Motivation Threshold Resummation Numerical Results Soft-gluon resummation for gluon-induced Higgs-Strahlung Vincent Theeuwes Institut für Theoretische Physik Westfälische Wilhelms-Universität Münster In Collaboration with: Robert Harlander, Anna Kulesza, Tom Zirke Firenze, 05.09.2014 Threshold resummation for gg → HZ 1 V. Theeuwes
Motivation Threshold Resummation Numerical Results Importance of gg → HZ • A Higgs boson found with a mass of 125 GeV • Precision study needed to determine if it is SM Higgs • One process is Higgs-strahlung (H+Z final state) • At LO pp → HZ is described by q ¯ q → HZ • Drell-Yan corrections up to NNLO [Hamberg, Neerven, Matsuura, ’91] [Harlander, Kilgore, ’02] [Brein, Djouadi, Harlander, ’04] • gg → HZ at NLO [Altenkamp, Dittmaier, Harlander, Rzehak, Zirke, ’12] - Large corrections (factor of 2) - Still has significant scale dependence ✛ ✘ H q H Z ∗ Z ∗ t Z Z ✚ ✙ q ¯ Threshold resummation for gg → HZ 2 V. Theeuwes
Motivation Threshold Resummation Numerical Results Results NLO gg → HZ [Altenkamp, Dittmaier, Harlander, Rzehak, Zirke, ’12] √ s [TeV] σ LO σ NLO m H [GeV] gg [fb] [fb] gg 19 . 8 +61% 39 . 3 +32% 8 115 − 34% − 24% 18 . 7 +61% 37 . 2 +32% 8 120 − 34% − 24% 17 . 7 +61% 35 . 1 +32% 8 125 − 34% − 24% 16 . 7 +61% 33 . 1 +32% 8 130 − 34% − 24% 79 . 1 +51% 152 +27% 14 115 − 31% − 21% 75 . 1 +51% 144 +27% 14 120 − 31% − 21% 71 . 1 +51% 136 +27% 14 125 − 31% − 21% 67 . 2 +51% 129 +27% 14 130 − 31% − 21% Threshold resummation for gg → HZ 3 V. Theeuwes
Motivation Threshold Resummation Numerical Results Importance of Resummation • Resummation up to NNLL already improved Higgs production results [Catani, de Florian, Grazzini, Nason, ’03] [de Florian, Grazzini, ’09] [de Florian, Grazzini, ’12] • gg → HZ similar loop induced process ⇒ threshold resummation could help further improve results ✻✵✳✵✵ σ ( gg → H + X ) ❬♣❜❪ √ ✺✵✳✵✵ S = 14 ❚❡❱ ✹✵✳✵✵ ✸✵✳✵✵ ✷✵✳✵✵ ▲❖ ✶✵✳✵✵ ◆▲❖ µ 0 = m H = 115 ●❡❱ ◆▲❖✰◆▲▲ ✵✳✵✵ ✵✳✷✺ ✵✳✺ ✶ ✷ ✹ µ/µ 0 Agrees with [Catani, de Florian, Grazzini, Nason, ’03] Threshold resummation for gg → HZ 4 V. Theeuwes
Motivation Threshold Resummation Numerical Results Definition of Threshold Q-approach M-approach (absolute threshold) ✛ ✘ ✛ ✘ Z ⇑ ✚ Q 2 ✙ ✚ ✙ τ Q = Q 2 τ M = M 2 Threshold variable ˆ Threshold variable ˆ s ˆ s ˆ Q 2 : the invariant mass final state particles M = m H + m Z τ M = 1 − M 2 τ Q = 1 − Q 2 1 − ˆ 1 − ˆ s ˆ s ˆ ∼ maximum energy of the emitted gluons ∼ energy of the emitted gluons total available energy total available energy √ ˆ s : the partonic center of mass energy Threshold resummation for gg → HZ 5 V. Theeuwes
