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11 and Feynman Diagrams

We now turn our attention to interacting quantum field theories. All of the results that we will derive in this section apply equally to both relativistic and non-relativistic theories with only minor changes. Herewewillusethe path integrals approach we developed in previous chapters. The properties of any field theory can be understood if the N-point cor- relation functions are known

GN x1,...,xN = 0 Tφ x1 ...φ xN 0 (11.1) Much of what we will do below can be adapted to any field theory ofinterest. ( ) ⟨ ∣ ( ) ( )∣ ⟩ We will discuss in detail the simplest case, the relativisticself-interacting scalar field theory. It is straightforward to generalize thistoothertheories of interest. We will only give a summary of results for the other cases.

11.1 The generating functional in perturbation theory The N-point function of a scalar field theory,

GN x1,...,xN = 0 Tφ x1 ...φ xN 0 , (11.2) can be computed from the generating functional Z J ( ) ⟨ ∣ ( ) ( )∣ ⟩ D i d xJ x φ x [ ] Z J = 0 Te ! 0 (11.3)

In D = d + 1-dimensional -time( ) (Z)J is given by the path [ ] ⟨ ∣ ∣ ⟩ integral [ ] iS φ + i dDxJ x φ x Z J = Dφe ! (11.4) ! [ ] ( ) ( ) [ ] 11.1 The generating functional in perturbation theory 359 where the action S φ is the action for a relativistic scalar field. The N-point function, Eq.(11.1), is obtained by functional differentiation, i.e.

[ ] N N 1 δ Z J G x ,...,x = −i (11.5) N 1 N = Z J δJ x1 ...δJ xN J 0 [ ] $ Similarly, the Feynman G x − x ,whichisessentiallythe$ ( ) ( ) F 1 2 $ 2-point function, is given by [ ] ( ) ( )$ ( ) 1 δ2Z J G x − x = i 0 Tφ x φ x 0 = −i (11.6) F 1 2 1 2 = Z J δJ x1 δJ x2 J 0 [ ] $ Thus, in principle, all we need to do is to compute Z J and,$ once it is ( ) ⟨ ∣ ( ) ( )∣ ⟩ $ determined, to use it to compute the N-point[ functions.] ( ) ( )$ We will derive an expression for Z J in the simplest theory,[ ] the relativis- tic real scalar field with a φ4 interaction, but the methods are very general. We will work in Euclidean space-time[ ] (i.e. in imaginary time)wherethe generating function takes the form

−S φ + dDxJ x φ x Z J = Dφe ! (11.7) ! [ ] ( ) ( ) where S φ now is [ ] 2 D 1 m λ [ ] S φ = d x ∂φ 2 + φ2 + φ4 (11.8) ! 2 2 4!

In the Euclidean theory[ ] the N-point% ( functions) are &

1 δN Z J G x ,...,x = φ x ...φ x = N 1 N 1 N = Z J δJ x1 ...δJ xN J 0 [ ] $ (11.9) ( ) ⟨ ( ) ( )⟩ $ Let us denote by Z0 J the generating action[ ] for the( free) scalar( field,)$ with action S0 φ .InChapter5wealreadycalculatedZ0 J ,andobtainedthe result (see eq. 5.141 and[ ] eq. 5.146) [ ] [ ] D −S0 φ + d xJ x φ x Z J = Dφe ! 0 ! [ ] 1( ) (D ) D [ ] −1 2 d x d yJ x G0 x − y J y = Det −∂2 + m2 e 2 ! ! / ( ) ( ) (11.10)( ) ' ( )* where ∂2 is the Laplacian operator in D-dimensional Euclidean space, and 360 Perturbation Theory and Feynman Diagrams

G0 x−y is the free field Euclidean propagator (i.e. the two-point correlation function) ( ) 1 G x − y = φ x φ y = x y (11.11) 0 0 −∂2 + m2 where the sub-index( label) 0 denotes⟨ ( ) ( a) free⟩ field⟨ ∣ expectation∣ ⟩ value. We can write the full generating function Z J in terms of the free field generating function Z0 J by noting that the interaction part of the ac- tion contributes with a weight of the path-integral[ ] that, upon expanding in powers of the [ ] λ takes the form

D λ 4 −S φ − d x φ x e int = e ! 4! ∞ −1 n λ (n ) [ ] = dDx ... dDx φ4 x ...φ4 x " n! 4! ! 1 ! n 1 n n=0 ( ) + , ( ) ( (11.12)) Hence, upon an expansion in powers of λ,thegeneratingfunctionZ J is

D −S0 φ + d xJ x φ x − Sint φ [ ] Z J = Dφe ! ! ∞ − n [λ] n ( ) ( ) [ ] [ ] = 1 " n! 4! n=0 ( ) + , D −S0 φ + d xJ x φ x × dDx ... dDx Dφφ4 x ...φ4 x e ! ! 1 ! n ! 1 n ∞ −1 n λ n δ4 [ ] δ4 ( ) ( ) = dDx ...( )dDx ( ) ... Z J " n! 4! ! 1 ! n 4 4 0 n=0 δJ x1 δJ xn ( ) 4 λ D+ , δ [ ] − d x ( ) ( ) 4! ! δJ x 4 ≡ e Z0 J (11.13) where the operator of the last line is defined by its power series expansion. ( ) [ ] We see that this amounts to the formal replacement δ S φ ↦ S (11.14) int int δJ

This expression allows us to write[ ] the generating+ , function ofthefulltheory Z J in terms of Z0 J ,thegeneratingfunctionofthefreefieldtheory, δ −S [ ] [ ] int δJ Z J = e Z0 J (11.15) + , [ ] [ ] 11.1 The generating functional in perturbation theory 361

Notice that this expression holds for any theory, not just a φ4 interaction. This result is the starting point of perturbation theory. Before we embark on explicit calculations in perturbation theory, it is worthwhile to see what assumptions we have made along the way.Weas- sumed, (a) that the fields obey Bose commutation relations, and (b) that the vacuum (or ) is non-degenerate. The restriction to Bose statistics was made at the level of thepathinte- gral of the scalar field. This approach is, however, of generalvalidity,and it also applies to theories with Fermi fields, which are path integrals over Grassmann fields. As we saw before, in all cases the generatingfunctional yields vacuum (or ground state) expectation values of time ordered prod- ucts of fields. The assumption of relativistic invariance is also not essential, although it simplifies the calculations significantly. The restriction to a non-degenerate vacuum state has more subtle physical consequences. We have mentioned that, in a number of cases, the vacuum may be degenerate if a global is spontaneously broken. Here, the thermodynamic limit plays an essential role. For example, ifthevacuum is doubly degenerate, we can do perturbation theory on one of the two vacuum states. If they are related by a global symmetry, the number of orders in perturbation theory which are necessary to have a mixing with its degenerate partner is proportional to the total number of degrees of freedom. Thus, in the thermodynamic limit, they will not mix unless thevacuum is unstable. Such an instability usually shows up in the form of infrared divergent contributions in the perturbation expansion. However, this is not asicknessoftheexpansion,buttheconsequenceofhavinganinadequate starting point for the ground state! In general, the “safe procedure” to deal with degeneracies that are due to symmetry is to add an additional term (like a source) to the Lagrangian that breaks the symmetry andtodoall the calculations in the presence of such symmetry breaking term. This term should be removed only after the thermodynamic limit is taken. The case of gauge symmetries has other and important subtleties. In the case of Maxwell’s Electrodynamics we saw that the ground state is locally gauge invariant and that, as a result, it is unique. It turns out that this is a generic feature of theories that are locally gauge invariantforallsymmetry groups and in all dimensions. There is a very powerful theorem(duetoS. Elitzur, which will be discussed later) that states that in theories with local gauge invariance, not only the ground state of theories with gauge invariance is unique and gauge-invariant, but that this restriction extends to the entire spectrum of the system. Thus only gauge-invariant operatorshavenon-zero 362 Perturbation Theory and Feynman Diagrams expectation values, and only such operators can generate thephysicalstates. To put it differently, a local symmetry cannot be broken spontaneously even in the thermodynamic limit. The physical reason behind this statement is that if the symmetry is local it takes only a finite order in perturbation theory to mix all symmetry related states. Thus, whatever mayhappenat the boundaries of the system has no consequence on what happens in the interior and the thermodynamic limit does not play a role any longer. This is why we can fix the gauge and remove the enormous redundancy of the description of the states. Nevertheless we have to be very careful about two issues. Firstly, the gauge fixing procedure must select one and only one state from each gauge class. Secondly, the perturbation theory is based on the propagatorofthegauge ′ ′ fields, Gµν x − x = −i 0 TAµ x Aν x 0 ,whichisnot gauge invariant and, unless a gauge is fixed, it vanishes. If a gauge is fixed, this propagator has contributions( ) that depend⟨ ∣ on( ) the( choice)∣ ⟩ of gauge, but the poles of this propagator do not depend on the choice of gauge since they describe physical excitations, e.g. . Furthermore although the propagator is gauge- dependent, it will only appear in combination with matter currents which are conserved.Thus,thegauge-dependenttermsofthepropagatordonot contribute to physical processes. Except for these caveats, we can now proceed to do perturbation theory for all the field theories of interest.

