Srednicki 5, 9, 10. 2. The LSZ ... REVIEW. 10 multiparticle states: is a Lorentz
invariant number in general, .... vacuum diagrams are omitted from the sum.
The LSZ reduction formula
W IE
based on S-5
In order to describe scattering experiments we need to construct appropriate initial and final states and calculate scattering amplitude.
Quantum Field Theory II
Summary of free theory:
PHYS-P 622
one particle state:
Radovan Dermisek, Indiana University
V E
vacuum state is annihilated by all a’s:
R
then, one particle state has normalization:
Notes based on: M. Srednicki, Quantum Field Theory Chapters: 13, 14, 16-21, 26-28, 51, 52, 61-68, 44, 53, 69-74, 30-32, 84-86, 75, 87-89, 29
normalization is Lorentz invariant! see e.g. Peskin & Schroeder, p. 23
1
3
Let’s define a time-independent operator:
W IE
Review of scalar field theory
wave packet with width !
V E
that creates a particle localized in the momentum space near
Srednicki 5, 9, 10
and localized in the position space near the origin. (go back to position space by Fourier transform)
is a state that evolves with time (in the Schrödinger picture), wave packet propagates and spreads out and so the particle is localized far from the origin in at .
R
for in the past.
is a state describing two particles widely separated
In the interacting theory 2
is not time independent 4
A guess for a suitable initial state:
Thus we have:
W IE
we can normalize the wave packets so that
Similarly, let’s consider a final state:
V E
where again
The scattering amplitude is then:
R
W IE
or its hermitian conjugate:
V E
we put in time ordering(without changing anything)
The scattering amplitude:
and
R
is then given as (generalized to n i- and n’ f-particles):
5
7
A useful formula:
R
V E
Integration by parts, surface term = 0, particle is localized, (wave packet needed).
W IE
W IE
Lehmann-Symanzik-Zimmermann formula (LSZ)
V E
Note, initial and final states now have delta-function normalization, multiparticle generalization of .
R
We expressed scattering amplitudes in terms of correlation functions! Now we need to learn how to calculate correlation functions in interacting quantum field theory.
is 0 in free theory, but not in interacting one!
E.g. 6
8
Comments:
multiparticle states:
we assumed that creation operators of free field theory would work comparably in the interacting theory ...
acting on ground state:
V E
we want
, so that
R
W IE
is a Lorentz invariant number
W IE
V E
is a Lorentz invariant number
in general, creates some multiparticle states. One can show that the overlap between a one-particle wave packet and a multiparticle wave packet goes to zero as time goes to infinity.
is a single particle state
R
otherwise it would create a linear combination of the ground state and a single particle state
see the discussion in Srednicki, p. 40-41
By waiting long enough we can make the multiparticle contribution to the scattering amplitude as small as we want.
we can always shift the field by a constant so that 9
11
Summary:
one particle state:
Scattering amplitudes can be expressed in terms of correlation functions of fields of an interacting quantum field theory:
W IE
W IE
is a Lorentz invariant number
V E
we want
,
since this is what it is in free field theory, correctly normalized one particle state.
R
V E
Lehmann-Symanzik-Zimmermann formula (LSZ)
creates a
R
provided that the fields obey:
we can always rescale (renormalize) the field by a constant so that .
these conditions might not be consistent with the original form of lagrangian! 10
12
Consider for example:
W IE
After shifting and rescaling we will have instead:
V E
R
W IE
it can be also written as:
R
V E
epsilon trick leads to additional factor; to get the correct normalization we require:
and for the path integral of the free field theory we have found:
13
15
Path integral for interacting field
W IE
W IE
based on S-9
Let’s consider an interacting “phi-cubed” QFT:
V E
with fields satisfying:
R
assumes
thus in the case of:
V E
the perturbing lagrangian is:
we want to evaluate the path integral for this theory:
R
counterterm lagrangian in the limit
14
we expect
and
we will find
and 16
Let’s look at Z( J ) (ignoring counterterms for now). Define:
V = 2, E = 0 ( P = 3 ):
W IE
!
exponentials defined by series expansion:
V E
! ! !
