A

Appendix

A.1 The Lie groups SU(N)

In this appendix we collect basic definitions and conventions for the Lie groupsSU(N) – the special unitary groups – and the corresponding Lie algebrassu(N). For a more detailed presentation we refer the reader to [1–3].

A.1.1 Basic properties

The defining representation of SU(N) is given by complex N × N matriceswhich are unitary and have determinant 1. This set of matrices is closed undermatrix multiplication: Let Ω1 and Ω2 be elements of SU(N), i.e., they obeyΩ†

i = Ω−1i and det[Ωi] = 1. Using standard linear algebra manipulations we

obtain

(Ω1Ω2)† = Ω†2Ω

†1 = Ω−1

2 Ω−11 = (Ω1Ω2)−1 ,

det[Ω1Ω2] = det[Ω1] det[Ω2] = 1 (A.1)

and have thus established that also the product of two SU(N) matrices isan SU(N) matrix. The unit matrix is also in SU(N) and for each matrix inSU(N) there exists an inverse (the hermitian conjugate matrix). Thus, the setSU(N) forms a group. Since the group operation – the matrix multiplication –is not commutative the groups SU(N) are non-abelian groups.

A.1.2 Lie algebra

Let us now count how many real parameters are needed to describe the ma-trices in SU(N). A complex N × N matrix has 2N2 real parameters. Therequirement of unitarity introduces N2 independent conditions which the pa-rameters have to obey. One more parameter is used for obeying the deter-minant condition such that one needs a total of N2 − 1 real parameters fordescribing SU(N) matrices.

Gattringer, C., Lang, C.B.: Appendix. Lect. Notes Phys. 788, 327–336 (2010)DOI 10.1007/978-3-642-01850-3 BM2 c© Springer-Verlag Berlin Heidelberg 2010

328 A Appendix

A convenient way of representing SU(N) matrices is to write them asexponentials of basis matrices Tj , the so-called generators. In particular, wewrite an element Ω of SU(N) as

Ω = exp

⎛⎝i

N2−1∑j=1

ω(j) Tj

⎞⎠ , (A.2)

where ω(j), j = 1, 2, . . . , N2 − 1, are the real numbers needed to parame-terize Ω. We remark that the parameters ω(j) can be changed continuously,making SU(N) so-called Lie groups that are groups whose elements dependcontinuously on their parameters. In order to cover all of the group space,the parameters have to be varied only over finite intervals, making the groupsSU(N) so-called compact Lie groups.

The generators Tj , j = 1, 2, . . . , N2 − 1, are chosen as traceless, complex,and hermitian N ×N matrices obeying the normalization condition

tr [Tj Tk] =12δjk . (A.3)

In addition, they are related among each other by an algebra of commutationrelations

[Tj , Tk ] = i fjkl Tl . (A.4)

The completely anti-symmetric coefficients fjkl are the so-called structureconstants. Below we will give an explicit representation of the generators forthe groups SU(2) and SU(3).

Let us verify that the representation (A.2) indeed describes elements ofSU(N). Using the facts that the generators are hermitian and that the ω(j)

are real, one finds that hermitian conjugation of the right-hand side of (A.2)simply produces an extra minus sign in the exponent (from the complex con-jugation of i). Thus, it is obvious that (A.2) implies Ω† = Ω−1. To show thatthe determinant equals 1, we use the equation

detΩ = exp (tr [lnΩ]) = exp

⎛⎝i

N2−1∑j=1

ω(j) tr Tj

⎞⎠ = e0 = 1 , (A.5)

where in first step we have used a formula for the determinant (see (A.54)below) and in the third step we have used the fact that the Tj are traceless.

Not only the group elements but also the exponents of our representation(A.2) have an interesting structure. The linear combinations

N2−1∑j=1

ω(j) Tj (A.6)

of the Tj form the so-called Lie algebra su(N). Their commutation propertiesare governed by the relations (A.4). Elements of su(N) are also complex N×N

A.1 The Lie groups SU(N) 329

matrices but have properties different from the elements of the group. Oneimportant difference is the fact that the unit matrix is contained in the group(all ω(j) = 0), while it is not an element of the algebra (all Tj are traceless).

A.1.3 Generators for SU(2) and SU(3)

The standard representation of the generators for SU(2) is given by

Tj =12σj , (A.7)

with the Pauli matrices

σ1 =[

0 11 0

], σ2 =

[0 −ii 0

], σ3 =

[1 00 −1

]. (A.8)

In this case the structure constants are particularly simple, given by the com-pletely anti-symmetric tensor, i.e., fjkl = εjkl.