11.2 Perturbative expansion for the two-point function Let us discuss the perturbative computation of the two-pointfunctioninφ4 field theory in D-dimensional Euclidean space-time. We recall that, under a , the analytic continuation to imaginary time ix0 ↦ xD,the two-point function in D-dimensional Minkowski space-time, 0 Tφ x1 φ x2 0 maps onto the correlation function of two fields in D-dimensional Euclidean space-time, ⟨ ∣ ( ) ( )∣ ⟩

0 Tφ x1 φ x2 0 ↦ φ x1 φ x2 (11.16)

2 Let us formally write⟨ ∣ the( ) two( point)∣ ⟩ function⟨ ( )G( )x⟩1 − x2 as a power series in the coupling constant λ, ( ) ( ) ∞ n 2 λ 2 G x1 − x2 = Gn x1 − x2 (11.17) "= n! ( ) n 0 ( ) ( ) ( ) 11.2 Perturbative expansion for the two-point function 363 Using the generating functional Z J we can write

1 δ2Z J G 2 x − x = [ ] (11.18) 1 2 = Z J δJ x1 δJ x2 J 0 ( ) [ ] $ where $ ( ) $ [ ] δ( ) ( ) $ −S int δJ Z J = e Z0 J (11.19) Hence, the two-point function can be+ expressed, as a ratio of two series [ ] [ ] expansions in powers of the coupling constant. The numeratorofEq.(11.18) is given by an expansion in powers of the coupling constant λ,whichcan formally be written as ∞ δ2Z J − n λ n = 1 δJ x δJ x J=0 " n! 4! 1 2 n=0 [ ] $ ( 2) 4 4 D $ D δ - . δ δ × d y ... $d y ... Z J ( ) 1 ( )$ n 4 4 0 = ! ! δJ x1 δJ x2 δJ y1 δJ yn J 0 $ (11.20) [ ]$ ( ) ( ) ( ) ( ) $ Likewise, the denominator of Eq.(11.18) is given by the expansion of Z 0 in powers of λ,whichhassimilarexpressionbutwithoutacontributiondue to the external legs, corresponding to the functional derivatives with respect[ ] to the source at the external points x1 and x2.Theequivalentexpansions in Minkowski space-time are obtained, term by term, by the replacement

−λ ↦ iλ (11.21) at every order in the expansion. We will now look at the form of the first few terms of the expansion of the two-point function in perturbation theory.

11.2.1 Zeroth order in λ To zeroth order in λ, O λ0 ,thenumeratorofEq.(11.18)reducesto δ δ δ δ Z J = Z J + O λ ( ) = 0 = δJ x1 δJ x2 J 0 δJ x1 δJ x2 J 0 $ = G x − x + O λ $ (11.22) [ ]$ 0 1 2 [ ]$ ( ) ( ) ( ) $ ( ) ( ) $ while the denominator of Eq.(11.18) is simply equal to one ( ) ( )

Z 0 = Z0 0 + O λ = 1 + O λ (11.23)

[ ] [ ] ( ) ( ) 364 Perturbation Theory and Feynman Diagrams where we dropped the contribution of the determinant in Z0 0 since it does not contribute to the expectation values of the local observables. Hence, we find that to zeroth order the two-point function is simply giv[ en] by

2 G x1 − x2 = G0 x1 − x2 + O λ (11.24) ( ) ( ) ( ) ( ) 11.2.2 First order in λ To first order in λ,thedenominatorZ 0 of Eq.(11.18) is given by

−1 λ D Z 0 = 1 + [ ] d (11.25) 1! 4! !

The expression inside the integrand[ ] can+ be, calculated from the Taylor ex- pansion of Z0 J in powers of J x .Tofindanon-zerocontributionwe need to bring down from the exponent enough factors of J so that they can be cancelled[ ] by the functional( derivatives.) Since the argument of the exponential factor in Z0 J is a bilinear functional of J x , 1 [ ] dDx dDyJ x G x − y (J )y 2 ! ! 0 Z0 J ∝ e , (11.26) ( ) ( ) ( ) only an even number[ ] of derivatives in J x can be cancelled to a given order. In particular, to first order in λ,wehavetocancelfourderivatives. This means that we need to expand the exponential( ) in Z0 J to second order in its argument to obtain the only non-vanishing contribution to first order in λ to Z J at J = 0, [ ]

−1 λ Z 0 = 1[+] 1! 4! ( ) 4 2 [ ] D δ+ , 1 1 D D × d x d y1 d y2J y1 G0 y1 − y2 J y2 ! δJ x 4 2! 2 ! ! J=0 + 2 $ O λ + ( ) ( ) ( ),(11.27)$ ( ) $ The derivatives( ) yield a set of delta-functions

δ4 4 J y1 ⋯J y4 = δ ypj − x (11.28) 4 J=0 " # δJ x P j=1 $ [ ( ) ( )]$ ( ) where P runs over( the) 4! permutations$ of the arguments y1,y2,y3 and y4. 11.2 Perturbative expansion for the two-point function 365 We can now write the first order correction to Z 0 ,intheform

−1 λ 1 1 2 [ ] Z 0 = 1 + 1! 4! 2! 2 4 [ ] × +dDx,+dD,y ...d+ ,Dy G y − y G y − y δ y − x ! ! 1 4 0 1 2 0 3 4 " # pj P j=1 + O λ2 ( ) ( ) ( )

−1 λ D = 1 + S d xG x, x G x, x + O λ2 (11.29) ( 1!) 4! ! 0 0 ( ) + , ( ) ( ) ( ) where S = 3isthemultiplicityfactorofthisdiagram,whichcountsthe number of times this picture is obtained by contracting all the lines in a pairwise fashion without changing the topology of the graph. It is useful to introduce a picture or diagram to represent this contribution. Let us mark four points y1,...,y4 and an additional point at x (which we will call a vertex)withfourlegscomingoutofit.Letusjoiny1 and y2 by alineandy3 with y4 by another line. To each line we assign a factor of G0 y1 − y2 and G0 y3 − y4 respectively,

( ) ( ) x y G0 x − y = (11.30) ( ) Next, because of the delta-functions, we have to identify each of the points y1,...,y4 with each one of the legs attached to y in all possible ways. The resulting expression has to be integrated over all values of the the coordinates and of x.Theresultis

λ D Z 0 = 1 − d x G x, x 2 + O λ2 (11.31) 8 ! 0

[ ] ( ( )) ( ) Physically, the first order contribution represents corrections to the ground state due to vacuum fluctuations.Thisexpressioncanberepresented more simply by the shown in Fig.11.1.

