Lattice Quantum Chromodynamics

Abstract

Concepts and applications of lattice quantum chromodynamics (LQCD) are introduced. After discussing how to define quarks and gluons on the Euclidean hypercubic lattice, the strong coupling expansion and the weak coupling expansions are reviewed to see the vital role played by the quantum fluctuations in QCD. Fundamental techniques for numerical LQCD simulations such as the Markov Chain Monte Carlo method and the Hybrid Monte Carlo method are discussed in some details. As examples of the high precision LQCD simulations, numerical results of the heavy quark potential and the hadron masses are shown. Recent LQCD results on the baryon-baryon interactions are briefly discussed.

Appendix

3.1.1 Four Vectors and Dirac Matrices

In the (3+1)-dimensional Minkowski spacetime, coordinates, derivatives and four vectors with μ = 0, 1, 2, 3 are

$$ \displaystyle\begin{array}{rcl} x^{\mu } = (t,\mathbf{x}),\ \ \partial ^{\mu } = (\partial _{t},-\nabla ),\ \ A^{\mu } = (A^{0},\mathbf{A}).& &{}\end{array}$$

(3.97)

In the 4-dimensional Euclidean space, we define the corresponding vectors for μ = 4, 1, 2, 3 as

$$\displaystyle{ (x_{\mu })^{\mathrm{E}} = (\tau = it,\mathbf{x}),\ \ (\partial _{\mu })^{\mathrm{E}} = (\partial _{\tau } = -i\partial _{ t},\nabla ),\ \ (A_{\mu })^{\mathrm{E}} = (A_{ 4} = iA^{0},\mathbf{A}). }$$

(3.98)

In the (3+1)-dimensional Minkowski spacetime with the metric g ^{μ ν} = diag(1, −1, −1, −1), the Dirac matrices satisfy the following relations for μ = 0, 1, 2, 3,

$$\displaystyle{ \{\gamma ^{\mu },\gamma ^{\nu }\} = 2g^{\mu \nu },\ \ \left (\gamma ^{\mu }\right )^{\dag } =\gamma ^{0}\gamma ^{\mu }\gamma ^{0},\ \ \gamma ^{5} = i\gamma ^{0}\gamma ^{1}\gamma ^{2}\gamma ^{3} =\gamma _{ 5} = \left (\gamma _{5}\right )^{\dag }. }$$

(3.99)

In the standard Dirac representation, we have

$$\displaystyle\begin{array}{rcl} \gamma ^{0} = \left (\begin{array}{cc} 1& 0 \\ 0& - 1\\ \end{array} \right ),\ \ \gamma ^{j} = \left (\begin{array}{cc} 0 & \sigma _{j} \\ -\sigma _{j}&0\\ \end{array} \right ),\ \ \gamma ^{5} = \left (\begin{array}{cc} 0&1 \\ 1&0\\ \end{array} \right ),& &{}\end{array}$$

(3.100)

where σ _j are the Pauli matrices; $\sigma _{1} = \left (\begin{array}{cc} 0&1\\ 1 &0\\ \end{array} \right ),\sigma _{2} = \left (\begin{array}{cc} 0& - i\\ i & 0\\ \end{array} \right ),\sigma _{3} = \left (\begin{array}{cc} 1& 0\\ 0 & -1\\ \end{array} \right ).$

In the 4-dimensional Euclidean space with the metric δ _{μ ν}  = diag(1, 1, 1, 1), we define the Euclidean Dirac matrices as

$$\displaystyle\begin{array}{rcl} \varGamma _{\mu } \equiv \left (\gamma _{4} =\gamma ^{0},-i\boldsymbol{\gamma }\right ),\ \ \varGamma _{ -\mu }\equiv -\varGamma _{\mu },\ \ \mathrm{and}\ \ \varGamma _{5} \equiv \gamma ^{5},& &{}\end{array}$$

(3.101)

which satisfy the relations,

$$\displaystyle\begin{array}{rcl} \{\varGamma _{\mu },\varGamma _{\nu }\} = 2\delta _{\mu \nu },\ \ \varGamma _{\mu }^{\dag } =\varGamma _{\mu },\ \ \ \ (\mathrm{for}\ \mu = 1,2,3,4,5)& &{}\end{array}$$