For SU(3) the generators are given by

Tj =12λj . (A.9)

The Gell–Mann matrices λj are 3 × 3 generalizations of the Pauli matrices:

λ1 =

⎡⎣

0 1 01 0 00 0 0

⎤⎦ , λ2 =

⎡⎣

0 −i 0i 0 00 0 0

⎤⎦ , λ3 =

⎡⎣

1 0 00 −1 00 0 0

⎤⎦ ,

λ4 =

⎡⎣

0 0 10 0 01 0 0

⎤⎦ , λ5 =

⎡⎣

0 0 −i0 0 0i 0 0

⎤⎦ , λ6 =

⎡⎣

0 0 00 0 10 1 0

⎤⎦ ,

λ7 =

⎡⎣

0 0 00 0 −i0 i 0

⎤⎦ , λ8 =

1√3

⎡⎣

1 0 00 1 00 0 −2

⎤⎦ . (A.10)

A.1.4 Derivatives of group elements

Let us now show an important property of derivatives of group elements. IfΩ(ω) is an element of SU(N) then

Mk(ω) = i∂Ω(ω)∂ω(k)

Ω(ω)† ∈ su(N) , (A.11)

i.e., the derivative times the hermitian conjugate is in the Lie algebra. In orderto prove this statement we have to show the defining properties of Lie algebraelements, i.e., we must show that Mk(ω) is hermitian and traceless.

Showing the hermiticity of Mk(ω) is straightforward. By differentiatingΩ(ω)Ω(ω)† = 1 with respect to ω(k) one finds

330 A Appendix

∂Ω(ω)∂ω(k)

Ω(ω)† + Ω(ω)∂Ω(ω)†

∂ω(k)= 0 . (A.12)

Thus

Mk(ω)† =(

i∂Ω(ω)∂ω(k)

Ω(ω)†)†

= −iΩ(ω)∂Ω(ω)†

∂ω(k)= i

∂Ω(ω)∂ω(k)

Ω(ω)† = Mk(ω) ,

(A.13)where we used (A.12) in the third step.

In order to show that Mk(ω) is traceless we use the fact that the deter-minant of a SU(N) matrix equals to 1 and we differentiate det[Ω(ω)] withrespect to ω(k). We obtain

0 =∂ det[Ω(ω)]

∂ω(k)=

∂ det[Ω(ω)]∂Ω(ω)ab

∂Ω(ω)ab

∂ω(k)

= det[Ω(ω)](Ω(ω)−1

)ba

∂Ω(ω)ab

∂ω(k)= tr

[∂Ω(ω)∂ω(k)

Ω(ω)†], (A.14)

where in the first step we applied the chain rule for derivatives. In the secondstep we used a standard formula for the derivative of the determinant det[Ω]with respect to an entry Ωab of the matrix Ω. In the third step we useddet[Ω] = 1 and Ω−1 = Ω†. Equation (A.14) establishes that Mk(ω) is alsotraceless and thus we have shown Mk(ω) ∈ su(N).

From (A.11) it follows that for the gauge transformation matrices Ω(x)with coefficients ω(k)(x), depending on the space–time coordinate x, the com-bination i (∂μΩ(x))Ω(x)† is in the Lie algebra, since

i (∂μΩ(x))Ω(x)† =∑

k

(i(

∂

∂ω(k)(x)Ω (ω(x))

)Ω (ω(x))†

)∂μω

(k)(x) ,

(A.15)and the right-hand side is a linear combination of su(N) elements with realcoefficients ∂μω

(k)(x).

A.2 Gamma matrices

The Euclidean gamma matrices γμ, μ = 1, 2, 3, 4 can be constructed from theMinkowski gamma matrices γM

μ , μ = 0, 1, 2, 3. The latter obey

{γMμ , γM

ν } = 2 gμν 1 , (A.16)

with the metric tensor given by gμν = diag(1,−1,−1,−1) and 1 is the 4 × 4unit matrix. Thus when we define the Euclidean matrices γμ by setting

γ1 = −iγM1 , γ2 = −iγM

2 , γ3 = −iγM3 , γ4 = γM

0 , (A.17)

we obtain the Euclidean anti-commutation relations

A.2 Gamma matrices 331

{γμ, γν} = 2 δμν 1 . (A.18)

