Loop corrections in Yukawa theorybased on S-51

Let’s consider the theory of a pseudoscalar field and a Dirac field:

and terms not allowed!the only couplings allowed by symmetries!

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A simple pole at with residue one implies:

We will calculate 1-loop corrections in the OS renormalization scheme:

(the LSZ formula is valid as it is; the lagrangian mass is the physical mass; propagators have appropriate poles with unit residue)

For the scalar propagator we found:we will assume

we use these conditions to fix and .

either or

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A simple pole at with residue one implies:

Similarly, the exact Dirac propagator can be written as:

a fermion plus a scalar

we use these conditions to fix and .

contains inverse matrices, but we can think of as an analytic function of The exact fermion propagator can be written as:

sum of 1PI diagrams with 2 external lines (and ext. propagators removed)

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Let’s evaluate diagrams contributing to the scalar propagator:

vertex factor:

was your homework

extra -1 for fermion loop (Homework S-51.1); and the trace

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Let’s evaluate the fermion loop:

numerator:

Combining the denominators we have:

changing the integration variable we get:

where

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where

the integral diverges in 4 spacetime dimensions and so we analytically continue it to ; we also make the replacement to keep the coupling dimensionless: 0

we use the usual formulas to get:

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0

Putting things together:

we find:

the divergent piece has the form that permits cancellation by the counterterms!

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Collecting contributions of all diagrams:

we get:

we can impose by writing:

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we can impose by writing:

fixed by imposing:

no correction in the OS scheme!

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Let’s evaluate diagrams contributing to the fermion propagator:

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combining the denominators we have:

the integral diverges in 4 spacetime dimensions and so we analytically continue it to ; we also make the replacement to keep the coupling dimensionless...

0

using the usual formulas we get:

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Adding contributions of both diagrams:

we get:

we can impose by writing:

fixed by imposing:

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Next, let’s evaluate the diagram contributing to the Yukawa vertex:

vertex factor:

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numerator:

combining the denominators we have:

using0

divergent finite

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finite piece fixed by imposing some condition, e.g.:

We proceed in a usual way, and get:

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Finally, let’s evaluate diagrams contributing to the 4-point vertex:

+ 5 others with permuted external lines

and

was your homework

+

+

extra -1 for fermion loop

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calculation is straightforward; let’s calculate the divergent part only:

divergent pieces are sufficient to find beta functions of the theory

we can set external momenta to zero (divergent piece doesn’t depend on these):

numerator

denominator

using the usual formula for the loop integral we find:

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Putting things together:

+ ... + + ...

we find:

that concludes the calculation of 1-loop corrections to the Yukawa theory

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Beta functions in Yukawa theorybased on S-52

The lagrangian in terms of renormalized fields and parameters:

can be also written in terms of bare fields and parameters (independent of )!

The dictionary for couplings:

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bare parameters do not depend on

must be finite as

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we obtained the beta functions of the Yukawa theory.

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Review of Feynman rules for QED

external lines:

incoming electron

outgoing electron

vertex and the rest of the diagram

incoming positron

outgoing positron

incoming photon

outgoing photonREVIEW

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vertex

draw all topologically inequivalent diagrams

for internal lines assign momenta so that momentum is conserved in each vertex (the four-momentum is flowing along the arrows)

propagators

for each internal photon

one arrow in and one out

the arrow for the photon can point both ways

for each internal fermionREVIEW

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sum over all the diagrams and get

spinor indices are contracted by starting at the end of the fermion line that has the arrow pointing away from the vertex, write or ; follow the fermion line, write factors associated with vertices and propagators and end up with spinors or .

assign proper relative signs to different diagrams

follow arrows backwards!

draw all fermion lines horizontally with arrows from left to right; with left end points labeled in the same way for all diagrams; if the ordering of the labels on the right endpoints is an even (odd) permutation of an arbitrarily chosen ordering then the sign of that diagram is positive (negative).

additional rules for counterterms and loops

The vector index on each vertex is contracted with the vector index on either the photon propagator or the photon polarization vector.

REVIEW

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and use covariant derivatives where:

Scalar electrodynamicsbased on S-61

Consider a theory describing interactions of a scalar field with photons:

is invariant under the global U(1) symmetry:

we promote this symmetry to a local symmetry:

so that

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A gauge invariant lagrangian for scalar electrodynamics is:

The Noether current is given by:

depends explicitly on the gauge field multiplied by e = electromagnetic current

New vertices:

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external lines:

incoming selectron

outgoing selectron

vertex and the rest of the diagram

incoming spositron

outgoing spositron

Additional Feynman rules:

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vertices:

incoming selectron outgoing selectron

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Let’s use our rules to calculate the amplitude for :

and we use to calculate the amplitude-squared, ...

