Computing of Charged Current DIS at three loops

Computing of Charged Current DIS at three loops Mikhail Rogal Mikhail.Rogal@desy.de DESY, Zeuthen, Germany ACAT 2007, Amsterdam, Netherlands, April 23-28, 2007 Computing of Charged CurrentDIS at three loops p.1

Content Introduction to the Deep Inelastic Scattering Charged current deep-inelastic scattering at three loops. S. Moch and M. R. arxiv:0704.1740v1 [hep-ph] Differences between CC coefficient functions, applications S. Moch, M. R. and A. Vogt. DESY 07-048 Computing of Charged CurrentDIS at three loops p.2

Content Introduction to the Deep Inelastic Scattering Related experiments Charged current deep-inelastic scattering at three loops. S. Moch and M. R. arxiv:0704.1740v1 [hep-ph] Differences between CC coefficient functions, applications S. Moch, M. R. and A. Vogt. DESY 07-048 Computing of Charged CurrentDIS at three loops p.2

Content Introduction to the Deep Inelastic Scattering Related experiments Perturbative QCD corrections at three loops in QCD Charged current deep-inelastic scattering at three loops. S. Moch and M. R. arxiv:0704.1740v1 [hep-ph] Differences between CC coefficient functions, applications S. Moch, M. R. and A. Vogt. DESY 07-048 Computing of Charged CurrentDIS at three loops p.2

Content Introduction to the Deep Inelastic Scattering Related experiments Perturbative QCD corrections at three loops in QCD Results and applications Charged current deep-inelastic scattering at three loops. S. Moch and M. R. arxiv:0704.1740v1 [hep-ph] Differences between CC coefficient functions, applications S. Moch, M. R. and A. Vogt. DESY 07-048 Computing of Charged CurrentDIS at three loops p.2

Content Introduction to the Deep Inelastic Scattering Related experiments Perturbative QCD corrections at three loops in QCD Results and applications Application of TFORM Charged current deep-inelastic scattering at three loops. S. Moch and M. R. arxiv:0704.1740v1 [hep-ph] Differences between CC coefficient functions, applications S. Moch, M. R. and A. Vogt. DESY 07-048 Computing of Charged CurrentDIS at three loops p.2

Introduction Deep-inelastic lepton-hadron scattering (e ± p, e ± n, νp, νp,... - collisions) Computing of Charged CurrentDIS at three loops p.3

Introduction Deep-inelastic lepton-hadron scattering (e ± p, e ± n, νp, νp,... - collisions) k lepton k lepton Gauge boson q P hadron X Computing of Charged CurrentDIS at three loops p.3

Introduction Deep-inelastic lepton-hadron scattering (e ± p, e ± n, νp, νp,... - collisions) k k lepton k lepton lepton k lepton Gauge boson hadron P q X = Gauge boson hadron P p q p Computing of Charged CurrentDIS at three loops p.3

Introduction Deep-inelastic lepton-hadron scattering (e ± p, e ± n, νp, νp,... - collisions) k k lepton k lepton lepton k lepton Gauge boson hadron P q X = Gauge boson hadron P p q p Gauge boson: γ, Z 0 - NC W ± - CC Kinematic variables momentum transfer Q 2 = q 2 > 0 Bjorken variable x = Q 2 /(2P q) Inelasticity y = (P q)/(p k) Computing of Charged CurrentDIS at three loops p.3

DIS experiments EW unification at HERA: neutral vs. charged current Computing of Charged CurrentDIS at three loops p.4

DIS experiments EW unification at HERA: neutral vs. charged current = Charged and neutral deep inelastic scattering cross sections become comparable when Q 2 reaches the electroweak scale Computing of Charged CurrentDIS at three loops p.4

Î ÕÙ Ö ¼ Î Î ÕÙ Ö ¼ Î Calculation Leading order diagrams at parton level Vector and axial-vector interaction aγ µ + bγ µ γ 5 Õ Õ Õ Õ Ô Õ Ô Õ Ô Ô Ô Ô Mellin moments with definite symmetry properties process dependent distinction even/odd N (from OPE) ÕÙ Ö ÕÙ Ö F i (N, Q 2 ) = F 3 (N, Q 2 ) = 1 0 1 0 ÕÙ Ö ÕÙ Ö dxx N 2 F i (x, Q 2 ), dxx N 1 F 3 (x, Q 2 ) i = 2, L Computing of Charged CurrentDIS at three loops p.5

