Quantum Field Theory II

Quantum ﬁeld theory II - Aspects of QCD

Prof. Dr. David Blaschke

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

Winter Semester 2013/2014

1 1 Introduction

The aim of this series of lectures 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 neu- tron 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 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 have also been performed at the Fermilab National Accelerator Facility (FermiLab), in Batavia, Illinois, the Stanford Linear Accel- erator Center (SLAC National Accelerator Laboratory), in Stanford, California, and at the Deutsches Elektronensynchrotron (DESY) in Hamburg. These facili- ties use high-energy probes and/or large momentum-transfer processes to explore the transition from the nonperturbative to the perturbative domain in QCD. 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.

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

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

1fm = 1/(0.197327 GeV) = 5.06773 GeV−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 , (3) µ 0 1 2 3 ≡ − − − µν where the metric tensor is 10 0 0  0 1 0 0  gµν = − . (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). (5) and (p,k)= p kν = E E ~p~k . (6) ν p k − Likewise, (x, p)= tE ~x~p. (7) − The momentum operator ∂ ∂ 1 pµ := i =(i , ~ ) =: i µ (8) ∂xµ ∂t i ∇ ∇ transforms as a contravariant four-vector, and I denote the four-vector analogue of the Laplacian as ∂ ∂ 2 pµp ∂ := µ = µ . (9) − ∂xµ ∂x The contravariant four-vector associated with the electromagnetic ﬁeld is

Aµ(x)=(φ(x), A~(x)) (10) with the electric and magnetic ﬁeld strengths obtained from

F µν = ∂ν Aµ ∂µAν . (11) − For example, ∂ E = F 0i; E~ = ~ φ A.~ (12) i −∇ − ∂t Similarly, Bi = εijkF jk; j, k =1, 2, 3. Analogous deﬁnitions hold in QCD for the chromomagnetic ﬁeld strengths.

2.2 Dirac Matrices The Dirac matrices are indispensable in a manifestly Poincarécovariant 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 (13) { } and one common 4 4 representation is [each entry represents a 2 2 matrix] × × 1 0 0 ~σ γ0 = , ~γ = , (14) 0 1 ~σ 0 " − # " − # where ~σ are the usual Pauli Matrices: 0 1 0 i 1 0 σ1 = , σ2 = , σ3 = , (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 , (16) 2 5 and I note that i γ5σµν = εµνρσσ , (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 = . (18) " 1 0 # Furthermore, γ ,γµ =0 , γ† = γ . (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,~ (20) µ − γµp =: p/ = γ0E ~γ~p, (21) µ − ∂ ∂ γµp =: i / i∂/ = iγ0 + i~γ ~ = iγµ . (22) µ ∇ ≡ ∂t ∇ ∂xµ (23) The following identities are important in evaluating the cross sections for decay and scattering processes:

trγ5 = 0 , (24) tr1 = 4 , (25) tra/b/ = 4(a, b) , (26) tra/ a/ a/ a/ = 4[(a , a )(a , a ) (a , a )(a , a )+(a , a )(a , a )] , (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 , (28) trγ5a/b/ = 0 , (29) µ ν ρ σ trγ5a/1a/2a/3a/4 = 4iεµνρσa1 a2a3a4 , (30) µ γµa/γ = 2a/, (31) µ − γµa/b/γ = 4(a, b) , (32) γ a/b/c/γµ = 2c/b/a/. (33) µ − (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 (35) 1 = a b tr[γµγν + γνγµ] (36) µ ν 2 1 = a b tr[2gµν1] (37) µ ν 2 1 = a b 2gµν4=4(a, b) . (38) µ ν 2 (39) 3 Dirac Equation

The unification of special relativity (Poincarécovariance) 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 Poincarégroup 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 p′, 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 , (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 , (41) − − which is obtained, as usual, via “minimal substitution:” pµ pµ eAµ in Eq. (40). These equations have a manifestly covariant appearanc→e but− proving their covariance requires the development of a representation of Lorentz transformations on spinors and Dirac matrices:

ψ′(x′) = S(Λ) ψ(x) , (42) ν µ −1 ν Λ µγ = S (Λ)γ S(Λ) , (43) i S(Λ) = exp[ σ ωµν] , (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].)

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) , (45) into Eq. (40) and obtains

(k/ m) u(k)=0 , (k/ + m) v(k)=0 . (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 . (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        (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 , (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) =   , (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) = ; (51) 0 ! 1 ! and, for negative energy, σ k · χα(m,~0) (α) k/ + m (α)  2m(m + E)  v (k) = − v (m,~0) = , (52) 1/2 2m(m + E)  qE + m   α ~   χ (m, 0)  q  2m    with χ(α) obvious analogues of φ(α) in Eq. (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. (46), and, for the ho- mogeneous 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 . (53) This equation yields the following free particle spinors in momentum space k/ + m u¯(α)(k) =u ¯(α)(m,~0) (54) 2m(m + E) q k/ + m v¯(α)(k) =v ¯(α)(m,~0) − , (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 . (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) . (57) + ⊗ αX=1,2

It is plain from the orthonormality relations, Eqs. (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 . (58) Now, since 1 0 1 + γ0 u(α)(m,~0) u¯(α)(m,~0) = = , (59) ⊗ 0 0 ! 2 αX=1,2 then 1 1 + γ0 Λ (k) = (k/ + m) (k/ + m) . (60) + 2m(m + E) 2 Noting that for k2 = m2; i.e., on shell,

(k/ + m) γ0 (k/ + m) = 2E (k/ + m) , (61) (k/ + m)(k/ + m) = 2m (k/ + m) , (62) one ﬁnally arrives at the simple closed form:

k/ + m Λ (k)= . (63) + 2 m The negative energy projection operator is

k/ + m Λ (k) := v(α)(k) v¯(α)(k)= − . (64) − − ⊗ 2 m αX=1,2 The projection operators have the following important properties:

2 Λ±(k) = Λ±(k) , (65)

trΛ±(k) = 2 , (66)

Λ+(k)+Λ−(k) = 1 . (67)

Such properties are a characteristic of projection operators. 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 (41): [i∂/ eA/(x) m] ψ(x)=0 , (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

′ ′ 4 ′ [i∂/ ′ eA/(x ) m] S(x , x)= 1 δ (x x) . (69) x − − − It is immediately apparent that if, at a given spacetime point x, ψ(x) is a solution of Eq. (68), then ψ(x′) := d4xS(x′, x) ψ(x) (70) Z is a solution of ′ ′ [i∂/ ′ eA/(x ) m] ψ(x )=0; (71) x − − i.e., S(x′, x) has propagated the solution at x to the point x′. 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 x′ is obtained by considering ψ(x) as a source of spherical waves that propagate outward from x. The amplitude of the wave at x′ is proportional to the amplitude of the original wave, ψ(x), and the constant of proportionality is the propagator (Green function), S(x′, x). The total amplitude of the wave at x′ is the sum over all the points on the wavefront; i.e., Eq. (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(x′, x) is impossible for all but the simplest cases.

