Introduction to Quantum Field Theory and Matter Under Extreme Conditions

Introduction to Quantum Field Theory and Matter under Extreme Conditions

Prof. Dr. David Blaschke

Institute for Theoretical Physics, University of Wroclaw, Poland Bogoliubov Laboratory for Theoretical Physics, JINR Dubna, Russia

Summer Semester 2007 Abstract

The series of lectures gives an introduction to the modern formulation of quantum field theories using Feynman path integrals. The formalism is developed first for the vacuum case and is then generalized to the conditions of finite temperatures, densities and strong fields with special emphasis on phase transitions, processes of particle creation in Quantum Field Theories of strong, electro-weak and gravitational interactions. Applications of the formalism are considered in the physics of condensed matter, plasma physics, ultrarelativistic heavy-ion collisions, high-intensity optical and X-ray lasers and the physics of compact objects such as neutron stars and black holes. Contents

1 Quantum Fields at Zero Temperature 1 1.1 Introduction ...... 1 1.2 Minkowski Space Conventions ...... 3 1.2.1 Four Vectors ...... 3 1.2.2 Dirac Matrices ...... 4 1.3 Dirac Equation ...... 6 1.3.1 Free Particle Solutions ...... 6 1.4 Green Functions ...... 9 1.4.1 Free-Fermion Propagator ...... 10 1.4.2 Green Function for the Interacting Theory ...... 13 1.4.3 Exercises ...... 14 1.5 Path Integral in Quantum Mechanics ...... 14 1.6 Functional Integral in Quantum Field Theory ...... 16 1.6.1 Scalar Field ...... 17 1.6.2 Lagrangian Formulation of Quantum Field Theory . . . . 19 1.6.3 Quantum Field Theory for a Free Scalar Field ...... 20 1.6.4 Scalar Field with Self-Interactions ...... 21 1.6.5 Exercises ...... 22 1.7 Functional Integral for Fermions ...... 23 1.7.1 Finitely Many Degrees of Freedom ...... 23 1.7.2 Fermionic Quantum Field ...... 26 1.7.3 Generating Functional for Free Dirac Fields ...... 28 1.7.4 Exercises ...... 31 1.8 Functional Integral for Gauge Field Theories ...... 31 1.8.1 Faddeev-Popov Determinant and Ghosts ...... 34 1.8.2 Exercises ...... 42 1.9 Dyson-Schwinger Equations ...... 42 1.9.1 Photon Vacuum Polarization ...... 42 1.9.2 Fermion Self Energy ...... 47 1.9.3 Exercises ...... 49 1.10 Perturbation Theory ...... 50 1.10.1 Quark Self Energy ...... 50

1 1.10.2 Dimensional Regularization ...... 53 1.10.3 Regularized Quark Self Energy ...... 57 1.10.4 Exercises ...... 59 1.11 Renormalized Quark Self Energy ...... 60 1.11.1 Renormalized Lagrangian ...... 60 1.11.2 Renormalization Schemes ...... 63 1.11.3 Renormalized Gap Equation ...... 66 1.11.4 Exercises ...... 68 1.12 Dynamical Chiral Symmetry Breaking ...... 68 1.12.1 Euclidean Metric ...... 68 1.12.2 Chiral Symmetry ...... 73 1.12.3 Mass Where There Was None ...... 75 References ...... 82

2 Quantum Fields at Finite Temperature and Density 84 2.1 Ensembles and Partition Function ...... 84 2.1.1 Partition function in Quantum Statistics and Quantum Field Theory ...... 85 2.1.2 Equivalence of Path Integral and Statistical Operator representation for the Partition function ...... 86 2.1.3 Bosonic Fields ...... 90 2.1.4 Fermionic Fields ...... 93 2.1.5 Gauge Fields ...... 95 2.2 Interacting Fermion Systems: Hubbard-Stratonovich Trick ...... 98 2.2.1 Walecka Model ...... 98 2.2.2 Nambu–Jona-Lasinio (NJL) Model ...... 98 2.2.3 Mesonic correlations at ﬁnite temperature ...... 106 2.2.4 Matsubara frequency sums ...... 107

3 Particle Production by Strong Fields 112 3.1 Introduction ...... 112 3.2 Dynamics of pair creation ...... 113 3.2.1 Creation of fermion pairs ...... 113 3.2.2 Creation of boson pairs ...... 118 3.3 Discussion of the source term ...... 119 3.3.1 Properties of the source term ...... 119 3.3.2 Numerical results ...... 121 3.4 Summary ...... 124

4 Problem sets - statistical QFT 127 4.1 Partition function - Introduction ...... 127 4.2 Partition function - Fermi systems ...... 128 4.3 Partition function - Quark matter ...... 129

5 Projects 130 5.1 Symmetry breaking. Goldstone-Theorem. Higgs-Eﬀect ...... 130 5.1.1 Spontaneous symmetry breaking: Complex scalar ﬁeld . . 130 5.1.2 Electroweak symmetry breaking: Higgs mechanism . . . . 131 Chapter 1

Quantum Fields at Zero Temperature

1.1 Introduction

This is the first part of a series of lectures whose aim is to provide the tools for the completion of a realistic calculation in quantum field theory (QFT) as it is relevant to Hadron Physics. Hadron Physics lies at the interface between nuclear and particle (high energy) physics. Its focus is an elucidation of the role played by quarks and gluons in the structure of, and interactions between, hadrons. This was once particle physics but that has since moved to higher energy in search of a plausible grand unified theory and extensions of the so-called Standard Model. The only high-energy physicists still focusing on hadron physics are those performing the numerical experiments necessary in the application of lattice gauge theory, and those pushing at the boundaries of applicability of perturbative QCD or trying to find new kinematic regimes of validity. There are two types of hadron: baryons and mesons: the proton and neutron are baryons; and the pion and kaon are mesons. Historically the names distinguished the particle classes by their mass but it is now known that there are structural differences: hadrons are bound states, and mesons and baryons are composed differently. Hadron physics is charged with the responsibility of providing a detailed understanding of the differences. To appreciate the difficulties inherent in this task it is only necessary to remember that even the study of two-electron atoms is a computational challenge. This is in spite of the fact that one can employ the Schrödinger equation for this problem and, since it is not really necessary to quantize the electromagnetic field, the underlying theory has few complications. The theory underlying hadron physics is quantum chromodynamics (QCD), and its properties are such that a simple understanding and simple calculations

1 are possible only for a very small class of problems. Even on the domain for which a perturbative application of the theory is appropriate, the final (observable) states in any experiment are always hadrons, and not quarks or gluons, so that complications arise in the comparison of theory with experiment. The premier experimental facility for exploring the physics of hadrons is the Thomas Jefferson National Accelerator Facility (TJNAF), in Newport News, Vir- ginia. Important experiments are also performed at the Fermilab National Ac- celerator Facility (FermiLab), in Batavia, Illinois, and at the Deutsches Elek- tronensynchrotron (DESY) in Hamburg. These facilities use high-energy probes and/or large momentum-transfer processes to explore the transition from the nonperturbative to the perturbative domain in QCD. On the basis of the introduction to nonperturbative methods in QFT in the vacuum, we will develop in the second part of the lecture series the tools for a generalization to the situation many-particle systems of hadrons at finite temperatures and densities in thermodynamical equilibrium within the Matsubara formalism. The third part in the series of lectures is devoted to applications for the QCD phase transition from hadronic matter to a quark-gluon plasma (QGP) in relativistic heavy-ion collisions and in the interior of compact stars. Data are provided from a completed program of experiments at the CERN-SPS and presently running programs at RHIC Brookhaven. The future of this direction will soon open a new domain of energy densities (temperatures) at CERN-LHC (2007) and at the future GSI facility FAIR, where construction shall be completed in 2015. The CBM experiment will then allow insights into the QCD phase transition at relatively low temperatures and high baryon densities, a situation which bears already similarities with the interior of compact stars, formed in supernova explosions and observed as pulsars in isolation or in binary systems. Modern astrophysical data have an unprecendented level of accuracy allowing for new stringent constraints on the behavior of the hadronic equation of state at high densities. The final part of lectures enters the domain of nonequilibrium QFT and will focus on a particular problem which, however, plays a central role: particle production in strong time-dependent external fields. The Schwinger mechanism for pair production as a strict result of quantum electrodynamics (QED) is still not experimentally verified. We are in the fortunate situation that developments of modern laser facilities in the X-ray energy domain with intensities soon reaching the Schwinger limit for electron- positron pair creation from vacuum will allow new insights and experimental tests of approaches to nonperturbative QFT in the strong field situation. These insights will allow generalizations for the other field theories such as the Standard Model and QCD, where still the puzzles of initial conditions in heavy-ion collisions and the origin of matter in the Universe remain to be solved. 1.2 Minkowski Space Conventions

In the ﬁrst part of this lecture series I will use the Minkowski metrics. Later I will employ a Euclidean metric because that is most useful and appropriate for nonperturbative calculations.

1.2.1 Four Vectors Normal spacetime coordinates are denoted by a contravariant four-vector:

xµ := (x0, x1, x2, x3) (t, x, y, z). (1.1) ≡ Throughout: c = 1 = h¯, and the conversion between length and energy is just:

1fm = 1/(0.197327 GeV) = 5.06773 GeV−1 . (1.2)

The covariant four-vector is obtained by changing the sign of the spatial components of the contravariant vector:

x := (x , x , x , x ) (t, x, y, z) = g xν , (1.3) µ 0 1 2 3 ≡ − − − µν where the metric tensor is 1 0 0 0  0 1 0 0  gµν = − . (1.4) 0 0 1 0    −   0 0 0 1   −  µ ν ν The contracted product of two four-vectors is (a, b) := gµνa b = aν b : i.e., a contracted product of a covariant and a contravariant four-vector. The Poincar´e- invariant length of any vector is x2 := (x, x) = t2 ~x2. − Momentum vectors are similarly deﬁned:

µ p = (E, px, py, pz) = (E, p~). (1.5) and (p, k) = p kν = E E p~~k . (1.6) ν p k − Likewise, (x, p) = tE ~xp~ . (1.7) − The momentum operator ∂ ∂ 1 pµ := i = (i , ~ ) =: i µ (1.8) ∂xµ ∂t i ∇ ∇ transforms as a contravariant four-vector, and I denote the four-vector analogue of the Laplacian as 2 µ ∂ ∂ ∂ := p pµ = µ . (1.9) − ∂xµ ∂x The contravariant four-vector associated with the electromagnetic ﬁeld is

Aµ(x) = (φ(x), A~(x)) (1.10) with the electric and magnetic ﬁeld strengths obtained from

F µν = ∂ν Aµ ∂µAν . (1.11) − For example, ∂ E = F 0i; E~ = ~ φ A.~ (1.12) i −∇ − ∂t Similarly, Bi = εijkF jk; j, k = 1, 2, 3. Analogous deﬁnitions hold in QCD for the chromomagnetic ﬁeld strengths.

1.2.2 Dirac Matrices The Dirac matrices are indispensable in a manifestly Poincaré covariant description of particles with spin; i.e., intrinsic angular momentum, such as fermions 1 with spin 2 . The Dirac matrices are defined by the Clifford Algebra γµ, γν = 2gµν (1.13) { } and one common 4 4 representation is [each entry represents a 2 2 matrix] × × 1 0 0 ~σ γ0 = , ~γ = , (1.14) 0 1 ~σ 0 " − # " − # where ~σ are the usual Pauli Matrices: 0 1 0 i 1 0 σ1 = , σ2 = , σ3 = , (1.15) 1 0 i −0 0 1 " # " # " − # and 1 = diag[1, 1]. Clearly γ† = γ ; and ~γ† = ~γ. NB. These properties are not 0 0 − specific to this representation; e.g., γ1γ1 = 1, an analogue of the properties of a purely imaginary number. − In discussing spin, two combinations of Dirac matrices frequently appear: i σµν = [γµ, γν] , γ5 = iγ0γ1γ2γ3 = γ , (1.16) 2 5 and I note that i γ5σµν = εµνρσσ , (1.17) 2 ρσ µνρσ 01223 with ε the completely antisymmetric Lévi-Civita tensor: ε = +1, εµνρσ = εµνρσ. In the representation introduced above, − 0 1 γ5 = . (1.18) " 1 0 # Furthermore, γ , γµ = 0 , γ† = γ . (1.19) { 5 } 5 5 γ5 plays a special role in the discussion of parity and chiral symmetry, two key aspects of the Standard Model. The “slash” notation is a frequently used shorthand: γµA =: A/ = γ0A0 ~γA~ , (1.20) µ − γµp =: /p = γ0E ~γp~ , (1.21) µ − ∂ ∂ γµp =: i / i∂/ = iγ0 + i~γ ~ = iγµ . (1.22) µ ∇ ≡ ∂t ∇ ∂xµ (1.23) The following identities are important in evaluating the cross sections for decay and scattering processes:

trγ5 = 0 , (1.24) tr1 = 4 , (1.25) tra//b = 4(a, b) , (1.26) tra/ a/ a/ a/ = 4[(a , a )(a , a ) (a , a )(a , a ) + (a , a )(a , a )] ,(1.27) 1 2 3 4 1 2 3 4 − 1 3 2 4 1 4 2 3 tra/1 . . . a/n = 0 , for n odd , (1.28) trγ5a//b = 0 , (1.29) µ ν ρ σ trγ5a/1a/2a/3a/4 = 4iεµνρσa1 a2a3a4 , (1.30) µ γµa/γ = 2a/ , (1.31) µ − γµa/b/γ = 4(a, b) , (1.32) γ a/b//cγµ = 2/bc/a/ . (1.33) µ − (1.34) They can all be derived using the fact that Dirac matrices satisfy the Cliﬀord algebra. For example (remember trAB = trBA): µ ν tra//b = aµbνtrγ γ (1.35) 1 = a b tr[γµγν + γνγµ] (1.36) µ ν 2 1 = a b tr[2gµν1] (1.37) µ ν 2 1 = a b 2gµν4 = 4(a, b) . (1.38) µ ν 2 (1.39) 1.3 Dirac Equation

The unification of special relativity (Poincaré covariance) and quantum mechanics took some time. Even today many questions remain as to a practical imple- mentation of a Hamiltonian formulation of the relativistic quantum mechanics of interacting systems. The Poincaré group has ten generators: the six associated with the Lorentz transformations – rotations and boosts – and the four associated with translations. Quantum mechanics describes the time evolution of a system with interactions, and that evolution is generated by the Hamiltonian. However, if the theory is formulated with an interacting Hamiltonian then boosts will almost always fail to commute with the Hamiltonian and thus the state vector calculated in one momentum frame will not be kinematically related to the state in another frame. That makes a new calculation necessary in every frame and hence the discussion of scattering, which takes a state of momentum p to another state with momentum p0, is problematic. (See, e.g., Ref. [2]). The Dirac equation provides the starting point for a Lagrangian formulation of the quantum field theory for fermions interacting via gauge boson exchange. For a free fermion [i∂/ m] ψ = 0 , (1.40) − where ψ(x) is the fermion’s “spinor” – a four component column vector, while in the presence of an external electromagnetic field the fermion’s wave function obeys [i∂/ eA/ m] ψ = 0 , (1.41) − − which is obtained, as usual, via “minimal substitution:” pµ pµ eAµ in Eq. (1.40). These equations have a manifestly covariant appearance→ but−proving their covariance requires the development of a representation of Lorentz transformations on spinors and Dirac matrices:

ψ0(x0) = S(Λ) ψ(x) , (1.42) ν µ −1 ν Λ µγ = S (Λ)γ S(Λ) , (1.43) i S(Λ) = exp[ σ ωµν] , (1.44) −2 µν where ωµν are the parameters characterising the particular Lorentz transformation. (Details can be found in the early chapters of Refs. [1, 3].)

1.3.1 Free Particle Solutions As usual, to obtain an explicit form for the free-particle solutions one substitutes a plane wave and ﬁnds a constraint on the wave number. In this case there are two qualitatively diﬀerent types of solution, corresponding to positive and negative energy. (An appreciation of the physical reality of the negative energy solutions led to the prediction of the existence of antiparticles.) One inserts

ψ(+) = e−i(k,x) u(k) , ψ(−) = e+i(k,x) v(k) , (1.45) into Eq. (1.40) and obtains

(k/ m) u(k) = 0 , (k/ + m) v(k) = 0 . (1.46) − Assuming that the particle’s mass in nonzero then working in the rest frame yields (γ0 1) u(m,~0) = 0 , (γ0 + 1) v(m,~0) = 0 . (1.47) − There are clearly (remember the form of γ0) two linearly-independent solutions of each equation:

1 0 0 0 0 1 0 0 u(1)(m,~0) =   , u(2)(m,~0) =   , v(1)(m,~0) =   , v(2)(m,~0) =   . 0 0 1 0                  0   0   0   1        (1.48)  The solution in an arbitrary frame can be obtained simply via a Lorentz boost but it is even simpler to observe that

(k/ m) (k/ + m) = k2 m2 = 0 , (1.49) − − with the last equality valid when the particles are on shell, so that the solutions for arbitrary kµ are: for positive energy (E > 0),

E + m 1/2 φα(m,~0) k/ + m 2m u(α)(k) = u(α)(m,~0) =  (1.50), σ k α 2m(m + E)  · φ (m,~0)     2m(m + E)  q    q  with the two-component spinors, obviously to be identiﬁed with the fermion’s spin in the rest frame (the only frame in which spin has its naive meaning)

1 0 φ(1) = , φ(2) = ; (1.51) 0 ! 1 ! and, for negative energy, σ k · χα(m,~0) (α) k/ + m (α)  2m(m + E)  v (k) = − v (m,~0) = (1.52), 1/2 2m(m + E)  qE + m   α ~   χ (m, 0)  q  2m    with χ(α) obvious analogues of φ(α) in Eq. (1.51). (This approach works because it is clear that there are two, and only two, linearly-independent solutions of the momentum space free-fermion Dirac equations, Eqs. (1.46), and, for the homogeneous equations, any two covariant solutions with the correct limit in the rest-frame must give the correct boosted form.) In quantum field theory, as in quantum mechanics, one needs a conjugate state to define an inner product. For fermions in Minkowski space that conjugate is ψ¯(x) = ψ†(x)γ0, which satisfies

← ψ¯(i ∂/ + m) = 0 . (1.53) This equation yields the following free particle spinors in momentum space k/ + m u¯(α)(k) = u¯(α)(m,~0) (1.54) 2m(m + E) q k/ + m v¯(α)(k) = v¯(α)(m,~0) − , (1.55) 2m(m + E) q where I have used γ0(γµ)†γ0 = γµ, which relation is easily derived and is particularly important in the discussion of intrinsic parity; i.e., the transformation properties of particles and antiparticles under space reﬂections (an improper Lorentz transformation). The momentum space free-particle spinors are orthonormalised u¯(α)(k) u(β)(k) = δ u¯(α)(k) v(β)(k) = 0 αβ . (1.56) v¯(α)(k) v(β)(k) = δ v¯(α)(k) u(β)(k) = 0 − αβ It is now possible to construct positive and negative energy projection operators. Consider Λ (k) := u(α)(k) u¯(α)(k) . (1.57) + ⊗ αX=1,2

It is plain from the orthonormality relations, Eqs. (1.56), that Λ+(k) projects onto positive energy spinors in momentum space; i.e.,

(α) (α) (α) Λ+(k) u (k) = u (k) , Λ+(k) v (k) = 0 . (1.58) Now, since 1 0 1 + γ0 u(α)(m,~0) u¯(α)(m,~0) = = , (1.59) ⊗ 0 0 ! 2 αX=1,2 then 1 1 + γ0 Λ (k) = (k/ + m) (k/ + m) . (1.60) + 2m(m + E) 2 Noting that for k2 = m2; i.e., on shell,

(k/ + m) γ0 (k/ + m) = 2E (k/ + m) , (1.61) (k/ + m) (k/ + m) = 2m (k/ + m) , (1.62) one ﬁnally arrives at the simple closed form:

k/ + m Λ (k) = . (1.63) + 2 m The negative energy projection operator is

k/ + m Λ (k) := v(α)(k) v¯(α)(k) = − . (1.64) − − ⊗ 2 m αX=1,2 The projection operators have the following important properties:

2 Λ(k) = Λ(k) , (1.65)

tr Λ(k) = 2 , (1.66)

Λ+(k) + Λ−(k) = 1 . (1.67)

Such properties are a characteristic of projection operators.

1.4 Green Functions

The Dirac equation is a partial differential equation. A general method for solving such equations is to use a Green function, which is the inverse of the differential operator that appears in the equation. The analogy with matrix equations is obvious and can be exploited heuristically. Equation (1.41): [i∂/ eA/(x) m] ψ(x) = 0 , (1.68) x − − yields the wave function for a fermion in an external electromagnetic field. Con- sider the operator obtained as a solution of the following equation

0 0 4 0 [i∂/ 0 eA/(x ) m] S(x , x) = 1 δ (x x) . (1.69) x − − − It is immediately apparent that if, at a given spacetime point x, ψ(x) is a solution of Eq. (1.68), then ψ(x0) := d4x S(x0, x) ψ(x) (1.70) Z is a solution of 0 0 [i∂/ 0 eA/(x ) m] ψ(x ) = 0 ; (1.71) x − − i.e., S(x0, x) has propagated the solution at x to the point x0. This eﬀect is equivalent to the application of Huygen’s principle in wave mechanics: if the wave function at x, ψ(x), is known, then the wave function at x0 is obtained by considering ψ(x) as a source of spherical waves that propagate outward from x. The amplitude of the wave at x0 is proportional to the amplitude of the original wave, ψ(x), and the constant of proportionality is the propagator (Green function), S(x0, x). The total amplitude of the wave at x0 is the sum over all the points on the wavefront; i.e., Eq. (1.70). This approach is practical because all physically reasonable external ﬁelds can only be nonzero on a compact subdomain of spacetime. Therefore the solution of the complete equation is transformed into solving for the Green function, which can then be used to propagate the free-particle solution, already found, to arbitrary spacetime points. However, obtaining the exact form of S(x0, x) is impossible for all but the simplest cases.

1.4.1 Free-Fermion Propagator In the absence of an external ﬁeld Eq. (1.69) becomes

0 4 0 [i∂/ 0 m] S(x , x) = 1 δ (x x) . (1.72) x − − Assume a solution of the form:

4 0 0 d p −i(p,x0−x) S0(x , x) = S0(x x) = 4 e S0(p) , (1.73) − Z (2π) so that substituting yields

/p + m (/p m) S (p) = 1 ; i.e., S (p) = . (1.74) − 0 0 p2 m2 − To obtain the result in conﬁguration space one must adopt a prescription for handling the on-shell singularities in S(p), and that convention is tied to the boundary conditions applied to Eq. (1.72). An obvious and physically sensible deﬁnition of the Green function is that it should propagate positive-energy- fermions and -antifermions forward in time but not backwards in time, and vice versa for negative energy states. As we have already seen, the wave function for a positive energy free-fermion is ψ(+)(x) = u(p) e−i(p,x) . (1.75) The wave function for a positive energy antifermion is the charge-conjugate of the negative-energy fermion solution:

(+) 0 i(p,x) ∗ T −i(p,x) ψc (x) = C γ v(p) e = C v¯(p) e , (1.76) where C = iγ2γ0 and ( )T denotes matrix transpose. (This deﬁnes the operation · 0 of charge conjugation.) It is thus evident that our physically sensible S0(x x) 0 − can only contain positive-frequency components for x0 x0 > 0. One can ensure this via a small modiﬁcation of the denominator− of Eq. (1.74):

/p + m /p + m S (p) = , (1.77) 0 p2 m2 → p2 m2 + iη − − with η 0+ at the end of all calculations. Inserting this form in Eq. (1.73) is equivalen→t to evaluating the p0 integral by employing a contour in the complex-p0 that is below the real-p0 axis for p0 < 0, and above it for p0 > 0. This prescription deﬁnes the Feynman propagator. To be explicit:

3 0 d p ip~·(~x0−~x) 1 S0(x x) = 3 e − Z (2π) 2 ω(p~) 0 ∞ dp 0 0 /p + m 0 0 /p + m e−ip (x0−x0) e−ip (x0−x0) , × −∞ 2π " p0 ω(p~) + iη − p0 + ω(p~) iη # Z − − (1.78) where the energy ω(p~) = √p~2 + m2. The integrals are easily evaluated using standard techniques of complex analysis, in particular, Cauchy’s Theorem. Focusing on the ﬁrst term of the sum inside the square brackets, it is apparent that the integrand has a pole in the fourth quadrant of the complex p0-plane. For 0 0 x0 x0 > 0 we can evaluate the p integral by considering a contour closed by a semicircle− of radius R in the lower half of the complex p0-plane: the integrand vanishes exponen→tially∞ along that arc, where p0 = iy, y > 0, because 0 0 − ( i) ( iy) (x0 x0) = y (x0 x0) < 0. The closed contour is oriented clockwise so−that− − − −

0 ∞ dp 0 0 /p + m 0 0 e−ip (x0−x0) = ( ) i e−ip (x0−x0)(/p + m) 0 + p0=ω(p~)−iη+ Z−∞ 2π p ω(p~) + iη − − 0 −iω(p~)(x −x0) 0 = i e 0 (γ ω(p~) γ p~ + m) − 0 − · = i e−iω(p~)(x0−x0) 2m Λ (p) . (1.79) − + 0 For x0 x0 < 0 the contour must be closed in the upper half plane but therein the integrand− is analytic and hence the result is zero. Thus

0 ∞ dp 0 0 /p + m 0 −ip (x0−x0) 0 −iω(p~)(x0−x) e = i θ(x0 x0) e 2m Λ+(p~) . −∞ 2π p0 ω(p~) + iη+ − − Z − Observe that the projection operator is only a function of p~ because p0 = ω(p~). Consider now the second term in the brackets. The integrand has a pole in the second quadrant. For x0 x > 0 the contour must be closed in the lower half 0 − 0 0 plane, the pole is avoided and hence the integral is zero. However, for x0 x0 < 0 the contour should be closed in the upper half plane, where the integrand− is obviously zero on the arc at inﬁnity. The contour is oriented anticlockwise so that

0 ∞ dp 0 0 /p + m 0 0 −ip (x0−x0) 0 −ip (x0−x) e = i θ(x0 x ) e (/p + m) 0 + 0 p0=−ω(p~)+iη+ Z−∞ 2π p + ω(p~) iη − − 0 0 +iω(p~)(x0−x) 0 = i θ(x0 x0) e ( γ ω( p~) γ p~ + m) − 0 − − · = i θ(x x0 ) e+iω(p~)(x0−x) 2m Λ ( p~) . (1.80) 0 − 0 − − Putting these results together, changing variables p~ p~ in Eq. (1.80), we → − have

3 0 d p m 0 −i(p,x˜ 0−x) S0(x x) = i 3 θ(x0 x0) e Λ+(p~) − − Z (2π) ω(p~) − h 0 + θ(x x0 ) ei(p,x˜ −x)Λ (p~) , (1.81) 0 − 0 − i µ [(p˜ ) = (ω(p~), p~)] which, using Eqs. (1.58) and its obvious analogue for Λ−(p~), manifestly propagates positive-energy solutions forward in time and negative energy solutions backward in time, and hence satisﬁes the physical requirement stipulated above. Another useful representation is obtained merely by observing that

0 0 0 ∞ dp 0 0 /p + m ∞ dp 0 0 γ ω(p~) ~γ p~ + m −ip (x0−x) −ip (x0−x) e 0 + = e 0 − · + Z−∞ 2π p ω(p~) + iη Z−∞ 2π p ω(p~) + iη − 0 − ∞ dp 0 0 1 −ip (x0−x) = e 2mΛ+(p~) , −∞ 2π p0 ω(p~) + iη+ Z − (1.82) and similarly

0 0 0 ∞ dp 0 0 /p + m ∞ dp 0 0 γ ω(p~) ~γ p~ + m −ip (x0−x) −ip (x0−x) e 0 + = e − 0 − · + Z−∞ 2π p + ω(p~) iη Z−∞ 2π p + ω(p~) iη − 0 − ∞ dp 0 0 1 −ip (x0−x) = e 2mΛ−( p~) , −∞ 2π − p0 + ω(p~) iη+ Z − (1.83) so that Eq. (1.78) can be rewritten

4 0 d p −i(p,x0−x) m 1 1 S0(x x) = e Λ+(p~) Λ−( p~) . − (2π)4 ω(p~) " p0 ω(p~) + iη − − p0 + ω(p~) iη # Z − − (1.84) The utility of this representation is that it provides a single Fourier amplitude for the free-fermion Green function; i.e., an alternative to Eq. (1.77):

m 1 1 S0(p) = Λ+(p~) Λ−( p~) , (1.85) ω(p~) " p0 ω(p~) + iη − − p0 + ω(p~) iη # − − which is indispensable in making a connection between covariant perturbation theory and time-ordered perturbation theory: the second term generates the Z- diagrams in loop integrals.

1.4.2 Green Function for the Interacting Theory We now return to Eq. (1.69):

0 0 4 0 [i∂/ 0 eA/(x ) m] S(x , x) = 1 δ (x x) , (1.86) x − − − which defines the Green function for a fermion in an external electromagnetic field. As mentioned, a closed form solution of this equation is impossible in all but the simplest field configurations. Is there, nevertheless, a way to construct a systematically improvable approximate solution? To achieve that one rewrites the equation:

0 4 0 0 0 [i∂/ 0 m] S(x , x) = 1 δ (x x) + eA/(x ) S(x , x) , (1.87) x − − which, it is easily seen by substitution, is solved by

0 0 4 0 S(x , x) = S0(x x) + e d y S0(x y)A/(y) S(y, x) (1.88) − Z − 0 4 0 = S0(x x) + e d y S0(x y)A/(y) S0(y x) − Z − − 2 4 4 0 +e d y1 d y2 S0(x y1)A/(y1) S0(y1 y2)A/(y2) S0(y2 x) + . . . Z Z − − − (1.89)

This perturbative expansion of the full propagator in terms of the free propagator provides an archetype for perturbation theory in quantum field theory. One obvious application is the scattering of an electron/positron by a Coulomb field, which is a worked example in Sec. 2.5.3 of Ref. [3]. Equation (1.89) is a first example of a Dyson-Schwinger equation [4]. This Green function has the following interpretation:

1. It creates a positive energy fermion (antifermion) at spacetime point x;

2. Propagates the fermion to spacetime point x0; i.e., forward in time;

3. Annihilates this fermion at x0. The process can equally well be viewed as 1. The creation of a negative energy antifermion (fermion) at spacetime point x0; 2. Propagation of the antifermion to the spacetime point x; i.e., backward in time; 3. Annihilation of this antifermion at x. Other propagators have similar interpretations.

1.4.3 Exercises 1. Prove these relations for on-shell fermions:

(k/ + m) γ0 (k/ + m) = 2E (k/ + m) , (k/ + m) (k/ + m) = 2m (k/ + m) .

2. Obtain the Feynman propagator for the free-ﬁeld Klein Gordon equation:

2 2 (∂x + m )φ(x) = 0 , (1.90) in forms analogous to Eqs. (1.81), (1.84).