In addition to the matrices γμ, μ = 1, 2, 3, 4 we define the matrix γ5 as theproduct

γ5 = γ1γ2γ3γ4 . (A.19)

The matrix γ5 anti-commutes with all other gamma matrices γμ, μ = 1, 2, 3, 4and obeys γ2

5 = 1.An explicit representation of the Euclidean gamma matrices can be ob-

tained from a representation of the Minkowski gamma matrices (see, e.g., [4]).Here we give the so-called chiral representation where γ5 (the chirality oper-ator) is diagonal:

γ1,2,3 =[

0 −iσ1,2,3

iσ1,2,3 0

], γ4 =

[0 12

12 0

], γ5 =

[12 00 −12

], (A.20)

where the σj are the Pauli matrices (A.8) and 12 is the 2 × 2 unit matrix.More explicitely the Euclidean gamma matrices read

γ1 =

⎡⎢⎢⎣

0 0 0 −i0 0 −i 00 i 0 0i 0 0 0

⎤⎥⎥⎦, γ2 =

⎡⎢⎢⎣

0 0 0 −10 0 1 00 1 0 0−1 0 0 0

⎤⎥⎥⎦, γ3 =

⎡⎢⎢⎣

0 0 −i 00 0 0 ii 0 0 00 −i 0 0

⎤⎥⎥⎦ ,

γ4 =

⎡⎢⎢⎣

0 0 1 00 0 0 11 0 0 00 1 0 0

⎤⎥⎥⎦ , γ5 =

⎡⎢⎢⎣

1 0 0 00 1 0 00 0 −1 00 0 0 −1

⎤⎥⎥⎦ . (A.21)

In addition to the anti-commutation relation (A.18) the gamma matrices obey(here μ = 1, . . . , 5)

γμ = γ†μ = γ−1μ . (A.22)

When we discuss charge conjugation, we need the charge conjugation ma-trix C defined through the relations (μ = 1, . . . , 4)

CγμC−1 = −γT

μ . (A.23)

Using the explicit form (A.21) it is easy to see that in the chiral representation(A.20) the charge conjugation matrix is given by

C = iγ2γ4 . (A.24)

It obeysC = C−1 = C† = −CT . (A.25)

We finally quote a simple formula for the inverse of linear combinations ofgamma matrices (a, bμ ∈ R):

(a1 + i

4∑μ=1

γμbμ

)−1

=a1− i

∑4μ=1 γμbμ

a2 +∑4

μ=1 b2μ

. (A.26)

This formula can be verified by multiplying both sides with a1 + i∑

μ γμbμ.

332 A Appendix

A.3 Fourier transformation on the lattice

The goal of this appendix is to discuss the Fourier transform f(p) of functionsf(n) defined on the lattice Λ. The lattice is given by

Λ = {n = (n1, n2, n3, n4) | nμ = 0, 1, . . . Nμ − 1} , (A.27)

and in most of our applications we have N1 = N2 = N3 = N , N4 = NT . Forthe total number of lattice points we introduce the abbreviation

|Λ| = N1N2N3N4 . (A.28)

We impose toroidal boundary conditions

f(n+ μNμ) = ei2πθμf(n) (A.29)

for each of the directions μ. Here μ denotes the unit vector in μ-direction. Di-rections with periodic boundary conditions have θμ = 0, anti-periodic bound-ary conditions correspond to θμ = 1/2.

The momentum space Λ, which corresponds to the lattice Λ with theboundary conditions (A.29), is defined as

Λ ={p = (p1, p2, p3, p4) | pμ =

2πaNμ

(kμ + θμ), kμ = −Nμ

2+ 1, . . . ,

Nμ

2

}.

(A.30)The boundary phases θμ have to be included in the definition of the momentapμ such that the plane waves

exp( i p · na) with p · n =4∑

μ=1

pμnμ (A.31)

also obey the boundary conditions (A.29).The basic formula, underlying Fourier transformation on the lattice, is

(here l is an integer with 0 ≤ l ≤ N − 1)

1N

N/2∑j=−N/2+1

exp(

i2πN

lj

)=

1N

N−1∑j=0

exp(

i2πN

lj

)= δl0 . (A.32)

For l = 0 this formula is trivial. For l = 0 (A.32) follows from applying thewell-known algebraic identity

N−1∑j=0

qj =1 − qN

1 − qto q = exp

(i2πNl

). (A.33)

