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3. Quantum Field Theory (QFT) — Renormalisation How do we measure particles • We can measure tracks of individual charged particles with – pixel detectors – wire chambers ⇒ we measure the 3-momentum p ~ • We can measure the energy of particles in – calorimeters ⇒ we measure the energy E • We can distinguish different kinds of particles by – sets of Cherenkov counters – comparing calorimeter- and track-data ⇒ reconstructing the 4-momentum: E 2 = p ~2 + m2 Thomas Gajdosik – Concepts of Modern Theoretical Physics 13.09.2012 1 3. Quantum Field Theory (QFT) — Renormalisation How do we measure intermediate particles • masses are measured for instance with ”invariant-mass distributions ” • example: e+ e− → Z → µ+µ− – the 4-momenta of the muons are reconstructed – these 4-momenta are summed up and squared: M 2 – events with 2 muons are distributed according to M 2: ∗ this gives a Breit-Wigner distribution (on top of a background) – measured mass = position of the peak in the Breit-Wigner curve • couplings are measured with the strength of the interaction • example: H → µ+µ− versus H → bb̄ – identifying the decaying particle in all events #events(H→µ+ µ− ) – relative coupling strength = #events(H→bb̄) Thomas Gajdosik – Concepts of Modern Theoretical Physics 13.09.2012 2 3. Quantum Field Theory (QFT) — Renormalisation How do we calculate • we generate all Feynman diagrams {f } for the process • we calculate the amplitude for each Feynman diagram Mf P • we add up all amplitudes M = f Mf • we square the modulus |M|2 • we integrate over the Lorentz invariant phase space dLips µ µ 2 . . . p |M| dLips(p dσ = symmetry n) 1 f lux µ µ – momentum-conservation P µ = Pin = i pi – and each particle i in the final state contributes, so X Y d3p ~i µ µ 4 4 dLips(p1 . . . pn ) = (2π) δ (P − pi) 3 2E (2π) p ~i i i P • in perturbation theory, we expand in orders of the coupling constant g: M= X k g M (k) where M (k) = k Thomas Gajdosik – Concepts of Modern Theoretical Physics X (k) Mf f 13.09.2012 3 3. Quantum Field Theory (QFT) — Renormalisation How do we calculate • When perturbation theory is applicable: |g nM(n)| > |g n+1M(n+1)| – the leading order (LO) is the smallest n with |M(n)| > 0 – the next-to leading order (NLO) is then n + 1 or n + 2. • the cross section is also a sum over powers of the coupling constant g: P where dσ (2n) = |g nM(n)|2dLips and dσ = k=2n g k dσ (k) h i dσ (2n+1) = (g nM(n))†(g n+1M(n+1)) + (g nM(n))(g n+1M(n+1))† dLips • we identified the ”physical” coupling g from the measurement of σ – LO: – NLO: R LO σ = dσ (2n) R NLO (2n) σ = (dσ + dσ (2n+1)) • if σ LO 6= σ NLO . . . what is now the right σ ? Thomas Gajdosik – Concepts of Modern Theoretical Physics ⇒ 13.09.2012 Renormalisation 4 3. Quantum Field Theory (QFT) — Renormalisation Example of Renormalisation: ABC-theory • we have 3 real scalar fields A, B, and C and 1 coupling g: 1 Φ (∂ 2 + m2 )Φ + g A B C L0 = − 2 0 0 0 0 0 0 0,Φ where Φ = A, B, C −1 – masses are the poles in the propagators i × [p2 − m2 0,Φ + iǫ] – the coupling connects all three different fields B C . • having the physical masses as mA > mB + mC , .A – A will decay into B and C: – at LO there is only one diagram, which gives g 1M(1) = g 3 3 d p ~B d p ~C dΩ κ – dLips = (2π)4δ 4(pA − pB − pC ) (2π) = 3 2E 3 (4π)2 ABC p ~B (2π) 2Ep ~C 1 σ LO(A → BC) = 2mA Z g 2 κABC κABC = |g| 2 (4π) 4π 2mA 2 dΩ g2 • g or α = 4π can be determined from the decay! Thomas Gajdosik – Concepts of Modern Theoretical Physics 13.09.2012 5 3. Quantum Field Theory (QFT) — Renormalisation Example of Renormalisation: ABC-theory we calculate now σ NLO(A → BC) B . C B . C B .C • there is no diagram of order g 2 • there are 4 diagrams of order g 3 . A . A . A – 3 of them have a ”bubble” on an external line ⇒ we will have to renormalise also the propagators! ∗ as it turns out, they will not contribute (on-shell scheme) B – the 4th diagram is just a triangle ∗ which is usually not zero ∗ but depends on incoming and outgoing momenta . .C A – so g 3M(3) = g 3tABC h i dΩ 1 2 3 † † 3 κABC |g| + g × (g tABC ) + g × (g tABC ) 2mA (4π)2 Z g 2 κABC 2 1 + 2g TABC with TABC = dΩ = 4π Re[tABC ] 4π 2mA NLO σ(A→BC) = Z Thomas Gajdosik – Concepts of Modern Theoretical Physics 13.09.2012 6 3. Quantum Field Theory (QFT) — Renormalisation Example of Renormalisation: ABC-theory • but we calculated with g0, not with the measured g ! • the same happens to the other parameters of the theory • writing g0 = g + δg , 2 2 m2 0,Φ = mΦ + δmΦ , and Φ0 = q ZΦ Φ = Φ + 1 2 δZΦ Φ we get a new Lagrangian q 2 2 2 1 L0 = − 2 ZΦΦ(∂ + mΦ + δmΦ)Φ + (g + δg) ZAZB ZC ABC = L + δL with 1 Φ(∂ 2 + m2 )Φ + gABC L = −2 Φ 1 Φ[δZ (∂ 2 + m2 ) + δm2 ]Φ + [ g (δZ + δZ + δZ ) + δg] ABC δL = − 2 Φ A B C Φ Φ 2 where only terms linear in the expansion of the parameters are kept. • We assume only one-loop accuracy for now. Thomas Gajdosik – Concepts of Modern Theoretical Physics 13.09.2012 7 3. Quantum Field Theory (QFT) — Renormalisation Example of Renormalisation: ABC-theory • Introducing abbreviations for the loops pi • The self energies iΠ(p2) = . pf g2 2 = i (4π)2 B0(p2; m2 0, m1 ) . • Using the Feynman rules we get to one loop iΠ(p2i )(2π)4δ 4(pi − pf ) Z d4 k1 d4k2 (ig)(2π)4δ 4 (pi − k1 − k2)i (ig)(2π)4δ 4(k1 + k2 − pf )i = (2π)4 (2π)4 k12 − m21 k22 − m20 Z 2 1 g 2 2 −1 4 2 2 −1 ] [(q + p ) − m d q [q − m = i(2π)4δ 4 (pi − pf ) i 1] 0 (4π)2 iπ 2 2 g 2 2 2 =: i(2π)4δ 4 (pi − pf ) (4π) 2 B0 (pi ; m0 , m1 ) pB 2 2 • In a similar way the vertex correction iT (p2 A , pB , pC ) is written with the function = g3 ). , m2 , m2 ; m2 , p2 , p2 C0(p2 2 C B A B A C (4π) Thomas Gajdosik – Concepts of Modern Theoretical Physics 13.09.2012 p. C pA 8 3. Quantum Field Theory (QFT) — Renormalisation Example of Renormalisation: ABC-theory • Now we have 7 counterterms that we should detemine: δm2 Φ , δZΦ , δg • they also give rise to new diagrams: – six counterterms are used by iδZΦ (p2 − m2Φ ) − iδm2Φ p .p = . pB p. C – the 7th counterterm by iCg = ig2 (δZA + δZB + δZC ) + iδg = . pA • We have to choose Renormalisation conditions for them. We can take (without proof): 1. the pole of the full propagator should be at m2 Φ ⇒ 2 δm2 Φ = Π(mΦ ) 2. the residuum of the full propagator should be 1 ⇒ d Π(x)| ′ (m2 ) = −Π δZΦ = − dx 2 Φ x=m Φ 3. the full decay width of A should be like in the tree level. ⇒ 2 , m2 ) − g (δZ + δZ + δZ ) , m δg = −T (m2 A B C C B A 2 Thomas Gajdosik – Concepts of Modern Theoretical Physics 13.09.2012 9 3. Quantum Field Theory (QFT) — Renormalisation Example of