1.5 Path Integral in Quantum Mechanics

Local gauge theories are the keystone of contemporary hadron and high-energy physics. Such theories are diﬃcult to quantise because the gauge dependence is an extra non-dynamical degree of freedom that must be dealt with. The modern approach is to quantise the theories using the method of functional integrals, attributed to Feynman and Kac. References [3, 5] provide useful overviews of the technique, which replaces canonical second-quantisation. It is customary to motivate the functional integral formulation by reviewing the path integral representation of the transition amplitude in quantum mechanics. Beginning with a state (q1, q2, . . . , qN ) at time t, the probability that one 0 0 0 0 will obtain the state (q1, q2, . . . , qN ) at time t is given by (remember, the time evolution operator in quantum mechanics is exp[ iHt], where H is the system’s Hamiltonian): −

N n n+1 0 0 0 0 dpα(ti) q , q , . . . , q ; t q1, q2, . . . , qN ; t = lim dqα(ti) h 1 2 N | i n→∞ 2π αY=1 Z iY=1 Z iY=1 n+1 qt + qt −1 exp i p (t )[q (t ) q (t )] H(p(t ), j j )(1.91), ×  α j α j − α j−1 − j 2  jX=1   0 0 where tj = t + j, = (t t)/(n + 1), t0 = t, tn+1 = t . A compact notation is commonly introduced to represen− t this expression:

t0 q0t0 qt J = [dq] [dp] exp i dτ [p(τ)q˙(τ) H(τ) + J(τ)q(τ)] , (1.92) h | i Z Z ( Zt − ) where J(t) is a classical external “source.” NB. The J = 0 exponent is nothing but the Lagrangian. Recall that in Heisenberg’s formulation of quantum mechanics it is the operators that evolve in time and not the state vectors, whose values are ﬁxed at a given initial time. Using the previous formulae it is a simple matter to prove that the time ordered product of n Heisenberg position operators can be expressed as

0 0 q t T Q(t1) . . . Q(tn) qt = h | { }| i 0 t [dq] [dp] q(t1) q(t2) . . . q(tn) exp i dτ [p(τ)q˙(τ) H(τ)] .(1.93) Z Z ( Zt − ) NB. The time ordered product ensures that the operators appear in chronological order, right to left. Consider a source that “switches on” at ti and “switches oﬀ” at tf , t < ti < 0 tf < t , then

0 0 J 0 0 J q t qt = dqidqf q t qf tf qf tf qiti qiti qt . (1.94) h | i Z h | i h | i h | i Alternatively one can introduce a complete set of energy eigenstates to resolve the Hamiltonian and write

0 0 0 0 −iEn(t −t) q t qt = q φn e φn q (1.95) h | i n h | i h | i X t0 i 0 t →−+i∞ 0 −iE0(t −t) →= ∞ q φ e φ q ; (1.96) h | 0i h 0| i i.e., in either of these limits the transition amplitude is dominated by the ground state. It now follows from Eqs. (1.94), (1.96) that

0 0 q t qt J lim 0 h | i = dq dq φ q t q t q t q t φ =: W [J] ; 0 −iE0(t −t) 0 i f 0 f f f f i i i i 0 t i e q φ0 φ0 q h | i h | i h | i t →−+i∞ h | i h | i Z → ∞ (1.97) i.e., the ground-state to ground-state transition amplitude (survival probability) in the presence of the external source J. From this it will readily be apparent that, with tf > t1 > t2 > . . . > tm > ti,

m δ W [J] m = i dqidqf φ0 qf tf qf tf T Q(t1) . . . Q(tn) qiti qiti φ0 , δJ(t ) . . . δJ(t ) h | i h | { }| i h | i 1 1 J=0 Z

(1.98) which is the ground state expectation value of a time ordered product of Heisen- berg position operators. The analogues of these expectation values in quantum field theory are the Green functions. δ The functional derivative introduce here: δJ(t) , is defined analogously to the derivative of a function. It means to write J(t) J(t) + (t); expand the functional in (t); and identify the leading order co→efficient in the expansion. Thus for 0 0 n Hn[J] = dt J(t ) (1.99) Z

0 0 n 0 0 0 n 0 0 n δHn[J] = δ dt J(t ) = dt [J(t ) + (t )] dt J(t ) (1.100) Z Z − Z = dt0 n J(t0)n−1 (t0) + [. . .]. (1.101) Z So that taking the limit δJ(t0) = (t0) δ(t t0) one obtains → − δH [J] n = n J(t)n−1 . (1.102) δJ(t) The limit procedure just described corresponds to defining δJ(t) = δ(t t0) . (1.103) δJ(t0) − This example, which is in fact the “product rule,” makes plain the very close analogy between functional differentiation and the differentiation of functions so that, with a little care, the functional differentiation of complicated functions is straightforward. Another note is in order. The fact that the limiting values in Eqs. (1.96), (1.97) are imaginary numbers is a signal that mathematical rigour may be found more easily in Euclidean space where t itE . Alternatively, at least in principle, the argument can be repeated and made→ − rigorous by making the replacement + En En iη, with η 0 . However, that does not help in practical calculations,→whic−h proceed via→Monte-Carlo methods; i.e., probability sampling, which is only effective when the integrand is positive definite.

1.6 Functional Integral in Quantum Field The- ory

The Euclidean functional integral is particularly well suited to a direct numerical evaluation via Monte-Carlo methods because it defines a probability measure. However, to make direct contact with the perturbation theory of canonical second- quantised quantum field theory, I begin with a discussion of the Minkowski space formulation. 1.6.1 Scalar Field I will consider a scalar field φ(t, x), which is customary because it reduces the number of indices that must be carried through the calculation. Suppose that a large but compact domain of space is divided into N cubes of volume 3 and label each cube by an integer α. Define the coordinate and momentum via

1 3 ˙ 1 3 ∂φ(t, x) qα(t) := φα(t) = 3 d x φ(t, x) , q˙α(t) := φα(t) = 3 d x ; (1.104) ZVα ZVα ∂t i.e., as the spatial averages over the cube denoted by α. The classical dynamics of the ﬁeld φ is described by a Lagrangian:

N L(t) = d3x L(t, x) 3 L (φ˙ (t), φ (t), φ (t)) , (1.105) → α α α αs Z αX=1 where the dependence on φαs(t); i.e., the coordinates in the neighbouring cells, is necessary to express spatial derivatives in the Lagrangian density, L(x). With the canonical conjugate momentum deﬁned as in a classical ﬁeld theory

∂L 3 ∂Lα 3 pα(t) := = =: πα(t) , (1.106) ∂φ˙α(t) ∂φ˙α(t) the Hamiltonian is obtained as

3 H = pα(t) q˙α(t) L(t) =: Hα , (1.107) α − α X X H (π (t), φ (t), φ (t)) = π (t) φ˙ (t) L . (1.108) α α α αs α α − α The ﬁeld theoretical equivalent of the quantum mechanical transition amplitude, Eq. (1.91), can now be written

N n n dπ (t ) n+1 φ (t ) φ (t ) lim dφ (t ) 3 α i exp i 3 π (t ) α j α j−1 + α i α j − n→∞,→0 2π  α ( αY=1 Z iY=1 Z iY=1 jX=1 X  φα(tj) + φα(tj−1) φαs(tj) + φαs(tj−1) Hα πα(tj), , − 2 2 !  t0 ∂φ(τ, ~x)  =: [ φ] [ π] exp i dτ d3x π(τ, ~x) H(τ, ~x) , (1.109) Z D Z D ( Zt Z " ∂τ − #) where, as classically, π(t, ~x) = ∂L(t, ~x)/∂φ˙(t, ~x) and its average over a spacetime cube is just πα(t). Equation (1.109) is the transition amplitude from an initial 0 field configuration φα(t0) := φα(t) to a final configuration φα(tn+1) := φα(t ). Generating Functional In quantum field theory all physical quantities can be obtained from Green functions, which themselves are determined by vacuum-to-vacuum transition amplitudes calculated in the presence of classical external sources. The physical or interacting vacuum is the analogue of the true ground state in quantum mechanics. And, as in quantum mechanics, the fundamental quantity is the generating functional:

1 4 ˙ 1 2 W [J] := [ φ] [ π] exp i d x [π(x)φ(x) H(x) + 2iηφ (x) + J(x)φ(x)] , D D − N Z Z Z (1.110) where is chosen so that W [0] = 1, and a real time-limit is implemented and made meaningfulN through the addition of the η 0+ term. (NB. This subtracts → a small imaginary piece from the mass.) It is immediately apparent that

n 1 δ W [J] 0˜ T φˆ(x1)φˆ(x2) . . . φˆ(xn) 0˜ = h | { }| i =: G(x1, x2, . . . , xn) , in δJ(x )δJ(x ) . . . δJ(x ) 0˜ 0˜ 1 2 n J=0 h | i (1.111) where 0˜ is the physical vacuum. G(x1, x2, . . . , xn) is the complete n-point Green function| ifor the scalar quantum field theory; i.e., the vacuum expectation value of a time-ordered product of n field operators. The appearance of the word “complete” means that this Green function includes contributions from products of lower-order Green functions; i.e., disconnected diagrams. The fact that the Green functions in a quantum field theory may be defined via Eq. (1.111) was first observed by Schwinger [6] and does not rely on the functional formula for W [J], Eq. (1.110), for its proof. However, the functional formulism provides the simplest proof and, in addition, a concrete means of calculating the generating functional: numerical simulations.

Connected Green Functions It is useful to have a systematic procedure for the a priori elimination of disconnected parts from a n-point Green function because performing unnecessary work, as in the recalculation of m < n-point Green functions, is ineﬃcient. A connected n-point Green function is given by

n n−1 δ Z[J] Gc(x1, x2, . . . , xn) = ( i) , (1.112) − δJ(x )δJ(x ) . . . δJ(x ) 1 2 n J=0

where Z[J], deﬁned via W [J] =: exp iZ[J] , (1.113) { } is the generating functional for connected Green functions. It is instructive to illustrate this for a simple case (recall that I have already proven the product rule for functional diﬀerentiation): δ2Z[J] δ2 ln W [J] Gc(x1, x2) = ( i) = − δJ(x ) δJ(x ) − δJ(x ) δJ(x ) 1 2 J=0 1 2 J=0

δ 1 δW [J] = − δJ(x1) "W [J] δJ(x2)# J=0

1 δW [J] δW [J] 1 δ2W [J] = + W 2[J] δJ(x ) δJ(x ) − W [J] δJ(x ) δJ(x ) 1 2 J=0 1 2 J=0

0˜ φˆ(x ) 0˜ 0˜ φˆ(x ) 0˜ 0˜ T φˆ(x )φˆ(x ) 0˜ = ih | 1 | i ih | 2 | i i2 h | { 1 2 }| i 0˜ 0˜ 0˜ 0˜ − 0˜ 0˜ h | i h | i h | i = G(x ) G(x ) + G(x , x ) . (1.114) − 1 2 1 2 1.6.2 Lagrangian Formulation of Quantum Field Theory The double functional integral employed above is cumbersome, especially since it involves the ﬁeld variable’s canonical conjugate momentum. Consider then a Hamiltonian density of the form 1 H(x) = π2(x) + f[φ(x), ~ φ(x)] . (1.115) 2 ∇ In this case Eq. (1.110) involves

4 1 2 ˙ [ π] exp i d x [ 2π (x) + π(x)φ(x)] Z D Z − {i d4x [φ˙(x)]2} i 4 ˙ 2 = e [ π] exp 2 d x [π(x) φ(x)] Z D − Z − R 4 2 = e{i d x [φ˙(x)] } N , (1.116) × R where “N” is simply a constant. (This is an example of the only functional integral than can be evaluated exactly: the Gaussian functional integral.) Hence, with a Hamiltonian of the form in Eq. (1.115), the generating functional can be written N 1 W [J] = [ φ] exp i d4x [L(x) + iηφ2(x) + J(x)φ(x)] .(1.117) D 2 N Z Z Recall now that the classical Lagrangian density for a scalar ﬁeld is

L(x) = L0(x) + LI (x) , (1.118) 1 L (x) = [∂ φ(x) ∂µφ(x) m2φ2(x)] , (1.119) 0 2 µ − with LI (x) some functional of φ(x) that usually does not depend on any derivatives of the field. Hence the Hamiltonian for such a theory has the form in Eq. (1.115) and so Eq. (1.117) can be used to define the quantum field theory. 1.6.3 Quantum Field Theory for a Free Scalar Field The interaction Lagrangian vanishes for a free scalar field so that the generating functional is, formally,

1 4 1 µ 2 2 2 W0[J] = ¯ [ φ] exp i d x 2[∂µφ(x) ∂ φ(x) m φ (x) + iηφ (x)] + J(x)φ(x) . D − N Z Z (1.120) The explicit meaning of Eq. (1.120) is

1 4 4 1 4 W0[J] = lim dφα exp i φα Kαβ φβ + Jαφα , + ¯  2  →0 α α α N Z Y  X Xβ X  (1.121)  4  where α, β label spacetime hypercubes of volume and Kαβ is any matrix that satisfies 2 2 4 lim Kαβ = [ ∂ m + iη]δ (x y) , (1.122) →0+ − − − + + + where α →0 x, β →0 y and 4 →0 d4x; i.e., K is any matrix whose → → α → αβ continuum limit is the inverse ofPthe FeynmanR propagator for a free scalar field. (NB. The fact that there are infinitely many such matrices provides the scope for “improving” lattice actions since one may choose Kαβ wisely rather than for simplicity.) Recall now that for matrices whose real part is positive definite

n 1 n n dxi exp xi Aij xj + bi xi Rn {−2 } Z iY=1 i,jX=1 Xi=1 n/2 n n/2 (2π) 1 −1 (2π) 1 t −1 = exp bi (A )ij bj = exp b A b .(1.123) √detA {2 } √detA 2 i,jX=1 Hence Eq. (1.121) yields

1 1 1 4 4 1 −1 W0[J] = lim exp Jα (K )αβ Jβ , (1.124) + ¯ 0 2 8 →0 √detA { α i } N X Xβ where, obviously, the matrix inverse is deﬁned via

−1 Kαγ (K )γβ = δαβ . (1.125) γ X Almost as obviously, consistency of limits requires 1 lim δ = δ4(x y) , lim 4 = d4x , (1.126) + 4 αβ + →0 − →0 α X Z so that, with 1 −1 O(x, y) := lim (K )αβ , (1.127) →0+ 8 the continuum limit of Eq. (1.125) can be understood as follows: 1 1 lim 4 K (K−1) = lim δ + αγ 8 γβ + 4 αβ →0 γ →0 X d4w [ ∂2 m2 + iη]δ4(x w) O(w, y) = δ4(x y) ⇒ − x − − − . Z [ ∂2 m2 + iη] O(x, y) = δ4(x y) . (1.128) ·· − x − − Hence O(x, y) = ∆ (x y); i.e., the Feynman propagator for a free scalar ﬁeld: 0 − d4p 1 ∆ (x y) = e−i(q,x−y) . (1.129) 0 − (2π)4 q2 m2 + iη Z − (NB. This makes plain the fundamental role of the “iη+” prescription in Eq. (1.77): it ensures convergence of the expression deﬁning the functional integral.) Putting this all together, the continuum limit of Eq. (1.124) is

1 i W [J] = exp d4x d4y J(x) ∆ (x y) J(y) . (1.130) ˆ −2 0 − N Z Z 1.6.4 Scalar Field with Self-Interactions

A nonzero interaction Lagrangian, LI [φ(x)], provides for a self-interacting scalar ﬁeld theory (from here on I will usually omit the constant, nondynamical normalisation factor in writing the generating functional):

4 W [J] = [ φ] exp i d x [L0(x) + LI (x) + J(x)φ(x)] Z D Z 4 1 δ 4 = exp i d x LI [ φ] exp i d x [L0(x) + J(x)φ(x)] (1.131) " Z i δJ(x)!# Z D Z 4 1 δ i 4 4 = exp i d x LI exp 2 d x d y J(x) ∆0(x y) J(y) , " Z i δJ(x)!# − Z Z − (1.132) where

∞ n n 4 1 δ i 1 δ exp i d x LI := LI . (1.133) " i δJ(x) !# n! " i δJ(x)!# Z nX=0 Equation (1.132) is the basis for a perturbative evaluation of all possible Green functions for the theory. As an example I will work through a ﬁrst-order calculation of the complete 2-point Green function in the theory deﬁned by λ L (x) = φ4(x) . (1.134) I −4! The generating functional yields

W [0] = 4 λ δ i 1 i d4x exp d4u d4v J(u) ∆ (u v) J(v)   2 0 − 4! Z δJ(x)! − Z Z −   J=0

 λ 4 2  = 1 i d x 3[i∆0(0)] . (1.135) − 4! Z The 2-point function is

2 δ W [J] λ 4 2 = i∆0(x1 x2) + i d x [i∆0(0)] [i∆0(x1 x2)] δJ(x1) δJ(x2) − − 8 Z − λ 4 +i d x [i∆0(0)][i∆0(x1 x)][i∆0(x x2)] . (1.136) 2 Z − − Using the deﬁnition, Eq. (1.111), and restoring the normalisation,

1 1 δ2W [J] G(x1, x2) = 2 (1.137) i W [0] δJ(x1) δJ(x2)

λ 4 = i∆0(x1 x2) i d x [i∆0(0)][i∆0(x1 x)][i∆0(x x(1.138)2)] , − − 2 Z − − where I have used a + λb = a + λ(b ac) + O(λ2) . (1.139) 1 + λc − This complete Green function does not contain any disconnected parts because the vacuum is trivial in perturbation theory; i.e.,

0˜ φˆ(x) 0˜ 1 δW [J] h | | i := G(x) J=0 = = 0 (1.140) 0˜ 0˜ | i δJ(x) J=0 h | i so that the ﬁeld does not have a nonzero vacuum exp ectation value. This is the simplest demonstration of the fact that dynamical symmetry breaking is a phenomenon inaccessible in perturbation theory.

1.6.5 Exercises

1. Repeat the derivation of Eq. (1.114) for Gc(x1, x2, x3). 2. Prove Eq. (1.131).

3. Derive Eq. (1.136).

4. Prove Eq. (1.140). 1.7 Functional Integral for Fermions

1.7.1 Finitely Many Degrees of Freedom Fermionic ﬁelds do not have a classical analogue: classical physics does not contain anticommuting ﬁelds. In order to treat fermions using functional integrals one must employ Grassmann variables. Reference [7] is the standard source for a rigorous discussion of Grassmann algebras. Here I will only review some necessary ideas. The Grassmann algebra GN is generated by the set of N elements, θ1 , . . . , θN which satisfy the anticommutation relations

θ , θ = 0 , i, j = 1, 2, . . . , N . (1.141) { i j} 2 It is clear from Eq. (1.141) that θi = 0 for i = 1, . . . , N. In addition, the elements θ provide the source for the basis vectors of a 2n-dimensional space, spanned { i} by the monomials:

1, θ1, . . . , θN , θ1θ2, . . . θN−1θN , . . . , θ1θ2 . . . θN ; (1.142)

N i.e., GN is a 2 -dimensional vector space. (NB. One can always choose the p- degree monomial in Eq. (1.142): θi1 θi2 . . . θIN , such that i1 < i2 < . . . < iN .) Obviously, any element f(θ) G can be written ∈ N f(θ) = f0+ f1(i1) θi1 + f2(i1, i2) θi1 θi2 +. . .+ fN (i1, i2, . . . , iN ) θ1θ2 . . . θN , Xi1 iX1,i2 i1,iX2,...,iN (1.143) where the coefficients fp(i1, i2, . . . , ip) are unique if they are chosen to be fully antisymmetric for p 2. ≥ Both “left” and “right” derivatives can be defined on GN . As usual, they are linear operators and hence it suffices to specify their operation on the basis elements:

∂ p−1 θi1 θi2 θip = δsi1 θi2 . . . θip δsi2 θi1 . . . θip + . . . + ( ) δsip θi1 θi2 . . . θ(1.144)ip−1 , ∂θs − − ← ∂ p−1 θi1 θi2 θip = δsip θi1 . . . θip−1 δsip−1 θi1 . . . θip−2 θip + . . . + ( ) δsi1 θi(1.145)2 θip . ∂θs − − The operation on a general element, f(θ) G , is easily obtained. It is also ∈ N obvious that ∂ ∂ ∂ ∂ f(θ) = f(θ) . (1.146) ∂θ1 ∂θ2 − ∂θ2 ∂θ1 A deﬁnition of integration requires the introduction of Grassmannian line elements: dθi, i = 1, . . . , N. These elements also satisfy Grassmann algebras:

dθ , dθ = 0 = θ , dθ , i, j = 1, 2, . . . N . (1.147) { i j} { i j} The integral calculus is completely deﬁned by the following two identities:

dθi = 0 , dθi θi = 1 , i = 1, 2, . . . , N . (1.148) Z Z For example, it is straightforward to prove, using Eq. (1.143),

dθN . . . dθ1 f(θ) = N! fN (1, 2, . . . , N) . (1.149) Z In standard integral calculus a change of integration variables is often used to simplify an integral. That operation can also be defined in the present context. Consider a nonsingular matrix (Kij), i, j = 1, . . . , N, and define new Grassmann variables ξ1, . . . , ξN via N θi = Kij ξj . (1.150) Xi=j With the definition N −1 dθi = (K )ji dξj (1.151) jX=1 one guarantees

dθi θj = δij = dξi ξj . (1.152) Z Z It follows immediately that

θ1θ2 . . . θN = (det K) ξ1ξ2 . . . ξN (1.153) −1 dθN dθN−1 . . . dθ1 = (det K ) dξN dξN−1 . . . dξ1 , (1.154) and hence

−1 dθN . . . dθ1 f(θ) = (det K ) dξN . . . dξ1 f(θ(ξ)) . (1.155) Z Z In analogy with scalar ﬁeld theory, for fermions one expects to encounter integrals of the type

N I := dθ . . . dθ exp θ A θ , (1.156) N 1  i ij j Z i,jX=1  where (Aij) is an antisymmetric matrix. NB.Any symmetric part of the matrix,A, cannot contribute since: relable θiAijθj = θjAjiθi (1.157) Xi,j Xj,i use sym. = θjAijθi (1.158) Xi,j anticom= . θ A θ . (1.159) − i ij j Xi,j · t .. θiAijθj = 0 for A = A . (1.160) Xi,j Assume for the moment that A is a real matrix. Then there is an orthogonal matrix S (SSt = I) for which

0 λ1 0 0 . . . λ 0 0 0 . . .  − 1  StAS = 0 0 0 λ . . . =: A˜ . (1.161)  2     0 0 λ2 0 . . .   −   ......      N Consequently, applying the linear transformation θi = i=1 Sij ξj and using Eq. (1.155), we obtain P

N I = dξ . . . dξ exp ξ A˜ ξ . (1.162) N 1  i ij j Z i,jX=1  Hence  

I = N/2 dξN . . . dξ1 exp 2 [λ1ξ1ξ2 + λ2ξ3ξ4 + . . . + λN/2ξN−1ξN ] = 2 λ1λ2 . . . λN , N even  R dξ . . . dξ exp n2 [λ ξ ξ + λ ξ ξ + . . . + λ ξ oξ ] = 0 , N odd  N 1 1 1 2 2 3 4 (N−1)/2 N−2 N−1 R n o  (1.163) i.e., since det A = det A˜, I = √det 2A , . (1.164) Equation (1.164) is valid for any real matrix, A. Hence, by the analytic function theorem, it is also valid for any complex matrix A. The Lagrangian density associated with the Dirac equation involves a field ψ¯, which plays the role of a conjugate to ψ. If we are express ψ as a vector in GN then we will need a conjugate space in which ψ¯ is defined. Hence it is necessary to define θ¯ , θ¯ , . . . , θ¯ such that the operation θ θ¯ is an involution of the 1 2 N i ↔ i algebra onto itself with the following properties:

i) (θ¯i) = θi ii) (θiθj) = θ¯j θ¯i (1.165) iii) λ θ = λ∗ θ¯ , λ C . i i ∈

The elements of the Grassmann algebra with involution are θ1, θ2, . . . , θN , θ¯1, θ¯2 . . . , θ¯N , each anticommuting with every other. Deﬁning integration via obvious analogy with Eq. (1.148) it follows that

N dθ¯ dθ . . . dθ dθ exp θ¯ B θ = det B , (1.166) N N 1 1 − i ij j Z  i,jX=1    for any matrix B. (NB. This is the origin of the fermion determinant in the quantum ﬁeld theory of fermions.) This may be compared with the analogous result for commuting real numbers, Eq. (1.123):

N N 1 dxi exp π xi Aij xj = . (1.167) RN {− } √det A Z iY=1 i,jX=1 1.7.2 Fermionic Quantum Field To describe a fermionic quantum ﬁeld the preceding analysis must be generalised to the case of inﬁnitely many generators. A rigorous discussion can be found in Ref. [7] but here I will simply motivate the extension via plausible but formal manipulations. Suppose the functions u (x) , n = 0, . . . , are a complete, orthonormal { n ∞} set that span a given Hilbert space and consider the Grassmann function

∞ θ(x) := un(x) θn , (1.168) nX=0 where θn are Grassmann variables. Clearly

θ(x), θ(y) = 0 . (1.169) { } The elements θ(x) are considered to be the generators of the “Grassman algebra” G and, in complete analogy with Eq. (1.143), any element of G can be uniquely written as

∞ f = dx1dx2 . . . dxN θ(x1)θ(x2) . . . θ(xN ) fn(x1, x2, . . . , xN ) , (1.170) nX=0 Z where, for N 2, the f (x , x , . . . , x ) are fully antisymmetric functions of their ≥ n 1 2 N arguments. In another analogy, the left- and right-functional-derivatives are deﬁned via their action on the basis vectors: δ θ(x )θ(x ) . . . θ(x ) = δθ(x) 1 2 n n−1 δ(x x1) θ(x2) . . . θ(xn) . . . + ( ) δ(x xn) θ(x1) . . . θ(xn−1)(1.171), − ← − − − δ θ(x )θ(x ) . . . θ(x ) = 1 2 n δθ(x) δ(x x ) θ(x ) . . . θ(x ) . . . + ( )n−1δ(x x ) θ(x ) . . . θ(x )(1.172), − n 1 n−1 − − − 1 2 n cf. Eqs. (1.144), (1.145). Finally, we can extend the deﬁnition of integration. Denoting

[ θ(x)] := lim dθN . . . dθ2dθ1 , (1.173) D N→∞ consider the “standard” Gaussian integral

I := [ θ(x)] exp dxdy θ(x)A(x, y)θ(y) (1.174) Z D Z where, clearly, only the antisymmetric part of A(x, y) can contribute to the result. Deﬁne Aij := dxdy ui(x)A(x, y)uj(y) , (1.175) Z then N θiAij θj I = lim dθN . . . dθ2dθ1 e i=1 (1.176) N→∞ Z P so that, using Eq. (1.164),

I = lim det 2AN , (1.177) N→∞ q where, obviously, AN is the N N matrix in Eq. (1.175). This provides a deﬁnition for the formal result: ×

I = [ θ(x)] exp dxdy θ(x)A(x, y)θ(y) = √Det2A , (1.178) Z D Z where I will subsequently identify functional equivalents of matrix operations as proper nouns. The result is independent of the the basis vectors since all such vectors are unitarily equivalent and the determinant is cylic. (This means that a new basis is always related to another basis via u0 = Uu, with UU † = I. Transforming to a new basis therefore introduces a modified exponent, now involving the matrix UAU †, but the result is unchanged because det UAU † = det A.) In quantum field theory one employs a Grassmann algebra with an involution. In this case, defining the functional integral via ¯ ¯ ¯ ¯ [ θ(x)][ θ(x)] := lim dθN dθN . . . dθ2dθ2 dθ1dθ1 , (1.179) D D N→∞ one arrives immediately at a generalisation of Eq. (1.166)

[ θ¯(x)][ θ(x)] exp dxdy θ¯(x) B(x, y) θ(x) = DetB , (1.180) Z D D − Z which is also a definition. The relation ln det B = tr ln B , (1.181) valid for any nonsingular, finite dimensional matrix, has a generalisation that is often used in analysing quantum field theories with fermions. Its utility is to make possible a representation of the fermionic determinant as part of the quantum field theory’s action via

DetB = exp Tr LnB . (1.182) { } I note that for an integral operator O(x, y)

Tr O(x, y) := d4x tr O(x, x) , (1.183) Z which is an obvious analogy to the definition for finite-dimensional matrices. Furthermore a functional of an operator, whenever it is well-defined, is obtained via the function’s power series; i.e., if

2 f(x) = f0 + f1 x + f2 x + [. . .] , (1.184) then

4 4 f[O(x, y)] = f0 δ (x y) + f1 O(x, y) + f2 d w O(x, w)O(w, y) + [. . .] . (1.185) − Z 1.7.3 Generating Functional for Free Dirac Fields The Lagragian density for the free Dirac ﬁeld is

ψ 4 ¯ L0 (x) = d x ψ(x) (i∂/ m) ψ(x) . (1.186) Z − Consider therefore the functional integral

W [ξ¯, ξ] = [ ψ¯(x)][ ψ(x)] exp i d4x ψ¯(x) i∂/ m + iη+ ψ(x) + ψ¯(x)ξ(x) + ξ¯(x)ψ(x) . D D − Z Z h i (1.187)

Here ψ¯(x), ψ(x) are identified with the generators of G, with the minor additional complication that the spinor degree-of-freedom is implicit; i.e., to be explicit, one should write 4 [ ψ¯ (x)] 4 [ ψ (x)]. This only adds a finite matrix r=1 D r s=1 D s degree-of-freedom Qto the problem,Q so that “Det A” will mean both a functional and a matrix determinant. This effect will be encountered again; e.g., with the appearance of fermion colour and flavour. In Eq. (1.187) I have also introduced anticommuting sources: ξ¯(x), ξ(x), which are also elements in the Grassmann algebra, G. The free-field generating functional involves a Gaussian integral. To evaluate that integral I write

O(x, y) = (i∂/ m + iη+)δ4(x y) (1.188) − − and observe that the solution of

d4w O(x, w) P (w, y) = I δ4(x y) (1.189) Z − i.e., the inverse of the operator O(x, y) is (see Eq. 1.72) precisely the free-fermion propagator: P (x, y) = S (x y) . (1.190) 0 − Hence I can rewrite Eq. (1.187) in the form

4 4 0 0 W [ξ¯, ξ] = [ ψ¯(x)][ ψ(x)] exp i d xd y ψ¯ (x)O(x, y)ψ (y) ξ¯(x)S0(x y)ξ(y) D D − − Z Z h (1.191) i where

ψ¯0(x) := ψ¯(x) + d4w ξ¯(w) S (w x) , ψ(x) := ψ(x) + d4w S (x w) ξ(w) . 0 − 0 − Z Z (1.192) Clearly, ψ¯0(x) and ψ0(x) are still in G and hence related to the original variables by a unitary transformation. Thus changing to the “primed” variables introduces a unit Jacobian and so

¯ 4 4 ¯ W [ξ, ξ] = exp i d xd y ξ(x) S0(x y) ξ(y) − Z − [ ψ¯0(x)][ ψ0(x)] exp i d4xd4y ψ¯0(x)O(x, y)ψ0(y) (1.193) × Z D D Z −1 4 4 ¯ = det[ iS0 (x y)] exp i d xd y ξ(x) S0(x y) ξ(y) (1.194) − − − Z − 1 = exp i d4xd4y ξ¯(x) S (x y) ξ(y) , (1.195) ψ − 0 − N0 Z where ψ := det[iS (x y)]. Clearly. N0 0 − ψ W [ξ¯, ξ] = 1 . (1.196) N0 ξ¯=0=ξ

The 2 point Green function for the free-fermion quantum ﬁeld theory is now easily obtained:

2 ¯ ˆ ˆ¯ ψ δ W [ξ, ξ] 0 T ψ(x)ψ(y) 0 0 = h | { }| i N iδξ¯(x) ( i)δξ(y) 0 0 ξ¯=0=ξ − h | i

= [ ψ¯(x)][ ψ(x )] ψ(x)ψ¯(y) exp i d4x ψ¯(x) i∂/ m + iη+ ψ(x) D D − Z Z = i S (x y) ; (1.197) 0 − i.e., the inverse of the Dirac operator, with exactly the Feynman boundary conditions. As in the example of a scalar quantum ﬁeld theory, the generating functional for connected n-point Green functions is Z[ξ¯, ξ], deﬁned via:

W [ξ¯, ξ] =: exp iZ[ξ¯, ξ] . (1.198) n o Hitherto I have not illustrated what is meant by “DetO,” where O is an integral operator. I will now provide a formal example. (Rigour requires a careful consideration of regularisation and limits.) Consider a translationally invariant operator 4 d p −i(p,x−y) O(x, y) = O(x y) = 4 O(p) e . (1.199) − Z (2π) Then, for f as in Eq. (1.185),

d4p f[O(x y)] = f + f O(p) + f O2(p) + [. . .] e−i(p,x−y) − (2π)4 0 1 2 Z n o 4 d p −i(p,x−y) = 4 f(O(p)) e . (1.200) Z (2π)

ψ I now apply this to 0 := Det[iS0(x y)] and observe that Eq. (1.182) means one can begin by consideringN TrLn iS (x− y). Writing 0 −

2 /p 2 1 S0(p) = m ∆0(p ) 1 + , ∆0(p ) = , (1.201) " m# p2 m2 + iη+ − the free fermion propagator can be re-expressed as a product of integral operators:

4 S0(x y) = d w m ∆0(x w) (w y) , (1.202) − Z − F − with ∆ (x y) given in Eq. (1.129) and 0 − 4 d p /p −i(p,x−y) (x y) = 4 1 + e . (1.203) F − Z (2π) " m # It follows that

TrLn iS (x y) = Tr Ln i m∆ (x y) + Ln δ4(x y) + (x y) . (1.204) 0 − 0 − − F − n h io Using Eqs. (1.183), (1.200), one can express the second term as

d4p TrLn δ4(x y) + (x y) = d4x tr ln [1 + (p)] − F − (2π)4 F h i Z Z 4 2 4 d p p = d x 4 2 ln 1 2 (1.205) Z Z (2π) " − m # and, applying the same equations, the ﬁrst term is

d4p 2 TrLn im∆ (x y) = d4x 2 ln i m∆ (p2) , (1.206) 0 − (2π)4 0 Z Z h i where in both cases d4x measures the (inﬁnite) spacetime volume. Combining these results one obtainsR 4 ψ 4 d p 2 Ln 0 = TrLn iS0(x y) = d x 4 2 ln ∆0(p ) , (1.207) N − Z Z (2π) where the factor of 2 reﬂects the spin-degeneracy of the free-fermion’s eigenvalues. (Including a “colour” degree-of-freedom, this would become “2Nc,” where Nc is the number of colours.)