We can combine four of the 1D sums in (A.32) to obtain the following iden-tities:

A.4 Wilson’s formulation of lattice QCD 333

1|Λ|

∑

p∈Λ

exp (i p · (n−n′)a) = δ(n− n′) = δn1n′1δn2n′

2δn3n′

3δn4n′

4, (A.34)

1|Λ|

∑n∈Λ

exp (i(p−p′) · na) = δ(p− p′) ≡ δk1k′1δk2k′

2δk3k′

3δk4k′

4. (A.35)

We stress that the right-hand side of (A.35) is a product of four Kroneckerdeltas for the integers kμ which label the momentum components pμ(compare(A.30)).

If we now define the Fourier transform

f(p) =1√|Λ|

∑n∈Λ

f(n) exp (−i p · na) , (A.36)

we find for the inverse transformation

f(n) =1√|Λ|

∑

p∈Λ

f(p) exp (i p · na) . (A.37)

The last equation follows immediately from inserting (A.36) in (A.37) andusing (A.34).

A.4 Wilson’s formulation of lattice QCD

In this appendix we collect the defining formulas for Wilson’s formulationof QCD on the lattice. The dynamical variables are the group-valued linkvariables Uμ(n) and the Grassmann-valued fermion fields ψ(f)(n)α

c, ψ

(f)(n)α

c.

They live on the links, respectively the sites of our lattice (A.27). Vacuumexpectation values are calculated according to

〈O〉 =1Z

∫D

[ψ,ψ

]D[U ] e−SF [ψ,ψ,U ]−SG[U ]O[ψ,ψ, U ] , (A.38)

where the partition function is given by

Z =∫

D[ψ,ψ

]D[U ] e−SF [ψ,ψ,U ]−SG[U ] . (A.39)

The measures over fermion and gauge fields are products over the measuresfor the individual field variables:

D[ψ,ψ

]=

∏n∈Λ

∏f,α,c

dψ(f)(n)αc

dψ(f)

(n)αc, D[U ] =

∏n∈Λ

4∏μ=1

dUμ(n) .

(A.40)

334 A Appendix

For the individual link variables Uμ(n) one uses the Haar measure discussed inSect. 3.1. For the fermions the rules for Grassmann integration from Sect. 5.1apply. The gauge field action for gauge group SU(N) is given by

SG[U ] =β

N

∑n∈Λ

∑μ<ν

Re tr [1− Uμν(n)] , (A.41)

where the plaquettes are defined as

Uμν(n) = Uμ(n)Uν(n+ μ)U−μ(n+ μ+ ν)U−ν(n+ ν)= Uμ(n)Uν(n+ μ)Uμ(n+ ν)† Uν(n)† . (A.42)

The fermion action is a sum over Nf flavors:

SF [ψ,ψ, U ] =Nf∑f=1

a4∑

n,m∈Λ

ψ(f)

(n)D(f)(n|m)ψ(f)(m) (A.43)

and the lattice Dirac operator is given by

D(f)(n|m)α βa b

=(m(f) +

4a

)δαβ δab δn,m− 1

2a

±4∑μ=±1

(1− γμ)αβ Uμ(n)ab δn+μ,m .

(A.44)In (A.42) and in the last equation we use the conventions

γ−μ = −γμ , U−μ(n) = Uμ(n− μ)† , μ = 1, 2, 3, 4 . (A.45)

We remark that Wilson’s Dirac operator (A.44) is γ5-hermitian, i.e., it obeys

γ5Dγ5 = D† . (A.46)

A.5 A few formulas for matrix algebra

In quantum mechanics one usually deals with hermitian or unitary matrices,while in lattice QCD often more general matrices occur. In this appendix welist a few results for general complex matrices together with short remarksconcerning their proof (for a more detailed account see, e.g., [5]).

