ELECTROMAGNETIC INTERACTION OF CHARGED PIONS

. (2.129)

The second term in D_V does not drop out of calculations and leads to bad ultraviolent (large k) behavior and thus to non-renormalizability. One also sees that the limitm→ 0 is not smooth.

A renormalizable gauge invariant theory of massive spin-1 fields, in which the mass is obtained by the Higgs mechanism¹² rather than as an elementary term in L, will be discussed in Chapter 4. In that case, (2.129) will still correspond to the unitary gauge, in which the physical degrees of freedom are manifest, but there are other gauges in which the m→0 limit is smooth.

2.6 ELECTROMAGNETIC INTERACTION OF CHARGED PIONS

We are now ready to combine the results of Sections 2.4and2.5. Consider the Lagrangian density

L(φ, A) = (∂µφ)^†∂^µφ−m²φ^†φ−1

4F_µνF^µν−V_I(φ, φ^†) +L^φA(φ, A) (2.130) for a complex scalar fieldφand the electromagnetic fieldA^µ. The scalar self-interactionV_I is defined in (2.89), andL^φA describes the electromagnetic interaction. Its form is dictated by the requirement of gauge invariance under (2.112). The only way to do this (without introducing non-renormalizable interactions) is the minimal electromagnetic substitution, familiar from classical and quantum mechanics. One replaces

p^µ→p^µ−qA^µ ⇐⇒i∂^µ−qA^µ (2.131) in the Lagrangian density in (2.88), whereq=e >0 is the charge of theπ⁺, and identifies the additional terms with L^φA. L will then be invariant under the generalized gauge (or local) transformation

A^µ →A^0µ =A^µ−1 e∂^µβ(x) φ→φ⁰=e^iqβ(x)/eφ=e^iβ(x)φ,

(2.132)

i.e.,L(φ, A) =L(φ⁰, A⁰). Equation (2.132) generalizes the global symmetry ofSection 2.4.1, i.e.,φ→e^iβφwhereβ= constant.

12It is also possible to construct aU(1) gauge invariant theory for a massive vector without the Higgs or other spontaneous symmetry breaking mechanism by the St¨uckelberg mechanism(Stueckelberg, 1938;

Cianfrani and Lecian, 2007).

Using the minimal substitution, L= [(∂_µ+iqA_µ)φ]^†

| {z }

(∂µ−iqAµ)φ^†

(∂^µ+iqA^µ)φ−m²φ^†φ−1

4F_µνF^µν−V_I(φ, φ^†)

= [Dµφ]^†[D^µφ]−m²φ^†φ−1

4F_µνF^µν−V_I(φ, φ^†),

(2.133)

where

D^µ≡∂^µ+iqA^µ (2.134)

is known as the gauge covariant derivative.

The gauge invariance of (2.133) is obvious except for the first term. Under (2.132), D^µφ→D^0µφ⁰=h

∂^µ+iqA^µ−iq

e∂^µβ(x)i

e^iβ(x)φ

=e^iβ(x)D^µφ (Dµφ)^†→(Dµφ)^†e^−iβ(x).

(2.135) That is, the shift inA^µcompensates for the change in∂^µφdue to the derivative ofβ, leaving Lgauge invariant. Thus, gauge invariance for the charged scalar requires the existence of a spin-1 field, dictates the form of the interaction with A^µ, forbids an elementary mass term (i.e., ¹₂m²_AA_µA^µis not gauge invariant), and restricts the form of the scalar self-interactions.

It turns out that the theory is then renormalizable.

From (2.133) one can read offiL^φA:

iL^φA=−q(∂^µφ)^†φA_µ+q(φ^†∂^µφ)A_µ+iq²A_µA^µφ^†φ

≡q(φ^†←→∂^µφ)Aµ+iq²AµA^µφ^†φ, (2.136) where iφ^†←→∂^µφ is the Noether current for the free complex scalar field¹³ in (2.108). The Feynman vertex rules can be found from (2.136), inserting the free-field expressions for φ andφ^† in (2.93) and recalling thata(~p) andb(~p) are the annihilation operators forπ⁺ and π⁻, respectively. For this purpose, it is convenient to explicitly display

∂^µφ0(x) =

Z d³~p (2π)³2Ep

−ip^µa(~p)e^−ip·x+ip^µb^†(~p)e^+ip·x

∂^µφ^†₀(x) =

Z d³~p (2π)³2Ep

−ip^µb(~p)e^−ip·x+ip^µa^†(~p)e^+ip·x ,

(2.137)

wherep= (Ep, ~p) withE_p≡p

~

p²+m² as usual.