4.1 Free-Fermion Propagator In the absence of an external ﬁeld Eq. (69) becomes

′ 4 ′ [i∂/ ′ m] S(x , x)= 1 δ (x x) . (72) x − − Assume a solution of the form:

4 ′ ′ d p −i(p,x′−x) S0(x , x)= S0(x x)= 4 e S0(p) , (73) − Z (2π) so that substituting yields p/ + m (p/ m) S (p)= 1 ; i.e., S (p)= . (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. (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) . (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)= Cγ v(p) e = C v¯(p) e , (76) where C = iγ2γ0 and ( )T denotes matrix transpose. (This deﬁnes the operation · ′ of charge conjugation.) It is thus evident that our physically sensible S0(x x) ′ − can only contain positive-frequency components for x0 x0 > 0. One can ensure this via a small modiﬁcation of the− denominator of Eq. (74):

p/ + m p/ + m S (p)= , (77) 0 p2 m2 → p2 m2 + iη − − with η 0+ at the end of all calculations. Inserting this form in Eq. (73) is equivalent→ 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 ′ d p i~p·(~x′−~x) 1 S0(x x)= 3 e − Z (2π) 2 ω(~p) 0 ∞ dp 0 ′ p/ + m 0 ′ p/ + m e−ip (x0−x0) e−ip (x0−x0) , × −∞ 2π " p0 ω(~p)+ iη − p0 + ω(~p) iη # Z − − (78) where the energy ω(~p) = √~p2 + 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 x′ x > 0 we can evaluate the p0 integral by considering a contour closed by 0 − 0 a semicircle of radius R in the lower half of the complex p0-plane: the integrand vanishes exponentially→ ∞ along that arc, where p0 = iy, y > 0, because ′ ′ − ( i)( iy)(x0 x0)= y (x0 x0) < 0. The closed contour is oriented clockwise so− that− − − − 0 ∞ dp 0 ′ p/ + m 0 ′ e−ip (x0−x0) = ( ) i e−ip (x0−x0)(p/ + m) 0 + p0=ω(~p)−iη+ Z−∞ 2π p ω(~p)+ iη − − ′ −iω(~p)(x −x0) 0 = i e 0 (γ ω(~p) γ ~p + m) − ′ − · = i e−iω(~p)(x0−x0) 2m Λ (p) . (79) − + For x′ x < 0 the contour must be closed in the upper half plane but therein 0 − 0 the integrand is analytic and hence the result is zero. Thus 0 ∞ dp 0 ′ p/ + m ′ −ip (x0−x0) ′ −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 x0 > 0 the contour must be closed in the lower half − ′ 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 integ−rand is obviously zero on the arc at infinity. The contour is oriented anticlockwise so that 0 ∞ dp 0 ′ p/ + m 0 ′ −ip (x0−x0) ′ −ip (x0−x) e = i θ(x0 x ) e (p/ + m) 0 + 0 p0=−ω(~p)+iη+ Z−∞ 2π p + ω(~p) iη − − ′ ′ +iω(~p)(x0−x) 0 = i θ(x0 x0) e ( γ ω( ~p) γ ~p + m) − ′ − − · = i θ(x x′ ) e+iω(~p)(x0−x) 2m Λ ( ~p) . (80) 0 − 0 − − Putting these results together, changing variables ~p ~p in Eq. (80), we → − have 3 ′ d p m ′ −i(˜p,x′−x) S0(x x) = i 3 θ(x0 x0) e Λ+(~p) − − Z (2π) ω(~p) − h ′ + θ(x x′ ) ei(˜p,x −x)Λ (~p) , (81) 0 − 0 − µ i [(˜p )=(ω(~p), ~p)] which, using Eqs. (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 ′ p/ + m ∞ dp 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 ′ 1 −ip (x0−x) = e 2mΛ+(~p) , −∞ 2π p0 ω(~p)+ iη+ Z − (82) and similarly

0 0 0 ∞ dp 0 ′ p/ + m ∞ dp 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 ′ 1 −ip (x0−x) = e 2mΛ−( ~p) , −∞ 2π − p0 + ω(~p) iη+ Z − (83) so that Eq. (78) can be rewritten

4 ′ d p −i(p,x′−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 − − (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. (77): m 1 1 S0(p)= Λ+(~p) Λ−( ~p) , (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.

4.2 Green Function for the Interacting Theory We now return to Eq. (69):

′ ′ 4 ′ [i∂/ ′ eA/(x ) m] S(x , x)= 1 δ (x x) , (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:

′ 4 ′ ′ ′ [i∂/ ′ m] S(x , x)= 1 δ (x x)+ eA/(x ) S(x , x) , (87) x − − which, it is easily seen by substitution, is solved by

′ ′ 4 ′ S(x , x) = S0(x x)+ e d yS0(x y)A/(y) S(y, x) (88) − Z − ′ 4 ′ = S0(x x)+ e d yS0(x y)A/(y) S0(y x) − Z − − 2 4 4 ′ +e d y1 d y2 S0(x y1)A/(y1) S0(y1 y2)A/(y2) S0(y2 x)+ ... Z Z − − − (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 (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 x′; i.e., forward in time;

3. Annihilates this fermion at x′.

The process can equally well be viewed as

1. The creation of a negative energy antifermion (fermion) at spacetime point x′;

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.

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 , (90)

in forms analogous to Eqs. (81), (84). 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 ′ ′ ′ ′ 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 ′ ′ ′ ′ 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 )] ǫH(p(t ), j j ) (91), ×  α j α j − α j−1 − j 2  jX=1   ′ ′ where tj = t + jǫ, ǫ = (t t)/(n + 1), t0 = t, tn+1 = t . A compact notation is commonly introduced to represent− this expression:

t′ q′t′ qt J = [dq] [dp] exp i dτ [p(τ)q ˙(τ) H(τ)+ J(τ)q(τ)] , (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

′ ′ q t T Q(t1) ...Q(tn) qt = h | { }| i ′ t [dq] [dp] q(t1) q(t2) ...q(tn) exp i dτ [p(τ)q ˙(τ) H(τ)] . (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 < ′ tf < t , then

′ ′ J ′ ′ J q t qt = dqidqf q t qf tf qf tf qiti qiti qt . (94) h | i Z h | ih | i h | i Alternatively one can introduce a complete set of energy eigenstates to resolve the Hamiltonian and write

′ ′ ′ ′ −iEn(t −t) q t qt = q φn e φn q (95) h | i n h | i h | i X t′ i ′ t →+−i∞ ′ −iE0(t −t) →= ∞ q φ e φ q ; (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. (94), (96) that

′ ′ q t qt J lim ′ h | i = dq dq φ q t q t q t q t φ =: W [J]; ′ −iE0(t −t) ′ i f 0 f f f f i i i i 0 t i e q φ0 φ0 q h | ih | i h | i t →+−i∞ h | ih | i Z → ∞ (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 | ih | { }| ih | i 1 1 J=0 Z

(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) + ǫ(t); expand the → functional in ǫ(t); and identify the leading order coefficient in the expansion. Thus for ′ ′ n Hn[J]= dt J(t ) (99) Z

′ ′ n ′ ′ ′ n ′ ′ n δHn[J] = δ dt J(t ) = dt [J(t )+ ǫ(t )] dt J(t ) (100) Z Z − Z = dt′ nJ(t′)n−1 ǫ(t′) + [...]. (101) Z So that taking the limit δJ(t′)= ǫ(t′) δ(t t′) one obtains → − δH [J] n = nJ(t)n−1 . (102) δJ(t) The limit procedure just described corresponds to defining δJ(t) = δ(t t′) . (103) δJ(t′) − 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. (96), (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, which→ proceed− via Monte-Carlo→ methods; i.e., probability sampling, which is only effective when the integrand is positive definite. 6 Functional Integral in Quantum Field Theory

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.

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 ǫ3 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 xφ(t, x) , q˙α(t) := φα(t)= 3 d x ; (104) ǫ ZVα ǫ 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) ǫ3 L (φ˙ (t),φ (t),φ (t)) , (105) → α α α α±s Z αX=1 where the dependence on φα±s(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) := = ǫ =: ǫ πα(t) , (106) ∂φ˙α(t) ∂φ˙α(t) the Hamiltonian is obtained as

3 H= pα(t)q ˙α(t) L(t) =: ǫ Hα , (107) α − α X X H (π (t),φ (t),φ (t)) = π (t) φ˙ (t) L . (108) α α α α±s α α − α The ﬁeld theoretical equivalent of the quantum mechanical transition amplitude, Eq. (91), can now be written

N n n dπ (t ) n+1 φ (t ) φ (t ) lim dφ (t ) ǫ3 α i exp i ǫ ǫ3 π (t ) α j α j−1 + α i α j − n→∞,ǫ→0 2π  α ( ǫ αY=1 Z iY=1 Z iY=1 jX=1 X  φα(tj)+ φα(tj−1) φα±s(tj)+ φα±s(tj−1) Hα πα(tj), , − 2 2 !  t′ ∂φ(τ, ~x)  =: [ φ] [ π] exp i dτ d3x π(τ, ~x) H(τ, ~x) , (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 (109) is the transition amplitude from an initial field ′ configuration φα(t0) := φα(t) to a final configuration φα(tn+1) := φα(t ).

