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 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 1 3 Review of scalar field theory Srednicki 5, 9, 10 Let s define a time-independent operator: that creates a particle localized in the momentum space near wave packet with width! 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 is a state describing two particles widely separated is not time independent 2 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 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) 5 7 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! 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. 6 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 that 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. 9 11 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. 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! these conditions might not be consistent with the original form of 10 12
Consider for example: After shifting and rescaling we will have instead: 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: 13 15 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: assumes thus in the case of: the perturbing lagrangian is: counterterm lagrangian in the limit we expect and we will find and 14 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 V = 2, E = 0 ( P = 3 ): =!!!!!!!!!!!!!!!! " " " " " " 1 8 3! 3! 3! 2 2! 6 6 3! 2 2 2 x 1 x 2 dx 1 dx 2 (iz g g) 2 1 i (x 1 x 1 ) 1 i (x 1 x 2 ) 1 i (x 1 x 1 ) symmetry factor 17 19 V = 2, E = 0 ( P = 3 ): =!!!!!!!!!!!!!!!! " " " " " 3! 3! 2 2 2 2! 6 6 3! 2 2 2 x 1 x 2 12 1 dx 1 dx 2 (iz g g) 2 1 i (x 1 x 2 ) 1 i (x 1 x 2 ) 1 i (x 1 x 2 ) symmetry factor " 18 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? What about those symmetry factors? e.g. for V = 1, E = 1 symmetry factors are related to symmetries of Feynman diagrams... 20
Symmetry factors: we can rearrange three derivatives without changing diagram we can rearrange two sources we can rearrange three vertices 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 23 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 24
25 27 26 28
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 29 31 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: 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 means we can omit vacuum diagrams vacuum diagrams are omitted from the sum 30 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) we used since we know which is not zero, as required for the LSZ; so we need counterterm 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... is purely imaginary. we can take the limit Y becomes infinite 33 35 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: Note, must be purely imaginary so that Y is real; and, in addition, the integral over k is ultraviolet divergent. 34 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: 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: 37 39 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 for every diagram with a propagator there is additional one with this vertex Summary: we have calculated in theory and expressed it as we used integration by parts where W is the sum of all connected diagrams with no tadpoles and at least two sources! 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. 38 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. 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: 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 y z putting together factors for all pieces of Feynman diagrams we get: 42 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. 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... 45 47 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: sum over all the diagrams and get 1 for each external line for each internal line with momentum k for each vertex 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 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 specified by the total three momentum k and other parameters: relative momenta,..., denoted symbolically by n then we have: 50 52
similarly: It is convenient to define to all orders via the geometric series: and we can plug them to the formula for time-ordered product: we get: was your homework One Particle Irreducible diagrams - 1PI (still connected after any one line is cut) or, in the momentum space: 1PI diagrams contributing at level: it has an isolated pole at Lehmann-Källén form of the exact propagator with residue one! 53 55 Loop corrections to the propagator The exact propagator: based on S-14 It is convenient to define to all orders via the geometric series: contributing diagrams at level: sum of connected diagrams One Particle Irreducible diagrams - 1PI (still connected after any one line is cut) following the Feynman rules we get: we can sum up the series and get: where, the self-energy is: we know that it has an isolated pole at and so we will require: to fix A and B. with residue one! 54 56
let s get back to calculation: (is divergent for and convergent for ) The first step: Feynman s formula to combine denominators 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: more general form: 14.1 homework 57 valid as far as faster than as. 59 in our case: In our case: and we can calculate the d-dimensional integral using for a= 0 and b = 2 other useful formulas: 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: not an actual parameter of the d = 6 theory, so no observable depends on it. then g is dimensionless for any d! 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 just numbers, do not depend on we fix them by requiring: or 62 64
Instead of calculating kappas directly we can obtain the result by noting : 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. The condition can be imposed by requiring: We also could have used Pauli-Villars regularization: replace Differentiating with respect to and requiring we find: 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? 65 67 The integral over Feynman variable can be done in closed form: We could have also calculated without explicitly calculating A and B: acquires an imaginary part for differentiate twice with respect to : square-root branch point at Re and Im parts of in units of : Im part is logarithmically divergent (we will discuss that later) 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? 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 71 Loop corrections to the vertex Let s consider loop corrections to the vertex: based on S-16 Wick rotation: (is divergent for and finite for ) for : 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. let s define and ; we get: E.g. for : the same behavior that we found for (we will discuss it later) 73 75 we can choose Other 1PI vertices At one loop level additional vertices can be generated, e.g. based on S-17 just a number, does not depend on or finite and independent of E.g. we can set 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: plus other two diagrams 74 76
Feynman s formula: 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? 77 79 Wick rotation... Let s discuss a general scalar field theory in d spacetime dimensions: finite for! Consider a Feynman diagram with E external lines, I internal lines, L closed loops and vertices that connect n lines: we get: finite and well defined! the same is true for one loop contribution to for. 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 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. Mass dimensions of both diagrams must be the same: there might be cancellations in the numerator, e.g. in QED dimension of a diagram = sum of dimensions of its parts finite 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: 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 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. but these divergences can be absorbed by. 82 84
Perturbation theory to all orders based on S-19 Procedure to calculate a scattering amplitude in theory in six 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: 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