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: REVIEW normalization is Lorentz invariant! see e.g. Peskin & Schroeder, p. 23 3
Let s define a time-independent operator: that creates a particle localized in the momentum space near wave packet with width σ 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. for in the past. In the interacting theory REVIEW is a state describing two particles widely separated is not time independent 4
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 REVIEW 5
A useful formula: Integration by parts, surface term = 0, particle is localized, (wave packet needed). REVIEW E.g. is 0 in free theory, but not in interacting one! 6
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) REVIEW 7
Lehmann-Symanzik-Zimmermann formula (LSZ) Note, initial and final states now have delta-function normalization, multiparticle generalization of. REVIEW 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: we want, so that we can always shift the field by a constant is a Lorentz invariant number is a single particle state otherwise it would create a linear combination of the ground state and a single particle state so thatreview 9
one particle state: we want, since this is what it is in free field theory, correctly normalized one particle state. is a Lorentz invariant number creates a REVIEW 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 REVIEW 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: provided that the fields obey: Lehmann-Symanzik-Zimmermann formula (LSZ) lagrangian!review these conditions might not be consistent with the original form of 12
Consider for example: After shifting and rescaling we will have instead: REVIEW 13
Path integral for interacting field Let s consider an interacting phi-cubed QFT: with fields satisfying: we want to evaluate the path integral for this theory: based on S-9 REVIEW 14
it can be also written as: epsilon trick leads to additional factor; to get the correct normalization we require: REVIEW 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 REVIEW 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): REVIEW 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 ): = 1 ¹ ¹ ² ² ¹ ¹ ¹ ¹ ¹ ¹ ² ² ² ² ² ² ³ ³ ³ ³ ³ dx 1 3! 3! 2 2 2 2! 6 6 3! 2 2 2 dx 2 symmetry factor x 1 x 2 12REVIEW (iz g g) 2 1 i (x 1 x 2 ) 1 i (x 1 x 2 ) 1 i (x 1 x 2 ) 18 ³
V = 2, E = 0 ( P = 3 ): = 1 8 ¹ ¹ ² ² ¹ ¹ ¹ ¹ ¹ ¹ ² ² ² ² ² ² ³ ³ ³ ³ ³ ³ dx 1 3! 3! 3! 2 2! 6 6 3! 2 2 2 dx 2 symmetry factor x 1 x 2 REVIEW (iz g g) 2 1 i (x 1 x 1 ) 1 i (x 1 x 2 ) 1 i (x 1 x 1 ) 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 What about those symmetry factors? e.g. for V = 1, E = 1 REVIEW 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 two sources we can rearrange three vertices REVIEW 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 REVIEW propagators can be rearranged in 3! ways, and all these rearrangements can be duplicated by exchanging the derivatives at the vertices 22
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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 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 REVIEW particular connected diagram 31
Now is given by summing all diagrams D: thus we have found that connected diagrams. imposing the normalization (those with no sources), thus we have: any D can be labeled by a set of n s is given by the exponential of the sum of REVIEW means we can omit vacuum diagrams 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: and we find: (the source is removed by the derivative) REVIEW 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 contributes: in order to satisfy we have to choose: REVIEW 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: and indeed, after choosing Y so that... we repeat the procedure at every order in g we will do this type of calculations later... REVIEW is purely imaginary. we can take the limit Y becomes infinite 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! REVIEW 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: REVIEW 37
finally, let s take a look at the other two counterterms: we get it results in a new vertex at which two lines meet, the corresponding vertex factor or the Feynman rule is Summary: we have calculated in theory and expressed it as we used integration by parts for every diagram with a propagator there is additional one with this vertex REVIEW 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:review 39
4-point function: we have dropped terms that contain Let s define connected correlation functions: and plug these into LSZ formula. does not correspond to any interaction; when plugged to LSZ, no scattering happens REVIEW 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: REVIEW 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: REVIEW 42
For two incoming and two outgoing particles the LSZ formula is: and we have just written in terms of propagators. We find: The LSZ formula highly simplifies due to: REVIEW 43
