Upload
vuongtruc
View
226
Download
4
Embed Size (px)
Citation preview
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!
128
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
129
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)
130
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
131
Let’s evaluate the fermion loop:
numerator:
Combining the denominators we have:
changing the integration variable we get:
where
132
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:
133
0
Putting things together:
we find:
the divergent piece has the form that permits cancellation by the counterterms!
134
Collecting contributions of all diagrams:
we get:
we can impose by writing:
135
we can impose by writing:
fixed by imposing:
no correction in the OS scheme!
136
Let’s evaluate diagrams contributing to the fermion propagator:
137
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:
138
Adding contributions of both diagrams:
we get:
we can impose by writing:
fixed by imposing:
139
Next, let’s evaluate the diagram contributing to the Yukawa vertex:
vertex factor:
140
numerator:
combining the denominators we have:
using0
divergent finite
141
finite piece fixed by imposing some condition, e.g.:
We proceed in a usual way, and get:
142
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
143
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:
144
Putting things together:
+ ... + + ...
we find:
that concludes the calculation of 1-loop corrections to the Yukawa theory
145
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:
146
147
bare parameters do not depend on
must be finite as
148
we obtained the beta functions of the Yukawa theory.
149
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
150
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
151
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
152
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
153
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:
154
external lines:
incoming selectron
outgoing selectron
vertex and the rest of the diagram
incoming spositron
outgoing spositron
Additional Feynman rules:
155
vertices:
incoming selectron outgoing selectron
156
Let’s use our rules to calculate the amplitude for :
and we use to calculate the amplitude-squared, ...
157
Loop corrections in QEDbased on S-62
Let’s calculate the loop corrections to QED:
adding interactions results in counterterms
158
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)
159
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
161
we ignore terms linear in q
162
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 :)
163
the integral over q is straightforward:
imposing fixes
and
164
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)
165
There is only one diagram contributing at one loop level:
fictitious photon mass
the photon propagator in the Feynman gauge:
166
following the usual procedure:
we get:
167
we can impose by writing:
we set Z’s to cancel divergent parts
fixed by imposing:
168
Finally, let’s evaluate the diagram contributing to the vertex:
169
combining denominators...
170
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.
171
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
172
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:
173
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:
174
we can set since these terms come from the finite piece
infrared regulator is needed
175
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
176
antisymmetric under , and so it doesn’t contribute when we integrate over Feynman’s parameterssymmetric under
Gordon identity
177
putting everything together
we get:
where the form factors are:
178
where the form factors are:
can be further simplified .... but we will be mostly interested in the values for :
the fine-structure constant
179
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
180
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:
181
Let’s evaluate it now:
we find:
and integrate by parts (the surface term will vanish thanks to wave packets)
182
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
183
we find:
184
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)
185
Loop corrections in scalar electrodynamicsbased on S-65
Let’s outline the calculation of loop corrections in scalar electrodynamics:
(in the scheme)
new vertices:
186
The one-loop corrections to the photon propagator:
in the scheme we find:
187
The one-loop corrections to the scalar propagator:
in the scheme we find:
188
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
189
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
190
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
191
Beta functions in quantum electrodynamicsbased on S-66
Let’s calculate the beta function in QED:
the dictionary:
Note !
192
following the usual procedure:
we find:
193
or equivalently:
For a theory with N Dirac fields with charges :
= 1
we find:
194
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
195
following the usual procedure we find:
Generalizing to the case of arbitrary number of complex scalar and Dirac fields:
196