6.1.1 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 (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 ˆ ˆ ˆ 1 δ W [J] 0˜ T φ(x1)φ(x2) ... φ(xn) 0˜ = h | { }| i =: G(x1, x2,...,xn) , in δJ(x )δJ(x ) ...δJ(x ) 0˜ 0˜ 1 2 n J=0 h | i (111) where 0˜ is the physical vacuum. G(x , x ,...,x ) is the complete n-point Green | i 1 2 n function for 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. (111) was first observed by Schwinger [6] and does not rely on the functional formula for W [J], Eq. (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.

6.1.2 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) , (112) − δJ(x )δJ(x ) ...δJ(x ) 1 2 n J=0

where Z[J], deﬁned via W [J] =: exp iZ[J] , (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 ) . (114) − 1 2 1 2 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)] . (115) 2 ∇ In this case Eq. (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 , (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. (115), the generating functional can be written N 1 W [J] = [ φ] exp i d4x [L(x)+ iηφ2(x)+ J(x)φ(x)] . (117) D 2 N Z Z Recall now that the classical Lagrangian density for a scalar ﬁeld is

L(x) = L0(x)+ LI (x) , (118) 1 L (x) = [∂ φ(x) ∂µφ(x) m2φ2(x)] , (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. (115) and so Eq. (117) can be used to define the quantum field theory.

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 1 W [J]= [ φ] exp i d4x [∂ φ(x) ∂µφ(x) m2φ2(x)+ iηφ2(x)] + J(x)φ(x) . 0 ¯ D 2 µ − N Z Z (120) The explicit meaning of Eq. (120) is

1 4 4 1 4 W0[J] = lim dφα exp i ǫ ǫ φα Kαβ φβ + ǫ Jαφα , (121) + ¯  2  ǫ→0 α α α N Z Y  X Xβ X  4 where α, β label spacetime hypercubes of volume ǫ and Kαβ is any matrix that satisfies 2 2 4 lim Kαβ = [ ∂ m + iη]δ (x y) , (122) ǫ→0+ − − − + + + where α ǫ→0 x, β ǫ→0 y and ǫ4 ǫ→0 d4x; i.e., K is any matrix whose → → α → αβ continuum limit is the inverse ofP the 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 . (123) √detA {2 } √detA 2 i,jX=1 Hence Eq. (121) yields

1 1 1 4 4 1 −1 W0[J] = lim exp ǫ ǫ Jα (K )αβ Jβ , (124) + ¯ ′ 2 8 ǫ→0 √detA { α iǫ } N X Xβ where, obviously, the matrix inverse is deﬁned via

−1 Kαγ (K )γβ = δαβ . (125) γ X Almost as obviously, consistency of limits requires 1 lim δ = δ4(x y) , lim ǫ4 = d4x , (126) + 4 αβ + ǫ→0 ǫ − ǫ→0 α X Z so that, with 1 −1 O(x, y) := lim (K )αβ , (127) ǫ→0+ ǫ8 the continuum limit of Eq. (125) can be understood as follows: 1 1 lim ǫ4 K (K−1) = lim δ + αγ 8 γβ + 4 αβ ǫ→0 γ ǫ ǫ→0 ǫ 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) . (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) . (129) 0 − (2π)4 q2 m2 + iη Z − (NB. This makes plain the fundamental role of the “iη+” prescription in Eq. (77): it ensures convergence of the expression deﬁning the functional integral.) Putting this all together, the continuum limit of Eq. (124) is

1 i W [J]= exp d4x d4yJ(x) ∆ (x y) J(y) . (130) ˆ −2 0 − N Z Z 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)] (131) " Z i δJ(x)!# Z D Z 4 1 δ i 4 4 = exp i d x LI exp 2 d x d yJ(x) ∆0(x y) J(y) , " Z i δJ(x)!# − Z Z − (132) where ∞ n n 4 1 δ i 1 δ exp i d x LI := LI . (133) " i δJ(x) !# n! " i δJ(x)!# Z nX=0 Equation (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) . (134) I −4! The generating functional yields

W [0] = 4 λ δ i 1 i d4x exp d4u d4vJ(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)] . (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)] . (136) 2 Z − − Using the deﬁnition, Eq. (111), and restoring the normalisation,

1 1 δ2W [J] G(x1, x2) = 2 (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 x2)](138), − − 2 Z − − where I have used a + λb = a + λ(b ac)+O(λ2) . (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 (140) 0˜ 0˜ | i δJ(x) J=0 h | i so that the ﬁeld does not have a nonzero vacuum expectation value. This is the simplest demonstration of the fact that dynamical symmetry breaking is a phenomenon inaccessible in perturbation theory.

6.5 Exercises

1. Repeat the derivation of Eq. (114) for Gc(x1, x2, x3). 2. Prove Eq. (131).

3. Derive Eq. (136).

4. Prove Eq. (140). 7 Functional Integral for Fermions

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. (141) { i j} 2 It is clear from Eq. (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 ; (142)

N i.e., GN is a 2 -dimensional vector space. (NB. One can always choose the p- degree monomial in Eq. (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 (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 ...θip−(144)1 , ∂θs − − ← ∂ p−1 θi1 θi2 θip = δsip θi1 ...θip−1 δsip−1 θi1 ...θip−2 θip + ... +( ) δsi1 θi2 θ(145)ip . ∂θs − − The operation on a general element, f(θ) G , is easily obtained. It is also ∈ N obvious that ∂ ∂ ∂ ∂ f(θ)= f(θ) . (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. (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. (148) Z Z For example, it is straightforward to prove, using Eq. (143),

dθN ...dθ1 f(θ)= N! fN (1, 2,...,N) . (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 . (150) Xi=j With the definition N −1 dθi = (K )ji dξj (151) jX=1 one guarantees

dθi θj = δij = dξi ξj . (152) Z Z It follows immediately that

θ1θ2 ...θN = (det K) ξ1ξ2 ...ξN (153) −1 dθN dθN−1 ...dθ1 = (det K ) dξN dξN−1 ...dξ1 , (154) and hence

−1 dθN ...dθ1 f(θ) = (det K ) dξN ...dξ1 f(θ(ξ)) . (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 , (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 (157) Xi,j Xj,i use sym. = θjAijθi (158) Xi,j anticom= . θ A θ . (159) − i ij j Xi,j · t .. θiAijθj = 0for A = A . (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.˜ (161)  2     0 0 λ2 0 ...   −   ......      N Consequently, applying the linear transformation θi = i=1 Sij ξj and using Eq. (155), we obtain P

N I = dξ ...dξ exp ξ A˜ ξ . (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  (163) i.e., since det A = det A˜, I = √det 2A,. (164) Equation (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 (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. (148) it follows that

N dθ¯ dθ ... dθ dθ exp θ¯ B θ = det B, (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. (123):

N N 1 dxi exp π xi Aij xj = . (167) RN {− } √det A Z iY=1 i,jX=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 , (168) nX=0 where θn are Grassmann variables. Clearly

θ(x), θ(y) =0 . (169) { } The elements θ(x) are considered to be the generators of the “Grassman algebra” G and, in complete analogy with Eq. (143), any element of G can be uniquely written as

∞ f = dx1dx2 ...dxN θ(x1)θ(x2) ...θ(xN ) fn(x1, x2,...,xN ) , (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) , (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 ) , (172) − n 1 n−1 − − − 1 2 n cf. Eqs. (144), (145). Finally, we can extend the deﬁnition of integration. Denoting

[ θ(x)] := lim dθN ...dθ2dθ1 , (173) D N→∞ consider the “standard” Gaussian integral

I := [ θ(x)] exp dxdy θ(x)A(x, y)θ(y) (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) , (175) Z then N θiAij θj I = lim dθN ...dθ2dθ1 e i=1 (176) N→∞ Z P so that, using Eq. (164),