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Let s define: four-momentum is conserved in scattering process scattering matrix element REVIEW 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: REVIEW sum over all the diagrams and get 1 for each external line for each internal line with momentum k for each vertex 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 REVIEW 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! The residue of a function that has a simple pole is given by: The residue of a function f(z)=g(z)/h(z) at a simple pole is given by: 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
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Let s define the spectral density: then we have: 52
similarly: and we can plug them to the formula for time-ordered product: we get: eq.(8.13) or, in the momentum space: it has an isolated pole at Lehmann-Källén form of the exact propagator with residue one! 53
Loop corrections to the propagator The exact propagator: based on S-14 contributing diagrams at level: sum of connected diagrams following the Feynman rules we get: where, the self-energy is: 54
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: 55
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: we know that it has an isolated pole at and so we will require: to fix A and B. with residue one! 56
let s get back to calculation: (is divergent for and convergent for ) The first step: Feynman s formula to combine denominators more general form: 14.1 homework 57
In our case: next we change the integration variables to q: 58
The second step: Wick s rotation to evaluate the integral over q It is convenient to define a d-dimensional euclidean vector: integration contour along the real axis can be rotated to the imaginary axis without passing through the poles and change the integration variables: valid as far as faster than as. 59
in our case: and we can calculate the d-dimensional integral using for a= 0 and b = 2 other useful formulas: prove it! (homework) is the Euler-Mascheroni constant 60
One complication: the coupling g is dimensionless for it has dimension where. To account for this, let s make the following replacement: and in general not an actual parameter of the d = 6 theory, so no observable depends on it. then g is dimensionless for any d! it will be important for when we discuss renormalization group later. 61
returning to our calculation: for a= 0 and b = 2 and we get: and for the self-energy we have: 62
The third step: take and evaluate integrals over Feynman s variables we get: or, in a rearranged way: 63
it is convenient to take with this choice we have finite and independent of just numbers, do not depend on we fix them by requiring: or 64
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: 65
The integral over Feynman variable can be done in closed form: acquires an imaginary part for square-root branch point at Re and Im parts of in units of : Re part is logarithmically divergent (we will discuss that later) 66
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 evaluate the integral as a function of conditions; take the limit. makes the integral convergent for ; fix A and B by imposing our Would we get the same result? 67
We could have also calculated without explicitly calculating A and B: differentiate twice with respect to : evaluate the integral; calculate and imposing our conditions. this integral is finite for by integrating it with respect to Would we get the same result? What happened with the divergence of the original integral? 68
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! 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 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) 70
We will follow the same procedure as for the propagator. Feynman s formula: 71
Wick rotation: (is divergent for and finite for ) for : for with the replacement we have: 72
take the limit : let s define and ; we get: 73
we can choose just a number, does not depend on or E.g. we can set 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. that corresponds to: 74
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 : the same behavior that we found for (we will discuss it later) 75
Other 1PI vertices At one loop level additional vertices can be generated, e.g. based on S-17 plus other two diagrams 76
Feynman s formula: 77
Wick rotation... we get: finite for! finite and well defined! the same is true for one loop contribution to for. 78
Higher-order corrections and renormalizability 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? 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: p is a linear combination of external and loop momenta Let s define the diagram s superficial degree of divergence: the diagram appears to be divergent if 80
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! 81
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. 82
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) 83
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. theories with spin 1 fields are renormalizable for spin 1 fields are associated with a gauge symmetry! if and only if theories of fields with spin greater than 1 are never renormalizable for. 84
Perturbation theory to all orders Procedure to calculate a scattering amplitude in dimensions to arbitrarily high order in g: 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: theory in six based on S-19 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 sum all diagrams to obtain the scattering amplitude external lines are assigned factor 1. 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