1.7.4 Exercises 1. Verify Eq. (1.149).

2. Verify Eqs. (1.153), (1.154).

3. Verify Eqs. (1.163), (1.164).

4. Verify Eqs. (1.166).

5. Verify Eq. (1.181).

6. Verify Eq. (1.197).

7. Verify Eq. (1.205).

1.8 Functional Integral for Gauge Field Theo- ries

To begin I will consider a non-Abelian gauge ﬁeld theory in the absence of couplings to matter ﬁeld, which is described by a Lagrangian density: 1 (x) = F a (x)F µν(x), (1.208) L −4 µν a F a (x) = ∂ Ba ∂ Ba + gf Bb Bc , (1.209) µν µ ν − ν µ abc µ ν where g is a coupling constant and fabc are the structure constants of SU(Nc): i.e., with T a : a = 1, . . . , N 2 1 denoting the generators of the group { c − }

[Ta, Tb] = ifabcTc . (1.210) λa a In the fundamental representation Ta 2 , where λ are the generalization of the eight Gell-Mann matrices, while{ } ≡in {the}adjoint represen{ } tation, relevant to the realization of transformations on the gauge ﬁelds,

T a = if . (1.211) { }bc − abc An obvious guess for the form of the generating functional is

a 4 µ a W [J] = [ Bµ] exp i d x[ (x) + Ja (x)Bµ] , (1.212) Z D Z L µ where, as usual, Ja (x) is a (classical) external source for the gauge field. It will immediately be observed that this is a Lagrangian-based expression for the generating functional, even though the Hamiltonian derived from Eq. (1.208) is not of the form in Eq. (1.115). It is nevertheless (almost) correct (I will motivate the modifications that need to be made to make it completely correct) and provides the foundation for a manifestly Poincaré covariant quantisation of the field theory. Alternatively, one could work with Coulomb gauge, build the Hamiltonian and construct W [J] in the canonical fashion, as described in Sec. ??, but then covariance is lost. The Coulomb gauge procedure gives the same S- matrix elements (Green functions) as the (corrected-) Lagrangian formalism and hence they are completely equivalent. However, manifest covariance is extremely useful as it often simplifies the allowed form of Green functions and certainly simplifies the calculation of cross sections. Thus the Lagrangian formulation is most often used. The primary fault with Eq. (1.212) is that the free-field part of the Lagrangian density is singular: i.e., the determinant encountered in evaluating the free-gauge- boson generating functional vanishes, and hence the operator cannot be inverted. This is easily demonstrated. Observe that

4 1 4 a a µ ν ν µ d x 0(x) = d x[∂µBν (x) ∂ν Bµ(x)][∂ Ba (x) ∂ Ba (x)](1.213) Z L −4 Z − − 1 = d4xd4yBa(x) [ gµν ∂2 + ∂µ∂ν ]δ4(x y) Ba((1.214)y) −2 µ − − ν Z n o 1 4 4 a µν a =: d xd yBµ(x)K (x, y)Bν (y) . (1.215) 2 Z The operator Kµν (x, y) thus determined can be expressed d4q Kµν (x y) = gµνq2 + qµqν e−i(p,x−y) (1.216) − (2π)4 − Z from which it is apparent that the Fourier amplitude is a projection operator; i.e., a key feature of Kµν (x, y) is that it projects onto the space of transverse gauge ﬁeld conﬁgurations:

q (gµνq2 qµqν) = 0 = (gµνq2 qµqν)q , (1.217) µ − − ν where qµ is the four-momentum associated with the gauge ﬁeld. It follows that the W0[J] obtained from Eq. (1.212) has no damping associated with longitudinal gauge ﬁelds and is therefore meaningless. A simple analogy is

∞ ∞ 2 dx dye−(x−y) , (1.218) Z−∞ Z−∞ in which the integrand does not damp along trajectories in the (x, y)-plane related via a spatial translation: (x, y) (x + a, y + a). Hence there is an overall divergence associated with the translation→ of the centre of momentum: X = (x + y)/2. Y = (x y): X X + (a, a), Y Y , − → → ∞ ∞ 2 ∞ ∞ 2 dx dye−(x−y) = dX dY e−Y (1.219) Z−∞ Z−∞ Z−∞ Z−∞ ∞ = dX √π = . (1.220) Z−∞ ∞ The underlying problem, which is signalled by the behavior just identiﬁed, is the gauge invariance of the action: (x); i.e., the action is invariant under local L ﬁeld transformation R B (x) := igBa(x)T a G(x)B (x)G−1(x) + [∂ G(x)]G−1(x) , (1.221) µ µ → µ µ G(x) = exp igT aΘa(x) . (1.222) {− }

This means that, given a reference field configuration Bˆµ(x), the integrand in the generating functional, Eq. (1.212), is constant along the path through the gauge field manifold traversed by the applying gauge transformations to Bˆµ(x). Since the parameters characterizing the gauge transformations, Θa, are continuous functions, each such gauge orbit contains an uncountable infinity of gauge field configurations. It is therefore immediately apparent that the generating functional, as written, is is undefined: it contains a multiplicative factor proportional to the length (or volume) of the gauge orbit. (NB. While there is, in addition, an uncountable infinity of distinct reference configurations, the action changes upon any shift orthogonal to a gauge orbit.) Returning to the example in Eq. (1.219), the analogy is that the “Lagrangian density” l(x, y) = (x y)2 is invariant under translations; i.e., the integrand is − invariant under the operation

∂ ∂ gs(x, y) = exp s + s (1.223) ( ∂x ∂y ) (x, y) (x0, y0) = g (x, y)(x, y) = (x + s, y + s) . (1.224) → s

Hence, given a reference point P = (x0, y0) = (1, 0), the integrand is constant along the path (x, y) = P0 + (s, s) through the (x, y)-plane. (This is a translation of the centre-of-mass: X = (x +y )/2 = 1/2 X +s.) Since s ( , ), this 0 0 0 → 0 ∈ −∞ ∞ path contains an uncountable infinity of points, and at each one the integrand has precisely the same value. The integral thus contains a multiplicative factor proportional to the length of the translation path, which clearly produces an infinite (meaningless) result for the integral. (NB. The value of the integrand changes upon a translation orthogonal to that just identified.)

1.8.1 Faddeev-Popov Determinant and Ghosts This problem with the functional integral over gauge ﬁelds was identiﬁed by Faddeev and Popov. They proposed to solve the problem by identifying and extracting the gauge orbit volume factor.

A Simple Model Before describing the procedure in quantum field theory it can be illustrated using the simple integral model. One begins by defining a functional of our “field variable”, (x, y), which intersects the centre-of-mass translation path once, and only once: f(x, y) = (x + y)/2 a = 0 . (1.225) − Now define a functional ∆f such that

∞ ∆f [x, y] daδ((x + y)/2 a) = 1 . (1.226) Z−∞ −

It is clear that ∆f [x, y] is independent of a:

∞ 0 0 −1 0 0 ∆f [x + a , y + a ] = daδ((x + a + y + a )/2 a) (1.227) Z−∞ − 0 ∞ a˜==a−a daδ˜ ((x + y)/2 a˜) (1.228) Z−∞ − −1 = ∆f [x, y] . (1.229)

Using ∆f the model generating functional can be rewritten

∞ ∞ ∞ ∞ ∞ −l(x,y) −l(x,y) dx dye = da dx dye ∆f [x, y]δ((x + y)/2 a) Z−∞ Z−∞ Z−∞ Z−∞ Z−∞ − (1.230) and now one performs a centre-of-mass translation: x x0 = x + a, y y0 = → → y + a, under which the action is invariant so that the integral becomes

∞ ∞ ∞ −l(x,y) da dx dye ∆f [x, y]δ((x + y)/2 a) (1.231) −∞ −∞ −∞ − Z ∞ Z ∞ Z ∞ −l(x,y) = da dx dye ∆f [x + a, y + a]δ((x + y)/2) . (1.232) Z−∞ Z−∞ Z−∞ Now making use of the a independence of ∆ [x, y], Eq. (1.227), − f ∞ ∞ ∞ −l(x,y) = da dx dye ∆f [x, y]δ((x + y)/2) . (1.233) Z−∞ Z Z−∞ Z−∞ In this last line the “volume” or “path length” has been explicitely factored out at the cost of introducing a δ-function, which fixes the centre-of-mass; i.e., a single point on the path of translationally equivalent configurations, and a functional ∆f , which, we will see, is the analogue in this simple model of the Fadeev-Popov determinant. Hence the “correct” definition of the generating functional for this model is ∞ ∞ −l(x,y)+jxx+jyy w(~j) = dx dye ∆f [x, y]δ((x + y)/2) . (1.234) Z−∞ Z−∞ Gauge Fixing Conditions To implement this idea for the real case of non-Abelian gauge fields one envisages an hypersurface, lying in the manifold of all gauge fields, which intersects each gauge orbit once, and only once. This means that if

f [Ba(x); x] = 0 , a = 1, 2, . . . , N 2 1 , (1.235) a µ c − is the equation describing the hypersurface, then there is a unique element in each gauge orbit that satisfies one Eq. (1.235), and the set of these unique elements, none of which cannot be obtained from another by a gauge transformation, forms a representative class that alone truly characterises the physical configuration of gauge fields. The gauge-equivalent, and therefore redundant, elements are absent. Equations (1.235) can also be viewed as defining a set of non-linear equations for G(x), Eq (??). This in the sense that for a given field configuration, Bµ(x), it is always possible to find a unique gauge transformation, G1(x), that yields a G1 gauge transformed field Bµ (x), from Bµ(x) via Eq. (??), which is the one and a,G1 only solution of fa[Bµ (x); x] = 0. Equation (1.235) therefore defines a gauge fixing condition. In order for Eqs. (1.235) to be useful it must be possible that, when given a configuration B (x), for which f [B (x); x] = 0, the equation µ a µ 6 G fa[Bµ (x); x] = 0 (1.236) can be solved for the gauge transformation G(x). To see a consequence of this b requirement consider a gauge configuration Bµ(x) that almost, but not quite, satisfies Eqs. (1.235). Applying an infinitesimal gauge transformation to this b Bµ(x) the requirement entails that it must be possible to solve

f [Bb (x) Bb (x) g f Bd(x) δΘc(x) ∂ δΘb(x); x] = 0 (1.237) a µ → µ − bdc µ − µ for the inﬁnitesimal gauge transformation parameters δΘa(x). Equation (1.237) can be written (using the chain rule)

b δfa[B (x); x] f [Bb (x)] d4y µ δcd∂ + gf Be(y) δΘd(y) = 0 . (1.238) a µ − δBc(y) ν ced ν Z ν h i This looks like the matrix equation f~ = θ~, which has a solution for θ~ if, and O only if, det = 0, and a similar constraint follows from Eq. (1.238): the gauge ﬁxing conditionsO 6 can be solved if, and only if,

b δfa[Bµ(x); x] cd e Det f := Det δ ∂ν gfcdeBν(y) M (− δBc(y) − ) ν h i b δfa[Bµ(x); x] =: Det c [Dν(y)]cd = 0 . (1.239) (− δBν(y) ) 6 Equation (1.239) is the so-called admissibility condition for gauge ﬁxing conditions. A simple illustration is provided by the lightlike (Hamilton) gauges, which are speciﬁed by nµBa(x) = 0 , n2 > 1 , a = 1, 2, . . . , (N 2 1) . (1.240) µ − Choosing (nµ) = (1, 0, 0, 0), the equation for G(x) is, using Eq. (??) ∂ G(t, ~x) = G(t, ~x) B0(t, ~x) , (1.241) ∂t − and this nonperturbative equation has the unique solution

t G(t, ~x) = T exp ds B0(s, ~x) , (1.242) − Z−∞ where T is the time ordering operator. One may compare this with Eq. (1.238), which only provides a perturbative [in g] solution. While that may be an advantage, Eq. (1.242) is not a Poincaré covariant constraint and that makes it difficult to employ in explicit calculations. A number of other commonly used gauge fixing conditions are

µ a ∂ Bµ(x) = 0 , Lorentz gauge µ a a ∂ Bµ(x) = A (x) , Generalised Lorentz gauge µ a 2 n Bµ(x) = 0 , n < 0 , Axial gauge (1.243) µ a 2 n Bµ(x) = 0 , n = 0 , Light-like gauge ~ B~a(x) = 0 , Coulomb gauge ∇ · and the generalised axial, light-like and Coulomb gauges, where an arbitrary function, Aa(x), features on the r.h.s. All of these choices satisfy the admissability condition, Eq. (1.239), for small gauge field variations but in some cases, such as Lorentz gauge, the uniqueness condition fails for large variations; i.e., those that are outside the domain of perturbation theory. This means that there are at least two solutions: G1, G2, of Eq. (1.236), and perhaps uncountably many more. Since no nonperturbative solution of any gauge field theory in four spacetime dimensions exists, the actual number of solutions is unknown. If it is infinite then the Fadde’ev-Popov definition of the generating functional fails in that gauge. These additional solutions are called Gribov Copies and their existence raises questions about the correct way to furnish a nonperturbative definition of the generating functional [9], which are currently unanswered.

Isolating and Eliminating the Gauge Orbit Volume To proceed one needs a little information about the representation of non-Abelian groups. Suppose u is an element of the group SU(N). Every such element can be characterised by (N 2 1) real parameters: Θa, a = 1, . . . , N 2 1 . Let G(u) be the representation of−u under which the gauge{ ﬁelds transform;− i.e.,} the adjoint representation, Eqs. (??), (??). For inﬁnitesimal tranformations

G(u) = I igT aΘa(x) + O(Θ2) (1.244) − where T a are the adjoint representation of the Lie algebra of SU(N), Eq. (??). Clearly{, if }u, u SU(N) then uu0 SU(n) and G(u)G(u0) = G(uu0). (These are basic properties0 ∈ of groups.) ∈ To define the integral over gauge fields we must properly define the gauge- field “line element”. This is the Hurwitz measure on the group space, which is invariant in the sense that du0 = d(u0u). In the neighbourhood of the identity one may always choose du = dΘa (1.245) a Y and the invariance means that since the integration represents a sum over all possible values of the parameters Θa, relabelling them as Θ˜ a cannot matter. It is now possible to quantise the gauge field; i.e., properly define Eq. (??). a Consider ∆f [Bµ] defined via, cf. Eq. (??),

b b,u ∆f [Bµ] du(x) δ[fa Bµ (x); x ] = 1 , (1.246) x x,a { } Z Y Y where Bb,u(x) is given by Eq. (??) with G(x) u(x). ∆ [Ba] is gauge invariant: µ → f µ

−1 b,u 0 b,u0u ∆f [Bµ ] = du (x) δ[fa Bµ (x); x ] x x,a { } Z Y Y 0 b,u0u = d(u (x)u(x)) δ[fa Bµ (x); x ] x x,a { } Z Y Y 00 b,u00 −1 b = du (x) δ[fa Bµ (x); x ] = ∆f [Bµ] . (1.247) x x,a { } Z Y Y Returning to Eq. (??), one can write

a a a,u 4 W [0] = du(x) [ Bµ] ∆f [Bµ] δ[fb Bµ (x); x ] exp i d x L(x) . x D { } { } Z Y Z Yx,b Z (1.248) a a,u−1 Now execute a gauge transformation: Bµ(x) Bµ (x), so that, using the invariance of the measure and the action, one has→

a a a 4 W [0] = [ du(x)] [ Bµ] ∆f [Bµ] δ[fb Bµ(x); x ] exp i d x L(x) , x D { } { } Z Y Z Yx,b Z (1.249) where now the integrand of the group measure no longer depends on the group element, u(x) (cf. Eq. (??)). The gauge orbit volume has thus been identiﬁed and can be eliminated so that one may deﬁne

W [J a] = [ Ba] ∆ [Ba] δ[f Ba(x); x ] exp i d4x [L(x) + J µ(x)Ba(x)] . µ D µ f µ b{ µ } { a µ } Z Yx,b Z (1.250) Neglecting for now the possible existence of Gribov copies, Eq. (1.250) is the foundation we sought for a manifestly Poincar´e covariant quantisation of the gauge ﬁeld. However, a little more work is needed to mould a practical tool.

Ghost Fields

a A first step is an explicit calculation of ∆f [Bµ]. Since it always appears multiplied by a δ-function it is sufficient to evaluate it for those field configurations that satisfy Eq. (1.235). Recalling Eqs. (1.238), (1.239), for infinitesimal gauge transformations

b,u b 4 fa[Bµ (x); x] = fa[Bµ(x); x] + d y ( f )acΘc(y) Z M 4 = 0 + d y ( f )acΘc(y) . (1.251) Z M Hence, using the deﬁnition, Eq. (1.246),

−1 b a 4 ∆f [Bµ] = dΘ (x) δ d y ( f )acΘc(y) (1.252) x,a M Z Y Z and changing variables: Θa Θ˜ a = ( ) Θ (y), this gives → Mf ac c −1 b ˜ a ˜ ∆f [Bµ] = Det f dΘ (x) δ Θa(y) , (1.253) M x,a Z Y n h io where the determinant is the Jacobian of the transformation, so that ∆ [Bb ] = Det . (1.254) f µ Mf Now recall Eq. (1.180). This means Eq. (1.254) can be expressed as a functional integral over Grassmann ﬁelds: φ¯ , φ , a, b = 1, . . . , (N 2 1), a b − b ¯ 4 4 ¯ ∆f [Bµ] = [ φb][ φa] exp d xd y φb(x) ( f )bc(x, y) φa(y) , . (1.255) Z D D − Z M Consequently, absorbing non-dynamical constants into the normalisation,

W [J a] = [ Ba] [ φ¯ ][ φ ] δ[f Ba(x); x ] µ D µ D b D a b{ µ } Z Yx,b 4 µ a 4 4 ¯ exp i d x [L(x) + Ja (x)Bµ(x)] + i d xd y φb(x) ( f )bc(x, y) φa(y) . × Z Z M (1.256) The Grassmann fields φ¯a, φb are the Fadde’ev-Popov Ghosts. They are an essential consequence of{gauge}fixing. As one concrete example, consider the Lorentz gauge, Eq. (1.243), for which ( ) = δ4(x y) δab ∂2 gf ∂µ Bc (x) (1.257) ML ab − − abc µ and therefore h i a a ¯ µ a 4 1 µν a W [Jµ] = [ Bµ] [ φb][ φa] δ[∂ Bµ(x)] exp i d x 4Fa (x) Fµν(x) D D D x,a − Z Y Z ¯ ν µ ¯ c µ a ∂µφa(x) ∂ φa(x) + gfabc[∂ φa(x)]φb(x)Bµ(x) + Ja (x)Bµ(x)(1.258). − This expression makes clear that a general consequence of the Fadde’ev-Popov procedure is to introduce a coupling between the gauge field and the ghosts. Thus the ghosts, and hence gauge fixing, can have a direct impact on the behaviour of gauge field Green functions. As another, consider the axial gauge, for which ( ) = δ4(x y) δab n ∂µ , . (1.259) MA ab − µ Expressing the related determinant via ghosth fields iti is immediately apparent that with this choice there is no coupling between the ghosts and the gauge field quanta. Hence the ghosts decouple from the theory and may be discarded as they play no dynamical role. However, there is a cost, as always: in this gauge µ a the effect of the delta-function, x,a δ[n Bµ(x)], in the functional integral is to µ complicate the Green functions bQy making them depend explicitly on (n ). Even the free-field 2-point function exhibits such a dependence. It is important to note now that this decoupling of the ghost fields is tied to an “accidental” elimination of the fabc term in Dµ(y), Eq. (1.239). That term is always absent in Abelian gauge theories, for which quantum electrodynamics is the archetype, because all generators commute and analogues of fabc must vanish. Hence in Abelian gauge theories ghosts decouple in every gauge. a To see how δ[fb Bµ(x); x ] influences the form of Green functions, consider the generalised Loren{ tz gauge:} ν a a ∂ Bµ(x) = A (x) , (1.260) where Aa(x) are arbitrary functions. The Fadde’ev-Popov determinant is the same in{ generalised} Lorentz gauges as it is in Lorentz gauge and hence a a ∆GL[Bµ] = ∆L[Bµ] , (1.261) where the r.h.s. is given in Eq. (1.257). The generating functional in this gauge is therefore a a ¯ µ a a 4 1 µν a W [Jµ ] = [ Bµ] [ φb][ φa] δ[∂ Bµ(x) A (x)] exp i d x 4Fa (x) Fµν(x) D D D x,a − − Z Y Z ¯ ν µ ¯ c µ a ∂µφa(x) ∂ φa(x) + gfabc[∂ φa(x)]φb(x)Bµ(x) + Ja (x)Bµ(x) . − Gauge invariance of the generating functional, Eq. (1.246), means that one can integrate over the Aa(x) , with a weight function to ensure convergence: { } i [ Aa] exp d4x Aa(x) Aa(x) , (1.262) Z D −2λ Z to arrive finally at the generating functional in a covariant Lorentz gauge, specified by the parameter λ:

a ¯a a a ¯ W [Jµ , ξg , ξg ] = [ Bµ] [ φb][ φa] Z D D D 4 1 µν a 1 µ a ν a exp i d x 4Fa (x) Fµν (x) [∂ Bµ(x)] [∂ Bν (x)] Z − − 2λ ∂ φ¯ (x) ∂ν φ (x) + gf [∂µφ¯ (x)]φ (x)Bc (x) − µ a a abc a b µ µ a ¯a a ¯a a +Ja (x)Bµ(x) + ξg (x)φ (x) + φ (x)ξg (x) , (1.263) ¯a a where ξg , ξg are anticommuting external sources for the ghost fields. (NB. To complete{ the}definition one should add convergence terms, iη+, for every field or, preferably, work in Euclidean space.) Observe now that the free gauge boson piece of the action in Eq. (1.263) is

1 4 4 a µν a 2 d x d y Bµ(x) K (x y; λ) Bν (y) Z − 1 4 4 a µν 2 µ ν 1 µν + 4 a := 2 d x d y Bµ(x) [g ∂ ∂ ∂ (1 ) ig η ]δ (x y) Bν(1.264)(y) Z − − λ − − The operator Kµν (x y; λ) thus defined can be expressed − 4 µν d q µν 2 + µ ν 1 −i(q,x−y) K (x y) = 4 g (q + iη ) + q q [1 ] e , (1.265) − Z (2π) − − λ cf. Eq. (??), and now 1 q Kµν = qν (1.266) µ − λ so that in this case the action does damp variations in the longitudinal components of the gauge field. Kµν(x y; λ) is the inverse of the free gauge boson propagator; i.e., the free gauge b−oson 2-point Green function, Dµν(x y), is − obtained via 4 µ ρν µν 4 d w Kρ (x w) D (w y) = g δ (x y) , (1.267) Z − − − and hence 4 µ ν µν d q µν q q 1 −i(q,x−y) D (x y) = 4 g + (1 λ) 2 + 2 + e . (1.268) − Z (2π) − − q + iη ! q + iη a The obvious λ dependence is a result of the presence of δ[fb Bµ(x); x ] in the generating functional, and this is one example of the δ-function’s{ direct }effect on the form of Green functions: they are, in general, gauge parameter dependent. 1.8.2 Exercises 1. Verify Eq. (1.257).

2. Verify Eq. (1.258).

3. Verify Eq. (1.259).

4. Verify Eq. (1.268).

1.9 Dyson-Schwinger Equations

It has long been known that, from the field equations of quantum field theory, one can derive a system of coupled integral equations interrelating all of a theory’s Green functions [6, 10]. This tower of a countable infinity of equations is called the complex of Dyson-Schwinger equations (DSEs). This intrinsically nonperturbative complex is vitally important in proving the renormalisability of quantum field theories and at its simplest level provides a generating tool for perturbation theory. In the context of quantum electrodynamics (QED) I will illustrate a nonperturbative derivation of two equations in this complex. The derivation of others follows the same pattern.

1.9.1 Photon Vacuum Polarization Generating Functional for QED

The action for QED with Nf ﬂavours of electomagnetically active fermion, is

Nf ¯ 4 ¯f f f f 1 µν 1 µ ν S[Aµ, ψ, ψ] = d x ψ i∂/ m0 + e0 A/ ψ Fµν F ∂ Aµ(x) ∂ Aν(x) ,  − − 4 − 2λ0  Z fX=1  (1.269)  where: ψ¯f (x), ψf (x) are the elements of the Grassmann algebra that describe f f the fermion degrees of freedom, m0 are the fermions’ bare masses and e0 their charges; and Aµ(x) describes the gauge boson [photon] ﬁeld, with

F = ∂ A ∂ A , (1.270) µν µ ν − ν µ and λ0 the bare covariant-Lorentz-gauge fixing parameter. (NB. To describe an electron the physical charge ef < 0.) The derivation of the generating functional in Eq. (1.263) can be employed with little change here. In fact, in this context it is actually simpler because ghost fields decouple. Combining the procedure for fermions and gauge fields, described in Secs. 1.7, ?? respectively, one arrives at ¯ ¯ W [Jµ, ξ, ξ] = [ Aµ] [ ψ][ ψ] Z D D D 4 1 µν 1 µ ν exp i d x 4F (x) Fµν(x) ∂ Aµ(x) ∂ Aν (x) Z − − 2λ0 Nf + ψ¯f i∂/ mf + ef A/ ψf +J µ(x)A (x) + ξ¯f (x)ψf (x) + ψ¯f (x)ξf (x) , − 0 0 µ fX=1 f ¯f where Jµ is an external source for the electromagnetic field, and ξ , ξ are external sources for the fermion field that, of course, are elements in the Grass- mann algebra. (NB. In Abelian gauge theories there are no Gribov copies in the covariant-Lorentz-gauges.)

Functional Field Equations As described in Sec. 1.6.1, it is advantageous to work with the generating functional of connected Green functions; i.e., Z[Jµ, ξ¯, ξ] deﬁned via

W [Jµ, ξ, ξ¯] =: exp iZ[Jµ, ξ, ξ¯] . (1.271) The derivation of a DSE now follows simplynfrom the observo ation that the integral of a total derivative vanishes, given appropriate boundary conditions. Hence, for example,

δ 4 f f f f µ 0 = [ Aµ] [ ψ][ ψ¯] exp i S[Aµ, ψ, ψ¯] + d x ψ ξ + ξ¯ ψ + AµJ D D D δAµ(x) Z Z h i = [ Aµ] [ ψ][ ψ¯] Z D D D δS 4 f f f f µ + Jµ(x) exp i S[Aµ, ψ, ψ¯] + d x ψ ξ + ξ¯ ψ + AµJ (δA (x) ) µ Z h i δS δ δ δ ¯ = , , + Jµ(x) W [Jµ, ξ, ξ] , (1.272) (δAµ(x) "iδJ iδξ¯ −iδξ # ) where the last line has meaning as a functional diﬀerential operator on the generating functional. Diﬀerentiating Eq. (1.269) gives

δS ρ 1 ν f f f = ∂ρ∂ gµν 1 ∂µ∂ν A (x) + e0 ψ (x)γµψ (x) (1.273) δAµ(x) − − λ0 Xf so that the explicit meaning of Eq. (1.272) is J (x) = − µ ρ 1 δZ f δZ δZ δ δ iZ ∂ρ∂ gµν 1 ∂µ∂ν + e0 f γµ f + f γµ f , − − λ0 δJν(x) −δξ (x) δξ¯ (x) δξ (x) " δξ¯ (x)#! Xf (1.274) where I have divided through by W [Jµ, ξ, ξ¯]. Equation (1.274) represents a compact form of the nonperturbative equivalent of Maxwell’s equations.