V E
!! !
number of surviving sources, (after taking all derivatives) E (for external) is
1 = 8
3V derivatives can act on 2P sources in (2P)! / (2P-3V)! different ways e.g. for V = 2, P = 3 there is 6! different terms
R
!
dx1
!
" " " " " "
x2
x1
2! 6 6 3! 2 2 2
E = 2P - 3V
!
! ! ! ! ! !
3! 3! 3! 2
let’s look at a term with particular values of P (propagators) and V (vertices):
R
W IE !
!
1 i
1 i
1 i
dx2 (iZg g)2 ∆(x1 − x1 ) ∆(x1 − x2 ) ∆(x1 − x1 )
symmetry factor
17
Feynman diagrams:
V = 2, E = 0 ( P = 3 ):
!
! ! !
1 = 12
!
dx1
!
1 i
1 i
vertex joining three line segments stands for
a filled circle at one end of a line segment stands for a source
" " " " " "
x1
2! 6 6 3! 2 2 2
W IE
a line segment stands for a propagator
!
! ! ! ! ! !
3! 3! 2 2 2
R
W IE !
!
V E
!! !
19
x2
R
V E
e.g. for V = 1, E = 1
What about those symmetry factors?
1 i
dx2 (iZg g)2 ∆(x1 − x2 ) ∆(x1 − x2 ) ∆(x1 − x2 )
symmetry factor 18
symmetry factors are related to symmetries of Feynman diagrams...
What about those symmetry factors?
20
Symmetry factors: we can rearrange three derivatives without changing diagram
V E
W IE
we can rearrange two sources
R
W IE
we can rearrange three vertices
R
we can rearrange propagators
this in general results in overcounting of the number of terms that give the same result; this happens when some rearrangement of derivatives gives the same match up to sources as some rearrangement of sources; this is always connected to some symmetry property of the diagram; factor by which we overcounted is the symmetry factor
V E
21
V E
W IE
the endpoints of each propagator can be swapped and the effect is duplicated by swapping the two vertices
R
23
R
propagators can be rearranged in 3! ways, and all these rearrangements can be duplicated by exchanging the derivatives at the vertices
22
W IE
V E
24
V E
R
W IE
R
W IE
V E
25
R
V E
W IE
27
R 26
W IE
V E
28
All these diagrams are connected, but Z( J ) contains also diagrams
V E
R
that are products of several connected diagrams:
W IE
W IE
e.g. for V = 4, E = 0 ( P = 6 ) in addition to connected diagrams we also have :
V E
A general diagram D can be written as:
R
the number of given C in D
additional symmetry factor not already accounted for by symmetry factors of connected diagrams; it is nontrivial only if D contains identical C’s:
particular connected diagram
29
All these diagrams are connected, but Z( J ) contains also diagrams
31
Now
that are products of several connected diagrams:
is given by summing all diagrams D: any D can be labeled by a set of n’s
W IE
W IE
e.g. for V = 4, E = 0 ( P = 6 ) in addition to connected diagrams we also have :
V E
and also:
and also:
R
V E
thus we have found that connected diagrams.
R
is given by the exponential of the sum of
imposing the normalization means we can omit vacuum diagrams (those with no sources), thus we have:
30
vacuum diagrams are omitted from the sum 32
If there were no counterterms we would be done:
to make sense out of it, we introduce an ultraviolet cutoff
W IE
in that case, the vacuum expectation value of the field is:
V E
R
(the source is “removed” by the derivative)
we used
V E
the integral is now convergent:
only diagrams with one source contribute:
and we find:
W IE
and in order to keep Lorentz-transformation properties of the propagator we make the replacement:
and indeed,
R
we will do this type of calculations later...
is purely imaginary.