Known NC (exchange via γ gauge boson) F2 ep CC (exchange via W ± gauge boson) F νp+ νp 2, F νp+ νp 3 even N for F 2, odd N for F 3 NLO Bardeen, Buras, Duke, Muta 78 N 2 LO Zijlstra, van Neerven 92 N 3 LO Moch, Vermaseren, Vogt 05/ 06 Unknown NC γ Z interference at N 3 LO still missing CC (exchange via W ± gauge boson) F νp νp 2, F νp νp 3 odd N for F 2, even N for F 3 order N 3 LO already known Moch, M. R. 07 best use: difference even-odd Moch, M. R. and Vogt. 07 Computing of Charged CurrentDIS at three loops p.6

In practice The calculation Big number of diagrams need of automatization e.g. DIS structure functions F νp± νp 2,L - 1076 diagrams, F νp± νp 3-1314 diagrams up to 3 loops Computing of Charged CurrentDIS at three loops p.7

In practice The calculation Big number of diagrams need of automatization e.g. DIS structure functions F νp± νp 2,L - 1076 diagrams, F νp± νp 3-1314 diagrams up to 3 loops latest version of FORM Vermaseren, FORM version 3.2 (Apr 16 2007) QGRAF generation of diagrams for DIS structure functions Nogueira 93 Computing of Charged CurrentDIS at three loops p.7

MINCER minces integrals Computing of Charged CurrentDIS at three loops p.8

Feynman diag s into MINCER Method of projection in pictures Identify scalar topologies Scalar diagram with external momenta P and Q = 3 n d D l n 1 (P l 1 ) 2 1 l 2 1...l2 8 N-th moment: coefficient of (2P Q) N = (2 P Q)N (Q 2 ) N+α C N Taylor expansion 1 (P l 1 ) 2 = i (2P l 1 ) i (l1 2 (2P l 1) N )i+1 (l1 2)N Feed scalar two-point functions in MINCER Computing of Charged CurrentDIS at three loops p.9

Mincer dp P µ [(P l j ) µ I(l 1,.., P,...)] = 0 - integration by part identities t Hooft, Veltman 72; Chetyrkin, Tkachov 81 Leibniz, Newton :-) Computing of Charged CurrentDIS at three loops p.10

Mincer dp P [(P l µ j ) µ I(l 1,.., P,...)] = 0 - integration by part identities t Hooft, Veltman 72; Chetyrkin, Tkachov 81 Leibniz, Newton :-) P 1 P P 2 α 1 α 0 α 2 Triangle rule Define I(α 0, β 1, β 2, α 1, α 2 ) = d D P 1 (P 2 ) α 0 ((P + P1 ) 2 ) β 1 (P 2 1 ) α 1 ((P + P2 ) 2 ) β 2 (P 2 2 ) α 2 and act the integrand with Recursion relation: P µ P µ = D + P µ P µ. Result I(α 0, β 1, β 2, α 1, α 2 ) (D 2α 0 β 1 β 2 ) = β 1 (I(α 0 1, β 1 + 1, β 2, α 1, α 2 ) I(α 0, β 1 + 1, β 2, α 1 1, α 2 )) β 2 (I(α 0 1, β 1, β 2 + 1, α 1, α 2 ) I(α 0, β 1, β 2 + 1, α 1, α 2 1)) Computing of Charged CurrentDIS at three loops p.10

Mincer dp P [(P l µ j ) µ I(l 1,.., P,...)] = 0 - integration by part identities t Hooft, Veltman 72; Chetyrkin, Tkachov 81 Leibniz, Newton :-) P 1 P P 2 α 1 α 0 α 2 Triangle rule Define I(α 0, β 1, β 2, α 1, α 2 ) = d D P 1 (P 2 ) α 0 ((P + P1 ) 2 ) β 1 (P 2 1 ) α 1 ((P + P2 ) 2 ) β 2 (P 2 2 ) α 2 and act the integrand with Recursion relation: P µ P µ = D + P µ P µ. Result I(α 0, β 1, β 2, α 1, α 2 ) (D 2α 0 β 1 β 2 ) = β 1 (I(α 0 1, β 1 + 1, β 2, α 1, α 2 ) I(α 0, β 1 + 1, β 2, α 1 1, α 2 )) β 2 (I(α 0 1, β 1, β 2 + 1, α 1, α 2 ) I(α 0, β 1, β 2 + 1, α 1, α 2 1)) In pictures = 1 ε Computing of Charged CurrentDIS at three loops p.10

