THE HERMITIAN SCALAR FIELD

dt L(t) = Z

d⁴xL(φ, ∂_µφ). (2.18) TheEuler-Lagrangeequations of motion forφare obtained by minimizing the action with respect to φ(x) andφ^†(x),

δL

δφ −∂_µ δL

δ∂µφ = 0, (2.19)

and similarly for φ^†. The fields φ are interpreted as operators in the Heisenberg picture, i.e., they are time-dependent while the states are time independent. Other quantities, such as the conjugate momentum, the Hamiltonian, and the canonical commutation rules, are summarized inAppendix A.

2.3 THE HERMITIAN SCALAR FIELD

A real (or more accurately, Hermitian) spin-0 (scalar) field satisfies φ(x) = φ^†(x). It is suitable for describing a particle such as theπ⁰that has no internal quantum numbers and is therefore the same as its antiparticle.

2.3.1 The Lagrangian and Equations of Motion The Lagrangian density for a Hermitian scalar is

L(φ, ∂µφ) =1 2

(∂µφ)²−m²φ²

−V_I(φ), (2.20)

where (∂µφ)² is a shorthand for (∂µφ)(∂^µφ). The first two terms correspond, respectively, to canonical kinetic energy and mass (the ¹₂ is special to Hermitian fields), while the last describes interactions.

The interactionpotentialis V_I(φ) =κφ³

3! +λφ⁴

4! +c+d₁φ+d₅φ⁵

5! +· · ·+ non−perturbative, (2.21) where thek! factors are for later convenience in cancelling combinatoric factors,²and “non-perturbative” allows for the possibility of non-polynomial interactions. The constant c is irrelevant unless gravity is included. A non-zero d1 (tadpole) term will induce a non-zero vacuum expectation value(VEV),h0|φ|0i 6= 0, suggesting that one is working in the wrong vacuum. Thed₁ term can be eliminated by a redefinition ofφ→φ⁰= constant +φ, as will be described inChapter 3.Landφhave dimensions of 4 and 1, respectively, in mass units, so the coefficient ofφ^khas the mass dimension 4−k. Thed_k terms withk≥5 are known as non-renormalizable orhigher-dimensional operators. They lead to new divergences in each order of perturbation theory, with d_k typically of the form d_k = c_k/M^k−4, where c_k is dimensionless andMis a large scale with dimensions of mass. Such terms would be absent in arenormalizable theory, but may occur in aneffective theory at low energy, where they describe the effects of the exchange of heavy particles (or other degrees of freedom) of mass M that are not explicitly taken into account in the field theory. (InChapters 7and 8 we will see that an example of this is the four-fermi operator that is relevant to describing the weak interactions at low energy.) Keeping just the renormalizable terms (and c=d1= 0), one has

V_I(φ) =κφ³ 3! +λφ⁴

4!, (2.22)

whereκ(dimensions of mass) andλ(dimensionless) describe three- and four-point interac-tions, respectively, as illustrated in Figure 2.1. From the Euler-Langrange equation (2.19), one obtains the field equation

2+m²

φ+∂VI

∂φ = 2+m²

φ+κφ² 2 +λφ³

6 = 0, (2.23)

where2+m²=∂_µ∂^µ+m²is theKlein-Gordonoperator. The expression for the Hamiltonian density is given inAppendix A.

2.3.2 The Free Hermitian Scalar Field

Letφ₀=φ^†₀ be the solution of (2.23) in thefree(or non-interacting) limitκ=λ= 0, i.e., 2+m²

φ0(x) = 0. (2.24)

Equation (2.24) can be solved exactly, and small values of the interaction parametersκand λcan then be treated perturbatively (as Feynman diagrams). The general solution is

φ0(x) =φ^†₀(x) =

Z d³~p (2π)³2E_p

a(~p)e^−ip·x+a^†(~p)e^+ip·x

, (2.25)

2Conventions for such factors may change, depending on the context.

− iκ − iλ

Figure 2.1

Three- and four-point interactions of a Hermitian scalar field

φ. The factor

is the coefficient of

φⁿ/n! iniL

, as described in

Appendix B.

where x = (t, ~x) and p = (Ep, ~p) with E_p ≡ p

~

p²+m², i.e., the four-momentum p in the Fourier transform is an on-shell momentum for a particle of mass m. The canonical commutation rules forφ0and its conjugate momentum given inAppendix Awill be satisfied if the Fourier coefficientsa(~p) satisfy the creation-annihilation operator rules in (2.2).

It is useful to define theFeynman propagatorforφ0,

i∆F(x−x⁰)≡ h0|T[φ0(x), φ0(x⁰)]|0i, (2.26) where

T[φ0(x), φ0(x⁰)]≡Θ(t−t⁰)φ0(x)φ0(x⁰) + Θ(t⁰−t)φ0(x⁰)φ0(x) (2.27) represents the time-ordered product of φ₀(x) and φ₀(x⁰). In (2.27) Θ(t−t⁰) is the step function, defined in (1.26). ∆F(x−x⁰) is just the Green’s function of the Klein-Gordon operator, i.e.,

2x+m²

∆F(x−x⁰) =−δ⁴(x−x⁰), (2.28) where2xrefers to derivatives w.r.t.x. The momentum space propagator is

∆F(k)≡ Z

d⁴xe^+ik·x∆F(x) = 1

k²−m²+i = 1

k²₀−~k²−m²+i. (2.29) kis an arbitrary four-momentum, i.e., it need not be on shell. The on-shell limit is correctly handled by theifactor in the denominator, where is a small positive quantity that can be taken to 0 at the end of the calculation.