Here, and below, we denote by a full line the bare propagator G0 x − y . δ2Z J Let us now compute the first order corrections to .We = δJ x1 δJ x2 (J 0 ) [ ] $ $ ( ) ( ) $ 366 Perturbation Theory and Feynman Diagrams

Figure 11.1 Vacuum fluctuations: first order correction to Z 0 . obtain [ ] −1 λ δ2 δ4 dDy Z J = 4 = 1! 4! ! δJ x1 δJ x2 δJ y J 0 ( ) 2 4 $ −1 λ D δ δ $ = d y [ ]$ ( ) ( ) ( )4 $ 1! 4! ! δJ x1 δJ x2 δJ y ( ) 3 1 1 D D × d z1 d z2J z1 G0 z1 − z2 J z2 (11.32) 3! 2 ! ( !) ( ) ( ) J=0 $ The non-vanishing contributions are obtained by matching$ the derivatives + ( ) ( ) ( ), $ in Eq.(11.32) with an equal number of powers of J.Weseethatwehave$ six factors of the source J at points z1,...,z6 and six derivatives, one at x1 and at x2,andfouraty.Tomatchderivativeswithpowersamountsto find all possible pair-wise contractions of these two sets of points. Once the delta functions have acted we are left with just one integral over the position y of the internal vertex. Hence, the result amounts to finding all possible contractions among the external legs at x1 and x2,witheachotherand/or with the internal vertex at y.Noticethatforeachcontractionwegetapower of the bare (unperturbed) propagator G0. The only non-vanishing terms resulting from this process arerepresented by the Feynman diagrams of Fig.11.2. y

x x 1 x2 x1 2 + y

Figure 11.2 First order contribution to the two-point function.

The first contribution, Fig.11.2(a), is the product of the bare propagator G0 x1 −x2 between the external points and the first order correction of the

( ) 11.2 Perturbative expansion for the two-point function 367 vacuum diagrams:

λ D 2 a = G x − x × − d y G y,y (11.33) 0 1 2 8 ! 0

The second( ) term,( the “”) / + diagram, of- Fig.11.2(b),( ). 0 is given by the expression

λ D b = − S d yG x ,y G y,y G y,x (11.34) 4! ! 0 1 0 0 2 = where the( multiplicity) + , factor is S ( 4 ×) 3.( It counts) ( the) number of ways (12) of contracting the external points to the internal vertex: there are four different ways of contracting one external point to one of the four lines attached to the internal vertex at y, and three different ways of contracting the remaining external point to the one of the three remaininglinesofthe internal vertex. There is only one way to contract the two leftover internal lines attached to the vertex at y. By collecting terms we get the result shown in Fig. 11.3.

+ + G(2) =

1 +

Figure 11.3 Diagrams for the two-point function to first order in λ.

To first order in λ,theexpansionofthetwo-pointfunctioncanalsobe written in the form shown in Fig. 11.4.

2 1+ O λ2 + +O(λ ) ! + ( )! ! ! G(2) =

1+ O λ2 ! + ( )!

Figure 11.4 Factorization of Feynman diagrams for the two-point function to first order in λ.

At least at this order in perturbation theory, a number of diagrams that contribute to the expansion of the numerator get exactly cancelled by the expansion of the denominator, the vacuum diagrams. The diagrams that get 368 Perturbation Theory and Feynman Diagrams cancelled are unlinked in the sense that one can split a diagram in two by drawing a line the does not cut any of the propagator lines. These diagrams contain a factor consisting of terms of the vacuum diagrams. We will see shortly that this is a feature of this expansion to all orders in λ. Thus, to first order in λ,thetwo-pointfunctionisgivenby(seefigure) λ G 2 x ,x = G 2 x ,x − dDyG2 x ,y G 2 y,y G 2 y,x +O λ2 1 2 0 1 2 2 ! 0 1 0 0 2 ( ) ( ) ( ) ( ) ( ) (11.35) as shown( in) Fig.11.5.( ) ( ) ( ) ( ) ( )

G(2) = + + ...

Figure 11.5 The two-point function to first order in λ.

11.3 Cancellation of the vacuum diagrams The cancellation of the vacuum diagrams is a general feature of perturbation theory. Let us reexamine this issue in more general terms. We will give the arguments for the case of the two-point function, but they aretrivialto generalize to any N-point function. This feature also holds in all theories, relativistic or not, bosonic or fermionic, provided the fields satisfy local canonical commutation (or anti-commutation) relations. The expansion of the two point function has the form ∞ 1 −1 n n φ x φ x = dDy ...dDy φ x φ x L φ y 1 2 Z " ! 1 n n! 1 2 # int j 0 0 n=0 j=1 ( ) ⟨ ( ) ( )⟩ ⟨ ( ) ( ) (11.36)1 ( )2⟩ The denominator[ factor] Z 0 has a similar expansion

∞ n N −1 D D Z 0 = [ d] y ...d y L φ y (11.37) " n! ! 1 n # int j 0 n=0 j=1 ( ) [ ] ⟨ 1 ( )2⟩ where O φ 0 denotes the expectation value of the operator O φ in free field theory. Let us⟨ consider[ ]⟩ first the numerator. Each expectation value involves[ ] a sum of products of pairwise contractions. If we assign a Feynman diagram to each contribution, it is clear that we can classify these terms into two classes: (a) linked and (b) unlinked diagrams. A diagram is said to be unlinked if it 11.3 Cancellation of the vacuum diagrams 369 contains a sub-diagram in which a set of internal vertices are linked with each other but not to an external vertex. The linked diagrams satisfy the opposite property. Since the vacuum diagrams by definition donotcontain any external vertices, they are unlinked. All the expectation values that appear in the numerator can bewrittenas asumofterms,eachoftheformofalinkeddiagramtimesavacuum graph, i.e.

