Feynman Diagrams, Momentum Space Feynman Rules, Disconnected Diagrams, Higher Correlation Functions

PHY646 - Quantum Field Theory and the Standard Model

Even Term 2020 Dr. Anosh Joseph, IISER Mohali

LECTURE 05

Tuesday, January 14, 2020

Topics: Feynman Diagrams, Momentum Space Feynman Rules, Disconnected Diagrams, Higher Correlation Functions.

Feynman Diagrams

We can use the Wick’s theorem to express the n-point function

h0|T {φ(x1)φ(x2) ··· φ(xn)}|0i as a sum of products of Feynman propagators. We will be interested in developing a diagrammatic interpretation of such expressions. Let us look at the expression containing four ﬁelds. We have

h0|T {φ1φ2φ3φ4} |0i = DF (x1 − x2)DF (x3 − x4)

+DF (x1 − x3)DF (x2 − x4)

+DF (x1 − x4)DF (x2 − x3). (1)

We can interpret Eq. (1) as the sum of three diagrams, if we represent each of the points, x1 through x4 by a dot, and each factor DF (x − y) by a line joining x to y. These diagrams are called Feynman diagrams. In Fig. 1 we provide the diagrammatic interpretation of Eq. (1). The interpretation of the diagrams is the following: particles are created at two spacetime points, each propagates to one of the other points, and then they are annihilated. This process can happen in three diﬀerent ways, and they correspond to the three ways to connect the points in pairs, as shown in the three diagrams in Fig. 1. Thus the total amplitude for the process is the sum of these three diagrams. PHY646 - Quantum Field Theory and the Standard Model Even Term 2020

⟨0|T {ϕ1ϕ2ϕ3ϕ4}|0⟩ = DF(x1 − x2)DF(x3 − x4) + DF(x1 − x3)DF(x2 − x4)

+DF(x1 − x4)DF(x2 − x3)

1 2 1 2 1 2 + + 3 4 3 4 3 4

Figure 1: Diagrammatic interpretation of Eq. (1). We have a sum of three Feynman diagrams for the four-point function.

The ﬁrst term is nothing but the free-ﬁeld result

h0|T {φ(x)φ(y)})0i = DF (x − y). (4)

The second term, in φ4, theory is

Z Z λ h0|T φ(x)φ(y)(−i) dt d3x φ4 |0i 4! −iλ Z = h0|T φ(x)φ(y) d4xφ(z)φ(z)φ(z)φ(z) |0i. (5) 4!

Applying Wick’s theorem we get

Z Z λ h0|T φ(x)φ(y)(−i) dt d3z φ4 |0i 4! −iλ Z = 3 · D (x − y) d4z D (z − z)D (z − z) 4! F F F −iλ Z + 12 · d4z D (x − z)D (y − z)D (z − z). (6) 4! F F F

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z x y + x z y

Figure 2: Feynman diagrams contained in Eq. (6).

x y x y

S = 2 S = 2 ⋅ 2 ⋅ 2 = 8 S = 3! = 6

Figure 3: Symmetry factors of Feynman diagrams.

The above expression is equal to the two diagrams given in Fig. 2. It is conventional to associate the expression R d4z(−iλ) with each vertex. The numbers 3 and 12 in Eq. (6) arise as combinatoric factors. We refer to the lines as propagators since they represent the propagation amplitude DF . The place where the four lines is called a vertex. The diagrams interpret the analytical formula as a process of particle creation, propagation, and annihilation which takes place in spacetime. We note that a diagram can look the same under the interchange of internal propagators and vertices. The total number of such interchanges is called the symmetry factor S. We need to divide by the symmetry factor as part of the Feynman rules. In Fig. 3 we provide some examples. Now we are in a position to summarize our rules for calculating the numerator of the expression for hΩ|T φ(x)φ(y)|Ωi. We have

Z h0|T φ(x)φ(y) exp −i dtHI (t) |0i X = (all possible diagrams with two external points) , (7)

3 / 6 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 where each diagram is built out of propagators, vertices, and external points. The rules for associating analytic expressions with pieces of diagrams are called Feynman rules. For the case of the φ4 theory, the Feynman rules are the following:

1. For each propagator: DF (x − y)

2. For each vertex: (−iλ) R d4z

3. For each external point: 1

4. Divide by symmetry factor S.

One way to interpret these rules is to think of the vertex factor (−iλ) as the amplitude for the emission and /or absorption of particles at a vertex. The integral R d4z instructs us to sum over all points where this process can occur. This is just the superposition principle of quantum mechanics. That is, when a process can happen in alternative ways, we add the amplitudes for each possible way. To compute the individual amplitudes, the Feynman rules tell us to multiply the amplitudes (propagators and vertex factors) for each independent part of the process.

