Quantum Field Theory II

Quantum Field Theory II PHYS-P 622

Radovan Dermisek, Indiana University

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 1

Review of scalar field theory Srednicki 5, 9, 10

2

The LSZ reduction formula based on S-5

In order to describe scattering experiments we need to construct appropriate initial and final states and calculate scattering amplitude.

Summary of free theory: one particle state: vacuum state is annihilated by all a’s: then, one particle state has normalization:

normalization is Lorentz invariant! see e.g. Peskin & Schroeder, p. 23 3

Let’s define a time-independent operator:

wave packet with width ! that creates a particle localized in the momentum space near and localized in the position space near the origin. (go back to position space by Furier transformation)

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 . for in the past.

is a state describing two particles widely separated

In the interacting theory

is not time independent 4

A guess for a suitable initial state: we can normalize the wave packets so that

Similarly, let’s consider a final state:

where again

and

The scattering amplitude is then:

5

A useful formula:

Integration by parts, surface term = 0, particle is localized, (wave packet needed).

is 0 in free theory, but not in interacting one!

E.g. 6

Thus we have:

or its hermitian conjugate:

The scattering amplitude:

we put in time ordering (without changing anything)

is then given as (generalized to n i- and n’ f-particles):

7

Lehmann-Symanzik-Zimmermann formula (LSZ) Note, initial and final states now have delta-function normalization, multiparticle generalization of . We expressed scattering amplitudes in terms of correlation functions! Now we need to learn how to calculate correlation functions in interacting quantum field theory.

8

Comments: we assumed that creation operators of free field theory would work comparably in the interacting theory ...

acting on ground state:

is a Lorentz invariant number

we want

, so that

is a single particle state

otherwise it would create a linear combination of the ground state and a single particle state

we can always shift the field by a constant so that 9

one particle state:

is a Lorentz invariant number

we want

,

since this is what it is in free field theory, correctly normalized one particle state.

creates a

we can always rescale (renormalize) the field by a constant so that .

10

multiparticle states:

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.

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.

11

Summary: Scattering amplitudes can be expressed in terms of correlation functions of fields of an interacting quantum field theory:

Lehmann-Symanzik-Zimmermann formula (LSZ) provided that the fields obey:

these conditions might not be consistent with the original form of lagrangian! 12

Consider for example:

After shifting and rescaling we will have instead:

13

Path integral for interacting field based on S-9

Let’s consider an interacting “phi-cubed” QFT:

with fields satisfying:

we want to evaluate the path integral for this theory:

14

it can be also written as:

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:

15

assumes

thus in the case of:

the perturbing lagrangian is:

counterterm lagrangian in the limit

we expect

and we will find

and 16

Let’s look at Z( J ) (ignoring counterterms for now). Define:

exponentials defined by series expansion:

let’s look at a term with particular values of P (propagators) and V (vertices):

number of surviving sources, (after taking all derivatives) E (for external) is

E = 2P - 3V 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 17

V = 2, E = 0 ( P = 3 ):

!

! ! !

!! !

!

!

! ! ! ! ! !

3! 3! 2 2 2 !

dx1

!

1 i

" " " " " "

x1

2! 6 6 3! 2 2 2 1 = 24

!

1 i

x2 1 i

dx2 (iZg g)2 ∆(x1 − x2 ) ∆(x1 − x2 ) ∆(x1 − x2 )

symmetry factor 18

V = 2, E = 0 ( P = 3 ):

!

! ! !

!! !

!

!

! ! ! ! ! !

3! 3! 3! 2 !

dx1

!

" " " " " "

x2

x1

2! 6 6 3! 2 2 2 1 = 8

!

1 i

1 i

1 i

dx2 (iZg g)2 ∆(x1 − x1 ) ∆(x1 − x2 ) ∆(x1 − x1 )

symmetry factor 19

Feynman diagrams: a line segment stands for a propagator vertex joining three line segments stands for a filled circle at one end of a line segment stands for a source e.g. for V = 1, E = 1

What about those symmetry factors? 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

we can rearrange three vertices

we can rearrange two sources 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 21

the endpoints of each propagator can be swapped and the effect is duplicated by swapping the two vertices propagators can be rearranged in 3! ways, and all these rearrangements can be duplicated by exchanging the derivatives at the vertices

