HW #6, due Oct Toy Dirac Model, Wick s theorem, LSZ reduction formula. Consider the following quantum mechanics Lagrangian,
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1 HW #6, due Oct 5. Toy Drac Model, Wck s theorem, LSZ reducton formula. Consder the followng quantum mechancs Lagrangan, L ψ(σ 3 t m)ψ, () where σ 3 s a Paul matrx, and ψ s defned by ψ ψ σ 3. ψ s a twocomponent varable. We quantze the dynamcal varable ψ and ts canoncal conjugate momentum ψ usng the canoncal ant-commutaton relaton {ψ α,ψ β } δ αβ for α, β,. () Show that the equaton of moton (σ 3 t m)ψ 0hasapostveand a negatve energy soluton, ( ) (σ 3 t m)ue mt 0, u () () We expand the operator ψ as (σ 3 t m)ve mt 0, v ( 0 0 ) (3) ψ(t) aue mt + b ve mt (4) Show that the operators a and b satsfy the algebra of creaton and annhlaton operators, {a, a } {b, b }, usng the canoncal antcommutaton relaton {ψ α,ψ β } δ αβ where α, β,. note The Hlbert space conssts of four states, the vacuum 0 defned by a 0 b 0 0, one-partcle states a a 0, b b 0 and the par state ab a b 0. (3) Show that the Hamltonan s gven by H 0 ψ mσ 3 ψ m(a a bb ) m(a a + b b)+constant.
2 (4) Show that the Feynman propagator s gven by S F (t t ) 0 Tψ α (t ) ψ β (t ) 0 ( ) θ(t t )e m(t t ) 0 + θ(t 0 0 t )e m(t t ) de π αβ (Eσ 3 + m) αβ E m + ɛ e E(t t ) ( (Hnt: Consder two cases t >t or t <t separately, and use contour ntegral n lower or upper half plane, respectvely. To locate the poles, the fgure on p. 3 of the textbook may help.) ) αβ (5) note The above Feynman propagator s often wrtten as S F (t t ) de π e E(t t ) Eσ 3 m + ɛ (6) (5) Calculate 0 Tψ(t ) ψ(t )ψ(t 3 ) ψ(t 4 ) 0 when t >t >t 3 >t 4 n two ways. () Use Wck s theorem. () Work t out explctly usng annhlaton and creaton operators. (6) Now we add a tme-dependent perturbaton to the system: H H 0 + V, V f(t)ψ σ ψ (7) where f(t) sac-number functon of tme, and assume f(t) 0 for both the nfnte past t< T and the nfnte future t>t.calculate the ampltude 0( ) 0() 0 I Te V I (t)dt 0 I at second order n perturbaton usng the Wck s theorem. note Because of the perturbaton, the Hesenberg operator ψ(t) does not follow the equaton of moton of the free feld. However, t does follow thefreeequaton fort < T and t > T because f(t) vanshes. Therefore, the followng Hesenberg operator e mt ūψ(t) (8) reduces to the annhlaton operator a for both t < T and t > T. Smlarly, another Hesenberg operator e mt ψ(t)v (9) reduces to b n these lmts.
3 (7) Consder the followng matrx element n the Hesenberg pcture dte mt ( )ū(σ 3 t m) 0 T O(t )ψ(t) 0 (0) where O(t ) s an arbtrary (bosonc) Hesenberg operator. partal ntegraton, show that t can be smplfed to Usng 0 a(t )O(t ) O(t )a(t ) 0 out a O(t ) 0. () (8) Consder the followng matrx element n the Hesenberg pcture dt 0 T O(t ) ψ β (t) 0 [( σ 3 t m)v] β e mt () where O(t ) s an arbtrary (bosonc) Hesenberg operator. partal ntegraton, show that t can be smplfed to Usng 0 (bo(t ) O(t )a) 0 out b O(t ) 0. (3) (9) Show that the par-creaton ampltude out ab 0 n can be rewrtten as out ab 0 n dt dt e mt [ ū(σ 3 t m)] α 0 Tψ α (t ) ψ β (t ) 0 [( σ 3 t m)v] β e mt (4) note Once we have the above expresson for the ampltude, we can calculate the correlaton functon 0 Tψ α (t ) ψ β (t ) 0 usng the tme-dependent perturbaton theory to obtan a perturbatve result for the ampltude. 3
