ハドロンコライダーにおけるnon-global logarithmsのNc=3での再足し上げ / Resumming non-global logarithms at hadron colliders for Nc = 3

QCDの関わる反応過程では，多くの場合その計算途中に赤外発散が現れる．それらの発散は実際の散乱断面積等を考えると相殺が起こって消えるが，場合によっては発散部分は消えるものの大きな有限の対数因子を残すことがある．大きな対数因子があると摂動展開の収束を悪化・破綻させてしまうため，全次数にわたっての再足し上げが必要となる．粒子の方向に制限を掛けた観測量(non-global observables)を考えると，よく知られたSudakov logarithmsに加えて，新たに赤外発散由来の対数因子が現れることがあり，non-global logarithms (NGL)と呼ばれている．

NGL の再足し上げは，主要対数(leading log)近似のみを考えたとしても，large-Nc極限を超えた取り扱いが極めて困難である．以前，我々はNGLのNc=3での主要対数項の再足し上げに対して，ランダムウォークを用いて実際に数値計算可能な手法を与えた[1]．これはsmall-x QCDに現れる対数因子の再足し上げの方法との類推が基となっている．この手法を用いて，これまでに電子陽電子消滅過程でのinterjet energy flow [1]とhemisphere jet mass distribution [2]を計算している．

今回，同手法を用いて(1)ヒッグス粒子のグルーオンへの崩壊(2)ハドロンコライダーでのヒッグス粒子の2ジェット随伴生成の二つの反応過程に関するラピディティギャップ残存確率について，Nc=3でのNGLの主要対数項の再足し上げを数値的に行った[3]．電子陽電子消滅過程では終状態のクォーク・反クォーク対から2つのSU(3)基本表現のWilson lineの相関が現れるのに対し，ハドロンコライダーの場合は最大で8つのWilson lineを含んだ相関関数が現れる(グルーオンは2つと数える)．本講演ではこれらの数値計算結果を示し，議論する．

[1] Y. Hatta and T. Ueda, Nucl. Phys. B 874 (2013) 808, arXiv:1304.6930 [hep-ph].
[2] Y. Hagiwara, Y. Hatta and T. Ueda, Phys. Lett. B 756 (2016) 254, arXiv:1507.07641 [hep-ph].
[3] Y. Hatta and T. Ueda, Nucl. Phys. B 962 (2021) 115273, arXiv:2011.04154 [hep-ph].

Transcript

1. ϋυϩϯίϥΠμʔʹ͓͚Δ non-global logarithms ͷ Nc = 3 Ͱͷ࠶଍্͛͠ ২ాߴ׮ (੒᪟େཧ޻) ڞಉݚڀऀ: ീాՂ޹ (BNL / ཧݚ BNL) NPB962 (2021) 115273, arXiv:2011.04154 [hep-ph] ‌ ‌ ೔ຊ෺ཧֶձୈ 76 ճ೥࣍େձ 2021 ೥ 3 ݄ 15 ೔@ΦϯϥΠϯ
2. ຊߨԋͷཁ໿ non-global logͷओཁର਺߲ (leading log; LL) ͷ (p. 3 Ͱઆ໌) શ࣍਺ʹΘͨΔ࠶଍্͛͠Λ Nc = 3Ͱ (large Nc → ∞ ͷۙࣅͳ͠ʹ) ߦ͏࿮૊Έ ϋυϩϯίϥΠμʔͰͷ൓Ԡաఔ (Χϥʔͷ଍͕ෳࡶ) ʹద༻ͯ͠਺஋ܭࢉͨ͠ͱ͜Ζɺڵຯਂ͍݁Ռ͕...!? 1/18
3. ಋೖ: ର਺Ҽࢠͱ࠶଍্͛͠ QCD ൓Ԡաఔͷઁಈܭࢉ (αs ≪ 1) O = c0 + c1αs + c2α2 s . . . LO NLO NNLO ੺֎ൃࢄʹىҼ͢Δ (େ͖ͳ) ର਺Ҽࢠ L (αsL ∼ 1) શ࣍਺ʹΘͨͬͯͷ࠶଍্͕͛͠ඞཁ εϥετ e.g., Abbate, Fickinger, Hoang, Mateu, Stewart ’10 ύʔτϯγϟϫʔ 2/18
4. ಋೖ: ର਺Ҽࢠͱ࠶଍্͛͠ QCD ൓Ԡաఔͷઁಈܭࢉ (αs ≪ 1) O = d00 + d11αsL + d10αs + d22α2 s L2 + d21α2 s L + d20α2 s . . . . . . . . . LO NLO NNLO ੺֎ൃࢄʹىҼ͢Δ (େ͖ͳ) ର਺Ҽࢠ L (αsL ∼ 1) શ࣍਺ʹΘͨͬͯͷ࠶଍্͕͛͠ඞཁ εϥετ e.g., Abbate, Fickinger, Hoang, Mateu, Stewart ’10 ύʔτϯγϟϫʔ 2/18
5. ಋೖ: ର਺Ҽࢠͱ࠶଍্͛͠ QCD ൓Ԡաఔͷઁಈܭࢉ (αs ≪ 1) O = d00 + d11αsL + d10αs + d22α2 s L2 + d21α2 s L + d20α2 s . . . . . . . . . = e0(αsL) + e1(αsL)αs + e2(αsL)α2 s +··· LO NLO NNLO LL NLL NNLL ੺֎ൃࢄʹىҼ͢Δ (େ͖ͳ) ର਺Ҽࢠ L (αsL ∼ 1) શ࣍਺ʹΘͨͬͯͷ࠶଍্͕͛͠ඞཁ εϥετ e.g., Abbate, Fickinger, Hoang, Mateu, Stewart ’10 ύʔτϯγϟϫʔ 2/18
6. ಋೖ: non-global log (NGL) ཻࢠͷํ޲ʹ੍ݶΛ՝ͨ͠؍ଌྔ (non-global observable) ؍ଌ͠ͳ͍ ؍ଌ͢Δ ৽ͨͳछྨͷର਺Ҽࢠ (non-global logarithm; NGL) Dasgupta, Salam ’01 (global observable ʹର͢Δ) ௨ৗͷ࠶଍্͕͛͠࢖͑ͳ͍ 3/18
7. ಋೖ: non-global log (NGL) ཻࢠͷํ޲ʹ੍ݶΛ՝ͨ͠؍ଌྔ (non-global observable) ؍ଌ͠ͳ͍ ؍ଌ͢Δ ৽ͨͳछྨͷର਺Ҽࢠ (non-global logarithm; NGL) Dasgupta, Salam ’01 (global observable ʹର͢Δ) ௨ৗͷ࠶଍্͕͛͠࢖͑ͳ͍ 3/18
8. ಋೖ: non-global log (NGL) ཻࢠͷํ޲ʹ੍ݶΛ՝ͨ͠؍ଌྔ (non-global observable) ؍ଌ͠ͳ͍ ؍ଌ͢Δ ৽ͨͳछྨͷର਺Ҽࢠ (non-global logarithm; NGL) Dasgupta, Salam ’01 (global observable ʹର͢Δ) ௨ৗͷ࠶଍্͕͛͠࢖͑ͳ͍ 3/18
9. ಋೖ: non-global log (NGL) ཻࢠͷํ޲ʹ੍ݶΛ՝ͨ͠؍ଌྔ (non-global observable) ؍ଌ͠ͳ͍ ؍ଌ͢Δ ৽ͨͳछྨͷର਺Ҽࢠ (non-global logarithm; NGL) Dasgupta, Salam ’01 (global observable ʹର͢Δ) ௨ৗͷ࠶଍্͕͛͠࢖͑ͳ͍ 3/18
10. ϥϐσΟςΟΪϟοϓ࢒ଘ֬཰ શํҐ֯Λ in ͱ out ͷྖҬʹ෼͚ɺout ྖҬʹ veto Λ՝͢ (“jet-gap-jet” ͷΑ͏ͳϥϐσΟςΟΪϟοϓΛཁٻ) θin in in out out ∑ i∈out Ei ≤ Eveto in ྖҬʹ͓͍ͯ (ྫ͑͹) Ωα ͱ Ωβ ʹ q ¯ q ͕ग़ͨͱͯ͠ɺ (ͦΕΒ͔ΒͷιϑτͳάϧʔΦϯͷ์ग़ʹ΋ؔΘΒͣ) ϥϐσΟςΟΪϟοϓ͕ੜ͖࢒Δ֬཰: Pαβ(τ) τ ∼ αs π ln Qhard Eveto ʹ͍ͭͯ࠶଍্͛͠ (NGL ΛؚΉ LL) (ݱ৅࿦@LHC: τ ≲ 0.5) 4/18
11. ϥϐσΟςΟΪϟοϓ࢒ଘ֬཰ શํҐ֯Λ in ͱ out ͷྖҬʹ෼͚ɺout ྖҬʹ veto Λ՝͢ (“jet-gap-jet” ͷΑ͏ͳϥϐσΟςΟΪϟοϓΛཁٻ) Ωα Ωβ in in out out ∑ i∈out Ei ≤ Eveto in ྖҬʹ͓͍ͯ (ྫ͑͹) Ωα ͱ Ωβ ʹ q ¯ q ͕ग़ͨͱͯ͠ɺ (ͦΕΒ͔ΒͷιϑτͳάϧʔΦϯͷ์ग़ʹ΋ؔΘΒͣ) ϥϐσΟςΟΪϟοϓ͕ੜ͖࢒Δ֬཰: Pαβ(τ) τ ∼ αs π ln Qhard Eveto ʹ͍ͭͯ࠶଍্͛͠ (NGL ΛؚΉ LL) (ݱ৅࿦@LHC: τ ≲ 0.5) 4/18
12. ϥϐσΟςΟΪϟοϓ࢒ଘ֬཰ શํҐ֯Λ in ͱ out ͷྖҬʹ෼͚ɺout ྖҬʹ veto Λ՝͢ (“jet-gap-jet” ͷΑ͏ͳϥϐσΟςΟΪϟοϓΛཁٻ) Ωα Ωβ in in out out ∑ i∈out Ei ≤ Eveto in ྖҬʹ͓͍ͯ (ྫ͑͹) Ωα ͱ Ωβ ʹ q ¯ q ͕ग़ͨͱͯ͠ɺ (ͦΕΒ͔ΒͷιϑτͳάϧʔΦϯͷ์ग़ʹ΋ؔΘΒͣ) ϥϐσΟςΟΪϟοϓ͕ੜ͖࢒Δ֬཰: Pαβ(τ) τ ∼ αs π ln Qhard Eveto ʹ͍ͭͯ࠶଍্͛͠ (NGL ΛؚΉ LL) (ݱ৅࿦@LHC: τ ≲ 0.5) 4/18
