4ループQCDパートン分岐関数について: 非一重項の場合 / On the 4-loop QCD splitting functions: the non-singlet case

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ループQCDパートン分岐関数について～非一重項の場合～植田高寛 (成蹊大理工) with Sven Moch (ハンブルク大), Ben Ruijl (チューリッヒ工科大), Jos Vermaseren (Nikhef), Andreas Vogt (リヴァプール大) Based on JHEP ( ) , arXiv: . [hep-ph] (cf. Phys.Lett. B ( ) , arXiv: . [hep-ph]) 日本物理学会年秋季大会素粒子論領域 aS - 信州大年月日 /

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Goal and Motivation N LO パートン分布関数(PDFs)のスケール発展を決めるループパートン分岐関数 (splitting functions) DGLAP (Dokshitzer-Gribov-Lipatov-Altarelli-Parisi) 方程式: d d ln µ fi (x, µ ) = j Pij as (µ ) ⊗ fj (µ ) (x) Pij = as P( ) ij + as P( ) ij + as P( ) ij + as P( ) ij + . . . /

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Goal and Motivation N LO パートン分布関数(PDFs)のスケール発展を決めるループパートン分岐関数 (splitting functions) なぜ? • 原子核・ハドロン物理: 精密な陽子構造 • 摂動論的量子場の理論: Challenging Feynman Integrals • 素粒子現象論: LHCでの精密物理 /

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背景: LHC Physics 標準模型を超えた物理のシグナルなし間接探索/精密測定が鍵に? ILCを建設する or じわじわ上がる(HL-)LHCデータの精度に期待 LHCでのQCD: 多くの反応過程でNNLO 一部ではN LO /

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gg → h + X (N LO) LO NLO NNLO NNNLO 0 10 20 30 40 50 σ/pb LHC pp→h+X gluon fusion MSTW08 68cl μ=μR=μF ∈ [mH/4,mH] Central scale: μ = mH/2 2 4 6 8 10 12 14 -0.2 -0.1 0.0 0.1 0.2 0.3 S /TeV Anastasiou et al., arXiv: . g g h t t t Anastasiou, Dulat, Duhr, Mistlberger ’ ; Anastasiou, Dulat, Duhr, Furlan, Gehrmann, Herzog, Lazopoulos, Mistlberger ’ ; Mistlberger ’ cf. qq → (qq gg) → qq h: Anzai, Hasselhuhn, Höschele, Hoff, Kilgore, Steinhauser, TU ’ /

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Missing N LO PDFs σpp→(gg+X)→h+X = . pb+ . pb(+ . %) − . pb(− . %) (theory) ± . pb( . %)(PDF αs) √ s = TeV, mh = GeV by N LO σgg→h+X with NNLO PDFs δ(scale) δ(trunc) δ(PDF-TH) δ(EW) δ(t, b, c) δ( /mt ) + . pb − . pb ± . pb ± . pb ± . pb ± . pb ± . pb + . % − . % ± . % ± . % ± % ± . % ± % Anastasiou et al. ’ Error by missing N LO PDFs ループパートン分岐関数 → N LO PDFs → 精密ヒッグス物理 • Singlet • Non-singlet /

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パートン分岐関数の計算 N-th Mellin moment: γab (N) = − dx xN− Pab (x) N-依存性を持つmassless propagator-type ファインマン積分の発散部分(後述) Q N 一般のNについての表式が得られれば Mellin逆変換でPab が求まる ( ループでは難しい...) /

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パートン分岐関数の計算 Fix N = , , , . . . → massless propagator-type ファインマン積分 Q F Ruijl, TU, Vermaseren ’ ループmassless propagator-type ファインマン積分の解析的計算 https://github.com/benruijl/forcer Fixed N = , , , . . . から近似的にPab を構成 / 600 800 1000 1200 0 0.2 0.4 0.6 0.8 1 x (1−x) P (2) (x) +,0 exact N = 2...12 -1000 1000 2000 ループでの例: exact vs. approx. non-singlet, nf -independent part From Moch, Vermaseren, Vogt NPB ( ) , arXiv:hep-ph/

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Method I Harmonic projection to probe-parton forward scattering Gorishnii, Larin, Tkachev ’ ; Gorishnii, Larin ’ Q{µ . . . QµN} N! ∂N ∂Pµ . . . ∂PµN Q = Q Q P P P= Setupが簡単 Nが大きくなると計算時間が急増 N = まで: 既存の結果を確認 Velizanin ’ ’ ; Baikov, Chetyrkin, Kühn, Rittinger N = まで計算 cf. Davies, Ruijl, TU, Vermaseren, Vogt NPB ( ) , arXiv: . / 0 0.1 0.2 0.3 0.4 0 5 10 15 20 γ( )+ NS : even N γ( )− NS : odd N N Method I SU( ), nf = expansion in αs

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Method II Matrix elements of twist- DIS operators ¯ ψγ{µ Dµ . . . DµN}ψ P P P = 計算時間の増加が緩やか Setupに難 (Feynman rule/renormalization) N = まで: Method Iの結果を確認 N = まで計算 / 0 0.1 0.2 0.3 0.4 0 5 10 15 20 γ( )+ NS : even N γ( )− NS : odd N N Method I SU( ), nf = expansion in αs