One-Particle-Irreducible Green Functions The next step is to introduce the generating functional for one-particle-irreducible (1PI) Green functions: Γ[Aµ, ψ, ψ¯], which is obtained from Z[Jµ, ξ, ξ¯] via a Leg- endre transformation ¯ ¯ 4 f f ¯f f µ Z[Jµ, ξ, ξ] = Γ[Aµ, ψ, ψ] + d x ψ ξ + ξ ψ + AµJ . (1.275) Z h i A one-particle-irreducible n-point function or “proper vertex” contains no contributions that become disconnected when a single connected m-point Green function is removed; e.g., via functional differentiation. This is equivalent to the statement that no diagram representing or contributing to a given proper vertex separates into two disconnected diagrams if only one connected propagator is cut. (A detailed explanation is provided in Ref. [3], pp. 289-294.) A simple generalisation of the analysis in Sec. 1.6.1 yields δZ δZ δZ = A (x) , = ψ(x) , = ψ¯(x) , (1.276) δJ µ(x) µ δξ¯(x) δξ(x) − where here the external sources are nonzero. Hence Γ in Eq. (1.275) must satisfy δΓ δΓ δΓ = J (x) , = ξf (x) , = ξ¯f (x) . (1.277) δAµ(x) − µ δψ¯f (x) − δψf (x) (NB. Since the sources are not zero then, e.g., δA A (x) = A (x; [J , ξ, ξ¯]) = 0 , (1.278) µ µ µ ⇒ δJ µ(x) 6 with analogous statements for the Grassmannian functional derivatives.) It is easy to see that setting ψ¯ = 0 = ψ after differentiating Γ gives zero unless there are equal numbers of ψ¯ and ψ derivatives. (This is analogous to the result for scalar fields in Eq. (1.140).) Now consider the product (with spinor labels r, s, t)

2 2 4 δ Z δ Γ d z f g ¯ . (1.279) − ¯h δψh(z)ψ (y) ξ = ξ = 0 Z δξr (x)ξt (z) t s ψ = ψ = 0

Using Eqs. (1.276), (1.277), this simpliﬁes as follows:

h g g 4 δψt (z) δξs (y) δξs (y) fg 4 = d z f h ¯ = f = δrsδ δ (x y) . δξr (x) δψt (z) ξ = ξ = 0 δξr (x) − Z ψ = ψ = 0 ψ = ψ = 0

(1.280) Now returning to Eq. (1.274) and setting ξ¯ = 0 = ξ one obtains

δΓ ρ 1 ν f f = ∂ρ∂ gµν 1 ∂µ∂ν A (x) i e0 tr γµS (x, x; [Aµ]) , δAµ(x) − − λ − ψ=ψ=0 0 f h i X (1.281) after making the identification δ2Z δ2Z Sf (x, y; [A ]) = = (no summation on f) , (1.282) µ − δξf (y)ξ¯f (x) δξ¯f (x)ξf (y) which is the connected Green function that describes the propagation of a fermion with flavour f in an external electromagnetic field Aµ (cf. the free fermion Green function in Eq. (1.197).) I observe that it is a direct consequence of Eq. (1.279) that δ2Γ Sf (x, y; [A])−1 = , (1.283) δψf (x)δψ¯f (y) ψ=ψ=0

and it is a general property that such functional deriv atives of the generating functional for 1PI Green functions are related to the associated propagator’s inverse. Clearly the vacuum fermion propagator or connected fermion 2-point function is f f S (x, y) := S (x, y; [Aµ = 0]) . (1.284) Such vacuum Green functions are of primary interest in quantum ﬁeld theory. To continue, one diﬀerentiates Eq. (1.281) with respect to Aν (y) and sets Jµ(x) = 0, which yields

2 δ Γ ρ 1 4 = ∂ρ∂ gµν 1 ∂µ∂ν δ (x y) δAµ(x)δAν(y) Aµ = 0 − − λ − 0 ψ = ψ = 0

−1 2 f δ δ Γ i e0 tr γµ  . − δA (y)  δψf (x)δψ¯f (x)  f ν ψ=ψ=0 X       (1.285) The l.h.s. is easily understood. Just as Eqs. (1.283), (1.284) deﬁne the inverse of the fermion propagator, so here is δ2Γ (D−1)µν (x, y) := . (1.286) δAµ(x)δAν(y) Aµ = 0

ψ = ψ = 0

The r.h.s., however, must be simpliﬁed and interpreted. First observe that

−1 δ δ2Γ = δA (y)  δψf (x)δψ¯f (x)  ν ψ=ψ=0

 

−1 −1 δ2Γ δ δ2Γ δ2Γ d4ud4w , −  δψf (x)δψ¯f (w)  δA (y) δψf (u)δψ¯f (w)  δψf (w)δψ¯f (x)  Z ψ=ψ=0 ν ψ=ψ=0

    (1.287) which is an analogue of the result for ﬁnite dimensional matrices:

d dA(x) dA−1(x) A(x)A−1(x) = I = 0 = A−1(x) + A(x) dx dx dx h i dA−1(x) dA(x) = A−1(x) A−1(x) .(1.288) ⇒ dx − dx Equation (1.287) involves the 1PI 3-point function

2 f f δ δ Γ e0 Γµ(x, y; z) := f f . (1.289) δAν (z) δψ (x)δψ¯ (y) This is the proper fermion-gauge-boson vertex. At leading order in perturbation theory Γf (x, y; z) = γ δ4(x z) δ4(y z) , (1.290) ν ν − − a result which can be obtained via the explicit calculation of the functional derivatives in Eq. (1.289). Now, deﬁning the gauge-boson vacuum polarisation:

f 2 4 4 f f f Πµν (x, y) = i (e0 ) d z1 d z2 tr γµS (x, z1)Γν (z1, z2; y)S (z2, x) , (1.291) Xf Z h i it is immediately apparent that Eq. (1.285) may be expressed as

−1 µν ρ 1 4 (D ) (x, y) = ∂ρ∂ gµν 1 ∂µ∂ν δ (x y) + Πµν (x, y) . (1.292) − − λ0 − In general, the gauge-boson vacuum polarisation, or “self-energy,” describes the modiﬁcation of the gauge-boson’s propagation characteristics due to the presence of virtual fermion-antifermion pairs in quantum ﬁeld theory. In particular, the photon vacuum polarisation is an important element in the description of process such as ρ0 e+e−. → The propagator for a free gauge boson was given in Eq. (1.268). In the presence of interactions; i.e., for Π = 0 in Eq. (1.292), this becomes µν 6 gµν + (qµqν/[q2 + iη]) 1 qµqν Dµν(q) = − λ , (1.293) q2 + iη 1 + Π(q2) − 0 (q2 + iη)2 where I have used the “Ward-Takahashi identity:”

qµ Πµν (q) = 0 = Πµν (q) qν , (1.294) which means that one can write

Πµν (q) = gµνq2 + qµqν Π(q2) . (1.295) − Π(q2) is the polarisation scalar and, in QED, it is independent of the gauge parameter, λ0. (NB. λ0 = 1 is called “Feynman gauge” and it is useful in perturbative calculations because it obviously simpliﬁes the Π(q2) = 0 gauge boson propagator enormously. In nonperturbative applications, however, λ0 = 0, “Landau gauge,” is most useful because it ensures the gauge boson propagator itself is transverse.) Ward-Takahashi identities (WTIs) are relations satisﬁed by n-point Green functions, relations which are an essential consequence of a theory’s local gauge invariance; i.e., local current conservation. They can be proved directly from the generating functional and have physical implications. For example, Eq. (1.295) ensures that the photon remains massless in the presence of charged fermions. (A discussion of WTIs can be found in Ref. [1], pp. 299-303, and Ref. [3], pp. 407-411; and their generalisation to non-Abelian theories as “Slavnov-Taylor” identities is described in Ref. [5], Chap. 2.) In the absence of external sources for fermions and gauge bosons, Eq. (1.291) can easily be represented in momentum space, for then the 2- and 3-point functions that appear explicitly must be translationally invariant and hence can be simply expressed in terms of Fourier amplitudes. This yields

d4` iΠ (q) = (ef )2 tr[(iγ )(iSf (`))(iΓf (`, ` + q))(iS(` + q))] . (1.296) µν − 0 (2π)d µ Xf Z It is the reduction to a single integral that makes momentum space representa- tions most widely used in continuum calculations. In QED the vacuum polarisation is directly related to the running coupling constant. This connection makes its importance obvious. In QCD the connection is not so direct but, nevertheless, the polarisation scalar is a key component in the evaluation of the strong running coupling. In the above analysis we saw that second derivatives of the generating func- ¯ tional, Γ[Aµ, ψ, ψ], give the inverse-fermion and -photon propagators and that the third derivative gave the proper photon-fermion vertex. In general, all derivatives of this generating functional, higher than two, produce the corresponding proper Green’s functions, where the number and type of derivatives give the number and type of proper Green functions that it can serve to connect.

1.9.2 Fermion Self Energy Equation (1.274) is a nonperturbative generalisation of Maxwell’s equation in quantum ﬁeld theory. Its derivation provides the model by which one can obtain an equivalent generalisation of Dirac’s equation. To this end consider that

δ g 0 = [ A ] [ ψ][ ψ¯] exp i S[A , ψ, ψ¯] + d4x ψ ξg + ξ¯gψg + A J µ D µ D D δψ¯f (x) µ µ Z Z h i = [ Aµ] [ ψ][ ψ¯] Z D D D δS f 4 g g g g µ + ξ (x) exp i S[Aµ, ψ, ψ¯] + d x ψ ξ + ξ¯ ψ + AµJ (δψ¯f (x) ) Z h i δS δ δ δ f ¯ = , , + η (x) W [Jµ, ξ, ξ] (1.297) (δψ¯f (x) "iδJ iδξ¯ −iδξ # )

f f f µ δ δ ¯ = ξ (x) + i∂/ m0 + e0 γ W [Jµ, ξ, ξ] . (1.298) " − iδJ µ(x)! iδξ¯f (x)# This is a nonperturbative, funcational equivalent of Dirac’s equation. One can proceed further. A functional derivative with respect to ξf : δ/δξf (y), yields

4 f f µ δ f δ (x y)W [Jµ] i∂/ m0 + e0 γ W [Jµ] S (x, y; [Aµ]) = (1.299)0 , − − − iδJ µ(x)! f ¯f after setting ξ = 0 = ξ , where W [Jµ] := W [Jµ, 0, 0] and S(x, y; [Aµ]) is deﬁned in Eq. (1.282). Now, using Eqs. (1.271), (1.277), this can be rewritten

4 f f f µ δ f δ (x y) i∂/ m0 + e0 A/(x; [J]) + e0 γ S (x, y; [Aµ]) =(1.300)0 , − − − iδJ µ(x)! which deﬁnes the nonperturbative connected 2-point fermion Green function (This is clearly the functional equivalent of Eq. (1.86).) The electromagentic four-potential vanishes in the absence of an external source; i.e., Aµ(x; [J = 0]) = 0, so it remains only to exhibit the content of the remaining functional diﬀerentiation in Eq. (1.300), which can be accomplished using Eq. (1.287): −1 2 δ f 4 δAν (z) δ δ Γ S (x, y; [Aµ]) = d z iδJ µ(x) iδJ µ(x) δA (z)  δψf (x)δψ¯f (y)  Z ν ψ=ψ=0

  f 4 4 4 δAν (z) f = e0 d z d u d w S (x, u) Γν(u, w; z) S(w, y) − Z iδJµ(x) f 4 4 4 f = e0 d z d u d w iDµν (x z) S (x, u) Γν(u, w; z) S(w, y) , − Z − (1.301) where, in the last line, I have set J = 0 and used Eq. (1.286). Hence, in the absence of external sources, Eq. (1.300) is equivalent to δ4(x y) = i∂/ mf Sf (x, y) − − 0 f 2 4 4 4 µν 4 i (e0 ) d z d u d w D (x, z) γµ S(x, u) Γν(u, w; z) S(w, y) = δ (x (1.302)y) . − Z − Just as the photon vacuum polarisation was introduced to simplify, or re- express, the DSE for the gauge boson propagator, Eq. (1.291), one can deﬁne a fermion self-energy:

f f 2 4 4 µν Σ (x, z) = i(e0 ) d u d w D (x, z) γµ S(x, u) Γν(u, w; z) , (1.303) Z so that Eq. (1.302) assumes the form

d4z i∂/ mf δ4(x z) Σf (x, z) S(z, y) = δ4(x y) . (1.304) x − 0 − − − Z h i Again using the property that Green functions are translationally invariant in the absence of external sources, the equation for the self-energy can be written in momentum space:

4 f f 2 d ` µν f f Σ (p) = i (e0 ) 4 D (p `) [iγµ] [iS (`)] [iΓν (`, p)] . (1.305) Z (2π) − In terms of the self-energy, it follows from Eq. (1.304) that the connected fermion 2-point function can be written in momentum space as 1 Sf (p) = . (1.306) /p mf Σf (p) + iη+ − 0 − Equation (1.305) is the exact Gap Equation. It describes the manner in which the propagation characteristics of a fermion moving through the ground state of QED (the QED vacuum) is altered by the repeated emission and reabsorp- tion of virtual photons. The equation can also describe the real process of Bremsstrahlung. Furthermore, the solution of the analogous equation in QCD its solution provides information about dynamical chiral symmetry breaking and also quark conﬁnement.

1.9.3 Exercises 1. Verify Eq. (1.277).

2. Verify Eq. (1.287).

3. Verify Eq. (1.296).

4. Verify Eq. (1.300). 1.10 Perturbation Theory

1.10.1 Quark Self Energy A key feature of strong interaction physics is dynamical chiral symmetry breaking (DCSB). In order to understand it one must ﬁrst come to terms with explicit chiral symmetry breaking. Consider then the DSE for the quark self-energy in QCD:

d4` i i Σ(p) = g2 Dµν (p `) λaγ S(`) Γa(`, p) , (1.307) 0 4 2 µ ν − − Z (2π) − where I have suppressed the ﬂavour label. The form is precisely the same as that in QED, Eq. (1.305), with the only diﬀerence being the introduction of the colour (Gell-Mann) matrices: λa; a = 1, . . . , 8 at the fermion-gauge-boson vertex. The interpretation of the sym{ bols is also analogous:} Dµν(`) is the connected gluon a 0 2-point function and Γν(`, ` ) is the proper quark-gluon vertex. The one-loop contribution to the quark’s self-energy is obtained by evaluating the r.h.s. of Eq. (1.307) using the free quark and gluon propagators, and the quark-gluon vertex: a (0) 0 i a Γν (`, ` ) = 2λ γν , (1.308) which appears to be a straightforward task. To be explicit, the goal is to calculate 4 µ ν (2) 2 d k µν k k 1 i Σ (p) = g0 4 g + (1 λ0) 2 + 2 + − − Z (2π) − − k + iη ! k + iη i 1 i λaγ λaγ (1.309) × 2 µ k + /p m + iη+ 2 µ 6 − 0 and one may proceed as follows. First observe that Eq. (1.309) can be re-expressed as d4k 1 1 i Σ(2)(p) = g2 C (R) − − 0 2 (2π)4 (k + p)2 m2 + iη+ k2 + iη+ Z − 0 µ (k, p)k γ (k + /p + m0) γµ (1 λ0) (k /p + m0) 2 (1 λ0) 6 × ( 6 − − 6 − − − k2 + iη+ ) (1.310) where I have used 1 1 N 2 1 a a I c 2λ 2λ = C2(R) c ; C2(R) = − , (1.311) 2Nc with Nc the number of colours (Nc = 3 in QCD) and Ic, the identity matrix in colour space. Now note that 2 (k, p) = [(k + p)2 m2] [k2] [p2 m2] (1.312) − 0 − − − 0 and hence d4k 1 1 i Σ(2)(p) = g2 C (R) − − 0 2 (2π)4 (k + p)2 m2 + iη+ k2 + iη+ Z − 0 µ γ (k + /p + m0) γµ + (1 λ0) (/p m0) ( 6 − −

2 2 k 2 2 k + (1 λ0) (p m0) 6 (1 λ0) [(k + p) m0] 6 . − − k2 + iη+ − − − k2 + iη+ ) (1.313) Focusing on the last term: d4k 1 1 k [(k + p)2 m2] 6 (2π)4 (k + p)2 m2 + iη+ k2 + iη+ − 0 k2 + iη+ Z − 0 d4k 1 k = 4 2 + 2 6 + = 0 (1.314) Z (2π) k + iη k + iη because the integrand is odd under k k, and so this term in Eq. (1.313) vanishes, leaving → − d4k 1 1 i Σ(2)(p) = g2 C (R) − − 0 2 (2π)4 (k + p)2 m2 + iη+ k2 + iη+ Z − 0 µ γ (k + /p + m0) γµ + (1 λ0) (/p m0) ( 6 − −

2 2 k + (1 λ0) (p m0) 6 . − − k2 + iη+ ) (1.315) Consider now the second term: d4k 1 1 (1 λ ) (/p m ) . − 0 − 0 (2π)4 (k + p)2 m2 + iη+ k2 + iη+ Z − 0 In particular, focus on the behaviour of the integrand at large k2: 1 1 2 1 k →∞ . (1.316) (k + p)2 m2 + iη+ k2 + iη+ ∼ (k2 m2 + iη+) (k2 + iη+) − 0 − 0 The integrand has poles in the second and fourth quadrant of the k0-plane but vanishes on any circle of radius R . That means one may rotate the contour anticlockwise to find → ∞ ∞ 0 1 dk 2 0 (k2 m + iη+) (k2 + iη+) Z − 0 i∞ 1 = dk0 0 0 2 ~ 2 2 + 0 2 ~ 2 + Z ([k ] k m0 + iη )([k ] k + iη ) 0 ∞ − − − k →ik4 1 = i dk4 . (1.317) 0 ( k2 ~k2 m2) ( k2 ~k2) Z − 4 − − 0 − 4 − 0 Performing a similar analysis of the −∞ part one obtains the complete result: d4k 1 R d3k ∞ dk 1 = i 4 . 4 2 2 + 2 + 3 2 2 2 2 2 (2π) (k m0 + iη ) (k + iη ) (2π) −∞ 2π ( ~k k m ) ( ~k k ) Z − Z Z − − 4 − 0 − − 4 (1.318) These two steps constitute what is called a “Wick rotation.” The integral on the r.h.s. is defined in a four-dimensional Euclidean space; i.e., k2 := k2 + k2 + k2 + k2 0 . . . k2 is nonnegative. A general vector in this 1 2 3 4 ≥ space can be written in the form: (k) = k (cos φ sin θ sin β, sin φ sin θ sin β, cos θ sin β, cos β) , (1.319) | | and clearly k2 = k 2. In this space the four-vector measure factor is | | ∞ π π 2π 4 1 2 2 2 dEk f(k1, . . . , k4) = dk k dβ sin β dθ sin θ dφ f(k, β, θ, φ) 2 0 0 0 0 Z Z Z Z Z (1.320) Returning now to Eq. (1.316) the large k2 behaviour of the integral can be determined via 4 ∞ d k 1 1 i 2 1 2 dk 2 (2π)4 (k + p)2 m + iη+ k2 + iη+ ≈ 16π2 0 (k2 + m ) Z − 0 Z 0 i Λ2 1 = lim dx 2 Λ→∞ 2 16π Z0 x + m0 i 2 2 = lim ln 1 + Λ /m0 ; 16π2 Λ→∞ → ∞ (1.321) i.e., after all this work, the result is meaningless: the one-loop contribution to the quark’s self-energy is divergent! Such “ultraviolet” divergences, and others which are more complicated, appear whenever loops appear in perturbation theory. (The others include “infrared” divergences associated with the gluons’ masslessness; e.g., consider what would happen in Eq. (1.321) with m 0.) In a renormalisable quantum field 0 → theory there exists a well-defined set of rules that can be used to render perturbation theory sensible. First, however, one must regularise the theory; i.e., introduce a cutoff, or some other means, to make finite every integral that appears. Then every step in the calculation of an observable is rigorously sensible. Renormalisation follows; i.e, the absorption of divergences, and the redefinition of couplings and masses, so that finally one arrives at -matrix amplitudes that are finite and physically meaningful. S The regularisation procedure must preserve the Ward-Takahashi identities (the Slavnov-Taylor identities in QCD) because they are crucial in proving that a theory can be sensibly renormalised. A theory is called renormalisable if, and only if, there are a finite number of different types of divergent integral so that only a finite number of masses and coupling constants need to be renormalised. 1.10.2 Dimensional Regularization The Pauli-Villars prescription is favoured in QED and that is described, for example, in Ref. [3]. In perturbative QCD, however, “dimensional regularisation” is the most commonly used procedure and I will introduce that herein. The key to the method is to give meaning to the divergent integrals by changing the dimension of spacetime. Returning to the exemplar, Eq. (1.316), this means we consider dDk 1 1 T = (1.322) (2π)D (k + p)2 m2 + iη+ k2 + iη+ Z − 0 where D is the dimension of spacetime and is not necessarily four. Observe now that 1 Γ(α + β) 1 xα−1 (1 x)β−1 = dx − , (1.323) aα bβ Γ(α) Γ(β) 0 [a x + b (1 x)]α+β Z − where Γ(x) is the gamma-function: Γ(n + 1) = n!. This is an example of what is commonly called “Feynman’s parametrisation,” and one can now write 1 dDk 1 T = dx 2 .(1.324) 0 (2π)D [(k xp)2 m (1 x) + p2x(1 x) + iη+]2 Z Z − − 0 − − (1.325)

The momentum integral is well-deﬁned for D = 1, 2, 3 but, as we have seen, not for D = 4. One proceeds under the assumption that D is such that the integral is convergent then a shift of variables is permitted: 1 dDk 1 T k→=k−xp dx , (1.326) 0 (2π)D [k2 a2 + iη+]2 Z Z − where a2 = m2(1 x) p2x(1 x). 0 − − − I will consider a generalisation of the momentum integral: dDk 1 I = , (1.327) n (2π)D [k2 a2 + iη+]n Z − and perform a Wick rotation to obtain

i n D 1 In = D ( 1) d k 2 2 n . (1.328) (2π) − Z [k + a ] The integrand has an O(D) spherical symmetry and therefore the angular integrals can be performed: 2πD/2 SD := dΩD = . (1.329) Z Γ(D/2) 2 Clearly, S4 = 2π , as we saw in Eq. (1.321). Hence ( 1)n 1 ∞ kD−1 In = i D−1 D/2 dk 2 2 n . (1.330) 2 π Γ(D/2) Z0 (k + a ) Writing D = 4 + 2 one arrives at

2 i 2 2−n a Γ(n 2 ) In = ( a ) − − . (1.331) (4π)2 − 4π ! Γ(n) (NB. Every step is rigorously justified as long as 2n > D.) The important point for continuing with this procedure is that the analytic continuation of Γ(x) is unique and that means one may use Eq. (1.331) as the definition of In whenever the integral is ill-defined. I observe that D = 4 is recovered via the limit 0− and the divergence of the integral for n = 2 in this case is encoded in → 1 Γ(n 2 ) = Γ( ) = γ + O() ; (1.332) − − − − E − i.e., in the pole in the gamma-function. (γE is the Euler constant.) Substituting Eq. (1.331) in Eq. (1.326) and setting n = 2 yields

1 2 2 2 i Γ( ) m0 p T = (gν ) 2 − dx 2 (1 x) 2 x(1 x) , (1.333) (4π) (4π) Z0 " ν − − ν − # wherein I have employed a nugatory transformation to introduce the mass-scale ν. It is the limit 0− that is of interest, in which case it follows that (x = exp ln x 1 + ln x→) ≈ 1 2 2 2 i 1 m0 p T = (gν ) 2 γE + ln 4π dx ln 2 (1 x) 2 x(1 x) (4π) (− − − Z0 " ν − − ν − #) 2 2 2 2 i 1 m0 m0 p = (gν ) 2 γE + ln 4π + 2 ln 2 1 2 ln 1 2 . (4π) (− − − ν − − p ! " − m0 #) (1.334)

It is important to understand the physical content of Eq. (1.334). While it is only one part of the gluon’s contribution to the quark’s self-energy, many of its properties hold generally.

2 2 2 1. Observe that T (p ) is well-defined for p < m0; i.e., for all spacelike mo- 2 2 menta and for a small domain of timelike momenta. However, at p = m0, T (p2) exhibits a ln-branch-point and hence T (p2) acquires an imaginary 2 2 part for p > m0. This imaginary part describes the real, physical process by which a quark emits a massless gluon; i.e., gluon Bremsstrahlung. In QCD this is one element in the collection of processes referred to as “quark fragmentation.” 2. The mass-scale, ν, introduced in Eq. (1.333), which is a theoretical artifice, does not affect the position of the branch-point, which is very good because that branch-point is associated with observable phenomena. While it may appear that ν affects the magnitude of physical cross sections because it modifies the coupling, that is not really so: in going to D = 2n di- − mensions the coupling constant, which was dimensionless for D = 4, has acquired a mass dimension and so the physical, dimensionless coupling constant is α := (gν)2/(4π). It is this dependence on ν that opens the door to the generation of a “running coupling constant” and “running masses” that are a hallmark of quantum field theory.

3. A number of constants have appeared in T (p2). These are irrelevant because they are eliminated in the renormalisation procedure. (NB. So far we have only regularised the expression. Renormalisation is another step.)

4. It is apparent that dimensional regularisation gives meaning to divergent integrals without introducing new couplings or new fields. That is a ben- 0 1 2 3 efit. The cost is that while γ5 = iγ γ γ γ is well-defined with particular properties for D = 4, a generalisation to D = 4 is difficult and hence so is the study of chiral symmetry. 6

D-dimensional Dirac Algebra When one employs dimensional regularisation all the algebraic manipulations must be performed before the integrals are evaluated, and that includes the Dirac algebra. The Cliﬀord algebra is unchanged in D-dimensions

γµ , γν = 2 gµν ; µ, ν = 1, . . . , D 1 , (1.335) { } − but now µν gµν g = D (1.336) and hence

ν γµ γ = D 1D , (1.337) γ γν γµ = (2 D) γν , (1.338) µ − γ γν γλ γµ = 4 gνλ1 + (D 4) γν γλ , (1.339) µ D − γ γν γλ γρ γµ = 2 γρ γλ γν + (4 D) γν γλ γρ , (1.340) µ − − where 1 is the D D-dimensional unit matrix. D × It is also necessary to evaluate traces of products of Dirac matrices. For a D-dimensional space, with D even, the only irreducible representation of the Cliﬀord algebra, Eq. (1.335), has dimension f(D) = 2D/2. In any calculation it is the (anti-)commutation of Dirac matrices that leads to physically important factors associated with the dimension of spacetime while f(D) always appears as a common multiplicative factor. Hence one can just set

f(D) 4 (1.341) ≡ in all calculations. Any other prescription merely leads to constant terms of the type encountered above; e.g., γE, which are eliminated in renormalisation. The D-dimensional generalisation of γ5 is a more intricate problem. However, I will not use it herein and hence will omit that discussion.

Observations on the Appearance of Divergences Consider a general Lagrangian density:

L(x) = L0(x) + giLi(x) , (1.342) Xi where L0 is the sum of the free-particle Lagrangian densities and Li(x) represents the interaction terms with the coupling constants, gi, written explicitly. Assume that Li(x) has fi fermion fields (fi must be even since fermion fields always appear ¯ ∂ in the pairs ψ, ψ), bi boson fields and ni derivatives. The action

S = d4x L(x) (1.343) Z must be a dimensionless scalar and therefore L(x) must have mass-dimension M 4. Clearly a derivative operator has dimension M. Hence, looking at the free- particle Lagrangian densities, it is plain that each fermion field has dimension M 3/2 and each boson field, dimension M 1. It follows that a coupling constant multiplying the interaction Lagrangian density Li(x) must have mass-dimension 3 4−di ∂ [gi] = M , di = 2fi + bi + ni . (1.344) It is a fundamentally important fact in quantum field theory that if there is even one coupling constant for which

[gi] < 0 (1.345) then the theory possesses infinitely many different types of divergences and hence cannot be rendered finite through a finite number of renormalisations. This defines the term nonrenormalisable. In the past nonrenormalisable theories were rejected as having no predictive power: if one has to remove infinitely many infinite constants before one can define a result then that result cannot be meaningful. However, the modern view is different. Chiral perturbation theory is a nonrenormalisable “effective theory.” However, at any finite order in perturbation theory there is only a finite number of undetermined constants: 2 at leading order; 6 more, making 8 in total, when one-loop effects are considered; and more than 140 new terms when two-loop effects are admitted. Nevertheless, as long as there is a domain in some physical external-parameter space on which the one-loop corrected Lagrangian density can be assumed to be a good approximation, and there is more data that can be described than there are undetermined constants, then the “effective theory” can be useful as a tool for correlating observables and elucidating the symmetries that underly the general pattern of hadronic behaviour.

1.10.3 Regularized Quark Self Energy I can now return and re-express Eq. (1.315):

dDk 1 1 1 i Σ(2)(p) = (g ν)2 C (R) − − 0 2 (2π)4 ν2 (k + p)2 m2 + iη+ k2 + iη+ Z − 0 µ γ (k + /p + m0) γµ + (1 λ0) (/p m0) ( 6 − −

2 2 k + (1 λ0) (p m0) 6 . − − k2 + iη+ ) (1.346)

It can be separated into a sum of two terms, each proportional to a diﬀerent Dirac structure: 2 2 Σ(/p) = ΣV (p ) /p + ΣS(p ) 1D , (1.347) that can be obtained via trace projection: 1 1 1 Σ (p2) = tr [/pΣ(/p)] , Σ (p2) = tr [Σ(/p)] . (1.348) V f(D) p2 D S f(D) D

These are the vector and scalar parts of the dressed-quark self-energy, and they are easily found to be given by

D 2 2 2 d k 1 1 p ΣV (p ) = i (g0ν ) C2(R) 4 2 2 2 + 2 + − Z (2π) ν ([(k + p) m0 + iη ] [k + iη ] − µ 2 µ 2 2 2 pµk (2 D)(p + pµk ) + (1 λ0)p + (1 λ0)(p m0) . × " − − − − k2 + iη+ #) (1.349) dDk 1 m (D 1 + λ ) Σ (p2) = i (g ν)2 C (R) 0 − 0 . (1.350) S − 0 2 (2π)4 ν2 [(k + p)2 m2 + iη+] [k2 + iη+] Z − 0 (NB. As promised, the factor of f(D) has cancelled.) These equations involve integrals of the general form dDk 1 1 I(α, β; p2, m2) = ,(1.351) (2π)4 ν2 [(k + p)2 m2 + iη+]α [k2 + iη+]β Z − dDk 1 kµ J µ(α, β; p2, m2) = ,(1.352) (2π)4 ν2 [(k + p)2 m2 + iη+]α [k2 + iη+]β Z − the ﬁrst of which we have already encountered in Sec. 1.10.2. The general results are (D = 4 + 2)

i 1 α+β−2 Γ(α + β 2 ) Γ(2 + β) I(α, β; p2, m2) = − − − (4π)2 p2 ! Γ(α) Γ(2 + ) p2 m2 2+−α−β 1 1 2F1(α + β 2 , 2 + β, 2 + ; ) , × −4πν2 ! − p2 ! − − − 1 (m2/p2) − (1.353)

i 1 α+β−2 Γ(α + β 2 ) Γ(3 + β) J µ(α, β; p2, m2) = pµ − − − (4π)2 p2 ! Γ(α) Γ(3 + ) p2 m2 2+−α−β 1 1 2F1(α + β 2 , 3 + β, 3 + ; ) × −4πν2 ! − p2 ! − − − 1 (m2/p2) − =: pµ J(α, β; p2, m2) , (1.354) where 2F1(a, b, c; z) is the hypergeometric function. Returning again to Eqs. (1.349), (1.350) it is plain that

Σ (p2) = i (g ν)2 C (R) 2(1 + ) J(1, 1) (1 + 2) I(1, 1) (p2 m2) J(1, 2) V − 0 2 − − − 0 nh i +λ (p2 m2) J(1, 2) I(1, 1) , (1.355) 0 − 0 − Σ (p2) = i (gh ν)2 C (R) m (3 + λ + 2i)oI(1, 1; p2, m2) , (1.356) S − 0 2 0 0 2 2 where I have omitted the (p , m0) component in the arguments of I, J. The integrals explicitly required are i 1 p2 I(1, 1) = + ln 4π γE ln + 2 (4π)2 (− − − −ν2 ! m2 m2 m2 m2 ln 1 ln 1 , (1.357) − p2 − p2 ! − − p2 ! − p2 !) i 1 1 p2 J(1, 1) = + ln 4π γE ln + 2 (4π)2 2 (− − − −ν2 ! m2 m2 m2 m2 m2 2 m2 m2 2 ln 1 2 + ln 1 , − p2 − p2 ! − p2 ! −  − p2 " p2 #  − p2 ! − p2      (1.358) i 1 m2 m2 m2 m2 J(1, 2) = ln + ln 1 + 1 . (1.359) (4π)2 p2 (− p2 − p2 ! p2 − p2 ! )

Using these expressions it is straightforward to show that the λ0-independent term in Eq. (1.355) is identically zero in D = 4 dimensions; i.e., for > 0−, and − hence 2 2 2 (gν ) 1 m0 ΣV (p ) = λ0 C2(R) ln 4π + γE + ln (4π)2 ( − ν2 m2 m4 p2 1 0 + 1 0 ln 1 . (1.360) − − p2 − p4 ! − m2 !)

It is obvious that in Landau gauge: λ0 = 0, in four spacetime dimensions: 2 ΣV (p ) 0, at this order. The scalar piece of the quark’s self-energy is also easily found:≡

2 2 2 (gν ) 1 m0 ΣS(p ) = m0 C2(R) (3 + λ0) ln 4π + γE + ln (4π)2 (− " − ν2 # 2 2 m0 p +2(2 + λ0) (3 + λ0) 1 2 ln 1 2 . (1.361) − − p ! − m0 !) Note that in Yennie gauge: λ = 3, in four dimensions, the scalar piece of the 0 − self-energy is momentum-independent, at this order. We now have the complete regularised dressed-quark self-energy at one-loop order in perturbation theory and its structure is precisely as I described in Sec. 1.10.2. Renormalisation must follow. One ﬁnal observation: the scalar piece of the self-energy is proportional to the bare current-quark mass, m0. That is true at every order in perturbation theory. Clearly then 2 2 lim ΣS(p , m0) = 0 (1.362) m0→0 and hence dynamical chiral symmetry breaking is impossible in perturbation theory.

1.10.4 Exercises 1. Verify Eq. (1.321). 2. Verify Eq. (1.328). 3. Verify Eq. (1.334). 4. Verify Eqs. (1.349) and (1.350). 5. Verify Eqs. (1.360) and (1.361). 1.11 Renormalized Quark Self Energy

Hitherto I have illustrated the manner in which dimensional regularisation is employed to give sense to the divergent integrals that appear in the perturbative calculation of matrix elements in quantum field theory. It is now necessary to renormalise the theory; i.e., to provide a well-defined prescription for the elimination of all those parts in the calculated matric element that express the divergences and thereby obtain finite results for Green functions in the limit 0− (or with the removal of whatever other parameter has been used to regularise→the divergences).

