4 4 and perturbation theory

Transcription

1 and perturbation theory We now consider interacting theories. A simple case is the interacting scalar field, with a interaction. This corresponds to a -body contact repulsive interaction between scalar bosons (the quanta of the field). The classical theory is given by the (Euclidean) action. Z apple S E [ ] = d d x (r x ) + m + g (.)! g>isthecouplingconstant. The /! factor is for convenience (but will be explained later). The real time action is Z apple S[ ] = d d x (@ µ m g ) (.)! Classical field configurations are solution of the non linear equation of motion + m + g 6 =, t ~x (.) Quantization can be performed in the canonical framework (see QFT I course). Here we look at the functional integral quantization scheme. Naively we would expect that the theory can be quantized by functional integration like for the free field, the free field action Z apple S [ ] = d d x (r x ) + m (.) being replaced by the action S[. For instance, the correlation functions of the interacting (Euclidean) theory should be given by (I omit the Euclidean suffix E when there is no ambiguity) R D [ ]exp ~ h (z ) (z N )i g = S[ ] (z ) (z N ) R D [ ]exp (.5) S[ ; g] ~ The functional measure being the same as that of the free field D [ ] = Y d (x) = Y apple / ~ (.6) d x nz d d n In particular the same lattice regularization scheme can be applied. If this calculation makes sense, on should recover via Wick rotation the real time Green functions, i.e. the vacuum expectation values of time ordered product of the field operators h (z ) (z N )i g! Wick rotation h T [ (z ) (z N )] i (.7) i is the vacuum state of the interacting theory. theory, different from the vacuum i of the free Perimeter Scholars International 9 Nov.

2 . Perturbation theory, Feynman diagrams As as been presented for QED, a natural scheme is to assume that g is small and perform a series expansion in powers of g. This amounts to consider that the interaction terms are small, and represents a small perturbation of the free theory. Thus we expand the interaction term in the functional integral apple Z g exp ~! d d x (x) = X K= K! g K x! ~ d d x...d D x K (x ) (x K ) (.8) Then we invert the perturbative series P K and the functional integral R D [ ]. Z D [ ] X = X Z D [ ] (.9) K K CAUTION! This inversion is highly non trivial. In particular it will transform a convergent series (the Taylor series of a exponential) into a divergent series, possibly asymptotic when g!, but with zero radius of convergence. Let us close our eyes, cross our fingers and carry on the calculation. The numerator in.5 is Z Z(z,,z N )= apple D [ ]exp ~ S[ ] (z ) (z N ) (.) and given by the series Z(z,,z N )= X g K x d d x ~ K!(!) K...d D x K h (z ) (z N ) (x ) (x K )i K (.) h i denoting the expectation value in the free field theory. Now these e.v. of products of fields in the free theory can be computed, using Wick theorem. They are given by a sum over all pairings between the M = N +K fields. (z ) (z N ) (x ) (x K ) = X h (y ) (y )i h (y M ) (y M )i (.) pairings Each term given by a product of L = M/ free field propagators h (y ) (y )i = ~ G (y,y ) (.) Applying the Feynman diagrammatic rules (represent each propagator by a line), we end up with a representation of. by a sum of Feynman integrals (Feynman amplitudes) I G associated with Feynman diagrams (or graphs) G, oftheform X Z (N) (z,,z N ; g) = ( g) K ~ L K c(g) I G (z,,z N ) (.) G with N legs Perimeter Scholars International Nov.

3 The amplitude I G is given by the integral over the positions of the K internal vertices (with legs) of the graph, of the integrand (the product of the propagators) x Y I G (z i ) = dx dx K G (y`,y `) (.5) lines `G We do not discuss if theses integrals make senses at this stage. The combinatorial coefficient c(g) comes from the factors K! and!, and from the fact that different pairings can give the same Feynman diagram G, with the same amplitude. These factors are called the symmetry factor of the diagram, since they are in fact given by the order of a symmetry group associated to each diagram. Rather than making a general theory, let me give a few explicit examples. remember the rules for the Euclidean theory:. one line: ~ and one propagator G. one internal vertex : afactor( g) ~,integrateoveritspositionx i. one external vertex #: positionz j fixed For the real-time theory they become. one line: ~ and one propagator G Feynman =ig. one internal vertex : afactor( i g) ~,integrateoveritspositionx i. one external vertex #: positionz j fixeds Vacuum diagrams ( N =) N =,K =:onediagram 8 N =,K =:threediagrams Perimeter Scholars International Nov.

