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

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