Renormalisation: ABC-theory • Calculating the NLO matrix element for the decay of A → B + C, we have to add 9 diagrams: B B C . . + b A C . . C + B . . C + c A B d . C . + A B C . B . C . B C . + h A A . + g A e A . + f . B C . + a B . i A . A 2 • adding c + g and Taylor expanding around p2 A = mA gives 2 2 2 2 2 ′ 2 2 2 ) − δm2 Π(mA) + Π (mA)(pA − mA) + O (pA − mA) δZA(p2 − m A A A − g −g 2 2 p2 p2 A − mA A − mA 2 2 = O (pA − mA) Thomas Gajdosik – Concepts of Modern Theoretical Physics 13.09.2012 10 3. Quantum Field Theory (QFT) — Renormalisation Example of Renormalisation: ABC-theory • a gives now g 1M(1) = g 2 2 • c + g , d + h , and e + i are all O (pA − mA) • so g 3M(3) = limp2 →m2 ( b + · · · + i ) = limp2 →m2 ( b + f ) Φ Φ Φ Φ g 2 2 ) + , m , m = T (m2 C B A 2 (δZA + δZB + δZC ) + δg = 0 • So now σ LO = σ NLO for this decay ! • g can be determined unambiguously. • but where do come new predictions? – A → 3B + C or A → B + 3C or . . . – scattering B + C → B + C: dσ(BC → BC) Thomas Gajdosik – Concepts of Modern Theoretical Physics 13.09.2012 11 3. Quantum Field Theory (QFT) — Renormalisation s Example of Renormalisation: ABC-theory, dσ(BC → BC) • with the same argument as before: no corrections on external legs • contributing diagrams to the s-channel, initial momenta k, final momenta p: C B. C B. + . B C a C B. + . B C b C B. + . B C c C B. + . B C d C B. + . B C e C B. + . B C f C B. + . B C g . B C h 2 = p2 = m2 , k2 = p2 = m2 , and (k + k )2 = (p + p )2 = s. • kB B C B C C C C B B • a = (ig)2 g2 i 2 (2) so g M =− s−m2 s−m2 A A • .b + c + d + e = 2 , k2 ) i3gT (s, p2B , p2C ) i3gT (s, kB i3gCg i3gCg C + + + 2 2 2 s − mA s − mA s − mA s − m2A T (s, m2B , m2C ) − T (m2A , m2B , m2C ) 2 0 = −2ig = O (s − mA ) s − m2A Thomas Gajdosik – Concepts of Modern Theoretical Physics 13.09.2012 12 3. Quantum Field Theory (QFT) — Renormalisation s Example of Renormalisation: ABC-theory, dσ(BC → BC) • .f + g = (ig)2i2iδZA (s − m2A ) − δm2A (ig)2i2iΠ(s) + (s − m2A )2 (s − m2A )2 = ig 2 Π(s) − δm2A + δZA (s − m2A ) 2 0 2 ) − m = O (p A A (s − m2A)2 2 , k 2 ; (−p − p )2 , (−p + k )2 ; m2 , m2 , m2 , m2 ) • h = g 4Dbox (p2B , p2C , kC B C C C B A C A B 4 (4) • .g M • so g 2[Π(s) − Π(m2A )] 2g[T (s) − T (m2A )] + g 2Π′ (m2A ) g2 − + Dbox =: ∆M = (s − m2A )2 s − m2A s − m2A dσ LO s (BC →BC) • and dσ NLOs (BC →BC) 1 = 2κsBC 2 2 −g 2 2 dΩ κsBC g dΩ = s − m2 (4π)2 2s 4π 4s(s − m2 )2 A A −g 2 g4 dΩ κsBC 1 4 (4) + · 2Re[g M ] = 2κsBC (s − m2A )2 s − m2A (4π)2 2s 2 2 dΩ g LO − 2Re[∆M]) = 6 dσ = (1 s (BC →BC) 4π 4s(s − m2A )2 Thomas Gajdosik – Concepts of Modern Theoretical Physics 13.09.2012 13 3. Quantum Field Theory (QFT) — Renormalisation Regularisation • is the prescription, how to deal with undefined integrals • examples: – only integrating up to p2 < Λ – subtracting the integrand with a regulator mass (Pauli-Villars) – making the dimension continuous (dimensional regularisation) • the physical result cannot depend on the regularisation • the counterterms can ”absorb” the regularisation constants ⇒ renormalisable theory Regularisation in ABC • only the selfenergies had ill defined integrals • δZΦ and δm2 Φ contain regularisation constants • but they drop out in all physical processes Thomas Gajdosik – Concepts of Modern Theoretical Physics 13.09.2012 14