PHY646 - Quantum Field Theory and the Standard Model

Even Term 2020 Dr. Anosh Joseph, IISER Mohali

LECTURE 34

Wednesday, March 18, 2020 (Note: This is an online lecture due to COVID-19 interruption.)

Topic: LSZ Reduction Formula and the S-Matrix.

LSZ Reduction Formula and the S-Matrix

The Lehmann-Symanzik-Zimmermann (LSZ) reduction formula is a general relation between cor- relation functions and the S-matrix elements. In the previous lecture we showed that the poles of the two-point correlation functions are related to the physical (renormalized) masses of particles and strengths of bare fields (wave function renormalization constants). The LSZ reduction formula for n > 2 correlation functions

Gn(x1, x2, ··· , xn) ≡ hΩ|T φ(x1)φ(x2) ··· φ(xn)|Ωi, relates the poles of their Fourier transforms Z Z 4 ip1x1 4 ipnxn Gn(p1, p2, ··· , pn) = d x1e ··· d xne Gn(x1, x2, ··· , xn), (1) to the S-matrix elements. 2 2 We encounter a pole when any of the n momenta pi approaches the mass shell, pi = Mphys. The most interesting pole, the simultaneous pole, occurs when all n momenta go on-shell. For simplicity, let us consider the case of scalar fields. Let us consider the following kinematics

0 0 pi → +E(pi), i = 1, 2, ··· n and kj → +E(kj), j = 1, 2, ··· m. (2) PHY646 - Quantum Field Theory and the Standard Model Even Term 2020

Then the LSZ reduction formula is

n Z m Z Y 4 ipi·xi Y 4 −ikj ·yj d xie d yje hΩ|T {φ(x1) ··· φ(xn)φ(y1) ··· φ(ym)}|Ωi i=1 j=1   n p ! m p Y i Zφ Y i Zφ ∼   hp1 ··· pn|S|k1 ··· kmi. (3) p2 − m2 + i k2 − m2 + i 0 i=1 i j=1 j each pi → +Epi 0 each kj → +Ekj

The LSZ formula tells us that we can compute the S-matrix elements as follows. Compute the appropriate Fourier-transformed correlation function, look at the region of momentum space where the external particles are near the mass shell, and identify the coefficient of the multiparticle pole. Let us express this procedure in the Language of Feynman diagrams. To be more precise, let us consider the case of 2-particle → 2-particle scattering. We will find the diagrammatic expansion of the scalar field four-point function and the S-matrix element. We will consider explicitly the fully connected Feynman diagrams contributing to the correlator. The exact four-point function

2 Z 2 Z Y 4 ipi·xi Y 4 −ikj ·yj d xie d yje hΩ|T {φ(x1)φ(x2)φ(y1)φ(y2)}|Ωi (4) i=1 j=1 has the general form shown in Fig. 1. In this figure, we have indicated explicitly the diagrammatic corrections on each leg; the shaded circle in the center represents the sum of all amputated four-point diagrams.

p1 p2

Amputated

k1 k2

Figure 1: Structure of the exact four-point function in scalar field theory.

If we expand each resummed propagator about the physical particle pole, we see that each

2 / 4 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 external leg of the four-point amplitude contributes

i iZ ∼ φ + (regular). (5) 2 2 2 0 2 2 p − m0 − M p →Ep p − m

Thus, the sum of diagrams contains a product of four poles

iZφ iZφ iZφ iZφ 2 2 · 2 2 · 2 2 · 2 2 . (6) p1 − m p2 − m k1 − m k2 − m

This is exactly the singularity given in the second line of Eq. (3). Comparing the coefficients of this product of poles, we find the relation ! p 4 amputated core with hp1 p2|S|k1 k2i = Zφ × . (7) 4 external lines

On the right hand side we have the amputated core with four external lines, see Fig. 2, multiplying the Z-factors.

p1 p2

Amputated

k1 k2

Figure 2: Amputated core with four external lines.

The LSZ formula distinguishes in and out particles only by the sign of the Fourier transform 0 0 2 momentum pi or ki . This means that, by analytically continuing the residue on the pole in p from positive to negative p0, we can convert the S-matrix element with φ(p) in the final state into the S-matrix element with the antiparticle φ∗(−p) in the initial state. This is exactly the statement of crossing symmetry:

h· · · φ(p)|S| · · · i = h· · · |S|φ∗(k) · · · i. (8) p=−k We will not prove the LSZ reduction formula in this lecture. (It is derived in P&S Chapter 7) The proof of the LSZ formula does not depend on perturbation theory, and thus we can infer that the crossing symmetry of the S-matrix is a general result of quantum theory, not merely a property of Feynman diagrams.

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References

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

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