Gavin Salam, CERN PSI Summer School Exothiggs, Zuoz, August 2016

INGREDIENTS FOR ACCURATE COLLIDER PHYSICS (2/2) Gavin Salam, CERN PSI Summer School Exothiggs, Zuoz, August 2016 1 8 1. Quantum chromodynamics The PDFs’ resulting dependence on µF is described by the Dokshitzer-Gribov-Lipatov- Altarelli-Parisi (DGLAP) equations [43], which to leading order (LO) read∗ 2 2 1 ∂fi/p x, µ αs µ dz x µ2 F = F P (1) (z) f ,µ2 , (1.14) F 2 2π z i←j j/p z F ∂!µF " ! " x #j $ % & (1) 2 2 with, for example, Pq←g(z)=TR(z +(1 z) ). The other LO splitting functions are listed − 2 3 in Sec. 16 of this Review,whileresultsuptoNLO,αs,andNNLO,αs,aregiveninRefs. 44 and 45 respectively. Beyond LO, the coefficient functions are also µF dependent, for µ2 example C(1)(x, Q2,µ2 ,µ2 )=C(1)(x, Q2,µ2 ,Q2) ln F 1 dz C(0)( x )P (1) (z). 2,i R F 2,i R − Q2 j x z 2,j z j←i As with the renormalization scale, the choice of factorizat! " 'ion( scale is arbitrary, but if one has an infinite number of terms in the perturbative series, the µF -dependences of the coefficient functions and PDFs will compensate each other fully. Given only N terms of the series, a residual (αN+1)uncertaintyisassociatedwiththeambiguityin O s the choice of µF .AswithµR,varyingµF provides an input in estimating uncertainties on predictions. In inclusive DIS predictions, the default choice for the scales is usually µR = µF = Q. As is the case for the running coupling, in DGLAP evolution one can introduce flavor thresholds near the heavy quark masses: below a given heavy quark’s mass, that quark is not considered to be part of the proton’s structure, while above it is considered to be part of the proton’s structure and evolves with massless DGLAP splitting kernels. With appropriate parton distribution matching terms at threshold, such a variable flavor number scheme (VFNS), when used with massless coefficient functions, gives the full heavy-quark contributions at high Q2 scales. For scales near the threshold, it is instead necessary to appropriately adapt the standard massive coefficient functions to account for theTUESDAY’S heavy-quark LECTURE contribution already included in the PDFs [46,47,48]. Hadron-hadron collisions. The extension to processes with two initial-state hadrons ➤ can beWe illustrated discussed with the the example“Master” of formula the total (inclusive) cross section for W boson production in collisions of hadrons h1 and h2,whichcanbewrittenas ∞ σ (h h W + X)= αn µ2 dx dx f x ,µ2 f x ,µ2 1 2 → s R 1 2 i/h1 1 F j/h2 2 F n#=0 % & #i,j $ % & % & 2 (n) 2 2 Λ σîj→W +X x1x2s, µR,µF + 4 , (1.15) × O )M * % & W ∗ ➤LO is generally taken to mean the lowest order at which a quantity is non-zero. This definitionand is its nearly main always inputs unambiguous, the one major exception being for the case of the etc. hadronic➤ branchingthe strong ratio coupling of virtual α photons, Z, τ, ,forwhichtwoconventionsexist: LO can either mean the lowest orders that contributes to the hadronic branching fraction, i.e. the➤ termParton “1” in Distribution Eq. (1.7); or Functions it can mean (PDFs) the lowest order at which the hadronic branching ratio becomes sensitive to the coupling, n =1inEq.(1.8), as is relevant when extracting➤ Today: the value we discuss of the coupling the actual from ascattering measurement cross of th sectionebranchingratio.Because of this ambiguity, we avoid use of the term “LO” in that context. May 5, 2016 21:57 2 8 1. Quantum chromodynamics The PDFs’ resulting dependence on µF is described by the Dokshitzer-Gribov-Lipatov- Altarelli-Parisi (DGLAP) equations [43], which to leading order (LO) read∗ 2 2 1 ∂fi/p x, µ αs µ dz x µ2 F = F P (1) (z) f ,µ2 , (1.14) F 2 2π z i←j j/p z F ∂!µF " ! " x #j $ % & (1) 2 2 with, for example, Pq←g(z)=TR(z +(1 z) ). The other LO splitting functions are listed − 2 3 in Sec. 16 of this Review,whileresultsuptoNLO,αs,andNNLO,αs,aregiveninRefs. 44 and 45 respectively. Beyond LO, the coefficient functions are also µF dependent, for µ2 example C(1)(x, Q2,µ2 ,µ2 )=C(1)(x, Q2,µ2 ,Q2) ln F 1 dz C(0)( x )P (1) (z). 