Motivation Threshold Resummation Numerical Results Logarithms Q-approach The IR divergences lead to logarithms: ✎ ☞ � log m (1 − ˆ τ Q ) � α n ≡ α n s D Q, m (ˆ τ Q ) , m ≤ 2 n − 1 s ✍ ✌ 1 − ˆ τ Q + In general logarithms of 1 − ˆ τ Q M-approach For 2 → 2 process: logarithms of 1 − ˆ τ M : ✞ ☎ ✝ ✆ α n s log m (1 − ˆ τ M ) ≡ α n s D M, m − 1 (ˆ τ M ) , m ≤ 2 n Logarithms become large in threshold: ˆ τ → 1 Threshold resummation for gg → HZ 6 V. Theeuwes
Motivation Threshold Resummation Numerical Results Mellin Transform Mellin transform is used with respect to τ (needed for factorization of ✤ ✜ phase space): � 1 ˜ dτ τ N − 1 Σ pp → HZ ( τ, m Z , m H , µ R , µ F ) Σ pp → HZ ( N ) ≡ 0 f j/p ( N + 1 , µ F )˜ � f i/p ( N + 1 , µ F ) ˜ ˜ ˆ = Σ ij → HZ ( N, µ R , µ F ) ✣ ✢ i,j • ˜ f i/p ( N + 1 , µ F ) : Mellin transform with respect to x • ˜ ˆ Σ ij → HZ ( N, µ R , µ F ) : Mellin transform with respect to ˆ τ d σ ij → HZ - Σ ij → HZ = in Q-approach d Q 2 - Σ ij → HZ = σ ij → HZ in M-approach τ ) ⇒ log n +1 N and threshold ˆ D n (ˆ τ → 1 ∼ N → ∞ Threshold resummation for gg → HZ 7 V. Theeuwes
Motivation Threshold Resummation Numerical Results Orders of Resummation Large logarithms log N ≡ L for N → ∞ Perturbation needs to be reordered in α s and L : [Kodaira, Trentadue, ’82][Sterman, ’87][Catani, d’Emilio, Trentadue, ’88][Catani, Trentadue, ’89] ✞ ☎ ✝ ✆ σ ∼ ˜ ˜ σ LO × C ( α s ) exp [ Lg 1 ( α s L ) + g 2 ( α s L ) + α s g 3 ( α s L ) + · · · ] ⇓ ⇓ ⇓ With orders of precision: LL NLL NNLL ⇓ ⇓ ⇓ s log n +1 ( N ) s log n ( N ) log n ( N ) α n α n α n +1 s Exponential functions are well known and the same as for gg → H [Catani, de Florian, Grazzini, Nason, ’03] Threshold resummation for gg → HZ 8 V. Theeuwes
Motivation Threshold Resummation Numerical Results Hard Matching Coefficient (Schematically) C ( α s ) = 1 + α s π C (1) + · · · Originates from NLO calculation. Using terms proportional to: ⇒ σ LO , σ LO D M, 0 , σ LO D M, 1 OR ⇒ σ LO δ ( Q 2 − ˆ s ) , σ LO D Q, 0 , σ LO D Q, 1 Mellin transform leads to: α s π [ C (1) ˜ Σ LO + O (˜ Σ LO log( N ) , ˜ Σ LO log 2 ( N )) + · · · ] ⇓ Expansion of exponential Threshold resummation for gg → HZ 9 V. Theeuwes
Motivation Threshold Resummation Numerical Results Hard Matching Coefficient (Schematically) C ( α s ) = 1 + α s π C (1) + · · · Originates from NLO calculation. Using terms proportional to: ⇒ σ LO , σ LO D M, 0 , σ LO D M, 1 OR ⇒ σ LO δ ( Q 2 − ˆ s ) , σ LO D Q, 0 , σ LO D Q, 1 Mellin transform leads to: α s π [ C (1) ˜ Σ LO + O (˜ Σ LO log( N ) , ˜ Σ LO log 2 ( N )) + · · · ] ⇓ Expansion of exponential Threshold resummation for gg → HZ 9 V. Theeuwes
Motivation Threshold Resummation Numerical Results Hard Matching Coefficient (Schematically) C ( α s ) = 1 + α s π C (1) + · · · Originates from NLO calculation. Using terms proportional to: ⇒ σ LO , σ LO D M, 0 , σ LO D M, 1 OR ⇒ σ LO δ ( Q 2 − ˆ s ) , σ LO D Q, 0 , σ LO D Q, 1 Mellin transform leads to: α s π [ C (1) ˜ Σ LO + O (˜ Σ LO log( N ) , ˜ Σ LO log 2 ( N )) + · · · ] ⇓ Expansion of exponential Threshold resummation for gg → HZ 9 V. Theeuwes
