE (2.2)
and such integral can be discretized and evaluated numerically, after replacing the Euclidean space-time with a discrete lattice. Since k1
bT ≡ β ≡ τ ≡ Nta, the
temper-ature may be modified by changing g (and therefore a) at fixed Nt or by changing
Nt at fixed a.
QCD thermodynamics at finite density can be studied by introducing a chemical potential µ into the partition function:
Z ≡ T r[e−β(H−µN )] . (2.3) Here N =R d3x ¯ψγ0ψ is the fermion number operator and, on the lattice, one might
think to modify the fermionic action as follows: SF 7→ SF + µa4
X
sites
¯
ψγ4ψ . (2.4)
Unfortunately, this replacement is not without issues, because it gives rise to a divergent energy density in the continuum limit:
lima→0(µ) ∝
µ a
2
. (2.5)
The key idea needed to solve this issue is realizing that the chemical potential acts as the fourth component of an imaginary gauge field [31]. Therefore, to introduce a
chemical potential in lattice QCD, one may multiply the forward (backward) time links by eaµ (e−aµ) in the fermionic action. For the staggered action one obtains SFstagg= a4X sites ¯ χ(x) ην(x) eaµδ4νU ν(x)χ(x + ν) − e−aµδ4νUν†(x − ν)χ(x − ν) 2a + mχ(x) ! , (2.6) where ην(x) ≡ (−1) P
ν<µxν are the staggered phases and χ(x) is the 1-component
staggered fermion field.
2.2
The phase diagram of QCD
In the 070, Wilczek, Gross and Politzer discovered an interesting property of QCD, asymptotic freedom: at high energies the strong coupling decreases and quarks behave as free particles. This property is experimentally verified, as can be seen from results of the measurement of the strong coupling illustrated in fig. 2.1.
Figure 2.1: Coupling constant αS as a function of the energy scale (Source: [33])
Because of asymptotic freedom, in extreme conditions of very high temperature or high density (hence short distances), like those found in the early Universe, in heavy-ion collisions or in neutron stars, hadronic matter may undergo a transition to a deconfined phase, called quark-gluon plasma [34]. A simple description of this phenomenon is provided by the bag model. In the bag model, quarks are thought as free particles confined inside a spherical bag whose total energy is given by
Ehadron= Ekin+ Epot=
A R +
4 3πR
3B . (2.7)
From P = −∂Ehadron
∂V =
A
4πR4 − B, one can see that B acts an inward pressure that
one finds A = 4πR4B, therefore
Ehadron=
16 3 πR
3B . (2.8)
Taking Ehadron ≈ 1GeV and R ≈ 0.8f m, one may determine an estimate for the
bag constant B ≈ 120M eV /f m3.
Using this estimate for the bag constant, the phase transition at µ = 0 can be described by considering the confined phase as composed of free massless pions and the deconfined phase as composed of free massless quarks and gluons. The energy density and pressure for the two phases are given respectively by
π = 3gπ π2 90T 4 pπ = gπ π2 90T 4 (2.9) and QGP = 3gQGP π2 90T 4+ B pQGP = gQGP π2 90T 4− B . (2.10)
The number of degrees of freedom for the confined phase is gπ = 3 (corresponding
to the three pions π+, π−, π0), while for the deconfined phase one has gQGP =
ggluons+ gquarks = 8 colors · 2 helicity states +78· 2 flavors · 3 colors · 2 spin states ·
2 charge states = 16 + 7824 = 37. By imposing the equilibrium condition pπ(Tc) =
pQGP(Tc), one finds Tc= B 90 π2(g QGP − gπ) 14 ≈ 140M eV . (2.11) From this estimate for the critical temperature Tc, one may also estimate the energy
density required to form a quark-gluon plasma as c= QGP(Tc) ≈ 1GeV /f m3.
The phase transition at finite µ can be represented on a T -µ phase diagram, as illustrated in fig. 1. At µ = 0, hadronic matter undergoes a transition to a quark-gluon plasma at Tc ≈ 150 ÷ 170 M eV (this is to be compared with the estimate
Tc≈ 140M eV derived from the bag model) and the transition is a crossover. It is
thought that at some critical value µ = µc the transition is of second oder and that
for µ > µc the transition is of first order. In presence of a phase transition, one
may ask himself if the transition is related to some broken symmetry and what is its order parameter.