4 Two point function ( N =) N =,K =:onediagram N =,K =:twodiagrams + 8 N =,K =:sevendiagrams Four point function ( N =) N =,K =:threediagrams + + Perimeter Scholars International Nov.

5 N =,K =:Tendiagrams These diagrammatic rules to construct and organize the perturbative expansion in terms of Feynman diagrams are the same than those obtained in the canonical formalism. This is not surprising since the basic rules (Wick theorem) are the same. The Feynam amplitudes are more easily represented and calculated in momentum space (taking the Fourier transform) than in position space. We shall discuss that later.. Connected functions One sees already that diagrams can be decomposed in connected components, and that the amplitudes of disconnected components of a diagram factorize. Similarily, a connected = Figure : Decomposition of a graph into its connected components Perimeter Scholars International Nov.

6 diagram can be decomposed into a tree structure, whose lines are the propagators, and the vertices the so called irreducible parts. These one particle irreducible diagrams (PI) are the connected diagrams which stay connected if any of its lines is removed (but the end-point vertices of the diagrams are not removed). First let us state (without demonstration) some = Figure : Decomposition of a connected graph C into a tree T whose vertices are its irreducible parts i basic results. Correlation functions: The correlation functions G(z,,z N ) = h (z ) (z N )i g = Z(z,,z N ) Z (.6) obtained from the functional integral.5, are given by the sum of diagrams which do not contain any connected vacuum diagram with no external legs. Indeed the contribution of these vacuum connected components in Z(z,,z N ) is cancelled by the denominator Z, which is precisely the sum over all vacuum diagrams. For instance, the two point function G(z,z ) is (I omit the ~) g (.7) The four point function G(z,z,z,z ) A g (.8) Perimeter Scholars International Nov.

7 Connected functions: We notice that the two point function is connected, but the four point function is not connected. More precisely, it is given by the products of the (connected) two points functions G(z,z )=W(z,z ),plusthesumoverallconnectedfourpointgraphs, which constitute the connected four point function W(z,,z ), so that we can write the correlation functions G in terms of the connected functions W G(z,z )=W(z,z ) (.9) G(z,z,z,z )=W(z,z )W(z,z )+W(z,z )W(z,z )+W(z,z )W(z,z )+W(z,z,z,z ) (.) The connected functions in QFT correspond to cumulants of a probability distribution in statistics, and to connected correlations in statistical mechanics. The zero point connected function W is defined as the sum of connected vacuum diagrams. It is given at order g by W = g 8 + g C A + (.) with the closed loop diagram representing the logarithm of the functional determinant tr log[ +m ], with its symmetry factor c( )=/ apple = trlog (.) +m This term represents the partition function of the free field, and is taken into account for consistency. At order g in the perturbative expansion, the connected two point functions and four point functions are: W(z,z ) = g + g C A (.) W(z,z,z,z ) = g C A + (.) Perimeter Scholars International 5 Nov.

