Quantum Field Theory IIPHYS-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

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Review of scalar field theory

Srednicki 5, 9, 10

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The LSZ reduction formulabased 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. 23REVIEW

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Let’s define a time-independent operator:

that creates a particle localized in the momentum space near

and localized in the position space near the origin.

wave packet with width !

(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 .

for is a state describing two particles widely separated

in the past.

In the interacting theory is not time independent REVIEW

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A guess for a suitable initial state:

Similarly, let’s consider a final state:

The scattering amplitude is then:

we can normalize the wave packets so that

where again and

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A useful formula:

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

E.g.

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

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Thus we have:

or its hermitian conjugate:

The scattering amplitude:

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

we put in time ordering(without changing

anything)

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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.REV

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we can always shift the field by a constant

we want ,

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

so that is a single particle state

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

so that REVIEW

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

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

we want ,

one particle state:

is a Lorentz invariant number

so that .REVIEW

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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.

By waiting long enough we can make the multiparticle contribution to the scattering amplitude as small as we want.

see the discussion in Srednicki, p. 40-41

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

Summary:

Lehmann-Symanzik-Zimmermann formula (LSZ)

provided that the fields obey:

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

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Consider for example:

After shifting and rescaling we will have instead:

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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:

REVIEW

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it can be also written as:

and for the path integral of the free field theory we have found:

epsilon trick leads to additional factor; to get the correct normalization we require:

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we will find and

in the limit we expect and

assumes

thus in the case of:

the perturbing lagrangian is:

counterterm lagrangianREVIEW

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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 - 3V3V derivatives can act on 2P sources in (2P)! / (2P-3V)! different ways

e.g. for V = 2, P = 3 there is 6! different termsREVIEW

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

! !

"! !

! !

!!!!! !!!! ! """""

=!

dx1

!dx2

3! 3! 2 2 2

2! 6 6 3! 2 2 2x1 x2

symmetry factor

(iZgg)21i!(x1 ! x2)

1i!(x1 ! x2)

1i!(x1 ! x2)

112 REV

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

! !

"! !

! !

!!!!! !!!! ! """""

=!

dx1

!dx2

3! 3! 3! 2

2! 6 6 3! 2 2 2x1 x2

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symmetry factor

(iZgg)21i!(x1 ! x1)

1i!(x1 ! x2)

1i!(x1 ! x1)REV

IEW

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Feynman diagrams:

vertex joining three line segments stands for

a line segment stands for a propagator

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?

What about those symmetry factors?

symmetry factors are related to symmetries of Feynman diagrams...REVIEW

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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

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propagators can be rearranged in 3! ways,and all these rearrangements can be duplicated by

exchanging the derivatives at the vertices

the endpoints of each propagator can be swapped and the effect is duplicated by swapping the two vertices

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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:

REVIEW

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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:

particular connected diagramadditional symmetry factornot 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

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imposing the normalization means we can omit vacuum diagrams(those with no sources), thus we have:

thus we have found that is given by the exponential of the sum of connected diagrams.

Now is given by summing all diagrams D:

any D can be labeled by a set of n’s

vacuum diagrams are omitted from the sumREV

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we used since we know

If there were no counterterms we would be done:

in that case, the vacuum expectation value of the field is:

and we find:(the source is “removed” by the derivative)

only diagrams with one source contribute:

which is not zero, as required for the LSZ; so we need countertermREVIEW

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in order to satisfy we have to choose:

Including term in the interaction lagrangian results in a new type of vertex on which a line segment ends

corresponding Feynman rule is:

e.g.

at the lowest order of g only contributes:

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

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after choosing Y so that we can take the limit

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

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.

Y becomes infiniteREVIEW

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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.

e.g. at we have to sum up:

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 whichhas no sources. = tadpoles

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

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we have calculated in theory and expressed it as

it results in a new vertex at which two lines meet, the corresponding vertex factor or the Feynman rule is

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

we get

we used integration by parts

for every diagram with a propagator there is additional one with this vertex

Summary:

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

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W contains diagrams with at least two sources + ...

Let’s define exact propagator:

based on S-10We 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.

Scattering amplitudes and the Feynman rules

short notation:

thus we find:REVIEW

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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.REV

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at the lowest order in g only one diagram contributes:

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

S = 8

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

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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

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

y z

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

and we have just written in terms of propagators.

The LSZ formula highly simplifies due to:

We find:

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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 .

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for each incoming and outgoing particle draw an external line and label it with four-momentum and an arrow specifying the momentum flow

Feynman rules to calculate :

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: for each external line

for each internal line with momentum k

1

for each vertex

sum over all the diagrams and get REV

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a diagram with L loops will have L internal momenta that are not fixed; integrate over all these momenta with measure

Additional rules for diagrams with loops:

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...REVIEW

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Lehmann-Källén form of the exact propagator

What can we learn about the exact propagator from general principles?

based on S-13

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

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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?

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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

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Let’s define the spectral density:

then we have:

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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!