The vertices are displayed inFigure 2.10 forq =e. The three-point vertices include a contribution−iep^µ_π+ for eachπ⁺and a +iep^µ_π− for eachπ⁻. These are always the physical momenta, whether initial or final. They can both be written as−ie¯p^µ_π±, where ¯p_π± =±p_π±

is the momentum in the direction of the arrow. There is also a four-point (seagull) vertex 2ie²g^µν, which is needed for gauge invariance. The Lorentz indices are contracted with _µ(k, λ) for an initial photon of momentum k, with ^∗_µ(k, λ) for a final photon, and with igµνDF(k) =−igµν/(k²+i) for an internal virtual photon line. Each internal pion has a propagator i∆F(k) = i/(k²−m²+i), and there is an integral R

d⁴k/(2π)⁴ over each unconstrained internal momentum.

13Interaction terms usually do not modify the form of the Noether currents. Gauge interactions of complex scalars are an exception because the interaction terms involve derivatives. For example, the conserved Noether current forLin (2.133) isJ^µ=iφ^†←→

∂^µφ−2qφ^†A^µφ, which appears in the field equation forAin the familiar Maxwell form∂µF^µν =qJ^ν.

π⁺(pi) π⁺(pf)

−ie(pi+pf)^µ γ

π⁻(pi) π⁻(pf)

+ie(pi+pf)^µ γ

π π

γ γ

+2ie²g^µν

γ

π⁺(p_π+) π⁻(pπ−)

−ie(p_π+−p_π−)^µ

π⁺(p_π+) π⁻(p_π−)

γ

−ie(p_π+−p_π−)^µ

Figure 2.10

Vertices involving charged pions and one or two photons. In the one-photon diagrams the initial pions enter from the bottom and the final leave from the top. Antiparticle (π

⁻

) vertices are obtained from particle ones by twisting the lines and replacing

p

by ¯

p ≡ −p, where p

is the physical four-momentum and ¯

p

follows the direction of the arrow. The wavy lines represent photons.

πKandππScattering

As a first example, consider the electromagnetic scatteringπ⁻K⁺ → π⁻K⁺, whereK^± is a complex field with the same electromagnetic couplings (but different mass) as π^±. They are introduced here to avoid identical particle effects. We ignore strong interactions of other non-electromagnetic couplings. There is a single tree-level diagram, as shown inFigure 2.11.

The Feynman rules lead to the transition amplitude Mf i=ie(p1+p3)^µ

−igµν

(p1−p3)²

(−ie)(p2+p4)^ν

=−4πiα(p₁+p₃)·(p₂+p₄)

(p1−p3)² =−4πiα s−u

t

,

(2.138)

where α≡e²/4πis the fine structure constant. The Mandelstam variabless, t, anduare defined in (2.31) and are related to the CM scattering angle in (2.40). From (2.57) the differential cross section in the CM is

dσ

dcosθ = 1

32πs|M_{f i}|²= πα² 2s

s/2p²+ 1 + cosθ 1−cosθ

²

−−−−−−→√ sm_K

πα² 2s

3 + cosθ 1−cosθ

² ,

(2.139)

→ associating the fields iniL^φA with the identical external particles, as shown inFigure 2.11.

The transition amplitude is

The differential cross section is again given by (2.57), but in this case there is an extra factor ¹₂ in the total cross section because the final particles are identical. At high energies,

√sm_π, one finds

14The integral is divergent for forward scattering due to the massless photon propagator pole, i.e., the long range Coulomb force. This divergence is already present for classical Coulomb scattering, and disappears for realistic situations in which screening by other charges or the finite resolution of the detector are taken into account. Similar comments apply toπ⁺π⁺for both forward and backward scattering and toπ⁻π⁺.

There are similarly two diagrams for π⁺π⁻ → π⁺π⁻, yielding

In this case, however, the final particles are not identical. Note that the amplitude forπ⁺π⁻ can be obtained from that for π⁺π⁺ by the formal substitutionsp₄→ −p₂ andp₂→ −p₄, an example ofcrossingsymmetry. That is, the amplitude for an outgoingπ⁺of momentum p is the same as that for an incoming π⁻ with momentum −p, as is apparent from the Feynman rules. Of course, the physical values ofpare different for the two cases, since p⁰ (−p⁰) must be positive for the first (second) one.

Figure 2.12

Diagrams for pion Compton scattering. The third (seagull) diagram is

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