13. Banfi-Marchesini-Smye (BMS)ํఔࣜ Banfi, Marchesini, Smye ’02 large-Nc Ͱͷ࠶଍্͛͠Λߦ͏ൃలํఔࣜ ∂Pαβ ∂τ = −2CF ∫ out dΩγ 4π 1−cosθαβ (1−cosθαγ)(1−cosθγβ) Pαβ +Nc ∫ in dΩγ 4π 1−cosθαβ (1−cosθαγ)(1−cosθγβ) (PαγPγβ − Pαβ) Ωα Ωβ in in out out CF = N2 c −1 2Nc = 4 3 ∼ Nc 2 = 3 2 for large Nc ॳظ৚݅: Pαβ(τ = 0) = 1 τ = 0 ⇒ Qhard = Eveto τ → ∞ ⇒ Qhard ≫ Eveto P τ 1 0 5/18
14. Banfi-Marchesini-Smye (BMS)ํఔࣜ Banfi, Marchesini, Smye ’02 large-Nc Ͱͷ࠶଍্͛͠Λߦ͏ൃలํఔࣜ ∂Pαβ ∂τ = −2CF ∫ out dΩγ 4π 1−cosθαβ (1−cosθαγ)(1−cosθγβ) Pαβ +Nc ∫ in dΩγ 4π 1−cosθαβ (1−cosθαγ)(1−cosθγβ) (PαγPγβ − Pαβ) Ωα Ωβ in in out out CF = N2 c −1 2Nc = 4 3 ∼ Nc 2 = 3 2 for large Nc ॳظ৚݅: Pαβ(τ = 0) = 1 τ = 0 ⇒ Qhard = Eveto τ → ∞ ⇒ Qhard ≫ Eveto P τ 1 0 5/18
15. large-Nc ͔Βfinite-Nc ΁ finite-Nc ޮՌΛऔΓೖΕ͍ͨ (∼ 1/N2 c ∼ 10%) For recent efforts to incorporate subleading-Nc terms into parton showers see, e.g., Hamilton, Medves, Salam, Scyboz, Soyez ’20 BMS ํఔࣜ͸ Balitsky-Kovchegov (BK) ํఔࣜ (large-Nc ) ʹࠅࣅ dSxy dτ = Nc ∫ d2z 2π (x− y)2 (x− z)2(z − y)2 (SxzSzy − Sxy) ࣮ࡍʹʮ౳Ձʯ: Hatta ’08; Caron-Huot ’15; Neill, Ringer ’20 BK ํఔࣜͷ finite-Nc ʹରԠͨ͠ఆࣜԽͰ͋Δ Jalilian-Marian-Iancu-McLerran-Weigert-Leonidov-Kovner (JIMWLK) ํఔࣜʹ͸ϥϯμϜ΢ΥʔΫΛ༻͍ͨ਺஋ղ๏͕ଘࡏ Blaizot, Iancu, Weigert ’03 BK JIMWLK ղ͚Δ BMS ? ղ͚Δ͸ͣ finite-Nc ౳Ձ finite-Nc Weigert ’03; Hatta, TU ’13 6/18
16. large-Nc ͔Βfinite-Nc ΁ finite-Nc ޮՌΛऔΓೖΕ͍ͨ (∼ 1/N2 c ∼ 10%) For recent efforts to incorporate subleading-Nc terms into parton showers see, e.g., Hamilton, Medves, Salam, Scyboz, Soyez ’20 BMS ํఔࣜ͸ Balitsky-Kovchegov (BK) ํఔࣜ (large-Nc ) ʹࠅࣅ dSxy dτ = Nc ∫ d2z 2π (x− y)2 (x− z)2(z − y)2 (SxzSzy − Sxy) ࣮ࡍʹʮ౳Ձʯ: Hatta ’08; Caron-Huot ’15; Neill, Ringer ’20 BK ํఔࣜͷ finite-Nc ʹରԠͨ͠ఆࣜԽͰ͋Δ Jalilian-Marian-Iancu-McLerran-Weigert-Leonidov-Kovner (JIMWLK) ํఔࣜʹ͸ϥϯμϜ΢ΥʔΫΛ༻͍ͨ਺஋ղ๏͕ଘࡏ Blaizot, Iancu, Weigert ’03 BK JIMWLK ղ͚Δ BMS ? ղ͚Δ͸ͣ finite-Nc ౳Ձ finite-Nc Weigert ’03; Hatta, TU ’13 6/18