φ x1 φ x2 Lint φ y1 ...Lint φ yn 0 = n k n n " ⟨ ( ) ( =) ( ( φ))x φ x ( ( L))⟩ φ y L φ y " k 1 2 # int j 0 # int j 0 k=0 j=1 j=k+1 3 4⟨ ( ) ( ) 1 ( )2⟩ ⟨ 1 ( (11.38))2⟩ where the super-index % denotes a linked factor, i.e. a factor that does not contain any vacuum sub-diagram. Thus, the numerator has the form

∞ n n k N −1 n " φ x φ x L y L φ y (11.39) " " n! k 1 2 # int k 0 # int j 0 n=0 k=0 j=1 j=k+1 ( ) which factorizes3 into4⟨ ( ) ( ) ( )⟩ ⟨ 1 ( )2⟩ ∞ −1 k k dDy ...dDy φ x φ x L φ y " " k! ! 1 k 1 2 # int j 0 k=0 j=1 ( ) ∞ - ⟨ ( )−(1 n) 1 ( )2⟩ .n × dDy ...dDy L φ y " n! ! 1 n # int j 0 n=0 j=1 ( ) - ⟨ 1 ((11.40))2⟩ We can clearly recognize that the second factor is exactly equal to the de- nominator Z 0 . Hence we find that we can write the two-point function as a sum oflinked Feynman diagrams:[ ] ∞ −1 n n φ x φ x = dDy ...dDy φ x φ x L φ y " 1 2 " n! ! 1 n 1 2 # int n 0 n=0 j=1 ( ) ⟨ ( ) ( )⟩ ⟨ ( ) ( ) ( ( (11.41)))⟩ This result is known as the linked-cluster theorem.Thistheorem,which proves that the vacuum diagrams cancel exactly out to all orders in pertur- bation theory, is valid for all the N-point functions (not just for two-point function) and for any local theory. It also holds in Minkowskispace-time upon the replacement −1 n ↔ in.Itholdsforalltheorieswithalocal canonical structure, relativistic or not. ( ) 370 Perturbation Theory and Feynman Diagrams 11.4 Summary of Feynman rules for φ4 theory 11.4.1 Position space The general rules to construct the diagrams for the N-point function 4 0 Tφ x1 ...φ xN 0 in φ theory in Minkowski space and φ x1 ...φ xN in Euclidean space, in position space are ⟨ ∣ ( ) ( )∣ ⟩ ⟨ ( ) ( )⟩ 1AgeneralgraphfortheN-point function has N external points and n internal interaction vertices, where n is the order in perturbation theory. Each vertex is a point with a coordinate label and 4 lines (for a φ4 theory) coming out of it. 2Drawalltopologicallydistinctgraphsobtainedbyconnecting the external points and the internal vertices in all possible ways. Discard all graphs that contain sub-diagrams not linked to at least one external point. 3Thefollowingweightisassignedtoeachgraph: −i λ − λ 1Foreveryvertexafactorof 4! in Minkowski space and 4! in Eu- clidean space .

2Foreverylineconnectingapairofpointsz1 and z2,apropagatorfactor = 2 of 0 Tφ z1 φ z2 0 0 −iG0 z1,z2 in Minkowski space, or φ z1 φ z2 0 = G0 z1 − z2 in( ) Euclidean space. ⟨ ∣ ( ) ( )∣ ⟩1 ( ) 3Anoverallfactorofn! . 4Amultiplicityfactorthatcountsthenumberofwaysinwhic⟨ ( ) ( )⟩ ( ) hthelines can be joined without changing the topology of the graph.

5Integrateoverallinternalcoordinates zi .

For example, the 4-point function { } 4 G x1,x2,x3,x4 = φ x1 φ x2 φ x3 φ x4 (11.42) ( ) has the two contributions at order λ2 shown in the diagram of figure 11.6. ( ) ⟨ ( ) ( ) ( ) ( )⟩ 2 = 2 These two diagrams have exactly the same weight (if G0 1, 2 G0 2, 1 ), ( ) ( ) x1 x x1 x 3 ( ) 3( )

y1 x y2 x y2 y1 2 x4 2 x4 (a) (b)

Figure 11.6 Two topologically distinct contributions to the four-point func- 2 tion to order λ that have the same analytic expression. 4 11.4 Summary of Feynman rules for φ theory 371 and their total contribution to the 4-point function is 1 −λ 2 S dDy dDy G 2 x ,y G 2 x ,y 2! 4! ! 1 ! 2 0 1 1 0 2 1 ( ) ( ) 2 2 2 2 + , × G0 y1,y(2 G)0 x3(,y2 G)0 x4,y2 (11.43) ( ) ( ) ( ) but are topologically distinct and, thus, count as separate contributions. The 5 ( )6 ( ) ( ) multiplicity factor is S = 4 × 3 2 × 2 × 2.

( ) 11.4.2 space

q

p p

Figure 11.7 First order contribution to the two-point function in momen- tum space.

The N-point functions can also be computed in momentum space. The rules for constructing Feynman diagrams in momentum space are: 1AgraphhasN external legs, labelled by a set of external momenta k1,...,kN ,flowingintothediagramandn internal vertices (for order n in perturbation theory). Each vertex has 4 lines (for φ4 theory) each carrying a momentum q1,...q4 (flowing out of the vertex). All lines must be connected in pairs. All vacuum terms have to be discarded. 2 Draw all the topologically different graphs. 3Weightofeachdiagram: • for each vertex, a factor of −λ 2π DδD 4 q . 4! ∑i=1 i • each line carries a momentum pµ and contributes to the weight with G 2 p = 1 ( )( ) ( ) afactor 0 p2+m2 ,inEuclideanspace.InMinkowskispaceit ( ) 2 −i becomes −iG = 2 2 . (0 ) p −m +i$ • all the numerical( ) factors are the same as in position space. • we must integrate over all the internal momenta. 372 Perturbation Theory and Feynman Diagrams For example, the first order contribution to the two-point function is the tadpole diagram shown in Fig.11.7. It has the algebraic weight

−λ 1 dDq 1 1 2 4 × 3 (11.44) 4! 1! ! 2π D q2 + m2 p2 + m2 + , ( ) + , ( ) 11.5 Feynman rules for theories with and gauge fields The Feynman rules that we derived for φ4 theory can be easily extended to other theories, relativistic or not. An important difference occurs in theories with fermions in which the rules need to be modified to account to the Fermi-Dirac statistics. As an example we will consider the case of (QED), the theory of and photons.Similarrules are formulated in (QCD) that we will discuss in alaterchapter.Feynmanrulesforself-interactingfermions are also devised in the case of the Gross-Neveu model (or the Nambu-Jona-Lasinio model) and in the case of the non-relativistic gas. The Lagrangian density of QED, in the Feynman-’t Hooft gauges, is

1 1 µ L = ψ¯iDψ − F 2 − ∂ A 2 (11.45) QED 4 µν 2α µ

The partition function in the presence/ of separate( sources) η ¯ x for the Dirac (spinor) field ψ x (and η x for the adjoint Dirac field ψ¯ x ,andJµ x for the electromagnetic gauge field Aµ x ,is ( ) ( ) ( ) ( ) ( ) i d4x L +J x Aµ x +η¯ x ψ x +ψ¯ x η x Z η, η¯,J = DψDψ¯DA e ∫( ) QED µ µ ! µ 7 ( ) ( ) ( ) ( ) ( ) ((11.46))8 where[ we dropped] the Faddeev-Popov determinant which is trivial in this abelian (see section 9.5). In what follows we drop the spinor labels to simplify the notation. 4 As in the case of φ theory, the partition function Z η, η¯,Jµ of Eq.(11.46) can be expressed in terms of the partition function of a free field theory with sources. In this case we have two partition functions[ of free]field theories: the partition function for free Dirac fields with sources,

Z η, η¯ = DψDψ¯ exp i d4x ψ¯ i∂ − m ψ + ηψ¯ + ψη¯ (11.47) Dirac ! ! and the partition[ ] function of the/ free Maxwell1 ( / field,) in the Feynman-’t20 Hooft 11.5 Feynman rules for theories with fermions and gauge fields 373

gauges, with sources Jµ given by

1 1 µ 2 µ Z J = DA exp i d4x − F 2 − ∂ A + J A Maxwell µ ! µ ! 4 µν 2α µ µ (11.48) where, here[ too,] we dropped the/ Faddeev-Popov+ determinant.1 2 , 0 In QED the interaction term of the Lagrangian is