1.11.1 Renormalized Lagrangian The bare QCD Lagrangian density is 1 1 L(x) = ∂ Ba(x)[∂µBνa(x) ∂ν Bµa(x)] ∂ Ba(x)∂ Ba(x) −2 µ ν − − 2λ µ ν µ ν 1 g f [∂µBνa(x) ∂ν Bµa(x)]Bb (x)Bc(x) −2 abc − µ ν 1 g2f f Bb (x)Bc(x)Bµd(x)Bνe(x) −4 abc ade µ ν a µ a a b µc ∂µφ¯ (x)∂ φ (x) + g fabc ∂µφ¯ (x) φ (x)B (x) − 1 +q¯f (x)i∂/qf (x) mf q¯f (x)qf (x) + g q¯f (x)λaB/ a(x)qf (x) ,(1.363) − 2 a ¯a a where: Bµ(x) are the gluon fields, with the colour label a = 1, . . . , 8; φ (x), φ (x) are the (Grassmannian) ghost fields; q¯f (x), qf (x) are the (Grassmannian) quark fields, with the flavour label f = u, d, s, c, b, t; and g, mf , λ are, respectively, the coupling, mass and gauge fixing parameter. (NB. The QED Lagrangian density is immediately obtained by setting f 0. It is clearly the non-Abelian nature of abc ≡ the gauge group, SU(Nc), that generates the gluon self-couplings, the triple-gluon and four-gluon vertices, and the ghost-gluon interaction.) The elimination of the divergent parts in the expression for a Green function can be achieved by adding “counterterms” to the bare QCD Lagrangian density, one for each different type of divergence in the theory; i.e., one considers the renormalised Lagrangian density

LR(x) := L(x) + Lc(x) , (1.364) with 1 1 L (x) = C ∂ Ba(x)[∂µBνa(x) ∂ν Bµa(x)] + C ∂ Ba(x)∂ Ba(x) c 3Y M 2 µ ν − 6 2λ µ ν µ ν 1 +C g f [∂µBνa(x) ∂ν Bµa(x)]Bb (x)Bc(x) 1Y M 2 abc − µ ν 1 2 b c µd νe +C5 4 g fabcfade Bµ(x)Bν (x)B (x)B (x) a µ a a b µc +C˜3 ∂µφ¯ (x)∂ φ (x) C˜1g fabc ∂µφ¯ (x) φ (x)B (x) − 1 C q¯f (x)i∂/qf (x) + C mf q¯f (x)qf (x) C g q¯f (x)λaB/ a(x)qf (x) . − 2F 4 − 1F 2 (1.365)

To prove the renormalisability of QCD one must establish that the coeﬃcients, Ci, each understood as a power series in g2, are the only additional terms necessary to remove all the ultraviolet divergences in the theory at every order in the perturbative expansion. In the example of Sec. 1.10.2 I illustrated that the divergent terms in the regularised self-energy are proportional to g2. This is a general property and hence 2 all of the Ci begin with a g term. The Ci-dependent terms can be treated just as the terms in the original Lagrangian density and yield corrections to the expressions we have already derived that begin with an order-g2 term. Returning to the example of the dressed-quark self-energy this means that we have an additional contribution: ∆Σ(2)(/p) = C /p C m , (1.366) 2F − 4 and one can choose C2F , C4 such that the total self-energy is ﬁnite. The renormalisation constants are introduced as follows:

Z := 1 C (1.367) i − i so that Eq. (1.365) becomes

Z3Y M Z6 L (x) = ∂ Ba(x)[∂µBνa(x) ∂ν Bµa(x)] ∂ Ba(x)∂ Ba(x) R − 2 µ ν − − 2λ µ ν µ ν Z1Y M g f [∂µBνa(x) ∂ν Bµa(x)]Bb (x)Bc(x) − 2 abc − µ ν Z5 g2f f Bb (x)Bc(x)Bµd(x)Bνe(x) − 4 abc ade µ ν Z˜ ∂ φ¯a(x)∂µφa(x) + Z˜ g f ∂ φ¯a(x) φb(x)Bµc(x) − 3 µ 1 abc µ Z1Y M +Z q¯f (x)i∂/qf (x) Z mf q¯f (x)qf (x) + g q¯f (x)λaB/ a(x)qf (x) , 2F − 4 2 (1.368)

NB. I have implicitly assumed that the renormalisation counterterms, and hence the renormalisation constants, are flavour independent. It is always possible to choose prescriptions such that this is so. I will now introduce the bare fields, coupling constants, masses and gauge fixing parameter: µa 1/2 µa f 1/2 f B0 (x) := Z3Y M B (x) , q0 (x) := Z2F q (x) , a ˜1/2 a ¯a ˜1/2 ¯a φ0(x) := Z3 φ (x) , φ0(x) := Z3 φ (x) , −3/2 ˜ ˜−1 −1/2 g0Y M := Z1Y M Z3Y M g , g˜0 := Z1 Z3 Z3Y M , (1.369) −1/2 −1 1/2 −1 g0F := Z1F Z3Y M Z2F g , g05 := Z5 Z3Y M g f −1 f −1 m0 := Z4 Z2F m , λ0 := Z6 Z3Y M λ . The fields and couplings on the r.h.s. of these definitions are called renormalised, and the couplings are finite and the fields produce Green functions that are finite even in D = 4 dimensions. ( NB. All these quantities are defined in D = 4 + 2- dimensional space. Hence one has for the Lagrangian density: [L(x)] = M D, and the field and coupling dimensions are [q(x)] = [q¯(x)] = M 3/2+ , [Bµ(x)] = M 1+ , [φ(x)] = [φ¯(x)] = M 1+ , [g] = M − , (1.370) [λ] = M 0 , [m] = M 1 .) The renormalised Lagrangian density can be rewritten in terms of the bare quantities: 1 a µ νa ν µa 1 a a LR(x) = ∂µB0ν (x)[∂ B0 (x) ∂ B0 (x)] ∂µB0ν(x)∂µB0ν(x) −2 − − 2λ0 1 g f [∂µBνa(x) ∂ν Bµa(x)]Ba (x)Ba (x) −2 0Y M abc 0 − 0 0µ 0ν 1 g2 f f Bb (x)Bc (x)Bµd(x)Bνe(x) −4 05 abc ade 0µ 0ν 0 0 ¯a µ a ¯a b µc ∂µφ0(x)∂ φ0(x) + g˜0 fabc ∂µφ0(x) φ0(x)B0 (x) − 1 +q¯f (x)i∂/qf (x) mf q¯f (x)qf (x) + g q¯f (x)λaB/ a(x)qf (x) . 0 0 − 0 0 0 2 0F 0 0 0 (1.371)

It is apparent that now the couplings are diﬀerent and hence LR(x) is not invariant under local gauge transformations (more properly BRST transformations) unless

g0Y M = g˜0 = g0F = g05 = g0 . (1.372) Therefore, if the renormalisation procedure is to preserve the character of the gauge theory, the renormalisation constants cannot be completely arbitrary but must satisfy the following “Slavnov-Taylor” identities:

Z3Y M Z˜3 g0Y M = g˜0 = , ⇒ Z1Y M Z˜1 Z3Y M Z2F g0Y M = g0F = , (1.373) ⇒ Z1Y M Z1F 2 Z1Y M g0Y M = g05 Z5 = . ⇒ Z3Y M In QED the second of these equations becomes the Ward-Takahashi identity: Z1F = Z2F .

1.11.2 Renormalization Schemes At this point we can immediately write an expression for the renormalised dressed- quark self-energy:

Σ(2)(/p) = Σ (p2) + C /p + Σ (p2) C m . (1.374) R V 2F S − 4 The subtraction constants are not yet determined and there are many ways one may choose them in order to eliminate the divergent parts of bare Green functions.

Minimal Subtraction In the minimal subtraction (MS) scheme one deﬁnes a dimensionless coupling

(gν)2 α := (1.375) 4π and considers each counterterm as a power series in α with the form

∞ j 1 α j C = C(2j) , (1.376) i i,k k π jX=1 kX=1 where the coeﬃcients in the expansion may, at most, depend on the gauge parameter, λ. Using Eqs. (1.360), (1.361) and (1.374) we have

2 (2) α 1 1 m ΣR (/p) = /p λ C2(R) ln 4π + γE + ln (1.377) ( π 4 " − ν2 m2 m4 p2 1 + 1 ln 1 + C2F (1.378) − − p2 − p4 ! − m2 !# ) α 1 1 m2 +m C2(R) (3 + λ) ln 4π + γE + ln (1.379) π 4 (− " − ν2 # m2 p2 +2(2 + λ) (3 + λ) 1 ln 1 C4 . (1.380) − − p2 ! − m2 ! − )

Now one chooses C2F , C4 such that they cancel the 1/ terms in this equation and therefore, at one-loop level,

α 1 1 Z = 1 C = 1 + λ C (R) , (1.381) 2F − 2F π 4 2 α 1 1 Z = 1 C = 1 + (3 + λ) C (R) , (1.382) 4 − 4 π 4 2 and hence Eq. (1.380) becomes

2 (2) α 1 m ΣR (/p) = /p λ C2(R) ln 4π + γE + ln (1.383) ( π 4 " − ν2 ! m2 m4 p2 1 + 1 ln 1 + C2F (1.384) − − p2 − p4 ! − m2 !# ) α 1 m2 +m C2(R) (3 + λ) ln 4π + γE + ln (1.385) π 4 (− − ν2 ! m2 p2 +2(2 + λ) (3 + λ) 1 ln 1 C4 , (1.386) − − p2 ! − m2 ! − ) which is the desired, ﬁnite result for the dressed-quark self-energy. It is not common to work explicitly with the counterterms. More often one uses Eq. (1.371) and the deﬁnition of the connected 2-point quark Green function (an obvious analogue of Eq. (1.282)):

2 a δ ZR[J , ξ, ξ¯] iSf (x, y; m, λ, α) = µ = 0 qf (x) q¯f (y) 0 = Z−1 0 qf (x) q¯f (y) 0 R − δξf (y)δξ¯f (x) h | | i 2F h | 0 0 | i (1.387) to write −1 SR(/p; m, λ, α) = lim Z S0(/p; m0, λ0, α0; ) , (1.388) →0 2F n o where in the r.h.s. m0, λ0, α0 have to be substituted by their expressions in terms of the renormalised quantities and the limit taken order by order in α. To illustrate this using our concrete example, Eq. (1.388) yields

1 Σ (p2; m, λ, α) /p m 1 + Σ (p2; m, λ, α)/m − V R − SR n o 2 n o 2 = lim Z2F 1 ΣV (p ; m0, λ0, α0) /p m0 1 + ΣS(p ; m0, λ0, α0)/m0 →0− − − nh i h (1.389)io

−1 and taking into account that m = Z4 Z2F m0 then

2 2 1 ΣV R(p ; m, λ, α) = lim Z2F 1 ΣV (p ; m0, λ0, α0) , (1.390) − →0− − 2 n 2 o 1 + ΣSR(p ; m, λ, α)/m = lim Z4 1 + ΣS(p ; m0, λ0, α0)/m0 .(1.391) →− n o Now the renormalisation constants, Z2F , Z4, are chosen so as to exactly cancel the 1/ poles in the r.h.s. of Eqs. (1.390), (1.391). Using Eqs. (1.360), (1.361), Eqs. (1.381), (1.382) are immediately reproduced.

Modified Minimal Subtraction The modified minimal substraction scheme MS is also often used in QCD. It takes advantage of the fact that the 1/ pole obtained using dimensional regularisation always appears in the combination 1 ln 4π + γ (1.392) − E so that the renormalisation constants are defined so as to eliminate this combination, in its entirety, from the Green functions. At one-loop order the renormalisation constants in the MS scheme are trivially related to those in the MS scheme. At higher orders there are different ways of defining the scheme and the relation between the renormalisation constants is not so simple.

Momentum Subtraction In the momentum subtraction scheme (µ-scheme) a given renormalised Green function, GR, is obtained from its regularised counterpart, G, by subtracting from G its value at some arbitrarily chosen momentum scale. In QCD that scale is always chosen to be a Euclidean momentum: p2 = µ2. Returning to our example of the dressed-quark self-energy, in this scheme −

Σ (p2; µ2) := Σ (p2; ) Σ (p2; ) ; A = V, S , (1.393) AR A − A and so

(2) 2 2 α(µ) 1 2 1 1 ΣV R(p ; µ ) = λ(µ) C2(R) m (µ) + π 4 (− p2 µ2 ! m4(µ) p2 m4(µ) µ2 + 1 ln 1 1 ln 1 + , − p4 ! − m(µ)2 ! − − µ4 ! m2(µ)!) (1.394) α(µ) 1 Σ(2) (p2; µ2) = m(µ) C (R) [3 + λ(µ)] SR π 4 2 {− m2(µ) p2 m2(µ) µ2 1 ln 1 1 + ln 1 + × " − p2 ! − m2(µ)! − µ2 ! m2(µ)!#) (1.395) where the renormalised quantities depend on the point at which the renormalisation has been conducted. Clearly, from Eqs. (1.390), (1.391), the renormalisation constants in this scheme are

Z(2) = 1 + Σ(2)(p2 = µ2; ) (1.396) 2F V − Z(2) = 1 Σ(2)(p2 = µ2; )/m(µ) . (1.397) 4 − S − It is apparent that in this scheme there is at least one point, the renormalisation mass-scale, µ, at which there are no higher order corrections to any of the Green functions: the corrections are all absorbed into the redefinitions of the coupling constant, masses and gauge parameter. This is valuable if the coefficients of the higher order corrections, calculated with the parameters defined through the momentum space subtraction, are small so that the procedure converges rapidly on a large momentum domain. In this sense the µ-scheme is easier to understand and more intuitive than the MS or MS schemes. Another advantage is the manifest applicability of the “decoupling theorem,” Refs. [11], which states that quark flavours whose masses are larger than the scale chosen for µ are irrelevant. This last feature, however, also emphasises that the renormalisation constants are flavour dependent and that can be a nuisance. Nevertheless, the µ-scheme is extremely useful in nonperturbative analyses of DSEs, especially since the flavour dependence of the renormalisation constants is minimal for light quarks when the Euclidean substraction point, µ, is chosen to be very large; i.e., much larger than their current-masses.

1.11.3 Renormalized Gap Equation Equation (1.307) is the unrenormalised QCD gap equation, which can be rewritten as

d4` i iS−1(p) = i(/p m ) + g2 Dµν (p `) λaγ S (`) Γa (`, p) . (1.398) 0 0 0 4 0 2 µ 0 0ν − − − Z (2π) − The renormalised equation can be derived directly from the generating functional deﬁned using the renormalised Lagrangian density, Eq. (1.368), simply by repeat- ing the steps described in Sec. 1.9.2. Alternatively, one can use Eqs. (1.371) to derive an array of relations similar to Eq. (1.387):

µν µν a −1 a D0 (k) = Z3Y M DR (k) , Γ0ν(k, p) = Z1F ΓRν (k, p) , (1.399) and others that I will not use here. (NB. It is a general feature that propagators are multiplied by the renormalisation constant and proper vertices by the inverse of renormalisation constants.) Now one can replace the unrenormalised couplings, masses and Green functions by their renormalised forms:

iZ−1 S−1(p) = ip/ + iZ−1 Z m − 2 R − 2F 4 R d4` i +Z2 Z−1 Z−2 g2 Z Dµν (p `) λaγ Z S (`) Z−1 Γa (`, p) , 1F 3Y M 2F R 4 3Y M R 2 µ 2F R 1F Rν Z (2π) − (1.400) which simpliﬁes to

iΣ (p) = i(Z 1) /p i(Z 1) m − R 2F − − 4 − R d4` i Z g2 Dµν(p `) λaγ S (`) Γa (`, p) 1F R 4 R 2 µ R Rν − Z (2π) − =: i(Z 1) /p i(Z 1) m iΣ0(p) , (1.401) 2F − − 4 − R − where Σ0(p) is the regularised self-energy. In the simplest application of the µ-scheme one would choose a large Euclidean mass-scale, µ2, and deﬁne the renormalisation constants such that

ΣR(p/ + µ = 0) = 0 (1.402) which entails

Z = 1 + Σ0 (/p + µ = 0) , Z = 1 Σ0 (/p + µ = 0)/m (µ) , (1.403) 2F V 4 − S R 0 0 0 where I have used Σ (p) = Σ (p) /p + ΣS(p). (cf. Eqs. (1.396), (1.397).) This is simple to implement, even nonperturbatively, and is always appropriate in QCD because conﬁnement ensures that dressed-quarks do not have a mass-shell.

On-shell Renormalization If one is treating fermions that do have a mass-shell; e.g., electrons, then an on-shell renormalisation scheme may be more appropriate. One ﬁxes the renormalisation constants such that

−1 SR (/p) = /p mR , (1.404) p/=mR −

which is interpreted as a constraint on the pole position and the residue at the pole; i.e., since

−1 d S (/p) = [p/ mR ΣR(/p)] p/=m [p/ mR] ΣR(/p) + . . . (1.405) − − | R − − "d/p # p/=mR

then Eq. (1.404) entails

d ΣR(/p) p/=m = 0 , ΣR(/p) = 0 . (1.406) | R d/p p/=mR

The second of these equations requires

0 2 2 d 0 2 d 0 2 Z2F = 1 + ΣV (mR) + 2 mR 2 ΣV (p ) 2 2 + 2 mR 2 ΣS (p ) (1.407) dp p =mR dp 2 2 p =mR

and the ﬁrst: Z = Z Σ0 (m2 ) Σ0 (m2 ) . (1.408) 4 2F − V R − S R 1.11.4 Exercises 1. Using Eqs. (1.360), (1.361) derive Eqs. (1.381), (1.382). 2. Verify Eqs. (1.396), (1.397). 3. Verify Eq. (1.401). 4. Verify Eq. (1.403). 5. Beginning with Eq. (1.404), derive Eqs. (1.407), (1.408). NB. /pp/ = p2 and d 2 d d 2 hence dp/ f(p ) = dp/ f(/p/p) = 2 /p dp2 f(p ).

1.12 Dynamical Chiral Symmetry Breaking

This phenomenon profoundly aﬀects the character of the hadron spectrum. How- ever, it is intrinsically nonperturbative and therefore its understanding is best sought via a Euclidean formulation of quantum ﬁeld theory, which I will now summarise.

1.12.1 Euclidean Metric I will formally describe the construction of the Euclidean counterpart to a given Minkowski space field theory using the free fermion as an example. The La- grangian density for free Dirac fields is given in Eq. (1.186): ∞ ψ 3 ¯ + L0 (x) = dt d x ψ(x) i∂/ m + iη ψ(x) . (1.409) −∞ − Z Z Recall now the observations at the end of Sec. 1.5 regarding the role of the iη+ in this equation: it was introduced to provide a damping factor in the generating functional. The alternative proposed was to change variables and introduce a Euclidean time: t itE . As usual ∞ → − ∞ ∞ dt f(t) = d( itE) f( itE) = i dtE f( itE) (1.410) Z−∞ Z−∞ − − − Z−∞ − if f(t) vanishes on the curve at infinity in the second and fourth quadrants of the complex-t plane and is analytic therein. To complete this Wick rotation, however, we must also determine its affect on i∂/, since that is part of the “f(t);” i.e., the integrand: ∂ ∂ ∂ ∂ i∂/ = iγ0 + iγi M→E iγ0 + iγi ∂t ∂xi → ∂( itE ) ∂xi − ∂ ∂ = γ0 ( iγi) (1.411) − ∂tE − − ∂xi E ∂ =: γµ E , (1.412) − ∂xµ where (xE) = (~x, x := it) and the Euclidean Dirac matrices are µ 4 − γE = γ0 γE = iγi . (1.413) 4 i − These matrices are Hermitian and satisfy the algebra

γE, γE = 2δ ; µ, ν = 1, . . . , 4, (1.414) { µ ν } µν where δµν is the four-dimensional Kroncker delta. Henceforth I will adopt the notation 4 aE bE = aEbE . (1.415) · µ ν µX=1 Now, assuming that the integrand is analytic where necessary, one arrives formally at

∞ dt d3x ψ¯(x) i∂/ m + iη+ ψ(x) −∞ − Z ∞ Z = i dtE d3x ψ¯(xE) γE ∂E m + iη+ ψ(xE) . (1.416) − −∞ − · − Z Z Intepreted naively, however, the action on the r.h.s. poses problems: it is not real E † E if one inteprets ψ¯(x ) = ψ (x ) γ4. Let us study its involution

4 E E E E + E AE = d x ψ¯(x ) ( γ ∂ m + iη ) ψ(x ) (1.417) Z − · − ¯ E E where ψr(x ) and ψs(x ) are intepreted as members of a Grassmann algebra with involution, as described in Sec. 1.7.1:

4 E ¯ E E E + E AE = d x ψr(x ) ( [γ ∂→]rs m δrs + iη δrs) ψs(x ) Z − · − 4 E ¯ E E E ∗ + E = d x ψs(x ) [γ ∂←]rs m δrs iη δrs ψr(x ) . (1.418) Z − · − − ! We know that the mass is real: m∗ = m, and that as an operator the gradient is anti-Hermitian; i.e., ∂E = ∂E (1.419) ← − → but that leaves us with

E E T [γµ ]rs = [(γµ ) ]sr . (1.420)

If we deﬁne (γE)T := C γE C† , (1.421) µ − E µ E where CE = γ2γ4 is the Euclidean charge conjugation matrix then

AE = AE (1.422) when ψ¯ and ψ are members of a Grassmann algebra with involution and we take the limit η+ = 0, which we will soon see is permissable. Return now to the generating functional of Eq. (1.187):

W [ξ¯, ξ] = [ ψ¯(x)][ ψ(x)] exp i d4x ψ¯(x) i∂/ m + iη+ ψ(x) + ψ¯(x)ξ(x) + ξ¯(x)ψ(x) . D D − Z Z h i (1.423)

A Wick rotation of the action transforms the source-independent part of the integrand, called the measure, into

exp d4xE ψ¯(xE) γ ∂ + m iη+ ψ(xE) (1.424) − · − Z h i so that the Euclidean generating functional

W E[ξ¯, ξ] := [ ψ¯(xE)][ ψ(xE)] Z D D exp d4x ψ¯(xE) (γ ∂ + m) ψ(xE) + ψ¯(xE)ξ(xE) + ξ¯(xE)ψ(xE) × − · Z h i (1.425) involves a positive-deﬁnite measure because the free-fermion action is real, as I have just shown. Hence the “+iη+” convergence factor is unnecessary here.

Euclidean Formulation as Definitive Working in Euclidean space is more than simply a pragmatic artifice: it is possible to view the Euclidean formulation of a field theory as definitive (see, for example, Refs. [12, 13, 14, 15]). In addition, the discrete lattice formulation in Euclidean space has allowed some progress to be made in attempting to answer existence questions for interacting gauge field theories [15]. The moments of the Euclidean measure defined by an interacting quantum field theory are the Schwinger functions:

n(xE,1, . . . , xE,n) (1.426) S which can be obtained as usual via functional diﬀerentiation of the analogue of Eq. (1.425) and subsequently setting the sources to zero. The Schwinger functions are sometimes called Euclidean space Green functions. Given a measure and given that it satisﬁes certain conditions [i.e., the Wight- man and Haag-Kastler axioms], then it can be shown that the Wightman func- n tions, (x1, . . . , xn), can be obtained from the Schwinger functions by analytic continuationW in each of the time coordinates:

n n 1 1 0 n n 0 (x1, . . . , xn) = lim ([~x , x + ix ], . . . , [~x , x + ix ]) (1.427) i 4 1 4 n W x4→0 S

0 0 0 with x1 < x2 < . . . < xn. These Wightman functions are simply the vacuum expectation values of products of ﬁeld operators from which the Green functions [i.e., the Minkowski space propagators] are obtained through the inclusion of step- [θ-] functions in order to obtain the appropriate time ordering. (This is described in some detail in Refs. [13, 14, 15].) Thus the Schwinger functions contain all of the information necessary to calculate physical observables. This notion is used directly in obtaining masses and charge radii in lattice simulations of QCD, and it can also be employed in the DSE approach since the Euclidean space DSEs can all be derived from the appropriate Euclidean generating functional using the methods of Sec. 1.9 and the solutions of these equations are the Schwinger functions. All this provides a good reason to employ a Euclidean formulation. Another is a desire to maintain contact with perturbation theory where the renormalisation group equations for QCD and their solutions are best understood [16].

Collected Formulae for Minkowski Euclidean Transcription ↔ To make clear my conventions (henceforth I will omit the supescript E used to denote Euclidean four-vectors): for 4-vectors a, b:

4 a b := a b δ := a b , (1.428) · µ ν µν i i Xi=1 2 so that a spacelike vector, Qµ, has Q > 0; the Dirac matrices are Hermitian and deﬁned by the algebra γ , γ = 2 δ ; (1.429) { µ ν} µν and I use γ := γ γ γ γ (1.430) 5 − 1 2 3 4 so that tr [γ γ γ γ γ ] = 4 ε , ε = 1 . (1.431) 5 µ ν ρ σ − µνρσ 1234 The Dirac-like representation of these matrices is:

0 i~τ τ 0 0 ~γ = , γ = , (1.432) i~τ −0 4 0 τ 0 ! − ! where the 2 2 Pauli matrices are: × 1 0 0 1 0 i 1 0 τ 0 = , τ 1 = , τ 2 = − , τ 3 = . (1.433) 0 1 1 0 i 0 0 1 ! ! ! − ! Using these conventions the [unrenormalised] Euclidean QCD action is

N 1 1 f 1 S[B, q, q¯] = d4x F a F a + ∂ Ba ∂ Ba + q¯ γ ∂ + m + ig λa γ Ba q ,  4 µν µν 2λ · · f · f 2 · f  Z fX=1  (1.434)  where F a = ∂ Ba ∂ Ba gf abcBb Bc. The generating functional follows: µν µ ν − ν µ − µ ν ¯ 4 ¯ a a W [J, ξ, ξ] = dµ(q¯, q, B, ω¯, ω) exp d x q¯ξ + ξ q + Jµ Aµ , (1.435) Z Z h i with sources: η¯, η, J, and a functional integral measure dµ(q¯, q, B, ω¯, ω) := (1.436) a a a g q¯φ(x) qφ(x) ω¯ (x) ω (x) Bµ(x) exp( S[B, q, q¯] S [B, ω, ω¯]) , x D D a D D µ D − − Y Yφ Y Y where φ represents both the flavour and colour index of the quark field, and ω¯ and ω are the scalar, Grassmann [ghost] fields. The normalisation

W [η¯ = 0, η = 0, J = 0] = 1 (1.437) is implicit in the measure. As we saw in Sec. 1.8.1, the ghosts only couple directly to the gauge ﬁeld:

S [B, ω, ω¯] = d4x ∂ ω¯ a ∂ ωa gf abc ∂ ω¯a ωb Bc , (1.438) g − µ µ − µ µ Z h i and restore unitarity in the subspace of transverse [physical] gauge fields. I note that, practically, the normalisation means that ghost fields are unnecessary in the calculation of gauge invariant observables using lattice-regularised QCD because the gauge-orbit volume-divergence in the generating functional, associated with the uncountable infinity of gauge-equivalent gluon field configurations in the continuum, is rendered finite by the simple expedient of only summing over a finite number of configurations. It is possible to derive every equation introduced above, assuming certain analytic properties of the integrands. However, the derivations can be sidestepped using the following transcription rules: Configuration Space Momentum Space

M E M E 1. d4xM i d4xE 1. d4kM i d4kE Z → − Z Z → Z 2. ∂/ iγE ∂E 2. k/ iγE kE → · → − · 3. A/ iγE AE 3. A/ iγE AE → − · → − · 4. A Bµ AE BE 4. k qµ kE qE µ → − · µ → − · 5. xµ∂ xE ∂E 5. k xµ kE xE µ → · µ → − · These rules are valid in perturbation theory; i.e., the correct Minkowski space integral for a given diagram will be obtained by applying these rules to the Euclidean integral: they take account of the change of variables and rotation of the contour. However, for the diagrams that represent DSEs, which involve dressed n-point functions whose analytic structure is not known as priori, the Minkowski space equation obtained using this prescription will have the right appearance but it’s solutions may bear no relation to the analytic continuation of the solution of the Euclidean equation. The diﬀerences will be nonperturbative in origin.

1.12.2 Chiral Symmetry Gauge theories with massless fermions have a chiral symmetry. Its effect can be visualised by considering the helicity: λ J p, the projection of the fermion’s ∝ · spin onto its direction of motion. λ is a Poincaré invariant spin observable that takes a value of 1. The chirality operator can be realised as a 4 4-matrix, γ5, and a chiral transformation is then represented as a rotation of the× 4 1-matrix × quark spinor field q(x) eiγ5θ q(x) . (1.439) → A chiral rotation through θ = π/2 has no effect on a λ = +1 quark, q q , λ= + → λ= + but changes the sign of a λ = 1 quark field, qλ= − qλ= −. In composite hadrons this is manifest as a flip−in their parity: J P = +→ −J P = −; i.e., a θ = π/2 ↔ chiral rotation is equivalent to a parity transformation. Exact chiral symmetry therefore entails that degenerate parity multiplets must be present in the spectrum of the theory. For many reasons, the masses of the u- and d-quarks are expected to be very small; i.e., m m Λ . Therefore chiral symmetry should only be weakly u ∼ d QCD broken, with the strong interaction spectrum exhibiting nearly degenerate parity partners. The experimental comparison is presented in Eq. (1.440):

1 + − − N( 2 , 938) π(0 , 140) ρ(1 , 770) 1 − + + . (1.440) N( 2 , 1535) a0(0 , 980) a1(1 , 1260) Clearly the expectation is very badly violated, with the splitting much too large to be described by the small current-quark masses. What is wrong? Chiral symmetry can be related to properties of the quark propagator, S(p). For a free quark (remember, I am now using Euclidean conventions) m iγ p S (p) = − · (1.441) 0 m2 + p2 and as a matrix iγ p m S (p) eiγ5θS (p)eiγ5θ = − · + e2iγ5θ (1.442) 0 → 0 p2 + m2 p2+m2 under a chiral transformation. As anticipated, for m = 0, S (p) S (p); i.e., the 0 → 0 symmetry breaking term is proportional to the current-quark mass and it can be measured by the “quark condensate” d4p d4p m q¯q := 4 tr [S(p)] 4 2 2 , (1.443) −h i Z (2π) ∝ Z (2π) p + m which is the “Cooper-pair” density in QCD. For a free quark the condensate vanishes if m = 0 but what is the eﬀect of interactions? As we have seen, interactions dress the quark propagator so that it takes the form 1 iγ pA(p2) + B(p2) S(p) := = − · , (1.444) iγ p + Σ(p) p2A2(p2) + B2(p2) · where Σ(p) is the self energy, expressed in terms of the scalar functions: A and B, which are p2-dependent because the interaction is momentum-dependent. On the valid domain; i.e., for weak-coupling, they can be calculated in perturbation theory and at one-loop order, Eq. (1.395) with p2 (pE)2 µ2, m2(µ), → − α p2 B(p2) = m 1 S ln , (1.445) − π " m2 #! which is m. This result persists: at every order in perturbation theory every mass-like∝correction to S(p) is m so that m is apparently the only source of chiral symmetry breaking and ∝q¯q m 0 as m 0. The current-quark h i ∝ → → masses are the only explicit chiral symmetry breaking terms in QCD. However, symmetries can be “dynamically” broken. Consider a point-particle in a rotationally invariant potential V (σ, π) = (σ2 + π2 1)2, where (σ, π) are the − particle’s coordinates. In the state depicted in Fig. 1.1, the particle is stationary at an extremum of the action that is rotationally invariant but unstable. In the ground state of the system, the particle is stationary at any point (σ, π) in the trough of the potential, for which σ2 + π2 = 1. There are an uncountable inﬁnity of such vacua, θ , which are related one to another by rotations in the (σ, π)- | i plane. The vacua are degenerate but not rotationally invariant and hence, in general, θ σ θ = 0. In this case the rotational invariance of the Hamiltonian is not exhibitedh | | ini 6 any single ground state: the symmetry is dynamically broken with interactions being responsible for θ σ θ = 0. h | | i 6 • 1.5

1 1

0.5 0.5

0 -1 0

-0.5 -0.5 0

0.5 -1 1 Figure 1.1: A rotationally invariant but unstable extremum of the Hamiltonian obtained with the potential V (σ, π) = (σ2 + π2 1)2. −

1.12.3 Mass Where There Was None The analogue in QCD is q¯q = 0 when m = 0. At any ﬁnite order in perturbation theory that is impossible.h Hoi 6 wever, using the Dyson-Schwinger equation [DSE] for the quark self energy [the QCD “gap equation”]:

iγ p A(p2) + B(p2) = Z iγ p + Z m · 2 · 4 Λ d4` λa 1 +Z g2 D (p `) γ Γa(`, p) , (1.446) 1 (2π)4 µν − µ 2 iγ À(`2) + B(`2) ν Z · depicted in Fig. 1.2, it is possible to sum infinitely many contributions. That allows one to expose effects in QCD which are inaccessible in perturbation theory. [NB. In Eq. (1.446), m is the Λ-dependent current-quark bare mass and Λ represents mnemonically a translationally-invariant regularisation of the integral,R with Λ the regularisation mass-scale. The final stage of any calculation is to remove the regularisation by taking the limit Λ . The quark-gluon-vertex and → ∞ 2 2 2 2 quark wave function renormalisation constants, Z1(ζ , Λ ) and Z2(ζ , Λ ), depend on the renormalisation point, ζ, and the regularisation mass-scale, as does the 2 2 2 2 −1 2 2 mass renormalisation constant Zm(ζ , Λ ) := Z2(ζ , Λ ) Z4(ζ , Λ ).] The quark DSE is a nonlinear integral equation for A and B, and it is the nonlinearity that makes possible a generation of nonperturbative effects. The Σ D = γ S Γ

Figure 1.2: DSE for the dressed-quark self-energy. The kernel of this equation is constructed from the dressed-gluon propagator (D - spring) and the dressed- quark-gluon vertex (Γ - open circle). One of the vertices is bare (labelled by γ) as required to avoid over-counting. kernel of the equation is composed of the dressed-gluon propagator:

2 2 2 kµkν (k ) 2 g g Dµν(k) = δµν G , (k ) := , (1.447) − k2 ! k2 G [1 + Π(k2)] where Π(k2) is the vacuum polarisation, which contains all the dynamical infor- a mation about gluon propagation, and the dressed-quark-gluon vertex: Γµ(k, p). The bare (undressed) vertex is

λa Γa (k, p) = γ . (1.448) µ bare µ 2

a Once Dµν and Γµ are known, Eq. (1.446) is straightforward to solve by iteration. One chooses an initial seed for the solution functions: 0A and 0B, and evaluates the integral on the right-hand-side (r.h.s.). The bare propagator values: 0A = 1 and 0B = m are often adequate. This ﬁrst iteration yields new functions: 1A and 1B, which are reintroduced on the r.h.s. to yield 2A and 2B, etc. The procedure is repeated until nA = n+1A and nB = n+1B to the desired accuracy. It is now easy to illustrate DCSB, and I will use three simple examples.