2.3
Chiral symmetry
In the massless limit, the QCD Lagrangian is invariant under transformations of the chiral group U (Nf)L× U (Nf)R: U (Nf)L× U (Nf)R: ψL 7→ ULψL ψR 7→ URψR UL , UR ∈ U (Nf) . (2.12)
These transformations act separately on the left and right spinors ψL,R≡ ˆPL,Rψ ≡ 1∓γ5
2 ψ through unitary Nf× Nf matrices and chiral symmetry follows from the fact
that the kinetic term of the Lagrangian couples spinors having the same chirality, ¯
ψ 6 Dψ = ¯ψL6 DψL+ ¯ψR6 DψR . (2.13)
The mass term, on the contrary, explicitly breaks chiral symmetry because it couples terms with different chirality,
¯
ψmψ = ¯ψLmψR+ ¯ψRmψL . (2.14)
The chiral group can be rewritten as
U (Nf)L× U (Nf)R= U (1)L× U (1)R× SU (Nf)L× SU (Nf)R, (2.15)
where transformations in U (1)L× U (1)R change the phases of the left and right
spinors, while transformations in SU (Nf)L × SU (Nf)R act on the left and right
spinors through matrices ∈ SU (Nf):
U (1)L× U (1)R: ψL 7→ eiαLψL ψR 7→ eiαRψR αL , αR∈ R (2.16) SU (Nf)L× SU (Nf)R: ψL 7→ ULψL ψR 7→ URψR UL , UR ∈ SU (Nf) . (2.17)
Furthermore, SU (Nf)L×SU (Nf)Rmay be rewritten as the direct product SU (Nf)L×
SU (Nf)R= SU (Nf)V × SU (Nf)Aof the vector and axial subgroups1:
SU (Nf)V : ψL 7→ V ψL ψR 7→ V ψR V ∈ SU (Nf) (2.18) SU (Nf)A: ψL 7→ AψL ψR 7→ A†ψR A ∈ SU (Nf) . (2.19)
After quantization, the SU (Nf)V × SU (Nf)Asymmetry is spontaneously broken to
SU (Nf)V, giving rise, in agreement with Goldstone’s theorem, to the pion triplet or
to the pseudoscalar meson octet (depending on whether one considers u, d or u, d, s as approximately massless). The order parameter associated with the spontaneous breaking of chiral symmetry is the chiral condensate h ¯ψψi: at low T the symmetry is spontaneously broken and h ¯ψψi is nonzero, while at high T the symmetry is restored and the chiral condensate is zero.
2.4
Z
3symmetry
Pure gauge QCD has a Z3 symmetry, namely the gauge action is invariant under
the following transformation2:
U4(~x, t) 7→ zU4(~x, t), where z ∈ Z3≡ {1, e±i 2π
3 } , (2.20) 1
This is a slight abuse of notation: SU (Nf)A is not actually a subgroup because the axial
transformations are not a closed set. 2
The center of a group G is defined as the set of all elements x ∈ G such that xg = gx ∀g ∈ G.
which multiplies every temporal link U4(~x, t) in a given time slice t = ˆt by the same
cubic root of 1. This can be seen by noting that the gauge action is built from plaquettes and that the plaquettes not starting from ˆt are trivially invariant, while, since zz†= 1, for x ≡ (~x, ˆt) one has
Uµ4 = Uµ(x)U4(x+µ)Uµ†(x+4)U † 4(x) 7→ Uµ(x)zU4(x+µ)Uµ†(x+4)z†U † 4(x) = Uµ4 . (2.21) The order parameter associated to the Z3 symmetry is the Polyakov loop,
L(~x) = T r "Nt−1 Y t=0 U4(~x, t) # , (2.22)
which transforms like L(~x) 7→ zL(~x), hence its mean value is not invariant unless it is zero. The physical meaning of the Polyakov loop is given by the relation [20]
hL(~x)L(~y)†i = e−βFqq¯ (r≡|~x−~y|) , (2.23)
where Fqq¯ (r) is the free energy of a ¯qq pair separated by a distance r. At large
distances, L(~x) and L(~y)† are uncorrelated and one may factorize the expectation value:
lim|~x−~y|→∞hL(~x)L(~y)†i = hL(~x)ihL(~y)†i = |hLi|2 . (2.24)
In the last step, translational invariance has been used to replace L(x) by the spatial average L ≡ V1 P
~xL(~x). Therefore one has
|hLi|2 = e−βFqq¯ , (2.25)
where Fqq¯ is the free energy of a ¯qq pair separated by a large distance. At low T , hLi
is zero, Fqq¯ is infinite and this signalizes confinement. On the other hand, at high
T the center symmetry is spontaneously broken, hLi is nonzero and this signalizes deconfinement.
