KIT – University of the State of Baden-Wuerttemberg and National Research Center of the Helmholtz Association

INSTITUTE OF EXPERIMENTAL PARTICLE PHYSICS (IEKP) – PHYSICS FACULTY

www.kit.edu

From Lagrangian Density to Observable

Roger Wolf19. Mai 2016

Institute of Experimental Particle Physics (IEKP)2

Schedule for today

Review of the QM model of scattering

Perturbative series

Introduction of the propagator

1

3

2

● What is a propagator?

● Is the following statement true: “the perturbative series is a Taylor expansion”?

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Lagrangian Density → Observable

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Scattering matrix transforms initial state wave function into scattering wave ( ).

QM model of particle scattering

● Consider incoming collimated beam of projectile particles on a target particle:

Initial particle: described by plain wave .

Observation (in ): projection of plain wave out of spherical scat-tering wave .

Localized potential.

Spherical scat-tering wave .

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Scattering matrix transforms initial state wave function into scattering wave ( ).

QM model of particle scattering

● Consider incoming collimated beam of projectile particles on a target particle:

Initial particle: described by plain wave .

Observation (in ): projection of plain wave out of spherical scat-tering wave .

Localized potential.

Spherical scat-tering wave .

Observation probability:

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Solution for

● In the case of fermion scattering the scattering wave is obtained as a solution of the inhomogeneous Dirac equation for an interacting field:

● The inhomogeneous Dirac equation is analytically not solvable.

(+)

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● In the case of fermion scattering the scattering wave is obtained as a solution of the inhomogeneous Dirac equation for an interacting field:

(+)

● The inhomogeneous Dirac equation is analytically not solvable. A formal solution can be obtained by the Green's Function :

Solution for

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● In the case of fermion scattering the scattering wave is obtained as a solution of the inhomogeneous Dirac equation for an interacting field:

(+)

● The inhomogeneous Dirac equation is analytically not solvable. A formal solution can be obtained by the Green's Function :

Solution for

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● In the case of fermion scattering the scattering wave is obtained as a solution of the inhomogeneous Dirac equation for an interacting field:

(+)

● The inhomogeneous Dirac equation is analytically not solvable. A formal solution can be obtained by the Green's Function :

● This is not a solution to (+), since appears on the left- and on the right-hand side of the equation. It turns the differential equation into an integral equation. It propagates the solution from the point to .

Solution for

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Green's function in Fourier space

● The best way to find the Green's function is to go to the Fourier space:

(Fourier transform)

Applying the Dirac equation to the Fourier transform of turns the derivative into a product operator:

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Green's function in Fourier space

● The best way to find the Green's function is to go to the Fourier space:

Applying the Dirac equation to the Fourier transform of turns the derivative into a product operator:

(Fourier transform)

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Green's function in Fourier space

● The best way to find the Green's function is to go to the Fourier space:

Applying the Dirac equation to the Fourier transform of turns the derivative into a product operator:

(Fourier transform)

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Green's function in Fourier space

● The best way to find the Green's function is to go to the Fourier space:

Applying the Dirac equation to the Fourier transform of turns the derivative into a product operator:

From the uniqueness of the Fourier transformation the solution for follows:

(Fourier transform)

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Fermion propagator

● The Fourier transform of the Green's function is called fermion propagator:

(fermion propagator)

● The fermion propagator is a matrix, which acts in the Spinor space.

● It is only defined for virtual fermions since .

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Fermion propagator ↔ Green's function

● The Green's function can be obtained from the propagator by inverse Fourier transformation:

● This integral can be solved with the methods of function theory.

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Fermion propagator ↔ Green's function

● The Green's function can be obtained from the propagator by inverse Fourier transformation:

● This integral can be solved with the methods of function theory.

● has two poles in the integration plane (at ).

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Excursion into function theory

cf. Freitag/Busam Funktionentheorie

http://www.springer.com/de/book/9783540317647

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Residual theorem

● When integrating a “well behaved” function w/o poles in the complex plain the path integral along any closed path is 0:

● When integrating a “well behaved” function w/ poles in the complex plain the solution is the sum of “residuals” of the poles surrounded by the path:

No matter how is chosen, as long as it includes .

Example:

Example:

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The Green's function (time integration for )

● For ( for ):→ close contour in lower plane & calculate integral from residual of enclosed pole.

pole at: residual:

● Choose path in complex plain to circumvent poles:

Sign due to sense of integration.

