The Partonic Cross Section at N 3 LO Introduction Methods Conclusions One-Loop Single-Real-Emission Contributions to pp → H + X at Next-to-Next-to-Next-to-Leading Order William B. Kilgore Brookhaven National Laboratory LoopFest XIII Brooklyn, NY June 20, 2014 based on WBK, Phys.Rev. D89 (2014) 073008. William Kilgore (BNL) One-Loop Single-Real April 30, 2014 1 / 26
The Partonic Cross Section at N 3 LO Introduction Methods Conclusions The Discovery of the Millennium ATLAS and CMS have discovered a scalar boson with mass ∼ 126 GeV! Is this the Higgs Boson TM ? William Kilgore (BNL) One-Loop Single-Real April 30, 2014 2 / 26
The Partonic Cross Section at N 3 LO Introduction Methods Conclusions It looks like a Higgs It’s still early in the LHC program and the measurements are not very precise, but the gross features look like the Higgs. William Kilgore (BNL) One-Loop Single-Real April 30, 2014 3 / 26
The Partonic Cross Section at N 3 LO Introduction Methods Conclusions How can we tell? How can we tell if this is the Higgs? Measure the cross section. Measure the couplings. Look for more Higgs bosons. Look for some other new physics. The cross section measurement is limited by a large theoretical uncertainty. The couplings measurements will require a lot of data and will never by “precision” measurements. The high energy runs at LHC will have great reach in the search for new physics, but the most “natural” models have already been excluded. William Kilgore (BNL) One-Loop Single-Real April 30, 2014 4 / 26
The Partonic Cross Section at N 3 LO Introduction Methods Conclusions Higgs production at the LHC The inclusive Higgs production cross section is dominated by the gluon fusion channel. 2 10 H+X) [pb] LHC HIGGS XS WG 2012 s = 8 TeV pp → H (NNLO+NNLL QCD + NLO EW) 10 → (pp pp → qqH (NNLO QCD + NLO EW) σ 1 pp pp → → WH (NNLO QCD + NLO EW) ZH (NNLO QCD +NLO EW) pp → ttH (NLO QCD) -1 10 -2 10 80 100 200 300 400 1000 M [GeV] H Cuts can bias the event selection toward the vector boson fusion channel, but there are also forward jets in gluon fusion so one can never get a pure sample. William Kilgore (BNL) One-Loop Single-Real April 30, 2014 5 / 26
The Partonic Cross Section at N 3 LO Introduction Methods Conclusions Theoretical Uncertainty Even at NNLO, the theoretical uncertainty is very large. The dominant contributions are scale uncertainty of the partonic cross sections and parton distribution uncertainty. William Kilgore (BNL) One-Loop Single-Real April 30, 2014 6 / 26
The Partonic Cross Section at N 3 LO Introduction Methods Conclusions Improving the theoretical uncertainty How can we improve the theoretical uncertainty? Reduce the uncertainty in the parton distributions. Collect more data to improve the fits. Compute the N 3 LO splitting kernels to improve parton evolution. Reduce the scale uncertainty in the partonic cross sections. Compute the partonic cross sections at N 3 LO. In this talk, I will be concerned with improving the theoretical prediction for the total cross section. William Kilgore (BNL) One-Loop Single-Real April 30, 2014 7 / 26
The Partonic Cross Section at N 3 LO Introduction Methods Conclusions Improving the partonic cross sections Improving the calculation of the partonic cross sections, however, is a real challenge. The partonic cross sections are already known at next-to-next-to-leading order. No hadronic scattering process is known to higher order and only a handful of processes are known at NNLO. The difference is that Higgs production through gluon fusion is anomalously slow to converge and has unusually large scale dependence. William Kilgore (BNL) One-Loop Single-Real April 30, 2014 8 / 26
The Partonic Cross Section at N 3 LO Introduction Methods Conclusions “Normal” Perturbative Convergence The theoretical predictions for both Drell-Yan and tt production show dramatic improvement when computed to NNLO. William Kilgore (BNL) One-Loop Single-Real April 30, 2014 9 / 26
The Partonic Cross Section at N 3 LO Introduction Methods Conclusions Higgs Perturbative Convergence Higgs production through gluon fusion converges much more slowly and has large scale dependence. The only way to improve this situation is to push to higher order. William Kilgore (BNL) One-Loop Single-Real April 30, 2014 10 / 26
The Partonic Cross Section at N 3 LO Introduction Methods Conclusions Contributions to Higgs Production at N 3 LO Virtual production through 3 loops, Single-real-emission through 2 loops, Double-real-emission through 1 loop and Triple-real-emission at tree-level. William Kilgore (BNL) One-Loop Single-Real April 30, 2014 11 / 26