2.5
Finite quark masses
The previous two paragraphs considered the limiting cases of zero and infinite quark masses, but physical quark masses are neither zero nor infinite. The Columbia plot, represented in fig. 2.2, illustrates what happens for different values of the quark masses. In the upper-left corner, the up and down quark are massless, while the strange quark has infinite mass. The system has a SU (2)×SU (2) ∼ O(4) symmetry, which is spontaneously broken at low temperatures, and the phase transition can be of second order, like what happens for the 3d Ising model. In the lower-left corner, the up down and strange quarks are massless, at low temperatures the SU (3)×SU (3) symmetry is spontaneously broken and from universality arguments it follows that the phase transition must be of first order [35, 36]. In the upper-right corner, the up down and strange quarks have infinite masses, the theory has center symmetry, which is spontaneously broken at high temperatures, and the phase transition is of first order [37]. In the central region of the Columbia plot, finite quark masses explicitly break both symmetries and the transition is a crossover. Along the boundaries between the first order transition and the crossover regions, the transition is of second order. Lattice simulations indicate that the physical point lies in the crossover region [3].
Figure 2.2: Columbia plot (Source: [32])
2.6
The sign problem
In lattice QCD, expected values hOi = Z1 R [DA] O detM (µ)e−SG are estimated
by Monte Carlo methods, generating gauge configurations with probability P ∝ detM (µ)e−SG and averaging O over those configurations: this technique requires
the probability density P to be real and positive. When the baryochemical poten-tial is zero, the fermionic matrix is γ5-hermitian, that is
γ5M γ5 = M†. (2.26)
From γ5-hermitianicity, it follows that for the characteristic polynomial one has
p(λ) = det(M − λI) = det(γ52(M − λI)) = det(γ5(M − λI)γ5) =
= det(γ5M γ5− λγ5Iγ5) = det(M†− λI) = (det(M − λ∗I))∗ = p(λ∗)∗ .
(2.27) Therefore, if λ is an eigenvalue, λ∗ is also an eigenvalue. Hence the fermionic deter-minant is real. When the baryochemical potential is nonzero, however, in general the fermionic matrix is not γ5-hermitian; indeed the baryochemical potential is
in-troduced by adding a term γ0µ to the fermionic matrix and one has
γ5γ0µγ5= −γ0µ = −(γ0µ∗)†. (2.28)
Unless µ is purely imaginary, γ5-hermitianicity is lost and the fermionic determinant
is in general a complex number. This is an issue in lattice QCD, since the afore-mentioned probability density becomes complex as well and one cannot trivially carry out Monte Carlo simulations as in the case of zero µ (sign problem). Many methods have been introduced to deal with the sign problem: reweighting, analytic continuation from imaginary µ, the Taylor expansion method, etc.
The reweighting method [5–7] consists in rewriting the partition function as the expected value of a reweighting factor with respect to a well-defined (real and positive) probability density, i.e.
Z = Z
[DA] detM (µ)e−SG =
Z
[DA]detM (µ)
detM (0)detM (0)e
−SG = Z(0) detM (µ) detM (0) µ=0 , (2.29)
hence the expected value of an observable O becomes hOi = D OdetM (µ)detM (0) E µ=0 D detM (µ) detM (0) E µ=0 . (2.30)
A drawback of the reweighting method is that one has DdetM (µ)detM (0)E
µ=0 = Z(µ) Z(0) =
e−VT∆f (µ,T ), where ∆f is the difference in the free energy density between the two
en-sembles, and for larger volumes this term becomes exponentially smaller and harder to estimate. Furthermore, as µ shifts away from 0, the sampled configurations don’t have sufficient overlap with the reweighted ensemble and exponentially large statis-tics is needed to estimate hOi.
Another technique to work around the sign problem is the Taylor expansion method [16–18], where one rewrites the expected values as a power series in the dimensionless parameter ˆµ = aµ,
hOi =X n=0 1 n! ∂nhOi ∂ ˆµn µ=0µˆ n≡X n=0 cnµˆn . (2.31)
An issue with this approach is that cn is the expected value of the trace of some
polynomial in M−1 and ∂dµmMm . When going to higher order coefficients, the order of
the polynomial increases and computing cnby noisy estimators becomes increasingly
expensive.