4 LSZ reducton formula and cross sectons. Real Klen Gordon feld ( Z ) n+m out p,,p n q, q m n n m d 4 x e p x ( x + m ) d 4 y j e q jy j ( yj + m ) j 0 Tφ(x ) φ(x n )φ(y ) φ(y m ) 0 (). Drac feld Z partcle(p, ±) n Z ant-partcle(p, ±) n Zout partcle(p, ±) Zout ant-partcle(p, ±) d 4 xe px T ψ(x) 0 ( )( m)u ± (p)() d 4 xe px v (p)( m)tψ(x) 0 (3) d 4 xe px ū ± (p)( )( m) 0 Tψ(x) (4) d 4 xe px 0 T ψ(x)( m)v (p) (5) Repeated applcaton of above formulae can convert all (ant)-partcles n ntal or fnal states nto the feld operators so that the tme-dependent perturbaton theory allows you to work out ampltudes. 3. Cross sectons M(π) 4 δ 4 ( p q q ) out p,,p n q,q n (6) where σ s β n d p M (π) 4 δ 4 ( p q q ) (7) d 3 p d p (8) (π) 3 E p s (q + q ) (9) (m β + m ( ) m + m ). (0) s s Here, m q and m q state. are the mass squareds of partcles n the ntal
5 An Explct Example: e + e µ + µ We would lke to calculate the cross secton of the process e + e µ + µ. The algorthm s the followng. () Wrte the matrx element out µ + µ e + e n n terms of Hesenberg feld operators usng the LSZ reducton formula. () Rewrte the correlaton functon of Hesenberg feld operators n terms of feld operators n the nteracton pcture. (3) Calculate the correlaton functon of the feld operators n the nteracton pcture usng the Wck s theorm. (4) Evaluate the ampltude. (5) Stck the ampltude nto the formula of the cross secton. (6) If you are computng hgher order correctons, compute the twopont functons to calculate Z factors. Below, we dscuss only the leadng order result so that we can set all Z. Let me show each of the steps brefly. The nteracton Hamltonan n QEDsgvenby H nt e d 4 x(ēγ µ e + µγ µ µ)a µ. () () LSZ reducton formula ) ( ) (Z µ Z e out µ (p,h )µ + (p,h ) e (p 3,h 3 )e + (p 4,h 4 ) n 4 d 4 x e (p x +p x p 3 x 3 p 4 x 4 ) [ū h (p )( )( x m µ )][ v h4 (p 4 )( x4 m e )] 0 Tµ(x ) µ(x )ē(x 3 )e(x 4 ) 0 [( x m µ )v h (p )][( )( x3 m e )u h3 (p 3 )] () () Interacton pcture. 0 Tµ(x ) µ(x )ē(x 3 )e(x 4 ) 0 0 Tµ I(x ) µ I (x )ē I (x 3 )e I (x 4 )e d 4 yh I (y) 0 0 Te d 4 yh I (y) 0 (3) (3) Wck s theorem 4 d 4 x e (p x +p x p 3 x 3 p 4 x 4 ) 0 Tµ(x ) µ(x )ē(x 3 )e(x 4 ) 0 4 d 4 x e (p x +p x p 3 x 3 p 4 x 4 ) ( e) d 4 y d 4 y!
6 0 Tµ I (x ) µ I (x )ē I (x 3 )e I (x 4 )H I (y )H I (y ) 0 ( e) d 4 y d 4 y 0 H I (y )H I (y ) 0 + O(e) 4 (! )( ) γ µ γ ν p m µ p m µ p 4 m e p 3 m e g µν q + ɛ (π)4 δ 4 (p + p p 3 p 4 )+O(e) 4 (4) Here, I took the Feynman gauge ξ for the photon propagator. Hereafter, IdropO(e) 4. q (p + p )(p 3 + p 4 ) s the four-momentum n the photon propagator. (4) Ampltude. Note that the LSZ reducton formula precsely cancels the extra fermon propagators. M ( e) [ū h (p )γ µ v h (p )] [ v h4 (p 4 )γ ν u h3 (p 3 )] g µν (p + p ) + ɛ (5) Then one can plug n the explct forms of u and v spnors to obtan the ampltude. (5) Cross secton. σ s β d p d p M (π) 4 δ 4 (p + p p 3 p 4 ). (6) In the center of momentum frame, the phase space ntegral can be drastcally smplfed: d p d p (π) 4 δ 4 (p + p p 3 p 4 ) β f d cos θ π dφ 8π 0 π, (7) where β f s defned by the same formula as β except the mass squareds are those of the fnal state partcles. In partcular, we have m m m µ n ths case and hence β f 4m µ s. (8)
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