(3.102)

3.1.2 SU(N) Algebra

Let $\mathcal{T}^{a}$ (a = 1, ⋯ , N ² − 1) are the Hermitian generators of the SU(N) group. They satisfy the Lie algebra

$$\displaystyle\begin{array}{rcl} \left [\mathcal{T}^{a},\mathcal{T}^{b}\right ] = if_{ abc}\mathcal{T}^{c},& &{}\end{array}$$

(3.103)

where f _abc is the structure constant being totally anti-symmetric in its indices. $(\mathcal{T}^{b})^{2}$ commutes with every generator $\mathcal{T}^{a}$ and is called the quadratic Casimir operator.

For N = 2, f _abc reduces to the anti-symmetric tensor ε _ijk with ε ₁₂₃ = 1. For N = 3, the non-vanishing components of f _abc read $f_{123} = 1,f_{147} = -f_{156} = f_{246} = f_{257} = f_{345} = -f_{367} = 1/2,f_{458} = f_{678} = \sqrt{3}/2$.

In the fundamental representation, $\mathcal{T}^{a}$ is written by the N × N matrices t ^a as

$$\displaystyle\begin{array}{rcl} t^{a} = \frac{1} {2}\lambda _{a},& &{}\end{array}$$

(3.104)

where λ _a for N = 2 reduce to the Pauli matrices σ _i , while those for N = 3 reduce to the Gell-Mann matrices.

Some useful relations of t ^a for general N are

$$\displaystyle{ \mathrm{tr}(t^{a}t^{b})\,=\,\frac{1} {2}\delta _{ab},\ \ \ t_{ij}^{a}t_{ kl}^{b}\,=\,\frac{1} {2}(\delta _{il}\delta _{jk} - \frac{1} {N}\delta _{ij}\delta _{kl}),\ \ \ (t^{a}t^{a})_{ ij} = C_{\mathrm{F}}\delta _{ij},\ \ \ \mathrm{with}\ C_{\mathrm{F}}\,=\,\frac{N^{2} - 1} {2N}. }$$

(3.105)

In the adjoint representation, $\mathcal{T}^{a}$ is written by (N ² − 1) × (N ² − 1) matrices T ^a as

$$\displaystyle\begin{array}{rcl} (T^{a})_{ bc} = -if_{abc},& &{}\end{array}$$

(3.106)

which satisfy the relations

$$\displaystyle\begin{array}{rcl} \mathrm{tr}(T^{a}T^{b})& =& N\delta _{ ab},\ \ \ (T^{a}T^{a})_{ bc} = C_{\mathrm{A}}\delta _{bc},\ \ \ \mathrm{with}\ C_{\mathrm{A}} = N.{}\end{array}$$

(3.107)

3.1.3 Gaussian and Grassmann Integrals

Basic Gaussian and Grassmann integrals are

$$\displaystyle\begin{array}{rcl} \int _{-\infty }^{+\infty }\frac{dx} {\sqrt{ 2\pi }}\ \mathrm{e}^{-ax^{2}/2 }& =& \frac{1} {\sqrt{a}},{}\end{array}$$

(3.108)

$$\displaystyle\begin{array}{rcl} \int \frac{dz^{{\ast}}dz} {2\pi i} \ \mathrm{e}^{-b\vert z\vert ^{2} }& =& \frac{1} {b},{}\end{array}$$

(3.109)

$$\displaystyle\begin{array}{rcl} \int d\bar{\xi }d\xi \ \mathrm{e}^{-c\bar{\xi }\xi }& =& c.{}\end{array}$$

(3.110)

Here x (z) is a real (complex) number, while $\bar{\xi }$ and ξ are anti-commuting Grassmann numbers ($\{\xi,\bar{\xi }\}= 0$, and $\xi ^{2} =\bar{\xi } ^{2} = 0$). a and b are assumed to be real and positive numbers, while c is an arbitrary complex number. Equation (3.109) can be shown by rewriting the integral in terms of the real and imaginary parts of z or in terms of the polar coordinates of z. Equation (3.110) can be shown by noting that $\mathrm{e}^{-c\bar{\xi }\xi } = 1 - c\bar{\xi }\xi$ and ∫ d ξ = ∂∕∂ ξ (integral = derivative) for Grassmann variables.