Nambu–Jona-Lasinio Model The Nambu–Jona-Lasinio model [17] has been popularised as a model of low- energy QCD. The commonly used gap equation is obtained from Eq. (1.446) via the substitution 2 1 2 2 g Dµν (p `) δµν 2 θ(Λ ` ) (1.449) − → mG − in combination with Eq. (1.448). The step-function in Eq. (1.449) provides a momentum-space cutoff. That is necessary to define the model, which is not renormalisable and hence can be regularised but not renormalised. Λ therefore persists as a model-parameter, an external mass-scale that cannot be eliminated. The gap equation is iγ p A(p2) + B(p2) · 4 1 d4` iγ À(`2) + B(`2) = iγ p + m + θ(Λ2 `2) γ γ(1.450), 3 2 4 µ − 2 ·2 2 2 2 µ · mG Z (2π) − ` A (` ) + B (` ) where I have set the renormalisation constants equal to one in order to complete the defintion of the model. Multiplying Eq. (1.450) by ( iγ p) and tracing over Dirac indices one obtains − · 8 1 d4` A(`2) p2 A(p2) = p2 + θ(Λ2 `2) p ` , (1.451) 3 2 4 2 2 2 2 2 mG Z (2π) − · ` A (` ) + B (` ) from which it is immediately apparent that A(p2) 1 . (1.452) ≡ This property owes itself to the the fact that the NJL model is defined by a four- fermion contact interaction in configuration space, which entails the momentum- independence of the interaction in momentum space. Simply tracing over Dirac indices and using Eq. (1.452) one obtains

16 1 d4` B(`2) B(p2) = m + θ(Λ2 `2) , (1.453) 3 2 4 2 2 2 mG Z (2π) − ` + B (` ) from which it is plain that B(p2) = constant = M is the only solution. This, too, is a result of the momentum-independence of the model’s interaction. Evaluating the angular integrals, Eq. (1.453) becomes

Λ2 1 1 M 1 1 2 2 M = m + 2 2 dx x 2 = m + M 2 2 (M , Λ )(1.454), 3π mG Z0 x + M 3π mG C (M 2, Λ2) = Λ2 M 2 ln 1 + Λ2/M 2 (1.455) C − h i Λ deﬁnes the model’s mass-scale and I will henceforth set it equal to one so that all other dimensioned quantities are given in units of this scale, in which case the gap equation can be written

1 1 2 M = m + M 2 2 (M , 1) . (1.456) 3π mG C

Irrespective of the value of mG, this equation always admits a solution M = 0 when the current-quark mass m = 0. 6 6 Consider now the chiral-limit, m = 0, wherein the gap equation is

1 1 2 M = M 2 2 (M , 1) . (1.457) 3π mG C This equation admits a solution M 0, which corresponds to the perturbative case considered above: when the bare≡mass of the fermion is zero in the beginning then no mass is generated via interactions. It follows from Eq. (1.443) that the condensate is also zero. This situation can be described as that of a theory without a mass gap: the negative energy Dirac sea is populated all the way up to E = 0. Suppose however that M = 0 in Eq. (1.457), then the equation becomes 6

1 1 2 1 = 2 2 (M , 1) . (1.458) 3π mG C It is easy to see that (M 2, 1) is a monotonically decreasing function of M with C a maximum value at M = 0: (0, 1) = 1. Consequently Eq. (1.458) has a M = 0 solution if, and only if, C 6 1 1 2 2 > 1 ; (1.459) 3π mG i.e., if and only if Λ2 m2 < (0.2 GeV )2 (1.460) G 3π2 ' for a typical value of Λ 1 GeV. Thus, even when the bare mass is zero, the NJL ∼ model admits a dynamically generated mass for the fermion when the coupling exceeds a given minimum value, which is called the critical coupling: the chiral symmetry is dynamically broken! (In this presentation the critical coupling is expressed via a maximum value of the dynamical gluon mass, mG.) At these strong couplings the theory exhibits a nonperturbatively generated gap: the ini- tially massless fermions and antifermions become massive via interaction with their own “gluon” field. Now the negative energy Dirac sea is only filled up to E = M, with the positive energy states beginning at E = +M; i.e., the theory − has a dynamically-generated, nonperturbative mass-gap ∆ = 2M. In addition the quark condensate, which was zero when evaluated perturbatively because m = 0, is now nonzero. Importantly, the nature of the solution of Eq. (1.456) also changes qualitatively when mG is allowed to fall below it’s critical value. It is in this way that dynamical chiral symmetry breaking (DCSB) continues to affect the hadronic spectrum even when the quarks have a small but nonzero current-mass.

Munczek-Nemirovsky Model The gap equation for a model proposed more recently [18], which is able to represent a greater variety of the features of QCD while retaining much of the simplicity of the NJL model, is obtained from Eq. (1.446) by using

(k2) G = (2π)4 G δ4(k) (1.461) k2 in Eq. (1.447) with the bare vertex, Eq. (1.448). Here G defines the model’s mass-scale. The gap equation is iγ p A(p2) + B(p2) iγ p A(p2) + B(p2) = iγ p + m + G γ − · γ (1.462), · · µ p2A2(p2) + B2(p2) µ where again the renormalisation constants have been set equal to one but in this case because the model is ultraviolet finite; i.e., there are no infinities that must be regularised and subtracted. The gap equation yields the following two coupled equations: A(p2) A(p2) = 1 + 2 (1.463) p2A2(p2) + B2(p2) B(p2) B(p2) = m + 4 , (1.464) p2A2(p2) + B2(p2) where I have set the mass-scale G = 1. Consider the chiral limit equation for B(p2): B(p2) B(p2) = 4 . (1.465) p2A2(p2) + B2(p2) The existence of a B 0 solution; i.e., a solution that dynamically breaks chiral symmetry, requires 6≡ p2A2(p2) + B2(p2) = 4 , (1.466) measured in units of G. Substituting this identity into equation Eq. (1.463) one finds 1 A(p2) 1 = A(p2) A(p2) 2 , (1.467) − 2 ⇒ ≡ which in turn entails B(p2) = 2 1 p2 . (1.468) − The physical requirement that the quarkq self energy be real in the spacelike region means that this solution is only acceptable for p2 1. For p2 > 1 one ≤ must choose the B 0 solution of Eq. (1.465), and on this domain we then find from Eq. (1.463) that≡ 2 1 A(p2) = 1 + A(p2) = 1 + 1 + 8/p2 . (1.469) p2 A(p2) ⇒ 2 q Putting this all together, the Munczek-Nemirovsky model exhibits the dynamical chiral symmetry breaking solution: 2 ; p2 1 A(p2) = 1 ≤ (1.470)  1 + 1 + 8/p2 ; p2 > 1  2 2q 2 2  √1 p ; p 1 B(p ) = − 2 ≤ (1.471) ( 0 ; p > 1 . which yields a nonzero quark condensate. Note that the dressed-quark self-energy is momentum dependent, as is the case in QCD. It is important to observe that this solution is continuous and defined for all p2, even p2 < 0 which corresponds to timelike momenta, and furthermore

p2 A2(p2) + B2(p2) > 0 , p2 . (1.472) ∀ This last fact means that the quark described by this model is confined: the propagator does not exhibit a mass pole! I also note that there is no critical coupling in this model; i.e., the nontrivial solution for B always exists. This exemplifies a contemporary conjecture that theories with confinement always exhibit DCSB. In the chirally asymmetric case the gap equation yields

2 B(p2) A(p2) = , (1.473) m + B(p2) 4 [m + B(p2)]2 B(p2) = m + . (1.474) B(p2)([m + B(p2)]2 + 4p2)

The second is a quartic equation for B(p2) that can be solved algebraically with four solutions, available in a closed form, of which only one has the correct p2 limit: B(p2) m. Note that the equations and their solutions always have→a ∞ → smooth m 0 limit, a result owing to the persistence of the DCSB solution. → Renormalization-Group-Improved Model Finally I have used the bare vertex, Eq. (1.448) and (Q) depicted in Fig. 1.3, in G solving the quark DSE in the chiral limit. If (Q = 0) < 1 then B(p2) 0 is the only solution. However, when (Q = 0) 1 theG equation admits an energetically≡ favoured B(p2) 0 solution; i.e.,G if the coupling≥ is large enough then even in the 6≡ absence of a current-quark mass, contrary to Eq. (1.445), the quark acquires a mass dynamically and hence

d4p B(p2) q¯q 4 2 2 2 2 2 = 0 for m = 0 . (1.475) h i ∝ Z (2π) p A(p ) + B(p ) 6

These examples identify a mechanism for DCSB in quantum ﬁeld theory. The nonzero condensate provides a new, dynamically generated mass-scale and if its magnitude is large enough [ q¯q 1/3 need only be one order-of-magnitude larger −h i than mu md] it can explain the mass splitting between parity partners, and many other∼ surprising phenomena in QCD. The models illustrate that DCSB is linked to the long-range behaviour of the fermion-fermion interaction. The same is true of conﬁnement. The question is then: How does the quark-quark interaction behave at large distances in QCD? It remains unanswered. 1.0

0.8

G(Q) 0.6

0.4

0.2

0.0 0 1 10 100 Q (GeV)

Figure 1.3: Illustrative forms of (Q): the behaviour of each agrees with perturbation theory for Q > 1 GeV. GThree possibilities are canvassed in Sec.1.12.3: (Q = 0) < 1; (Q = 0) = 1; and (Q = 0) > 1. G G G Bibliography

[1] J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964).

[2] B.D. Keister and W.N. Polyzou, “Relativistic Hamiltonian dynamics in nuclear and particle physics,” Adv. Nucl. Phys. 20, 225 (1991).

[3] C. Itzykson and J.-B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980).

[4] C.D. Roberts and A.G. Williams, “Dyson-Schwinger Equations and their Application to Hadronic Physics,” Prog. Part. Nucl. Phys. 33, 477 (1994).

[5] P. Pascual and R. Tarrach, Lecture Notes in Physics, Vol. 194, QCD: Renor- malization for the Practitioner (Springer-Verlag, Berlin, 1984).

[6] J.S. Schwinger, “On The Green’s Functions Of Quantized Fields: 1 and 2,” Proc. Nat. Acad. Sci. 37 (1951) 452; ibid 455.

[7] F.A. Berezin, The Method of Second Quantization (Academic Press, New York, 1966).

[8] L.D. Faddeev and V.N. Popov, “Feynman Diagrams For The Yang-Mills Field,” Phys. Lett. B 25 (1967) 29.

[9] V.N. Gribov, “Quantization Of Nonabelian Gauge Theories,” Nucl. Phys. B 139 (1978) 1.

[10] F.J. Dyson, “The S Matrix In Quantum Electrodynamics,” Phys. Rev. 75 (1949) 1736.

[11] K. Symanzik, “Infrared Singularities And Small Distance Behavior Analy- sis,” Commun. Math. Phys. 34 (1973) 7; T. Appelquist and J. Carazzone, “Infrared Singularities And Massive Fields,” Phys. Rev. D 11 (1975) 2856.

[12] K. Symanzik in Local Quantum Theory (Academic, New York, 1969) edited by R. Jost.

82 [13] R.F. Streater and A.S. Wightman, A.S., PCT, Spin and Statistics, and All That, 3rd edition (Addison-Wesley, Reading, Mass, 1980).

[14] J. Glimm and A. Jaﬀee, Quantum Physics. A Functional Point of View (Springer-Verlag, New York, 1981).

[15] E. Seiler, Gauge Theories as a Problem of Constructive Quantum Theory and Statistical Mechanics (Springer-Verlag, New York, 1982).

[16] D.J. Gross, “Applications Of The Renormalization Group To High-Energy Physics,” in Proc. of Les Houches 1975, Methods In Field Theory (North Holland, Amsterdam, 1976) pp. 141-250.

[17] Y. Nambu and G. Jona-Lasinio, “Dynamical Model Of Elementary Particles Based On An Analogy With Superconductivity. I,II” Phys. Rev. 122 (1961) 345, 246

[18] H.J. Munczek and A.M. Nemirovsky, Phys. Rev. D 28 (1983) 181. Chapter 2

Quantum Fields at Finite Temperature and Density

2.1 Ensembles and Partition Function

In equilibrium statistical mechanice, one normally encounters three types of ensemble. The microcanonical ensemble is used to describe an isolated system which has a fixed energy E, a fixed particle number N, and a fixed volume V . The canonical ensemble is used to describe a system in contact with a heat reservoir at temperature T . The system can freely exchange energy with the reservoir. Thus T , N, and V are fixed. In the grand canonical ensemble, the system can exchange particles as well as energy with a reservoir. In this ensemble, T , V , and the chemical potential µ are fixed variables. In the latter two ensembles T −1 = β may be thought of as a Lagrange multiplier which determines the mean energy of the system. Similarly, µ may be thought of as a Lagrange multiplier which determines the mean number of particles in the system. In a relativistic quantum system where particles can be created and destroyed, it is most straightforward to compute observables in the grand canonical ensemble. So we therefore use this ensemble. This is without loss of generality since one can pass over to either of the other ensembles by performing a Laplace transform on the variable µ and/or the variable β. Consider a system described by a Hamiltonian Hˆ and a set of conserved number operators Nî. In relativistic QED, for example, the number of electrons minus the number of positrons is a conserved quantity, not the electron number or the positron number separately. These number operators must be Hermitian and must commute with H as well as with each other. Also, the number operators must be extensive in order that the usual macroscopic thermodynamic limit can be taken. The statistical density matrix is

ρˆ = exp β(Hˆ µ Nˆ ) , (2.1) − − i i h i

84 where a summation over i is implied. The ensemble average of an operator Aˆ is

TrρˆAˆ A = . (2.2) Trρˆ The grand canonical partition function is

Z = Trρˆ . (2.3)

The function Z = Z(T, V, µ1, µ2, . . .) is the single most important function in thermodynamics. From it all other standard thermodynamic properties may be determined. For example the equations of state for, e.g., the pressure P , the particle number N, the entropy S, and the energy E are, in the inﬁnite volume limit, ∂ ln Z P = T , (2.4) ∂V ∂ ln Z Ni = T , (2.5) ∂µi ∂(T ln Z) P = , (2.6) ∂T E = P V + T S + µ N . (2.7) − i i Note that the notion of ensembles as introduced here for the situation in thermodynamical equilibrium can be extended to the nonequilibrium situation, where a generalized Gibbs ensemble can be introduced for systems which are in a nonequilibrium situation that can be characterized by further observables such as currents or reaction variables. These additional observables shall be accounted for by an enlarged set of Lagrange multipliers thus arriving at a statistical operator of the nonequilibrium state, also called relevant statistical operator within the Zubarev formalism.

2.1.1 Partition function in Quantum Statistics and Quan- tum Field Theory In order to calculate the partition using methods of quantum ﬁeld theory, we recall that in quantum statistics

−β(Hˆ −µiNî) −β(Hˆ −µiNî) Z = Tr e = dφa φa e φa , (2.8) Z h | | i where the sum runs over all (eigen-)states. This has an appearance very similar to the transition amplitude (time evolution operator) in Quantum Field Theory whne one switches to an imaginary time variable τ = i t and limit the integration over τ to the region between 0 and β. The trace operation means that we have to integrate over all fields φa. Finally, if the system admits some conserved charge, then we must make the replacement

(π, φ) (pi, φ) = (π, φ) µ (π, φ) , (2.9) H → K H − N where (π, φ) is the conserved charge density. In fact,N we can express the partition function Z as a functional integral over ﬁelds and their conjugate momenta. This fundamental formula reads

β ∂φ Z = π φ exp d3x i π (π, φ) + µ (π, φ) . (2.10) Z D Zperiodic D (Z0 Z ∂τ − H N !) The term “periodic” means that the integration over the field is constrained so that φ(~x, 0) = φ(~x, β). This is a consequence of the trace operation, setting φa(~x) = φ(~x, 0) = φ(~x, β). There is no restriction on the π integration. The generalization of (2.10) to an arbitrary number of fields and conserved charges is obvious. In the following subsection we show the equivalence of the expressions (2.10 ) and (2.8) for the partition function. The key lesson is that the quantization in the Path integral representation is provided by the integration over all alternative classical field configurations (under given constraints) whereas in the statistical operator representation the notion of field operators has to be introduced.

2.1.2 Equivalence of Path Integral and Statistical Opera- tor representation for the Partition function Be φˆ(~x, 0) a field operator in the Schrödinger picture at time t = 0 and πˆ(~x, 0) the corresponding canonically conjugated field momentum operator. For eigenstates φ of the field holds the eigenvalue equation | i φˆ(~x, 0) φ = φ(~x) φ , (2.11) | i | i where φ(~x) is the “eigenvalue” corresponding to the field operator. For the eigenstates of the fields completeness and orthonormality shall hold

dφ(~x) φ φ = 1 (2.12) Z | ih | φ φ = δ [φ (~x) φ (~x)] . (2.13) h a | bi a − b For the field momentum operator and its eigenstates π holds analogously | i πˆ(~x, 0) π = π(~x) π (2.14) | i | i dπ(~x) π π = 1 (2.15) Z 2π | ih | π π = δ [π (~x) π (~x)] . (2.16) h a | bi a − b The transition amplitude between coordinate and momentum eigenstates in quantum mechanics is (¯h = 1) x p = eipx . (2.17) h | i The generalization to the quantum field theory case follows by going over to an infinite number of degrees of freedom p x d3x π(~x)φ(~x) thus arriving at i i → P R φ π = exp i d3x π(~x)φ(~x) . (2.18) h | i Z For a dynamical description of the system we require the Hamiltonian operator

Hˆ = d3x π, φ . (2.19) H ∧ ∧ Z Consider the state φ at t = 0. At a later time t this state has evolved to | ai f −Hˆ tf e φa . The transition amplitude of the state φa to the state φb at time | i −Hˆ tf | i | i tf is therefore given by φb e φa . In order to express theh |quantum| statisticali partition function, we are inter- ested in the case that the system returns at t = tf to the initial state at t = 0. The sign of the state is not an observable and is left undetermined at this stage.

ˆ e−iHtf φ φ (2.20) | ai → | ai

In order to evaluate the transition amplitude the time interval (0, tf ) is decomposed into equidistant parts of the length t = tf /N. At each time step we introduce a complete set of ﬁeld and ﬁeld-momen4 tum states

φ φ = δ (φ φ ) (2.22) h 1 | ai 1 − a 3 φi+1 πi = exp i d x πi(~x)φi+1(~x) . (2.23) h | i Z For t 0 the exponential function can be expanded 4 → π e−iHˆ 4t φ π (1 Hˆ t) φ h i | | ii ' h i | − 4 | ii = π φ (1 H t) h i | ii − i4 3 = (1 Hi t) exp i d x πi(~x)φi(~x) , (2.24) − 4 Z where H = d3x π (~), φ (~) . (2.25) i H i § i § Z Taken all expressions together yields

N ˆ dπidφi −iHtf φa e φa = lim δ (φ1 φa) h | | i N→∞ 2π ! − Z iY=1 N exp i t d3x[ (π , φ ) × {− 4 H | | − jX=1 Z φ φ π j+1 − j ] (2.26) − j t } 4 Here holds φN+1 = φa = φ1. In the continuum limit we obtain

φ(~x,tf )=φa tf

−iHˆ tf 3 ∂φ φa e φa = π φ exp i dt d ~x π (φ, π)  h | | i D D ∂t − H ! Z Z Z0 Z φ(~x,0)=φa   φ(~x,tf )=φa  tf  = π φ exp i dt d3~x (φ, π) (2.27) D D  L  Z Z Z0 Z φ(~x,0)=φa     The notations π and φ stand for the Functional Integration over ﬁelds and D D their conjugate momenta. The partition function of a quantum statistical system is deﬁned as

ˆ ˆ Z = T r e−β(H−µiNi) { } −β(Hˆ −µiNˆi) = dφa φa e φa (2.28) Z h | | i The expression under the integral shows formal equivalence to the time evolution operator (2.27) except for the factor i in the exponent. The formal equivalence can be made still closer by introducing the imaginary time τ = it. Nˆi is an operator corresponding to conserved charges in the system, such as baryon number. These conservation laws can be incorporated into the formalism as constraints by the method of Lagrangian multipliers. This is done by the replacement (π, φ) (π, φ) = (π, φ) µ (π, φ) (2.29) H → K H − iNi The result for the partition function reads

β 3 ∂φ Z = [ dπ] [dφ] exp  dτ d x iπ (π, φ) + µi i(π, φ)  (2.30) ∂τ − H N ! Z Z Z0 Z     The index stands for the symmetry of the ﬁelds at the borders of the imaginary time interval which is determined up to a phase factor φ(~x, 0) = φ(~x, β), where the upper sign stands for Bosons, while the lower is for Fermions. The next step is the transition to the Fourier representation

1 2 ∞ β i(p·x+ωnτ) φ(x, τ) = e φn(p) (2.31) V ! n=−∞ p X X which allows to get rid of differential operators in the action functional and transform it to an algebraic expression of momenta and frequencies. Due to the (anti)periodicity on the imaginary time interval the conjugate frequencies become discrete. For bosonic fields φ(~x, 0) = φ(~x, β) is fulfilled for

ωn = 2nπT (2.32)

The ωn are denoted as Matsubara-Frequencies. For fermionic fields a similar argument leads to the fermionic Matsubara frequencies ωn = (2n + 1)πT . 2.1.3 Bosonic Fields Neutral Scalar Field The most general renormalizable Lagrangian for a neutral scalar field is 1 1 = ∂ φ∂µφ m2φ2 U(φ) , (2.33) L 2 µ − 2 − where the potential is U(φ) = gφ3 + λφ4 , (2.34) and λ 0 for stability of the vacuum. The momentum conjugate to the field is ≥ ∂ ∂φ π = L = , (2.35) ∂(∂0φ) ∂t and the Hamiltonian is ∂φ 1 1 1 = π = π2 + ( φ)2 + m2φ2 + U(φ) . (2.36) H ∂t − L 2 2 ∇ 2 There is no conserved charge. The first step in evaluating the partition function is to return to the discretized version ∞ N dπi Z = lim Πi=1 dφi (2.37) N→∞ Z−∞ 2π Zperiodic ! N 1 1 1 exp d3x iπ (φ φ ) ∆τ π2 + ( φ )2 + m2φ2 + U(φ ) .  j j+1 − j − 2 j 2 ∇ j 2 j j  jX=1 Z  The momentum integrations can be done immediately since they are just products  of Gaussian integrals. Divide position space into M 3 little cubes with V = L3, L = aM, a 0, M , M an integer. For conv→enience,→and∞to ensure that Z remains explicitely dimensionless at 3 1/2 each step in the calculation, we write πj = Aj/(a ∆τ) and integrate Aj from to + . For each cube we obtain −∞ ∞ ∞ dA 1 a3 1/2 j exp A2 + i (φ φ )A  j j+1 j j Z−∞ 2π −2 ∆τ ! −  a3(φ φ )2  = (2π)−1/2 exp − j+1 − j . (2.38) " 2∆τ # Thus far we have

−M 3N/2 N Z = lim (2π) Πi=1dφi M,N→∞ Z h i N 2 3 1 (φj+1 φj) 1 2 1 2 2 exp ∆τ d x − ( φj) m φj U(φj(2.39)) .  −2 ∆τ ! − 2 ∇ − 2 −   jX=1 Z      Returning to the continuum limit, we obtain

β Z = N 0 φ exp dτ d3x . (2.40) Zperiodic D Z0 Z L! The Lagrangian is expressed as a functional of φ and its first derivatives. The formula (2.39) expresses Z as a functional integral over φ of the exponential of the action in imaginary time. The normalization constant is irrelevant, since multiplication of Z by any constant does not change the thermodynamics. Next, we turn to the case of noninteracting fields with U(φ) = 0. Interactions will be discussed separately. Define

β 1 β ∂φ 2 S = dτ d3x = dτ d3x + ( φ)2 + m2φ2 . (2.41)   Z0 Z L −2 Z0 Z ∂τ ! ∇   Integrating by parts and taking note of the periodicity of φ, we obtain

β 2 1 3 ∂ 2 2 S = dτ d xφ 2 + m φ . (2.42) −2 Z0 Z −∂τ − ∇ ! The ﬁeld can be decomposed into a Fourier series according to

1/2 ∞ β i(p~~x+ωnτ) φ(~x, τ) = e φn(p~) , (2.43) V ! n=−∞ X Xp~ where ωn = 2πnT , due to the constraint of periodicity that φ(~x, β) = φ(~x, 0) for all ~x. The normalization of (2.43) is chosen conveniently so that each Fourier amplitude is dimensionless. Substituting (2.43) into (2.42), and noting that φ ( p~) = φ∗ (p~) as required by the reality of φ (p~), we ﬁnd −n − n n

1 2 2 2 ∗ S = β (ωn + ω )φn(p~)φn(p~) (2.44) −2 n X Xp~ with ω = √p~2 + m2. The integrand depends only on the amplitude of φ and not its phase. The phases can be integrated out to get

∞ 0 1 2 2 2 2 Z = N ΠnΠp~ dAn(p~) exp β (ωn + ω )An(p~) Z−∞ −2 0 2 2 2 1/2 = N ΠnΠp~ 2π/(β (ωn + ω )) . (2.45) h i Ignoring an overall multiplicative factor independent of β and V , which does not aﬀect the thermodynamics, we arrive at

2 2 2 −1/2 Z = ΠnΠp~ β (ωn + ω ) . (2.46) h i More formally one can arrive at this result by using the general rules for Gaussian functional integrals over commuting (bosonic) variables, derived before in the QFT chapter, since (2.40) and (2.42) can be expressed as 1 Z = N 0 φ exp (φ, Dφ) = N 0constant(det D)−1/2 , (2.47) Z D −2 2 2 2 where D = β (ωn + ω ) in (p,~ ωn) space and (φ, Dφ) denotes the inner product on the function space. Thus far we have

1 2 2 2 ln Z = ln β (ωn + ω ) . (2.48) −2 n X Xp~ h i Note that the sum over n is divergent. This unfortunate feature stems from our careless handling of the integration measure φ. A more rigorous treatment D using the proper definition of φ gives a finite result. In order to handle (2.48), we make use of D β2ω2 dΘ2 ln (2πn)2 + β2ω2) = + ln 1 + (2πn)2 , (2.49) 1 Θ2 + (2πn)2 h i Z h i where the last term is β independent and thus can be ignored. Furthermore, − ∞ 1 2π2 2 2 2 = 1 + Θ , (2.50) −∞ n + (Θ/2π) Θ e 1 X − hence βω 1 1 ln Z = dΘ + . (2.51) − 1 2 eΘ 1 Xp~ Z − Carrying out the Θ integral, and throwing away a β independent piece, we − finally arrive at

3 d p 1 −βω ln Z = V 3 βω ln(1 e ) , (2.52) Z (2π) −2 − − from which we obtain immediately the well-known expression for the ideal Bose gas (µ = 0), once we subtract the divergent expressions for the zero-point energy ∂ d3p ω E0 = ln Z0 = V 3 , (2.53) −∂β Z (2π) 2 and for the zero-point pressure ∂ E P = T ln Z = 0 , (2.54) 0 ∂V 0 − V which are typical for the quantum field-theoretical treatment. With this subtraction the vacuum is defined as the state with zero energy and pressure. 2.1.4 Fermionic Fields Dirac fermions are descibed by a four-spinor field ψ with a Lagrangian density

= ψ¯(i∂/ m)ψ L − ∂ = ψ†γ0 iγ0 + i~γ ~ m ψ . (2.55) ∂t · ∇ − !