¯ ¯ µ Lint ψ, ψ,Aµ = eAµψγ ψ (11.49)

Then, the full generating function[ of eq.(11.46)] is expressed as

4 δ δ δ Z η, η¯,Jµ = exp i d xLint −i , −i , −i ZDirac η, η¯ ZMaxwell Jµ ! δη¯ δη δJµ (11.50) where[ we] made the+ replacements/ 0, [ ] [ ]

δ δ δ ψ x → −i ,,ψ¯ x → −i ,Ax → −i (11.51) δη¯ x δη x µ δJµ x ( ) ( ) ( ) inside the interaction( ) term of the lagrangian.( ) ( ) The Feynman rules then follow by the same line of argument thatweused in φ4 theory. The main differences are that

1. Now we have two : the propagator of the free massive Dirac field, SF x − y = −i 0 Tψ x ψ¯ y 0 ,whichwasgivenexplicitlyinsec- tion 7.2, and the propagator of the Maxwell field (in the Feynman-’t Hooft gauges) G( µν x)− y,α⟨ =∣ i 0( TA) (µ )x∣ A⟩ ν y 0 ,givenexplicitlyinsection 9.6. 2. The interaction( vertex) involves⟨ ∣ two( ) Dirac( )∣ fermions⟩ and a gauge field, shown in Fig. 11.8. Each bare vertex has a weight in a Feynman diagram of eγµ. 3. Each closed Dirac loop contributes with a minus sign, and each diagram now has a weight −1 F ,whereF is the number of fermion loops. This rule follows from the form of the path integral forfermions (see, chapter 7). It applies to( all) theories of fermions, relativistic or not.

Aside from these changes the Feynman rules are the same. 374 Perturbation Theory and Feynman Diagrams p + q

q p

Figure 11.8 The bare interaction vertex of QED: the full lines are Dirac fermion propagators and the wavy line is a propagator. The bare vertex has a weight eγµ.

11.6 The two-point function and the self-energy in φ4 theory We now return to φ4 theory. To first order in λ,andinmomentumspace, the two point function in Euclidean space is

1 −λ dDq 1 1 2 G 2 p = + 4 × 3 + O λ2 p2 + m2 4! ! 2π D q2 + m2 p2 + m2 ( ) (11.52) ( ) 2 ( )3 4 + , ( ) Let us define by µ the effective or renormalized( ) mass (squared) such that

1 1 λ dDq 1 1 = 1 − + O λ2 p2 + µ2 p2 + m2 2 ! 2π D q2 + m2 p2 + m2 (11.53) 9 3 4 + , ( ): Again, to first order in λ,wecanwritetheequivalentexpression( ) 1 G 2 p = + O λ2 (11.54) λ dDq 1 ( ) 2 + 2 + p m D 2 2 ( ) 2 ! 2π q + m ( ) This expression leads us to define µ2 to be ( ) λ dDq 1 µ2 = m2 + + ... (11.55) 2 ! 2π D q2 + m2 This equation is equivalent to a sum of a large number of diagrams with higher order in λ.Howdoweknowthatitisconsistent?Letusfirstnotethat( ) we have summed diagrams of the form shown in figure 11.9. These diagrams have the very special feature that it is possible to split the diagram into two sub-diagrams by cutting only a single internal line. Momentum conservation requires that the momentum of that line be equal to the momentum on the incoming external leg. Thus, once again we have two types of diagrams: (a) 4 11.6 The two-point function and the self-energy in φ theory 375

q1 q2 q3 qn

...

p p p p p p

Figure 11.9 The set of all one-particle reducible diagrams of the two-point function to leading order in λ. one-particle reducible diagrams (which satisfy the property defined above) and (b) the one-particle irreducible graphs that do not. Hence, the total contribution to the two-point function is the solution of theequationshown in Fig 11.10.

= +

Figure 11.10 The Dyson equation for the two-point function.

Here the thick lines are the full propagator, the thin line is the bare prop- agator, and the shaded blob represents all the irreducible diagrams, i.e. dia- grams with amputated external legs. We represent the blob by the self-energy operator Σ p ,showninFig.11.11

( ) = + + + ...

Figure 11.11 Feynman diagrams summed by the Dyson equation.

Thus, the total sum satisfies the Dyson equation

2 = 2 2 2 G p G0 p + G0 p Σ p G p (11.56) ( ) ( ) ( ) ( ) 2 2 The inverse of G p(,)Γ p ,satisfies( ) ( ) ( ) ( ) ( ) ( ) 2 = 2 −1 = 2 2 Γ (p) G0( )p − Σ p p + m − Σ p (11.57) ( ) ( ) To first order in λ, (Σ )p is just( ) the tadpole( ) term ( ) λ dDq 1 Σ (p ) = − + O λ2 (11.58) 2 ! 2π D q2 + m2 ( ) ( ) ( ) 376 Perturbation Theory and Feynman Diagrams which, to this order in λ,happenstobeindependentoftheexternalmo- mentum pµ.Ofcourse,thehigherordertermsingeneralwillbefunctions of pµ. In terms of the renormalized mass µ2,toorderone-loop(i.e.O λ )weget

D 2 2 2 λ d q 1 2 ( ) µ = m − Σ p = m + + O λ (11.59) 2 ! 2π D q2 + m2 ( ) ( ) Thus, we conclude that, to O λ ,vacuumfluctuationsrenormalizethemass.( ) However, a quick look at Eq.(11.59) reveals that this is very large renormal- ization. Indeed, fluctuations of( all) momenta,rangingfromlongwave-lengths (and low ) with q ∼ 0, to short wave-lengths (or high energies) con- tribute to the mass . In fact, the high-energy fluctuations, with q2 ≫ m2,yieldthelargest contributions to Eq.(11.59), since the mass effectively cuts-offthe contributions in the infrared, IR, q → 0. Moreover, for all dimensions D ≥ 2thehigh-energy(orultraviolet,UV,q → ∞)con- tribution is divergent.Ifweweretocutofftheintegralatahigh-momentum scale Λ, in general space-time dimension D the the contribution from this − diagram will diverge as ΛD 2.Inparticular,thetadpolecontributionto the mass renormalization is logarithmically divergent for D = 2(i.e.1+ 1) dimensions, and quadratically divergent for D = 4dimensions. Thus, although it is consistent (and quite physical) to regard the leading effect of fluctuations as a mass renormalization, they amount toadivergent change. The reason for this divergence is that all wavelengths contribute, from the IR to the UV. This happens because space-time is continuous, and we assumed that there is no intrinsic short-distance scale below which local field theory should not be valid. There is a way to think about this problem. The problem of how toun- derstand the of these singular contributions, indeed of how the con- tinuum limit (a theory without cutoff) of quantum field theory is defined is the central purpose of the (RG). We will study this approach in detail in a later chapter. Here we will discuss some qualitative features. From the point of view of the RG the problem is that the contin- uum theory (i.e. defining a theory without a UV cutoff) cannot bedefined naively. We will see that for such a procedure to work it is necessary to be able to define the theory in a regime in which there is no scale, i.e. in a scale-invariant regime. This requirement means that one should look at a regime in which the renormalized mass becomes arbitrarily small, µ2 → 0. As we will see below, this requires to fine tune the bare coupling constant and the bare mass to some determined critical values.Itturnsoutthat, 4 11.6 The two-point function and the self-energy in φ theory 377 near such a critical point acontinuumfieldtheory(withoutaUVcutoff) can be defined. The RG point of view relates the problem of the definition of a Quantum Theory to that of finding a continuous phase transition, which a central problem in Statistical Physics. However, there are alternative descriptions, such as StringTheory,that postulate that local field theory is not the correct description at short dis- −33 % = hG̵ ∼ tances, typically near the Planck scale, Planck c3 10 cm (!), where G is Newton’s gravitational constant. From this; view point, these singular contributions at high energies signal a breakdown of the theory at those scales. Before we try to compute Σ p ,itisworthtomentiontheHartree Ap- proximation.Itconsistsinsummingupalltadpolediagrams(andonlythe tadpole diagrams) to all orders in( )λ.Atypicalgraphisshowninfigure11.12.