The analytic continuation method [8–14] exploits the fact that, for a purely imaginary chemical potential µ = iµi, the fermionic matrix is still γ5-hermitian and
the fermionic determinant is real, therefore observables can be directly determined by Monte Carlo methods. The resulting estimates may then be analytically continued to real values of the chemical potential. A drawback of this technique is that the partition function for imaginary chemical potentials is periodic in µi
T, Z( µi T) = Z( µi T + 2π
3 ), and the region useful to perform an extrapolation to real µ is limited to µi T . 1.
2.7
Previous research on the pseudocritical line
The study of the pseudocritical line is an active area of research in lattice QCD and many numerical investigations have been conducted in order to determine the cur-vature coefficient kB, attacking the sign problem with the Taylor expansion and the
analytic continuation methods. In refs. [11, 13], kB= 0.018(4) and kB = 0.0126(22)
have been reported by studying the chiral susceptibility with the analytic continua-tion method, ref. [11] also found kB= 0.0132(18) by studying the chiral condensate.
These estimates are in agreement with the updated estimates obtained by the same groups after including the results of simulations performed on finer lattices (refs. [12, 14] found respectively kB = 0.020(4) and kB = 0.0132(10) by studying the
chiral susceptibility, ref. [12] also found kB = 0.0134(13) by studying the chiral
condensate). Ref. [15], combining the results obtained by analytic continuation for various observables, found kB = 0.0149(21).
Using the Taylor expansion method, ref. [18] estimated kB = 0.0066(20) by
studying the chiral condensate, while ref. [17] estimated kB= 0.00655(50) from the
expansion (shown in red and magenta) are somewhat lower than those obtained by analytic continuation (shown in blue, cyan and green).
0 0.005 0.01 0.015 0.02 0.025 kB arXiv:1102.1356 [hep-lat] arXiv:1011.3130 [hep-lat] arXiv:1410.5758 [hep-lat] arXiv:1410.5758 [hep-lat] arXiv:1403.0821 [hep-lat] arXiv:1507.07510 [hep-lat]
Figure 2.3: Curvature coefficient kB: previous determinations. From the left to the
right: ref. [18], ref. [17], ref. [11], ref. [11], ref. [13] and ref. [15]. The red, magenta, blue, cyan and green colors indicate that the estimate has been obtained respectively by Taylor expansion + chiral condensate, Taylor expansion + chiral susceptibility, analytic continuation + chiral condensate and analytic continuation + chiral sus-ceptibility, analytic continuation + combined analysis of various observables.
Since the transition is a crossover, a discrepancy in the results obtained by study-ing different observables is not unexpected, but refs. [18] and [11] studied the same observable, the chiral condensate, and adopted the same discretization, the tree-level Symanzik gauge action and the stout-staggered fermion action, hence their results allow a direct comparison. The former study found kB ≈ 0.0066 by Taylor
expan-sion, while the latter found kB ≈ 0.013 by analytic continuation: in light of this
discrepancy, further study is needed in order to investigate the systematics. Specif-ically, the cause of this discrepancy may lie in the different Ns/Ntratio of the finite
lattices used in the simulations by the two groups, in the different renormalization chosen for the chiral condensate, in the different prescription adopted to define Tc
or in the different method adopted to tackle the sign problem.