Motivation Threshold Resummation Numerical Results Hard Matching Coefficient (Schematically) C ( α s ) = 1 + α s π C (1) + · · · Originates from NLO calculation. Using terms proportional to: ⇒ σ LO , σ LO D M, 0 , σ LO D M, 1 OR ⇒ σ LO δ ( Q 2 − ˆ s ) , σ LO D Q, 0 , σ LO D Q, 1 Mellin transform leads to: α s π [ C (1) ˜ Σ LO + O (˜ Σ LO log( N ) , ˜ Σ LO log 2 ( N )) + · · · ] ⇓ Expansion of exponential Threshold resummation for gg → HZ 9 V. Theeuwes
Motivation Threshold Resummation Numerical Results Hard Matching Coefficient Σ NLO = ˆ Σ R + ˆ Σ V + ˆ ˆ Σ C � � � � � � � Σ V + dˆ Σ R | ǫ =0 − dˆ Σ A | ǫ =0 dˆ dˆ Σ A + ˆ Σ C = + 3 2 1 ǫ =0 Threshold resummation for gg → HZ 10 V. Theeuwes
Motivation Threshold Resummation Numerical Results Hard Matching Coefficient Σ NLO = ˆ Σ R + ˆ Σ V + ˆ ˆ Σ C ✘ � � ✘✘✘✘✘✘✘✘✘✘ � � � � � Σ V + dˆ Σ R | ǫ =0 − dˆ Σ A | ǫ =0 dˆ dˆ Σ A + ˆ Σ C = + 3 2 1 ǫ =0 ⇓ Supressed in threshold limit Threshold resummation for gg → HZ 10 V. Theeuwes
Motivation Threshold Resummation Numerical Results Hard Matching Coefficient Σ NLO = ˆ Σ R + ˆ Σ V + ˆ ˆ Σ C ✘ � � ✘✘✘✘✘✘✘✘✘✘ � � � � � Σ V + dˆ Σ R | ǫ =0 − dˆ Σ A | ǫ =0 dˆ dˆ Σ A + ˆ Σ C = + 3 2 1 ǫ =0 ⇓ Supressed in threshold limit ⇒ C (1) calculated by: ˆ Σ V + ˆ Σ A + ˆ Σ C In agreement with: [Catani, Cieri, de Florian, Ferrera, Grazzini, ’13] Threshold resummation for gg → HZ 10 V. Theeuwes
Motivation Threshold Resummation Numerical Results Hard Matching Coefficient (Result) � µ 2 � 2 � 11 � � � C (1) = ˆ σ virt π + 3 T R n l − 6 − 2 γ E C A log ˆ σ L O α s W 2 9 − 2 π 2 � 50 � C A + 16 − 2 γ 2 − 9 T R n l E 3 σ L O , W 2 = Q 2 • Q-approach: Absolute threshold expansion ˆ σ v irt and ˆ • W-approach: W 2 = M 2 Threshold resummation for gg → HZ 11 V. Theeuwes
Motivation Threshold Resummation Numerical Results Hard Matching Coefficient (Result) � µ 2 � 2 � 11 � � � C (1) = ˆ σ virt π + 3 T R n l − 6 − 2 γ E C A log ˆ σ L O α s W 2 9 − 2 π 2 � 50 � C A + 16 − 2 γ 2 − 9 T R n l E 3 σ L O , W 2 = Q 2 • Q-approach: Absolute threshold expansion ˆ σ v irt and ˆ • W-approach: W 2 = M 2 Threshold resummation for gg → HZ 11 V. Theeuwes
Motivation Threshold Resummation Numerical Results Hard Matching Coefficient (Result) � µ 2 � 2 � 11 � � � C (1) = ˆ σ virt π + 3 T R n l − 6 − 2 γ E C A log ˆ σ L O α s W 2 9 − 2 π 2 � 50 � C A + 16 − 2 γ 2 − 9 T R n l E 3 σ L O , W 2 = Q 2 • Q-approach: Absolute threshold expansion ˆ σ v irt and ˆ • W-approach: W 2 = M 2 Threshold resummation for gg → HZ 11 V. Theeuwes
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