22

23

24

25

26

27

28

29

All these diagrams are connected, but Z( J ) contains also diagrams that are products of several connected diagrams:

e.g. for V = 4, E = 0 ( P = 6 ) in addition to connected diagrams we also have :

and also:

and also:

30

All these diagrams are connected, but Z( J ) contains also diagrams that are products of several connected diagrams:

e.g. for V = 4, E = 0 ( P = 6 ) in addition to connected diagrams we also have :

A general diagram D can be written as:

additional symmetry factor not already accounted for by symmetry factors of connected diagrams; it is nontrivial only if D contains identical C’s:

the number of given C in D

particular connected diagram

31

Now

is given by summing all diagrams D: any D can be labeled by a set of n’s

thus we have found that connected diagrams.

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:

vacuum diagrams are omitted from the sum 32

If there were no counterterms we would be done: in that case, the vacuum expectation value of the field is:

only diagrams with one source contribute:

(the source is “removed” by the derivative)

and we find: we used

since we know

which is not zero, as required for the LSZ; so we need counterterm 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: at the lowest order of g only

in order to satisfy

contributes:

we have to choose:

Note, must be purely imaginary so that Y is real; and, in addition, the integral over k is ultraviolet divergent. 34

to make sense out of it, we introduce an ultraviolet cutoff and in order to keep Lorentz-transformation properties of the propagator we make the replacement:

the integral is now convergent:

we will do this type of calculations later...

and indeed,

is purely imaginary.

after choosing Y so that

we can take the limit Y becomes infinite

... we repeat the procedure at every order in g 35

e.g. at

we have to sum up:

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! 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 36

all that is left with up to 4 sources and 4 vertices is:

37

finally, let’s take a look at the other two counterterms:

we get

we used integration by parts

it results in a new vertex at which two lines meet, the corresponding vertex factor or the Feynman rule is for every diagram with a propagator there is additional one with this vertex

Summary: we have calculated

in

theory and expressed it as

where W is the sum of all connected diagrams with no tadpoles and at least two sources! 38

Scattering amplitudes and the Feynman rules 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: short notation:

W contains diagrams with at least two sources

+ ...

thus we find: 39

4-point function:

we have dropped terms that contain does not correspond to any interaction; when plugged to LSZ, no scattering happens

Let’s define connected correlation functions:

and plug these into LSZ formula. 40

at the lowest order in g only one diagram contributes:

S=8 derivatives remove sources in 4! possible ways, and label external legs in 3 distinct ways:

each diagram occurs 8 times, which nicely cancels the symmetry factor. 41

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

y

z

putting together factors for all pieces of Feynman diagrams we get:

42

For two incoming and two outgoing particles the LSZ formula is:

and we have just written terms of propagators.

in The LSZ formula highly simplifies due to:

We find:

43

44

four-momentum is conserved in scattering process

Let’s define:

scattering matrix element From this calculation we can deduce a set of rules for computing

.

45

Feynman rules to calculate

:

for each incoming and outgoing particle draw an external line and label it with four-momentum and an arrow specifying the momentum flow draw all topologically inequivalent diagrams for internal lines draw arrows arbitrarily but label them with momenta so that momentum is conserved in each vertex assign factors:

1

for each external line for each internal line with momentum k for each vertex

sum over all the diagrams and get 46

Additional rules for diagrams with loops: a diagram with L loops will have L internal momenta that are not fixed; integrate over all these momenta with measure

divide by a symmetry factor include diagrams with counterterm vertex that connects two propagators, each with the same momentum k; the value of the vertex is

now we are going to use

to calculate cross section...

47

Lehmann-Källén form of the exact propagator based on S-13

What can we learn about the exact propagator from general principles? Let’s define the exact propagator: The field is normalized so that Normalization of a one particle state in d-dimensions: The d-dimensional completeness statement:

identity operator in one-particle subspace

Lorentz invariant phase-space differential 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

Let’s insert the complete set of energy eigenstates between the two fields; for we have:

ground state, 0 - energy one particle states

multiparticle continuum of states specified by the total three momentum k and other parameters: relative momenta, ..., denoted symbolically by n 50

51

Let’s define the spectral density:

then we have:

52

similarly: and we can plug them to the formula for time-ordered product:

was your homework

we get: or, in the momentum space:

Lehmann-Källén form of the exact propagator it has an isolated pole at

with residue one! 53