The Partonic Cross Section at N 3 LO Introduction Methods Conclusions One-Loop Single-Real-Emissions Contributions I will discuss only one set of contributions to the N 3 LO cross-section: those arising from one-loop single-real-emission Although drawn as a cut loop integral, I will compute this contribution as a squared amplitude integrated over phase space. One-loop amplitudes can be computed in closed form. The phase-space element for single-real-emission is very simple. This gives strong analytic control of the functions involved and allows one to attack the problem from multiple directions. William Kilgore (BNL) One-Loop Single-Real April 30, 2014 12 / 26
The Partonic Cross Section at N 3 LO Introduction Methods Conclusions Single-real-emission Amplitudes The single-real-emission amplitudes ( H ggg , H gqq ) can be written, at any loop order, in terms of linearly independent gauge invariant tensors. 3 M ( H ; g 1 , g 2 , g 3 ) = g v C 1 ( α s ) f ijk ε i 1 µ ε j 2 ν ε k A n Y µνρ ∑ , 3 ρ n n = 0 M ( H ; g , q , q ) = ig B 1 X µ 1 + B 2 X µ v C 1 ( α s )( T g ) ¯ ı � � j ε µ ( p g ) , 2 I calculate the amplitudes as follows: I generate the Feynman diagrams with QGRAF and use FORM to implement the Feynman rules and contract the diagrams with projectors onto the gauge invariant tensors. I use REDUZE2 to reduce Feynman integrals to master integrals by means of Integration-by-Parts [Chetyrkin, Tkachov] identities coupled with the LaPorta algorithm. William Kilgore (BNL) One-Loop Single-Real April 30, 2014 13 / 26
The Partonic Cross Section at N 3 LO Introduction Methods Conclusions One-loop Master Integrals There are only two master Feynman integrals at one loop, the one-loop bubble and the one-loop box with a single massive external leg. � µ 2 � ε ic Γ I ( 1 ) ( Q 2 ) = 2 − Q 2 ε ( 1 − 2 ε ) �� µ 2 � ε � � H ) = 2 ic Γ 1 1 , − ε ; 1 − ε ; − s 31 I ( 1 ) ( s 12 , s 23 ; M 2 2 F 1 4 ε 2 − s 12 s 12 s 23 s 23 � µ 2 � µ 2 � ε � ε �� 1 , − ε ; 1 − ε ; − M 2 � � � 1 , − ε ; 1 − ε ; − s 31 H s 31 + 2 F 1 − 2 F 1 − s 23 − M 2 s 12 s 12 s 23 H William Kilgore (BNL) One-Loop Single-Real April 30, 2014 14 / 26
The Partonic Cross Section at N 3 LO Introduction Methods Conclusions Computing the Partonic Cross Sections With the preliminaries out of the way, I will now describe the computation of the partonic cross section. This proceeds as follows: Squaring the amplitude and integrating over phase space Use Integration-by-Parts to map the full set of phase space integrals onto a small set of master phase space integrals Evaluate the master integrals. The evaluation of the master integrals is greatly simplified by taking advantage of the nice mathematical properties of the Hypergeometric functions. William Kilgore (BNL) One-Loop Single-Real April 30, 2014 15 / 26
The Partonic Cross Section at N 3 LO Introduction Methods Conclusions Phase Space Integration The partonic cross section is computed by squaring the production amplitudes, averaging (summing) over initial- (final-) state colors and spins, and integrating over phase space. � ε ( s 23 s 31 ) ε � � 4 π µ 2 1 1 Γ ( 1 − ε ) | M | 2 ds 23 ∑ σ = 16 π s 12 S s 12 spin / color s = s 12 , I introduce dimensionless parameters x = M 2 H / ˆ x = 1 − x , Defining ˆ s , ¯ y = 1 2 ( 1 − cos θ ∗ ) , ¯ y = 1 − y , to get M 2 s 12 = ˆ s , H = x ˆ s , s 23 = ¯ s , s 31 = ¯ s , x y ˆ x ¯ y ˆ � ε � 1 � 4 π µ 2 x 1 − 2 ε 1 1 ¯ y − ε | M | 2 . dy y − ε ¯ ∑ σ = 16 π ˆ Γ ( 1 − ε ) s S s ˆ 0 spin / color William Kilgore (BNL) One-Loop Single-Real April 30, 2014 16 / 26
The Partonic Cross Section at N 3 LO Introduction Methods Conclusions Integration-by-Parts for Phase Space Computing the partonic cross section involves a large number of phase space integrals. To simplify the problem, I borrow the idea of using IBP identities. For example, � 1 dy d y − ε ¯ y − 2 ε 2 F 1 ( 1 , − ε ; 1 − ε ; ¯ � � x y ) = 0 dy 0 � 1 0 dyy − ε ¯ y − 2 ε 2 F 1 ( 1 , − ε ; 1 − ε ; ¯ = ( 1 − 2 ε ) x y ) � 1 0 dyy − ε ¯ y − 1 − 2 ε 2 F 1 ( 1 , − ε ; 1 − ε ; ¯ + 2 ε x y ) � 1 y − 2 ε ( 1 − ¯ 0 dyy − ε ¯ x y ) − 1 − ε I get non-trivial relations among the phase space integrals allowing me to map the full calculation onto a small number of master phase space integrals. William Kilgore (BNL) One-Loop Single-Real April 30, 2014 17 / 26
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