PHY646 - Quantum Field Theory and the Standard Model

Even Term 2020Dr. Anosh Joseph, IISER Mohali

LECTURE 09

Monday, January 20, 2020

Topics: S-Matrix Elements from Feynman Diagrams.

S-Matrix Elements from Feynman Diagrams

We have found expressions for cross sections and decay rates in terms of the invariant matrix elementM. The remaining task is to find a way of computingM for various processes in various interactingfield theories.

Let us write down a formula forM in terms of Feynman diagrams.The S-matrix is simply the time-evolution operator,

exp(−iHt),

in the limit of very large t:

〈p1p2 · · · |S|kAkB〉 = limT→∞

〈p1p2 · · · |e−iH(2T )|kAkB〉. (1)

To compute this quantity we need to replace the external plane-wave states in Eq. (1), whichare eigenstates of H, with their counterparts in the unperturbed theory, which are eigenstates ofH0.

We successfully made such a replacement for the vacuum state |Ω〉 before

|Ω〉 = limT→∞(1−iε)

(e−iE0T 〈Ω|0〉

)−1e−iHT |0〉. (2)

Let us find a relation of the form

|kAkB〉 ∝ limT→∞(1−iε)

e−iHT |kAkB〉0, (3)

PHY646 - Quantum Field Theory and the Standard Model Even Term 2020

where we have omitted some unknown phases and overlap factors. (Note the factor e−iHT insteadof e−iH(2T ).)

Let us keep in mind that interactions affect not only the scattering of distinct particles but alsothe form of the single-particle states themselves.

If the above formula Eq. (3) could be justified, then we could use it to rewrite the right-handside of Eq. (1) as

limT→∞(1−iε)

0〈p1 · · ·pn|e−iH(2T )|pApB〉0

∝ limT→∞(1−iε)

0〈p1 · · ·pn|T(

exp

[−i∫ T

−TdtHI(t)

])|pApB〉0. (4)

Here also, it can be shown that the awkward proportionality factors between free and interactingvacuum states cancel out of the final formula.

The formula for the non-trivial part of the S-matrix can be simplified to the following form

〈p1 · · ·pn|iT |pApB〉

= limT→∞(1−iε)

(0〈p1 · · ·pn|T

(exp

[−i∫ T

−TdtHI(t)

])|pApB〉0

)CA

, (5)

where CA refers to “connected and amputated". They are there to indicate restrictions on the classof possible Feynman diagrams.

We will learn more about it when we look at cluster decomposition and the LSZ reduction later.Let us expand the exponential as a power series.

The Lowest Order Term

The first term is

0〈p1p2|pApB〉0 =√

2E12E22EA2EB〈0|a1a2a†Aa†B|0〉

= 2EA2EB(2π)6(δ(3)(pA − p1)δ

(3)(pB − p2)

+ δ(3)(pA − p2)δ(3)(pB − p1)

). (6)

The delta functions force the final state to be identical to the initial state. Thus, this term ispart of the ‘1’ in S = 1 + iT . The instruction “connected" tells us that we should eliminate it fromthe scattering amplitude. Thus this term does not contribute to the scattering matrix elementM.We can represent it diagrammatically as shown in Fig. 1.

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PHY646 - Quantum Field Theory and the Standard Model Even Term 2020

1 2

𝒜 ℬ 𝒜 ℬ

1 2

+

Figure 1: Diagrammatic interpretation of Eq. (6).

Linear in λ Term

The next term in 〈p1p2|S|pApB〉 is

0〈p1p2|T(−i λ

4!

∫d4z φ4(z)

)|pApB〉0

= 0〈p1p2|N(−i λ

4!

∫d4z φ4(z) + contractions

)|pApB〉0, (7)

using Wick’s theorem.We note that the terms that are not fully contracted do not necessarily vanish, since the external

states are not |0〉.We can use an annihilation operator from φ(z) to annihilate an initial-state particle, or a creation

operator from φ(z) to produce a final-state particle.For example,

φ+(z)|p〉0 =

∫d3k

(2π)31√2Ek

ake−ikz√2Epa

†p|0〉

=

∫d3k

(2π)31√2Ek

e−ikz√

2Ep(2π)3δ(3)(k− p)|0〉

= e−ipz |0〉. (8)

Let us define contractions of field operators with external states

φ(z) |p〉 = e−ipz |0〉, 〈p|φ(z) = 〈0|e+ipz. (9)

To compute the S-matrix element we simply need to write down all possible full contractions ofthe φ operators and the external-state momenta.

The normal ordered product contains terms of the form

φφφφ; φφφφ; φφφφ

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PHY646 - Quantum Field Theory and the Standard Model Even Term 2020

The last term, in which the φ operators are fully contracted with each other,

−i λ4!

∫d4z 0〈p1p2|φφφφ|pApB〉0 (10)

is equal to a vacuum bubble diagram times the sum of the two disconnected diagrams computed inthe previous section. This is shown in Fig. 2. This is just another contribution to the trivial partof the S-matrix. Thus let us ignore it.

1 2

𝒜 ℬ 𝒜 ℬ

1 2

+×

Figure 2: Diagrammatic interpretation of Eq. (10).

Next we look at the term in which two of the four φ operators are contracted. Here, one of theφ’s must be contracted with an initial-state |p〉, the other with a final-state 〈p|. The un-contracted|p〉 and 〈p| will give delta function. To represent these quantities diagrammatically, we introduceexternal lines to our Feynman rules. See Fig. 3.

pz p z

Figure 3: Diagrammatic interpretation of (Left) φ(z) |p〉 = e−ipz |0〉 and (Right) 〈p|φ(z) = 〈0|e+ipz.

Feynman diagrams for S-matrix elements will always contain external lines. Rather than theexternal points of diagrams for correlation functions.

The second type of contraction thus gives the four diagrams given in Fig. (4). The integration∫d4z produces a momentum-conserving delta function at each vertex, including external momenta.

So these diagrams again describe trivial processes in which the initial and final states are identical.The general principle is: Only fully connected diagrams, in which all external lines are connected

to each other, contribute to the T -matrix.Now let us consider the term in which none of the φ operators are contracted with each other.

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PHY646 - Quantum Field Theory and the Standard Model Even Term 2020

+ + +

Figure 4: Four disconnected diagrams of leading order in λ, containing loops.

We can contract two of the φ’s with |pApB〉 and the other two with 〈p1p2|. We can do this in 4!

ways.The expression is

(4!) ·(−i λ

4!

)∫dz e−i(pA+pB−p1−p2)x = −iλ(2π)4δ(4)(pA + pB − p1 − p2). (11)

The diagram is shown in Fig. 5

1 2

ℬ𝒜

Figure 5: A fully connected diagram, leading order in λ.

The above expression is exactly of the form

iM(2π)4δ(4)(pA + pB − p1 − p2), (12)

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PHY646 - Quantum Field Theory and the Standard Model Even Term 2020

withM = −λ. (13)

This would be a suitable time to pause and turn this expression into a cross section.For scattering in the center-of-mass frame, we get

|M|2 = λ2 (14)

into the equation for the cross section to obtain(dσ

dΩ

)CM

=λ2

64π2E2cm

. (15)

This is our first calculation on quantum field theory cross section.Integrating over dΩ, and dividing by 2 since there are two identical particles int he final state,

we find the total corss section

σtotal =λ2

32πE2cm

. (16)

In practice we can use this result to measure the value of λ.

References

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

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