Classification of loop integrals Classify according to topology of underlying two-point function top-level topology types ladder, benz, non-planar Computing of Charged CurrentDIS at three loops p.11

Classification of loop integrals Classify according to topology of underlying two-point function top-level topology types ladder, benz, non-planar Using IBP identities more complicated topologies are reduced to simpler topologies,,, Computing of Charged CurrentDIS at three loops p.11

Results arxiv:0704.1740v1 [hep-ph] C3,10 ns 1953379 = 1 + a s C F 138600 + a2 s C F n f 537659500957277 «597399446375524589 + a 2 s 15975002736000 C F 2 14760902528064000 + 7202 «5832602058122267 105 ζ 3 + a 2 sc A C F 29045459520000 99886 «1155 ζ 3 + a 3 2 51339756673194617191 sc F n f 996360920644320000 + 48220 «18711 ζ 3 + a 3 s C F 2 n f 125483817946055121351353 209235793335307200000 + a 3 3 sc F 744474223606695878525401307 59829376 3274425 ζ 3 + 24110 693 ζ 4 7088908678200207936000000 + 28630985464358 24960941775 ζ 3 + 151796299 8004150 ζ 4 53708 «99 ζ 5 + a 3 s C A C F n f 185221350045507487753 + 8071097 226445663782800000 39690 ζ 3 24110 «693 ζ 4 + a 3 2 19770078729338607732075449 sc A C F 619383700181 8369431733412288000000 5546875950 ζ 3 «151796299 5336100 ζ 4 37322 99 ζ 5 93798719639056648125143 + a 3 s C A 2 C F 36231306205248000000 + 151796299 16008300 ζ 4 + 195422 «231 ζ 5. 43202630363 20582100 ζ 3 «Computing of Charged CurrentDIS at three loops p.12

Checks Known Mellin moments for F νp+ νp 2,L (even) and F νp+ νp 3 (odd) recalculated Computing of Charged CurrentDIS at three loops p.13

Checks Known Mellin moments for F νp+ νp 2,L (even) and F νp+ νp 3 (odd) recalculated All calculations with gauge parameter ξ for gluon propagator (Up to 10 th MM) i gµν + (1 ξ)q µ q ν q 2 Computing of Charged CurrentDIS at three loops p.13

Checks Known Mellin moments for F νp+ νp 2,L (even) and F νp+ νp 3 (odd) recalculated All calculations with gauge parameter ξ for gluon propagator (Up to 10 th MM) i gµν + (1 ξ)q µ q ν q 2 Adler sum rule for DIS structure functions C2,1 ns = 1 1 0 dx( F νp 2 (x, Q 2 ) F2 νn (x, Q 2 ) ) = 2 x measures isospin of the nucleon in the quark-parton model neither perturbative nor non-perturbative corrections in QCD Computing of Charged CurrentDIS at three loops p.13

Applications Gottfried sum rule (charged lepton(l)-proton(p ) or neutron(n) DIS) 1 0 dx( F lp x 2 (x, Q 2 ) F2 ln (x, Q 2 ) ) Conjecture: difference of non-singlet coefficient functions for even and odd Mellin moments suppressed by [C F C A /2] 1/N c Broadhurst, Kataev, Maxwell 04 Computing of Charged CurrentDIS at three loops p.14

δc ns i,n = CνP+ νp i,n e.g. νp νp Ci,n with color coefficient [C F C A /2] δc2,3 ns = +a 2 s C F [C F C A /2] 4285 96 122ζ 3 + 671 9 ζ 2 + 128 «5 ζ 2 2 + a 3 sc F [C F C A /2] 2 1805677051 466560 7787113 1944 ζ 2 + 55336 9 + a 3 s C F 2 [C F C A /2] 5165481803 1399680 2648 9 ζ 5 + 10093427 ζ 3 1472 810 3 ζ 3 2 ζ 2 ζ 3 378838 45 + 40648 9 ζ 5 9321697 810 ζ 2 2 8992 63 ζ 2 3 «ζ 3 + 1456 3 ζ 3 2 + 8046059 1944 ζ 2 4984ζ 2 ζ 3 + 798328 135 ζ 2 2 56432 «315 ζ 2 3 20396669 + a 3 sn f C F [C F C A /2] 116640 1792 9 ζ 5 + 405586 405 ζ 3 139573 486 ζ 2 + 1408 9 ζ 2ζ 3 50392 «135 ζ 2 2. Computing of Charged CurrentDIS at three loops p.15