17. large-Nc ͔Βfinite-Nc ΁ finite-Nc ޮՌΛऔΓೖΕ͍ͨ (∼ 1/N2 c ∼ 10%) For recent efforts to incorporate subleading-Nc terms into parton showers see, e.g., Hamilton, Medves, Salam, Scyboz, Soyez ’20 BMS ํఔࣜ͸ Balitsky-Kovchegov (BK) ํఔࣜ (large-Nc ) ʹࠅࣅ dSxy dτ = Nc ∫ d2z 2π (x− y)2 (x− z)2(z − y)2 (SxzSzy − Sxy) ࣮ࡍʹʮ౳Ձʯ: Hatta ’08; Caron-Huot ’15; Neill, Ringer ’20 BK ํఔࣜͷ finite-Nc ʹରԠͨ͠ఆࣜԽͰ͋Δ Jalilian-Marian-Iancu-McLerran-Weigert-Leonidov-Kovner (JIMWLK) ํఔࣜʹ͸ϥϯμϜ΢ΥʔΫΛ༻͍ͨ਺஋ղ๏͕ଘࡏ Blaizot, Iancu, Weigert ’03 BK JIMWLK ղ͚Δ BMS ? ղ͚Δ͸ͣ finite-Nc ౳Ձ finite-Nc Weigert ’03; Hatta, TU ’13 6/18
18. large-Nc ͔Βfinite-Nc ΁ finite-Nc ޮՌΛऔΓೖΕ͍ͨ (∼ 1/N2 c ∼ 10%) For recent efforts to incorporate subleading-Nc terms into parton showers see, e.g., Hamilton, Medves, Salam, Scyboz, Soyez ’20 BMS ํఔࣜ͸ Balitsky-Kovchegov (BK) ํఔࣜ (large-Nc ) ʹࠅࣅ dSxy dτ = Nc ∫ d2z 2π (x− y)2 (x− z)2(z − y)2 (SxzSzy − Sxy) ࣮ࡍʹʮ౳Ձʯ: Hatta ’08; Caron-Huot ’15; Neill, Ringer ’20 BK ํఔࣜͷ finite-Nc ʹରԠͨ͠ఆࣜԽͰ͋Δ Jalilian-Marian-Iancu-McLerran-Weigert-Leonidov-Kovner (JIMWLK) ํఔࣜʹ͸ϥϯμϜ΢ΥʔΫΛ༻͍ͨ਺஋ղ๏͕ଘࡏ Blaizot, Iancu, Weigert ’03 BK JIMWLK ղ͚Δ BMS ? ղ͚Δ͸ͣ finite-Nc ౳Ձ finite-Nc Weigert ’03; Hatta, TU ’13 6/18
19. ϥϯμϜ΢ΥʔΫʹΑΔNGLͷ࠶଍্͛͠ Uα(τ+ϵ) = eiS(2) α eiAαUα(τ)eiBα eiS(1) α S(i) α = √ ϵ 4π ∫ out dΩγ (nα − nγ)k 1− nα · nγ taξ(i)k γa (i = 1,2) Aα = − √ ϵ 4π ∫ in dΩγ (nα − nγ)k 1− nα · nγ UγtaU† γ ξ(1)k γa Bα = √ ϵ 4π ∫ in dΩγ (nα − nγ)k 1− nα · nγ taξ(1)k γa Uα : Ωα ํ޲΁ͷ Wilson line (SU(Nc ) جຊදݱ) ξ(i)k γα : Ψ΢γΞϯന৭ϊΠζ 7/18
20. ྫ: q ¯ q ͷ৔߹ Hatta, TU ’13 e+e− → γ∗ → q ¯ q q ¯ q i,Ωα j,Ωβ Pαβ = ⟨ 1 Nc tr(UαU† β ) ⟩ U: جຊදݱͷ Wilson line θin = 60◦ Ωα Ωβ 8/18
21. ྫ: q ¯ q ͷ৔߹ Hatta, TU ’13 e+e− → γ∗ → q ¯ q q ¯ q i,Ωα j,Ωβ Pαβ = ⟨ 1 Nc tr(UαU† β ) ⟩ U: جຊදݱͷ Wilson line θin = 60◦ Ωα Ωβ 8/18
