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Lehmann–Symanzik–Zimmermann (LSZ) Reduction Formula

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Lehmann–Symanzik–Zimmermann (LSZ) Reduction Formula Lehmann–Symanzik–Zimmermann (LSZ) Reduction Formula Earlier in class (see my notes) I explained the two-point correlation functions and showed how their poles are related to the physical masses of particles and the strengths of bare fields. The LSZ reduction formula focuses on the n> 2 correlation functions def (x ,...,x ) = Ω TΦ(ˆ x ) Φ(ˆ x ) Ω (1) Fn 1 n h | 1 ··· n | i — where the quantum fields Φ(ˆ x) are in the Heisenberg picture of quantum mechanics — and relates the poles of their Fourier transforms (p ,...,p ) = d4x eip1x1 d4x eipnxn (x ,...,x ), (2) Fn 1 n 1 ··· n ×Fn 1 n Z Z to the S–matrix elements. The poles happen when any of the n momenta pi approaches the mass shell, p2 M 2 , and the most interesting pole is the simultaneous pole when all n i → phys momenta go on-shell. Specifically, let p0 +E(p ) = + p + M 2, ..., p0 +E(p ) (3) 1 → 1 1 k → k p for some k < n, but p0 E(p ), ..., p0 E(p ). (4) k+1 → − k+1 n → − n In this limit, Lehmann–Symanzik–Zimmermann formula (published in 1955) gives us n √ i Z ˆ n(p1,...,pn) 2 2 pk+1,..., pn S p1,...,pk . (5) F −−−−→on shell pi M + iǫ × h− − | | i Yi=1 − The field-strength factors √Z in this formula stem from the in eq. (1) being the correlation Fn of the bare fields. If we re-define it as the correlation function of the renormalized fields — or equivalently, use the counterterm perturbation theory to calculate the correlation function, — 1 then we would get renormalized = Z−n/2 bare (6) Fn Fn and consequently n renormalized i ˆ n (p1,...,pn) 2 2 pk+1,..., pn S p1,...,pk . (7) F −−−−→on shell pi M + iǫ × h− − | | i Yi=1 − Perturbation Theory for the Correlation Functions Before deriving the LSZ reduction formula, let me show how the poles at p2 M 2 arise i → from formal resummation of the perturbation theory. For simplicity, let’s focus on the connected correlation functions, which in perturbation theory obtain as all connected diagrams conn(p ,...,p ) = . (8) Fn 1 n with n external vertices X Topologically, a general diagram of this kind has an amputated core, plus any number of external leg bubbles on any of the n external legs, thus 2 1PI 1PI 1PI 1PI amputated core 1PI 1PI 1PI 1PI (9) Each external leg bubble here is one-particle irreducible (1PI) — if we cut any propagator internal to the bubble, it stays connected. As to the amputated core, if we cut any propagator internal to the core, it may stay connected or break into two disconnected parts, but if it breaks then each part must remain connected to at least two external legs. A general diagram (9) contributing to the connected n-point function may have: Any kind of amputated core with n external legs. • Any number N =0, 1, 2,... of leg bubbles in each of the n legs. • i And any such bubble may be any kind of 1PI subgraph with 2 external legs. • ⋆ Most importantly, we may chose any amputated core we like and any leg bubbles we like completely independently from each other. 3 Consequently, when we formally sum over all connected Feynman diagrams with n external vertices, the sum factorizes into a product of a sum over the cores and a sum over the bubbles in each leg, n connected amputated external conn(p ,...,p ) = = . (10) Fn 1 n diagrams cores × leg factors X X Yi=1 where each external leg factors includes the blue propagators and the leg bubbles, and should be summed over all numbers of bubbles of any kinds. In general, for Ni bubbles we have Ni +1 blue propagators with fixed momentum pi, thus ∞ i i external i N +1 single N = 2 2 leg factor i pi mb + iǫ × bubbles NXi=0 − X ∞ Ni+1 Ni i 2 (11) = 2 2 iΣ(pi ) pi mb + iǫ × − NXi=0 − h i i 2 = = 2(p ), p2 m2 Σ(p2)+ iǫ F i − b − exactly as in the two-point correlation function. Therefore, n amputated conn(p ,...,p ) = (p2) . (12) Fn 1 n F2 i × cores Yi=1 X This formula explains where the poles in the correlation functions come from: When any of the momenta p goes