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Loop corrections to the propagator

The exact propagator:

based on S-14

sum of connected diagramscontributing diagrams at level:

following the Feynman rules we get:

where, the self-energy is:

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It is convenient to define to all orders via the geometric series:

One Particle Irreducible diagrams - 1PI(still connected after any one line is cut)

1PI diagrams contributing at level:

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to fix A and B.

we know that it has an isolated pole at with residue one!

It is convenient to define to all orders via the geometric series:

One Particle Irreducible diagrams - 1PI(still connected after any one line is cut)

we can sum up the series and get:

and so we will require:

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(is divergent for and convergent for )

let’s get back to calculation:

The first step: Feynman’s formula to combine denominators

more general form:

14.1 homework57

In our case:

next we change the integration variables to q:

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The second step: Wick’s rotation to evaluate the integral over q

It is convenient to define a d-dimensional euclidean vector:

and change the integration variables:

integration contour along the real axis can be

rotated to the imaginary axis without passing

through the poles

valid as far as faster than as .

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in our case:

and we can calculate the d-dimensional integral using

prove it! (homework)

is the Euler-Mascheroni constant

for a= 0 and b = 2

other useful formulas:

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One complication: the coupling g is dimensionless for and in general it has dimension where .

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.

it will be important for when we discuss renormalization group later.

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returning to our calculation:

for a= 0 and b = 2 and we get:

and for the self-energy we have:

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The third step: take and evaluate integrals over Feynman’s variables

we get:

or, in a rearranged way:

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it is convenient to take

with this choice we have

just numbers, do not depend on or

finite and independent of we fix them by requiring:

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Instead of calculating kappas directly we can obtain the result by noting :

The condition can be imposed by requiring:

Differentiating with respect to and requiring we find:

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The integral over Feynman variable can be done in closed form:

in units of :Re and Im parts of

square-root branch point at

acquires an imaginary part for

Im part is logarithmically divergent (we will discuss that later)

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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 .

We also could have used Pauli-Villars regularization:

replace

makes the integral convergent for

evaluate the integral as a function of ; fix A and B by imposing our conditions; take the limit .

Would we get the same result?

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We could have also calculated without explicitly calculating A and B:

differentiate twice with respect to :

this integral is finite for

evaluate the integral; calculate by integrating it with respect to and imposing our conditions.

Would we get the same result?

What happened with the divergence of the original integral?

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To understand this better let’s make a Taylor expansion of

about :

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 !

For the procedure breaks down, the theory is non-renormalizable!

It turns out that the theory is renormalizable only for .

And it does not matter which regularization scheme we use!

(due to higher order corrections; we will discuss it later)69

Loop corrections to the vertex

Let’s consider loop corrections to the vertex:

based on S-16

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)

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We will follow the same procedure as for the propagator.

Feynman’s formula:

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(is divergent for and finite for )

Wick rotation:

for :

for with the replacement we have:

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take the limit :

let’s define and ; we get:

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just a number, does not depend on or

we can choose

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:

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E.g. for :

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 .

the same behavior that we found for (we will discuss it later)

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Other 1PI vertices

At one loop level additional vertices can be generated, e.g.

based on S-17

plus other two diagrams

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Feynman’s formula:

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finite for !

Wick rotation...

we get:

finite and well defined!

the same is true for one loop contribution to for .

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Higher-order corrections and renormalizability

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.

based on S-18

What are the necessary conditions for renormalizability?

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 .

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Consider a Feynman diagram with E external lines, I internal lines, L closed loops and vertices that connect n lines:

Let’s define the diagram’s superficial degree of divergence:

p is a linear combination of external and loop momenta

the diagram appears to be divergent if

Let’s discuss a general scalar field theory in d spacetime dimensions:

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There is also a contributing tree level diagram with E external legs:

Mass dimensions of both diagrams must be the same:

dimension of a diagram = sum of dimensions of its parts

thus we find a useful formula:

if any we expect uncontrollable divergences, since D increases with every added vertex of this type.

A theory with any is nonrenormalizable!

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the dimension of couplings:

and so

E.g. in four dimensions terms with higher powers than make the theory nonrenormalizable; (in six dimensions only is allowed).

If all couplings have positive or zero dimensions, the only dangerous diagrams are those with

but these divergences can be absorbed by .

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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

finite divergent

divergent subdiagram (it always can be absorbed by adjusting Z-factors)

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theories of fields with spin greater than 1 are never renormalizable for .

theories with spin 1 fields are renormalizable for if and only if spin 1 fields are associated with a gauge symmetry!

Summary and comments:

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.

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Perturbation theory to all orders

Procedure to calculate a scattering amplitude in theory in six dimensions to arbitrarily high order in g:

based on S-19

sum all 1PI diagrams with two external lines; obtain

sum all 1PI diagrams with three external lines; obtain

construct n-point vertex functions with :

Order by order in g adjust A, B, C so that:

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)

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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.

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