Generalization of the above results to the case of multiple variables is straightforward. For x = (x ₁, ⋯ , x _n ), z = (z ₁, ⋯ , z _n ), ξ = (ξ ₁, ⋯ , ξ _n ), and $\bar{\xi }= (\bar{\xi }_{1},\cdots \,,\bar{\xi }_{n})$ with $\{\xi _{k},\xi _{l}\} =\{\bar{\xi } _{k},\bar{\xi }_{l}\} =\{\xi _{k},\bar{\xi }_{l}\} = 0$, we have

$$\displaystyle\begin{array}{rcl} \int \prod _{l=1}^{n}\frac{dx_{l}} {\sqrt{2\pi }}\ \mathrm{e}^{-\frac{1} {2} xAx}& =& \frac{1} {\sqrt{\mathrm{Det }\ A}},{}\end{array}$$

(3.111)

$$\displaystyle\begin{array}{rcl} \int \prod _{l=1}^{n}\frac{dz_{l}^{{\ast}}dz_{ l}} {2\pi i} \ \mathrm{e}^{-z^{{\ast}}Bz }& =& \frac{1} {\mathrm{Det}\ B},{}\end{array}$$

(3.112)

$$\displaystyle\begin{array}{rcl} \int \prod _{l=1}^{n}d\bar{\xi }_{ l}d\xi _{l}\ \mathrm{e}^{-\bar{\xi }C\xi }& =& \mathrm{Det}\ C.{}\end{array}$$

(3.113)

Here A is a non-singular and real-symmetric matrix whose eigenvalues a _l satisfy a _l  > 0 for all l. B is a non-singular complex matrix whose complex eigenvalues b _l obtained by the biunitary transformation (UBV ^†) satisfy Re b _l  > 0 for all l. C is an arbitrary complex matrix with no conditions. Note that B and C do not have to be Hermitian matrices. In field theories, the label “l” summarizes all possible indices including spin, flavor, color, spacetime points etc. and “Det” denotes the determinant for all these indices.

3.1.4 Method of Characteristics

We need to construct a general solution of the following partial differential equation,

$$\displaystyle\begin{array}{rcl} \left (\lambda \frac{\partial } {\partial \lambda } +\beta (g) \frac{\partial } {\partial g}\right )f(\lambda,g) = 0.& &{}\end{array}$$

(3.114)

For this purpose, we introduce the running coupling $\bar{g}(\lambda )$ through $\lambda d\bar{g}/d\lambda = -\beta (\bar{g})$ whose formal solution reads

$$\displaystyle\begin{array}{rcl} \lambda =\exp \left (-\int _{g}^{\bar{g}(\lambda )} \frac{dg'} {\beta (g')}\right ).& &{}\end{array}$$

(3.115)

Then the solution of Eq. (3.114) can be written as

$$\displaystyle\begin{array}{rcl} f(\lambda,g) = f(1,\bar{g}(\lambda )).& &{}\end{array}$$

(3.116)

This can be explicitly checked by applying the partial derivative on both sides,

$$\displaystyle\begin{array}{rcl} \lambda \partial _{\lambda }f(\lambda,g)& =& \lambda (\partial _{\lambda }\bar{g})(\partial _{\bar{g}}f) = -\beta (\bar{g})(\partial _{\bar{g}}f), \\ \beta \partial _{g}f(\lambda,g)& =& \beta (g)\left.(\partial \bar{g}/\partial g)\right \vert _{\lambda }(\partial _{\bar{g}}f) =\beta (\bar{g})(\partial _{\bar{g}}f).{}\end{array}$$

(3.117)

where we have used the relation $\partial \bar{g}/\partial g =\beta (\bar{g})/\beta (g)$ obtained from Eq. (3.115).