Momentum Space Feynman Rules

The rules given above are written in terms of the spacetime points x, y, etc. They are sometimes called the position-space Feynman rules. In most calculations it is simpler to express the Feynman rules in terms of momenta, by introducing the Fourier expansion of each propagator:

Z d4p i D (x − y) = e−ip(x−y). (8) F (2π)4 p2 − m2 + i

We can represent this in the diagram by assigning a 4-momentum p to each propagator, indicating the direction with an arrow. Note that since DF (x − y) = DF (y − x), the direction of p is arbitrary. The vertex factor, takes the form Z 4 −ip1z −ip2z −ip3z ip4z 4 (4) d ze e e e = (2π) δ (p1 + p2 + p3 − p4), (9) where the delta-function ensures that the momentum is conserved at each vertex. The momentum space Feynman rules are the following:

i 1. For each propagator: p2−m2+i 2. For each vertex −iλ

3. For each external point: e−ipx

4. Impose momentum conservation at each vertex

R d4p 5. Integrate over each undetermined momentum: (2π)4 6. Divide by the symmetry factor.

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Disconnected Diagrams

A disconnected diagram is a diagram that is not connected to an external point. Such a diagram will have momenta running inside the loops contributing to a factor ! 4 (4) X 4 (4) (2π) δ pi = (2π) δ (0). (10) i

In position space this is just the integral over a constant over the spacetime volume d4w: Z d4w(const) ∝ (2T ) · volume of space. (11)

Thus we have (2π)4δ(4)(0) = 2T · V, (12) where V is the volume of space and T appears as the range of time [−T,T ]. Every disconnected piece of such a diagram will have one such contributing factor. Such diagrams are also called vacuum bubbles or vacuum to vacuum transition diagrams. Space- time processes given by vacuum bubbles can happen at any place in space, and at any time between −T and T . We can show that the sum of all diagrams given by

Z T lim h0|T φ(x)φ(y) exp −i dtHI (t) |0i T →∞(1−i) −T is equal to the sum of all connected diagrams, times the exponential of the sum of all disconnected diagrams. The expression Z T h0|T exp −i dtHI (t) |0i −T is equal to the exponential of the sum of all disconnected diagrams. Thus we have the expression hΩ|T [φ(x)φ(y)]|Ωi representing the sum of all connected diagrams with two external points. We can also show that " # X exp Vi ∝ exp [−iE0(2T )] , (13) i where Vi represents various possible disconnected pieces. The elements of Vi are connected internally, but disconnected from external points. We can ﬁnd an expression for the energy density of the vacuum, relative to the zero energy set

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Figure 4: Disconnected diagrams that are not part of vacuum bubbles.

by H0|0i = 0. We have X E0 i V = × 2T · V, (14) i V i where V is the volume of space. Then the energy density is

E i P V 0 = i i (15) V (2π)4δ(4)(0)

Higher Correlation Functions

The generalization to higher correlation functions is straightforward. We have

hΩ|T [φ(x1)φ(x2) ··· φ(xn)] |Ωi representing the sum of all connected diagrams with n external points. In higher correlation functions, the diagrams can also be disconnected in another sense. There can be diagrams in which the external points are disconnected from each other. Such diagrams contribute to the amplitude just as do the fully connected diagrams. (Fully connected means that any point can be reached from any other by traveling along the lines.) See Fig. 4 for examples of disconnected diagrams that are not part of vacuum bubbles.

References

[1] M. E. Peskin and D. Schroeder, Introduction to Quantum Field Theory, Westview Press (1995).

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