8 And for completeness the connected six point function (you start to see the tree structure of the connected functions) W(z, z 6 ) = g 6 C 5 + ( terms) A g 6 5 C + ( terms) A ( terms) C A (6 terms) C A (5 terms) CC AA (.5). Generating functionals These diagrammatic manipulations are a bit tedious, but it is a good training to compute these functions at first order. However the functional integral formalism provides a very powerful tool to manipulate the correlation functions through generating functionals. Generating functions have been invented long ago by L. Euler. They allow to translate combinatorial manipulations of objects into algebraic calculations on functions (often easier). The generating functional for correlation functions is defined (Euclidean theory) as Z apple Z[j] = D [ ]exp (S[ ] j ) (.6) ~ j = {j(x); x M d } is a classical source term (a real function over space-time), and Z[j] is a functional of this function. = { (x); x M d } is the random field variable integrated over in the functional integral, and represents the quantum field. j is a compact notation for the scalar product of the functions j and (considered as vectors in L (M d ) Z j = d d xj(x) (x) (.7) This definition for Z[j] is a compact way to manipulate all the N point functions Z(z, z N ) defined in., since expanding in j Z[j] = X ~ N x d d z d d z N j(z ) j(z N ) Z(z, z N ) (.8) N! N Equivalently, the Z(z, z N ) are the functional derivatives of the functional Z[j] Z(z, z N ) = ~ N j(z ) j(z N ) Z[j] j= (.9) Perimeter Scholars International 6 Nov.

9 For the real time theory we define the generating function as Z apple i Z[j] = D [ ]exp (S[ ]+j ) (.) ~ The generating functional for the correlation functions (the Green functions) G(z,,z N ).6 is obviously G[j] = Z[j] (.) Z[] Now comes the power of the formalism. functions W(z,,z N ),definedas W[j] = X N ~ N+ N! x (note the additional ~ factor) is simply In other words Z[j] =exp The generating functional for the connected d d z d d z N j(z ) j(z N ) W(z, z N ) (.) W[j] =~ log (Z[j]) (.) ~ W[j] =+~ W[j]+ ~ W[j] + 6 ~ W[j] + (.) The term W[j] P constructs all the diagrams with P connected components. We shall see later how to obtain all the irreducible amplitudes.. Perturbation theory and semiclassical expansion Asimplebutimportantobservation. Withournormalizationforthedefinitionoftheconnected functions, let us look at the ~ factor for a single diagram. Each propagator is proportional to ~, eachinteractionvertexgivesafactor~. Finally each external line (attached to an external vertex) gives also a factor ~,andthereisafinal~ factor per connected component, hence the final factor is ~ V +L+ V = number of vertices, L = number of lines (.5) Now a celebrated formula of Euler states that for any connected graph V + L + = B number of internal independent loops of the graph (.6) For instance the graph of Fig. has V =and L =7,henceB =. This number B is the number of independent internal momenta k for a Feynman amplitude in the momentum representation, hence the number on momentum integrations to be performed in the evaluation of the amplitude. Perimeter Scholars International 7 Nov.

10 Figure : A famous graph with V =vertices or islands, L =7links or bridges and B =loops or cycles This formula is valid for any kind of graph, not only the diagrams of the theory. This justifies the fact that the perturbative expansions in QFT are also semiclassical expansions (expansions in ~, andareoftendenotedthe loop expansion. For the theory, one has the specific relation (a link has two end-points, an internal vertex has legs, an external vertex only ) L = N +K (.7) where N = number of external vertices and K = number of interaction vertices, so that B = N/+K (.8) So the ~ or loop expansion is also the interaction coupling expansion in g. Butnotethatfor instance in QED, the loop expansion is an expansion in e (i.e. the fine structure constant ), not in e..5 The effective action at loop The effective action is an important concept of QFT. Firstly it is the generating functional for the one particle irreducible amplitudes. But the effective action plays also an important role when discussing renormalisation and renormalisation group in QFT, gauge symmetries and anomalies, QFT in non trivial gravitational backgrounds, etc. The idea is to look at the properties and observables of the theory, expressed as a function of the v.e.v. of the field h i. LetusstartfromthegenerationfunctionalW[j]. The background field '(x) is the e.v. of the field,forgeneralvaluesofthesourceterm(or external classical field) j(x). '(x) = W[j] j(x) = h (x)i j (.9) The background field ' is a classical field, which is a functional of the classical source term j. Itisanon-localfunctional,sincealocalchangeinj at a given x leads to changes of '(x) for x far away from x. The background field' is the response to the external source j. Perimeter Scholars International 8 Nov.