The basic result for general complex matrices is that they are unitarilyequivalent to upper triangular matrices: Let M be a complex-valued N ×Nmatrix. Then there exits a unitary matrix U and an upper triangular matrixT , such that

U†M U = T . (A.47)

This result can be proven by induction in N . The elements tj on the diagonalof T are the roots of the characteristic polynomial of M since

A.5 A few formulas for matrix algebra 335

P (λ) = det[M − λ1] = det[T − λ1] =N∏

j=1

(tj − λ) . (A.48)

An important consequence of this result is a unique classification of ma-trices that can be diagonalized with a unitary transformation. A complexmatrix M is called normal if it commutes with its hermitian conjugate, i.e.,[M,M†] = 0. It is obvious that hermitian or unitary matrices are normal. Theannounced result is: If and only if M is normal, then there exists a unitarymatrix U such that

U†M U = D , (A.49)

where D is diagonal. It is straightforward to see that a matrix M which isunitarily equivalent to a diagonal matrix is normal. To prove the other di-rection we first note that the normality of M implies the normality of thetriangular matrix T corresponding to M . When evaluating explicitly the twosides of the normality condition, T †T = TT †, for the upper triangular ma-trix T , one concludes that T must be diagonal and the statement is proven.Equations (A.49) and (A.48) imply that a normal matrix has a complete setof orthonormal eigenvectors, the columns of U .

The existence of a complete orthonormal set of eigenvectors v(j) with eigen-values λ(j) can be used to represent the matrix M in the form

M =N∑

j=1

λ(j) v(j) v(j)† , (A.50)

the so-called spectral representation. On the right-hand side of this equationmatrix/vector notation was used to write the dyadic product v(j)v(j)†. Thespectral representation of the matrix can be used to define a function of M interms of the function for the eigenvalues, if this exists. This gives rise to thespectral theorem

f (M) =N∑

j=1

f(λ(j)

)v(j) v(j)† . (A.51)

We finally discuss a formula for the expansion of the determinant:

det[1−M ] = exp (tr[ ln(1−M) ] ) . (A.52)

In this equation M is a complex matrix and the logarithm (where it exists) isdefined through its series expansion. The proof of (A.52) applies (A.47):

det[1−M ] = det[1− T ] =N∏

j=1

(1 − tj) = exp

⎛⎝

N∑j=1

ln (1 − tj)

⎞⎠ (A.53)

= exp

⎛⎝−

N∑j=1

∞∑n=1

1n

(tj)n

⎞⎠ = exp

(−

∞∑n=1

1n

tr[Tn]

)= exp (tr[ln(1−M)] ) .

336 A Appendix

In the fifth step we have used the fact that when evaluating powers of atriangular matrix the diagonal elements do not mix with other entries of thematrix. In the last step we used tr[Tn] = tr[Mn] which follows from (A.47).

Since a matrix A may always be written as A = 1−M , the result (A.52)is often stated as

det[A] = exp (tr[ lnA ] ) . (A.54)

References

1. H. Georgi: Lie Algebras in Particle Physics (Benjamin/Cummings, Reading,Massachusetts 1982) 327

2. H. F. Jones: Groups, Representations and Physics (Hilger, Bristol 1990) 3273. M. Hamermesh: Group Theory and Its Application to Physical Problems

(Addison-Wesley, Reading, Massachusetts 1964) 3274. M. E. Peskin and D. V. Schroeder: An Introduction to Quantum Field Theory

(Addison-Wesley, Reading, Massachusetts 1995) 3315. P. Lancaster: Theory of Matrices (Academic Press, New York 1969) 334

Index

Z3 symmetry, 304, 307, 322β-function, 67γ5-hermiticity, 121, 135, 166, 167, 187,

201, 202, 315MS-scheme, 160, 173, 175, 260, 267–2695D theory, 252

accept-reject step, 194, 197acceptance probability, 190, 195action, 9adjoint operator, 2anomaly, 238, 239, 271APE plot, 150APE-smearing, 142area preservation, 192asymptotic freedom, 68autocorrelation, 94autocorrelation time

exponential, 94integrated, 95

AWI, 273AWI-mass, 220, 279axial anomaly, 157, 160, 170axial symmetry, 160axial vector field operator, 268axial Ward identity, 273

Baker-Campbell-Hausdorff formula, 38balance equation, 77Banks-Casher relation, 175, 308bare mass, 279bare parameter, 67baryon

interpolator, 129

interpolator examples, 130baryon chemical potential, 312baryon number, 312basis, 2Bayesian analysis, 147Bi-CGStab, 140Bi-Conjugate Gradient, 139blocking, 213, 222, 223, 228, 236, 248blocking method

data, 96Boltzmann factor, 1, 21, 74bootstrap

statistical, 97boundary conditions, 82, 120

Cabibbo-Kobayashi-Maskawa matrix,294

canonical partition function, 322canonical quantization, 7, 11, 12Cayley-Hamilton relation, 198center symmetry, 304, 307, 322center transformation, 304charge conjugation, 117chemical potential, 301, 312

