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Field Theory for the Standard Model

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Field Theory for the Standard Model Field theory for the Standard Model R. Kleiss Radboud University, Nijmegen, The Netherlands 1 0+0 field theory and Feynman diagrams Among the ingredients of quantum field theory most directly relevant to particle phenomenology are the Feynman diagrams. These are most easily introduced in a context that is as simple as possible. To this end, it is useful to reduce the size of the universe to a single point, i.e., we start with quantum field theory in 0+0 dimensions. 1.1 Perturbation theory The path integral describing quantum field theory for a zero-dimensional real (scalar) field ' is simply an integral involving a probability density. The measure is usually given by exp( S(')), where − µ2 λ S(') = '2 + V (') ; V (') = n 'n ; lim S(') = ; (1) 2 n! ' 1 n 3 !1 X≥ where ' denotes the single field value, µ is real and nonzero, and the real coupling constants λn may be zero or nonzero. In practice, often λ = 0 for n 5. In particle phenomenology one is usually n ≥ interested in the Green's functions, that is, the expectation values of powers of ': 1 1 1 − n S(') n S(') G = ' = e− ' d' e− d' : (2) n h i 0 1 0 1 Z Z @−1 A @−1 A The set of all Green's functions is combined in the path integral: J n 1 Z(J) = G = exp( S(') + J') d' G = Z(n)(0)=Z(0) : (3) n! n − ! n n 0 Z X≥ −1 1 Even more significant are the connected Green's functions Cn: J n Z(J) C W (J) = log : (4) n! n ≡ Z(0) n 1 X≥ The path integral satisfies a Schwinger–Dyson (SD) equation: n 1 @ @ λn @ − S0 Z(J) = µ + Z(J) = JZ(J) ; (5) @J 0 @J (n 1)! (@J)n 1 1 n 3 − X≥ − @ A where Z(J) is the solution that is real for real J. The common approach to computing Green's functions is perturbation theory, in which the coupling constants in V (') are treated as small2: 1 µ2'2=2 1 n Z(J) e− (J' V (')) d' ; (6) ! n! − n 0 Z X≥ −1 1 The relation between Gn and Cn is that between moments and cumulants of a probability distribution: they encode the same information in different ways. 2In this expression, integral and summation have been cavalierly interchanged. As a result, the perturbation series is not convergent, and has at most a formal meaning. However, it seems to work pretty well in practice. 1 R. KLEISS and the various powers of ' in the expansion integrate to 1 1 µ2'2=2 2n (2n) 1 2π µ2'2=2 2n+1 e− ' d' = ; e− ' d' = 0 : (7) n!2n µ2n µ2 Z r Z −1 −1 1.2 Feynman diagrams and Dyson summation A practical way of computing Green's functions is to use Feynman diagrams, which consist of vertices and lines (`propagators') connecting them. They correspond to algebraic quantities by way of Feynman rules, which in this case read 1 ; J ; λ : (8) ! µ2 ! ! − n n The propagators may or may not end in a vertex: propagator endings that do not are called external lines. Diagrams may be connected or disconnected: a disconnected diagram is considered as the product of its connected parts (and therefore the `empty' diagram equals unity). Each Feynman diagram carries an additional symmetry factor, which is the product of a factor 1=p! for each set of p equivalent internal lines, and a factor 1=q! for each set of q equivalent vertices in the diagram: `equivalent' means that these objects can be interchanged without affecting the topology of the diagram. This also implies a symmetry factor 1=r! for each set of r identical diagrams in a product. Here are two examples of diagrams, with their symmetry factors: λ λ λ λ J 2 = 3 4 ; = 3 4 : (9) (2!)µ10 (2!)(2!)µ10 Let us now consider the (infinite) set φ(J) of all connected diagrams with precisely one external leg and any number of J vertices in addition to other allowed vertices. This is called the amplitude-generating function (AGF), and is denoted by a shaded blob. The set of all connected diagrams with more external legs is given by derivatives: @ @2 φ(J) = ; = φ(J) ; = φ(J) : : : (10) @J (@J)2 Let us specialize to a theory in which λ4 is the only nonzero coupling. By inspection, it is clear that φ(J) satisfies a diagrammatic recursion relation: = + + + : (11) By starting with = and iterating, we can thus build up the whole set of connected diagrams. Fol- lowing the Feynman rules (including the symmetry factors for equivalent lines and blobs) this translates into the SD equation for the AGF: J λ λ @ λ @2 φ(J) = 4 φ(J)3 4 φ(J) φ(J) 4 φ(J) : (12) µ2 − (3!)