11 If one is interested in the properties of the theory as a function of ' one must consider the Legendre transform of W, instead of W. One makes the change of variables '(x) j! j(x) ' (.) The effective action is the functional of ' given by the Legendre transform of the functional W[j] Z ['] = j ' W [j], with j ' = d d xj(x)'(x) (.) This effective action contains all the information of the quantum theory. In fact one can consider the effective action as the classical action of a non-local theory, whose equations of motion give the correlation functions of the quantum theory. For instance, finding the minimum of the effective action, ' such that ['] '(x) ' = (.) amounts to find the v.e.v. of the quantum field for the quantum theory h (x)i = ' (x) locus of the minimum of ['] (.) Similarity, the two point (connected) function is the inverse of the Hessian of the effective action (we shall use this later) W(x, y) =h (x) (y)i c =( [' ]) x,y, ['] x,y = ['] '(x) '(y) (.) To see the advantage of the functional integral formalism, let us compute explicitly at one loop the quantum effective action for a general theory of a scalar field with action S[ ] (not necessarily ). We stay in the Euclidean case. There is a simple beautiful formula ['] =S[']+ ~ tr [log (S ['])] + O(~ ) (.5) where S ['] is the operator (acting of functions (x)) givenbythefunctionalderivativeof order of the classical action S['] (the Hessian), whose integral kernel is S ['](x,x )= S['] (.6) '(x ) '(x ) In other words: at tree level, the effective action equals the classical action; the one-loop term in the effective action is given by the quantum fluctuations around the background field configuration ', likeforthepathintegral,buthere' is a general configuration, it does not need to be a classical solution of the equations of motion. Perimeter Scholars International 9 Nov.

12 Proof: To show.5, let us start from the generating functional Z apple Z[j] = D [ ]exp (S[ ] j ) (.7) ~ and use the saddle point method. If j 6= the saddle point c[j] is solution of S x[ c] j = (.8) i.e. explicitely We rewrite S x[ c] = S[ ] (x) in the functional integral as c = j(x) (.9) (x) = c (x)+~ / (x) (.5) represents the quantum fluctuations, and the normalization factor ~ / is such that in the functional integral the typical are of order 'O() (.5) Now we expand the action S j around c.weobtainatorder~ S[ ] j = S[ c] j c + ~ S [ c] + (.5) Note that the linear term in disappears, thank to.8. Again we use the compact notation S [ c] x = d d xd d y (x)sxy[ c] (y) with S xy[ ]= S[ ] (x) (y) the Hessian of S[ ] We now integrate over to compute Z[j]. At one loop level it is just a Gaussian integral around the saddle point c, andweobtain Z[j] =(det[s [ c]]) / exp ~ (j c S[ c]) ( + O(~)) (.5) The generating functional for the connected functions is therefore W[j] = ~ log(z[j]) = j c S[ c] + ~ tr log [S [ c]] + O(~ ) (.5) We now compute the background field ' '(x) = W[j] Z j(x) = c(y) c(x)+ j(x) (y) y apple j S[ ] ~ tr log [S [ ]] + O(~ ) j fixed, = c (.55) Perimeter Scholars International 5 Nov.

13 But (y) [j S[ ]] j fixed, = c = j(y) S y[ c] = so that.55 becomes '(x) = c (x) Z ~ y c(y) j(x) (y) h i tr log [S ]] + O(~ ) (.56) j fixed, = c We need only to keep the O(~ ) order, which is simply '(x) = c (x) +O(~) (.57) Thus, at tree level, we can identify the background field ' with the saddle point c. This of course was to be expected. Now we can compute the effective action ['] at one loop. It is the Legendre transform of W[j], definedas ['] =j ' W[j] (.58) and using.5 it is at one loop ['] = S[ c]+j (' c) + ~ tr log [S [ c]] + O(~ ) (.59) The first term can be rewritten as S[']. Indeed,since' = first order around c S['] =S[ c]+(' c) S [ c]+o(~ ) (.6) c + O(~) we can expand S['] at so that, using again the saddle point equation.8, we obtain S[ c]+j (' c) =S[']+(j S [ c])(' c)+o(~ )=S[']+O(~ ) (.6) It is sufficient to use again.57 to have tr log [S [ c]] = tr log [S [']] + O(~) (.6) So we obtain the announced result.5 for the one loop effective action ['] = S['] + ~ tr log S ['] + O(~ ) (.6) This is a very simple and important result. It is independent of the precise form of the action S[ ], and extends to any theory with bosonic fields. Now two important remarks: Perimeter Scholars International 5 Nov.