The momentum conjugate to this ﬁeld is ∂ Π = L = iψ† , (2.56) ∂(∂ψ/∂t) because γ0γ0 = 1. Thus, somewhat paradoxically, ψ and ψ† must be treated as independent entities in the Hamiltonian formulation. The Hamiltonian density is found by standard procedures,

∂ψ ∂ = Π = ψ† i ψ = ψ¯( i~γ ~ + m)ψ , (2.57) H ∂t − L ∂t ! − L − · ∇ and the partition function is

Z = Tr e−β(Hˆ −µQˆ) , (2.58) with the conserved charge Q = d3xψ†ψ. The functional integral representation reads R β ∂ Z = ψ† ψ exp dτ d3xψ† γ0 + i~γ ~ m + µγ0 ψ (2.59) Z D D "Z0 Z − ∂τ · ∇ − ! #

As with bosons, it is most convenient to work in (p~, ωn) space instead of (~x, τ) space, i.e., 1/2 ∞ β i(p~~x+ωnτ) ψα(~x, τ) = e ψ˜α;n(p~) , (2.60) V ! n=−∞ X Xp~ where now ωn = (2n + 1)πT due to the antiperiodicity of the (Grassmannian) Fermion ﬁeld at the borders of the fundamental strip 0 τ β in the imaginary time, ψ(~x, 0) = ψ(~x, β). ≤ ≤ − Now we are ready to evaluate the fermionic partition function (2.59),

˜† ˜ S Z = ΠnΠp~Πα idψα;n(p~)dψα;n(p~) e , Z † S = iψα;n(p~)Dαρψρ;n(p~) , n X Xp~ D = iβ ( iω + µ) γ0~γ p~ mγ0 , (2.61) − − n − · − h i using our knowledge about Grassmannian integration of Gaussian functional integrals, resulting in Z = det D . (2.62) Employing the identity ln det D = Tr ln D , (2.63) and evaluating the determinant in Dirac space explicitly (Exercise !), one ﬁnds

2 2 2 ln Z = 2 ln β (ωn + iµ) + ω . (2.64) n X Xp~ n h io Since both positive and negative frequencies have to be summed over, the latter expression can be put in a form analogous to the above expression in the bosonic case,

2 2 2 2 2 2 ln Z = ln β ωn + (ω µ) + ln β ωn + (ω + µ) . (2.65) n − X Xp~ n h i h io In the further evaluation we can go similar steps as in the bosonic case, with two exceptions: (1) the presence of a chemical potential, splitting the contributions of particles and antiparticles; (2) the Matsubara frequencies are now odd multiples of πT , so that the inﬁnite sum to be exploited reads

∞ 1 1 1 1 2 2 2 = Θ . (2.66) n=−∞ (2n + 1) π + Θ Θ 2 − e + 1 X Integrating over the auxiliary variable Θ, and dropping terms independent of β and µ, we ﬁnally obtain

d3p ln Z = 2V βω + ln(1 + e−β(ω−µ)) + ln(1 + e−β(ω+µ)) . (2.67) (2π)3 Z h i 1 Notice that the factor 2 corresponding to the spin- 2 nature of the fermions comes out automatically. Separate contributions from particles (µ) and antiparticles (-µ) are evident. Finally, the zero-point energy of the vacuum also appears in this formula. 2.1.5 Gauge Fields Quantizing the Electromagnetic Field In this chapter we would like to understand how the Faddeev-Popov ghosts as introduced to eliminate divergences of the gauge freedom will act in the pure gauge theory to eliminate unphysical degrees of freedom and allow to derive the blackbody radiation law with the correct number of physical degrees of freedom. We start with recalling the path integral formulation of QED for the photon ﬁeld Aµ(x) with the ﬁeld strength tensor F = ∂ A ∂ A (2.68) µν µ ν − ν µ and the free action functional

1 4 µν S = d xFµν F . (2.69) −4 Z This action is invariant under gauge transformations A (x) A (x) = A0 (x) + ∂ ω(x) , (2.70) µ → µ µ µ where ω(x) is a scalar function which parametrizes the gauge transformations. The momenta conjugate to the space components of Ai(x) are, up to a sign, the components Ei(x) = Ei(x) of the electric ﬁeld π = E = F , (2.71) i − i − 0i while the magnetic ﬁeld B(x) is

k Bi = εijk∂jA . (2.72)

3 We work in an axial gauge, A = 0 to be speciﬁc. The momenta π1 and π2 are independent variables; E3 is not an independnet variable, but it is a function of π1 and π2, which may be computed from Gauss’s law E = 0. There are thus two 1 2 ∇ · dynamical variables A and A with conjugate momenta π1 and π2. We deﬁne π = E (π , π ), but π is not to be interpreted as a conjugate momentum. The 3 − 3 1 2 3 partition function is written as a Hamiltonian path integral

β 1 2 4 1 2 Z = (π1, π2) (A , A ) exp d x(iπ1∂τ A + iπ2∂τ A ) (2.73), i i Z D ZA (0)=A (β) D "Z0 − H # where we have used the notation β β d4x = dτ d3x , (2.74) Z0 Z0 Z while the Hamiltonian density is H 1 2 2 1 2 2 2 2 = (E + B ) = π + π + E (π1, π2) + B (2.75) H 2 2 1 2 3 Equation (2.73) is then transformed by using

1 = π3δ(π3 + E3(π1, π2)) , (2.76) Z D and ∂( π) δ(π3 + E3(π1, π2)) = δ( π) det ∇ · ∇ · ∂π3 ! = det ∂ δ3(x y) δ( π) . (2.77) 3 − ∇ · h i In the following step, one inserts an integral representation of δ( π) ∇ · β 4 δ( π) = A4 exp i d xA4( π) , (2.78) ∇ · Z D " Z0 ∇ · # where A4 = iA0, and we work in Euclidean space now: xµ = (x, x4) = (x, τ), A = (A, A4). Performing the π integration, we are left with 4 − β 3 4 1 2 1 2 Z = (A1, A2, A4) det ∂3δ (x y) exp d x (i∂τ A i A4) B , D − " 0 2 − ∇ − 2 # Z h i Z (2.79) where A = (A1, A2, 0). Note that the argument of the exponential is 1 1 E2 B2 = . (2.80) 2 − 2 L The A integration is rendered more aesthetic by inserting −

1 = A3δ(A3) , (2.81) Z D and the partition function assumes the form β µ 3 4 Z = A δ(A3) det ∂3δ (x y) exp d x . (2.82) D − 0 L! Z h i Z The axial gauge A3 = 0 is not a particularly convenient gauge to use for practical computations. Furthermore, it is not immediately apparent that (2.82) is a gauge invariant expression for Z. Take an arbitrary gauge specified by F = 0, where F is some function of Aµ and its derivatives. For the gauge above, F = A3. For this gauge, (2.82) is given by ∂F β Z = Aµδ(F ) det exp d4x . (2.83) Z D ∂α ! Z0 L! Equation (2.83) is manifestly gauge invariant: is invariant, the gauge fixing factor times the Jacobian of the transformationLδ(F ) det(∂F/∂α) is invariant, and the integration is over all four components of the vector potential. Equation (2.83) reduces to (2.82) in the case of the axial gauge A3 = 0. We know this is correct since it was derived from first principles in the Hamiltonian formulation ˆ of the gauge theory, Z = Tr e−βH Blackbody radiation It is important to verify that (2.83) describes blackbody radiation with two polarization degrees of freedom. We will do this here in the axial gauge A3 = 0, the Feynman gauge is left as an exercise. In the axial gauge, we rewrite (2.79) as

S0 Z = (A0, A1, A2) det(∂3)e Z D 1 3 S0 = dτ d x(A0, A1, A2) 2 Z Z 2 ∂ ∂ ∂1 ∂τ ∂2 ∂τ A0 ∇ ∂ 2 − 2 ∂2 −  ∂1 ∂τ ∂2 + ∂3 + ∂τ 2 ∂1∂2   A1  . (2.84) × − ∂ 2 2 ∂2 ∂ ∂ ∂ ∂ + ∂ + 2 A2  2 ∂τ 1 2 1 3 ∂τ     − −    We can express the determinant of ∂3 as a functional integral over a complex ghost ﬁeld C: that is, a Grassmann ﬁeld with spin-0,

β 3 det(∂3) = C¯ C exp dτ d xC¯∂3C . (2.85) Z D D Z0 Z ! These ghost fields C¯ and C are not physical fields since they do not appear in the Hamiltonian. Furthermore, since they are anticommuting scalar fields they violate the connection between spin and statistics. It is simply a convenient functional integral representation of the determinant of an operator. The greatest applicability of these ficticious ghost fields will be to non-Abelian gauge theories, see also Sect. 5.2.1. of Le Bellac [2]. In frequency-momentum space the partition function is expressed as 1 ln Z = ln det(βp ) ln det(D) , 3 − 2 2 p ωnp1 ωnp2 2 2 − 2 2 − D = β  ωnp1 ωn + p2 + p3 p1p2  . −ω p p p ω2 +− p2 + p2  n 1 1 2 n 1 3   − −  Carrying out the determinantal operation, 1 1 ln Z = Tr ln(β2p2) Tr ln β6p2(ω2 + p2)2 2 3 − 2 3 n h i 2 2 2 −1 = ln ΠnΠp β (ωn + p ) 3 h i d p 1 −βω = 2V 3 βω ln(1 e ) . (2.86) Z (2π) −2 − − Here, ω = p . Comparison with the result for the scalar field case shows that (2.86) describ| |es massless bosons with two spin degrees of freedom in thermal equilibrium; in other words, blackbody radiation. 2.2 Interacting Fermion Systems: Hubbard-Stratonovich Trick

So far we have dealt with free quantum fields in the absence of interactions and have obtained nice closed expressions for the thermodynamic potential, i.e., therefore also for the generating functionals of the thermodynamic Green functions. However, once we switch on the interactions in our model field theories, there is only a very limited class of soluble models, in general we have to apply approx- imations. The most common technique is based on perturbation theory, which requires a small parameter. For strong interactions at low momentum transfer (the infrared region), the coupling is nonperturbatively strong and alternative, nonperturbative methods have to be invoked. One of the strategies, which is especially suitable for the treatment of quantum field theories within the path integral formulation is based on the introduction of collective variables (auxialiary fields) by an exact integral transformation due to Stratonovich and Hubbard which allows to eliminate (integrate out) the elementary degrees of freedom. Generally, the (dual) coupling of the auxialiary fields is weak so that perturbative expansions of the nonlinear effective action make sense and provide useful results already at low orders of this expansion. A general class of interactions for which the Hubbard-Stratonovich (HS) transformation is immediately applicable, are four-fermion couplings of the current- current type = G(ψ¯ψ)2 . (2.87) Lint A Fermi gas with this typ of interaction serves as a model for electronic superconductivity (Bardeen-Cooper-Schrieffer (BCS) model, 1957) or for chiral symmetry breaking in quark matter (Nambu–Jona-Lasinio (NJL) model, 1961). The HS-transformation for (2.87) reads

σ2 exp G(ψ¯ψ)2 = σ exp + ψ¯ψσ (2.88) N D "4G # h i Z and allows to bring the functional integral over fermionic ﬁelds into a quadratic (Gaussian) form so that fermions can be integrated out. This is also called Bosonization procedure.

2.2.1 Walecka Model This notes for this subsection are provided separately.

2.2.2 Nambu–Jona-Lasinio (NJL) Model Here we will present an application of the HS technique to the NJL model for quark matter at ﬁnite densities and temperatures. This is possible since the interaction of this model is of the current-current form and therefore the HS trick for the bosonization of 4-fermion interactions applies. The Lagrangian density is given by

= q¯ (i∂/δ δ M 0 δ + µ γ0)q L iα ij αβ − ij αβ ij,αβ jβ 8 a 2 a 2 + GS (q¯τf q) + (q¯iγ5τf q) aX=0 h i C C 0 0 0 0 0 0 + GD (q¯iαijkαβγ qjβ)(q¯i0α0 i j kα β γqj β ) Xk,γ h C C 0 0 0 0 0 0 + (q¯iαiγ5ijkαβγ qjβ)(q¯i0α0 iγ5i j kα β γ qj β ) , (2.89) i where from here on the quark spinor is qiα, with the flavor index i = u, d, s and 0 0 0 0 α = r, g, b stands for the color degree of freedom. Mij = diag(mu, md, ms) is the current quark mass matrix in flavor space and µij,αβ is the chemical potential matrix in color and flavor space. The grand cononical thermodynamical potential Ω = T ln Z is known once we manage to evaluate the partition function in a reasonable approximation. A closed solution, even for the simple model Lagrangion (2.89), is not possible. The bosonization of the partition function proceeds as follows

Z = q¯ q exp d4x Z D D Z L † 4 HS = q¯ qΠi=u,d,s φiΠkγ=ur,dg,sb ∆kγ ∆kγ exp d x (2.90) Z D D D D D Z L whehre the eﬀective ’linearized’ Lagrangian is given by

HS = q¯ i∂/δ δ (M 0 φ δ )δ + µ γ0 q L iα ij αβ − ij − i ij αβ ij,αβ jβ h 2 2 i † φi ∆kγ ∆kγ C C ∆kγ | | + q¯iα qjβ + q¯iα qjβ, (2.91) − 8GS − 4GD 2 2 i=Xu,d,s kγ=Xur,dg,sb The fermionic quark spinor fields can be grouped into bi-spinors, formed by the original and the charge-conjugated, transposed anti-spinor. This way, the action functional can be given the quadratic form which allows to integrate out the elementary quark fields using the Gaussian Functional Integration rule, thus leaving us after this exact Hubbard-Stratonovich transformation with an alternative representation of the partition function in terms of collective fields φi and ∆kγ. The next step is the mean-field approximation, which consists of replacing the collective fields by those values which make the action functional extremal and neglecting the fluctuations around them, i.e. dropping the functional integration over these collective fields. This yields the thermodynamic potential φ2 + φ2 + φ2 ∆ 2 + ∆ 2 + ∆ 2 Ω(T, µ) = u d s + | ud| | us| | ds| 8GS 4GD 3 d p 1 1 −1 T 3 Tr ln S (iωn, p~) − n (2π) 2 T X Z + Ω Ω . (2.92) e − 0 Here S−1(p) is the inverse propagator of the quark fields at four momentum p = (iωn, p~), 0 −1 /p M + µγ ∆kγ S (iωn, p~) = − † 0 , (2.93) " ∆ /p M µγ # kγ − − and ωn = (2n + 1)πT are the Matsubara frequencies for fermions. The thermodynamic potential of ultrarelativistic electrons, 1 1 7 Ω = µ4 µ2 T 2 π2T 4, (2.94) e −12π2 Q − 6 Q − 180 has been added to the potential, and the vacuum contribution, 2 2 2 φ0u + φ0d + φ0s Ω0 = Ω(0, 0) = 8GS d3p 2N M 2 + p2, (2.95) − c (2π)3 i Xi Z q has been subtracted in order to get zero pressure in vacuum. Using the identity Tr(ln(D)) = ln(det(D)) and evaluating the determinant (see Appendix A), we obtain 18 2 2 1 −1 ωn + λa(p~) ln det S (iωn, p~) = 2 ln . (2.96) T T 2 ! aX=1 The quasiparticle dispersion relations, λa(p~), are the eigenvalues of the Hermitian matrix, 0 0 0 γ ~γ p~ γ M + µ γ ∆kγC = − · 0 − † 0 T 0 , (2.97) M " γ C∆ γ ~γ p~ + γ M µ# kγ − · − in color, flavor, and Nambu-Gorkov space. This result is in agreement with [?, ?]. Finally, the Matsubara sum can be evaluated on closed form [?], 2 2 ωn + λa −λa/T T ln 2 = λa + 2T ln(1 + e ), (2.98) n T ! X leading to an expression for the thermodynamic potential on the form φ2 + φ2 + φ2 ∆ 2 + ∆ 2 + ∆ 2 Ω(T, µ) = u d s + | ud| | us| | ds| 8GS 4GD d3p 18 λ + 2T ln 1 + e−λa/T − (2π)3 a Z aX=1 + Ω Ω . (2.99) e − 0 It should be noted that (14) is an even function of λa, so the signs of the quasiparticle dispersion relations are arbitrary. In this paper, we assume that there are no trapped neutrinos. This approximation is valid for quark matter in neutron stars, after the short period of deleptonization is over. Equations (2.94), (2.95), (2.97), and (2.99) form a consistent thermodynamic model of superconducting quark matter. The independent variables are µ and T . The gaps, φi, and ∆ij, are variational order parameters that should be determined by minimization of the grand canonical thermodynamical potential, Ω. Also, quark matter should be locally color and electric charge neutral, so at the physical minima of the thermodynamic potential the corresponding number densities should be zero ∂Ω nQ = = 0, (2.100) −∂µQ ∂Ω n8 = = 0, (2.101) −∂µ3 ∂Ω n3 = = 0. (2.102) −∂µ8 The pressure, P , is related to the thermodynamic potential by P = Ω at the − global minima of Ω. The quark density, entropy and energy density are then obtained as derivatives of the thermodynamical potential with respect to µ, T and 1/T , respectively. The numerical solutions to be reported in this Section are obtained with the following set of model parameters, taken from Table 5.2 of Ref. [?] for vanishing ’t Hooft interaction,

0 mu,d = 5.5 MeV , (2.103) 0 ms = 112.0 MeV , (2.104) 2 GSΛ = 2.319 , (2.105) Λ2 = 602.3 MeV . (2.106)

With these parameters, the following low-energy QCD observables can be reproduced: mπ = 135 MeV, mK = 497.7 MeV, fπ = 92.4 MeV. The value of the diquark coupling strength GD = ηGS is considered as a free parameter of the model. Here we present results for η = 0.75 (intermediate coupling) and η = 1.0 (strong coupling).

Quark masses and pairing gaps at zero temperature The dynamically generated quark masses and the diquark pairing gaps are determined selfconsistently at the absolute minima of the thermodynamic potential, in the plane of temperature and quark chemical potential. This is done for both 600 600 Ms 500 M 500 M u s M M d u 400 M , M M u d 400 d ∆ Mu, Md ud ∆ 300 ∆ , ∆ 300 ud us ds ∆ , ∆ us ds , M [MeV] ∆ , M [MeV]

∆ 200 200

100 100

300 350 400 450 500 550 300 350 400 450 500 550 µ [MeV] µ [MeV]

Figure 2.1: Gaps and dynamical quark masses as a function of µ at T=0 for intermediate diquark coupling, η = 0.75 (left) and for strong diquark coupling, η = 1 (right).

the strong and the intermediate diquark coupling strength. In Figs. 1 and 2 we show the dependence of masses and gaps on the quark chemical potential at T = 0 for η = 0.75 and η = 1.0, resp. A characteristic feature of this dynamical quark model is that the critical quark chemical potentials where light and strange quark masses jump from their constituent mass values down to almost their current mass values do not coincide. With increasing chemical potential the system undergoes a sequence of two transitions: (1) vacuum two-flavor quark → matter, (2) two-flavor three-flavor quark matter. The intermediate two-flavor quark matter phase occurs→ within an interval of chemical potentials typical for compact star interiors. While at intermediate coupling the asymmetry between of up and down quark chemical potentials leads to a mixed NQ-2SC phase below temperatures of 20-30 MeV, at strong coupling the pure 2SC phase extends down to T=0. Simultaneously, the limiting chemical potentials of the two-flavor quark matter region are lowered by about 40 MeV. Three-flavor quark matter is always in the CFL phase where all quarks are paired. The robustness of the 2SC condensate under compact star constraints, with respect to changes of the coupling strength, as well as to a softening of the momentum cutoff by a formfactor, has been recently investigated within a different parametrization [?] with similar trend: for η = 0.75 and NJL formfactor the 2SC condensate does not occur for moderate chemical potentials while for η = 1.0 it occurs simultaneously with chiral symmetry restoration. Fig. 3 shows the corresponding dependences of the chemical potentials conjugate to electric (µQ) and color (µ8) charges. All phases considered in this work have zero n3 color charge for µ3 = 0. 500 0 ug-dr ub-sr ub-sr, db-sg db-sg -50 400 ur-dg-sb

µ -100 Q, η=0.75 µ 300 8, η=0.75

[MeV] µ 8 Q, η=1 µ

, -150 µ η E [MeV] Q 8, =1 200 µ

-200 100

-250 0 300 350 400 450 500 550 600 0 200 400 0 200 400 600 µ [MeV] p [MeV] p [MeV]

Figure 2.2: Left: Chemical potentials µQ and µ8 at T=0 for both values of the diquark coupling η = 0.75 and η = 1. All phases considered in this work have zero n3 color charge for µ3 = 0, hence µ3 is omitted in the plot. Right: Quark-quark quasiparticle dispersion relations. For η = 0.75, T = 0, and µ = 480 MeV (left panel) there is a forbidden energy band above the Fermi surface. All dispersion relations are gapped at this point in the µ T plane, see Fig. 5. There is no − forbidden energy band for the ub sr, db sq, and ur dg sb quasiparticles at η = 1, T = 84 MeV, and µ = 500− MeV (righ− t panel). −This −point in the µ T − plane constitute a part of the gapless CFL phase.

Dispersion relations and gapless phases In Fig. 4 we show the quasiparticle dispersion relations of different excitations at two points in the phase diagram: (I) the CFL phase (left panel), where there is a finite energy gap for all dispersion relations. (II) the gCFL phase (right panel), where the energy spectrum is shifted due to the assymetry in the chemical potentials, such that the CFL gap is zero and (gapless) excitations with zero energy are possible. In the present model, this phenomenon occurs only at rather high temperatures, where the condensates are diminished by thermal fluctuations.

Phase diagram The thermodynamical state of the system is characterized by the values of the order parameters and their dependence on T and µ. Here we illustrate this dependency in a phase diagram. We identify the following phases:

1. NQ: ∆ud = ∆us = ∆ds = 0; 2. NQ-2SC: ∆ = 0, ∆ = ∆ = 0, 0¡χ ¡1; ud 6 us ds 2SC 3. 2SC: ∆ = 0, ∆ = ∆ = 0; ud 6 us ds 80 120 NQ 70 NQ g2SC gCFL 100 g2SC gCFL 60 guSC 80 50 guSC 2SC 40 60 T [MeV] T [MeV] 2SC CFL 30 χ = 1.0 2SC 40 0.9 20 0.8

0.7 175 =200 MeV 20 175 150 s =200 MeV

10 s NQ-2SC M CFL M 0 0 350 400 450 500 550 300 350 400 450 500 550 µ [MeV] µ [MeV]

Figure 2.3: Left: Phase diagram of neutral three-ﬂavor quark matter for intermediate diquark coupling η = 0.75. First-order phase transition boundaries are indicated by bold solid lines, while thin solid lines correspond to second-order phase boundaries. The dashed lines indicate gapless phase boundaries. The volume fraction, χ2SC , of the 2SC component of the mixed NQ-2SC phase is denoted with thin dotted lines, while the constituent strange quark mass is denoted with bold dotted lines. Right: Phase diagram of neutral three-ﬂavor quark matter for strong diquark coupling η = 1. Line styles as in previous Figures.

4. uSC: ∆ = 0, ∆ = 0, ∆ = 0; ud 6 us 6 ds 5. CFL: ∆ = 0, ∆ = 0, ∆ = 0; ud 6 ds 6 us 6 and their gapless versions. The resulting phase diagrams for intermediate and strong coupling are given in Figs. 5 and 6, resp. and constitute the main result of this work, which is summarized in the following statements:

1. Gapless phases occur only at high temperatures, above 50 MeV (intermediate coupling) or 60 MeV (strong coupling).

2. CFL phases occur only at rather high chemical potential, well above the chiral restoration transition, i.e. above 464 MeV (intermediate coupling) or 426 MeV (strong coupling).

3. Two-ﬂavor quark matter for intermediate coupling is at low temperatures (T¡20-30 MeV) in a mixed NQ-2SC phase, at high temperatures in the pure 2SC phase.

4. Two-flavor quark matter for strong coupling is in the 2SC phase with rather high critical temperatures of 100 MeV. ∼ 5. The critical endpoint of first order chiral phase transitions is at (T,µ)=(44 MeV, 347 MeV) for intermediate coupling and at (92 MeV, 305 MeV) for strong coupling. 2.2.3 Mesonic correlations at finite temperature In the previous section we have seen how the concept of order parameters can be introduced in quantum field theory in the mean-field approximation. By analysis of the gap equations describing the minima of the thermodynamical potential in the space of order parameters we have we could investigate the phenomenon of spontaneous symmetry breaking, indicated by a nonvanishing value of the order parameter (gap). Important examples being: chiral symmetry breaking (mass gap) and superconductivity (energy gap). At finite temperature and density the values of these gaps change and their vanishing indicates the restoration of a symmetry. As the symmetries prevailing under given thermodynamical conditions of temperature and chemical potential (density) characterize a phase of the system, we have thus acquainted ourselves with a powerful quantum field theoretical method of analysing phase transitions. The results of such an analysis are summarized in phase diagrams. A prominent example is the phase diagram of QCD in the temperature- density plane, shown schematically in Fig. ??. It exhibits two major domains: Hadronic matter (confined quarks and gluons) and the Quark-gluon plasma (QGP), separated by the phenomenon of quark (and gluon) deconfinement under investigation in heavy-ion collisions, but also in the physics of compact stars and in simulations of Lattice-gauge QCD on modern Ter- aflop computers. The hadronic phase is subdivided into a nuclear matter phase at low temperatures with a gas-liquid transition (similar to the van-der-Waals treatment of real gases) and a ’plasma phase’ of a hot hadron gas with a multi- tude of hadronic resonances. The QGP phase is subdivided into a quark matter phase at low temperatures where most likely gluons are still condensed (confined) and the strongly correlated Fermi-liquid of quarks shoild exhibit the phenomenon of superconductivity/ superfludity with a Bose condensate of Cooper pairs of . diquarks. At asymptotic temperatures and densities one expects a system of free quarks and gluons (due to asymptotic freedom of QCD). In any real situation in terrestrial experiments or in astrophysics, one expects to be quite far from this ideal gas state. As the nature of the confinement-deconfinement transition is not yet clarified, we must speculate what to expect in the vicinity of the conjectured deconfinement phase transition. In the lattice-QCD simulations one has found evidence for strong correlations even above the critical temperature Tc 170 MeV, which is defined by an increase of the effective number of degrees ∼ 4 of freedom ε(T )/T by about one order of magnitude in a close vicinity of Tc. Indications for strong correlations in the QGP one has also found in recent RHIC experiments, where from particle production and flow one has found fast ther- malization and an extremely low viscosity (’perfect fluid’). One speaks about an

’sQGP phase’ at temperatures Tc < T < 2 Tc. In order to investigate the phase∼diagram∼ experimentally, one has to correlate the observables with the regions. As the detectors are situated in the ’vacuum’ at zero temperature and chemical potential where conﬁnement prevails, no direct Big Bang Quark-Gluon-Plasma 1.5 RHIC, LHC (construction)

DECONFINEMENT CERN-SPS FAIR (Project) AGS Brookhaven

H Hadron gas [T =140 MeV]

SIS Darmstadt

Quark Matter

QCD - Lattice Gauge Theory Super-

Temperature Novae Heavy Ion Collisions CONFINEMENT COLOR SUPERCONDUCTIVITY Nuclear matter 0.1 Neutron / Quark Stars -3 1 Baryon Density 3 [n ο =0.16 fm ]

Figure 2.4: The conjectured phase diagram of QCD and places where to investigate it. detection of quarks and gluons from the plasma phase is possible and one has to conjecture about the properties of the sQGP phase(s) from the traces which they might leave in the hadronic and electromagnetic spectra emitted by the hadronizing fireball formed in a heavy-ion collision or by the neutrino-emitting cooling processes of compact stars. As a prerequisite for discussing hadronic spectra and their modifications, we will study the mesonic spectral function(s), and for their analysis within effective models like the NJL or MN models, we need to evaluate loop diagrams at finite temperature. A basic ingredient are Matsubara frequency sums.

2.2.4 Matsubara frequency sums In computing Feynman diagrams with internal fermion lines we shall encounter frequency sums, and we have to learn how to evaluate them. Let us denote the 2 2 −1 quantity (ωn + Ep ) , which appears in the free Fermion propagator

∞ dp0 ρF(p0, p) m p/ SF (iωn, p) = = 2 − 2 (2.107) − −∞ 2π iωn p0 ω E Z − n − p by ∆˜ (iωn, Ep), and its mixed representation by ∆(˜ τ, Ep), suppressing the sub- script F for simplicity

−iωnτ ∆˜ (τ, Ep) = T e ∆(˜ iωn, Ep) . (2.108) n X It can easily be checked (Exercise !) that

1 −Epτ Epτ ∆(˜ τ, Ep) = (1 n˜(Ep))e n˜(Ep)e 2Ep − − h s i ˜ −sEpτ = (1 f(sEp))e , (2.109) 2Ep − sX=1 where the Fermi-Dirac distribution n˜(p0) is 1 n˜(p0) = . (2.110) eβ|p0| + 1

One should note the absolute value of p0 in (2.110), in contrast with the deﬁnition of f˜(p0) = 1/[exp(βp0) + 1]. Note also that

f˜(E) = n˜(E) , f˜( E) = 1 n˜(E) , − − 1 f˜(E) f˜( E) = 0 . (2.111) − − − In frequency space, the formula corresponding to (2.109) is

˜ 1 ˜ ∆(iωn, Ep) = 2 2 = ∆s(iωn, Ep ωn Ep − sX=1 s 1 = . (2.112) −2Ep iωn sEp sX=1 − The frequency sums are performed by following the methods of analytic continuation iωn k0 and contour integration with a function having simple poles at the discrete frequencies→ k = iω with unit residuum and convergence for k . 0 n | 0| → ∞ Let us give two general examples for Matsubara sums of one-loop diagrams with two external (amputated) legs.

Fermion-boson case, see Fig. 2.2.4: • ˜ ˜ s1s2 1 + f(s1E1) f(s2E2) T ∆s1 (iωn, E1)∆s2 (i(ω ωn), E2) = − n − −4E1E2 iω s1E1 s2E2 X − − (2.113) ˜ ˜ is2 1 + f(s1E1) f(s2E2) T ωn∆s1 (iωn, E1)∆s2 (i(ω ωn), E2) = − n − −4E2 iω s1E1 s2E2 X − − (2.114) φ 1 k

q φ

φ q−k 2 Figure 2.5: One-loop diagram for the fermion-antifermion polarization function.

φ q

φ 1 φ q−k 2 Figure 2.6: One-loop diagram for the fermion self-energy. Fermion-antifermion case, see Fig. 2.2.4: • ˜ ˜ ˜ ˜ s1s2 1 f(s1E1) f(s2E2) T ∆s1 (iωn, E1)∆s2 (i(ω ωn), E2) = − − n − −4E1E2 iω s1E1 s2E2 X − − (2.115) ˜ ˜ ˜ ˜ is2 1 f(s1E1) f(s2E2) T ωn∆s1 (iωn, E1)∆s2 (i(ω ωn), E2) = − − n − −4E2 iω s1E1 s2E2 X − − (2.116)

Note that you can obtain unify these formula using the rule: f(sE) f˜(sE) when replacing a bosonic by a fermionic line. → − Exercise: Verify these expressions! Bibliography

[1] J.I. Kapusta, Finite-temperature Field Theory, Cambridge University Press, 1989

[2] M. Le Bellac, Thermal Field Theory, Cambridge University Press, 1996

[3] D. Blaschke, et al., Phys. Rev. D 72 (2005) 065020.

111 Chapter 3

Particle Production by Strong Fields

3.1 Introduction

Particle production under the influence of strong, time-dependent external fields as in ultrarelativistic heavy-ion collisions and high-intensity lasers raises a number of challenging problems. One of these interesting questions is how to formulate a kinetic theory that incorporates the mechanism of particle creation [1, 2, 3, 4, 5, 6, 7, 8, 9]. Within the framework of a flux tube model[4, 10, 11, 12], a lot of promising research has been carried out during the last years. In the scenario where a chromo-electric field is generated by a nucleus-nucleus collision, the production of parton pairs can be described by the Schwinger mechanism[13, 14, 15]. The produced charged particles generate a field which, in turn, modifies the initial electric field and may cause plasma oscillations. The interesting question of the back reaction of this field has been analyzed within a field theoretical approach[16, 17, 18]. The results of a simple phenomenological consideration based on kinetic equations and the field-theoretical treatment[5, 19, 20, 21] agree with each other. The source term which occurs in such a modified Boltzmann equation was derived phenomenologically in Ref.[22]. However, the systematic derivation of this source term in relativistic transport theory is not yet fully carried out. For example, recently it was pointed out that the source term may have non-Markovian character[6, 23] even for the case of a constant electric field. In the present lecture, a kinetic equation is derived in a consistent field theoretical treatment for the time evolution of the pair creation in a time-dependent and spatially homogeneous electric field. This derivation is based on the Bogoli- ubov transformation for field operators between the asymptotic in-state and the instantaneous state. In contrast with phenomenological approaches, but in agreement with[6, 23], the source term of the particle production is of non-Markovian character. The kinetic equation derived reproduces the Schwinger result in the

112 low density approximation in the weak ﬁeld limit.

3.2 Dynamics of pair creation

3.2.1 Creation of fermion pairs In this section we demonstrate the derivation of a kinetic equation with a source term for fermion-antifermion production. As an illustrative example we consider electron-positron creation, however the generalization to quark-antiquark pair creation in a chromoelectric field in Abelian approximation is straightforward. For the description of e+e− production in an electric field we start from the QED lagrangian 1 = ψ¯iγµ(∂ + ieA )ψ mψ¯ψ F F µν , (3.1) L µ µ − − 4 µν where F µν is the field strength, the metric is taken as gµν = diag(1, 1, 1, 1) − − − and for the γ– matrices we use the conventional definition [24]. In the following we consider the electromagnetic field as classical and quantize only the matter field. Then the Dirac equation reads (iγµ∂ eγµA m)ψ(x) = 0 . (3.2) µ − µ − We use a simple field-theoretical model to treat charged fermions in an external electric field charactarized by the vector potential Aµ = (0, 0, 0, A(t)) with A(t) = A3(t). The electric field E(t) = E (t) = A˙(t) = dA(t)/dt (3.3) 3 − − is assumed to be time-dependent but homogeneous in space (E1 = E2 = 0). This quasi-classical electric field interacts with a spinor field ψ of fermions. We look for solutions of the Dirac equation where eigenstates are represented in the form

() 0 k 3 () ip¯x¯ ψpr¯ (x) = iγ ∂0 + γ pk eγ A(t) + m χ (p,¯ t) Rr e , (3.4) − where k = 1, 2, 3 and the superscript ( ) denotes eigenstates with the positive and negative frequencies. Herein Rr (r = 1, 2) is an eigenvector of the matrix γ0γ3 0 1 1 0 R =   , R =   , (3.5) 1 0 2 1    −   1   0   −    +   ()   so that Rr Rs = 2δrs . The functions χ (p,¯ t) are related to the oscillator-type equation

χ¨()(p,¯ t) = ω2(p,¯ t) + ieA˙(t) χ()(p,¯ t) , (3.6) − 2 2 2 where we define the total energy ω (p,¯ t) = ε⊥ + Pk (t), the transverse energy 2 2 2 () ε⊥ = m + p¯⊥ and Pk(t) = pk eA(t). The solutions χ (p,¯ t) of Eq. (3.6) for positive and negative frequencies− are defined by their asymptotic behavior at t = t , where A(t ) = 0. We obtain 0 → −∞ 0 χ()(p,¯ t) exp ( iω (p¯) t) , (3.7) ∼ 0 where the total energy in asymptotic limit is given as ω0(p¯) = ω0(p,¯ t0) = lim ω(p¯, t). Note that the system of the spinor functions (3.4) is complete t→−∞ and orthonormalized. The field operators ψ(x) and ψ¯(x) can be decomposed in the spinor functions (3.4) as follows:

(−) (+) + ψ(x) = ψp¯r (x) bpr¯ (t0) + ψpr¯ (x) d−p¯r(t0) . (3.8) r,p¯ X + + The operators bpr¯ (t0), bpr¯ (t0) and dp¯r(t0), dp¯r(t0) describe the creation and annihilation of electrons and positrons in the in-state 0in > at t = t0, and satisfy the anti-commutation relations [24] | + + b (t ), b 0 0 (t ) = d (t ), d 0 0 (t ) = δ 0 δ 0 . (3.9) { pr¯ 0 p¯ r 0 } { p¯r 0 p¯ r 0 } rr p¯p¯ The evolution affects the vacuum state and mixes states with positive and negative energies resulting in non-diagonal terms that are responsible for pair creation. The diagonalization of the hamiltonian corresponding to a Dirac-particle (Eq. (3.2)) in the homogeneous electric field (3.3) is achieved by a time-dependent Bogoliubov transformation + bpr¯ (t) = αp¯(t) bpr¯ (t0) + βp¯(t) d−p¯r(t0) , (3.10) d (t) = α (t) d (t ) β (t) b+ (t ) pr¯ −p¯ p¯r 0 − −p¯ −pr¯ 0 with the condition α (t) 2 + β (t) 2 = 1 . (3.11) | p¯ | | p¯ | Here, the operators bpr¯ (t) and dp¯r(t) describe the creation and annihilation of quasiparticles at the time t with the instantaneous vacuum 0t >. Clearly, the + + | operator system b(t0), b (t0); d(t0), d (t0) is unitary non-equivalent to the system b(t), b+(t); d(t), d+(t). The substitution of Eqs. (3.10) into Eq. (3.8) leads to the new representation of the field operators

(−) (+) + ψ(x) = Ψpr¯ (x) bpr¯ (t) + Ψpr¯ (x) d−p¯r(t) . (3.12) r,p¯ X () The link between the new Ψp¯r (x) and the former (3.4) basis functions is given by a canonical transformation ψ(−)(x) = α (t) Ψ(−)(x) β∗(t) Ψ(+)(x) , pr¯ p¯ pr¯ − p¯ p¯r (3.13) (+) ∗ (+) (−) ψpr¯ (x) = αp¯(t) Ψ (x)pr¯ + βp¯(t) Ψp¯r (x) . () Therefore it is justiﬁed to assume that the functions Ψpr¯ have a spin structure (+) similar to that of ψpr¯ in Eq. (3.4),

() 0 k 3 () iΘ(t) ip¯x¯ Ψpr¯ (x) = iγ ∂0 + γ pk eγ A(t) + m φp¯ (x) Rr e e , (3.14) − where the dynamical phase is deﬁned as t Θ(p,¯ t) = dt0ω(p¯, t0) . (3.15) Zt0 () The functions φp¯ are yet unknown. The substitution of Eq. (3.14) into Eqs. (3.13) leads to the relations χ(−)(p,¯ t) = α (t) φ(−)(t) e−iΘ(p,t¯ ) β∗(t) φ(+)(t) eiΘ(p¯,t) , p¯ p¯ − p¯ p¯ (3.16) (+) ∗ (+) iΘ(p¯,t) (−) −iΘ(p,t¯ ) χ (p,¯ t) = αp¯(t) φp¯ (t) e + βp¯(t) φp¯ (t) e .