2.8
Phenomenology of heavy-ion collisions
The small µB region of the QCD phase diagram can be experimentally explored
by heavy-ion collision experiments, like those conducted at RHIC in Brookhaven, where beams collide with a center-of-mass energy per nucleon pair up to hundreds GeV. The energy density found inside the fireball resulting from the collisions can be estimated from the Bjorken’s formula [39]:
Bjorken= 1 S 1 cτ dE⊥ dy (2.32)
where S is an effective nuclear area that can be approximated by S ≈ π(1f m×A13)2,
τ is an effective thermalization time that may be approximated by τ ≈ 1f m/c and
dE⊥
dy is the transverse energy per unit of rapidity. For RHIC Au − Au collisions
experiments at √sN N = 200 GeV , A ≈ 200 and dEdy⊥ has been measured to be
≈ 600 GeV [40]. Thus, the energy density is Bjorken ≈ 5 ÷ 6 GeV /f m3 1
GeV /f m3, high enough to allow the formation of a quark-gluon plasma. The fireball then expands, cools down and rehadronizes until it reaches a point where hadronic multiplicities are fixed (chemical freezeout). The baryochemical potential µB and
the freezeout temperature TF Z can be determined by fitting the measured particle
yields and ratios with statistical models [41]. The simplest approach is to describe the system using the grand canonical partition function of a hadron resonance gas, lnZ(T, V, ~µ) =P ilnZi(T, V, ~µ), where lnZi(T, V, ~µ) = V gi 2π2 Z ∞ 0 ±p2dp ln(1 ± e−β( √ p2+m2 i−µi)) . (2.33)
Here ~µ ≡ (µB, µQ, µS) are the chemical potentials coupled to the conserved charges,
mi is the mass of the i-th particle specie, gi is its spin degeneracy factor, µi ≡
µBBi+ µQQi+ µSSi and the sign ± is + for a fermion and − for a boson. From
the partition function, one may derive the density of the i-th specie ni = gi 2π2 Z ∞ 0 p2dp eβ( √ p2+m2 i−µi)± 1 (2.34) and TF Z, µB may be extrapolated by minimizing
χ2=X i (Rexpi − Rstat i )2 σi , (2.35)
where Riexp is the yield for the i-th specie measured from experiment (with uncer-tainty σi) and Rstati is the yield resulting from the statistical model. Assuming that
TF Z . Tc, a comparison can be made with lattice determinations of the critical
Chapter 3
Curvature of the pseudocritical
line
This chapter discusses how to determine the curvature coefficient of the pseudocrit-ical line by Taylor expansion and illustrates the numerpseudocrit-ical set-up adopted for the numerical simulations, whose results are reported in the following chapter.
With the intention of checking the effects of the prescription used to locate the transition on the determination of kB, two different methods to determine kB are
illustrated, corresponding to two possible definitions of the pseudocritical tempera-ture at finite µB. The first technique, adopted in ref. [18], relies on the hypothesis
of constant value at Tc for the chiral condensate; this hypothesis is an
approxima-tion, whose validity is to be assessed. The second technique defines Tcas the actual
inflection point of the chiral condensate. Note that this prescription is faithful, as it remains valid in presence of a real phase transition, though it leads to noisier results, as it requires the evaluation of higher order derivatives. Both techniques require to evaluate the derivatives of the chiral condensate with respect to µB, hence a brief
discussion is given on how these derivatives can be measured.
The numerical set-up has been chosen so as to allow a direct comparison with previous determinations reported in the literature. Having this in mind, in order to investigate the systematic uncertainty due to the renormalization chosen for the chiral condensate, the two renormalization prescriptions used in [11, 18] have been adopted. For the same reason, the simulations have been carried out, as in these two studies, using the tree level Symanzik gauge action and the stout-smeared staggered fermion action.
3.1
Calculation of the curvature coefficient by Taylor
expansion
Due to the sign problem, studying the phase transition at nonzero baryochemical potential by numerical simulations is a difficult task. To make the problem tractable, one may focus on what happens at small µB. In a neighborhood of µB = 0, the
pseudocritical line can be parametrized as Tc(µB) = Tc0 1 − kB µB T0 c 2 + O(µ4B) ! , (3.1)
where Tc0 is the critical temperature at µB = 0, Tc0 ≡ Tc(0), and kBis the curvature
coefficient.
A suitable technique to determine kBfrom lattice simulations is the one adopted
in [18]; in this work, the critical temperature is defined under the hypothesis of constant value at Tcfor some observable O (i.e. the chiral condensate). Under this
hypothesis, the critical temperature at nonzero µB is defined by the relation:
O(T, µ2B)|T =Tc(µ2 B)≡ O(T 0 c, 0) . (3.2) The differential of O is dO = ∂O ∂T µ B=0 dT + ∂O ∂(µ2) µ B=0 d(µ2B) . (3.3) Since O is constant along the curve Tc(µ2B) by definition, one has dO = 0 and it
follows that dTc d(µ2 B) = − ∂O ∂(µ2 B) |µB=0,T =T0 c ∂O ∂T|µB=0,T =Tc0 , (3.4)
therefore it is possible to determine k by evaluating derivatives in µB = 0:
kB = −Tc0 dTc d(µ2B) = T 0 c ∂O ∂(µ2 B) |µ B=0,T =Tc0 ∂O ∂T|µB=0,T =Tc0 . (3.5)
Alternatively, in order to not rely on the constant O at Tc hypothesis, one may
Taylor expand O around µB= 0,
O(T, µB) = A(T ) + B(T )µ2B , (3.6)
where A(T ) ≡ O(T, 0) and B(T ) ≡ ∂(µ∂O2 B)
(T, 0). The critical temperature can then be defined as the maximum of O, i.e. for the chiral susceptibility, or as the inflection point of O, i.e. for the chiral condensate. The maximum of O can be located by requiring that
0 = O0(T, µB) = A0(T ) + B0(T )µ2B= A0(Tc0) + A00(Tc0)t + (B0(Tc0) + B00(Tc0)t)µ2B .