µ2 − (2!)µ2 @J − (3!)µ2 (@J)2 If we now write @ 1 @ φ(J) = W (J) = Z(J) ; (13) @J Z(J) @J 2 FIELD THEORY FOR THE STANDARD MODEL precisely the SD equation (5) for Z(J) is recovered. Since J 2 J Z(J) = exp φ(J) = ; (14) λ4=0 2µ2 ! λ4=0 µ2 we have thus proven that 1. Cn equals the set of all connected diagrams with precisely n external lines and no J vertices; 2. W (J) equals the set of all connected Feynman diagrams with at least one J vertex and no external lines; 3. Z(J)=Z(0) equals the set of all (connected and disconnected) Feynman diagrams without external lines and at least one J vertex. For other theories the treatment is exactly the same, with the same results. The special roˆle of the diagrams without any external lines or J vertices, the so-called vacuum bubbles, stems from the fact that they make up precisely Z(0) and are therefore cancelled by the normalization of the path integral. It may seem illogical that the two-point vertex (µ2) is treated differently from the interaction ones 2 (λn). To show that the treatment is actually consistent, consider a theory in which, in addition to the µ 2 term we add another term λ2' =2. The SD equation then becomes J λ 1 φ(J) = 2 φ(J) (interaction terms) : (15) µ2 − µ2 − µ2 It is easy to see that this can be rewritten as J 1 φ(J) = 2 2 (interaction terms) ; (16) µ + λ2 − −µ + λ2 so that the addition of the quadratic interaction term is simply a modification of the µ2 in the Feynman rules. This is Dyson summation, where any two-point interaction may be subsumed in the definition of the propagator: + + + 1 1 1 ·1· · 1 1 = + ( λ ) + + ( λ ) ( λ ) + µ2 µ2 − 2 µ2 µ2 − 2 µ2 − 2 µ2 · · · 1 1 1 = 2 2 = 2 : (17) µ 1 + λ2=µ µ + λ2 1.3 The loop expansion In actually performing the (diagrammatic) perturbation expansion one has to decide how to truncate the perturbation series. When several different coupling constants are involved, this is to some extent ambiguous. It is therefore customary to order the diagrams topologically, according to the number of closed loops contained. Diagrams without closed loops are called tree diagrams. We can conveniently implement this by inserting, in the SD equation (12), a factor ~ for each closed loop involving the explicit vertex, so that (12) becomes J λ ~λ @ ~2λ @2 φ(J) = 4 φ(J)3 4 φ(J) φ(J) 4 φ(J) : (18) µ2 − (3!)µ2 − (2!)µ2 @J − (3!)µ2 (@J)2 This so-called loop expansion requires a modification of what we have seen so far: we need to write @ 1 1 φ(J) = ~ log Z(J) ; Z(J) = exp (S(') J') d' : (19) @J −~ − Z −1 3 R. KLEISS The SD equation then reads @ S0 ~ Z(J) = JZ(J) : (20) @J Accordingly, the Feynman rules now read ~ J λ ; ; n : (21) ! µ2 ! ~ ! − ~ n A diagram containing E external lines, I internal lines, Vn vertices of n-point type, and consisting of P connected parts, obeys the following two topological sum rules: V = I L + P ; nV = 2I + E : (22) n − n n n X X E P +L The ~-dependence of a connected diagram (P = 1) is therefore given as ~ − so that indeed each additional loop in a given Cn brings in another factor ~. The loop expansion also resolves the truncation 1+n=2 ambiguity by making each λn equivalent to ~− . 1.4 Classical limit, instantons, and effective action The limit ~ 0 is of interest by itself. On the one hand, the diagrams involved in the SD equation (12) ! for φc(J) = lim~ 0 φ(J) are those without any closed loops, so that it reads ! S0 (φc(J)) = J : (23) On the other hand, we may take ~ 0 in the path integral. Then, the dominant contribution comes from ! that value 'c where the integrand has a maximum: S0(' ) J = 0 : (24) c − It is seen that φc(J) = 'c as expected. In more realistic theories, the classical equation (24) is for instance the Klein–Gordon, Dirac or Maxwell equation. The only theory with a single minimum point is the free theory, in which λ = 0, n 3. Nonfree n ≥ theories contain several solutions to the classical field equation, of which often a single one corresponds to the global minimum of the action. Let us denote this value by '0. Another, local minimum value ' will give to the physics of the theory a contribution that is suppressed by a factor exp( (S(' ) 1 − 1 − S('0))=~): such contributions do not admit of a series expansion in powers of ~ and are therefore nonperturbative.
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