In general, the first-order partial differential equation (PDE) can be transformed to a set of ordinary differential equations (ODEs) and can be solved by the method of characteristics. As an illustration, let us consider the following PDE,

$$\displaystyle\begin{array}{rcl} a(t,x)\partial _{t}u(t,x) + b(t,x)\partial _{x}f(t,x) = c(t,x).& &{}\end{array}$$

(3.118)

This is equivalent to the coupled ODEs,

$$\displaystyle\begin{array}{rcl} \frac{d\bar{t}} {ds} = a(\bar{t},\bar{x}),\ \ \frac{d\bar{x}} {ds} = b(\bar{t},\bar{x}),\ \ \frac{df(\bar{t},\bar{x})} {ds} = a(\bar{t},\bar{x}),& &{}\end{array}$$

(3.119)

where s parametrizes the “flow” of the coordinates. $(\bar{t}(s),\bar{x}(s))$. This is called the characteristic curve as shown in Fig. 3.15.

Fig. 3.15

Schematic illustration of the characteristic curve

The function f can be obtained by integrating the last equation of Eq. (3.119) on the characteristic curve from the initial point (t _i, x _i) to the final point (t _F, x _F) ≡ (t, x),

$$\displaystyle\begin{array}{rcl} f(t,x) = f(t_{\mathrm{i}},x_{\mathrm{i}}) + h(t,x,t_{\mathrm{i}},x_{\mathrm{i}})& &{}\end{array}$$

(3.120)

where h stands for an integration of the known function $c(\bar{t},\bar{x})$ on the characteristic curve. Equation (3.120) implies that the desired function at (t, x) is obtained essential by a “pullback” of the point to (t _i, x _i) along the characteristic curve. It is a straightforward exercise to generalize the above derivation to the system with more coordinates, (t, x).

3.1.5 Leapfrog Integrator in Molecular Dynamics

Let us start with a Tayler expansion of the field ϕ:

$$\displaystyle\begin{array}{rcl} \phi (s+\varepsilon )& =& \phi (s) +\varepsilon \dot{\phi } (s) + \frac{\varepsilon ^{2}} {2}\ddot{\phi }(s) + O(\varepsilon ^{3}), \\ & =& \phi (s) +\varepsilon \pi (s) + \frac{\varepsilon ^{2}} {2}\dot{\pi }(s) + O(\varepsilon ^{3}), \\ & =& \phi (s) +\varepsilon \pi (s +\varepsilon /2) + O(\varepsilon ^{3}),{}\end{array}$$

(3.121)

where we have used the equation of motion, $\dot{\phi }(s) \equiv d\phi (s)/ds =\pi (s)$. To evaluate π(s +ɛ∕2), we take the midpoint prescription which does not have O(ɛ ²) error,

$$\displaystyle\begin{array}{rcl} \pi (s +\varepsilon /2)& =& \pi (s -\varepsilon /2) +\varepsilon \dot{\pi } (s) + O(\varepsilon ^{3}) \\ & =& \pi (s -\varepsilon /2) -\varepsilon \frac{\delta S(\phi )} {\delta \phi (s)} + O(\varepsilon ^{3}).{}\end{array}$$

(3.122)

Equations (3.121) and (3.122) give a procedure to move the molecular dynamics one-step forward, (ϕ(s), π(s −ɛ∕2)) → (ϕ(s +ɛ), π(s +ɛ∕2)). The initial and final steps need to receive special care,

$$\displaystyle{ \pi (\varepsilon /2) =\pi (0) -\frac{1} {2}\varepsilon \frac{\delta S(\phi )} {\delta \phi (s)} + O(\varepsilon ^{2}),\ \ \ \pi (s_{\mathrm{ F}}) =\pi (s_{\mathrm{F}} -\varepsilon /2) -\frac{1} {2}\varepsilon \frac{\delta S(\phi )} {\delta \phi (s_{\mathrm{F}})} + O(\varepsilon ^{2}), }$$

(3.123)

which have only O(ε ²) accuracy. An illustration of this leapfrog integrator is shown in Fig. 3.16. Since the initial and final steps introduce O(ɛ ²) error irrespective of the length of the MD trajectory, and the intermediate steps introduce O(ɛ ³) ×ɛ ⁻¹ = O(ɛ ²) error as a whole, one finds Δ H = O(ɛ ²) after one MD trajectory before the Metropolis test.

The leapfrog integrator satisfies the reversibility and symplectic property, which can be checked explicitly by using the above definitions (Exercise 3.13).

Fig. 3.16

The leapfrog integrator