baryon, 312imaginary, 316, 321isospin, 316

chiral condensate, 161, 172chiral fermions, 252chiral limit, 153, 158chiral perturbation theory, 154, 274

partially quenched, 173chiral rotation, 158

338 Index

chiral symmetry, 157continuum, 157lattice, 164spontaneous breaking, 160

chirally improved Dirac operator, 235ChPT, 154, 173, 274CKM matrix, 294classical field, 21Clifford algebra, 234clover improvement, 217clover term, 217cold start, 91compact Lie group, 328condensate

bare, 173physical, 173

condition number, 181confinement, 43, 63conjugate gradient method, 139connected piece, 128conserved charge, 315conserved current, 274, 276, 277continuum limit, 21, 69, 153, 205

naive, 36correlation length, 205Coulomb gauge, 51, 53Coulomb part of static potential, 62covariant current, 276, 277covariant derivative, 29CP-violation, 295critical point, 205critical slowing down, 96critical temperature, 303crossover, 308current algebra, 267, 268

data blocking method, 96decay

strong, 284, 287, 289weak, 270, 295

decay constants, 267, 268deconfinement phase, 304, 305detailed balance condition, 78diquark, 129Dirac equation, 27Dirac operator, 223

domain wall, 249, 250fixed point, 164, 235Fourier transformation, 111

naive, 32, 110overlap, 164, 177staggered, 243, 248twisted mass, 253Wilson, 110, 112, 113

Dirac operator spectrum, 166disconnected pieces, 128discrete symmetries, 117discretization, 16, 19, 32domain wall fermions, 249doubler fermions, 103, 112, 113doubling problem, 110, 111dynamical critical exponent, 96dynamical fermions, 185, 307

Edinburgh plot, 150effective action, 216effective fermion action, 188effective mass, 144, 145eigenstate, 2eigenvalue density, 176eigenvector, 2electromagnetic form factor, 267electromagnetic pion form factor, 290Elitzur’s theorem, 53energy density, 310energy levels, 6energy spectrum

two particles, 288equilibration, 77, 90, 91equilibrium distribution, 77ergodicity, 77estimator

noisy, 203Euclidean action, 9, 17, 19, 20Euclidean correlator, 1, 4Euclidean time transporter, 14Euler-Lagrange equations, 11even-odd preconditioning, 202exceptional configurations, 141, 153,

208, 249excited states, 147extended source, 136

fat link, 143Fermi statistics, 103, 104fermion action, 26

naive, 34fermion contraction, 128

Index 339

fermion determinant, 104, 108, 117hopping expansion, 117observable, 186weight factor, 186

fermion line, 114fermion loop, 114, 117fermions

domain wall, 249fixed point, 164Ginsparg-Wilson, 208Kogut-Susskind, 243naive, 32, 110overlap, 164, 177staggered, 243twisted mass, 253Wilson, 110, 112

field strength tensor, 29Fierz transformation, 246finite size effects, 152, 284, 287fixed point, 231fixed point action, 232, 234, 235fixed point fermions, 164flavor, 26Fock space, 3form factor

pion, 290Fourier transformation, 222, 302, 332

Dirac operator, 111fugacity, 312

expansion, 312, 318, 322, 323full QCD calculations, 209full twist, 255functional, 18

gamma matrices, 27, 330gauge action, 30, 36, 37gauge fixing, 49, 51

Landau, 282gauge invariance, 28, 44, 49, 53gauge transformation

fermions, 28, 33gauge fields, 29link variables, 33

gauge transporter, 34Gauss-Seidel iteration, 139Gaussian integral, 8, 108Gell-Mann matrices, 329Gell-Mann–Oakes–Renner relation, 274generalized eigenvalue problem, 149

generalized parton distribution, 290generating functional, 236, 238

for fermions, 109generator, 328, 329Gibbs measure, 75Ginsparg-Wilson

circle, 167equation, 163, 164, 166, 226, 240fermions, 208

global update, 190glueballs, 56, 308gluodynamics, 43gluon fields, 26GMOR, 274Goldstone bosons, 162GPD, 290Grassmann algebra, 103, 105

differentiation, 105integration, 106

Grassmann numbers, 40, 103, 105group integrals

SU(N), 49SU(3), 46

group integration, 45

Haar measure, 40, 44, 45Lie group, 45SU(2), 86

hadroncorrelators, 132decay, 284interpolator, 124masses, 143spectroscopy, 123structure, 267