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Loop corrections in QEDbased on S-62

Let’s calculate the loop corrections to QED:

adding interactions results in counterterms

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The exact photon propagator:

the sum of 1PI diagrams with two external photon lines (and the external propagators removed)

we saw that we can add or ignore terms containing

the free photon propagator in a generalized Feynman gauge or gauge:

Feynman gauge

Lorentz (Landau) gauge

The observable amplitudes^2 cannot depend on which suggests:

(we will prove that later)

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In the OS scheme we choose:

and so we can write it as:

is the projection matrixwe can write the propagator as:

summing 1PI diagrams we get:

has a pole at with residue

to have properly normalized states in the LSZ160

Let’s now calculate the at one loop:

extra -1 for fermion loop; and the trace

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we ignore terms linear in q

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the integral diverges in 4 spacetime dimensions and so we analytically continue it to ; we also make the replacement to keep the coupling dimensionless:

see your homework

is transverse :)

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the integral over q is straightforward:

imposing fixes

and

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Let’s now calculate the fermion propagator at one loop:

the exact propagator in the Lehmann-Källén form:

no isolated pole with well defined residue

it is a signal of an infrared divergence associated with the massless photon; a simple way out is to introduce a fictitious photon mass. After adding contributions to the cross section from processes that are indistinguishable due to detector inefficiencies it is safe to take ; it turns out that in QED we do not have to abandon the OS scheme.

using this procedure we can write the exact propagator as:

a simple pole at with residue one implies: ,

we use these conditions to fix and .

sum of 1PI diagrams with 2 external lines (and ext. propagators removed)

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There is only one diagram contributing at one loop level:

fictitious photon mass

the photon propagator in the Feynman gauge:

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following the usual procedure:

we get:

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we can impose by writing:

we set Z’s to cancel divergent parts

fixed by imposing:

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Finally, let’s evaluate the diagram contributing to the vertex:

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combining denominators...

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continuing to d dimensions

evaluating the loop integral we get:

the infinite part can be absorbed by Z

the finite part of the vertex function is fixed by a suitable condition.

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The vertex function in QEDbased on S-63

For the vertex function we can impose a physically meaningful condition:

momentum conservation allows all three particles to be on shell:

and so we can define the electron charge via:

consistent with the definition given by Coulomb’s law

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exact propagator and exact vertex approach their tree level values as

Consider electron-electron scattering:

finite when

physically, means that the electron’s momentum changes very little during the scattering; measuring the slight deflection in the trajectory of the electron is how we can measure the coefficient in the Coulomb’s law.

Our on-shell condition enforces and so the condition imposed on the vertex function can be written as:

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Now we use our condition to completely determine the vertex function:

we can use the freedom to choose the finite part of

fixed by imposing:

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we can set since these terms come from the finite piece

infrared regulator is needed

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To calculate e-e scattering we need the vertex function for arbitrary ;

we need to calculate:

using

we can rewrite it in terms of and

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antisymmetric under , and so it doesn’t contribute when we integrate over Feynman’s parameterssymmetric under

Gordon identity

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putting everything together

we get:

where the form factors are:

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where the form factors are:

can be further simplified .... but we will be mostly interested in the values for :

the fine-structure constant

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The magnetic moment of the electronbased on S-64

For the vertex function we have found (at one loop):

momentum of an incoming photon

where the form factors for are:

We can obtain the vertex function from the quantum action:

incoming photon

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the wave packet (rotationally invariant) and sharply peaked at , with

We define the magnetic moment in the following way:

we take the photon field to be a classical field that corresponds to a constant magnetic field in the z direction:

all other components are zero

the magnetic moment of a normalized quantum state with definite angular momentum in the direction is defined as:

normalized state of an electron at rest, with spin along the z axis:

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Let’s evaluate it now:

we find:

and integrate by parts (the surface term will vanish thanks to wave packets)

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f is rotationally invariant and so the derivative is odd in ; in addition it is also

odd in due to ;

thus this term doesn’t contribute!

expand for small p, take derivative and set

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we find:

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we find that the magnetic moment of the electron is:

Bohr magnetonLandé g factor

corrections of order were calculated!

anomalous magnetic moment of the electron:

exp. value is: .0011659208 (6)

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Loop corrections in scalar electrodynamicsbased on S-65

Let’s outline the calculation of loop corrections in scalar electrodynamics:

(in the scheme)

new vertices:

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The one-loop corrections to the photon propagator:

in the scheme we find:

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The one-loop corrections to the scalar propagator:

in the scheme we find:

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The one-loop corrections to the three-point vertex:

in the scheme we find:

it is convenient to work in the Lorentz gauge and choosing the momentum of incoming scalar = 0 if we are interested in the divergent part only

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The one-loop corrections to the scalar-scalar-photon-photon vertex:

in the scheme we find:

plus many diagrams that do not contribute in the Lorentz gauge with external momenta = 0

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The one-loop corrections to the four-scalar vertex:

in the scheme we find:

plus many diagrams that do not contribute in the Lorentz gauge with external momenta = 0

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Beta functions in quantum electrodynamicsbased on S-66

Let’s calculate the beta function in QED:

the dictionary:

Note !

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following the usual procedure:

we find:

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or equivalently:

For a theory with N Dirac fields with charges :

= 1

we find:

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For completeness, let’s calculate the beta functions in scalar ED:

the dictionary:

Note !

needed for consistency of two different relations between renormalized and bared couplings

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following the usual procedure we find:

Generalizing to the case of arbitrary number of complex scalar and Dirac fields:

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