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N LO NS splitting functions 1 x (1-x) P ns (1-x) P (3)± (x) exp. in α S , n f = 4 appr. + appr. − large n c -4 -2 0 2 4 6 8 10 -4 10 -3 10 -2 10 -1 1 P = αs P( ) + αs P( ) + αs P( ) +αs P( ) + . . . Large-nc part: analytically reconstructed Rest: approximation /

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q± NS evolution: NNLO vs N LO Logarithmic derivatives w.r.t. the factorization scale ˙ q ≡ d ln q d ln µ f Reference input: xq(x, µ ) = x . ( − x) with αs(µ ) = . , nf = 0.96 0.98 1 1.02 1.04 10 -4 10 -3 10 -2 10 -1 1 0.96 0.98 1 1.02 1.04 10 -4 10 -3 10 -2 10 -1 1 x q . + NS N2LO / NLO N3LO / N2LO µr = µf x q . − NS N2LO / NLO N3LO / N2LO µr = µf x q . v NS N2LO / NLO N3LO / N2LO µr = µf 0.9 0.95 1 1.05 1.1 10 -4 10 -3 10 -2 10 -1 /

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q+ NS evolution: renormalization scale dependence -0.26 -0.25 -0.24 -0.23 -0.22 -0.14 -0.135 -0.13 -0.125 -0.12 -0.036 -0.034 -0.032 -0.03 -0.028 0.042 0.044 0.046 0.048 0.05 0.052 10 -1 1 10 0.06 0.065 0.07 0.075 0.08 10 -1 1 10 x = 0.8 d ln q + / d ln µf 2 NS x = 0.5 NLO N2LO N3LO A,B x = 0.15 x = 0.01 µr 2 / µf 2 x = 10 −3 µr 2 / µf 2 x = 10 −4 µr 2 / µf 2 0.065 0.07 0.075 0.08 0.085 0.09 10 -1 1 10 N LO: stable scale uncertainty below for µf / ≤ µr ≤ µf /

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Summary and Outlook (HL-)LHCでの更なるヒッグス精密物理にはループパートン分岐関数が必要 Fixed-N momentsを計算することにより Non-singlet part P( ) NS をLHCでの現象論に十分な精度で得た Moch, Ruijl, TU, Vermaseren, Vogt JHEP ( ) , arXiv: . Singlet: 計算途上 cf. Moch, Ruijl, TU, Vermaseren, Vogt PLB ( ) , arXiv: . この計算が完了すれば → N LO PDFs → LHCでのヒッグスセクターに対する精密物理への道(∼ ) /

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Backup Appendix - /

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Parton evolutions DGLAP equation d d ln µ f fa (x, µf ) = b Pab (αs (µf )) ⊗ fb (µf ) (x) nf (anti-)quark distributions decomposed as • q± ns,ik = (qi ± ¯ qi ) − (qk ± ¯ qk ), flavor non-singlet, (nf − ) components, evolving with P± ns • qv ns = nf i= (qi − ¯ qi ): flavor non-singlet (“valence”), evolving with Pv ns = P− ns + O(αs ) • qs = nf i= (qi + ¯ qi ): flavor singlet, mixing with gluons. Pqq = P+ ns + O(αs ) d d ln µ f qs g = Pqq Pqg Pgq Pgg ⊗ qs g Appendix - /

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The F program F program for -loop massless propagator-type (scalar) Feynman integrals Ruijl, TU, Vermaseren ’ ; https://github.com/benruijl/forcer Works with F . Extends the Mincer approach to -loops Chetyrkin, Tkachov ’ ; Schoonschip version: Gorishny, Larin, Surguladze, Tkachov ’ ; F version: Larin, Tkachov, Vermaseren ’ When possible (determined by the topology), one-loop integration/triangle rule → Chetyrkin, Kataev, Tkachov ’ ; Chetyrkin, Tkachov ’ → and/or Chetyrkin, Tkachov ’ ; Diamond extension: Ruijl, TU, Vermaseren ’ Appendix - /

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Reconstructing full-N expression Up to N = for non-singlet, large-nc γ± NS (N): (qi ± ¯ qi ) − (qk ± ¯ qk ) Ansatz: if analogous to lower orders γ(n) NS (N) = n+ w= c w Sw (N) + a n+ k= n+ −k w= cakw Sw (N) (N + a)k γNS: constrained by ‘self-tuning’ (conjecture, conformal symmetry) γ+ NS = γ− NS for large-nc Large-N and small-x limits Small-x ressumation: known coefficients N ≤ Diophantine eqs. to fix remaining coefficients Checked by N = , Appendix - /

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Approximation for large-nc suppressed terms Remaining large-nc suppressed terms (non-singlet) resulting trial functions, parameters ﬁxed from the ﬁrst moments, two representatives chosen that indicate the remaining uncertainty -8000 -6000 -4000 -2000 0 10 -4 10 -3 10 -2 10 -1 1 x x0.4(1-x) P N,0 x0.4(1-x) P (3)+(x) A, B x P N,0 P (3)+(x) / ref. ref. = (A + B) / 2 A, B 0.97 0.98 0.99 1 1.01 1.02 1.03 0 0.2 0.4 0.6 0.8 1 Checked by the th moment, e.g., P+( ) N, (N = ): . B < . ...exact < . A Appendix - /