22. ྫ: q ¯ q ͷ৔߹ Hatta, TU ’13 e+e− → γ∗ → q ¯ q q ¯ q i,Ωα j,Ωβ Pαβ = ⟨ 1 Nc tr(UαU† β ) ⟩ U: جຊදݱͷ Wilson line θin = 60◦ Ωα Ωβ 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 80×60 lattice, ϵ = 5×10−5, first 50 trajectories 8/18
23. ྫ: q ¯ q ͷ৔߹ Hatta, TU ’13 e+e− → γ∗ → q ¯ q q ¯ q i,Ωα j,Ωβ Pαβ = ⟨ 1 Nc tr(UαU† β ) ⟩ U: جຊදݱͷ Wilson line θin = 60◦ Ωα Ωβ 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 P(Nc =3) 80×60 lattice, ϵ = 5×10−5, 3000 trajectories, stat. err. + syst. err. 3000 ճͷي੻ͷฏۉ (࣮෦ͷΈɺڏ෦͸ฏۉΛऔΔͱམͪΔ) 8/18
24. ྫ: q ¯ q ͷ৔߹ Hatta, TU ’13 e+e− → γ∗ → q ¯ q q ¯ q i,Ωα j,Ωβ Pαβ = ⟨ 1 Nc tr(UαU† β ) ⟩ U: جຊදݱͷ Wilson line θin = 60◦ Ωα Ωβ 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 P(Nc =3) P(large-Nc ,CF =4/3) P(large-Nc ,CF =3/2) 80×60 lattice, ϵ = 5×10−5, 3000 trajectories, stat. err. + syst. err. BMS(large-Nc , CF = 4/3) Ͱ ಘΒΕΔղͱॏͳΔ (ฏۉ৔ۙࣅ͕ྑ͍ۙࣅͱͳ͍ͬͯΔ) ܭࢉ࣌ؒ: BMS ∼ ਺࣌ؒ ϥϯμϜ΢ΥʔΫ ∼ 1 ͔݄ 8/18
25. ຊߨԋͷཁ໿ non-global logͷओཁର਺߲ (leading log; LL) ͷ (p. 3 Ͱઆ໌) શ࣍਺ʹΘͨΔ࠶଍্͛͠Λ Nc = 3Ͱ (large Nc → ∞ ͷۙࣅͳ͠ʹ) ߦ͏࿮૊Έ ͜͜·Ͱऴྃ ϋυϩϯίϥΠμʔͰͷ൓Ԡաఔ (Χϥʔͷ଍͕ෳࡶ) ʹద༻ͯ͠਺஋ܭࢉͨ͠ͱ͜Ζɺ࣮ʹڵຯਂ͍݁Ռ͕...!! ۩ମతʹ͸ H0 → gg ͱ pp → H0 +2j ͰͷϥϐσΟςΟΪϟοϓ࢒ଘ֬཰ 9/18
26. gg ͷ৔߹ H0 → gg a,Ωα b,Ωβ PH = ⟨ 1 N2 c −1 tr( ˜ Uα ˜ U† β ) ⟩ ˜ U: ਵ൐දݱͷ Wilson line ˜ Uab = 2tr(U†taUtb) Λ࢖͏ θin = 60◦ Ωα Ωβ 10/18
27. gg ͷ৔߹ H0 → gg a,Ωα b,Ωβ PH = ⟨ 1 N2 c −1 tr( ˜ Uα ˜ U† β ) ⟩ ˜ U: ਵ൐දݱͷ Wilson line ˜ Uab = 2tr(U†taUtb) Λ࢖͏ θin = 60◦ Ωα Ωβ 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 PH (Nc =3) 80×60 lattice, ϵ = 5×10−5, 3000 trajectories, stat. err. + syst. err. 10/18
28. gg ͷ৔߹ H0 → gg a,Ωα b,Ωβ PH = ⟨ 1 N2 c −1 tr( ˜ Uα ˜ U† β ) ⟩ ˜ U: ਵ൐දݱͷ Wilson line ˜ Uab = 2tr(U†taUtb) Λ࢖͏ θin = 60◦ Ωα Ωβ 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 PH (Nc =3) [P(large-Nc ,CF =3/2)]2 80×60 lattice, ϵ = 5×10−5, 3000 trajectories, stat. err. + syst. err. q ¯ q ͷ৔߹ͷ BMS(large-Nc , CF = 3/2) (p. 8 ͷᒵͷ఺ઢ) ͷ 2 ৐ͱ ॏͳΔ NGL ͷ exponentiation Λࣔࠦʁ eC2 A = ( eCACF )2 for CA = Nc, CF = Nc/2 Observation by Gavin Salam 10/18