after choosing Y so that
since we know
which is not zero, as required for the LSZ; so we need counterterm
we can take the limit Y becomes infinite
... we repeat the procedure at every order in g 33
Including term in the interaction lagrangian results in a new type of vertex on which a line segment ends e.g. corresponding Feynman rule is:
V E
at the lowest order of g only
R
in order to satisfy
35
e.g. at
W IE
contributes:
we have to sum up:
W IE
V E
and add to Y whatever
term is needed to maintain
...
this way we can determine the value of Y order by order in powers of g. Adjusting Y so that means that the sum of all connected diagrams with a single source is zero!
R
we have to choose:
In addition, the same infinite set of diagrams with source replaced by ANY subdiagram is zero as well. Rule: ignore any diagram that, when a single line is cut, fall into two parts, one of which has no sources. = tadpoles
Note, must be purely imaginary so that Y is real; and, in addition, the integral over k is ultraviolet divergent. 34
36
Scattering amplitudes and the Feynman rules
all that is left with up to 4 sources and 4 vertices is:
V E
R
W IE
W IE
based on S-10 We have found Z( J ) for the “phi-cubed” theory and now we can calculate vacuum expectation values of the time ordered products of any number of fields.
Let’s define exact propagator:
R
thus we find:
V E
short notation:
W contains diagrams with at least two sources
37
finally, let’s take a look at the other two counterterms:
we get
V E
W IE
R
in
W IE
V E
we have dropped terms that contain
we used integration by parts
R
for every diagram with a propagator there is additional one with this vertex
we have calculated
39
4-point function:
it results in a new vertex at which two lines meet, the corresponding vertex factor or the Feynman rule is
Summary:
+ ...
does not correspond to any interaction; when plugged to LSZ, no scattering happens
Let’s define connected correlation functions:
theory and expressed it as
where W is the sum of all connected diagrams with no tadpoles and at least two sources!
and plug these into LSZ formula. 38
40
For two incoming and two outgoing particles the LSZ formula is:
W IE
W IE
at the lowest order in g only one diagram contributes:
V E
S=8
derivatives remove sources in 4! possible ways, and label external legs in 3 distinct ways:
R
V E
and we have just written terms of propagators.
We find:
each diagram occurs 8 times, which nicely cancels the symmetry factor.
R
in
The LSZ formula highly simplifies due to:
41
43
General result for tree diagrams (no closed loops): each diagram with a distinct endpoint labeling has an overall symmetry factor 1. Let’s finish the calculation of
z
y
V E
W IE
putting together factors for all pieces of Feynman diagrams we get:
R
R 42
W IE
V E
44
four-momentum is conserved in scattering process
Additional rules for diagrams with loops:
Let’s define:
V E
R
W IE
divide by a symmetry factor
V E
include diagrams with counterterm vertex that connects two propagators, each with the same momentum k; the value of the vertex is
scattering matrix element
From this calculation we can deduce a set of rules for computing
W IE
a diagram with L loops will have L internal momenta that are not fixed; integrate over all these momenta with measure
R
.
now we are going to use
to calculate cross section...
45
47
Lehmann-Källén form of the exact propagator
Feynman rules to calculate
:
V E
based on S-13
W IE
What can we learn about the exact propagator from general principles? Let’s define the exact propagator: The field is normalized so that
for each incoming and outgoing particle draw an external line and label it with four-momentum and an arrow specifying the momentum flow
Normalization of a one particle state in d-dimensions:
draw all topologically inequivalent diagrams
R
for internal lines draw arrows arbitrarily but label them with momenta so that momentum is conserved in each vertex assign factors:
1
The d-dimensional completeness statement:
for each external line for each internal line with momentum k for each vertex
identity operator in one-particle subspace
Lorentz invariant phase-space differential
sum over all the diagrams and get 46
48
Let’s also define the exact propagator in the momentum space:
In free field theory we found:
it has an isolated pole at
with residue one!