(3.7) In the above equation, quantities are derived with respect to T , A(T ) and B(T ) have been Taylor expanded around Tc0 and t ≡ T − Tc0. Since, by definition, A0(Tc0) = 0, it follows that t = −B 0(T0 c) A00(T0 c) + B00(Tc0)µ2B µ2B= −B 0(T0 c) A00(T0 c) µ2B+ O(µ4B) , (3.8) therefore one finds
kB =
B0(Tc0) A00(T0
c)
Tc0 . (3.9)
Likewise, to locate the inflection point of some observable O, one may impose the condition 0 = O00(T, µB), obtaining kB= B00(Tc0) A000(T0 c) Tc0 . (3.10)
Since one of the objectives of this work is to investigate the validity of the constant h ¯ψψri at Tc hypothesis, the critical temperature is defined as the inflection point of
the chiral condensate. Thus, the quantities needed to determine kB are
Tc0 A000(Tc0) = ∂T∂33O(T, 0)|T =T0 c B00(T0 c) = ∂ 2 ∂T2( ∂O(T,µB) ∂(µ2 B) |µB=0)|T =Tc0 . (3.11)
3.2
Calculation of derivatives with respect to µ
BTo find an expression for the derivatives with respect to µB of some observable O,
one starts from the partition function in the staggered formalism Z = Z [DU ]e−SG[U ] Y f =uds (detMf(U, mf, µf)) 1 4 (3.12) and calculates ∂logZ ∂µB = 1 Z ∂Z ∂µB = = Z1 R [DU ]e−SG[U ]{[ ∂ ∂µB(detMu) 1 4](detMd) 1 4(detMs) 1 4+ +[∂µ∂ B(detMd) 1 4](detMu) 1 4(detMs) 1 4+ +[∂µ∂ B(detMs) 1 4](detMu) 1 4(detMd) 1 4} = = Z1 R [DU ]e−SG[U ]Q f =uds(detMf) 1 4P uds14T r[M −1 f M 0 f] = hni , (3.13) where Mf0 ≡ ∂Mf
∂µB, the third step follows from ∂ ∂µB(detM ) 1 4 = 1 4(detM ) 1 4−1 ∂detM ∂µB = 1 4(detM ) 1 4∂logdetM ∂µB = = 14(detM )14∂T r[logM ] ∂µB = 1 4(detM ) 1 4T r[∂logM ∂µB ] = = 14(detM )14T r[M−1M0] (3.14)
and, in the last step, n has been defined as n ≡ X f =uds 1 4T r[M −1 f M 0 f] . (3.15)
Now, by choosing a generic expectation value hφi as observable, one obtains for the first derivative with respect to µB
∂hφi ∂µB = ∂ ∂µB 1 ZR [DU ]φe −SG[U ]Q
f =uds(detMf(U, mf, µf)) 1 4 =
= − 1
Z2∂Z∂µ R [DU ]φe−SG[U ]
Q
f =uds(detMf(U, mf, µf)) 1 4+
+Z1∂µ∂ R [DU ]φe−SG[U ]Q
f =uds(detMf(U, mf, µf)) 1 4 =
= −hnihφi + hφ0i + hnφi .