I = lim det 2AN , (177) N→∞ q where, obviously, A is the N N matrix in Eq. (175). This provides a deﬁnition N × for the formal result:

I = [ θ(x)] exp dxdy θ(x)A(x, y)θ(y) = √Det2A, (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 u′ = 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 dθ¯N dθN ...dθ¯2dθ2 dθ¯1dθ1 , (179) D D N→∞ one arrives immediately at a generalisation of Eq. (166)

[ θ¯(x)][ θ(x)] exp dxdy θ¯(x) B(x, y) θ(x) = DetB, (180) Z D D − Z which is also a definition. The relation ln det B = tr ln B, (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 . (182) { } I note that for an integral operator O(x, y)

Tr O(x, y) := d4x tr O(x, x) , (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 + [...] , (184) then

4 4 f[O(x, y)] = f0 δ (x y)+ f1 O(x, y)+ f2 d wO(x, w)O(w,y) + [...] . (185) − Z 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) . (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 (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-freedomQ to 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. (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) (188) − − and observe that the solution of

d4wO(x, w) P (w,y)= I δ4(x y) (189) Z − i.e., the inverse of the operator O(x, y) is (see Eq. 72) precisely the free-fermion propagator: P (x, y)= S (x y) . (190) 0 − Hence I can rewrite Eq. (187) in the form

4 4 ′ ′ 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 (191) i where

ψ¯′(x) := ψ¯(x)+ d4w ξ¯(w) S (w x) , ψ(x) := ψ(x)+ d4wS (x w) ξ(w) . 0 − 0 − Z Z (192) Clearly, ψ¯′(x) and ψ′(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¯′(x)][ ψ′(x)] exp i d4xd4y ψ¯′(x)O(x, y)ψ′(y) (193) × Z D D Z −1 4 4 ¯ = det[ iS0 (x y)] exp i d xd y ξ(x) S0(x y) ξ(y) (194) − − − Z − 1 = exp i d4xd4y ξ¯(x) S (x y) ξ(y) , (195) ψ − 0 − N′ Z where ψ := det[iS (x y)]. Clearly. N0 0 − ψ W [ξ,ξ¯ ] =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 = iS (x y); (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¯ ] . (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 . (199) − Z (2π) Then, for f as in Eq. (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 . (200) Z (2π)

ψ I now apply this to 0 := Det[iS0(x y)] and observe that Eq. (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 )= , (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) , (202) − Z − F − with ∆ (x y) given in Eq. (129) and 0 − 4 d p p/ −i(p,x−y) (x y)= 4 1+ e . (203) F − Z (2π) " m # It follows that

TrLn iS (x y) = Tr Ln i m∆ (x y)+Ln δ4(x y)+ (x y) . (204) 0 − 0 − − F − n h io Using Eqs. (183), (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 (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) , (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 ) , (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.)

7.4 Exercises 1. Verify Eq. (149).

2. Verify Eqs. (153), (154).

3. Verify Eqs. (163), (164).

4. Verify Eqs. (166).

5. Verify Eq. (181).

6. Verify Eq. (197).

7. Verify Eq. (205). 8 Functional Integral for Gauge Field Theories

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), (208) L −4 µν a F a (x) = ∂ Ba ∂ Ba + gf Bb Bc , (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 . (210) a In the fundamental representation T λ , where λa are the generalization { a}≡{ 2 } { } of the eight Gell-Mann matrices, while in the adjoint representation, relevant to the realization of transformations on the gauge fields, T a = if . (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] , (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. (208) is not of the form in Eq. (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 Poincarécovariant quantisation of the field theory. Alternatively, one could work with Coulomb gauge, build the Hamilto- nian 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. (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)] (213) Z L −4 Z − − 1 = d4xd4yBa(x) [ gµν ∂2 + ∂µ∂ν ]δ4(x y) Ba(y)(214) −2 µ − − ν Z n o 1 4 4 a µν a =: d xd yBµ(x)K (x, y)Bν (y) . (215) 2 Z The operator Kµν (x, y) thus determined can be expressed

d4q Kµν (x y)= gµνq2 + qµqν e−i(p,x−y) (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 , (217) µ − − ν where qµ is the four-momentum associated with the gauge ﬁeld. It follows that the W0[J] obtained from Eq. (212) has no damping associated with longitudinal gauge ﬁelds and is therefore meaningless. A simple analogy is

∞ ∞ 2 dx dye−(x−y) , (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 (219) −∞ −∞ −∞ −∞ Z Z Z ∞ Z = dX √π = . (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) , (221) µ µ → µ µ G(x) = exp igT aΘa(x) . (222) {− }

This means that, given a reference field configuration Bˆµ(x), the integrand in the generating functional, Eq. (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. (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 (223) ( ∂x ∂y ) (x, y) (x′,y′) = g (x, y)(x, y)=(x + s,y + s) . (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: X0 =(x0 +y0)/2=1/2 X0 +s.) Since s ( , ), this 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.)

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.

8.1.1 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 . (225) − Now define a functional ∆f such that

∞ ∆f [x, y] daδ((x + y)/2 a)=1 . (226) Z−∞ −

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

∞ ′ ′ −1 ′ ′ ∆f [x + a ,y + a ] = daδ((x + a + y + a )/2 a) (227) Z−∞ − ′ ∞ a˜==a−a daδ˜ ((x + y)/2 a˜) (228) Z−∞ − −1 = ∆f [x, y] . (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 (230) and now one performs a centre-of-mass translation: x x′ = x + a, y y′ = 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) (231) −∞ −∞ −∞ − Z ∞ Z ∞ Z ∞ −l(x,y) = da dx dye ∆f [x + a, y + a]δ((x + y)/2) . (232) Z−∞ Z−∞ Z−∞ Now making use of the a independence of ∆ [x, y], Eq. (227), − f ∞ ∞ ∞ −l(x,y) = da dx dye ∆f [x, y]δ((x + y)/2) . (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) . (234) Z−∞ Z−∞ 8.1.2 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 , (235) a µ c − is the equation describing the hypersurface, then there is a unique element in each gauge orbit that satisfies one Eq. (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 (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 (235) therefore defines a gauge fixing condition. In order for Eqs. (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 (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, b satisfies Eqs. (235). Applying an infinitesimal gauge transformation to this 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 (237) a µ → µ − bdc µ − µ for the inﬁnitesimal gauge transformation parameters δΘa(x). Equation (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 . (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 only if, det = 0, and a similar constraintO follows from Eq. (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 . (239) (− δBν(y) ) 6 Equation (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) . (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) , (241) ∂t − and this nonperturbative equation has the unique solution

t G(t, ~x)= T exp ds B0(s, ~x) , (242) − Z−∞ where T is the time ordering operator. One may compare this with Eq. (238), which only provides a perturbative [in g] solution. While that may be an advantage, Eq. (242) is not a Poincarécovariant 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 (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. (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. (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.

8.1.3 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) (244) − where T a are the adjoint representation of the Lie algebra of SU(N), Eq. (??). { } Clearly, if u, u SU(N) then uu′ SU(n) and G(u)G(u′) = G(uu′). (These are basic properties′ ∈ 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 du′ = d(u′u). In the neighbourhood of the identity one may always choose du = dΘa (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 , (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 ′ b,u′u ∆f [Bµ ] = du (x) δ[fa Bµ (x); x ] x x,a { } Z Y Y ′ b,u′u = d(u (x)u(x)) δ[fa Bµ (x); x ] x x,a { } Z Y Y ′′ b,u′′ −1 b = du (x) δ[fa Bµ (x); x ] = ∆f [Bµ] . (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 (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 (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 (250) Neglecting for now the possible existence of Gribov copies, Eq. (250) is the foundation we sought for a manifestly Poincar´ecovariant quantisation of the gauge ﬁeld. However, a little more work is needed to mould a practical tool.