What about the exact propagator in the interacting theory?
49
51
Let’s insert the complete set of energy eigenstates between the two fields; for we have:
ground state, 0 - energy
Let’s define the spectral density:
one particle states
multiparticle continuum of states
then we have:
specified by the total three momentum k and other parameters: relative momenta, ..., denoted symbolically by n 50
52
It is convenient to define
similarly:
to all orders via the geometric series:
and we can plug them to the formula for time-ordered product: One Particle Irreducible diagrams - 1PI (still connected after any one line is cut) was your homework
we get: 1PI diagrams contributing at
or, in the momentum space:
level:
Lehmann-Källén form of the exact propagator it has an isolated pole at
with residue one! 53
Loop corrections to the propagator
55
It is convenient to define
to all orders via the geometric series:
based on S-14
The exact propagator:
contributing diagrams at
level:
sum of connected diagrams
One Particle Irreducible diagrams - 1PI (still connected after any one line is cut)
we can sum up the series and get:
following the Feynman rules we get: where, the self-energy is:
we know that it has an isolated pole at and so we will require:
with residue one! to fix A and B.
54
56
let’s get back to
calculation:
The second step: Wick’s rotation to evaluate the integral over q integration contour along the real axis can be rotated to the imaginary axis without passing through the poles
(is divergent for
and convergent for
)
It is convenient to define a d-dimensional euclidean vector:
The first step: Feynman’s formula to combine denominators
and change the integration variables: more general form: valid as far as
faster than
as
.
14.1 homework 57
59
in our case: In our case: and we can calculate the d-dimensional integral using
other useful formulas:
for a= 0 and b = 2
prove it! (homework)
next we change the integration variables to q: is the Euler-Mascheroni constant 58
60
One complication: the coupling g is dimensionless for it has dimension where .
and in general
The third step: take
and evaluate integrals over Feynman’s variables
To account for this, let’s make the following replacement:
then g is dimensionless for any d! not an actual parameter of the d = 6 theory, so no observable depends on it.
we get:
it will be important for when we discuss renormalization group later.
or, in a rearranged way:
61
63
returning to our calculation:
it is convenient to take for a= 0 and b = 2 and
we get: with this choice we have
and for the self-energy we have:
finite and independent of
62
just numbers, do not depend on we fix them by requiring:
or
64
The procedure we have followed is known as dimensional regularization: evaluate the integral for (for which it is convergent); analytically continue the result to arbitrary d; fix A and B by imposing our conditions; take the limit .
Instead of calculating kappas directly we can obtain the result by noting :
The condition
can be imposed by requiring: We also could have used Pauli-Villars regularization: replace
makes the integral convergent for
Differentiating with respect to
and requiring
evaluate the integral as a function of conditions; take the limit .
we find:
; fix A and B by imposing our
Would we get the same result?
65
67
We could have also calculated
The integral over Feynman variable can be done in closed form:
differentiate
acquires an imaginary part for
without explicitly calculating A and B:
twice with respect to
:
square-root branch point at Re and Im parts of
in units of
: this integral is finite for
evaluate the integral; calculate and imposing our conditions.
by integrating it with respect to
Would we get the same result? What happened with the divergence of the original integral?
Im part is logarithmically divergent (we will discuss that later) 66
68
To understand this better let’s make a Taylor expansion of about :
We will follow the same procedure as for the propagator. Feynman’s formula:
divergent for divergent for divergent for
but we have only two parameters that can be fixed to get finite
.