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The Green's function (time integration for )

● For ( for ):→ close contour in lower plane & calculate integral from residual of enclosed pole.

● Choose path in complex plain to circumvent poles:

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The Green's function (time integration for )

● For ( for ):→ close contour in upper plane & calculate integral from residual of enclosed pole.

● Choose path in complex plain to circumvent poles:

pole at: residual:Sign due to sense of integration.

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The Green's function (time integration for )

● For ( for ):→ close contour in upper plane & calculate integral from residual of enclosed pole.

● Choose path in complex plain to circumvent poles:

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The Green's function (time integration for )

● For ( for ):→ close contour in upper plane & calculate integral from residual of enclosed pole.

● Choose path in complex plain to circumvent poles:

Sign due to in integral kernel.

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The Green's function (Nota Bene)

● The bending of the integration path can be avoided by shifting the poles by .

● Choose path in complex plain to circumvent poles:

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The Green's function (Nota Bene)

● The bending of the integration path can be avoided by shifting the poles by .

● Choose path in complex plain to circumvent poles:

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The Green's function (Nota Bene)

● The bending of the integration path can be avoided by shifting the poles by .

● Choose path in complex plain to circumvent poles:

(fermion propagator)

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Summary of time evolution

(Fermion propagator in momentum space)

● Green's function (for , forward evolution):

● Green's function (for , backward evolution):

● But why did I choose explicitly THIS integration path and not another one?

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Summary of time evolution

(Fermion propagator in momentum space)

● The chosen integration path defines the time evolution of the solution.

particle w/ pos. energy traveling forward in time.

particle w/ pos. energy traveling backward in time.

● General solution to (inhomogeneous) Dirac equation:

particle w/ neg. energy traveling forward in time.

particle w/ neg. energy traveling backward in time.

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The perturbative series

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The perturbative series

● The integral equation can be solved iteratively:

● 0th order perturbation theory:

( = solution of the homogeneous Dirac equation)

● Just take as solution (→ boring).

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The perturbative series

● The integral equation can be solved iteratively:

● 0th order perturbation theory:

● 1st order perturbation theory:

( = solution of the homogeneous Dirac equation)

● Just take as solution (→ boring).

● Assume that is close enough to actual solution on RHS.

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The perturbative series

● The integral equation can be solved iteratively:

● 0th order perturbation theory:

● 1st order perturbation theory:

● 2nd order perturbation theory:

( = solution of the homogeneous Dirac equation)

● Just take as solution (→ boring).

● Assume that is close enough to actual solution on RHS.

● Take as better approximation at RHS to solve inhomogene-ous equation.

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The perturbative series

● The integral equation can be solved iteratively:

● 0th order perturbation theory:

● 1st order perturbation theory:

● 2nd order perturbation theory:

( = solution of the homogeneous Dirac equation)

● Just take as solution (→ boring).

● Assume that is close enough to actual solution on RHS.

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The perturbative series

● The integral equation can be solved iteratively:

● 0th order perturbation theory:

● 1st order perturbation theory:

● 2nd order perturbation theory:

( = solution of the homogeneous Dirac equation)

This procedure is justified since .

● Just take as solution (→ boring).

● Assume that is close enough to actual solution on RHS.

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The matrix element

● is obtained from the projection of the scattering wave on :

● 1st order perturbation theory:

“LO” “NLO”

cf. slide 7

cf. slide 28

For and respectively.

NB: the time integration has already been carried out for the backward evolution from to to arrive at the equation of slide 28.

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The matrix element

● is obtained from the projection of the scattering wave on :

● 1st order perturbation theory:

This corresponds exactly to the IA term in , including the multiplication by (cf. Lecture-05 slide 39).

“LO” “NLO”

(1st order matrix element)

http://www-ekp.physik.uni-karlsruhe.de/~quast/vorlesung/TP2HiggsSS16/Higgs-VL-05-HiggsMechanism.pdf

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Concluding Remarks

● Amplitude of scattering processes can be obtained from a QM model via perturbation theory.

● Introduced propagator as formal solution of the equation of motion for fermion case.

(1st order matrix element)

● Derived 1st order matrix element.

We are not yet done: since projectile is back-scattered also evolves! This part will be discussed during the next lecture.

● In the next lecture we will complete the picture of Feynman rules for the simple example of electron scattering.

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Backup

Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38