8.1.4 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. (235). Recalling Eqs. (238), (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) . (251) Z M Hence, using the deﬁnition, Eq. (246),

−1 b a 4 ∆f [Bµ] = dΘ (x) δ d y ( f )acΘc(y) (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) , (253) M x,a Z Y n h io where the determinant is the Jacobian of the transformation, so that

∆ [Bb ] = Det . (254) f µ Mf Now recall Eq. (180). This means Eq. (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) ,. (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 (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. (243), for which

( ) = δ4(x y) δab ∂2 gf ∂µ Bc (x) (257) ML ab − − abc µ h i and therefore

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) (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 ,. (259) MA ab − µ h i Expressing the related determinant via ghost fields it 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 byQ 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. (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. To see how δ[f Ba(x); x ] influences the form of Green functions, consider b{ µ } the generalised Lorentz gauge:

ν a a ∂ Bµ(x)= A (x) , (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] , (261) where the r.h.s. is given in Eq. (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. (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) , (262) Z D −2λ Z to arrive ﬁnally at the generating functional in a covariant Lorentz gauge, speci- ﬁed 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) , (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. (263) is

1 4 4 a µν a 2 d xd y Bµ(x) K (x y; λ) Bν (y) Z − 1 4 4 a µν 2 µ ν 1 µν + 4 a := 2 d xd y Bµ(x) [g ∂ ∂ ∂ (1 ) ig η ]δ (x y) Bν (y)(264) 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 , (265) − Z (2π) − − λ cf. Eq. (??), and now 1 q Kµν = qν (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 boson− 2-point Green function, Dµν(x y), is obtained via − 4 µ ρν µν 4 d wKρ (x w) D (w y)= g δ (x y) , (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 . (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.

8.2 Exercises 1. Verify Eq. (257).

2. Verify Eq. (258).

3. Verify Eq. (259).

4. Verify Eq. (268). 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.

9.1 Photon Vacuum Polarization 9.1.1 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  (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 , (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. (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. 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.) 9.1.2 Functional Field Equations As described in Sec. 6.1.2, 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¯] . (271) n o The derivation of a DSE now follows simply from the observation 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¯] , (272) (δAµ(x) "iδJ iδξ¯ −iδξ # ) where the last line has meaning as a functional diﬀerential operator on the generating functional. Diﬀerentiating Eq. (269) gives

δS ρ 1 ν f f f = ∂ρ∂ gµν 1 ∂µ∂ν A (x)+ e0 ψ (x)γµψ (x) (273) δAµ(x) − − λ0 Xf so that the explicit meaning of Eq. (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 (274) ¯ where I have divided through by W [Jµ, ξ, ξ]. Equation (274) represents a compact form of the nonperturbative equivalent of Maxwell’s equations.

9.1.3 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 . (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. 6.1.2 yields δZ δZ δZ = A (x) , = ψ(x) , = ψ¯(x) , (276) δJ µ(x) µ δξ¯(x) δξ(x) − where here the external sources are nonzero. Hence Γ in Eq. (275) must satisfy δΓ δΓ δΓ = J (x) , = ξf (x) , = ξ¯f (x) . (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 , (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. (140).) Now consider the product (with spinor labels r, s, t)

2 2 4 δ Z δ Γ d z f g ¯ . (279) − ¯h δψh(z)ψ (y) ξ = ξ =0 Z δξr (x)ξt (z) t s ψ = ψ =0

Using Eqs. (276), (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

(280) Now returning to Eq. (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 (281) after making the identification δ2Z δ2Z Sf (x, y;[A ]) = = (no summation on f) , (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. (197).) I observe that it is a direct consequence of Eq. (279) that δ2Γ Sf (x, y;[A])−1 = , (283) δψf (x)δψ¯f (y) ψ=ψ=0

and it is a general property that such functional derivatives 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]) . (284) Such vacuum Green functions are of primary interest in quantum ﬁeld theory. To continue, one diﬀerentiates Eq. (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       (285) The l.h.s. is easily understood. Just as Eqs. (283), (284) deﬁne the inverse of the fermion propagator, so here is δ2Γ (D−1)µν (x, y) := . (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

    (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) . (288) ⇒ dx − dx Equation (287) involves the 1PI 3-point function

2 f f δ δ Γ e0 Γµ(x, y; z) := f f . (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) , (290) ν ν − − a result which can be obtained via the explicit calculation of the functional derivatives in Eq. (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) , (291) Xf Z h i it is immediately apparent that Eq. (285) may be expressed as

−1 µν ρ 1 4 (D ) (x, y)= ∂ρ∂ gµν 1 ∂µ∂ν δ (x y) + Πµν (x, y) . (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. (268). In the presence of interactions; i.e., for Π = 0 in Eq. (292), this becomes µν 6 gµν +(qµqν/[q2 + iη]) 1 qµqν Dµν(q)= − λ , (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 , (294) which means that one can write

Πµν (q)= gµνq2 + qµqν Π(q2) . (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. (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. (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 d4ℓ iΠ (q)= (ef )2 tr[(iγ )(iSf (ℓ))(iΓf (ℓ,ℓ + q))(iS(ℓ + q))] . (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 functional, 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.

9.2 Fermion Self Energy Equation (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¯] (297) (δψ¯f (x) "iδJ iδξ¯ −iδξ # )

f f f µ δ δ ¯ = ξ (x)+ i∂/ m0 + e0 γ W [Jµ, ξ, ξ] . (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]) = 0(299), − − − 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. (282). Now, using Eqs. (271), (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]) = 0(300), − − − iδJ µ(x)! which deﬁnes the nonperturbative connected 2-point fermion Green function (This is clearly the functional equivalent of Eq. (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. (300), which can be accomplished using Eq. (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 zd ud w S (x, u)Γν(u,w; z) S(w,y) − Z iδJµ(x) f 4 4 4 f = e0 d zd ud w iDµν (x z) S (x, u)Γν(u,w; z) S(w,y) , − Z − (301) where, in the last line, I have set J = 0 and used Eq. (286). Hence, in the absence of external sources, Eq. (300) is equivalent to

δ4(x y)= i∂/ mf Sf (x, y) − − 0 f 2 4 4 4 µν 4 i (e0 ) d zd ud w D (x, z) γµ S(x, u)Γν(u,w; z) S(w,y)= δ (x (302)y) . − Z − Just as the photon vacuum polarisation was introduced to simplify, or re- express, the DSE for the gauge boson propagator, Eq. (291), one can deﬁne a fermion self-energy:

f f 2 4 4 µν Σ (x, z)= i(e0 ) d ud w D (x, z) γµ S(x, u)Γν(u,w; z) , (303) Z so that Eq. (302) assumes the form

d4z i∂/ mf δ4(x z) Σf (x, z) S(z,y)= δ4(x y) . (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 ℓ µν f f Σ (p)= i (e0 ) 4 D (p ℓ)[iγµ][iS (ℓ)] [iΓν (ℓ,p)] . (305) Z (2π) − In terms of the self-energy, it follows from Eq. (304) that the connected fermion 2-point function can be written in momentum space as 1 Sf (p)= . (306) p/ mf Σf (p)+ iη+ − 0 − Equation (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.