Thus the whole procedure is well defined only for ! And it does not matter which regularization scheme we use! For
the procedure breaks down, the theory is non-renormalizable! It turns out that the theory is renormalizable only for . (due to higher order corrections; we will discuss it later) 69
Loop corrections to the vertex
71
Wick rotation: based on S-16
Let’s consider loop corrections to the vertex: (is divergent for for :
for
and finite for
with the replacement
)
we have:
Exact three-point vertex function: defined as the sum of 1PI diagrams with three external lines carrying incoming momenta so that . (this definition allows
to have either sign) 70
72
take the limit
:
The integral over Feynman parameters cannot be done in closed form, but it is easy to see that the magnitude of the one-loop correction to the vertex function increases logarithmically with when . E.g. for let’s define
and
:
; we get:
the same behavior that we found for
(we will discuss it later)
73
75
Other 1PI vertices based on S-17
we can choose
At one loop level additional vertices can be generated, e.g.
just a number, does not depend on
or
finite and independent of
What condition should we impose to fix the value of
?
Any condition is good!
Different conditions correspond to different definitions of the coupling.
E.g. we can set
that corresponds to: plus other two diagrams 74
76
Higher-order corrections and renormalizability
Feynman’s formula:
based on S-18
We were able to absorb divergences of one-loop diagrams (for phi-cubed theory in 6 dimensions) by the coefficients of terms in the lagrangian. If this is true for all higher order contributions, then we say that the theory is renormalizable! If further divergences arise, it may be possible to absorb them by adding some new terms to the lagrangian. If the number of terms required is finite, the theory is still renormalizable. If an infinite number of new terms is required, then the theory is said to be nonrenormalizable.
such a theory can still make useful predictions at energies below some ultraviolet cutoff .
What are the necessary conditions for renormalizability? 77
Wick rotation...
79
Let’s discuss a general scalar field theory in d spacetime dimensions:
Consider a Feynman diagram with E external lines, I internal lines, L closed loops and vertices that connect n lines: finite for
!
we get: p is a linear combination of external and loop momenta
Let’s define the diagram’s superficial degree of divergence: finite and well defined! the same is true for one loop contribution to
for
the diagram appears to be divergent if
.
78
80
There is also a contributing tree level diagram with E external legs:
Note on superficial degree of divergence: a diagram might diverge even if , or it might be finite even if
.
there might be cancellations in the numerator, e.g. in QED
Mass dimensions of both diagrams must be the same: finite
dimension of a diagram = sum of dimensions of its parts
divergent
thus we find a useful formula: divergent subdiagram (it always can be absorbed by adjusting Z-factors)
if any we expect uncontrollable divergences, since D increases with every added vertex of this type. A theory with any is nonrenormalizable! 81
83
Summary and comments: the dimension of couplings:
theories with couplings whose mass dimensions are all positive or zero are renormalizable. This turns out to be true for theories that have spin 0 and spin 1/2 fields only.
and so
theories with spin 1 fields are renormalizable for spin 1 fields are associated with a gauge symmetry! E.g. in four dimensions terms with higher powers than make the theory nonrenormalizable; (in six dimensions only is allowed).
theories of fields with spin greater than 1 are never renormalizable for .
If all couplings have positive or zero dimensions, the only dangerous diagrams are those with but these divergences can be absorbed by
if and only if
. 82
84
Perturbation theory to all orders based on S-19
Procedure to calculate a scattering amplitude in dimensions to arbitrarily high order in g:
theory in six
sum all 1PI diagrams with two external lines; obtain sum all 1PI diagrams with three external lines; obtain Order by order in g adjust A, B, C so that:
construct n-point vertex functions
with
:
draw all the contributing 1PI diagrams but omit diagrams that include either propagator or three-point vertex corrections - skeleton expansion. Take propagators and vertices in these diagrams to be given by the exact propagator and the exact vertex. Sum all the contributing diagrams to get . (this procedure is equivalent to computing all 1PI diagrams) 85
draw all tree-level diagrams that contribute to the process of interest (with E external lines) including not only 3-point vertices but also n-point vertices. evaluate these diagrams using the exact propagator and exact vertices external lines are assigned factor 1. sum all diagrams to obtain the scattering amplitude order by order in g this procedure is equivalent to summing all the usual contributing diagrams
This procedure is the same for any quantum field theory.
86