29. pp → H0 +2j લํͱޙํʹδΣοτɺதԝϥϐσΟςΟྖҬʹώοάε ઁಈͷ࠷௿࣍ɺΞΠίφʔϧۙࣅ QCD resum. w/o NGL: Forshaw, Sjödahl ’07 1+2 → 3+4+ H 1 2 3 4 θin = 60◦ θ13 = θ24 = 30◦ 4 ͭͷύʔτϯ͔ΒͷιϑτͳάϧʔΦϯͷ์ࣹ • qq → qqH0 (q ¯ q → q ¯ qH0 ΋ಉ༷) • qg → qgH0 ( ¯ qg → ¯ qgH0 ΋ಉ༷) • gg → ggH0 11/18
30. qq → qqH0 VBF (Z boson) color singlet exchange 1, i 2, j 3, k 4, l ggF color octet exchange 1, i 2, j 3, k 4, l Mi jkl = M1δkiδl j + M8ta ki ta l j = ( M1 − M8 2Nc ) δkiδl j + M8 2 δk jδli → ( M1 − M8 2Nc ) (U3U† 1 )ki(U4U† 2 )l j + M8 2 (U3U† 2 )k j(U4U† 1 )li ta ki ta l j = 1 2 [ δk jδli − 1 Nc δkiδl j ] (W Λަ׵͢ΔμΠΞάϥϜ͸ ggF ͱׯবͤͣɺZ ަ׵ͱΧϥʔҼࢠ͸ಉ͡ͳͷͰ M8 = 0 ͱஔ͚͹Α͍) 12/18
31. qq → qqH0 VBF (Z boson) color singlet exchange 1, i 2, j 3, k 4, l ggF color octet exchange 1, i 2, j 3, k 4, l Mi jkl = M1δkiδl j + M8ta ki ta l j = ( M1 − M8 2Nc ) δkiδl j + M8 2 δk jδli → ( M1 − M8 2Nc ) (U3U† 1 )ki(U4U† 2 )l j + M8 2 (U3U† 2 )k j(U4U† 1 )li ta ki ta l j = 1 2 [ δk jδli − 1 Nc δkiδl j ] (W Λަ׵͢ΔμΠΞάϥϜ͸ ggF ͱׯবͤͣɺZ ަ׵ͱΧϥʔҼࢠ͸ಉ͡ͳͷͰ M8 = 0 ͱஔ͚͹Α͍) 12/18
32. qq → qqH0 1 N2 c ∑ i jkl M′ 2 = ( M1 − M8 2Nc ) M8 Nc Re { 1 Nc tr ( U3U† 1 U4U† 2 )} + ( M1 − M8 2Nc )2 1 N2 c tr ( U3U† 1 ) tr ( U4U† 2 ) + M2 8 4 1 N2 c tr ( U3U† 2 ) tr ( U4U† 1 ) 13/18
33. qq → qqH0 1 N2 c ∑ i jkl M′ 2 = ( M1 − M8 2Nc ) M8 Nc Re { 1 Nc tr ( U3U† 1 U4U† 2 )} + ( M1 − M8 2Nc )2 1 N2 c tr ( U3U† 1 ) tr ( U4U† 2 ) + M2 8 4 1 N2 c tr ( U3U† 2 ) tr ( U4U† 1 ) 0.0 0.2 0.4 0.6 0.8 1.0 1 Nc tr(U3 U1 U4 U2 ) 1 N2 c tr(U3 U1 )tr(U4 U2 ) 1 N2 c tr(U3 U2 )tr(U4 U1 ) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.95 1.00 1.05 ratio 80×60 lattice, ϵ = 5×10−5, 3000 trajectories, stat. err. + syst. err. ਺஋తʹ 1 Nc tr ( U3U† 1 U4U† 2 ) ͱ 1 N2 c tr ( U3U† 1 ) tr ( U4U† 2 ) ͕Ұக M1M8 ͷ ׯব߲ͷ܎਺͕Ωϟϯηϧ 13/18
34. qq → qqH0 ׯব߲͕ফ͑ͯ M2 1 P1 qq + N2 c −1 4N2 c M2 8 P8 qq P1 qq ≡ 1 N2 c ⟨ tr ( U3U† 1 ) tr ( U4U† 2 )⟩ P8 qq ≡ 1 N2 c −1 ⟨ tr ( U3U† 2 ) tr ( U4U† 1 ) − 1 N2 c tr ( U3U† 1 ) tr ( U4U† 2 )⟩ (P1 qq ͱ P8 qq ͸ τ = 0 Ͱ 1 ͱͳΔΑ͏ʹن֨Խͯ͋͠Δ) 14/18