9.3 Exercises 1. Verify Eq. (277).

2. Verify Eq. (287).

3. Verify Eq. (296).

4. Verify Eq. (300). 10 Perturbation Theory

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: d4ℓ i i Σ(p)= g2 Dµν(p ℓ) λaγ S(ℓ)Γa(ℓ,p) , (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. (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 symbols is also analogous: Dµν(ℓ) is the connected gluon a ′ 2-point function and Γν(ℓ,ℓ ) 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. (307) using the free quark and gluon propagators, and the quark- gluon vertex: a (0) ′ i a Γν (ℓ,ℓ )= 2λ γν , (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 (309) × 2 µ k + p/ m + iη+ 2 µ 6 − 0 and one may proceed as follows. First observe that Eq. (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+ ) (310) where I have used 1 1 N 2 1 a a I c 2λ 2λ = C2(R) c ; C2(R)= − , (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] (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+ ) (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 (314) Z (2π) k + iη k + iη because the integrand is odd under k k, and so this term in Eq. (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+ ) (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 →±∞ . (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 . (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 (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) , (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 (320) Returning now to Eq. (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→∞ →∞ (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. (321) with m 0.) In a renormalisable quantum field the- 0 → ory 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. Renormalisa- tion 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. 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. (316), this means we consider dDk 1 1 T = (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 − , (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 . (324) 0 (2π)D [(k xp)2 m (1 x)+ p2x(1 x)+ iη+]2 Z Z − − 0 − − (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:

D k→k−xp 1 d k 1 T = dx , (326) 0 (2π)D [k2 a2 + iη+]2 Z Z − 2 2 2 where a = m0(1 x) p x(1 x). I will consider− a generalisation− − of the momentum integral:

dDk 1 I = , (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 . (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 = . (329) Z Γ(D/2) 2 Clearly, S4 =2π , as we saw in Eq. (321). Hence ( 1)n 1 ∞ kD−1 In = i D−1 D/2 dk 2 2 n . (330) 2 π Γ(D/2) Z0 (k + a ) Writing D =4+2ǫ one arrives at

2 ǫ i 2 2−n a Γ(n 2 ǫ) In = ( a ) − − . (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. (331) as the definition of In whenever the integral is ill-defined. I observe that D = 4 is recovered via the limit ǫ 0− and the divergence of the integral for n = 2 in this case is encoded in → 1 Γ(n 2 ǫ)=Γ( ǫ)= γ + O(ǫ); (332) − − − ǫ − E − i.e., in the pole in the gamma-function. (γE is the Euler constant.) Substituting Eq. (331) in Eq. (326) and setting n = 2 yields

1 2 2 ǫ ǫ 2 i Γ( ǫ) m0 p T =(gν ) 2 − ǫ dx 2 (1 x) 2 x(1 x) , (333) (4π) (4π) Z0 " ν − − ν − # wherein I have employed a nugatory transformation to introduce the mass-scale ν. It is the limit ǫ 0− that is of interest, in which case it follows that (xǫ = exp ǫ ln x 1+ ǫ ln x→) ≈ 1 2 2 ǫ 2 i 1 m0 p T = (gν ) 2 γE + ln 4π dx ln 2 (1 x) 2 x(1 x) (4π) (− ǫ − − Z0 " ν − − ν − #) 2 2 2 ǫ 2 i 1 m0 m0 p = (gν ) 2 γE + ln 4π +2 ln 2 1 2 ln 1 2 . (4π) (− ǫ − − ν − − p ! " − m0 #) (334)

It is important to understand the physical content of Eq. (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. (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 ǫ 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 := (gνǫ)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

10.2.1 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 , (335) { } − but now µν gµν g = D (336) and hence

ν γµ γ = D 1D , (337) γ γν γµ = (2 D) γν , (338) µ − γ γν γλ γµ = 4 gνλ1 +(D 4) γν γλ , (339) µ D − γ γν γλ γρ γµ = 2 γρ γλ γν + (4 D) γν γλ γρ , (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. (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 (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.

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

L(x)= L0(x)+ giLi(x) , (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) (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 . (344) It is a fundamentally important fact in quantum field theory that if there is even one coupling constant for which

[gi] < 0 (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.

10.3 Regularized Quark Self Energy I can now return and re-express Eq. (315):

dDk 1 1 1 i Σ(2)(p) = (g νǫ)2 C (R) − − 0 2 (2π)4 ν2ǫ (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+ ) (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 , (347) that can be obtained via trace projection: 1 1 1 Σ (p2)= tr [p/Σ(p/)] , Σ (p2)= tr [Σ(p/)] . (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 ǫ 2 d k 1 1 p ΣV (p ) = i (g0ν ) C2(R) 4 2ǫ 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+ #) (349) dDk 1 m (D 1+ λ ) Σ (p2) = i (g νǫ)2 C (R) 0 − 0 . (350) S − 0 2 (2π)4 ν2ǫ [(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) = , (351) (2π)4 ν2ǫ [(k + p)2 m2 + iη+]α [k2 + iη+]β Z − dDk 1 kµ J µ(α, β; p2, m2) = , (352) (2π)4 ν2ǫ [(k + p)2 m2 + iη+]α [k2 + iη+]β Z − the ﬁrst of which we have already encountered in Sec. 10.2. The general results are (D =4+2ǫ)

i 1 α+β−2 Γ(α + β 2 ǫ)Γ(2+ ǫ β) I(α, β; p2, m2)= − − − (4π)2 p2 ! Γ(α)Γ(2+ ǫ) p2 ǫ m2 2+ǫ−α−β 1 1 2F1(α + β 2 ǫ, 2+ ǫ β, 2+ ǫ; ) , × −4πν2 ! − p2 ! − − − 1 (m2/p2) − (353)

i 1 α+β−2 Γ(α + β 2 ǫ)Γ(3 + ǫ β) J µ(α, β; p2, m2)= pµ − − − (4π)2 p2 ! Γ(α)Γ(3+ ǫ) p2 ǫ m2 2+ǫ−α−β 1 1 2F1(α + β 2 ǫ, 3+ ǫ β, 3+ ǫ; ) × −4πν2 ! − p2 ! − − − 1 (m2/p2) − =: pµ J(α, β; p2, m2) , (354) where 2F1(a, b, c; z) is the hypergeometric function. Returning again to Eqs. (349), (350) it is plain that

Σ (p2) = i (g νǫ)2 C (R) 2(1 + ǫ) J(1, 1) (1+2ǫ) I(1, 1) (p2 m2) J(1, 2) V − 0 2 − − − 0 nh i +λ (p2 m2) J(1, 2) I(1, 1) , (355) 0 − 0 − Σ (p2) = i (gh νǫ)2 C (R) m (3 + λ +2ǫio) I(1, 1; p2, m2) , (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 (− ǫ − − −ν2 ! m2 m2 m2 m2 ln 1 ln 1 , (357) − p2 − p2 ! − − p2 ! − p2 !) i 1 1 p2 J(1, 1) = + ln 4π γE ln +2 (4π)2 2 (− ǫ − − −ν2 ! m2 m2 m2 m2 m2 2 m2 m2 2 ln 1 2 + ln 1 , − p2 − p2 ! − p2 ! −  − p2 " p2 #  − p2 ! − p2      (358) i 1 m2 m2 m2 m2 J(1, 2) = ln + ln 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. (355) is identically zero in D = 4 dimensions; i.e., for ǫ > 0−, and hence − ǫ 2 2 2 (gν ) 1 m0 ΣV (p ) = λ0 C2(R) ln 4π + γE + ln (4π)2 ( ǫ − ν2 m2 m4 p2 1 0 + 1 0 ln 1 . (360) − − p2 − p4 ! − m2 !)

It is obvious that in Landau gauge: λ0 = 0, in four spacetime dimensions: Σ (p2) 0, at this order. The scalar piece of the quark’s self-energy is also V ≡ easily found: ǫ 2 2 2 (gν ) 1 m0 ΣS(p ) = m0 C2(R) (3 + λ0) ln 4π + γE + ln (4π)2 (− " ǫ − ν2 # 2 2 m0 p +2(2 + λ0) (3 + λ0) 1 2 ln 1 2 . (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. 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 (362) m0→0 and hence dynamical chiral symmetry breaking is impossible in perturbation theory.

10.4 Exercises 1. Verify Eq. (321). 2. Verify Eq. (328). 3. Verify Eq. (334). 4. Verify Eqs. (349) and (350). 5. Verify Eqs. (360) and (361). 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 ǫ 0− (or with the removal of whatever other parameter has been used to regularise→ the divergences).