Figure 11.12 A typical tree diagram.

The sum of all the tadpole diagrams can be done by means of a very simple trick. Let us modify the expression for the self-energy to make it self consistent, i.e. λ dDq 1 = − Σ0 p D 2 2 (11.60) 2 ! 2π q + m − Σ0 q This formula is equivalent( ) to a Dyson equation in which the internal propa- gator is replaced by the full propagator,( ) as in figure 11.15.( ) This approxima- tion becomes exact for a theory of an N-component real scalar field φa x (a = 1,...N), with O N symmetry, and interaction (shown in Fig.11.13)

N 2 ( ) λ 2 λ L φ (= ) φ 2 = φ x φ x (11.61) int 4! 4! " a a a=1 in the large N limit,[ ]N → ∞((.Otherwise,thesolutionofthisintegralequa-) ) 3 ( ) ( )4 378 Perturbation Theory and Feynman Diagrams tion just yields the leading correction. The reason why this works can be seen a b

a b

Figure 11.13 The vertex for the real scalar field with O N symmetry; here a, b = 1,...,N. ( ) by a direct counting of the powers of N in the expansion of the two-point function to order one-loop, as shown in Fig.11.14.

b a

a a a a

Figure 11.14 The 2-point function to one-loop order in the O N theory; here a, b = 1,...,N. The diagram on the left is of order N while the diagram on the right is of order 1. ( )

G(0) G(2)

!→

Figure 11.15 The self-energy in the one-loop (Hartree) approximation.

Eq.(11.60) is an integral non-linear equation for Σ0 p .Equationsofthis type are common in many-body physics. For example, the gap equation of the BCS theory of superconductivity has a similar form( ) (although in BCS theory one sums ladder diagrams, instead of bubble diagrams.) Let us now evaluate the integral in the equation for Σ0 p ,Eq.(11.60). Clearly Σ0 p is a correction due to virtual fluctuations with momenta qµ ranging from zero to infinity. These “off-shell” fluctuations( do)not obey the 2 2 mass shell( condition) p = m .Noticethat,atthislevelofapproximation, Σ p is independent of the momentum.Thisisonlycorrecttoorderone-loop.

( ) 4 11.6 The two-point function and the self-energy in φ theory 379 Before computing the integral, let us rewrite Eq.(11.60) in terms of the effective or renormalized mass µ2,

λ dDq 1 µ2 = m2 − Σ p = m2 + (11.62) 2 ! 2π D q2 + µ2

Let us denote by m2 the value( ) of the bare mass such that the renormalized c ( ) mass vanishes, µ2 = 0:

D 2 λ d q 1 0 = mc + (11.63) 2 ! 2π D q2

2 Clearly, mc is IR divergent for D ≤ 2andUVdivergentforD ≥ 2. Let us now 2 ( ) 2 2 2 2 express the renormalized mass µ in terms of mc,anddefineδm = m −mc. We find,

D 2 2 2 λ d q 1 µ =δm + mc + 2 ! 2π D q2 + µ2 λ dDq 1 1 =δm2 + − 2 ! 2π D( q)2 + µ2 q2 λ dDq 1 =δm2 − µ2 + , (11.64) 2 !( 2)π D q2 q2 + µ2 In D = 4dimensionstheintegralcontributeswithalogarithmicdivergence (c.f. Eq.(11.99) of section 11.8): ( ) ( ) λ 1 Λ γ µ2 = δm2 − µ2 ln − (11.65) 2 8π2 µ 16π2 Thus, although the stronger, quadratic+ divergence,+ , was abs, orbed by a renor- malization of the mass, a weaker logarithmic singularity remains. It turns out, as we will discuss in a later chapter, that this remainingsingularcontri- bution can absorbed only by a renormalization of the couplingconstantλ. In any case, what is clear is that, already at the lowest order in perturbation theory, the leading corrections can (and do) yield a larger contribution to the behavior of physical quantities than the bare, unperturbed, values, and that these corrections are not small. We will end with a discussion of what these large perturbativecorrections mean in the context of the theory of phase transitions since, after all, it is also described by a theory with the same form. We saw before that in the Landau theory of phase transitions the mass (squared) parameter, the coefficient of 2 2 the φ term in the action, is related to the difference m ≡ T − T0,i.e.the 380 Perturbation Theory and Feynman Diagrams temperature T measured from the mean field critical temperature, T0.We 2 can think of mc ≡ Tc − T0,definingacorrectedcriticaltemperature,Tc,

λ dDq 1 Tc = T0 − (11.66) 2 ! 2π D q2

It shows that fluctuations suppress T downwards from T due to (precisely) c ( ) 0 the large contributions from fluctuations at short distances. Thus, the UV singular effects can be absorbed in a new (and lower) Tc.Thesubtracted 2 mass, δm ≡ T − Tc,isnowthenewcontrolparameter,asshowninthelast line of Eq.(11.64). Returning to the mass renormalization of Eq.(11.64), we can use the in- tegral of Eq.(11.98) (of section 11.8), to find the result

2 D −1 D µ 2 Γ 2 − µ2 = δm2 − λ 2 (11.67) 4π D 2 D − 2 ( ) ( ) 2 = / Since δm T − Tc,wecanwritethisresultinthesuggestiveform( ) D λ Γ 2 − D − 2 2 2 2 2 1 δm = T − Tc = µ + µ (11.68) π D 2 D − 2 ⎡ 4 ⎤ ⎢ ( )⎥ ⎢ / ⎥ ( ) We will now look at the IR behavior,⎢ where the renormalized⎥ mass µ2 → 0. ⎣⎢( ) ⎦⎥ We see that if D > 4, the exponent of µ2 in the second term of the r.h.s. of Eq.(11.68) is always larger than 1 and hence this contribution is small compared to the first term. However, for D < 4thereverseistrueandforµ2 small enough the fluctuations dominate. hence, perturbationtheorybreaks down for some small value of µ2 where the two contributions are comparable. Hence perturbation theory is reliable (in the IR) for

D 2 4−D λ Γ 2 − ≳ 2 ≈ 2 δT µ /( ) (11.69) 4π D 2 D − 2 ⎛ ( )⎞ / ⎜2 ⎟ and breaks down for δT (or ⎝µ()smallerthanthatvalue.Thisisknownas) ⎠ the Ginzburg Criterion which gives an estimate of the width ofthecrit- ical region: the range temperatures (or renormalized mass) for which the perturbation theory breaks down in the IR and does not describe the true behavior. Let us recall that the relation between the susceptibility and the effective − (or renormalized) mass: µ2 = χ 1. Hence, at one-loop order, we would predict 11.7 The four point function and the effective coupling constant 381 that −2 D−2 T − Tc ∶ D < 4 −1 χ T ∝ T − T /( ) ∶ D > 4 (11.70) ⎧ c ⎪ −1 ⎪(T − T ) × small logarithmic corrections ∶ D = 4 ⎪ c ( ) ⎪⎨( ) We see that⎪ one effect of these fluctuations can be to change the dependence ⎪( ) of a physical⎩ quantity, such as the susceptibility, on the control parameter, T − Tc,whichsetshowclosethetheoryisthemassless or critical regime. Akeypurposeoftherenormalizationgroupprogramistheprediction of critical exponents such as the one we fund in Eq.(11.70). We will see that this result is actually the beginning of a set of controlled approximations to the exact values. Just as important will be the fact that the renormalization group will give a deeper interpretation of the meaning of renormalization.