(3.16)
Likewise, for the second derivative one finds ∂2hφi ∂µ2B |µB=0= h(n 2+ n0)φi − hn2+ n0ihφi + h2nφ0+ φ00i , (3.17) where n0 ≡ X f =uds 1 4T r[M −1 f M 00 f − Mf−1M 0 fMf−1M 0 f] . (3.18)
3.3
Observable used to locate the transition
The observable chosen to study the phase transition is the chiral condensate h ¯ψψli ≡ T V ∂logZ ∂ml = T Vh X f =ud 1 4T r[M −1 f ]i . (3.19)
The chiral condensate is subject to both an additive and a multiplicative divergence, hence it needs to be renormalized. The following renormalizations will be used:
h ¯ψψr1i = (h ¯ψψli−2mlmsh ¯ψψsi)(T ) (h ¯ψψli−2mlmsh ¯ψψsi)(0) h ¯ψψr2i = mm4l π(h ¯ψψli(T ) − h ¯ψψli(0)) . (3.20)
The former is the renormalization introduced by [38] and used by [11], the additive divergence is eliminated by choosing a suitable combination of the light and strange quark condensates and the multiplicative divergence is eliminated by dividing with the subtracted condensate at zero T . The latter is the renormalization adopted by [18], the additive divergence is removed by subtracting the light condensate at zero T and the multiplicative divergence is removed by multiplying by the bare quark mass ml.
The second derivative with respect to µB of the light chiral condensate is given
by
∂2h ¯ψψli
∂µ2B |µB=0 = h(n
2+ n0) ¯ψψ
li − hn2+ n0ih ¯ψψli + h2n ¯ψψl0+ ¯ψψ00li , (3.21)
where the derivatives of the operator ¯ψψl are determined by using the property
(M−1)0 = −M−1M0M−1: ¯ ψψl0 = P f =ud14T r[−M −1 f M 0 fM −1 f ] ¯ ψψl00 = P f =ud14T r[2M −1 f Mf0M −1 f Mf0M −1 f − M −1 f Mf00M −1 f ] . (3.22) Similar expressions are obtained for the strange chiral condensate.
3.4
Numerical set-up
Simulations have been carried out using the tree-level Symanzik gauge action and the stout-smeared staggered fermion action:
Z = R [DU ]e−SG[U ]Q
f =uds(detMf(U, mf, µf)) 1 4 SG = −β3 P i,µ6=ν(56ReT rU 1×1 µν (i) −121 ReT rU 1×2 µν (i))
(Mf)i,j = mfδi,j+12ην(i)(eµδ4νUν(2)(i)δi,j−ν− e−µδ4νUν(2)†(i)δi,j+ν) .
(3.23)
Here Uµν1×1(i) and Uµν1×2(i) are respectively the plaquettes and the rectangular 1 × 2 loops starting from the i-th site and orientend along the µν plane, while Uν(2)(i)
denotes the gauge links, twice stout-smeared with an isotropic smearing parameter ρ = 0.15. The lines of constant physics adopted in the simulations are summarized in table 3.1: they have been obtained by spline interpolation of the values reported in refs. [30, 43], where the bare quark masses have been determined from the ratios
mK fK ,
mK
β ms ml= 28.15ms a (f m) 3.49 0.132 0.0046891652 0.2556 3.51 0.121 0.0042984014 0.2425 3.52 0.116 0.0041207815 0.2361 3.525 0.11350 0.0040319716 0.23297 3.53 0.111 0.0039431616 0.2297 3.535 0.10873 0.0038625222 0.22663 3.54 0.10643 0.0037808171 0.2235 3.545 0.10419 0.0037012433 0.22039 3.55 0.10200 0.0036234458 0.2173 3.555 0.099864 0.0035475666 0.21424 3.56 0.09779 0.0034738899 0.2112 3.565 0.095750 0.0034014209 0.20820 3.57 0.09378 0.0033314387 0.2052 3.58 0.08998 0.0031964476 0.1994 3.60 0.08296 0.0029470693 0.1881 3.62 0.07668 0.0027239787 0.1773 3.63 0.07381 0.0026220249 0.1722 3.635 0.07240 0.0025719360 0.1697 3.64 0.07110 0.0025257549 0.1672 3.645 0.06978 0.0024788632 0.1648 3.655 0.06731 0.0023911190 0.1601 3.66 0.06615 0.0023499112 0.1579 3.665 0.06500 0.0023090586 0.1557 3.67 0.06390 0.0022699822 0.1535 3.675 0.06284 0.0022323268 0.1514 3.68 0.06179 0.0021950266 0.1493 3.69 0.05982 0.0021250444 0.1453 3.71 0.05624 0.0019978686 0.1379