Hamiltonian, 11heat bath algorithm, 85

for SU(2), 85for SU(3), 88

heavy quark effective theory, 260, 263,280

heavy quarks, 261Heisenberg picture, 6Hilbert space, 2HMC, see hybrid Monte Carlohopping expansion, 114, 117

fermion determinant, 117quark propagator, 114

hopping matrix, 114

340 Index

hopping parameter, 114hot start, 91HQET, 260, 263, 280hybrid algorithms, 192, 199hybrid Monte Carlo, 85, 190, 192, 195

computational effort, 199HYP-smearing, 143

imaginary chemical potential, 316, 321importance sampling, 74, 75improvement, 213index theorem, 157, 168initialization, 90, 91instanton, 169, 176integration measure, 17, 20interpolator

baryon, 129hadron, 124meson, 124

inverse coupling, 44, 70inversion algorithms, 138isospin, 125, 129isospin chemical potential, 316

jackknife, 97Jacobi determinant, 46Jacobi iteration, 138Jacobi smearing, 137

Klein-Gordonequation, 11field, 10

Kogut-Susskind fermions, 199, 243Krylov space, 139

Luscher-Weisz gauge action, 217Lagrangian density, 10Landau gauge, 51Langevin, 85, 192, 199lattice, 16, 32lattice constant, 19lattice fermions, 103, 243lattice regularization, 12lattice spacing, 65, 66lattice units, 144, 151leapfrog integration, 192Legendre transformation, 11Lie algebra, 327, 328Lie group, 44, 327, 328

compact, 328Haar measure, 45

link variable, 33locality, 178low energy constants, 267, 278

Markov chain, 73, 75mass

AWI, 220, 279bare, 279PCAC, 220, 279residual, 279

mass-preconditioning, 202matrix elements, 289Matsubara frequencies, 303Matthews-Salam formula, 108maximal tree, 49maximal twist, 255, 257, 258maximum entropy method, 148Maxwell construction, 323measure, 20

gauge field, 44Haar, 44, 45

mesoncorrelator, 127interpolator, 124operator, 124quantum numbers, 124

Metropolis algorithm, 78gauge action, 79

microcanonical, 85molecular dynamics, 191

leapfrog, 191trajectory, 192

momentum projection, 131Monte Carlo method, 74

for SU(3), 80Monte Carlo step, 76Monte Carlo update, 76multi-histogram technique, 319multi-hit algorithm, 80multi-mass solver, 141multi-particle state, 3multi-pseudofermions, 201

n-point function, 110naive currents, 274naive fermion action, 32, 34, 110natural units, 151

Index 341

Nielsen-Ninomiya theorem, 162Noether current, 159, 273, 275noisy estimator, 203noisy pseudofermions, 188, 203

estimators, 188non-abelian group, 28, 327non-leptonic weak decays, 296non-singlet Axial Ward identity, 273norm, 115normal matrix, 167, 335NRQCD, 260, 263nucleon, 129numerical simulation, 73

operator, 2adjoint, 2self-adjoint, 2sink, 131source, 131trace, 3

operator product expansion, 267, 290,295

Osterwalder-Schrader reconstruction,21

overlap fermions, 164, 177, 316overlap operator, 177

eigenmode reduction, 180, 182numerics, 179polynomial approximation, 180rational approximation, 182Zolotarev approximation, 181

overrelaxation, 85, 88

parity, 119partial conservation of axial current, see

PCACpartial quenching, 134partial-global update, 190, 200particle–anti-particle asymmetry, 316partition function, 4, 16, 20parton distributions, 290path integral, 1, 16path integral quantization, 7, 19, 21Pauli matrices, 329Pauli principle, 40Pauli term, 216Pauli-Villars fields, 252PCAC, 219, 268, 270PCAC-mass, 220, 279

phase diagram, 205, 317Ginsparg-Wilson fermions, 208Wilson fermions, 206

phase shift, 287phase transitions, 205physical units, 151pion decay constant, 270pion field normalization, 269pion field operator, 268pion form factor, 290plane wave, 8, 14plaquette, 37point source, 136polar mass, 255Polyakov loop, 54, 57, 304, 305, 307polynomial HMC, 200Pontryagin index, 169potential

