Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results Soft Gluon Resummation for associated t ¯ tH Production at the LHC Vincent Theeuwes University at Buffalo The State University of New York In Collaboration with: Anna Kulesza, Leszek Motyka, Tomasz Stebel arXiv:1509.02780 and in preperation Loopfest, 08-15-2016 Threshold resummation for t ¯ 1 V. Theeuwes tH
Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results Importance of pp → t ¯ tH • A Higgs boson with a mass close to 125 GeV • Precision study needed to determine if it is SM Higgs • t ¯ tH direct way to access Yukawa coupling LHC HIGGS XS WG 2016 H+X) [pb] 2 10 M(H)= 125 GeV W ) E L O + N C D Q N L L + N L O N N → H ( p p 10 → qqH (NNLO QCD + NLO EW) (pp → pp W ) O E N L C D + O Q N N L σ → H ( 1 W W ) p p O E + N L bbH (NNLO QCD in 5FS, NLO QCD in 4FS) C D O Q N N L → Z H ( p p → pp W ) E L O N + D C Q − O 1 L ( N 10 H → t t p p tH (NLO QCD) → pp − 2 10 6 7 8 9 10 11 12 13 14 15 s [TeV] [LHCHXSWG] Threshold resummation for t ¯ 2 V. Theeuwes tH
Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results Current status of pp → t ¯ tH • QCD Corrections up to NLO [Beenakker et al. , ’02] [Dawson et al. , ’02] • Matched to parton showers by: aMC@NLO [Frederix et al. , ’11] , PowHel [Garzelli et al. , ’11] , Sherpa [Hoeche et al., ’12] , POWHEG-BOX [Hartanto et al. , ’14] • Electroweak correction [Frixione et al. , ’14,’15][Zhang, ’14] • Including top decays [Denner, Feger, ’15] • Absolute threshold at NLL [Kulesza, Motyka, Stebel, VT, ’15] • Expansion of NNLL in SCET [Broggio et al., ’15] ✞ ✟ ✒ ✒ ❴ s ✭ � � ✺ ✥ ✥ ❍ ✁ ✂ ✄ ▼ ❢ ☎ ◆ ✞ ✑ ✒ ✒ ❘ ✆ ✝ ✞ ✟ ✠ ✡☛ ✎ ✝ ✞ ✑ ✒ ✓ ✡☛ ✏ ✌ ✍ ♠ ✝ ✔ ✕ ✎ ✴ ✑ ✞ ✒ ✒ ✒ ✵ t ✏ ✽ ✒ ✒ ☞✌ ✍ ✻ ✒ ✒ ✟ ✒ ✒ ✑ ✒ ✒ ✒ ✖ ✑ ✒ ✖ ✗ ✞ ✑ ✗ ♠ ✴ ♠ ✵ [Beenakker et al. , ’02] Threshold resummation for t ¯ 3 V. Theeuwes tH
Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results Why resummation for t ¯ tH ? Gains • NNLO corrections out of reach • Resummation can help reduce scale uncertainty • Good process to start: • Simple color structure • Massive particles → no final state collinear divergences Threshold resummation for t ¯ 4 V. Theeuwes tH
Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results Why resummation for t ¯ tH ? Gains • NNLO corrections out of reach • Resummation can help reduce scale uncertainty • Good process to start: • Simple color structure • Massive particles → no final state collinear divergences Pitfalls • 2 → 3 phase space supressed near threshold ( σ ∝ β 4 ) • Small corrections from near absolute threshold Threshold resummation for t ¯ 4 V. Theeuwes tH
Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results Definition of Threshold Q-approach M-approach (absolute threshold) 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 : the sum of final state masses 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 t ¯ 5 V. Theeuwes tH
Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results Logarithms Q-approach The IR divergences lead to logarithms: � � � log(1 − ˆ � τ Q ) − 1 − 2 ǫ = − 1 1 τ Q ) (1 − ˆ 2 ǫδ (1 − ˆ τ Q ) + − 2 ǫ 1 − ˆ τ Q 1 − ˆ τ Q ✎ ☞ + + � log m (1 − ˆ τ Q ) � α n ✍ ✌ s 1 − ˆ τ Q + In general logarithms of 1 − ˆ τ Q M-approach ✞ ☎ After integration over ˆ τ Q : logarithms of 1 − ˆ τ M : ✝ ✆ α n s log m (1 − ˆ τ M ) Logarithms become large in threshold limit: ˆ τ → 1 Threshold resummation for t ¯ 6 V. Theeuwes tH
Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient 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 → t ¯ ˜ σ pp → t ¯ tH ( N ) ≡ tH ( τ, µ R , µ F ) 0 f i/p ( N + 1 , µ F ) ˜ ˜ f j/p ( N + 1 , µ F )˜ � = ˆ σ ij → t ¯ tH ( N, µ R , µ F ) ✣ ✢ i,j • ˜ f i/p ( N + 1 , µ F ) : Mellin transform with respect to x • ˜ ˆ σ ij → t ¯ tH ( N, µ R , µ F ) : Mellin transform with respect to ˆ τ τ ) ⇒ log n N and threshold ˆ log n (1 − ˆ τ → 1 ∼ N → ∞ First application for 2 → 3 in Mellin space Threshold resummation for t ¯ 7 V. Theeuwes tH
Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results General factorization In general cross section factorizes into: σ ij → kl... = H ij → kl... ,IJ ⊗ ψ i ⊗ ψ j ⊗ S JI ⊗ J k ⊗ J l . . . ˆ • H ij → kl,IJ Hard function ψ i • ψ i,j Initial state collinear emission J k • J k,l,... Final state collinear H ∗ H S emission J l • S JI Soft emission ψ j Each of these functions is computed through renormalization group equations Threshold resummation for t ¯ 8 V. Theeuwes tH
Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results Differential formalism General formalism developed for 2 → 2 . [Kidonakis,Oderda,Sterman’98] , [Laenen,Oderda,Sterman’98] Using a infra-red safe weight ( ω ) to describe the soft limit of the emission. For 2 → 2 : • Pair invariant mass ω PIM = z = 1 − Q 2 / ˆ s • One particle inclusive ω 1PI = s 4 = (( p 1 + p 2 − p 3 ) 2 − m 2 4 ) / ˆ s Threshold resummation for t ¯ 9 V. Theeuwes tH
Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results Resummation for differential distributions ✞ ☎ ✝ ✆ W = ω 1 c 1 + ω 2 c 2 + ω s + ω k + ω l + . . . � dWe − NW dσ ij → kl... d ˆ Π n � � � dWe − NW H IJ � ˆ = Π n dω 1 dω 2 dω s dω k dω l . . . δ ( W − ω 1 c 1 − ω 2 c 2 − ω s − ω k − ω l − . . . ) ψ i/i ( ω 1 ) ψ j/j ( ω 2 ) S JI ( ω s ) J k ( ω k ) J l ( ω l ) . . . � � ˆ ψ i/i ( Nc 1 ) ˜ ˜ ψ j/j ( Nc 2 ) ˜ S JI ( N ) ˜ J k ( N ) ˜ = H IJ Π n J l ( N ) . . . • ω 1 / 2 Initial state collinear weights • ω s Soft-wide angle weight • ω k/l... Final state collinear weights Threshold resummation for t ¯ 10 V. Theeuwes tH
Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results 2 → 3 weights for differential notation pp → Q ¯ ⇒ Q 2 QB + X z 5 = ˆ s − s 345 ˆ s Threshold resummation for t ¯ 11 V. Theeuwes tH
Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results 2 → 3 weights for differential notation pp → Q ¯ ⇒ Q 2 QB + X z 5 = ˆ s − s 345 ˆ s pp → B + X [ Q ¯ Q ] ⇒ P T, 5 , y 5 v 5 = ( p 1 + p 2 − p 5 ) 2 − s 34 s + ˜ t 15 + ˜ t 25 + m 2 = ˆ 5 − s 34 s ˆ ˆ s Threshold resummation for t ¯ 11 V. Theeuwes tH
Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results 2 → 3 weights for differential notation pp → Q ¯ ⇒ Q 2 QB + X z 5 = ˆ s − s 345 ˆ s pp → B + X [ Q ¯ Q ] ⇒ P T, 5 , y 5 v 5 = ( p 1 + p 2 − p 5 ) 2 − s 34 s + ˜ t 15 + ˜ t 25 + m 2 = ˆ 5 − s 34 s ˆ ˆ s pp → Q ¯ ⇒ P T, 34 , y 34 , Q 2 Q + X [ B ] 34 s 5 = ( p 1 + p 2 − p 3 − p 4 ) 2 − m 2 s + ˜ t 13 + ˜ t 14 + ˜ t 23 + ˜ t 24 + s 34 − m 2 = ˆ 5 5 s ˆ ˆ s Threshold resummation for t ¯ 11 V. Theeuwes tH
Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results Orders of Resummation Large logarithms log N ≡ L for N → ∞ Perturbation needs to be reordered in α s and L : ✞ ☎ ✝ ✆ 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 ⇓ ⇓ ⇓ α n s log n +1 ( N ) α n s log n ( N ) α n +1 log n ( N ) s Exponential functions are universal for initial state emission [Kodaira, Trentadue, ’82][Sterman, ’87][Catani, d’Emilio, Trentadue, ’88][Catani, Trentadue, ’89] Threshold resummation for t ¯ 12 V. Theeuwes tH
Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results Color space Need to project the matrix element onto a color basis. Use the s-channel color basis: gg q ¯ q δ A 1 A 2 δ a 3 a 4 1 : 1 : δ a 2 a 1 δ a 3 a 4 T D a 3 a 4 d DA 1 A 2 8 S : T D a 2 a 1 T D 8 : a 3 a 4 iT D a 3 a 4 f DA 1 A 2 8 A : Basis for diagonalization of soft anomalous dimension in absolute threshold limit. Same as for t ¯ t production Threshold resummation for t ¯ 13 V. Theeuwes tH
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