11.7 The four point function and the effective coupling constant We will now discuss briefly the perturbative contributions tothefour-point function, 4 G x1,x2,x3,x4 = φ x1 φ x2 φ x3 φ x4 (11.71) ( ) which is also known as the “two-particle” correlation function. We will dis- ( ) ⟨ ( ) ( ) ( ) ( )⟩ cuss its connection with the effective (or renormalized) coupling constant. To zeroth order in perturbation theory, O λ0 ,thefourpointfunction factorizes into a product of all (three) possible two point functions obtained by pair-wise contractions of the four field operators( ) (shown in Fig.11.16)

1 2 1 2 1 2

+ +

3 4 3 4 3 4

4 0 Figure 11.16 The four point function G 1, 2, 3, 4 to order O λ . ( ) ( ) ( ) 4 G x1,x2,x3,x4 =

( =)G0 x1,x3 G0 x2,x4 + G0 x1,x2 G0 x3,x4 + G0 x1,x4 G0 x2,x3 + O λ ( ) (11.72) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 382 Perturbation Theory and Feynman Diagrams As it is apparent, to zeroth order in λ,thefourpointfunctionreducesto just products of bare two-point functions and hence nothing new is learned from it. We will show in the next chapter, Eq.(12.40), that, toallorderson perturbation theory, the four-point function has the following structure:

4 2 2 G x1,x2,x3,x4 =G x1,x3 G x2,x4 ( ) ( ) 2 ( ) 2 2 2 + G x1,x2 G x3,x4 + G x1,x4 G x2,x3 ( ) ( ) ( ) ( ) ( ) ( ) ( ) − dDy ... dDy G 2 x ,y G 2 x ,y G 2 x ,y G 2 x ,y ! 1 ! 4 1( 1 ) 2( 2 ) 3 3( ) 4 4( ) ( ) 4 ( ) ( ) ( ) × Γ( y1,y)2,y3,y( 4 ) ( ) ( )(11.73) ( ) 2 ′ where the factors of G x,( x represent) the exact two-point function,and 4 the new four-point function,( ) Γ y1,y2,y3,y4 ,isknownasthefour-point .Thevertexfunctionisdefinedasthesetofone-particle( )( ) irreducible (1PI) Feynman diagrams,( i.e. diagrams) that cannot be split in two by cutting a single propagator line, with the external lines “amputated” (they are already accounted for in the propagator factors). In momentum space, due to momentum conservation at the vertex, Γ 4 has the form ( ) 4 D D Γ 4 p ,...,p = 2π δ p Γ 4 p ,...,p (11.74) 1 4 " i 1 4 i=1 ( ) ( ) ( ) ( ) 3 4 4 ( ) The lowest order contribution to Γ y1,y2,y3,y4 appears at order λ ( ) 4 y ,y ,y ,y = λ + O λ2 Γ 1 2 3 4 ( ) (11.75) ( ) depicted by the tree level diagrams of Fig.11.17. In momentum space the ( ) ( ) Γ(4) λ

=

Figure 11.17 The one-particle irreducible four-point (vertex) function at the tree level. one-particle irreducible four-point function is

4 2 Γ p1,...,p4 = λ + O λ (11.76) ( ) ( ) ( ) 11.7 The four point function and the effective coupling constant 383

To one-loop order, O λ2 ,thefour-pointvertexfunctionisasumof(three) Feynman diagrams of the form of Fig.11.18. The total contribution to the ( ) p1 q p2

p4 p3 p1 + p2 − q

Figure 11.18 The one-loop contribution to the one-particle irreducible four- point (vertex) function. vertex function Γ 4 ,toorderone-loop,is

( ) 4 Γ p1,...,p4 = ( ) λ2 dDq 1 λ − + ( ) D 2 2 2 2 two permutations 2 ! 2π q + m p1 + p2 − q + m 3 L + O λ (11.77)M ( ) 1 2 1( ) 2 This expression has a logarithmic UV divergence in D = 4, and more severe ( ) divergencies for D > 4. To address this problem let us proceed by analogy with the mass renormalization and define the physical or renormalized cou- 4 pling constant g by the value of Γ p1,...,p4 at zero external momenta, = = = p1 ... p4 0. (It is up to us( to) define it at any momentum scale we wish.) ( )

4 4 g ≡ lim Γ p1,...,p4 = Γ 0,...,0 (11.78) pi→0 ( ) ( ) This definition is convenient and( simple) but it is( problemati) ciftherenor- malized mass µ2 vanishes (i.e. in the massless or critical theory). To order one-loop, the renormalized coupling constant g is

λ2 dDq 1 g = λ − 3 + O λ3 (11.79) 2 ! 2π D q2 + m2 2 Using the same line of argument we used to define( the) self energy, Σ, ( ) 1 2 we will now sum all the one loop diagrams, as shown in Fig.11.19. This “bubble” sum is a geometric series, and it is equivalent to thereplacement 384 Perturbation Theory and Feynman Diagrams

1 2 q q q q1 q2 q3 + + + ...

Figure 11.19 The sum of bubble diagrams. of Eq.(11.79) by

g2 dDq 1 g = λ − 3 + O λ3 (11.80) 2 ! 2π D q2 + m2 2 or, alternatively, to write the bare coupling constant λ (in terms) of the renor- ( ) malized coupling g as 1 2

3 dDq 1 λ = g + g2 + O g3 (11.81) 2 ! 2π D q2 + µ2 2 where we have replaced the bare mass m2 with the renormalized( ) mass µ2. ( ) 1 2 This amounts to add the insertions of tadpole diagrams in the internal prop- agators. This is consistent at this order in perturbation theory. Written in terms of the renormalized coupling constant g and of the renor- malized mass µ2,thevertexfunctionbecomes

4 Γ p1,...,p4 = ( ) g2 dDq 1 1 ( = g −) − D 2 2 2 2 2 2 2 2 ! 2π q + µ p1 + p2 − q + µ q + µ + two permutations% + O g3 (11.82)& ( ) 1 2 1( ) 2 1 2 which is UV finite for D < 6. Thus, the renormalization of the coupling ( ) constant leads to a subtraction of the singular expression for the vertex function. After the renormalization of the coupling constant, the singular behavior of the integral now appears only in the relation between the bare coupling constant λ and the renormalized coupling constant g,giveninEq.(11.81). Clearly there are a number of ways to interpret the meaning of this relation. One interpretation is to say that at a fixed value of the bare coupling con- stant λ,Eq.(11.81)relatestheregulatorΛ(whichisamomentumscale) and the renormalized coupling constant. In other terms, the effective of renormal- ized coupling constant g has become a function of a momentum (or energy) scale!. Conversely, we can fix the renormalized coupling and ask how do we 11.7 The four point function and the effective coupling constant 385 have to change the bare coupling constant λ as we send the regulator to infinity. It will be useful to work with dimensionless quantities. Since the bare − and the renormalized coupling constants have units of Λ4 D,wedefinethe dimensionless bare coupling constant u by −D λ = Λ4 u (11.83) Then D D− 3 d q 1 u =Λ 4 g + g2 + O g3 2 ! 2π D q2 2