2,i R F 2,i R − Q2 j x z 2,j z j←i As with the renormalization scale, the choice of factorizat! " 'ion( scale is arbitrary, but if one has an infinite number of terms in the perturbative series, the µF -dependences of the coefficient functions and PDFs will compensate each other fully. Given only N terms of the series, a residual (αN+1)uncertaintyisassociatedwiththeambiguityin O s the choice of µF .AswithµR,varyingµF provides an input in estimating uncertainties on predictions. In inclusive DIS predictions, the default choice for the scales is usually µR = µF = Q. As is the case for the running coupling, in DGLAP evolution one can introduce flavor thresholds near the heavy quark masses: below a given heavy quark’s mass, that quark is not considered to be part of the proton’s structure, while above it is considered to be part of the proton’s structure and evolves with massless DGLAP splitting kernels. With appropriate parton distribution matching terms at threshold, such a variable flavor number scheme (VFNS), when used with massless coefficient functions, gives the full heavy-quark contributions at high Q2 scales. For scales near the threshold, it is instead necessary to appropriately adapt the standard massive coefficient functions to account for theTUESDAY’S heavy-quark LECTURE contribution already included in the PDFs [46,47,48]. Hadron-hadron collisions. The extension to processes with two initial-state hadrons ➤ can beWe illustrated discussed with the the example“Master” of formula the total (inclusive) cross section for W boson production in collisions of hadrons h1 and h2,whichcanbewrittenas ∞ σ (h h W + X)= αn µ2 dx dx f x ,µ2 f x ,µ2 1 2 → s R 1 2 i/h1 1 F j/h2 2 F n#=0 % & #i,j $ % & % & 2 (n) 2 2 Λ σîj→W +X x1x2s, µR,µF + 4 , (1.15) × O )M * % & W ∗ ➤LO is generally taken to mean the lowest order at which a quantity is non-zero. This definitionand is its nearly main always inputs unambiguous, the one major exception being for the case of the etc. hadronic➤ branchingthe strong ratio coupling of virtual α photons, Z, τ, ,forwhichtwoconventionsexist: LO can either mean the lowest orders that contributes to the hadronic branching fraction, i.e. the➤ termParton “1” in Distribution Eq. (1.7); or Functions it can mean (PDFs) the lowest order at which the hadronic branching ratio becomes sensitive to the coupling, n =1inEq.(1.8), as is relevant when extracting➤ Today: the value we discuss of the coupling the actual from ascattering measurement cross of th sectionebranchingratio.Because of this ambiguity, we avoid use of the term “LO” in that context. May 5, 2016 21:57 2 8 1. Quantum chromodynamics The PDFs’ resulting dependence on µF is described by the Dokshitzer-Gribov-Lipatov- Altarelli-Parisi (DGLAP) equations [43], which to leading order (LO) read∗ 2 2 1 ∂fi/p x, µ αs µ dz x µ2 F = F P (1) (z) f ,µ2 , (1.14) F 2 2π z i←j j/p z F ∂!µF " ! " x #j $ % & (1) 2 2 with, for example, Pq←g(z)=TR(z +(1 z) ). The other LO splitting functions are listed − 2 3 in Sec. 16 of this Review,whileresultsuptoNLO,αs,andNNLO,αs,aregiveninRefs. 44 and 45 respectively. Beyond LO, the coefficient functions are also µF dependent, for µ2 example C(1)(x, Q2,µ2 ,µ2 )=C(1)(x, Q2,µ2 ,Q2) ln F 1 dz C(0)( x )P (1) (z). 2,i R F 2,i R − Q2 j x z 2,j z j←i As with the renormalization scale, the choice of factorizat! " 'ion( scale is arbitrary, but if one has an infinite number of terms in the perturbative series, the µF -dependences of the coefficient functions and PDFs will compensate each other fully. Given only N terms of the series, a residual (αN+1)uncertaintyisassociatedwiththeambiguityin O s the choice of µF .AswithµR,varyingµF provides an input in estimating uncertainties on predictions. In inclusive DIS predictions, the default choice for the scales is usually µR = µF = Q. As is the case for the running coupling, in DGLAP evolution one can introduce flavor thresholds near the heavy quark masses: below a given heavy quark’s mass, that quark is not considered to be part of the proton’s structure, while above it is considered to be part of the proton’s structure and evolves with massless DGLAP splitting kernels. With appropriate parton distribution matching terms at threshold, such a variable flavor number scheme (VFNS), when used with massless coefficient functions, gives the full heavy-quark contributions at high Q2 scales. For scales near the threshold, it is instead necessary to appropriately adapt the standard massive coefficient functions to account for theTUESDAY’S heavy-quark LECTURE contribution already included in the PDFs [46,47,48]. Hadron-hadron collisions. The extension to processes with two initial-state hadrons ➤ can beWe illustrated discussed with the the example“Master” of formula the total (inclusive) cross section for W boson production in collisions of hadrons h1 and h2,whichcanbewrittenas ∞ σ (h h W + X)= αn µ2 dx dx f x ,µ2 f x ,µ2 1 2 → s R 1 2 i/h1 1 F j/h2 2 F n#=0 % & #i,j $ % & % & 2 (n) 2 2 Λ σîj→W +X x1x2s, µR,µF + 4 , (1.15) × O )M * % & W ∗ ➤LO is generally taken to mean the lowest order at which a quantity is non-zero.

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