14 Real time: We defined the effective action for an Euclidean QFT. In Minkowski space (real-time) W[j] and ['] are defined as W[j] = ~ Z apple i log D [ ]exp (S[ ]+j ) (.6) i ~ At one loop we get ['] = j ' + W[j], '(x) = W[j] j(x) (.65) ['] = S['] +i ~ tr log S ['] + O(~ ) (.66) Theories with fermions and gauge fields: This one loop result is modified when there are fermionic fields, since fermionic fields anticommute. We shall see later how. When gauge fields are present, gauge invariance require some care, we shall see later what has to be done..6 Application to What does this generic compact formula means for the theory? We consider the Euclidean theory for simplicity. At tree level the classical action S['] can be represented graphically as the inverted propagator plus the interaction vertex Z S['] = dx (r x '(x)) + m ' (x) + g Z dx ' (x)! tree['] = + (.67)! With the convention: barring with a apropagator(truncation)meansapplyingtheoperator +m to it, so that the truncated propagator is just ( +m )G = d (x y), while the twice-truncated propagator is just the inverse propagator, i.e. the operator +m =( +m ) =, =( +m ) =( +m ) (.68) At one loop (order ~) wehave loop['] = tr log [S [']] = tr log h x + m + g ' i (.69) The Hessian: The Hessian operator S ['] = x + m + g ' is an elliptic differential operator acting on the functions (x) as S ['] (x) = x (x)+m (x)+ g ' (x) (x) (.7) Perimeter Scholars International 5 Nov.

15 We expand in g the tr log[ ] of this operator, using the factorization of determinants (its a bit formal since we are dealing with infinite dimensional determinants, which are in fact infinite...) tr log(ab) =logdet(ab) =log(deta det B) =trloga +trlogb We obtain loop['] = h( tr log x + m ) + g i ( x + m ) ' = tr log x + m + h tr log + g i ( x + m ) ' = tr log x + m + X K= g K ( )K (K+) K tr h ( x + m ) ' Ki The first term of order g gives the total vacuum energy for the free field. We represent it graphically (the factor is chosen for consistency) as a closed loop tr log x + m = (.7) Since the integral kernels of ( ' adiracdistribution, x + m ) and ' are respectively the propagator G and ( x + m ) xy = G (x y), (' ) xy = '(x) (x y) (.7) the trace in the term of order g K is h tr ( x + m ) ' Ki x = dx dx K ' (x )G (x x x )' (x ) ' (x K )G (x K x ) = dx dx K ' (x ) ' (x K ) G (x x ) G (x K x ) (.7) Graphically this product of K free field propagators in the integral is descriped by a closed K-sided polygon 7 K 6 5 (.7) It gives the contribution of the one-loop irreducible diagram with K vertices and external amputated lines attached to eack vertex. One can check that the factor ( ) K /( K+ K) is precisely the combinatoric (or symmetry) factor attached to each irreducible part, coming from the application of the Feynman diagrammatic rules and of the Wick theorem. Perimeter Scholars International 5 Nov.

16 So at one loop, the effective action is indeed given by the sum of all the PI (one particle irreducible) diagrams, as depicted here loop = (.75) Each external represents a ' field. The truncated propagator represents simply a (x y) function, enforcing two ' to sit at each internal vertex. This result extends to general theories and to higher loop diagrams. The effective action is the generating functional of the irreducible amplitudes at all order. This means that ['] = X N= x N! d d z d d z N '(z ) '(z N ) (z, z N ) (.76) The N point function (z, z N ) is given by the sum of irreducible diagrams amplitude with N truncated external legs (z, z N )= X ~ B ( g) K c(g)i G (z,,z N ) (.77) PI G B being the number of loops of the diagram and K the number of vertices (for the theory). Perimeter Scholars International 5 Nov.