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 +¯qf (x)i∂/qf (x) mf q¯f (x)qf (x)+ g q¯f (x)λa/Ba(x)qf (x) , (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) , (364) with 1 1 L (x) = C ∂ Ba(x)[∂µBνa(x) ∂ν Bµa(x)] + C ∂ Ba(x)∂ Ba(x) c 3YM 2 µ ν − 6 2λ µ ν µ ν 1 +C g f [∂µBνa(x) ∂ν Bµa(x)]Bb (x)Bc(x) 1YM 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)λa/Ba(x)qf (x) . − 2F 4 − 1F 2 (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. 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 , (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 (367) i − i so that Eq. (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)λa/Ba(x)qf (x) , 2F − 4 2 (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) := Z3YM 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 g0YM := Z1YM Z3YM g , g˜0 := Z1 Z3 Z3YM , (369) −1/2 −1 1/2 −1 g0F := Z1F Z3YM Z2F g, g05 := Z5 Z3YM g f −1 f −1 m0 := Z4 Z2F m , λ0 := Z6 Z3YM 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+2ǫ- 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+ǫ , [Bµ(x)] = M 1+ǫ , [φ(x)] = [φ¯(x)] = M 1+ǫ , [g]= M −ǫ , (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 0YM 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)+˜g0 fabc ∂µφ0(x) φ0(x)B0 (x) − 1 +¯qf (x)i∂/qf (x) mf q¯f (x)qf (x)+ g q¯f (x)λa/Ba(x)qf (x) . 0 0 − 0 0 0 2 0F 0 0 0 (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

g0YM =g ˜0 = g0F = g05 = g0 . (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:

Z3YM Z˜3 g0YM =g ˜0 = , ⇒ Z1YM Z˜1 Z3YM Z2F g0YM = g0F = , (373) ⇒ Z1YM Z1F 2 Z1YM g0YM = g05 Z5 = . ⇒ Z3YM In QED the second of these equations becomes the Ward-Takahashi identity: Z1F = Z2F .

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 . (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.

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

(gνǫ)2 α := (375) 4π and considers each counterterm as a power series in α with the form

∞ j 1 α j C = C(2j) , (376) i i,k ǫk π jX=1 kX=1 where the coeﬃcients in the expansion may, at most, depend on the gauge parameter, λ. Using Eqs. (360), (361) and (374) we have

2 (2) α 1 1 m ΣR (p/) = p/ λ C2(R) ln 4π + γE + ln (377) ( π 4 " ǫ − ν2 m2 m4 p2 1 + 1 ln 1 + C2F (378) − − p2 − p4 ! − m2 !# ) α 1 1 m2 +m C2(R) (3 + λ) ln 4π + γE + ln (379) π 4 (− " ǫ − ν2 # m2 p2 +2(2 + λ) (3 + λ) 1 ln 1 C4 . (380) − − p2 ! − m2 ! − )

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

α 1 1 Z =1 C = 1+ λ C (R) , (381) 2F − 2F π 4 2 ǫ α 1 1 Z =1 C = 1+ (3 + λ) C (R) , (382) 4 − 4 π 4 2 ǫ and hence Eq. (380) becomes

2 (2) α 1 m ΣR (p/) = p/ λ C2(R) ln 4π + γE + ln (383) ( π 4 " − ν2 ! m2 m4 p2 1 + 1 ln 1 + C2F (384) − − p2 − p4 ! − m2 !# ) α 1 m2 +m C2(R) (3 + λ) ln 4π + γE + ln (385) π 4 (− − ν2 ! m2 p2 +2(2 + λ) (3 + λ) 1 ln 1 C4 , (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. (371) and the deﬁnition of the connected 2-point quark Green function (an obvious analogue of Eq. (282)):

2 a δ ZR[J , ξ, ξ¯] iSf (x, y; m,λ,α)= µ = 0 qf (x)¯qf (y) 0 = Z−1 0 qf (x)¯qf (y) 0 R − δξf (y)δξ¯f (x) h | | i 2F h | 0 0 | i (387) to write −1 SR(p/; m,λ,α) = lim Z S0(p/; m0,λ0,α0; ǫ) , (388) ǫ→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. (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 ǫ→0− − − nh i h (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) , (390) − ǫ→0− − 2 n 2 o 1 + ΣSR(p ; m,λ,α)/m = lim Z4 1 + ΣS(p ; m0,λ0,α0)/m0 . (391) ǫ→ǫ− n o Now the renormalisation constants, Z2F , Z4, are chosen so as to exactly cancel the 1/ǫ poles in the r.h.s. of Eqs. (390), (391). Using Eqs. (360), (361), Eqs. (381), (382) are immediately reproduced.

11.2.2 Modified Minimal Subtraction The modified minimal substraction scheme MS is also often used in QCD. It takes advantage of the fact that the 1/ǫ pole obtained using dimensional regularisation always appears in the combination 1 ln 4π + γ (392) ǫ − 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.

11.2.3 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; ǫ) Σ (p2; ǫ); A = V,S, (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)!) (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)!#) (395) where the renormalised quantities depend on the point at which the renormalisation has been conducted. Clearly, from Eqs. (390), (391), the renormalisation constants in this scheme are

Z(2) = 1+Σ(2)(p2 = µ2; ǫ) (396) 2F V − Z(2) = 1 Σ(2)(p2 = µ2; ǫ)/m(µ) . (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.

11.3 Renormalized Gap Equation Equation (307) is the unrenormalised QCD gap equation, which can be rewritten as

d4ℓ i iS−1(p)= i(p/ m )+ g2 Dµν (p ℓ) λaγ S (ℓ)Γa (ℓ,p) . (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. (368), simply by repeating the steps described in Sec. 9.2. Alternatively, one can use Eqs. (371) to derive an array of relations similar to Eq. (387):

µν µν a −1 a D0 (k)= Z3YM DR (k) , Γ0ν(k,p)= Z1F ΓRν (k,p) , (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 d4ℓ i +Z2 Z−1 Z−2 g2 Z Dµν (p ℓ) λaγ Z S (ℓ) Z−1 Γa (ℓ,p) , 1F 3YM 2F R 4 3YM R 2 µ 2F R 1F Rν Z (2π) − (400) which simpliﬁes to

iΣ (p) = i(Z 1) p/ i(Z 1) m − R 2F − − 4 − R d4ℓ i Z g2 Dµν(p ℓ) λaγ S (ℓ)Γa (ℓ,p) 1F R 4 R 2 µ R Rν − Z (2π) − =: i(Z 1) p/ i(Z 1) m iΣ′(p) , (401) 2F − − 4 − R − where Σ′(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 (402) which entails

Z =1+Σ′ (p/ + µ = 0) , Z =1 Σ′ (p/ + µ = 0)/m (µ) , (403) 2F V 4 − S R ′ ′ ′ where I have used Σ (p) = Σ (p) p/ + ΣS (p). (cf. Eqs. (396), (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.

11.3.1 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 , (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/) + ... (405) − − | R − − "d/p # /p=mR

then Eq. (404) entails

d ΣR(p/) /p=m =0 , ΣR(p/) =0 . (406) | R d/p /p=mR

The second of these equations requires

′ 2 2 d ′ 2 d ′ 2 Z2F =1+ΣV (mR)+2 mR 2 ΣV (p ) 2 2 +2 mR 2 ΣS (p ) (407) dp p =mR dp 2 2 p =mR

and the ﬁrst: Z = Z Σ′ (m2 ) Σ′ (m2 ) . (408) 4 2F − V R − S R 11.4 Exercises 1. Using Eqs. (360), (361) derive Eqs. (381), (382).

2. Verify Eqs. (396), (397).

3. Verify Eq. (401).

4. Verify Eq. (403).

5. Beginning with Eq. (404), derive Eqs. (407), (408). NB. p/p/ = p2 and hence d 2 d d 2 dp/ f(p )= dp/ f(p/p/)=2 p/ dp2 f(p ). 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.

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. (186): ∞ ψ 3 ¯ + L0 (x)= dt d x ψ(x) i∂/ m + iη ψ(x) . (409) −∞ − Z Z Recall now the observations at the end of Sec. 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) (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) (411) − ∂tE − − ∂xi E ∂ =: γµ E , (412) − ∂xµ where (xE)=(~x, x := it) and the Euclidean Dirac matrices are µ 4 − γE = γ0 γE = iγi . (413) 4 i − These matrices are Hermitian and satisfy the algebra γE,γE =2δ ; µ, ν =1,..., 4, (414) { µ ν } µν where δµν is the four-dimensional Kroncker delta. Henceforth I will adopt the notation 4 aE bE = aEbE . (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) . (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 ) (417) Z − · − E E where ψ¯r(x ) and ψs(x ) are intepreted as members of a Grassmann algebra with involution, as described in Sec. 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 ) . (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 (419) ← − → but that leaves us with

E E T [γµ ]rs = [(γµ ) ]sr . (420)

If we deﬁne (γE)T := C γE C† , (421) µ − E µ E where CE = γ2γ4 is the Euclidean charge conjugation matrix then

AE = AE (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. (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 (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) (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 (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.

