…from collisions to the Higgs boson Fabrizio Caola Rudolf Peierls Centre for Theoretical Physics & Wadham College
how do quark and gluons or: interact and create a Higgs?
The subatomic world at high energies Q F T + = u h i a e e n o l t r d special relativity u y m quantum mechanics • Structure of fundamental interactions: very constrained • Only freedom ~ particle content and symmetries first principles calculations
Quantum chromodynamics To study any process at the LHC: we need to understand how quark and gluons interact Quantum field theory for strong interactions → quantum chromodynamics (QCD) A well-defined, well-established theory…
Quantum chromodynamics To study any process at the LHC: we need to understand how quark and gluons interact Quantum field theory for strong interactions → quantum chromodynamics (QCD) A well-defined, well-established theory… extremely hard to deal with
<latexit sha1_base64="MvxP8zcA+dM/hc7QtFsPDs34SAo=">AB/HicdVDLSgMxFM34rPU12qWbYBHqZsj0YXVXdOygn1AW4ZMmpmGJjNDkhHKUH/FjQtF3Poh7vwb04egogcuHM65l3v8RPOlEbow1pZXVvf2Mxt5bd3dvf27YPDtopTSWiLxDyWXR8rylEW5pTruJpFj4nHb8dXM79xRqVgc3epJQgcChxELGMHaSJ5d6GOejLCnSsJrnsJ+GELXs4vIqSH3DF1A5CBUrVQXpFyv1aBryAxFsETs9/7w5ikgkacKxUz0WJHmRYakY4neb7qaIJmMc0p6hERZUDbL58VN4YpQhDGJpKtJwrn6fyLBQaiJ80ymwHqnf3kz8y+ulOjgfZCxKUk0jslgUpBzqGM6SgEMmKdF8YgmkplbIRlhiYk2eVNCF+fwv9Ju+y4Fad8Uy02Lpdx5MAROAYl4I6aIBr0AQtQMAEPIAn8GzdW4/Wi/W6aF2xljMF8APW2ydLX5Pl</latexit> <latexit sha1_base64="vUcGLbfH3unEsBTWAT4+u/N0Xxk=">AB/3icdVDLSgMxFM34rPVFdy4CRahboZMH1Z3RTdVrAPaMuQSdM2NMkMSUYoYxf+ihsXirj1N9z5N6YPQUPXDg515y7wkizrRB6MNZWl5ZXVtPbaQ3t7Z3djN7+w0dxorQOgl5qFoB1pQzSeuGU5bkaJYBJw2g9HV1G/eUqVZKG/MOKJdgQeS9RnBxkp+5rCDeTEvs4Jv3oKO5oJiFzPz2SRW0LeGbqwT4SKheKc5MulEvQsmSILFqj5mfdOLySxoNIQjrVueygy3Qrwink3Qn1jTCZIQHtG2pxILqbjLbfwJPrNKD/VDZkgbO1O8TCRZaj0VgOwU2Q/3bm4p/e3Y9M+7CZNRbKgk84/6MYcmhNMwYI8pSgwfW4KJYnZXSIZYWJsZGkbwtel8H/SyLtewc1fF7OVy0UcKXAEjkEOeKAMKqAKaqAOCLgD+AJPDv3zqPz4rzOW5ecxcwB+AHn7RMAZTU</latexit> QCD for Higgs studies: weakly coupled At high energies, QCD becomes weakly coupled April 2016 α s ( m P ) � 1 (Q 2 ) τ decays (N 3 LO) α s DIS jets (NLO) Heavy Quarkonia (NLO) e + e – jets & shapes (res. NNLO) 0.3 e.w. precision fits (NNLO) ( – ) pp –> jets (NLO ) pp –> tt (NNLO) 0.2 m H ~ 125 GeV 0.1 α s ( m H ) ∼ 0 . 1 QCD α s (M z ) = 0.1181 ± 0.0011 1 10 100 1000 Q [GeV] Study interactions as perturbation around theory of free quarks/gluons σ = σ 0 + α s σ 1 + α s2 σ 2 +… → perturbative QFT: Feynman diagrams
Perturbation theory and Feynman diagrams I → F σ = σ 0 + α s σ 1 + α s2 σ 2 +… •For a given QFT → set of basic building blocks QCD, only gluons
Perturbation theory and Feynman diagrams I → F σ = σ 0 + α s σ 1 + α s2 σ 2 +… •For a given QFT → set of basic building blocks QCD, only gluons • σ 0 : connect I to F , in all possible ways, minimising closed loops + + + →
Perturbation theory and Feynman diagrams I → F σ = σ 0 + α s σ 1 + α s2 σ 2 +… •For a given QFT → set of basic building blocks QCD, only gluons QCD, only gluons • σ 0 : connect I to F , in all possible ways, minimising closed loops + + + → •Higher orders: dress with real and virtual quark and gluons +… one extra leg/loop per order +…
Perturbation theory and Feynman diagrams σ = σ 0 + α s σ 1 + α s2 σ 2 +… • Feynman rules : associate to each diagram an analytic formula ``T ree’’ diagrams: simple rational ``Loop’’: must integrate over momenta of functions of momenta / polarisations particles in the loop → non trivial transcendental functions •Very well-understood procedure since the `60s •Fully algorithmic
The punch-line: To compute any precise theoretical prediction for any LHC process → need to compute Feynman diagrams with many legs/loops How many? •We want to test Higgs interactions at the few percent… σ = σ 0 + α s σ 1 + α s2 σ 2 +…, α s ~ 0.1 Leading Order (LO) → very imprecise
The punch-line: To compute any precise theoretical prediction for any LHC process → need to compute Feynman diagrams with many legs/loops How many? •We want to test Higgs interactions at the few percent… σ = σ 0 + α s σ 1 + α s2 σ 2 +…, α s ~ 0.1 Leading Order (LO) → very imprecise Next-to-Leading Order (NLO) → ~ 10%
The punch-line: To compute any precise theoretical prediction for any LHC process → need to compute Feynman diagrams with many legs/loops How many? •We want to test Higgs interactions at the few percent… σ = σ 0 + α s σ 1 + α s2 σ 2 +…, α s ~ 0.1 Leading Order (LO) → very imprecise Next-to-Leading Order (NLO) → ~ 10% Need Next-to-next-to-leading order (NNLO) and beyond for precision …in principle: compute a bunch of diagrams with extra legs/loops
… in practice: back to our example Dress with one real gluon + 22 similar terms Rational function of momenta and polarisations
… in practice: back to our example Dress with one real gluon + 22 similar terms 98 pages analytic formula!