D−4 3 1 6 ( D−)4 2 3 =Λ g + ( ) 1 2 Λ g + O g (11.84) 4π D 2 D − 2 D − 4 We will vary the3 momentum/ scale Λand the dimensionless( )4 bare coupling ( )( ) constant u at fixed g(. The) differential change of the dimensionless coupling constant is known as the renormalization group beta-function, ∂u β u = −Λ (11.85) ∂Λ g $ where I have used the sign convention standard$ in StatisticalPhysics(op- ( ) $ posite to that in High Energy Physics!). Hence$ (for D → 4) 3 β u = 4 − D u − u2 + O u3 (11.86) 16π2 The behavior of the( Renormalization) ( ) group beta-function( ) (orGell-Mann- Low) for D < 4, D = 4andD > 4isshowninFig.11.20.Theinfrared (IR) RG flows,i.e.theflowofthedimensionlesscouplingconstantu as the momentum scale λ is decreased,isalsoshowninFig.11.20(bottom).The UV flows, i.e. the flow of u as the momentum scale is increased,isobtained by reversing the direction of the RG flows. Clearly for D ≥ 4thedimensionlesscouplingconstantflowsto0inthe infrared (at low energies and long distances). That means that in the IR regime, Λ → 0, the theory becomes weakly coupled,andperturbationtheory becomes reliable in that regime. However, at short distances(orathighen- ergies), Λ → ∞,theoppositehappens:thedimensionlesscouplingbecomes large and perturbation theory breaks down. Four dimensions is special in that the approach and departure from the decoupled limit, u = 0, is very slow, and leads to logarithmic corrections to the free field values. However, for D < 4somethingnewhappens:thereisanon-trivial fixed ∗ ∗ point at u where β u = 0. At the fixed point,thecouplingconstantdoes

( ) 386 Perturbation Theory and Feynman Diagrams β u β u β u

( ) ( ) ( ) ∗ u u u u

u u u

(a) D < 4 (b) D = 4 (c) D > 4

Figure 11.20 The beta-function (top) and the IR RG flows (bottom) for D < 4, D = 4andD > 4. The UV RG flows are the reverse. not flow as the momentum scale changes. Hence, at a fixed point it is possible to send the momentum scale Λ → ∞ and effectively have a theory without a cutoff. Notice that even infinitesimally away from the fixed point the UV flows are unstable. Also, and for the same reason, at the fixed point it is also possible to go into the deep infrared regime and have a theory with ∗ afinitecouplingconstantu .Itturnsoutthatthisbehavioriscentralfor the theory of phase transitions. We will come back to the problem of the renormalization group in another chapter where we will develop it in detail, and discuss its application to different theories.

11.8 One-loop Integrals Here we will sketch the computation of one-loop integrals (inEuclidean space-time). We introduce a momentum cutoffΛand to suppress the con- tributions at large momenta, q ≫ Λofintegralsoftheform

q2 2 D − µ d q 1 2 ID = e Λ (11.87) Λ2 ! 2π D q2 + µ2 3 4 where we used a Gaussian cutoff(function) (or regulator). We will only be interested in the regime µ2 ≪ Λ2.UsingaFeynman-Schwingerparametriza- 11.8 One-loop Integrals 387 tion we can write 2 q 2 2 µ2 ∞ dDq − − α q + µ I = dα e Λ2 D 2 D Λ !0 ! 2π ( ) 3 4 1 2 ∞ 2 dDq − + α q = −α(µ ) Λ2 dαe D e !0 ! 2π + , D −1 2 2 2 ∞ µ µ − D −t = e Λ2 ( dt) t 2 e D 4π 2 !µ2 Λ2 ( ) (11.88) / / ( ) Hence D −1 2 2 2 2 µ 2 µ µ D µ 2 ID = Γ 1 − , e Λ (11.89) Λ2 4π D 2 2 Λ2 ( ) / 2 where Γ ν, z is the incomplete3 4 gamma function,3 with4 z = µ and ν = 1− D , ( ) Λ2 2 ∞ ν−1 −t ( ) Γ ν, z = dt t e (11.90) !z = and Γ ν, 0 Γ ν is the Gamma( ) function ∞ = ν−1 −t ( ) ( ) Γ ν dt t e (11.91) !0 2 If the regulator Λis removed( ) (i.e. if we take the limitΛ → ∞), ID µ formally becomes: D − 2 2 1 ( ) 2 µ D ID µ = Γ 1 − (11.92) 4π D 2 2 ( ) For general D,Γ 1−D 2 (is a) meromorphic/ function+ , of the complex variable D,andhassimplepolesforν = (0orany) negative integer, Γ 1 − D 2 has poles for D = 2, 4(, 6,.../ ) In D = 4dimensions,ν = −1whereΓν has a pole, the( incomplete/ ) Gamma function at ν = −1is(asz → 0)

1 1 ( )− − = − z + Γ 1,z z ln z e γ (11.93) where γ is the Euler-Mascheroni( ) constant+ , ∞ −t γ = − dt e ln t = 0.5772 ... (11.94) !0 388 Perturbation Theory and Feynman Diagrams 2 2 Hence, for µ ≪ Λ , I4 is

2 2 2 µ Λ µ Λ γ 2 I4 = − ln + µ (11.95) Λ2 16π2 8π2 µ 16π2

Here we see that the3 leading4 singularity is quadratic+ , in the regulator Λ, with asub-leadinglogarithmicpiece. 2 2 In two dimensions I2 has instead a logarithmic singularity for µ ≪ Λ

µ2 1 µ2 I2 = Γ 0, Λ2 4π Λ2 1 Λ γ 3 4 = ln3 −4 2π µ 4π (11.96) + , Asecondintegralofinterestis

µ2 dDq 1 JD = Λ2 ! 2π D q2 q2 + µ2 3 4 1 µ2 = ( ID) 0 −1ID 2 (11.97) µ2 Λ2

2 2 3 ( ) 3 44 JD µ Λ is UV finite if D < 4, where it is given by

D −2 µ2 2 D − ( / ) 2 Γ 2 2 JD µ = (11.98) 4π D 2 D − 1 ( ) (2 ) In four dimensions, J ,hasalogarithmicdivergence( ) / 4 ( ) µ2 1 Λ γ J4 = ln − (11.99) Λ2 8π2 µ 16π2

Athirdusefulintegralis3 4 + ,

q2 2 D − ′ µ d q 1 2 ∂ID ID = e Λ = − (11.100) Λ2 ! 2π D q2 + µ2 2 ∂µ2

In the massless( limit) it becomes ( ) 1 2 ′ 1 4 D−4 ID 0 = Λ (11.101) 4π D 2 D − 2 D − 4 ( ) / ( ) ( )( ) 11.8 One-loop Integrals 389 In four dimensions it becomes 2 ′ µ 1 Λ γ + 1 I = ln − (11.102) 4 Λ2 8π2 µ 16π2 ( ) + , + ,