on-shell, p2 M 2 , the corresponding (p2) has a pole, i i → phys F2 i 2 iZ 2(p ) = + finite, (13) F p2 M 2 + iǫ i − which translates into the pole of the whole product (12). When all of the n momenta go on 4 shell at the same time, all n of the (p2) factors develop poles, thus F2 i n conn iZ amputated n (p1,...,pn) 2 2 ) . (14) F −→ pi M + iǫ × cores Yi=1 − X Note the combined residue of this n-fold pole, conn n amputated Residue [ (p1,...,pn)] 2 2 = (iZ) . (15) Fn all pi →M × cores X Thus far, we have explained the poles of the Lehmann–Symanzik–Zimmermann formula (5). In a moment, we should compare the residues. But first let’s cluster-expand the LHS of the LSZ formula into connected correlation functions while the S-matrix element on the RHS likewise expands into connected and disconnected pieces. For example, for n = 4 we have (p ,p ,p ,p ) = conn(p ,p ,p ,p ) F4 1 2 3 4 F4 1 2 3 4 + (p ,p ) (p ,p ) + (p ,p ) (p ,p ) (16) F2 1 2 ×F2 3 4 F2 1 3 ×F2 2 4 + (p ,p ) (p ,p ), F2 1 4 ×F2 2 3 and at the same time p , p Sˆ p ,p = (2π)4δ(4)(p + p + p + p ) p , p i p ,p h− 3 − 4| | 1 2i 1 2 3 4 × h− 3 − 4| M| 3 4i + p Sˆ p p Sˆ p + p Sˆ p p Sˆ p (17) h− 2| | 1i × h− 4| | 3i h− 3| |c1i×h− 4| | 2i + p Sˆ p p Sˆ p . h− 4| | 1i × h− 3| | 2i In terms of the LSZ formula for n = 4, we get 4 terms — 1 connected and 3 disconnected — on each side of the equation. The corresponding disconnected terms on the left-hand and the right hand sides match each other by the LSZ formula for n = 2, so the connected terms should also match each other, 4 conn i√Z 4 (4) 4 (p1,p2,p3,p4) 2 2 (2π) δ (pnet) p3, p4 i p1,p2 , F −−−−→on shell pi M + iǫ! × × h− − | M| i Yi=1 − c (18) 5 Likewise, for n> 4 n conn i√Z 4 (4) n (p1,...,p4) 2 2 (2π) δ (pnet) F −−−−→on shell pi M + iǫ! × (19) Yi=1 − p ,..., p i p ,...,p × h− k+1 − n| M| 1 ki c 0 0 0 0 for p1,...,pk > 0 while pk+1,...,pn < 0. Now let’s compare the residue of the combined pole here, conn √ n 4 (4) Residue [ n (p1,...,pn)]on shell = (i Z) (2π) δ (pnet) F × (20) p ,..., p i p ,...,p , × h− k+1 − n| M| 1 ki c to the residue (15) of the same connected correlation function which obtains from the Feynman rules. Matching the two expressions, we see that the LSZ formula implies amputated (2π)4δ(4)(p ) p ,..., p i p ,...,p = Zn/2 . (21) net × h− k+1 − n| M| 1 ki × cores X c And this is why we calculate the scattering amplitudes using only the amputated Feynman diagrams! The factor Zn/2 in eq. (21) follows from using the correlation functions for bare fields and hence the bare perturbation theory for the amputated diagrams. In the counterterm perturbation theory, this factor goes away and we are left with amputated cores (2π)4δ(4)(p ) i (on shell p ,...,p ) = . (22) net × M 1 n with n external lines X 6 Deriving the LSZ Reduction Formula Now that we know what is the LSZ reduction formula good for, let’s prove it. Let’s start by focusing on a single momentum, say p1, and look for the quantum origin of the poles when that momentum goes on shell, p0 E(p ). To simplify our notations, we keep the (x ,...,x ) 1 → ± 1 2 n coordinates of the n-point correlation function in the coordinate basis, only the x1 gets Fourier transformed to the momentum basis, thus (p ; x ,...,x ) = d4x e−ip1x1 (x , x ,...,x ). (23) Fn 1 2 n 1 ×Fn 1 2 n Z Let’s split the time integral here over t = x0 into 3 integration ranges: Range (i) from to 1 1 −∞ some very early but finite time T1; range (II) from T1 to some very late but finite time T2; and range (III) from T to + . Thus, 2 ∞ 3 0 0 (p ; x ,...,x ) = dt e−ix1p1 d3x e−ix1p1 (x , x ,...,x ). (24) Fn 1 2 n 1 × 1 ×Fn 1 2 n Xi=1range#Z i Z Note that in the coordinate space, the (x , x ,...,x ) is an analytic function of the xµ. Fn 1 2 n 1 Consequently, integrating the phase over a finite range#2 of time cannot possibly produce Fn × a pole — all such integrals are analytic and finite. Instead, the poles at p0 = E(p ) must 1 ± 1 come from integrating over the semi-infinite time ranges #1 and #3. So let’s take a closer look at these time ranges.
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