12.1.1 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) (426) S which can be obtained as usual via functional diﬀerentiation of the analogue of Eq. (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 functions, n(x ,...,x ), can be obtained from the Schwinger functions by analytic W 1 n continuation in each of the time coordinates:

n(x ,...,x ) = lim n([~x1, x1 + ix0],..., [~xn, xn + ix0 ]) (427) 1 n i 4 1 4 n W x4→0 S

0 0 0 with x1 < x2 <...

12.1.2 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 , (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 ; (429) { µ ν} µν and I use γ := γ γ γ γ (430) 5 − 1 2 3 4 so that tr [γ γ γ γ γ ]= 4 ε , ε =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 = , (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 = . (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  (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 qξ¯ + ξ q + Jµ Aµ , (435) Z Z h i with sources:η ¯, η, J, and a functional integral measure dµ(¯q, q, B, ω,ω¯ ) := (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, andω ¯ and ω are the scalar, Grassmann [ghost] fields. The normalisation W [¯η =0, η =0,J =0]=1 (437) is implicit in the measure. As we saw in Sec. 8.1.4, the ghosts only couple directly to the gauge field: S [B, ω, ω¯]= d4x ∂ ω¯ a ∂ ωa gf abc ∂ ω¯a ωb Bc , (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. 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 Poincaréinvariant∝ · 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) . (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., mu md ΛQCD. Therefore chiral symmetry should only be weakly broken, with the∼ strong≪ interaction spectrum exhibiting nearly degenerate parity partners. The experimental comparison is presented in Eq. (440):

1 + − − N( 2 , 938) π(0 , 140) ρ(1 , 770) 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)= − · (441) 0 m2 + p2 and as a matrix iγ p m S (p) eiγ5θS (p)eiγ5θ = − · + e2iγ5θ (442) 0 → 0 p2 + m2 p2+m2 under a chiral transformation. As anticipated, for m = 0, S0(p) S0(p); i.e., the symmetry breaking term is proportional to the current-quark ma→ss and it can be measured by the “quark condensate”

d4p d4p m qq¯ := 4 tr [S(p)] 4 2 2 , (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? • 1.5

1 1

0.5 0.5

0 -1 0

-0.5 -0.5 0

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

As we have seen, interactions dress the quark propagator so that it takes the form 1 iγ pA(p2)+ B(p2) S(p) := = − · , (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. (395) with p2 (pE)2 µ2, m2(µ), → − ≫ α p2 B(p2)= m 1 S ln , (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 qq¯ m 0 as m 0. The current-quark masses are the only explicit chiralh symmetryi ∝ → 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,− 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 Σ D = γ S Γ

Figure 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. 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)- plane. The vacua| i are degenerate but not rotationally invariant and hence, in general, θ σ θ = 0. In this case the rotational invariance of the Hamiltonian is h | | i 6 not exhibited in any single ground state: the symmetry is dynamically broken with interactions being responsible for θ σ θ = 0. h | | i 6 12.3 Mass Where There Was None The analogue in QCD is qq¯ = 0 when m = 0. At any ﬁnite order in perturbation theory that is impossible.h However,i 6 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 d4ℓ λa 1 +Z g2 D (p ℓ) γ Γa(ℓ,p) , (446) 1 (2π)4 µν − µ 2 iγ ℓA(ℓ2)+ B(ℓ2) ν Z · depicted in Fig. 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. (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 quark → ∞ 2 2 2 2 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 mass 2 2 2 2 −1 2 2 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 kernel of the equation is composed of the dressed-gluon propagator:

2 2 2 kµkν (k ) 2 g g Dµν(k)= δµν G , (k ) := , (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 . (448) µ bare µ 2 a Once Dµν and Γµ are known, Eq. (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.

12.3.1 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. (446) via the substitution 2 1 2 2 g Dµν (p ℓ) δµν 2 θ(Λ ℓ ) (449) − → mG − in combination with Eq. (448). The step-function in Eq. (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 d4ℓ iγ ℓA(ℓ2)+ B(ℓ2) = iγ p + m + θ(Λ2 ℓ2) γ γ ,(450) 3 2 4 µ − 2 ·2 2 2 2 µ · mG Z (2π) − ℓ A (ℓ )+ B (ℓ ) where I have set the renormalisation constants equal to one in order to complete the defintion of the model. Multiplying Eq. (450) by ( iγ p) and tracing over Dirac indices one obtains − · 8 1 d4ℓ A(ℓ2) p2 A(p2)= p2 + θ(Λ2 ℓ2) p ℓ , (451) 3 2 4 2 2 2 2 2 mG Z (2π) − · ℓ A (ℓ )+ B (ℓ ) from which it is immediately apparent that A(p2) 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. (452) one obtains

16 1 d4ℓ B(ℓ2) B(p2)= m + θ(Λ2 ℓ2) , (453) 3 2 4 2 2 2 mG Z (2π) − ℓ + B (ℓ ) 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. (453) becomes

Λ2 1 1 M 1 1 2 2 M = m + 2 2 dx x 2 = m + M 2 2 (M , Λ ) ,(454) 3π mG Z0 x + M 3π mG C (M 2, Λ2) = Λ2 M 2 ln 1+Λ2/M 2 (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) . (456) 3π mG C Irrespective of the value of m , this equation always admits a solution M = 0 G 6 when the current-quark mass m = 0. Consider now the chiral-limit,6 m = 0, wherein the gap equation is

1 1 2 M = M 2 2 (M , 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. (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. (457), then the equation becomes 6

1 1 2 1= 2 2 (M , 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. (458) has a M = 0 solution if, and only if, C 6 1 1 2 2 > 1; (459) 3π mG i.e., if and only if Λ2 m2 < (0.2 GeV )2 (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. (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.

12.3.2 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. (446) by using (k2) G = (2π)4 G δ4(k) (461) k2 in Eq. (447) with the bare vertex, Eq. (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 + Gγ − · γ , (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 (463) p2A2(p2)+ B2(p2) B(p2) B(p2) = m +4 , (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 . (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 , (466) measured in units of G. Substituting this identity into equation Eq. (463) one finds 1 A(p2) 1= A(p2) A(p2) 2 , (467) − 2 ⇒ ≡ which in turn entails B(p2)=2 1 p2 . (468) − The physical requirement that the quark selfq 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. (465), and on this domain we then find from Eq. (463) that ≡ 2 1 A(p2)=1+ A(p2)= 1+ 1+8/p2 . (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 ≤ (470)  1+ 1+8/p2 ; p2 > 1  2 2q 2 2  √1 p ; p 1 B(p ) = − 2 ≤ (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 . (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) = , (473) m + B(p2) 4 [m + B(p2)]2 B(p2) = m + . (474) B(p2)([m + B(p2)]2 +4p2) 1.0

0.8

G(Q) 0.6

0.4

0.2

0.0 0 1 10 100 Q (GeV)

Figure 3: Illustrative forms of (Q): the behaviour of each agrees with perturbation theory for Q > 1 GeV.G Three possibilities are canvassed in Sec.12.3.3: (Q = 0) < 1; (Q =0) =1; and (Q = 0) > 1. G G G

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. → 12.3.3 Renormalization-Group-Improved Model Finally I have used the bare vertex, Eq. (448) and (Q) depicted in Fig. 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;G i.e., if the≥ coupling is large enough then even in 6≡ the absence of a current-quark mass, contrary to Eq. (445), the quark acquires a mass dynamically and hence

d4p B(p2) qq¯ 4 2 2 2 2 2 =0 for m =0 . (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 [ qq¯ 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. References

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