35. qq → qqH0 color singlet exchange 0.0 0.2 0.4 0.6 0.8 1.0 P1 qq P1 qq (MFA) P1 qq (large-Nc ;CF =3/2) P1 qq (large-Nc ;CF =4/3) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.95 1.00 1.05 ratio large-Nc ͱͷͣΕ͕ݟ͑Δ 80×60 lattice, ϵ = 5×10−5, 3000 trajectories, stat. err. + syst. err. ੨͍࣮ઢ: ϥϯμϜ΢ΥʔΫ (Nc = 3) ࠇ͍఺ઢͱ੺͍఺ઢ: large-Nc (1/Nc Λແࢹ) Ͱɺฏۉ৔ۙࣅ (〈AB〉 → 〈A〉〈B〉) Λ༻͍ɺ BMS ͷ਺஋ղ (ͦΕͧΕ CF = 3/2 ͱ CF = 4/3) ͰධՁͨ͠΋ͷ 15/18
36. qq → qqH0 color singlet exchange 0.0 0.2 0.4 0.6 0.8 1.0 P1 qq P1 qq (MFA) P1 qq (large-Nc ;CF =3/2) P1 qq (large-Nc ;CF =4/3) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.95 1.00 1.05 ratio large-Nc ͱͷͣΕ͕ݟ͑Δ color octet exchange 0.0 0.2 0.4 0.6 0.8 1.0 P8 qq P8 qq (large-Nc +) P8 qq (MFA) P8 qq (large-Nc ;CF =3/2) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 2 4 ratio 80×60 lattice, ϵ = 5×10−5, 3000 trajectories, stat. err. + syst. err. large-Nc (CF = 3/2) ͱͳ͔ͥҰக ੨͍࣮ઢ: ϥϯμϜ΢ΥʔΫ (Nc = 3) ࠇ͍఺ઢͱ੺͍఺ઢ: large-Nc (1/Nc Λແࢹ) Ͱɺฏۉ৔ۙࣅ (〈AB〉 → 〈A〉〈B〉) Λ༻͍ɺ BMS ͷ਺஋ղ (ͦΕͧΕ CF = 3/2 ͱ CF = 4/3) ͰධՁͨ͠΋ͷ 15/18
37. qg → qgH0 ggF ͷΈ 1, i 2, a 3, k 4, b M ab ik = tc ki Tc ba M8 → ( U3tcU† 1 ) ki ( ˜ U4Tc ˜ U† 2 ) ba M8 M2 8 2 Pqg Pqg ≡ 1 2Nc(N2 c −1) ⟨ tr ( U2U † 1 ) tr ( U3U † 4 ) tr ( U4U † 2 ) +tr ( U4U † 1 ) tr ( U2U † 4 ) tr ( U3U † 2 ) −tr ( U4U † 2 U3U † 4 U2U † 1 ) −tr ( U4U † 1 U2U † 4 U3U † 2 )⟩ 0.0 0.2 0.4 0.6 0.8 1.0 Pqg Pqg (large-Nc +) Pqg (MFA) Pqg (large-Nc ;CF =3/2) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 2 4 ratio 80×60 lattice, ϵ = 5×10−5, 3000 trajectories, stat. err. + syst. err. large-Nc (CF = 3/2) ͱ·ͨҰக 16/18
38. gg → ggH0 ggF ͷΈ 1, a 2, a′ 3, b 4, b′ Maa′bb′ = M8Tc ba Tc b′a′ → M8 ( ˜ U3Tc ˜ U† 1 ) ba ( ˜ U4Tc ˜ U2 ) b′a′ N2 c N2 c −1 M2 8 Pgg Pgg ≡ 1 2N2 c (N2 c −1) ⟨ Re { tr ( U1U † 3 )[ tr ( U2U † 4 ) tr ( U4U † 1 ) tr ( U3U † 2 ) +tr ( U4U † 2 ) tr ( U2U † 1 ) tr ( U3U † 4 )] −tr ( U1U † 3 )[ tr ( U4U † 2 U3U † 