static, 43, 54, 56, 58, 64, 99preconditioning, 141pressure, 310propagator

quark, 112, 114pseudo heat bath

for SU(3), 88pseudofermion fields, 188, 252pseudofermions, 187, 188

noisy, 203pure gauge theory, 43

QCDcontinuum action, 25

QCD phase diagram, 317Quantum Chromodynamics, see QCDquark fields, 26quark number, 312

density, 312quark propagator, 112, 141

continuum limit, 112hopping expansion, 114

quark sources, 135quark-gluon plasma, 310quenched approximation, 133, 153, 185quenching

partial, 134

R-algorithm, 199random number generator, 84rational approximation, 182

342 Index

rational HMC, 200real space renormalization group, 227reflection

Euclidean, 119, 120time, 120

regularization independent scheme, 281renormalization, 281renormalization constants, 268, 279renormalization group, 67, 213, 227

equation, 232, 233flow, 231

residual mass, 279resonance, 287resonance decay, 284, 289reversibility, 192reweighting method, 319RI scheme, 281RI/MOM scheme, 281Rome-Southampton method, 281rotational invariance, 56running coupling, 67

saddle point, 232scalar field theory, 10scalar product, 2scale setting, 63, 65, 151scaling, 67scaling analysis, 69scattering, 287

amplitude, 287phase shift, 287

Schrodinger functional, 220Schwarz alternating procedure, 203screening masses, 311sea quarks, 133, 134selection probability, 190self interaction, 31self-adjoint operator, 2sequential source method, 292series expansion, 321Sheikholeslami-Wohlert coefficient, 216simple sampling, 74simulation, 89

fermions, 185, 190pure gauge theory, 73

smeared source, 137smearing, 142, 248

APE, 142HYP, 143

stout, 143smearing functions, 137smoothing, 142Sommer parameter, 64–66, 151source

extended, 136point, 136quark, 135smeared, 137

spectral theorem, 335spectroscopy, 123spin system, 22staggered Dirac operator, 248staggered fermions, 199, 243, 244

tastes, 245staggered transformation, 244standard deviation, 94staples, 79static potential, 43, 54, 56, 58, 64, 99

Coulomb part, 62parametrization, 59strong coupling expansion, 59

statistical analysis, 93statistical bootstrap, 97statistical mechanics, 22step scaling function, 264stochastic differential equation, 85stout-smearing, 143strange quark mass, 151string breaking, 63, 307string tension, 59, 62strong coupling expansion, 59structure constants, 328structure functions, 267SU(N)

gauge group, 69generators, 329group integrals, 49Lie group, 327

SU(2)Haar measure, 86random element, 83representation, 81unitarization, 82

SU(3), 28group integrals, 46random element, 84representation, 81unitarization, 82

Index 343

susceptibility, 310sweep, 91Symanzik improvement, 213–215symmetries, 236

γ5-hermiticity, 121, 135, 166, 167,187, 201, 202

charge conjugation, 117parity, 119reflection, 119

tadpole improvement, 219tastes of staggered fermions, 245, 248temperature, 301

deconfinement phase, 310phase diagram, 307, 308

temporal gauge, 51, 55thermal Wilson line, 305thermodynamic limit, 69thermodynamic quantities, 310time reflection, 120time slice, 56, 124, 136tmQCD, see twisted mass QCDtopological charge, 168topological sector, 173topological susceptibility, 170trace, 3trace class, 3Trotter formula, 8, 15truncated overlap operator, 253twist angle, 255twisted mass, 254

physical basis, 256twisted basis, 256

twisted mass fermions, 253twisted mass QCD, 253, 256, 258

U(1)gauge group, 69

random element, 83representation, 81unitarization, 82

U(1)A chiral symmetry, 160ultralocal, 189universality class, 248, 249UV-filtering, 201, 202

vacuum, 4vacuum energy, 5valence quarks, 133variational analysis, 148variational method, 288vector field operator, 268vector meson dominance, 291

wall-source, 137Ward identities, 219, 271

continuum formulation, 270weak decay constant, 270weak form factor, 267weak matrix elements, 294Weyl fermion, 240Wick rotation, 7Wick’s theorem, 109Wilson fermion action, 110, 112

discrete symmetries, 117Wilson gauge action, 36, 37Wilson line, 57, 305Wilson loop, 54, 64Wilson term, 113Witten-Veneziano formula, 170, 171Wolfenstein parameters, 295

Yang-Mills theory, 28

zero mode, 168Zolotarev approximation, 181