NuTeV experiment - Paschos-Wolfenstein relation Exact relation for massless quarks and isospin zero target Paschos,Wolfenstein 73, Llewelin Smith 83 R = σ(ν µn ν µ X) σ( ν µ N ν µ X) σ(ν µ N µ X) σ( ν µ N µ + X) = 1 2 sin2 θ W measurement of sin 2 θ W NuTeV 01 with large deviations from Standard model expectations QCD corrections to the Paschos-Wolfenstein relation second moments of valence PDFs q = dxx(q q) expansion in isoscalar combination u + d Davidson, Forte, Gambino, Rius, Strumia 01; Dobrescu, Ellis 03; Moch, McFarland 03 R = 1 2 sin2 θ W» + 1 7 3 sin2 θ W + 8α s 9π 1 + αs 1.689 + α 2 s(3.661 ± 0.002) 1 «2 sin2 θ W u d u + d s u + d + c «u + d Computing of Charged CurrentDIS at three loops p.16

NuTeV experiment - Paschos-Wolfenstein relation Exact relation for massless quarks and isospin zero target Paschos,Wolfenstein 73, Llewelin Smith 83 R = σ(ν µn ν µ X) σ( ν µ N ν µ X) σ(ν µ N µ X) σ( ν µ N µ + X) = 1 2 sin2 θ W measurement of sin 2 θ W NuTeV 01 with large deviations from Standard model expectations QCD corrections to the Paschos-Wolfenstein relation second moments of valence PDFs q = dxx(q q) expansion in isoscalar combination u + d Davidson, Forte, Gambino, Rius, Strumia 01; Dobrescu, Ellis 03; Moch, McFarland 03 R = 1 2 sin2 θ W» + 1 7 3 sin2 θ W + 8α s 1 + αs 1.689 + α 2 s(3.661 ± 0.002) 1 «9π 2 sin2 θ W u d u + d s u + d + c «u + d main uncertainties in s Martin, Roberts, Stirling, Thorne 04; Lai, Nadolsky, Pumplin, Stump, Tung, Yuan 07 QCD corrections under control Moch, M. R., Vogt 07 Computing of Charged CurrentDIS at three loops p.16

Application of TFORM TEST 8MM of 3-loop diagram of the ladder topology Computing of Charged CurrentDIS at three loops p.17

Application of TFORM TEST 8MM of 3-loop diagram of the ladder topology IT WORKS!!! Computing of Charged CurrentDIS at three loops p.17

time(n threads) / time(1 thread) Application of TFORM TEST 8MM of 3-loop diagram of the ladder topology IT WORKS!!! TFORM on Opteron machine with 2 processors 2.2 GHz time(normal FORM) time(tform 2 threads) = 1.63 On Xeon machine with 4 Dual Core processors 3.4 GHz 1 thread 598 seconds 1,0 0,8 0,6 0,4 0,2 Without options with threads 0 1 2 3 4 5 6 7 8 Number of threads Computing of Charged CurrentDIS at three loops p.17

Summary New results for fixed Mellin moments at order α 3 s F νp νp 2,L (odd) and F νp νp 3 (even) and differences even-odd 1/N c suppression of even-odd : conjecture of Broadhurst, Kataev, Maxwell 04 verified at three loops Stability of QCD α s expansion for Paschos-Wolfenstein relation approximations in x-space for even-odd differences available sufficient for HERA-CC, ν - DIS (e.g. Alekhin makes use of it) We master the whole chain of the calculation can be used for other applications Generation of diagrams with QGRAF (topologies) Generation of database of Feynman graphs with a FORM program (depend on the process considered) Automatic evaluation of diag s from this database using MINCER program Computing of Charged CurrentDIS at three loops p.18