2.3.3 The Feynman Rules

The Feynman rules allow a systematic diagrammatic representation of the terms in the perturbative expansion (in κ and λ) of the transition amplitude Mf i between an initial state i and a final state f. The derivation is beyond the scope of this book, but can be found in any standard field theory text. (The derivation of a simple example is sketched in Appendix B.) Heuristic derivations may also be found, e.g., in (Bjorken and Drell, 1964;

Renton, 1990). For the Hermitian scalar field with the potential (2.22), the rules are Draw each connected topologically distinct diagram in momentum space correspond-ing to initial (final) statesi (f), with internal lines corresponding to virtual (inter-mediate) particles. The internal and external lines are joined at three- and four-point vertices corresponding to the interactions inV_I. Each external and internal line has an associated four-momentum, which is off-shell for the virtual particles. It is conve-nient to put an arrow on the line to indicate the direction of momentum flow. This

direction is only a convention, and for the Hermitian scalar field (with no internal quantum numbers) there is no restriction on how many arrows flow into or out of a diagram.

There is a factor of −iκat every three-point vertex and a factor −iλ at each four-point vertex, as inFigure 2.1. These correspond to the coefficients ofφ³/3! andφ⁴/4!, respectively, iniL, with the 1/3! forκcancelling against 3! ways to associate the three lines with the three fields inφ³, and similarly for theλterm.

There is a factor ofi∆F(k) =_k2−mⁱ²+i for each internal line with four-momentumk.

Four-momentum is conserved at each vertex, implying that the overall four-momentum is conserved (i.e.,M_{f i} is only defined for Σp_i= Σp_f).

Integrate over each unconstrained internal momentum (there will be one for each internal loop in the diagram), with a factorR _d⁴_k

(2π)⁴.

There may be additional combinatoric factors³ associated with the interchange of internal lines for fixed vertices, e.g., a factor of 1/n! if n internal lines connect the same pair of vertices, as inFigure 2.2.

Figure 2.2

Diagrams with additional factors of 1/2! (left) and 1/3! (right), required because not all of the (4!)

²

ways of associating the four fields at each vertex with four lines lead to distinct diagrams.

The ordering and arrangement of the external lines in a Feynman diagram is usually irrele-vant, although the relative ordering between two diagrams does matter for fermions. In this book we will usually, but not always, place the initial state particles at the bottom and the final particles at the top.

As a simple example, the tree-level diagrams for the 2 → 2 scattering amplitude M_{f i}=hp~₃~p₄|M|~p₁~p₂iare shown inFigure 2.3. Applying the Feynman rules, these diagrams correspond to the expression

M_{f i}=−iλ+ (−iκ)² i

s−m² + i

t−m² + i u−m²

, (2.30)

wheres, t, anduare theMandelstam variables

s≡(p1+p2)²= (p3+p4)² t≡(p1−p3)²= (p4−p2)² u≡(p₁−p₄)²= (p₃−p₂)².

(2.31)

3In general, there may be subtleties involving combinatorial factors (or, signs when fermions are involved), especially in higher-order diagrams, which are best resolved by returning to the original derivation.

For the present (equal mass) case, s = E²_CM ≥ 4m², t ≤ 0, and u ≤ 0, where E_CM is the total energy in the center of mass. The internal lines are never on-shell (s, t, u6=m²) for physical (on-shell) external momenta, so one can drop the +i. It is implicit that the external momenta satisfy the four-momentum conservationp1+p2=p3+p4. The second, third, and fourth diagrams inFigure 2.3are said to haves-channel,t-channel, andu-channel poles, respectively.

p1 p3

p2 p4

=

p1 p3

p2 p4

+ i∆F(p1 +p2)

p1 p3

p2 p4

−iκ

−iκ

i∆F(p13) p1

p3

p2 p4

+ −iκ −iκ

i∆F(p14) p1

p4

p2 p3

+ −iκ −iκ

−iλ

Figure 2.3

Tree level diagrams for

M_{f i}

=

hp~3~p4|M|p~1~p2i

for a Hermitian scalar field.

The arrows label the directions of momentum flow, and

p_ij ≡p_i−p_j

. The arrange-ment of lines in the last diagram is modified to allow the diagram to be drawn without crossing lines.

2.3.4 Kinematics and the Mandelstam Variables

Let us digress to generalize to the case of a 2→2 scattering process 1 + 2 → 3 + 4, where we allow for the possibility of inelastic scattering with unequal masses for 1, 2, 3, and 4.

In the absence of spin, the scattering amplitude can be expressed in terms of the Lorentz invariant Mandelstam variables defined in (2.31).s, t, and uare not independent, but are related by

s+t+u=m²₁+m²₂+m²₃+m²₄. (2.32) Of course,s=m²₁+m²₂+ 2p1·p₂, etc.

The kinematics is simplest in thecenter of mass(CM) frame, which is more accurately the center of momentum, in which the total three-momentum of the initial and final state vanishes:

p1= (E1, ~p_i), p2= (E2,−~p_i), p3= (E3, ~p_f), p4= (E4,−~p_f), (2.33) where ~p_i and ~p_f are, respectively, the initial and final three-momenta; the energy and velocity of particle 1 are

E₁²=~p_i²+m²₁, β~1= ~p_i

E₁, (2.34)

and similarly for 2, 3,and 4; and the CM scattering angleθis related by

~

p_i·~p_f =p_ip_fcosθ, (2.35) as shown inFigure 2.4. In the CM frame, s= (E1+E2)² = (E3+E4)² is just the square

1 2

3

4 pi

−pi

pf

−pf

θ

(i)

1 2

3

4 θ3

(ii)

1

2

3 4

(iii)

Figure 2.4

Scattering kinematics in (i) the center of mass, (ii) the lab, and (iii) the

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