4 U2U † 1 ) +tr ( U2U † 4 U3U † 2 U4U † 1 )] −tr ( U2U † 4 )[ tr ( U1U † 3 U4U † 1 U3U † 2 ) +tr ( U3U † 1 U4U † 3 U1U † 2 )] +tr ( U3U † 2 U4U † 1 ) tr ( U2U † 4 U1U † 3 ) +tr ( U1U † 2 U4U † 3 ) tr ( U2U † 4 U3U † 1 )}⟩ 0.0 0.2 0.4 0.6 0.8 1.0 Pgg Pgg (large-Nc +) Pgg (MFA) Pgg (large-Nc ;CF =3/2) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 2 4 ratio 80×60 lattice, ϵ = 5×10−5, 3000 trajectories, stat. err. + syst. err. large-Nc (CF = 3/2) ͱ΍͸ΓҰக 17/18
39. ·ͱΊ H0 → gg ͱ pp → H0 +2j ʹ͓͚Δ ϥϐσΟςΟΪϟοϓ࢒ଘ֬཰ʹର͠ NGL ΛؚΊͨ࠶଍্͛͠ (LL) Λ Nc = 3 Ͱ਺஋తʹߦͬͨ P1 qq Ͱ͸ large-Nc ͱͷͣΕ ඇࣗ໌ͳؔ܎ࣜ • M1 ͱ M8 ͷׯব߲͕Ωϟϯηϧ • PH , P8 qq , Pqg , Pgg ͕ BMS ͷղ (CF = 3/2) ͰදͤΔ কདྷύʔτϯγϟϫʔʹ NGL ΛऔΓೖΕΔࡍ full-Nc ͷޮՌ͕ܭࢉίετͷ௿͍ large-Nc Ͱग़ͤΔʁ 18/18
40. Backup
41. H0 → gg: CF = 4/3 vs. 3/2 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 [P(large-Nc ,CF =4/3)]2 PH (Nc =3) [P(large-Nc ,CF =3/2)]2 80×60 lattice, ϵ = 5×10−5, 3000 trajectories, stat. err. + syst. err. Appendix - 1/3
42. Asymmetric jets configurations θin = 60◦, θ24 = 30◦ θ13 = 15◦ 0.0 0.2 0.4 0.6 0.8 1.0 Pgg Pgg (large-Nc +) Pgg (MFA) Pgg (large-Nc ;CF =3/2) Pgg (large-Nc ;CF =4/3) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 2 4 ratio θ13 = 45◦ 0.0 0.2 0.4 0.6 0.8 1.0 Pgg Pgg (large-Nc +) Pgg (MFA) Pgg (large-Nc ;CF =3/2) Pgg (large-Nc ;CF =4/3) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 2 4 ratio 80×60 lattice, ϵ = 5×10−5, 3000 trajectories, stat. err. + syst. err. Appendix - 2/3
43. Single-hemisphere jet mass distribution Hagiwara, Hatta, TU ’15 ୈ 71 ճ೥࣍େձ 20pAC-6 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 <gLR > τ this work(2600tr) DS(Cf=3/2) DS(Cf=4/3) SZ-KD KD(resum) PLB 756 (2016) 254, arXiv:1507.07641 80×80 lattice, ϵ = 5×10−5, 2600 trajectories, stat. err. ¯ q q L R ࠶଍্͛͠ͷ݁Ռ͸ɺfinite-Nc 4 ϧʔϓ + large-Nc 5 ϧʔϓ (SZ-KD) ͱ Schwartz, Zhu ’14; Khelifa-Kerfa, Delenda ’15 τ ≲ 0.5 ͰҰக (SZ-KD ͸ઁಈల։ͳͷͰ τ ͕େ͖͍ͱ͜ΖͰ blow up) Appendix - 3/3