Now we are able to find explicit expressions for the coefficients αp¯(t) and βp¯(t). Taking into account that the functions χ()(p,¯ t) are defined by Eq. (3.6), we introduce additional conditions to Eqs. (3.16) according to the Lagrange method

(−) (−) −iΘ(p,t¯ ) ∗ (+) iΘ(p¯,t) χ˙ (p¯, t) = iω(p,¯ t) αp¯(t) φp¯ (t) e + βp¯(t) φp¯ (t) e , − (3.17) (+) ∗ (+) iΘ(p¯,t) (−) −iΘ(p,t¯ ) χ˙ (p¯, t) = iω(p¯, t) αp¯(t) φp¯ (t) e βp¯(t) φp¯ (t) e . − Diﬀerentiating these equations with respect to time, using Eqs. (3.6) and (3.16) and then choosing as an Ansatz

ω(p,¯ t) P (t) φ()(t) = k , (3.18) p¯ v u ω(p¯, t) u t we obtain the following differential equations for the coefficients eE(t)ε α˙ (t) = ⊥ β∗(t) e2iΘ(p¯,t) , p¯ 2ω2(p,¯ t) p¯ (3.19) eE(t)ε β˙ ∗(t) = ⊥ α (t) e−2iΘ(p,t¯ ) . p¯ −2ω2(p,¯ t) p¯ As the result of the Bogoliubov transformation we obtained the new coefficients of the instantaneous state at the time t. The relations between them read

1 (−) (−) iΘ(p,t¯ ) αp¯(t) = ω(p¯, t) χ (p,¯ t) + i χ˙ (p,¯ t) e , 2 ω(p¯, t) (ω(p¯, t) Pk(t)) − q (3.20) ∗ 1 (−) (−) −iΘ(p,t¯ ) βp¯(t) = ω(p,¯ t) χ (p,¯ t) i χ˙ (p,¯ t) e . −2 ω(p,¯ t) (ω(p¯, t) Pk(t)) − − q It is convenient to introduce new operators which absorb the dynamical phase

−iΘ(p,t¯ ) Bpr¯ (t) = bpr¯ (t) e , (3.21) −iΘ(p,t¯ ) Dpr¯ (t) = dp¯r(t) e (3.22) satisfying the anti-commutation relations:

+ + B (t), B 0 0 (t) = D (t), D 0 0 (t) = δ 0 δ 0 . (3.23) { pr¯ p¯ r } { p¯r p¯ r } rr p¯p¯ It is easy to show that these operators satisfy the Heisenberg-like equations of motion dB (t) eE(t)ε p¯r = ⊥ D+ (t) + i [H(t), B (t)] , dt −2ω2(p,¯ t) −p¯r pr¯ (3.24) dD (t) eE(t)ε p¯r = ⊥ B+ (t) + i [H(t), D (t)] , dt 2ω2(p,¯ t) −pr¯ p¯r where H(t) is the hamiltonian of the quasiparticle system

+ + H(t) = ω(p¯, t) Bpr¯ (t) Bpr¯ (t) D−p¯r(t) D−p¯r(t) . (3.25) r,p¯ − X The ﬁrst term on the r.h.s. of Eqs. (3.24) is caused by the unitary non-equivalence of the in-representation and the quasiparticle one. Now we explore the evolution of the distribution function of electrons with the momentum p¯ and spin r deﬁned as

f (p¯, t) =< 0 b+ (t) b (t) 0 >=< 0 B+ (t) B (t) 0 > . (3.26) r in| pr¯ pr¯ | in in| p¯r p¯r | in According to the charge conservation the distribution functions for electrons and ¯ positrons are equal fr(p,¯ t) = fr(p,¯ t), where

f¯ (p,¯ t) = < 0 d+ (t) d (t) 0 > = < 0 D+ (t) D (t) 0 > . (3.27) r in| −p¯r −p¯r | in in| −p¯r −p¯r | in The distribution functions (3.26) and (3.27) are normalized to the total number of pairs N(t) of the system at a given time t

fr(p,¯ t) = f¯r(p,¯ t) = N(t) . (3.28) r,p¯ r,p¯ X X Time diﬀerentiation of Eq. (3.26) leads to the following equation df (p,¯ t) eE(t)ε r = ⊥ Re Φ (p,¯ t) . (3.29) dt − ω2(p¯, t) { r } Herein, we have used the equation of motion (3.24) and evaluated the occurring commutator. The function Φr(p¯, t) in Eq. (3.29) describes the creation and annihilation of an electron-positron pair in the external electric ﬁeld E(t) and is given as Φ (p,¯ t) =< 0 D (t) B (t) 0 > . (3.30) r in| −p¯r pr¯ | in It is straightforward to evaluate the derivative of this function. Applying the equations of motion (3.24), we obtain

dΦr(p,¯ t) eE(t)ε⊥ = 2 2fr(p,¯ t) 1 2iω(p¯, t) Φr(p,¯ t) , (3.31) dt 2ω (p¯, t) − − where because of charge neutrality of the system, the relation fr(p,¯ t) = f¯r(p,¯ t) is used. The solution of Eq. (3.31) may be written in the following integral form

t 0 ε⊥ 0 eE(t ) 0 2i[Θ(p,t¯ 0)−Θ(p,t¯ )] Φr(p,¯ t) = dt 2 0 2fr(p,¯ t ) 1 e . (3.32) 2 Zt0 ω (p,¯ t ) − 0 The functions Θ(p,¯ t) and Θ(p¯, t ) in Eq. (3.32) can be taken at t0 (see the deﬁnition (3.15)). Hence with A(t0) = 0, the function Φr(p,¯ t) vanishes at t0. Inserting Eq. (3.32) into the r.h.s of Eq. (3.29) we obtain

t 0 dfr(p¯, t) eE(t)ε⊥ 0 eE(t )ε⊥ 0 0 = 2 dt 2 0 1 2fr(p,¯ t ) cos 2[Θ(p,¯ t) Θ(p,¯ t )] . dt 2ω (p,¯ t) Z−∞ ω (p,¯ t ) − − (3.33) Since the distribution function obviously does not depend on spin (3.33), we can define: f = f. With the substitution f(p,¯ t) F (P¯, t), where the 3-momentum r → is now defined as P¯(p⊥, Pk(t)), the kinetic equation (3.33) is reduced to its final form: dF (P¯, t) ∂F (P¯, t) ∂F (P¯, t) = + eE(t) = (−)(P¯, t) , (3.34) dt ∂t ∂Pk(t) S with the Schwinger source term

t 0 ¯ eE(t)ε⊥ 0 eE(t )ε⊥ ¯ 0 0 (−)(P , t) = 2 dt 2 0 1 2F (P , t ) cos 2[Θ(p¯, t) Θ(p,¯ t )] . S 2ω (p,¯ t) Z−∞ ω (p,¯ t ) − − (3.35) Recently in Ref.[6], a kinetic equation similar to (3.34) has been derived within a projection operator formalism for the case of a time-independent electric field where it was first noted that this source term has non-Markovian character. As well as in this method the multiple pair creation is not considered. The presence of the Pauli blocking factor [1 2F (P¯, t)] in the source term has been − obtained earlier in Ref.[18]. We would like to emphasize the closed form of the kinetic equation in the present work where the source term does not include the anomalous distribution functions (3.30) for fermion-antifermion pair creation (annihilation). 3.2.2 Creation of boson pairs In this subsection we derive the kinetic equation with source term for the case of pairs of charged bosons in a strong electric field. The Klein-Gordon equation reads

µ µ 2 (∂ + ieA )(∂µ + ieAµ) + m φ(x) = 0 . (3.36) The solution of the Klein-Gordon equation in the presence of the electric ﬁeld deﬁned by the vector potential Aµ = (0, 0, 0, A(t)) is taken in the form [24]

() −1/2 ix¯p¯ () φp¯ (x) = [2ω(p)] e g (p¯, t) , (3.37) where the functions g()(p,¯ t) satisfy the oscillator-type equation with a variable frequency g¨()(p,¯ t) + ω2(p,¯ t) g()(p,¯ t) = 0 . (3.38) Solutions of Eq. (3.38) for positive and negative frequencies are defined by their asymptotic behavior at t = t similarly to Eq. (3.7). 0 → −∞ The field operator in the in-state is defined as

3 (−) (+) + φ(x) = d p [ φp¯ (x) ap¯(t0) + φp¯ (x) bvp(t0) ] . (3.39) Z The diagonalization of the hamiltonian corresponding to the instantaneous stateis achieved by the transition to the quasiparticle representation. The Bogoliubov transformation for creation and annihilation operators of quasiparticles has the following form

+ ap¯(t) = αp¯(t) ap¯(t0) + βp¯(t) b−p¯(t0) , (3.40) + b−p¯(t) = α−p¯(t) bp¯(t0) + β−p¯(t) a−p¯(t0) with the condition α (t) 2 β (t) 2 = 1 . (3.41) | p¯ | − | p¯ | The derivation of the Bogoliubov coeﬃcients α and β is similar to that given in the previous subsection. We obtain the equations of motion for the coeﬃcients of the canonical transformation (3.40) as follows

ω˙ (p,¯ t) α˙ (t) = β∗(t) e2iΘ(p¯,t) , (3.42) p¯ 2ω(p¯, t) p¯ ω˙ (p,¯ t) β˙ (t) = α∗(t) e2iΘ(p¯,t) . (3.43) p¯ 2ω(p¯, t) p¯ Following the derivation for the case of fermion production, we arrive at the final result for the source term in the bosonic case t 0 ¯ eE(t)ε⊥ 0 eE(t )ε⊥ ¯ 0 0 (+)(P , t) = 2 dt 2 0 1 + 2F (P , t ) cos 2[Θ(p¯, t) Θ(p,¯ t )] , S 2ω (p,¯ t) Z−∞ ω (p,¯ t ) − (3.44) which differs from the fermion case just by the sign in front of the distribution function due to the different statistics of the produced particles.

3.3 Discussion of the source term

3.3.1 Properties of the source term We can combine the results for fermions (3.35) and for bosons (3.44) into a single kinetic equation

dF()(P¯, t) ∂F()(P¯, t) ∂F()(P¯, t) = + eE(t) = ()(P¯, t) . (3.45) dt ∂t ∂P3 S Here, the upper (lower) sign corresponds to the Bose-Einstein (Fermi-Dirac) statistics. Based on microscopic dynamics, these kinetic equations are exact within the approximation of a time-dependent homogeneous electric field and the neglect of collisions. The Schwinger source terms (3.35) and (3.44) are characterized by the following features: 1. The kinetic equations (3.45) are of non-Markovian type due to the explicit dependence of the source terms on the whole pre-history via the statistical factor 1 2F (P¯, t) for fermions or bosons, respectively. The memory effect is expected to lead to a modification of particle pair creation as compared to the (Markovian) low-density limit, where the statistical factor is absent. 2. The difference of the dynamical phases, Θ(p,¯ t) Θ(p¯, t0), under the integrals (3.35) and (3.44) generates high frequency oscillations.− 3. The appearance of such a source term leads to entropy production due to pair creation (see also Rau[6]) and therefore the time reversal symmetry should be violated, but it does not result in any monotonic entropy increase (in absence of collisions). 4. The source term and the distribution functions have a non-trivial momentum dependence resulting in the fact that particles are produced not only at rest as assumed in previous studies, e.g. Ref.[19]. 5. In the low density limit and in the simple case of a constant electric field we reproduce the pair production rate given by Schwinger’s formula e2E2 πm2 cl = lim (2π)−3g d3P (P¯, t) = exp . (3.46) S t→+∞ S 4π3 − eE Z | | As noted above, Rau[6] found that the source term has a non-Markovian behaviour by deriving the production rate within a projector method. In the limit of a constant field our results agree with Ref.[6]. In our approach the electric field is treated as a general time dependent field and hence there is no a priori limitation to constant fields. However our result allows to explore the influence of any time-dependent electric field on the pair creation process. It is important to note that in general this time dependence should be given by a selfconsistent solution of the coupled field equations, namely the Dirac (Klein-Gordon) equation and the Maxwell equation. This would incorporate back reactions as mentioned in the introduction. However, the solution of such a system of equations is beyond the scope of this work. Herein we will restrict ourselves to the study of some features of the new source term. Finally we remark that the source term is characterized by two time scales: the memory time ε τ ⊥ (3.47) mem ∼ eE and the production interval

τprod = 1/ < S() > , (3.48) with < S() > denoting the time averaged production rate. As long as E << 2 2 m /e < ε⊥/e, the particle creation process is Markovian: τmem << τprod. This results for constant ﬁelds agree with those of Rau[6].

30.0

20.0

10.0

ε 0.0 S/ -10.0

1.0

0.0 ε S/ -1.0

-2.0 -3.0 -1.0 1.0 3.0 ε p|| /

Figure 3.1: The pair production rate as a function of parallel momentum for a constant, strong field (upper plot: E˜0 = 1.5) and a weak field (lower plot: E˜0 = 0.5) at t˜= 0. 3.3.2 Numerical results In order to study the new source term, we consider two different cases, namely a constant field and a time dependent electric field. For the numerical evaluation we start with Eq. (3.35) assuming a dilute system, F = 0, and

S(p˜ , t˜) E˜(t˜) t˜ E˜(t˜0) ˜(p˜ , t˜) = k = dt˜0 cos 2[Θ(p˜ , t˜) Θ(p˜ , t˜0)] , k 2 ˜ 2 ˜0 k k S ε⊥ 2ω˜ (p˜k, t) Z−∞ ω˜ (p˜k, t ) − (3.49) where we have introduced dimensionless variables

˜ ˜ ˜ 2 ˜ E(t) = eE(t)/ε⊥ , t = tε⊥ , (3.50)

p˜k = pk/ε⊥ , ω˜ = ω/ε⊥ . (3.51) This notation is convenient to distinguish the weak field (E˜ < 1) and strong field (E˜ > 1) limits. A particularly simple result is obtained if we assume a constant field, A˜(t˜) = A(t˜)/ε⊥ = t˜E˜0/e , (3.52) where the electric field does not depend on time and the energy is given as

ω˜2(p˜ , t˜) = 1 + (p˜ E˜ t˜)2 . (3.53) 0 k k − 0 For the source term we obtain

˜2 t˜ t˜ E0 0 1 00 00 ˜(p˜k, t˜) = dt˜ cos 2 dt˜ ω˜0(p˜k, t˜ ) . (3.54) 2 ˜ 2 ˜0 ˜0 S 2ω˜0(p˜k, t) Z−∞ ω˜0(p˜k, t ) Zt In Fig. 3.1 we plot the particle production rate as function of the parallel momentum pk for a weak constant field and a strong field, respectively. The rates are normalized to be of the order of one for E˜0 = 0.5 at zero values of both momentum and time. The production rate is positive when particles are produced with positive momenta. Pairs with negative momenta are moving against the field and hence get annihilated, clearly to be seen in the negative production rate. These results mainly agree with those of Ref.[6], since no time dependence was considered. Note that using the prefactor and field strengths chosen by Rau, we reproduce exactly the results given in Ref.[6]. In considering the time dependence of the source term, we go beyond the analysis of Ref.[6]. In Fig. 3.2, we display the time dependence of the production rate at zero momentum. The maximum of the production rate is concentrated around zero and shows an oscillating behaviour for large times. Indeed it is possible to write Eq. (3.54) in terms of the Airy function because the constant field provides an analytical solution for the dynamical phase difference using the energy given in Eq. (3.53). The production of pairs for strong fields is larger 30.0

20.0

10.0

ε 0.0 S/ -10.0

1.0

0.0 ε S/ -1.0

-2.0 -3.0 2.0 7.0 tε

Figure 3.2: The pair production rate as a function of time for a constant, strong ﬁeld (upper plot: E˜0 = 1.5) and a weak ﬁeld (lower plot: E˜0 = 0.5) at zero parallel momentum.

Figure 3.3: The pair production rate, S˜(p˜k, t˜), as a function of time and parallel momentum for a time dependent weak electric field charactarized by E˜0 = 0.5, σ˜ = 1 and τ = 0 . All plotted values are dimensionless as described in the text. Figure 3.4: The pair production rate, S˜(p˜k, t˜), as a function of time and parallel momentum for a time dependent strong electric field charactarized by E˜0 = 1.5, σ˜ = 1 and τ = 0 . All plotted values are dimensionless as described in the text. than that of weak fields, and the typical time period of the oscillations is smaller.

The situation changes if we allow the electric field to be time dependent. We assume a Gaussian field at the dimensionsless time τ with the width σ˜ = σε⊥, −(t˜−τ)2/σ˜2 E˜(t˜) = E˜0e (3.55) and obtain √π A˜(t˜) = E˜0σ Erf[(t˜ τ)/σ˜] Erf[( τ t0)/σ˜] . (3.56) − 2 − − − − The occuring error function is defined as

2 z 2 Erf(z) = e−x dx . (3.57) √π Z0 Using this Ansatz for the field strength in Eq. (3.49) we obtain the numerical results plotted in Figs. 3.3 and 3.4. Therein all occuring values are dimensionsless. The electric field is non zero within a certain width σ˜ = 1 for τ = 0 around t˜ = 0. Therefore the oscillations for times beyond the time where the electrical field is finite are damped out. The pair production rate is peaked around small momenta for both weak and strong fields. For strong fields the distribution is shifted to positive momenta but remains still close to small parallel momenta. It is important to point out that the production of particles happens not only at rest (pk = 0) what is assumed in many works also addressing the back reaction problem, e.g.[19]. In contrary we find a non-trivial momentum dependence of the pair creation rate depending on the field strength and on time. 3.4 Summary

We have derived a quantum kinetic equation within a consistent field theoretical treatment which contains the creation of particle-antiparticle pairs in a time- dependent homogeneous electric field. For both fermions and bosons we obtain a source term providing a kinetic equation of non-Markovian character. The source term is characterized by a pair production rate that contains a time integration over the evolution of the distribution function and therefore involves memory effects. For the simple case of a constant electric field in low density limit and Marko- vian approximation, we analytically and numerically reproduce the results of earlier works[6]. Since our approach is not restricted to constant fields, we have explored the dependence of the source term on a time dependent (model) electric field with a Gaussian shape. Within these two different Ansätze for the field, we have performed investigations of the time structure of the source term. The production of pairs does not happen at rest only. We observe a non-trivial momentum dependence of the source term depending on the field strength and on time. The particle production source is dominated by two time scales: the memory time and the production time. The numerical results mainly show oscillations due to the dynamical phase and urge the need to include the Maxwell equation to determine the electric field by physical boundary conditions (back reactions), work in this direction is in progress. Furthermore, it would be of great interest to extend this approach to the QCD case to explore the impact of a non-Markovian source term on the pre-equilibrium physics in ultrarelativistic heavy-ion collisions. Bibliography

[1] N.B. Naroshni and A.I. Nikishov, Yad. Fiz. 11 (1970) 1072 (Sov. J. Nucl. Phys. 11 (1970) 596); V.S. Popov and M.S. Marinov, Yad. Fiz. 16 (1972) 809 (Sov. J. Nucl. Phys. 16 (1974) 449); V.S. Popov, Zh. Eksp. Teor. Fiz. 62 (1972) 1248 (Sov. Phys. JETP 35 (1972) 659); M.S. Marinov and V.S. Popov, Fort. d. Phys. 25 (1977) 373.

[2] A. Bialas and W. Czyz,˙ Phys. Rev. D 30 (1984) 2371; 31 (1985) 198; Z. Phys. C 28 (1985) 255; Nucl. Phys. B 267 (1985) 242; Acta Phys. Pol. B17 (1986) 635.

[3] K. Kajantie and T. Matsui, Phys. Lett. B 164 (1985) 373.

[4] G. Gatoﬀ, A.K. Kerman, and T. Matsui, Phys. Rev. D 36 (1987) 114.

[5] F. Cooper, J.M. Eisenberg, Y. Kluger, E. Mottola, and B. Svetitsky, Phys. Rev. D 48 (1993) 190.

[6] J. Rau, Phys.Rev. D 50 (1994) 6911.

[7] R.S. Bhalerao and V. Ravishankar, Phys.Lett. B 409 (1997) 38.

[8] W. Greiner, B. Muller,¨ and J. Rafelski, Quantum Electrodynamics of Strong Fields (Springer-Verlag, Berlin, 1985).

[9] A.A. Grib, S.G. Mamaev, and V.M. Mostepanenko, Vacuum quantum eﬀects in strong external ﬁelds, (Atomizdat, Moscow, 1988).

[10] S. Nussinov, Phys. Rev. Lett. 34 (1975) 1286.

[11] B. Andersson, G. Gustafson, G. Ingelman, and T. Sjostrand, Phys. Rep. 97 (1993) 31.

[12] N.K. Glendenning and T. Matsui, Phys. Rev. D 28 (1983) 2890.

[13] F. Sauter, Z. Phys. 69 (1931) 742.

[14] W. Heisenberg and H. Euler, Z. Phys. 98 (1936) 714.

125 [15] J. Schwinger, Phys. Rev. 82 (1951) 664.

[16] F. Cooper and E. Mottola, Phys. Rev. D 40 (1989) 456.

[17] Y. Kluger, J.M. Eisenberg, B. Svetitsky, F. Cooper, and E. Mottola, Phys. Rev. Lett. 67 (1991) 2427.

[18] Y. Kluger, J.M. Eisenberg, B. Svetitsky, F. Cooper, and E. Mottola, Phys. Rev. D 45 (1992) 4659.

[19] Y. Kluger, J.M. Eisenberg, and B. Svetitsky, Int. J. Mod. Phys. E 2 (1993) 333 (here further references may be found).

[20] F. Cooper, S. Habib, Y. Kluger, E. Mottola, J.P. Paz, and P.R. Anderson, Phys. Rev. D 50 (1994) 2848.

[21] J.M. Eisenberg, Phys. Rev. D 51 (1995) 1938.

[22] Ch. Best and J.M. Eisenberg, Phys. Rev. D 47 (1993) 4639.

[23] J. Rau and B. Muller,¨ Phys. Rep. 272 (1996) 1.

[24] J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964). Chapter 4

Problem sets - statistical QFT

4.1 Partition function - Introduction

1.1. Construct the statistical operator ρ for the the grand canonical ensemble from the principle of maximum information entropy (Jaynes) S = Tr ρ ln ρ Max. − { } −→ 1.2. The Hamiltonian form of the Path Integral representation of the partition function for a scalar ﬁeld theory is given by (τ = it)

β ∂φ Z(T, V, µ) = π φ exp dτ d3x iπ (π, φ) µ , Z D Zφ(0)=φ(β) D (Z0 ZV " ∂τ − H − N #) 1 1 (π, φ) = π2 + m2φ2 + U(φ) H 2 2 Find the Lagrangian formulation by evaluating the path integral over the ﬁeld momentum π in the case µ = 0.

1.3. The pressure for a noninteracting scalar ﬁeld is

1 1 2 2 2 p(T, V, µ) = ln Z0(T, V, µ) = ln[β (ωn + ω (p))] . βV 2βV n p X X

Perform the summation over Matsubara frequencies ωn = 2πnT, β = 1/T explicitely!

127 4.2 Partition function - Fermi systems

2.1 The partition function of a Fermi-gas is given by Z = det D, where D = iβγ0[γ0( iω + µ) ~γ p~ m]. Use the representation of the 4x4 − − n − · − Dirac gamma matrices and evaluate the determinant to show the result:

4 2 2 2 det D = β [(ωn + iµ) + ω (p)] , Dirac

where ω2(p) = p2 + m2.

2.2 The pressure of a noninteracting Fermi-gas is given by (β = 1/T )

1 2 2 2 2 p(T, V, µ) = ln Z0(T, V, µ) = ln β [(ωn + iµ) + ω (p)] . βV βV n p { } X X

perform the summation over the Matsubara- Frequencies ωn = (2n + 1)πT , and show the result

3 d p −β(ω(p)−µ) −β(ω(p)+µ) p(T, V, µ) = 2 3 ω(p) + T ln[1 + e ] + T ln[1 + e ] Z (2π) { }

2.3 Evaluate the pressure of a massless ideal Fermi-gas (e.g. Neutrinos) by performing the momentum integration with the result:

µ4 µ2T 2 7π2T 4 p = Ω = + + . − 12π2 6 180 Use the relation ε = T s p + µn with s = ∂Ω/∂T and n = ∂Ω/∂µ to prove − ε = 3p for this case.

2.4 The pressure of a cold, interacting neutron gas in the relativistic mean-ﬁeld (Walecka) model is given by

∗ 1 2 ∗ 3 ∗ 2 ∗ ∗ 4 EF + pF P = 2 EF pF mn EF pF + mn ln ( ∗ ) 8π 3 − mn 2 2 1 gω 2 1 gσ 2 + 2 n 2 ns 2 mω ! − 2 mσ ! p3 g2 n = F , E = p2 + m∗ 2 , m∗ = m σ n 3π2 F ∗ F n n n − m2 s q σ ! ∗ ∗ mn ∗ ∗ 2 EF + pF ns = 2 EF pF mn ln ( ∗ ) . 2π − mn Show that this system of equations can be given a closed form! Derive the expression for the energy density! 4.3 Partition function - Quark matter

The grand canonical thermodynamical potential for quark matter within the nonlocal chiral quark model is given by

2 2 ∞ φ φ0 6 2 Ω(φ; µ, T ) = − 2 dq q Eφ(q) Eφ0 (q) 4 G1 − π Z0 − E (q) µ E (q) + µ +T ln 1 + exp φ − + T ln 1 + exp φ (4.1). " − T !# " − T !#

2 2 The dispersion relation is Eφ(q) = q + (m + φ g(q)) where the mass gap (order parameter, chiral gap) φ = φ(T,qµ) depends on temperature T and chemical potential µ and φ0 = φ(0, 0) is the value of the order parameter in the vacuum 2 at T = µ = 0. The coupling constant is G1 = 3.761/Λ where Λ = 1.025 GeV is the range of the separable interaction with the Gaussian formfactor g(q) = exp( q2/Λ2) and m = 2.41 MeV is the current quark mass. − 3.1 Find the T = 0 limit of Eq. (1) and derive the condition for the minimum of the thermodynamical potential with respect to a variation of the order parameter φ (gap equation)! Solve this gap equation for µ = 0 in order to ﬁnd the value of φ0! 3.2 Solve the gap equation φ(µ) for 0 µ 500 MeV and ﬁnd the critical ≤ ≤ chemical potential µc where the mass gap jumps to a very low value (chiral symmetry restoration).

3.3 Insert the solution of the gap equation φ(µ) in Eq. (1) and evaluate the pressure p(µ) = Ω(φ; µ, T = 0) by momentum integration for 0 µ 500 − ≤ ≤ MeV.

3.4 Perform a ﬁt of the quark matter pressure for µ > µc to a bag model

µ4 p (µ) = B (4.2) Bag 2π2 − and discuss whether a stable quark matter core inside a neutron star is possible by comparing with solutions from the Diploma thesis by Thomas Kl¨ahn!

3.5 Perform the same steps for Lorentzian and cut-oﬀ (Nambu–Jona-Lasinio) form factor models! The parameters can be taken from the reference http://arXiv.org/abs/hep-ph/0602238. What is the inﬂuence of the form of the interaction on the structure of a neutron star with quark matter core? Chapter 5

Projects

5.1 Symmetry breaking. Goldstone-Theorem. Higgs-Eﬀect

5.1.1 Spontaneous symmetry breaking: Complex scalar ﬁeld Spontaneous symmetry breaking is a very general phenomenon in nature, best explained on the example of a simple model, the complex scalar ﬁeld with negative mass-squared = ∂ Φ∗∂µΦ m2Φ∗Φ λ(Φ∗Φ)2 (5.1) L µ − − The potential energy density corresponding to this problem depends on the values of the parameters λ and m. Because of stability, λ has to be positive. Then the shape of the potential depends on the sign of the mass term. For positive m2 it is given in the upper graph of Fig. 5.1.1, while for m2 = c2 < 0 it has the shape − of a Mexican hat (bottom of a wine-bottle), see lower graph in Fig. 5.1.1. The Lagrangian (5.1) has a global U(1) symmetry, i.e. the replacement Φ Φeiα with a real phase α, independent on the location x, leaves it invariant. But→ any ground state at the bottom of the potential given by

Φ0 = Φ1,0 + iΦ2,0 (5.2) will break this rotational symmetry, see lower graph in Fig. 5.1.1.

1. Find the ground state of the system at T = 0 and the masses m1 and m2 of the elementary excitations in the mean field approximation! 2. Evaluate the thermodynamical potential (and the pressure) at finite temperature! 3. Show that the symmetry gets restored at finite temperature in a second order phase transition!

130 (a)

(y,π)

(x,σ)

(b) (y,π)

(x,σ)

Figure 5.1: Potential with spontaneous symmetry breaking.

4. Formulate and prove the Goldstone theorem!

5.1.2 Electroweak symmetry breaking: Higgs mechanism Modern gauge theories attempt to unify forces of nature, such as weak and electromagnetic forces are unified in the Weinberg-Salam model. In this model, a scalar field (Higgs field) breaks spontaneously the gauge symmetry and gives mass to the vector bosons (observed as W and Z0). Describe this phenomenon in the high-temperature mean-field approximation to the Lagrangian 1 = (∂µ ie Aµ)Φ∗(∂ + ie A )Φ + c2Φ∗Φ λ(Φ∗Φ)2 F µν F , (5.3) L − µ µ − − 4 µν where F = ∂ A ∂ A . µν µ ν − ν µ Bibliography

[1] J.I. Kapusta, Finite-temperature ﬁeld theory, Cambridge University Press (1989), chapter 7 and chapter 9.

132