More legs… Explosion of Final state gluons 2 3 4 5 6 . . . n N − 2 4 25 220 2485 34300 # diagrams 4 45 510 5040 40320 . . . terms (1984) (1984) calculations 30y ago Prospect for theoretical
Loops? •Very difficult integrals… •… when you compute them you get infinity!
Loops? in principle, last century physics tells us what to do… … in practice, we don’t go very far •Very difficult integrals… •… when you compute them you get infinity!
T ypical expectation: hopeless My first interaction with a Nobel Prize winner… •Him, to friend A: what are you working on? •My friend: X and Y […] •Him: Fascinating […] Keep on the good work! Similar pattern with friend B, C… until it is my turn •Him: and you, what are you doing? •Me: I am trying to do precision physics at the LHC •Him, genuinely worried for me: my dear boy, no! You should change topic, the LHC is a messy environment, we are going to discover stuff but we cannot do precision physics there! if we kept relying on `70s understanding of QFT, he had a point…
… but in the meantime + 22 similar terms Massive simplification! = Same physical content
Massive simplifications in tree amplitudes Similar simplification persist at higher multiplicity Simplest helicity configuration: Sum of 220 Feynman diagrams! Sum of ~n! diagrams
Massive simplifications in tree amplitudes Similar simplification persist at higher multiplicity Simplest helicity configuration: What is going on? Sum of 220 Feynman diagrams! Sum of ~n! diagrams
QFT in the XXI century Structure of QFT extremely rigid → very much constrained by special relativity and quantum mechanics •Special relativity: everything is local → tree-level results can only have a well-defined set of simple singularities •Quantum mechanics: unitarity → what happens at singular points entirely determined by trees with lower multiplicity Completely hidden in Feynman diagrams! Trees have a natural recursive nature, that can be fully exploited to reconstruct the result for n+1 legs from the n -leg one.
QFT in the XXI century Structure of QFT extremely rigid → very much constrained by special relativity and quantum mechanics •Special relativity: everything is local → tree-level results can only Trees: problem solved have a well-defined set of simple singularities •Quantum mechanics: unitarity → what happens at singular points entirely determined by trees with lower multiplicity Completely hidden in Feynman diagrams! Trees have a natural recursive nature, that can be fully exploited to reconstruct the result for n+1 legs from the n -leg one. Any process .
… what about loops? es Similar ideas: • ``cut open a loop’’ → tree • use smart trees → simplify dramatically the function to integrate Integrals? • one loop → solved long time ago • higher loop → very complicated, but they seem to have nice geometrical structures… can we understand this? ? One-loop Multi-loop
… what about loops? es Similar ideas: • ``cut open a loop’’ → tree • use smart trees → simplify dramatically the function to integrate A lot of progress Integrals? • one loop → solved long time ago • higher loop → very complicated, but they seem to have nice geometrical structures… can we understand this? ? One-loop Multi-loop
Precision physics at the LHC: timeline W/Z total, H total, Harlander, Kilgore σ = σ 0 + α s σ 1 + α s2 σ 2 +… H total, Anastasiou, Melnikov H total, Ravindran, Smith, van Neerven [ thanks to Gavin for the graphics ] Second-order ( → few percent) WH total, Brein, Djouadi, Harlander LHC predictions H diff., Anastasiou, Melnikov, Petriello H diff., Anastasiou, Melnikov, Petriello W diff., Melnikov, Petriello W/Z diff., Melnikov, Petriello H diff., Catani, Grazzini W/Z diff., Catani et al. VBF total, Bolzoni, Maltoni, Moch, Zaro WH diff., Ferrera, Grazzini, Tramontano γ - γ , Catani et al. ZH diff., Ferrera, Grazzini, Tramontano 2002 2004 2006 2008 2010 2012 2014 2016 2018
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